Diffraction patterns formed by an off-axis paraboloid surface

We derive a general, closed-form expression for the diffraction patterns including the aberrations that are due to off-axis alignment and positioning, of a paraboloid mirror. The diffraction patterns obtained in the focal plane of an off-axis paraboloidal mirror suffer modifications by the aberrations that are inherent in these surfaces: astigmatism and coma. Different magnifications in two perpendicular spatial directions indicate astigmatism. The Airy function, which affects a single spatial coordinate, describes the coma aberration. We identified the coma by the increased number of intensity zeros within adjacent lobules.     

Inclined toroidal surface that fits an off-axis conic section

We developed a formulation using the continuous least-squares method to determine the inclined toroidal surface that best fits a given off-axis conic section. A toroid with a known curvature is used to obtain an analytic equation for the angle of inclination of the axis with respect to the normal to the center of the off-axis section.     

The paraboloid failure surface for the general orthotropic material

Summary It has been established that the elliptic paraboloid failure criterion (EPFC) is a potential means of describing the yielding and failure behavior of transversely isotropic (transtropic) materials presenting also strength differential effects along all principal stress axes. The cases studied correspond to symmetric forms of loading which are parallel to the principal strength directions of the material.In this paper the general case of an orthotropic material not presenting any kind of strength symmetry was encountered. It has been shown that an elloptic parabloid surface may continue to represent the failure surface of such materials. However the symmetry of this surface relatively to the principal diagonal plane containing the strong ₃-axis and the bisector of the two remaining axes is destroyed. For these materials the elliptic paraboloid is leaning toward the intermediate strength ₂-axis and the angle of inclination of the plane of symmetry of the paraboloid with the ₁₂-plane depends on the difference in strengths between the ₂- and ₁-axes. The characteristic properties of the inclined paraboloid were studied and important differences between the inclined and the symmetric surfaces were established. Examples with orthotropic polymers and composites were included exemplifying the characteristic properties of orthotropic materials.With 7 Figures     

Design of a low-loss off-axis beam expander

A method is given for the build up of simple off-axis beam expanders for Gaussian beams with low aberrations. In particular we show that it is possible to correct exactly the astigmatism in such a system. A realistic example is given.     

Coherent and incoherent off-axis Hermite-Gaussian beam combinations

A detailed study of the coherent and the incoherent combinations of two-dimensional off-axis Hermite-Gaussian beams with rectangular symmetry is made. The closed-form propagation formulas of the resulting beam are derived, and the resulting beam quality in terms of the M(2) factor and power in the bucket is discussed and compared for the coherent and the incoherent combinations. In addition, it is shown that the resulting astigmatic beam can be symmetrized in the sense of the second-moment definition of beam width. However, the symmetrizing transformation of the resulting astigmatic beams is incomplete, because there exist different irradiance profiles.     

Fractional Fourier transform for partially coherent off-axis Gaussian Schell-model beam

The fractional Fourier transform (FRT) is applied to a partially coherent off-axis Gaussian Schell-model (GSM) beam, and an analytical formula is derived for the FRT of a partially coherent off-axis GSM beam. The corresponding tensor ABCD law for performing the FRT of a partially coherent off-axis GSM beam is also obtained. As an application example, the FRT of a partially coherent linear laser array that is expanded as a sum of off-axis GSM beams is studied. The derived formulas are used to provide numerical examples. The formulas provide a convenient way to analyze and calculate the FRT of a partially coherent off-axis GSM beam.     

Lattice design for subaperture stitching test of a concave paraboloid surface

Lattice design is subtle and complicated for the subaperture stitching test of aspheric surfaces. Methods are described in this paper for the collection and arrangement of subapertures, and calculation of the best-fit sphere for each subaperture. The best-fit sphere is determined by minimizing the mean-square aspheric deviations in the form of a surface integral. Finally, a numerical example is given to illustrate the procedure, and also to verify the validity of our proposed methods.     

High-aperture diffraction of a scalar, off-axis Gaussian beam

A scalar treatment for Gaussian beams offset from the optic axis and then focused by a high-numerical-aperture lens is presented. Such a theory is required for describing certain types of Doppler microscopes, i.e., when the measurement is simultaneously performed by more than a single beam axially offset and then focused by a lens. Analytic expressions for the intensity in the focal region of the high-aperture lens are derived. From these expressions we calculate the intensity in the focal region with parameters of beam size, beam offset, and the numerical aperture of the lens. The relative location and variation of the intensity around the focal region are discussed in detail. We show that for small-diameter Gaussian beams the Strehl ratio increases above unity as the beam is offset from the optic axis. This is explained by the increase in the effective numerical aperture of the offset beam compared with the one collinear with the optic axis. From examining the focal distribution, we conclude that it rotates for small beam size and that increasing beam diameter causes the focused distribution to rotate and shear, i.e., to distort. We also show that the distortion of the distribution increases with increasing numerical aperture.     

Evanescent wave coupling in light scattering of an off-axis normally incident Gaussian beam by two parallel nonabsorbing cylinders

A theoretical and numerical investigation is devised for resonant light scattering of an off-axis normally incident Gaussian beam by two parallel nonabsorbing cylinders based on the related beam theory developed in J. Opt. Soc. Am. A 14, 640 (1997). By varying the half-beam width, we show that the multireflection process between the two scatterers can be minimized. Moreover, the study is an attempt to understand the underlying physics present in the process of resonance excitation by evanescent wave coupling.     

Far-field propagation of an off-axis Gaussian wave

Based on a far-field asymptotic solution of the Helmholtz equation a vector theory to describe the propagation of an off-axis Gaussian wave is developed, the accurate formulas represented in terms of elementary functions are derived, and the propagation properties such as wave spot size and divergence angle are discussed in detail. The applicable range of scalar theory is also presented. A relative error criterion of optical intensity is given by epsilon = sin(2) alpha.     

Electrical simulation of contact stresses for an eccentrically loaded paraboloid of rotation entering an elastic half-space

A method is proposed for solving three-dimensional contact problems by electrical simulation. There is an analogy between the equations describing the reactive-pressure distribution over the contact area and the charge at the surface of a current-carrying plate. A method is given for determining the indentation by a nonplanar plunger driven by an eccentric load and entering an elastic half-space, for which a special electrical simulator is used. This device is briefly described. The error of the method is estimated. A detailed example is considered for a plunger in the form of a second-degree paraboloid of rotation. The inclination of the plunger affects the reactive-pressure distribution on the contact area. Pressure patterns are given for several angles of inclination.Translated from Problemy Prochnosti, No. 1, pp. 57–60, January, 1992.     

Exact solution for the steady-state surface profile of a liquid metal in an external electric field

An analysis is made of the equilibrium profile of the free surface of a liquid metal in an external electric field neglecting gravitational forces. It is shown that a conformal mapping method can be used to find a wide range of exact solutions corresponding to the case of planar symmetry. Pis’ma Zh. Tekh. Fiz. 25, 79–83 (November 26, 1999)     

Freeform lens arrays for off-axis illumination in an optical lithography system

We propose a method of designing a freeform lens array for off-axis illumination (OAI) in optical lithography to produce desired OAI patterns and improve efficiency. Based on the Snell law and the conservation law of energy, a set of first-order partial differential equations are derived and the coordinate relations for each OAI pattern are established. The contours of the freeform lens unit are calculated numerically by solving the partial differential equations, and the freeform lens array is obtained by arraying the lens units. Moreover, the optical performance for each OAI pattern is simulated and analyzed by software. Simulation results show that the irradiance distribution of each OAI pattern can be well controlled with a maximum uniformity of 92.45% and a maximum efficiency of 99.35%. Also, analysis indicates that this method has the advantages of reducing the complexity of the exposure system and having good tolerance to the input intensity variations of the laser beam.     

Far-field scattering of a non-Gaussian off-axis axisymmetric laser beam by a spherical particle

Experimental laser beam profiles often deviate somewhat from the ideal Gaussian shape of the axisymmetric TEM(00) laser mode. To take these deviations into account when calculating light scattering of an off-axis beam by a spherical particle, we use our phase-modeling method to approximate the beam-shape coefficients in the partial wave expansion of an experimental laser beam. We then use these beam-shape coefficients to compute the near-forward direction scattering of the off-axis beam by the particle. Our results are compared with laboratory data, and we give a physical interpretation of the various features observed in the angular scattering patterns.     

Vibration compensating beam scanning interferometer for surface measurement

Light beam scanning using a dispersive element and wavelength tuning is coupled with fiber-optic interferometry to realize a new surface measurement instrument. The instrument is capable of measuring nanoscale surface structures and form deviations. It features active vibration compensation and a small optical probe size that may be placed remotely from the main apparatus. Active vibration compensation is provided by the multiplexing of two interferometers with near common paths. Closed loop control of a mirror mounted on a piezoelectric transducer is used to keep the path length stable. Experiments were carried out to deduce the effectiveness of the vibration compensation and the ability to carry out a real measurement in the face of large environmental disturbance.     

Exact integration of surface and volume potentials

A method for exact analytical integration of potentials from sources distributed on planar and volume elements is presented. The method is based on reduction of the surface integrals to a function similar to an incomplete elliptic integral, giving the integrals in closed form as functions of geometric properties of the surface or volume element. Explicit formulae and recursions are given for the integrals, allowing the evaluation of the potential for arbitrary polynomial sources. Volume integrals are derived from the surface integrals using a simple coordinate transformation which gives the volume integral with little more effort than that required for the surface calculation.