Characterizing flash-radiography source spots

Carl Ekdahl

doi:10.1364/JOSAA.28.002501

Characterizing flash-radiography source spots

Carl Ekdahl

Journal of the Optical Society of America A
Vol. 28,
Issue 12,
pp. 2501-2509
(2011)
•https://doi.org/10.1364/JOSAA.28.002501

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Carl Ekdahl, "Characterizing flash-radiography source spots," J. Opt. Soc. Am. A 28, 2501-2509 (2011)

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Related Topics
Table of Contents Category
- X-ray Optics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Image quality assessment (110.3000)
- Modulation transfer function (110.4100)
- Transforms (350.6980)
- X-ray imaging (340.7440)

About this Article
History
- Original Manuscript: August 2, 2011
- Manuscript Accepted: August 24, 2011
- Published: November 11, 2011

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Open Access

Abstract

Flash radiography of large hydrodynamic experiments driven by high explosives is a venerable diagnostic technique in use at many laboratories. The size of the radiographic source spot is often quoted as an indication of the resolving power of a particular flash-radiography machine. A variety of techniques for measuring spot size have evolved at the different laboratories, as well as different definitions of spot size. Some definitions are highly dependent on the source spot intensity distributions, and not necessarily well correlated with resolution. The concept of limiting resolution based on bar target measurements is introduced, and shown to be equivalent to the spatial wavenumber at a modulation transfer function value of 5%. This resolution is shown to be better correlated with the full width at half-maximum of the spot intensity distribution than it is with other definitions of spot size.

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References

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|
Year
|
Author
|
Publication

T. J. Boyd, B. T. Rogers, R. R. Tesche, and Douglas Venable, “PHERMEX—A high-current electron accelerator for use in dynamic radiography,” Rev. Sci. Instrum. 36, 1401–1408 (1965).
[Crossref]
C. Ekdahl, “Modern electron accelerators for radiography,” IEEE Trans. Plasma Sci. 30, 254–261 (2002).
[Crossref]
K. Mueller, “Measurement and characterization of x-ray spot size,” Los Alamos National Laboratory Report LA-UR-89-1886 (August 1989).
D. W. Forster, “Radiographic spot size definitions,” AWE Report HWH/9201/R8 (January 1992).
“Modulation transfer function of screen-film systems,” ICRU Report 41 (International Commission on Radiation Units and Measurements, 1986).
“Military standard photographic lenses,” MIL-STD-150A (U.S. Government Printing Office, 1959), p. 21.
E. Rose and T. Beery, Los Alamos National Laboratory (personal communication, 2006).
T. J. Goldsack, T. F. Bryant, P. F. Beech, S. J. Clough, G. M. Cooper, R. Davitt, R. D. Edwards, N. Kenna, J. McLean, A. G. Pearce, M. J. Phillips, K. P. Pullinger, D. J. Short, M. A. Sinclair, K. J. Thomas, J. R. Threadgold, M. C. Williamson, and K. Krushelnick, “Multimegavolt multiaxis high-resolution flash x-ray source development for a new hydrodynamics research facility at AWE Aldermaston,” IEEE Trans. Plasma Sci. 30, 239–253 (2002).
[Crossref]

2002 (2)

C. Ekdahl, “Modern electron accelerators for radiography,” IEEE Trans. Plasma Sci. 30, 254–261 (2002).
[Crossref]

T. J. Goldsack, T. F. Bryant, P. F. Beech, S. J. Clough, G. M. Cooper, R. Davitt, R. D. Edwards, N. Kenna, J. McLean, A. G. Pearce, M. J. Phillips, K. P. Pullinger, D. J. Short, M. A. Sinclair, K. J. Thomas, J. R. Threadgold, M. C. Williamson, and K. Krushelnick, “Multimegavolt multiaxis high-resolution flash x-ray source development for a new hydrodynamics research facility at AWE Aldermaston,” IEEE Trans. Plasma Sci. 30, 239–253 (2002).
[Crossref]

1965 (1)

T. J. Boyd, B. T. Rogers, R. R. Tesche, and Douglas Venable, “PHERMEX—A high-current electron accelerator for use in dynamic radiography,” Rev. Sci. Instrum. 36, 1401–1408 (1965).
[Crossref]

Beech, P. F.

Beery, T.

E. Rose and T. Beery, Los Alamos National Laboratory (personal communication, 2006).

Boyd, T. J.

T. J. Boyd, B. T. Rogers, R. R. Tesche, and Douglas Venable, “PHERMEX—A high-current electron accelerator for use in dynamic radiography,” Rev. Sci. Instrum. 36, 1401–1408 (1965).
[Crossref]

Bryant, T. F.

Clough, S. J.

Cooper, G. M.

Davitt, R.

Edwards, R. D.

Ekdahl, C.

C. Ekdahl, “Modern electron accelerators for radiography,” IEEE Trans. Plasma Sci. 30, 254–261 (2002).
[Crossref]

Forster, D. W.

D. W. Forster, “Radiographic spot size definitions,” AWE Report HWH/9201/R8 (January 1992).

Goldsack, T. J.

Kenna, N.

Krushelnick, K.

McLean, J.

Mueller, K.

K. Mueller, “Measurement and characterization of x-ray spot size,” Los Alamos National Laboratory Report LA-UR-89-1886 (August 1989).

Pearce, A. G.

Phillips, M. J.

Pullinger, K. P.

Rogers, B. T.

T. J. Boyd, B. T. Rogers, R. R. Tesche, and Douglas Venable, “PHERMEX—A high-current electron accelerator for use in dynamic radiography,” Rev. Sci. Instrum. 36, 1401–1408 (1965).
[Crossref]

Rose, E.

E. Rose and T. Beery, Los Alamos National Laboratory (personal communication, 2006).

Short, D. J.

Sinclair, M. A.

Tesche, R. R.

T. J. Boyd, B. T. Rogers, R. R. Tesche, and Douglas Venable, “PHERMEX—A high-current electron accelerator for use in dynamic radiography,” Rev. Sci. Instrum. 36, 1401–1408 (1965).
[Crossref]

Thomas, K. J.

Threadgold, J. R.

Venable, Douglas

T. J. Boyd, B. T. Rogers, R. R. Tesche, and Douglas Venable, “PHERMEX—A high-current electron accelerator for use in dynamic radiography,” Rev. Sci. Instrum. 36, 1401–1408 (1965).
[Crossref]

Williamson, M. C.

IEEE Trans. Plasma Sci. (2)

C. Ekdahl, “Modern electron accelerators for radiography,” IEEE Trans. Plasma Sci. 30, 254–261 (2002).
[Crossref]

Rev. Sci. Instrum. (1)

T. J. Boyd, B. T. Rogers, R. R. Tesche, and Douglas Venable, “PHERMEX—A high-current electron accelerator for use in dynamic radiography,” Rev. Sci. Instrum. 36, 1401–1408 (1965).
[Crossref]

Other (5)

K. Mueller, “Measurement and characterization of x-ray spot size,” Los Alamos National Laboratory Report LA-UR-89-1886 (August 1989).

D. W. Forster, “Radiographic spot size definitions,” AWE Report HWH/9201/R8 (January 1992).

“Modulation transfer function of screen-film systems,” ICRU Report 41 (International Commission on Radiation Units and Measurements, 1986).

“Military standard photographic lenses,” MIL-STD-150A (U.S. Government Printing Office, 1959), p. 21.

E. Rose and T. Beery, Los Alamos National Laboratory (personal communication, 2006).

Cited By

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Fig. 1 Comparison of PSFs for source intensity distributions having the same Los Alamos spot size,

d_{LANL} = 2.0 mm

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Fig. 2 Comparison of LSFs for source intensity distributions having the same Los Alamos spot size,

d_{LANL} = 2.0 mm

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Fig. 3 Comparison of MTFs for source intensity distributions having the same Los Alamos spot size,

d_{LANL} = 2.0 mm

Download Full Size | PPT Slide | PDF

Fig. 4 Comparison of PSFs for source intensity distributions having the same LR,

LR = 1.00 lp / mm

Download Full Size | PPT Slide | PDF

Fig. 5 Comparison of LSFs for source intensity distributions having the same LR,

LR = 1.00 lp / mm

Download Full Size | PPT Slide | PDF

Fig. 6 Comparison of MTFs for source intensity distributions having the same LR,

LR = 1.00 lp / mm

Download Full Size | PPT Slide | PDF

Fig. 7 LSF for the DARHT-I distribution with

d_{LANL} = 2.00 mm

(solid black curve) compared with that for a quasi-Bennett with

d_{LANL} = 0.724 \times 2.00 = 1.448 mm

(dashed green curve).

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Tables (7)

Table 1 Example Source Distributions

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Table 2 Spot-Size Metrics for the Example Source Distributions

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Table 3 Comparison of Source Spots Having the Same Los Alamos Spot Size

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Table 4 Comparison of Source Spots Having the Same FWHM

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Table 5 Comparison of Source Spots Having the Same AWE Size

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Table 6 Comparison of Source Spots Having the Same Limiting Resolution

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Table 7 Results of Calculation of AWE Spot for a Gaussian PSF

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Equations (51)

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C = \frac{\max (I_{f}) - \min (I_{f})}{\max (I_{f}) + \min (I_{f})},

\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} PSF (x, y) d x d y = 1

\int_{0}^{\infty} PSF (r) 2 π r d r = 1

PSF (x, y) = I (x, y) / I_{TOTAL} .

LSF (x) = \int_{- \infty}^{+ \infty} PSF (x, y) d y .

LSF (x) = 2 \int_{x}^{\infty} \frac{r PSF (r)}{\sqrt{r^{2} - x^{2}}} d r .

PSF (r) = - \frac{1}{π} \int_{r}^{\infty} \frac{d LSF / d x}{\sqrt{x^{2} - r^{2}}} d x .

ESF (x) = \int_{- \infty}^{x} LSF (x^{'}) d x^{'} .

MTF (ν_{x}, ν_{y}) = | H (ν_{x}, ν_{y}) |,

H (ν_{x}, ν_{y}) = \int_{- \infty}^{\infty} d x \int_{- \infty}^{\infty} PSF (x, y) \exp [- 2 π j (ν_{x} x + ν_{y} y)] d y .

MTF (ν_{x}, 0) = | H (ν_{x}, 0) | = | \int_{- \infty}^{\infty} d x \int_{- \infty}^{\infty} PSF (x, y) \exp [- 2 π j (ν_{x} x)] d y | .

MTF (ν_{x}, 0) = | \int_{- \infty}^{\infty} LSF (x) \exp [- 2 π j (ν_{x} x)] d x | .

x = r \cos (θ), y = r \sin (θ), 2 π ν_{x} = k \cos (φ), 2 π ν_{y} = k \sin (φ) .

2 π (ν_{x} x + ν_{y} y) = k r (\cos (θ) \cos (φ) + \sin (θ) \sin (φ)) = k r \cos (θ - φ),

H (k, φ) = \int_{0}^{\infty} PSF (r) r d r \int_{0}^{2 π} \exp [- j k r \cos (θ - φ)] d θ = \int_{0}^{\infty} PSF (r) r d r \int_{0}^{2 π} \exp [- j k r \cos (ϑ)] d ϑ = 2 \int_{0}^{\infty} PSF (r) r d r \int_{0}^{π} \cos (k r \cos (ϑ)) d ϑ,

MTF (k) = H (k) = 2 π \int_{0}^{\infty} PSF (r) J_{0} (k r) r d r .

PSF (r) = \frac{1}{2 π} \int_{0}^{\infty} MTF (k) J_{0} (k r) k d k .

Im = \frac{Raw - dark}{Flat - dark},

d_{AWE} = 2.47541 Δ_{x} .

d_{LANL} = 0.70508 / ν_{1 / 2} = 4.4301 / k_{1 / 2} .

\begin{matrix} PSF (r) = \frac{1}{π a^{2}} & r \leq a \\ = 0 & r > a \end{matrix},

LSF (x) = \frac{2}{π a^{2}} \sqrt{a^{2} - x^{2}},

ESF (x) = \frac{1}{π a^{2}} [x \sqrt{a^{2} - x^{2}} + a^{2} \sin^{- 1} (x / a) + π a^{2} / 2] .

MTF (k) = 2 π \int_{0}^{a} J_{0} (r) r d r = 2 \frac{| J_{1} (k a) |}{k a} .

k_{1 / 2} a = 2.21508937,

d_{LANL} = 4.43017874 / k_{1 / 2} .

PSF (r) = \frac{1}{π a^{2}} \exp (- r^{2} / a^{2}) .

FWHM = 2 a \sqrt{\ln 2} = 1.6651092 a

a = \frac{FWHM}{2 \sqrt{\ln 2}} = 0.6005613 FWHM .

MTF (k) = \frac{2}{a^{2}} \int_{0}^{\infty} \exp (- r^{2} / a^{2}) J_{0} (k r) r d r = \exp (- k^{2} a^{2} / 4) .

k_{1 / 2} a = 2 π ν_{1 / 2} a = 2 \sqrt{\ln 2},

LSF (x) = \frac{1}{a \sqrt{π}} \exp (- x^{2} / a^{2}),

ESF (x) = \frac{1}{2} [\erf (x / a) + 1] .

ESF (x) = q @ \erf (x / a) = 2 q - 1

d_{AWE} = 2.082 a .

ν_{LR} = \frac{\sqrt{\ln 20}}{π a} = 0.55094 / a .

PSF (r) = \frac{1}{π a^{2}} \frac{1}{[1 + (r / a)^{2}]^{2}} .

FWHM = 2 \sqrt{\sqrt{2} - 1} a = 1.2872 a .

MTF (k) = \frac{2}{a^{2}} \int_{0}^{\infty} \frac{r}{[1 + (r / a)^{2}]^{2}} J_{0} (k r) d r = k a K_{1} (k a) .

LSF (x) = \frac{a^{2}}{2 (a^{2} + x^{2})^{3 / 2}},

ESF (x) = \frac{1}{2} [\frac{x}{\sqrt{a^{2} + x^{2}}} + 1] .

d_{AWE (25 - 75)} = 2.85836 a .

PSF (r) = \frac{1}{2 π a^{2}} \frac{1}{[1 + (r / a)^{2}]^{3 / 2}} .

FWHM = 2 \sqrt{2^{2 / 3} - 1} a = 1.5328 a .

MTF (k) = \frac{1}{a^{2}} \int_{0}^{\infty} \frac{r}{[1 + (r / a)^{2}]^{3 / 2}} J_{0} (k r) d r = e^{- k a}

LSF (x) = \frac{a}{π (a^{2} + x^{2})},

ESF (x) = \frac{1}{π} \tan^{- 1} (x / a) + \frac{1}{2} .

LSF (x) = \frac{A}{2 a} e^{- | x | / a} + \frac{1 - A}{2 q a} e^{- | x | / (q a)},

MTF (k) = \frac{A}{1 + (k a)^{2}} + \frac{1 - A}{1 + (k q a)^{2}} .

PSF (r) = \frac{A}{2 π a^{2}} K_{0} (r / a) + \frac{1 - A}{2 π q^{2} a^{2}} K_{0} (r / (q a)),

\begin{matrix} ESF (x) = \frac{A}{2} e^{x / a} + \frac{1 - A}{2} e^{x / (q a)} & x \leq 0 \\ = \frac{A}{2} [2 - e^{- x / a}] + \frac{1 - A}{2} [2 - e^{- x / (q a)}] & x \geq 0 \end{matrix} .

Source Distribution	$PSF (r)$	$LSF (x)$	$MTF (k)$
Uniform disk	$\begin{matrix} \frac{1}{π a^{2}} & r \leq a \\ 0 & r > a \end{matrix}$	$\begin{matrix} \frac{2}{π a^{2}} \sqrt{a^{2} - x^{2}} & x < a \\ 0 & x \geq a \end{matrix}$	$2 \frac{\| J_{1} (k a) \|}{k a}$
Gaussian	$\frac{1}{π a^{2}} \exp (- r^{2} / a^{2})$	$\frac{1}{a \sqrt{π}} \exp (- x^{2} / a^{2})$	$\exp (- k^{2} a^{2} / 4)$
Bennett	$\frac{1}{π a^{2}} \frac{1}{[1 + (r / a)^{2}]^{2}}$	$\frac{a^{2}}{2 (a^{2} + x^{2})^{3 / 2}}$	$k a K_{1} (k a)$
Quasi-Bennett	$\frac{1}{2 π a^{2}} \frac{1}{[1 + (r / a)^{2}]^{3 / 2}}$	$\frac{a}{π (a^{2} + x^{2})}$	$e^{- k a}$
DARHT-I	$\frac{0.93}{2 π a^{2}} K_{0} (r / a) + \frac{0.07}{2 π (10.9 a)^{2}} K_{0} (r / (10.9 a))$	$\frac{0.93}{2 a} e^{- \| x \| / a} + \frac{0.07}{2 (10.9 a)} e^{- \| x \| / (10.9 a)}$	$\frac{0.93}{1 + (k a)^{2}} + \frac{0.07}{1 + (10.9 k a)^{2}}$

Source Distribution	FWHM	$d_{LANL}$	$d_{AWE}$	LR
Uniform disk	$2 a$	$2 a$	$2 a$	na
Gaussian	$1.665 a$	$2.661 a$	$2.082 a$	$0.551 / a$
Bennett	$1.287 a$	$3.474 a$	$2.858 a$	$0.636 / a$
Quasi-Bennett	$1.533 a$	$6.351 a$	$4.951 a$	$0.477 / a$
DARHT-I	na	$4.770 a$	$3.502 a$	$0.668 / a$

Source Distribution	$FWHM$ (mm)	$d_{LANL}$ (mm)	$d_{AWE}$ (mm)	$LR$ (lp/mm)
Gaussian	1.251	2.000	1.565	0.733
Bennett	0.741	2.000	1.645	1.105
Quasi-Bennett	0.483	2.000	1.559	1.515
DARHT-I	na	2.000	1.468	1.593
Variation coefficient, %	47.4%	0%	4.6%	32.2%

Source Distribution	$FWHM$ (mm)	$d_{LANL}$ (mm)	$d_{AWE}$ (mm)	$LR$ (lp/mm)
Gaussian	1.000	1.598	1.250	0.917
Bennett	1.000	2.699	2.221	0.819
Quasi-Bennett	1.000	4.143	3.230	0.731
Variation coefficient, %	0%	45.4%	44.3%	11.3%

Source Distribution	$FWHM$ (mm)	$d_{LANL}$ (mm)	$d_{AWE}$ (mm)	$LR$ (lp/mm)
Gaussian	1.599	2.556	2.000	0.574
Bennett	0.901	2.431	2.000	0.909
Quasi-Bennett	0.619	2.566	2.000	1.181
DARHT-I	na	2.724	2.000	1.170
Variation coefficient, %	48.5%	4.7%	0%	32.1%

Source Distribution	$FWHM$ (mm)	$d_{LANL}$ (mm)	$d_{AWE}$ (mm)	$LR$ (lp/mm)
Gaussian	0.917	1.466	1.147	1.000
Bennett	0.819	2.209	1.818	1.000
Quasi-Bennett	0.731	3.029	2.362	1.000
DARHT-I	na	3.186	2.339	1.000
Variation coefficient, %	11.3%	32.2%	29.8%	0%

q	$2 q - 1$	$x / a$
0.25	$- 0.5$	$- 0.5204998778$
0.75	0.5	$+ 0.5204998778$

q	$2 q - 1$	$x / a$
0.25	$- 0.5$	$- 0.5204998778$
0.75	0.5	$+ 0.5204998778$

Abstract

References

2002 (2)

1965 (1)

Beech, P. F.

Beery, T.

Boyd, T. J.

Bryant, T. F.

Clough, S. J.

Cooper, G. M.

Davitt, R.

Edwards, R. D.

Ekdahl, C.

Forster, D. W.

Goldsack, T. J.

Kenna, N.

Krushelnick, K.

McLean, J.

Mueller, K.

Pearce, A. G.

Phillips, M. J.

Pullinger, K. P.

Rogers, B. T.

Rose, E.

Short, D. J.

Sinclair, M. A.

Tesche, R. R.

Thomas, K. J.

Threadgold, J. R.

Venable, Douglas

Williamson, M. C.

IEEE Trans. Plasma Sci. (2)

Rev. Sci. Instrum. (1)

Other (5)

Cited By

Figures (7)

Tables (7)

Equations (51)

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