Advanced laser phase plates for electron microscopy

Abstract

The present disclosure relates to robust optical cavities designed to stabilize optical systems through geometric, structural, and / or power-dependent stabilization methods. The disclosed optical cavities enhance system performance by minimizing optical distortions and improving alignment stability. Additionally, the invention encompasses systems integrating such optical cavities, including electron beam imaging systems, to achieve improved precision and operational robustness.

Description

ADVANCED LASER PHASE PLATES FOR ELECTRON MICROSCOPYFIELD OF THE INVENTION

[0001] The presently disclosed subject matter relates to the field of robust optical cavities configured to stabilize optical systems by means of geometric, structural and / or power dependence stabilization methods. The invention further relates to systems comprising optical cavities including electron beam imaging systems.BACKGROUND

[0002] Optical resonators (also referred to as optical cavities), such as, e.g., Fabry-Perot resonators, are widely used for frequency stabilization, mode filtering, resonant enhancement of laser beam power and intensity, etc. They are also used for enhancement of the interaction strength between light and a particular system of interest.

[0003] In some applications it is advantageous to have a cavity supporting one or more modes of light that have a tight focus inside the cavity. Such conditions can be referred to as the cavity mode having a small waist diameter. A small mode waist diameter increases intensity in a focal location inside the cavity, and therefore increases the strength of interaction of the light inside the cavity with a system of interest positioned in the focal location.

[0004] Despite the advantages of cavities with a small mode waist, making a cavity with a waist below a certain limit is challenging. The difficulties stem from the fact that such cavities approach degeneracy. Cavities that are close to degeneracy are critically sensitive to perturbations and imperfections such as, e.g., imperfection of the shapes of optical elements comprising a cavity, deviation of the cavity element in position or angle from their ideal configuration, thermally or mechanically induced stresses and deformations; photo-thermal effect, etc. The sensitivity of the cavities to such perturbations makes it increasingly challenging to decrease the mode waist below a certain size. In practice, this limits the numerical aperture (NA) of optical cavities to around 0.05 to 0.1.

[0005] Several important applications require optical cavities that, in addition to having a tight internal focus, are able to support high circulating powers. One application is generation of X- rays via inverse Compton scattering of laser light by an electron beam. Another is the generation of deep UV and soft X-ray via high harmonic generation process inside an optical cavity. Further uses are envisioned in future fusion reactors. A key application that motivates this work is phaseretardation of electron beams in transmission electron microscopy. Such applications require optical cavities that can support circulating laser power in the range of tens of kW to MW.

[0006] The necessity to support high circulating power further exacerbates the challenge of building and operating of cavities with a tight internal focus. In this case, the laser beam circulating inside a cavity applies a significant heat load on the surface of the cavity mirrors, which causes the cavity mirrors to bulge locally. Since the mode shape depends on the exact shape of the mirrors, this creates conditions of interdependence between the shape of the modes and the deformation of the mirror surfaces. This mutual dependence leads to unstable cavity behavior.

[0007] Furthermore, high-power cavities are susceptible to another type of instability that is related to radiation pressure on the mirror surface.SUMMARY

[0008] In one embodiment the invention provides an optical cavity comprising: a first segment with a numerical aperture (NAi) and a length (L1); and a second segment with a numerical aperture (NA2) and a length (L2); wherein the first segment and the second segment each comprise two optical elements; wherein NAi is at least 0.05; and wherein L2 is greater than L1by a factor ranging between about 2 and 10,000.

[0009] In one embodiment the mode size on each of the one or more optical elements ranges between 0.5 mm to 1 m. In one embodiment the ratio NA1 / NA2 is at least 2. In one embodiment the ratio NA1 / NA2 ranges between about 2 and 10,000. In one embodiment NAi ranges between 0.05 and 0.99 and NA2 ranges between 0.0001 and 0.04. In one embodiment the L1ranges between 1 mm and 1 m and L2 ranges between 1 cm and 10 m. In one embodiment the at least one of the at least two optical elements are astigmatic. In one embodiment the two or more optical elements are selected from: reflective element, transmissive element, lens-mirror element, or a combination thereof. In one embodiment the reflective element is selected from: a mirror with a concave reflective surface, a mirror with a convex reflective surface, a mirror with a planar reflective surface, a lens-mirror element or a combination thereof. In one embodiment the reflective element is at least partially transmissive. In one embodiment the reflective element comprises a reflective coating. In one embodiment the reflective element has a reflectivity of at least 90%. In one embodiment the transmissive element is selected from: lens, planar optical window, or a combination thereof. In one embodiment the transmissive element comprises ananti -reflective coating. In one embodiment the transmissive element is at least 99% transmissive. In one embodiment the lens is biconcave, biconvex, plano-convex, plano-concave, spherical, aspherical, concave-convex, or a combination thereof. In one embodiment the first segment has a tight focus with a mode waist size ranging between 0.1 pm and 10 pm. In one embodiment the optical cavity further comprises at least one thermal conductor in contact with at least part of the first segment, the second segment, or a combination thereof. In one embodiment the optical cavity further comprises at least one additional optical segment with at least one of the optical elements being the same in at least one other optical segment in the optical cavity. In one embodiment the optical cavity is in a standing wave cavity configuration or a ring cavity configuration. In one embodiment the optical cavity comprises the first segment comprising a first mirror and a lens; and the second segment comprising the lens and a second mirror. In one embodiment the first mirror has a smaller radius of curvature than the second mirror. In one embodiment the first mirror, the lens and the second mirror are arranged to share the same optical axis. In one embodiment the first mirror, the second mirror or a combination thereof, are at least partially transmissive. In one embodiment the optical cavity further comprises an anti -reflective coating disposed on at least one side of the lens.

[0010] In one embodiment of the optical cavity: the first segment comprises a first mirror and a second mirror; the second segment comprises the second mirror and a third mirror; and wherein the first mirror, the second mirror and the third mirror form the standing wave cavity.

[0011] In one embodiment the first mirror and the second mirror have a smaller radius of curvature than the third mirror. In one embodiment the first mirror, the second mirror or the third mirror, or a combination thereof, are at least partially transmissive. In one embodiment of the optical cavity at least three mirrors form in a ring cavity configuration.

[0012] In one embodiment of the optical cavity: the first segment comprises a first mirror and a second mirror; the second segment comprises the second mirror and a third mirror; and wherein the first mirror, the second mirror and the third mirror form the ring cavity configuration.

[0013] In one embodiment the first mirror or the second mirror have a smaller radius of curvature than the third mirror. In one embodiment the first mirror, the second mirror or the thirdmirror, or a combination thereof, are at least partially transmissive. In one embodiment the present invention provides an optical cavity comprising an optical segment with an NA greater than 0.05, wherein the optical segment or an additional optical segment comprises at least one nonlinear optical element with a focusing power responsivity ranging between 0.001 m-1kW-1and 100 m-1kW’1. In one embodiment the NA responsivity of the cavity ranges between 0.0001 / kW to 1 / kW. In one embodiment the optical cavity further comprises a laser beam coupled to the optical cavity which has an intracavity circulating power ranging between IkW to 10 MW. In one embodiment of the optical cavity the NA increases by an amount ranging between about 0.001 to 0.9 upon coupling the laser beam to the optical cavity. In one embodiment the nonlinear optical element comprises a material which has: a negative thermal expansion coefficient, a positive thermal coefficient of the refractive index, a positive Kerr coefficient, or a combination thereof. In one embodiment the optical segments comprise optical elements selected from: reflective element, transmissive element, lens-mirror element, or a combination thereof. In one embodiment the reflective element is selected from: a mirror with a concave reflective surface, a mirror with a convex reflective surface, a mirror with a planar reflective surface, or a combination thereof. In one embodiment the reflective element is at least partially transmissive. In one embodiment the optical cavity further comprises a thermal conductor in contact with at least part of the optical segment. In one embodiment the lens of the lens-mirror element further comprises an anti -reflective coating. In one embodiment the lens and the mirror of the lens-mirror element are joined by an adhesive, an adhesive-free bonding method, or optical contacting. In one embodiment the lens-mirror element comprises a lens or a planar optical window having a first optical surface and a second optical surface, wherein an antireflective coating is disposed on the first optical surface and a reflective coating is disposed on the second optical surface. In one embodiment the optical cavity further comprises at least one additional optical segment with at least one of the optical elements being shared in at least one other optical segment in the optical cavity.

[0014] In one embodiment the present invention provides: an optical segment comprising a first reflecting element and a second reflecting element, wherein the first reflecting element, the second reflecting element, or a combination thereof, comprise a negative thermal expansion coefficient (NTEC) material.

[0015] In one embodiment the NTEC material is selected from: optical glass, optical crystal, fused silica, Ultra-Low Expansion (ULE) glass, diamond, silicon, or a combination thereof. In one embodiment the first reflecting element and the second reflective element are selected from:a mirror with a concave reflective surface, a mirror with a convex reflective surface, a mirror with a planar reflective surface, or a combination thereof. In one embodiment the first reflecting element, the second reflecting element, or a combination thereof, is at least partially transmissive. In one embodiment the optical cavity further comprises a thermal conductor in contact with at least part of the optical segment. In one embodiment the optical cavity further comprises at least one additional optical segment with at least one of the optical elements being the same in at least one other optical segment in the optical cavity.

[0016] In one embodiment the present invention provides an optical cavity system comprising: at least one of the optical cavities disclosed herein; at least one laser; and a plurality of coupling optics and electronic components configured to couple and frequency -lock at least one laser beam from the at least one laser to the optical cavity.

[0017] In one embodiment the optical cavity system further comprises a temperature control system configured to maintain a preset temperature of at least one of the elements in the system and a vacuum system configured to establish a vacuum in the at least one optical cavity. In one embodiment the laser beam coupled into the at least one optical cavity is resonantly enhanced by the cavity by a factor ranging between 3 and 1,000,000. In one embodiment the optical cavity system further comprises an adjustable suspension including piezoelectric actuators or thermomechanical actuators to adjust the angle or position of at least one of the optical elements of the optical cavity. In one embodiment the optical cavity system further comprises a fiber-optic member configured to couple the laser to the optical cavity. In one embodiment the temperature control system is a Peltier system.

[0018] In one embodiment the invention provides an electron beam imaging system comprising: a transmission electron microscope (TEM), comprising an electron source configured to provide an electron beam; and the optical cavity system disclosed herein, configured such that the electron beam passes through the at least one optical cavity.

[0019] In one embodiment the laser beam coupled into the optical cavity is configured to shift the phase of the part of the electron beam passing through the laser beam.

[0020] In one embodiment the at least one optical cavity is positioned: at about a back focal plane of the TEM; orat about a plane that is conjugate to the back focal plane.

[0021] In one embodiment the at least one laser is a continuous-wave laser. In one embodiment the focal spot of the cavity is positioned at about the back focal plane of the TEM, or at about the conjugate plane, and wherein the cavity is positioned to allow an unscattered electron wave of the electron beam to pass through the focal spot. In one embodiment the material structure of the optical cavity system is configured and positioned such that the electron beam of the TEM is at least 0.1 mm away from any material element of the optical cavity system. In one embodiment the TEM further comprises an electron detector positioned at an image plane of the TEM and configured to receive the electron beam.

[0022] In one embodiment each of the at least one optical cavities: is positioned at about the back focal plane of the TEM, or at about a plane that is conjugate to the back focal plane of the TEM; is configured to allow the electron beam, provided by the TEM, to pass through it.

[0023] In one embodiment the at least one laser beam circulating inside the cavity has a power ranging between 10 kW to 10 MW. In one embodiment the at least one laser beam provided by the laser is non-monochromatic. In one embodiment the non-monochromatic at least one laser beam comprises two or more laser beams having different wavelengths. In one embodiment the at least one optical cavity comprises two or more optical cavities with overlapping focal spots.

[0024] In one embodiment of the system: the at least one optical cavity is tilted with respect to the optical axis of the TEM; or the optical axis of the at least one optical cavity is tilted with respect to the optical axis of the TEM; or the direction of the propagation of the at least one laser beam in the at least one optical cavity is tilted with respect to the optical axis of the TEM.

[0025] In one embodiment the electron beam imaging system further comprises an electron camera or one or more sensors, positioned at the image plane and operable to analyze a Ronchigram and provide feedback for automatic control of the electron beam position relative to the optical cavity.

[0026] In one embodiment the invention provides a method for electron beam imaging or electron beam spectroscopy comprising: providing the system according to any one of claims 52 to 57;shifting the phase of a pre-determined portion of the electron beam, via the optical cavity system; and detecting TEM images using a detector.BRIEF DESCRIPTION OF THE DRAWINGS

[0027] In order to better understand the subject matter that is disclosed herein and to exemplify how it may be carried out in practice, embodiments will now be described, by way of non-limiting example only, with reference to the accompanying drawings, in which:

[0028] Figures 1A-1D show graphs comparing the tolerance of a mirror-lens-mirror cavity and a Fabry -Perot cavity to misalignment, as a function of the NA of the high -NA segment of the cavity. The tolerance is defined as the amount of misalignment that reduces the overlap between the perturbed mode (with the misalignment) with the unperturbed mode (without the misalignment) to 0.9. For the mirror-lens-mirror cavity, tolerance to misalignment is shown for each of its three elements. For the Fabry-Perot, only one tolerance is shown for each type of misalignment due to symmetry. Figure 1A shows the tolerance of the two cavities with respect to axial shift of the elements. Figure IB shows the tolerance of the two cavities with respect to tilt of the elements. Figure 1C shows the tolerance of the two cavities with respect to the change of the radius of curvature of the elements. Figure ID shows the tolerance of the two cavities with respect to lateral (transversal) shift of the elements.

[0029] Figures 2A-2G illustrates the enhancement of cavity robustness stemming from the inverse cavity nonlinearity. Figure 2A shows a checkmark cavity configuration. Figure 2B shows a ring cavity configuration. Figure 2C shows the mirror-lens-mirror cavity configuration. Figure 2D shows a Fabry-Perot configuration with lenses attached to adjacent mirrors i.e., lensmirror elements. Figure 2E shows a Fabry -Perot configuration with lenses and reflective coating on their external side. Figure 2F shows a Fabry-Perot configuration with mirrors comprising of a material with a negative thermal expansion coefficient. Figure 2G shows a Fabry-Perot configuration with one mirror comprising of a material with a negative thermal expansion coefficient (NTEC) and the other optical element is a lens-mirror.

[0030] Figure 3 shows a schematic of an improved cavity design operating as a laser phase plate for a transmission electron microscope.

[0031] Figures 4A-4C Illustrates, for comparison purposes, the properties of a conventional Fabry-Perot cavity. Figure 4A shows a schematic of a conventional Fabry-Perot cavity comprising two mirrors, with the mode contour indicated at a low power (blue line) and at a high power (red line). The smaller angle of divergence of the red lines indicates a lower NA of thecavity at a high power compared to its low-power NA. Figure 4B shows the tolerance of a conventional Fabry-Perot cavity to tilt misalignment of one of the cavity mirrors, as a function of the laser power circulating in the cavity. The tolerance is defined as the amount of tilt that reduces the overlap between the perturbed mode (with the tilt) with the unperturbed mode (without the tilt) to 0.9. The cavity is configured to have an NA of 0.1 at each circulating power. Figure 4C shows the initial, or low-power NA of a conventional Fabry-Perot cavity needed to achieve the NA of 0.1, as a function of the circulating laser power at which the NA of 0.1 is to be achieved.

[0032] Figures 5A-5C illustrate the properties of a mirror-lens-mirror cavity. Figure 5A shows a schematic of a mirror-lens-mirror cavity with the mode contour indicated at a low power (blue line) and at a high power (red line). The higher angle of divergence of the red lines indicates a higher NA of the cavity at a high power compared to its low-power NA. Figure 5B shows the tolerance of a mirror-lens-mirror cavity to tilt misalignment of one of the cavity mirrors, as a function of the laser power circulating in the cavity. The tolerance is defined as the amount of tilt that reduces the overlap between the perturbed mode (with the tilt) with the unperturbed mode (without the tilt) to 0.9. The cavity is configured to have an NA of 0.1 at each circulating power. Figure 5C shows the initial, or low-power NA of a mirror-lens-mirror cavity needed to achieve the NA of 0.1, as a function of the circulating laser power at which the NA of 0.1 is to be achieved.

[0033] Figures 6A-6C Illustrate the properties of a Fabry-Perot cavity comprising two mirrors with substrates composed of a material with a negative thermal expansion coefficient. Figure 6A shows a schematic of the NTEC Fabry-Perot cavity with the mode contour indicated at a low power (blue line) and at a high power (red line). The higher angle of divergence of the red lines indicates a higher NA of the cavity at a high power compared to its low-power NA. Figure 6B shows the tolerance of a the NTEC Fabry -Perot cavity to tilt misalignment of one of the cavity mirrors, as a function of the laser power circulating in the cavity. The tolerance is defined as the amount of tilt that reduces the overlap between the perturbed mode (with the tilt) with the unperturbed mode (without the tilt) to 0.9. The cavity is configured to have an NA of 0.1 at each circulating power. Figure 6C shows the initial, or low-power NA of a the NTEC Fabry-Perot cavity needed to achieve the NA of 0.1, as a function of the circulating laser power at which the NA of 0.1 is to be achieved.

[0034] Figure 7 illustrates the enhanced robustness of the new types of cavities.

[0035] Figure 8 further illustrates the enhanced robustness of the new types of cavities.

[0036] Figures 9A and 9B show schematic representations of the system for electron beam imaging or for electron beam spectroscopy, according to some embodiments of the invention.

[0037] Figure 10A shows a schematic of an optical cavity. Figure 10B shows one example of an optomechanical support for the high-NA arm of the optical cavity.

[0038] For simplicity and clarity of illustration, elements shown in the figures are not necessarily drawn to scale, and the dimensions of some elements may be exaggerated relative to other elements. In addition, reference numerals may be repeated among the figures to indicate corresponding or analogous elements.DETAILED DESCRIPTION

[0039] Here, we describe a device, which we refer to as a Stabilized Focusing Cavity, that provides a solution to the problems listed above. One aspect of the invention allows the cavity to have a tight focus (small mode waist) while being less sensitive to various types of imperfections and perturbations. The invention thus makes it possible to build cavities with a smaller mode waist than possible in conventional cavities. Another aspect of the invention makes positive use of the laser beam-induced thermal effects on the cavity mode shapes, so that the cavity robustness is enhanced by the thermal effects rather than diminished. Both aspects if the invention contribute to the capability of creating optical cavities that have a tight internal focus and operate at a high power.

[0040] Additionally, this disclosure provides a system comprising a transmission electron microscope and a Stabilized Focusing Cavity, which enables imaging with improved contrast and signal-to-noise ratio in transmission electron microscopy.

[0041] This disclosure provides also a method of using a Stabilized Focusing Cavity in a transmission electron microscope to achieve imaging with improved contrast and signal-to-noise ratio in electron microscopy.

[0042] An optical cavity can be described as having a tight internal focus or a small mode waist if it has at least one plane (which can be called a focal plane) where the width of one of the cavity modes is small. As used herein, the term “tight focus”, “tight waist” “tight internal focus” and “focus” are understood interchangeably. The cavity mode of interest is usually a fundamental mode of the cavity. The width of the mode in the focal plane, where the mode width is the smallest, is known as the “mode waist size” wo. That waist size can be defined as the root-meansquare width of the laser intensity distribution in the focal plane, or the full width at half maximum, or a width at a specific intensity level relative to the peak intensity, or using another definition. The mode can have two different sizes in the two transverse directions (and can also have two different corresponding focal planes). In general, the cavity mode can be considered having a small mode waist if the mode waist is small in at least one of the transverse directions.

[0043] The smallness of the mode waist can be understood in comparison to the wavelength λ of light that the cavity is designed for: it can be considered small if it does not exceed, e.g., 100 times λ, or 10 times λ, or λ, or 0.5 times λ.

[0044] One way to build a cavity with a small mode waist is by making its length L sufficiently small. However, many applications require a certain minimal distance between the focal point of the cavity and the nearest optical element. Furthermore, a key constraint of designing a cavity for a high optical power is that the mode size at every optical element (also referred to as the laser spot size at that element) must not be too small, to avoid laser damage. The required spot size differs depending on materials, wavelength of light, and the required power, but is usually at least 100 times λ, or at least 1000 times λ, or at least 10000 times λ. The high power requirement or the minimum distance requirement often preclude the use of cavities with a very small length L.

[0045] Therefore, another useful point of comparison for the cavity mode waist size wo is the mode waist wcof a confocal cavity of the same total length L, given by the formulaA cavity can be considered as having a small waist if it has a mode waist size much smaller than the confocal waist: wo«wc. Much smaller can mean at least 10 times smaller, at least 100 times smaller, at least 1000 times smaller, or at least 10000 times smaller.

[0046] A convention approach to achieving a small mode waist satisfying the condition wo«wcis to introduce an intra-cavity telescope so as to produce a cavity segment with a small mode diameter. However, this solution requires a small laser spot size at least on some of the optical elements comprising the cavity, and is therefore not suitable for a cavity where the spot size needs to be large to avoid laser damage.

[0047] Typically, a small mode waist combined with a large spot size on all elements requires a cavity that is close to the edge of stability. This means that a small change in one of the cavity parameters can turn the cavity from stable to unstable. Under such conditions, the cavity is necessarily close to mode degeneracy, which is a condition where different cavity modes have the same frequency. Under near-degenerate conditions, the cavity modes become critically sensitive to perturbations and imperfections such as, e.g., imperfection of the shapes of optical elements comprising the cavity; deviation of the cavity element in position or angle from their ideal configuration; thermally or mechanically induced stresses and deformations; photo-thermal effects, etc. This means that the smaller the waist size (and the closer the cavity is to degeneracy), the more difficult it is to manufacture and operate a cavity meeting the exceedingly tight tolerances imposed by the degeneracy.

[0048] These difficulties have severely constrained the focus size and power achievable in conventional cavity configurations, such as a near-concentric cavity or a bow-tie cavity design. To overcome these constraints, we introduce a Stabilized Focusing Cavity (SFC) design that enables a small mode waist while limiting the sensitivity of the cavity to perturbations and imperfections. The idea of the SFC is to combine a tightly focusing degenerate arm, with a much longer, non-degenerate stabilizing arm. Thus, the SFC is characterized by having a first segment with a small mode waist and a second segment that has a length much larger than the length of the first segment. The resulting cavity combines the best qualities of the constituent arms: it has a tight focus and is non-degenerate.

[0049] The enhanced robustness of the SFC can be explained by considering the Gouy phase accumulated by the fundamental mode of the cavity in one round trip. A tightly focused cavity arm contributes a close to it phase shift on each pass, whereas a cavity arm with a length much smaller than the Rayleigh range ZR provides a phase contribution close to zero. Thus, a conventional cavity comprising a few arms of comparable length, some tightly focused and some not, will have a roundtrip Gouy phase shift close to πN, resulting in near-degeneracy of the cavity modes. In the SFC, the stabilizing arm length is comparable to the Rayleigh range, providing a finite frequency separation between cavity modes even in the limit of high NA of the focused arm.

[0050] The enhanced robustness of this cavity design to perturbation is illustrated in Figure 1. Figure 2 shows several example configurations of the SFC, having a short focusing segment and (at least one) much longer stabilizing arm. Figure 1 illustrates the enhanced robustness of the SFC by showing the allowed perturbation to cavity alignment parameters as a function of the cavity’s numerical aperture (NA). The NA is a parameter reciprocal to the mode waist size w, according to the formula The graphs show that the SFC is more robust againstmisalignments than a conventional Fabry-Perot cavity having the same waist size. Furthermore, the robustness advantage of the SFC is more pronounced at smaller mode waist size.

[0051] Another aspect of the invention addresses modifications of the optical elements comprising the cavity due to the laser beam. One example of such effect is the deformation of the cavity elements by the heat load produced by the high-power laser beam. The resulting thermo-elastic deformation (e.g. bulging) changes the shape of the mirrors, which in turn changes the mode shape and redistributes the heat load. Such thermal effects lead to increased cavity sensitivity to alignment and mirror shape imperfections and reduce the NA, leading to increased focus size. More importantly, though, the resulting dynamics lead to unstable behavior, whichlimits the attainable power in near-concentric resonators. Since focusing cavities get more sensitive to mirror shape perturbations as the focus tightness increases, the thermoelastic effects also limit the minimum mode waist size at a given laser beam power, or, conversely, limits the maximum operating power at a given mode waist size.

[0052] Furthermore, similarly detrimental effects can occur due to other mechanisms whereby the laser beam affects the optical properties of the optical elements making up the cavity. Such mechanisms can include other thermal effects, such as, e.g., thermally induced changes in the refractive index (thermo-refractive effect), or thermally induced stress in the optical element, or non-thermal mechanisms such as electrostriction, Kerr effect, etc.

[0053] Even though such laser beam-induced changes in the properties of the optical elements comprising the cavity are detrimental in many frequently used cavity designs, here we introduce cavity designs based on inverse cavity nonlinearity that turn such phenomena into a crucial advantage. The inverse cavity nonlinearity (also referred to here as negative, or anomalous nonlinearity) occurs when the cavity comprises at least one optical element that increases its focusing power in response to illumination by a laser beam. In the following, we will refer to such an optical element as a nonlinear element. The term “nonlinear” here refers to the fact that the properties of the nonlinear optical element depend on the properties of the incident laser beam (power, intensity, shape, etc.). This breaks the linear relations between the input and output electromagnetic waves that are typical in conventional optical elements, and in conventional (linear) optics. The inverse cavity nonlinearity acts to reduce the sensitivity of the cavity to small perturbations, such as imperfections in the optical elements or small misalignments, making the cavity easier to manufacture and more robust in operation.

[0054] Increasing the focusing power of an optical element means, for a positive focusing element (e.g., a plano-convex lens), that its focal length is decreased in absolute value; and for a negative focusing element (e.g., a plano-concave lens), that its focal length is increased in absolute value. For a non-focusing element such as a planar plate (window), it means that such element acquires a positive focusing property.

[0055] In addition, instead of leading to mode shape instability, the inverse cavity nonlinearity provides the cavity with a dynamic stabilization mechanism. Thus, the inverse cavity nonlinearity can be used to remove a key constraint on the mode waist size or circulating power in optical cavities.

[0056] Furthermore, in some embodiments, the inverse cavity leads to a phenomenon where the mode waist size decreases (or, equivalently, the NA increases) with the power of the laser beam coupled into the cavity. Then, the cavity can initially be aligned at a low laser power, andat a moderate NA. As the laser power is increased, the mode waist size decreases to the desired predefined value without a dramatic increase in sensitivity to small misalignments or optics imperfections. This may obviate the need for active alignment in such a cavity. This highly beneficial property of cavities with inverse nonlinearity can be called a self-alignment property. It can also be described as a mode freeze-in property as the mode configuration is pinned to its initial configuration, with only the NA changing significantly as the laser power is ramped up. This is in contrast with conventional cavities where typically mode configuration varies significantly with power and becomes unstable at high power.

[0057] Yet another advantage of inverse cavity nonlinearity is that in some embodiments, it enables dynamic control of the aspherical properties of the optical elements. The negative nonlinearity perturbs the fundamental mode of the cavity in a way that enables tighter focusing at a given geometric footprint of the cavity mode. This property is particularly helpful for applications where a cavity must operate in a geometrically constrained space, such as inside a transmission electron microscope.

[0058] Figures 7 and 8 illustrate the enhancement of cavity robustness stemming from the inverse cavity nonlinearity. Figure 2 (C, D, E, F and G) shows examples of cavity configurations that can be used to achieve the inverse nonlinearity. Figure 8 shows the tolerance of the cavity to mirror tilt misalignments as a function of laser power, at a given NA achieved at high power. Whereas for a conventional cavity the tolerance greatly decreases with the laser power (making the cavity very sensitive to alignment), a cavity with inverse nonlinearity shows tolerance to misalignment increasing with laser power, thus demonstrating enhanced robustness of high- power operation.

[0059] The inverse cavity nonlinearity can be achieved e.g. by utilizing mirror substrates made of materials with negative coefficient of thermal expansion. In this case it can be referred to as negative thermal nonlinearity or anomalous thermal nonlinearity.

[0060] The mirror substrates are made of materials having a negative coefficient of thermal expansion (CTE). For example, in certain temperature ranges, materials such as fused silica, Corning Ultra-Low Expansion glass (ULE), or other optical glasses, silicon, diamond, or other optical crystals, have negative CTE.

[0061] In some embodiments, the cavity is a multi-segment cavity (such as, e.g., a bow-tie cavity) where the mirror substrates are made of material having a negative CTE. In some embodiments, the cavity is a multi-segment cavity. In one embodiment the cavity is a bow-tie cavity.

[0062] In some embodiments, the cavity comprises a transmissive optical element such as a lens or a flat plate with a positive CTE, so that its focusing power increases upon illumination by a laser beam.

[0063] In some embodiments, the cavity comprises a transmissive element such as a lens or a flat plate, with a positive thermo-refractive coefficient (meaning that the refractive index increases with temperature), so that its focal length decreases with laser power, or, in other words, the focusing power of that optical element increases with the laser power.

[0064] In some embodiments, the cavity includes an optical element that combines the functionality of a lens with the functionality of a mirror. The lens-mirror element consists of a body of material transparent to laser light, coated on one side with a reflective coating and on the other side with an anti -reflective coating. The lens-mirror element is positioned so that its anti- reflective coated side is internal to the cavity, so that the laser light passes through the transparent material and is reflected from the reflective coating. In this embodiment, e.g., thermo-elastic deformation and thermo-refractive effect lead to increased focusing power.

[0065] The SFC design, which provides enhanced cavity robustness, can be combined with the inverse cavity nonlinearity for compound robustness of operation at a high power and with a small mode waist size.

[0066] The strategies of cavity design described above can provide an improved cavity that can be used as a laser phase plate in a transmission electron microscope (TEM) to achieve improved image contrast and signal -to-noise ratio. A conceptual schematic of using one of the example cavity designs in an electron microscope is shown in Figure 3.

[0067] illustrates the enhanced robustness of the new types of cavities. Figure 7 illustrates the enhanced robustness of the new types of cavities. The graphs show the initial, or low -power NA of the cavities needed to achieve the NA of 0.1 in the high-NA segment of the cavity, as a function of the circulating laser power at which the NA of 0.1 is to be achieved. The blue line shows the low-power NA of the mirror-lens-mirror cavity of Figure 2C. The green line shows the low- power NA of the Fabry -Perot cavity of Figure 2F, comprising two mirrors with a negative thermal expansion coefficient. The orange line shows the low-power NA of a conventional symmetric Fabry-Perot cavity. The graphs show that in the conventional cavity (orange line) the low-power NA needed to achieve the NA of 0.1 at a given circulating laser power, increases with the circulating laser power. Since coupling of the laser power into a cavity necessarily starts with a low power which is then increased, such behavior reflects the difficulty of achieving a high NA with such cavities at a high circulating power. Furthermore, the graph shows that at a certain circulating power, the required low-power NA increases dramatically, meaning that a certainpower ceiling cannot be exceeded in such cavities. In contrast, in the cavity comprising mirrors with negative CTE (green line), the low-power NA needed to achieve the NA of 0.1 at a given circulating laser power decreases with circulating laser power. This indicates enhanced robustness of such cavities at a high circulating power. The graph indicates that such a cavity does not have a ceiling of circulating power. In the mirror-lens-mirror cavity (blue line), the low- power NA needed to achieve the NA of 0.1 at a given circulating laser power also decreases with the laser power, demonstrating enhanced robustness of such cavities at a high circulating power and the absence of a ceiling of laser power.

[0068] Figure 8 further illustrates the enhanced robustness of the new types of cavities. The graphs show the tolerance of the cavities to tilt misalignment of the cavity mirror as a function of the laser power circulating in the cavity. The tolerance is defined as the amount of tilt that reduces the overlap between the perturbed mode (with the tilt) with the unperturbed mode (without the tilt) to 0.9. The comparison is made at the same NA of the high-NA segment for all the cavities, at the NA of 0.1. The blue line shows the tolerance of the mirror-lens-mirror cavity of Figure 2C with respect to the tilt of the mirror 204, belonging to the high-NA segment. The green line shows the tolerance of the Fabry -Perot cavity of Figure 2F, comprising two mirrors with a negative thermal expansion coefficient. The tolerance is shown with respect to the tilt of either of the two mirrors forming the cavity. The orange line shows the tolerance of a conventional symmetric Fabry-Perot cavity with respect to the tilt of either of the two mirrors forming the cavity.

[0069] The graphs show that the conventional cavity (orange line) has a tolerance that decreases with the circulating laser power. That reflects the known tendency of such cavities to become increasingly sensitive and less robust at a high circulating power. Furthermore, the graph shows that at a certain circulating power, the tolerance drops precipitously, meaning that a certain power ceiling cannot be exceeded in such cavity. In contrast, the cavity comprising mirrors with negative CTE (green line) has a tolerance that increases with the circulating laser power, indicating enhanced robustness at high power. The graph indicates that such a cavity does not have a ceiling of circulating power. The mirror-lens-mirror cavity (blue line) also shows tolerance increasing with the laser power and the absence of ceiling of laser power. Furthermore, due to its stabilizing arm, the mirror-lens-mirror cavity has about four orders of magnitude higher tolerance at all powers compared to the negative-CTE Fabry-Perot cavity. In other words, the mirror-lens-mirror cavity combines the benefits of geometrical stabilization of the cavity and power-dependent stabilization.Optical Cavities of the Invention

[0070] Embodiments of the presently disclosed subject matter are directed, inter alia, to laserbased control of electron beams / waves, with the goal of improving various aspects of electronbased microscopy and spectroscopy.

[0071] As used herein, in some embodiments the terms “cavity”, “optical cavity”, “resonator”, “optical resonator” may be used interchangeably having all the same meanings and qualities.

[0072] As used herein, in some embodiments, the term "wave" and "beam" may be used interchangeably having the same meanings and qualities.

[0073] As used herein, in some embodiments, the term "electron gun" and "electron source" may be used interchangeably having the same meanings and qualities.

[0074] As used herein, in some embodiments, the term "object" and "sample" may be used interchangeably having the same meanings and qualities.

[0075] As used herein, in some embodiments, the term “mirror”, “reflector” and “reflecting optical element” may be used interchangeably having the same meanings and qualities. As understood herein, reflecting optical elements can be any optical element that serve the function of reflecting light when light is incident on its surface. As such, the reflectivity of such a mirror can range from non-zero reflection to total reflection, in various embodiments.

[0076] As used herein an "optical segment” is also referred to as an “arm” or simply a “segment” and is defined as a part of a cavity which includes two adjacent optical elements. In one embodiment an optical segment comprises at least two optical elements. In one embodiment the optical segment comprises a plurality of optical elements. Typically, and in one embodiment the optical segment consists of two optical elements. The optical segment is configured to comprise a laser beam therein. In various embodiments the invention provides an optical cavity comprising at least one optical segment. In some embodiments the optical cavity comprises two optical segments.

[0077] In one embodiment the optical cavity comprises: a first segment with a numerical aperture (NAi) and a length (L1); and a second segment with a numerical aperture (NA2) and a length (L2); wherein the first segment and the second segment each comprise at least two optical elements; and wherein NAi is greater than NA2.

[0078] In one embodiment the optical cavity comprises: a first segment with a numerical aperture (NAi) and a length (L1); and a second segment with a numerical aperture (NA2) and a length (L2);wherein the first segment and the second segment each comprise two optical elements.

[0079] In one embodiment the optical cavity comprises: a first segment with a numerical aperture (NAi) and a length (L1); and a second segment with a numerical aperture (NA2) and a length (L2); wherein the first segment and the second segment each comprise two optical elements; wherein NAi is greater than NA2; and wherein L2 is greater than L1.

[0080] In one embodiment the optical cavity comprises: a first segment with a numerical aperture (NAi) and a length (L1); and a second segment with a numerical aperture (NA2) and a length (L2); wherein the first segment and the second segment each comprise two optical elements; and wherein NAi is at least 0.05.

[0081] In one embodiment the optical cavity comprises: a first segment with a numerical aperture (NAi) and a length (L1); a second segment with a numerical aperture (NA2) and a length (L2); wherein the first segment and the second segment each comprise two optical elements; wherein NAi is at least 0.05; and wherein L2 is greater than L1by a factor ranging between about 2 and 10,000.

[0082] In one embodiment the first segment and the second share at least one optical element. In one embodiment the first and second segment share no optical elements. As understood herein the NA is not a direct property of just the dimensions and structure of the cavity. A cavity can typically support a series of resonant modes, some of which are called “fundamental” modes, and the NA is an attribute of each of those modes. Each mode has a specific frequency (and thus wavelength). The NA of the modes depends on the wavelength, in addition to the cavity configuration. Usually, a cavity is designed to operate in a particular narrow spectral range, or in a vicinity of a particular wavelength, so the wavelength(s) of the relevant modes are more or less predefined, with some accuracy. With respect to a particular spectral range, the NA of each segment is a function of the cavity configuration. Thus, as referred to herein the “NA of a segment” is the NA of the fundamental mode in this segment, at a certain wavelength or in a certain spectral range, in various embodiments. Furthermore, if the mode shape within a segment is elliptic or astigmatic, the NA of that segment can have two distinct values corresponding to the mode waist size in two directions orthogonal to the optical axis of that segment.

[0083] In one embodiment the optical cavity comprises:a first segment with a numerical aperture (NAi) and a length (L1); a second segment with a numerical aperture (NA2) and a length (L2); wherein the first segment and the second segment each comprise two optical elements; wherein the numerical aperture of the first segment (NAi) is at least 0.05; wherein the length of the second segment (L2) is greater than the length of the first segment (L1) by a factor ranging between about 2 and 10,000; and wherein the mode size on each of the one or more optical elements ranges between 0.5 mm to 1 m.

[0084] In various embodiments, three criteria are required to define the optical cavity of the present invention:1) one segment with a high NA. Although the ratio of NA1 / NA2 partly describes the higher NA of the first segment, the upper limit on NAi is by definition equal to 1;2) In some embodiments the length is best specified as a relative factor between L1and L2;3) the spot size on all optics is relatively large.

[0085] In some embodiments the length of the second segment (L2) is greater than the length of the first segment (L1) by a factor ranging between about 2 and 10,000. In some embodiments the length of the second segment (L2) is greater than the length of the first segment (L1) by a factor ranging between about 2 and 5,000. In some embodiments the length of the second segment (L2) is greater than the length of the first segment (L1) by a factor ranging between about 2 and 1000. In some embodiments the length of the second segment (L2) is greater than the length of the first segment (L1) by a factor ranging between about 2 and 500. In some embodiments the length of the second segment (L2) is greater than the length of the first segment (L1) by a factor ranging between about 2 and 100. In some embodiments the length of the second segment (L2) is greater than the length of the first segment (L1) by a factor ranging between about 2 and 50. In some embodiments the length of the second segment (L2) is greater than the length of the first segment (L1) by a factor ranging between about 2 and 10.

[0086] In some embodiments the length of the second segment (L2) is greater than the length of the first segment (L1) by a factor ranging between about 10 and 10,000. In some embodiments the length of the second segment (L2) is greater than the length of the first segment (L1) by a factor ranging between about 100 and 10,000. In some embodiments the length of the second segment (L2) is greater than the length of the first segment (L1) by a factor ranging between about1,000 and 10,000. In some embodiments the length of the second segment (L2) is greater than the length of the first segment (L1) by a factor ranging between about 5,000 and 10,000.

[0087] In some embodiments the length of the second segment (L2) is greater than the length of the first segment (L1) by a factor ranging between about 1.5 and 100.

[0088] In some embodiments L1is greater than 1mm. In some embodiments L1ranges between 1 mm and 1 m. In some embodiments L1ranges between 1 mm and 50 cm. In some embodiments L1ranges between 1 mm and 10 cm. In some embodiments L1ranges between 1 mm and 5 cm. In some embodiments L1ranges between 1 mm and 1 cm. In some embodiments L1ranges between 1 cm and 50 cm. In some embodiments L1ranges between 1 cm and 10 cm. In some embodiments L1ranges between 1 cm and 5 cm.

[0089] In some embodiments L2 is greater than 2cm. In some embodiments L2 ranges between 1 cm and 10 m. In some embodiments L2 ranges between 1 cm and 5 m. In some embodiments L2 ranges between 1 cm and 1 m. In some embodiments L2 ranges between 50 cm and 10 m. In some embodiments L2 ranges between 50 cm and 1 m.

[0090] Any of the length ranges disclosed for L1and L2 are considered within the scope of the present invention, on the condition that L1< L2. In various embodiments L1« L2.

[0091] As understood herein the “mode size” (understood and used interchangeably with ‘laser spot size’) refers to the size of the laser beam at any point in the optical cavity. Furthermore, the “size” of the laser beam refers to at least one of the dimensions used to describe the spot e.g., the diameter. In some embodiments the “mode size” refers to the full width half maximum (FWHM) of the laser. Typically, the mode size refers to the spot size on any of the optical elements within the optical cavity. In various embodiments, the mode size changes throughout the cavity according to the laser beam propagation across the optical elements of the cavity. The mode size can also refer to the size of the laser spot size on at least one of the optical elements in the optical cavity. Typically, the spot is approximately circular in shape. The spot shape of the laser beam on any of the optical elements can be any shape. In one embodiment the laser beam shape on an optical element is selected from: circular, elliptical, etc. As the laser passes through optical elements the shape of the laser beam may change. As the laser passes through optical elements the shape of the laser spot may change. In some embodiments the mode size on each of the one or more optical elements ranges between 0.5 mm to 1 m. In some embodiments the mode size on each of the one or more optical elements ranges between 0.5 mm to 0.5 m. In some embodiments the mode size on each of the one or more optical elements ranges between 0.5 mm to 10 cm. In some embodiments the mode size on each of the one or more optical elements ranges between 0.5 mm to 1 cm. In some embodiments the mode size on each of the one or more opticalelements ranges between 0.5 mm to 1 mm. In some embodiments the mode size on each of the one or more optical elements ranges between 1 mm to 1 m. In some embodiments the mode size on each of the one or more optical elements ranges between 10 mm to 1 m.

[0092] In some embodiments the laser has a Gaussian beam profile.

[0093] This should not be confused with width of the mode in the focal plane, where the mode width is the smallest, and is known as the “mode waist size”, wo. Generally, the invention provides an optical cavity to achieve a small mode waist size, as will be shown.

[0094] The numerical aperture (NA) is labelled according to each optical segment e.g., NAi refers to the NA of the first segment. As such, and in some embodiments, the ratio NA1 / NA2 is at least 2. In some embodiments the ratio NA1 / NA2 is at least 5. In some embodiments the ratio NA1 / NA2 is at least 10. In some embodiments the ratio NA1 / NA2 is at least 100.

[0095] In some embodiments the ratio NA1 / NA2 ranges between 1.01 and 10,000. In some embodiments the ratio NA1 / NA2 ranges between 1.01 and 1,000. In some embodiments the ratio NA1 / NA2 ranges between 1.01 and 500. In some embodiments the ratio NA1 / NA2 ranges between 1.01 and 100. In some embodiments the ratio NA1 / NA2 ranges between 1.01 and 10.

[0096] In some embodiments the ratio NA1 / NA2 ranges between 2 and 10,000. In some embodiments the ratio NA1 / NA2 ranges between 2 and 5,000. In some embodiments the ratio NA1 / NA2 ranges between 2 and 1,000. In some embodiments the ratio NA1 / NA2 ranges between 2 and 100. In some embodiments the ratio NA1 / NA2 ranges between 2 and 50. In some embodiments the ratio NA1 / NA2 ranges between 2 and 10.

[0097] The optical segments of the optical cavity can have a range of NAs that are configured to achieve the desired outcomes of the present invention. Namely, at least, one segment providing a tight focus and a second arm being a stabilizing arm. In some embodiments, NAi ranges between 0.05 and 0.99. In some embodiments, NAi ranges between 0.05 and 0.5. In some embodiments, NAi ranges between 0.05 and 0.4. In some embodiments, NAi ranges between 0.05 and 0.3. In some embodiments, NAi ranges between 0.1 and 0.99. In some embodiments, NAi ranges between 0.2 and 0.99. In some embodiments, NAi ranges between 0.3 and 0.99. In some embodiments, NAi ranges between 0.4 and 0.99. In some embodiments, NAi ranges between 0.5 and 0.99. In some embodiments, NAi ranges between 0.6 and 0.99. In some embodiments, NAi ranges between 0.7 and 0.99. In some embodiments, NAi ranges between 0.8 and 0.99. In some embodiments, NAi ranges between 0.9 and 0.99.

[0098] In some embodiments NA2 ranges between 0.0001 and 0.04. In some embodiments NA2 ranges between 0.001 and 0.04. In some embodiments NA2 ranges between 0.01 and 0.04. In some embodiments NA2 ranges between 0.1 and 0.04.

[0099] In some embodiments NAi ranges between 0.05 and 0.99 and NA2 ranges between 0.0001 and 0.04. In some embodiments NAi > NA2. In some embodiments NAi » NA2.[000100] The segments of the optical cavities comprise optical elements. Optical elements are any physical structure that interacts with light. More generally understood an “optical element” is any component or device that interacts with light to manipulate its properties, such as direction, intensity, polarization, or phase, etc. Such components can include, but are not limited to: lenses, mirrors, reflective coating, prisms, filters, polarizers, gratings, waveplates, collimators, etc. The segments of embodiments of the present optical cavity comprise at least one optical element. In some embodiments at least one optical element is astigmatic. In other embodiments at least one optical element is spherical or aspherical.[000101] In some embodiments the optical elements are selected from: at least one reflective element, at least one transmissive element, at least one lens-mirror element, or a combination thereof. In some embodiments the optical elements are selected from: a reflective element, a transmissive element, a lens-mirror element, or a combination thereof. In some embodiments, the optical cavity comprises a plurality of optical elements.[000102] In some embodiments the lens-mirror element comprises a lens or a planar optical window having a first optical surface and a second optical surface, wherein an antireflective coating is disposed on the first optical surface and a reflective coating is disposed on the second optical surface. The addition of antireflective coatings on any of the optical elements can be selected according to the particular optical cavity setup and is not limited to one configuration or position.[000103] In some embodiments the reflective element is selected from: a mirror with a concave reflective surface, a mirror with a convex reflective surface, a mirror with a planar reflective surface, a lens-mirror or a combination thereof. In some embodiments the reflective element is at least partially transmissive. In one embodiment the optical cavity further comprises an optomechanical support configured such that the first segment is rigid. In some embodiments the optical elements detailed herein are selected from: partially transmissive or fully transmissive. The corresponding terms for transmissivity and reflectivity are often used interchangeably herein. In some embodiments the reflective coating comprises a reflective coating. The coatings disclosed herein comprise at least one layer, in various embodiments. In some embodiments the reflective coatings are selected from: metallic, dielectric, or a combination thereof. Metals for reflective coatings are selected from, but are not limited to: aluminum, silver, gold, rhodium, chromium, titanium, platinum, etc. In some embodiments the reflective element has a reflectivity of at least 90%. In some embodiments the reflective element has a reflectivity of at least 95%. Insome embodiments the reflective element has a reflectivity of at least 98%. In some embodiments the reflective element has a reflectivity of at least 99%.[000104] In some embodiments the transmissive element is selected from: lens, planar optical window, or a combination thereof. In some embodiments the transmissive element comprises an anti -reflective coating. In some embodiments the transmissive element is at least 99% transmissive. In some embodiments the transmissive element is at least 95% transmissive. In some embodiments the transmissive element is at least 90% transmissive.[000105] Optical elements can include lenses of all varieties, according to the requirements of a particular configuration and optical cavity. In various embodiments a lens shape is selected from: biconcave, biconvex, plano-convex, plano-concave, spherical, aspherical, concave-convex, or a combination thereof. In various embodiments mirrors and / or reflective surfaces can have correspondingly similar shapes to those of lenses. In some embodiments a reflective element is selected from: concave, convex, plano-convex, plano-concave, spherical, aspherical, concave- convex, or a combination thereof. “Elements” are also referred to herein as “optical elements”.[000106] One of the principle aims of the present invention is to achieve an optical cavity with a tight focus. This is, at least, facilitated by the multiple segment nature of the cavity configuration. In one embodiment at least one optical segment has a mode waist size ranging between 0.1 pm and 10 pm. In one embodiment at least one optical segment has a mode waist size ranging between 0.1 pm and 5 pm. In one embodiment at least one optical segment has a mode waist size ranging between 0.1 pm and 1 pm. In one embodiment at least one optical segment has a mode waist size ranging between 0.5 pm and 10 pm. In one embodiment at least one optical segment has a mode waist size ranging between 0.5 pm and 5 pm. In one embodiment at least one optical segment has a mode waist size ranging between 0.1 pm and 10 pm. In some embodiments one segment of the optical cavity has a tight mode waist. In other embodiments, at least one segment has a tight mode waist.[000107] As will become clear, cavity robustness is another key aim of the present invention, which is facilitated by the configuration of segments of the optical cavities disclosed herein. Fabry-Perot cavity configurations commonly exhibit high sensitivity, making them particularly vulnerable to the impact of slight changes on the functionality of the cavity e.g., tilt, mechanical distortions, bulging due to heating / movement of optical elements. The optical cavity configurations of the present invention ensure improved robustness of the optical cavity. In terms of robustness the cavity is able to maintain performance characteristics and functionality under varying conditions such as dynamic motion of optical elements, thermally induced changes tooptical elements, other environmental factors, component configurations, or external disturbances without any significant degradation or deterioration in performance.[000108] In some embodiments the cavity further comprises at least one thermal conductor in contact with at least part of at least one optical segment. In one embodiment a thermal conductor is in contact with at least one optical element. In one embodiment a thermal conductor is in contact with an optical segment. In one embodiment all thermal conductors are in contact with one another. In one embodiment the thermal conductor is in contact with a temperature regulation system, as will be described. Any of the optical cavities described herein can comprise the thermal conductor feature.[000109] As will become clear, the number of optical segments comprised within the optical cavity is non-limiting. Although some embodiments describe only one, or perhaps two optical segments, at least one other optical segment can be added to the optical cavity configuration. For example, additional mirrors and lenses can be used to ‘fold’ an optical segment to enable it to fit in a particular volumetric space such as in an electron microscope; or at least partially. In one configuration a beam may exhibit astigmatism, which can be intentionally introduced by employing elements in an asymmetric configuration, e.g. by reflection at an angle from a concave or convex mirror. One method to achieve astigmatism is by using non-cylindrically symmetric elements, effectively introducing astigmatism through the geometry of the optical elements.[000110] In one embodiment the optical cavities disclosed herein further comprise at least one additional optical segment. This additional optical segment is optically connected with at least one other optical segment, directly or indirectly. The terms “optically connected” and “in optical contact” are to be understood interchangeably where optical segments are linked or share optical interfaces, optical elements, or the likes, either directly or indirectly. In other embodiments, when two optical segments are in communication with one another they share at least one optical element e.g., a mirror. This additional optical segment is comprised of at least one optical element. In one embodiment the additional optical segment consists of one additional optical element. In some embodiments the additional optical segment is comprised of a plurality of optical elements.[000111] In one embodiment the optical cavity further comprises at least one additional optical segment with at least one of the optical elements being the shared in at least one other optical segment in the optical cavity. For example, where a first and second segment are already present, a third additional segment can be added which shares a mirror with either the first, second, or both first and second, segments.[000112] The optical cavities generally disclosed herein can be categorized into at least two types:1) Standing wave cavities - a plurality of mirrors with a laser beam oscillating therebetween; the laser beam reflecting back and forth between the elements generating a standing / resonating wave e.g., ‘checkmark’ cavity, mirror-lens-mirror cavity, Fabry -Perot cavities;2) Ring cavity - where a laser beam circulates in a closed loop, forming a running wave, with any number of optical elements arranged in a ring generating a resonating laser beam between the optical elements e.g., triangle cavity, bow-tie cavity.[000113] Optical cavities are built to comprise a laser beam therein. As understood herein, reference is made to a “laser beam” and to “at least one laser beam”. In various embodiments, these are to be understood interchangeably since embodiments of the application can be extended from the use of a single laser (e.g., a single laser beam) and to one or more laser beam (i.e., at least one laser) within a single optical cavity and / or a plurality of optical cavities. In various embodiments a “laser beam” is a coherent beam of light, and a “laser”, is a machine which produces such beams.[000114] In one embodiment an optical cavity comprises a standing wave cavity configuration. In one embodiment the optical cavity comprises a ring cavity configuration. In one embodiment the cavity configuration is selected from: standing wave or ring cavity. As understood herein, and in some embodiments, a ring cavity and running wave cavity are understood interchangeably. In one embodiment the cavity configuration is selected from: standing wave, ring, or a combination thereof. In one embodiment the optical cavity comprises at least one standing wave cavity. In one embodiment the optical cavity comprises at least one ring cavity. In one embodiment the optical cavity comprises at least one ring cavity, or at least one standing wave cavity, or a combination thereof.[000115] A number of embodiments of both types will now be described in further detail.Mirror-Lens-Mirror Configuration for Optical Cavity of this Invention[000116] A mirror-lens-mirror optical cavity configuration 220 will now be described and is depicted in Figure 2C. The optical cavity 220 shows a first segment 221 and a second segment 222. The first segment comprises a plano-concave mirror 204 and a convex lens 205. In various embodiments the lens 205 has an additional anti -reflective coating 206 (not explicitly shown in the Figure). Furthermore, typically the anti -reflective coating 206 is present on both sides of the lens 205. The resonating beam 202 (shown extended throughout the optical cavity) has thenarrowest waist within the first segment 221. The second segment 222 is comprised of the shared lens 205 of the first segment 221 as well as another plano-concave mirror 203 which is partially transmissive to couple the laser beam 201 to the optical cavity. As depicted in Figure 2C, the mirror 204 of the first segment (with the tight focus) 221 has a smaller radius of curvature in comparison with the radius of curvature of the concave mirror 203 in the second segment 222. [000117] In some embodiments the optical cavity is a mirror-lens-mirror configuration. In one embodiment the optical cavity comprises: a first segment comprising a first mirror and a lens; and a second segment comprising the lens and a second mirror.[000118] In one embodiment the first mirror has a smaller radius of curvature than the second mirror. The labelling of the first segment as being the ‘first’ is merely used here as convention to depict the segment which has the tighter focus. In one embodiment the first mirror, the lens and the second mirror are arranged to share the same optical axis. In various embodiments, the mirrorlens-mirror configuration comprises at least one additional optical element. In some embodiments the first mirror, the second mirror or a combination thereof, are at least partially transmissive. In various embodiments, any of the mirrors depicted herein can be the partially transmissive element which allows laser light to enter the optical cavity. In one embodiment an anti -reflective coating is disposed on at least one side of the lens. In one embodiment an anti- reflective coating is disposed on one side of the lens. In one embodiment an anti -reflective coating is disposed on both sides of the lens. In principle this forms a cylindrically-symmetric configuration for a standing wave cavity. However, the introduction of astigmatic elements could render this configuration not cylindrically symmetric (astigmatic). All such cylindrically symmetric, or near-cylindrically symmetric, and non-cylindrically-symmetric configurations are considered within the scope of this present embodiment.[000119] In some embodiments, the mirror-lens-mirror configuration further comprises at least one additional optical segment, sharing an optical element with at least one other optical segment in the optical cavity. As with other variations of the optical cavity, thermal conductors can be added to at least one of the optical elements comprised therein.“Checkmark” Standing Wave Optical Cavity of the Invention[000120] An optical cavity in a standing wave configuration is now provided, as shown in Figure 2A. The “checkmark” cavity 200 comprises a first segment 221 and a second segment 222. The first segment 221 is comprised of two high radius of curvature mirrors 204. The second segment is comprised of one of the small radius of curvature mirrors 204 of the first segment 221and a third mirror 203 with a larger radius of curvature. The resonating beam 202 (shown extended throughout the optical cavity) has the narrowest waist within the first segment 221. The second segment 222 is comprised of the mirror 204 of the first segment 221 as well as another plano-concave mirror 203 which is partially transmissive to enable a laser beam 201 to enter the optical cavity. In this configuration the mirrors are angled such that the laser beam reflects between the three mirrors as a standing wave cavity. Namely, the laser beam 201 is coupled to the cavity 200, and enters at the third mirror 203, reflects off the lower positioned of the small radius of curvature mirrors 204, then onto the second small radius of curvature mirror 204, back to the first small radius of curvature mirror 204 and then back to the large radius of curvature mirror 203; the standing wave is thus formed therein.[000121] In one embodiment the optical cavity comprises a first segment and a second segment configured in a standing wave configuration. In one embodiment the optical cavity comprises: the first segment comprising a first mirror and a second mirror; the second segment comprising the second mirror and a third mirror; and wherein the first mirror, the second mirror and the third mirror form a standing wave cavity.[000122] In one embodiment the first and second mirror have a smaller radius of curvature than the third mirror. The narrow waist is seen in the first segment 221 of this configuration with the second segment 222 being much longer than the first segment 221.[000123] In various embodiments, the first mirror, the second mirror or the third mirror, or a combination thereof, are at least partially transmissive. In various embodiments, any of the mirrors depicted herein can be the partially transmissive element which couples the laser light to the optical cavity.[000124] As with other configurations, and in some embodiments, the cavity further comprises at least one additional optical segment, optically in contact with at least one other optical segment in the optical cavity. As with other variations of the optical cavity, thermal conductors can be added to at least one of the optical elements comprised therein. In principle any number of additional optical elements can be added to this configuration without compromising the ability to achieve a tight focus in one segment whilst having a longer stabilizing segment in another.Ring Cavities of the Invention[000125] An optical ring cavity is now provided, as shown in Figure 2B. The ring cavity 210 comprises a first segment 221 and a second segment 222. The first segment 221 is comprised of two small radius of curvature mirrors 204. The second segment is comprised of either one of thesmall radius of curvature mirrors 204 of the first segment 221 and a third mirror 203 with a larger radius of curvature. The resonating beam 202 (shown extended throughout the optical cavity) has the narrowest waist within the first segment 221. Importantly, the ring cavity 210 comprises a resonating laser 202 in a closed loop, thus any number of optical elements can be used to achieve a ring cavity configuration. The second segment 222 is comprised of the mirrors 204 of the first segment 221 as well as another plano-concave mirror 203 which is partially transmissive to enable a laser beam 201 to enter the optical cavity. In various embodiments, any of the mirrors depicted herein can be the partially transmissive element which couples the laser light to the optical cavity. In this configuration the mirrors are angled such that the laser beam reflects between the three mirrors as a ring cavity. Namely, the laser beam 201 enters the cavity 200, reflects off the one of the high radius of curvature mirrors 204, then onto the second high radius of curvature mirror 204, back to the first high radius of curvature mirror 204 and then back to the low radius of curvature mirror 203, a closed loop ring cavity is thus formed therein.[000126] In one embodiment the optical cavity comprises a first segment and a second segment configured in a ring cavity configuration. In one embodiment the ring cavity comprises at least three mirrors. In one embodiment the ring cavity comprises at least three segments. In one embodiment the optical cavity comprises: the first segment comprising a first mirror and a second mirror; the second segment comprising the second mirror and a third mirror; and wherein the first mirror, the second mirror and the third mirror form a ring cavity configuration.[000127] In this embodiment, the first mirror or the second mirror have a smaller radius of curvature than the third mirror. In one embodiment the first mirror, the second mirror, the third mirror or a combination thereof, are at least partially transmissive. Although only one mirror 203 shows a laser beam 201 entering the optical cavity 210, any one (or more) of the optical elements in the optical cavity 210 can be partially transmissive and adapted to couple the laser beam to the optical cavity 210.Power Dependence Stabilization - Optical Cavities of the Invention[000128] One aspect of the invention makes positive use of the laser beam -induced thermal or non-thermal effects on the cavity mode shapes, so that the cavity robustness is enhanced by the thermal effects rather than diminished. Two configurations are now presented which achieve this effect. However, the possible configurations that embody the essential characteristics of this invention are limitless.[000129] Figure 2D shows an optical cavity 230 with two adjacent optical elements each comprising a plano-concave mirror 203 and a lens 205; otherwise referred to as a lens-mirror (or mirror-lens). In some embodiments, the lens-mirror element is formed by optical contacting. “Optical contacting” is a method of joining two optical elements, bringing their surfaces into direct physical contact without the use of adhesives. In various embodiments, the lens-mirror interface further comprises an adhesive element e.g., a polymer, glue, resin, gel, etc. The resonating beam 202 (shown extended throughout the optical cavity) has the narrowest waist within the optical segment extending between the two adjacent lens-mirror elements. In various embodiments at least one lens 205 has an additional anti -reflective coating 206 on the side facing the adjacent lens-mirror. Although only the right lens-mirror 203,205 shows a laser beam 201 entering the optical cavity 230; any one of the lens-mirrors of the optical cavity 230 can be partially transmissive and adapted to couple the laser beam to the optical cavity 230. In some embodiments a similar optical cavity can be made of a lens-mirror element and a regular concave mirror, achieving the same effect.[000130] Figure 2E shows a similar optical cavity to Figure 2D. In Figure 2E the optical cavity 240 with two adjacent optical elements each comprising a lens 205 with a reflective coating 207 on the exterior side of the lens; otherwise referred to as a lens-mirror (or mirror-lens). In various embodiments the reflective coating is deposited on to the lens by any thin-film deposition technique. The resonating beam 202 (shown extended throughout the optical cavity 240) has the narrowest waist within the optical segment extending between the two lens-mirror elements. In various embodiments at least one lens 205 has an additional anti -reflective coating 206 on the side facing the adjacent lens-mirror. Although only the right lens-mirror 203,205 shows a laser beam 201 entering the optical cavity 240; any one of the lens-mirrors of the optical cavity 240 can be partially transmissive and adapted to allow the laser light in.[000131] As will be understood by an expert in optics, one of the lens-mirror elements of Figure 2D can be used with an adjacent lens-mirror element of Figure 2E forming an optical cavity achieving a similar effect.[000132] In one embodiment the optical cavity comprises: an optical segment comprising at least two optical elements; wherein the optical segment has a numerical aperture (NAi) of at least 0.05; and wherein the at least two optical elements are configured to increase the focusing power of the at least two optical elements upon illumination by a laser beam.[000133] In one embodiment the optical cavity comprises:an optical segment comprising at least two optical elements; wherein the optical segment has a numerical aperture (NAi) of at least 0.05; and wherein the at least two optical elements comprise at least one nonlinear optical element; wherein the focusing power of the at least one nonlinear optical element is configured to increase upon coupling of a laser beam to the optical cavity.[000134] As used herein “nonlinear optical element” (also referred to as interchangeably as “nonlinear element”) is an optical element that exhibits a nonlinear response to the intensity of light incident thereon. It changes its own focusing power and not of other elements.[000135] As used herein “coupling”, as referred to the laser beam in an optical cavity, refers to the process or method of introducing a laser beam into the optical cavity such that the laser beam resonates and / or oscillates and / or circulates within the cavity.[000136] In a nonlinear optical element, the focusing power increase with incident laser power can be quantified by a focusing power responsivity, defined as the ratio of the focusing power increase to the laser power applied. The focusing power responsivity of a nonlinear element is quantified in the units of inverse length times inverse power. The focusing power responsivity typically ranges between 0.001 m-1kW-1and 100 m-1kW-1.[000137] In some embodiments the optical cavity comprises an optical segment with an NA greater than 0.05, wherein the optical segment or an additional optical segment comprises at least one nonlinear optical element with a focusing power responsivity ranging between 0.01 m-1kW’1and 100 m-1kW-1. In one embodiment the nonlinear optical element is in the high-NA segment. In one embodiment the nonlinear optical element is not in the high-NA segment.[000138] In one embodiment the power responsivity of the nonlinear optical element ranges between 0.001 m-1kW-1and 100 m-1kW-1. In one embodiment the power responsivity of the nonlinear optical element ranges between 0.1 m-1kW-1and 100 m-1kW-1. In one embodiment the power responsivity of the nonlinear optical element ranges between 1 m-1kW-1and 100 m-1kW’1. In one embodiment the power responsivity of the nonlinear optical element ranges between 10 m-1kW-1and 100 m-1kW-1. In one embodiment the power responsivity of the nonlinear optical element ranges between 0.01 m-1kW-1and 10 m-1kW-1. In one embodiment the power responsivity of the nonlinear optical element ranges between 0.01 m-1kW-1and 1 m-1kW-1. In one embodiment the power responsivity of the nonlinear optical element ranges between 0.01 m’1kW-1and 0.1 m-1kW-1.[000139] A cavity comprising a nonlinear element in general has mode shapes depending on the power of the laser beam coupled into the cavity. In particular, in a cavity having a high-NA segment, the NA of that segment will generally depend on the coupled laser power. It is possible to define for that segment a coefficient of proportionality between the change of NA in the high- NA segment of the cavity and the laser power, termed “NA responsivity” of the cavity. The NA responsivity of the cavity has the units of inverse power. It is typically possible to position the nonlinear optical element within the cavity to achieve a positive NA responsivity of the cavity. [000140] Alternatively, the optical cavity can be defined as having a high -power regime and a low power-regime, in various embodiments. In this sense, the nonlinear optical element and the optical cavity responds differently to a coupled laser depending on whether the optical cavity is in the high- or low-power regime. In various embodiments the nonlinear optical element has a threshold power below which the nonlinear optical element (and the NA of the resulting segment) exhibits low-power regime with its corresponding properties. Conversely, above a threshold power, the nonlinear optical element exhibits a high-power regime with its corresponding properties.[000141] The nonlinear optical element can operate in two regimes: a low-power regime, where its nonlinear properties do not significantly manifest and it operates as a regular (linear) optical element; and a high-power regime, where the focusing power of the nonlinear optical element is modified by a laser beam incident on it. In some embodiments the low-power regime can be defined as occurring at the incident laser power of 1 mW or less. In some embodiments the low- power regime is defined as occurring at the incident laser power of 10 mW or less. In some embodiments the low-power regime is defined as occurring at the incident laser power of 100 mW or less. In some embodiments the low-power regime is defined as occurring at the incident laser power of 1 W or less. In some embodiments the high-power regime is defined as occurring at the incident laser power of 10 W or higher. In some embodiments the high -power regime is defined as occurring at the incident laser power of 100 W or higher. In some embodiments the high-power regime is defined as occurring at the incident laser power of 1 kW or higher. In some embodiments the high-power regime is defined as occurring at the incident laser power of 10 kW or higher.[000142] Correspondingly, a cavity comprising a nonlinear optical regime can operate in two regimes: a low-power regime where the mode properties do not depend on laser power, and a high power regime, where mode properties (including NAs in the segments of the cavity) depend on laser power. In some embodiments the low-power regime is defined as occurring at the circulating laser power of 1 mW or less. In some embodiments the low-power regime is definedas occurring at the circulating laser power of 10 mW or less. In some embodiments the low- power regime is defined as occurring at the circulating laser power of 100 mW or less. In some embodiments the low-power regime is defined as occurring at the circulating laser power of 1W or less. In some embodiments the high-power regime is defined as occurring at the circulating laser power of 10 W or higher. In some embodiments the high -power regime is defined as occurring at the circulating laser power of 100 W higher. In some embodiments the high -power regime is defined as occurring at the circulating laser power of 1 kW or higher.[000143] It is typically possible to position the nonlinear optical element within a cavity comprising a nonlinear element and having a high-NA segment, to achieve a higher NA in that segment in the high-power regime compared to the low-power regime. All configurations that achieve this result are considered within the scope of this invention.[000144] In some embodiments of the optical cavity the NA responsivity of the cavity ranges between 0.0001 / kW to 1 / kW. In some embodiments of the optical cavity the NA responsivity of the cavity ranges between 0.001 / kW to 1 / kW. In some embodiments of the optical cavity the NA responsivity of the cavity ranges between 0.01 / kW to 1 / kW. In some embodiments of the optical cavity the NA responsivity of the cavity ranges between 0.1 / kW to 1 / kW. In some embodiments of the optical cavity the NA responsivity of the cavity ranges between 0.0001 / kW to 0.1 / kW. In some embodiments of the optical cavity the NA responsivity of the cavity ranges between 0.0001 / kW to 0.01 / kW. In some embodiments of the optical cavity the NA responsivity of the cavity ranges between 0.0001 / kW to 0.001 / kW.[000145] In some embodiments the optical cavity further comprises a laser beam coupled to the optical cavity which has an intracavity circulating power ranging between 1 kW to 10 MW. In some embodiments the optical cavity further comprises a laser beam coupled to the optical cavity which has an intracavity circulating power ranging between 1 kW to 1 MW. In some embodiments the optical cavity further comprises a laser beam coupled to the optical cavity which has an intracavity circulating power ranging between 1 kW to 100 kW. In some embodiments the optical cavity further comprises a laser beam coupled to the optical cavity which has an intracavity circulating power ranging between 1 kW to 10 kW.[000146] In some embodiments the NA increases an amount ranging between about 0.001 to 0.9 upon coupling of a laser beam to the optical cavity. In some embodiments the NA increases by an amount ranging between about 0.001 to 0.1 upon coupling of a laser beam to the optical cavity. In some embodiments the NA increases an amount ranging between about 0.001 to 0.01 upon coupling of a laser beam to the optical cavity. In some embodiments the NA increases by an amount ranging between about 0.001 to 10 upon coupling of a laser beam to the optical cavity.In some embodiments the NA increases by at least 0.01 upon coupling of a laser beam to the optical cavity. In some embodiments the NA increases by at least 0.1 upon coupling of a laser beam to the optical cavity.[000147] In some embodiments the nonlinear optical element comprises a material which has: a negative thermal expansion coefficient, a positive thermal coefficient of the refractive index, a positive Kerr coefficient, or a combination thereof.[000148] As used herein the “focusing power” refers to the ability of an optical element to focus light. As such, the increase in focusing power refers to the increase of focusing power of at least one of the elements in the optical cavity. Thus ‘their focusing power’ can refer to any one of the optical elements e.g., at least one of the nonlinear optical elements. The increase in focusing power of individual optical elements can result in an increase of NA for the optical cavity as a whole. In one embodiment the increase in focusing power ranges between about 0.01 m-1and 100 m-1. In one embodiment the increase in focusing power ranges between about 0.01 m-1and 10 m-1. In one embodiment the increase in focusing power ranges between about 0.01 m-1and 1 m-1. In one embodiment the increase in focusing power ranges between about 0.1 m-1and 10 m’ f In one embodiment the increase in focusing power ranges between about 1 m-1and 100 m-1.[000149] As a result of the addition of at least one nonlinear optical element, the NA of the optical cavity increases when a laser beam is coupled into it. Thus, in various embodiments the optical segment has a numerical aperture (NAi) of at least 0.05 i.e., prior to coupling of a laser beam into the cavity. In some embodiments NA increases by an amount ranging between about 0.001 to 0.9. In some embodiments NA increases by an amount ranging between about 0.001 to 0.5. In some embodiments NA increases by an amount ranging between about 0.001 to 0.3. In some embodiments NA increases by an amount ranging between about 0.001 to 0.1. In some embodiments NA increases by an amount ranging between about 0.001 to 0.01.[000150] In one embodiment the laser beam has an intracavity circulating power in the range of IkW to 10 MW. In one embodiment the laser beam has an intracavity circulating power in the range of IkW to 1 MW. In one embodiment the laser beam has an intracavity circulating power in the range of IkW to 100 kW.[000151] In one embodiment any of the optical elements disclosed herein comprise a material which has: a negative thermal expansion coefficient, a positive thermal coefficient of refractive index, a positive Kerr coefficient, or a combination thereof. In one embodiment any of the optical elements disclosed herein comprise a material which has at least one of: a negative thermal expansion coefficient, a positive thermal coefficient of refractive index and a positive Kerr coefficient. In one embodiment any of the optical elements disclosed herein comprises a materialwhich has a negative thermal expansion coefficient. In one embodiment any of the optical elements disclosed herein comprises a material which has a positive thermal coefficient of refractive index. In one embodiment any of the optical elements disclosed herein comprises a material which has a positive Kerr coefficient. It is understood that a combination of any of these properties can be utilized by any single optical element e.g., those comprising more than one element. For example, a mirror-lens element can comprise two separate materials, each with its own material coefficient. Furthermore, a plurality of optical elements can be selected, together with their optical material properties to achieve any one of the stated aims of the optical cavities disclosed as embodiments herein. For example, in an optical segment comprising two optical elements, each optical element can have a different material property, or the same material property but of different values, or the same material property of the same value; all according to the requirements of a particular optical cavity setup.[000152] In some embodiments the negative thermal expansion coefficient is for reflective elements. In some embodiments, a positive thermal expansion coefficient achieves the correct sign of nonlinearity for transmissive elements.[000153] In one embodiment the at least one of the at least two optical elements comprise a material which has: a negative or positive thermal expansion coefficient, a negative or positive thermal coefficient of the refractive index, a positive or negative Kerr coefficient, optical electrostriction effect or a combination thereof.[000154] In one embodiment the increase of focusing power ranges between 0.001 m-1and 100 m-1per 1 kW of incident laser power. In one embodiment the increase of focusing power ranges between 0.01 m-1and 50 m-1per 1 kW of incident laser power. In one embodiment the increase of focusing power ranges between 0.01 m-1and 10 m-1per 1 kW of incident laser power.[000155] In one embodiment the at least two optical elements are selected from: a reflective element, a transmissive element, a lens-mirror element, or a combination thereof. In one embodiment the at least two optical elements are selected from: reflective element, transmissive element, lens-mirror element, or a combination thereof.[000156] In one embodiment the reflective element is selected from: a mirror with a concave reflective surface, a mirror with a convex reflective surface, a mirror with a planar reflective surface, or a combination thereof. In one embodiment the reflective element is at least partially transmissive. In some embodiments the reflective element is at least 99% reflective. In some embodiments the reflective element is at least 95% reflective. In some embodiments the reflective element is at least 90% reflective.[000157] In some embodiments the cavity further comprises at least one thermal conductor in contact with at least part of at least one optical element. In one embodiment all thermal conductors are in contact with one another to remove heat from the optical cavity element. In one embodiment the thermal conductors comprise elements to actively cool the optical elements. In one embodiment the thermal conductor is in contact with a temperature regulation system. In various embodiments the optical cavity further comprises a heating system coupled to a temperature regulation system, as will be described.[000158] In one embodiment any of the lenses of the lens-mirror element further comprise an anti -reflective coating.[000159] In one embodiment the cavity further comprises at least one additional optical segment. This additional optical segment is optically connected with at least one other optical segment, directly or indirectly. In one embodiment the additional optical segment shares at least one optical element with at least one other optical segment. The terms “optically connected” and “in optical contact” are to be understood interchangeably where optical segments are linked or share optical interfaces, optical elements, or the likes, either directly or indirectly. This additional optical segment is comprised of at least one optical element. In some embodiments the additional optical segment comprises two optical elements. In some embodiments the additional optical segment is comprised of a plurality of optical elements.[000160] In one embodiment the lens and mirror of a lens-mirror element are joined by an adhesive, an adhesive-free bonding method, optical contacting, or a combination thereof. In one embodiment the lens and mirror of the lens-mirror element are joined by an adhesive, an adhesive-free bonding method, or optical contacting.Optical Cavity Comprising Components of Negative Thermal Expansion Coefficient Material [000161] Another variation of the invention makes positive use of laser beam -induced thermal effects on the cavity mode shapes. This enhances the cavity robustness is induced by thermal effects. Typically, such effects serve to diminish the performance of optical cavities. The present examples disclose how optical cavities can utilize this effect to increase robustness.[000162] Figure 2F shows an optical cavity 250 with two adjacent optical elements 208 each comprising a negative thermal expansion coefficient (NTEC) material. In one embodiment the two adjacent elements 208 consist of NTEC material. In one embodiment the two adjacent elements 208 comprise a NTEC material. In various embodiments the two adjacent optical elements 208 are the same, but in other embodiments they are different, as will be shown. In some embodiments the two adjacent optical elements 208 comprise the same NTEC material. Insome embodiments the two adjacent optical elements 208 comprise a different NTEC material. The resonating beam 202 (shown extended throughout the optical cavity) has the narrowest waist within the optical segment extending between the two adjacent optical elements 208. In various embodiments the adjacent optical elements 208 are reflective elements e.g., with a concave reflective surface. In various embodiments at least one of the adjacent optical elements 208 has an additional anti -reflective coating (not shown). Although only the optical element on the right of Figure 2F shows a laser beam 201 entering the optical cavity 250; any one (or both) of the adjacent optical elements 208 of the optical cavity 250 can be partially transmissive and adapted to couple the laser beam to the optical cavity 250.[000163] As stated previously, an optical segment comprises two adjacent optical elements. Essentially, any of the optical elements described herein can be coupled together to form an optical segment, a further example of which is shown in Figure 2G. In Figure 2G the left optical element 208 comprises a NTEC material and the right optical element 203,205 comprises a lensmirror with an optional reflective coating 206 thereon. The resonating beam 202 (shown extended throughout the optical cavity) has the narrowest waist within the optical segment extending between the two adjacent optical elements. Although only the right optical element 203,205 of Figure 2G shows a laser beam 201 entering the optical cavity 250; any one (or both) of the adjacent optical elements of the optical cavity 250 can be partially transmissive and adapted to couple the laser beam to the optical cavity 250. In various embodiments an optical cavity comprises an NTEC mirror and a concave mirror.[000164] In one embodiment the optical cavity comprises: an optical segment comprising a first reflecting element and a second reflecting element, wherein the first reflecting element, the second reflecting element, or a combination thereof, comprise a negative thermal expansion coefficient (NTEC) material.[000165] In one embodiment either the first reflecting element or the second reflecting element comprise a NTEC material. In one embodiment either the first optical element or the second optical element consist of a NTEC material. In various embodiments the NTEC material is selected from: optical glass, optical crystal, fused silica, Ultra-Low Expansion (ULE) glass, diamond, silicon, or a combination thereof. As used herein the terms “reflecting element” and “optical element” are understood interchangeably where appropriate.[000166] In some embodiments the first reflecting element and the second reflective element are selected from: a mirror with a concave reflective surface, a mirror with a convex reflective surface, a mirror with a planar reflective surface, or a combination thereof. In one embodimentthe first reflecting element and the second reflective element are selected from: a mirror with a concave reflective surface, a mirror with a planar reflective surface, or a combination thereof. In one embodiment the first reflecting element, the second reflecting element, or a combination thereof, are at least partially transmissive.The Optical Cavity System of the Invention[000167] In various embodiments, the optical cavities of the invention are comprised within systems. Any of the optical cavities described herein can be comprised within the optical cavity systems as now described. Typically, an optical cavity is coupled to at least one laser.[000168] In one embodiment the optical cavity system comprises: at least one of the optical cavities as described herein; at least one laser; and a plurality of coupling optics and electronic components configured to couple and frequency-lock at least one laser beam from the at least one laser to the optical cavity.[000169] In one embodiment the optical cavity system further comprises a temperature control system configured to maintain a preset temperature of at least one of the elements in the system. [000170] In one embodiment the optical cavity system further comprises a vacuum system configured to establish a vacuum in the at least one optical cavity.[000171] In various embodiments, the system can further comprise any components to enable the frequency-locking element. For example, the system can further comprise any of the following: coupling optics, beam splitters, mirrors, lenses, frequency stabilization electronics, feedback systems, piezoelectric actuators, frequency shifting elements, control electronics, phase-lock loop (PLL) system, controller, lock-in amplifier, PID controller, sensors, photodetectors, frequency references, actuators and drivers, data logging devices, power supply, current source, voltage source, spectrum analyzer, etc.[000172] In various embodiments, the optical cavities of the invention are comprised within systems. In one embodiment the optical cavity system comprises: at least one of the optical cavities as described herein; at least one laser; a plurality of coupling optics and electronic components configured to couple and frequency -lock at least one laser beam from the at least one laser to the optical cavity; a temperature control system configured to maintain a preset temperature of at least one of the elements in the system; and a vacuum system configured to establish a vacuum in the at least one optical cavity.[000173] In one embodiment the optical cavity system comprises: an optical cavity as described herein; at least one laser; a plurality of coupling optics and electronic components configured to couple and frequency -lock at least one laser beam from the at least one laser to the optical cavity; a temperature control system configured to maintain a preset temperature of at least one of the elements in the system; a vacuum system configured to establish a vacuum in the at least one optical cavity. [000174] In one embodiment the temperature control system is configured to heat and / or cool. In various embodiments the optical cavity system can further comprise a cavity housing wherein the at least one optical cavity is positioned in the cavity housing. In one embodiment the optical cavity system further comprises a controller.[000175] In one embodiment the laser beam coupled into the at least one optical cavity is resonantly enhanced by the optical cavity by a factor of at least 3. In one embodiment the laser beam coupled into the at least one optical cavity is resonantly enhanced by the optical cavity by a factor of at least 10. In one embodiment the laser beam coupled into the at least one optical cavity is resonantly enhanced by the optical cavity by a factor of at least 100. In one embodiment the laser beam coupled into the at least one optical cavity is resonantly enhanced by the optical cavity by a factor of at least 10,000. In one embodiment the laser beam coupled into the at least one optical cavity is resonantly enhanced by the optical cavity by a factor of at least 100,000. In one embodiment the laser beam coupled into the at least one optical cavity is resonantly enhanced by the optical cavity by a factor of at least 1,000,000.[000176] In one embodiment the laser beam coupled into the at least one optical cavity is resonantly enhanced by the optical cavity by a factor ranging between 3 and 1,000,000. In one embodiment the laser beam coupled into the at least one optical cavity is resonantly enhanced by the optical cavity by a factor ranging between 3 and 100,000. In one embodiment the laser beam coupled into the at least one optical cavity is resonantly enhanced by the optical cavity by a factor ranging between 3 and 10,000. In one embodiment the laser beam coupled into the at least one optical cavity is resonantly enhanced by the optical cavity by a factor ranging between 3 and 1,000. In one embodiment the laser beam coupled into the at least one optical cavity is resonantly enhanced by the optical cavity by a factor ranging between 3 and 100. In one embodiment the laser beam coupled into the at least one optical cavity is resonantly enhanced by the optical cavity by a factor ranging between 100 and 1,000,000. In one embodiment the laser beam coupled into the at least one optical cavity is resonantly enhanced by the optical cavity by a factor rangingbetween 1,000 and 1,000,000. In one embodiment the laser beam coupled into the at least one optical cavity is resonantly enhanced by the optical cavity by a factor ranging between 1,000 and 1,000,000. In one embodiment the laser beam coupled into the at least one optical cavity is resonantly enhanced by the optical cavity by a factor ranging between 10,000 and 1,000,000. In one embodiment the laser beam coupled into the at least one optical cavity is resonantly enhanced by the optical cavity by a factor ranging between 100,000 and 1,000,000.[000177] In various embodiments, when there is more than one optical cavity, each optical cavity is housed within a separate housing. In other embodiments all optical cavities are comprised within the same housing.[000178] In some embodiments at least one optical element is mounted on adjustable suspensions. In various embodiments the adjustable suspension is operable for adjustment of the angle and position of the optical elements. The adjustable suspension includes piezoelectric actuators, in various embodiments. The adjustable suspension includes thermomechanical actuators, in various embodiments. A corresponding electrical system configured to carry out the function of the adjustable suspension is included.[000179] In various embodiments the laser comprises any of the following selected from: gain medium, laser source, pump source, coupling optics, output coupler, control electronics, controller, cooling system, vacuum system, etc. The laser can comprise any type of laser selected from: solid-state, gas, chemical, semiconductor, dye, metal -vapor, a fiber laser, a diode laser, a diode pumped solid state laser, a diode pumped semiconductor laser, a MOPA laser system, etc. In some embodiments more than one laser source is used in the optical cavity system. In one embodiment the laser comprises a semiconductor diode laser. In one embodiment the laser comprises at least one semiconductor diode laser. As understood herein, the terms “laser-based device” and “laser” are used interchangeably.[000180] Examples of gas lasers include, but are not limited to: helium -neon, argon, krypton, xenon ion, nitrogen, carbon dioxide, excimer, etc.[000181] Examples of chemical lasers include, but are not limited to: hydrogen fluoride, deuterium fluoride, COIL, Agil, etc.[000182] Examples of dye lasers include, but are not limited to: rhodamine, fluorescein, coumarin, stilbene, umbelliferone, tetracene, malachite green, etc.[000183] Examples of metal-vapor lasers include, but are not limited to: helium-cadmium, helium-mercury, helium-silver, strontium, neon-copper, copper, gold, manganese (Me / MnCL), etc. Examples of solid-state lasers include, but are not limited to: ruby, Nd:YAG, Nd:YAP, Nd:Cr:YAG, Er:YAG, Nd:YLF, Nd:YVO4, Nd:YCOB, Nd:YCOB, Ti:sapphire, Tm:YAG,Yb:YAG, Ytterbium:2O3Ytterbium-doped glass, Ho:YAG, Cr:ZnSe, Ce:LiSAF, Ce:LiCAF,147Pm+3: Glass, alexandrite, Erbium-doped and erbium-ytterbium codoped glass, U:CaF2, Sm:CaF2, F-center, etc.[000184] Examples of semiconductor lasers include, but are not limited to: GaN, InGaN, AlGalnP, AlGaAs, InGaAsP, vertical -cavity surface-emitting laser (VCSEL), quantum cascade laser, quantum dot, quantum well, hybrid silicon laser, etc.[000185] In some embodiments the temperature control system is a Peltier system. In various embodiments the temperature control system comprises at least one of any of the following selected from: a temperature sensor, temperature controller, heating elements, insulation, temperature regulation software, vacuum chamber, temperature bath, heat transfer fluid, temperature control valves, thermal insulation covers, power supply, or any combination thereof. In various embodiments the temperature cooling system is coupled to any one of the thermal conductors that are in contact with any one of the optical elements of the optical cavities disclosed herein. The temperature control system is configured to control the temperature of at least one component in the optical cavity system.[000186] In various embodiments the vacuum system comprises at least one of any of the following selected from: vacuum chamber, pump, pressure gauges, valves, flanges, feedthroughs, filters, traps, heating system, pressure control system, cryopump, cryotrap, measurement system, vacuum interlocks, controller, gas inlet system, or any combination thereof. In some embodiments the vacuum system comprises an ultra-high vacuum (UHV) system. In some embodiments the pressure established by the vacuum system ranges between 10'5torr to 10-12torr. In some embodiments the pressure established by the vacuum system ranges between 10'7to 10-12torr. In some embodiments the pressure established by the vacuum system ranges between 10-10to 10-12torr. In some embodiments the pressure established by the vacuum system ranges between 10'5to 10'9torr. These vacuum conditions are established in any parts of the optical cavity system e.g., the optical cavity itself.[000187] In various embodiments the optical cavity system further comprises at least one fiberoptic member configured to couple the laser to the optical cavity. The fiber-optic member is selected from: single mode, hollow-core fiber, multi-mode, plastic optic fiber, step-index, graded index, etc.[000188] In some embodiments the optical cavity system further comprises a cryogenic system. In such systems cryogenic fluids or gases are used for active cooling. Examples of cryogenic fluids or gases used for the present invention include, but are not limited to: liquid nitrogen and liquid helium.Electron Microscopy Imaging Systems of the Invention[000189] A conceptual schematic of using one of the example optical cavity designs in an electron microscope is shown in Figure 3. The electron microscope system of Figure 3 shows an electron gun, electron optics, the sample, cavity, cavity spacer (i.e., cavity housing), optical elements (e.g., mirror, lens, etc.), thermal conductors and a camera. The optical cavity is positioned at least partly inside the TEM. A laser system is schematically shown in Figure 3. Not all components are shown, for clarity. As used herein the term “laser system” and “laserbased device” can be used and understood interchangeably. The laser system comprises: a laser source, amplifier, isolator, polarizing beam splitter (PBS), filter, guiding mirrors, optics, mode matching lens, photodetector, phase shifter, oscillator, phase modulator, mixer, low pass filer, high pass filter, PID controller, feedback controller, PDH system, etc. Any number of these elements can be combined in the laser system in combination.[000190] In some embodiments the invention provides an electron beam imaging system comprising: a transmission electron microscope (TEM), comprising an electron source configured to provide an electron beam; and the optical cavity system of the invention, configured such that the electron beam passes through the at least one optical cavity.[000191] In one embodiment the laser beam coupled into the optical cavity is configured to shift the phase of the part of the electron beam passing through the laser beam.[000192] In one embodiment the at least one optical cavity is positioned at about a back focal plane of the TEM; or at about a plane that is conjugate to the back focal plane. In one embodiment the laser is a continuous-wave laser. In one embodiment the laser is a pulsed laser. In one embodiment the focal spot of the optical cavity is positioned at about the back focal plane of the TEM, or at about the conjugate plane, and wherein the cavity is positioned to allow an unscattered electron wave of the electron beam to pass through the focal spot.[000193] In one embodiment the material structure of the optical cavity system is configured and positioned such that the electron beam of the TEM is at least 0.1 mm from any material element of the optical cavity system. In one embodiment the material structure of the optical cavity system is configured and positioned such that the electron beam of the TEM is at least 1 mm from any material element of the optical cavity system. In one embodiment the material structure of the optical cavity system is configured and positioned such that the electron beam ofthe TEM is at least 10 mm from any material element of the optical cavity system. In one embodiment the material structure of the optical cavity system is configured and positioned such that the electron beam of the TEM is at least 0.1 mm or at least 1 mm or at least 10 mm away from any material element of the optical cavity system.[000194] In one embodiment the TEM further comprises an electron detector positioned at an image plane of the TEM and configured to receive the electron beam. In one embodiment the TEM further comprises at least one electron detector positioned at an image plane of the TEM and configured to receive the electron beam.[000195] In various embodiments the optical cavity is positioned at about the back focal plane of the TEM, or at about a plane that is conjugate to the back focal plane of the TEM; is configured to allow the electron beam, provided by the TEM, to pass through it. In one embodiment the one laser beam circulating inside the cavity has a power ranging between 10 kW to 10 MW. In one embodiment the laser beam provided by the laser is non-monochromatic. In one embodiment the laser beam provided by the laser is monochromatic. In some embodiments the non- monochromatic laser beam comprises two or more laser beams having different wavelengths.[000196] According to some embodiments, the TEM comprises a plurality of lenses [103A- 103D], configured to condition and direct the electron beam

[0104] to illuminate an object or a sample being imaged; the object / sample can be supported and / or positioned via a sample holder

[0106] ,[000197] According to some embodiments, the plurality of lenses are magnetic lenses.[000198] Figure 9 demonstrates that, due to the interaction of the illumination electron beam

[0104] with the sample

[0105] , a portion of the illumination electron beam is scattered, forming a plurality of scattered waves

[0108] , while the other portion of the illumination beam, which is not scattered by the interaction with the object, forms an unscattered wave

[0107] ,[000199] Figures 9A and 9B further demonstrate that, according to some embodiments, the unscattered wave

[0107] is focused by magnetic lenses [103B,103C] in the center [HOC] of the back focal plane

[0110] , It is also demonstrated that the scattered waves

[0108] are focused elsewhere [110S] in the back focal plane

[0110] ,[000200] According to some embodiments, the laser-based device

[0115] is positioned: at about the back focal plane

[0110] of the TEM, meaning at a range of ±lmm or ±0.1mm or ±0.01mm or ±0.001mm, of the back focal plane; or at about a plane that is conjugate to the back focal plane, meaning at a range of 1mm or ±0.1mm or ±0.01 / 7777? or ±0.001 / 77 / 77, of the conjugate plane.[000201] According to some embodiments, the laser beam / s is / are further configured to provide a predetermined phase shift to a portion of the electron beam. According to some embodiments, the phase shift is applied / provided to the whole unscattered wave

[0107] ,[000202] According to some embodiments and as demonstrated in Figure 9, the laser-based device

[0115] is configured to create a laser focal spot [HOC], and to locate the laser focal spot at: about the back focal plane

[0110] of the TEM, meaning at a range of ±lmm or ±0.1mm or ±0.01 mm or ±0.001mm, of the back focal plane, or about the conjugate plane, meaning at a range of ± 1mm or ±0.1mm or 0.01mm or 0.001mm, of the conjugate plane, and wherein the laser-based device is positioned to allow the unscattered electron wave

[0107] to pass through the focal spot [110C],[000203] According to some embodiments, the laser beam / s is / are selected to have wavelengths between 350 nm and 2000 nm, or between 2000 nm and 15 pm.[000204] According to some embodiments, the material structure of the laser-based device is configured and positioned such that the electron beam of the TEM is at least 0.1mm or at least 1mm or at least 10mm away from any material element of the laser-based device, to avoid electron beam distortion, scattering, and / or dephasing by the interaction with the physical structure of the laser device.[000205] In some embodiments the system further comprises an electron camera or one or more sensors, positioned at the image plane and operable to analyze a Ronchigram and provide feedback for automatic control of the electron beam position relative to the optical cavity.[000206] According to some embodiments each optical cavity: is configured to admit / couple in a laser beam

[0109] ; is positioned at about the back focal plane

[0110] of the TEM, meaning at a range of ±1 mm or ±0.1mm or 0.01mm or 0.001mm, of the back focal plane, or at about a plane that is conjugate to the back focal plane of the TEM, meaning at a range of ±1 mm or ±0.1mm or 0.01mm or 0.001mm, of the conjugate plane; is configured to allow the electron beam, provided by the TEM, to pass through it.[000207] According to some embodiments and as demonstrated in Figure 9, the resonator

[0112] is positioned in the TEM, such that the laser beam inside it

[0109] has a focal spot [110C] at aboutthe center of the back focal plane [1 IOC] of the TEM, meaning at a range of ±1 mm or ±0.1mm or iO.Olmm or ±0.001mm, of the center of the back focal plane [1 IOC],[000208] According to some embodiments, the laser-based device

[0115] is configured to provide the laser beam

[0114] with light at two or more different wavelengths.[000209] According to some embodiments and as demonstrated in Figure 9, the TEM is equipped with a laser port

[0113] , wherein the laser beam

[0114] passes through the laser port

[0113] , to enter into the TEM

[0101] and is further routed, or coupled, into the cavity

[0112] ,[000210] According to some embodiments, the energy provided to the electron beams, by the TEM’s electron gun

[0102] is selected between 10 keV and 500 keV.[000211] According to some embodiments, and as demonstrated in Figure 9, the scattered waves

[0108] and the unscattered waves

[0107] are then recombined by the TEM and the recombined beam, forming an image of the sample / object, is then directed to the electron detector [H8].[000212] According to some embodiments, the laser beam

[0114] provided by the laser -based device

[0115] is non-monochromatic. According to some embodiments, the non-monochromatic laser beam is provided by a configuration of coupling two or more laser beams having different wavelengths into one optical resonator. According to some other embodiments, the non- monochromatic laser beam is provided by a configuration of two or more optical cavities with overlapping focal spots.[000213] According to some embodiments, the cavity

[0112] is tilted with respect to the optical axis of the TEM; or the optical axis of the cavity is tilted with respect to the optical axis of the TEM; or the direction of the laser beam propagation in the cavity is tilted with respect to the optical axis of the TEM.[000214] In one embodiment the invention provides a method for electron beam imaging or electron beam spectroscopy comprising: providing the electron beam system according to embodiments of the invention; shifting the phase of a pre-determined portion of the electron beam, via the optical cavity system; and detecting TEM images using a detector.[000215] In one embodiment the optical cavity further comprises at least one optomechanical system (also referred to herein as “optomechanical support” or simply “support” or “spacer”). Inone embodiment the optical cavity further comprises an optomechanical support configured such that the first segment is rigid. In one embodiment the optical cavity further comprises an optomechanical support configured such that the first segment is at least partially rigid. As understood herein the terms ‘high -NA arm’ and ‘first segment’ are to be understood interchangeably. Thus, in one embodiment the first segment of the optical cavity further comprises an optomechanical system. In one embodiment the optomechanical support is configured with zero degrees of freedom. In one embodiment the optomechanical support is configured between two optical elements. The optomechanical support could also be a rigid frame, such as a system of rods, or some other rigid support. In this case, the optomechanical support of the high -NA arm can be made in the form of a monolithic spacer or a rigid multielement spacer, wherein the length of the arm can be tuned either by controlling the temperature of (some part of) the spacer, or by applying a mechanical force (e.g. using a piezo element) to some part of the spacer to controllably deform it. Alternatively, the optomechanical support of the high-NA arm can be implemented with only one articulated degree of freedom to control the length of the arm. In this case, the optomechanical support of the high-NA arm can be implemented as a monolithic flexure with one degree of freedom or as a conventional optomechanical system.[000216] In one embodiment the short arm of the cavity (the high-NA arm) can be made mechanically solid, with no articulated degrees of freedom other than optional thermal or stress- induced expansion. The optomechanical system of the optical resonator ensures that the optical elements comprising it can be positioned and oriented in their proper position and proper orientation. While having many active (adjustable) degrees of freedom provides more flexibility, a compact and robust design is desirable for the laser phase plate. To this end, two non-limiting designs are proposed with the following features:1. The short arm of the optical resonator may have no adjustable degrees of freedom, or may have just one degree of freedom - the distance between the lens and the mirror of the short arm. This degree of freedom may be actuated thermally (by applying a thermal load to a spacer) or by applying a force e.g. with a piezo element. The short arm holder may be monolithic. It could be e.g. a tube-shaped spacer, with the lens attached to one end and the mirror attached to the other end. It can also be a plate to which the mirror and the lens are attached. The attachment can be accomplished e.g. by use of an adhesive, by optical contacting, by silicate bonding, or by using an optical frit;2. The mechanical system housing the long arm of the optical resonator functionally consists of a kinematic mount for the mirror of the long arm (the "input" mirror of the resonator)and a spacer that connects it to the short arm. The mirror mount may have two to five degrees of freedom: tip / tilt, or tip / tilt / axial shift, or the same plus two lateral shifts of the mirror. The mount may be designed as a conventional mirror mount design or as a flexure. The flexure may be monolithic. The spacer and the kinematic mirror mount can be a single monolithic piece, or they can be separate pieces joined mechanically.[000217] Examples of the materials used for such designs is now provided. One design makes use of the positive thermal coefficient of the refractive index of the lens to generate inverse cavity nonlinearity. The lens can be made of fused silica, sapphire, or another highly transparent optical material. In one implementation, a sapphire lens is used because of the higher refractive index than other similarly transparent materials, which allows for smaller angles of incidence on the lens at a given NA which leads to lower surface reflectivity.[000218] The inverse cavity nonlinearity generated by the lens overcomes the positive cavity nonlinearity arising from thermo-elastic deformations of the mirrors, which means that the material of the mirrors does not need to be chosen to minimize thermo-elastic effects (unlike, e.g., in laser phase plate designs based on near-concentric cavities). This enables using mirrors with substrates made of fused silica rather than ULE glass that is used to achieve high circulating power in near-concentric cavities. Fused silica is a material of choice for high-grade optical polishing, so being able to use fused silica substrates enables higher quality of mirrors [000219] One implementation provides a resonator made of two mirrors and a lens as shown in Fig. 2C, with a short arm and a long arm. The lens is designed to have different radii of curvature (ROC) on its two sides, to minimize reflection from the surfaces at a given numerical aperture (NA). The anti -reflective coatings are designed to provide the least possible reflection, but it is challenging to provide low reflectivity for a laser beam that covers a range of angles of incidence. Minimizing the range of angles enables lower average reflectivity. Reflectivity values can reach in the order of 100 parts per million with the best available coatings produced by ion beam sputtering.[000220] Surface roughness of the optical elements: the presently disclosed designs having a relatively high NA of >0.1 or >0.2 reduces the intra-cavity circulating power requirements for optimal laser phase plate operation (inversely proportionally to the NA). This enables making the cavity out of optical elements with a higher surface roughness, e.g., a "normal" high grade optical polish (about 5 Angstroms RMS surface roughness), which are easier to make, or which can be made in a larger variety of shapes, than superpolished elements (< 2 Angstroms RMS). [000221] The optical cavities of the invention can tolerate high cavity losses, which means that the optimal mirror coatings have relatively high transmissivities. One embodiment calls for theinput mirror transmissivity of 300 to 1000 parts per million, and the output mirror transmissivity of about 100 parts per million.[000222] In one embodiment the optical cavity further comprises an optomechanical support configured to maintain a fixed relative position between at least two optical elements of the short arm. In one embodiment the optomechanical support is comprised in the first segment (i.e., the high-NA arm) and configured to maintain a fixed relative position between the two optical elements. In one embodiment the optomechanical support is comprised in the first segment configured to maintain a fixed relative position between at least two optical elements. In one embodiment the first segment further comprises an optomechanical support configured to maintain a fixed relative position between the two optical elements. In one embodiment the first segment further comprises an optomechanical support configured to maintain a fixed relative position between at least two optical elements.[000223] All parts of the cavity are held in place by its optomechanical system. Normally, all parts of the optomechanical support are designed with adjustment degrees of freedom (i.e., articulated degrees of freedom). In one embodiment the fixed relative position of some of the optical elements is permanent. For example, unlike conventional designs with adjustable degrees of freedom, a possible configuration ensures that once assembled, the relative positioning remains unchanged over time and under varying conditions.[000224] A cavity or a part of a cavity (e.g., an arm of a cavity) can be described as rigid if its optomechanical support system does not have articulated connections. Optical elements comprised in a rigid cavity or in a rigid part of a cavity generally have fixed relative positions and orientations. However, in a rigid cavity (or in a rigid part of a cavity), relative positions and orientations of elements can be adjusted within a limited range (typically, nanometers to a few micrometers of linear motion, and microradians to a few milliradians of angular motion) by controllable deformation of the rigid support system. Such controllable deformation can be achieved by applying a force or athermal load to parts of the support system. The optomechanical support of a rigid cavity (or a rigid part of a cavity) can be monolithic, i.e., comprising a single, continuous piece of material. It can also be composed of multiple elements connected with fixed connections.[000225] As used herein, an “articulated” or “adjustable” connections refer to any connection that allows for some degrees of freedom to be adjusted, while other degrees of freedom to be constrained. Examples of articulated connections include, but are not limited to: a ball-and-V- groove joint, a flexure hinge, a sliding joint, crossed roller bearing connection, etc. This is incontrast to fixed connections, which do not allow relative motion between connected parts, such as, e.g., screw connections, or connections using adhesives or welding.[000226] In one embodiment the optomechanical support is a rigid spacer. Examples of rigid spacers include: monolithic spacer e.g., a base plate or a hollow tube, multi -element spacer e.g., a rod system, a cage system or multi -element frame. A monolithic spacer is a single, continuous piece of material. A base plate is a flat, solid structure. In one embodiment the base plate is attached to optical elements by optical contacting, silicate bonding, adhesive, or an optical frit. Examples of base plate materials include, but are not limited to: glass-like material such as fused silica, BK7 glass, ultra-low thermal expansion glass, a crystalline material such as YAG, sapphire, or CVD diamond, or a metal (or alloy) such as copper, bronze, aluminum or titanium. [000227] In one embodiment the optomechanical support is further configured to adjust the length between the at least two optical elements by: controlling the temperature, mechanical deformation, or a combination thereof. For example, the axial distance is adjusted. Using temperature control, thermal expansion or contraction of the material is induced by heating or cooling regions of the support. By selecting materials with suitable thermal expansion coefficients, fine adjustments in length can be achieved. For example, localized heating can cause controlled expansion, increasing the distance between optical elements, while cooling contracts the structure to reduce the distance. Using mechanical deformation, piezoelectric actuators, flexure mechanisms, or controlled mechanical stress can be applied to specific parts of the support. In one embodiment temperature-induced and mechanical-induced deformation are combined. In one embodiment the controlling of temperature comprises heating or cooling at least a portion of the optomechanical support to induce thermal expansion or contraction, thereby adjusting the length between the at least two optical elements.[000228] In one embodiment the optomechanical support of the high-NA arm is configured with one articulated degree of freedom, wherein the one articulated degree of freedom is limited to controlling the length of the high-NA arm. In this case, the optomechanical support of the high- NA arm can be implemented as a monolithic flexure with one degree of freedom (e.g., translational motion) or as a conventional optomechanical system. In one embodiment the range of motion of the one articulated degree of freedom ranges between lOOnm to 10mm. In one embodiment the range of motion of the one articulated degree of freedom ranges between 1 pm to 1mm. In one embodiment the range of motion of the one articulated degree of freedom ranges between 10pm to 10mm.[000229] The optomechanical support allows for a compact and robust design, while the necessary adjustment of the cavity alignment can be achieved by controlling the articulateddegrees of freedom of other optical elements (which are not part of the high-NA arm). All the optical elements making up the cavity are held by its optomechanical system. Thus, proper alignment of the cavity can be maintained by only using active degrees of freedom in other parts of the cavity, while the short arm support is rigid. This is important because the high-NA arm needs to be compact, as it needs to be inserted into the electron beam path where space is confined. The other parts of the cavity are less restricted in the amount of space available.[000230] Figure 10B shows an example of an optomechanical support 500 for the high-NA arm of an optical cavity (not fully shown). This is an example of a monolithic set-up, where the two small optical elements (the small mirror 204 and the lens 205) are fixed to a base plate (between 50pm to 1mm thick), made of a rigid, strong and thermally conductive material, such as CVD diamond or sapphire. The distance of the lens 205 from the big mirror 506 (arrow points to where the big mirror is located, far away) is very large compared to the distance to the small mirror 204. The optomechanical system 500, and correspondingly also the optical mode, has different dimensions in the two axes normal to the optical axis: it is short in the vertical axis, and long in the horizontal axis. The electron beam 504 passes through an opening in the base plate, through the tight focus of the beam 202 in the short arm.[000231] In one embodiment, the term “a” or “one” or “an” refers to at least one. In one embodiment the phrase “two or more” may be of any denomination, which will suit a particular purpose. In one embodiment, “about” or "approximately" may comprise a deviance from the indicated term of + 1 %, or in some embodiments, - 1 %, or in some embodiments, ± 2.5 %, or in some embodiments, ± 5 %, or in some embodiments, ± 7.5 %, or in some embodiments, ± 10 %, or in some embodiments, ± 15 %, or in some embodiments, ± 20 %, or in some embodiments, ± 25 %.[000232] Those skilled in the art to which this invention pertains will readily appreciate that numerous changes, variations, and modifications can be made without departing from the scope of the presently disclosed subject matter, mutatis mutandis.EXAMPLES EXAMPLE 1Geometry and Parameters of Optical Cavity[000233] Figures 10A shows a schematic diagram of the optical cavity 220 of Figure 2C. The optical cavity 220 comprises three elements: 1) small mirror 204 denoted by mi, 2) lens 205 denoted by / , and 3) big mirror 203 denoted by m2. The long arm 222 is denoted by L2 and theshort arm 221 is denoted by Li. The distance Li between the small mirror mi and the lens I is significantly smaller than the distance L2 between the lens I and the big mirror m2. The beam has a gaussian profile in the plane normal to its optical axis.[000234] The rays 202 for which the intensity drops below e-1of the intensity at the optical axis are shown. The angle of incidence of those lines with the lens are equal on both sides (10.8 degrees in the value for “Marginal incidence angle” in Table 1). This feature is designed to minimize the maximal value of the incidence angles, thereby minimizing reflection losses. The spot size (w) diameter is a measure for the distance of the marginal rays from the optical axis on the different elements. The optical mode has a large divergence angle (corresponding to a large NAi) in the left, short arm 221 and a smaller divergence angle (corresponding to a smaller NA2 in the long arm 222.Table 1 - parameters and values of optical cavity related to geometry.[000235] Results shown in Table 2 show an example of the material and optical properties in an optical cavity set up. The scratch-dig parameter is a measure for the frequency and typical size of flaws on the surface of the element (small mirror, lens, big mirror). The surface figure parameters is a measure of how close the face of the element is to the face of an ideal sphere. Thesurface roughness parameter is a measure for how smooth the surface of the element is. The transmissivity is a measure of the optical power transmitted through the optical element.Table 2 - parameters and values of optical cavity based on materials and optical properties.

Claims

CLAIMS1. An optical cavity comprising: a first segment with a numerical aperture (NAi) and a length (L1); and a second segment with a numerical aperture (NA2) and a length (L2); wherein the first segment and the second segment each comprise two optical elements; wherein NAi is at least 0.05; and wherein L2 is greater than L1by a factor ranging between about 2 and 10,000.

2. The optical cavity of claim 1 wherein the mode size on each of the one or more optical elements ranges between 0.5 mm to 1 m.

3. The optical cavity of claim 1 wherein the ratio NA1 / NA2 is at least 2.

4. The optical cavity of claim 1 wherein the ratio NA1 / NA2 ranges between about 2 and 10,000.

5. The optical cavity of claim 1 wherein NAi ranges between 0.05 and 0.99 and NA2 ranges between 0.0001 and 0.04.

6. The optical cavity of claim 1 wherein L1ranges between 1 mm and 1 m and L2 ranges between 1 cm and 10 m.

7. The optical cavity of claim 1 wherein at least one of the at least two optical elements are astigmatic.

8. The optical cavity of claim 1 wherein the two or more optical elements are selected from: reflective element, transmissive element, lens-mirror element, or a combination thereof.

9. The optical cavity of claim 8 wherein the reflective element is selected from: a mirror with a concave reflective surface, a mirror with a convex reflective surface, a mirror with a planar reflective surface, a lens-mirror element or a combination thereof.

10. The optical cavity of claim 8 wherein the reflective element is at least partially transmissive.

11. The optical cavity of claim 8 wherein the reflective element comprises a reflective coating.

12. The optical cavity of claim 8 wherein the transmissive element is selected from: lens, planar optical window, or a combination thereof.

13. The optical cavity of claim 8 wherein the transmissive element comprises an anti- reflective coating.

14. The optical cavity of claim 8 wherein the transmissive element is at least 99% transmissive.

15. The optical cavity of claim 12 wherein the lens is biconcave, biconvex, plano-convex, plano-concave, spherical, aspherical, concave-convex, or a combination thereof.

16. The optical cavity of claim 1 wherein the first segment has a tight focus with a mode waist size ranging between 0.1 pm and 10 pm.

17. The optical cavity of claim 1 further comprising at least one thermal conductor in contact with at least part of the first segment, the second segment, or a combination thereof.

18. The optical cavity of claim 1 further comprising at least one additional optical segment with at least one of the optical elements being the same in at least one other optical segment in the optical cavity.

19. The optical cavity of claim 1 in a standing wave cavity configuration or a ring cavity configuration.

20. The optical cavity of claim 1 wherein: the first segment comprises a first mirror and a lens; and the second segment comprises the lens and a second mirror.

21. The optical cavity of claim 20 wherein the first mirror has a smaller radius of curvature than the second mirror.

22. The optical cavity of claim 20 wherein the first mirror, the lens and the second mirror are arranged to share the same optical axis.

23. The optical cavity of claim 20 wherein the first mirror, the second mirror or a combination thereof, are at least partially transmissive.

24. The optical cavity of claim 20 further comprising an anti -reflective coating disposed on at least one side of the lens.

25. The optical cavity of claim 1 wherein: the first segment comprises a first mirror and a second mirror; the second segment comprises the second mirror and a third mirror; and wherein the first mirror, the second mirror and the third mirror form the standing wave cavity.

26. The optical cavity of claim 25 wherein the first mirror and the second mirror have a smaller radius of curvature than the third mirror.

27. The optical cavity of claim 25 wherein the first mirror, the second mirror or the third mirror, or a combination thereof, are at least partially transmissive.

28. The optical cavity of claim 1 comprising at least three mirrors in a ring cavity configuration.

29. The optical cavity of claim 28 wherein: the first segment comprises a first mirror and a second mirror; the second segment comprises the second mirror and a third mirror; and wherein the first mirror, the second mirror and the third mirror form the ring cavity configuration.

30. The optical cavity of claim 28 wherein the first mirror or the second mirror have a smaller radius of curvature than the third mirror.

31. The optical cavity of claim 28 wherein the first mirror, the second mirror or the third mirror, or a combination thereof, are at least partially transmissive.

32. The optical cavity of claim 1 further comprising an optomechanical support configured such that the first segment is rigid.

33. An optical cavity comprising an optical segment with an NA greater than 0.05, wherein the optical segment or an additional optical segment comprises at least one nonlinearoptical element with a focusing power responsivity ranging between 0.001 m-1kW-1and 100 m’1kW’1.

34. The optical cavity of claim 33 wherein the NA responsivity of the cavity ranges between 0.0001 / kW to 1 / kW.

35. The optical cavity of claim 33 further comprising a laser beam coupled to the optical cavity which has an intracavity circulating power ranging between IkW to 10 MW.

36. The optical cavity of claim 35 wherein NA increases by an amount ranging between about 0.001 to 0.9 upon coupling the laser beam to the optical cavity.

37. The optical cavity of claim 33 wherein the nonlinear optical element comprises a material which has: a negative thermal expansion coefficient, a positive thermal coefficient of the refractive index, a positive Kerr coefficient, or a combination thereof.

38. The optical cavity of claim 33 wherein the optical segments comprise optical elements selected from: reflective element, transmissive element, lens-mirror element, or a combination thereof.

39. The optical cavity of claim 38 wherein the reflective element is selected from: a mirror with a concave reflective surface, a mirror with a convex reflective surface, a mirror with a planar reflective surface, or a combination thereof.

40. The optical cavity of claim 38 wherein the reflective element is at least partially transmissive.

41. The optical cavity of claim 33, further comprising a thermal conductor in contact with at least part of the optical segment.

42. The optical cavity of claim 38 wherein the lens of the lens-mirror element further comprises an anti -reflective coating.

43. The optical cavity of claim 38 wherein the lens and the mirror of the lens-mirror element are joined by an adhesive, an adhesive-free bonding method, or optical contacting.

44. The optical cavity of claim 38 wherein the lens-mirror element comprises a lens or a planar optical window having a first optical surface and a second optical surface, wherein an antireflective coating is disposed on the first optical surface and a reflective coating is disposed on the second optical surface.

45. The optical cavity of claim 33 further comprising at least one additional optical segment with at least one of the optical elements being shared in at least one other optical segment in the optical cavity.

46. An optical cavity comprising: an optical segment comprising a first reflecting element and a second reflecting element, wherein the first reflecting element, the second reflecting element, or a combination thereof, comprise a negative thermal expansion coefficient (NTEC) material.

47. The optical cavity of claim 46 wherein the NTEC material is selected from: optical glass, optical crystal, fused silica, Ultra-Low Expansion (ULE) glass, diamond, silicon, or a combination thereof.

48. The optical cavity of claim 46 wherein the first reflecting element and the second reflective element are selected from: a mirror with a concave reflective surface, a mirror with a convex reflective surface, a mirror with a planar reflective surface, or a combination thereof.

49. The optical cavity of claim 46 wherein the first reflecting element, the second reflecting element, or a combination thereof, is at least partially transmissive.

50. The optical cavity of claim 46, further comprising a thermal conductor in contact with at least part of the optical segment, at least one additional optical segment with at least one of the optical elements being shared with at least one other optical segment in the optical cavity, or a combination thereof.

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