Abstract

Multiple-scale analysis is employed for the analysis of plane-wave refraction at a nonlinear slab. It will be demonstrated that the perturbation method will lead to a nonuniformly valid approximation to the solution of the nonlinear wave equation. To construct a uniformly valid approximation, we will exploit multiple-scale analysis. Using this method, we will derive the zeroth-order approximation to the solution of the nonlinear wave equation analytically. This approximate solution clearly shows the effects of self-phase modulation (SPM) and cross-phase modulation (XPM) on plane-wave refraction at the nonlinear slab. As will be shown, the proposed method can be generalized to the rigorous study of nonlinear wave propagation in one-dimensional photonic band-gap structures.

© 2005 Optical Society of America

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References

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  1. H. G. Winful, J. H. Marburger, and E. Garmire, �??Theory of bistability in nonlinear distributed feedback structures,�?? Appl. Phys. Lett. 35, 379-381 (1979).
    [CrossRef]
  2. D. N. Christodoulides and R. I. Joseph, �??Slow Bragg solitons in nonlinear periodic structures,�?? Phys. Rev. Lett. 62, 1746-1749 (1989).
    [CrossRef] [PubMed]
  3. N. G. R. Broderick, D. J. Richardson, and M. Ibsen, �??Nonlinear switching in a 20-cm long fiber Bragg grating,�?? Opt. Lett. 25, 536�??538 (2000).
    [CrossRef]
  4. K. Senthilnathan, P. Malathi, and K. Porsezian, �??Dynamics of nonlinear pulse propagation through a fiber Bragg grating with linear coupling,�?? J. Opt. Soc. Am. B 20, 366�??372 (2003).
    [CrossRef]
  5. R. E. Slusher and B. J. Eggleton, eds., Nonlinear Photonic Crystals (Springer-Verlag Berlin Heidelberg, Berlin, 2003).
  6. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, �??Interaction between light waves in a nonlinear dielectric,�?? Phys. Rev. 127, 1918�??1939 (1962).
    [CrossRef]
  7. N. Bloembergen and P. S. Pershan, �??Light waves at the boundary of nonlinear media,�?? Phys. Rev. 128, 606�??622 (1962).
    [CrossRef]
  8. M. M. Carroll, �??Plane waves of constant amplitude in nonlinear dielectrics,�?? Phys. Rev. A 6, 1977�??1980 (1972).
    [CrossRef]
  9. Th. Peschel, P. Dannberg, U. Langbein, and F. Lederer, �??Investigation of optical tunneling through nonlinear films,�?? J. Opt. Soc. Am. B 5, 5�??36 (1988).
    [CrossRef]
  10. K. Hayata, M. Nagai, and M. Koshiba, �??Finite-element formalism for noninear slab-guided waves,�?? IEEE Trans. Microwave Theory Tech. 36, 1207�??1215 (1988).
    [CrossRef]
  11. S. V. Polstyanko, R. Dyczij-Edlinger, and J. F. Lee, �??A full vectorial analysis of a nonlinear slab waveguide based on the nonlinear hybrid vector finite-element method,�?? Opt. Lett. 21, 98�??100 (1996).
    [CrossRef] [PubMed]
  12. V. Van and S. K. Chaudhuri, �??A hybrid implicit-explicit FDTD scheme for nonlinear optical waveguide modeling,�?? IEEE Trans. Microwave Theory Tech. 47, 540�??545 (1999).
    [CrossRef]
  13. R. M. Joseph and A. Taflove, �??FDTD Maxwell�??s equations models for nonlinear electrodynamics and optics,�?? IEEE Trans. Microwave Theory Tech. 45, 364�??374 (1997).
  14. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, Singapore, 1978).
  15. K. Ogusu, �??Self-switching in hollow waveguides with a Kerrlike nonlinear permittivity,�?? IEEE J. Lightwave Technol. 8, 1541�??1547 (1990).
    [CrossRef]
  16. U. Trutschel, F. Lederer, and M. Golz, �??Nonlinear guided waves in multilayer systems,�?? IEEE J. Quantum Electron. 25, 194�??200 (1989).
    [CrossRef]

Appl. Phys. Lett. (1)

H. G. Winful, J. H. Marburger, and E. Garmire, �??Theory of bistability in nonlinear distributed feedback structures,�?? Appl. Phys. Lett. 35, 379-381 (1979).
[CrossRef]

IEEE J. Lightwave Technol. (1)

K. Ogusu, �??Self-switching in hollow waveguides with a Kerrlike nonlinear permittivity,�?? IEEE J. Lightwave Technol. 8, 1541�??1547 (1990).
[CrossRef]

IEEE J. Quantum Electron. (1)

U. Trutschel, F. Lederer, and M. Golz, �??Nonlinear guided waves in multilayer systems,�?? IEEE J. Quantum Electron. 25, 194�??200 (1989).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (3)

K. Hayata, M. Nagai, and M. Koshiba, �??Finite-element formalism for noninear slab-guided waves,�?? IEEE Trans. Microwave Theory Tech. 36, 1207�??1215 (1988).
[CrossRef]

V. Van and S. K. Chaudhuri, �??A hybrid implicit-explicit FDTD scheme for nonlinear optical waveguide modeling,�?? IEEE Trans. Microwave Theory Tech. 47, 540�??545 (1999).
[CrossRef]

R. M. Joseph and A. Taflove, �??FDTD Maxwell�??s equations models for nonlinear electrodynamics and optics,�?? IEEE Trans. Microwave Theory Tech. 45, 364�??374 (1997).

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

Th. Peschel, P. Dannberg, U. Langbein, and F. Lederer, �??Investigation of optical tunneling through nonlinear films,�?? J. Opt. Soc. Am. B 5, 5�??36 (1988).
[CrossRef]

K. Senthilnathan, P. Malathi, and K. Porsezian, �??Dynamics of nonlinear pulse propagation through a fiber Bragg grating with linear coupling,�?? J. Opt. Soc. Am. B 20, 366�??372 (2003).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. (2)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, �??Interaction between light waves in a nonlinear dielectric,�?? Phys. Rev. 127, 1918�??1939 (1962).
[CrossRef]

N. Bloembergen and P. S. Pershan, �??Light waves at the boundary of nonlinear media,�?? Phys. Rev. 128, 606�??622 (1962).
[CrossRef]

Phys. Rev. A (1)

M. M. Carroll, �??Plane waves of constant amplitude in nonlinear dielectrics,�?? Phys. Rev. A 6, 1977�??1980 (1972).
[CrossRef]

Phys. Rev. Lett. (1)

D. N. Christodoulides and R. I. Joseph, �??Slow Bragg solitons in nonlinear periodic structures,�?? Phys. Rev. Lett. 62, 1746-1749 (1989).
[CrossRef] [PubMed]

Other (2)

R. E. Slusher and B. J. Eggleton, eds., Nonlinear Photonic Crystals (Springer-Verlag Berlin Heidelberg, Berlin, 2003).

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, Singapore, 1978).

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Figures (4)

Fig. 1.
Fig. 1.

A plane wave incident on a slab with Kerr-type nonlinearity.

Fig. 2.
Fig. 2.

(a) Variation of magnitude of reflection coefficient at z=0 as a function of ao . (b) Variation of magnitude of transmission coefficient at z=L as a function of ao . (c) Normalized reflected (blue line) and transmitted intensities (red line) as a function of ao . The green line shows the summation of normalized reflected and transmitted intensities.

Fig. 3.
Fig. 3.

Magnitude of reflection coefficient at z=0 versus wavelength in linear regime.

Fig. 4.
Fig. 4.

(a) Normalized magnitude of electric field phasor, |Ex /ao |, for three different values of ao . (b) Normalized electric field at t=0 (real part of Ex /ao ) for three different values of ao . Blue line: ao =7×106 V/m. Green line: ao =12×107 V/m . Red line: ao=22×107 V/m.

Equations (34)

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ε r = ( n o + n 2 E x 2 ) 2 n o 2 + 2 n o n 2 E x 2
dE x dz = jωμ o H y
dH y dz = jωε o ε r E x .
d 2 E x dz 2 + k o 2 ( n o 2 + α E x 2 ) E x = 0
E x = m = 0 α m E m = E o + α E 1 + α 2 E 2 +
d 2 E o dz 2 + k o 2 n o 2 E o = 0
d 2 E 1 dz 2 + k o 2 n o 2 E 1 = k o 2 E o 2 E o .
E o = a exp ( jβz ) + b exp ( jβz ) .
E o 2 = a 2 + b 2 + ab * exp ( 2 jβz ) + a * b exp ( 2 jβz ) .
E x ( z ) = E o z ζ + α E 1 z ζ +
dE x dz = ( E o z + E o ζ dz ) + α ( E 1 z + E 1 ζ dz ) +
dE x dz = E o z + α ( E o ζ + E 1 ζ ) + O ( α 2 ) .
d 2 E x dz 2 = 2 E o z 2 + α ( 2 2 E o ζ z + 2 E 1 z 2 ) + O ( α 2 ) .
2 E o z 2 + β 2 E o = 0
2 E 1 z 2 + β 2 E 1 = k o 2 E o 2 E o 2 2 E o ζ z .
E o z ζ = a ( ζ ) exp ( jβz ) + b ( ζ ) exp ( jβz ) .
( 2 da k o 2 a ( a 2 + 2 b 2 ) ) exp ( jβz )
( 2 db k o 2 b ( 2 a 2 + b 2 ) ) exp ( jβz )
k o 2 a 2 b * exp ( 3 jβz ) k o 2 a * b 2 exp ( 3 jβz ) .
{ da = j k o 2 2 β a ( a 2 + 2 b 2 ) db = j k o 2 2 β b ( 2 a 2 + b 2 )
a ( ζ ) = R 1 ( ζ ) exp ( 1 ( ζ ) )
b ( ζ ) = R 2 ( ζ ) exp ( 2 ( ζ ) ) .
{ 1 = k o 2 n o ( R 1 2 + 2 R 2 2 ) 2 = k o 2 n o ( 2 R 1 2 + R 2 2 )
a ( ζ ) = R 1 ( 0 ) exp ( 1 ( 0 ) ) exp ( j k o 2 n o ( R 1 2 ( 0 ) + 2 R 2 2 ( 0 ) ) ζ )
b ( ζ ) = R 2 ( 0 ) exp ( 2 ( 0 ) ) exp ( j k o 2 n o ( 2 R 1 2 ( 0 ) + R 2 2 ( 0 ) ) ζ ) .
a ( ζ ) = a 1 exp ( j k o 2 n o ( a 1 2 + 2 b 1 2 ) ζ )
b ( ζ ) = b 1 exp ( j k o 2 n o ( 2 a 1 2 + b 1 2 ) ζ )
H o z ζ = a 1 Z c [ 1 + α 2 n o 2 ( a 1 2 + 2 b 1 2 ) ] exp ( j k 1 ζ ) exp ( jβz ) b 1 Z c [ 1 + α 2 n o 2 ( 2 a 1 2 + b 1 2 ) ] exp ( j k 2 ζ ) exp ( jβz )
a o + b o = a 1 + b 1
a o b o Z o = a 1 Z c [ 1 + α 2 n o 2 ( a 1 2 + 2 b 1 2 ) ] b 1 Z c [ 1 + α 2 n o 2 ( 2 a 1 2 + b 1 2 ) ] .
E o ( L , αL ) H o ( L , αL ) = Z L .
b 1 a 1 exp ( j 3 k o 2 n o αL ( a 1 2 + b 1 2 ) ) = Z ¯ ( 1 + α 2 n o 2 ( a 1 2 + 2 b 1 2 ) ) 1 Z ¯ ( 1 + α 2 n o 2 ( 2 a 1 2 + b 1 2 ) ) + 1 exp ( 2 jβL )
a ¯ 1 { 1 + n o [ 1 + α 2 n o 2 a o 2 ( a ¯ 1 2 + 2 b ¯ 1 2 ) ] } + b ¯ 1 { 1 n o [ 1 + α 2 n o 2 a o 2 ( 2 a ¯ 1 2 + b ¯ 1 2 ) ] } = 2
b ¯ 1 a ¯ 1 exp ( j 3 k o 2 n o αL ( a ¯ 1 2 + b ¯ 1 2 ) ) = Z ¯ ( 1 + α 2 n o 2 a o 2 ( a ¯ 1 2 + 2 b ¯ 1 2 ) ) 1 Z ¯ ( 1 + α 2 n o 2 a o 2 ( 2 a ¯ 1 2 + b ¯ 1 2 ) ) + 1 exp ( 2 jβL ) .

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