Maximum-likelihood methods in wavefront sensing: stochastic models and likelihood functions

Maximum-likelihood (ML) estimation in wavefront sensing requires careful attention to all noise sources and all factors that influence the sensor data. We present detailed probability density functions for the output of the image detector in a wavefront sensor, conditional not only on wavefront parameters but also on various nuisance parameters. Practical ways of dealing with nuisance parameters are described, and final expressions for likelihoods and Fisher information matrices are derived. The theory is illustrated by discussing Shack-Hartmann sensors, and computational requirements are discussed. Simulation results show that ML estimation can significantly increase the dynamic range of a Shack-Hartmann sensor with four detectors and that it can reduce the residual wavefront error when compared with traditional methods.     

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.     

Study of a MEMS-based Shack-Hartmann wavefront sensor with adjustable pupil sampling for astronomical adaptive optics

We introduce a Shack-Hartmann wavefront sensor for adaptive optics that enables dynamic control of the spatial sampling of an incoming wavefront using a segmented mirror microelectrical mechanical systems (MEMS) device. Unlike a conventional lenslet array, subapertures are defined by either segments or groups of segments of a mirror array, with the ability to change spatial pupil sampling arbitrarily by redefining the segment grouping. Control over the spatial sampling of the wavefront allows for the minimization of wavefront reconstruction error for different intensities of guide source and different atmospheric conditions, which in turn maximizes an adaptive optics system's delivered Strehl ratio. Requirements for the MEMS devices needed in this Shack-Hartmann wavefront sensor are also presented.     

Misalignment effects of the Shack-Hartmann sensor

The Shack-Hartmann sensor uses a microlens array and a CCD camera for wave-front measurements. To obtain wave-front measurements with high accuracy, an accurate relative alignment of both is essential. The different states of misalignment of the Shack-Hartmann sensor are divided into groups and are treated theoretically and experimentally. Their effect on the accuracy of wave-front measurements is evaluated. In addition, a practical method for proper alignment of the Shack-Hartmann sensor is proposed.     

Experimental comparison of a Shack-Hartmann sensor and a phase-shifting interferometer for large-optics metrology applications

We performed a direct side-by-side comparison of a Shack-Hartmann wave-front sensor and a phase-shifting interferometer for the purpose of characterizing large optics. An expansion telescope of our own design allowed us to measure the surface figure of a 400-mm-square mirror with both instruments simultaneously. The Shack-Hartmann sensor produced data that closely matched the interferometer data over spatial scales appropriate for the lenslet spacing, and much of the <20-nm rms systematic difference between the two measurements was due to diffraction artifacts that were present in the interferometer data but not in the Shack-Hartmann sensor data. The results suggest that Shack-Hartmann sensors could replace phase-shifting interferometers for many applications, with particular advantages for large-optic metrology.     

Large diameter radial semiconductor drift chambers for low noise radiation detection

A semiconductor drift chamber with cylindrical symmetry has been investigated theoretically for its suitability as a low noise radiation detector. The signal/noise of a silicon drift chamber at room temperature is likely to be determined by the detector dark current and the drift chamber should be superior in performance to a conventional pn junction detector of equivalent dimensions. A device of 48 mm diameter and 300 μm thickness is predicted to be capable of an energy resolution of less than 5 keV fwhm at 300 K. This is about seven times better than that predicted for its conventional counterpart.     

Analysis of optimal centroid estimation applied to Shack-Hartmann sensing

The problem of estimating the centroid of an incoherently imaged point with a CCD array is analyzed. An exact analysis is presented that uses the actual short-exposure function at the CCD instead of the traditional Gaussian approximation. The analysis shows that, for Poisson noise, the centroid variance depends on the CCD size and that truncation effects play a significant part in determining the optimum CCD size. The effects of this on a wave-front reconstruction formed by a Shack-Hartmann sensor are described.     

Chemical sensing systems using xerogel-based sensor elements and CMOS photodetectors

We present the first example of an integrated complementary metal-oxide-semiconductor (CMOS) photodetector coupled with a solid-state xerogel-based thin-film sensor to produce a compact chemical sensor system. We compare results using two different CMOS-based detector systems to results obtained by using a standard photomultiplier tube (PMT) or charge-coupled device (CCD) detector. Because the chemical sensor elements are governed by a Stern-Volmer relationship, the Stern-Volmer quenching constant is used as the primary comparator between the different detectors. All of the systems yielded Stern-Volmer constants from 0.042 to 0.049 O/sub 2/%/sup -1/. The results show that the CMOS detector system yields analytical data that are comparable to the CCD- and PMT-based systems. The disparity between the data obtained from each detector is primarily associated with the difference in how the signals are obtained by each detector as they presently exist. We have also observed satisfactory reversibility in the operation of the sensor system. The CMOS-based system exhibits a response time that is faster than the chemical sensor element's intrinsic response time, making the CMOS suitable for time-dependent measurements. The CMOS array detector also uses less than 0.1% the power in comparison to a standard PMT or CCD. The combined xerogel/CMOS system represents an important step toward the development of a portable, efficient sensor system.     

Wave-front sensing from subdivision of the focal plane with a lenslet array

A wave-front sensing scheme based on placing a lenslet array at the focal plane of the telescope with each lenslet reimaging the aperture is analyzed. This wave-front sensing arrangement is the dual of the Shack-Hartmann sensor, with the wave front partitioned in the focal plane rather than in the aperture plane. This arrangement can be viewed as the generalization of the pyramid sensor and allows direct comparisons of this sensor with the Shack-Hartmann sensor. We show that, as with the Shack-Hartmann sensor, when subdividing in the focal plane, the quality of the wave-front estimate is a trade-off between the quality of the slope measurements over each region in the aperture and the resolution to which the slope measurements are obtained. Open-loop simulation results demonstrate that the performance of the lenslet array at the focal plane is equivalent to that of the Shack-Hartmann sensor when no modulation is applied to the lenslet array. However, when the array is modulated in a manner akin to that of the pyramid sensor, subdivision at the focal plane provides advantages when compared with the Shack-Hartmann sensor.     

Detection of phase singularities with a Shack-Hartmann wavefront sensor

While adaptive optical systems are able to remove moderate wavefront distortions in scintillated optical beams, phase singularities that appear in strongly scintillated beams can severely degrade the performance of such an adaptive optical system. Therefore the detection of these phase singularities is an important aspect of strong-scintillation adaptive optics. We investigate the detection of phase singularities with the aid of a Shack-Hartmann wavefront sensor and show that, in spite of some systematic deficiencies inherent to the Shack-Hartmann wavefront sensor, it can be used for the reliable detection of phase singularities, irrespective of their morphologies. We provide full analytical results, together with numerical simulations of the detection process.     

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.     

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.     

Equivalent Magnetic Noise Limit of Low-Cost GMI Magnetometer

We present a noise analysis of a giant magnetoimpedance (GMI) sensor using a peak detector at the optimal magnetic field working bias point of a sensor wire, by considering internal noise sources (intrinsic GMI device associated noise sources and conditioning electronic noise sources). An expression is obtained for the theoretical expected noise for known electronic design parameters and physical characteristics of the GMI wire. The most significant contributions to noise in a GMI measurement, using two basic oscillators (either a simple discrete RC oscillator or quartz oscillator) along with a peak detector, are presented. We discuss the expected extrinsic equivalent magnetic noise limit.      

Sorting method to extend the dynamic range of the Shack-Hartmann wave-front sensor

We propose a simple and powerful algorithm to extend the dynamic range of a Shack-Hartmann wave-front sensor. In a conventional Shack-Hartmann wave-front sensor the dynamic range is limited by the f-number of a lenslet, because the focal spot is required to remain in the area confined by the single lenslet. The sorting method proposed here eliminates such a limitation and extends the dynamic range by tagging each spot in a special sequence. Since the sorting method is a simple algorithm that does not change the measurement configuration, there is no requirement for extra hardware, multiple measurements, or complicated algorithms. We not only present the theory and a calculation example of the sorting method but also actually implement measurement of a highly aberrated wave front from nonrotational symmetric optics.     

Cumulative Reconstructor: fast wavefront reconstruction algorithm for Extremely Large Telescopes

The Cumulative Reconstructor (CuRe) is a new direct reconstructor for an optical wavefront from Shack-Hartmann wavefront sensor measurements. In this paper, the algorithm is adapted to realistic telescope geometries and the transition from modified Hudgin to Fried geometry is discussed. After a discussion of the noise propagation, we analyze the complexity of the algorithm. Our numerical tests confirm that the algorithm is very fast and accurate and can therefore be used for adaptive optics systems of Extremely Large Telescopes.     

An interface circuit for measuring capacitance changes based upon capacitance-to-duty cycle (CDC) converter

We present a direct-to-digital capacitive sensor readout circuit that converts capacitance changes of a sensor element to changes of the duty cycle of a square-wave oscillator, which, in turn, is converted to a digital output by a counter. The readout circuit resembles a single-slope analog-to-digital converter structure. There are several advantages of this readout scheme. First, due to its simplicity and low number of components, the power consumption of the circuit is expected to be significantly smaller than in similar digital readout designs. Furthermore, linearization of the output may be achieved using an EEPROM lookup table. Another advantage is the possibility of performing adaptive measurements where the sensor resolution and bandwidth may be changed via the readout circuit software. Finally, we present a theory of the adaptive measurement and an analysis of the design tradeoffs. The capacitance-to-duty cycle readout circuit may achieve large bandwidth and high resolution in a modern low-voltage, low-power CMOS implementation. The performance of a prototype readout circuit built from discrete components is 13-bit effective resolution with a 1-kHz bandwidth.     

Dynamic magneto-optic detector for analog readout

The addition of a polarization azimuth vibrator, such as a Faraday cell, to a conventional Kerr apparatus yields a dynamic detector with a high SNR capability. The method described here has been utilized in the past as a null detector to observe Kerr rotation. It can be used equally well for analog readout of the magnetic state of the film element. The improved Kerr apparatus is well-suited to a common-mode noise rejection configuration. The common-mode configuration is usable both as a measuring instrument and as a high-speed analog readout system. Dynamic single- and double-ended Kerr apparatus are discussed and typical photodetector output waveforms are shown.     

Centroid error analysis of Shack–Hartmann wave-front sensor induced by the sensitivity non-uniformity

Centroid error is the crucial limit to the performance of the Shack–Hartmann wave-front sensor. The previous studies showed that the centroid error was affected by the discrete sample, the photon noise and the readout noise of the CCD. For many special characters, a CMOS camera is also used in the Shack–Hartmann wave-front sensor. Compared to a CCD, the sensitivity of a CMOS camera is obviously non-uniform. In this paper, a detailed analysis of the centroid error induced by the sensitivity non-uniformity of pixels is presented. The theoretical results are represented and demonstrated through an experiment. The results show that the error induced by the sensitivity non-uniformity is only dependent upon the detector, and has no relationship with the other factors. For the CMOS camera mentioned in this paper, the centroid error is less than 0.015 pixels.     

Hierarchical wave-front sensing

We present a new wave-front sensing technique for adaptive optics based on use of several wave-front sensors dedicated to the sensing of a different range of spatial frequencies. We call it a hierarchical wave-front sensor. We present the concept of a hierarchical wave-front sensor and apply it to the Shack-Hartmann sensor. We show the gain that is expected with two Shack-Hartmann sensors. We obtain a gain that increases with the size of the largest sensor, and we detail the application of hierarchical wave-front sensing to extreme adaptive optics and extremely large telescopes.     

Kalman filtering for adaptive nonuniformity correction in infrared focal-plane arrays

A novel statistical approach is undertaken for the adaptive estimation of the gain and bias nonuniformity in infrared focal-plane array sensors from scene data. The gain and the bias of each detector are regarded as random state variables modeled by a discrete-time Gauss-Markov process. The proposed Gauss-Markov framework provides a mechanism for capturing the slow and random drift in the fixed-pattern noise as the operational conditions of the sensor vary in time. With a temporal stochastic model for each detector's gain and bias at hand, a Kalman filter is derived that uses scene data, comprising the detector's readout values sampled over a short period of time, to optimally update the detector's gain and bias estimates as these parameters drift. The proposed technique relies on a certain spatiotemporal diversity condition in the data, which is satisfied when all detectors see approximately the same range of temperatures within the periods between successive estimation epochs. The performance of the proposed technique is thoroughly studied, and its utility in mitigating fixed-pattern noise is demonstrated with both real infrared and simulated imagery.