Simulation of a Kolmogorov phase screen

Abstract

AbstracL W O new methods for madelling Kolmogomv phase fluctuations oyer a finite apenure are described. Ihe firs1 method relies on the incorporation of subharmonim in order to model accurately the low frequencies of the Kolmogomv SpeclNm. Ihc second method provides a less accurate, but much faster method for simulating the Kolmogorw spectrum ty using a midpoint displacement algorithm used in computer graphics. BackgroundThe simulation of atmospherically distorted wavefronts is an important tool for studying light propagation and imaging. The work in this paper is motivated by the need to develop efficient and effective methods of imaging astronomical objects through the turbulent atmosphere. Although the true test of any imaging algorithm is always provided by actual data, a good simulation is needed to be able to test different algorithms both in a controlled manner and under a wide variety of conditions. This paper outlines new methods for simulating the effects of static atmospheric turbulence, which is an essential component of any atmospheric imaging simulation.The starting point for nearly all analyses of atmospheric turbulence has been the assumption that atmospheric turbulence follows a Kolmogorov spectrum and has a phase that is statistically uniform Over the interval -T to T . The fluctuations induced by the turbulence then cause a distortion of both the magnitude and phase of the wavefront incident on the atmosphere. In practice the phase distortion has considerably more effect on the quality of images formed from light passing through the turbulence than those effects due to the magnitude distortion. In many situations, an adequate approximation is a single phase screen located at the entrance pupil of the optical system, although this can not account for non-isoplanatic effects [l].A typical short exposure image formed by viewing a point source through turbulence does not consist of a single diffraction pattern with a diameter fixed by the diffraction limit of the telescope, but rather the image consists of a number of superimposed speckles distributed over a diameter determined by the severity of the turbulence. Each individual bright speckle has a diameter given approximately by the diffraction limit of the telescope. It is important that in addition to the production of a speckled distortion of the image, the simulated turbulence should also shift the centroid of the image formed. This effect of centroid motion is primarily due to the low frequencies in the Kolmogorov spectrum, which are often not modelled well in conventional F F l procedures.Whilst it has been proposed that the centroid motion can be compensated after the generation of the speckles, this is not the only part of the information present

Simulation of Time-evolving Speckle Patterns Using Kolmogorov Statistics

“…The difficulties in simulating Kolmogorov turbulence arise from the necessary involvement of the very low frequencies and are described in detail in a previous paper [21] . The two methods used there to simulate Kolmogorov turbulence are briefly summarized here and are extended to time evolution of the turbulent layer .…”

“…This restriction can be circumvented by the addition of subharmonics as shown in [21 ] .The Fourier transform of a discrete array containing a sine wave with a period length longer than the size of the array has non-zero values at all frequencies . This is because the frequency of the sine wave cannot be represented by discrete values of the spectrum as this frequency is smaller than unity .…”

“…function is defined as sine (x) =sin (x)/x . This process is described in more detail in [21] and the advantage over the conventional method is demonstrated for static turbulence .…”

“…In the following we describe our model for evolving speckle patterns using a single layer that is evolving as well as moving . A new algorithm is used to include low frequencies with a period length much larger than the phase screen according to Kolmogorov statistics [21] . We produce a sequence of speckle images and describe them by statistical means .…”

Propagation of low-quality laser beams in turbulent air: new fast simulation methods and experimental results

“…1), are then added to the phases of the small screen. This procedure is a kind of technique called "addition of subharmonics" [13]. The method has been successfully tested with reference to the theoretical phase structure function…”