Abstract

The problem of diffraction at a sinusoidally stratified dielectric grating is treated. The analysis of the reconstruction process from a hologram formed in a “thick” photographic emulsion leads to this problem, but it may also occur in other areas. A rigorous solution of this problem is presented for both polarizations of the electric wave. The solution is evaluated numerically for values of the different parameters that are typical for holograms. It is shown that the amplitude of the diffracted light has a maximum if the light is incident at the Bragg angle, a feature already observed experimentally. Results are given for different values of the period as well as the thickness of the grating and for both polarizations.

© 1966 Optical Society of America

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References

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  1. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962).
    [Crossref]
  2. K. S. Pennington and L. H. Lin, Appl. Phys. Letters 7, 56 (1965).
    [Crossref]
  3. T. Tamir, H. C. Wang, and A. A. Oliner, IEEE Trans. Microwave Theory Tech. MTT-12, 323 (1964).
    [Crossref]
  4. C. Yeh, K. F. Casey, and Z. A. Kaprielian, IEEE Trans. Microwave Theory Tech. MTT-13, 297 (1965).
    [Crossref]
  5. M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), 2nd ed.
  6. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, New York, 1940), 4th ed.
  7. R. W. Hamming, Numerical Methods for Scientists and Engineers (McGraw-Hill Book Co., Inc., New York, 1962).
  8. P. Penfield, J. Franklin Inst. 273, 107 (1962).
    [Crossref]
  9. A. Ralston and H. S. Wilf, Mathematical Methods for Digital Computers (John Wiley & Sons, Inc., New York, 1960).
  10. S. Ramo and J. R. Whinnery, Fields and Waves in Modern Radio (John Wiley & Sons, Inc., New York, 1960), 2nd ed.
  11. R. M. Fano, L. J. Chu, and R. B. Adler, Electromagnetic Fields, Energy, and Forces (John Wiley & Sons, Inc., New York, 1960).
  12. M. G. Cohen and E. I. Gordon, Bell System Tech. J. 44, 693 (1965), formula on p. 701 without the approximation sinx≈ x.
    [Crossref]

1965 (3)

K. S. Pennington and L. H. Lin, Appl. Phys. Letters 7, 56 (1965).
[Crossref]

C. Yeh, K. F. Casey, and Z. A. Kaprielian, IEEE Trans. Microwave Theory Tech. MTT-13, 297 (1965).
[Crossref]

M. G. Cohen and E. I. Gordon, Bell System Tech. J. 44, 693 (1965), formula on p. 701 without the approximation sinx≈ x.
[Crossref]

1964 (1)

T. Tamir, H. C. Wang, and A. A. Oliner, IEEE Trans. Microwave Theory Tech. MTT-12, 323 (1964).
[Crossref]

1962 (2)

Adler, R. B.

R. M. Fano, L. J. Chu, and R. B. Adler, Electromagnetic Fields, Energy, and Forces (John Wiley & Sons, Inc., New York, 1960).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), 2nd ed.

Casey, K. F.

C. Yeh, K. F. Casey, and Z. A. Kaprielian, IEEE Trans. Microwave Theory Tech. MTT-13, 297 (1965).
[Crossref]

Chu, L. J.

R. M. Fano, L. J. Chu, and R. B. Adler, Electromagnetic Fields, Energy, and Forces (John Wiley & Sons, Inc., New York, 1960).

Cohen, M. G.

M. G. Cohen and E. I. Gordon, Bell System Tech. J. 44, 693 (1965), formula on p. 701 without the approximation sinx≈ x.
[Crossref]

Fano, R. M.

R. M. Fano, L. J. Chu, and R. B. Adler, Electromagnetic Fields, Energy, and Forces (John Wiley & Sons, Inc., New York, 1960).

Gordon, E. I.

M. G. Cohen and E. I. Gordon, Bell System Tech. J. 44, 693 (1965), formula on p. 701 without the approximation sinx≈ x.
[Crossref]

Hamming, R. W.

R. W. Hamming, Numerical Methods for Scientists and Engineers (McGraw-Hill Book Co., Inc., New York, 1962).

Kaprielian, Z. A.

C. Yeh, K. F. Casey, and Z. A. Kaprielian, IEEE Trans. Microwave Theory Tech. MTT-13, 297 (1965).
[Crossref]

Leith, E. N.

Lin, L. H.

K. S. Pennington and L. H. Lin, Appl. Phys. Letters 7, 56 (1965).
[Crossref]

Oliner, A. A.

T. Tamir, H. C. Wang, and A. A. Oliner, IEEE Trans. Microwave Theory Tech. MTT-12, 323 (1964).
[Crossref]

Penfield, P.

P. Penfield, J. Franklin Inst. 273, 107 (1962).
[Crossref]

Pennington, K. S.

K. S. Pennington and L. H. Lin, Appl. Phys. Letters 7, 56 (1965).
[Crossref]

Ralston, A.

A. Ralston and H. S. Wilf, Mathematical Methods for Digital Computers (John Wiley & Sons, Inc., New York, 1960).

Ramo, S.

S. Ramo and J. R. Whinnery, Fields and Waves in Modern Radio (John Wiley & Sons, Inc., New York, 1960), 2nd ed.

Tamir, T.

T. Tamir, H. C. Wang, and A. A. Oliner, IEEE Trans. Microwave Theory Tech. MTT-12, 323 (1964).
[Crossref]

Upatnieks, J.

Wang, H. C.

T. Tamir, H. C. Wang, and A. A. Oliner, IEEE Trans. Microwave Theory Tech. MTT-12, 323 (1964).
[Crossref]

Watson, G. N.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, New York, 1940), 4th ed.

Whinnery, J. R.

S. Ramo and J. R. Whinnery, Fields and Waves in Modern Radio (John Wiley & Sons, Inc., New York, 1960), 2nd ed.

Whittaker, E. T.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, New York, 1940), 4th ed.

Wilf, H. S.

A. Ralston and H. S. Wilf, Mathematical Methods for Digital Computers (John Wiley & Sons, Inc., New York, 1960).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), 2nd ed.

Yeh, C.

C. Yeh, K. F. Casey, and Z. A. Kaprielian, IEEE Trans. Microwave Theory Tech. MTT-13, 297 (1965).
[Crossref]

Appl. Phys. Letters (1)

K. S. Pennington and L. H. Lin, Appl. Phys. Letters 7, 56 (1965).
[Crossref]

Bell System Tech. J. (1)

M. G. Cohen and E. I. Gordon, Bell System Tech. J. 44, 693 (1965), formula on p. 701 without the approximation sinx≈ x.
[Crossref]

IEEE Trans. Microwave Theory Tech. (2)

T. Tamir, H. C. Wang, and A. A. Oliner, IEEE Trans. Microwave Theory Tech. MTT-12, 323 (1964).
[Crossref]

C. Yeh, K. F. Casey, and Z. A. Kaprielian, IEEE Trans. Microwave Theory Tech. MTT-13, 297 (1965).
[Crossref]

J. Franklin Inst. (1)

P. Penfield, J. Franklin Inst. 273, 107 (1962).
[Crossref]

J. Opt. Soc. Am. (1)

Other (6)

A. Ralston and H. S. Wilf, Mathematical Methods for Digital Computers (John Wiley & Sons, Inc., New York, 1960).

S. Ramo and J. R. Whinnery, Fields and Waves in Modern Radio (John Wiley & Sons, Inc., New York, 1960), 2nd ed.

R. M. Fano, L. J. Chu, and R. B. Adler, Electromagnetic Fields, Energy, and Forces (John Wiley & Sons, Inc., New York, 1960).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), 2nd ed.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, New York, 1940), 4th ed.

R. W. Hamming, Numerical Methods for Scientists and Engineers (McGraw-Hill Book Co., Inc., New York, 1962).

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Figures (4)

Fig. 1
Fig. 1

Configuration for forming the grating.

Fig. 2
Fig. 2

Model used for computing diffraction.

Fig. 3
Fig. 3

Relative amplitude of the first-order diffracted wave as a function of angle of the incident light for the H mode and the following parameters: grating period = 0.6328 μ, d = 15 μ, a = 0.0035, r0 = 2.3225, λ = 0.6328 μ.

Fig. 4
Fig. 4

Relative amplitude of the first-order diffracted wave as a function of angle of the incident light for the H mode and the following parameters: grating period = 0.6328 μ, μ = 50 μ, a = 0.0035, r0 = 2.3225, λ = 0.6328 μ.

Tables (1)

Tables Icon

Table I Summary of results of computations for various parameters.

Equations (75)

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r ( x ) = r 0 ( 1 + a cos 2 π b x ) .
2 E μ 0 ( 2 E / t 2 ) + grad [ ( E / ) · grad ] = 0.
2 E y + k 2 r 0 ( 1 + a cos 2 π b x ) E y = 0 ,
E y ( x , z ) = Z ( z ) · X ( x ) ,
( d 2 Z / d z 2 ) / Z = α 2
( d 2 X / d x 2 ) + k 2 r 0 ( 1 + a cos 2 π b x ) X + α 2 X = 0 ,
X ( x ) = l = β l exp [ j 2 π ( ξ + l b ) x ] .
ξ = ( sin θ ) / λ ,
( D ) β = α 2 β .
d l , m = k 2 r 0 + 4 π 2 ( ξ + l b ) 2 for l = m ,
d l , m = 1 2 k 2 r 0 a for l = m + 1 or l = m 1 ,
d l , m = 0 for l > m + 1 or l < m 1.
L 1 l L 2 ,
Z ( z ) = A e ± α z ,
E y = n = L 1 L 2 { [ A n exp ( α n z ) + A n exp ( α n z ) ] l = L 1 L 2 β l , n × exp { j 2 π ( ξ + l b ) x } } .
E y 1 = E + 01 exp ( j 2 π ξ x ) exp [ j k ( 1 λ 2 ξ 2 ) 1 2 z ] + l = L 1 L 2 E l 1 exp { j 2 π ( ξ + l b ) x } × exp { j k [ 1 λ 2 ( ξ + l b ) 2 ] 1 2 z } ,
H x = ( 1 / j μ 0 ω ) ( E y / z ) ,
H x 1 = 1 μ 0 ω [ K 0 E + 01 exp ( j 2 π ξ x ) exp ( j K 0 z ) + l = L 1 L 2 E l 1 K l exp { j 2 π ( ξ + l b ) x } exp ( j K l z ) ] ,
K l = + k [ 1 λ 2 ( ξ + l b ) 2 ] 1 2 .
H x 2 = 1 j μ 0 ω n = L 1 L 2 { [ A n α n exp ( α n z ) A n α n exp ( α n z ) ] × l = L 1 L 2 β l , n exp { j 2 π ( ξ + l b ) x } } .
E y 3 = l = L 1 L 2 E l 3 exp { j 2 π ( ξ + l b ) x } exp ( j K l z )
H x 3 = 1 μ 0 ω [ l = L 1 L 2 K l E l 3 exp { j 2 π ( ξ + l b ) x } exp ( j K l z ) ] .
E + l 1 + E l 1 = n = L 1 L 2 ( A n + A n ) β l , n for l = 0
E l 1 = n = L 1 L 2 ( A n + A n ) β l , n for l 0 , L 1 l L 2 .
K l E + l 1 + K l E l 1 = j n = L 1 L 2 ( A n α n A n α n ) β l , n for l = 0
K l E l 1 = j n = L 1 L 2 ( A n α n A n α n ) β l , n for l 0 , L 1 l L 2 .
( M ) A + ( M ) A = R .
R l = 2 K 0 E + for l = 0 = 0 for l 0.
m l , n = ( K l + j α n ) β l , n ,
m l , n = ( K l j α n ) β l , n .
( N ) A + ( N ) A = 0 ,
n l , n = [ K l exp ( α n d ) j α n exp ( α n d ) ] β l , n ,
n l , n = [ K l exp ( α n d ) + j α n exp ( α n d ) ] β l , n .
( ( M ) ( M ) ( N ) ( N ) ) · ( A A ) = ( R 0 ) .
E l 3 exp ( j K l d ) = n = L 1 L 2 [ A n exp ( α n d ) + A n exp ( α n d ) ] β l , n .
| E l 3 | = { [ Re ( E l 3 ) ] 2 + [ Im ( E l 3 ) ] 2 } 1 2
2 H μ 0 ( 2 H / t 2 ) + grad × ( curl H ) = 0.
H = H y ( x , z ) e j ω t ,
2 H y + [ a 2 π b sin 2 π b x / ( 1 + a cos 2 π b x ) ] ( H y / x ) + r 0 ( 1 + a cos 2 π b x ) k 2 H y = 0.
( d 2 Z / d z 2 ) / Z = α 2
d 2 X / d x 2 + [ a 2 π b sin 2 π b x / ( 1 + a cos 2 π b x ) ] · ( d X / d x ) + r 0 k 2 ( 1 + α 2 / r 0 k 2 + a cos 2 π b x ) X = 0 ,
U ( x ) = X ( x ) / ( 1 + a cos 2 π b x ) 1 2 .
d 2 U / d x 2 + U · f ( x ) + α 2 U = 0 ,
f ( x ) = 2 π 2 a b 2 cos 2 π b x 1 + a cos 2 π b x + r 0 k 2 ( 1 + a cos 2 π b x ) a 2 3 π 2 b 2 sin 2 2 π b x ( 1 + a cos 2 π b x ) 2 .
f ( x ) = n = ϑ n exp ( 2 π j n b x ) .
U ( x ) = l = β l exp { j 2 π ( ξ + l b ) x } .
( D ) β α 2 β = 0.
d l , i = 4 π 2 ( ξ + l b ) 2 π 2 b 2 + π 2 b 2 ( 1 a 2 ) 1 2 r 0 k 2 for l = i ,
d l , i = a r 0 k 2 / 2 π 2 b 2 ( 4 τ 3 2 τ ) / ( τ 2 1 ) for i = l + 1 or i = l 1 ,
d l , i = π 2 b 2 [ ( 3 n + 1 ) τ n + 2 ( 3 n 1 ) τ n ] / ( τ 2 1 ) with n = | l i | for i l + 2 or i l 2 ,
τ = 1 / a + 1 / [ a ( 1 + a 2 ) 1 2 ] .
H y 2 = X ( x ) Z ( z ) = ( 1 + a cos 2 π b x ) 1 2 × n = L 1 L 2 { [ A n exp ( α n z ) + A n exp ( α n z ) ] · l = L 1 L 2 β l , n exp { j 2 π ( ξ + l b ) x } } ,
E x 2 = ( 1 / j ω 0 r ) ( H y / z ) ,
E x 2 = 1 j ω 0 r 0 ( 1 + a cos 2 π b x ) 1 2 × n = L 1 L 2 { [ A n α n exp ( α n z ) A n α n exp ( α n z ) ] × l = L 1 L 2 β l , n exp { j 2 π ( ξ + l b ) x } } .
H y 1 = H + 01 exp ( + j 2 π ξ x ) exp ( j K 0 z ) + l = L 1 L 2 H l 1 exp { j 2 π ( ξ + l b ) x } exp ( + j K l z ) ,
E x 1 = 1 ω 0 [ K 0 H + 01 exp ( j 2 π ξ x ) exp ( j K 0 z ) l = L 1 L 2 K l H l 1 exp { j 2 π ( ξ + l b ) x } exp ( j K l z ) ] .
H y 3 = l = L 1 L 2 H y 3 exp { j 2 π ( ξ + l b ) x } exp ( j K l z )
E x 3 = 1 ω 0 l = L 1 L 2 K l H l 3 × exp { j 2 π ( ξ + l b ) x } exp ( j K l z ) .
( 1 + a cos 2 π b x ) 1 2 = m = Γ m exp ( j 2 π b m x ) ,
1 ( 1 + a cos 2 π b x ) 1 2 = m = Λ m exp ( j 2 π b m x ) .
H y 2 = n = L 1 L 2 { [ A n exp ( α n z ) + A n exp ( α n z ) ] × l = L 1 L 2 γ l , n exp { j 2 π ( ξ + l b ) x }
γ l , n = L 1 l L 2 l Γ m β l + m , n .
E x 2 = j ω 0 0 r { n = L 1 L 2 [ A n α n exp ( α n z ) A n α n exp ( α n z ) ] × l = L 1 L 2 δ l , n exp { j 2 π ( ξ + l b ) x } } ,
δ l , n = m = L 1 l L 2 l Λ m · β l + m , n .
( ( M ) ( M ) ( N ) ( N ) ) · ( A n A n ) = ( R 0 ) ,
m l , n = K l γ l , n + ( j / r 0 ) α n δ l , n ,
m l , n = K l γ l , n ( j / r 0 ) α n δ l , n ,
n l , n = K l exp ( α n d ) γ l , n ( j / r 0 ) α n exp ( α n d ) δ l , n ,
n l , n = K l exp ( α n d ) γ l , n + ( j / r 0 ) α n exp ( α n d ) δ l , n ,
R l = 2 K 0 H + 01 for l = 0 ,
R l = 0 for l 0.
n = L 1 L 2 [ A n exp ( α n d ) + A n exp ( α n d ) ] γ l , n = H l 3 exp ( j K l d ) L 1 l L 2 .
H l 1 = n = L 1 L 2 ( A n + A n ) γ l , n l 0 , L 1 l L 2 .
E 1 d cos θ 2 = E 2 d cos θ 1 .
I = sin 2 { 1 4 [ a ( r 0 ) 1 2 ] ( 2 π / λ ) ( d / cos θ ) } ,