We shall conclude this chapter where we started it by looking again at the atmospheric degradation of a stellar image. Does the K41 model of turbulence give a good picture of the point spread function of a large-aperture telescope? Although long-exposure images of point stars have long been recorded, and have been used in some ways to investigate this question, it becomes much more interesting when applied to instantaneous images which show the effect of a "frozen" statistical state of the atmosphere. This became possible with the development of image intensi-fiers having amplifications of the order of 106 and reasonable quantum efficiencies (about 10%). The resulting speckle images, of which we showed an example in figure 5.1, are the basis of the technique of speckle interferometry, initially suggested and developed by Labeyrie (1970) and discussed in detail in chapter 6. We have already discussed (section 5.5.1) the frequency spectrum of fluctuations, which determines how short an exposure is necessary to freeze an "instantaneous" image; now we shall apply the atmospheric structure analysis to modeling a speckle image.

Fig. 5.11. Simulated speckle images, using the structure function (5.28), with r0 = 7 units. (a) The phase field across a circular aperture, radius 64 units. Phase, modulo 2n, is indicated by gray level from white to black. (b) The point spread function corresponding to the phase field (a). (c) The ideal point spread function for the same circular aperture. (d) Long-exposure average of 50 random simulations like (b).

Fig. 5.11. Simulated speckle images, using the structure function (5.28), with r0 = 7 units. (a) The phase field across a circular aperture, radius 64 units. Phase, modulo 2n, is indicated by gray level from white to black. (b) The point spread function corresponding to the phase field (a). (c) The ideal point spread function for the same circular aperture. (d) Long-exposure average of 50 random simulations like (b).

The speckle image is that of a point source degraded by the phase distortions produced by the atmosphere, described in section 5.4 by the function ^(x, h) (5.10). Assuming that the amplitude variations are relatively insignificant, we can describe the image as the Fraunhofer diffraction pattern (i.e. Fourier transform, F(u)) of the function f (r) = 5"0(r)exp[i0(r)]. Here 5"0(r) describes the telescope aperture, i.e. has value 1 within it and 0 outside, and would generally be circ(r/R) or maybe an annular function to allow for a central obscuration (secondary mirror).

Of course, the observed image, which is the instantaneous point spread function, is |F(u)|2.

We describe here a simple model which has the right features and illustrates the physics. The atmospheric fluctuations are modeled by representing the phase 0(r) as a random function which changes by amounts in the range of ±n from point to point on the distance scale of Fried's parameter r0. The simulation allows any wavefront structure function to be used, although it appears that the differences between the results for the Kolmogorov structure function (5.28) and a random Gaussian approximation are very minor.

In this model, we start with the facts that the wavefront f (r) in the aperture plane has unit modulus and unknown phase 0(r), but its autocorrelation function has the Kolmogorov form

Since the intensity of the Fourier transform* F(u) of f (r) is the transform of its autocorrelation function, but its phase is arbitrary, we use the known B(r), transform it to b(u) and then take the most general form of the transform F(u) to be V|b(u)| exp[i$(u)] where $(u) is a random phase function. We then transform this back to the aperture plane, where it gives us a function f1(r) with random phases and amplitudes having the right correlation function. We then constrain this function by the bounding aperture S0. Finally, we transform this back to the image plane u to see the speckle image or atmospherically degraded point spread function.

Figure5.11 shows an example of results with a 512 x 512 pixel input field. The figure shows the phase field, in which the Fried parameter can readily be appreciated as the average size of a uniform patch, a typical speckle image, the diffraction-limited image [n R22 /1 (uR)/(uR)]2 and a simulated long-exposure image obtained by superimposing 50 independent calculations.

Two further details which are of interest can easily be seen from the same simulation. First, if the range of the random 0 is less than 2n, there is in addition a stronger spot at the center (u = 0) into which a larger and larger fraction of the energy is concentrated as the range of 0 gets smaller (figure 5.12a). The fraction of the wave energy concentrated in this central spot, defined by the size of the diffraction-limited spot for the complete aperture S0, is called the Strehl ratio. Clearly, when the random phase range approaches zero, the Strehl ratio becomes unity*. Second, the form

* We use the convention (Appendix A) that a function and its Fourier transform are indicated by capital and lower-case symbols.

* This property is employed in the spatial filters commonly used with laser systems in the laboratory. The wavefront from a laser has small spatially varying deviations (in both amplitude and phase) from a plane wave. We focus the wave to as small a region as possible and insert a pinhole around the central spot. The transmitted wave is a much closer approximation to an ideal spherical wave because the components of the original wavefront with non-zero spatial frequencies have been rejected by the pinhole filter.

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