Abstract

The spectral performances of nonideal rectangular Bragg gratings, integrated in a rib waveguide, are analyzed by a multilayer approach based on the effective-index method. The effects of errors on the photolithographic definition of the grating, that is, period and shape, and of errors on the control of etching depth are investigated. Also the influence of the stitching error, which is unavoidable when the grating is realized by means of electron-beam photolithography, is addressed. A novel analytical approach that extends coupled-mode theory to the analysis of real gratings is also presented.

© 1999 Optical Society of America

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  1. D. Eisert, G. Bacher, N. Mais, J. P. Reithmaier, A. Forchel, B. Jobst, D. Hommel, “First order gain and index coupled distributed feedback lasers in ZnSe structures with finely tuned emission wavelengths,” Appl. Phys. Lett. 68, 599–601 (1996).
    [CrossRef]
  2. T. F. Krauss, R. M. De La Rue, P. J. R. Laybourn, B. Vögele, C. R. Stanley, “Efficient semiconductor ring lasers made by a simple self-aligned fabrication process,” IEEE J. Sel. Top. Quantum Electron. 1, 757–760 (1995).
    [CrossRef]
  3. M. Fallahi, M. Dion, F. Chatenoud, I. M. Temoleton, R. Barber, “High temperature operation of circular grating surface emitting DBR lasers fabricated on an InGaAs/GaAs structure,” IEEE Photon. Technol. Lett. 6, 326–329 (1994).
    [CrossRef]
  4. S. L. McCall, P. M. Platzman, “An optimized π/2 distributed feedback laser,” IEEE J. Quantum Electron. QE-21, 1899–1904 (1985).
    [CrossRef]
  5. A. Cutolo, M. Iodice, A. Irace, P. Spirito, L. Zeni, “An electrically controlled Bragg reflector integrated in a silicon rib SOI waveguide,” Appl. Phys. Lett. 71, 199–201 (1997).
    [CrossRef]
  6. M. Y. Liu, S. Y. Chou, “High modulation depth and short cavity length silicon Fabry–Perot modulator with two Bragg reflectors,” Appl. Phys. Lett. 68, 170–172 (1996).
    [CrossRef]
  7. M. Okai, I. F. Lealman, J. Rivers, C. Dix, M. J. Robertson, “In line Fabry–Perot optical waveguide filter with quasi-chirped grating,” Electron. Lett. 32, 108–109 (1996).
    [CrossRef]
  8. H. J. Lee, N. A. Olsson, H. Henry, R. F. Kazarinov, K. J. Orlowski, “Narrowband Bragg reflector filter at 1.52 µm,” Appl. Opt. 27, 211–213 (1988).
    [CrossRef] [PubMed]
  9. J. Martin, F. Ouellette, “Novel writing technique of long highly reflective in-fiber gratings,” Electron. Lett. 30, 811–812 (1994).
    [CrossRef]
  10. C. H. Lin, Z. H. Zhu, Y. H. Lo, “New grating fabrication technology for optoelectronic devices: cascaded self-induced holography,” Appl. Phys. Lett. 67, 3072–3074 (1995).
    [CrossRef]
  11. P. Unger, V. Boegli, P. Buchmann, R. Germann, “High resolution electron-beam lithography for fabricating visible semiconductor lasers with curved mirrors and integrated holograms,” Microelectron. Eng. 23, 461–464 (1994).
    [CrossRef]
  12. T. Kjellberg, S. Nilsson, T. Klinga, B. Broberg, R. Schatz, “Investigation on the spectral characteristics of DFB lasers with different grating configurations made by electron-beam lithography,” J. Lightwave Technol. 11, 1405–1415 (1993).
    [CrossRef]
  13. T. Kjellberg, R. Schatz, “The effect of stitching errors on the spectral characteristics of DFB lasers fabricated using electron-beam lithography,” J. Lightwave Technol. 10, 1256–1266 (1992).
    [CrossRef]
  14. V. V. Wong, J. Ferrara, J. N. Damask, T. E. Murphy, H. I. Smith, H. A. Haus, “Distributed Bragg grating integrated-optical filters: synthesis and fabrication,” J. Vac. Sci. Technol. B 13, 2859–2864 (1995).
    [CrossRef]
  15. A. Basu, J. M. Ballantyne, “Random fluctuations in first-order waveguide grating filters,” Appl. Opt. 18, 2575–2579 (1979).
    [CrossRef] [PubMed]
  16. A. Yariv, “Coupled mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
    [CrossRef]
  17. A. Hardy, “Exact derivation of coupling coefficients in corrugated waveguides with rectangular tooth shape,” IEEE J. Quantum Electron. QE-20, 1132–1139 (1984).
    [CrossRef]
  18. M. J. Li, S. I. Najafi, “Polarization dependence of grating assisted waveguide Bragg reflectors,” Appl. Opt. 32, 4517–4521 (1993).
    [CrossRef] [PubMed]
  19. K. A. Winick, “Effective-index method and coupled theory for almost-periodic waveguide gratings: a comparison,” Appl. Opt. 31, 757–764 (1992).
    [CrossRef] [PubMed]
  20. P. Blair, M. R. Taghizadeh, W. Parkers, C. D. W. Wilkinson, “High-efficiency binary fan-out gratings by modulation of high-frequency carrier grating,” Appl. Opt. 34, 2406–2413 (1995).
    [CrossRef] [PubMed]
  21. P. Verly, R. Tremblay, J. W. Y. Lit, “Application of the effective-index method to the study of distributed feedback in corrugated waveguides. TE polarization,” J. Opt. Soc. Am. 70, 964–968 (1980).
    [CrossRef]
  22. P. Verly, R. Tremblay, J. W. Y. Lit, “Application of the effective-index method to the study of distributed feedback in corrugated waveguides. TM polarization,” J. Opt. Soc. Am. 70, 1218–1221 (1980).
    [CrossRef]
  23. R. W. Gruhlke, D. G. Hall, “Comparison of two approaches to the waveguide scattering problem: TM polarization,” Appl. Opt. 23, 127–133 (1984).
    [CrossRef] [PubMed]
  24. W. Streifer, D. R. Scrifres, R. D. Burnham, “TM-mode coupling coefficients in guided-wave distributed feedback lasers,” IEEE J. Quantum Electron. QE-12, 74–78 (1976).
    [CrossRef]
  25. W. Streifer, D. R. Scrifres, R. D. Burnham, “Coupling coefficients for distributed feedback single and double heterostructure diode lasers,” IEEE J. Quantum Electron. QE-11, 867–873 (1975).
    [CrossRef]
  26. G. Weitman, A. Hardy, “Reduction of the coupling coefficients for distributed Bragg reflectors in corrugated narrow rib waveguide,” IEE Proc. Optoelectron. 144, 101–103 (1997).
    [CrossRef]

1997 (2)

A. Cutolo, M. Iodice, A. Irace, P. Spirito, L. Zeni, “An electrically controlled Bragg reflector integrated in a silicon rib SOI waveguide,” Appl. Phys. Lett. 71, 199–201 (1997).
[CrossRef]

G. Weitman, A. Hardy, “Reduction of the coupling coefficients for distributed Bragg reflectors in corrugated narrow rib waveguide,” IEE Proc. Optoelectron. 144, 101–103 (1997).
[CrossRef]

1996 (3)

M. Y. Liu, S. Y. Chou, “High modulation depth and short cavity length silicon Fabry–Perot modulator with two Bragg reflectors,” Appl. Phys. Lett. 68, 170–172 (1996).
[CrossRef]

M. Okai, I. F. Lealman, J. Rivers, C. Dix, M. J. Robertson, “In line Fabry–Perot optical waveguide filter with quasi-chirped grating,” Electron. Lett. 32, 108–109 (1996).
[CrossRef]

D. Eisert, G. Bacher, N. Mais, J. P. Reithmaier, A. Forchel, B. Jobst, D. Hommel, “First order gain and index coupled distributed feedback lasers in ZnSe structures with finely tuned emission wavelengths,” Appl. Phys. Lett. 68, 599–601 (1996).
[CrossRef]

1995 (4)

T. F. Krauss, R. M. De La Rue, P. J. R. Laybourn, B. Vögele, C. R. Stanley, “Efficient semiconductor ring lasers made by a simple self-aligned fabrication process,” IEEE J. Sel. Top. Quantum Electron. 1, 757–760 (1995).
[CrossRef]

C. H. Lin, Z. H. Zhu, Y. H. Lo, “New grating fabrication technology for optoelectronic devices: cascaded self-induced holography,” Appl. Phys. Lett. 67, 3072–3074 (1995).
[CrossRef]

V. V. Wong, J. Ferrara, J. N. Damask, T. E. Murphy, H. I. Smith, H. A. Haus, “Distributed Bragg grating integrated-optical filters: synthesis and fabrication,” J. Vac. Sci. Technol. B 13, 2859–2864 (1995).
[CrossRef]

P. Blair, M. R. Taghizadeh, W. Parkers, C. D. W. Wilkinson, “High-efficiency binary fan-out gratings by modulation of high-frequency carrier grating,” Appl. Opt. 34, 2406–2413 (1995).
[CrossRef] [PubMed]

1994 (3)

J. Martin, F. Ouellette, “Novel writing technique of long highly reflective in-fiber gratings,” Electron. Lett. 30, 811–812 (1994).
[CrossRef]

P. Unger, V. Boegli, P. Buchmann, R. Germann, “High resolution electron-beam lithography for fabricating visible semiconductor lasers with curved mirrors and integrated holograms,” Microelectron. Eng. 23, 461–464 (1994).
[CrossRef]

M. Fallahi, M. Dion, F. Chatenoud, I. M. Temoleton, R. Barber, “High temperature operation of circular grating surface emitting DBR lasers fabricated on an InGaAs/GaAs structure,” IEEE Photon. Technol. Lett. 6, 326–329 (1994).
[CrossRef]

1993 (2)

T. Kjellberg, S. Nilsson, T. Klinga, B. Broberg, R. Schatz, “Investigation on the spectral characteristics of DFB lasers with different grating configurations made by electron-beam lithography,” J. Lightwave Technol. 11, 1405–1415 (1993).
[CrossRef]

M. J. Li, S. I. Najafi, “Polarization dependence of grating assisted waveguide Bragg reflectors,” Appl. Opt. 32, 4517–4521 (1993).
[CrossRef] [PubMed]

1992 (2)

K. A. Winick, “Effective-index method and coupled theory for almost-periodic waveguide gratings: a comparison,” Appl. Opt. 31, 757–764 (1992).
[CrossRef] [PubMed]

T. Kjellberg, R. Schatz, “The effect of stitching errors on the spectral characteristics of DFB lasers fabricated using electron-beam lithography,” J. Lightwave Technol. 10, 1256–1266 (1992).
[CrossRef]

1988 (1)

1985 (1)

S. L. McCall, P. M. Platzman, “An optimized π/2 distributed feedback laser,” IEEE J. Quantum Electron. QE-21, 1899–1904 (1985).
[CrossRef]

1984 (2)

A. Hardy, “Exact derivation of coupling coefficients in corrugated waveguides with rectangular tooth shape,” IEEE J. Quantum Electron. QE-20, 1132–1139 (1984).
[CrossRef]

R. W. Gruhlke, D. G. Hall, “Comparison of two approaches to the waveguide scattering problem: TM polarization,” Appl. Opt. 23, 127–133 (1984).
[CrossRef] [PubMed]

1980 (2)

1979 (1)

1976 (1)

W. Streifer, D. R. Scrifres, R. D. Burnham, “TM-mode coupling coefficients in guided-wave distributed feedback lasers,” IEEE J. Quantum Electron. QE-12, 74–78 (1976).
[CrossRef]

1975 (1)

W. Streifer, D. R. Scrifres, R. D. Burnham, “Coupling coefficients for distributed feedback single and double heterostructure diode lasers,” IEEE J. Quantum Electron. QE-11, 867–873 (1975).
[CrossRef]

1973 (1)

A. Yariv, “Coupled mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

Bacher, G.

D. Eisert, G. Bacher, N. Mais, J. P. Reithmaier, A. Forchel, B. Jobst, D. Hommel, “First order gain and index coupled distributed feedback lasers in ZnSe structures with finely tuned emission wavelengths,” Appl. Phys. Lett. 68, 599–601 (1996).
[CrossRef]

Ballantyne, J. M.

Barber, R.

M. Fallahi, M. Dion, F. Chatenoud, I. M. Temoleton, R. Barber, “High temperature operation of circular grating surface emitting DBR lasers fabricated on an InGaAs/GaAs structure,” IEEE Photon. Technol. Lett. 6, 326–329 (1994).
[CrossRef]

Basu, A.

Blair, P.

Boegli, V.

P. Unger, V. Boegli, P. Buchmann, R. Germann, “High resolution electron-beam lithography for fabricating visible semiconductor lasers with curved mirrors and integrated holograms,” Microelectron. Eng. 23, 461–464 (1994).
[CrossRef]

Broberg, B.

T. Kjellberg, S. Nilsson, T. Klinga, B. Broberg, R. Schatz, “Investigation on the spectral characteristics of DFB lasers with different grating configurations made by electron-beam lithography,” J. Lightwave Technol. 11, 1405–1415 (1993).
[CrossRef]

Buchmann, P.

P. Unger, V. Boegli, P. Buchmann, R. Germann, “High resolution electron-beam lithography for fabricating visible semiconductor lasers with curved mirrors and integrated holograms,” Microelectron. Eng. 23, 461–464 (1994).
[CrossRef]

Burnham, R. D.

W. Streifer, D. R. Scrifres, R. D. Burnham, “TM-mode coupling coefficients in guided-wave distributed feedback lasers,” IEEE J. Quantum Electron. QE-12, 74–78 (1976).
[CrossRef]

W. Streifer, D. R. Scrifres, R. D. Burnham, “Coupling coefficients for distributed feedback single and double heterostructure diode lasers,” IEEE J. Quantum Electron. QE-11, 867–873 (1975).
[CrossRef]

Chatenoud, F.

M. Fallahi, M. Dion, F. Chatenoud, I. M. Temoleton, R. Barber, “High temperature operation of circular grating surface emitting DBR lasers fabricated on an InGaAs/GaAs structure,” IEEE Photon. Technol. Lett. 6, 326–329 (1994).
[CrossRef]

Chou, S. Y.

M. Y. Liu, S. Y. Chou, “High modulation depth and short cavity length silicon Fabry–Perot modulator with two Bragg reflectors,” Appl. Phys. Lett. 68, 170–172 (1996).
[CrossRef]

Cutolo, A.

A. Cutolo, M. Iodice, A. Irace, P. Spirito, L. Zeni, “An electrically controlled Bragg reflector integrated in a silicon rib SOI waveguide,” Appl. Phys. Lett. 71, 199–201 (1997).
[CrossRef]

Damask, J. N.

V. V. Wong, J. Ferrara, J. N. Damask, T. E. Murphy, H. I. Smith, H. A. Haus, “Distributed Bragg grating integrated-optical filters: synthesis and fabrication,” J. Vac. Sci. Technol. B 13, 2859–2864 (1995).
[CrossRef]

De La Rue, R. M.

T. F. Krauss, R. M. De La Rue, P. J. R. Laybourn, B. Vögele, C. R. Stanley, “Efficient semiconductor ring lasers made by a simple self-aligned fabrication process,” IEEE J. Sel. Top. Quantum Electron. 1, 757–760 (1995).
[CrossRef]

Dion, M.

M. Fallahi, M. Dion, F. Chatenoud, I. M. Temoleton, R. Barber, “High temperature operation of circular grating surface emitting DBR lasers fabricated on an InGaAs/GaAs structure,” IEEE Photon. Technol. Lett. 6, 326–329 (1994).
[CrossRef]

Dix, C.

M. Okai, I. F. Lealman, J. Rivers, C. Dix, M. J. Robertson, “In line Fabry–Perot optical waveguide filter with quasi-chirped grating,” Electron. Lett. 32, 108–109 (1996).
[CrossRef]

Eisert, D.

D. Eisert, G. Bacher, N. Mais, J. P. Reithmaier, A. Forchel, B. Jobst, D. Hommel, “First order gain and index coupled distributed feedback lasers in ZnSe structures with finely tuned emission wavelengths,” Appl. Phys. Lett. 68, 599–601 (1996).
[CrossRef]

Fallahi, M.

M. Fallahi, M. Dion, F. Chatenoud, I. M. Temoleton, R. Barber, “High temperature operation of circular grating surface emitting DBR lasers fabricated on an InGaAs/GaAs structure,” IEEE Photon. Technol. Lett. 6, 326–329 (1994).
[CrossRef]

Ferrara, J.

V. V. Wong, J. Ferrara, J. N. Damask, T. E. Murphy, H. I. Smith, H. A. Haus, “Distributed Bragg grating integrated-optical filters: synthesis and fabrication,” J. Vac. Sci. Technol. B 13, 2859–2864 (1995).
[CrossRef]

Forchel, A.

D. Eisert, G. Bacher, N. Mais, J. P. Reithmaier, A. Forchel, B. Jobst, D. Hommel, “First order gain and index coupled distributed feedback lasers in ZnSe structures with finely tuned emission wavelengths,” Appl. Phys. Lett. 68, 599–601 (1996).
[CrossRef]

Germann, R.

P. Unger, V. Boegli, P. Buchmann, R. Germann, “High resolution electron-beam lithography for fabricating visible semiconductor lasers with curved mirrors and integrated holograms,” Microelectron. Eng. 23, 461–464 (1994).
[CrossRef]

Gruhlke, R. W.

Hall, D. G.

Hardy, A.

G. Weitman, A. Hardy, “Reduction of the coupling coefficients for distributed Bragg reflectors in corrugated narrow rib waveguide,” IEE Proc. Optoelectron. 144, 101–103 (1997).
[CrossRef]

A. Hardy, “Exact derivation of coupling coefficients in corrugated waveguides with rectangular tooth shape,” IEEE J. Quantum Electron. QE-20, 1132–1139 (1984).
[CrossRef]

Haus, H. A.

V. V. Wong, J. Ferrara, J. N. Damask, T. E. Murphy, H. I. Smith, H. A. Haus, “Distributed Bragg grating integrated-optical filters: synthesis and fabrication,” J. Vac. Sci. Technol. B 13, 2859–2864 (1995).
[CrossRef]

Henry, H.

Hommel, D.

D. Eisert, G. Bacher, N. Mais, J. P. Reithmaier, A. Forchel, B. Jobst, D. Hommel, “First order gain and index coupled distributed feedback lasers in ZnSe structures with finely tuned emission wavelengths,” Appl. Phys. Lett. 68, 599–601 (1996).
[CrossRef]

Iodice, M.

A. Cutolo, M. Iodice, A. Irace, P. Spirito, L. Zeni, “An electrically controlled Bragg reflector integrated in a silicon rib SOI waveguide,” Appl. Phys. Lett. 71, 199–201 (1997).
[CrossRef]

Irace, A.

A. Cutolo, M. Iodice, A. Irace, P. Spirito, L. Zeni, “An electrically controlled Bragg reflector integrated in a silicon rib SOI waveguide,” Appl. Phys. Lett. 71, 199–201 (1997).
[CrossRef]

Jobst, B.

D. Eisert, G. Bacher, N. Mais, J. P. Reithmaier, A. Forchel, B. Jobst, D. Hommel, “First order gain and index coupled distributed feedback lasers in ZnSe structures with finely tuned emission wavelengths,” Appl. Phys. Lett. 68, 599–601 (1996).
[CrossRef]

Kazarinov, R. F.

Kjellberg, T.

T. Kjellberg, S. Nilsson, T. Klinga, B. Broberg, R. Schatz, “Investigation on the spectral characteristics of DFB lasers with different grating configurations made by electron-beam lithography,” J. Lightwave Technol. 11, 1405–1415 (1993).
[CrossRef]

T. Kjellberg, R. Schatz, “The effect of stitching errors on the spectral characteristics of DFB lasers fabricated using electron-beam lithography,” J. Lightwave Technol. 10, 1256–1266 (1992).
[CrossRef]

Klinga, T.

T. Kjellberg, S. Nilsson, T. Klinga, B. Broberg, R. Schatz, “Investigation on the spectral characteristics of DFB lasers with different grating configurations made by electron-beam lithography,” J. Lightwave Technol. 11, 1405–1415 (1993).
[CrossRef]

Krauss, T. F.

T. F. Krauss, R. M. De La Rue, P. J. R. Laybourn, B. Vögele, C. R. Stanley, “Efficient semiconductor ring lasers made by a simple self-aligned fabrication process,” IEEE J. Sel. Top. Quantum Electron. 1, 757–760 (1995).
[CrossRef]

Laybourn, P. J. R.

T. F. Krauss, R. M. De La Rue, P. J. R. Laybourn, B. Vögele, C. R. Stanley, “Efficient semiconductor ring lasers made by a simple self-aligned fabrication process,” IEEE J. Sel. Top. Quantum Electron. 1, 757–760 (1995).
[CrossRef]

Lealman, I. F.

M. Okai, I. F. Lealman, J. Rivers, C. Dix, M. J. Robertson, “In line Fabry–Perot optical waveguide filter with quasi-chirped grating,” Electron. Lett. 32, 108–109 (1996).
[CrossRef]

Lee, H. J.

Li, M. J.

Lin, C. H.

C. H. Lin, Z. H. Zhu, Y. H. Lo, “New grating fabrication technology for optoelectronic devices: cascaded self-induced holography,” Appl. Phys. Lett. 67, 3072–3074 (1995).
[CrossRef]

Lit, J. W. Y.

Liu, M. Y.

M. Y. Liu, S. Y. Chou, “High modulation depth and short cavity length silicon Fabry–Perot modulator with two Bragg reflectors,” Appl. Phys. Lett. 68, 170–172 (1996).
[CrossRef]

Lo, Y. H.

C. H. Lin, Z. H. Zhu, Y. H. Lo, “New grating fabrication technology for optoelectronic devices: cascaded self-induced holography,” Appl. Phys. Lett. 67, 3072–3074 (1995).
[CrossRef]

Mais, N.

D. Eisert, G. Bacher, N. Mais, J. P. Reithmaier, A. Forchel, B. Jobst, D. Hommel, “First order gain and index coupled distributed feedback lasers in ZnSe structures with finely tuned emission wavelengths,” Appl. Phys. Lett. 68, 599–601 (1996).
[CrossRef]

Martin, J.

J. Martin, F. Ouellette, “Novel writing technique of long highly reflective in-fiber gratings,” Electron. Lett. 30, 811–812 (1994).
[CrossRef]

McCall, S. L.

S. L. McCall, P. M. Platzman, “An optimized π/2 distributed feedback laser,” IEEE J. Quantum Electron. QE-21, 1899–1904 (1985).
[CrossRef]

Murphy, T. E.

V. V. Wong, J. Ferrara, J. N. Damask, T. E. Murphy, H. I. Smith, H. A. Haus, “Distributed Bragg grating integrated-optical filters: synthesis and fabrication,” J. Vac. Sci. Technol. B 13, 2859–2864 (1995).
[CrossRef]

Najafi, S. I.

Nilsson, S.

T. Kjellberg, S. Nilsson, T. Klinga, B. Broberg, R. Schatz, “Investigation on the spectral characteristics of DFB lasers with different grating configurations made by electron-beam lithography,” J. Lightwave Technol. 11, 1405–1415 (1993).
[CrossRef]

Okai, M.

M. Okai, I. F. Lealman, J. Rivers, C. Dix, M. J. Robertson, “In line Fabry–Perot optical waveguide filter with quasi-chirped grating,” Electron. Lett. 32, 108–109 (1996).
[CrossRef]

Olsson, N. A.

Orlowski, K. J.

Ouellette, F.

J. Martin, F. Ouellette, “Novel writing technique of long highly reflective in-fiber gratings,” Electron. Lett. 30, 811–812 (1994).
[CrossRef]

Parkers, W.

Platzman, P. M.

S. L. McCall, P. M. Platzman, “An optimized π/2 distributed feedback laser,” IEEE J. Quantum Electron. QE-21, 1899–1904 (1985).
[CrossRef]

Reithmaier, J. P.

D. Eisert, G. Bacher, N. Mais, J. P. Reithmaier, A. Forchel, B. Jobst, D. Hommel, “First order gain and index coupled distributed feedback lasers in ZnSe structures with finely tuned emission wavelengths,” Appl. Phys. Lett. 68, 599–601 (1996).
[CrossRef]

Rivers, J.

M. Okai, I. F. Lealman, J. Rivers, C. Dix, M. J. Robertson, “In line Fabry–Perot optical waveguide filter with quasi-chirped grating,” Electron. Lett. 32, 108–109 (1996).
[CrossRef]

Robertson, M. J.

M. Okai, I. F. Lealman, J. Rivers, C. Dix, M. J. Robertson, “In line Fabry–Perot optical waveguide filter with quasi-chirped grating,” Electron. Lett. 32, 108–109 (1996).
[CrossRef]

Schatz, R.

T. Kjellberg, S. Nilsson, T. Klinga, B. Broberg, R. Schatz, “Investigation on the spectral characteristics of DFB lasers with different grating configurations made by electron-beam lithography,” J. Lightwave Technol. 11, 1405–1415 (1993).
[CrossRef]

T. Kjellberg, R. Schatz, “The effect of stitching errors on the spectral characteristics of DFB lasers fabricated using electron-beam lithography,” J. Lightwave Technol. 10, 1256–1266 (1992).
[CrossRef]

Scrifres, D. R.

W. Streifer, D. R. Scrifres, R. D. Burnham, “TM-mode coupling coefficients in guided-wave distributed feedback lasers,” IEEE J. Quantum Electron. QE-12, 74–78 (1976).
[CrossRef]

W. Streifer, D. R. Scrifres, R. D. Burnham, “Coupling coefficients for distributed feedback single and double heterostructure diode lasers,” IEEE J. Quantum Electron. QE-11, 867–873 (1975).
[CrossRef]

Smith, H. I.

V. V. Wong, J. Ferrara, J. N. Damask, T. E. Murphy, H. I. Smith, H. A. Haus, “Distributed Bragg grating integrated-optical filters: synthesis and fabrication,” J. Vac. Sci. Technol. B 13, 2859–2864 (1995).
[CrossRef]

Spirito, P.

A. Cutolo, M. Iodice, A. Irace, P. Spirito, L. Zeni, “An electrically controlled Bragg reflector integrated in a silicon rib SOI waveguide,” Appl. Phys. Lett. 71, 199–201 (1997).
[CrossRef]

Stanley, C. R.

T. F. Krauss, R. M. De La Rue, P. J. R. Laybourn, B. Vögele, C. R. Stanley, “Efficient semiconductor ring lasers made by a simple self-aligned fabrication process,” IEEE J. Sel. Top. Quantum Electron. 1, 757–760 (1995).
[CrossRef]

Streifer, W.

W. Streifer, D. R. Scrifres, R. D. Burnham, “TM-mode coupling coefficients in guided-wave distributed feedback lasers,” IEEE J. Quantum Electron. QE-12, 74–78 (1976).
[CrossRef]

W. Streifer, D. R. Scrifres, R. D. Burnham, “Coupling coefficients for distributed feedback single and double heterostructure diode lasers,” IEEE J. Quantum Electron. QE-11, 867–873 (1975).
[CrossRef]

Taghizadeh, M. R.

Temoleton, I. M.

M. Fallahi, M. Dion, F. Chatenoud, I. M. Temoleton, R. Barber, “High temperature operation of circular grating surface emitting DBR lasers fabricated on an InGaAs/GaAs structure,” IEEE Photon. Technol. Lett. 6, 326–329 (1994).
[CrossRef]

Tremblay, R.

Unger, P.

P. Unger, V. Boegli, P. Buchmann, R. Germann, “High resolution electron-beam lithography for fabricating visible semiconductor lasers with curved mirrors and integrated holograms,” Microelectron. Eng. 23, 461–464 (1994).
[CrossRef]

Verly, P.

Vögele, B.

T. F. Krauss, R. M. De La Rue, P. J. R. Laybourn, B. Vögele, C. R. Stanley, “Efficient semiconductor ring lasers made by a simple self-aligned fabrication process,” IEEE J. Sel. Top. Quantum Electron. 1, 757–760 (1995).
[CrossRef]

Weitman, G.

G. Weitman, A. Hardy, “Reduction of the coupling coefficients for distributed Bragg reflectors in corrugated narrow rib waveguide,” IEE Proc. Optoelectron. 144, 101–103 (1997).
[CrossRef]

Wilkinson, C. D. W.

Winick, K. A.

Wong, V. V.

V. V. Wong, J. Ferrara, J. N. Damask, T. E. Murphy, H. I. Smith, H. A. Haus, “Distributed Bragg grating integrated-optical filters: synthesis and fabrication,” J. Vac. Sci. Technol. B 13, 2859–2864 (1995).
[CrossRef]

Yariv, A.

A. Yariv, “Coupled mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

Zeni, L.

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Figures (6)

Fig. 1
Fig. 1

Channel waveguide Bragg grating and its elementary cell.

Fig. 2
Fig. 2

Deformation of the reflectivity spectrum for different errors in the grating’s period.

Fig. 3
Fig. 3

Reduction in peak reflectivity as a function of the error made in determining the period length.

Fig. 4
Fig. 4

Scaling effect on the peak reflectivity of the grating.

Fig. 5
Fig. 5

Error in the grating period seen as a series of rectangular pulses.

Fig. 6
Fig. 6

Comparison between the numerical and the analytical approach.

Tables (1)

Tables Icon

Table 1 Summary of the Physical Geometric Properties of the Waveguide and Grating

Equations (15)

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E0+H0+=M1EΛ1-HΛ1-, EΛ1+HΛ1+=M2EΛ-HΛ-,
Mi=cosβiΛi-j z0ni sinβiΛi-j niz0 sinβiΛicosβiΛi,
MT=ABCD,
A=cosβ1Λ1cosβ2Λ2-n2n1 sinβ1Λ1sinβ2Λ2,  B=-jZ01n2 cosβ1Λ1sinβ2Λ2+1n1 sinβ1Λ1cosβ2Λ2,  C=-j 1Z0n1 sinβ1Λ1cosβ2Λ2+n2 cosβ1Λ1sinβ2Λ2,  D=cosβ1Λ1cosβ2Λ2-n1n2 sinβ1Λ1sinβ2Λ2.
M=n=1N MTn=ANBNCNDN.
M=MTn=u1 u2η1N00η2Nu1 u2-1,
r=AN+noZ0 BN-Z0niCN+noZ0 DNAN+noZ0 BN+Z0niCN+noZ0 DN,  t=2AN+noZ0 BN+Z0niCN+noZ0 DN.
Teff=noni |t|2,  Reff=|r|2,
Λ1R=Λ1+ε1,  Λ2R=Λ2+ε2,
errz=an=0N-1 signwnΠz-nΛ+wn2|wn|+j=0N-1 signνjΠz-jΛ+Λ2-νj2|νj|,
Δn2x, z=Δn2xΔrealn2z=Δn2xΔidealnz+errz2.
Δrealn2z=k=-+ γk exp2kπNΛ z,
kreal=ω04 γl- Δn2xEysx2dx
kreal=kideal1-1Nn=0N-12-cos2πΛ wn-cos2πΛνn.
kideal=k02nc2-nf24πrβN2 sinrπ wΛ×1qexp(-2qg2)-1-g1-sin2hg12h-qh2g1-sin2hg12h-qh21-cos2hg1,

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