Wavefront error caused by centroid position random error

The wavefront error caused by centroid position random error is derived in detail, and an exact formula, which evaluates the wavefront error associated with the centroid position random error, is obtained when the Zernike modes are used as the basis for wavefront reconstruction. The formula is proved by two Shack–Hartmann wavefront sensors with different wavefront reconstruction matrices, and it can precisely describe the wavefront measuring error caused by centroid position random error.     

An ocular wavefront sensor based on binary phase element: design and analysis

A modal wavefront sensor for ocular aberrations exhibits two main advantages compared to a conventional Shack–Hartmann sensor. As the wavefront is detected in the Fourier plane, the method is robust against local loss of information (e.g. local opacity of ocular lens as in the case of cataract), and is not dependent on the spatial distribution of wavefront sampling. We have proposed a novel method of wavefront sensing for ocular aberrations that directly detects the strengths of Zernike aberrations. A multiplexed Fourier computer-generated hologram has been designed as the binary phase element (BPE) for the detection of second-order and higher-order ocular aberrations (HOAs). The BPE design has been validated by comparing the simulated far-field pattern with the experimental results obtained by displaying it on a spatial light modulator. Simulation results have demonstrated the simultaneous wavefront detection with an accuracy better that ～λ/30 for a measurement range of ±2.1λ with reduced cross-talk. Sensor performance is validated by performing a numerical experiment using the City data set for test waves containing second-order and HOAs and measurement errors of 0.065?µm peak-to-valley (PV) and 0.08?µm (PV) have been obtained, respectively.     

Numerical Evaluation of Diffraction Integrals

This paper describes a simple numerical integration method for diffraction integrals which is based on elementary geometrical considerations of the manner in which different portions of the incident wavefront contribute to the diffracted field. The method is applicable in a wide range of cases as the assumptions regarding the type of integral are minimal, and the results are accurate even when the wavefront is divided into only a relatively small number of summation elements. Higher accuracies can be achieved by increasing the number of summation elements and/or incorporating Simpson’s rule into the basic integration formula. The use of the method is illustrated by numerical examples based on Fresnel’s diffraction integrals for circular apertures and apertures bounded by infinite straight lines (slits, half planes). In the latter cases, the numerical integration formula is reduced to a simple recursion formula, so that there is no need to perform repetitive summations for every point of the diffraction profile.     

Noniterative boundary-artifact-free wavefront reconstruction from its derivatives

Wavefront sensors are usually based on measuring the wavefront derivatives. The most commonly used approach to quantitatively reconstruct the wavefront uses discrete Fourier transform, which leads to artifacts when phase objects are located at the image borders. We propose here a simple approach to avoid these artifacts based on the duplication and antisymmetrization of the derivatives data, in the derivative direction, before integration. This approach completely erases the border effects by creating continuity and differentiability at the edge of the image. We finally compare this corrected approach to the literature on model images and quantitative phase images of biological microscopic samples, and discuss the effects of the artifacts on the particular application of dry mass measurements.     

Wavefront reconstruction by spatial-phase-shift imaging interferometry

Common-path imaging interferometers offer some advantages over other interferometers, such as insensitivity to vibrations and the ability to be attached to any optical system to analyze an imaged wavefront. We introduce the spatial-phase-shift imaging interferometry technique for surface measurements and wavefront analysis in which different parts of the wavefront undergo certain manipulations in a certain plane along the optical axis. These manipulations replace the reference-beam phase shifting of existing interferometry methods. We present the mathematical algorithm for reconstructing the wavefront from the interference patterns and detail the optical considerations for implementing the optical system. We implemented the spatial phase shift into a working system and used it to measure a variety of objects. Measurement results and comparison with other measurement methods indicate that this approach improves measurement accuracy with respect to existing quantitative phase-measurement methods.     

Direct demodulation of Hartmann-Shack patterns

Hartmann-Shack wave-front sensors produce a distorted grid of spots whose deviation from perfection is linear with the wave-front gradient. Usually, the centroid of each spot is calculated to provide that deviation, but it is also possible to perform the calculation by Fourier demodulation of the spot pattern [Opt. Commun. 215, 285, 2003]. We show that this demodulation can be performed directly on the grid, without reverting to Fourier transforms. Tracking the motion of each centroid individually is limited to well-defined spots with motions smaller than their pitch. In contrast, our method treats the image as a whole, is not limited to non-overlapping or sharp spots, and allows large spot motions. By replicating the array of spots slightly beyond the edge of the aperture, we reduce the chance for boundary phase dislocations in the reconstruction of the wave front. The method is especially suited to very large arrays.     

A comparison of ultrasonic wavefront distortion and compensation inone-dimensional and two-dimensional apertures

A comparison of wavefront distortion and compensation in one-dimensional and two-dimensional apertures is made using two-dimensional transmission measurements through 14 different specimens of human abdominal wall. The measurements were employed to emulate data in one dimension by summing waveforms in the elevation direction after a geometric correction was performed using a fourth-order polynomial fit to the surface of arrival time. Distortion calculation and time-shift compensation were performed independently on waveforms in the one-dimensional and two-dimensional apertures and the waveforms were focussed using a Fourier transform method to obtain the point-spread function in the central elevation plane. The results show that distortion is smoothed increasingly by one-dimensional apertures as the elevation dimension grows and that the smoothing is substantial for elevations similar to those employed in present clinical imagers. The results also show that one-dimensional compensation becomes less effective than two-dimensional compensation as the size of the elevation increases but that one-dimensional compensation performs almost as well as two-dimensional compensation in apertures with elevations like those in current imaging systems     

Spatial phase-shift interferometry--a wavefront analysis technique for three-dimensional topometry

We describe a new wavefront analysis method, in which certain wavefront manipulations are applied to a spatially defined area in a certain plane along the optical axis. These manipulations replace the reference-beam phase shifting of existing methods, making this method a spatial phase-shift interferometry method. We demonstrate the system's dependence on a defined spatial Airy number, which is the ratio of the characteristic dimension of the manipulated area and the Airy disk diameter of the optical system. We analytically obtain the resulting intensity data of the optical setup and develop various methods to accurately reconstruct the inspected wavefront out of the data. These reconstructions largely involve global techniques, in which the entire wavefront's pattern affects the reconstruction of the wavefront in any given position. The method's noise sensitivity is analyzed, and actual reconstruction results are presented.     

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.     

Adaptive optics implementation with a Fourier reconstructor

Adaptive optics takes its servo feedback error cue from a wavefront sensor. The common Hartmann-Shack spot grid that represents the wavefront slopes is usually analyzed by finding the spot centroids. In a novel application, we used the Fourier decomposition of a spot pattern to find deviations from grid regularity. This decomposition was performed either in the Fourier domain or in the image domain, as a demodulation of the grid of spots. We analyzed the system, built a control loop for it, and tested it thoroughly. This allowed us to close the loop on wavefront errors caused by turbulence in the optical system.     

Measuring the centroid gain of a Shack-Hartmann quad-cell wavefront sensor by using slope discrepancy

Shack-Hartmann wavefront sensors (SH WFS) are used by many adaptive optics (AO) systems to measure the wavefront. In this WFS, the centroid of the spots is proportional to the wavefront slope. If the detectors consist of 2 x 2 quad cells, as is the case in most astronomical AO systems, then the centroid measurement is proportional to the centroid gain. This quantity varies with the strength of the atmospheric turbulence and the angular extent of the beacon. The benefits of knowing the centroid gain and current techniques to measure it are discussed. A new method is presented, which takes advantage of the fact that, in a SH-WFS-based AO system, there are usually more measurements than actuators. Centroids in the null space of the wavefront reconstructor, called slope discrepancy measurements, contain information about the centroid gain. Tests using the W. M. Keck Observatory AO system demonstrate the accuracy of the algorithm.     

Wavefront measurements of a laser-induced breakdown spark in still air

Experimental measurements of the wavefronts of the light from a laser-induced breakdown (LIB) spark in non-moving air are presented and compared to spark dimensional data acquired from photographic measurements of the spark. The data show that the variation in the spark emitted wavefront between ignitions can be directly related to the motion of the spark volumetric centroid. The dominant modal components of the emitted wavefront variations are presented, as well as quantitative results for the magnitude of the wavefront variations. The results are relevant to the use of LIB as a light source for the measurement of optical aberrations such as those caused by compressible (i.e., "aero-optic") flows around an aircraft in flight, and data are shown indicating that LIB could be successfully used to measure the aberrating effect of compressible shear layers and boundary layers at typical cruise Mach numbers.     

Tomographic wavefront error using multi-LGS constellation sensed with Shack-Hartmann wavefront sensors

Noise effects induced by laser guide star (LGS) elongation have to be considered globally in a multi-LGS tomographic reconstruction analysis. This allows a fine estimation of performance and the comparison of different launching options. We present a modal analysis of the wavefront error with Shack-Hartmann wavefront sensors based on quasi-analytical matrix formalism. Including spot elongation and the Rayleigh fratricide effect, edge launching produces similar performance to central launching and avoids the risk of possible underestimation of fratricide scatter. Performance improves slightly with an optimized centroid estimator and is not affected by a slight field-of-view truncation of the subapertures. Finally we discuss detector characteristics for a LGS Shack-Hartmann wavefront sensor.     

Purely numerical compensation for microscope objective phase curvature in digital holographic microscopy: influence of digital phase mask position

Introducing a microscope objective in an interferometric setup induces a phase curvature on the resulting wavefront. In digital holography, the compensation of this curvature is often done by introducing an identical curvature in the reference arm and the hologram is then processed using a plane wave in the reconstruction. This physical compensation can be avoided, and several numerical methods exist to retrieve phase contrast images in which the microscope curvature is compensated. Usually, a digital array of complex numbers is introduced in the reconstruction process to perform this curvature correction. Different corrections are discussed in terms of their influence on the reconstructed image size and location in space. The results are presented according to two different expressions of the Fresnel transform, the single Fourier transform and convolution approaches, used to propagate the reconstructed wavefront from the hologram plane to the final image plane.     

Nonredundant array of apertures to measure the spatial coherence in two dimensions with only one interferogram

We propose to use a mask with a nonredundant array (NRA) of multiple apertures to measure spatial coherence in two dimensions. The spatial distribution of the apertures in the mask is made in such a way that we obtain a quasi-uniform sampling in the coherence domain. The spatial coherence is obtained by Fourier transform of the interferogram generated by the mask when it is illuminated by the light field under analysis.     

Shack-Hartmann wavefront sensing with elongated sodium laser beacons: centroiding versus matched filtering

We describe modeling and simulation results for the Thirty Meter Telescope on the degradation of sodium laser guide star Shack-Hartmann wavefront sensor measurement accuracy that will occur due to the spatial structure and temporal variations of the mesospheric sodium layer. By using a contiguous set of lidar measurements of the sodium profile, the performance of a standard centroid and of a more refined noise-optimal matched filter spot position estimation algorithm is analyzed and compared for a nominal mean signal level equal to 1000 photodetected electrons per subaperture per integration time, as a function of subaperture to laser launch telescope distance and CCD pixel readout noise. Both algorithms are compared in terms of their rms spot position estimation error due to noise, their associated wavefront error when implemented on the Thirty Meter Telescope facility adaptive optics system, their linear dynamic range, and their bias when detuned from the current sodium profile.     

Optimal modal fourier-transform wavefront control

Optimal modal Fourier-transform wavefront control combines the speed of Fourier-transform reconstruction (FTR) with real-time optimization of modal gains to form a fast, adaptive wavefront control scheme. Our modal basis is the real Fourier basis, which allows direct control of specific regions of the point-spread function. We formulate FTR as modal control and show how to measure custom filters. Because the Fourier basis is a tight frame, we can use it on a circular aperture for modal control even though it is not an orthonormal basis. The modal coefficients are available during reconstruction, greatly reducing computational overhead for gain optimization. Simulation results show significant improvements in performance in low-signal-to-noise-ratio situations compared with nonadaptive control. This scheme is computationally efficient enough to be implemented with off-the-shelf technology for a 2.5 kHz, 64 x 64 adaptive optics system.     

High-resolution retinal imaging with micro adaptive optics system

Based on the dynamic characteristics of human eye aberration, a microadaptive optics retina imaging system set is established for real-time wavefront measurement and correction. This paper analyzes the working principles of a 127-unit Hartmann-Shack wavefront sensor and a 37-channel micromachine membrane deformable mirror adopted in the system. The proposed system achieves wavefront reconstruction through the adaptive centroid detection method and the mode reconstruction algorithm of Zernike polynomials, so that human eye aberration can be measured accurately. Meanwhile, according to the adaptive optics aberration correction control model, a closed-loop iterative aberration correction algorithm based on Smith control is presented to realize efficient and real-time correction of human eye aberration with different characteristics, and characteristics of the time domain of the system are also optimized. According to the experiment results tested on a USAF 1951 standard resolution target and a living human retina (subject ZHY), the resolution of the system can reach 3.6?LP/mm, and the human eye wavefront aberration of 0.728λ (λ=785?nm) can be corrected to 0.081λ in root mean square (RMS) so as to achieve the diffraction limit (Strehl ratio is 0.866), then high-resolution retina images are obtained.     

小波滤波在Shack-Hartmann传感器上的应用

在使用Shack-Hartmann传感器进行大口径非球面镜面检测中,外部环境的各种振动影响以及气流、温差的干扰都会使检测精度下降。针对这个问题,提出了一种新的时域小波滤波技术。这项技术可以对传感器的信号干扰在时间域上进行不同层次的小波分析,提取干扰信号的先验特征,对测量数据进行有效的滤波,减小波前的扰动起伏,以更准确地探测质心。实验结果表明,采用这种技术后,Shack-Hartmann波前传感器对光学镜面检测的静态测量精度提高了50%以上,离散性减少到原来的20%-30%。     

Time-domain calculation of spectral centroid from backscattered ultrasound signals

Spectral centroid estimation from backscattered ultrasound RF signals is the preliminary step for quantitative ultrasound analysis in many medical applications. The traditional approach of estimating the spectral centroid in the frequency domain takes a long time because discrete Fourier transform (DFT) processing for each RF segment is required. To avoid this, we propose time-domain methods to estimate the spectral centroid in this paper. First, we derive the continuous-time-domain equations for the spectral centroid estimation using Parseval's theorem and Hilbert transform theory. Then, we extend the method to the discrete-time domain to ease the implementation while maintaining the same accuracy as the calculation in the frequency domain. From the result, we observe that it is not practical to apply the discrete-time equations directly, because a high sampling rate is needed to approximate the time derivative in the discrete-time domain. Therefore, we also derive the feasible version of the discrete-time equations using a circular autocorrelation function, which has no constraints on the sampling rate for real RF signals acquired from pulse-echo ultrasound systems. Simulation results using numerical phantoms show that the time-domain calculation is approximately 4.4 times faster on average than the frequency-domain method when the software's built-in functions were used. The average estimation error compared with that of the frequency-domain method using DFT is less than 0.2% for the entire propagation depths. The proposed time-domain approach to estimate the spectral centroid can be easily implemented in real-time ultrasound systems.