Abstract

The problem of diffraction of optical beams with arbitrary profiles by a periodically modulated layer is studied for incidence at normal or at the first Bragg angle. It is shown that the far-field patterns of the nth diffracted order of the transmitted and reflected waves are simply the algebraic multiplications of the angular spectral amplitude of the beam profile and the transmission and reflection coefficients for the nth-order diffracted plane wave. Numerical results are illustrated for six different beam profiles.

© 1980 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), Chap. 12, p. 579.
  2. R. S. Chu and T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 485–504 (1970).
  3. W. Klein and B. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
    [Crossref]
  4. R. S. Chu and J. A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).
  5. J. A. Kong, “Second-order coupled-mode equations for spatially periodic media,” J. Opt. Soc. Am. 67, 825–829 (1977).
    [Crossref]
  6. D. L. Lee, An Integral Equation Formulation for Wave Propagation in Spatially-Periodic Media, Ph.D. dissertation, Massachusetts Institute of Technology, 1977 (unpublished).
  7. R. S. Chu and T. Tamir, “Bragg diffraction of Gaussian beams by periodically modulated media,” J. Opt. Soc. Am. 66, 220–226 (1976).
    [Crossref]
  8. R. S. Chu and T. Tamir, “Diffraction of Gaussian beams by periodically modulated media for incidence close to a Bragg angle,” J. Opt. Soc. Am. 66, 1438–1440 (1976).
    [Crossref]
  9. R. S. Chu, J. A. Kong, and T. Tamir, “Diffraction of Gaussian beams by a periodically modulated layer,” J. Opt. Soc. Am. 67, 1555–1561 (1977).
    [Crossref]
  10. M. R. B. Forshaw, “Diffraction of a narrow laser beam by a thick hologram: Experimental results,” Opt. Commun. 12, 279–281 (1974).
    [Crossref]
  11. L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960), Sec. 8, p. 100.

1977 (3)

1976 (2)

1974 (1)

M. R. B. Forshaw, “Diffraction of a narrow laser beam by a thick hologram: Experimental results,” Opt. Commun. 12, 279–281 (1974).
[Crossref]

1970 (1)

R. S. Chu and T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 485–504 (1970).

1967 (1)

W. Klein and B. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), Chap. 12, p. 579.

Brekhovskikh, L. M.

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960), Sec. 8, p. 100.

Chu, R. S.

Cook, B.

W. Klein and B. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[Crossref]

Forshaw, M. R. B.

M. R. B. Forshaw, “Diffraction of a narrow laser beam by a thick hologram: Experimental results,” Opt. Commun. 12, 279–281 (1974).
[Crossref]

Klein, W.

W. Klein and B. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[Crossref]

Kong, J. A.

Lee, D. L.

D. L. Lee, An Integral Equation Formulation for Wave Propagation in Spatially-Periodic Media, Ph.D. dissertation, Massachusetts Institute of Technology, 1977 (unpublished).

Tamir, T.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), Chap. 12, p. 579.

IEEE Trans. Microwave Theory Tech. (2)

R. S. Chu and T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 485–504 (1970).

R. S. Chu and J. A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).

IEEE Trans. Sonics Ultrason. (1)

W. Klein and B. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[Crossref]

J. Opt. Soc. Am. (4)

Opt. Commun. (1)

M. R. B. Forshaw, “Diffraction of a narrow laser beam by a thick hologram: Experimental results,” Opt. Commun. 12, 279–281 (1974).
[Crossref]

Other (3)

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960), Sec. 8, p. 100.

D. L. Lee, An Integral Equation Formulation for Wave Propagation in Spatially-Periodic Media, Ph.D. dissertation, Massachusetts Institute of Technology, 1977 (unpublished).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), Chap. 12, p. 579.

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Figures (6)

FIG. 1
FIG. 1

Geometrical configuration of the problem.

FIG. 2
FIG. 2

Transmitted field amplitudes for a normally incident Gaussian beam. 1 = 2 = 3 = 1.0, λ = 0.6328 μm, d = 6.328 μm, k0L = 700, W ¯ 0 = π W 0 / d = 500, Q = 7.

FIG. 3
FIG. 3

Reflected field amplitudes for a normally incident Gaussian beam. 1 = 2 = 3 = 1.0, λ = 0.6328 μm, d= 6.328 μm, k0L = 700, W ¯ 0 = 500, Q = 7.

FIG. 4
FIG. 4

Amplitudes of the first-order transmitted beams for six different beam profiles. 1 = 2 = 3 = 1.0, λ = 0.6328 μm, d = 6.328 μm, k0L = 700, M = 5 × 10−3, W 0 ¯ = 500, Q = 7.

FIG. 5
FIG. 5

Zeroth-order transmitted beams for L = D1/2, M = 1 × 10−4, 1 = 1.0, 2 = 2.25, 3 = 3.0, λ = 0.6328 μm, d= 1.2656 μm, D1 = 8.3193 mm, W 0 ¯ = 500 , θ 0 ( 1 ) = 14.4775 ° , θ 0 ( 3 ) = 8.2989 °.

FIG. 6
FIG. 6

Bragg-centered beams for L = D1/2, M = 1 × 10−4, 1 = 1.0, 2 = 2.25, 3 = 3.0 λ = 0.6328 μm, d = 1.2656 μm, D1 = 8.3193 mm, W 0 ¯ = 500 , θ 0 ( 1 ) = 14.4775 ° and θ 0 ( 3 ) = 8.2989 °.

Tables (1)

Tables Icon

TABLE I Beam profiles F(z) and their corresponding angular spectral functions G(β0).

Equations (58)

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( z ) = 2 ( 1 + M cos 2 π z / d ) ,
E inc ( x , z ) = G ( β 0 ) exp i ( ξ 0 ( 1 ) x + β 0 z ) d β 0 ,
β 0 = ( 2 π / λ ) 1 sin θ
ξ 0 ( 1 ) = ( 2 π / λ ) 1 cos θ = ( 2 π / λ ) 2 1 β 0 2 .
G ( β 0 ) = 1 2 π E inc ( 0 , z ) e i β 0 z d z .
E inc ( 0 , z ) = F ( z ) e i b z ,
b ( 2 π / λ ) 1 sin θ 0 ( 1 ) = ( 2 π / λ ) 3 sin θ 0 ( 3 )
G ( β 0 ) = 1 2 π F ( z ) e i ( β 0 b ) z d z .
exp i ( ξ 0 ( 1 ) x + β 0 z )
E t n ( x , z ) = G ( β 0 ) T n ( β 0 ) exp i [ ξ 0 ( 3 ) ( x L ) + β n z ] d β 0 , x L
β n = β 0 + n 2 π / d
ξ n ( 3 ) = ( 2 π / λ ) 2 3 β n 2 .
E r n ( x , z ) = G ( β 0 ) R n ( β 0 ) exp i ( ξ n ( 1 ) x + β n z ) d β 0 , x 0
ξ n ( 1 ) = ( 2 π / λ ) 2 1 β n 2 .
P t n ( θ ) = E t n ( L , z ) exp { i [ ( 2 π / λ ) 3 sin θ ] z } d z .
P t n ( θ ) = G ( β 0 ) T n ( β 0 ) × exp [ i ( β 0 + n 2 π d 2 π λ 3 sin θ ) z ] d β 0 d z = G ( β 0 ) T n ( β 0 ) × { exp [ i ( β 0 + n 2 π d 2 π λ 3 sin θ ) z ] d z } d β 0 = 2 π G ( 2 π λ 3 sin θ n 2 π d ) T n ( 2 π λ 3 sin θ n 2 π d ) .
P r n ( θ ) = E r n ( 0 , z ) exp [ i ( 2 π λ 1 sin θ ) z ] d z = G ( β 0 ) R n ( β 0 ) × { exp [ i ( β 0 + n 2 π d 2 π λ 1 sin θ ) z ] d z } d β 0 = 2 π G ( 2 π λ 1 sin θ n 2 π d ) × R n ( 2 π λ 1 sin θ n 2 π d ) .
P r 0 ( θ ) = 2 π G ( 2 π λ 1 sin θ ) R 0 ( 2 π λ 1 sin θ ) ,
P t 0 ( θ ) = 2 π G ( 2 π λ 3 sin θ ) T 0 ( 2 π λ 3 sin θ ) .
P r n ( Δ θ ) 2 π G ( 2 π λ 1 sin Δ θ cos θ n ( 1 ) ) × R n ( 2 π λ 1 sin Δ θ cos θ n ( 1 ) ) ,
P t n ( Δ θ ) 2 π G ( 2 π λ 3 sin Δ θ cos θ n ( 3 ) ) × T n ( 2 π λ 3 sin Δ θ cos θ n ( 3 ) ) ,
θ n ( 1 ) sin 1 ( sin θ 0 ( 1 ) + n λ d 1 1 ) = sin 1 ( n λ d 1 1 ) ,
θ n ( 3 ) sin 1 ( sin θ 0 ( 3 ) + n λ d 1 3 ) = sin 1 ( n λ d 1 3 ) ,
δ n 0 + R n = α = 1 3 [ ( m α 0 ) + D α n ( 1 + e i k ˜ α L ) + ( m α 0 ) D α n ( 1 e i k ˜ α L ) ] ,
T n = e i k t n L = α = 1 3 [ ( m α 0 ) + D α n ( 1 + e i k ˜ α L ) ( m α 0 ) D α n ( 1 e i k ˜ α L ) ] , n = 0 , ± 1
D α n 2 ( k ˜ α 2 k ¯ x n 2 ) / k 1 2 ,
k 1 = ( 2 π / λ ) 1 = k 0 1 , k 0 = 2 π / λ ,
k ¯ x n ξ n 2 + k 1 2 ( 2 1 ) / 1 ,
k t n ξ n 2 + k 1 2 ( 3 1 ) / 1 ,
ξ n = k 0 2 1 β n 2 ,
β n = β 0 + n 2 π / d , β 0 = ( 2 π / λ ) 1 sin θ ,
( k ˜ α 2 ) 3 ( k ¯ x 1 2 + k ¯ x 0 2 + k ¯ x 1 2 ) ( k ˜ α 2 ) 2 + [ k ¯ x 0 2 k ¯ x 1 2 + k ¯ x 1 2 ( k ¯ x 1 2 + k ¯ x 0 2 ) 2 ( k 0 2 2 M / 2 ) 2 ] ( k ˜ α 2 ) k ¯ x 0 2 k ¯ x 1 2 k ¯ x 1 2 + ( 1 / 2 ) k 0 2 2 M ( k ¯ x 1 2 + k ¯ x 1 2 ) = 0
( m 1 0 ) ± = ( ξ 0 / Δ ± ) [ A ± ( 1 , 2 ) A ± ( 1 , 3 ) D 2 1 D 3 1 A ± ( 1 , 2 ) A ± ( 1 , 3 ) D 2 1 D 3 1 ] ,
( m 2 0 ) ± = ( ξ 0 / Δ ± ) [ A ± ( 1 , 1 ) A ± ( 1 , 3 ) D 1 1 D 3 1 A ± ( 1 , 1 ) A ± ( 1 , 3 ) D 1 1 D 3 1 ] ,
( m 3 0 ) ± = ( ξ 0 / Δ ± ) [ A ± ( 1 , 1 ) A ± ( 1 , 2 ) D 1 1 D 2 1 A ± ( 1 , 1 ) A ± ( 1 , 2 ) D 1 1 D 2 1 ] ,
A ± ( n , α ) = ( ξ n + k ˜ α ) ± ( ξ n k ˜ α ) e i k ˜ α L ,
Δ ± = A ± ( 1 , 1 ) D 1 1 [ A ± ( 0 , 2 ) A ± ( 1 , 3 ) D 3 1 A ± ( 0 , 3 ) A ± ( 1 , 2 ) D 2 1 ] A ± ( 1 , 2 ) D 2 1 [ A ± ( 0 , 1 ) A ± ( 1 , 3 ) D 3 1 A ± ( 0 , 3 ) A ± ( 1 , 1 ) D 1 1 ] + A ± ( 1 , 3 ) D 3 1 [ A ± ( 0 , 1 ) A ± ( 1 , 2 ) D 2 1 A ± ( 0 , 2 ) A ± ( 1 , 1 ) D 1 1 ] .
Q = 2 π λ L / 2 d 2 = 7.
b ( 2 π / λ ) 1 sin θ B ( 1 ) = ( 2 π / λ ) 3 sin θ B ( 3 ) = π / d .
P t 0 ( θ ) = 2 π G ( 2 π λ 3 sin θ ) T 0 ( 2 π λ 3 sin θ ) ,
P t 1 ( θ ) = 2 π G ( 2 π λ 3 sin θ + 2 π d ) T 1 ( 2 π λ 3 sin θ + 2 π d ) ,
T 0 = 4 k 1 x a ( α 2 α 1 ) ( α 1 A b b α 2 B b b ) exp ( i k 3 x a L ) / Det ,
T 1 = 4 k 1 x a ( α 2 α 1 ) ( A b a B b a ) exp ( i k 3 x b L ) / Det ,
Det = ( α 2 A a a α 1 B a a ) ( α 1 A b b α 2 B b b ) α 1 α 2 ( A a b B a b ) ( A b a B b a )
α 1 = 1 M k 2 2 { β 1 2 β 0 2 + [ ( β 1 2 β 0 2 ) 2 + ( M k 0 2 2 ) 2 ] 1 / 2 } ,
α 2 = 1 M k 2 2 { β 1 2 β 0 2 [ ( β 1 2 β 0 2 ) 2 + ( M k 0 2 2 ) 2 ] 1 / 2 } ,
A ρ σ = k 2 x a ( 1 + k 1 x ρ k 2 x a ) ( 1 + k 3 x σ k 2 x a ) × [ exp ( i k 2 x a L ) R 21 a ρ R 23 a σ exp ( i k 2 x a L ) ] ,
B ρ σ = k 2 x b ( 1 + k 1 x ρ k 2 x b ) ( 1 + k 3 x σ k 2 x b ) × [ exp ( i k 2 x b L ) R 21 b ρ R 23 b σ exp ( i k 2 x b L ) ] ,
R i j ρ σ = k i x ρ k j x σ k i x ρ + k j x σ ρ , σ = a , b , i , j = 1 , 2 , 3 ,
k 1 x a = k 0 1 cos θ = ( 2 π / λ ) 2 1 β 0 2 = ξ 0 ( 1 ) ,
k 1 x b = ( k 0 2 1 β 1 2 ) 1 / 2 = ξ 1 ( 1 ) ,
k 2 x a = { [ 1 ( 1 2 ) M α 1 ] k 0 2 2 β 0 2 } 1 / 2 ,
k 2 x b = { [ 1 ( 1 2 ) M α 2 ] k 0 2 2 β 0 2 } 1 / 2 ,
k 3 x a = ( k 0 2 3 β 0 2 ) 1 / 2 = ξ 0 ( 3 ) ,
k 3 x b = ( k 0 2 3 β 1 2 ) 1 / 2 = ξ 1 ( 3 ) .
D 1 = 2 d q cot θ ¯ B ( 2 ) = 2 d q ξ ¯ 0 d π ,
q = 2 ( d / λ ) 2 M 2 , θ ¯ B ( 2 ) = sin 1 ( λ 2 d 1 2 ) ,
ξ ¯ 0 = ( 2 π / λ ) 2 2 β 0 2 .