Abstract

Monochromatic energy exchanges in thin-film dielectric waveguides can be understood in terms of a zig-zag ray picture that includes Goos–Haenchen shifts that occur at the film boundaries, as pointed out by Burke. Recently, Kogelnik and Weber have shown that, in order to predict the correct modal group velocity, the ray model should also include the time delay associated with the ray shifts at the film boundaries. In this paper, we explore in detail the ray-optical descriptions of anisotropic film waveguides that consist of three-layered biaxial materials with one of the principal axes of the crystal in each of the three media oriented in the direction of propagation and the other two parallel and perpendicular to the plane of the film. We show that the Burke–Kogelnik–Weber (BKW) ray model provides a complete description of energy flow in such anisotropic structures. Expressions for the ray shift, group velocity, power flow, and stored energy, applicable to both TE and TM modes, are presented.

© 1974 Optical Society of America

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References

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  1. G. B. Airy, Philos. Mag. 2, 20 (1833).
  2. A. Sommerfeld, Optics, (Academic, New York, 1954), p. 47 ff.
  3. H. K. V. Lotsch, Optik 27, 239 (1968).
  4. P. K. Tien and R. Ulrich, J. Opt. Soc. Am. 60, 1325 (1970).
    [CrossRef]
  5. J. J. Burke, J. Opt. Soc. Am. 61, 676 (1971); Opt. Sci. Newsletter (U. Arizona) 5, 66 (1971).
  6. N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 79.
  7. H. Kogelnik and V. Ramaswamy, Appl. Opt. 13, 1857 (1974).
    [CrossRef] [PubMed]
  8. H. Kogelnik and H. P. Weber, J. Opt. Soc. Am. 64, 174 (1974).
    [CrossRef]
  9. V. Ramaswamy, Appl. Phys. Lett. 21, 183 (1972).
    [CrossRef]
  10. D. Hall, A. Yariv, and E. Garmire, Opt. Commun. 1, 403 (1970).
    [CrossRef]
  11. I. P. Kaminow and J. R. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
    [CrossRef]
  12. J. M. Hammer and W. Phillips, Appl. Phys. Lett. 24, 545 (1974).
    [CrossRef]
  13. P. K. Tien, R. J. Martin, S. L. Blank, S. H. Wemple, and L. J. Varnerin, Appl. Phys. Lett. 21, 207 (1972).
    [CrossRef]
  14. S. Yamamoto, Y. Koyamada, and T. Makimoto, J. Appl. Phys. 43, 5090 (1972).
    [CrossRef]
  15. D. P. GiaRusso and J. H. Harris, J. Opt. Soc. Am. 63, 138 (1973).
    [CrossRef]
  16. V. Ramaswamy, Appl. Opt. 13, 1363 (1974).
    [CrossRef] [PubMed]
  17. M. S. Kharusi, J. Opt. Soc. Am. 64, 27 (1974).
    [CrossRef]
  18. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1958), p. 670, Eq. (41); p. 678, Eq. (6).
  19. Reference 18, p. 666, Eq. (9).
  20. H. L. Bertoni and A. Hessel, IEEE Trans. Antennas Propag. 14, 344 (1966).
    [CrossRef]
  21. J. A. Arnaud and A. A. M. Saleh, Proc. IEEE Lett. 60, 639 (1972).
    [CrossRef]

1974 (5)

1973 (2)

D. P. GiaRusso and J. H. Harris, J. Opt. Soc. Am. 63, 138 (1973).
[CrossRef]

I. P. Kaminow and J. R. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
[CrossRef]

1972 (4)

J. A. Arnaud and A. A. M. Saleh, Proc. IEEE Lett. 60, 639 (1972).
[CrossRef]

P. K. Tien, R. J. Martin, S. L. Blank, S. H. Wemple, and L. J. Varnerin, Appl. Phys. Lett. 21, 207 (1972).
[CrossRef]

S. Yamamoto, Y. Koyamada, and T. Makimoto, J. Appl. Phys. 43, 5090 (1972).
[CrossRef]

V. Ramaswamy, Appl. Phys. Lett. 21, 183 (1972).
[CrossRef]

1971 (1)

J. J. Burke, J. Opt. Soc. Am. 61, 676 (1971); Opt. Sci. Newsletter (U. Arizona) 5, 66 (1971).

1970 (2)

P. K. Tien and R. Ulrich, J. Opt. Soc. Am. 60, 1325 (1970).
[CrossRef]

D. Hall, A. Yariv, and E. Garmire, Opt. Commun. 1, 403 (1970).
[CrossRef]

1968 (1)

H. K. V. Lotsch, Optik 27, 239 (1968).

1966 (1)

H. L. Bertoni and A. Hessel, IEEE Trans. Antennas Propag. 14, 344 (1966).
[CrossRef]

1833 (1)

G. B. Airy, Philos. Mag. 2, 20 (1833).

Airy, G. B.

G. B. Airy, Philos. Mag. 2, 20 (1833).

Arnaud, J. A.

J. A. Arnaud and A. A. M. Saleh, Proc. IEEE Lett. 60, 639 (1972).
[CrossRef]

Bertoni, H. L.

H. L. Bertoni and A. Hessel, IEEE Trans. Antennas Propag. 14, 344 (1966).
[CrossRef]

Blank, S. L.

P. K. Tien, R. J. Martin, S. L. Blank, S. H. Wemple, and L. J. Varnerin, Appl. Phys. Lett. 21, 207 (1972).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1958), p. 670, Eq. (41); p. 678, Eq. (6).

Burke, J. J.

J. J. Burke, J. Opt. Soc. Am. 61, 676 (1971); Opt. Sci. Newsletter (U. Arizona) 5, 66 (1971).

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 79.

Carruthers, J. R.

I. P. Kaminow and J. R. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
[CrossRef]

Garmire, E.

D. Hall, A. Yariv, and E. Garmire, Opt. Commun. 1, 403 (1970).
[CrossRef]

GiaRusso, D. P.

Hall, D.

D. Hall, A. Yariv, and E. Garmire, Opt. Commun. 1, 403 (1970).
[CrossRef]

Hammer, J. M.

J. M. Hammer and W. Phillips, Appl. Phys. Lett. 24, 545 (1974).
[CrossRef]

Harris, J. H.

Hessel, A.

H. L. Bertoni and A. Hessel, IEEE Trans. Antennas Propag. 14, 344 (1966).
[CrossRef]

Kaminow, I. P.

I. P. Kaminow and J. R. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
[CrossRef]

Kapany, N. S.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 79.

Kharusi, M. S.

Kogelnik, H.

Koyamada, Y.

S. Yamamoto, Y. Koyamada, and T. Makimoto, J. Appl. Phys. 43, 5090 (1972).
[CrossRef]

Lotsch, H. K. V.

H. K. V. Lotsch, Optik 27, 239 (1968).

Makimoto, T.

S. Yamamoto, Y. Koyamada, and T. Makimoto, J. Appl. Phys. 43, 5090 (1972).
[CrossRef]

Martin, R. J.

P. K. Tien, R. J. Martin, S. L. Blank, S. H. Wemple, and L. J. Varnerin, Appl. Phys. Lett. 21, 207 (1972).
[CrossRef]

Phillips, W.

J. M. Hammer and W. Phillips, Appl. Phys. Lett. 24, 545 (1974).
[CrossRef]

Ramaswamy, V.

Saleh, A. A. M.

J. A. Arnaud and A. A. M. Saleh, Proc. IEEE Lett. 60, 639 (1972).
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, Optics, (Academic, New York, 1954), p. 47 ff.

Tien, P. K.

P. K. Tien, R. J. Martin, S. L. Blank, S. H. Wemple, and L. J. Varnerin, Appl. Phys. Lett. 21, 207 (1972).
[CrossRef]

P. K. Tien and R. Ulrich, J. Opt. Soc. Am. 60, 1325 (1970).
[CrossRef]

Ulrich, R.

Varnerin, L. J.

P. K. Tien, R. J. Martin, S. L. Blank, S. H. Wemple, and L. J. Varnerin, Appl. Phys. Lett. 21, 207 (1972).
[CrossRef]

Weber, H. P.

Wemple, S. H.

P. K. Tien, R. J. Martin, S. L. Blank, S. H. Wemple, and L. J. Varnerin, Appl. Phys. Lett. 21, 207 (1972).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1958), p. 670, Eq. (41); p. 678, Eq. (6).

Yamamoto, S.

S. Yamamoto, Y. Koyamada, and T. Makimoto, J. Appl. Phys. 43, 5090 (1972).
[CrossRef]

Yariv, A.

D. Hall, A. Yariv, and E. Garmire, Opt. Commun. 1, 403 (1970).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (4)

V. Ramaswamy, Appl. Phys. Lett. 21, 183 (1972).
[CrossRef]

I. P. Kaminow and J. R. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
[CrossRef]

J. M. Hammer and W. Phillips, Appl. Phys. Lett. 24, 545 (1974).
[CrossRef]

P. K. Tien, R. J. Martin, S. L. Blank, S. H. Wemple, and L. J. Varnerin, Appl. Phys. Lett. 21, 207 (1972).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

H. L. Bertoni and A. Hessel, IEEE Trans. Antennas Propag. 14, 344 (1966).
[CrossRef]

J. Appl. Phys. (1)

S. Yamamoto, Y. Koyamada, and T. Makimoto, J. Appl. Phys. 43, 5090 (1972).
[CrossRef]

J. Opt. Soc. Am. (5)

Opt. Commun. (1)

D. Hall, A. Yariv, and E. Garmire, Opt. Commun. 1, 403 (1970).
[CrossRef]

Optik (1)

H. K. V. Lotsch, Optik 27, 239 (1968).

Philos. Mag. (1)

G. B. Airy, Philos. Mag. 2, 20 (1833).

Proc. IEEE Lett. (1)

J. A. Arnaud and A. A. M. Saleh, Proc. IEEE Lett. 60, 639 (1972).
[CrossRef]

Other (4)

A. Sommerfeld, Optics, (Academic, New York, 1954), p. 47 ff.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 79.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1958), p. 670, Eq. (41); p. 678, Eq. (6).

Reference 18, p. 666, Eq. (9).

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Figures (4)

Fig. 1
Fig. 1

Zig-zag-ray (BKW) model of light propagation in three-layer biaxially anisotropic waveguide. The principal axes of the crystal in each medium are oriented along the coordinate axes, as shown. The ray is shifted (Goos–Haenchen shift) by 2zi and the corresponding penetration depth is given by xi.

Fig. 2
Fig. 2

Ray and wave-normal directions in the film for the TM mode, illustrated by use of the index ellipsoid.

Fig. 3
Fig. 3

The wave-normal velocity vw, the phase velocity vp, and the ray velocity vr of the TM mode and the interconnecting relations used in the zig-zag-ray description. v ˜ g equals the group velocity only when the energy in the cover and substrate region can be neglected.

Fig. 4
Fig. 4

The angle of incidence of the zig-zag ray θ, as a function of the angle of incidence of the wave normal θ′. In the case of the TM mode, Δ2 is the normalized index difference between the x components of the film and the substrate. K is the anisotropy ratio given by nz/nx; K = 1 represents the TE mode; Δ2 then defines the difference between the y components in refractive index.

Equations (95)

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x 2 n x 2 + y 2 n y 2 + z 2 n z 2 = 1 ,
n = n x n z ( n z 2 sin 2 θ + n x 2 cos 2 θ ) - 1 2 ,             TM waves
n = n y ,         TE waves .
β = k n sin θ ,
κ = k n cos θ ,
tan θ = β / κ .
v w = c / n .
v p = ω / β = c / N ,
N = n sin θ .
v r = v w / cos α ,
v r = c / n r ,
n r = n cos ( θ - θ ) .
m r = n r + ω ( d n r / d ω ) .
v ˜ g = c sin θ / n r .
z = f tan θ .
τ = z / v ˜ g .
z i = ϕ i / β ;             i = 1 , 2
τ i = - ϕ i / ω ;             i = 1 , 2.
v g = z + z 1 + z 2 τ + τ 1 + τ 2 ,
x i = z i / tan θ ;             i = 1 , 2.
tan δ = - d z d x | ( z 0 , x 0 ) .
tan θ = K 2 tan θ ,             TM waves
θ = θ ,             TE waves ,
K = n z / n x
θ c = arcsin [ 1 + K 2 Δ 2 ( 2 - Δ 2 ) ( 1 - Δ 2 ) 2 ] - 1 2 ,             TM waves ,
Δ 2 = 1 - n 2 x / n x .
θ c = arcsin [ 1 - Δ 2 ]             for TE waves ,
Δ 2 = 1 - n 2 y / n y .
z i = 1 γ i β κ ;             i = 1 , 2.
z i = 1 γ i tan θ i ;             i = 1 , 2.
x i = 1 γ i ;             i = 1 , 2.
τ i = β ω z i = N c z i ;             i = 1 , 2.
q i = K 2 [ N 2 n i x 2 n i z 2 - N 2 n x 2 n z 2 - ( 1 n i z 2 - 1 n z 2 ) ] ( 1 n i x 2 - 1 n x 2 ) - 1 ,             i = 1 , 2
z i = K 2 γ i q i tan θ = 1 γ i q i tan θ ;             i = 1 , 2
x i = 1 γ i q i ;             i = 1 , 2.
τ i = β ω γ i K 2 q i tan θ = β ω z i = N c z i ;             i = 1 , 2.
f eff = f + x 1 + x 2 .
v g = d ω / d β = P / W ,
P = 1 2 - + ( E × H * ) z d x = P 0 f eff ,
P 0 = 1 4 A 2 β ω 0 x ,             TM modes
P 0 = 1 4 A 2 β ω μ 0 ,             TE modes ,
P f = 1 2 - t t ( E × H * ) z d x
P s = P - P f .
W = 1 4 E · ɛ · E * d x d y
W μ = 1 4 μ 0 H · H * d x d y .
W = W t + W z ,
W μ = W t μ + W z μ .
v ˜ g = c N / n x 2 ,             TM waves
v ˜ g = c N / n y 2 ,             TE waves ,
P s / W s = c / N = v p .
P f / W f v ˜ g .
P = P κ + P γ
W = W κ + W γ ,
P κ = P f / f eff ,
P γ = ( i = 1 2 1 γ i q i ) P / f eff ,
W κ = ( n x 2 / c N ) P f / f eff ,
W γ = ( i = 1 2 1 γ i q i ) ( N c ) P / f eff .
P κ / W κ = v ˜ g ,
P γ / W γ = v p ,
P κ / P γ = z / ( z 1 + z 2 ) ,
W κ / W γ = τ / ( τ 1 + τ 2 ) .
v g = P κ + P γ W κ + W γ = W κ v ˜ g + W γ v p W κ + W γ .
v g = z + z 1 + z 2 τ + τ 1 + τ 2 ,
E y = A cos ϕ 1 e - γ 1 ( x - t ) , Region 1 x t = A cos ( κ x - ψ ) , Film - t x t = A cos ϕ 2 e γ 2 ( x + t ) , Region 2 x - t H x = β ω μ 0 E y , H z = - 1 j ω μ 0 E y x ,
γ i 2 = β 2 - k 2 n i y 2 ,             i = 1 , 2
κ 2 = k 2 n y 2 - β 2 .
ϕ i = arctan { γ i κ } ,             i = 1 , 2 ,
ψ = κ t - ϕ 1 = ϕ 2 - κ t .
2 κ t = ϕ 1 + ϕ 2 .
H y = A cos ϕ 1 e - γ 1 ( x - t ) . Region 1 = A cos ( κ x - ψ ) , Film = A cos ϕ 2 e γ 2 ( x + t ) , Region 2 E x = ( β / ω 0 i x ) H y ,             i = 1 , 2 Region 1 , 2 = ( β / ω 0 x ) H y , Film E z = ( 1 j ω 0 i z ) H y x ,             i = 1 , 2 Region 1 , 2 = ( 1 j ω 0 z ) H y x , Film ,
γ i 2 = ( β 2 - k 2 n i x 2 ) K i 2 ,             i = 1 , 2
κ 2 = ( k 2 n x 2 - β 2 ) K 2 ,
K i = n i z / n i x ;             K = n z / n x ,             i = 1 , 2 ,
ϕ i = arctan { ( n z / n i z ) 2 ( γ i / κ ) ,             i = 1 , 2.
P f = P f eff ( f + i = 1 2 1 γ i q i K 2 γ i 2 K i 2 κ 2 + K 2 γ i 2 ) .
P s = P f eff ( i = 1 2 1 γ i q i K i 2 κ 2 K i 2 κ 2 + K 2 γ i 2 ) .
W t f = ( f + i = 1 2 1 γ i q i K 2 γ i 2 K i 2 κ 2 + K 2 γ i 2 ) N P 2 c N f eff ,
W z f = ( f - i = 1 2 1 γ i q i K 2 γ i 2 K i 2 κ 2 + K 2 γ i 2 ) ( n x 2 - N 2 ) P 2 c N f eff ,
W t f = ( f + i = 1 2 1 γ i q i K 2 γ i 2 K i 2 κ 2 + K 2 γ i 2 ) n x 2 P 2 c N f eff ,
W f = W t f + W z f + W t f = ( n x 2 f + i = 1 2 N 2 γ i q i K 2 γ i 2 K i 2 κ 2 + K 2 γ i 2 ) P c N f eff .
W z s = ( i = 1 2 1 γ i q i K i 2 κ 2 K i 2 κ 2 + K 2 γ i 2 ( N 2 - n i x 2 ) ) P 2 c N f eff ,
W t s = ( i = 1 2 1 γ i q i K i 2 κ 2 K i 2 κ 2 + K 2 γ i 2 ) N 2 P 2 c N f eff ,
W t s = ( i = 1 2 1 γ i q i K i 2 κ 2 K i 2 κ 2 + K 2 γ i 2 · n i x 2 ) P 2 c N f eff ,
W s = W z s + W t s + W t s μ = ( i = 1 2 N 2 γ i q i K i 2 κ 2 K i 2 κ 2 + K 2 γ i 2 ) P c N f eff .
W = W f + W s = { n x 2 f + i = 1 2 N 2 γ i q i } P c N f eff .
W = W μ = [ n x 2 f + i = 1 2 N 2 γ i q i ] P 2 c N f eff ,
W t = N P / 2 c ,
W t = ( n x 2 - N 2 ) f P 2 c N f eff + N P c ,
W z = W z = ( n x 2 - N 2 ) f P 2 c N f eff ,
W t - W z = N P / c .
q i = K 2 [ N 2 n i x 2 n i z 2 - N 2 n x 2 n z 2 - ( 1 n i z 2 - 1 n z 2 ) ] ( 1 n i x 2 - 1 n x 2 ) - 1 ,             i = 1 , 2.
τ = m r f / c cos θ .
κ = k n r / cos θ - β tan θ .
τ d ω d β = f tan θ + d ( ϕ 1 + ϕ 2 ) d β .
d ω d β = z + z 1 + z 2 τ + τ 1 + τ 2 .