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, 29–36 (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]

2003 (1)

2000 (1)

1999 (1)

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]

1997 (1)

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).

1996 (1)

1990 (1)

K. Ogusu, “Self-switching in hollow waveguides with a Kerrlike nonlinear permittivity,” IEEE J. Lightwave Technol. 8, 1541–1547 (1990).
[Crossref]

1989 (2)

U. Trutschel, F. Lederer, and M. Golz, “Nonlinear guided waves in multilayer systems,” IEEE J. Quantum Electron. 25, 194–200 (1989).
[Crossref]

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[Crossref] [PubMed]

1988 (2)

Th. Peschel, P. Dannberg, U. Langbein, and F. Lederer, “Investigation of optical tunneling through nonlinear films,” J. Opt. Soc. Am. B 5, 29–36 (1988).
[Crossref]

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]

1979 (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]

1972 (1)

M. M. Carroll, “Plane waves of constant amplitude in nonlinear dielectrics,” Phys. Rev. A 6, 1977–1980 (1972).
[Crossref]

1962 (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]

Armstrong, J. A.

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]

Bender, C. M.

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

Bloembergen, N.

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
[Crossref]

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]

Broderick, N. G. R.

Carroll, M. M.

M. M. Carroll, “Plane waves of constant amplitude in nonlinear dielectrics,” Phys. Rev. A 6, 1977–1980 (1972).
[Crossref]

Chaudhuri, S. K.

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]

Christodoulides, D. N.

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[Crossref] [PubMed]

Dannberg, P.

Ducuing, J.

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]

Dyczij-Edlinger, R.

Garmire, E.

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]

Golz, M.

U. Trutschel, F. Lederer, and M. Golz, “Nonlinear guided waves in multilayer systems,” IEEE J. Quantum Electron. 25, 194–200 (1989).
[Crossref]

Hayata, K.

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]

Ibsen, M.

Joseph, R. I.

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[Crossref] [PubMed]

Joseph, R. M.

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).

Koshiba, M.

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]

Langbein, U.

Lederer, F.

U. Trutschel, F. Lederer, and M. Golz, “Nonlinear guided waves in multilayer systems,” IEEE J. Quantum Electron. 25, 194–200 (1989).
[Crossref]

Th. Peschel, P. Dannberg, U. Langbein, and F. Lederer, “Investigation of optical tunneling through nonlinear films,” J. Opt. Soc. Am. B 5, 29–36 (1988).
[Crossref]

Lee, J. F.

Malathi, P.

Marburger, J. H.

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]

Nagai, M.

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]

Ogusu, K.

K. Ogusu, “Self-switching in hollow waveguides with a Kerrlike nonlinear permittivity,” IEEE J. Lightwave Technol. 8, 1541–1547 (1990).
[Crossref]

Orszag, S. A.

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

Pershan, P. S.

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
[Crossref]

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]

Peschel, Th.

Polstyanko, S. V.

Porsezian, K.

Richardson, D. J.

Senthilnathan, K.

Taflove, A.

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).

Trutschel, U.

U. Trutschel, F. Lederer, and M. Golz, “Nonlinear guided waves in multilayer systems,” IEEE J. Quantum Electron. 25, 194–200 (1989).
[Crossref]

Van, V.

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]

Winful, H. G.

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]

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)

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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