Abstract

A ring resonator based on Bragg reflection is studied in detail. Closed form expressions for the field and dispersion curves for radial Bragg gratings and photonic crystals based resonators are derived and compared to FDTD simulations. For strong confinement, the required gratings exhibit a chirped period and a varying index profile. Small bending radii and low radiation losses are shown to be possible due to the Bragg confinement. The sensitivity of the resonator characteristics to fabrication errors is analyzed quantitatively. A mixed confinement configuration utilizing both Bragg reflection and total internal reflection is also suggested and analyzed.

© 2003 Optical Society of America

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References

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Appl. Opt. (1)

Appl. Phys. Lett. (1)

S. Kim, H. Ryu, H. Park, G. Kim, Y. Choi, Y. Lee and J. Kim, �??Two-dimensional photonic crystal hexagonal waveguide ring laser,�?? Appl. Phys. Lett. 81, 2499-2501 (2002).
[CrossRef]

IEEE J. Quantum Electron. (2)

M. Heiblum and J. H. Harris, �??Analysis of curved optical waveguides by conformal transformation,�?? IEEE J. Quantum Electron. QE-11, 75-83 (1975).
[CrossRef]

L. Djaloshinski and M. Orenstein, �??Disk and Ring Microcavity Lasers and Their Concentric Coupling,�?? IEEE J. Quantum Electron. 35, 737-744, 1999.
[CrossRef]

IEEE Photonics Technol. Lett. (1)

A. Yariv, �??Critical Coupling and Its Control in Optical Waveguide-Ring Resonator Systems,�?? IEEE Photonics Technol. Lett. 14, 483-485 (2002).
[CrossRef]

J. Opt. Soc. Am. B. (1)

J. Scheuer and A. Yariv, �??Annular Bragg-defect-mode Resonators,�?? J. Opt. Soc. Am. B. 20 2285-2291 (2003).
[CrossRef]

Opt. Commun. (1)

P. Yeh and A. Yariv, �??Bragg Reflection Waveguides,�?? Opt. Commun. 19, 427-430 (1976).
[CrossRef]

Opt. Lett. (3)

Other (2)

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley- Interscience Publications, New-York 1999).

K. Sakoda, Optical Properties of Photonic Crystals (Springer, New-York 1999).

Supplementary Material (2)

» Media 1: MPG (1124 KB)     
» Media 2: MPG (1756 KB)     

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Figures (10)

Fig. 1.
Fig. 1.

A Bragg reflection based ring resonator. (A) Reflection by annular Bragg gratings; (B) Realization using an inhomogeneous hole density photonic crystal

Fig. 2.
Fig. 2.

Refractive index profile in (I) the (U, V) plane and (II) in the (ρ, θ) plane. R=479µm.

Fig. 3.
Fig. 3.

Field propagation (absolute value) in I) (1.1MB) a line defect waveguide and in II) (1.5MB) the corresponding annular PC resonator.

Fig. 4.
Fig. 4.

Comparison between the transformed modal field profile of a line defect PC waveguide (green) and the modal field profile of the corresponding annular PC resonator (blue).

Fig. 5.
Fig. 5.

Resonance wavelengths (circles) and a quadratic fit (solid line) of the resonator shown in Fig. 3II.

Fig. 6.
Fig. 6.

Resonance wavelengths when random shifts in the holes positions are introduced: optimal structure (blue) and maximal error of 50nm (green) 100nm (blue) and 200nm (purple).

Fig. 7.
Fig. 7.

Resonance wavelengths when random errors in the holes radii are introduced: optimal structure (blue) and maximal error of 50nm (green).

Fig. 8.
Fig. 8.

Refractive index profile in (I) the (U, V) plane and (II) in the (ρ, θ) plane for a mixed confinement-methods structure. R=479µm.

Fig. 9.
Fig. 9.

FDTD simulation of an annular PC resonator employing mixed confinement-methods.

Fig. 10.
Fig. 10.

Modal field profile of the resonator shown in Fig. 9.

Equations (16)

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1 ρ ρ ( ρ E ρ ) + 1 ρ 2 2 E θ 2 + k 0 2 n 2 ( ρ ) E = 0
ρ = R · exp ( U R ) ; θ = V R
2 E U 2 + 2 E V 2 + k 0 2 n eq 2 ( U ) E = 0
U = R · ln ( ρ R ) , V = θ · R , n ( ρ ) = n eq ( ρ ) · R ρ
E ( U , V ) = E ¯ ( U ) · exp ( iβV )
E ¯ ( U ) = { c 1 cos ( q U ) + c 2 sin ( q U ) U L < U < U R E K ( U ) exp ( iKU ) U < U L , U > U R
E ¯ ( U ) = { E 0 cos ( π b ( U U cent ) ) U U cent W 2 E 0 cos ( π b ( U U cent ) ) exp [ κ 1 ( U U cent W 2 ) ] ( U U cent ) W 2 E 0 cos ( π b ( U U cent ) ) exp [ κ 1 ( U U cent + W 2 ) ] ( U U cent ) W 2
β = k 0 2 ε eq , 0 ( l π b ) 2 ; l = 1 , 2 , 3
βR = k 0 2 · ( n eq min ) 2 ( l · π b ) · R = m m = 1 , 2 , 3
λ m = 2 n defect ρ defect ( m π ) 2 ( l b ) 2
FSR = dm = c · ( 2 n eq min · ν ) 2 ( c · l b ) 2 ( 2 n eq min ) 2 R π · ν
FSR c 2 n Defect π ρ Defect
2 π R k 0 n eff = 2 π m , m = 1 , 2 , 3
E ¯ ( U < U 0 W ) = J m ( n L k 0 R · exp ( U R ) )
E ¯ ( U ) = { J m ( n L k 0 R · exp ( U R ) ) U U 0 W A 1 sin ( π b ( U U 0 ) ) U 0 W U U 0 A 1 sin ( π b ( U U 0 ) ) exp [ κ 1 ( U U 0 ) ] U U 0
exp ( ( U 0 W ) R ) · J m ( n L k 0 R · exp ( ( U 0 W ) R ) ) J m ( n L k 0 R · exp ( ( U 0 W ) R ) ) = λ 2 n L b cot ( π W b )

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