Showing papers on "Paraxial approximation published in 1995"

Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation

Abstract: The diffraction of electromagnetic waves for light focused by a high numerical aperture lens from a first material into a second material is treated. The second material has a different refractive index from that of the first material and introduces spherical aberration. We solve the diffraction problem for the case of a planar interface between two isotropic and homogeneous materials with this interface perpendicular to the optical axis. The solution is obtained in a rigorous mathematical manner, and it satisfies the homogeneous wave equation. The electric and magnetic strength vectors are determined in the second material. The solution is in a simple form that can be readily used for numerical computation. A physical interpretation of the results is given, and the paraxial approximation of the solution is derived.

Optical Design Fundamentals for Infrared Systems

Abstract: Radiometric considerations. Basic optics. Primary aberrations. Wave aberrations. Special optical surfaces and components. Design examples. Thermal effects. Optical coatings. Image evaluation. Diamond turning. Appendix A.1: Paraxial ray tracing. Appendix A.2: Spherical aberration of a thin lens.

Beyond ray tomography: Wavepaths and Fresnel volumes

Abstract: Two techniques that account for the band-limited nature of seismic data are incorporated into tomographic traveltime inversion schemes. The first technique, the wavepath algorithm, is based upon the wave equation, the Born approximation, and an adjoint method for computing Frechet derivatives. Computation of a single wavepath requires the forward propagation of the seismic wavefield, as well as the reverse propagation of a residual wavefield. The second technique, the Fresnel volume approach, is based upon the paraxial ray approximation. The Fresnel volume algorithm requires little more computation than does conventional ray tracing and an order of magnitude less computer time than our calculation of wavepaths. When the Fresnel volume sensitivity functions are normalized by the area of the Fresnel ellipse perpendicular to the ray, the sensitivity estimates are very similar to the wavepaths. In particular, there is heightened sensitivity to velocity structure near the source and receiver locations. The normalization by the Fresnel ellipse area is necessary to ensure ray theoretical results in the limit of infinite frequency. Tomographic inversion based upon wavepaths or Fresnel volumes is more appropriate when considering the arrival time of the peak of the initial pulse rather than the first-arrival time. Furthermore, using the traveltime of the peak instead of the first-arrival time reduces the bias of tomograms to high velocity anomalies. The raypath, wavepath, and Fresnel volume techniques were applied to a set of cross-borehole traveltime observations gathered at the Grimsel Rock Laboratory. All methods imaged a low velocity fracture zone in the granitic site, in agreement with independent well information. Estimates of model parameter resolution are similar for the wavepath and Fresnel volume schemes. The source-receiver regions are the most well resolved areas. However, the model parameter resolution computed using a conventional ray-based formalism is more evenly distributed over the cross-borehole area.

Gaussian beam depth migration for anisotropic media

Abstract: Gaussian beam migration (GBM), as it is implemented today, efficiently handles isotropic inhomogeneous media. The approach is based on the solution of the wave equation in ray‐centered coordinates. Here, I extend the method to work for 2-D migration in generally anisotropic inhomogeneous media. Extension of the Gaussian‐beam method from isotropic to anisotropic media involves modification of the kinematics and dynamics in the required ray tracing. While the accuracy of the paraxial expansion for anisotropic media is comparable to that for isotropic media, ray tracing in anisotropic media is much slower than that in isotropic media. However, because ray tracing is just a small portion of the computation in GBM, the increased computational effort in general anisotropic GBM is typically only about 40%. Application of this method to synthetic examples shows successful migration in inhomogeneous, transversely isotropic media for reflector dips up to and beyond 90°. Further applications to synthetic data of lay...

Atom-optical properties of a standing-wave light field

Abstract: The focusing of atoms to nanometer-scale dimensions by a near-resonant standing-wave light field is examined from a particle optics perspective. The classical equation of motion for atoms traveling through the lens formed by a node of the standing wave is derived and converted to a spatial trajectory equation. A paraxial solution is obtained, which results in simple expressions for the focal properties of the lens, useful for estimating its behavior. Aberrations are also discussed, and an exact numerical solution of the trajectory equation is presented. The effects on focal linewidth of angular collimation and velocity spread in the atomic beam are investigated, and it is shown that angular collimation has a much more significant effect than velocity spread, even when the velocity spread is thermal.

Decentered Gaussian beams

Abstract: A generalization of the Gaussian beam is obtained by introducing a complex-valued shift in the transverse dimension. The resulting beam has a Gaussian intensity distribution with width varying as an ordinary Gaussian beam, but whose peak traces an inclined linear trajectory. The wave fronts are displaced laterally in a sheared fashion. This generalized beam preserves its form after passing through arbitrary paraxial optical components, even if they are decentered. The peak-intensity line is modified by such systems as if it were a ray.

Slow-wave finite-difference beam propagation method

Abstract: By invoking the slow-wave approximation, the wave equation resumes the form of the Fresnel equation. Codes developed previously for the paraxial beam propagation can be extended to simulate the backward reflection and diffraction at any angle. Results of planar waveguide gratings and a beveled corner bend are presented. >

Scattering of an off-axis Gaussian beam by a dielectric cylinder compared with a rigorous electromagnetic approach

Abstract: For the development of new kinds of refractive-index sensors for capillary electrophoresis, we investigated the scattering of an off-axis incident Gaussian beam by a homogeneous dielectric cylinder at normal incidence. The numerical calculations are based on the exact solution of the Helmholtz equation in circular cylindrical coordinates. Contrary to the procedures with geometrical and paraxial models, this procedure gives accurate results even when the beam dimensions are of the order of the wavelength of light and when the beam diameter is greater than the diameter of the dielectric cylinder. This rigorous electromagnetic treatment is verified by experimental measurements for cylinders with diameters from 5 to 100 μm.

Topological phase in Gaussian beam optics

Abstract: The evolution of a complex beam width parameter g describing focusing or defocusing of a paraxial Gaussian beam is considered as the cause of an additional topological (Berry) phase of the electromagnetic field associated with the beam. It is pointed out that the well-known Gouy phase is a special case of such a phase that should arise in general in all symplectic systems.

Splitting of operators, alternate directions, and paraxial approximations for the three-dimensional wave equation

Abstract: We design a family of new three-dimensional (3-D) paraxial equations well-adapted to numerical solution by the alternate directions method. These equations do not suffer from bad anisotropic effects and the cost for their numerical integration remains cheap in comparison with classical paraxial equations.

Stability issues in vector electric field propagation

Abstract: Demonstrates that although standard paraxial and wide-angle vector field propagation techniques lead to divergences for sufficiently small grid-point spacings and large refractive index differences, stability may be restored through either certain Pade approximates to the propagation operator or suitable boundary conditions. The authors also introduce a novel alternating directional implicit method applicable to less divergent discretizations of the vector wave equation. >

Paraxial wave theory of second and third harmonic generation in uniaxial crystals. I: Narrowband pump fields

Abstract: We present a paraxial wave formulation of three-wave mixing in a negative uniaxial crystal, including effects of diffraction and transverse walkoff. The theory, though general, is applied specifically to second harmonic generation in KDP with Nd:glass laser radiation, followed by the mixing of the second harmonic with the fundamental in a second KDP crystal to produce third harmonic radiation near 0·35 μm. For applications of interest, which can involve third harmonic conversion efficiencies approaching 90%, walkoff is potentially much more deleterious to conversion than diffraction. However, walkoff is negligible when the angular spectrum of the pump field does not have substantial contributions from spatial frequencies ≳ (aL)−1, where L is the crystal length and a depends upon the derivative of each extraordinary refractive index with respect to the angle between the optic axis and the direction of wave propagation. A similar result, scaling as L− 1/2 rather than L −1, holds for diffraction. We...

Multiple scattering of narrow light beams in aerosols

Abstract: A multiple scattering propagation model of narrow light beams in aerosol media is described. It is based on a paraxial approximation of the radiative transfer equation in which the flux normal to the incident beam direction is modeled by a diffusion process. The model solutions are the forward- and backscattered intensity profiles for the specified geometry and receiver aperture and field of view. The required inputs are the system parameters, and the aerosol single scattering angular phase function and extinction and scattering coefficients which are allowed to vary along the beam axis. Good agreement is shown with measurements performed in the laboratory over scales ranging from a few tens of mm to a few m, and in the atmosphere over a scale of the order of 1 km. The solutions are valid for optical depths smaller than ≈ 10, for phase functions corresponding to average size parameters of order one or greater, and for off-axis positions not exceeding ≈ 25% of the reciprocal of the scattering coefficient.

Chaos and low-order corrections to classical mechanics or geometrical optics.

Abstract: Based on simple first-order quantum corrections to classical equations of motion, which we show to be closely related to Gaussian wave-packet dynamics (GWD) and a time-dependent variational principle (TDVP), we deduce that quantum corrections to classical dynamics should typically become most pronounced when the classical system becomes chaotic. The time duration over which classical dynamics, GWD, or TDVP may provide good approximations is much shorter when the classical dynamics are chaotic. However, for certain situations involving very short laser pulses, these approximations can be very accurate. The same concepts are applicable to paraxial wave optics, which may offer simpler experimental studies of ``quantum chaos'': the distinction between classical and ``quantum'' chaos is in large part the distinction between ray versus wave behavior.

Processing method and processor using the same

Abstract: PURPOSE: To obtain a processing method adequate for paraxial calculation of an optical system for calculating the paraxial quantity developed around the reference axis of the optical system including a curved surface (off-axial curved surface) which is not the plane having the plane normal not aligned to the reference axis at the point where the optical path (reference axis) of the reference wavelength from an object plane to an image plane intersects with the curved surface and a processor using the same. CONSTITUTION: At least one among A, D, B, Φ of the Gaussian bracket by which the calculation equation is obtained, focal length, two principal point positions, magnification β and the paraxial quantity of back focus are calculated for each of the respective faces or over the entire system by the technique developed around the reference axis of the optical system including the off-axial curved surface which is not the plane having the plane normal not aligned to the reference axis at the point where the reference axis of the reference wavelength from the object plane to an image plane intersects with the curved surface.

Single-pass and double-pass propagation through complex paraxial optical systems

Abstract: A generalized form of spectral representation theory is developed and used with the ABCD formulation of the Huygens–Fresnel integral for studying optical wave propagation through a random medium in the presence of any complex paraxial optical system that can be characterized by an ABCD ray matrix. Formal expressions are developed for the basic optical field moments and various related second-order statistical quantities in terms of three fundamental moments of the first- and second-order complex phase perturbations. Special propagation environments include line-of-sight propagation, single-pass propagation through arbitrary ABCD optical systems, and double-pass propagation through the same random medium in the presence of an ABCD optical system. For illustrative purposes the method is used in the development of expressions for the mean and the normalized variance of the irradiance associated with the Fourier-transform-plane geometry of a lens and the enhanced backscatter effect (EBS) associated with irradiance and phase fluctuations of a reflected Gaussian-beam wave from a Gaussian mirror. The EBS analysis accounts for both finite size and finite focal length of the mirror.

Generalized acoustic diffraction tomogra phy1

Abstract: A generalized diffraction tomography algorithm is developed, which in principle can handle irregularly spaced data, curved acquisition lines and non-uniform background models. By direct comparison with medical diffraction tomography, it is shown that the generalized method involves the same two processing steps: data filtering and back-propagation. The filter handles the irregular sampling of the model space and the uneven energy coverage, while the back-propagation operator removes the wave propagation effects. Paraxial ray-tracing techniques are employed to compute both these quantities. In medical diffraction tomography, the resolution vector (i.e. the Fourier vector of the model space) is defined by the incident and scattered plane-wave directions. It is shown here that a similar relationship exists for a non-uniform background, where the resolution vector at a particular image point is defined by the incident and scattered ray directions. Consequently, the impulse response of the generalized algorithm becomes space variant. Finally, a general processing procedure for transmission mode seismic data, based on this generalized algorithm, is proposed. The potential of the method is demonstrated using synthetic cross-hole data.

Time-dependent perturbation formalism for calculating the aberrations of systems with large ray gradients

Abstract: A time-dependent perturbation formalism is outlined which enables the calculation of aberrations for systems with large gradients of the particle trajectories in a consistent way. The position of a particle is referred to that of an axial reference particle making the axial coordinate a small quantity. The initial conditions of the trajectory are incorporated by transforming the Lorentz equations in a set of coupled inhomogeneous integral equations for the three positional coordinates of the particle. The integral equations are solved by an iteration procedure which starts from the paraxial approximation. The number of required iteration steps is equal to the rank of the aberration minus one. As an example, the primary and secondary axial aberrations of mirrors are investigated in detail.

Snake beam: a paraxial arbitrary focal line.

Abstract: The creation of paraxial arbitrary focal lines by a Fourier computer-generated hologram is demonstrated. The desired focal line is represented by a series of connected straight line segments, each of which is implemented by a radial harmonic function located on a different radial portion of the entire hologram. Each subhologram is multiplied by appropriate linear and quadratic phase functions and is shifted by some distance from the center. The two phase factors determine the location of each line segment, while the in-plane shift determines the tilt angle of the segment.

Spatiotemporal effects in double phase conjugation

Abstract: Spatial and temporal effects arising in photorefractive crystals during the process of double phase conjugation are analyzed numerically with a novel beam-propagation method. Slowly varying envelope wave equations in the paraxial approximation are solved under the appropriate boundary conditions. Our analysis includes dynamical effects caused by the buildup of diffraction gratings in the crystal and the turn-on of phase-conjugate beams as well as spatial effects caused by the finite transverse spread of beams and by the propagation directions of the beams. Various phenomena are observed, such as self-bending of phase-conjugate beams, convective flow of energy out of the interaction region, mode oscillations, critical slowing down at the oscillation threshold, and irregular spatial pattern formation. For a real beam-coupling constant and constructive interaction of interference fringes in the crystal we find steady or periodic behavior. For a complex coupling constant and/or induced phase mismatch in the grating a transition to spatiotemporal chaos is observed. We believe that under stable operating conditions the transverse double phase-conjugate mirror in the paraxial approximation is a convective oscillator, rather than an amplifier. Improved agreement with experimental results is obtained.

Measurement of the Wigner distribution function based on the inverse Radon transformation

Abstract: The Wigner Distribution Function (WDF) has been well known in quantum mechanics since the thirties. Applied to paraxial optics it can be considered as a phase-space representation of quasi-monochromatic, partially coherent beams. For a two-dimensional beam (one transverse dimension) it depends on one spatial and one angular coordinate. Due to some properties of the WDF, it may be considered as an intensity distribution of geometrical rays, depending on position and direction, although this analogy is limited. Once the WDF of a laser beam in one transverse plane is known, the intensity distribution behind any optical system can be derived from it. The measurement of the WDF cannot be connected with a single intensity measurement as can be shown easily. Several suggestions for measurement procedures involving different kinds of apertures have been made. But all of them suffer from the influences of those apertures. Here we present a method which uses only a single focusing lens and several intensity measurements. It is based on a known mathematical procedure called the inverse radon transformation. An experimental result of applying this method to a laser beam emerged by an unstable resonator is presented, too.

Focusing of ultrashort laser pulses by the combination of diffractive and refractive elements

Abstract: Hybrid elements, containing optical power with both diffractive (holographic) and refractive components, are shown to be able to eliminate the effect of propagation time difference. The consideration is provided through a paraxial approximation of diffraction theory.

Synthesis of paraxial-domain diffractive elements by rigorous electromagnetic theory

Abstract: In the paraxial domain of diffractive optics diffractive elements are, in general, designed by scalar theory. The resulting elements possess transverse features that are large compared with the wavelength. We demonstrate some advantages of using wavelength-scale features and hence rigorous diffraction theory in the paraxial domain of diffractive optics. Our design procedure leads to elements with efficiencies that exceed by a considerable margin the upper bounds of diffraction efficiency predicted by scalar synthesis theory.

Analysis of a three-dimensional computer optical scheme based on bidirectional free-space optical interconnects

Abstract: We describe the optical design of an optoelectronic 3-D system that is being developed by the Optoelectronic Computing Systems Center at the University of Colorado to prove the utility and viability of 3-D computers that use free-space optical interconnects to achieve a high degree of global connectivity among the PEs of a fine-grained parallel computer The features of the VCSEL array as a source of coherent emission for hologram reconstruction and the CGH design procedure are discussed An optical design in paraxial approximation of the 3-D computer with bidirectional 8 x 8 holographic interconnects is presented The effect of VCSEL wavelength variation on diftraction crosstalk is estimated The aberration in optical system based on the shelf objective is calculated, and a distortion compensation procedure is proposed

Aberration analysis of Wien filters and design of an electron energy-selective imaging system

Abstract: Focusing-imaging, dispersive and aberration performances of Wien filters have been studied. The first order chromatic aberration and second order geometric aberration coefficients for Wien filters not satisfying the stigmatic focusing condition have been derived. An aberration correction method has been proposed using the symmetry of fields and fundamental paraxial rays to correct all the second order aberrations. A design of an energy-selective imaging system using the symmetry and consisting of four identical cascading Wien filters with an octupole electrical magnetic structure is described, and the imaging properties of the energy-selective imaging system has been computed.

3D free electron laser gain for an electron beam with finite emittance and energy spread

Abstract: A generalization of the analytical 3-D gain derivation for an arbitrary free electron laser (FEL) with a Hermite-Gaussian TEM wave is presented for the low gain, small signal regime. The finite emittance and energy spread of the electron beam as well as the orbit distortions and misalignments are rigorously taken into account. The paraxial approximation is used both for the electron beam and the optical wave. The expression for FEL gain is analytically reduced from an integral over eight dimensions to a two dimensional integral along the FEL. Some peculiar FEL modes are used for illustration of this method.