Abstract

A coupled-mode formulation is presented for parallel cavities of a laser array. The interaction between neighboring elements is via diffraction resulting from perturbation of the boundary conditions. Each high-reflectivity cavity mirror is surrounded by a much-lower-reflectivity background, which, for practical purposes, is considered to be zero. The formulation applies both to ordinary laser arrays and to vertical-cavity laser arrays with high-contrast laterally patterned mirrors. Some examples are considered and compared against exact numerical calculations.

© 1998 Optical Society of America

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  1. P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991); P. L. Gourley, M. E. Warren, G. A. Wawter, T. M. Brennan, B. E. Hammons, “Optical Bloch waves in a semiconductor photonic lattice,” Appl. Phys. Lett. 60, 2714–2716 (1992).
    [CrossRef]
  2. R. A. Morgan, K. Kojima, “Optical characteristics of two-dimensional coherently coupled vertical-cavity surface-emitting laser arrays,” Opt. Lett. 18, 352–354 (1993).
    [CrossRef] [PubMed]
  3. M. Orenstein, A. von Lehmen, C. Chang-Hasnain, N. G. Stoffel, J. P. Harbison, L. T. Florez, “Matrix addressable vertical cavity surface emitting laser array,” Electron. Lett. 27, 437–438 (1991).
    [CrossRef]
  4. M. Orenstein, E. Kapon, N. G. Stoffel, J. P. Harbison, L. T. Florez, J. Wullert, “Two-dimensional phase locked arrays of vertical cavity semiconductor lasers by mirror reflectivity modulation,” Appl. Phys. Lett. 58, 804–806 (1991); M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, N. G. Stoffel, “Large two-dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
    [CrossRef]
  5. J. M. Catchmark, L. E. Rogers, R. A. Morgan, M. T. Asom, G. D. Guth, D. N. Christodoulides, “Optical characteristics of multitransverse mode two-dimensional vertical-cavity top surface-emitting laser arrays,” IEEE J. Quantum Electron. 32, 986–995 (1996).
    [CrossRef]
  6. M. Orenstein, E. Kapon, L. T. Florez, J. P. Harbison, N. G. Stoffel, “Effect of dimensionality on the spatial coherence of phase-locked vertical-cavity semiconductor laser arrays,” in Integrated Photonics Research, Vol. 8 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 50, Paper Tu F5.
  7. H. Pier, E. Kapon, “Photon localization in lattices of coupled vertical-cavity surface-emitting lasers with dimensionalities between one and two,” Opt. Lett. 22, 546–548 (1997).
    [CrossRef] [PubMed]
  8. H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. van der Gaag, J. R. Hayes, A. von Lehmen, E. Kapon, Y.-S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
    [CrossRef]
  9. H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, Y.-S. Kwon, “Array mode analysis of two-dimensional phased arrays of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 26, 1039–1051 (1990).
    [CrossRef]
  10. A. Hardy, E. Kapon, “Coupled-mode formulations for parallel-laser resonators with application to vertical-cavity semiconductor-laser arrays,” IEEE J. Quantum Electron. 32, 966–971 (1996).
    [CrossRef]
  11. T. Fishman, A. Hardy, E. Kapon, “Formulations for calculating the eigenmodes of vertical-cavity laser arrays,” IEEE J. Quantum Electron. 33, 1756–1762 (1997).
    [CrossRef]
  12. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).
  14. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
    [CrossRef]
  15. A. Hardy, W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” J. Lightwave Technol. LT-4, 90–99 (1986).
    [CrossRef]
  16. E. Kapon, J. Katz, C. P. Lindsey, S. Margalit, A. Yariv, “Control of the mutual phase locking of monolithically integrated semiconductor lasers,” Appl. Phys. Lett. 43, 421–423 (1983).
    [CrossRef]
  17. R. F. M. Hendriks, M. P. van Exter, J. P. Woerdman, C. J. van der Poel, “Phase coupling of two optically pumped vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 69, 869–871 (1996).
    [CrossRef]
  18. F. B. Hildebrand, Methods of Applied Mathematics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N. J., 1965).
  19. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).
  20. I. S. Gradshteyn, I. W. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).
  21. T. Li, “Diffraction loss and selection of modes in maser resonators with circular mirrors,” Bell Syst. Tech. J. 44, 917–932 (1965).
    [CrossRef]

1997 (2)

T. Fishman, A. Hardy, E. Kapon, “Formulations for calculating the eigenmodes of vertical-cavity laser arrays,” IEEE J. Quantum Electron. 33, 1756–1762 (1997).
[CrossRef]

H. Pier, E. Kapon, “Photon localization in lattices of coupled vertical-cavity surface-emitting lasers with dimensionalities between one and two,” Opt. Lett. 22, 546–548 (1997).
[CrossRef] [PubMed]

1996 (3)

A. Hardy, E. Kapon, “Coupled-mode formulations for parallel-laser resonators with application to vertical-cavity semiconductor-laser arrays,” IEEE J. Quantum Electron. 32, 966–971 (1996).
[CrossRef]

J. M. Catchmark, L. E. Rogers, R. A. Morgan, M. T. Asom, G. D. Guth, D. N. Christodoulides, “Optical characteristics of multitransverse mode two-dimensional vertical-cavity top surface-emitting laser arrays,” IEEE J. Quantum Electron. 32, 986–995 (1996).
[CrossRef]

R. F. M. Hendriks, M. P. van Exter, J. P. Woerdman, C. J. van der Poel, “Phase coupling of two optically pumped vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 69, 869–871 (1996).
[CrossRef]

1993 (1)

1991 (3)

P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991); P. L. Gourley, M. E. Warren, G. A. Wawter, T. M. Brennan, B. E. Hammons, “Optical Bloch waves in a semiconductor photonic lattice,” Appl. Phys. Lett. 60, 2714–2716 (1992).
[CrossRef]

M. Orenstein, A. von Lehmen, C. Chang-Hasnain, N. G. Stoffel, J. P. Harbison, L. T. Florez, “Matrix addressable vertical cavity surface emitting laser array,” Electron. Lett. 27, 437–438 (1991).
[CrossRef]

M. Orenstein, E. Kapon, N. G. Stoffel, J. P. Harbison, L. T. Florez, J. Wullert, “Two-dimensional phase locked arrays of vertical cavity semiconductor lasers by mirror reflectivity modulation,” Appl. Phys. Lett. 58, 804–806 (1991); M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, N. G. Stoffel, “Large two-dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
[CrossRef]

1990 (2)

H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. van der Gaag, J. R. Hayes, A. von Lehmen, E. Kapon, Y.-S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, Y.-S. Kwon, “Array mode analysis of two-dimensional phased arrays of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 26, 1039–1051 (1990).
[CrossRef]

1986 (1)

A. Hardy, W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” J. Lightwave Technol. LT-4, 90–99 (1986).
[CrossRef]

1983 (1)

E. Kapon, J. Katz, C. P. Lindsey, S. Margalit, A. Yariv, “Control of the mutual phase locking of monolithically integrated semiconductor lasers,” Appl. Phys. Lett. 43, 421–423 (1983).
[CrossRef]

1973 (1)

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

1965 (1)

T. Li, “Diffraction loss and selection of modes in maser resonators with circular mirrors,” Bell Syst. Tech. J. 44, 917–932 (1965).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Asom, M. T.

J. M. Catchmark, L. E. Rogers, R. A. Morgan, M. T. Asom, G. D. Guth, D. N. Christodoulides, “Optical characteristics of multitransverse mode two-dimensional vertical-cavity top surface-emitting laser arrays,” IEEE J. Quantum Electron. 32, 986–995 (1996).
[CrossRef]

Brennan, T. M.

P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991); P. L. Gourley, M. E. Warren, G. A. Wawter, T. M. Brennan, B. E. Hammons, “Optical Bloch waves in a semiconductor photonic lattice,” Appl. Phys. Lett. 60, 2714–2716 (1992).
[CrossRef]

Catchmark, J. M.

J. M. Catchmark, L. E. Rogers, R. A. Morgan, M. T. Asom, G. D. Guth, D. N. Christodoulides, “Optical characteristics of multitransverse mode two-dimensional vertical-cavity top surface-emitting laser arrays,” IEEE J. Quantum Electron. 32, 986–995 (1996).
[CrossRef]

Chang-Hasnain, C.

M. Orenstein, A. von Lehmen, C. Chang-Hasnain, N. G. Stoffel, J. P. Harbison, L. T. Florez, “Matrix addressable vertical cavity surface emitting laser array,” Electron. Lett. 27, 437–438 (1991).
[CrossRef]

Christodoulides, D. N.

J. M. Catchmark, L. E. Rogers, R. A. Morgan, M. T. Asom, G. D. Guth, D. N. Christodoulides, “Optical characteristics of multitransverse mode two-dimensional vertical-cavity top surface-emitting laser arrays,” IEEE J. Quantum Electron. 32, 986–995 (1996).
[CrossRef]

Fishman, T.

T. Fishman, A. Hardy, E. Kapon, “Formulations for calculating the eigenmodes of vertical-cavity laser arrays,” IEEE J. Quantum Electron. 33, 1756–1762 (1997).
[CrossRef]

Florez, L. T.

M. Orenstein, E. Kapon, N. G. Stoffel, J. P. Harbison, L. T. Florez, J. Wullert, “Two-dimensional phase locked arrays of vertical cavity semiconductor lasers by mirror reflectivity modulation,” Appl. Phys. Lett. 58, 804–806 (1991); M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, N. G. Stoffel, “Large two-dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
[CrossRef]

M. Orenstein, A. von Lehmen, C. Chang-Hasnain, N. G. Stoffel, J. P. Harbison, L. T. Florez, “Matrix addressable vertical cavity surface emitting laser array,” Electron. Lett. 27, 437–438 (1991).
[CrossRef]

H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. van der Gaag, J. R. Hayes, A. von Lehmen, E. Kapon, Y.-S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

M. Orenstein, E. Kapon, L. T. Florez, J. P. Harbison, N. G. Stoffel, “Effect of dimensionality on the spatial coherence of phase-locked vertical-cavity semiconductor laser arrays,” in Integrated Photonics Research, Vol. 8 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 50, Paper Tu F5.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

Gourley, P. L.

P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991); P. L. Gourley, M. E. Warren, G. A. Wawter, T. M. Brennan, B. E. Hammons, “Optical Bloch waves in a semiconductor photonic lattice,” Appl. Phys. Lett. 60, 2714–2716 (1992).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. W. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Guth, G. D.

J. M. Catchmark, L. E. Rogers, R. A. Morgan, M. T. Asom, G. D. Guth, D. N. Christodoulides, “Optical characteristics of multitransverse mode two-dimensional vertical-cavity top surface-emitting laser arrays,” IEEE J. Quantum Electron. 32, 986–995 (1996).
[CrossRef]

Hadley, G. R.

P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991); P. L. Gourley, M. E. Warren, G. A. Wawter, T. M. Brennan, B. E. Hammons, “Optical Bloch waves in a semiconductor photonic lattice,” Appl. Phys. Lett. 60, 2714–2716 (1992).
[CrossRef]

Hammons, B. E.

P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991); P. L. Gourley, M. E. Warren, G. A. Wawter, T. M. Brennan, B. E. Hammons, “Optical Bloch waves in a semiconductor photonic lattice,” Appl. Phys. Lett. 60, 2714–2716 (1992).
[CrossRef]

Harbison, J. P.

M. Orenstein, A. von Lehmen, C. Chang-Hasnain, N. G. Stoffel, J. P. Harbison, L. T. Florez, “Matrix addressable vertical cavity surface emitting laser array,” Electron. Lett. 27, 437–438 (1991).
[CrossRef]

M. Orenstein, E. Kapon, N. G. Stoffel, J. P. Harbison, L. T. Florez, J. Wullert, “Two-dimensional phase locked arrays of vertical cavity semiconductor lasers by mirror reflectivity modulation,” Appl. Phys. Lett. 58, 804–806 (1991); M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, N. G. Stoffel, “Large two-dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
[CrossRef]

H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. van der Gaag, J. R. Hayes, A. von Lehmen, E. Kapon, Y.-S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

M. Orenstein, E. Kapon, L. T. Florez, J. P. Harbison, N. G. Stoffel, “Effect of dimensionality on the spatial coherence of phase-locked vertical-cavity semiconductor laser arrays,” in Integrated Photonics Research, Vol. 8 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 50, Paper Tu F5.

Hardy, A.

T. Fishman, A. Hardy, E. Kapon, “Formulations for calculating the eigenmodes of vertical-cavity laser arrays,” IEEE J. Quantum Electron. 33, 1756–1762 (1997).
[CrossRef]

A. Hardy, E. Kapon, “Coupled-mode formulations for parallel-laser resonators with application to vertical-cavity semiconductor-laser arrays,” IEEE J. Quantum Electron. 32, 966–971 (1996).
[CrossRef]

A. Hardy, W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” J. Lightwave Technol. LT-4, 90–99 (1986).
[CrossRef]

Hayes, J. R.

H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. van der Gaag, J. R. Hayes, A. von Lehmen, E. Kapon, Y.-S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, Y.-S. Kwon, “Array mode analysis of two-dimensional phased arrays of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 26, 1039–1051 (1990).
[CrossRef]

Hendriks, R. F. M.

R. F. M. Hendriks, M. P. van Exter, J. P. Woerdman, C. J. van der Poel, “Phase coupling of two optically pumped vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 69, 869–871 (1996).
[CrossRef]

Hildebrand, F. B.

F. B. Hildebrand, Methods of Applied Mathematics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N. J., 1965).

Kapon, E.

H. Pier, E. Kapon, “Photon localization in lattices of coupled vertical-cavity surface-emitting lasers with dimensionalities between one and two,” Opt. Lett. 22, 546–548 (1997).
[CrossRef] [PubMed]

T. Fishman, A. Hardy, E. Kapon, “Formulations for calculating the eigenmodes of vertical-cavity laser arrays,” IEEE J. Quantum Electron. 33, 1756–1762 (1997).
[CrossRef]

A. Hardy, E. Kapon, “Coupled-mode formulations for parallel-laser resonators with application to vertical-cavity semiconductor-laser arrays,” IEEE J. Quantum Electron. 32, 966–971 (1996).
[CrossRef]

M. Orenstein, E. Kapon, N. G. Stoffel, J. P. Harbison, L. T. Florez, J. Wullert, “Two-dimensional phase locked arrays of vertical cavity semiconductor lasers by mirror reflectivity modulation,” Appl. Phys. Lett. 58, 804–806 (1991); M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, N. G. Stoffel, “Large two-dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
[CrossRef]

H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. van der Gaag, J. R. Hayes, A. von Lehmen, E. Kapon, Y.-S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

E. Kapon, J. Katz, C. P. Lindsey, S. Margalit, A. Yariv, “Control of the mutual phase locking of monolithically integrated semiconductor lasers,” Appl. Phys. Lett. 43, 421–423 (1983).
[CrossRef]

M. Orenstein, E. Kapon, L. T. Florez, J. P. Harbison, N. G. Stoffel, “Effect of dimensionality on the spatial coherence of phase-locked vertical-cavity semiconductor laser arrays,” in Integrated Photonics Research, Vol. 8 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 50, Paper Tu F5.

Katz, J.

E. Kapon, J. Katz, C. P. Lindsey, S. Margalit, A. Yariv, “Control of the mutual phase locking of monolithically integrated semiconductor lasers,” Appl. Phys. Lett. 43, 421–423 (1983).
[CrossRef]

Kojima, K.

Kwon, Y.-S.

H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, Y.-S. Kwon, “Array mode analysis of two-dimensional phased arrays of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 26, 1039–1051 (1990).
[CrossRef]

H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. van der Gaag, J. R. Hayes, A. von Lehmen, E. Kapon, Y.-S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

Li, T.

T. Li, “Diffraction loss and selection of modes in maser resonators with circular mirrors,” Bell Syst. Tech. J. 44, 917–932 (1965).
[CrossRef]

Lindsey, C. P.

E. Kapon, J. Katz, C. P. Lindsey, S. Margalit, A. Yariv, “Control of the mutual phase locking of monolithically integrated semiconductor lasers,” Appl. Phys. Lett. 43, 421–423 (1983).
[CrossRef]

Margalit, S.

E. Kapon, J. Katz, C. P. Lindsey, S. Margalit, A. Yariv, “Control of the mutual phase locking of monolithically integrated semiconductor lasers,” Appl. Phys. Lett. 43, 421–423 (1983).
[CrossRef]

Morgan, R. A.

J. M. Catchmark, L. E. Rogers, R. A. Morgan, M. T. Asom, G. D. Guth, D. N. Christodoulides, “Optical characteristics of multitransverse mode two-dimensional vertical-cavity top surface-emitting laser arrays,” IEEE J. Quantum Electron. 32, 986–995 (1996).
[CrossRef]

R. A. Morgan, K. Kojima, “Optical characteristics of two-dimensional coherently coupled vertical-cavity surface-emitting laser arrays,” Opt. Lett. 18, 352–354 (1993).
[CrossRef] [PubMed]

Orenstein, M.

M. Orenstein, A. von Lehmen, C. Chang-Hasnain, N. G. Stoffel, J. P. Harbison, L. T. Florez, “Matrix addressable vertical cavity surface emitting laser array,” Electron. Lett. 27, 437–438 (1991).
[CrossRef]

M. Orenstein, E. Kapon, N. G. Stoffel, J. P. Harbison, L. T. Florez, J. Wullert, “Two-dimensional phase locked arrays of vertical cavity semiconductor lasers by mirror reflectivity modulation,” Appl. Phys. Lett. 58, 804–806 (1991); M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, N. G. Stoffel, “Large two-dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
[CrossRef]

M. Orenstein, E. Kapon, L. T. Florez, J. P. Harbison, N. G. Stoffel, “Effect of dimensionality on the spatial coherence of phase-locked vertical-cavity semiconductor laser arrays,” in Integrated Photonics Research, Vol. 8 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 50, Paper Tu F5.

Paek, E. G.

H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, Y.-S. Kwon, “Array mode analysis of two-dimensional phased arrays of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 26, 1039–1051 (1990).
[CrossRef]

H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. van der Gaag, J. R. Hayes, A. von Lehmen, E. Kapon, Y.-S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

Pier, H.

Rogers, L. E.

J. M. Catchmark, L. E. Rogers, R. A. Morgan, M. T. Asom, G. D. Guth, D. N. Christodoulides, “Optical characteristics of multitransverse mode two-dimensional vertical-cavity top surface-emitting laser arrays,” IEEE J. Quantum Electron. 32, 986–995 (1996).
[CrossRef]

Ryzhik, I. W.

I. S. Gradshteyn, I. W. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Scherer, A.

H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, Y.-S. Kwon, “Array mode analysis of two-dimensional phased arrays of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 26, 1039–1051 (1990).
[CrossRef]

H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. van der Gaag, J. R. Hayes, A. von Lehmen, E. Kapon, Y.-S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

Siegman, A. E.

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

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Stoffel, N. G.

M. Orenstein, A. von Lehmen, C. Chang-Hasnain, N. G. Stoffel, J. P. Harbison, L. T. Florez, “Matrix addressable vertical cavity surface emitting laser array,” Electron. Lett. 27, 437–438 (1991).
[CrossRef]

M. Orenstein, E. Kapon, N. G. Stoffel, J. P. Harbison, L. T. Florez, J. Wullert, “Two-dimensional phase locked arrays of vertical cavity semiconductor lasers by mirror reflectivity modulation,” Appl. Phys. Lett. 58, 804–806 (1991); M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, N. G. Stoffel, “Large two-dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
[CrossRef]

M. Orenstein, E. Kapon, L. T. Florez, J. P. Harbison, N. G. Stoffel, “Effect of dimensionality on the spatial coherence of phase-locked vertical-cavity semiconductor laser arrays,” in Integrated Photonics Research, Vol. 8 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 50, Paper Tu F5.

Streifer, W.

A. Hardy, W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” J. Lightwave Technol. LT-4, 90–99 (1986).
[CrossRef]

van der Gaag, B. P.

H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. van der Gaag, J. R. Hayes, A. von Lehmen, E. Kapon, Y.-S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

van der Poel, C. J.

R. F. M. Hendriks, M. P. van Exter, J. P. Woerdman, C. J. van der Poel, “Phase coupling of two optically pumped vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 69, 869–871 (1996).
[CrossRef]

van Exter, M. P.

R. F. M. Hendriks, M. P. van Exter, J. P. Woerdman, C. J. van der Poel, “Phase coupling of two optically pumped vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 69, 869–871 (1996).
[CrossRef]

Vawter, G. A.

P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991); P. L. Gourley, M. E. Warren, G. A. Wawter, T. M. Brennan, B. E. Hammons, “Optical Bloch waves in a semiconductor photonic lattice,” Appl. Phys. Lett. 60, 2714–2716 (1992).
[CrossRef]

von Lehmen, A.

M. Orenstein, A. von Lehmen, C. Chang-Hasnain, N. G. Stoffel, J. P. Harbison, L. T. Florez, “Matrix addressable vertical cavity surface emitting laser array,” Electron. Lett. 27, 437–438 (1991).
[CrossRef]

H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. van der Gaag, J. R. Hayes, A. von Lehmen, E. Kapon, Y.-S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

Warren, M. E.

P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991); P. L. Gourley, M. E. Warren, G. A. Wawter, T. M. Brennan, B. E. Hammons, “Optical Bloch waves in a semiconductor photonic lattice,” Appl. Phys. Lett. 60, 2714–2716 (1992).
[CrossRef]

Woerdman, J. P.

R. F. M. Hendriks, M. P. van Exter, J. P. Woerdman, C. J. van der Poel, “Phase coupling of two optically pumped vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 69, 869–871 (1996).
[CrossRef]

Wullert, J.

M. Orenstein, E. Kapon, N. G. Stoffel, J. P. Harbison, L. T. Florez, J. Wullert, “Two-dimensional phase locked arrays of vertical cavity semiconductor lasers by mirror reflectivity modulation,” Appl. Phys. Lett. 58, 804–806 (1991); M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, N. G. Stoffel, “Large two-dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
[CrossRef]

Yariv, A.

E. Kapon, J. Katz, C. P. Lindsey, S. Margalit, A. Yariv, “Control of the mutual phase locking of monolithically integrated semiconductor lasers,” Appl. Phys. Lett. 43, 421–423 (1983).
[CrossRef]

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

Yoo, H.-J.

H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. van der Gaag, J. R. Hayes, A. von Lehmen, E. Kapon, Y.-S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, Y.-S. Kwon, “Array mode analysis of two-dimensional phased arrays of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 26, 1039–1051 (1990).
[CrossRef]

Appl. Phys. Lett. (5)

H.-J. Yoo, A. Scherer, J. P. Harbison, L. T. Florez, E. G. Paek, B. P. van der Gaag, J. R. Hayes, A. von Lehmen, E. Kapon, Y.-S. Kwon, “Fabrication of a two-dimensional phased array of vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 56, 1198–1200 (1990).
[CrossRef]

E. Kapon, J. Katz, C. P. Lindsey, S. Margalit, A. Yariv, “Control of the mutual phase locking of monolithically integrated semiconductor lasers,” Appl. Phys. Lett. 43, 421–423 (1983).
[CrossRef]

R. F. M. Hendriks, M. P. van Exter, J. P. Woerdman, C. J. van der Poel, “Phase coupling of two optically pumped vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 69, 869–871 (1996).
[CrossRef]

M. Orenstein, E. Kapon, N. G. Stoffel, J. P. Harbison, L. T. Florez, J. Wullert, “Two-dimensional phase locked arrays of vertical cavity semiconductor lasers by mirror reflectivity modulation,” Appl. Phys. Lett. 58, 804–806 (1991); M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, N. G. Stoffel, “Large two-dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
[CrossRef]

P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991); P. L. Gourley, M. E. Warren, G. A. Wawter, T. M. Brennan, B. E. Hammons, “Optical Bloch waves in a semiconductor photonic lattice,” Appl. Phys. Lett. 60, 2714–2716 (1992).
[CrossRef]

Bell Syst. Tech. J. (1)

T. Li, “Diffraction loss and selection of modes in maser resonators with circular mirrors,” Bell Syst. Tech. J. 44, 917–932 (1965).
[CrossRef]

Electron. Lett. (1)

M. Orenstein, A. von Lehmen, C. Chang-Hasnain, N. G. Stoffel, J. P. Harbison, L. T. Florez, “Matrix addressable vertical cavity surface emitting laser array,” Electron. Lett. 27, 437–438 (1991).
[CrossRef]

IEEE J. Quantum Electron. (5)

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

J. M. Catchmark, L. E. Rogers, R. A. Morgan, M. T. Asom, G. D. Guth, D. N. Christodoulides, “Optical characteristics of multitransverse mode two-dimensional vertical-cavity top surface-emitting laser arrays,” IEEE J. Quantum Electron. 32, 986–995 (1996).
[CrossRef]

H.-J. Yoo, J. R. Hayes, E. G. Paek, A. Scherer, Y.-S. Kwon, “Array mode analysis of two-dimensional phased arrays of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 26, 1039–1051 (1990).
[CrossRef]

A. Hardy, E. Kapon, “Coupled-mode formulations for parallel-laser resonators with application to vertical-cavity semiconductor-laser arrays,” IEEE J. Quantum Electron. 32, 966–971 (1996).
[CrossRef]

T. Fishman, A. Hardy, E. Kapon, “Formulations for calculating the eigenmodes of vertical-cavity laser arrays,” IEEE J. Quantum Electron. 33, 1756–1762 (1997).
[CrossRef]

J. Lightwave Technol. (1)

A. Hardy, W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” J. Lightwave Technol. LT-4, 90–99 (1986).
[CrossRef]

Opt. Lett. (2)

Other (6)

M. Orenstein, E. Kapon, L. T. Florez, J. P. Harbison, N. G. Stoffel, “Effect of dimensionality on the spatial coherence of phase-locked vertical-cavity semiconductor laser arrays,” in Integrated Photonics Research, Vol. 8 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 50, Paper Tu F5.

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

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

F. B. Hildebrand, Methods of Applied Mathematics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N. J., 1965).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

I. S. Gradshteyn, I. W. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

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Figures (12)

Fig. 1
Fig. 1

Schematic illustration of the vertical cavity laser array.

Fig. 2
Fig. 2

Schematic of the unfolded laser array of Fig. 1. For convenience, only a single Fabry–Perot cavity of the array is shown.

Fig. 3
Fig. 3

Variation of the system eigenvalue |γˆ| for an array of two identical cavities with one-dimensional Gaussian reflectivity mirrors, as a function of the separation s=D-2a between the two cavities. Solid curves, exact array modes; dashed curves, approximate analytical coupled-Gaussian mode solution; circles, numerical coupled-mode solution. Since the analytical single-cavity eigenvalue (|γ||γˆ|) differs slightly from the exact single-cavity eigenvalue, owing to the truncation (in this example |γexact|=0.8361, |γanalytical|=0.8366, the effects of the coupling on the system eigenvalue |γˆ|-|γˆ| are compared. In this example, a=2 µm, λ0=0.8 µm, n=3.5, 2L=30λ0/n, and x0=a/2.

Fig. 4
Fig. 4

Same as in Fig. 3, but with x0=a/3. Owing to the excellent agreement between the exact (solid curves) and the analytical coupled-mode (dashed curves) solutions, the numerical coupled-mode calculation is omitted. In this example |γexact|=0.67697, |γanalytical|=0.67732.

Fig. 5
Fig. 5

Rectangular mirrors’ configuration in the x plane for the two coupled cavities considered in Subsection 3.B. Each mirror is centered at Pj=(Pj1, 0), j=1, 2.

Fig. 6
Fig. 6

Variation of the system eigenvalue |γˆ| for an array of identical circular cavities as a function of the separation s=D-2a. Solid curves, exact array modes; circles, coupled-modes. In this example, a=4 µm, 2L=70λ0/n, λ0=1 µm, n=3.5.

Fig. 7
Fig. 7

Contour plot of the lowest-order symmetric dual-cavity-system mode. Note the excellent agreement between the exact (a) and the coupled-mode (b) solutions. The system parameters are as in Fig. 6. The mirror separation is s=2 µm.

Fig. 8
Fig. 8

Contour plot of the lowest-order antisymmetric dual-cavity-system mode. Note the excellent agreement between the exact (a) and the coupled-mode (b) solutions. The system parameters are as in Fig. 6. The mirror separation is s=2 µm.

Fig. 9
Fig. 9

Cross section (at y1=0) of the symmetric (a) and the antisymmetric (b) supermodes of two coupled identical cavities with circular mirrors. The system parameters are as in Fig. 6 for separation distances s=1 µm (solid curves) and s=5 µm (dashed curves).

Fig. 10
Fig. 10

Schematic diagram of a symmetrical 2D array of four identical symmetrical cavities.

Fig. 11
Fig. 11

Variation of the system eigenvalue |γˆ| for a symmetrical 2D array of four identical symmetrical cavities, as a function of the separation s=D-2a. Solid curves, exact array modes; circles, coupled modes. In this example, a=2 µm, 2L=35λ0/n, λ0=1 µm, n=3.5.

Fig. 12
Fig. 12

Intensity contour plots of the lowest-order modes of a symmetrical 2D array of four identical symmetrical cavities. Three basic lowest-order modes are presented: (a) symmetric about both y1 and y2 axes, (b) symmetric about y1=0 and antisymmetric about y2=0, (c) antisymmetric about both y1 and y2 axes. The normalized intensity contour values are [1 0.8 0.6 0.2 0.1 0.05 0.01]. The system parameters are as in Fig. 11, with s=1 µm.

Equations (100)

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ρ(ξ)=j=1Nρj(ξ-Pj),
γˆvˆ(y)=Xρ(x)K(y, x)vˆ(x)dx,
K(y, x)=K(|y-x|)=K(x, y).
K(y, x)=2Lniλ0exp{-i(2πn/λ0)[(x1-y1)2+(x2-y2)2+(2L)2]1/2}(x1-y1)2+(x2-y2)2+(2L)2,
K(y, x)=in2Lλ0exp-iπn2Lλ0(|x-y|2),
fj(x-Pj)=vˆ(x)forallxonthejthmirror0otherwise,
gj(y-Pj)=γˆvˆ(y)forallyonthejthmirror0otherwise.
l=1Ngl(y-Pl)=j=1Nρj(x-Pj)×Kj(y-Pj, x-Pj)fj(x-Pj)dx,
fj(x-Pj)=ν=1aν(j)uν(j)(x-Pj),j=1, , N,
gl(y-Pl)=μ=1bμ(l)uμ(l)(y-Pl),l=1, , N,
l=1Nμ=1nlbμ(l)uμ(l)(y-Pl)=j=1Nν=1njaν(j)hν(j)(y),
hν(j)(y)jthmirrorapertureρj(x-Pj)Kj(y-Pj, x-Pj)×uν(j)(x-Pj)dx,
bη(m)=j=1Nν=1njMη,ν(m, j)aν(j),
Mη,ν(m, j)=mthmirrorapertureρm(y-Pm)uη(m)(y-Pm)hν(j)(y)dy.
Mη,ν(j, j)=γν(j)δη,ν.
Mη,ν(m, j)=Mν,η(j, m),
γˆaη(m)=j=1Nν=1njMη,ν(m, j)aν(j)
γˆA=MA,
ρj(x-Pj)=exp-(x-Pj)2x02for|x-Pj|a0for|x-Pj|>a,
j=1, 2.
hν(j)(y)-ρj(x-Pj)Kj(y-Pj, x-Pj)×uν(j)(x-Pj)dx=γν(j)uν(j)(y-Pj),
Mη,ν(m, j)γν(j)Pm-aPm+aρm(y-Pm)×uη(m)(y-Pm)uν(j)(y-Pj)dyγν(j)-ρm(y-Pm)uη(m)(y-Pm)×uν(j)(y-Pj)dy
M=ΓSS˜Γ,
Γη,ν=γνδην,η=0, 1, , N-1;
ν=0, 1, , N-1,
Sη,νMη,ν(1, 2),η=0, 1, , N-1;
ν=0, 1, , N-1,
γ0+s00-s01s02-s01γ1-s11s12s02s12γ2+s22a0a1a2=γˆ(s)a0a1a2
γ0-s00s01-s02s01γ1+s11-s12-s02-s12γ2-s22a0a1a2=γˆ(A)a0a1a2
M(2)η2,ν2(m, j)=M(2)η2,ν2(m, m)=M(2)η2,ν2(j, j)=δη2,ν2γ(2)ν2,
m=1, 2,j=1, 2,
γˆaη1,η2(m)=j=12ν1M(1)η1,ν1(m, j)γ(2)η2aν1,η2(j),m=1, 2.
(γˆ/γ(2)s)aη1,s(m)=j=12ν1=1νmaxM(1)η1,ν1(m, j)aν1,s(j),
ρj(1)(x1-Pj1)=exp-(x1-Pj1)2x012|x1-Pj1|a0|x1-Pj1|>a,
ρj(2)(x2-Pj2)=exp-(x2-Pj2)2x022|x2-Pj2|b0|x2-Pj2|>b.
γmum(y)=mirrorapertureρ(x)K(y, x)um(x)dx,
γmψm(y)=mirrorapertureT(y, x)ψm(x)dx,
mirrorapertureψm(x)ψn(x)dx=0.
mirrorapertureρ(x)um(x)un(x)dx=δm,n
f(x)=mamum(x),
ρj(x)=ρj(1)(x1)ρj(2)(x2),j=1, , N.
ρ(x)=j=1Nρj(1)(x1-Pj1)ρj(2)(x2-Pj2),
K(y, x)=K(|y-x|)=K(1)(|y1-x1|)K(2)(|y2-x2|)=K(1)(y1, x1)K(2)(y2, x2),
γ(1)m1(j)u(1)m1(j)(y1-Pj1)=ρj(1)(x1-Pj1)K(y1, x1)u(1)m1(j)(x1-Pj1)dx1,
γ(2)m2(j)u(2)m2(j)(y2-Pj2)=ρj(2)(x2-Pj2)K(y2, x2)u(2)m2(j)(x2-Pj2)dx2,
γm(j)γm1,m2(j)=γ(1)m1(j)γ(2)m2(j),
um(j)(x)=u(1)m1(j)(x1)u(2)m2(j)(x2).
hν(j)(y)=jthmirrorρj(1)(x1-Pj1)Kj(1)(y1-Pj1, x1-Pj1)u(1)ν1(j)(x1-Pj1)dx1×jthmirrorρj(2)(x2-Pj2)Kj(2)(y2-Pj2, x2-Pj2)u(2)ν2(j)(x2-Pj2)dx2h(1)ν1(j)(y1)h(2)ν2(j)(y2).
Mη,ν(m, j)=mthmirrorρm(1)(y1-Pm1)u(1)η1(m)(y1-Pmi)h(1)ν1(j)(y1)dy1×mthmirrorρm(2)(y2-Pm2)u(2)η2(m)(y2-Pm2)h(2)ν2(j)(y2)dy2M(1)η1,ν1(m, j)M(2)η2,ν2(m, j).
Mη,ν(j, j)=M(1)η1ν1(j, j)M(2)η2ν2(j, j)=(γ(1)ν1(j)δη1,ν1)(γ(2)ν2(j)δη2,ν2),
M(1)η1ν1(m, j)M(2)η2ν2(m, j)=M(1)ν1η1(j, m)M(2)ν2η2(j, m).
γˆaη1,η2(m)=j=1Nν1ν2M(1)η1ν1(m, j)M(2)η2ν2(m, j)aν1,ν2(j)
ρ(x)=exp[-(x/x0)2]
uν(x)=QνHν2xv˜exp-ik0x22qˆ,ν=0, 1, 2,,
1qˆnR-iλnπω2=nR-iλ0πω2.
1qˆ=D-A2B±1BA+D22-11/2,
v˜2=i2Bk0A+D22-1-1/2,
γν=(A+B/qˆ)-ν-1/2.
A=1-iλ0πx022Ln,
B=2L/n,
C=-iλ0πx02,
D=1.
1qˆ=iλ02πx021-1+i2πx02nλ0L1/2,
v˜2=2x021+i2πx02nλ0L-1/2,
γν=1-iλ0Lnπx021+1+i2πx02nλ0L1/2-ν-1/2,
v˜2(1/x02+ik0/qˆ)=2,
-γνiv˜k02qˆ=γν+1iv˜k02qˆ-2v˜,
IˆQνQη- exp[-(x/x0)2]Hη2xv˜Hν2xv˜×exp-ik0x2qˆdx
Hσ(x)=lims0 σsσ[exp(2xs-s2)],σ=ν, η.
Qν=2πv˜2νν!1/2
Mη,ν(m, j)=γν(j)QηQν- exp-yx02×Hη2yv˜Hν2(y-α)v˜×exp-ik02qˆ[y2+(y-α)2]dy
Mη,ν(m, j)=γν(j) exp-α2k02qˆv˜2k04qˆ+i(2ηη!)1/2(2νν!)1/2×limt0s0 ηtηνsν[exp(2st+Pt+Rs)],
P(i/2)2v˜(k0/qˆ)α,
RP-2(2/v˜)α.
M0,0(m, j)=γ0 exp-α2k02qˆv˜2k04qˆ+i,
M1,0(m, j)=γ0 exp-α2k02qˆv˜2k04qˆ+iiv˜k02qˆα,
M0,1(m, j)=γ1 exp-α2k02qˆv˜2k04qˆ+iiv˜k02qˆ-2v˜α,
M1,1(m, j)=γ1 exp-α2k02qˆv˜2k04qˆ+i×1+iv˜k02qˆiv˜k02qˆ-2v˜α2,
M2,0(m, j)=γ0 exp-α2k02qˆv˜2k04qˆ+iiv˜k02qˆ2 α22,
M0,2(m, j)=γ2 exp-α2k02qˆv˜2k04qˆ+iiv˜k02qˆ-2v˜2 α22,
M2,1(m, j)=γ1 exp-α2k02qˆv˜2k04qˆ+iiv˜k02qˆ×2+iv˜k02qˆiv˜k02qˆ-2v˜α2 α2,
M1,2(m, j)=γ2 exp-α2k02qˆv˜2k04qˆ+iiv˜k02qˆ-2v˜×2+iv˜k02qˆiv˜k02qˆ-2v˜α2 α2,
M2,2(m, j)=γ2 exp-α2k02qˆv˜2k04qˆ+i×1+iv˜k02qˆiv˜k02qˆ-2v˜2α2+iv˜k02qˆ2iv˜k02qˆ-2v˜2 α42.
ρ(x)=ρ(r)ra0r>a.
γmum(r2, ϕ2)=0a02πρ(r1)K(r2, ϕ2; r1, ϕ1)×um(r1, ϕ1)r1dr1dϕ1,
K(r2, ϕ2; r1, ϕ1)=i2Lλexp-iπ2Lλ[r12+r22-2r1r2 cos(ϕ1-ϕ2)]
exp[in(π/2-β)]Jn(xy)=12π02πexp{i[xy cos(α-β)-nα]}dα
un,l(r, ϕ)=Rn,l(r)exp(-inϕ)
γnlψn,l(r2)=0aTn(r2, r1)ψn,l(r1)dr1,
ψnl(r)ρ(r)rRnl(r),
Tn(r2, r1)=in+1πλLJnπr1r2λLr1r2×exp-iπ2Lλ(r12+r22)ρ(r1)ρ(r2)
0aρ(r)Rnk(r)Rnl(r)rdr=0forkl.
0a02πρ(r)umk(r, ϕ)un,l(r, ϕ)rdrdϕ=δmnδkl.
sunl(r, ϕ)Rnl(r)sin(nϕ),
cunl(r, ϕ)Rnl(r)cos(nϕ),
0aρ(r)[Rnl(r)]2rdr=1πn=1, 2, 3,,12πn=0
0a02πρ(r)sum,k(r, ϕ)sun,l(r, ϕ)rdrdϕ=δmnδkl,
0a02πρ(r)cum,k(r, ϕ)cun,l(r, ϕ)rdrdϕ=δmnδkl,
0a02πρ(r)cum,k(r, ϕ)sun,l(r, ϕ)rdrdϕ=0.
f(r, ϕ)=n=0l[anlcunl(r, ϕ)+bnlsunl(r, ϕ)]

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