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

Optics Express, Vol. 19, Issue 7, pp. 6923-6937 (2011)

http://dx.doi.org/10.1364/OE.19.006923

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### Abstract

We investigate statistical properties of collective optical excitations in disordered chains of microspheres using transfer-matrix method based on nearest-neighbors approximation. Radiative losses together with transmission and reflection coefficients of optical excitations are studied numerically. We found that for the macroscopically long chain, the transmission coefficient demonstrates properties typical for a one dimensional strongly localized system: log-normal distribution with parameters obeying standard scaling relation. At the same time, we show that the distribution function of the radiative losses behaves very differently from other lossy optical systems. We also studied statistical properties of the optical transport in short chains of resonators and demonstrated that even small disorder results in significant drop of transmission coefficient acompanied by strong enhancement of the radiative losses.

© 2011 OSA

## 1. Introduction

19. P. Pradhan and N. Kumar, “Localization of light in coherently amplifying random media,” Phys. Rev. B **50**, 9644–9647 (1994). [CrossRef]

25. D. V. Savin and H.-J. Sommers, “Distribution of reflection eigenvalues in many-channel chaotic cavities with absorption,” Phys. Rev. E **69**, 035201 (2004). [CrossRef]

*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

3. V. N. Astratov, J. P. Franchak, and S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. **85**, 5508–5510 (2004). [CrossRef]

4. Y. Hara, T. Mukaiyama, K. Takeda, and M. Kuwata-Gonokami, “Heavy photon states in photonic chains of resonantly coupled cavities with supermonodispersive microspheres,” Phys. Rev. Lett. **94**, 203905 (2005). [CrossRef] [PubMed]

4. Y. Hara, T. Mukaiyama, K. Takeda, and M. Kuwata-Gonokami, “Heavy photon states in photonic chains of resonantly coupled cavities with supermonodispersive microspheres,” Phys. Rev. Lett. **94**, 203905 (2005). [CrossRef] [PubMed]

26. C.-S. Deng, H. Xu, and L. Deych, “Optical transport and statistics of radiative losses in disordered chains of microspheres,” Phys. Rev. A **82**, 041803 (2010). [CrossRef]

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

*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

*n*th sphere whose center is located at point

**r**

*, one presents the incident and internal fields as linear combinations of vector spherical harmonics (VSH)*

_{n}**M**

*(*

_{m,l}**r**) (TE polarization) and

**N**

*(*

_{m,l}**r**) (TM polarization) defined in the coordinate system with origin at

**r**

*: 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: 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*

_{n}*n*th sphere, one can derive a system of equations relating coefficients

*n*th sphere,

*x*, can be presented as the following combination of real-valued functions of frequency

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

*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

28. H. Miyazaki and Y. Jimba, “Ab initio tight-binding description of morphology-dependent resonance in a bi-sphere,” Phys. Rev. B **62**, 7976–7997 (2000). [CrossRef]

*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

11. L. I. Deych and O. Roslyak, “Photonic band mixing in linear chains of optically coupled microspheres,” Phys. Rev. E **73**, 036606 (2006). [CrossRef]

*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

4. Y. Hara, T. Mukaiyama, K. Takeda, and M. Kuwata-Gonokami, “Heavy photon states in photonic chains of resonantly coupled cavities with supermonodispersive microspheres,” Phys. Rev. Lett. **94**, 203905 (2005). [CrossRef] [PubMed]

*l*= 29,

*m*= 1, and

*s*= 1 in the chain of polystyrene microspheres with radius of about 2.5 μ

*m*were shown to be well described within the single-mode approach.

*l*of their parent single-sphere modes, and the system of equations for the coefficients of the scattered field takes the following form: 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

**94**, 203905 (2005). [CrossRef] [PubMed]

*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.

*a*∝ exp(

_{n}*iqnd*) with the Bloch wave number

*q*satisfying standard dispersion relation where

*ω*

_{0}

*±*2

*γU*. Exponential dependence of translation coefficient

*U*upon inter-sphere distance

*d*results in fast decrease of the band width Δ = 4

*γU*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.

*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.

29. A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Collective emission and absorption in a linear resonator chain,” Opt. Express **17**, 15210 (2009). [CrossRef] [PubMed]

*a*and adding the resulting equations, we obtain after summing up the result over all spheres: This expression is the energy conservation statement for the system under consideration with

_{n}*J*, defined as representing the energy flux along the chain. This can be confirmed by direct calculation of the complex Poynting vector

_{n}**P**

_{s}of the scattered field

**E**

_{s},

**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: 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*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

_{n}*a*.

_{n}*T*is the transfer matrix which relates the fields on one side of the

_{n}*n*th 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.

*q*is the complex-valued wave number in the leads defined by Eq. (8). Expression for flux

*J*in the plane wave representation take the following form

_{n}*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*

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

*A*defined as is naturally interpreted as the flux of radiative losses.

30. A. D. Stone, J. D. Joannopoulos, and D. J. Chadi, “Scaling studies of the resistance of the one-dimensional anderson model with general disorder,” Phys. Rev. B **24**, 5583–5596 (1981). [CrossRef]

*P*relating amplitudes (

_{n}## 3. Transport in asymptotically long chains

### 3.1. Numerical procedure

*P*has, in terms of these parameters, the following generic form Comparing Eq. (19) with Eq. (21), one can find expressions for

_{n}*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: 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

*ξ*

^{−1}= –lim

_{N}_{→∞}ln

*T/*(2

*N*). For systems of finite size the localization length can be computed as an ensemble average

*ξ*

^{−}^{1}=

*–*〈ln

*T*〉

*/*(2

*N*). 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.

*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

31. Z.-Q. Zhang, “Light amplification and localization in randomly layered media with gain,” Phys. Rev. B **52**, 7960–7964 (1995). [CrossRef]

32. X. Jiang and C. M. Soukoulis, “Transmission and reflection studies of periodic and random systems with gain,” Phys. Rev. B **59**, 6159–6166 (1999). [CrossRef]

*ξ*and

*ξ*

_{0}was found to be given by where

*ξ*

_{0}is the localization length of the “lossless” system, and

*l*is the length characterizing the losses in the system. In the case considered here

_{d}*l*= 1/|

_{d}*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

33. G. Czycholl, B. Kramer, and A. MacKinnon, “Conductivity and localization of electron states in one dimensional disordered systems: Further numerical results,” Z. Phys. B **43**, 5–11 (1981). [CrossRef]

34. B. Kramer and A. MacKinnon, “Localization: theory and experiment,” Rep. Prog. Phys. **56**, 1469 (1993). [CrossRef]

*ξ*

_{0}∝

*δ*

^{−}^{2}

^{/}^{3}for the states near the band edges [35

35. B. Derrida and E. Gardner, “Lyapounov exponent of the one dimensional Anderson model : weak disorder expansions,” J. Phys. France **45**, 1283–1295 (1984). [CrossRef]

36. F. M. Izrailev, S. Ruffo, and L. Tessieri, “Classical representation of the one-dimensional Anderson model,” J. Phys. A: Math. Gen. **31**, 5263 (1998). [CrossRef]

*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.

10. V. N. Astratov, “Fundamentals and Applications of Microsphere Resonator Circuits,” in *Photonic Microresonator Research and Applications*, I. Chremmos, O. Schwelb, and N. Uzunoglu, eds., (Springer Series in Optical Sciences156, 2010), pp. 423–457. [CrossRef]

**94**, 203905 (2005). [CrossRef] [PubMed]

*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

37. A. V. Kanaev, V. N. Astratov, and W. Cai, “Optical coupling at a distance between detuned spherical cavities,” Appl. Phys. Lett. **88**, 111111 (2006). [CrossRef]

### 3.3. Scaling properties of transport coefficients

*λ*, defined as

*λ*= –〈ln

*T*〉/(2

*N*). 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*and

_{d}*τ*=

*σ*

^{2}

*Nξ*. It was established in Ref. [20

20. V. Freilikher, M. Pustilnik, and I. Yurkevich, “Effect of absorption on the wave transport in the strong localization regime,” Phys. Rev. Lett. **73**, 810–813 (1994). [CrossRef] [PubMed]

*τ*is a universal function of

*ρ*, whose explicit form can be found in [20

20. V. Freilikher, M. Pustilnik, and I. Yurkevich, “Effect of absorption on the wave transport in the strong localization regime,” Phys. Rev. Lett. **73**, 810–813 (1994). [CrossRef] [PubMed]

*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

20. V. Freilikher, M. Pustilnik, and I. Yurkevich, “Effect of absorption on the wave transport in the strong localization regime,” Phys. Rev. Lett. **73**, 810–813 (1994). [CrossRef] [PubMed]

*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

**73**, 810–813 (1994). [CrossRef] [PubMed]

### 3.4. Probability distribution of losses

*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.

*a*|

_{n}^{2}∝ exp[–

*nd*/

*ξ*]. Summing up this expression over all spheres, one has 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

*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.

*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.

*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.

*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*).

*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*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 〈

_{d}*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.

## 5. Conclusions

*ab initio*form along the lines of approach utilized in Ref. [38

38. C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, T. Pertsch, V. Shuvayev, and L. I. Deych, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A **80**, 043841 (2009). [CrossRef]

## Acknowledgments

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38. | C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, T. Pertsch, V. Shuvayev, and L. I. Deych, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A |

**OCIS Codes**

(290.4020) Scattering : Mie theory

(230.4555) Optical devices : Coupled resonators

**ToC Category:**

Scattering

**History**

Original Manuscript: January 18, 2011

Revised Manuscript: March 3, 2011

Manuscript Accepted: March 7, 2011

Published: March 25, 2011

**Citation**

Chao-Sheng Deng, Hui Xu, and Lev Deych, "Effect of size disorder on the optical transport in chains of coupled microspherical resonators," Opt. Express **19**, 6923-6937 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-6923

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