Abstract

An analysis is presented of the nonlinear effects of holographically recording discrete image points on a phase recording material. The analysis is restricted to thin, two- and three-beam holographic gratings recorded on a material that exhibits a linear phase shift vs exposure. Harmonics, intermodulation noise, and small signal effects are considered. Experimental measurements were carried out for three-beam holographic gratings and diffuse object holograms recorded on photopolymer recording materials. Intermodulation noise is found to be a serious limitation for discrete image point holograms, because this noise cannot be spatially separated from the desired image points. Intermodulation noise can be reduced by increasing the reference-to-object beam irradiance ratio and by reducing the diffraction efficiency. Photopolymer gratings with 50-dB signal-to-intermodulation noise ratio were obtained with diffraction efficiency greater than 110% at beam irradiance ratios of 400:1. The image contrast of photopolymer holograms of diffuse objects is compared with the image contrast reported for bleached silver halide emulsions.

© 1972 Optical Society of America

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References

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  1. See for example, J. W. Goodman, Introduction to Fourier Optics (Polytechnic Press, Brooklyn, 1967), p. 155.
  2. A. Kozma, J. Opt. Soc. Am. 56, 428 (1966).
    [CrossRef]
  3. A. A. Friesem, J. S. Zelenka, Appl. Opt. 6, 1755 (1967).
    [CrossRef] [PubMed]
  4. O. Bryngdahl, A. Lohmann, J. Opt. Soc. Am. 58, 1325 (1968).
    [CrossRef]
  5. J. W. Goodman, G. R. Knight, J. Opt. Soc. Am. 58, 1276 (1968).
    [CrossRef]
  6. F. J. Tischer, Appl. Opt. 9, 1369 (1970).
    [CrossRef] [PubMed]
  7. J. Upatnieks, C. D. Leonard, J. Opt. Soc. Am. 60, 297 (1970).
    [CrossRef]
  8. H. Dammann, J. Opt. Soc. Am. 60, 1635 (1970).
    [CrossRef]
  9. R. S. Sirohi, C. S. Vikram, Opt. Commun. (Netherlands) 3, 122 (1971).
    [CrossRef]
  10. M. King, J. Opt. Soc. Am. 60, 513 (1970).
    [CrossRef]
  11. J. A. Jenney, J. Opt. Soc. Am. 60, 1155 (1970).
    [CrossRef]
  12. J. B. Rust, L. J. Miller, J. D. Margerum, Polym. Eng. Sci. 9, 40 (1969).
    [CrossRef]
  13. H. Kogelnik, in Proceedings of Symposium on Modern Optics (Polytechnic Press, Brooklyn, 1967), p. 605.
  14. Ref. 1, p. 240.
  15. J. A. Jenney, J. Opt. Soc. Am. 61, 1116 (1971).
    [CrossRef]

1971 (2)

R. S. Sirohi, C. S. Vikram, Opt. Commun. (Netherlands) 3, 122 (1971).
[CrossRef]

J. A. Jenney, J. Opt. Soc. Am. 61, 1116 (1971).
[CrossRef]

1970 (5)

1969 (1)

J. B. Rust, L. J. Miller, J. D. Margerum, Polym. Eng. Sci. 9, 40 (1969).
[CrossRef]

1968 (2)

1967 (1)

1966 (1)

Bryngdahl, O.

Dammann, H.

Friesem, A. A.

Goodman, J. W.

J. W. Goodman, G. R. Knight, J. Opt. Soc. Am. 58, 1276 (1968).
[CrossRef]

See for example, J. W. Goodman, Introduction to Fourier Optics (Polytechnic Press, Brooklyn, 1967), p. 155.

Jenney, J. A.

King, M.

Knight, G. R.

Kogelnik, H.

H. Kogelnik, in Proceedings of Symposium on Modern Optics (Polytechnic Press, Brooklyn, 1967), p. 605.

Kozma, A.

Leonard, C. D.

Lohmann, A.

Margerum, J. D.

J. B. Rust, L. J. Miller, J. D. Margerum, Polym. Eng. Sci. 9, 40 (1969).
[CrossRef]

Miller, L. J.

J. B. Rust, L. J. Miller, J. D. Margerum, Polym. Eng. Sci. 9, 40 (1969).
[CrossRef]

Rust, J. B.

J. B. Rust, L. J. Miller, J. D. Margerum, Polym. Eng. Sci. 9, 40 (1969).
[CrossRef]

Sirohi, R. S.

R. S. Sirohi, C. S. Vikram, Opt. Commun. (Netherlands) 3, 122 (1971).
[CrossRef]

Tischer, F. J.

Upatnieks, J.

Vikram, C. S.

R. S. Sirohi, C. S. Vikram, Opt. Commun. (Netherlands) 3, 122 (1971).
[CrossRef]

Zelenka, J. S.

Appl. Opt. (2)

J. Opt. Soc. Am. (8)

Opt. Commun. (Netherlands) (1)

R. S. Sirohi, C. S. Vikram, Opt. Commun. (Netherlands) 3, 122 (1971).
[CrossRef]

Polym. Eng. Sci. (1)

J. B. Rust, L. J. Miller, J. D. Margerum, Polym. Eng. Sci. 9, 40 (1969).
[CrossRef]

Other (3)

H. Kogelnik, in Proceedings of Symposium on Modern Optics (Polytechnic Press, Brooklyn, 1967), p. 605.

Ref. 1, p. 240.

See for example, J. W. Goodman, Introduction to Fourier Optics (Polytechnic Press, Brooklyn, 1967), p. 155.

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Figures (9)

Fig. 1
Fig. 1

The ratio of the second harmonic intensity to the signal intensity vs the signal diffraction efficiency.

Fig. 2
Fig. 2

Three-beam hologram recording geometry.

Fig. 3
Fig. 3

Point images in the Fourier transform plane of a three-beam hologram.

Fig. 4
Fig. 4

The calculated signal-to-intermodulation noise ratio vs diffraction efficiency for a three-beam hologram. K is the reference-to-signal beam irradiance ratio.

Fig. 5
Fig. 5

A diagram of the optical system used to record diffraction efficiency vs exposure curves.

Fig. 6
Fig. 6

Diffraction efficiency vs exposure for a 12-μm thick grating exposed at 110 cycles/mm and with equal beam irradiances.

Fig. 7
Fig. 7

Phase shift vs average exposure response calculated from the curve in Fig. 6. An induction exposure of 1.01 mJ/cm2 was subtracted from the average exposure values.

Fig. 8
Fig. 8

Diffraction efficiency and image contrast (S/N) c as a function of beam ratio K for photopolymer holograms of a diffuse object: ▲-·-▲ diffraction efficiency for an average exposure of 2.5 mJ/cm2 at K = 1, •-·-• diffraction efficiency for an average exposure of 0.5 mJ/cm2 at K = 1, ▲---▲ (S/N) c for an average exposure of 2.5 mJ/cm at K = 1, •---• (S/N) c for an average exposure of 0.5 mJ/cm2 at K = 1.

Fig. 9
Fig. 9

Diffraction efficiency and image contrast (S/N) c as a function of beam ratio K for photopolymer holograms of a specular object: ▲-·-▲ diffraction efficiency for an average exposure of 2.5 mJ/cm2 at K = 1, •-·-• diffraction efficiency for an average exposure of 0.5 mJ/cm2 at K = 1, ▲---▲ (S/N) c for an average exposure of 2.5 mJ/cm2 at K = 1, •---• (S/N) c for an average exposure of 0.5 mJ/cm2 at K = 1.

Tables (1)

Tables Icon

Table I Comparison of Measured Signal-to-Intermodulation Noise Ratio with Values Calculated from Eq. (17)

Equations (24)

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T ( x , y ) = [ τ ( x , y ) ] 1 2 exp [ i ϕ ( x , y ) ] ,
T A ( x ) = exp ( i ϕ 0 ) n = + i n J n ( ϕ 1 ) exp ( i n k x ) ,
J n 2 ( ϕ 1 ) / J 1 2 ( ϕ 1 ) .
E ( x ) = E 0 exp ( i Ω 0 ) + E 1 exp [ i ( k 1 x + Ω 1 ) ] + E 2 exp [ i ( k 2 x + Ω 3 ) ] ,
k 1 = ( 2 π / λ ) sin θ 1 , k 2 = ( 2 π / λ ) sin θ 2 .
ϕ ( x ) = β t E ( x ) E * ( x ) ,
ϕ ( x ) = ϕ 0 + ϕ 1 cos ( k 1 x + Ω 1 ) + ϕ 2 cos ( k 2 x + Ω 2 ) + ϕ 3 cos ( k 3 x + Ω 3 ) ,
ϕ 0 = β t ( E 0 2 + E 1 2 + E 2 2 ) , ϕ 1 = 2 β t ( E 0 E 1 ) , ϕ 2 = 2 β t ( E 0 E 2 ) , ϕ 3 = 2 β t ( E 1 E 2 ) , Ω 1 = Ω 1 Ω 0 , Ω 2 = Ω 2 Ω 0 , Ω 3 = Ω 2 Ω 1 , k 3 = k 2 k 1 .
T A ( x ) = exp ( i ϕ 0 ) l = + i l J l ( ϕ 1 ) exp [ i l ( k 1 x + Ω 1 ) ] × m = + i m J m ( ϕ 2 ) exp [ i m ( k 2 x + Ω 2 ) ] × n = + i n J n ( ϕ 3 ) exp [ i n ( k 3 x + Ω 3 ) ] .
D ( ω ) = aperture T A ( x ) exp ( i ω x ) d x .
D ( ω ) = exp ( i ϕ 0 ) l = + i l J l ( ϕ 1 ) exp ( i l Ω 1 ) × m = + + i m J m ( ϕ 2 ) exp ( i m Ω 2 ) n = + i n J n ( ϕ 3 ) × exp ( i n Ω 3 ) δ [ ( l n ) k 1 + ( m + n ) k 2 ω ] ,
D ( ω ) = exp ( i ϕ 0 ) l m n } = + J l ( ϕ 1 ) J m ( ϕ 2 ) J n ( ϕ 3 ) × exp { i [ ( l n ) Ω 1 + ( m + n ) Ω 2 + ( l + m + n ) ( π / 2 ) ] } × δ [ ( l n ) k 1 + ( m + n ) k 2 ω ] .
ϕ 1 = ϕ 2 , ϕ 3 = ϕ 1 / K , Ω 1 = Ω 2 = 0 ,
D ( ω ) = exp ( i ϕ 0 ) l m n } = + J l ( ϕ 1 ) J m ( ϕ 1 ) J n ( ϕ 1 / K ) × exp [ i ( π / 2 ) ( l + m + n ) ] × δ [ ( l n ) k 1 + ( m + n ) k 2 ω ] .
J 0 ( ϕ 1 ) = 1 ( ϕ 1 2 / 4 ) + , J 1 ( ϕ 1 ) = ( ϕ 1 / 2 ) ( ϕ 1 3 / 16 ) + , J 2 ( ϕ 1 ) = ( ϕ 1 2 / 8 ) + .
η ~ ( ϕ 1 2 / 4 )
η I ~ ( ϕ 1 4 / 16 K ) + ( ϕ 6 / 256 ) [ 1 + ( 1 / K ) ] 2
S / N I = ( η / η I ) = 4 K / [ 4 η + η 2 ( K + 1 ) 2 / K ] .
ϕ 2 + ϕ 1 / ( R ) 1 2 , ϕ 3 = ϕ 1 / ( K R ) 1 2 , Ω 1 = Ω 2 = 0.
I 1 I 2 = R [ 1 + ϕ 1 2 4 ( 1 1 K 1 R + 1 K R 2 ) + ] .
ϕ ( x ) = ϕ 0 ϕ 1 cos k x ϕ 2 cos 2 k x ,
T ( x ) = exp ( i ϕ 0 ) n = + i n J n ( ϕ 1 ) × exp ( i n k x ) m = + i m J m ( ϕ 2 ) exp ( i 2 m k x ) .
η = J 1 2 ( ϕ 1 ) J 0 2 ( ϕ 2 ) .
ϕ 1 = a t I γ ,

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