## High-order pulse front tilt caused by high-order angular dispersion

Optics Express, Vol. 11, Issue 25, pp. 3365-3376 (2003)

http://dx.doi.org/10.1364/OE.11.003365

Acrobat PDF (208 KB)

### Abstract

We have found general expressions relating the high-order pulse front tilt and the high-order angular dispersion in an ultrashort pulse, for the first time to our knowledge. The general formulae based on Fermat’s principle are applicable for any ultrashort pulse with angular dispersion in the limit of geometrical optics. By virtue of these formulae, we can calculate the high-order pulse front tilt in the sub-20-fs UV pulse generated in a novel scheme of sum-frequency mixing in a nonlinear crystal accompanied by angular dispersion. It is also demonstrated how the high-order angular dispersion can be eliminated in the calculation.

© 2003 Optical Society of America

## 1. Introduction

1. S. Sartania, Z. Cheng, M. Lenzner, G. Tempea, C. Spielmann, F. Krausz, and K. Ferencz, “Generation of 0.1-TW 5-fs optical pulses at a 1-kHz repetition rate,” Opt. Lett. **22**, 1562–1564 (1997). [CrossRef]

4. B. Schenkel, J. Biegert, U. Keller, C. Vozzi, M. Nisoli, G. Sansone, S. Stagira, S. D. Silvestri, and O. Svelto, “Generation of 3.8-fs pulses from adaptive compression of a cascaded hollow fiber supercontinuum,” Opt. Lett. **28**, 1987–1989 (2003). [CrossRef] [PubMed]

1. S. Sartania, Z. Cheng, M. Lenzner, G. Tempea, C. Spielmann, F. Krausz, and K. Ferencz, “Generation of 0.1-TW 5-fs optical pulses at a 1-kHz repetition rate,” Opt. Lett. **22**, 1562–1564 (1997). [CrossRef]

2. J. Seres, A. Müller, E. Seres, K. O’Keeffe, M. Lenner, R. F. Herzog, D. Kaplan, C. Spielmann, and F. Krausz, “Sub-10-fs, terawatt-scale Ti:sapphire laser system,” Opt. Lett. **28**, 1832–1834 (2003). [CrossRef] [PubMed]

3. K. Yamane, Z. Zhang, K. Oka, R. Morita, M. Yamashita, and A. Suguro, “Optical pulse compression to 3.4 fs in the monocycle region by feedback phase compensation,” Opt. Lett. **28**, 2258–2260 (2003). [CrossRef] [PubMed]

*ε*, the angle from the angularly dispersive element, and

*λ*, the wavelength as a variable. Subscript

_{0}denotes the substitution of the fixed wavelength of

*λ*

_{0}characterizing the ultrashort pulse.

*et al.*[8

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

9. 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]

10. K. Osvay and I. N. Ross, “On a pulse compressor with gratings having arbitrary orientation,” Opt. Commun. **105**, 271–278 (1994). [CrossRef]

*et al.*measured the external noncollinear angles of the idler pulse for each wavelength in a noncollinear OPA system and showed that the residual dependence of the angle on the wavelength corrected by the angular-dispersion compensator consisting of a grating and a telescope is nonlinear or somewhat sinusoidal with an amplitude of ~400

*µ*radians in the modulation, although they were successful in obtaining an almost transform-limited pulse with a duration of 9 fs [11

11. A. Shirakawa, I. Sakane, and T. Kobayashi, “Pulse-front-matched optical parametric amplification for sub-10-fs pulse generation tunable in the visible and near infrared,” Opt. Lett. **23**, 1292–1294 (1998). [CrossRef]

12. T. Hofmann, K. Mossavi, F. K. Tittel, and G. Szabó, “Spectrally compensated sum-frequency mixing scheme for generation of broadband radiation at 193 nm,” Opt. Lett. **17**, 1691–1693 (1992). [CrossRef]

13. M. Hacker, T. Feurer, R. Sauerbrey, T. Lucza, and G. Szabó, “Programmable femtosecond laser pulses in the ultraviolet,” J. Opt. Soc. Am. B **18**, 866– 871 (2001). [CrossRef]

14. Y. Nabekawa and K. Midorikawa, “Broadband sum frequency mixing using noncollinear angularly dispersed geometry for indirect phase control of sub-20-femtosecond UV pulses,” Opt. Express **11**, 324–338 (2003). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-4-324 [CrossRef] [PubMed]

12. T. Hofmann, K. Mossavi, F. K. Tittel, and G. Szabó, “Spectrally compensated sum-frequency mixing scheme for generation of broadband radiation at 193 nm,” Opt. Lett. **17**, 1691–1693 (1992). [CrossRef]

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

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

## 2. Formulation of the high-order pulse front tilt

*ω*is diffracted using an optical device such as a prism or a grating, as shown in Fig. 1, where two rays at different positions are named the ‘A’ ray and the ‘B’ ray, respectively. Because the angular dispersion originates from one point at the entrance of the diffractive device for each ray, we can define the fictitious origin, which does not need to lie on the real ray, of the diffracted ray as one point from which rays for any

*ω*component extend in the free space, as is indicated by

*O*and

_{A}*O*in Fig. 1. Although the point of

_{B}*O*in Fig. 1 is occasionally also the real origin of the diffracted ray, we can proceed with this argument of angular dispersion without losing generality because the fictitious origins of any rays from different entrances inevitably are on the line

_{B}*θ*

_{0}for the fixed angular frequency of

*ω*

_{0}, which is identified with a near-center angular frequency in an ultrashort pulse. A deviation of the angle from

*θ*

_{0}induced by a variation of the angular frequency from

*ω*

_{0}, which is Δ

*ω*in our definition, is denoted by Δ

*θ*.

*O*on the ‘A’ ray for the angular frequency

_{A}*ω*

_{0}is equivalent to that at point

*P*on the ‘B’ ray, which is determined by drawing an intersecting line from

_{B}*O*perpendicularly to the ‘B’ ray, namely,

_{A}*ϕ*(

_{A}*ω*) and

*ϕ*(

_{B}*ω*) are the spectral phases of the ‘A’ ray and the ‘B’ ray, respectively, for the angular frequency of

*ω*at the equivalent points on each ray, which are

*O*(=

_{A}*P*) and

_{A}*P*if

_{B}*ω*is equal to

*ω*

_{0}.

*P*′

_{A}on the deviated ‘B’ ray for the varying angular frequency of

*ω*

_{0}+Δ

*ω*by a similar procedure. Point

*P*′

_{A}reflects the deviated spectral phase of the ‘A’ ray on the ‘B’ ray, which is denoted by

*ϕ*(

_{A}*ω*

_{0}+Δ

*ω*). On the other hand, the phase-equivalent point on the deviated ‘B’ ray to point

*P*on the fixed ‘B’ ray is determined by drawing an intersecting line from

_{B}*P*perpendicularly to the deviated ‘B’ ray, shown as

_{B}*P*′

_{B}in Fig. 1, which reflects the deviated spectral phase of the ‘B’ ray, namely,

*ϕ*(

_{B}*ω*

_{0}+Δ

*ω*). Since the optical path difference of the ‘B’ ray to the ‘A’ ray for the angular frequency of

*ω*

_{0}+Δ

*ω*is equivalent to

*c*is the velocity of light,

*ℓ*is the length of the line

*x*

_{0}is defined as a position on the ‘B’ ray from

*O*along the axis of

_{A}*ω*is finite, not infinitesimal, this equation gives the exact spectral phase difference of the ray spatially separated by a distance of

*x*

_{0}from the other ray. We note that the separation must be defined for the rays for the fixed angular frequency of

*ω*

_{0}.

*ω*. By comparing the coefficients in the series at each order of Δ

*ω*, we can obtain the relation denoted by Eq. (2) at the zeroth order and also obtain the following equations.

*τ*(

_{B}*ω*

_{0})=

*dϕ*|

_{B}/dω_{0}) from that in the ‘A’ ray (

*τ*(

_{A}*ω*

_{0})

*=dϕ*|

_{A}/dω_{0}) and represents the (first order) pulse front tilt, which is clearly seen by converting the variable of the angular frequency into the wavelength such as,

*c*(

*τ*(

_{B}*ω*

_{0})-

*τ*(

_{A}*ω*

_{0}))/

*x*

_{0}, we can reproduce the formula of Eq. (1) from Eq. (7), except for the notation of the angular dispersion.

*x*

_{0}, Eq. (8) and Eq. (9) include information on how the pulse shape varies. The left-hand sides of Eq. (8) and Eq. (9) are the GDD difference and the TOD difference, respectively, which make the temporal profile different at each separated position,

*x*

_{0}. Thus they are regarded as high-order effects in the pulse front tilt. We can find that both the GDD difference and the TOD difference are proportional to

*x*

_{0}and the GDD difference is also proportional to the second order angular dispersion, while the right-hand side of Eq. (9) contributes all the angular dispersions, up to the third order, to the TOD difference.

*x*

_{0}depend on how the pulse passes thorough the diffractive element and also on the dispersion suffered due to the other optical devices before entering and after leaving the diffractive element.

## 3. Examples: deep UV pulse generated with GDD-matched SFM

*α*(

_{ab}*ω*) between the quasi-monochromatic input pulse A and the broadband input pulse B, which is assumed to be a sub-20-fs Ti:sapphire laser pulse with an angular frequency of

_{b}*ω*, the output angle from the optical axis in the generated pulse,

_{b}*θ*, varies, following the change of the angular frequency of

_{c}*ω*such as

_{c}*α*(

_{ac}*ω*

_{c}) is the noncollinear angle between the input pulse A and the generated pulse C obtained from

*k*(

_{b}*ω*) are independent of polar angles to the optical axis. Thus Eq. (11) explicitly describes the dependence of the output angle on the generated pulse with the photon-energy conservation of

_{b}16. Y. Nabekawa, Y. Shimizu, and K. Midorikawa, “Sub-20-fs terawatt-class laser system with a mirrorless regenerative amplifier and an adaptive phase controller,” Opt. Lett. **27**, 1265–1267 (2002). [CrossRef]

*ϕ*=

_{A}*const*. in Eq. (3).

*ℓ*/mm and replace the telescope with that having a transverse magnification factor of 2.5/1.4, which means that the angular dispersion in the generated DUV pulse is reduced to 1.4/2.5. This hybrid configuration with the grating and the prism enables us to control the ratios of both the second- and third-order angular dispersion to the first-order one, which results in the compensation of almost all angular dispersions up to the third order with the incident angles of 66.00° to the grating and 79.06° to the prism, as shown by solid curves in Fig. 4.

## 4. Summary and discussion

7. O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. **59**, 229–232 (1986). [CrossRef]

17. Z. Bor and Z. L. Horváth, “Distortion of femtosecond pulses in lenses.Wave optical description,” Opt. Commun. **94**, 249–258 (1992). [CrossRef]

*et al.*, is very useful [8

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

*µ*radians in these calculations.

## Acknowledgments

## References and links

1. | S. Sartania, Z. Cheng, M. Lenzner, G. Tempea, C. Spielmann, F. Krausz, and K. Ferencz, “Generation of 0.1-TW 5-fs optical pulses at a 1-kHz repetition rate,” Opt. Lett. |

2. | J. Seres, A. Müller, E. Seres, K. O’Keeffe, M. Lenner, R. F. Herzog, D. Kaplan, C. Spielmann, and F. Krausz, “Sub-10-fs, terawatt-scale Ti:sapphire laser system,” Opt. Lett. |

3. | K. Yamane, Z. Zhang, K. Oka, R. Morita, M. Yamashita, and A. Suguro, “Optical pulse compression to 3.4 fs in the monocycle region by feedback phase compensation,” Opt. Lett. |

4. | B. Schenkel, J. Biegert, U. Keller, C. Vozzi, M. Nisoli, G. Sansone, S. Stagira, S. D. Silvestri, and O. Svelto, “Generation of 3.8-fs pulses from adaptive compression of a cascaded hollow fiber supercontinuum,” Opt. Lett. |

5. | Z. Bor, B. Rácz, G. Szabó, M. Hilbert, and H. A. Hazim, “Femtosecond pulse front tilt caused by angular dispersion,” Opt. Eng. |

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

7. | O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. |

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

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

10. | K. Osvay and I. N. Ross, “On a pulse compressor with gratings having arbitrary orientation,” Opt. Commun. |

11. | A. Shirakawa, I. Sakane, and T. Kobayashi, “Pulse-front-matched optical parametric amplification for sub-10-fs pulse generation tunable in the visible and near infrared,” Opt. Lett. |

12. | T. Hofmann, K. Mossavi, F. K. Tittel, and G. Szabó, “Spectrally compensated sum-frequency mixing scheme for generation of broadband radiation at 193 nm,” Opt. Lett. |

13. | M. Hacker, T. Feurer, R. Sauerbrey, T. Lucza, and G. Szabó, “Programmable femtosecond laser pulses in the ultraviolet,” J. Opt. Soc. Am. B |

14. | Y. Nabekawa and K. Midorikawa, “Broadband sum frequency mixing using noncollinear angularly dispersed geometry for indirect phase control of sub-20-femtosecond UV pulses,” Opt. Express |

15. | Y. Nabekawa and K. Midorikawa, “Group-delay-dispersion matched sum-frequency mixing for the generation of deep ultraviolet in sub-20-fs regime,” submitted to Appl. Phys. B. |

16. | Y. Nabekawa, Y. Shimizu, and K. Midorikawa, “Sub-20-fs terawatt-class laser system with a mirrorless regenerative amplifier and an adaptive phase controller,” Opt. Lett. |

17. | Z. Bor and Z. L. Horváth, “Distortion of femtosecond pulses in lenses.Wave optical description,” Opt. Commun. |

**OCIS Codes**

(140.3610) Lasers and laser optics : Lasers, ultraviolet

(190.7110) Nonlinear optics : Ultrafast nonlinear optics

(320.5540) Ultrafast optics : Pulse shaping

**ToC Category:**

Research Papers

**History**

Original Manuscript: October 31, 2003

Revised Manuscript: November 25, 2003

Published: December 15, 2003

**Citation**

Yasuo Nabekawa and Katsumi Midorikawa, "High-order pulse front tilt caused by high-order angular dispersion," Opt. Express **11**, 3365-3376 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-25-3365

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### References

- S. Sartania, Z. Cheng, M. Lenzner, G. Tempea, C. Spielmann, F. Krausz, and K. Ferencz, �??Generation of 0.1-TW 5-fs optical pulses at a 1-kHz repetition rate,�?? Opt. Lett. 22, 1562�??1564 (1997). [CrossRef]
- J. Seres, A. Müller, E. Seres, K. O�??Keeffe, M. Lenner, R. F. Herzog, D. Kaplan, C. Spielmann, and F. Krausz, �??Sub-10-fs, terawatt-scale Ti:sapphire laser system,�?? Opt. Lett. 28, 1832�??1834 (2003). [CrossRef] [PubMed]
- K. Yamane, Z. Zhang, K. Oka, R. Morita, and M. Yamashita, A. Suguro, �??Optical pulse compression to 3.4 fs in the monocycle region by feedback phase compensation,�?? Opt. Lett. 28, 2258�??2260 (2003). [CrossRef] [PubMed]
- B. Schenkel, J. Biegert, U. Keller, C. Vozzi, M. Nisoli, G. Sansone, S. Stagira, S. D. Silvestri, and O. Svelto, �??Generation of 3.8-fs pulses from adaptive compression of a cascaded hollow fiber supercontinuum,�?? Opt. Lett. 28, 1987�??1989 (2003). [CrossRef] [PubMed]
- Z. Bor, B. Rácz, G. Szabó, M. Hilbert, and H. A. Hazim, �??Femtosecond pulse front tilt caused by angular dispersion,�?? Opt. Eng. 32, 2501�??2504 (1993). [CrossRef]
- Z. Bor and B. Rácz, �??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]
- K. Varjú, A. P. Kovács, G. Kurdi, and K. Osvay, �??High-precision measurement of angular dispersion in a CPA laser,�?? Appl. Phys. B [Suppl.] 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]
- K. Osvay and I. N. Ross, �??On a pulse compressor with gratings having arbitrary orientation,�?? Opt. Commun. 105, 271�??278 (1994). [CrossRef]
- A. Shirakawa, I. Sakane, and T. Kobayashi, �??Pulse-front-matched optical parametric amplification for sub-10-fs pulse generation tunable in the visible and near infrared,�?? Opt. Lett. 23, 1292�??1294 (1998). [CrossRef]
- T. Hofmann, K. Mossavi, F. K. Tittel, and G. Szabó, �??Spectrally compensated sum-frequency mixing scheme for generation of broadband radiation at 193 nm,�?? Opt. Lett. 17, 1691�?? 1693 (1992). [CrossRef]
- M. Hacker, T. Feurer, R. Sauerbrey, T. Lucza, and G. Szabó, �??Programmable femtosecond laser pulses in the ultraviolet,�?? J. Opt. Soc. Am. B 18, 866�?? 871 (2001). [CrossRef]
- Y. Nabekawa and K. Midorikawa, �??Broadband sum frequency mixing using noncollinear angularly dispersed geometry for indirect phase control of sub-20-femtosecond UV pulses,�?? Opt. Express 11, 324�??338 (2003). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-4-324">href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-4-324</a> [CrossRef] [PubMed]
- Y. Nabekawa and K. Midorikawa, �??Group-delay-dispersion matched sum-frequency mixing for the generation of deep ultraviolet in sub-20-fs regime,�?? submitted to Appl. Phys. B.
- Y. Nabekawa, Y. Shimizu, and K. Midorikawa, �??Sub-20-fs terawatt-class laser system with a mirrorless regenerative amplifier and an adaptive phase controller,�?? Opt. Lett. 27, 1265�??1267 (2002). [CrossRef]
- Z. Bor and Z. L. Horváth, �??Distortion of femtosecond pulses in lenses. Wave optical description,�?? Opt. Commun. 94, 249�??258 (1992). [CrossRef]

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