Diffraction and Filtering

1. Introduction

This is a review of diffraction basics and how diffraction filters the timestreams.

2. Aperture Illumination, PSF, and Fourier Plane

Fraunhofer diffraction theory relates the illumination of the telescope aperture and the far-field beam pattern. Consider the case of a uniformly-illuminated circular aperture with 0 phase across the aperture:
telescope illumination

The far-field electric field distribution is given by the two-dimensional Fourier transform of the aperture illumination function:
far-field E distribution

The far-field power distribution (or point-spread function, or PSF) is the squared magnitude of the electric field distribution (point-spread function, or PSF):
far-field P distribution

For this case, the far-field power pattern is the well-known Airy disk, with peak-to-first-minimum size of 1.22 lambda/D, where lambda is the wavelength and D is the telescope aperture diameter. Finally, the Fourier transform of the power distribution is the transfer function of the telescope:
telescope transfer function

Convolution of a map with the telescope PSF is essentially the same operation as multiplying its 2-D F.T. by the telescope transfer function. We observe that the maximum spatial frequency in the transfer function is D/lambda.

In general, the telescope illumination, far-field E, and telescope transfer function are complex functions. Only the far-field P is real only.

We consider another case with a central blockage in the telescope aperture:
telescope illumination
far-field E distribution
far-field P distribution
telescope transfer function

Again, we observe that the maximum spatial frequency in the transfer function is D/lambda. (There must be a general proof of this.)

KEY CONCLUSION: The maximum spatial frequency in the map is given by D/lambda.

3. One-Dimensional Approximations

Consider a 1-D scan across a point source. Without diffraction, the point source produces equal power at all angular scales. Diffraction, however, acts as a low pass filter to the data. The curves below show 1-D scans through (red curve) and near (green, blue, and purple curves) a point source following diffraction from a circular aperture.
cuts through PSF
Fourier transforms of cuts through PSF

We can see the same hard filter edge acting on all four scans, but the distribution of power at lower frequencies varying among the scans.

The plot below shows the diffraction filtering for the case of the annular telescope aperture:
Fourier transforms of cuts through PSF

4. Scanning and Filtering of Timestreams

In this last section, we document the maximum temporal frequency in a diffraction-limited timestream resulting from scanned observing: 
f_max = velocity*D/lambda
For example, at the shortest wavelength of the SPIRE photometer: Therefore: f_max = 2.0 Hz (nominal) to 4.1 Hz (fast)
CDD, 2008 Oct 27