## 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.

© 2011 Optical Society of America

## 1. Introduction

The propagation of electromagnetic waves is strongly influenced by their polarization. A well-known example is the Brewster effect, which states that when a *p*-polarized wave is incident on a uniform dielectric medium with refractive index *n* at the Brewster angle *θ _{B}* satisfying tan

*θ*=

_{B}*n*, it is completely transmitted. If the dielectric medium is inhomogeneous, the elementary argument is not applicable and Brewster’s law needs to be modified [1,2]. In this paper, we consider a situation where there is one-dimensional random inhomogeneity. This problem has close similarity to the Anderson localization problem of quantum particles and classical waves in one dimension [3–5].

The localization effect is stronger in one dimension than in three dimensions. It is widely known that noninteracting electrons and waves in one dimension are localized in an arbitrarily weak random potential [3, 4]. There exist, however, several interesting exceptions where waves are extended in one-dimensional random systems [6–9]. In this study, we consider a phenomenon called Brewster anomaly. It has been shown that *p* waves incident on a stratified random medium can be delocalized at a certain critical angle in some special cases [10]. In spite of many papers devoted to this problem, an exact theoretical treatment is still lacking [10–17].

In this paper, we revisit the Brewster anomaly problem and calculate the localization length and the disorder-averaged transmittance in a numerically precise manner using the invariant imbedding method [18–22]. We find that Brewster’s law is modified greatly in stratified random media. The localization length is found to take a very large maximum value at some incident angle, which we call the generalized Brewster angle. The disorder-averaged transmittance also takes a maximum close to one at the same angle. Even in the presence of an arbitrarily weak disorder, we find that the generalized Brewster angle is substantially different from the Brewster angle in uniform media. We discover 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 make a surprising observation that the Anderson localization of incident *p* waves can be substantially weakened by disorder in a certain parameter regime, when the average dielectric permittivity is less than one. This phenomenon can be of great interest in connections with metamaterials.

## 2. Wave equation

We consider the propagation of a plane electromagnetic wave of vacuum wave number *k* =*ω*/*c*. The wave is incident on a stratified random medium, where the dielectric permittivity *ɛ* varies only in the *z* direction. We assume that the random medium lies in 0 ≤ *z* ≤ *L* and the wave propagates in the *xz* plane. In the *s* wave case, the complex amplitude of the electric field, *E*, satisfies

*q*is the

*x*component of the wave vector. When

*θ*is the angle of incidence,

*q*is equal to

*k*sin

*θ*. In the

*p*wave case, the magnetic field amplitude,

*H*, satisfies

We assume that the wave is incident from the region where *z* > *L* and *ɛ* = 1 and transmitted to the region where *z* < 0 and *ɛ* = 1. In 0 ≤ *z* ≤ *L*, *ɛ* is given by *ɛ*(*z*) = *ɛ̄* + *δɛ*(*z*), where *ɛ̄* is the disorder-averaged relative dielectric permittivity of the random medium and *δɛ*(*z*) is a Gaussian random function satisfying

*g̃*measures the strength of randomness and 〈···〉 denotes statistical averaging over disorder.

## 3. Invariant imbedding equations

We consider a *p* wave of unit magnitude *H̃*(*x,z*) = *H*(*z*)exp(*iqx*) = exp[*ip*(*L – z*) + *iqx*], where *p* = *k*cos*θ*, incident on the medium. The quantities of main interest are the reflection and transmission coefficients, *r* = *r*(*L*) and *t* = *t*(*L*), defined by

*r*and

*t*[18–20]

*l*= 0 to

*l*=

*L*using the initial conditions,

*r*(0) = 0 and

*t*(0) = 1. We will use Eq. (5) in calculating the disorder averages of various quantities consisting of

*r*and

*t*. In this paper, we are mainly interested in the disorder-averaged transmittance 〈

*T*〉 (=〈|

*t*|

_{2}〉) and the localization length

*ξ*defined by

*ξ*= −lim

_{L}_{→∞}(

*L*/〈ln

*T*〉). In the absence of dissipation, 〈

*T*〉 is equal to 1 – 〈

*R*〉, where

*R*(= |

*r*|

^{2}) is the reflectance.

## 4. Disorder-averaged transmittance and the localization length

An infinite number of coupled *nonrandom* ordinary differential equations satisfied by the moments of the reflectance are obtained using Eq. (5) and Novikov’s formula [23, 24]. We introduce the definition *Z _{nñ}* ≡ 〈

*r*〉. It turns out that in order to obtain 〈

^{n}r^{*ñ}*R*〉 = 〈

^{n}*r*〉 for

^{n}r^{*n}*n*> 0, one needs to compute the moments

*Z*= 〈

_{nñ}*r*〉 for all nonnegative integers

^{n}r^{*ñ}*n*and

*ñ*. In other words, the moments

*Z*with

_{nñ}*n*=

*ñ*are coupled to those with

*n*≠

*ñ*. In order to be able to use Novikov’s formula in the

*p*wave case, we assume that the inhomogeneity is weak and use the approximation

*τ*

_{1}= (tan

^{2}

*θ*)/

*ɛ̄*and

*τ*

_{2}= (tan

^{2}

*θ*)/

*ɛ̄*

^{2}. We emphasize that this is the only approximation used in the present study.

The differential equations satisfied by *Z _{nñ}* are derived by a straightforward generalization of the method developed in [24] and have the form

The initial conditions for *Z _{nñ}*’s are

*Z*

_{00}= 1 and

*Z*(

_{nñ}*l*= 0) = 0 for

*n*> 0 or

*ñ*> 0. The moments

*Z*with

_{nñ}*n, ñ*≥ 0 are coupled to one another and their magnitudes decay as either

*n*or

*ñ*increases. Based on this crucial observation, we solve the infinite number of coupled differential equations, Eq. (7), by a simple truncation method. We assume

*Z*= 0 for either

_{nñ}*n*or

*ñ*greater than some large positive integer

*N*and solve the

*finite*number [= (

*N*+ 1)

^{2}] of coupled differential equations numerically for given values of

*kL*,

*θ*,

*ɛ̄*and

*g*. We increase the cutoff

*N*, repeat a similar calculation, and then compare the newly obtained

*Z*with the value of the previous step. If there is no change in the values of

_{nñ}*Z*within an allowed numerical error, we conclude that we have obtained the exact solution of

_{nñ}*Z*.

_{nñ}In order to compute the localization length, we need to compute the average 〈ln*T*〉 in the *l* → ∞ limit. The nonrandom differential equation satisfied by 〈ln*T*〉 is obtained using Eq. (5) and Novikov’s formula in a straightforward manner

*l*→ ∞ limit.

In [10], an approximate analytical expression for the localization length has been derived. In terms of our notation, it can be written as

*θ*smaller than the critical angle

*θ*defined by $\text{sin}{\theta}_{c}=\sqrt{\overline{\varepsilon}}$. We notice that the localization length given by Eq. (11) is inversely proportional to

_{c}*g*and diverges at the angle

*θ*defined by Except for the special case where

_{B}*ɛ̄*= 1 and

*θ*= 45°, the divergence of

_{B}*ξ*is an artifact of the approximation used in [10]. In the next section, we will also show that

*ξ*is not strictly proportional to

*g*

^{−1}.

## 5. Results

In Fig. 1, we plot the normalized localization length, *kξ*, as a function of the 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* is equal to 1, the Anderson localization is destroyed and *ξ* diverges at *θ* = 45° for all nonzero values of *g*, as is expected from the theory of Brewster anomaly [10]. We also find numerically that when *ɛ̄* is different from 1, the localization length becomes extremely large at some critical incident angle, though it is not strictly divergent unlike when *ɛ̄* = 1. We call this angle the generalized Brewster angle and denote it by *θ _{B}*. We observe that as

*ɛ̄*increases from zero,

*θ*increases from 0 to 90° rapidly in a monotonic manner.

_{B}In Fig. 2, we show the dependence of *θ _{B}* on

*ɛ̄*obtained by solving Eq. (7) in a numerically exact way. We find that the curve obtained when

*g*= 0.1 is indistinguishable from that obtained when

*g*= 0.05. This result is compared with the analytical formula in the uniform case, $\text{tan}{\theta}_{B}=\sqrt{\overline{\varepsilon}}$. We find that except for the cases where

*ɛ̄*= 1 or

*ɛ̄*≈ 0, the generalized Brewster angle in random media is substantially different from the ordinary Brewster angle in uniform media, even when the disorder parameter

*g*is arbitrarily small. This leads to a remarkable conclusion that in stratified random media with weak uncorrelated Gaussian disorder, the dependence of

*θ*on

_{B}*ɛ̄*is

*universal*in the sense that it is independent of the strength of disorder. To the best of our knowledge, this kind of universality has never been reported previously.

Somewhat surprisingly, we find that the dependence of *θ _{B}* on

*ɛ̄*in the random case obtained numerically agrees almost perfectly with the approximate analytical formula, Eq. (12), in the interval 0 <

*ɛ̄*< 2. According to the theory leading to Eq. (12),

*θ*is equal to 90° for

_{B}*ɛ̄*≥ 2, whereas our numerical result shows that

*θ*is slightly smaller than 90° when

_{B}*ɛ̄*= 2 and approaches to 90° in a logarithmically slow manner as

*ɛ̄*increases above 2.

In Fig. 3, we compare 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. When *ɛ̄* is sufficiently larger than 1 and *g* is sufficiently small, our numerical result agrees extremely well with Eq. (11), as can be seen in Fig. 3(a). As *ɛ̄* decreases below 1, the disagreement between our result and Eq. (11) becomes significant. We note that the localization length given by Eq. (11) vanishes at the critical angle *θ _{c}* such that
$\text{sin}{\theta}_{c}=\sqrt{\overline{\varepsilon}}$ and is undefined above

*θ*. On the contrary, the localization length obtained numerically using our method is well-defined for all incident angles and does not show any distinct feature at

_{c}*θ*=

*θ*. The discrepancy between our result and Eq. (11) is especially large when

_{c}*ɛ̄*is very small and

*θ*is larger than

*θ*, as can be seen clearly in Fig. 3(d).

_{B}A careful examination of Fig. 3(d) reveals that when *ɛ̄* is sufficiently small and *θ* is somewhat larger than *θ _{B}*, the localization length actually increases as the disorder strength

*g*increases. This implies a very strange phenomenon that the Anderson localization becomes weaker due to stronger disorder. In Fig. 4, we plot the normalized localization length as a function of the disorder strength

*g*for three different values of the incident angle,

*θ*= 30°, 45° and 60°, when

*ɛ̄*= 0.6,

*θ*= 33.2° and

_{B}*θ*= 50.8°. In order to clarify the influence of wave polarization, we show the results for incident

_{c}*s*waves as well as those for incident

*p*waves. In the

*p*wave case, we observe three different kinds of behavior depending on the incident angle. When

*θ*is smaller than

*θ*, the localization length decreases monotonically as

_{B}*g*increases in the weak disorder regime. When

*θ*is larger than

*θ*but smaller than

_{B}*θ*, the localization length shows a surprising nonmonotonic behavior with

_{c}*g*such that it initially decreases to a minimum and then increases as

*g*increases. In other words, a disorder-enhanced delocalization phenomenon occurs above a small critical value of

*g*, which depends on the incident angle. Finally, when

*θ*is above

*θ*, we observe that

_{c}*ξ*increases monotonically as

*g*increases from zero. In this case, it is more appropriate to call

*ξ*the tunneling decay length, rather than the localization length. The monotonic increase of the tunneling decay length with disorder strength is also a novel phenomenon, which has never been reported before.

The behavior in the *s* wave case is completely different. Since there is no Brewster effect in this case, the Brewster angle does not exist. Below the critical angle *θ _{c}*, we find that the localization length is always a decreasing function of

*g*, as is usually expected. Above

*θ*, however,

_{c}*ξ*initially increases to a maximum and then decreases as

*g*increases. The enhancement of

*ξ*due to disorder when

*g*is very small is related to the disorder-enhanced tunneling transmission phenomenon, which has been studied in detail previously [25–28].

From the comparison between the two cases, it is clear that the disorder-enhanced delocalization phenomenon occurring when the incident angle is between *θ _{B}* and

*θ*is unique only for the

_{c}*p*waves. The monotonic increase of the tunneling decay length with disorder when

*θ*>

*θ*is also unique for the

_{c}*p*waves. We have verified that these phenomena occur only when

*ɛ̄*is smaller than 1 and there is critical angle. This implies that they occur due to an interplay among the Brewster effect, the total reflection and the Anderson localization in random media.

We have also calculated the disorder-averaged transmittance 〈*T*〉 by solving Eq. (7). In Fig. 5, we plot 〈*T*〉 versus *θ* for *p* waves, when *kL* = 20, *g* = 0.1 and *ɛ̄* = 0.6, 1, 1.4. Even though the medium is one-dimensionally disordered, there exists an incident angle at which the *p* waves are almost completely transmitted and 〈*T*〉 is very close to 1. If *ɛ̄* = 1, 〈*T*〉 is strictly equal to 1 at *θ* = 45° for all values of *kL*. When *ɛ̄* ≠ 1, however, the maximum value of 〈*T*〉 is slightly smaller than 1 and decreases gradually as |*ɛ̄* – 1| increases. The angle at which 〈*T*〉 takes a maximum value, *θ _{m}*, is similar to

*θ*plotted in Fig. 2, though there is a small discrepancy when

_{B}*ɛ̄*≠ 1. As

*kL*increases to infinity,

*θ*converges to

_{m}*θ*in a logarithmically slow manner.

_{B}The disorder-enhanced delocalization phenomenon is also manifested in the behavior of the disorder-averaged transmittance. In Fig. 6, we plot 〈*T*〉 versus *θ*, when *kL* = 20, *ɛ̄* = 0.6 and *g* = 0.1, 0.2. We notice that the behavior above *θ _{m}* (≈

*θ*) is opposite to that below

_{B}*θ*. Above

_{m}*θ*, the value of 〈

_{m}*T*〉 is larger for

*g*= 0.2 than for

*g*= 0.1. In other words, transmission is enhanced due to stronger disorder.

## 6. Conclusion

In summary, we have studied the propagation and the Anderson localization of *p* waves incident obliquely on randomly stratified media with weak uncorrelated Gaussian disorder. We have confirmed that the Anderson localization of *p* waves is almost destroyed at a critical incident angle called the generalized Brewster angle. Even in the presence of an arbitrarily weak disorder, we have found that the generalized Brewster angle is substantially different from the Brewster angle in uniform media. We have discovered a new kind of universality such that the dependence of the generalized Brewster angle on the average dielectric permittivity is universal. In addition, we have made a surprising observation that the Anderson localization of incident *p* waves can be substantially weakened by disorder in a certain parameter regime. Attempts to confirm these predictions in experiments on carefully designed multilayered random media will be highly desirable.

## Acknowledgments

This work has been supported by the Korea Science and Engineering Foundation grant (No. R0A-2007-000-20113-0) funded by the Korean Government.

## References and links

**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]