Observing Complex Dispersion in a Vapor
Problems related to an optics experiment
PHYS 352, Optics, Spring 2013
The Mach-Zehnder Interferometer is a configuration of mirrors and beam splitters
arranged in a rectangle as shown in Figure 1. A laser beam is divided into two equal intensity
halves at the top left beam splitter. When light hits a 50:50 beam splitter, half of the irradiance
passes straight through, and half is reflect away according to the law of reflection. Now, one half
of the laser beam travels a distance,
L1, and the other half travels a
distance L2. The two beams are
reunited at the bottom right beam
splitter and then proceed to a
photodiode detector, which is a
semiconductor that produces a DC
voltage that is proportional to the
net irradiance incident upon it. The
distance that each beam travels
might be referred to as the optical
path length, but use this term a
little carefully since optical path Figure 1. A Mach-Zehnder interferometer splits a beam and then
length is a combination of recombines it after the two beams travelled different path lengths.
geometry and index of refraction.
Problem 1. Show that the irradiance received at the photodiode is given by the expression
I
Io
1  coskL 
2
where Io is the irradiance of the initial beam and L=L2-L1. Attach a graph of I/Io vs. kL made
using your favorite CAS software. The maxima on the graph are referred to as “fringes” in the
spatial distribution of the irradiance.
The complex index of refraction can be generally written as n=no(1+i), where  is the
loss factor and no is the real part of the index of refraction. Around resonance, both  and no are
strongly frequency dependent. This is a very general statement, applicable to any system: solid,
liquid, vapor. A vapor can be characterized first by its atomic or molecular absorption
resonances that can be excited by electromagnetic waves, and second by an index of refraction
that tends to real and unity away from these resonances. In class, we modeled the complex index
Rev 1, January 4, 2013
of refraction using a 2nd order differential equation based on Newton’s second law. The
dissipation term, which is proportional to velocity in Newton’s second law, includes the
dissipation constant .
Problem 2. In class, it was shown that (oscillator strength aside) around resonance
Re( n)  no  1 
Im(n)  no 
e2 N


m o o 4 2   2
and
e2 N

.

2m o o 4 2   2
Show that the real and imaginary parts are related through
no  1  
2( )no

which is a form of the Kramers-Kronig relation, which is a set of relations that relate the real and
imaginary parts of physical properties. Re-write this expression in terms of Re(n) and Im(n).
The Mach-Zehnder interferometer in Figure 1 is modified in an experiment by inserting a sample
of length z in the path of the beam along the lower portion as shown in Figure 2.
Problem 3. Show that the phase
shift experienced by the beam
passing through the sample with
complex index of refraction is
    i(kz   )
where =knoz and =k(no-1)z.
Explain the physical meanings of
,  and kz.
Figure 2. A Mach-Zehnder interferometer with a sample placed in one
of the beams.
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Problem 4. Show that the irradiance received by the photodiode detector is given by the
expression
I


Io
1  e 2  2e  cos(kL   ) .
4
For experimental convenience, it would be helpful to consider changes in kL to be insignificant.
We can achieve this by keeping L small. Let’s plan on this when we do an experiment.
Problem 5. Let’s suppose that the laser is swept through a range /2=5 GHz. How small
must L be such that I/Io goes through one half of an interference fringe?
You probably learned somewhere that resonance peaks are modeled using a Lorentzian,
also called a Cauchy distribution function, f(x)=fo/(1+x2). (See Bevington and Robinson, Section
2.4 for more details.) So, a way to write the frequency dependence of the absorption is

o
  

1  
  
2
where for a dimensionless frequency variable, we use the ratio of frequency deviation from
resonance to absorption coefficient.
In an experiment, the control variable will be , adjusted by sweeping a diode laser through a
very small range of frequencies. Implicitly then, it is  which becomes the varied quantity. An
experimenter can vary the laser frequency, thus vary , and measure the irradiance I/Io seen at the
photodiode.
Problem 6. Using your favorite CAS, make two graphs of I/Io versus /, one for each value
of o=0.5, and o=10. Vary / from -20 to +20. Make 5 plots on each graph for different
values of the phase, kL, ranging from 0 to  rad. Note that through the result of Problem 2, the
independent variable / is also (1-no)/2no.
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The Experiment
You can vary o by varying the density of the vapor which is being used as the sample. One
common experimental arrangement is to use an airtight, evacuated glass-walled cylinder for the
sample. Inside the cylinder is a small piece of metal attached to a heater. As the temperature of
the metal is varied, atoms will vaporize, changing the density of the gas in the cylinder. This gas
will be the sample of length z. When the heater temperature is low, o is small (perhaps about
0.5). When the heater temperature is high, o is large (perhaps about 10).
For our experiment we will use rubidium, which has significant vapor pressure at moderate
temperatures in the range of 40oC to 80oC. Observe the photodiode on an oscilloscope as the
laser is swept around one of the resonant frequencies of rubidium. A Doppler broadened
resonance line is about 5 GHz wide (hence Problem 5), and that will be about the width of our
sweep. When the temperature of the rubidium cell is low (around 40oC), the oscilloscope display
should resemble one of the peaks that you produced in Problem 6 using o=0.5. When the
temperature is high (around 80oC) the oscilloscope display should resemble one of the peaks that
you produced using o=10.
(This procedure is based on a lab handout made available from CalTech.)
Rev 1, January 4, 2013