Thickness dependence of piezo-bimorph adaptive mirror bending.

A new generation of adaptive x-ray optics (AXO) is being installed on high-coherent-flux x-ray beamlines worldwide to correct and control the optical wavefront with sub-nm precision. These ultra-smooth mirrors achieve high reflectivities at glancing angles of incidence and can be hundreds of mm long. One type of adaptive x-ray mirror relies on piezoelectric ceramic strips which are segmented into channels and actuated to induce local, longitudinal bending, generating one-dimensional shape changes in the mirror substrate. A recently described mirror model uses a three-layer geometry with parallel actuators on the front and back surfaces of a thicker mirror substrate. By analogy to a solved problem in the thermal actuation of a tri-metal strip, we show that the achievable bending radius varies approximately as the square of the substrate thickness. We provide an analytic solution and simulate bending using a finite-element model.

Real-time machine-learning-driven control system of a deformable mirror for achieving aberration-free X-ray wavefronts.

A neural-network machine learning model is developed to control a bimorph adaptive mirror to achieve and preserve aberration-free coherent X-ray wavefronts at synchrotron radiation and free electron laser beamlines. The controller is trained on a mirror actuator response directly measured at a beamline with a real-time single-shot wavefront sensor, which uses a coded mask and wavelet-transform analysis. The system has been successfully tested on a bimorph deformable mirror at the 28-ID IDEA beamline of the Advanced Photon Source at Argonne National Laboratory. It achieved a response time of a few seconds and maintained desired wavefront shapes (e.g., a spherical wavefront) with sub-wavelength accuracy at 20 keV of X-ray energy. This result is significantly better than what can be obtained using a linear model of the mirror's response. The developed system has not been tailored to a specific mirror and can be applied, in principle, to different kinds of bending mechanisms and actuators.

Off-axis representation of hyperbolic mirror shapes for X-ray beamlines.

Mirror-centered, closed-form expressions for hyperbolic surfaces used in X-ray beamlines have been derived. Hyperbolic mirrors create a virtual focus or source point and can be used to lengthen or shorten the effective focal distance of a compound optical system. The derivations here express off-axis segments of a hyperbolic surface in terms of the real and virtual focal distances and the incident glancing angle at the center of the mirror. Conventional mathematical expressions of hyperbolic shapes describe the surfaces in Cartesian or polar coordinates centered on an axis of symmetry, necessitating cumbersome rotation and translation to mirror-centered coordinates. The representation presented here, with zero slope and the origin at the central point, is most convenient for modeling, metrology, aberration correction, and general surface analysis of off-axis configurations. The direct derivation avoids the need for nested coordinate transforms. A series expansion provides a helpful approximation; the coefficients of the implicit equation are also provided.

Data-driven modeling and control of an X-ray bimorph adaptive mirror.

Adaptive X-ray mirrors are being adopted on high-coherent-flux synchrotron and X-ray free-electron laser beamlines where dynamic phase control and aberration compensation are necessary to preserve wavefront quality from source to sample, yet challenging to achieve. Additional difficulties arise from the inability to continuously probe the wavefront in this context, which demands methods of control that require little to no feedback. In this work, a data-driven approach to the control of adaptive X-ray optics with piezo-bimorph actuators is demonstrated. This approach approximates the non-linear system dynamics with a discrete-time model using random mirror shapes and interferometric measurements as training data. For mirrors of this type, prior states and voltage inputs affect the shape-change trajectory, and therefore must be included in the model. Without the need for assumed physical models of the mirror's behavior, the generality of the neural network structure accommodates drift, creep and hysteresis, and enables a control algorithm that achieves shape control and stability below 2 nm RMS. Using a prototype mirror and ex situ metrology, it is shown that the accuracy of our trained model enables open-loop shape control across a diverse set of states and that the control algorithm achieves shape error magnitudes that fall within diffraction-limited performance.

Analytic descriptions of parabolic X-ray mirrors.

On X-ray beamlines and telescopes, glancing-incidence mirrors with parabolic profiles are used to collimate, focus, and collect light. Here, analytic descriptions for paraboloidal, plane-parabolic, and parabolic cylindrical mirrors in several congruent geometries that are commonly used in fabrication, metrology, and modeling are provided. The exact expressions are derived directly from Fermat's principle, without coordinate transformations, in several mirror-centered coordinate systems, including one with the surface tangent to the central point of intersection. Coefficients for a sixth-order polynomial series approximation are calculated for that coordinate system.

Derivation of closed-form ellipsoidal X-ray mirror shapes from Fermat's principle.

Ellipsoidal and plane-elliptical surfaces are widely used as reflective, point-to-point focusing elements in many optical systems, including X-ray optics. Here the classical optical path function approach of Fermat is applied to derive a closed-form expression for these surfaces that are uniquely described by the object and image distances and the angle of incidence at a point on a mirror surface. A compact description facilitates design, modeling, fabrication, and testing to arbitrary accuracy. Congruent surfaces in two useful coordinate systems - a system centered on the ellipsoid's axes of symmetry and a mirror-centered or `vertex' system with the surface tangent to the xy plane at the mirror's center - are presented. Expressions for the local slope and radii of curvature are derived from the result, and the first several terms of the Maclauren series expansion are provided about the mirror center.

Diaboloidal mirrors: algebraic solution and surface shape approximations.

A new type of optical element that can focus a cylindrical wave to a point focus (or vice versa) is analytically described. Such waves are, for example, produced in a beamline where light is collimated in one direction and then doubly focused by a single optic. A classical example in X-ray optics is the collimated two-crystal monochromator, with toroidal mirror refocusing. The element here replaces the toroid, and in such a system provides completely aberration free, point-to-point imaging of rays from the on-axis source point. We present an analytic solution for the mirror shape in its laboratory coordinate system with zero slope at the centre, and approximate solutions, based on bending an oblique circular cone and a bent right circular cylinder, that may facilitate fabrication and metrology.

Simulations of applications using diaboloid mirrors.

The diaboloid is a reflecting surface that converts a spherical wave to a cylindrical wave. This complex surface may find application in new Advanced Light Source bending-magnet beamlines or in other beamlines that now use toroidal optics for astigmatic focusing. Here, the numerical implementation of diaboloid mirrors is described, and the benefit of this mirror in beamlines exploiting diffraction-limited storage rings is studied by ray tracing. The use of diaboloids becomes especially interesting for the new low-emittance storage rings because the reduction of aberration becomes essential for such small sources. The validity of the toroidal and other mirror surfaces approximating the diaboloid, and the effect of the mirror magnification, are discussed.

Final 5-Year Outcomes of the Multicenter Randomized Sham-Controlled Trial of a Water Vapor Thermal Therapy for Treatment of Moderate to Severe Lower Urinary Tract Symptoms Secondary to Benign Prostatic Hyperplasia.

PURPOSE: We present final 5-year outcomes of the multicenter randomized sham-controlled trial of a water vapor therapy (Rezum™) for treatment of moderate to severe lower urinary tract symptoms due to benign prostatic hyperplasia. MATERIALS AND METHODS: A total of 197 subjects >50 years of age with International Prostate Symptom Score ≥13, maximum flow rate ≤15 ml/second and prostate volume 30 to 80 cc were randomized and followed for 5 years. From the control arm of 61 subjects, a subset of 53 subjects requalified and after 3 months received treatment as part of the crossover group and were also followed for 5 years. The total number of vapor treatments to each lobe of the prostate was determined by length of prostatic urethra and included middle lobe treatment per physician discretion. RESULTS: Significant improvement of lower urinary tract symptoms was observed at <3 months post-thermal therapy, remaining durable through 5 years in the treatment group (International Prostate Symptom Score reduced 48%, quality of life increased 45%, maximum flow rate improved 44%, Benign Prostatic Hyperplasia Impact Index decreased 48%). Surgical re-treatment rate was 4.4% with no reports of device or procedure related sexual dysfunction or sustained de novo erectile dysfunction. Results within the crossover group were similar through 5 years. CONCLUSIONS: Minimally invasive treatment with water vapor thermal therapy provides significant and durable symptom relief as well as flow rate improvements through 5 years, with low surgical re-treatment rates and without impacting sexual function. It is a versatile therapy, providing successful treatment to obstructive lateral and middle lobes.

Pseudo-gray-scale halftone gratings for shearing and Hartmann wavefront sensors.

Now in use on x-ray beamlines worldwide, shearing interferometry and Hartmann wavefront sensing provide effective feedback for measuring and optimizing high-quality beams. Conventionally, both approaches spatially modulate the beam properties (amplitude or phase) using two-tone, binary patterns, leading to deleterious diffraction effects that must be mitigated. In shearing, the presence of multiple diffraction orders affects measurement near boundaries. In Hartmann, diffraction limits the measurement point density. We demonstrate that the use of pseudo-gray-scale halftone patterns in the diffracting elements can improve the performance of both techniques.

Binary Amplitude Reflection Gratings for X-ray Shearing and Hartmann Wavefront Sensors.

New, high-coherent-flux X-ray beamlines at synchrotron and free-electron laser light sources rely on wavefront sensors to achieve and maintain optimal alignment under dynamic operating conditions. This includes feedback to adaptive X-ray optics. We describe the design and modeling of a new class of binary-amplitude reflective gratings for shearing interferometry and Hartmann wavefront sensing. Compact arrays of deeply etched gratings illuminated at glancing incidence can withstand higher power densities than transmission membranes and can be designed to operate across a broad range of photon energies with a fixed grating-to-detector distance. Coherent wave-propagation is used to study the energy bandwidth of individual elements in an array and to set the design parameters. We observe that shearing operates well over a ±10% bandwidth, while Hartmann can be extended to ±30% or more, in our configuration. We apply this methodology to the design of a wavefront sensor for a soft X-ray beamline operating from 230 eV to 1400 eV and model shearing and Hartmann tests in the presence of varying wavefront aberration types and magnitudes.

Reflective binary amplitude grating for soft x-ray shearing and Hartmann wavefront sensing.

We demonstrate a reflective wavefront sensor grating suitable for the characterization of high-quality x-ray beamlines and optical systems with high power densities. Operating at glancing incidence angles, the optical element is deeply etched with a two-level pattern of shearing interferometry gratings and Hartmann wavefront sensor grids. Transverse features block unwanted light, enabling binary amplitude in reflection with high pattern contrast. We present surface characterization and soft x-ray reflectometry of a prototype grating array to demonstrate function prior to wavefront measurement applications. A simulation of device performance is shown.

Compensation of heat load deformations using adaptive optics for the ALS upgrade: a wave optics study.

A realistic wave optics simulation method has been developed to study how wavefront distortions originating from heat load deformations can be corrected using adaptive X-ray optics. Several planned soft X-ray and tender X-ray insertion-device beamlines in the Advanced Light Source upgrade rely on a common design principle. A flat, first mirror intercepts the white beam; vertical focusing is provided by a variable-line-space monochromator; and horizontal focusing comes from a single, pre-figured, adaptive mirror. A variety of scenarios to cope with thermal distortion in the first mirror are studied by finite-element analysis. The degradation of the intensity distribution at the focal plane is analyzed and the adaptive optics that correct it is modeled. The range of correctable wavefront errors across the operating range of the beamlines is reported in terms of mirror curvature and spatial frequencies. The software developed is a one-dimensional wavefront propagation package made available in the OASYS suite, an adaptable, customizable and efficient beamline modeling platform.

Adaptive shape control of wavefront-preserving X-ray mirrors with active cooling and heating.

This article describes the development and testing of a novel, water-cooled, active optic mirror system (called "REAL: Resistive Element Adjustable Length") that combines cooling with applied auxiliary heating, tailored to the spatial distribution of the thermal load generated by the incident beam. This technique is theoretically capable of sub-nanometer surface figure error control even at high power density. Tests conducted in an optical metrology laboratory and at synchrotron X-ray beamlines showed the ability to maintain the mirror profile to the level needed for the next generation storage rings and FEL mirrors.

Extreme ultraviolet microscope characterization using photomask surface roughness.

We demonstrate a method for characterizing the field-dependent aberrations of a full-field synchrotron-based extreme ultraviolet microscope. The statistical uniformity of the inherent, atomic-scale roughness of readily-available photomask blanks enables a self-calibrating computational procedure using images acquired under standard operation. We characterize the aberrations across a 30-um field-of-view, demonstrating a minimum aberration magnitude of smaller than [Formula: see text] averaged over the center 5-um area, with a measurement accuracy better than [Formula: see text]. The measured field variation of aberrations is consistent with system geometry and agrees with prior characterizations of the same system. In certain cases, it may be possible to additionally recover the illumination wavefront from the same images. Our method is general and is easily applied to coherent imaging systems with steerable illumination without requiring invasive hardware or custom test objects; hence, it provides substantial benefits when characterizing microscopes and high-resolution imaging systems in situ.

Amplitude versus phase effects in extreme ultraviolet lithography mask scattering and imaging.

It is now well established that extreme ultraviolet (EUV) mask multilayer roughness leads to wafer-plane line-width roughness (LWR) in the lithography process. Analysis and modeling done to date has assumed, however, that the roughness leading to scatter is primarily a phase effect and that the amplitude can be ignored. Under this assumption, simple scattering measurements can be used to characterize the statistical properties of the mask roughness. Here, we explore the implications of this simplifying assumption by modeling the imaging impacts of the roughness amplitude component as a function of the balance between amplitude and phase induced scatter. In addition to model-based analysis, we also use an EUV microscope to compare experimental through focus data to modeling in order to assess the actual amount of amplitude roughness on a typical EUV multilayer mask. The results indicate that amplitude roughness accounts for less than 1% of the total scatter for typical EUV masks.

Convective Thermal Therapy: Durable 2-Year Results of Randomized Controlled and Prospective Crossover Studies for Treatment of Lower Urinary Tract Symptoms Due to Benign Prostatic Hyperplasia.

PURPOSE: We report 2-year outcomes of a multicenter randomized controlled trial plus 1-year results of a crossover trial after treatment with convective radiofrequency water vapor thermal energy for lower urinary tract symptoms due to benign prostatic hyperplasia. MATERIALS AND METHODS: A total of 197 men at least 50 years old with I-PSS (International Prostate Symptom Score) 13 or greater, maximum flow rate 15 ml per second or less and prostate size 30 to 80 cc were randomized 2:1 to thermal therapy with the Rezum® System or a control group. Rigid cystoscopy with simulated active treatment sounds served as the control procedure. After unblinding at 3 months control subjects could requalify for crossover study. Convectively delivered radiofrequency thermal energy was delivered into obstructive prostate tissue, including the median lobe as needed. The primary efficacy end point was a change in severity of symptom scores. RESULTS: Convective radiofrequency thermal therapy improved urinary symptoms significantly over controls at 3 months and provided a sustained 51% reduction from baseline at 24 months (p <0.0001). This produced a 5 and 8-point or greater score decrease in 84% and 74% of subjects, respectively, at 24 months. Crossover subject symptoms, flow rate and quality of life measures were markedly improved after thermal therapy compared to after the control procedure (p = 0.024 to <0.0001). No de novo erectile dysfunction was reported. CONCLUSIONS: Convective radiofrequency water vapor thermal therapy is a minimally invasive office or outpatient procedure that provides early effective symptom relief that remains durable for 2 years and is applicable to the median lobe.

Optimized mirror shape tuning using beam weightings based on distance, angle of incidence, reflectivity, and power.

For glancing-incidence optical systems, such as short-wavelength optics used for nano-focusing, incorporating physical factors in the calculations used for shape optimization can improve performance. Wavefront metrology, including the measurement of a mirror's shape or slope, is routinely used as input for mirror figure optimization on mirrors that can be bent, actuated, positioned, or aligned. Modeling shows that when the incident power distribution, distance from focus, angle of incidence, and the spatially varying reflectivity are included in the optimization, higher Strehl ratios can be achieved. Following the works of Maréchal and Mahajan, optimization of the Strehl ratio (for peak intensity with a coherently illuminated system) occurs when the expectation value of the phase error's variance is minimized. We describe an optimization procedure based on regression analysis that incorporates these physical parameters. This approach is suitable for coherently illuminated systems of nearly diffraction-limited quality. Mathematically, this work is an enhancement of the methods commonly applied for ex situ alignment based on uniform weighting of all points on the surface (or a sub-region of the surface). It follows a similar approach to the optimization of apodized and non-uniformly illuminated optical systems. Significantly, it reaches a different conclusion than a more recent approach based on minimization of focal plane ray errors.