Abstract

The reconstruction properties of thin phase holograms of diffuse objects with large reference–object irradiance ratios are investigated, using the statistical method proposed by Goodman and Knight [ J. Opt. Soc. Am. 58, 1276 ( 1968)]. Three types of phase-recording characteristics are treated: the linear-phase characteristic (the phase modulation being proportional to exposure), a nonlinear-phase characteristic, and the binary characteristic. For the maximum reconstruction efficiency of one of the ideal twin images, we obtain 18.4% for the linear-phase hologram and 31.8% for the binary-phase hologram.

© 1970 Optical Society of America

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References

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  1. J. W. Goodman and G. R. Knight, J. Opt. Soc. Am. 58, 1276 (1968).
    [Crossref]
  2. W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw–Hill, New York, 1958), pp. 277 ff.
  3. J. Upatnieks and C. Leonard, J. Opt. Soc. Am. 60, 297 (1970).
    [Crossref]
  4. W. Gröbner and N. Hofreiter, Integraltafel, Zweiter Teil (Springer, Wien, 1961), pp. 131, 132.
  5. Reference 4, p. 143.
  6. Reference 4, p. 66.

1970 (1)

1968 (1)

Davenport, W. B.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw–Hill, New York, 1958), pp. 277 ff.

Goodman, J. W.

Gröbner, W.

W. Gröbner and N. Hofreiter, Integraltafel, Zweiter Teil (Springer, Wien, 1961), pp. 131, 132.

Hofreiter, N.

W. Gröbner and N. Hofreiter, Integraltafel, Zweiter Teil (Springer, Wien, 1961), pp. 131, 132.

Knight, G. R.

Leonard, C.

Root, W. L.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw–Hill, New York, 1958), pp. 277 ff.

Upatnieks, J.

J. Opt. Soc. Am. (2)

Other (4)

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw–Hill, New York, 1958), pp. 277 ff.

W. Gröbner and N. Hofreiter, Integraltafel, Zweiter Teil (Springer, Wien, 1961), pp. 131, 132.

Reference 4, p. 143.

Reference 4, p. 66.

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Figures (2)

Fig. 1
Fig. 1

First-order efficiencies for linear-phase (φ2 = 0) and nonlinear-phase hologram.

Fig. 2
Fig. 2

Flux contrast Q3 for linear-phase (φ2 = 0) and nonlinear-phase hologram.

Equations (56)

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t ( E ) = { exp ( - i γ 1 ) for E E 0 exp ( - i γ 2 ) for E > E 0 }             ( γ 1 , γ 2     real     constants ) .
L m = κ ( h m 2 / m ! ) [ R E ( 0 ) ] m ,
κ R t ( 0 ) = 0 L m .
L m = L m / κ
R 0 ( 0 ) = a ( x ) 2
R E ( 0 ) = σ 2 .
L m = ( h m 2 / m ! ) σ 2 m .
R E ( Δ x ) m = ( τ 2 K 2 ) m l = 0 m ( m l ) R 0 ( Δ x ) m - l R 0 * ( Δ x ) l × exp [ - 2 π i α Δ x ( m - 2 l ) ] .
N m = κ h m 2 m ! ( τ 2 K 2 ) m [ ( m l 1 ) + ( m l 2 ) ] ) α ( x ) 2 m .
N 2 q + 1 = [ ( 2 q + 1 ) ! / 2 2 q q ! ( q + 1 ) ! ] L 2 q + 1 .
N = q = 1 N 2 q + 1 .
f ( i v ) = - + exp [ - i ( c + v ) E ] d E = 2 π δ ( c + v ) .
h m 2 = c 2 m exp ( - σ 2 c 2 ) .
L m = ( σ 2 c 2 ) m exp ( - σ 2 c 2 ) / m ! .
L m = ( 1 / m ! ) φ 1 2 m exp ( - φ 1 2 ) .
N = L 1 q = 1 1 q ! ( q + 1 ) ! ( φ 1 4 4 ) q = L 1 { I 1 ( φ 1 2 ) ( 1 2 φ 1 2 ) - 1 } ,
f ( i v ) = - + exp { - i ( ( c + v ) E ± d 2 2 E 2 ) } d E     .
f ( i v ) = [ ( 1 i ) π 1 2 / d ] exp [ ± i ( c + v ) 2 / 2 d 2 ] .
h m = 1 i 2 d π 1 2 - + exp [ ± i ( c + v ) 2 2 d 2 ] ( i v ) m exp ( - 1 2 σ 2 v 2 ) d v ,
h m 2 = [ ( 2 d 2 ) m / π ] A m 2 ,
A m = - + y m exp [ i ( y + a ) 2 ] exp ( - b y 2 ) d y .
( b - i ) A m = i a A m - 1 + [ ( m - 1 ) / 2 ] A m - 2 ,
A 0 2 = π ( b 2 + 1 ) - 1 2 exp ( - 2 b a 2 b 2 + 1 ) = π ( σ 4 d 4 + 1 ) - 1 2 exp ( - σ 2 c 2 σ 4 d 4 + 1 ) .
L m = [ ( 2 d 2 σ 2 ) m / π m ! ] A m 2 .
f ( w ) = a 1 - E 0 e - w E d E + α 2 E 0 e - w E d E .
f ( w ) = a 1 e - w E 0 0 e w x d x + a 2 e - w E 0 0 e - w x d x .
w + = - + i v     for     the     first     integral , w - = + i v     for     the     second .
f ( w ) = f + ( w + ) + f - ( w - ) = a 1 exp ( - w + E 0 ) w + - a 2 exp ( - w - E 0 ) w - .
h m ( + ) = a 1 2 π i - i + i exp ( - w E 0 ) w m - 1 exp ( 1 2 σ 2 w 2 ) d w .
h m ( + ) = a 1 i m 2 π i - + v m - 1 exp ( - 1 2 σ 2 v 2 ) × [ cos v E 0 - i sin v E 0 ] d v .
h 1 ( + ) = a 1 2 π - + cos ( v E 0 ) exp ( - 1 2 σ 2 v 2 ) d v = a 1 σ ( 2 π ) 1 2 exp ( - E 0 2 2 σ 2 ) .
and             h 1 = [ ( a 1 - a 2 ) / σ ( 2 π ) 1 2 ] exp ( - E 0 2 / 2 σ 2 ) L 1 = ( a 1 - a 2 2 / 2 π ) exp ( - E 0 2 / 2 σ 2 ) .
h 2 l ( + ) = - a 1 E 0 ( - 1 ) l 2 π - + v 2 l exp ( - 1 2 σ 2 v 2 ) d v     ( l 1 )
h 2 l + 1 ( + ) = a 1 ( - 1 ) l 2 π - + v 2 l exp ( - 1 2 σ 2 v 2 ) d v     ( l 1 ) .
- + v 2 l exp ( - 1 2 σ 2 v 2 ) d v = ( 1 2 σ 2 ) - ( l + 1 2 ) Γ ( l + 1 2 ) .
h 0 ( + ) = a 1 2 π i - i + i w - 1 exp { 1 2 σ 2 w 2 - E 0 } d w .
w = + i v : - v - w = e i φ : - π / 2 φ π / 2 w = i v : v .
h 0 = ( a 1 + a 2 ) / 2 + [ ( a 1 - a 2 ) / 2 ) ]     erf ( E 0 / 2 1 2 σ ) L 0 = h 0 2 .
Q = L 1 / N .
Q 3 = L 1 / N 3 = 4 3 L 1 / L 3 .
η 0 = exp ( - φ 1 2 ) , η 1 = 1 2 φ 1 2 exp ( - φ 1 2 ) , η 2 = 1 4 φ 1 4 exp ( - φ 1 2 ) , η 3 = 1 12 φ 1 6 exp ( - φ 1 2 ) .
φ 1 = 1 ,             max ( η 1 ) = 1 / 2 e 18.4 % .
Q = φ 1 2 / [ 2 I 1 ( φ 1 2 ) - φ 1 2 ] .
Q 3 = 8 / φ 1 4 .
η 0 = ( 1 + φ 2 2 ) - 1 2 exp ( - φ 1 2 1 + φ 2 2 ) η 1 = 1 2 φ 1 2 ( 1 + φ 2 2 ) - 3 2 exp ( - φ 1 2 1 + φ 2 2 ) η 2 = 1 4 [ ( φ 2 2 - φ 1 2 ) + φ 2 2 ] ( 1 + φ 2 2 ) - 5 2 × exp ( - φ 1 2 1 + φ 2 2 ) η 3 = 3 4 φ 1 2 [ ( φ 2 2 - 1 3 φ 1 2 ) + φ 2 2 ] ( 1 + φ 2 2 ) - 7 / 2 × exp ( - φ 1 2 1 + φ 2 2 ) . }
φ 1 2 = 1 + φ 2 2 ,             max ( η 1 ) = 1 / 2 e ( 1 + φ 2 2 ) 1 2 ,
Q 3 = η 1 3 4 η 3 = 8 9 ( 1 + φ 2 2 ) 2 ( φ 2 2 - 1 3 φ 1 2 ) 2 + φ 2 2 .
Q 3 ( 8 / 9 ) [ ( 1 + φ 2 2 ) / φ 2 2 ]     for     φ 1 2 3 φ 2 2 Q 3 ( 8 / φ 1 4 ) ( 1 + φ 2 2 ) 2             for     φ 1 2 3 φ 2 2 .
φ 1 2 = 3 φ 2 2 ,             max ( Q 3 ) = ( 8 / 9 ) [ ( 1 + φ 2 2 ) 2 / φ 2 2 ] .
η 1 = 1 - cos ( γ 2 - γ 1 ) 2 π exp ( - E 0 2 σ 2 ) = sin 2 [ ( γ 2 - γ 1 ) / 2 ] π exp ( - E 0 2 σ 2 ) ,
γ 2 - γ 1 = π ,             E 0 = 0 :     max ( η 1 ) = 1 / π 31.8 % .
η 0 = cos 2 γ 2 - γ 1 2 + sin 2 γ 2 - γ 1 2 [ erf ( E 0 2 1 2 σ ) ] 2 .
N / 2 + η 1 = ( 4 / π 2 )     sin 2 [ ( γ 2 - γ 1 ) / 2 ] = 4 / π 2 40.6 %             for     γ 2 - γ 1 = π .
Q = ( 1 / π ) / ( 4 / π 2 - 1 / π ) = π / ( 4 - π ) = 3.66.
Q 3 = 6.
( flux     contrast ) × total     area     of     image     frame actually     illuminated     area ,