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Optics Express

Optics Express

  • Editor: C. Martijin de Sterke
  • Vol. 19, Iss. 7 — Mar. 28, 2011
  • pp: 6923–6937
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Effect of size disorder on the optical transport in chains of coupled microspherical resonators

Chao-Sheng Deng, Hui Xu, and Lev Deych  »View Author Affiliations


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

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

16. D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008). [CrossRef]

18

18. S. Mookherjea, “Spectral characteristics of coupled resonators,” J. Opt. Soc. Am. B 23, 1137–1145 (2006). [CrossRef]

]. Fussell et al. [16

16. D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008). [CrossRef]

] addressed this issue for a linear chain of photonic crystal defects while Mookherjea et al. [17

17. S. Mookherjea and A. Oh, “Effect of disorder on slow light velocity in optical slow-wave structures,” Opt. Lett. 32, 289–291 (2007). [CrossRef] [PubMed]

, 18

18. S. Mookherjea, “Spectral characteristics of coupled resonators,” J. Opt. Soc. Am. B 23, 1137–1145 (2006). [CrossRef]

] 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

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]

], while the authors of Ref. [5

5. B. M. Möller, U. Woggon, and M. V. Artemyev, “Bloch modes and disorder phenomena in coupled resonator chains,” Phys. Rev. B 75, 245327 (2007). [CrossRef]

] 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 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

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]

]) 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 rn satisfies inequality r (1 – δ) ≤ rnr (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 = 〈rn〉, 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.

The collective excitations of this structure are described using ab initio multisphere Mie theory [27

27. M. Mishchenko, L. Travis, and A. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University Press, Cambridge, 2002).

], 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:
Einc(n )=l,m[ζl,m(n)Nm,l(1)(rrn)+ηl,m(1)Mm,l(1)(rrn)],
(1)
Eint(n )=l,m[cl,m(n)Nm,l(1)(rrn)+dl,m(1)Mm,l(1)(rrn)].
(2)
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:
Es=n=1Nl,m[al,m(n)Nm,l(3)(rrn)+bl,m(n)Mm,l(3)(rrn)],
(3)
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

27. M. Mishchenko, L. Travis, and A. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University Press, Cambridge, 2002).

] to set up Maxwell boundary conditions at the surface of nth sphere, one can derive a system of equations relating coefficients al,m(n) characterizing scattering field of TM polarization to the coefficients of the incident field ζl,m(n):
al,m(n)=αl(n){ζl,m(n)+jnl,m[al,m(j)Ul,ml,m(rjrn)+bl,m(j)Vl,ml,m(rjrn)]}.
(4)
Single-sphere scattering amplitude for TM polarization of the nth sphere, αl(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 βl(n)(ω) and gl(n)(ω) (their explicit expressions can be found, for instance in Ref. [27

27. M. Mishchenko, L. Travis, and A. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University Press, Cambridge, 2002).

]):
αl(n)=iβl(n)gl(n)iβl(n).
(5)
The spectral positions of the WGM resonances are determined by equation gl(n)[ωl,s(n)]=0, where index s distinguishes resonances with different radial dependencies, so that at the exact resonance one has αl(n)[ωl,s(n)]=1. In the vicinity of a single resonance αl(n) has the following approximate representation, which takes into account this important property:
αl(n)=iγl,s(n)ωωl,s(n)+iγl,s(n),
(6)
where γl,s(n) determines simultaneously the width of the resonance and its “strength”.

Electromagnetic interaction between the spheres, described by translation coefficients Ul,ml,m and Vl,ml,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 Vl,ml,m, which are much smaller than Ul,ml,m and assume that the latter are purely imaginary [28

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]

]. 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

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

]. 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

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]

], 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:
1αn(l)an(l,m)=Un,n1(l,m)an1(l,m)+Un,n+1(l,m)an+1(l,m).
(7)
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

27. M. Mishchenko, L. Travis, and A. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University Press, Cambridge, 2002).

]. Having in mind experimental results 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]

], 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 an ∝ exp(iqnd) with the Bloch wave number q satisfying standard dispersion relation
ωω0+iγ=2γUcos(qd),
(8)
where U=Im[Ul,ml,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

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]

]. 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 an*, its complex conjugated version by an and adding the resulting equations, we obtain after summing up the result over all spheres:
n=1N|an|2=JN+1+J1.
(9)
This expression is the energy conservation statement for the system under consideration with Jn, defined as
Jn=i2Un,n1(an1an*an1*an),
(10)
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, Ps=Es×Bs*, 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:
Ws=14πdφdθsinθn·12Re(Es×Hs*),
(11)
which with help of Eq. (3) and translation theorem [27

27. M. Mishchenko, L. Travis, and A. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University Press, Cambridge, 2002).

] can be in the nearest-neighbors approximation reduced to WsJnJn +1, confirming our identification of Jn 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 βl,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 an.

The tight-binding Eq. (7) has a natural transfer-matrix representation defined in the site representation:
(an+1an)=Tn(anan1),Tn=(1αnUn,n+1Un,n1Un,n+110),
(12)
where Tn 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 an+ and an respectively, according to
an=an+eiqnd+aneiqndan1=an+eiq(n1)d+aneiq(n1)d,
(13)
where q is the complex-valued wave number in the leads defined by Eq. (8). Expression for flux Jn in the plane wave representation take the following form
Jn=Un,n1sin(kd)[e(2n1)βd|an|2e(2n1)βd|an+|2]+Un,n1isinh(βd)[(an+)*anei(2n1)kd(an)*an+ei(2n1)kd],
(14)
where k and β are the real and imaginary part of wave number q respectively: q = k + . In order to define transmission and reflection coefficients we introduce amplitudes of incident, reflected and transmitted waves according to a1=1, a1+=RNl, aN+1+=0, aN+1=TNl. By identifying incident and transmitted waves with a rather than with a + we take into account the sign of translation coefficients Un,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=UN+1,NU1,0e2Nβd|TN+1l|2,
(15)
R=e2βd|RN+1l|2isinh(βd)[RN+1l*eikdRN+1leikd]eβdsin(kd)
(16)
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,
(17)
where A defined as
A=n=1N|an|2U1,0eβdsin(kd)
(18)
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

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]

], we find the transfer-matrix Pn relating amplitudes ( an+1+, an+1) to ( an+, an) have the following form
Pn=12isin(qd)(1αneiqd(Un,n+1+Un,n1)αnUn,n+1e2iqnd1αn(eiqdUn,n+1+eiqdUn,n1)αnUn,n+1e2iqnd1αn(eiqdUn,n+1+eiqdUn,n1)αnUn,n+11αneiqd(Un,n+1+Un,n1)αnUn,n+1).
(19)

3. Transport in asymptotically long chains

3.1. Numerical procedure

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 = –limN →∞ 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.

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

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

, 32

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]

]. For weak disorder, the relation between ξ and ξ 0 was found to be given by
1ξ=1ξ0+1ld,
(23)
where ξ 0 is the localization length of the “lossless” system, and ld is the length characterizing the losses in the system. In the case considered here ld = 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 0δ 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

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

] and ξ 0δ 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

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]

]. Taking this into account we rewrite Eq. (23) as
ldξ=ldξ0+1{δ2+1band centerδ2/3+1band edges .
(24)

We verified the relationships given by Eq. (24) by numerically calculating ld/ξ for different frequencies and disorder strengths in the interval 0.5×10−3δ ≤ 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 ld/ξ as a function of δ 2 for different frequencies around the band center while the inset frame in this figure shows relationship between ld/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: ld/ξ as a function of δ 2 for different frequencies around the band center. Inset: ld/ξ as a function of δ 2/3 for different frequencies around the band edge. The straight lines show the linear relationships between ld/ξ and δ 2, ld/ξ and δ 2/3.

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

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]

]. 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

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]

]. 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

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]

]. 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 Nξ 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/ld and τ = σ 2 . 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]

] that in a lossy optical system with uniform absorption, 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]

]. 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

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]

] 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

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]

], 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.

3.4. Probability distribution of losses

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.

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.

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.

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 (Nξ).

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.

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 ld 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.

5. Conclusions

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

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]

]. 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.

References and links

1.

A. B. Matsko and V. S. Ilchenko, “Optical Resonators With Whispering-Gallery Modes—Part I: Basics,” IEEE J. OF Sel. Topics in Q. El. 12, 3–14 (2006). [CrossRef]

2.

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999). [CrossRef]

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]

5.

B. M. Möller, U. Woggon, and M. V. Artemyev, “Bloch modes and disorder phenomena in coupled resonator chains,” Phys. Rev. B 75, 245327 (2007). [CrossRef]

6.

K. Grujic and O. G. Hellesø, “Dielectric microsphere manipulation and chain assembly by counter-propagating waves in a channel waveguide,” Opt. Express 15, 6470–6477 (2007). [CrossRef] [PubMed]

7.

J. Goeckeritz and S. Blair, “Optical characterization of coupled resonator slow-light rib waveguides,” Opt. Express 18, 18190–18199 (2010). [CrossRef] [PubMed]

8.

S. Yang and V. N. Astratov, “Photonic nanojet-induced modes in chains of size-disordered microspheres with an attenuation of only 0.08 db per sphere,” Appl. Phys. Lett. 92, 261111 (2008). [CrossRef]

9.

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Express 18, 26505–26516 (2010). [CrossRef] [PubMed]

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]

11.

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

12.

Z. Chen, A. Taflove, and V. Backman, “Highly efficient optical coupling and transport phenomena in chains of dielectric microspheres,” Opt. Lett. 31, 389–391 (2006). [CrossRef] [PubMed]

13.

G. S. Blaustein, M. I. Gozman, O. Samoylova, I. Y. Polishchuk, and A. L. Burin, “Guiding optical modes in chains of dielectric particles,” Opt. Express 15, 17380–17391 (2007). [CrossRef] [PubMed]

14.

M. Gozman, I. Polishchuk, and A. Burin, “Light propagation in linear arrays of spherical particles,” Phys. Lett. A 372, 5250 – 5253 (2008). [CrossRef]

15.

A. Petrov, M. Krause, and M. Eich, “Backscattering and disorder limits in slow light photonic crystal waveguides,” Opt. Express 17, 8676–8684 (2009). [CrossRef] [PubMed]

16.

D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008). [CrossRef]

17.

S. Mookherjea and A. Oh, “Effect of disorder on slow light velocity in optical slow-wave structures,” Opt. Lett. 32, 289–291 (2007). [CrossRef] [PubMed]

18.

S. Mookherjea, “Spectral characteristics of coupled resonators,” J. Opt. Soc. Am. B 23, 1137–1145 (2006). [CrossRef]

19.

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

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]

21.

J. Heinrichs, “Transmission, reflection and localization in a random medium with absorption or gain,” J. Phys.: Condens. Matter 18, 4781 (2006). [CrossRef]

22.

J. C. J. Paasschens, T. S. Misirpashaev, and C. W. J. Beenakker, “Localization of light: Dual symmetry between absorption and amplification,” Phys. Rev. B 54, 11887–11890 (1996). [CrossRef]

23.

V. Freilikher and M. Pustilnik, “Phase randomness in a one-dimensional disordered absorbing medium,” Phys. Rev. B 55, R653–R655 (1997). [CrossRef]

24.

S. K. Joshi, D. Sahoo, and A. M. Jayannavar, “Modeling of stochastic absorption in a random medium,” Phys. Rev. B 62, 880–885 (2000). [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]

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]

27.

M. Mishchenko, L. Travis, and A. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University Press, Cambridge, 2002).

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]

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]

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]

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]

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]

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]

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]

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]

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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References

  1. A. B. Matsko and V. S. Ilchenko, “Optical Resonators With Whispering-Gallery Modes—Part I: Basics,” IEEE J. OF Sel. Topics in Q. El. 12, 3–14 (2006). [CrossRef]
  2. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999). [CrossRef]
  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]
  5. B. M. Möller, U. Woggon, and M. V. Artemyev, “Bloch modes and disorder phenomena in coupled resonator chains,” Phys. Rev. B 75, 245327 (2007). [CrossRef]
  6. K. Grujic and O. G. Hellesø, “Dielectric microsphere manipulation and chain assembly by counter-propagating waves in a channel waveguide,” Opt. Express 15, 6470–6477 (2007). [CrossRef] [PubMed]
  7. J. Goeckeritz and S. Blair, “Optical characterization of coupled resonator slow-light rib waveguides,” Opt. Express 18, 18190–18199 (2010). [CrossRef] [PubMed]
  8. S. Yang and V. N. Astratov, “Photonic nanojet-induced modes in chains of size-disordered microspheres with an attenuation of only 0.08 db per sphere,” Appl. Phys. Lett. 92, 261111 (2008). [CrossRef]
  9. M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Express 18, 26505–26516 (2010). [CrossRef] [PubMed]
  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]
  11. L. I. Deych and O. Roslyak, “Photonic band mixing in linear chains of optically coupled microspheres,” Phys. Rev. E 73, 036606 (2006). [CrossRef]
  12. Z. Chen, A. Taflove, and V. Backman, “Highly efficient optical coupling and transport phenomena in chains of dielectric microspheres,” Opt. Lett. 31, 389–391 (2006). [CrossRef] [PubMed]
  13. G. S. Blaustein, M. I. Gozman, O. Samoylova, I. Y. Polishchuk, and A. L. Burin, “Guiding optical modes in chains of dielectric particles,” Opt. Express 15, 17380–17391 (2007). [CrossRef] [PubMed]
  14. M. Gozman, I. Polishchuk, and A. Burin, “Light propagation in linear arrays of spherical particles,” Phys. Lett. A 372, 5250 – 5253 (2008). [CrossRef]
  15. A. Petrov, M. Krause, and M. Eich, “Backscattering and disorder limits in slow light photonic crystal waveguides,” Opt. Express 17, 8676–8684 (2009). [CrossRef] [PubMed]
  16. D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008). [CrossRef]
  17. S. Mookherjea and A. Oh, “Effect of disorder on slow light velocity in optical slow-wave structures,” Opt. Lett. 32, 289–291 (2007). [CrossRef] [PubMed]
  18. S. Mookherjea, “Spectral characteristics of coupled resonators,” J. Opt. Soc. Am. B 23, 1137–1145 (2006). [CrossRef]
  19. P. Pradhan and N. Kumar, “Localization of light in coherently amplifying random media,” Phys. Rev. B 50, 9644–9647 (1994). [CrossRef]
  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]
  21. J. Heinrichs, “Transmission, reflection and localization in a random medium with absorption or gain,” J. Phys.: Condens. Matter 18, 4781 (2006). [CrossRef]
  22. J. C. J. Paasschens, T. S. Misirpashaev, and C. W. J. Beenakker, “Localization of light: Dual symmetry between absorption and amplification,” Phys. Rev. B 54, 11887–11890 (1996). [CrossRef]
  23. V. Freilikher and M. Pustilnik, “Phase randomness in a one-dimensional disordered absorbing medium,” Phys. Rev. B 55, R653–R655 (1997). [CrossRef]
  24. S. K. Joshi, D. Sahoo, and A. M. Jayannavar, “Modeling of stochastic absorption in a random medium,” Phys. Rev. B 62, 880–885 (2000). [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]
  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]
  27. M. Mishchenko, L. Travis, and A. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University Press, Cambridge, 2002).
  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]
  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]
  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]
  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]
  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]
  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]
  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]
  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]

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