SURFACE PROFILES

CLAIM OF PRIORITY

[0001] This application claims the benefit of priority to U.S. Provisional Appl. No. 62/972,236 filed on February 10, 2020 and incorporated in its entirety by reference herein.

BACKGROUND

Field

[0002] The present application relates generally to x-ray mirror optics.

Description of the Related Art

[0003] Conventional x-ray sources generate x-ray beams that originate and diverge from in a target region that is bombarded by energetic electron beams. In some applications, the x-rays from the x-ray source are collected and directed (e.g., in an extended, focused x-ray beam or in a collimated x-ray beam) to irradiate a sample for analysis, sometimes with additional x-ray optical components, such as a crystal or multilayer monochromator. The x-rays can be directed to irradiate a sample (e.g., the whole sample or a small portion of the sample) to perform one or more forms of x-ray analysis (e.g., imaging; crystallography; absorption spectroscopy; emission spectroscopy; elemental or chemical analysis).

[0004] In some other applications, x-rays (e.g., fluorescence x-rays, elastically or inelastically scattered x-rays) resulting from irradiation of ionizing radiation (e.g., x-rays, energetic electrons, or ions) in a sample are collected and directed to a detector for analysis.

[0005] Various forms of x-ray optic elements/systems are used to collect and condition (e.g., focus; collimate) the x-rays. Grazing incidence x-ray mirrors are achromatic and can provide an advantage over diffractive and refractive optics for many applications. Generally, the grazing incidence angles for desired reflectivities are equal to or less than the critical angle. For each reflection, the angle between the reflected x-ray and the incident x-ray is equal to two times the incidence angle, as measured from the tangent to the surface. SUMMARY

[0006] In certain implementations, an x-ray mirror optic comprises a plurality of surface segments with quadric cross-sections having differing quadric parameters. The quadric cross-sections of the surface segments share a common axis and are configured to reflect x-rays in a plurality of reflections along a single optical axis or in a scattering plane defined as containing an incident x-ray and a corresponding reflected x-ray.

[0007] In certain implementations, an x-ray mirror optic comprises a first plurality of non-axially symmetric mirror sections and a second plurality of non-axially symmetric mirror sections. The first plurality of non-axially symmetric mirror sections and/or the second plurality of non-axially symmetric mirror sections comprises pairs of mirror sections, each pair comprising two mirror sections that have substantially the same quadric surface shape and quadric surface parameters as one another and are on opposite sides of an optical axis of the x-ray optic.

[0008] In certain implementations, a method of fabricating an x-ray mirror optic is provided. The x-ray mirror optic comprises a first plurality of non-axially symmetric mirror sections and a second plurality of non-axially symmetric mirror sections. The first plurality of non-axially symmetric mirror sections and/or the second plurality of non-axially symmetric mirror sections comprises pairs of mirror sections, each pair comprising two mirror sections that have substantially the same quadric surface shape and quadric surface parameters as one another and are on opposite sides of an optical axis of the x-ray optic. The method comprises etching the first plurality of non-axially symmetric mirror sections and/or the second plurality of non-axially symmetric mirror sections into a substrate.

[0009] In certain implementations, an x-ray mirror optic system comprises a first substrate comprising a first plurality of reflective surfaces that are curved in a first cross- sectional plane and that are substantially not curved along a direction substantially perpendicular to the first cross-sectional plane. The x-ray mirror optic system further comprises a second substrate comprising a second plurality of reflective surfaces that are curved in a second cross-sectional plane and that are substantially not curved along a direction substantially perpendicular to the second cross-sectional plane. The second substrate is positioned relative to the first substrate such that the second cross-sectional plane is substantially perpendicular to the first cross-sectional plane. [0010] In certain implementations, an x-ray mirror optic system comprises a pair of first substrates, one substrate of the pair of first substrates comprising a first plurality of reflective surfaces that are curved in a first cross-sectional plane and that are substantially not curved along a direction substantially perpendicular to the first cross-sectional plane. The other substrate of the pair of first substrates comprises a second plurality of reflective surfaces that are curved in a second cross-sectional plane and that are substantially not curved along a direction substantially perpendicular to the second cross-sectional plane. The x-ray mirror optic system further comprises a pair of second substrates, one substrate of the pair of second substrates comprising a third plurality of reflective surfaces that are curved in a third cross-sectional plane and that are substantially not curved along a direction substantially perpendicular to the third cross-sectional plane. The other substrate of the pair of second substrates comprises a fourth plurality of reflective surfaces that are curved in a fourth cross-sectional plane and that are substantially not curved along a direction substantially perpendicular to the fourth cross-sectional plane. The pair of first substrates are positioned such that the second cross-sectional plane is substantially parallel to the first cross-sectional plane, and the pair of second substrates are positioned such that the fourth cross-sectional plane is substantially parallel to the third cross-sectional plane, and the third cross-sectional plane is substantially perpendicular to the first cross-sectional plane.

BRIEF DESCRIPTION OF THE DRAWINGS

[0011] FIG. 1 schematically illustrates an axially symmetric Wolter Type I optic configured for two-dimensional (2D) imaging and comprising a confocal hyperboloidal mirror section followed by an ellipsoidal mirror section.

[0012] FIG. 2 schematically illustrates an example polycapillary x-ray optic.

[0013] FIG. 3A includes various equations relevant to a hyperbolic surface portion (e.g., in 2D) and a hyperboloidal surface portion (e.g., in 3D) in accordance with certain implementations described herein.

[0014] FIG. 3B schematically illustrates an example x-ray optic having hyperbolic / hyperboloidal surface portions in the x-y plane in accordance with certain implementations described herein.

[0015] FIG. 4 schematically illustrates a cross-sectional view of an example x-ray optic comprising at least one first mirror section having a hyperboloidal surface shape and at least one second mirror section having a paraboloidal surface shape in accordance with certain implementations described herein.

[0016] FIG. 5A schematically illustrates a perspective view and a one-half cut away view of an example cylindrical x-ray optic in accordance with certain implementations described herein.

[0017] FIG. 5B schematically illustrates a perspective view and a one-quarter cut away view of an example rotational symmetric x-ray optic in accordance with certain implementations described herein.

[0018] FIG. 6A schematically illustrates first and second substrates in accordance with certain implementations described herein.

[0019] FIG. 6B schematically illustrates a pair of first substrates and a pair of second substrates in accordance with certain implementations described herein.

DETAILED DESCRIPTION

Overview

[0020] X-ray mirrors can be used to focus or collimate x-rays from an x-ray source or from a sample irradiated by ionizing radiation (e.g., x-rays; electrons; ions), and to produce an image of an object irradiated by an x-ray beam (e.g., in an x-ray microscope). Conventional x-ray mirror optics are limited in numerical aperture, optical aperture, and/or focal length, and these limitations can be problematic for various applications. Examples of conventional x-ray mirror optics to be used with an x-ray source (e.g., a primary x-ray source or a secondary x-ray source) include x-ray mirror lenses having a Wolter-type x-ray optic configuration. A Wolter-type x-ray optic configuration is a compound optic configuration comprising two mirrors of different reflecting surface profiles, and such configurations have been used previously for x-ray telescopes. There are several different types of Wolter x-ray optics that have been specifically designed for x-ray astronomy (e.g., the Wolter x-ray optics focus an extended parallel incident x-ray beam), and examples include:

• Type I: Comprising an ellipsoidal mirror section and a hyperboloidal mirror section with x-rays reflected in the same direction (e.g., for demagnifying use);

• Type II: Comprising a hyperboloidal mirror section and an ellipsoidal mirror section, with x-rays reflected in opposite directions; and • Type III: Comprising an ellipsoidal mirror section and a paraboloidal mirror section, with x-rays reflected in opposite directions.

[0021] FIG. 1 schematically illustrates an axially symmetric Wolter Type I optic 10 configured for two-dimensional (2D) imaging and comprising a confocal hyperboloidal mirror section 12 followed by (e.g., upstream from) an ellipsoidal mirror section 14. At least some of the radiation (e.g., neutrons; x-rays) emitted from a source positioned substantially at a focus 16 of the hyperboloidal mirror section 12 is reflected by the hyperboloidal mirror section 12 and is subsequently reflected by the ellipsoidal mirror section 14. The radiation 18 reflected from the hyperboloidal mirror section 12 appears to come from a conjugate focus 20 of a virtual hyperbola 22 corresponding to the hyperboloidal mirror section 12 (denoted in FIG. 1 by dotted line 24). When this conjugate focus 20 of the virtual hyperbola 22 is substantially coincident with one of the two foci 26a of the ellipsoid 28 corresponding to the ellipsoidal mirror section 14, the radiation 30 reflected from the hyperboloidal mirror section 12 and subsequently from the ellipsoidal mirror section 14 produces a real, magnified image 32 of the object, the magnified image 32 at the other of the two foci 26b of the ellipsoid 28 (e.g., where a radiation detector can be positioned). The magnified image 32 can have a magnification M equal to the ratio of the distance ft between the Wolter optic 10 and the image 32 and the distance f ₀ between the Wolter optic 10 and the focus 16 (e.g., M = 10). Other configurations in which the positions of the object and the image are switched with one another (e.g., resulting in a demagnification of the object) are also possible. In addition, other configurations which utilize a confocal hyperbolic mirror section followed by (e.g., upstream from) an elliptic mirror section can be used for one-dimensional (ID) imaging. In addition to the Wolter optic 10, an aperture stop 34 can be configured (e.g., positioned along the optical axis 36 of the Wolter optic 10) to absorb x-rays that are not reflected by the Wolter optic 10. While FIG. 1 shows a single Wolter optic 10, nesting of many mirrors can increase flux collection.

[0022] FIG. 2 schematically illustrates an example polycapillary x-ray optic 50, which is considered to be a non-imaging optic. Each capillary of the polycapillary x-ray optic 50 serves as a light-guide, with the x-rays 52 from an x-ray source 54 received by a first end 56a of the polycapillary x-ray optic 50 being reflected many times on the inside surface of the hollow capillaries. The points on the capillary surfaces where an incident x-ray is reflected are not deterministic. As seen in FIG. 2, x-rays 58 emerging from the second end 56b of the polycapillary x-ray optic 50 diverge, with an angular extent that is determined by the critical angle of the capillary surfaces (e.g., in a x-ray spot having a micron-scale width). Since the critical angle is proportional to the wavelength, soft x-rays spread rapidly beyond the second end 56b of the polycapillary x-ray optic 50, while hard x-rays generally retain their directionality. However, the critical angle also determines the acceptance at the first end 56a of the polycapillary x-ray optic, 50 with the acceptance much higher for soft x-rays than for hard x-rays.

[0023] While it is desirable to collect and direct a larger fraction of the x-rays from an x-ray source to speed the analysis (e.g., using an optical system that captures a large fraction of the x-rays emerging from the x-ray source and to direct the captured x-rays onto the small sample), conventional axially symmetric x-ray mirror optic designs (e.g., for ID and 2D focusing, collimating, and/or imaging applications) have limited numerical apertures (NA) and are therefore limited in the amount of radiation that can be collected (e.g., from a primary source). This limitation of the NA constrains the collection efficiency of the x-ray optic of x-rays from a primary x-ray source.

[0024] For example, an axially symmetric single-reflection x-ray optic (e.g., ellipsoidal x-ray optic) has an NA that is generally limited to about the critical angle. Axially symmetric Wolter optics which provide two reflections for collimating optics (see, e.g., FIG. 1) and four reflections for a focusing double Wolter optic can be configured to improve the radiation collection greatly (e.g., four times the critical angle for a focusing Wolter optic, in which two mirrored Wolter optics are used). In certain implementations described herein, the NA is increased further by increasing the number of reflections from the x-ray optic.

[0025] In addition, a beam stop (e.g., aperture stop 34) can be used to intercept (e.g., block) unreflected x-rays emitted from the object from propagating to the location of the image through the central region of the hollow cylindrical x-ray optic, as schematically illustrated by FIG. 1. These unreflected x-rays propagating through the central region can be a substantial fraction of the emitted x-rays and these x-rays are lost (e.g., not used in the focusing, collimating, and/or imaging by the x-ray optic), even for x-ray optics having a large diameter. However, designs which include nested x-ray optics (e.g., multiple x-ray optic components arranged concentrically about a common optical axis), can be used to reduce the “wasted” central area of a large diameter optic. While axially symmetric nested x-ray optics can be very difficult to make using conventional technology (e.g., glass capillary shaping through mandrels or by glass blowing), certain implementations described herein can utilize KB-type arrangements of sets of one-dimensional x-ray optic elements (e.g., sets of x-ray mirrors that are substantially curved in one plane along the optical axis and substantially flat in another plane substantially perpendicular to the optical axis).

[0026] The limited NA (e.g., limited optical aperture) can be problematic in applications in which these conventional x-ray mirror optics are used. For example, the x-ray mirror optic can limit the x-rays collected from a laboratory x-ray source (e.g., which can have a brightness that is orders of magnitude lower than that of a synchrotron x-ray source) and directed to illuminate a sample. For another example, the x-ray mirror optic can limit the x-rays collected for analysis (e.g., by an energy dispersive detector, wavelength spectrometer, and/or confocal XRF analyzer) from a sample irradiated by ionizing radiation (e.g., x-rays; electrons; protons), the x-rays generated in response to (e.g., induced by) the ionizing radiation. For still another example, the x-ray mirror optic can limit the spatial resolution and/or the x-ray collection efficiency when used as either a condenser or an objective lens in a full-field x-ray imaging microscope.

[0027] The focal length of conventional x-ray mirror optics can also be limited. For example, the focal length can be constrained to be longer than either the length of the x-ray mirror optic (e.g., for a single x-ray mirror optic) or the length of the downstream x-ray mirror optic (e.g., for two or more x-ray mirror optics, such as Wolter type I x-ray mirror optics, twin paraboloidal mirror lenses, and/or KB mirror pairs). The long focal length can lead directly to long distances between the sample and the detector (e.g., camera) (e.g., when used as an objective in a full-field x-ray transmission microscope or as an focusing optic in a confocal x-ray fluorescence system). These long distances, resulting from the long focal lengths, can become problematic for practical reasons when fabricating x-ray analysis systems (e.g., less stable performance in response to thermal fluctuations and/or utilizing excessive laboratory space).

Example implementations

[0028] Certain implementations described herein provide an x-ray mirror optic having a plurality of surface segments with quadric (e.g., hyperbolic or hyperboloidal) cross- sections having differing quadric (e.g., hyperbolic or hyperboloidal) parameters. The quadric (e.g., hyperbolic or hyperboloidal) cross-sections of the surface segments share a common axis, and are configured to reflect x-rays in a plurality of reflections along a single direction or in a scattering plane (e.g., defined as containing the incident x-ray on the surface and the reflected x-ray from the surface). Certain such implementations advantageously provide larger numerical apertures, larger optical apertures, and/or shorter focal lengths than do conventional x-ray mirror optics. Certain implementations can be used in place of conventional x-ray optics (e.g., condensers; KB mirror pairs) and/or can be used in place of a zone plate x-ray optic. While various implementations are described herein as comprising reflective surface portions that are curved in two dimensions (e.g., paraboloidal; ellipsoidal; hyperboloidal) (e.g., having a first curved cross-section in a first plane and having a second curved cross-section in a second plane perpendicular to the first plane), other implementations can comprise reflective surface portions that are curved in only one direction (e.g., parabolic; elliptic; hyperbolic) (e.g., having a first curved cross-section in a first plane and having a second substantially flat cross-section in a second plane perpendicular to the first plane).

[0029] Certain implementations described herein provide axially symmetric “Super Wolter” x-ray optics comprising three or more x-ray mirror sections having quadric surface profiles (e.g., parabolic; elliptic; hyperbolic; paraboloidal; ellipsoidal; hyperboloidal). In certain implementations, these three or more x-ray mirror sections are produced from replicas of one or more mandrels. For example, the surfaces of multiple mandrels can be individually shaped to have the quadric surface profiles of the axially symmetric mirror sections and aligned with one another. In certain implementations, mandrels can be used to fabricate the sets of x-ray mirror sections (e.g., x-ray mirror sections substantially curved in one dimension or two dimensions) in multiple parts (e.g., two halves; two or more parts) and the sets of x-ray mirror sections can be assembled together to cover a predetermined range of angles around the optical axis (e.g., at least 60 degrees; at least 90 degrees; at least 180 degrees; at least 270 degrees; up to 360 degrees; less than 360 degrees).

[0030] For another example, different portions of an inner surface of a continuous glass tube can be directly shaped to form the quadric surface profiles of the axially symmetric mirror sections. In certain implementations, portions of the shaped axially symmetric capillaries are cut and aligned (e.g., by a holder or by adhering the portions together) to form the x-ray optic. In certain implementations, the inner surfaces of the glass capillary portions can comprise at least one layer (e.g., at least one coating; a plurality of layers; multilayers) comprising at least one high atomic number element (e.g., platinum; iridium), the at least one layer configured to improve the x-ray reflectivity of the inner surfaces of the glass capillary portions (e.g., in a predetermined range of x-ray energies).

[0031] FIG. 3A includes various equations relevant to a hyperbolic surface portion (e.g., in 2D) and a hyperboloidal surface portion (e.g., in 3D) in accordance with certain implementations described herein. FIG. 3B schematically illustrates an example x-ray optic 100 having hyperbolic / hyperboloidal surface portions 110, 120 in the x-y plane in accordance with certain implementations described herein. In certain implementations, a first hyperbolic / hyperboloidal surface portion 110 has one of its foci 112a aligned with a small target region 130 of an x-ray source which generates x-rays, the first hyperbolic / hyperboloidal surface portion 110 configured to create a reflected x-ray beam that appears to originate from, and to be diverging from, the conjugate focus 112b of the first hyperbolic / hyperboloidal surface portion 110. While the discussion herein is valid for both hyperbolic mirrors in 2D and hyperboloidal mirrors in 3D, only hyperboloidal mirror optics are referred to explicitly for simplicity sake.

[0032] A hyperboloid is defined by the lengths a and c, as shown in FIG. 3B, and the slope of the asymptote b/a, where b ² = c ² - a ² . As shown in FIG. 3B, an x-ray source 130 is positioned at a first focus (C, 0) (e.g., a primary focus 112a) of a first hyperboloidal surface portion 110 of a first hyperboloidal x-ray mirror 114, the first hyperboloidal surface portion 110 having a first hyperboloidal shape that is symmetric about the y-axis. At least some of the x-rays 140 emitted from the x-ray source (e.g., at an angle a relative to the A-axis) are reflected by the first hyperboloidal surface portion 110 (e.g., at a position (A, y)), and the reflected x-rays 142 appear to originate from a second focus (-C, 0) (e.g., the conjugate focus 112b) of the first hyperboloidal surface portion 110. As the reflection point (A, y) moves closer to the vertex (a, 0) of the first hyperboloidal surface portion 110, the grazing angle O _{gm- g} (e.g., the angle between the emitted x-ray 140 and the tangent of the first hyperboloidal surface portion 110 at the reflection point (A, y)) increases. At the position at which the grazing angle Ogm-mg reaches the critical angle 0 _cr iticai, the first hyperboloidal surface portion 110 no longer reflects the x-rays 140 from the x-ray source 130. With the angle a between the emitted x-ray 140 and a first axis of the first hyperboloidal surface portion 110 (e.g., the A-axis), the angle between the reflected x-ray 142 and the first axis (e.g., the A-axis) is equal to (a - 2- Ogm-mg)·

[0033] FIG. 4 schematically illustrates a cross-sectional view of an example x-ray optic 200 comprising at least one first mirror section 210 having a hyperboloidal surface shape and at least one second mirror section 220 having a paraboloidal surface shape in accordance with certain implementations described herein. For example, the at least one first mirror section 210 can comprise a unitary axially symmetric mirror section and the at least one second mirror section 220 can comprise a unitary axially symmetric mirror section. In certain implementations, these unitary axially symmetric mirror sections are portions of a single monolithic reflecting element (e.g., surface of an axially symmetric substrate or tube). The focus 222 of the paraboloidal surface shape is at (e.g., substantially coincident with) the conjugate focus 212b of the hyperboloidal surface shape. In certain implementations in which the x-ray source is at the primary focus 212a of the hyperboloidal surface shape, the at least one second mirror section 220 can collimate the x-rays that are emitted by the x-ray source and that are reflected by the at least one first mirror section 210, which appear to be emitted from a virtual x-ray source positioned at the conjugate focus 212b of the hyperboloidal surface shape. In certain other implementations in which the incident x-rays are collimated and incident from the right-side of FIG. 4, the example x-ray optic 200 focuses the x-rays reflected by the at least one first mirror section 210 and the at least one second mirror section 220 to the primary focus of the hyperboloidal surface shape (e.g., acts as a telescope). In other example x-ray optics comprising a plurality of hyperboloidal surface sections from which x-rays are sequentially reflected, the focus of the paraboloidal surface shape can be at (e.g., substantially coincident with) the conjugate focus 212b of the last hyperboloidal surface shape of the plurality of hyperboloidal surface shapes. While not shown in FIGs. 3B or 4, certain implementations comprise one or more beam stops configured to intercept (e.g., block) unreflected x-rays emitted from the x-ray source from propagating downstream through the central region of the axially symmetric x-ray optic.

[0034] FIG. 5A schematically illustrates a perspective view and a one-half cut away view of an example cylindrical Wolter x-ray optic 300 in accordance with certain implementations described herein. FIG. 5B schematically illustrates a perspective view and a one-quarter cut-away view of an example rotational symmetric Wolter x-ray optic 300 in accordance with certain implementations described herein. The example Wolter x-ray optics 300 of FIGs. 5A and 5B comprises a first plurality of non-axially symmetric mirror sections 310 (e.g., two, three, four, five or more) each having a quadric (e.g., hyperbolic) surface shape and a second plurality of non-axially symmetric mirror sections 320 (e.g., two, three, four, five or more) each having a quadric (e.g., hyperbolic) surface shape. In FIG. 5A, the mirror sections 310 are curved in only one first plane (e.g., along an optical axis 330 of the x-ray optic 300) and the mirror sections 320 are curved in only one second plane (e.g., along the optical axis 330 of the x-ray optic 300 and substantially perpendicular to the first plane) in accordance with certain implementations described herein. In FIG. 5B, the mirror sections 310 are axially symmetric (e.g., curved around the optical axis 330 of the x-ray optic 300) and the mirror sections 320 are axially symmetric (e.g., curved around the optical axis 330 of the x-ray optic 300) in accordance with certain implementations described herein. In certain implementations, as schematically illustrated by FIG. 5A, the first plurality of mirror sections 310 and/or the second plurality of mirror sections 320 can comprise pairs of mirror sections, each pair comprising two mirror sections that have substantially the same quadric (e.g., hyperbolic) surface shape and substantially the same quadric surface parameters as one another and are on opposite sides of the optical axis 330 of the x-ray optic 300. In certain other implementations, the first plurality of mirror sections 310 and/or the second plurality of mirror sections 320 can comprise multiple mirror sections that have substantially the same quadric (e.g., hyperbolic) surface shape and substantially the same quadric surface parameters as one another and are positioned around the optical axis 330 of the x-ray optic 300. In certain other implementations, other quadric shapes (e.g., elliptic) can be used. In certain implementations, the first plurality of mirror sections 310 can provide focusing (e.g., one-dimensional) along a first direction (e.g., line), and the second plurality of mirror sections 320 can provide focusing (e.g., one-dimensional) along a second direction (e.g., line) substantially perpendicular to the first direction, resulting in two-dimensional focusing (e.g., point focusing), in a manner similar to that of Kirkpatrick-Baez (KB) x-ray optics.

[0035] The first plurality of mirror sections 310 can be nested (e.g., spaced apart from and extending generally along one another; arranged concentrically about a common optical axis; substantially parallel with one another) and/or the second plurality of mirror sections 320 can be nested (e.g., spaced apart from and extending generally along one another; arranged concentrically about a common optical axis; substantially parallel with one another) and can be oriented substantially perpendicularly to the first plurality of mirror sections (e.g., similar to a KB mirror configuration). In certain implementations, adjacent mirror sections of the first plurality of mirror sections (e.g., two mirror sections without another mirror section therebetween) are spaced from one another by a distance in a range 1 micron to 10 microns. In certain implementations, adjacent mirror sections of the second plurality (e.g., two mirror sections without another mirror section therebetween) are spaced from one another by a distance in a range of 1 micron to 10 microns. In certain other implementations, the first plurality of mirror sections 310 and the second plurality of mirror sections 320 are interweaved with one another (e.g., the mirrors sections 310, 320 of the first plurality and the second plurality alternate with one another along an optical axis 330 of the x-ray optic 300. In certain implementations (see, e.g., FIG. 5B), the x-ray optic 300 comprises a plurality of nested mirror portions 310, 320 and a support structure 340 (e.g., spacers; struts; braces) configured to support and align the mirror portions 310, 320 relative to one another.

[0036] In certain implementations, the mirror sections 310, 320 of the first plurality and/or the second plurality can be etched into a substrate (e.g., silicon) (e.g., using metal assisted etching or LIGA). FIG. 6A schematically illustrates first and second substrates 410, 420 in accordance with certain implementations described herein. The first substrate 410 comprises a first plurality of reflective surfaces 412 (e.g., etched into the first substrate 410) that are curved in a first cross-sectional plane 414 (schematically illustrated by a dotted line) and that are substantially not curved (e.g., straight) along a direction substantially perpendicular to the first cross-sectional plane 414. The second substrate 420 comprises a second plurality of reflective surfaces 422 (e.g., etched into the second substrate 420) that are curved in a second cross-sectional plane 424 (schematically illustrated by a dotted line) and that are substantially not curved (e.g., straight) along a direction substantially perpendicular to the second cross-sectional plane 424. The second substrate 420 is positioned relative to the first substrate 410 such that the second cross-sectional plane 424 is substantially perpendicular to the first cross-sectional plane 414. In certain implementations, the first plurality of reflective surfaces 412 and the second plurality of reflective surfaces 422 are configured to focus incident x-rays to a focal point (e.g., in a KB mirror configuration). In certain other implementations, the first plurality of reflective surfaces 412 and the second plurality of reflective surfaces 422 are configured to collimate incident x-rays (e.g., collimate in two dimensions). In certain implementations, the first plurality of reflective surfaces 412 and the second plurality of reflective surfaces 422 are separated from one another by a distance in a range of less than one micron, a range of 1 micron to 5 microns, a range of 5 microns to 20 microns, or a range of 20 microns to 100 microns. In certain implementations, the reflective surfaces 412, 422 are KB -mirror- like to focus in a point-like manner, while in certain other implementations, the reflective surfaces 412, 422 both have a collimating shape to collimate in two dimensions.

[0037] FIG. 6B schematically illustrates a pair of first substrates 410 and a pair of second substrates 420 in accordance with certain implementations described herein. One substrate of the pair of first substrates 410 comprises a first plurality of reflective surfaces 412a (e.g., etched into the substrate) that are curved in a first cross-sectional plane 414a and that are substantially not curved (e.g., straight) along a direction substantially perpendicular to the first cross-sectional plane 414a and the other substrate of the pair of first substrates 410 comprises a second plurality of reflective surfaces 412b (e.g., etched into the substrate) that are curved in a second cross-sectional plane 414b and that are substantially not curved (e.g., straight) along a direction substantially perpendicular to the second cross-sectional plane 414b. One substrate of the pair of second substrates 420 comprises a third plurality of reflective surfaces (not shown)(e.g., etched into the substrate) that are curved in a third cross- sectional plane 424a and that are substantially not curved (e.g., straight) along a direction substantially perpendicular to the third cross-sectional plane 424a and the other substrate of the pair of second substrates 420 comprises a fourth plurality of reflective surfaces (not shown)(e.g., etched into the substrate) that are curved in a fourth cross-sectional plane 424b and that are substantially not curved (e.g., straight) along a direction substantially perpendicular to the fourth cross-sectional plane 424b. The pair of first substrates 410 can be positioned such that the second cross-sectional plane 414b is substantially parallel to the first cross-sectional plane 414a, and the pair of second substrates 420 can be positioned such that the fourth cross-sectional plane 424b is substantially parallel to the third cross-sectional plane 424a, and the third cross-sectional plane 424a is substantially perpendicular to the first cross- sectional plane 414a. For example, as schematically illustrated by FIG. 6B, the pair of first substrates 410 can form a Wolter x-ray optic and the pair of second substrates 420 can form a Wolter x-ray optic in an orthogonal direction to that of the pair of first substrates 410. In both FIGs. 6A and 6B, the separation between adjacent and substantially parallel reflective surfaces of a plurality of reflective surfaces can be on the order of microns (e.g., in a range of 1 micron to 10 microns), and the surface roughness can be less than 2 nanometers (e.g., by using thermal annealing or another process to reduce surface roughness of the etched surfaces).

[0038] In certain implementations, the example x-ray optic can be used in place of a zone plate x-ray optic downstream from a sample under analysis. The example x-ray optic can provide achromatic focusing, so that multiple zone plates for different x-ray wavelengths are not needed. For example, in x-ray systems utilizing multiple x-ray wavelengths, instead of using multiple zone plates, each designed for one x-ray energy, a single x-ray optic in accordance with certain implementations described herein can be used if the x-ray wavelengths are sufficiently close together such that the critical angles for reflection are not substantially different from one another.

[0039] For other x-ray systems in which the x-ray wavelengths are not sufficiently close together (e.g., TriLambda x-ray system marketed by Sigray, Inc. of Concord CA), multiple x-ray optics in accordance with certain implementations described herein can be used. In certain implementations, the x-ray optic can be used as a zone plate replacement for hard x-ray energy imaging (e.g., 17 keV with molybdenum x-rays or 11 keV with Au x-rays).

[0040] In addition, zone plates exhibit substantial chromatic aberration, so that laboratory x-ray systems are limited to x-ray sources that produce a characteristic x-ray line that can be separated from other characteristic x-ray lines (e.g., the K _a x-ray line can be filtered from the K _b x-ray line). By using the achromatic performance of the x-ray optic of certain implementations instead of using zone plates, other laboratory x-ray sources (e.g., molybdenum; silver; rhodium K-line) can be used. Example implementation #1

[0041] In certain implementations, one or more x-ray mirrors having a plurality of hyperbolic / hyperboloidal surface portions can be used to collect x-rays from an x-ray source with a large emission angle. For example, referring to FIG. 3B, the one or more x-ray mirrors comprises a first hyperboloidal surface portion 110, as described above, and a second hyperboloidal surface portion 120 that has a second axis common (e.g., shared; substantially coaxial) with the first axis (e.g., the c-axis) but having a smaller value of the asymptote slope b/a, and with a first focus (e.g., a primary focus 122a) substantially coincident with the conjugate focus 112b at (-c, 0) of the first hyperboloidal surface portion 110. The first hyperboloidal surface portion 110 is configured to intercept (e.g., reflect) x-rays 140 emitted from the x-ray source 130 at the primary focus 112a at (c, 0) of the first hyperboloidal surface portion 110. These singly-reflected x-rays 142 appear to originate from the conjugate focus 112b at (-c, 0) of the first hyperboloidal surface portion 110 (e.g., a virtual x-ray source). The second hyperboloidal surface portion 120 is configured to intercept (e.g., reflect) x-rays 142 reflected by the first hyperboloidal surface portion 110, resulting in an additional reduction of the angle between the twice-reflected x-rays 144 with the shared hyperboloidal mirror axis (e.g., the A-axis). The twice-reflected x-rays 144 appear to originate from a second focus 122b at {-Cl, 0) (e.g., the conjugate focus 122b) of the second hyperboloidal surface portion 120.

[0042] In certain implementations, additional hyperboloidal surface portions can be used (e.g., 3, 4, 5, or more hyperboloidal surface portions), with each hyperboloidal surface portion serving as a virtual x-ray source for the subsequent hyperboloidal surface portion. The sequential hyperboloidal surface portions have successively smaller asymptote slope values b/a, to multiply reflect and to obtain larger reductions of the angle between the initial x-ray emitted from the x-ray source with the hyperboloidal mirror axis common to (e.g., shared by; substantially coaxial with) all the hyperboloidal mirror portions (e.g., the A-axis). For example, the angle between the last reflected x-ray and the shared hyperboloidal mirror axis can be reduced to be close to zero. In certain implementations, an x-ray optic comprises a plurality of hyperboloidal mirror surface portions that are configured to collect x-rays from an x-ray source with large angles with respect to the shared hyperboloidal axis (e.g., up to 2· N O _criticai , where N is the number of successive hyperboloidal surface portions. In certain implementations, the x-rays emitted from the x-ray source at (C, 0) that impinge and are not reflected by the first hyperboloidal surface are intercepted by a central beam stop and/or the nested set of additional hyperboloidal surfaces.

Example implementation #2

[0043] In certain implementations, according to the reciprocal principle, incident x-rays are reflected by and focused by the hyperbolic / hyperboloidal surface portions disclosed in the example implementation #1 to converge at the point (C, 0). Certain such implementations utilize a carefully tailored converging beam of x-rays to match the reversed output from example implementations #1.

Example implementation #3

[0044] In certain implementations, one or more x-ray mirrors having a plurality of nested hyperbolic / hyperboloidal surface portions can be used to collect x-rays in a large collection angle from the x-ray source. For example, to collect x-rays from an x-ray source over a large angular range, a nested array of hyperboloidal mirrors having a common (e.g., shared; substantially coincident) primary focus and a common (e.g., shared; substantially coaxial) hyperboloidal axis, each hyperboloidal mirror of the nested array having a different value of the hyperbolic parameter b (or a different value of the ratio bla, since the nested set shares the same value of c, so if b changes, bla also changes). In certain implementations, the thickness of the hyperboloidal mirror is less than the gap distance between two neighboring hyperboloidal mirrors.

Example implementation #4

[0045] In certain implementations, one or more x-ray mirrors having a plurality of nested hyperbolic / hyperboloidal surface portions can be used in combination with a nested set of parabolic / paraboloidal mirror surfaces to produce a collimated x-ray beam. For example, the nested set of parabolic / paraboloidal mirror surfaces can be substantially coaxial with the common (e.g., shared) hyperbolic / hyperboloidal axis and the foci of the parabolic / paraboloidal mirror surfaces can be at the conjugate foci of the last hyperbolic / hyperboloidal mirror surfaces of the nested hyperbolic / hyperboloidal mirror surfaces (e.g., provided that the angle of the x-rays reflected by the last hyperbolic / hyperboloidal mirror surface and the shared axis is less than the critical angle of the parabolic / paraboloidal mirror surface). Example implementation #5

[0046] In certain implementations, one or more x-ray mirrors having a plurality of hyperbolic / hyperboloidal surface portions can be used in combination with an optic train following the plurality of hyperbolic / hyperboloidal surface portions, to produce a focused x-ray beam. The optic train can comprise at least one of the following:

• An elliptical / ellipsoidal mirror surface (elliptical if following a hyperbolic mirror surface, ellipsoidal if following a hyperboloidal mirror surface), when the x-ray beam divergence angle is less than the critical angle of the reflection surface of the elliptical / ellipsoidal mirror surface;

• A first parabolic / paraboloidal mirror surface (parabolic if following a hyperbolic mirror surface, paraboloidal if following a hyperboloidal mirror surface) configured to collect and collimate the x-rays followed by a second parabolic / paraboloidal mirror surface configured to focus the collimated beam;

• An elliptical / ellipsoidal mirror surface (elliptical if following a hyperbolic mirror surface, ellipsoidal if following a hyperboloidal mirror surface) followed by a second set of hyperbolic / hyperboloidal surface portions having a reversed orientation relative to the first set of hyperbolic / hyperboloidal surface portions; and

• A cylindrical mirror, when the x-ray beam divergence angle, after the plurality of hyperbolic / hyperboloidal surface portions, is smaller than the critical angle of the cylindrical mirror. Certain such implementations provide a symmetric focusing system with 1:1 magnification when the second set of hyperbolic / hyperboloidal surface portions is the same as the first set of hyperbolic / hyperboloidal surface portions.

Example implementation #6

[0047] In certain implementations, the x-ray optic is a compound optic and comprises a first plurality of orthogonal ID focusing optics comprising hyperbolic surface portions and corresponding elliptical surface portions following the hyperbolic surface portions and a second plurality of orthogonal ID focusing optics comprising hyperbolic surface portions and corresponding elliptical surface portions following the hyperbolic surface portions, configured to provide point focusing. In certain implementations, the first plurality can be nested with one another and/or the second plurality can be nested with one another. The x-ray optic can be a compound optic configured to collect x-rays from a point source over a large acceptance angle.

Example implementation #7

[0048] In certain implementations, the x-ray optic is a compound optic and comprises a first plurality of orthogonal ID focusing optics comprising hyperbolic surface portions and corresponding parabolic surface portions following the hyperbolic surface portions (e.g., configured to provide ID collimation) and a second plurality of orthogonal ID focusing optics comprising hyperbolic surface portions and corresponding parabolic surface portions following the hyperbolic surface portions (e.g., configured to provide ID collimation), all configured to provide 2D collimation. In certain implementations, the first plurality can be nested with one another and/or the second plurality can be nested with one another. The x-ray optic can be a compound optic configured to focus a collimated x-ray beam to a point focus.

Example implementation #8

[0049] In certain implementations, the x-ray optic of example implementation #6 can be followed by another x-ray optic of example implementation #6 in reverse order, so as to collect x-rays from an x-ray source and to focus the x-rays to a point focus.

Example implementation #9

[0050] In certain implementations, the x-ray optic of example implementation #7 can be followed by another x-ray optic of example implementation #7 in reverse order, so as to collect collimated x-rays and to focus the x-rays to a point focus.

[0051] Although commonly used terms are used to describe the systems and methods of certain implementations for ease of understanding, these terms are used herein to have their broadest reasonable interpretations. Although various aspects of the disclosure are described with regard to illustrative examples and implementations, the disclosed examples and implementations should not be construed as limiting. Conditional language, such as “can,” “could,” “might,” or “may,” unless specifically stated otherwise, or otherwise understood within the context as used, is generally intended to convey that certain implementations include, while other implementations do not include, certain features, elements, and/or steps. Thus, such conditional language is not generally intended to imply that features, elements, and/or steps are in any way required for one or more implementations. In particular, the terms “comprises” and “comprising” should be interpreted as referring to elements, components, or steps in a non-exclusive manner, indicating that the referenced elements, components, or steps may be present, or utilized, or combined with other elements, components, or steps that are not expressly referenced.

[0052] Conjunctive language such as the phrase “at least one of X, Y, and Z,” unless specifically stated otherwise, is to be understood within the context used in general to convey that an item, term, etc. may be either X, Y, or Z. Thus, such conjunctive language is not generally intended to imply that certain implementations require the presence of at least one of X, at least one of Y, and at least one of Z.

[0053] Language of degree, as used herein, such as the terms “approximately,” “about,” “generally,” and “substantially,” represent a value, amount, or characteristic close to the stated value, amount, or characteristic that still performs a desired function or achieves a desired result. For example, the terms “approximately,” “about,” “generally,” and “substantially” may refer to an amount that is within ± 10% of, within ± 5% of, within ± 2% of, within ± 1% of, or within ± 0.1% of the stated amount. As another example, the terms “generally parallel” and “substantially parallel” refer to a value, amount, or characteristic that departs from exactly parallel by ± 10 degrees, by ± 5 degrees, by ± 2 degrees, by ± 1 degree, or by ± 0.1 degree, and the terms “generally perpendicular” and “substantially perpendicular” refer to a value, amount, or characteristic that departs from exactly perpendicular by ± 10 degrees, by ± 5 degrees, by ± 2 degrees, by ± 1 degree, or by ± 0.1 degree. The ranges disclosed herein also encompass any and all overlap, sub-ranges, and combinations thereof. Language such as “up to,” “at least,” “greater than,” less than,” “between,” and the like includes the number recited. As used herein, the meaning of “a,” “an,” and “said” includes plural reference unless the context clearly dictates otherwise. While the structures and/or methods are discussed herein in terms of elements labeled by ordinal adjectives (e.g., first, second, etc.), the ordinal adjectives are used merely as labels to distinguish one element from another, and the ordinal adjectives are not used to denote an order of these elements or of their use.

[0054] Various configurations have been described above. It is to be appreciated that the implementations disclosed herein are not mutually exclusive and may be combined with one another in various arrangements. Although this invention has been described with reference to these specific configurations, the descriptions are intended to be illustrative of the invention and are not intended to be limiting. Various modifications and applications may occur to those skilled in the art without departing from the tme spirit and scope of the invention. Thus, for example, in any method or process disclosed herein, the acts or operations making up the method/process may be performed in any suitable sequence and are not necessarily limited to any particular disclosed sequence. Features or elements from various implementations and examples discussed above may be combined with one another to produce alternative configurations compatible with implementations disclosed herein. Various aspects and advantages of the implementations have been described where appropriate. It is to be understood that not necessarily all such aspects or advantages may be achieved in accordance with any particular implementation. Thus, for example, it should be recognized that the various implementations may be carried out in a manner that achieves or optimizes one advantage or group of advantages as taught herein without necessarily achieving other aspects or advantages as may be taught or suggested herein.