Abstract

This paper is concerned with the general theory of wave propagation in layered biaxially anisotropic media. Details are presented for the calculation of the induced waves that are due to an arbitrarily polarized and obliquely incident wave impinging on a three-layer structure. The total numbers of partial waves with their respective phase velocities, direction of phase propagation, and polarization are determined by the use of the Fresnel equation and Snell’s law applied to each layer. The vector amplitudes of the partial waves are found by proper matching of the field components at the interfaces. The expressions thus found are shown to reduce to known results for a uniaxial three-layer structure. An extension of this theory to an arbitrary number of layers is also presented.

© 1972 Optical Society of America

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References

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  1. E. O. Ammann, in Progress in Optics, IX, edited by E. Wolf (North–Holland, Amsterdam, 1971), pp. 125–177.
  2. R. E. Collins, Field Theory of Guided Waves (McGraw–Hill, New York, 1960), pp. 101–106.
  3. C. E. Curry, Electromagnetic Theory of Light (Macmillan, London, 1905), pp. 369–383.
  4. G. Szivessy, in Handbuch per Physik, 20, edited by H. Geiger and K. Scheel (Springer, Berlin, 1928).
  5. Reference 3, pp. 356–369.
  6. H. Schopper, Z. Physik 132, 146 (1952), condensed by O. S. Heavens, in Optical Properties of Thin Solid Films(Butterworths, London, 1955), pp. 92–95.
    [Crossref]
  7. A. B. Winterbottom, Kgl. Norske Videnskab. Selskab, Skrifters 1, 27 (1955); Kgl. Norske Videnskab. Selskab, Skrifters 1, 37 (1955).
  8. A. M. Goncharenko and F. J. Federov, Opt. Spectrosk. 14, 94, (1962) [Opt. Spectrosc. 14, 48 (1963)].
  9. D. A. Holmes and D. L. Feucht, J. Opt. Soc. Am. 56, 1763 (1966).
    [Crossref]
  10. G. N. Ramachandran and S. Ramaseshan, in Handbuch der Physik, 25, 1, edited by S. Flügge (Springer, Berlin, 1961), p. 118.
  11. Reference 2, pp. 340–342.
  12. Reference 1, p. 97.
  13. Reference 2, pp. 350–351.
  14. M. Javid and P. M. Brown, Field Analysis and Electromagnetics (McGraw–Hill, New York, 1963), pp. 167–177.
  15. F. R. Gantmacher, Theory of Matrices, Vol. 1 (Chelsea, New York, 1960), pp. 23–28.
  16. Reference 1, pp. 161–66, and Ref. 7.
  17. Reference 1, p. 97.
  18. Reference 2, pp. 349–350.

1966 (1)

1962 (1)

A. M. Goncharenko and F. J. Federov, Opt. Spectrosk. 14, 94, (1962) [Opt. Spectrosc. 14, 48 (1963)].

1955 (1)

A. B. Winterbottom, Kgl. Norske Videnskab. Selskab, Skrifters 1, 27 (1955); Kgl. Norske Videnskab. Selskab, Skrifters 1, 37 (1955).

1952 (1)

H. Schopper, Z. Physik 132, 146 (1952), condensed by O. S. Heavens, in Optical Properties of Thin Solid Films(Butterworths, London, 1955), pp. 92–95.
[Crossref]

Ammann, E. O.

E. O. Ammann, in Progress in Optics, IX, edited by E. Wolf (North–Holland, Amsterdam, 1971), pp. 125–177.

Brown, P. M.

M. Javid and P. M. Brown, Field Analysis and Electromagnetics (McGraw–Hill, New York, 1963), pp. 167–177.

Collins, R. E.

R. E. Collins, Field Theory of Guided Waves (McGraw–Hill, New York, 1960), pp. 101–106.

Curry, C. E.

C. E. Curry, Electromagnetic Theory of Light (Macmillan, London, 1905), pp. 369–383.

Federov, F. J.

A. M. Goncharenko and F. J. Federov, Opt. Spectrosk. 14, 94, (1962) [Opt. Spectrosc. 14, 48 (1963)].

Feucht, D. L.

Gantmacher, F. R.

F. R. Gantmacher, Theory of Matrices, Vol. 1 (Chelsea, New York, 1960), pp. 23–28.

Goncharenko, A. M.

A. M. Goncharenko and F. J. Federov, Opt. Spectrosk. 14, 94, (1962) [Opt. Spectrosc. 14, 48 (1963)].

Holmes, D. A.

Javid, M.

M. Javid and P. M. Brown, Field Analysis and Electromagnetics (McGraw–Hill, New York, 1963), pp. 167–177.

Ramachandran, G. N.

G. N. Ramachandran and S. Ramaseshan, in Handbuch der Physik, 25, 1, edited by S. Flügge (Springer, Berlin, 1961), p. 118.

Ramaseshan, S.

G. N. Ramachandran and S. Ramaseshan, in Handbuch der Physik, 25, 1, edited by S. Flügge (Springer, Berlin, 1961), p. 118.

Schopper, H.

H. Schopper, Z. Physik 132, 146 (1952), condensed by O. S. Heavens, in Optical Properties of Thin Solid Films(Butterworths, London, 1955), pp. 92–95.
[Crossref]

Szivessy, G.

G. Szivessy, in Handbuch per Physik, 20, edited by H. Geiger and K. Scheel (Springer, Berlin, 1928).

Winterbottom, A. B.

A. B. Winterbottom, Kgl. Norske Videnskab. Selskab, Skrifters 1, 27 (1955); Kgl. Norske Videnskab. Selskab, Skrifters 1, 37 (1955).

J. Opt. Soc. Am. (1)

Kgl. Norske Videnskab. Selskab, Skrifters (1)

A. B. Winterbottom, Kgl. Norske Videnskab. Selskab, Skrifters 1, 27 (1955); Kgl. Norske Videnskab. Selskab, Skrifters 1, 37 (1955).

Opt. Spectrosk. (1)

A. M. Goncharenko and F. J. Federov, Opt. Spectrosk. 14, 94, (1962) [Opt. Spectrosc. 14, 48 (1963)].

Z. Physik (1)

H. Schopper, Z. Physik 132, 146 (1952), condensed by O. S. Heavens, in Optical Properties of Thin Solid Films(Butterworths, London, 1955), pp. 92–95.
[Crossref]

Other (14)

E. O. Ammann, in Progress in Optics, IX, edited by E. Wolf (North–Holland, Amsterdam, 1971), pp. 125–177.

R. E. Collins, Field Theory of Guided Waves (McGraw–Hill, New York, 1960), pp. 101–106.

C. E. Curry, Electromagnetic Theory of Light (Macmillan, London, 1905), pp. 369–383.

G. Szivessy, in Handbuch per Physik, 20, edited by H. Geiger and K. Scheel (Springer, Berlin, 1928).

Reference 3, pp. 356–369.

G. N. Ramachandran and S. Ramaseshan, in Handbuch der Physik, 25, 1, edited by S. Flügge (Springer, Berlin, 1961), p. 118.

Reference 2, pp. 340–342.

Reference 1, p. 97.

Reference 2, pp. 350–351.

M. Javid and P. M. Brown, Field Analysis and Electromagnetics (McGraw–Hill, New York, 1963), pp. 167–177.

F. R. Gantmacher, Theory of Matrices, Vol. 1 (Chelsea, New York, 1960), pp. 23–28.

Reference 1, pp. 161–66, and Ref. 7.

Reference 1, p. 97.

Reference 2, pp. 349–350.

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Figures (4)

Fig. 1
Fig. 1

Structure of media with the directions and angles of phase propagation and polarization of the incident displacement field.

Fig. 2
Fig. 2

Directions and angles of phase propagation and polarization of the reflected and transmitted displacement fields in the media.

Fig. 3
Fig. 3

Directions and angles of phase propagation and polarization of the reflected and transmitted displacement field for the isotropic–uniaxial–isotropic media.

Fig. 4
Fig. 4

Components of the electric field.

Equations (41)

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ɛ ˜ = ( x 0 0 0 y 0 0 0 z ) .
A 0 e - j k n ˆ · r ,
r = x â x + y â y + z â z ,
n ˆ = cos θ x â x + cos θ y â y + cos θ z â z ,
D = D p ˆ e - j k n ˆ · r ,
cos 2 θ x k x 2 k 2 - k x 2 + cos 2 θ y k y 2 k 2 - k y 2 + cos 2 θ z k z 2 k 2 - k z 2 = 0 ,
p ˆ = cos θ x k x 2 k 2 - k x 2 â x + cos θ y k y 2 k 2 - k y 2 â y + cos θ z k z 2 k 2 - k z 2 â z .
cos γ = k - 2 [ cos 2 ϕ x k x 4 + cos 2 ϕ y k y 4 + cos 2 ϕ z k z 4 ] - 1 2 .
D n = D n p ˆ n e - j k n n ˆ n · r ,
p ˆ n = sin ϕ n â x + cos ϕ n sin θ n â y ± cos ϕ n cos θ n â z ,
n ˆ n = - cos θ n â y ± sin θ n â z ,
k 0 sin θ 0 = k n sin θ n ,             n = 1 , , m
H n = ω D n k n n ˆ n × p ˆ n e - j k n n ˆ n · r ,
E 0 x = μ 0 ω 2 k 0 2 D 0 sin ϕ 0 e - j k 0 n ˆ 0 · r , E 0 z = - μ 0 ω 2 k 0 2 ( cos ϕ 0 cos θ 0 - tan γ 0 sin θ 0 ) D 0 e - j k 0 n ˆ 0 · r .
n = 0 2 D n a n - n = 3 6 D n a n = 0 ,
n = 0 2 D n b n - n = 3 6 D n b n = 0 ,
D 0 c 0 - n = 1 4 D n c n + n = 5 6 D n c n = 0 ,
D 0 d 0 + - n = 1 2 D n d n - - n = 3 4 D n d n + + n = 5 6 D n d n - = 0 ,
a n = sin θ n cos ϕ n ,             b n = sin 2 θ n sin ϕ n , c n = sin θ n cos θ n sin ϕ n ,
d n ± = sin 2 θ n ( cos θ n cos ϕ n ± tan γ n sin θ n ) .
n = 3 4 D n a n f n + n = 5 6 D n a n f n - 1 - n = 7 8 D n a n f n = 0 ,
n = 3 4 D n b n f n + n = 5 6 D n b n f n - 1 - n = 7 8 D n b n f n = 0 ,
n = 3 4 D n c n f n - n = 5 6 D n c n f n - 1 - n = 7 8 D n c n f n = 0 ,
n = 3 4 D n d n + f n - n = 5 6 D n d n - f n - 1 - n = 7 8 D n d n + f n = 0 ,
( k 2 - k x 2 ) [ k 2 k x 2 ( 1 - sin 2 θ ) + k 2 k z 2 sin 2 θ - k x 2 k z 2 ] = 0.
k 1 = k x ,             sin θ 1 = k 0 sin θ 0 k x ,
k 2 = k x [ ( k x 2 - k z 2 ) k 0 2 sin θ 0 + k x 2 k z 2 ] 1 2 , sin θ 2 = k 0 sin θ 0 k z .
( D 3 + D 5 ) sin θ 2 = sin θ 0 [ D 1 cos ϕ 1 + D 0 cos ϕ 0 ] ,
( D 2 + D 4 ) sin 2 θ 1 = sin 2 θ 0 [ D 1 sin ϕ 1 + D 0 sin ϕ 0 ] ,
( D 2 - D 4 ) sin θ 1 cos θ 1 = sin θ 0 cos θ 0 [ D 0 sin ϕ 0 - D 1 sin ϕ 1 ] ,
D 3 sin 2 θ 2 ( cos θ 2 + sin θ 2 tan γ 2 ) - D 5 sin 2 θ 2 ( cos θ 2 - tan γ 2 ) = sin 2 θ 0 cos θ 0 ( D 0 cos ϕ 0 - D 1 cos ϕ 1 ) ,
( D 3 f 2 + D 5 f 2 - 1 ) sin θ 2 = D 6 f 0 sin θ 0 cos ϕ 6 ,
( D 2 f 1 + D 4 f 1 - 1 ) sin 2 θ 1 = D 6 f 0 sin 2 θ 0 sin ϕ 6 ,
( D 2 f 1 - D 4 f 1 - 1 ) sin θ 1 cos θ 1 = D 6 f 0 sin θ 0 cos θ 0 sin ϕ 6 ,
D 3 sin 2 θ 2 ( cos θ 2 + tan γ 2 sin θ 2 ) f 2 = D 5 sin 2 θ 2 ( cos θ 2 - tan γ 2 sin θ 2 ) f 2 - 1 + D 6 f 0 sin 2 θ 0 cos θ 0 cos ϕ 6 .
D = k 2 ω 2 μ 0 [ E - n ˆ ( n ˆ · E ) ] .
D = k 2 ω 2 μ 0 E cos γ ,
E = E ( cos γ p ˆ + sin γ n ˆ ) e - j k n ˆ · r .
n ˆ 0 = - sin θ 0 â z - cos θ 0 â y ,
p ˆ 0 = sin ϕ 0 â x + cos ϕ 0 sin θ 0 â y - cos ϕ 0 cos θ 0 â z .
E 0 x = ω 2 μ 0 k 0 2 sin ϕ 0 D 0 e - j k 0 n ˆ 0 · r , E 0 y = ω 2 μ 0 k 0 2 ( cos ϕ 0 sin θ 0 - tan γ 0 cos θ 0 ) D 0 e - j k 0 n ˆ 0 · r , E 0 z = - ω 2 μ 0 k 0 2 ( cos ϕ 0 cos θ 0 - tan γ 0 sin θ 0 ) D 0 e - j k 0 n ˆ 0 · r .