Measurement of Berry’s Phase in a single-mode optical fiber
Christopher D. McFarland
University of Rochester, Rochester, New York 14627
The phenomenon of Berry’s Phase can be observed in a helically-twisted,
single-mode fiber-optic.
Berry’s Phase is a change in polarization angle of
plane-polarized light that occurs when the momentum of a beam of light is
changed adiabatically, returning to its original direction. A central focus of this
study was to explore the sources of experimental error in previous measurements
of Berry’s Phase in a helical fiber-optic and formally test the validity of Berry's
Phase. Our data failed to refute our hypothesized change in polarization angle
due to Berry’s Phase (p > 0.05, N = 13).
I.
Introduction
Berry’s phase is a manifestation of a more general geometric property called geometric phase.
Geometric phase is the phase difference that is acquired when a system cycles adiabatically
through the parameter-space of its Hamiltonian. An adiabatic change in this context is any
change which occurs slowly. More specifically, the path that a system traces through its
parameter space of its Hamiltonian must be smooth for it to be adiabatic. In geometric phase, the
system must be confined to move along a curvilinear surface. When a system traces out a closed
path through its parameter space on the confining surface it can gain a change of phase despite
returning to its original location on the surface. Thus, moving a system through a closed loop
around two parameters of its Hamiltonian can cause changes in other parameters of the system,
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even when the system returns to its original position with respect to the only parameters being
explicitly varied.
In Berry’s phase, light enters an optical fiber, traces out a path and exits the fiber in the
same direction that it enters. Despite exiting in the same direction as it entered, the polarization
angle of light when it exits differs from when it enters. So in Berry’s phase, the azimuthal and
polar components of a photon’s momentum go around a closed loop causing a change in the
angle of polarization. Momentum can be adiabatically changed because the fiber functions like a
waveguide and the momentum of the light is at all times tangent to the fiber. The polarization
angle is an orientation of the electric and magnetic components of the wave, which is
perpendicular to the momentum vector and direction of propagation. Since the amplitude of the
wave does not change while in the fiber, its momentum vector must remain constant in
magnitude. Thus, the momentum vector is confined to move over a spherical surface of constant
magnitude. As the optical fiber waveguide traces out a smooth path, the momentum vector of
the light traces out a path in momentum space confined to a sphere. Because the beam exits in
the same direction that it enters, the path is closed. The polarization angle, defined by the
orientation of the electric and magnetic fields will remain perpendicular to the momentum vector
as it propagates around the sphere of momentum space. When it returns to its original position
the polarization angle will point in a new direction. The change in phase is equal to the solid
angle circumscribed by the closed path, thus:
=  (C)
(1)
where is the helicity spin number of the photon, which equals +/- 1 depending upon which
direction the photon takes around our contour, and (C) is the solid angle in momentum-space
circumscribed by the closed contour C (Tomita and Chiao, 1986).
2
I.
Hypothesis
To test Berry’s Phase we will send light around a helical path in the aforementioned fiber-optic
waveguide. The helical path of the photons can be parameterized as follows:
z
r
r tan
(2, 3, 4)
where = n c t sin. n is the index of refraction, c is the speed of light, t is time, and θ is the
helical angle of our waveguide. Basically, is time under a dilation that simplifies our
equations. For one helical turn:
= [= 0, = 2] = [0, 2 r tan]
(5)
Converting these equations in cylindrical coordinates to Cartesian coordinates gives:
x
cos
r cos
y
sin
r sin
z
r tan
r tan
z
(6, 7, 8)
The momentum of a photon in each direction is proportional to the change in its position with
respect to time. So lets find that:
x
cot sin
y
cot cos
z
1
r tan
r tan
(9, 10, 11)
Converting to spherical coordinates:
3
x2
y2
cot2
cos
1
tan
1
z2
1
z
y
x
cot2
sin2
r tan
cot2
cos2
1
r tan
csc
cos
1
tan
1
1
csc
2
cot cos
r tan
cot sin
tan
1
cot
r tan
r tan
r tan
(12, 13, 14, 15, 16)
Because δ depends on , we can employ equation (5) to restrict δ to any value on the domain:
0
r tan
,
2
r tan
r tan
0, 2
These equations for δ and  describe a cone with apex angle 2θ. Since we are only concerned
with the solid angle traced out in momentum space, not the entire description of momentum we
only care about the solid angle traced out by our cone, not its height. Notice that the photon’s
path in momentum space is circular. Therefore, its path is smooth and satisfies our condition that
the change be adiabatic. The solid angle traced out by a cone with apex angle 2 is:
(C) = 2(1 - cos)
(17)
Using equation (1), assuming the photons travel in the +z direction, and generalizing our
problem to N helical turns, we get:
=  (C) = (1) (2(1 - cos)) = 2(1-cos)
= N 2(1 - cos)
(for 1helical turn)
(for N helical turns)
(18, 19)
This relationship between the change in polarization angle and helical angle was used as the
theoretical hypothesis for testing Berry’s Phase in a helical fiber-optic.
II.
Experimental Design
4
Light from a Helium-Neon laser (wavelength = 633 nm) is passed through a plane polarizer. The
beam is fed via a coupler into a single-mode optical fiber, which loops around a cylindrical
mount. A single-mode optical fiber is chosen to prevent dispersion of the beam. Extreme care is
taken to ensure that the fiber traces out a flat path before passing around the cylindrical mount, as
a curved path will induce further change in polarization angle due to unintended Berry’s Phase.
The helical angle of the fiber as it propagates around the cylinder is recorded and kept constant.
After making several revolutions, the fiber exits the cylindrical mount and the beam leaves the
fiber-optic in the same direction that it enters. The beam passes through a second polarizer and
then is incident upon a photo-detector. Data from this detector is read by an oscilloscope. The
path of the beam is surrounded by rigid boxes to minimize the amount of dust that may interfere
with the beam.
Data is recorded by varying the helical angle of the fiber over a range of 25º to 90º. The
cylindrical mount contained markings to guide the fiber for different helical angles. The fiber
was taped to this mount because touching the fiber, even with steady hands, corrupted the data.
For large angles, it was possible to make three revolutions around the helix, for angles of 60º or
less only two revolutions could be made, and only one revolution was used to record data for 30º
and 25º. The change in polarization angle was determined by varying the orientation angle of the
second polarizer until a local minimum in the photo-detector output was observed. The angle of
this minimum was recorded. The changes in polarization angles were calculated by subtracting
these angles from the angle obtain for 90º. Hence, it was assumed that no change in polarization
angle occurred for a helical angle of 90º. Local minima in the photo-detector output were
identified using the oscilloscope.
III.
Description of Experimental Error
5
A central goal of this report was to account for sources of experimental error and ascertain
whether-or-not experimental error is sufficient to explain any deviations from predicted values.
While collecting data, it became abundantly clear that there were three main sources of error: the
initially plane polarized beam which exits the fiber elliptically polarized, dust particles which
obstructed the beam’s path, and accidental curvature in the fiber before it passed around the
cylindrical mount that causes Berry’s Phase in unintended locations.
Plane polarized light traveling through an isotropic medium should not change its
polarization state. The single mode optical fiber is made of glass, and true glasses are
amorphous. Thus, they should be isotropic. Nonetheless, in experimentation there may be
aberrations in the glass that would cause the glass to become anisotropic. It would seem that
aberrations can arise from at least two possible sources: defects or crystal-like structures present
in the optical fiber when it was produced, and sharp bends introduced into the fiber as we
directed the fiber along its helical path. If the aberrations are because of the former, i.e. intrinsic
to the fiber, then our error should be systematic and equal for every helical angle. Therefore, we
will remove error introduced in this way when we subtract the measured polarization angle from
our 90º reference angle. If the aberrations are because of the later, i.e. introduced by bends we
make during the experiment, then the error will not be systematic. Non-systematic error is a
more serious concern because it will alter our calculated polarization angles even after
subtracting the reference angle.
An elliptically polarized beam can be decomposed into x and y components out of phase
by 90º with different magnitudes as follows:
E z, t
A cos kz
t x B cos k z
2
t y
(20)
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After passing through a polarizer, only the components parallel to the polarizer’s polarization
angle of each of these sub-components will survive; however, these components now point in the
same direction, the direction of polarization of the second polarizer. Thus the equation for this
wave will be as follows after passing through a polarizer:
E z, t
cos A cos kz
t
sin B cos k z
t
2
p
(21)
where θ is the angle formed between the x axis and the polarization angle of our polarizer. The
intensity of this beam (in CGS) is then:
I
1
2
E z, t
2
2
B z, t
t
(22)
In a vacuum, this can be reduced, and we can assume that the energy contribution from the
magnetic field equals the contribution from the electric field, thus:
I
A2 cos2
I
cos2 kz
B2 sin2
t
A B cos sin cos kz
I
2
2
2
A cos
2
2
cos
kz
2
E z, t
t
t
t
(23)
cos2 k z
t cos k z
2
2
2
B sin
2
0
t
2
t
2
t
(24)
cos2 k z
2
0
t
t
2
A B cos sin
I
1
2
2
cos kz
2
t cos k z
0
2
A cos
2
2
B sin
AB
2
cos
k
2
t
2
sin
t
(25)
2
(26)
Here we can see that if B = 0, our elliptical wave becomes a plane-polarized wave and our
equation for intensity reduces to the more familiar Malus’ Law:
I
2
I0 cos
(27)
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where I0 = ½A2. The point of this derivation that is relevant to our discussion of error is that we
can use equation (26) to determine the ratio of A to B by measuring the intensity of various
polarizer angles θ for the elliptical beams that exit our fiber optic. This ratio is basically a
measure of the eccentricity of the light’s elliptical path. If the eccentricity remains the same,
then the error is probably systematic. If the eccentricity varies for various helical angles, then
our error is not systematic.
Data collected by M Mikel-Stites and G Voronov (2007) allow us to compare the A to B
ratios for two helical angles in our apparatus: 90º and 75º (Figure 1). Using this data we can fit
our equation of intensity, equation (26), to both of these curves by minimizing the mean-squared
deviation of theory from experimental values using the Gauss-Newton Method (Before fitting the
curve, a third parameter Φ was introduced to determine the phase shift between θ and the
polarizer angle, since this was unknown). The ratio of A to B for 90º was 0.8290, while A:B for
75º was 0.8003. This represents a relative difference of 3.59%. Hence, experimental error
introduced from anisotropic behavior was believed to be primarily systematic. Based on this
conclusion, this study decided that other sources of error, besides anisotropathy, were primarily
responsible for variation in experimental data.
Significant error in data was introduced through dust particles interfering with the laser’s
path and unintentional bends in the fiber optic. For the purpose of this study, both errors were
approximated as gaussian distributions with a mean change in polarization angle of zero and unassumed variance. Since a sum of gaussian distributions is another gaussian distribution, this
approximation enabled the use of parametric methods to test the hypothesis. The distribution of
randomly floating dust particles of negligible volume is a Poisson process, with a Possion
distribution of dust particles in the beam: P(r | λ) = e-λ λr / r!, where r = 0, 1, 2, … and λ = mean
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number of dust particles in the beam. As λ approaches infinity, P(r) approaches a gaussian
distribution, since there was a very large number of dust particles interfering with the beam we
can approximate this error with a gaussian curve. Bends in the fiber-optic are harder to model;
however, if we thought of such bends as a random-walk process, then the distribution would still
approach a gaussian distribution for large iterations of the walk.
IV.
Results and Discussion
Plane-polarized light was sent through a single mode optical fiber, which made several helical
turns before the light exited the fiber-optic. The change in polarization angle for various helical
angles was recorded, and compared to theoretical predictions, equation (19), based on Berry’s
Phase (Figure 2). The data closely resembled theoretical predictions. A correlational coefficient
was not calculated, as it is unwarranted with nonlinear relationships. The standard deviation of
experimental values from theoretical values was 11.30°. With an average change in helical angle
of 169° for the data collected, the standard error is 6.68%. While, in general, this is relatively
low, most of the error was concentrated in measurements at low helical angles where the
standard error would be much larger. This phenomenon may have occurred because a fewer
number of turns in the helical cable at low helical angles means that aberrations in one turn
would not be weighed out by all the other turns.
Using our detailed analysis of experimental error above, we can make a more formal test
of the validity of Berry’s Phase. We expected a gaussian distribution of experimental values
about our theoretical values with a mean of zero (because experiment should match theory on
average) and unknown variance. The unbiased, fasted-converging estimator of variance then
becomes:
2
s
Xi
N
M
2
(28)
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where Xi = (experimental value) – (theoretical value) for each point, M <Xi> = 0, and N = (# of
degrees of freedom (the sample size)) = 13. There is not N – 1 degrees of freedom because we
have already constrained the system by expecting that M = 0. The sampling distribution for this
situation is a t-distribution whose density function is:
N 1
2
f t
1
N
2
N
t2
N 1
2
N
(29)
where Γ(x) is a Gamma Function. The parameter t, can be calculated from:
t
Xi
s
N
(30)
In our experiment, t = -1.423. Using the cumulative distribution of f(t), we find that the
probability of this event and all events less common than this event (assuming that our
hypothesis is true) is ~0.18. Statistical significance normally requires that p < 0.05. Therefore,
we cannot refute this study’s hypothesis based on Berry’s Phase from the data obtained.
V.
Conclusions
The phenomenon of Berry’s Phase can be observed in a helically-twisted, single-mode fiberoptic. Berry’s Phase is a change in polarization angle that occurs when the momentum vector of
a beam of light is changed adiabatically, despite returning to its original momentum. While the
fiber-optic itself altered the polarization state of the beam, most of this error was systematic and
did not prevent further testing of Berry’s Phase. Our data failed to refute the hypothesized
change in polarization angle due to Berry’s Phase (p > 0.05, N = 13).
VI.
Figures
10
Intesity (Volts)
5
4
90 degree
helical angle
3
2
75 degree
helical angle
1
0
0
100
200
300
400
Angle of Second Polarizer (degrees)
FIGURE 1. Identifying the elliptical content of light exiting the fiber optic. The relationship
between intensity and angle of the second polarizer indicates that light exiting the fiber optic is
elliptically polarized. While local minima of intensity for the two helical angles varied, the
eccentricity of the elliptically polarized beams was about the same, suggesting that this elliptical
Change in Polarization (degrees)
transformation in the fiber-optic is intrinsic to the optical fiber and systematic.
400
350
300
250
200
150
100
50
0
0
20
40
60
80
100
Helical Angle (degrees)
FIGURE 2. Berry’s Phase in a single-mode fiber optic traveling through a helical path. A
plane polarized beam was sent through a fiber-optic waveguide that twisted around a cylinder in
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a helical fashion at various helical angles. The solid line represents theoretically predicted
changes in polarization angles based on Berry’s Phase.
VII.
References
Mikel-Stites, M. and Voronov, G. “Berry’s phase.” Physics 243W: Advanced Lab Course
Schedule. 10 Dec. 2007 http://www.pas.rochester.edu/~advlab/class2007.html
Tomita, A. and Chiao, R. Y. (1986) “Observation of Berry’s Topological Phase by Use of an
Optical Fiber.” Phys. Rev. Letters 57:937-940.
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