Effect of size disorder on the optical transport in chains of coupled microspherical resonators

Chao-Sheng Deng; Hui Xu; Lev Deych

doi:10.1364/OE.19.006923

1. Introduction

Propagation of optical excitations in chains of optical microresonators due to evanescent coupling of high-Q whispering gallery modes (WGM) [1] has recently attracted significant interest. The initial proposal to use such structures as low loss waveguides in Ref. [2] gave rise to a number of experimental [3–10] and theoretical [11–15] papers, in which properties of collective optical excitations in linear chains of coupled microspheres and microdisks have been studied. The main impediment to practical realization of the proposal of Ref. [2] comes from unavoidable fluctuations in size and position of the individual resonators. Therefore, understanding the effects of disorder on optical properties of these structures is crucial for assessing their potential for applications in practical devices.

Theoretical efforts in this direction have been so far limited to analysis of effects of disorder on group velocity of optical excitations in various types of coupled microresonators [16–18]. Fussell et al. [16] addressed this issue for a linear chain of photonic crystal defects while Mookherjea et al. [17, 18] treated the similar problem using a more general phenomenological model. Experimental studies, on the other hand, were mostly concerned with transport properties and/or normal mode structure of collective excitation of microresonator chains. For instance, the attenuation of light intensity along a chain of coupled microspheres with significant size dispersion was studied experimentally in Ref. [3], while the authors of Ref. [5] used doping of microspheres with CdSe quantum dots to excite and visualize normal modes in such structures. The latter paper also offered a theoretical analysis of the effects of the disorder on the modes of the studied structure within a coupled oscillator model.

In addition to being of interest for applications, disordered chains of optical microresonators also present a more general interest for physics of disordered systems as a previously unexplored type of one-dimensional disordered systems, demonstrating Anderson localization. Two features of this system sets apart from disordered optical structures studied in the past. First, this system supports optical excitations defined on a discrete lattice, and second, it is characterized by inherent radiative losses intimately connected with dispersion properties of collective excitations of the system. While in previous studies of Anderson localization in systems with dissipation, the losses were modeled via a phenomenologically introduced parameter independent of other characteristics of the system [19–25], in the structure under consideration in this work such a parameter does not exist, and, therefore, the limit of a lossless system is not easily defined.

In this work we focus on statistical properties of optical transport through a disordered chains of microspheres characterized by transmission, reflection and loss coefficients. Unlike previous works we employ and approach that, while relying on a certain number of approximations, still remains of an ab initio type explicitly derived from Maxwell equations. This approach is used to study transport properties of two types of structures: asymptotically long chains whose properties can be described by scaling relations, and relatively short chains, which are of more importance for practical applications. In realistic experimental situations such a chain would be formed by placing the individual microspheres on a substrate [3, 4], which can affect properties of the chain by damping some of its modes. If, however, the points of contact between the spheres and the substrate are significantly separated from the plane, in which WGMs of the chain are excited, (see, for instance, Ref. [4]) the effect of the substrate can be neglected. In this case, the analysis can be based on the transfer-matrix approach to numerical simulation of the transport properties of the system.

In Sec. 2 we present a description of our model and define transport coefficients by considering fluxes of incoming, reflected and transmitted wave. Sec. 3 deals with fundamental localization and scaling properties of asymptotically long chains (partially results of this section has been published in recent Rapid Communication [26]) and in Sec. 4 we discuss the effects of disorder on the optical properties of short chains focusing on the behavior in the spectral region of “slow light”. Finally, we summarize our results in Sec. 5.

2. Ab initio description of a disordered chain of coupled microspheres

We consider a chain of N microspheres characterized by identical refractive index n, whose centers are aligned along a straight line (see Fig. 1) and positioned at equal distance, d, from each other. The disorder is introduced into the system by allowing the radius of the spheres to fluctuate. It is assumed that the radii can be represented by non-correlated random variables obeying uniform statistical distribution of width δ so that the radius of the nth sphere r_n satisfies inequality r (1 – δ) ≤ r_n ≤ r (1 +δ). In this model the disorder only affect the on-site resonance frequency (diagonal disorder) while keeping inter-sphere coupling coefficients constant. The ab initio nature of our approach allows us to incorporate coupling disroder and consider, for instance, the case of microspheres positioned in contact with each other. Numerical analysis showed, however, that fluctuations of the coupling coefficient have very little effect on the transport properties, and, therefore, we limit our consideration here to the case of diagonal disorder. Generalization to the more general situation is quite trivial. To model the input and output leads, we suppose that this disordered segment is connected to two semi-infinite ordered segments comprised of spheres with radius r = 〈r_n〉, where 〈⋯〉 indicates averaging over the uniform distribution.

Fig. 1 Schematic of a disordered chain composed of N coupled microspheres with a finite length L = Nd. R and T are the reflection and transmission coefficient of the fundamental Bloch mode.

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The collective excitations of this structure are described using ab initio multisphere Mie theory [27], which consists in separating the electromagnetic field associated with the structure into incident field E _inc, scattered field E _s and internal field E _int of each sphere, all characterized by the same frequency ω. For the nth sphere whose center is located at point r _n, one presents the incident and internal fields as linear combinations of vector spherical harmonics (VSH) M _m,l(r) (TE polarization) and N _m,l(r) (TM polarization) defined in the coordinate system with origin at r _n:

E_{inc}^{(n
)} = \sum_{l, m} [ζ_{l, m}^{(n)} N_{m, l}^{(1)} (r - r_{n}) + η_{l, m}^{(1)} M_{m, l}^{(1)} (r - r_{n})],

E_{int}^{(n
)} = \sum_{l, m} [c_{l, m}^{(n)} N_{m, l}^{(1)} (r - r_{n}) + d_{l, m}^{(1)} M_{m, l}^{(1)} (r - r_{n})] .

The scattered field contains contributions of the fields scattered by all spheres in the structure, each defined in the coordinate systems centered at the respective spheres:

E_{s} = \sum_{n = 1}^{N} \sum_{l, m} [a_{l, m}^{(n)} N_{m, l}^{(3)} (r - r_{n}) + b_{l, m}^{(n)} M_{m, l}^{(3)} (r - r_{n})],

VSHs with upper index (1) have their dependence on the radial coordinate presented by Beseel functions of the 1st kind, while the upper index (3) refers to the radial dependence described by outgoing Hankel functions. Using the addition theorem for the VSH [27] to set up Maxwell boundary conditions at the surface of nth sphere, one can derive a system of equations relating coefficients

a_{l, m}^{(n)}

characterizing scattering field of TM polarization to the coefficients of the incident field

ζ_{l, m}^{(n)}

:

a_{l, m}^{(n)} = α_{l}^{(n)} {ζ_{l, m}^{(n)} + \sum_{j \neq n} \sum_{l^{'}, m'} [a_{l', m'}^{(j)} U_{l, m}^{l', m'} (r_{j} - r_{n}) + b_{l', m'}^{(j)} V_{l, m}^{l', m'} (r_{j} - r_{n})]} .

Single-sphere scattering amplitude for TM polarization of the nth sphere,

α_{l}^{(n)}

, which has poles at the specific values of the dimensionless frequency parameter x, can be presented as the following combination of real-valued functions of frequency

β_{l}^{(n)} (ω)

and

g_{l}^{(n)} (ω)

(their explicit expressions can be found, for instance in Ref. [27]):

α_{l}^{(n)} = \frac{i β_{l}^{(n)}}{g_{l}^{(n)} - i β_{l}^{(n)}} .

The spectral positions of the WGM resonances are determined by equation

g_{l}^{(n)} [ω_{l, s}^{(n)}] = 0

, where index s distinguishes resonances with different radial dependencies, so that at the exact resonance one has

α_{l}^{(n)} [ω_{l, s}^{(n)}] = - 1

. In the vicinity of a single resonance

α_{l}^{(n)}

has the following approximate representation, which takes into account this important property:

α_{l}^{(n)} = \frac{- i γ_{l, s}^{(n)}}{ω - ω_{l, s}^{(n)} + i γ_{l, s}^{(n)}},

where

γ_{l, s}^{(n)}

determines simultaneously the width of the resonance and its “strength”.

Electromagnetic interaction between the spheres, described by translation coefficients $U_{l, m}^{l^{'}, m^{'}}$ and $V_{l, m}^{l^{'}, m^{'}}$ , is responsible not only for optical coupling between different spheres, but also for coupling between WGMs with different orbital l and azimuthal m numbers as well as between modes with different polarizations. As a result, the collective modes of the chain cannot be characterized by orbital and azimuthal numbers of individual resonators. However, for a linear chain of spheres, choosing the polar axes of all coordinate systems along the chain, one can make translation coefficients to be diagonal with respect to m, restoring, thereby, one’s ability to classify collective modes of the chain by the same azimuthal number m as individual spheres. In the case of high-Q WGMs characterized by large orbital numbers l ≫ 1 additional simplifications are possible. First, one can neglect the cross-polarization translation coefficients $V_{l, m}^{l^{'}, m^{'}}$ , which are much smaller than $U_{l, m}^{l^{'}, m^{'}}$ and assume that the latter are purely imaginary [28]. Second, since the optical coupling between WGM with l ≫ 1 is of evanescent nature, one can introduce the nearest-neighbors approximation, and keep in the sum over the spheres in Eq. (4) only terms with j = n ± 1[11]. Finally, since the strength of interaction between single-sphere modes with different polar numbers l significantly diminishes with increased spectral separation between modes, one can neglect this interaction for modes with different l, which are well separated spectrally. Such modes are easier to find for smaller spheres with a larger mean free spectral range. For instance, in experiments of Ref. [4], collective excitations originating from single sphere modes with l = 29, m = 1, and s = 1 in the chain of polystyrene microspheres with radius of about 2.5 μm were shown to be well described within the single-mode approach.

In this approximation, collective excitations of the chain can again be characterized by the polar number l of their parent single-sphere modes, and the system of equations for the coefficients of the scattered field takes the following form:

\frac{1}{α_{n}^{(l)}} a_{n}^{(l, m)} = U_{n, n - 1}^{(l, m)} a_{n - 1}^{(l, m)} + U_{n, n + 1}^{(l, m)} a_{n + 1}^{(l, m)} .

The inter-mode coupling becomes more important in the case of overlapping modes and can result in additional interesting effects such as spectral diffusion, which will be discussed in a subsequent work. We have also assumed that the incident field excites only spheres in an incoming lead and omitted it in Eq. (7). It should be emphasized that the remaining inter-sphere coupling parameters in this equation are calculated using exact representation for the translation coefficients, which can be found, for instance, in Ref. [27]. Having in mind experimental results of Ref. [4], we choose for our calculations modes with l = 29, m = 1, s = 1 and, from now on, abridge our notations by dropping azimuthal, radial and polar indexes since all calculation are carried out for these fixed values.

For an ideal chain this approximation yields a typical tight-binding equation describing harmonic waves a_n ∝ exp(iqnd) with the Bloch wave number q satisfying standard dispersion relation

ω - ω_{0} + i γ = 2 γ U cos (q d),

where

U = Im [U_{l, m}^{l, m} (d)]

, and we took advantage of the approximation for the scattering amplitude given by Eq. (6). This equation obviously describes a single band of excitations with band boundaries at ω ₀ ± 2γU. Exponential dependence of translation coefficient U upon inter-sphere distance d results in fast decrease of the band width Δ = 4γU with increasing d. The collective excitation exist, of course, only while the band width is much larger than the width of single-sphere WGMs, γ, which is satisfied as long as U ≫ 1.

An important feature of dispersion relation given by Eq. (8) is that the band width Δ described by this equation depends not only on U, but also on γ. This reflects a radiative nature of the coupling between the spheres: if the WGM did not leak outside of the spheres and had zero width, no coupling between them would have been possible. This property establishes a significant (and often overlooked) difference between optical coupling of resonators and chemical bonding of atoms: while the latter is determined by the overlap of their wavefunctions, the former depends not only on the overlap of the fields in the evanescent (near-field) region, but also on their radiative (far-field) properties.

The same circumstance also results in a peculiar form of a transport coefficient describing radiative losses in the system, which does not show an explicit dependence of the radiative widths of the coupled resonances [29]. In order to see this, we apply a standard procedure for deriving expressions for energy fluxes to the tight-binding Eq. (7). Multiplying Eq. (7) by the complex conjugate of the respective coefficient $a_{n}^{*}$ , its complex conjugated version by a_n and adding the resulting equations, we obtain after summing up the result over all spheres:

\sum_{n = 1}^{N} {| a_{n} |}^{2} = - J_{N + 1} + J_{1} .

This expression is the energy conservation statement for the system under consideration with J_n, defined as

J_{n} = - \frac{i}{2} U_{n, n - 1} (a_{n - 1} a_{n}^{*} - a_{n - 1}^{*} a_{n}),

representing the energy flux along the chain. This can be confirmed by direct calculation of the complex Poynting vector P _s of the scattered field E _s,

P_{s} = E_{s} \times B_{s}^{*}

, where B _s is the scattered magnetic field. The power scattered by a single resonator can be found by integrating the Poynting vector over an infinitely large spherical surface:

W_{s} = \frac{1}{4 π} \int d φ d θ sin θ n \cdot \frac{1}{2} Re (E_{s} \times H_{s}^{*}),

which with help of Eq. (3) and translation theorem [27] can be in the nearest-neighbors approximation reduced to W_s ∝ J_n – J_n ₊₁, confirming our identification of J_n as a quantity proportional to the one-dimensional energy flux. Eq. (9) will be used in the subsequent sections of the paper to define reflection, transmission and radiation loss coefficients for the chain. The latter is defined by the left-hand side of this equation and does not show any explicit dependence on the loss parameter, determined, in general, by functions

β_{l, s}^{(n)}

. This distinguishes the system under consideration from other optical systems, in which global losses of the system are directly proportional to a whatever local loss coefficient is introduced. In the system under consideration, the radiative losses are manifested in a more subtle way through spatial distribution of the coefficients a_n.

The tight-binding Eq. (7) has a natural transfer-matrix representation defined in the site representation:

(\begin{matrix} a_{n + 1} \\ a_{n} \end{matrix}) = T_{n} (\begin{matrix} a_{n} \\ a_{n - 1} \end{matrix}), T_{n} = (\begin{matrix} \frac{1}{α_{n} U_{n, n + 1}} & - \frac{U_{n, n - 1}}{U_{n, n + 1}} \\ 1 & 0 \end{matrix}),

where T_n is the transfer matrix which relates the fields on one side of the nth sphere with the fields on the another side. It is necessary to notice that this transfer matrix is not unimodular (with the standard definition of the inner product) and does not have time reversal symmetry because the system is inherently leaky.

In order to properly introduce transport coefficients we need to rewrite the transfer-matrix equation in the plane-wave representation by replacing on-site scattering coefficients with amplitudes of forward- and backward- traveling waves $a_{n}^{+}$ and $a_{n}^{-}$ respectively, according to

\begin{array}{l} a_{n} = a_{n}^{+} e^{iqnd} + a_{n}^{-} e^{- iqnd} \\ a_{n - 1} = a_{n}^{+} e^{iq (n - 1) d} + a_{n}^{-} e^{- iq (n - 1) d}, \end{array}

where q is the complex-valued wave number in the leads defined by Eq. (8). Expression for flux J_n in the plane wave representation take the following form

\begin{array}{l} J_{n} = & U_{n, n - 1} sin (kd) [e^{(2 n - 1) β d} {| a_{n}^{-} |}^{2} - e^{- (2 n - 1) β d} {| a_{n}^{+} |}^{2}] \\ + U_{n, n - 1} i sinh (β d) [(a_{n}^{+}) * a_{n}^{-} e^{- i (2 n - 1) kd} - (a_{n}^{-}) * a_{n}^{+} e^{i (2 n - 1) kd}], \end{array}

where k and β are the real and imaginary part of wave number q respectively: q = k + iβ. In order to define transmission and reflection coefficients we introduce amplitudes of incident, reflected and transmitted waves according to

a_{1}^{-} = 1

,

a_{1}^{+} = R_{N}^{l}

,

a_{N + 1}^{+} = 0

,

a_{N + 1}^{-} = T_{N}^{l}

. By identifying incident and transmitted waves with a ⁻ rather than with a ⁺ we take into account the sign of translation coefficients U_n,n± ₁, which result in negative group velocity of the modes described by Eq. (8). Defining transmission and reflection coefficients T, R via ratios of transmitted and reflected fluxes to the incident flux we have

T = \frac{U_{N + 1, N}}{U_{1, 0}} e^{2 N β d} {| T_{N + 1}^{l} |}^{2},

R = e^{- 2 β d} {| R_{N + 1}^{l} |}^{2} - \frac{i sinh (β d) [R_{N + 1}^{l *} e^{- ikd} - R_{N + 1}^{l} e^{ikd}]}{e^{β d} sin (kd)}

Taking into account definitions of the transport coefficients in terms of fluxes, we can rewrite Eq. (9) in the standard flux conservation form

A + R + T = 1,

where A defined as

A = \frac{\sum_{n = 1}^{N} | a_{n} |^{2}}{U_{1, 0} e^{β d} sin (kd)}

is naturally interpreted as the flux of radiative losses.

Computation of these coefficients requires that transfer-matrix be also rewritten in the plane wave approximation. Following Ref. [30], we find the transfer-matrix P_n relating amplitudes ( $a_{n + 1}^{+}$ , $a_{n + 1}^{-}$ ) to ( $a_{n}^{+}$ , $a_{n}^{-}$ ) have the following form

P_{n} = \frac{1}{2 i sin (q d)} (\begin{matrix} \frac{1 - α_{n} e^{- iqd} (U_{n, n + 1} + U_{n, n - 1})}{α_{n} U_{n, n + 1}} & e^{- 2 iqnd} \frac{1 - α_{n} (e^{- iqd} U_{n, n + 1} + e^{iqd} U_{n, n - 1})}{α_{n} U_{n, n + 1}} \\ - e^{2 iqnd} \frac{1 - α_{n} (e^{iqd} U_{n, n + 1} + e^{- iqd} U_{n, n - 1})}{α_{n} U_{n, n + 1}} & - \frac{1 - α_{n} e^{iqd} (U_{n, n + 1} + U_{n, n - 1})}{α_{n} U_{n, n + 1}} \end{matrix}) .

3. Transport in asymptotically long chains

3.1. Numerical procedure

In the case of long chains, calculation of the transport coefficients via direct multiplication of transfer-matrices defined in Eq. (19) is numerically unstable. This problem can be circumvented by rewriting the transfer-matrix in terms of on-site amplitude transmission and reflection coefficients (not to be confused with transport coefficients introduced in the previous section) defined as

r_{n}^{l} = \frac{a_{n}^{+}}{a_{n}^{-}}, r_{n}^{r} = \frac{a_{n + 1}^{-}}{a_{n + 1}^{+}}, t_{n}^{l} = \frac{a_{n + 1}^{-}}{a_{n}^{-}}, t_{n}^{r} = \frac{a_{n}^{+}}{a_{n + 1}^{+}} .

The on-site transfer-matrices P_n has, in terms of these parameters, the following generic form

P_{n} = (\begin{matrix} 1 / t_{n}^{r} & - r_{n}^{l} / t_{n}^{r} \\ r_{n}^{r} / t_{n}^{r} & t_{n}^{l} - r_{n}^{l} r_{n}^{r} / t_{n}^{r} \end{matrix}) .

Comparing Eq. (19) with Eq. (21), one can find expressions for

t_{n}^{r}

,

r_{n}^{r}

,

r_{n}^{l}

, and

t_{n}^{l}

in terms of microscopic parameters of the structure under consideration. Taking into account that the transfer-matrix for a chain consisting of N spheres has the same form as Eq. (21) one can derive recurrence relations relating amplitude reflection/transmission coefficients of the chains with N – 1 and N spheres:

T_{N}^{r} = \frac{t_{N}^{r} T_{N - 1}^{r}}{1 - r_{N}^{l} R_{N - 1}^{r}}, R_{N}^{r} = r_{N}^{r} + \frac{R_{N - 1}^{r} t_{N}^{l} t_{N}^{r}}{1 - r_{N}^{l} R_{N - 1}^{r}}, T_{N}^{l} = \frac{t_{N}^{l} T_{N - 1}^{l}}{1 - r_{N}^{l} R_{N - 1}^{r}}, R_{N}^{l} = R_{N - 1}^{l} + \frac{r_{N}^{l} T_{N - 1}^{l} T_{N - 1}^{r}}{1 - r_{N}^{l} R_{N - 1}^{r}} .

This recursion allows one to calculate transport coefficients for arbitrary long structures. Statistical analysis of these coefficient was carried out by calculating a large number of statistically independent realizations of the structure by choosing radii of each sphere from a statistical ensemble as described above.

3.2. Localization length

Transport properties of one-dimensional disordered systems are characterized by the phenomenon of Anderson localization (provided the distribution of disorder does not have any long-range correlations). This property is characterized by localization length defined as ξ ⁻¹ = –lim_N _→∞ lnT/(2N). For systems of finite size the localization length can be computed as an ensemble average ξ⁻ ¹ = –〈lnT〉/(2N). Localization length depends on the strength of disorder, frequency and the band width, which is determined by the inter-sphere distance d. The latter dependence, however, is rather trivial: decreasing band width results in overall decrease of the localization length, therefore, we focus mainly on frequency and disorder dependence. Figure 2 plots the localization length ξ as a function of frequency for a chain with disorder strength δ = 10⁻ ³. As expected, the localization length is largest at the band center and decreases significantly toward the band edges. It should be noted that within the “slow light” spectral region localization length becomes less than N = 10 indicating that in this region even relatively short chains are characterized by strongly localized behavior.

Fig. 2 The localization length ξ as a function of frequency for a chain of N = 1000 spheres and for disorder strength δ = 10⁻ ³.

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Of particular interest is the evolution of the localization length ξ with the increasing disorder strength δ. It was suggested that in the presence of losses, the localization length ξ of a long disordered chain becomes smaller [31, 32]. For weak disorder, the relation between ξ and ξ ₀ was found to be given by

\frac{1}{ξ} = \frac{1}{ξ_{0}} + \frac{1}{l_{d}},

where ξ ₀ is the localization length of the “lossless” system, and l_d is the length characterizing the losses in the system. In the case considered here l_d = 1/|β| is well defined, but the meaning of ξ ₀ is not clear since the losses are inherent to the system and cannot be eliminated as discussed above. It is interesting to verify if this relation remains valid for the system under consideration. It is known that in localized lossless systems the localization length in the case of weak disorder behaves as ξ ₀ ∝ δ⁻ ² for the frequencies near the center of the band [33, 34] and ξ ₀ ∝ δ⁻ ² ^/ ³ for the states near the band edges [35, 36]. Taking this into account we rewrite Eq. (23) as

\frac{l_{d}}{ξ} = \frac{l_{d}}{ξ_{0}} + 1 \propto {\begin{array}{l} δ^{2} + 1 & band center \\ δ^{2 / 3} + 1 & band edges
. \end{array}

We verified the relationships given by Eq. (24) by numerically calculating l_d/ξ for different frequencies and disorder strengths in the interval 0.5×10⁻³ ≤ δ ≤ 2 × 10⁻ ³, where the system is in the localized phase for both band center and band edge for a chain of N = 1000 spheres. In the main frame of Fig. 3, we plot l_d/ξ as a function of δ ² for different frequencies around the band center while the inset frame in this figure shows relationship between l_d/ξ and δ ^2/3 for the frequencies around band edge. In all cases the data points form straight lines consistent with Eq. (24) (they all cross the vertical axes at unity when extrapolated to zero disorder). This result is somewhat surprising given the fact that the lossless localization length ξ ₀ is not readily definable in the system under consideration and its physical meaning is not immediately clear.

Fig. 3 Main frame: l_d/ξ as a function of δ ² for different frequencies around the band center. Inset: l_d/ξ as a function of δ ^2/3 for different frequencies around the band edge. The straight lines show the linear relationships between l_d/ξ and δ ², l_d/ξ and δ ^2/3.

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These calculations also show that the localization length in the chain of microspheres is very sensitive to increase in disorder: it becomes of the order of several inter-sphere distances even at the center of the band for size mismatch of 3 ÷ 5%, which is typical for many experimental works with such structures [10]. This means that formation and possible utilization of collective single-mode optical excitations in the chains of coupled microspheres requires much smaller disorder of the order of 0.05% achieved, for instance, in Ref. [4]. The situation, however, can be improved by working in the spectral regions where WGM modes with different values of orbital number l are spectrally close to each other. In this case, size mismatch can actually create additional propagating channels by facilitating spectral overlap of single-sphere WGMs with different l numbers [37]. More detailed consideration of this situation, however, is outside of the scope of this paper.

3.3. Scaling properties of transport coefficients

Statistical properties of transmission in one-dimensional disordered systems are usually described via probability distribution function of Lyapunov exponent, λ, defined as λ = –〈lnT〉/(2N). Our calculations showed that in the structure under consideration this distribution is normal (provided N ≫ ξ as is expected in a strongly localized one-dimensional disordered system). The mean of this distribution (which is given by inverse localization length ξ⁻ ¹) and its variance, σ ², obey a scaling relation defined in terms of scaling parameters ρ = ξ/l_d and τ = σ ² Nξ. It was established in Ref. [20] that in a lossy optical system with uniform absorption, τ is a universal function of ρ, whose explicit form can be found in [20]. We used numerical data obtained in our model for various values of disorder strength and frequencies (all in the vicinity of the band center) to verify this relation. Results of our calculations shown in Fig. 4 demonstrates not only that τ indeed is a function of a single variable ρ, but that this function has exactly the same form as the one obtained in Ref. [20] for a much simpler model. In order to emphasize that this scaling is not just a visual effect, we show in the inset in this figure τ as a function of localization length ξ, in which data points corresponding to different frequencies form clearly different curves. These results demonstrate universal nature of the scaling relation of Ref. [20], which was originally derived for a rather particular model of a one-dimensional lossy system.

Fig. 4 Scaling relation between τ and ρ. The inset shows τ as a function of ξ.

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3.4. Probability distribution of losses

While in the distribution of transmission coefficient, the chain of coupled microspheres behaves as any other one-dimensional localized system, it is quite surprising that it deviates from generic behavior in the distribution of losses. In asymptotically long disordered systems, transmission is exponentially small, and the losses in the system can be approximately related to the reflection coefficient: A ≈ 1 – R. Distribution of reflection in optical systems with uniform absorption was intensively studied [19, 22, 24] and was found to be of the following form

f_{u} (A) \propto A^{- 2} exp (- a_{u} / A),

where a_u is a parameter determined by the local decay rate. We computed distribution of the radiative losses in our model directly from transmission and reflection coefficients using Eq. (17) as well as from Eq. (18) with identical results. The results of our simulations, obtained for frequency 21.954 and disorder strength 10⁻³, are presented in Fig. 5(b), where numerically obtained histogram is shown together with a fitting function f(A) of the following form:

f_{u} (A) \propto A^{- 2} exp [- {(a - A)}^{2} / (b^{2} A^{2})] .

Fig. 5 (a) Linear relationship between A and ρ for a single realization and for a wide range of frequencies. (b) Distribution function of A: numerical histogram and its fit by the distribution of ξ.

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The distribution function given by Eq. (26) differs significantly from the one found in systems with uniform local loss, Eq. (25). Parameters a and b of this distribution relate to the decay rate in the uniform system via their dependence on the single scaling variable ρ. In order to demonstrate this scaling it is technically more convenient to compute mean 〈A〉 and variance var(A) of the loss coefficient, which are functions of a and b. Fig. 6 presents results of these calculations, where 〈A〉 and var(A) are plotted versus ρ for several different values of frequency, confirming the assumed scaling.

Fig. 6 Scaling of mean value and variance of the losses A for several values of frequencies in the vicinity of band center.

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One can explain qualitatively origin of Eq. (26) by appealing to Eq. (18), which relates radiation losses to the coefficients of the scattered field. Since our system is in a strongly localized regime, one can assume that the envelope of these coefficients decays exponentially as one moves away from the boundary of the chain, i.e., |a_n|² ∝ exp[–nd/ξ]. Summing up this expression over all spheres, one has

\sum_{n} {| a_{n} |}^{2} = 1 / (1 - exp [- d / ξ]) \approx ξ,

where at the last step it was assumed that ξ ≫ d. Then combining Eq. (27) with Eq. (18) one obtains A ∝ ξ. In the light of this relation, the origin of distribution function given by Eq. (26) is quite clear: this is a normal distribution of Lyapunov exponent λ rewritten in terms of distribution of ξ = λ ⁻¹. In order to better understand the nature of this distribution, we studied its dependence on the number of spheres in the chain N and found that for N ≫ ξ it is independent of N. This result is consistent with the previous analysis since it shows that this distribution is formed by the segment of the chain of the order of localization length. Since absorption coefficient A is dimensionless quantity, while ξ has dimension of length, and in the absence of dependence of A upon N, one can assume that radiative loss coefficient is a function of a single scaling parameter ρ. The validity of this prediction is directly confirmed by plotting losses A versus scaling parameter ρ for a single realization of disorder and a wide range of frequencies in the vicinity of band center. Fig. 5(a) clearly demonstrates that all data points obtained for various frequencies in the interval between 21.864 and 21.978 form a single straight line. This result obviously agrees with scaling properties of the coefficients of f(A), reported earlier, but it presents actually a much stronger statement implying that not just distribution function of losses, but the radiative loss coefficient in a single realization depends only on a single scaling variable.

4. Transport in short chains

When the length of the system becomes smaller than the localization length it looses all its universal scaling properties. This regime, however, of most importance is these structures are to have practical applications. For analysis of this situation we choose the length of the chain as N = 6. This choice will let the system stays in intermediate regime for both band edge and band center frequencies and thus avoid the effects of localization.

In Fig. 7(a), we show average loss coefficient 〈A〉 as a function of frequency for several values of disorder strength, δ = 1.0 × 10⁻³, 1.5 × 10⁻³, and 2 × 10⁻³. This figure demonstrates that for a given strength of disorder losses grow substantially as the frequency approaches the band edges entering the slow light regime. On another hand, we see from both Fig. 7(a) and Fig. 7(b) that 〈A〉 at the band center is much less sensitive to disorder changing when the disorder strength increases four-fold from 0.5 × 10⁻³ to 2 × 10⁻³. At the band edges, however, the effect of disorder on the losses is significant. For both lower and upper band edges, we find that losses decrease monotonically as the disorder strength increases. This decrease of radiative losses from the interior of the chain is a reflection of a much smaller amount of electromagnetic energy propagating through the chain due to increased reflection at the chain boundary.

Fig. 7 Evolution of 〈A〉 for short chain N = 6. (a) 〈A〉 as a function of frequency for various values of disorder strength δ. (b) 〈A〉 as a function of disorder strength δ for various values of frequency.

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This assertion is confirmed by Fig. 8, which shows average transmission T and reflection R coefficients as functions of frequency x for disorder strength δ = 10⁻³ (a) and dependence of transmission on disorder strength for several frequencies representing band-center and two band-edges (b). One can see from Fig. 8 that the average transmission T decreases rapidly when frequency approaches the “slow-light” region while the reflection increases. As a function of disorder average transmission decreases throughout the entire propagating band, but at the band-edges this increase occurs at a faster rate than at the band center, which confirms again that this spectral region is much more sensitive to fabrication uncertainties.

Fig. 8 Evolution of 〈T〉 and 〈R〉 for short chain N = 6. (a) 〈T〉 and 〈R〉 as functions of frequency for δ = 10⁻³. The frequencies are all around upper band edge frequency. (b) 〈T〉 as a function of δ for various values of frequency.

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To complete the analysis of transport properties of short chains, we calculated dependence of average transmission, reflection and loss coefficients upon the number of chains in the system. Fig. 9(a) shows transmission 〈T〉 and reflection 〈R〉 as functions of the length of the chain for a single disorder strength δ = 10⁻³. Both 〈T〉 and 〈R〉 show moderate change as the chain becomes longer with expected increase in the former and decrease in the latter. At the same time, the average loss coefficient 〈A〉 increase with N almost linearly as shown in Fig. 9(b) for several values of disorder strength, δ = 1.0 × 10⁻³, 1.5 × 10⁻³, and 2 × 10⁻³. The found behavior of all three transport coefficients is consistent with the fact that the the system is in quasi-ballistic regime (N ≪ ξ).

Fig. 9 (a) Dependence of 〈T〉 and 〈R〉 on the length of the chain for N = 6 and δ = 10⁻³ at band edge frequency x = 21.838. (b) Dependence of 〈A〉 on the length of the chain for N = 6 and for several values of disorder strength at band edge frequency x = 21.838.

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While the probability distributions f(A) was found to be size independent in asymptotically long chains, this is no longer the case for small N. This point is illustrated in Fig. 10 showing the probability distributions f(A) for different N and disorder strengths. All calculations are carried out at the same value of frequency x = 21.954 so that the dissipation length l_d is fixed. Fig. 10(a) shows that with increasing length of the chain the probability distribution broadens while it’s peak moves to the right, reflecting increase in 〈A〉. Fig. 10(b) illustartes that the strength of disorder does not affect the position of the peak of the distribution in agreement with the results for the average loss coefficient (Fig. 7(b)), but it significantly increases its width. This broadening of the loss distribution would have an additional detrimental effect for waveguiding characterstics of the coupled resonators.

Fig. 10 Probability distribution f (A) of losses A for (a) different chain length but fixed disorder strength δ = 10⁻³, (b) different disorder strength but fixed chain length N = 6. The frequency was kept as x = 21.954 for all of these calculations.

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5. Conclusions

In conclusion, we investigated statistical properties of collective optical excitations in disordered chains of microspheres in strong localization (system size is much larger than the localization length) and quasi-ballistic (localization length is much larger than the system size) regimes. For strongly localized regime, we found that while the structure under study behaves in some aspects as a typical Anderson localized system, its radiative loss statistics is strongly different from that observed for other lossy optical systems. We ascribe this difference to that radiative nature of the losses and provide qualitative analytical explanation to the obtained numerical results. For the shorter chain we confirm that the band edges are much more sensitive to disorder than the band center. However, unlike previous studies of the issue, we show that this affects, first of all, capability to couple light of the band edge frequency to the modes of the chain as disorder strongly enhances reflection of the incident signal. While this conclusion can be somewhat affected by our choice of the leads for this particular study, its seems justified to assume that while numerical values of reflection and transmission coefficients might change for a different choice of the leads, the general trend related to the effect of the disorder on these coefficients will remain intact. It is also important that at a given disorder strength, when frequency of the incident signal approaches the band edge radiative losses from the system increase simultaneously with reflection resulting in even more significant drop in transmitted power. Our calculations were carried out for very small disorder strength corresponding to the size dispersion of ≈ 0.1%, which shows that these structures can realistically be used as waveguides only if fabrication precision becomes significantly better than this number.

The results obtained in this paper can, in principle, be extended to other types of coupled resonators such as microdisks or toroids. In the single mode and nearest neighbors approximations any type of coupled resonators would be described by Eq. (7), with corrected expressions for single resonator scattering amplitudes and coupling coefficients. The former can often be presented in the form of Eq. (6) while the latter can be considered as a phenomenological parameter. In the case of disk resonaotrs, the theory can be developed again in ab initio form along the lines of approach utilized in Ref. [38]. Extension of this thoery is also possible for two- and three-dimensional structures. It is clear, however, that in this case WGMs with different azimtuhal numbers will be coupled making numerical analysis more computationally costly.

Acknowledgments

This research was supported in part by a grant of computer time from the City University of New York’s High Performance Computing Research Center. Financial support provided by China Scholarship Council (No. 2008637023) is acknowledged.

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Effect of size disorder on the optical transport in chains of coupled microspherical resonators

Abstract

1. Introduction

2. Ab initio description of a disordered chain of coupled microspheres

3. Transport in asymptotically long chains

3.1. Numerical procedure

3.2. Localization length

3.3. Scaling properties of transport coefficients

3.4. Probability distribution of losses

4. Transport in short chains

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (10)

Equations (27)

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