Matrix Modeling and Testing of An Optical Modulation System
Truitt Wiensz, Kyle Cheston, Jaret Rennie
Apr. 6, 2001
1
Objectives
There are two main objectives in this experiment:
1. To build and test an optical modulator which modulates the output power of a laser
signal using a half-wave plate.
2. Find the locations of the expected maxima and minima of the modulation system
using the system matrix approach, and compare with observed results.
2
Theory
2.1
Introduction
The polarization of an electromagnetic wave is defined as the direction of the electric field,
and is characterized by the profile traced out by the electric field, as a function of time, in a
plane transverse to the direction of travel ak of the wave. Several types of polarization are
possible: linear, circular, and elliptic. It will be seen later that the first two are simply special
cases of the latter.
2.2
The Jones Vector Representation of Polarization States
The Jones vector representation provides a convenient means of retaining amplitude and
phase information for studying polarization systems.
2.2.1
‘Bra’ and ‘Ket’ Vector Representation
Polarization of an optical signal is conveniently represented in terms of the components of
the electric field along basis vectors. In the lab system, a polarized signal E0 may be
represented as follows:
E 0  a x e i x x  a y e
1
i y
y
(1)
Here ax and ay are the magnitudes of the signal components along the basis vectors, and x
and y define the phases of the two components. The basis vectors x and y are defined
as follows:
1
0 
x   , y   
0 
1
(2)
These vectors form a two-dimensional basis in the lab frame, usually defined in terms of the
horizontal and vertical directions. Transformations from the lab system to other coordinate
systems, such as circular or principal-axis system basis vectors, are performed by
multiplication by a 22 coordinate-transformation matrix. Another useful basis is the rightand left-circular components, defined as follows:


1 e i 2 
1 e  i 2 
 , l 


r 
2 1 
2 1 
 


(3)
The electric field vector E0 defined in equation 1 above is known as the “bra” vector, as
seen in the Dirac formulation of quantum mechanics. The “ket” vector E0 is the adjoint
of the bra vector. The scalar product of the two is known as the bracket product, and is
defined as:
E0 E0  a x2  a y2
(4)
Given this definition of the bra and ket vectors, the simple transformation from lab to
circular components is given by the following matrix multiplication:
 a R e  R   r x


 L 
a
e
 L
  l x
2.2.2
r y   a x e  x 

 
l y  a y e y 
(5)
Signal Intensity
Intensity of an optical signal is, in general, proportional to the square of the electric field.
Having the advantage of the Jones vector representation for a polarization state, the intensity
of an optical signal E0 is given by the bracket product, as follows:
I P  E0 E0  a x2  a y2
2
(6)
This quantity is termed the normalized intensity; the actual signal intensity I is easily found
from this by:
I  12 c 0 nr I P
(7)
In this experiment, IP will be used for intensity calculations. The concept of orthogonal
states may be introduced with the bracket product notation. A state E0  orthogonal to
the state E0 is defined such that:
E 0 E 0  0 .
(8)
For the case of two-source interference, the measured intensity may be formulated in a
similar manner. The total electric field from two sources is expressed as ET  E1  E2 ,
and the resulting intensity from the two sources is given by:
I P  ET ET   E1  E 2
 E
1
 E2

 E1 E1  E 2 E 2  2  Re E1 E 2
(9)
The first two terms in this equation are the individual signal intensities, and the third term is
called the mutual coherency function
12  E1 E2  12 e i ,
(10)
which provides a measure of the interaction between signals 1 and 2. The coherency phase 
provides a measure of interference. The result 12  0 corresponds to two incoherent
sources.
In this experiment, the mutual coherency will provide a measure of the
effectiveness of the modulation system.
2.3
System Matrix Representation
3
The effect of polarization-changing systems may be described by matrix operators, which act
on a given input optical signal. The input signal Ein , polarization-modifying system
S SYS  , and output optical signal
Eout are related by the following:
S
E out  S SYS   E out   11
S 21
S12 
 E out
S 22 
(11)
This representation is used in the description of all optical components of interest in this
experiment. Elements of the system matrix S SYS  are complex, in general, and represent
the individual transfer functions from each input component to each output component.
For example, S11 describes the output x component to the x component of the input
signal.
2.4
The Poincaré Sphere
The most general type of polarization is the elliptical type.
This section provides a
generalized presentation of a geometric representation of elliptical states based on ellipse
parameters. The polarization vector of an optical signal may be described by its components
ax and ay along the lab axes, and the phase difference  between these components. An
alternate representation, which lends itself to a more geometrical interpretation, is to
characterize the polarization ellipse by the following: the length of its major and minor axes,
the orientation angle  of the ellipse, and by the ellipticity angle , as shown in Fig. 1.
Fig. 1: Ellipse Parameters
4
In Figure 1, the x and y axes are a system in the lab, and the frame at angle  is the principal
axis system of the ellipse. By recognizing that the orientation angle of the ellipse lies in the
range   [0, 180), and that the ellipticity angle is defined for   [-45, 45]; a sphere
representing the polarization may be constructed, having latitude 2  [-90, 90] and
longitude 2  [0, 360). This sphere is named the Poincaré sphere, and its radius is simply
the normalized intensity of the signal, IP = a2 + b2 = ax2 + ay2. A typical polarization state is
shown in the Poincaré sphere representation below in Figure 2.
Fig. 2: The Poincaré Sphere
The coordinates (Ip, 2, 2) represent the coordinates of a polarization state in spherical
coordinates on the Poincaré sphere, and the axes (S1, S2, S3) shown above provide the
Cartesian axis system for the space in which the sphere lies. The coordinate transformations
from the spherical coordinates (Ip, 2, 2) to the Stokes parameters (S1, S2, S3) are as follows:
S1  I P cos 2  cos 2
S 2  I P cos 2  sin 2
S 3  I P sin 2 
5
(12)
The special cases of linear and circular polarization occupy unique positions on the Poincaré
sphere. Physically, linear polarization corresponds to the x and y components being inphase, resulting in zero ellipticity angle. Thus linear polarization states lie in the S1-S2 plane,
with  giving the orientation angle of the linear state. Circular polarization states have equal
x and y components, with a phase difference of +/- 90, for right and left circular
signals, respectively. The right-hand sense circular state lies at the zenith of the sphere, with
 = 45, and the left-have sense circular state is at the opposite end, having  = -45.
Orthogonal states also have a simple representation on the Poincaré sphere. Recalling that
the definition of the orthogonal state E0  to state E0 is defined as in equation 8, it can
be seen that orthogonal states lie on directly opposite points of the Poincaré sphere.
An intuitive discussion of the specific functions of polarization-changing systems may now
be given through this geometric representation of the effect of polarization systems and
through the Jones vector representation.
6
3
Apparatus
The apparatus of the experiment used retarders to operate on the phase of an incoming light
beam to implement optical modulation. The apparatus consisted of, two quarter wave
plates, a half wave plate, two polarizers, two mirrors, an He-Ne laser, and a photodetector.
A schematic diagram of the setup can be seen in Figure 3.
Fig. 3: Schematic of Optical Modulator System
The laser light was incident upon the first linear polarizer, P1 , which was set to produce an
optical axis at 45. The system matrix for a Polaroid is shown below
Polaroid:
Eout
lab


polaroid
 Slab
Ein
Eout
Eout
lab
lab
(S = System Matrix):
lab
cos

 sin 
 sin   1 0  cos
cos  0 0  sin 
 cos 2 

cos sin 
cos sin  
 Ein
sin 2  
lab
sin  
Ein
cos 
lab
(13)
For a Polaroid set at  = 45:
Eout
lab

1 1 1
Ein
2 1 1
lab
(14)
This linear polarization makes both the horizontal and vertical amplitudes equal before they
become incident upon the beam-splitter. The beam-splitter allows two beams to travel in
7
the primary (p) and secondary (s) paths as shown in Figure 3. These two beams will travel
the same distance throughout the setup eliminating any phase differences due to different
path lengths. When the two beams are traveling between the two mirrors they come into
contact with two QWP and a HWP. The equations for a beam passing through the QWP
and the HWP are shown below:
QWP:
Eout
lab


QWP
 S lab
Ein
Eout
lab
lab
cos

 sin 
 sin   1 0   cos
i 
cos  0 e 2   sin 
i

cos 2   sin 2 e 2

i 2
cos sin   cos sin e

Eout
lab
sin  
Ein
cos 
lab
cos sin   cos sin e
i 
sin 2   cos 2 e 2
(15)
i 2

 Ein

lab
For a QWP set at  = 45:
Eout
HWP:
Eout
lab
lab


1 1  i 1  i 
Ein
2 1  i 1  i 

HWP
 S lab
Ein
Eout
Eout
lab
lab
lab
lab
cos

 sin 
 sin   1 0   cos
cos  0 e i   sin 

cos 2   sin 2 e i

i
cos sin   cos sin e
sin  
Ein
cos 
lab
cos sin   cos sin e i 
 Ein
sin 2   cos 2 e i

(16)
 i
Since e  1 , and using triq identities then a HWP of arbitrary rotation is shown as:
Eout
lab
cos 2

 sin 2
sin 2 
Ein
 cos 2 
lab
(17)
The QWP-HWP-QWP combination seen in Figure 3 allowed the phase modulation to
occur. A wave polarized at 45 to the optical axis of the QWP, and will become circular.
When the wave passes through the HWP the sense of the circular wave will become
reversed. Finally, when the wave passes through the second QWP the result will be a
polarized wave that is orthogonal to the original wave entering the first QWP. The process
can be seen below in Figure 4.
8
lab
Figure 4: QWP-HWP-QWP Combination
The modulation of the experiment was controlled by rotating the HWP through a full
rotation relative to the orientation of the QWPs. Also, as seen in Figure X, a photodetector
measured the intensity of the signal leaving the modulation system.
9
4
Procedure
System alignment was checked to ensure proper transmission of signals along the paths.
Some calibration was performed on the He-Ne laser, in order to determine the polarization
of the laser. A plot of the laser signal intensity as a function of angle is shown below in
Figure 5.
Fig. 5: Laser Polarization
The combined modulator output signal was measured at the photodetector for a full rotation
of the HWP. Intensity measurements were taken for rotation angles  from -180 to 180 in
increments of 5. Full tabulation of the data obtained can be seen in the Appendices.
10
5
Observations
Intensity measurements were taken as a function of half-wave plate orientation angle, with
the observed measurements shown below in Figure 6. Horizontal and vertical error bars ( 
1 and  0.005 mW/cm2, respectively) are too small to be displayed.
Fig. 6: Modulation System Output Intensity
A comparison is shown in Figure 6 with the expected results, as is outlined in the Analysis
section.
11
6
Analysis
A schematic of the modulation system, including the signals from the polarizing beam
splitter, is shown below in Fig. 7.
Fig. 7: Modulation System Signal Paths
Following is analysis of the change in polarization of the signals due to the signals E p and
E s passing through the system shown, in opposite directions.
6.1
Passage of E p  a p x1 Through Path QWP1HWPQWP2
As shown in the supporting material for this experiment, the output polarization for this
signal is given by:
E po u t  a xp e i   2  x .
(18)
This will be used, along with the polarization of the secondary output signal, E so u t , to find
the expected output intensity of the system.
6.2
Passage of E s  a s y 2 Through Path QWP2HWPQWP1
In this experiment, the HWP is offset by 45. By substituting an angle
180  45     135  
12
(19)
for the angle of the previously derived matrix, the system becomes the following, using the
system matrix of a HWP with optic axis offset by 45:
Eout
lab
Eout
lab
Eout
lab
cos 2135    sin 2135    

 Ein
 sin 2135     cos 2135   
cos270  2  sin 270  2  

 Ein
 sin 270  2   cos270  2 
 sin 2

 cos 2
 cos 2 
Ein
 sin 2 
(20)
lab
lab
(21)
lab
The QWP-HWP-QWP combination gives the following system matrix:
Eout
lab
Eout
lab
Eout
lab

1 1  i 1  i   sin 2
2 1  i 1  i   cos 2
 cos 2  1  i 1  i  1
Ein
 sin 2  1  i 1  i  2

1  4 cos 2  4i sin 2
0
4 

Ein
 4 cos 2  4i sin 2 
 cos 2  i sin 2

0

lab
(22)
0

Ein
 cos 2  i sin 2 
0
lab
lab
(23)
The transmitted beam that passes through the QWP-HWP-QWP combination gives (where
a xp is the amplitude of the wave transmitted through the beam splitter):
 cos 2  i sin 2

0

 a xp
 cos 2  i sin 2  0
0
Etrans
lab
Etrans
lab
 a xp cos 2  i sin 2  x
Etrans
lab
 a xp e i 2 x
(24)
The reflected beam that passes through the QWP-HWP-QWP combination gives (where a yp
is the amplitude of the wave reflected from the beam splitter):
 cos 2  i sin 2

0

 0
 cos 2  i sin 2  a yp
Erefl
lab
Erefl
lab
 a yp cos 2  i sin 2  y
Erefl
lab
 a yp e i 2 y
0
(25)
13
Also, since the first polarizer was oriented at 45 the beam splitter splits the transmitted and
reflected waves into two equal amounts ( a xp = a yp = a ). The output that is traveling towards
the second polarizer is a wave consisting of both the reflected and transmitted portions.
This is shown as:
Eout
Eout
lab
lab

 a e i 2 x  e i 2 y

 e i 2 
 a  i 2 
e 
(26)
Finally, the wave passes through the second polarizer set at 45. This is shown by:
Eout
Eout
Eout
Eout
lab
lab
lab
lab
a 1 1  e i 2 
 


2 1 1 e i 2 
a e i 2  e i 2 
   i 2

2 e  e i 2 
a 2 cos 2 
 
2 2 cos 2 
cos 2 
 a 

cos 2 
(27)
The intensity of the wave, as in equation 6 in Section 2.2.2, is expressed as follows:
I  E x2  E y2
I  a 2 cos 2 2  a 2 cos 2 2
I  2a 2 cos 2 2
(28)
The expected minima will occur when:
cos 2 2  0 , or cos 2  0 .
Therefore: 2  
(29)
2n  1 and    2n  1 at an expected minimum.
2
4
The expected maxima will occur when:
cos 2 2  1 or cos 2  1 .
Therefore: 2  n and   
n
at an expected maximum.
2
At an expected maxima, the intensity is:
14
(30)
I max  2a 2
(31)
The relative intensity of the modulated output is:
I rel 
I
I max

2a 2 cos 2 2
2a 2
I rel  cos 2 2
(32)
The error in the modulated output is:
I rel 
I rel 
I  I max  I  I max
2
I max

16a 3 cos 2 2 a  8a 4 cos2  sin 2 
4a 4
4 cos 2 2 a  a sin 2 
a
(33)
Finally, the modulated output is:
I rel  cos 2 
2
4 cos 2 2 a  a sin 2 
a
(34)
Shown below in Figure 8 is a plot of the system output intensity, as predicted by the systems
matrix derivation of the output signal polarizations, as described in equation 34. This result
is plotted in Figure 6 above.
7
Discussion
The design of the optical modulator worked well to modulate the phase of an incoming laser
signal. The laser that was used in the experiment had an intrinsic polarization that appeared
slightly elliptical. To maximize our results, the laser was lined up at 45 to allow the most
amount of light to pass through the first polarizer. After the original signal was polarized it
was then passed through a beam splitter. The testing of the setup required that the split
beams have equal amplitudes. This reduced the error of the experiment. The amplitudes
were adjusted by rotating the beam splitter clockwise until they became equal. The rest of
the design was dependent on the setup of a QWP-HWP-QWP combination and a second
polarizer. The distance that the two beams traveled was nearly equal. This reduced the error
produced by environmental effects and mechanical disturbances. The experimental values
received after the second polarizer, plotted in Figure 6, fit well with the expected cosine
curve. The experimental and expected values fit within a phase difference of only a few
15
degrees. The expected maxima were located at   
located at   
n
and the expected minima were
2
2n  1 . The experimental results met these expectations.
4
References
[1]
Hecht, Eugene. Optics. Chapter 8: Polarization, pp. 346-350
[2]
Sofko, George. Supporting material for Optical Modulation Experiment
16