## Pulse-front tilt caused by spatial and temporal chirp

Optics Express, Vol. 12, Issue 19, pp. 4399-4410 (2004)

http://dx.doi.org/10.1364/OPEX.12.004399

Acrobat PDF (585 KB)

### Abstract

Pulse-front tilt in an ultrashort laser pulse is generally considered to be a direct consequence of, and equivalent to, angular dispersion. We show, however, that, while this is true for certain types of pulse fields, simultaneous temporal chirp and spatial chirp also yield pulse-front tilt, even in the absence of angular dispersion. We verify this effect experimentally using GRENOUILLE.

© 2004 Optical Society of America

## 1. Introduction

1. R. L. Fork, O. E. Martinez, and J. P. Gordon, “Negative Dispersion Using Pairs of Prisms,” Opt. Lett. **9**, 150–152 (1984). [CrossRef] [PubMed]

*et al*. [4

4. Z. Bor, B. Racz, G. Szabo, M. Hilbert, and H. A. Hazim, “Femtosecond Pulse Front Tilt Caused by Angular-Dispersion,” Optical Engineering **32**, 2501–2504 (1993). [CrossRef]

5. J. Hebling, “Derivation of the pulse front tilt caused by angular dispersion,” Opt. and Quantum Electron. **28**, 1759–1763 (1996). [CrossRef]

*et al*. [6

6. C. Dorrer, E. M. Kosik, and I. A. Walmsley, “Spatio-temporal characterization of ultrashort optical pulses using two-dimensional shearing interferometry,” Appl. Phys. B **74** [suppl.], 209–219 (2002). [CrossRef]

*p*is the PFT. We have suppressed the

*y*-dependence and assumed that, apart from PFT,

*E*(

*x*,

*z*,

*t*) has no coupling of its coordinates, so it can be separated into

*E*(

_{xz}*x*,

*z*) and

*E*(

*t*). (This is a more rigorous expression than that given in Ref. [6

6. C. Dorrer, E. M. Kosik, and I. A. Walmsley, “Spatio-temporal characterization of ultrashort optical pulses using two-dimensional shearing interferometry,” Appl. Phys. B **74** [suppl.], 209–219 (2002). [CrossRef]

*x*-

*t*domain to the

*k*-

*ω*domain and using two applications of the Shift Theorem, we have:

*dk*/d

_{x}*ω*=

*p*, or the AD is d

*θ*

_{0}/d

*ω*=

*p*/

*k*

_{0}, where

*θ*

_{0}is the propagation angle, and

*k*

_{0}is the nominal wave-number in vacuum.

*not*equivalent, and we provide an additional (and rather common!) source of pulse-front tilt, in which no angular dispersion occurs. We point out that the “proof” of AD/PFT equivalence only holds for fields of the above form, and our counterexample incorporates a beam with spatial chirp, which cannot be written in the above form.

7. I. Z. Kozma, G. Almasi, and J. Hebling, “Geometrical optical modeling of femtosecond setups having angular dispersion,” Appl. Phys. B **76**, 257–261 (2003). [CrossRef]

8. Z. Bor and B. Racz, “Group-Velocity Dispersion in Prisms and Its Application to Pulse-Compression and Traveling-Wave Excitation,” Opt. Commun. **54**, 165–170 (1985). [CrossRef]

9. O. E. Martinez, “Pulse Distortions in Tilted Pulse Schemes for Ultrashort Pulses,” Opt. Commun. **59**, 229–232 (1986). [CrossRef]

## 2. AD and PFT in the presence of SC

*x*-

*ω*domain for the electric field of a pulse with linear SC and AD.

*k*

_{0}is the nominal wave-number,

*ω*is the offset from the center angular frequency, and

*q*is the complex

*q*parameter of a Gaussian beam:

*d*is the position of the beam waist and

*w*is the spot size. The SC and AD are parameterized by

*x*

_{0}is the beam center position of the

*ω*-component of the beam and

*θ*

_{0}is the propagation angle of this component.

*x*-direction only and therefore neglect the beam’s

*y*-dependence. Generalization to both

*x*and

*y*dependences is straightforward.

*υ*, Eq. (7) may be rewritten as

*w*due to spatial chirp,

*τ*

_{0}due to the reduced locally available bandwidth.

*t*

_{0}as the pulse-front (maximum intensity contour) arrival time, and the PFT may be characterized by the derivative of

*t*

_{0}with respect to

*x*,

*z*—is then given by

*p*

_{AD}is the well known angular-dispersion term, as derived by Bor

*et al*. [4

4. Z. Bor, B. Racz, G. Szabo, M. Hilbert, and H. A. Hazim, “Femtosecond Pulse Front Tilt Caused by Angular-Dispersion,” Optical Engineering **32**, 2501–2504 (1993). [CrossRef]

5. J. Hebling, “Derivation of the pulse front tilt caused by angular dispersion,” Opt. and Quantum Electron. **28**, 1759–1763 (1996). [CrossRef]

*p*

_{SC+TC}is a PFT effect caused by the combination of SC, which is characterized by the frequency gradient

*υ*and temporal chirp, which is characterized by group-delay dispersion

*φ*

^{(2)}. This new PFT effect is clearly the cause of the PFT in the scenario shown in Fig. 1 (right), in which no AD exists.

*t*yields a field in the

*x*-

*ω*domain of the form:

*x*-

*ω*domain of the form:

*x*and

*ω*beyond the simple complex exponential. An example of this coupling is Eq. (3). The presence of SC in the form of frequency gradient requires an expression in the

*x*-

*ω*domain of the form:

## 3. Propagation of ultrashort-pulse beams with angular dispersion and spatial chirp

*z*

_{0}. In this section, we propagate the field to an arbitrary position

*z*, and discuss how the spatial-temporal coupling parameters, including SC and PFT, evolve. To accomplish this, we use the Fresnel-Kirchoff integral formula [11]:

*z*

_{0}= 0 ,

*z*-evolved values:

*υ*and PFT

*p*:

12. A. G. Kostenbauder, “Ray-Pulse Matrices: A Rational Treatment for Dispersive Optical Systems,” IEEE J. Quantum Electron. **26**, 1148–1157 (1990). [CrossRef]

## 4. Experiment

*ξ*, or the slope of the center frequency vs. position, which yields the frequency gradient

*υ*. The same beam was also sent to a Swamp Optics GRENOUILLE [13

13. P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, “Highly simplified ultrashort pulse measurement,” Opt. Lett. **26**, 932 (2001). [CrossRef]

14. R. Trebino, *Frequency-Resolved Optical Gating* (Kluwer Academic Publishers, Boston, 2002). [CrossRef]

15. 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]

16. 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]

17. K. Varjú, A. P. Kovács, G. Kurdi, and K. Osvay, “High-precision measurement of angular dispersion in a CPA laser,” Appl. Phys, B **74**, S259–S263 (2002). [CrossRef]

18. K. Varjú, A. P. Kovács, K. Osvay, and G. Kurdi, “Angular dispersion of femtosecond pulses in a Gaussian beam,” Opt. Lett. **27**, 2034–2036 (2002). [CrossRef]

6. C. Dorrer, E. M. Kosik, and I. A. Walmsley, “Spatio-temporal characterization of ultrashort optical pulses using two-dimensional shearing interferometry,” Appl. Phys. B **74** [suppl.], 209–219 (2002). [CrossRef]

19. B. S. Prade, J. M. Schins, E. T. J. Nibbering, M. A. Franco, and A. Mysyrowickz, “A Simple Method for the Determination of the Intensity and Phase of Ultrashort Optical Pulses,” Opt. Commun. **113**, 79–84 (1995). [CrossRef]

15. 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]

*p*vs. GDD

*φ*

^{(2)}yields the frequency gradient

*υ*. Figure 4 shows such a plot. We measured the slope of this plot to be 8.78×10

^{-3}(rad·fs

^{-1})/mm

^{-3}(rad·fs

^{-1})/mm

## 5. Conclusion

## Appendix

20. O. E. Martinez, “Matrix Formalism for Dispersive Laser Cavities,” IEEE J. Quantum Electron. **25**, 296–300 (1989). [CrossRef]

21. O. E. Martinez, “Matrix Formalism for Pulse Compressors,” IEEE J. Quantum Electron. **24**, 2530–2536 (1988). [CrossRef]

12. A. G. Kostenbauder, “Ray-Pulse Matrices: A Rational Treatment for Dispersive Optical Systems,” IEEE J. Quantum Electron. **26**, 1148–1157 (1990). [CrossRef]

*ξ*is spatial dispersion and φ

^{(2)}is the GDD.

*K*can be obtained either by calculating the system ray-pulse matrix for a two-prism pulse compressor separated by

*L*or for a fictitious system that introduces only spatial chirp followed by a dispersive slab of thickness

*nL*(where

*n*=

*n*(

*ω*) is the index of refraction). In both cases, GDD is the total GDD due to both the material and angular dispersions. Note that this approach describes only rays or plane waves (which has the form given in Eqs. (1) and (2)), so the matrix shows no pulse-front tilt

*Q*matrix, as illustrated by Kostenbauder in Ref. [12

12. A. G. Kostenbauder, “Ray-Pulse Matrices: A Rational Treatment for Dispersive Optical Systems,” IEEE J. Quantum Electron. **26**, 1148–1157 (1990). [CrossRef]

*Q*

^{-1}indicate spatial-temporal coupling. If we write the magnitude of electric field in terms of the local pulse length and the pulse-front tilt as

## Acknowledgments

## References and links

1. | R. L. Fork, O. E. Martinez, and J. P. Gordon, “Negative Dispersion Using Pairs of Prisms,” Opt. Lett. |

2. | J. P. Gordon and R. L. Fork, “Optical resonator with negative dispersion,” Opt. Lett. |

3. | O. E. Martinez, J. P. Gordon, and R. L. Fork, “Negative group-velocity dispersion using refraction,” J. Opt. Soc. Am. B |

4. | Z. Bor, B. Racz, G. Szabo, M. Hilbert, and H. A. Hazim, “Femtosecond Pulse Front Tilt Caused by Angular-Dispersion,” Optical Engineering |

5. | J. Hebling, “Derivation of the pulse front tilt caused by angular dispersion,” Opt. and Quantum Electron. |

6. | C. Dorrer, E. M. Kosik, and I. A. Walmsley, “Spatio-temporal characterization of ultrashort optical pulses using two-dimensional shearing interferometry,” Appl. Phys. B |

7. | I. Z. Kozma, G. Almasi, and J. Hebling, “Geometrical optical modeling of femtosecond setups having angular dispersion,” Appl. Phys. B |

8. | Z. Bor and B. Racz, “Group-Velocity Dispersion in Prisms and Its Application to Pulse-Compression and Traveling-Wave Excitation,” Opt. Commun. |

9. | O. E. Martinez, “Pulse Distortions in Tilted Pulse Schemes for Ultrashort Pulses,” Opt. Commun. |

10. | X. Gu, S. Akturk, and R. Trebino, “Parameterizations of Spatial Chirp in Ultrafast Optics,” Opt. Commun. (to be published). |

11. | E. Hecht |

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

13. | P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, “Highly simplified ultrashort pulse measurement,” Opt. Lett. |

14. | R. Trebino, |

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

16. | 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 |

17. | K. Varjú, A. P. Kovács, G. Kurdi, and K. Osvay, “High-precision measurement of angular dispersion in a CPA laser,” Appl. Phys, B |

18. | K. Varjú, A. P. Kovács, K. Osvay, and G. Kurdi, “Angular dispersion of femtosecond pulses in a Gaussian beam,” Opt. Lett. |

19. | B. S. Prade, J. M. Schins, E. T. J. Nibbering, M. A. Franco, and A. Mysyrowickz, “A Simple Method for the Determination of the Intensity and Phase of Ultrashort Optical Pulses,” Opt. Commun. |

20. | O. E. Martinez, “Matrix Formalism for Dispersive Laser Cavities,” IEEE J. Quantum Electron. |

21. | O. E. Martinez, “Matrix Formalism for Pulse Compressors,” IEEE J. Quantum Electron. |

**OCIS Codes**

(320.5550) Ultrafast optics : Pulses

(320.7100) Ultrafast optics : Ultrafast measurements

**ToC Category:**

Research Papers

**History**

Original Manuscript: July 22, 2004

Revised Manuscript: September 3, 2004

Published: September 20, 2004

**Citation**

Selcuk Akturk, Xun Gu, Erik Zeek, and Rick Trebino, "Pulse-front tilt caused by spatial and temporal chirp," Opt. Express **12**, 4399-4410 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-19-4399

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### References

- 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. P. Gordon and R. L. Fork, "Optical resonator with negative dispersion," Opt. Lett. 9, 153-155 (1984) [CrossRef] [PubMed]
- O. E. Martinez, J. P. Gordon, and R. L. Fork, "Negative group-velocity dispersion using refraction," J. Opt. Soc. Am. B 1, 1003-1006 (1984). [CrossRef]
- Z. Bor, B. Racz, G. Szabo, M. Hilbert, and H. A. Hazim, "Femtosecond Pulse Front Tilt Caused by Angular-Dispersion," Optical Engineering 32, 2501-2504 (1993). [CrossRef]
- J. Hebling, "Derivation of the pulse front tilt caused by angular dispersion," Opt. and Quantum Electron. 28, 1759-1763 (1996) [CrossRef]
- C. Dorrer, E. M. Kosik, and I. A. Walmsley, "Spatio-temporal characterization of ultrashort optical pulses using two-dimensional shearing interferometry," Appl. Phys. B 74 , 209-219 (2002) [CrossRef]
- I. Z. Kozma, G. Almasi, and J. Hebling, "Geometrical optical modeling of femtosecond setups having angular dispersion," Appl. Phys. B 76, 257-261 (2003) [CrossRef]
- Z. Bor and B. Racz, "Group-Velocity Dispersion in Prisms and Its Application to Pulse-Compression and Traveling-Wave Excitation," Opt. Commun. 54, 165-170 (1985) [CrossRef]
- O. E. Martinez, "Pulse Distortions in Tilted Pulse Schemes for Ultrashort Pulses," Opt. Commun. 59, 229- 232 (1986) [CrossRef]
- X. Gu, S. Akturk, and R. Trebino, "Parameterizations of Spatial Chirp in Ultrafast Optics," Opt. Commun. (to be published).
- E. Hecht, Optics, 3rd ed. (Addison Wesley Longman, Inc., 1998)
- A. G. Kostenbauder, "Ray-Pulse Matrices: A Rational Treatment for Dispersive Optical Systems," IEEE J. Quantum Electron. 26, 1148-1157 (1990) [CrossRef]
- P. O'Shea, M. Kimmel, X. Gu, and R. Trebino, "Highly simplified ultrashort pulse measurement," Opt. Lett. 26, 932 (2001) [CrossRef]
- R. Trebino, Frequency-Resolved Optical Gating (Kluwer Academic Publishers, Boston, 2002) [CrossRef]
- 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]
- 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]
- K. Varjú, A. P. Kovács, G. Kurdi, and K. Osvay, "High-precision measurement of angular dispersion in a CPA laser," Appl. Phys. B 74, S259-S263 (2002) [CrossRef]
- K. Varjú, A. P. Kovács, K. Osvay, and G. Kurdi, "Angular dispersion of femtosecond pulses in a Gaussian beam," Opt. Lett. 27, 2034-2036 (2002) [CrossRef]
- B. S. Prade, J. M. Schins, E. T. J. Nibbering, M. A. Franco, and A. Mysyrowickz, "A Simple Method for the Determination of the Intensity and Phase of Ultrashort Optical Pulses," Opt. Commun. 113, 79-84 (1995) [CrossRef]
- O. E. Martinez, "Matrix Formalism for Dispersive Laser Cavities," IEEE J. Quantum Electron. 25, 296-300 (1989). [CrossRef]
- O. E. Martinez, "Matrix Formalism for Pulse Compressors," IEEE J. Quantum Electron. 24, 2530-2536 (1988) [CrossRef]

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