1.1.

Two-Dimensional Shearing Interferometer

The shearing interferometer uses orthogonal phase gratings, which can be designed as either two crossed phase gratings or a single checkerboard pattern as it’s two-dimensional wavefront sensor, as shown in Fig. 1. The phase gratings are designed such that the even orders of the grating are eliminated. In order for the efficiency of the even orders, greater than the $m=0$ order, of a transmission grating to go to zero at x-ray wavelengths, the width of the slits must be half of the grating pitch.^8,9 In addition, for the efficiency of the $m=0$ order of the grating to go to zero, there must be negligible absorption and the bar structure of the grating must produce a shift of $\pi$ radians relative to the slits of the grating.

The coherency requirements for the two-dimensional x-ray shearing interferometer are such that the source is required to be nearly spatially coherent. This is consistent with using a spatially filtered x-ray source. The requirements are such that the pinhole in front of the x-ray source be sufficiently small so that the diffractive spreading of the x-rays exceed the pitch of the gratings or $L\lambda /D>p$, where L is the distance between the source and the grating, $\lambda$ is the wavelength, $D$ is the diameter of the x-ray source and $p$ is the pitch of the grating. For a grating pitch of $p=4\mu \mathrm{m}$, an x-ray wavelength of $\lambda =4$ angstrom and a separation of $L=20\mathrm{cm}$ between the x-ray source and the crossed phase grating, the requirements on the pinhole size are that it be less than $D\sim 10\mu \mathrm{m}$ in size. Extended x-ray sources can also be used when the source is appropriately made periodic. By placing a Ronchi ruling or grating in front of the extended source, it can be made to appear as a spatially coherent source as photons from the different regions of the source can be made to align the peaks of the diffraction pattern in the same location on the detector thereby forming good contrast fringes.¹⁹ This has the advantage of greatly increasing the amount of x-rays impinging on the sample but the disadvantage of convolving the measurement with the nearest neighbors, which will affect the high spatial frequency information.

1.2.

Hartmann Sensors

There are several potential implementations for an x-ray Hartmann sensor. This can be simply an array of holes,^7,8 zone plates,⁷ multilayer mirrors,⁹ or refractive lenses.¹⁰ The Hartmann wavefront sensor based on an array of holes is shown schematically in Fig. 2. The displacement of the spots on the detector is proportional to the wavefront gradient across the corresponding hole in the Hartmann mask. This approach uses an amplitude mask, which throws the majority of the signal away. In the visible regime all of the signal is used by modifying the Hartmann mask to utilize a lenslet array. In the x-ray regime this signal loss can be avoided using an array of phase zone plates as shown in Fig. 3. This can be accomplished using either circular or two crossed one-dimensional zone plates. In either case, the x-ray spot on the detector must be larger than a pixel size such that a very poor resolution zone plate would be required. Assuming a subaperture size of 20 μm, a charge coupled device (CCD) pixel size of 5 μm and two zones, inner zone has radius of 7.07 μm and outer zone has a radius of 10 μm. The focal length at 0.8 keV (8 keV) would be 0.032 m (0.32 m) and the spot size on the camera would be $\sim 6\mu \mathrm{m}$.

Fig. 1

Two-dimensional phase grating used to implement a shearing interferometer.

Fig. 2

Hartmann wavefront sensor implemented with an array of holes. The displacement of the spots on the detector is proportional to the wavefront gradient across the corresponding hole in the Hartmann mask.

Fig. 3

Hartmann wavefront sensor implemented with an array of zone plates.

The requirements placed on the transverse spatial coherence for a Hartmann sensor are not as stringent as for the shearing interferometer. In its simplest implementation, the Hartmann screen would consist of a regular array of holes with the displacement of the x-rays traveling through the holes providing the phase gradient information and the amplitude of the x-rays providing the absorption information. The sensitivity expected from a Hartmann sensor can be calculated analytically as

2.1.

Characterization of a Stationary Deuterium-Tritium Fusion Capsule Phase Profile

The experimental measurement setup for the characterization of the fusion capsules consists primarily of a micro-focus x-ray source, the fusion capsule itself, a phase flattener, the wavefront sensor, a filter, and the detector as shown in Fig. 4. The micro-focus x-ray source will be assumed to contain a source size of approximately 5 μm in diameter. Current experimental work with phase contrast imaging uses a micro-focus x-ray source which has a source size of 5 μm in diameter and a tungsten anode operating at 50 kV.²² The L shell emission is in the 8 to 11 keV x-ray range. Operating the source at 50 kV will result in significant bremsstrahlung radiation at higher x-ray energies.^23,24 The detector, however, is an x-ray CCD camera, which becomes optically thin to the higher-energy photons.²² The lower-energy x-rays can be removed and a narrower energy range within the L-shell emission can be selected by filtering with a thin foil such as copper.

The fusion capsules are spherical in shape and consist of an outer ablator shell with an inner layer of DT fuel. The fusion capsules have an overall diameter of approximately 2 mm. In the case of an outer CH ablator, the thickness of the ablator shell is approximately 190 μm and the DT layer on the inside of the capsule is approximately 68 μm thick. At x-ray wavelengths the index of refraction is expressed as $n=(1-\delta )+i\beta$, where $1-\delta$ gives rise to a phase shift as the x-rays pass through the sample and the $\beta$ term results in absorption. The length for a $\pi$ phase shift, $x_{\pi }$, is expressed as $x_{\pi }=\lambda /(2\delta )$ and the absorption length, $x_{\mu }$, is written as $x_{\mu }=\lambda /(4\pi \beta )$. At an x-ray energy of 10 keV, the x-rays have a wavelength of $\lambda =1.24\times {10}^{-10}\mathrm{m}$. The phase shift due to the CH ablator, $1.1\mathrm{g}/{\mathrm{cm}}^3$, at this wavelength is $x_{\pi }=\lambda /(2\delta )$, $\delta =2.5\times {10}^{-6}$, or a $\pi$ phase shift over a distance of 25 μm. The phase shift due to the DT, $0.101\mathrm{g}/{\mathrm{cm}}^3$, at this wavelength is $x_{\pi }=\lambda /(2\delta )$, $\delta =4\times {10}^{-7}$, or a $\pi$ phase shift over a distance of 150 μm. The CH represents a line-integrated depth of $\sim 380\mu \mathrm{m}$ or 47.8 radians and the DT represents a line-integrated depth of 136 μm or 2.8 radians. The contribution from both sides of the CH ablator and from both DT ice layers yields a combined phase shift of $\sim 50.6\text{radians}$. The DT ice layer in the fusion capsule forms grain boundaries, which range between 1 to 10 μm in depth. Based on experimental data, it is believed that the maximum grain boundary depth that can be tolerated on a fusion shot is $\sim 5\mu \mathrm{m}$ without unacceptably impacting the yield. In addition the maximum cross-sectional area for a given grain boundary that can be tolerated is $200{\mu \mathrm{m}}^2$. It is therefore desired to reject all targets that have grain boundary depths that exceed the 5 μm depth and that have a cross-sectional area greater than $200{\mu \mathrm{m}}^2$. A quantitative method is therefore needed to measure the phase profile of a given capsule to determine if any of the grain boundaries present in the DT layer exceed these parameters. A 5 μm depth at the grain boundary in the DT ice layer would then make a difference in the phase of 5 μm out of the line-integrated DT depth of 130 μm. This is in addition to an $\sim 0.5\mu \mathrm{m}$ RMS surface roughness for the DT ice layer. That represents only a 3.8% difference in the path length or 0.098 rad in the phase shift for the crevice, and 0.4% or 0.0098 rad RMS due to the surface roughness. The wavefront sensors therefore must be able to detect the gradients from a phase shift of only $\sim 0.1\text{radians}$ representing the peak of the ice grain boundary against the background phase, which would be approximately 50.6 radians in the center, to greater than double that at the edge of the capsule 1 mm away.

For this article, an analytical model for the grain boundary is assumed. The phase contribution due to the grain boundary, ${\varphi }_{gb}$, is represented in analytical form by the expression ${\varphi }_{gb}={\varphi }_o[|\tanh (2\pi x/50)|-1]$, where ${\varphi }_o$ is the peak amplitude of the phase from the grain boundary and $x$ is the spatial coordinate in μm. The analytic phase gradient is expressed as $d{\varphi }_{gb}/dx=(8\pi {\varphi }_o/50)/{[\exp (2\pi x/50)+\exp (-2\pi x/50)]}^2$ for x greater than 0. Based on these analytic expressions, the refraction angle, ${\varphi }_{\mathrm{refr}}$, of x-rays passing through the ice grain boundary can be approximated as ${\varphi }_{\mathrm{refr}}=\arctan \lfloor (4\lambda {\varphi }_o/50)/{[\exp (2\pi x/50)+\exp (-2\pi x/50)]}^2\rfloor$ for $x$ greater than 0.

A phase flattener is proposed to reduce the low-order phase response from the capsule geometry. The phase of the fusion capsule itself will vary from approximately 50 radians in the center to more than double that at the edge of the capsule a millimeter away. This is not critical as the measurement will primarily concentrate on the central region of the fusion capsule with two additional orthogonal views to provide information on grain boundaries over the entire capsule. With this phase flattener in, however, the difference between the reference spot locations measured before the fusion capsule and flattener are inserted and the spot locations measured after the fusion capsule and flattener are inserted will provide a direct measurement of the rms surface roughness and the depth and cross-sectional area of the grain boundaries. The phase flattener would then represent an inverse phase to the fusion capsule and could be placed immediately before or after the fusion capsule.

For the simulations, the following assumptions are made; 5.4 μm CCD pixel pitch, 43.2 μm x-ray grating pitch (in collimated space), $f_c=0.1\mathrm{m}$, $f_x=0.367\mathrm{m}$ and $f=1.1\mathrm{m}$, where $f_c$, $f_x$, and $f$ are the fusion capsule plane, the x-ray mask plane and the focal length of the “lens,” respectively. In an actual experiment the gratings would have a pitch of $\sim 4\mu \mathrm{m}$ in the spherically expanding x-ray beam and the x-ray CCD would have $\sim 6\mu \mathrm{m}$ pixel size. Given these assumptions, the fusion capsule is magnified by a factor of 3.67 onto the x-ray mask and a factor of 11 onto the x-ray CCD camera. The capsule is 2.2 mm in diameter, which would require a CCD with an active area of at least 2.4 cm. This is consistent with 4096 pixels at $6\mu \mathrm{m}/\text{pixel}$. The x-ray CCD will measure the wavefront gradient at a scale of 4 μm on the capsule (43.2 μm in collimated space). The simulations are performed on one-quarter of the fusion capsule, $1.05\times 1.05{\mathrm{mm}}^2$, such that across the simulation box there are 2048 pixels on the x-ray CCD. Each grating feature with 21.6 μm pitch spacing represents an area on the x-ray CCD of $4\times 4$ CCD pixels or a total of 512 simulation pixels. The simulations are performed with both read noise and Poisson noise. For the simulations shown the read noise was assumed to be 2 e-RMS, and the x-rays were assumed to produce several hundred thousand photoelectrons per every CCD pixel.

The simulations begin by defining a uniform field at the image plane of the fusion capsule located at the left-hand side of Fig. 5 and assuming a point x-ray source. The simulations were performed with $2048\times 2048$ simulation pixels covering a range on the camera of $\sim 5\mu \mathrm{m}$ per simulation pixel, representing $\sim 0.5\mu \mathrm{m}/\mathrm{sim.pix}$. on the fusion capsule itself. One quarter of the fusion capsule sphere was simulated with an initial phase profile of 0.026 rad RMS, $\sim 0.5\mu \mathrm{m}$ RMS, placed on the fusion capsule to simulate surface roughness. A Kolmogorov turbulence profile was assumed for the surface roughness of the DT ice layer. In addition to the phase profile representing the surface roughness of the DT ice, six DT ice grain boundaries were placed across the fusion capsule with each one having a width of 50 μm (500 μm in collimated space) but different lengths, depths and angles, as shown in Fig. 7(a). In particular, a long horizontal and vertical ice grain boundary were introduced, which had a 5 μm depth. The phase representing the fusion capsule and phase flattener were then used to construct a new field, which was then Fresnel propagated to the two-dimensional x-ray transmission grating shown in Fig. 6(a). The periodic phase pattern representing the crossed phase grating is then added to the field, and the field is propagated to the x-ray CCD camera. When a periodic structure is placed in a beam, images of that structure will appear downstream of the object as discovered by Talbot.²⁵ More precisely, if a phase grating is placed in the beam composed of alternating equal width bars of 0 and $\pi$ phases, then the field at the location of the phase structure will be reproduced a distance $d_T=d^2/2\lambda$ downstream of the phase structure. In this expression, $d_T$ is the Talbot distance, d represents the pitch of the phase grating and $\lambda$ is the wavelength of the source. At distances of $d_T/4$ and $3d_T/4$, the initial phase pattern across the beam has become uniform and the initially uniform intensity has acquired the periodic structure of the initial phase pattern with the pitch of the intensity pattern equal to half that of the original phase grating. At a distance of $d_T/2$, the phase pattern is reversed from the original phase grating, and the intensity pattern is uniform such that this particular location cannot be used for wavefront sensing.

Fig. 6

The phase mask used for the shear interferometer (a) results in the spatial profile of the x-rays impinging on the simulated detector (b). The amplitude mask used for the Hartmann sensor (c) produces the spatial profile of the x-rays impinging on the simulated detector (d).

Fig. 7

The applied phase pattern representing the fuel capsule (a) the reconstructed phase pattern from the shear interferometer (b) and the reconstructed phase pattern for the Hartmann mask (c).

The x-ray masks for the shearing interferometer are shown in Fig. 6(a) and 6(c), respectively. The respective two-dimensional array of spots from the masks are then shown in Fig. 6(b) and 6(d) for the shearing interferometer and the Hartmann mask, respectively. A comparison between these images shows how similar these wavefront sensors are from the perspective of analyzing the phase and amplitude of the object. In Fig. 6(b) and 6(d), the images represent the number of x-rays impinging upon the detector and not the number of photoelectrons generated in the detector.

The two-dimensional reconstructed phase from the shearing interferometer and the Hartmann sensor is displayed in Fig. 7 along with the phase profile imparted on the object for the simulation. In particular, the applied phase is shown in Fig. 7(a), the reconstructed phase from the shearing interferometer in Fig. 7(b), and the reconstructed phase from the Hartmann sensor in Fig. 7(c). For both the shearing interferometer and the Hartmann sensor the four largest phase amplitudes resulting from the grain boundaries in the DT layer are visible in the reconstruction, but the two smallest phase amplitude DT ice grain boundaries are not obviously identifiable.

Figure 8 shows a comparison between the reconstructed spatial phase and the applied spatial phase across a DT ice grain boundary. This spatial profile is averaged along the length of the DT ice grain boundary and will be referred to as a line out across the ice grain boundary. Specifically, Fig. 8(a) shows line outs across two of the reconstructed DT ice grain boundaries, averaged along the length of the grain boundary for the shearing interferometer. The solid black line represents the analytic phase profile applied to the vertical and horizontal ice grain boundaries in Fig. 7(a). The light gray dashed line represents the reconstruction of the long horizontal DT ice grain boundary seen in Fig. 7(a), and the dark gray dashed line represents the reconstruction of the vertical DT ice grain boundary seen in Fig. 7(a). Figure 8(b) shows line outs across two of the reconstructed DT ice grain boundaries, averaged along the length of the grain boundary for the Hartmann sensor. The solid black line represents the analytic phase profile applied to the vertical and horizontal ice grain boundaries in Fig. 7(a). The light gray dashed line represents the reconstruction of the long horizontal DT ice grain boundary seen in Fig. 7(a), and the dark gray dashed line represents the reconstruction of the vertical DT ice grain boundary seen in Fig. 7(a).

Fig. 8

Line outs of the applied phase (solid black line) compared with the reconstructed phases from the main horizontal DT ice grain boundary in Fig. 7(a) (dashed light gray) and the main vertical DT ice grain boundary in Fig. 7(a) (dashed dark gray) for the shearing interferometer (a) and the Hartmann mask (b).

2.2.

Characterization of an Imploding Deuterium-Tritium Fusion Capsule: Detection of High-Frequency Spikes of Ablator Material Seen in DT Fuel Capsule Implosions

This section compares the phase reconstruction of a shearing interferometer and Hartmann sensor for an imploding DT fusion capsule. This is modeled with an “asterisk” phase profile to test the ability of these wavefront sensors to detect spikes of ablator material seen in DT fuel capsule implosions.^26,27 In both simulations, x-rays were assumed to be diverging from a point projection x-ray source, a laser-driven x-ray backlighter, with an f/# of 110, where the f/# is defined as the focal length of the focusing optic divided by the diameter of the x-ray beam. The simulations were performed with 10 keV x-rays, and the geometry of the simulation is shown in Fig. 4. The object was placed 0.1 m from the point projection x-ray source and the mask, phase or amplitude was placed 0.367 m from the focus with the detector placed 1.1 m from the focus. As a consequence the object was magnified a factor of 3.67 onto the mask, and the mask was, in turn, magnified a factor of three onto the detector. The phase gratings for the shearing interferometer were simulated with bar and trough widths equal to four simulation pixels or 21.6 μm. In the case of the Hartmann sensor, the amplitude Hartmann mask was also simulated with the pitch of the holes equal to four simulation pixels or 21.6 μm. As in the previous section, the simulations utilized wave optics to transport the electric field between the various planes containing phase or amplitude objects. The grating structure or Hartmann mask and the phase object are added to the electric field after the field has been propagated to their respective locations. The wavefront is reconstructed from the simulated spots by first locating the displacement of each of the spots with a center-of-mass centroider²⁸ and then reconstructing the resulting gradients with a multigrid wavefront reconstructor.²⁹

Figure 9(a) and 9(b) represents the intensity pattern at the detector with an “asterisk-shaped” phase object in the beam. Based on the spot patterns in Fig. 9(a) and 9(b), the local gradients were determined, the phase reconstructed, and the amplitude solved for. The two phases were then reconstructed using a multigrid algorithm²⁶ to determine the phase of the object. The results of this phase recovery process are shown in Fig. 10(b) and 10(c) for the shearing interferometer and the Hartmann sensor, respectively, with the applied phase being displayed in Fig. 10(a).

Fig. 9

Intensity profiles at the detector for both the two-dimensional shearing interferometer (a) and the Hartmann sensor (b). The size of the simulation box shown, 200 pixels, represents 1.08 mm in collimated space and 98 μm at the capsule plane.

Fig. 10

Retrieved phase object with the two-dimensional shearing interferometer and the Hartmann sensor; the actual phase of the object placed in the expanding x-ray beam (a) the reconstructed phase for the shearing interferometer (b), and the reconstructed phase for the Hartmann sensor (c). The outer diameter of the “asterisk” pattern/spikes, 1000 pixels, represents 5.4 mm in collimated space and 490 μm at the capsule plane.