Universal shift of the Brewster angle and disorder-enhanced delocalization of p waves in stratified random media

Kwang Jin Lee; Kihong Kim

doi:10.1364/OE.19.020817

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 tanθ_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

\frac{d^{2} E}{d z^{2}} + [k^{2} ɛ (z) - q^{2}] E = 0,

where q is the x component of the wave vector. When θ is the angle of incidence, q is equal to ksinθ. In the p wave case, the magnetic field amplitude, H, satisfies

\frac{d^{2} H}{d z^{2}} - \frac{1}{ɛ (z)} \frac{d ɛ}{d z} \frac{d H}{d z} + [k^{2} ɛ (z) - q^{2}] H = 0 .

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

〈 δ ɛ (z) δ ɛ (z^{'}) 〉 = \tilde{g} δ (z - z^{'}), 〈 δ ɛ (z) 〉 = 0 .

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 = kcosθ, incident on the medium. The quantities of main interest are the reflection and transmission coefficients, r = r(L) and t = t(L), defined by

\tilde{H} (x, z) = {\begin{array}{l} [e^{ip (L - z)} + r e^{ip (z - L)}] e^{iqx}, & z > L \\ t e^{- ipz + iqx}, & z < 0 \end{array} .

Using the invariant imbedding method of wave propagation in stratified media, we derive exact differential equations satisfied by r and t [18–20]

\begin{array}{l} \frac{1}{ik cos θ} \frac{d r}{d l} & = & 2 [\bar{ɛ} + δ ɛ (l)] r (l) - \frac{\bar{ɛ} - 1 + δ ɛ (l)}{2} [1 - \frac{{tan}^{2} θ}{\bar{ɛ} + δ ɛ (l)}] {[1 + r (l)]}^{2}, \\ \frac{1}{ik cos θ} \frac{dt}{d l} & = & [\bar{ɛ} + δ ɛ (l) t (l)] - \frac{\bar{ɛ} - 1 + δ ɛ (l)}{2} [1 - \frac{{tan}^{2} θ}{\bar{ɛ} + δ ɛ (l)}] [1 + r (l)] t (l) . \end{array}

These equations are integrated from 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/〈lnT〉). 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 ≡ 〈rⁿr^*ñ〉. It turns out that in order to obtain 〈Rⁿ〉 = 〈rⁿr^*n〉 for n > 0, one needs to compute the moments Z_nñ = 〈rⁿr^*ñ〉 for all nonnegative integers n and ñ. In other words, the moments Z_nñ with 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

[\bar{ɛ} - 1 + δ ɛ (l)] [1 - \frac{{tan}^{2} θ}{\bar{ɛ} + δ ɛ (l)}] \approx (\bar{ɛ} - 1) (1 - τ_{1}) + (1 - τ_{2}) δ ɛ (l),

where τ₁ = (tan²θ)/ɛ̄ and τ₂ = (tan²θ)/ɛ̄². 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

\begin{array}{l} \frac{1}{k} \frac{d Z_{n \tilde{n}}}{dl} = [i c_{1} (n - \tilde{n}) - g_{3} {(n - \tilde{n})}^{2} - g_{2} (n^{2} + {\tilde{n}}^{2})] Z_{n \tilde{n}} \\ + [- i c_{2} + g_{1} (2 n - 2 \tilde{n} + 1)] n Z_{n + 1, \tilde{n}} + [i c_{2} - g_{1} (2 n - 2 \tilde{n} - 1)] \tilde{n} Z_{n, \tilde{n} + 1} \\ + [- i c_{2} + g_{1} (2 n - 2 \tilde{n} - 1)] n Z_{n - 1, \tilde{n}} + [i c_{2} - g_{1} (2 n - 2 \tilde{n} + 1)] \tilde{n} Z_{n, \tilde{n} - 1} \\ + g_{2} n \tilde{n} Z_{n + 1, \tilde{n} + 1} + g_{2} n \tilde{n} Z_{n - 1, \tilde{n} - 1} + g_{2} n \tilde{n} Z_{n + 1, \tilde{n} - 1} + g_{2} n \tilde{n} Z_{n - 1, \tilde{n} + 1} \\ - \frac{g_{2}}{2} n (n + 1) Z_{n + 2, \tilde{n}} - \frac{g_{2}}{2} \tilde{n} (\tilde{n} + 1) Z_{n, \tilde{n} + 2} - \frac{g_{2}}{2} n (n - 1) Z_{n - 2, \tilde{n}} - \frac{g_{2}}{2} \tilde{n} (\tilde{n} - 1) Z_{n, \tilde{n} - 2}, \end{array}

where

\begin{array}{l} c_{1} = [2 + (\bar{ɛ} - 1) (1 + τ_{1})] cos θ, c_{2} = [(\bar{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 = \tilde{g} k / 4 . \end{array}

The initial conditions for Z_nñ’s are Z₀₀ = 1 and Z_nñ(l = 0) = 0 for n > 0 or ñ > 0. The moments Z_nñ with 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_nñ = 0 for either 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_nñ with the value of the previous step. If there is no change in the values of Z_nñ within an allowed numerical error, we conclude that we have obtained the exact solution of Z_nñ.

In order to compute the localization length, we need to compute the average 〈lnT〉 in the l → ∞ limit. The nonrandom differential equation satisfied by 〈lnT〉 is obtained using Eq. (5) and Novikov’s formula in a straightforward manner

\frac{1}{k} \frac{d 〈 ln T 〉}{d l} = - g_{2} - Re [(i c_{2} - 2 g_{1}) Z_{10} + g_{2} Z_{20}],

which reduces to

\frac{1}{k ξ} = g_{2} + Re [(i c_{2} - 2 g_{1}) Z_{10} (l \to \infty) + g_{2} Z_{20} (l \to \infty)],

in the 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

k ξ = \frac{{\bar{ɛ}}^{2}}{g} \frac{\bar{ɛ} - {sin}^{2} θ}{{(\bar{ɛ} - 2 {sin}^{2} θ)}^{2}},

which is applied for θ smaller than the critical angle θ_c defined by

sin θ_{c} = \sqrt{\bar{ɛ}}

. We notice that the localization length given by Eq. (11) is inversely proportional to g and diverges at the angle θ_B defined by

sin θ_{B} = {(\frac{\bar{ɛ}}{2})}^{1 / 2} .

Except for the special case where ɛ̄ = 1 and θ_B = 45°, the divergence of 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_B increases from 0 to 90° rapidly in a monotonic manner.

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.

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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, $tan θ_{B} = \sqrt{\bar{ɛ}}$ . 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_B on 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.

Fig. 2 Generalized Brewster angle θ_B, 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, $tan θ_{B} = \sqrt{\bar{ɛ}}$ , and Eq. (12).

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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_B is equal to 90° for ɛ̄ ≥ 2, whereas our numerical result shows that θ_B is slightly smaller than 90° when 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 $sin θ_{c} = \sqrt{\bar{ɛ}}$ and is undefined above θ_c. 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 θ = θ_c. The discrepancy between our result and Eq. (11) is especially large when ɛ̄ is very small and θ is larger than θ_B, as can be seen clearly in Fig. 3(d).

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.

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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_B = 33.2° and θ_c = 50.8°. In order to clarify the influence of wave polarization, we show the results for incident 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_B, the localization length decreases monotonically as g increases in the weak disorder regime. When θ is larger than θ_B but smaller than θ_c, the localization length shows a surprising nonmonotonic behavior with 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_c, we observe that 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.

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°.

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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_c, however, 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_c is unique only for the p waves. The monotonic increase of the tunneling decay length with disorder when θ > θ_c is also unique for the 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_B plotted in Fig. 2, though there is a small discrepancy when ɛ̄ ≠ 1. As kL increases to infinity, θ_m converges to θ_B in a logarithmically slow manner.

Fig. 5 Disorder-averaged transmittance versus incident angle for p waves, when kL = 20, g = 0.1 and ɛ̄ = 0.6, 1, 1.4.

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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_B) is opposite to that below θ_m. Above θ_m, the value of 〈T〉 is larger for g = 0.2 than for g = 0.1. In other words, transmission is enhanced due to stronger disorder.

Fig. 6 Disorder-averaged transmittance versus incident angle for p waves, when kL = 20, ɛ̄ = 0.6 and g = 0.1 and 0.2.

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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.

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Universal shift of the Brewster angle and disorder-enhanced delocalization of p waves in stratified random media

Abstract

1. Introduction

2. Wave equation

3. Invariant imbedding equations

4. Disorder-averaged transmittance and the localization length

5. Results

6. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Equations (12)

Optics Express