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Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation

M. Sumetsky

Open Access Open Access

Abstract

The coil optical resonator (COR) is an optical microfiber coil tightly wound on an optical rod. The resonant behavior of this all-pass device is determined by evanescent coupling between the turns of the microfiber. This paper investigates the uniform COR with N turns. Its transmission characteristics are surprisingly different from those of the known types of resonators and of photonic crystal structures. It is found that for certain discrete sequences of propagation constant and interturn coupling, the light is completely trapped by the resonator. For N →∞, the COR spectrum experiences a fractal collapse to the points corresponding to the second order zero of the group velocity. For a relatively small coupling between turns, the COR waveguide behavior resembles that of a SCISSOR (side-coupled integrated spaced sequence of resonators), while for larger coupling it resembles that of a CROW (coupled resonator optical waveguide).

©2005 Optical Society of America

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Figures (6)
Fig. 1.
Fig. 1. (a) Illustration of a COR; (b)illustration of a layered structure and a CROW; (c) illustration of a SCISSOR. Arrows show possible directions of light propagation.

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Fig. 2.
Fig. 2. Surface plots of the time delay in the plane (B,K) for the number of turns N equal to 2 through 7, 10, and 20. Brighter points correspond to larger time delay. For N=2,3,4,5, the points corresponding to the COR eigenmodes are marked by black dots. Upper row of plots show the time delay for 0<K<20, while the lower plots show it for 0<K<2.5 with higher resolution. In the lower plots, the blue and green circles mark similar features, V-shaped and W-shaped, respectively. For N→∞, all similar features tend to the collapse point. The ordinate of collapse points, K=0.5, is marked by a dotted red line.

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Fig. 3.
Fig. 3. Comparison of the time delay dependencies on the propagation constant for COR, CROW, and SCISSOR. (a) time delay spectrum for a COR with N=4 turns for K=0.5, 1.5, 2.5, and 3.5; (b) characteristic time delay spectrum of a CROW consisting of 4 rings; (c) characteristic time delay spectrum of a SCISSOR.

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Fig. 4.
Fig. 4. Surface plots of time delay for N=5,10,15,20,25 and 30. Each plot has different scale enhanced proportionally to N 2. The lower value of coupling parameter for all plots is K=1/2. Lines connect the upper left eigenvalue points of the V-shaped features at each plot.

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Fig. 5.
Fig. 5. Spatial distribution of the electromagnetic filed for the first COR eigenmodes corresponding to g 1=1 through 5 for N=2,3,4, and 10. For each N, the surface plot shows the corresponding time delay profile, similar to the ones in Fig. 2. The arrows link eigenvalues on the (B,K) plane and the corresponding eigenmodes. In case of N=10, for better visualization, the scale of the surface plot is enhanced in the neighborhood of K=1/2. The enhanced region is indicated by curved arrows. The numbers shown in each eigenmode plot is the corresponding value of the coupling parameter, K.

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Fig. 6.
Fig. 6. Characteristic dispersion relation for a COR, a CROW, and a SCISSOR. (a) dispersion relations for a COR having the coupling parameter K=0.25 (a1), K=0.5, (a2), and K=1.5 (a3); (b) dispersion relation of a CROW; (c) dispersion relation of a SCISSOR.

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Tables (1)

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Table 1. Smallest eigenvalue coupling parameters of the COR, K(1,1)(N) , as a function of number of turns, N

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Equations (11)

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d A 1 d s = κ A 2
d A m d s = κ ( A m − 1 + A m + 1 ) , m = 1,2 , … , N − 1
d A N d s = κ A N − 1
A m + 1 ( 0 ) = A m ( S ) exp ( i β S ) , m = 1,2 , … , N − 1 .
A m ( s ) = ∑ n = 1 N a ¯ n 1 A m n ( s ) , ∥ a ¯ m n ∥ = ∥ a m n ∥ − 1 ,
a m n = exp [ i S ( β + 2 κ cos ⁡ π n N + 1 ) ] sin ⁡ ( π ( m − 1 ) n N + 1 ) − sin ⁡ ( π m n N + 1 ) ,
A m n ( s ) = exp ( 2 i κ s cos ⁡ π n N + 1 ) sin ⁡ ( π m n N + 1 ) , m , n = 1,2 , … , M
T = ( A 1 ( 0 ) ) − 1 ∑ n = 1 N a ¯ n , 1 A N n ( S ) ,
t d = ( n eff ⁄ c ) Im [ d ln ( T ) ⁄ d β ] ,
t d ( B , K ) ≈ n eff S c 2 Im ( d g ( N ) ) ( K − K g ( N ) ) 2 [ B − B g ( N ) + c g ( N ) ( K − K g ( N ) ) ] 2 + [ Im ( d g ( N ) ) ] 2 [ K − K g ( N ) ] 2 .
ω ( ξ ) = c n eff ( ξ − 2 κ cos ⁡ ( ξ S ) )
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