Radius Of Wavefront Curvature Research Articles

Characteristics of Spiral Patterns Formed by Coaxial Interference between Two Vortex Beams with Different Radii of Wavefront Curvatures

Spiral pattern is formed for coaxial interference between two vortex beams with different radii of wavefront curvatures and different topological charges (TCs). A theoretical model considering various parameters (such as phase difference, radius of wavefront curvature, and TCs) is established to predict all kinds of interference patterns. An improved Mach-Zehnder interferometer is set up in an experiment to generate different kinds of spiral patterns and verify the theoretical model. The number of spiral lobes is determined by the absolute value of TCs’ difference between two vortex beams, and the twist direction relates to the sign of TCs’ difference and the difference of reciprocals for the radii of wavefront curvature, clockwise for the same sign, and counterclockwise for the opposite signs. The twist direction of the spiral pattern reverses and the lobes direction near the core of the pattern changes obviously when the spherical wave changes from convergence to divergence.

Single-shot measurements by Fresnel diffraction of divergent waves from a phase plate.

Recently, Fresnel diffraction (FD) of a plane wave from phase steps has been studied and applied for precise measurements of the light wavelength, and height and refractive index of the step, by changing the angle of incidence or step height to induce phase shifts. In this study, we formulate the FD of cylindrical and spherical wavefronts as 1D and 2D divergent waves from a phase plate. Since the phase difference of the divergent wave varies continuously along the edge of the phase plate, it can be applied for single-shot measurements. It is shown that the diffracted intensity distribution is a periodic function along the lines parallel to the plate edge. The phase distribution in this direction is a linearly varying function of the position squared, with a slope dependent on the light wavelength, plate thickness and refractive index, and the radius of wavefront curvature (RWC) on the observation plane. The diffraction patterns are simulated and experimentally verified. Also, the RWC and displacement are determined as examples of applications in the experimental part of the report.

Changes in echo-pulse instantaneous frequency when scanning the transducer

A theoretical model has been proposed that makes it possible to explain peculiarities in the frequency characteristics of echo signals at the output of a receiving transducer depending on the geometrical parameters of a reflecting spherical surface when the transducer axis is displaced with respect to the surface’s center of curvature, that is, during scanning. It has been demonstrated that both with and without displacement of the transducer’s geometrical axis with respect to the center of curvature of the reflecting surface, the type of changes in the temporal dependences of the instantaneous frequency of signals at the output of the receiving transducer essentially depends on the geometrical parameters of a wavefront reflected from the surface: the smaller the radius of wavefront curvature at the aperture of the receiving transducer, the stronger the changes of the instantaneous frequency in the echo pulse. It has also been shown that the larger the curvature of this surface, the stronger the changes in the frequency characteristics of the echo pulse as the transducer axis is displaced with respect to the center of curvature of the reflecting spherical surface.

Analysis of surface deformation in thin-film coatings by carrier frequency interferometry.

We demonstrate a method based on carrier frequency interferometry (CFI) that measures surface deformation with high accuracy. The method is applied to assess the deformation of thin-film dielectrics deposited on thick substrates. CFI measured the wavefront radius of curvature R with an accuracy of 0.2% for an R smaller than 500 m and 2% for an R between 500 and 2000 m (flat reference substrate). We show the method has a significantly larger dynamic range and sensitivity than Twyman-Green and comparable sensitivity to white light interferometry.

English

Some characteristic parameters of the Bessel- Gaussian beam were determined theoretically by direct analytical calculations. Variation of the final beam radius (ω) with the starting beam radius (ωo) was studied. The Rayleigh range of a Bessel-Gauss beam was calculated for each value of the minimum starting beam radius. The beam wavefront radius of curvature (R) was calculated as a function of starting beam radius at different distances from the source. A modified Bessel-Gauss beam was also considered. The beam intensity was calculated in the waist plane. The variation of the intensity near the center depends on whether the radius (a) equal, less than or greater than the starting beam radius (ωo). Key words: Bessel- Gauss beam, beam radius, waist plane, intensity.

Ghost reflections of Gaussian beams in anamorphic optical systems with an application to Michelson interferometer.

In this paper, a methodology is developed to model and analyze the effect of undesired (ghost) reflections of Gaussian beams that are produced by anamorphic optical systems. The superposition of these beams with the nominal beam modulates the nominal power distribution at the recording plane. This modulation may cause contrast reduction, veiling parts of the nominal image, and/or the formation of spurious interference fringes. The developed methodology is based on synthesizing the beam optical paths into nominal and ghost optical beam paths. Similar to the nominal beam, we present the concept that each ghost beam is characterized by a beam size, wavefront radius of curvature, and Gouy phase in the paraxial regime. The nominal and ghost beams are sequentially traced through the system and formulas for estimating the electric field magnitude and phase of each ghost beam at the recording plane are presented. The effective electric field is the addition of the individual nominal and ghost electric fields. Formulas for estimating Gouy phase, the shape of the interference fringes, and the central interference order are introduced. As an application, the theory of the formation of the interference fringes by Michelson interferometer is presented. This theory takes into consideration the ghost reflections that are formed by the beam splitter. To illustrate the theory and to show its wide applicability, simulation examples that include a Mangin mirror, a Michelson interferometer, and a black box optical system are provided.

Speckle dynamics for dual-beam optical illumination of a rotating structure

An out-of-plane rotating object is illuminated with two spatially separated coherent beams, giving rise to fully developed speckles, which will translate and gradually decorrelate in the observation plane, located in the far field. The speckle pattern is a compound structure, consisting of random speckles modulated by a smaller and repetitive structure. Generally, these two components of the compound speckle structure will move as rigid structures with individual velocities determined by the characteristics of the two illuminating beams. Closed-form analytical expressions are found for the space- and time-lagged covariance of irradiance and the corresponding power spectrum for the two spatially separated illuminating beams. The present analysis is valid for propagation through an arbitrary ABCD system, though the focus for the experimental evaluation is far-field observations using an optical Fourier transform system. It is shown that the compound speckle structures move as two individual structures with the same decorrelation length. The velocity of the random speckles is a combination of angular and peripheral velocity, where the peripheral velocity is inversely proportional to the radius of the wavefront curvature of the incident beams. The velocity of the repetitive structure is a combination of angular and peripheral velocity, where the peripheral velocity is proportional to the ratio of the angle to the distance between the beams in the object plane. Experimental data demonstrate good agreement between theory and measurements for selected combinations of beam separation, angle between beams, and radius of wavefront curvature at the object.

Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media

The formation and the evolution of the far-field patterns of the Gaussian beams passing through a thin self-focusing medium and a thin self-defocusing medium are studied using the Fresnel–Kirchhoff diffraction theory, and the effects of the different Kerr media and the different Gaussian beams on the far-field pattern formation and evolution are analysed by taking into consideration both the change in the additional phase shift induced by the refractive index and the change in the sign of the radius of wavefront curvature of the laser beam passing through the nonlinear medium. Our results show that, when either the divergent Gaussian beam passes through the self-defocusing medium or the convergent Gaussian beam passes through the self-focusing medium, the far-field intensity distribution pattern is a series of thick diffraction rings with a central dark spot; while, when either the divergent Gaussian beam traverses the self-focusing medium or the convergent Gaussian beam traverses the self-defocusing medium, the far-field intensity distribution pattern is a series of thin diffraction rings with a central bright spot. Our results also show that whether the central dark spot or the central bright spot occurs depends mainly on the sign of the product of the additional phase shift and the radius of the wavefront curvature. When the sign is negative, that is, when the divergent Gaussian beam passes through the self-defocusing media, or the convergent Gaussian beam passes through the self-focusing media, the central dark spot surrounded by the thick diffraction rings will emerge in the far field; while, when the sign is positive, that is, when the divergent Gaussian beam traverses the self-focusing media, or the convergent Gaussian beam passes through the self-defocusing media, the central bright spot with the thin diffraction rings will occur in the far field. The difference between the diffraction patterns is attributed to the interplay of the strong spatial self-phase modulation caused by the refractive index change of the medium and the changes in sign of the radius of the wavefront curvature of the incident Gaussian beam. The results obtained in this paper are of significance to the design of practical nonlinear optical limiters for the eye or sensor protection and many other applications.

Estimation of kinematic wavefront characteristics and their use for multiple attenuation

Kinematic wavefront characteristics of primary reflected waves can be used to predict and attenuate surface-, as well as interbed multiples. In order to estimate kinematic wavefront characteristics directly from unstacked data the common-shot-point homeomorphic-imaging (CSP HI) method can be applied. This macro-model-independent method is based on a new local moveout correction that depends on two wavefront parameters: the emergence angle and the radius of wavefront curvature of a reflected primary wavefront. These parameters can be estimated by optimizing the semblance correlation measure, calculated in the common-shot-gather along the travel time curve defined by the new moveout correction. In the case of local maxima in the semblance functional, an automatic maximization procedure might lead to a wrong estimation of these parameters. In order to avoid such a situation we propose an interactive, horizon-based implementation of the CSP HI method, which allows manual picking of the optimal wavefront parameters along the seismic line. Afterwards, we use the estimated emergence angles for the prediction and attenuation of multiples, based on the simple but powerful idea that any multiple event can be represented as a combination of primaries. A synthetic example shows the viability of the parameter estimation, prediction and attenuation procedure.

Accumulated Gouy phase shift in Gaussian beam propagation through first-order optical systems

We define the accumulated Gouy phase shift as the on-axis phase accumulated by a Gaussian beam in passing through an optical system, in excess of the phase accumulated by a plane wave. We give an expression for the accumulated Gouy phase shift in terms of the parameters of the system through which the beam propagates. This quantity complements the beam diameter and the wave-front radius of curvature to constitute three parameters that uniquely characterize the beam with respect to a reference point in the system. Measurement of these parameters allows one to uniquely recover the parameters characterizing the first-order system through which the beam propagates.

Propagation of Optical Beams in Active Media Inside Gaussian Cavities

The evolution of relevant beam parameters of a light beam propagating through an iterated sequence of spherical lenses and Gaussian apertures immersed in an active medium is investigated. This structure is a classical model for an optical resonator. The effective size of the field distribution, the beam quality factor and the radius of wave front curvature are calculated as functions of the number of transits. Steady-state values of these parameters are also determined and the dependence of the results on the gain of the medium, the aperture width and the focal length is illustrated. A comparison with the case without active medium is also discussed.

Propagation of partially coherent beams in a periodic sequence of lenses and Gaussian apertures

The evolution of the cross-spectral density of an initially spatially incoherent beam propagating through an iterated sequence of spherical lenses and Gaussian apertures is studied. The global degree of spatial coherence, the spot size and the radius of wavefront curvature are calculated as functions of the number of transits. Steady-state values of the beam parameters are also given in a closed form. The dependence of the results on the aperture width and the focal length is discussed.

Depth‐focusing analysis using a wavefront‐curvature criterion

Depth‐focusing analysis (DFA), a method of refining velocities for prestack depth migration, relies on amplitude buildups at zero offset to determine the extrapolation depths that best focus the migrated data. Unfortunately, seismic energy from dipping interfaces, diffractions, and noise often produce spurious amplitude indications of focusing. To reduce possible ambiguity in the DFA interpretation process, we introduce a new attribute for determining focusing that is relatively independent of amplitude. Our approach is based on estimates of the radius of wavefront curvature. The estimates are derived from normal moveout analysis of nonzero‐offset data saved during migration. By relating steeper moveout to smaller radius of wavefront curvature, focusing is defined by a wavefront curvature of zero radius. Additionally, we show that applying inverse‐radius weights to the amplitude data attenuates nonfocused events due to their large radius of curvature. Using the Marmousi data set, our weighting scheme resulted in reduced spurious focusing and enhanced velocity resolution in DFA.

Optical beam wave propagation through complex optical systems

20We describe a novel formulation of light beam propagation through any complex optical system that can be described by an ABCD ray-transfer matrix. Within the paraxial approximation, optical propagation can be formulated in terms of a Huygens principle expressed in terms of the ray-transfer ABCD matrix elements of the optical system. We extend and generalize previous treatments to include the effects of finite-sized limiting apertures (i.e., diffractive screens) in the optical train, tilt and random jitter of the optical elements, and distributed random inhomogeneities along the optical path (e.g., clear air turbulence and aerosols). In the presence of limiting apertures the ABCD matrix elements of the optical system are complex. For the case of laser beam propagation and Gaussian-shaped limiting apertures in the optical train, we obtain analytical expressions for both the spot radius and the wave-front radius of curvature at an arbitrary observation plane and give illustrative examples of practical concern. In particular, analytical expressions for the fringe visibility obtained in a coherent laser interferometric system are presented. An analytical expression for the mean spot radius of a laser beam propagating through an optical system in the presence of tilt and random jitter is obtained. We also consider the propagation of partially coherent light through optical systems. In particular, we derive a generalized van Cittert–Zernike theorem that is valid for an arbitrary optical system that can be characterized by an ABCD ray-transfer matrix. Finally, the propagation of laser beams through a general optical system in the presence of distributed random inhomogeneities is considered. An explicit expression for the mean irradiance distribution of a Gaussian-shaped beam is derived that is valid for an arbitrary optical system. In addition, we derive an expression for the mutual-coherence function for wave propagation through an arbitrary optical system. In all cases the results are expressed in terms of the ABCD matrix elements of the complete optical system. The formulation of optical propagation presented here is a rather simple and straightforward way of determining the effects of finite-sized optical elements, tilt and random jitter, and distributed random inhomogeneities along the optical path. It is merely necessary first to multiply the relevant ray matrices together to find the complete system matrix and then to substitute this matrix into the expressions given in this paper.

Propagation parameters of gaussian Schell-model beams

The propagation of partially coherent light beams generated by gaussian Schell-model sources is found to be conveniently characterized by two parameters that are analogous to the beam radius and the radius of wavefront curvature of fully coherent gaussian laser beams. Gaussian quasi-homogeneous beams are examined as a limiting case.

WAVEFRONT CURVATURE IN A LAYERED MEDIUM

For a layered elastic medium consisting of horizontal layers with constant velocity as shown in Figure 1, the wavefront radius of curvature is derived as a function of the source‐receiver separation. Newman (1973) derived the radius of curvature as a function of the ray parameter. We shall express the radius of curvature and the source‐receiver separation as functions of the ray parameter which will subsequently be eliminated to give the desired result. A similar technique has been used by Taner and Koehler (1969) to derive an approximate expression for the traveltime of the reflected waves in the same geometric model. The derived expression for the wavefront radius of curvature may be used to compensate for divergence effects in order to derive amplitude information from seismic data, as discussed by Newman (1973).

Fluctuating Properties of Gaussian Light Beams in an Atmosphere

Fluctuations of a Gaussian light beam propagating through an atmosphere, whose refractive index inhomogeneities are assumed to obey the Kolmogorov's law, are discussed statistically from the viewpoint of mode conversion. The results obtained here are concerned with the dependence of the mode conversion on beam shape parameters (beam spot size and radius of wave front curvature), propagation distance, and medium turbulence features. The mode conversion increases proportionally to the propagation distance z in the near-field region of a transmitting aperture and then to 8/3 powers of z in the far-field region. The mode conversion also increases according to 5/3 powers of the beam spot size s at a short distance, while it decreases for a large spot size at a long distance. Studies on wave front curvature are also done. Near the focal point of a focused beam, the reduction effect of the mode conversion appears.