Showing papers on "Wavefront published in 1974"

Digital Wavefront Measuring Interferometer for Testing Optical Surfaces and Lenses

Abstract: A self-scanned 1024 element photodiode array and minicomputer are used to measure the phase (wavefront) in the interference pattern of an interferometer to lambda/100. The photodiode array samples intensities over a 32 x 32 matrix in the interference pattern as the length of the reference arm is varied piezoelectrically. Using these data the minicomputer synchronously detects the phase at each of the 1024 points by a Fourier series method and displays the wavefront in contour and perspective plot on a storage oscilloscope in less than 1 min (Bruning et al. Paper WE16, OSA Annual Meeting, Oct. 1972). The array of intensities is sampled and averaged many times in a random fashion so that the effects of air turbulence, vibrations, and thermal drifts are minimized. Very significant is the fact that wavefront errors in the interferometer are easily determined and may be automatically subtracted from current or subsequent wavefrots. Various programs supporting the measurement system include software for determining the aperture boundary, sum and difference of wavefronts, removal or insertion of tilt and focus errors, and routines for spatial manipulation of wavefronts. FFT programs transform wavefront data into point spread function and modulus and phase of the optical transfer function of lenses. Display programs plot these functions in contour and perspective. The system has been designed to optimize the collection of data to give higher than usual accuracy in measuring the individual elements and final performance of assembled diffraction limited optical systems, and furthermore, the short loop time of a few minutes makes the system an attractive alternative to constraints imposed by test glasses in the optical shop.

Method for Evaluating Lateral Shearing Interferograms

Abstract: Wavefront shearing interferometers have inherent advantages over more conventional interferometers because they do not need a separate reference wavefront. However, the fringe patterns are less directly related to the wavefront shape. In this paper, a method is described that uses data obtained from two lateral shear interferograms sheared in orthogonal directions to describe a wavefront of any arbitrary shape. Analysis of the data defines the wavefront on a regular grid, using a least squares criterion to match the measured data to the reconstructed shear data. Because each point on the final wavefront is involved in at most four measurements, the matrices tend to have many zero elements, making them easily solvable by simple numerical techniques, even for several hundred points. An error analysis indicates that the accuracy of the final results can be as good as the accuracy of the measured data. The procedure is described, and results of a typical analysis are shown.

On the position of the plane of stationarity in laser refraction.

Abstract: In the laser technique for subjective refraction the coherent laser beam illuminates the diffusing surface of a slowly rotating drum. The patient sees a moving speckle pattern which can be rendered stationary by the use of appropriate correcting lenses. It is shown that the patient's retina is not then, in general, conjugate with the frontal plane passing through the nearest point on the drum surface, but rather with a “plane of stationarity” whose position is a function both of the drum radius and of the curvature and obliquity of the incident coherent illuminating wavefront. An expression is derived for the position of this plane, and representative practical configurations are discussed, particularly with regard to the employment of the laser technique in accommodation studies.

Real-time wavefront correction system

Abstract: An optical imaging system having the capability of detecting and eliminating in real-time phase distortions in a wavefront being imaged by the optical system. The resolution of ground based telescopes is severely limited by random wavefront phase changes and tilts produced by atmospheric turbulence. The disclosed invention was designed to overcome this problem. In the disclosed invention, an AC, lateral shearing interferometer measures in real-time the relative phase differences of the wavefront being imaged by the optical system. Phase differences measured by the shearing interferometer are directed to an analog data processor which, in combination with other circuitry, generates a plurality of electrical signals proportional to the required phase corrections at different areas of the wavefront. The electrical signals are applied to a phase corrector upon which the wavefront is incident to change the relative phase at various locations of the wavefront to achieve a wavefront in which the phase distortion is removed. In one embodiment the phase correction device consists of a mirror having an array of piezoelectric elements which function to selectively deform the mirror to correct phase distortions in the wavefront. In a second embodiment the phase correction device consists of a refractive device which has the capability of having its index of refraction selectively changed in different areas to correct phase distortions in the wavefront.

Monolithic piezoelectric mirror for wavefront correction

Abstract: Experimental and theoretical analysis show that a spatially varying potential applied to one surface of a thick block of piezoelectric material will induce a piezoelectric deformation of that surface proportional to the potential at each point of the surface. There is essentially no dispersion in the piezoelectric response as a function of the spatial frequency of the applied potential even at high resolutions. This principle has been used to construct an electrically addressed monolithic piezoelectric mirror (MPM) which has high accuracy, stability, and speed of response. The MPM has been used successfully in a demonstration of real‐time correction of image phase distortions caused by a turbulent atmosphere.

Method and apparatus to determine spatial distribution of magnitude and phase of electro-magnetic fields especially optical fields

Abstract: A method and apparatus for numerical determination of the distribution of magnitude and phase of a time harmonic electromagnetic, particularly an optical, wave field by using the wavefront of a reference wave as the optical reference surface and by introducing additional information to the recording process such that the field becomes uniquely determined and this is realized by the recording of three interference fringe patterns which belong to variations of the reference field distribution. A reference field is chosen whose spatial distribution is similar to the unknown field whereby fringes are obtained the position of which varies slowly with the recording coordinates. The recorded data are numerically evaluated and this may be done conveniently in a computer. After these measurements are made the reference beam is shifted by an optical element e.g. a prism, gaseous wedge, electro-optical diffraction device placed in the reference optical path and a second set of measurements is made. In this way a reference field whose spatial distribution is not known precisely is introduced but the system is such that the reference spatial distribution information is used and then eliminated from the information finally obtained on the unknown wavefield.

Application of the boundary-diffraction-wave theory to gaussian beams*

Abstract: Within the limits of the paraxial approximation used in treating gaussian beams, the ordinary boundary-diffraction-wave theory is also applicable to diffraction problems that involve gaussian incident beams. The total field diffracted by an aperture is thus given by the interference of two component waves: a boundary-diffraction wave and, if allowed by geometrical considerations, the unperturbed wave that would propagate freely to the observation point in the absence of the diffracting aperture. To this end, the gaussian field distribution must be described by properly defined complex amplitude and phase functions. Examples are calculated for gaussian beams with cylindrical symmetry. The general equation for the ray paths associated with the gaussian beams is also derived; it is used to show that the shadow boundary behind the diffracting screen follows a hyperbola.

Maréchal intensity criteria modified for gaussian beams.

Abstract: The Marechal evaluation of the Strehl definition is reexamined for a Gaussian aperture where we have included the primary aberrations and all orders of spherical aberration. The result is particularly useful for evaluating the far-field peak intensity degradation, due to aberrations, for a well-corrected optical system when the wavefront distortion at the exit pupil is known. Further, the resulting equations have the same form as Marechal's equations for a uniform beam with the exception of additional factors that are the consequence of the Gaussian beam. These factors approach unity as the Gaussian beam approaches a uniform beam. As a consequence, the effects of the Gaussian illumination are readily identified. It is also shown how various aberrations may be balanced against one another in order to obtain the best peak intensity in the presence of a truncated Gaussian beam.

A New Theory for the Practical Interpretation of Holographic Interference Patterns Resulting from Static Surface Displacements

Abstract: It is re-established, in a suitable form, that live holographic interference patterns are the result of the interference occurring between the wavefronts diffracted from corresponding points in the reconstructed image and the displaced real object. This result is employed to determine the path difference equations for the wavefront forming the interference pattern as observed in three orthogonal directions. A set of linear equations is derived which is shown to relate the components of a small general surface displacement to the fringe spacing and localization plane of the observed interference pattern. The practical application of the theory is then discussed. Numerical equations are derived for various geometries of interferometer and in particular the measurements of rigid-body translations and plane strain are dealt with. In addition it is shown how problems that arise in the measurement of complicated surface displacements may be considerably simplified.

Mathematical interpretation of radial shearing interferometers.

Abstract: The procedure for computing a radial shearing interferometric pattern is given. The interferometric pattern is analyzed to obtain the wavefront shape. Restricting the discussion to wavefronts having rotational symmetry, we give two different methods of finding the wavefront. One approach is to scan along a diameter of the interferometric pattern and the other is to examine the shape of the fringes. The relative sensitivity of a radial shearing interferometer with respect to that of a Twyman-Green interferometer is also analyzed.

Second-order focusing conditions for ruled concave gratings

Abstract: The problem of image evaluation in vuv optical mountings of ruled concave gratings has been treated by means of the method of phase-error balancing. In the classical diffraction limit, the tolerances on the aberrations of normal-incidence mounts are evaluated by means of a modified Strehl criterion. For large phase errors, a quality factor that implies an association of ray density with energy flux, leads to determination of the tolerated magnitude of large aberrations. In addition, we give the related generalized focusing conditions that include shifts, off the gaussian image plane, as functions of the amounts of aberrations.

Elastic constant and strain measurements using a three beam speckle pattern interferometer

Abstract: The theoretical interpretation of speckle pattern interference fringes produced by small surface displacements is considered. New arrangements for a speckle pattern interferometer are described. Firstly the object is illuminated by plane wavefronts propagating at the same angle to orthogonal axes in the plane of the object. Secondly a 'composite speckle interferogram' is obtained when adjacent regions in the object are illuminated by different configurations of wavefront. Part of the object is illuminated as previously described, the remainder is illuminated by two wavefronts propagated at equal angles to a single axis in the surface. Interferograms made using this system can be employed to determine the components of plane strain tensors. A method for the measurement of elastic constants is experimentally verified and white light projection of frozen speckle pattern interference fringes is briefly discussed.

Analog data processor

Abstract: An optical imaging system having the capability of detecting and eliminating in real-time phase distortions in a wavefront being imaged by the optical system. The resolution of ground based telescopes is severely limited by random wavefront phase changes and tilts produced by atmospheric turbulence. the disclosed system was designed to overcome this problem. In the disclosed system, an AC, lateral shearing interferometer measures in realtime the relative phase differences of the wavefront being imaged by the optical system. Phase differences measured by the shearing interferometer are directed to an analog data processor which, in combination with other circuitry, generates a plurality of electrical signals proportional to the required phase corrections at different areas of the wavefront. The electrical signals are applied to a phase corrector upon which the wavefront is incident to change the relative phase at various locations of the wavefront to achieve a wavefront in which the phase distortion is removed. In one embodiment the phase correction device consists of a mirror having an array of piezoelectric elements which function to selectively deform the mirror to correct phase distortions in the wavefront. In a second embodiment the phase correction device consists of a refractive device which has the capability of having its index of refraction selectively changed in different areas to correct phase distortions in the wavefront.

Real-time wavefront correction through Bragg diffraction of light by sound waves

Abstract: The direction of propagation and phase of a light wave diffracted successively by two sound waves of the same frequency but different wavelengths can be controlled by varying the acoustic frequency and the relative phase of the sound waves. Because the Doppler frequency shifts produced by the two sound waves are of equal magnitude but opposite signs, they cancel each other, with the consequence that the diffracted-diffracted wave is coherent with the incident (or undiffracted) light. If a wavefront distorted by atmospheric turbulence is divided into subwavefronts, each subwavefront being planar, it can be corrected in real time if it is allowed to pass through an array of Bragg cells. Each cell carries two pairs of orthogonal sound waves to correct for tilts and phase errors of a corresponding subwavefront.

Wavefront visualisation of acoustic surface waves

Abstract: Acoustic surface waves (a.s.w.) propagating on yz-cut lithium niobate have been visualised by two imaging techniques. The wavefronts of straight-crested and convergent a.s.w. have been rendered visible on the screen of a storage oscilloscope.

Optical reflector for lasers

Abstract: Optical surface distorting thermal stresses are significantly reduced in an optical reflector by a structure consisting of a plurality of hollow reflector cells. Each cell has a light reflecting surface arranged to provide a composite continuous reflecting surface for an incident optical wavefront. Each reflector cell includes an internally located feeder tube which directs a flow of coolant from a manifold distribution system against internal cell walls.

Analytical evaluation of image formation in optical systems

Abstract: The methods of modern theoretical physics, in particular eigenfunction expansions and complex integration are used to analyse image formation in optical systems. The problem of the refraction of a wavefront in a spherical boundary between two media is solved. The fields are expanded in terms of spherical harmonics and the amplitude distribution in the image is then evaluated by the method of stationary phase. Both aberrations and diffraction are included.

Analysis of wave-front aberrations caused by deformation of hologram media

Abstract: Analytical expressions for the wave-front aberration caused by deformation of the hologram medium have been derived. These expressions are used for the determination of allowable tolerance of deformation. In the case of nearly diffraction-limited imaging, the tolerance is determined by evaluating the wave-front aberration. For other cases, in which more aberration is allowed, the tolerance is determined by evaluating the geometrical aberration. The dependence of deformation tolerance upon the type of hologram, the size of hologram, and the recording arrangement, especially that of the object and the reference beam, is shown. For small-size holograms, which are attractive for high-density information storage, the tolerance depends mainly upon the recording arrangement, and not upon the type of hologram. For larger-sized holograms, Fourier-transform holograms with parallel reference beams are the best, with regard to deformation tolerance. Numerical examples of the tolerance are given for nearly diffraction-limited imaging, using the Rayleigh criterion.

Wavefront visualization of bulk acoustic waves

Abstract: Bulk acoustic waves at frequencies up to several hundred megahertz and acoustic powers down to the nanowatt range have been visualized via acousto‐optic Bragg diffraction. Using double‐optical and electrical‐heterodyne mixing, the acoustic wavefronts have been displayed over several wavelengths along the propagation path. Possible applications of the reported experiments are briefly reviewed.

Collimated light acoustooptic lateral shearing interferometer.

Abstract: In this Letter, a collimated light acoustooptic lateral shearing interferometer (AO/LSI) capable of variable shear is described for use with either visual or ac electronic real-time phase measurements. The principle of operation of the AO/LSI is shown in Figs. 1 and 2. An input beam of diameter W is incident at Bragg’s angle upon two acoustooptic modulators (AOM’s) in series. They are shown schematically as thin gratings in Fig. 1, where the Bragg angles of the two AOM’s are indicated as θ B and θ 2. The first AOM is driven at an acoustic frequency f1 (typically in the tens of megahertz), the second AOM at frequency f2 = f1 + ∆ f. If ∆ f << f1, θ2 is nearly the same as θ B, as assumed in Fig. 2. It may be seen from Figs. 1 and 2 that some light is diffracted out of the zero-order beam by each AOM. The separation x between each AOM is adjusted until the two overlapping diffracted beams, which produce an interference pattern, are sheared (i.e., shifted) by an amount s. (A convenient value of s is 0.1 W.) The amount of shear can be varied by adjusting f1 too. In Fig. 2, it may be seen that the diffracted wave is assumed to originate from the center of each AOM. Actually, the wave is diffracted from the volume of the AOM, and the resultant wavefront exits the AOM at the angle θ B. However, for purposes of this discussion, it is convenient to assume that the central ray of the incident wavefront is reflected (i.e., diffracted) from the center of the AOM. From the geometry of Fig. 2 together with the Bragg diffraction equation, an equation relating the various parameters of the collimated light AO/LSI can be derived:

Gradient index optics: aspects of design, testing,tolerancing, and fabrication

Abstract: The properties of gradient index optical components were studied. Lens systems composed of materials whose index of refraction varied in a prescribed manner were designed using both third order aberration theory and raytrace analysis. The lenses contained either an axial gradient, where the index varies in the optical axis direction or a radial gradient where the variation is orthogonal to the optical axis. The raytrace was based on a polynomial expansion of the ray path in the gradient index media. Single element collimators with wavefront errors of less than 1/20 were designed. The equivalence of an aspheric surface and a spherical surface bounded by an axial gradient was shown in the design of corrector plates for a spherical primary mirror. Tolerancing analysis of the gradient index profile and the effects of its decentration, tilt, and displacement were also studied for these lenses. The index profiles were measured automatically by an instrument that utilized AC interferometry. Measurements of the gradient, of the dispersion of the gradient, and of the birefringence were made on samples manufactured by Bausch and Lomb. A single element lens with an axial gradient was fabricated from the glass of one sample. The measured wavefront error was 0.75λ at 0.5145μ and 1.0λ at 0.6328μ. These errors were in excellent agreement with those predicted by the theoretical design.

Two new optical techniques to measure strain

Abstract: A grating engraved on the surface to be analyzed is utilized to measure strain. The grating is illuminated by a monochromatic, plane wavefront. A lens produces the diffraction spectrum of the grating, and an opaque screen is located in the plane of the spectrum. A window allows a single diffraction order to go through the screen. Two techniques are presented.

High resolution velocity/depth spectra estimation for seismic profiling

Abstract: High resolution methods such as the so called maximum likelihood or maximum entropy procedures have found extensive use in spectral analysis and array processing. In all of these applications the spectral signals have been sinusoids or planewaves respectively. This led to a very close coupling to harmonic analysis as spectra or wave number functions. If one examines the general structure of these methods, the procedures are more closely related to parameter estimation so that one can consider the analysis nonstationary, or nonhomogeneous signal fields. One such situation exists in seismic profiling when the reflected signal structure presents a hyperbolic wavefront. Results will be presented showing the application of the generalized from of these high resolution methods to a proposed six channel array system for work to be done with the Woods Hole Oceanographic Institute - International Decade of Ocean Exploration Program.

Ultrasonic holography at the critical angle

Abstract: Ultrasonic holography is a recording technique preserving phase and amplitude of a wavefront on a two‐dimensional surface. This paper explores the possibility of using this technique to map phase changes occuring on an otherwise featureless surface by illuminating the surface at the Rayleigh critical angle. Since phase response at the critical angle is extremely sensitive to such things as stress concentrations, subsurface defects, or work hardening, it appears that this technique may offer a way to detect and map such distributions. We present preliminary experimental results that substantiate this claim.

Phase Difference and Angle-of-Arrival Fluctuations in Tracking a Moving Point Source

Abstract: The correlation function and power spectrum of phase difference fluctuations and angle-of-arrival fluctuations are calculated for tracking a moving spherical wave optical source through atmospheric turbulence. For Kolmogorov turbulence, it is shown that, in order to freeze effectively the received wavefront, the scanning frequency must be essentially equal to the highest frequency contained in the power spectrum.

Optics For Laser Scanning

Abstract: Although laser scanning is a relatively new technology, it has already found many applications in recording systems, OCR, film readers and data digitizers. A primary reason is that the monochromaticity and coherence of the laser beam makes it possible to focus the wavefront to a diffraction limited point image which then becomes the recording or analyzing spot.

Microphonically balanced pyroelectric detectors.

Abstract: Because pyroeiectric detectors are often fabricated from the same materials used to make microphones, it is not surprising to find that the former frequently exhibit characteristics of the latter. In fact, the sensitivity of these elements to mechanically and acoustically coupled disturbances frequently renders it impossible to realize the excellent noise performance that theory predicts for these simple, broadband optical detectors. Figure 1 shows a remedy that has yielded more than 20 dB of microphonic noise suppression over a broad band of acoustical frequencies. Assembled in pairs, the pyroeiectric elements are mounted in close proximity so that they experience the same mechanical/acoustical accelerations. However, they are connected to an amplifier in such a way that the microphonically induced signals tend to cancel. Depending on the structure chosen, one of the elements may be blinded to radiation by shielding or baffling. (When blinding is employed, the design is somewhat encumbered by the added thermal noise and capacitance of the second element.) Typical of the configurations tested, the cell shown consists of a pair of 2 mm X 2 mm silvered piezoelectric chips, 0.25 mm thick, indium-soldered to a brass substrate. Electrical connection is by means of No. 30 AWG copper wires soldered to the exposed faces. The poled ceramic elements are mounted with the same polarity terminal grounded to the backplate. This results in oppos-