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NASA NTRS2000

Freely Suspended Smectic Filaments and the Structure of the B7 Phase of MHOBOW

Our recent discovery of the spontaneous formation of chiral domains in fluid smectic phases of achiral bow-shaped molecules opens up a wide variety of possibilities for new liquid crystal phases and phenomena. The basic, spontaneously chiral layer structure of the highest temperature fluid smectic phases, the B2 and B7, are shown. One of the most intriguing aspects of this structure is the plethora of possible phases coming from different stacking sequences of the polar ordering and tilt directions. The four possibilities of next-nearest neighbor alternation are shown. In the original material studied, NOBOW, the ground states found are antiferroelectric, either the racemic SmC(sub S)P(sub A) or the chiral SmC(sub A)P(sub A). We are currently studying MHOBOW, synthesized by D. Walba which, by virtue of its methyl hexyloxy tail has a tendency to form anticlinic layer interfaces, in the hope of finding a phase with a ferroelectric ground state, either SmC(sub A)P(sub S) or SmC(sub S)P(sub A), which can be obtained in NOBOW only by applying a field. Preliminary observations of MHO-BOW have made its study, from the point of view of understanding novel LC structures, extremely high priority. The following truly remarkable characteristics have been revealed: (i) The smectic phase grows out of the isotropic in the form of helical ribbons. The resulting planar aligned textures of focal conics with layers normal to glass plates exhibit bizarre modulations, including stripes and checker-boards. These have also been seen in other materials suggesting that this is a new phase (tentatively called B7), which is a fluid smectic with some kind of in-layer structure. (ii) It is virtually impossible to make freely suspended films of MHOBOW. Rather it makes the freely suspended filaments which preliminary x-ray scattering experiments reveal to have the nested cylinder layer structure indicated; (iii) The powder x-ray diffraction exhibits four resolution-limited smectic layering peaks, very close in layer spacing, which vary continuously with T. This is further evidence for a more complex three dimensional structure than NOBOW, which has a typical single layering reflection. (iv) The x-ray structure factor of the layering peak of the filaments is extraordinarily complex and rich. Varying in qL (the scattering vector component along the filament axis) from a double slit-like pattern to modulated layer-like patterns, as qH (the scattering vector component normal to the filament axis) is varied over the range where the four powder peaks are located. These results suggest some kind of mosaic structure, perhaps with different layer spacings corresponding to the different stacking sequences. Recent x-ray diffraction experiments show that the peaks are modulated in intensity upon translation along a filament, in domains of several hundred microns dimension. These preliminary experiments suggest that the B7 is a fluid smectic with extremely unusual and fascinating structures. Of all of the many hundreds of fluid smectic materials we have attempted to study in the freely suspended film geometry over the years, only a few have failed to form films, and none showed any great tendency to form filaments, although this clearly should be a possible freely suspended smectic LC morphology. On several occasions in the past we have intentionally tried to make filaments from a variety of smectics without success. Thus the smectic filament formation property makes the B7 phase unique. It seems quite likely that the stability of filaments is related to the in-plane structure. The filaments exhibit other interesting structural and optical features. They are birefringent with a local optic axis which is oblique and which can vary continuously along filament and which can be manipulated with an electric field applied normal to the fiber, as if the field were causing a rotation of the optic axis about the fiber axis. Rapid displacement of the ends of the fiber toward one another causes a macroscopic helixing at low T and causes thick regions to transiently appear at high T, a 1D analog of island formation on a rapidly compressed film.

Clark, N.↗
NASA NTRS2001

Characterization of a Multilayered Dielectric Transmissive Phase Modulator

We describe a multilayered dielectric stack configuration designed specifically for use as a transmissive phase modulator for broadband optical signals. Applications for this device range from full aperture wavefront correction to nonmechanical beam steering arrays for free space optical communication links. Our implementation employs alternating GaAs and AlAs layers of varying thickness on a GaAs substrate to create a bandpass region of high average transmission centered about the one micrometer wavelength. Within this transmission bandpass, the phase component of the complex transmission coefficient varies in a near-linear fashion with respect to wavelength. The transmission bandpass is designed to have a bandwidth of 21.0 nm (or 6.3THz frequency bandwidth) and to have an edge-to-edge phase change of greater than 47T radians. Modification of the stack materials' optical properties causes the transmission profile to shift spectrally, resulting in a phase modulation for bands of transmitted frequencies. Our broadband phase modulator imparts up to a full-cycle of phase modulation with low loss and low group velocity dispersion. We identify several methods for implementing the requisite modulation, including refractive index modulation through free carrier injection and optical path length modulation through variation in angle of incidence. At least one sample comprising 91 alternating layers has been fabricated to exhibit the bandpass properties required for optical signal phase modulation. We experimentally characterize the sample using an interferometer and spectrometer to measure the transmitted signal spectrum and relative phase modulation. We compare the experimental data to computational predictions and discuss the results.

Keys, Andrew S.↗
NASA NTRS2003

GPS-Like Phasing Control of the Space Solar Power System Transmission Array

The problem of phasing of the Space Solar Power System's transmission array has been addressed by developing a GPS-like radio navigation system. The goal of this system is to provide power transmission phasing control for each node of the array that causes the power signals to add constructively at the ground reception station. The phasing control system operates in a distributed manner, which makes it practical to implement. A leader node and two radio navigation beacons are used to control the power transmission phasing of multiple follower nodes. The necessary one-way communications to the follower nodes are implemented using the RF beacon signals. The phasing control system uses differential carrier phase relative navigation/timing techniques. A special feature of the system is an integer ambiguity resolution procedure that periodically resolves carrier phase cycle count ambiguities via encoding of pseudo-random number codes on the power transmission signals. The system is capable of achieving phasing accuracies on the order of 3 mm down to 0.4 mm depending on whether the radio navigation beacons operate in the L or C bands.

Psiaki, Mark L.↗
NASA NTRS2002

Designing Phase 2 for the Double-Lunar Swingby of the Magnetospheric Multiscale Mission (MMS)

The Magnetospheric Multiscale (MMS) mission is a formation flying mission that consists of four distinct phases: phases 1 and 2 are low-inclination highly elliptical orbits (HEO), phase 3 is a double-lunar swingby which transfers phase 2 to phase 4, a high-inclination HE0 orbit. Phase 2 is designed to reach the first lunar swingby of phase 3 in the most efficient fashion. It is shown that when the orientation of the line of apsides of phase 2 is properly chosen, no extra Delta-V is required beyond what is typically needed to raise apogee to lunar distance.

Edery, Ariel↗
NASA NTRS2003

Genetic Algorithm Phase Retrieval for the Systematic Image-Based Optical Alignment Testbed

Phase retrieval requires calculation of the real-valued phase of the pupil fimction from the image intensity distribution and characteristics of an optical system. Genetic 'algorithms were used to solve two one-dimensional phase retrieval problem. A GA successfully estimated the coefficients of a polynomial expansion of the phase when the number of coefficients was correctly specified. A GA also successfully estimated the multiple p h e s of a segmented optical system analogous to the seven-mirror Systematic Image-Based Optical Alignment (SIBOA) testbed located at NASA s Marshall Space Flight Center. The SIBOA testbed was developed to investigate phase retrieval techniques. Tiphilt and piston motions of the mirrors accomplish phase corrections. A constant phase over each mirror can be achieved by an independent tip/tilt correction: the phase Conection term can then be factored out of the Discrete Fourier Tranform (DFT), greatly reducing computations.

Taylor, Jaime↗
NASA NTRS1997

Kinematic principles of primate rotational vestibulo-ocular reflex. I. Spatial organization of fast phase velocity axes

The spatial organization of fast phase velocity vectors of the vestibulo-ocular reflex (VOR) was studied in rhesus monkeys during yaw rotations about an earth-horizontal axis that changed continuously the orientation of the head relative to gravity ("barbecue spit" rotation). In addition to a velocity component parallel to the rotation axis, fast phases also exhibited a velocity component that invariably was oriented along the momentary direction of gravity. As the head rotated through supine and prone positions, torsional components of fast phase velocity axes became prominent. Similarly, as the head rotated through left and right ear-down positions, fast phase velocity axes exhibited prominent vertical components. The larger the speed of head rotation the greater the magnitude of this fast phase component, which was collinear with gravity. The main sequence properties of VOR fast phases were independent of head position. However, peak amplitude as well as peak velocity of fast phases were both modulated as a function of head orientation, exhibiting a minimum in prone position. The results suggest that the fast phases of vestibulo-ocular reflexes not only redirect gaze and reposition the eye in the direction of head motion but also reorient the eye with respect to earth-vertical when the head moves relative to gravity. As further elaborated in the companion paper, the underlying mechanism could be described as a dynamic, gravity-dependent modulation of the coordinates of ocular rotations relative to the head.

Non-NASA Center↗
NASA NTRS2004

Los Angeles International Airport Runway Incursion Studies: Phase III--Center-Taxiway Simulation

Phase III of the Los Angeles International Airport Runway Incursion Studies was conducted, under an agreement with HNTB Corporation, at the NASA Ames FutureFlight Central (FFC) facility in June 2003. The objective of the study was the evaluation of a new center-taxiway concept at LAX. This study is an extension of the Phase I and Phase II studies previously conducted at FFC. This report presents results from Phase III of the study, in which a center-taxiway concept between runways 25L and 25R was simulated and evaluated. Phase III data were compared objectively against the Baseline data. Subjective evaluations by participating LAX controllers were obtained with regard to workload, efficiency, and safety criteria. To facilitate a valid comparison between Baseline and Phase III data, the same scenarios were used for Phase III that were tested during Phases I and II. This required briefing participating controllers on differences in airport and airline operations between 2001 and today.

Madson, Michael D.↗
NASA NTRS1998

Apparatus and method for measuring relative phase of signals in a multiple-echo system

An apparatus and method for measuring the relative phase of echo signals in a multiple-echo system. A signal generator generates an exciting tone burst and subsequent tone bursts delayed in phase from the exciting tone burst. The exciting tone burst is gated into a transducer coupled to the multiple-echo system. Each subsequent tone burst is converted into a series of pulses. Each pulse triggers a sample/hold circuit to sample each echo signal. The samples of the echo signal are averaged and then compared to a reference value. The signal generator is controlled to vary the subsequent tone burst phase delay to determine for each echo signal a subsequent tone burst phase delay that causes the average of the samples of the echo signal to be substantially equal to the reference value wherein the determined subsequent tone burst phase delay is the phase delay of the echo signal with respect to the exciting tone burst. The control circuit then determines the difference in phase delay between echo signals to determine the relative phase delay of the echo signals.

Froggatt, Mark E.↗
NASA NTRS2006

The Phase Curve Survey of the Irregular Saturnian Satellites: A Possible Method of Physical Classification

During its 2005 January opposition, the saturnian system could be viewed at an unusually low phase angle. We surveyed a subset of Saturn's irregular satellites to obtain their true opposition magnitudes, or nearly so, down to phase angle values of 0.01 deg. Combining our data taken at the Palomar 200-inch and Cerro Tololo Inter-American Observatory's 4-m Blanco telescope with those in the literature, we present the first phase curves for nearly half the irregular satellites originally reported by Gladman et al. [2001. Nature 412, 163-166], including Paaliaq (SXX), Siarnaq (SXXIX), Tarvos (SXXI), Ijiraq (SXXII), Albiorix (SXVI), and additionally Phoebe's narrowest angle brightness measured to date. We find centaur-like steepness in the phase curves or opposition surges in most cases with the notable exception of three, Albiorix and Tarvos, which are suspected to be of similar origin based on dynamical arguments, and Siarnaq.During its 2005 January opposition, the saturnian system could be viewed at an unusually low phase angle. We surveyed a subset of Saturn's irregular satellites to obtain their true opposition magnitudes, or nearly so, down to phase angle values of 0.01 deg. Combining our data taken at the Palomar 200-inch and Cerro Tololo Inter-American Observatory's 4-m Blanco telescope with those in the literature, we present the first phase curves for nearly half the irregular satellites originally reported by Gladman et al. [2001. Nature 412, 163-166], including Paaliaq (SXX), Siarnaq (SXXIX), Tarvos (SXXI), Ijiraq (SXXII), Albiorix (SXVI), and additionally Phoebe's narrowest angle brightness measured to date. We find centaur-like steepness in the phase curves or opposition surges in most cases with the notable exception of three, Albiorix and Tarvos, which are suspected to be of similar origin based on dynamical arguments, and Siarnaq.

satellites↗
NASA NTRS2007

Magnetic Field Observations of Partial Ring Current during Storm Recovery Phase

We present results of an extensive survey of the magnetic field observations in the inner magnetosphere using 30 years of magnetospheric magnetic field data from Polar, Cluster, ISEE, and AMPTE/CCE missions. The purpose of this study is to understand the magnetic field evolution during the recovery phase of geomagnetic storms, and its implication to the ring current recovery and loss mechanisms of ring current particles. Our previous work on global ring current distribution [Le et al., 2004] has shown that a significant partial ring current is always present at all Dst levels (regardless of storm phases) even for quiet time ring current. The total current carried by the partial ring current is much stronger than (during stormtime) or at least comparable to (during quiet time) the symmetric ring current. It is now commonly believed that a strong partial ring current is formed during the storm main phase due to the enhanced earthward convection of energetic ions from nightside plasma sheet. But the presence of a strong partial ring current throughout the recovery phase remains controversial. The magnetic field generated by the ring current inflates the inner magnetosphere and causes magnetic field depressions in the equatorial magnetosphere. During the storm recovery phase, we find that the distribution of the equatorial magnetic field depression exhibits similar local time dependence as the ring current distribution obtained from the combined dataset in the earlier study. It shows that a strong partial ring current is a permanent feature throughout the recovery phase. In the early recovery phase, the partial ring current peaks near the dusk terminator as indicated by the peak of the magnetic field depression. As the recovery phase progresses, the partial ring current decays most quickly near the dusk and results in a dusk-to-midnight moving of the peak of the partial ring current. Thus the loss mechanisms work most effectively near the dusk. The magnetic field depression increases the gyroradius of ring current protons to a scale greater or comparable to the thickness of the magnetopause, which increases the chance of ion drift loss near the dusk magnetopause at larger L-shell (L>5). But the drift loss mechanism alone cannot explain the loss of ring current ions especially in the smaller L-shell (L<5). The precipitation loss due to wave-particle interaction is most likely the dominant loss mechanism in the small L-shell as it works most effectively at the same local time.

Le, Guan↗
NASA NTRS2008

L-Band Transmit/Receive Module for Phase-Stable Array Antennas

Interferometric synthetic aperture radar (InSAR) has been shown to provide very sensitive measurements of surface deformation and displacement on the order of 1 cm. Future systematic measurements of surface deformation will require this capability over very large areas (300 km) from space. To achieve these required accuracies, these spaceborne sensors must exhibit low temporal decorrelation and be temporally stable systems. An L-band (24-cmwavelength) InSAR instrument using an electronically steerable radar antenna is suited to meet these needs. In order to achieve the 1-cm displacement accuracy, the phased array antenna requires phase-stable transmit/receive (T/R) modules. The T/R module operates at L-band (1.24 GHz) and has less than 1- deg absolute phase stability and less than 0.1-dB absolute amplitude stability over temperature. The T/R module is also high power (30 W) and power efficient (60-percent overall efficiency). The design is currently implemented using discrete components and surface mount technology. The basic T/R module architecture is augmented with a calibration loop to compensate for temperature variations, component variations, and path loss variations as a function of beam settings. The calibration circuit consists of an amplitude and phase detector, and other control circuitry, to compare the measured gain and phase to a reference signal and uses this signal to control a precision analog phase shifter and analog attenuator. An architecture was developed to allow for the module to be bidirectional, to operate in both transmit and receive mode. The architecture also includes a power detector used to maintain a transmitter power output constant within 0.1 dB. The use of a simple, stable, low-cost, and high-accuracy gain and phase detector made by Analog Devices (AD8302), combined with a very-high efficiency T/R module, is novel. While a self-calibrating T/R module capability has been sought for years, a practical and cost-effective solution has never been demonstrated. By adding the calibration loop to an existing high-efficiency T/R module, there is a demonstrated order-of-magnitude improvement in the amplitude and phase stability.

Andricos, Constantine↗
NASA NTRS2008

Role of Hf on Phase Formation in Ti45Zr(38-x)Hf(x)Ni17 Liquids and Solids

Hafnium and zirconium are very similar, with almost identical sizes and chemical bonding characteristics. However, they behave differently when alloyed with Ti and Ni. A sharp phase formation boundary near 18-21 at.% Hf is observed in rapidly-quenched and as-cast Ti45Zr38-xHfxNi17 alloys. Rapidly-quenched samples that contain less than 18 at.% Hf form the icosahedral quasicrystal phase, whiles samples containing more than 21 at.% form the 3/2 rational approximant phase. In cast alloys, a C14 structure is observed for alloys with Hf lower than the boundary concentration, while a large-cell (11.93 ) FCC Ti2Ni-type structure is found in alloys with Hf concentrations above the boundary. To better understand the role of Hf on phase formation, the structural evolution with supercooling and the solidification behavior of liquid Ti45Zr38-xHfxNi17 alloys (x=0, 12, 18, 21, 38) were studied using the Beamline Electrostatic Levitation (BESL) technique using 125keV x-rays on the 6ID-D beamline at the Advanced Photon Source, Argonne National Laboratory. For all liquids primary crystallization was to a BCC solid solution phase; interestly, an increase in Hf concentration leads to a decrease in the BCC lattice parameter in spite of the chemical similarity between Zr and Hf. A Reitveld analysis confirmed that as in the cast alloys, the secondary phase that formed was the C14 below the phase formation boundary and a Ti2Ni-type structure at higher Hf concentrations. Both the liquidus temperature and the reduced undercooling change sharply on traversing the phase formation boundary concentration, suggesting a change in the liquid structure. Structural information from a Honeycutt-Anderson index analysis of reverse Monte Carlo fits to the S(q) liquid data will be presented to address this issue.

Wessels, V.↗
NASA NTRS2007

Compensation for Phase Anisotropy of a Metal Reflector

A method of compensation for the polarization- dependent phase anisotropy of a metal reflector has been proposed. The essence of the method is to coat the reflector with multiple thin alternating layers of two dielectrics that have different indices of refraction, so as to introduce an opposing polarization-dependent phase anisotropy. The anisotropy in question is a phenomenon that occurs in reflection of light at other than normal incidence: For a given plane wave having components polarized parallel (p) and perpendicular (s) to the plane of incidence, the phase of s-polarized reflected light differs from the phase p-polarized light by an amount that depends on the angle of incidence and the complex index of refraction of the metal. The magnitude of the phase difference is zero at zero angle of incidence (normal incidence) and increases with the angle of incidence. This anisotropy is analogous to a phase anisotropy that occurs in propagation of light through a uniaxial dielectric crystal. In such a case, another uniaxial crystal that has the same orientation but opposite birefringence can be used to cancel the phase anisotropy. Although it would be difficult to prepare a birefringent material in a form suitable for application to the curved surface of a typical metal reflector in an optical instrument, it should be possible to effect the desired cancellation of phase anisotropy by exploiting the form birefringence of multiple thin dielectric layers. (The term "form birefringence" can be defined loosely as birefringence arising, in part, from a regular array of alternating subwavelength regions having different indices of refraction.)

Hong, John↗
NASA NTRS2007

Calibration Test Set for a Phase-Comparison Digital Tracker

An apparatus that generates four signals at a frequency of 7.1 GHz having precisely controlled relative phases and equal amplitudes has been designed and built. This apparatus is intended mainly for use in computer-controlled automated calibration and testing of a phase-comparison digital tracker (PCDT) that measures the relative phases of replicas of the same X-band signal received by four antenna elements in an array. (The relative direction of incidence of the signal on the array is then computed from the relative phases.) The present apparatus can also be used to generate precisely phased signals for steering a beam transmitted from a phased antenna array. The apparatus (see figure) includes a 7.1-GHz signal generator, the output of which is fed to a four-way splitter. Each of the four splitter outputs is attenuated by 10 dB and fed as input to a vector modulator, wherein DC bias voltages are used to control the in-phase (I) and quadrature (Q) signal components. The bias voltages are generated by digital-to-analog- converter circuits on a control board that receives its digital control input from a computer running a LabVIEW program. The outputs of the vector modulators are further attenuated by 10 dB, then presented at high-grade radio-frequency connectors. The attenuation reduces the effects of changing mismatch and reflections. The apparatus was calibrated in a process in which the bias voltages were first stepped through all possible IQ settings. Then in a reverse interpolation performed by use of MATLAB software, a lookup table containing 3,600 IQ settings, representing equal amplitude and phase increments of 0.1 , was created for each vector modulator. During operation of the apparatus, these lookup tables are used in calibrating the PCDT.

Boas, Amy↗
NASA NTRS2010

Objective Lightning Probability Forecasting for Kennedy Space Center and Cape Canaveral Air Force Station, Phase III

The AMU created new logistic regression equations in an effort to increase the skill of the Objective Lightning Forecast Tool developed in Phase II (Lambert 2007). One equation was created for each of five sub-seasons based on the daily lightning climatology instead of by month as was done in Phase II. The assumption was that these equations would capture the physical attributes that contribute to thunderstorm formation more so than monthly equations. However, the SS values in Section 5.3.2 showed that the Phase III equations had worse skill than the Phase II equations and, therefore, will not be transitioned into operations. The current Objective Lightning Forecast Tool developed in Phase II will continue to be used operationally in MIDDS. Three warm seasons were added to the Phase II dataset to increase the POR from 17 to 20 years (1989-2008), and data for October were included since the daily climatology showed lightning occurrence extending into that month. None of the three methods tested to determine the start of the subseason in each individual year were able to discern the start dates with consistent accuracy. Therefore, the start dates were determined by the daily climatology shown in Figure 10 and were the same in every year. The procedures used to create the predictors and develop the equations were identical to those in Phase II. The equations were made up of one to three predictors. TI and the flow regime probabilities were the top predictors followed by 1-day persistence, then VT and Ll. Each equation outperformed four other forecast methods by 7-57% using the verification dataset, but the new equations were outperformed by the Phase II equations in every sub-season. The reason for the degradation may be due to the fact that the same sub-season start dates were used in every year. It is likely there was overlap of sub-season days at the beginning and end of each defined sub-season in each individual year, which could very well affect equation performance.

Crawford, Winifred C.↗
NASA NTRS2011

On the Importance of the Flare's Late Phase for the Solar Extreme Ultraviolet Irradiance

The new solar extreme ultraviolet (EUV) irradiance observations from NASA Solar Dynamics Observatory (SDO) have revealed a new class of solar flares that are referred to as late phase flares. These flares are characterized by the hot 2-5 MK coronal emissions (e.g., Fe XVI 33.5 nm) showing large secondary peaks that appear many minutes to hours after an eruptive flare event. In contrast, the cool 0.7-1.5 MK coronal emissions (e.g., Fe IX 17.1 nm) usually dim immediately after the flare onset and do not recover until after the delayed second peak of the hot coronal emissions. We refer to this period of 1-5 hours after the fl amrea sin phase as the late phase, and this late phase is uniquely different than long duration flares associated with 2-ribbon flares or large filament eruptions. Our analysis of the late phase flare events indicates that the late phase involves hot coronal loops near the flaring region, not directly related to the original flaring loop system but rather with the higher post-eruption fields. Another finding is that space weather applications concerning Earth s ionosphere and thermosphere need to consider these late phase flares because they can enhance the total EUV irradiance flare variation by a factor of 2 when the late phase contribution is included.

Woods, Thomas N.↗
NASA NTRS2011

Testing and Calibration of Phase Plates for JWST Optical Simulator

Three phase plates were designed to simulate the JWST segmented primary mirror wavefront at three on-orbit alignment stages: coarse phasing, intermediate phasing, and fine phasing. The purpose is to verify JWST's on-orbit wavefront sensing capability. Amongst the three stages, coarse alignment is defined to have piston error between adjacent segments being 30 m to 300 m, intermediate being 0.4 m to 10 m, and fine is below 0.4 m. The phase plates were made of fused silica, and were assembled in JWST Optical Simulator (OSIM). The piston difference was realized by the thickness difference of two adjacent segments. The two important parameters to phase plates are piston and wavefront errors. Dispersed Fringe Sensor (DFS) method was used for initial coarse piston evaluation, which is the emphasis of this paper. Point Diffraction Interferometer (PDI) is used for fine piston and wavefront error. In order to remove piston's 2 pi uncertainty with PDI, three laser wavelengths, 640nm, 660nm, and 780nm, are used for the measurement. The DHS test setup, analysis algorithm and results are presented. The phase plate design concept and its application (i.e. verifying the JWST on-orbit alignment algorithm) are described. The layout of JWST OSIM and the function of phase plates in OSIM are also addressed briefly.

Gong, Qian↗
NASA NTRS2012

Sub-pixel spatial resolution wavefront phase imaging

A phase imaging method for an optical wavefront acquires a plurality of phase images of the optical wavefront using a phase imager. Each phase image is unique and is shifted with respect to another of the phase images by a known/controlled amount that is less than the size of the phase imager's pixels. The phase images are then combined to generate a single high-spatial resolution phase image of the optical wavefront.

Stahl, H. Philip↗