Analytic solution for the peak reflectivity of an asymmetric mirror

Asymmetric Bragg reflectors have been shown to optimize mirror performance in strained-layer material systems. Although the theory behind the reflectivity of symmetric mirrors has been well studied, little is known about asymmetric mirror designs. We present a closed-form solution for the peak reflectivity of an asymmetric mirror that results from applying a tanh substitution. This elegant technique has been shown to yield a markedly simplified calculation of the peak reflectivity of a symmetric mirror. These analytic expressions will be useful in mirror design by providing a straightforward way to compare the trade-offs between asymmetric and symmetric mirror designs.     

Vibration of a thickness-twist mode piezoelectric resonator with asymmetric, nonuniform electrodes

We studied the effect of electrodes with varying thicknesses on thickness-twist modes in a piezoelectric plate resonator of crystals of 6mm symmetry. The focus is on the effects of asymmetric electrodes, which do not seem to have been examined before. A theoretical analysis is performed using the theory of linear piezoelectricity. A trigonometric series solution is obtained that is relatively rare from the equations of piezoelectricity. Numerical calculations are made based on the series solution. Results show that asymmetric, nonuniform electrodes have a strong effect on vibration mode shapes. This effect offers the possibility of using nonuniform electrodes in design to achieve various goals. The quantitative results in this paper are useful for the purpose.     

Optical design of freeform two-mirror beam-shaping systems

The problem of design of a two-mirror optical system for reshaping the irradiance distribution of a laser beam in a prescribed manner is considered in the geometrical optics approximation. The presented design equations are derived in a rigorous manner and are applicable to two-mirror optical systems not limited to radiance profiles and beam cross sections that are rotational or rectangular symmetric. The resulting mirrors are free-form surfaces not restricted by a priori constraints. Moreover, the presented approach shows also that even in the general case two different designs are available for the same data. In one of these designs the first mirror is always concave and the second is convex, while in the second design the resulting mirrors may be neither convex nor concave. Since, in general, the surface mirrors are aspherical, the availability of a design with convex and concave mirrors is particularly important for fabrication.     

Extreme-ultraviolet Mo/Si multilayer mirrors deposited by radio-frequency-magnetron sputtering

Mo/Si multilayer mirrors with a high reflectance at normal incidence in the 130-135-? spectral region have been deposited by rf magnetron sputtering. A new and quick technique was used to calibrate the deposition rates. Characterization by transmission electron microscopy (TEM) and grazing-incidence x-ray diffraction (XRD) indicated good thickness control of the deposition process and low interface roughness. However, the TEM and, indirectly, the XRD reflectance measurements, indicated that the interfaces are asymmetric. A brief review discussing the origin of the modulation of the Bragg peak intensities in the XRD reflectance is given. An analytical formula was derived for periodic multilayers that describes the effect of asymmetric interfaces on the amplitude of the Bragg peak modulation. Theoretically, in XRD reflectance measurements, any asymmetry in the interdiffusion of the Mo-Si interfaces results in a decrease of the usual amplitude modulation of the Bragg peaks. Extreme-ultraviolet (XUV) reflectance measurements were also made with synchrotron radiation on a new high-resolution reflectometer. The near-normal-incidence peak reflectances measured at λ = 134 ? were ~ 59% for the best multilayer mirrors. Good fits to both XRD and XUV reflectance measurements have been obtained with a model that allows for interface asymmetry.     

Optical tests of nanoengineered liquid mirrors

We describe a new technology for the fabrication of inexpensive high-quality mirrors. We begin by chemically producing a large number of metallic nanoparticles coated with organic ligands. The partides are then spread on a liquid substrate where they self-assemble to give optical quality reflective surfaces. Since liquid surface can be modified by a variety of means (e.g., rotation, electromagnetic fields), this opens the possibility of making a new class of versatile and inexpensive optical elements that can have complex shapes and that can be modified within short time scales. Interferommetric measurements show optical quality surfaces. We have obtained reflectivity curves that show 80% peak reflectivities. We are confident that we can improve the reflectivity curves because theoretical models predict higher values. We expect nanoengineered liquid mirrors to be useful for scientific and engineering applications. The technology is interesting for large optics, such as large rotating parabolic mirrors, because of its low cost. Furthermore, because the surfaces of of ferrofluids can be shaped with magnetic fields, one can generate complex, time-varying surfaces that are difficult to make with conventional techniques.     

Fractal Roughness Structure of Diamond-Turned Copper Mirrors

Single-point diamond turning of suitable materials (copper, aluminum, and electroless-plated nickel, etc.) can generate high precision mirrors and aspherical optics rapidly and repeatedly. We have examined the diamond-turned copper mirrors that exhibit a high-frequency periodic feature with an atomic force microscope on various scales. Like other super-smooth surfaces, such as super-ground surfaces, single-point-diamond-turned surfaces may also be identified as a self-affine fractal in the stochastic sense with its correlation length. The exponent for root-mean-square roughness was obtained to be 0.28 for a feed rate of 1.4 μm/rev. With this value, we can predict the roughness value of the single-point-diamond-turned copper mirrors at any scale length, if the scale is within the correlation length, and provided that one such value of a scale is known. The roughness exponent is a very useful parameter, knowledge of which helps to improve the quality of the mirror surface and evaluate its performance in optical instruments.     

Phase Conjugation

Abstract

The background mechanisms responsible for the production of phase-conjugate waves are presented in a unified way. One set of wave equations can be used to describe both four-wave mixing and stimulated backscattering phase-conjugate mirrors. The nonlinear-optical mechanism of photorefraction is described in detail and several of the most exciting potential applications of phase-conjugate mirrors are highlighted.     

Theory of quarter-wave-stack dielectric mirrors used in a thin fabry-perot filter

I present a new derivation of the analytic form for the phase shift near resonance and the optical penetration length upon reflection from a distributed dielectric mirror consisting of a quarter-wave stack. The requirement of proper termination to achieve high reflectivity is suspended to investigate large optical penetration depths. Separate equations, derived for N and N + 1/2 layer pairs, are convenient for the design of tunable Fabry-Perot filters with a specified tuning range. The analysis is also applicable to distributed Bragg reflectors, vertical-cavity surface-emitting lasers, and resonant photodiodes. I show that the penetration length can sharply reduce the overly broad free spectral range of an ultrathin Fabry-Perot filter that might be useful in applications such as tunable wavelength filters for wavelength division multiplexing applications. The results also demonstrate regimes of zero dispersion and of superluminal reflection in the dielectric mirrors, which are of particular interest in photonic bandgap structures.     

Exact Design of Aplanatic Microscope Objectives consisting of Two Conic Mirrors

The necessary equations are derived for the design of aplanatic microscope objectives consisting of two mirrors, one concave and the other convex. The first-order parameters are calculated along with the conic constants of the mirrors, determined by means of an exact analysis to arrive at an aplanatic system.     

Static and free vibration analyses of continuously graded fiber-reinforced cylindrical shells using generalized power-law distribution

In this study, based on the three-dimensional theory of elasticity, static and free vibration characteristics of continuously graded fiber-reinforced (CGFR) cylindrical shells are considered by making use of a generalized power-law distribution. In the present formulation, the cylindrical shell is assumed to be made of an orthotropic material. The CGFR cylindrical shells have a smooth variation of matrix volume fraction in the radial direction. Symmetric and asymmetric volume fraction profiles are presented in this paper. Suitable displacement functions that identically satisfy the boundary conditions at the simply supported edges are used to reduce the equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which can be solved by a generalized differential quadrature method. The fast rate of convergence of the method is demonstrated, and comparison studies are carried out to establish its very high accuracy and versatility. The main contribution of this work is to illustrate useful results for a cylindrical shell continuously graded fiber reinforced in the radial direction. Finally, these results are compared with a similar discrete laminated composite cylindrical shell.     

Fabrication and testing of large steering mirrors for a scanning lidar

Large. inexpensive scanning mirrors for a lidar have been designed and built out of common float glass mirrors and aluminum honeycomb.The flatness of the scanning mirrors has been characterized with a modified Foucault knife-edge test.The peak-to-peak surface slope error over the entire surface of the mirrors was found to be less than 1 mrad, with slope errors of 0.15 mrad over small areas. This performance was sufficient for use of the mirrors in a scanning CO(2) lidar system that uses direct detection.     

Unraveling mirror properties in time-delayed quantum feedback scenarios

Abstract

We derive in the Heisenberg picture a widely used phenomenological coupling element to treat feedback effects in quantum optical platforms. Our derivation is based on a microscopic Hamiltonian, which describes the mirror-emitter dynamics based on a dielectric, a mediating fully quantized electromagnetic field and a single two-level system in front of the dielectric. The dielectric is modelled as a system of identical two-state atoms. The Heisenberg equation yields a system of describing differential operator equations, which we solve in the Weisskopf–Wigner limit. Due to a finite round-trip time between emitter and dielectric, we yield delay differential operator equations. Our derivation motivates and justifies the typical phenomenologicalassumed coupling element and allows, furthermore, a generalization to a variety of mirrors, such as dissipative mirrors or mirrors with gain dynamics.     

Narrow-bandpass filters with broad rejection band for single-mode waveguides

Fabry-Perot bandpass filters made of mirrors with both high- and low-Dn (refractive-index modulation) have simultaneously a broad rejection band and a narrow passband. The higher Dn's are obtained with multilayer mirrors and the lower with Bragg gratings (BG's). Implementation of a sampling calculation technique based on the characteristic matrix formalism used for interference coatings allows for simulation of hybrid filters constructed from multilayer mirrors and BG's. The possible defects of hybrid filters are extensively analyzed. Bandpass filters made purely of both high- and low-Dn BG's are also simulated. All these filters are useful for wavelength division multiplexing applications.     

Wolter type I x-ray focusing mirror using multilayer coatings

A multilayer coating is a useful addition to a mirror in the x-ray region and has been applied to normal incidence mirrors used with soft x rays. When a multilayer coating is used on grazing incidence optics, higher performance can be achieved than without it. Cr/Sc multilayers coated on a Wolter type I mirror substrate for a soft x-ray microscope are considered. The reflectivity and effective solid angle are calculated for Wolter type I mirrors with uniform and laterally graded multilayer coatings. The laterally graded multilayer mirror showed superior x-ray performance, and the multilayer tolerances were relaxed. This multilayer mirror could be especially useful in the soft x-ray microscope intended for biological applications.     

n-Hindle-sphere arrangement with an exact ray trace for testing hyperboloid convex mirrors

We present calculations with an exact ray trace to determine the dimensions that define one or two Hindle spheres, since the paraxial theory is incongruent for convex hyperboloid mirrors with small f numbers. The equations are generalized to calculate the dimensions of n Hindle spheres, since in this way it is possible to reduce the dimensions of the spheres more. Actual calculations are done for the secondary mirrors of the Benemerita Universidad Autonoma de Puebla and Large Milimetric Telescopes; experimental results are shown for the latter.     

Simple yet accurate noncontact device for measuring the radius of curvature of a spherical mirror

An easily reproducible device is demonstrated to be capable of measuring the radii of curvature of spherical mirrors, both convex and concave, without resorting to high-end interferometric or tactile devices. The former are too elaborate for our purposes, and the latter cannot be used due to the delicate nature of the coatings applied to mirrors used in high-power CO(2) laser applications. The proposed apparatus is accurate enough to be useful to anyone using curved optics and needing a quick way to assess the values of the radii of curvature, be it for entrance quality control or trouble shooting an apparently malfunctioning optical system. Specifically, the apparatus was designed for checking 50 mm diameter resonator (typically flat or tens of meters concave) and telescope (typically some meters convex and concave) mirrors for a high-power CO(2) laser, but it can easily be adapted to any other type of spherical mirror by a straightforward resizing.     

Improving the accuracy of the caustic test

The caustic or Gaviola test, used for testing aspheric mirrors, has been studied with reference to the research of Schroader ["The caustic test," in Amateur Telescope Making III, A. G. Ingalls, ed. (Kingsport Press, Kingsport, Tenn., 1953), pp. 429-456]. It is shown that the two central formulas, which are used to determine the mirror quality, give significant errors for the large-diameter, short-focal-length mirrors commonly used in astronomical optics. We derive analytically two alternative equations, which are more exact, and show that they lead to significant improvement.     

Second-harmonic generation and cascaded second-order processes in a counterpropagating quasi-phase-matched device

Numerical solutions to the nonlinear coupled-wave equations of a counterpropagating quasi-phase-matched device are analyzed by numerical methods for both second-harmonic generation and cascaded processes. Normalized derivations for second-harmonic generation efficiency are also presented. The nonlinear phase shifts acquired in this device by cascaded second-order processes are promising in all-optical-switching applications. Specifically, a pi/2 phase shift is shown to be achievable with 42 times less input intensity than the standard Type I configuration and 100% throughput. The effects of metallic mirrors are also presented. Careful use of the phase mismatch is shown to compensate for nonideal mirrors. Finally, conservation of power in this configuration is briefly investigated.     

Simple dense-pattern optical multipass cells

Multiple-pass optical cells with dense spot patterns are useful for many applications, especially when the cell volume must be minimized relative to the optical path length. Present methods to achieve these dense patterns require expensive, highly precise astigmatic mirrors and complex alignment procedures. This work describes a new, simpler, and less demanding mirror system, comprising either a pair of cylindrical mirrors or one cylindrical and one spherical mirror.     

Energy flow of electric dipole radiation in between parallel mirrors

We have studied the energy flow patterns of the radiation emitted by an electric dipole located in between parallel mirrors. It appears that the field lines of the Poynting vector (the flow lines of energy) can have very intricate structures, including many singularities and vortices. The flow line patterns depend on the distance between the mirrors, the distance of the dipole to one of the mirrors and the angle of oscillation of the dipole moment with respect to the normal of the mirror surfaces. Already for the simplest case of a dipole moment oscillating perpendicular to the mirrors, singularities appear at regular intervals along the direction of propagation (parallel to the mirrors). For a parallel dipole, vortices appear in the neighbourhood of the dipole. For a dipole oscillating under a finite angle with the surface normal, the radiating tends to swirl around the dipole before travelling off parallel to the mirrors. For relatively large mirror separations, vortices appear in the pattern. When the dipole is off-centred with respect to the midway point between the mirrors, the flow line structure becomes even more complicated, with numerous vortices in the pattern, and tiny loops near the dipole. We have also investigated the locations of the vortices and singularities, and these can be found without any specific knowledge about the flow lines. This provides an independent means of studying the propagation of dipole radiation between mirrors.