Observation of enhanced backscattering from a plane mirror viewed through polymer-film-dispersed liquid crystals

We report experimental results on enhanced backscattering from a plane mirror that is viewed through polymer-film-dispersed nematic liquid crystals. The distribution of the averaged intensity of the light reflected from the mirror placed behind the polymer film is investigated with an image-processing system when a Gaussian beam wave is incident. The enhanced light peak is observed in an incident beam direction, the result of which is predicted by a theory based on the circular Gaussian statistic random-phase-screen model. We pay attention to the enhancement dependence on parameters such as the distance between the polymer film and the flat mirror. The observed result is similar to a previous study by Jakeman et al. in which a random diffusive glass plate was used as a random-phase screen [J. Phys. D 21, 32 (1988)].     

Basic elliptical Gaussian wave and beam in a uniaxial crystal

Electromagnetic beams in a uniaxial crystal are treated with emphasis on the extraordinary mode. A virtual source that generates a basic elliptical Gaussian wave propagating obliquely to the optic axis is identified. An exact expression is obtained for this basic elliptical Gaussian wave that simplifies to the corresponding basic elliptical Gaussian beam in the appropriate limit. In the direction of amplitude propagation, the paraxial result becomes identical to the exact result and the sum of all the nonparaxial contributions vanish. The characteristics of the basic elliptical Gaussian beam are illustrated with a numerical example. From the spectral representation of the basic Gaussian wave, the first three orders of nonparaxial corrections for the basic elliptical Gaussian beam are determined. The nonparaxial results reduce correctly to those of the fundamental Gaussian beam in an isotropic medium.     

Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam

The truncated fractional Fourier transform (FRT) is applied to a partially coherent Gaussian Schell-model (GSM) beam. The analytical propagation formula for a partially coherent GSM beam propagating through a truncated FRT optical system is derived by using a tensor method. Furthermore, we report the experimental observation of the truncated FRT for a partially coherent GSM beam. The experimental results are consistent with the theoretical results. Our results show that initial source coherence, fractional order, and aperture width (i.e., truncation parameter) have strong influences on the intensity and coherence properties of the partially coherent beam in the FRT plane. When the aperture width is large, both the intensity and the spectral degree of coherence in the FRT plane are of Gaussian distribution. As the aperture width decreases, the diffraction pattern gradually appears in the FRT plane, and the spectral degree of coherence becomes of non-Gaussian distribution. As the coherence of the initial GSM beam decreases, the diffraction pattern for the case of small aperture widths gradually disappears.     

Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics

We report the experimental observation of the fractional Fourier transform (FRT) for a partially coherent optical beam with Gaussian statistics [i.e., partially coherent Gaussian Schell-model (GSM) beam]. The intensity distribution (or beam width) and the modulus of the square of the spectral degree of coherence (or coherence width) of a partially coherent GSM beam in the FRT plane are measured, and the experimental results are analyzed and agree well with the theoretical results. The FRT optical system provides a convenient way to control the properties, e.g., the intensity distribution, beam width, spectral degree of coherence, and coherence width, of a partially coherent beam.     

Effects of finite inner and outer scales on the scintillation index of turbulent slant path

In this paper, the effects of nonzero inner scales and finite outer scales are investigated, in the context of Gaussian beam propagation along a slant path under general turbulence conditions. Theoretical expressions for the cut-off spatial frequencies are derived with an approach method, and thereby a modified scintillation model is developed to incorporate inner scale and outer scale parameters in the analysis. Then, inner and outer scale effects on the downlink are analysed with respect to the zenith angle, the altitude of the transmitter, the initial beam radius, as well as the turbulence strength. Numerical results indicate that the effects of a finite outer scale mainly influence transmission that occurs at large zenith angles or high altitudes, while the inner scale effects are more prevalent. This study may be helpful to improve the accuracy of calculation of slant-path scintillation index, and thus benefit the characterization and optimization of space/air-ground laser communication systems.     

Noncollimated bidirectional shearing interferometer for measuring a long radius of curvature

A method for measuring a long radius of curvature with a modified half-aperture bidirectional shearing interferometer is described. A plane mirror of the interferometer is replaced with a test mirror, and the incident beam is decollimated by shifting of the source to equalize the widths of the bidirectional shearing fringes reflected from the plane and test mirrors. This method can be applied to concave and convex surfaces.     

焦区附近球形粒子对高斯光束的弹性散射

本文应用复源球面波理论,将高斯光束场按矢量球波函数展开,对高斯光束入射到单个球形粒子上时的弹性散射问题进行了理论分折。并对球形粒子在波束传播轴上时的远场散射光强角分布进行了数值计算,同时还与平面波散射的结果进行了比较。     

Dynamics of the linearly polarized fundamental Gaussian light wave

A continuous planar array of dipoles that are oriented in a particular direction and have an amplitude distribution that is Gaussian in the paraxial limit is introduced as a source for the fundamental Gaussian light wave. The radiation intensity of the Gaussian light wave is determined and its characteristics are analyzed. The universal Gaussian beam factor is deduced and identified as the radiation intensity of the scalar Gaussian wave. The total radiated power, the mean center of the localized wave, and the beam widths of the intensity distribution are obtained. The ratio of the power in the Gaussian wave to that in the corresponding paraxial Gaussian beam is used as a measure of the quality of the paraxial beam approximation. A limiting factor for the power ratio is introduced as an indicator for the acceptability of the paraxial beam approximation. The cross section and the beam widths of the localized light wave are investigated in the large and small kw0 limits, where k is the wavenumber and w0 is the beam waist at the input plane. The beam width of the full Gaussian wave is found to be less than that of the corresponding paraxial Gaussian beam both for the scalar Gaussian wave and for the Gaussian light wave.     

Reflection of a Beam of Finite Size from a Corrugated Waveguide

Abstract

The paper is concerned with the influence of the finite size of an incident beam upon both the anomalous-reflection spectrum and the shape of the energy distribution in a reflected beam. It is proved experimentally that the use of corrugated waveguides as a laser-resonator selective mirror improves the spatial coherence of radiation.     

Derivation of a Monte Carlo method for modeling heterodyne detection in optical coherence tomography systems

A Monte Carlo (MC) method for modeling optical coherence tomography (OCT) measurements of a diffusely reflecting discontinuity embedded in a scattering medium is presented. For the first time to the authors' knowledge it is shown analytically that the applicability of an MC approach to this optical geometry is firmly justified, because, as we show, in the conjugate image plane the field reflected from the sample is delta-correlated from which it follows that the heterodyne signal is calculated from the intensity distribution only. This is not a trivial result because, in general, the light from the sample will have a finite spatial coherence that cannot be accounted for by MC simulation. To estimate this intensity distribution adequately we have developed a novel method for modeling a focused Gaussian beam in MC simulation. This approach is valid for a softly as well as for a strongly focused beam, and it is shown that in free space the full three-dimensional intensity distribution of a Gaussian beam is obtained. The OCT signal and the intensity distribution in a scattering medium have been obtained for several geometries with the suggested MC method; when this model and a recently published analytical model based on the extended Huygens-Fresnel principle are compared, excellent agreement is found.     

Effective beam parameters and the turbulent beam waist for convergent Gaussian beams

Expressions are developed for the location and the size of the beam waist for a convergent Gaussian beam in statistically homogeneous and isotropic atmospheric turbulence. Subsidiary expressions are presented that lead to the maximum distance from the transmitter at which the beam waist can be located under given optical turbulence conditions and the optimal initial radius of curvature required for placing the beam waist at a desired location. The free-space beam radius W of a Gaussian beam satisfies the relationship ?W/?z = - W/R, where z represents the path length and R is the phase-front radius of curvature at z. By enforcing this relation on the effective beam spot size in turbulence W(e), we can define an effective radius of curvature R(e). In addition to specifying the beam waist, R(e) leads to a pair of effective beam parameters θ(e) and Λ(e) that provide a natural extension to the complex amplitude plane. Within this context, general propagation characteristics may be described, including the coherence properties of a Gaussian beam in both weak and strong optical turbulence.     

Multilayer-coating-induced aberrations in extreme-ultraviolet lithography optics

A multilayer coating alters the amplitude and phase of a reflected wave front. The amplitude effects are multiplicative and well understood. We present a mathematical formalism that can be used to describe the phase effects of coating in a general case. On the basis of this formalism we have developed an analytical method of estimating the wave-front aberrations introduced by the multilayer coating. For the case of field-independent aberrations, we studied both uniform and graded multilayer coatings. For the case of field-dependent aberrations, we studied only the effects of a uniform multilayer coating. Our analysis is based on a coated plane mirror tilted with respect to an incident converging beam. Altogether we have found, up to the second order, the following aberrations: a field-dependent piston, a field-squared-dependent piston, defocus, field-independent tilt, field-independent astigmatism, and anamorphic magnification. To obtain numerical results we apply our analysis to the specific case of a plane mirror tilted 8.2 deg with respect to an incident converging beam with a numerical aperture of 0.1. We find that the magnitudes of the field-independent aberration coefficients for the graded coating are approximately ten times smaller than those for the uniform coating. We show that a coating can introduce anamorphic magnification.     

Diffraction and focusing of extremely short optical pulses: generalization of the Sommerfeld integral

The Sommerfeld diffraction theory is extended to the case of extremely short pulses. We show that in the far field the energy-density distribution of the diffractional pattern is transformed into a Gaussian distribution, when the plane wave with uniform radial amplitude distribution and one oscillation period falls upon the circular aperture. When one focuses the Gaussian beam with one oscillation period, the energy-density distribution in the focal plane differs from the Gaussian distribution.     

Plane gratings for high-resolution grazing-incidence monochromators: holographic grating versus mechanically ruled varied-line-spacing grating

Comparative studies have been made on the holographic plane grating and the ruled varied-line-spacing (VLS) plane grating designed for two kinds of objective Monk-Gillieson type high-resolution grazing incidence monochromator, I and II. The ray-traced performance of monochromator types I and II on a synchrotron radiation beam line was evaluated in terms of resolving power and spectral purity by the introduction of new concepts of effective Gaussian line and purity profiles. The resolving power defined on the basis of the effective Gaussian profile is consistent with the spectral purity of the beam emerging from the exit slit and is more realistic as compared with those defined in the conventional manner, especially when spectral images have asymmetric profiles. It is concluded that holographic plane gratings recorded with a spherical and an aspheric wave front are capable of providing high resolution with high spectral purity and are fully interchangeable with the corresponding ruled VLS plane gratings. This interchangeability provides more flexibility for users in choosing a proper grating for a high-resolution grazing incidence monochromator of the Monk-Gillieson type.     

Spatial fidelity of image amplification in photorefractive crystals

We numerically studied the spatial fidelity of coherent image amplification by two-wave mixing configurations in photorefractive BaTiO(3):Ce using a three-dimensional analysis. The results are given for the case when the input one-dimensional rectangular amplitude of the image-bearing extraordinary beam is finite in the plane of incidence and when the amplitude is finite in the orthogonal plane. The fidelity and the gain versus the angle between the propagating direction of the image-bearing beam and the crystal c axis, the pump-to-image intensity ratio, and the input beamwidth, are analyzed.     

Electromagnetic Beam Fields

An electromagnetic beam is defined using mathematical properties of the associated angular spectrum of plane waves. It is found that the usual paraxial theory for the Hermite Gaussian or Laguerre Gaussian beams, produced by some lasers, can be replaced by a more general theory which is precise according to Maxwell's equations. In this theory the beams exhibit an amplitude distribution over any plane normal to the direction of propagation which can be described using prolate spheroidal wave functions. As the degree of collimation is increased, these beams asymptotically take on the familiar Gaussian amplitude cross section. However, as the divergence from focus is increased, these beams asymptotically approach modified dipole fields. It is found that two, mutually exclusive, classes of beam fields exist. For each beam in one class there is always a complementary beam in the other class. As the degree of collimation is increased, complementary beams become almost identical. Complementary beams contain electromagnetic components which are related to one another in the same manner as between the fields of similar electric and magnetic multipoles.     

Laser beam intensity profile transformation with a fabricated mirror

A nonaxisymmetric mirror is designed by the same method as a computer-generated hologram for laser beam intensity profile transformation and is fabricated by plasma chemical vaporization machining. We successfully transformed a circular Gaussian beam of a He-Ne laser into a rectangular uniform beam maintaining spatial coherence and using a nonaxisymmetric surface profile mirror. There are ripples in the intensity profile of the transformed rectangular beam. These ripples in the intensity profile result from small ripples on the mirror surface. These results show that we can perform coordinate transformation using these fabricated mirrors, which has so far been possible only by using computer-generated holograms.     

Stochastic electromagnetic beams focused by a bifocal lens

In this paper, we study the focusing of a stochastic electromagnetic beam by a bifocal lens. By taking the electromagnetic Gaussian Schell-model (EGSM) beam as an example, the changes in the spectral density, in the spectral degree of coherence, and in the spectral degree of polarization of the EGSM beam as the beam is focused by an unapertured bifocal lens are investigated. It is shown that the spectral density, the spectral degree of coherence, and the spectral degree of polarization of the focused electromagnetic EGSM beams depend upon the coherence lengths and focal lengths of the bifocal lens. The influence of the coherence lengths and the focal lengths on the focused spectral density, the spectral degree of coherence, and the spectral degree of polarization are investigated in great detail.     

Family of hypergeometric laser beams

We discuss a three-parameter family of paraxial coherent light fields that originate from a complex amplitude composed of the following four cofactors: the Gaussian beam, a logarithmic axicon, a spiral phase plate (angular harmonic), and an amplitude power function with a possible singularity at the origin of coordinates. For such types of beams, the near-field complex amplitude is proportional to the degenerate hypergeometric function, prompting the beams' name--hypergeometric (HyG) beams. When the Gaussian beam is replaced by a plane wave, the above beams change to generalized HyG modes that preserve their structure up to scale upon propagation. The intensity profile of the HyG beams is similar to that of the Bessel modes, forming a set of alternating bright and dark rings. However, the thickness of the rings of the HyG beams decreases with increasing ring number.     

Unified theory for parallel and focused beam nonspecular reflection at liquid-solid interfaces

Summary This paper presents a theoretical model based on wave mechanics for the reflection of focused and parallel ultrasonic beams from a liquid-solid interface. The incident beam is defined by a Gaussian velocity distribution along a plane emitter, and the reflected beam is described through its pressure field by means of asymptotic analysis based on the short wave hypothesis. In the case of a focused reflected beam, nonspecular phenomena are observed for any angle of incidence. However, in the case of a parallel reflected beam, nonspecular reflection only occurs if the incidence of the beam is in the neighborhood of the Rayleigh angle. The objective of this paper is to discuss in detail the physical mechanisms of nonspecular reflection for both focused and unfocused beams and to provide an understanding of the complex phenomena related to nonspecular reflection of ultrasonic beams, which plays an important role in scanning acustic microscopy for materials nondestructive characterization.