Shack-Hartmann sensor for fast infrared wave-front testing

Abstract

A Shack-Hartmann sensor has been designed for testing the wave front of CO₂ lasers. Fabrication of a lens array and a detector array with tight tolerances on position accuracy are essential steps. Parallel electronics allow for high-speed wave-front measurements with 1 kHz sampling frequency. The device has been used to investigate the behaviour of a high-power CO₂ laser. Besides the expected thermal drifts of beam direction at the beginning of laser action, periodic changes of beam direction, have been detected. The Shack-Hartmann sensor seems the appropriate device for controlling adaptive optics in high-power laser applications.     

Iterative zonal wave-front estimation algorithm for optical testing with general-shaped pupils

An iterative zonal wave-front estimation algorithm for slope or gradient-type data in optical testing acquired with regular or irregular pupil shapes is presented. In the mathematical model proposed, the optical surface, or wave-front shape estimation, which may have any pupil shape or size, shares a predefined wave-front estimation matrix that we establish. Owing to the finite pupil of the instrument, the challenge of wave front shape estimation in optical testing lies in large part in how to properly handle boundary conditions. The solution we propose is an efficient iterative process based on Gerchberg-type iterations. The proposed method is validated with data collected from a 15 x 15-grid Shack-Hartmann sensor built at the Nanjing Astronomical Instruments Research Center in China. Results show that the rms deviation error of the estimated wave front from the original wave front is less than lambda/130-lambda/150 after approximately 12 iterations and less than lambda/100 (both for lambda = 632.8 nm) after as few as four iterations. Also, a theoretical analysis of algorithm complexity and error propagation is presented.     

Measurement of wave-front aberration in soft contact lenses by use of a Shack-Hartmann wave-front sensor

Lower- and higher-order wave-front aberrations of soft contact lenses were accurately measured with a Shack-Hartmann wave-front sensor. The soft contact lenses were placed in a wet cell filled with lens solution to prevent surface deformation and desiccation during measurements. Aberration measurements of conventional toric and multifocal soft contact lenses and a customized soft contact lens have proved that this method is reliable. A Shack-Hartmann wave-front sensor can be used to assess optical quality of both conventional and customized soft contact lenses and to assist in enhancing lens quality control.     

Method for measuring small optical absorption coefficients with use of a Shack-Hartmann wave-front detector

We present a method for measuring absorption at the 1 x 10(-5) cm(-1) level in high-quality optical materials. Using a Shack-Hartmann wave-front detector, thermal lensing in these materials may be measured. Then, the absorption coefficient may be estimated by fitting the observed deformation to a thermal lensing model based on the temperature dependences of the refractive index and the thermal expansion coefficient. For a particular sample of fused silica, the absorption coefficient was determined to be 1.8 +/- 0.4 x 10(-5) cm(-1). Obtaining this result requires a resolution in the optical path length better than +/- 0.1 nm.     

Large-scale wave-front reconstruction for adaptive optics systems by use of a recursive filtering algorithm

We propose a new recursive filtering algorithm for wave-front reconstruction in a large-scale adaptive optics system. An embedding step is used in this recursive filtering algorithm to permit fast methods to be used for wave-front reconstruction on an annular aperture. This embedding step can be used alone with a direct residual error updating procedure or used with the preconditioned conjugate-gradient method as a preconditioning step. We derive the Hudgin and Fried filters for spectral-domain filtering, using the eigenvalue decomposition method. Using Monte Carlo simulations, we compare the performance of discrete Fourier transform domain filtering, discrete cosine transform domain filtering, multigrid, and alternative-direction-implicit methods in the embedding step of the recursive filtering algorithm. We also simulate the performance of this recursive filtering in a closed-loop adaptive optics system.     

Breadboard testing of a phase-conjugate engine with an interferometric wave-front sensor and a microelectromechanical systems-based spatial light modulator

Laboratory breadboard results of a high-speed adaptive-optics system are presented. The wave-front sensor for the adaptive-optics system is based on a quadrature interferometer, which directly measures the turbulence-induced phase aberrations. The spatial light modulator used in the phase-conjugate engine was a microelectromechanical systems-based piston-only correction device with 1024 actuators. Laboratory experiments were conducted with this system utilizing Kolmogorov phase screens to simulate atmospheric phase distortions. The adaptive-optics system achieved correction speeds in excess of 800 Hz and Strehl ratios greater than 0.5 with the Kolmogorov phase screens.     

Improvement of resolution for phase retrieval by use of a scanning slit

An improved method for resolution of object reconstruction using phase retrieval by use of a scanning slit aperture is proposed. The reconstruction is based on measurements of the Fraunhofer diffraction intensities of wave fields transmitted through a scanning slit in the Fresnel-zone plane of an object. In the improved method, the measurement coordinates of the intensities depend not only on the slit's position used in a previous method but also on the slit's position scaled by the ratio between two distances among the object, Fresnel-zone, and detector planes. The spatial-frequency band for the object reconstruction, which is limited in a previous method by the extent of the Fourier transform of the slit function, can be extended to the bandwidth dependent on the scanning area with the slit. In addition, even in the measurement of the Fresnel diffraction intensities of wave fields transmitted through the slit, the improved resolution can be retained by compensation for a transverse shift of the intensities.     

Practical issues in wave-front sensing by use of phase diversity

We present the results of the phase-diversity algorithm applied to simulated and laboratory data. We show that the exact amount of defocus distance does not need to be known exactly for the phase-diversity algorithm on extended scene imaging. We determine, through computer simulation, the optimum diversity distance for various scene types. Using laboratory data, we compare the aberrations recovered with the phase-diversity algorithm and those measured with a Fizeau interferometer that uses a He-Ne laser. The two aberration sets agree with a Strehl ratio of over 0.9. The contrast of the recovered object is found to be ten times that of the raw image.     

Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform

Wave-front reconstruction with the use of the fast Fourier transform (FFT) and spatial filtering is shown to be computationally tractable and sufficiently accurate for use in large Shack-Hartmann-based adaptive optics systems (up to at least 10,000 actuators). This method is significantly faster than, and can have noise propagation comparable with that of, traditional vector-matrix-multiply reconstructors. The boundary problem that prevented the accurate reconstruction of phase in circular apertures by means of square-grid Fourier transforms (FTs) is identified and solved. The methods are adapted for use on the Fried geometry. Detailed performance analysis of mean squared error and noise propagation for FT methods is presented with the use of both theory and simulation.     

Phase-diversity wave-front sensing with a distorted diffraction grating

We describe a novel wave-front sensor comprising a distorted diffraction grating, simple optics, and a single camera. A noniterative phase-diversity algorithm is used for wave-front reconstruction. The sensor concept and practical implementation are described in detail, and performance is validated against different Zernike modes and a representative atmospheric phase map.     

A Radial-shear Interferometer for Use with a Laser Source

An interferometer is described that combines the design simplicity of a laser interferometer and the insensitivity to vibration of a common-path instrument. It is used to test wave aberration in ‘single passage’.     

Highly sensitive wave-front sensor with a non-zero-order phase plate

We propose a novel highly sensitive wave front detection method for a quick check of a flat wave front by taking advantage of a non-zero-order pi phase plate that yields a non-zero-order diffraction pattern. When a light beam with a flat wave front illuminates a phase plate, the zero-order intensity is zero. When there is a slight distortion of the wave front, the zero-order intensity increases. The ratio of first-order intensity to that of zero-order intensity is used as the criterion with which to judge whether the wave front under test is flat, eliminating the influence of background light. Experimental results demonstrate that this method is efficient, robust, and cost-effective and should be highly interesting for a quick check of a flat wave front of a large-aperture laser beam and adaptive optical systems.     

Rainbow holography with a synthesized double slit

Rainbow holography with a synthesized double slit is proposed. Diffuse three-dimensional objects are translated uniformly in the x(0)-y(0) plane. The propagation direction of the coherent plane wave illuminating the objects is situated in the x(0)-z(0) plane. As a result of this process, a sinc function that modulates the complex-amplitude distribution of the objects is presented on the back focal plane of the lens, and the synthesized slit is formed. The central position of the synthesized slit depends on both the direction of motion of the object and the spatial frequency of the illuminating wave in the x(0) direction. Therefore the synthesized double slit is generated with a two-exposure method that has two illuminating waves of different spatial frequencies. The theoretical analysis and some experimental results are presented.     

Focal plane position detection with a diffractive optic for shack-hartmann wave-front sensor fabrication

During construction of a Shack-Hartmann wave-front sensor it is critical that the spacing between the lens array and the detector array be equal to the lens array focal length to obtain accurate and precise measurements of a wave front. This separation is often difficult to determine with large f/# lenses, because their focal spot diameter does not change substantially for small displacements on either side of the focal plane. We describe a method with an array of off-axis lens segments for determining the location of the focal plane. Because the lenses are off axis, changes in the distance from the optic to the detector array result in transverse focal spot position variations as a function of their separation from the lenses. By analyzing the focal spot pattern on a CCD, we achieved 12-mum rms error in the axial position measurement while moving a 4-mm-focal-length optic over 1 mm.     

Practical configuration for wave-front sensing with two-beam phase retrieval

A wave-front sensing approach based on two-beam phase retrieval is described. Light from an aberrated beam is split into two paths. A random phase and amplitude perturbation is applied to the beam in one path, and the interference patterns of the resultant two beams are measured in two planes along the axis of propagation. By modulation of one of the two beams, the intensity of each beam and the phase difference between the two beams are recovered in each plane. A rapidly convergent phase-retrieval algorithm is formulated by the method of sequential projections onto constraint sets. Examples are given illustrating the convergence properties of the approach.     

Phase retrieval from diffraction intensities by use of a scanning slit aperture

A noniterative method of retrieving the phase of a wave field from intensities measured during scanning of a slit aperture is proposed. In the measurements, one records the diffraction intensities of wave fields transmitted through a slit that is scanned along two directions in the Fresnel-zone plane of an object's field. From these intensities, the phase in the Fresnel-zone plane can be retrieved by a method in which a novel phase calculation technique that uses Fourier transforms is included. Because the method does not require lens systems, it provides a potentially useful means for coherent imaging by use of x rays, electrons, or nuclear particles.     

Spatial spectrum analysis of wave-front correction with a segmented mirror

An expression is derived for the spatial power spectrum of wave-front errors after correction with a segmented mirror. This includes estimates of the spectral contributions of segment piston and tilt corrections and spatial aliasing by a regular array of segments. The approach allows rapid computation of wave-front error spectra in systems with highly segmented mirrors.     

Generation of Phase-Modulated Periodic Optical Signals by use of a Polarization Interferometer

Polarization phase-shifting interferometry is an established technique in optical metrology. In the present study it is shown that, by use of this technique, not only is it possible to realize any discrete magnitude of a predetermined phase difference (from 0 to 2pi) between two light beams but also phase-modulated periodic optical signals can be generated simply by rotation of a polarizer or a retarder or both placed at the input of a conventional two-beam interferometer. Some representative linear and nonlinear periodic polarization-induced phase-modulated optical signals are shown. A linear phase modulation of 0-2pi with constant output intensity is obtained in some cases. The Poincaré sphere representation is introduced as a convenient tool for visualizing the dynamics involved in the generation of polarization-phase-modulated waveforms and as a possible aid to intelligent modification of the generated waveform as required. This all-optical technique of continuous and periodic phase variation should prove useful for introducing phase modulation without the need for electro-optic devices.     

Resolution of a reflection hologram recorded with a slit

Resolution of the reconstructed image is evaluated for the reflection hologram recorded by use of a slit. Sharp and deep images are obtained because the resolution in the vertical direction is higher than that of the conventional reflection holograms and is independent of the size of the illuminating light source. In contrast, the resolution in the horizontal direction depends on the light-source size in this direction. The optimum source size is discussed in connection with the balance in the resolutions for both directions. A method for obtaining the vertical parallax by use of multiple slits is proposed, and application of the proposed hologram to the heads-up display is also demonstrated.