A fast-Fourier-transform method of topography and interferometry is proposed. By computer processing of a noncontour type of fringe pattern, automatic discrimination is achieved between elevation and depression of the object or wave-front form, which has not been possible by the fringe-contour-generation techniques. The method has advantages over moire topography and conventional fringe-contour interferometry in both accuracy and sensitivity. Unlike fringe-scanning techniques, the method is easy to apply because it uses no moving components.

Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry

A self-scanned 1024 element photodiode array and minicomputer are used to measure the phase (wavefront) in the interference pattern of an interferometer to lambda/100. The photodiode array samples intensities over a 32 x 32 matrix in the interference pattern as the length of the reference arm is varied piezoelectrically. Using these data the minicomputer synchronously detects the phase at each of the 1024 points by a Fourier series method and displays the wavefront in contour and perspective plot on a storage oscilloscope in less than 1 min (Bruning et al. Paper WE16, OSA Annual Meeting, Oct. 1972). The array of intensities is sampled and averaged many times in a random fashion so that the effects of air turbulence, vibrations, and thermal drifts are minimized. Very significant is the fact that wavefront errors in the interferometer are easily determined and may be automatically subtracted from current or subsequent wavefrots. Various programs supporting the measurement system include software for determining the aperture boundary, sum and difference of wavefronts, removal or insertion of tilt and focus errors, and routines for spatial manipulation of wavefronts. FFT programs transform wavefront data into point spread function and modulus and phase of the optical transfer function of lenses. Display programs plot these functions in contour and perspective. The system has been designed to optimize the collection of data to give higher than usual accuracy in measuring the individual elements and final performance of assembled diffraction limited optical systems, and furthermore, the short loop time of a few minutes makes the system an attractive alternative to constraints imposed by test glasses in the optical shop.

Digital wavefront measuring interferometer for testing optical surfaces and lenses

Recent progresses on real-time 3D shape measurement using digital fringe projection techniques

Manufacturing and measurement of freeform optics

Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations

Fringe scanning techniques, now renamed heterodyning or phase shift interferometry, are covered in a completely rewritten chapter. New chapters have been added to cover wavefront fitting and evaluation as well as holographic and Moire methods. The chapter on parameter measurements has been completely rewritten and an appendix added suggesting appropriate tests for typical optical surfaces.

Optical Shop Testing

(condensed). Review and Comparison of the Main Interferometric Systems: Two-Wave Interferometers and Configurations Used in Optical Testing. Twyman-Green Interferometer. Fizeau Interferometers. Typical Interferograms in Twyman-Green and Fizeau Interferometers. Lateral Shear Interferometers. Ronchi Test. Hartmann Test. Fringe Projection. Talbot Interferometry and Moire Deflectometry. Common Light Sources Used in Interferometry. Aspherical Compensators and Aspheric Wavefronts. Imaging of the Pupil on the Observation Plane. Multiple-Wavelength Interferometry. Fourier Theory Review: Introduction. Fourier Series. Fourier Transforms. The Convolution of Two Functions. The Cross-Correlation of Two Functions. Sampling Theorem. Sampling of a Periodical Function. Sampling of a Periodical Function with Interval Averaging. Fast Fourier Transform. Digital Image Processing: Introduction. Histogram and Gray-Scale Transformations. Space and Frequency Domain of Interferograms. Digital Processing of Images. Some Useful Spatial Filters. Square Window Filter. Hamming and Hanning Window Filters. Cosinusoidal and Sinusoidal Window Filters. Extrapolation of Fringes Outside of the Pupil. Light Detectors Used To Digitize Images. Fringe Contouring and Polynomial Fitting: Fringe Detection Using Manual Digitizers. Fringe Tracking and Fringe Skeletonizing. Global Polynomial Interpolation. Local Interpolation by Segments. Wavefront Representation by an Array of Gaussians.References. Periodic Signal Phase Detection and Algorithms Analysis: Least Squares Phase Detection of a Sinusoidal Signal. Quadrature Phase Detection of a Sinusoidal Signal. Discrete Low-Pass Filtering Functions. Fourier Description of Synchronous Phase Detection. Synchronous Detection Using a Few Sampling Points. Signal Amplitude Measurement. Characteristic Polynomial of a Sampling Algorithm. General Error Analysis of Synchronous Phase-Detection Algorithms. Some Sources of Phase Error. Shifting Algorithms with Respect to the Phase Origin. Optimization of Phase-Detection Algorithms. Influence of Window Function of Sampling Algorithms. Conclusions. Appendix: Derivative of the Amplitude of the Fourier Transform of the Reference Sampling Functions. References. Phase-Detection Algorithms: General Properties of Synchronous Phase-Detection Algorithms. Three-Step Algorithms To Measure the Phase. Four-Step Algorithms To Measure the Phase. Five-Step Algorithm. Algorithms with Symmetrical N +1 Phase Steps. Combined Algorithms in Quadrature. Detuning-Insensitive Algorithms for Distorted Signals. Algorithms Corrected for Nonlinear Phase-Shifting Error. Continuous Sampling in a Finite Interval. Asynchronous Phase-Detection Algorithms. Algorithm Summary. References. Phase-Shifting Interferometry: Phase-Shifting Basic Principles. An Introduction to Phase Shifting. Phase-Shifting Schemes and Phase Measurement. Heterodyne Interferometry. Phase-lock Detection. Sinusoidal Phase Oscillation Detection. Practical Sources of Phase Error. Selection of the Reference Sphere in Phase-Shifting Interferometry. Paraxial Focus. Best Focus. Marginal Focus. Optimum Tilt and Defocusing in Phase-Shifting Interferometry. References. Spatial Linear and Circular Carrier Analysis: Spatial Linear Carrier Analysis. Space-Domain Phase Demodulation with a Linear Carrier. Basic Space-Domain Phase Demodulation Theory. Circular Spatial Carrier Analysis. Phase Demodulation with a Circular Carrier. Fourier Transform Phase Demodulation with a Linear Carrier. Fourier Transform Phase Demodulation with a Circular Carrier. References. Interferogram Analysis with Moire Methods: Moire Techniques. Moire Formed by Two Interferograms with a Linear Carrier. Moire Formed by Two Interferograms with a Circular Carrier. Summary of Moire Effects. Holographic Interpretation of Moire Patterns. Conclusion. References. Interferogram Analysis without a Carrier: Introduction. Mathematical Model of the Fringes. The Phase Tracker. The N-Dimensional Quadrature Transform. Conclusion. References. Phase Unwrapping: The Phase Unwrapping Problem. Unwrapping Consistent Phase Maps Unwrapping Noisy Phase Maps. Unwrapping Subsampled Phase Maps. Conclusions. References. Wavefront Curvature Sensing: Wavefront Determination by Slope Sensing. Wavefront Curvature Sensing. Wavefront Determination with Defocused Images. Conclusions. References. Index. Short TOC

Interferogram analysis for optical testing

Optical shop testing

Geometrical Optics Principles Wave Nature of Light and Fermat's Principle Reflection and Refraction Laws Basic Meridional Ray Tracing Equations Gaussian or First-Order Optics Image Formation Stop, Pupils, and Principal Ray Delano's Relation Optical Sine Theorem Lagrange Invariant Herschel Invariant and Image Magnifications Thin Lenses and Spherical Mirrors Thin Lenses Formulas for Image Formation with Thin Lenses Nodal Points of A Thin Lens Image Formation with Converging Lenses Image Formation with Diverging Lenses Systems of Several Lenses and Thick Lenses Focal Length and Power of A Lens System Image Formation with Thick Lenses or Systems of Lenses Cardinal Points Image Formation with A Tilted or Curved Object Thick Lenses Systems of Thin Lenses The Lagrange Invariant in A System of Thin Lenses Effect of Object or Stop Shifting The Delano y - y Diagram Chromatic Aberrations Introduction Axial Chromatic Aberration Conrady's D - d Method of Achromatization Secondary Color Aberration Magnification Chromatic Aberration Spherical Aberration Spherical Aberration Calculation Primary Spherical Aberration Aspherical Surfaces Spherical Aberration of Aspherical Surfaces Surfaces without Spherical Aberration Aberration Polynomial for Spherical Aberration High-Order Spherical Aberration Spherical Aberration Correction with Gradient Index Monochromatic Off-Axis Aberrations Introduction Petzval Curvature Coma Astigmatism Aplanatic Surfaces Distortion Off-Axis Aberrations in Aspherical Surfaces The Symmetrical Principle and the Bow-Sutton Conditions Stop Shift Equations Aberrations of the Pupil Aberration Polynomials and High-Order Aberrations Wavefronts in an Optical System Ray Aberrations and Wavefront Aberrations Wavefront Aberration Polynomial Zernike Polynomials Fitting of Wavefront Deformations to A Polynomial Wavefront Representation by an Array of Gaussians Wavefront Aberrations in Refractive Surfaces Wavefront Aberrations in Reflective Surfaces Aldis Theorem Computer Evaluation of Optical Systems Transverse Aberration Polynomials Transverse Aberrations with H.H. Hopkins, Seidel, and Buchdahl Coefficients Meridional Ray Tracing and Stop Position Analysis Spot Diagram Wavefront Deformation Point and Line Spread Function Optical Transfer Function Tolerance to Aberrations Diffraction in Optical Systems Huygens-Fresnel Theory Fresnel Diffraction Fraunhofer Diffraction Diffraction Images with Aberrations Strehl Ratio Optical Transfer Function Resolution Criteria Gaussian Beams Prisms Tunnel Diagram Deflecting A Light Beam Transforming an Image Deflecting and Transforming Prisms Nondeflecting Transforming Prisms Beam-Splitting Prisms Chromatic Dispersing Prisms Nonimaging Prisms Basic Optical Systems and Simple Photographic Lenses Optical Systems Diversity Magnifiers and Single Imaging Lens Landscape Lenses Periscopic Lens Achromatic Landscape Lens Doublets Laser Light Collimators Spherical and Paraboloidal Mirrors Some Catoptric and Catadioptric Systems F-Theta Lenses Fresnel Lenses and Gabor Plates Complex Photographic Lenses Introduction Asymmetrical Systems Symmetrical Anastigmat Systems Varifocal and Zoom Lenses The Human Eye and Ophthalmic Lenses The Human Eye Ophthalmic Lenses Ophthalmic Lens Design Prismatic Lenses Spherocylindrical Lenses Astronomical Telescopes Resolution and Light-Gathering Power Reflecting Two-Mirror Cameras and Telescopes Catadioptric Cameras Astronomical Telescopes Field Correctors Multiple-Mirror Telescopes Active and Adaptive Optics Visual Systems and Afocal Systems Visual Optical Systems Basic Telescopic System Afocal Systems Visual And Terrestrial Telescopes Telescope Eyepieces Relays and Periscopes Microscopes Compound Microscope Microscope Objectives Microscope Eyepieces Microscope Illuminators Projection Systems Image Projectors Main Projector Components Coherence Effects in Projectors Anamorphic Projection Slide and Movie Projectors Overhead Projectors Profile Projectors Television Projectors LCD Computer and Home Theater Projectors Lens Design Optimization Basic Principles Optimization Methods Glatzel Adaptive Method Constrained Damped Least-Squares Optimization Method Merit Function and Boundary Conditions Modern Trends in Optical Design Flowchart for a Lens Optimization Program Practical Tips for the Use of Lens Evaluation Programs Some Commercial Lens Design Programs Appendices Appendix: Notation and Primary Aberration Coefficients Summary Appendix: Mathematical Representation of Optical Surfaces Appendix: Optical Materials Appendix: Exact Ray Tracing of Skew Rays Appendix: General Bibliography on Lens Design Index Chapters include references.

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Handbook of optical design

Basic ray optics basic wave optics basic photon optics refractive optical components reflective optical components diffractive optical components some lens optical devices telescopes spectrometers wavefront slope measurements in optical testing basic interferometers modern fringe pattern analysis in interferometry optical metrology: point methods optical metrology of diffuse objects: full-field methods holography Fourier optics and optical image processing electro-optical and acousto-optical devices radiometry, incoherent light sources lasers spatial and spectral filters optical fibers and accessories isotropic amorphous optical materials anisotropic materials light-sensitive material optical fabrication.

Daniel Malacara

Papers

Optical Shop Testing

Interferogram analysis for optical testing

Optical shop testing

Handbook of optical design

Handbook of Optical Engineering