Abstract

The static radiation attenuator consists of one or several layers of absorbing material into which randomly distributed holes are drilled. The attenuator needs no moving parts and reduces not only the time averaged but also the instantaneous brightness, and reduction by an arbitrarily large factor can be easily achieved. The transmitted intensity fluctuates randomly as a function of spatial position, but the expected fluctuation can be evaluated and can be made arbitrarily small. Various properties of the device are explored.

© 1983 Optical Society of America

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References

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  1. A device of this type was recently designed by R. O. Tatchyn to be used in conjunction with synchrotron radiation x rays.
  2. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), Chap. 9.
  3. Exposure is defined in Sec. II.G.
  4. It is assumed that σi ≫ λ. To a good approximation the whole beam is normally incident.
  5. Because the effective illuminated area will not deviate by more than a factor of ≈4, even for points near the edges of the illuminated detector area, for points beyond the boundary of the illuminated area the fluctuation can be larger but is of little interest since hardly any photons will arrive there.
  6. Here we assume that each pinhole receives the same amount of exposure. If that is not the case, each has to be weighted according to the exposure it receives.
  7. Diffraction by previous attenuator layers will cause part of the radiation to deviate from normal incidence. This deviation is governed by λ/ri. We continue to assume λ/ri ≪ 1 and neglect it.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), Chap. 9.

Other

A device of this type was recently designed by R. O. Tatchyn to be used in conjunction with synchrotron radiation x rays.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), Chap. 9.

Exposure is defined in Sec. II.G.

It is assumed that σi ≫ λ. To a good approximation the whole beam is normally incident.

Because the effective illuminated area will not deviate by more than a factor of ≈4, even for points near the edges of the illuminated detector area, for points beyond the boundary of the illuminated area the fluctuation can be larger but is of little interest since hardly any photons will arrive there.

Here we assume that each pinhole receives the same amount of exposure. If that is not the case, each has to be weighted according to the exposure it receives.

Diffraction by previous attenuator layers will cause part of the radiation to deviate from normal incidence. This deviation is governed by λ/ri. We continue to assume λ/ri ≪ 1 and neglect it.

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Figures (9)

Fig. 1
Fig. 1

X rays of wavelength λ are incident from the left. The attenuator consists of a shield of thickness h made of a material which (at this thickness) does not allow x rays to pass through. Pinholes of radius r are drilled in this material to allow a fraction of the incident x rays to pass. The position of the pinholes is random. On the average the number of pinholes per unit surface of shield is N. The average distance between the centers of neighboring pinholes is D. As x rays pass through pinholes, diffraction causes the beam to spread with a typical half-angle θ. At a distance l beyond the pinhole, the diffraction pattern produced by the pinhole has its first minimum at a radius 0.609rF as measured from the axis, where rF ≡ λl/r.

Fig. 2
Fig. 2

Plane x-ray beam is incident on the attenuator from the left. The portion of the wave incident on a pinhole is transmitted through the entrance of the pinhole. Diffraction causes the transmitted beam to spread with a half-angle θ ≈ λ/r for λ ≪ r. As a result, part of the transmitted beam will hit the sidewalls of the pinhole. For θ ≪ 1, the ratio of photons transmitted directly by the exit from the pinhole (on the right) to photons entering the pinhole (from the left) is of the order of 1 − (hλ/r2)2. The rest of the entering photons hit the sidewalls of the pinhole and are either absorbed there or reflected.

Fig. 3
Fig. 3

Point P located on the detector surface is assumed to be illuminated by all pinholes located within a circle of radius rD centered on P0. The P0 is the normal projection of P onto the attenuator surface. So far we assumed σirD, as shown in the figure. When rD > σi, all pinholes located within a circle of radius σi centered on P0 are assumed to illuminate P.

Fig. 4
Fig. 4

Incident beam illuminates a circular are of radius σi on the attenuator surface centered around K0. Any point S within this circular area will contribute some amount to the brightness at P on the detector surface. That contribution depends on rs, the distance between S and P0. P0 is the normal projection of P onto the attenuator surface. Thus any point S lying on the circular are of radius rs centered on P0 between points C and C′ will contribute the same amount. The angle between the lines P0K0 and P0C is φM. That between P0K0 and P0C′ is −φM. The incident beam axis intersects the detector surface at K. The distance between K and P (or K0 and P0) is r0.

Fig. 5
Fig. 5

B(ui,u0) plotted as a function of r0/rF = u0/2π when σi/rF = ui/2π assumes the value (a) 10, (b) 1, (c) 0.1.

Fig. 6
Fig. 6

B(ui,0) plotted as a function of r0/rF = u0/2π.

Fig. 7
Fig. 7

Incident beam illuminates the area inside the solid circle of radius σi centered on K0 on the attenuator surface. Point S is a point chosen inside this circle. The normal projections of points P and P′ (located on the detector surface) onto the attenuator surface are P0 and P 0, respectively. The dashed circle is the normal projection onto the detector surface of the circle of radius σd drawn around P. P 0 lies inside the dashed circle. The distance between K0 and P0 is r0, that between S and P0 is rs, and that between S and P 0 is r s. The angle between lines SP0 and K0P0 is φ, that between SP0 and S P 0 is φ′.

Fig. 8
Fig. 8

C(ui,u0,ud) plotted as a function of r0/rF = u0/2π for σd/rF = ud/2π = 10,3,1,0.3,0.1 when σi/rF = ui/2π assumes the value 10 [Fig. 8(a)], 1 [Fig. 8(b)], and 0.1 [Figs. 8(c) and (d)].

Fig. 9
Fig. 9

Radius of all pinholes drilled in the ith attenuator layer is ri, the number of such holes per unit surface of the ith attenuator layer is Ni, the distance between the ith and (i + 1)st attenuator layer is li.

Tables (1)

Tables Icon

Table I ΔI/I Obtainable with a Single Attenuatora

Equations (66)

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f = N π r 2 ,
N = f / ( π r 2 ) .
( λ h 2 r 2 ) 2 + λ h r 2
δ f f ( λ h 2 r 2 ) 2 + λ h r 2 .
b d ( r d ) = [ J 1 ( 2 π r d / r F ) ] 2 ( π r d 2 ) 1 b i ( π r 2 ) ,
r F = ( λ l ) / r .
b max = ( π r r F ) 2 b i .
r D = 0.609 r F .
r F > r ,
Δ f f = ( N π σ i 2 ) 1 / 2 = 1 f r σ i .
N π r D 2 1 4 f ( r F r ) 2 .
Δ b ( P ) b ( P ) 2 f r r F ; if 2 f r r F 1 ,
Δ b ( P ) b ( P ) 1 f r σ i ; if 1 f r σ i 1 .
Δ b ( r F , σ i , r 0 ) b ( r F , σ i , r 0 ) = π / 2 f r r F B ( u i , u 0 ) , if RHS is 1 ,
u i = 2 π σ i / r F , u 0 = 2 π r 0 / r F ,
B ( u i u 0 + 1 , u 0 ) 2 [ 0 u i + u 0 d u J 1 4 ( u ) / u 3 ] 1 / 2 0 u i + u 0 d u J 1 2 ( u ) / u ,
B ( u i 1,0 ) 2 [ 0 u i d u J 1 4 ( u ) / u 3 ] 1 / 2 0 u i d u J 1 2 ( u ) / u .
Δ I ( r F , σ i , r 0 , σ d ) I ( r F , σ i , r 0 , σ d ) 1 N π ( σ d + r F 2 ) = 2 f r ( 2 σ d + r F ) ; if σ i > r 0 + σ d + r 0 , if right-hand side 1.
Δ I ( r F , σ i , r 0 , σ d ) I ( r F , σ i , r 0 , σ d ) = π 2 f r r F C ( u i , u 0 , u d ) = π f r ( 2 σ d + r F ) ( 2 σ d r F + 1 2 ) C ( u i , u 0 , u d ) ; if right-hand side is 1 , u d 2 π σ d / r F ,
f i = N i π r i 2 .
f = f 1 f 2 f n = N 1 N 2 N n π n ( r 1 r 2 r n ) 2 .
b max , 1 = ( π r 1 r F 1 ) 2 b i .
b max , 2 = ( π r 2 r F 2 ) 2 ( π r 1 r F 1 ) 2 b i .
b max , n = π 2 n ( r 1 r 2 r n r F 1 r F 2 r F n ) 2 b i .
r D , n = ( r D 1 2 + r D 2 2 + + r D n 2 ) 1 / 2 .
b ( P ) = 0 d r s r s π π d φ [ J 1 ( 2 π r s / r F 2 ) ] 2 π r s 2 × f 2 [ J 1 ( 2 π r s 0 / r F 1 ) ] 2 π ( r s 0 ) 2 ( π r i 2 ) b i .
r s 0 = ( r 0 2 + r s 2 2 r 0 r s cos φ ) 1 / 2 .
Δ I I = { i = 1 n [ ( Δ i I I ) 2 + 1 ] 1 } 1 / 2 .
Δ I I [ i = 1 n ( Δ i I I ) 2 ] 1 / 2 .
Δ i I I ( N i A i ) 1 / 2 ; if Δ i I I 1 .
r E , j + 1 2 r E , j 2 + ( 1 2 r F , j ) 2 .
r E , j 2 σ i 2 + 1 4 R = 1 j 1 r F , k 2 .
r P , j 2 σ d 2 + 1 4 k = j n r F , k 2 .
Δ I I = [ i = 1 n ( r i 2 f i r E , i 2 + 1 ) 1 ] 1 / 2 r i / f i r E , i 0 ( i = 1 n r i 2 f i r E , i 2 ) 1 / 2 ; if RHS 1.
Δ I I = [ i = 1 n ( r i 2 f i r P , i 2 + 1 ) 1 ] 1 / 2 r i / f i r D , i 0 ( i = 1 n r i 2 f i r P , i 2 ) 1 / 2 .
Δ I I = [ j = 1 n ( 1 f j r 2 σ j 2 + j r F 2 4 + 1 ) 1 ] 1 / 2 r i / f j r F 0 ( j = 1 n 1 f j r 2 σ d 2 + j 1 4 r F 2 ) 1 / 2 .
f = j = 1 n f j
f j σ i 2 + 1 4 j r F 2 .
Δ I I r σ d 1 f 1 / 2 n n ( n ! 2 n ! ) 1 / 2 n ; if Δ I I 1.
Δ I I r σ d 0.82 ( 2 f ) 1 / 2 n .
Δ 1 b ( P ) b ( P ) = 1 N 1 ( 0 σ i d r r π π d φ 2 π { 0 d r π π d φ 2 π [ J 1 ( 2 π r / r F 1 ) ] 2 π r [ J 1 ( 2 π r s / r F 2 ) ] 2 π r s 2 } 2 ) 1 / 2 0 σ i d r r π π d φ 2 π 0 d r π π d φ 2 π [ J 1 ( 2 π r / r F 1 ) ] 2 π r [ J 1 ( 2 π r s / r F 2 ) ] 2 π r s 2 , r s 2 ( 2 r d cos φ ) r s + r 0 2 r 2 + 2 r r cos φ = 0 ; Δ 2 b ( P ) b ( P ) = 1 N 2 ( 0 d r r π π d φ 2 π { r m r M d r φ M + φ M d φ 2 π [ J 1 ( 2 π r / r F 1 ) ] 2 π r [ J 1 ( 2 π r / r F 2 ) ] 2 π r 2 } 2 ) 1 / 2 0 d r r π π d φ 2 π r m r M d r φ M + φ M d φ 2 π [ J 1 ( 2 π r / r F 1 ) ] 2 π r [ J 1 ( 2 π r / r F 2 ) ] 2 π r 2 , ( φ M , r m , r M ) = L ( σ i , r A , r ) , r A 2 = r 0 2 + r 2 2 r 0 r cos φ ; if right-hand sides 1 .
Δ I I r σ d 0.82 ( 2 f ) 1 / 2 n .
b ( r F , σ i , r 0 , r s ) = d r s φ M + φ M d φ r s [ J 1 ( 2 π r s / r F ) ] 2 π r s 2 N b i π r 2 .
b ( r F , σ i , r 0 ) = r s m r s , M d r s 2 φ M r s [ J 1 ( 2 π r s / r F ) ] 2 π r s 2 N b i π r 2 .
φ M = | arccos ( r 0 2 + r s 2 σ i 2 ) / 2 r 0 r s | .
φ M = { | arccos ( r 0 2 + r s 2 σ i 2 ) / 2 r 0 r s | ; if 0 r 0 σ d , and | r 0 σ d | r s r 0 + σ d π ; if r 0 < σ d , and r s < σ d r 0 0 ; otherwise ,
r s , m = { r 0 σ i ; if σ i < r 0 , 0 ; if σ i r 0 ,
r s , M = σ i + r 0 .
( r s , m , r s , M , φ M ) = L ( σ i , r 0 , r s )
N d r s r s d φ ( b i π r 2 ) 2 [ J 1 2 ( 2 π r s / r F ) π r s 2 ] 2 ,
[ Δ b ( r F , σ i , r 0 ) ] 2 = N ( b i π r 2 ) 2 r s , m r s , M d r s 2 φ M [ J 1 ( 2 π r s / r F ) ] 4 π 2 r s 3 ,
Δ b ( r F , σ i , r 0 ) b ( r F , σ i , r 0 ) = 1 2 π f r r F B ( u i , u 0 ) ,
u i = 2 π σ i / r F , u 0 = 2 π r 0 / r F ,
B ( u i , u 0 ) 2 [ u m u M d u φ M π J 1 4 ( u ) / u 3 ] 1 / 2 u m u M d u φ M π J 1 2 ( u ) / u ;
d r s r s d φ N { ( b i π r 2 ) r s , m r s , M d r s r s × φ M + φ M d φ [ J 1 ( 2 π r s / r F ) ] 2 / π ( r s ) 2 } ,
( r s , m , r s , M , φ M ) = L ( σ d , r s , r s ) ,
I ( r F , σ i , r 0 , σ d ) = N b i π r 2 r s , m r s , M d r s r s φ M φ M d φ × r s , m r s , M d r s r s φ M + φ M d φ [ J 1 ( 2 π r s / r F ) ] 2 π ( r s ) 2 ,
( r s , m r s , M , φ M ) = L ( δ i , r 0 , r s ) .
Δ I ( r F , σ i , r 0 , σ d ) I ( r F , σ i , r 0 , σ d ) = π / 2 f r r F C ( u i , u 0 , u d ) ,
C ( u i , u 0 , u d ) = 2 { υ m υ M d υ υ φ π [ u m u M d u φ π J 1 2 ( u ) / u ] 2 } 1 / 2 υ m υ M d υ υ φ π u m u M d u φ π J 1 2 ( u ) / u ,
u i = 2 π σ i / r F ; u = 2 π r s / r F , u 0 = 2 π r 0 / r F ; υ = 2 π r s / r F , u d = 2 π σ d / r F ,
( φ , u m , u M ) = L ( u d , υ , u ) ; ( φ , υ m , υ M ) = L ( u i , u 0 , υ ) .
Δ a i = a i a 0 , Δ b j = b j b 0 , Δ x k = x k x 0 .
δ a 2 = i ( Δ a i ) 2 / a 0 2 , δ b 2 = j ( Δ b j ) 2 / b 0 2 , , δ x 2 = k ( Δ x k ) 2 / x 0 2 .
i Δ a i = j Δ b j = = k Δ x k = 0.
δ a , b 2 , , x = i , j , k ( a i b j x k a 0 b 0 x 0 ) 2 / ( a 0 b 0 x 0 ) 2 = i , j , k [ ( Δ a i + a 0 ) ( Δ b j + b 0 ) ( Δ x k + x 0 ) a 0 b 0 x 0 ] 2 / ( a 0 b 0 x 0 ) 2 = ( δ a 2 + 1 ) ( δ b 2 + 1 ) ( δ x 2 + 1 ) 1 ,

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