1. Detecting Pancharatnams Phase Using LG Vortex Beams
Charlotte Welch
Mentor: Surendra Singh
University of Arkansas, Physics Department
(Dated: July 23, 2015)
An experiment was conducted to detect Pancharatnam’s phase. Two quarter waveplates with
a half waveplate in between were placed in the path of a plane wave. The waveplates dictated
the cyclic evolution of polarization. The plane wave was interfered with a Laguerre-Gauss vortex
beam and the interference pattern was observed. As the half waveplate rotated, the fringes in the
interference pattern shifted. The experiment verified the prediction of Pancharatnam’s phase due
to cyclic evolution of polarization in agreement with the theory.
I. INTRODUCTION
As an electromagnetic wave propagates through space,
an additional phase beyond its dynamical phase can arise
from polarization depending on the geometry of the path
followed by the polarization. This is an example of geo-
metric phase [2]. This phase is known as Pancharatnam’s
phase, and it depends on the evolution of polarization as
the wave propagates [1]. There have been several ways of
observing Pancharatnam’s phase of light [2]. Pancharat-
nam’s phase can be derived using Jones matrices, and
it was first derived by using the spinor representation of
polarization states [3].
Many lasers usually emit light as a Hermite-Gauss
(HG) beam. HG beams are solutions of the wave equa-
tion in Cartesian coordinates, and Laguerre-Gauss (LG)
beams are solutions of the wave equation in circular cylin-
drical coordinates [4]. LG beams have a null at the center
and carry an average orbital angular momentum of per
photon. HG beams do not carry angular momentum.
In this paper, Section II provides background informa-
tion about polarization. Section III describes the exper-
imental setup section and experimental methods used to
detect Pancharatnam’s phase. Section IV presents re-
sults and discussion reporting the findings of the experi-
ment. Finally, the conclusion is presented in Section IV.
II. THEORY
The equation for the electric field of an electromagnetic
wave propagating in the z-direction is given by
E(z, t) = [ ˆexEx + ˆeyEyeiδ
]ei(φ+φ0)
, (1)
where the dynamical phase
φ = kz − ωt (2)
and wave number
k =
nω
c
. (3)
Here n is the refractive index of the medium and c
is the speed of light in a vacuum. The behavior of the
electric field vector depends on the x and y components
of the electric field in time t, the relative phase difference
δ between the two, and the dynamic phase of the wave,
φ. If polarization undergoes a cyclic evolution after the
beam has traveled a distance ∆z, then the equation for
the field becomes
E(z, t) = [ ˆexEx + ˆeyEyeiδ
]ei∆φ
e2iα
, (4)
where
φ = k∆z. (5)
Note the added term e2iα
, which represents the phase
contribution made by polarization. This term is called
Pancharatnam’s phase and is usually neglected because it
is not widely known, or the polarization does not undergo
a cyclic evolution.
For a wave propagating in the z direction, the electric
field vector lies in the x-y plane. In the most general
case, the electric field vector traces an ellipse (FIG 1).
The shape traced by the vector depends on Ex, Ey and
δ. If the light is circularly polarized, then the resulting
electric field vector traces a circle in x-y plane as the
wave propagates (FIG 2). The ellipse or circle may be
traced clockwise or counterclockwise. In linearly polar-
ized light, the vector points along a fixed direction as the
wave propagates (FIG 3).
FIG. 1: Elliptical Polarization

2. 2
FIG. 2: Circular polarization
FIG. 3: Linear polarization
A convenient representation of polarization, especially
for situations involving evolution of polarization, is the
Poincare sphere. Each point on the sphere corresponds
to a state of polarization.
FIG. 4: Poincare sphere [3]
Using the parameters Ex, Ey and δ, a point P on the
sphere is determined by the following equations.
tan(2ψ) =
2ExEy cos δ
E2
x − E2
y
(6)
sin(2χ) =
2ExEy sin δ
E2
x + E2
y
(7)
Linearly polarized light is represented by a point on
the equator, points A or B, and circularly polarized light
is plotted at the poles: left circularly polarized light, at
point L and right circularly polarized light at point R
(FIG 4). A point on any other part of the sphere is
elliptically polarized. The area enclosed by the circuit
subtends a solid angle 4α at the center of the sphere,
and Pancharatnam’s phase is half of that angle, 2α.
III. EXPERIMENTAL SETUP
The experimental setup was a modified Mach-Zehnder
interferometer and is shown below.
FIG. 5: Experimental Setup
The light source was an Argon ion laser operating at
514 nm. The beam was linearly polarized from the laser.
A fine fiber in the laser cavity generated higher order laser
modes. The mode generation is further described in the
Vickers paper [4]. The laser output was a Hermite-Gauss
beam, and operated at 2 modes: HG10 and HG20.

3. 3
FIG. 6: HG10 beam
FIG. 7: HG20 beam
A beam splitter, BS, split the beam from the laser into
2 paths: A and B. In path A, the Hermite-Gauss (HG)
beam was expanded by a factor of 3 and collimated by a
15 cm lens and a 50 cm lens (represented by L1 and L2 in
FIG. 5) separated by approximately 32.5 cm. After this
collimation, a lobe of the HG beam can approximated by
a plane wave. The collimated beam traveled through a
quarter waveplate, a half waveplate, then another quarter
waveplate. Both of the quarter waveplates were oriented
with their fast axis 45 degrees from the x axis. The half
waveplate was attached to a motor. The speed of rotation
was controlled by the voltage input.
In path B, the beam traveled through an astigmatic
mode converter that consisted of L3 with focal length
54.6 cm and a pair of cylindrical lenses, C1 and C2, each
with focal length 15 cm separated by a distance
√
2fC1
The distance between L3 and the center of C1 and C2
was approximately 63.5 cm [5]. The mode converter con-
verted the laser beam from HG to LG. The LG beam is
shown below
FIG. 8: LG01 beam
Paths A and B joined at the beam combiner. The
LG beam from path B interfered with an expanded lobe
of the HG beam in path A. Two interference patterns
were used for data collection: a spiral and a fork. The
interference patterns are shown below.
FIG. 9: spiral fringes(left), fork fringes(right)
If the expanded HG beam and LG beam were parallel
to one another, a spiral pattern was seen. As the HG
beam and LG beam were less parallel, the curvature of
the fringes decreased and the fork pattern could be seen.
As shown in FIG. 9, one of the fringes of the fork starts
from the lower center and is discontinuous towards the
middle of the image. The discontinuous fringe is the mid-
dle prong of a 3 prong fork.
The frequency of the motorized half waveplate was cal-
ibrated using the curve in FIG 10. For 4 different voltages
the time to complete 5 rotations was recorded. This was
repeated 4 times for each voltage setting. The average
period was determined and used to calculate the corre-
sponding frequency for each voltage. The line of best fit
was used to calculate the frequencies of the 3 voltage set-
tings: 8 V, 10 V and 12 V with corresponding frequencies
280 mHz, 350 mHz, and 430 mHz, respectively. The line
of best fit was y = 38.1x − 31.96 with an uncertainty of
0.41 in the slope. The error bars on calibration curve
represent standard deviation for each voltage setting.
4 5 6 7 8 9 10 11 12 13 14 15
50
100
150
200
250
300
350
400
450
500
550
600
y = mx + b
m = 38.01 +/- 0.41
b = -31.96 +/- 4.27
Frequency Calibration Curve
Frequency(mHz)
Voltage (V)
FIG. 10: Frequency Calibration Curve
The half waveplate’s rotation causes a cyclic evolution

4. 4
of polarization. The evolution of polarization is traced
on the Poincare sphere in FIG. 11. The beam begins
as horizontally linearly polarized at point A. The first
quarter waveplate left circularly polarizes the beam, and
the circuit on the sphere traces to point L. The rotating
half waveplate right circularly polarize beam, connecting
point L to point R. The second quarter waveplate lin-
early polarizes the beam, and the circuit traces back to
point A, back to its initial polarization state. The circuit
encloses a solid angle 4α. The size of the angle is propor-
tional to the area enclosed by the circuit, and the area
enclosed by the circuit is dependent upon the rotation of
the half waveplate.
FIG. 11: Poincare sphere showing cyclic evolution of po-
larization
As the half waveplate rotated, the fringes shifted. The
fringe shift in the spiral after a 180 degree rotation of the
half waveplate is shown below.
FIG. 12: original position of spiral fringes (left) spiral
fringe position after 180 degree rotation of half waveplate
(right)
Data was collected by recording the time for 100 fringe
shifts in the interference pattern while the half waveplate
was rotating at a fixed frequency. The cumulative time
was recorded for 100 cycles in increments of 10 cycles.
The different sets of data with LG beams of = 1 and
= 2 were collected for clockwise and counterclockwise
rotation of the half waveplate.
IV. RESULTS AND DISCUSSION
The experimental results are shown in Table I and Ta-
ble II. Column 1 indicates the order of the laser beam,
column 2 indicates the frequency of rotation of the half
waveplate, column 3 indicates the direction of the rota-
tion along the axis about φ, and column 4 indicates the
slope calculated from the line of best fit after plotting
φ (the phase shift observed in the interference pattern)
vs. α (the angle of rotation of the half waveplate. Since
Pancharatnam’s phase in this case is 2α, the expected
value of the slope for each data set is 2.
TABLE I: Spiral Data
= 1 208 mHz counterclockwise 1.96
clockwise 1.98
350 mHz counterclockwise 2.02
clockwise 2.27
430 mHz counterclockwise 2.00 ; 1.99 ; 2.03 ; 2.00
clockwise 1.91
= 2 280 mHz counterclockwise 1.96
clockwise 1.98
350 mHz counterclockwise 2.02
clockwise 2.27
430 mHz counterclockwise 2.00
clockwise 1.91
TABLE II: Fork Data
= 1 280 mHz counterclockwise 2.43
clockwise 2.39
350 mHz counterclockwise 2.14
clockwise 2.22
430 mHz counterclockwise 1.85
clockwise 1.84
= 2 280 mHz counterclockwise 2.13
clockwise 2.16
350 mHz counterclockwise 2.04 ; 2.02 ; 2.03 ; 2.04
clockwise 2.02
430 mHz counterclockwise 2.02
clockwise 1.82
The average value of all the slopes was 2.06 ± 0.16.
The average value for the slopes from the spiral data
was 2.03 ± 0.09. The average value for the slopes from
the fork data was 2.10 ± 0.21. The variability of slope
values in the fork data was larger than the variability of
values in the spiral data. Because the phase change in
the fork interference pattern was more subtle than the

5. 5
phase change in the spiral pattern, we expect the fork
data to have more error.
The average slope for the = 1 data was 2.09 ± 0.22
and the average slope for the = 2 data was 2.03 ± 0.09.
The variability of slope values was larger in the = 1
data than the = 2 data.
A large source of error in the experiment was the judge-
ment in the phase shift counting. Phase shifts were ob-
served visually. The phase shifts were difficult to ob-
serve because of the instability of the interference pat-
tern. The pattern not only fluctuated in brightness, but
also in phase.
Relative phase fluctuation in the two arms of the inter-
ferometer due to air movement and vibration contributed
to uncertainty in fringe counts. Imperfections in HG to
LG mode conversion led to less than ideal fringe pattern
may also contribute to uncertainty in fringe count as well
as imperfections in the alignment of waveplates.
4 trials were conducted for the = 1, 430 mHz fre-
quency, counterclockwise spiral fringes and 4 trials were
conducted for the = 2, 350 mHz frequency, counter-
clockwise fork fringes. The data for those orientations
was used for the graphs is shown below.
0 4 8 12 16 20
0
1
2
3
4
y = mx + b
m = 2.007 +/- 0.0031
b = -133.39 +/- 34.66
phi vs. alpha
phi(degreesx10
4
)
alpha (degrees x 10
3
)
FIG. 13: = 1 spiral fringe data with error bars. The
error bars are covered by the data points.
0 4 8 12 16 20
0
1
2
3
4
y = mx + b
m = 2.031 +/- 0.0037
b = 12.77 +/- 40.67
phi vs. alpha
phi(degreesx10
4
)
alpha (degrees x 10
3
)
FIG. 14: = 2 fork fringe data with error bars. The
error bars are covered by the data points.
The data points on FIG. 13 and 14 are the average
values of α vs φ for four trials. Both slopes, 2.007 and
2.031 agreed with expected value, 2. The experiment
could be improved by performing multiple trials for each
orientation of data collection. The y-intercept on the
graph shows the initial phase of the wave.
V. CONCLUSION
The existence of Pancharatnam’s phase was verified in
this experiment. The expected phase contribution made
by the evolution of cyclic polarization was 2α. The re-
sults agreed with the theory within experimental uncer-
tainty.
VI. ACKNOWLEDGEMENT
This research was made possible by the National Sci-
ence Foundation funded Research Experiences for Un-
dergraduate program (NSF Award 1460754), and was
supported by the University of Arkansas.
[1] Pancharatnam, S. Generalized theory of interference. A44:
247, 1956.
[2] Galvez, E. Applications of geometric phase in optics. 165-
6, 170-7, 2002.
[3] Singh, S. Vyas, R. Geometrical Phase of Light via Jones
Matrices. Unpublished, 2014.
[4] Vickers, J. Burch, M. Vyas, R. Singh, S. Phase and inter-
ference properties of optical vortex beams. 25, 2008.
[5] Conry, J. Properties of Polarized Maxwell-Gauss Laser
Beams. University of Arkansas. 2012.