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Articles 61 - 90 of 349
Full-Text Articles in Optics
Temperature-Immune Self-Referencing Fabry–Pérot Cavity Sensors, Hengky Chandrahalim, Jonathan W. Smith
Temperature-Immune Self-Referencing Fabry–Pérot Cavity Sensors, Hengky Chandrahalim, Jonathan W. Smith
AFIT Patents
A passive microscopic Fabry-Pérot Interferometer (FPI) sensor an optical fiber a three-dimensional microscopic optical structure formed on a cleaved tip of an optical fighter that reflects a light signal back through the optical fiber. The reflected light is altered by refractive index changes in the three-dimensional structure that is subject to at least one of: (i) thermal radiation; and (ii) volatile organic compounds.
Characterization Of Infrared Metasurface Optics With An Optical Scatterometer, Matthew R. Miller
Characterization Of Infrared Metasurface Optics With An Optical Scatterometer, Matthew R. Miller
Theses and Dissertations
An optical scatterometer is used to characterize the infrared scatter of a dielectric metasurface cylindrical lens and two variants of that design. The design uses dielectric nanopillars to create the parabolic phase delay required for lensing; the variants change the length of the nanopillars from the design length of 4 microns to 0.9 and 5.2 microns. Scatter measurements were made at the design wavelength of 4 microns, and at 3.39 and 5 microns. These measurements showed wide-angle scatter greater than that measured for a conventional refractive optic, and that these metasurfaces perform their optical function best at the design wavelength …
Uncertainty Analysis For Ccd-Augmented Casi® Brdf Measurement System, Todd V. Small, Samuel D. Butler, Michael A. Marciniak
Uncertainty Analysis For Ccd-Augmented Casi® Brdf Measurement System, Todd V. Small, Samuel D. Butler, Michael A. Marciniak
Faculty Publications
This work presents a measurement uncertainty analysis for a system designed to simultaneously capture specular in-plane and out-of-plane bidirectional reflectance distribution function (BRDF) data with high spatial resolution by augmenting the Complete Angle Scatter Instrument (CASI®) with a charge-coupled device (CCD) camera. Various scatter flux, incident flux, scatter angle, and detector solid angle uncertainty contributions are considered and evaluated based on imperfectly known system parameters. In particular, incident flux temporal fluctuation, detector noise and non-linearity, and out-of-plane aperture misalignment considerations each require significant adjustment from original CASI® uncertainty analysis, and expressions for neutral density (ND) filter, scatter angle, and solid …
Spatiotemporal Non-Uniformly Correlated Beams, Milo W. Hyde Iv
Spatiotemporal Non-Uniformly Correlated Beams, Milo W. Hyde Iv
Faculty Publications
We present a new partially coherent source with spatiotemporal coupling. The stochastic light, which we call a spatiotemporal (ST) non-uniformly correlated (NUC) beam, combines space and time in an inhomogeneous (shift- or space-variant) correlation function. This results in a source that self-focuses at a controllable location in space-time, making these beams potentially useful in applications such as optical trapping, optical tweezing, and particle manipulation. We begin by developing the mutual coherence function for an ST NUC beam. We then examine its free-space propagation characteristics by deriving an expression for the mean intensity at any plane z ≥ 0. To validate …
Method Of Making Temperature-Immune Self-Referencing Fabry–Pérot Cavity Sensors, Hengky Chandrahalim, Jonathan W. Smith
Method Of Making Temperature-Immune Self-Referencing Fabry–Pérot Cavity Sensors, Hengky Chandrahalim, Jonathan W. Smith
AFIT Patents
A method of making passive microscopic Fabry-Pérot Interferometer (FPI) sensor includes forming a three-dimensional microscopic optical structure on a cleaved tip of an optical fiber that reflects a light signal back through the optical fiber. The reflected light is altered by refractive index changes in the three-dimensional structure that is subject to at least one of: (i) thermal radiation; and (ii) volatile organic compounds.
Solar Cell Brdf Measurement And Modeling With Out-Of-Plane Data, Todd V. Small, Samuel D. Butler, Michael A. Marciniak
Solar Cell Brdf Measurement And Modeling With Out-Of-Plane Data, Todd V. Small, Samuel D. Butler, Michael A. Marciniak
Faculty Publications
In this work, a CCD-augmented complete angle scatter instrument (CASI) with a visible red laser source was used to measure the BRDF of a commercially available solar cell designed for small satellites, simultaneously capturing both in-plane and out-of-plane data with high angular resolution surrounding the specular direction. The measurements exhibited three distinct scatter features: a central specular peak, an offset specular peak, and a diffraction pattern. The two peaks were caused by different material surfaces with slightly different normal directions, and the diffraction pattern arose from periodically-spaced metal conducting bars running in one direction across the solar cell surface. The …
Analysis Of Space To Ground Ladar Performance With Non-Traditional Optics, Prayant P.S. Hanjra
Analysis Of Space To Ground Ladar Performance With Non-Traditional Optics, Prayant P.S. Hanjra
Theses and Dissertations
Two major obstacles to space-based LADAR systems are low power returns from targets and limitations on size and weight for transporting large optics into orbit. Signals incur significant losses during roundtrip propagation through the atmosphere and from diffuse scattering off of targets. Models, such as the Laser Environmental Effects Definition and Reference (LEEDR) simulator and High Energy Laser End to End Operational Simulation (HELEEOS) can predict these losses due to the atmosphere and optical components for a variety of atmospheric and environmental conditions across the globe. A transmissometer is used to validate these models. These losses are used to determine …
Improved Out-Of-Plane Brdf Measurement And Modeling, Todd V. Small
Improved Out-Of-Plane Brdf Measurement And Modeling, Todd V. Small
Theses and Dissertations
The bi-directional reflectance distribution function (BRDF) describes the directional (spatial) nature of light’s reflectance from a material surface. When incident light of a particular wavelength strikes a material surface from a particular direction, portions of that incident light will be reflected into various directions in various amounts, depending on the material’s surface characteristics. Historically, the vast majority of BRDF measurement and modeling research has focused on reflection within the plane-of incidence (in-plane) and dealt primarily with simplified isotropic BRDFs. Remote sensing applications, such as satellite light curve analysis, typically rely on closed-form microfacet models for efficiency. There are many factors, …
Data-Driven Algorithm To Classify The Degree Of Isotropy In The Bidirectional Reflectance Distribution Function, Anne W. Werkley, Samuel D. Butler, Todd V. Small, Michael A. Marciniak
Data-Driven Algorithm To Classify The Degree Of Isotropy In The Bidirectional Reflectance Distribution Function, Anne W. Werkley, Samuel D. Butler, Todd V. Small, Michael A. Marciniak
Faculty Publications
The bidirectional reflectance distribution function (BRDF) is used to describe reflectances of materials by calculating the ratio of the reflected radiance to the incident irradiance. While it was found that the isotropic models maintained symmetry about ϕs = π, such symmetry was not maintained about the θs = θi axis, except for close to the specular peak. This led to the development of a data-driven metric for how isotropic a BRDF measurement is. Research efforts centered around developing an algorithm that could determine material anisotropy without having to fit to models. This algorithm was tested using high …
Robust Method Of Determining Microfacet Brdf Parameters In The Presence Of Noise Via Recursive Optimization, Michael W. Bishop, Samuel D. Butler, Michael A. Marciniak
Robust Method Of Determining Microfacet Brdf Parameters In The Presence Of Noise Via Recursive Optimization, Michael W. Bishop, Samuel D. Butler, Michael A. Marciniak
Faculty Publications
Accurate bidirectional reflectance distribution function (BRDF) models are essential for computer graphics and remote sensing performance. The popular microfacet class of BRDF models is geometric-optics-based and computationally inexpensive. Fitting microfacet models to scatterometry measurements is a common yet challenging requirement that can result in a model being fit as one of several unique local minima. Final model fit accuracy is therefore largely based on the quality of the initial parameter estimate. This makes for widely varying material parameter estimates and causes inconsistent performance comparisons across microfacet models, as will be shown with synthetic data. We proposed a recursive optimization method …
Re-Visiting Acoustic Sounding To Advance The Measurement Of Optical Turbulence, Steven T. Fiorino, Santasri Bose-Pillai, Kevin J. Keefer
Re-Visiting Acoustic Sounding To Advance The Measurement Of Optical Turbulence, Steven T. Fiorino, Santasri Bose-Pillai, Kevin J. Keefer
Faculty Publications
Optical turbulence, as determined by the widely accepted practice of profiling the temperature structure constant, C2T, via the measurement of ambient atmospheric temperature gradients, can be found to differ quite significantly when characterizing such gradients via thermal-couple differential temperature sensors as compared to doing so with acoustic probes such as those commonly used in sonic anemometry. Similar inconsistencies are observed when comparing optical turbulence strength derived via C2T as compared to those through direct optical or imaging measurements of small fluctuations of the index of refraction of air (i.e., scintillation). These irregularities are especially apparent …
Multi-Gaussian Random Variables For Modeling Optical Phenomena, Olga Korotkova, Milo W. Hyde Iv
Multi-Gaussian Random Variables For Modeling Optical Phenomena, Olga Korotkova, Milo W. Hyde Iv
Faculty Publications
A generalization of the classic Gaussian random variable to the family of multi-Gaussian (MG) random variables characterized by shape parameter M > 0, in addition to the mean and the standard deviation, is introduced. The probability density function (PDF) of the MG family members is an alternating series of Gaussian functions with suitably chosen heights and widths. In particular, for integer values of M, the series has a finite number of terms and leads to flattened profiles, while reducing to the classic Gaussian PDF for M = 1. For non-integer, positive values of M, a convergent infinite series of …
Zernike Integrated Partial Phase Error Reduction Algorithm, Stephen C. Cain
Zernike Integrated Partial Phase Error Reduction Algorithm, Stephen C. Cain
Faculty Publications
A modification to the error reduction algorithm is reported in this paper for determining the prescription of an imaging system in terms of Zernike polynomials. The technique estimates the Zernike coefficients of the optical prescription as part of a modified Gerchberg-Saxton iteration combined with a new gradient-based phase unwrapping algorithm. Zernike coefficients are updated gradually as the error reduction algorithm converges by recovering the partial pupil phase that differed from the last known pupil phase estimate. In this way the wrapped phase emerging during each iteration of the error reduction algorithm does not represent the entire wrapped phase of the …
Beam Formation And Vernier Steering Off Of A Rough Surface, Eric K. Nagamine, Kenneth W. Burgi, Samuel D. Butler
Beam Formation And Vernier Steering Off Of A Rough Surface, Eric K. Nagamine, Kenneth W. Burgi, Samuel D. Butler
Faculty Publications
Wavefront shaping can refocus light after it reflects from an optically rough surface. One proposed use case of this effect is in indirect imaging; if any rough surface could be turned into an illumination source, objects out of the direct line of sight could be illuminated. In this paper, we demonstrate the superior performance of a genetic algorithm compared to other iterative feedback-based wavefront shaping algorithms in achieving reflective inverse diffusion for a focal plane system. Next, the ability to control the pointing direction of the refocused beam with high precision over a narrow angular range is demonstrated, though the …
Estimating Turbulence Distribution Over A Heterogeneous Path Using Time‐Lapse Imagery From Dual Cameras, Benjamin Wilson, Santasri Bose-Pillai, Jack E. Mccrae, Kevin J. Keefer, Steven T. Fiorino
Estimating Turbulence Distribution Over A Heterogeneous Path Using Time‐Lapse Imagery From Dual Cameras, Benjamin Wilson, Santasri Bose-Pillai, Jack E. Mccrae, Kevin J. Keefer, Steven T. Fiorino
Faculty Publications
Knowledge of turbulence distribution along an experimental path can help in effective turbulence compensation and mitigation. Although scintillometers are traditionally used to measure the strength of turbulence, they provide a path-integrated measurement and have limited operational ranges. A technique to profile turbulence using time-lapse imagery of a distant target from spatially separated cameras is presented here. The method uses the turbulence induced differential motion between pairs of point features on a target, sensed at a single camera and between cameras to extract turbulence distribution along the path. The method is successfully demonstrated on a 511 m almost horizontal path going …
Twisted Spatiotemporal Optical Vortex Random Fields, Milo W. Hyde Iv
Twisted Spatiotemporal Optical Vortex Random Fields, Milo W. Hyde Iv
Faculty Publications
We present twisted spatiotemporal optical vortex (STOV) beams, which are partially coherent light sources that possess a coherent optical vortex and a random twist coupling their space and time dimensions. These beams have controllable partial coherence and transverse orbital angular momentum (OAM), which distinguishes them from the more common spatial vortex and twisted beams (known to carry longitudinal OAM) in the literature and should ultimately make them useful in applications such as optical communications and optical tweezing. We present the mathematical analysis of twisted STOV beams, deriving the mutual coherence function and linear and angular momentum densities. We simulate the …
Achieving The Shot-Noise Limit Using Experimental Multi-Shot Digital Holography Data, Douglas E. Thornton, Cameron J. Radosevich, Samuel Horst, Mark F. Spencer
Achieving The Shot-Noise Limit Using Experimental Multi-Shot Digital Holography Data, Douglas E. Thornton, Cameron J. Radosevich, Samuel Horst, Mark F. Spencer
Faculty Publications
In this paper, we achieve the shot-noise limit using straightforward image-post-processing techniques with experimental multi-shot digital holography data (i.e., off-axis data composed of multiple noise and speckle realizations). First, we quantify the effects of frame subtraction (of the mean reference-only frame and the mean signal-only frame from the digital-hologram frames), which boosts the signal-to-noise ratio (SNR) of the baseline dataset with a gain of 2.4 dB. Next, we quantify the effects of frame averaging, both with and without the frame subtraction. We show that even though the frame averaging boosts the SNR by itself, the frame subtraction and the stability …
Temperature-Immune Self-Referencing Fabry–Pérot Cavity Sensors, Hengky Chandrahalim, Jonathan W. Smith
Temperature-Immune Self-Referencing Fabry–Pérot Cavity Sensors, Hengky Chandrahalim, Jonathan W. Smith
AFIT Patents
A passive microscopic Fabry-Pérot Interferometer (FPI) sensor an optical fiber a three-dimensional microscopic optical structure formed on a cleaved tip of an optical fighter that reflects a light signal back through the optical fiber. The reflected light is altered by refractive index changes in the three-dimensional structure that is subject to at least one of: (i) thermal radiation; and (ii) volatile organic compounds.
Optical Study Of 2-D Detonation Wave Stability, Eulaine T. Grodner
Optical Study Of 2-D Detonation Wave Stability, Eulaine T. Grodner
Theses and Dissertations
Fundamental optical detonation study of detonations constricted to a 2-d plane propagation, and detonations propagating around a curve. All images were processed using modern image processing techniques. The optical techniques used were shadowgraph, Schlieren, and chemiluminescence. In the 2-Dstraight channels, it was determined wave stability was a factor of cell size. It was also determined the detonation wave thickness (area between the combustion and shockwave) was a factor of how much heat available for the detonation. For the detonations propagating around a curve, it was determined the three main classifications of wave stability were stable, unstable, and detonation wave restart. …
A Comparative Evaluation Of The Fast Optical Pulse Response Of Event-Based Cameras, Tyler J. Brewer
A Comparative Evaluation Of The Fast Optical Pulse Response Of Event-Based Cameras, Tyler J. Brewer
Theses and Dissertations
Event cameras use biologically inspired readout circuit architecture to offer a faster and more efficient method of imaging than traditional frame-based detectors. The asynchronous event reporting circuit timestamps events to 1 microsecond resolution, but latency increases when many pixels are stimulated simultaneously. To characterize this variability, the DAVIS240, DAVIS346, DVXPlorer, and Prophesee Gen3M VGA-CD 1.1 cameras were exposed to single step-function flashes with amplitudes from 9.3-771cd/m2, stimulating from 0.0042-100 of pixels. The Median Absolute Deviation of pixel response times ranged between 0 and 6086µs, increasing with the percent of pixels stimulated (PSP). The number of events generated per …
Examination Of A Lenseless Setup For Reflective Inverse Diffusion Of Light, John Nguyen
Examination Of A Lenseless Setup For Reflective Inverse Diffusion Of Light, John Nguyen
Theses and Dissertations
Reflective inverse diffusion uses spatial light modulators to shape an incident wavefront so that when the wavefront interacts with some diffuse scattering sample, the reflected light will constructively interfere at a single focus. This thesisexamines the efficacy of using a lensless setup against a focal plane setup in achieving reflective inverse diffusion while simultaneously beamsteering. The lensless setup outperformed its counterpart by being able to focus more energy in a region, but failed to achieve the same beamsteering capabilities as the focal plane system. Understanding the flaws behind the lensless setup will be instrumental in creating a setup that can …
Dynamic Holography In Resonant Nonlinear Media: Theory And Application, Jonathan E. Slagle
Dynamic Holography In Resonant Nonlinear Media: Theory And Application, Jonathan E. Slagle
Theses and Dissertations
Two beam coupling (TBC) is a coherent interaction in which energy is transferred from one laser beam to another and has promising applications in real-time holography and coherent beam combing. We have recently shown efficient degenerate frequency TBC for counter-propagation geometries in isotropic two-photon absorbing media pumped with a nanosecond pulsed laser. When an interference pattern is generated in this media, single and two photon absorption initiates a population redistribution resulting in a holographic grating with the same modulation period and phase initially. However, due to temporal convolution of self- and cross-phase modulation, the grating will begin to shift in …
Neuromorphic Vision Sensors For Space-Based Applications, Jessica L. Horn
Neuromorphic Vision Sensors For Space-Based Applications, Jessica L. Horn
Theses and Dissertations
This research examines the viability of event-based vision (neuromorphic) sensors for future use on satellites. Outputting single pixel events only when a change in intensity is detected, event-based vision sensors offer a potential low power, low latency, high temporal resolution, and high dynamic range solution as compared to traditional CCD or CMOS camera technology. The commercial off-the-shelf DVS240C sensor, is analyzed to determine if operability is affected by launch conditions and the low Earth space environment through vacuum, thermal vacuum, and vibration tests. No appreciable changes in sensor operation was observed throughout testing. Additionally, the sensor was able to detect …
Independently Controlling Stochastic Field Realization Magnitude And Phase Statistics For The Construction Of Novel Partially Coherent Sources, Milo W. Hyde Iv
Independently Controlling Stochastic Field Realization Magnitude And Phase Statistics For The Construction Of Novel Partially Coherent Sources, Milo W. Hyde Iv
Faculty Publications
In this paper, we present a method to independently control the field and irradiance statistics of a partially coherent beam. Prior techniques focus on generating optical field realizations whose ensemble-averaged autocorrelation matches a specified second-order field moment known as the cross-spectral density (CSD) function. Since optical field realizations are assumed to obey Gaussian statistics, these methods do not consider the irradiance moments, as they, by the Gaussian moment theorem, are completely determined by the field’s first and second moments. Our work, by including control over the irradiance statistics (in addition to the CSD function), expands existing synthesis approaches and allows …
On-Chip Silicon Photonic Controllable 2 × 2 Four-Mode Waveguide Switch, Cao Dung Truong, Duy Nguyen Thi Hang, Hengky Chandrahalim, Minh Tuan Trinh
On-Chip Silicon Photonic Controllable 2 × 2 Four-Mode Waveguide Switch, Cao Dung Truong, Duy Nguyen Thi Hang, Hengky Chandrahalim, Minh Tuan Trinh
Faculty Publications
Multimode optical switch is a key component of mode division multiplexing in modern high-speed optical signal processing. In this paper, we introduce for the first time a novel 2 × 2 multimode switch design and demonstrate in the proof-of-concept. The device composes of four Y-multijunctions and 2 × 2 multimode interference coupler using silicon-on-insulator material with four controllable phase shifters. The shifters operate using thermo-optic effects utilizing Ti heaters enabling simultaneous switching of the optical signal between the output ports on four quasi-transverse electric modes with the electric power consumption is in order of 22.5 mW and the switching time …
Impact Of The Liquid Crystal Director Twisting On Two-Beam Energy Exchange In A Hybrid Photorefractive Inorganic-Liquid Crystal Cell, Victor Y. Reshetnyak, Igor P. Pinkevych, Michael E. Mcconney, Jonathan E. Slagle, Dean R. Evans
Impact Of The Liquid Crystal Director Twisting On Two-Beam Energy Exchange In A Hybrid Photorefractive Inorganic-Liquid Crystal Cell, Victor Y. Reshetnyak, Igor P. Pinkevych, Michael E. Mcconney, Jonathan E. Slagle, Dean R. Evans
Student Publications
We studied the energy transfer between light beams on the director grating in a hybrid photorefractive liquid crystal (LC) cell assuming the propagation of light waves in the cell to be in the Mauguin regime. This approach makes it possible to trace the change of the gain coefficient dependence on the director grating spacing with the change of the LC director twist. Conditions for the LC flexoelectric parameters and the director helix pitch necessary for transformation the gain coefficient dependence from the nematic to cholesteric type are obtained. The influence of the director splay and bend deformations on the gain …
2 × 2 Anisotropic Transfer Matrix Approach For Optical Propagation In Uniaxial Transmission Filter Structures, Hammid Al-Ghezi, Rudra Gnawali, Partha P. Banerjee, Lirong Sun, Jonathan E. Slagle, Dean Evans
2 × 2 Anisotropic Transfer Matrix Approach For Optical Propagation In Uniaxial Transmission Filter Structures, Hammid Al-Ghezi, Rudra Gnawali, Partha P. Banerjee, Lirong Sun, Jonathan E. Slagle, Dean Evans
Student Publications
Multi-layered metamaterial structures show promise in a wide variety of optical applications such as superlenses, electromagnetic cloaking, tunable filters, sensors, and spatial light modulators. Optical transmission analysis of multilayer metallo-dielectric stacks with overall thickness less than the wavelength of light can be modeled using effective medium theory and the Berreman matrix method. For multilayer anisotropic stacks of arbitrary thickness, a rigorous 4 × 4 transfer matrix embodiment is typically used. In this work, a 2 × 2 anisotropic transfer matrix method is developed to analyze optical propagation through multilayer uniaxial stacks of arbitrary thicknesses. Optical transmission of a multilayer silver–zinc …
Generation Of Vector Partially Coherent Optical Sources Using Phase-Only Spatial Light Modulators, Milo W. Hyde Iv, Santasri R. Bose-Pillai
Generation Of Vector Partially Coherent Optical Sources Using Phase-Only Spatial Light Modulators, Milo W. Hyde Iv, Santasri R. Bose-Pillai
AFIT Patents
A vector partially coherent source (VPCS) generator includes a laser that emits coherent light; an interferometer consisting of polarizing beam splitters (PBSs) to split the laser light into its vertical and horizontal polarization components;] first and second spatial light modulators (SLMs) that respectively control the vertical and horizontal polarization components; and a control system communicatively coupled to the first and second SLMs to adjust beam shape and coherence without physically moving or removing optical elements in the interferometer.
Artificial Neural Network Discovery Of A Switchable Metasurface Reflector, J. R. Thompson, J. A. Burrow, P. J. Shah, Jonathan E. Slagle, E. S. Harper, A. Van Rynbach, I. Agha, M. S. Mills
Artificial Neural Network Discovery Of A Switchable Metasurface Reflector, J. R. Thompson, J. A. Burrow, P. J. Shah, Jonathan E. Slagle, E. S. Harper, A. Van Rynbach, I. Agha, M. S. Mills
Student Publications
Optical materials engineered to dynamically and selectively manipulate electromagnetic waves are essential to the future of modern optical systems. In this paper, we simulate various metasurface configurations consisting of periodic 1D bars or 2D pillars made of the ternary phase change material Ge2Sb2Te5 (GST). Dynamic switching behavior in reflectance is exploited due to a drastic refractive index change between the crystalline and amorphous states of GST. Selectivity in the reflection and transmission spectra is manipulated by tailoring the geometrical parameters of the metasurface. Due to the immense number of possible metasurface configurations, we train deep neural networks capable of exploring …
Measurements Of Optical Turbulence Over 149-Km Path, Jack E. Mccrae, Santasri Bose-Pillai, Steven T. Fiorino, Aaron J. Archibald, Joel Meoak, Brannon Elmore, Thomas Kesler, Christopher A. Rice
Measurements Of Optical Turbulence Over 149-Km Path, Jack E. Mccrae, Santasri Bose-Pillai, Steven T. Fiorino, Aaron J. Archibald, Joel Meoak, Brannon Elmore, Thomas Kesler, Christopher A. Rice
Faculty Publications
An experiment was conducted to study turbulence along a 149-km path between the Mauna Loa and Haleakala mountain tops using digital cameras and light-emitting diode (LED) beacons. Much of the path is over the ocean, and a large portion of the path is 3 km above sea level. On the Mauna Loa side, six LED beacons were placed in a roughly linear array with pair spacings from 7 to 62 m. From the Haleakala side, a pair of cameras separated by 83.8 cm observed these beacons. Turbulence along the path induces tilts on the wavefronts, which results in displacements of …