## Simultaneous analytical characterisation of two ultrashort laser pulses using spectrally resolved interferometric correlations

Optics Express, Vol. 14, Issue 10, pp. 4538-4551 (2006)

http://dx.doi.org/10.1364/OE.14.004538

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### Abstract

In this paper we discuss in detail the underlying theory of a novel method that allows the characterizing of ultrashort laser pulses to be achieved in an analytical way. MEFISTO, (measuring the electric field by interferometric spectral trace observation) is based on a Fourier analysis of the information contained in a spectrally resolved interferometric correlation and can be applied to both situations: the characterization of an unknown pulse (MEFISTO) or to the simultaneous characterization of two different unknowns pulses (Blind-MEFISTO). The theoretical development and experimental practical implications are discussed in both situations.

© 2006 Optical Society of America

## 1. Introduction

2. J.-C. Diels, E. W. Van Stryland, and G. Benedict, “Generation and measurement of 200 femtosecond optical pulses,” Opt. Commun. **25**, 93–95 (1978). [CrossRef]

3. 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]

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

5. D. J. Kane and R. Trebino, “Single-shot measurement of the intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating,” Opt. Lett. **18**, 823–825 (1993). [CrossRef] [PubMed]

6. 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]

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

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

9. I. Amat-Roldán, I. G. Cormack, P. Loza-Alvarez, and D. Artigas, “Starch-based second-harmonic-generated collinear frequency-resolved optical gating pulse characterization at the focal plane of a high-numerical-aperture lens,” Opt. Lett. **29**, 2282–2284 (2004). [CrossRef] [PubMed]

10. I. Amat-Roldán, I. G. Cormack, P. Loza-Alvarez, and D. Artigas, “Measurement of electric field by interferometric spectral trace observation,” Opt. Lett. **30**, 1063–1065 (2005). [CrossRef] [PubMed]

## 2. Theory

*τ*and the frequency

*f*. An example of the resulting trace can be seen in Fig. 2(a). In a general case, this trace can be mathematically described as

*τ*axis, i.e.,

*Y*

^{SHG}(

*f*,

*κ*)=

*F*

_{τ}{

*I*

^{SHG}(

*f*,

*τ*)}. The resulting expression consist of 5 main spectral components [see Fig. 2(b)] at delay-frequencies

*κ*=0, ±

*f*

_{0}and ±2

*f*

_{0}. Since the interferometric trace [Fig. 2(a)] is real, the negative spectral components in Fig. 2(b) are the complex conjugate of the positive ones. We therefore only need to focus upon one of the components (either the positive or the negative) to analyze the information enclosed in the transformed trace. Each of these terms contain information of the pulse phase and intensity and their use will depend on the particular experimental conditions. To highlight the information enclosed in

*Y*

^{SHG}(

*f*,

*κ*), we will separately focus on the mathematical expression for each component and analyze their possibilities.

*κ*=0, which can be written as:

*Y*

^{SHG}(

*f*,

*κ*) at

*κ*≈2

*f*

_{0}. This component can be written as

*κ*=

*f*+2

*f*

_{0}. This term can be used for different purposes. Firstly, in an experimental trace, the divergence from an ideal delta function gives a measure of the pulse jitter, as suggested in Ref. [7

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

*κ*(real frequency) and

*f*(base band frequency) will indicate a spectrograph miss-calibration. This component can also determine

*E*(

*t*) when the gating function,

*G*(

*t*), is already known. It can therefore be used as a further verification of the blind-MEFISTO retrieval by checking that the two pulses fulfill

*κ*=0 and ±2

*f*

_{0}are useful to obtain some pulse information, it is the component at

*κ*≈

*f*

_{0}of

*Y*

^{SHG}(

*f*,

*κ*) that allows the simultaneous analytical characterization of two unknown ultrashort pulses. The mathematical expression for this component can be written as:

*E*(

*f*)=

*E*

_{0}

*U*(

*f*)exp(

*iϕ*(

*f*)),

*G*(

*f*)=

*G*

_{0}

*V*(

*f*) exp(

*iγ*(

*f*)), and equivalently for the harmonic pulses

*E*

_{SHG}(

*f*)=

*U*

_{SHG}(

*f*)exp(

*iϕ*

_{SHG}(

*f*)) and

*G*

_{SHG}(

*f*)=

*V*

_{SHG}(

*f*)exp(

*iϕ*

_{SHG}(

*f*)). Here, for convenience, all spectral profiles

*U*(

*f*) are normalized at the central base-band frequency (

*U*(

*f*=0)=1). Then, taking

*Y*

^{SHG}(

*f*,

*κ*)=

*R*(

*f*,

*κ*) exp(

*iθ*(

*f*,

*κ*)), Eq. (5) can be written as

*χ*

_{1}=

*E*

_{0}

*G*

_{0}and

*χ*

_{2}=

*E*

_{0}

*G*

_{0}, which can be understood as effective conversion efficiency parameters. In section 4, we will go on to describe a procedure to obtain

*χ*

_{1}and

*χ*

_{2}from the experimental trace. In what follows, and for clarity purposes, we will assume that these parameters are known. Under typical lab conditions the normalized spectra profiles,

*U*(

*f*),

*V*(

*f*),

*U*

_{SHG}(

*f*) and

*V*

_{SHG}(

*f*) can be easily measured. As

*R*(

*f*,

*κ*) and

*θ*(

*f*,

*κ*) are obtained from the interferometric trace, therefore, the only unknowns in Eq. (6) corresponds to the phase of the fundamental and second harmonic pulse, i.e.,

*ϕ*(

*f*),

*γ*(

*f*),

*ϕ*

_{SHG}(

*f*) and

*γ*

_{SHG}(

*f*) . In order to fully characterize the pulses these unknowns must be calculated. This can be achieved by first taking two different slices in the transformed space of the interferometric trace, e.g., at

*κ*=

*f*

_{0}and

*κ*=

*f*

_{0}-Δ

*f*. Then, by taking the real and imaginary parts in Eq. (6), we can isolate the phase component at

*κ*=

*f*

_{0}, obtaining

*κ*=

*f*

_{0}-Δ

*f*

*E*(

*f*), i.e.,

*ϕ*(

*f*) can be determined by taking an arbitrary origin

*ϕ*(0) and adding up Δ

*ϕ*

_{k}(

*f*) as

*G*(

*f*) in terms of the frequency as

*γ*(0) find the spectral phase components as

*γ*(0)-

*γ*(-Δ

*f*) and

*ϕ*(0)-

*ϕ*(-Δ

*f*) in eqs. (8) and (10) are unknown constants. These terms add a linear spectral phase shift that only affects the electric field time origin and therefore they can be decided arbitrarily without affecting the shape of the temporal pulse envelope. Equations (8–11) are the principal result of this work showing that to simultaneously characterize two different pulses in an analytical way is possible.

## 3. Simultaneous characterization of two unknown pulses: blind-MEFISTO

*κ*=

*f*and at

*κ*=

*f*

_{0}+Δ

*f*) of the Fourier transformed interferometric trace shown in Fig. 2(b). The amplitude of the trace

*R*(

*f*,

*κ*) and the phases

*θ*(

*f*,

*κ*) are respectively shown in Fig. 3 (a) and 3(b) at

*κ*=

*f*

_{0}and

*κ*=

*f*

_{0}-Δ

*f*. In addition, to evaluate the Ω

_{1}(

*f*,

*κ*) and Ω

_{2}(

*f*,

*κ*) functions, the two fundamental pulses spectra are needed [Fig. 3(c)], which must be obtained experimentally. The two spectra of the second harmonic pulses [Fig. 3(d)] and the parameters

*χ*

_{1}and

*χ*

_{2}can be either obtained experimentally or directly from the interferometric trace using the method outlined in section 4 (Eqs. 15–16).

_{1}(

*f*,

*κ*) and Ω

_{2}(

*f*,

*κ*) are therefore evaluated using solely experimental data. The resulting Ω

_{1}(

*f*,

*κ*) and Ω

_{2}(

*f*,

*κ*) function are shown, respectively, in Figures 4(a) and 4(b), at

*κ*=

*f*

_{0}and

*κ*=

*f*

_{0}-Δ

*f*.

*κ*≈

*f*

_{0}in Fig. 2(b) but it is not solution of the complete interferometric trace. In particular, the pulses in Figs. 6(c)–6(d) do not fulfil Eq. (4), that, as commented before, can be used as error-checking procedure. Experimentally, any source of errors, including the spurious solution, can be detected by numerically generating the interferometric trace using the retrieved pulses and comparing the result with the experimental trace. Equivalently, we can compare the interferometric correlations of the two solutions with the experimental one, which can be directly obtained from the interferometric trace time marginal. We have demonstrated this in Fig. 7. Here, only the interferometric correlation corresponding to the solution in Figs. 6(a)–6(b) is identical to the trace time marginal obtained by integrating the frequency axis of the interferometric trace in Fig. 2(a). Apart from the spurious solution, which can be rejected using the two checking procedures described above, we have not been able to detect other ambiguities as the ones present in XFROG and discussed in Ref. [14

14. B. Seifert, H. Stolz, and M. Tasche, “Nontrivial ambiguities for blind frequency-resolved optical gating and the problem of uniqueness,” J. Opt. Soc. Am. B **21**, 1089–1097 (2004). [CrossRef]

## 4. Experimental and practical considerations

### 4.1. Pulse bandwidth

*κ*=

*f*

_{0}and

*κ*=

*f*

_{0}-Δ

*f*must not be affected by the tails of the

*κ*=0 component. In principle, this suggests that the available bandwidth could be equal to the optical carrier, i.e., Δ

*λ*

_{max}≈

*λ*

_{0}. However, in practice, this can be affected by the particular spectral shape.

### 4.2. Frequency resolution

*f*and

*κ*axis coincides (Δ

*κ*=Δ

*f*). Initially, the frequency resolution is in fact given by the time-delay span

*τ*

_{span}, i.e., Δ

*κ*=1/

*τ*

_{span}, which experimentally can differ from the spectrograph resolution. However, by using interpolating techniques to fulfil Δ

*κ*=Δ

*f*, the frequency resolution of the method can be extended to Δ

*f*.

*f*,

*κ*) function can result in errors in determining the SSP. This source of errors can be reduced by increasing the frequency resolution. We must ensure, therefore, that the sampling, and therefore the resolution, results in a reasonable number of points inside the bandwidth.

### 4.3. Delay resolution.

*τ*. This is elected following Nyquist criterion to resolve the inteferometric fringes: Δ

*τ*<1/2

*f*

_{max}<1/4

*f*

_{0}. From the experimental point of view this results in large data sets and acquisition times (of the order of seconds) [10

10. I. Amat-Roldán, I. G. Cormack, P. Loza-Alvarez, and D. Artigas, “Measurement of electric field by interferometric spectral trace observation,” Opt. Lett. **30**, 1063–1065 (2005). [CrossRef] [PubMed]

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

### 4.4. Determining the conversion efficiency parameters χ_{1} and χ_{2}

*E*

_{0},

*G*

_{0}and

*χ*are included in the conversion efficiency parameters

*χ*

_{1}and

*χ*

_{2}. Here we will outline how these parameters can be calculated using the

*κ*=0 and

*κ*=2

*f*

_{0}components of the transformed interferometric trace. To achieve this we must first define three separate equations, two that are derived from the

*κ*=0 term and one that is derived from the

*κ*=2

*f*

_{0}+

*f*term. Firstly, using the same procedure outlined in Ref. [8

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

*κ*=0 term in Eq. (2) can be separated. The first part, involving the delta function, when

*κ*=0 gives

*κ*=0 and

*f*=0, yields,

*P*=

*df′U*

^{2}(

*f′*)

*V*

^{2}(-

*f′*) can be experimentally determined from the experimental spectra

*I*

_{E}(

*f*)=

*U*

^{2}(

*f*) and

*I*

_{G}(

*f*)=

*V*

^{2}(

*f*). The third equation is obtained by looking at the component at

*κ*=2

*f*

_{0}+

*f*and taking its modulus;

*χ*

_{1}>

*χ*

_{2}. By determining Eqs. (15) and (16) we have shown how it is possible to evaluate the functions Ω

_{1}(

*f*,

*κ*) and Ω

_{2}(

*f*,

*κ*) without the need of experimentally measuring the intensities of the second harmonic spectra

*U*

_{SHG}(

*f*) and

*V*

_{SHG}(

*f*).

### 4.5. Pectral calibration

*f*

_{0}and 2

*f*

_{0}places extreme demands and importance on the spectral calibration. These demands can be relaxed significantly by evaluating the second term of Eq. (2) at

*κ*=0, and using it to directly relate

*f*

_{0}and 2

*f*

_{0}with one another:

### 4.6. Noise.

*R*(

*f*,

*κ*≅

*f*

_{0}), shown in Fig. 3(a). Under these conditions it was still possible to detect the SSPs and we were able to obtain the spectral phase inside the spectrum bandwidth [Fig. 8(a)]. A further rising in the noise level, increased the difficulty in distinguishing the SSPs. To overcome this problem, we reject the high-frequency noise components by filtering the Ω

_{1}(

*f*,

*κ*) and Ω

_{2}(

*f*,

*κ*) functions in the

*f*axis. This procedure allows the phase to be obtained at noise levels as high as 13 dB SNR, as shown in Fig. 8(b).

## 5. Characterization of an unknown single pulse: degenerate case

10. I. Amat-Roldán, I. G. Cormack, P. Loza-Alvarez, and D. Artigas, “Measurement of electric field by interferometric spectral trace observation,” Opt. Lett. **30**, 1063–1065 (2005). [CrossRef] [PubMed]

*τ*thus its Fourier transform must be real. This has two consequences. First, the upper term in the Ω(

*f*,

*κ*) functions is directly

*Y*

^{SHG}(

*f*,

*κ*), where Ω(

*f*,

*κ*) now is written as

*χ*

_{eff}=

*Y*

^{SHG}(

*f*,

*κ*) can be attributed to experimental errors. This is because the interferometric nature of this technique requires having a perfectly centered trace, exact delay steps, no laser instabilities, etc. In this situation, when the center of the delay axis is accurately known, the imaginary part can be omitted. This is the equivalent to the symmetrization process performed in the FROG technique. Retaining the absolute value is however preferred in most of the cases since experimental measurement are never performed under an ideal conditions and the delay origin is unknown. The second consequence of the trace being symmetric is that a positive and negative sign in front of Eq. (18) will give the as result

*E*(

*f*) and

*E**(

*f*). This is equivalent to the intrinsic ambiguity that appears in SHG-FROG measurements.

*f*,

*κ*) is the effective conversion efficiency parameter

*χ*

_{eff}. In the degenerate case, the procedure to obtain

*χ*

_{eff}is very much simplified. In contrast to blind-MEFISTO where an external measurement of the fundamental spectra

*U*

^{2}(

*f*) is required, the degenerate case allows all the information to be directly obtained from the experimental interferometric trace. This is achieved as follows. First, the amplitude of the second harmonic pulse are obtained from the term at

*κ*=2

*f*

_{0}+

*f*, resulting in

*U*

_{SHG}(

*f*=0), the spectral profile

*U*

_{SHG}(

*f*) and the factor √

*χ*·

*χ*

_{eff}are obtained. Second, the term at

*κ*=0 in the degenerate case results in

*U*(

*f*=0)=1, the spectral profile of the fundamental pulse

*U*(

*f*) and the factor √

*χ*·

*χ*

_{eff}are obtained. By not requiring an extra spectral measurement, all the data used for the pulse measurement is contained within a single set of data. This helps keep everything self-consistent and as a consequence reduces the risk of erroneous errors entering into the final results.

## 5. Conclusions

14. B. Seifert, H. Stolz, and M. Tasche, “Nontrivial ambiguities for blind frequency-resolved optical gating and the problem of uniqueness,” J. Opt. Soc. Am. B **21**, 1089–1097 (2004). [CrossRef]

## Acknowledgment

## References and links

1. | J.-C. Diels, E. W. Van Stryland, and D. Gold, “Investigation of the parameters affecting subpicosecond pulse duration of passively mode-locked dye laser,” in |

2. | J.-C. Diels, E. W. Van Stryland, and G. Benedict, “Generation and measurement of 200 femtosecond optical pulses,” Opt. Commun. |

3. | C. Iaconis, V. Wong, and I. A. Walmsley, “Direct interferometric techniques for characterizing ultrashort optical pulses,” IEEE J. Sel. Top. Quantum Electron. |

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

5. | D. J. Kane and R. Trebino, “Single-shot measurement of the intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating,” Opt. Lett. |

6. | J. L. A. Chilla and O. E. Martinez, “Direct determination of the amplitude and the phase of femtosecond light pulses,” Opt. Lett. |

7. | G. Stibenz and G. Steinmeyer, “Interferometric frequency-resolved optical gating,” Opt. Express |

8. | I. Amat-Roldán, I. G. Cormack, P. Loza-Alvarez, E. J. Gualda, and D. Artigas, “Ultrashort pulse characterisation with SHG collinear-FROG,” Opt. Express |

9. | I. Amat-Roldán, I. G. Cormack, P. Loza-Alvarez, and D. Artigas, “Starch-based second-harmonic-generated collinear frequency-resolved optical gating pulse characterization at the focal plane of a high-numerical-aperture lens,” Opt. Lett. |

10. | I. Amat-Roldán, I. G. Cormack, P. Loza-Alvarez, and D. Artigas, “Measurement of electric field by interferometric spectral trace observation,” Opt. Lett. |

11. | 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. |

12. | K. W. DeLong, R. Trebino, and W. E. White, “Simultaneous recovery of two ultrashort laser pulses from a single spectrogram,” J. Opt. Soc. Am. B |

13. | A. V. Oppenheim and R. W. Schafer, |

14. | B. Seifert, H. Stolz, and M. Tasche, “Nontrivial ambiguities for blind frequency-resolved optical gating and the problem of uniqueness,” J. Opt. Soc. Am. B |

**OCIS Codes**

(100.5070) Image processing : Phase retrieval

(190.1900) Nonlinear optics : Diagnostic applications of nonlinear optics

(320.7100) Ultrafast optics : Ultrafast measurements

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: March 7, 2006

Revised Manuscript: April 28, 2006

Manuscript Accepted: May 2, 2006

Published: May 15, 2006

**Citation**

Ivan Amat-Roldan, David Artigas, Iain G. Cormack, and Pablo Loza-Alvarez, "Simultaneous analytical characterisation of two ultrashort laser pulses using spectrally resolved interferometric correlations," Opt. Express **14**, 4538-4551 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-10-4538

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### References

- J.-C. Diels, E. W. Van Stryland, and D. Gold, "Investigation of the parameters affecting subpicosecond pulse duration of passively mode-locked dye laser," in Proceedings, First International Conference on Picosecond Phenomena, (Springer-Verlag, New York, 1978), pp. 117-120.
- J.-C. Diels, E. W. Van Stryland, and G. Benedict, "Generation and measurement of 200 femtosecond optical pulses," Opt. Commun. 25, 93-95 (1978). [CrossRef]
- 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]
- C. Iaconis and I. A. Walmsley, "Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses," Opt. Lett. 23, 792-794 (1998). [CrossRef]
- D. J. Kane and R. Trebino, "Single-shot measurement of the intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating," Opt. Lett. 18, 823-825 (1993). [CrossRef] [PubMed]
- 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]
- G. Stibenz, G. Steinmeyer, "Interferometric frequency-resolved optical gating," Opt. Express 13, 2617-2626 (2005). [CrossRef] [PubMed]
- I. Amat-Roldán, I. G. Cormack, P. Loza-Alvarez, E. J. Gualda, and D. Artigas, "Ultrashort pulse characterisation with SHG collinear-FROG," Opt. Express 12, 1169-1178 (2004). [CrossRef] [PubMed]
- I. Amat-Roldán, I. G. Cormack, P. Loza-Alvarez, and D. Artigas, "Starch-based second-harmonic-generated collinear frequency-resolved optical gating pulse characterization at the focal plane of a high-numerical-aperture lens," Opt. Lett. 29, 2282-2284 (2004). [CrossRef] [PubMed]
- I. Amat-Roldán, I. G. Cormack, P. Loza-Alvarez, and D. Artigas, "Measurement of electric field by interferometric spectral trace observation," Opt. Lett. 30, 1063-1065 (2005). [CrossRef] [PubMed]
- 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]
- K. W. DeLong, R. Trebino and W. E. White, "Simultaneous recovery of two ultrashort laser pulses from a single spectrogram," J. Opt. Soc. Am. B 12, 2463-2466 (1995). [CrossRef]
- A. V. Oppenheim and R. W. Schafer, Digital Signal Processing (Prentice-Hall, 1975).
- B. Seifert, H. Stolz, M. Tasche, "Nontrivial ambiguities for blind frequency-resolved optical gating and the problem of uniqueness," J. Opt. Soc. Am. B 21, 1089-1097 (2004). [CrossRef]

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