## Describing first-order spatio-temporal distortions in ultrashort pulses using normalized parameters

Optics Express, Vol. 15, Issue 1, pp. 242-251 (2007)

http://dx.doi.org/10.1364/OE.15.000242

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### Abstract

We develop a first-order description of spatio-temporal distortions in ultrashort pulses using normalized parameters that allow for a direct assessment of their severity, and we give intuitive pictures of pulses with different amounts of the various distortions. Also, we provide an experimental example of the use of these parameters in the case of spatial chirp monitored in real-time during the alignment of an amplified laser system.

© 2007 Optical Society of America

## 1. Introduction

1. C. B. Schaffer, A. Brodeur, J. F. García, and E. Mazur, “Micromachining bulk glass by use of femtosecond laser pulses with nanojoule energy,” Opt. Lett. **26**,93–95 (2001). [CrossRef]

2. W. Denk, J. H. Strickler, and W. W. Webb, “Two-Photon Laser Scanning Fluorescence Microscopy,” Science **248**,73–76 (1990). [CrossRef] [PubMed]

3. R. L. Fork, O. E. Martinez, and J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. **9**,150–152 (1984). [CrossRef] [PubMed]

*spatio-temporal distortions*, which include angular dispersion,spatial chirp, pulse-front tilt, and angular delay, to name a few. While in theory perfect alignment of a compressor guarantees that the output pulse is free of any of these distortions, in practice residual distortions are often present.

4. 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]

7. S. Akturk, M. Kimmel, P. O'Shea, and R. Trebino, “Measuring spatial chirp in ultrashort pulses using
single-shot Frequency-Resolved Optical Gating,” Opt. Express **11**,68–78 (2003). [CrossRef] [PubMed]

12. M. Kempe, U. Stamm, B. Wilhelmi, and W. Rudolph, “Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems,” J. Opt. Soc. Am. B **9**,1158–1165 (1992). [CrossRef]

15. S. Akturk, X. Gu, P. Gabolde, and R. Trebino, “The general theory of first-order spatio-temporal
distortions of Gaussian pulses and beams,” Opt. Express **13**,8642–8661 (2005). [CrossRef] [PubMed]

## 2. Formal definitions of spatial chirp and other spatio-temporal couplings

15. S. Akturk, X. Gu, P. Gabolde, and R. Trebino, “The general theory of first-order spatio-temporal
distortions of Gaussian pulses and beams,” Opt. Express **13**,8642–8661 (2005). [CrossRef] [PubMed]

*x*-

*ω*domain.Generalization to the other spatio-temporal couplings, namely pulse-front tilt, angular dispersion, and angular delay, is immediate by considering the

*x*-

*t*,

*k*-

_{x}*ω*and

*k*-

_{x}*t*domains.Extension to the

*y*coordinate is also immediate.

*I*(

*x*,

*ω*) ≡ ∣

*E*(

*x*,

*ω*)∣

^{2}the (spatio-spectral) intensity distribution of the pulse, where

*x*and

*ω*are measured with respect to the beam center and the carrier frequency (that is, have the mean position and mean frequency subtracted off). The intensity

*I*(

*x*,

*ω*) is normalized such that its integral over space and frequency is 1. We now define the normalized spatial chirp parameter ρ

*xω*as the first mixed moment of

*I*(

*x*,

*ω*), divided by the global beam size Δ

*x*and the global bandwidth Δω:

*ω*/d

*x*and spatial dispersion d

*x*/d

*ω*parameters introduced in Ref. 15 for Gaussian pulses, in the sense that:

*ρ*is calculated from ∣

_{xω}*E*(

*x*,

*ω*)∣

^{2}, it does not include a coupling between

*x*and

*ω*that may appear in the

*phase*of

*E*(

*x*,

*ω*). This coupling essentially amounts to angular dispersion [15

15. S. Akturk, X. Gu, P. Gabolde, and R. Trebino, “The general theory of first-order spatio-temporal
distortions of Gaussian pulses and beams,” Opt. Express **13**,8642–8661 (2005). [CrossRef] [PubMed]

- (1) It is an extension to arbitrary pulses and beams that is consistent with previous definitions of frequency gradient and spatial dispersion.
- (2) It is symmetric: when
*I*(*x*,*ω*) is recorded using a camera, it does not matter whether the position axis is vertical and the frequency axis horizontal, or vice-versa. - (3) It is scale-invariant: except for a possible change of sign, it is unaffected by the transformations
*x*→*αx*or*ω*→*βω*. Thus, beam magnification does not affect the result. An important practical implication is that experimental trace need not be calibrated: the variables*x*and*ω*can represent pixel numbers on a camera, and not necessarily physical quantities with proper units. - (4) It is a dimensionless number.
- (5) Because
*ρ*can be identified with the linear correlation of the joint distribution_{xω}*I*(*x*,*ω*) [16], it is even possible to show that: - (6) Conveniently,
*ρ*= 0 corresponds to the absence of the distortion to first order,while an increased value of ∣_{xω}*ρ*∣ indicates an increase in the magnitude of spatial chirp (see Fig. 1)._{xω} - (7) The sign of
*ρ*simply reveals whether the beam center position increases or decreases with_{xω}*ω*. - (8) Also, for all but near-single-cycle pulses, the change from frequency
*ω*to wavelength λ is a linear transformation:*λ*=-λ_{0}^{2}*ω*/(2πc); again,*λ*; is measured with respect to the central wavelength*λ*_{0}. Written in this form, the change from*ω*(or*v*) to*λ*is just a change of scale and sign, and therefore: - (9) Finally,
*ρ*is equal to the eccentricity of an elliptical beam caused by spatial chirp._{xω}

*x*direction (for example, a misaligned stretcher). We take the input beam to have the same size in the

*x*and

*y*directions: Δ

*x*= Δ

*y*. Because of spatial chirp, the size of the output beam in the

*x*direction increases to Δ

*x*′. The output beam is therefore elliptical, and can be characterized by its eccentricity

*e*:

_{xy}*ρ*∣ =

_{xω}*e*. Although it is easy and intuitive to

_{xy}*think*of

*ρ*in terms of the eccentricity of the spatial profile, for precise measurements it is preferable to rely on

_{xω}*ρ*obtained from the intensity distribution

_{xω}*I*(

*x*,

*ω*). In addition, note that if spatial chirp results in a spatial broadening of the beam, and therefore in an elliptical beam, it also results in a

*temporal*broadening of the pulse, because of the decrease of available bandwidth at each point

*x*in the beam. Thus, in the presence of spatial chirp, the duration of a pulse with a flat spectral phase does

*not*reach its Fourier limit, as can be clearly seen on Fig. 1(d).

*ρ*– and more generally any correlation coefficient

_{xω}*ρ*that appears in this paper – is very sensitive to small amounts of spatio-temporal coupling, but saturates to a near-unity value for extremely large amounts of coupling (this situation is explored in more details in section 5).

## 3. Experimental determination of *ρ*_{xλ} and *ρ*_{yλ}

_{xλ}

_{yλ}

*I*(

*x*,

*λ*) and

*I*(

*y*,

*λ*), and we show how to calculate

*ρ*and

_{xλ}*ρ*from experimental data.

_{yλ}_{1},and the diffracted order

*m*

_{1}= 1, focused by a cylindrical lens, illuminates a digital camera. Simultaneously, the specular reflection (

*m*

_{1}=0) from G

_{1}is sent onto a second grating G

_{2}that disperses the beam

*vertically*in a Littrow configuration so that all the beams of interest are contained in the same horizontal plane; the first order (

*m*

_{2}=-1) of G

_{2}is focused by a second cylindrical lens and illuminates the same digital camera. By blocking the order

*m*

_{1}=1 from G

_{1}, the camera records

*I*(

*x*,

*λ*), while by blocking the order

*m*

_{1}=0, the camera records

*I*(

*y*,

*λ*).

*ρ*and

_{xλ}*ρ*from

_{yλ}*I*(

*x*,

*λ*) and

*I*(

*y*,

*λ*) is a direct application of Eq. (1), as long as the integrals are replaced by discrete sums. As stated in section 2, it is not necessary to calibrate the axes of the digital camera:

*x*,

*y*and

*λ*can simply refer to pixel numbers. Additionally, we use the fact that the wavelength axis can be either horizontal or vertical. However, Eq. (1) does require that the function

*I*(

*x*,

*λ*) be centered with respect to its axes. When pixel numbers are used, this is never the case, and therefore it is easier to rewrite Eq. (1) in the case of un-centered, discrete distributions. To do so, we first introduce the moments

*μ*of the intensity distribution

_{pq}*I*(

*x*,

*λ*):

*ρ*is then computed using the following equation, which is a convenient form of Eq. (1) that does not require the data

_{xλ}*I*(

*x*,

*λ*) to be centered:

*I*(

*x*,

*λ*) [7

7. S. Akturk, M. Kimmel, P. O'Shea, and R. Trebino, “Measuring spatial chirp in ultrashort pulses using
single-shot Frequency-Resolved Optical Gating,” Opt. Express **11**,68–78 (2003). [CrossRef] [PubMed]

*ρ*may be calculated using Eq. (2

_{xλ}2. W. Denk, J. H. Strickler, and W. W. Webb, “Two-Photon Laser Scanning Fluorescence Microscopy,” Science **248**,73–76 (1990). [CrossRef] [PubMed]

7. S. Akturk, M. Kimmel, P. O'Shea, and R. Trebino, “Measuring spatial chirp in ultrashort pulses using
single-shot Frequency-Resolved Optical Gating,” Opt. Express **11**,68–78 (2003). [CrossRef] [PubMed]

*I*(

*x*,

*λ*) is low. To mitigate these effects, it is desirable to apply a threshold to

*I*(

*x*,

*λ*) before calculating

*ρ*, by setting to 1 any values of the intensity that are above a pre-defined threshold, and setting the others to 0 (see Fig. 3 for an example). As a simple alternative, it is possible to let the camera saturate a large portion of the trace, and only retain the saturated values (i.e., setting non-saturated values to zero) before applying Eq. (7). We found both methods to be consistent and equivalently robust to noise,and numerical simulations show that they yield the same result as a direct application of Eq. (7).

_{xλ}## 4. Experimental results

*ρ*and

_{xλ}*ρ*were calculated and displayed in real time.

_{yλ}17. K. Osvay, A. P. Kovács, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatári, “Angular Dispersion and Temporal Change of Femtosecond Pulses From Misaligned Pulse Compressors,” IEEE J. Sel. Top. Quant. Electron. **10**,213–220 (2004). [CrossRef]

*x*and

*y*as we aligned the system (Fig. 3). Table 1 shows typical values of spatial chirp that we found during this procedure. A misaligned stretcher exhibits typical values of

*ρ*=0.50-0.60, and occasionally values as high as 0.80 or 0.90. Realignment of a retro-reflector inside the unit brought

*ρ*to values typically below 0.20. Even smaller values are obtained after amplification and re-compression, which we attribute to the spectral clipping that happens in our compressor unit. During these alignment procedures, beam pointing changes resulted in deviations of

*ρ*on the order of 0.01, which can be roughly considered as the experimental detection limit of our setup.

_{xλ}## 5. Analogy with pulse broadening in dispersive media and extension to other spatio-temporal distortions

*τ*/d

*ω*, although this can also be considered as a temporal variation of the instantaneous frequency

*ω*

_{inst}at a constant rate d

*ω*

_{inst}/d

*t*. In analogy with Eq. (2

2. W. Denk, J. H. Strickler, and W. W. Webb, “Two-Photon Laser Scanning Fluorescence Microscopy,” Science **248**,73–76 (1990). [CrossRef] [PubMed]

*ρ*, normalized by the pulse duration Δ

_{ωt}*t*and the bandwidth Δ

*ω*, and that satisfies:

*ρ*can also be defined in a form similar to Eq. (1) by considering the Wigner distribution of the pulse

_{ωt}*I*(

_{W}*ω*,

*t*). As an example, consider a chirped Gaussian pulse with a bandwidth Δ

*ω*and a group-delay dispersion d

*τ*/d

*ω*:

*t*– (d

*τ*/d

*ω*)

*ω*, whose

*ω*-dependent term becomes important when d

*τ*/d

*ω*≠ 0 (i.e.,

*ρ*≠ 0), and which is similar to the coupling term,

_{ωt}*x*– (d

*x*/d

*ω*)

*ω*, that arises in the case of spatial chirp (

*ρ*≠ 0).

_{xω}*ρ*and the pulse duration Δ

_{ωt}*t*because dispersion effects are easily and intuitively interpreted in the time domain. Figure 4(a) shows the dependence of the pulse duration (normalized to its Fourier limit) with

*ρ*. It is obvious that the parameter

_{ωt}*ρ*is very sensitive to

_{ωt}*small amounts of dispersion*: a value of

*ρ*= 0.30 corresponds to a pulse stretched by only 5%, which is acceptable in many situations. On the other hand, very large stretching ratios, such as those obtained by pulse stretchers in CPA systems, correspond to values of

_{ωt}*ρ*very close to 1, and rapidly become indistinguishable.Thus, these correlation coefficients are ideal for monitoring ultrafast systems that must approach the Fourier limit, but less than ideal for cases in which one is deliberately attempting to introduce massive amounts of these distortions.

_{ωt}*degree of spatio-temporal uniformity μ*[10

10. C. Dorrer and I. A. Walmsley, “Simple linear technique for the measurement of space-time coupling in ultrashort optical pulses,” Opt. Lett. **27**, (2002). [CrossRef]

*amplitude*:

*μ*may be measured experimentally using linear techniques, and it describes

*all*possible spatio-temporal couplings, which can be convenient in some cases:

*μ*=1 corresponds to a pulse free of spatio-temporal distortions, while 0<μ<1 indicates that some distortions are present. However, the parameter μ is not very sensitive to small amounts of spatio-temporal distortions. As shown in Fig. 4(b) in the case of spatial chirp, there is little change in μ in the region of small distortions (

*ρ*≈ 0).

_{xω}*ρ*and

_{ωt}*ρ*. In practice, ∣

_{xω}*ρ*∣ ≤ 0.30 or 0.40 seems a reasonable condition to aim for. These considerations are also valid for the parameters

*ρ*,

_{xt}*ρ*and

_{kω}*ρ*,which can be used to measure pulse-front tilt, angular dispersion, and angular delay, respectively, as long as the intensity distributions

_{kt}*I*(

*x*,

*t*),

*I*(

*k*,

_{x}*ω*) and

*I*(

*k*,

_{x}*t*) are known:

19. Z. Bor and B. Racz, “Group velocity dispersion in prisms and its application to pulse compression and travelling-wave excitation,” Opt. Commun. **54**,165–170 (1985). [CrossRef]

20. S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Opt. Express **12**,4399–4410 (2004). [CrossRef] [PubMed]

*t*/d

*x*=(d

*τ*/d

*ω*)∙(dω/d

*x*). This formula can be expressed in terms of normalized ρ-parameters as well:

*ρ*=

_{xt}*ρ*∙

_{xω}*ρ*(see Fig. 6 for an example). For more complex pulses however, closed-form expressions for relationships between spatio-temporal distortions become difficult to establish, and from a practical point of view it is preferable to aim at maintaining all the various

_{ωt}*ρ*-parameters below a certain threshold (e.g., 0.30) that eventually depends on the overall spatio-temporal pulse quality that is sought.

*ρ*-parameters also seem to offer the possibility to describe spatio-temporal distortions beyond the first order, such as chromatic aberrations in lenses, or pulse-front curvature, by considering higher-order cross moments

*μ*of the relevant intensity distributions.

_{pq}## 6. Conclusions

1. C. B. Schaffer, A. Brodeur, J. F. García, and E. Mazur, “Micromachining bulk glass by use of femtosecond laser pulses with nanojoule energy,” Opt. Lett. **26**,93–95 (2001). [CrossRef]

1. C. B. Schaffer, A. Brodeur, J. F. García, and E. Mazur, “Micromachining bulk glass by use of femtosecond laser pulses with nanojoule energy,” Opt. Lett. **26**,93–95 (2001). [CrossRef]

*severity*of these distortions. These parameters are especially sensitive to small amounts of distortion. We also presented a simple, practical apparatus allowing the real-time monitoring of the corresponding spatial-chirp parameters

*ρ*and

_{xλ}*ρ*,. We believe that these parameters will help better understand spatio-temporal distortions and their consequences, and will be used as a benchmark enabling the comparison of the performance of ultrafast lasers.

_{yλ}## Acknowledgments

## References and links

1. | C. B. Schaffer, A. Brodeur, J. F. García, and E. Mazur, “Micromachining bulk glass by use of femtosecond laser pulses with nanojoule energy,” Opt. Lett. |

2. | W. Denk, J. H. Strickler, and W. W. Webb, “Two-Photon Laser Scanning Fluorescence Microscopy,” Science |

3. | R. L. Fork, O. E. Martinez, and J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. |

4. | 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. |

5. | R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbuegel, and D. J. Kane, “Measuring Ultrashort Laser Pulses in the Time-Frequency Domain Using Frequency-Resolved Optical Gating,” Rev. Sci. Instrum. |

6. | P. O'Shea, M. Kimmel, X. Gu, and R. Trebino, “Highly simplified device for ultra-short measurement,” Opt. Lett. |

7. | S. Akturk, M. Kimmel, P. O'Shea, and R. Trebino, “Measuring spatial chirp in ultrashort pulses using
single-shot Frequency-Resolved Optical Gating,” Opt. Express |

8. | S. Akturk, M. Kimmel, P. O'Shea, and R. Trebino, “Measuring pulse-front tilt in ultrashort pulses using GRENOUILLE,” Opt. Express |

9. | C. Dorrer, E.M. Kosik, and I. A. Walmsley, “Spatio-temporal characterization of the electric field of ultrashort pulses using two-dimensional shearing interferometry,” Applied Physics B (Lasers and Optics) |

10. | C. Dorrer and I. A. Walmsley, “Simple linear technique for the measurement of space-time coupling in ultrashort optical pulses,” Opt. Lett. |

11. | K. Varju, A. P. Kovacs, G. Kurdi, and K. Osvay, “High-precision measurement of angular dispersion in a CPA laser,” Appl. Phys. B Suppl.,259–263 (2002). [CrossRef] |

12. | M. Kempe, U. Stamm, B. Wilhelmi, and W. Rudolph, “Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems,” J. Opt. Soc. Am. B |

13. | X. Gu, S. Akturk, and R. Trebino, “Spatial chirp in ultrafast optics,” Opt. Commun. |

14. | A. G. Kostenbauder, “Ray-Pulse Matrices: A Rational Treatment for Dispersive Optical Systems,” IEEE J. Quantum Electron. |

15. | S. Akturk, X. Gu, P. Gabolde, and R. Trebino, “The general theory of first-order spatio-temporal
distortions of Gaussian pulses and beams,” Opt. Express |

16. | R. V. Hogg and A. Craig, |

17. | K. Osvay, A. P. Kovács, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatári, “Angular Dispersion and Temporal Change of Femtosecond Pulses From Misaligned Pulse Compressors,” IEEE J. Sel. Top. Quant. Electron. |

18. | L. Cohen, |

19. | Z. Bor and B. Racz, “Group velocity dispersion in prisms and its application to pulse compression and travelling-wave excitation,” Opt. Commun. |

20. | S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Opt. Express |

**OCIS Codes**

(320.5550) Ultrafast optics : Pulses

(320.7100) Ultrafast optics : Ultrafast measurements

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: November 15, 2006

Revised Manuscript: December 15, 2006

Manuscript Accepted: December 18, 2006

Published: January 8, 2007

**Citation**

Pablo Gabolde, Dongjoo Lee, Selcuk Akturk, and Rick Trebino, "Describing first-order spatio-temporal distortions in ultrashort pulses using normalized parameters," Opt. Express **15**, 242-251 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-1-242

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### References

- C. B. Schaffer, A. Brodeur, J. F. García, and E. Mazur, "Micromachining bulk glass by use of femtosecond laser pulses with nanojoule energy," Opt. Lett. 26, 93-95 (2001). [CrossRef]
- W. Denk, J. H. Strickler, and W. W. Webb, "Two-Photon Laser Scanning Fluorescence Microscopy," Science 248, 73-76 (1990). [CrossRef] [PubMed]
- R. L. Fork, O. E. Martinez, and J. P. Gordon, "Negative dispersion using pairs of prisms," Opt. Lett. 9, 150-152 (1984). [CrossRef] [PubMed]
- 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]
- R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbuegel, and D. J. Kane, "Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating," Rev. Sci. Instrum. 38, 3277-3295 (1997). [CrossRef]
- P. O'Shea, M. Kimmel, X. Gu, and R. Trebino, "Highly simplified device for ultra-short measurement," Opt. Lett. 26, 932-934 (2001).
- S. Akturk, M. Kimmel, P. O'Shea, and R. Trebino, "Measuring spatial chirp in ultrashort pulses using single-shot Frequency-Resolved Optical Gating," Opt. Express 11, 68-78 (2003). [CrossRef] [PubMed]
- 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]
- C. Dorrer, E. M. Kosik, and I. A. Walmsley, "Spatio-temporal characterization of the electric field of ultrashort pulses using two-dimensional shearing interferometry," Appl. Phys. B: Lasers Opt. 74 [Suppl.], S209-S217 (2002). [CrossRef]
- C. Dorrer, and I. A. Walmsley, "Simple linear technique for the measurement of space-time coupling in ultrashort optical pulses," Opt. Lett. 27, 1947-1949 (2002). [CrossRef]
- K. Varju, A. P. Kovacs, G. Kurdi, and K. Osvay, "High-precision measurement of angular dispersion in a CPA laser," Appl. Phys. B Suppl., 259-263 (2002). [CrossRef]
- M. Kempe, U. Stamm, B. Wilhelmi, and W. Rudolph, "Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems," J. Opt. Soc. Am. B 9, 1158-1165 (1992). [CrossRef]
- X. Gu, S. Akturk, and R. Trebino, "Spatial chirp in ultrafast optics," Opt. Commun. 242, 599-604 (2004). [CrossRef]
- A. G. Kostenbauder, "Ray-Pulse Matrices: A Rational Treatment for Dispersive Optical Systems," IEEE J. Quantum Electron. 26, 1148-1157 (1990). [CrossRef]
- S. Akturk, X. Gu, P. Gabolde, and R. Trebino, "The general theory of first-order spatio-temporal distortions of Gaussian pulses and beams," Opt. Express 13, 8642-8661 (2005). [CrossRef] [PubMed]
- R. V. Hogg, and A. Craig, Introduction to Mathematical Statistics (Prentice Hall, 1994).
- K. Osvay, A. P. Kovács, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatári, "Angular Dispersion and Temporal Change of Femtosecond Pulses From Misaligned Pulse Compressors," IEEE J. Sel. Top. Quantum Electron. 10, 213-220 (2004). [CrossRef]
- L. Cohen, Time-frequency analysis (Prentice Hall, 1995).
- Z. Bor, and B. Racz, "Group velocity dispersion in prisms and its application to pulse compression and travelling-wave excitation," Opt. Commun. 54, 165-170 (1985). [CrossRef]
- S. Akturk, X. Gu, E. Zeek, and R. Trebino, "Pulse-front tilt caused by spatial and temporal chirp," Opt. Express 12, 4399-4410 (2004). [CrossRef] [PubMed]

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