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Phase-Shifting Liquid Crystal Point-Diffraction Interferometry

Microgravity fluid physics experiments frequently measure concentration and temperature. Interferometers such as the Twyman Green illustrated have performed full-field measurement of these quantities. As with most such devices, this interferometer uses a reference path that is not common with the path through the test section. Recombination of the test and reference wavefronts produces interference fringes. Unfortunately, in order to obtain stable fringes, the alignment of both the test and reference paths must be maintained to within a fraction of the wavelength of the light being used for the measurement. Otherwise, the fringes will shift and may disappear. Because these interferometers are extremely sensitive to bumping, jarring and transmitted vibration, they are typically mounted on optical isolation tables. Schlieren deflectometers or the more recent Shack-Hartmann wavefront sensors also measure concentration and temperature in laboratory fluid flows. Ray optics describe the operation of both devices. In a schlieren system, an expanded, collimated beam passes through a test section where refractive index gradients deflect rays. A lens focuses the beam to a filter placed in the rear focal plane of the decollimating lens. In a quantitative color schlieren system, gradients in the index of refraction appear as colors in the field of view due to the action of the color filter. Since sensitivity is a function of the focal length of the decollimating lens, these systems are rather long and filter fabrication and calibration is rather difficult. A Shack-Hartmann wavefront sensor is an array of small lenslets. Typical diameters are on the order of a few hundred microns. Since these lenslets divide the test section into resolution elements, the spatial resolution can be no smaller than an individual lenslet. Such a device was recently used to perform high-speed tomography of heated air exiting a 1.27 cm diameter nozzle. While these wavefront sensors are very compact, the limited spatial resolution and the methods required for data reduction suggest that a more useful instrument needs to be developed. The category of interferometers known as common path interferometers can eliminate much of the vibration sensitivity associated with traditional interferometry as described above. In these devices, division of the amplitude of the wavefront following the test section produces the reference beam. Examples of these instruments include shearing and point diffraction interferometers. In the latter case, shown schematically, a lens focuses light passing through the test section onto a small diffracting object. Such objects are typically either a circle of material on a high quality glass plate or a small sphere in a glass cell. The size of the focused spot is several times larger than the object so that the light not intercepted by the diffracting object forms the test beam while the diffracted light generates a spherical reference beam. While this configuration is mechanically stable, phase shifting one beam with respect to the other is difficult due to the common path. Phase shifting enables extremely accurate measurements of the phase of the interferogram using only gray scale intensity measurements and is the de facto standard of industry. Mercer and Creath 2 demonstrated phase shifting in a point diffraction interferometer using a spherical spacer in a liquid crystal cell as the diffracting object. By changing the voltage across the cell, they were able to shift the phase of the undiffracted beam relative to the reference beam generated by diffraction from the sphere. While they applied this technology to fluid measurements, the device shifted phase so slowly that it was not useful for studying transient phenomena. We have identified several technical problems that precluded operation of the device at video frame rates and intend to solve them to produce a phase-shifting liquid crystal point-diffraction interferometer operating at video frame rates. The first task is to produce high contrast fringes. Since the diffracted beam is much weaker than the transmitted beam, interferograms have poor contrast unless a dye is added to the liquid crystal to reduce the intensity of the undiffracted light. Dyes previously used were not rigorously characterized and suffered from hysteresis in both the initial alignment state of the device and the electro-optic switching characteristics. Hence, our initial effort will identify and characterize dyes that do not suffer from these difficulties and are readily soluble in the liquid crystal host. Since the ultimate goal of this research is to produce interferometers capable of phase shifting at video frame rates, we will quantify the difference in switching times between ferroelectric and nematic liquid crystals. While we have more experience with nematic crystals, they typically switch more slowly than ferroelectric cells. As part of that effort, we will investigate the difference in the modulation of the interferograms as a function of the type of liquid crystal in the cell. Because the temporal switching response of a liquid crystal cell is directly related its thickness, we intend to explore techniques required to produce cells that are as thin as possible. However, the cells must still produce a total phase shift of two pi radians.

Griffin, DeVon W.

Diffraction hardware testbed and model validation

Optical systems, which operate over a wide range of Fresnel numbers, are often times performance-limited by diffraction effects. In order to characterize such effects at the 40-100 picometer level, a diffraction testbed has been built which has the capability of measuring diffraction effects at this level. Concurrently, mathematical diffraction modeling tools have been developed that propagate an input wavefront through an optical train, while retaining amplitude and phase information at a grid resolution sufficient for yielding picometer-resolution diffraction test data. This paper contains a description of this diffraction hardware testbed, the diffraction modeling approach, and a comparison of the modeled and hardware test results, which then serves as validation of the diffraction modeling methodology.

optical

A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface

A compact dyadic diffraction coefficient for electromagnetic waves obliquely incident on a curved edge formed by perfectly conducting curved or plane surfaces is obtained. This diffraction coefficient remains valid in the transition regions adjacent to shadow and reflection boundaries, where the diffraction coefficients of Keller's original theory fail. Our method is based on Keller's method of the canonical problem, which in this case is the perfectly conducting wedge illuminated by plane, cylindrical, conical, and spherical waves. When the proper ray-fixed coordinate system is introduced, the dyadic diffraction coefficient for the wedge is found to be the sum of only two dyads, and it is shown that this is also true for the dyadic diffraction coefficients of higher order edges. One dyad contains the acoustic soft diffraction coefficient; the other dyad contains the acoustic hard diffraction coefficient. The expressions for the acoustic wedge diffraction coefficients contain Fresnel integrals, which ensure that the total field is continuous at shadow and reflection boundaries.

Kouyoumjian, R. G.

Diffractive optics fabricated by direct write methods with an electron beam

State-of-the-art diffractive optics are fabricated using e-beam lithography and dry etching techniques to achieve multilevel phase elements with very high diffraction efficiencies. One of the major challenges encountered in fabricating diffractive optics is the small feature size (e.g. for diffractive lenses with small f-number). It is not only the e-beam system which dictates the feature size limitations, but also the alignment systems (mask aligner) and the materials (e-beam and photo resists). In order to allow diffractive optics to be used in new optoelectronic systems, it is necessary not only to fabricate elements with small feature sizes but also to do so in an economical fashion. Since price of a multilevel diffractive optical element is closely related to the e-beam writing time and the number of etching steps, we need to decrease the writing time and etching steps without affecting the quality of the element. To do this one has to utilize the full potentials of the e-beam writing system. In this paper, we will present three diffractive optics fabrication techniques which will reduce the number of process steps, the writing time, and the overall fabrication time for multilevel phase diffractive optics.

Kress, Bernard

High Pressure X-Ray Diffraction Studies of Nanocrystalline Materials

Experimental evidence obtained for a variety of nanocrystalline materials suggest that the crystallographic structure of a very small size particle deviates from that in the bulk crystals. In this paper we show the effect of the surface of nanocrystals on their structure by the analysis of generation and distribution of macro- and micro-strains at high pressures and their dependence on the grain size in nanocrystalline powders of Sic. We studied the structure of Sic nanocrystals by in-situ high-pressure powder diffraction technique using synchrotron and neutron sources and hydrostatic or isostatic pressure conditions. The diffraction measurements were done in HASYLAB at DESY using a Diamond Anvil Cell (DAC) in the energy dispersive geometry in the diffraction vector range up to 3.5 - 4/A and under pressures up to 50 GPa at room temperature. In-situ high pressure neutron diffraction measurements were done at LANSCE in Los Alamos National Laboratory using the HIPD and HIPPO diffractometers with the Paris-Edinburgh and TAP-98 cells, respectively, in the diffraction vector range up to 26 Examination of the response of the material to external stresses requires nonstandard methodology of the materials characterization and description. Although every diffraction pattern contains a complete information on macro- and micro-strains, a high pressure experiment can reveal only those factors which contribute to the characteristic diffraction patterns of the crystalline phases present in the sample. The elastic properties of powders with the grain size from several nm to micrometers were examined using three methodologies: (l), the analysis of positions and widths of individual Bragg reflections (used for calculating macro- and micro-strains generated during densification) [I], (2). the analysis of the dependence of the experimental apparent lattice parameter, alp, on the diffraction vector Q [2], and (3), the atomic Pair Distribution Function (PDF) technique [3]. The results of our studies show, that Sic nanocrystals have the features of two phases, each with its distinct elastic properties. and under pressures up to 8 GPa.

Palosz, B.

DXRD : a user-friendly suite of two- and multiple-beam dynamical X-ray diffraction programs

The DXRD program suite consisting of a series of dynamical theory programs is introduced for computing dynamical X-ray diffraction from single crystals. Its interactive graphical user interfaces (GUIs) allow general users to make complicated calculations with minimal effort. It can calculate plane-wave Darwin curves of single crystals (or multiple crystals) for both the Bragg and Laue cases, including grazing-incidence diffraction and backward diffraction (with Bragg angles approaching 90°). It is also capable of simulating rocking curves for divergent incident X-ray beams with finite bandwidths. A unique feature of DXRD is that it provides a convenient GUI-based multiple-beam diffraction program that can accurately compute arbitrary N-beam diffraction of any geometry using a universal 4N × 4N matrix method. DXRD also provides a mapping program for plotting all the multiple-beam diffraction lines (monochromator glitches) in the azimuth–energy coordinate system. All these functions make DXRD a convenient and powerful software tool for designing crystal-based synchrotron/X-ray optics (monochromators, analyzers, polarizers, phase plates etc.) and for crystal characterization, X-ray spectroscopy and X-ray diffraction teaching.

Bragg reflection

Deconvoluting thermomechanical effects in X-ray diffraction data using machine learning

X-ray diffraction is ideal for probing the sub-surface state during complex or rapid thermomechanical loading of crystalline materials. However, challenges arise as the size of diffraction volumes increases due to spatial broadening and because of the inability to deconvolute the effects of different lattice deformation mechanisms. Here, we present a novel approach that uses combinations of physics-based modeling and machine learning to deconvolve thermal and mechanical elastic strains for diffraction data analysis. The method builds on a previous effort to extract thermal strain distribution information from diffraction data. The new approach is applied to extract the evolution of the thermomechanical state during laser melting of an Inconel 625 wall specimen which produces significant residual stress upon cooling. A combination of heat transfer and fluid flow, elasto-plasticity and X-ray diffraction simulations is used to generate training data for machine-learning (Gaussian process regression, GPR) models that map diffracted intensity distributions to underlying thermomechanical strain fields. First-principles density functional theory is used to determine accurate temperature-dependent thermal expansion and elastic stiffness used for elasto-plasticity modeling. The trained GPR models are found to be capable of deconvoluting the effects of thermal and mechanical strains, in addition to providing information about underlying strain distributions, even from complex diffraction patterns with irregularly shaped peaks.

36 MATERIALS SCIENCE

Large-Angle Rocking Beam Electron Diffraction of Large Unit Cell Crystals Using Direct Electron Detector

We report a large-angle rocking beam electron diffraction (LARBED) technique for electron diffraction analysis. Diffraction patterns are recorded in a scanning transmission electron microscope (STEM) using a direct electron detector with large dynamical range and fast readout. We use a nanobeam for diffraction and perform the beam double rocking by synchronizing the detector with the STEM scan coils for the recording. Using this approach, large-angle convergent beam electron diffraction (LACBED) patterns of different reflections are obtained simultaneously. By using a nanobeam, instead of a focused beam, the LARBED technique can be applied to beam-sensitive crystals as well as crystals with large unit cells. Here, this paper describes the implementation of LARBED and evaluates the performance using silicon and gadolinium gallium garnet crystals as test samples. We demonstrate that our method provides an effective and robust way for recording LARBED patterns and paves the way for quantitative electron diffraction of large unit cell and beam-sensitive crystals.

4DSTEM

Effective field theory factorization for diffraction

We derive a factorization formula for coherent and incoherent ep diffraction using the soft collinear effective theory, utilizing multiple power expansion parameters to handle different kinematic regions. This goes beyond the known hard-collinear diffractive factorization to address the small-x Regge dynamics and Pomeron exchange from first principles. The effective field theory analysis also uncovers and factorizes an important irreducible incoherent background generated by color-nonsinglet exchange, dubbed “quasi-diffraction”, for which we calculate the associated Sudakov suppression. For unpolarized scattering we show that there are four diffractive structure functions at leading power, and point out the importance of studying ${F}_{3,4}^D$ through asymmetries, in addition to ${F}_{2,L}^D$. For the quasi-diffractive background, we make model independent predictions for ratios of the corresponding structure functions in a perturbative kinematic region. Our analysis also makes predictions for six leading-power spin-dependent structure functions. Finally, we provide connections to diffractive parton distributions, and assess the Ingelman-Schlein model. Our work lays a path for further QCD-based studies of diffraction.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

Optimizing structured surfaces for diffractive waveguides

We introduce universal diffractive waveguide designs that can match the performance of conventional dielectric waveguides and achieve various functionalities. Optimized using deep learning, diffractive waveguides can be cascaded to form any desired length and are comprised of transmissive diffractive surfaces that permit the propagation of desired modes with low loss and high mode purity. In addition to guiding the targeted modes through cascaded diffractive units, we also developed various waveguide components and introduced bent diffractive waveguides, rotating the direction of mode propagation, as well as spatial and spectral mode filtering and mode splitting diffractive waveguide designs, and mode-specific polarization control. This framework was experimentally validated in the terahertz spectrum to selectively pass certain spatial modes while rejecting others. Without the need for material dispersion engineering diffractive waveguides can be scaled to operate at different wavelengths, including visible and infrared spectrum, covering potential applications in, e.g., telecommunications, imaging, sensing and spectroscopy.

Applied optics

Developing time-resolved x-ray diffraction diagnostics at the National Ignition Facility (invited)

As part of a program to measure phase transition timescales in materials under dynamic compression, we have designed new x-ray imaging diagnostics to record multiple x-ray diffraction measurements during a single laser-driven experiment. Our design places several ns-gated hybrid CMOS (hCMOS) sensors within a few cm of a laser-driven target. The sensors must be protected from an extremely harsh environment, including debris, electromagnetic pulses, and unconverted laser light. Another key challenge is reducing the x-ray background relative to the faint diffraction signal. Building on the success of our predecessor (Target Diffraction In Situ), we implemented a staged approach to platform development. First, we built a demonstration diagnostic (Gated Diffraction Development Diagnostic) with two hCMOS sensors to confirm we could adequately protect them from the harsh environment and also acquire acceptable diffraction data. This allowed the team to quickly assess the risks and address the most significant challenges. Here, we also collected scientifically useful data during development. Leveraging what we learned, we recently developed a much more ambitious instrument (Flexible Imaging Diffraction Diagnostic for Laser Experiments) that can field up to eight hCMOS sensors in a flexible geometry and participate in back-to-back shots at the National Ignition Facility (NIF). The design also allows for future iterations, such as faster hCMOS sensors and an embedded x-ray streak camera. The enhanced capabilities of the new instrument required a much more complex design, and the unexpected issues encountered on the first few shots at NIF remind us that complexity has consequences. Our progress in addressing these challenges is described herein, as is our current focus on improving data quality by reducing x-ray background and quantifying the uncertainties of our diffraction measurements.

36 MATERIALS SCIENCE

Surface wave diffraction by a truncated dielectric slab recessed in a perfectly conducting surface

The diffraction of a TM sub 0 surface wave by a terminated dielectric slab which is flush mounted in a perfectly conducting surface is studied. The incident surface wave gives rise to waves reflected and diffracted by the termination; these reflected and diffracted fields may be expressed in terms of the geometrical theory of diffraction by introducing surface wave reflection and diffraction coefficients which are associated with the termination. In this investigation the surface wave reflection and diffraction coefficients have been deduced from a formally exact solution to this canonical problem. The solution is obtained via a combination of the generalized scattering matrix technique and the Wiener-Hopf procedure. Expressions for the reflection and diffraction coefficients contain integrals which can be evaluated numerically.

Pathak, P. H.

Radar cross section of an open-ended circular waveguide Calculation of second-order diffraction terms

The radar cross section of an open-ended, semi-infinite circular waveguide is calculated by the geometrical theory of diffraction and the equivalent-current (EC) method. Both single- and double-diffraction terms are included. It is found that the double-diffraction term is stronger than the single-diffraction term for the horizontal polarization at wide-angle incidence. Its inclusion is necessary in order to check with experimental data and the asymptotic expansion of the exact Wiener-Hopf solution. The EC method is used for the single diffraction. On the other hand the double-diffraction term is obtained by using the EC method for axial incidence and geometrical theory of diffraction for wide-angle incidence. These two solutions are matched with a proper matching function. This technique is computationally more efficient than the matching technique of integration over the equivalent currents.

Lee, C. S.

Wavelength-division multiplexed optical integrated circuit with vertical diffraction grating

A semiconductor optical integrated circuit for wave division multiplexing has a semiconductor waveguide layer, a succession of diffraction grating points in the waveguide layer along a predetermined diffraction grating contour, a semiconductor diode array in the waveguide layer having plural optical ports facing the succession of diffraction grating points along a first direction, respective semiconductor diodes in the array corresponding to respective ones of a predetermined succession of wavelengths, an optical fiber having one end thereof terminated at the waveguide layer, the one end of the optical fiber facing the succession of diffraction grating points along a second direction, wherein the diffraction grating points are spatially distributed along the predetermined contour in such a manner that the succession of diffraction grating points diffracts light of respective ones of the succession of wavelengths between the one end of the optical fiber and corresponding ones of the optical ports.

Lang, Robert J.

MANN: A program to transfer designs for diffractive optical elements to a MANN photolithographic mask generator

There are two basic areas of interest for diffractive optics. In the first, the property of wavefront division is exploited for achieving optical fanout, analogous to the more familiar electrical fanout of electronic circuitry. The basic problem here is that when using a simple uniform diffraction grating the energy input is divided unevenly among the output beams. The other area of interest is the use of diffractive elements to replace or supplement standard refractive elements such as lenses. Again, local grating variations can be used to control the amount of bending imparted to optical rays, and the efficiency of the diffractive element will depend on how closely the element can be matched to the design requirements. In general, production restrictions limit how closely the element approaches the design, and for the common case of photolithographic production, a series of binary masks is required to achieve high efficiency. The actual design process is much more involved than in the case of elements for optical fanout, as the desired phase of the optical wavefront over some reference plane must be specified and the phase alteration to be introduced at each point by the diffraction element must be known. This generally requires the utilization of a standard optical design program. Two approaches are possible. In the first approach, the diffractive element is treated as a special type of lens and the ordinary optical design equations are used. Optical design programs tend to follow a second approach, namely, using the equations of optical interference derived from holographic theory and then allowing the introduction of phase front corrections in the form of polynomial equations. By using either of these two methods, diffractive elements can be used not only to compensate for distortions such as chromatic or spherical aberration, but also to perform the work of a variety of other optical elements such as null correctors, beam shapers, etc. The main focus of the project described in this report is how the design information from the lens design program is incorporated into the photolithographic process. It is shown that the MANN program, a photolithographic mask generator, fills the need for a link between lens design programs and mask generation controllers.The generated masks can be used to expose a resist-coated substrate which is etched and then must be re-coated, re-exposed, and re-etched for making copies, just as in the electronics industry.

Matthys, Donald R.

Transmissive Diffractive Optical Element Solar Concentrators

Solar-thermal-radiation concentrators in the form of transmissive diffractive optical elements (DOEs) have been proposed as alternatives to mirror-type solar concentrators now in use. In comparison with functionally equivalent mirror-type solar concentrators, the transmissive, diffractive solar concentrators would weigh and cost less, and would be subject to relaxed mechanical tolerances. A DOE concentrator would be made from a thin, flat disk or membrane of a transmissive material having a suitable index of refraction. By virtue of its thinness, the DOE concentrator would have an areal mass density significantly less than that of a functionally equivalent conventional mirror. The DOE concentrator would have a relatively wide aperture--characterized by a focal-length/aperture-diameter ratio ('f number') on the order of 1. A kinoform (a surface-relief phase hologram) of high diffractive order would be microfabricated onto one face of the disk. The kinoform (see figure) would be designed to both diffract and refract incident solar radiation onto a desired focal region, without concern for forming an image of the Sun. The high diffractive order of this kinoform (in contradistinction to the low diffractive orders of some other kinoforms) would be necessary to obtain the desired f number of 1, which, in turn, would be necessary for obtaining a desired concentration ratio of 2,500 or greater. The design process of optimizing the concentration ratio of a proposed DOE solar concentrator includes computing convolutions of the optical bandwidth of the Sun with the optical transmission of the diffractive medium. Because, as in the cases of other non-imaging, light-concentrating optics, image quality is not a design requirement, the process also includes trading image quality against concentration ratio. A baseline design for one example calls for an aperture diameter of 1 m. This baseline design would be scalable to a diameter as large as 10 m, or to a smaller diameter for a laboratory test article. Initial calculations have indicated that the characteristics of the test article would be readily scalable to a full-size unit.

Baron, Richard

Suppressing Ghost Diffraction in E-Beam-Written Gratings

A modified scheme for electron-beam (E-beam) writing used in the fabrication of convex or concave diffraction gratings makes it possible to suppress the ghost diffraction heretofore exhibited by such gratings. Ghost diffraction is a spurious component of diffraction caused by a spurious component of grating periodicity as described below. The ghost diffraction orders appear between the main diffraction orders and are typically more intense than is the diffuse scattering from the grating. At such high intensity, ghost diffraction is the dominant source of degradation of grating performance. The pattern of a convex or concave grating is established by electron-beam writing in a resist material coating a substrate that has the desired convex or concave shape. Unfortunately, as a result of the characteristics of electrostatic deflectors used to control the electron beam, it is possible to expose only a small field - typically between 0.5 and 1.0 mm wide - at a given fixed position of the electron gun relative to the substrate. To make a grating larger than the field size, it is necessary to move the substrate to make it possible to write fields centered at different positions, so that the larger area is synthesized by "stitching" the exposed fields.

Wilson, Daniel

Amorphous Phase Characterization Through X-Ray Diffraction Profile Modeling: Implications for Amorphous Phases in Gale Crater Rocks and Soils

The CheMin X-ray diffraction instrument on the Mars Science Laboratory rover has analyzed 18 rock and soil samples in Gale crater. Diffraction data allow for the identification of major crystalline phases based on the positions and intensities of well-defined peaks and also provides information regarding amorphous and poorly-ordered materials based on the shape and positions of broad scattering humps. The combination of diffraction data, elemental chemistry from APXS (Alpha Particle X-ray Spectrometer) and evolved gas analyses (EGA) from SAM (Sample Analysis at Mars) help constrain possible amorphous materials present in each sample (e.g., glass, opal, iron oxides, sulfates) but are model dependent. We present a novel method to characterize amorphous material in diffraction data and, through this approach, aim to characterize the phases collectively producing the amorphous profiles in CheMin diffraction data. This method may be applied to any diffraction data from samples containing X-ray amorphous materials, not just CheMin datasets, but we re-strict our discussion to Martian-relevant amorphous phases and diffraction data measured by CheMin or CheMin-like instruments.

Achilles, C. N.