Abstract

A laser beam can be coupled with high efficiency into a light-guiding thin film by means of a prism–film coupler. Basically this device is a totally reflecting prism, the light-guiding film being separated from the reflecting prism face by a narrow gap of reduced refractive index. This coupling scheme is analyzed in detail by the method of plane-wave expansion. It is shown how the coupling efficiency is determined by the competition between the desired coupling effect and the reverse effect of leakage. A general condition is derived under which the transverse profile of the input beam continues undistorted into the guide. The theory is illustrated for a gaussian beam, which allows a maximum coupling efficiency of 0.80.

© 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. R. Shubert and J. H. Harris, IEEE Trans. MTT-16, 1048 (1968).
  3. P. K. Tien, R. Ulrich, and R. J. Martin, Appl. Phys. Letters 14, 291 (1969).
    [CrossRef]
  4. J. E. Midwinter and F. Zernike, Appl. Phys. Letters 16, 198 (1970).
    [CrossRef]
  5. J. E. Midwinter, IEEE J. QE-6, 583 (1970).
    [CrossRef]
  6. P. K. Tien and R. Ulrich, J. Opt. Soc. Am. 60, 1325 (1970).
    [CrossRef]
  7. J. Kane and H. Osterberg, J. Opt. Soc. Am. 54, 347 (1964).
    [CrossRef]
  8. E. R. Schineller, R. F. Flam, and D. W. Wilmot, J. Opt. Soc. Am. 58, 1171 (1968).
    [CrossRef]
  9. M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1964).
  10. W. N. Hansen, J. Opt. Soc. Am. 58, 380 (1968).
    [CrossRef]
  11. G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
    [CrossRef]
  12. H. Kogelnik and T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  13. Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun, Natl. Bur. Stds. (U.S.) Handb. Appl. Math. Ser. (U. S. Govt. Printing Office, Washington, D. C., 1964; Dover, New York, 1965).
  14. C. W. Martz, Tables of the Complex Fresnel Integral, NASA Spec. Publ. 3010 (NASA Scientific and Tech. Division, Washington, D. C., 1964).
  15. D. Marcuse, Bell System Tech. J. 49, 273 (1970).
    [CrossRef]
  16. H. Osterberg and L. W. Smith, J. Opt. Am. 54, 1078 (1964).
    [CrossRef]
  17. L. Bergstein and C. Shulman, Appl. Opt. 5, 9 (1966).
    [CrossRef] [PubMed]

1970 (4)

J. E. Midwinter and F. Zernike, Appl. Phys. Letters 16, 198 (1970).
[CrossRef]

J. E. Midwinter, IEEE J. QE-6, 583 (1970).
[CrossRef]

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

D. Marcuse, Bell System Tech. J. 49, 273 (1970).
[CrossRef]

1969 (2)

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

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

1968 (3)

1966 (2)

1964 (2)

H. Osterberg and L. W. Smith, J. Opt. Am. 54, 1078 (1964).
[CrossRef]

J. Kane and H. Osterberg, J. Opt. Soc. Am. 54, 347 (1964).
[CrossRef]

1961 (1)

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
[CrossRef]

Bergstein, L.

Born, M.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1964).

Boyd, G. D.

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
[CrossRef]

Flam, R. F.

Gordon, J. P.

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
[CrossRef]

Hansen, W. N.

Harris, J. H.

R. Shubert and J. H. Harris, IEEE Trans. MTT-16, 1048 (1968).

Kane, J.

Kogelnik, H.

Li, T.

Marcuse, D.

D. Marcuse, Bell System Tech. J. 49, 273 (1970).
[CrossRef]

Martin, R. J.

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

Martz, C. W.

C. W. Martz, Tables of the Complex Fresnel Integral, NASA Spec. Publ. 3010 (NASA Scientific and Tech. Division, Washington, D. C., 1964).

Midwinter, J. E.

J. E. Midwinter and F. Zernike, Appl. Phys. Letters 16, 198 (1970).
[CrossRef]

J. E. Midwinter, IEEE J. QE-6, 583 (1970).
[CrossRef]

Miller, S. E.

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

Osterberg, H.

H. Osterberg and L. W. Smith, J. Opt. Am. 54, 1078 (1964).
[CrossRef]

J. Kane and H. Osterberg, J. Opt. Soc. Am. 54, 347 (1964).
[CrossRef]

Schineller, E. R.

Shubert, R.

R. Shubert and J. H. Harris, IEEE Trans. MTT-16, 1048 (1968).

Shulman, C.

Smith, L. W.

H. Osterberg and L. W. Smith, J. Opt. Am. 54, 1078 (1964).
[CrossRef]

Tien, P. K.

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

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

Ulrich, R.

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

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

Wilmot, D. W.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1964).

Zernike, F.

J. E. Midwinter and F. Zernike, Appl. Phys. Letters 16, 198 (1970).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Letters (2)

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

J. E. Midwinter and F. Zernike, Appl. Phys. Letters 16, 198 (1970).
[CrossRef]

Bell System Tech. J. (3)

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

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
[CrossRef]

D. Marcuse, Bell System Tech. J. 49, 273 (1970).
[CrossRef]

IEEE J. (1)

J. E. Midwinter, IEEE J. QE-6, 583 (1970).
[CrossRef]

IEEE Trans. (1)

R. Shubert and J. H. Harris, IEEE Trans. MTT-16, 1048 (1968).

J. Opt. Am. (1)

H. Osterberg and L. W. Smith, J. Opt. Am. 54, 1078 (1964).
[CrossRef]

J. Opt. Soc. Am. (4)

Other (3)

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1964).

Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun, Natl. Bur. Stds. (U.S.) Handb. Appl. Math. Ser. (U. S. Govt. Printing Office, Washington, D. C., 1964; Dover, New York, 1965).

C. W. Martz, Tables of the Complex Fresnel Integral, NASA Spec. Publ. 3010 (NASA Scientific and Tech. Division, Washington, D. C., 1964).

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Figures (7)

Fig. 1
Fig. 1

Prism–film coupler, schematic.

Fig. 2
Fig. 2

Four parallel-layered dielectric media, forming the prism–film coupler. The y axis is directed into the paper.

Fig. 3
Fig. 3

Peak energy density W 1 in the film relative to the energy density W 3 of the field in the incident wave in the prism, plotted as a function of β for three different coupling strengths. λ= 6328Å, ZnS film on glass substrate (n0=1.52; n1=2.35; W =5000Å), rutile prism (extraordinary ray n3 =2.87), TE polarization. The width S of the coupling gap (n2=1.00) is the parameter at the curves.

Fig. 4
Fig. 4

Field distribution in the film of a prism–film coupler. The incident beam has a gaussian profile of width 2w (dotted curve) and is centered at x=0 at the prism base. The normalized power η(x) in the film builds up over the illuminated region, and then decreases in x≫w by leakage. The parameter of the curves is the relative coupling length lm/w. A large value of lm/w means weak coupling.

Fig. 5
Fig. 5

Peak coupling efficiency η(xM) and position xM of this peak, plotted vs the coupling parameter lm/w.

Fig. 6
Fig. 6

Methods of interrupting the coupling in order to retain the power in the guide. Of special interest is method (c), which eliminates the need for a gap layer beyond xB. ABS is an absorbing layer to remove the reflected beam in an orderly manner from the prism–substrate.

Fig. 7
Fig. 7

Coupling of a two-dimensional beam, schematic. For clarity, the coupling gap and the reflected beam are not shown. The Y contours of the guided beam are obtained by projecting the Y contours of the extrapolated input beam on the plane of the guide. (The labels z and Z at the coordinate axes must be interchanged.)

Equations (83)

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n 2 n 0 < n 1 n 3 .
β n 3 sin θ 3
ξ j ( β ) ( n j 2 - β 2 ) 1 2 = n j cos θ j .
n 0 < β < n 1 ,
q k = { 1 for TE polarization n k 2 for TM polarization ,
r j k ( β ) ( q k ξ j - q j ξ k ) / ( q k ξ j + q j ξ k ) .
r j k exp [ - 2 i ϕ j k ( β ) ] ,
ϕ j k ( β ) = arctan ( q j ξ k / i q k ξ j ) .
1 ( β ) k ξ 1 W
h exp ( i k ξ 2 S ) .
12 k ξ 1 S
V 3 ( x , y , z ) = v 3 ( β ) exp ( i k β x - i k ξ 3 z )
v 3 ( β ) τ 321 ( β ) exp ( i k β x - i k ξ 1 z ) ,
τ 321 ( β ) h ( 1 + r 32 ) ( 1 + r 21 ) exp ( - i 12 ) / ( 1 + h 2 r 32 r 21 ) .
r 123 ( β ) ( r 12 + h 2 r 23 ) ( 1 + h 2 r 32 r 21 ) .
V 1 ( x , y , z ) = v 3 ( β ) u ( β ) τ 321 ( β ) a ( β , z ) exp ( i k β x ) ,
u ( β ) [ 1 - r 10 r 123 exp ( 2 i 1 ) ] - 1
a ( β , z ) = 2 cos ( k ξ 1 z + 1 + 12 - ϕ 10 ) · exp ( i 1 + i 12 - i ϕ 10 ) .
V ˆ 1 ( x ) = t ( β ) v 3 ( β ) exp ( i k β x ) ,
t ( β ) u ( β ) τ 321 ( β ) â ( β )
â ( β ) 2 exp [ i ( 1 + 12 - ϕ 10 - m π ) ] .
V R ( x , y , z ) = ρ ( β ) v 3 ( β ) exp [ i k ( β x + ξ 3 z ) ]
ρ ( β ) r 32 + h 2 ( 1 - r 32 2 ) r 210 1 + h 2 r 32 r 210 .
r 210 r 21 + r 10 exp ( 2 i 1 ) 1 + r 21 r 10 exp ( 2 i 1 )
ψ ( β ) 2 k ξ 1 ( β ) W - 2 ϕ 10 ( β ) - 2 ϕ 12 ( β ) .
ψ ( N m ) = 2 m π             m = 0 , 1 , 2 , .
W ( N m ) = [ ϕ 10 ( N m ) + ϕ 12 ( N m ) + m π ] / [ k ξ 1 ( N m ) ] .
χ ( β ) - ψ / β .
P 1 ( x ) = ( 16 π k q 1 ) - 1 μ c N m V ˆ 1 ( m ) ( x ) 2 ,
μ 0 ( i q 1 / ξ 0 q 0 ) cos 2 ϕ 10 μ 1 k W + ( 1 / 2 ξ 1 ) ( sin 2 ϕ 10 + sin 2 ϕ 12 ) μ 2 ( i q 1 / ξ 2 q 2 ) cos 2 ϕ 12 . }
2 μ = χ ξ 1 / N m = χ cot θ 1 .
W j ( n j 2 / 4 π q j ) V ˆ j 2 .
W 1 ( β ) / W 3 = ( n 1 2 q 3 / 4 n 3 2 q 1 ) t ( β ) 2 .
γ m N m ( p ) + i K m .
h 1.
r 123 = r 12 ( 1 - 2 i h 2 r 32 sin 2 ϕ 12 )
τ 321 ( β ) = 4 i h sin ϕ 12 cos ϕ 32 exp [ - i ( ϕ 12 + ϕ 32 + 12 ) ] .
N m ( p ) = N m - 2 ( h 2 / χ ) sin 2 ϕ 12 cos 2 ϕ 32 ,
K m = [ 1 - r 321 ( N m ) ] / χ = 2 ( h 2 / χ ) sin 2 ϕ 12 sin 2 ϕ 32 .
u ( β ) u m ( β ) [ 1 / i χ ( N m ) ] [ 1 / ( β - γ m ) ] ,
t m ( β ) τ 321 ( N m ) â ( N m ) u m ( β ) ,
t ˆ m t m ( N m ( p ) ) = 2 i C h - 1 exp ( - i ϕ 32 ) ,
C [ 2 cos ϕ 12 ( N m ) sin ϕ 32 ( N m ) ] - 1 .
ρ ( β ) ρ m ( β ) r 32 ( β - γ m * ) / ( β - γ m ) ,
V 3 ( x , y , z ) = - + v 3 ( β ) exp [ i k ( β x - ξ 3 z ) ] d β
V 3 ( x ) = v 3 ( β ) exp ( i k β x ) d β
v 3 ( β ) = k 2 π V 3 ( x ) exp ( - i k β x ) d x .
V 1 ( x , y , z ) = v 3 ( β ) u ( β ) τ 321 ( β ) a ( β , z ) exp ( i k β x ) d β .
B I < π / χ ( β I ) ,
V ˆ 1 ( x ) = v 3 ( β ) t m ( β ) exp ( i k β x ) d β ,
V ˆ 1 ( x ) = k 2 π T m ( x - ζ ) V 3 ( ζ ) d ζ .
T m ( x ) = { 0 for x < 0 2 π t ˆ m K m exp ( i k γ m x ) for x > 0.
V ˆ 1 ( x ) = k K m t ˆ m - x V 3 ( ζ ) exp [ i k γ m ( x - ζ ) ] d ζ .
V R ( x ) = v 3 ( β ) ρ ( β ) exp ( i k β x ) d β .
V R ( x ) = r 32 [ V 3 ( x ) - 2 k K m - x V 3 ( ζ ) × exp [ i k γ m ( x - ζ ) ] d ζ ] .
d V ˆ 1 ( x ) / d x - i k γ m V ˆ 1 ( x ) = k K m t ˆ m V 3 ( x ) .
V R ( x ) = r 32 V 3 ( x ) - 2 ( r 32 / t ˆ m ) V ˆ 1 ( x ) .
d d x V ˆ 1 ( x ) 2 = 4 h C D m V 3 ( x ) V ˆ 1 ( x ) sin ( Ψ 1 - Ψ 3 + ϕ 32 ) - 2 h 2 D m V ˆ 1 ( x ) 2 .
D m k K m h - 2 = 2 ( k / χ ) sin 2 ϕ 12 sin 2 ϕ 32 .
[ - d P 1 ( x ) / d x ] / P 1 ( x ) = 2 k K m = 2 h 2 D m .
l m ( k K m ) - 1 = D m - 1 h - 2 .
β I n 3 sin θ I = N m ( p ) .
η ( x ) P 1 ( x ) / P I .
p 3 ( x ) ( 8 π q 3 ) - 1 c ξ 3 V 3 ( x ) 2 .
V 3 ( x ) = A exp ( i k N m ( p ) x - x 2 / w 2 ) .
V ˆ 1 ( x ) = A w 2 l m t ˆ m π 1 2 [ 1 + erf ( x w - w 2 l m ) ] exp ( i k γ m x + w 2 4 l m 2 ) .
w 2 = w 0 2 ( 1 + 2 i L / b ) cos 2 θ I .
V ( X , Y , Z ) = F ( X , Y ) G ( X , Z ) exp ( i k n 3 X ) .
E 3 ( X , Y ) = V ( X , Y , Z ) / V ( X , 0 , Z ) = F ( X , Y ) / F ( X , 0 ) ,
E 1 ( x , y ) = E 3 ( x / sin θ I , y ) .
V ( X , Y , Z ) = f ( X , β Y ) g ( X , β Z ) × exp [ i k ( n 3 X + β Y Y + β Z Z ) ] d β Y d β Z .
V ( X , Y , Z ) = f ( 0 , β Y ) g ( 0 , β Z ) · exp { i k [ β Y Y + β Z Z + ( n 3 2 - β Y 2 - β Z 2 ) 1 2 X ] } d β Y d β Z ,
f ( X , β Y ) = f ( 0 , β Y ) exp ( - i k X β Y 2 / 2 n 3 )
g ( X , β Z ) = g ( 0 , β Z ) exp ( - i k X β Z 2 / 2 n 3 )
β Z = β x cos θ I + β z sin θ I = ( β 2 - β y 2 ) 1 2 cos θ I - ( n 3 2 - β 2 ) 1 2 sin θ I ,
V 3 ( x , y , 0 ) = v 3 ( β , β y ) exp [ i k ( β x + β y y ) ] d β d β y ,
v 3 ( β , β y ) = f ( x / sin θ I , β y ) g [ 0 , Q 1 ( β ) - Q 2 β y 2 ] × [ J ( β , 0 ) + Q 2 β y 2 / β I ]
Q 1 ( β ) = β - β I cos θ I + ( β - β I cos θ I ) 2 tan θ I 2 n 3
Ω k w Z Q 2 B Y 2 π Q 2 B Y 2 / B Z .
v 3 ( β , β y ) = f ( x / sin θ I , β y ) { g [ 0 , Q 1 ( β ) ] J ( β , 0 ) - [ J g - g / β I ] β y 2 Q 2 } .
V ˆ 1 ( x , y ) = F ( x / sin θ I , y ) t m ( β ) g [ 0 , Q 1 ( β ) ] J ( β , 0 ) × exp ( i k β x ) d β + V ˆ 1 [ 1 ] ( x , y )
V ˆ 1 [ 1 ] ( x , y ) = Q 2 k - 2 F Y Y ( x / sin θ I , y ) × ( J g - g / β I ) t m exp ( i k β x ) d β .
π Q 2 B Y 2 / B z 1.