Download Homework Set #6 Due: 3-28-14

Homework Set #6
Due: 3-28-14
(1) This problem introduces a vital component in the laboratory,
the waveplate. Waveplates allow easy modification of the
polarization of light without significant modification to the
beam path. “Half-wave” plates rotate the polarization
direction and “quarter-wave” plates can transform linear
polarization to elliptical polarization to circular polarization
and back.
Ordinary axis.
Extraordinary axis.
To summarize what we’re learned recently: Anisotropic
media will in general have an index of refraction that depends
on the polarization direction. For a given crystal and direction of the light, two polarization modes
exist for which there is a definite index of refraction. These are referred to as the extraordinary and
ordinary modes. When light is incident on an anisotropic medium at non-normal incidence it will in
general refract into two beams, each at a different angle, inside the medium. This is because, so long
as the input polarization of the light is a mixture of the two modes, each mode will refract according
to its own index. This is called double refraction.
Suppose we have a thin plate made of an anisotropic crystal (quartz is usually employed) oriented so
that, for linearly polarized light at normal incidence, vertical polarization is ordinary (n = no) and
horizontal polarization is extraordinary (n = ne) with no ≠ ne (for quartz, ne < no). If the polarization is
not initially parallel to either axis, then the polarization of the light will not be preserved as the light
propagates through the crystal. We can take advantage of this effect. (Note that in Type I SHG we
avoided it and the polarization of all fields was well-defined and preserved.) To proceed, we can
always break the polarization vector of the input light into components parallel to either axis. Thus,
we take the electric field at the input face of the crystal to be:

E  E o (cos eˆ  sin  oˆ )
where ê and ô are unit vectors pointing along the extraordinary and ordinary axes (to the right and
up, respectively, in the figure), and  is the angle the polarization vector makes with the extraordinary
axis. In practice, the waveplate is in a rotary mount so this angle can be varied.
The following problems can be solved as is or you can switch to a complex representation for the
electric field (I prefer to use the complex representation).
(a) Half-wave plate. Suppose we have monochromatic light, of wavelength , as given above. At the
output of the crystal, the ordinary and extraordinary components will have a phase difference
between them because they traveled through different optical pathlengths. Suppose the thickness
of the crystal is selected such that, after propagating through it, this phase difference is
(N + ½) * 2 where N is a positive integer. For ease of construction and for strength the plates
are usually ~1 mm thick, so N is large.
(i)
Find an expression for the length of the crystal taking no and ne to be given.
(ii) Show that the effect of the crystal is to rotate the polarization vector and determine how the
output polarization angle is related to .
(iii) How does the output polarization depend on N?
(b) Qualitatively describe what will happen if a different wavelength, ’ ≠ , is used. How does this
answer depend on N?
(c) Quarter-wave plate. Now let the phase shift be (N+¼ ) * . Show that the effect of the crystal
is to produce elliptical polarization and that for  = 0o, 45o, 90o, 135o the resultant polarization is
linear, circular, linear, circular (with opposite handedness) respectively.
Discussion.
Waveplates are made from flats that have sides that are flat and parallel to high precision. This helps avoid
wavefront distortion. Unfortunately, this means waveplates will also act as Fabry-Perot etalons (with R  5%)
and thus have a wavelength dependent transmission. To avoid this, as well as to avoid loss of energy and stray
beams, waveplates are always anti-reflection (AR) coated on both sides. Note that a beam sent through a
waveplate and telescope, for example, will lose ~27% of its power if all the surfaces are uncoated.
Another difficulty is that the index of refraction seen by the light will vary if the light is not normally incident
upon the waveplate. The ordinary and extradinary polarization directions and the extraordinary wave index of
refraction depend on input angle. If normal incidence is not used, a half-wave plate will produce elliptically
polarized light: the larger the deviation, the worse the effect. If the beam is diverging or converging when it
travels through the waveplate, then there is no well-defined angle of incidence; a range of angles is present. The
output polarization will now vary spatially. For example, for a half-wave plate, the center of the beam would be
linearly polarized, but off-center light would be slightly elliptically polarized. Waveplates require careful
alignment. In practice, they usually degrade the polarization of the beam somewhat.
For short pulse work, we must also take into account the consequence of a range of wavelengths being present
at the same time, as you can see from your answer to part (b).
All of the above effects are minimized if N = 0! Such a waveplate is called a “zero-order waveplate”. They
generally aren’t used for monochromatic light, but they are mandatory for short pulse work. The question is,
how do you make one? The most direct approach, using a very thin plate, is impractical. You should convince
yourself that the flat would have to be a few 10’s of wavelengths long. (This can be done. The optic will be
expensive and fragile, however.)
One clever way to do this is to sandwich two flats together, the extraordinary axis of one aligned to the
ordinary axis of the other. If the two plates were of identical thickness, there would be no polarization effect at
all. By making one of the plates slightly thicker, an effective zero-order half- or quarter-waveplate is obtained.
This is usually what is done.


(2) In this problem we consider the mismatch, k = k p  k o , for FWDM (four-wave
degenerate mixing) in BK7 glass using 514.5 nm light for the geometry shown. As
in the class notes, kp is the non-linear polarization k-vector and ko is the k-vector
associated with the output wave generated by the polarization. The output wave has
the same frequency and direction as the “polarization-wave”. There are three input
beams (#1-3), all the same frequency, and we will focus on the output beam that is
not parallel to any of the input beams. Beams #1 and #2 make a 10o angle with
respect to the normal and beam #3 makes a 5o angle.
z
L
x
#1
#3
Note: The angles used in the figure are exaggerated for clarity. As discussed in class, there will be a
delay “sweep” because of the noncollinear geometry and the beams’ finite widths that will change the
effective path length. You do not need to take this into account.


(a) Find k p . Use the specified coordinate system. Remember that k p must be specified inside the
medium, but the input rays are given outside the medium, so take refraction into account.
(b) Find k.
(c) Suppose that we wish to use this process in an experiment. We can limit the phase error due to the
k-vector mismatch by keeping the sample thin. What sample thickness, L, corresponds to a phase
error of ? As you can see from your answer, we’ll probably use a thicker sample and depend on
focusing to effectively limit the sample interaction length.
(3) What is the phase matching angle for collinear sum frequency generation of 355 nm light from 1064
nm and 532 nm light in KD*P? Specify which of the fields are to be ordinary and which are to be
extraordinary. (The resulting equation cannot be solved explicitly. A graphical solution will work fine
or you can use a numerical equation solver such as that provided in Mathematica.) This calculation is
similar to that for SHG in the notes. You do not need to write full equations of motion based on the
wave equation. Just find the phase matching condition and then solve for KD*P and the specific
wavelengths for this problem. KD*P information is available from the public class web site in the
companies section.
#2