Abstract

We demonstrate, in both theory and experiment, 4-port, electrically tunable photonic filters using silicon contra-directional couplers (contra-DCs) with uniform and phase-shifted waveguide Bragg gratings. Numerical analysis, including both intra- and inter-waveguide couplings, is performed using coupled-mode theory and the transfer-matrix method. The contra-DC devices were fabricated by a CMOS-photonics manufacturing foundry and are electrically tunable using free-carrier injection. A 4-port, grating-based photonic resonator has been obtained using the phase-shifted contra-DC, showing a resonant peak with a 3-dB bandwidth of 0.2 nm and an extinction ratio of 24 dB. These contra-DC devices enable on-chip integration of Bragg-grating-defined functions without using circulators and have great potential for applications such as wavelength-division multiplexing networks and optical signal processing.

© 2013 OSA

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2012 (5)

L. Chrostowski, S. Grist, J. Flueckiger, W. Shi, X. Wang, E. Ouellet, H. Yun, M. Webb, B. Nie, Z. Liang, K. C. Cheung, A. S. S, D. M. Ratner, and N. A. F. Jaeger, “Silicon photonic resonator sensors and devices,” Proceedings of SPIE8236, 823620 (2012).
[CrossRef]

W. Shi, X. Wang, W. Zhang, H. Yun, C. Lin, L. Chrostowski, and N. A. F. Jaeger, “Grating-coupled silicon microring resonators,” Appl. Phys. Lett.100, 121118 (2012).
[CrossRef]

J. Yao, “A tutorial on microwave photonics,” IEEE Photonics Society Newsletter26, 5–12 (2012).

X. Wang, W. Shi, H. Yun, S. Grist, N. A. F. Jaeger, and L. Chrostowski, “Narrow-band waveguide Bragg gratings on SOI wafers with CMOS-compatible fabrication process,” Opt. Express20, 15547–15558 (2012).
[CrossRef] [PubMed]

V. Veerasubramanian, G. Beaudin, A. Giguère, B. L. Drogoff, V. Aimez, and A. G. Kirk, “Waveguide-coupled drop filters on SOI using quarter-wave shifted sidewalled grating resonators,” Opt. Express20, 15983–15990 (2012).
[CrossRef] [PubMed]

2011 (6)

D. T. H. Tan, K. Ikeda, S. Zamek, A. Mizrahi, M. P. Nezhad, A. V. Krishnamoorthy, J. E. C. K. Raj, X. Zheng, I. Shubin, Y. Luo, and Y. Fainman, “Wide bandwidth, low loss 1 by 4 wavelength division multiplexer on silicon for optical interconnects,” Opt. Express19, 2401–2409 (2011).
[CrossRef] [PubMed]

S. Khan, M. A. Baghban, and S. Fathpour, “Electronically tunable silicon photonic delay lines,” Opt. Express19, 11780–11785 (2011).
[CrossRef] [PubMed]

W. Shi, X. Wang, W. Zhang, L. Chrostowski, and N. A. F. Jaeger, “Contradirectional couplers in silicon-on-insulator rib waveguides,” Opt. Lett.36, 3999–4001 (2011).
[CrossRef] [PubMed]

A. Mekis, S. Gloeckner, G. Masini, A. Narasimha, T. Pinguet, S. Sahni, and P. D. Dobbelaere, “A grating-coupler-enabled CMOS photonics platform,” Selected Topics in Quantum Electronics, IEEE Journal of17, 597–608 (2011).
[CrossRef]

A. D. Simard, N. Ayotte, Y. Painchaud, S. Bédard, and S. LaRochelle, “Impact of sidewall roughness on integrated Bragg gratings,” J. Lightwave Tech.29, 3693–3704 (2011).
[CrossRef]

X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, “Uniform and sampled Bragg gratings in SOI strip waveguides with sidewall corrugations,” IEEE Photon. Technol. Lett.23, 290–292 (2011).

2010 (3)

G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics4, 518 – 526 (2010).
[CrossRef]

D. T. H. Tan, P. Sun, and Y. Fainman, “Monolithic nonlinear pulse compressor on a silicon chip,” Nat. Commun.1, 116 (2010).
[CrossRef] [PubMed]

W. A. Zortman, D. C. Trotter, and M. R. Watts, “Silicon photonics manufacturing,” Opt. Express18, 23598–23607 (2010).
[CrossRef] [PubMed]

2009 (2)

D. T. H. Tan, K. Ikeda, and Y. Fainman, “Cladding-modulated Bragg gratings in silicon waveguides,” Opt. Lett.34, 1357–1359 (2009).
[CrossRef] [PubMed]

D. T. H. Tan, K. Ikeda, and Y. Fainman, “Coupled chirped vertical gratings for on-chip group velocity dispersion engineering,” Appl. Phys. Lett.95, 141109 (2009).
[CrossRef]

2008 (1)

2006 (1)

2003 (1)

M. Qiu, M. Mulot, M. Swillo, S. Anand, B. Jaskorzynska, and A. Karlsson, “Photonic crystal optical filter based on contra-directional waveguide coupling,” Appl. Phys. Lett.83, 5121–5124 (2003).
[CrossRef]

2002 (1)

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

2001 (2)

C. Riziotis and M. N. Zervas, “Design considerations in optical add/drop multiplexers based on grating-assisted null couplers,” J. Lightwave Tech.19, 92–104 (2001).
[CrossRef]

T. E. Murphy, J. T. Hastings, and H. I. Smith, “Fabrication and characterization of narrow-band Bragg-reflection filters in silicon-on-insulator ridge waveguides,” J. Lightwave Tech.19, 1938–1942 (2001).
[CrossRef]

1993 (2)

J.-P. Weber, “Spectral characteristics of coupled-waveguide Bragg-reflection tunable optical filter,” Optoelectronics, IEEE Proceedings J140, 275–284 (1993).
[CrossRef]

J. Hong and W. Huang, “Coupled-waveguide exchange-Bragg resonator filters: Coupled-mode analysis with loss and gain,” J. Lightwave Tech.11, 226–233 (1993).
[CrossRef]

1980 (1)

Aimez, V.

Alexandre, I.

J.-F. Cliche, Y. Painchaud, C. Latrasse, M.-J. Picard, I. Alexandre, and M. Têtu, “Ultra-narrow Bragg grating for active semiconductor laser linewidth reduction through electrical feedback,” Proc. of BGPP’07, paper BTuE2, Quebec City, Canada (2007).

Anand, S.

M. Qiu, M. Mulot, M. Swillo, S. Anand, B. Jaskorzynska, and A. Karlsson, “Photonic crystal optical filter based on contra-directional waveguide coupling,” Appl. Phys. Lett.83, 5121–5124 (2003).
[CrossRef]

Ayazi, A.

T. Baehr-Jones, R. Ding, A. Ayazi, T. Pinguet, M. Streshinsky, N. Harris, J. Li, L. He, M. Gould, Y. Zhang, A. E.-J. Lim, T.-Y. Liow, S. H.-G. Teo, G.-Q. Lo, and M. Hochberg, “A 25 Gb/s silicon photonics platform,” arXiv preprint arXiv:1203.0767 (2012).

Ayotte, N.

A. D. Simard, N. Ayotte, Y. Painchaud, S. Bédard, and S. LaRochelle, “Impact of sidewall roughness on integrated Bragg gratings,” J. Lightwave Tech.29, 3693–3704 (2011).
[CrossRef]

Azaña, J.

Baehr-Jones, T.

T. Baehr-Jones, R. Ding, A. Ayazi, T. Pinguet, M. Streshinsky, N. Harris, J. Li, L. He, M. Gould, Y. Zhang, A. E.-J. Lim, T.-Y. Liow, S. H.-G. Teo, G.-Q. Lo, and M. Hochberg, “A 25 Gb/s silicon photonics platform,” arXiv preprint arXiv:1203.0767 (2012).

W. Shi, X. Wang, C. Lin, H. Yun, Y. Liu, T. Baehr-Jones, M. Hochberg, N. A. F. Jaeger, and L. Chrostowski, “Electrically tunable resonant filters in phase-shifted contra-directional couplers,” IEEE Group IV Photonics Conference (San Diego, CA, USA2012), paper WP2.
[CrossRef]

Baghban, M. A.

Beaudin, G.

Bédard, S.

A. D. Simard, N. Ayotte, Y. Painchaud, S. Bédard, and S. LaRochelle, “Impact of sidewall roughness on integrated Bragg gratings,” J. Lightwave Tech.29, 3693–3704 (2011).
[CrossRef]

Berger, N. K.

Bowers, J. E.

Cheung, K. C.

L. Chrostowski, S. Grist, J. Flueckiger, W. Shi, X. Wang, E. Ouellet, H. Yun, M. Webb, B. Nie, Z. Liang, K. C. Cheung, A. S. S, D. M. Ratner, and N. A. F. Jaeger, “Silicon photonic resonator sensors and devices,” Proceedings of SPIE8236, 823620 (2012).
[CrossRef]

Chrostowski, L.

L. Chrostowski, S. Grist, J. Flueckiger, W. Shi, X. Wang, E. Ouellet, H. Yun, M. Webb, B. Nie, Z. Liang, K. C. Cheung, A. S. S, D. M. Ratner, and N. A. F. Jaeger, “Silicon photonic resonator sensors and devices,” Proceedings of SPIE8236, 823620 (2012).
[CrossRef]

X. Wang, W. Shi, H. Yun, S. Grist, N. A. F. Jaeger, and L. Chrostowski, “Narrow-band waveguide Bragg gratings on SOI wafers with CMOS-compatible fabrication process,” Opt. Express20, 15547–15558 (2012).
[CrossRef] [PubMed]

W. Shi, X. Wang, W. Zhang, H. Yun, C. Lin, L. Chrostowski, and N. A. F. Jaeger, “Grating-coupled silicon microring resonators,” Appl. Phys. Lett.100, 121118 (2012).
[CrossRef]

W. Shi, X. Wang, W. Zhang, L. Chrostowski, and N. A. F. Jaeger, “Contradirectional couplers in silicon-on-insulator rib waveguides,” Opt. Lett.36, 3999–4001 (2011).
[CrossRef] [PubMed]

X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, “Uniform and sampled Bragg gratings in SOI strip waveguides with sidewall corrugations,” IEEE Photon. Technol. Lett.23, 290–292 (2011).

X. Wang, W. Shi, M. Hochberg, K. A. E. Schelew, J. F. Young, N. A. F. Jaeger, and L. Chrostowski, “Lithography simulation for the fabrication of silicon photonic devices with deep-ultraviolet lithography,” IEEE Group IV Photonics Conference (San Diego, CA, USA2012), paper ThP17.
[CrossRef]

W. Shi, M. Greenberg, X. Wang, C. Lin, N. A. F. Jaeger, and L. Chrostowski, “Single-band add-drop filters using anti-reflection, contradirectional couplers,” IEEE Group IV Photonics Conference (San Diego, CA, USA2012), paper WA7.
[CrossRef]

X. Wang, W. Shi, S. Grist, H. Yun, N. A. F. Jaeger, and L. Chrostowski, “Narrow-band transmission filter using phase-shifted Bragg gratings in SOI waveguide,” IEEE Photonics Conference (Arlington, VA, 2011), paper ThZ1.

W. Shi, X. Wang, C. Lin, H. Yun, Y. Liu, T. Baehr-Jones, M. Hochberg, N. A. F. Jaeger, and L. Chrostowski, “Electrically tunable resonant filters in phase-shifted contra-directional couplers,” IEEE Group IV Photonics Conference (San Diego, CA, USA2012), paper WP2.
[CrossRef]

Chrowtowski, L.

W. Shi, X. Wang, H. Yun, W. Zhang, L. Chrowtowski, and N. A. F. Jaeger, “Add-drop filters in silicon grating-assisted asymmetric couplers,” OFC/NFOEC (2012) paper OTh3D.3.

Cliche, J.-F.

J.-F. Cliche, Y. Painchaud, C. Latrasse, M.-J. Picard, I. Alexandre, and M. Têtu, “Ultra-narrow Bragg grating for active semiconductor laser linewidth reduction through electrical feedback,” Proc. of BGPP’07, paper BTuE2, Quebec City, Canada (2007).

Ding, R.

T. Baehr-Jones, R. Ding, A. Ayazi, T. Pinguet, M. Streshinsky, N. Harris, J. Li, L. He, M. Gould, Y. Zhang, A. E.-J. Lim, T.-Y. Liow, S. H.-G. Teo, G.-Q. Lo, and M. Hochberg, “A 25 Gb/s silicon photonics platform,” arXiv preprint arXiv:1203.0767 (2012).

Dobbelaere, P. D.

A. Mekis, S. Gloeckner, G. Masini, A. Narasimha, T. Pinguet, S. Sahni, and P. D. Dobbelaere, “A grating-coupler-enabled CMOS photonics platform,” Selected Topics in Quantum Electronics, IEEE Journal of17, 597–608 (2011).
[CrossRef]

Drogoff, B. L.

Fainman, Y.

Fang, A. W.

Fathpour, S.

Fischer, B.

Flueckiger, J.

L. Chrostowski, S. Grist, J. Flueckiger, W. Shi, X. Wang, E. Ouellet, H. Yun, M. Webb, B. Nie, Z. Liang, K. C. Cheung, A. S. S, D. M. Ratner, and N. A. F. Jaeger, “Silicon photonic resonator sensors and devices,” Proceedings of SPIE8236, 823620 (2012).
[CrossRef]

Gajda, A.

I. Giuntoni, D. Stolarek, A. Gajda, J. B. G. Winzer, B. Tillack, K. Petermann, and L. Zimmerman, “Integrated drop-filter for dispersion compensation based on SOI rib waveguides,” Optical Fiber Communication Conference, OSA Technical Digest (San Diego, CA, 2010), paper OThJ5.

Gardes, F. Y.

G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics4, 518 – 526 (2010).
[CrossRef]

Giguère, A.

Giuntoni, I.

I. Giuntoni, D. Stolarek, A. Gajda, J. B. G. Winzer, B. Tillack, K. Petermann, and L. Zimmerman, “Integrated drop-filter for dispersion compensation based on SOI rib waveguides,” Optical Fiber Communication Conference, OSA Technical Digest (San Diego, CA, 2010), paper OThJ5.

Gloeckner, S.

A. Mekis, S. Gloeckner, G. Masini, A. Narasimha, T. Pinguet, S. Sahni, and P. D. Dobbelaere, “A grating-coupler-enabled CMOS photonics platform,” Selected Topics in Quantum Electronics, IEEE Journal of17, 597–608 (2011).
[CrossRef]

Gould, M.

T. Baehr-Jones, R. Ding, A. Ayazi, T. Pinguet, M. Streshinsky, N. Harris, J. Li, L. He, M. Gould, Y. Zhang, A. E.-J. Lim, T.-Y. Liow, S. H.-G. Teo, G.-Q. Lo, and M. Hochberg, “A 25 Gb/s silicon photonics platform,” arXiv preprint arXiv:1203.0767 (2012).

Greenberg, M.

W. Shi, M. Greenberg, X. Wang, C. Lin, N. A. F. Jaeger, and L. Chrostowski, “Single-band add-drop filters using anti-reflection, contradirectional couplers,” IEEE Group IV Photonics Conference (San Diego, CA, USA2012), paper WA7.
[CrossRef]

Grist, S.

L. Chrostowski, S. Grist, J. Flueckiger, W. Shi, X. Wang, E. Ouellet, H. Yun, M. Webb, B. Nie, Z. Liang, K. C. Cheung, A. S. S, D. M. Ratner, and N. A. F. Jaeger, “Silicon photonic resonator sensors and devices,” Proceedings of SPIE8236, 823620 (2012).
[CrossRef]

X. Wang, W. Shi, H. Yun, S. Grist, N. A. F. Jaeger, and L. Chrostowski, “Narrow-band waveguide Bragg gratings on SOI wafers with CMOS-compatible fabrication process,” Opt. Express20, 15547–15558 (2012).
[CrossRef] [PubMed]

X. Wang, W. Shi, S. Grist, H. Yun, N. A. F. Jaeger, and L. Chrostowski, “Narrow-band transmission filter using phase-shifted Bragg gratings in SOI waveguide,” IEEE Photonics Conference (Arlington, VA, 2011), paper ThZ1.

Harris, N.

T. Baehr-Jones, R. Ding, A. Ayazi, T. Pinguet, M. Streshinsky, N. Harris, J. Li, L. He, M. Gould, Y. Zhang, A. E.-J. Lim, T.-Y. Liow, S. H.-G. Teo, G.-Q. Lo, and M. Hochberg, “A 25 Gb/s silicon photonics platform,” arXiv preprint arXiv:1203.0767 (2012).

Hastings, J. T.

T. E. Murphy, J. T. Hastings, and H. I. Smith, “Fabrication and characterization of narrow-band Bragg-reflection filters in silicon-on-insulator ridge waveguides,” J. Lightwave Tech.19, 1938–1942 (2001).
[CrossRef]

He, L.

T. Baehr-Jones, R. Ding, A. Ayazi, T. Pinguet, M. Streshinsky, N. Harris, J. Li, L. He, M. Gould, Y. Zhang, A. E.-J. Lim, T.-Y. Liow, S. H.-G. Teo, G.-Q. Lo, and M. Hochberg, “A 25 Gb/s silicon photonics platform,” arXiv preprint arXiv:1203.0767 (2012).

Hochberg, M.

T. Baehr-Jones, R. Ding, A. Ayazi, T. Pinguet, M. Streshinsky, N. Harris, J. Li, L. He, M. Gould, Y. Zhang, A. E.-J. Lim, T.-Y. Liow, S. H.-G. Teo, G.-Q. Lo, and M. Hochberg, “A 25 Gb/s silicon photonics platform,” arXiv preprint arXiv:1203.0767 (2012).

X. Wang, W. Shi, M. Hochberg, K. A. E. Schelew, J. F. Young, N. A. F. Jaeger, and L. Chrostowski, “Lithography simulation for the fabrication of silicon photonic devices with deep-ultraviolet lithography,” IEEE Group IV Photonics Conference (San Diego, CA, USA2012), paper ThP17.
[CrossRef]

W. Shi, X. Wang, C. Lin, H. Yun, Y. Liu, T. Baehr-Jones, M. Hochberg, N. A. F. Jaeger, and L. Chrostowski, “Electrically tunable resonant filters in phase-shifted contra-directional couplers,” IEEE Group IV Photonics Conference (San Diego, CA, USA2012), paper WP2.
[CrossRef]

Hong, J.

J. Hong and W. Huang, “Coupled-waveguide exchange-Bragg resonator filters: Coupled-mode analysis with loss and gain,” J. Lightwave Tech.11, 226–233 (1993).
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X. Wang, W. Shi, M. Hochberg, K. A. E. Schelew, J. F. Young, N. A. F. Jaeger, and L. Chrostowski, “Lithography simulation for the fabrication of silicon photonic devices with deep-ultraviolet lithography,” IEEE Group IV Photonics Conference (San Diego, CA, USA2012), paper ThP17.
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X. Wang, W. Shi, M. Hochberg, K. A. E. Schelew, J. F. Young, N. A. F. Jaeger, and L. Chrostowski, “Lithography simulation for the fabrication of silicon photonic devices with deep-ultraviolet lithography,” IEEE Group IV Photonics Conference (San Diego, CA, USA2012), paper ThP17.
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W. Shi, X. Wang, C. Lin, H. Yun, Y. Liu, T. Baehr-Jones, M. Hochberg, N. A. F. Jaeger, and L. Chrostowski, “Electrically tunable resonant filters in phase-shifted contra-directional couplers,” IEEE Group IV Photonics Conference (San Diego, CA, USA2012), paper WP2.
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G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics4, 518 – 526 (2010).
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X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, “Uniform and sampled Bragg gratings in SOI strip waveguides with sidewall corrugations,” IEEE Photon. Technol. Lett.23, 290–292 (2011).

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W. Shi, X. Wang, W. Zhang, H. Yun, C. Lin, L. Chrostowski, and N. A. F. Jaeger, “Grating-coupled silicon microring resonators,” Appl. Phys. Lett.100, 121118 (2012).
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Figures (11)

Fig. 1
Fig. 1

Schematic drawing of a contra-DC-based add-drop filter. The field, E(z), in the coupler region, can be decomposed into the transverse modes, E1 and E2, as shown by Eq. (3).

Fig. 2
Fig. 2

Schematic drawing of a contra-directional coupler. A uniform grating is formed between two different-sized waveguides. The arrows indicate the optical waves in the coupled-mode analysis.

Fig. 3
Fig. 3

Schematic drawing of a phase-shifted contra-directional coupler. The red arrows indicate the resonant loop of the optical cavity. The transfer matrixes, C(z0, z1), P(z1, z2), and C(z1, z2), are labeled as C01, P12, and C23, respectively. The dashed arrows indicate the optical waves incident from the add port, which are present in the coupled-mode equations but were not excited in our experiment (i.e., there was no input from the add port in our experiment). The optical waves due to the intra-waveguide back reflections (i.e., A and B+) are not shown.

Fig. 4
Fig. 4

Cross-sections of the high-index section (top) and the low-index section (bottom) in each grating period of the contra-DC.

Fig. 5
Fig. 5

(a) Calculated intensity distributions of the electric fields for the fundamental TE-like modes of the individual waveguides; (b) Calculated intensity distributions of the electric fields for the first and second TE-like modes of the contra-DC; (c) Calculated effective indices of the modes with the phase-match conditions and corresponding wavelengths labeled.

Fig. 6
Fig. 6

Simulated spectra of the contra-DC with a uniform grating: (a) through-port and drop-port responses and (b) phase, ϕ, and group delay, τ, of the drop-port, without the intra-waveguide reflections considered; (c) through-port and drop-port responses and (d) input-waveguide reflection and reflection-caused add-port response with the intra-waveguide re-flections considered assuming ideal mode transitions; (e) through-port and drop-port responses and (f) input-waveguide reflection and reflection-caused add-port response, with the intra-waveguide reflections and mode transitions considered (assuming the worst case where no taper is used). In all the calculations, α = 5 dB/cm has been assumed.

Fig. 7
Fig. 7

Simulated spectra of the phase-shifted contra-DC: (a) ideal through-port and drop-port power responses and (b) ideal drop-port phase response, without intra-waveguide reflection or optical loss considered (the drop-port response at the resonant wavelength is zero); (c) through-port and drop-port power responses and (d) drop-port phase response, with α = 5 dB/cm but without intra-waveguide reflection considered; (e) through-port and drop-port power responses and (f) drop-port phase response, with α = 5 dB/cm and the intra-waveguide back reflections and mode-transitions considered.

Fig. 8
Fig. 8

Measured and fit spectra of the through-port and drop-port responses of the contra-DC with a uniform grating.

Fig. 9
Fig. 9

Measured and fit spectra of the through-port and drop-port responses of the phase-shifted contra-DC: (a) entire measured spectral range; (b) zoomed spectra near the resonant peak.

Fig. 10
Fig. 10

(a) Measured drop-port spectra for various currents; (b) Measured and simulated resonant-wavelength shift, Δλ, as a function of current; (c) Measured and simulated I–V curves; (d) Measured small-signal frequency response.

Fig. 11
Fig. 11

Simulated spectral responses of contra-DCs, one with a uniform grating and one with an apodized grating, illustrating the side-lobe suppression in the apodized grating design.

Tables (1)

Tables Icon

Table 1 Parameters used in the simulation (Fig. 6 and 7) and the fit with experiment (Fig. 8 and 9). The effective indices are slightly tuned for wavelength alignment. The coupling coefficients have a unit of m−1.

Equations (37)

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β 1 + β 2 2 π Λ = 0
2 β 1 , 2 2 π Λ = 0
E ( x , y , z ) = [ A + ( z ) e j β ^ 1 z + A ( z ) e j β ^ 1 z ] E 1 ( x , y ) + [ B + ( z ) e j β ^ 2 z + B ( z ) e j β ^ 2 z ] E 2 ( x , y )
d A + d z = j κ 11 A e j 2 Δ β ^ 1 z j κ 12 B e j ( Δ β ^ 1 + Δ β ^ 2 ) z
d B + d z = j κ 12 A e j ( Δ β ^ 1 + Δ β ^ 2 ) z j κ 22 B e j 2 Δ β ^ 2 z
d A d z = j κ 11 * A + e j 2 Δ β ^ 1 z + j κ 12 * B + e j ( Δ β ^ 1 + Δ β ^ 2 ) z
d B d z = j κ 12 * A + e j ( Δ β ^ 1 + Δ β ^ 2 ) z + j κ 22 * B + e j 2 Δ β ^ 2 z
κ 11 = ω 4 E 1 * ( x , y ) Δ ε 1 ( x , y ) E 1 ( x , y ) d x d y
κ 12 = κ 21 * = ω 4 E 1 * ( x , y ) Δ ε 1 ( x , y ) E 2 ( x , y ) d x d y
κ 22 = ω 4 E 2 * ( x , y ) Δ ε 1 ( x , y ) E 2 ( x , y ) d x d y
E ( z ) = [ A + ( z ) B + ( z ) A ( z ) B ( z ) ]
E ( z 0 ) = C ( z 0 , z 1 ) E ( z 1 )
C ( z 0 , z 1 ) = e S 1 ( z 1 z 0 ) e S 2 ( z 1 z 0 )
S 1 = [ j Δ β ^ 1 0 0 0 0 j Δ β ^ 2 0 0 0 0 j Δ β ^ 1 0 0 0 0 j Δ β ^ 2 ]
S 2 = [ j Δ β ^ 1 0 j κ 11 e j 2 Δ β ^ 1 z 1 j κ 12 e j ( Δ β ^ 1 + Δ β ^ 2 ) z 1 0 j Δ β ^ 2 j κ 12 e j ( Δ β ^ 1 + Δ β ^ 2 ) z 1 j κ 22 e j 2 Δ β ^ 2 z 1 j κ 11 * e j 2 Δ β ^ 1 z 1 j κ 12 * e j ( Δ β ^ 1 + Δ β ^ 2 ) z 1 j Δ β ^ 1 0 j κ 12 * e j ( Δ β ^ 1 + Δ β ^ 2 ) z 1 j κ 22 * e j 2 Δ β ^ 2 z 1 0 j Δ β ^ 2 ]
η c = | B 0 | 2 | A 0 + | 2
E ( z 0 ) = C ( z 0 , z 1 ) P ( z 1 , z 2 ) C ( z 2 , z 3 ) E ( z 3 )
P ( z 1 , z 2 ) = [ e j β ^ 1 ( z 2 z 1 ) 0 0 0 0 e j β ^ 2 ( z 1 z 1 ) 0 0 0 0 e j β ^ 1 ( z 2 z 1 ) 0 0 0 0 e j β ^ 2 ( z 2 z 1 ) ]
δ r t = δ c 1 + ( β 1 + β 2 ) Λ + δ c 2
δ c 1 ( λ D ) = δ c 2 ( λ D ) = [ β 1 ( λ D ) + β 2 ( λ D ) ] Λ = 2 π
η c 1 = A r t η c 2
λ D = 2 Λ n a v = Λ ( n 1 + n 2 )
λ r 1 = 2 Λ n 1
λ r 2 = 2 Λ n 2
k i j 2 = Re { d S E i × H j * d S E j × H i * d S E i × H i * d S E j × H j * } ; i = a , b ; j = 1 , 2
τ = d ϕ d ω
κ 12 ( n ) = κ max e a ( n 0.5 N ) 2 N 2
E ( z ) = [ E + ( z ) E ( z ) ]
E + ( z ) = [ A + ( z ) B + ( z ) ]
E ( z ) = [ A ( z ) B ( z ) ]
E ( z 0 ) = [ E + ( z 0 ) E ( z 0 ) ] = M × E ( z m ) = [ M + + M + M + M ] [ E + ( z m ) E ( z m ) ]
[ A + ( z m ) B + ( z m ) A ( z 0 ) B ( z 0 ) ] = M [ A + ( z 0 ) B + ( z 0 ) A ( z m ) B ( z m ) ]
M = [ M + + M + ( M ) 1 M + M + ( M ) 1 ( M ) 1 M + ( M ) 1 ]
[ A + ( z 0 ) B + ( z 0 ) A ( z m ) B ( z m ) ] = E in [ k a 1 k a 2 0 0 ]
[ A + ( z m ) B + ( z m ) A ( z 0 ) B ( z 0 ) ] = M E in [ k a 1 k a 2 0 0 ]
[ E Thru E Add E R E Drop ] = K [ A + ( z m ) B + ( z m ) A ( z 0 ) B ( z 0 ) ] = K M E in [ k a 1 k a 2 0 0 ]
K = [ k a 1 k a 2 0 0 k b 1 k b 2 0 0 0 0 k a 1 k a 2 0 0 k b 1 k b 2 ]

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