Physical (diffraction) optics

Up until now, we have avoided considering the wave nature of light which introduces diffraction from interference of light coming from different parts of the aperture. Because of diffraction, images of a point source will be slightly blurred. From simple geometric arguments, we can estimate the size of the blur introduced from diffraction:

We find that:

$$\theta \sim {\lambda\over D}$$

Using this, we find that the diffraction blur is smaller than the blur introduced by seeing for $D\mathrel{\vcenter{\offinterlineskip \hbox{$>$} \kern 0.3ex \hbox{$\sim$}}}0.2$ meters at 5500 Å, even for the excellent seeing conditions of 0.5 arcsecond images. However, the study of diffraction has become important recently because of several reasons: 1) the existence of the Hubble Space Telescope, which is diffraction limited (no seeing), 2) the increasing use of infrared observations, where diffraction is more important than in the optical, and 3) the development of adaptive optics, which attempts to remove some of the distortions caused by seeing. Consequently, it's now worthwhile to understand some details about diffraction.

To work out in detail the shape of the images formed from diffraction involves understanding wave propagation. Basically, one integrates over all of the source points in the aperture (or exit pupil for an optical system), determining the contribution of each point at each place in the image plane. The contributions are all summed taking into account phase differences at each image point, which causes reinforcment at some points and cancellation at others. The expression which sums all of the individual source points is called the diffraction integral. When the details are worked out, one finds that the intensity in the image plane is related to the intensity and phase at the exit pupil. In fact the wavefront is described at any plane by the optical transfer function, which gives the intensity and phase of the wave at all locations in that plane. The OTF at the pupil plane and at the image plane are a Fourier transform pair. Consequently, we can determine the light distribution in the image plane by taking the Fourier transform of the pupil plane; the light distribution, or point spread function, is just the modulus-squared of the OTF at the image plane. Symbolically, we have

$$PSF = \left\vert FT(OTF(pupil))\right\vert^2$$

$$OTF(pupil) = P(x,y)\exp{i k \phi(x,y)}$$

$P(x,y)$ is the pupil function, which gives the transmission properties of the pupil, and usually consists of ones and zeros for locations where light is either transmitted or blocked (e.g., for a circular lens, the pupil function is unity within the radius of lens, and zero outside; for a typical telescope the pupil function includes obscuration by the secondary and secondary support structure). $\phi$ is the phase in the pupil. More relevantly, $\phi$ can be taken to be the optical path difference in the pupil with some fiducial phase, since only OPDs matter, not the absolute phase. Finally the wavenumber $k$ is just ${2\pi\over \lambda}$.

For the simple case of a plane wave with no phase errors, the diffraction integral can be solved analytically. The result for a circular aperture with a central obscuration, when the fractional radius of the obscuration is given by $\epsilon$, the expression for the PSF is:

$$PSF \propto \left[{2J_1(v)\over v} - \epsilon^2{2J_1(\epsilon v)\over \epsilon v}\right]^2$$

$$v = {\pi r \over \lambda F}$$

where $J_1$ is a first order Bessel function, $r$ is the distance in the image plane, $\lambda$ is the wavelength, and $F$ is the focal ratio ($F=f/D$).

This expression gives the so-called Airy pattern which has a central disk surrounded by concentric dark and bright rings. One finds that the radius of the first dark ring is at the physical distance $r=1.22\lambda F$, or alternatively, the angular distance $\alpha = 1.22\lambda/D$. This gives the size of the Airy disk.

For more complex cases, the diffraction integral is solved numerically by doing a Fourier transform. The pupil function is often more complex than a simple circle, because there are often additional items which block light in the pupil, such as the support structures for the secondary mirror.

This figure shows the Airy pattern, both without obscurations, and with a central obscuration and spiders in a setup typical of a telescope.

In addition, there may be phase errors in the exit pupil, because of the existence of any one of the sources of aberration discussed above. For general use, $\phi$ is often expressed as an series, where the expansion is over a set of orthogonal polynomials for the aperture which is being used. For circular apertures with (or without) a central obscuration (the case most often found in astronomy), the appropriate polynomials are called Zernike polynomials. The lowest order terms are just uniform slopes of phase across the pupil, called tilt, and simply correspond to motion in the image plane. The next terms correspond to the expressions for the OPD which we found above for focus, astigmatism, coma, and spherical aberration, generalized to allow any orientation of the phase errors in the pupil. Higher order terms correspond to higher order aberrations.

This figure shows the form of some of the low order Zernike terms: the first corresponds to focus aberration, the next two to astigmatism, the next two to coma, the next two to trefoil aberration, and the last to spherical aberration.

A wonderful example of the application of all of this stuff was in the diagnosis of spherical aberration in the Hubble Space Telescope, which has been corrected in subsequent instruments in the telescope, which introduce spherical aberration of the opposite sign. To perform this correction, however, required and accurate understanding of the amplitude of the aberration. This was derived from analysis of on-orbit images, as shown in this figure. Note that it is possible in some cases to try to recover the phase errors from analysis of images. This is called phase retrieval. There are several ways of trying to do this, some of which are complex, so we won't go into them, but it's good to know that it is possible. But an accurate amplitude of spherical aberration was derived from these images. This derived value was later found to correspond almost exactly to the error expected from an error which was made in the testing facility for the HST primary mirror, and the agreement of these two values allowed the construction of new corrective optics to proceed...