Abstract

By using a discontinuous spectral element method, we analyze evanescent wave coupling of whispering gallery modes (WGMs) in microcylinder coupled resonator optical waveguides (CROWs). We demonstrate successful light propagation by WGMs through a chain of coupled cylinder resonators, and that the speed of such propagation is strongly dependent on the inter-resonator gap sizes. Our simulations also show that light propagates slower by WGMs with bigger azimuthal numbers than by those with smaller azimuthal numbers. On the other hand, the light propagation by WGMs of the same azimuthal number appears to have the same speed in CROWs regardless of the size and the material of the resonators, indicating that the tail (the part of a WGM outside the resonator) determines inter-resonator coupling strength.

© 2004 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]

Appl. Numer. Math.

S. Abarbanel and D. Gottlieb, �??On the construction and analysis of absorbing layers in CEM,�?? Appl. Numer. Math. 27, 331-340 (1998).
[CrossRef]

Appl. Phys. Lett.

T. D. Happ, M. Kamp, A. Forchel, J.-L. Gentner, and L. Goldstein, �??Two-dimensional photonic crystal coupled-defect laser diode,�?? Appl. Phys. Lett. 82, 4-6 (2003)
[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, 5508-5510 (2004).
[CrossRef]

Comput. Phys. Comm.

A. H. Mahammadian, V. Shankar, andW. F. Hall, �??Computation of electromagnetic scattering and radiation using a time-domain finite-volume discretization procedure,�?? Comput. Phys. Comm. 8, 175-196 (1991).
[CrossRef]

ICASE Technical Report, No. 99-30

A. Yefet and P. G. Petropoulos, �??A non-dissipative staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell�??s equations,�?? ICASE Technical Report, No. 99-30, NASA/CR-1999-209514, 1999.

IEEE J. Sel. Top. Quamtum Electron.

S. Mookherjea and A. Yariv, �??Coupled resonator optical waveguides,�?? IEEE J. Sel. Top. Quamtum Electron. 8, 448-456 (2002).
[CrossRef]

IEEE Trans. Antennas Propag.

K. S. Yee, �??Numerical solution of initial boundary value problems in solving Maxwell�??s equations in isotropic media,�?? IEEE Trans. Antennas Propag. P-14, 302-307 (1966).

Int. J. Numer. Meth. Eng.

S. J. Sherwin and G. E. Karniadakis, �??A new triangular and tetrahedral basis for high-order (hp) finite element methods,�?? Int. J. Numer. Meth. Eng. 38, 3775-3802 (1995).
[CrossRef]

D. A. Kopriva, S. L. Woodruff, and M. Y. Hussaini, �??Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method,�?? Int. J. Numer. Meth. Eng. 53, 105-122 (2002).
[CrossRef]

J. Comput. Phys.

T. Lu, P. Zhang, andW. Cai, �??Discontinuous Galerkin methods for dispersive and lossy Maxwell�??s equations and PML boundary conditions,�?? J. Comput. Phys. 200, 549-580 (2004).
[CrossRef]

J. Berenger, �??A perfectly matched layer for the absorption of electromagnetic waves,�?? J. Comput. Phys. 14, 185-200 (1994).
[CrossRef]

B. Yang, D. Gottlieb, and J. S. Hesthaven, �??Spectral simulation of electromagnetic wave scattering,�?? J. Comput. Phys. 34, 216-230 (1997).
[CrossRef]

J. Lightwave Technol.

X. Ji, T. Lu, W. Cai, and P. Zhang, �??Discontinuous Galerkin time domain (DGTD) methods for the study of waveguide coupled microring resonators,�?? J. Lightwave Technol. (submitted, October 2004).

S. C. Hagness, D. Rafizadeh, S. T. Ho, and A. Taflove, �??FDTD microcavity simulations: design and experimental realization of waveguide-coupled single-mode ring and whispering-gallery-mode disk resonators,�?? J. Lightwave Technol. 15, 2154-2165 (1997).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, �??Microring resonator channel dropping filters,�?? J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

J. Opt. Soc. Am. A

A. Martinez, A. Garcia, P. Sanchis, and J. Marti, �??Group velocity and dispersion model of coupled-cavity waveguides in photonic crystals,�?? J. Opt. Soc. Am. A 20, 147-150 (2003).
[CrossRef]

S. Deng and W. Cai, �??Discontinuous spectral element method modelling of optical coupling by whispering gallery modes between microcylinders,�?? J. Opt. Soc. Am. A (to be published).

J. Opt. Soc. Am. B

Opt. and Quantum Electron.

A. Melloni, F. Morichetti, and M. Martinelli, �??Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,�?? Opt. and Quantum Electron. 35, 365-379 (2003).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. B

B. M. Möller, U.Woggon, M. V. Artemyev, and R.Wannemacher, �??Photonic molecules doped with semiconductor nanocrystals,�?? Phys. Rev. B 70, 115323 (2004).
[CrossRef]

M. Bayindir and E. Ozbay, �??Heavy photons at coupled-cavity waveguide band edges in a three-dimensional photonic crystal,�?? Phys. Rev. B 62, R2247-R2250 (2000).
[CrossRef]

N. Stefanou and A. Modinos, �??Impurity bands in photonic insulators,�?? Phys. Rev. B 57, 12127-12133 (1998).
[CrossRef]

A. D. Bristow, D. M. Whittaker, V. N. Astratov, M. S. Skolnick, A. Tahraoui, T. F. Krauss, M. Hopkinson, M. P. Groucher, and G. A. Gehring, �??Defect states and commensurability in dual-period AlxGa1-xAs photonic crystal waveguides, �?? Phys. Rev. B 68, 033303 (2003).
[CrossRef]

Phys. Rev. E

J. E. Heebner, R. W. Boyd, and Q-H. Park, �??Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonant-array waveguide,�?? Phys. Rev. E 65, 036619 (2002).
[CrossRef]

Phys. Rev. Lett.

M. Bayer, T. Gutbrod, A. Forchel, T. L. Reinecke, P. A. Knipp, R. Werner, and J. P. Reithmaier, �??Optical demonstration of a crystal band structure formation,�?? Phys. Rev. Lett. 83, 5374-5377 (1999).
[CrossRef]

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, �??Tight-binding photonic molecule modes of resonant bispheres,�?? Phys. Rev. Lett. 82, 4623-4626 (1999).
[CrossRef]

Radio Sci.

J. R. Wait, �??Electromagnetic whispering gallery modes in a dielectric rod,�?? Radio Sci. 2, 1005-1017 (1967).

Other

D. Gottlieb and S. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (SIAM-CBMS, Philadelphia, 1977).
[CrossRef]

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zhang, Spectral Methods in Fluid Dynamics (Springer-Verlag, New York, 1987).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, Mass., 2000).

B. Szabo and I. Babuska, Finite Element Analysis (John Wiley & Sons, New York, 1991).

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Figures (7)

Fig. 1.
Fig. 1.

DSEM simulation of light propagation through 6 coupled microcylinder resonators. The field plotted here is the Ez component. The radius of the cylinders is r=4.023µm, the material index is n=3.2, and the inter-resonator gap size is w=4%r. The initial state of the system is represented by a WGM of azimuthal number 16 in the most left cylinder, and the frequency of the WGM is assumed to be 100THz. The five sequential snapshots demonstrate the successful wave propagation all the way through the chain to the most right cylinder. (a) t=3520fs; (b) t=5500fs; (c) t=7480fs; (d) t=8800fs; and (e) t=12100fs.

Fig. 2.
Fig. 2.

Temporal history of energy distribution in individual microcylinders during the light propagation through the CROW described in Fig. 1. The energies are normalized to the electromagnetic energy of the initial WGM in the most left cylinder.

Fig. 3.
Fig. 3.

Temporal history of energy distribution in individual microcylinders during the light propagation by a WGM through a 10 cylinder long CROW. The radius of the cylinders is r=2.283µm, the material index is n=3.2, and the inter-resonator gap size is w=4%r. The initial WGM in the most left cylinder has azimuthal number 10 and time frequency 100THz. The energies are normalized to the electromagnetic energy of the initial WGM in the most left cylinder.

Fig. 4.
Fig. 4.

Temporal history of energy distribution in individual microcylinders during the light propagation through the CROW described in Fig. 3. However, light will be coupled out of the CROW via an output waveguide attached to the end of the CROW. The energies are normalized to the electromagnetic energy of the initial WGM in the most left cylinder.

Fig. 5.
Fig. 5.

The cross-center sections of the field component Ez of the two WGMs described in Figs. 1 and 3, respectively. The fields are normalized to the maximum field amplitude in the system. For the WGM described in Fig. 1, 98.5% of the electromagnetic energy is confined inside the cylinder, while for the WGM described in Fig. 3, only 96.2% of the energy is confined inside the cylinder.

Fig. 6.
Fig. 6.

The speed of light propagating through 10 cylinder long CROWs. The radius and the index of refraction of the cylinders are r=1.7325µm and n=3.2, respectively. The initial WGM’s azimuthal number is 8. And the inter-cylinder gap size varies from 15%r to 40%r.

Fig. 7.
Fig. 7.

The speed of light propagating through 10 cylinder long CROWs. The initial WGM’s azimuthal number and the inter-cylinder gap size are fixed at 8 and w=16%r, respectively. However, the material index of refraction varies from n=3.6 to n=1.6, and accordingly the radius of the cylinder varies from r=1.53µm to r=4.05µm.

Equations (23)

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H r = [ a n n k 2 μ ω λ 2 r G n ( λ r ) + b n ih λ G n ( λ r ) ] F n ,
H θ = [ a n i k 2 μ ω λ G n ( λ r ) b n nh λ 2 r G n ( λ r ) ] F n ,
H z = b n G n ( λ r ) F n ,
E r = [ a n ih λ G n ( λ r ) b n μ ω n λ 2 r G n ( λ r ) ] F n ,
E θ = [ a n n h λ 2 r G n ( λ r ) + b n i μ ω λ G n ( λ r ) ] F n ,
E z = a n G n ( λ r ) F n ,
[ μ 1 u J n ( u ) J n ( u ) μ 2 v H n ( 1 ) ( v ) H n ( 1 ) ( v ) ] [ k 1 2 μ 1 u J n ( u ) J n ( u ) k 2 2 μ 2 v H n ( 1 ) ( v ) H n ( 1 ) ( v ) ] = n 2 h 2 ( 1 v 2 1 u 2 ) 2 ,
H ( x , y , z , t ) = H ( x , y , t ) exp ( ihz ) , E ( x , y , z , t ) = E ( x , y , t ) exp ( ihz ) .
μ H t = × E , ε E t = × H ,
Q t + A ( ε , μ ) Q x + B ( ε , μ ) Q y = S ,
Q = [ μ H ε E ] ,
A ( ε , μ ) = [ 0 0 0 0 0 0 0 0 0 0 0 1 ε 0 0 0 0 1 ε 0 0 0 0 0 0 0 0 0 1 μ 0 0 0 0 1 μ 0 0 0 0 ] , B ( ε , μ ) = [ 0 0 0 0 0 1 ε 0 0 0 0 0 0 0 0 0 1 ε 0 0 0 0 1 μ 0 0 0 0 0 0 0 0 0 1 μ 0 0 0 0 0 ] ,
S = [ ih E y ih E x 0 ih H y ih H x 0 ] .
Q t + · F = S ,
Q ̂ t + ξ · F ̂ = S ̂ .
Q ̂ = J Q , S ̂ = J S , F ̂ 1 = ( y η , x η ) · F , F ̂ 2 = ( y ξ , x ξ ) · F ,
ψ mn ( ξ , η ) = ϕ m ( ξ ) ϕ n ( η ) , 0 m , n M ,
ϕ i ( ξ ) = j = 0 , j i M ( ξ τ j ) ( τ i τ j )
Q ̂ ( ξ , η , t ) Q ̂ M ( ξ , η , t ) = m , n = 0 M Q ̂ mn ( t ) ψ mn ( ξ , η ) ,
( Q ̂ M t , ψ ij ) + Ω ψ ij F ̂ · n d s ( F ̂ · ψ ij ) = ( S ̂ , ψ ij ) , i , j = 0 , 1 , 2 , , M ,
F · n = [ n × ( Y E n × H ) + ( Y E + n × H ) + Y + Y + n × ( Z H + n × E ) + ( Z H n × E ) + Z + Z + ] .
Q t + A ( ε , μ ) Q x + B ( ε , μ ) Q y = S ( σ x + σ y ) Q ,
P = V ( ε E 2 2 + μ H 2 2 ) d v .

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