Abstract

Mode coupling caused by the presence of an oxide dielectric aperture is addressed. For rays passing through the dielectric, we apply the complex reflection coefficient that corresponds to a Fabry–Perot subcavity formed by a slab dielectric and the nearest distributed Bragg reflector (DBR) mirror. The DBR reflectivity applies to rays passing through the aperture opening. The diffraction effects for curved wave-front incidence are also included. The expansion coefficients of the cavity eigenmodes into pure Gauss–Laguerre (GL) modes are obtained for the lowest eigenmodes in the optimum waist representation. The effects of the aperture location in the cavity standing wave are addressed. Over the entire range of aperture diameters, higher cavity losses and higher threshold currents result for antinode placement, owing to wide-angle scattering. This agrees with experimental results and earlier scattering analysis by use of pure GL (uncoupled) eigenmodes. Mixing with higher modes increases round-trip losses at small apertures, compared with uncoupled-mode results. In the limit of a large aperture diameter, the cavity eigenmodes decouple to pure GL modes.

© 2003 Optical Society of America

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References

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  1. S. Riyopoulos, D. Dialeti s, J-M. Inman, and A. Phillips, “Active cavity vertical-cavity surface-emitting laser eigenmodes with simple analytic representation,” J. Opt. Soc. Am. B 18, 1268–1284 (2001).
    [CrossRef]
  2. S. Riyopoulos and D. Dialetis, “Radiation scattering by apertures in vertical-cavity surface-emitting laser cavities and its effects on mode structure,” J. Opt. Soc. Am. B 18, 1497–1511 (2001).
    [CrossRef]
  3. G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. Corzine, “Comprehensive numerical modeling of VCSELs,” IEEE J. Quantum Electron. 32, 607 (1989).
    [CrossRef]
  4. B. Demeulenaere, P. Bienstman, B. Dhoedt, and R. G. Baets, “Detailed study of AlAs-oxidized apertures in VCSEL cavities for optimized modal performance,” IEEE J. Quantum Electron. 35, 358 (1999).
    [CrossRef]
  5. S. Riyopoulos, D. Dialetis, J. Liu, and B. Riely, “Generic representation of active cavity VCSEL eigenmodes by optimized waist gain guided Gauss–Laguerre modes,” IEEE J. Sel. Top. Quantum Electron. (to be published).
  6. K. L. Lear, K. D. Choquette, R. P. Schneider, and S. P. Kilcoyne, Appl. Phys. Lett. 66, 2616 (1994).
    [CrossRef]
  7. Y. G. Zhao and J. McInerney, “Transverse-mode control of VCSELs,” IEEE J. Quantum Electron. 32, 1950 (1996).
    [CrossRef]
  8. A. Bond, P. D. Dapkus, and J. D. O’Brien, “Design of low-loss, single-mode VCSELs,” IEEE J. Sel. Top. Quantum Electron. 5, 574 (1999).
    [CrossRef]
  9. See, for example, C. C. Davis in Lasers and Electro-optics (Cambridge U. Press, Cambridge, UK, 1996), pp. 73–79.
  10. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), pp. 108–118.
  11. S. Mukai, “Non-monotonic threshold size dependence of buried post VCLs,” IEEE J. Quantum Electron. 5, 552 (2001).
    [CrossRef]
  12. D. I. Babic and S. W. Corzine, “Analytic expressions for reflection delay, penetration depth and absorptance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28, 514 (1992).
    [CrossRef]
  13. D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, “Modal reflection of quarter-wave mirrors in VCSELs,” IEEE J. Quantum Electron. 29, 1950 (1993).
    [CrossRef]
  14. The standard FP etalon theory yields reflection maxima for ΔΦ΄=lπ+π/2. In that case, however, ΔΦ΄ gives the phase shift not between the optical element edges but between two reference surfaces located so that T=i|T| and R=−|R|, i.e., T is pure imaginary and R is pure negative (see Ref. 6).
  15. P. Bienstman, R. G. Baets, J. Vukusic, A. Larsson, M. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. Bava, H. Wenzel, B. Klein, R. Pregla, S. Riyopoulos, J.-F. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models of the simulation of position dependent effects of thin oxide apertures,” IEEE J. Quantum Electron. 37, 1618 (2001).
    [CrossRef]
  16. K. D. Choquette and R. E. Liebenquth, “Control of VCSEL polarization with anisotropic transverse cavity geometries,” IEEE Photonics Technol. Lett. 6, 40–42 (1994).
    [CrossRef]
  17. E. R. Hegblom, D. I. Babic, B. J. Thibeault, and L. A. Coldren, “Scattering losses from dielectric apertures in VCSELs,” IEEE J. Sel. Top. Quantum Electron. 3, 379 (1997).
    [CrossRef]
  18. See K. Okamoto, Fundamentals of Optical Waveguides (Academic, San Diego, Calif., 2000), pp. 35–39 and references therein.

2001 (4)

S. Riyopoulos, D. Dialeti s, J-M. Inman, and A. Phillips, “Active cavity vertical-cavity surface-emitting laser eigenmodes with simple analytic representation,” J. Opt. Soc. Am. B 18, 1268–1284 (2001).
[CrossRef]

S. Riyopoulos and D. Dialetis, “Radiation scattering by apertures in vertical-cavity surface-emitting laser cavities and its effects on mode structure,” J. Opt. Soc. Am. B 18, 1497–1511 (2001).
[CrossRef]

S. Mukai, “Non-monotonic threshold size dependence of buried post VCLs,” IEEE J. Quantum Electron. 5, 552 (2001).
[CrossRef]

P. Bienstman, R. G. Baets, J. Vukusic, A. Larsson, M. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. Bava, H. Wenzel, B. Klein, R. Pregla, S. Riyopoulos, J.-F. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models of the simulation of position dependent effects of thin oxide apertures,” IEEE J. Quantum Electron. 37, 1618 (2001).
[CrossRef]

1999 (2)

A. Bond, P. D. Dapkus, and J. D. O’Brien, “Design of low-loss, single-mode VCSELs,” IEEE J. Sel. Top. Quantum Electron. 5, 574 (1999).
[CrossRef]

B. Demeulenaere, P. Bienstman, B. Dhoedt, and R. G. Baets, “Detailed study of AlAs-oxidized apertures in VCSEL cavities for optimized modal performance,” IEEE J. Quantum Electron. 35, 358 (1999).
[CrossRef]

1997 (1)

E. R. Hegblom, D. I. Babic, B. J. Thibeault, and L. A. Coldren, “Scattering losses from dielectric apertures in VCSELs,” IEEE J. Sel. Top. Quantum Electron. 3, 379 (1997).
[CrossRef]

1996 (1)

Y. G. Zhao and J. McInerney, “Transverse-mode control of VCSELs,” IEEE J. Quantum Electron. 32, 1950 (1996).
[CrossRef]

1994 (2)

K. L. Lear, K. D. Choquette, R. P. Schneider, and S. P. Kilcoyne, Appl. Phys. Lett. 66, 2616 (1994).
[CrossRef]

K. D. Choquette and R. E. Liebenquth, “Control of VCSEL polarization with anisotropic transverse cavity geometries,” IEEE Photonics Technol. Lett. 6, 40–42 (1994).
[CrossRef]

1993 (1)

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

1992 (1)

D. I. Babic and S. W. Corzine, “Analytic expressions for reflection delay, penetration depth and absorptance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28, 514 (1992).
[CrossRef]

1989 (1)

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. Corzine, “Comprehensive numerical modeling of VCSELs,” IEEE J. Quantum Electron. 32, 607 (1989).
[CrossRef]

Babic, D. I.

E. R. Hegblom, D. I. Babic, B. J. Thibeault, and L. A. Coldren, “Scattering losses from dielectric apertures in VCSELs,” IEEE J. Sel. Top. Quantum Electron. 3, 379 (1997).
[CrossRef]

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

D. I. Babic and S. W. Corzine, “Analytic expressions for reflection delay, penetration depth and absorptance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28, 514 (1992).
[CrossRef]

Baets, R. G.

P. Bienstman, R. G. Baets, J. Vukusic, A. Larsson, M. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. Bava, H. Wenzel, B. Klein, R. Pregla, S. Riyopoulos, J.-F. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models of the simulation of position dependent effects of thin oxide apertures,” IEEE J. Quantum Electron. 37, 1618 (2001).
[CrossRef]

B. Demeulenaere, P. Bienstman, B. Dhoedt, and R. G. Baets, “Detailed study of AlAs-oxidized apertures in VCSEL cavities for optimized modal performance,” IEEE J. Quantum Electron. 35, 358 (1999).
[CrossRef]

Bava, G.

P. Bienstman, R. G. Baets, J. Vukusic, A. Larsson, M. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. Bava, H. Wenzel, B. Klein, R. Pregla, S. Riyopoulos, J.-F. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models of the simulation of position dependent effects of thin oxide apertures,” IEEE J. Quantum Electron. 37, 1618 (2001).
[CrossRef]

Bienstman, P.

P. Bienstman, R. G. Baets, J. Vukusic, A. Larsson, M. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. Bava, H. Wenzel, B. Klein, R. Pregla, S. Riyopoulos, J.-F. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models of the simulation of position dependent effects of thin oxide apertures,” IEEE J. Quantum Electron. 37, 1618 (2001).
[CrossRef]

B. Demeulenaere, P. Bienstman, B. Dhoedt, and R. G. Baets, “Detailed study of AlAs-oxidized apertures in VCSEL cavities for optimized modal performance,” IEEE J. Quantum Electron. 35, 358 (1999).
[CrossRef]

Bond, A.

A. Bond, P. D. Dapkus, and J. D. O’Brien, “Design of low-loss, single-mode VCSELs,” IEEE J. Sel. Top. Quantum Electron. 5, 574 (1999).
[CrossRef]

Bowers, J. E.

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

Brunner, M.

P. Bienstman, R. G. Baets, J. Vukusic, A. Larsson, M. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. Bava, H. Wenzel, B. Klein, R. Pregla, S. Riyopoulos, J.-F. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models of the simulation of position dependent effects of thin oxide apertures,” IEEE J. Quantum Electron. 37, 1618 (2001).
[CrossRef]

Choquette, K. D.

K. D. Choquette and R. E. Liebenquth, “Control of VCSEL polarization with anisotropic transverse cavity geometries,” IEEE Photonics Technol. Lett. 6, 40–42 (1994).
[CrossRef]

K. L. Lear, K. D. Choquette, R. P. Schneider, and S. P. Kilcoyne, Appl. Phys. Lett. 66, 2616 (1994).
[CrossRef]

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. Corzine, “Comprehensive numerical modeling of VCSELs,” IEEE J. Quantum Electron. 32, 607 (1989).
[CrossRef]

Chuang, S. L.

P. Bienstman, R. G. Baets, J. Vukusic, A. Larsson, M. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. Bava, H. Wenzel, B. Klein, R. Pregla, S. Riyopoulos, J.-F. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models of the simulation of position dependent effects of thin oxide apertures,” IEEE J. Quantum Electron. 37, 1618 (2001).
[CrossRef]

Chung, Y.

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

Coldren, L. A.

E. R. Hegblom, D. I. Babic, B. J. Thibeault, and L. A. Coldren, “Scattering losses from dielectric apertures in VCSELs,” IEEE J. Sel. Top. Quantum Electron. 3, 379 (1997).
[CrossRef]

Corzine, S.

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. Corzine, “Comprehensive numerical modeling of VCSELs,” IEEE J. Quantum Electron. 32, 607 (1989).
[CrossRef]

Corzine, S. W.

D. I. Babic and S. W. Corzine, “Analytic expressions for reflection delay, penetration depth and absorptance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28, 514 (1992).
[CrossRef]

Dagli, N.

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

Dapkus, P. D.

A. Bond, P. D. Dapkus, and J. D. O’Brien, “Design of low-loss, single-mode VCSELs,” IEEE J. Sel. Top. Quantum Electron. 5, 574 (1999).
[CrossRef]

Debernardi, P.

P. Bienstman, R. G. Baets, J. Vukusic, A. Larsson, M. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. Bava, H. Wenzel, B. Klein, R. Pregla, S. Riyopoulos, J.-F. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models of the simulation of position dependent effects of thin oxide apertures,” IEEE J. Quantum Electron. 37, 1618 (2001).
[CrossRef]

Demeulenaere, B.

B. Demeulenaere, P. Bienstman, B. Dhoedt, and R. G. Baets, “Detailed study of AlAs-oxidized apertures in VCSEL cavities for optimized modal performance,” IEEE J. Quantum Electron. 35, 358 (1999).
[CrossRef]

Dhoedt, B.

B. Demeulenaere, P. Bienstman, B. Dhoedt, and R. G. Baets, “Detailed study of AlAs-oxidized apertures in VCSEL cavities for optimized modal performance,” IEEE J. Quantum Electron. 35, 358 (1999).
[CrossRef]

Dialeti, D.

Dialetis, D.

Fratta, L.

P. Bienstman, R. G. Baets, J. Vukusic, A. Larsson, M. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. Bava, H. Wenzel, B. Klein, R. Pregla, S. Riyopoulos, J.-F. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models of the simulation of position dependent effects of thin oxide apertures,” IEEE J. Quantum Electron. 37, 1618 (2001).
[CrossRef]

Gulden, K.

P. Bienstman, R. G. Baets, J. Vukusic, A. Larsson, M. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. Bava, H. Wenzel, B. Klein, R. Pregla, S. Riyopoulos, J.-F. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models of the simulation of position dependent effects of thin oxide apertures,” IEEE J. Quantum Electron. 37, 1618 (2001).
[CrossRef]

Hadley, G. R.

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. Corzine, “Comprehensive numerical modeling of VCSELs,” IEEE J. Quantum Electron. 32, 607 (1989).
[CrossRef]

Hegblom, E. R.

E. R. Hegblom, D. I. Babic, B. J. Thibeault, and L. A. Coldren, “Scattering losses from dielectric apertures in VCSELs,” IEEE J. Sel. Top. Quantum Electron. 3, 379 (1997).
[CrossRef]

Inman, J-M.

Kilcoyne, S. P.

K. L. Lear, K. D. Choquette, R. P. Schneider, and S. P. Kilcoyne, Appl. Phys. Lett. 66, 2616 (1994).
[CrossRef]

Klein, B.

P. Bienstman, R. G. Baets, J. Vukusic, A. Larsson, M. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. Bava, H. Wenzel, B. Klein, R. Pregla, S. Riyopoulos, J.-F. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models of the simulation of position dependent effects of thin oxide apertures,” IEEE J. Quantum Electron. 37, 1618 (2001).
[CrossRef]

Larsson, A.

P. Bienstman, R. G. Baets, J. Vukusic, A. Larsson, M. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. Bava, H. Wenzel, B. Klein, R. Pregla, S. Riyopoulos, J.-F. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models of the simulation of position dependent effects of thin oxide apertures,” IEEE J. Quantum Electron. 37, 1618 (2001).
[CrossRef]

Lear, K. L.

K. L. Lear, K. D. Choquette, R. P. Schneider, and S. P. Kilcoyne, Appl. Phys. Lett. 66, 2616 (1994).
[CrossRef]

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. Corzine, “Comprehensive numerical modeling of VCSELs,” IEEE J. Quantum Electron. 32, 607 (1989).
[CrossRef]

Liebenquth, R. E.

K. D. Choquette and R. E. Liebenquth, “Control of VCSEL polarization with anisotropic transverse cavity geometries,” IEEE Photonics Technol. Lett. 6, 40–42 (1994).
[CrossRef]

McInerney, J.

Y. G. Zhao and J. McInerney, “Transverse-mode control of VCSELs,” IEEE J. Quantum Electron. 32, 1950 (1996).
[CrossRef]

Mukai, S.

S. Mukai, “Non-monotonic threshold size dependence of buried post VCLs,” IEEE J. Quantum Electron. 5, 552 (2001).
[CrossRef]

Noble, M.

P. Bienstman, R. G. Baets, J. Vukusic, A. Larsson, M. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. Bava, H. Wenzel, B. Klein, R. Pregla, S. Riyopoulos, J.-F. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models of the simulation of position dependent effects of thin oxide apertures,” IEEE J. Quantum Electron. 37, 1618 (2001).
[CrossRef]

O’Brien, J. D.

A. Bond, P. D. Dapkus, and J. D. O’Brien, “Design of low-loss, single-mode VCSELs,” IEEE J. Sel. Top. Quantum Electron. 5, 574 (1999).
[CrossRef]

Phillips, A.

Pregla, R.

P. Bienstman, R. G. Baets, J. Vukusic, A. Larsson, M. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. Bava, H. Wenzel, B. Klein, R. Pregla, S. Riyopoulos, J.-F. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models of the simulation of position dependent effects of thin oxide apertures,” IEEE J. Quantum Electron. 37, 1618 (2001).
[CrossRef]

Riyopoulos, S.

P. Bienstman, R. G. Baets, J. Vukusic, A. Larsson, M. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. Bava, H. Wenzel, B. Klein, R. Pregla, S. Riyopoulos, J.-F. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models of the simulation of position dependent effects of thin oxide apertures,” IEEE J. Quantum Electron. 37, 1618 (2001).
[CrossRef]

S. Riyopoulos and D. Dialetis, “Radiation scattering by apertures in vertical-cavity surface-emitting laser cavities and its effects on mode structure,” J. Opt. Soc. Am. B 18, 1497–1511 (2001).
[CrossRef]

S. Riyopoulos, D. Dialeti s, J-M. Inman, and A. Phillips, “Active cavity vertical-cavity surface-emitting laser eigenmodes with simple analytic representation,” J. Opt. Soc. Am. B 18, 1268–1284 (2001).
[CrossRef]

Schneider, R. P.

K. L. Lear, K. D. Choquette, R. P. Schneider, and S. P. Kilcoyne, Appl. Phys. Lett. 66, 2616 (1994).
[CrossRef]

Scott, J. W.

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. Corzine, “Comprehensive numerical modeling of VCSELs,” IEEE J. Quantum Electron. 32, 607 (1989).
[CrossRef]

Seurin, J.-F.

P. Bienstman, R. G. Baets, J. Vukusic, A. Larsson, M. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. Bava, H. Wenzel, B. Klein, R. Pregla, S. Riyopoulos, J.-F. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models of the simulation of position dependent effects of thin oxide apertures,” IEEE J. Quantum Electron. 37, 1618 (2001).
[CrossRef]

Thibeault, B. J.

E. R. Hegblom, D. I. Babic, B. J. Thibeault, and L. A. Coldren, “Scattering losses from dielectric apertures in VCSELs,” IEEE J. Sel. Top. Quantum Electron. 3, 379 (1997).
[CrossRef]

Vukusic, J.

P. Bienstman, R. G. Baets, J. Vukusic, A. Larsson, M. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. Bava, H. Wenzel, B. Klein, R. Pregla, S. Riyopoulos, J.-F. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models of the simulation of position dependent effects of thin oxide apertures,” IEEE J. Quantum Electron. 37, 1618 (2001).
[CrossRef]

Warren, M. E.

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. Corzine, “Comprehensive numerical modeling of VCSELs,” IEEE J. Quantum Electron. 32, 607 (1989).
[CrossRef]

Wenzel, H.

P. Bienstman, R. G. Baets, J. Vukusic, A. Larsson, M. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. Bava, H. Wenzel, B. Klein, R. Pregla, S. Riyopoulos, J.-F. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models of the simulation of position dependent effects of thin oxide apertures,” IEEE J. Quantum Electron. 37, 1618 (2001).
[CrossRef]

Zhao, Y. G.

Y. G. Zhao and J. McInerney, “Transverse-mode control of VCSELs,” IEEE J. Quantum Electron. 32, 1950 (1996).
[CrossRef]

Appl. Phys. Lett. (1)

K. L. Lear, K. D. Choquette, R. P. Schneider, and S. P. Kilcoyne, Appl. Phys. Lett. 66, 2616 (1994).
[CrossRef]

IEEE J. Quantum Electron. (7)

Y. G. Zhao and J. McInerney, “Transverse-mode control of VCSELs,” IEEE J. Quantum Electron. 32, 1950 (1996).
[CrossRef]

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. Corzine, “Comprehensive numerical modeling of VCSELs,” IEEE J. Quantum Electron. 32, 607 (1989).
[CrossRef]

B. Demeulenaere, P. Bienstman, B. Dhoedt, and R. G. Baets, “Detailed study of AlAs-oxidized apertures in VCSEL cavities for optimized modal performance,” IEEE J. Quantum Electron. 35, 358 (1999).
[CrossRef]

S. Mukai, “Non-monotonic threshold size dependence of buried post VCLs,” IEEE J. Quantum Electron. 5, 552 (2001).
[CrossRef]

D. I. Babic and S. W. Corzine, “Analytic expressions for reflection delay, penetration depth and absorptance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28, 514 (1992).
[CrossRef]

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

P. Bienstman, R. G. Baets, J. Vukusic, A. Larsson, M. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. Bava, H. Wenzel, B. Klein, R. Pregla, S. Riyopoulos, J.-F. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models of the simulation of position dependent effects of thin oxide apertures,” IEEE J. Quantum Electron. 37, 1618 (2001).
[CrossRef]

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

E. R. Hegblom, D. I. Babic, B. J. Thibeault, and L. A. Coldren, “Scattering losses from dielectric apertures in VCSELs,” IEEE J. Sel. Top. Quantum Electron. 3, 379 (1997).
[CrossRef]

A. Bond, P. D. Dapkus, and J. D. O’Brien, “Design of low-loss, single-mode VCSELs,” IEEE J. Sel. Top. Quantum Electron. 5, 574 (1999).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

K. D. Choquette and R. E. Liebenquth, “Control of VCSEL polarization with anisotropic transverse cavity geometries,” IEEE Photonics Technol. Lett. 6, 40–42 (1994).
[CrossRef]

J. Opt. Soc. Am. B (2)

Other (5)

S. Riyopoulos, D. Dialetis, J. Liu, and B. Riely, “Generic representation of active cavity VCSEL eigenmodes by optimized waist gain guided Gauss–Laguerre modes,” IEEE J. Sel. Top. Quantum Electron. (to be published).

See, for example, C. C. Davis in Lasers and Electro-optics (Cambridge U. Press, Cambridge, UK, 1996), pp. 73–79.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), pp. 108–118.

See K. Okamoto, Fundamentals of Optical Waveguides (Academic, San Diego, Calif., 2000), pp. 35–39 and references therein.

The standard FP etalon theory yields reflection maxima for ΔΦ΄=lπ+π/2. In that case, however, ΔΦ΄ gives the phase shift not between the optical element edges but between two reference surfaces located so that T=i|T| and R=−|R|, i.e., T is pure imaginary and R is pure negative (see Ref. 6).

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Figures (17)

Fig. 1
Fig. 1

Schematic illustration of diffraction and scattering in an oxide aperture cavity: (a) amplitude and (b) wave fronts.

Fig. 2
Fig. 2

Schematic illustration of ray paths for reflection and transmission: (a) FP cavity, (b) slab, and (c) curved envelope effects. Only incidence from the left is considered.

Fig. 3
Fig. 3

(a) Model of the dielectric slab–DBR mirror subcavity. (b) Dielectric aperture–DBR mirror subcavity with a<am. (c) No subcavity forms for a>am.

Fig. 4
Fig. 4

(a) Reflection (solid curve) and transmission coefficients through a dielectric slab versus thickness kd. (b) FP reflection coefficients from a slab–DBR cavity versus center slab separation from the DBR edge kD, for various slab thicknesses kd. Maximum for matched quarter pair kD=kd=π/2.

Fig. 5
Fig. 5

Reflection coefficients for TM polarization versus edge-to-edge separation from DBR kD for various thin slab thicknesses kd: (a) magnitude |ζ| and (b) cosine of the phase.

Fig. 6
Fig. 6

Magnitude of the reflection coefficient from an aperture in front of a DBR versus edge-to-edge separation from DBR kD for various thin slab thicknesses kd: (a) TM polarization and (b) TE polarization.

Fig. 7
Fig. 7

Round-trip amplification factor versus aperture radius for antinode placement. The two coupled-mode branches are compared with the uncoupled fundamental (00) and first radial (01) modes.

Fig. 8
Fig. 8

Expansion coefficients of the two Ψ+ and Ψ- modes into the fundamental (00) and first radial (01) GL modes versus the aperture radius (antinode).

Fig. 9
Fig. 9

Relative frequency shift for the two Ψ+ and Ψ- from the nominal resonant frequency without the aperture (antinode).

Fig. 10
Fig. 10

Round-trip amplification factor versus the aperture radius for node placement. The two coupled-mode branches are compared with the uncoupled fundamental (00) and first radial (01) modes.

Fig. 11
Fig. 11

Expansion coefficients of the two Ψ+ and Ψ- modes into the fundamental (00) and first radial (01) GL modes versus the aperture radius (node).

Fig. 12
Fig. 12

Relative frequency shift for the two Ψ+ and Ψ- from the nominal resonant frequency without the aperture (node).

Fig. 13
Fig. 13

Intensity profile for antinode placement of the fundamental axisymmetric eigenmode |Ψ+|2 for two aperture radii a=0.60 μm (solid curve) and a=3.00  μm (dashed curve).

Fig. 14
Fig. 14

Envelope phase of pure GL modes (end of round trip) for two aperture radii a=0.60 μm (solid curve) and a=3.00μm (dashed curve).

Fig. 15
Fig. 15

Envelope phase (end of round trip) of the fundamental axisymmetric eigenmode |Ψ+|2 for node placement and aperture radii a=0.60 μm (solid curve) and a=3.00 μm (dashed curve).

Fig. 16
Fig. 16

Envelope phase (end of round trip) of the fundamental axisymmetric eigenmode |Ψ+|2 for antinode placement and aperture radii a=0.60 μm (solid curve) and a=3.00 μm (dashed curve).

Fig. 17
Fig. 17

Gain threshold versus aperture distance from the active layer, for various aperture diameters.

Equations (66)

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A(x, y, z)=Ψ(x, y, z)exp(ikz-iωt).
2ik Ψz+2Ψ=0.
ψmp(ρ, θ, z)=Ump(Υ)exp(ipθ)expik ρ22R×exp[-i(m+2p+1)φ],
Ump(Υ)=2πW21/2m!(m+p)!1/2Υp/2Lmp(Υ)×exp(-Υ/2),
A(j)(ρ, θ, z)=Ψ(j)(ρ, θ, z)exp(ikjz-iωjt)=m,pCmp(j)ψmp(ρ, θ, z)exp(ikjz-iωjt).
T(z2)ψ[W(z1), R(z1), φ(z1)]ψ[W(z1+z2),
R(z1+z2), φ(z1+z2)].
R=rQ,
G=I+g*Q.
Qmpnq(a/W)=02πdθ0adρρψpm(r; z)ψqn*(r; z),
Pmpnq(L/b)=02πdθ0dρρψpm(r; z=L)ψqn*(r; z=0),
NbaπLˆ=ka22πLˆ,
Σmpmp=1-12m!(m+p)! (k1deff)21-2×Ump2(2ν2) 1+ν/ka[1+(2ν/ka)2]212 ς*.
Λ=P1Σ1R1Σ1GcPrΣrRrΣrGc.
det|Λ-ζ(j)I|=0.
Ψ(m)p(r)ψmp(r; wo),
ζ(m)pΛmpmp(wo).
w |ψmp|Λ|ψmp|=w |Λ0000(w)|=0.
|Λ0000(w)|=|P0000|Rl0000Gc0000|P0000|Rr0000Gc0000+O(ε2);
RFP=Rd+TdRmTdl=0[RmRdexp(2iΔ)]l=Rd+Td2Rmexp(2iΔ)1-RdRmexp(2iΔ),
TFP=TdTmexp(iΔ)l=0[RmRdexp(2iΔ)]l=TdTmexp(iΔ)1-RdRmexp(2iΔ),
Rd=rd+tdrdtdl=0[rdrdexp(2iδ)]l=rd+rdtdtdexp(2iδ)1-rdrdexp(2iδ),
Td=tdtdexp(iδ)l=0[rdrdexp(2iδ)]l=tdtdexp(iδ)1-rdrdexp(2iδ),
rd=±n-nn+n=-rd,
td=nn+n,td=nn+n,
RFP=Rd+TdRmTdPFPl=0[RmRdPFPexp(2iΔ)]l=Rd+Td2PFPRmexp(2iΔ)1-PFPRdRmexp(2iΔ),
TFP=TdTmexp(iΔ)l=0[RmRdPFP*exp(2iΔ)]l=TdTmexp(iΔ)1-PFP*RdRmexp(2iΔ).
PFP=ψ(r; Ld)|ψ(r; Ld+2Dˆ)ψ(r; 0)|ψ(r; 2Dˆ)P(Dˆ),
Θ=rQa+(Qm-Qa)R˜FP+Rd(I-Qm),a<am,
Θ=rQm+Rd(I-Qa),a>am,
rPLˆQa+PLˆ-Dˆ[(Qm-Qa)R˜FP+Rd(I-Qm)],
Θr(I-Δ)+ς*Δ,
ς*=ς exp[-i2k(D-d/2)]
ς=Rd+(Td2-Rd2)Rmexp(2iΔ)1-RdRmexp(2iΔ),
Θ1OOΘ2. . .. . .,
Θp=r+Δ0p0p(ς*-r)Δ1p0p(ς*-r)Δ2p0p(ς*-r):Δ0p1p(ς*-r)r+Δ1p1p(ς*-r)Δ2p1p(ς*-r):Δ2p2p(ς*-r)Δ1p2p(ζ*-r)r+Δ2p2p(ς*-r):.........
Rdmx=±2rd1+rd2,Tdmx=td21+rd2=1-rd21+rd2.
R=Rdmx-Rmexp(2iΔ)1-RdmxRmexp(2iΔ).
|Rmx|2=(Rdmx+Rm)2(1+RdmxRm)2
Θ00=Rm[1-Δ00(μ)]+Δ00(μ)|ς*|exp(iΔϕ).
|Θ00|=[Rm2(1-Δ00)2+2|ς|2Δ002+2|ς|(1-Δ00)Δ00cos Δϕ]1/2,
Z=PlΣlRlΣlGcPrΣrRrΣrGcΘ=ΛΘ,
Z0-ζO: OZ1-ζ: ....:=0,
Zp=Λ0p0pΘ0p0pΛ0p0pΘ1p0pΛ0p0pΘ2p0p:Λ1p1pΘ0p1pΛ1p1pΘ1p1pΛ1p1pΘ2p1p:Λ2p2pΘ0p2pΛ2p2pΘ1p2pΛ2p2pΘ2p2p:.........
Λ00Θ00-ζΛ00Θ10Λ11Θ01Λ11Θ11-ζ=0.
ζ˜±=Λ00Θ00+Λ11Θ112±12[(Λ00Θ00-Λ11Θ11)2-4Λ00Θ10Λ11Θ01]1/2.
Ψ+=1(1+|C˜01|2)1/2 (ψ0+C˜01ψ1),
Ψ-=1(1+|C˜10|2)1/2 (ψ1+C˜10ψ0),
C˜01=-Λ00Θ00-ζ+Λ00Θ10,C˜10=-Λ11Θ11-ζ-Λ11Θ01.
|ζ±|μ=0.
|Ψ+(ρ, z)|={[U01(ρ¯2)+|C˜01|U11(ρ¯2)cos Δφ]2+[|C˜01|U11(ρ¯2)sin Δφ]2}1/2(1+|C˜01|2)1/2,
Φ(ρ, z)=kρ22R(z)-φ01(z)+tan-1|C˜01|U10(ρ¯2)sin ΔφU01(ρ¯2)+|C˜01|U11(ρ¯2)cos Δφ,
(1+2g¯)=2(S00+S11)2±[(S00-S11)2+4S01S10]1/2.
R±11+2go=11+2g¯G00+G112  ,
Qmpnq=2παmpαnqδpqexp{-i[(m-n)+2(p-q)]φ}1402νdX×Lpm(X)Lqn*(X)exp(-X),
Pmpnq=2παmpαnqδpqexp{-i[(m-n)+2(p-q)]φ}14ξ0dX×Lpm(ξX)exp[i(b/2R)X]Lqn*(X)exp(-X),
Fmpnq(ν, ξ, γ)=2παmpαnqδpqexp{-i[(m-n)+2(p-q)]φ}×14ξ02νdXLpm(ξX)Lqn*(X)×exp[-(γ/2)X].
Fmpnq(ν, ξ, γ)
=δpqexp{-i[(m-n)+2(p-q)]φ}×[m!n!(m+p)!(n+q)!]1/22γp+1ξ(p+1)/2×k=0ml=0nξ2k/22γk+l×(-1)k+l+p(k+l+p)!k!l!(m-k)!(n-l)!(p+k)!(p+l)!×1-exp(-2νγ/2)i=0k+l+p(2ν)k+l+p-i(k+l+p-i)!.
Qmpnq(μ, N)=Fmpnq for νμ21+1πN2ξ1γ2,
Pmpnq(μ, N)=Fmpnqforνξ11+1πN2γ1+1-i 1πN1+1πN2.
R(θ)-R(θ)R(θ)
=2 sin2 θ(1-ν)[ν2cos θ+(ν2-sin2 θ)1/2][cos θ+(ν2-sin2 θ)1/2]θ21/ν-1(1+ν)2,
T(θ)-T(θ)T(θ)
=2 cos θ(ν-1)[(ν2-sin2 θ)1/2-ν cos θ][ν2cos θ+(ν2-sin2 θ)1/2][cos θ+(ν2-sin2 θ)1/2]θ2ν-1/ννν-1(1+ν)2,
R(0)-R(θ)R(0)=2 ν cos θ-(ν2-sin2 θ)1/2(ν-1)[cos θ+(ν2-sin2 θ)1/2]θ21/ν-νν2-1.

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