Abstract

An efficient numerical algorithm is presented for the numerical modeling of the propagation of ultrashort pulses with arbitrary spatial and temporal characteristics through linear homogeneous dielectrics. The consequence of proper sampling of the spectral phase in pulse propagation and its influence on the efficiency of computation are discussed in detail. The numerical simulation presented here is capable of analyzing the pulse in the space–frequency domain and its on- and off-axis evolution in the space–time domain in any arbitrary plane. As an example, pulse propagation effects such as temporal and spectral shifts, pulse broadening effects, asymmetry, and chirping in dispersive media are demonstrated for axial and off-axial time evolution of a TEM01 pulse. Snapshots of dynamic pulse evolution are presented at various planes transverse to the direction of propagation. The pulse profiles at arbitrary points in the transverse planes are also obtained as a function of time.

© 2006 Optical Society of America

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References

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  1. I. P. Christov, "Propagation of femtosecond light pulses," Opt. Commun. 53, 364-366 (1985).
    [CrossRef]
  2. R. W. Ziolkowski and J. B. Judkins, "Propagation characteristics of ultrawide-bandwidth pulsed Gaussian beams," J. Opt. Soc. Am. A 9, 2021-2030 (1992).
    [CrossRef]
  3. C. J. R. Sheppard and X. Gan, "Free-space propagation of femtosecond light pulses," Opt. Commun. 133, 1-6 (1997).
    [CrossRef]
  4. G. P. Agrawal, "Spectrum-induced changes in diffraction of pulsed optical beams," Opt. Commun. 157, 52-56 (1998).
    [CrossRef]
  5. A. E. Kaplan, "Diffraction-induced transformation of near-cycle and subcycle pulses," J. Opt. Soc. Am. B 15, 951-956 (1998).
    [CrossRef]
  6. G. P. Agrawal, "Far-field diffraction of pulsed optical beams in dispersive media," Opt. Commun. 167, 15-22 (1999).
    [CrossRef]
  7. M. A. Porras, "Propagation of single-cycle pulse light beams in dispersive media," Phys. Rev. A 60, 5069-5073 (1999).
    [CrossRef]
  8. H. Ichikawa, "Analysis of femtosecond-order optical pulses diffracted by periodic structure," J. Opt. Soc. Am. A 16, 299-304 (1999).
    [CrossRef]
  9. R. Piestun and D. A. B. Miller, "Spatiotemporal control of ultrashort optical pulses by refractive-diffractive-dispersive structured optical elements," Opt. Lett. 26, 1373-1375 (2001).
    [CrossRef]
  10. 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]
  11. G. O. Mattei and M. A. Gil, "Spherical aberration in spatial and temporal transforming lenses of femtosecond laser pulses," Appl. Opt. 38, 1058-1064 (1999).
    [CrossRef]
  12. U. Fuchs, U. D. Zeitner, and A. Tünnermann, "Ultrashort pulse propagation in complex optical systems," Opt. Express 13, 3852-3861 (2005).
    [CrossRef] [PubMed]
  13. L. Sereda, A. Ferrari, and M. Bertolotti, "Diffraction of a time Gaussian-shaped pulsed plane wave from a slit," Pure Appl. Opt. 5, 349-353 (1996).
    [CrossRef]
  14. H.-E. Hwang and G.-H. Yang, "Far-field diffraction characteristics of a time-variant Gaussian pulsed beam propagating from a circular aperture," Opt. Eng. 41, 2719-2727 (2002).
    [CrossRef]
  15. Z. Liu and B. Lü, "Spectral shifts and spectral switches in diffraction of ultra-short pulsed beams passing through a circular aperture," Optik 115, 447-454 (2004).
    [CrossRef]
  16. H.-E. Hwang and G.-H. Yang, "Near-field diffraction characteristics of a time-dependent Gaussian pulsed beam propagating from a circular aperture," Opt. Eng. 42, 2719-2727 (2003).
  17. M. Lefrancois and S. F. Pereira, "Time evolution of the diffraction pattern of an ultrashort laser pulse," Opt. Express 11, 1114-1122 (2003).
    [CrossRef] [PubMed]
  18. F. Wyrowski and J. Turunen, "Wave-optical engineering," in International Trends in Applied Optics, A.H.Guenther, ed. (SPIE, 2002), pp. 471-491.
  19. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
    [CrossRef]
  20. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).
  21. R. Gonzales and P. Wintz, Digital Image Processing (Addison-Wesley, 1993).
  22. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  23. International Organization for Standardization, "Laser and laser-related equipment—test methods for laser beam parameters: beam width, divergence angle and beam propagation factor," ISO 11146:1999 (1999).
  24. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).
  25. S. P. Veetil, C. Vijayan, D. K. Sharma, H. Schimmel, and F. Wyrowski, "Diffraction induced space-time splitting effects in ultra short pulse propagation," J. Mod. Opt., doi: 10.1080/09500340600647339.

2005 (1)

2004 (1)

Z. Liu and B. Lü, "Spectral shifts and spectral switches in diffraction of ultra-short pulsed beams passing through a circular aperture," Optik 115, 447-454 (2004).
[CrossRef]

2003 (2)

H.-E. Hwang and G.-H. Yang, "Near-field diffraction characteristics of a time-dependent Gaussian pulsed beam propagating from a circular aperture," Opt. Eng. 42, 2719-2727 (2003).

M. Lefrancois and S. F. Pereira, "Time evolution of the diffraction pattern of an ultrashort laser pulse," Opt. Express 11, 1114-1122 (2003).
[CrossRef] [PubMed]

2002 (1)

H.-E. Hwang and G.-H. Yang, "Far-field diffraction characteristics of a time-variant Gaussian pulsed beam propagating from a circular aperture," Opt. Eng. 41, 2719-2727 (2002).
[CrossRef]

2001 (1)

1999 (4)

G. O. Mattei and M. A. Gil, "Spherical aberration in spatial and temporal transforming lenses of femtosecond laser pulses," Appl. Opt. 38, 1058-1064 (1999).
[CrossRef]

H. Ichikawa, "Analysis of femtosecond-order optical pulses diffracted by periodic structure," J. Opt. Soc. Am. A 16, 299-304 (1999).
[CrossRef]

G. P. Agrawal, "Far-field diffraction of pulsed optical beams in dispersive media," Opt. Commun. 167, 15-22 (1999).
[CrossRef]

M. A. Porras, "Propagation of single-cycle pulse light beams in dispersive media," Phys. Rev. A 60, 5069-5073 (1999).
[CrossRef]

1998 (2)

G. P. Agrawal, "Spectrum-induced changes in diffraction of pulsed optical beams," Opt. Commun. 157, 52-56 (1998).
[CrossRef]

A. E. Kaplan, "Diffraction-induced transformation of near-cycle and subcycle pulses," J. Opt. Soc. Am. B 15, 951-956 (1998).
[CrossRef]

1997 (1)

C. J. R. Sheppard and X. Gan, "Free-space propagation of femtosecond light pulses," Opt. Commun. 133, 1-6 (1997).
[CrossRef]

1996 (1)

L. Sereda, A. Ferrari, and M. Bertolotti, "Diffraction of a time Gaussian-shaped pulsed plane wave from a slit," Pure Appl. Opt. 5, 349-353 (1996).
[CrossRef]

1992 (2)

1985 (1)

I. P. Christov, "Propagation of femtosecond light pulses," Opt. Commun. 53, 364-366 (1985).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, "Far-field diffraction of pulsed optical beams in dispersive media," Opt. Commun. 167, 15-22 (1999).
[CrossRef]

G. P. Agrawal, "Spectrum-induced changes in diffraction of pulsed optical beams," Opt. Commun. 157, 52-56 (1998).
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

Bertolotti, M.

L. Sereda, A. Ferrari, and M. Bertolotti, "Diffraction of a time Gaussian-shaped pulsed plane wave from a slit," Pure Appl. Opt. 5, 349-353 (1996).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Christov, I. P.

I. P. Christov, "Propagation of femtosecond light pulses," Opt. Commun. 53, 364-366 (1985).
[CrossRef]

Ferrari, A.

L. Sereda, A. Ferrari, and M. Bertolotti, "Diffraction of a time Gaussian-shaped pulsed plane wave from a slit," Pure Appl. Opt. 5, 349-353 (1996).
[CrossRef]

Fuchs, U.

Gan, X.

C. J. R. Sheppard and X. Gan, "Free-space propagation of femtosecond light pulses," Opt. Commun. 133, 1-6 (1997).
[CrossRef]

Gil, M. A.

Gonzales, R.

R. Gonzales and P. Wintz, Digital Image Processing (Addison-Wesley, 1993).

Hwang, H.-E.

H.-E. Hwang and G.-H. Yang, "Near-field diffraction characteristics of a time-dependent Gaussian pulsed beam propagating from a circular aperture," Opt. Eng. 42, 2719-2727 (2003).

H.-E. Hwang and G.-H. Yang, "Far-field diffraction characteristics of a time-variant Gaussian pulsed beam propagating from a circular aperture," Opt. Eng. 41, 2719-2727 (2002).
[CrossRef]

Ichikawa, H.

Judkins, J. B.

Kaplan, A. E.

Kempe, M.

Lefrancois, M.

Liu, Z.

Z. Liu and B. Lü, "Spectral shifts and spectral switches in diffraction of ultra-short pulsed beams passing through a circular aperture," Optik 115, 447-454 (2004).
[CrossRef]

Lü, B.

Z. Liu and B. Lü, "Spectral shifts and spectral switches in diffraction of ultra-short pulsed beams passing through a circular aperture," Optik 115, 447-454 (2004).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Mattei, G. O.

Miller, D. A. B.

Pereira, S. F.

Piestun, R.

Porras, M. A.

M. A. Porras, "Propagation of single-cycle pulse light beams in dispersive media," Phys. Rev. A 60, 5069-5073 (1999).
[CrossRef]

Rudolph, W.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

Schimmel, H.

S. P. Veetil, C. Vijayan, D. K. Sharma, H. Schimmel, and F. Wyrowski, "Diffraction induced space-time splitting effects in ultra short pulse propagation," J. Mod. Opt., doi: 10.1080/09500340600647339.

Sereda, L.

L. Sereda, A. Ferrari, and M. Bertolotti, "Diffraction of a time Gaussian-shaped pulsed plane wave from a slit," Pure Appl. Opt. 5, 349-353 (1996).
[CrossRef]

Sharma, D. K.

S. P. Veetil, C. Vijayan, D. K. Sharma, H. Schimmel, and F. Wyrowski, "Diffraction induced space-time splitting effects in ultra short pulse propagation," J. Mod. Opt., doi: 10.1080/09500340600647339.

Sheppard, C. J. R.

C. J. R. Sheppard and X. Gan, "Free-space propagation of femtosecond light pulses," Opt. Commun. 133, 1-6 (1997).
[CrossRef]

Stamm, U.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

Tünnermann, A.

Turunen, J.

F. Wyrowski and J. Turunen, "Wave-optical engineering," in International Trends in Applied Optics, A.H.Guenther, ed. (SPIE, 2002), pp. 471-491.

Veetil, S. P.

S. P. Veetil, C. Vijayan, D. K. Sharma, H. Schimmel, and F. Wyrowski, "Diffraction induced space-time splitting effects in ultra short pulse propagation," J. Mod. Opt., doi: 10.1080/09500340600647339.

Vijayan, C.

S. P. Veetil, C. Vijayan, D. K. Sharma, H. Schimmel, and F. Wyrowski, "Diffraction induced space-time splitting effects in ultra short pulse propagation," J. Mod. Opt., doi: 10.1080/09500340600647339.

Wilhelmi, B.

Wintz, P.

R. Gonzales and P. Wintz, Digital Image Processing (Addison-Wesley, 1993).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Wyrowski, F.

F. Wyrowski and J. Turunen, "Wave-optical engineering," in International Trends in Applied Optics, A.H.Guenther, ed. (SPIE, 2002), pp. 471-491.

S. P. Veetil, C. Vijayan, D. K. Sharma, H. Schimmel, and F. Wyrowski, "Diffraction induced space-time splitting effects in ultra short pulse propagation," J. Mod. Opt., doi: 10.1080/09500340600647339.

Yang, G.-H.

H.-E. Hwang and G.-H. Yang, "Near-field diffraction characteristics of a time-dependent Gaussian pulsed beam propagating from a circular aperture," Opt. Eng. 42, 2719-2727 (2003).

H.-E. Hwang and G.-H. Yang, "Far-field diffraction characteristics of a time-variant Gaussian pulsed beam propagating from a circular aperture," Opt. Eng. 41, 2719-2727 (2002).
[CrossRef]

Zeitner, U. D.

Ziolkowski, R. W.

Appl. Opt. (1)

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (2)

Opt. Commun. (4)

I. P. Christov, "Propagation of femtosecond light pulses," Opt. Commun. 53, 364-366 (1985).
[CrossRef]

C. J. R. Sheppard and X. Gan, "Free-space propagation of femtosecond light pulses," Opt. Commun. 133, 1-6 (1997).
[CrossRef]

G. P. Agrawal, "Spectrum-induced changes in diffraction of pulsed optical beams," Opt. Commun. 157, 52-56 (1998).
[CrossRef]

G. P. Agrawal, "Far-field diffraction of pulsed optical beams in dispersive media," Opt. Commun. 167, 15-22 (1999).
[CrossRef]

Opt. Eng. (2)

H.-E. Hwang and G.-H. Yang, "Far-field diffraction characteristics of a time-variant Gaussian pulsed beam propagating from a circular aperture," Opt. Eng. 41, 2719-2727 (2002).
[CrossRef]

H.-E. Hwang and G.-H. Yang, "Near-field diffraction characteristics of a time-dependent Gaussian pulsed beam propagating from a circular aperture," Opt. Eng. 42, 2719-2727 (2003).

Opt. Express (2)

Opt. Lett. (1)

Optik (1)

Z. Liu and B. Lü, "Spectral shifts and spectral switches in diffraction of ultra-short pulsed beams passing through a circular aperture," Optik 115, 447-454 (2004).
[CrossRef]

Phys. Rev. A (1)

M. A. Porras, "Propagation of single-cycle pulse light beams in dispersive media," Phys. Rev. A 60, 5069-5073 (1999).
[CrossRef]

Pure Appl. Opt. (1)

L. Sereda, A. Ferrari, and M. Bertolotti, "Diffraction of a time Gaussian-shaped pulsed plane wave from a slit," Pure Appl. Opt. 5, 349-353 (1996).
[CrossRef]

Other (8)

F. Wyrowski and J. Turunen, "Wave-optical engineering," in International Trends in Applied Optics, A.H.Guenther, ed. (SPIE, 2002), pp. 471-491.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

R. Gonzales and P. Wintz, Digital Image Processing (Addison-Wesley, 1993).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

International Organization for Standardization, "Laser and laser-related equipment—test methods for laser beam parameters: beam width, divergence angle and beam propagation factor," ISO 11146:1999 (1999).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

S. P. Veetil, C. Vijayan, D. K. Sharma, H. Schimmel, and F. Wyrowski, "Diffraction induced space-time splitting effects in ultra short pulse propagation," J. Mod. Opt., doi: 10.1080/09500340600647339.

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Figures (7)

Fig. 1
Fig. 1

(a) Spectral and (b) pulse shapes at on-axis ( x = 0 , y = 6 μ m ) and off-axis ( x = 0 , y = 500 μ m ) locations in the initial plane of the pulse. The central vertical line in (a) shows the spectral maximum at 800 nm ( 3.75 × 10 5 GHz ) . Pulse width remains the same.

Fig. 2
Fig. 2

On-axis ( x = 0 , y = 40 μ m ) and off-axis ( x = 0 , y = 900 μ m ) (a) spectral and (b) pulse shapes on propagation to a distance of 500 mm . On-axis spectra show a blueshift to 782 nm ( 3.8 × 10 5 GHz ) and off-axis spectra show a redshift to 830 nm ( 3.6 × 10 5 GHz ) .

Fig. 3
Fig. 3

On-axis ( x = 0 , y = 400 μ m ) and off-axis ( x = 0 , y = 9.4 mm ) (a) spectral and (b) pulse shapes on propagation to a distance of 5 m . On-axis spectra are blueshifted to 769 nm ( 3.8 × 10 5 GHz ) and off-axis spectra are redshifted to 861 nm ( 3.48 × 10 5 GHz ) .

Fig. 4
Fig. 4

Transverse profile of the pulse in space and time in the (a) initial plane, at (b) 500 mm , (c) 1 m , (d) 5 m . The pulse front curvature increases on increasing the propagation distance since the off-axis pulse arrives much later in the spatially broadened pulse.

Fig. 5
Fig. 5

Snapshots of three-dimensional detection of an ultrashort pulse in the observation plane in the far field at different instants of time. (a) t = 0 fs , (b) t = 0.8 fs , (c) t = 3.2 fs , (d) t = 5.2 fs , (e) t = 6.5 fs . The gray scale on the right side of each graph shows the amplitude at that instant in time.

Fig. 6
Fig. 6

(a) Spectral and (b) pulse shapes on propagation to a distance 500 mm through dispersive media. On-axis spectra at ( x = 0 , y = 10 μ m ) show a blueshift to 781 nm ( 3.83 × 10 5 GHz ) and off-axis spectra at ( x = 0 , y = 800 μ m ) show a redshift to 854 nm ( 3.5 × 10 5 GHz ) . The off-axis pulse width is slightly wider than the on-axis pulse width, and the pulse shape becomes asymmetric.

Fig. 7
Fig. 7

(a) Transverse profile of the pulse in the plane at 500 mm in a dispersive medium. The pulse has broadened in time and the trailing edge of the pulse shows that the pulse is asymmetric. (b) The variation of carrier frequency with time—a positive chirp developed on propagation through the dispersive medium.

Equations (45)

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

U ( x , y , z , t ) = u ( x , y , z ) P ( t ) exp [ j ω 0 t ] .
U ̃ ( x , y , z , ω ) = FT { U ( x , y , z , t ) } ,
U ̃ ( x , y , z , ω i ) = { U ̃ 1 ( x , y , z , ω 1 ) , U ̃ 2 ( x , y , z , ω 2 ) , , , U ̃ n ( x , y , z , ω I ) } .
U ( x , y , z , t i ) = IFT U ̃ ( x , y , z , ω i ) ,
U ̃ ( x , y , z , ω i ) out = P U ̃ ( x , y , z , ω i ) in
= { P U ̃ 1 ( x , y , z , ω 1 ) , P U ̃ 2 ( x , y , z , ω 2 ) , , , , P U ̃ n ( x , y , z , ω i ) } ,
U ̃ ( x , y , z , ω ) = A ( k x , k y , 0 , ω ) exp [ j k z z ] exp [ j ( k x x + k y y ) ] d k x d k y ,
k z = [ k 2 ( k x 2 + k y 2 ) ] 1 2 .
U ( x , y , z , t ) = 1 2 π A ( k x , k y , 0 , ω ) exp [ j { k 2 ( k x 2 + k y 2 ) } 1 2 z ] × exp [ j ( k x x + k y y ) ] exp [ j ω t ] d ω d k x d k y .
[ k 2 ( k x 2 + k y 2 ) ] 1 2 z = ω c [ 1 ( k x 2 + k y 2 ) k 2 ] 1 2 z ,
t = t + 1 c [ 1 ( k x 2 + k y 2 ) k 2 ] 1 2 z = t + t prop
δ ϕ = ω c [ 1 ( k x 2 + k y 2 ) k 2 ] 1 2 z ω c z ,
δ t = z c [ 1 ( k x 2 + k y 2 ) k 2 ] 1 2 z c .
U ̃ ( x , y , z , ω ) 2 π j k ( z r ) A ( k x , k y , 0 , ω ) exp [ j k r ] r ,
ϕ ( r , ω ) = k r = ω c r .
r = [ x 2 + y 2 + z 2 ] 1 2 .
r = [ x 2 + y 2 + R 2 ( z ) ] 1 2 .
ϕ ( x , y , z , ω ) = ω c z + ω c [ x 2 + y 2 + R 2 ( ω , z ) ] 1 2 ω c R ( ω , z ) ,
= ω c z + ω c D ( x , y , z , ω ) .
ϕ ( x , y , z , ω ) = ω c z ,
ϕ ( x , y , z , ω ) = ω c [ x 2 + y 2 + R 2 ( z ) ] 1 2 .
ϕ ( x , y , z , ω ) = ϕ L ( x , y , z , ω ) + ϕ ̂ ( x , y , z , ω ) ,
ϕ L ( x , y , z , ω ) = ω c z + ω c D ( x , y , z , ω 0 ) .
U ( x , y , z , t ) = 1 2 π 2 π j k ( z r 2 ) A ( k x , k y , 0 , ω ) exp [ j ϕ L ( x , y , z , ω ) ] × exp [ j ϕ ̂ ( x , y , z , ω ) ] exp [ j ω t ] d ω ,
= F ( ω ) exp [ j ( z c + D ( x , y , z , ω 0 ) c + t ) ω ] d ω ,
t = t + z c + D ( x , y , z , ω 0 ) c .
δ ϕ = ϕ L 2 ( 0 , 0 , z , ω 2 ) ϕ L 1 ( 0 , 0 , z , ω 1 ) = ( ω 2 ω 1 ) z c = δ ω z c = π .
I = Δ ω δ ω .
R ( z , ω ) = z [ 1 + ( π W 0 2 z λ M 2 ) 2 ] ,
U ̃ ( x , y , z , ω i ) out = U ̃ ( x , y , z , ω i ) exp [ j ω c z ] exp [ j ω c D ( x , y , z , ω 0 ) ] out .
U ̃ ( x , y , z , ω i ) out = U ̃ ( x , y , z , ω i ) exp [ j ω c D ( x , y , z , ω 0 ) ] out .
U ( x , y , z , t i ) out = IFT { U ̃ ( x , y , z , ω i ) exp [ j ω c D ( x , y , z , ω 0 ) ] out } .
t i = t + D ( x , y , z , ω 0 ) c ,
ϕ L ( x , y , z , ω ) = ω c n ( ω ) z + ω c n ( ω ) D ( x , y , z , ω 0 ) = ω c n ( ω ) z 0 ,
ϕ L ( x , y , z , ω ) = ϕ 0 + ( ω ω 0 ) ϕ L ω = ω 0 + ( ω ω 0 ) 2 2 ϕ L ω = ω 0 + ( ω ω 0 ) 3 6 ϕ L ω = ω 0 + ,
ϕ L = z 0 v g ( ω 0 ) = z 0 c [ n ( λ 0 ) λ 0 d n d λ ] .
t = t + ϕ L = t + z 0 v g ( ω 0 ) .
( ω 2 ω 0 ) 2 k z ( ω 1 ω 0 ) 2 k z = π ,
δ ω = ( ω 1 ω 0 ) ± [ ( ω 1 ω 0 ) 2 + π k z ] 1 2 ,
U ̃ ( x , y , z , ω i ) out = U ̃ ( x , y , z , ω i ) exp [ j ω ϕ L ( x , y , z , ω ) ] out .
ϕ R ( x , y , z , ω ) = ϕ L ( x , y , z , ω ) ϕ 0 ϕ L .
U ̃ ( x , y , z , ω k ) out = U ̃ ( x , y , z , ω k ) exp [ j ω ϕ R ( x , y , z , ω ) ] out .
U ̃ ( x , y , z , ω k ) out = U ̃ ( x , y , z , ω k ) exp [ j ω c D ( x , y , z , ω 0 ) ] out ,
U ( x , y , z , t k ) out = IFT { U ̃ ( x , y , z , ω k ) exp [ j ω c D ( x , y , z , ω 0 ) ] out } .
t k = t + D ( x , y , z , ω 0 ) c ,

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