Abstract

Waveguide mode propagation in optical waveguides containing uniaxial media is investigated. When the optical axis of each uniaxial medium lies in the plane containing the waveguide normal and the propagation direction, waveguide propagation is described by uncoupled TE and TM modes. The phenomenon of wave-front tilt is discussed and exact mode dispersion equations are derived. The experimentally convenient situation of the optic axis lying in the film plane is also analyzed. Here mode coupling between TE and TM polarizations occurs and the propagating modes are hybrid. The wave-vector components in the uniaxial media transverse to the waveguide are shown to have ordinary or extraordinary characteristics. Exact mode-dispersion equations for the hybrid modes are derived and are shown to differ substantially from the TE and TM mode approximations when the latter are near degeneracy.

© 1974 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. F. Nelson and J. McKenna, J. Appl. Phys. 38, 4075 (1967).
    [CrossRef]
  2. S. Wang, M. Shah, and J. D. Crow, J. Appl. Phys. 43, 1861 (1972).
    [CrossRef]
  3. R. A. Andrews, IEEE J. Quantum Electron. 7, 523 (1971).
    [CrossRef]
  4. D. P. Gia Russo and J. H. Harris, J. Opt. Soc. Am. 63, 138 (1973).
    [CrossRef]
  5. M. Shah, J. D. Crow, and S. Wang, Appl. Phys. Lett. 20, 66 (1972).
    [CrossRef]
  6. T. P. Sosnowski and H. P. Weber, Optics Commun. 7, 47 (1973).
    [CrossRef]
  7. J. E. Bjorkholm, T. P. Sosnowski, and C. V. Shank, Appl. Phys. Lett. 22, 132 (1973).
    [CrossRef]
  8. H. Kogelnik, C. V. Shank, and J. E. Bjorkholm, Appl. Phys. Lett. 22, 135 (1973).
    [CrossRef]
  9. T. P. Sosnowski, Optics Commun. 4, 408 (1972).
    [CrossRef]
  10. W. K. Burns and R. A. Andrews, Appl. Opt. 12, 2249 (1973).
    [CrossRef] [PubMed]
  11. T. P. Sosnowski and H. P. Weber, Appl. Phys. Lett. 21, 310 (1972).
    [CrossRef]
  12. J. P. Sheridan, J. M. Schnur, and T. G. Giallorenzi, Appl. Phys. Lett. 22, 560 (1973).
    [CrossRef]
  13. J. M. Hammer, D. J. Channin, M. T. Duffy, and J. P. Whittke, Appl. Phys. Lett. 21, 358 (1972).
    [CrossRef]
  14. M. Françon, in Advanced Optical Techniques, edited by A. C. S. van Heel (Wiley, New York, 1967), p. 39.
  15. A. Yariv, in Quantum Electronics (Wiley, New York, 1967), p. 299.
  16. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 668.
  17. A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).
    [CrossRef]

1973 (7)

T. P. Sosnowski and H. P. Weber, Optics Commun. 7, 47 (1973).
[CrossRef]

J. E. Bjorkholm, T. P. Sosnowski, and C. V. Shank, Appl. Phys. Lett. 22, 132 (1973).
[CrossRef]

H. Kogelnik, C. V. Shank, and J. E. Bjorkholm, Appl. Phys. Lett. 22, 135 (1973).
[CrossRef]

J. P. Sheridan, J. M. Schnur, and T. G. Giallorenzi, Appl. Phys. Lett. 22, 560 (1973).
[CrossRef]

A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).
[CrossRef]

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

W. K. Burns and R. A. Andrews, Appl. Opt. 12, 2249 (1973).
[CrossRef] [PubMed]

1972 (5)

J. M. Hammer, D. J. Channin, M. T. Duffy, and J. P. Whittke, Appl. Phys. Lett. 21, 358 (1972).
[CrossRef]

T. P. Sosnowski, Optics Commun. 4, 408 (1972).
[CrossRef]

T. P. Sosnowski and H. P. Weber, Appl. Phys. Lett. 21, 310 (1972).
[CrossRef]

S. Wang, M. Shah, and J. D. Crow, J. Appl. Phys. 43, 1861 (1972).
[CrossRef]

M. Shah, J. D. Crow, and S. Wang, Appl. Phys. Lett. 20, 66 (1972).
[CrossRef]

1971 (1)

R. A. Andrews, IEEE J. Quantum Electron. 7, 523 (1971).
[CrossRef]

1967 (1)

D. F. Nelson and J. McKenna, J. Appl. Phys. 38, 4075 (1967).
[CrossRef]

Andrews, R. A.

Bjorkholm, J. E.

J. E. Bjorkholm, T. P. Sosnowski, and C. V. Shank, Appl. Phys. Lett. 22, 132 (1973).
[CrossRef]

H. Kogelnik, C. V. Shank, and J. E. Bjorkholm, Appl. Phys. Lett. 22, 135 (1973).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 668.

Burns, W. K.

Channin, D. J.

J. M. Hammer, D. J. Channin, M. T. Duffy, and J. P. Whittke, Appl. Phys. Lett. 21, 358 (1972).
[CrossRef]

Crow, J. D.

S. Wang, M. Shah, and J. D. Crow, J. Appl. Phys. 43, 1861 (1972).
[CrossRef]

M. Shah, J. D. Crow, and S. Wang, Appl. Phys. Lett. 20, 66 (1972).
[CrossRef]

Duffy, M. T.

J. M. Hammer, D. J. Channin, M. T. Duffy, and J. P. Whittke, Appl. Phys. Lett. 21, 358 (1972).
[CrossRef]

Françon, M.

M. Françon, in Advanced Optical Techniques, edited by A. C. S. van Heel (Wiley, New York, 1967), p. 39.

Gia Russo, D. P.

Giallorenzi, T. G.

J. P. Sheridan, J. M. Schnur, and T. G. Giallorenzi, Appl. Phys. Lett. 22, 560 (1973).
[CrossRef]

Hammer, J. M.

J. M. Hammer, D. J. Channin, M. T. Duffy, and J. P. Whittke, Appl. Phys. Lett. 21, 358 (1972).
[CrossRef]

Harris, J. H.

Kogelnik, H.

H. Kogelnik, C. V. Shank, and J. E. Bjorkholm, Appl. Phys. Lett. 22, 135 (1973).
[CrossRef]

McKenna, J.

D. F. Nelson and J. McKenna, J. Appl. Phys. 38, 4075 (1967).
[CrossRef]

Nelson, D. F.

D. F. Nelson and J. McKenna, J. Appl. Phys. 38, 4075 (1967).
[CrossRef]

Schnur, J. M.

J. P. Sheridan, J. M. Schnur, and T. G. Giallorenzi, Appl. Phys. Lett. 22, 560 (1973).
[CrossRef]

Shah, M.

S. Wang, M. Shah, and J. D. Crow, J. Appl. Phys. 43, 1861 (1972).
[CrossRef]

M. Shah, J. D. Crow, and S. Wang, Appl. Phys. Lett. 20, 66 (1972).
[CrossRef]

Shank, C. V.

H. Kogelnik, C. V. Shank, and J. E. Bjorkholm, Appl. Phys. Lett. 22, 135 (1973).
[CrossRef]

J. E. Bjorkholm, T. P. Sosnowski, and C. V. Shank, Appl. Phys. Lett. 22, 132 (1973).
[CrossRef]

Sheridan, J. P.

J. P. Sheridan, J. M. Schnur, and T. G. Giallorenzi, Appl. Phys. Lett. 22, 560 (1973).
[CrossRef]

Sosnowski, T. P.

J. E. Bjorkholm, T. P. Sosnowski, and C. V. Shank, Appl. Phys. Lett. 22, 132 (1973).
[CrossRef]

T. P. Sosnowski and H. P. Weber, Optics Commun. 7, 47 (1973).
[CrossRef]

T. P. Sosnowski, Optics Commun. 4, 408 (1972).
[CrossRef]

T. P. Sosnowski and H. P. Weber, Appl. Phys. Lett. 21, 310 (1972).
[CrossRef]

Wang, S.

M. Shah, J. D. Crow, and S. Wang, Appl. Phys. Lett. 20, 66 (1972).
[CrossRef]

S. Wang, M. Shah, and J. D. Crow, J. Appl. Phys. 43, 1861 (1972).
[CrossRef]

Weber, H. P.

T. P. Sosnowski and H. P. Weber, Optics Commun. 7, 47 (1973).
[CrossRef]

T. P. Sosnowski and H. P. Weber, Appl. Phys. Lett. 21, 310 (1972).
[CrossRef]

Whittke, J. P.

J. M. Hammer, D. J. Channin, M. T. Duffy, and J. P. Whittke, Appl. Phys. Lett. 21, 358 (1972).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 668.

Yariv, A.

A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).
[CrossRef]

A. Yariv, in Quantum Electronics (Wiley, New York, 1967), p. 299.

Appl. Opt. (1)

Appl. Phys. Lett. (6)

T. P. Sosnowski and H. P. Weber, Appl. Phys. Lett. 21, 310 (1972).
[CrossRef]

J. P. Sheridan, J. M. Schnur, and T. G. Giallorenzi, Appl. Phys. Lett. 22, 560 (1973).
[CrossRef]

J. M. Hammer, D. J. Channin, M. T. Duffy, and J. P. Whittke, Appl. Phys. Lett. 21, 358 (1972).
[CrossRef]

M. Shah, J. D. Crow, and S. Wang, Appl. Phys. Lett. 20, 66 (1972).
[CrossRef]

J. E. Bjorkholm, T. P. Sosnowski, and C. V. Shank, Appl. Phys. Lett. 22, 132 (1973).
[CrossRef]

H. Kogelnik, C. V. Shank, and J. E. Bjorkholm, Appl. Phys. Lett. 22, 135 (1973).
[CrossRef]

IEEE J. Quantum Electron. (2)

R. A. Andrews, IEEE J. Quantum Electron. 7, 523 (1971).
[CrossRef]

A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).
[CrossRef]

J. Appl. Phys. (2)

D. F. Nelson and J. McKenna, J. Appl. Phys. 38, 4075 (1967).
[CrossRef]

S. Wang, M. Shah, and J. D. Crow, J. Appl. Phys. 43, 1861 (1972).
[CrossRef]

J. Opt. Soc. Am. (1)

Optics Commun. (2)

T. P. Sosnowski, Optics Commun. 4, 408 (1972).
[CrossRef]

T. P. Sosnowski and H. P. Weber, Optics Commun. 7, 47 (1973).
[CrossRef]

Other (3)

M. Françon, in Advanced Optical Techniques, edited by A. C. S. van Heel (Wiley, New York, 1967), p. 39.

A. Yariv, in Quantum Electronics (Wiley, New York, 1967), p. 299.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 668.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Coordinate system for two-dimensional waveguide composed of top layer (region 1), film of thickness W (region 2), and substrate (region 3). Propagation is in the z direction.

Fig. 2
Fig. 2

(a) c axis orientation for uncoupled mode operation, case I. The c axis lies in the x, z plane, at an angle ϕ from the z axis. (b) c axis orientation for coupled-mode operation with the c axis in the film (y,z) plane, at an angle θ from the z axis.

Fig. 3
Fig. 3

Transverse wave-vector solutions γTE, γTM, γo, and γe vs propagation angle θ. The uniaxial substrate has indices no = 1.858 and ne = 1.716, k = 5.5 μm−1, and β = 2.0k.

Fig. 4
Fig. 4

Hybrid mode dispersion (+ and −) in a GGG/YIG/LiIO3 waveguide when the uncoupled TE1 and TM1 modes are degenerate. Wavelength is 1.152 μm.

Tables (1)

Tables Icon

Table I TE and TM field components in isotropic top layer and film

Equations (57)

Equations on this page are rendered with MathJax. Learn more.

× × E = - c - 2 ɛ · E ¨ .
ɛ = ( n x 2 0 δ 0 n y 2 0 δ 0 n z 2 ) ,
n x 2 = n o 2 cos 2 φ + n e 2 sin 2 φ ,
n y 2 = n o 2 ,
n z 2 = n e 2 cos 2 φ + n o 2 sin 2 φ ,
δ = ( n e 2 - n o 2 ) cos φ sin φ .
( β 2 - k 2 n x 2 0 - i β ( ± i ) γ - k 2 δ 0 β 2 - ( ± i ) 2 γ 2 - k 2 n y 2 0 - i β ( ± i ) γ - k 2 δ 0 - ( ± i ) 2 γ 2 - k 2 n z 2 ) ( E x E y E z ) = 0 ,
( ± i ) 2 γ TE 2 = β 2 - k 2 n y 2 .
γ TM = n e n o n x 2 ( k 2 n x 2 - β 2 ) 1 2 ± β δ n x 2 ,
γ TM = n e n o n x 2 ( β 2 - k 2 n x 2 ) 1 2 + i β δ n x 2 .
× H = c - 1 ɛ · E ˙ ,
× E = - c - 1 H ˙ .
( k 2 n y 2 2 - β 2 ) 1 2 W = tan - 1 ( β 2 - k 2 n y 3 2 k 2 n y 2 2 - β 2 ) 1 2 + tan - 1 ( β 2 - k 2 n 1 2 k 2 n y 2 2 - β 2 ) 1 2 + m π
n e 2 n o 2 n x 2 2 ( k 2 n x 2 2 - β 2 ) 1 2 W = tan - 1 [ n e 2 n o 2 n e 3 n o 3 ( β 2 - k 2 n x 3 2 k 2 n x 2 2 - β 2 ) 1 2 ] + tan - 1 [ n e 2 n o 2 n 1 2 ( β 2 - k 2 n 1 2 k 2 n x 2 2 - β 2 ) 1 2 ] + m π
ɛ = ( n x 2 0 0 0 n y 2 δ 0 δ n z 2 ) ,
n x 2 = n o 2 ,
n y 2 = n o 2 cos 2 θ + n e 2 sin 2 θ ,
n z 2 = n e 2 cos 2 θ + n o 2 sin 2 θ ,
δ = ( n e 2 - n o 2 ) cos θ sin θ ,
( β 2 - k 2 n x 2 0 - i β ( ± i ) γ 0 β 2 - ( ± i ) 2 γ 2 - k 2 n y 2 - k 2 δ - i β ( ± i ) γ - k 2 δ - ( ± i ) 2 γ 2 - k 2 n z 2 ) × ( E x E y E z ) = 0.
( ± i ) 2 γ TE 2 = ( β 2 - k 2 n y 2 )
( ± i ) 2 γ TM 2 = ( n z / n x ) 2 ( β 2 - k 2 n x 2 ) .
γ o 2 = ( a a + b ) γ TE 2 + ( b a + b ) γ TM 2
γ e 2 = ( b a + b ) γ TE 2 + ( a a + b ) γ TM 2 ,
( ± i ) 2 γ o 2 = β 2 - k 2 n o 2 ,
( ± i ) 2 γ e 2 = ( n z / n x ) 2 [ β 2 - k 2 n e 2 ( θ ) ] .
1 n c 2 ( θ ) = cos 2 θ n o 2 + sin 2 θ n c 2 .
E z = - tan θ E y ,
E z = - ( a b ) tan θ E y = - ( β 2 - k 2 n x 2 ) k 2 n x 2 tan θ E y .
D y = ( n y 2 + δ A ) E y ,
D z = ( δ + n z 2 A ) E y .
tan α E = A ,
tan α D = ( δ + n z 2 A ) / ( n y 2 + δ A ) .
tan α = δ A 2 + A ( n y 2 - n z 2 ) - δ n z 2 A 2 + 2 δ A + n y 2 .
tan α = δ C / ( n o 2 sin 2 θ + C 2 n e 2 cos 2 θ ) ,
C = 1 - [ k n x / ( β cos θ ) ] 2 .
E y = E y o e γ o x + E y e e γ e x ,
H z = ( i / k ) ( γ o E y o e γ o x + γ e E y e e γ e x ) ,
H y = - i [ ( k n x 2 tan θ γ o ) E y o e γ o x + ( γ e k tan θ ) E y e e γ e x ] ,
E z = - [ tan θ E y o e γ o x + ( γ o 2 k 2 n x 2 tan θ ) E y e e γ e x ] .
ϕ e 2 = 2 γ 2 W - 2 tan - 1 ( γ 1 / γ 2 ) ,
ϕ h 2 = 2 γ 2 W - 2 tan - 1 [ n 2 2 γ 1 / n 1 2 γ 2 ] .
R TE = - [ γ e - γ 2 tan ( ϕ e 2 / 2 ) γ o - γ 2 tan ( ϕ e 2 / 2 ) ] ,
R TM = - γ o 2 k 2 n x 2 tan 2 θ [ 1 - ( n x 2 γ 2 γ e / n o 2 γ o 2 ) tan ( ϕ h 2 / 2 ) 1 - ( n x 2 γ 2 / n 2 2 γ o ) tan ( ϕ h 2 / 2 ) ] .
a tan 2 ( γ 2 W ) + b tan ( γ 2 W ) + c = 0.
a = ( 1 / γ 2 2 ) [ A ( s t ) cos 2 θ - ( w x ) sin 2 θ ] ,
b = ( 1 / γ 2 ) [ - A ( u t + s v ) cos 2 θ + ( y x + w z ) sin 2 θ ] ,
c = A ( u v ) cos 2 θ - ( y z ) sin 2 θ ,
s = γ 2 2 - γ 1 γ o ,
t = ( n x 2 / n 2 2 ) ( γ e / γ o ) γ 2 2 - ( n 2 2 / n 1 2 ) γ o γ 1 ,
u = γ 1 + γ o ,
v = ( n x 2 / n 1 2 ) γ 1 ( γ e / γ o ) + γ o ,
w = γ 2 2 - γ e γ 1 ,
x = ( n x 2 / n 2 2 ) γ 2 2 - ( n 2 2 / n 1 2 ) γ o γ 1 ,
y = γ 1 + γ e ,
z = ( n x 2 / n 1 2 ) γ 1 + γ o .
W = ( 1 / γ 2 ) { tan - 1 [ ( - b ± [ b 2 - 4 a c ] 1 2 ) / 2 a ] + m π } ,