Anderson localization of electromagnetic waves in randomly-stratified magnetodielectric media with uniform impedance

Kihong Kim

doi:10.1364/OE.23.014520

1. Introduction

Anderson localization of electromagnetic (EM) waves in random media is a topic of continuing high research interest [1–5]. A recent flurry of activity in the localization of EM waves in complex media and metamaterials is especially noteworthy [6–13]. Despite many similarities between the localization of EM waves in random dielectric media and that of electron matter waves in disordered conductors, they have several crucial differences. One aspect which has been studied in detail is concerned about the strong dependence of Anderson localization of EM waves on their polarization. The Brewster anomaly phenomenon, which refers to the de- localization of p-polarized EM waves incident at a certain critical angle on stratified random dielectric media, has been studied extensively [14–19].

Another aspect is that in the case of EM wave propagation, there can be two separate kinds of random potentials, due to the fact that the dielectric permittivity ε and the magnetic permeability μ both affect EM waves. In several previous papers, it has been pointed out that when the wave impedance $Z = (= \sqrt{μ / ε})$ is uniform, while the refractive index $n (= \sqrt{ε μ})$ is periodic in a periodically-layered slab of magnetodielectric media, the photonic band gap (PBG) is not formed when the wave is incident perpendicularly [20–22]. When the EM wave is incident obliquely, the PBGs are formed and their sizes increase as the incident angle increases. The magnitudes of the PBGs, however, are much smaller than those of the corresponding band gaps in the cases where the wave impedance is periodic [22]. A very similar phenomena also occur in photonic quasicrystals.

In the present paper, I study the characteristics of Anderson localization of EM waves in a randomly-stratified slab, where both ε and μ depend on one spatial coordinate z in a random manner. I am especially interested in the case where the wave impedance Z is uniform, while the refractive index n is random. Using the invariant imbedding theory of wave propagation generalized to random media [23], I calculate the transmittance and the localization length of s and p waves incident obliquely on the slab, as a function of the incident angle θ and the strength of randomness in a numerically precise manner. I find that the waves incident perpendicularly on the slab are delocalized, as is expected in impedance-matched cases. Obliquely incident waves, however, are found to be localized. As the incident angle increases from zero, the localization length decreases from infinity monotonically to some finite value at θ = 90°. Based on accurate numerical results, I find that the localization length depends on the incident angle as θ⁻⁴ in the small θ region and derive a simple analytical formula, which works quite well for weak disorder and small incident angles. I find that the localization length does not depend on the wave polarization, but the disorder-averaged transmittance generally does.

A similar problem has been studied previously in [7], where the authors have calculated the localization length and the transmittance as a function of the incident angle for a multilayered system consisting of two different kinds of layers with the same impedance, but different refractive indices. These two kinds of layers are alternatingly arranged, with a random distribution of thicknesses. A large number of random configurations have been considered and numerical averages have been taken. The results reported in that work are consistent with those obtained here.

2. Model

I am interested in the propagation and the Anderson localization of a plane electromagnetic wave of vacuum wave number k₀ = ω/c in isotropic random magnetodielectric media with uniform wave impedance. The wave is incident on a stratified random medium, where both the dielectric permittivity and the magnetic permeability vary randomly only in the z direction. I assume that the random medium lies in 0 ≤ z ≤ L and the wave propagates in the xz plane. In the s wave case, then, the complex amplitude of the electric field, E, satisfies

\frac{d^{2} E}{d z^{2}} - \frac{1}{μ (z)} \frac{d μ}{d z} \frac{d E}{d z} + [k_{0}^{2} ε (z) μ (z) - q^{2}] E = 0,

where q is the x component of the wave vector. In the p wave case, the magnetic field amplitude, H, satisfies

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

I assume that the wave is incident from the region where ε = ε₁, μ = μ₁ and z > L and transmitted to the region where ε = ε₁, μ = μ₁ and z < 0. When θ is the angle of incidence, the quantity q is equal to ksinθ, where $k = k_{0} \sqrt{ε_{1} μ_{1}}$ . In the inhomogeneous region where 0 ≤ z ≤ L, ε and μ are given by

ε = \bar{ε} + δ ε (z), μ = \bar{μ} + δ μ (z),

where δε(z) and δμ(z) are δ-correlated Gaussian random functions with zero averages. In dielectric media, it is convenient to introduce the refractive index n and the wave impedance Z, which satisfy n² = εμ and Z² = μ/ε. In random media, these quantities are also random and expressed as

\begin{array}{l} n^{2} = (\bar{ε} + δ ε) (\bar{μ} + δ μ) = \bar{ε} \bar{μ} (1 + \frac{δ ε}{\bar{ε}} + \frac{δ μ}{\bar{μ}} + \frac{δ ε}{\bar{ε}} \frac{δ μ}{\bar{μ}}), \\ Z^{2} = \frac{\bar{μ} + δ μ}{\bar{ε} + δ ε} = \frac{\bar{μ}}{\bar{ε}} \frac{1 + \frac{δ μ}{\bar{μ}}}{1 + \frac{δ ε}{\bar{ε}}} . \end{array}

In this paper, I am interested in the special case where δε(z) and δμ(z) are related to each other by

\frac{δ ε}{\bar{ε}} = \frac{δ μ}{\bar{μ}} .

The averages $\bar{ε}$ and $\bar{μ}$ are assumed to be nonzero. In this case, it is easy to see that the wave impedance is always uniform, while the refractive index is random:

n^{2} \approx \bar{ε} \bar{μ} (1 + 2 \frac{δ ε}{\bar{ε}}), Z^{2} = \frac{\bar{μ}}{\bar{ε}},

where I have assumed that the randomness is sufficiently weak so that the magnitudes of

δ ε / \bar{ε}

and

δ μ / \bar{μ}

are much smaller than 1.

It is convenient to introduce the normalized variables $\tilde{ε}$ and $\tilde{μ}$ defined by

\tilde{ε} \equiv \frac{ε}{ε_{1}} = a + δ \tilde{ε} (z), \tilde{μ} \equiv \frac{μ}{μ_{1}} = b + δ \tilde{μ} (z),

where

\begin{matrix} a = \frac{\bar{ε}}{ε_{1}}, & b = \frac{\bar{μ}}{μ_{1}}, & δ \tilde{ε} (z) = \frac{δ ε (z)}{ε_{1}}, & δ \tilde{μ} (z) = \frac{δ μ (z)}{μ_{1}} . \end{matrix}

Then I can eliminate $δ \tilde{μ}$ using

δ \tilde{μ} = \frac{b}{a} δ \tilde{ε} .

The random variable $δ \tilde{ε}$ satisfies

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

where 〈⋯〉 denotes averaging over disorder and

\tilde{g}

is the parameter measuring the strength of disorder.

3. Invariant imbedding method

Let us consider a p wave of unit magnitude $\tilde{H} (x, z) = H (z) e^{i q x} = e^{i p (L - z) + i q x}$ , where p = k cos θ, incident obliquely on the randomly-stratified medium. The quantities of main interest in the invariant imbedding method are the reflection and transmission coefficients, r = r(L) and t = t(L), defined by

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

Using the invariant imbedding method [22–27], I derive exact differential equations satisfied by r and t:

\begin{array}{l} \frac{d r}{d l} = 2 i p \tilde{ε} r + \frac{i}{2} p [\tilde{μ} - \tilde{ε} + (\tilde{μ} - \frac{1}{\tilde{ε}}) \tan^{2} θ] {(1 + r)}^{2}, \\ \frac{d t}{d l} = i p \tilde{ε} t + \frac{i}{2} p [\tilde{μ} - \tilde{ε} + (\tilde{μ} - \frac{1}{\tilde{ε}}) \tan^{2} θ] (1 + r) t, \end{array}

starting from Eq. (2). These equations are integrated numerically from l = 0 to l = L using the initial conditions, r(0) = 0 and t(0) = 1. I will use Eq. (12) in calculating the disorder averages of various quantities consisting of r and t. In this paper, I am mainly interested in the disorder-averaged transmittance 〈T〉 (=〈|t|²〉) and the localization length ξ defined by

ξ = - \lim_{L \to \infty} (\frac{L}{〈 \ln T 〉}) .

When there is no dissipation, 〈T〉 is equal to 1 − 〈R〉, where R (= |r|²) is the reflectance. The corresponding set of equations for the s wave is obtained by exchanging $\tilde{ε}$ and $\tilde{μ}$ in Eq. (12):

\begin{array}{l} \frac{d r}{d l} = 2 i p \tilde{μ} r + \frac{i}{2} p [\tilde{ε} - \tilde{μ} + (\tilde{ε} - \frac{1}{\tilde{μ}}) \tan^{2} θ] {(1 + r)}^{2}, \\ \frac{d t}{d l} = i p \tilde{μ} t + \frac{i}{2} p [\tilde{ε} - \tilde{μ} + (\tilde{ε} - \frac{1}{\tilde{μ}}) \tan^{2} θ] (1 + r) t . \end{array}

The invariant imbedding equations for r and t are stochastic differential equations with random coefficients. In order to handle the random terms occurring in the denominators of the coefficients, I need to assume that the disorder is sufficiently weak so that

\frac{1}{\tilde{ε}} = \frac{1}{a + δ \tilde{ε}} \approx \frac{1}{a} - \frac{δ \tilde{ε}}{a^{2}}, \frac{1}{\tilde{μ}} = \frac{1}{b + δ \tilde{μ}} \approx \frac{1}{b} - \frac{δ \tilde{μ}}{b^{2}} .

I point out that this is the only approximation used in the present work. In the p wave case, by substituting the first of Eq. (15) into Eq. (12), I obtain

\begin{array}{l} \frac{1}{i k \cos θ} \frac{d r}{d l} = 2 (a + δ \tilde{ε}) r + \frac{1}{2} [\frac{b}{\cos^{2} θ} + \frac{δ \tilde{μ}}{\cos^{2} θ} - (a + \frac{\tan^{2} θ}{a}) - (1 - \frac{\tan^{2} θ}{a^{2}}) δ \tilde{ε}] {(1 + r)}^{2}, \\ \frac{1}{i k \cos θ} \frac{d t}{d l} = (a + δ \tilde{ε}) t + \frac{1}{2} [\frac{b}{\cos^{2} θ} + \frac{δ \tilde{μ}}{\cos^{2} θ} - (a + \frac{\tan^{2} θ}{a}) - (1 - \frac{\tan^{2} θ}{a^{2}}) δ \tilde{ε}] (1 + r) t . \end{array}

The equations for the s wave case are obtained by exchanging a and b and also $δ \tilde{ε}$ and $δ \tilde{μ}$ in Eq. (16).

In impedance-matched random media, I substitute $δ \tilde{μ} = (b / a) δ \tilde{ε}$ into Eq. (16) and obtain

\begin{array}{l} \frac{1}{i k \cos θ} \frac{d r}{d l} = 2 (a + δ \tilde{ε}) r + \frac{1}{2} [\frac{b}{\cos^{2} θ} - a - \frac{\tan^{2} θ}{a} + (\frac{b}{a \cos^{2} θ} - 1 + \frac{\tan^{2} θ}{a^{2}}) δ \tilde{ε}] {(1 + r)}^{2}, \\ \frac{1}{i k \cos θ} \frac{d t}{d l} = (a + δ \tilde{ε}) t + \frac{1}{2} [\frac{b}{\cos^{2} θ} - a - \frac{\tan^{2} θ}{a} + (\frac{b}{a \cos^{2} θ} - 1 + \frac{\tan^{2} θ}{a^{2}}) δ \tilde{ε}] (1 + r) t . \end{array}

An infinite number of coupled nonrandom differential equations satisfied by the moments of the reflection coefficient are obtained using Eq. (17) and Novikov’s formula [23, 28]. I introduce the quantity Z_nñ ≡ 〈rⁿr^*^ñ〉, where n and ñ are arbitrary nonnegative integers. In order to obtain 〈Rⁿ〉 = 〈rⁿr^*n〉 for n > 0, I need to compute the moments Z_nñ = 〈rⁿr^*^ñ〉 for all nonnegative integers n and ñ because the moments Z_nñ with n = ñ are coupled to those with n ≠ ñ. The coupled differential equations satisfied by Z_nñ in the p wave case have the form

\begin{array}{l} \frac{1}{k} \frac{d Z_{n \tilde{n}}}{d l} = i (n - \tilde{n}) \cos θ (a - \frac{\tan^{2} θ}{a} + \frac{b}{\cos^{2} θ}) Z_{n \tilde{n}} \\ - \frac{i}{2} n C_{0} (Z_{n + 1, \tilde{n}} + Z_{n - 1, \tilde{n}}) + \frac{i}{2} \tilde{n} C_{0} (Z_{n, \tilde{n} + 1} + Z_{n, \tilde{n} - 1}) \\ - g {[2 (1 + \frac{\tan^{2} θ}{a^{2}}) {(n - \tilde{n})}^{2} + {(1 - \frac{\tan^{2} θ}{a^{2}})}^{2} (n^{2} + {\tilde{n}}^{2})] \cos^{2} θ \\ + \frac{b}{a} [(3 n^{2} - 4 n \tilde{n} + 3 {\tilde{n}}^{2}) (\frac{2 \tan^{2} θ}{a^{2}} + \frac{b}{a \cos^{2} θ}) + 2 n^{2} + 2 {\tilde{n}}^{2} - 8 n \tilde{n}]} Z_{n \tilde{n}} \\ + g (2 n - 2 \tilde{n} + 1) n C_{1} Z_{n + 1, \tilde{n}} - g (2 n - 2 \tilde{n} - 1) \tilde{n} C_{1} Z_{n, \tilde{n} + 1} \\ + g (2 n - 2 \tilde{n} - 1) n C_{1} Z_{n - 1, \tilde{n}} - g (2 n - 2 \tilde{n} + 1) \tilde{n} C_{1} Z_{n, \tilde{n} - 1} \\ - \frac{g}{2} n (n + 1) C_{2} Z_{n + 2, \tilde{n}} - \frac{g}{2} \tilde{n} (\tilde{n} + 1) C_{2} Z_{n, \tilde{n} + 2} \\ - \frac{g}{2} n (n - 1) C_{2} Z_{n - 2, \tilde{n}} - \frac{g}{2} \tilde{n} (\tilde{n} - 1) C_{2} Z_{n, \tilde{n} - 2} \\ + g n \tilde{n} C_{2} (Z_{n + 1, \tilde{n} + 1} + Z_{n - 1, \tilde{n} - 1} + Z_{n + 1, \tilde{n} - 1} + Z_{n - 1, \tilde{n} + 1}), \end{array}

where the dimensionless disorder parameter g is defined by

g = \frac{\tilde{g} k}{4}

and the parameters C₀, C₁ and C₂ are given by

\begin{array}{l} C_{0} = (a + \frac{\tan^{2} θ}{a} - \frac{b}{\cos^{2} θ}) \cos θ, \\ C_{1} = (1 - \frac{\tan^{2} θ}{a^{4}}) \cos^{2} θ - \frac{b}{a} (\frac{2 \tan^{2} θ}{a^{2}} + \frac{b}{a \cos^{2} θ}), \\ C_{2} = {(1 - \frac{\tan^{2} θ}{a^{2}} - \frac{b}{a \cos^{2} θ})}^{2} \cos^{2} θ . \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 decrease rapidly as either n or ñ increases. Based on this observation, I solve the infinite number of coupled differential equations, Eq. (18), by a truncation method [23]. I 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, a, b and g. I 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, I conclude that I have obtained the exact solution of Z_nñ. In the absence of dissipation, the disorder-averaged transmittance is given by 〈T〉 = 1 − 〈R〉 = 1 − Z₁₁.

In order to obtain the localization length, I need to compute the average 〈lnT〉 in the l → ∞ limit. The nonrandom differential equation satisfied by 〈lnT〉 is obtained using Eq. (17) and Novikov’s formula in a straightforward manner:

\frac{1}{k} \frac{d 〈 \ln T 〉}{d l} = - g C_{2} - \frac{i}{2} C_{0} (Z_{10} + Z_{01}) + g C_{1} (Z_{10} + Z_{01}) - \frac{g}{2} C_{2} (Z_{20} + Z_{02}),

which reduces to

\frac{1}{k ξ} = g C_{2} + Re [(i C_{0} - 2 g C_{1}) Z_{10} (l \to \infty) + g C_{2} Z_{20} (l \to \infty)]

in the l → ∞ limit. It is straightforward to verify that the equations for the s wave case corresponding to Eqs. (17)–(22) are obtained by exchanging a and b and replacing g by gb²/a².

4. Results

In order to calculate the localization length from Eq. (22), one needs to obtain the quantities Z₁₀ and Z₂₀ in the l → ∞ limit. In that limit, I expect dZ_nñ/dl = 0, then the left-hand side of Eq. (18) vanishes and I get an infinite number of coupled algebraic equations. These equations are solved numerically by the truncation method described in Sec. 3 [23].

In Fig. 1, I plot the normalized localization length, kξ, and its inverse as a function of the incident angle when a = b = 1. The value of the disorder parameter g is equal to 0.001 and 0.01. There is a perfect symmetry between s waves and p waves, therefore the localization length is identical for both polarizations. I compare the numerical result obtained using the invariant imbedding method with the approximate analytical formula obtained from Eq. (22). In that equation, I assume that the magnitudes of Z₁₀ and Z₂₀ are sufficiently small and approximate (kξ)⁻¹ by gC₂, which gives

\frac{1}{k ξ} \approx 4 g \sin^{2} θ \tan^{2} θ,

when a = b = 1. The agreement is seen to be very good for θ ≤ 75° when g = 0.001 and for θ ≤ 60° when g = 0.01. I have checked numerically that approximating (kξ)⁻¹ by gC₂ is valid only when a = b = 1.

Fig. 1 (a) Normalized localization length, kξ, and (b) its inverse versus incident angle, θ, when a = b = 1 and g = 0.001, 0.01. Due to symmetry, the localization length is identical for both s and p waves. In (a), the numerical results obtained using the invariant imbedding method are compared with the approximate analytical formula, Eq. (23).

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In the present model where the wave impedance is uniform, but the refractive index is random, the wave incident on the random medium perpendicularly is always delocalized, as is expected in impedance-matched media. This phenomenon can be thought as a special kind of Brewster anomaly phenomenon, where the critical angle is zero. As the incident angle increases from zero, I notice that the localization length decreases monotonically from infinity, following the dependence ξ ∝ θ⁻⁴ in the region where θ is not too large.

Next, I consider the case where a is not equal to b and the explicit symmetry between s waves and p waves is broken. In Fig. 2, I plot the normalized localization length and its inverse versus incident angle when a = 2 and b = 1. The disorder parameters are the same as in Fig. 1. I obtain a behavior qualitatively similar to that of Fig. 1. The localization length decreases monotonically from infinity as θ increases from zero. From numerical results, I have succeeded in deducing an approximate analytical formula of the form

k ξ \approx \frac{a^{3} b}{4 g} θ^{- 4},

which is valid in the small θ region for any positive values of a and b, when g is sufficiently small. From this form, one can see explicitly that the localization length is the same for both s and p waves. If ξ for p waves is given by Eq. (24), then the expression for s waves is obtained by exchanging a and b and replacing g by gb²/a², which recovers Eq. (24). In Fig. 2, I observe that this formula works quite well for θ ≤ 45° In addition, I find numerically that the invariant imbedding results for s and p waves are completely identical, even though the explicit symmetry between s waves and p waves is broken. This result is fully consistent with that reported in [7].

Fig. 2 (a) Normalized localization length and (b) its inverse plotted versus incident angle when a = 2 and b = 1. The disorder parameter g is equal to 0.001 and 0.01. The localization length is identical for both s and p waves. In (a), the numerical results obtained using the invariant imbedding method are compared with the approximate analytical formula, Eq. (24).

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In Fig. 3, I show more examples of the validity of Eq. (24) in a log-log plot of the normalized localization length versus incident angle, for three different sets of values of a and b when g = 0.01. I find a very good agreement between the invariant imbedding calculation and Eq. (24) for θ ≤ 30°. I have also verified for many other sets of values of a and b that Eq. (24) works well when g is sufficiently small and θ is not too large. In all cases, the localization length is found to be identical for both s and p waves.

Fig. 3 Normalized localization length versus incident angle in a log-log plot for three different sets of values of a and b, when g = 0.01. The localization length is identical for both s and p waves in all cases. The results obtained using the invariant imbedding method are compared with the approximate analytical formula, Eq. (24).

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Next, I consider the frequency dependence of the localization length in the low-frequency region. In Fig. 4, I show the normalized localization length as a function of the dimensionless disorder parameter, $g (= k \tilde{g} / 4)$ , in a log-log plot when a = b = 1 and the incident angle is 5° and 50°. In the region where g ≤ 0.02, both curves are linear with slopes approximately equal to −1. This suggests that kξ ∝ g⁻¹, and therefore

ξ \propto {\tilde{g}}^{- 1} k^{- 2} \propto {\tilde{g}}^{- 1} ω^{- 2},

when

\tilde{g}

and ω are sufficiently small. The ω⁻² dependence of the localization length in the low-frequency region is consistent with the behavior in other localization phenomena [23].

Fig. 4 Normalized localization length versus disorder parameter, g, in a log-log plot, when a = b = 1 and θ = 5°, 50°. The localization length is identical for both s and p waves. The results obtained using the invariant imbedding method are compared with the approximate analytical formula, Eq. (24).

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I have also calculated the disorder-averaged transmittance 〈T〉 by solving the coupled differential equations, Eq. (18), numerically. In Fig. 5, I plot 〈T〉 versus incident angle when a = b = 1 and kL = 40. The disorder parameter g is equal to 0.001 and 0.01. When θ is zero, the average transmittance is identically equal to 1, as is expected in impedance-matched media. In the small θ region, it is very close to 1, but as θ increases to 90°, it decreases monotonically to a small value. This behavior is consistent with the monotonic decrease of the localization length with θ shown in Fig. 1. When a is equal to b, the results are identical for both s and p waves due to symmetry.

Fig. 5 Disorder-averaged transmittance, 〈T〉, versus incident angle, when a = b = 1, kL = 40 and g = 0.001, 0.01. Due to symmetry, the results are identical for both s and p waves.

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When a is not equal to b and the explicit symmetry between s and p waves is broken, the disorder-averaged transmittances for s and p waves are generally unequal. Furthermore, when a or b is not equal to 1, a more complicated nonmonotonic dependence of 〈T〉 is observed. In Fig. 6(a), I plot 〈T〉 versus incident angle when a = 2, b = 1, kL = 40 and g = 0.001. The average transmittance decreases from a value close to 1 at θ = 0 to a small value in an oscillatory manner, as θ increases to 90°. I have verified that the oscillatory behavior is stronger for s (p) waves if a > b (a < b). This dependence is a consequence of the Fabry-Perot-type resonance in disordered media. As the disorder strength increases, the oscillatory behavior is smoothed out, as shown in Fig. 6(b).

Fig. 6 Disorder-averaged transmittance versus incident angle, when a = 2, b = 1, kL = 40 and g = 0.001, 0.01. The results for the s wave case are compared with those for the p wave case.

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5. Conclusion

In the present paper, I have studied the propagation and the Anderson localization of EM waves in a randomly-stratified slab, where both the dielectric permittivity and the magnetic permeability depend on one spatial coordinate in a random manner. I have especially considered the case where the wave impedance is uniform, while the refractive index is random. Using the invariant imbedding method, I have calculated the localization length and the disorder-averaged transmittance for s and p waves incident obliquely on the slab, as a function of the incident angle and the strength of randomness in a numerically precise manner. I have found that the waves incident perpendicularly on the slab are delocalized, while those incident obliquely are localized. As the incident angle increases from zero, the localization length decreases from infinity monotonically to some finite value at θ = 90°. I have found that the localization length ξ depends on the incident angle as θ⁻⁴ and derived a simple analytical formula for ξ, which works quite well for weak disorder and relatively small incident angles. I have also found that the localization length does not depend on the wave polarization, but the disorder-averaged transmittance generally does. The incident angle dependence of the localization length can be qualitatively understood by examining the form of Eq. (17), where the coefficient of $δ \tilde{ε}$ in the second terms is a monotonically increasing function of θ. This implies that the effective disorder strength increases monotonically as θ increases.

The theoretical results obtained here can be tested experimentally using a random multilayered system composed of alternating dielectric and magnetic layers. In addition to the study of Anderson localization of propagating waves reported here, I have also studied the tunneling transmission of evanescent waves in the situation where the total internal reflection occurs and found intriguing and unique disorder-enhanced tunneling behaviors [29–32]. These results will be presented in a later publication.

Acknowledgments

This work has been supported by the National Research Foundation of Korea Grant ( NRF-2012R1A1A2044201) funded by the Korean Government.

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Anderson localization of electromagnetic waves in randomly-stratified magnetodielectric media with uniform impedance

Abstract

1. Introduction

2. Model

3. Invariant imbedding method

4. Results

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Equations (25)

Optics Express