Showing papers on "Wavefront published in 1985"

Analysis and applications of optical diffraction by gratings

Abstract: Diffraction characteristics of general dielectric planar (slab) gratings and surface-relief (corrugated) gratings are reviewed. Applications to laser-beam deflection, guidance, modulation, coupling, filtering, wavefront reconstruction, and distributed feedback in the fields of acoustooptics, integrated optics, holography, and spectral analysis are discussed. An exact formulation of the grating diffraction problem without approximations (rigorous coupled-wave theory developed by the authors) is presented. The method of solution is in terms of state variables and this is presented in detail. Then, using a series of fundamental assumptions, this rigorous theory is shown to reduce to the various existing approximate theories in the appropriate limits. The effects of these fundamental assumptions in the approximate theories are quantified and discussed.

Multiple-wavelength phase-shifting interferometry

Abstract: This paper describes a method to enhance the capability of two-wavelength phase-shifting interferometry. By introducing the phase data of a third wavelength, one can measure the phase of a very steep wave front. Experiments have been performed using a linear detector array to measure surface height of an off-axis parabola. For the wave front being measured the optical path difference between adjacent detector pixels was as large as 3.3 waves. After temporal averaging of five sets of data, the repeatability of the measurement is better than 25-A rms (λ = 6328 A).

Shear-wave polarizations on a curved wavefront at an isotropic free surface

Abstract: Summary. We present polarization diagrams of the particle motions at the free surface of an isotropic half-space generated by incident shear waves from a local buried point source. The reflectivity technique is used to calculate synthetic seismograms from which the particle motions are plotted. The particle motions are examined over a range of epicentral distances in a uniform isotropic half-space for different source frequencies and polarization angles, and for different Poisson's ratios. The particle motions due to a curved wavefront possess different characteristics from those generated by plane wavefronts at corresponding incidence angles. A curved wavefront generates a local SP-phase: a P-headwave which propagates along the free surface, and arrives shortly before the direct S-wave. These two arrivals give rise to cruciform particle motions in the sagittal and horizontal planes, which could be misinterpreted as anisotropy-induced shear-wave splitting. An examination of the particle motion in the transverse plane, mutually orthogonal to the sagittal and horizontal planes, can be used to discriminate between isotropic and anisotropic interpretations. The amplitude of the SP-phase is enhanced when it propagates in a low-velocity surface layer overlying the source layer, and may then become the dominant phase on radial-component seismograms. The presence of even a single surface layer may introduce considerable complexity into the seismogram, and we examine the effects of layer thickness, velocity contrast, and source depth on the corresponding polarization diagrams. Reliable information on the source and propagation path characteristics of shear waves from a buried local point source can only be obtained from free-surface records if they are recorded within a very limited epicentral distance range.

Phase shifter calibration in phase-shifting interferometry.

Abstract: This paper describes some practical methods to calibrate the phase shifter in phase-shifting interferometry (PSI). The phase shifter used in the experiment is a piezoelectric transducer (PZT) that has a nonlinearity of <1%. Using the quantitative method described in this paper, the repeatability in the measurement of the phase-shifting angle is ~0.046° rms, and the 3σ value is 0.139°. A calibration-insensitive phase calculation algorithm is discussed and compared with other synchronous detection equations (e.g., the three-bucket or the four-bucket method). Experimental results verify the calibration-insensitive mechanism of the self-calibrating algorithm.

Shearing interferometer for phase shifting interferometry with polarization phase shifter.

Abstract: A cyclic interferometer can be used for both lateral and radial shearing. Being common-path this interferometer is quite stable. The use of a cyclic shear interferometer for phase shifting interferometry is discussed. The phase shifting is provided here by polarization techniques. Some useful cyclic phase shifting interferometer arrangements are discussed, and the working of a polarization phase shifting cyclic shear interferometer is demonstrated.

Contouring Aspheric Surfaces Using Two-wavelength Phase-shifting Interferometry

Abstract: Two-wavelength holography and phase-shifting interferometry are combined to measure the phase contours of deep wavefronts and surfaces, such as those produced by aspherics, with a variable sensitivity. When interference fringes are very closely spaced, the phase data contain high frequencies where 2 ~ ambiguities cannot be resolved. In this technique, the surface is tested at a synthesized longer equivalent wavelength. The phase of the wavefront is calculated modulo 2φ using phase-shifting techniques at each of two visible wavelengths. The difference between these two phase sets is the phase of the wavefront as it would be measured at λeq=λ1λ2/|λ1 − λ2 |, assuming that 2π ambiguities can be removed at λeq. This technique enables surfaces to be contoured to an accuracy of λeq/100.

Testing aspheric surfaces with computer-generated holograms: analysis of adjustment and shape errors.

Abstract: Various interferometric methods can be used for testing aspheric surfaces with high accuracy. Using a partially compensating lens in a Twyman-Green arrangement together with a computer-generated hologram instead of a null corrector gives great flexibility. For quantitative analysis a high-resolution automatic fringe analysis is necessary. The interferometric arrangement and the setup for the interferogram data acquisition are described along with different algorithms for calculating the wave front data. Tilt and decentering of the aspheric surface and the synthetic hologram as well as the actual aspherical parameters can be derived from the calculated wave front using Zernike polynomials to communicate with a ray tracing program. For small adjustment and shape errors the linearity will hold, leading to fast correction and numerical compensation of their effects on the surface error. The method is verified by several examples. Experimental results agree with the proposed model.

Quasi- heterodyne hologram interferometry

Abstract: Wider use of hologram interferometry for quantitative measure-ments has been delayed by the fact that interpolation between the fringe maxima and minima to obtain the optical path difference at a particular point in the field is laborious and inaccurate. A solution to this problem is quasi-hetero-dyne interferometry, which permits rapid and accurate measurements simultaneously at a number of points distributed over the interference pattern. In this technique a television camera is used in conjunction with digital electronics to measure and store the irradiance values at points on a rectangular sampling grid covering the real-time interference fringes. The phase difference between the interfering wavefronts at each point is then calculated from the irradiance values obtained from successive scans of the camera made while the phase of one of the wavefronts is shifted either continuously or in steps. A practical system is described with which values of the optical path difference for 10,000 data points can be obtained with an accuracy of ± A/200 in less than 10 s. The application of quasi-heterodyne hologram interferometry to the measurement of vector displacements and to holographic contouring is discussed.

Beam cleanup using photorefractive two-wave mixing

Abstract: Photorefractive two-wave mixing exhibits energy transfer without phase cross talk. Such a phenomenon can be used to clean up wave-front aberrations of laser beams. We report the cleanup of spatial and temporal phase aberration using photorefractive two-wave mixing in a strontium barium niobate crystal.

Range of validity of seismic ray and beam methods in general inhomogeneous media - I. General theory

Abstract: Summary. The limitations of asymptotic wave theory and its geometrical manifestations are newly formalized and scrutinized. Necessary and sufficient conditions for the existence of acoustic and seismic rays and beams in general inhomogeneous media are expressed in terms of new physical parameters: the threshold frequency ω0 associated with the P/S decoupling condition, the cut-off frequency ωc associated with the radiation-zone condition, the total curvature of the wavefront and the Fresnel-zone radius. The analysis is facilitated with the introduction of a new ancillary functional – the hypereikonal which is capable of representing ordinary as well as evanescent waves. The hypereikonal is the natural extension of the eikonal theory. With the aid of the above new parameters, simple conditions are obtained for the decoupled far field, the decoupled near field, two point dynamic ray tracing, paraxial wavefields and Gaussian beams.

A wavefront interpretation of the singularity expansion method

Abstract: The singularity expansion method (SEM) represents transient scattering by superposition of damped oscillatory fields corresponding to the complex resonant frequencies of the scatterer. The series of these global wave fields, which encompass the scattering object as a whole, is slowly convergent at early observation times and even deficient at very early times when portions of the object are as yet unexcited. Thus, the resonance series representation must generally be augmented by an entire function in the complex frequency domain. The choice of the entire function is relatively arbitrary but affects the excitation coefficients, called coupling coefficients, of individual resonances and also the "turn-on" and "switch-on" times of the SEM series. Moreover, it contains essential (intrinsic) and nonessential (removable) portions which have been subjected to various interpretations. By formulating the transient problem in terms of traveling (progressing) incident, reflected and diffracted wavefronts, these constructs in the SEM can be interpreted in a precise and physical manner. Furthermore, the analysis clarifies the evolution of resonances as collective summations of multiple wavefront fields which are caused by successive reflections or diffractions at the surfaces and scattering centers comprising the object. By combining wavefronts and resonances self-consistently, one may construct a hybrid field that avoids the difficulties at early times in the SEM formulation. The systematic exploration of the interplay between wavefronts and resonances is facilitated through use of a flow diagram, as introduced in system theory. These concepts are developed in broad generality and are illustrated for two-dimensional scattering by various special configurations.

Integrated adaptive optical wavefront sensing and compensating system

Abstract: A wavefront sensing and compensating system for detecting and correcting for distortion in light wavefronts is described in which the wavefront is divided into a plurality of subapertures and light intensified and imaged as spots of light from each subaperture onto a detector array. The individual detector elements of the array form a plurality of electrical signals proportional to the local divergence of the vector gradient field (∇ 2 φ). This ∇ 2 φ signal after interfacing or reconstruction is applied to corrective mirrors which may be of the deformable or membrane type.

Applications of the tandem component: an element with optimum light efficiency.

Abstract: The tandem component is a holographic component which permits generation of a general complex wave front with 100% light efficiency. It consists of two phase-only elements in two Fourier conjugate planes and of a lens. Applications of this component are discussed such as complex wave front reconstruction, beam shaping, and correlation type measurements.

Binary Lenses For Use At 10.6 Micrometers

Abstract: By combining advances in lithography and electromagnetic grating theory, we recently have demonstrated the ability to produce highly efficient binary gratings and binary lenses for use at 10.6 um. Electromagnetic theory predicts that binary gratings with the proper parameters can achieve a first-order diffraction efficiency of nearly 100%. If the periodicity of the grating is on the order of the radiation wavelength, all of the orders become evanescent except for the zero and positive first orders. By choosing the depth-to-period ratio and duty cycle properly, the zero order can be suppressed, placing virtually all of the incident radiation into the first diffracted order. Theoretical calculations have been done only for constant period gratings. However, assuming a lens pattern to be a minor perturbation of a grating, we succeeded in producing an f/5 binary lens with a diffraction efficiency of 96% at 10.6 um. Furthermore, because of the high efficiency of these elements, it becomes practical to consider using more than a single diffractive element in a system. We have constructed a simple afocal telescope from two binary lenses. The telescope has a 2 in. entrance aperture and a magnification of 5. A final point to be considered is the wavefront quality of these elements. Electron beam machines, which are used to write the lens patterns, are designed to draw the pattern in a raster fashion. This quantization sets a limit on the quality of the lens pattern.

Phase conjugation method

Abstract: The use of large sensor and phase shifter arrays in phase conjugation adaptive systems is currently limited by the computational power conventionally employed to solve small arrays. Noting that in large arrays many of the derivatives of wavefront slopes are zero, then the properties of sparse matrix mathematics may be used to efficiently calculate the extent of movement of a particular actuator to deform a mirror and thus compensate for phase distortions in an incoming wavefront. Apparatus and methods are provided for handling such large arrays.

Holographic multi-combiner for head-up display

Abstract: A head-up display which combines an internal image, produced by a cathode ray tube image source, at substantially a wavelength of light λ, with an external image, using a holographic optical element. The holographic optical element consists of two or more holographic optical subelements in series, each satisfying the Bragg condition with respect to wavefronts at the wavelength λ from a particular direction. While wavefronts satisfying these conditions are reflected to the observer's field of view, all others are transmitted through the holographic subelements. The holographic optical subelements largely overlap in the direction of their illumination by the internal image source, and may slightly overlap in the direction their images are to be presented to the observer. Wavelength λ is chosen at the peak of the phosphor response curve of the cathode ray tube image source.

Wavefront sensing deconvolution and adaptive optics

Abstract: For large telescopes, the fundamental limit set by the diffraction to the angular resolution can be reached if the aberrations are perfectly corrected. Two solutions are practicable: image restoration by numerical processing or real time correction by adaptive optics. These two processes require the determination of the wavefront or of the optical transfer function. New large detectors arrays can be used in wavefront sensing which then allow the measurement of the wavefront at a large number of point. The general lay-out of such an instrument is given and its accuracy evaluated. The wavefront sensing contribution to numerical image processing is studied mainly for the case of degradation by atmospheric turbulence. The improvement of numerical processing by a partial active correction is discussed.

Spatial processing for single or dual shear wavefront sensor

Abstract: A shearing interferometer produces a matrix of loops of wavefront phase difference vector signals derived from the interferogram indicative of wavefront slopes, where each loop corresponds to a sub-aperture portion of the wavefront under examination. Curl operators are applied to sum the vector signals in one direction about the loops to provide sums which are set equal adjacent loops are identified having such modified sums differing from each other by at least one integer, and in certain cases having modified sums of integers of opposite sign indicative of phase difference measurements greater than the dynamic range of the interferometer, and thus having error components therein. The phase difference values at the boundary of each of such particular identified adjacent loops are operated upon to set the modified sums to zero, to in turn eliminate the error components due to operation of the interferometer outside of the limited dynamic range. Particular edge loops are also identified at the outer edges of the matrix having a positive or negative modified sum, and also having all three loops bounding the particular loops having modified sums of zero. The edge position phase difference values are modified in one or more steps, until the positive or negative modified sums of the particular edge loop are reduced to zero. Eight algorithms for executing the above stated functions are setforth herein.

White-light viewable, cylindrical holograms and method of spatially filtering wavefronts

Abstract: A white-light viewable, cylindrical hologram is made by recording on a cylindrical recording medium the interference between a radial reference beam and an object wavefront emanating from the center of the cylindrical recording medium. In a second step, the cylindrical hologram is placed in a flat configuration and a second hologram is recorded by recording the interference between a reference beam and an object wavefront formed by directing coherent light through the first hologram. Finally, a third hologram is formed by recording the interference between a reference beam and an object wavefront formed by directing coherent light through a narrow strip of the second hologram. The third hologram is viewed by placing it in a cylindrical configuration and directing white light onto the hologram from a point source located on the axis of the cylindrical hologram.

Visualization of the strain wave front of a progressive acoustic wave based on the Talbot effect.

Abstract: How to visualize a strain wave front of a progressive acoustic wave by means of conventional Fresnel imaging is shown. The theory suggests that for a Raman-Nath parameter of 0.1 ≤ υ ≤ 1, the stroboscopic illumination automatically produces the strain wave front image in a limited detection zone, because the spatial harmonic images of the original sinusoidal field coming from the interference among the higher-order diffracted light, are appropriately superimposed. The experiment showed good coincidence with theory and its usefulness in observing surface acoustic waves.

Real-time measurement of large liquid surface deformation using a holographic shearing interferometer.

Abstract: A new holographic lateral shear interferometer with two three-beam holograms is described which is useful for the real-time measurement of large liquid surface deformations. In the interferometer, it is shown theoretically and experimentally that the aberrations of the optical system can be cancelled and the amount of shear changed easily. The shearing interferograms of the liquid surface deformations caused by a water strider, etc. are demonstrated.

Phase-Shifting Speckle Interferometry

Abstract: Speckle patterns have high-frequency phase data, which make finding the absolute phase of a single speckle pattern difficult. However, the phase of the difference between two correlated speckle patterns can be determined by applying phase-shifting techniques to speckle interferometry, which will quantitatively determine the phase of double-exposure speckle measurements. The technique uses computer control to take data and calculate phase without an intermediate recording step. The randomness of the speckle causes noisy data points that are removed by data processing routines. A study of the phase errors attributable to decorrelation of the speckle patterns shows a 33° rms error for 10 waves of tilt. One application of this technique is finding the phase of deformations, where up to 10 waves of wavefront deformation can easily be measured. Results of deformations caused by tilt of a metal plate and a disbond in a honeycomb structure brazed to an aluminum plate are shown.

Realization of a uniform circular source using a two-dimensional binary filter.

Abstract: A uniform-intensity circular light source has been realized from a Gaussian laser beam by using an area-modulated binary filter. The technique preserves the wave front, and as a result the uniform circular source can be located at a plane perpendicular to the axis of propagation over a wide area.

Device for analyzing and correcting wavefront surfaces in real time using a polarization interferometer

Abstract: An optical system for analyzing and correcting wave fronts comprising a deformable mirror for correcting wave fronts and a system for analyzing and detecting phase distortion, an interferometer with lateral duplication constituting the analysis system, receiving the wave front for analysis and duplicating it and deducing from two neighboring wave fronts obtained signals to control deformation of the deformable mirror, wherein this lateral duplication interferometer is a polarization interferometer consisting of a Wollaston double-refractive biprism with an angle θ, the two prisms being assembled head to tail and cut parallel to the crystallographic axis such that the respective axes are parallel and perpendicular to the edges of the prisms, a polarizer and an analyzer on either side of the biprism and an oscillating optical member on the path of the biprism.

Laser wavefront measurement device utilizing crossed Ronchi gratings

Abstract: A wavefront, such as from a laser, is sampled, preferably at a beam splitter to break off only a fraction of the total light energy. The sampled beam is adjusted and sized--either by expansion or contraction at a conventional telescope--and thereafter sent through paired crossed Ronchi gratings and onto a CCD camera located at the first wave distance of constructive interference from the crossed Ronchi gratings. The crossed Ronchi gratings--in the order of 200 lines per inch and preferably with 200 equally spaced gaps per inch--project an array of square spots onto the focus of the CCD camera. The image of the array of square spots at the CCD camera is frozen via a frame grabber and digitized in a conventional format. The digitized image is played through a disclosed computer program to locate the spots. Thereafter, the located spots are compared to a standard spots. By knowing the difference between spot location of the standard spots and spot location of the suspect spots, wavefront analysis in phase can be analyzed. There results a simple wavefront analysis without the use of precision and moving optical parts which provides for measurement of the wavefront.

Lasers with wavefront-reversing mirrors (review)

Abstract: The principles of construction of lasers with wavefront reversing (WFR) mirrors are considered. Descriptions are given of methods for the determination of the transverse and longitudinal mode structures in such lasers. The advantages and shortcomings of various systems with WFR mirrors are reviewed. A comparison is made of the various systems with one another and with conventional mirrors. The future trends and the main lines of further research are forecast.

The Effects Of Rigid Body Motion In Interferometric Tests Of Large-Aperture, Off-Axis, Aspheric Optics

Abstract: We present a simple, general approach for ascertaining the aberrations due to rigid body motions that are encountered in optical tests. The expressions used are generally applicable to the displacement of an aspheric optic, relative to the aligned position. However, we apply these expressions to two cases of particular interest in interferometric testing. In the first case, the optic is tested in autocollimation at its focal point. We derive an equation for the wavefront aberrations induced by a small displacement from this focal position. Next we calculate the sensitivity to motion in a center-of-curvature test using nulling optics. We obtain the aberration function for a displacement of the test element from an aligned position. For each case, we analyze the sensitivities to the various degrees of freedom for particular off-axis aspheres of interest. In all such testing situations, in particular for high-precision work, displacements are present to some degree. The key requirement is then to remove the induced errors. We develop an optimum procedure for such removal and we give some example calculations. This procedure allows the metrologist to unambiguously identify surface correction profiles needed in the fabrication of precision, off-axis, aspheric segments.

Dual shear wavefront sensor

Abstract: A wavefront of light is focused upon a first shearing interferometer having a relatively large shear and small dynamic range and a relatively minor portion of the light is focused upon a second shearing interferometer having a relatively small shear but large dynamic range. Owing to the limited dynamic range of the first shearing interferometer a plurality of a plurality of virtual candidate measurements are manifest which are compared with the temporally and/or spatially averaged values of each measurement produced by the second shearing interferometer and the closest match is employed to obtain a highly accurate unambiguous reading of the wavefront slope measurements.

A practical comparison of the systolic and wavefront array processing architectures

Abstract: The development of a wavefront array processor for recursive least-squares minimization is described. It comprises a triangular array of processors each of which emulates the INMOS transputer chip. The array is programmed in occam - an associated high level language also developed by INMOS Ltd. Preliminary experiments have been carried out to compare the performance of the wavefront array with that of the associated systolic array. The results are presented.