Abstract

Under certain conditions, ultrashort pulse dynamics in a linear dispersive medium with absorption result in the appearance of optical precursors that dominate the pulse evolution for large propagation distances as the peak amplitude in the initial pulse spectrum decays exponentially. The effects of a nonlinear medium response on this precursor formation is considered using the split-step Fourier method. Comparison of the nonlinear pulse evolution when the full dispersion is used to that when a quadratic Taylor series approximation of the wave number is used shows that the group velocity approximation misses the precursor fields entirely.

© 2010 Optical Society of America

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References

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  1. A. Sommerfeld, “Über die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys. (Leipzig)  44, 4, 177–202 (1914).
  2. L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys. (Leipzig)  44, 4, 177–202 (1914).
  3. L. Brillouin, Wave Propagation and Group Velocity (Academic Press, New York, 1960).
  4. P. Debye, “Näherungsformeln für die zylinderfunktionen für grosse werte des arguments und unbeschränkt verander liche werte des index,” Math. Ann. 67, 535–558 (1909).
    [CrossRef]
  5. J. A. Stratton, Electromagnetic Theory, (McGraw-Hill, New York, 1941), §5.18.
  6. K. E. Oughstun, and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, Berlin-Heidelberg, 1994).
  7. K. E. Oughstun, Electromagnetic and Optical Pulse Propagation1: Spectral Representations in Temporally Dispersive Media (Springer, New York, 2006).
    [PubMed]
  8. K. E. Oughstun, Electromagnetic and Optical Pulse Propagation2: Temporal Pulse Dynamics in Dispersive, Attenuative Media (Springer, New York, 2009).
  9. K. E. Oughstun, and G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5(4), 817–849 (1988).
    [CrossRef]
  10. N. A. Cartwright, and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Rev. 49(4), 628–648 (2007).
    [CrossRef]
  11. K. E. Oughstun, and C. M. Balictsis, “Gaussian Pulse Propagation in a Dispersive, Absorbing Dielectric,” Phys. Rev. Lett. 77(11), 2210–2213 (1996).
    [CrossRef]
  12. T. H. Havelock, “The propagation of groups of waves in dispersive media,” Proc. R. Soc. London, Ser. A LXXXI, 398 (1908).
    [CrossRef]
  13. T. H. Havelock, The Propagation of Disturbances in Dispersive Media (Cambridge U. Press, Cambridge, 1914).
  14. L. Kelvin, “On the waves produced by a single impulse in water of any depth, or in a dispersive medium,” Proc. Roy. Soc., London XLII, 80 (1887).
  15. K. E. Oughstun, and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78(4), 642–645 (1997).
    [CrossRef]
  16. H. Xiao, and K. E. Oughstun, “Failure of the group velocity description for ultrawideband pulse propagation in a double resonance Lorentz model dielectric,” J. Opt. Soc. Am. B 16(10), 1773–1785 (1999).
    [CrossRef]
  17. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 1989).
  18. R. H. Hardin, and F. D. Tappert, “Applications of the split-step Fourier method to the solution of nonlinear and variable coefficient wave equations,” SIAM Rev. 15(2), 423 (1973).
  19. T. A. Laine, and A. T. Friberg, “Self-guided waves and exact solutions of the nonlinear Helmholtz equation,” J. Opt. Soc. Am. B 17(5), 751–757 (2000).
    [CrossRef]

2007

N. A. Cartwright, and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Rev. 49(4), 628–648 (2007).
[CrossRef]

2000

1999

1997

K. E. Oughstun, and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78(4), 642–645 (1997).
[CrossRef]

1996

K. E. Oughstun, and C. M. Balictsis, “Gaussian Pulse Propagation in a Dispersive, Absorbing Dielectric,” Phys. Rev. Lett. 77(11), 2210–2213 (1996).
[CrossRef]

1988

1973

R. H. Hardin, and F. D. Tappert, “Applications of the split-step Fourier method to the solution of nonlinear and variable coefficient wave equations,” SIAM Rev. 15(2), 423 (1973).

1914

A. Sommerfeld, “Über die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys. (Leipzig)  44, 4, 177–202 (1914).

L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys. (Leipzig)  44, 4, 177–202 (1914).

1909

P. Debye, “Näherungsformeln für die zylinderfunktionen für grosse werte des arguments und unbeschränkt verander liche werte des index,” Math. Ann. 67, 535–558 (1909).
[CrossRef]

1908

T. H. Havelock, “The propagation of groups of waves in dispersive media,” Proc. R. Soc. London, Ser. A LXXXI, 398 (1908).
[CrossRef]

1887

L. Kelvin, “On the waves produced by a single impulse in water of any depth, or in a dispersive medium,” Proc. Roy. Soc., London XLII, 80 (1887).

Balictsis, C. M.

K. E. Oughstun, and C. M. Balictsis, “Gaussian Pulse Propagation in a Dispersive, Absorbing Dielectric,” Phys. Rev. Lett. 77(11), 2210–2213 (1996).
[CrossRef]

Brillouin, L.

L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys. (Leipzig)  44, 4, 177–202 (1914).

Cartwright, N. A.

N. A. Cartwright, and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Rev. 49(4), 628–648 (2007).
[CrossRef]

Debye, P.

P. Debye, “Näherungsformeln für die zylinderfunktionen für grosse werte des arguments und unbeschränkt verander liche werte des index,” Math. Ann. 67, 535–558 (1909).
[CrossRef]

Friberg, A. T.

Hardin, R. H.

R. H. Hardin, and F. D. Tappert, “Applications of the split-step Fourier method to the solution of nonlinear and variable coefficient wave equations,” SIAM Rev. 15(2), 423 (1973).

Havelock, T. H.

T. H. Havelock, “The propagation of groups of waves in dispersive media,” Proc. R. Soc. London, Ser. A LXXXI, 398 (1908).
[CrossRef]

Kelvin, L.

L. Kelvin, “On the waves produced by a single impulse in water of any depth, or in a dispersive medium,” Proc. Roy. Soc., London XLII, 80 (1887).

Laine, T. A.

Oughstun, K. E.

N. A. Cartwright, and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Rev. 49(4), 628–648 (2007).
[CrossRef]

H. Xiao, and K. E. Oughstun, “Failure of the group velocity description for ultrawideband pulse propagation in a double resonance Lorentz model dielectric,” J. Opt. Soc. Am. B 16(10), 1773–1785 (1999).
[CrossRef]

K. E. Oughstun, and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78(4), 642–645 (1997).
[CrossRef]

K. E. Oughstun, and C. M. Balictsis, “Gaussian Pulse Propagation in a Dispersive, Absorbing Dielectric,” Phys. Rev. Lett. 77(11), 2210–2213 (1996).
[CrossRef]

K. E. Oughstun, and G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5(4), 817–849 (1988).
[CrossRef]

Sherman, G. C.

Sommerfeld, A.

A. Sommerfeld, “Über die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys. (Leipzig)  44, 4, 177–202 (1914).

Tappert, F. D.

R. H. Hardin, and F. D. Tappert, “Applications of the split-step Fourier method to the solution of nonlinear and variable coefficient wave equations,” SIAM Rev. 15(2), 423 (1973).

Xiao, H.

H. Xiao, and K. E. Oughstun, “Failure of the group velocity description for ultrawideband pulse propagation in a double resonance Lorentz model dielectric,” J. Opt. Soc. Am. B 16(10), 1773–1785 (1999).
[CrossRef]

K. E. Oughstun, and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78(4), 642–645 (1997).
[CrossRef]

Ann. Phys.

A. Sommerfeld, “Über die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys. (Leipzig)  44, 4, 177–202 (1914).

L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys. (Leipzig)  44, 4, 177–202 (1914).

J. Opt. Soc. Am. B

Math. Ann.

P. Debye, “Näherungsformeln für die zylinderfunktionen für grosse werte des arguments und unbeschränkt verander liche werte des index,” Math. Ann. 67, 535–558 (1909).
[CrossRef]

Phys. Rev. Lett.

K. E. Oughstun, and C. M. Balictsis, “Gaussian Pulse Propagation in a Dispersive, Absorbing Dielectric,” Phys. Rev. Lett. 77(11), 2210–2213 (1996).
[CrossRef]

K. E. Oughstun, and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78(4), 642–645 (1997).
[CrossRef]

Proc. R. Soc. London, Ser. A

T. H. Havelock, “The propagation of groups of waves in dispersive media,” Proc. R. Soc. London, Ser. A LXXXI, 398 (1908).
[CrossRef]

Proc. Roy. Soc., London

L. Kelvin, “On the waves produced by a single impulse in water of any depth, or in a dispersive medium,” Proc. Roy. Soc., London XLII, 80 (1887).

SIAM Rev.

N. A. Cartwright, and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Rev. 49(4), 628–648 (2007).
[CrossRef]

R. H. Hardin, and F. D. Tappert, “Applications of the split-step Fourier method to the solution of nonlinear and variable coefficient wave equations,” SIAM Rev. 15(2), 423 (1973).

Other

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 1989).

T. H. Havelock, The Propagation of Disturbances in Dispersive Media (Cambridge U. Press, Cambridge, 1914).

L. Brillouin, Wave Propagation and Group Velocity (Academic Press, New York, 1960).

J. A. Stratton, Electromagnetic Theory, (McGraw-Hill, New York, 1941), §5.18.

K. E. Oughstun, and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, Berlin-Heidelberg, 1994).

K. E. Oughstun, Electromagnetic and Optical Pulse Propagation1: Spectral Representations in Temporally Dispersive Media (Springer, New York, 2006).
[PubMed]

K. E. Oughstun, Electromagnetic and Optical Pulse Propagation2: Temporal Pulse Dynamics in Dispersive, Attenuative Media (Springer, New York, 2009).

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Figures (19)

Fig. 1
Fig. 1

Exact (blue curves) and quadratic Taylor series approximation (green curves) of the scaled propagation factor (c/ω)β(ω) (upper plot) and the attenuation coefficient α(ω) of the complex wave number for a single resonance Lorentz model dielectric with ω0 = 2.4 × 1015r/s, δ = 6.0 × 1013r/s, ωp = 3.05 × 1012r/s.

Fig. 2
Fig. 2

Heaviside step-function signal response of the dispersive material at 5 absorption depths with (green curve) and without (blue curve) the cubic nonlinearity included.

Fig. 3
Fig. 3

Propagated spectra of the Heaviside step function signal illustrated in Fig. 2 for the linear (solid blue curve) and nonlinear (solid green curve) dispersion cases.

Fig. 4
Fig. 4

Comparison of the spectral magnitude |ũ(ω)| of the gaussian pulse envelope for the ultrawideband 2τ0 ≃ 3.14 fs (magenta), the wideband 2τ0 ≃ 18.8 fs (red), and the narrow-band 2τ0 ≃ 628 fs (blue) pulse cases. The linear material phase dispersion (green curve) and ultrawideband spectrum |ωωc|–1 (dashed curve) plots are included for reference.

Fig. 5
Fig. 5

Comparison of the propagated linear (blue curve) and nonlinear (green curve) gaussian pulses at 5 absorption depths (z = 5zd) in a single resonance Lorentz-model dielectric with below resonance carrier frequency ωc = 0.416̄ω0.

Fig. 6
Fig. 6

Comparison of the nonlinear gaussian pulse distortion using the full (blue curve) and quadratic approximation (green curve) of the linear material dispersion at 5 absorption depths for the Nosc = 0.5 gaussian envelope case (2τ0 ≃ 3.14 fs) with τ0/τc ≃ 0.11.

Fig. 7
Fig. 7

Comparison of the nonlinear gaussian pulse distortion using the full (blue curve) and quadratic approximation (green curve) of the linear material dispersion at 5 absorption depths for the Nosc = 3 gaussian envelope case (2τ0 ≃ 18.8 fs) with τ0/τc ≃ 0.66.

Fig. 8
Fig. 8

Comparison of the nonlinear gaussian pulse distortion using the full (blue curve) and quadratic approximation (green curve) of the linear material dispersion at 5 absorption depths for the Nosc = 5 gaussian envelope case (2τ0 ≃ 31.4 fs) with τ0/τc ≃ 1.10.

Fig. 9
Fig. 9

Comparison of the nonlinear gaussian pulse distortion using the full (blue curve) and quadratic approximation (green curve) of the linear material dispersion at 5 absorption depths for the Nosc = 10 gaussian envelope case (2τ0 ≃ 62.8 fs) with τ0/τc ≃ 2.20.

Fig. 10
Fig. 10

Comparison of the nonlinear gaussian pulse distortion using the full (blue curve) and quadratic approximation (green curve) of the linear material dispersion at 5 absorption depths for the Nosc = 15 gaussian envelope case (2τ0 ≃ 94.3 fs) with τ0/τc ≃ 3.30.

Fig. 11
Fig. 11

Comparison of the nonlinear gaussian pulse distortion using the full (blue curve) and quadratic approximation (green curve) of the linear material dispersion at 5 absorption depths for the Nosc = 20 gaussian envelope case (2τ0 ≃ 125.7 fs) with τ0/τc ≃ 4.39.

Fig. 12
Fig. 12

Comparison of the nonlinear gaussian pulse distortion using the full (blue curve) and quadratic approximation (green curve) of the linear material dispersion at 5 absorption depths for the Nosc = 100 gaussian envelope case (2τ0 ≃ 628.3 fs) with τ0/τc ≃ 22.0.

Fig. 13
Fig. 13

Filtered and unfiltered spectra for the initial 100 oscillation gaussian envelope pulse.

Fig. 14
Fig. 14

Initial nonlinear space-time evolution of the envelope of a 3 oscillation gaussian pulse with initial pulse width 2τ0 ≃ 18.8 fs using the full material dispersion. The time axis spans the interval Δt = 125 fs.

Fig. 15
Fig. 15

Initial nonlinear space-time evolution of the envelope of a 3 oscillation gaussian pulse with initial pulse width 2τ0 ≃ 18.8 fs using the quadratic approximation of the linear material dispersion. The spurious side lobes are due to the manner in which the envelope is numerically constructed from the pulse shape.

Fig. 16
Fig. 16

Nonlinear space-time evolution of the envelope of a 3 oscillation gaussian pulse with initial pulse width 2τ0 ≃ 18.8 fs using the full material dispersion. The time axis spans the interval Δt = 125 fs.

Fig. 17
Fig. 17

Nonlinear space-time evolution of the envelope of a 3 oscillation gaussian pulse with initial pulse width 2τ0 ≃ 18.8 fs using the quadratic approximation of the linear material dispersion. The time axis spans the interval Δt = 125 fs.

Fig. 18
Fig. 18

Peak amplitude decay of a 3 oscillation gaussian pulse with initial pulse width 2τ0 ≃ 18.8 fs using the full material dispersion (blue curve) and using the quadratic approximation of the linear material dispersion (green curve). The black dashed line describes the Beer’s law exponential decay limit ez/zd for comparison.

Fig. 19
Fig. 19

Comparison of the nonlinear hyperbolic secant pulse distortion using the full (blue curve) and quadratic approximation (green curve) of the linear material dispersion at 5 absorption depths for the Nosc = 3 envelope case (2τ0 ≃ 18.8 fs) with τ0/τc ≃ 0.66.

Equations (19)

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

E ( z , t ) = 1 2 π i a i a + f ˜ ( ω ) e z c ϕ ( ω , θ ) d ω ,
( 2 + k ˜ 2 ( ω ) ) E ˜ ( z , ω ) = 0
ϕ ( ω , θ ) i c z [ k ˜ ( ω ) z ω t ] = i ω [ n ( ω ) θ ]
n ( ω ) = ( 1 ω p 2 ω 2 ω 0 2 + 2 i δ ω ) 1 / 2 ,
E ( z , t ) = E S ( z , t ) + E B ( z , t ) + E c ( z , t ) ,
k ˜ ( ω ) k ˜ ( ω c ) + k ˜ ( ω c ) ( ω ω c ) + ( 1 / 2 ) k ˜ ( ω c ) ( ω ω c ) 2 ,
2 E ( r , t ) 1 c 2 2 E ( r , t ) t 2 = 4 π c 2 2 P ( r , t ) t 2
2 E ω c ( r , t ) n 2 ( ω c ) c 2 2 E ω c ( r , t ) t 2 = 0 .
2 E ω c ( r , t ) n 2 ( ω c ) c 2 2 E ω c ( r , t ) t 2 + γ | E ω c ( r , t ) | 2 E ω c ( r , t ) = 0 ,
2 E ω c ( r , t ) n 2 ( ω c ) c 2 2 E ω c ( r , t ) t 2 = 0 ,
2 E NL ( r , t ) + γ | E NL ( r , t ) | 2 E N L ( r , t ) = 0 ,
( 2 + k ˜ 2 ( ω ) ) E ˜ D ( r , ω ) = 0 ,
E NL ( z + h , t ) = e ι γ | E NL ( z , t ) | h E NL ( z , t ) ,
E ˜ D ( z + h , ω ) = e ι k ˜ ( ω ) h E ˜ D ( z , ω ) ,
E g ( 0 , t ) = exp ( t 2 / τ 0 2 ) sin ( ω c t + φ ) ,
u ( t ) = e ( t 2 / τ 0 2 )
u ( t ) = 2 t τ 0 2 e ( t 2 / τ 0 2 ) , u ( t ) = 2 τ 0 2 e ( t 2 / τ 0 2 ) ( 2 t 2 τ 0 2 1 ) .
τ c = δ 1 2 / e 1.43 × 10 14 s ,
E sech ( 0 , t ) = sech ( 2 t / τ 0 ) cos ( ω c t )

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