Abstract

We analyze modal properties of dielectric optical bent slot waveguides by using the multilayer formulation of the well-known classical analytical model of bent waveguides based on the Bessel–Hankel functions. Unlike the previously studied approximate model based on the Airy functions, this model is valid for all values of bend radii. The present approach allows quick and accurate computations of propagation constants, mode profiles, and field-power densities for the 2D bent slot waveguides with very small radii. Using this model we characterize the optimal slot position inside the bent core to maximize the field enhancement in the slot. Such modal analysis is quite useful for the design of devices involving bent slot waveguides. Moreover the results obtained by the present 2D rigorous analytical model can also be used for benchmarking other numerical tools.

© 2009 Optical Society of America

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References

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  1. V. R. Almeida, Q. Xu, C. A. Barios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29, 1209-1211 (2004).
    [CrossRef] [PubMed]
  2. P. A. Anderson, B. S. Schmidt, and M. Lipson, “High confinement in silicon slot waveguides with sharp bends,” Opt. Express 14, 9197-9202 (2006).
    [CrossRef] [PubMed]
  3. N.-N. Feng, J. Michel, and L. C. Kimerling, “Optical field concentration in low-index waveguides,” IEEE J. Quantum Electron. 42, 885-890 (2006).
    [CrossRef]
  4. C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216-219 (2009).
    [CrossRef]
  5. C.-Y. Chao, “Simple and effective calculation of modal properties of bent slot waveguides,” J. Opt. Soc. Am. B 24, 2373-2377 (2007).
    [CrossRef]
  6. J. Lu, S. He, and V. G. Romanov, “A simple and effective method for calculating the bending loss and phase enhancement of a bent planar waveguide,” Fiber Integr. Opt. 24, 25-26 (2005).
    [CrossRef]
  7. E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103-2132 (1969).
  8. L. Lewin, D. C. Chang, and E. F. Kuester, Electromagnetic Waves and Curved Structures (Peter Peregrinus, 1977).
  9. S. Kawakami, M. Miyagi, and S. Nishida, “Bending losses of dielectric slab optical waveguide with double or multiple claddings: theory,” Appl. Opt. 14, 2588-2597 (1975).
    [CrossRef] [PubMed]
  10. E. C. M. Pennings, “Bends in optical ridge waveguides, modelling and experiment,” Ph.D. thesis (Delft University, The Netherlands, 1990).
  11. K. R. Hiremath, M. Hammer, S. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37-61 (2005).
    [CrossRef]
  12. J. S. Gu, P. A. Besse, and H. Melchior, “Method of lines for the analysis of the propagation characteristics of curved optical rib waveguides,” IEEE J. Quantum Electron. 27, 531-537 (1991).
    [CrossRef]
  13. W. Pascher and R. Pregla, “Vectorial analysis of bends in optical strip waveguides by the method of lines,” Radio Sci. 28, 1229-1233 (1993).
    [CrossRef]
  14. L. Prkna, M. Hubálek, and J. Čtyroký, “Vectorial eigenmode solver for bent waveguides based on mode matching,” IEEE Photon. Technol. Lett. 16, 2057-2059 (2004).
    [CrossRef]
  15. L. Prkna, M. Hubálek, and J. Čtyroký, “Field modeling of circular microresonators by film mode matching,” IEEE J. Sel. Top. Quantum Electron. 11, 217-223 (2005).
    [CrossRef]
  16. W. Pascher, “Modelling of rib waveguide bends for sensor applications,” Opt. Quantum Electron. 33, 433-449 (2001).
    [CrossRef]
  17. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D. C., 1964).
  18. D. E. Amos, “A portable package for Bessel functions of a complex argument and nonnegative order,” (1983). Http://www.netlib.org/amos/.
  19. T.Tamir (editor), Integrated Optics, Vol. 7 of Topics in Applied Physics (Springer-Verlag, 1982).
  20. T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q optical resonators in silicon-on-insulator-based slot waveguides,” Appl. Phys. Lett. 86, 081101:1-3 (2005).
    [CrossRef]
  21. K. R. Hiremath, R. Stoffer, and M. Hammer, “Modeling of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Commun. 257, 277-297 (2006).
    [CrossRef]

2009 (1)

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216-219 (2009).
[CrossRef]

2007 (1)

2006 (3)

P. A. Anderson, B. S. Schmidt, and M. Lipson, “High confinement in silicon slot waveguides with sharp bends,” Opt. Express 14, 9197-9202 (2006).
[CrossRef] [PubMed]

N.-N. Feng, J. Michel, and L. C. Kimerling, “Optical field concentration in low-index waveguides,” IEEE J. Quantum Electron. 42, 885-890 (2006).
[CrossRef]

K. R. Hiremath, R. Stoffer, and M. Hammer, “Modeling of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Commun. 257, 277-297 (2006).
[CrossRef]

2005 (4)

T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q optical resonators in silicon-on-insulator-based slot waveguides,” Appl. Phys. Lett. 86, 081101:1-3 (2005).
[CrossRef]

L. Prkna, M. Hubálek, and J. Čtyroký, “Field modeling of circular microresonators by film mode matching,” IEEE J. Sel. Top. Quantum Electron. 11, 217-223 (2005).
[CrossRef]

K. R. Hiremath, M. Hammer, S. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37-61 (2005).
[CrossRef]

J. Lu, S. He, and V. G. Romanov, “A simple and effective method for calculating the bending loss and phase enhancement of a bent planar waveguide,” Fiber Integr. Opt. 24, 25-26 (2005).
[CrossRef]

2004 (2)

V. R. Almeida, Q. Xu, C. A. Barios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29, 1209-1211 (2004).
[CrossRef] [PubMed]

L. Prkna, M. Hubálek, and J. Čtyroký, “Vectorial eigenmode solver for bent waveguides based on mode matching,” IEEE Photon. Technol. Lett. 16, 2057-2059 (2004).
[CrossRef]

2001 (1)

W. Pascher, “Modelling of rib waveguide bends for sensor applications,” Opt. Quantum Electron. 33, 433-449 (2001).
[CrossRef]

1993 (1)

W. Pascher and R. Pregla, “Vectorial analysis of bends in optical strip waveguides by the method of lines,” Radio Sci. 28, 1229-1233 (1993).
[CrossRef]

1991 (1)

J. S. Gu, P. A. Besse, and H. Melchior, “Method of lines for the analysis of the propagation characteristics of curved optical rib waveguides,” IEEE J. Quantum Electron. 27, 531-537 (1991).
[CrossRef]

1975 (1)

1969 (1)

E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103-2132 (1969).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D. C., 1964).

Almeida, V. R.

Amos, D. E.

D. E. Amos, “A portable package for Bessel functions of a complex argument and nonnegative order,” (1983). Http://www.netlib.org/amos/.

Anderson, P. A.

Baehr-Jones, T.

T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q optical resonators in silicon-on-insulator-based slot waveguides,” Appl. Phys. Lett. 86, 081101:1-3 (2005).
[CrossRef]

Baets, R.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216-219 (2009).
[CrossRef]

Barios, C. A.

Besse, P. A.

J. S. Gu, P. A. Besse, and H. Melchior, “Method of lines for the analysis of the propagation characteristics of curved optical rib waveguides,” IEEE J. Quantum Electron. 27, 531-537 (1991).
[CrossRef]

Biaggio, I.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216-219 (2009).
[CrossRef]

Bogaerts, W.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216-219 (2009).
[CrossRef]

Chang, D. C.

L. Lewin, D. C. Chang, and E. F. Kuester, Electromagnetic Waves and Curved Structures (Peter Peregrinus, 1977).

Chao, C.-Y.

Ctyroký, J.

K. R. Hiremath, M. Hammer, S. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37-61 (2005).
[CrossRef]

L. Prkna, M. Hubálek, and J. Čtyroký, “Field modeling of circular microresonators by film mode matching,” IEEE J. Sel. Top. Quantum Electron. 11, 217-223 (2005).
[CrossRef]

L. Prkna, M. Hubálek, and J. Čtyroký, “Vectorial eigenmode solver for bent waveguides based on mode matching,” IEEE Photon. Technol. Lett. 16, 2057-2059 (2004).
[CrossRef]

Diederich, F.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216-219 (2009).
[CrossRef]

Dumon, P.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216-219 (2009).
[CrossRef]

Esembeson, B.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216-219 (2009).
[CrossRef]

Feng, N.-N.

N.-N. Feng, J. Michel, and L. C. Kimerling, “Optical field concentration in low-index waveguides,” IEEE J. Quantum Electron. 42, 885-890 (2006).
[CrossRef]

Freude, W.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216-219 (2009).
[CrossRef]

Gu, J. S.

J. S. Gu, P. A. Besse, and H. Melchior, “Method of lines for the analysis of the propagation characteristics of curved optical rib waveguides,” IEEE J. Quantum Electron. 27, 531-537 (1991).
[CrossRef]

Hammer, M.

K. R. Hiremath, R. Stoffer, and M. Hammer, “Modeling of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Commun. 257, 277-297 (2006).
[CrossRef]

K. R. Hiremath, M. Hammer, S. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37-61 (2005).
[CrossRef]

He, S.

J. Lu, S. He, and V. G. Romanov, “A simple and effective method for calculating the bending loss and phase enhancement of a bent planar waveguide,” Fiber Integr. Opt. 24, 25-26 (2005).
[CrossRef]

Hiremath, K. R.

K. R. Hiremath, R. Stoffer, and M. Hammer, “Modeling of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Commun. 257, 277-297 (2006).
[CrossRef]

K. R. Hiremath, M. Hammer, S. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37-61 (2005).
[CrossRef]

Hochberg, M.

T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q optical resonators in silicon-on-insulator-based slot waveguides,” Appl. Phys. Lett. 86, 081101:1-3 (2005).
[CrossRef]

Hubálek, M.

L. Prkna, M. Hubálek, and J. Čtyroký, “Field modeling of circular microresonators by film mode matching,” IEEE J. Sel. Top. Quantum Electron. 11, 217-223 (2005).
[CrossRef]

L. Prkna, M. Hubálek, and J. Čtyroký, “Vectorial eigenmode solver for bent waveguides based on mode matching,” IEEE Photon. Technol. Lett. 16, 2057-2059 (2004).
[CrossRef]

Kawakami, S.

Kimerling, L. C.

N.-N. Feng, J. Michel, and L. C. Kimerling, “Optical field concentration in low-index waveguides,” IEEE J. Quantum Electron. 42, 885-890 (2006).
[CrossRef]

Koos, C.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216-219 (2009).
[CrossRef]

Kuester, E. F.

L. Lewin, D. C. Chang, and E. F. Kuester, Electromagnetic Waves and Curved Structures (Peter Peregrinus, 1977).

Leuthold, J.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216-219 (2009).
[CrossRef]

Lewin, L.

L. Lewin, D. C. Chang, and E. F. Kuester, Electromagnetic Waves and Curved Structures (Peter Peregrinus, 1977).

Lipson, M.

Lu, J.

J. Lu, S. He, and V. G. Romanov, “A simple and effective method for calculating the bending loss and phase enhancement of a bent planar waveguide,” Fiber Integr. Opt. 24, 25-26 (2005).
[CrossRef]

Marcatili, E. A. J.

E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103-2132 (1969).

Melchior, H.

J. S. Gu, P. A. Besse, and H. Melchior, “Method of lines for the analysis of the propagation characteristics of curved optical rib waveguides,” IEEE J. Quantum Electron. 27, 531-537 (1991).
[CrossRef]

Michel, J.

N.-N. Feng, J. Michel, and L. C. Kimerling, “Optical field concentration in low-index waveguides,” IEEE J. Quantum Electron. 42, 885-890 (2006).
[CrossRef]

Michinobu, T.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216-219 (2009).
[CrossRef]

Miyagi, M.

Nishida, S.

Pascher, W.

W. Pascher, “Modelling of rib waveguide bends for sensor applications,” Opt. Quantum Electron. 33, 433-449 (2001).
[CrossRef]

W. Pascher and R. Pregla, “Vectorial analysis of bends in optical strip waveguides by the method of lines,” Radio Sci. 28, 1229-1233 (1993).
[CrossRef]

Pennings, E. C. M.

E. C. M. Pennings, “Bends in optical ridge waveguides, modelling and experiment,” Ph.D. thesis (Delft University, The Netherlands, 1990).

Pregla, R.

W. Pascher and R. Pregla, “Vectorial analysis of bends in optical strip waveguides by the method of lines,” Radio Sci. 28, 1229-1233 (1993).
[CrossRef]

Prkna, L.

K. R. Hiremath, M. Hammer, S. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37-61 (2005).
[CrossRef]

L. Prkna, M. Hubálek, and J. Čtyroký, “Field modeling of circular microresonators by film mode matching,” IEEE J. Sel. Top. Quantum Electron. 11, 217-223 (2005).
[CrossRef]

L. Prkna, M. Hubálek, and J. Čtyroký, “Vectorial eigenmode solver for bent waveguides based on mode matching,” IEEE Photon. Technol. Lett. 16, 2057-2059 (2004).
[CrossRef]

Romanov, V. G.

J. Lu, S. He, and V. G. Romanov, “A simple and effective method for calculating the bending loss and phase enhancement of a bent planar waveguide,” Fiber Integr. Opt. 24, 25-26 (2005).
[CrossRef]

Scherer, A.

T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q optical resonators in silicon-on-insulator-based slot waveguides,” Appl. Phys. Lett. 86, 081101:1-3 (2005).
[CrossRef]

Schmidt, B. S.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D. C., 1964).

Stoffer, R.

K. R. Hiremath, R. Stoffer, and M. Hammer, “Modeling of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Commun. 257, 277-297 (2006).
[CrossRef]

Stoffer, S.

K. R. Hiremath, M. Hammer, S. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37-61 (2005).
[CrossRef]

Vallaitis, T.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216-219 (2009).
[CrossRef]

Vorreau, P.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216-219 (2009).
[CrossRef]

Walker, C.

T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q optical resonators in silicon-on-insulator-based slot waveguides,” Appl. Phys. Lett. 86, 081101:1-3 (2005).
[CrossRef]

Xu, Q.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q optical resonators in silicon-on-insulator-based slot waveguides,” Appl. Phys. Lett. 86, 081101:1-3 (2005).
[CrossRef]

Bell Syst. Tech. J. (1)

E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103-2132 (1969).

Fiber Integr. Opt. (1)

J. Lu, S. He, and V. G. Romanov, “A simple and effective method for calculating the bending loss and phase enhancement of a bent planar waveguide,” Fiber Integr. Opt. 24, 25-26 (2005).
[CrossRef]

IEEE J. Quantum Electron. (2)

N.-N. Feng, J. Michel, and L. C. Kimerling, “Optical field concentration in low-index waveguides,” IEEE J. Quantum Electron. 42, 885-890 (2006).
[CrossRef]

J. S. Gu, P. A. Besse, and H. Melchior, “Method of lines for the analysis of the propagation characteristics of curved optical rib waveguides,” IEEE J. Quantum Electron. 27, 531-537 (1991).
[CrossRef]

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

L. Prkna, M. Hubálek, and J. Čtyroký, “Field modeling of circular microresonators by film mode matching,” IEEE J. Sel. Top. Quantum Electron. 11, 217-223 (2005).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

L. Prkna, M. Hubálek, and J. Čtyroký, “Vectorial eigenmode solver for bent waveguides based on mode matching,” IEEE Photon. Technol. Lett. 16, 2057-2059 (2004).
[CrossRef]

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

Nat. Photonics (1)

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216-219 (2009).
[CrossRef]

Opt. Commun. (1)

K. R. Hiremath, R. Stoffer, and M. Hammer, “Modeling of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Commun. 257, 277-297 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Opt. Quantum Electron. (2)

W. Pascher, “Modelling of rib waveguide bends for sensor applications,” Opt. Quantum Electron. 33, 433-449 (2001).
[CrossRef]

K. R. Hiremath, M. Hammer, S. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37-61 (2005).
[CrossRef]

Radio Sci. (1)

W. Pascher and R. Pregla, “Vectorial analysis of bends in optical strip waveguides by the method of lines,” Radio Sci. 28, 1229-1233 (1993).
[CrossRef]

Other (5)

E. C. M. Pennings, “Bends in optical ridge waveguides, modelling and experiment,” Ph.D. thesis (Delft University, The Netherlands, 1990).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D. C., 1964).

D. E. Amos, “A portable package for Bessel functions of a complex argument and nonnegative order,” (1983). Http://www.netlib.org/amos/.

T.Tamir (editor), Integrated Optics, Vol. 7 of Topics in Applied Physics (Springer-Verlag, 1982).

L. Lewin, D. C. Chang, and E. F. Kuester, Electromagnetic Waves and Curved Structures (Peter Peregrinus, 1977).

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Figures (6)

Fig. 1
Fig. 1

Bent slot waveguide: A slot of low refractive index n l and width g is formed between two regions of high refractive index n h and widths η w and ( 1 η ) w , respectively, where w is the total width of the regions with the high refractive index, and 0 η 1 is the asymmetry parameter ( η = 0.5 corresponds to the symmetrical waveguide setting). The total width of the waveguide core is w tot = w + g . The bend radius R is defined as the outermost material interface of the core.

Fig. 2
Fig. 2

Effect of inclusion of the slot on modal solutions H ̃ y (first row) and E ̃ r (second row). A slot of width 25 nm (second column), 50 nm (third column), and 100 nm (forth column) is introduced in a three-layer bent waveguide (first column) made up of a core of refractive index n h = 2.914406 and width w tot = 0.475 μ m , embedded in the surrounding medium of refractive index n l = 1 , R = 2 μ m . λ = 1.55 μ m , and the slot position is given by η = 0.5 . N eff of the modes are given in Table 2. The field profiles are power normalized.

Fig. 3
Fig. 3

Effect of the bend radius on the effective index of a bent slot waveguide. The bent slot waveguide configuration is the same as that of Fig. 2.

Fig. 4
Fig. 4

Effect of the slot position on the real part (left) and and the imaginary part (center) of the effective index of the bent slot waveguide with R = 2.725 μ m . n h = 2.914406 , n l = 1 , w = 0.4 μ m , λ = 1.55 μ m . For a fixed slot width g = 50 nm , the right-most plot shows influence of the bend radius on the symmetry of the curves of real part of N eff . R = corresponds to the straight waveguide.

Fig. 5
Fig. 5

Effect of position of the slot. From left to right, the plots show the modal solution for H ̃ y (first row) and for E ̃ r (second row) with no slot and with slot positioned for η = 0.3 , 0.5 , 0.7 , respectively. The effective indices for these modal solutions are 2.25671 i 2.81308 × 10 14 , 1.74200 i 2.64315 × 10 7 , 1.53877 i 2.28484 × 10 5 , 1.69136 i 1.90260 × 10 7 respectively. The waveguide configuration is n h = 2.914406 , n l = 1 , w tot = 0.45 μ m , g = 50 nm , R = 2.725 , λ = 1.55 μ m .

Fig. 6
Fig. 6

Effect of the slot position on the power in the slot (left) and the slot power density (right). Bent slot waveguide is as in Fig. 5.

Tables (2)

Tables Icon

Table 1 Comparison of Simulation Results for the Effective Index N eff = γ k of TM 0 Mode of the Slot Waveguide in Section 3 a

Tables Icon

Table 2 Effect of the Slot Width g on the Effective Index a

Equations (2)

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( E H ) ( r , θ , t ) = ( ( E ̃ r , E ̃ y , E ̃ θ ) ( H ̃ r , H ̃ y , H ̃ θ ) ) ( r ) exp ( i ( ω t γ R θ ) ) ,
ϕ ( r ) = { A 0 J γ R ( n l k r ) , if 0 r R w g , A 1 J γ R ( n h k r ) + B 1 Y γ R ( n h k r ) , if R w g r R ( 1 η ) w g , A 2 J γ R ( n l k r ) + B 2 Y γ R ( n l k r ) , if R ( 1 η ) w g r R ( 1 η ) w , A 3 J γ R ( n h k r ) + B 3 Y γ R ( n h k r ) , if R ( 1 η ) w r R , A 4 H γ R ( 2 ) ( n l k r ) , for r R , }

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