FIELD

The disclosure pertains to interferometric x-ray imaging.

ACKNOWLEDGMENT OF GOVERNMENT SUPPORT

This invention was made by an agency of the United States Government or under a contract with an agency of the United States Government. The name of the U.S. Government agency is the Department of Health and Human Services, National Institutes of Health.

BACKGROUND

Hard x-ray differential phase contrast imaging has flourished with the development of Talbot grating interferometers. In these interferometers, a wave propagation distance from a phase grating to a detector or third grating is set to an appropriate fraction of the Talbot distance so that strong intensity fringes are produced at a detector plane. Talbot interferometer configures are suitable for use with polychromatic sources, and fractional Talbot distances for grating periods of a few microns are a good match for most bench-top macro imaging applications. Talbot

interferometers have inherent sensitivity limitations since differential phase contrast generally scales with a ratio of wave propagation distance to grating period. Because the Talbot distance is a quadratic function of the grating period, higher sensitivity requires larger propagation distances, which can quickly exceed the practical size of an imager. The opposite approach of reducing the grating period while maintaining the wave propagation distance runs into the situation where the propagation distance becomes many times the Talbot distance. This approach forces the interferometer into a far field regime, where chromatic dispersion and higher order optical aberrations result in low fringe amplitude and poor efficiency.

For visible light, Mach-Zehnder interferometers permit achromatic, far field interferometry. Such interferometers define two interfering light paths that are balanced with each other, so that chromatic dispersion is eliminated. In x-ray optics, the Bonse-Hart interferometer is a type of Mach-Zehnder interferometer using x-ray Laue diffraction in monolithic crystals to split and deflect x-ray beams. Momose et al. have used the Bonse-Hart interferometer to obtain absolute phase images. See Momose and Fukuda, "Phase-contrast radiographs of nonstained rat cerebellar specimen," Medical Physics 22:375-379 (1995). Imaging with Bonse-Hart interferometers requires narrow line width x-ray radiation due to the energy selectivity of Laue diffraction in crystals for any given incident angle. Thus, Bonse-Hart interferometers require prolonged imaging time with compact x-ray sources in practical applications, and different approaches are needed.

SUMMARY

Disclosed herein are methods and apparatus pertaining to x-ray imaging using grating-based interferometers. As discussed below, these methods and apparatus permit use of broadband x-ray radiation and provide far-field imagery at convenient working distances.

In some examples, x-ray imaging systems comprising at least two x-ray gratings situated along an axis so that diffraction orders of the at least two diffraction gratings define a first optical path and a second optical path that extend from an input location to an output location. A first portion and second portion of an input x-ray beam propagate along the first optical path and the second optical path, respectively, and a specimen is situated in at least one of the first optical path and the second optical path. A third grating is situated so as to receive the first portion and the second portion of the input x-ray beam from the output location, and direct the first portion and the second portion so as to propagate along a common axis. A detector is situated at a distance from the third grating to receive the first portion and the second portion from the third grating and allow propagation of the first portion and second portion over that distance, and establish a fringe pattern based on optical interference between the first and second portions. The fringe pattern is established by adjusting the position of the first grating or the third grating along the beam axis, or the angle between the plane of the first or third grating and the beam axis, or the period of the first grating or the third grating. The fringe pattern can also be established by placing an aperture stop before the first grating to limit the transverse size of the beam, such that a diffraction path of the gratings falls on a limited area on the detector and diffraction orders of the third grating do not overlap on the detector. In another embodiment, the third grating is an absorption mask grating placed at an output location where beams propagating along the first and the second paths coherently interfere to produce an interference fringe pattern. The periodicity and orientation of the mask pattern of the analyzer grating is typically nearly the same as the interference fringe pattern. The pattern of the transmitted intensity through the third grating either contains broad moire fringes that are directly resolved by an x-ray detector which is placed immediately behind the third grating, or a uniform intensity that fluctuates with the stepping of one of the three gratings, which can be detected by a detector behind the grating. In some embodiments, the first optical path and the second optical path are balanced optical paths, and a fringe processor is configured to produce a specimen image based on the fringe pattern. In some examples, the specimen image is a phase contrast image or a differential phase contrast image. In other examples, the specimen image is a de-coherence or scattering image, based on attenuation of fringe pattern amplitude in addition to intensity attenuation. In a representative example, the at least two gratings include a first grating and a second grating, wherein a grating pitch of the first grating is twice a grating pitch of the second grating. In other examples, the at least two gratings include a first grating and a second grating, wherein a grating pitch of the first grating is that same as a grating pitch of the second grating. According to other embodiments, an x-ray source is configured to deliver an x-ray beam to the at least two gratings along the axis. In additional examples, an aperture stop is secured to at least one of the at least two x-ray gratings so as to intercept at least one diffraction order. The apertures stop the diffraction orders that are not part of the balanced optical paths. In still further embodiments, a specimen holder configured to position the specimen in the first optical path and the second optical path. In some embodiments, the at least two gratings are phase gratings situated at a common angle of at least 10, 20, 30, or 40 degrees with respect to the axis.

Methods comprise dividing an x-ray beam into first and second portions with a diffraction grating, and directing the first and second portions along first and second paths, respectively, so as to combine to form interference fringes. The interference fringes are detected, and associated fringe data stored in a computer-readable medium. Typically, the first and second paths are balanced paths and a specimen is situated in at least one of the first path or the second path, or in both paths. According to some examples, the first and second paths are defined by diffraction orders of at least a first diffraction grating and a second diffraction grating, and at least one of the first and second diffraction gratings is a phase grating having a grating pitch of less than 200 nm. In further examples, x-ray beams propagating along the balanced paths are directed to an x-ray detector with a third diffraction grating, allowing a distance of propagation between the third grating and the detector. In still further examples, the balanced paths pass through a third absorption mask grating at the output location, and the transmitted intensity contains broad moire fringes at the x-ray detector. In still further examples, the method comprises stepping at least one of the first and second diffraction gratings to a plurality of positions in direction perpendicular to a grating period. In other examples, the method comprises moving the spot of an x-ray cone beam to a plurality of positions in a direction perpendicular to the axis of the beam, by electromagnetic or mechanical methods. In still further examples, the method comprises moving the sample to a plurality of positions in a direction perpendicular to the axis of the x-ray beam. Interference fringes are detected at the plurality of positions, and based on the detected fringes, a specimen image is produced. In typical examples, the first and second paths are defined by diffraction orders of a first diffraction grating and a second diffraction grating situated along an axis, and x-radiation propagating along the first and second paths is directed so as to form interference fringes at an x- ray detector. Interference fringes are acquired and at a plurality of offset locations of the first or the second diffraction grating and an image is produced based on the acquired interference fringes. In some examples, at least one of the first and second diffraction gratings is defined by a periodic arrangement of stacked layers have alternating less and more density with respect to x-radiation. In additional alternatives, the first and second paths are defined by a first set of diffraction orders, and the detected fringes are associated with the first set of diffraction orders. The first and second portions are also directed in part along third and fourth paths, respectively, so as to combine to form interference fringes, wherein the third and fourth paths are defined by a second set of diffraction orders. Interference fringes associated with the second set of diffraction orders are detected as well. In some examples, at least one of the first and second paths overlaps at least a portion of the third and fourth paths. In other examples, a specimen in at least one of the first path and the second path, and at least one of the third path and the fourth path. At least one of the first and second gratings is stepped, the focal spot of the x-ray cone beam is stepped, or the position of the sample itself is stepped, and an associated plurality of interference fringes associated with at least one of the first set of diffraction orders and the second set of diffraction orders is detected. An image is formed based on the interference fringes. In a typical example, a period of the second grating is one half a period of the first grating and the third grating.

In some examples, x-ray imaging methods comprise obtaining a phase image and a de- coherence image of a specimen based on a plurality of sets of detected interference fringes formed by interference of x-ray beams propagating along balanced paths in a diffraction grating based interferometer. An x-ray amplitude image of the specimen is obtained and an image spatial frequency is selected. The phase image and the amplitude image are combined so that at spatial frequencies above the selected image spatial frequency the combined image tends to be a phase image and at spatial frequencies below the selected image spatial frequency the combined image tends to be an amplitude image. In such a combined image, in areas of large de-coherence, the amplitude image is weighted more than the phase image.

The foregoing and other features and advantages of the disclosed technology will become more apparent from the following detailed description, which proceeds with reference to the accompanying figures. BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A-1B illustrate a grating based X-ray interferometer using gratings having the same pitch and placed equidistant along an axis.

FIG. 2 illustrates a grating based X-ray interferometer using two gratings having a pitch P and an intermediate grating having a pitch P/2.

FIGS. 3-4 are schematic views of representative x-ray transmission gratings.

FIG. 5A illustrates specimen placement for obtaining fringes based on absolute phase.

FIG. 5B illustrate fringe changes associated with a specimen situated as shown in FIG. 5A.

FIG. 5C illustrates specimen placement for obtaining fringes based on differential phase.

FIG. 5D illustrates a differential phase image produced using the configuration of FIG. 5A.

FIG. 6A includes specimen images based on absorption and phase.

FIG. 6B is graph illustrating absorption and phase shifts along a section of the images of FIG. 6A.

FIG. 7 is an image of displaced fringes associated with several combinations of diffraction orders.

FIG. 8 is a block diagram of a representative x-ray imaging method.

FIGS. 9A-9D illustrate amplitude and phase imaging of a hair.

FIGS. 10A-10B illustrate amplitude and phase imaging of a sample comprising polystyrene spheres.

FIG. 11 is a schematic view of a diffraction grating based x-ray interferometer that includes apertures defined on grating surfaces.

FIG. 12 illustrates a grating interferometer that receives beams of different wavelengths propagating along different axes so that fringes of a common phase and period are produced.

FIG. 13 illustrates a grating interferometer based on π phase gratings of pitches 2P-P-2P.

FIG. 14 illustrates diffraction intensity of a phase grating for use in a grating interferometer.

FIG. 15 is an image of diffracted light from an interferometer such as shown in FIG. 13 showing three bands in which a central band contains fringes of 78% visibility.

FIG. 16 illustrates a fringe shift associated with a phase object produced by an

interferometer such as shown in FIG. 14.

FIG. 17 illustrates an alternate configuration of an x-ray grating interferometer.

FIG. 18 illustrates a configuration of an x-ray grating interferometer with normal incidence gratings.

FIG. 19 illustrates oblique fringes produces with an apparatus such as shown in FIG. 18. DETAILED DESCRIPTION

As used in this application and in the claims, the singular forms "a," "an," and "the" include the plural forms unless the context clearly dictates otherwise. Additionally, the term "includes" means "comprises." Further, the term "coupled" does not exclude the presence of intermediate elements between the coupled items.

The systems, apparatus, and methods described herein should not be construed as limiting in any way. Instead, the present disclosure is directed toward all novel and non-obvious features and aspects of the various disclosed embodiments, alone and in various combinations and sub- combinations with one another. The disclosed systems, methods, and apparatus are not limited to any specific aspect or feature or combinations thereof, nor do the disclosed systems, methods, and apparatus require that any one or more specific advantages be present or problems be solved. Any theories of operation are to facilitate explanation, but the disclosed systems, methods, and apparatus are not limited to such theories of operation.

Although the operations of some of the disclosed methods are described in a particular, sequential order for convenient presentation, it should be understood that this manner of description encompasses rearrangement, unless a particular ordering is required by specific language set forth below. For example, operations described sequentially may in some cases be rearranged or performed concurrently. Moreover, for the sake of simplicity, the attached figures may not show the various ways in which the disclosed systems, methods, and apparatus can be used in conjunction with other systems, methods, and apparatus. Additionally, the description sometimes uses terms like "produce" and "provide" to describe the disclosed methods. These terms are high- level abstractions of the actual operations that are performed. The actual operations that correspond to these terms will vary depending on the particular implementation and are readily discernible by one of ordinary skill in the art.

In some examples, values, procedures, or apparatus' are referred to as "lowest", "best", "minimum," or the like. It will be appreciated that such descriptions are intended to indicate that a selection among many used functional alternatives can be made, and such selections need not be better, smaller, or otherwise preferable to other selections.

The examples disclosed below pertain to x-ray interferometers using phase gratings for x- ray beam splitting and combining. As used herein, x-rays refers to electromagnetic radiation at wavelengths between of 0.01 and 10 nm, frequencies between 30 PHz and 45 EHz, or energies between 100 eV and 150 keV. Diffraction gratings for such x-rays have grating periods between 10 nm and 10000 nm, 50 nm and 500 nm, or 100 nm and 250 nm. For examples, x-rays at a central energy of 22.5 keV and gratings having a grating period of 200 nm and a grating spacing along an axis of 65 cm can be used. This representative configuration produces a beam displacements of about 180 μιη. As used herein, "image" refers to a viewable image of a specimen as well as stored representations of such viewable images. In some cases, image also refers to a detected, displayed, or stored fringe pattern.

With reference to FIGS. 1A-1B, a representative x-ray grating interferometer 100 includes x-ray gratings 102, 104, 106 arranged along an axis 110. As shown in FIGS. 1A-1B, the gratings 102, 104, 106 are equally spaced along the axis 110, and spacings of at least as much as 100 cm can be used. An x-ray source 108 delivers an x-ray beam to the gratings 102, 104, 106 and diffracted x- ray beams are incident to a detector 140 that generates an electrical image signal associated with interference of at least some beams associated with selected x-ray diffraction orders. The x-ray beam is coupled through an aperture 109 that serves to block undesired x-ray radiation and/or to partially collimate the x-ray beam. The detector 140 is coupled to a fringe processor 161 that produces one or more images of a specimen 150 based on detected fringes. The gratings 102, 104, 106 are tilted with respect to the axis 110, but normal incidence can be used with suitable gratings. Slits 170, 171 can be situated to block any unwanted diffraction orders. Separations of diffraction orders are generally exaggerated in FIG. 1 for convenient illustration.

Each of the gratings can direct any input x-ray beam along a variety of paths corresponding to diffraction orders. These paths are at angles of ηλ/Λ, wherein λ is a wavelength associated with the input x-ray beam, n is an integer, and A is a grating pitch or period. In the example of FIGS.

1A-1B, the grating period is the same for gratings 102, 104, 106, and the gratings 102, 104, 106 are phase gratings. In most examples, less x-ray power is diffracted into higher diffraction orders, i.e., those paths associated with larger values of n or -n. For convenience, upwardly directed diffraction orders shown in FIG. 1A are associated with n > 0, and downwardly directed diffraction orders are associated with n < 0. For example paths 111, 112, 113 correspond to +1, 0, and -1 diffraction orders of an x-ray beam input along the axis 110 to the grating 102. The 0 order propagating along path 112 is diffracted by the grating 104, and +1, -1 orders are shown as paths 122, 123, respectively. The -1 order propagating along path 113 is diffracted by the grating 104, and +1 order is shown as path 124. Other orders are not shown. At grating 106, a diffracted beam propagating along the path 123 is diffracted onto +1, 0 order paths 133, 134, respectively. A diffracted beam propagating along the path 124 is diffracted onto 0, -1 order paths which are the same as the paths 133, 134. Similarly diffracted beams are directed to the detector along paths 131, 132. As shown in FIG. 1A, the dual diffraction path 151 (shown with heavier lines) defined by paths 112, 113, 123, 124 provides substantially the same total phase delay for beams exiting along paths 133, 134 so that interference fringes are formed at the detector 140 even for x-ray beams with limited temporal coherence, or for x-ray beams of somewhat dispersed angles of incidence. A path 152 (shown in FIG. IB) also provides substantially equal phase delays and produces fringes as well. Typically, any paths between the gratings 102, 104 that form a parallelogram produce acceptable fringes. As used herein, paths that have substantially the same total phase delay are referred to as "balanced." Unbalanced paths can also be used to generate fringes, but can impose difficult coherence requirements on the input x-ray beam. The x-ray beam produced by the x-ray source 108 also exhibits lateral or transverse coherence, at least to some extent. Typically, a lateral coherence distance that is equal to or greater than about ¼ of the pitch of the grating 170 is satisfactory. Fringe contrast increases as coherence distance increases. Fringe visibility VM can be defined as VM = (Imax - Imin)/ (Imax + Imin), wherein I _max and 7 _mi „ are measured maximum and minimum fringe intensities. For lateral coherence distances of about 1 grating period or ¼ grating period fringe visibilities VM can be about 0.7 and 0.12, respectively. Thus, relatively low lateral coherence distances are satisfactory.

The specimen 150 is shown as situated between the gratings 104, 106 and interacting only along the path 123 and not the path 124. In other examples, the specimen is situated between gratings 102, 104 and is situated to intercept paths associated with both of interfering beams. If the specimen intercepts only one of the interfering paths, fringes are associated with absolute phase differences. If a specimen intercepts both interfering paths, fringes are associated with phase differences between different portions of the specimen. In one example, the gratings 102, 104, 106 have a common period of 200 nm and are spaced apart by 65 cm on the axis 110. At least one grating (such as the grating 104) is stepped or scanned in a direction 160 to produce a plurality of fringe patterns that are acquired and processed to form phase based images.

In a grating interferometer, interference fringe patterns produced by a symmetric pair of diffraction pathways is fully determined by relative positions of the grating lines, independent of wavelength and angle of incidence. With a broadband, extended source, fringe patterns from different wavelengths and angles of incidence are phase-locked to each other. Referring to FIG. 12, beams 1202, 1204 are incident at different angles of incidence to a grating interferometer that includes gratings 1206-1208. The beams 1202, 1204 are split and recombined, and associated respective fringe patterns 1212, 1214 are directed to a detector 1210. As shown in FIG. 12, the fringe patterns 1202, 1204 are in phase with each other and can be summed to increase image contrast. The beams 1202, 1204 can also be at different wavelengths and the associated fringe patterns remain phase-locked.

Another example is illustrated in FIG. 2. As above, only selected diffraction orders are illustrated, and apertures used to block one or more unwanted diffraction orders are not shown. An x-ray source 202 is configured to direct an x-ray beam 203 to diffraction gratings 204, 206, 208 that are situated along an axis 201. The gratings 204, 208 have a grating period A and the grating 206 has a grating period A/2. With these grating pitches, balanced paths can be obtained. The grating 204 diffracts the input beam 203 into a plurality of diffraction orders, but only the +1 and -1 orders are shown in FIG. 2 as paths 210, 212, respectively. The grating 206 directs portions of the diffracted x-ray beam on the paths 210, 212 to produce a -1 diffraction order along a path 214 and a +1 diffraction order along a path 216, respectively. The paths 210, 214 and 212, 216 are balanced, and form sides of a parallelogram 220. The grating 208 directs beams from these paths to from an output beam 220 that is directed to a detector and fringe analysis system. An aperture plate 222 is situated to block unwanted diffraction orders.

A sample 250 can be situated in paths 210, 212 as shown to produce fringes based on phase differences. Alternatively, the sample 250 can be situated in one of the paths 210, 212, 214, 216 or in the paths 214, 216. The aperture stop 222 is situated to block other diffraction orders, but additional diffraction orders can be transmitted and the associated fringes detected as well.

Aperture stops can also be situated before and/or after any of the gratings 204, 206, 208, and can be secured surfaces of the gratings 204, 206, 208.

A suitable grating 300 is shown in FIG. 3. A substrate 302 is configured to retain multilayer stacks 304, 306, 308, 310, 312 that include pluralities of layers of different refractive indices. The grating is configured to be used with radiation incident along a direction 301 that is generally not orthogonal to an entrance surface 312 of the substrate 302. The substrate can include one or more layers or materials that serve to encapsulate and or protect the multilayer stacks, and the substrate 302 can be formed of two or more pieces that are bonded together.

A more detailed illustration of a representative transmission x-ray grating 400 is provided in FIG. 4. A stepped substrate 402 such as an anisotropically etched silicon substrate includes a plurality of steps 404-409 that are provided with respective multilayer coatings 414-419. For purposes of illustration, an uncoated step 410 is also shown in order to illustrate a grating tilt angle φ. In the etched silicon substrate 402, the steps 404-410 correspond to sides of isosceles triangles having a base of length B. The multilayer coatings 414-419 are preferably configured to have a height H that is substantially the same as the associated step height. The multilayer coatings 414- 419 generally comprise a number of alternating bilayers of relatively less dense and more dense materials. In the disclosed example, the etched silicon substrate 402 has a step height of about 8.16 μιη, and the multilayer coatings 414-419 include 20 Si/W bilayers with each layer having a thickness of about 200 nm. A silicon filler layer 422 is provided over the etched silicon substrate 402 and the multilayer coatings 414-419. This layer can be formed by depositing a silicon layer followed by polishing. The x-ray grating 400 then has a uniform thickness with parallel exterior surfaces 424, 426. The combination of constant multilayer coating height matching the step height, and the silicon filler layer 422 eliminates or reduces grating substrate envelope modulation in interfering diffraction orders.

Transmission gratings such as that of FIG. 4 can exhibit diffraction with little or no envelope modulation due to the substrate if the bilayers are configured to have a height that is substantially the same as the step height H of the substrate steps and the filler layer 426 is provided. However, gratings with substrate modulation can be used as the substrate modulation is fixed and can be compensated in image analysis. In some examples, phase gratings are configured to provide greater intensities in diffraction orders that are to be used in fringe formation. For phase gratings, selection of a suitable phase or phases can be used. Some aspects of X-ray grating fabrication are found in Lynch et al., "Fabrication of 200 nm period centimeter area hard x-ray absorption gratings by multilayer deposition," J. Micromechanics and Microengineering 22: 105007 (2012).

Transmission gratings with a continuous uniform structure are also fabricated to eliminate any envelope modulation and allow normal incidence, as described in Miao et al., "Fabrication of 200 nm Period Hard X-ray Phase Gratings," Nano Letters 14(6):3453-3458 (2014).

FIG. 5A illustrates a portion of a grating based x-ray interferometer used to produce fringes based on an absolute phase difference. An object 522 (such as a dielectric sphere) is situated in a sample optical path 514. Radiation propagating along the sample optical path 514 and the object 522 is combined with radiation from a reference path 516 at an third grating 508. The combined radiation from optical paths 514, 516 is directed by the grating 508 to a detector 522 that produces an electrical or other image signal that can be processed to obtain an image of the object.

FIG. 5B illustrates typical fringe patterns received by the detector 522. Fringe pattern 502 is associated with fringes produced by radiation propagating along sample and reference paths in the absence of a sample. Fringe pattern 504 is associated with fringes produced by radiation propagating along sample path with the object 522 in radiation from the reference path 516. Fringe distortions associated with the object 522 are apparent in a region 506 of the fringe pattern 502.

FIG. 5C illustrates a configuration similar to that of FIG. 5A, but with an object 523 situated in both the reference path 516 and the object path 514. A representative image of an object (a spider) obtained in this manner is illustrated in FIG. 5D along with a cross sectional image intensity. The image of FIG. 5D was acquired in number of steps, each step covering a 150 μιη wide band over the sample. The bands were tiled together to give a full image. In this phase image, twin images of the spider of opposite phase values can be seen, vertically spaced by 141 μιη, corresponding to the separation of the two interfering x-ray beam paths at the location of the spider. The magnitude of the phase shifts exceeds π in many locations, resulting in frequent phase wrapping

While fringes can be used to produce phase-based specimen images, phase and amplitude (absorption) data can be combined to produce specimen images. Referring to FIG. 6 A, an x-ray absorption image 602 and a fringe-shift based differential phase contrast image 604 are obtained for a wing tip of a fruit fly. Phase shift and attenuation along a line 606 are shown in FIG. 6B. The images 602 and 604 can be combined to use higher spatial frequency components from the phase contrast image and lower spatial frequency components from the attenuation image.

The attenuation of the fringe amplitudes can be higher than an average intensity attenuation, due to a loss of phase coherence of the beam as it propagates through a sample. Excess attenuation is represented as de-coherence, or a scattering image and contains information on structures in the sample that are smaller than the resolution of the imaging system.

In the disclosed examples, phase gratings are used to define x-ray interferometer systems, and these systems are configured so that different diffraction orders (associated with balanced or unbalanced paths) are laterally separated on a detector, typically a camera. Overlapping of diffraction orders tends to reduce fringe visibility, and such reduction can be substantial.

Separation of diffraction orders is achieved by selecting an x-ray beam width and angular divergence such that x-ray beam width is less than a lateral separation between adjacent diffraction orders. For highly collimated beams (i.e., beams with very low angular divergence), the angular beam spread need not be considered, and a required or preferred lateral separation depends only on incident beam width.

FIG. 7 illustrates multiple fringes associated with non-overlapping diffraction orders obtained using an interferometer such as that of FIGS. 1A-1B with an x-ray line beam at an x-ray photon energy of 22.5 keV. Gratings with a grating period of 200 nm and a grating separation of 65 cm were used. Gratings were oriented horizontally. A width of the incident x-ray beam was limited to 160 μιη by a slit. Balanced diffraction paths produce interference fringes associated with +1, +2 and -1, -2 bands 701, 702, 703, 704, respectively. A 0 ^th order band 708 exhibits reduced fringe visibility due to unbalanced x-ray paths.

With reference to FIG. 8, images based on fringe patterns are obtained by recording a reference phase image at 801. A specimen is inserted into at least one x-ray beam path at 802. At 804, a first fringe pattern is recorded. At 806, a determination as to acquisition of additional fringe patterns is made. If an additional fringe pattern is needed, a grating (in a three grating system, typically the second grating) is stepped at 805 and additional fringe patterns are recorded. In other examples, the x-ray beam (typically a cone beam) or the sample is stepped to obtain additional fringe patterns, or one or more of gratings, x-ray beam, or the sample is stepped. For horizontal gratings, the stepped grating is moved vertically by, for example, about 20 to 40 nm. Data for one or more diffraction bands can be obtained at each grating step. Fourier analysis of fringe intensity can be used to determine an estimated actual position of the grating in each phase step in order to reduce the effects of instrumental drifts. Once phase stepping positions are determined, a least squares fitting routine can be used to determine the phase of each pixel at 808. At 810, a phase image is store or displayed, or combined with an amplitude image, and then stored or displayed.

Images acquired from a sample of two intersecting hairs are shown in FIGS. 9A-9D. FIG. 9A is an image based on direct x-ray projection having edges that are enhanced by Fresnel diffraction over a sample-camera distance, FIGS. 9B-9C are images acquired with a grating based interferometer, using a single fringe band. FIG. 9B shows reference fringes obtained without the sample, and FIG. 9C shows fringes with the sample in place. Using a series of such fringes obtained by stepping at least one of the interferometer diffraction gratings, an absolute phase image shown in FIG. 9D is produced. The vertical hair shown in FIG. 9A intersected both interferometer paths, resulting in twin images of opposite phase as shown in FIG. 9C. Phase wrapping is seen in both hairs due to the fact that the phase shifts exceeded π.

FIGS. 10A-10B are images of a distribution of polystyrene spheres. FIG. 10A is a direct projection, and edge enhancement is due to Fresnel diffraction over a sample-camera distance. FIG. 10B is a phase image acquired with a grating interferometer. Phase values associated with each bead can be positive or negative, depending on in which of the two interfering beams it is located. In addition, there are overlapping phase images of two beads of opposite phases from beads that were physically in separate paths.

In examples using phase gratings, diffracted radiation from all or many diffraction orders may reach a detector. In the examples above, aperture plates are provided to block some diffraction orders. Such aperture plates can be provided along with and secured to one or more gratings as shown in FIG. 11. Gratings 1102, 1104, 1106 are situated along an axis 1110 and include respective aperture layers 1103, 1105, 1107 that are configured to selectively transmit and attenuate x-rays as desired. As a result, x-radiation from only selected diffraction orders reaches a detector 1120. The aperture layers 1103, 1105, 1107 can include apertures defined in various ways. For example, as shown in FIG. 11, the aperture layer 1103 includes apertures 1113-1115 defined by voids or specially coated or thinned regions.

FIG. 13 illustrates a grating interferometer configured to produce fringes based substantially on a single pair of symmetric beam paths, achieving near ideal fringe visibility. As shown in FIG. 13, an aperture 1302 is situated to admit an optical beam 1314 so as to be incident to and be diffracted by gratings 1304, 1306, 1308 having periods 2P, P, 2P, respectively. Each of the gratings is a π phase grating. The beam 1314 is diffracted into diffraction orders 1316, 1317 at the grating 1306. These orders are diffracted into diffraction orders 1318, 1319, respectively at the grating 1306 so as to be combined and directed by the grating 1308 as a common beam 1319 to a detector 1312. An aperture 1307 is situated at an output side of the grating 1306 so as to block unwanted diffraction orders. A sample 1324 is situated so as to interact with both of the diffraction orders 1318, 1319, but other placements of the sample 1324 into one or both beams can be used.

FIGS. 14-16 illustrate operation of such a grating interferometer using white light. FIG. 14 shows diffraction intensity in the +1 and -1 orders of a grating such as the grating 1306, demonstrating efficient coupling of power into these two orders. FIG. 15 is an image of bands formed with a white light interferometer such as shown in FIG. 13. FIG. 16 shows fringe shifts associated with a phase object placed in an interferometer path.

In the examples above, X-ray interferometers include gratings that are scanned or translated in a direction perpendicular to an axis of an X-ray beam and slits or other apertures are situated to block selected diffraction orders. In some examples, such slits or apertures are not used. As shown in FIG. 17, an x-ray source 1702 is configured to direct an x-ray beam 1703 to diffraction gratings 1704, 1706, 1708 that are situated along an axis 1701. The gratings 1704, 1708 have a grating period A and the grating 1706 has a grating period A/2. With these grating pitches, balanced paths can be obtained. The grating 1704 diffracts the input beam 1703 into a plurality of diffraction orders, but only the +1 and -1 orders are shown in FIG. 17 as paths 1710, 1712, respectively. The grating 1706 directs portions of the diffracted x-ray beam on the paths 1710, 1712 to produce a -1 diffraction order along a path 1714 and a +1 diffraction order along a path 1716, respectively. The paths 1710, 1714 and 1712, 1716 are balanced, and form sides of a parallelogram 1720. The grating 1708 directs beams from these paths to form an output beam 1720 that is directed to a detector and fringe analysis system. Although not shown in FIG. 17, beams associated with other diffraction orders are also incident to the detector 1740 and the associated fringe patterns may have difference phases from the pattern of 1720. A spacing between the grating 1708 and the detector 1740 is set to allow the diffraction orders to propagate over a distance. The positions of the first or third grating are adjusted to make the fringe patterns of the diffraction orders sum constructively on the detector. The resulting overall fringe pattern is processed with the fringe processor 1761. A sample 1750 is situated in paths 1710, 1712 as shown to produce fringes based on phase differences. As noted above, other sample placements in any one of the paths 1710, 1712, 1714, or 1716 can be used. Fringe spacing can be selected by translating one or more of the gratings in a direction 1770, or a direction perpendicular to the direction 1770.

In the example of FIG. 17, the X-ray beam need not be limited by one or more slits so as to limit beam extent in a direction perpendicular to grating lines. Thus, a cone beam can be used, and need not be aperture to produce a fan beam. In some examples, a slit or other aperture 1709 is provided to stop scattered x-rays in the source. Superior results are obtained if a distance between a final grating (such as the grating 1708) and a detector (such as the detector 1740) is sufficiently large so as to allow diffraction orders to displace relative to each other perpendicularly to the beam axis, before they overlap on the detector, and the positions, tilt angles or periods of the gratings are adjusted by small amounts to make the fringe patterns of the diffraction orders sum constructively on the detector.

In an example shown in FIG. 18, a source 1802 provides a cone beam to gratings 1804,

1806, 1808 so that diffracted beams are incident to a detector 1818. With a cone beam there are divergent incident rays at the first grating 1804, each forming its own set of diffraction orders and paths. Two such paths are shown in FIG. 18. A ray 1820 produces a set of diffraction orders 1821,

1822, 1823 on the detector 1818. The central diffraction order 1822 contains fringes, and the diffraction orders 1821, 1823 also contain fringes but of the opposite phase. The diffraction orders 1831, 1832, 1833 are associated with an angle of incidence of a ray 1830. With some spacing between the third grating 1808 and the detector 1818, diffraction order 1831 overlaps diffraction orders 1822, 1823 and so forth. With perfectly symmetric grating placement (grating periods exactly P, P/2 and P, respectfully and equally spaced gratings), fringes associated with central paths (diffraction orders 1822, 1832) are all phase aligned, and fringes of the "wing" paths (diffraction orders 1821, 1823, 1831, 1833) all have the opposite phase relative to the central paths. Orders 1822 and 1831 overlap destructively, orders 1823, 1832 overlap destructively etc., so that a sum of all paths has no fringes. However, by adjusting the position of the third grating 1808 as shown at 1810, a slope of phase modulation is introduced across the front of the cone beam on the surface of the third grating 1808, which is in a direction perpendicular to the grating lines (vertical as shown in FIG. 18). The result is that the diffraction orders 1821-1833 are shifted by π phase relative to the diffraction orders 1831-1833. In this case, orders 1822, 1831 sum constructively, as well as orders

1823, 1832. More generally, overlapping orders from different incident angles now sum constructively to some degree, resulting in visible horizontal fringes. This can also be done by tilting the grating planes to effectively change the grating period, or by pre-determined small adjustments of the grating periods during grating fabrication. For convenience, one grating can be rotated about a beam axis slightly to make oblique fringes. An image of such a fringe pattern is shown in FIG. 19. This fringe pattern is produced by adjusting the position of the third grating (grating 1808). In this example, the third grating is 1 mm tall and fringes are seen in all of its area.

In view of the many possible embodiments to which the principles of the disclosed invention may be applied, it should be recognized that the illustrated embodiments are only preferred examples of the invention and should not be taken as limiting the scope of the invention. Rather, the scope of the invention is defined by the following claims. We therefore claim as our invention all that comes within the scope and spirit of these claims.