Birefringence measurements in gradient-index rod lenses

Stress birefringence can be found in gradient-index (GRIN) materials because they contain a variation in composition. GRIN glass fabricated by ion exchange may contain stress from two different processes. These include a size difference between the exchange and the diffusing ions and a variation in the thermal-expansion coefficient across the gradient region. The optical properties of the stressed material are polarization dependent, and therefore image quality is directly affected. We examine birefringence in GRIN rod lenses that have lengths more than ten times greater than their diameters. The effects are more easily observed in long rod lenses because of the large optical path lengths.     

模板匹配算法在高精度压电微位移测量中的应用

提出了一种压电微位移器的微位移测量方法,即通过模板匹配算法测量Twyman-Green干涉系统中对应干涉条纹的移动量,进而实现对应位移量的高精度测量.为确保测量结果的可靠性,同时利用匹配模板的归一化转动惯量特征对测量结果进行实时验证.应用该测量方法对自制的压电微位移器的电压位移特性曲线进行了实测,并详细分析了模板大小的选取对于测量结果的影响,实验中的归一化转动惯量相对误差均小于5%,且模板匹配互相关系数大于0.97;同时运用一高精度纳米微位移平台对测量方法的可行性进行了实验验证,测量精度达到了λ/50.实验结果表明,运用本方法测量压电微位移,具有较好的噪声抑制能力和可靠性,可实现纳米量级的测量精度.     

High-precision shape measurement by white-light interferometry with real-time scanner error correction

White-light interferometric techniques allow high-precision shape measurement of objects with discontinuous structures by detecting the peak of the coherence envelope. These techniques assume a specific change in the optical path difference (OPD) between the interfering beams; however, the scanning device effecting that change often introduces OPD errors that are carried over to the measurements. We present a technique for measuring OPD changes from the collected interference fringes during each measurement. Information about the scan is directly fed into the algorithm, which compensates for the errors, resulting in improved measurement accuracy. The method corrects not only the scanner errors but also slowly varying vibrations. In addition, this technique can be easily adapted to any existing low-coherence interferometer because no large data storage or postprocessing is required.     

Use of the fast Fourier transform method for analyzing linear and equispaced Fizeau fringes

The Fourier transform method is applied to analyze the initial phase of linear and equispaced Fizeau fringes. We develop an algorithm for high-precision phase measurement by using the Fourier coefficient that corresponds to the spatial frequency of the Fizeau fringes, and we describe methods for determining the fringe carrier frequency. Errors caused by carrier frequency fluctuation and data truncation are studied theoretically and by computer simulation. To demonstrate the method we apply it to the real-time calibration of a piezoelectric transducer mirror in a Twyman-Green interferometer.     

Investigation of a half-wave method for birefringence or thickness measurements of a thick, semitransparent, uniaxial, anisotropic substrate by use of spectroscopic ellipsometry

A half-wave method of measurement of wafer birefringence that is based on interference fringes recorded from a uniaxial wafer by use of a standard phase-modulated spectroscopic ellipsometer is investigated. The birefringence of uniaxial wafers is calculated from the extremal points in the recorded oscillating intensities. A formalism is developed to incorporate the change in birefringence with wavelength as a correction factor. The correction explains the overestimation of the birefringence from previous similar research on thick uniaxial sapphire substrates. The enhanced derivative of the birefringence that is due to polarization-dependent intraconduction band transitions is detected. Furthermore, for well-characterized wafers it is shown that this method can be used in wafer-thickness mapping of 4H-SiC and similar uniaxial high-bandgap semiconductors.     

Approach for the evaluation of speckle deformation measurements by application of the wavelet transformation

We introduce a new, to our knowledge, method using wavelets and probability theory for the evaluation of speckle interference patterns for quantitative out-of-plane deformation measurements of rough surfaces of nontransparent solids. The experiment uses a conventional Twyman-Green interferometer setup. The speckle interference patterns are obtained by the common method of subtraction of images taken before and after a surface deformation. The data are processed by a wavelet transformation, which analyzes the image structures on different length scales. Thus it is possible to separate the interference fringes from the noise. From the locations of the interference fringes, the deformation of the surface can be reconstructed by means of probability theory.     

Hologram Interferometry: Identification of the Sign of Surface Displacements

It is difficult to test aspheric surfaces with a Twyman-Green interferometer because the interferogram frequently contains too many fringes. A simple way of overcoming this problem is to use a lateral-shearing interferometer, in which case the number of fringes in the interferogram can be controlled by varying the shear. However, this has the drawback that two interferograms with orthogonal directions of shear are required; in addition, the accuracy with which the shape of the surface can be evaluated from measurements on photographs of the fringes is limited. In this paper it is shown how these difficulties can be overcome by using a microcomputer-controlled digital radial-shear interferometer. The values of the phase difference in the interferogram at a matrix of points covering the pupil are processed directly in the same microcomputer to give the actual shape of the surface. Typical results obtained with an off-axis paraboloid are presented.     

Birefringent properties of diametrically loaded gradient-index lenses

Gradient-index (GRIN) lenses have been widely used as collimators in various fiber-optic sensors and as optical coupling devices in components designed for optical communication systems. However, relatively little attention has been paid to the birefringent properties of GRIN lenses and the potential for using them as photoelastic sensing elements in optical transducers. Analytical and experimental results are described that were obtained for the intensity distribution produced by studying a GRIN lens by using a polariscope. The residual birefringence inherent in an unloaded lens is initially studied. The lens is then assumed to be diametrically loaded and the superposition is studied by the method of ray tracing. When the results obtained from the simulation for a Selfoc, 0.25-pitch lens are compared with experimental data, an excellent agreement is obtained. Intensity increases monotonically with load, confirming that the lens would be a good choice for the sensing element of an optical transducer designed as part of a strain or acceleration measurement system. The numerical simulation is then used to study the influence of residual stress on sensitivity.     

Gradient-index contact lens

A gradient-index (GRIN) contact lens (CL) is proposed to decrease spherical aberration and to increase the diopter. A plastic radial GRIN rod was successfully obtained by using the vapor-phase diffusion copolymerization technique. The resulting index distribution of the GRIN rod was almost parabolic against the distance from the center axis, and the Δn value was -0.030. The GRIN CL was prepared by grinding and polishing the rod. It is theoretically and experimentally concluded that using the radial GRIN material can significantly improve the optical properties of CL's.     

Thermally induced strain and birefringence calculations for a Nd:YAG rod encapsulated in a solid pump light collector

Calculations and experimental measurements of the thermally induced strain and birefringence are presented for a diode-pumped Nd:YAG rod that is encapsulated in a prismatic pump light collector. A numerical model is developed to determine the spatiotemporal stress-induced strain distribution across the prism, index-matching fixant, and laser rod, and the birefringence that arises from the stress-induced strain within the laser rod. Calculations of the birefringence are compared with polarscopic measurements and display good agreement. Support for the rod on all sides is provided by the prism and fixant, and the distribution and degree of the stress-induced strain (and birefringence) within the laser rod are therefore influenced by the geometry and composition of the prism and fixant. These strains are thermomechanical in origin and are primarily a function of the elastic modulus of the fixant and the temperature of the system. Such stress-induced strains are additional to those strains that are produced from temperature gradients across the laser rod and result from the laser rod being constrained from expanding. Collectors utilizing index-matching fluid as the encapsulant display the smallest measure of birefringence relating to the temperature gradients in the rod. However, for collectors utilizing solid fixants (with significant elastic modulus), an increase in the birefringence results. In this case collector designs that have the laser rod located in a symmetrically shaped prism are effective in reducing the nonuniform pressures on the sides of the rod and therefore the birefringence.     

Polarization phase-shifting interferometric technique for complete evaluation of birefringence

A full-field technique for simultaneous measurement of the magnitude of birefringence and its orientation is presented. This is achieved using a monolithic birefringence sensitive interferometer where the interference fringes carry the information of both the birefringence phase and the orientation of the fast axis of an optically transmissive anisotropic material placed at the output of the interferometer. The interferometer consists of a suitably polarization-masked cube beam splitter, orientated as in the Gates interferometer, which serves to generate a pair of orthogonally polarized and collinearly propagating light beams. Experimental results are obtained through an algorithm incorporating eight polarization phase-shifted interferograms.     

Self-interference patterns and their application to inertial-fusion target characterization

The sphericity and wall-thickness uniformity requirements of direct-drive inertial-fusion targets are of the order of less than 1%. These shells display self-interference patterns (SIP's) when irradiated with a spatially incoherent, narrow-bandwidth light source and viewed with a compound microscope. These patterns are distinct concentric fringes when the target is uniform, whereas faint, distorted, or discontinuous fringes indicate a nonuniform target. We determined the wall thickness to within +/-0.5 microm by counting the number of fringes in the SIP, independent of the outside diameter. Thickness uniformity is verified to an accuracy better than 0.05 microm. The wall thickness of gas-filled targets can be determined to this accuracy without knowledge of the type of gas or its pressure. The SIP fringe technique is used to select polymer shells typically of 800- to 1000-microm diameter and 5- to 12-microm wall thickness. The fringe locations have been modeled by use of ray tracing and agree well with actual measurements of well-characterized shells. Details of the formation of the SIP fringes, a theoretical model, and the method used for quantitative measurement of the shell-wall thickness with the SIP are presented with validation examples.     

Alternative method for measuring the full-field refractive index of a gradient-index lens with normal incidence heterodyne interferometry

A linearly/circularly polarized heterodyne light beam coming from a heterodyne light source with an electro-optic modulator in turn enters a modified Twyman-Green interferometer to measure the surface plane of a GRIN lens. Two groups of periodic sinusoidal segments recorded by a fast complementary metal-oxide semiconductor camera are modified, and their associated phases are derived with the unique technique. The data are substituted into the special equations derived from the Fresnel equations, and the refractive index can be obtained. When the processes are applied to other pixels, the full-field refractive-index distribution can be obtained similarly. Its validity is demonstrated.     

Correction of chromatic aberrations in GRIN endoscopes

A gradient-index (GRIN) endoscope can be constructed by substituting for the usual objective and relay sections suitable cylindrical index-distribution rod lenses. Currently available GRIN lenses exhibit large amounts of chromatic aberration. Axial color arises mostly from the relay lens, while lateral color is due to the objective lens. A negative lens cemented to a shortened GRIN relay lens can simultaneously correct axial and lateral chromatic aberrations with commercially available components. This correction system reduces the requirements for mechanical centration better than do color correctors that are incorporated into the ocular design. Monochromatic aberrations are also considered.     

Measurement of the surface profile of an axicon lens with a polarization phase-shifting shearing interferometer

We present a Twyman-Green interferometer (TGI)-based polarization phase-shifting shearing interferometric technique for testing the conical surface of an axicon (AX) lens. In this technique, the annular beam generated due to the passing of an expanded collimated laser beam traveling along the axis of revolution of the transparent glass AX element is split up into its reflected and transmitted components, having the plane of polarization in the orthogonal planes, by the polarization beam splitter (PBS) cube of the TGI-based optical setup. The split-up components are made to travel unequal paths along the two arms of the TGI and are recombined by the PBS. Because of the difference in path lengths traveled by the annular conical beams, a linear shear is introduced along the radial direction between the interfering components. Thus, the resulting interference pattern gives a map of the optical path difference (OPD) between two successive close points along a radial direction on the conical surface of the AX lens. The OPD map along radial directions, and hence the slopes/profiles of the conical surface, are obtained by applying polarization phase-shifting interferometry. Results obtained for an AX lens are presented.     

Uneven fringe projection for efficient calibration in high-resolution 3D shape metrology

A novel uneven fringe projection technique is presented whereby nonuniformly spaced fringes are generated at a digital video projector to give evenly spaced fringes in the measurement volume. The proposed technique simplifies the relation between the measured phase and the object's depth independent of pixel position. This method needs just one coefficient set for calibration and depth calculation. With uneven fringe projection the shape data are referenced to a virtual plane instead of a physical reference plane, so an improved measurement with lower uncertainty is achieved. Further, the method can be combined with a radial lens distortion model. The theoretical foundation of the method is presented and experimentally validated to demonstrate the advantages of the uneven fringe projection approach compared with existing methods. Measurement results on a National Physical Laboratory (UK) "step standard" confirm the measurement uncertainty using the proposed method.     

Studies on the light-focusing plastic rod. 14: GRIN rod of CR-39-trifluoroethyl methacrylate copolymer by a vapor-phase transfer process

The gel rod, partially polymerized CR-39 [diethylene glycol-bis-(allyl carbonate)], is placed in the atmosphere of trifluoroethyl methacrylate (3FMA) vapor, followed by heat-treatment to yield the GRIN rod. Examining the relationship between the preparation conditions and the optical properties of the GRIN rod, the plastic GRIN rods with a quadratic-index distribution up to their periphery and with higher than 0.36 N.A. were fabricated.     

Performance analyses of an infrared single-mode all-fiber-optic Fourier-transform spectrometer

Stretching one of a pair of fiber arms can be done to realize optical phase modulation for an IR single-mode all-fiber-optic Fourier-transform spectrometer (FTS). But this operation will inevitably limit the physical performance of a FTS. We study these limits theoretically and experimentally. The optical path difference (OPD) will be dispersive. At the first-order approximation, this OPD dispersion will result in a shift in the recovered spectra. The spectral resolution and the sampling distance will also be dispersive. Linear birefringence introduced when a curved fiber is stretched will affect the final spectra. This effect can be eliminated by real-time compensation and (or) by system design. Errors encountered uniquely in the all-fiber-optic FTS in the optical phase domain, such as the fiber-parameter errors, nonlinearity of the piezoelectric cylinder, and their effects on the spectra are analyzed, from which we deduce the requirements for calibration. Finally, calibration methods for optical phase modulation are discussed.     

Interferometric chromatic dispersion measurements of long, gradient-index rod lenses in the visible and the near infrared

A new procedure is presented for testing long, radial-gradient-index rod lenses in a phase-stepping Twyman-Green interferometer. In addition, a method for converting the measured optical phase data to the rod's index of refraction profile is reviewed. This procedure was used to measure refractive-index profile dispersions over the wavelength range of 441 to 1064 nm. These measurements provide the first data for refractive-index profiles of rod lenses in the infrared and show the existence of positive dispersion, negative dispersion, and near-infrared dispersion reversal in some samples. The dispersion results suggest the need for modifications to the current glass model.     

Polynomial fitting of interferograms with Gaussian errors on fringe coordinates. 1: Computer simulations

A series of simulations were made for an ideal Twyman-Green interferogram of equally spaced straight fringes having tilt only about x. It was found that fitting polynomials to the interferometric data resulted in biased estimates of some of the fitting coefficients to the optical path difference. The acceptance of the Seidel aberrations grows with the noise level and diminishes when the number of fringes is increased.