Abstract

A study of vertical coupling conditions between microdisk based resonators and waveguides is presented using an analytical model. The coupling efficiency optimization is investigated and compared to 3D FDTD computations. We also demonstrate that coupling losses can be exploited to favor high quality factor modes in circular resonators. In addition, we propose to modify the shape of the coupled waveguide to enhance mode selectivity and obtain a very compact structure with mode hoping capacities. Lower thresholds and modulation are also expected.

© 2010 Optical Society of America

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References

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  1. D. Yang, Y. Li, F. Sun, S. Chen, and J. Yu, "Fabrication of a 4×4 strictly nonblocking SOI switch matrix," Opt. Commun. 250, 48-53 (2005).
    [CrossRef]
  2. I. Kiyat, A. Aydinli, and N. Dagli, "High-Q silicon-on-insulator optical rib waveguide racetrack resonators," Opt. Express 13, 1900-1905 (2005).
    [CrossRef] [PubMed]
  3. A. Kaźmierczak, W. Bogaerts, D. Van Thourhout, E. Drouard, P. Rojo Romeo, D. Giannone, and F. Gaffiot, "Analysis of silicon on insulator (SOI) optical microring add-drop filter based on waveguide intersections," Proc. SPIE 6996, 69960D (2008).
    [CrossRef]
  4. L. Vivien, D. Marris-Morini, J. Mangeney, P. Crozat, E. Cassan, S. Laval, J.-M. Fedeli, J. Damlencourt, and Y. Lecunff, "42 GHz waveguide Germanium-on-silicon vertical PIN photodetector," 5th IEEE International Conference on Group IV Photonics (2008).
  5. P. Binetti, J. Van Campenhout, X. Leijtens, M. Nikoufard, T. de Vries, Y. Oei, L. Di Cioccio, J. Fedeli, C. Lagahe, R. Orobtchouk et al., "An optical interconnect layer on silicon," Proceedings of the 13th European Conference on Integrated Optics (ECIO), 1-3 (2007).
  6. J. Van Campenhout, P. Rojo Romeo, P. Regreny, C. Seassal, D. Van Thourhout, S. Verstuyft, L. Di Cioccio, J. Fedeli, C. Lagahe, and R. Baets, "Electrically pumped InP-based microdisk lasers integrated with a nanophotonic silicon-on-insulator waveguide circuit," Opt. Express 15, 6744-6749 (2007).
    [CrossRef] [PubMed]
  7. L. Liu, J. VanCampenhout, G. Roelkens, R. Soref, D. VanThourhout, P. Rojo Romeo, P. Regreny, C. Seassal, J. Fedeli, and R. Baets, "Compact optical modulator based on carrier induced gain of an InP-InGaAsP microdisk cavity integrated on SOI," Proc. SPIE 7135, 71351l (2008).
  8. P. Rojo Romeo, J. Van Campenhout, F. Mandorlo, C. Seassal, X. Letartre, P. Regreny, D. Van Thourhout, R. Baets, L. DiCioccio, and J. Fedeli, "Integration of an Electrically Driven InGaAsP Based Microdisk laser with a Silicon based Passive Photonic Circuit," Optical Society of America-CLEO/QELS Conference (2007).
  9. L. Zhang and E. Hu, "Lasing from InGaAs quantum dots in an injection microdisk," Appl. Phys. Lett. 82, 319 (2003).
    [CrossRef]
  10. J. Van Campenhout, P. Rojo Romeo, D. Van Thourhout, C. Seassal, P. Regreny, L. Cioccio, J. Fedeli, and R. Baets, "Design and Optimization of Electrically Injected InP-Based Microdisk lasers Integrated on and Coupled to a SOI Waveguide Circuit," J. Lightwave Technol. 26, 52-63 (2008).
    [CrossRef]
  11. F. Mandorlo, P. Rojo Romeo, X. Letartre, and P. Viktorovitch, "A simple perturbative analysis for fast design of an electrically pumped micro-disk laser," Opt. Express 17, 70-79 (2009).
    [CrossRef] [PubMed]
  12. F. Mandorlo, P. Rojo Romeo, X. Letartre, J. Fedeli, and P. Viktorovitch, "Improving contact design for micro-disc based lasers in integrated circuits," 5th IEEE International Conference on Group IV Photonics (2008).
  13. F. Mandorlo, J. Fedeli, and P. Rojo Romeo, "Microdisc system with gallery modes for electrically pumped optical sources," European Patent Office (EPO) Patent EP2 101 380 (2009).
  14. K. Chiang, "Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides," IEEE Trans. Microwave Theory Tech. 44, 692-700 (1996).
    [CrossRef]
  15. M. Fujita, A. Sakai, and T. Baba, "Ultrasmall and ultralow threshold GaInAsP-InP microdisk injectionlasers: design, fabrication, lasing characteristics, and spontaneous emission factor," IEEE J. Sel. Top. Quantum Electron. 5, 673-681 (1999).
    [CrossRef]
  16. P. Rojo Romeo, J. Van Campenhout, P. Regreny, A. Kazmierczak, C. Seassal, X. Letartre, G. Hollinger, D. Van Thourhout, R. Baets, J. Fedeli et al., "Heterogeneous integration of electrically driven microdisk based laser sources for optical interconnects and photonic ICs," Opt. Express 14, 3864-3871 (2006).
    [CrossRef]
  17. M. Borselli, K. Srinivasan, P. Barclay, and O. Painter, "Rayleigh scattering, mode coupling, and optical loss in silicon microdisks," Appl. Phys. Lett. 85, 3693 (2004).
    [CrossRef]
  18. F. Mandorlo, P. Rojo Romeo, X. Letartre, P. Regreny, P. Viktorovitch, J. Fedeli, and P. Grosse, "Contacting InP based micro disk lasers on 200 mm Si wafers," 20th International Conference on Indium Phosphide and Related Materials (IPRM), 1-4 (2008).
  19. L. Coldren and S. Corzine, Diode lasers and photonic integrated circuits, 36 (1997).
  20. C. Manolatou, M. Khan, S. Fan, P. Villeneuve, H. Haus, and J. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322-1331 (1999).
    [CrossRef]
  21. TessaFDTD , http://alioth.debian.org/projects/tessa/
  22. Harminv, http://ab-initio.mit.edu/harminv/
  23. R. Orobtchouk, S. Jeannot, B. Han, T. Benyattou, J. Fedeli, and P. Mur, "Ultra compact optical link made in amorphous silicon waveguide," Proc. SPIE 6183, 618304 (2006).
    [CrossRef]
  24. K. P. Huy, "Etude de micro structures utilisant le guidage r’efractif `a fort confinement de la lumi`ere," Ph.D. dissertation, Institut de Micro’electronique, Electromagn’etisme et Photonique (2005).
  25. M. Fujita and T. Baba, "Microgear laser," Appl. Phys. Lett. 80, 2051 (2002).
    [CrossRef]

2009 (1)

2008 (2)

J. Van Campenhout, P. Rojo Romeo, D. Van Thourhout, C. Seassal, P. Regreny, L. Cioccio, J. Fedeli, and R. Baets, "Design and Optimization of Electrically Injected InP-Based Microdisk lasers Integrated on and Coupled to a SOI Waveguide Circuit," J. Lightwave Technol. 26, 52-63 (2008).
[CrossRef]

A. Kaźmierczak, W. Bogaerts, D. Van Thourhout, E. Drouard, P. Rojo Romeo, D. Giannone, and F. Gaffiot, "Analysis of silicon on insulator (SOI) optical microring add-drop filter based on waveguide intersections," Proc. SPIE 6996, 69960D (2008).
[CrossRef]

2007 (1)

2006 (1)

2005 (2)

D. Yang, Y. Li, F. Sun, S. Chen, and J. Yu, "Fabrication of a 4×4 strictly nonblocking SOI switch matrix," Opt. Commun. 250, 48-53 (2005).
[CrossRef]

I. Kiyat, A. Aydinli, and N. Dagli, "High-Q silicon-on-insulator optical rib waveguide racetrack resonators," Opt. Express 13, 1900-1905 (2005).
[CrossRef] [PubMed]

2004 (1)

M. Borselli, K. Srinivasan, P. Barclay, and O. Painter, "Rayleigh scattering, mode coupling, and optical loss in silicon microdisks," Appl. Phys. Lett. 85, 3693 (2004).
[CrossRef]

2003 (1)

L. Zhang and E. Hu, "Lasing from InGaAs quantum dots in an injection microdisk," Appl. Phys. Lett. 82, 319 (2003).
[CrossRef]

2002 (1)

M. Fujita and T. Baba, "Microgear laser," Appl. Phys. Lett. 80, 2051 (2002).
[CrossRef]

1999 (2)

C. Manolatou, M. Khan, S. Fan, P. Villeneuve, H. Haus, and J. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

M. Fujita, A. Sakai, and T. Baba, "Ultrasmall and ultralow threshold GaInAsP-InP microdisk injectionlasers: design, fabrication, lasing characteristics, and spontaneous emission factor," IEEE J. Sel. Top. Quantum Electron. 5, 673-681 (1999).
[CrossRef]

1996 (1)

K. Chiang, "Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides," IEEE Trans. Microwave Theory Tech. 44, 692-700 (1996).
[CrossRef]

Aydinli, A.

Baba, T.

M. Fujita and T. Baba, "Microgear laser," Appl. Phys. Lett. 80, 2051 (2002).
[CrossRef]

M. Fujita, A. Sakai, and T. Baba, "Ultrasmall and ultralow threshold GaInAsP-InP microdisk injectionlasers: design, fabrication, lasing characteristics, and spontaneous emission factor," IEEE J. Sel. Top. Quantum Electron. 5, 673-681 (1999).
[CrossRef]

Baets, R.

Barclay, P.

M. Borselli, K. Srinivasan, P. Barclay, and O. Painter, "Rayleigh scattering, mode coupling, and optical loss in silicon microdisks," Appl. Phys. Lett. 85, 3693 (2004).
[CrossRef]

Bogaerts, W.

A. Kaźmierczak, W. Bogaerts, D. Van Thourhout, E. Drouard, P. Rojo Romeo, D. Giannone, and F. Gaffiot, "Analysis of silicon on insulator (SOI) optical microring add-drop filter based on waveguide intersections," Proc. SPIE 6996, 69960D (2008).
[CrossRef]

Borselli, M.

M. Borselli, K. Srinivasan, P. Barclay, and O. Painter, "Rayleigh scattering, mode coupling, and optical loss in silicon microdisks," Appl. Phys. Lett. 85, 3693 (2004).
[CrossRef]

Chen, S.

D. Yang, Y. Li, F. Sun, S. Chen, and J. Yu, "Fabrication of a 4×4 strictly nonblocking SOI switch matrix," Opt. Commun. 250, 48-53 (2005).
[CrossRef]

Chiang, K.

K. Chiang, "Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides," IEEE Trans. Microwave Theory Tech. 44, 692-700 (1996).
[CrossRef]

Cioccio, L.

Dagli, N.

Di Cioccio, L.

Drouard, E.

A. Kaźmierczak, W. Bogaerts, D. Van Thourhout, E. Drouard, P. Rojo Romeo, D. Giannone, and F. Gaffiot, "Analysis of silicon on insulator (SOI) optical microring add-drop filter based on waveguide intersections," Proc. SPIE 6996, 69960D (2008).
[CrossRef]

Fan, S.

C. Manolatou, M. Khan, S. Fan, P. Villeneuve, H. Haus, and J. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

Fedeli, J.

Fujita, M.

M. Fujita and T. Baba, "Microgear laser," Appl. Phys. Lett. 80, 2051 (2002).
[CrossRef]

M. Fujita, A. Sakai, and T. Baba, "Ultrasmall and ultralow threshold GaInAsP-InP microdisk injectionlasers: design, fabrication, lasing characteristics, and spontaneous emission factor," IEEE J. Sel. Top. Quantum Electron. 5, 673-681 (1999).
[CrossRef]

Gaffiot, F.

A. Kaźmierczak, W. Bogaerts, D. Van Thourhout, E. Drouard, P. Rojo Romeo, D. Giannone, and F. Gaffiot, "Analysis of silicon on insulator (SOI) optical microring add-drop filter based on waveguide intersections," Proc. SPIE 6996, 69960D (2008).
[CrossRef]

Giannone, D.

A. Kaźmierczak, W. Bogaerts, D. Van Thourhout, E. Drouard, P. Rojo Romeo, D. Giannone, and F. Gaffiot, "Analysis of silicon on insulator (SOI) optical microring add-drop filter based on waveguide intersections," Proc. SPIE 6996, 69960D (2008).
[CrossRef]

Haus, H.

C. Manolatou, M. Khan, S. Fan, P. Villeneuve, H. Haus, and J. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

Hollinger, G.

Hu, E.

L. Zhang and E. Hu, "Lasing from InGaAs quantum dots in an injection microdisk," Appl. Phys. Lett. 82, 319 (2003).
[CrossRef]

Joannopoulos, J.

C. Manolatou, M. Khan, S. Fan, P. Villeneuve, H. Haus, and J. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

Kazmierczak, A.

A. Kaźmierczak, W. Bogaerts, D. Van Thourhout, E. Drouard, P. Rojo Romeo, D. Giannone, and F. Gaffiot, "Analysis of silicon on insulator (SOI) optical microring add-drop filter based on waveguide intersections," Proc. SPIE 6996, 69960D (2008).
[CrossRef]

P. Rojo Romeo, J. Van Campenhout, P. Regreny, A. Kazmierczak, C. Seassal, X. Letartre, G. Hollinger, D. Van Thourhout, R. Baets, J. Fedeli et al., "Heterogeneous integration of electrically driven microdisk based laser sources for optical interconnects and photonic ICs," Opt. Express 14, 3864-3871 (2006).
[CrossRef]

Khan, M.

C. Manolatou, M. Khan, S. Fan, P. Villeneuve, H. Haus, and J. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

Kiyat, I.

Lagahe, C.

Letartre, X.

Li, Y.

D. Yang, Y. Li, F. Sun, S. Chen, and J. Yu, "Fabrication of a 4×4 strictly nonblocking SOI switch matrix," Opt. Commun. 250, 48-53 (2005).
[CrossRef]

Mandorlo, F.

Manolatou, C.

C. Manolatou, M. Khan, S. Fan, P. Villeneuve, H. Haus, and J. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

Painter, O.

M. Borselli, K. Srinivasan, P. Barclay, and O. Painter, "Rayleigh scattering, mode coupling, and optical loss in silicon microdisks," Appl. Phys. Lett. 85, 3693 (2004).
[CrossRef]

Regreny, P.

Rojo Romeo, P.

Sakai, A.

M. Fujita, A. Sakai, and T. Baba, "Ultrasmall and ultralow threshold GaInAsP-InP microdisk injectionlasers: design, fabrication, lasing characteristics, and spontaneous emission factor," IEEE J. Sel. Top. Quantum Electron. 5, 673-681 (1999).
[CrossRef]

Seassal, C.

Srinivasan, K.

M. Borselli, K. Srinivasan, P. Barclay, and O. Painter, "Rayleigh scattering, mode coupling, and optical loss in silicon microdisks," Appl. Phys. Lett. 85, 3693 (2004).
[CrossRef]

Sun, F.

D. Yang, Y. Li, F. Sun, S. Chen, and J. Yu, "Fabrication of a 4×4 strictly nonblocking SOI switch matrix," Opt. Commun. 250, 48-53 (2005).
[CrossRef]

Van Campenhout, J.

Van Thourhout, D.

Verstuyft, S.

Viktorovitch, P.

Villeneuve, P.

C. Manolatou, M. Khan, S. Fan, P. Villeneuve, H. Haus, and J. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

Yang, D.

D. Yang, Y. Li, F. Sun, S. Chen, and J. Yu, "Fabrication of a 4×4 strictly nonblocking SOI switch matrix," Opt. Commun. 250, 48-53 (2005).
[CrossRef]

Yu, J.

D. Yang, Y. Li, F. Sun, S. Chen, and J. Yu, "Fabrication of a 4×4 strictly nonblocking SOI switch matrix," Opt. Commun. 250, 48-53 (2005).
[CrossRef]

Zhang, L.

L. Zhang and E. Hu, "Lasing from InGaAs quantum dots in an injection microdisk," Appl. Phys. Lett. 82, 319 (2003).
[CrossRef]

Appl. Phys. Lett. (3)

L. Zhang and E. Hu, "Lasing from InGaAs quantum dots in an injection microdisk," Appl. Phys. Lett. 82, 319 (2003).
[CrossRef]

M. Borselli, K. Srinivasan, P. Barclay, and O. Painter, "Rayleigh scattering, mode coupling, and optical loss in silicon microdisks," Appl. Phys. Lett. 85, 3693 (2004).
[CrossRef]

M. Fujita and T. Baba, "Microgear laser," Appl. Phys. Lett. 80, 2051 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

C. Manolatou, M. Khan, S. Fan, P. Villeneuve, H. Haus, and J. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

M. Fujita, A. Sakai, and T. Baba, "Ultrasmall and ultralow threshold GaInAsP-InP microdisk injectionlasers: design, fabrication, lasing characteristics, and spontaneous emission factor," IEEE J. Sel. Top. Quantum Electron. 5, 673-681 (1999).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

K. Chiang, "Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides," IEEE Trans. Microwave Theory Tech. 44, 692-700 (1996).
[CrossRef]

J. Lightwave Technol. (1)

Opt. Commun. (1)

D. Yang, Y. Li, F. Sun, S. Chen, and J. Yu, "Fabrication of a 4×4 strictly nonblocking SOI switch matrix," Opt. Commun. 250, 48-53 (2005).
[CrossRef]

Opt. Express (4)

Proc. SPIE (1)

A. Kaźmierczak, W. Bogaerts, D. Van Thourhout, E. Drouard, P. Rojo Romeo, D. Giannone, and F. Gaffiot, "Analysis of silicon on insulator (SOI) optical microring add-drop filter based on waveguide intersections," Proc. SPIE 6996, 69960D (2008).
[CrossRef]

Other (12)

L. Vivien, D. Marris-Morini, J. Mangeney, P. Crozat, E. Cassan, S. Laval, J.-M. Fedeli, J. Damlencourt, and Y. Lecunff, "42 GHz waveguide Germanium-on-silicon vertical PIN photodetector," 5th IEEE International Conference on Group IV Photonics (2008).

P. Binetti, J. Van Campenhout, X. Leijtens, M. Nikoufard, T. de Vries, Y. Oei, L. Di Cioccio, J. Fedeli, C. Lagahe, R. Orobtchouk et al., "An optical interconnect layer on silicon," Proceedings of the 13th European Conference on Integrated Optics (ECIO), 1-3 (2007).

L. Liu, J. VanCampenhout, G. Roelkens, R. Soref, D. VanThourhout, P. Rojo Romeo, P. Regreny, C. Seassal, J. Fedeli, and R. Baets, "Compact optical modulator based on carrier induced gain of an InP-InGaAsP microdisk cavity integrated on SOI," Proc. SPIE 7135, 71351l (2008).

P. Rojo Romeo, J. Van Campenhout, F. Mandorlo, C. Seassal, X. Letartre, P. Regreny, D. Van Thourhout, R. Baets, L. DiCioccio, and J. Fedeli, "Integration of an Electrically Driven InGaAsP Based Microdisk laser with a Silicon based Passive Photonic Circuit," Optical Society of America-CLEO/QELS Conference (2007).

F. Mandorlo, P. Rojo Romeo, X. Letartre, P. Regreny, P. Viktorovitch, J. Fedeli, and P. Grosse, "Contacting InP based micro disk lasers on 200 mm Si wafers," 20th International Conference on Indium Phosphide and Related Materials (IPRM), 1-4 (2008).

L. Coldren and S. Corzine, Diode lasers and photonic integrated circuits, 36 (1997).

TessaFDTD , http://alioth.debian.org/projects/tessa/

Harminv, http://ab-initio.mit.edu/harminv/

R. Orobtchouk, S. Jeannot, B. Han, T. Benyattou, J. Fedeli, and P. Mur, "Ultra compact optical link made in amorphous silicon waveguide," Proc. SPIE 6183, 618304 (2006).
[CrossRef]

K. P. Huy, "Etude de micro structures utilisant le guidage r’efractif `a fort confinement de la lumi`ere," Ph.D. dissertation, Institut de Micro’electronique, Electromagn’etisme et Photonique (2005).

F. Mandorlo, P. Rojo Romeo, X. Letartre, J. Fedeli, and P. Viktorovitch, "Improving contact design for micro-disc based lasers in integrated circuits," 5th IEEE International Conference on Group IV Photonics (2008).

F. Mandorlo, J. Fedeli, and P. Rojo Romeo, "Microdisc system with gallery modes for electrically pumped optical sources," European Patent Office (EPO) Patent EP2 101 380 (2009).

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Figures (11)

Fig. 1.
Fig. 1.

Overview of an electrically driven microdisk coupled to a single waveguide (white arrows represent losses).

Fig. 2.
Fig. 2.

Spectral properties for TE modes between 1.35 and 1.8 µm for a R=2.5 µm radius microdisk of InP (550 nm thick) with l<3 and n<2. Curve fitting is done using m>13.

Fig. 3.
Fig. 3.

Description of the model to study the vertical coupling between a WGM (effective radius Reff ) and a waveguided mode (at position xg ).

Fig. 4.
Fig. 4.

Schematic representation (top view) of the additional diffraction losses at the edge of the resonator (in red) due to the presence of the waveguide.

Fig. 5.
Fig. 5.

Comparisons between 3D FDTD (red dots) and the model (blue line) for the TE(0,45,0) mode with H=0.55 µm, Hg =0.2 µm,Wg =0.3 µm and R=4 µm.

Fig. 6.
Fig. 6.

Evolution of total and coupling loss rates for the TE(0,26,0) mode (λr =1.51 µm) of a microdisk with a slab for the bottom contact [6].

Fig. 7.
Fig. 7.

Influence of the vertical confinement when coupling to a 200×500 nm straight waveguide with a separation distance from the microdisk (R=4 µm, 550 nm height) of 100 nm, a representative value used for vertical coupling [6, 10]. When xg > Reff , losses rates are balanced using the energy density.

Fig. 8.
Fig. 8.

Overview of a ring resonator vertically coupled to a silicon waveguide.

Fig. 9.
Fig. 9.

Comparison between 2D FTDT and our model based on the CMT for different sizes of the feedback loop, and optimization for the TE(0,26) mode using the model.

Fig. 10.
Fig. 10.

Wavelength selection using a feedback loop. For figures (a), (b), and (c), the blue comb corresponds to frequencies for which ΔϕAB ≡0 and black lines represent WGMs.

Fig. 11.
Fig. 11.

Description of the structure with a ring resonator (top view).

Tables (2)

Tables Icon

Table 1. Evolution of the effective coupling loss rate, frequency and global quality factor in the waveguide assuming Q −1 c ε·Q −1 c Q −1 0

Tables Icon

Table 2. Type of interferences in the Y coupler depending on the symmetry at the coupling areas and the azimuthal order for stationary modes

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

I t = I τ c I t τ c 1 = η c · FSR ( l , n )
{ C d g = C d g · cos ( θ x g ) C g d = C g d · cos ( θ x g ) and { ( γ ) 2 = ( β d · cos ( θ x g ) β g 2 ) 2 + C d g · C g d η c = ( C d g γ ) 2 · sin 2 ( γ · L )
C i j = π · ( n j 2 n clad 2 ) λ r · n eff i m H i u i ( x ) · u j ( x ) · dx with + u i , j 2 ( x ) · dx = 1
x g Adapt = β g β d · R eff
ϕ AB = 2 · ( m · θ x g β g · R eff 2 x g 2 )
{ ( τ c ) 1 = τ c 1 · ( 1 + cos ( ϕ AB ) ) ω r = ω 0 1 2 · τ c 1 · sin ( ϕ AB ) Q r = ω r τ 0 1 + ( τ c ) 1
e m t + div ( S ) = 0 Σ S · ds = V e m t · dv = E t
( τ c ) 1 = P g ( ω r ) E ( ω r ) = P g ( ω r ) t Σ S ( ω r ) · ds · dt a g 2 ( ω r ) Σ S ( ω r ) · ds
{ ( τ c ) 1 = τ c 1 · ( 1 + ( 1 ε ) · cos ( ϕ AB ) ) ω r = ω 0 1 ε 2 · τ c 1 · sin ( ϕ AB ) ϕ AB = m · π i β i · d i

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