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Advances in Optics and Photonics

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  • Editor: Bahaa E. A. Saleh
  • Vol. 1, Iss. 2 — Apr. 15, 2009
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Characterization of ultrashort electromagnetic pulses

Ian A. Walmsley and Christophe Dorrer  »View Author Affiliations


Advances in Optics and Photonics, Vol. 1, Issue 2, pp. 308-437 (2009)
http://dx.doi.org/10.1364/AOP.1.000308


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Abstract

Ultrafast optics has undergone a revolution in the past two decades, driven by new methods of pulse generation, amplification, manipulation, and measurement. We review the advances made in the latter field over this period, indicating the general principles involved, how these have been implemented in various experimental approaches, and how the most popular methods encode the temporal electric field of a short optical pulse in the measured signal and extract the field from the data.

© 2009 Optical Society of America

1. Introduction

1.1. Need for Ultrafast Metrology

The development of mode-locked lasers in the mid-1960s gave rise to the problem of ultrashort pulse measurement, since optical pulses generated by this class of lasers were of significantly shorter duration than any photodetector response time. Despite the vastly increased capabilities of modern photodetectors in terms of both speed of response and sensitivity, the equally dramatic improvements in laser technology have sustained this disparity, and, indeed, with the emergence of attosecond pulse trains have extended it.

The need for metrology has increased along with the development of new sources and their application in a wide range of new fields. Of course, determining the pulse durations remains critical, both because this parameter is an important specification of the laser output needed for other applications and because it provides a diagnosis of the system operation.

Modern mode-locked lasers, for example, generate pulses with spectral bandwidths exceeding one octave, corresponding to pulses the brevity of which is well beyond anything that can be characterized by means of fast photodetectors. The operation of such lasers relies on a complex combination of linear pulse propagation, influenced by the chromatic dispersion of the laser material, the mirrors, and the intracavity dispersion-compensating devices, together with nonlinear effects, such as self-phase modulation of the pulse in the laser material or saturation of an intracavity absorption, such as in a semiconductor saturable absorber mirror (SESAM), as well as, in some cases, space–time coupling. The optimization of a mode-locked laser is made practicable by means of a diagnostic providing the electric field as a function of time or frequency, or at least providing some temporal information such as the second-order intensity autocorrelation [1

1. R. L. Fork, C. H. Brito Cruz, P. C. Becker, and C. V. Shank, “Compression of optical pulses to six femtoseconds by using cubic phase compensation,” Opt. Lett. 12, 483–485 (1987). [CrossRef] [PubMed]

, 2

2. G. Taft, A. Rundquist, M. M. Murnane, H. C. Kapteyn, K. W. Delong, R. Trebino, and I. P. Christov, “Ultrashort optical waveform measurements using frequency-resolved optical gating,” Opt. Lett. 20, 743–745 (1995). [CrossRef] [PubMed]

, 3

3. P. G. Bollond, L. P. Barry, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Characterization of nonlinear switching in a figure-of-eight fiber laser using frequency-resolved optical gating,” IEEE Photon. Technol. Lett. 10, 343–345 (1998). [CrossRef]

, 4

4. A. Kasper and K. J. Witte, “Contrast and phase of ultrashort laser pulses from Ti:sapphire ring and Fabry–Perot resonators based on chirped mirrors,” J. Opt. Soc. Am. B 15, 2490–2495 (1998). [CrossRef]

, 5

5. L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, C. Iaconis, and I. A. Walmsley, “Characterization of sub-6-fs optical pulses with spectral phase interferometry for direct electric-field reconstruction,” Opt. Lett. 24, 1314–1316 (1999). [CrossRef]

, 6

6. D. H. Sutter, G. Steinmeyer, L. Gallmann, N. Matuschek, F. Morier-Genoud, U. Keller, V. Scheuer, G. Angelow, and T. Tschudi, “Semiconductor saturable-absorber mirror-assisted Kerr-lens mode-locked Ti:sapphire laser producing pulses in the two-cycle regime,” Opt. Lett. 24, 631–633 (1999). [CrossRef]

, 7

7. J. M. Dudley, S. F. Boussen, D. M. J. Cameron, and J. D. Harvey, “Complete characterization of a self-mode-locked Ti:sapphire laser in the vicinity of zero group-delay dispersion by frequency-resolved optical gating,” Appl. Opt. 38, 3308–3315 (1999). [CrossRef]

]. Figure 1 displays the characterization results of the output pulse from a Ti:sapphire oscillator. One of the primary limits at present to the generation of few-cycle pulses directly from a laser is the dispersion of the intracavity mirrors and other optical elements. Historically, detailed measurements of laser output were able to identify this as a major obstacle to generating pulses of greater brevity.

Chirped pulse amplification (CPA) operates by lowering the peak power of the pulses in the amplifier gain medium, which would otherwise induce nonlinear phase distortion of the pulse or damage to the amplification medium [8

8. D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219–221 (1985). [CrossRef]

, 9

9. S. Backus, C. G. Durfee III, M. M. Murnane, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. 69, 1207–1223 (1998). [CrossRef]

]. To achieve this, the pulses are stretched in time by means of a dispersive delay line, often based on angular dispersion from diffraction gratings or prisms. After amplification, the pulse is temporally recompressed by using an inverse dispersive delay line, or compressor, that compensates for the dispersion introduced by the stretcher and the propagation through the other amplifier elements. Obtaining peak performance from such a scheme requires a reliable and rapid method to characterize the output. Accurate characterization of the output pulses enables the optimization of the parameters of the system, such as the distance between the two gratings of the compressor and the angle of incidence of the input beam on the gratings. The usual optimization parameters in such an application are the duration of the recompressed pulses, since the peak power scales as the ratio of the energy per pulse to the duration, and the temporal contrast, since prepulses can hinder the control or observation of the physical processes of interest, for example the ionization of a target. Examples of optimization of CPA systems can be found in [10

10. B. Kohler, V. V. Yakovlev, K. R. Wilson, J. A. Squier, K. W. Delong, and R. Trebino, “Phase and intensity characterization of femtosecond pulses from a chirped-pulse amplifier by frequency-resolved optical gating,” Opt. Lett. 20, 483–485 (1995). [CrossRef] [PubMed]

, 11

11. C. Dorrer, B. de Beauvoir, C. Le Blanc, S. Ranc, J.-P. Rousseau, P. Rousseau, and J.-P. Chambaret, “Single-shot real-time characterization of chirped-pulse amplification systems by spectral phase interferometry for direct electric-field reconstruction,” Opt. Lett. 24, 1644–1646 (1999). [CrossRef]

]. Figure 2 presents an example of CPA optimization obtained with spectral phase interferometry for direct electric-field reconstruction (SPIDER) [11

11. C. Dorrer, B. de Beauvoir, C. Le Blanc, S. Ranc, J.-P. Rousseau, P. Rousseau, and J.-P. Chambaret, “Single-shot real-time characterization of chirped-pulse amplification systems by spectral phase interferometry for direct electric-field reconstruction,” Opt. Lett. 24, 1644–1646 (1999). [CrossRef]

]. The spectral phase of the output pulse from a Ti:sapphire CPA system is plotted before and after optimization. The compressor optimization consisted in adjusting the angle of the diffraction gratings relative to the input beam and the relative distance between the two gratings. The large cubic spectral phase gives rise to significant prepulses, and the compressor optimization leads to a better pulse shape with a higher peak intensity.

The bandwidth of an optical pulse can be increased while maintaining a deterministic phase relation between different spectral components by means of various nonlinear optical processes such as self-phase modulation and harmonic generation. All of these require careful compensation of the spectral phase in order to lead to an output pulse with a shorter duration than the input. Further, these processes are dynamically complicated and sensitive to the details of the input pulse shape. Therefore, even characterizing the raw output pulse before recompression can be a difficult task [12

12. A. Baltuska, M. S. Pshenichnikov, and D. A. Wiersma, “Amplitude and phase characterization of 4.5-fs pulses by frequency-resolved optical gating,” Opt. Lett. 23, 1474–1476 (1998). [CrossRef]

, 13

13. Z. Cheng, A. Furbach, S. Sartania, M. Lenzner, C. Spielmann, and F. Krausz, “Amplitude and chirp characterization of high-power laser pulses in the 5-fs regime,” Opt. Lett. 24, 247–249 (1999). [CrossRef]

, 14

14. S. A. Diddams, H. K. Eaton, A. A. Zozulya, and T. S. Clement, “Characterizing the nonlinear propagation of femtosecond pulses in bulk media,” IEEE J. Sel. Top. Quantum Electron. 4, 306–316 (1998). [CrossRef]

, 15

15. X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O’Shea, A. P. Shreenath, R. Trebino, and R. S. Windeler, “Frequency-resolved optical gating and single-shot spectral measurements reveal fine structure in microstructure-fiber continuum,” Opt. Lett. 27, 1174–1176 (2002). [CrossRef]

]. Figure 3 shows the characteristics of a filament pulse compressor that allows the generation of high-energy ultrashort optical pulses [16

16. G. Stibenz, N. Zhavoronkov, and G. Steinmeyer, “Self-compression of millijoule pulses to 7.8fs duration in a white-light filament,” Opt. Lett. 31, 274–276 (2006). [CrossRef] [PubMed]

]. The output pulses have complicated spectral and temporal structures, and correlation between time and frequency can be visualized in the chronocyclic time–frequency space by calculating the spectrogram of the output electric field.

Shaped pulses, sometimes of a quite complex temporal structure, are now commonly used to both probe and manipulate fundamental processes in atoms and molecules [17

17. A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929–1960 (2000). [CrossRef]

, 18

18. D. Goswami, “Optical pulse shaping approaches to coherent control,” Phys. Rep. 374, 385–481 (2003). [CrossRef]

]. For instance, the study of primary processes in biologically relevant systems via ultrafast microscopy is now quite common. The details of the pulse shapes usually contain important information about the dynamical process under study, and this information, residing in both the temporal amplitude and the temporal phase of the field, can be extracted only by using modern techniques of metrology. For example, the important phenomenon of the self-action of intense optical pulses in nonlinear media gives rise to a complicated set of dynamics that has analogs in many branches of physics. The study of the changes in the shape of pulses propagating through such media provides access to these dynamics. Optical pulse shaping can also be used to generate trains of pulses useful in optical telecommunications or to generate shaped electrical waveforms after optical-to-electrical conversion by a photodetector. Figure 4 displays the intensity of a train of pulses generated by an optical pulse shaper based on a liquid crystal spatial light modulator placed in a zero-dispersion line.

There are also important technological applications of metrology. In optical telecommunication systems, such metrology is used to characterize modulators and the dispersion of fiber links. The propagation of light pulses carrying bits of information from transmitter to receiver demands long-distance transmission through various passive and active elements. Both linear processes (e.g., the frequency-dependent transmission and phase of the medium) and nonlinear processes (e.g., the intensity-dependent index and absorption) modify the electric field of the pulses, and these effects must be quantified in order to maximize the overall system performance. There is also a need to optimize the shape of the pulses that are used, which are typically carved from a continuous-wave (cw) quasi-monochromatic source by a modulator. Although telecommunication pulses typically have durations ranging from 1ps to 1ns, the deleterious effects of propagation are significant because propagation distances of the order of 1000km in a physical medium can be involved. Moreover, the pulses used in state-of-the-art commercial and research optical telecommunication systems are beyond the reach of all-electronic characterization. Further, temporal phase information is also needed. A review of high-speed diagnostics for optical telecommunication systems is presented in [19

19. C. Dorrer, “High-speed measurements for optical telecommunication systems,” IEEE J. Sel. Top. Quantum Electron. 12, 843–858 (2006). [CrossRef]

], and some examples of diagnostics used in the telecommunication environment can be found in [20

20. J. Debeau, B. Kowalski, and R. Boittin, “Simple method for the complete characterization of an optical pulse,” Opt. Lett. 23, 1784–1786 (1998). [CrossRef]

, 21

21. K. Taira and K. Kikuchi, “Optical sampling system at 1.55  micron for the measurement of pulse waveform and phase employing sonogram characterization,” IEEE Photon. Technol. Lett. 13, 505–507 (2001). [CrossRef]

, 22

22. L. P. Barry, S. Del Burgo, B. C. Thomsen, R. T. Watts, D. A. Reid, and J. D. Harvey, “Optimization of optical data transmitters for 40-Gbs lightwave systems using frequency resolved optical gating,” IEEE Photon. Technol. Lett. 14, 971–973 (2002). [CrossRef]

, 23

23. C. Dorrer and I. Kang, “Simultaneous temporal characterization of telecommunication optical pulses and modulators using spectrograms,” Opt. Lett. 27, 1315–1317 (2002). [CrossRef]

]. Figure 5 presents results of a pulse carver optimization using a real-time pulse characterization diagnostic based on linear spectrograms [24

24. C. Dorrer and I. Kang, “Real-time implementation of linear spectrograms for the characterization of high bit-rate optical pulse trains,” IEEE Photon. Technol. Lett. 16, 858–860 (2004). [CrossRef]

]. The electric field of the output of a Mach–Zehnder modulator driven by a 20GHz sinusoidal RF drive depends on the phase difference between the two arms of the interferometer, which is controlled by a continuous voltage. The modulation format can be set to 33% return to zero (8ps pulses with identical phases) or 67% carrier-suppressed return to zero (17ps pulses with a π phase shift between adjacent pulses). Data-encoded optical signals require diagnostics that can acquire an invertible experimental trace in a single shot or can gather statistically significant samples of an optical waveform. Intensity sampling diagnostics use nonlinear cross-correlation schemes, while sampling systems based on homodyne detection are sensitive to the electric field of the waveform under test. These diagnostics are not detailed in this review, and relevant references can be found in [19

19. C. Dorrer, “High-speed measurements for optical telecommunication systems,” IEEE J. Sel. Top. Quantum Electron. 12, 843–858 (2006). [CrossRef]

].

1.2. Historical Developments

Considerable insight into the field can be gained from a look at the history of ultrafast metrology. Therefore we outline in a more or less chronological order the major advances over the past nearly four decades, since the invention of mode locking. To prefigure the structure of the review, the chronology is given in terms of several threads that have led to distinct techniques.

Of course, much has been written on the subject in recent years, and a number of excellent reviews of a few methods exist. A review of pulse measurement methods prior to 1974 can be found in the article by Bradley and New [25

25. D. J. Bradley and G. New, “Ultrashort pulse measurements,” Proc. IEEE 62, 313–345 (1974). [CrossRef]

]. A review of concepts for shaping and analysis of short optical pulses can be found in a 1983 article by Froehly and coworkers [26

26. C. Froehly, B. Colombeau, and M. Vampouille, “Shaping and analysis of picosecond light pulses,” in Progress in Optics. Vol. XX, E. Wolf ed. (North-Holland, 1983), pp. 63–153. [CrossRef]

], and a summary of methods available up to 1990 in the chapter by Laubereau in the book edited by Kaiser [27

27. A. Laubereau, “Optical nonlinearities with ultrashort pulses,” in Ultrashort Laser Pulses and Applications, W. Kaiser ed., Vol. 60 of Topics in Applied Physics (Springer-Verlag, 1988), pp. 35–112.

]. In more recent developments, a comprehensive description of frequency-resolved optical gating (FROG) is given in a book edited by Trebino [28

28. R. Trebino, ed., Frequency Resolved Optical Gating: the Measurement of Ultrashort Optical Pulses (Kluwer Academic, 2002).

], and a broader treatment of the field in the context of ultrafast optics is to be found in the book by Diels and Rudolph [29

29. J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena: Fundamentals, Techniques and Applications on the Femtosecond Time Scale, 2nd ed. (Academic, 2006).

].

For pulses in the range of several picoseconds or longer, the temporal intensity can be measured by using a streak camera or a photodiode. Combined with a measurement of the spectrum of the pulse, this information can be used to provide a reasonable characterization. For pulses in the femtosecond and indeed attosecond range such methods are not possible, in part because detectors that can absorb across the spectral range of these pulses are not always available, but mostly because direct photodetection is not fast enough.

Thus a different approach is required, one that avoids the need for fast detectors. Nonetheless, it is clear that something with a response time as brief as the pulse itself is needed, and the initial work in the field made use of the most obvious short event to hand—the pulse itself. This was used to synthesize a rapidly responding material excitation by means of the nonlinear optical responses of several common processes and materials. This trend has continued, though it is now understood that measurements using linear systems may also provide sufficient information to measure a pulse field.

Historically, the lack of fast detectors led to the adoption of nonlinear optical processes for the purposes of pulse characterization. An early technique, and one that dominated the field for many years, was the measurement of the intensity autocorrelation of a pulse. This relies on the observation that the efficiency of a nonlinear process (such as second-harmonic generation) is higher for higher input intensities. Thus the second-harmonic signal when a pair of pulses is incident on a nonlinear crystal is greater when they arrive at the same time, as opposed to when they arrive separately. Therefore a measurement of the second-harmonic power as a function of the delay gives an estimate of the pulse duration. Although it is possible to determine something of the time dependence of the phase of the pulse from more sophisticated versions of the autocorrelation, it is not possible to get a complete map. Nonetheless, it was also understood that the temporal structure of the pulses was strongly dependent on the spectral phase, and various representations of the pulse were developed to help visualize this and to develop new methods of measurement. Among the most fruitful of these was the spectrogram, consisting of a time–frequency map that plotted what was called the instantaneous frequency of the pulse.

Many current methods of metrology borrow heavily from methods developed in other branches of optics, notably imaging and testing. The strong analogy between space and time in Maxwell’s equations was employed first in metrology in the concept of the “time lens.” This employs a temporal phase modulator (a time-domain analog to a spatial phase modulator—a lens) and a dispersive delay line (a frequency-domain analog to free-space propagation) to generate a time-stretched replica of the input pulse whose temporal intensity can be measured by using a relatively slow detector.

The ideas of time–frequency representations have proved to be one of the most lasting in metrology, through both spectrography and its cousin sonography. The former builds on the original notion by developing methods to measure the spectrum of sequential time slices of the test pulse. The latter, in contrast, measures the time dependence of adjacent spectral slices. The relationship of these time–frequency (or chronocyclic) representations to various pulse measurement schemes has been an important source of ideas in ultrafast metrology.

A second analogy from optics that has proved equally fruitful for pulse characterization is interferometry. This is a well-known and sensitive method for extracting phase information about an optical field and is commonly employed in precision metrology. The measurement of the time-dependent phase of an optical pulse was first demonstrated by interfering it with a reference pulse of known character. In this case, a source with a narrowband spectrum provides a usable reference. This is analogous to using a point source as a reference wave in optical testing. It is possible to make this into a self-referencing interferometer by selecting the narrow frequency band for the reference from the test pulse spectrum. This is analogous to the generation of a spatial reference wave in optical testing by selecting a single point from the input beam. The temporal interference pattern obtained by combining different frequencies from the test pulse spectrum, recorded for example by using a cross-correlation (synthesized by using the same nonlinear mechanisms as had been developed for the autocorrelator), enables the relative phases of the two spectral components to be determined.

A different approach to interferometry avoids the need for a fast detector and is instead based on measurements made in the frequency domain. In spectral interferometry, a test pulse is gauged by using a known reference, and the phase difference extracted by using a noniterative algorithm. This method, first applied to the measurement of pulse distortions through propagation, was shown to be an extremely sensitive tool for pulse characterization, capable of attaining the quantum limit for photodetection. The proposal of self-referencing spectral interferometry showed how the lack of a known reference pulse could be circumvented by interfering the test pulse with a frequency-shifted (or spectrally sheared) version of itself and measuring the resulting two-pulse spectrum. A nonlinear implementation of this idea—SPIDER—retains the direct and unique inversion characteristic of interferometry with the ability to acquire and process data on individual laser pulses at rates up to 1kHz. This opens the way to measurements of statistical properties of pulse trains.

The final optical imaging analogy that has proved useful in pulse measurement is chronocyclic tomography. In this approach, the pulse field is reconstructed from a set of spectra after phase modulation. The name comes from the idea that these spectra represent projections of the chronocyclic phase-space representation of the field in the form of a Wigner function. The phase modulation serves to rotate the phase space, thus giving a series of one-dimensional sections of this two-dimensional entity. Further analogs from optical imaging have been used to develop simplified versions of this method that require significantly fewer measurements at the cost of some assumptions about the character of the input pulses. One of the earliest attempts to reconstruct pulse fields this way was to use the dispersive properties of glass to temporally stretch the pulse, then to determine the time dependence of the stretched pulse intensity by using an intensity cross-correlation with the unstretched pulse. This idea was further extended by using phase retrieval algorithms. Several approaches to tomographic pulse reconstruction have also been made by using self- and cross-phase modulation to achieve the phase-space rotation, coupled with measurements of the spectrum of the modulated pulse. Because the nonlinear mechanisms involve an ancillary pulse that must have a known (if not precisely controlled) shape, the best approach to inverting such measurements also makes use of iterative image processing algorithms.

Although most of the development of metrology for ultrafast pulses has made use of nonlinear optical processes, this turns out to be an artifact of the time scales involved rather than a fundamental restriction. In fact, it has been shown that complete characterization can be achieved by using entirely linear optical filters, such as spectrometers and temporal modulators. The only requirements are that the apparatus consist of at least one filter with a time-stationary response (e.g., a spectrometer) and at least one with a time-nonstationary response (e.g., a phase modulator) [30

30. All nonstationary optical elements require some reference signal that determines the timing of the modulation. In practice, a number of practically useful nonstationary elements are based on the mixing of an independent control signal with the optical pulse by means of a material nonlinear response. We wish however, to emphasize that self-activated nonlinear responses are not necessary, and that passive elements without nonlinear responses are not sufficient for pulse characterization.

]. All of the above-mentioned classes of measurement can be formulated in this way. It is only in recent years, however, that it has been possible to get active optical elements that have time and modulation scales appropriate for operating on subpicosecond pulses. Nonetheless, since modulation and photodetection with sub-100ps response times are required for a 40Gbits optical telecommunications system, these elements are also available to build linear methods that have proved very useful in assessing the performance of systems and components in this application. Interferometric, spectrographic, and tomographic methods have been implemented by using linear temporal modulators and spectral measurements, with time-integrating or “slow” detectors (i.e., with electrical bandwidths much less than 40GHz).

The subsequent sections of this review deal with each of these approaches in detail, providing both an analysis of the methods and a description of the current state of the art. To begin, a general analysis of pulse characterization is described in Section 2. This covers all known methods and indicates the necessary minimum conditions that all apparatuses must satisfy in order to operate successfully. The following sections describe each of the major approaches in turn: spectrography in Section 3, tomography in Section 4, and interferometry in Section 5. Some current areas of research are described in Section 6, together with the conclusions.

2. General Principles and Concepts of Pulse Characterization

2.1. Concepts and Protocols

2.1a. Representation of Pulsed Fields

Before describing how pulse measurement methods operate, it will be well to set out some definitions and to delineate exactly what we mean by pulse characterization. An electromagnetic pulse may be specified by its electric field alone, at least below intensities that give rise to fields that will accelerate electrons to relativistic energies. Thus a useful notation is that of the analytic signal, whose amplitude and phase we seek to determine via measurement. The (real) electric field of the pulse is given in terms of the analytic signal by
ɛ(t)=E(t)+E*(t),
(2.1)
where E(t) is an analytic function of time (and space, although we suppress other arguments here for clarity). The signal E is taken to have compact support in the domain [T,T], and we shall refer to it henceforth as the “field of the ultrashort pulse.” The analytic signal is complex and therefore can be expressed uniquely in terms of an amplitude and phase,
E(t)=|E(t)|exp[iψ(t)]exp(iψ0)exp(iω0t),
(2.2)
where |E(t)| is the time-dependent envelope, ω0 is the carrier frequency (usually chosen near the center of the pulse spectrum), ψ(t) is the time-dependent phase, and ψ0 a constant, known as the “carrier-envelope offset phase.” The square of the envelope, I(t)=|E(t)|2, is the time-dependent instantaneous power of the pulse, which can be measured if a detector of sufficient bandwidth is available (note that absolute measurement of the instantaneous power is usually not required, and most pulse characterization diagnostics return a normalized representation of this quantity). The derivative of the time-dependent phase accounts for the occurrence of different frequencies at different times, i.e., Ω(t)=ψt is the instantaneous frequency of the pulse that describes the oscillations of the electric field around that time. The frequency representation of the analytic signal is defined by the Fourier transform
Ẽ(ω)=|Ẽ(ω)|exp[iϕ(ω)]=TTdtE(t)eiωt,
(2.3)
so that ɛ̃(ω)=Ẽ(ω)+Ẽ*(ω). Note that Ẽ contains only positive frequency components, since E(t)=0(dω2π)Ẽ(ω)eiωt. This is therefore a reasonable description for the field of pulses propagating in charge-free regions of space, for which the pulse area TTdtɛ(t)=ɛ̃(0) must be zero. Here |Ẽ(ω)| is the spectral amplitude and ϕ(ω) is the spectral phase. The square of the spectral amplitude, Ĩ(ω)=|Ẽ(ω)|2, is the spectral intensity (strictly speaking this quantity is the spectral density—the quantity measured in the familiar way by means of a spectrometer followed by a photodetector). The spectral phase describes the relative phases of the optical frequencies composing the pulse, and its derivative ϕω is the group delay T(ω) at the corresponding frequency, i.e., the time of arrival of a subset of optical frequencies of the pulse around ω. A pulse with a constant group delay, i.e., a linear spectral phase, is said to be Fourier-transform limited because it is the shortest pulse that can be obtained for a given optical spectrum [33

33. I. A. Walmsley, L. Waxer, and C. Dorrer, “The role of dispersion in ultrafast optics,” Rev. Sci. Instrum. 72, 1–29 (2001). [CrossRef]

].

A single pulse is said to be completely characterized if the function E(t) is known on the domain [T,T]. In practice one usually adopts the approximation that the pulse is also characterized by the function Ẽ(ω) on the domain [ω0Ω,ω0+Ω], where Ω is a frequency that is large compared with the bandwidth of the source (i.e., large compared with the inverse of the coherence time of the source). The sampling theorem prevents a function from having compact support in both domains, but it is usually a reasonable approximation to truncate the spectral function at large frequencies, where the spectral density falls below the noise level of the detector. With this approximation, all integrals are usually formally extended from to +. Figure 6 presents the temporal and spectral representations of a Gaussian pulse with flat spectral phase, a quadratic spectral phase ϕ(2)ω22, and a cubic spectral phase ϕ(3)ω36. The impact of these different phases on the temporal profile of the pulse can also be seen.

2.1b. Correlation Functions and Chronocyclic Representations

The analytic signal describing a pulse field is not sufficient to specify the character of an ensemble of pulses. For example, each pulse from an amplifier system may be, indeed probably is, slightly different from its predecessor and successor, and thus each pulse represents a different realization of the ensemble. A complete specification of the ensemble is given by the probability distribution of the field at each point in time. However, it is usually sufficient to specify a set of correlation functions of the field, since experiments can be described in terms of a fairly small number of such functions.

The lowest order of these is the two-time correlation function C(t,t)=E(t)E*(t), where the brackets indicate either a time average over the pulse train or an ensemble average over repeated experiments. Note that C(t,t) is not the same as the correlation function that is derived from the pulse spectral intensity |Ẽ(ω)|2 via the Wiener–Khintchine theorem. In that case, the Fourier transform yields the reduced correlation
C(τ)=dtC(t,t+τ)=dω2π|Ẽ(ω)|2eiωτ.
(2.4)
This obviously contains no more information than the spectrum itself, in contrast to C(t,t), which encodes dynamical correlations in the electric field across the pulse.

A knowledge of the two-time correlation function allows one to determine whether the pulses in the ensemble are coherent, that is, to determine whether they have the same pulse field. A useful number characterizing the similarity of the pulses in the ensemble is the degree of temporal coherence, defined by [34

34. C. Iaconis, V. Wong, and I. A. Walmsley, “Direct interferometric techniques for characterizing ultrashort optical pulses,” IEEE J. Sel. Top. Quantum Electron. 4, 285–294 (1998). [CrossRef]

]
μ=dtdt|C(t,t)|2[dtC(t,t)]2.
(2.5)
When this number is unity, all pulses are the same, and values smaller than 1indicate various degrees of statistical variations in the pulse ensemble. In the case of identical pulses, the correlation function factorizes and the ensemble may be characterized by a single pulsed field. The analytic signal may be extracted from a single line of the correlation function, since E(t)C(t,t0).

Similarly, one can define a two-frequency correlation function C͌, which is the double Fourier transform of its temporal counterpart, by
C͌(ω,ω)=Ẽ(ω)Ẽ*(ω).
(2.6)
For application to interferometry, it is most useful to consider correlation functions written in terms of center- and difference-frequency variables,
C͌(Δω,ωC)=Ẽ(ωC+Δω2)Ẽ*(ωCΔω2),
(2.7)
and similarly in the time domain,
C(tC,Δt)=E(tC+Δt2)E*(tCΔt2),
(2.8)
where ωC=(ω+ω)2, Δω=ωω, tC=(t+t)2, and Δt=tt. An obvious way to measure correlation functions is to make repeated measurements of the electric field of the individual pulses that make up the realizations of the ensemble. From a large set of such measurements it is possible to estimate the statistics of the pulse field of the ensemble, or at least to determine some of the lower correlation functions. This has been done in several cases, and the fluctuations in pulse shape from a chirped-pulse amplifier system have been systematically characterized [35

35. C. Dorrer, B. de Beauvoir, C. Le Blanc, J.-P. Rousseau, S. Ranc, P. Rousseau, J.-P. Chambaret, and F. Salin, “Characterization of chirped-pulse amplification systems with spectral phase interferometry for direct electric-field reconstruction,” Appl. Phys. B 70, S77–S84 (2000). [CrossRef]

, 36

36. C. Dorrer and I. A. Walmsley, “Measurement of the statistical properties of a train of ultrashort light pulses,” in Summaries of Papers Presented at the Conference on Lasers and Electro-Optics, 2002. CLEO '02. Technical Digest (IEEE, 2002), Vol.1, pp. 85–86.

].

Beyond this approach, though, the correlation functions are difficult to measure. The reason is that the measured signals are functionals of the two-time correlation function and cannot always be simply inverted. This problem is usually ignored, and it is assumed from the beginning that the pulse train may be described in terms of a field. This makes possible more or less straightforward inversion algorithms.

Whether or not the pulse train is coherent, it is nevertheless useful to consider metrologic schemes in the two-dimensional space of the correlation function. The reason is that the output of all absorptive detectors is proportional to bilinear functional of the pulse field, and thus a linear functional of the two-time correlation function. However, it is frequently productive to work with a variation of the correlation function that uses the two-dimensional chronocyclic space (t,ω). The intuitive concept of time-dependent frequency can be most easily seen within this space.

Some examples of chronocyclic Wigner functions for common pulse shapes are shown in Fig. 7. The Wigner function of a Gaussian pulse with a flat spectral phase does not show a correlation between time and frequency. However, with a quadratic spectral phase (i.e. a linear group delay), the Wigner function acquires a slope indicating the correlation between time and frequency, and its contours provide some intuition about the pulse chirp via a graph of the time-dependent frequency. The Wigner function of a pair of phase-locked Gaussian pulses and a Gaussian pulse with cubic spectral phase take some negative values, although their marginals are, as expected, positive quantities.

2.2. General Strategies for Pulse Characterization

2.2a. Linear Systems Model and Photodetection

The oscillations of the electric field ɛ are too fast to be directly resolved by photodetection. Photodetectors are intrinsically square-law detectors, sensitive to the intensity of optical waves but not to their phase. Indirect approaches are therefore used to provide phase sensitivity with square-law photodetectors and to resolve the shape of short optical pulses. The basic elements required for the complete characterization of optical pulses are quite simple: at least one fast shutter or phase modulator, a spectrometer or an element to temporally stretch the pulse via dispersion, and one or two beam splitters. One can think of all elements except the beam splitters as two-port devices: a pulse enters at one port and exits at another. There may be ancillary ports for control signals, such as the timing signal for the shutter opening, for example, but these are essentially linear systems, in that the output pulse field scales linearly with the input pulse field. Thus the input–output relations for these devices are all of the kind
EOUTPUT(t)=dtH(t,t)EINPUT(t),
(2.13)
where EINPUT and EOUTPUT are the analytic signals of the input and output field, and H is the (causal) response function of the device. We will specify the functional forms of the common linear filters given above in subsequent paragraphs.

The beam splitter is a four-port device, having two input and two output ports. The input–output relations for this device are well known, and the main utility in pulse measurement applications is either in providing a means to generate a replica of a pulse (one input and two outputs) or to combine the unknown pulse with a reference pulse (two inputs and two outputs), or as elements of an interferometer in which phase-to-amplitude conversion takes place.

Linear filters may be separated into two classes: those with time-stationary response functions and those with time-nonstationary responses. For the former class, which includes the spectrometer and dispersive delay line, the shape of the output pulse does not depend on the arrival time of the pulse. For the latter class, which includes the phase modulator and the shutter, the output pulse shape clearly depends on the timing of the input pulse with respect to the shutter opening or the modulator drive signal.

Time-stationary filters are characterized by response functions of the form H(t,t)=S(tt), and a particularly useful class of time-nonstationary filters by H(t,t)=N(t)δ(tt). Equivalently, in the frequency domain, these stationary filters take the general form H͌(ω,ω)=S̃(ω)δ(ωω), and the nonstationary the form H͌(ω,ω)=Ñ(ωω), where the tilde represents a Fourier transform.

We may postulate a general linear filter function in the form of a temporal Fresnel kernel:
H(t,t)=12πbexp[i2b(at22tt+dt2)],
(2.15)
where a, b, and d are complex numbers (though real for phase-only filters). H is unitary and satisfies
dtH(t,t)H*(t,t)=δ(tt).
(2.16)
Most common manipulations can be described by such a filter function; indeed, an arbitrary response function may be constructed piecewise by concatenating several such filters. Representative response functions for the various elements named above, that facilitate analysis of all pulse measurement apparatuses, are particular cases of this general kernel with elements a, b, d determined by the action of the filter. Some particularly useful examples are the response functions for
Shutter (time gate),NA(tτ;τg)=exp[(tτ)2τg2],
(2.17)
Linear phase modulator,NLP(t;ψ(1))=exp(iψ(1)t),
(2.18)
Quadratic phase modulator,NQP(t;ψ(2),τ)=exp[iψ(2)(tτ)22],
(2.19)
Spectrometer,S̃A(ωΩ;Γ)=exp[(ωΩ)2Γ2],
(2.20)
Delay line,S̃LP(ω;ϕ(1))=exp(iϕ(1)ω),
(2.21)
Dispersive delay line,S̃QP(ω;ϕ(2),ωR)=exp[iϕ(2)(ωωR)22],
(2.22)
where N and S̃ indicate that the response functions are associated with nonstationary and stationary filters, the superscripts A and P denote amplitude-only and phase-only filters, and for phase-only filters, L and Q denote linear phase modulation and quadratic phase modulation. Although the spectrometer’s response function is not strictly causal, it can be made so by the introduction of a suitable delay that has no physical significance in the measurement protocol [39

39. V. Wong and I. A. Walmsley, “Ultrashort-pulse characterization from dynamic spectrograms by iterative phase retrieval,” J. Opt. Soc. Am. B 14, 944–949 (1997). [CrossRef]

]. The various parameters characterizing these filters are the following.
  • Gate: opening time τ, and duration of opening window τg,
  • Linear temporal phase modulator: frequency shift ψ(1),
  • Quadratic temporal phase modulator: amplitude of quadratic phase modulation ψ(2), and time of phase modulation extremum τ
  • Spectrometer: center frequency of passband Ω, and bandwidth Γ
  • Delay line: delay ϕ(1)
  • Dispersive delay line: group-delay dispersion ϕ(2) at reference frequency ωR

Some of these parameters become variables in the measured signal function (for example, the opening time of the gate or the center frequency of the spectrometer passband), while other parameters might be constant (for example the gate duration). The variable parameters are those on which the inversion is based. It is therefore important to ensure that the number and type of filters are adequate to the task.

2.2b. Measurement of the Marginals of the Wigner Function

Pulse energy spectrum. Possibly the simplest quantity that can be measured for an isolated pulse is its spectrum. It is therefore also one of the most important, since it can be used as a consistency check for all pulse characterization techniques: the reconstructed spectrum must match an independent direct measurement of the spectrum. The pulse spectrum is usually determined by the obvious expedient of sending the pulse into a spectrometer (usually a grating spectrometer is necessary to yield the necessary dispersion) and recording the output as a function of the setting of the passband of the instrument, Ω. Then the spectrometer output is
S(Ω;Γ)=dt|dtSA(tt;Ω,Γ)E(t)|2=dω2π|S̃A(ωΩ)|2Ĩ(ω),
(2.23)
with S̃A(ω) given by Eq. (2.20). When the bandwidth of the spectrometer is small relative to the variations of the spectrum, the measured signal is simply the optical spectrum of the source. It is clear that the signal measured in this way contains no information about the spectral phase of the pulse and can at best lead to the optical spectrum when the filter passband is significantly narrower than the features of the spectrum of the pulse under test.

It is of particular importance that an equation equivalent to Eq. (2.23) can be written for all stationary filters; i.e., the output signal of a device built entirely with stationary amplitude and/or phase filters does not depend on the spectral phase of the pulse. The implication is that stationary-only filters are insufficient to gather information on the spectral phase of an optical pulse and can at best return information on the spectral intensity of the pulse.

In terms of the Wigner representation, the measurement of a pulse spectrum is written as
S(Ω;Γ)=dtdω2πW(t,ω)WS(t,ω;Ω,Γ),
(2.24)
where WS(t,ω;Ω,Γ) is the Wigner chronocyclic representation of the spectrometer response function, defined by
WS(t,ω;Ω,Γ)=dtSA(tt2;Ω,Γ)SA(t+t2;Ω,Γ)eiωt.
(2.25)
Equation (2.24) gives the same result as Eq. (2.23); that is, the measured signal is the frequency marginal of the pulse chronocyclic Wigner function, which is the spectrum of the source.

The important point is that all measurement techniques can be represented in terms of the overlap of the Wigner function of the test pulse (or pulse ensemble) and that of the apparatus. This provides an important insight into ways that the experimental data may be inverted to obtain the pulse field itself, as discussed in subsequent sections.

Measurement of the temporal intensity. The measurement of the temporal intensity of an optical pulse is in some sense the conjugate operation of the measurement of its optical spectrum. If the pulse under test is sent to a fast square-law detector (or equivalently, a fast shutter followed by a time-integrating detector), the measured output is
S(τ;τg)=dt|NA(tτ;τg)|2I(t).
(2.26)
Because of the relatively slow response time of photodetectors, the measured signal is usually only a blurred representation of the actual temporal intensity of an ultrashort optical pulse. However, direct photodetection is commonly used with longer pulses, such as the pulses used in optical telecommunication systems.

2.2c. Autocorrelations and Cross-Correlations

Intensity autocorrelation. The simplest technique for gathering at least moderate quantitative information about the temporal structure of an ultrashort pulse is the intensity autocorrelation. In a conventional autocorrelator, two pulse replicas are mixed in a nonlinear material, and the average power of the generated beam (measured with an integrating detector) is recorded as a function of the relative delay between the two replicas. By assuming a functional form for the temporal shape of the test pulse, one can estimate its duration from the autocorrelation trace. Because of its simplicity, autocorrelation is by far the most common method of measuring ultrashort optical pulses. However, the autocorrelation trace by itself provides little more than an estimate of the pulse duration.

A variety of schemes based on intensity correlation measurements were demonstrated during the late 1960s and early 1970s [40

40. J. A. Armstrong, “Measurement of picosecond laser pulse widths,” Appl. Phys. Lett. 10, 16–18 (1967). [CrossRef]

, 41

41. H. P. Weber, “Method for pulsewidth measurement of ultrashort light pulses generation by phase-locked lasers using nonlinear optics,” J. Appl. Phys. 38, 2231–2234 (1967). [CrossRef]

, 42

42. J. A. Giordmaine, P. M. Rentzepis, S. L. Shapiro, and K. W. Wecht, “Two-photon excitation of fluorescence by picosecond light pulses,” Appl. Phys. Lett. 11, 216–218 (1967). [CrossRef]

, 43

43. S. L. Shapiro, “Second harmonic generation in LiNbO3 by picosecond pulses,” Appl. Phys. Lett. 13, 19–21 (1968). [CrossRef]

, 44

44. D. H. Auston, “Measurement of picosecond pulse shape and background level,” Appl. Phys. Lett. 18, 249–251 (1971). [CrossRef]

]. One particular form, the second-order intensity autocorrelation function (AC) became one of the standard techniques in the field for nearly two decades and is still in use today. This technique uses the lowest-order nonlinear process available, and therefore operates at the lowest power possible for a nonlinear process. This is important for making measurements of pulse trains from mode-locked laser oscillators, whose energy is in the picojoule to nanojoule range. The most common approach to extracting information from this AC data, however, involves fitting an AC calculated from a specific pulse shape.

Interferometric autocorrelation. The AC is often extended to its so-called fringe-resolved form [48

48. T. Mindl, P. Hefferle, S. Schneider, and F. Dörr, “Characterisation of a train of subpicosecond laser pulses by fringe resolved autocorrelation measurements,” Appl. Phys. B 31, 201–207 (1983). [CrossRef]

] by using a collinear setup [Fig. 8(d)]. One advantage of the interferometric autocorrelation (IAC) is that it is sensitive to the phase of the electric field. Another advantage is that the quickly varying fringes lead to a natural calibration of the temporal axis, which is useful when characterizing few-cycle pulses. The upconverted signal is given by
IAC(τ)=dt|E(t)+E(tτ)|4.
(2.32)
The interferometric autocorrelation contains the intensity autocorrelation as well as correlation terms of E(t) and E(tτ). Since the field of the input pulse oscillates at a frequency ω0, the interferometric autocorrelation contains oscillating terms at the frequencies ω0 and 2ω0. These terms are phase sensitive and can in theory be used to estimate the temporal phase present on an optical pulse [49

49. J.-C. M. Diels, J. J. Fontaine, I. C. McMichael, and F. Simoni, “Control and measurement of ultrashort pulse shapes (in amplitude and phase) with femtosecond accuracy,” Appl. Opt. 24, 1270–1282 (1985). [CrossRef] [PubMed]

, 50

50. K. Naganuma, K. Mogi, and H. Yamada, “General method for ultrashort light-pulse chirp measurement,” IEEE J. Quantum Electron. 25, 1225–1233 (1989). [CrossRef]

]. Figures 8(e), 8(f) display the interferometric autocorrelations corresponding to a Gaussian Fourier-transform-limited pulse and a Gaussian pulse with a quadratic spectral phase. The interferometric autocorrelation is sensitive to the temporal phase of the pulse, and the two autocorrelations have different structures, although the corresponding intensity autocorrelations are similar.

Another area where cross-correlations have been used extensively is pulse shaping. Pulse shapers can transform an input pulse into a temporally shaped waveform with a temporal support much larger than the input pulse duration. In many cases, a description of the intensity of the output field is sufficient, and this can be obtained by cross correlating the output waveform with a replica of the input pulse (Fig. 4).

Finally, recovery of the intensity of a test pulse from triple cross-correlations of the intensity has been attempted [57

57. T. Feurer, S. Niedermeier, and R. Sauerbrey, “Measuring the temporal intensity of ultrashort laser pulses by triple correlation,” Appl. Phys. B 66, 163–168 (1998). [CrossRef]

]. The temporal intensity can in theory be reconstructed unambiguously from the two-dimensional triple correlation function measured as a function of the two relative delays between replicas of the test intensity.

2.2d. Classes of Pulse Characterization Devices

Measurement devices may be categorized by the arrangements of filters through which the test pulse is passed before being detected. A second classification involves the type of algorithm that is used to extract the pulsed field from the experimental data. The data is a function only of the filter parameters, and there should be a sufficient number of these, and of the right kind, that complete information about the test pulse is encoded in the data. The requirements that this places on the apparatus will be laid out in this subsection.

The filters are characterized by a set of parameters {pi}. For example. the shutter transmits any portion of the pulse that falls within a time window of duration τg near the opening time τ. Likewise the spectrometer transmits any portion of the pulse that falls within a spectral window of width Γ near the passband center frequency Ω. The modulator adds a time-dependent phase onto the pulse, whose magnitude depends on the modulation index ψ(2), and the time of arrival of the pulse compared with the peak of the modulation at time τ. The dispersive line adds a spectrally dependent phase onto the pulse, whose magnitude depends on the second-order dispersion ϕ(2) and the position of the pulse spectrum with respect to the reference frequency ωR.

Although these are not the most general linear response functions possible, they are sufficient for our purposes. Moreover, the categories they represent are complete, in that any linear filter may be synthesized from a sequence of such filters. A pulse measurement apparatus therefore consists of a sequence of filters in series or in parallel, or both, followed by an integrating detector, as shown in Fig. 9.

Within this framework, the signal measured by a detector following a sequence of such filters is a function of the filter parameters. It may be written as the overlap of the Wigner function of the pulse with a chronocyclic window function F(t,ω;{pi}) depending only on the properties of the arrangement of linear filters:
D({pi})=dtdω2πW(t,ω)F(t,ω;{pi}).
(2.34)
The action of F is to smooth the pulse Wigner function to yield a positive signal measurable by a square-law detector. The trick is to design F such that W can be recovered from the experimental data D. If F is able, by suitable choices of the pi, to explore all of the phase space occupied by the pulse, then D contains sufficient information to reconstruct the pulse field. Indeed, this is both a necessary and sufficient condition for characterizing the pulse. The window function formed by a sequence of time-stationary filters can be shown to be dependent only on frequency ω, a window function formed using time-nonstationary filters on t alone. They do not generate a window function that can move throughout the phase space. Therefore all apparatuses must contain at least one time-stationary filter and one time-nonstationary filter. This is a necessary, but not sufficient condition. These elements may be combined in a number of different ways for pulse measurement. It is clear that the final filter (that is, the one immediately preceding the detector), must be an amplitude filter (or at least not be a phase-only filter), as phase-only filters will not change the detected signal. This restricts the number of configurations of filters that are allowed.

If arranged in series, there are four combinations. Two of these belong to the class of spectrographic measurements, and two to the class of tomographic measurements. If arranged in parallel, these elements give another four schemes. All of the latter are based on interferometry: two in the time domain, and two in the spectral domain. A final amplitude filter that is either a shutter (a time gate) or a spectrometer (a frequency gate) enables a slow detector to measure the interferogram. The full catalog of possible configurations is shown in Fig. 10: we describe each separately below.

It is instructive to revisit the autocorrelation in light of the chronocyclic representation. It consists of a delay [time-stationary filter S̃LP(ω;τ)=eiωτ] followed by a shutter (time-nonstationary filter NA(t)), so that the detected signal is
D(τ)=dtdω2πW(t,ω)F(t;τ)=dtI(t)F(t;τ).
(2.35)
The shutter response function, however, is the pulse field itself, so that NA=E. It is clear that F does not provide the necessary phase-space coverage and that this particular arrangement of filters is inadequate to characterize the electric field of an optical pulse in a general manner.

2.2e. In-Series Filtering Measurements

Spectrography. Spectrography refers to schemes in which a simultaneous measurement of the spectral and temporal intensity of the pulse is made. In particular, methods of this type are based on the measurement of the spectra of different temporal sections of the pulse, or on the measurement of the temporal intensity of different spectral sections (in which case it is known as a “sonogram”). For the former, one needs a fast shutter opening at time τ (with a speed comparable with, though not necessarily as fast as, the test pulse itself) followed by a high-resolution spectrometer with passband at frequency Ω [Fig. 10(a)].

For this arrangement, the Wigner function of the measurement apparatus is
WM(t,ω;{Ω,τ})=dω2π|S̃A(ωΩ)|2dtNA(t+t2τ)NA*(tt2τ)exp[i(ωω)t].
(2.36)
In the limit of narrowband filtering, i.e., |S̃A(ω;Ω)|2δ(ωΩ), the apparatus function occupies the minimum volume of phase space allowed by Fourier’s theorem and therefore smooths the pulse Wigner function by the least possible amount. In this limit, the signal may be written as
D(Ω,τ)=dω2πdtW(t,ω)WM(tτ,ωΩ)=|dtE(t)NA(tτ)exp(iΩt)|2.
(2.37)
In this case, the experimental trace is the Gabor spectrogram with a window NA. Provided the gate function NA is known with sufficient precision, the signal is directly invertible to the pulse field, although an iterative deconvolution algorithm is usually required.

The elements may also be used in the reverse order [Fig. 10(b)]. In this case, referred to as sonography, the test pulse first encounters a low-resolution spectrometer, then a very fast shutter. The Gabor sonogram is defined in an analogous manner to Eq. (2.37) by
D(Ω,τ)=|dω2πS̃A(ωΩ)Ẽ(ω)eiωτ|2,
(2.38)
where this time the spectral gate is of the form of Eq. (2.20). Again, a fast shutter may also be synthesized by a nonlinear optical process. In fact, it is clear from the form of the integral kernel of the Gabor spectrogram why this is so: the gate function is a time-shifted replica of the test pulse, and the product of the test pulse with itself can be realized by sum-frequency generation in a second-order nonlinear interaction.

The first method to provide complete information about an ultrashort optical pulse used a shutter based on upconversion of the spectrally filtered (and therefore temporally stretched) test pulse with the test pulse itself. The shutter speed is then equal to the duration of the test pulse, and something close to a sonogram of the test pulse can be measured [58

58. J. L. A. Chilla and O. E. Martinez, “Direct determination of the amplitude and the phase of femtosecond light pulses,” Opt. Lett. 16, 39–41 (1991). [CrossRef] [PubMed]

].

In measuring either spectrograms or sonograms, it is important that the first filter encountered by the test pulse have low resolution in its appropriate domain (a nominally slow shutter for spectrography, and a low-resolution spectrometer for sonography), and that the second filter have high resolution (a high-resolution spectrometer for spectrography and a fast shutter for sonography). This makes the measured spectrogram or sonogram most similar to the true Gabor-type spectrogram or sonogram of the test pulse.

As discussed previously, nonlinear optics is not a necessity for pulse characterization. Its use in spectrographic techniques when characterizing sub-100fs pulses is required because there is no other way to build a shutter with a similar responses time. The test pulse itself is, ipso facto, the shortest-duration entity to which the experimenter has access, thereby setting a lower limit on the shutter speed. Because of this constraint, measurements of a sonogram of a femtosecond optical pulse always have lower resolution than the corresponding spectrogram.

When nonlinear optics is used, the measured spectrograms are nonlinear functionals of the test pulse Wigner function. A true spectrogram, such as the Gabor spectrogram, is a linear functional of the test pulse and the known shutter response Wigner function. Reconstruction of the pulse field from the Gabor spectrogram requires a deconvolution, but has in most cases a unique solution. Reconstruction of the pulse from nonlinear spectrograms requires an iterative nonlinear deconvolution where the convolution function depends on the unknown pulse. This problem might have multiple solutions. As a consequence, much of the effort devoted to these techniques has concentrated on devising robust iterative algorithms for extracting the field from the measured quantity, and this is discussed in Section 3.

An alternative approach to pulse reconstruction is to ensure that the apparatus operates with parameters that allow approximate direct inversion. This is possible with sonography [59

59. J. L. A. Chilla and O. E. Martinez, “Analysis of a method of phase measurement of ultrashort pulses in the frequency domain,” IEEE J. Quantum Electron. 27, 1228–1235 (1991). [CrossRef]

], for example, and methods have been suggested for spectrography [60

60. J. Paye, M. Ramaswamy, J. G. Fujimoto, and E. P. Ippen, “Measurement of the amplitude and phase of ultrashort light pulses from spectrally resolved autocorrelation,” Opt. Lett. 18, 1946–1948 (1993). [CrossRef] [PubMed]

] as well as the spectrally and temporally resolved upconversion technique (STRUT) [61

61. J.-P. Foing, J.-P. Likforman, M. Joffre, and A. Migus, “Femtosecond pulse phase measurement by spectrally resolved up-conversion: application to continuum compression,” IEEE J. Quantum Electron. 28, 2285–2290 (1992). [CrossRef]

, 62

62. J. Rhee, T. Sosnowski, A. C. Tien, and T. Norris, “Real-time dispersion analyzer of femtosecond laser pulses with use of a spectrally and temporally resolved upconversion technique,” J. Opt. Soc. Am. B 13, 1780–1785 (1996). [CrossRef]

].

The combination of the temporal modulator and dispersive stretcher allows one to perform an operation called “temporal imaging” by analogy to the operation performed by an optical imaging system in the spatial domain. Consider a standard optical imaging device, consisting of an object placed some distance before a lens, and an image plane (at which is placed a detector) some distance after the lens. The underlying physics of image formation is that light from the object undergoes diffraction in free space for a prescribed distance, then refraction by the lens, then further diffraction before being detected. For the appropriate adjustment of the distances and power of the lens, a magnified image of the object can be formed. The time–frequency analog is that the grating stretcher plays the role of diffraction and the temporal modulator the role of the lens. Using such a setup, a temporally magnified image of the input short pulse can be constructed, which is easy to measure by using detectors with response times much longer than the input pulse.

2.2f. In-Parallel Filtering Measurements

Interferometry refers to the situation where the phase of the test pulse is encoded into the intensity by means of mixing with a second pulse, which may be an ancillary reference pulse or the test pulse itself. These two categories are known as “test-plus-reference” and “self-referencing” interferometry, respectively. They are both direct techniques, in that it is possible to reconstruct the correlation function in either the time domain or the frequency domain directly (i.e., noniteratively) from the recorded intensity distributions. A general model of this category of measurement devices may be developed in terms of a sequence of in-parallel linear filters. In this model each pulse in the ensemble is split into two replicas at a beam splitter, and each replica is independently filtered before being recombined. The interference of the field from the parallel pathways introduces structure on the output intensity distribution, which then carries information about both the amplitude and the phase of the correlation function of the input field. If the ancillary port of the input beam splitter is empty, then the interferometer is said to be self-referencing. Alternatively, if the ancillary port is used to inject a characterized reference pulse, then it is possible to reconstruct the electric field of the test pulse in a rather straightforward manner [63

63. L. Lepetit, G. Chériaux, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B 12, 2467–2474 (1995). [CrossRef]

, 64

64. D. N. Fittinghoff, J. L. Bowie, J. N. Sweetser, R. T. Jennings, M. A. Krumbügel, K. W. Delong, R. Trebino, and I. A. Walmsley, “Measurement of the intensity and phase of ultraweak, ultrashort laser pulses,” Opt. Lett. 21, 884–886 (1996). [CrossRef] [PubMed]

]. Of course, this approach requires one to first obtain a well-characterized reference pulse.

One significant advantage of direct techniques compared with phase-space techniques is that the entire space over which the phase-space or correlation functions are defined need not be explored if the pulse train is assumed to consist of identical pulses. Only a single section of one quadrature of the (complex) correlation function is necessary to obtain the electric field amplitude and phase, and these are precisely what is recorded by direct techniques.

Test-plus-reference interferometry. The most common form of test-plus-reference interferometry is Fourier-transform spectral interferometry (FTSI) [63

63. L. Lepetit, G. Chériaux, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B 12, 2467–2474 (1995). [CrossRef]

, 64

64. D. N. Fittinghoff, J. L. Bowie, J. N. Sweetser, R. T. Jennings, M. A. Krumbügel, K. W. Delong, R. Trebino, and I. A. Walmsley, “Measurement of the intensity and phase of ultraweak, ultrashort laser pulses,” Opt. Lett. 21, 884–886 (1996). [CrossRef] [PubMed]

]. In this approach, the test and the reference pulse are delayed in time with respect to each other by τ before combining at the input beam splitter. The detected signal (interferogram) is then S(ω;τ)=|Ẽ(ω)+ẼR(ω)eiωτ|2, where ẼR and Ẽ are the spectral representations of the analytic signal of the reference and the test pulse. The spectral phase difference between test and reference pulses is encoded in the relative positions of the spectral fringes with respect to the nominal spacing of 2πτ and can be extracted by using a three-step algorithm involving a Fourier transform to the time domain, a filtering operation, and an inverse Fourier transform. The phase of the reference pulse must then be subtracted, leaving the spectral phase of the test pulse as required. A measurement of the test pulse spectrum then provides sufficient information to characterize the pulse. In common with all interferometric methods, the data set has one parameter, frequency, and may therefore be collected by using a one-dimensional detector array. This leaves the second dimension of a camera, for example, available for coding information about other degrees of freedom of the test pulse, such as the spatial phase. This method is therefore easily extended to full space–time characterization of the test field, again provided that a suitable (i.e., fully space–time characterized) reference pulse is available.

Self-referencing interferometry. It is possible to extract the phase of a field without a known reference pulse by gauging one spectral or temporal component of the field with another component. This is known as “self-referencing interferometry.” In this approach, the goal is to reconstruct the correlation function in either the time domain or the frequency domain directly (i.e., noniteratively) from one or several recorded intensity distributions. In fact, when the pulse train is coherent, it is necessary only to measure a section of the two-time or two-frequency correlation function in order to reconstruct the pulse field [34

34. C. Iaconis, V. Wong, and I. A. Walmsley, “Direct interferometric techniques for characterizing ultrashort optical pulses,” IEEE J. Sel. Top. Quantum Electron. 4, 285–294 (1998). [CrossRef]

].

The in-parallel amplitude-only filters select either two frequency or two time slices of the pulse that beat together at the output of the interferometer. These are the time-domain analogs of Young’s double-slit interferometer [Figs. 10(e), 10(f)].

In the spectral domain [Fig. 10(e)], the center frequencies of the spectral filters are ωC1 and ωC2, and each has the same bandwidth Γ. The selected frequency components are recombined, giving rise to temporal fringes—or a time-dependent modulation of the intensity—at the output. These are resolved by using a time gate or a fast shutter. The signal recorded by the square-law detector is a function of the spectral filter center frequencies as well as the time of maximum transmission τ of the time gate,
D(ωC1,ωC2,τ)=dt|NA(tτ)dω2π[S̃A(ωωC1)+S̃A(ωωC2)]Ẽ(ω)exp(iωt)|2.
(2.41)
The detected signal takes on a particularly useful form when the passband of the spectral filters is much narrower than the spectrum of the input pulses and the time gate is short. In this case, the functions NA(t) and S̃A(ω) may be replaced by Dirac δ functions in the appropriate domains, and Eq. (2.41) simplifies to
D(ω+Δω2,ωΔω2,τ)=Ĩ(ω+Δω2)+Ĩ(ωΔω2)+2|C͌(Δω,ω)|cos{arg[C͌(Δω,ω)]Δωτ},
(2.42)
where ω=(ωC1+ωC2)2 and Δω=ωC1ωC2. This is an interferogram, for which the visibility of the fringes, occurring with nominal temporal period 2πΔω, provides a measure of the magnitude of C͌(Δω,ω). The location of the fringes along the delay axis τ provides a relative measure of the phase of C͌(Δω,ω). Each temporal beat note in the fringe pattern supplies enough information to reconstruct the two-frequency correlation function at the single point (Δω,ω).

A complementary form of interferometer consists of an in-parallel fast time-gate (time-nonstationary amplitude-only filters) followed by a spectral filter (time-stationary amplitude-only filter), as pictured in Fig. 10(f). The two replicas of the pulse are independently sampled with variable times,τ1 and τ2, before being recombined. The spectral beats, resulting from the overlap of the two time slices, are resolved by a spectrometer. The resulting signal, for the case of a very fast time gate and a very high-resolution spectrometer, written in terms of the center-time (t=(τ1+τ2)2) and difference-time (Δt=τ1τ2) coordinates, is the temporal interferogram
D(t+Δt2,tΔt2,Ω)=I(t+Δt2)+I(tΔt2)+2|C(t,Δt)|cos{arg[C(t,Δt)]+ΔtΩ}.
(2.43)
The visibility of the spectral fringes, occurring at the spectral period 2πΔt, is a measure of the magnitude of the two-time correlation function at the point (t,Δt), while the position of the fringes is linked to the phase of the correlation function.

An entirely analogous argument may be made for temporal shearing interferometers [Fig. 10(f)]. In this case, the delay line in one arm of the interferometer causes the pulses on recombining at the second beam splitter to exhibit temporal beats in their intensity that may be resolved by a fast time gate. This latter element is the amplitude-only filter that replaces the spectrometer required in the spectral shearing interferometer. In this arrangement, a temporal linear phase modulator may be used to provide a temporal carrier for the two-time correlation function in the interference term. This is accomplished by frequency shifting one of the pulses with respect to the other by a shear ψ(1) and by introducing a relative delay Δt between the pulses. The signal detected as a function of τ, the delay of the time-nonstationary gate, which is assumed to be of infinitesimal duration, is then
D(τ,Δt;ψ(1))=I(tC+Δt2)+I(tCΔt2)+2|C(tC,Δt)|cos{arg[C(tC,Δt)]ψ(1)(tCΔt2)},
(2.46)
where tC=τ+Δt2 is the center time. A similar algorithm as described for the spectral shearing interferogram may be used to extract the temporal phase of the test pulse in this case. In practice, however, it is very difficult to provide a short enough time gate to enable this method to work. Nonlinear optical interactions that cross correlate the interferogram with the test pulse will not provide enough temporal resolution to resolve the fringes. Therefore this method is restricted to pulses whose duration is long enough that an externally controlled time gate, such as a telecommunication pulse carver, may be used. This is typically in the regime of several tens of picoseconds or longer.

2.2g. Joint Measurements

There are several modifications to the methods that have allowed some headway. The spectrum of the pulse helps in determining whether the pulse is close to the Fourier-transform limit and is an obvious second piece of data that is relatively easy to measure. Several iterative schemes have been developed to extract the pulse shape from a correlation and the spectrum [50

50. K. Naganuma, K. Mogi, and H. Yamada, “General method for ultrashort light-pulse chirp measurement,” IEEE J. Quantum Electron. 25, 1225–1233 (1989). [CrossRef]

, 65

65. J. W. Nicholson, J. Jasapara, W. Rudolph, F. G. Omenetto, and A. J. Taylor, “Full-field characterization of femtosecond pulses by spectrum and cross-correlation measurements,” Opt. Lett. 24, 1774–1776 (1999). [CrossRef]

, 66

66. J. W. Nicholson, M. Mero, J. Jasapara, and W. Rudolph, “Unbalanced third-order correlations for full characterization of femtosecond pulses,” Opt. Lett. 25, 1801–1803 (2000). [CrossRef]

, 67

67. K. H. Hong, Y. S. Lee, and C. H. Nam, “Electric-field reconstruction of femtosecond laser pulses from interferometric autocorrelation using an evolutionary algorithm,” Opt. Commun. 271, 169–177 (2007). [CrossRef]

]. They provide varying degrees of success in extracting the pulse fields, but all share the same characteristic that they are very sensitive to noise in the data [46

46. J. H. Chung and A. M. Weiner, “Ambiguity of ultrashort pulse shapes retrieved from the intensity autocorrelation and the power spectrum,” IEEE J. Sel. Top. Quantum Electron. 7, 656–666 (2001). [CrossRef]

].

Attempts at retrieving the electric field of the pulse from a set of intensity autocorrelations measured after various amounts of second-order dispersion have been made [68

68. R. G. M. P. Koumans and A. Yariv, “Time-resolved optical gating based on dispersive propagation: a new method to characterize optical pulses,” IEEE J. Quantum Electron. 36, 137–144 (2000). [CrossRef]

, 69

69. R. G. M. P. Koumans and A. Yariv, “Pulse characterization at 1.5  micron using time-resolved optical gating based on dispersive propagation,” IEEE Photon. Technol. Lett. 12, 666–668 (2000). [CrossRef]

]. The use of the intensity autocorrelation in the temporally resolved optical gating (TROG) technique, instead of a direct intensity measurement, significantly increases the complexity of the retrieval compared with tomographic techniques.

Deterministic changes of the spectral phase of the pulse with a pulse shaper have also attracted some attention. For a given spectrum, the autocorrelation signal at τ=0 is maximized by a flat spectral phase. Since this signal can be measured directly by doubling the pulse and measuring the energy of the converted pulse with a photodetector, an iterative algorithm can be used to modify the spectral phase and maximize the measured signal. For a given pulse, the spectral phase introduced by the pulse shaper when the maximum is reached corresponds to the opposite of the spectral phase of the input pulse, therefore leading to a measurement of the phase by adaptive pulse shaping [70

70. D. Meshulach, D. Yelin, and Y. Silberberg, “Adaptive ultrashort pulse compression and shaping,” Opt. Commun. 138, 345–348 (1997). [CrossRef]

]. In a multiphoton intrapulse interference phase scan (MIIPS), the spectrum of the upconverted signal is used as a feedback mechanism when a spectral phase is scanned across the spectral support of the pulse with a pulse shaper [71

71. B. Xu, J. M. Gunn, J. M. Dela Cruz, V. V. Lozovoy, and M. Dantus, “Quantitative investigation of the multiphoton intrapulse interference phase scan method for simultaneous phase measurement and compensation of femtosecond laser pulses,” J. Opt. Soc. Am. B 23, 750–759 (2006). [CrossRef]

]. Iterations are required for accurately determining the spectral phase of the input pulse: the pulse shaper is also used to introduce a static spectral phase that attempts to compensate the spectral phase of the input pulse, and a specific multiphoton intrapulse interference phase scan trace is obtained when the shaper output pulse is Fourier-transform limited.

2.3. Conclusions

There are three general classes of measurement techniques for characterizing ultrashort optical pulses—spectrography, tomography, and interferometry—which lead to eight devices consisting of the smallest possible number of optical elements (two spectrographic, two tomographic, and four interferometric). All of these devices contain at least one time-stationary and one time-nonstationary filter, which may be linear in the input field.

The two spectrographic methods measure a smoothed version of the chronocyclic Wigner function by using sequential amplitude filters to make a simultaneous measurement of time and frequency. Tomographic methods measure projections of the chronocyclic Wigner function onto the frequency variable, following the application of a quadratic phase modulator to the input pulse. This serves to rotate the phase-space distribution of the pulse, so that a measurement of its modified spectrum reveals information about its initial orientation, and hence chirp. The data, consisting of a set of projections of the Wigner function for a range of phase-space rotations, can be deterministically inverted to retrieve the Wigner function itself.

Self-referencing interferometric techniques measure a point or a section of the two-frequency or two-time correlation function. A single section of either function is adequate for reconstructing the underlying electric field. Interferometers work by splitting each pulse in the ensemble into two replicas at a beam splitter, independently filtering the replicas, and then recombining them at a second beam splitter. The interference of the parallel pathways introduces structure on the output intensity distribution, which then carries information about both the amplitude and the phase of the correlation function. There are two interferometric devices that are analogs of Young’s double slits: either two spectral or two temporal slices are taken of the input pulse and fringes recorded in the temporal or spectral domains, respectively. Since these devices require time gates that are short compared with the input pulse duration, they are difficult to implement for femtosecond pulses. A more useful approach is via shearing interferometry. In this case, one of the pulses is shifted in frequency (or in time) with respect to the other, and an interference pattern recorded in the spectral (or temporal) domain. The simple and direct inversion algorithm gives a provably unique solution to the problem of pulsed field reconstruction.

3. Spectrography

3.1. Introduction

Some of the earliest attempts at measuring the chirp of optical pulses were based on spectrographic concepts. Such ideas also underpinned the first attempts at precisely characterizing the electric field of pulses. The concept of “chirp” was developed for microwave pulses and refers to the existence of a time-dependent instantaneous frequency—or, equivalently, a frequency-dependent group delay—in which all of the frequencies of the pulse do not arrive at the observer simultaneously. In 1971, Treacy quantified this quantity for pulses from a mode-locked Nd:glass laser by measuring the time of arrival of spectral slices of the pulse [72

72. E. B. Treacy, “Measurement and interpretation of dynamic spectrograms of picosecond light pulses,” J. Appl. Phys. 42, 3848–3858 (1971). [CrossRef]

]. This recording allowed the first evaluation of the optical chirp, which had been known for microwaves for some time. A description of early developments can be found in [26

26. C. Froehly, B. Colombeau, and M. Vampouille, “Shaping and analysis of picosecond light pulses,” in Progress in Optics. Vol. XX, E. Wolf ed. (North-Holland, 1983), pp. 63–153. [CrossRef]

]. Different implementations of the same concept were then developed and are usually referred to as time-resolved spectroscopy [73

73. A. S. L. Gomes, A. S. Gouveia-Neto, and J. R. Taylor, “Direct measurement of chirped optical pulses with picosecond resolution,” Electron. Lett. 22, 41–42 (1986). [CrossRef]

, 74

74. C. M. Olsen and H. Izadpanah, “Time-resolved chirp evaluations of Gbit/s NRZ and gain-switched DFB laser pulses using narrowband Fabry–Perot spectrometer,” Electron. Lett. 25, 1018–1019 (1989). [CrossRef]

, 75

75. K. Mori, T. Morioka, and M. Saruwatari, “Group velocity dispersion measurement using supercontinuum picosecond pulses generated in an optical fibre,” Electron. Lett. 29, 987–989 (1993). [CrossRef]

]. Another precursor to the spectrographic techniques used nowadays is the measurement of optical spectra of the upconverted signal in an intensity autocorrelator [76

76. A. Watanabe, H. Saito, Y. Ishida, and T. Yajima, “Computer-assisted spectrum-resolved SHG autocorrelator for monitoring phase characteristics of femtosecond pulses,” Opt. Commun. 63, 320–324 (1987). [CrossRef]

]. Chilla and Martinez’s implementation of sonograms using nonlinear wave mixing in a crystal has inspired most setups for sonographic measurements of femtosecond pulses [58

58. J. L. A. Chilla and O. E. Martinez, “Direct determination of the amplitude and the phase of femtosecond light pulses,” Opt. Lett. 16, 39–41 (1991). [CrossRef] [PubMed]

]. Spectrograms and sonograms are now widely used in ultrafast optics, and the development of phase retrieval algorithms enables full recovery of the amplitude and phase of the electric field without prior assumptions as to its functional form. The best-known example of this class of measurements is frequency-resolved optical gating (FROG) [28

28. R. Trebino, ed., Frequency Resolved Optical Gating: the Measurement of Ultrashort Optical Pulses (Kluwer Academic, 2002).

]. In this section, we describe the principles of spectrography, the apparatuses required for measuring spectrograms and sonograms, and the approaches available for extracting the pulse field from the experimental data. We also give some experimental implementations of these concepts adapted to ultrafast optics.

3.2. General Implementation of Spectrography

3.2a. Definitions

Figure 12 shows two typical arrangements for spectrographic measurements consisting of two sequential filters. In Fig. 12(a) the first filter is a time-nonstationary device modulating the electric field with a gating function g, and the second filter is a time-stationary filter described by R̃. The gating function can be delayed in time by a delay τ relative to the pulse under test, and the stationary filter can be scanned in frequency, Ω describing a parameter relevant to this filter, for example the center of its passband. The signal measured by a time-integrating detector is
S(τ,Ω)=dω2π|R̃(ωΩ)|2|dtE(t)g(tτ)exp(iωt)|2,
(3.1)
where the two integrals extend from to + in the time and frequency domains. The second filter is chosen to have a high resolution, which in this case implies a spectrometer capable of resolving all the features of the optical spectra after they pass through the first filter. This choice leads to minimal blurring of the spectrogram, and therefore more reliable inversion. From a formal point of view, the transfer function of the stationary filter may be replaced by a Dirac function, so that the measured experimental trace becomes
S(τ,Ω)=|dtE(t)g(tτ)exp(iΩt)|2.
(3.2)
This quantity is by definition the spectrogram of the electric field E measured with the window or gate g [37

37. L. Cohen, Time–Frequency Analysis, Prentice Hall Signal Processing Series (Prentice Hall PTR, 1994).

].

The order of the stationary and nonstationary filters can be inverted, so that the measurement is that of the temporal intensity of the pulse after spectral filtering [Fig. 12(b)]. The signal measured by a time-integrating detector is in this case
S(τ,Ω)=dt|g(tτ)|2|dω2πẼ(ω)R̃(ωΩ)exp(iωt)|2,
(3.3)
where g is the impulse response of the time gate and R̃ is the transfer function of the spectral filter. If the time-gating nonstationary filter has sufficient resolution to reveal all the temporal features of the spectrally filtered pulse, its response function can be formally replaced by a Dirac function and the experimental trace is
S(τ,Ω)=|dω2πẼ(ω)R̃(ωΩ)exp(iωτ)|2.
(3.4)
This quantity is by definition the sonogram of the electric field Ẽ measured with the spectral filter R̃. In practice, the measured sonograms are often given by Eq. (3.3) (where the nonstationary filter can be a function of the test pulse) instead of Eq. (3.4). For example, nonstationary filtering of femtosecond pulses is often provided by cross-correlation with another pulse, usually the unknown pulse under test itself [77

77. D. T. Reid, “Algorithm for complete and rapid retrieval of ultrashort pulse amplitude and phase from a sonogram,” IEEE J. Quantum Electron. 35, 1584–1589 (1999). [CrossRef]

]. Spectrograms and sonograms should be understood as making simultaneous measurements of the time and frequency degrees of freedom of the test pulse. Note that the spectrogram and sonogram given by Eqs. (3.2, 3.4) are mathematically equivalent, and the spectrogram calculated from the temporal representations E(t) and g(t) is the sonogram calculated from the spectral representations Ẽ(ω) and g̃(ω). Mathematical properties of these time–frequency distributions can be found, for example, in [37

37. L. Cohen, Time–Frequency Analysis, Prentice Hall Signal Processing Series (Prentice Hall PTR, 1994).

].

3.2b. Wigner Representation

The spectrogram of Eq. (3.2) and the sonogram of Eq. (3.4) can be written as a double convolution of the Wigner function of the test pulse WE with the Wigner function of the apparatus Wg:
S(τ,Ω)=dtdω2πWE(t,ω)Wg(tτ,Ωω).
(3.5)
The spectrogram is the result of the measurement of the Wigner function of the pulse in the chronocyclic space (ω,τ) with a measurement device having an instrument function equal to the Wigner function of the time or frequency gate. (Note that although a Wigner function may have negative values, the convolution of two Wigner functions is always nonnegative, so that the signal is always a physically realizable quantity.) Varying the delay τ and frequency Ω is equivalent to moving the instrument function around the chronocyclic space. It is clear that this motion must encompass the portion of the chronocyclic space where the Wigner function of the pulse under test has nonzero values. It is usually desirable to have an instrument function of area as small as possible in the chronocyclic space to provide minimal blurring of the measured Wigner function. However, the size of the support of any Wigner function has a lower bound; i.e., the area of the chronocyclic space where it is nonzero is larger than π. This lower bound arises from Fourier’s principle; if it were not so, there could exist an apparatus Wigner function that was highly localized in both time and frequency and that would, therefore, be able to measure with high precision the time and frequency variables. This contradicts Fourier’s theorem concerning conjugate variables. In fact, a rapid time-nonstationary filter realizes good temporal resolution but provides little spectral information about the test pulse. Its Wigner function has a correspondingly small extension in the temporal variable, but large spread in the spectral variable. In contrast, a narrowband time-stationary filter as used in the sonogram provides good spectral resolution but little temporal resolution, and its Wigner function has small extension in the spectral variable but large spread in the temporal variable. The spectrogram and sonogram are therefore always blurred versions of the Wigner function of the pulse under test, in the way described by Eq. (3.5).

3.2c. Chirp Representation

The first-order moments of the Wigner function can be linked to the group delay and instantaneous frequencies defined from the first derivatives of the spectral phase and the temporal phase of the electric field. The first-order moments of the spectrogram are by definition
ΩS(τ)=dΩΩS(τ,Ω)dΩS(τ,Ω),
(3.6)
TS(Ω)=dττS(τ,Ω)dτS(τ,Ω).
(3.7)
One can show that
ΩS(τ)=dtIE(t)Ig(tτ)[ΩE(t)+Ωg(tτ)]dtIE(t)Ig(tτ),
(3.8)
TS(Ω)=dωIE(ω)Ig(Ωω)[TE(ω)Tg(Ωω)]dωIE(ω)Ig(Ωω),
(3.9)
where the subscripts E and g refer to the test pulse and the time-nonstationary filter, so that, for example, IE(t) is the temporal intensity of the test pulse, and Tg(ω) the frequency-dependent group delay of the response function of the nonstationary filter. Because of the symmetric role played by E and g in the definition of the spectrogram, these moments depend identically upon the properties of the pulse and the gate, and the ability of a spectrogram or sonogram to represent chirp in the test pulse is linked to the properties of the gate. The first-order frequency moment of a spectrogram measured with a rapid time gate (with real response function) is the instantaneous frequency of the pulse, given by
ΩS(τ)=ΩE(τ)=ψt(τ).
(3.10)
A spectrogram implemented with a gate that is narrowband and real in the spectral domain leads to the equivalence of the spectrogram group delay and the test pulse group delay:
TS(Ω)=TE(Ω)=ϕω(Ω).
(3.11)
Figures 13(a), 13(b) display the spectrogram of a Gaussian pulse with second- and third-order dispersion calculated with a real gate. Note that the ridge of the spectrogram follows a curve corresponding to the group delay in the pulse, which is a straight line for second-order dispersion and a parabola for third-order dispersion. As expected, the negative values of the Wigner function in the latter case have been washed out in the convolution process. The ability of the spectrogram and sonogram to represent chirp in an intuitive manner finds application in signal representation and processing. They are time-tested concepts and are still in use today.

3.3. Inversion Procedures for Spectrographic Techniques

The basic problem behind the inversion of the spectrogram is the determination of a relevant quantity describing the train of pulses under test (e.g., chirp, electric field, or Wigner function) from the measured time–frequency distribution. In some implementations of spectrographic techniques, the gate is unknown; for example it can be a function of the pulse under test itself in FROG, where the time-nonstationary filter is synthesized by a nonlinear interaction with a replica of the unknown pulse under test. The inversion approaches are classified here as chirp retrieval, Wigner deconvolution, and phase retrieval.

3.3a. Chirp Retrieval

A quantitative assessment of the chirp of the test pulse can be obtained from a spectrogram or sonogram by calculating its first-order moments by using Eqs. (3.10, 3.11), or simply by locating the delay at which the spectrogram has a maximum for each frequency or, equivalently, the frequency at which the spectrogram has a maximum for each delay (assuming that the pulse structure is simply enough that the maxima are unique). These properties were understood very early on and are at the basis of the works of Treacy [72

72. E. B. Treacy, “Measurement and interpretation of dynamic spectrograms of picosecond light pulses,” J. Appl. Phys. 42, 3848–3858 (1971). [CrossRef]

] and Chilla and Martinez [58

58. J. L. A. Chilla and O. E. Martinez, “Direct determination of the amplitude and the phase of femtosecond light pulses,” Opt. Lett. 16, 39–41 (1991). [CrossRef] [PubMed]

], who determined the group delay as a function of frequency for an optical pulse by spectrally filtering the pulse and determining the time of arrival of the wave packets centered at the corresponding frequencies. Precise estimation of the chirp is made difficult by the fact that the second-order moments of the time–frequency distribution along one axis increase significantly when tight filtering is performed along the conjugate variable [37

37. L. Cohen, Time–Frequency Analysis, Prentice Hall Signal Processing Series (Prentice Hall PTR, 1994).

]. For example, a spectrogram measured by using a short nonstationary filter leads to a large spread of the spectrogram along the frequency axis, which means that the practical determination of the instantaneous frequency requires a very high signal-to-noise ratio. Another limitation of this approach, in common with many methods, is that the chirp and the group delay are measures of the derivative of the phase with respect to time or frequency, respectively. To extract the full phase of the test pulse field, it is necessary to integrate the measured quantities, which can be done when the support of the field is continuous but is difficult otherwise. Such an approach would not perform well, for instance, in the characterization of pulses with disjoint spectral or temporal support or pulses with phase jumps (an example of the latter is a train of pulses used in telecommunication, such as carrier-suppressed return-to-zero pulses where adjacent pulses differ by a π phase shift [78

78. P. Winzer, C. Dorrer, R. J. Essiambre, and I. Kang, “Chirped return-to-zero modulation by imbalanced pulse carver driving signals,” IEEE Photon. Technol. Lett. 16, 1379–1381 (2004). [CrossRef]

]).

3.3b. Wigner Deconvolution

There is in principle a more direct way to extract the field from the spectrogram. When the gate response function is known, the corresponding apparatus Wigner function Wg is known, and the Wigner function of the pulse under test can in principle be obtained by inverting the convolution of Eq. (3.5) [79

79. V. Wong and I. A. Walmsley, “Phase retrieval in time-resolved spectral phase measurement,” Proc. SPIE 2377, 178–186 (1995). [CrossRef]

]. The steps necessary to perform such an operation are the calculation of the double Fourier transform of the measured spectrogram or sonogram, the division of this quantity by the double Fourier transform of Wg, and the calculation of the inverse double Fourier transform of the obtained quantity, which leads to the Wigner function of the test pulse, followed by the calculation of the electric field of the test pulse from its Wigner function. However, direct deconvolution is highly sensitive to the precision with which the gate response function is known and to the signal-to-noise ratio and is prone to error at points in the phase space where the Fourier transform of the Wigner function of the gate takes zero values. Further, this approach does not take into account additional information such as the degree of coherence of the test pulse ensemble, nor can it easily include any assumptions about this. Therefore this approach is not widely used in practice.

3.3c. Phase Retrieval

The retrieval of E and g is equivalent to the retrieval of the phase of the short-term Fourier transform dtE(t)g(tτ)exp(iΩt). The spectrogram is by definition the modulus square of the latter quantity, and once the short-term Fourier transform is known, both E and g can be obtained by Fourier transformation. Spectrogram inversion therefore falls into the category of phase retrieval problems. These problems have been studied extensively in optics, owing to the fact that common square-law detectors, such as charge-coupled device arrays (CCDs) used in imaging, provide only intensity information. Various phase retrieval algorithms have been used for such inversion in the context of imaging, and phase retrieval for ultrafast optical metrology can be traced back to the spectrogram inversion by Kane and Trebino [80

80. D. J. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993). [CrossRef]

] and the later sonogram inversion by Wong and Walmsley [39

39. V. Wong and I. A. Walmsley, “Ultrashort-pulse characterization from dynamic spectrograms by iterative phase retrieval,” J. Opt. Soc. Am. B 14, 944–949 (1997). [CrossRef]

]. The general approach to iterative inversion is to locate the intersections of two sets of two-dimensional functions corresponding to two constraints. The first constraint is that the modulus square of the short-term Fourier transform must match the experimentally measured spectrogram. The second constraint is that the experimental signal should be consistent with the functional form of spectrogram of a pulse gated by a gate; i.e., it can be written as Eq. (3.2) or (3.4). There can also be additional constraints, such as the functional dependence between the pulse and gate, or the spectral characteristics of the field or gate. Since the two sets of constraints are not convex, convergence is not guaranteed, but iterating by projecting the solution at each step onto each set has proved a robust way of inverting the spectrogram. Projection on the set of functions satisfying the modulus constraint is easily performed by replacement of the modulus with the square-root of the measured spectrogram. Projection on the set of functions satisfying the spectrogram mathematical form was initially performed by using an error minimization algorithm [81

81. K. W. Delong, R. Trebino, and B. White, “Simultaneous recovery of two ultrashort laser pulses from a single spectrogram,” J. Opt. Soc. Am. B 12, 2463–2466 (1995). [CrossRef]

]. A more efficient algorithm to achieve this task is the principal component generalized projection algorithm (PCGPA) [82

82. D. J. Kane, G. Rodriguez, A. J. Taylor, and T. S. Clement, “Simultaneous measurement of two ultrashort laser pulses from a single spectrogram in a single shot,” J. Opt. Soc. Am. B 14, 935–943 (1997). [CrossRef]

, 83

83. D. J. Kane, “Recent progress toward real-time measurement of ultrashort laser pulses,” IEEE J. Quantum Electron. 35, 421–431 (1999). [CrossRef]

] (Fig. 14).

3.3d. Ambiguities, Accuracy, Precision, and Consistency

Ambiguities. An ambiguity in phase retrieval arises when more than one phase function can be assigned to the reconstructed field while satisfying all constraints of the inversion problem. Consider, for example, the spectrogram
S(τ,Ω)=|dtE(t)g(tτ)exp(iΩt)|2=|dtg*(t)E*[(tτ)]exp(iΩt)|2.
(3.12)
It is clear that the pairs [E(t),g(t)] and [g*(t),E*(t)] are always solutions of the same phase retrieval problem, regardless of the inversion algorithm. This is called the time-reversal ambiguity, because both a specific pulse and gate and their time-reversed versions produce the same spectrogram. Ambiguities may arise particularly in blind deconvolution when no prior or side information about the pulse or the gate is available. Often it is possible to obtain such information experimentally by measuring the optical spectrum of the pulse, for example. Further, in cases where the gate is a prescribed function, say using a temporal modulator with an external drive signal unrelated to the test pulse, it is possible to make use of the independently measured gate function for a range of different test pulses.

Equation (3.12) demonstrates the important direction-of-time ambiguity of SHG-FROG, as explained below in Subsection 3.5b. In this case, the gate is derived from the test pulse so that g=E, and the inversion will yield either E(t) or E*(t). This implies that a single SHG-FROG measurement cannot determine the direction of time unless one has additional information about the test pulse structure. Various studies on ambiguities for specific implementations of FROG can be found in [28

28. R. Trebino, ed., Frequency Resolved Optical Gating: the Measurement of Ultrashort Optical Pulses (Kluwer Academic, 2002).

, 86

86. D. Keusters, H.-S. Tan, P. O’Shea, E. Zeek, R. Trebino, and W. S. Warren, “Relative-phase ambiguities in measurements of ultrashort pulses with well-separated multiple frequency components,” J. Opt. Soc. Am. B 20, 2226–2237 (2003). [CrossRef]

, 87

87. B. Yellampalle, K. Kim, and A. J. Taylor, “Amplitude ambiguities in second-harmonic generation frequency-resolved optical gating,” Opt. Lett. 32, 3558 (2007). [CrossRef] [PubMed]

, 88

88. B. Yellampalle, K. Kim, and A. J. Taylor, “Amplitude ambiguities in second-harmonic generation frequency-resolved optical gating: erratum,” Opt. Lett. 33, 2854 (2008). [CrossRef]

, 89

89. L. Xu, E. Zeek, and R. Trebino, “Simulations of frequency-resolved optical gating for measuring very complex pulses,” J. Opt. Soc. Am. B 25, A70–A80 (2008). [CrossRef]

].

Accuracy. The accuracy of a diagnostic quantifies the similarity between the measured quantity and the physical quantity. This can be specified only if a well-known test pulse is available or by means of numerical simulations. There have been no extensive studies of these for modern ultrafast spectrographic methods, especially in the presence of noise or nonoptimal experimental conditions. Some simulations relevant to this issue can be found in [90

90. D. N. Fittinghoff, K. W. Delong, R. Trebino, and C. L. Ladera, “Noise sensitivity in frequency-resolved-optical-gating measurements of ultrashort optical pulses,” J. Opt. Soc. Am. B 12, 1955–1967 (1995). [CrossRef]

].

Consistency. The consistency of the inversion of a spectrogram or sonogram specifies the degree to which the data reconstructed from the solution matches the experimental data; i.e., it tells how well the inversion algorithm for the problem worked. The two constraints that are used in inverting the data yield their own consistency criterion:
  • The rms difference between the measured experimental trace and the trace calculated from the retrieved solution, also known as the “FROG error,” quantifies the match with the experimentally measured trace.
  • The relative magnitude of the singular values given by the singular value decomposition quantifies the match with the outer-product form in the decomposition of the spectrogram: a single nonzero singular value corresponds to a perfect decomposition as an outer product. The distribution of singular values may be used to evaluate the convergence of the algorithm [91

    91. D. J. Kane, F. G. Omenetto, and A. J. Taylor, “Convergence test for inversion of frequency-resolved optical gating spectrograms,” Opt. Lett. 25, 1216–1218 (2000). [CrossRef]

    ].

Precision. Evaluating the precision of a measurement device requires the ability to compare several retrievals of the same quantity by the same device. In the case of ultrafast pulse characterization, this may done in principle by characterizing the same test pulse several times, with the underlying assumption that the ensemble of test pulses is coherent, so that the electric field of each is the same. In practice, it is much more useful to be able to evaluate the precision of a given measurement from a single experimental trace. This obviously requires some redundancy. In the case of spectrographic techniques, such redundancy is likely to be present because of the size mismatch between the experimental trace and the measured quantities. Redundant data can be used to evaluate precision in conjunction with a statistical technique called bootstrapping, where multiple inversions of the data are performed after removal of some of the data points [92

92. M. Munroe, D. H. Christensen, and R. Trebino, “Error bars in intensity and phase measurements of ultrashort laser pulses,” in Summaries of papers presented at the Conference on Lasers and Electro-Optics, 1998. CLEO 98. Technical Digest. (IEEE, 1998), pp. 462–463. [CrossRef]

, 93

93. Z. Wang, E. Zeek, R. Trebino, and P. Kvam, “Determining error bars in measurements of ultrashort laser pulses,” J. Opt. Soc. Am. B 20, 2400–2405 (2003). [CrossRef]

].

3.4. Specific Implementation of Sonograms

3.4a. Treacy’s Sonogram

Among the earliest sonographic methods, Treacy’s sonogram [72

72. E. B. Treacy, “Measurement and interpretation of dynamic spectrograms of picosecond light pulses,” J. Appl. Phys. 42, 3848–3858 (1971). [CrossRef]

] (also known as the “dynamic spectrogram”) was implemented by using the angular dispersion of a diffraction grating to spread the spectrum of the pulse in space. Temporal information about the arrival time of each spectral component was obtained by using two-photon fluorescence in a dye cell. In the original experiment, the dispersed pulse was correlated with a spatially inverted copy of itself, therefore comparing the time of arrival of optical frequencies symmetrically located on each side of a reference frequency. More recent implementations instead correlate the spatially dispersed pulse with a short pulse (for example, a replica of the pulse under test).

3.4b. Measurement of Sonograms with Nonlinear Crystals

Sonogram measurement of ultrashort pulses nowadays makes use of nonlinear crystals, which provide both reasonable signal amplitudes and appropriate temporal resolution [39

39. V. Wong and I. A. Walmsley, “Ultrashort-pulse characterization from dynamic spectrograms by iterative phase retrieval,” J. Opt. Soc. Am. B 14, 944–949 (1997). [CrossRef]

, 58

58. J. L. A. Chilla and O. E. Martinez, “Direct determination of the amplitude and the phase of femtosecond light pulses,” Opt. Lett. 16, 39–41 (1991). [CrossRef] [PubMed]

, 59

59. J. L. A. Chilla and O. E. Martinez, “Analysis of a method of phase measurement of ultrashort pulses in the frequency domain,” IEEE J. Quantum Electron. 27, 1228–1235 (1991). [CrossRef]

, 94

94. J. L. A. Chilla and O. E. Martinez, “Frequency domain phase measurement of ultrashort light pulses. Effect of noise,” Opt. Commun. 89, 434–440 (1992). [CrossRef]

]. A typical setup is illustrated in Fig. 15(a). There, the test pulse is split into two replicas. One of the replicas is sent to the spectral filter that acts as the stationary filter, for example a slit in a zero-dispersion line. The output of this filter is cross correlated with the other (short) replica in a nonlinear crystal. The complete sonogram can be measured by scanning the frequency of the spectral filter and the delay between the filtered replica and the replica of the input pulse. One advantage of this implementation of the sonogram, as well as all subsequent implementations based on SHG, is that the experimental trace usually gives directly a good picture of the chirp. This is clearly seen in Fig. 15(b), which shows the correlation between time and frequency in the sonogram of a chirped pulse. As is shown below, implementations of spectrograms with SHG (SHG-FROG) do not benefit from such an intuitive structure. Martinez’s approach led to the measurement of the chirp of a colliding-pulse mode-locking laser by using the determination of the group delay as a function of the optical frequency.

Two-photon absorption has also been used to measure sonograms with high sensitivity [95

95. D. T. Reid, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Sonogram characterisation of picosecond pulses at 1.5  micron using waveguide two photon absorption,” Electron. Lett. 36, 1141–1142 (2000). [CrossRef]

]. The setup is similar to the setup shown schematically in Fig. 15, but it is necessary to remove the constant background arising from one-photon excitation that is present on the traces. Real-time implementations of the sonogram based on a scanning Fabry–Perot filter and a two-photon detector have also been demonstrated [96

96. I. G. Cormack, R. Ortega-Martinez, W. Sibbett, and D. T. Reid, “Ultrashort pulse characterisation using a scanning Fabry–Perot etalon to rapidly acquire and retrieve a sonogram,” Summaries of Papers Presented at the Conference on Lasers and Electro-Optics, 2001. CLEO '01. Technical Digest (IEEE, 2001), pp. 272–273. [CrossRef]

].

3.4c. Spectrally Resolved Cross-Correlation

A spectrally resolved cross-correlation approach has also been used to characterize femtosecond pulses. In this method, a narrowband reference pulse is generated from the test pulse by spectral filtering (for example, using a slit in a zero-dispersion line). The field resulting from the cross-correlation between the reference pulse and the unfiltered test pulse in a nonlinear crystal is then spectrally resolved by using a spectrometer. The resulting two-dimensional trace is
S(τ,Ω)=|dtE(t)ER(tτ)exp(iΩt)|2,
(3.13)
and is therefore the spectrogram of the pulse under test measured with a gate equal to the field of the reference pulse. While such a trace is identical to a cross-correlation FROG (X-FROG) trace (see Subsection 3.5b), it also appears that if the gate response function is narrowband and real, the first moment of the spectrogram will lead to the group delay in the pulse following Eq. (3.11). This property was used in [61

61. J.-P. Foing, J.-P. Likforman, M. Joffre, and A. Migus, “Femtosecond pulse phase measurement by spectrally resolved up-conversion: application to continuum compression,” IEEE J. Quantum Electron. 28, 2285–2290 (1992). [CrossRef]

], and a setup providing real-time measurements was demonstrated in [62

62. J. Rhee, T. Sosnowski, A. C. Tien, and T. Norris, “Real-time dispersion analyzer of femtosecond laser pulses with use of a spectrally and temporally resolved upconversion technique,” J. Opt. Soc. Am. B 13, 1780–1785 (1996). [CrossRef]

].

3.4d. Measurement of Sonograms with Fast Photodetection

Chirp measurements are important for optical telecommunications because of the detrimental effect of chromatic dispersion and self-phase modulation in optical fibers and the presence of time-varying phase modulation on the pulses generated by lasers and modulators. Telecommunication pulses have low peak power, and their polarization state can vary quickly, which makes diagnostics based on nonlinear optics difficult to implement. Since these pulses often have durations longer than a few picoseconds, time-resolved information can be obtained by using a streak camera, for example. A streak camera can display a two-dimensional image where one spatial direction corresponds to time (calibration of the space-to-time correspondence is, of course, required) and the other direction corresponds to a physical spatial coordinate. A sonogram can therefore be recorded by mapping the optical frequency onto a spatial coordinate at the Fourier plane of a monochromator [Fig. 16(a)]. The pulse under test goes into the monochromator (diffraction grating and imaging system) that maps the optical frequency onto the spatial coordinate x. The streak camera maps the temporal intensity onto spatial intensity along the y direction. The (x,y) image therefore corresponds to the sonogram as a function of Ω and τ. Sonograms measured with fast photodetection have been used, for example, in the chirp evaluation of various externally modulated lasers [74

74. C. M. Olsen and H. Izadpanah, “Time-resolved chirp evaluations of Gbit/s NRZ and gain-switched DFB laser pulses using narrowband Fabry–Perot spectrometer,” Electron. Lett. 25, 1018–1019 (1989). [CrossRef]

, 97

97. D. A. Fishman, “Design and performance of externally modulated 1.5  micron laser transmitter in the presence of chromatic dispersion,” J. Lightwave Technol. 11, 624–632 (1993). [CrossRef]

, 98

98. A. Bresson, N. Stelmakh, J.-M. Lourtioz, A. Shen, and C. Froehly, “Chirp measurement of multimode Q-switched laser diode pulses by use of a streak camera and a grating monochromator,” Appl. Opt. 37, 1022–1025 (1998). [CrossRef]

], the measurement of the chromatic dispersion of optical fibers [73

73. A. S. L. Gomes, A. S. Gouveia-Neto, and J. R. Taylor, “Direct measurement of chirped optical pulses with picosecond resolution,” Electron. Lett. 22, 41–42 (1986). [CrossRef]

, 75

75. K. Mori, T. Morioka, and M. Saruwatari, “Group velocity dispersion measurement using supercontinuum picosecond pulses generated in an optical fibre,” Electron. Lett. 29, 987–989 (1993). [CrossRef]

], and the characterization of Raman radiation generated by propagation of optical pulses in a fiber [99

99. A. S. L. Gomes, V. L. Silva, and J. R. Taylor, “Direct measurement of nonlinear frequency chirp of Raman radiation in single-mode optical fibers using a spectral window method,” J. Opt. Soc. Am. B 5, 373–379 (1988). [CrossRef]

].

A recent implementation of the sonogram for trains of pulses in the telecommunication environment is based on phase comparison in the RF domain [100

100. Y. Ozeki, Y. Takushima, H. Yoshimi, K. Kikuchi, H. Yamauchi, and H. Taga, “Complete characterization of picosecond optical pulses in long-haul dispersion-managed transmission systems,” IEEE Photon. Technol. Lett. 17, 648–650 (2005). [CrossRef]

]. These trains of pulses usually have repetition rates f of the order of 10GHz. As shown in Fig. 16(b), the train of pulses under test is spectrally filtered at the optical frequency Ω (upper part of the setup) and detected by a photodetector with bandwidth greater than f. This gives a RF signal whose phase is proportional to the group delay for the group of frequencies around Ω selected by the spectral filter. This phase can be measured by comparison with another RF signal at the same frequency, generated by sending the unfiltered train of pulses to an identical photodetector (lower part of the setup); the two RF signals are downconverted by mixing with an identical local oscillator running at a frequency close to f. The measurement of the RF phase as a function of the filtered optical frequency then yields the group delay in the pulse composing the pulse train. This implementation uses conventional telecommunication and RF components, is polarization insensitive, and generates its own temporal reference, all of which are significant advantages. Further, telecommunication signals can have large amounts of incoherent amplified spontaneous emission, and it has been suggested that the measurement process is insensitive to this noise background.

3.4e. Sonogram with Phase Retrieval

3.4f. Single-Shot Sonograms

Single-shot sonograms require the acquisition of the two-dimensional sonogram where the frequency and time variable are simultaneously scanned. Two experimental implementations have been demonstrated.

The thick nonlinear crystal approach [102

102. D. T. Reid and I. G. Cormack, “Single-shot sonogram: a real-time chirp monitor for ultrafast oscillator,” Opt. Lett. 27, 658–660 (2002). [CrossRef]

] follows lines similar to those developed in the poor man’s FROG and the GRENOUILLE devices, which are presented in Subsection 3.5 [103

103. C. Radzewicz, P. Wasylczyk, and J. S. Krasinski, “A poor man’s FROG,” Opt. Commun. 186, 329–333 (2000). [CrossRef]

, 104

104. P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, “Highly-simplified device for ultrashort pulse measurement,” Opt. Lett. 26, 932–934 (2001). [CrossRef]

]. Phase-matching conditions in a nonlinear crystal can be used to provide strong spectral filtering, therefore enabling, for example, angular dispersion, while the nonlinearity itself provides the gating mechanism. This was implemented in a type II crystal, following Fig. 17(a). The combination of a cylindrical and a spherical lens magnifies the input beam in the horizontal transverse direction and focuses it in the vertical direction. A Wollaston prism is then used to split the incoming pulse into two orthogonally polarized beams that propagate at an angle. This therefore encodes the relative delay between the two pulses on the horizontal spatial axis. The tight vertical focusing in a type II KDP crystal (KH2PO4) leads to a SHG process that is mostly narrowband along the extraordinary axis and broadband along the ordinary axis, with a one-to-one relation between the output angle and the wavelength of the extraordinary wave being phase matched. In other words, the crystal essentially performs the narrowband spectral filtering along the extraordinary axis and gates the broadband pulse under test with the filtered pulse. Another combination of cylindrical and spherical lenses is used to map the vertical angle and the horizontal position on a two-dimensional detector to allow the single-shot measurement of the sonogram as a function of optical frequency and time.

Single-shot sonograms have also been measured by using the encoding of the optical frequency on a spatial coordinate, using a zero-dispersion line and spatially dependent time gating by correlation with the short pulse under test in a noncollinear geometry on a two-photon detector [105

105. D. Panasenko and Y. Fainman, “Single-shot sonogram generation for femtosecond laser pulse diagnsotics by use of two-photon conductivity in a silicon CCD camera,” Opt. Lett. 27, 1475–1477 (2002). [CrossRef]

, 106

106. D. Panasenko, P. C. Sun, N. Alic, and Y. Fainman, “Single-shot generation of a sonogram by time gating of a spectrally decomposed ultrashort laser pulse,” Appl. Opt. 41, 5185–5190 (2002). [CrossRef] [PubMed]

]. Following Fig. 17(b), the pulse under test is spatially dispersed by using a diffraction grating and a cylindrical lens so that the optical frequency is encoded on the horizontal variable. A short reference pulse (in practice, a replica of the pulse under test) is incident at the Fourier plane of the zero-dispersion line and produces a cross-correlation signal on a two-photon CCD array, where the relative delay between the two interacting pulses is encoded onto the vertical direction.

3.5. Specific Implementations of Spectrograms

Spectrogram measurement apparatuses can be classified by their implementation of the time-nonstationary filter and the specific measurement geometry.

3.5a. Early Attempts

3.5b. Frequency-Resolved Optical Gating

Second-harmonic generation frequency-resolved optical gating. In SHG-FROG, the pulse is mixed with a delayed replica in a nonlinear crystal with large spectral acceptance, as indicated in Fig. 18 [60

60. J. Paye, M. Ramaswamy, J. G. Fujimoto, and E. P. Ippen, “Measurement of the amplitude and phase of ultrashort light pulses from spectrally resolved autocorrelation,” Opt. Lett. 18, 1946–1948 (1993). [CrossRef] [PubMed]

, 110

110. J. Paye, “How to measure the amplitude and phase of an ultrashort light pulse with an autocorrelator and a spectrometer,” IEEE J. Quantum Electron. 30, 2693–2697 (1994). [CrossRef]

, 111

111. K. W. Delong, R. Trebino, J. Hunter, and W. E. White, “Frequency-resolved optical gating with the use of second-harmonic generation,” J. Opt. Soc. Am. B 11, 2206–2215 (1994). [CrossRef]

]. The experimental spectrogram is
S(τ,Ω)=|dtE(t)E(tτ)exp(iΩt)|2.
(3.15)
SHG-FROG is more sensitive than most FROG techniques, and it can be used to characterize ultrafast pulses from Ti:sapphire oscillators [7

7. J. M. Dudley, S. F. Boussen, D. M. J. Cameron, and J. D. Harvey, “Complete characterization of a self-mode-locked Ti:sapphire laser in the vicinity of zero group-delay dispersion by frequency-resolved optical gating,” Appl. Opt. 38, 3308–3315 (1999). [CrossRef]

] and pulse trains in the telecommunication environment [22

22. L. P. Barry, S. Del Burgo, B. C. Thomsen, R. T. Watts, D. A. Reid, and J. D. Harvey, “Optimization of optical data transmitters for 40-Gbs lightwave systems using frequency resolved optical gating,” IEEE Photon. Technol. Lett. 14, 971–973 (2002). [CrossRef]

, 112

112. L. P. Barry, J. M. Dudley, P. G. Bollond, J. D. Harvey, and R. Leonhardt, “Complete characterisation of pulse propagation in optical fibres using frequency-resolved optical gating,” Electron. Lett. 32, 2339–2340 (1996). [CrossRef]

, 113

113. L. P. Barry, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Characterization of 1.55-μm pulses from a self-seeded gain-switched Fabry–Perot laser diode using frequency-resolved optical gating,” IEEE Photon. Technol. Lett. 10, 935–937 (1998). [CrossRef]

]. Sensitivity can be enhanced by using nonlinear interaction in waveguide structures [114

114. H. Miao, S.-D. Yang, C. Langrock, R. V. Roussev, M. M. Fejer, and A. M. Weiner, “Ultralow-power second-harmonic generation frequency-resolved optical gating using aperiodically poled lithium niobate waveguides,” J. Opt. Soc. Am. B 25A41–A53 (2008). [CrossRef]

]. SHG-FROG can also be used to characterize extremely short pulses when various deleterious effects such as the dispersion of the nonlinear crystal and the nonuniform response of the wave-mixing and spectral detection system are taken into account [115

115. A. Baltuska, M. S. Pshenichnikov, and D. A. Wiersma, “Second-harmonic generation frequency-resolved optical gating in the single-cycle regime,” IEEE J. Quantum Electron. 35, 459–478 (1999). [CrossRef]

, 116

116. S. Akturk, C. D’Amico, and A. Mysyrowicz, “Measuring ultrashort pulses in the single-cycle regime using frequency-resolved optical gating,” J. Opt. Soc. Am. B 25, A63–A69 (2008). [CrossRef]

], and, since second-order nonlinearities are widely available, it can be used in the mid-IR [117

117. B. A. Richman, M. A. Krumbügel, and R. Trebino, “Temporal characterization of mid-IR free-electron-laser pulses by frequency-resolved optical gating,” Opt. Lett. 22, 721–723 (1997). [CrossRef] [PubMed]

].

SHG-FROG has a major drawback that is derived from the fact that the gate is the electric field of the test pulse itself. This leads to a spectrogram that is rather unintuitive, and the sign of a chirp is, for example, not visible on a SHG-FROG trace. Inversion of a SHG-FROG trace is ambiguous in the direction of time, as explained previously. The spectrograms of a Gaussian pulse with second- and third-order dispersion are shown in Fig. 18. Determination of the chirp from first-order moments of the SHG-FROG trace is not possible, and the traces are both symmetric with respect to the relative delay τ. Issues due to the direction-of-time ambiguity can be alleviated in various ways. Some prior knowledge of the electric field of the pulse under test (e.g., knowing that the pulse is positively chirped) or the introduction of a recognizable feature (e.g., a trailing pulse using multiple reflections in a piece of glass) can break this ambiguity. Another approach is to perform an additional measurement of the SHG-FROG trace after addition of chromatic dispersion of a known sign, but this is rather impractical in applications requiring single-shot operation. The deleterious effects of limited spectral acceptance of the nonlinear crystal can be compensated by numerical correction of the experimental trace or by crystal dithering [118

118. P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, “Increased-bandwidth in ultrashort-pulse measurement using an angle-dithered nonlinear-optical crystal,” Opt. Express 7, 342–349 (2000). [CrossRef] [PubMed]

]. While SHG-FROG is usually implemented in a noncollinear geometry, some collinear SHG-FROG setups, useful when the experimental setup has some spatial constraints, have been investigated [119

119. D. N. Fittinghoff, A. C. Millard, J. A. Squier, and M. Muller, “Frequency-resolved optical gating measurement of ultrashort pulses passing through a high numerical aperture objective,” IEEE J. Quantum Electron. 35, 479–486 (1999). [CrossRef]

, 120

120. L. Gallmann, G. Steinmeyer, D. H. Sutter, N. Matuschek, and U. Keller, “Collinear type II second-harmonic-generation frequency-resolved optical gating for the characterization of sub-10-fs optical pulses,” Opt. Lett. 25, 269–271 (2000). [CrossRef]

, 121

121. I. Amat-Roldán, I. G. Cormack, P. Loza-Alvarez, E. J. Gualda, and D. Artigas, “Ultrashort pulse characterization with SHG collinear-FROG,” Opt. Express 12, 1169–1178 (2004). [CrossRef]

]. Collinearity of the two replicas of the pulse in the SHG crystal leads to an interferometric modulation of the experimental trace, which can be used independently of the SHG-FROG trace for phase retrieval [121

121. I. Amat-Roldán, I. G. Cormack, P. Loza-Alvarez, E. J. Gualda, and D. Artigas, “Ultrashort pulse characterization with SHG collinear-FROG,” Opt. Express 12, 1169–1178 (2004). [CrossRef]

, 122

122. G. Stibenz and G. Steinmeyer, “Interferometric frequency-resolved optical gating,” Opt. Express 13, 2617–2626 (2005). [CrossRef] [PubMed]

, 123

123. G. Stibenz and G. Steinmeyer, “Structures of interferometric frequency-resolved optical gating,” IEEE J. Sel. Top. Quantum Electron. 12, 286–296 (2006). [CrossRef]

]. A recent example of the application of SHG-FROG to the determination of pulse formation in a nonlinear optical process is shown in Fig. 19 [124

124. J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597–603 (2007). [CrossRef]

].

Self-diffraction frequency-resolved optical gating. Self-diffraction FROG (SD-FROG) uses the diffracting properties of an index grating written in a material by the interference of two replicas of the pulse under test via the Kerr effect [80

80. D. J. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993). [CrossRef]

]. The efficiency of diffraction of each replica on the grating is proportional to the interference term between the electric field of the two replicas. The SD-FROG trace [Fig. 21(a)] is obtained by delaying one replica with respect to the other and spectrally resolving one of the replicas and can be written as
S(τ,Ω)=|dtE2(t)E*(tτ)exp(iΩt)|2.
(3.17)
SD FROG traces are rather intuitive, although the relation between group delay and moments depends on the order of the phase distortions because the temporal gate is pulse-dependent and not necessarily real. The self-diffraction effect is not phase matched, and the phase mismatch is wavelength dependent. This constrains the nonlinear medium to be thin and makes the technique difficult for broadband pulses. However, SD-FROG does not require high-extinction polarizers and can therefore be used to characterize short-wavelength pulses [127

127. T. S. Clement, A. J. Taylor, and D. J. Kane, “Single-shot measurement of the amplitude and phase of ultrashort laser-pulses in the violet,” Opt. Lett. 20, 70–72 (1995). [CrossRef] [PubMed]

]. SD-FROG has also been implemented by using cascaded second-order nonlinearities [126

126. A. Kwok, L. Jusinski, M. A. Krumbügel, J. N. Sweetser, D. N. Fittinghoff, and R. Trebino, “Frequency-resolved optical gating using cascaded second-order nonlinearities,” IEEE J. Sel. Top. Quantum Electron. 4, 271–277 (1998). [CrossRef]

].

Third-harmonic generation frequency-resolved optical gating. Third-harmonic generation FROG (THG-FROG) has been implemented by using surface harmonic generation in a glass plate [128

128. T. Tsang, M. A. Krumbügel, K. W. Delong, D. N. Fittinghoff, and R. Trebino, “Frequency-resolved optical-gating measurements of ultrashort pulses using surface third-harmonic generation,” Opt. Lett. 21, 1381–1383 (1996). [CrossRef] [PubMed]

] and more recently in organic films [129

129. G. Ramos-Ortiz, M. Cha, S. Thayumanavan, J. Mendez, S. R. Marder, and B. Kippelen, “Ultrafast-pulse diagnostic using third-order frequency-resolved optical gating in organic films,” Appl. Phys. Lett. 85, 3348–3350 (2004). [CrossRef]

]. Two replicas of the pulse under test are mixed noncollinearly in a medium, so that they overlap spatially at one of the surfaces of the medium [Fig. 21(b)]. This is actually the most sensitive FROG setup based on third-order nonlinear effects for femtosecond pulses, and it has the advantage of a large phase-matching bandwidth. The traces of THG-FROG can be difficult to interpret; for example, the THG-FROG trace of a Gaussian pulse with second-order dispersion does not show the familiar correlation between time and frequency of linear chirp. However, the THG-FROG traces usually have some asymmetry; for example, the THG-FROG trace of a Gaussian pulse with third-order dispersion has a more familiar shape. Since there is no third-harmonic generation from a beam with circular polarization, a collinear fringe-free THG-FROG setup can be built by using two beams with opposite circular polarizations [130

130. R. Chadwick, E. Spahr, J. A. Squier, C. G. Durfee, B. C. Walker, and D. N. Fittinghoff, “Fringe-free, background-free, collinear third-harmonic generation frequency-resolved optical gating measurements for multiphoton microscopy,” Opt. Lett. 31, 3366–3368 (2006). [CrossRef] [PubMed]

].

Transient grating frequency-resolved optical gating. Transient gradient FROG (TG-FROG) is based on a three-beam geometry similar to what is called BOXCARS in nonlinear spectroscopy [131

131. J. N. Sweetser, D. N. Fittinghoff, and R. Trebino, “Transient-grating frequency-resolved optical gating,” Opt. Lett. 22, 519–521 (1997). [CrossRef] [PubMed]

]. As in Figs. 21(c), 21(d), the input test pulse E(t) is split into three replicas E1(t), E2(t), and E3(t). The field generated by four-wave mixing is proportional to E1(t)E2(t)E3*(t). Depending on the choice of the delayed pulse (either E2(t) or E3(t)), the TG-FROG trace obtained by frequency resolving the generated field is, respectively
S(τ,Ω)=|dtE(t)I(tτ)exp(iΩt)|2
(3.18)
or
S(τ,Ω)=|dtE2(t)E*(tτ)exp(iΩt)|2,
(3.19)
which are, respectively, mathematically equivalent to the PG-FROG trace or the SD-FROG trace. The nonlinear process of TG-FROG is phase matched, and therefore a long nonlinear medium can be used to increase the sensitivity.

Four-wave mixing frequency-resolved optical gating. The strong wave-mixing effects in semiconductor optical amplifiers (SOAs) have been used to characterize pulses in the telecommunication environment. The pulse is once again split into two replicas, where one is used as a strong pump, depleting the carriers in a semiconductor optical amplifier, therefore temporally modulating the gain and phase of the SOA. The second replica, with a lower power, is used as a probe, and one spectrally resolves this replica after the SOA. This approach is extremely sensitive, as it usually suffices to have peak powers of the order of 1mW to induce significant changes in the SOA transmission [132

132. P.-A. Lacourt, J. M. Dudley, J.-M. Merolla, H. Porte, J.-P. Goedgebuer, and W. T. Rhodes, “Milliwatt-peak-power pulse characterization at 1.55 μm by wavelength-conversion frequency-resolved optical gating,” Opt. Lett. 27, 863–865 (2002). [CrossRef]

]. Spectrograms measured by using a SOA depleted by a strong pump pulse as the gate acting on a probe pulse from a different source have also been measured, leading to the characterization of dynamical processes in SOAs [133

133. I. Kang, C. Dorrer, L. Zhang, M. Dinu, M. Rasras, L. L. Buhl, S. Cabot, A. Bhardwaj, X. Liu, M. A. Cappuzzo, L. Gomez, A. Wong-Foy, Y. F. Chen, N. K. Dutta, S. S. Patel, D. T. Neilson, C. R. Giles, A. Piccirilli, and J. Jaques, “Characterization of the dynamical processes in all-optical signal processing using semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. 14, 758–769 (2008). [CrossRef]

].

Two-photon absorption frequency-resolved optical gating. The high sensitivity of the two-photon absorption in an indium phosphide crystal has been used to characterize pulses in the telecommunication environment [134

134. K. Ogawa and M. D. Pelusi, “High-sensitivity pulse spectrogram measurement using two-photon absorption in a semiconductor at 1.5-μm wavelength,” Opt. Express 7, 135–140 (2000). [CrossRef] [PubMed]

, 135

135. K. Ogawa, “Real-time intuitive spectrogram measurement of ultrashort optical pulses using two-photon absorption in a semiconductor,” Opt. Express 10, 262–267 (2002). [CrossRef] [PubMed]

]. This is a two-pulse mixing, but it is not background free, since the pump induces absorption on the probe. A background-free trace can be obtained by proper modulation of the interacting beams.

Cross-correlation frequency-resolved optical gating. Nonlinear mixing of the pulse under test with another optical pulse has also been used. In this cross-correlation FROG (X-FROG), the pulse and the gate do not have an explicit mathematical link, and the X-FROG trace can be written as function of the field of the pulse E(t) and the field of the gate E(t) as
S(τ,Ω)=|dtE(t)E(tτ)exp(iΩt)|2.
(3.20)
X-FROG is not self-referencing, as it requires an additional pulse. It is particularly useful when characterizing pulses in a wavelength range where regular FROG setups would be difficult to implement, often because of the absence of phase-matched nonlinear interactions. For example, X-FROG has been used to characterize blue pulses around 400nm by downconversion with a pulse at 800nm [136

136. S. Linden, J. Kuhl, and H. Giessen, “Amplitude and phase characterization of weak blue ultrashort pulses by downconversion,” Opt. Lett. 24, 569–571 (1999). [CrossRef]

] and to characterize infrared pulses at 4μm by frequency mixing with a pulse at 770nm [137

137. D. T. Reid, P. Loza-Alvarez, C. T. A. Brown, T. Beddard, and W. Sibbett, “Amplitude and phase measurement of mid-infrared femtosecond pulses by using cross-correlation frequency-resolved optical gating,” Opt. Lett. 25, 1478–1480 (2000). [CrossRef]

]. As the measured signal increases with the energy of the ancillary pulse, low-energy pulses can be characterized with a high-energy ancillary pulse. Parametric gain can be used in a X-FROG configuration to provide high-sensitivity measurements, though at the expense of background noise [138

138. J. Y. Zhang, A. P. Shreenath, M. Kimmel, E. Zeek, and R. Trebino, “Measurement of the intensity and phase of attojoule femtosecond light pulses using optical-parametric-amplification cross-correlation frequency-resolved optical gating,” Opt. Express 11, 601–609 (2003). [CrossRef] [PubMed]

, 139

139. J. Y. Zhang, C.-K. Lee, J. Y. Huang, and C.-L. Pan, “Sub femto-joule sensitive single-shot OPA-XFROG and its application in study of white-light supercontinuum generation,” Opt. Express 12, 574–581 (2004). [CrossRef] [PubMed]

]. X-FROG is particularly useful for characterizing the broadband pulses generated by nonlinear propagation in optical fibers and therefore for interpreting the combination of linear and nonlinear effect that are operative in the formation of solitons [15

15. X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O’Shea, A. P. Shreenath, R. Trebino, and R. S. Windeler, “Frequency-resolved optical gating and single-shot spectral measurements reveal fine structure in microstructure-fiber continuum,” Opt. Lett. 27, 1174–1176 (2002). [CrossRef]

, 140

140. J. M. Dudley, X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O’Shea, R. Trebino, S. Coen, and R. S. Windeler, “Cross-correlation frequency resolved optical gating analysis of broadband continuum generation in photonic crystal fiber: simulations and experiments,” Opt. Express 10, 1215–1221 (2002). [CrossRef] [PubMed]

, 141

141. A. Efimov and A. J. Taylor, “Supercontinuum generation and solition timing jitter in SF6 soft glass photonic crystal fibers,” Opt. Express 16, 5942–5953 (2008). [CrossRef] [PubMed]

].

Single-shot frequency-resolved optical gating. The single-shot acquisition of a FROG trace relies on encoding the time and frequency variables into the two transverse spatial coordinates, making use of a two-dimensional detector to record the spectrogram [142

142. D. J. Kane, A. J. Taylor, R. Trebino, and K. W. Delong, “Single-shot measurement of the intensity and phase of a femtosecond UV laser-pulse with frequency-resolved optical gating,” Opt. Lett. 19, 1061–1063 (1994). [CrossRef] [PubMed]

]. Mapping the optical frequency to one spatial coordinate is performed with a conventional grating-based spectrometer, which can record the optical spectrum of a pulse in a single shot by using a CCD array. A relative delay between two pulses can be mapped into the other spatial coordinate by using a geometry similar to the single-shot autocorrelator [143

143. F. Salin, P. Georges, G. Roger, and A. Brun, “Single-shot measurement of a 52-fs pulse,” Appl. Opt. 26, 4528–4531 (1987). [CrossRef] [PubMed]

]. In this configuration, the convergence of two unfocused beams means that the delay is a linear function of the distance across the interaction plane of the beams. The field at spatial position x after nonlinear interaction is E(t)E(tαx). This plane can be reimaged onto the input slit of an imaging spectrometer, which then allows the measurement of the optical spectrum as a function of the optical frequency ω and spatial position x. The measured intensity is therefore S(τ,ω), where the relation between x and τ has been calibrated. Since delay is now coded in a transverse spatial coordinate, it is necessary to avoid spatial distortions of the beam, as they can appear as temporal distortions on the FROG trace. Some corrections of the FROG trace taking into account the beam spatial profile can in general be performed [144

144. D. Lee, Z. Wang, X. Gu, and R. Trebino, “Effect—and removal—of an ultrashort pulse's spatial profile on the single-shot measurement of its temporal profile,” J. Opt. Soc. Am. B 25, A93–A100 (2008). [CrossRef]

]. Single-shot operation of FROG has been used to characterize the output of large laser systems based on chirped-pulse amplification [10

10. B. Kohler, V. V. Yakovlev, K. R. Wilson, J. A. Squier, K. W. Delong, and R. Trebino, “Phase and intensity characterization of femtosecond pulses from a chirped-pulse amplifier by frequency-resolved optical gating,” Opt. Lett. 20, 483–485 (1995). [CrossRef] [PubMed]

].

3.5c. Poor Man’s Frequency-Resolved Optical Gating and GRENOUILLE

In FROG, the stationary filter allowing the optical spectrum measurement after a nonlinear interaction is usually a standard optical spectrum analyzer. It is, however, known that nonlinear interactions in a thick nonlinear crystal having limited spectral acceptance lead to a coupling between the optical frequency of the converted field and its wave vector. That is, the upconversion of a particular frequency occurs only in a particular direction. It was pointed out that such coupling could be used to provide angular dispersion of the upconverted spectrum, allowing, by means of suitable imaging optics and a detector, implementation of the frequency-resolving element of a FROG diagnostic [103

103. C. Radzewicz, P. Wasylczyk, and J. S. Krasinski, “A poor man’s FROG,” Opt. Commun. 186, 329–333 (2000). [CrossRef]

, 104

104. P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, “Highly-simplified device for ultrashort pulse measurement,” Opt. Lett. 26, 932–934 (2001). [CrossRef]

]. This property can be used in concert with the time-to-space mapping configuration of single-shot autocorrelators and FROG to provide complete mapping of the two transverse spatial coordinates into the time and frequency coordinates of the FROG trace (Fig. 22). Such an arrangement, nicknamed grating eliminated no-nonsense observation of ultrafast incident laser light electric fields (i.e., GRENOUILLE, the French word for “frog”), is composed of a cylindrical lens, a Fresnel biprism, a thick nonlinear crystal, and a combination of two cylindrical lenses, so that a spatially extended input pulse is focused in the vertical direction in the crystal, where the spatially dependent phase matching leads to angular dispersion in the upconverted beam [104

104. P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, “Highly-simplified device for ultrashort pulse measurement,” Opt. Lett. 26, 932–934 (2001). [CrossRef]

]. To characterize shorter pulses some of the transmissive optics may be replaced by reflective optics [145

145. S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, “Extremely simple device for measuring 20-fs pulses,” Opt. Lett. 29, 1025–1027 (2004). [CrossRef] [PubMed]

]. The biprism generates two beams that cross one another from a single input beam, so that the relative time between the two waves is mapped into the horizontal spatial coordinate. The two cylindrical lenses after the crystal are used to map the horizontal position and the vertical dispersion into the horizontal and vertical coordinates in a plane where a two-dimensional detector can be set. The GRENOUILLE trace is fundamentally a SHG-FROG trace; it therefore suffers from a direction-of-time ambiguity, and the trace does not encode chirp information in a direct visual manner. Nonetheless, it has been shown that some types of space–time coupling can be evaluated by using some prior assumptions [146

146. S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, “Measuring pulse-front tilt in ultrashort pulses using GRENOUILLE,” Opt. Express 11, 491–501 (2003). [CrossRef] [PubMed]

].

3.5d. Linear Spectrograms

A spectrogram can be measured by gating the test pulse with a fast temporal modulator that acts as the nonstationary filter [23

23. C. Dorrer and I. Kang, “Simultaneous temporal characterization of telecommunication optical pulses and modulators using spectrograms,” Opt. Lett. 27, 1315–1317 (2002). [CrossRef]

, 147

147. C. Dorrer and I. Kang, “Linear self-referencing techniques for short-optical-pulse characterization,” J. Opt. Soc. Am. B 25, A1–A12 (2008). [CrossRef]

]. There are several advantages to this approach, which is particularly well suited to optical telecommunication pulse durations and wavelengths [148

148. B. C. Thomsen, M. A. F. Roelens, R. T. Watts, and D. J. Richardson, “Comparison between nonlinear and linear spectrographic techniques for the complete characterization of high bit-rate pulses used in optical telecommunications,” IEEE Photon. Technol. Lett. 17, 1914–1916 (2005). [CrossRef]

]. The technique is highly sensitive, since no nonlinear interaction is involved, and pulse trains with as low as 100nW of average power have been characterized. The choice of appropriate electroabsorption modulator renders the method fairly insensitive to both the polarization and the wavelength. Another technical advantage is that both the electric field of the pulse and the transfer function of the modulator can be retrieved by using the principal component generalized projection algorithm . Therefore, such an experiment can also be used to characterize the amplitude and phase transfer function of an unknown modulator. Since the gate is independent of the pulse, deconvolution of the spectrogram can be performed with a known or approximate transfer function for the gate. While the gate must have a bandwidth similar to the bandwidth of the pulse under test, this condition is rather loose, and subpicosecond pulses have been characterized by using a 30ps gate. The action of the modulator must be synchronized to the pulse under test. In the telecommunication environment, a clock synchronized to the source of pulses is usually available, and can therefore be used to drive the modulator. This approach has been implemented, for example, with an electroabsorption modulator or an electro-optic modulator [149

149. D. Reid and J. D. Harvey, “Linear spectrograms using electrooptic modulators,” IEEE Photon. Technol. Lett. 19, 535–537 (2007). [CrossRef]

, 150

150. K. T. Vu, A. Malinovski, M. A. F. Roelens, M. Ibsen, P. Petropoulos, and D. J. Richardson, “Full characterization of low-power picosecond pulses from a gain-switched diode laser using electrooptic modulation-based linear FROG,” IEEE Photon. Technol. Lett. 20, 505–507 (2008). [CrossRef]

] driven by a RF sine wave, in which the relative delay between the pulse under test and the gate was controlled by using a RF phase shifter, so that no free-space optical delay line was needed [Fig. 23(a)]. Such an arrangement was also used to characterize the output of various pulse carvers [78

78. P. Winzer, C. Dorrer, R. J. Essiambre, and I. Kang, “Chirped return-to-zero modulation by imbalanced pulse carver driving signals,” IEEE Photon. Technol. Lett. 16, 1379–1381 (2004). [CrossRef]

, 151

151. X. Wei, J. Leuthold, C. Dorrer, D. M. Gill, and X. Liu, “Chirp reduction of π2 alternate-phase pulses by optical filtering,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2005), paper JWA42.

, 152

152. R. Maher, P. M. Anandarajah, A. D. Ellis, D. Reid, and L. P. Barry, “Optimization of a 42.7Gbs wavelength tunable RZ transmitter using a linear spectrogram technique,” Opt. Express 16, 11281–11288 (2008). [CrossRef] [PubMed]

], to characterize various pulse shapers [153

153. D. M. Marom, C. Dorrer, I. Kang, C. R. Doerr, M. A. Cappuzzo, L. Gomez, E. Chen, A. Wong-Foy, E. Laskowski, F. Klemens, C. Bolle, R. Cirelli, E. Ferry, T. Sorsch, J. Miner, E. Bower, M. E. Simon, F. Pardo, and D. Lopez, “Compact spectral pulse shaping using hybrid planar lightwave circuit and free-space optics with MEMS piston micro-mirrors and spectrogram feedback control,” in The 17th Annual Meeting of the IEEE Lasers and Electro-Optics Society, 2004. LEOS 2004. (IEEE LEOS, 2004), Vol. 2, pp. 585–586.

, 154

154. M. A. F. Roelens, J. A. Bolger, D. Williams, and B. J. Eggleton, “Multi-wavelength synchronous pulse burst generation with a wavelength selective switch,” Opt. Express 16, 10152–10157 (2008). [CrossRef] [PubMed]

], and to characterize subpicosecond pulses [147

147. C. Dorrer and I. Kang, “Linear self-referencing techniques for short-optical-pulse characterization,” J. Opt. Soc. Am. B 25, A1–A12 (2008). [CrossRef]

, 155

155. C. Dorrer, “Investigation of the spectrogram technique for the characterization of picosecond optical pulses,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2005), paper OTuB3.

]. Figure 23(b) presents the spectral representation of the 2.4ps pulse from a mode-locked laser diode. This pulse was sent in a nonlinear fiber, where self-phase modulation was used to broaden its spectrum, followed by a dispersive fiber to compensate for the nonlinear chirp. The output pulse [represented in Fig. 23(c)] is recompressed to 900fs. The characterization of the temporal behavior of a gain-depleted SOA was also performed by gating a probe pulse with the SOA under test [133

133. I. Kang, C. Dorrer, L. Zhang, M. Dinu, M. Rasras, L. L. Buhl, S. Cabot, A. Bhardwaj, X. Liu, M. A. Cappuzzo, L. Gomez, A. Wong-Foy, Y. F. Chen, N. K. Dutta, S. S. Patel, D. T. Neilson, C. R. Giles, A. Piccirilli, and J. Jaques, “Characterization of the dynamical processes in all-optical signal processing using semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. 14, 758–769 (2008). [CrossRef]

]. It is possible to operate a linear spectrogram system close to video-rate retrieval, effectively in real time, by using a Fabry–Perot etalon to measure the optical spectra after modulation has demonstrated update rates of the order of 10Hz [24

24. C. Dorrer and I. Kang, “Real-time implementation of linear spectrograms for the characterization of high bit-rate optical pulse trains,” IEEE Photon. Technol. Lett. 16, 858–860 (2004). [CrossRef]

].

3.6. Conclusions

Spectrographic techniques have been used for a large variety of pulses in very diverse configurations. Nonlinear optics plays a central role in most implementations to provide a gating process for subpicosecond pulses, although linear implementations have also been used to characterize longer pulses, e.g., in optical telecommunications. Spectrographic techniques lead in some cases to intuitive experimental traces capable of displaying the time-to-frequency correlation in a chirped pulse. In most cases, the electric field reconstruction must be iterative.

4. Tomography and Imaging Concepts

4.1. Introduction

Means for measuring the phase of the spatial electric field describing a quasi-monochromatic beam of light have been known for a long time. The concepts developed for imaging and wavefront sensing have inspired approaches to characterizing the temporal field variation of short optical pulses. For example, imaging of a pulse using temporal magnification is analogous to imaging of a spatially localized, pointlike object with spatial magnification by means of a conventional imaging system. Similarly, approaches to the measurement of spatially extended wave field by means of tomography have parallels in temporal pulse characterization. In this section the analogy between the spatial and the temporal domains in wave propagation is outlined. We then focus our attention on specific aspects of imaging and wave-field tomography as applied to the characterization of short optical pulses. Finally, we describe in detail various experimental implementations of these concepts.

4.1a. Analogy between Space and Time

4.1b. Wigner Formalism

The temporal and spectral intensities of the pulse are the time and frequency marginals of the Wigner function, i.e., the projections of that function onto the time and frequency axes:
I(t)=dω2πW(t,ω),
(4.7)
Ĩ(ω)=dtW(t,ω).
(4.8)
It is usually experimentally easier for short pulses to measure the frequency marginal than the time marginal. Indeed, the frequency marginal can be measured accurately by using a spectrometer or monochromator of sufficient resolution, while the temporal marginal requires a detector with time resolution much better than the duration of the pulse under test.

4.1c. Tomography

Tomography broadly relates to the reconstruction of an object in N dimensions from a set of projections onto lower-dimensional data sets [195

195. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1987).

]. This concept has been used in medical imaging for a long time [196

196. S. Webb, From the Watching of Shadows: the Origins of Radiological Tomography (Institute of Physics, 1990).

] and has also been applied to quantum state reconstruction [197

197. U. Leonhardt, Measuring the quantum state of light (Cambridge University Press, 1997).

], where a high-dimensional entity is estimated from a set of probability distributions. In imaging applications, the three-dimensional reconstruction of an object is obtained from a set of two-dimensional measurements of the attenuation of a probe beam through the object taken along different directions through the object. Restricting our attention to a two-dimensional object with attenuation specified by the function a(x,y), we can define Pθ(u) as the projection of a nondiffracting source of uniform spatial intensity orthogonal to an axis making an angle θ with the y axis (Fig. 25). Noting that the line Δθ(u) satisfies y=xtan(θ)+ucos(θ), one has
Pθ(u)=dxdya(x,y)δ[yxtan(θ)ucos(θ)],
(4.9)
where δ is the Dirac delta function. Integrating over y gives
Pθ(u)=dxa[x,ucos(θ)+xtan(θ)].
(4.10)
Gathering a set of projections Pθ(u) for different angles θ (a set usually referred to as the Radon transform of the function a), one can then attempt the reconstruction of a(x,y). In the context of ultrashort optical pulses, tomographic reconstruction implies estimating the two-dimensional chronocyclic Wigner function representing the pulse train. The approach is similar to that described above: measure a set of projections of the chronocyclic Wigner function, from which the Wigner function itself can be obtained. As noted above, if the train of pulses is coherent, its electric field, within some arbitrary constants, can be obtained directly. If the train of pulses is partially coherent, the description by an electric field is inappropriate, and the Wigner function is the next-lowest-order description. The procedure is known as “chronocyclic tomography” [198

198. M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett. 18, 2041–2043 (1993). [CrossRef] [PubMed]

], as it applies tomography to the chronocyclic space (t,ω). The time-to-frequency converter mentioned above and simplified chronocyclic tomography [199

199. C. Dorrer and I. Kang, “Complete temporal characterization of short optical pulses by simplified chronocyclic tomography,” Opt. Lett. 28, 1481–1483 (2003). [CrossRef] [PubMed]

] are variations of the complete tomographic technique that use a restricted set of projections.

4.2. Chronocyclic Tomography

4.2a. Principle

4.2b. Inversion for Chronocyclic Tomography

Tomographic data inversion has been studied extensively, both theoretically and experimentally. We deal here with inversion methods for a common configuration: parallel projections measured with a nondiffracting source. For data sets taken in this configuration, inversion can be performed by using the filtered backprojection algorithm. We briefly discuss this algorithm in order to illustrate its principle and to emphasize that such inversion is algebraic and noniterative. More complete treatments of tomographic imaging can be found in the literature [195

195. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1987).

]. The key to the inversion of tomographic data is the Fourier slice theorem. Using the notation of Subsection 4.1c, we can define the Fourier transform of the attenuation function a(x,y) by
ã(u,v)=dxdya(x,y)exp[2πi(ux+vy)].
(4.15)
This Fourier transform taken at u=0 gives
ã(0,v)=dy[dxa(x,y)]exp(2πivy),
(4.16)
where the quantity between brackets is by definition the projection of the attenuation along a line of constant y, i.e., P0(y). This leads to
ã(0,v)=P̃0(v).
(4.17)
The Fourier transform of the image at u=0 can therefore be calculated from the Fourier transform of the projection measured for θ=0. As this is independent of the angle between the object and the axis, any one-dimensional slice of the Fourier transform of the object can be obtained from the one-dimensional Fourier transform of the projection measured at the appropriate angle, and the Fourier slice theorem can be written with our definitions as
ã[ρcos(θ),ρsin(θ)]=P̃π2θ(ρ).
(4.18)
This is used to derive the filtered backprojection algorithm. The object function is written as
a(x,y)=dudvã(u,v)exp[2πi(ux+vy)],
(4.19)
which can be expressed in circular coordinates (ρ,θ) as
a(x,y)=0πdρdθã[ρcos(θ),ρsin(θ)]|ρ|exp{2πiρ[xcos(θ)+ysin(θ)]}.
(4.20)
Using the Fourier slice theorem, this leads to
a(x,y)=0πdρdθP̃π2θ(ρ)|ρ|exp{2πiρ[xcos(θ)+ysin(θ)]}.
(4.21)

The image can therefore be directly reconstructed from the set of projections Pθ. This involves a filtering operation, represented by the product with |ρ|, and the operation is equivalent to a projection back from the set of projections to the image, hence the name “filtered backprojection” given to this reconstruction procedure. When applied to pulse characterization, this approach to tomography has several advantages. It allows the algebraic noniterative reconstruction of the Wigner function independently of the state of coherence. However, in all tomographic reconstruction procedures, proper sampling of the experimental trace is critical, and the quality of the reconstruction varies greatly with the number of projections [195

195. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1987).

].

4.3. Time-to-Frequency Conversion

4.3a. Exact Time-to-Frequency Converter

A device that permits estimation of the temporal intensity of a pulse by using temporal and spectral modulations followed by a measurement of the pulse spectrum is known as the “time-to-frequency converter” [172

172. M. Vampouille, A. Barthélémy, B. Colombeau, and C. Froehly, “Observation et applications des modulations de fréquence dans les fibres unimodales,” J. Opt. (Paris) 15, 385–390 (1984). [CrossRef]

, 173

173. M. Vampouille, J. Marty, and C. Froehly, “Optical frequency intermodulation between two picosecond laser pulses,” IEEE J. Quantum Electron. QE-22, 192–194 (1986). [CrossRef]

, 174

174. M. T. Kaufman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994). [CrossRef]

, 175

175. L. Kh. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795–801 (2000). [CrossRef]

, 200

200. N. L. Markaryan and L. Kh. Mouradian, “Determination of the temporal profiles of ultrashort pulses by a fibre-optic compression technique,” Quantum Electron. 25, 668–670 (1995). [CrossRef]

]. From Eq. (4.14), for (1ψ)ϕ=0 (i.e., θ=π2), one measures Ĩπ2(ω)=I(ϕω). The successive action of the quadratic temporal and spectral phase modulation is depicted in Fig. 26. The key point is that the frequency marginal of the output Wigner distribution is a scaled version of the temporal marginal of the input Wigner distribution, and temporal intensity measurements can therefore be performed by using a spectrometer. Under these conditions, a quadratic spectral phase modulation remains after the temporal phase modulation, since the Wigner function is W[ϕω,ω+ψt]. This is apparent from the orientation of the interference fringes of the function in the chronocyclic space. Therefore, the spectral field after the two modulations is not a mapping of the input temporal field to the output, just as the image of an object in a simple telescope does not arise from a replication of the object field in the image space. Rather, it is a mapping of the moduli of the two fields only. Similarly to the imaging system, this does not affect the recovery of the temporal intensity. As in the spatial domain, a complete Fourier-transform setup can be obtained by combining dispersion ϕ, quadratic temporal modulation 1ϕ, and dispersion ϕ. This would make the interference fringes observed in Fig. 26 vertical.

4.3b. Chirped Pulse Modulation

The intensity of the modulation (e.g., the intensity of an optical test pulse, or the intensity transmission of a modulator) can be measured in a single shot by using a spectrometer that records the optical spectrum of the ancilla after modulation. A similar formalism can be used to describe frequency-to-time conversion by a single time lens followed by dispersive propagation. The optical spectral density of the test pulse is found by measuring the temporal intensity of the ancilla [176

176. J. Azaña, N. K. Berger, B. Levit, and B. Fischer, “Spectral Fraunhofer regime: time-to-frequency conversion by the action of a single time lens on an optical pulse,” Appl. Opt. 43, 483–490 (2004). [CrossRef] [PubMed]

, 204

204. T. Jannson, “Real-time Fourier transformation in dispersive optical fibers,” Opt. Lett. 8, 232–234 (1983). [CrossRef] [PubMed]

, 205

205. J. Azaña and M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. 36, 517–526 (2000). [CrossRef]

, 206

206. N. K. Berger, B. Levit, S. Atkins, and B. Fischer, “Time-lens-based spectral analysis of optical pulses by electrooptic phase modulation,” Electron. Lett. 36, 1644–1645 (2000). [CrossRef]

].

4.4. Simplified Chronocyclic Tomography

Provided a number of prior assumptions about the coherent nature of the test pulse ensemble are accepted, the electric field of the pulse can be obtained from a limited number of projections of its Wigner function [199

199. C. Dorrer and I. Kang, “Complete temporal characterization of short optical pulses by simplified chronocyclic tomography,” Opt. Lett. 28, 1481–1483 (2003). [CrossRef] [PubMed]

, 207

207. T. Alieva, M. J. Bastiaans, and L. Stankovic, “Signal reconstruction from two close fractional Fourier power spectra,” IEEE Trans. Signal Process. 51, 112–123 (2003). [CrossRef]

]. More specifically, the electric field can be retrieved by using one projection of W, i.e., Iα, and its angular derivative, i.e., Iαα. Because the latter can be obtained as a finite difference, i.e., as the difference between two projections for two different angles when the difference between the angles tends to zero, the required number of projections is equal to two (Fig. 27). Following Eq. (4.10), suppose that one can measure Ĩθ(ω)=dtW[t,ωcos(θ)tan(θ)t]. The angular derivative of this function with respect to θ is
Ĩθθ(ω)=dt[ωθ(1cos(θ))tθ(tan(θ))]Wω[t,ωcos(θ)tan(θ)t].
(4.24)
Calculating the derivatives at θ=0 gives
|Ĩθθ|θ=0(ω)=dttWω(t,ω)=ωdttW(t,ω).
(4.25)
Finally, since ϕω is equal to the first-order temporal moment of W, one obtains
|Ĩθθ|θ=0(ω)=ω[Ĩ(ω)ϕω].
(4.26)

The spectral phase of the pulse ϕ(ω) can therefore be reconstructed by using the angular derivative of the frequency marginal of the rotated Wigner function. The spectral intensity of the pulse, which is needed for the reconstruction of the field as well as for the reconstruction of the phase, can be obtained directly as the marginal for no rotation, i.e., Ĩ0(ω). It is therefore possible to reconstruct the field by using the frequency marginal of its Wigner function and the angular derivative of its frequency marginal taken at θ=0. Note that if one uses the frequency marginal at a finite angle θ and its derivative, the reconstructed field is the field corresponding to the Wigner function rotated by θ, from which the field in the absence of rotation can be reconstructed algebraically as long as θ is known precisely.

It turns out that the proposed reconstruction is also valid when one uses only a quadratic temporal phase modulation (in which case the operation on the Wigner function is not a rotation but a shear) [199

199. C. Dorrer and I. Kang, “Complete temporal characterization of short optical pulses by simplified chronocyclic tomography,” Opt. Lett. 28, 1481–1483 (2003). [CrossRef] [PubMed]

]. Indeed, the projection on the frequency axis of the Wigner distribution for a quadratic temporal phase modulation ψ is
Ĩψ(ω)=dtW[t,ω+ψt].
(4.27)
The derivative of this quantity with respect to ψ is
Ĩψψ(ω)=dtW(t,ω+ψt)ψ=dttW(t,ω+ψt)ω=ωdttW(t,ω+ψt).
(4.28)
Applying this relation at ψ=0 gives
|Ĩψψ|ψ=0(ω)=ωdttW(t,ω)=ω[Ĩ(ω)ϕω].
(4.29)
The group delay ϕω can be obtained by dividing the derivative of the optical spectrum of the modulated pulse with respect to the amplitude of the temporal phase modulation by the optical spectrum of the pulse. The derivative can be obtained experimentally as a finite difference, i.e., as the difference between two spectra measured for small finite quadratic temporal phase modulations. These spectra correspond to the projections of the Wigner function after small rotations in the chronocyclic space, since tan(θ)=ψ is small. Figure 27 represents the Wigner function of a pulse having a Gaussian spectrum and a cubic spectral phase after small quadratic temporal phase modulations of opposite signs. The difference between the resulting spectral marginals is plotted in Fig. 27(c). The electric field can be reconstructed in the spectral domain by using the optical spectrum and measured difference, and since the optical spectrum can be estimated by the average of the two spectral marginals, the electric field is completely reconstructed in the spectral domain by using only two projections of the Wigner function. This measurement requires a modulator of smaller bandwidth than the time-to-frequency converter and yet provides a complete measurement of the electric field by using only two one-dimensional spectra [199

199. C. Dorrer and I. Kang, “Complete temporal characterization of short optical pulses by simplified chronocyclic tomography,” Opt. Lett. 28, 1481–1483 (2003). [CrossRef] [PubMed]

], or, more practically, by directly measuring the spectrum and its derivative by using synchronous detection [208

208. I. Kang and C. Dorrer, “Highly sensitive differential tomographic technique for real-time ultrashort pulse characterization,” Opt. Lett. 30, 1545–1547 (2005). [CrossRef] [PubMed]

].

Simplified chronocyclic tomography can also be implemented with projections of the Wigner function onto the temporal axis [209

209. C. Dorrer, “Characterization of nonlinear phase shifts by use of the temporal transport-of-intensity equation,” Opt. Lett. 30, 3237–3239 (2005). [CrossRef] [PubMed]

]. The temporal phase of the pulse can be reconstructed from the derivative of the temporal intensity of the pulse under test with respect to an applied quadratic spectral phase modulation. The derivative can be approximated by the finite difference of two measured temporal intensities after different amounts of quadratic spectral phase modulation.

Simplified chronocyclic tomography is analogous to wavefront reconstruction from spatial intensity distribution measured in various planes, as first studied by Teague [210

210. M. R. Teague, “Irradiance moments: their propagation and use for unique retrieval of phase,” J. Opt. Soc. Am. 72, 1199–1209 (1982). [CrossRef]

] and Roddier [211

211. F. Roddier, “Wavefront sensing and the irradiance transport equation,” Appl. Opt. 29, 1402–1403 (1990). [CrossRef] [PubMed]

]. In this case, the transport-of-intensity equation, which is a two-dimensional equivalent of Eq. (4.29), is used to reconstruct the wavefront of a monochromatic beam. A similar approach has also been proposed for the temporal characterization of attosecond x-ray pulses [212

212. R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. V. Yakovlev, F. Bammer, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature 427, 817–821 (2004). [CrossRef] [PubMed]

].

4.5. Temporal Imaging

4.5a. Exact Temporal Imaging

Temporal imaging is the process of generating an optical pulse field that is a temporally magnified version of the input test pulse field (usually up to a constant phase, which is of no importance when square-law detection is performed) [164

164. B. H. Kolner and M. Nazarathy, “Temporal imaging with a time-lens,” Opt. Lett. 14, 630–632 (1989). [CrossRef] [PubMed]

, 165

165. I. P. Christov, “Theory of a time telescope,” Opt. Quantum Electron. 22, 473–480 (1990). [CrossRef]

, 177

177. A. A. Godil, B. A. Auld, and D. M. Bloom, “Time-lens producing 1.9ps optical pulses,” Appl. Phys. Lett. 62, 1047–1049 (1992). [CrossRef]

, 178

178. C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100Gbs optical data with an up-conversion time microscope,” Appl. Phys. Lett. 65, 2513–2515 (1994). [CrossRef]

, 179

179. C. V. Bennett and B. H. Kolner, “Upconversion time microscope demonstrating 103× magnification of femtosecond waveforms,” Opt. Lett. 24, 783–785 (1999). [CrossRef]

]. The bandwidth of the imaged pulse is smaller than that of the input pulse, so that registering temporal structure requires a smaller bandwidth detector than for the input pulse. Temporal magnifications of the order of 100 have been experimentally demonstrated.

The temporal magnification equations are analogous to spatial imaging of an object with a combination of free-space propagation along a distance z1, propagation through a lens of focal length f, and free-space propagation along a distance z2. The well-known condition for imaging is
1z1+1z2=1f,
(4.37)
and the magnification of such a system is given by
M=z2z1.
(4.38)
It should be noted that in the time–frequency domain a dispersion of arbitrary sign can be implemented, and therefore it is possible to obtain positive magnification by simply combining dispersion, lens, and dispersion.

The reason why temporal magnification is attractive is that fast photodetectors may be able to temporally resolve the time-magnified intensity. The bandwidth requirement for the photodetector is decreased compared with that required to measure the test pulse temporal intensity directly, owing to the reduction in bandwidth of the pulse provided by the temporal magnification setup. Also, in contrast to sampling oscilloscopes, which may have sufficient bandwidth, time-magnification systems can operate in single-shot mode. By reducing the bandwidth of the test pulse, one can potentially use real-time oscilloscopes to measure the temporal intensity.

4.5b. Time-Stretch Technique

4.6. Cross-Phase Modulation and Self-Phase Modulation with an Unknown Pulse

Tomographic ideas can also be applied in self-referencing temporal field characterization, when there is no reference pulse, and for which there is no common spatial equivalent. Suppose one implements a temporal phase modulation by using cross-phase modulation (XPM) or self-phase modulation (SPM). In the case of XPM with a bell-shaped pulse (i.e., one with quadratic temporal intensity profile), a temporal lens is obtained. However, if the temporal intensity profile of the pulse is unknown, then the induced phase modulation using either XPM or SPM is unknown. This is generally the case for self-referencing techniques, unless one has modified the pulse under test in a controlled manner. Nonetheless, XPM and SPM can be used for pulse characterization.

4.6a. Two-Spectra Technique

4.6b. Multispectra Technique

One way of obtaining redundancy in the experimental trace is to use XPM with a delayed replica of the test pulse [219

219. M. A. Franco, H. R. Lange, J.-F. Ripoche, B. S. Prade, and A. Mysyrowicz, “Characterization of ultrashort pulses by cross-phase modulation,” Opt. Commun. 140, 331–340 (1997). [CrossRef]

, 220

220. H. R. Lange, M. A. Franco, J.-F. Ripoche, B. S. Prade, P. Rousseau, and A. Mysyrowicz, “Reconstruction of the time profile of femtosecond laser pulses through cross-phase modulation,” IEEE J. Sel. Top. Quantum Electron. 4, 295–300 (1998). [CrossRef]

]. The spectrum of the pulse after XPM with a pump pulse of intensity IPUMP(t) can be measured for various relative delays between the two pulses in order to build a two-dimensional trace, as a function of the optical frequency ω and the delay τ,
S(τ,ω)=|dtE(t)exp[iαIPUMP(tτ)]exp(iωt)|2.

The electric field can be reconstructed from this trace by means of a generalization of the Gerchberg–Saxton algorithm. Backpropagation between the spectra measured for each delay is made possible by the fact that the gate is purely a phase gate, so that division by the gate response function does not make the algorithm unstable. There is a strong similarity of this approach to spectrography. Indeed, time scanning the unknown nonstationary filter obtained with XPM across the test pulse is equivalent to the temporal scanning of the gate in spectrography. Although semantically one would expect a gate to be an amplitude-varying filter, nothing precludes the use of a phase-only nonstationary filter in spectrography. Inversion based on generalized projections has also been used for the XPM technique [221

221. M. D. Thomson, J. M. Dudley, L. P. Barry, and J. D. Harvey, “Complete pulse characterization at 1.5 μm by cross-phase modulation in optical fibers,” Opt. Lett. 23, 1582–1584 (1998). [CrossRef]

]. Note that, in this case, the gate function can be unambiguously calculated from the pulse electric field provided that the nonlinearity α is known.

4.7. Practical Implementations of Tomography

4.7a. Quadratic Spectral Phase Modulation

Quadratic spectral phase modulation can be obtained by linear propagation of the test pulse through a dispersive device with specific phase transfer function. Useful devices include dispersive materials far from absorption resonances, waveguides, interferometers, devices based on diffraction, such as the two-grating compressor, or devices based on refraction, such as the two-prism compressor. The spectral phase added by linear propagation is a quadratic spectral phase modulation provided that all other terms in the Taylor development of the phase can be neglected over the bandwidth of the pulse under test. Such a modulation is independent of the pulse under test and can be accurately calibrated by using linear techniques. Techniques that require a large number of different spectral phase modulations benefit from the use of the two-grating or two-prism compressor, which can be tuned to vary the amount of quadratic spectral phase modulation.

4.7b. Quadratic Temporal Phase Modulation

Three approaches to generating a quadratic temporal phase modulation on a short optical pulse have been demonstrated.

Electro-optic phase modulation. Electro-optic phase modulators, based, for example, on lithium niobate waveguides, rely on the index change induced by a voltage via the electro-optic effect. Quadratic temporal phase modulation is obtained by synchronizing the optical pulse with one of the extrema of the modulation induced by a narrowband RF sine wave [177

177. A. A. Godil, B. A. Auld, and D. M. Bloom, “Time-lens producing 1.9ps optical pulses,” Appl. Phys. Lett. 62, 1047–1049 (1992). [CrossRef]

, 199

199. C. Dorrer and I. Kang, “Complete temporal characterization of short optical pulses by simplified chronocyclic tomography,” Opt. Lett. 28, 1481–1483 (2003). [CrossRef] [PubMed]

], as depicted in Fig. 29 for simplified chronocyclic tomography. The sinusoidal drive voltage V(t)=V0cos(Ωt) induces the phase modulation
ψ(t)=πV0Vπcos(Ωt)=ψ0πV0Ω22Vπt2
(4.42)
around t=0, where Vπ is the voltage needed to obtain a π phase shift. This leads to a quadratic temporal phase modulation with amplitude πV0Ω22Vπ at the maximum of the phase modulation, and synchronization with a minimum of the modulation leads to the opposite amplitude πV0Ω22Vπ. For the simplified chronocyclic tomography setup of Fig. 29(a), alternation of the relative delay between the train of pulses under test and the modulation allows synchronization with either the positive or negative quadratic temporal phase modulation. This alternation is performed at frequency f, and the spectral density around the optical frequency ω is therefore modulated at the same frequency. Lock-in detection allows extraction of the time average of the modulation Ĩ0(ω) (i.e., the optical spectrum) and the oscillating component Ĩf(ω) (i.e., the difference between the optical spectra obtained for the two quadratic temporal phase modulations). The results obtained with this setup included pulse compression by nonlinear propagation and dispersion compensation, as shown in Figs. 29(b), 29(c). Real-time operation allowed quick optimization of this nonlinear compressor. For a sine wave at a 10GHz frequency, with V0=10V, the quadratic temporal modulation achievable in a modulator with Vπ=7V is 8×1021s2. The temporal phase modulation is quadratic only over a limited temporal window, e.g., for a 10GHz sine wave with a 100ps period, but the temporal window over which the modulation is within 1% of its parabolic approximation is approximately 10ps. This approach, although still limited in terms of bandwidth, allows a pulse-independent temporal modulation, which can be accurately determined from the measurement of the parameters of the drive voltage and modulator. This is also a completely linear modulating scheme. The development of these modulators has benefited from the developments of high-speed lithium niobate modulators, and progress in this area is likely to occur.

Wave-mixing. Wave-mixing with a chirped pulse can provide a quadratic temporal phase modulation of arbitrary sign [172

172. M. Vampouille, A. Barthélémy, B. Colombeau, and C. Froehly, “Observation et applications des modulations de fréquence dans les fibres unimodales,” J. Opt. (Paris) 15, 385–390 (1984). [CrossRef]

, 173

173. M. Vampouille, J. Marty, and C. Froehly, “Optical frequency intermodulation between two picosecond laser pulses,” IEEE J. Quantum Electron. QE-22, 192–194 (1986). [CrossRef]

, 179

179. C. V. Bennett and B. H. Kolner, “Upconversion time microscope demonstrating 103× magnification of femtosecond waveforms,” Opt. Lett. 24, 783–785 (1999). [CrossRef]

, 228

228. R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Optical time lens based on four-wave mixing on a silicon chip,” Opt. Lett. 33, 1047–1049 (2008). [CrossRef] [PubMed]

]. The electric field of a short pulse EPUMP(t) after large quadratic spectral phase modulation ϕPUMPω22 is EPUMP(t)=ẼPUMP(tϕPUMP)exp[it2(2ϕPUMP)] within some multiplicative factors. Nonlinear mixing of the test pulse with an ancillary chirped pump pulse is therefore formally equivalent to quadratic temporal phase modulation, provided that the amplitude of the latter is constant over the temporal support of the pulse under test. A schematic of a time-magnification setup based on wave mixing is shown in Fig. 31(a). The input pump pulse is sent into a dispersive delay line (for example, a two-grating compressor or an optical fiber), leading to the second-order dispersion ϕPUMP. The waveform under test is first dispersed by a dispersive delay line introducing a second-order dispersion ϕ1, then interacts with the chirped pump pulse in a nonlinear medium, and the product of this interaction is sent into an additional dispersive delay line with second-order dispersion ϕ2. The imaging condition relating ϕ1, ϕ2, and ϕPUMP depends on the nonlinear interaction.

With three-wave mixing, [172

172. M. Vampouille, A. Barthélémy, B. Colombeau, and C. Froehly, “Observation et applications des modulations de fréquence dans les fibres unimodales,” J. Opt. (Paris) 15, 385–390 (1984). [CrossRef]

, 173

173. M. Vampouille, J. Marty, and C. Froehly, “Optical frequency intermodulation between two picosecond laser pulses,” IEEE J. Quantum Electron. QE-22, 192–194 (1986). [CrossRef]

, 179

179. C. V. Bennett and B. H. Kolner, “Upconversion time microscope demonstrating 103× magnification of femtosecond waveforms,” Opt. Lett. 24, 783–785 (1999). [CrossRef]

] the electric field of the generated signal is proportional to EEPUMP, i.e., the product of the electric field of the input chirped signal and the chirped pump, and the temporal phase modulation induced by the time lens is t2(2ϕPUMP). Equation (4.34) is directly applicable, with ϕ1, ϕ2 and ψ=1ϕPUMP, leading to 1ϕ1+1ϕ2=1ϕPUMP and M=ϕ2ϕ1. For example, [179

179. C. V. Bennett and B. H. Kolner, “Upconversion time microscope demonstrating 103× magnification of femtosecond waveforms,” Opt. Lett. 24, 783–785 (1999). [CrossRef]

] uses a stretcher to provide the dispersion ϕ1=0.17606ps2 on the waveform under test, and a compressor to provide the dispersion ϕPUMP=0.17784ps2 on the pump. After sum-frequency generation, the signal propagates in a compressor providing the dispersion ϕ2=17.606ps2. This setup theoretically leads to a temporal magnification M=100, and was indeed shown experimentally to magnify the temporal intensity of pairs of pulses by 103. This allowed waveform measurements with a 300fs resolution by use of direct photodetection and a sampling oscilloscope.

With four-wave mixing [228

228. R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Optical time lens based on four-wave mixing on a silicon chip,” Opt. Lett. 33, 1047–1049 (2008). [CrossRef] [PubMed]

], the electric field of the generated idler is proportional to EPUMP2E*, i.e., the square of the field of the chirped pump and the conjugate of the field of the signal. Because of this, Eq. (4.34) must be applied to ϕ1, ϕ2, and ψ=2ϕPUMP, leading to 1ϕ1+1ϕ2=2ϕPUMP and M=ϕ2ϕ1. This has, for example, been implemented in a silicon chip [228

228. R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Optical time lens based on four-wave mixing on a silicon chip,” Opt. Lett. 33, 1047–1049 (2008). [CrossRef] [PubMed]

]. In this case, the pump and waveform under test are, respectively, dispersed by 1900 and 1000  m of standard single-mode fiber, leading to dispersions ϕ1=21.6ps2 and ϕPUMP=41ps2. The idler generated by four-wave mixing is sent into a dispersion-compensating module with dispersion ϕ2=434ps2. This setup demonstrates a temporal magnification M=20. An example of a setup for time-to-frequency conversion using four-wave mixing in a silicon waveguide is shown in Fig. 31(b) [229

229. M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456, 81–85 (2008). [CrossRef] [PubMed]

]. The chromatic dispersion for the pump and test waveform is provided by optical fibers. Four-wave mixing occurs in a silicon waveguide, and wave-mixing with a pump pulse that has twice the dispersion of the test waveform leads to time-to-frequency conversion. An additional span of fiber identical to the input fiber span allows a full time-to-frequency conversion of the electric field, although this is not required for intensity measurements. Time-to-frequency conversion enabled measurement of the intensity of high-speed optical waveforms over a time interval longer than 100ps with a 220fs temporal resolution. Single-shot operation is provided by using a spectrometer capable of measuring the entire spectrum after modulation in a single shot. Examples of waveforms measured with this setup are shown in Fig. 32. The left column corresponds to results from the time-to-frequency conversion ultrafast oscilloscope. The right column corresponds to the intensity measured by cross-correlation of the test waveform with a short optical pulse. Very good agreement is obtained for a number of different waveforms, even in single-shot operation.

4.8. Conclusions

Chronocyclic tomography provides a means by which the full two-time correlation function of a pulse ensemble can be determined. It has proved difficult to implement full tomographic reconstruction of femtosecond pulses in practice because of the difficulty in modulating pulses with sufficient bandwidth. However, a number of subtomographic approaches have been implemented successfully, and the most common of these, the temporal imaging system, allows direct measurement (and indeed simple visualization) of subpicosecond waveforms by using single-shot data acquisition by means of a fast photodetector and sampling oscilloscope. Such devices are useful for low-repetition-rate systems, or for systems where the pulse shape is changing rapidly from shot to shot and from which samples can be taken.

5. Interferometry

5.1. Introduction

Interferometry provides a very sensitive and accurate means to measure the phase of an optical field. This approach has a long pedigree in the field of optical testing [230

230. M. Françon, Optical Interferometry (Academic, 1966).

, 231

231. D. Malacara, Optical Shop Testing (Wiley-Interscience, 1991).

]. The conversion of phase to amplitude information that is the hallmark of interferometric measurement allows deterministic and robust extraction of the phase from the measured data.

The earliest suggestions for using interferometric methods for pulse characterization made use of the concept of test-plus-reference interferometry in the spectral domain to show how the phase of an optical pulse was changed during propagation in a linear or nonlinear optical medium [232

232. C. Froehly, A. Lacourt, and J. C. Vienot, “Notions de réponse impulsionelle et de fonction de transfert temporelle des pupilles optiques, justifications expérimentales et applications,” Nouv. Rev. Opt. 4, 183–196 (1973). [CrossRef]

, 233

233. J. Piasecki, B. Colombeau, M. Vampouille, C. Froehly, and J. A. Arnaud, “Nouvelle méthode de mesure de la réponse impulsionnelle des fibres optiques,” Appl. Opt. 19, 3749–3755 (1980). [CrossRef] [PubMed]

, 234

234. F. Reynaud, F. Salin, and A. Barthélémy, “Measurement of phase shifts introduced by nonlinear optical phenomena on subpicosecond pulses,” Opt. Lett. 14, 275–277 (1989). [CrossRef] [PubMed]

]. This implementation of spectral interferometry (SI) made use of the fact that the measured signal reflected the difference in the spectral phase between the test and reference pulses, so that differential measurements (say, of the input to the output fields) could be made with great precision.

It was soon realized that detailed knowledge of the reference pulse enabled extraction of the complete spectral phase of the test pulse, which, together with a measurement of its spectrum, constitutes a complete characterization of the pulse [63

63. L. Lepetit, G. Chériaux, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B 12, 2467–2474 (1995). [CrossRef]

, 64

64. D. N. Fittinghoff, J. L. Bowie, J. N. Sweetser, R. T. Jennings, M. A. Krumbügel, K. W. Delong, R. Trebino, and I. A. Walmsley, “Measurement of the intensity and phase of ultraweak, ultrashort laser pulses,” Opt. Lett. 21, 884–886 (1996). [CrossRef] [PubMed]

]. Of course, this begs the question, since it depends on the availability of a known pulse of appropriate character. However, the extremely high sensitivity of this method, which is entirely linear in the test pulse electric field, has enabled it to continue as a viable method in certain applications.

Methods of interferometry that do not require a reference pulse have been developed and are now in wide use. The basic feature of self-referencing methods is to measure the temporal beats that arise when one replica of the pulse is interfered with a second, time-shifted, replica, or, equivalently, the spectral fringes that occur when one replica of the pulse is interfered with a second, frequency-shifted (or spectrally sheared) replica. The latter case has strong analogies to SI and is known as “spectral shearing interferometry” (SSI). In both cases the fringes patterns reveal the relative phase between two adjacent parts of the pulse field (either two time slots or two frequency components), from which the complete phase function may be reconstructed.

The key features of interferometry that make it useful for pulse characterization are the rapidity of data acquisition, the direct and fast reconstruction of the field from the data, and the insensitivity of the measurement to wavelength-dependent apparatus response. These properties are important for characterizing sources for which the pulse shape fluctuates and the pulses have large bandwidths. In this Section, we discuss interferometric methods of pulse characterization and show how these are implemented in ultrafast optics.

5.2. General Considerations and Implementations

5.2a. Definitions

Two classes of interferometer can be identified. In the first, measurements are made in the frequency domain, and in the second in the time domain. The signals are then proportional to sections of the two-frequency and two-time correlation functions, respectively. Because available detectors are slow compared with the pulses themselves, measurement in the frequency domain is usually preferred. In both of these classes there are two species of interferometer, self-referencing and test plus reference.

A typical spectral-domain test-plus-reference interferometer is illustrated in Fig. 33(a). The test pulse is shifted in time by the delay τ with respect to the reference pulse using a delay line. It is mixed with the reference pulse at a beam splitter, and the resulting spectrum is measured. This exhibits fringes in the spectral domain, whose spacing is inversely proportional to the delay τ. The detected signal is
D(Ω;τ)=|ẼR(Ω)+Ẽ(Ω)eiΩτ|2
(5.1)
where ẼR is the reference field (the Fourier transform of the reference pulse analytic signal) and Ẽ the test pulse field. The spectral phase difference between test and reference pulses is encoded in the relative positions of the spectral fringes with respect to the nominal spacing of 2πτ.

The time-domain analog of this interferometer shifts the test pulse in frequency with respect to the reference pulse by an amount Ω, before mixing the two at a beam splitter [Fig. 33(b)]. The resulting temporal interference pattern is measured by passing the signal through a fast shutter and recording the transmitted energy. In this case the signal is given by
D(τ;Ω)=|ER(τ)+E(τ)eiΩτ|2,
(5.2)
where in this case the relative temporal phase of the two pulses is encoded in the relative positions of the temporal fringes with respect to the nominal spacing of 2πΩ.

In the spectral case, the final spectrometer must have a resolution that is high compared with the nominal spectral fringe spacing. In the temporal case, the shutter must be open for a time short compared with the period of the temporal beat pattern.

The class of self-referencing interferometers similarly has implementations both in the time and the frequency domains. Self-referencing interferometers are based on spectral or temporal shearing. This type of interferometer uses modulators and delay lines to generate two modified versions of the input test pulse, shifted in time and frequency with respect to each other, that are then interfered. The resulting interferogram may be measured in the time or the frequency domain. When characterizing ultrashort pulses by using slow square-law detectors, the most common sort of interferometer makes use of spectral shearing.

Schematic apparatuses for shearing interferometry are shown in Fig. 34. The test pulse enters the interferometer, experiencing a frequency shift in one arm (by means of a linear temporal phase modulator) and a time-delay (or temporal shift) in the other (effected using a simple delay line, which may be considered in linear filter terms as a linear spectral phase modulator). The two modified pulses are combined at the exit ports of the interferometer, and the output is measured in the time domain or frequency domain by passing it through a fast time gate or narrow spectral filter, respectively.

The field after the interferometer in both apparatuses is
EOUT(t)=E(t)eiΩt+E(tτ)=FT[Ẽ(ω+Ω)+Ẽ(ω)eiωτ],
(5.3)
where FT[] represents the Fourier transform, τ is the delay imposed in one of the arms of the interferometer, and Ω the frequency shift imposed in the other. These may be thought of as lateral shears in their respective domains.

In the case of a temporal shearing interferometer, the signal is measured directly in the time domain by means of a very fast time gate or shutter, followed by the usual integrating square-law detector. When the shutter response is very fast with respect to the variations in the pulse temporal field, the detected signal is
D(t;Ω,τ)=|EOUT(t)|2=I(t)+I(tτ)+2Re[E(t)E*(tτ)eiΩt],
(5.4)
where I(t)=|E(t)|2 is the intensity of the pulse. In the case of the spectral shearing interferometer, the detected signal is measured in the frequency domain by means of a high-resolution spectrometer, with a slow detector. The spectrometer passband must be narrow with respect to variations in the pulse spectral field, in which case the detected signal is
Ĩ(ω;Ω,τ)=|ẼOUT(ω)|2=Ĩ(ω)+Ĩ(ω+Ω)+2Re[Ẽ(ω)Ẽ*(ω+Ω)eiωτ],
(5.5)
where Ĩ(ω)=|Ẽ(ω)|2 is the spectrum of the pulse. In both cases, it is clear that the interferogram encodes the derivative of the temporal or spectral phase function in the fringe spacing. For example, in the spectral shearing interferometer, the fringe extrema are located at frequencies ω satisfying
ϕ(ω)ϕ(ω+Ω)+ωτ=mπ,
(5.6)
where ϕ(ω)=arg[Ẽ(ω)] is the spectral phase function and m is an integer. Therefore, as with phase-space methods, it is possible to determine ϕ(ω) to within a constant (the carrier-envelope offset phase) and a linear term (the overall delay of the pulse with respect to an external clock). For most applications, this is sufficient to characterize the pulse, as long as the spectrum of the pulse is known.

5.2b. Interpretation of Interferograms

Interferometry measures the two-frequency (or two-point, in the case of spatial interferometry) correlation function of a field. (Since, as with all methods, it derives from measurements made with square-law detectors, it must yield some bilinear functional of the input field.) In the most general case, it yields a two-dimensional complex function, whose arguments are both time variables or both frequency variables. To this extent, it is quite different from spectrographic techniques, which work in a time–frequency representation, and tomographic methods, which use a set of one-dimensional functions parameterized by an external variable often unrelated to time or frequency.

As a particular example, it is useful to see how a spectral shearing interferogram relates to the correlation function. Starting from Eq. (5.5), variables are changed to the mean- and difference-frequency coordinates, ωC=ω+Ω2, Δω=Ω. Then the interferogram may be written as
Ĩ(ωC;Δω,τ)=|Ẽ(ωC+Δω2)+ei(ωCΔω2)τẼ(ωCΔω2)|2,
(5.10)
or, in terms of the correlation function,
Ĩ(ωC;Δω,τ)=Ĩ(ωC+Δω2)+Ĩ(ωCΔω2)+2|C͌(Δω,ωC)|cos{arg[C͌(Δω,ωC)]+τ(ωCΔω2)}.
(5.11)
This shows that the interferogram maps out a line of the two-frequency correlation function, taken as a function of ωC, keeping Δω fixed. Since C͌(Δω,ωC) is a complex function and Ĩ(ωC;Δω,τ) a real one, two measurements are required. The interferogram is a superposition of the two quadratures of the two-frequency correlation function, so that these can be individually retrieved by using only two values of the delay τ. In the case of a coherent ensemble, we have seen that the complete correlation function need not be measured in order to extract the field. This is in contrast to phase-space methods, in which the entire distribution must be measured. Importantly in this common experimental situation, only a single line of either quadrature of the two-frequency correlation function is sufficient for reconstructing the electric field, making interferometry inherently more economical of data than other methods.

5.3. Inversion

The inversion methods for interferometric measurements are direct, and therefore robust and reliable. The basic element of the inversion algorithms is the extraction of the correlation function, which is a complex entity, from a purely real and positive detected signal. Though there are a number of ways of doing this, the simplest and most commonly used is based on a Fourier analysis of the signal, accompanied by filtering to remove the symmetry in the Fourier domain that arises from its real character. This is the first step of all interferometric phase retrieval methods; the subsequent steps depend on whether the method is self-referencing. In this section we first describe this algorithmic element as applied to methods with known reference pulses, then describe the additional steps needed for self-referencing methods.

5.3a. Fourier-Transform Spectral Interferometry

Fourier-transform spectral interferometry (FTSI) is a version of test-plus-reference interferometry where the signal is measured in the frequency domain relative to a reference pulse [232

232. C. Froehly, A. Lacourt, and J. C. Vienot, “Notions de réponse impulsionelle et de fonction de transfert temporelle des pupilles optiques, justifications expérimentales et applications,” Nouv. Rev. Opt. 4, 183–196 (1973). [CrossRef]

]. Typically this is recorded with a detector array placed in the focal plane of a flat-field grating spectrometer, to yield a spectral interferogram. The schematic apparatus is shown in Fig. 35(a) in the case when the pulse under test is derived from the reference pulse by propagation in a device under test. The data set is a function of only a single variable—the frequency—rather than of two variables as in time–frequency methods. This means that the second dimension of a two-dimensional detector array may be used to record spatial variations in the spectral phase, for example.

The spectral phase is extracted via a direct inversion that is both rapid and robust. The test and reference pulse are delayed in time with respect to one another by τ by using a linear time-stationary filter S̃LP(ω)=eiωτ. The detected signal (interferogram) is then D(ω;τ)=|ẼR(ω)+Ẽ(ω)eiωτ|2, where ẼR is the reference field and Ẽ the test pulse field. The spectral phase difference between test and reference pulses is encoded in the relative positions of the spectral fringes with respect to the nominal spacing of 2πτ. Examples of interferograms corresponding to identical reference and test pulses and to a test pulse with a quadratic spectral phase are plotted in Fig. 35(b). In the first case no change in the spacing of the fringes is observed, while for the quadratic spectral phase the fringe spacing is clearly a function of the optical frequency, revealing that the spectral phase difference between the two pulses is quadratic. The phase difference can be extracted by using a three-step algorithm involving a Fourier transform to the time domain, a filtering operation, and an inverse Fourier transform [63

63. L. Lepetit, G. Chériaux, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B 12, 2467–2474 (1995). [CrossRef]

, 235

235. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe- pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982). [CrossRef]

]. The interferogram may be written as
D(ω;τ)=D(dc)(ω)+D(ac)(ω)eiωτ+[D(ac)(ω)eiωτ]*,
(5.12)
where
D(dc)(ω)=Ĩ(ω)+ĨR(ω),
(5.13)
D(ac)(ω)=|Ẽ(ω)ẼR(ω)|ei[ϕ(ω)ϕR(ω)].
(5.14)
The dc portion of the interferogram, Eq. (5.13), is the sum of the individual spectra of the pulses and contains no phase information. The ac term, Eq. (5.14), contain all of the relative phase information.

The above analysis pertains to an idealized version of an experiment. The spectrometer has a finite spectral resolution that depends on its optics and detector, which leads to a decreased fringe contrast when the fringe period becomes comparable with the spectral resolution. Furthermore, sampling of the interferogram of Eq. (5.12) is performed at a finite rate (e.g., with the array of finite-size photodetectors that compose the detector located at the Fourier plane of the spectrometer). The interferogram is sampled at frequencies that are not necessarily evenly spaced. Finally, the quickly varying fringes that allow the extraction of the spectral phase difference from a single interferogram can make FTSI sensitive to frequency calibration of the optical spectrum analyzer. These effects are not detrimental to most applications of SI, and can be accounted for [236

236. C. Dorrer, “Influence of the calibration of the detector on spectral interferometry,” J. Opt. Soc. Am. B 16, 1160–1168 (1999). [CrossRef]

, 237

237. C. Dorrer, N. Belabas, J.-P. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B 17, 1795–1802 (2000). [CrossRef]

].

5.3b. Concatenation

Shearing interferometry may be implemented in the optical frequency domain and thus be used to measure the spectral phase function of the input pulse using itself as a reference [31

31. V. Wong and I. A. Walmsley, “Analysis of ultrashort pulse-shape measurement using linear interferometers,” Opt. Lett. 19, 287–289 (1994). [CrossRef] [PubMed]

, 250

250. V. A. Zubov and T. I. Kuznetsova, “Solution of the phase problem for time-dependent optical signals by an interference system,” Sov. J. Quantum Electron. 21, 1285–1286 (1991). [CrossRef]

]. Two delayed replicas of the (unknown) test pulse are generated in an interferometer, and one is frequency shifted with respect to the other. The combined spectrum of the pulse pair is measured by using a spectrometer and a detector array, can be measured simultaneously with the interferogram [251

251. C. Dorrer, “Implementation of spectral phase interferometry for direct electric-field reconstruction with a simultaneously recorded reference interferogram,” Opt. Lett. 24, 1532–1534 (1999). [CrossRef]

], or can be extracted from the shearing interferogram itself [252

252. A. Müller and M. Laubscher, “Spectral phase and amplitude interferometry for direct electric-field reconstruction,” Opt. Lett. 26, 1915–1917 (2001). [CrossRef]

]. The important feature is that the frequency shift, or spectral shear, allows two adjacent frequencies in the original pulse spectrum to interfere on an integrating detector. The resulting fringe pattern thus reflects the spectral phase difference between spectral components of the pulse separated by the shear. Extracting the spectral phase of the input pulse therefore requires additional steps. The simplicity of the inversion mean that such characterization can be done at very rapid rates—up to a 1kHz refresh rate has been reported to date, limited only by the detector readout time [253

253. T. M. Shuman, M. E. Anderson, J. Bromage, C. Iaconis, L. Waxer, and I. A. Walmsley, “Real-time SPIDER: ultrashort pulse characterization at 20Hz,” Opt. Express 5, 134–143 (1999). [PubMed]

, 254

254. W. Kornelis, J. Biegert, J. W. G. Tisch, M. Nisoli, G. Sansone, C. Vozzi, S. De Silvestri, and U. Keller, “Single-shot kilohertz characterization of ultrashort pulses by spectral phase interferometry for direct electric-field reconstruction,” Opt. Lett. 28, 281–283 (2003). [CrossRef] [PubMed]

].

The SSI interferogram has a similar form to the FTSI interferogram [Eq. (5.12)] except that the dc and ac terms contain different frequency arguments:
D(dc)(ω)=Ĩ(ω+Ω)+Ĩ(ω),
(5.17)
D(ac)(ω)=|Ẽ(ω+Ω)Ẽ(ω)|ei[ϕ(ω+Ω)ϕ(ω)].
(5.18)

The spectral phase difference θ(ω)=ϕ(ω+Ω)ϕ(ω) between two frequencies separated by the spectral shear Ω is extracted from the interferogram by using the FTSI algorithm previously described. It is then concatenated into the spectral phase of the pulse under test, ϕ, by following the formula
ϕ(0)=0,
ϕ[(n+1)Ω]=ϕ(nΩ)+θ(nΩ).
(5.19)
An interpolation of the spectrum on the same grid completes the measurement in the spectral domain. This then gives the electric field in the spectral domain at frequencies [0,Ω,2Ω,,NΩ].

In this method, therefore, a sampling of the spectral phase (to within a constant) at intervals of the shear Ω across the pulse spectrum is obtained. According to the Shannon theorem, all pulses with compact support in the domain [T2,T2] may be completely characterized by a sampling of their spectral representation every 2πT. Thus SSI is able to reconstruct all pulses that have support (i.e., that do not have energy outside this domain) only in the temporal window [πΩ,πΩ]. Moreover, the inversion is unique.

5.3c. Ambiguities, Accuracy and Precision in Phase Extraction

Ambiguities. Difficulties in reconstruction arise in SSI when the spectrum goes to zero over a region that is large compared with the spectral shear, in which case the spectral phase is not defined for several samples of the SSI phase. In this case, two interferograms measured by using two different values of the shear are needed to reconstruct the pulse. The first returns the spectral phase across each continuous region of the spectrum; the second returns the relative phase between the two discontinuous pieces. Note that zeros of intensity in the time domain do not lead to ambiguities, unless they are associated with zeros in the spectral domain.

When a nonlinear interaction is used to spectrally shear an optical pulse, this difficulty can result in an undeterminable phase. For example, in spectral phase interferometry for direct electric field reconstruction (SPIDER) the single known case for which the data is incomplete is that of a pulse whose spectrum consists of no more than two well-separated components, when the measurement is made using only the pulse itself (i.e., not by means of a separate uncharacterized chirped pulse). By “well separated”, we mean that the spectral intensity is below the noise level of the detection system over a domain that is larger than the shear [86

86. D. Keusters, H.-S. Tan, P. O’Shea, E. Zeek, R. Trebino, and W. S. Warren, “Relative-phase ambiguities in measurements of ultrashort pulses with well-separated multiple frequency components,” J. Opt. Soc. Am. B 20, 2226–2237 (2003). [CrossRef]

]. For spectra with several such components, it is still possible to obtain the relative phases between them by using several different values of the shear. When a separate independent pulse is used as an ancillary for inducing the spectral shear, even the case of two-component spectra is possible.

Accuracy and precision. For any measurement, testing the accuracy of the reconstruction, i.e., how close the measurement result from the apparatus is to the actual physical quantity, is of primary importance. This is mainly a theoretical task, relying on simulations or equations, for the obvious reason that in most experimental situations the measured field is unknown before the measurement. A measure of the difference between the input target field and the output retrieved field provides the criterion of accuracy. The choice of a measure is, however, somewhat subjective [255

255. M. E. Anderson, L. E. E. de Araujo, E. M. Kosik, and I. A. Walmsley, “The effects of noise on ultrashort-optical-pulse measurement using spectral phase interferometry for direct electric-field reconstruction,” Appl. Phys. B 70, S85–S93 (2000). [CrossRef]

, 256

256. C. Dorrer and I. A. Walmsley, “Accuracy criterion for ultrashort pulse characterization techniques: application to spectral phase interferometry for direct electric-field reconstruction,” J. Opt. Soc. Am. B 19, 1019–1029 (2001). [CrossRef]

].

In SSI, the accuracy of the reconstruction of the spectral phase is perfect in the absence of noise, when the spectral phase function on the sampling interval Ω is represented by a polynomial, and the sampling criterion is satisfied. Therefore it is possible to reconstruct very sharp spectral phase functions, especially those produced by a Fourier-plane pulse shaper. Beyond the sampling limit for the pulse spectrum, the accuracy depends somewhat upon the details of the reconstruction algorithm. In practice, those that use integration over the measured spectral phase give the most accurate results.

In practice the accuracy must be evaluated for each implementation, on the basis of the parameter settings for that piece of apparatus. It is therefore impossible to make a general statement about the accuracy of the spectral shearing method as a whole. However, an instrument using a simple integration algorithm for which the spectral phase is oversampled by a factor of 2 has an accuracy that scales roughly as ten times the noise fraction, where this is defined as the ratio of the variance of the noise to the maximum signal of the interferogram [256

256. C. Dorrer and I. A. Walmsley, “Accuracy criterion for ultrashort pulse characterization techniques: application to spectral phase interferometry for direct electric-field reconstruction,” J. Opt. Soc. Am. B 19, 1019–1029 (2001). [CrossRef]

]. Although the SSI experimental trace is a single one-dimension interferogram, its robustness to noise was found similar to spectrographic techniques requiring the acquisition of a two-dimensional experimental trace [255

255. M. E. Anderson, L. E. E. de Araujo, E. M. Kosik, and I. A. Walmsley, “The effects of noise on ultrashort-optical-pulse measurement using spectral phase interferometry for direct electric-field reconstruction,” Appl. Phys. B 70, S85–S93 (2000). [CrossRef]

].

One useful feature of the direct inversion possible in SSI is that it is possible to determine analytically the effect of systematic errors in the apparatus on the estimation of the test field. The primary systematic errors arise from miscalibration of the delay τ that gives rise to the carrier fringes. For example, miscalibration of the spectrometer can lead to some error in the delay calibration, although efficient and simple calibration procedures have been devised [236

236. C. Dorrer, “Influence of the calibration of the detector on spectral interferometry,” J. Opt. Soc. Am. B 16, 1160–1168 (1999). [CrossRef]

]. Ultrabroadband pulses also require a carefully designed interferometer [257

257. J. R. Birge, R. Ell, and F. X. Kärtner, “Two-dimensional spectral shearing interferometry for few-cycle pulse characterization,” Opt. Lett. 31, 2063–2065 (2006). [CrossRef] [PubMed]

, 258

258. J. R. Birge and F. X. Kärtner, “Analysis and mitigation of systematic errors in spectral shearing interferometry of pulses approaching the single-cycle limit,” J. Opt. Soc. Am. B 25, A111–A119 (2008). [CrossRef]

]. A delay calibration error δτ leads to an additive linear component ωδτ on the spectral phase before concatenation. The integrated phase has an additional component that is quadratic in frequency, δτω2(2Ω), i.e., an error δτΩ is made in the retrieved second-order dispersion. This may alter the duration of the reconstructed pulse compared with the actual pulse. A simple example illustrates the main issues. Consider a Gaussian test pulse with bandwidth Δω, corresponding to a Fourier-transform-limited pulse duration ΔtFTL and to an actual pulse duration Δt0. In the presence of second-order dispersion ϕ(2) and error on the second-order dispersion δτΩ, the actual pulse duration is
Δt02=ΔtFTL2[1+Δω2ΔtFTL2(ϕ(2))2],
(5.20)
and the measured pulse duration is
Δt2=ΔtFTL2[1+Δω2ΔtFTL2(ϕ(2)+δτΩ)2].
(5.21)
If the actual pulse is Fourier-transform limited (i.e., ϕ(2)=0 and Δt0=ΔtFTL), Eq. (5.21) can be written simply as
ɛΔt=1+(Nɛτ)21,
(5.22)
where ɛΔt is the relative error on the pulse duration (ΔtΔt0)Δt0, ɛτ is the relative error on the delay δτΔtFTL, and N=ΔωΩ is the degree above the Fourier-transform limit chosen for the sampling window. If the pulse is far from the Fourier-transform limit, Eq. (5.21) can be written simply as
ɛΔt=NɛτΔtFTLΔt0.
(5.23)
Orders of magnitude can be obtained from a simple example. For N=20 (i.e., a setup that is arranged to measure pulses up to 20 times the Fourier-transform limit) a 1% delay calibration error corresponds to a 2% error in the estimated pulse duration for a Fourier-transform-limited pulse. Taking the example of a chirped pulse with Δt0=10ΔtFTL, a 1% delay error still leads to a 2% error in the estimated pulse duration. This is not a severe constraint. For example, even for the most extreme case of pulses in the single-cycle regime, where ΔtFTL=2.5fs, the delay must be calibrated to within a path length of λ100, which requires a well-designed interferometer and proper alignment procedure. In that extreme case, however, all methods have some form of systematic error that must be dealt with carefully. Unfortunately, it is not so simple to ascertain the severity of calibration errors for most methods—and not at all in analytic form.