Abstract

A combination of the plane-wave expansion method with the effective-index approximation is presented, which allows vectorial properties of electromagnetic field in three-dimensional structures to be considered. I formulate mathematical equations that are the basis of this combined approach, and then I show the simulation of the optical properties of oxide-confined vertical-cavity surface-emitting lasers (VCSELs). The results are compared with the ones obtained with the classical effective-index method and with the fully vectorial plane-wave admittance method. Next the semivectorial properties of the new model are utilized to analyze a birefringent VCSEL structure.

© 2009 Optical Society of America

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References

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  1. K. Panajotov, M. Peeters, and I. Veretennicoff, “Self-consistent modelling of VCSELs with noncylindrical geometry,” in Digest of the 2000 International Conference on Laser and Electro-Optics (CLEO 2000) (IEEE, 2000) p. 232.
  2. G. R. Hadley, “Effective index model for vertical-cavity surface-emitting lasers,” Opt. Lett. 20, 1483-1485 (1995).
    [CrossRef] [PubMed]
  3. M. Dems, R. Kotynski, and K. Panajotov, “Plane-wave admittance method--a novel approach for determining the electromagnetic modes in photonic structures,” Opt. Express 13, 3196-3207 (2005).
    [CrossRef] [PubMed]
  4. M. Dems, T. Czyszanowski, and K. Panajotov “Numerical analysis of high Q-factor photonic-crystal VCSELs with plane-wave admittance method,” Opt. Quantum Electron. 39, 419-426 (2007).
    [CrossRef]
  5. M. Dems, “Plane-wave admittance method and its applications to modeling semiconductor lasers and planar photonic-crystal structures,” Ph.D. thesis, Technical University of Lodz (2007).
  6. T. Czyszanowski, M. Dems, H. Thienpont, and K. Panajotov, “Full vectorial electromagnetic modeling of vertical-cavity surface-emitting diode lasers by the plane wave admittance method,” Proc. SPIE 6185, 61,850Y (2006).
  7. P. Bienstman, R. Baets, J. Vukusic, A. Larsson, M. J. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. P. Bava, W. Wenzel, K. Klein, C. Conradi, R. Pregla, S. A. Riyopoulos, J.-F. P. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models on the simulation of oxide-confined devices,” IEEE J. Quantum Electron. 37, 1618-1631 (2001).
    [CrossRef]
  8. H. Wenzel and H. J. Wünsche, “The effective frequency method in the analysis of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 1156-1162 (1997).
    [CrossRef]
  9. T. Czyszanowski and W. Nakwaski, “Validity of scalar approaches to radiation modes of the GaAs-based 1.3-μm diode lasers designed for the optical-fibre communication,” Opt. Quantum Electron. 38, 349-360 (2006).
    [CrossRef]
  10. T. Czyszanowski and W. Nakwaski, “Usability limits of the scalar effective frequency method used to determine modes distributions in oxide-confined vertical-cavity surface-emitting diode lasers,” J. Phys. D 39, 30-35 (2006).
    [CrossRef]
  11. T. Mukaihara, N. Ohnoki, Y. Hayashi, N. Hatori, F. Koyama, and K. Iga, “Polarization control of vertical-cavity surface emitting lasers using a birefringent metal/dielectric polarizer loaded on top distributed Bragg reflector,” IEEE J. Sel. Top. Quantum Electron. 1, 667-673 (1995).
    [CrossRef]

2007 (2)

M. Dems, T. Czyszanowski, and K. Panajotov “Numerical analysis of high Q-factor photonic-crystal VCSELs with plane-wave admittance method,” Opt. Quantum Electron. 39, 419-426 (2007).
[CrossRef]

M. Dems, “Plane-wave admittance method and its applications to modeling semiconductor lasers and planar photonic-crystal structures,” Ph.D. thesis, Technical University of Lodz (2007).

2006 (3)

T. Czyszanowski, M. Dems, H. Thienpont, and K. Panajotov, “Full vectorial electromagnetic modeling of vertical-cavity surface-emitting diode lasers by the plane wave admittance method,” Proc. SPIE 6185, 61,850Y (2006).

T. Czyszanowski and W. Nakwaski, “Validity of scalar approaches to radiation modes of the GaAs-based 1.3-μm diode lasers designed for the optical-fibre communication,” Opt. Quantum Electron. 38, 349-360 (2006).
[CrossRef]

T. Czyszanowski and W. Nakwaski, “Usability limits of the scalar effective frequency method used to determine modes distributions in oxide-confined vertical-cavity surface-emitting diode lasers,” J. Phys. D 39, 30-35 (2006).
[CrossRef]

2005 (1)

2001 (1)

P. Bienstman, R. Baets, J. Vukusic, A. Larsson, M. J. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. P. Bava, W. Wenzel, K. Klein, C. Conradi, R. Pregla, S. A. Riyopoulos, J.-F. P. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models on the simulation of oxide-confined devices,” IEEE J. Quantum Electron. 37, 1618-1631 (2001).
[CrossRef]

2000 (1)

K. Panajotov, M. Peeters, and I. Veretennicoff, “Self-consistent modelling of VCSELs with noncylindrical geometry,” in Digest of the 2000 International Conference on Laser and Electro-Optics (CLEO 2000) (IEEE, 2000) p. 232.

1997 (1)

H. Wenzel and H. J. Wünsche, “The effective frequency method in the analysis of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 1156-1162 (1997).
[CrossRef]

1995 (2)

G. R. Hadley, “Effective index model for vertical-cavity surface-emitting lasers,” Opt. Lett. 20, 1483-1485 (1995).
[CrossRef] [PubMed]

T. Mukaihara, N. Ohnoki, Y. Hayashi, N. Hatori, F. Koyama, and K. Iga, “Polarization control of vertical-cavity surface emitting lasers using a birefringent metal/dielectric polarizer loaded on top distributed Bragg reflector,” IEEE J. Sel. Top. Quantum Electron. 1, 667-673 (1995).
[CrossRef]

Baets, R.

P. Bienstman, R. Baets, J. Vukusic, A. Larsson, M. J. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. P. Bava, W. Wenzel, K. Klein, C. Conradi, R. Pregla, S. A. Riyopoulos, J.-F. P. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models on the simulation of oxide-confined devices,” IEEE J. Quantum Electron. 37, 1618-1631 (2001).
[CrossRef]

Bava, G. P.

P. Bienstman, R. Baets, J. Vukusic, A. Larsson, M. J. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. P. Bava, W. Wenzel, K. Klein, C. Conradi, R. Pregla, S. A. Riyopoulos, J.-F. P. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models on the simulation of oxide-confined devices,” IEEE J. Quantum Electron. 37, 1618-1631 (2001).
[CrossRef]

Bienstman, P.

P. Bienstman, R. Baets, J. Vukusic, A. Larsson, M. J. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. P. Bava, W. Wenzel, K. Klein, C. Conradi, R. Pregla, S. A. Riyopoulos, J.-F. P. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models on the simulation of oxide-confined devices,” IEEE J. Quantum Electron. 37, 1618-1631 (2001).
[CrossRef]

Brunner, M.

P. Bienstman, R. Baets, J. Vukusic, A. Larsson, M. J. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. P. Bava, W. Wenzel, K. Klein, C. Conradi, R. Pregla, S. A. Riyopoulos, J.-F. P. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models on the simulation of oxide-confined devices,” IEEE J. Quantum Electron. 37, 1618-1631 (2001).
[CrossRef]

Chuang, S. L.

P. Bienstman, R. Baets, J. Vukusic, A. Larsson, M. J. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. P. Bava, W. Wenzel, K. Klein, C. Conradi, R. Pregla, S. A. Riyopoulos, J.-F. P. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models on the simulation of oxide-confined devices,” IEEE J. Quantum Electron. 37, 1618-1631 (2001).
[CrossRef]

Conradi, C.

P. Bienstman, R. Baets, J. Vukusic, A. Larsson, M. J. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. P. Bava, W. Wenzel, K. Klein, C. Conradi, R. Pregla, S. A. Riyopoulos, J.-F. P. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models on the simulation of oxide-confined devices,” IEEE J. Quantum Electron. 37, 1618-1631 (2001).
[CrossRef]

Czyszanowski, T.

M. Dems, T. Czyszanowski, and K. Panajotov “Numerical analysis of high Q-factor photonic-crystal VCSELs with plane-wave admittance method,” Opt. Quantum Electron. 39, 419-426 (2007).
[CrossRef]

T. Czyszanowski, M. Dems, H. Thienpont, and K. Panajotov, “Full vectorial electromagnetic modeling of vertical-cavity surface-emitting diode lasers by the plane wave admittance method,” Proc. SPIE 6185, 61,850Y (2006).

T. Czyszanowski and W. Nakwaski, “Validity of scalar approaches to radiation modes of the GaAs-based 1.3-μm diode lasers designed for the optical-fibre communication,” Opt. Quantum Electron. 38, 349-360 (2006).
[CrossRef]

T. Czyszanowski and W. Nakwaski, “Usability limits of the scalar effective frequency method used to determine modes distributions in oxide-confined vertical-cavity surface-emitting diode lasers,” J. Phys. D 39, 30-35 (2006).
[CrossRef]

Debernardi, P.

P. Bienstman, R. Baets, J. Vukusic, A. Larsson, M. J. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. P. Bava, W. Wenzel, K. Klein, C. Conradi, R. Pregla, S. A. Riyopoulos, J.-F. P. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models on the simulation of oxide-confined devices,” IEEE J. Quantum Electron. 37, 1618-1631 (2001).
[CrossRef]

Dems, M.

M. Dems, “Plane-wave admittance method and its applications to modeling semiconductor lasers and planar photonic-crystal structures,” Ph.D. thesis, Technical University of Lodz (2007).

M. Dems, T. Czyszanowski, and K. Panajotov “Numerical analysis of high Q-factor photonic-crystal VCSELs with plane-wave admittance method,” Opt. Quantum Electron. 39, 419-426 (2007).
[CrossRef]

T. Czyszanowski, M. Dems, H. Thienpont, and K. Panajotov, “Full vectorial electromagnetic modeling of vertical-cavity surface-emitting diode lasers by the plane wave admittance method,” Proc. SPIE 6185, 61,850Y (2006).

M. Dems, R. Kotynski, and K. Panajotov, “Plane-wave admittance method--a novel approach for determining the electromagnetic modes in photonic structures,” Opt. Express 13, 3196-3207 (2005).
[CrossRef] [PubMed]

Fratta, L.

P. Bienstman, R. Baets, J. Vukusic, A. Larsson, M. J. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. P. Bava, W. Wenzel, K. Klein, C. Conradi, R. Pregla, S. A. Riyopoulos, J.-F. P. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models on the simulation of oxide-confined devices,” IEEE J. Quantum Electron. 37, 1618-1631 (2001).
[CrossRef]

Gulden, K.

P. Bienstman, R. Baets, J. Vukusic, A. Larsson, M. J. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. P. Bava, W. Wenzel, K. Klein, C. Conradi, R. Pregla, S. A. Riyopoulos, J.-F. P. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models on the simulation of oxide-confined devices,” IEEE J. Quantum Electron. 37, 1618-1631 (2001).
[CrossRef]

Hadley, G. R.

Hatori, N.

T. Mukaihara, N. Ohnoki, Y. Hayashi, N. Hatori, F. Koyama, and K. Iga, “Polarization control of vertical-cavity surface emitting lasers using a birefringent metal/dielectric polarizer loaded on top distributed Bragg reflector,” IEEE J. Sel. Top. Quantum Electron. 1, 667-673 (1995).
[CrossRef]

Hayashi, Y.

T. Mukaihara, N. Ohnoki, Y. Hayashi, N. Hatori, F. Koyama, and K. Iga, “Polarization control of vertical-cavity surface emitting lasers using a birefringent metal/dielectric polarizer loaded on top distributed Bragg reflector,” IEEE J. Sel. Top. Quantum Electron. 1, 667-673 (1995).
[CrossRef]

Iga, K.

T. Mukaihara, N. Ohnoki, Y. Hayashi, N. Hatori, F. Koyama, and K. Iga, “Polarization control of vertical-cavity surface emitting lasers using a birefringent metal/dielectric polarizer loaded on top distributed Bragg reflector,” IEEE J. Sel. Top. Quantum Electron. 1, 667-673 (1995).
[CrossRef]

Klein, K.

P. Bienstman, R. Baets, J. Vukusic, A. Larsson, M. J. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. P. Bava, W. Wenzel, K. Klein, C. Conradi, R. Pregla, S. A. Riyopoulos, J.-F. P. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models on the simulation of oxide-confined devices,” IEEE J. Quantum Electron. 37, 1618-1631 (2001).
[CrossRef]

Kotynski, R.

Koyama, F.

T. Mukaihara, N. Ohnoki, Y. Hayashi, N. Hatori, F. Koyama, and K. Iga, “Polarization control of vertical-cavity surface emitting lasers using a birefringent metal/dielectric polarizer loaded on top distributed Bragg reflector,” IEEE J. Sel. Top. Quantum Electron. 1, 667-673 (1995).
[CrossRef]

Larsson, A.

P. Bienstman, R. Baets, J. Vukusic, A. Larsson, M. J. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. P. Bava, W. Wenzel, K. Klein, C. Conradi, R. Pregla, S. A. Riyopoulos, J.-F. P. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models on the simulation of oxide-confined devices,” IEEE J. Quantum Electron. 37, 1618-1631 (2001).
[CrossRef]

Mukaihara, T.

T. Mukaihara, N. Ohnoki, Y. Hayashi, N. Hatori, F. Koyama, and K. Iga, “Polarization control of vertical-cavity surface emitting lasers using a birefringent metal/dielectric polarizer loaded on top distributed Bragg reflector,” IEEE J. Sel. Top. Quantum Electron. 1, 667-673 (1995).
[CrossRef]

Nakwaski, W.

T. Czyszanowski and W. Nakwaski, “Validity of scalar approaches to radiation modes of the GaAs-based 1.3-μm diode lasers designed for the optical-fibre communication,” Opt. Quantum Electron. 38, 349-360 (2006).
[CrossRef]

T. Czyszanowski and W. Nakwaski, “Usability limits of the scalar effective frequency method used to determine modes distributions in oxide-confined vertical-cavity surface-emitting diode lasers,” J. Phys. D 39, 30-35 (2006).
[CrossRef]

Noble, M. J.

P. Bienstman, R. Baets, J. Vukusic, A. Larsson, M. J. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. P. Bava, W. Wenzel, K. Klein, C. Conradi, R. Pregla, S. A. Riyopoulos, J.-F. P. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models on the simulation of oxide-confined devices,” IEEE J. Quantum Electron. 37, 1618-1631 (2001).
[CrossRef]

Ohnoki, N.

T. Mukaihara, N. Ohnoki, Y. Hayashi, N. Hatori, F. Koyama, and K. Iga, “Polarization control of vertical-cavity surface emitting lasers using a birefringent metal/dielectric polarizer loaded on top distributed Bragg reflector,” IEEE J. Sel. Top. Quantum Electron. 1, 667-673 (1995).
[CrossRef]

Panajotov, K.

M. Dems, T. Czyszanowski, and K. Panajotov “Numerical analysis of high Q-factor photonic-crystal VCSELs with plane-wave admittance method,” Opt. Quantum Electron. 39, 419-426 (2007).
[CrossRef]

T. Czyszanowski, M. Dems, H. Thienpont, and K. Panajotov, “Full vectorial electromagnetic modeling of vertical-cavity surface-emitting diode lasers by the plane wave admittance method,” Proc. SPIE 6185, 61,850Y (2006).

M. Dems, R. Kotynski, and K. Panajotov, “Plane-wave admittance method--a novel approach for determining the electromagnetic modes in photonic structures,” Opt. Express 13, 3196-3207 (2005).
[CrossRef] [PubMed]

K. Panajotov, M. Peeters, and I. Veretennicoff, “Self-consistent modelling of VCSELs with noncylindrical geometry,” in Digest of the 2000 International Conference on Laser and Electro-Optics (CLEO 2000) (IEEE, 2000) p. 232.

Peeters, M.

K. Panajotov, M. Peeters, and I. Veretennicoff, “Self-consistent modelling of VCSELs with noncylindrical geometry,” in Digest of the 2000 International Conference on Laser and Electro-Optics (CLEO 2000) (IEEE, 2000) p. 232.

Pregla, R.

P. Bienstman, R. Baets, J. Vukusic, A. Larsson, M. J. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. P. Bava, W. Wenzel, K. Klein, C. Conradi, R. Pregla, S. A. Riyopoulos, J.-F. P. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models on the simulation of oxide-confined devices,” IEEE J. Quantum Electron. 37, 1618-1631 (2001).
[CrossRef]

Riyopoulos, S. A.

P. Bienstman, R. Baets, J. Vukusic, A. Larsson, M. J. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. P. Bava, W. Wenzel, K. Klein, C. Conradi, R. Pregla, S. A. Riyopoulos, J.-F. P. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models on the simulation of oxide-confined devices,” IEEE J. Quantum Electron. 37, 1618-1631 (2001).
[CrossRef]

Seurin, J.-F. P.

P. Bienstman, R. Baets, J. Vukusic, A. Larsson, M. J. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. P. Bava, W. Wenzel, K. Klein, C. Conradi, R. Pregla, S. A. Riyopoulos, J.-F. P. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models on the simulation of oxide-confined devices,” IEEE J. Quantum Electron. 37, 1618-1631 (2001).
[CrossRef]

Thienpont, H.

T. Czyszanowski, M. Dems, H. Thienpont, and K. Panajotov, “Full vectorial electromagnetic modeling of vertical-cavity surface-emitting diode lasers by the plane wave admittance method,” Proc. SPIE 6185, 61,850Y (2006).

Veretennicoff, I.

K. Panajotov, M. Peeters, and I. Veretennicoff, “Self-consistent modelling of VCSELs with noncylindrical geometry,” in Digest of the 2000 International Conference on Laser and Electro-Optics (CLEO 2000) (IEEE, 2000) p. 232.

Vukusic, J.

P. Bienstman, R. Baets, J. Vukusic, A. Larsson, M. J. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. P. Bava, W. Wenzel, K. Klein, C. Conradi, R. Pregla, S. A. Riyopoulos, J.-F. P. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models on the simulation of oxide-confined devices,” IEEE J. Quantum Electron. 37, 1618-1631 (2001).
[CrossRef]

Wenzel, H.

H. Wenzel and H. J. Wünsche, “The effective frequency method in the analysis of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 1156-1162 (1997).
[CrossRef]

Wenzel, W.

P. Bienstman, R. Baets, J. Vukusic, A. Larsson, M. J. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. P. Bava, W. Wenzel, K. Klein, C. Conradi, R. Pregla, S. A. Riyopoulos, J.-F. P. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models on the simulation of oxide-confined devices,” IEEE J. Quantum Electron. 37, 1618-1631 (2001).
[CrossRef]

Wünsche, H. J.

H. Wenzel and H. J. Wünsche, “The effective frequency method in the analysis of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 1156-1162 (1997).
[CrossRef]

IEEE J. Quantum Electron. (2)

P. Bienstman, R. Baets, J. Vukusic, A. Larsson, M. J. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. P. Bava, W. Wenzel, K. Klein, C. Conradi, R. Pregla, S. A. Riyopoulos, J.-F. P. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models on the simulation of oxide-confined devices,” IEEE J. Quantum Electron. 37, 1618-1631 (2001).
[CrossRef]

H. Wenzel and H. J. Wünsche, “The effective frequency method in the analysis of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 1156-1162 (1997).
[CrossRef]

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

T. Mukaihara, N. Ohnoki, Y. Hayashi, N. Hatori, F. Koyama, and K. Iga, “Polarization control of vertical-cavity surface emitting lasers using a birefringent metal/dielectric polarizer loaded on top distributed Bragg reflector,” IEEE J. Sel. Top. Quantum Electron. 1, 667-673 (1995).
[CrossRef]

J. Phys. D (1)

T. Czyszanowski and W. Nakwaski, “Usability limits of the scalar effective frequency method used to determine modes distributions in oxide-confined vertical-cavity surface-emitting diode lasers,” J. Phys. D 39, 30-35 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Opt. Quantum Electron. (2)

T. Czyszanowski and W. Nakwaski, “Validity of scalar approaches to radiation modes of the GaAs-based 1.3-μm diode lasers designed for the optical-fibre communication,” Opt. Quantum Electron. 38, 349-360 (2006).
[CrossRef]

M. Dems, T. Czyszanowski, and K. Panajotov “Numerical analysis of high Q-factor photonic-crystal VCSELs with plane-wave admittance method,” Opt. Quantum Electron. 39, 419-426 (2007).
[CrossRef]

Proc. SPIE (1)

T. Czyszanowski, M. Dems, H. Thienpont, and K. Panajotov, “Full vectorial electromagnetic modeling of vertical-cavity surface-emitting diode lasers by the plane wave admittance method,” Proc. SPIE 6185, 61,850Y (2006).

Other (2)

K. Panajotov, M. Peeters, and I. Veretennicoff, “Self-consistent modelling of VCSELs with noncylindrical geometry,” in Digest of the 2000 International Conference on Laser and Electro-Optics (CLEO 2000) (IEEE, 2000) p. 232.

M. Dems, “Plane-wave admittance method and its applications to modeling semiconductor lasers and planar photonic-crystal structures,” Ph.D. thesis, Technical University of Lodz (2007).

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Figures (6)

Fig. 1
Fig. 1

Schematic structure of oxide-confined VCSEL.

Fig. 2
Fig. 2

The resonant wavelength (a) LP 01 and (b) LP 11 modes in an oxide-confined VCSEL as a function of an oxide position: 1 means location of the oxide in the node of the standing wave inside the cavity and 5 is the anti-node. Circles mean the semivectorial method and triangles a reference vectorial method. In addition the results of effective index and effective frequency calculation are shown in (a).

Fig. 3
Fig. 3

The cold-cavity modal loss for (a) LP 01 and (b) LP11 modes in an oxide-confined VCSEL as a function of an oxide position.

Fig. 4
Fig. 4

(a) The cold-cavity resonant wavelength and (b) modal loss for a LP 01 mode in an oxide-confined VCSEL as a function of an oxide position computed with semivectorial method with two choices of k 0 .

Fig. 5
Fig. 5

Schematic structure of the top of polarization-controlled VCSEL.

Fig. 6
Fig. 6

(a) The cold-cavity resonant wavelength and (b) modal loss in polarization-controlled VCSEL as a function of the GaAs cap thickness, computed for the polarization parallel ( x ) and perpendicular ( y ) to the grating. The results are obtained with both semivectorial (lines) and fully vectorial (dots) models.

Equations (23)

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

z E ( z ) = i R ¯ H H ( z ) ,
z H ( z ) = i R ¯ E E ( z ) ,
R ¯ H = η 0 k [ y ε z 1 y + μ x k 2 y ε z 1 x x ε z 1 y x ε z 1 x + μ y k 2 ] ,
R ¯ E = 1 η 0 k [ x μ z 1 x + ε y k 2 x μ z 1 y y μ z 1 x y μ z 1 y + ε x k 2 ] ,
E ( z ) = f E ( z ) E ,
E z f E = i R ¯ H H ( z ) ,
z H ( z ) = i f E R ¯ E E .
[ R ¯ H R ¯ E f E + I z 2 f E ] E = 0 ,
[ layer f E * R ¯ H R ¯ E f E d z + I layer f E * z 2 f E d z ] E = 0 .
z 2 f E = κ 2 n ( z ) 2 f E ,
n 2 = l layer f E * n ( z ) 2 f E d z ,
[ R ¯ H R ¯ E κ 2 n 2 I ] E = 0 ,
Ψ l layer f E * Ψ f E d z ,
Ψ = l Ψ layer f E * f E d z = l Ψ Φ l ,
Φ l = layer f E * f E d z ,
R ¯ E = k 1 R E + k d E ,
R ¯ H = k 1 R H + k d H ,
R ¯ H R ¯ E = k 2 R H R E + R H d E + d H R E + k 2 d H d E .
k 2 = k 0 2 + 1 τ 2 ,
k 2 k 0 2 k 0 4 τ 2 .
k 0 4 R H R E d E d E E = τ 2 [ k 0 2 ( R H + k 0 2 d H ) ( R E + k 0 2 d E ) κ 2 n 2 I ] E .
ε = ( t 1 ε 1 1 + t 2 ε 2 1 t 1 + t 2 ) 1 ,
ε = t 1 ε 1 + t 2 ε 2 t 1 + t 2 ,

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