Abstract

I apply the method of characteristics to both bidirectional and unidirectional pulse propagation in dispersionless media containing nonlinearity of arbitrary order. The differing analytic predictions for the shocking distance quantify the effects of the unidirectional approximation used in many pulse propagation models. Results from numerical simulations support the theoretical predictions and reveal the nature of the coupling between forward and backward waves.

© 2007 Optical Society of America

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  1. T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
    [CrossRef]
  2. P. Kinsler and G. H. C. New, "Few cycle pulse propagation," Phys. Rev. A 67, 023813 (2003).
    [CrossRef]
  3. P. Kinsler, S. B. P. Radnor, and G. H. C. New, "Theory of directional pulse propagation," Phys. Rev. A 72, 063807 (2005).
    [CrossRef]
  4. M. Kolesik, J. V. Moloney, and M. Mlejnek, "Unidirectional optical pulse propagation equation," Phys. Rev. Lett. 89, 283902 (2002).
    [CrossRef]
  5. J. C. A. Tyrrell, P. Kinsler, and G. H. C. New, "Pseudospectral spatial-domain: A new method for nonlinear pulse propagation in the few-cycle regime with arbitrary dispersion," J. Mod. Opt. 52, 973-986 (2005).
    [CrossRef]
  6. K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
    [CrossRef]
  7. A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and Á. Montero, "Forward-backward equations for nonlinear propagation in axially invariant optical systems," Phys. Rev. E 71, 016601 (2005).
    [CrossRef]
  8. G. Genty, P. Kinsler, B. Kibler, and J. M. Dudley, "Nonlinear envelope equation modeling of sub-cycle dynamics and harmonic generation in nonlinear waveguides," Opt. Express 15, 5382-5387 (2007).
    [CrossRef] [PubMed]
  9. P. Kinsler, "Theory of directional pulse propagation: detailed calculations," arXiv:physics/0611216.
  10. If the forward field has a wave vector k0 and evolves as exp(+ik0z), the generated backward component will evolve as exp(−ik0z). This gives a very rapid relative oscillation exp(−2ik0z), which will quickly average to zero.
  11. S. E. Harris, "Proposed backward wave oscillation in the infrared," Appl. Phys. Lett. 9, 114-116 (1966).
    [CrossRef]
  12. J. U. Kang, Y. J. Ding, W. K. Burns, and J. S. Melinger, "Backward second-harmonic generation in periodically poled bulk LiNbO3," Opt. Lett. 22, 862-864 (1997).
    [CrossRef] [PubMed]
  13. Y. J. Ding and J. B. Khurgin, "Backward optical parametric oscillators and amplifiers," IEEE J. Quantum Electron. 32, 1574-1582 (1996).
    [CrossRef]
  14. Y. J. Ding, J. U. Kang, and J. B. Khurgin, "Theory of backward second-harmonic and third-harmonic generation using laser pulses in quasi-phase-matched second-order nonlinear medium," IEEE J. Quantum Electron. 34, 966-974 (1998).
    [CrossRef]
  15. J. Z. Sanborn, C. Hellings, and T. D. Donnelly, "Breakdown of the slowly-varying-amplitude approximation: generation of backward-traveling, second-harmonic light," J. Opt. Soc. Am. B 20, 152-157 (2003).
    [CrossRef]
  16. G. Rosen, "Electromagnetic shocks and the self-annihilation of intense linearly polarized radiation in an ideal dielectric material," Phys. Rev. A 139, A539-A543 (1965).
  17. G. B. Whitham, Lectures on Wave Propagation (Wiley, 1979).
  18. P. Kinsler, S. B. P. Radnor, J. C. A. Tyrrell, and G. H. C. New, "Optical carrier wave shocking: detection and dispersion," Phys. Rev. E 75, 066603 (2007).
    [CrossRef]
  19. S. B. P. Radnor (Department of Physics, Imperial College London, London, personal communication, 2006).
  20. S. B. P. Radnor, L. E. Chipperfield, P. Kinsler, and G. H. C. New, "Carrier wave self-steepening and application to high harmonic generation," submitted to Phys. Rev. A .
  21. L. W. Casperson, "Field-equation approximations and amplification in high-gain lasers: numerical results," Phys. Rev. A 44, 3291-3304 (1991).
    [CrossRef] [PubMed]
  22. P. Kinsler, "Pulse propagation methods in nonlinear optics," arXiv:0707.0982.
  23. K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).
    [CrossRef]

2007 (2)

G. Genty, P. Kinsler, B. Kibler, and J. M. Dudley, "Nonlinear envelope equation modeling of sub-cycle dynamics and harmonic generation in nonlinear waveguides," Opt. Express 15, 5382-5387 (2007).
[CrossRef] [PubMed]

P. Kinsler, S. B. P. Radnor, J. C. A. Tyrrell, and G. H. C. New, "Optical carrier wave shocking: detection and dispersion," Phys. Rev. E 75, 066603 (2007).
[CrossRef]

2005 (3)

A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and Á. Montero, "Forward-backward equations for nonlinear propagation in axially invariant optical systems," Phys. Rev. E 71, 016601 (2005).
[CrossRef]

J. C. A. Tyrrell, P. Kinsler, and G. H. C. New, "Pseudospectral spatial-domain: A new method for nonlinear pulse propagation in the few-cycle regime with arbitrary dispersion," J. Mod. Opt. 52, 973-986 (2005).
[CrossRef]

P. Kinsler, S. B. P. Radnor, and G. H. C. New, "Theory of directional pulse propagation," Phys. Rev. A 72, 063807 (2005).
[CrossRef]

2003 (2)

2002 (1)

M. Kolesik, J. V. Moloney, and M. Mlejnek, "Unidirectional optical pulse propagation equation," Phys. Rev. Lett. 89, 283902 (2002).
[CrossRef]

1998 (1)

Y. J. Ding, J. U. Kang, and J. B. Khurgin, "Theory of backward second-harmonic and third-harmonic generation using laser pulses in quasi-phase-matched second-order nonlinear medium," IEEE J. Quantum Electron. 34, 966-974 (1998).
[CrossRef]

1997 (2)

J. U. Kang, Y. J. Ding, W. K. Burns, and J. S. Melinger, "Backward second-harmonic generation in periodically poled bulk LiNbO3," Opt. Lett. 22, 862-864 (1997).
[CrossRef] [PubMed]

T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
[CrossRef]

1996 (1)

Y. J. Ding and J. B. Khurgin, "Backward optical parametric oscillators and amplifiers," IEEE J. Quantum Electron. 32, 1574-1582 (1996).
[CrossRef]

1991 (1)

L. W. Casperson, "Field-equation approximations and amplification in high-gain lasers: numerical results," Phys. Rev. A 44, 3291-3304 (1991).
[CrossRef] [PubMed]

1989 (1)

K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[CrossRef]

1966 (2)

S. E. Harris, "Proposed backward wave oscillation in the infrared," Appl. Phys. Lett. 9, 114-116 (1966).
[CrossRef]

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).
[CrossRef]

1965 (1)

G. Rosen, "Electromagnetic shocks and the self-annihilation of intense linearly polarized radiation in an ideal dielectric material," Phys. Rev. A 139, A539-A543 (1965).

Appl. Phys. Lett. (1)

S. E. Harris, "Proposed backward wave oscillation in the infrared," Appl. Phys. Lett. 9, 114-116 (1966).
[CrossRef]

IEEE J. Quantum Electron. (3)

Y. J. Ding and J. B. Khurgin, "Backward optical parametric oscillators and amplifiers," IEEE J. Quantum Electron. 32, 1574-1582 (1996).
[CrossRef]

Y. J. Ding, J. U. Kang, and J. B. Khurgin, "Theory of backward second-harmonic and third-harmonic generation using laser pulses in quasi-phase-matched second-order nonlinear medium," IEEE J. Quantum Electron. 34, 966-974 (1998).
[CrossRef]

K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).
[CrossRef]

J. Mod. Opt. (1)

J. C. A. Tyrrell, P. Kinsler, and G. H. C. New, "Pseudospectral spatial-domain: A new method for nonlinear pulse propagation in the few-cycle regime with arbitrary dispersion," J. Mod. Opt. 52, 973-986 (2005).
[CrossRef]

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

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (5)

G. Rosen, "Electromagnetic shocks and the self-annihilation of intense linearly polarized radiation in an ideal dielectric material," Phys. Rev. A 139, A539-A543 (1965).

P. Kinsler and G. H. C. New, "Few cycle pulse propagation," Phys. Rev. A 67, 023813 (2003).
[CrossRef]

P. Kinsler, S. B. P. Radnor, and G. H. C. New, "Theory of directional pulse propagation," Phys. Rev. A 72, 063807 (2005).
[CrossRef]

S. B. P. Radnor, L. E. Chipperfield, P. Kinsler, and G. H. C. New, "Carrier wave self-steepening and application to high harmonic generation," submitted to Phys. Rev. A .

L. W. Casperson, "Field-equation approximations and amplification in high-gain lasers: numerical results," Phys. Rev. A 44, 3291-3304 (1991).
[CrossRef] [PubMed]

Phys. Rev. E (2)

A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and Á. Montero, "Forward-backward equations for nonlinear propagation in axially invariant optical systems," Phys. Rev. E 71, 016601 (2005).
[CrossRef]

P. Kinsler, S. B. P. Radnor, J. C. A. Tyrrell, and G. H. C. New, "Optical carrier wave shocking: detection and dispersion," Phys. Rev. E 75, 066603 (2007).
[CrossRef]

Phys. Rev. Lett. (2)

M. Kolesik, J. V. Moloney, and M. Mlejnek, "Unidirectional optical pulse propagation equation," Phys. Rev. Lett. 89, 283902 (2002).
[CrossRef]

T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
[CrossRef]

Other (5)

P. Kinsler, "Theory of directional pulse propagation: detailed calculations," arXiv:physics/0611216.

If the forward field has a wave vector k0 and evolves as exp(+ik0z), the generated backward component will evolve as exp(−ik0z). This gives a very rapid relative oscillation exp(−2ik0z), which will quickly average to zero.

S. B. P. Radnor (Department of Physics, Imperial College London, London, personal communication, 2006).

G. B. Whitham, Lectures on Wave Propagation (Wiley, 1979).

P. Kinsler, "Pulse propagation methods in nonlinear optics," arXiv:0707.0982.

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Figures (6)

Fig. 1
Fig. 1

Progressive distortion of an initially sinusoidal wave profile as the shocking distance is approached for (a) χ ( 3 ) and (b) χ ( 2 ) nonlinearities in a dispersionless medium. In the spectral domain, the distortion corresponds to a buildup of significant quantities of higher-order harmonic content.

Fig. 2
Fig. 2

Here two points A and B on the field profile (with fields E A , E B ) follow their characteristics as the wave propagates. Separated initially by a time difference d t , they travel at different speeds (v and v d v ) and meet at point C.

Fig. 3
Fig. 3

Scaled shocking distances for χ ( 3 ) as a function of nonlinearity: MOC predictions for the exact bidirectional case (solid curve) and unidirectional approximation (dotted line). Simulation results are denoted by symbols (E, H, ◻; G ± , +; unidirectional G + ∗), with distances determined by using LDD shock detection.

Fig. 4
Fig. 4

Scaled shocking distances for χ ( 2 ) as a function of nonlinearity: MOC predictions for the exact bidirectional case (solid curve) and unidirectional approximation (dotted line). Simulation results are denoted by symbols (E, H, ◻; G ± , +; unidirectional G + , ∗), with distances determined by using LDD shock detection.

Fig. 5
Fig. 5

Comparison of forward ( G + ) and backward ( G ) field contributions at the LDD shocking point for a χ ( 3 ) nonlinearity with C 3 1.32 . The G field has been scaled up to enhance detail.

Fig. 6
Fig. 6

Comparison of forward ( G + ) and backward ( G ) field contributions at the LDD shocking point for a χ ( 2 ) nonlinearity with C 2 0.71 . The G field has been scaled up to enhance detail.

Equations (16)

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D = ϵ 0 ( E + χ ( 1 ) E + χ ( m ) E m ) .
c 2 2 E z 2 = n 0 2 2 E t 2 + χ ( m ) 2 E m t 2 ,
E t + v m ( E ) E z = 0 ,
v m ( E ) = c n 0 [ 1 + m χ ( m ) E m 1 n 0 2 ] 1 2 .
d v d t = v t = v 2 L ,
d v m d t = m c χ ( m ) n 0 2 2 ( 1 + m χ ( m ) E m 1 n 0 2 ) 3 2 ( E m 1 ) t
= m χ ( m ) 2 c 2 v m 3 ( E m 1 ) t ,
L m = v m 2 d v m d t = 2 c n 0 1 + m χ ( 3 ) E m 1 n 0 2 m χ ( m ) ( E m 1 t ) .
S m = 2 c n 0 m Min [ C m χ ( m ) ( E m 1 t ) ] ,
C m = 1 + m χ ( m ) E m 1 n 0 2 .
S m 2 c n 0 m Min [ 1 χ ( m ) ( E m 1 t ) ] .
E ± z + n 0 c E ± t = ± χ ( m ) 2 c ( E + + E ) m t .
E + t + v m + ( E + ) E + z = 0
u m ( E + ) = c n 0 [ 1 + m χ ( m ) E + m 1 2 n 0 2 ] 1 .
d u m d t = n 0 c m c χ ( m ) 2 n 0 2 ( 1 + m χ ( m ) E + m 1 2 n 0 2 ) 2 ( E + m 1 ) t ,
S m + = 2 c n 0 m Min [ 1 χ ( m ) ( E m 1 t ) ] .

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