Abstract

A ray model for the treatment of a planar anisotropic dielectric waveguide is presented. The attenuation coefficient of the guided wave predicted by this model is consistent with the energy-conservation requirement.

© 1998 Optical Society of America

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References

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  1. H. Kogelnik, H. P. Weber, “Rays, stored energy, and power flow in dielectric waveguides,” J. Opt. Soc. Am. 64, 174–185 (1974).
    [CrossRef]
  2. S. R. Seshadri, “Coupling of guided modes in thin films with surface corrugation,” J. Appl. Phys. 63, R115–R146 (1988).
    [CrossRef]
  3. P. K. Tien, R. Ulrich, “Theory of prism-film coupler and thin-film light guides,” J. Opt. Soc. Am. 60, 1325–1337 (1970).
    [CrossRef]
  4. S. R. Seshadri, “Quasi-optics of the coupling of guided modes in a corrugated dielectric waveguide,” presented at the National Radio Science Meeting, Los Angeles, Calif., June 16–19, 1981.
  5. M. T. Wlodarczyk, S. R. Seshadri, “Analysis of grating couplers in planar waveguides for waves at oblique incidence,” J. Opt. Soc. Am. A 2, 171–185 (1985).
    [CrossRef]
  6. S. R. Seshadri, “Quasi-optics of the coupling of guided modes in two parallel, identical dielectric waveguides,” J. Opt. Soc. Am. A 4, 1030–1036 (1987).
    [CrossRef]
  7. Z. H. Wang, S. R. Seshadri, “Quasi-optics of the evanescent wave excitation of planar dielectric waveguides,” J. Opt. Soc. Am. A 4, 2141–2149 (1987).
    [CrossRef]
  8. D. W. C. So, S. R. Seshadri, “Thin dielectric waveguide with a phased array of two gratings on its surfaces,” J. Appl. Phys. 75, 4851–4872 (1994).
    [CrossRef]
  9. D. W. C. So, S. R. Seshadri, “Effect of phase shifter on the characteristics of the distributed feedback laser,” J. Appl. Phys. 78, 5244–5252 (1995).
    [CrossRef]
  10. D. W. C. So, S. R. Seshadri, “Quasi-optics of second-order Bragg interaction in a thin film optical waveguide,” J. Appl. Phys. 79, 6741–6749 (1996).
    [CrossRef]
  11. H. Kogelnik, T. P. Sosnowski, H. P. Weber, “A ray-optical analysis of thin film polarization converters,” IEEE J. Quantum Electron. QE-9, 795–800 (1973).
    [CrossRef]
  12. M. A. Sletten, S. R. Seshadri, “Quasi-optics of a thin film polarization converter,” J. Opt. Soc. Am. A 6, 748–757 (1989).
    [CrossRef]
  13. V. Ramaswamy, “Ray model of energy and power flow in anisotropic film waveguide,” J. Opt. Soc. Am. 64, 1313–1320 (1974).
    [CrossRef]
  14. A. Sommerfeld, Optics (Academic, New York, 1954), pp. 122–123.
  15. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960), pp. 269–271, 315–321.
  16. P. K. Tien, “Light waves in thin films and integrated optics,” Appl. Opt. 10, 2395–2413 (1971).
    [CrossRef] [PubMed]
  17. H. K. V. Lotsch, “Reflection and refraction of a beam of light at a plane interface,” J. Opt. Soc. Am. 58, 551–561 (1968).
    [CrossRef]
  18. N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972), pp. 73–87.

1996 (1)

D. W. C. So, S. R. Seshadri, “Quasi-optics of second-order Bragg interaction in a thin film optical waveguide,” J. Appl. Phys. 79, 6741–6749 (1996).
[CrossRef]

1995 (1)

D. W. C. So, S. R. Seshadri, “Effect of phase shifter on the characteristics of the distributed feedback laser,” J. Appl. Phys. 78, 5244–5252 (1995).
[CrossRef]

1994 (1)

D. W. C. So, S. R. Seshadri, “Thin dielectric waveguide with a phased array of two gratings on its surfaces,” J. Appl. Phys. 75, 4851–4872 (1994).
[CrossRef]

1989 (1)

1988 (1)

S. R. Seshadri, “Coupling of guided modes in thin films with surface corrugation,” J. Appl. Phys. 63, R115–R146 (1988).
[CrossRef]

1987 (2)

1985 (1)

1974 (2)

1973 (1)

H. Kogelnik, T. P. Sosnowski, H. P. Weber, “A ray-optical analysis of thin film polarization converters,” IEEE J. Quantum Electron. QE-9, 795–800 (1973).
[CrossRef]

1971 (1)

1970 (1)

1968 (1)

Burke, J. J.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972), pp. 73–87.

Kapany, N. S.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972), pp. 73–87.

Kogelnik, H.

H. Kogelnik, H. P. Weber, “Rays, stored energy, and power flow in dielectric waveguides,” J. Opt. Soc. Am. 64, 174–185 (1974).
[CrossRef]

H. Kogelnik, T. P. Sosnowski, H. P. Weber, “A ray-optical analysis of thin film polarization converters,” IEEE J. Quantum Electron. QE-9, 795–800 (1973).
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960), pp. 269–271, 315–321.

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960), pp. 269–271, 315–321.

Lotsch, H. K. V.

Ramaswamy, V.

Seshadri, S. R.

D. W. C. So, S. R. Seshadri, “Quasi-optics of second-order Bragg interaction in a thin film optical waveguide,” J. Appl. Phys. 79, 6741–6749 (1996).
[CrossRef]

D. W. C. So, S. R. Seshadri, “Effect of phase shifter on the characteristics of the distributed feedback laser,” J. Appl. Phys. 78, 5244–5252 (1995).
[CrossRef]

D. W. C. So, S. R. Seshadri, “Thin dielectric waveguide with a phased array of two gratings on its surfaces,” J. Appl. Phys. 75, 4851–4872 (1994).
[CrossRef]

M. A. Sletten, S. R. Seshadri, “Quasi-optics of a thin film polarization converter,” J. Opt. Soc. Am. A 6, 748–757 (1989).
[CrossRef]

S. R. Seshadri, “Coupling of guided modes in thin films with surface corrugation,” J. Appl. Phys. 63, R115–R146 (1988).
[CrossRef]

S. R. Seshadri, “Quasi-optics of the coupling of guided modes in two parallel, identical dielectric waveguides,” J. Opt. Soc. Am. A 4, 1030–1036 (1987).
[CrossRef]

Z. H. Wang, S. R. Seshadri, “Quasi-optics of the evanescent wave excitation of planar dielectric waveguides,” J. Opt. Soc. Am. A 4, 2141–2149 (1987).
[CrossRef]

M. T. Wlodarczyk, S. R. Seshadri, “Analysis of grating couplers in planar waveguides for waves at oblique incidence,” J. Opt. Soc. Am. A 2, 171–185 (1985).
[CrossRef]

S. R. Seshadri, “Quasi-optics of the coupling of guided modes in a corrugated dielectric waveguide,” presented at the National Radio Science Meeting, Los Angeles, Calif., June 16–19, 1981.

Sletten, M. A.

So, D. W. C.

D. W. C. So, S. R. Seshadri, “Quasi-optics of second-order Bragg interaction in a thin film optical waveguide,” J. Appl. Phys. 79, 6741–6749 (1996).
[CrossRef]

D. W. C. So, S. R. Seshadri, “Effect of phase shifter on the characteristics of the distributed feedback laser,” J. Appl. Phys. 78, 5244–5252 (1995).
[CrossRef]

D. W. C. So, S. R. Seshadri, “Thin dielectric waveguide with a phased array of two gratings on its surfaces,” J. Appl. Phys. 75, 4851–4872 (1994).
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, Optics (Academic, New York, 1954), pp. 122–123.

Sosnowski, T. P.

H. Kogelnik, T. P. Sosnowski, H. P. Weber, “A ray-optical analysis of thin film polarization converters,” IEEE J. Quantum Electron. QE-9, 795–800 (1973).
[CrossRef]

Tien, P. K.

Ulrich, R.

Wang, Z. H.

Weber, H. P.

H. Kogelnik, H. P. Weber, “Rays, stored energy, and power flow in dielectric waveguides,” J. Opt. Soc. Am. 64, 174–185 (1974).
[CrossRef]

H. Kogelnik, T. P. Sosnowski, H. P. Weber, “A ray-optical analysis of thin film polarization converters,” IEEE J. Quantum Electron. QE-9, 795–800 (1973).
[CrossRef]

Wlodarczyk, M. T.

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

H. Kogelnik, T. P. Sosnowski, H. P. Weber, “A ray-optical analysis of thin film polarization converters,” IEEE J. Quantum Electron. QE-9, 795–800 (1973).
[CrossRef]

J. Appl. Phys. (4)

S. R. Seshadri, “Coupling of guided modes in thin films with surface corrugation,” J. Appl. Phys. 63, R115–R146 (1988).
[CrossRef]

D. W. C. So, S. R. Seshadri, “Thin dielectric waveguide with a phased array of two gratings on its surfaces,” J. Appl. Phys. 75, 4851–4872 (1994).
[CrossRef]

D. W. C. So, S. R. Seshadri, “Effect of phase shifter on the characteristics of the distributed feedback laser,” J. Appl. Phys. 78, 5244–5252 (1995).
[CrossRef]

D. W. C. So, S. R. Seshadri, “Quasi-optics of second-order Bragg interaction in a thin film optical waveguide,” J. Appl. Phys. 79, 6741–6749 (1996).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (4)

Other (4)

A. Sommerfeld, Optics (Academic, New York, 1954), pp. 122–123.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960), pp. 269–271, 315–321.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972), pp. 73–87.

S. R. Seshadri, “Quasi-optics of the coupling of guided modes in a corrugated dielectric waveguide,” presented at the National Radio Science Meeting, Los Angeles, Calif., June 16–19, 1981.

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Figures (2)

Fig. 1
Fig. 1

Geometry of the planar anisotropic dielectric waveguide: O.A., optic axis.

Fig. 2
Fig. 2

Ray model for the synthesis of the guided mode.

Equations (79)

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ε=xˆxˆεxx+xˆzˆεxz+yˆyˆε+zˆxˆεzx+zˆzˆεzz
εxx=ε cos2 α+ε sin2 α,
εxz=εzx=(ε-ε)cos α sin α,
εzz=ε sin2 α+ε cos2 α.
iωHy(x, z)=zEx(x, z)-xEz(x, z),
-iω[εxxEx(x, z)+εxzEz(x, z)]=-zHy(x, z),
-iω[εzxEx(x, z)+εzzEz(x, z)]=xHy(x, z).
Hy(x, z)iscontinuous,
Ez(x, z)iscontinuous.
εxxεzz-εxz2=εε.
Exf(x, z)=1iωεεεxz x+εzz zHyf(x, z),
Ezf(x, z)=-1iωεεεxx x+εxz zHyf(x, z).
εxx 2x2+2εxz 2xz+εzz 2z2+ω2εε
Hyf(x, z)=0.
εxxkf2+2εxzβkf+εzzβ2-ω2εε=0.
kf+=(-εxzβ+εkf0)/εxx,
kf-=(-εxzβ-εkf0)/εxx,
kf0=εε(ω2εxx-β2)1/2.
ωkf=1ωεε(εxxkf+εxzβ).
ωkf+=-ωkf-=kf0ωε.
νgz=ωβ=1ωεε(εxzkf+εzzβ).
νgz,+=ωβ=1ωεεxx(εβ+εxzkf0)forkf=kf+,
νgz,-=ωβ=1ωεεxx(εβ-εxzkf0)forkf=kf-.
tan θgi=ωβ/ωkfforkf=kf+=(εβ+εxzkf0)εxxkf0,
tan θgr=ωβ/-ωkfforkf=kf-=(εβ-εxzkf0)εxxkf0.
tan θgi+tan θgr=2εβεxxkf0=-kf+β-kf-β.
Hyc(x, z)=A˜ exp[-αc(x-b)]exp(iβz)
forb<x<,
Hyf(x, z)={B˜ exp[ikf+(x+a)]+C˜ exp[ikf-(x-b)]}exp(iβz)
for-a<x<b,
Hys(x, z)=D˜ exp[αs(x+a)]exp(iβz)
for-<x<-a,
αν=(β2-ω2εν)1/2forν=c, s,
B˜=exp{-i[kf+(b+a)-ϕc]}2 cos ϕcA˜,
C˜=exp(-iϕc)2 cos ϕcA˜,
C˜=B˜ exp[ikf+(b+a)]exp(-i2ϕc),
B˜=exp(-iϕs)2 cos ϕsD˜,
C˜=exp{i[kf-(b+a)+ϕs]}2 cos ϕsD˜,
B˜=C˜ exp[-ikf-(b+a)]exp(-i2ϕs),
tan ϕν=εανkf0ενforν=c, s.
zν=2 ϕνβ,tν=-2 ϕνωforν=c, s.
zν=ωβtν=2ενε2(εxx-εν)ω2βεανkf0(εv2kf02+αν2ε2)
forν=c, s.
D(ω, β)=2π(j-1),j=1, 2, 3,  ,
D(ω, β)=2(b+a) εεxxkf0-2ϕc-2ϕs
Vgz=ZB/TB,
ZB=-Dβ=(b+a)(tan θgi+tan θgr)+zc+zs,
TB=Dω=(b+a)tan θgiνgz,++tan θgrνgz,-+tc+ts.
Hyf(x, z)=A˜ cosεεxxkf0(x-b)+ϕccos ϕc×exp-iβ εxzεxx(x-b)exp(iβz)
for-a<x<b,
Hys(x, z)=A˜(-1)(j-1) cos ϕscos ϕcexp[αs(x+a)]×expi(b+a)β εxzεxxexp(iβz)
for-<x<-a.
A˜=ANg,
Ng=4ωεxx cos2 ϕcwβ(b+a)eff1/2,
(b+a)eff=(b+a)+1αcqc+1αsqs,
qv=β2(εε-εν2)ω2εxxεν(εxx-εν)-(εε-ενεxx)εxx(εxx-εν)
forν=c, s.
ZB=(tan θgi+tan θgr)(b+a)eff.
B˜(z)=C˜(z-ZB)exp{-i[kf-(b+a)+2ϕs]}.
C˜(z)=C˜(z-ZB)exp[iD(ω, β)].
D(ω, β, ε˜, ε˜)=2π(j-1)+iεi Dε+iεi Dε.
C˜(z-ZB)=C˜(z)-ZB zC˜(z),
exp[iD(ω, β, ε˜, ε˜)]=1-εi Dε-εi Dε.
zA(z)=-αgA(z),
αg=1ZBεi Dε+εi Dε.
αg=1ZBεi εkf0(b+a)εxx2(kf02 sin2 α+β2 cos2 α)-(ω2ε sin2 α-β2)2ε2kf0(sin 2ϕc+sin 2ϕs)+εi 1kf0(b+a)εxx2ε(ε2kf02 cos2 α+ε2β2 sin2 α)-(ω2ε cos2 α-β2)2εkf0×(sin 2ϕc+sin 2ϕs).
-z|A(z)|2=2αg|A(z)|2.
Pdis(x)=12ω{εxxi|Exf(x, z)|2+2εxzi×Re[Exf(x, z)Ezf*(x, z)]+εzzi|Ezf(x, z)|2},
|Exf(x, z)|2=|A|2Ng2ω2εxx2ε2 cos2 ϕcε2β2×cos2εεxxkf0(x-b)+ϕc+εxz2kf02 sin2εεxxkf0(x-b)+ϕc,
Re[Exf(x, z)Ezf*(x, z)]=-|A|2Ng2 εxzkf02ω2εxxε2 cos2 ϕc×sin2εεxxkf0(x-b)+ϕc,
|Ezf(x, z)|2=|A|2Ng2 kf02ω2ε2 cos2 ϕc×sin2εεxxkf0(x-b)+ϕc.
PD=12ωw|A(z)|2Ng2εxxiβ2ω2εxx2 cos2ϕc×(b+a)2+εxx4εkf0(sin 2ϕc+sin 2ϕs)+kf02ω2ε2 cos2 ϕcεxz2εxx2εxxi-2 εxzεxxεxzi+εzzi×(b+a)2-εxx4εkf0(sin 2ϕc+sin 2ϕs).
D(ω, β˜, ε˜, ε˜)=2π(j-1)-iαgZB+iεi Dε+iεi Dε.
whf=14|Hyf(x, z)|2,
wef=14{εxx|Exf(x, z)|2+2εxz×Re[Exf(x, z)Ezf*(x, z)]+εzz|Ezf(x, z)|2}.
W=w|A˜|24 cos2 ϕc(b+a)+β2εxxω21αcqc+1αsqs.
TB=2ωεkf0(b+a)+β2εxxω21αcqc+1αsqs.
|A˜|2=|A|2ZB4 cos2 ϕcw2ωεkf0.
W=|A|2 TBZB.

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