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

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References

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  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]
  2. J. P. Gordon and R. L. Fork, “Optical resonator with negative dispersion,” Opt. Lett. 9, 153–155 (1984).
    [CrossRef] [PubMed]
  3. 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]
  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]
  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]
  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]
  10. X. Gu, S. Akturk, and R. Trebino, “Parameterizations of Spatial Chirp in Ultrafast Optics,” Opt. Commun. (to be published).
  11. E. HechtOptics, 3rd ed. (Addison Wesley Longman, Inc., 1998).
  12. A. G. Kostenbauder, “Ray-Pulse Matrices: A Rational Treatment for Dispersive Optical Systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
    [CrossRef]
  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]
  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]
  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]

2003 (3)

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]

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]

2002 (2)

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]

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]

2001 (1)

1996 (1)

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

1995 (1)

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]

1993 (1)

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]

1990 (1)

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

1989 (1)

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

1988 (1)

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

1986 (1)

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

1985 (1)

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]

1984 (3)

Akturk, S.

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]

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

Almasi, G.

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]

Bor, Z.

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]

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]

Dorrer, C.

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]

Fork, R. L.

Franco, M. A.

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]

Gordon, J. P.

Gu, X.

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

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

Hazim, H. A.

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]

Hebling, J.

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]

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

Hecht, E.

E. HechtOptics, 3rd ed. (Addison Wesley Longman, Inc., 1998).

Hilbert, M.

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]

Kimmel, M.

Kosik, E. M.

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]

Kostenbauder, A. G.

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

Kovács, A. P.

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]

Kozma, I. Z.

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]

Kurdi, G.

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]

Martinez, O. E.

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]

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

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]

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

Mysyrowickz, A.

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]

Nibbering, E. T. J.

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’Shea, P.

Osvay, K.

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]

Prade, B. S.

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]

Racz, B.

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]

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]

Schins, J. M.

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]

Szabo, G.

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]

Trebino, R.

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]

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]

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

Varjú, K.

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]

Walmsley, I. A.

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]

Appl. Phys, B (1)

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]

Appl. Phys. B (2)

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]

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]

IEEE J. Quantum Electron. (3)

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

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

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]

Opt, Express (1)

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]

Opt. and Quantum Electron. (1)

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

Opt. Commun. (3)

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]

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]

Opt. Express (1)

Opt. Lett. (4)

Optical Engineering (1)

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]

Other (3)

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

E. HechtOptics, 3rd ed. (Addison Wesley Longman, Inc., 1998).

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

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Figures (4)

Fig. 1.
Fig. 1.

Two sources of pulse-front tilt. Left: The well-known angular dispersion. Right: The combination of spatial and temporal chirp.

Fig. 2.
Fig. 2.

Apparatus to introduce constant spatial chirp, variable temporal chirp and no angular dispersion.

Fig. 3.
Fig. 3.

GRENOUILLE traces of a beam that has a constant spatial chirp and variable temporal chirp. Note that the amount of shift of the center (a measure of pulse-front tilt) increases with increasing temporal chirp.

Fig. 4.
Fig. 4.

Experimental measurements (plus-sign symbols) of pulse-front tilt for different amounts of GDD. The red line shows the linear fit.

Equations (56)

Equations on this page are rendered with MathJax. Learn more.

E ( x , z , t ) = E xz ( x , z ) E t ( t px )
E ˜ ̂ ( k x , k z , ω ) = E ˜ ̂ k x k z ( k x , k z ) E ˜ ̂ ω ( ω )
E ( x , ω ) = E ( ω ) exp [ i k ( x ζω ) 2 2 q ] exp ( i k 0 βωx )
q ( z ) = ( z + d ) + i π w 2 λ = ( z + d ) + i k 0 w 2 2
E ( ω ) = E 0 exp ( ω 2 τ 0 2 4 ) exp ( i φ ( 2 ) 2 ω 2 )
q ( z ) q 0 ( d ) = i k w 2 2
E ( x , ω ) = E 0 exp ( ω 2 τ 0 2 4 ) exp [ ( x ζω ) 2 w 2 ] exp ( i φ ( 2 ) 2 ω 2 ) exp ( i k 0 βωx )
υ = ζ ζ 2 + w 2 τ 0 2 4
E ( x , ω ) = E 0 exp [ ( x w′ ) 2 ] exp ( ( τ′ ) 2 4 ( ω υx ) 2 ) exp ( i φ ( 2 ) 2 ω 2 ) exp ( i k 0 βωx )
E ( x , ω ) = E 0 exp [ ( x w′ ) 2 ] exp [ i ( k 0 β + φ ( 2 ) 2 ) υ x 2 ]
× exp [ ( τ′ ) 2 4 ( ω υx ) 2 ] exp [ i φ ( 2 ) 2 ( ω υx ) 2 ]
× exp [ i ( k 0 β + φ ( 2 ) υ ) x ( ω υx ) ]
E 0 ( x , t ) = f ( x ) exp [ ( t t 0 ) 2 τ 2 ] exp { i [ ϕ ( 1 ) ( t t 0 ) + ϕ ( 2 ) 2 ( t t 0 ) 2 ] }
f ( x ) = 1 π [ ( τ′ ) 2 + i 2 φ 2 ] 1 / 2 E 0 exp [ ( x w′ ) 2 ] exp ( i φ ( 2 ) 2 υ 2 x 2 )
t 0 = ( k 0 β + φ ( 2 ) υ ) x
τ = [ ( τ′ ) 2 + 4 ( φ ( 2 ) ) 2 ( τ′ ) 2 ] 1 / 2 = [ τ 0 2 + 4 ζ 2 w 2 + 4 ( φ ( 2 ) ) 2 τ 0 2 + 4 ζ 2 w 2 ] 1 / 2
ϕ ( 1 ) = υ x
ϕ ( 2 ) = ϕ ( 2 ) ( τ′ ) 2 4 + ( φ ( 2 ) ) 2 = φ ( 2 ) 1 4 ( τ 0 2 + 4 ζ 2 σ 2 ) 2 + ( φ ( 2 ) ) 2
p d t 0 d x
tan ψ = pc
p = p AD + p SC + TC
p AD = k 0 β
p SC + TC = φ ( 2 ) υ
E ˜ ( x , z , ω ) = E xz ( x , z ) E ˜ ω ( ω ) exp ( i pxω )
E ˜ ( x , z , ω ) = E xz ( x ζω , z ) E ˜ ω ( ω ) exp ( i pxω )
E ˜ ( x , z , ω ) = E xz ( x , z ) E ˜ ω ( ω υx ) exp ( i pxω )
E ( x , ω , z ) = i λz E ( x , ω , z = 0 ) exp [ i π λz ( x x′ ) 2 ] d x′
E ( x , ω , z = 0 ) = E ( ω , z = 0 ) exp [ i k 0 ( x ζ 0 ω ) 2 2 q 0 ] exp ( i k 0 βωx )
= E 0 exp ( ω 2 τ 0 2 4 ) exp ( i φ 0 ( 2 ) 2 ω 2 ) exp [ i k 0 ( x ζ 0 ω ) 2 2 q 0 ] exp ( i k 0 βωx )
E ( x , ω , z ) = i k 0 2 πz E 0 exp ( ω 2 τ 0 2 4 ) exp ( i φ 0 ( 2 ) 2 ω 2 )
× exp [ i k 0 ( x ) 2 2 q ( 0 ) ] exp [ i k 0 βω ( x′ + ζ 0 ω ) ] exp [ i k 0 2 z ( x′ + ζ 0 ω x ) 2 ] d x′
= [ i k 0 2 πz q ( 0 ) q ( z ) ] 1 / 2 E 0 exp ( ω 2 τ 0 2 4 ) exp ( i φ 0 ( 2 ) 2 ω 2 )
× exp { i k 0 z 2 q ( 0 ) q ( 0 ) ( x ζ 0 ω z βω ) 2 i k 0 [ β ζ 0 ω 2 + ( x ζ 0 ω ) 2 2 z ] }
q ( 0 ) q ( z ) = d + i k 0 w 2 2 z + d + i k 0 w 2 2 = 1 + i 2 z k 0 w 2
E ( x , ω , z ) = ( i k 0 2 πz ) 1 / 2 ( 1 + i 2 z k 0 w 2 ) 1 / 2 E 0 exp ( ω 0 τ 0 2 4 ) exp [ i 2 ( φ 0 ( 2 ) k 0 β 2 z ) ω 2 ]
× exp { [ x ( ζ 0 + βz ) ω ] 2 w 2 } exp ( i k 0 βωx )
ζ ( z ) = ζ 0 + βz
φ ( 2 ) ( z ) = φ 0 ( 2 ) k 0 β 2 z
υ ( z ) = ζ 0 + βz ( ζ 0 + βz ) 2 + w 2 τ 0 2 4
p = k 0 β + ( φ 0 ( 2 ) k 0 β 2 z ) υ ( z )
K = [ A B 0 E C D 0 F G H 1 I 0 0 0 1 ] = [ 1 L 0 2 πζ 0 1 0 0 0 2 πζ / λ 0 1 2 π φ ( 2 ) 0 0 0 1 ]
E ( x , t ) = exp { i π λ 0 ( x t ) T Q 1 ( x t ) } =
exp [ i π λ 0 ( ( Q 1 ) 11 x 2 + ( Q 1 ) 12 xt ( Q 1 ) 21 xt ( Q 1 ) 22 t 2 ) ]
E ( x , t ) exp [ ( t px ) 2 τ 2 ]
τ = [ π λ 0 Im { ( Q 1 ) 22 } ] 1 2
p = π τ 2 2 λ 0 Im { ( Q 1 ) 12 ( Q 1 ) 21 } = Im { ( Q 1 ) 12 ( Q 1 ) 21 } 2 Im { ( Q 1 ) 22 }
( Q in 1 ) 11 = 1 q
( Q in 1 ) 22 = i λ 0 π τ 0 2
Q in = [ q 0 0 i π τ 0 2 λ 0 ]
Q out = { [ A 0 G 1 ] Q in + [ B E / λ 0 H I / λ 0 ] } · { [ C 0 0 1 ] Q in + [ D F / λ 0 0 1 ] } 1
Q out = [ q + L 2 πζ λ 0 2 πζ λ 0 2 πφ ( 2 ) λ 0 i π τ 0 2 λ 0 ]
Q out 1 = [ λ 0 ( φ ( 2 ) i 2 τ 0 2 ) φ ( 2 ) ( L + q ) λ 0 + 2 π ζ 2 i 2 ( L + q ) λ 0 τ 0 2 λ 0 ζ φ ( 2 ) ( L + q ) λ 0 + 2 π ζ 2 i 2 ( L + q ) λ 0 τ 0 2 λ 0 ζ φ ( 2 ) ( L + q ) λ 0 + 2 π ζ 2 i 2 ( L + q ) λ 0 τ 0 2 1 2 π ( L + q ) λ 0 2 φ ( 2 ) ( L + q ) λ 0 + 2 π ζ 2 i 2 ( L + q ) λ 0 τ 0 2 ]
q ( L ) = L + q i π w 2 λ 0
Q out 1 = [ λ 0 π 2 φ ( 2 ) i τ 0 2 4 ζ 2 + w 2 τ 0 2 + i 2 φ ( 2 ) w 2 2 λ 0 π ζ 4 ζ 2 + w 2 τ 0 2 + i 2 φ ( 2 ) w 2 2 λ 0 π ζ 4 ζ 2 + w 2 τ 0 2 + i 2 φ ( 2 ) w 2 i λ 0 π w 2 4 ζ 2 + w 2 τ 0 2 + i 2 φ ( 2 ) w 2 ]
τ = [ τ 0 2 + 4 ζ 2 w 2 + 4 ( φ ( 2 ) ) 2 τ 0 2 + 4 ζ 2 w 2 ] 1 / 2
p = φ ( 2 ) ζ ζ 2 + 1 4 w 2 τ 0 2

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