Showing papers on "Wavefront published in 1971"

Propagation of a Finite Optical Beam in an Inhomogeneous Medium

Abstract: The first part of this paper is devoted to extending the Huygens-Fresnel principle to a medium that exhibits a spatial (but not temporal) variation in index of refraction. Utilizing a reciprocity theorem for a monochromatic disturbance in a weakly inhomogeneous medium, it is shown that the secondary wavefront will be determined by the envelope of spherical wavelets from the primary wavefront, as in the vacuum problem, but that each wavelet is now determined by the propagation of a spherical wave in the refractive medium. In the second part, the above development is applied to the case in which the index of refraction is a random variable; a further application of the reciprocity theorem results in a formula for the mean intensity distribution from a finite aperture in terms of the complex disturbance in the aperture and the modulation transfer function (MTF) for a spherical wave in the medium. The results are applicable for an arbitrary complex disturbance in the transmitting aperture in both the Fresnel and Fraunhofer regions of the aperture. Using a Kolmogorov spectrum for the index of refraction fluctuations and a second-order expression for the MTF, the formula is used to calculate the mean intensity distribution for a plane wave diffracting from a circular aperture and to give approximate expressions for the beam spreading at various ranges.

Signal-to-Noise Relationships for Coaxial Systems that Heterodyne Backscatter from the Atmosphere.

Abstract: This paper analyzes the signal-to-noise ratio for a coaxial laser system that heterodynes the signal backscattered from the atmospheric aerosol. The laser radiation, which is assumed to have a wavefront with a gaussian amplitude distribution, is transmitted into the atmosphere through a telescope. Radiation is collected by the same telescope and directed onto a detector where it is mixed with a local oscillator beam originating from the same laser source. The signal-to-noise ratio at the output of the detector is calculated under shot noise limited conditions. The calculation is general and applies for both near and far fields and for focused and unfocused systems. Three specific cases are considered. These are a pulsed system, a cw system illuminating an infinite target, and a cw system illuminating a target of finite extent.

Speckle pattern and holographic techniques in engineering metrology

Abstract: Some of the benefits of holographic interferometry in metrology are explored and related to difficulties experienced in practical methods. A new type of speckle pattern application is introduced which effectively samples the wavefront phase, enabling the interferometric measuring process to be carried out with low density data recording.

Computer Generated Holograms for Testing Optical Elements

Abstract: The application of computer generated holograms to the interferometric testing of aspheric optical elements has been investigated, and it has been shown that they provide a convenient and practical method of producing an aspheric reference wavefront.

Shearing interferometer using the grating as the beam splitter.

Abstract: The theoretical interpretation of the shearing interferometer based on the moire method using the fourier image of the grating is described. To obtain a pattern with good contrast, the observing plane must coincide with the normal fourier image plane of the grating or with the reversed fourier image plane. The information obtained by this method is the first partial derivative and under certain conditions the second partial derivative of the distortion from the reference wavefront, which is planar or spherical. Applications to measurement of the phase gradient and lens aberration are shown.

Computer-Based Analysis of Holography Using Ray Tracing

Abstract: The application of a ray-tracing methodology to holography is presented. Emphasis is placed on establishing a very general foundation from which to build a general computer-based implementation. As few restrictions as possible are placed on the recording and reconstruction geometry. The necessary equations are established from the construction and reconstruction parameters of the hologram. The aberrations are defined following H. H. Hopkins, and these aberration specification techniques are compared with those used previously to analyze holography. Representative of the flexibility of the ray-tracing approach, two examples are considered. The first compares the answers between a wavefront matching and the ray-tracing analysis in the case of aberration balancing to compensate for chromatic aberrations. The results are very close and establish the basic utility of aberration balancing. Further indicative of the power of a ray tracing, a thick media analysis is included in the computer programs. This section is then used to perform a study of the effects of hologram emulsion shrinkage and methods for compensation. The results of compensating such holograms are to introduce aberrations, and these are considered in both reflection and transmission holograms.

Approximate Treatment of Elastic Body Waves in Media with Spherical Symmetry

Abstract: Summary An Earth-flattening approximation for body waves is given which transforms a spherically symmetrical Earth into a vertically inhomogeneous half-space. This approximation includes a depth transformation and a transformation of the velocities of compressional and shear waves. It is an optimum approximation in the sense that the displacement amplitudes of corresponding seismic rays, i.e. of rays leaving a point source in the Earth and the image source in the half-space under the same radiation angle, show maximum agreement in their geometrical optics approximation. The epicentral distances of these rays are identical, and likewise the travel times provided that source and receiver are at the same depth. The amplitude difference is 10 (20,30) per cent at the epicentral distance 60° (82°, 96°). The main field of application of this Earth-flattening approximation is the computation of theoretical body wave seismograms with methods originally devised for half-spaces with plane layers. Theoretical P wave seismograms are given for the upper mantle models of Jeffreys, Gutenberg, Lehmann, Johnson and Mayer-Rosa. The method of calculation is an extension of the ray-theoretical method which was developed in a previous paper. Even without wave front approximations, it permits rather fast computations. Finally, the Earth-flattening approximation is applied to vertical reflections from a spherically symmetrical medium, and a method is briefly described which accounts for the Earth's curvature in computations of crustal and mantle transfer functions, respectively.

Reconstruction of Ultrasonic Images by Backward Propagation

Abstract: In a previous paper1, the authors discussed two methods for image construction by computer from ultrasonic data. The first method involved recording the intensity distribution of ultrasonic holograms, and using this data in a computer simulation of the wavefront reconstruction procedure of holography2. The second involved recording the amplitude and phase of the ultrasonic wavefront and computing the image directly by backward propagation3. Successful reconstructions were obtained by both methods, but the backward propagation technique, which permits a considerable reduction in the spatial frequency bandwidth (and therefore computer time) required, seemed the more promising for further effort. Reconstructions using the backward propagation method have also been reported by Sondhi9.

Acoustic Waves Transmitted Through Solid Elastic Cylinders

Abstract: The acoustic field of a plane wave incident on an elastic cylinder may be decomposed into two parts, one representing circumferential waves that propagate around the cylinder, the other part obeying in the high‐frequency limit the laws of geometrical optics. The latter part is studied in this paper, and is decomposed into the specularly reflected wave and into a series of transmitted waves which traverse the cylinder either directly along a secant, or undergo additional internal reflections with and without mode convention (compressional to shear type, or vice versa). Wavefront loci and amplitudes of these waves are calculated using the saddle‐point method, and the corresponding reflection and transmission coefficients are shown to reduce, for the case of a large cylinder, to the known expressions for a flat elastic half‐space in contact with a liquid.

Wavefront Distortion of Acoustic Surface Waves from Apodized Interdigital Transducers

Abstract: The acoustic surface wave launched by an interdigital transducer is experimentally shown to have a distorted wavefront when the transducer has a varying finger overlap (apodization) and is on a high‐coupling piezoelectric substrate, LiNbO3. Results are shown for different transducer geometries, when an acoustic wave passes under the comb and when the apodized comb itself is driven. One geometry having extra metal fingers significantly reduces the wavefront distortion. The origin of the distortion is a retardation of the wave passing under metal fingers.

Efficient and aberration-free wavefront reconstruction from holograms illuminated at wavelengths differing from the forming wavelength.

Abstract: Recording materials producing higher efficiency, low absorption holograms are usually photosensitive only to blue and uv light. If a hologram is formed at a blue wavelength in such a material and is illuminated at a red wavelength, generally the reconstructed wavefront is aberrated, and the resolution of the image is reduced. A method of hologram formation is described that can alleviate this problem. In this method, a hologram H(1) is formed first at the red wavelength lambda(1) in a photographic emulsion. This hologram is then illuminated at the blue wavelength lambda(2) The diffracted wave from H(1) is used as the subject wave for forming the desired high efficiency hologram H(2). If certain requirements are met, illumination of H(2) at lambda(1) can produce an aberration-free reconstruction of the original subject wave. Experimental results of forming H(2) of a point source on dichromated gelatin film are presented. The hologram was formed at lambda(2) = 4800 A but could reconstruct an aberration-free wavefront at lambda(1) = 6328 A. The hologram had about 80% diffraction efficiency and essentially no absorption loss.

The Influence of Random Wavefront Errors on the Imaging Characteristics of an Optical System

Abstract: The influence of random wavefront errors on the transfer function and point spread function of an optical system is studied theoretically. The stochastic part of the aberration function is assumed to be Gaussian and spatially stationary (although the requirement of stationarity is relaxed in § 5). The stochastic transformation from wavefront to transfer function is non-linear. The consequences of this non-linear transformation are two-fold: first, the statistics of the transfer function are non-Gaussian, second, the transfer function is non-stationary. (The same statements hold for the point spread function.) Therefore the characterization of these processes requires an infinite number of averaged products (moments), not just the first two if the processes were Gaussian. These averaged products are obtained in the form of multiple integrals involving the characteristic function of the wavefront and are suitable for calculation on a high speed computer. Some numerical results for the mean of both processes...

Theory of Certain Diffraction Gratings Produced by the Holographic Method

Abstract: It is now well known that a system of interference fringes photographed on a photographic plate has very interesting properties when used as a diffraction grating. This paper considers two cases. One is when the photographic plate is a plane surface; the aberration properties of this are worked out as a function of wavelength. There are three positions for which the spherical aberration is zero. Of these, one is of little interest as this is simply zero-order position. Another case considered was that of a photographic plate in the form of a concave spherical surface. In this case, there are three positions at which the spherical aberration is zero. The nature of variation of spherical aberration between these zero-aberration positions is presented in the form of curves computed from the theory of these gratings.

Dose-distribution measurement of high-intensity pulsed radiation by means of holographic interferometry.

Abstract: A novel calorimetric method has been developed for absolute determination of the spatial distribution of radiation absorbed dose in transparent liquids, particularly for dosimetry of pulsed, high-intensity electron beams. Local temperature change is determined without the introduction of temperature sensors that may disturb the radiation field. The change in a liquid's refractive index, which is a function of temperature, is measured by an interferometric method employing holographic wavefront reconstruction. For radiation beams essentially symmetric to the direction of propagation, the three-dimensional dose distribution can be derived from the information given by the interferogram. Dose distributions in water have been evaluated for short electron pulses of different energy spectra.

Temporal reference wavefront reconstruction process and system

Abstract: An improved process and method of wavefront reconstruction is disclosed in which one of the reference and object illuminating beams is shifted in phase relative to the other for a subsequent illumination. When applied to acoustical holography, the phase shift is lambda /4 at a time relative to the acoustical frequency that is 1/2 cycle later. In other applications, the phase shift may be limited to 0.1 lambda and can modulate a one of the beams at the acoustical frequency. Temporal reference holography also is disclosed in which a hologram is created without the use of a reference beam.

Optical differentiation of the displacement patterns using shearing interferometry by wavefront reconstruction: The purpose of this paper is to describe the application of holographic interferometry to the differentiation of moiré patterns

Abstract: Two techniques of moire-pattern differentiation are described and implemented. One of the techniques is extremely simple since the grid printed in the model provides both the displacement information and the shearing mechanism required to display the displacement derivatives. This technique can only be applied to the derivatives in the direction parallel to the projected displacement. To find the derivativs in the direction perpendicular to the projected displacement, a double-exposure technique is utilized.

Compensation of Aberrations due to a Wavelength Shift in Holography

Abstract: A shift in wavelength between the recording and reconstruction beams of a hologram introduces aberrations in the image wavefront and decreases the diffraction efficiency. The holograms investigated in this paper are recorded on dichromate gelatin film with either plane waves or point sources and are designed to image to a point source when illuminated with a plane wave. A computer program is employed to calculate the aberrations present in the image wavefront and to determine an optimized recording geometry necessary for minimum aberrations. It is shown that the computer program, which was designed for surface-type holograms, has some application in thick transmission-type phase holograms. The experimental results are in close agreement with the computed results indicating that compensation for the wavelength shift may be obtained by using a computer program to calculate the proper recording geometry.

Effect of emulsion thickness variations on wavefront reconstruction.

Abstract: In this paper an approach will be developed for wavefront recording and reconstruction with a photographic plate when emulsion thickness is not uniform. It is shown that in a first approximation an emulsion thickness variation will not affect the wavefront recording. However, thickness variation does affect the hologram image reconstruction. It is found that a first-order thickness variation (wedge-shaped), except for a lateral translation of image coordinates, will not affect the precision of the reconstruction. However, a second-order variation not only affects the precise dimension of the image reconstruction but also causes an astigmatic effect in the hologram image.

Sensor array acoustic detection system

Abstract: An acoustic detection system comprising two arrays of omnidirectional sensors or hydrophones wherein the sensors in each array are disposed along a curved line, such as an arc of a cylindrical surface. Accordingly an acoustic signal or wavefront propagating along any vector within the arc is registered by the respective sensors as phase lags determined by the locations thereof. The respective sensor output signals are each connected to a preselected electrical delay circuit such that the registered phase lags are further increased. The delayed sensor signals are then summed for each array, forming two directional beams and the signal sum for each array is then connected to identical narrow bandpass filters. The output of each filter is fed to a hard limiter, then to a differentiator and the output signals of the differentiators control a bistable flip-flop producing a pulse train where the width of each pulse corresponds to the phase lag difference between the beam signals. The pulse widths are then sorted into discrete bins of phase in a coincidence processor and read out when any one bin registers a number of pulses exceeding a predetermined number.

Moiré Patterns and Hologram Interferometry

Abstract: A DOUBLE exposure hologram can be produced if an object (C) is illuminated by a spherical coherent wavefront originating from a point source (A). Observations can be made from another point (B), at a distance L from A and a photographic plate (P) between C and B can be used to record all the optical information from C. P also must be illuminated by a reference beam1–6.

Acoustic surface wave transmission device

Abstract: A surface wave integratable filter includes input and output transducers spaced apart on an acoustic wave propagating medium. The input transducer launches acoustic surface waves along a path in which the wavefronts diverge. Disposed on the medium between the input and output transducers is an acoustic lens formed of a material that exhibits an acoustical refractive index greater than one. The lens acts to change the width of the acoustic wavefront and thereby enables appropriate selection of the physical size of the output transducer so as to obtain desired input and output impedances while securing increased efficiency of interaction at the output transducer.

Application of Holographic Interferometry to Mechanical Experiments

Abstract: One of the merits of holographic interferometry (1,2) is that distinct interference fringes appear even on a rough surface. The use of interferometry is not restricted only to an object with specularly reflecting surfaces. Another merit is the capability of time shearing interferometry. A wave front from an object can be compared not with any reference wave front but with that of the object itself in different states. Such advantages make the interferometric method applicable extensively to engineering problems. Moreover, the interferometry enables us to measure essentially the longitudinal displacement more easily than the lateral one. In the present paper, some new attempts are made mainly for bending experiments where the measurement of deflection is a fundamental task. The following three experiments ----- the first two are static and the third dynamic ----- are carried out; 1) Determination of Poisson’s ratio of plate; 2) Analysis of moments in bending slab; 3) Flexural vibration of plate. Further the following experiment is supplemented; 4) Analysis of moving object.

High Precision 10-cm Aperture Penta and Roof-Penta Mirror Assemblies.

Abstract: 10-cm aperture penta and penta-roof-type mirror assemblies of unusually high precision and stability for this aperture have been designed and fabricated by Perkin-Elmer Corporation. These assemblies, which are part of the internal optical alignment monitoring subsystem of a large (122-cm) aperture telescope, provide 90° deviations within ±1 sec of arc. In order to minimize degradation of the reflected wavefront due to variations of the thermal environment, each assembly is constructed of optical quality Cer-Vit mirror plates contacted to Cer-Vit blocks which establish and preserve the proper angular relationships between the reflecting surfaces. Thermal, vibration, and shock tests of prototype units have confirmed their inherent resistance to adverse environmental influences. In this paper, the method of construction of these prototype units is described and the results of various tests are given. Since the reflecting surfaces of the mirrors are dielectric coatings which maximize reflectivity in the 0.9–0.95-μ, wavelength region but have very low reflectivity at visible wavelengths, a real-time, electrooptical interferometric technique was developed for evaluating with near-infrared radiation the wavefront deformation caused by one of these assemblies. This experimental technique is also described.

Techniques of holographic data reduction utilizing an additional diffusing structure during reconstruction

Abstract: An off-axis optical hologram is constructed of an object with an object modified light being passed through an array of small lenses or some other light modulator. Upon reconstruction of the hologram, a diffracted wavefront is passed back through the array of small lenses or other light modulator. Additionally, during reconstruction, a separate light diffusing element is placed in the path of the diffracted image-carrying beam prior to its striking the light modulator.

Optical angle measuring system

Abstract: An optical system is disclosed for automatically measuring corner angles of graphic patterns. In a preferred embodiment, the image of a graphic pattern is illuminated by a light source, slowly rotated by a rotating K-mirror assembly and rapidly scanned by a rotating mirror drum which sequentially projects predetermined incremental portions of the graphic pattern through a slit into a light sensor for detection. An angle mark generator is utilized to generate angle marks as the K-mirror assembly rotates. A very steep transient pulse or wavefront is generated by the light sensor whenever either edge of the corner angle of the pattern is parallel to the slit. A digital processor is responsive to the steep transient pulses and the angle marks for detecting the angular difference, during one revolution of the image, between consecutive appearances of two steep transient pulses.

Electrical Recording of the Amplitude and Phase Fluctuations of a Wave Front

Abstract: A method to determine the amplitude and phase fluctuations in each point of a wavefront is presented. The basic idea of the method is the possibility of investigating the amplitude and phase at each point of a wavefront by superposing a spherical reference wavefront that has its origin in the point under investigation. The information on the fluctuations is extracted in the form of an electrical signal. Application to the study of phase fluctuations produced by atmospheric turbulence is considered.

Electromagnetic Image Formation with Holograms of Arbitrary Shape

Abstract: A diffraction-limited, electromagnetic theory of image formation is presented for a point-reference hologram whose recording arrangement consists of a surface of arbitrary shape, a point-reference source, and the object. The hologram is illuminated by a spherical electromagnetic wave during reconstruction. The electromagnetic hologram is assumed to have recorded two components of the field scattered from the object so that the vector field is completely reconstructed. The vector hologram is modeled by electric and magnetic surface currents determined from the irradiance of each of two orthogonal components of the object field on the film. The image field is described by a dyadic kernel, the system response to a point object, which is related to the scalar kernel by Π(r,r′) = DK(r,r′), where D is the dyadic operator D= (I+k−2∇∇). It is shown that the conjugate-image field produced by a point-reference electromagnetic hologram approximates the field produced by the ideal system, which forms the image of a point object by launching a spherically converging wave.