Optical
Physics and Quantum Electronics

Applied Physics 216
Spring Term 19992000
Assignment 1(pdf)
Due Tuesday, March 14, 2000.

A little "warm up" with some geometric optics and ABCD matrices:
The image magnification or "effective focal length" of a zoom lens may
be varied, in a certain range, without changing the objecttoimage distance
and, thus, a continuous scaling of the image may be produced. A simple,
three element, reasonably good varifocal system is illustrated below.
Your task here is to analyze this system in the paraxial approximation
and to examine its varifocal characteristics.

Background: For brevity, let us refer to the ABCD matrix as a
system
matrix
which transforms the ray from location
to location .
With this convention we may write the ray transformation equations in the
form

More background: We illustrate below a lens formed from two spherical surfaces
with radii of curvature
and .
For this lens find, in the paraxial approximation, the matrix which links
ray parameters defined in the plane of the vertex
to those defined in the plane of the vertex
 viz
where the set of matrix elements
are the so called Gaussian constants of the lens. Find this matrix
by multiplying component matrices which account for refraction at the two
surfaces and the translation between surfaces. Hint: In the paraxial limit,
is approximately equal to the intervertex distance.

More background: Using
from above, show that the matrix which links ray
parameters defined in the front focal plane ()
of the lens to those defined in the back focal plane ()
is given by
and find the constant(s).

Finally, the main task: Find the system matrix connecting
points in the object and image plane for the varifocal system illustrated
above. Show that image magnification can be varied by moving the middle
lens while the distance between object and image planes remains relatively
constant.

A light ray is incident upon a layered halfspace as illustrated below.
Let us suppose that the index of refraction for
is constant and equal to .
For
we suppose that it fluctuates sinusoidally and is given by
with .

In the paraxial approximation, find an expression for ray trajectories
in the layer medium. Carefully specify the conditions under which the expression
is valid.

Using your favorite math processor (i.e., Mathematica, MATLAB,
etc.),
plot a sample set of such trajectories for the case when .

We may gain considerable insight by revisiting Problem 2 as a coupled wave
problem. In particular, treat the reflected wave as a field which is coupled
to an incident plane wave by the spatial varying dielectric constant .
Suppose that driving incident field propagates at an angle [[theta]] with
respect to the zaxis.

Following the development in the lecture set Nonlinear Optics  Classical
Picture, generate a set of coupled Maxwell equations for the growing
reflected wave and the diminishing driving field.

Find the condition(s) for "phase matching." Discuss.

Solve the coupled Maxwell equations for the phase matched condition(s).

In optics a common practical problem is the focusing of the output of a
laser beam onto the end face of an optical fiber located at a distance
L from the laser. Assume that the laser beam is of fundamental gaussian
beam form with a beam waist (of radius w1) at the output of the laser.
For most effective coupling, the beam should be focused at the face of
the fiber to a waist of radius w2 which is smaller than the radius of the
fiber. Find the distance d from the laser at which a thin lens (of focal
length f) should be located to achieve the requisite transformation. (A
hint which should be taken seriously: work forward from the laser and backward
from the fiber end until the gaussian beams interect.)

Suppose that a Gaussian beam is propagating along a quadratic GRIN
rod.

If the beam has a qvalue of
at some particular position ,
show that the qvalue at some position is given by
[1]
.

Using your favorite math processor, plot the the beam curvature and width
as function of position.

Find an expression for dispersion relationship of the GRINguided Gaussian
beam  i.e. an expression for the longitudinal phase constant,
b,
as a function of frequency, w.

In turn, find an expression for the group velocity of the GRINguided Gaussian
beam  i.e .

The frequency separation of the modes of a cavity is very important issue
in laser physics. The following exercises is a attempt to shed light on
that issue.

a. Following the discussion of asymmetric spherical (mirror) resonators
in the lecture notes, show that the resonance frequencies may be written

For a symmetric spherical resonator, use your favorite math processor's
graphics to prepare a plot of how the following frequency differences vary
with the parameter u over the range of stable values  i.e.,
from
0 to 2.

In the lecture set Descriptions of Polarized Light we defined the
Stokes vector representation of the polarization state of a light beam.
As we asserted there, perhaps the most valuable attribute of this representation
is its use in connection with Mueller matrices. The general idea is that
effect of a given device or process on the polarization state of can be
treated as a vector transformation of the form ("Mueller calculus")
where the asterisk indicates Stokes parameters transformed
by propagation through a device or physical process. Of course, the Stokes
parameters are defined with respect to a particular reference frame and,
in particular, a frame in which the initial beam has the simplest representation.
The initially defined reference frame may not be the most useful for representing
the effect of a given device or physical process. You should convince yourself
that the following transformation can be used to convert the initial Stokes
parameters to a new reference frame at an angle
with respect to the in intial frame:
(It should be noted that this transform can be used to represent
the effect of "optical activity")
Thus, the compound effect of the change of reference frame and device
interaction can be represented as
Of course, the true value of the Mueller calculus lies in the
fact that propagation through a long series of devices or physical processes
is systematical tracked through a series of matrix multiplications.
As a ramp up to the main task, you should convince yourself
that the following tabulation makes sense:
Device

Mueller Matrix

Partial polarizer
and
are the intensity transmission
coefficients in two orthogonal directions.


Pure retarder
 i.e.,devices which alter the phase relationship
between two orthogonal components of light
(D is the differential retardation produced)


Isotropic absorber
is the intensity
transmission coefficient.


Perfect depolarizer


Finally, your tasks are to:

Find the Mueller matrix for specular reflection from a planar interface
between two different isotropic dielectric materials.

Find the Mueller matrix for transmission through a planar interface between
two different isotropic dielectric materials

Find the Mueller matrix for a beam propagating at an angle [[theta]] with
respect to the optic axis of uniaxial optical material.

Find the Mueller matrix for a beam propagating at an angle [[theta]] with
respect to the magnetic axis in a simple magnetooptical substance.

8. In the lecture set Optical Pulse Propagation,we established that
the optical pulse envelope for a pulse propagation in low dispersion media
is described by the following firstorder pulse dispersion differential
equation:
.
where .
For many problems a pulse dispersion integration equation is more
useful in doing calculations.

Therefore derive the following pulse dispersion integration equation (I
can think of two distinct ways to establish this equation):
where ,,
and
is the "impulse response" function  i.e. the response for a sharp
spike at
 given by
.

In studies of pulse propagation it is useful to compare the behavior of
Gaussian and superGaussian pulse shapes. The mthorder superGaussians
are generalizations of the Gaussian pulse shape discussed in lecture 
i.e.,
 that have steeper leading and trailing edges and take the
form
.
where in both cases, C is the initial "chirp parameter."
Using your favorate math processor to do numerical integrations
of the pulse dispersion integration equation above, plot the pulse envelope
shape for an initially unchirped, thirdorder superGaussian (m
= 3) at three distance  viz.,
where .
Compare these results with analytic expression derived in lecture for an
initially unchirped firstorder Gaussian
[1] Electrical engineers in the crowd will recognize
that this is the formula for wave impedance transformation in transmission
line theory and, thus, Smith Charts may be used to find the values
of . If you don't understand this comment, just forget about it!
This page was prepared and is maintained
by R. Victor Jones, jones@deas.harvard.edu
Last updated March 1, 2000