Abstract

A Fresnel zone plate, used as a coded aperture, offers a great advantage in geometric collection efficiency over a conventional pinhole or collimator. We present a detailed analysis of the signal-to-noise ratio (SNR) of a quantum-limited zone plate camera. The magnitude and spatial distribution of the noise field and its dependence on the source distribution and the characteristics of the optical processing system are derived. It is shown that the largest SNR advantage occurs for a point source, while for very large, uniform sources there may be a slight net disadvantage to using a zone plate. It is also shown that optical processing does not give the highest possible SNR.

© 1974 Optical Society of America

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References

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  1. N. D. Young, Sky & Telescope 25, 8 (1963).
  2. L. Mertz, N. D. Young, Proc. Int. Conf. Opt. Instrum., London (1961), p. 305.
  3. H. Barrett, J. Nucl. Med. 13, 382 (1972).
    [PubMed]
  4. W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, J. Nuc. Med. 13, 612 (1972).
  5. H. H. Barrett, G. D. DeMeester, D. T. Wilson, M. H. Farmelant, Medical Radioisotope Scintigraphy 1972 (IAEA, Vienna, 1937, Vol. 1, p. 281.
  6. H. H. Barrett, F. A. Horrigan, Appl. Opt. 12, 2686 (1973).
    [CrossRef] [PubMed]
  7. J. W. Goodman, in Modern OpticsJ. Fox, Ed. (Polytechnic Press, Brooklyn, 1967), p. 573.
  8. S. O. Rice, Bell Syst. Tech. J. 23, 297 (1944).
  9. S. O. Rice, Bell Syst. Tech. J. 23, 331 (1944).

1973 (1)

1972 (2)

H. Barrett, J. Nucl. Med. 13, 382 (1972).
[PubMed]

W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, J. Nuc. Med. 13, 612 (1972).

1963 (1)

N. D. Young, Sky & Telescope 25, 8 (1963).

1944 (2)

S. O. Rice, Bell Syst. Tech. J. 23, 297 (1944).

S. O. Rice, Bell Syst. Tech. J. 23, 331 (1944).

Barrett, H.

H. Barrett, J. Nucl. Med. 13, 382 (1972).
[PubMed]

Barrett, H. H.

H. H. Barrett, F. A. Horrigan, Appl. Opt. 12, 2686 (1973).
[CrossRef] [PubMed]

H. H. Barrett, G. D. DeMeester, D. T. Wilson, M. H. Farmelant, Medical Radioisotope Scintigraphy 1972 (IAEA, Vienna, 1937, Vol. 1, p. 281.

Beierwaltes, W. H.

W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, J. Nuc. Med. 13, 612 (1972).

DeMeester, G. D.

H. H. Barrett, G. D. DeMeester, D. T. Wilson, M. H. Farmelant, Medical Radioisotope Scintigraphy 1972 (IAEA, Vienna, 1937, Vol. 1, p. 281.

Farmelant, M. H.

H. H. Barrett, G. D. DeMeester, D. T. Wilson, M. H. Farmelant, Medical Radioisotope Scintigraphy 1972 (IAEA, Vienna, 1937, Vol. 1, p. 281.

Goodman, J. W.

J. W. Goodman, in Modern OpticsJ. Fox, Ed. (Polytechnic Press, Brooklyn, 1967), p. 573.

Han, K. S.

W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, J. Nuc. Med. 13, 612 (1972).

Horrigan, F. A.

Jones, L. W.

W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, J. Nuc. Med. 13, 612 (1972).

Mertz, L.

L. Mertz, N. D. Young, Proc. Int. Conf. Opt. Instrum., London (1961), p. 305.

Rice, S. O.

S. O. Rice, Bell Syst. Tech. J. 23, 297 (1944).

S. O. Rice, Bell Syst. Tech. J. 23, 331 (1944).

Rogers, W. L.

W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, J. Nuc. Med. 13, 612 (1972).

Wilson, D. T.

H. H. Barrett, G. D. DeMeester, D. T. Wilson, M. H. Farmelant, Medical Radioisotope Scintigraphy 1972 (IAEA, Vienna, 1937, Vol. 1, p. 281.

Young, N. D.

N. D. Young, Sky & Telescope 25, 8 (1963).

L. Mertz, N. D. Young, Proc. Int. Conf. Opt. Instrum., London (1961), p. 305.

Appl. Opt. (1)

Bell Syst. Tech. J. (2)

S. O. Rice, Bell Syst. Tech. J. 23, 297 (1944).

S. O. Rice, Bell Syst. Tech. J. 23, 331 (1944).

J. Nuc. Med. (1)

W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, J. Nuc. Med. 13, 612 (1972).

J. Nucl. Med. (1)

H. Barrett, J. Nucl. Med. 13, 382 (1972).
[PubMed]

Sky & Telescope (1)

N. D. Young, Sky & Telescope 25, 8 (1963).

Other (3)

L. Mertz, N. D. Young, Proc. Int. Conf. Opt. Instrum., London (1961), p. 305.

H. H. Barrett, G. D. DeMeester, D. T. Wilson, M. H. Farmelant, Medical Radioisotope Scintigraphy 1972 (IAEA, Vienna, 1937, Vol. 1, p. 281.

J. W. Goodman, in Modern OpticsJ. Fox, Ed. (Polytechnic Press, Brooklyn, 1967), p. 573.

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Figures (6)

Fig. 1
Fig. 1

Schematic of a Fresnel zone-plate camera. The gamma rays emitted by the source are efficiently transmitted by the zone-plate aperture and detected by a scintillation camera or other image detector. Each individual detected gamma-ray photon produces a spot on the photographic transparency that is the ultimate output of the image detector. The complete pattern of spots is the coded image or hologram that is to be decoded in a coherent optical system.

Fig. 2
Fig. 2

Diagram showing the definition of various coordinates in the hologram or film plane.

Fig. 3
Fig. 3

Diagram of a simple coherent optical reconstruction system using a plane wave of light.

Fig. 4
Fig. 4

A practical reconstruction employing an iris to restrict the spatial frequency passband. The light amplitude distribution in plane A, a distance fl from the hologram, is proportional to the Fourier transform of the hologram transmission. Plane B, just beyond the iris, is the filtered Fourier transform. The reconstruction illustrated here for a single point source appears in plane C, a distance z from the hologram.

Fig. 5
Fig. 5

The signal-to-noise ratio in the reconstructed image as a function of the total number of quanta collected. The object is assumed to be uniform over an area of M resolution elements and to be small compared to the zone-plate shadow.

Fig. 6
Fig. 6

Illustration of the Relative SNR advantage of a zone-plate compared to a pinhole of the same resolution and spacings (s1 and s2). The quantities ηzp and ηph are the geometric collection efficiencies of the zone plate and pinhole, respectively. The ratio ηzp:ηph is typically about 1000.

Tables (1)

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Table I a

Equations (90)

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T ( r ) = m = 1 M n = 1 K m Z ( r n m - ζ m ) g ( r - r n m ) .
Z ( r ) = { 1 if sin α ρ 2 0 and ρ D / 2 , 0 if sin α ρ 2 > 0 and ρ D / 2 , 0 if ρ > D / 2.
T ( r ) = m = 1 M Z ( r - ζ m ) n = 1 K m g ( r - r n m ) .
N m = K m ( π D 2 / 8 a f ) ,
A ( r ) = d 2 r T ( r ) [ exp ( i k R ) / R ] ,
R z [ 1 + ( 1 / 2 z 2 ) ( ρ 2 + ρ 2 - 2 ρ ρ cos ( θ - θ ) ] .
Z ( r ) = ½ - ( 2 / π ) sin α ρ 2 - ( 2 / 3 π ) sin 3 α ρ 2 - ( 2 / 5 π ) sin 5 α ρ 2 ( ρ D / 2 )
Z ( r ) = ½ + i π l = - 1 2 l - 1 exp [ ( 2 l - 1 ) i α ρ 2 ] .
A ( r ) = z - 1 exp ( i ϕ ) d 2 r Z ( r ) × exp [ i k ( ρ 2 2 z - ρ ρ cos θ z ) ] n = 1 K g ( r - r n ) ,
ϕ = ( k ρ 2 / 2 z ) + k z .
z = k / 2 α .
A ( r ) = exp ( i ϕ ) i π z 0 D / 2 ρ d ρ 0 2 π d θ × exp ( - i k ρ ρ cos θ / z ) n = 1 K g ( r - r n ) .
n = 1 K g ( r - r n ) = ( a s / a f ) K = 8 a s N / π D 2 ,
a s = d 2 r g ( r ) .
A ( r ) = [ exp ( i ϕ ) / i π z ] × 2 a s N [ 2 J 1 ( k D ρ / 2 z ) / ( k D ρ / 2 z ) ] .
A ¯ A ( r ) .
( Δ A ) 2 A ( r ) - A ¯ 2 = A ( r ) 2 - A ¯ 2 .
A ( r ) 2 = d 2 r d 2 r Z ( r ) Z ( r ) W ( r , r ) W * ( r , r ) × m , n K g ( r - r m ) g ( r - r n ) ,
W ( r , r ) = z - 1 exp [ ( i k ρ 2 / 2 z ) - ( i k ρ ρ cos θ / z ) ] .
m , n K g ( r - r m ) g ( r - r n ) = ( K / a f ) R g ( τ ) + ( a s K / a f ) 2 ,
R g ( τ ) g ( r ) g ( r + τ ) d 2 r ,
τ = r - r .
g ( r ) = a s δ ( r ) ,
R g ( τ ) = a s 2 δ ( τ ) .
A ( r ) 2 = K a s 2 a f d 2 r [ Z ( r ) ] 2 W ( r , r ) 2 + ( a s K a f ) 2 d 2 r Z ( r ) W ( r , r ) 2 .
W ( r , r ) 2 = z - 2
[ Z ( r ) ] 2 = Z ( r ) .
d 2 r Z ( r ) = ½ ( π D 2 / 4 ) .
( Δ A ) δ 2 = N a s 2 / z 2 .
A ( r = 0 ) δ / ( Δ A ) δ = ( 2 / π ) N 1 / 2 .
T ( r ) = n Z ( r n ) g ( r - r n ) = d 2 r 0 Z ( r 0 ) n δ ( r 0 - r n ) g ( r - r 0 ) = [ Z ( r ) n δ ( r - r n ) ] g ( r ) ,
A ( r ) = A ( r ) δ g ( r ) .
( Δ A ) 2 = K d 2 r d 2 r W ( r , r ) W * ( r , r ) × d 2 r n a f [ Z ( r n ) ] 2 g ( r - r n ) g ( r - r n ) .
W ( r , r ) W * ( r , r ) = z - 2 exp [ ( i k / 2 z ) ( r - r 2 - r - r 2 ) ] .
r - r n = r 0 , r - r n = r 0 = r 0 + τ .
r - r 2 - r - r 2 = r 0 2 - r 0 2 + 2 ( r 0 - r 0 ) · ( r n - r ) .
( Δ A ) 2 = K a f z 2 [ Z ( r n ) ] 2 g ( r 0 ) g ( r 0 + τ ) × exp [ i k z ( r 0 - r n ) · τ ] d 2 r 0 d 2 r n d 2 τ .
( Δ A ) 2 = K a f z 2 [ Z ( r n ) ] 2 R g ( τ ) exp [ i k z ( r - r n ) · τ ] d 2 r n d 2 τ .
S g ( ω ) d 2 τ R g ( τ ) exp ( - i ω · τ ) .
( Δ A ) 2 = ( K / a f z 2 ) [ Z ( r ) ] 2 S g ( k r / z ) .
D iris = ( Δ ν ) λ f l ,
( Δ ν ) z p = D / λ f z ,
D iris = D f l / f z .
1 / z = ( 1 / f l ) - ( 1 / f z ) .
D iris / ( z - f l ) = D / z .
( Δ A ) 2 = ( 4 d s 2 N / π D 2 z 2 ) circ ( 2 r / D ) circ ( 2 r / D ) ,
circ ( r / r 0 ) { 1 if r r 0 , 0 if r > r 0 , D D [ ( z - f l ) / f l ] .
A ( r ) δ = ( 2 i a s / π z ) exp ( i k z ) m = 1 M N m P ( r - ζ m )
( Δ A ) δ 2 = ( a s 2 / z 2 ) m = 1 M N m ,
P ( r ) = [ 2 J 1 ( k D r / 2 z ) / ( k D r / 2 z ) ] exp [ ( i k / 2 z ) r 2 ] .
A ( r = ζ m ) δ / ( Δ A ) δ = ( 2 / π ) N m / N t 1 / 2 ( 2 / π ) N 1 / 2 / M ,
N t m = 1 M N m M N .
A ( r ) = ( 2 i / π z ) exp ( i k z ) [ n ( r ) P ( r ) g ( r ) ] .
( Δ A ) 2 = ( 8 / π D 2 z 2 ) { n ( r ) [ Z ( r ) ] 2 S g ( k r / z ) } .
( Δ A ) 2 = ( 8 / π D 2 z 2 ) { [ ( n Z 2 ) · circ ( r / R d ) ] S g } .
I i / σ A 2 / [ A 4 - A 2 2 ] 1 / 2 .
A A s + A n = A s + x + i y ,
A s 2 = A s 2 = A ¯ 2 , A s 4 = A s 4 = A ¯ 4 , A n = 0 , A n 2 = ( Δ A ) 2 .
A 4 = A s 4 + 4 A s 2 A n 2 + A n 2 ,
A n 4 = x 4 + y 4 + 2 x 2 y 2 = 2 x 4 + 2 x 2 2 .
x 4 = y 4 = 3 x 2 2 .
A n 4 = 8 x 2 2 = 2 A n 2 2 ,
A 4 = A s 4 + 4 A s 2 A n 2 + 2 A n 2 2 .
I i / σ = w / ( 1 + 2 w ) 1 / 2 ,
w = A s 2 / A n 2 = A ¯ 2 / ( Δ A ) 2 .
I i / σ ( 1 / 2 ) [ A ¯ / ( Δ A ) ] .
( SNR ) z p I i / σ = ( D z p / π D p h ) ( SNR ) p h ( point source ) .
( SNR ) z p = ( 2 D p h / π D z p ) N 1 / 2 = ( 1 / π ) ( SNR ) p h ( infinite source ) .
G = n = 1 K g ( r - r n )
G 2 = n = 1 K m = 1 K g ( r - r n ) g ( r - r m ) .
G = 0 p ( K ) d K d 2 r 1 a f d 2 r K a f n = 1 K g ( r - r n ) ,
G = 0 p ( K ) d K n = 1 K d 2 r n a f g ( r - r n ) .
d 2 r n a f g ( r - r n ) = a s a f .
G = ( a s / a f ) 0 K p ( K ) d K = ( a s / a f ) K ,
G 2 = 0 ρ ( K ) d K n = 1 K m = 1 K d 2 r 1 a f d 2 r k a f × g ( r - r n ) g ( r - r m ) .
d 2 r n a f g ( r n ) g ( r n + τ ) a f - 1 R g ( τ ) ,
d 2 r n a f g ( r - r n ) d 2 r m a f g ( r - r m ) = a f - 2 [ d 2 r g ( r ) ] 2 = a s 2 / a f 2 .
G 2 = 0 p ( K ) d K [ K a f R g ( τ ) + ( K 2 - K ) a s 2 a f 2 ] = K a f R g ( τ ) + K 2 - K a s 2 a f 2
K 2 - K = K 2 ,
r n = r 1 n .
α = π / r 1 2 .
f z = ± r 1 2 / λ .
ν ( ρ ) = ρ / r 1 2 .
( Δ ν ) z p = D / r 1 2 .
( Δ r ) min = 1 / 2 ν max = r 1 2 / D = r 1 / 2 n z ,
D / 2 = r 1 n z ,
n z = ( D / 2 r 1 ) 2 .
exp [ ( i k / 2 z ) r 0 2 ] ,
( k / 2 z ) r 0 2 ( k / 2 f z ) ( Δ r ) min 2 .
k r 0 2 / 2 z π / 4 n z .

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