Abstract

We discuss the construction of field transfer matrices and derive expressions for the modal condition and elliptically polarized fields of anisotropic multilayer waveguides.

© 1990 Optical Society of America

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References

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  1. S. Teitler, B. W. Henvis, “Refraction in Anisotropic Stratified Media,” J. Opt. Soc. Am. 60, 830–834 (1966).
    [CrossRef]
  2. D. W. Berreman, “Optics in Stratified and Anisotropic Media,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  3. K. Eidner, “Light Propagation in Stratified Anisotropic Media: Orthogonality and Symmetry Properties of 4 × 4 Matrix Formalisms,” J. Opt. Soc. Am. A 6, 1657–1660 (1989).
    [CrossRef]
  4. J. T. Chilwell, I. J. 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]
  5. P. Yeh, “Electromagnetic Propagation in Birefringent Layered Media,” J. Opt. Soc. Am. 69, 742–756 (1979).
    [CrossRef]
  6. R. Ulrich, “Theory of the Prism-Film Coupler by Plane-Wave Analysis,” J. Opt. Soc. Am. 60, 1337–1350 (1970).
    [CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Coordinate axes for the anisotropic multilayer waveguide, the basis of the field vector a within a layer, and the basis of a in the substrate.

Equations (23)

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m T = ( E 2 , - H y , H z , E y ) ,
E x , y , z ( j ) ± exp [ i k α ( j ) x ] exp ( i k β y ) exp ( - i ω t ) , H x , y , z ( j ) ± exp [ i k α ( j ) x ] exp ( i k β y ) exp ( - i ω t ) ,
E x , y , z = j = 1 4 a j E x , y , z ( j ) ± ,             H x , y , z = j = 1 4 a j H x , y , z ( j ) ± ,
F = [ E z ( 1 ) + E z ( 2 ) + E z ( 3 ) - E z ( 4 ) - - H y ( 1 ) + - H y ( 2 ) + - H y ( 3 ) - - H y ( 4 ) - H z ( 1 ) + H z ( 2 ) + H z ( 3 ) - H z ( 4 ) - E y ( 1 ) + E y ( 2 ) + E y ( 3 ) - E y ( 4 ) - ] .
m = Fa             a = F - 1 m .
a = Aa ,             m = Mm .
A d = [ exp [ - i ϕ ( 1 ) ] 0 0 0 0 exp [ - i ϕ ( 2 ) ] 0 0 0 0 exp [ - i ϕ ( 3 ) ] 0 0 0 0 exp [ - i ϕ ( 4 ) ] ] ,
M = F A d F - 1 ,
A = F c - 1 M F s ,
F s = [ 0 1 0 1 0 γ s 0 - γ s 1 / γ s 0 - 1 / γ s 0 1 0 1 0 ]
F c - 1 = 1 2 [ 0 1 γ c 1 0 1 / γ c 0 0 0 0 - γ c 1 1 - 1 / γ c 0 0 ] .
a c = A a s
a c T = ( E c y + , E c z + , E c y - , E c z - ) ,             a s T = ( E s y + , E s z + , E s y - , E s z - ) .
E c y - = ( a 22 a 31 - a 21 a 32 ) E c y + + ( a 11 a 32 - a 12 a 31 ) E c z + a 11 a 22 - a 12 a 21 , E s y + = a 22 E c y + - a 12 E c z + a 11 a 22 - a 12 a 21 , E c z - = ( a 22 a 41 - a 21 a 42 ) E c y + + ( a 11 a 42 - a 12 a 41 ) E c z + a 11 a 22 - a 12 a 21 , E s z + = - a 21 E c y + + a 11 E c z + a 11 a 22 - a 12 a 21 .
r p p = a 22 a 31 - a 21 a 32 a 11 a 22 - a 12 a 21 .
[ 0 0 E c y - E c z - ] = A [ E s y + E s y + 0 0 ] ,
χ A ( β ) = a 11 a 22 - a 12 a 21 = 0 ,
χ M ( β ) = ( γ c m 11 + γ c γ s m 12 + m 21 + γ s m 22 ) × ( γ c m 33 + γ c γ s m 34 + m 34 + γ s m 44 ) - ( γ c m 13 + γ c γ s m 14 + m 23 + γ c m 24 ) × ( γ c m 31 + γ c γ s m 32 + m 41 + γ s m 42 ) = 0.
E c - = E c y - j ^ + E c z - k ^ = ( a 11 a 32 - a 12 a 31 ) j ^ + ( a 11 a 42 - a 12 a 41 ) k ^ , E s + = E s y + j ^ + E s z + k ^ = - a 12 j ^ + a 11 k ^ .
= 1 2 tan - 1 ( E y E z * + E y * E z E y E y * - E z E z * ) ,             E y E z = E y cos + E z sin E z cos + E y sin
m s T = ( E s z + , γ s E s z + , E s y + / γ s , E s y + ) ,
m T = ( E z , - H y , H z , E y ) .
p y = ½ ( E z H x * - E x H y * ) ,

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