Class Activity: Optics Problem 3

Telescopes work much the same way as a series of slits in a wall letting light through onto a screen. We know that the pattern of light on the screen is related to the size, spacing, and number of slits in the wall, and if there are several slits next to each other with infinitesimal widths between them, the effect is essentially the same as having one large slit (see image below).

We know that the intensity of the light on the screen is the Fourier transform of the intensity of the light coming through the slits, which, for several slits right next to each other (one big slit), is a top-hat function. The Fourier transform of a top-hat function is a sinc function, or decaying sine function, as shown above. The only difference is that the intensity of light on the screen is the square of the light wave hitting the screen, so the actual sinc function includes negative values. In addition, if the width of the top-hat is W, then in the Fourier transform, the sinc function equals zero at the points 1/W, 2/W, 3/W, and so on.

The width is also related to the wavelength of the light coming through the large slit, since there will be destructive interference where the light from the bottom of the slit and the light from the top of the slit are out of phase by λ/2. Therefore, the width of the center peak of intensity will be:

This is the angular resolution, and it simplifies to:

Therefore, the angular resolution of CCAT, which is a 25-meter telescope that detects wavelengths of 850 microns = 8.5 * 10^-4 m, will have an angular resolution of:

where the angular resolution is measured in radians. The Keck 10-meter telescope observes in the J-band, which is defined as wavelengths between 1.1 and 1.4 micrometers = 1.1 * 10^-6  to 1.4 * 10^-6 m. Its angular resolution, using the maximum observation wavelength, is:

with D in radians, again. A smaller angular resolution means smaller sources can be resolved, since anything with an angular size larger than the angular resolution can be resolved by the telescope. This means that the Keck telescope, despite being smaller than CCAT, can resolve smaller objects in its observational range.

Secondary author: Joanna Robaszewski