Abstract

The existence of localized or defect modes in a periodic array of symmetric and lossless micro-ring resonators is demonstrated for both finite and infinite structures using the transfer matrix method, for two types of array: one consisting of a cascade of coupled resonators coupled to an input and an output waveguide, and another consisting of uncoupled resonators periodically coupled between two bus waveguides. The defect can be introduced either by removing one ring, or by making one ring bigger or smaller. The 1-D periodic dielectric waveguide structures consisting of micro-ring resonators can exhibit photonic bandgaps, and when point defects are introduced defect states can form within the bandgaps, giving rise to donor and acceptor modes similar to other photonic crystals. The results based on the transfer matrix model agree with the finite-difference time-domain method, and are compared with those of a quarter-wave mirror stack.

© 2005 Optical Society of America

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References

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IEEE J. Quantum Electron.

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IEEE J. Quantum Electronics.

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IEEE Photonics Technol. Lett.

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P-T. Ho, �??Higher order filter response in coupled microring resonators,�?? IEEE Photonics Technol. Lett. 12, 320-322 (2000).
[CrossRef]

B.E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V.Van, O.King, and M. Trakalo, �??Very high-order microring resonator filters for WDM applications,�?? IEEE Photonics Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

IEEE Photonics. Technol. Lett.

R. Orta, P. Savi, R. Tascone, and D. rinchero, �??Synthesis of multiple-ring-resonator filters for optical systems,�?? IEEE Photonics. Technol. Lett. 7, 1447-1449 (1995).
[CrossRef]

G. Griffel, �??Synthesis of optical filters using ring resonator arrays,�?? IEEE Photonics. Technol. Lett. 12, 810-812 (2000).
[CrossRef]

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, �??Microring resonator arrays for VLSI photonics,�?? IEEE Photonics. Technol. Lett. 12, 323-325 (2000).
[CrossRef]

S. T. Chu, W.Pan, S. Sato, T. Kaneko, Y. Kokubun, and B. E. Little, �??An eight-channel add/drop filter using vertically coupled microring resonators over a cross grid,�?? IEEE Photonics. Technol. Lett. 11, 691- 693 (1999).
[CrossRef]

J. Lightwave Technol.

R. Grover, V. Van, T.A. Ibrahim, P.P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, �??Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters,�?? J. Lightwave Technol. 20, 900-905 (2002).
[CrossRef]

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

J. Opt. Soc. Am. A.

A. A. Tovar and L. W. Casperson, �??Generalized Sylvester theorems for periodic applications in matrix optics,�?? J. Opt. Soc. Am. A. 12, 578-590 (1995).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Phys. Rev. B

K. Sakoda and H. Shiroma, �??Numerical method for localized defect modes in photonic lattices,�?? Phys. Rev. B 56, 4830 (1997).
[CrossRef]

G. Gutroff., M. Bayer., J. P. Reithmaier., A. Forchel., P. A. Knipp., T. L. Reinecke., �??Photonic Defect States in chains of coupled microresonators,�?? Phys. Rev. B 64, 155313 (2001).
[CrossRef]

Phys. Rev. Lett.

E. Yablonovitch, �??Inhibited spontaneous emission in solid-state physics and electronics,�?? Phys. Rev. Lett. 58, 2059�??2062 (1987).
[CrossRef] [PubMed]

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, �??Microwave propagation in twodimensional dielectric lattices,�?? Phys. Rev. Lett. 67, 2017 (1991).
[CrossRef] [PubMed]

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Figures (12)

Fig. 1.
Fig. 1.

(a) Type I and (b) Type II linear arrays. Type II is equivalent to a dielectric mirror stack.

Fig. 2.
Fig. 2.

A single (point) defect embedded in a (a) Type I, and (b) Type II microring arrays.

Fig. 3.
Fig. 3.

The Fabry-Perot cavity equivalent of the linear array with a defect.

Fig. 4.
Fig. 4.

The unit cell for Type I periodic array. a = input field, R = Reflection, T = Transmission.

Fig. 5.
Fig. 5.

Left: The dispersion diagram for an infinite Type I structure, showing the photonic bandgap regions. Right: The transmission spectrum for a finite structure with N = 20.

Fig. 6.
Fig. 6.

The reflection amplitude and phase responses for the side-coupled linear cascade for N = 1, 5, and 10. The π phase jumps outside the reflection band occur when R = 0 and correspond to the sidelobes in the amplitude response.

Fig. 7.
Fig. 7.

The frequencies of the defect modes vs. the defect length for (a) an infinite Type I periodic structure, and (b) a finite structure with N = 10 and r2 = 0.9. (c) shows the cross sectional view for L defect = 0.75 (i.e., L c = 3/4 of the regular ring circumference).

Fig. 8.
Fig. 8.

Absolute amplitude of the field distribution in and around the defect. The amplitude is flat within the defect L defect.

Fig. 9.
Fig. 9.

Left: Dispersion diagram showing the real (solid lines) and imaginary parts (dashed lines) of the Bloch eigenvalues. Right: The transmission spectra (amplitude squared and phase) for a waveguide-coupled CROW with 5 rings. The bandgaps are centered at δ/2π = m+1/2.

Fig. 10.
Fig. 10.

The frequencies of the defect modes vs. the defect radius (normalized by the normal radius) for (a) an infinite Type II periodic structure, and (b) a finite structure with M = 5 and r2 = 0.3. (c) shows the transmission amplitude squared and phase (ϕT ) response for the case of RD = 2/3R. (i.e., L defect =2/3)

Fig. 11.
Fig. 11.

FDTD simulations of (a) the power distribution near the defect ring, and (b) the transmission function (green) through the Type II structure, showing the defect modes near 173 and 198 THz (circled). The splitting in the defect mode near 173 THz is not real. The transfer matrix results (with r=0.3) are also shown (blue).

Fig. 12.
Fig. 12.

Contour map of the “defect states” with wavelength λ, and the continuous bands for a periodic quarter-wave stack with a central defect layer. L defect is the total thickness of the defect unit cell, and Λ is the period of the regular unit cell. λ o is the Bragg wavelength; λ o /λ is equivalent to the normalized frequency f/f o.

Equations (26)

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T = T 1 T 2 exp ( j δ D ) m = 0 m = { R 1 R 2 exp ( j 2 δ D ) } m
= T 1 T 2 exp ( j δ D ) 1 R 1 R 2 exp ( j 2 δ D )
ϕ R = δ D
[ c a ] n + 1 = [ m 11 m 12 m 21 m 22 ] [ c a ] n
[ c a ] n + 1 = exp ( jk Λ ) [ c a ] n
[ m 11 e jk Λ m 12 m 21 m 22 e jk Λ ] [ c a ] n = 0
cos ( k Λ ) = 1 2 cos ( m 11 + m 22 )
[ c a ] = [ m 12 m 11 e jk Λ ]
R = R exp ( j ϕ R ) = c a = m 12 m 11 exp ( jk Λ )
R = b a = r ( 1 τ e ) 1 r 2 τ e
T = c a = t 2 τ 1 / 2 e / 2 1 r 2 τ e
δ = 2 π f FSR = 2 π ( m + Δ f FSR )
[ d b ] = 1 R [ 1 T T ( R 2 T 2 ) ] [ c a ]
[ c n + 1 a n + 1 ] = 1 R [ e L b T e L b T e L b ( R 2 T 2 ) e L b ] [ c n a n ] [ m 11 m 12 m 21 m 22 ] [ c n a n ]
cos k Λ = 1 R cos ( φ R )
[ c N + 1 a N + 1 ] = [ m 11 m 12 m 21 m 22 ] N [ c 1 a 1 ] [ M 11 M 12 M 21 M 22 ] [ c 1 a 1 ]
[ M 11 M 12 M 21 M 22 ] = 1 sin θ [ m 11 sin ( ) sin [ ( N 1 ) θ ] m 12 sin ( ) m 21 sin ( ) m 22 sin ( ) sin [ ( N 1 ) θ ] ]
R I = c 1 / a 1 exp ( j ϕ R ) = M 12 / M 11
T I = a N + 1 / a 1 exp ( j ϕ T ) = 1 / M 11
[ a n + 1 b n + 1 ] = 1 jt [ e / 2 r e / 2 r e / 2 e / 2 ] [ a n b n ] [ p 11 p 12 p 21 p 22 ] [ a n b n ]
cos ( k Λ ) = ( 1 / t ) sin ( δ / 2 )
B = ( 2 / π ) FSR sin 1 ( t )
[ a M + 1 b M + 1 ] = 1 ( jt ) M [ e / 2 r e / 2 r e / 2 e / 2 ] M [ a 1 b 1 ] [ P 11 P 12 P 21 P 22 ] [ a 1 b 1 ]
R c = b 1 a 1 exp ( j ϕ R ) = r P 11 + P 21 r P 12 + P 22
T C = c M + 1 a 1 exp ( j ϕ T ) = jt r P 12 + P 22
| R | = | m 12 1 / 2 ( m 22 m 11 ) i m a g i n a r y sinh ( γ Λ ) r e a l | | R 2 | = | m 12 | 2 ( 1 / 4 ) { i ( m 22 m 11 ) } 2 + sinh 2 ( γ Λ ) = | m 12 | 2 ( 1 / 4 ) { i ( m 22 m 11 ) } 2 + cosh 2 ( γ Λ ) 1 = | m 12 | 2 ( 1 / 4 ) { i ( m 22 m 11 ) } 2 + ( 1 / 4 ) ( m 11 + m 22 ) 2 1 = | m 12 | 2 ( 1 / 4 ) ( m 22 m 11 ) 2 + ( 1 / 4 ) ( m 11 + m 22 ) 2 1 = | m 12 | 2 ( 1 / 4 ) ( 2 m 22 ) ( 2 m 11 ) 1 = | m 12 | 2 m 22 m 11 ( m 11 m 22 m 12 m 21 ) = | m 12 | 2 m 12 m 21 = | m 12 | 2 | m 12 | 2 = 1

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