Abstract

A thick hologram grating with modulation that decreases along the direction perpendicular to the grating vector has been analyzed using coupled-wave theory developed by Kogelnik. General solutions of two predominant waves, undiffracted and first-order diffracted waves, are given for transmission and reflection types of hologram gratings. Calculated efficiency of diffraction for some special cases is compared with that for the uniform grating. The influence of the attenuation of grating modulation upon the validity of the theory when applied to the Bragg reflection is also discussed.

© 1973 Optical Society of America

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References

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  1. E. N. Leith, A. Kozma, J. Upatnieks, J. Marks, and N. Massey, Appl. Opt. 5, 1303 (1966).
    [CrossRef] [PubMed]
  2. C. B. Burckhardt, J. Opt. Soc. Am. 56, 1502 (1966);J. Opt. Soc. Am. 57, 601 (1967).
    [CrossRef]
  3. D. Gabor and G. W. Stroke, Proc. R. Soc. (Lond.) A304, 275 (1968).
    [CrossRef]
  4. H. Kogelnik, in Proceedings of the Symposium on Modern Optics, edited by J. Fox (Polytechnic Press, Brooklyn, 1967), pp. 605–617.
  5. H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
    [CrossRef]
  6. P. Phariseau, Proc. Indian Acad. Sci. A44, 165 (1956).
  7. F. S. Chen, J. T. LaMacchia, and D. B. Fraser, Appl. Phys. Lett. 13, 223 (1968).
    [CrossRef]
  8. G. E. Peterson, A. M. Glass, and T. J. Negran, Appl. Phys. Lett. 19, 130 (1971).
    [CrossRef]
  9. J. J. Amodei, W. Phillips, and D. L. Staebler, Appl. Opt. 11, 390 (1972).
    [CrossRef] [PubMed]
  10. See, for instance, E. Kamke, Differentialgleichungen, Lösungs-methoden und Lösungen (Chelsea, New York, 1959), pp. 437–440.
  11. N. Uchida, Japan. J. Appl. Phys. 8, 935 (1969).
    [CrossRef]
  12. C. F. Quate, C. D. W. Wilkinson, and D. K. Winslow, Proc. Inst. Electr. Eng. 53, 1604 (1965).
    [CrossRef]
  13. G. W. Willard, J. Acoust. Soc. Am. 21, 101 (1949).
    [CrossRef]
  14. W. R. Klein and B. D. Cook, IEEE Trans. Sonics Ultrason. 14, 123 (1967).
    [CrossRef]

1972 (1)

1971 (1)

G. E. Peterson, A. M. Glass, and T. J. Negran, Appl. Phys. Lett. 19, 130 (1971).
[CrossRef]

1969 (2)

N. Uchida, Japan. J. Appl. Phys. 8, 935 (1969).
[CrossRef]

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
[CrossRef]

1968 (2)

D. Gabor and G. W. Stroke, Proc. R. Soc. (Lond.) A304, 275 (1968).
[CrossRef]

F. S. Chen, J. T. LaMacchia, and D. B. Fraser, Appl. Phys. Lett. 13, 223 (1968).
[CrossRef]

1967 (1)

W. R. Klein and B. D. Cook, IEEE Trans. Sonics Ultrason. 14, 123 (1967).
[CrossRef]

1966 (2)

1965 (1)

C. F. Quate, C. D. W. Wilkinson, and D. K. Winslow, Proc. Inst. Electr. Eng. 53, 1604 (1965).
[CrossRef]

1956 (1)

P. Phariseau, Proc. Indian Acad. Sci. A44, 165 (1956).

1949 (1)

G. W. Willard, J. Acoust. Soc. Am. 21, 101 (1949).
[CrossRef]

Amodei, J. J.

Burckhardt, C. B.

Chen, F. S.

F. S. Chen, J. T. LaMacchia, and D. B. Fraser, Appl. Phys. Lett. 13, 223 (1968).
[CrossRef]

Cook, B. D.

W. R. Klein and B. D. Cook, IEEE Trans. Sonics Ultrason. 14, 123 (1967).
[CrossRef]

Fraser, D. B.

F. S. Chen, J. T. LaMacchia, and D. B. Fraser, Appl. Phys. Lett. 13, 223 (1968).
[CrossRef]

Gabor, D.

D. Gabor and G. W. Stroke, Proc. R. Soc. (Lond.) A304, 275 (1968).
[CrossRef]

Glass, A. M.

G. E. Peterson, A. M. Glass, and T. J. Negran, Appl. Phys. Lett. 19, 130 (1971).
[CrossRef]

Kamke, E.

See, for instance, E. Kamke, Differentialgleichungen, Lösungs-methoden und Lösungen (Chelsea, New York, 1959), pp. 437–440.

Klein, W. R.

W. R. Klein and B. D. Cook, IEEE Trans. Sonics Ultrason. 14, 123 (1967).
[CrossRef]

Kogelnik, H.

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
[CrossRef]

H. Kogelnik, in Proceedings of the Symposium on Modern Optics, edited by J. Fox (Polytechnic Press, Brooklyn, 1967), pp. 605–617.

Kozma, A.

LaMacchia, J. T.

F. S. Chen, J. T. LaMacchia, and D. B. Fraser, Appl. Phys. Lett. 13, 223 (1968).
[CrossRef]

Leith, E. N.

Marks, J.

Massey, N.

Negran, T. J.

G. E. Peterson, A. M. Glass, and T. J. Negran, Appl. Phys. Lett. 19, 130 (1971).
[CrossRef]

Peterson, G. E.

G. E. Peterson, A. M. Glass, and T. J. Negran, Appl. Phys. Lett. 19, 130 (1971).
[CrossRef]

Phariseau, P.

P. Phariseau, Proc. Indian Acad. Sci. A44, 165 (1956).

Phillips, W.

Quate, C. F.

C. F. Quate, C. D. W. Wilkinson, and D. K. Winslow, Proc. Inst. Electr. Eng. 53, 1604 (1965).
[CrossRef]

Staebler, D. L.

Stroke, G. W.

D. Gabor and G. W. Stroke, Proc. R. Soc. (Lond.) A304, 275 (1968).
[CrossRef]

Uchida, N.

N. Uchida, Japan. J. Appl. Phys. 8, 935 (1969).
[CrossRef]

Upatnieks, J.

Wilkinson, C. D. W.

C. F. Quate, C. D. W. Wilkinson, and D. K. Winslow, Proc. Inst. Electr. Eng. 53, 1604 (1965).
[CrossRef]

Willard, G. W.

G. W. Willard, J. Acoust. Soc. Am. 21, 101 (1949).
[CrossRef]

Winslow, D. K.

C. F. Quate, C. D. W. Wilkinson, and D. K. Winslow, Proc. Inst. Electr. Eng. 53, 1604 (1965).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

F. S. Chen, J. T. LaMacchia, and D. B. Fraser, Appl. Phys. Lett. 13, 223 (1968).
[CrossRef]

G. E. Peterson, A. M. Glass, and T. J. Negran, Appl. Phys. Lett. 19, 130 (1971).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
[CrossRef]

IEEE Trans. Sonics Ultrason. (1)

W. R. Klein and B. D. Cook, IEEE Trans. Sonics Ultrason. 14, 123 (1967).
[CrossRef]

J. Acoust. Soc. Am. (1)

G. W. Willard, J. Acoust. Soc. Am. 21, 101 (1949).
[CrossRef]

J. Opt. Soc. Am. (1)

Japan. J. Appl. Phys. (1)

N. Uchida, Japan. J. Appl. Phys. 8, 935 (1969).
[CrossRef]

Proc. Indian Acad. Sci. (1)

P. Phariseau, Proc. Indian Acad. Sci. A44, 165 (1956).

Proc. Inst. Electr. Eng. (1)

C. F. Quate, C. D. W. Wilkinson, and D. K. Winslow, Proc. Inst. Electr. Eng. 53, 1604 (1965).
[CrossRef]

Proc. R. Soc. (Lond.) (1)

D. Gabor and G. W. Stroke, Proc. R. Soc. (Lond.) A304, 275 (1968).
[CrossRef]

Other (2)

H. Kogelnik, in Proceedings of the Symposium on Modern Optics, edited by J. Fox (Polytechnic Press, Brooklyn, 1967), pp. 605–617.

See, for instance, E. Kamke, Differentialgleichungen, Lösungs-methoden und Lösungen (Chelsea, New York, 1959), pp. 437–440.

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Figures (8)

F. 1
F. 1

Model of a grating used for the analysis.

F. 2
F. 2

Contours of constant F as functions of θ0 and ϕ at the Bragg angle of incidence and in the case for which αg = 5. Case I.

F. 3
F. 3

Contours of constant F as functions of θ0 and ϕ at Bragg angle (αg = 5). Case II.

F. 4
F. 4

Angular sensitivity of the normalized diffraction efficiency for various values of aL. These curves are applicable to both transmission and reflection types of the hologram gratings.

F. 5
F. 5

Diffraction efficiency of lossless dielectric grating as a function of modulation for various values of αgL. ——Transmission hologram.

F. 6
F. 6

Square root of diffraction efficiency of absorption grating as a function of modulation D1 for various values of αgL and modulation depth D1/D0. ——Transmission hologram.

F. 7
F. 7

Diffraction efficiency of lossless dielectric grating as a function of bRL for various values of aL. ——Reflection hologram.

F. 8
F. 8

Normalized irradiance ratio of the second-order diffracted light to the first order as a function of Q′ for various values of G(=αgL).

Equations (50)

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c 0 d A 0 d z + α A 0 = j κ 0 ( a 0 a 1 ) e α g z / sin | ϕ | e j Δ k z z A 1
c 1 d A 1 d z + α A 1 = j κ 0 ( a 0 a 1 ) e α g z / sin | ϕ | e j Δ k z z A 0 ,
c 0 = cos θ , c 1 = cos ( 2 ϕ 2 θ 0 + θ ) , Δ k z = K [ cos ( ϕ θ 0 + θ ) cos ϕ ] ,
κ 0 = 1 2 ( k Δ n 0 + j Δ α 0 ) .
k 1 = k 0 K Δ k ,
d 2 A 1 d z 2 + ( a + d ) d A 1 d z + ( b 2 e 2 a z + f ) A 1 = 0 ,
a = α g / sin | ϕ | , b = κ 0 ( a 0 a 1 ) / ( c 0 c 1 ) 1 2 , d = ( 1 / c 0 + 1 / c 1 ) α j Δ k z ,
f = ( α / c 1 ) ( a + α / c 0 j Δ k z ) .
ξ ( ζ ) = A 1 ζ and ζ = e a z ,
ξ = ζ ( a d ) / 2 a [ C 1 J ν ( b a ζ 1 ) + C 2 J ν ( b a ζ 1 ) ] ,
ν = ( a + g ) / 2 a
g = ( 1 / c 0 1 / c 1 ) α j Δ k z .
A 0 ( z ) = b π 2 a sin ν π e ( a + d * ) z / 2 [ J ν + 1 ( b a e a z ) J ν ( b a ) + J ν 1 ( b a e a z ) J ν ( b a ) ] ,
A 1 ( z ) = j ( c 0 / c 1 ) 1 2 b π 2 a sin ν π e ( a + d ) z / 2 × [ J ν ( b a e a z ) J ν ( b a ) J ν ( b a e a z ) J ν ( b a ) ] ,
A 0 ( z ) = b π 2 a sin μ π e a L + ( a d * ) z / 2 × [ J μ + 1 ( b a e a ( L z ) ) J μ ( b a e a L ) + J μ 1 ( b a e a ( L z ) ) J μ ( b a e a L ) ]
A 1 ( z ) = j ( c 0 / c 1 ) 1 2 b π 2 a sin μ π e a L + ( a d ) z / 2 × [ J μ ( b a e a L ) J μ ( b a e a ( L z ) ) J μ ( b a e a L ) J μ ( b a e a ( L z ) ) ] ,
μ = ( g a ) / 2 a .
A 0 ( z ) = e ( a + d * ) z / 2 [ J ν + 1 ( b a e a z ) J ν ( b a e a L ) + J ν 1 ( b a e a z ) J ν ( b a e a L ) ] / B ,
A 1 ( z ) = j ( c 0 / c 1 ) 1 2 e ( a + d ) z / 2 × [ J ν ( b a e a z ) J ν ( b a e a L ) J ν ( b a e a z ) J ν ( b a e a L ) ] / B ,
B = J ν + 1 ( b a ) J ν ( b a e a L ) + J ν 1 ( b a ) J ν ( b a e a L ) .
A 0 ( L ) = 2 a sin ( ν π ) e ( a d * ) L / 2 / b π B
A 1 ( 0 ) = j ( c 0 / c 1 ) 1 2 [ J ν ( b a ) J ν ( b a e a L ) J ν ( b a ) J ν ( b a e a L ) ] / B .
A 0 ( z ) = e ( a d * ) z / 2 [ J μ ( b a ) J μ 1 ( b a e a ( L z ) ) + J μ ( b a ) J μ + 1 ( b a e a ( L z ) ) ] / B
A 1 ( z ) = j ( c 0 / c 1 ) 1 2 e ( a d ) z / 2 × [ J μ ( b a ) J μ ( b a e a ( L z ) ) J μ ( b a ) J μ ( b a e a ( L z ) ) ] / B .
A 1 ( z ) = j ( c 0 / c 1 ) 1 2 bz e ( a + d ) z / 2 sinh a ν z a ν z
A 1 ( z ) = j ( c 0 / c 1 ) 1 2 bz e a L e ( a d ) z / 2 sinh a μ z a μ z .
η = ( c 1 / c 0 ) A 1 A 1 * ,
η = ( b b * L 2 ) exp { [ a + ( 1 / c 0 + 1 / c 1 ) α ] L } × sinh 2 { [ a ± ( 1 / c 0 1 / c 1 ) α ] L / 2 } + sin 2 ( Δ k z L / 2 ) { [ a ± ( 1 / c 0 1 / c 1 ) α ] L / 2 } 2 + ( Δ k z L / 2 ) 2 ,
F = α g / ( α sin | ϕ | ) ± ( 1 / c 0 1 / c 1 ) ,
η = e 2 α L / cos θ B { sin 2 [ b R α g ( 1 e α g L ) ] + sinh 2 [ b I α g ( 1 e α g L ) ] } ,
b R = k Δ n 0 ( a 0 a 1 ) / 2 cos θ B ,
b I = Δ α 0 ( a 0 a 1 ) / 2 cos θ B ,
D 0 = α L / cos θ B
D 1 = Δ α 0 ( a 0 a 1 ) L / cos θ B .
η = ( b b * L 2 ) exp { [ a + ( 1 / c 0 1 / c 1 ) α ] L } × sinh 2 { [ a ± ( 1 / c 0 1 / c 1 ) α ] L / 2 } + sin 2 ( Δ k z L / 2 ) { [ a ± ( 1 / c 0 1 / c 1 ) α ] L / 2 } 2 + ( Δ k z L / 2 ) 2 ,
A 0 ( L ) = sech [ b R a ( 1 e a L ) ]
A 1 ( 0 ) = j ( c 0 / | c 1 | ) 1 2 tanh [ b R a ( 1 e a L ) ] ,
b R = k Δ n 0 ( a 0 a 1 ) / 2 ( c 0 | c 1 | ) 1 2 .
Q = K 2 L / n k c 0
k 2 = k 0 2 K Δ k ,
c 2 d A 2 d z + α A 2 = j κ 0 ( a 1 a 2 ) e α g z / sin | ϕ | e j ( Δ k z Δ k z ) z A 1 ,
c 2 = cos ( 2 ϕ 2 θ 0 + θ )
Δ k z = 2 K [ cos ( ϕ θ 0 + θ ) cos ϕ ] .
A 1 ( z ) = j b R ( 1 e α g z ) / α g ,
A 2 ( L ) = ( b R L ) 2 G [ 1 e 2 G + j Q 2 G j Q 1 e G + j Q G j Q ] ,
b R 2 = k 2 Δ n 0 2 ( a 0 a 1 ) ( a 1 a 2 ) / 4 cos θ B cos ( 2 θ B θ B ) , G = α g L , Q = Q cos θ B ( 2 cos θ B cos θ B ) ,
ρ = A 2 A 2 * A 1 A 1 * = ( b R 2 / b R ) 2 L 2 Q 2 ( 1 + 4 G 2 / Q 2 ) ( 1 + G 2 / Q 2 ) × [ R 2 S sin Q Q + T sin 2 ( Q / 2 ) ( Q / 2 ) 2 ] ,
R = e 2 G + G 2 ( 1 e G ) 2 / Q 2 , S = G e G / ( 1 e G ) ,
T = G 2 e G ( 2 e G ) / ( 1 e G ) 2 .
ρ / ρ 0 ~ 1 2 G ;