Imaging Through Turbulence

Michael C. Roggemann, Byron M. Welsh

• Publisher: CRC Press
• Year: 2018
• ISBN:
  9781351439305
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Electrical Engineering & Telecommunications
• Index: Technology & Engineering

Learn how to overcome resolution limitations caused by atmospheric turbulence in Imaging Through Turbulence. This hands-on book thoroughly discusses the nature of turbulence effects on optical imaging systems, techniques used to overcome these effects, performance analysis methods, and representative examples of performance. Neatly pulling together widely scattered material, it covers Fourier and statistical optics, turbulence effects on imaging systems, simulation of turbulence effects and correction techniques, speckle imaging, adaptive optics, and hybrid imaging. Imaging Through Turbulence is written in tutorial style, logically guiding you through these essential topics. It helps you bring down to earth the complexities of coping with turbulence.

1 Introduction

1.1 Overview of the problem area

The simple experiment of turning a telescope with a diameter of 20 cm or larger towards a bright star on a clear night and measuring the point spread function of the combined atmosphere-telescope system at visible wavelengths reveals a critical defect in the measured image. Even if the telescope is completely free of aberrations, the point spread function (PSF) measured over an exposure time on the order of several seconds will be much broader than the PSF predicted by diffraction alone. A short exposure image, with exposure time of a few tens of milliseconds, demonstrates a different manifestation of the same phenomenon - images which have a speckled appearance. Examples of these image defects are shown in Fig. 1.1. The physical origin of these image defects is known to be atmospheric turbulence. In the absence of compensating techniques, atmospheric turbulence imposes a fundamental limit on the angular resolution of many optical systems. Resolution limits imposed by turbulence profoundly limit the performance of imaging systems which must look through the atmosphere.

Atmospheric turbulence affects imaging systems by virtue of wave propagation through a medium with a nonuniform index of refraction. Light in a narrow spectral band approaching the atmosphere from a distant light source, such as a star, is well modeled by a plane wave. The planar nature of this wave remains unchanged as long as the wave propagates through free space, which has a uniform index of refraction. The atmosphere, however, does not have a uniform index of refraction. Rather, the atmosphere contains a multitude of randomly distributed regions of uniform index of refraction, referred to as turbulent eddies. The index of refraction varies from eddy to eddy. As a result, the light wave that reaches the surface of the Earth is not planar. Excursions of this wave from a plane are manifested as random aberrations in astronomical imaging systems. Anyone who wears glasses understands the general effects of aberrations on imaging systems; they generally broaden the point spread function of the image, lowering the resolution which can be achieved compared to an ideal system. Imagine now the situation in which these aberrations are random and evolve with time. In this situation glasses (i.e., fixed correcting optics) are no longer capable of correcting for the random aberrations. It is the randomness and time evolution of the random aberrations caused by atmospheric turbulence that make imaging through the Earth’s atmosphere a difficult and challenging problem. These aberrations are the underlying causes of the image defects shown in Fig. 1.1.

The practical consequence of atmospheric turbulence is that resolution is generally limited by turbulence rather than by the optical design and optical quality of a telescope. Even at the best observatory sites, the angular resolution is limited to approximately 1 arcsec (~ 5μ rad) at visible wavelengths regardless of the size of the telescope. Compare this resolution to the theoretically achievable resolution of 0.013 arcsec for a 8 meter telescope. The Hubble Space Telescope achieves a diffraction limited resolution of 0.05 arcsec due to the fact it is above the Earth’s atmosphere. Very large telescopes, on the order of four to eight meters in diameter, have been successfully built for astronomical imaging. These telescopes have the full light gathering capability of their large aperture, allowing extremely dim objects to be observed. However, at visible and near infrared wavelengths, the angular resolution achieved is equivalent to that obtained with a much smaller telescope, on the order of a few tens of centimeters in diameter.

Figure 1.1 Simulated star images: (a) short exposure image; (b) long exposure image; (c) diffraction-limited image. For these results the telescope diameter is D = 1 meter, the atmospheric turbulence conditions represent average seeing at a good observatory location (i.e., the atmospheric coherence diameter r₀ = 10 centimeters) and the mean wavelength is λ = 550 nanometers.

Since the effects of turbulence on imaging systems were widely recognized in the 1950’s and 1960’s, a number of methods have been proposed to mitigate these effects. Three broad classes of techniques to mitigate turbulence effects exist: 1) pure post processing techniques, such as the so-called speckle imaging methods, which use specialized image measurements and image post processing; 2) adaptive optics techniques, which use mechanical means to sense and correct for turbulence effects as they occur; and 3) hybrid methods, which combine elements of the post processing techniques and the adaptive optics techniques. Within each of these broad classes there exists one or more imaging techniques for overcoming turbulence effects. Each of these techniques has its own set of performance limits, hardware requirements, and software requirements.

The goal of this book is to provide a single reference on how turbulence affects imaging systems, and the on various techniques for overcoming of the effects of turbulence on imaging systems. Even though we focus on applications involving ground-based systems looking up through the Earth’s atmosphere, most of the theoretical wave propagation and imaging system results can also be applied to applications involving horizontal propagation. Presently, this information is scattered across a wide body of technical literature. The discussion presented here integrates this large body of technical literature, and provides information regarding the tradeoffs between the various imaging methods. Every effort has been made to accurately represent the state of the art as it exists at this writing. However, the area of imaging through turbulence is an active research area - future innovations will no doubt lead to greater advances.

1.2 Historical overview of imaging through turbulence

This section provides a brief discussion of the history of understanding turbulence effects on imaging systems, and the efforts to overcome the limits imposed by atmospheric turbulence. The literature in this area is far too extensive to cite all contributions. A topical survey of key results is provided below.

1.2.1 Recognition of turbulence effects

Issac Newton was aware that, in the absence of any correction, it is impossible to attain diffraction limited performance at visible wavelengths with a ground-based telescope bigger than a few tens of centimeters in diameter [1], regardless of the design and optical quality of the telescope. In Newton’s day some of the optical consequences of atmospheric turbulence were known. The twinkling of the stars was well known, and it had also been noted that the planets did not twinkle. Further, by Newton’s time it was known that the point spread function of a telescope obtained by looking at a star was significantly broader than the point spread function which could be observed under laboratory conditions. Newton correctly attributed these effects to “tremors” in the atmosphere [1, page 423]:

“If the theory of making Telescopes could at length he fully brought into Practice, yet would there be certain Bounds beyond which Telescopes could not perform. For the air through which we look upon the Stars, is in perpetual Tremor; as may be seen by the tremulous Motion of Shadows cast from high Towers, and by the twinkling of the fix’d stars. “

Newton was also able to explain qualitatively why stars twinkle when viewed with the naked eye, but do not twinkle when viewed with telescopes:

“But these Stars do not twinkle when viewed through Tele s с op e s which have large apertures. For the Rays of Light which pass through divers parts of the aperture, tremble each of them apart, and by means of their various and sometimes contrary Tremors, fall at one and the same time upon different points at the bottom of the Eye, and their trembling Motions are too quick and confused to be perceived severally. “

Though we would use more modern terms to describe this phenomenon today, Newton’s insight that atmospheric turbulence was the cause of this effect was correct. Newton also noted that the point spread function of a telescope looking through turbulence is broader than would be expected in the absence of the atmosphere. As a result, large telescopes could be used to measure dim objects by virtue of the light gathering capability of a large aperture, but a large telescope alone could not overcome the effects of atmospheric turbulence:

“And all these illuminated P oints constitute one broad lucid P oint, composed ofthose many trembling Points confusedly and insensibly mixed with one another by very short and swift Tremors, and thereby cause the Star to appear broader than it is, and without any trembling of the whole. Long Telescopes may cause Objects to appear brighter and larger than short ones can do, but they cannot be so formed as to take away the confusion of the Rays which arises from the Tremors of the Atmosphere. “

Newton’s suggestion that observatories be placed atop high mountains to partially mitigate the effects of atmospheric turbulence remains the standard wisdom for choosing observatory sites:

“The only Remedy is a most serene and quiet Air, such as may perhaps be found on the tops of the highest Mountains above the grosser Clouds. “

Understanding the origin of the optical effects of atmospheric turbulence did little to improve the state-of-the-art of astronomy until modern times. In Newton’s day the only available light detector, and the only processor of optical signals was the human visual system. The invention of photographic film in the early 1800’s eventually resulted in the ability to permanently record images measured through turbulence, but the combined effects of poor film sensitivity and an interest in viewing dim objects resulted in long exposure image measurements. These long exposure images contained the result of a very large number of realizations of the random turbulence effects averaged into a single measurement. The resulting images were similar in character to those shown in Fig. 1.1b - the images of stars were much broader than the images that would arise due to diffraction alone. By the 1950’s, film systems had progressed to the point where it was possible to measure short exposure images of bright objects, essentially freezing the turbulence effects during the image measurement time. The first short exposure images were reported to look like a “bunch of grapes” [2], containing what are now called “speckles” (see Fig. 1.1a). The speckles were observed to be approximately diffraction limited in extent. These first short exposure images provided a hint that high resolution information was somehow encoded in short exposure image measurements. Further advances in turbulence understanding, light detection devices, and computerized signal processing were required to exploit this insight.

1.2.2 Understanding turbulence effects on wave propagation and imaging systems

Atmospheric turbulence arises from heating and cooling of the Earth’s surface by the sun. Sunlight warms large land masses during daylight hours, and these warm land masses heat the air. During the night the Earth’s surface gradually cools, and this heat is also coupled into the air. Heating the air in this manner results in large spatial scale motions. This air motion eventually becomes turbulent, with the result that the large spatial scale motions break up into progressively smaller scale motions, eventually giving rise to randomly sized and distributed pockets of air, each having a characteristic temperature. These pockets of air are the turbulent eddies referred to earlier. The index of refraction of air is sensitive to temperature, and hence, the atmosphere exhibits variations in the index of refraction. Plane waves propagating through the atmosphere are no longer planar when they arrive at the surface of the Earth.

The study of turbulent air motion is a problem in the field of fluid mechanics. During the 1940’s, Kolmogorov [3] developed a model for how energy is transported from large scale turbulent eddies to small scale turbulent eddies. Kolmogorov’s model provides a spatial power spectrum for the index of refraction fluctuations. Tatarskii applied Kolmogorov’s model to solve the wave equation for propagation through regions of weak random index fluctuations [4]. Fried used Tatarskii’s results to describe turbulence effects in terms of Zernike polynomials [5], and to derive a useful single parameter, 7*0, referred to as the atmospheric coherence diameter, to describe the effects of turbul...

Table of contents

• Cover Page
• Half Title
• Title Page
• Copyright
• Contents
• Preface
• 1 Introduction
• 2 Background: Fourier and Statistical Optics
• 3 Turbulence Effects on Imaging Systems
• 4 Speckle Imaging Techniques
• 5 Adaptive Optical Imaging Systems
• 6 Hybrid Imaging Techniques
• Index