Abstract

The Wigner-distribution-function representation of the source’s and the receiver’s light fields is used to express the coupling efficiency. The symmetries of the Wigner-distribution graphical representations are connected with the amount of coupled light.

© 1995 Optical Society of America

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References

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  1. W. L. Emkey, “Optical coupling between single-mode semiconductor lasers and strip waveguides,” J. Lightwave Technol. 1, 436–443 (1983).
    [CrossRef]
  2. W. B. Joyce, B. G. DeLoach, “Alignment of Gaussian beams,” Appl. Opt. 23, 4187–4196 (1984).
    [CrossRef] [PubMed]
  3. J. T. Horng, D. C. Chang, “Coupling an elliptical Gaussian beam into a multimode step-index fiber,” Appl. Opt. 22, 3887–3891 (1983).
    [CrossRef] [PubMed]
  4. D. Onciul, “Efficiency of light launching into waveguides: a phase-space approach,” Optik 96, 20–24 (1994).
  5. D. Onciul, “Waveguide launching efficiency for multimoded and partially coherent light sources,” Optik 97, 76–77 (1994).
  6. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  7. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [CrossRef]
  8. D. Dragoman, “Phase-space representation of modes in optical waveguides,” J. Mod. Opt. (to be published).
  9. K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1-D signals,” Opt. Commun. 42, 310–314 (1982).
    [CrossRef]
  10. H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
    [CrossRef]
  11. H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).
    [CrossRef]
  12. N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase-space beam analyser,” Opt. Commun. 24, 927–949 (1992).
  13. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  14. M. T. Camargo Silva, M. A. G. Martinez, P. R. Herczfeld, “MCQW intensity optical modulator for InP based MMIC/photonics integrated circuits,” in IEEE Microwave Theory and Techiques Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 229–232.
    [CrossRef]

1994 (2)

D. Onciul, “Efficiency of light launching into waveguides: a phase-space approach,” Optik 96, 20–24 (1994).

D. Onciul, “Waveguide launching efficiency for multimoded and partially coherent light sources,” Optik 97, 76–77 (1994).

1992 (2)

H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).
[CrossRef]

N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase-space beam analyser,” Opt. Commun. 24, 927–949 (1992).

1984 (1)

1983 (2)

W. L. Emkey, “Optical coupling between single-mode semiconductor lasers and strip waveguides,” J. Lightwave Technol. 1, 436–443 (1983).
[CrossRef]

J. T. Horng, D. C. Chang, “Coupling an elliptical Gaussian beam into a multimode step-index fiber,” Appl. Opt. 22, 3887–3891 (1983).
[CrossRef] [PubMed]

1982 (1)

K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1-D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

1980 (1)

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

1979 (1)

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Bartelt, H. O.

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Bastiaans, M. J.

Brenner, K. H.

K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1-D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Camargo Silva, M. T.

M. T. Camargo Silva, M. A. G. Martinez, P. R. Herczfeld, “MCQW intensity optical modulator for InP based MMIC/photonics integrated circuits,” in IEEE Microwave Theory and Techiques Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 229–232.
[CrossRef]

Chang, D. C.

DeLoach, B. G.

Dragoman, D.

D. Dragoman, “Phase-space representation of modes in optical waveguides,” J. Mod. Opt. (to be published).

Emkey, W. L.

W. L. Emkey, “Optical coupling between single-mode semiconductor lasers and strip waveguides,” J. Lightwave Technol. 1, 436–443 (1983).
[CrossRef]

Haase, T.

N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase-space beam analyser,” Opt. Commun. 24, 927–949 (1992).

Herczfeld, P. R.

M. T. Camargo Silva, M. A. G. Martinez, P. R. Herczfeld, “MCQW intensity optical modulator for InP based MMIC/photonics integrated circuits,” in IEEE Microwave Theory and Techiques Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 229–232.
[CrossRef]

Hodgson, N.

N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase-space beam analyser,” Opt. Commun. 24, 927–949 (1992).

Horng, J. T.

Joyce, W. B.

Kostka, R.

N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase-space beam analyser,” Opt. Commun. 24, 927–949 (1992).

Lohmann, A. W.

K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1-D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Martinez, M. A. G.

M. T. Camargo Silva, M. A. G. Martinez, P. R. Herczfeld, “MCQW intensity optical modulator for InP based MMIC/photonics integrated circuits,” in IEEE Microwave Theory and Techiques Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 229–232.
[CrossRef]

Onciul, D.

D. Onciul, “Waveguide launching efficiency for multimoded and partially coherent light sources,” Optik 97, 76–77 (1994).

D. Onciul, “Efficiency of light launching into waveguides: a phase-space approach,” Optik 96, 20–24 (1994).

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Weber, H.

N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase-space beam analyser,” Opt. Commun. 24, 927–949 (1992).

H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).
[CrossRef]

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Appl. Opt. (2)

J. Lightwave Technol. (1)

W. L. Emkey, “Optical coupling between single-mode semiconductor lasers and strip waveguides,” J. Lightwave Technol. 1, 436–443 (1983).
[CrossRef]

J. Mod. Opt. (1)

H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (3)

K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1-D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase-space beam analyser,” Opt. Commun. 24, 927–949 (1992).

Optik (2)

D. Onciul, “Efficiency of light launching into waveguides: a phase-space approach,” Optik 96, 20–24 (1994).

D. Onciul, “Waveguide launching efficiency for multimoded and partially coherent light sources,” Optik 97, 76–77 (1994).

Phys. Rev. (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Other (3)

D. Dragoman, “Phase-space representation of modes in optical waveguides,” J. Mod. Opt. (to be published).

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

M. T. Camargo Silva, M. A. G. Martinez, P. R. Herczfeld, “MCQW intensity optical modulator for InP based MMIC/photonics integrated circuits,” in IEEE Microwave Theory and Techiques Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 229–232.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

WDF of (a) fundamental (l = 0), (b) first (l = 1), (c) second (l = 2), (d) third (l = 3), and (e) fourth (l = 4) mode of a planar waveguide.

Fig. 2
Fig. 2

Coupling efficiency of a pure Gaussian light source into (a) first, (b) second, (c) third, and (d) fourth mode of a planar waveguide as a function of the displacement for x s = x 0.

Fig. 3
Fig. 3

Same as in Fig. 2 but as a function of the tilt.

Fig. 4
Fig. 4

Coupling efficiency of a first WM-like source as a function of the displacement in (a) second and (b) third mode of a planar waveguide, and as a function of the tilt in (c) second and (d) third mode of a planar waveguide.

Fig. 5
Fig. 5

WDF of a modulator.

Fig. 6
Fig. 6

Coupling efficiency between the modulator and (a) fundamental, (b) first, (c) second, (d) third, and (e) fourth mode of a planar waveguide as a function of the displacement.

Equations (7)

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η = | φ s ( x , y ) φ r * ( x , y ) d x d y | 2 | φ s ( x , y ) | 2 d x d y | φ r ( x , y ) | 2 d x d y ,
η = 4 π 2 W s ( x , y , p x , p y ) W r ( x , y , p x , p y ) d x d y d p x d p y W s ( x , y , p x , p y ) d x d y d p x d p y W r ( x , y , p x , p y ) d x d y d p x d p y ,
W i ( x , y , p x , p y ) = φ i ( x + x 2 , y + y 2 ) × φ * i ( x x 2 , y y 2 ) × exp ( i p x x + i p y y ) d x d y , i = r , s .
φ l ( x ) = ( x x 0 ) l exp ( x 2 / 2 x 0 2 ) ,
W l ( x , p ) = 2 x 0 π exp ( x 2 / x 0 2 p 2 x 0 2 ) × k = 0 l ( x x 0 ) 2 k C l k ( p x 0 ) 2 ( l k ) × j = 0 l k ( 1 ) j j ! C 2 j j C 2 ( l k ) 2 j ( 2 p x 0 ) 2 j ,
C l k = l ! k ! ( l k ) !
φ s ( x ) = exp ( x 2 / 2 x s 2 ) .

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