Abstract

The fields of radiation modes in the semi-infinite regions above and below a dielectric multilayer can be completely specified once the reflection and transmission coefficients of the stack are known. Given these fields in the substrate or cover regions, the fields in the layers may be generated by successive application of two simple algebraic relationships that give the amplitude and phase of the standing wave in each layer in terms of the amplitude and phase of the layer above it or below it.

© 1994 Optical Society of America

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References

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  1. D. Marcuse, Theory of Dielectric Waveguides (Academic, New York, 1974).
  2. H. Kogelnik, “Theory of optical waveguides,” in Guided-Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1988).
    [CrossRef]
  3. D. L. Lee, Electromagnetic Principles of Integrated Optics (Wiley, New York, 1986).
  4. H. Nishihara, M. Haruna, T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York, 1989).
  5. S. Miyanaga, T. Asakura, M. Imai, “Scattering characteristics of a beam mode in a dielectric-slab optical waveguide,” Opt. Quantum Electron. 11, 205–215 (1979).
    [CrossRef]
  6. P. Benech, D. A. M. Khalil, F. Saint André, “An exact simplified method for the normalization of radiation modes in planar multilayer structures,” Opt. Commun. 88, 96–100 (1992).
    [CrossRef]
  7. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), p. 114.
  8. N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972), pp. 58–64.
  9. To calculate the cross power, one must evaluate integrals of products of two cosine functions, each of the form of Eq. (8) but with, in general, a different set of parameters κi, ϕi, β. If we then define a function F(x) in such a way that its derivative with respect to xis equal to the integrand in the integrals that we must evaluate, we can show, using the continuity of the tangential fields at the interfaces, that the sum of the finite terms obtained by integrating over all xvanishes. Specifically, in an n-layer structure this sum takes the form F(0) + F(b1) − F(0) + F(b2) − F(b1) + … +F(bn) − F(bn−1) + F(c) − F(bn) − F(c). The first and last terms are contributed by the integrals over the substrate and the cover. The two remaining terms in the integration involve infinite limits and sum to 2πAs2δ(κs− κs′) for modes with ϕs= ϕ+(or ϕ−) in both cosine functions in the integrand. One can also show by direct integration that the integral also vanishes in the degenerate case where κs′ = κsbut ϕs= ϕ+in one field and ϕ−in the other. Alternatively, one may use Kogelnik’s Eq. (2.3.33) (Ref. 2) and our field expressions to demonstrate orthogonality. One may also demonstrate orthogonality by employing the arguments of Ref. 6, in which it is concluded that only the incoming plane waves from the sources at infinity are needed for determining the power in a mode and normalizing it. Accordingly, I note that the incoming plane waves of the degenerate modes are in phase in the substrate but πout of phase in the cover. This means that the cross power of the degenerate modes are of equal magnitude but opposite sign in these two regions. It is remarkable that the integrals over the multilayers make no contribution to the total power, even when κs= κs′. This is consistent with the findings of Ref. 6.

1992 (1)

P. Benech, D. A. M. Khalil, F. Saint André, “An exact simplified method for the normalization of radiation modes in planar multilayer structures,” Opt. Commun. 88, 96–100 (1992).
[CrossRef]

1979 (1)

S. Miyanaga, T. Asakura, M. Imai, “Scattering characteristics of a beam mode in a dielectric-slab optical waveguide,” Opt. Quantum Electron. 11, 205–215 (1979).
[CrossRef]

Asakura, T.

S. Miyanaga, T. Asakura, M. Imai, “Scattering characteristics of a beam mode in a dielectric-slab optical waveguide,” Opt. Quantum Electron. 11, 205–215 (1979).
[CrossRef]

Benech, P.

P. Benech, D. A. M. Khalil, F. Saint André, “An exact simplified method for the normalization of radiation modes in planar multilayer structures,” Opt. Commun. 88, 96–100 (1992).
[CrossRef]

Burke, J. J.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972), pp. 58–64.

Haruna, M.

H. Nishihara, M. Haruna, T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York, 1989).

Imai, M.

S. Miyanaga, T. Asakura, M. Imai, “Scattering characteristics of a beam mode in a dielectric-slab optical waveguide,” Opt. Quantum Electron. 11, 205–215 (1979).
[CrossRef]

Kapany, N. S.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972), pp. 58–64.

Khalil, D. A. M.

P. Benech, D. A. M. Khalil, F. Saint André, “An exact simplified method for the normalization of radiation modes in planar multilayer structures,” Opt. Commun. 88, 96–100 (1992).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Theory of optical waveguides,” in Guided-Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1988).
[CrossRef]

Lee, D. L.

D. L. Lee, Electromagnetic Principles of Integrated Optics (Wiley, New York, 1986).

Marcuse, D.

D. Marcuse, Theory of Dielectric Waveguides (Academic, New York, 1974).

Miyanaga, S.

S. Miyanaga, T. Asakura, M. Imai, “Scattering characteristics of a beam mode in a dielectric-slab optical waveguide,” Opt. Quantum Electron. 11, 205–215 (1979).
[CrossRef]

Nishihara, H.

H. Nishihara, M. Haruna, T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York, 1989).

Saint André, F.

P. Benech, D. A. M. Khalil, F. Saint André, “An exact simplified method for the normalization of radiation modes in planar multilayer structures,” Opt. Commun. 88, 96–100 (1992).
[CrossRef]

Suhara, T.

H. Nishihara, M. Haruna, T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York, 1989).

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), p. 114.

Opt. Commun. (1)

P. Benech, D. A. M. Khalil, F. Saint André, “An exact simplified method for the normalization of radiation modes in planar multilayer structures,” Opt. Commun. 88, 96–100 (1992).
[CrossRef]

Opt. Quantum Electron. (1)

S. Miyanaga, T. Asakura, M. Imai, “Scattering characteristics of a beam mode in a dielectric-slab optical waveguide,” Opt. Quantum Electron. 11, 205–215 (1979).
[CrossRef]

Other (7)

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), p. 114.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972), pp. 58–64.

To calculate the cross power, one must evaluate integrals of products of two cosine functions, each of the form of Eq. (8) but with, in general, a different set of parameters κi, ϕi, β. If we then define a function F(x) in such a way that its derivative with respect to xis equal to the integrand in the integrals that we must evaluate, we can show, using the continuity of the tangential fields at the interfaces, that the sum of the finite terms obtained by integrating over all xvanishes. Specifically, in an n-layer structure this sum takes the form F(0) + F(b1) − F(0) + F(b2) − F(b1) + … +F(bn) − F(bn−1) + F(c) − F(bn) − F(c). The first and last terms are contributed by the integrals over the substrate and the cover. The two remaining terms in the integration involve infinite limits and sum to 2πAs2δ(κs− κs′) for modes with ϕs= ϕ+(or ϕ−) in both cosine functions in the integrand. One can also show by direct integration that the integral also vanishes in the degenerate case where κs′ = κsbut ϕs= ϕ+in one field and ϕ−in the other. Alternatively, one may use Kogelnik’s Eq. (2.3.33) (Ref. 2) and our field expressions to demonstrate orthogonality. One may also demonstrate orthogonality by employing the arguments of Ref. 6, in which it is concluded that only the incoming plane waves from the sources at infinity are needed for determining the power in a mode and normalizing it. Accordingly, I note that the incoming plane waves of the degenerate modes are in phase in the substrate but πout of phase in the cover. This means that the cross power of the degenerate modes are of equal magnitude but opposite sign in these two regions. It is remarkable that the integrals over the multilayers make no contribution to the total power, even when κs= κs′. This is consistent with the findings of Ref. 6.

D. Marcuse, Theory of Dielectric Waveguides (Academic, New York, 1974).

H. Kogelnik, “Theory of optical waveguides,” in Guided-Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1988).
[CrossRef]

D. L. Lee, Electromagnetic Principles of Integrated Optics (Wiley, New York, 1986).

H. Nishihara, M. Haruna, T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York, 1989).

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Figures (1)

Fig. 1
Fig. 1

Radiation modes may be constructed from the superposition of the plane waves illustrated in (a) and (b) by their wave normals.

Equations (18)

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S x , s ( incoming ) = ( κ s / ω μ ) A 2 ,             S x , c ( incoming ) = ( κ c / ω μ ) A 2 ,
κ i = ( 2 π / λ ) n i cos θ i
κ s A 2 = κ c A 2 .
A 2 = r A + t A 2 .
r 2 A 2 + t 2 A 2 + 2 r t A A 2 × cos ( ϕ A - ϕ A + ϕ r - ϕ t ) ,
ϕ A - ϕ A + ϕ r - ϕ t = ± π / 2.
ϕ A - ϕ A + ϕ r - ϕ t = ± π / 2.
ϕ r + ϕ r - ϕ t - ϕ t = ± π .
2 ( ϕ A - ϕ A ) + ϕ r - ϕ t - ϕ r + ϕ t = 0.
ϕ A - ϕ A = ϕ r - ϕ r 2 = ϕ t - ϕ r ± π / 2.
r A + t A = A exp [ i ( ϕ A + ϕ r ) ] ( R i T ) .
A s cos ( κ s x + ϕ ) exp [ i ( ω t - β z ) ] ,
ϕ = ϕ r arctan T / R 2 ,             A s = 2 A exp ( i ϕ ) .
κ s / κ c A s cos [ κ c ( x - c ) + ϕ ± - ϕ t π / 2 ] ,
E y i = A i cos [ κ i ( x - b i ) + ϕ i ] exp [ i ( ω t - β z ) ] ,
d i = b i + 1 - b i ,
tan ϕ i + 1 = ( κ i / κ i + 1 ) tan ( κ i d i + ϕ i ) , A i + 1 = A i cos ( κ i d i + ϕ i ) cos ϕ i + 1 = A i [ cos 2 ( κ i d i + ϕ i ) + ( κ i 2 / κ i + 1 2 ) × sin 2 ( κ i d i + ϕ i ) ] 1 / 2 .
tan ϕ 1 = ( κ s / κ 1 ) tan ϕ , A 1 = A s cos ϕ cos ϕ 1 = A s [ cos 2 ϕ + ( κ s 2 / κ 1 2 ) sin 2 ϕ ] 1 / 2 .

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