Abstract

The holographic grating coupler was used experimentally by H. Kogelnik and T. P. Sosnowski to obtain high coupling efficiency in thin film waveguides. This paper presents a perturbation analysis of that grating coupler. The calculated results agree very well with the experimental data. The analysis indicates that the maximum efficiency will depend on the effective length of the evanescent tail of the guided wave mode in the gelatin. The longer the evanescent tail is, the higher will be the excitation efficiency; the maximum possible efficiency is, of course, 81%. The maximum efficiency does not depend significantly on the refractive index of the grating, or the slant angle of the grooves, or the thickness of the gelatin (provided that the thickness is larger than the length of the evanescent tail).

© 1973 Optical Society of America

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References

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  1. H. Kogelnik, T. P. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).
  2. K. Ogawa, W. S. C. Chang, B. Sopori, F. J. Rosenbaum, IEEE, J. of Quantum Electron. QE-9, 29 (1973).
    [CrossRef]
  3. D. Marcuse, Bell Syst. Tech. J. 48, 318 (1969).
  4. D. Marcuse, Light Transnssion Optics (Van Nostrand, Princeton, 1972).

1973

K. Ogawa, W. S. C. Chang, B. Sopori, F. J. Rosenbaum, IEEE, J. of Quantum Electron. QE-9, 29 (1973).
[CrossRef]

1970

H. Kogelnik, T. P. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).

1969

D. Marcuse, Bell Syst. Tech. J. 48, 318 (1969).

Chang, W. S. C.

K. Ogawa, W. S. C. Chang, B. Sopori, F. J. Rosenbaum, IEEE, J. of Quantum Electron. QE-9, 29 (1973).
[CrossRef]

Kogelnik, H.

H. Kogelnik, T. P. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).

Marcuse, D.

D. Marcuse, Bell Syst. Tech. J. 48, 318 (1969).

D. Marcuse, Light Transnssion Optics (Van Nostrand, Princeton, 1972).

Ogawa, K.

K. Ogawa, W. S. C. Chang, B. Sopori, F. J. Rosenbaum, IEEE, J. of Quantum Electron. QE-9, 29 (1973).
[CrossRef]

Rosenbaum, F. J.

K. Ogawa, W. S. C. Chang, B. Sopori, F. J. Rosenbaum, IEEE, J. of Quantum Electron. QE-9, 29 (1973).
[CrossRef]

Sopori, B.

K. Ogawa, W. S. C. Chang, B. Sopori, F. J. Rosenbaum, IEEE, J. of Quantum Electron. QE-9, 29 (1973).
[CrossRef]

Sosnowski, T. P.

H. Kogelnik, T. P. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).

Bell Syst. Tech. J.

H. Kogelnik, T. P. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).

D. Marcuse, Bell Syst. Tech. J. 48, 318 (1969).

IEEE, J. of Quantum Electron.

K. Ogawa, W. S. C. Chang, B. Sopori, F. J. Rosenbaum, IEEE, J. of Quantum Electron. QE-9, 29 (1973).
[CrossRef]

Other

D. Marcuse, Light Transnssion Optics (Van Nostrand, Princeton, 1972).

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Figures (4)

Fig. 1
Fig. 1

The holographic grating coupler.

Fig. 2
Fig. 2

Optimum coupling distance and attenuation rate of the TE0 mode.

Fig. 3
Fig. 3

Optimum coupling distance and attenuation rate of the TE0 mode.

Fig. 4
Fig. 4

Maximum coupling efficiency of the TE0 mode.

Tables (1)

Tables Icon

Table I Table of fa(x′), fs(x′), and fss(x′)

Equations (18)

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{ [ 2 / ( x 2 ) ] + [ 2 / ( z 2 ) ] + k 2 n 2 ( x ) } E y = k 2 Δ n g 2 ( x , z ) [ u ( x - t - T ) - u ( x - t ) ] · [ u ( z ) - u ( z - ξ L ) ] E y 0 ,
Δ n g 2 ( x , z ) = p D p exp ( - j 2 π p cos ϕ x / L - j 2 π p sin ϕ z / L ) ,
D p = ( n g 2 - n ¯ g 2 ) sin ( p π / 2 ) / ( p π ) for rectangular grating , D ± 1 = 1 4 ( n g 2 - n ¯ g 2 ) D p = O ( p ǂ ± 1 ) for sinusoidal grating .
α m = k 4 16 ω μ ( i { C a ( i ) 2 [ k 2 - ( β ( i ) ) 2 ] 1 / 2 + C s ( i ) 2 [ n s 2 k 2 - ( β ( i ) ) 2 ] 1 / 2 } + l C s ( l ) 2 [ n s 2 k 2 - ( β ( l ) ) 2 ] 1 / 2 ) ,
β ( i ) = k sin θ = β m - ( 2 π i sin ϕ / L ) , 0 < β ( i ) < k .
β ( l ) = n s k sin θ l = β m - ( 2 π l sin ϕ / L ) , k < β ( l ) < n s k .
C a ( i ) = D i t t + T f a ( x ) E m ( x ) exp [ ( + j ( 2 π i cos ϕ / L ) x ) d x ] ,
C s ( i ) = D i t t + T f s ( x ) E m ( x ) exp [ ( + j ( 2 π i cos ϕ / L ) x ) d x ] ,
C s ( l ) = D l t t + T f s s ( x ) E m ( x ) exp [ + j ( 2 π l cos ϕ / L ) x ) d x ] ,
E y I N C = A o f a ( x ) exp ( - j β o z ) for excitation from the air , β o < k ; E y I N C = A o f s ( x ) exp ( - j β o z ) for excitation from the substrate , β o < k ; and E y I N C = A o f s s ( x ) exp ( - j β o z ) for excitation from the substrate , k < β o < k n s .
β 0 = k sin θ 0 = β m - ( 2 π N sin ϕ / L ) for excitation from the air , β 0 = n s k sin θ 0 = β m - ( 2 π N sin ϕ / L ) for excitation from the substrate .
η = η o [ 1 - exp ( - α m ξ L / sin ϕ ) ] ( α m ξ L / sin ϕ ) ;
η o = k 4 8 ω μ α m S a ( N ) 2 ( k 2 - β o 2 ) 1 / 2 for excitation from the air , β o < k ; = k 4 8 ω μ α m S s ( N ) 2 / ( k 2 n s 2 - β o 2 ) 1 / 2 for excitation from the substrate , β o < k ; = k 4 8 ω μ α m S s s ( N ) 2 / ( k 2 n s 2 - β o 2 ) 1 / 2 for excitation from the substrate , k < β o < n s k ;
S a ( N ) = D N t t + T f a ( x ) E m ( x ) exp [ - j ( 2 π N cos ϕ / L ) x ] d x ;
S s ( N ) = D N t t + T f s ( x ) E m ( x ) exp [ - j ( 2 π N cos ϕ / L ) x ] d x ;
S s s ( N ) = D N t t + T f s s ( x ) E m ( x ) exp [ - j ( 2 π N cos ϕ / L ) x ] d x .
ξ L = ξ L o p = 1.25 sin ϕ / α m ,
η max = 0.407 η o .

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