PHYSICS 106
Spring 2000
Homework set #3
Due March 7
Geometrical optics
1. Space shuttle optics
In microgravity (as inside an orbiting
space shuttle) it is possible to have a sphere of water suspended in mid-air.
Where are its principal points? Thinking of it as a lens, what is its focal
length? What happens to the focal length as the water evaporates?
Solution to problem #1
2. Beam expander
I want to construct a beam expander,
i.e. a device which will take a narrow (1 mm diameter) laser beam and expand
it to a parallel beam about 1 cm in diameter. I begin with a thick biconcave
lens with unequal radii of curvature R1
and R2 for the two surfaces.
-
Construct the matrix for such a lens.
What is its focal length in terms of R1
and R2 and the center thickness?
What effect do you expect such a lens to have on the laser beam (i.e. diverge,
converge, or what)? Does your matrix give the correct result?
-
Now make a beam expander by adding a
biconvex lens at some distance from the first lens. Choose reasonable values
for the radii, thicknesses, and indices of refraction of the lenses and
design a beam expander which will be practical. (Note, for example, that
I would not consider a device 5 m long to be practical.)
Solution to problem #2
3. Spherical aberration
Many street lights equipped with
photosensors to turn them on and off contain a simple thick lens consisting
of one spherical surface and one planar one as shown below. The lens serves
to focus light from a source at infinity (i.e. the sky) on to the back
plane of the lens, which is attached to the photodetector.
-
In the paraxial approximation, what
must the index of refraction m
of the lens be so that light from infinity is focused on the back plane
of the lens?
In fact the rays will not all be paraxial
(some of them will strike the front of the lens at points far from the
optic axis and thus make large angles with the lens normal), and rays parallel
to the optic axis that strike the front surface of the lens at different
distances h from the optic axis will be focused at different points,
i.e. they will cross the optic axis at different distances from the back
plane (ignoring refraction at the back plane). This is longitudinal
spherical aberration, which we express in terms of z, the distance
from point A (the intersection of the back plane with the optic
axis) at which a ray crosses the optic axis. Which brings us to the second
(longer!) part of the exercise:
-
Derive an exact relation between
z and h (i.e. without the paraxial approximation), neglecting
refraction at the back plane. [Do not worry about grinding through the
algebra necessary to get z(h) explicitly, just carry it to
the point where you have an equation that contains only z, h,
R and m.]
What is z for the paraxial rays? For marginal rays (rays that strike
the lens at its edge)? How could you reduce the effects of this spherical
aberration?
Solution
to problem 3
4. GRIN lens
A GRadient INdex or GRIN lens is
made by varying the index of refraction, rather than the thickness, of
the optical element. That is, it consists of a glass plate of uniform thickness
t whose index varies with the distance r from the center according
to 
where m
2 and a
are constants. (This is accomplished by changing the composition of the
glass from the center to the edge.) Define the optical path length
from point A to point B in a medium as
so that the time it takes light to travel from A to B is 
.
Fermat's Principle states that light travels in a medium along
the path that takes the minimum time. Use this principle to show
that the focal length of the GRIN lens is f=1/2ta.
If you prefer, you may instead approach this problem by formulating the appropriate matrix for the passage of light through the GRIN lens. Note that mirages arise from a similar phenomenon, in which the refractive
index of air over a hot surface (such as the desert floor) varies
with height due to the temperature gradient. The self-focussing of powerful laser beams in materials due to electric-field-induced changes in the refractive index is another example.
Solution to problem 4