Abstract

We study theoretically the propagation and the Anderson localization of p-polarized electromagnetic waves incident obliquely on randomly stratified dielectric media with weak uncorrelated Gaussian disorder. Using the invariant imbedding method, we calculate the localization length and the disorder-averaged transmittance in a numerically precise manner. We find that the localization length takes an extremely large maximum value at some critical incident angle, which we call the generalized Brewster angle. The disorder-averaged transmittance also takes a maximum very close to one at the same incident angle. Even in the presence of an arbitrarily weak disorder, the generalized Brewster angle is found to be substantially different from the ordinary Brewster angle in uniform media. It is a rapidly increasing function of the average dielectric permittivity and approaches 90° when the average relative dielectric permittivity is slightly larger than two. We make a remarkable observation that the dependence of the generalized Brewster angle on the average dielectric permittivity is universal in the sense that it is independent of the strength of disorder. We also find, surprisingly, that when the average relative dielectric permittivity is less than one and the incident angle is larger than the generalized Brewster angle, both the localization length and the disorder-averaged transmittance increase substantially as the strength of disorder increases in a wide range of the disorder parameter. In other words, the Anderson localization of incident p waves can be weakened by disorder in a certain parameter regime.

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  1. J.-J. Greffet, “Theoretical model of the shift of the Brewster angle on a rough surface,” Opt. Lett. 17, 238–240 (1992).
    [CrossRef] [PubMed]
  2. T. Kawanishi, “The shift of Brewster’s scattering angle,” Opt. Commun. 186, 251–258 (2000).
    [CrossRef]
  3. P. A. Lee and T. V. Ramakrishnan, “Disordered electronic systems,” Rev. Mod. Phys. 57, 287–337 (1985).
    [CrossRef]
  4. I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, Introduction to the Theory of Disordered Systems (Wiley, 1988).
  5. P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Academic Press, 1995).
  6. F. Delyon, B. Simon, and B. Souillard, “From power-localized to extended states in a class of one-dimensional disordered systems,” Phys. Rev. Lett. 52, 2187–2189 (1984).
    [CrossRef]
  7. N. Hatano and D. R. Nelson, “Localization transitions in non-Hermitian quantum mechanics,” Phys. Rev. Lett. 77, 570–573 (1996).
    [CrossRef] [PubMed]
  8. F. A. B. F. de Moura and M. L. Lyra, “Delocalization in the 1D Anderson model with long-range correlated disorder,” Phys. Rev. Lett. 81, 3735–3738 (1998).
    [CrossRef]
  9. J. Heinrichs, “Absence of localization in a disordered one-dimensional ring threaded by an Aharonov-Bohm flux,” J. Phys. Condens. Matter 21, 295701 (2009).
    [CrossRef] [PubMed]
  10. J. E. Sipe, P. Sheng, B. S. White, and M. H. Cohen, “Brewster anomalies: a polarization-induced delocalization effect,” Phys. Rev. Lett. 60, 108–111 (1988).
    [CrossRef] [PubMed]
  11. A. G. Aronov, V. M. Gasparian, and U. Gummich, “Transmission of waves throgh one-dimensional random layered systems,” J. Phys. Condens. Matter 3, 3023–3039 (1991).
    [CrossRef]
  12. W. Kohler, G. Papanicolaou, M. Postel, and B. White, “Reflection of pulsed electromagnetic waves from a randomly stratified half-space,” J. Opt. Soc. Am. A 8, 1109–1125 (1991).
    [CrossRef]
  13. A. Kondilis, “Combined effect of periodicity, disorder, and absorption on wave propagation through stratified media: an approximate analytical solution,” Phys. Rev. B 55, 14214–14221 (1997).
    [CrossRef]
  14. W. Deng and Z.-Q. Zhang, “Amplification and localization behaviors of obliquely incident light in randomly layered media,” Phys. Rev. B 55, 14230–14235 (1997).
    [CrossRef]
  15. X. Du, D. Zhang, X. Zhang, B. Feng, and D. Zhang, “Localization and delocalization of light under oblique incidence,” Phys. Rev. B 56, 28–31 (1997).
    [CrossRef]
  16. K. Kim, F. Rotermund, D.-H. Lee, and H. Lim, “Propagation of p-polarized electromagnetic waves obliquely incident on stratified random media: random phase approximation,” Wave Random Complex 17, 43–53 (2007).
    [CrossRef]
  17. D. Mogilevtsev, F. A. Pinheiro, R. R. dos Santos, S. B. Cavalcanti, and L. E. Oliveira, “Suppression of Anderson localization of light and Brewster anomalies in disordered superlattices containing a dispersive metamaterial,” Phys. Rev. B 82, 081105 (2010).
    [CrossRef]
  18. R. Rammal and B. Doucot, “Invariant imbedding approach to localization. I. general framework and basic equations,” J. Phys. (Paris) 48, 509–526 (1987).
    [CrossRef]
  19. V. I. Klyatskin, “The imbedding method in statistical boundary-value wave problems,” Prog. Opt. 33, 1–127 (1994).
    [CrossRef]
  20. K. Kim, H. Lim, and D.-H. Lee, “Invariant imbedding equations for electromagnetic waves in stratified magnetic media: applications to one-dimensional photonic crystals,” J. Korean Phys. Soc. 39, L956–L960 (2001).
  21. 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]
  22. K. Kim, D. K. Phung, F. Rotermund, and H. Lim, “Propagation of electromagnetic waves in stratified media with nonlinearity in both dielectric and magnetic responses,” Opt. Express 16, 1150–1164 (2008).
    [CrossRef] [PubMed]
  23. E. A. Novikov, “Functionals and the random-force method in turbulence theory,” Sov. Phys. JETP 20, 1290–1294 (1965).
  24. K. Kim, “Reflection coefficient and localization length of waves in one-dimensional random media,” Phys. Rev. B 58, 6153–6160 (1998).
    [CrossRef]
  25. V. Freilikher, M. Pustilnik, and I. Yurkevich, “Enhanced transmission through a disordered potential barrier,” Phys. Rev. B 53, 7413–7416 (1996).
    [CrossRef]
  26. J. M. Luck, “Non-monotonic disorder-induced enhanced tunnelling,” J. Phys. A 37, 259–271 (2004).
    [CrossRef]
  27. K. Kim, F. Rotermund, and H. Lim, “Disorder-enhanced transmission of a quantum mechanical particle through a disordered tunneling barrier in one dimension: exact calculation based on the invariant imbedding method,” Phys. Rev. B 77, 024203 (2008).
    [CrossRef]
  28. J. Heinrichs, “Enhanced quantum tunnelling induced by disorder,” J. Phys. Condens. Matter 20, 395215 (2008).
    [CrossRef]

2010 (1)

D. Mogilevtsev, F. A. Pinheiro, R. R. dos Santos, S. B. Cavalcanti, and L. E. Oliveira, “Suppression of Anderson localization of light and Brewster anomalies in disordered superlattices containing a dispersive metamaterial,” Phys. Rev. B 82, 081105 (2010).
[CrossRef]

2009 (1)

J. Heinrichs, “Absence of localization in a disordered one-dimensional ring threaded by an Aharonov-Bohm flux,” J. Phys. Condens. Matter 21, 295701 (2009).
[CrossRef] [PubMed]

2008 (3)

K. Kim, F. Rotermund, and H. Lim, “Disorder-enhanced transmission of a quantum mechanical particle through a disordered tunneling barrier in one dimension: exact calculation based on the invariant imbedding method,” Phys. Rev. B 77, 024203 (2008).
[CrossRef]

J. Heinrichs, “Enhanced quantum tunnelling induced by disorder,” J. Phys. Condens. Matter 20, 395215 (2008).
[CrossRef]

K. Kim, D. K. Phung, F. Rotermund, and H. Lim, “Propagation of electromagnetic waves in stratified media with nonlinearity in both dielectric and magnetic responses,” Opt. Express 16, 1150–1164 (2008).
[CrossRef] [PubMed]

2007 (1)

K. Kim, F. Rotermund, D.-H. Lee, and H. Lim, “Propagation of p-polarized electromagnetic waves obliquely incident on stratified random media: random phase approximation,” Wave Random Complex 17, 43–53 (2007).
[CrossRef]

2005 (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]

2004 (1)

J. M. Luck, “Non-monotonic disorder-induced enhanced tunnelling,” J. Phys. A 37, 259–271 (2004).
[CrossRef]

2001 (1)

K. Kim, H. Lim, and D.-H. Lee, “Invariant imbedding equations for electromagnetic waves in stratified magnetic media: applications to one-dimensional photonic crystals,” J. Korean Phys. Soc. 39, L956–L960 (2001).

2000 (1)

T. Kawanishi, “The shift of Brewster’s scattering angle,” Opt. Commun. 186, 251–258 (2000).
[CrossRef]

1998 (2)

F. A. B. F. de Moura and M. L. Lyra, “Delocalization in the 1D Anderson model with long-range correlated disorder,” Phys. Rev. Lett. 81, 3735–3738 (1998).
[CrossRef]

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

1997 (3)

A. Kondilis, “Combined effect of periodicity, disorder, and absorption on wave propagation through stratified media: an approximate analytical solution,” Phys. Rev. B 55, 14214–14221 (1997).
[CrossRef]

W. Deng and Z.-Q. Zhang, “Amplification and localization behaviors of obliquely incident light in randomly layered media,” Phys. Rev. B 55, 14230–14235 (1997).
[CrossRef]

X. Du, D. Zhang, X. Zhang, B. Feng, and D. Zhang, “Localization and delocalization of light under oblique incidence,” Phys. Rev. B 56, 28–31 (1997).
[CrossRef]

1996 (2)

N. Hatano and D. R. Nelson, “Localization transitions in non-Hermitian quantum mechanics,” Phys. Rev. Lett. 77, 570–573 (1996).
[CrossRef] [PubMed]

V. Freilikher, M. Pustilnik, and I. Yurkevich, “Enhanced transmission through a disordered potential barrier,” Phys. Rev. B 53, 7413–7416 (1996).
[CrossRef]

1994 (1)

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

1992 (1)

1991 (2)

W. Kohler, G. Papanicolaou, M. Postel, and B. White, “Reflection of pulsed electromagnetic waves from a randomly stratified half-space,” J. Opt. Soc. Am. A 8, 1109–1125 (1991).
[CrossRef]

A. G. Aronov, V. M. Gasparian, and U. Gummich, “Transmission of waves throgh one-dimensional random layered systems,” J. Phys. Condens. Matter 3, 3023–3039 (1991).
[CrossRef]

1988 (1)

J. E. Sipe, P. Sheng, B. S. White, and M. H. Cohen, “Brewster anomalies: a polarization-induced delocalization effect,” Phys. Rev. Lett. 60, 108–111 (1988).
[CrossRef] [PubMed]

1987 (1)

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

1985 (1)

P. A. Lee and T. V. Ramakrishnan, “Disordered electronic systems,” Rev. Mod. Phys. 57, 287–337 (1985).
[CrossRef]

1984 (1)

F. Delyon, B. Simon, and B. Souillard, “From power-localized to extended states in a class of one-dimensional disordered systems,” Phys. Rev. Lett. 52, 2187–2189 (1984).
[CrossRef]

1965 (1)

E. A. Novikov, “Functionals and the random-force method in turbulence theory,” Sov. Phys. JETP 20, 1290–1294 (1965).

Aronov, A. G.

A. G. Aronov, V. M. Gasparian, and U. Gummich, “Transmission of waves throgh one-dimensional random layered systems,” J. Phys. Condens. Matter 3, 3023–3039 (1991).
[CrossRef]

Cavalcanti, S. B.

D. Mogilevtsev, F. A. Pinheiro, R. R. dos Santos, S. B. Cavalcanti, and L. E. Oliveira, “Suppression of Anderson localization of light and Brewster anomalies in disordered superlattices containing a dispersive metamaterial,” Phys. Rev. B 82, 081105 (2010).
[CrossRef]

Cohen, M. H.

J. E. Sipe, P. Sheng, B. S. White, and M. H. Cohen, “Brewster anomalies: a polarization-induced delocalization effect,” Phys. Rev. Lett. 60, 108–111 (1988).
[CrossRef] [PubMed]

de Moura, F. A. B. F.

F. A. B. F. de Moura and M. L. Lyra, “Delocalization in the 1D Anderson model with long-range correlated disorder,” Phys. Rev. Lett. 81, 3735–3738 (1998).
[CrossRef]

Delyon, F.

F. Delyon, B. Simon, and B. Souillard, “From power-localized to extended states in a class of one-dimensional disordered systems,” Phys. Rev. Lett. 52, 2187–2189 (1984).
[CrossRef]

Deng, W.

W. Deng and Z.-Q. Zhang, “Amplification and localization behaviors of obliquely incident light in randomly layered media,” Phys. Rev. B 55, 14230–14235 (1997).
[CrossRef]

dos Santos, R. R.

D. Mogilevtsev, F. A. Pinheiro, R. R. dos Santos, S. B. Cavalcanti, and L. E. Oliveira, “Suppression of Anderson localization of light and Brewster anomalies in disordered superlattices containing a dispersive metamaterial,” Phys. Rev. B 82, 081105 (2010).
[CrossRef]

Doucot, B.

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

Du, X.

X. Du, D. Zhang, X. Zhang, B. Feng, and D. Zhang, “Localization and delocalization of light under oblique incidence,” Phys. Rev. B 56, 28–31 (1997).
[CrossRef]

Feng, B.

X. Du, D. Zhang, X. Zhang, B. Feng, and D. Zhang, “Localization and delocalization of light under oblique incidence,” Phys. Rev. B 56, 28–31 (1997).
[CrossRef]

Freilikher, V.

V. Freilikher, M. Pustilnik, and I. Yurkevich, “Enhanced transmission through a disordered potential barrier,” Phys. Rev. B 53, 7413–7416 (1996).
[CrossRef]

Gasparian, V. M.

A. G. Aronov, V. M. Gasparian, and U. Gummich, “Transmission of waves throgh one-dimensional random layered systems,” J. Phys. Condens. Matter 3, 3023–3039 (1991).
[CrossRef]

Gredeskul, S. A.

I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, Introduction to the Theory of Disordered Systems (Wiley, 1988).

Greffet, J.-J.

Gummich, U.

A. G. Aronov, V. M. Gasparian, and U. Gummich, “Transmission of waves throgh one-dimensional random layered systems,” J. Phys. Condens. Matter 3, 3023–3039 (1991).
[CrossRef]

Hatano, N.

N. Hatano and D. R. Nelson, “Localization transitions in non-Hermitian quantum mechanics,” Phys. Rev. Lett. 77, 570–573 (1996).
[CrossRef] [PubMed]

Heinrichs, J.

J. Heinrichs, “Absence of localization in a disordered one-dimensional ring threaded by an Aharonov-Bohm flux,” J. Phys. Condens. Matter 21, 295701 (2009).
[CrossRef] [PubMed]

J. Heinrichs, “Enhanced quantum tunnelling induced by disorder,” J. Phys. Condens. Matter 20, 395215 (2008).
[CrossRef]

Kawanishi, T.

T. Kawanishi, “The shift of Brewster’s scattering angle,” Opt. Commun. 186, 251–258 (2000).
[CrossRef]

Kim, K.

K. Kim, F. Rotermund, and H. Lim, “Disorder-enhanced transmission of a quantum mechanical particle through a disordered tunneling barrier in one dimension: exact calculation based on the invariant imbedding method,” Phys. Rev. B 77, 024203 (2008).
[CrossRef]

K. Kim, D. K. Phung, F. Rotermund, and H. Lim, “Propagation of electromagnetic waves in stratified media with nonlinearity in both dielectric and magnetic responses,” Opt. Express 16, 1150–1164 (2008).
[CrossRef] [PubMed]

K. Kim, F. Rotermund, D.-H. Lee, and H. Lim, “Propagation of p-polarized electromagnetic waves obliquely incident on stratified random media: random phase approximation,” Wave Random Complex 17, 43–53 (2007).
[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, H. Lim, and D.-H. Lee, “Invariant imbedding equations for electromagnetic waves in stratified magnetic media: applications to one-dimensional photonic crystals,” J. Korean Phys. Soc. 39, L956–L960 (2001).

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

Klyatskin, V. I.

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

Kohler, W.

Kondilis, A.

A. Kondilis, “Combined effect of periodicity, disorder, and absorption on wave propagation through stratified media: an approximate analytical solution,” Phys. Rev. B 55, 14214–14221 (1997).
[CrossRef]

Lee, D.-H.

K. Kim, F. Rotermund, D.-H. Lee, and H. Lim, “Propagation of p-polarized electromagnetic waves obliquely incident on stratified random media: random phase approximation,” Wave Random Complex 17, 43–53 (2007).
[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, H. Lim, and D.-H. Lee, “Invariant imbedding equations for electromagnetic waves in stratified magnetic media: applications to one-dimensional photonic crystals,” J. Korean Phys. Soc. 39, L956–L960 (2001).

Lee, P. A.

P. A. Lee and T. V. Ramakrishnan, “Disordered electronic systems,” Rev. Mod. Phys. 57, 287–337 (1985).
[CrossRef]

Lifshits, I. M.

I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, Introduction to the Theory of Disordered Systems (Wiley, 1988).

Lim, H.

K. Kim, F. Rotermund, and H. Lim, “Disorder-enhanced transmission of a quantum mechanical particle through a disordered tunneling barrier in one dimension: exact calculation based on the invariant imbedding method,” Phys. Rev. B 77, 024203 (2008).
[CrossRef]

K. Kim, D. K. Phung, F. Rotermund, and H. Lim, “Propagation of electromagnetic waves in stratified media with nonlinearity in both dielectric and magnetic responses,” Opt. Express 16, 1150–1164 (2008).
[CrossRef] [PubMed]

K. Kim, F. Rotermund, D.-H. Lee, and H. Lim, “Propagation of p-polarized electromagnetic waves obliquely incident on stratified random media: random phase approximation,” Wave Random Complex 17, 43–53 (2007).
[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, H. Lim, and D.-H. Lee, “Invariant imbedding equations for electromagnetic waves in stratified magnetic media: applications to one-dimensional photonic crystals,” J. Korean Phys. Soc. 39, L956–L960 (2001).

Luck, J. M.

J. M. Luck, “Non-monotonic disorder-induced enhanced tunnelling,” J. Phys. A 37, 259–271 (2004).
[CrossRef]

Lyra, M. L.

F. A. B. F. de Moura and M. L. Lyra, “Delocalization in the 1D Anderson model with long-range correlated disorder,” Phys. Rev. Lett. 81, 3735–3738 (1998).
[CrossRef]

Mogilevtsev, D.

D. Mogilevtsev, F. A. Pinheiro, R. R. dos Santos, S. B. Cavalcanti, and L. E. Oliveira, “Suppression of Anderson localization of light and Brewster anomalies in disordered superlattices containing a dispersive metamaterial,” Phys. Rev. B 82, 081105 (2010).
[CrossRef]

Nelson, D. R.

N. Hatano and D. R. Nelson, “Localization transitions in non-Hermitian quantum mechanics,” Phys. Rev. Lett. 77, 570–573 (1996).
[CrossRef] [PubMed]

Novikov, E. A.

E. A. Novikov, “Functionals and the random-force method in turbulence theory,” Sov. Phys. JETP 20, 1290–1294 (1965).

Oliveira, L. E.

D. Mogilevtsev, F. A. Pinheiro, R. R. dos Santos, S. B. Cavalcanti, and L. E. Oliveira, “Suppression of Anderson localization of light and Brewster anomalies in disordered superlattices containing a dispersive metamaterial,” Phys. Rev. B 82, 081105 (2010).
[CrossRef]

Papanicolaou, G.

Pastur, L. A.

I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, Introduction to the Theory of Disordered Systems (Wiley, 1988).

Phung, D. K.

Pinheiro, F. A.

D. Mogilevtsev, F. A. Pinheiro, R. R. dos Santos, S. B. Cavalcanti, and L. E. Oliveira, “Suppression of Anderson localization of light and Brewster anomalies in disordered superlattices containing a dispersive metamaterial,” Phys. Rev. B 82, 081105 (2010).
[CrossRef]

Postel, M.

Pustilnik, M.

V. Freilikher, M. Pustilnik, and I. Yurkevich, “Enhanced transmission through a disordered potential barrier,” Phys. Rev. B 53, 7413–7416 (1996).
[CrossRef]

Ramakrishnan, T. V.

P. A. Lee and T. V. Ramakrishnan, “Disordered electronic systems,” Rev. Mod. Phys. 57, 287–337 (1985).
[CrossRef]

Rammal, R.

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

Rotermund, F.

K. Kim, D. K. Phung, F. Rotermund, and H. Lim, “Propagation of electromagnetic waves in stratified media with nonlinearity in both dielectric and magnetic responses,” Opt. Express 16, 1150–1164 (2008).
[CrossRef] [PubMed]

K. Kim, F. Rotermund, and H. Lim, “Disorder-enhanced transmission of a quantum mechanical particle through a disordered tunneling barrier in one dimension: exact calculation based on the invariant imbedding method,” Phys. Rev. B 77, 024203 (2008).
[CrossRef]

K. Kim, F. Rotermund, D.-H. Lee, and H. Lim, “Propagation of p-polarized electromagnetic waves obliquely incident on stratified random media: random phase approximation,” Wave Random Complex 17, 43–53 (2007).
[CrossRef]

Sheng, P.

J. E. Sipe, P. Sheng, B. S. White, and M. H. Cohen, “Brewster anomalies: a polarization-induced delocalization effect,” Phys. Rev. Lett. 60, 108–111 (1988).
[CrossRef] [PubMed]

P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Academic Press, 1995).

Simon, B.

F. Delyon, B. Simon, and B. Souillard, “From power-localized to extended states in a class of one-dimensional disordered systems,” Phys. Rev. Lett. 52, 2187–2189 (1984).
[CrossRef]

Sipe, J. E.

J. E. Sipe, P. Sheng, B. S. White, and M. H. Cohen, “Brewster anomalies: a polarization-induced delocalization effect,” Phys. Rev. Lett. 60, 108–111 (1988).
[CrossRef] [PubMed]

Souillard, B.

F. Delyon, B. Simon, and B. Souillard, “From power-localized to extended states in a class of one-dimensional disordered systems,” Phys. Rev. Lett. 52, 2187–2189 (1984).
[CrossRef]

White, B.

White, B. S.

J. E. Sipe, P. Sheng, B. S. White, and M. H. Cohen, “Brewster anomalies: a polarization-induced delocalization effect,” Phys. Rev. Lett. 60, 108–111 (1988).
[CrossRef] [PubMed]

Yurkevich, I.

V. Freilikher, M. Pustilnik, and I. Yurkevich, “Enhanced transmission through a disordered potential barrier,” Phys. Rev. B 53, 7413–7416 (1996).
[CrossRef]

Zhang, D.

X. Du, D. Zhang, X. Zhang, B. Feng, and D. Zhang, “Localization and delocalization of light under oblique incidence,” Phys. Rev. B 56, 28–31 (1997).
[CrossRef]

X. Du, D. Zhang, X. Zhang, B. Feng, and D. Zhang, “Localization and delocalization of light under oblique incidence,” Phys. Rev. B 56, 28–31 (1997).
[CrossRef]

Zhang, X.

X. Du, D. Zhang, X. Zhang, B. Feng, and D. Zhang, “Localization and delocalization of light under oblique incidence,” Phys. Rev. B 56, 28–31 (1997).
[CrossRef]

Zhang, Z.-Q.

W. Deng and Z.-Q. Zhang, “Amplification and localization behaviors of obliquely incident light in randomly layered media,” Phys. Rev. B 55, 14230–14235 (1997).
[CrossRef]

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. Korean Phys. Soc. (1)

K. Kim, H. Lim, and D.-H. Lee, “Invariant imbedding equations for electromagnetic waves in stratified magnetic media: applications to one-dimensional photonic crystals,” J. Korean Phys. Soc. 39, L956–L960 (2001).

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

J. Phys. (Paris) (1)

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

J. Phys. A (1)

J. M. Luck, “Non-monotonic disorder-induced enhanced tunnelling,” J. Phys. A 37, 259–271 (2004).
[CrossRef]

J. Phys. Condens. Matter (3)

J. Heinrichs, “Enhanced quantum tunnelling induced by disorder,” J. Phys. Condens. Matter 20, 395215 (2008).
[CrossRef]

A. G. Aronov, V. M. Gasparian, and U. Gummich, “Transmission of waves throgh one-dimensional random layered systems,” J. Phys. Condens. Matter 3, 3023–3039 (1991).
[CrossRef]

J. Heinrichs, “Absence of localization in a disordered one-dimensional ring threaded by an Aharonov-Bohm flux,” J. Phys. Condens. Matter 21, 295701 (2009).
[CrossRef] [PubMed]

Opt. Commun. (1)

T. Kawanishi, “The shift of Brewster’s scattering angle,” Opt. Commun. 186, 251–258 (2000).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (7)

K. Kim, F. Rotermund, and H. Lim, “Disorder-enhanced transmission of a quantum mechanical particle through a disordered tunneling barrier in one dimension: exact calculation based on the invariant imbedding method,” Phys. Rev. B 77, 024203 (2008).
[CrossRef]

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

V. Freilikher, M. Pustilnik, and I. Yurkevich, “Enhanced transmission through a disordered potential barrier,” Phys. Rev. B 53, 7413–7416 (1996).
[CrossRef]

A. Kondilis, “Combined effect of periodicity, disorder, and absorption on wave propagation through stratified media: an approximate analytical solution,” Phys. Rev. B 55, 14214–14221 (1997).
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Figures (6)

Fig. 1
Fig. 1

Normalized localization length versus incident angle for p waves, when kL = ∞, ɛ̄ = 0.2, 0.6, 1, 1.4, 1.8 and g = 0.01, 0.05. When ɛ̄ = 1, ξ diverges at θ = 45° for all nonzero values of g.

Fig. 2
Fig. 2

Generalized Brewster angle θB, at which the localization length takes a maximum value, versus ɛ̄. The curve obtained when g = 0.1 agrees precisely with that obtained when g = 0.05. The universal curve obtained in a numerically exact way is compared with the analytical formula in the uniform case, tan θ B = ɛ ¯, and Eq. (12).

Fig. 3
Fig. 3

Comparison between the localization lengths obtained using our numerical method with the approximate analytical formula, Eq. (11), when ɛ̄ = 0.2, 0.6, 1, 1.4 and g = 0.01, 0.05.

Fig. 4
Fig. 4

Normalized localization length versus the disorder strength g for (a) p and (b) s waves, when ɛ̄ = 0.6 and θ = 30°, 45°, 60°.

Fig. 5
Fig. 5

Disorder-averaged transmittance versus incident angle for p waves, when kL = 20, g = 0.1 and ɛ̄ = 0.6, 1, 1.4.

Fig. 6
Fig. 6

Disorder-averaged transmittance versus incident angle for p waves, when kL = 20, ɛ̄ = 0.6 and g = 0.1 and 0.2.

Equations (12)

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d 2 E d z 2 + [ k 2 ɛ ( z ) q 2 ] E = 0 ,
d 2 H d z 2 1 ɛ ( z ) d ɛ d z d H d z + [ k 2 ɛ ( z ) q 2 ] H = 0 .
δ ɛ ( z ) δ ɛ ( z ) = g ˜ δ ( z z ) , δ ɛ ( z ) = 0 .
H ˜ ( x , z ) = { [ e ip ( L z ) + r e ip ( z L ) ] e iqx , z > L t e ipz + iqx , z < 0 .
1 ik cos θ d r d l = 2 [ ɛ ¯ + δ ɛ ( l ) ] r ( l ) ɛ ¯ 1 + δ ɛ ( l ) 2 [ 1 tan 2 θ ɛ ¯ + δ ɛ ( l ) ] [ 1 + r ( l ) ] 2 , 1 ik cos θ dt d l = [ ɛ ¯ + δ ɛ ( l ) t ( l ) ] ɛ ¯ 1 + δ ɛ ( l ) 2 [ 1 tan 2 θ ɛ ¯ + δ ɛ ( l ) ] [ 1 + r ( l ) ] t ( l ) .
[ ɛ ¯ 1 + δ ɛ ( l ) ] [ 1 tan 2 θ ɛ ¯ + δ ɛ ( l ) ] ( ɛ ¯ 1 ) ( 1 τ 1 ) + ( 1 τ 2 ) δ ɛ ( l ) ,
1 k d Z n n ˜ dl = [ i c 1 ( n n ˜ ) g 3 ( n n ˜ ) 2 g 2 ( n 2 + n ˜ 2 ) ] Z n n ˜ + [ i c 2 + g 1 ( 2 n 2 n ˜ + 1 ) ] n Z n + 1 , n ˜ + [ i c 2 g 1 ( 2 n 2 n ˜ 1 ) ] n ˜ Z n , n ˜ + 1 + [ i c 2 + g 1 ( 2 n 2 n ˜ 1 ) ] n Z n 1 , n ˜ + [ i c 2 g 1 ( 2 n 2 n ˜ + 1 ) ] n ˜ Z n , n ˜ 1 + g 2 n n ˜ Z n + 1 , n ˜ + 1 + g 2 n n ˜ Z n 1 , n ˜ 1 + g 2 n n ˜ Z n + 1 , n ˜ 1 + g 2 n n ˜ Z n 1 , n ˜ + 1 g 2 2 n ( n + 1 ) Z n + 2 , n ˜ g 2 2 n ˜ ( n ˜ + 1 ) Z n , n ˜ + 2 g 2 2 n ( n 1 ) Z n 2 , n ˜ g 2 2 n ˜ ( n ˜ 1 ) Z n , n ˜ 2 ,
c 1 = [ 2 + ( ɛ ¯ 1 ) ( 1 + τ 1 ) ] cos θ , c 2 = [ ( ɛ ¯ 1 ) ( 1 τ 1 ) cos θ ] / 2 , g 1 = g ( 1 τ 2 2 ) cos 2 θ , g 2 = g ( 1 τ 2 ) 2 cos 2 θ , g 3 = 2 g ( 1 + τ 2 ) 2 cos 2 θ , g = g ˜ k / 4 .
1 k d ln T d l = g 2 Re [ ( i c 2 2 g 1 ) Z 10 + g 2 Z 20 ] ,
1 k ξ = g 2 + Re [ ( i c 2 2 g 1 ) Z 10 ( l ) + g 2 Z 20 ( l ) ] ,
k ξ = ɛ ¯ 2 g ɛ ¯ sin 2 θ ( ɛ ¯ 2 sin 2 θ ) 2 ,
sin θ B = ( ɛ ¯ 2 ) 1 / 2 .

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