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

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References

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  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, 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]
  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]
  5. 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]
  6. 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]
  7. O. E. Martinez, �??Pulse distortions in tilted pulse schemes for ultrashort pulses,�?? Opt. Commun. 59, 229�??232 (1986).
    [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]
  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]
  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). <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]
  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. 27, 1265�??1267 (2002).
    [CrossRef]
  17. Z. Bor and Z. L. Horváth, �??Distortion of femtosecond pulses in lenses. Wave optical description,�?? Opt. Commun. 94, 249�??258 (1992).
    [CrossRef]

Appl. Phys. B [Suppl.] (1)

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]

J. Opt. Soc. Am. B (1)

Opt. Commun. (4)

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

Z. Bor and Z. L. Horváth, �??Distortion of femtosecond pulses in lenses. Wave optical description,�?? Opt. Commun. 94, 249�??258 (1992).
[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]

Opt. Eng. (1)

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]

Opt. Express (1)

Opt. Lett. (8)

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]

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]

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]

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]

Other (1)

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.

Supplementary Material (4)

» Media 1: GIF (641 KB)     
» Media 2: GIF (488 KB)     
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Figures (8)

Fig. 1.
Fig. 1.

Schematic figure of two rays passing through a diffractive device. The argument of the pulse front tilt in this section is general for any diffractive devices although we show a prism as an example in this figure.

Fig. 2.
Fig. 2.

Wave-vector matching in sum-frequency mixing with a noncollinear angularly dispersed geometry.

Fig. 3.
Fig. 3.

Spectrum of the generated DUV with the GDD-matched SFM (light blue hatched area on bottom graph) obtained in the experiment, and calculated output angle (blue curve on top graph). An acceptable power spectrum is also shown as the red hatched area in the bottom graph.

Fig. 4.
Fig. 4.

Residual angular dispersions at the zeroth (angle), first, and second orders. Dotted curves in each graph correspond to those compensated with an inverse angular disperser consisting of a one-to-one telescope and a fused silica prism. Solid curves are the ones corrected by adding a grating to the inverse angular disperser and replacing the one-to-one telescope with another one having a magnification factor of 2.4/1.5.

Fig. 5.
Fig. 5.

(2.1 MB) Change of the pulse front tilt caused by the residual high-order angular dispersions compensated with the inverse angular disperser of a prism. The color bar at the top left indicates the incident angle relative to the prism.

Fig. 6.
Fig. 6.

(1.6 MB) Change of the pulse front tilt caused by the residual high-order angular dispersions compensated with the hybrid compensation of a prism and a grating. The color bar at the top left indicates the relative incident angle to the prism. The incident angle to the grating is fixed in the calculation.

Fig. 7.
Fig. 7.

(2.2 MB) Change of the interferogram corresponding the change of the pulse front tilt in Fig. 5.

Fig. 8.
Fig. 8.

(1.6 MB) Change of the interferogram corresponding the change of the pulse front tilt in Fig. 6.

Equations (13)

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tan γ = λ 0 | 0 ,
ϕ A ( ω 0 ) = ϕ B ( ω 0 ) ,
ϕ B ( ω 0 + Δ ω ) ϕ A ( ω 0 + Δ ω )
= ( ω 0 + Δ ω c ) sin θ 0 cos Δ ω ( ω 0 + Δ ω c ) sin ( θ 0 + Δ θ )
= ( ω 0 + Δ ω c ) x 0 sin Δ θ ,
d ϕ B d ω | 0 d ϕ A d ω | 0 = x 0 c ω 0 d θ d ω | 0
d 2 ϕ B d ω 2 | 0 d 2 ϕ A d ω 2 | 0 = x 0 c ( ω 0 d 2 θ d ω 2 | 0 + 2 d θ d ω | 0 )
d 3 ϕ B d ω 3 | 0 d 3 ϕ A d ω 3 | 0 = x 0 c [ ω 0 { d 3 θ d ω 3 | 0 ( d θ d ω | 0 ) 3 } + 3 d 2 θ d ω 2 | 0 ]
τ B ( ω 0 ) τ A ( ω 0 ) = x 0 c λ 0 d θ d λ | 0 ,
d 2 ϕ B d ω 2 | 0 d 2 ϕ A d ω 2 | 0 = x 0 2 π c 2 λ 0 3 d 2 θ d λ 2 | 0 ,
d 3 ϕ B d ω 3 | 0 d 3 ϕ A d ω 3 | 0 = x 0 4 π 2 c 3 λ 0 2 { 3 λ 0 2 d 2 θ d λ 2 | 0 + λ 0 3 d 3 θ d λ 3 | 0 + λ 0 3 ( d θ d λ | 0 ) 3 } .
θ c = θ a 0 + α ac ( ω c ) ,
α ac ( ω c ) = arctan { k b ( ω b ) sin α ( ω b ) k a 0 ( ω a 0 ) + k b ( ω b ) cos α ( ω b ) } ,

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