Abstract

The diffraction characteristics of phase, absorption, and mixed transmission gratings of arbitrary thickness, modulation, and shape are treated. Single-wave, two-wave, and multiwave diffraction regimes are quantified and discussed. Boundaries between these regimes and their relationships to elementary “thin” and “thick” grating theories are presented.

© 1978 Optical Society of America

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References

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  1. R. Magnusson and T. K. Gaylord, “Analysis of multiwave diffraction of thick gratings,” J. Opt. Soc. Am. 67, 1165–1170 (1977).
    [Crossref]
  2. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [Crossref]
  3. L. Bergstein and D. Kermisch, “Image storage and reconstruction in volume holography,” Proc. Symp. Modern Opt. 17, 655–680 (1967).
  4. R. Alferness, “Analysis of propagation at the second-order Bragg angle of a thick holographic grating,” J. Opt. Soc. Am. 66, 353–362 (1976).
    [Crossref]
  5. R. S. Chu and T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Techn. MTT-18, 486–504 (1970).
  6. S. F. Su and T. K. Gaylord, “Calculation of arbitrary-order diffraction efficiencies of thick gratings with arbitrary grating shape,” J. Opt. Soc. Am. 65, 59–64 (1975).
    [Crossref]
  7. H. M. Smith, “Effect of emulsion thickness on the diffraction efficiency of amplitude holograms,” J. Opt. Soc. Am. 62, 802–806 (1972).
    [Crossref]
  8. W. R. Klein, B. D. Cook, and W. G. Mayer, “Light diffraction by ultrasonic gratings,” Acustica 15, 67–74 (1965).
  9. C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,” J. Opt. Soc. Am. 56, 1502–1509 (1966).
    [Crossref]
  10. F. G. Kaspar, “Diffraction by thick, periodically stratified gratings with complex dielectric constant,” J. Opt. Soc. Am. 63, 37–45 (1973).
    [Crossref]
  11. W. R. Klein and B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Son. Ultrason. SU-14, 123–134 (1967).
    [Crossref]
  12. R. Alferness, “Analysis of optical propagation in thick holographic gratings,” Appl. Phys. 7, 29–33 (1975).
    [Crossref]
  13. R. Alferness, “Optical propagation in holographic gratings,” Ph. D. thesis (University of Michigan, 1976) (unpublished).
  14. G. L. Fillmore and R. F. Tynan, “Sensitometric characteristics of hardened dichromated gelatin films,” J. Opt. Soc. Am. 61, 199–203 (1971).
    [Crossref]
  15. D. Kermisch, “Nonuniform sinusoidally modulated dielectric gratings,” J. Opt. Soc. Am. 59, 1409–1414 (1969).
    [Crossref]
  16. N. Uchida, “Calculation of diffraction efficiency in hologram gratings attenuated along the direction perpendicular to the grating vector,” J. Opt. Soc. Am. 63, 280–287 (1973).
    [Crossref]
  17. M. Chang and N. George, “Holographic dielectric grating: Theory and practice,” Appl. Opt. 9, 713–719 (1970).
    [Crossref] [PubMed]

1977 (1)

1976 (1)

1975 (2)

1973 (2)

1972 (1)

1971 (1)

1970 (2)

M. Chang and N. George, “Holographic dielectric grating: Theory and practice,” Appl. Opt. 9, 713–719 (1970).
[Crossref] [PubMed]

R. S. Chu and T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Techn. MTT-18, 486–504 (1970).

1969 (2)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[Crossref]

D. Kermisch, “Nonuniform sinusoidally modulated dielectric gratings,” J. Opt. Soc. Am. 59, 1409–1414 (1969).
[Crossref]

1967 (2)

L. Bergstein and D. Kermisch, “Image storage and reconstruction in volume holography,” Proc. Symp. Modern Opt. 17, 655–680 (1967).

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

1966 (1)

1965 (1)

W. R. Klein, B. D. Cook, and W. G. Mayer, “Light diffraction by ultrasonic gratings,” Acustica 15, 67–74 (1965).

Alferness, R.

R. Alferness, “Analysis of propagation at the second-order Bragg angle of a thick holographic grating,” J. Opt. Soc. Am. 66, 353–362 (1976).
[Crossref]

R. Alferness, “Analysis of optical propagation in thick holographic gratings,” Appl. Phys. 7, 29–33 (1975).
[Crossref]

R. Alferness, “Optical propagation in holographic gratings,” Ph. D. thesis (University of Michigan, 1976) (unpublished).

Bergstein, L.

L. Bergstein and D. Kermisch, “Image storage and reconstruction in volume holography,” Proc. Symp. Modern Opt. 17, 655–680 (1967).

Burckhardt, C. B.

Chang, M.

Chu, R. S.

R. S. Chu and T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Techn. MTT-18, 486–504 (1970).

Cook, B. D.

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

W. R. Klein, B. D. Cook, and W. G. Mayer, “Light diffraction by ultrasonic gratings,” Acustica 15, 67–74 (1965).

Fillmore, G. L.

Gaylord, T. K.

George, N.

Kaspar, F. G.

Kermisch, D.

D. Kermisch, “Nonuniform sinusoidally modulated dielectric gratings,” J. Opt. Soc. Am. 59, 1409–1414 (1969).
[Crossref]

L. Bergstein and D. Kermisch, “Image storage and reconstruction in volume holography,” Proc. Symp. Modern Opt. 17, 655–680 (1967).

Klein, W. R.

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

W. R. Klein, B. D. Cook, and W. G. Mayer, “Light diffraction by ultrasonic gratings,” Acustica 15, 67–74 (1965).

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[Crossref]

Magnusson, R.

Mayer, W. G.

W. R. Klein, B. D. Cook, and W. G. Mayer, “Light diffraction by ultrasonic gratings,” Acustica 15, 67–74 (1965).

Smith, H. M.

Su, S. F.

Tamir, T.

R. S. Chu and T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Techn. MTT-18, 486–504 (1970).

Tynan, R. F.

Uchida, N.

Acustica (1)

W. R. Klein, B. D. Cook, and W. G. Mayer, “Light diffraction by ultrasonic gratings,” Acustica 15, 67–74 (1965).

Appl. Opt. (1)

Appl. Phys. (1)

R. Alferness, “Analysis of optical propagation in thick holographic gratings,” Appl. Phys. 7, 29–33 (1975).
[Crossref]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[Crossref]

IEEE Trans. Microwave Theory Techn. (1)

R. S. Chu and T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Techn. MTT-18, 486–504 (1970).

IEEE Trans. Son. Ultrason. (1)

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

J. Opt. Soc. Am. (9)

Proc. Symp. Modern Opt. (1)

L. Bergstein and D. Kermisch, “Image storage and reconstruction in volume holography,” Proc. Symp. Modern Opt. 17, 655–680 (1967).

Other (1)

R. Alferness, “Optical propagation in holographic gratings,” Ph. D. thesis (University of Michigan, 1976) (unpublished).

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Figures (4)

FIG. 1
FIG. 1

Diffraction regimes of a sinusoidal phase grating. The readout wave is incident at the first Bragg angle. The transition boundaries for diffraction efficiency deviations of 0.1% to 1.0% from elementary “thin” and “thick” theories are represented by the dot pattern.

FIG. 2
FIG. 2

Diffraction efficiencies as functions of γ for Q′ = 5 according to “thick” two-wave theory (sin2γ) and according to multiwave theory.

FIG. 3
FIG. 3

Diffraction efficiencies as functions of Q′ showing the transition behaviors for the sinusoidal, square, and sawtooth phase gratings. The readout wave is incident at the second Bragg angle. A single transmitted wave regime results for the sinusoidal and square gratings at large-Q′ values due to the absence of a second harmonic grating component.

FIG. 4
FIG. 4

Diffraction regimes of a sinusoidal absorption grating. The readout wave is incident at the first Bragg angle. The transition boundaries for diffraction efficiency deviations of 0.01% to 0.1% from elementary “thin” and “thick” theories are represented by the dot pattern.

Equations (19)

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Δ n = h = 1 ( n c h cos h K x + n s h sin h K x ) , Δ α = h = 1 ( α c h cos h K x + α s h sin h K x ) ,
S i z + ( α 0 d cos θ + j i ( m - i ) Q 2 ) S i + j d 2 cos θ h = 1 ( S i - h C h + S i + h D h ) = 0 ,
S i z + α 0 d cos θ S i + j d 2 cos θ h = 1 ( S i - h C h + S i + h D h ) = 0.
τ ( x , z ) = exp [ - ( α 0 + Δ α + j 2 π Δ n / λ ) z d / cos θ ] ,
τ ( x , z ) = i = - S i ( z ) exp ( j i K x ) .
S i ( z ) = ( 2 π ) - 1 exp ( - ψ 0 z ) × 0 2 π exp { - [ ψ 1 z g ( ξ ) + j 2 γ z f ( ξ ) ] } exp ( - j i ξ ) d ξ ,
S i ( z ) = ( - 1 ) i exp ( - ψ 0 z ) J i ( 2 γ z - j ψ 1 z ) ,
S i ( z ) = ( - 1 ) i exp ( - ψ 0 z ) k = - j k J i + k ( 2 γ z ) I k ( ψ 1 z ) ,
η i = exp ( - 2 ψ 0 ) J i 2 ( 2 γ ) .
η i = exp ( - 2 ψ 0 ) I i 2 ( ψ 1 ) .
S 0 z + ψ 0 S 0 + j d 2 cos θ D m S m = 0 , S m z + ψ 0 S m + j d 2 cos θ C m S 0 = 0.
S m ( z ) = - j exp ( - ψ 0 z ) ( C m / D m ) 1 / 2 × sin [ d ( C m D m ) 1 / 2 z / 2 cos θ ] .
η m = exp ( - 2 ψ 0 ) [ sin 2 ( π d n c m / λ cos θ ) + sinh 2 ( d α c m / 2 cos θ ) ] .
S i z + j i ( m - i ) Q 2 S i + γ ( S i - 1 - S i + 1 ) = 0 ,
η 1 - η ,
S i z + ( ψ 0 + j i ( m - i ) Q 2 ) S i + j ψ 1 2 ( S i + 1 - S i - 1 ) = 0 ,
S 0 / z + j ( γ 1 S 1 + γ 2 S 2 ) = 0 , S 1 / z + ( j Q / 2 ) S 1 + j γ 1 ( S 0 + S 2 ) = 0 , S 2 / z + j ( γ 1 S 1 + γ 2 S 0 ) = 0 ,
η m = sin 2 { 2 m - 1 γ m / [ ( m - 1 ) ! ] 2 ( Q ) m - 1 } ,
Δ ϕ = ϕ 0 - ϕ i = i ( m - i ) Q .