## 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.

© 2011 Optical Society of America

## 1. INTRODUCTION

Flash radiography of large hydrodynamic experiments driven by high explosives is a venerable diagnostic technique in use at many laboratories. Typically multi-million-electron-volt radiation pulses with tens of nanseconds pulse widths are used to radiograph metals driven by high explosives in order to obtain a stopped-motion image [1]. Most often the source of radiation is bremsstrahlung from electrons striking a heavy- metal target, and there has been a substantial investment in high-power electron accelerators for this purpose [2]. This is point-projection radiography, in which the object is back lit with a small source of penetrating radiation. A highly valued aspect of this diagnostic is the ability to resolve fine detail in the object. Since resolution depends on the size of the radiographic source spot, “spot size” is often quoted as an indication of the resolving power of a particular flash- radiography machine. Over the years, a variety of techniques for measuring spot size has evolved at the different laboratories, with a multiplicity of “spot size” definitions [3, 4]. These definitions all depend on the distribution of radiation over the spot, which can be vastly different for different machines. Moreover, inference of resolving power from some spot-size definitions can be misleading when applied to different intensity distributions. In this article, I review the most common of the techniques and definitions in use, with the intent of identifying the best predictor of resolution, independent of the shape of the intensity distribution.

The approach is to analyze examples of source spot intensity distributions that have been used to characterize experimental measurements on various flash radiograph machines. The most commonly quoted spot-size metrics are calculated and compared, and also compared with the commonly accepted measure of resolving power: the limiting resolution (LR). Then some general conclusions are drawn about each metric.

Only the main results and conclusions are presented in the body of this article. The detailed definitions and calculations are relegated to the appendices. Appendix A reviews fundamental definitions of the concepts used in spot-size measurements. Typical experimental methods for spot- size and resolution measurements are briefly described in Appendix B. Appendix C reviews the definitions of the most commonly used spot-size metrics. Although there may be others, the only metrics that we consider here are the full width at half-maximum (FWHM), the Los Alamos National Laboratory (LANL) definition of spot size (${d}_{\mathrm{LANL}}$), the Atomic Weapons Establishment (AWE) definition of spot size (${d}_{\mathrm{AWE}}$), and the LR. Finally, Appendix D details the calculation of the spot-size metrics for spot distributions used as analytic examples.

## 2. ANALYSIS

The normalized spot intensity distribution (profile or shape) is known as the point-spread function (PSF). The analytic tool that enables a quantification of resolving power is the modulation transfer function (MTF), which is the two-dimensional Fourier transform of the PSF [5].

The radiographic source spot can be considered to be a spatial frequency filter applied to the image of the object, and the MTF of the spot is analogous to the response function of a frequency filter in a temporal data recording system. The blurring of the radiograph image resulting from the source spot is analogous to the suppression of high frequencies in temporal data by the cables and other bandwidth-limiting components of the data acquisition system.

The MTF is a powerful tool that can greatly simplify radiographic system analysis, because the image obtained is a convolution of the blurring of each system component, so the system MTF is simply the product of the MTFs of each component. For example, to find the MTF of a spot imaged with a pinhole camera, one need not deconvolve pinhole camera data, so long as the MTF of the pinhole is known. Moreover, analysis of penumbral imaging (roll-bar) data can be simplified using the MTF. These, and other properties of the MTF, are reviewed in Appendix A.

#### 2A. Resolving Power and Limiting Resolution

A convenient object for determining the resolving power of a radiography system is a set of dense (opaque) bars with different spacing. The PSF of such an opaque grating is a square wave with unnormalized maximum of unity (full transmission) and minimum of zero (no transmission). This is equivalent to the Air Force bar target used to quantify the resolving power of military optical systems [6]. The resolving power of an imaging system is usually based on human vision, and is given by the highest spatial frequency of bars that can be resolved by eye. The human eye cannot resolve images with less than $\sim 5\%$ contrast, and this value is often referred to as the “limiting resolution.” So, the LR of a radiographic source spot is the spatial frequency at which the contrast of the image of an opaque grating is 5%, with contrast given by

When using the MTF, the LR prescription to quantify the resolving power of the source spot of a particular radiography machine is conceptually straightforward: simply calculate the MTF from roll bar or pinhole measurements, and then find the spatial frequency at which the MTF is 5%. This is equivalent to using a bar pattern object with various frequencies to determine the LR. The practical problem, which lies in extracting an accurate 5% from noisy data typical of spot measurements, is extraordinarily difficult, if not impossible. This is complicated by the fact that, at 5%, the MTF is slowly varying, so the error in determining the 5% frequency is magnified. Therefore, it would be convenient to have a more easily measured metric that is well correlated with the LR across all spot distributions considered in this article. That is the quest of this article, and we will show that the FWHM comes close to this ideal.

#### 2B. Examples

To illustrate the effect that the spot intensity distribution has on resolution, we compare spot-size metrics calculated for several different PSF intensity distributions. We begin with some examples that can be treated analytically, yielding easily calculated formulas for the metrics. These are listed in Table 1, and were chosen to cover the range of central tendencies that one encounters in real experimental data, from broad dis tributions to sharply peaked ones with large wings. Three of these examples (uniform spot, Gaussian, and Bennett) have been used in the past to describe experimental data [3, 4]. Here we have added the sum of exponentials used to describe the line-spread function (LSF) of the spot produced by the first accelerator at the Los Alamos Dual Axis Radiographic Hydrodynamic Testing (DARHT) facility [7], and an approximation to it (quasi-Bennett). This approximation is motivated by the fact that, although the sum of exponentials may fit the DARHT-I data quite well, its PSF cannot be produced by a physically realistic electron beam, whereas the quasi-Bennett PSF can. These analytic examples and the resulting formulas for spot-size metrics are summarized in Tables 1, 2. (Details of the calculations are given in Appendix D.)

First we compare spots that have the same Los Alamos spot size, ${d}_{\mathrm{LANL}}$. This sort of comparison might be made when comparing spots produced by different types of machines, for example, in a comparison of different types of pulsed-power diodes, or diodes with linear accelerators. The source distributions for these kinds of comparisons can be expected to be quite different, so assessing the relative resolving powers among several machines requires a metric that correlates well with LR, independent of the spot distribution.

Figures 1, 2, 3 show plots of the PSFs, LSFs, and MTFs for ${d}_{\mathrm{LANL}}=2.00$.

The metrics for these distributions are compared in Table 3, where it is seen how the metrics for the quasi-Bennett closely approximate those of the DARHT-I, showing its utility for representing the DARHT-I PSF.

Table 3 also shows that the LR varies by more than a factor of 2 for these different intensity distributions, whereas they all have the same LANL spot size. Therefore, the LANL spot size is not a very good way to predict resolution, independent of spot intensity distribution. Furthermore, there is only 4.6% variation in the AWE spot size for the 32.2% variation in LR. The correlation coefficient between the sets of FWHM and LR values is $|r(\mathrm{LR},\mathrm{FWHM})|=0.977$, but between ${d}_{\mathrm{AWE}}$ and the LR it is only $|r(\mathrm{LR},\mathrm{AWE})|=0.377$.

Neither ${d}_{\mathrm{LANL}}$ nor ${d}_{\mathrm{AWE}}$ are very well correlated with the LR, so they are not reliable metrics for comparing the resolv ing power of different source distributions. It follows that they are not very good for comparing the potential resolving power of different machines that produce different source distributions.

Next we compare spots having the same FWHM. The DARHT-I PSF is ill-behaved at the origin, so there is no well-defined FWHM for this distribution and it is excluded from this comparison. As shown in Table 4, it is clear that the FWHM has a much better correlation with the LR. Thus, the FWHM is a preferred metric for comparing different machines, because it is better correlated with the LR than either ${d}_{\mathrm{LANL}}$ or ${d}_{\mathrm{LANL}}$.

The next comparison is between spots with the same AWE size. This simply reiterates that the AWE definition correlates well with the LANL definition, but not with the LR. Again, the correlation between FWHM and LR is excellent ($|r(\mathrm{LR},\mathrm{FWHM})|=0.984$), but the LR is poorly correlated with the Los Alamos spot size, $|r(\mathrm{LR},\mathrm{LANL})|=0.299$.

Finally, we compare spots having the same LR, $\mathrm{LR}=1.0\text{\hspace{0.17em}}\mathrm{lp}/\mathrm{mm}$. This comparison simply reinforces the previous results. Figures 4, 5, 6 show plots of the PSFs, LSFs, and MTFs for this choice of LR. A comparison of metrics for spots having $\mathrm{LR}=1.00\text{\hspace{0.17em}}\mathrm{lp}/\mathrm{mm}$ is presented in Table 6. Again, the traditional metrics of the quasi-Bennett closely approximate those used for DARHT-I, showing its value as an alternate, physically realistic intensity distribution.

## 3. DISCUSSION

The metrics for the sum of exponentials used to characterize the DARHT-I LSF are well approximated by the physically realistic quasi-Bennett PSF. The best (least squares) agreement between the metrics for the two distributions is a quasi- Bennett with ${d}_{\mathrm{LANL}}$ equal to $\sim 0.724$ times ${d}_{\mathrm{LANL}}$ for the DARHT-I distribution. Then the rms error for ${d}_{\mathrm{LANL}}$, ${d}_{\mathrm{AWE}}$, and LR is $<4.6\%$. Figure 7 compares the LSFs for these two distributions when the DARHT-I distribution has ${d}_{\mathrm{LANL}}=2.00\text{\hspace{0.17em}}\mathrm{mm}$.

It is clear from Tables 3, 4, 5, 6 that there is excellent correlation between the Los Alamos spot-size definition and the AWE definition, with correlation coefficient $|r(\mathrm{LANL},\mathrm{AWE})|>0.98$. The AWE spot size is about 78% of the LANL spot size for all cases. On the other hand, the correlation of the Los Alamos definition with the LR is tenuous, at best. The best correlation with LR is the FWHM. This should not be surprising, since the LR is largely due to the concentration of intensity near the axis.

It is recommended that the LR be used for comparisons of flash-radiography source spots whenever possible to eliminate the ongoing confusion between the various definitions of “spot size.” We have shown that it can be deduced from pinhole or roll-bar data, as well as directly measured from resolution bar targets. However, it is very difficult to measure accurately from pinhole data, because of its sensitivity to backgrounds. If LR measurements are not practical, then the next best direct comparison is the FWHM, which has the best correlation with LR of any of the spot-size definitions currently in use. A convenient rule of thumb is that the LR is about $0.8/\mathrm{FWHM}$, regardless of the distribution. This estimate is accurate for the range of distributions considered here to within $\sim \pm 11\%$. In particular, the LR is $\sim 0.73/\mathrm{FWHM}$ for the quasi-Bennett approximation to the DARHT-I spot intensity distribution.

## APPENDIX A: DEFINITIONS AND DERIVATIONS

An imaging system can be characterized by a PSF, which is normalized so that

orfor the symmetric intensity distributions considered here.For point-projection flash radiography with the resolution limited by the source spot, rather than the imaging optics, the PSF is just the normalized intensity distribution of the spot:

Experimentally, the PSF is the result of a pinhole measurement of the spot after deblurring by deconvolving the pinhole PSF.The LSF is the projection of the PSF onto a plane; for example,

The LSF can also be computed by using the forward Abel transform:The MTF is defined as the modulus of the two-dimensional Fourier transform of the PSF of the x-ray imaging system:

whereA useful analytic tool is the projection-slice theorem, which states that the Fourier transform of the projection (LSF) yields a slice of the MTF orthogonal to the projection. That is, a slice of the two-dimensional MTF can be calculated by projecting the PSF onto a plane to produce the LSF, and then taking the one-dimensional Fourier transform of the LSF. This can be verified by inspection of Eq. (A9) by setting one of the spatial frequencies to zero. For example, consider the slice of the MTF in the ${k}_{x}$ direction, which is given by

It is worth noting that the MTF can be found directly from a known (measured) LSF by taking its Fourier transform. The integral over *y* in Eq. (A10) is just the LSF, $\mathrm{LSF}(x)$, according to the definition in Eq. (A4), so it follows that

It can be shown that the MTF of an azimuthally symmetric PSF can be expressed as a zeroth-order Hankel transform. Let

*k*, and then one can transform Eq. (A9) by using

*ϑ*is just

*π*times the integral representation of the zero-order Bessel function, and so Eq. (A14) becomes

## APPENDIX B: EXPERIMENTAL METHODS

The three most commonly used experimental methods for finding the spot size or resolving power of a radiographic source are pinhole imaging, penumbral imaging, and imaging of a resolution target.

## Pinhole Imaging

The origins of image formation using pinholes are lost in antiquity, but it is commonly assumed that some of the master painters of the 17th century used a form of pinhole camera, or camera obscura, to aid in portraiture. The ideal pinhole is an infinitesimally small hole through an infinitesimally thin sheet of perfectly opaque material. The PSF of such a pinhole is a delta function, and the image formed has no blurring or distortion, regardless of the size of the object. This cannot be realized in practice, especially when imaging x rays, gamma rays, or other penetrating radiation from a source spot, because of the thickness of material needed to make the “opaque” sheet. So, the practical pinhole suffers from bleed through of the radiation, as well as vignetting.

In practice, pinholes constructed for high-energy spot-size measurements are usually tapered so that the image of a flat field is flat over the dimension of the spot, and the system can considered to be isoplanatic over the dimensions of interest. In this case, and the pinhole can be represented by a PSF, which can be convolved with the PSF of the source spot to produce the blurred image of the source. A close appro ximation to isoplanatism for the purpose of spot distribution measurements can usually be achieved by careful pinhole design.

Pinhole blurring can frequently be estimated by assuming that both the spot and pinhole can be approximated by Gaussian PSFs, in which case the size of the spot image is just the quadrature sum of the spot size and the size of the pinhole. For example, a spot with $1.0\text{\hspace{0.17em}}\mathrm{mm}$ FWHM imaged through a $\mathrm{200}\text{\hspace{0.17em}}\mathrm{\mu m}$ diameter pinhole will have an apparent size of $\sqrt{{1.0}^{2}+{0.2}^{2}}=1.02\text{\hspace{0.17em}}\mathrm{mm}$ in the image. Thus the rule of thumb that, to keep pinhole blurring to less than 2%, the pinhole diameter must be less than 20% of the spot size.

## Penumbral Imaging

The penumbral imaging technique uses thick, opaque shields to block part of the radiation from the source, and infers the size of the source from the density distribution of the shadow. The simplest version of penumbral imaging is the roll-bar measurement, which employs a thick, heavy-metal bar with an edge that has a large radius of curvature ($\sim 1\text{\hspace{0.17em}}\mathrm{m}$) to ease alignment. This is the radiographic analogue of the knife- edge test of an optical lens, and directly yields the ESF. The roll-bar technique has the advantages of simplicity, and not needing a pinhole PSF deconvolution. As will be shown in what follows, differentiation of roll-bar data to find the PSF can be avoided through the use of Fourier and Hankel transforms.

The simple roll-bar cannot measure azimuthal asymmetry of the spot, although multiple bars have been used to infer gross asymmetry. For example, roll-bar measurements of the DARHT-I spot are satisfactory; little azimuthal asym metry is expected, because that accelerator employs only solenoidal focusing elements. On the other hand, DARHT-II uses quadrupole elements in the downstream transport, so pinhole imaging to fully resolve ellipticity is the preferred technique.

## Bar Charts (Resolution Targets)

It can be argued that the best way to determine the resolving power of an imaging system is to measure the ability to resolve fine details. For optical systems, charts with opaque bar patterns with varying spacing (Air Force charts) are used for this. Thick tungsten bar patterns are used to assess the flash-radiography sources at the Los Alamos DARHT facility. In addition, thick sheets of material are also inserted so that the measurements are relevant to the linear density of material in the object to be radiographed. Final tuning of the DARHT accelerators before a hydrotest is based on the best resolution visually obtained with this method. This technique relates directly to the LR metric defined in this paper.

Before analyzing images obtained with any of these methods (Appendix B, Subsections B.1–B.3), they should be dewarped to account for distortion of the optics, flat fielded to account for spatial nonlinearity of sensitivity (pixel to pixel variations), and dark frame subtracted to account for light leakage, dark count, and radiation bleed-through contribution to the background. The usual prescription for these corrections is

where Raw is the dewarped raw image, $\text{dark}$ is the dewarped image with the radiation source blocked, and $\text{Flat}$ is the dewarped image of a uniformly irradiated converter. To the extent that any of these effects is small and neglected, there is a commensurate increase in uncertainty of the measurement.## APPENDIX C: DEFINITIONS OF THE METRICS

## Full Width at Half-Maximum

Quite frequently, the spot size is characterized by its width at 50% of peak value, which is known as the FWHM. A related, infrequently quoted metric is the full width at some other percentage of the peak value. For example, the 10% width is sometimes used.

## AWE Spot Size

This metric was developed at the AWE as a means for rapid characterization of roll-bar (penumbral imaging) data [4, 8], to avoid having to differentiate the data (thereby adding noise) in order to find the PSF. The AWE spot size is defined in terms of the horizontal separation ${\mathrm{\Delta}}_{x}$ between two values of the measured ESF. This distance is then multiplied by a number that would have given the same separation for a uniformly filled disk of diameter ${d}_{\mathrm{AWE}}$ [4, 8]. Traditionally, the two values of the ESF are chosen to be 25% and 75% of the maximum. For this choice, the multiplier is 2.47541 (see Appendix D, Subsection D.1) and the AWE spot size is

This is frequently rounded off to ${d}_{\mathrm{AWE}}=2.5{\mathrm{\Delta}}_{x}$, which has an easily visualized geometrical interpretation for data analysis.Differentiation of roll-bar data to find the PSF can be avoided through the use of Fourier and Hankel transforms. From Eq. (A7) it follows that, if $\mathcal{F}(\mathrm{ESF})$ is the Fourier transform of the measured ESF (the data), then the Fourier transform of the LSF is simply $\mathcal{F}(\mathrm{LSF})=\mathrm{k}\mathcal{F}(\mathrm{ESF})$. But, we have seen [Eq. (A11)] that the MTF is just the Fourier transform of the LSF, so $\mathrm{MTF}(\mathrm{k})=\mathrm{k}\mathcal{F}(\mathrm{ESF})$, and the inverse Hankel transform [Eq. (A16)] can be used to find the PSF. These are integral transforms, so add no numerical noise to the analysis.

## Los Alamos Spot Size

In flash radiography at LANL, the source spot size is frequently characterized by its “50% MTF size.” Simply stated, the LANL spot size, ${d}_{\mathrm{LANL}}$, is the diameter of a uniformly illuminated disk that has the same spatial frequency at $\mathrm{MTF}=0.5$ as the actual source spot [3]. In practice, one computes the MTF of the source spot, finds the spatial frequency at which the $\mathrm{MTF}=0.5$, ${\nu}_{1/2}={k}_{1/2}/2\pi $, and divides this into 0.70508 to get the equivalent disk diameter (see Appendix D, Subsection D.1). That is,

In the context of the spot as a spatial filter, the 50% MTF spot size is analogous to the “$6\text{-}\mathrm{dB}$-down frequency” often quoted for frequency filters used in time-history data recording systems. As with frequency filters, this single number does not tell the whole story. High frequency data can be more easily recovered from recording systems with a $6\text{\hspace{0.17em}}\mathrm{dB}/\text{octave}$ roll off than from systems with a $12\text{\hspace{0.17em}}\mathrm{dB}/\text{octave}$ roll off. Thus, in addition to their $6\text{\hspace{0.17em}}\mathrm{dB}$ frequency, filters are usually described by the number of poles in their transfer functions, which determine how fast the filter rolls off (approximately $6\text{\hspace{0.17em}}\mathrm{dB}/\text{pole}$).

## Limiting Resolution

Using the MTF, the LR prescription is straightforward: from measurements or theory, simply calculate the MTF, and then find the spatial frequency at which the MTF is 5%. That is, $\mathrm{LR}\equiv {\nu}_{\mathrm{LR}}={k}_{\mathrm{LR}}/2\pi \text{\hspace{0.17em}}(\mathrm{lp}/\mathrm{mm})$, where $\mathrm{MTF}({k}_{\mathrm{LR}})=0.05$. This is equivalent to the separation of line pairs in an Air Force target that are barely distinguishable by normal human vision.

## APPENDIX D: CALCULATION OF THE METRICS

## Uniform Disk

A disk with uniform intensity has often been used as a reference standard. The PSF of a disk with diameter *d* is

The LSF and ESF of the uniform spot are only defined for $-a\le x\le a$ and are

Equations (D1, D2, D3, D4) can be used to find the traditional metrics, as well as the LR, in terms of the radius *a*. The FWHM is just $\mathrm{FWHM}=2a$. The AWE and Los Alamos spot sizes are ${d}_{\mathrm{AWE}}={d}_{\mathrm{LANL}}=2a$ by definition (see Appendix C). The horizontal separation between 25% and 75% points of the ESF is just ${\mathrm{\Delta}}_{x}=2{x}_{75}$, where $\mathrm{ESF}({x}_{75})=0.75$. The resulting transcendental equation can be solved to show that ${x}_{75}=0.403973a$, thus $2a={d}_{\mathrm{AWE}}=2.47541{\mathrm{\Delta}}_{x}$.

The Los Alamos spot size definition is based on the 50% value of this MTF. For reference, the solution to Eq. (D4) for $\mathrm{MTF}({k}_{1/2})=0.5$ is

and the diameter is ${d}_{\mathrm{LANL}}=2a$, or Substituting the wavenumber at the 50% MTF of any other distribution gives the Los Alamos spot size [e.g., Eq. (C2)].## Gaussian Spot

The Gaussian intensity distribution is often used for illustrative purposes, because the calculations are easy, and in some cases it approximates the measured spot. Consider a Gaussian distribution with *e*-fold radius *a*. The PSF of this spot is

The MTF of the Gaussian spot is given by

The Los Alamos spot size is found from the MTF, Eq. (D10). The 50% MTF occurs at ${k}_{1/2}$, which is found by solving

which yields ${\nu}_{1/2}=0.6501/a$. Dividing this into 0.70508 gives ${d}_{\mathrm{LANL}}=2.661a$ for the diameter of the equivalent uniform disk.For the AWE analysis of spot size, one needs the ESF given by Eq. (A7). The LSF (projection) of the Gaussian spot is found by integrating the PSF over one dimension:

and integrating again gives the ESF: From the ESF, one finds the AWE spot size by solving the equation for values that are 25% and 75% of maximum. The results are shown in Table 7.From these results, one finds the required horizontal separations ${\mathrm{\Delta}}_{x}$, from which the AWE spot size is

Finally, from the 5% value of the MTF, one finds the LR:

## Bennett Spot

The Bennett profile is sometimes used to characterize spots that are sharply peaked and have wings that slowly fall off in radius. The PSF of the Bennett is

The FWHM isThe MTF can be calculated directly from Eqs. (A15, D17) and is given by

The AWE spot size is derived from the ESF in the same way as for the Gaussian distribution. The LSF for the Bennett is

and the ESF is From the ESF, the horizontal separation between 25% and 75% points is ${\mathrm{\Delta}}_{x}=2a/\sqrt{3}=1.1547a$, so## Quasi-Bennett Spot

Another example of a concentrated spot distribution with large wings has a PSF given by

For lack of a better name, we will call this distribution quasi-Bennett. It has easily calculated formulas for the LSF, ESF, and MTF.The quasi-Bennett PSF is useful because it has values for ${d}_{\mathrm{LANL}}$, ${d}_{\mathrm{AWE}}$, and LR that closely approximate the same values for the sum of exponentials used to represent the LSF of the DARHT-I spot. This is evident by comparing these values in Tables 2, 5, 6. Moreover, it has a more realistic PSF than the DARHT-I LSF, including a physically realizable value at the origin, and a FWHM that is actually defined and can be calculated from data. This PSF is also useful because it happens to characterize the pinhole used in time-resolved spot measurements at DARHT, so it is worthwhile including in our analysis.

The FWHM of the quasi-Bennett distribution is

The MTF can be calculated directly from Eq. (A15, D23) and is given byThe LSF of this distribution is

and the ESF is The AWE spot sizes are derived from the ESF as in the previous examples. Coincidentally, the horizontal separation is ${\mathrm{\Delta}}_{x}=2a$ exactly for this distribution for the AWE spot size, which is then ${d}_{\mathrm{AWE}}=4.9508a$.## DARHT-I Sum of Exponentials

As our final example, we consider the spot distribution that has a LSF given by the sum of two exponentials. This is also sharply peaked and has broad wings, but it often can be more accurately fit to source distribution data because there are two independent fitting parameters; the *e*-fold radii of each exponential. A sum of two exponentials is frequently used at Los Alamos as the LSF of the source distribution when analyzing radiographic images from DARHT-I [7]. Analysis of this distribution illustrates how the transfer function and associated transforms can be effectively used to analyze data obtained with the roll-bar method. The LSF is characterized by three parameters, and can be expressed as

*a*. This is the distribution used for the worked examples in the main text.

The MTF can be found from the Fourier transform of the LSF and is given by

Solving $\mathrm{MTF}=1/2$ with the DARHT-I spot parameters gives ${d}_{\mathrm{LANL}}=4.77017a$. Solving $\mathrm{MTF}=1/20$ with the DARHT-I spot parameters gives ${\nu}_{\mathrm{LR}}=0.667929/a$.The PSF of this distribution can be found from the inverse Hankel transform of the MTF:

The LSF is integrated to find the ESF for calculating the AWE spot size:

## ACKNOWLEDGMENTS

The author acknowledges stimulating discussions with Tom Beery, Evan Rose, B. Trent McCuistian, and Scott Watson on these, and other, topics. This work was supported by the United States Department of Energy (DOE) under contract number W-7405-ENG-36.

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