Abstract

A novel method for computation of guided mode reflectivity from optical waveguide end-facet is presented. The method is based on the characteristic Green’s function (CGF) technique formulation combined with the complex images (CIs) method for dielectric planar waveguides. By separability assumption of the structure, a uniform and closed-form expression of spatial Green’s function is obtained. Derived expression consists of discrete and continuous spectrum contributions which denote guided and radiation modes effects, respectively. Having a full-wave solution, efficient optimization procedure is then used to calculate the exact reflection coefficients of guided modes at the end-facets. To find mode coupling coefficients, CGF-CI formulation is modified to incorporate coupling matrix of abruptly terminated waveguides. The main advantage of this method lies in its simple implementation as well as accuracy for any refractive index contrast. Excellent agreements with rigorous methods are illustrated in several examples.

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  1. B.-K. Kang, Y. H. Park, S. Lee, S. S. Choi, J. Lee, T. Kamiya, S.-H. Park, "Measurement of facet reflectivity of antireflection-coated electroabsorption modulator using induced photocurrent," IEEE Photon. Technol. Lett. 13, 112-114 (2001).
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  7. P. Gelin, M. Petenzi, J. Citerne, "Rigorous analysis of the scattering of surface waves in an abruptly ended slab dielectric waveguide," IEEE Trans. Microw. Theory Techn. MTT-29, 107-114 (1981).
  8. T. Rozzi, G. in’t Veld, "Variational treatment of the diffraction at the facet of d.h. lasers and of dielectric millimeter wave antennas," IEEE Trans. Microw. Theory Techn. MTT-28, 61-73 (1980).
  9. C. Herzinger, C. Lu, T. DeTemple, W. Chew, "The semiconductor waveguide facet reflectivity problem ," IEEE J. Quantum Electron. 29, 2273-2281 (1993).
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  14. H. Derudder, F. Olyslager, D. De Zutter, S. van den Berghe, "Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers," IEEE Trans. Antennas Propag. 49, 185- 195 (2001).
  15. M. Akbari, H. Burkhard, K. Schünemann, "Rigorous analysis of the reflection/transmission at the end-facets of DFB laser diodes," Opt. Quantum Electron. 31, 893- 907 (1999).
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  17. S. H. Wei, Y. Y. Lu, "Application of Bi-CGSTAB to waveguide discontinuity problems," IEEE Photon. Technol. Lett. 14, 645-647 (2002).
  18. D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, "Lanczos reduction techniques for electric field reflection ," J. Lightw. Technol. 10, 1234-1237 (1992).
  19. H. El-Refaei, I. Betty, D. Yevick, "The application of complex Pade approximants to reflection at optical waveguide facets," IEEE Photon. Technol. Lett. 12, 158-160 (2000).
  20. H. Jamid, M. Khan, "3-D full-vectorial analysis of strong optical waveguide discontinuities using Pade approximants ," IEEE J. Quantum Electron. 43, 343-349 (2007).
  21. C. Yu, D. Yevick, "Application of the bidirectional parabolic equation method to optical waveguide facets ," J. Opt. Soc. Amer. A 14, 1448-1450 (1997).
  22. H. Jamid, M. Z. M. Khan, "A noniterative formulation for 2-D optical waveguide discontinuity problems based on Pade approximants," IEEE Photon. Technol. Lett. 20, 469-471 (2008 ).
  23. N.-N. Feng, W.-P. Huang, "A simple noniterative solution scheme for facet reflectivity of optical waveguides," J. Lightw. Technol. 22, 664 -668 (2004).
  24. A. Parsa, R. Paknys, "Interior Green’s function solution for a thick and finite dielectric slab," IEEE Trans. Antennas Propag. 55, 3504-3514 (2007).
  25. H. Zhang, Q. Guo, W.-P. Huang, "Analysis of waveguide discontinuities by a fourth-order finite-difference reflective scheme," J. Lightw. Technol. 25, 556-561 (2007).
  26. M. Vahidpour, A. A. Shishegar, "Reflection analysis of the end-facet dielectric slab waveguide by FDTD method ," Proc. 3rd Int. Conf. Comput. Electromagn. Appl. (2004) pp. 453-456.
  27. S. Maci, L. Borselli, L. Rossi, "Diffraction at the edge of a truncated grounded dielectric slab," IEEE Trans. Antenn. Propag. 44, 863-873 (1996).
  28. V. Volski, G. Vandenbosch, "Diffraction of a surface wave at the truncation of a dielectric structure ," IEEE Trans. Antennas Propag. 50, 1779-1785 (2002).
  29. S. S. A. Obayya, "Novel finite element analysis of optical waveguide discontinuity problems," J. Lightw. Technol. 22, 1420-1425 (2004 ).
  30. R. Faraji-Dana, “An efficient and accurate Green’s function analysis of packaged microwave integrated circuits,” Ph.D. dissertation, Dept. Elect. Eng., Univ. Waterloo, Waterloo, ON, Canada, 1993..
  31. A. A. Shishegar, R. Faraji-Dana, "A closed-form spatial Green’s function for finite dielectric structures ," Electromagnetics 23, 579-594 (2003).
  32. K. Michalski, J. Mosig, "Multilayered media Green’s functions in integral equation formulations ," IEEE Trans. Antenn. Propag. 45, 508-519 (1997).
  33. Y. Hua, T. Sarkar, "Generalized pencil-of-function method for extracting poles of an EM system from its transient response," IEEE Trans. Antenn. Propag. 37, 229-234 (1989 ).
  34. A. A. Shishegar, R. Faraji-Dana, S. Safavi-Naeini, " Finite substrate microstrip transmission line analysis using the characteristic Green’s function-complex images technique," Proc. 32nd Eur. Microw. Conf. (2002) pp. 1-4.
  35. T. F. Coleman, Y. Li, "An interior trust region approach for nonlinear minimization subject to bounds," SIAM J. Optim. 6, 418-445 (1996).
  36. R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. Philadelphia, PA, USA: SIAM, 1987, no. 43..
  37. E. A. Marcatili, "Dielectric rectangular waveguide and directional coupler for integrated optics," Bell Syst. Tech. J. 48, 2071-2102 (1969).
  38. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic , 1991).
  39. R. Zia, A. Chandran, M. L. Brongersma, "Dielectric waveguide model for guided surface polaritons," Opt. Lett. 30, 1473-1475 (2005).
  40. R. D. Kekatpure, A. C. Hryciw, E. S. Barnard, and M. L. Brongersma, “Solving dielectric and plasmonic waveguide dispersion relations on a pocket calculator,” Opt. Exp., vol. 17, no. 26, pp. 24 112–24 129, Dec. 2009. .
  41. S. Al-Bader, "Optical transmission on metallic wires-fundamental modes," IEEE J. Quantum Electron. 40, 325-329 (2004).

2012 (1)

Y. Chen, Y. Lai, T. C. Chong, S.-T. Ho, "Exact solution of facet reflections for guided modes in high-refractive-index-contrast sub-wavelength waveguide via a fourier analysis and perturbative series summation: Derivation and applications," J. Lightw. Technol. 30, 2455-2471 (2012).

2007 (3)

H. Jamid, M. Khan, "3-D full-vectorial analysis of strong optical waveguide discontinuities using Pade approximants ," IEEE J. Quantum Electron. 43, 343-349 (2007).

A. Parsa, R. Paknys, "Interior Green’s function solution for a thick and finite dielectric slab," IEEE Trans. Antennas Propag. 55, 3504-3514 (2007).

H. Zhang, Q. Guo, W.-P. Huang, "Analysis of waveguide discontinuities by a fourth-order finite-difference reflective scheme," J. Lightw. Technol. 25, 556-561 (2007).

2005 (1)

2004 (2)

S. Al-Bader, "Optical transmission on metallic wires-fundamental modes," IEEE J. Quantum Electron. 40, 325-329 (2004).

N.-N. Feng, W.-P. Huang, "A simple noniterative solution scheme for facet reflectivity of optical waveguides," J. Lightw. Technol. 22, 664 -668 (2004).

2003 (1)

A. A. Shishegar, R. Faraji-Dana, "A closed-form spatial Green’s function for finite dielectric structures ," Electromagnetics 23, 579-594 (2003).

2002 (2)

V. Volski, G. Vandenbosch, "Diffraction of a surface wave at the truncation of a dielectric structure ," IEEE Trans. Antennas Propag. 50, 1779-1785 (2002).

S. H. Wei, Y. Y. Lu, "Application of Bi-CGSTAB to waveguide discontinuity problems," IEEE Photon. Technol. Lett. 14, 645-647 (2002).

2001 (2)

H. Derudder, F. Olyslager, D. De Zutter, S. van den Berghe, "Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers," IEEE Trans. Antennas Propag. 49, 185- 195 (2001).

B.-K. Kang, Y. H. Park, S. Lee, S. S. Choi, J. Lee, T. Kamiya, S.-H. Park, "Measurement of facet reflectivity of antireflection-coated electroabsorption modulator using induced photocurrent," IEEE Photon. Technol. Lett. 13, 112-114 (2001).

2000 (1)

H. El-Refaei, I. Betty, D. Yevick, "The application of complex Pade approximants to reflection at optical waveguide facets," IEEE Photon. Technol. Lett. 12, 158-160 (2000).

1999 (1)

M. Akbari, H. Burkhard, K. Schünemann, "Rigorous analysis of the reflection/transmission at the end-facets of DFB laser diodes," Opt. Quantum Electron. 31, 893- 907 (1999).

1997 (2)

K. Michalski, J. Mosig, "Multilayered media Green’s functions in integral equation formulations ," IEEE Trans. Antenn. Propag. 45, 508-519 (1997).

C. Yu, D. Yevick, "Application of the bidirectional parabolic equation method to optical waveguide facets ," J. Opt. Soc. Amer. A 14, 1448-1450 (1997).

1996 (2)

S. Maci, L. Borselli, L. Rossi, "Diffraction at the edge of a truncated grounded dielectric slab," IEEE Trans. Antenn. Propag. 44, 863-873 (1996).

T. F. Coleman, Y. Li, "An interior trust region approach for nonlinear minimization subject to bounds," SIAM J. Optim. 6, 418-445 (1996).

1993 (2)

C. Herzinger, C. Lu, T. DeTemple, W. Chew, "The semiconductor waveguide facet reflectivity problem ," IEEE J. Quantum Electron. 29, 2273-2281 (1993).

P. Kendall, D. A. Roberts, P. Robson, M. Adams, M. Robertson, "New formula for semiconductor laser facet reflectivity," IEEE Photon. Technol. Lett. 5, 148-150 (1993).

1992 (1)

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, "Lanczos reduction techniques for electric field reflection ," J. Lightw. Technol. 10, 1234-1237 (1992).

1991 (1)

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, "Split-operator electric field reflection techniques ," IEEE Photon. Technol. Lett. 3, 527-529 (1991).

1986 (1)

D. Handelman, A. Hardy, A. Katzir, "Reflectivity of TE modes at the facets of buried heterostructure injection lasers," IEEE J. Quantum Electron. QE-22, 498-500 (1986).

1985 (1)

C. Vassallo, "Rigorous and approximate calculations of antireflection layer parameters for travelling-wave diode laser amplifiers," Electron. Lett. 21, 333-334 (1985).

1981 (1)

P. Gelin, M. Petenzi, J. Citerne, "Rigorous analysis of the scattering of surface waves in an abruptly ended slab dielectric waveguide," IEEE Trans. Microw. Theory Techn. MTT-29, 107-114 (1981).

1980 (1)

T. Rozzi, G. in’t Veld, "Variational treatment of the diffraction at the facet of d.h. lasers and of dielectric millimeter wave antennas," IEEE Trans. Microw. Theory Techn. MTT-28, 61-73 (1980).

1979 (1)

K. Morishita, S. Inagaki, N. Kumagai, "Analysis of discontinuities in dielectric waveguides by means of the least squares boundary residual method," IEEE Trans. Microw. Theory Techn. MTT-27, 310-315 (1979).

1975 (1)

L. Lewin, "A method for the calculation of the radiation-pattern and mode-conversion properties of a solid-state heterojunction laser," IEEE Trans. Microw. Theory Techn. MTT-23, 576- 585 (1975).

1972 (1)

T. Ikegami, "Reflectivity of mode at facet and oscillation mode in double-heterostructure injection lasers," IEEE J. Quantum Electron. QE-8, 470-476 (1972).

1971 (1)

F. K. Reinhart, I. Hayashi, M. Panish, "Mode reflectivity and waveguide properties of double-heterostructure injection lasers," J. Appl. Phys. 42, 4466-4479 (1971).

1969 (1)

E. A. Marcatili, "Dielectric rectangular waveguide and directional coupler for integrated optics," Bell Syst. Tech. J. 48, 2071-2102 (1969).

Bell Syst. Tech. J. (1)

E. A. Marcatili, "Dielectric rectangular waveguide and directional coupler for integrated optics," Bell Syst. Tech. J. 48, 2071-2102 (1969).

Electromagnetics (1)

A. A. Shishegar, R. Faraji-Dana, "A closed-form spatial Green’s function for finite dielectric structures ," Electromagnetics 23, 579-594 (2003).

Electron. Lett. (1)

C. Vassallo, "Rigorous and approximate calculations of antireflection layer parameters for travelling-wave diode laser amplifiers," Electron. Lett. 21, 333-334 (1985).

IEEE Photon. Technol. Lett. (1)

S. H. Wei, Y. Y. Lu, "Application of Bi-CGSTAB to waveguide discontinuity problems," IEEE Photon. Technol. Lett. 14, 645-647 (2002).

IEEE J. Quantum Electron. (1)

D. Handelman, A. Hardy, A. Katzir, "Reflectivity of TE modes at the facets of buried heterostructure injection lasers," IEEE J. Quantum Electron. QE-22, 498-500 (1986).

IEEE J. Quantum Electron. (4)

C. Herzinger, C. Lu, T. DeTemple, W. Chew, "The semiconductor waveguide facet reflectivity problem ," IEEE J. Quantum Electron. 29, 2273-2281 (1993).

T. Ikegami, "Reflectivity of mode at facet and oscillation mode in double-heterostructure injection lasers," IEEE J. Quantum Electron. QE-8, 470-476 (1972).

H. Jamid, M. Khan, "3-D full-vectorial analysis of strong optical waveguide discontinuities using Pade approximants ," IEEE J. Quantum Electron. 43, 343-349 (2007).

S. Al-Bader, "Optical transmission on metallic wires-fundamental modes," IEEE J. Quantum Electron. 40, 325-329 (2004).

IEEE Photon. Technol. Lett. (2)

H. El-Refaei, I. Betty, D. Yevick, "The application of complex Pade approximants to reflection at optical waveguide facets," IEEE Photon. Technol. Lett. 12, 158-160 (2000).

P. Kendall, D. A. Roberts, P. Robson, M. Adams, M. Robertson, "New formula for semiconductor laser facet reflectivity," IEEE Photon. Technol. Lett. 5, 148-150 (1993).

IEEE Photon. Technol. Lett. (3)

B.-K. Kang, Y. H. Park, S. Lee, S. S. Choi, J. Lee, T. Kamiya, S.-H. Park, "Measurement of facet reflectivity of antireflection-coated electroabsorption modulator using induced photocurrent," IEEE Photon. Technol. Lett. 13, 112-114 (2001).

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, "Split-operator electric field reflection techniques ," IEEE Photon. Technol. Lett. 3, 527-529 (1991).

H. Jamid, M. Z. M. Khan, "A noniterative formulation for 2-D optical waveguide discontinuity problems based on Pade approximants," IEEE Photon. Technol. Lett. 20, 469-471 (2008 ).

IEEE Trans. Antennas Propag. (1)

H. Derudder, F. Olyslager, D. De Zutter, S. van den Berghe, "Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers," IEEE Trans. Antennas Propag. 49, 185- 195 (2001).

IEEE Trans. Microw. Theory Techn. (1)

P. Gelin, M. Petenzi, J. Citerne, "Rigorous analysis of the scattering of surface waves in an abruptly ended slab dielectric waveguide," IEEE Trans. Microw. Theory Techn. MTT-29, 107-114 (1981).

IEEE Trans. Antenn. Propag. (3)

S. Maci, L. Borselli, L. Rossi, "Diffraction at the edge of a truncated grounded dielectric slab," IEEE Trans. Antenn. Propag. 44, 863-873 (1996).

K. Michalski, J. Mosig, "Multilayered media Green’s functions in integral equation formulations ," IEEE Trans. Antenn. Propag. 45, 508-519 (1997).

Y. Hua, T. Sarkar, "Generalized pencil-of-function method for extracting poles of an EM system from its transient response," IEEE Trans. Antenn. Propag. 37, 229-234 (1989 ).

IEEE Trans. Antennas Propag. (2)

A. Parsa, R. Paknys, "Interior Green’s function solution for a thick and finite dielectric slab," IEEE Trans. Antennas Propag. 55, 3504-3514 (2007).

V. Volski, G. Vandenbosch, "Diffraction of a surface wave at the truncation of a dielectric structure ," IEEE Trans. Antennas Propag. 50, 1779-1785 (2002).

IEEE Trans. Microw. Theory Techn. (1)

L. Lewin, "A method for the calculation of the radiation-pattern and mode-conversion properties of a solid-state heterojunction laser," IEEE Trans. Microw. Theory Techn. MTT-23, 576- 585 (1975).

IEEE Trans. Microw. Theory Techn. (2)

T. Rozzi, G. in’t Veld, "Variational treatment of the diffraction at the facet of d.h. lasers and of dielectric millimeter wave antennas," IEEE Trans. Microw. Theory Techn. MTT-28, 61-73 (1980).

K. Morishita, S. Inagaki, N. Kumagai, "Analysis of discontinuities in dielectric waveguides by means of the least squares boundary residual method," IEEE Trans. Microw. Theory Techn. MTT-27, 310-315 (1979).

J. Appl. Phys. (1)

F. K. Reinhart, I. Hayashi, M. Panish, "Mode reflectivity and waveguide properties of double-heterostructure injection lasers," J. Appl. Phys. 42, 4466-4479 (1971).

J. Lightw. Technol. (1)

H. Zhang, Q. Guo, W.-P. Huang, "Analysis of waveguide discontinuities by a fourth-order finite-difference reflective scheme," J. Lightw. Technol. 25, 556-561 (2007).

J. Lightw. Technol. (4)

S. S. A. Obayya, "Novel finite element analysis of optical waveguide discontinuity problems," J. Lightw. Technol. 22, 1420-1425 (2004 ).

N.-N. Feng, W.-P. Huang, "A simple noniterative solution scheme for facet reflectivity of optical waveguides," J. Lightw. Technol. 22, 664 -668 (2004).

Y. Chen, Y. Lai, T. C. Chong, S.-T. Ho, "Exact solution of facet reflections for guided modes in high-refractive-index-contrast sub-wavelength waveguide via a fourier analysis and perturbative series summation: Derivation and applications," J. Lightw. Technol. 30, 2455-2471 (2012).

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, "Lanczos reduction techniques for electric field reflection ," J. Lightw. Technol. 10, 1234-1237 (1992).

J. Opt. Soc. Amer. A (1)

C. Yu, D. Yevick, "Application of the bidirectional parabolic equation method to optical waveguide facets ," J. Opt. Soc. Amer. A 14, 1448-1450 (1997).

J. Opt. Soc. Amer. A. (1)

A. Hardy, "Formulation of two-dimensional reflectivity calculations for transverse-magnetic-like modes," J. Opt. Soc. Amer. A. 1 , 760-763 (1984 ).

Opt. Lett. (1)

Opt. Quantum Electron. (1)

M. Akbari, H. Burkhard, K. Schünemann, "Rigorous analysis of the reflection/transmission at the end-facets of DFB laser diodes," Opt. Quantum Electron. 31, 893- 907 (1999).

SIAM J. Optim. (1)

T. F. Coleman, Y. Li, "An interior trust region approach for nonlinear minimization subject to bounds," SIAM J. Optim. 6, 418-445 (1996).

Other (6)

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. Philadelphia, PA, USA: SIAM, 1987, no. 43..

M. Vahidpour, A. A. Shishegar, "Reflection analysis of the end-facet dielectric slab waveguide by FDTD method ," Proc. 3rd Int. Conf. Comput. Electromagn. Appl. (2004) pp. 453-456.

A. A. Shishegar, R. Faraji-Dana, S. Safavi-Naeini, " Finite substrate microstrip transmission line analysis using the characteristic Green’s function-complex images technique," Proc. 32nd Eur. Microw. Conf. (2002) pp. 1-4.

R. Faraji-Dana, “An efficient and accurate Green’s function analysis of packaged microwave integrated circuits,” Ph.D. dissertation, Dept. Elect. Eng., Univ. Waterloo, Waterloo, ON, Canada, 1993..

R. D. Kekatpure, A. C. Hryciw, E. S. Barnard, and M. L. Brongersma, “Solving dielectric and plasmonic waveguide dispersion relations on a pocket calculator,” Opt. Exp., vol. 17, no. 26, pp. 24 112–24 129, Dec. 2009. .

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic , 1991).

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