Abstract

Optical mode discrimination in vertical-cavity surface-emitting lasers that contain distributed Bragg reflectors (DBRs) and a spatially limited gain medium is analyzed numerically. It is assumed that the output field is linearly polarized owing to gain selectivity. The analysis employs a three-dimensional model and an angular spectrum of plane-wave decomposition with the proper polarizations. Two types of round aperture are considered, namely, a Gaussian aperture and a ring-peak aperture that represents gain saturation. Coupled with the DBRs, the former aperture yields nearly Laguerre–Gaussian modes, whereas the latter aperture significantly distorts the mode shapes. In both cases, narrowband DBRs provide the best mode discrimination.

© 2005 Optical Society of America

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References

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  1. P. Bienstman, R. Baets, A. Larsson, M. J. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. P. Bava, H. Wenzel, B. Klein, O. Conradi, R. Pregla, S. A. Riyopoulos, J.-F. P. Seurin, S. L. Chuang, “Comparison of optical VCSEL models on the simulation of oxide-confined devices,” IEEE J. Quantum Electron. 37, 1618–1631 (2001).
    [CrossRef]
  2. A. Valle, “Selection and modulation of higher-order transverse modes in vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 34, 1924–1932 (1998).
    [CrossRef]
  3. G. P. Bava, P. Debernardi, L. Fratta, “Three-dimensional model for vectorial fields in vertical-cavity surface-emitting lasers,” Phys. Rev. A 63, 23816 (2001).
    [CrossRef]
  4. P. Debernardi, G. P. Bava, “Coupled mode theory: a powerful tool for analyzing complex VCSELs and designing advanced device features,” IEEE J. Sel. Top. Quantum Electron. 9, 905–917 (2003).
    [CrossRef]
  5. D. I. Babic, Y. Chung, N. Dagli, J. E. Bowers, “Modal reflection of quarter-wave mirrors in vertical-cavity lasers,” IEEE J. Quantum Electron. 29, 1950–1962 (1993).
    [CrossRef]
  6. M. Achtenhagen, A. A. Hardy, E. Kapon, “Design of distributed Bragg reflector structures for transverse-mode discrimination in vertical-cavity surface-emitting lasers,” IEEE J. Lightwave Technol. 22, 1962–1967 (2004).
    [CrossRef]
  7. A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
    [CrossRef]
  8. J. Martin-Regalado, F. Prati, M. San Miguel, N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
    [CrossRef]
  9. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  10. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).
  11. S. H. Friedberg, A. J. Insel, Introduction to Linear Algebra with Applications (Prentice-Hall, Englewood Cliffs, N.J., 1986).
  12. H. K. Bissessur, F. Koyama, K. Iga, “Modeling of oxide-confined vertical-cavity surface-emitting lasers,” IEEE J. Sel. Top. Quantum Electron. 3, 344–352 (1997).
    [CrossRef]
  13. M. Jungo, D. Erni, W. Baechtold, “Quasi-analytic steady-state solution of VCSEL rate equations including spatial hole burning and carrier diffusion losses,” Int. J. Numer. Model. 16, 143–159 (2003).
    [CrossRef]
  14. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  15. T. F. Johnston, “M2concept characterizes beam quality,” Laser Focus 5, 173–183 (1990).
  16. A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
    [CrossRef]

2004 (1)

M. Achtenhagen, A. A. Hardy, E. Kapon, “Design of distributed Bragg reflector structures for transverse-mode discrimination in vertical-cavity surface-emitting lasers,” IEEE J. Lightwave Technol. 22, 1962–1967 (2004).
[CrossRef]

2003 (2)

P. Debernardi, G. P. Bava, “Coupled mode theory: a powerful tool for analyzing complex VCSELs and designing advanced device features,” IEEE J. Sel. Top. Quantum Electron. 9, 905–917 (2003).
[CrossRef]

M. Jungo, D. Erni, W. Baechtold, “Quasi-analytic steady-state solution of VCSEL rate equations including spatial hole burning and carrier diffusion losses,” Int. J. Numer. Model. 16, 143–159 (2003).
[CrossRef]

2001 (2)

G. P. Bava, P. Debernardi, L. Fratta, “Three-dimensional model for vectorial fields in vertical-cavity surface-emitting lasers,” Phys. Rev. A 63, 23816 (2001).
[CrossRef]

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

1998 (1)

A. Valle, “Selection and modulation of higher-order transverse modes in vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 34, 1924–1932 (1998).
[CrossRef]

1997 (2)

J. Martin-Regalado, F. Prati, M. San Miguel, N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[CrossRef]

H. K. Bissessur, F. Koyama, K. Iga, “Modeling of oxide-confined vertical-cavity surface-emitting lasers,” IEEE J. Sel. Top. Quantum Electron. 3, 344–352 (1997).
[CrossRef]

1993 (1)

D. I. Babic, Y. Chung, N. Dagli, J. E. Bowers, “Modal reflection of quarter-wave mirrors in vertical-cavity lasers,” IEEE J. Quantum Electron. 29, 1950–1962 (1993).
[CrossRef]

1991 (1)

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

1990 (1)

T. F. Johnston, “M2concept characterizes beam quality,” Laser Focus 5, 173–183 (1990).

1961 (1)

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Abraham, N. B.

J. Martin-Regalado, F. Prati, M. San Miguel, N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[CrossRef]

Achtenhagen, M.

M. Achtenhagen, A. A. Hardy, E. Kapon, “Design of distributed Bragg reflector structures for transverse-mode discrimination in vertical-cavity surface-emitting lasers,” IEEE J. Lightwave Technol. 22, 1962–1967 (2004).
[CrossRef]

Babic, D. I.

D. I. Babic, Y. Chung, N. Dagli, J. E. Bowers, “Modal reflection of quarter-wave mirrors in vertical-cavity lasers,” IEEE J. Quantum Electron. 29, 1950–1962 (1993).
[CrossRef]

Baechtold, W.

M. Jungo, D. Erni, W. Baechtold, “Quasi-analytic steady-state solution of VCSEL rate equations including spatial hole burning and carrier diffusion losses,” Int. J. Numer. Model. 16, 143–159 (2003).
[CrossRef]

Baets, R.

P. Bienstman, R. Baets, A. Larsson, M. J. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. P. Bava, H. Wenzel, B. Klein, O. Conradi, R. Pregla, S. A. Riyopoulos, J.-F. P. Seurin, 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. Debernardi, G. P. Bava, “Coupled mode theory: a powerful tool for analyzing complex VCSELs and designing advanced device features,” IEEE J. Sel. Top. Quantum Electron. 9, 905–917 (2003).
[CrossRef]

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

G. P. Bava, P. Debernardi, L. Fratta, “Three-dimensional model for vectorial fields in vertical-cavity surface-emitting lasers,” Phys. Rev. A 63, 23816 (2001).
[CrossRef]

Bienstman, P.

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

Bissessur, H. K.

H. K. Bissessur, F. Koyama, K. Iga, “Modeling of oxide-confined vertical-cavity surface-emitting lasers,” IEEE J. Sel. Top. Quantum Electron. 3, 344–352 (1997).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

Bowers, J. E.

D. I. Babic, Y. Chung, N. Dagli, J. E. Bowers, “Modal reflection of quarter-wave mirrors in vertical-cavity lasers,” IEEE J. Quantum Electron. 29, 1950–1962 (1993).
[CrossRef]

Brunner, M.

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

Chung, Y.

D. I. Babic, Y. Chung, N. Dagli, J. E. Bowers, “Modal reflection of quarter-wave mirrors in vertical-cavity lasers,” IEEE J. Quantum Electron. 29, 1950–1962 (1993).
[CrossRef]

Conradi, O.

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

Dagli, N.

D. I. Babic, Y. Chung, N. Dagli, J. E. Bowers, “Modal reflection of quarter-wave mirrors in vertical-cavity lasers,” IEEE J. Quantum Electron. 29, 1950–1962 (1993).
[CrossRef]

Debernardi, P.

P. Debernardi, G. P. Bava, “Coupled mode theory: a powerful tool for analyzing complex VCSELs and designing advanced device features,” IEEE J. Sel. Top. Quantum Electron. 9, 905–917 (2003).
[CrossRef]

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

G. P. Bava, P. Debernardi, L. Fratta, “Three-dimensional model for vectorial fields in vertical-cavity surface-emitting lasers,” Phys. Rev. A 63, 23816 (2001).
[CrossRef]

Erni, D.

M. Jungo, D. Erni, W. Baechtold, “Quasi-analytic steady-state solution of VCSEL rate equations including spatial hole burning and carrier diffusion losses,” Int. J. Numer. Model. 16, 143–159 (2003).
[CrossRef]

Fox, A. G.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Fratta, L.

G. P. Bava, P. Debernardi, L. Fratta, “Three-dimensional model for vectorial fields in vertical-cavity surface-emitting lasers,” Phys. Rev. A 63, 23816 (2001).
[CrossRef]

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

Friedberg, S. H.

S. H. Friedberg, A. J. Insel, Introduction to Linear Algebra with Applications (Prentice-Hall, Englewood Cliffs, N.J., 1986).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gulden, K.

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

Hardy, A. A.

M. Achtenhagen, A. A. Hardy, E. Kapon, “Design of distributed Bragg reflector structures for transverse-mode discrimination in vertical-cavity surface-emitting lasers,” IEEE J. Lightwave Technol. 22, 1962–1967 (2004).
[CrossRef]

Iga, K.

H. K. Bissessur, F. Koyama, K. Iga, “Modeling of oxide-confined vertical-cavity surface-emitting lasers,” IEEE J. Sel. Top. Quantum Electron. 3, 344–352 (1997).
[CrossRef]

Insel, A. J.

S. H. Friedberg, A. J. Insel, Introduction to Linear Algebra with Applications (Prentice-Hall, Englewood Cliffs, N.J., 1986).

Johnston, T. F.

T. F. Johnston, “M2concept characterizes beam quality,” Laser Focus 5, 173–183 (1990).

Jungo, M.

M. Jungo, D. Erni, W. Baechtold, “Quasi-analytic steady-state solution of VCSEL rate equations including spatial hole burning and carrier diffusion losses,” Int. J. Numer. Model. 16, 143–159 (2003).
[CrossRef]

Kapon, E.

M. Achtenhagen, A. A. Hardy, E. Kapon, “Design of distributed Bragg reflector structures for transverse-mode discrimination in vertical-cavity surface-emitting lasers,” IEEE J. Lightwave Technol. 22, 1962–1967 (2004).
[CrossRef]

Klein, B.

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

Koyama, F.

H. K. Bissessur, F. Koyama, K. Iga, “Modeling of oxide-confined vertical-cavity surface-emitting lasers,” IEEE J. Sel. Top. Quantum Electron. 3, 344–352 (1997).
[CrossRef]

Larsson, A.

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

Li, T.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Martin-Regalado, J.

J. Martin-Regalado, F. Prati, M. San Miguel, N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[CrossRef]

Noble, M. J.

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

Prati, F.

J. Martin-Regalado, F. Prati, M. San Miguel, N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[CrossRef]

Pregla, R.

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

San Miguel, M.

J. Martin-Regalado, F. Prati, M. San Miguel, N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[CrossRef]

Seurin, J.-F. P.

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

Siegman, A. E.

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Valle, A.

A. Valle, “Selection and modulation of higher-order transverse modes in vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 34, 1924–1932 (1998).
[CrossRef]

Wenzel, H.

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

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

Bell Syst. Tech. J. (1)

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

IEEE J. Lightwave Technol. (1)

M. Achtenhagen, A. A. Hardy, E. Kapon, “Design of distributed Bragg reflector structures for transverse-mode discrimination in vertical-cavity surface-emitting lasers,” IEEE J. Lightwave Technol. 22, 1962–1967 (2004).
[CrossRef]

IEEE J. Quantum Electron. (5)

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

J. Martin-Regalado, F. Prati, M. San Miguel, N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[CrossRef]

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

A. Valle, “Selection and modulation of higher-order transverse modes in vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 34, 1924–1932 (1998).
[CrossRef]

D. I. Babic, Y. Chung, N. Dagli, J. E. Bowers, “Modal reflection of quarter-wave mirrors in vertical-cavity lasers,” IEEE J. Quantum Electron. 29, 1950–1962 (1993).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

P. Debernardi, G. P. Bava, “Coupled mode theory: a powerful tool for analyzing complex VCSELs and designing advanced device features,” IEEE J. Sel. Top. Quantum Electron. 9, 905–917 (2003).
[CrossRef]

H. K. Bissessur, F. Koyama, K. Iga, “Modeling of oxide-confined vertical-cavity surface-emitting lasers,” IEEE J. Sel. Top. Quantum Electron. 3, 344–352 (1997).
[CrossRef]

Int. J. Numer. Model. (1)

M. Jungo, D. Erni, W. Baechtold, “Quasi-analytic steady-state solution of VCSEL rate equations including spatial hole burning and carrier diffusion losses,” Int. J. Numer. Model. 16, 143–159 (2003).
[CrossRef]

Laser Focus (1)

T. F. Johnston, “M2concept characterizes beam quality,” Laser Focus 5, 173–183 (1990).

Phys. Rev. A (1)

G. P. Bava, P. Debernardi, L. Fratta, “Three-dimensional model for vectorial fields in vertical-cavity surface-emitting lasers,” Phys. Rev. A 63, 23816 (2001).
[CrossRef]

Other (4)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

S. H. Friedberg, A. J. Insel, Introduction to Linear Algebra with Applications (Prentice-Hall, Englewood Cliffs, N.J., 1986).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

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Figures (6)

Fig. 1
Fig. 1

(a) Schematic illustration of the VCSEL structure. (b) Multilayer structure used to model the DBRs.

Fig. 2
Fig. 2

Lowest-order TEMpq modes for a cavity with a Gaussian aperture and flat-band mirrors.

Fig. 3
Fig. 3

Absolute value squared of the eigenvalues for the TEMpq modes presented in Fig. 2 as functions of Fresnel number NF = ωap2L.

Fig. 4
Fig. 4

Absolute value squared of the eigenvalues of the lowest-order TEM00 mode for the flat-band mirror (dotted curves) and the broadband (dashed curves) and the narrowband (solid curves) DBR mirrors. The aperture shape and size are as in Fig. 3. Also shown is the difference |γ0|2 − |γ0|2 between the lowest-order TEM00 mode and the next-higher-order TEM01 mode.

Fig. 5
Fig. 5

Same as in Fig. 4 but now with the ring-peaked aperture.

Fig. 6
Fig. 6

Output beam quality of the TEM00 mode versus Fresnel number NF. Mirrors and aperture are as in Fig. 5.

Tables (1)

Tables Icon

Table 1 Specifications of the Three DBR Mirror Pairs Used in the Simulation

Equations (27)

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

E ( x , y , z ) = 1 ( 2 π ) 2 A ( k x , k y , z ) × exp [ j ( k x x + k y y ) ] d k x d k y ,
A ( k x , k y , z ) = E ( x , y , z ) exp [ j ( k x x + k y y ) ] d x d y ,
A ( k x , k y , z 2 ) = A ( k x , k y , z 1 ) exp [ j ( k 0 2 n c 2 k x 2 k y 2 ) 1 / 2 × ( z 2 z 1 ) ] .
E ( x , y , z ap ± ) = E ( x , y , z ap ) t a ( x , y ) ,
A s = A ( k x , k y , z 0 ) k y k x 2 + k y 2 [ k y , k x , 0 ] ,
A p = A ( k x , k y , z 0 ) k x k x 2 + k y 2 [ k x , k y , 0 ] .
r ̂ ( θ 0 ) = r 01 + r ( θ 1 ) exp ( j 2 β 1 ) 1 + r 01 r ( θ 1 ) exp ( j 2 β 1 ) ,
t ̂ ( θ 0 ) = t 01 t ( θ 1 ) exp ( j β 1 ) 1 + r 01 r ( θ 1 ) exp ( j 2 β 1 ) ,
A ref = A ( k x , k y , z 0 ) ( r ̂ s k y 2 + r ̂ p k x 2 k x 2 + k y 2 ) .
G ( N + 1 ) = | E ( x , y , 0 ) N + 1 E ( x , y , 0 ) N | 2 d x d y ,
γ = E ( x , y , 0 ) N + 1 E ( x , y , 0 ) N ,
t G ( x , y ) = exp ( x 2 + y 2 ω ap 2 ) .
t R ( x , y ) = t G ( x , y ) 0.2 U ( x , y ) .
r ( θ 1 ) = [ m 11 + m 12 P 4 ( s , p ) ] P 1 ( s , p ) [ m 21 + m 22 P 4 ( s , p ) ] [ m 11 + m 12 P 4 ( s , p ) ] P 1 ( s , p ) + [ m 21 + m 22 P 4 ( s , p ) ] ,
t ( θ 1 ) = 2 P 1 ( s , p ) [ m 11 + m 12 P 4 ( s , p ) ] P 1 ( s , p ) + [ m 21 + m 22 P 4 ( s , p ) ] ,
r 01 = P 0 ( s , p ) P 1 ( s , p ) P 0 ( s , p ) + P 1 ( s , p ) ,
t 01 = 2 P 0 ( s , p ) P 0 ( s , p ) + P 1 ( s , p ) .
P j ( s ) = n j cos ( θ j ) ,
P j ( p ) = ( 1 / n j ) cos ( θ j ) ,
sin ( θ j ) = ( n 0 / n j ) sin ( θ 0 ) .
β j = 2 π λ 0 n j h j cos ( θ j ) , j = 0 , , 3
m 11 = [ cos ( β 2 ) cos ( β 3 ) P 3 ( s , p ) P 2 ( s , p ) sin ( β 2 ) sin ( β 3 ) ] U M 1 ( a ) U M 2 ( a ) ,
m 12 = i [ 1 P 3 ( s , p ) cos ( β 2 ) sin ( β 3 ) + 1 P 2 ( s , p ) sin ( β 2 ) cos ( β 3 ) ] U M 1 ( a )
m 21 = i [ P 2 ( s , p ) sin ( β 2 ) cos ( β 3 ) + P 3 ( s , p ) cos ( β 2 ) sin ( β 3 ) ] U M 1 ( a ) ,
m 22 = [ cos ( β 2 ) cos ( β 3 ) P 2 ( s , p ) P 3 ( s , p ) sin ( β 2 ) sin ( β 3 ) ] U M 1 ( a ) U M 2 ( a ) ,
a = cos ( β 2 ) cos ( β 3 ) 1 2 [ P 2 ( s , p ) P 3 ( s , p ) + P 3 ( s , p ) P 2 ( s , p ) ] × sin ( β 2 ) sin ( β 3 ) ,
U M ( x ) = sin [ ( M + 1 ) cos 1 ( x ) ] ( 1 x 2 ) 1 / 2 .

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