Abstract

Microcavities with subwavelength features, such as oxide apertures and etched-mesa edges, suffer wide-angle scattering that cannot be captured by the paraxial propagation limit; hence the scattering cannot be fully accounted for by index guiding or by lenslike phase shifts. We present a systematic treatment by using the Born approximation in the vector Maxwell equations. We then introduce the scattering losses in the cavity round-trip matrix, by using a Gauss–Laguerre representation of the cavity modes. Optimization of the round-trip coefficient including confinement, diffraction, and scattering losses yields the mode waist in laterally open vertical cavity surface emitting laser (VCSEL) cavities. A simple equation relating the current aperture to mode spot size is obtained. The analytic results are applied to etched-mesa and oxide-confined VCSEL designs. Predictions for the mode waist and threshold current are comparable with experimental results in oxide-confined VCSELs.

© 2001 Optical Society of America

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  1. D. L. Huffaker, D. G. Deppe, K. Kumar, and T. J. Rogers, “Native-oxide defined ring contact for low threshold VCSELs,” Appl. Phys. Lett. 65, 97–99 (1994).
    [CrossRef]
  2. K. D. Choquette, R. P. Schneider, K. L. Lear, and K. M. Geib, “Low threshold VCSELs fabricated by selective oxidization,” Electron. Lett. 30, 2043–2044 (1994).
    [CrossRef]
  3. Y. Hayashi, T. Mukaihara, N. Hatori, N. Ohnoki, A. Matsutani, F. Koyama, and K. Iga, “Record low threshold index-guided InGaAs/GaAlAs VCSELs with native oxide confinement structure,” Electron. Lett. 31, 560–561 (1995).
    [CrossRef]
  4. G. M. Yang, M. H. MacDougal, and P. D. Dapkus, “Ultralow threshold current VCSELs obtained with selective oxidization,” Electron. Lett. 31, 886–887 (1995).
    [CrossRef]
  5. B. J. Thibeault, K. Bertilsson, E. R. Hegblom, E. Strzelecka, P. D. Floyd, R. Naone, and L. A. Coldren, “High-speed characteristics of low optical loss oxide-apertured VCSELs,” IEEE Photon. Technol. Lett. 9, 11–13 (1997).
    [CrossRef]
  6. H. J. Unold, S. W. Mahmoud, F. Eberhard, R. Jaeger, M. Kicherer, F. Mederer, M. C. Riedl, and K. J. Ebeling, “Large-area, single-mode selectively oxidized VCSELs: approaches and experimental,” in Vertical-Cavity Surface Emitting Lasers IV, K. D. Choquette and C. Lei, eds. Proc. SPIE 3946, 207–217 (2000).
    [CrossRef]
  7. 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, 607–616 (1996).
    [CrossRef]
  8. 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, 379–389 (1997).
    [CrossRef]
  9. D. Burak and R. Binder, “Cold-cavity vectorial eigenmodes of VCSEL’s,” IEEE J. Quantum Electron. 33, 1205–1215 (1997).
    [CrossRef]
  10. K. D. Choquette, W. W. Chow, G. R. Hadley, H. Q. Chow, and K. M. Gab, “Scalability of small-aperture selectively oxidized VCSELs,” Appl. Phys. Lett. 70, 823–825 (1997).
    [CrossRef]
  11. G. Liu, J.-F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barnes, and T. N. Tiouririne, “Mode-selectivity study of VCSELs,” Appl. Phys. Lett. 73, 726–729 (1998).
    [CrossRef]
  12. A. E. Bond, P. D. Dapkus, and J. D. O’Brien, “Design of low-loss single-mode VCSELs,” IEEE J. Sel. Top. in Quantum Electron. 5, 574–581 (1999).
    [CrossRef]
  13. 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, 358–367 (1999).
    [CrossRef]
  14. P. Bienstman, R. G. Baets, J. Vukusic, A. Larsson, M. Noble, M. Brunner, K. Gulden, P. Debernardi, H. Wenzel, and B. Klein, “Comparison of optical VCSEL models of the simulation of position dependent effects of thin oxide apertures,” Rep. COST268 (1999).
  15. S. Riyopoulos, D. Dialetis, J-M. Inman, and A. Phillips, “Active-cavity vertical-cavity surface-emiting laser eigenmodes with simple analytic representation,” J. Opt. Soc. Am. B 18, 1268–1284 (2001).
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  18. 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).
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  20. See, for example, C. C. Davis, Lasers and Electro-optics (Cambridge, U. Press, Cambridge, UK, 1996), pp. 416–423.
  21. S. P. Hegarty, G. Huyet, P. Porta, J. G. McInerney, K. D. Choquette, K. M. Geib, and H. Q. Hou, “Transverse mode structure and pattern formation in oxide-confined VCSELs,” J. Opt. Soc. Am. B 16, 2060–2071 (2000).
    [CrossRef]
  22. J. D. Lambkin, T. Calvert, B. Corbett, J. Woodhead, S. M. Pinches, A. Onischenko, T. E. Sale, J. Hosea, P. Van Daele, K. Van de Putte, A. Van Hove, A. Valster, J. G. McInerney, and P. Porta, “Development of a red VCSEL-to-plastic fiber module for use in parallel optical links,” (in Vertical-Cavity Surface-Emitting Lasers IV, K. D. Choquette and C. Lei eds.) Proc. SPIE 3946, 95–105 (2000).
    [CrossRef]

2001 (1)

2000 (3)

J. D. Lambkin, T. Calvert, B. Corbett, J. Woodhead, S. M. Pinches, A. Onischenko, T. E. Sale, J. Hosea, P. Van Daele, K. Van de Putte, A. Van Hove, A. Valster, J. G. McInerney, and P. Porta, “Development of a red VCSEL-to-plastic fiber module for use in parallel optical links,” (in Vertical-Cavity Surface-Emitting Lasers IV, K. D. Choquette and C. Lei eds.) Proc. SPIE 3946, 95–105 (2000).
[CrossRef]

S. P. Hegarty, G. Huyet, P. Porta, J. G. McInerney, K. D. Choquette, K. M. Geib, and H. Q. Hou, “Transverse mode structure and pattern formation in oxide-confined VCSELs,” J. Opt. Soc. Am. B 16, 2060–2071 (2000).
[CrossRef]

H. J. Unold, S. W. Mahmoud, F. Eberhard, R. Jaeger, M. Kicherer, F. Mederer, M. C. Riedl, and K. J. Ebeling, “Large-area, single-mode selectively oxidized VCSELs: approaches and experimental,” in Vertical-Cavity Surface Emitting Lasers IV, K. D. Choquette and C. Lei, eds. Proc. SPIE 3946, 207–217 (2000).
[CrossRef]

1999 (2)

A. E. Bond, P. D. Dapkus, and J. D. O’Brien, “Design of low-loss single-mode VCSELs,” IEEE J. Sel. Top. in Quantum Electron. 5, 574–581 (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, 358–367 (1999).
[CrossRef]

1998 (1)

G. Liu, J.-F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barnes, and T. N. Tiouririne, “Mode-selectivity study of VCSELs,” Appl. Phys. Lett. 73, 726–729 (1998).
[CrossRef]

1997 (4)

B. J. Thibeault, K. Bertilsson, E. R. Hegblom, E. Strzelecka, P. D. Floyd, R. Naone, and L. A. Coldren, “High-speed characteristics of low optical loss oxide-apertured VCSELs,” IEEE Photon. Technol. Lett. 9, 11–13 (1997).
[CrossRef]

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, 379–389 (1997).
[CrossRef]

D. Burak and R. Binder, “Cold-cavity vectorial eigenmodes of VCSEL’s,” IEEE J. Quantum Electron. 33, 1205–1215 (1997).
[CrossRef]

K. D. Choquette, W. W. Chow, G. R. Hadley, H. Q. Chow, and K. M. Gab, “Scalability of small-aperture selectively oxidized VCSELs,” Appl. Phys. Lett. 70, 823–825 (1997).
[CrossRef]

1996 (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, 607–616 (1996).
[CrossRef]

1995 (2)

Y. Hayashi, T. Mukaihara, N. Hatori, N. Ohnoki, A. Matsutani, F. Koyama, and K. Iga, “Record low threshold index-guided InGaAs/GaAlAs VCSELs with native oxide confinement structure,” Electron. Lett. 31, 560–561 (1995).
[CrossRef]

G. M. Yang, M. H. MacDougal, and P. D. Dapkus, “Ultralow threshold current VCSELs obtained with selective oxidization,” Electron. Lett. 31, 886–887 (1995).
[CrossRef]

1994 (2)

D. L. Huffaker, D. G. Deppe, K. Kumar, and T. J. Rogers, “Native-oxide defined ring contact for low threshold VCSELs,” Appl. Phys. Lett. 65, 97–99 (1994).
[CrossRef]

K. D. Choquette, R. P. Schneider, K. L. Lear, and K. M. Geib, “Low threshold VCSELs fabricated by selective oxidization,” Electron. Lett. 30, 2043–2044 (1994).
[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, 514–524 (1992).
[CrossRef]

Babic, D. I.

G. Liu, J.-F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barnes, and T. N. Tiouririne, “Mode-selectivity study of VCSELs,” Appl. Phys. Lett. 73, 726–729 (1998).
[CrossRef]

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, 379–389 (1997).
[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, 514–524 (1992).
[CrossRef]

Baets, R. G.

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, 358–367 (1999).
[CrossRef]

Barnes, D. C.

G. Liu, J.-F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barnes, and T. N. Tiouririne, “Mode-selectivity study of VCSELs,” Appl. Phys. Lett. 73, 726–729 (1998).
[CrossRef]

Bertilsson, K.

B. J. Thibeault, K. Bertilsson, E. R. Hegblom, E. Strzelecka, P. D. Floyd, R. Naone, and L. A. Coldren, “High-speed characteristics of low optical loss oxide-apertured VCSELs,” IEEE Photon. Technol. Lett. 9, 11–13 (1997).
[CrossRef]

Bienstman, P.

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, 358–367 (1999).
[CrossRef]

Binder, R.

D. Burak and R. Binder, “Cold-cavity vectorial eigenmodes of VCSEL’s,” IEEE J. Quantum Electron. 33, 1205–1215 (1997).
[CrossRef]

Bond, A. E.

A. E. Bond, P. D. Dapkus, and J. D. O’Brien, “Design of low-loss single-mode VCSELs,” IEEE J. Sel. Top. in Quantum Electron. 5, 574–581 (1999).
[CrossRef]

Burak, D.

D. Burak and R. Binder, “Cold-cavity vectorial eigenmodes of VCSEL’s,” IEEE J. Quantum Electron. 33, 1205–1215 (1997).
[CrossRef]

Calvert, T.

J. D. Lambkin, T. Calvert, B. Corbett, J. Woodhead, S. M. Pinches, A. Onischenko, T. E. Sale, J. Hosea, P. Van Daele, K. Van de Putte, A. Van Hove, A. Valster, J. G. McInerney, and P. Porta, “Development of a red VCSEL-to-plastic fiber module for use in parallel optical links,” (in Vertical-Cavity Surface-Emitting Lasers IV, K. D. Choquette and C. Lei eds.) Proc. SPIE 3946, 95–105 (2000).
[CrossRef]

Choquette, K. D.

S. P. Hegarty, G. Huyet, P. Porta, J. G. McInerney, K. D. Choquette, K. M. Geib, and H. Q. Hou, “Transverse mode structure and pattern formation in oxide-confined VCSELs,” J. Opt. Soc. Am. B 16, 2060–2071 (2000).
[CrossRef]

K. D. Choquette, W. W. Chow, G. R. Hadley, H. Q. Chow, and K. M. Gab, “Scalability of small-aperture selectively oxidized VCSELs,” Appl. Phys. Lett. 70, 823–825 (1997).
[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, 607–616 (1996).
[CrossRef]

K. D. Choquette, R. P. Schneider, K. L. Lear, and K. M. Geib, “Low threshold VCSELs fabricated by selective oxidization,” Electron. Lett. 30, 2043–2044 (1994).
[CrossRef]

Chow, H. Q.

K. D. Choquette, W. W. Chow, G. R. Hadley, H. Q. Chow, and K. M. Gab, “Scalability of small-aperture selectively oxidized VCSELs,” Appl. Phys. Lett. 70, 823–825 (1997).
[CrossRef]

Chow, W. W.

K. D. Choquette, W. W. Chow, G. R. Hadley, H. Q. Chow, and K. M. Gab, “Scalability of small-aperture selectively oxidized VCSELs,” Appl. Phys. Lett. 70, 823–825 (1997).
[CrossRef]

Chuang, S. L.

G. Liu, J.-F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barnes, and T. N. Tiouririne, “Mode-selectivity study of VCSELs,” Appl. Phys. Lett. 73, 726–729 (1998).
[CrossRef]

Coldren, L. A.

B. J. Thibeault, K. Bertilsson, E. R. Hegblom, E. Strzelecka, P. D. Floyd, R. Naone, and L. A. Coldren, “High-speed characteristics of low optical loss oxide-apertured VCSELs,” IEEE Photon. Technol. Lett. 9, 11–13 (1997).
[CrossRef]

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, 379–389 (1997).
[CrossRef]

Corbett, B.

J. D. Lambkin, T. Calvert, B. Corbett, J. Woodhead, S. M. Pinches, A. Onischenko, T. E. Sale, J. Hosea, P. Van Daele, K. Van de Putte, A. Van Hove, A. Valster, J. G. McInerney, and P. Porta, “Development of a red VCSEL-to-plastic fiber module for use in parallel optical links,” (in Vertical-Cavity Surface-Emitting Lasers IV, K. D. Choquette and C. Lei eds.) Proc. SPIE 3946, 95–105 (2000).
[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, 607–616 (1996).
[CrossRef]

Corzine, S. W.

G. Liu, J.-F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barnes, and T. N. Tiouririne, “Mode-selectivity study of VCSELs,” Appl. Phys. Lett. 73, 726–729 (1998).
[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, 514–524 (1992).
[CrossRef]

Dapkus, P. D.

A. E. Bond, P. D. Dapkus, and J. D. O’Brien, “Design of low-loss single-mode VCSELs,” IEEE J. Sel. Top. in Quantum Electron. 5, 574–581 (1999).
[CrossRef]

G. M. Yang, M. H. MacDougal, and P. D. Dapkus, “Ultralow threshold current VCSELs obtained with selective oxidization,” Electron. Lett. 31, 886–887 (1995).
[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, 358–367 (1999).
[CrossRef]

Deppe, D. G.

D. L. Huffaker, D. G. Deppe, K. Kumar, and T. J. Rogers, “Native-oxide defined ring contact for low threshold VCSELs,” Appl. Phys. Lett. 65, 97–99 (1994).
[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, 358–367 (1999).
[CrossRef]

Dialetis, D.

Ebeling, K. J.

H. J. Unold, S. W. Mahmoud, F. Eberhard, R. Jaeger, M. Kicherer, F. Mederer, M. C. Riedl, and K. J. Ebeling, “Large-area, single-mode selectively oxidized VCSELs: approaches and experimental,” in Vertical-Cavity Surface Emitting Lasers IV, K. D. Choquette and C. Lei, eds. Proc. SPIE 3946, 207–217 (2000).
[CrossRef]

Eberhard, F.

H. J. Unold, S. W. Mahmoud, F. Eberhard, R. Jaeger, M. Kicherer, F. Mederer, M. C. Riedl, and K. J. Ebeling, “Large-area, single-mode selectively oxidized VCSELs: approaches and experimental,” in Vertical-Cavity Surface Emitting Lasers IV, K. D. Choquette and C. Lei, eds. Proc. SPIE 3946, 207–217 (2000).
[CrossRef]

Floyd, P. D.

B. J. Thibeault, K. Bertilsson, E. R. Hegblom, E. Strzelecka, P. D. Floyd, R. Naone, and L. A. Coldren, “High-speed characteristics of low optical loss oxide-apertured VCSELs,” IEEE Photon. Technol. Lett. 9, 11–13 (1997).
[CrossRef]

Gab, K. M.

K. D. Choquette, W. W. Chow, G. R. Hadley, H. Q. Chow, and K. M. Gab, “Scalability of small-aperture selectively oxidized VCSELs,” Appl. Phys. Lett. 70, 823–825 (1997).
[CrossRef]

Geib, K. M.

S. P. Hegarty, G. Huyet, P. Porta, J. G. McInerney, K. D. Choquette, K. M. Geib, and H. Q. Hou, “Transverse mode structure and pattern formation in oxide-confined VCSELs,” J. Opt. Soc. Am. B 16, 2060–2071 (2000).
[CrossRef]

K. D. Choquette, R. P. Schneider, K. L. Lear, and K. M. Geib, “Low threshold VCSELs fabricated by selective oxidization,” Electron. Lett. 30, 2043–2044 (1994).
[CrossRef]

Hadley, G. R.

K. D. Choquette, W. W. Chow, G. R. Hadley, H. Q. Chow, and K. M. Gab, “Scalability of small-aperture selectively oxidized VCSELs,” Appl. Phys. Lett. 70, 823–825 (1997).
[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, 607–616 (1996).
[CrossRef]

Hatori, N.

Y. Hayashi, T. Mukaihara, N. Hatori, N. Ohnoki, A. Matsutani, F. Koyama, and K. Iga, “Record low threshold index-guided InGaAs/GaAlAs VCSELs with native oxide confinement structure,” Electron. Lett. 31, 560–561 (1995).
[CrossRef]

Hayashi, Y.

Y. Hayashi, T. Mukaihara, N. Hatori, N. Ohnoki, A. Matsutani, F. Koyama, and K. Iga, “Record low threshold index-guided InGaAs/GaAlAs VCSELs with native oxide confinement structure,” Electron. Lett. 31, 560–561 (1995).
[CrossRef]

Hegarty, S. P.

Hegblom, E. R.

B. J. Thibeault, K. Bertilsson, E. R. Hegblom, E. Strzelecka, P. D. Floyd, R. Naone, and L. A. Coldren, “High-speed characteristics of low optical loss oxide-apertured VCSELs,” IEEE Photon. Technol. Lett. 9, 11–13 (1997).
[CrossRef]

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, 379–389 (1997).
[CrossRef]

Hosea, J.

J. D. Lambkin, T. Calvert, B. Corbett, J. Woodhead, S. M. Pinches, A. Onischenko, T. E. Sale, J. Hosea, P. Van Daele, K. Van de Putte, A. Van Hove, A. Valster, J. G. McInerney, and P. Porta, “Development of a red VCSEL-to-plastic fiber module for use in parallel optical links,” (in Vertical-Cavity Surface-Emitting Lasers IV, K. D. Choquette and C. Lei eds.) Proc. SPIE 3946, 95–105 (2000).
[CrossRef]

Hou, H. Q.

Huffaker, D. L.

D. L. Huffaker, D. G. Deppe, K. Kumar, and T. J. Rogers, “Native-oxide defined ring contact for low threshold VCSELs,” Appl. Phys. Lett. 65, 97–99 (1994).
[CrossRef]

Huyet, G.

Iga, K.

Y. Hayashi, T. Mukaihara, N. Hatori, N. Ohnoki, A. Matsutani, F. Koyama, and K. Iga, “Record low threshold index-guided InGaAs/GaAlAs VCSELs with native oxide confinement structure,” Electron. Lett. 31, 560–561 (1995).
[CrossRef]

Inman, J-M.

Jaeger, R.

H. J. Unold, S. W. Mahmoud, F. Eberhard, R. Jaeger, M. Kicherer, F. Mederer, M. C. Riedl, and K. J. Ebeling, “Large-area, single-mode selectively oxidized VCSELs: approaches and experimental,” in Vertical-Cavity Surface Emitting Lasers IV, K. D. Choquette and C. Lei, eds. Proc. SPIE 3946, 207–217 (2000).
[CrossRef]

Kicherer, M.

H. J. Unold, S. W. Mahmoud, F. Eberhard, R. Jaeger, M. Kicherer, F. Mederer, M. C. Riedl, and K. J. Ebeling, “Large-area, single-mode selectively oxidized VCSELs: approaches and experimental,” in Vertical-Cavity Surface Emitting Lasers IV, K. D. Choquette and C. Lei, eds. Proc. SPIE 3946, 207–217 (2000).
[CrossRef]

Koyama, F.

Y. Hayashi, T. Mukaihara, N. Hatori, N. Ohnoki, A. Matsutani, F. Koyama, and K. Iga, “Record low threshold index-guided InGaAs/GaAlAs VCSELs with native oxide confinement structure,” Electron. Lett. 31, 560–561 (1995).
[CrossRef]

Kumar, K.

D. L. Huffaker, D. G. Deppe, K. Kumar, and T. J. Rogers, “Native-oxide defined ring contact for low threshold VCSELs,” Appl. Phys. Lett. 65, 97–99 (1994).
[CrossRef]

Lambkin, J. D.

J. D. Lambkin, T. Calvert, B. Corbett, J. Woodhead, S. M. Pinches, A. Onischenko, T. E. Sale, J. Hosea, P. Van Daele, K. Van de Putte, A. Van Hove, A. Valster, J. G. McInerney, and P. Porta, “Development of a red VCSEL-to-plastic fiber module for use in parallel optical links,” (in Vertical-Cavity Surface-Emitting Lasers IV, K. D. Choquette and C. Lei eds.) Proc. SPIE 3946, 95–105 (2000).
[CrossRef]

Lear, K. L.

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, 607–616 (1996).
[CrossRef]

K. D. Choquette, R. P. Schneider, K. L. Lear, and K. M. Geib, “Low threshold VCSELs fabricated by selective oxidization,” Electron. Lett. 30, 2043–2044 (1994).
[CrossRef]

Liu, G.

G. Liu, J.-F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barnes, and T. N. Tiouririne, “Mode-selectivity study of VCSELs,” Appl. Phys. Lett. 73, 726–729 (1998).
[CrossRef]

MacDougal, M. H.

G. M. Yang, M. H. MacDougal, and P. D. Dapkus, “Ultralow threshold current VCSELs obtained with selective oxidization,” Electron. Lett. 31, 886–887 (1995).
[CrossRef]

Mahmoud, S. W.

H. J. Unold, S. W. Mahmoud, F. Eberhard, R. Jaeger, M. Kicherer, F. Mederer, M. C. Riedl, and K. J. Ebeling, “Large-area, single-mode selectively oxidized VCSELs: approaches and experimental,” in Vertical-Cavity Surface Emitting Lasers IV, K. D. Choquette and C. Lei, eds. Proc. SPIE 3946, 207–217 (2000).
[CrossRef]

Matsutani, A.

Y. Hayashi, T. Mukaihara, N. Hatori, N. Ohnoki, A. Matsutani, F. Koyama, and K. Iga, “Record low threshold index-guided InGaAs/GaAlAs VCSELs with native oxide confinement structure,” Electron. Lett. 31, 560–561 (1995).
[CrossRef]

McInerney, J. G.

S. P. Hegarty, G. Huyet, P. Porta, J. G. McInerney, K. D. Choquette, K. M. Geib, and H. Q. Hou, “Transverse mode structure and pattern formation in oxide-confined VCSELs,” J. Opt. Soc. Am. B 16, 2060–2071 (2000).
[CrossRef]

J. D. Lambkin, T. Calvert, B. Corbett, J. Woodhead, S. M. Pinches, A. Onischenko, T. E. Sale, J. Hosea, P. Van Daele, K. Van de Putte, A. Van Hove, A. Valster, J. G. McInerney, and P. Porta, “Development of a red VCSEL-to-plastic fiber module for use in parallel optical links,” (in Vertical-Cavity Surface-Emitting Lasers IV, K. D. Choquette and C. Lei eds.) Proc. SPIE 3946, 95–105 (2000).
[CrossRef]

Mederer, F.

H. J. Unold, S. W. Mahmoud, F. Eberhard, R. Jaeger, M. Kicherer, F. Mederer, M. C. Riedl, and K. J. Ebeling, “Large-area, single-mode selectively oxidized VCSELs: approaches and experimental,” in Vertical-Cavity Surface Emitting Lasers IV, K. D. Choquette and C. Lei, eds. Proc. SPIE 3946, 207–217 (2000).
[CrossRef]

Mukaihara, T.

Y. Hayashi, T. Mukaihara, N. Hatori, N. Ohnoki, A. Matsutani, F. Koyama, and K. Iga, “Record low threshold index-guided InGaAs/GaAlAs VCSELs with native oxide confinement structure,” Electron. Lett. 31, 560–561 (1995).
[CrossRef]

Naone, R.

B. J. Thibeault, K. Bertilsson, E. R. Hegblom, E. Strzelecka, P. D. Floyd, R. Naone, and L. A. Coldren, “High-speed characteristics of low optical loss oxide-apertured VCSELs,” IEEE Photon. Technol. Lett. 9, 11–13 (1997).
[CrossRef]

O’Brien, J. D.

A. E. Bond, P. D. Dapkus, and J. D. O’Brien, “Design of low-loss single-mode VCSELs,” IEEE J. Sel. Top. in Quantum Electron. 5, 574–581 (1999).
[CrossRef]

Ohnoki, N.

Y. Hayashi, T. Mukaihara, N. Hatori, N. Ohnoki, A. Matsutani, F. Koyama, and K. Iga, “Record low threshold index-guided InGaAs/GaAlAs VCSELs with native oxide confinement structure,” Electron. Lett. 31, 560–561 (1995).
[CrossRef]

Onischenko, A.

J. D. Lambkin, T. Calvert, B. Corbett, J. Woodhead, S. M. Pinches, A. Onischenko, T. E. Sale, J. Hosea, P. Van Daele, K. Van de Putte, A. Van Hove, A. Valster, J. G. McInerney, and P. Porta, “Development of a red VCSEL-to-plastic fiber module for use in parallel optical links,” (in Vertical-Cavity Surface-Emitting Lasers IV, K. D. Choquette and C. Lei eds.) Proc. SPIE 3946, 95–105 (2000).
[CrossRef]

Phillips, A.

Pinches, S. M.

J. D. Lambkin, T. Calvert, B. Corbett, J. Woodhead, S. M. Pinches, A. Onischenko, T. E. Sale, J. Hosea, P. Van Daele, K. Van de Putte, A. Van Hove, A. Valster, J. G. McInerney, and P. Porta, “Development of a red VCSEL-to-plastic fiber module for use in parallel optical links,” (in Vertical-Cavity Surface-Emitting Lasers IV, K. D. Choquette and C. Lei eds.) Proc. SPIE 3946, 95–105 (2000).
[CrossRef]

Porta, P.

J. D. Lambkin, T. Calvert, B. Corbett, J. Woodhead, S. M. Pinches, A. Onischenko, T. E. Sale, J. Hosea, P. Van Daele, K. Van de Putte, A. Van Hove, A. Valster, J. G. McInerney, and P. Porta, “Development of a red VCSEL-to-plastic fiber module for use in parallel optical links,” (in Vertical-Cavity Surface-Emitting Lasers IV, K. D. Choquette and C. Lei eds.) Proc. SPIE 3946, 95–105 (2000).
[CrossRef]

S. P. Hegarty, G. Huyet, P. Porta, J. G. McInerney, K. D. Choquette, K. M. Geib, and H. Q. Hou, “Transverse mode structure and pattern formation in oxide-confined VCSELs,” J. Opt. Soc. Am. B 16, 2060–2071 (2000).
[CrossRef]

Riedl, M. C.

H. J. Unold, S. W. Mahmoud, F. Eberhard, R. Jaeger, M. Kicherer, F. Mederer, M. C. Riedl, and K. J. Ebeling, “Large-area, single-mode selectively oxidized VCSELs: approaches and experimental,” in Vertical-Cavity Surface Emitting Lasers IV, K. D. Choquette and C. Lei, eds. Proc. SPIE 3946, 207–217 (2000).
[CrossRef]

Riyopoulos, S.

Rogers, T. J.

D. L. Huffaker, D. G. Deppe, K. Kumar, and T. J. Rogers, “Native-oxide defined ring contact for low threshold VCSELs,” Appl. Phys. Lett. 65, 97–99 (1994).
[CrossRef]

Sale, T. E.

J. D. Lambkin, T. Calvert, B. Corbett, J. Woodhead, S. M. Pinches, A. Onischenko, T. E. Sale, J. Hosea, P. Van Daele, K. Van de Putte, A. Van Hove, A. Valster, J. G. McInerney, and P. Porta, “Development of a red VCSEL-to-plastic fiber module for use in parallel optical links,” (in Vertical-Cavity Surface-Emitting Lasers IV, K. D. Choquette and C. Lei eds.) Proc. SPIE 3946, 95–105 (2000).
[CrossRef]

Schneider, R. P.

K. D. Choquette, R. P. Schneider, K. L. Lear, and K. M. Geib, “Low threshold VCSELs fabricated by selective oxidization,” Electron. Lett. 30, 2043–2044 (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, 607–616 (1996).
[CrossRef]

Seurin, J.-F.

G. Liu, J.-F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barnes, and T. N. Tiouririne, “Mode-selectivity study of VCSELs,” Appl. Phys. Lett. 73, 726–729 (1998).
[CrossRef]

Strzelecka, E.

B. J. Thibeault, K. Bertilsson, E. R. Hegblom, E. Strzelecka, P. D. Floyd, R. Naone, and L. A. Coldren, “High-speed characteristics of low optical loss oxide-apertured VCSELs,” IEEE Photon. Technol. Lett. 9, 11–13 (1997).
[CrossRef]

Tan, M.

G. Liu, J.-F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barnes, and T. N. Tiouririne, “Mode-selectivity study of VCSELs,” Appl. Phys. Lett. 73, 726–729 (1998).
[CrossRef]

Thibeault, B. J.

B. J. Thibeault, K. Bertilsson, E. R. Hegblom, E. Strzelecka, P. D. Floyd, R. Naone, and L. A. Coldren, “High-speed characteristics of low optical loss oxide-apertured VCSELs,” IEEE Photon. Technol. Lett. 9, 11–13 (1997).
[CrossRef]

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, 379–389 (1997).
[CrossRef]

Tiouririne, T. N.

G. Liu, J.-F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barnes, and T. N. Tiouririne, “Mode-selectivity study of VCSELs,” Appl. Phys. Lett. 73, 726–729 (1998).
[CrossRef]

Unold, H. J.

H. J. Unold, S. W. Mahmoud, F. Eberhard, R. Jaeger, M. Kicherer, F. Mederer, M. C. Riedl, and K. J. Ebeling, “Large-area, single-mode selectively oxidized VCSELs: approaches and experimental,” in Vertical-Cavity Surface Emitting Lasers IV, K. D. Choquette and C. Lei, eds. Proc. SPIE 3946, 207–217 (2000).
[CrossRef]

Valster, A.

J. D. Lambkin, T. Calvert, B. Corbett, J. Woodhead, S. M. Pinches, A. Onischenko, T. E. Sale, J. Hosea, P. Van Daele, K. Van de Putte, A. Van Hove, A. Valster, J. G. McInerney, and P. Porta, “Development of a red VCSEL-to-plastic fiber module for use in parallel optical links,” (in Vertical-Cavity Surface-Emitting Lasers IV, K. D. Choquette and C. Lei eds.) Proc. SPIE 3946, 95–105 (2000).
[CrossRef]

Van Daele, P.

J. D. Lambkin, T. Calvert, B. Corbett, J. Woodhead, S. M. Pinches, A. Onischenko, T. E. Sale, J. Hosea, P. Van Daele, K. Van de Putte, A. Van Hove, A. Valster, J. G. McInerney, and P. Porta, “Development of a red VCSEL-to-plastic fiber module for use in parallel optical links,” (in Vertical-Cavity Surface-Emitting Lasers IV, K. D. Choquette and C. Lei eds.) Proc. SPIE 3946, 95–105 (2000).
[CrossRef]

Van de Putte, K.

J. D. Lambkin, T. Calvert, B. Corbett, J. Woodhead, S. M. Pinches, A. Onischenko, T. E. Sale, J. Hosea, P. Van Daele, K. Van de Putte, A. Van Hove, A. Valster, J. G. McInerney, and P. Porta, “Development of a red VCSEL-to-plastic fiber module for use in parallel optical links,” (in Vertical-Cavity Surface-Emitting Lasers IV, K. D. Choquette and C. Lei eds.) Proc. SPIE 3946, 95–105 (2000).
[CrossRef]

Van Hove, A.

J. D. Lambkin, T. Calvert, B. Corbett, J. Woodhead, S. M. Pinches, A. Onischenko, T. E. Sale, J. Hosea, P. Van Daele, K. Van de Putte, A. Van Hove, A. Valster, J. G. McInerney, and P. Porta, “Development of a red VCSEL-to-plastic fiber module for use in parallel optical links,” (in Vertical-Cavity Surface-Emitting Lasers IV, K. D. Choquette and C. Lei eds.) Proc. SPIE 3946, 95–105 (2000).
[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, 607–616 (1996).
[CrossRef]

Woodhead, J.

J. D. Lambkin, T. Calvert, B. Corbett, J. Woodhead, S. M. Pinches, A. Onischenko, T. E. Sale, J. Hosea, P. Van Daele, K. Van de Putte, A. Van Hove, A. Valster, J. G. McInerney, and P. Porta, “Development of a red VCSEL-to-plastic fiber module for use in parallel optical links,” (in Vertical-Cavity Surface-Emitting Lasers IV, K. D. Choquette and C. Lei eds.) Proc. SPIE 3946, 95–105 (2000).
[CrossRef]

Yang, G. M.

G. M. Yang, M. H. MacDougal, and P. D. Dapkus, “Ultralow threshold current VCSELs obtained with selective oxidization,” Electron. Lett. 31, 886–887 (1995).
[CrossRef]

Appl. Phys. Lett. (3)

D. L. Huffaker, D. G. Deppe, K. Kumar, and T. J. Rogers, “Native-oxide defined ring contact for low threshold VCSELs,” Appl. Phys. Lett. 65, 97–99 (1994).
[CrossRef]

K. D. Choquette, W. W. Chow, G. R. Hadley, H. Q. Chow, and K. M. Gab, “Scalability of small-aperture selectively oxidized VCSELs,” Appl. Phys. Lett. 70, 823–825 (1997).
[CrossRef]

G. Liu, J.-F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barnes, and T. N. Tiouririne, “Mode-selectivity study of VCSELs,” Appl. Phys. Lett. 73, 726–729 (1998).
[CrossRef]

Electron. Lett. (3)

K. D. Choquette, R. P. Schneider, K. L. Lear, and K. M. Geib, “Low threshold VCSELs fabricated by selective oxidization,” Electron. Lett. 30, 2043–2044 (1994).
[CrossRef]

Y. Hayashi, T. Mukaihara, N. Hatori, N. Ohnoki, A. Matsutani, F. Koyama, and K. Iga, “Record low threshold index-guided InGaAs/GaAlAs VCSELs with native oxide confinement structure,” Electron. Lett. 31, 560–561 (1995).
[CrossRef]

G. M. Yang, M. H. MacDougal, and P. D. Dapkus, “Ultralow threshold current VCSELs obtained with selective oxidization,” Electron. Lett. 31, 886–887 (1995).
[CrossRef]

IEEE J. Quantum Electron. (4)

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, 607–616 (1996).
[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, 514–524 (1992).
[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, 358–367 (1999).
[CrossRef]

D. Burak and R. Binder, “Cold-cavity vectorial eigenmodes of VCSEL’s,” IEEE J. Quantum Electron. 33, 1205–1215 (1997).
[CrossRef]

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

A. E. Bond, P. D. Dapkus, and J. D. O’Brien, “Design of low-loss single-mode VCSELs,” IEEE J. Sel. Top. in Quantum Electron. 5, 574–581 (1999).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron (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, 379–389 (1997).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

B. J. Thibeault, K. Bertilsson, E. R. Hegblom, E. Strzelecka, P. D. Floyd, R. Naone, and L. A. Coldren, “High-speed characteristics of low optical loss oxide-apertured VCSELs,” IEEE Photon. Technol. Lett. 9, 11–13 (1997).
[CrossRef]

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

Proc. SPIE (2)

H. J. Unold, S. W. Mahmoud, F. Eberhard, R. Jaeger, M. Kicherer, F. Mederer, M. C. Riedl, and K. J. Ebeling, “Large-area, single-mode selectively oxidized VCSELs: approaches and experimental,” in Vertical-Cavity Surface Emitting Lasers IV, K. D. Choquette and C. Lei, eds. Proc. SPIE 3946, 207–217 (2000).
[CrossRef]

J. D. Lambkin, T. Calvert, B. Corbett, J. Woodhead, S. M. Pinches, A. Onischenko, T. E. Sale, J. Hosea, P. Van Daele, K. Van de Putte, A. Van Hove, A. Valster, J. G. McInerney, and P. Porta, “Development of a red VCSEL-to-plastic fiber module for use in parallel optical links,” (in Vertical-Cavity Surface-Emitting Lasers IV, K. D. Choquette and C. Lei eds.) Proc. SPIE 3946, 95–105 (2000).
[CrossRef]

Other (5)

P. Bienstman, R. G. Baets, J. Vukusic, A. Larsson, M. Noble, M. Brunner, K. Gulden, P. Debernardi, H. Wenzel, and B. Klein, “Comparison of optical VCSEL models of the simulation of position dependent effects of thin oxide apertures,” Rep. COST268 (1999).

J. D. Jackson Classical Electrodynamics 2nd. ed. (Wiley, New York, 1975), pp. 391–397.

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).

D. I. Babic, “Double-fused long wavelength VCSELs,” Ph.D. dissertation (University of California, Santa Barbara, Calif. 1995), pp. 67–212.

See, for example, C. C. Davis, Lasers and Electro-optics (Cambridge, U. Press, Cambridge, UK, 1996), pp. 416–423.

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Figures (12)

Fig. 1
Fig. 1

Illustrations of typical (a) oxide-aperture VCSEL cavities, (b) proton-implanted mesa structure.

Fig. 2
Fig. 2

Geometry and coordinates for the scattered radiation computation, depicting the aperture segment of the cavity.

Fig. 3
Fig. 3

Plots of (a) the scattering factor normalized to the rim intensity σ/Ump2; (b) the normalized scattering factor ς versus kw for oxide aperture. A fixed ratio a/w=2 is used.

Fig. 4
Fig. 4

Contour plots demonstrating the near-invariance of the normalized scattering factor ς over the operational VCSEL parameter space; maximum 0.485, minimum 0.430, contour interval 0.0055.

Fig. 5
Fig. 5

Scattered radiation fraction versus aperture-to-waist ratio a2/w2 for the first four GL modes: (a) oxide aperture, (b) etched mesa.

Fig. 6
Fig. 6

Comparison of finite mirror self-interference and scattering losses versus the ratio w/a for (a) oxide aperture, (b) etched mesa.

Fig. 7
Fig. 7

Ratio of cavity mode waist to aperture radius versus oxide aperture radius for the lowest-three cavity eigenmodes. For comparison, dashed lines show the mode waist that would result by ignoring scattering effects.

Fig. 8
Fig. 8

Ratio of cavity mode waist to etched mesa radius a versus mesa radius for the lowest-three cavity eigenmodes. For comparison, dashed lines show the mode waist that would result by ignoring scattering effects.

Fig. 9
Fig. 9

Total round-trip losses vs. oxide-aperture radius for the lowest-three cavity eigenmodes. For comparison, dashed lines show the mode waist that would result by ignoring scattering effects.

Fig. 10
Fig. 10

Total round trip losses versus mesa radius for the lowest-three cavity eigenmodes. For comparison, dashed lines show the mode waist that would result by ignoring scattering effects.

Fig. 11
Fig. 11

Threshold current: (solid) theory and (open) experiment squares for the etched mesa 1.55-µm VCSEL detailed in Ref. 19, Table 7.3, design F170. Crosses mark the predicted threshold if the mesa edge were placed at a maximum (antinode) of the standing wave.

Fig. 12
Fig. 12

Theory–experiment comparison for an oxide aperture VCSEL. (a) Fundamental mode near-field FWHM versus aperture size; solid curve for theory; points, experimental values; dashed curves, the weak-guiding theory. (b) Threshold current density versus aperture size; solid curves, theory; points, experiment.

Equations (77)

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

Atot(r)=Ain(r)+1c1 Vd3rJ(r)exp(ik1|r-r|)|r-r|,
Atot(r)=Ain(r)+exp(ik1r)c1r Vd3rJ(r)×exp(-ik1n·r),
Atot(r)=Ain(r)+exp(ik1r)c1r V1d3r14πχ1Etott×exp(-ik1n·r)+exp(ik1r)c1r
×V2d3r14πχ2χ1Etott exp(-ik1n·r).
Atot(r)=Ain(r)+exp(ik1r)c1r V1+V2d3r14π χ1Etott×exp(-ik1n·r)+exp(ik1r)c1r×V2d3r14πχ2χ1-1×Etott exp(-ik1n·r).
Asc(r)=-iωc1r exp(ik1r)2-14π1×V2d3rEin exp(-ik1n·r),
Ein=[ExΥx(r)xˆ, EyΥy(r)yˆ, EzΥz(r)zˆ]mp exp(-ik1z),
Aj(r)=-ik1Ejexp(ik1r)r2-14π1 Fj,
FjV2d3rΥmp(r)exp(ik1z-ik1n·r).
dPscdΩ=c18πk14Eo22-14π12|(nˆ×F)×nˆ|2.
dPscdΩ=c18πk14Eo22-14π12|1-(nˆ·xˆ)|2|F|2.
Υmp(ρ, θ, z)=αmpUmp(2ρ2/W2)sin pθcos pθexpik1ρ22Rexp[-i(2m+p+1)φ], Ump(X)=χp/2Lmp(X)exp(-X/2), αmp=2/(1+δp0)πW21/2m!(m+p)!1/2
W(z)=w(1+z2/b2)1/2,
R(z)=z2+b2z,
φ=tan-1zb,
b=12kw2.
F(θ, ϕ)=adρ02πdϕρUmp2ρ2W2sin pϕcos pϕ×exp(-ik1ρ2/2R)×exp[-ikρ sin θ cos(ϕ-ϕ)×za-d/2za+d/2dz exp(-ik1z(cos θ-1)].
Eo[exp(ik1z)+r exp(-ik1z)exp(-iωt)]
Zdza-d/2za+d/2dz exp[ik1z(1-cos θ)]+exp[-ik1z(1+cos θ)].
Smp=12m!(m+p)!(k1d)2(k1W)22-11212σmp,
σmp=0πdθ sin θ|Z|2I21-sin2 θ1+δp121-sin2 θ1-δp12,
|Z|2cos2(k1za)1+sin k1dk1d2+sin2(k1za)1-sin k1dk1d2,
I=a/wdρ¯ρ¯Ump(2ρ¯2)Jp(k1Wρ¯sin θ),
Ump2(2ν)k1W(k1W+ν)[(k1W)2+(2ν)2]2.
Smp=12m!(m+p)!(k1d)22-112×Ump2(2ν)νk1a+11+2νk1a2212ς,
Eo[exp(-κz+ik1z)+r exp(-κz-ik1z)]
Lˆ/2dz exp(-kz)[(exp(ikz-ik1z cos θ)+r exp(-ikz-ikz cos θ)]deffZ,
Smp=12m!(m+p)!k1κ2(k1W)22-11212σmp,
|Z|2=(1+r)2+(1-r)2(k1/κ)2(1-cos θ)2[1+(k1/κ)2(1-cos θ)2]2cos2k1L2.
deff=1κ,
Smp=12m!(m+p)!(k1deff)22-112×Ump2(2ν)νk1a+11+2νk1a2212ς*,
Λ=PlTclRlTlcGcPrTcrRrTrcGc.
R=rQ(1, ν),
G=1+g*Q(1, ν),
P=Q(ξ, )×1-1212Lˆ/b1+(Lˆ/b)22(p+2)!p!21+ξ22+iQ(ξ, )12Lˆ/b1+(Lˆ/b)2(p+1)!p!21+ξ2,
NbaπLˆ=k1a22πLˆ.
mpmp=1-Smp(k1W, a/W),
Λˆmpmp=Λmpmp(w)[1-Smp(w)]1/2Λmpmp(w)×1-12Smp(w).
w|ψmp|Λˆ|ψmp|=0.
RΛˆ(w)=RP2r2Q2(1+gQ2)(1-S2)P1r1Q1×(1-S1)(1+gQ1), IΛˆ(w)=IP1+IP2+gα(Q2+Q1).
Λˆ(μ)1+ΔG1+Δ1+ΔR1+Δ2+ΔG2+ΔR2+ΔP1+ΔP2.
ΔR1,2=r1,2-1,
ΔG1,2=g(1+iα)[1-exp(-2µ)],
ΔP1,2=-μ21πN214-581πN2μ2-i12μ1πN
Δj=-18(k1d)22-112ς1+μk1a11+(zj¯/πN)2μ2×1-exp-2µ1+(zj¯/πN)2μ2,
ΔR1r11-exp-2µ1+1/4(1/πN)2μ2-1,
Δj=-18m!(m+p)!(k1d)22-112ς*×1+μk1a11+(1/2πN)2μ2×1-exp-2µ1+(1/2πN)2μ2.
4g+2r1+14(k1deff)21-212ςexp(-2µ)
=12μ1πN2,
RmpΛˆmp(wo)(1+g)2,
go lnJJtr=-ln R002Mdqwα00,
Jth(a)=76.6 expα(a)843.
2koΨτ=(2+ko2Δ)Ψ
(x, y)dzΦ*(x, y, z)Φ/dzΦ*Φ.
KaJ-1(Ka)J0(κa)+κaI-1(κa)I0(κa)=0.
F(θ,ϕ)=Zd2πadρρUmp2ρ2W2Jp(k1ρ sin θ)cos pϕsin pϕ,
Z=exp(-ik cos θza)cos(k1za)sin1-cos θ2k1d1-cos θ2k1d+sin1+cos θ2k1d1+cos θ2k1d-i sin(k1za)×sin1-cos θ2k1d1-cos θ2k1d-sin1+cos θ2k1d1+cos θ2k1d.
Psc=02πdϕ0π sin θdθdPscdΩ=c18πk14Eo22-14π12d2(2π)2×0πdθ sin θ|Z(k1d, θ)|2αmp2×adρρUmp2ρ2W2Jp(k1ρ sin θ)2×02πdϕcos p2ϕsin p2ϕ(1-sin2 θ cos2 ϕ).
Ssc=k14d22-14π1212αmp2(2π)3×0πdθ sin θ|Z(k1d, θ)|2×adρρUmp(2ρ2/W2)Jp(k1ρ sin θ)2×12-sin2 θ1+δp1412-sin2 θ1-δp14.
|Z|2cos2(k1za)sink1d sin2θ2k1d sin2θ2+sink1d cos2θ2k1d cos2θ22+sin2(k1za)sink1d sin2θ2k1d sin2θ2-sink1d cos2θ2k1d cos2θ22cos2(k1za)1+sin k1dk1d2+sin2(k1za)1-sin k1dk1d2.
Lˆ/2dz exp(-κz)[exp(ik1z-ik1z cos θ)
+r exp(-ik1z-ik1z cos θ)]
=(1+r)κ+i(1-r)k1(1-cos θ)κ2+k12(1-cos θ)2 cosk1L2.
κ-1ZdeffZ
Qmpnq=02πdθ0adρρψpmψqn*.
Qmpnq=2πw2αmp2δpq exp{-i[2(m-n)+p-q)]φ}×1402νdXUpm(ξ2X)Uqn*(X),
Qmpnq=exp{-i[2(m-n)+(p-q)]φ}|Qmpnq(ξ,ν)|,
|Qmpnq(ξ,ν)|
=δpq[m!n!(m+p)!(n+q)!]1/2(2ξ)p+1(1+ξ2)p+1×k=0ml=0nξ2k21+ξ2k+l×(-1)k+l+p(k+l+p)!k!l!(m-k)!(n-l)!(p+k)!(p+l)!×1-exp(-2ν)i=0k+l+p(2ν)k+l+p-i(k+l+p-i)!,
Dmpnq=2πw2αmp2δpq exp{-i[2(m-n)+(p-q)]φ}×140dXUpm(ξ2X)exp(-ik1ρ2/R)Uqn*(X).
0dX1-12b2R2X2+ib2RXUpm(ξ2X)Uqn*(X).
Pmpnq=|Qmpnq(ξ,)|-12b2R2[Qm,p](2)+ib2R1[Qm,p](1),
[Qmpnq](j)(ξ,ν)=δpq[m!n!(m+p)!(n+q)!]1/2×(2ξ)p+1(1+ξ2)p+1k=0ml=0nξ2k21+ξ2k+l+j
×(-1)k+l+p(k+l+p+j)!k!l!(m-k)!(n-l)!(p+k)!(p+l)!.
P0p0p=Q0p0p(ξ,)1-12L/2b1+(L/b)22(p+2)!p!×21+ξ22,iL/2b1+(L/b)2(p+1)!p!21+ξ2,
|Λˆ|μ4g exp[-2µ]-μ1πN214-1πN2μ2+j=1,214(k1d)22-112ς21+μk1a11+(zj¯/πN)2μ2-1k1a×exp-2µ1+(zj¯/πN)2μ21-(zj¯/πN)2μ2(1+(zj¯/πN)2μ2)2=0,
|Λˆ|μ2r11-(1/2πN)2μ2[1+(1/2πN)2μ2]2 exp-2µ1+(1/2πN)2μ2+4g exp(-2µ)-μ(1/πN)214-1πN2μ2+14(k1deff)22-112ς*21+μk1a11+(1/2πN)2μ2-1k1a×exp-2µ1+(1/2πN)2μ21-(1/2πN)2μ2[1+(1/2πN)2μ2]2=0.

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