Abstract

Lossy dielectric gratings have been analyzed using a Raman-Nath formalism modified to incorporate losses. Four second-order coupled wave equations are retained for computation of the zero, first- and second-order diffracted beams for a multitude of practical cases. Significant differences are found in comparison with computations in which only two coupled waves are retained. The entire range of losses and thicknesses encountered for holograms in film emulsions has been studied using this unified approach. Graphs have been prepared to show the efficiency, i.e., power diffracted in the first-order relative to the total incident power, vs the index modulation for a wide range of thicknesses and losses. At a given thickness, optimum frequency requires a specific exposure. The efficiency for an optimum exposure is plotted vs the loss factor with thickness as a parameter. New experimental data are presented for bleached gratings in which several diffracted orders are measured and compared to our theory for a wide range of index modulation and loss factors.

© 1970 Optical Society of America

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  1. E. N. Leith, A. Kozma, J. Upatnieks, J. Marks, N. Massey, Appl. Opt. 5, 1301 (1966).
    [CrossRef]
  2. C. B. Burckhardt, J. Opt. Soc. Amer. 57, 601 (1967).
    [CrossRef]
  3. H. Kogelnik, in Modern Optics. J. Fox, Ed. (Polytechnic Press, Brooklyn, 1967).
  4. N. George, J. W. Matthews, Appl. Phys. Lett. 9, 212 (1966).
    [CrossRef]
  5. M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), p. 651.
  6. Ref. 5, p. 87.
  7. J. Brown, Microwave Lenses (Methuen and Co. Ltd., London, 1953).
  8. M. M. T. Chang, Holographic Dielectric Gratings (California Institute of Technology, Pasadena, 1969), Ph.D. thesis, pp. 18–19.
  9. C. E. Kenneth Mees, T. H. James, The Theory of the Photographic Process (The Macmillan Company, New York, 1952), Chap. 23, p. 499.
  10. H. Hannes, J. Opt. Soc. Amer. 58, 140 (1968).
    [CrossRef]
  11. C. V. Raman, N. S. N. Nath, Proc. Indian Acad. Sci. A2, 406, 413 (1935); Proc. Indian Acad. Sci. A3, 75, 119 (1936); also N. S. N. Nath, Proc. Indian Acad. Sci. A4, 222 (1936); Proc. Indian Acad. Sci. A8, 499 (1938).
  12. Ref. 5, Chap. 12.
  13. S. P. Gill, U.S. Office of Naval Research Contract Nonr 1866 (24) NR-384-903, Tech. Memo. 58 (1964).
  14. M. V. Berry, The Diffraction of Light by Ultrasound (Academic Press, Inc., London, 1966).
  15. W. R. Klein, B. D. Cook, IEEE Trans. Sonic Ultrasonics SU-14, 123 (1967).
    [CrossRef]
  16. Ref. 5, pp. 597, 598.
  17. H. Kogelnik, J. Opt. Soc. Amer. 57, 431 (1967).
    [CrossRef]
  18. Ref. 8, p. 61.
  19. P. Phariseau, Proc. Indian Acad. Sci. A44, 165 (1956).
  20. Ref. 8, p. 67.
  21. W. C. Miller, Publ. Astron. Soc. Pacific 76, 328 (1964).
    [CrossRef]
  22. J. C. Wyant, M. P. Givens, J. Opt. Soc. Amer. 58, 357 (1968).
    [CrossRef]
  23. 10 g potassium dichromate, 18 cc concentrated HCl in 1 liter of water.
  24. F. D. Perrin, J. H. Altman, J. Opt. Soc. Amer. 42, 462 (1952).
    [CrossRef]
  25. “Method for Producing Phase Holograms with Conventional Photographic Materials” Eastman Kodak Company, Special Applications, New York (1968).
  26. The unfixed plate is immersed in a shallow basin of water. Pour 1: 1 concentrated nitric acid into the basin until the plate starts to clear. The plate is rinsed in water and dried.
  27. J. Upatnieks, C. Leonard, Appl. Opt. 8, 85 (1969).
    [CrossRef] [PubMed]
  28. P. Glafkides, Photographic Chemistry (Fountain Press, London, 1960) Vol. 2, Chap. 42.

1969 (1)

1968 (2)

H. Hannes, J. Opt. Soc. Amer. 58, 140 (1968).
[CrossRef]

J. C. Wyant, M. P. Givens, J. Opt. Soc. Amer. 58, 357 (1968).
[CrossRef]

1967 (3)

W. R. Klein, B. D. Cook, IEEE Trans. Sonic Ultrasonics SU-14, 123 (1967).
[CrossRef]

H. Kogelnik, J. Opt. Soc. Amer. 57, 431 (1967).
[CrossRef]

C. B. Burckhardt, J. Opt. Soc. Amer. 57, 601 (1967).
[CrossRef]

1966 (2)

1964 (1)

W. C. Miller, Publ. Astron. Soc. Pacific 76, 328 (1964).
[CrossRef]

1956 (1)

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

1952 (1)

F. D. Perrin, J. H. Altman, J. Opt. Soc. Amer. 42, 462 (1952).
[CrossRef]

1935 (1)

C. V. Raman, N. S. N. Nath, Proc. Indian Acad. Sci. A2, 406, 413 (1935); Proc. Indian Acad. Sci. A3, 75, 119 (1936); also N. S. N. Nath, Proc. Indian Acad. Sci. A4, 222 (1936); Proc. Indian Acad. Sci. A8, 499 (1938).

Altman, J. H.

F. D. Perrin, J. H. Altman, J. Opt. Soc. Amer. 42, 462 (1952).
[CrossRef]

Berry, M. V.

M. V. Berry, The Diffraction of Light by Ultrasound (Academic Press, Inc., London, 1966).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), p. 651.

Brown, J.

J. Brown, Microwave Lenses (Methuen and Co. Ltd., London, 1953).

Burckhardt, C. B.

C. B. Burckhardt, J. Opt. Soc. Amer. 57, 601 (1967).
[CrossRef]

Chang, M. M. T.

M. M. T. Chang, Holographic Dielectric Gratings (California Institute of Technology, Pasadena, 1969), Ph.D. thesis, pp. 18–19.

Cook, B. D.

W. R. Klein, B. D. Cook, IEEE Trans. Sonic Ultrasonics SU-14, 123 (1967).
[CrossRef]

George, N.

N. George, J. W. Matthews, Appl. Phys. Lett. 9, 212 (1966).
[CrossRef]

Gill, S. P.

S. P. Gill, U.S. Office of Naval Research Contract Nonr 1866 (24) NR-384-903, Tech. Memo. 58 (1964).

Givens, M. P.

J. C. Wyant, M. P. Givens, J. Opt. Soc. Amer. 58, 357 (1968).
[CrossRef]

Glafkides, P.

P. Glafkides, Photographic Chemistry (Fountain Press, London, 1960) Vol. 2, Chap. 42.

Hannes, H.

H. Hannes, J. Opt. Soc. Amer. 58, 140 (1968).
[CrossRef]

James, T. H.

C. E. Kenneth Mees, T. H. James, The Theory of the Photographic Process (The Macmillan Company, New York, 1952), Chap. 23, p. 499.

Kenneth Mees, C. E.

C. E. Kenneth Mees, T. H. James, The Theory of the Photographic Process (The Macmillan Company, New York, 1952), Chap. 23, p. 499.

Klein, W. R.

W. R. Klein, B. D. Cook, IEEE Trans. Sonic Ultrasonics SU-14, 123 (1967).
[CrossRef]

Kogelnik, H.

H. Kogelnik, J. Opt. Soc. Amer. 57, 431 (1967).
[CrossRef]

H. Kogelnik, in Modern Optics. J. Fox, Ed. (Polytechnic Press, Brooklyn, 1967).

Kozma, A.

Leith, E. N.

Leonard, C.

Marks, J.

Massey, N.

Matthews, J. W.

N. George, J. W. Matthews, Appl. Phys. Lett. 9, 212 (1966).
[CrossRef]

Miller, W. C.

W. C. Miller, Publ. Astron. Soc. Pacific 76, 328 (1964).
[CrossRef]

Nath, N. S. N.

C. V. Raman, N. S. N. Nath, Proc. Indian Acad. Sci. A2, 406, 413 (1935); Proc. Indian Acad. Sci. A3, 75, 119 (1936); also N. S. N. Nath, Proc. Indian Acad. Sci. A4, 222 (1936); Proc. Indian Acad. Sci. A8, 499 (1938).

Perrin, F. D.

F. D. Perrin, J. H. Altman, J. Opt. Soc. Amer. 42, 462 (1952).
[CrossRef]

Phariseau, P.

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

Raman, C. V.

C. V. Raman, N. S. N. Nath, Proc. Indian Acad. Sci. A2, 406, 413 (1935); Proc. Indian Acad. Sci. A3, 75, 119 (1936); also N. S. N. Nath, Proc. Indian Acad. Sci. A4, 222 (1936); Proc. Indian Acad. Sci. A8, 499 (1938).

Upatnieks, J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), p. 651.

Wyant, J. C.

J. C. Wyant, M. P. Givens, J. Opt. Soc. Amer. 58, 357 (1968).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

N. George, J. W. Matthews, Appl. Phys. Lett. 9, 212 (1966).
[CrossRef]

IEEE Trans. Sonic Ultrasonics (1)

W. R. Klein, B. D. Cook, IEEE Trans. Sonic Ultrasonics SU-14, 123 (1967).
[CrossRef]

J. Opt. Soc. Amer. (5)

C. B. Burckhardt, J. Opt. Soc. Amer. 57, 601 (1967).
[CrossRef]

H. Kogelnik, J. Opt. Soc. Amer. 57, 431 (1967).
[CrossRef]

H. Hannes, J. Opt. Soc. Amer. 58, 140 (1968).
[CrossRef]

J. C. Wyant, M. P. Givens, J. Opt. Soc. Amer. 58, 357 (1968).
[CrossRef]

F. D. Perrin, J. H. Altman, J. Opt. Soc. Amer. 42, 462 (1952).
[CrossRef]

Proc. Indian Acad. Sci. (2)

C. V. Raman, N. S. N. Nath, Proc. Indian Acad. Sci. A2, 406, 413 (1935); Proc. Indian Acad. Sci. A3, 75, 119 (1936); also N. S. N. Nath, Proc. Indian Acad. Sci. A4, 222 (1936); Proc. Indian Acad. Sci. A8, 499 (1938).

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

Publ. Astron. Soc. Pacific (1)

W. C. Miller, Publ. Astron. Soc. Pacific 76, 328 (1964).
[CrossRef]

Other (16)

P. Glafkides, Photographic Chemistry (Fountain Press, London, 1960) Vol. 2, Chap. 42.

“Method for Producing Phase Holograms with Conventional Photographic Materials” Eastman Kodak Company, Special Applications, New York (1968).

The unfixed plate is immersed in a shallow basin of water. Pour 1: 1 concentrated nitric acid into the basin until the plate starts to clear. The plate is rinsed in water and dried.

10 g potassium dichromate, 18 cc concentrated HCl in 1 liter of water.

Ref. 8, p. 67.

Ref. 8, p. 61.

H. Kogelnik, in Modern Optics. J. Fox, Ed. (Polytechnic Press, Brooklyn, 1967).

Ref. 5, pp. 597, 598.

Ref. 5, Chap. 12.

S. P. Gill, U.S. Office of Naval Research Contract Nonr 1866 (24) NR-384-903, Tech. Memo. 58 (1964).

M. V. Berry, The Diffraction of Light by Ultrasound (Academic Press, Inc., London, 1966).

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), p. 651.

Ref. 5, p. 87.

J. Brown, Microwave Lenses (Methuen and Co. Ltd., London, 1953).

M. M. T. Chang, Holographic Dielectric Gratings (California Institute of Technology, Pasadena, 1969), Ph.D. thesis, pp. 18–19.

C. E. Kenneth Mees, T. H. James, The Theory of the Photographic Process (The Macmillan Company, New York, 1952), Chap. 23, p. 499.

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Figures (6)

Fig. 1
Fig. 1

(a) Recording of Holographic Grating. (b) Model used for calculating the diffraction intensities.

Fig. 2
Fig. 2

(a) MTF of Kodak High Contrast Copy film. (b) Normalized m calculated using Eq. (7).

Fig. 3
Fig. 3

Computer solution of four coupled wave equations, with n1 = 1.5, λ = 6328 Å, m = 1, and 2θ = 15°, for thicknesses in microns, μ.

Fig. 4
Fig. 4

Maximum efficiency, obtained from graphs such as Fig. 3, plotted against the loss factor for various thicknesses.

Fig. 5
Fig. 5

H-D curve of normal and preflashed 649F plates.

Fig. 6
Fig. 6

Experimental result using 649F microflat, preflashed and bleached in chromium intensifier. Exposure unit ≅ 150 ergs/cm2.

Equations (33)

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α = a 3 1 ( 2 1 ) / ( 2 + 2 1 )
/ 1 = [ 1 + ( 8 π N α / 3 ) ] / [ 1 4 π N α / 3 ) ] .
1 = 1 + 3 f [ ( 2 + 2 1 ) / ( 2 1 ) ] f ,
n = n 1 { 1 + 1.5 f [ ( 2 1 ) / ( 2 + 2 1 ) ] } ,
f Δ n 2 = 1.5 f n 1 ( 2 1 ) / ( 2 + 2 1 ) .
n = n 2 + i f n 2
n 2 = n 1 [ 1 + 1.5 f ( 2 2 + 2 2 2 1 2 + 1 2 ) ( 2 + 2 1 ) 2 + 2 2 ]
f n 2 = 4.5 f n 1 1 2 ( 2 + 2 1 ) 2 + 2 2 .
D 2 = 0.869 k 0 f n 2 L
n 3 = n 1 [ 1 + 1.73 ζ D 2 n 2 k 0 L ( 3 1 3 + 2 1 ) ] ,
M = ( T max T min ) / ( T max + T min ) ,
M = tanh ( k 0 L n 2 f 0 m ) .
Φ MTF = ( Φ max Φ min ) / ( Φ max + Φ min )
Φ MTF = m .
E Z = l = i l exp [ i ( k ¯ sin θ ¯ x + k ¯ cos θ ¯ y ) + i l K x ] U l ,
i m Δ sec 2 θ ¯ 2 ( 1 + Δ ) U ¨ l ( χ ) 2 U ˙ l ( χ ) U l + 1 ( χ ) + U l 1 ( χ ) 2 i ( 1 + Δ ) / ( m Δ ) { 2 l K sin θ ¯ / k ¯ + l 2 K 2 / k ¯ 2 } U l ( χ ) = 0
U 0 ( 0 ) = 1 , U l 0 ( 0 ) = 0 , U ˙ l = ± 1 ( 0 ) = ± 1 2 and U ˙ l ± 1 ( 0 ) = 0.
P l ( y ) = U l U l * | exp [ i ( k ¯ x sin θ ¯ + k ¯ y cos θ ¯ ) ] | 2
P l ( y ) = U l U l * exp [ 2 k R q ( κ cos γ + sin γ ) y ] ,
R = ( n 1 + 1 2 Δ / n 1 ) κ = Δ / ( 2 n 1 2 + Δ ) q = [ 1 2 ( 1 κ 2 ) R 2 ( 1 + κ 2 ) 2 sin 2 θ + ( 1 + κ 2 ) 2 R 4 ( 1 + κ 2 ) 4 sin 4 θ ] 1 4 γ = 1 2 tan 1 { 2 κ R 2 ( 1 + κ 2 ) 2 sin 2 θ / [ 1 1 κ 2 R 2 ( 1 + κ 2 ) 2 sin 2 θ ] } ,
Δ = Δ + i Δ , χ = χ + i χ = 1 2 y k ¯ m Δ sec θ ¯ / ( 1 + Δ ) .
| 8 i ( 1 + Δ ) K m Δ k ¯ sin θ ¯ χ | = | 4 L K tan θ ¯ | 1 ,
| 2 i ( 1 + Δ ) K 2 m Δ k ¯ 2 χ | = | K 2 L sec θ ¯ / k ¯ | 1 ,
| 4 i ( 1 + Δ ) K 2 m Δ k ¯ 2 χ 2 | = | K 2 L 2 m Δ sec 2 θ ¯ / ( 1 + Δ ) | 1 ,
| 1 2 i m Δ sec θ ¯ / ( 1 + Δ ) | 1 ,
U l ( χ ) = ( 1 ) l J l ( χ ) ,
| 1 2 m Δ sec 2 θ ¯ / ( 1 + Δ ) | 1 ,
| 2 ( 1 + Δ ) K 2 / ( k ¯ 2 m Δ ) | 1.
U 1 ( χ ) = 1 2 sin [ ( σ 2 + 1 4 ) 1 2 χ ] ( σ 2 + 1 4 ) 1 2 exp [ i σ χ ]
U 0 ( χ ) = { cos [ ( σ + 1 4 ) 1 2 χ ] i σ sin [ ( σ + 1 4 ) 1 2 χ ] / ( σ + 1 4 ) 1 2 } × exp [ i σ χ ] ,
U 0 ( χ ) = cos ( χ / 2 ) , U 1 ( χ ) = sin ( χ / 2 ) .
Q = | 8 ( 1 + Δ ) sin 2 θ ¯ / ( m Δ ) | 1.
Q = | 8 L n 1 sin θ ¯ tan θ ¯ / λ | 1.

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