Abstract

We describe postfabrication trimming of coupling in both laterally and vertically coupled polymer microring resonators (MRRs), using photobleaching. For both cases, a tapered directional-coupler-based simple analytical model is developed to simulate the change in coupling due to a bleaching-induced decrease in refractive index. A tightly focused laser beam spot (a few kilowatts per square centimeter) is used to precisely bleach the coupling region alone. Coupling control is achieved for (1) high-Q passive rings by bleaching the vertically coupled chromophore-doped bus waveguide, and for (2) laterally coupled electro-optic ring modulators, by bleaching both the ring and the waveguide in the coupling region. The power coupling ratio (PCR) of an undercoupled high-Q MRR filter is reduced by 0.54 percentage points for the TE mode, causing the MRR finesse to increase from a value of 72 to 108. For a ring modulator, the PCR was increased by 3.5 percentage points for the TM mode, causing a 6dB increase in extinction ratio, to achieve a final value of nearly 25dB. Phase/group-delay characterization confirmed that the ring was trimmed toward critical coupling.

© 2009 Optical Society of America

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2008 (1)

G. Gupta, W. H. Steier, Y. Liao, L. R. Dalton, J. D. Luo, and A. K.-Y. Jen, “Modeling photobleaching of optical chromophores: light-intensity effects in precise trimming of integrated polymer devices,” J. Phys. Chem. C 112, 8051-8060 (2008).
[CrossRef]

2006 (4)

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microring resonator,” Appl. Phys. Lett. 88, 111107 (2006).
[CrossRef]

H. Tazawa, Y. H. Kuo, I. Dunayevskiy, J. Luo, A. K. Y. Jen, H. R. Fetterman, and W. H. Steier, “Ring resonator-based electrooptic polymer traveling wave modulator,” J. Lightwave Technol. 24, 3514-3519 (2006).
[CrossRef]

C.-Y. Chao, W. Fung, and L. J. Guo, “Polymer microring resonators for biochemical sensing applications,” J. Sel. Top. Quantum Electron. 12, 134-142 (2006).
[CrossRef]

A. Stapleton, S. Farell, H. Akhavan, R. Shafiiha, Z. Peng, S. J. Choi, J. O'Brien, P. D. Dapkus, and W. Marshall, “Optical phase characterization of active semiconductor microdisk resonators in transmission,” Appl. Phys. Lett. 88, 031106(2006).
[CrossRef]

2005 (5)

P. Rabiei, “Calculation of losses in micro-ring resonators with arbitrary refractive index or shape profile and its applications,” J. Lightwave Technol. 23, 1295-1301 (2005).
[CrossRef]

Y.-C. Hung and H. R. Fetterman, “Polymer-based directional coupler modulator with high linearity,” IEEE Photon. Technol. Lett. 17, 2565-2567 (2005).
[CrossRef]

B. Bhola, H.-C. Song, H. Tazawa, and W. H. Steier, “Polymer microresonator strain sensors,” IEEE Photon. Technol. Lett. 17, 867-869 (2005).
[CrossRef]

M. C. M. Lee and M. C. Wu, “MEMS-actuated microdisk resonators with variable power coupling ratios,” IEEE Photon. Technol. Lett. 17, 1034-1036 (2005).
[CrossRef]

W. M. J. Green, R. K. Lee, G. A. DeRose, A. Scherer, and A. Yariv, “Hybrid InGaSP-InP Mach-Zehnder racetrack resonator for thermooptic switching and coupling control,” Opt. Express 13, 1651-1659 (2005).
[CrossRef] [PubMed]

2004 (4)

2003 (1)

Y. H. Kuo, W. H. Steier, S. Dubovitsky, and B. Jalali, “Demonstration of wavelength-insensitive biasing using an electrooptic polymer modulator,” IEEE Photon. Technol. Lett. 15, 813-815 (2003).
[CrossRef]

2002 (4)

A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. 14, 483-485 (2002).
[CrossRef]

B. Liu, A. Shakouri, and J. E. Bowers, “Wide tunable double ring resonator coupled lasers,” IEEE Photon. Technol. Lett. 14, 600-602 (2002).
[CrossRef]

P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, “Polymer micro-ring filters and modulators,” J. Lightwave Technol. 20, 1968-1975 (2002).
[CrossRef]

Z. V. Vardeny, “A boost for fibre optics,” Nature 416, 489(2002).
[CrossRef] [PubMed]

2001 (1)

2000 (1)

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321-322 (2000).
[CrossRef]

1999 (2)

1997 (1)

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]

1987 (1)

D. Marcuse, “Directional couplers made of nonidentical asymmetric slabs. Part I: synchronous couplers,” J. Lightwave Technol. 5, 113-118 (1987).
[CrossRef]

Akhavan, H.

A. Stapleton, S. Farell, H. Akhavan, R. Shafiiha, Z. Peng, S. J. Choi, J. O'Brien, P. D. Dapkus, and W. Marshall, “Optical phase characterization of active semiconductor microdisk resonators in transmission,” Appl. Phys. Lett. 88, 031106(2006).
[CrossRef]

Bhola, B.

B. Bhola, H.-C. Song, H. Tazawa, and W. H. Steier, “Polymer microresonator strain sensors,” IEEE Photon. Technol. Lett. 17, 867-869 (2005).
[CrossRef]

Borreman, A.

C. G. H. Roeloffzen, L. Zhuang, R. G. Heideman, A. Borreman, and W. van Etten, “Ring resonator-based tunable optical delay line in LPCVD waveguide technology,” in 2005 IEEE/LEOS Symposium Benelux Chapter Proceedings (IEEE, 2005), pp. 79-82.

Bowers, J. E.

B. Liu, A. Shakouri, and J. E. Bowers, “Wide tunable double ring resonator coupled lasers,” IEEE Photon. Technol. Lett. 14, 600-602 (2002).
[CrossRef]

Boyd, R. W.

Brener, I.

Bruce, A. J.

Campbell, K.

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microring resonator,” Appl. Phys. Lett. 88, 111107 (2006).
[CrossRef]

Capuzzo, M. A.

Chao, C.-Y.

C.-Y. Chao, W. Fung, and L. J. Guo, “Polymer microring resonators for biochemical sensing applications,” J. Sel. Top. Quantum Electron. 12, 134-142 (2006).
[CrossRef]

Choi, J. M.

Choi, S. J.

A. Stapleton, S. Farell, H. Akhavan, R. Shafiiha, Z. Peng, S. J. Choi, J. O'Brien, P. D. Dapkus, and W. Marshall, “Optical phase characterization of active semiconductor microdisk resonators in transmission,” Appl. Phys. Lett. 88, 031106(2006).
[CrossRef]

Chu, S. T.

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]

Dalton, L. R.

Dapkus, P. D.

A. Stapleton, S. Farell, H. Akhavan, R. Shafiiha, Z. Peng, S. J. Choi, J. O'Brien, P. D. Dapkus, and W. Marshall, “Optical phase characterization of active semiconductor microdisk resonators in transmission,” Appl. Phys. Lett. 88, 031106(2006).
[CrossRef]

DeRose, G. A.

Dubovitsky, S.

Y. H. Kuo, W. H. Steier, S. Dubovitsky, and B. Jalali, “Demonstration of wavelength-insensitive biasing using an electrooptic polymer modulator,” IEEE Photon. Technol. Lett. 15, 813-815 (2003).
[CrossRef]

Dunayevskiy, I.

Fainman, Y.

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microring resonator,” Appl. Phys. Lett. 88, 111107 (2006).
[CrossRef]

Farell, S.

A. Stapleton, S. Farell, H. Akhavan, R. Shafiiha, Z. Peng, S. J. Choi, J. O'Brien, P. D. Dapkus, and W. Marshall, “Optical phase characterization of active semiconductor microdisk resonators in transmission,” Appl. Phys. Lett. 88, 031106(2006).
[CrossRef]

Fetterman, H. R.

Foresi, J.

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]

Forest, S. R.

V. M. Menon, W. Tong, and S. R. Forest, “Control of quality factor and critical coupling in microring resonators through integration of a semiconductor optical amplifier,” IEEE Photon. Technol. Lett. 16, 1343-1345 (2004).
[CrossRef]

Fung, W.

C.-Y. Chao, W. Fung, and L. J. Guo, “Polymer microring resonators for biochemical sensing applications,” J. Sel. Top. Quantum Electron. 12, 134-142 (2006).
[CrossRef]

Gomez, L. T.

Green, W. M. J.

Groisman, A.

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microring resonator,” Appl. Phys. Lett. 88, 111107 (2006).
[CrossRef]

Guo, L. J.

C.-Y. Chao, W. Fung, and L. J. Guo, “Polymer microring resonators for biochemical sensing applications,” J. Sel. Top. Quantum Electron. 12, 134-142 (2006).
[CrossRef]

Gupta, G.

G. Gupta, W. H. Steier, Y. Liao, L. R. Dalton, J. D. Luo, and A. K.-Y. Jen, “Modeling photobleaching of optical chromophores: light-intensity effects in precise trimming of integrated polymer devices,” J. Phys. Chem. C 112, 8051-8060 (2008).
[CrossRef]

G. Gupta, “Microring resonator based filters and modulators: optical coupling control and applications to digital communications,” Ph.D. dissertation (University of Southern California, 2008).

Haus, H. A.

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]

Heebner, J. E.

Heideman, R. G.

C. G. H. Roeloffzen, L. Zhuang, R. G. Heideman, A. Borreman, and W. van Etten, “Ring resonator-based tunable optical delay line in LPCVD waveguide technology,” in 2005 IEEE/LEOS Symposium Benelux Chapter Proceedings (IEEE, 2005), pp. 79-82.

Huang, Y.

Hung, Y.-C.

Y.-C. Hung and H. R. Fetterman, “Polymer-based directional coupler modulator with high linearity,” IEEE Photon. Technol. Lett. 17, 2565-2567 (2005).
[CrossRef]

Y.-C. Hung, University of California, Los Angeles, Calif. (personal communication, 2006).

Jalali, B.

Y. H. Kuo, W. H. Steier, S. Dubovitsky, and B. Jalali, “Demonstration of wavelength-insensitive biasing using an electrooptic polymer modulator,” IEEE Photon. Technol. Lett. 15, 813-815 (2003).
[CrossRef]

Jen, A. K. Y.

Jen, A. K.-Y.

G. Gupta, W. H. Steier, Y. Liao, L. R. Dalton, J. D. Luo, and A. K.-Y. Jen, “Modeling photobleaching of optical chromophores: light-intensity effects in precise trimming of integrated polymer devices,” J. Phys. Chem. C 112, 8051-8060 (2008).
[CrossRef]

Kuo, Y. H.

H. Tazawa, Y. H. Kuo, I. Dunayevskiy, J. Luo, A. K. Y. Jen, H. R. Fetterman, and W. H. Steier, “Ring resonator-based electrooptic polymer traveling wave modulator,” J. Lightwave Technol. 24, 3514-3519 (2006).
[CrossRef]

Y. H. Kuo, W. H. Steier, S. Dubovitsky, and B. Jalali, “Demonstration of wavelength-insensitive biasing using an electrooptic polymer modulator,” IEEE Photon. Technol. Lett. 15, 813-815 (2003).
[CrossRef]

Laine, J.-P.

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]

Lee, M. C. M.

M. C. M. Lee and M. C. Wu, “MEMS-actuated microdisk resonators with variable power coupling ratios,” IEEE Photon. Technol. Lett. 17, 1034-1036 (2005).
[CrossRef]

Lee, R. K.

Lenz, G.

Levy, U.

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microring resonator,” Appl. Phys. Lett. 88, 111107 (2006).
[CrossRef]

Liao, Y.

G. Gupta, W. H. Steier, Y. Liao, L. R. Dalton, J. D. Luo, and A. K.-Y. Jen, “Modeling photobleaching of optical chromophores: light-intensity effects in precise trimming of integrated polymer devices,” J. Phys. Chem. C 112, 8051-8060 (2008).
[CrossRef]

Little, B. E.

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]

Liu, B.

B. Liu, A. Shakouri, and J. E. Bowers, “Wide tunable double ring resonator coupled lasers,” IEEE Photon. Technol. Lett. 14, 600-602 (2002).
[CrossRef]

Luo, J.

Luo, J. D.

G. Gupta, W. H. Steier, Y. Liao, L. R. Dalton, J. D. Luo, and A. K.-Y. Jen, “Modeling photobleaching of optical chromophores: light-intensity effects in precise trimming of integrated polymer devices,” J. Phys. Chem. C 112, 8051-8060 (2008).
[CrossRef]

Madsen, C. K.

Marcuse, D.

D. Marcuse, “Directional couplers made of nonidentical asymmetric slabs. Part I: synchronous couplers,” J. Lightwave Technol. 5, 113-118 (1987).
[CrossRef]

Marshall, W.

A. Stapleton, S. Farell, H. Akhavan, R. Shafiiha, Z. Peng, S. J. Choi, J. O'Brien, P. D. Dapkus, and W. Marshall, “Optical phase characterization of active semiconductor microdisk resonators in transmission,” Appl. Phys. Lett. 88, 031106(2006).
[CrossRef]

Marz, R.

R. Marz, Integrated Optics: Design and Modeling (Artech House, 1995).

Menon, V. M.

V. M. Menon, W. Tong, and S. R. Forest, “Control of quality factor and critical coupling in microring resonators through integration of a semiconductor optical amplifier,” IEEE Photon. Technol. Lett. 16, 1343-1345 (2004).
[CrossRef]

Mookherjea, S.

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microring resonator,” Appl. Phys. Lett. 88, 111107 (2006).
[CrossRef]

Nielsen, T. N.

O'Brien, J.

A. Stapleton, S. Farell, H. Akhavan, R. Shafiiha, Z. Peng, S. J. Choi, J. O'Brien, P. D. Dapkus, and W. Marshall, “Optical phase characterization of active semiconductor microdisk resonators in transmission,” Appl. Phys. Lett. 88, 031106(2006).
[CrossRef]

Paloczi, G. T.

Peng, Z.

A. Stapleton, S. Farell, H. Akhavan, R. Shafiiha, Z. Peng, S. J. Choi, J. O'Brien, P. D. Dapkus, and W. Marshall, “Optical phase characterization of active semiconductor microdisk resonators in transmission,” Appl. Phys. Lett. 88, 031106(2006).
[CrossRef]

Poon, J. K. S.

Rabiei, P.

Roeloffzen, C. G. H.

C. G. H. Roeloffzen, L. Zhuang, R. G. Heideman, A. Borreman, and W. van Etten, “Ring resonator-based tunable optical delay line in LPCVD waveguide technology,” in 2005 IEEE/LEOS Symposium Benelux Chapter Proceedings (IEEE, 2005), pp. 79-82.

Scherer, A.

Scheuer, J.

Schwelb, O.

Shafiiha, R.

A. Stapleton, S. Farell, H. Akhavan, R. Shafiiha, Z. Peng, S. J. Choi, J. O'Brien, P. D. Dapkus, and W. Marshall, “Optical phase characterization of active semiconductor microdisk resonators in transmission,” Appl. Phys. Lett. 88, 031106(2006).
[CrossRef]

Shakouri, A.

B. Liu, A. Shakouri, and J. E. Bowers, “Wide tunable double ring resonator coupled lasers,” IEEE Photon. Technol. Lett. 14, 600-602 (2002).
[CrossRef]

Song, H.-C.

B. Bhola, H.-C. Song, H. Tazawa, and W. H. Steier, “Polymer microresonator strain sensors,” IEEE Photon. Technol. Lett. 17, 867-869 (2005).
[CrossRef]

Stapleton, A.

A. Stapleton, S. Farell, H. Akhavan, R. Shafiiha, Z. Peng, S. J. Choi, J. O'Brien, P. D. Dapkus, and W. Marshall, “Optical phase characterization of active semiconductor microdisk resonators in transmission,” Appl. Phys. Lett. 88, 031106(2006).
[CrossRef]

Steier, W. H.

G. Gupta, W. H. Steier, Y. Liao, L. R. Dalton, J. D. Luo, and A. K.-Y. Jen, “Modeling photobleaching of optical chromophores: light-intensity effects in precise trimming of integrated polymer devices,” J. Phys. Chem. C 112, 8051-8060 (2008).
[CrossRef]

H. Tazawa, Y. H. Kuo, I. Dunayevskiy, J. Luo, A. K. Y. Jen, H. R. Fetterman, and W. H. Steier, “Ring resonator-based electrooptic polymer traveling wave modulator,” J. Lightwave Technol. 24, 3514-3519 (2006).
[CrossRef]

B. Bhola, H.-C. Song, H. Tazawa, and W. H. Steier, “Polymer microresonator strain sensors,” IEEE Photon. Technol. Lett. 17, 867-869 (2005).
[CrossRef]

Y. H. Kuo, W. H. Steier, S. Dubovitsky, and B. Jalali, “Demonstration of wavelength-insensitive biasing using an electrooptic polymer modulator,” IEEE Photon. Technol. Lett. 15, 813-815 (2003).
[CrossRef]

P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, “Polymer micro-ring filters and modulators,” J. Lightwave Technol. 20, 1968-1975 (2002).
[CrossRef]

Tazawa, H.

Tong, W.

V. M. Menon, W. Tong, and S. R. Forest, “Control of quality factor and critical coupling in microring resonators through integration of a semiconductor optical amplifier,” IEEE Photon. Technol. Lett. 16, 1343-1345 (2004).
[CrossRef]

van Etten, W.

C. G. H. Roeloffzen, L. Zhuang, R. G. Heideman, A. Borreman, and W. van Etten, “Ring resonator-based tunable optical delay line in LPCVD waveguide technology,” in 2005 IEEE/LEOS Symposium Benelux Chapter Proceedings (IEEE, 2005), pp. 79-82.

Vardeny, Z. V.

Z. V. Vardeny, “A boost for fibre optics,” Nature 416, 489(2002).
[CrossRef] [PubMed]

Wu, M. C.

M. C. M. Lee and M. C. Wu, “MEMS-actuated microdisk resonators with variable power coupling ratios,” IEEE Photon. Technol. Lett. 17, 1034-1036 (2005).
[CrossRef]

Xu, Y.

Yariv, A.

Zhang, C.

Zhao, J. H.

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

Zhuang, L.

C. G. H. Roeloffzen, L. Zhuang, R. G. Heideman, A. Borreman, and W. van Etten, “Ring resonator-based tunable optical delay line in LPCVD waveguide technology,” in 2005 IEEE/LEOS Symposium Benelux Chapter Proceedings (IEEE, 2005), pp. 79-82.

Appl. Phys. Lett. (2)

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microring resonator,” Appl. Phys. Lett. 88, 111107 (2006).
[CrossRef]

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Figures (23)

Fig. 1
Fig. 1

Single ring coupled to (a) single waveguide; (b) two waveguides; (c) serial cascade of rings coupled to single waveguide; (d) series of rings coupled to each other, with each end ring coupled to a bus waveguide.

Fig. 2
Fig. 2

(a) Chromophore-doped bus waveguide vertically coupled to passive MRR, bleached by a circular green laser beam spot. (b) Lateral coupling of ring and waveguide, both made of chromophore-doped polymer, bleached simultaneously by an elliptical red laser beam spot.

Fig. 3
Fig. 3

Single (racetrack) ring coupled to a bus waveguide, with relevant parameters.

Fig. 4
Fig. 4

(a) Transmittance and (b) phase characteristics of undercoupled, critically coupled, and overcoupled rings.

Fig. 5
Fig. 5

Schematic supporting the geometric derivation of the effective coupling length L eff , for the case of a circular ring vertically coupled to a bus waveguide.

Fig. 6
Fig. 6

(a) Cross section through the coupling region of a vertically coupled ring. (b) Schematic of equivalent slab directional coupler.

Fig. 7
Fig. 7

(a) Cross section through the coupling region of a laterally coupled ring. (b) Schematic of equivalent slab directional coupler.

Fig. 8
Fig. 8

Cross section through coupling region of fabricated vertically coupled ring.

Fig. 9
Fig. 9

(a) Effect of bleaching on Δ β and, hence, on PCR, for the case of a MRR vertically coupled to a bus waveguide of height h = 0.7 μm . (b) PCR versus n curves for different waveguide heights.

Fig. 10
Fig. 10

(a) Optical field overlap for the case of lateral coupling. (b) Cross section through coupling region of ring modulator.

Fig. 11
Fig. 11

(a) Effect of bleaching on K and, hence, on PCR, for the case of a laterally coupled inverted-rib ring and bus waveguides of trench depth t = 1.2 μm . (b) PCR versus n curves for different trench depths.

Fig. 12
Fig. 12

Fabrication flow diagram for a passive polymer ring vertically coupled to a chromophore-doped waveguide.

Fig. 13
Fig. 13

Experimental setup used for characterizing vertically coupled rings.

Fig. 14
Fig. 14

Phase characterization setup used for laterally coupled rings.

Fig. 15
Fig. 15

Measured (a) transmittance and (b) phase/group-delay characteristics of near critically coupled ring ( L = 150 μm ).

Fig. 16
Fig. 16

Measured phase and group-delay characteristics of overcoupled ring ( L = 200 μm ).

Fig. 17
Fig. 17

Laser photobleaching of vertically coupled YLD161b/APC waveguide.

Fig. 18
Fig. 18

Optical microscope image of bleached coupling region for SU-8 ring vertically coupled to YLD161b/APC waveguide.

Fig. 19
Fig. 19

Solid curve is the simulated bleaching curve for a vertically coupled YLD161b/APC waveguide of height 0.85 μm and misalignment factor of 0.4 μm . The dots correspond to the experimental values of PCR before and after bleaching. The extracted Δ n is indicated.

Fig. 20
Fig. 20

Laser photobleaching of laterally coupled electro-optic ring.

Fig. 21
Fig. 21

Measured (a) transmittance and (b) group-delay spectra of the laterally coupled ring, before and after bleaching, with respective ER and group-delay values highlighted.

Fig. 22
Fig. 22

Solid curve is the simulated bleaching curve for laterally coupled ring and bus waveguides of trench depth 1.2 μm . The dots correspond to the experimental values of PCR before and after bleaching. The extracted Δ n is indicated.

Fig. 23
Fig. 23

Bleached coupling region for laterally coupled AJL8/APC ring modulator.

Tables (3)

Tables Icon

Table 1 Measured and Derived Values of Relevant Quantities for Laterally Coupled Electro-Optic Racetrack Rings Obtained from Transmittance and Phase Measurements

Tables Icon

Table 2 Measured and Derived Parameters for Photobleach Trimming of a Vertically Coupled High-Q MRR

Tables Icon

Table 3 Measured and Derived Parameters for Photobleach Trimming of a Laterally Coupled MRR Modulator

Equations (19)

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T = a 2 + | t | 2 2 a t cos ( ϕ ) 1 2 a t cos ( ϕ ) + a 2 t 2 ,
E out E in = exp [ i · ( π + ϕ ) ] · a t · exp ( i ϕ ) 1 a t · exp ( i ϕ ) ,
Q U = 2 π n eff α λ 0 ,
a = exp ( α ( π r + L ) ) .
Q C = 2 π n eff ( 2 π r + 2 L ) κ 2 λ 0 .
1 Q L = 1 Q U + 1 Q C .
Q L = f 0 BW .
[ ( a t ) · ( 1 + a · t ) ( a + t ) · ( 1 + a · t ) ] 2 = T min ,
π 2 · sin 1 [ ( 1 a · t ) 2 + 2 · a 2 · t 2 ] = FSR BW = F ,
κ = K ( z ) e j Δ β z d z ,
κ 2 = K 2 K 2 + δ 2 · sin 2 ( K 2 + δ 2 L eff ) ,
L eff = 2 x = ( r + w R 2 ) 2 ( r w WG 2 + d ) 2 .
K = 2 η 2 η 4 γ 3 exp ( 2 g γ 3 ) k 2 β { ( n 2 2 n 3 2 ) ( n 4 2 n 3 2 ) ( d R + 1 / γ 1 + 1 / γ 3 ) ( d WG + 1 / γ 5 + 1 / γ 3 ) } 1 / 2 ,
β = β R + β WG 2 ;
η j = n j 2 k 2 β R 2 , j = 2 , 4 ; γ j = β WG 2 n j 2 k 2 , j = 1 , 3 , 5 ;
L eff = ( 2 π r s ) 1 / 2 ,
L eff = L + ( 2 π r s ) 1 / 2 ,
K = 2 η 2 2 γ 1 exp ( γ 1 g ) β ( w + 2 / γ 1 ) ( η 2 2 + γ 1 2 ) ,
τ g = ( ϕ ω ) .

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