Abstract

A prism–film coupler has been discussed recently by Tien, Ulrich, and Martin as a device to couple efficiently a laser beam into thin-film dielectric light guides. This coupler also allows an accurate measurement of the spectrum of propagating modes from which the refractive index and the thickness of the film can be determined. We present here a theory of the prism–film coupler. The physical principles involved are illustrated by a method that combines wave and ray optics. We study the modes in the thin-film light guide and their modification by the effect of coupling. We also calculate the field distributions in the prism and the film, the power transfer between the prism and the film, and derive a condition of optimum operation. In one example, 81% of the laser power can be fed into any desired mode of propagation in the film.

© 1970 Optical Society of America

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References

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  1. S. E. Miller, Bell System Tech. J. 48, 2059 (1969).
    [Crossref]
  2. P. K. Tien, R. Ulrich, and R. J. Martin, Appl. Phys. Letters 14, 291 (1969).
    [Crossref]
  3. Bell Telephone Laboratories, Incorporated, Murray Hill, N. J.
  4. J. E. Goell and R. D. Standley, Bell System Tech. J. 48, 3445 (1969).
    [Crossref]
  5. M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965).
  6. D. Marcuse, Bell System Tech. J. 48, 3187 (1969).
    [Crossref]
  7. W. N. Hansen, J. Opt. Soc. Am. 58, 380 (1967).
    [Crossref]
  8. Reference 5, pp. 327 and 344.

1969 (4)

S. E. Miller, Bell System Tech. J. 48, 2059 (1969).
[Crossref]

P. K. Tien, R. Ulrich, and R. J. Martin, Appl. Phys. Letters 14, 291 (1969).
[Crossref]

J. E. Goell and R. D. Standley, Bell System Tech. J. 48, 3445 (1969).
[Crossref]

D. Marcuse, Bell System Tech. J. 48, 3187 (1969).
[Crossref]

1967 (1)

Born, M.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965).

Goell, J. E.

J. E. Goell and R. D. Standley, Bell System Tech. J. 48, 3445 (1969).
[Crossref]

Hansen, W. N.

Marcuse, D.

D. Marcuse, Bell System Tech. J. 48, 3187 (1969).
[Crossref]

Martin, R. J.

P. K. Tien, R. Ulrich, and R. J. Martin, Appl. Phys. Letters 14, 291 (1969).
[Crossref]

Miller, S. E.

S. E. Miller, Bell System Tech. J. 48, 2059 (1969).
[Crossref]

Standley, R. D.

J. E. Goell and R. D. Standley, Bell System Tech. J. 48, 3445 (1969).
[Crossref]

Tien, P. K.

P. K. Tien, R. Ulrich, and R. J. Martin, Appl. Phys. Letters 14, 291 (1969).
[Crossref]

Ulrich, R.

P. K. Tien, R. Ulrich, and R. J. Martin, Appl. Phys. Letters 14, 291 (1969).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965).

Appl. Phys. Letters (1)

P. K. Tien, R. Ulrich, and R. J. Martin, Appl. Phys. Letters 14, 291 (1969).
[Crossref]

Bell System Tech. J. (3)

J. E. Goell and R. D. Standley, Bell System Tech. J. 48, 3445 (1969).
[Crossref]

D. Marcuse, Bell System Tech. J. 48, 3187 (1969).
[Crossref]

S. E. Miller, Bell System Tech. J. 48, 2059 (1969).
[Crossref]

J. Opt. Soc. Am. (1)

Other (3)

Reference 5, pp. 327 and 344.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965).

Bell Telephone Laboratories, Incorporated, Murray Hill, N. J.

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Figures (16)

Fig. 1
Fig. 1

Prism–film coupler.

Fig. 2
Fig. 2

Experimental arrangement for observation of the mode spectrum of a thin-film waveguide.

Fig. 3
Fig. 3

Photograph of the m lines.

Fig. 4
Fig. 4

Experimental arrangement for feeding a laser beam into a thin film.

Fig. 5
Fig. 5

Photograph of a streak of guided light excited in a semiconductor film.

Fig. 6
Fig. 6

Thin-film prism constructed by depositing a triangular portion thicker than the background film.

Fig. 7
Fig. 7

Photograph of a light beam deflected by the thin-film prism.

Fig. 8
Fig. 8

(a) Total reflection from the lower film boundary, (b) total reflection from the upper film boundary, and (c) optics in a thin-film waveguide.

Fig. 9
Fig. 9

W vs β/k for n0=2.190, n1=3.275, n2= 1.00, and laser wavelength 10.6 μm.

Fig. 10
Fig. 10

(a) Symmetric waveguide having a phase constant β for a mode of order m =0, (b) symmetric waveguide having a phase constants β for a mode of order m′=2, and (c) asymmetric waveguide constructed by combining the lower half of (a) with the upper half of (b). The dotted curves are the field distributions, Ev for the TE waves and Hv for the TM waves.

Fig. 11
Fig. 11

The thickness of a thin-film waveguide may be considered as the sum of W10, W12, and mW1.

Fig. 12
Fig. 12

kn1W10 (or kn1W12) vs β/kn1. The parameters on the curves are the values of n0/n1 (or n2/n1).

Fig. 13
Fig. 13

kn1W1β/kn1. Parameters as in Fig. 12.

Fig. 14
Fig. 14

(a) Thin film n1 coupled to a semi-infinite medium n3 through a gap S, (b) two semi-infinite media, n3 and n1, coupled through a gap S, and (c) waves in a prism–film coupler.

Fig. 15
Fig. 15

Distributions of wave amplitudes in a prism–film coupler for (a) incident beam, (b) A1 or B1 wave in the film, and (c) reflected beam in the prism.

Fig. 16
Fig. 16

δ vs θ1 and θ3.

Equations (78)

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2 E / x 2 + 2 E / z 2 = - ( k n j ) 2 E ,             j = 0 , 1 , 2 , or 3 ,
H x = ( i / k ) ( / z ) E y             and             H z = - ( i / k ) ( / x ) E y E x = - ( i / k n j 2 ) ( / z ) H y
E z = ( i / k n j 2 ) ( / x ) H y .
exp ( - i ω t - i b 1 z + i β x ) ,
b 1 = k n 1 cos θ 1             and             β = k n 1 sin θ 1 .
exp ( i ω t + i b 1 z + i β x ) .
E y = A 1 ( or A 1 ) e - i b 1 z ;             H x = n 1 cos θ 1 A 1 ( or A 1 ) e - i b 1 z
E y = B 1 e i b 1 z ;             H x = n 1 cos θ 1 B 1 e i b 1 z ,
E y = C 0 e p 0 z ;             H x = i p 0 k C 0 e p 0 z ;             z < 0 ,
E y = D 2 e - p 2 ( z - W ) ;             H x = i p 2 k D 2 e - p 2 ( z - W ) ;             z > w .
β = k n 1 sin θ 1 ;             b 1 = k n 1 cos θ 1 , b 1 2 = ( k n 1 ) 2 - β 2 , p 0 2 = β 2 - ( k n 0 ) 2 , p 2 2 = β 2 - ( k n 2 ) 2 .
B 1 / A 1 = e - i 2 Φ 10 .
A 1 / B 1 = e - i 2 Φ 12 ,
tan Φ 10 = p 0 / b 1 ,             tan Φ 12 = p 2 / b 1
tan Φ 10 = ( n 1 / n 0 ) 2 p 0 / b 1 ,             tan Φ 12 = ( n 1 / n 2 ) 2 p 2 / b 1 .
- ω t + β x c .
- ω t + β x a + β ( x c - x a ) + 2 b 1 W - 2 Φ 10 - 2 Φ 12 .
2 b 1 W - 2 Φ 10 - 2 Φ 12 = 2 m π .
b 1 k ( n 1 2 - n 0 2 ) 1 2 ,             p 0 0 ,             Φ 10 0 , p 2 k ( n 0 2 - n 2 2 ) 1 2 .
W min = 1 k [ m π + tan - 1 ( n 0 2 - n 2 2 n 1 2 - n 0 2 ) 1 2 ] / ( n 1 2 - n 0 2 ) 1 2
W = W 10 + W 12 + m W 1 ,
W 10 = Φ 10 / b 1 ;             W 12 = Φ 12 / b 1 ;             W 1 = π / b 1 .
( c / 8 π ) Re ( E y H z * )
P = ( c / 4 π ) A 1 A 1 * n 1 sin θ 1 ( W + 1 / p 0 + 1 / p 2 ) .
E y = C 2 e p 2 ( z - W ) + D 2 e - p 2 ( z - W ) , H x = ( i p 2 / k ) ( - C 2 e p 2 ( z - W ) + D 2 e - p 2 ( z - W ) ) ,
β = k n 3 sin θ 3 ,             b 3 = k n 3 cos θ 3 , b 3 2 = ( k n 3 ) 2 - β 2 .
tan Φ 32 = p 2 / b 3 ;             Φ 32 π / 2
tan Φ 32 = ( n 3 / n 2 ) 2 p 2 / b 3 ;             Φ 32 π / 2
A 1 / B 1 = R 1 , ( n 3 cos θ 3 ) 1 2 B 3 / ( n 1 cos θ 1 ) 1 2 B 1 = T ,
R 1 = r 1 e - i 2 Φ 02 = tan Φ 32 - tan Φ 12 - i ( tanh p 2 S ) ( 1 + tan Φ 32 tan Φ 12 ) tan Φ 32 + tan Φ 12 - i ( tanh p 2 S ) ( 1 - tan Φ 32 tan Φ 12 ) ,
T = 1 / cosh p s S · 2 ( tan Φ 32 tan Φ 12 ) 1 2 tan Φ 32 + tan Φ 12 - i ( tanh p 2 S ) ( 1 - tan Φ 32 tan Φ 12 ) ,
2 b 1 W - 2 Φ 10 - 2 Φ 12 = 2 m π ,
T 2 4 e - 2 p 2 S sin 2 Φ 12 sin 2 Φ 32
Φ 12 Φ 12 + e - 2 p 2 S sin 2 Φ 12 cos 2 Φ 32 .
A 1 A 1 * n 1 cos θ 1 B 1 B 1 * n 1 cos θ 1 = A 1 A 1 * B 1 B 1 * = R 1 2 = r 1 2
R 1 2 + T 2 = 1.
B 3 / A 3 = R 3 ,
( n 1 cos θ 1 ) 1 2 A 1 / ( n 3 cos θ 3 ) 1 2 A 3 = T ,
R 3 = r 3 e - i 2 Φ 32 = tan Φ 12 - tan Φ 32 - i ( tanh p 2 S ) ( 1 + tan Φ 32 tan Φ 12 ) tan Φ 32 + tan Φ 12 + i ( tanh p 2 S ) ( 1 - tan Φ 32 tan Φ 12 ) .
R 3 2 + T 2 = 1.
( A j ) ( A j ) * = A j A j * n j cos θ j ( B j ) ( B j ) * = B j B j * n j cos θ j             j = 1 , 3
( A j ) ( A j ) * = A j A j * cos θ j / n j ( B j ) ( B j ) * = B j B j * cos θ j / n j             j = 1 , 3
( B 3 ) n = ( B 3 ) n + ( B 3 ) n = R 3 ( A 3 ) n + T ( B 1 ) n
( A 1 ) n = ( A 1 ) n + ( A 1 ) n = T ( A 3 ) n + R 1 ( B 1 ) n ,
( B 1 ) n = ( A 1 ) n - 1 exp ( i 2 b 1 W - i 2 Φ 10 ) .
( A 1 ) n = T ( A 3 ) n + r 1 ( A 1 ) n - 1 × exp ( i 2 b 1 W - i 2 Φ 10 - i 2 Φ 12 ) .
( A 1 ) n = T ( A 3 ) n + r 1 ( A 1 ) n - 1 ,
( A 1 ) n - ( A 1 ) n - 1 = T ( A 3 ) n + ( 1 - r 1 ) ( A 1 ) n - 1 .
d [ A 1 ( x ) ] / d x = ( 1 / 2 W tan θ 1 ) × { T ( A 3 ) - ( 1 - r 1 ) [ A 1 ( x ) ] } .
[ A 1 ( x ) / [ A 3 ( x ) ] = T / ( 1 - r 1 ) ,
T / ( 1 - r 1 ) 2 / T .
[ A 1 ( x ) ] = T ( A 3 ) ( 1 - r 1 ) { 1 - exp [ - ( 1 - r 1 ) x 2 W tan θ 1 ] } ,             x < l .
T 2 / R 1 R 3 = - T T * / R 1 R 1 * = - T T * / ( 1 - T T * ) .
[ B 3 ( x ) ] = R 3 ( A 3 ) { 1 - T T * 1 - T T * 1 1 - r 1 × [ r - exp ( - ( 1 - r 1 ) x 2 W tan θ 1 ) ] } ,             x < l .
1 - r 1 = ( 1 - r 1 2 ) / ( 1 + r 1 ) T T * / 2 = T 2 / 2.
| [ A 1 ( x ) ] ( A 3 ) | 2 T [ 1 - exp ( - T 2 x 4 W tan θ 1 ) ] ,             x < l ,
| [ B 3 ( x ) ] ( A 3 ) | [ - 1 + 2 exp ( - T 2 x 4 W tan θ 1 ) ] ,             x < l .
B 1 ( x ) A 1 ( x ) .
[ B 3 ( x ) ] = [ B 3 ( x ) ] + [ B 3 ( x ) ] .
Δ P = ( c / 8 π ) { ( A 3 ) ( A 3 ) * - [ B 3 ( x ) ] [ B 3 ( x ) ] * } Δ x .
[ A 1 ( x ) ] = [ A 1 ( l ) ] exp [ - ( 1 - r 1 ) ( x - l ) 2 W tan θ 1 ]
[ A 1 ( l ) ] exp [ - T 2 ( x - l ) 4 W tan θ 1 ] ;             x > l .
[ B 3 ( x ) ] = T R 1 [ A 1 ( x ) ] = - 2 R 3 ( A 3 ) [ 1 - exp ( - T 2 l 4 W tan θ 1 ) ] × exp [ - T 2 ( x - l ) 4 W tan θ 1 ] ,             x > l .
( c / 4 π ) [ A 1 ( l ) ] [ A 1 ( l ) ] * W tan θ 1 .
( c l / 8 π ) ( A 3 ) ( A 3 ) * .
coupling efficiency = [ A 1 ( l ) ] [ A 1 ( l ) ] * ( A 3 ) ( A 3 ) * 2 W tan θ 1 l .
coupling efficiency = 8 W tan θ 1 l T 2 [ 1 - exp ( - T 2 l 4 W tan θ 1 ) ] 2 .
l T 2 / 4 W tan θ 1 1.25 ,
R 1 ( n 1 cos θ 1 - n 3 cos θ 3 ) / ( n 1 cos θ 1 + n 3 cos θ 3 ) , R 3 ( n 3 cos θ 3 - n 1 cos θ 1 ) / ( n 1 cos θ 1 + n 3 cos θ 3 ) , T [ 2 ( n 1 n 3 cos θ 1 cos θ 3 ) 1 2 ] / ( n 1 cos θ 1 + n 3 cos θ 3 ) .
2 b 1 W - 2 Φ 10 - 2 Φ 12 = 2 m π + δ ,
( A 1 ) n = T ( A 3 ) + r 1 e i δ ( A 1 ) n - 1 .
[ A 1 ( q ) ] = T ( A 3 ) [ 1 + r 1 e i δ + r 1 2 e j 2 δ + + r 1 q - 1 e i ( q - 1 ) δ ] = T ( A 3 ) ( 1 - r 1 q e i q δ ) 1 - r 1 e i δ .
[ A 1 ( q ) ] [ A 1 ( q ) ] * ( A 3 ) ( A 3 ) * = T 2 ( 1 - r 1 q ) 2 ( 1 - r 1 ) 2 1 + G q sin 2 ( q δ / 2 ) 1 + F sin 2 ( δ / 2 ) ,
G q = 4 r 1 q / ( 1 - r 1 q ) 2 ;             F = 4 r 1 / ( 1 - r 1 ) 2 .
F sin 2 ( δ / 2 ) = 1.
δ ( half-power ) T 2 / 2.
finesse = ( π / 2 ) ( F ) 1 2 π / T 2 .
δ ( half-power ) = T 2 / 2 + α l .