Abstract

General coupling equations are derived describing field amplitude transfer from a prism to an anisotropic, uniaxial waveguide, having its optical axis in the plane of incidence. The influence of optical axis orientation on coupling efficiency and selectivity is discussed for a Gaussian incident beam.

© 1981 Optical Society of America

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References

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  1. P. K. Tien, R. Ulrich, J. Opt. Soc. Am. 60, 1325 (1970).
    [CrossRef]
  2. R. Ulrich, J. Opt. Soc. Am. 60, 1337 (1970).
    [CrossRef]
  3. R. Ulrich, J. Opt. Soc. Am. 61, 1467 (1971).
    [CrossRef]
  4. J. H. Harris, R. Shubert, J. N. Polky, J. Opt. Soc. Am. 60, 1007 (1970).
    [CrossRef]
  5. D. P. GiaRusso, J. H. Harris, J. Opt. Soc. Am. 63, 138 (1973).
    [CrossRef]
  6. V. Ramaswamy, Appl. Opt. 13, 1363 (1974).
    [CrossRef] [PubMed]
  7. R. A. Steinberg, T. G. Giallorenzi, J. Opt. Soc. Am. 67, 523 (1977).
    [CrossRef]
  8. W. K. Burns, J. Werner, J. Opt. Soc. Am. 64, 441 (1974).
    [CrossRef]
  9. D. Marcuse, IEEE J. Quantum Electron. QE-14, 736 (1978).
    [CrossRef]
  10. I. V. Jogansen, V. V. Malov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 21, 1060 (1978).
  11. As in Ref. 2, the analysis may be extended to a 2-D Gaussian incident beam (corresponding to a laser TEM00 mode), because the coupling process leaves the profile in the beam unchanged in the direction perpendicular to the principal plane of the incidence.
  12. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1968), Chap. 7.
  13. See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), Chap. 1.6.2.
  14. See, for example, Ref. 13, Chap. 14.3.

1978

D. Marcuse, IEEE J. Quantum Electron. QE-14, 736 (1978).
[CrossRef]

I. V. Jogansen, V. V. Malov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 21, 1060 (1978).

1977

1974

1973

1971

1970

Born, M.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), Chap. 1.6.2.

Burns, W. K.

Giallorenzi, T. G.

GiaRusso, D. P.

Harris, J. H.

Jogansen, I. V.

I. V. Jogansen, V. V. Malov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 21, 1060 (1978).

Malov, V. V.

I. V. Jogansen, V. V. Malov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 21, 1060 (1978).

Marcuse, D.

D. Marcuse, IEEE J. Quantum Electron. QE-14, 736 (1978).
[CrossRef]

Polky, J. N.

Ramaswamy, V.

Shubert, R.

Steinberg, R. A.

Tien, P. K.

Ulrich, R.

Werner, J.

Wolf, E.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), Chap. 1.6.2.

Appl. Opt.

IEEE J. Quantum Electron.

D. Marcuse, IEEE J. Quantum Electron. QE-14, 736 (1978).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved. Radiofiz.

I. V. Jogansen, V. V. Malov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 21, 1060 (1978).

J. Opt. Soc. Am.

Other

As in Ref. 2, the analysis may be extended to a 2-D Gaussian incident beam (corresponding to a laser TEM00 mode), because the coupling process leaves the profile in the beam unchanged in the direction perpendicular to the principal plane of the incidence.

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1968), Chap. 7.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), Chap. 1.6.2.

See, for example, Ref. 13, Chap. 14.3.

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Figures (9)

Fig. 1
Fig. 1

Geometry of the system.

Fig. 2
Fig. 2

Modal curves Nm vs angle of optical axis inclination α for the following material parameters: n = 1.531; n|| = 1.696; n0 = n2 = 1.51; and d/λ = 4.

Fig. 3
Fig. 3

Modal curves ( N m - N m p ) ( α ) for s = 0.4λ, n3 = 1.8, and for the same parameters as in Fig. 2. The quantity N m p is the real part of resonance value γm in the presence of the prism. The parameter of the curves is the mode number.

Fig. 4
Fig. 4

Leakage length lm = (kKm)−1 for different modes vs angle α for λ = 0.6328 μm and for the same remaining parameters as in Fig. 3.

Fig. 5
Fig. 5

Coupling efficiency η vs the normalized parameter (y/w) for different values a = k w ( N m p - β I ).

Fig. 6
Fig. 6

Dependence of coupling efficiency η on the departure from the phase condition described by the parameter a = k w ( N m p - β I ).

Fig. 7
Fig. 7

Parameter (Δθ3)m characterizing the directional selectivity of the modes vs angle α for the same material constants as in Fig. 4 and a = 5. The dotted lines correspond to modes which exist in a limited range of α.

Fig. 8
Fig. 8

Curves ϕNm/∂α plotted as a function of α for the same parameters as in Fig. 2.

Fig. 9
Fig. 9

Dependence of the (Δα)m characterizing rotational selectivity on angle α for the same parameters as in Fig. 7. The decreasing dotted curves for m = 3,4,5 result from the fact that the leakage length lm grows to infinity near the angle of appearance of these modes.

Equations (33)

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[ x x , 0 , 0 0 , y y , y z 0 , z y , z z ]             [ , 0 , 0 0 , , 0 0 , 0 , ] ,
x x = , y y = sin 2 α + cos 2 α , y z = z y = ( - ) sin α cos α , z z = cos 2 α + sin 2 α .
V 3 ( y , z ) = v 3 ( β ) exp [ j k ( β y - b 3 z ) ] ,
b 1 ( ± ) = ρ A ± ρ B ,
ρ A = 1 z z ( z z - β 2 ) ;             ρ B = y z z z β .
V ( - ) = v 3 τ 321 exp [ - j k b 1 ( - ) s ] 1 - r 10 r 123 exp ( j k 2 ρ A d ) exp [ j k ( β y - b 1 ( - ) z ) ] ,
V ( + ) = v 3 τ 321 r 10 exp [ - j k b 1 ( - ) s ] exp [ j 2 k ρ A ( s + d ) ] 1 - r 10 r 123 exp ( j k 2 ρ A d ) × exp [ j k ( β y + b 1 ( + ) z ) ] ,
V 1 ( y , z ) = 2 v 3 τ 321 cos ( k ρ A z + δ 2 - ϕ 10 ) exp [ j ( δ 1 - ϕ 10 ) ] 1 - r 10 r 123 exp ( 2 j δ 1 ) × exp { j k β [ y + y z z z ( z + s ) ] } ,
ψ ( β ) = - 2 ϕ 10 - 2 ϕ 12 + 2 δ 1 = 2 m π ,
K m = [ - 2 h 2 sin ( 2 ϕ 12 ) sin ( 2 ϕ 32 ) ψ / β ] β = N m ,
N m - N m p = [ K m / tan ( 2 ϕ 32 ) ] β = N m .
V 1 ( y , z ) = v 3 ( β ) t ( z , β ) exp { j k β [ y + y z z z ( z + s ) ] } ,
t ( z , β ) = 2 τ 321 cos ( k ρ A z + δ 2 - ϕ 10 ) 1 - r 10 r 123 exp ( 2 j δ 1 ) exp [ j ( δ 1 - ϕ 10 ) ] ,
P 1 ( y ) = β 4 c k z z V ^ 1 ( y ) 2 { k d + 1 2 ρ A [ sin ( 2 ϕ 12 ) + sin ( 2 ϕ 10 ) ] + z z cos 2 ϕ 12 n 2 2 ( β 2 - n 2 2 ) 1 / 2 + z z cos 2 ϕ 10 n 0 2 ( β 2 - n 0 2 ) 1 / 2 } ,
t m ( z 0 , β ) = [ - 8 h sin ϕ 12 cos ϕ 32 ( ψ / β ) exp ( - j ϕ 32 ) ] ( β - γ m ) - 1 ,
V 1 ( y , z 0 ) = A ( γ m , z 0 ) - y V 3 ( ξ ) exp [ j k γ m ( y - ξ ) ] d ξ ,
A ( γ m , z 0 ) = k K m t m ( N m ) exp ( j k N m y 0 ) ,
d V 1 ( y , z 0 ) d y = j k γ m V 1 ( y , z 0 ) + A ( γ m , z 0 ) V 3 ( y ) ,
V 3 ( y ) = B 3 exp ( j k β I y - y 2 / w 2 ) ,
P 3 = w B 3 2 2 c π 2 n 3 2 - β I 2 n 3 2 .
η = π 2 w l m exp ( w 2 2 l m 2 - 2 y w - a 2 4 ) × | 1 + erf ( y w - w 2 l m + j a 2 ) + erf ( j a 2 ) | 2 ,
erf ( ξ ) = 2 π 0 ξ exp ( - t 2 ) d t ,
β I = N m p ,
l m W = l m ( β , α ) cos θ 3 W 0 = 1.48.
( Δ θ 3 ) m 1.48 a k l m ( n 3 2 - N m 2 ) - 1 / 2 .
( Δ α ) m = 1.48 a k l m / N m α .
r 1 i = sin θ 1 cos θ 1 - sin θ i cos θ i sin θ 1 cos θ 1 + sin θ i cos θ i ,
cos θ 1 = cos θ 1 + p S y sin α 1 - p S z ,
τ 321 = h ( r 32 + 1 ) ( r 21 + 1 ) 1 + h 2 r 32 r 21 ,
r 123 = r 12 + h 2 r 23 1 + h 2 r 23 r 12 ,
r i j = exp ( - j 2 ϕ i j ) .
n 1 2 = 1 - p S y 2 .
ϕ 1 i = tan - 1 ( n i 2 β 2 - n i 2 n z z 2 - β 2 ) .

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