Abstract

We develop a generalized version of the invariant imbedding method, which allows us to solve the electromagnetic wave equations in arbitrarily inhomogeneous stratified media where both the dielectric permittivity and magnetic permeability depend on the strengths of the electric and magnetic fields, in a numerically accurate and efficient manner. We apply our method to a uniform nonlinear slab and find that in the presence of strong external radiation, an initially uniform medium of positive refractive index can spontaneously change into a highly inhomogeneous medium where regions of positive or negative refractive index as well as metallic regions appear. We also study the wave transmission properties of periodic nonlinear media and the influence of nonlinearity on the mode conversion phenomena in inhomogeneous plasmas. We argue that our theory is very useful in the study of the optical properties of a variety of nonlinear media including nonlinear negative index media fabricated using wires and split-ring resonators.

© 2008 Optical Society of America

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References

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  1. R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, 2003).
  2. W. S. Weiglhofer and A. Lakhtakia, eds., Introduction to Complex Mediums for Optics and Electromagnetics (SPIE Press, 2003).
    [CrossRef]
  3. K. Huang, L. M. Kahn, and D. L. Mills, "Optical transmissivity of superlattices with nonlinear magnetic susceptibility and absorption," Phys. Rev. B 41, 7981-7987 (1990).
    [CrossRef]
  4. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
    [CrossRef] [PubMed]
  5. V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photonics 1, 41-48 (2007).
    [CrossRef]
  6. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  7. R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
    [CrossRef] [PubMed]
  8. A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, "Nonlinear properties of left-handed metamaterials," Phys. Rev. Lett. 91, 037401 (2003).
    [CrossRef] [PubMed]
  9. S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, "Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials," Phys. Rev. B 69, 241101(R) (2004).
  10. N. A. Zharova, I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, "Nonlinear transmission and spatiotemporal solitons in metamaterials with negative refraction," Opt. Express 13, 1291-1298 (2005).
    [CrossRef] [PubMed]
  11. I. V. Shadrivov and Y. S. Kivshar, "Spatial solitons in nonlinear left-handed metamaterials," J. Opt. A: Pure Appl. Opt. 7, S68-S72 (2005).
    [CrossRef]
  12. M. Scalora, G. D’Aguanno, M. Bloemer, M. Centini, D. de Ceglia, N. Mattiucci, and Y. S. Kivshar, "Dynamics of short pulses and phase matched second harmonic generation in negative index materials," Opt. Express 14, 4746-4756 (2006).
    [CrossRef] [PubMed]
  13. G. I. Babkin and V. I. Klyatskin, "Theory of wave propagation in nonlinear inhomogeneous media," Sov. Phys. JETP 52, 416-420 (1980).
  14. H. F. Arnoldus and T. F. George, "Theory of optical phase conjugation in Kerr media," Phys. Rev. A 51, 4250-4263 (1995).
    [CrossRef] [PubMed]
  15. R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding (Wiley, 1976).
  16. V. I. Klyatskin, "The imbedding method in statistical boundary-value wave problems," Prog. Opt. 33, 1-127 (1994).
    [CrossRef]
  17. R. Rammal and B. Doucot, "Invariant-imbedding approach to localization. I. General framework and basic equations," J. Phys. (Paris) 48, 509-526 (1987).
    [CrossRef]
  18. B. Doucot and R. Rammal, "Invariant-imbedding approach to localization. II. Non-linear random media," J. Phys. (Paris) 48, 527-546 (1987).
    [CrossRef]
  19. K. Kim, D.-H. Lee, and H. Lim, "Theory of the propagation of coupled waves in arbitrarily inhomogeneous stratified media," Europhys. Lett. 69, 207-213 (2005).
    [CrossRef]
  20. K. Kim, "Reflection coefficient and localization length of waves in one-dimensional random media," Phys. Rev. B 58, 6153-6160 (1998).
    [CrossRef]
  21. K. Kim and D.-H. Lee, "Invariant imbedding theory of mode conversion in inhomogeneous plasmas. I. Exact calculation of the mode conversion coefficient in cold, unmagnetized plasmas," Phys. Plasmas 12, 062101 (2005).
    [CrossRef]
  22. K. Kim and D.-H. Lee, "Invariant imbedding theory of mode conversion in inhomogeneous plasmas. II. Mode conversion in cold, magnetized plasmas with perpendicular inhomogeneity," Phys. Plasmas 13, 042103 (2006).
    [CrossRef]
  23. H. Chang and L. C. Chen, "Simple numerical approach for determining the optical response of a nonlinear dielectric film for both TE and TM waves," Phys. Rev. B 43, 9436-9441 (1991).
    [CrossRef]
  24. W. Chen and D. L. Mills, "Optical behavior of a nonlinear thin film with oblique S-polarized incident wave," Phys. Rev. B 38, 12814-12822 (1988).
    [CrossRef]
  25. W. Chen and D. L. Mills, "Optical response of nonlinear multilayer structures: Bilayers and superlattices," Phys. Rev. B 36, 6269-6278 (1987).
    [CrossRef]
  26. A. V. Kochetov and A. M. Feigin, "Bleaching of dense plasma by an intense TM wave," Sov. J. Plasma Phys. 14, 423-429 (1988).
  27. V. B. Gil’denburg, A. V. Kochetov, A. G. Litvak, and A. M. Feigin, "Self-sustaining waveguide channels in a plasma," Sov. Phys. JETP 57, 28-34 (1983).
  28. V. L. Ginzburg, The Propagation of Electromagnetic Waves in Plasmas (Pergamon, 1970).

2007 (1)

V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photonics 1, 41-48 (2007).
[CrossRef]

2006 (2)

M. Scalora, G. D’Aguanno, M. Bloemer, M. Centini, D. de Ceglia, N. Mattiucci, and Y. S. Kivshar, "Dynamics of short pulses and phase matched second harmonic generation in negative index materials," Opt. Express 14, 4746-4756 (2006).
[CrossRef] [PubMed]

K. Kim and D.-H. Lee, "Invariant imbedding theory of mode conversion in inhomogeneous plasmas. II. Mode conversion in cold, magnetized plasmas with perpendicular inhomogeneity," Phys. Plasmas 13, 042103 (2006).
[CrossRef]

2005 (4)

K. Kim, D.-H. Lee, and H. Lim, "Theory of the propagation of coupled waves in arbitrarily inhomogeneous stratified media," Europhys. Lett. 69, 207-213 (2005).
[CrossRef]

K. Kim and D.-H. Lee, "Invariant imbedding theory of mode conversion in inhomogeneous plasmas. I. Exact calculation of the mode conversion coefficient in cold, unmagnetized plasmas," Phys. Plasmas 12, 062101 (2005).
[CrossRef]

I. V. Shadrivov and Y. S. Kivshar, "Spatial solitons in nonlinear left-handed metamaterials," J. Opt. A: Pure Appl. Opt. 7, S68-S72 (2005).
[CrossRef]

N. A. Zharova, I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, "Nonlinear transmission and spatiotemporal solitons in metamaterials with negative refraction," Opt. Express 13, 1291-1298 (2005).
[CrossRef] [PubMed]

2004 (1)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

2003 (1)

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, "Nonlinear properties of left-handed metamaterials," Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef] [PubMed]

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

2000 (1)

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

1998 (1)

K. Kim, "Reflection coefficient and localization length of waves in one-dimensional random media," Phys. Rev. B 58, 6153-6160 (1998).
[CrossRef]

1995 (1)

H. F. Arnoldus and T. F. George, "Theory of optical phase conjugation in Kerr media," Phys. Rev. A 51, 4250-4263 (1995).
[CrossRef] [PubMed]

1994 (1)

V. I. Klyatskin, "The imbedding method in statistical boundary-value wave problems," Prog. Opt. 33, 1-127 (1994).
[CrossRef]

1991 (1)

H. Chang and L. C. Chen, "Simple numerical approach for determining the optical response of a nonlinear dielectric film for both TE and TM waves," Phys. Rev. B 43, 9436-9441 (1991).
[CrossRef]

1990 (1)

K. Huang, L. M. Kahn, and D. L. Mills, "Optical transmissivity of superlattices with nonlinear magnetic susceptibility and absorption," Phys. Rev. B 41, 7981-7987 (1990).
[CrossRef]

1988 (2)

A. V. Kochetov and A. M. Feigin, "Bleaching of dense plasma by an intense TM wave," Sov. J. Plasma Phys. 14, 423-429 (1988).

W. Chen and D. L. Mills, "Optical behavior of a nonlinear thin film with oblique S-polarized incident wave," Phys. Rev. B 38, 12814-12822 (1988).
[CrossRef]

1987 (3)

W. Chen and D. L. Mills, "Optical response of nonlinear multilayer structures: Bilayers and superlattices," Phys. Rev. B 36, 6269-6278 (1987).
[CrossRef]

R. Rammal and B. Doucot, "Invariant-imbedding approach to localization. I. General framework and basic equations," J. Phys. (Paris) 48, 509-526 (1987).
[CrossRef]

B. Doucot and R. Rammal, "Invariant-imbedding approach to localization. II. Non-linear random media," J. Phys. (Paris) 48, 527-546 (1987).
[CrossRef]

1983 (1)

V. B. Gil’denburg, A. V. Kochetov, A. G. Litvak, and A. M. Feigin, "Self-sustaining waveguide channels in a plasma," Sov. Phys. JETP 57, 28-34 (1983).

1980 (1)

G. I. Babkin and V. I. Klyatskin, "Theory of wave propagation in nonlinear inhomogeneous media," Sov. Phys. JETP 52, 416-420 (1980).

Arnoldus, H. F.

H. F. Arnoldus and T. F. George, "Theory of optical phase conjugation in Kerr media," Phys. Rev. A 51, 4250-4263 (1995).
[CrossRef] [PubMed]

Babkin, G. I.

G. I. Babkin and V. I. Klyatskin, "Theory of wave propagation in nonlinear inhomogeneous media," Sov. Phys. JETP 52, 416-420 (1980).

Bloemer, M.

Centini, M.

Chang, H.

H. Chang and L. C. Chen, "Simple numerical approach for determining the optical response of a nonlinear dielectric film for both TE and TM waves," Phys. Rev. B 43, 9436-9441 (1991).
[CrossRef]

Chen, L. C.

H. Chang and L. C. Chen, "Simple numerical approach for determining the optical response of a nonlinear dielectric film for both TE and TM waves," Phys. Rev. B 43, 9436-9441 (1991).
[CrossRef]

Chen, W.

W. Chen and D. L. Mills, "Optical behavior of a nonlinear thin film with oblique S-polarized incident wave," Phys. Rev. B 38, 12814-12822 (1988).
[CrossRef]

W. Chen and D. L. Mills, "Optical response of nonlinear multilayer structures: Bilayers and superlattices," Phys. Rev. B 36, 6269-6278 (1987).
[CrossRef]

D’Aguanno, G.

de Ceglia, D.

Doucot, B.

B. Doucot and R. Rammal, "Invariant-imbedding approach to localization. II. Non-linear random media," J. Phys. (Paris) 48, 527-546 (1987).
[CrossRef]

R. Rammal and B. Doucot, "Invariant-imbedding approach to localization. I. General framework and basic equations," J. Phys. (Paris) 48, 509-526 (1987).
[CrossRef]

Feigin, A. M.

A. V. Kochetov and A. M. Feigin, "Bleaching of dense plasma by an intense TM wave," Sov. J. Plasma Phys. 14, 423-429 (1988).

V. B. Gil’denburg, A. V. Kochetov, A. G. Litvak, and A. M. Feigin, "Self-sustaining waveguide channels in a plasma," Sov. Phys. JETP 57, 28-34 (1983).

George, T. F.

H. F. Arnoldus and T. F. George, "Theory of optical phase conjugation in Kerr media," Phys. Rev. A 51, 4250-4263 (1995).
[CrossRef] [PubMed]

Gil’denburg, V. B.

V. B. Gil’denburg, A. V. Kochetov, A. G. Litvak, and A. M. Feigin, "Self-sustaining waveguide channels in a plasma," Sov. Phys. JETP 57, 28-34 (1983).

Huang, K.

K. Huang, L. M. Kahn, and D. L. Mills, "Optical transmissivity of superlattices with nonlinear magnetic susceptibility and absorption," Phys. Rev. B 41, 7981-7987 (1990).
[CrossRef]

Kahn, L. M.

K. Huang, L. M. Kahn, and D. L. Mills, "Optical transmissivity of superlattices with nonlinear magnetic susceptibility and absorption," Phys. Rev. B 41, 7981-7987 (1990).
[CrossRef]

Kim, K.

K. Kim and D.-H. Lee, "Invariant imbedding theory of mode conversion in inhomogeneous plasmas. II. Mode conversion in cold, magnetized plasmas with perpendicular inhomogeneity," Phys. Plasmas 13, 042103 (2006).
[CrossRef]

K. Kim, D.-H. Lee, and H. Lim, "Theory of the propagation of coupled waves in arbitrarily inhomogeneous stratified media," Europhys. Lett. 69, 207-213 (2005).
[CrossRef]

K. Kim and D.-H. Lee, "Invariant imbedding theory of mode conversion in inhomogeneous plasmas. I. Exact calculation of the mode conversion coefficient in cold, unmagnetized plasmas," Phys. Plasmas 12, 062101 (2005).
[CrossRef]

K. Kim, "Reflection coefficient and localization length of waves in one-dimensional random media," Phys. Rev. B 58, 6153-6160 (1998).
[CrossRef]

Kivshar, Y. S.

Klyatskin, V. I.

V. I. Klyatskin, "The imbedding method in statistical boundary-value wave problems," Prog. Opt. 33, 1-127 (1994).
[CrossRef]

G. I. Babkin and V. I. Klyatskin, "Theory of wave propagation in nonlinear inhomogeneous media," Sov. Phys. JETP 52, 416-420 (1980).

Kochetov, A. V.

A. V. Kochetov and A. M. Feigin, "Bleaching of dense plasma by an intense TM wave," Sov. J. Plasma Phys. 14, 423-429 (1988).

V. B. Gil’denburg, A. V. Kochetov, A. G. Litvak, and A. M. Feigin, "Self-sustaining waveguide channels in a plasma," Sov. Phys. JETP 57, 28-34 (1983).

Lee, D.-H.

K. Kim and D.-H. Lee, "Invariant imbedding theory of mode conversion in inhomogeneous plasmas. II. Mode conversion in cold, magnetized plasmas with perpendicular inhomogeneity," Phys. Plasmas 13, 042103 (2006).
[CrossRef]

K. Kim and D.-H. Lee, "Invariant imbedding theory of mode conversion in inhomogeneous plasmas. I. Exact calculation of the mode conversion coefficient in cold, unmagnetized plasmas," Phys. Plasmas 12, 062101 (2005).
[CrossRef]

K. Kim, D.-H. Lee, and H. Lim, "Theory of the propagation of coupled waves in arbitrarily inhomogeneous stratified media," Europhys. Lett. 69, 207-213 (2005).
[CrossRef]

Lim, H.

K. Kim, D.-H. Lee, and H. Lim, "Theory of the propagation of coupled waves in arbitrarily inhomogeneous stratified media," Europhys. Lett. 69, 207-213 (2005).
[CrossRef]

Litvak, A. G.

V. B. Gil’denburg, A. V. Kochetov, A. G. Litvak, and A. M. Feigin, "Self-sustaining waveguide channels in a plasma," Sov. Phys. JETP 57, 28-34 (1983).

Mattiucci, N.

Mills, D. L.

K. Huang, L. M. Kahn, and D. L. Mills, "Optical transmissivity of superlattices with nonlinear magnetic susceptibility and absorption," Phys. Rev. B 41, 7981-7987 (1990).
[CrossRef]

W. Chen and D. L. Mills, "Optical behavior of a nonlinear thin film with oblique S-polarized incident wave," Phys. Rev. B 38, 12814-12822 (1988).
[CrossRef]

W. Chen and D. L. Mills, "Optical response of nonlinear multilayer structures: Bilayers and superlattices," Phys. Rev. B 36, 6269-6278 (1987).
[CrossRef]

Pendry, J. B.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Rammal, R.

B. Doucot and R. Rammal, "Invariant-imbedding approach to localization. II. Non-linear random media," J. Phys. (Paris) 48, 527-546 (1987).
[CrossRef]

R. Rammal and B. Doucot, "Invariant-imbedding approach to localization. I. General framework and basic equations," J. Phys. (Paris) 48, 509-526 (1987).
[CrossRef]

Scalora, M.

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Shadrivov, I. V.

I. V. Shadrivov and Y. S. Kivshar, "Spatial solitons in nonlinear left-handed metamaterials," J. Opt. A: Pure Appl. Opt. 7, S68-S72 (2005).
[CrossRef]

N. A. Zharova, I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, "Nonlinear transmission and spatiotemporal solitons in metamaterials with negative refraction," Opt. Express 13, 1291-1298 (2005).
[CrossRef] [PubMed]

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, "Nonlinear properties of left-handed metamaterials," Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef] [PubMed]

Shalaev, V. M.

V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photonics 1, 41-48 (2007).
[CrossRef]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Smith, D. R.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

Zharov, A. A.

Zharova, N. A.

Europhys. Lett. (1)

K. Kim, D.-H. Lee, and H. Lim, "Theory of the propagation of coupled waves in arbitrarily inhomogeneous stratified media," Europhys. Lett. 69, 207-213 (2005).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

I. V. Shadrivov and Y. S. Kivshar, "Spatial solitons in nonlinear left-handed metamaterials," J. Opt. A: Pure Appl. Opt. 7, S68-S72 (2005).
[CrossRef]

J. Phys. (Paris) (2)

R. Rammal and B. Doucot, "Invariant-imbedding approach to localization. I. General framework and basic equations," J. Phys. (Paris) 48, 509-526 (1987).
[CrossRef]

B. Doucot and R. Rammal, "Invariant-imbedding approach to localization. II. Non-linear random media," J. Phys. (Paris) 48, 527-546 (1987).
[CrossRef]

Nat. Photonics (1)

V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photonics 1, 41-48 (2007).
[CrossRef]

Opt. Express (2)

Phys. Plasmas (2)

K. Kim and D.-H. Lee, "Invariant imbedding theory of mode conversion in inhomogeneous plasmas. I. Exact calculation of the mode conversion coefficient in cold, unmagnetized plasmas," Phys. Plasmas 12, 062101 (2005).
[CrossRef]

K. Kim and D.-H. Lee, "Invariant imbedding theory of mode conversion in inhomogeneous plasmas. II. Mode conversion in cold, magnetized plasmas with perpendicular inhomogeneity," Phys. Plasmas 13, 042103 (2006).
[CrossRef]

Phys. Rev. A (1)

H. F. Arnoldus and T. F. George, "Theory of optical phase conjugation in Kerr media," Phys. Rev. A 51, 4250-4263 (1995).
[CrossRef] [PubMed]

Phys. Rev. B (5)

K. Kim, "Reflection coefficient and localization length of waves in one-dimensional random media," Phys. Rev. B 58, 6153-6160 (1998).
[CrossRef]

H. Chang and L. C. Chen, "Simple numerical approach for determining the optical response of a nonlinear dielectric film for both TE and TM waves," Phys. Rev. B 43, 9436-9441 (1991).
[CrossRef]

W. Chen and D. L. Mills, "Optical behavior of a nonlinear thin film with oblique S-polarized incident wave," Phys. Rev. B 38, 12814-12822 (1988).
[CrossRef]

W. Chen and D. L. Mills, "Optical response of nonlinear multilayer structures: Bilayers and superlattices," Phys. Rev. B 36, 6269-6278 (1987).
[CrossRef]

K. Huang, L. M. Kahn, and D. L. Mills, "Optical transmissivity of superlattices with nonlinear magnetic susceptibility and absorption," Phys. Rev. B 41, 7981-7987 (1990).
[CrossRef]

Phys. Rev. Lett. (2)

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, "Nonlinear properties of left-handed metamaterials," Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef] [PubMed]

Prog. Opt. (1)

V. I. Klyatskin, "The imbedding method in statistical boundary-value wave problems," Prog. Opt. 33, 1-127 (1994).
[CrossRef]

Science (2)

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

Sov. J. Plasma Phys. (1)

A. V. Kochetov and A. M. Feigin, "Bleaching of dense plasma by an intense TM wave," Sov. J. Plasma Phys. 14, 423-429 (1988).

Sov. Phys. JETP (2)

V. B. Gil’denburg, A. V. Kochetov, A. G. Litvak, and A. M. Feigin, "Self-sustaining waveguide channels in a plasma," Sov. Phys. JETP 57, 28-34 (1983).

G. I. Babkin and V. I. Klyatskin, "Theory of wave propagation in nonlinear inhomogeneous media," Sov. Phys. JETP 52, 416-420 (1980).

Other (5)

R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding (Wiley, 1976).

V. L. Ginzburg, The Propagation of Electromagnetic Waves in Plasmas (Pergamon, 1970).

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, 2003).

W. S. Weiglhofer and A. Lakhtakia, eds., Introduction to Complex Mediums for Optics and Electromagnetics (SPIE Press, 2003).
[CrossRef]

S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, "Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials," Phys. Rev. B 69, 241101(R) (2004).

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Figures (11)

Fig. 1.
Fig. 1.

(a) Transmittance of a uniform slab with the Kerr-type nonlinearity only in ε plotted versus the nonlinearity parameter αw/εL , in the (a) s and (b) p wave cases. The parameters used are k 0 L=2.2π, εL =16, μL =ε 1=ε2 =μ1 =μ2 =1, θ=45° and β=0.

Fig. 2.
Fig. 2.

(a) Transmittance of a nonlinear slab plotted versus the nonlinearity parameter |α|w. The parameters used are k 0 L=20, εL =μL =ε 1=ε 2=μ 1=μ 2=1, θ=0 and β=0.8α (< 0). (b) Dependence of w on the initial value w 0. The points A, B, C and D correspond to the same value of |α|w (≈ 1.12579). The distance between B and C is too small for them to be distinguished in (a) and (b) (see the inset of (b)).

Fig. 3.
Fig. 3.

Spatial distributions of (a) the normalized electric field intensity, (b) the effective dielectric permittivity and magnetic permeability, and (c) the effective refractive index, corresponding to the point A in Fig. 2 (|α|w 0=0.42191892). The dashed line in (c) is a plot of ε μ when εμ<0.

Fig. 4.
Fig. 4.

Spatial distributions of (a) the normalized electric field intensity, (b) the effective dielectric permittivity and magnetic permeability, and (c) the effective refractive index, corresponding to the point C in Fig. 2 (|α|w 0=0.500849).

Fig. 5.
Fig. 5.

Spatial distributions of (a) the normalized electric field intensity, (b) the effective dielectric permittivity and magnetic permeability, and (c) the effective refractive index, corresponding to the point D in Fig. 2 (|α|w 0=1.0559815).

Fig. 6.
Fig. 6.

Transmittance of a nonlinear slab plotted versus the nonlinearity parameter |α|w in the s wave case. The parameters used are k 0 L=20, εL =μL =ε 1=ε 2=μ 1=μ 2=1, θ=1° and β=0.8α(<0). The point P corresponds to αw 0=-0.5982.

Fig. 7.
Fig. 7.

Spatial distributions of (a) the normalized electric field intensity, (b) the effective dielectric permittivity, and (c) the effective magnetic permeability, corresponding to the point P in Fig. 6. Note that μ changes discontinuously wherever it passes through zero.

Fig. 8.
Fig. 8.

(a) Scatter plot of the transmittance of a nonlinear slab with sinusoidal spatial variations of εL (=1+0.3sin(2πz/Λ)) and μL (=1-0.2sin(2πz/Λ)) versus the nonlinearity parameter |α|w. The parameters used are k 0Λ=2.85, L/Λ=20, ε 1=ε 2=μ 1=μ 2=1, θ=0 and β=0.8α (< 0). (b) Scatter plot of the dependence of w on the initial value w 0. The point Q corresponds to the third resonant transmission peak with αw 0=-0.03547.

Fig. 9.
Fig. 9.

Spatial distribution of the normalized electric field intensity corresponding to the point Q in Fig. 8.

Fig. 10.
Fig. 10.

(a) Absorptance of a nonlinear slab with the Kerr-type nonlinearity only in ε and with a parabolic spatial variation of εL (=1+8z(z-L)/L 2+0.00001i), plotted versus the nonlinearity parameter αw (> 0) in the p wave case. The parameters used are k 0 L=20, μL =ε 1=ε 2=μ 1=μ 2=1, θ=1° and β=0. The point R corresponds to αw 0=0.0001156.

Fig. 11.
Fig. 11.

Spatial distributions of (a) the normalized magnetic field intensity and (b) the effective dielectric permittivity, corresponding to the point R in Fig. 10. ε changes discontinuously wherever it passes through zero. The discontinuities at k 0 z=2.95 and 17.05 are too small to be noticed.

Equations (33)

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d 2 E d z 2 1 μ ( z ) d μ d z d E d z + [ k 0 2 ε ( z ) μ ( z ) q 2 ] E = 0 .
ε ( z , E ( z ) 2 ) = ε L ( z ) + α ( z ) f ( E ( z ) 2 ) ,
μ ( z , H ( z ) 2 ) = μ L ( z ) + β ( z ) g ( H ( z ) 2 ) ,
E ~ ( x , y ) = { v exp [ i p ( L z ) + iqx ] + vr ( L ) exp [ ip ( z L ) + iqx ] , z > L vt ( L ) exp ( ip z + iqx ) , z < 0 ,
u ( z , L , w ) = G ( z , L , L , w )
+ ip 2 0 L d z G ( z , z , L , w ) { ε ( z , wI ( z , L , w ) ) ε 1 μ ( z , wJ ( z , L , w ) ) μ 1
+ q 2 p 2 [ ε ( z , w I ( z , L , w ) ) ε 1 μ 1 μ ( z , w J ( z , L , z ) ) ] } u ( z , L , w ) ,
G ( z , z , L , w ) = exp [ ip sgn ( z z ) z z d z μ ( z , wJ ( z , L , w ) ) μ 1 ] ,
J ( z , L , w ) = ε 1 μ 1 μ ( z , wJ ( z , L , w ) ) 2 [ 1 k 2 u ( z , L , w ) z 2 + q 2 k 2 u ( z , L , w ) 2 ] .
u ( z , L , w ) L = A ( L , w ) u ( z , L , w ) + w [ A ( L , w ) + A * ( L , w ) ] u ( z , L , w ) w ,
A ( L , w ) = ip μ ( L , wJ ( L , L , w ) ) μ 1 + ip 2 { ε ( L , wI ( L , L , w ) ) ε 1 μ ( L , wJ ( L , L , w ) ) μ 1
+ q 2 p 2 [ ε ( L , wI ( L , L , w ) ) ε 1 μ 1 μ ( L , wJ ( L , L , w ) ) ] } u ( L , L , w ) .
u ( L , L , w ) = 1 + r ( L , w ) ,
u ( 0 , L , w ) = t ( L , w ) ,
r ( l , w ) l = 2 ip μ ( l , wJ ( l , l , w ) ) μ 1 r ( l , w ) + ip 2 { ε ( l , wI ( l , l , w ) ) ε 1 μ ( l , wJ ( l , l , w ) ) μ 1
+ q 2 p 2 [ ε ( l , wI ( l , l , w ) ) ε 1 μ 1 μ ( l , wJ ( l , l , w ) ) ] } [ 1 + r ( l , w ) ] 2
+ w [ A ( l , w ) + A * ( l , w ) ] r ( l , w ) w ,
t ( l , w ) l = ip μ ( l , wJ ( l , l , w ) ) μ 1 t ( l , w ) + ip 2 { ε ( l , wI ( l , l , w ) ) ε 1 μ ( l , wJ ( l , l , w ) ) μ 1
+ q 2 p 2 [ ε ( l , wI ( l , l , w ) ) ε 1 μ 1 μ ( l , wJ ( l , l , w ) ) ] } [ 1 + r ( l , w ) ] t ( l , w )
+ w [ A ( l , w ) + A * ( l , w ) ] t ( l , w ) w ,
1 p d r ( l ) d l = 2 i μ ( l ) μ 1 r ( l ) + i 2 a ( l ) [ 1 + r ( l ) ] 2 ,
1 p d t ( l ) d l = i μ ( l ) μ 1 t ( l ) + i 2 a ( l ) [ 1 + r ( l ) ] t ( l ) ,
1 p d w ( l ) d l = [ Im b ( l ) ] w ( l ) ,
a ( l ) = ε ( l ) ε 1 μ ( l ) μ 1 + [ ε ( l ) ε 1 μ 1 μ ( l ) ] tan 2 θ ,
b ( l ) = 2 μ ( l ) μ 1 + a ( l ) [ 1 + r ( l ) ] ,
ε ( l ) = ε L ( l ) + α ( l ) f ( w ( l ) 1 + r ( l ) 2 ) ,
μ ( l ) = μ L ( l ) + β ( l ) g ( w ( l ) J ~ ( l ) ) ,
J ~ ( l ) = ε 1 μ 1 μ ( l ) 2 1 + r ( l ) 2 sin 2 θ + ε 1 μ 1 1 r ( l ) 2 cos 2 θ .
ε ( l ) = ε L ( l ) + α ( l ) w ( l ) 1 + r ( l ) 2 ,
μ ( l ) = μ L ( l ) + β ( l ) w ( l ) J ~ ( l ) .
r ( 0 ) = μ 2 ε 1 μ 1 cos θ μ 1 ( ε 2 μ 2 ε 1 μ 1 sin 2 θ ) 1 2 μ 2 ε 1 μ 1 cos θ + μ 1 ( ε 2 μ 2 ε 1 μ 1 sin 2 θ ) 1 2 ,
t ( 0 ) = 2 μ 2 ε 1 μ 1 cos θ μ 2 ε 1 μ 1 cos θ + μ 1 ( ε 2 μ 2 ε 1 μ 1 sin 2 θ ) 1 2 .
1 p u ( z , l ) l = i μ ( l ) μ 1 u ( z , l ) + i 2 a ( l ) [ 1 + r ( l ) ] u ( z , l ) ,

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