1051-455-20073 Homework #7 Due 15 May 2008 (Th)
Do the following problems; SHOW YOUR WORK
1. Consider monochromatic light of wavelength λ0incident on a single slit of width balong the x-axis
and infinite length along y.The light is observed in the Fraunhofer diffraction region at a distance
L. Derive an approximate expression for the angular width at half-maximum irradiance of the central
peak of the diffraction spot (i.e., the FWHM).
2. The Fraunhofer diffraction pattern of a pair of rectangular slits of width bseparated by the center-to-
center distance dis illuminated by monochromatic light with λ0= 650 nm. The Fraunhofer diffraction
pattern is viewed at the back focal plane of a lens with focal length f= 800 mm. The center-to=center
separation between fringe maxima is observed to be D=1.04 mm.Thefifth maximum of the inter-
ference pattern on each side is “missing,” which means that it coincides with a zero in the diffraction
“envelope” due to the width of the slit. Determine band d.
3. Monochromatic light with wavelength λ0is incident on a circular aperture of diameter d. The circularly
symmetric diffraction pattern observed at a distance Lin the Fraunhofer diffraction region has radius
rfrom the central maximum to the first zero:
r∼
=1.22 Lλ0
d
(this distance is the separation required of the diffraction patterns from two point sources for them to
be distinguished under the Rayleigh criterion for resolution)
(a) Compare this result to the linear distance from the central maximum to the first zero for a square
aperture of width equal to the diameter dof the circular aperture.
(b) If the monochromatic light at λ0illuminates a circular lens of diameter dand focal length f,the
Fraunhofer diffraction pattern is observed at the focal plane so that L∼
=fand :
r∼
=1.22 fλ0
d
If observed in blue visible light, find an approximate relation between the diameter of the diffrac-
tion spot and the focal length of the lens. This is a very convenient “rule of thumb” for imaging
systems.
4. Compare the diameters of the diffraction spots for telescopes with primary optics having diameters
d1= 200 in (Hale Telescope on Palomar Mountain) and d2=90mm (Questar).
1