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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 25 — Dec. 10, 2007
  • pp: 17380–17391
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Guiding optical modes in chains of dielectric particles

Gail S. Blaustein, Michael I. Gozman, Olga Samoylova, I. Ya. Polishchuk, and Alexander L. Burin  »View Author Affiliations


Optics Express, Vol. 15, Issue 25, pp. 17380-17391 (2007)
http://dx.doi.org/10.1364/OE.15.017380


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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 r >2 and the quality factor of the most bound mode scales by N3. 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

Fig. 1. (a) Linear chain of scattering particles; (b) Possible values of the quasi-wavevector of propagating polariton modes with domain of guiding modes indicated. All notations are described within the text.

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

8. C. Kittel, Introduction to Solid State Physics, Wiley, New York, 1996.

] 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

(πa<k<ωc),(ωc<q<πa),
(1)

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

a<λ2.
(2)

Our derivation is equivalent to the classical work[9

9. H. W. Ehrespeck and H. Poehler, “A new method for obtaining maximum gain from Yagi antennas,” IEEE Trans. Antennas Propag. AP-7, 379–386 (1959).

] where a light cone constraint was applied and later used in Refs. [1

1. R. A. Shore and A. D. Yaghjian, “Traveling electromagnetic waves on linear periodic arrays of lossless spheres,” Electron. Lett. 41, 578–580 (2005). [CrossRef]

, 4

4. A. L. Burin, “Bound whispering gallery modes in circular arrays of dielectric spherical particles,” Phys. Rev. E 73, 066614 (2006). [CrossRef]

, 10

10. R. W. P. King, G. J. Fikioris, and R. B. Mask, Cylindrical Antennas and Arrays, Cambridge University Press, Cambridge, 2005.

, 11

11. A. L. Burin, G. C. Schatz, H. Cao, and M. A. Ratner, “High quality optical modes in low-dimensional arrays of nanoparticles. Application to random lasers,” J. Opt. Soc. Am. B 21, 121–131 (2004). [CrossRef]

] 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

6. S. A. Mayer, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nature Mater. 2, 229–232 (2003). [CrossRef]

, 7

7. S. A. Mayer, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, B. E. Koel, and H. A. Atwater, “Plasmonics A route to nanoscale optical devices,” Adv. Mater. 15, 562–562 (2003). [CrossRef]

] 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

11. A. L. Burin, G. C. Schatz, H. Cao, and M. A. Ratner, “High quality optical modes in low-dimensional arrays of nanoparticles. Application to random lasers,” J. Opt. Soc. Am. B 21, 121–131 (2004). [CrossRef]

] 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 TiO2, 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.

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

23. Y. L. Xu, “Electromagnetic scattering by an aggregate of spheres: far field,” Appl. Opt. 36, 9496–9508 (1997). [CrossRef]

, 24

24. Y. L. Xu, “Scattering Mueller matrix of an ensemble of variously shaped small particles,” J. Opt. Soc. Am. A 20, 2093–2105 (2003). [CrossRef]

] (see also [4

4. A. L. Burin, “Bound whispering gallery modes in circular arrays of dielectric spherical particles,” Phys. Rev. E 73, 066614 (2006). [CrossRef]

]), which has a distinct advantage compared to the standard finite difference time domain (FDTD) approach [25

25. A. Taflove and S. C. Hagness, Computational Electrodynamics: The finite-difference time-domain method, 3rd ed. Artech House Publishers, 2005.

] 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

amnla¯n+j=1(l)Nn=1nmaxm=nnAmnμνjlaμνj+j=1(l)Nn=1nmaxm=nnBmnμνjlbμνj=pmnl;
bmnlb¯n+j=1(l)Nn=1nmaxm=nnAmnμνjlbμνj+j=1(l)Nn=1nmaxm=nnBmnμνjlaμνj=qmnl.
(3)

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

4. A. L. Burin, “Bound whispering gallery modes in circular arrays of dielectric spherical particles,” Phys. Rev. E 73, 066614 (2006). [CrossRef]

], 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 max=∞. Remember that Eq. (3) is written in the frequency domain for a given frequency z. Mie scattering coefficients a n, b n are functions of the product qd l/2 of photon wavevector q=z/c and l-th particle radius d l/2. They also depend on particle refractive index n r, while vector translation coefficients for spheres j and l separated by the distance r jl depend only on the dimensionless product qr jl.

Following Ref. [4

4. A. L. Burin, “Bound whispering gallery modes in circular arrays of dielectric spherical particles,” Phys. Rev. E 73, 066614 (2006). [CrossRef]

], we will study quasi-states of light, which are eigen-modes of Eq. (3) defined by the homogeneous equation

amnla¯n+j=1(l)Nn=1nmaxm=nnAmnμνjlaμνj+j=1(l)Nn=1nmaxm=nnBmnμνjlbμνj=0;
bmnlb¯n+j=1(l)Nn=1nmaxm=nnAmnμνjlbμνj+j=1(l)Nn=1nmaxm=nnBmnμνjlaμνj=0,
(4)

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- a 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 can be defined in the usual way as

Qa=ωa2γa,
(5)

which is our topic of interest in this work. A similar approach was taken in Refs. [4

4. A. L. Burin, “Bound whispering gallery modes in circular arrays of dielectric spherical particles,” Phys. Rev. E 73, 066614 (2006). [CrossRef]

, 11

11. A. L. Burin, G. C. Schatz, H. Cao, and M. A. Ratner, “High quality optical modes in low-dimensional arrays of nanoparticles. Application to random lasers,” J. Opt. Soc. Am. B 21, 121–131 (2004). [CrossRef]

] 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

4. A. L. Burin, “Bound whispering gallery modes in circular arrays of dielectric spherical particles,” Phys. Rev. E 73, 066614 (2006). [CrossRef]

]). Four other choices are defined by setting either n max=1 (C), n max=2 (D), n max=3 (E) and n max=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 n max=3 provided a sufficiently accurate approximation of the quality factor. Therefore we did not consider n max>4.

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), TiO2 (rutile, n r≈2.7) and GaAs (n r≈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

11. A. L. Burin, G. C. Schatz, H. Cao, and M. A. Ratner, “High quality optical modes in low-dimensional arrays of nanoparticles. Application to random lasers,” J. Opt. Soc. Am. B 21, 121–131 (2004). [CrossRef]

], where lasing modes were investigated for interacting dipolar oscillators.

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 mnµv Eq. (4) for both centers of spheres j and l belonging to the z-axis are equal to zero if mµ. 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

M̂(z)x=0,
(6)

where M̂ is the matrix of size N×N in cases (A) and (B), 2N×2N in case (C), 4N×4N 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 M̂ has a zero eigenvalue. Thus, we need to find a value for z which forces one eigenvalue of matrix M̂(z) to be zero and possesses a minimal imaginary part. We define frequency z as a limit of a generalized Newton-Raphson algorithm

zn+1=znf(zn)(df(zn)dz).
(7)

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

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/(2d) 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, TiO2 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

12. Y. Hara, “Heavy photon states in photonic chains of resonantly coupled cavities with supermonodispersive microspheres,” Phys. Rev. Lett.94, Art. No. 203905 (2005). [CrossRef] [PubMed]

, 13

13. A.M. Kapitonov and V. N. Astratov, “Observation of nanojet-induced modes with small propagation losses in chains of coupled spherical cavities,” Opt. Lett. 32, 409–411 (2007). [CrossRef] [PubMed]

, 14

14. A. V. Kanaev, V. N. Astratov, and W. Cai, “Optical coupling at a distance between detuned spherical cavities,” Appl. Phys. Lett.88, Art. No. 111111 (2006). [CrossRef]

, 15

15. 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 (2004). [CrossRef]

, 16

16. L. I. Deych and O. Roslyak, “Photonic band mixing in linear chains of optically coupled microspheres,” Phys. Rev. B 73, art no 036606 (2006)

]). It can be concluded from table 1 that all low frequency a- and b-modes for GaAs and TiO2 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

4. A. L. Burin, “Bound whispering gallery modes in circular arrays of dielectric spherical particles,” Phys. Rev. E 73, 066614 (2006). [CrossRef]

] 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

4. A. L. Burin, “Bound whispering gallery modes in circular arrays of dielectric spherical particles,” Phys. Rev. E 73, 066614 (2006). [CrossRef]

] for details). Particularly for refractive index n r≈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 2 so its Fourier transform perfectly converges. Therefore there are indeed no longitudinal guiding modes for n r<1.9.

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

2. S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B 12, pp. 1267–72 (1995). [CrossRef]

, 3

3. R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434 (1993). [CrossRef]

] and also more recent work on 2-dimensional systems having rotational symmetry [27

27. E. I. Smotrova and A. I. Nosich, “Mathematical study of the two-dimensional lasing problem for the whispering gallery modes in a circular dielectric microcavity,” Opt. Quantum Electron. 36, 213–221 (2004). [CrossRef]

, 28

28. S. V. Boriskina, “Theoretical prediction of a dramatic Q-factor enhancement and degeneracy removal of WG modes in symmetrical photonic molecules,” Opt. Lett. 31, 338–340 (2006). [CrossRef] [PubMed]

]). 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.

Table 1. Guiding parameter for low frequency modes.

table-icon
View This Table

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 TiO2, we believe that n max=3 gives a sufficiently accurate estimate of the quality factor in all situations. The results below are given in the approximation using n max=2 for ZnO and TiO2, where the convergence is faster than in GaAs where we used n max=3.

Our study of all b-modes in GaAs and TiO2 shows that their quality factor obeys the law

Q(N)N3
(8)

Fig. 2. Quality factor of a transverse b-mode for the chain of GaAs particles versus number of particles N, calculated in various approaches. The results for the approximation (F) corresponding to n max=4 are not shown because they will be not distinguishable with the approximation (E) different from it by less than 1%.

Following Ref. [11

11. A. L. Burin, G. C. Schatz, H. Cao, and M. A. Ratner, “High quality optical modes in low-dimensional arrays of nanoparticles. Application to random lasers,” J. Opt. Soc. Am. B 21, 121–131 (2004). [CrossRef]

], we propose a model to estimate the dependence of that lifetime τ (and the mode quality factor Qt) 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) QL 3, which is equivalent to Eq. (8).

Thus we obtained and interpreted the dependence of the quality factor on the number of particles QN 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

21. J. T. Shen, M. L. Povinelli, S. Sandhu, and S. H. Fan, “Stopping single photons in one-dimensional circuit quantum electrodynamics systems,” Phys. Rev. B75, Art. No. 035320, (2007). [CrossRef]

, 22

22. A. Figotin and I. Vitebskiy, “Frozen light in photonic crystals with degenerate band edge,” Phys. Rev. E74, Art. No. 066613 (2006). [CrossRef]

], which can be an interesting addition to other existing realizations [17

17. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005). [CrossRef] [PubMed]

, 18

18. P. C. Ku, F. Sedgwick, C. J. Chang-Hasnain, P. Palinginis, T. Li, H. Wang, S.-W. Chang, and S.-L. Chuang, “Slow light in semiconductor quantum wells,” Opt. Lett. 29, 22912293 (2004). [CrossRef]

, 19

19. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200202 (2003). [CrossRef]

, 20

20. J. T. Mok and B. J. Eggleton, “Expect more delays,” Nature 433, 811812 (2005). [CrossRef]

].

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, TiO2, 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 max=2 for TiO2 and ZnO, where this approximation is already sufficiently accurate.

Fig. 3. Quality factor of transverse and longitudinal b-modes for the chain of GaAs and TiO2 particles versus number of particles N, calculated for n max=3.

It is clear that both GaAs and TiO2 longitudinal modes possess a higher quality factor. The difference in quality factors is by a factor of 4 for TiO2 and by a factor of 10 for GaAs. This result differs from the results for circular arrays [4

4. A. L. Burin, “Bound whispering gallery modes in circular arrays of dielectric spherical particles,” Phys. Rev. E 73, 066614 (2006). [CrossRef]

], 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 TiO2 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), TiO2 (n r=2.7, rutile phase) and ZnO (n r=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 b̄1. 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 max=3 corresponding to simultaneous consideration of dipoles, quadrupoles and octupoles.

Fig. 4. Quality factor of transverse and longitudinal b-modes for the chain of ZnO particles versus number of particles N, calculated for n max=2.

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(x)={U>0,L<x<L0,otherwise

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

Ψ0=C0cos(EUx),x<L, Ψ1=C1eiEx,x>L.

The boundary condition is selected as the outgoing wave as it has to be for quasistates. [31

31. L. D. Landau and E.M. Lifshitz, QuantumMechanics: Non-Relativistic Theory, Pergamon Press, Oxford, New York, 1977.

] 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 Ψ0(L)=Ψ1(L), Ψ 0(L)=Ψ 1(L). Making the appropriate substitutions yields the following system of equations

C0cos(EUL)=C1eiEL,
(9)
C0EUsin(EUL)=iEC1eiEL.
(10)

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

1EUtan(EUL)=iE
(11)

In our case, L→∞, EU 0, so in order for Eq. (11) to be valid, we must have

EU=a,aL=π2δ,δ1
(12)

Using Eq. (12) and substituting δδ=iπ2U12 we obtain

π2aLiπ2LU12,a=π2L(1iLU)

a2=π24L2(12iLU)

EU+π24L2iπ24L3U

Hence

ImE=iπ24L3U
ReE=U+π24L2U
(13)

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

Q=ReE2ImE=L3π2U32,
(14)

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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2.

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3.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434 (1993). [CrossRef]

4.

A. L. Burin, “Bound whispering gallery modes in circular arrays of dielectric spherical particles,” Phys. Rev. E 73, 066614 (2006). [CrossRef]

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Z. Y. Tang and N. A. Kotov, “One-dimensional assemblies of nanoparticles: Preparation, properties, and promise,” Adv. Mater. 17, 951–962 (2005). [CrossRef]

6.

S. A. Mayer, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nature Mater. 2, 229–232 (2003). [CrossRef]

7.

S. A. Mayer, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, B. E. Koel, and H. A. Atwater, “Plasmonics A route to nanoscale optical devices,” Adv. Mater. 15, 562–562 (2003). [CrossRef]

8.

C. Kittel, Introduction to Solid State Physics, Wiley, New York, 1996.

9.

H. W. Ehrespeck and H. Poehler, “A new method for obtaining maximum gain from Yagi antennas,” IEEE Trans. Antennas Propag. AP-7, 379–386 (1959).

10.

R. W. P. King, G. J. Fikioris, and R. B. Mask, Cylindrical Antennas and Arrays, Cambridge University Press, Cambridge, 2005.

11.

A. L. Burin, G. C. Schatz, H. Cao, and M. A. Ratner, “High quality optical modes in low-dimensional arrays of nanoparticles. Application to random lasers,” J. Opt. Soc. Am. B 21, 121–131 (2004). [CrossRef]

12.

Y. Hara, “Heavy photon states in photonic chains of resonantly coupled cavities with supermonodispersive microspheres,” Phys. Rev. Lett.94, Art. No. 203905 (2005). [CrossRef] [PubMed]

13.

A.M. Kapitonov and V. N. Astratov, “Observation of nanojet-induced modes with small propagation losses in chains of coupled spherical cavities,” Opt. Lett. 32, 409–411 (2007). [CrossRef] [PubMed]

14.

A. V. Kanaev, V. N. Astratov, and W. Cai, “Optical coupling at a distance between detuned spherical cavities,” Appl. Phys. Lett.88, Art. No. 111111 (2006). [CrossRef]

15.

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 (2004). [CrossRef]

16.

L. I. Deych and O. Roslyak, “Photonic band mixing in linear chains of optically coupled microspheres,” Phys. Rev. B 73, art no 036606 (2006)

17.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005). [CrossRef] [PubMed]

18.

P. C. Ku, F. Sedgwick, C. J. Chang-Hasnain, P. Palinginis, T. Li, H. Wang, S.-W. Chang, and S.-L. Chuang, “Slow light in semiconductor quantum wells,” Opt. Lett. 29, 22912293 (2004). [CrossRef]

19.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200202 (2003). [CrossRef]

20.

J. T. Mok and B. J. Eggleton, “Expect more delays,” Nature 433, 811812 (2005). [CrossRef]

21.

J. T. Shen, M. L. Povinelli, S. Sandhu, and S. H. Fan, “Stopping single photons in one-dimensional circuit quantum electrodynamics systems,” Phys. Rev. B75, Art. No. 035320, (2007). [CrossRef]

22.

A. Figotin and I. Vitebskiy, “Frozen light in photonic crystals with degenerate band edge,” Phys. Rev. E74, Art. No. 066613 (2006). [CrossRef]

23.

Y. L. Xu, “Electromagnetic scattering by an aggregate of spheres: far field,” Appl. Opt. 36, 9496–9508 (1997). [CrossRef]

24.

Y. L. Xu, “Scattering Mueller matrix of an ensemble of variously shaped small particles,” J. Opt. Soc. Am. A 20, 2093–2105 (2003). [CrossRef]

25.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The finite-difference time-domain method, 3rd ed. Artech House Publishers, 2005.

26.

E. D. Palik, “Handbook of Optical Constants in Solids,” Acad. Press Handbook Series, Academic Press INC. 1985.

27.

E. I. Smotrova and A. I. Nosich, “Mathematical study of the two-dimensional lasing problem for the whispering gallery modes in a circular dielectric microcavity,” Opt. Quantum Electron. 36, 213–221 (2004). [CrossRef]

28.

S. V. Boriskina, “Theoretical prediction of a dramatic Q-factor enhancement and degeneracy removal of WG modes in symmetrical photonic molecules,” Opt. Lett. 31, 338–340 (2006). [CrossRef] [PubMed]

29.

J.M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite one-dimensional, photonic band-gap structures,” Phys. Rev. B 53, 4107–4121 (1996). [CrossRef]

30.

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band-edge laser a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994). [CrossRef]

31.

L. D. Landau and E.M. Lifshitz, QuantumMechanics: Non-Relativistic Theory, Pergamon Press, Oxford, New York, 1977.

OCIS Codes
(060.5295) Fiber optics and optical communications : Photonic crystal fibers

ToC Category:
Novel Concepts and Theory

History
Original Manuscript: October 8, 2007
Revised Manuscript: November 9, 2007
Manuscript Accepted: November 9, 2007
Published: December 10, 2007

Virtual Issues
Physics and Applications of Microresonators (2007) Optics Express

Citation
Gail S. Blaustein, Michael I. Gozman, Olga Samoylova, I. Ya. Polishchuk, and Alexander L. Burin, "Guiding optical modes in chains of dielectric particles," Opt. Express 15, 17380-17391 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-25-17380


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References

  1. R. A. Shore and A. D. Yaghjian, "Traveling electromagnetic waves on linear periodic arrays of lossless spheres," Electron. Lett. 41,578-580 (2005). [CrossRef]
  2. S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade and J. D. Joannopoulos, "Guided and defect modes in periodic dielectric waveguides," J. Opt. Soc. Am. B 12,1267-72 (1995). [CrossRef]
  3. R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, "Accurate theoretical analysis of photonic band-gap materials," Phys. Rev. B 48,8434 (1993). [CrossRef]
  4. A. L. Burin, "Bound whispering gallery modes in circular arrays of dielectric spherical particles," Phys. Rev. E 73,066614 (2006). [CrossRef]
  5. Z. Y. Tang and N. A. Kotov, "One-dimensional assemblies of nanoparticles: Preparation, properties, and promise," Adv. Mater. 17,951-962 (2005). [CrossRef]
  6. S. A. Mayer, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, A. A. G. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nat. Mater. 2,229-232 (2003). [CrossRef]
  7. S. A. Mayer, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, B. E. Koel, and H. A. Atwater, "Plasmonics A route to nanoscale optical devices," Adv. Mater. 15,562-562 (2003). [CrossRef]
  8. C. Kittel, Introduction to Solid State Physics, (Wiley, New York, 1996).
  9. H. W. Ehrespeck and H. Poehler, "A new method for obtaining maximum gain from Yagi antennas," IEEE Trans. Antennas Propag. AP- 7,379-386 (1959).
  10. R. W. P. King, G. J. Fikioris, and R. B. Mask, Cylindrical Antennas and Arrays, (Cambridge University Press, Cambridge, 2005).
  11. A. L. Burin, G. C. Schatz, H. Cao, and M. A. Ratner, "High quality optical modes in low-dimensional arrays of nanoparticles. Application to random lasers," J. Opt. Soc. Am. B 21,121-131 (2004). [CrossRef]
  12. Y. Hara, "Heavy photon states in photonic chains of resonantly coupled cavities with supermonodispersive microspheres," Phys. Rev. Lett. 94, Art. No. 203905 (2005). [CrossRef] [PubMed]
  13. A.M. Kapitonov and V. N. Astratov, "Observation of nanojet-induced modes with small propagation losses in chains of coupled spherical cavities," Opt. Lett. 32,409-411 (2007). [CrossRef] [PubMed]
  14. A. V. Kanaev, V. N. Astratov, and W. Cai, "Optical coupling at a distance between detuned spherical cavities," Appl. Phys. Lett. 88, Art. No. 111111 (2006). [CrossRef]
  15. 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 (2004). [CrossRef]
  16. L. I. Deych and O. Roslyak, "Photonic band mixing in linear chains of optically coupled microspheres," Phys. Rev. B 73, art no 036606 (2006)
  17. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438,65-69 (2005). [CrossRef] [PubMed]
  18. P. C. Ku, F. Sedgwick, C. J. Chang-Hasnain, P. Palinginis, T. Li, H. Wang, S.-W. Chang, and S.-L. Chuang, "Slow light in semiconductor quantum wells," Opt. Lett. 29,22912293 (2004). [CrossRef]
  19. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Superluminal and slow light propagation in a room-temperature solid," Science 301,200202 (2003). [CrossRef]
  20. J. T. Mok and B. J. Eggleton, "Expect more delays," Nature 433,811812 (2005). [CrossRef]
  21. J. T. Shen, M. L. Povinelli, S. Sandhu, and S. H. Fan, "Stopping single photons in one-dimensional circuit quantum electrodynamics systems," Phys. Rev. B 75, Art. No. 035320, (2007). [CrossRef]
  22. A. Figotin and I. Vitebskiy, "Frozen light in photonic crystals with degenerate band edge," Phys. Rev. E 74, Art. No. 066613 (2006). [CrossRef]
  23. Y. L. Xu, "Electromagnetic scattering by an aggregate of spheres: far field," Appl. Opt. 36,9496-9508 (1997). [CrossRef]
  24. Y. L. Xu, "Scattering Mueller matrix of an ensemble of variously shaped small particles," J. Opt. Soc. Am. A 20,2093-2105 (2003). [CrossRef]
  25. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-DifferenceTtime-Domain Method, 3rd ed. (Artech House Publishers, 2005).
  26. E. D. Palik, Handbook of Optical Constants in Solids, (Acad. Press Handbook Series, Academic Press INC. 1985).
  27. E. I. Smotrova and A. I. Nosich, "Mathematical study of the two-dimensional lasing problem for the whisperinggallery modes in a circular dielectric microcavity," Opt. Quantum Electron. 36,213-221 (2004). [CrossRef]
  28. S. V. Boriskina, "Theoretical prediction of a dramatic Q-factor enhancement and degeneracy removal of WG modes in symmetrical photonic molecules," Opt. Lett. 31,338-340 (2006). [CrossRef] [PubMed]
  29. J. M. Bendickson, J. P. Dowling, and M. Scalora, "Analytic expressions for the electromagnetic mode density in finite one-dimensional, photonic band-gap structures," Phys. Rev. B 53,4107-4121 (1996). [CrossRef]
  30. J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, "The photonic band-edge laser a new approach to gain enhancement," J. Appl. Phys. 75,1896-1899 (1994). [CrossRef]
  31. L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, (Pergamon Press, Oxford, New York, 1977).

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