## Abstract

We have investigated low frequency guiding polariton modes in finite linear chains of closely packed dielectric spherical particles of different optical materials. These guiding (chain bound) modes cannot decay radiatively, because photon emission cannot take place with simultaneous conservation of energy and momentum. For extending previous work on infinite chains of spherical particles[1] and infinite rods[2, 3], we were able to apply the multisphere Mie scattering formalism to finite chains of dielectric particles to calculate quality factors of most bound modes originating from the first two Mie resonances depending on the number of particles *N* and the material’s refractive index *n*
* _{r}*. We found that, in agreement with the earlier work [4], guiding modes exist for

*n*

*>2 and the quality factor of the most bound mode scales by*

_{r}*N*

^{3}. We interpreted this behavior as the property of “frozen” modes near the edges of guiding bands with group velocity vanishing as

*N*increases. In contrast with circular arrays, longitudinal guiding modes in particle chains possess a higher quality factor compared to the transverse ones.

© 2007 Optical Society of America

## 1. Introduction

One-dimensional chains of scattering optical particles are attracting increasing interest because they can be used in various micro and nano-systems where optical energy is received, transferred and converted on a subwavelength scale. [5, 6, 7] To our knowledge, the first practical nano-waveguide capable of transferring optical energy within a distance of 100 nm was constructed by Atwater and coworkers[6, 7] as a linear chain of spherical silver or gold particles (see Fig. 1). Through the excitation of the surface plasmon resonance in a particle at one end of the waveguide, energy transfers to all particles in turn through the excitation of plasmon resonances in adjacent particles down the chain. The transferred energy can stimulate a number of processes such as the photoexcitation of a molecule or a chemical reaction in the other end of a waveguide. However in a circuit of metal particles the efficiency of energy transfer greatly decreases with the length of the waveguide due to absorption related energy loss.

Optical energy is also vulnerable to radiative losses associated with photon emission and absorption of light by the material. Losses due to photon emission can be almost completely suppressed if the energy used for particle excitation belongs to the energy domain of the guiding modes. This energy domain exists where photon emission is forbidden by the momentum conservation law. In the infinite chain with period *a* (Fig. 1), all optical excitations can be classified by their quasi-momentum projection *q* to the chain axis *z*, which belongs to domain (-*π*/*a*<*q*<*π*/*a*). [8] If the resonant photon wavevector *k*=*ω*/*c* is less then the maximum excitation wavevector *π*/*a*, then two branches of guiding modes are formed, namely

where photon emission is forbidden by the momentum conservation law (Fig. 1). The criterion *ω*/*c*<*π*/*a* is equivalent to the condition that the interparticle distance *a* should be less than half of the resonant wavelength

Our derivation is equivalent to the classical work[9] where a light cone constraint was applied and later used in Refs. [1, 4, 10, 11] to study long-living modes in chains and rings of particles.

Metals can easily satisfy Eq. (2) because their plasmon resonance frequencies correspond to the visible light spectrum as their wavelengths can be as long as a few hundred nanometers with a particle diameter of as low as 30nm, which is the minimum distance between particles. That is why metal particles can be used to build waveguides free of radiative loss for transfering optical energy on a subwavelength scale. [6, 7] However, all metals possess conducting electrons, which absorb optical energy. That is to say electrons have a continuous spectrum that consumes arbitrary amounts of photon energy greatly reducing energy transmission in nano-waveguides. Therefore, it may be more convenient to use dielectric materials having very weak light absorption. Then, instead of relying on surface plasmon resonance, one can make use of Mie scattering resonances to form polariton modes where oscillations of material polarizations are coupled with electromagnetic waves. According to rudimentary estimates of Ref. [11] made within the framework of the simple dipolar oscillator model, guiding modes satisfying Eq. (2) can indeed be formed in chains of particles made of optical materials like TiO_{2}, ZnO or GaAs. In contrast with metals, dielectric particles have negligible absorption of light which allow dielectric particle chains to transfer optical energy particularly efficiently.

One should notice that there exists extensive experimental investigation regarding the propagation of light in chains of mono-dispersive dielectric particles. [12, 13, 14, 15, 16] However, the size of particles in those studies remarkably exceed all optical wavelengths and consequently guiding modes are not formed Eq. (2) in the sense that the resulting quality factor does not approach infinity as the number of particles increases. Although the quality factor is not limitless as the number of particles increases, a high quality factor can be attained because of the individual particle whispering gallery modes. Nevertheless, we are interested in a different regime where wavelengths are comparable or larger than particle size. In the case of Ref. [12, 13, 14, 15, 16], our modes correspond to the infrared spectral domain. An extension of our formalism to the regime of interest in Ref. [12, 13, 14, 15, 16] is possible and we plan to investigate it in future. We again wish to emphasize that in contrast with [12, 13, 14, 15, 16] in our study true guiding modes are formed allowing the quality factor to approach infinity as the number of spheres increases. In addition guiding polariton modes form an energy band and the modes near the band edge possess zero group velocity permitting us to slow down the propagation of light as discussed below. *It is remarkable that the particle chains discussed here also demonstrate slow light modes in addition to other systems investigated experimentally and theoretically* (cf. Ref. [17, 18, 19, 20, 21, 22]).

Optical modes, or “polaritons,” are collective excitations created from the superposition of material polarizations and photons. In this paper, we will study polaritons formed within long finite chains of dielectric spherical particles possessing the longest possible lifetime (highest quality factor) and, for that reason, having the greatest relevance regarding energy manipulations using particle chains. We will investigate optical excitations within a one-dimensional chain of particles (Fig. 1) using the multi-sphere Mie scattering formalism [23, 24] (see also [4]), which has a distinct advantage compared to the standard finite difference time domain (FDTD) approach [25] in that the Mie scattering formalism represents a scattering particle polarization by a discrete set of multipole moments. Our solution for polariton modes is valid everywhere in space; therefore, our results are independent of boundary conditions that critically affect the FDTD method. As we show here, the analysis of the quality factor (not necessarily field amplitudes) of low energy optical modes for reasonably large refractive indices (*n*
* _{r}*>2) can be performed using a simple multipole approach, which means that numerical calculations can be performed quickly. Also it can be difficult to apply absorbing boundary conditions to the guiding mode within the chain of particles because the chain field is quasi-evanescent in the three-dimensional nearly spherical domain surrounding the chain of particles.

This paper is organized as follows. In Section 2, we describe the multi-sphere Mie scattering formalism and its application to the investigation of eigensolutions of Maxwell’s equations for the particle chain. In Section 3, results for the quality factor of particle chains using various approaches are presented. We also discuss the numerical results for quality factor dependence on the number of particles and the physical nature of obtained dependencies. In Section 4, we present our conclusions.

## 2. Multisphere Mie scattering formalism in the study of particle chains

The multisphere Mie scattering formalism [23, 24] has been developed to study the scattering of optical waves in aggregates of spherical dielectric particles. In our other work, [4] we used this formalism to study quality factors of whispering gallery modes in circular arrays of spherical particles. This formalism uses the spherical vector function expansion of solutions to Maxwells equations in the frequency domain. The scattering wave is characterized by partial amplitudes *a*
^{l}* _{mn}*,

*b*

^{l}*for an*

_{mn}*l*

*sphere*

^{th}*l*=1,2,3, ..

*N*and the spherical wave with angular momentum

*n*=1,2, … and its projection

*m*=-

*n*,-

*n*+1, …

*n*. There are two amplitudes

*a*and

*b*because electromagnetic waves are represented by a transverse vector field so these amplitudes express two projections of that field. Partial amplitudes of the scattering wave can be expressed through partial amplitudes

*p*

^{l}*,*

_{mn}*q*

^{l}*of the incident wave with the help of Mie scattering amplitudes*

_{mn}*a*ā

*,*

_{n}*b*̄

*and matrices*

_{n}**A**,

**B**of vector translation coefficients as [23, 24]

$$\frac{{b}_{\mathrm{mn}}^{l}}{{\overline{b}}_{n}}+\sum _{j=1\left(\ne l\right)}^{N}\sum _{n=1}^{{n}_{\mathrm{max}}}\sum _{m=-n}^{n}{A}_{\mathrm{mn}\mu \nu}^{\mathrm{jl}}{b}_{\mu \nu}^{j}+\sum _{j=1\left(\ne l\right)}^{N}\sum _{n=1}^{{n}_{\mathrm{max}}}\sum _{m=-n}^{n}{B}_{\mathrm{mn}\mu \nu}^{\mathrm{jl}}{a}_{\mu \nu}^{j}={q}_{\mathrm{mn}}^{l}.$$

The expansion over partial amplitudes is identical to a standard multipole expansion and vector translation coefficients behave like multipole interactions. Particularly, *A*
^{jl}_{m1µ1} is equivalent to the “retarded” dipole-dipole interaction [4], while *B*
^{jl}_{m1µ1} reflects the interaction of electric and magnetic dipoles. An upper bound is placed on the maximum angular momentum *n*
* _{max}* to make the problem numerically solvable. In the exact formalism

*n*

*=∞. Remember that Eq. (3) is written in the frequency domain for a given frequency*

_{max}*z*. Mie scattering coefficients

*a*

*,*

_{n}*b*

*are functions of the product*

_{n}*qd*

*/2 of photon wavevector*

_{l}*q*=

*z*/

*c*and

*l*-

*th*particle radius

*d*

*/2. They also depend on particle refractive index*

_{l}*n*

*, while vector translation coefficients for spheres*

_{r}*j*and

*l*separated by the distance

*r*

*depend only on the dimensionless product*

_{jl}*qr*

*.*

_{jl}Following Ref. [4], we will study quasi-states of light, which are eigen-modes of Eq. (3) defined by the homogeneous equation

$$\frac{{b}_{\mathrm{mn}}^{l}}{{\overline{b}}_{n}}+\sum _{j=1\left(\ne l\right)}^{N}\sum _{n=1}^{{n}_{\mathrm{max}}}\sum _{m=-n}^{n}{A}_{\mathrm{mn}\mu \nu}^{\mathrm{jl}}{b}_{\mu \nu}^{j}+\sum _{j=1\left(\ne l\right)}^{N}\sum _{n=1}^{{n}_{\mathrm{max}}}\sum _{m=-n}^{n}{B}_{\mathrm{mn}\mu \nu}^{\mathrm{jl}}{a}_{\mu \nu}^{j}=0,$$

obtained from Eq. (3) by setting partial amplitudes of the incident wave to be equal zero. Then our solution has only outgoing wave asymptotic behavior at infinity. This homogeneous equation has a nontrivial solution only at a discrete set of frequencies *z*
* _{a}*=

*ω*

*-*

_{a}*iγ*

*thus making the determinant of Eq. (4) equal zero. Generally, these solutions have a finite imaginary part due to radiative losses. The quality factor of mode*

_{a}*a*can be defined in the usual way as

which is our topic of interest in this work. A similar approach was taken in Refs. [4, 11] and we will report and discuss these results in greater detail.

Below we have six choices for the maximum number of coupled equations in Eq. (4). The first two simplest choices are such that we set all partial amplitudes equal to zero except for either *a*
^{l}_{m1} (A) or *b*
^{l}_{m1} (B) (cf. [4]). Four other choices are defined by setting either *n*
* _{max}*=1 (C),

*n*

*=2 (D),*

_{max}*n*

*=3 (E) and*

_{max}*n*

*=4 (F) in Eq. (4). By comparing results of these six approaches to calculate frequencies and quality factors we found that approximations (A) and (B) are sufficiently accurate for the definition of mode frequency, while using*

_{max}*n*

*=3 provided a sufficiently accurate approximation of the quality factor. Therefore we did not consider*

_{max}*n*

*>4.*

_{max}We will use Eq. (4) to find low frequency modes possessing the highest quality factor in a finite linear chain of identical, equally separated and closely packed dielectric spheres composed of various optical materials including ZnO (*n*
* _{r}*≈1.9), TiO

_{2}(rutile,

*n*

*≈2.7) and GaAs (*

_{r}*n*

*≈3.5). Below we describe in some details the method of our study which incorporates the modified Newton-Raphson method in determining solutions to the transcendental equations. This method is partially inherited from Ref. [11], where lasing modes were investigated for interacting dipolar oscillators.*

_{r}We will ignore the absorption of light by the dielectric materials although it becomes important when the radiative losses are comparable or smaller than the absorption related losses. The simple numerical estimate shows that if the material is characterized by complex refractive index *n*=*n*
_{1}+*in*
_{2} then for particle arrays with a small quality factor *Q*<*Q*
* _{max}*~

*n*

_{1}/

*n*

_{2}defined in the absence of absorption we can neglect absorption of light because the radiative losses are more significant. If

*Q*>

*Q*

*the actual losses are defined by absorption and the overall quality factor is saturated at*

_{max}*Q*

*. We have considered the possible constraints for the quality factor associated with absorption using the Handbook [26]. For the visible and infrared light one can specify three domains of photon energies*

_{max}*E*=

*h*̄

*w*corresponding to different absorption mechanisms including the low frequency domain 0.01

*eV*<

*E*<0.15

*eV*, where there is direct absorption by phonons and 10

^{-4}<

*n*

_{2}<10

^{-2}[26]; intermediate domain 0.15

*eV*<

*E*<

*E*

*(*

_{c}*E*

*is the width of the forbidden band,*

_{c}*E*

*≈3*

_{c}*eV*in TiO

_{2}and

*E*

*≈1.42eV in GaAs) where absorption reaches a deep minimum around 10*

_{c}^{-7}[26]; and the domain of absorption by electrons

*E*>

*E*

*where*

_{c}*n*

_{2}>0.1. [26] According to the results of our later consideration (see e. g. Fig. 2) in the intermediate domain of photon energies we can neglect absorption of light for chains of several hundreds of particles. At lower frequencies the quality factor cannot remarkably exceed 1000, while at higher frequency it cannot be made much larger than 10.

Because of the symmetry of the chain with respect to rotation about the *z*-direction, projection *m* of the excitation angular momentum with respect to the *z*-axis is conserved and *m*=-1, 0, or 1 can be selected to be constant. Formally, this conservation results from the fact that interactions *A*
* _{mnµv}* and

*B*

*Eq. (4) for both centers of spheres*

_{mnµv}*j*and

*l*belonging to the

*z*-axis are equal to zero if

*m*≠

*µ*. Therefore, Eq. (4) splits into three independent sets of equations where there are two identical transverse (

*t*) modes,

*m*=1 or

*m*=-1, with polarization perpendicular to the chain and one longitudinal (

*l*) mode characterized by the angular momentum projection

*m*=0 polarized parallel to the chain. Any equation describing the particular mode in the chain of

*N*ordered or disordered spheres with centers along the

*z*-axis can be written as

where *M*̂ is the matrix of size *N*×*N* in cases (A) and (B), 2*N*×2*N* in case (C), 4*N*×4*N* in case (D), etc. This matrix is extracted from Eq. (2). The diagonal elements of this matrix are inverse Mie scattering coefficients and its off-diagonal elements are defined by vector translation coefficients. Vector **x** represents partial amplitudes. A nontrivial solution of Eq. (4) exists when matrix *M*̂ has a zero eigenvalue. Thus, we need to find a value for *z* which forces one eigenvalue of matrix *M*̂(*z*) to be zero and possesses a minimal imaginary part. We define frequency *z* as a limit of a generalized Newton-Raphson algorithm

The function *f*(*z*) is defined as an eigenvalue of the matrix *M*̂(*z*) (cf. [4]) that minimizes the absolute value of the imaginary part of *z* in the next iteration defined by Eq. (7). If the initial value *z*
_{0} is chosen to be real and smaller than 2.4/*d*, then the procedure converges to the same result *z*
* _{fin}*. To test whether

*z*

*corresponds to the quasi-state possessing the minimum decay rate, we also used a modified algorithm to look for a solution in a certain frequency domain. This algorithm consists of choosing the function*

_{fin}*f*(

*z*) as the eigenvalue of the matrix

*M*̂(

*z*) with a real-part closest to the center of the domain of interest. Using this approach, we were able to find all low frequency modes in a wavevector domain from 0.5 to 2 (with sphere diameter chosen to be

*d*=2). We have verified for 10 and 20 spheres and refractive indices 1.9 and 3.5 that the original Newton-Raphson algorithm [4] practically always converges to the correct frequency, possessing the smallest decay rate. An alternative verification of mode frequencies was made by comparison with semi-analytical results for mode frequencies in circular arrays. [4] This comparison has confirmed that our solution is the correct mode. We used all algorithms described above to study modes related to the two lowest Mie resonances described by Mie scattering amplitudes

*a*ā

_{1}and

*b*̄

_{1}.

## 3. Investigation of particle chains

Below we report the analysis of frequencies and quality factors of ordered linear chains of identical equally spaced particles. It is clear that the regime of the most strongly bound polariton modes is realized when particles are as close as possible to each other. This takes place when the distance between centers of particles is equal to their diameter (cf. the criterion Eq. (2)) or when *d*=*a*. In numerical calculations, we have chosen the interparticle distance *a*=2. Since mode frequencies and decay rates are inversely proportional to the particle size, one can easily recalculate them for any size *a*.

## 3.1. Mode frequencies

The study of quasi-states possessing highest quality factors in cases (A) and (B) leads to two different modes corresponding to the lowest frequency Mie resonances in Mie scattering coefficients *a*ā_{1} and *b*̄_{1}, respectively. We shall conveniently call those modes *a*-mode and *b*-mode. The frequency of *b*-mode is less than *a*-mode corresponding to previous findings. [4] Both modes are also clearly distinguishable in a more advanced analysis using cases (C)-(F) corresponding to *n*
* _{max}*=1,2,3,4. For both modes an increase of

*n*

*always leads to a reduction in their frequencies, which can be explained by level repulsion. [4] However this reduction in frequency is very small. Comparing the approximations (A) and (B) with the most accurate case (F) corresponding to*

_{max}*n*

*=4 shows that the error in the frequency definition by the simplest approximations (A) and (B) does not exceed 1%.*

_{max}Frequencies of both *a*- and *b*- modes also depend weakly on the number of particles. Therefore, we were able to estimate frequencies using only *N*=10 spheres. In table 1 the dimensionless parameter λ/(2*d*) is given for *b*- and *a*- transverse and longitudinal modes and particle refractive indices *n*
* _{r}*=3.5,2.7,1.9 corresponding to GaAs, TiO

_{2}and ZnO, respectively. Recall that when this parameter is greater than unity, the mode is a guiding mode which means that the quality factor approaches infinity with increasing number of particles. Otherwise, the mode cannot transfer energy towards very long chains of particles (see, however, [12, 13, 14, 15, 16]). It can be concluded from table 1 that all low frequency

*a*- and

*b*-modes for GaAs and TiO

_{2}are guiding, while in

*ZnO*only the transverse

*b*-mode is definitely guiding. Since the transverse

*a*-mode and longitudinal

*b*-mode for ZnO are very close to the threshold, we cannot be 100% sure whether they are guiding or not. The longitudinal

*a*-mode in ZnO is definitely not guiding. As it was discussed in [4] we do expect that there always exists guiding transverse modes because of the long-range radiative field decreasing inversely proportional to the distance. This leads to the logarithmic divergency of the Fourier transform of the intersphere interaction. Due to this divergency, one can always satisfy Eq. (1) although the number of spheres needed to obtain the remarkable increase in the quality factor grows exponentially with the reduction of the refractive index (see [4] for details). Particularly for refractive index

*n*

*≈1.5, one would need to have a chain of tens of thousand of spheres which is difficult practically. This is not the case for longitudinal mode, because longitudinal coupling of far separated spheres decreases with the distance as 1/*

_{r}*r*

^{2}so its Fourier transform perfectly converges. Therefore there are indeed no longitudinal guiding modes for

*n*

*<1.9.*

_{r}One should notice that guiding modes have been studied previously in 2-dimensional systems using both frequency domain and time domain investigation (See Refs. [2, 3] and also more recent work on 2-dimensional systems having rotational symmetry [27, 28]). These studies proved the presence of guiding modes in linear chains made of infinite rods. One should note that there is difference between interactions of spheres in 3-dimensional systems and rods in quasi-2-dimensional system. The interaction of rods decreases with the distance *r* as 1/√*r* which makes the divergence of the interaction Fourier transform much stronger then in a 3-dimensional case. Therefore it is much easier to observe guiding modes in a 2-dimensional case at small refractive index then in a corresponding 3-dimensional case which is the main target of the present paper.

## 3.2. Quality Factors

In contrast to frequencies, quality factors for *a*- and *b*- modes show quantitative sensitivity to approximation, while their qualitative behavior has certain universal properties in the guiding regime. To illustrate what we mean in Fig. 2, we show how the quality factor depends on the number of particles *N* in the transverse *b*-mode in GaAs. Irrespective to approximation, the dependence *Q*(*N*) can be expressed algebraically as *Q*(*N*)≈*CN*
^{3} (see Fig. 2). The coefficient *C* depends only on the approximation. Since the difference between approximations *n*
_{max}=3 and *n*
* _{max}*=4 is as small as 0.3% in GaAs and is even smaller in ZnO and TiO

_{2}, we believe that

*n*

*=3 gives a sufficiently accurate estimate of the quality factor in all situations. The results below are given in the approximation using*

_{max}*n*

*=2 for ZnO and TiO*

_{max}_{2}, where the convergence is faster than in GaAs where we used

*n*

*=3.*

_{max}Our study of all *b*-modes in GaAs and TiO_{2} shows that their quality factor obeys the law

similarly to previous findings [11]. The physical nature of this dependence has been interpreted based on analysis of the lifetime of the mode in a finite photonic one-dimensional system near the band edge, following previous work by Dowling and coworkers[29, 30] (see also recent theoretical and experimental work [17, 18, 19, 20, 21, 22]). The main difference between our present work and Ref. [11] is that here we are using the explicit Mie resonance following Ref. [4], while in Ref. [11] dielectric spheres were modeled as harmonic oscillators. Although our considerations are quite similar, our justification for the *Q*∝*N*
^{3} law is given in greater detail here. The most bound mode for all systems of interest is realized at the maximum polariton quasi-wavevector *q*=*π*/*a*. This was clearly demonstrated in Ref. [4] for circular arrays, where the modes are defined by their conserving quasi-angular momenta and the value of quasi-angular momentum for the most bound mode clearly corresponds to this maximum wavevector. There is no translational or rotational invariance in the finite chain, which permits us to find the exact momentum of the mode, although the most bound mode frequencies *ω* (wavevectors *k*=*ω*/*c*) are nearly identical in circular arrays and linear chains. In circular arrays the point *q*=*π*/*a* is a local or global minimum or maximum in the polariton energy band (cf. Fig. 1) defined by the dispersion function *ω*(*q*) because of the relationship *ω*(*π*/*a*-*x*)=*ω*(*π*/*a*+*x*) (cf. Ref. [11]). Therefore the point *k*=*π*/*a* in a linear array corresponds to the top or the bottom of the energy band where the group velocity of the mode has a minimum leading to the maximum in its lifetime in a finite sample of the length *L*. This lifetime maximum is because the radiative losses occur at the edges of the particle chain and it takes the longest time for the slowest mode to travel to the edge of the chain.

Following Ref. [11], we propose a model to estimate the dependence of that lifetime *τ* (and the mode quality factor *Q*∝*t*) in a linear array of length *L*=*Na* (or number of particles *N*). In this model we consider quantum mechanical quasistates of a particle with a unit mass within the material placed in the domain (-*L*<*x*<*L*) and the potential energy *U*
_{0}>0. The quantum states with energy *E* close to the potential minimum inside the material |*E*-*U*
_{0}|≪*E* are used to model polariton modes near the band edge. Calculations performed in the Appendix show that the quality factor of the mode with the lowest energy increases with the size of the system as (see Eq. (14) in Appendix) *Q*∝*L*
^{3}, which is equivalent to Eq. (8).

Thus we obtained and interpreted the dependence of the quality factor on the number of particles *Q*∝*N*
^{3} for low-frequency most bound modes. This result has one important consequence. *It is clear that those modes possess the minimal group velocity that vanishes with increasing the number of particles as* 1/*N. Therefore the low frequency modes in particle chains can be used to slow down the propagation of light.* Our model gives a one-dimensional realization for the “frozen light” described in Ref. [21, 22], which can be an interesting addition to other existing realizations [17, 18, 19, 20].

The quality factor of *a*-modes does not necessarily increase with the number of particles as fast as *N*
^{3} even for guiding modes. However, this is not true for the transverse *a*-mode in a chain of GaAs particles. Probably this is due to the hybridization of this *a*-mode with rapidly decaying modes in the same frequency range associated with the Mie scattering amplitude *b*. On the other hand, we see *N*
^{3} dependence of the quality factor for the longitudinal *a*-mode in GaAs possibly because thismode has no frequency overlap with other longitudinal modes. In all cases of *a*-modes we found either that the quality factor increases with the number of particles *N* slower than *N*
^{3} or that the quality factor is smaller than for the corresponding *b*-mode. Therefore, the analysis below is restricted to the *b*- modes for GaAs, TiO_{2}, and ZnO. The quality factors for transverse and longitudinal *b*-modes for these materials are shown in Figs. 3, 4. Data in graphs have been calculated using *n*
* _{max}*=3 for GaAs and

*n*

*=2 for TiO*

_{max}_{2}and ZnO, where this approximation is already sufficiently accurate.

It is clear that both GaAs and TiO_{2} longitudinal modes possess a higher quality factor. The difference in quality factors is by a factor of 4 for TiO_{2} and by a factor of 10 for *GaAs*. This result differs from the results for circular arrays [4], where we found that the transverse *b*-mode possesses the highest quality factor. This is not surprising however, because in the case of circular arrays the quality factor increases exponentially *Q*(*N*)∝exp(*kN*) with the number of particles *N* and the factor *k* increases with the reduction of frequency. Therefore, in the case of circular arrays, low-frequency modes always possess the highest quality factor for sufficiently large number *N* of particles. This analysis is not applicable to our case where the quality factor depends algebraically on the number of particles (*Q*(*N*)∝*N*
^{3}). The reason why the longitudinal mode has a longer life time than the transverse mode is not easy to clarify. Perhaps this is because the vacuum photon field is transverse, and therefore couples better to the transverse polariton mode then to the longitudinal one.

Since the transverse *b*-mode in ZnO is slightly above the guiding threshold and the longitudinal mode is below it (see table 1), we do not see the *N*
^{3} dependence for the quality factor of either mode. At a small number of particles *N*<22, longitudinal modes have a higher quality factor similar to TiO_{2} and GaAs, while at larger number of particles a transverse mode “wins” because it is guiding while a longitudinal mode is not.

## 4. Conclusion

In this paper we studied low frequency optical modes possessing highest quality factors formed in one-dimensional finite chains of dielectric particles including GaAs (refractive index *n*
* _{r}*=3.5), TiO

_{2}(

*n*

*=2.7, rutile phase) and ZnO (*

_{r}*n*

*=1.9). The study was performed using the multi-sphere Mie-scattering formalism. We investigated quasi-states of light possessing lowest decay rates, i. e. highest quality factor, for transverse and longitudinal modes with frequency corresponding to the firstMie resonance associated with theMie scattering amplitude*

_{r}*b*̄

_{1}. It was shown that mode frequencies can be defined accurately using the simplest approach of dipolar interaction between spheres, while an accurate estimate of quality factor requires the use of

*n*

*=3 corresponding to simultaneous consideration of dipoles, quadrupoles and octupoles.*

_{max}We demonstrated that the quality factor of most bound modes indeed increases with the number of particles within the chain approximately as *Q*(*N*)∝*N*
^{3} for GaAs and TiO_{2} possessing higher refractive indices. It turns out that in these materials the mode possessing the highest quality factor is the longitudinal mode which contrasts to the situation in circular arrays. [4] The theoretical justification for the *N*
^{3} law is supported by modeling the modes near the bandedge of a one-dimensional finite system. It is interesting that modes possessing the highest quality factor have a group velocity vanishing with increasing chain length and can be considered regarding slow light related applications. [21, 22] The universal behavior *Q* µ *N*
^{3} does not take place in ZnO, which has the smallest refractive index *n*
* _{r}*=1.9 because it is is near the threshold for formation of bound guiding modes and therefore the quality factor behavior there is quite non-universal. The best spectrum domain where absorption of light by materials is negligible corresponds to intermediate photon frequencies ranging from 0.15eV to the electronic forbidden band width.

## Appendix

We model the quality factor dependence on the number of particles using the mode having the longest lifetime, which is the mode nearest to the band edge. The simplest model for such mode can be formulated using the one-dimensional system with the potential

$U\left(x\right)=\{\begin{array}{c}U>0,-L<x<L\\ 0,\mathrm{otherwise}\end{array}$

The quasistate with the lowest energy *E*>*U*
_{0} can be used to estimate the quality factor following Eq. (5). We let the mass to be equal 1 because it does not influence the quality factor dependence on the size *L*. Then one can represent the solution for the quasistate at *x*>0 in the form

${\Psi}_{0}={C}_{0}\mathrm{cos}\left(\sqrt{E-U}x\right),x<L,$ ${\Psi}_{1}={C}_{1}{e}^{i\sqrt{E}x},x>L.$

The boundary condition is selected as the outgoing wave as it has to be for quasistates. [31] Due to the symmetry of the model, it is sufficient to satisfy the boundary conditions at *x*=*L*, where the boundary conditions can be expressed as the continuity requirements for the wavefunction and its derivative Ψ_{0}(*L*)=Ψ_{1}(*L*), Ψ^{′}
_{0}(*L*)=Ψ^{′}
_{1}(*L*). Making the appropriate substitutions yields the following system of equations

where Eq. (9) is for the wave function and Eq. (10) is for its derivatives. Dividing Eq.(9) by Eq.(10) gives us

In our case, *L*→∞, *E* →*U*
_{0}, so in order for Eq. (11) to be valid, we must have

Using Eq. (12) and substituting $\delta \approx \delta =i\frac{\pi}{2}{U}^{-1\u20442}$ we obtain

$\frac{\pi}{2}-\mathrm{aL}\approx i\frac{\pi}{2L}{U}^{-1\u20442},a=\frac{\pi}{2L}\left(1-\frac{i}{L\sqrt{U}}\right)$

${a}^{2}=\frac{{\pi}^{2}}{4{L}^{2}}\left(1-\frac{2i}{L\sqrt{U}}\right)$

$E\approx U+\frac{{\pi}^{2}}{4{L}^{2}}-i\frac{{\pi}^{2}}{4{L}^{3}\sqrt{U}}$

Hence

$$\mathrm{ReE}=U+\frac{{\pi}^{2}}{4{L}^{2}}\approx U$$

and the quality factor can be expressed as Eq. (5)

which agrees with our numerical findings.

## Acknowledgement

This work is supported by the U.S. Air Force Office of Scientific Research (Grant No. FA9550-06-1-0110). We are grateful to Arthur Yaghjian, Svetlana Boriskina, Hui Cao, Vasily Astratov, Dmitry Uskov and Alexei Yamilov for useful discussions and suggestions. AB also acknowledges organizers and participants of the workshop on Physics of Microresonators, June 6–9, 2007, in Charlotte for useful discussions, which had a positive impact on our group research.

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