Abstract

What we believe to be a new optical waveguide is introduced consisting of direct coupled diatomic microresonators, we call the diatomic coupled-resonator optical waveguide (CROW), where the inter-resonator coupling and/or the optical length of adjacent resonators alternate so that the unit cell comprises two resonators. We investigate this device analytically and numerically to find new transmission, group delay, dispersion, and switching characteristics, including a subsidiary stop band within the passband, a result of the alternating resonator parameters, whose width and extinction ratio are directly related to the parameter perturbation, displaying the signature characteristics associated with a finite Bragg grating. Analytical expressions are derived for the band-edge frequencies and the subsidiary stop band width, and numerical simulations illustrate the extent of versatility of the diatomic CROW design, including dispersion slope manipulation. The use of a simple matching structure terminating the diatomic CROW is found to significantly improve the device performance. The sensitivity of the diatomic CROW to the resonator loss is investigated and fabrication issues are also addressed.

© 2010 Optical Society of America

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2009 (2)

2008 (1)

2007 (1)

2006 (1)

2005 (6)

G. N. Nielson, D. Seneviratne, F. Lopez-Royo, P. T. Rakich, Y. Avrahami, M. R. Watts, H. A. Haus, H. L. Tuller, and G. Barbastathis, “Integrated wavelength-selective optical MEMS switching using ring resonator filters,” IEEE Photon. Technol. Lett. 17, 1190–1192 (2005).
[CrossRef]

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005).
[CrossRef] [PubMed]

G. Cusmai, F. Morichetti, P. Rosotti, R. Costa, and A. Melloni, “Circuit-oriented modeling of ring-resonators,” Opt. Quantum Electron. 37, 343–358 (2005).
[CrossRef]

Y. Landobasa and M. Chin, “Defect modes in micro-ring resonator arrays,” Opt. Express 13, 7800–7815 (2005).
[CrossRef] [PubMed]

M. Cherchi, “Bloch analysis of finite periodic microring chains,” Appl. Phys. B 80, 109–113 (2005).
[CrossRef]

S. J. Emelett and R. Soref, “Design and simulation of silicon microring optical routing switches,” J. Lightwave Technol. 23, 1800–1807 (2005).
[CrossRef]

2004 (1)

2003 (2)

1999 (1)

1997 (1)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

Avrahami, Y.

G. N. Nielson, D. Seneviratne, F. Lopez-Royo, P. T. Rakich, Y. Avrahami, M. R. Watts, H. A. Haus, H. L. Tuller, and G. Barbastathis, “Integrated wavelength-selective optical MEMS switching using ring resonator filters,” IEEE Photon. Technol. Lett. 17, 1190–1192 (2005).
[CrossRef]

Barbastathis, G.

G. N. Nielson, D. Seneviratne, F. Lopez-Royo, P. T. Rakich, Y. Avrahami, M. R. Watts, H. A. Haus, H. L. Tuller, and G. Barbastathis, “Integrated wavelength-selective optical MEMS switching using ring resonator filters,” IEEE Photon. Technol. Lett. 17, 1190–1192 (2005).
[CrossRef]

Capmany, J.

Chak, P.

Chang, H.

Cheben, P.

Cherchi, M.

M. Cherchi, “Bloch analysis of finite periodic microring chains,” Appl. Phys. B 80, 109–113 (2005).
[CrossRef]

Chin, M.

Costa, R.

G. Cusmai, F. Morichetti, P. Rosotti, R. Costa, and A. Melloni, “Circuit-oriented modeling of ring-resonators,” Opt. Quantum Electron. 37, 343–358 (2005).
[CrossRef]

Cusmai, G.

G. Cusmai, F. Morichetti, P. Rosotti, R. Costa, and A. Melloni, “Circuit-oriented modeling of ring-resonators,” Opt. Quantum Electron. 37, 343–358 (2005).
[CrossRef]

Delâge, A.

Densmore, A.

Domenech, J. D.

Doménech, J. D.

Eggleton, B. J.

Emelett, S. J.

Erdogan, T.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

Ferrari, C.

Fuller, K. A.

Haus, H. A.

G. N. Nielson, D. Seneviratne, F. Lopez-Royo, P. T. Rakich, Y. Avrahami, M. R. Watts, H. A. Haus, H. L. Tuller, and G. Barbastathis, “Integrated wavelength-selective optical MEMS switching using ring resonator filters,” IEEE Photon. Technol. Lett. 17, 1190–1192 (2005).
[CrossRef]

Huang, Y.

Janz, S.

Landobasa, Y.

Lapointe, J.

Lee, R. K.

Lipson, M.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005).
[CrossRef] [PubMed]

Lopez-Royo, F.

G. N. Nielson, D. Seneviratne, F. Lopez-Royo, P. T. Rakich, Y. Avrahami, M. R. Watts, H. A. Haus, H. L. Tuller, and G. Barbastathis, “Integrated wavelength-selective optical MEMS switching using ring resonator filters,” IEEE Photon. Technol. Lett. 17, 1190–1192 (2005).
[CrossRef]

Ma, R.

Martinelli, M.

Melloni, A.

A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. 33, 2389–2391 (2008).
[CrossRef] [PubMed]

G. Cusmai, F. Morichetti, P. Rosotti, R. Costa, and A. Melloni, “Circuit-oriented modeling of ring-resonators,” Opt. Quantum Electron. 37, 343–358 (2005).
[CrossRef]

Mookherjea, S.

Morichetti, F.

A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. 33, 2389–2391 (2008).
[CrossRef] [PubMed]

G. Cusmai, F. Morichetti, P. Rosotti, R. Costa, and A. Melloni, “Circuit-oriented modeling of ring-resonators,” Opt. Quantum Electron. 37, 343–358 (2005).
[CrossRef]

Muñoz, P.

Muriel, M. A.

Nielson, G. N.

G. N. Nielson, D. Seneviratne, F. Lopez-Royo, P. T. Rakich, Y. Avrahami, M. R. Watts, H. A. Haus, H. L. Tuller, and G. Barbastathis, “Integrated wavelength-selective optical MEMS switching using ring resonator filters,” IEEE Photon. Technol. Lett. 17, 1190–1192 (2005).
[CrossRef]

Paloczi, G.

Poon, J.

Pradhan, S.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005).
[CrossRef] [PubMed]

Rakich, P. T.

G. N. Nielson, D. Seneviratne, F. Lopez-Royo, P. T. Rakich, Y. Avrahami, M. R. Watts, H. A. Haus, H. L. Tuller, and G. Barbastathis, “Integrated wavelength-selective optical MEMS switching using ring resonator filters,” IEEE Photon. Technol. Lett. 17, 1190–1192 (2005).
[CrossRef]

Rosotti, P.

G. Cusmai, F. Morichetti, P. Rosotti, R. Costa, and A. Melloni, “Circuit-oriented modeling of ring-resonators,” Opt. Quantum Electron. 37, 343–358 (2005).
[CrossRef]

Scherer, A.

Scheuer, J.

Schmid, J. H.

Schmidt, B.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005).
[CrossRef] [PubMed]

Seneviratne, D.

G. N. Nielson, D. Seneviratne, F. Lopez-Royo, P. T. Rakich, Y. Avrahami, M. R. Watts, H. A. Haus, H. L. Tuller, and G. Barbastathis, “Integrated wavelength-selective optical MEMS switching using ring resonator filters,” IEEE Photon. Technol. Lett. 17, 1190–1192 (2005).
[CrossRef]

Sipe, J. E.

Smith, D. D.

Soref, R.

Sumetsky, M.

Tuller, H. L.

G. N. Nielson, D. Seneviratne, F. Lopez-Royo, P. T. Rakich, Y. Avrahami, M. R. Watts, H. A. Haus, H. L. Tuller, and G. Barbastathis, “Integrated wavelength-selective optical MEMS switching using ring resonator filters,” IEEE Photon. Technol. Lett. 17, 1190–1192 (2005).
[CrossRef]

Vachon, M.

Watts, M. R.

G. N. Nielson, D. Seneviratne, F. Lopez-Royo, P. T. Rakich, Y. Avrahami, M. R. Watts, H. A. Haus, H. L. Tuller, and G. Barbastathis, “Integrated wavelength-selective optical MEMS switching using ring resonator filters,” IEEE Photon. Technol. Lett. 17, 1190–1192 (2005).
[CrossRef]

Xu, D. -X.

Xu, Q.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005).
[CrossRef] [PubMed]

Xu, Y.

Yariv, A.

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, 1988), Sec. 6.

Appl. Phys. B (1)

M. Cherchi, “Bloch analysis of finite periodic microring chains,” Appl. Phys. B 80, 109–113 (2005).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

G. N. Nielson, D. Seneviratne, F. Lopez-Royo, P. T. Rakich, Y. Avrahami, M. R. Watts, H. A. Haus, H. L. Tuller, and G. Barbastathis, “Integrated wavelength-selective optical MEMS switching using ring resonator filters,” IEEE Photon. Technol. Lett. 17, 1190–1192 (2005).
[CrossRef]

J. Lightwave Technol. (2)

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

Nature (1)

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005).
[CrossRef] [PubMed]

Opt. Express (6)

Opt. Lett. (3)

Opt. Quantum Electron. (1)

G. Cusmai, F. Morichetti, P. Rosotti, R. Costa, and A. Melloni, “Circuit-oriented modeling of ring-resonators,” Opt. Quantum Electron. 37, 343–358 (2005).
[CrossRef]

Other (1)

P. Yeh, Optical Waves in Layered Media (Wiley, 1988), Sec. 6.

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Figures (12)

Fig. 1
Fig. 1

Schematic of the diatomic CROW consisting of ring resonators with alternating power coupling coefficients K i , perimeter lengths L i , and/or propagation constants β i ( i = 1 , 2 ) caused by index difference. The dashed rectangle indicates the diatomic unit cell. The waves entering ( a , c ) and exiting ( b , d ) the unit cell are also shown. When the number of rings is odd (even) the last ring is L 1 ( L 2 ) , the last coupler is K 2 ( K 1 ) , and the output is at port 3 (port 4).

Fig. 2
Fig. 2

Drop port intensity as a function of normalized detuning and K 2 for a lossless diatomic CROW with N u = 6 (12 resonators) and a fixed K 1 = 0.2 .

Fig. 3
Fig. 3

Drop port intensity as a function of normalized detuning and K 2 for a lossless diatomic CROW with N u = 6.5 (13 resonators) and a fixed K 1 = 0.2 .

Fig. 4
Fig. 4

Drop port intensity as a function of normalized detuning for a lossless diatomic CROW of 13 resonators versus the mode number perturbation N r δ ( K 1 = K 2 = 0.2 ) indicating a shift in the distribution of the ripples but retaining the location of the stop band center.

Fig. 5
Fig. 5

Drop port intensity as a function of normalized detuning for a lossless diatomic CROW of 12 resonators versus the average mode number perturbation N r ( δ 1 + δ 2 ) / 2 holding the differential perturbation fixed at N r ( δ 1 δ 2 ) = 0.1 ( K 1 = K 2 = 0.2 ) . Note the gradual shift of the stop band center in contrast to the result of Fig. 4.

Fig. 6
Fig. 6

(a) Drop port intensity of a ten unit cell (20-resonator) lossy diatomic CROW where K 1 = K 2 and N r 1 = N r ( 1 + δ ) , N r 2 = N r ( 1 δ ) with N r = 40 and δ = 5 × 10 3 . The parameter is the common coupling coefficient which increases from 0.5 to 0.9 in steps of 0.1. Increasing coupling strength decreases the extinction and the width of the dip. (b) Relative group delay at the drop port of the same diatomic CROW as in (a), but here the parameter is δ, increasing from 1 × 10 3 to 5 × 10 3 in steps of 1 × 10 3 , while the common coupling strength is fixed at 0.9. Larger δ corresponds to a deeper minimum. When δ is kept constant but K 1 = K 2 decreases, the width of this subsidiary stop band increases, its minimum descends, and the peaks marking its edges rise sharply.

Fig. 7
Fig. 7

(a) Surface plot of lossless four-resonator diatomic CROW where K 1 = 0.54 . Critical coupling is at K 2 = 0.28 . (b) Normalized group delay of the same CROW for four K 2 values ranging from 0.22 to 0.28 in steps of 0.02. Critical coupling is at K 2 = 0.22 .

Fig. 8
Fig. 8

(a) Group delay dispersion of a four-resonator diatomic CROW as a function of detuning; K 1 is fixed at 0.54. These plots apply to both a lossless as well as to a lossy device with a waveguide attenuation of α = 8   dB / cm ( σ = 0.99873 ) . (b) Group delay dispersion of a diatomic CROW with K 1 = K 2 = 0.3 , an effective perimeter perturbed according to N r 1 = N r ( 1 + δ ) , N r 2 = N r ( 1 δ ) , δ = 1 × 10 3 , and two-ring AR structures at both ends of an eight-resonator chain using matching coupling coefficients K a = 0.92 , K b = 0.72 , and K c = 0.45 . The slope of the red dotted line is 0.9 × 10 15 .

Fig. 9
Fig. 9

Switching in diatomic CROW with ten identical resonators and alternating coupling coefficients, where K 1 = 0.14 in both switch settings. (a) Solid lines: I 21 ; dotted lines: I 41 . (b) Normalized group velocities. Loss of up to σ = 0.99873 ( α = 8   dB / cm ) has minimal effect on the characteristics.

Fig. 10
Fig. 10

Switching in lossy ( σ = 0.99873 , α = 8   dB / cm ) diatomic CROW with identical coupling coefficients ( K 1 = K 2 = 0.8 ) , consisting of 20 resonators ( N r = 40 ) . (a),(b) The sweep covers the passband situated between approximately Δ f / F S R = ± 0.37 plus the adjacent stop bands. (c),(d) A subsidiary stop band develops in the middle of the passband when the effective indices are perturbed by δ = 4 × 10 3 to cause N r 1 = N r ( 1 + δ ) and N r 2 = N r ( 1 δ ) . The solid lines represent I 21 .

Fig. 11
Fig. 11

(a),(b) Drop port intensity and normalized group delay of a uniform CROW ( K = 0.4 , light dashed-dotted line) of a uniform CROW with two-ring AR structures (solid line), and a diatomic CROW with two-ring AR structures ( K 1 = 0.36 , K 2 = 0.42 , dotted line). The CROW consists of 70 lossless resonators ( N u = 35 ) and the coupling coefficients of the AR structure are K a = 0.92 , K b = 0.62 , and K c = 0.44 . When K 1 and K 2 are interchanged the sidelobe excursions increase, i.e., the AR device must be re-optimized. The baseline of the dashed-dotted plot is τ 41 / τ 0 = N u , and the midband ripple of the matched I 41 curve is 0.015 dB.

Fig. 12
Fig. 12

Lateral offset technique proposed in [15] to achieve coupling strength perturbation to fabricate the diatomic CROW.

Equations (19)

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T = 1 K 1 K 2 [ ( 1 K 1 ) ( 1 K 2 ) e j δ θ e j θ ¯ 1 K 2 e j δ θ 1 K 1 e j θ ¯ 1 K 2 e j δ θ 1 K 1 e j θ ¯ ( 1 K 1 ) ( 1 K 2 ) e j δ θ e j θ ¯ ] .
D = [ e j 2 k Λ 0 0 e j 2 k Λ ] ,     P = [ T 12 T 12 A + A ] ,
e j 2 k Λ = H ± H 2 1 ,
A ± = e j 2 k Λ T 11 ,
H = 1 2 Tr ( T ) = ( 1 K 1 ) ( 1 K 2 ) cos ( δ θ ) cos   θ ¯ K 1 K 2 ,
T K 2 = 1 j K 2 [ 1 1 K 2 1 K 2 1 ] ,     T L 1 = [ e j θ 1 / 2 0 0 e j θ 1 / 2 ] ,
[ a d 0 ] = M [ a i n a t ] ,     where   M = { T K 2 T L 1 T N u T K 1 , for   N u + 1 / 2 T N u T K 1 , for   N u , }
S 21 = M 21 M 22 = { ( A + 1 K 1 + T 21 ) e j k 2 N u Λ ( A 1 K 1 + T 21 ) e j k 2 N u Λ ( A + + 1 K 1 T 21 ) e j k 2 N u Λ ( A + 1 K 1 T 21 ) e j k 2 N u Λ , for   N u ( A + 1 K 1 e j θ 1 / 2 + A 1 K 2 e j θ 1 / 2 U ) e j k 2 N u Λ ( A + 1 K 2 e j θ 1 / 2 + A 1 K 1 e j θ 1 / 2 U ) e j k 2 N u Λ ( A + e j θ 1 / 2 + A ( 1 K 1 ) ( 1 K 2 ) e j θ 1 / 2 + V ) e j k 2 N u Λ ( A + ( 1 K 1 ) ( 1 K 2 ) e j θ 1 / 2 + A e j θ 1 / 2 + V ) e j k 2 N u Λ , for   N u + 1 / 2 , }
{ S 41 S 31 } = 1 M 22 = { ( A + A ) j K 1 ( A + 1 K 1 T 21 ) e j k 2 N u Λ ( A + + 1 K 1 T 21 ) e j k 2 N u Λ , for   N u ( A + A ) K 1 K 2 ( A + ( 1 K 1 ) ( 1 K 2 ) e j θ 1 / 2 + A e j θ 1 / 2 + V ) e j k 2 N u Λ ( A + e j θ 1 / 2 + A ( 1 K 1 ) ( 1 K 2 ) e j θ 1 / 2 + V ) e j k 2 N u Λ , for   N u + 1 / 2 , }
U = ( 1 K 1 ) ( 1 K 2 ) T 12 e j θ 1 / 2 T 21 e j θ 1 / 2 ,
V = 1 K 1 T 21 e j θ 1 / 2 1 K 2 T 12 e j θ 1 / 2 ,
τ j 1 = ϕ j 1 ω = Im [ 1 S j 1 S j 1 ω ] ,     j = 2 , 3 , 4
n g , CROW = c v g = c τ 41 N u L = c N u L Im ( 1 S 41 S 41 ω ) .
cos ( 2 k Λ ) = ( 1 K 1 ) ( 1 K 2 )   cos ( δ θ ) cos   θ ¯ K 1 K 2 .
1 2 π | sin 1 ( K 1 ) sin 1 ( K 2 ) | | Δ f | F S R 1 2 π [ sin 1 ( K 1 ) + sin 1 ( K 2 ) ] .
Δ f SB = F S R 1 π | sin 1 ( K 1 ) sin 1 ( K 2 ) | .
1 π sin 1 [ 1 K | sin ( π N r δ ) | ] | Δ f F S R | 1 π sin 1 [ K + ( 1 K ) sin 2 ( π N r δ ) ] .
Δ f SB = F S R 2 π sin 1 [ 1 K | sin ( π N r δ ) | ]
D i j = 1 N u L τ i j λ = 2 π c λ 2 N u L τ i j ω ,

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