Abstract

Edge coupling of a focused partially coherent Gaussian Schell-model beam into a planar dielectric waveguide is examined. The incident field is decomposed into a sum of coherent modes that are expressed as a discrete superposition of plane-wave components. A model based on the rigorous diffraction theory of gratings is used to replace the waveguide with a corresponding periodic multilayer structure to determine the coupling efficiencies. Numerical simulations are presented for single and multimode planar waveguides and for a graded index waveguide. The results are compared with the predictions of the overlap integral method.

© 2004 Optical Society of America

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References

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  3. V. Ramaswamy and P. G. Suchoski, “Power loss at step discontinuity in an asymmetrical dielectric slab waveguide,” J. Opt. Soc. Am. 1, 754–759 (1984).
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  4. C. N. Capsalis, J. G. Fikioris, and N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” IEEE J. Lightwave Technol. LT-3, 408–415 (1985).
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  6. N. K. Uzunoglu, C. N. Capsalis, and I. Tigelis, “Scattering from an abruptly terminated single-mode-fiber waveguide,” J. Opt. Soc. Am. 4, 2150–2157 (1987).
    [Crossref]
  7. K. Hirayama and M. Koshiba, “Analysis of discontinuities in an open dielectric slabe waveguide by combination of finite and boundary elements,” IEEE Trans. Microwave Theory Technol. MTT-37, 761–768 (1989).
    [Crossref]
  8. S. S. Patrick and K. J. Webb, “A variational vector finite difference analysis for dielectric waveguides,” IEEE Trans. Microwave Theory Technol. MTT-40, 692–698 (1992).
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  15. E. Silberstein, Ph. Lalanne, J-P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875, (2001).
    [Crossref]
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2004 (1)

2001 (2)

E. Silberstein, Ph. Lalanne, J-P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875, (2001).
[Crossref]

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001)
[Crossref]

2000 (1)

Ph. Lalanne and E. Silberstein, “Fourier-modal methods applied to waveguide computational problems,” Opt. Lett. 25, 1902–1904 (2000).
[Crossref]

1998 (1)

P. Vahimaa, M. Kuittinen, J. Turunen, J. Saarinen, R.-P. Salmio, E. Lopez Lago, and J. Liñares, “Guided-mode propagation through an ion-exchanged graded-index boundary,” Opt. Commun. 147, 247–253 (1998).
[Crossref]

1996 (3)

1995 (2)

1993 (1)

K. Hirayama and M. Koshiba, “Rigorous analysis of coupling between laser and passive waveguide in multilayer slab waveguide,” J. Lightwave Technol. LT-11, 1353–1358 (1993).
[Crossref]

1992 (2)

S. S. Patrick and K. J. Webb, “A variational vector finite difference analysis for dielectric waveguides,” IEEE Trans. Microwave Theory Technol. MTT-40, 692–698 (1992).
[Crossref]

X. Wang, Z. Wang, and Z. Huang, “Propagation constant of a planar dielectric waveguide with arbitrary refractive-index variation,” Opt. Lett. 18, 805–807 (1992).
[Crossref]

1991 (1)

1990 (1)

K.-H. Schlereth and M. Tacke, “The complex propagation constant of multilayer waveguides: an algorithm for a personal computer,” J. Quantum Electron. 26, 627–630 (1990)
[Crossref]

1989 (1)

K. Hirayama and M. Koshiba, “Analysis of discontinuities in an open dielectric slabe waveguide by combination of finite and boundary elements,” IEEE Trans. Microwave Theory Technol. MTT-37, 761–768 (1989).
[Crossref]

1988 (1)

B. M. A. Rahman and J. B. Davies, “Analyses of optical waveguide discontinuities,” J. Lightwave Technol. LT-6, 52–57 (1988).
[Crossref]

1987 (1)

N. K. Uzunoglu, C. N. Capsalis, and I. Tigelis, “Scattering from an abruptly terminated single-mode-fiber waveguide,” J. Opt. Soc. Am. 4, 2150–2157 (1987).
[Crossref]

1985 (2)

C. N. Capsalis, J. G. Fikioris, and N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” IEEE J. Lightwave Technol. LT-3, 408–415 (1985).
[Crossref]

L. M. Walpita, “Solutions for planar optical waveguide equations by selecting zero elements in a characteristic matrix,” J. Opt. Soc. Am. A 2, 595–602 (1985).
[Crossref]

1984 (3)

J. Chilwell and I. Hodgkinson, “Thin-films field-transfer matrix theory of planar multilayer waveguides and reflection from prism-loaded waveguides,” J. Opt. Soc. Am. A 1, 742–753 (1984).
[Crossref]

B. M. A. Rahman and J. B. Davies, “Finite-element analysis of optical and microwave waveguide problems,” IEEE Trans. Microwave Theory Technol. MTT-32, 20–28 (1984).
[Crossref]

V. Ramaswamy and P. G. Suchoski, “Power loss at step discontinuity in an asymmetrical dielectric slab waveguide,” J. Opt. Soc. Am. 1, 754–759 (1984).
[Crossref]

1982 (3)

1981 (2)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[Crossref]

P. Gelin, M. Petenzi, and J. Citerne, “Rigorous analysis of the scattering of surface waves in an abruptly ended slab electric waveguide,” IEEE Trans. Microwave Theory Technol. MTT-29, 107–114 (1981).
[Crossref]

1978 (1)

Aalto, T.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001)
[Crossref]

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[Crossref]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[Crossref]

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[Crossref]

Cao, Q.

Capsalis, C. N.

N. K. Uzunoglu, C. N. Capsalis, and I. Tigelis, “Scattering from an abruptly terminated single-mode-fiber waveguide,” J. Opt. Soc. Am. 4, 2150–2157 (1987).
[Crossref]

C. N. Capsalis, J. G. Fikioris, and N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” IEEE J. Lightwave Technol. LT-3, 408–415 (1985).
[Crossref]

Chilwell, J.

Citerne, J.

P. Gelin, M. Petenzi, and J. Citerne, “Rigorous analysis of the scattering of surface waves in an abruptly ended slab electric waveguide,” IEEE Trans. Microwave Theory Technol. MTT-29, 107–114 (1981).
[Crossref]

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[Crossref]

Davies, J. B.

B. M. A. Rahman and J. B. Davies, “Analyses of optical waveguide discontinuities,” J. Lightwave Technol. LT-6, 52–57 (1988).
[Crossref]

B. M. A. Rahman and J. B. Davies, “Finite-element analysis of optical and microwave waveguide problems,” IEEE Trans. Microwave Theory Technol. MTT-32, 20–28 (1984).
[Crossref]

Fikioris, J. G.

C. N. Capsalis, J. G. Fikioris, and N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” IEEE J. Lightwave Technol. LT-3, 408–415 (1985).
[Crossref]

Forbes, G. W.

Gaylord, T. K.

Gelin, P.

P. Gelin, M. Petenzi, and J. Citerne, “Rigorous analysis of the scattering of surface waves in an abruptly ended slab electric waveguide,” IEEE Trans. Microwave Theory Technol. MTT-29, 107–114 (1981).
[Crossref]

Granet, G.

Grann, E. B.

Guizal, B.

Heimala, P.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001)
[Crossref]

Hinata, T.

T. Hosono, T. Hinata, and A. Inoue, “Numerical analysis of the discontinuities in slab dielectric waveguides,” Radio Science 17, 75–83 (1982).
[Crossref]

Hirayama, K.

K. Hirayama and M. Koshiba, “Rigorous analysis of coupling between laser and passive waveguide in multilayer slab waveguide,” J. Lightwave Technol. LT-11, 1353–1358 (1993).
[Crossref]

K. Hirayama and M. Koshiba, “Analysis of discontinuities in an open dielectric slabe waveguide by combination of finite and boundary elements,” IEEE Trans. Microwave Theory Technol. MTT-37, 761–768 (1989).
[Crossref]

Hodgkinson, I.

Hosono, T.

T. Hosono, T. Hinata, and A. Inoue, “Numerical analysis of the discontinuities in slab dielectric waveguides,” Radio Science 17, 75–83 (1982).
[Crossref]

Houde-Walter, S. N.

Huang, Z.

Hugonin, J-P.

Inoue, A.

T. Hosono, T. Hinata, and A. Inoue, “Numerical analysis of the discontinuities in slab dielectric waveguides,” Radio Science 17, 75–83 (1982).
[Crossref]

Knop, K.

Koshiba, M.

K. Hirayama and M. Koshiba, “Rigorous analysis of coupling between laser and passive waveguide in multilayer slab waveguide,” J. Lightwave Technol. LT-11, 1353–1358 (1993).
[Crossref]

K. Hirayama and M. Koshiba, “Analysis of discontinuities in an open dielectric slabe waveguide by combination of finite and boundary elements,” IEEE Trans. Microwave Theory Technol. MTT-37, 761–768 (1989).
[Crossref]

Kuittinen, M.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001)
[Crossref]

P. Vahimaa, M. Kuittinen, J. Turunen, J. Saarinen, R.-P. Salmio, E. Lopez Lago, and J. Liñares, “Guided-mode propagation through an ion-exchanged graded-index boundary,” Opt. Commun. 147, 247–253 (1998).
[Crossref]

Lalanne, Ph.

Leppihalme, M.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001)
[Crossref]

Li, L.

Liñares, J.

P. Vahimaa, M. Kuittinen, J. Turunen, J. Saarinen, R.-P. Salmio, E. Lopez Lago, and J. Liñares, “Guided-mode propagation through an ion-exchanged graded-index boundary,” Opt. Commun. 147, 247–253 (1998).
[Crossref]

Lopez Lago, E.

P. Vahimaa, M. Kuittinen, J. Turunen, J. Saarinen, R.-P. Salmio, E. Lopez Lago, and J. Liñares, “Guided-mode propagation through an ion-exchanged graded-index boundary,” Opt. Commun. 147, 247–253 (1998).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).

McPhedran, R. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[Crossref]

Moharam, M. G.

Morf, R. H.

Morris, G. M.

Obayya, S. S. A.

Patrick, S. S.

S. S. Patrick and K. J. Webb, “A variational vector finite difference analysis for dielectric waveguides,” IEEE Trans. Microwave Theory Technol. MTT-40, 692–698 (1992).
[Crossref]

Petenzi, M.

P. Gelin, M. Petenzi, and J. Citerne, “Rigorous analysis of the scattering of surface waves in an abruptly ended slab electric waveguide,” IEEE Trans. Microwave Theory Technol. MTT-29, 107–114 (1981).
[Crossref]

Pommet, D. A.

Rahman, B. M. A.

B. M. A. Rahman and J. B. Davies, “Analyses of optical waveguide discontinuities,” J. Lightwave Technol. LT-6, 52–57 (1988).
[Crossref]

B. M. A. Rahman and J. B. Davies, “Finite-element analysis of optical and microwave waveguide problems,” IEEE Trans. Microwave Theory Technol. MTT-32, 20–28 (1984).
[Crossref]

Ramaswamy, V.

V. Ramaswamy and P. G. Suchoski, “Power loss at step discontinuity in an asymmetrical dielectric slab waveguide,” J. Opt. Soc. Am. 1, 754–759 (1984).
[Crossref]

Saarinen, J.

P. Vahimaa, M. Kuittinen, J. Turunen, J. Saarinen, R.-P. Salmio, E. Lopez Lago, and J. Liñares, “Guided-mode propagation through an ion-exchanged graded-index boundary,” Opt. Commun. 147, 247–253 (1998).
[Crossref]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[Crossref]

Salmio, R.-P.

P. Vahimaa, M. Kuittinen, J. Turunen, J. Saarinen, R.-P. Salmio, E. Lopez Lago, and J. Liñares, “Guided-mode propagation through an ion-exchanged graded-index boundary,” Opt. Commun. 147, 247–253 (1998).
[Crossref]

Schlereth, K.-H.

K.-H. Schlereth and M. Tacke, “The complex propagation constant of multilayer waveguides: an algorithm for a personal computer,” J. Quantum Electron. 26, 627–630 (1990)
[Crossref]

Silberstein, E.

E. Silberstein, Ph. Lalanne, J-P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875, (2001).
[Crossref]

Ph. Lalanne and E. Silberstein, “Fourier-modal methods applied to waveguide computational problems,” Opt. Lett. 25, 1902–1904 (2000).
[Crossref]

Smith, R. E.

Suchoski, P. G.

V. Ramaswamy and P. G. Suchoski, “Power loss at step discontinuity in an asymmetrical dielectric slab waveguide,” J. Opt. Soc. Am. 1, 754–759 (1984).
[Crossref]

Tacke, M.

K.-H. Schlereth and M. Tacke, “The complex propagation constant of multilayer waveguides: an algorithm for a personal computer,” J. Quantum Electron. 26, 627–630 (1990)
[Crossref]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[Crossref]

Tervo, J.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001)
[Crossref]

Tigelis, I.

N. K. Uzunoglu, C. N. Capsalis, and I. Tigelis, “Scattering from an abruptly terminated single-mode-fiber waveguide,” J. Opt. Soc. Am. 4, 2150–2157 (1987).
[Crossref]

Turunen, J.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001)
[Crossref]

P. Vahimaa, M. Kuittinen, J. Turunen, J. Saarinen, R.-P. Salmio, E. Lopez Lago, and J. Liñares, “Guided-mode propagation through an ion-exchanged graded-index boundary,” Opt. Commun. 147, 247–253 (1998).
[Crossref]

P. Vahimaa and J. Turunen, in Diffractive Optics and Micro-Optics, Vol 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 69.

Uzunoglu, N. K.

N. K. Uzunoglu, C. N. Capsalis, and I. Tigelis, “Scattering from an abruptly terminated single-mode-fiber waveguide,” J. Opt. Soc. Am. 4, 2150–2157 (1987).
[Crossref]

C. N. Capsalis, J. G. Fikioris, and N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” IEEE J. Lightwave Technol. LT-3, 408–415 (1985).
[Crossref]

Vahimaa, P.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001)
[Crossref]

P. Vahimaa, M. Kuittinen, J. Turunen, J. Saarinen, R.-P. Salmio, E. Lopez Lago, and J. Liñares, “Guided-mode propagation through an ion-exchanged graded-index boundary,” Opt. Commun. 147, 247–253 (1998).
[Crossref]

P. Vahimaa and J. Turunen, in Diffractive Optics and Micro-Optics, Vol 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 69.

Walpita, L. M.

Wang, X.

Wang, Z.

Webb, K. J.

S. S. Patrick and K. J. Webb, “A variational vector finite difference analysis for dielectric waveguides,” IEEE Trans. Microwave Theory Technol. MTT-40, 692–698 (1992).
[Crossref]

Wolf, E.

IEEE J. Lightwave Technol. (1)

C. N. Capsalis, J. G. Fikioris, and N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” IEEE J. Lightwave Technol. LT-3, 408–415 (1985).
[Crossref]

IEEE Trans. Microwave Theory Technol. (4)

P. Gelin, M. Petenzi, and J. Citerne, “Rigorous analysis of the scattering of surface waves in an abruptly ended slab electric waveguide,” IEEE Trans. Microwave Theory Technol. MTT-29, 107–114 (1981).
[Crossref]

B. M. A. Rahman and J. B. Davies, “Finite-element analysis of optical and microwave waveguide problems,” IEEE Trans. Microwave Theory Technol. MTT-32, 20–28 (1984).
[Crossref]

K. Hirayama and M. Koshiba, “Analysis of discontinuities in an open dielectric slabe waveguide by combination of finite and boundary elements,” IEEE Trans. Microwave Theory Technol. MTT-37, 761–768 (1989).
[Crossref]

S. S. Patrick and K. J. Webb, “A variational vector finite difference analysis for dielectric waveguides,” IEEE Trans. Microwave Theory Technol. MTT-40, 692–698 (1992).
[Crossref]

J. Lightwave Technol. (3)

K. Hirayama and M. Koshiba, “Rigorous analysis of coupling between laser and passive waveguide in multilayer slab waveguide,” J. Lightwave Technol. LT-11, 1353–1358 (1993).
[Crossref]

S. S. A. Obayya, “Novel finite element analysis of optical waveguide discontinuity problems,” J. Lightwave Technol. 22, 1420–1425 (2004).
[Crossref]

B. M. A. Rahman and J. B. Davies, “Analyses of optical waveguide discontinuities,” J. Lightwave Technol. LT-6, 52–57 (1988).
[Crossref]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (8)

J. Quantum Electron. (1)

K.-H. Schlereth and M. Tacke, “The complex propagation constant of multilayer waveguides: an algorithm for a personal computer,” J. Quantum Electron. 26, 627–630 (1990)
[Crossref]

Opt. Acta (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[Crossref]

Opt. Commun. (2)

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001)
[Crossref]

P. Vahimaa, M. Kuittinen, J. Turunen, J. Saarinen, R.-P. Salmio, E. Lopez Lago, and J. Liñares, “Guided-mode propagation through an ion-exchanged graded-index boundary,” Opt. Commun. 147, 247–253 (1998).
[Crossref]

Opt. Lett. (3)

Radio Science (1)

T. Hosono, T. Hinata, and A. Inoue, “Numerical analysis of the discontinuities in slab dielectric waveguides,” Radio Science 17, 75–83 (1982).
[Crossref]

Other (3)

P. Vahimaa and J. Turunen, in Diffractive Optics and Micro-Optics, Vol 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 69.

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B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[Crossref]

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Figures (5)

Fig. 1.
Fig. 1.

The periodic waveguide structure of period d used to model the real non-periodic structure of Eq. (4).

Fig. 2.
Fig. 2.

Coupling of a set of Gaussian Schell-model beams from a uniform medium into a periodically modulated waveguide used to model single-beam coupling into non-periodic waveguide.

Fig. 3.
Fig. 3.

Coupling efficiencies η 1 into the fundamental waveguide mode as a function of beam center position and beam width w when the global degree of coherence δ = 1. (a) Overlap integral method. (b) Quasi-rigorous method.

Fig. 4.
Fig. 4.

Convergence of the coupling efficiency into the fundamental mode of the single mode waveguide in the case δ = 1 as a function of the number of eigenmodes retained in the calculation. The curves represent the periods d = 80d g, d = 125d g, d = 150d g, and d = 175d g respectively, starting from the lowermost.

Fig. 5.
Fig. 5.

The refractive index distribution of the graded index waveguide as a function of the guiding layer thickness d g.

Tables (4)

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Table 1. Optimum coupling efficiencies of the GSM beam into a single mode waveguide with guiding layer thickness dg = 0.21 μm.

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Table 2. Optimum coupling efficiencies of the GSM beam into a three-mode waveguide with guiding layer thickness dg = 0.7 μm.

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Table 3. Optimum coupling efficiencies of the GSM beam into a graded index waveguide mode m = 1 with guiding layer thickness dg = 5.0 μm.

Tables Icon

Table 4. Optimum coupling efficiencies of the GSM beam into a graded index waveguide mode m = 3 with guiding layer thickness dg = 5.0 μm.

Equations (31)

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W ( r 1 , r 2 ) = n = 1 λ n ϕ n * ( r 1 ) ϕ n ( r 2 ) .
D W ( r 1 , r 2 ) ϕ n ( r 1 ) d 3 r 1 = λ n ϕ n ( r 2 ) ,
D ϕ m * ( r ) ϕ n ( r ) d 3 r = δ mn ,
n ( x ) = { n c when x > d g n l when x l x < x l + 1 , n s when x < 0
n max = max { n l }
U ( x , z ) = X ( x ) [ a exp ( z ) + b exp ( z ) ] .
X ( x + d ) = X ( x ) exp ( i α 0 d ) ,
{ guided modes : [ k max ( n s , n c ) ] 2 < γ 2 < ( k n max ) 2 evanescent modes : γ 2 < 0 radiation modes : otherwise .
E y in ( x , z ) = m = M M A m exp [ i ( α m x + r m z ) ] ,
E y r ( x , z ) = m = R m exp [ i ( α m x + r m z ) ] ,
α m = α 0 + 2 π m d ,
r m = { ( k 2 n 2 α m 2 ) 1 2 if α m k n i ( α m 2 k 2 n 2 ) 1 2 otherwise ,
E y t ( x , z ) = m = 1 a m exp ( i γ m z ) X m ( x ) ,
E y in ( x , z ) + E y r ( x , z ) = E y t ( x , z ) ,
z E y in ( x , z ) + z E y r ( x , z ) = z E y t ( x , z ) .
A + R = Pa ,
r ( A R ) = P Γ a ,
P m q = 1 d o d X m ( x ) exp ( i 2 π q x d ) d x .
a = 2 ( + rP ) 1 rA .
W y in ( x 1 , x 2 ) = exp [ ( x 1 x ¯ ) 2 + ( x 2 x ¯ ) 2 w 0 2 ] exp [ ( x 1 x 2 ) 2 2 σ 0 2 ] ,
ϕ n ( x , z ) = ( 2 c π ) 1 4 1 2 n n ! H n [ ( x x ¯ ) 2 c ] exp [ c ( x x ¯ ) 2 ] ,
λ n = ( π a + b + c ) 1 2 ( b a + b + c ) n ,
{ a = w 0 2 b = σ 0 2 2 c = ( a 2 + 2 a b ) 1 2 ,
η m = P m P in .
P = 1 2 0 d { E y ( x , z ) H x * ( x , z ) } d x ,
P in = d 2 ω μ 0 n = 1 λ n m = M M r m A m n 2 ,
P m = d 2 ω μ 0 { γ m } n = 1 λ n a m n 2 q = P m q 2 .
P r = d 2 ω μ 0 { γ m } n = 1 λ n q = R m n 2 .
η m = n = 1 λ n X m * ( x ) ϕ n ( x ) d x 2 X m ( x ) 2 d x n = 1 λ n ϕ n ( x ) 2 d x .
W F = λ f π w 0 β ,
σ F = σ 0 w F w 0 .

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