Abstract

Formulas have been derived for calculating the modulation transfer function of x-ray intensifying screens. Emphasis has been placed on simplified models where results can be obtained in closed form and a physical understanding more easily realized. For selected cases, the MTF’s of transparent screens are calculated by direct integration. The MTF’s of diffusing screens are determined from suitable approximate solutions of the Boltzmann equation. The intermediate case and the transition from scattering to nonscattering are discussed. As a byproduct, formulas for light output and signal-to-noise ratio are obtained.

© 1973 Optical Society of America

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References

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  1. J. Gasper, J. Opt. Soc. Am. 62, 720A (1972); complete paper to be published in J. Opt. Soc. Am.
  2. R. E. Marshak, H. Brooks, H. Hurwitz, Nucleonics 4 (5), 10 (1949). Our Eq. (9) is their Eq. (20).
    [PubMed]
  3. σ and τ may be related to frequently used scattering parameters as follows: Let λa = absorption mean free path; λtr = transport mean free path. Then τ = (3/λaλtr)1/2 = (2τ/λa)1/2.
  4. H. C. Hamaker, Philips Res. Rept. 2, 55 (1947). Our Eq. (23) is his Eq. (27) except for the following differences: his σ0, β0, ρ1, ρ2, D are represented in our formula by σ, σ/τ, ρ0, ρ1, T, respectively. In addition, our formula is divided by the factor N0K(1 − e−μD). (his terminology) relative to his, making the result measure the fraction of generated light that reaches the photocathode, independent of x-ray flux absorption, etc. Finally, our formula contains a factor 2ρ2/(1 + ρ2) relative to his, which is the transmission factor of the phosphor–photocathode interface. The use of this factor may be open to question. If the dominant influence in ρ2 is reflection at the photocathode–vacuum interface, this factor should probably be omitted. Note that this factor equals unity when the cathode is nonreflecting, as was assumed in most of the results presented here. In any event, the relative MTF curve is not affected by this factor.
  5. R. K. Swank, to be published in J. Appl. Phys.Sept1973. GE Report 72CRD322, Schenectady, N.Y. (1972).
  6. R. E. Marshak, H. Brooks, H. Hurwitz, Nucleonics 4 (6), 43 (1949).

1972 (1)

J. Gasper, J. Opt. Soc. Am. 62, 720A (1972); complete paper to be published in J. Opt. Soc. Am.

1949 (2)

R. E. Marshak, H. Brooks, H. Hurwitz, Nucleonics 4 (5), 10 (1949). Our Eq. (9) is their Eq. (20).
[PubMed]

R. E. Marshak, H. Brooks, H. Hurwitz, Nucleonics 4 (6), 43 (1949).

1947 (1)

H. C. Hamaker, Philips Res. Rept. 2, 55 (1947). Our Eq. (23) is his Eq. (27) except for the following differences: his σ0, β0, ρ1, ρ2, D are represented in our formula by σ, σ/τ, ρ0, ρ1, T, respectively. In addition, our formula is divided by the factor N0K(1 − e−μD). (his terminology) relative to his, making the result measure the fraction of generated light that reaches the photocathode, independent of x-ray flux absorption, etc. Finally, our formula contains a factor 2ρ2/(1 + ρ2) relative to his, which is the transmission factor of the phosphor–photocathode interface. The use of this factor may be open to question. If the dominant influence in ρ2 is reflection at the photocathode–vacuum interface, this factor should probably be omitted. Note that this factor equals unity when the cathode is nonreflecting, as was assumed in most of the results presented here. In any event, the relative MTF curve is not affected by this factor.

Brooks, H.

R. E. Marshak, H. Brooks, H. Hurwitz, Nucleonics 4 (5), 10 (1949). Our Eq. (9) is their Eq. (20).
[PubMed]

R. E. Marshak, H. Brooks, H. Hurwitz, Nucleonics 4 (6), 43 (1949).

Gasper, J.

J. Gasper, J. Opt. Soc. Am. 62, 720A (1972); complete paper to be published in J. Opt. Soc. Am.

Hamaker, H. C.

H. C. Hamaker, Philips Res. Rept. 2, 55 (1947). Our Eq. (23) is his Eq. (27) except for the following differences: his σ0, β0, ρ1, ρ2, D are represented in our formula by σ, σ/τ, ρ0, ρ1, T, respectively. In addition, our formula is divided by the factor N0K(1 − e−μD). (his terminology) relative to his, making the result measure the fraction of generated light that reaches the photocathode, independent of x-ray flux absorption, etc. Finally, our formula contains a factor 2ρ2/(1 + ρ2) relative to his, which is the transmission factor of the phosphor–photocathode interface. The use of this factor may be open to question. If the dominant influence in ρ2 is reflection at the photocathode–vacuum interface, this factor should probably be omitted. Note that this factor equals unity when the cathode is nonreflecting, as was assumed in most of the results presented here. In any event, the relative MTF curve is not affected by this factor.

Hurwitz, H.

R. E. Marshak, H. Brooks, H. Hurwitz, Nucleonics 4 (6), 43 (1949).

R. E. Marshak, H. Brooks, H. Hurwitz, Nucleonics 4 (5), 10 (1949). Our Eq. (9) is their Eq. (20).
[PubMed]

Marshak, R. E.

R. E. Marshak, H. Brooks, H. Hurwitz, Nucleonics 4 (5), 10 (1949). Our Eq. (9) is their Eq. (20).
[PubMed]

R. E. Marshak, H. Brooks, H. Hurwitz, Nucleonics 4 (6), 43 (1949).

Swank, R. K.

R. K. Swank, to be published in J. Appl. Phys.Sept1973. GE Report 72CRD322, Schenectady, N.Y. (1972).

J. Opt. Soc. Am. (1)

J. Gasper, J. Opt. Soc. Am. 62, 720A (1972); complete paper to be published in J. Opt. Soc. Am.

Nucleonics (2)

R. E. Marshak, H. Brooks, H. Hurwitz, Nucleonics 4 (5), 10 (1949). Our Eq. (9) is their Eq. (20).
[PubMed]

R. E. Marshak, H. Brooks, H. Hurwitz, Nucleonics 4 (6), 43 (1949).

Philips Res. Rept. (1)

H. C. Hamaker, Philips Res. Rept. 2, 55 (1947). Our Eq. (23) is his Eq. (27) except for the following differences: his σ0, β0, ρ1, ρ2, D are represented in our formula by σ, σ/τ, ρ0, ρ1, T, respectively. In addition, our formula is divided by the factor N0K(1 − e−μD). (his terminology) relative to his, making the result measure the fraction of generated light that reaches the photocathode, independent of x-ray flux absorption, etc. Finally, our formula contains a factor 2ρ2/(1 + ρ2) relative to his, which is the transmission factor of the phosphor–photocathode interface. The use of this factor may be open to question. If the dominant influence in ρ2 is reflection at the photocathode–vacuum interface, this factor should probably be omitted. Note that this factor equals unity when the cathode is nonreflecting, as was assumed in most of the results presented here. In any event, the relative MTF curve is not affected by this factor.

Other (2)

R. K. Swank, to be published in J. Appl. Phys.Sept1973. GE Report 72CRD322, Schenectady, N.Y. (1972).

σ and τ may be related to frequently used scattering parameters as follows: Let λa = absorption mean free path; λtr = transport mean free path. Then τ = (3/λaλtr)1/2 = (2τ/λa)1/2.

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Figures (10)

Fig. 1
Fig. 1

Diagram for calculating the MTF of a transparent phosphor.

Fig. 2
Fig. 2

Effect of x-ray absorption on MTF of a transparent phosphor.

Fig. 3
Fig. 3

Effect of a transparent index-matched nonphosphor layer of thickness D between phosphor and photocathode.

Fig. 4
Fig. 4

Effect of bulk light absorption in a transparent phosphor. G(0) is the fraction of the emitted light that reaches the photocathode.

Fig. 5
Fig. 5

Photon density function ψk for a filamentary light source at z = z0.

Fig. 6
Fig. 6

Special cases of scattering and nonscattering phosphors. Curve A, transparent phosphor, reflective backing; B, transparent phosphor, black backing; C, diffusion limit, no light absorption in bulk or backing; D, diffusion limit, reflective backing, 50% light absorption in bulk; E, diffusion limit, black backing, no bulk absorption. Nonreflecting photocathode assumed in all cases.

Fig. 7
Fig. 7

Effect of scattering constant on MTF, according to Eq. (23).

Fig. 8
Fig. 8

Check of diffusion theory by integral equation calculation for isotropic scattering.

Fig. 9
Fig. 9

Effect of cathode reflectivity and scattering constant on MTF.

Fig. 10
Fig. 10

Relative light output vs back reflectance for several values of scattering constant. Note that when τT is large, all backings tend to behave like black backings.

Equations (47)

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F ( x ) = 1 T 0 T - z 4 π r 3 d y d z ,
F ( x ) = ( 1 / 4 π T ) ln ( 1 + T 2 / x 2 ) .
G ( ν ) = ( 1 - e - 2 π ν T ) / 4 π ν T .
G ( 0 ) = - F ( x ) d x
G ( ν ) = 1 2 [ μ / ( e μ T - 1 ) ] { [ 1 - e - ( 2 π ν - μ ) T ] / ( 2 π ν - μ ) } .
G ( ν ) = e - 2 π ν D ( 1 - e - 2 π ν T ) / 4 π ν T .
G ( ν ) = 0 2 T z d z - e - i k x d x - e - A r 4 π r 3 d y ,
G ( ν ) = 1 2 0 2 T z d z z e - A r r 2 J 0 [ k ( r 2 - z 2 ) 1 / 2 ] d r ,
- 2 ϕ ( r ) + σ 2 ϕ ( r ) = S ( r ) ,
S ( r ) = δ ( x ) δ ( z - z 0 ) = [ δ ( z - z 0 ) / 2 π ] - cos k x d k ,
ϕ ( x , z ) = - ψ k ( z ) cos k x d k .
[ - ( d 2 ψ k / d z 2 ) ] + ( σ 2 + k 2 ) ψ k = δ ( z - z 0 ) ,
ψ k = A e q z + B e - q z ,
q = ( σ 2 + k 2 ) 1 / 2 .
j left = 1 2 [ ϕ τ + ( d z / d ϕ ) ] ,
j right = 1 2 [ ϕ τ - ( d ϕ / d z ) ] ,
j right = r 0 j left .
d ϕ / d z z = 0 = ρ 0 τ ϕ z = 0 ,
ρ 0 ( 1 - r 0 ) / ( 1 + r 0 )
d ϕ / d z z = T = ρ 1 τ ϕ z = T ,
lim 0 [ ( d ϕ / d z ) z 0 - + ( d ϕ / d z ) z 0 + ] = 1.
G ( ν , z 0 ) = [ ρ 1 / ( 1 + ρ 1 ) ] [ ψ k τ - ( d ψ k / d z ) ] z = T .
G ( ν , z 0 ) = τ ρ 1 { [ ( q + τ ρ 0 ) e q z 0 + ( q - τ ρ 0 ) e - q z 0 ] / [ ( q + τ ρ 0 ) ( q + τ ρ 1 ) e q T - ( q - τ ρ 0 ) ( q - τ ρ 1 ) e - q T ] } .
G ( ν ) = [ ( μ τ ρ 1 e - μ T ) / ( 1 - e - μ T ) ( μ 2 - q 2 ) ] × { [ - ( μ + q ) ( q + τ ρ 0 ) e q T - ( μ - q ) ( q - τ ρ 0 ) e - q T + 2 q ( μ + τ ρ 0 ) e μ T ] / [ ( q + τ ρ 0 ) ( q + τ ρ 1 ) e - q T - ( q - τ ρ 0 ) ( q - τ ρ 1 ) e - q T ] } ,
E ( z 0 ) G ( 0 , z 0 )
A N = A N - τ ,
I = M 1 2 / M 2 M 0
M j = 0 T N ( z 0 ) E j ( z 0 ) d z 0 ,
M 2 = { [ μ τ 2 ρ 1 2 / ( 1 - e - μ T ) ] / [ ( σ + τ ρ 0 ) ( σ + τ ρ 1 ) e σ T - ( σ - τ ρ 0 ) ( σ - τ ρ 1 ) e - σ T ] 2 } × { [ ( σ + τ ρ 0 ) 2 ( 1 - e - ( μ - 2 σ ) T ) / ( μ - 2 σ ) ] + [ ( σ - τ ρ 0 ) 2 ( 1 - e - ( μ + 2 σ ) T ) / ( μ + 2 σ ) ] + + [ 2 ( σ 2 - τ 2 ρ 0 2 ) ( 1 - e - μ T ) / μ ] }
I = G 2 ( 0 ) / M 2 .
G ( ν , z 0 ) = cosh 2 π ν z 0 / cosh 2 π ν T
G ( ν ) = tanh 2 π ν T / 2 π ν T .
E ( z 0 ) = G ( 0 ) = I = 1.
G ( ν , z 0 ) = sinh 2 π ν z 0 / sinh 2 π ν T ,
G ( ν ) = ( cosh 2 π ν T - 1 ) / 2 π ν T sinh 2 π ν T ,
E ( z 0 ) = z 0 / T ,
G ( 0 ) = 0.5 ,
I = 0.75.
G ( ν , z 0 ) = cosh ( 4 π 2 ν 2 + σ 2 ) 1 / 2 z 0 / cosh ( 4 π 2 ν 2 + σ 2 ) 1 / 2 T ,
G ( ν ) = tanh ( 4 π 2 ν 2 + σ 2 ) 1 / 2 T / ( 4 π 2 ν 2 + σ 2 ) 1 / 2 T ,
E ( z 0 ) = ( cosh σ z 0 ) / cosh σ T .
G ( 0 ) = 0.482 , I = 0.841.
η k ( z ) = 0 1 K k ( z - z ) η k ( z ) d z + 1 ,
K k ( z ) = τ 3 0 1 exp - ( s 2 + k 2 ) 1 / 2 z ( s 2 + k 2 ) 1 / 2 d s .
G ( ν ) = 0 1 η k ( z ) I k ( z ) d z
I k ( z ) = z 2 z ( e - 2 r r / 3 / r 2 ) J 0 [ k ( r 2 - z 2 ) 1 / 2 ] d r
k = 2 π ν     as before .

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