Modeling the combined effect of static and varying phase distortions on the performance of adaptive optical systems

The end-to-end performance achieved by an adaptive optical (AO) imaging system is determined by a combination of the residual time-varying phase distortions associated with atmospheric turbulence and the quasi-static unsensed and uncorrectable aberrations in the optical system itself. Although the effects of these two errors on the time-averaged Strehl ratio and the time-averaged optical transfer function (OTF) of the AO system are not formally separable, such an approximation is found to be accurate to within a few percent for a range of representative residual wave-front errors. In these calculations, we combined static optical system aberrations and time-varying residual phase distortion characteristics of a deformable mirror fitting error, wave-front sensor noise, and anisoplanatism. The static aberrations consist of focus errors of varying magnitudes as well as a combination of unsensed and uncorrectable mirror figure errors derived from modeling by the Gemini 8-Meter Telescopes Project. The overall Strehl ratios and OTF's that are due to the combined effect of these error sources are well approximated as products of separate factors for the static and time-varying aberrations, as long as the overall Strehl ratio that is due to both errors is greater than approximately 0.1. For lower Strehl ratios, the products provide lower bounds on the actual values of the Strehl ratio and the OTF. The speckle transfer function is also well approximated by a product of two functions, but only where AO compensation is sufficiently good that speckle imaging techniques are usually not required.     

Strehl ratio versus defocus for noncentrally obscured pupils

We discuss from the viewpoint of the Strehl ratio versus defocus, or the normalized axial-irradiance distribution, the influence of decentering the dark mask of an annular pupil. Our treatment, which is valid for pupil apertures with any Fresnel number, permits us to infer that the axial behavior of a noncentrally obscured pupil is equivalent to that of an apodizer with continuous amplitude variations. Hence the Strehl ratio versus defocus of an optical system can be shaped by use of noncentered dark masks that act as continuous gray apodizers. Several numerically evaluated examples are presented.     

Image-quality metrics for characterizing adaptive optics system performance

Adaptive optics system (AOS) performance is a function of the system design, seeing conditions, and light level of the wave-front beacon. It is desirable to optimize the controllable parameters in an AOS to maximize some measure of performance. For this optimization to be useful, it is necessary that a set of image-quality metrics be developed that vary monotonically with the AOS performance under a wide variety of imaging environments. Accordingly, as conditions change, one can be confident that the computed metrics dictate appropriate system settings that will optimize performance. Three such candidate metrics are presented. The first is the Strehl ratio; the second is a novel metric that modifies the Strehl ratio by integration of the modulus of the average system optical transfer function to a noise-effective cutoff frequency at which some specified image spectrum signal-to-noise ratio level is attained; and the third is simply the cutoff frequency just mentioned. It is shown that all three metrics are correlated with the rms error (RMSE) between the measured image and the associated diffraction-limited image. Of these, the Strehl ratio and the modified Strehl ratio exhibit consistently high correlations with the RMSE across a broad range of conditions and system settings. Furthermore, under conditions that yield a constant average system optical transfer function, the modified Strehl ratio can still be used to delineate image quality, whereas the Strehl ratio cannot.     

The Solution of Some Inverse Diffraction Problems in Terms of the Diffraction Coefficients

The form of the recently developed series representation of the axially symmetrical diffraction fields enables us to establish the pupil function from the series coefficients _γk (u). A new interpretation of the Strehl intensity ratio is given. It represents the ratio of the energy coming to the image focus and that coming to the other foci of the filter. Further, by the use of the three-term approximation, the transmissivity of the filter giving the desired two-point resolution and attaining the maximum value of the Strehl criterion is found and compared with the exact solution proposed by Luneberg. Finally, it is shown that the Luneberg filter transmissivity represents a common approximative solution of the apodization problems that utilize the integral criteria of apodization.     

Strehl ratio of a Gaussian beam

We discuss the Strehl ratio of systems with a Gaussian pupil and determine the range of validity of its approximate expression based on the aberration variance. The results given are equally applicable to propagation of Gaussian beams. The uniform and weakly truncated pupils are considered as limiting cases of a Gaussian pupil. We show that the approximate expression for Strehl ratio in terms of the aberration variance yields a good estimate of the true value for a strongly truncated pupil but a much smaller value for a weakly truncated pupil.     

Zernike annular polynomials and atmospheric turbulence

Imaging through atmospheric turbulence by systems with annular pupils is discussed using the Zernike annular polynomials. Fourier transforms of these polynomials are derived analytically to facilitate the calculation of variance and covariance of the aberration coefficients. Zernike annular shape functions are derived and used to calculate the Strehl ratio and the residual phase structure and mutual coherence functions when a certain number of modes are corrected using, say, a deformable mirror. Special cases of long- and short-exposure images are also considered. The results for systems with a circular pupil are obtained as a special case of the annular pupil.     

Dependence of Strehl ratio on f-number of optical system

Formulas for a minimum of wave aberration variance and a maximum of the Strehl ratio in the optimal image point are derived using the third- and fifth-order aberration theory. Moreover, relations for the calculation of the optimal value of f-number of the optical system were derived, which enabled us to theoretically analyze real optical systems and their image quality. The optimal f-number corresponds to such a value of f-number when the image quality of a real optical system is comparable to an aberration-free optical system. This value may also serve as an auxiliary criterion of the image quality of the optical system, for example, in photography.     

Influence of thermal deformations of the output windows of high-power laser systems on beam characteristics

By using the well-known Green's function methods, we study the three-dimensional temperature distributions and thermal deformations of the output windows of unstable optical resonators induced by an incident annular laser beam. Some expressions and theoretical profiles of the temperature distributions and thermal deformations as functions of the radius and of the thickness of optical windows are obtained. Moreover, the influence of the thermal deformations of sapphire, silica, and silicon windows within unstable optical resonators on the Strehl ratio and on the far-field laser intensity distribution is also discussed. Under conditions of 50-kW intense laser irradiation during 5 s, the maximum thermal deformation in sapphire, silica, and silicon substrates is 1.993, 0.393, and 6.251 microm, respectively. Under the same conditions the Strehl ratio of sapphire is higher than that of silica.     

Modal compensation and the atmospheric-turbulence outer scale: computer simulations

The influence of the turbulence outer scale on the Strehl ratio obtained with low-order adaptive optics systems is examined by numerical simulation. The Karhunen-Loeve approach is used to generate wave-front samples. A method that allows construction of the outer-scale-dependent Karhunen-Loeve functions is described. It is shown that the Strehl ratio produced by a second-order adaptive optics correction (tip-tilt, defocus, and astigmatism) is affected quite strongly by the finite outer scale. For the higher-order correction, the effect under study is weak and appears only when the outer-scale magnitude becomes less than the aperture diameter. It is also shown that the finite outer scale has a positive effect on the Strehl ratio of the uncorrected long-exposure image.     

A Study of the Possibility of Image Optimization by Apodization Filters in Optical Systems with Residual Aberrations

A systematic study to optimize the image in real optical systems, by fitting the filter transmission function on the exit pupil to different polynomial types is developed. The systems we have treated, whose residual aberrations are larger than the tolerance values, could be carried to tolerance by the use of an adequate filter, making the Strehl ratio higher than 0·8. The influence of the apodization filter on the optical transfer function is also studied.     

Theoretical study of near-field optical storage with a solid immersion lens

Both the reflection inside a hemisphere solid immersion lens (SIL) and the reflection inside the gap between the SIL and the optical recording medium are considered. The near-field SIL imaging theory for high numerical aperture is developed by using the vector diffraction and thin-film optics. Numerical results show that the spot size, Strehl ratio, and sidelobe intensity have an oscillatory behavior with the change of thickness of the air gap, which results from the interference effect of the transmitted field. We find that for smaller spot size, the Strehl ratio is smaller but the sidelobe intensity is larger. A certain thickness of air gap is useful for optical storage, which is less than 63 nm for the system in the simulated examples.     

Aberration and the Strehl ratio

An exact expression is derived for the Strehl ratio as a function of the minimum root-mean-square aberration value that may produce it. The result is applicable to any aperture and is shown to imply a similar expression for the Strehl ratio as a function of the minimum amplitude range of the aberration.     

Effect of wavefront corrugations on fringe motion in an astronomical interferometer with spatial filters

Numerical simulations of atmospheric turbulence and adaptive optics (AO) wavefront correction are performed to investigate the time scale for fringe motion in optical interferometers with spatial filters. These simulations focus especially on partial AO correction, where only a finite number of Zernike modes are compensated. The fringe motion is found to depend strongly on both the aperture diameter and the level of AO correction used. In all the simulations the coherence time scale for interference fringes is found to decrease dramatically when the Strehl ratio provided by the AO correction is < or = 30%. For AO systems that give perfect compensation of a limited number of Zernike modes, the aperture size that gives the optimum signal for fringe phase tracking is calculated. For AO systems that provide noisy compensation of Zernike modes (but are perfectly piston neutral), the noise properties of the AO system determine the coherence time scale of the fringes when the Strehl ratio is < or = 30%.     

Residual phase variance in partial correction: application to the estimate of the light intensity statistics

Although the wave-front correction provided by an adaptive optics system should be as complete as possible, only a partial compensation is attainable in the visible. An estimate of the residual phase variance in the compensated wave front can be used to calibrate system performance, but it is not a simple task when errors affect the compensation process. We propose a simple method for estimation of the residual phase variance that requires only the measurement of the Strehl ratio value. It provides good results over the whole range of compensation degrees. The estimate of the effective residual phase variance is useful not only for system calibration but also for determining the light intensity statistics to be expected in the image as a function of the degree of compensation introduced.     

Far-field optical degradation due to near-field transmission through a turbulent heated jet

When a laser beam traverses an optically active, turbulent flow field, the laser wave front is aberrated by the flow. Density variations in a heated two-dimensional jet, for example, correspond to index-of-refraction variations, and this modulation of the index in the fluid can imprint an optical phase disturbance, or phase error, onto the laser wave front. Adaptive-optic systems seek to correct the phase error of the wave front, and thus restore the integrity of the far-field irradiance pattern. Given a near-field spatial mapping of a phase disturbance, the far-field irradiance pattern of the affected wave front can be calculated with Fourier-optics techniques. A Fourier-optics computer code was used to study the far-field irradiance patterns arising from actual time-varying measurements of a fluid-induced phase error. The time-averaged Strehl ratio was studied to provide insight into the spatial and temporal design requirements for adaptive-optic systems applied to the time series of near-field spatial phase-error maps.     

Modeling turbulent wave-front phase as a fractional Brownian motion: a new approach

We introduce a new, general formalism to model the turbulent wave-front phase by using fractional Brownian motion processes. Moreover, it extends results to non-Kolmogorov turbulence. In particular, generalized expressions for the Strehl ratio and the angle-of-arrival variance are obtained. These are dependent on the dynamic state of the turbulence.     

Adaptive optics for in-orbit aberration correction: spherical aberration feasibility study

We investigate the feasibility of using an adaptive mirror for in-orbit aberration corrections. The advantage of an in situ aberration correction of optical components in the space environment is that the mirror shape can be adjusted in an iterative fashion until the best image is obtained. Using the actuator spacing, corresponding to one half of the Nyquist frequency, the Strehl ratio of the corrected wave front improves to 0.95 when the mirror is fabricated with 6.5 waves of spherical aberration. The Strehl ratio decreases to 0.86 when the number of actuators is reduced by a factor of 4, in a two-dimensional adaptive optics model.     

Calculation of retinal image quality for polychromatic light

Although the retinal image is typically polychromatic, few studies have examined polychromatic image quality in the human eye. We begin with a conceptual framework including the formulation of a psychophysical linking hypothesis that underlies the utility of image quality metrics based on the polychromatic point-spread function. We then outline strategies for computing polychromatic point-spread functions of the eye when monochromatic aberrations are known for only a single wavelength. Implementation problems and solutions for this strategy are described. Polychromatic image quality is largely unaffected by wavelength-dependent diffraction and higher-order chromatic aberration. However, accuracy is found to depend critically upon spectral sampling. Using typical aberrations from the Indiana Aberration Study, we assessed through-focus image quality for model eyes with and without chromatic aberrations using a polychromatic metric called the visual Strehl ratio. In the presence of typical levels of monochromatic aberrations, the effect of longitudinal chromatic aberration is greatly reduced. The effect of typical levels of transverse chromatic aberration is virtually eliminated in the presence of longitudinal chromatic aberration and monochromatic aberrations. Clinical value and limitations of the method are discussed.     

遥感相机像移速度的滤波处理

在遥感相机对地面目标摄影的过程中 ,由于地面目标在相机象面上的象是运动的 ,即存在象移 ,这将导致相机分辨力下降。为获得较高的动态照象分辨力 ,必须对象移进行补偿 ,而补偿的关键是准确地获得当前的象移速度。本文采用统计实验方法 ,分析了采用计算法得到的象移速度的分布特性 ,给出了实用的象移速度滤波预处理、卡尔曼滤波、野值剔除方法。仿真结果表明 ,所给出的滤波处理方法效果明显 ,简单实用     

Zernike representation and Strehl ratio of optical systems with variable numerical aperture

We consider optical systems with variable numerical aperture (NA) on the level of the Zernike coefficients of the correspondingly scalable pupil function. We thus present formulas for the Zernike coefficients and their first two derivatives as a function of the scaling factor ε ≤ 1, and we apply this to the Strehl ratio and its derivatives of NA-reduced optical systems. The formulas for the Zernike coefficients of NA-reduced optical systems are also useful for the forward calculation of point-spread functions and aberration retrieval within the Extended Nijboer–Zernike (ENZ) formalism for optical systems with reduced NA or systems that have a central obstruction. Thus, we retrieve a Gaussian, comatic pupil function on an annular set from the intensity point-spread function in the focal region under high-NA conditions.