Abstract

A procedure for calculating the mode structure in optical waveguides made up of multilayers of optically anisotropic media is derived. This procedure is applicable to a waveguide of any arbitrary number of layers, with each layer homogeneous but otherwise having arbitrary dielectric, magnetic-permeability, and optical-rotation tensors. The substrate is included as an end layer of large, but finite, thickness. When programmed for the computer, the method provides a powerful and useful tool for analyzing a waveguide consisting of any complex array of anisotropic layers, requiring simply specification of the relevant material tensors and layer thicknesses as inputs. The sequence of steps involved in the procedure is illustrated by two examples that can be carried out explicitly, viz., that of an isotropic dielectric layer and that of a uniaxial crystal layer.

© 1974 Optical Society of America

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References

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  1. P. K. Tien, Appl. Opt. 10, 2395 (1971).
    [CrossRef] [PubMed]
  2. J. H. Harris, R. Shubert, and J. N. Polky, J. Opt. Soc. Am. 60, 1007 (1970).
    [CrossRef]
  3. R. Shubert and J. H. Harris, J. Opt. Soc. Am. 61, 154 (1971).
    [CrossRef]
  4. S. Wang, M. L. Shah, and J. D. Crow, IEEE J. Quantum Electron. 8, 212 (1972).
    [CrossRef]
  5. D. P. Gia Russo and J. H. Harris, J. Opt. Soc. Am. 63, 138 (1973).
    [CrossRef]
  6. D. O. Smith, Opt. Acta 12, 13 (1965).
    [CrossRef]
  7. S. Teitler and B. W. Henvis, J. Opt. Soc. Am. 60, 830 (1970).
    [CrossRef]
  8. D. W. Berreman, J. Opt. Soc. Am. 62, 502 (1972).
    [CrossRef]
  9. D. O. Smith, Opt. Acta 12, 195 (1960); Opt. Acta 13, 121 (1966); Opt. Acta 13, 195 (1966); Opt. Acta 14, 351 (1967).
  10. For a general exposition see, for example, P. H. Lissberger, Rep. Prog. Phys. 33, 197 (1970).
    [CrossRef]
  11. R. C. Jones, J. Opt. Soc. Am. 31, 488 (1941); J. Opt. Soc. Am. 31, 500 (1941).
    [CrossRef]
  12. F. Abelès, Ann. Phys. (Paris) 5, 598 (1950).
  13. Except for changes in notation designed to make the results more concise, this development is the same as in Sec. 2 of Ref. 8.
  14. See, for example, R. M. Thrall and L. Tornheim, Vector Spaces and Matrices (Wiley, New York, 1957), Ch. 10.
  15. See the discussion in Ref. 5 pertaining to Eqs. (7) of that reference.

1973 (1)

1972 (2)

D. W. Berreman, J. Opt. Soc. Am. 62, 502 (1972).
[CrossRef]

S. Wang, M. L. Shah, and J. D. Crow, IEEE J. Quantum Electron. 8, 212 (1972).
[CrossRef]

1971 (2)

1970 (3)

1965 (1)

D. O. Smith, Opt. Acta 12, 13 (1965).
[CrossRef]

1960 (1)

D. O. Smith, Opt. Acta 12, 195 (1960); Opt. Acta 13, 121 (1966); Opt. Acta 13, 195 (1966); Opt. Acta 14, 351 (1967).

1950 (1)

F. Abelès, Ann. Phys. (Paris) 5, 598 (1950).

1941 (1)

Abelès, F.

F. Abelès, Ann. Phys. (Paris) 5, 598 (1950).

Berreman, D. W.

Crow, J. D.

S. Wang, M. L. Shah, and J. D. Crow, IEEE J. Quantum Electron. 8, 212 (1972).
[CrossRef]

Gia Russo, D. P.

Harris, J. H.

Henvis, B. W.

Jones, R. C.

Lissberger, P. H.

For a general exposition see, for example, P. H. Lissberger, Rep. Prog. Phys. 33, 197 (1970).
[CrossRef]

Polky, J. N.

Shah, M. L.

S. Wang, M. L. Shah, and J. D. Crow, IEEE J. Quantum Electron. 8, 212 (1972).
[CrossRef]

Shubert, R.

Smith, D. O.

D. O. Smith, Opt. Acta 12, 13 (1965).
[CrossRef]

D. O. Smith, Opt. Acta 12, 195 (1960); Opt. Acta 13, 121 (1966); Opt. Acta 13, 195 (1966); Opt. Acta 14, 351 (1967).

Teitler, S.

Thrall, R. M.

See, for example, R. M. Thrall and L. Tornheim, Vector Spaces and Matrices (Wiley, New York, 1957), Ch. 10.

Tien, P. K.

Tornheim, L.

See, for example, R. M. Thrall and L. Tornheim, Vector Spaces and Matrices (Wiley, New York, 1957), Ch. 10.

Wang, S.

S. Wang, M. L. Shah, and J. D. Crow, IEEE J. Quantum Electron. 8, 212 (1972).
[CrossRef]

Ann. Phys. (Paris) (1)

F. Abelès, Ann. Phys. (Paris) 5, 598 (1950).

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

S. Wang, M. L. Shah, and J. D. Crow, IEEE J. Quantum Electron. 8, 212 (1972).
[CrossRef]

J. Opt. Soc. Am. (6)

Opt. Acta (2)

D. O. Smith, Opt. Acta 12, 195 (1960); Opt. Acta 13, 121 (1966); Opt. Acta 13, 195 (1966); Opt. Acta 14, 351 (1967).

D. O. Smith, Opt. Acta 12, 13 (1965).
[CrossRef]

Rep. Prog. Phys. (1)

For a general exposition see, for example, P. H. Lissberger, Rep. Prog. Phys. 33, 197 (1970).
[CrossRef]

Other (3)

Except for changes in notation designed to make the results more concise, this development is the same as in Sec. 2 of Ref. 8.

See, for example, R. M. Thrall and L. Tornheim, Vector Spaces and Matrices (Wiley, New York, 1957), Ch. 10.

See the discussion in Ref. 5 pertaining to Eqs. (7) of that reference.

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Figures (2)

Fig. 1
Fig. 1

Field of a wave guided by a dielectric plate. The boundaries of the plate and the coordinate axes (x,z) are shown in the upper portion of the figure. The propagation direction is denoted by x. The dotted lines are planes of constant amplitude and phase. A representative electric field profile (cross-sectional distribution) of a mode in the plate is drawn in the lower portion of the figure. The profile is shown dotted within the waveguide and solid outside of it.

Fig. 2
Fig. 2

Geometrical configuration of the multilayer.

Equations (67)

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{ E ( r , t ) ( r , t ) } = { ( z ) ( z ) } exp { i ω c ( α x - c t ) } .
{ - ( z ) - ( z ) } = { A - B - } exp { ω c β 0 z }
{ + ( z ) + ( z ) } = { A + B + } exp { ω c β 0 ( z l - z ) } ,
k ± = ω c ( α , 0 , ± i β 0 ) n 0 ω c s ± ,
s ± · ± = 0
± = n 0 s ± × ± .
z ± = ± i α β 0 x ±
± = ( i β 0 y ± , ± i β 0 x ± - α z ± , α y ± ) ,
x ± = i β 0 y ± ;             y ± = i n 0 2 β 0 x ± .
Ψ ( z ) = ( x y y x ) ,
Ψ ( z l ) = T Ψ ( z 1 ) .
Ψ ( z l ) = ( A x + - i n 0 2 β 0 A x + A y + - i β 0 A y + ) ;             Ψ ( z 1 ) = ( A x - i n 0 2 β 0 A x - A y - i β 0 A y - ) .
[ T 21 + i n 0 2 β 0 ( T 11 + T 22 ) - n 0 4 β 0 2 T 12 ] × [ T 43 + i β 0 ( T 33 + T 44 ) - β 0 2 T 34 ] = [ T 23 + i ( β 0 T 24 + n 0 2 β 0 T 13 ) - n 0 2 T 14 ] × [ T 41 + i ( β 0 T 31 + n 0 2 β 0 T 42 ) - n 0 2 T 32 ] = 0.
T = j = 1 l - 1 U j .
and             [ E H ] = Γ ( z ) exp { i ω c ( α x - c t ) } [ D B ] = Λ ( z ) exp { i ω c ( α x - c t ) } ,
R Γ = - i ω c Λ ,
R = [ 0 R 1 R 1 0 ]
R 1 = ( 0 - d d z 0 d d z 0 - i ω c α 0 i ω c α 0 ) .
Λ = M Γ ,
M = [ ɛ ϱ ϱ μ ] .
Γ σ = ν = 1 4 a σ m ν Γ m ν             for             σ = 3 , 6 ,
{ m ν } { 1 , 2 , 4 , 5 } .
a 3 m ν = [ M ˆ 6 m ν M ˆ 36 - M ˆ 3 m ν M ˆ 66 ] / d
a 6 m ν = [ M ˆ 63 M ˆ 3 m ν - M ˆ 33 M ˆ 6 m ν ] / d
M ˆ σ ν = M σ ν + α [ δ σ 3 δ ν 5 + δ σ 5 δ ν 3 - δ σ 2 δ ν 6 - δ σ 6 δ ν 2 ] .
d = M 33 M 66 - M 36 M 63 ,
( P d d z - i ω c S ) ( x y x y ) = 0.
( 0 0 0 1 0 0 - 1 0 0 - 1 0 0 1 0 0 0 )
S σ ν = M ˆ m σ m ν + τ = 3 , 6 M ˆ m σ τ a τ m ν .
Q = ( 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 )
( x y x y ) = Q Ψ ,
d Ψ d z = i ω c Δ Ψ ,
Δ = Q - 1 PSQ .
Ψ ( z ) = exp [ i ω c ( z - z ) Δ ] Ψ ( z ) ,
C ( z - z ) = U ( z - z ) .
U ζ ν = λ ν ζ ν .
UV = VK .
U ( z - z ) = VKV - 1 .
Δ ζ ν = γ ν ζ ν
λ ν = exp [ i ω c ( z - z ) γ ν ] .
Ψ ( z ) = U ( z - z j ) Ψ ( z j ) .
and             a 35 = - α / n 1 2 a 62 = α .
S = ( n 1 2 0 0 0 0 β 1 2 0 0 0 0 1 0 0 0 0 β 1 2 n 1 2 ) ,
Δ = ( 0 β 1 2 n 1 2 0 0 n 1 2 0 0 0 0 0 0 - 1 0 0 - β 1 2 0 ) .
( 0 ± n 1 2 β 1 0 0 )             and             ( 0 0 1 β 1 )             for             γ = ± β 1 .
V = ( 1 1 0 0 n 1 2 β 1 - n 1 2 β 1 0 0 0 0 1 1 0 0 - β 1 β 1 ) ,
K = ( e i θ 0 0 0 0 e - i θ 0 0 0 0 e i θ 0 0 0 0 e - i θ ) ,
T = ( cos θ i β 1 n 1 2 sin θ 0 0 i n 1 2 β 1 sin θ cos θ 0 0 0 0 cos θ - i β 1 sin θ 0 0 - i β 1 sin θ cos θ ) .
Z = 1 n 1 sec φ 1 = β 1 n 1 2
Z = 1 n 1 cos φ 1 = 1 β 1
tan θ = { 2 n 0 2 n 1 2 β 0 β 1 n 0 4 β 1 2 - n 1 4 β 0 2 for TM waves 2 β 0 β 1 β 1 2 - β 0 2 for TE waves .
θ - φ 10 = m π ,
tan φ 10 = { β 0 β 1 TE n 1 2 β 0 n 0 2 β 1 TM .
ɛ = ( 3 0 2 0 0 0 2 0 1 ) ,
1 = 0 cos 2 ξ + e sin 2 ξ , 2 = ( e - 0 ) sin ξ cos ξ , 3 = 0 sin 2 ξ + e cos 2 ξ .
a 31 = - ( 2 / 1 ) , a 35 = - ( α / 1 ) , a 62 = α
S = ( 3 - 2 2 / 1 0 0 - α 2 / 1 0 0 - α 2 0 0 0 0 1 0 - α 2 / 1 0 0 1 - α 2 / 1 ) .
Δ = ( - α 2 / 1 1 - α 2 / 1 0 0 3 - 2 2 / 1 - α 2 / 1 0 0 0 0 0 - 1 0 0 α 2 - 0 0 ) .
( 1 ± p 0 0 )             belonging to the eigenvalues u ± v
( 0 0 1 w )             belonging to the eigenvalues ± w .
V = ( 1 1 0 0 p - p 0 0 0 0 1 1 0 0 - w w ) ,
K = ( exp { i ω c ( u + v ) h } 0 0 0 0 exp { i ω c ( u - v ) h } 0 0 0 0 exp { i ω c w h } 0 0 0 0 exp { i ω c w h } ) .
T = ( exp { i ω c u h } cos ( ω c v h ) i p exp { i ω c u h } sin ( ω c v h ) 0 0 i p exp { i ω c u h } sin ( ω c v h ) exp { i ω c u h } cos ( ω c v h ) 0 0 0 0 cos ( ω c w h ) - i w sin ( ω c w h ) 0 0 - i w sin ( ω c w h ) cos ( ω c w h ) ) .
tan ( ω c v h ) = - 2 n 0 2 β 0 p n 0 4 p 2 - β 0 2             for TM modes ,
tan ( ω c w h ) = 2 β 0 w 1 - β 0 2 w 2             for TE modes .
ω c v h + 2 tan - 1 ( n 0 2 p β 0 ) = m π             for TM modes
ω c w h - 2 tan - 1 ( β 0 w ) = m π             for TE modes .