## Self-referenced characterization of femtosecond laser pulses by chirp scan |

Optics Express, Vol. 21, Issue 21, pp. 24879-24893 (2013)

http://dx.doi.org/10.1364/OE.21.024879

Acrobat PDF (1696 KB)

### Abstract

We investigate a variant of the d-scan technique, an intuitive pulse characterization method for retrieving the spectral phase of ultrashort laser pulses. In this variant a ramp of quadratic spectral phases is applied to the input pulses and the second harmonic spectra of the resulting pulses are measured for each chirp value. We demonstrate that a given field envelope produces a unique and unequivocal chirp-scan map and that, under some asymptotic assumptions, both the spectral amplitude and phase of the measured pulse can be retrieved analytically from only two measurements. An iterative algorithm can exploit the redundancy of the information contained in the chirp-scan map to discard experimental noise, artifacts, calibration errors and improve the reconstruction of both the spectral intensity and phase. This technique is compared to two reference characterization techniques (FROG and SRSI). Finally, we perform d-scan measurements with a simple grating-pair compressor.

© 2013 OSA

1. I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon. **1**, 308–437 (2009). [CrossRef]

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

3. A. S. Radunsky, I. A. Walmsley, S.-P. Gorza, and P. Wasylczyk, “Compact spectral shearing interferometer for ultrashort pulse characterization,” Opt. Lett. **32**, 181–183 (2007). [CrossRef]

4. R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. **68**, 3277–3295 (1997). [CrossRef]

5. C. Iaconis and I. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. **23**, 792–794 (1998). [CrossRef]

6. V. V. Lozovoy, I. Pastirk, and M. Dantus, “Multiphoton intrapulse interference. iv. ultrashort laser pulse spectral phase characterization and compensation,” Opt. Lett. **29**, 775–777 (2004). [CrossRef] [PubMed]

8. B. Xu, J. M. Gunn, J. M. D. 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]

12. M. Miranda, T. Fordell, C. Arnold, A. L’Huillier, and H. Crespo, “Simultaneous compression and characterization of ultrashort laser pulses using chirped mirrors and glass wedges,” Opt. Express **20**, 688 (2012). [CrossRef] [PubMed]

*E*(

*t*) and for each test phase the second harmonic spectrum is recorded. The test functions are usually parameterized by a single scalar parameter

*p*and the experimental data take the form of a two-dimensional map

*I*

_{SHG}(

*ω*,

*p*). For MIIPS in its original version [6

6. V. V. Lozovoy, I. Pastirk, and M. Dantus, “Multiphoton intrapulse interference. iv. ultrashort laser pulse spectral phase characterization and compensation,” Opt. Lett. **29**, 775–777 (2004). [CrossRef] [PubMed]

8. B. Xu, J. M. Gunn, J. M. D. 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]

*p*is a phase offset. For d-scan and some variants of MIIPS [7

7. V. V. Lozovoy, B. Xu, Y. Coello, and M. Dantus, “Direct measurement of spectral phase for ultrashort laser pulses,” Opt. Express **16**, 592–597 (2008). [CrossRef] [PubMed]

15. I. Pastirk, B. Resan, A. Fry, J. MacKay, and M. Dantus, “No loss spectral phase correction and arbitrary phase shaping of regeneratively amplified femtosecond pulses using miips,” Opt. Express **14**, 9537–9543 (2006). [CrossRef] [PubMed]

*i*) there is a one-to-one relationship between E(t) and

*I*

_{SHG}(

*ω*,

*ϕ*

_{2}) where

*ϕ*

_{2}is the chirp coefficient, (

*ii*)

*E*(

*t*) can be analytically retrieved from only two SH measurements if the accessible range of

*ϕ*

_{2}values is large enough, (

*iii*)

*E*(

*t*) can alternatively be retrieved by an iterative algorithm. In section 3 we demonstrate and compare two pulse retrieval algorithms (CRT and 2D-fit). In section 4 we compare d-scan measurements with two well-established reference techniques (bFROG [16

16. N. Forget, V. Crozatier, and T. Oksenhendler, “Pulse-measurement techniques using a single amplitude and phase spectral shaper,” J. Opt. Soc. Am. B **27**, 742–756 (2010). [CrossRef]

17. S. Grabielle, A. Moulet, N. Forget, V. Crozatier, S. Coudreau, R. Herzog, T. Oksenhendler, C. Cornaggia, and O. Gobert, “Self-referenced spectral interferometry cross-checked with spider on sub-15fs pulses,” Nucl. Instrum. Methods Phys. Res., Sect. A **653**, 121–125 (2011). [CrossRef]

## 1. Second harmonic of chirped pulses

### 1.1. Second harmonic spectrum

*E*(

*ω*) stands for the complex spectral amplitude of the input pulse the SH signal collected at the angular frequency 2

*ω*can be written as This simple model of SH generation assumes no absorption, an instantaneous nonlinearity, negligible higher-order nonlinear effects, a thin nonlinear medium and a perfect phase-matching over the bandwidth of interest. With broadband pulses the latter assumption is seldom fulfilled and a more appropriate expression for the SH power spectrum can be derived by including the wavevector-mismatch and other frequency-dependent parameters of the SH and detection stages by adding a spectral filter

*R*(2

*ω*) [18

18. G. Taft, A. Rundquist, M. M. Murnane, I. P. Christov, H. C. Kapteyn, K. W. DeLong, D. N. Fittinghoff, M. Krumbugel, J. N. Sweetser, and R. Trebino, “Measurement of 10-fs laser pulses,” IEEE J. Quantum Electron. **2**, 575–585 (1996). [CrossRef]

19. 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]

*R*(2

*ω*) in the following sections.

_{ℝ}

*a*(

*x*)exp[

*iϕ*(

*x*)]

*dx*, where

*a*(

*x*)

*>*0 and

*ϕ*(

*x*) are real valued functions, originate from stationary points, i.e. from the vicinity of values

*x*such that

*ϕ′*(

*x*) = 0. For the SH spectrum, the relevant amplitude and phase functions

*a*and

*ϕ*to be considered are, respectively,

*a*(Ω) = |

*E*(

*ω*− Ω)

*E*(

*ω*+ Ω)| and

*ϕ*(Ω) = arg[

*E*(

*ω*− Ω)

*E*(

*ω*+ Ω)]. If

*E*(

*ω*) = |

*E*(

*ω*)|exp[

*iφ*(

*ω*)], then the usual stationary phase approximation states that

*I*

_{SHG}(2

*ω*) reaches a maximum at angular frequencies fulfilling

*φ′*(

*ω*+ Ω) −

*φ′*(

*ω*− Ω) ≃ 0. In particular, this condition is locally fulfilled if

*φ″*(

*ω*) = 0. Consequently, the SH spectrum is expected to exhibit a local or global maximum at 2

*ω*if

*φ″*(

*ω*) cancels.

### 1.2. Quadratic test functions

*φ*(

*ω*) +

*ϕ*

_{2}(

*ω*−

*ω*

_{0})

^{2}/2. The measured SH spectrum then becomes: Equation (2) can be equivalently expressed using the Fourier-transform operator

*FT*:

*I*

_{SHG}(2

*ω*,

*ϕ*

_{2}) can be seen to be proportional to the square modulus of the Fresnel transform of

*E*

^{2}(

*t*). The analogy between wave propagation and spatial focusing, on one side, and dispersion and temporal focusing on the other side was pointed out as earlier as 1968 [20

20. S. Akhmanov, A. Chirkin, K. Drabovich, A. Kovrigin, R. Khokhlov, and A. Sukhorukov, “Nonstationary nonlinear optical effects and ultrashort light pulse formation,” IEEE J. Quantum Electron ., **4**, 598–605 (1968). [CrossRef]

*ϕ*

_{2}is equivalent to recording the spatial profile at several locations along the beam propagation direction. However, this analogy is not exact since Eq. (3) involves the square of the electric field amplitude instead of the field amplitude itself. Nevertheless, some key features remain valid and this analogy will be used later in this paper. In particular, small and large |

*ϕ*

_{2}| values can be associated to respectively near-field and far-field regimes.

### 1.3. Far-field approximation

*ϕ*

_{2}| values. Indeed the integral ∫

_{ℝ}

*a*(

*x*)exp[

*iϕ*(

*x*)]

*dx*admits a remarkable asymptotic approximation if

*ϕ*(

*x*) exhibits a single and well-defined non degenerate stationary point (i.e.

*ϕ′*(

*x*

_{0}) = 0 and

*ϕ″*(

*x*

_{0}) ≠ 0) [21

21. E. Chassande-mottin and P. Flandrin, “On the time-frequency detection of chirps,” Appl. Comput. Harmon. Anal. **281**, 252–281 (1999). [CrossRef]

*h*is a scaling factor used to express the condition that

*ϕ′*(

*x*) takes small values in a very narrow range around

*x*

_{0}. In practice,

*h*is a mute factor and can be ignored. If

*ϕ*possesses a finite number of non degenerate stationary points, the integral can be approximated by the sum of the individual contributions. Besides, the residual error

*a*(

*x*) and of the higher derivatives of

*ϕ″*(

*x*) [21

21. E. Chassande-mottin and P. Flandrin, “On the time-frequency detection of chirps,” Appl. Comput. Harmon. Anal. **281**, 252–281 (1999). [CrossRef]

*ϕ*

_{2}|, the overall spectral phase is such that the corresponding group delay is strictly monotonic with frequency and there is a single non-degenerate stationary point for

*ϕ*. The asymptotic expression of the SH spectrum is simply As can be noted, the SH signal tends to diverge when

*ϕ*

_{2}≃ −

*φ″*(

*ω*). In this particular case, the stationary point becomes degenerate and Eq. (4) is not valid anymore. In practice, the harmonic signal tends to reach a local maximum.

### 1.4. Uniqueness

*ϕ*

_{2}+

*φ″*(

*ω*)|

^{−1}for large positive and negative values of

*ϕ*

_{2}(i.e. such that |

*ϕ*

_{2}| ≫ max(|

*φ″*(

*ω*)|)). It follows from (5) that it is theoretically possible to extract analytically

*φ″*(

*ω*) from only two measurements. Indeed, let

*ω*, and one then gets:

*I*

_{min}(2

*ω*) =

*I*

_{SHG}(

*ω*) and

*I*

_{max}(2

*ω*) =

*I*

_{SHG}(

*ω*). In other words, for a given angular frequency, the local chirp

*φ″*(

*ω*) is the arithmetic mean of

*I*

_{min}(2

*ω*) and

*I*

_{max}(2

*ω*). By integrating

*φ″*(

*ω*) twice, the spectral phase

*φ*(

*ω*) can be unambiguously retrieved up to a linear phase (the absolute group delay and absolute phase are not measurable in the slow varying envelope approximation). In principle, the input spectrum can also be retrieved by this method if the second harmonic spectra are acquired with a sufficiently high dynamic range (the root square operation performed on the SH spectra tends to amplify the noise). As a result, both the spectral amplitude and phase can be unambiguously measured if at least two far-field SH spectra are acquired (formulas similar to Eq. (6) can be derived for chirp coefficients of the same sign). To the best of our knowledge, equation (6) is the first mathematical proof of the bijective character of the chirp-scan technique.

*φ″*(

*ω*)/

*ϕ*

_{2}≪ 1 for all angular frequencies. Although some complex mathematical condition may be established [21

21. E. Chassande-mottin and P. Flandrin, “On the time-frequency detection of chirps,” Appl. Comput. Harmon. Anal. **281**, 252–281 (1999). [CrossRef]

*τ*

_{0}is the pulse duration of the Fourier-Transform-limited pulse and

*σ*

_{t}

*σ*is the expected time-bandwidth product of the input pulses, expressed as the product of the temporal and spectral standard deviations.

_{ω}## 2. Chirp-scan maps: examples and properties

*I*

_{SHG}(2

*ω*,

*ϕ*

_{2}) is obtained by scanning

*ϕ*

_{2}from

*ϕ*

_{2,}

*to*

_{min}*ϕ*

_{2,}

*and acquiring*

_{max}*N*second harmonic spectra.

*ϕ*

_{2,}

*and*

_{min}*ϕ*

_{2,}

*are chosen to both encompass the SH maxima and extend to the farfield regime.*

_{max}### 2.1. Examples

*φ″*(

*ω*) = 0) a typical chirp-scan map is shown in Fig. 1(a): for all wavelengths the SH signal is maximum at

*ϕ*

_{2}= 0 and decreases symmetrically with respect to the maximum (at

*ϕ*

_{2}= 0). For a Gaussian fundamental spectrum the shape of the SH spectra is also Gaussian for all applied

*ϕ*

_{2}values and the SH signal decreases as [1 + (

*ϕ*

_{2}Δ

*ω*

^{2}/4ln(2))

^{2}]

^{−1/2}where Δ

*ω*is the FWHM spectral width of the input pulse. For all angular frequencies, the SH drops by a factor of 2 for

*ϕ*

_{2}|.

*φ″*(

*ω*) =

*φ″*(

*ω*

_{0}), Fig. 1(b)), the maxima are simply shifted toward the

*ϕ*

_{2}that compensates the chirp of the input pulse (i.e. to

*ϕ*

_{2}= −

*φ″*(

*ω*

_{0})).

*I*

_{SHG}(2

*ω*,

*ϕ*

_{2}) are aligned along the white dashed line revealing

*φ″*(

*ω*). As stated by [7

7. V. V. Lozovoy, B. Xu, Y. Coello, and M. Dantus, “Direct measurement of spectral phase for ultrashort laser pulses,” Opt. Express **16**, 592–597 (2008). [CrossRef] [PubMed]

*ω*is reached when

*ϕ*

_{2}= −

*φ″*(

*ω*) (stationary phase approximation, see section 1.1). For the third order phase, the second order phase is a linear function of

*ω*and the SH maxima are located at

*ϕ*

_{2}= −

*φ*

^{(3)}(

*ω*

_{0})(

*ω*−

*ω*

_{0}).

*ϕ*

_{2}= −

*φ*

^{(3)}(

*ω*

_{0})(

*ω*−

*ω*

_{0}) −

*φ*

^{(4)}(

*ω*

_{0})(

*ω*−

*ω*

_{0})

^{2}/2. The stationary phase approximation is however not exact and the

*ϕ*

_{2}value for which

*I*

_{SHG}(

*ϕ*

_{2}, 2

*ω*) is maximized is not exactly on this parabola.

*ϕ*

_{2}on 2

*ω*) can still be recognized.

*φ″*(

*ω*). Here the maxima are all aligned at

*ϕ*

_{2}= 0. In such a case, the stationary phase approximation completely fails. Large parts of the spectrum indeed contribute to the SH signal owing to the broadband intrapulse frequency mixing. The approximation

*ϕ*

_{2,}

*≃ −*

_{max}*φ″*(

*ω*) nevertheless holds in some specific cases: when a very small portion of the spectrum contributes to the signal at 2

*ω*(spectral edges for example), spectral phase of large amplitude with slow variations (like in [7

7. V. V. Lozovoy, B. Xu, Y. Coello, and M. Dantus, “Direct measurement of spectral phase for ultrashort laser pulses,” Opt. Express **16**, 592–597 (2008). [CrossRef] [PubMed]

### 2.2. Symmetry

*ϕ*

_{2}axis is when

*φ″*(

*ω*) is constant i.e. for pure quadratic spectral phases. Such feature is extremely practical, since any departure from a perfect “left-right” symmetry indicates the presence of some higher-order spectral phase. From equation (2) one can also deduce that if the spectrum is symmetric with respect to the carrier frequency

*ω*

_{0}then then an even spectral phase leads to an axial symmetry with respect to

*ω*= 2

*ω*

_{0}whereas anti-symmetric spectral phases lead to centro-symmetric diagrams (Fig. 1(c), (e) and (f)).

### 2.3. Marginals

*I*

_{SHG}(2

*ω*,

*ϕ*

_{2}) along

*ω*is expected to decrease as 1/|

*ϕ*

_{2}| in the far-field approximation. This hyperbolic decay can be deduced from Eq. (5) and can be used to check if the far-field regime is reached. Secondly, the integral of

*I*

_{SHG}(2

*ω*,

*ϕ*

_{2}) along

*ϕ*

_{2}is independent of

*φ*(

*ω*) for a given scanning range and a given fundamental spectrum

*I*(

*ω*), provided that the far-field is reached on the edges of the scan. Equation (2) can indeed be rewritten as: The integration over

*ϕ*

_{2}with the change of variables Ω = (

*u*+

*v*)/2 and Ω

*′*= (

*u*−

*v*)/2 then leads to: and finally: where

*f*(

*u*,

*v*) is a complex-valued function, Δ

*φ*

_{2}=

*ϕ*

_{2,}

*−*

_{max}*ϕ*

_{2,}

*and 〈*

_{min}*φ*

_{2}〉 = (

*ϕ*

_{2,}

*+*

_{max}*ϕ*

_{2,}

*)/2. Since the spectral support of the fundamental spectrum is experimentally bounded, the integral limits are also bounded and the sinc(Δ*

_{min}*φ*

_{2}

*uv*/2) term behaves as a sampling function (Dirac function) if Δ

*φ*

_{2}is large enough. More precisely, this condition is Δ

*φ*

_{2}Δ

*ω*

^{2}≫

*π*and corresponds to the far-field approximation. As a result, the integral becomes, to the first order, independent of the spectral phase: where

*α*is a constant coefficient depending on Δ

*φ*

_{2}.

## 3. Experimental demonstration

### 3.1. Experimental setup

*λ*≈ 50 nm FWHM centered at

*λ*

_{0}= 800 nm which supports sub-25 fs pulses. The experimental setup consists of a beam sampling plate, an acousto-optic programmable dispersive filter (AOPDF [22

22. P. Tournois, “Acousto-optic programmable dispersive filter for adaptive compensation of group delay time dispersion in laser systems,” Opt. Commun. **140**, 245–249 (1997). [CrossRef]

*μ*m-tick

*β*-barium borate (BBO) crystal cut for a type I second harmonic generation at 800 nm (

*θ*= 29°), a coloured filter (BG37, Schott) to isolate the SH signal, a focusing lens and a high-resolution spectrometer covering 350–450 nm with a 0.1 nm resolution (AvaSpec2048, Avantes, with a 10 μm input slit). The spectral response of both the spectrometer and the colored filter were calibrated with a blackbody source. The AOPDF is a 25 mm HR-cut TeO

_{2}crystal (HR800, Fastlite) and is used simultaneously to compensate for the dispersion of the AOPDF itself and add the ramp of quadratic phases required for the chirp-scan measurements and, optionally, compensate the measured spectral phase. After self-compensation the AOPDF was able to add quadratic phases with chirp coefficients ranging from −9000 fs

^{2}to +9000 fs

^{2}with a precision better than 1%. Such a span correspond to a pulse broadening is up to about 50 times the FTL pulse duration. Given the thickness of the BBO crystal the finite phase-matching bandwidth does not distort the SH spectrum by more than a few percents at the edges of the SH spectrum (the beam is not focused on the BBO crystal). Pulse durations as given at FWHM with a precision of

*±*0.5 fs limited by the shot-to-shot fluctuations.

### 3.2. Chirp-reversal technique

^{2}and +5000 fs

^{2}. The corresponding SH spectra are shown in Fig. 3(a). The reconstructed spectra and the second derivative of the spectral phase are displayed in Figs. 3(b) and 3(c) (blue solid curves) and are compared to the spectrum and phase retrieved by the 2D-fit described in the sub-section 3.3. The noise level on the retrieved spectrum is intrinsic to the technique since any noise on the measured SH spectra (or any defect of the phase applied by the AOPDF) will be directly transferred on the retrieved spectrum. To avoid numerical artifacts the SH spectra were smoothed and apodized before applying formula 7. The analytic reconstruction is still very close to the real spectrum. The retrieved spectral phase is complex and is composed of high-order polynomial phase and oscillating features. Although this phase structure was confirmed by further experiments, its physical origin was not explored.

### 3.3. Iterative algorithm

*N*of SH spectra recorded is typically 64). The fitting procedure is based on a standard Levenberg-Marquardt least-square fit procedure. Unlike[12

12. M. Miranda, T. Fordell, C. Arnold, A. L’Huillier, and H. Crespo, “Simultaneous compression and characterization of ultrashort laser pulses using chirped mirrors and glass wedges,” Opt. Express **20**, 688 (2012). [CrossRef] [PubMed]

*R*(2

*ω*) is known (i.e. calibrated spectrometer, perfect phase-matching) and we reconstruct simultaneously

**both**the spectral amplitude

**and**the spectral phase. Such an assumption can be experimentally justified for pulse durations longer than ≃10–15 fs. The main advantage of this approach is to avoid an independent measurement of the spectrum and potential artifacts such as beam inhomogeneities or amplified spontaneous emission do not come into account. Our procedure proceeds all the data in a single loop. The fit parameters are directly the sampled spectral amplitude and the second derivative of the spectral phase (|

*E*(

*ω*)| and

_{i}*φ″*(

*ω*) for

_{i}*i*ranging from 1 to

*N*). Working with the second derivative (i.e. local chirp) rather than with the phase directly is warranted by the fact that, in the slowly varying envelop approximation, the SH signal is insensitive to the absolute and linear phase. We use the result of the CRT algorithm as initial conditions to start the fit with reasonable initial conditions for the amplitude and phase parameters. Using this strategy, the convergence of the algorithm is very stable and we found numerically that it always converges towards the same solution. With

_{s}*N*≃

*N*≃ 64 and about 200–300 spectral points a typical convergence time is about 10 s with a standard laptop computer.

_{s}### 3.4. Example 1: complex chirped pulse

*ϕ*

_{2}= 0 and is almost symmetric with respect to it. As mentioned in subsection 2.2 this is a strong indication of an FTL pulse. The spectrum, phase and time profile before and after feedback are compared in Figs. 4(c) and 4(f). The higher order spectral phase is corrected and the pulse satellites are efficiently suppressed.

### 3.5. Example 2: near-FTL pulse

### 3.6. Calibration of the ϕ_{2} axis

12. M. Miranda, T. Fordell, C. Arnold, A. L’Huillier, and H. Crespo, “Simultaneous compression and characterization of ultrashort laser pulses using chirped mirrors and glass wedges,” Opt. Express **20**, 688 (2012). [CrossRef] [PubMed]

*ϕ*

_{2}values can be accurately retrieved by the fit algorithm. A precise knowledge of the chirp values

*ϕ*

_{2}is indeed not required and the measurement is somehow “self-calibrated”. This striking property can be intuitively explained by the fact that, as in the case of the FROG, the two coordinates (wavelength and chirp) are not independent: a trace corresponding to a physical signal has to fulfill some constraints in the frequency-chirp space. For example, the SH signal cannot vanish within tens of fs

^{2}if the pulse bandwidth is only a few nanometers wide. As a result, if the spectrometer is assumed to be perfectly calibrated, the chirp values

*ϕ*

_{2}introduced by the AOPDF can be retrieved. In practice, it is necessary to set at least two

*ϕ*

_{2}conditions: the global

*ϕ*

_{2}offset (i.e. the origin of the

*ϕ*

_{2}axis), the direction of the

*ϕ*

_{2}axis (otherwise the algorithm might reconstruct −

*φ″*(

*ω*) instead of +

*φ″*(

*ω*)). We chose to set

*ϕ*

_{2}= 0 when no spectral phase is introduced by the AOPDF.

*ϕ*

_{2}reconstruction, we try to reconstruct the experimental map displayed in Fig. 6 with different set of initial conditions for

*ϕ*

_{2}. The initial conditions are obtained by multiplying the expected

*ϕ*

_{2}values by 1 (black solid line), 1.5 (red solid line) and 0.75 (blue solid line). As done previously, the initial guess of the spectral amplitude and spectral phase are determined with the CRT method. Independently of the initial conditions, the

*ϕ*

_{2}values retrieved by the algorithm are almost identical (blue, red and black dots are superposed on Fig. 6) and almost match the expected values (black solid line). A small discrepancy can however be noticed for large negative

*ϕ*

_{2}values. This discrepancy may originate from a small calibration error of the AOPDF and is stronger for negative value because of the dispersion bias (the AOPDF also compensates for its own dispersion of about −13000fs

^{2}).

## 4. Comparison with reference techniques

16. N. Forget, V. Crozatier, and T. Oksenhendler, “Pulse-measurement techniques using a single amplitude and phase spectral shaper,” J. Opt. Soc. Am. B **27**, 742–756 (2010). [CrossRef]

17. S. Grabielle, A. Moulet, N. Forget, V. Crozatier, S. Coudreau, R. Herzog, T. Oksenhendler, C. Cornaggia, and O. Gobert, “Self-referenced spectral interferometry cross-checked with spider on sub-15fs pulses,” Nucl. Instrum. Methods Phys. Res., Sect. A **653**, 121–125 (2011). [CrossRef]

23. T. Oksenhendler, S. Coudreau, N. Forget, V. Crozatier, S. Grabielle, R. Herzog, O. Gobert, and D. Kaplan, “Self-referenced spectral interferometry,” Appl. Phys. B **99**, 7–12 (2010). [CrossRef]

24. A. Moulet, S. Grabielle, C. Cornaggia, N. Forget, and T. Oksenhendler, “Single-shot, high-dynamic-range measurement of sub-15 fs pulses by self-referenced spectral interferometry,” Opt. Lett. **35**, 3856–3858 (2010). [CrossRef] [PubMed]

### 4.1. Comparison with b-FROG

## 5. Dispersion-scan with a grating compressor

**20**, 688 (2012). [CrossRef] [PubMed]

*et al.*[13

13. M. Miranda, C. L. Arnold, T. Fordell, F. Silva, B. Alonso, R. Weigand, A. L’Huillier, and H. Crespo, “Characterization of broadband few-cycle laser pulses with the d-scan technique,” Opt. Express **20**, 18732–18743 (2012). [CrossRef] [PubMed]

*<*50 nm FWHM at 800nm): the dispersion of a prism pair is indeed rather low and large propagation distances might be required. In this section we show that, at the cost of a dispersion calibration step, the compressor of the laser system itself can be used directly for the scan.

*ϕ*

_{3}to

*ϕ*

_{2}is almost constant and given by: The theoretical value (groove density of 1500 g/mm and incidence angle of 50°) is −2.09 fs at 800 nm and it agrees well with the experimental value. Calculations also give a

*ϕ*

_{4}to

*ϕ*

_{2}ratio of 7.0 fs

^{2}at 800 nm, which is also in agreement with the measurements. Once the compressor is calibrated, the third order dispersion is taken into account by the chirp-scan algorithm. As we previously demonstrated (Figure 6) it is not necessary to accurately know the real value of the applied chirp, it is possible to simply scan the gratings separation distance without even measuring the actual displacements.

*ϕ*

_{3}correction. It is worth noting that the retrieved chirp values are not perfectly regular as shown in Fig. 12(d) (the translation speed wasn’t perfectly constant due to the manual operation), which shows that, as stated before, the

*φ*

_{2}values can be accurately retrieved.

## 6. Conclusions

## Acknowledgments

## References and links

1. | I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon. |

2. | P. O’shea, M. Kimmel, X. Gu, and R. Trebino, “Highly simplified device for ultrashort-pulse measurement,” Opt. Lett. |

3. | A. S. Radunsky, I. A. Walmsley, S.-P. Gorza, and P. Wasylczyk, “Compact spectral shearing interferometer for ultrashort pulse characterization,” Opt. Lett. |

4. | R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. |

5. | C. Iaconis and I. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. |

6. | V. V. Lozovoy, I. Pastirk, and M. Dantus, “Multiphoton intrapulse interference. iv. ultrashort laser pulse spectral phase characterization and compensation,” Opt. Lett. |

7. | V. V. Lozovoy, B. Xu, Y. Coello, and M. Dantus, “Direct measurement of spectral phase for ultrashort laser pulses,” Opt. Express |

8. | B. Xu, J. M. Gunn, J. M. D. 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 |

9. | S. Grabielle, N. Forget, S. Coudreau, T. Oksenhendler, D. Kaplan, J.-F. Hergott, and O. Gobert, “Local spectral compression method for cpa lasers,” in “ |

10. | S. Grabielle, “Manipulation et caractrisation du champ lectrique optique: applications aux impulsions femtosecondes,” Ph.D. thesis, Ecole Polytechnique (2011). |

11. | “Application note 38 - automated control of amplitied pulse duration using the dazzler/dazcope solution,” Tech. rep., Newport Corporation (2009). |

12. | M. Miranda, T. Fordell, C. Arnold, A. L’Huillier, and H. Crespo, “Simultaneous compression and characterization of ultrashort laser pulses using chirped mirrors and glass wedges,” Opt. Express |

13. | M. Miranda, C. L. Arnold, T. Fordell, F. Silva, B. Alonso, R. Weigand, A. L’Huillier, and H. Crespo, “Characterization of broadband few-cycle laser pulses with the d-scan technique,” Opt. Express |

14. | M. Miranda, P. Rudawski, C. Guo, F. Silva, C. L. Arnold, T. Binhammer, H. Crespo, and A. L’Huillier, “Ultrashort laser pulse characterization from dispersion scans: a comparison with spider,” in “ |

15. | I. Pastirk, B. Resan, A. Fry, J. MacKay, and M. Dantus, “No loss spectral phase correction and arbitrary phase shaping of regeneratively amplified femtosecond pulses using miips,” Opt. Express |

16. | N. Forget, V. Crozatier, and T. Oksenhendler, “Pulse-measurement techniques using a single amplitude and phase spectral shaper,” J. Opt. Soc. Am. B |

17. | S. Grabielle, A. Moulet, N. Forget, V. Crozatier, S. Coudreau, R. Herzog, T. Oksenhendler, C. Cornaggia, and O. Gobert, “Self-referenced spectral interferometry cross-checked with spider on sub-15fs pulses,” Nucl. Instrum. Methods Phys. Res., Sect. A |

18. | G. Taft, A. Rundquist, M. M. Murnane, I. P. Christov, H. C. Kapteyn, K. W. DeLong, D. N. Fittinghoff, M. Krumbugel, J. N. Sweetser, and R. Trebino, “Measurement of 10-fs laser pulses,” IEEE J. Quantum Electron. |

19. | 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. |

20. | S. Akhmanov, A. Chirkin, K. Drabovich, A. Kovrigin, R. Khokhlov, and A. Sukhorukov, “Nonstationary nonlinear optical effects and ultrashort light pulse formation,” IEEE J. Quantum Electron ., |

21. | E. Chassande-mottin and P. Flandrin, “On the time-frequency detection of chirps,” Appl. Comput. Harmon. Anal. |

22. | P. Tournois, “Acousto-optic programmable dispersive filter for adaptive compensation of group delay time dispersion in laser systems,” Opt. Commun. |

23. | T. Oksenhendler, S. Coudreau, N. Forget, V. Crozatier, S. Grabielle, R. Herzog, O. Gobert, and D. Kaplan, “Self-referenced spectral interferometry,” Appl. Phys. B |

24. | A. Moulet, S. Grabielle, C. Cornaggia, N. Forget, and T. Oksenhendler, “Single-shot, high-dynamic-range measurement of sub-15 fs pulses by self-referenced spectral interferometry,” Opt. Lett. |

**OCIS Codes**

(320.5540) Ultrafast optics : Pulse shaping

(320.7100) Ultrafast optics : Ultrafast measurements

**ToC Category:**

Ultrafast Fiber Lasers

**History**

Original Manuscript: August 8, 2013

Revised Manuscript: September 26, 2013

Manuscript Accepted: September 26, 2013

Published: October 10, 2013

**Citation**

Vincent Loriot, Gregory Gitzinger, and Nicolas Forget, "Self-referenced characterization of femtosecond laser pulses by chirp scan," Opt. Express **21**, 24879-24893 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-21-24879

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### References

- I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon.1, 308–437 (2009). [CrossRef]
- P. O’shea, M. Kimmel, X. Gu, and R. Trebino, “Highly simplified device for ultrashort-pulse measurement,” Opt. Lett.26, 932–934 (2001). [CrossRef]
- A. S. Radunsky, I. A. Walmsley, S.-P. Gorza, and P. Wasylczyk, “Compact spectral shearing interferometer for ultrashort pulse characterization,” Opt. Lett.32, 181–183 (2007). [CrossRef]
- R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum.68, 3277–3295 (1997). [CrossRef]
- C. Iaconis and I. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett.23, 792–794 (1998). [CrossRef]
- V. V. Lozovoy, I. Pastirk, and M. Dantus, “Multiphoton intrapulse interference. iv. ultrashort laser pulse spectral phase characterization and compensation,” Opt. Lett.29, 775–777 (2004). [CrossRef] [PubMed]
- V. V. Lozovoy, B. Xu, Y. Coello, and M. Dantus, “Direct measurement of spectral phase for ultrashort laser pulses,” Opt. Express16, 592–597 (2008). [CrossRef] [PubMed]
- B. Xu, J. M. Gunn, J. M. D. 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. B23, 750–759 (2006). [CrossRef]
- S. Grabielle, N. Forget, S. Coudreau, T. Oksenhendler, D. Kaplan, J.-F. Hergott, and O. Gobert, “Local spectral compression method for cpa lasers,” in “CLEO Europe-EQEC 2009” (IEEE, 2009).
- S. Grabielle, “Manipulation et caractrisation du champ lectrique optique: applications aux impulsions femtosecondes,” Ph.D. thesis, Ecole Polytechnique (2011).
- “Application note 38 - automated control of amplitied pulse duration using the dazzler/dazcope solution,” Tech. rep., Newport Corporation (2009).
- M. Miranda, T. Fordell, C. Arnold, A. L’Huillier, and H. Crespo, “Simultaneous compression and characterization of ultrashort laser pulses using chirped mirrors and glass wedges,” Opt. Express20, 688 (2012). [CrossRef] [PubMed]
- M. Miranda, C. L. Arnold, T. Fordell, F. Silva, B. Alonso, R. Weigand, A. L’Huillier, and H. Crespo, “Characterization of broadband few-cycle laser pulses with the d-scan technique,” Opt. Express20, 18732–18743 (2012). [CrossRef] [PubMed]
- M. Miranda, P. Rudawski, C. Guo, F. Silva, C. L. Arnold, T. Binhammer, H. Crespo, and A. L’Huillier, “Ultrashort laser pulse characterization from dispersion scans: a comparison with spider,” in “CLEO: QELS Fundamental Science,” (OSA, 2013).
- I. Pastirk, B. Resan, A. Fry, J. MacKay, and M. Dantus, “No loss spectral phase correction and arbitrary phase shaping of regeneratively amplified femtosecond pulses using miips,” Opt. Express14, 9537–9543 (2006). [CrossRef] [PubMed]
- N. Forget, V. Crozatier, and T. Oksenhendler, “Pulse-measurement techniques using a single amplitude and phase spectral shaper,” J. Opt. Soc. Am. B27, 742–756 (2010). [CrossRef]
- S. Grabielle, A. Moulet, N. Forget, V. Crozatier, S. Coudreau, R. Herzog, T. Oksenhendler, C. Cornaggia, and O. Gobert, “Self-referenced spectral interferometry cross-checked with spider on sub-15fs pulses,” Nucl. Instrum. Methods Phys. Res., Sect. A653, 121–125 (2011). [CrossRef]
- G. Taft, A. Rundquist, M. M. Murnane, I. P. Christov, H. C. Kapteyn, K. W. DeLong, D. N. Fittinghoff, M. Krumbugel, J. N. Sweetser, and R. Trebino, “Measurement of 10-fs laser pulses,” IEEE J. Quantum Electron.2, 575–585 (1996). [CrossRef]
- 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]
- S. Akhmanov, A. Chirkin, K. Drabovich, A. Kovrigin, R. Khokhlov, and A. Sukhorukov, “Nonstationary nonlinear optical effects and ultrashort light pulse formation,” IEEE J. Quantum Electron., 4, 598–605 (1968). [CrossRef]
- E. Chassande-mottin and P. Flandrin, “On the time-frequency detection of chirps,” Appl. Comput. Harmon. Anal.281, 252–281 (1999). [CrossRef]
- P. Tournois, “Acousto-optic programmable dispersive filter for adaptive compensation of group delay time dispersion in laser systems,” Opt. Commun.140, 245–249 (1997). [CrossRef]
- T. Oksenhendler, S. Coudreau, N. Forget, V. Crozatier, S. Grabielle, R. Herzog, O. Gobert, and D. Kaplan, “Self-referenced spectral interferometry,” Appl. Phys. B99, 7–12 (2010). [CrossRef]
- A. Moulet, S. Grabielle, C. Cornaggia, N. Forget, and T. Oksenhendler, “Single-shot, high-dynamic-range measurement of sub-15 fs pulses by self-referenced spectral interferometry,” Opt. Lett.35, 3856–3858 (2010). [CrossRef] [PubMed]

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