## Guiding optical modes in chains of dielectric particles

Optics Express, Vol. 15, Issue 25, pp. 17380-17391 (2007)

http://dx.doi.org/10.1364/OE.15.017380

Acrobat PDF (149 KB)

### 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[*N* and the material’s refractive index *n**
_{r}
*. We found that, in agreement with the earlier work [

*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

*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

*ω*/

*c*<

*π*/

*a*is equivalent to the condition that the interparticle distance

*a*should be less than half of the resonant wavelength

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

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]

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]

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]

_{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.

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]

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]

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]

**32**, 409–411 (2007). [CrossRef] [PubMed]

**85**, 5508 (2004). [CrossRef]

*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

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]

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

*n*

*>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.*

_{r}## 2. Multisphere Mie scattering formalism in the study of particle chains

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]

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

*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

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]

*A*

^{jl}_{m1µ1}is equivalent to the “retarded” dipole-dipole interaction [4

**73**, 066614 (2006). [CrossRef]

*B*

^{jl}_{m1µ1}reflects the interaction of electric and magnetic dipoles. An upper bound is placed on the maximum angular momentum

*n*

*to make the problem numerically solvable. In the exact formalism*

_{max}*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}**73**, 066614 (2006). [CrossRef]

*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

**73**, 066614 (2006). [CrossRef]

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]

*a*

^{l}_{m1}(A) or

*b*

^{l}_{m1}(B) (cf. [4

**73**, 066614 (2006). [CrossRef]

*n*

*=1 (C),*

_{max}*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}*n*

*≈1.9), TiO*

_{r}_{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*

_{r}**21**, 121–131 (2004). [CrossRef]

*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*

*and*

_{mnµv}*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

*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

## 3. Investigation of particle chains

*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

*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

**73**, 066614 (2006). [CrossRef]

*n*

*=1,2,3,4. For both modes an increase of*

_{max}*n*

*always leads to a reduction in their frequencies, which can be explained by level repulsion. [4*

_{max}**73**, 066614 (2006). [CrossRef]

*n*

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

_{max}*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*

*=3.5,2.7,1.9 corresponding to GaAs, TiO*

_{r}_{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

**32**, 409–411 (2007). [CrossRef] [PubMed]

**85**, 5508 (2004). [CrossRef]

*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

**73**, 066614 (2006). [CrossRef]

**73**, 066614 (2006). [CrossRef]

*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}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. 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]

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]

*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

*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*

*=4 is as small as 0.3% in GaAs and is even smaller in ZnO and TiO*

_{max}_{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}*b*-modes in GaAs and TiO

_{2}shows that their quality factor obeys the law

**21**, 121–131 (2004). [CrossRef]

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]

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]

**21**, 121–131 (2004). [CrossRef]

**73**, 066614 (2006). [CrossRef]

**21**, 121–131 (2004). [CrossRef]

*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

**73**, 066614 (2006). [CrossRef]

*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

**21**, 121–131 (2004). [CrossRef]

*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.

**21**, 121–131 (2004). [CrossRef]

*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).

*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

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]

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]

*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*

*=3 for GaAs and*

_{max}*n*

*=2 for TiO*

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

_{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

**73**, 066614 (2006). [CrossRef]

*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.

*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

*n*

*=3.5), TiO*

_{r}_{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}## Appendix

*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

*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

## Acknowledgement

## References and links

1. | R. A. Shore and A. D. Yaghjian, “Traveling electromagnetic waves on linear periodic arrays of lossless spheres,” Electron. Lett. |

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 |

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 |

4. | A. L. Burin, “Bound whispering gallery modes in circular arrays of dielectric spherical particles,” Phys. Rev. E |

5. | Z. Y. Tang and N. A. Kotov, “One-dimensional assemblies of nanoparticles: Preparation, properties, and promise,” Adv. Mater. |

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. |

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. |

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. |

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 |

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. |

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. |

16. | L. I. Deych and O. Roslyak, “Photonic band mixing in linear chains of optically coupled microspheres,” Phys. Rev. B |

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 |

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. |

19. | M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science |

20. | J. T. Mok and B. J. Eggleton, “Expect more delays,” Nature |

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. |

24. | Y. L. Xu, “Scattering Mueller matrix of an ensemble of variously shaped small particles,” J. Opt. Soc. Am. A |

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. |

28. | S. V. Boriskina, “Theoretical prediction of a dramatic Q-factor enhancement and degeneracy removal of WG modes in symmetrical photonic molecules,” Opt. Lett. |

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 |

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. |

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

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- 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]
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- A. L. Burin, "Bound whispering gallery modes in circular arrays of dielectric spherical particles," Phys. Rev. E 73,066614 (2006). [CrossRef]
- Z. Y. Tang and N. A. Kotov, "One-dimensional assemblies of nanoparticles: Preparation, properties, and promise," Adv. Mater. 17,951-962 (2005). [CrossRef]
- 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]
- 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]
- C. Kittel, Introduction to Solid State Physics, (Wiley, New York, 1996).
- 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).
- R. W. P. King, G. J. Fikioris, and R. B. Mask, Cylindrical Antennas and Arrays, (Cambridge University Press, Cambridge, 2005).
- 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]
- 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]
- 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]
- 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]
- 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]
- L. I. Deych and O. Roslyak, "Photonic band mixing in linear chains of optically coupled microspheres," Phys. Rev. B 73, art no 036606 (2006)
- 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]
- 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]
- 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]
- J. T. Mok and B. J. Eggleton, "Expect more delays," Nature 433,811812 (2005). [CrossRef]
- 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]
- A. Figotin and I. Vitebskiy, "Frozen light in photonic crystals with degenerate band edge," Phys. Rev. E 74, Art. No. 066613 (2006). [CrossRef]
- Y. L. Xu, "Electromagnetic scattering by an aggregate of spheres: far field," Appl. Opt. 36,9496-9508 (1997). [CrossRef]
- Y. L. Xu, "Scattering Mueller matrix of an ensemble of variously shaped small particles," J. Opt. Soc. Am. A 20,2093-2105 (2003). [CrossRef]
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-DifferenceTtime-Domain Method, 3rd ed. (Artech House Publishers, 2005).
- E. D. Palik, Handbook of Optical Constants in Solids, (Acad. Press Handbook Series, Academic Press INC. 1985).
- 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]
- 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]
- 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]
- 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]
- L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, (Pergamon Press, Oxford, New York, 1977).

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