1. Supposing that the following object is imaged by a lens system suffering spherical aberration only, make a sketch of the image.
2. The
Arecibo radio telescope has a spherical primary mirror, called dish by
radio astronomers, fixed in the ground in a valley in Puerto Rico. The disk has
a diameter of 305m, a radius of curvature of 265m, and a depth of 51m. A
schematic of the telescope is visible at this page.
(a) What is the focal length of the dish (for rays originating at infinity
and reflecting close to the principal axis of the telescope)?
(b) Like any spherical reflecting surface, Arecibo's dish is subject to spherical
aberration. In this part, you will find where a ray that
reflects from the dish's edge focuses. To do this, you will need to make a careful
diagram. Consider a ray parallel to the
principal axis that hits the dish at its edge. Trace the path of this ray as it is
reflected. Now draw a line showing the radius of the dish at this point. Calculate,
using geometry and trigonometry, the distance from the focal point at which rays
from the outer edge of the disk will focus. Also indicate whether this point is
closer or further from the dish than the focal point.
(c) Look at this
picture showing a closeup of the Arecibo platform. At right is the 96-foot long 430 MHz
line-feed, one of the detectors, designed to correct spherical aberration from the dish.
Explain how it works and compare the number that you found in (b).
(d) While the line feed you investigated in (c) does correct spherical
aberration, it is subject to other effects, such as noise from radio sources on
the ground. So in the mid-1990s, the dome structure at left in
the picture
was added to correct for spherical aberration in a more efficient manner.
The dome is the height of a 6-story building and is suspended 137m above the dish.
It uses two reflecting surfaces to correct the spherical aberration.
It also contains Arecibo's detectors and the radar transmitter.
You can see inside the dome in
this picture.
Schematics of the reflectors and the light paths are shown at
this site. Explain how the combination of the two reflectors corrects for
spherical aberration. Speculate why the system is designed using these two particular
(complex) shapes and arrangement.
Note: A telescope design that places the secondary reflector behind the focal point of
the primary reflector is called 'Gregorian', because it was designed by James Gregory.
Arecibo's dome is called the Gregorian dome because the optics are based on this design.
Last updated September 17, 2009
