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The Poincaré-sphere approach to polarization: Formalism and new labs with Poincaré
beams
Joshua A. Jones, Anthony J. D’Addario and Enrique J. Galvez∗
Department of Physics and Astronomy, Colgate University, Hamilton, New York, 13346, U.S.A.
(Dated: March 16, 2016)
We present a geometric-analytic introductory treatment of polarization based on the circular
polarization basis, which connects directly to the Poincaré sphere. This enables a more intuitive way
to arrive at the polarization ellipse from the components of the field. We also present an advanced
optics lab that uses Poincaré beams, which have a polarization that is spatially variable. The
physics of this lab reinforces students’ understanding of all states of polarization, and in particular,
elliptical polarization. In addition, it exposes them to Laguerre-Gauss modes, the spatial modes used
in creating Poincaré beams, which have unique physical properties. In performing this lab, students
gain experience in experimental optics, such as aligning and calibrating optical components, using
and programming a spatial light modulator, building an interferometer, and performing polarimetry
measurements. We present the apparatus for doing the experiments, detailed alignment instructions
and lower-cost alternatives.
I.
INTRODUCTION
Polarization, the vectorial aspect of light, is a core
topic in physical optics that is the subject of much fundamental research today. It is an important dimension
in studies of light in all of its complexity, as well as in
the interaction of light fields with matter. It is also an
important component in applications, such as metrology, sensing, communications and display technologies.
New technologies, materials and devices, such as liquid
crystals,1 have advanced our understanding of the subtle
ways in which vector fields interact with matter, creating at the same time new ways to harness the properties
of light. Thus it is sensible to find ways to enhance the
teaching of polarization optics. One of the goals of this
article is to revisit a fundamental treatment of polarization and to provide more tools to explain and illustrate
it to students. Along with the theoretical formalism, we
describe an intermediate to advanced optics laboratory
experiment to prepare light beams in Poincaré modes,
which have spatially variable polarization. The determination of the polarization states of the beam is used
as an exercise to help students understand all states of
polarization and their measurement.
The standard way to describe polarization optics is
in terms of the Cartesian components of the electric
field.2–4 It works well to describe interactions with materials. However, the description of elliptical polarization,
in terms of the linear components of the fields (the linear
basis), is neither easy nor intuitive. Here we propose a
treatment of polarization in terms of the circular states
of polarization. This is because the student can get a
more direct connection between the properties of the ellipse and the coefficients of the basis states. We use the
Poincaré sphere5 as a geometric tool for this treatment,
discussed in Sec. II.
Teaching optics is facilitated by the easy implementation of tabletop laboratory experiments, the replacement of optical rails by optical tables or breadboards,
and the growth of the optics hardware industry. The
availability of devices that manipulate the light on a
table-top gives students a setting to explore optical phenomena via simple experiments, alowing them to define
their own measurements and experiments, understand
the concepts at a fundamental level, and rid themselves of
misconceptions.6 Yet, advanced laboratories should not
be ignored altogether. They provide students with a
glimpse of the subtleties and depth of optical phenomena, and exposure to sophisticated equipment.
In the understanding of elliptical polarization, there
are a number of proposals of experiments to measure
it via a rotating polarizer.7–12 Such a simple experiment reveals, for example, a non-intuitive result: that
while the electric field describes an ellipse, the square
of the field (intensity) describes a dumbell-type shape
as a function of polarizer angle. Other approaches to
polarization laboratories use the interaction of the vectorial nature of light with birefringent materials, such
as cello-tape,13–15 cellophane,16 Plexiglass17 and photo
elastic gel.18 In particular, the study of the wavelength
dependence in some of these14,16 adds an additional
variable not available with narrow-band lasers. The
study of conoscopic patterns, fringe patterns that appear
when light passes through birefringent media in between
crossed polarizers, is a challenging but rich topic that
has much to offer in teaching not only vectorial nature
of light, but its interaction with birefringent media.19–21
These patterns are widely used in geology for the optical characterization of minerals,22 but they can easily
be reproduced with cellophane,23 plastics24 and overhead
transparencies.25–27 Along the same lines, liquid crystals
also provide a setting for understanding polarization and
birefringence.28–30
In this article we propose an experiment that focuses
exclusively on the light: the generation and diagnosis of
optical beams that have a state of polarization that varies
from point to point in the transverse plane of the beam.
Conoscopic patterns already contain these rich structures, but they are ignored altogether by projecting them
with a polarizer. The particular beams we use are also
2
known as Poincaré beams.31–33 In general, space-variant
polarized light fields are ubiquitous but complex, so they
are often avoided. However, they are appealing when the
pattern of states of polarization follows a certain symmetry or organization. Previous discussions of these in
the educational context include their generation by passage through a birefringent media34 or by simply crossing optical beams with orthogonal polarizations.35 An
early incarnation of Poincaré beams are vector beams.36
They contain spatially-variable linear polarization states.
They can be produced simply using a variable-fast-axis
half-wave plate made of cellophane tape,37 or its liquidcrystal counterpart, the q-plate.38
We prepared the Poincaré beams using a simple polarization interferometer, whereby the split beams are produced by a phase grating using a low-cost spatial light
modulator (SLM). The latter have been used recently for
undergraduate experiments.39,40 One of the beams generated by the SLM was in a Laguerre-Gauss (LG) mode, a
spatial mode of light with interesting optical properties,41
which can be used as a topic of experimentation in the
undergraduate laboratory on its own right.40,42–44 This
combination of components gives rise to an experiment
rich in optics and polarization.
In Sec. III we give a quick presentation of LG and
Poincaré modes and describe the details of the experiment, which includes the generation and detection of the
beams. We show our results in Sec. IV. In two appendices we give details on the programming of the SLM and
in alignment procedures. A third appendix has a discussion of equipment costs and experiments performed with
inexpensive wave plates and polarizers.
II.
to the lab environment, we label horizontal (H) and vertical (V ) the x and y directions of the field, respectively.
A general expression for the field is then given by2,3
E = E0 cos α êH + sin α e−i2δ êV ei(kz−ωt)
(1)
= E0 ê ei(kz−ωt) ,
where α ∈ [0, π/2] is an angle that specifies the relative
magnitudes of the H and V components of the field, and
δ ∈ [0, π] is half of their relative phase. (We use the factor of 2 in the relative phase for reasons that will become
clear later.) Other constants in Eq. 1 are the wavenumber k and the angular frequency ω. Without any loss
of generality, we set the phase of the field to be zero at
z = 0 and time t = 0.
We denote the polarization basis states by êH and êV .
As can be seen in Eq. 1, we specify the vectorial part
of the field, the polarization, by ê. Linear polarization
states aligned along the H (x) and V (y) orientations are
given respectively when α = 0 and α = π/2, with δ = 0.
The circular polarization states correspond to α = π/4
and 2δ = ±π/2. They are states where the tip of the
electric field vector describes a circle as a function of time
at a given value of z, a motion defined as right-handed
(êR ) when the field rotates clockwise when looking into
the beam of light (along the negative z direction), for
which 2δ = +π/2. Conversely, left-handed circular (êL )
corresponds to a counter-clockwise rotation of the field,
for which 2δ = −π/2. We formalize these definitions with
the relations (and ignoring an overall phase):2
1
êR = √ (êH − i êV )
2
1
êL = √ (êH + i êV ) .
2
FORMALISM
The traditional way to explain polarization is to start
with the linear states and describe the other states (circular, elliptical) in terms of the linear components of the
field.2–4 Understanding the circular states is straight forward, but understanding elliptical states is not. This is
because it is not easy to visualize the form of the elliptical
state in terms of the relative amplitudes and phases of the
linear components. For this reason, discussion about elliptical states of polarization in basic treatments of optics
is often minimized, deferred to more advanced courses,
or even avoided altogether. To provide a more accessible way to describe elliptical states, here we propose
a change in the description of polarization by using the
circular basis representation after a quick introduction of
polarization in terms of the linear basis. The new treatment gives a more accessible way to describe elliptical
states. In addition, we use the Poincaré sphere5 as a way
to get a clearer understanding of all states of polarization
and their relations.
We start by defining a state of polarization in terms
of the linear x and y components of the electric field of
amplitude E0 traveling along the z axis. To relate easily
(2)
(3)
For completeness, we define two other useful basis states,
those with diagonal linear polarization (D) and antidiagonal linear polarization (A):
1
êD = √ (êH + êV )
2
1
êA = √ (êH − êV ) .
2
(4)
(5)
In the most general case, the tip of the electric field
describes an ellipse of semi-major axis a and semi-minor
axis b, oriented at an angle θ. How are these parameters
related to α and δ? The answer to the question is: nottrivially, as we cannot easily visualize the ellipse based
on the values of α and δ in Eq. 1. In contrast, if we
express the state of polarization in terms of the circular
components, obtaining the ellipse parameters is straight
forward, as shown next.
A.
The Poincaré sphere and the circular basis
It is useful to describe the Poincaré sphere, shown in
Fig 1(a), before we present the representation in terms
3
of the circular basis. Each point on the surface of the
sphere represents a pure state of polarization, with the
north and south poles representing right-circular (R) and
left-circular (L) polarization, respectively. Points along
the equator represent linearly polarized states of varying
orientation, with notable cases at the axes: horizontal
(H), vertical (V ), diagonal (D) and anti-diagonal (A).
Other points in the northern hemisphere represent righthanded ellipses, and correspondingly, points in the southern hemisphere represent left-handed ellipses. The polar
angle is defined as 2χ and the azimuthal angle is 2θ.
general equation of the state of polarization is given by
ê = cos χ êR + sin χ e−i2θ êL .
Note the simplicity of the description: the relative magnitude of the circular components determines exclusively
the ellipticity (related to χ) and their relative phase determines exclusively the orientation of the ellipse (θ).
Thus, when the polarization is expressed in the form of
Eq. 7, one can directly visualize the state of polarization
by inspecting these two independent variables.
B.
FIG. 1: (a) Poincaré sphere representing all the states of polarization; (b) dependence of ellipticity and orientation on
the half polar and azimuthal angles of the sphere. Rightand left-handed states are denoted by red and blue figures,
respectively.
45
The ellipticity of the polarization state is given by
=±
b
= tan (π/4 − χ) ,
a
(6)
which is directly related to the latitude on the sphere,
and with its sign representing the handedness of the polarization ellipse. The simplicity of the Poincaré sphere
for visualizing polarization is that points of the same latitude have the same ellipticity, and points with the same
longitude have the same orientation of their semi-major
axis. Thus, as shown in Fig. 1(b), the ellipticity and orientation are independently controlled by angles χ and θ,
respectively.
In terms of the circular basis states êR and êL , the
(7)
Linear basis and a geometric transformation
Note again the correlation between the Poincaré sphere
and Eq. 7: the relative magnitude of the components
specifies the polar angle on the sphere, and the relative phase specifies the azimuthal angle. The linear basis
states êH and êV are antipodes along the equator of the
sphere. In fact, the geometry of the sphere is such that
any pair of points that are antipodes constitute an orthogonal basis set. If we think of the (H, V ) basis as a
rotated sphere, with H and V being the poles, then α
in Eq. 1 is half of the “polar” angle and 2δ is the corresponding azimuthal angle, with 2δ = 0 being on the
equatorial plane of the (R, L) polar sphere, as shown in
Fig. 2.
Let us look at particular cases. When 2δ = π/2, 3π/2,
ellipses have semi-major axes that are either horizontal
(for 2α < π/2) or vertical (for 2α > π/2), regardless of
ellipticity (see Fig. 2). Similarly, when 2α = π/2 the
semi-major axes are either diagonal (for 2δ < π/2 and
2δ > 3π/2) or anti-diagonal (for π/2 < 2δ < 3π/2),
regardless of the ellipticity.
The circular basis can easily help us find the state of
polarization via χ and θ. If we could relate α and δ to χ
and θ we could find a path to identify the state of polarization in the linear basis. We can get to this relation via
spherical trigonometric relations.46 Applying the spherical Pythagorean theorem to the spherical triangles WPY
and UPZ shown in Fig. 2, yield respectively
cos 2α = cos 2θ sin 2χ
cos 2χ = sin α sin 2δ
(8)
(9)
We can use these relations to transform the state of
polarization of Eq. 1, with α and δ given, to the circular
basis (Eq. 7):
r
1 + sin 2α sin 2δ
cos χ =
(10)
2
r
1 − sin 2α sin 2δ
sin χ =
(11)
2
!
cos 2α
2θ = cos−1 p
(12)
1 − sin2 2α sin2 2δ
4
If the polar angle in the sphere with the D − A polar axis
is β (angle subtended by points P and T in Fig. 2), then
s2 = cos 2β = cos2 β − sin2 β = (ID − IA )/I0 .
(16)
Once the normalized Stokes parameters are obtained
through measurements, we can then derive the polarization ellipse parameters defined earlier (see also Fig. 1):
1
χ = cos−1
2
s3
p
!
s21 + s22 + s23
(17)
and
θ=
FIG. 2: Poincaré sphere showing two spherical triangles:
WPY of angular sides 2α (WP, the spherical hypotenuse),
π/2 − 2χ (PY) and 2θ (YW); and UPZ of angular sides 2χ
(UP, the spherical hypotenuse), π/2 − 2α (PZ), and π/2 − 2δ
(ZU).
For example, suppose α = π/4 and 2δ = π/4. From
identifying these angles on the Poincaré sphere we deduce
that it is a right-handed ellipse oriented somewhere between 0 and 45 degrees. Using the above relations we get
◦
χ = 30◦ , which gives
p an ellipticity = tan 15 = 0.27;
and 2θ = cos−1 2/3 = 35.26◦ , or a right-handed ellipse forming an angle of 17.63◦ with the horizontal.
These equations are related to other relations between
the linear basis parameters and the polarization ellipse
parameters,45 but are more convenient to use in our geometrical analysis.
C.
Determination of the polarization and Stokes
parameters from measurements
Pure states are represented by points on the surface of
the Poincaré sphere, shown in Fig. 1. The coordinates of
these points are known as the Stokes parameters. The
component along the R − L axis is
s3 = cos 2χ = cos2 χ − sin2 χ = (IR − IL )/I0 ,
(13)
where IR and IL are the intensities of the light in the
right and left circular states, and
I0 = IR + IL = IH + IV = ID + IA .
(14)
Likewise, IH , IV , ID and IA the intensities in the corresponding states. In the laboratory they are obtained by
passing the light through the corresponding polarization
filters, described below. Similarly, using the linear basis,
and its description in terms of the H − V polar axis,
s1 = cos 2α = cos2 α − sin2 α = (IH − IV )/I0 .
(15)
1
tan−1 (s2 /s1 ).
2
(18)
These definitions account for the case when the light is
not in a pure
p state, or having an unpolarized component,
for which s21 + s22 + s23 < 1.
The presentation given in this section can be incorporated into one or two days of class of an undergraduate
course in optics. The lab should follow one or more class
periods where the polarization is defined in terms of the
motion of the electric field in time and space expressed
in the linear basis. Additionally, the student should be
familiar with exponential notation to represent phases.
Once this is done, then the concept of polarization as
a state of the light can be introduced as described in
this section. The main thrust of this section is to introduce Eq. 7, from which we can easily deduce the elliptical polarization parameters and represent the possible
polarization states using the Poincaré sphere. Relations
10-12 provide a way to relate the parameters of Eq. 1 to
those of Eq. 7. Finally, an introduction to the Poincaré
sphere also serves to prepare students to understand the
Bloch sphere, which is used in quantum mechanics to
understand two-level systems. The Bloch sphere has experienced renewed interest in quantum information to
represent the state of a qubit.47
III.
A NEW LAB ON POLARIZATION
Research on singular optics with Poincaré beams
has opened new possibilities for interesting undergraduate laboratories on polarization optics.
Poincaré
beams are a class of beams that have spatially variable
polarization,31–33,48 which correspond to a mapping of
states of polarization from the Poincaré sphere onto the
transverse mode of the beam. In this article we present
an experiment that illustrates all states of polarization
and their detection. We propose a simple experimental
arrangement, which uses a modern diffractive-optical device: a spatial light modulator (SLM). In this section
we present the various components of the experiments.
Low-cost alternatives to the SLM and wave plates are
also presented.
5
This laboratory experience can be presented at several levels. We have used it as a semester-long capstoneproject level, or as a summer experience to introduce
the student to more advanced topics. The starting point
for these experiences involve the student and the faculty
member working first on understanding the theory and
formalism as presented in the previous section. It should
also include the determination of the Stokes parameters
from measurements, as presented in this section, as a way
to determine the state of the light. Although we have not
done so, for curricular reasons, it can be introduced as a
several-week-long advanced lab experiment. We are also
introducing it as an advanced optics-course laboratory.
Beyond polarization, this laboratory introduces spatial
modes of light. In particular, Laguerre-Gauss modes,
which are a requirement in producing the Poincaré
beams. Thus, the lab experience should be preceded by
a reading and discussion of the subsection that follows.
Finally, if the student is involved in setting up the apparatus, he or she should be familiar with fine steering of
optical beams.43
A.
Laguerre-Gauss and Poincaré modes
In this lab we make Poincaré modes of various types.
All involve a superposition of spatial modes in orthogonal states of polarization. The spatial modes that
we use are of a very interesting class: Laguerre-Gauss
(LG), which have been subject of previous articles in
this journal.42,43,49 There are also numerous formal studies on them,41,50 so here we present only what is needed
to define the problem. We will focus on a subset of LG
modes that have a singly-ringed, or “doughnut,” intensity distribution in the transverse plane. The normalized
amplitude of the field of the light propagating along the
z axis is given by the function50,51
LG`0 = A` r|`| ei`φ GW,
(19)
parameterized by `, known as the topological charge. The
variables r and φ are the transverse polar coordinates. A`
is a normalization constant that depends on `, given by
A` =
1 |`|+1
2|`|+1 2 1
.
π|`|!
w
(20)
w is a parameter representing the half-width of the mode.
The defining characteristic of these modes is the term
ei`φ , which denotes a phase that varies with the angular
coordinate. Thus ` is the number of times that the phase
advances by 2π in one turn about the center of the mode.
The factor r|`| gives the mode the distinctive doughnut
shape for ` 6= 0. The function G provides a Gaussian decrease of the amplitude with r, and specifying the limited
transverse extent of the mode:
G = e−r
2
/w2
.
(21)
The term represented by W accounts for propagation
phase effects, such as the z dependence of the phase, the
radius of curvature of the expanding the wavefront, and
an evolving phase known as the Gouy phase. It is given
by
W = ei[kz+kr
2
/(2R)−ϕ]
,
(22)
where k is the wave vector, R the radius of curvature of
the wavefront, and ϕ the Gouy phase, given by50
ϕ = (|`| + 1) tan−1 (z/zR ),
(23)
where zR is the Rayleigh range. LG modes are in general multi-ringed: in their most general form, LG`p , the
additional parameter p specifies the radial structure.41,50
The φ- and z-dependence of the phase yields a wavefront
that consists of ` intertwined helices. Due to the slanted
wavefronts, LG modes carry orbital angular momentum.
Each photon occupying this mode carries an angular momentum of `~.41,50 This is in addition to the spin angular
momentum due to polarization, which carry an angular
momentum of −~ and +~ per photon for right- and leftcircular states, respectively.
To form a Poincaré mode we create a superposition of
LG modes in orthogonal states of polarization. Here we
focus on a particular case given by32
1
U (r, φ) = √ LG`0 êR + LG00 e−i2γ êL
2
(24)
where 2γ is the relative phase between the two modes.
We can rewrite Eq. 24 in a way that has the form of
Eq. 7:
U (r, φ) = U0 (r, φ) cos χ(r) êR + sin χ(r) e−i2θ(φ) êL ,
(25)
where U0 (r, φ) is a scalar function that factors out. We
are interested in the term within parenthesis, which describes the vectorial part of the state of the light with
A0
χ(r) = tan−1
(26)
A` r `
and
θ(φ) = `φ/2 + γ.
(27)
Since χ specifies the ellipticity of the polarization and θ
the orientation of the semi-major axis, Eqs. 25, 26 and
27 specify a mode where the ellipticity depends on the
radial distance from the center of the beam and the orientation depends on the angular coordinate. They constitute a mapping of the Poincaré sphere onto the transverse mode. Figure 3 shows the polarization patterns for
` = ±1 and ` = +2. In the following section we show
how to make these beams with a simple apparatus. Its
importance is that it is an optical mode rich in polarization, which we can then use to teach about polarization
within the context of an advanced optics experiment.
6
zontal axis. We inserted two mirrors (M1 and M2 ) after
the HeNe source as steering optics to let the beam travel
enough distance to expand and fill the diffractive element. We passed the beam through two apertures for
alignment purposes. A polarizer (P1 ) was placed after
the steering optics to “clean up” the polarization of the
beam from any changes in polarization caused by the
mirrors or any slight ellipticity from the laser. The SLM
acted on only one polarization component: the one parallel to the long side of the Cambridge SLM active area.
We loaded a “forked” diffraction pattern onto the SLM
(see Appendix A) to generate a first-order beam that was
“doughnut shaped” (i.e., an LG mode).
FIG. 3: Patterns showing four notable cases of Poincaré
modes. According to Eq. 24, they correspond to ` = 1 for
(a), ` = −1 for (b); and ` = 2 with γ = 0 for (c), and γ = π/2
for (d). The false color encodes the orientation of the semimajor axis of the ellipse. The saturation encodes the intensity
of the mode.
The modes shown in Fig. 3 are important in characterizing modes with spatially-variable polarization.
They contain polarization singularities, also known as
C-points.52–54 Around the center of the mode the orientation of the polarization ellipse rotates. Thus, as we
approach the center we reach a singularity in ellipse orientation, which is precisely the circular state, for which
the orientation parameter is undefined. The mode of
Fig. 3(c) is known as the radial mode because all the
orientations point to the center of the mode. The patterns of Fig. 3 (a) and (b), known respectively as lemon
and star.53–55 The shown patterns correspond to a relative phase γ = 0. If we change γ (something that is hard
to avoid in the interferometer shown below), the states
of polarization rotate. This results in a rotation of patterns (a) and (b), but a transformation of mode (c) into,
for example, mode (d). We challenge the student reader
to specify a value of γ (say, π) and predict the resulting
pattern.
B.
Apparatus Arrangement
A schematic of the experimental setup is shown in
Fig. 4(a), and a photo of it is shown in Fig. 5(a). We
used a polarized Helium-Neon (HeNe) laser with an output wavelength 632.8 nm. Our HeNe source was oriented
to emit light that was linearly polarized along the hori-
FIG. 4: (a) Schematic of the apparatus to generate Poncaré
beams. Optical components include mirrors (M), a spatial
light modulator (SLM) with insert showing an example of a
programmed forked blazed-phase grating (with line spacing
increased for illustration only), half-wave plates (H), a polarizing beam splitter (PBS), quarter-wave plates (Q), and
polarizers (P). (b) Array of wave plates used in the detection
of states of polarization.
The zero-order and first-order beams were reflected off
two large (> 2 inch) rectangular mirrors (M3 and M4 )
and passed through another horizontal linear polarizer
(P2 ; again, for cleaning up the polarization). The zeroorder beam was passed through a half-wave plate (H1 ),
to rotate its polarization. The first order beam missed
the wave plate, traveling directly to a polarizing beam
splitter (PBS). Thus, we had to machine a portion of the
wave plate mount (the 1-inch collar of a 1/2-inch optic)
so that the first-order beam passed through the mount
unobstructed without going through the half-wave plate
(see Fig. 5(b) for a close-up of our mount). The half-wave
plate in combination with the PBS served to equalize the
intensities of the two beams when they were superposed.
In practice, we found that the PBS, which is expected to
reflect only the vertical component, still reflected a signif-
7
icant amount of the horizontal component. Therefore, we
added a fixed polarizer (P3 in Fig. 4) with transmission
axis vertical to make sure that the zero-order component
was vertically polarized.
Past P3 , the zero-order beam was reflected off a rectangular mirror (M5 ) and directed to the PBS. A close-up of
the two components is shown in Fig. 5(c). We mounted
the mirror on a translation stage to adjust its position
relative to the PBS. The two beams were merged together by the PBS, which transmitted the horizontallypolarized first-order beam and reflected the verticallypolarized zero-order beam. The SLM, M5 and the PBS
formed a polarization interferometer. A critical part of
the set up involved aligning the two beams to leave the interferometer collinear (see Appendix B). The combined
beam, now in a superposition of horizontal first-order
and vertical zero-order, was sent through a quarter-wave
plate (Q1 ; with axis at 45◦ to the horizontal). The latter converted the horizontal and vertical states into the
right- and left-circular polarization states, respectively.
The resulting beam was in the Poincaré mode specified
by Eq. 24.
The Poincaré beam was then directed to a polarimetry setup consisting of a rotatable half-wave plate (H2 ),
a fixed quarter-wave plate (Q2 ), a second rotatable halfwave plate (H3 ), and a fixed polarizer (P4 ). A lens was
used to focus the beam onto a digital camera. Alternatively, we also expanded the beam with a diverging lens
and projected it on a screen after a mirror (to effect a mirror inversion). When imaging with the digital camera we
added a second polarizer (P5 ) to adjust the intensity of
the light incident on the digital camera. A photo of the
apparatus is shown in Fig. 5. The optical components
seen in the photo were on a 20 × 40 optical breadboard.
The camera was just outside the breadboard, so we recommend at least a 20 × 50 breadboard for fitting all the
components. In our experience, the full assembly of the
optical layout, starting with a clean breadboard, should
not take longer than a 3-hour lab period.
C.
Detecting the states of polarization.
There are several ways to detect an arbitrary state of
polarization. The simplest uses two elements: a quarterwave plate and a polarizer. In the first part of the laboratory exercise presented here we null the state of interest.
Because Poincaré beams have a mostly uniform intensity, with a state of polarization that varies from point to
point, it is visually compelling to null the state of polarization under study, which creates an intensity minimum
centered at the location of the state of interest. Our approach uses more wave plates than necessary, but only to
provide a clearer understanding of the polarization pattern of the beam and the rationale behind the detection
of elliptical states. In Fig. 4(b) we show the approach.
We use two half-wave plates, one quarter-wave plate and
a fixed polarizer.
FIG. 5: Photos of the apparatus: (a) The layout of the experimental setup with lines denoting the path of the HeNe
beam; (b) Detail of the half-wave plate (H1 ) with a cut-out in
its mount; and (c) Close-up of the PBS-mirror combination,
with insert showing the paths of the beams.
The motion of the electric field around the ellipse can
be understood in a simple way: The components of the
field along the semi-major and semi-minor axes are 90◦
out of phase from each other.53 Thus if we pass the light
through a quarter-wave plate with fast axis along either
axis, the state will be converted to a linear state. The
linear state can then be blocked by a polarizer. This is
the principle of the two-element null detector. In our case
we split this process into four steps. In the first step, the
light with orientation θ and ellipticity goes through a
half-wave plate with fast axis forming h2 = θ/2 with the
horizontal. This rotates the polarization so that the semimajor axis is horizontal. It does not affect the absolute
value of the ellipticity but it does change its sign. A
quarter-wave plate with its fast axis fixed in the vertical
direction then converts the elliptical state to a linear state
oriented relative to the horizontal by the angle (π/4 − χ),
which is directly related to the ellipticity of the state via
Eq. 6. A second half-wave plate with fast axis at h3 =
(π/4−χ)/2 makes the polarization orientation horizontal.
A final polarizer with transmission axis vertical blocks
the light in this state. This method is attractive because
the first and third wave plates control the orientation
and ellipticity of the nulled state independently, allowing
an easy mapping of the polarization pattern of the beam
(below). By reading the values of the two wave plates
we can extract the full state of polarization of the light,
giving:
θ = 2h2
(28)
= tan 2h3
(29)
for |h2 | ≤ π/2 and
8
for |h3 | ≤ π/8. Equations 28 and 29 do not hold outside
the bounds given. For other angles they must be modified
slightly, but this may lead to confusion.
An alternative method of detecting the polarization
state of the entire beam involved imaging polarimetry.56
In this case we rotated the last polarizer so that its axis
was horizontal, thus making the system a polarization filter (i.e., transmitting the selected state). We then took
bit-mapped images with each of the six filter settings (H,
V , D, A, R and L). The images were read as matrices
by a program written in Matlab (using “imread” instruction), and using the relations of Eqs. 13, 15, 16, 17 and
18 we found the state of polarization (θ and ) of each
point of the image (Matlab easily operates on matrices
for this purpose). Table I shows the settings of the optical
elements of the polarization filter.
TABLE I: The settings for each optical element in the setup
to detect the the states of polarization. Polarizer P4 has two
settings: as a filter (to transmit the state) or as a null (to
block the state).
State
H
D
V
A
R
L
h2
0◦
22.5◦
45◦
67.5◦
0◦
0◦
D.
q2
90◦
90◦
90◦
90◦
90◦
90◦
h3
0◦
0◦
0◦
0◦
−22.5◦
+22.5◦
p4 (filter/null)
0◦ /90◦
0◦ /90◦
0◦ /90◦
0◦ /90◦
0◦ /90◦
0◦ /90◦
Experiments without the SLM
The experiments presented here can be performed
without the SLM. We have done the same experiments
using a plastic binary forked grating.43 See Sec. A 4 for
details on how to generate the pattern. Once the pattern was generated in the computer, it was printed and
then photographed with black and white film. The developed negative is the actual grating. The apparatus
must change slightly because the diffraction is now in
transmission mode.
IV.
RESULTS
Here we present the results of the experiments with
the Poincaré beams. For all cases we prepared the mode
according to Eq. 24, superimposing an LG mode ` 6= 0
in right-circular polarization with the fundamental laser
mode (also an LG mode with ` = 0) in left-circular polarization. We present cases with ` = +1 and ` = +2.
Other possibilities that students can try include ` = −1
and ` = −2 or even larger values of |`| (below), which
show different patterns. Below we present the results obtained by detection via state nulling as well as imaging
polarimetry. The latter method finds the state of polarization of each imaged point. For fun we conclude with
a discussion of the case ` = 4, to show that any value of
` can be investigated.
A.
The Challenge: Case ` = 1
A first part of the student exercise is to respond to a
challenge question. Once the SLM is programed and the
beams are superimposed and aligned, we set p4 = 90◦ for
state nulling measurements. The student was asked to
map (that is, to draw carefully in a sheet of paper) the
polarization pattern of the light, by varying h2 and h3
and using Eqs. 28 and 29. The answer is the pattern of
Fig. 3(a), or a rotated version of it (effected by a relative
phase between the component LG modes).
We observed the light via two methods: by expanding
the beam onto a screen, or by imaging it with a camera.
For students, the former is the simplest. Past the polarization elements, the beam appears with a dark spot (and
often, depending on the alignment, with a slight tail due
to the slight difference in radii of curvature of the two
component modes). Figure 6 shows photos of the expanded beam projected on a screen for different values
of h2 and h3 . The location of the blocked state rotates
about the center of the beam when h2 is increased, indicating that the orientation of the polarization changes
smoothly about the center of the beam. For example,
when h2 = 22.5◦ the dark spot is on the upper side of the
beam, indicating that the orientation of the semi-major
axis of the nulled state is diagonal (i.e. θ = 45◦ ). The second half-wave plate adjusted the ellipticity of the nulled
state. When h3 is increased we see that the dark spot
approaches the center of the beam (see Fig. 6), and reaching it when h3 = 22.5◦ , which corresponds to = −1 or
left-circular polarization. Indeed, when h3 = 22.5◦ we
are nulling the component mode with ` = 0, leaving only
the ` = 1 mode, the LG mode produced on first order by
the SLM. Similarly, for h3 = 0 the null state has = 0 or
linear polarization. In our lab, the student was not asked
to make the mosaic of Fig. 6, but was free to decide what
approach to take to map the polarization of each point.
The results were good, but only after the student was
confronted with thoroughly understanding the meaning
of the experimental variables and how their variation related to the observations. We concluded that the exercise
was successful in bringing the forefront of the lab an understanding of the concept of elliptical polarization.
A second part of the experiment entailed finding the
exact pattern and comparing it to the nulling measurements. Here we needed to set p3 = 0, to transmit the
selected state of polarization instead of blocking it. We
then collected images with each of the six filtered states
of polarization of Table I. The bitmapped images were
captured using a digital camera and downloaded to a
PC. We used a Matlab program to generate a map of the
polarization across the beam profile. We then generated
9
FIG. 7: Polarimetry measurements of the combined ` = 0
and ` = +1 modes. The ellipses denote the polarization, the
false color denotes orientation, and the saturation denotes
the intensity. The polarization is a lemon pattern (compare
to Fig. 3) with a C-point at the center.
B.
FIG. 6: Changing the position of the state of polarization that
is being blocked using the polarimetry elements H2 and H3 .
The columns and rows determine, respectively, the angular
orientation and ellipticity of the blocked state.
images that graphed in false color the polarization-ellipse
parameters. Figure 7 shows an image where the false
color encodes the orientation of the ellipses. The saturation of the color that is related to the intensity of the
beam, specified by I0 (see Eq. 14). In addition, our program had a routine that drew small ellipses at periodic
intervals, representing the state at the point where they
were drawn. This program and sample files are available
on our website.57 These helped in understanding the abstraction of the false color. The measurements reveal the
richness of polarization information within the beam, including a circular polarization singularity at the center
(C-point). Compare this result to the calculated polarization pattern seen in Fig. 3(a). We made the program
available to the student. However, depending on the level
of independence of the student exercise, and time devoted
to it, one could provide the student with a skeleton of a
program that does the graphing, and then ask the student to enter the instructions that calculate the Stokes
parameters, and the state of polarization of each imaged
point.
Note that in Eq. 27 the orientation angle θ depends on
the relative phase γ between the two beams. The student
can be asked to predict the effect of changing γ by means
of putting a tilted glass plate in the path of one of the
beams.
The Prediction: Case ` = 2 and beyond
When we combined the higher-order LG mode with
` = +2, with the fundamental ` = 0 Gaussian mode, the
number of dark spots increased to two. As with the previous case, the location of the blocked polarization state
depends on the setting of the wave plates H2 and H3 : the
radial distance is varied by changing h3 , consistent with
changing the ellipticity of the blocked state; and the angular position is varied by changing h2 , consistent with
changing the orientation of the blocked states. Note that
from Eq. 27 the orientation also depends on `. For the
case of ` = 2, θ = φ + γ. Images demonstrating these
dependencies are shown in Fig. 8.
Like before, students can perform a mapping by hand
using the blocked state. This second part of the experience could involve in principle any LG mode. If students
are allowed to choose from several SLM patterns that encode distinct values of ` (see Appendix A), they can be
challenged to predict the pattern that is encoded. In general, using an LG mode of topological charge ` produces
|`| dark spots (i.e., blocked states) arranged symmetrically about the center of the beam. As an example, we
show the case for ` = 4 in Fig. 9. As we change h2 , the
dark spots rotate about the center, and as we change h3
they move radially, merging into one at h3 = 22.5◦ , as
seen in Fig. 8 for the case of ` = 2.
V.
CONCLUSION
In summary, we present a new analytical treatment
for understanding polarization using the variables that
define the coefficients of the circular polarization components. We believe that this treatment leads to a clearer
presentation of elliptical polarization, going beyond current treatments that are limited to making the student
aware that it exists. Our approach embraces elliptical
10
dents start with little to no optics experience, and so the
exercise helps students get up-to-speed on the basics of
polarization-optics in a short period.
Appendix A: Programming of the SLM
FIG. 8: Changing the position of the blocked state of polarization using the polarimetry elements H2 and H3 . Moving from
left to right represents changes in the angle h2 and moving
from top to bottom represents changes in the angle h3 .
FIG. 9: Image obtained by blocking the state of polarization
of an elliptical state of ellipticity = +0.58 and orientation
θ = 55◦ .
polarization via the definition of elliptical polarization
states and their two relevant parameters (ellipticity and
orientation), including exposure to the Poincaré sphere.
We also propose a new intermediate to advanced laboratory that examines the states of polarization of Poincaré
beams, which have a polarization that is spatially variable. This proposed experiment reinforces the proposed
treatment of polarization. It also exposes students to
a technique to measure an unknown state of polarization.The proposed apparatus is the product of our experience with other more elaborate designs,32,58,59 which
served either as capstone research projects for undergraduates or as summer research projects. Some research stu-
A phase-only SLM consists of a thin liquid-crystal cell
that has a pixelated electrode. A voltage applied between
the pixel electrode and a back-plane electrode generates
an electric field that changes the index of refraction of
the liquid crystal medium. Thus, light going through the
liquid crystal medium gains a phase that depends on the
applied voltage. The SLM is set up like a display monitor,
with a standard pixel resolution. The SLM acts as an
external monitor when connected to a personal computer
(PC). The color information for each pixel is used by the
SLM circuitry to generate a number that is converted into
the voltage applied to the liquid-crystal pixel. When the
number is 8-bits or less, the SLM reads only one of the
8-bit color pixels. In SLMs with higher bit-resolution,
SLMs concatenate the bits of two or more colors to form
the number that gets converted into the applied voltage.
From the user standpoint, we need to generate an image where the intensity of the colors encodes the phase.
This image gets sent to the SLM via the secondary monitor output of the PC. The easiest method is to set the
computer screen to have the same resolution as the SLM,
duplicate the external monitor to the main monitor, and
display the pattern in full screen.
Commercial SLMs with full phase control can be quite
expensive, with prices in the range of $5,000-$20,000, but
there are inexpensive options. SLMs are at the core of
some types of classroom projectors, so they can in principle be removed from the projector, programmed and
used.40 We used a low-cost (∼$1,200) commercial spatial
light modulator from Cambridge Correlators. Since it
has 8-bit resolution, we created images in gray scale. Its
shortcoming is that it does not encode the full 2π phase
(the vendor claims 0 − 0.8π at 633 nm). This reduced
phase depth is a shortcoming for some applications but
not for our purposes.
The patterns for programming the SLM can be generated using available commercial software, such as Matlab,
Labview or Mathematica. We used Matlab. In our program we created a matrix of dimensions 768 × 1024 × 3,
where the first two numbers were respectively the number of columns and rows of pixels of the SLM, and the
third dimension specified each color. Once we calculated
the values of each matrix element (below), the program
output the matrix into a file. This program is available
on our website.57
We were able to generate the beam that we wished by
encoding a phase-blazed diffraction grating. The blazing
action consists of generating fringes with a phase that
ramps up from a minimum value (white) to the maximum value (black) within each fringe. Thus, the phase
that is encoded along the rows of pixels has the cross
11
section of a sawtooth. The discontinuous jump in phase
from minimum to maximum diffracts the light. Therefore, by manipulating the shape of the fringes, we can
generate the spatial mode that we desire. The blazing action also helps to concentrate the light onto the diffracted
orders. We optimized the diffracted beams by reducing
the efficiency of the grating in certain regions, a form of
amplitude modulation of the diffracted light. We also
made some corrections to the sawtooth shape that we
describe below.
1.
Grating Pattern
The ideal diffraction grating can be thought of as a
hologram: it is the interference pattern between a plane
wave and the wave that we wish to generate in first order.
The fringes are directly related to the phase difference
between the two interfering beams. Consider a reference
frame on the plane of the SLM and centered on it. In
the case of the interference of the mode LG`0 and a plane
wave forming an angle Φ, the phase difference at the SLM
is60
y 2πx sin Φ
Θ = ` tan−1
−
,
(A1)
x
λ
where x and y are the coordinates of pixels on the SLM.
If for a moment we pick ` = 0, the points of equal phase
difference are points of equal value of x (i.e., forming vertical lines). The fringe separation corresponds to points
separated by ∆Θ = 2π, or
∆x =
λ
.
sin Φ
(A2)
If we turn the argument around, a grating with fringe
separation ∆x will produce a first-order diffraction at an
angle Φ away from the zero order. If we set ` 6= 0, the
diffracted beams of order n will be a close approximation
to LG beams with topological charge `n = n`. If we
focus only on the first order, then it will have the same
topological charge as the grating. Thus all we need to do
is to program a set of fringes onto the SLM to get our
desired beam in first-order diffraction. In our experiment
∆x ∼ 60 ± 5 µm. With the pixel separation of the SLM
being about 9.1 µm, each fringe was about 6.6 pixels
on the SLM. This corresponds to an angular separation
between the zero and first orders of about 0.6◦ .
The next step in programming the SLM is to encode
this phase onto the normalized gray level g of the image.
This is obtained with
g = (Θ mod 2π)/2π.
FIG. 10: Computer-generated holograms to create LG beams
with ` = 1 in first order: (a) plain phase-blazed pattern; (b)
with additional amplitude modulation; (c) with the blazing
action corrected for maximizing the intensity in the first order;
and (d) binary pattern for making a photographic grating.
2.
Amplitude modulation
The grating described in the previous section will generate a beam that has the same phase structure as an LG
mode, but it will not be a pure mode because the SLM
only does phase modulation. A better approximation to
a pure mode is accomplished by a combination of amplitude and phase modulation.61 This operation consisted
of multiplying the phase with the absolute value of the
normalized amplitude of the mode, given by
f=
r
r`
|`|
exp
−r2 + r`2
w2
(A4)
p
where w is the half width of the mode and r` = w `/2
is the radius at which the amplitude is maximum. The
amplitude modulation then gets programmed onto the
SLM with
h = gf.
(A5)
Figure 10(b) shows the resulting amplitude-modulated
pattern programmed with Eq. A5. We found this to be
a worthwhile modification, as the quality of the mode
improved substantially with this correction.
(A3)
3.
Figure 10(a) shows the pattern that is obtained when
programming the SLM by combining equations A1 and
A3. For sake of clarity we increased the fringe separation
by a factor of 10.
Phase-blaze correction
There is one more correction that we can make. The
previous sections described a method that works well
with the ideal SLM, where in changing from h = 0 to
12
h = 1 results in a 2π phase shift. However, the commercial SLM that we had went as far as 0.8π. The purpose of
the blazing action is to concentrate most of the intensity
of the light onto the first diffracted order. This is accomplished with a phase blaze of 2π/fringe. Since we do
not have the full phase range, what we do is to program
40% of each fringe from 0 to 0.8π and leave at a constant
phase the other 60% of the fringe. We do so following
the procedure outlined previously,61 where g = 0 for the
first 30% of the fringe and g = 1 for the last 30% of the
fringe, for every fringe. A simple programming algorithm
takes care of it. The final pattern that was programmed
onto the SLM looked like the one shown in Fig. 10(c).
4.
b.
Aligning the polarizing beam splitter (PBS)
The PBS was placed close to the edge of its mount to
allow for mirror M5 to be placed as close to its surface as
possible (see Fig. 11(a)). The PBS and mount were also
placed so that the first order beam was as close as possible
to the edge of the PBS, adjacent to M5 (see Fig. 5(c)).
We temporarily set polarizer P2 with transmission axis
forming a non-zero angle with the horizontal. This way,
the first-order beam had a vertical component that was
reflected by the PBS. With the zero-order beam blocked,
we aligned the PBS so that the reflection off the PBS was
parallel to the rows of holes of the breadboard. Estimated
assembly time: 30 minutes.
Passive binary grating option
We can use the same procedures to generate a passive
binary (black and white) diffraction grating. To do this
we follow the rationale of Sec. A 1 but instead of defining
g as in Eq. A3, we define it as
(
0, if (Θ mod 2π) < π
gbin =
(A6)
1, if (Θ mod 2π) ≥ π
The resulting binary grating is shown in Fig. 10(d). It is
one of the options of the program that we have posted in
our website.57 The next step then involves generating a
photographic black and white film/plate of the pattern,
which serves as the grating. An even cheaper alternative,
but one of less quality, is to print photo-reduced grating
onto a transparency paper.
FIG. 11: (a) Schematics for a coarse first step in the alignment of the beams: to make sure both reflected beams are
parallel to the holes of the breadboard. (b) Schematic of the
second fine step in the adjustment of the collinearity of the
two modes. It includes a tilting piece of glass (G) and placing
polarizer P5 . Inserts show images of the superposition of the
two beams when they are non-collinear (left) and collinear
(right). Arrows show the direction of motion of the interference maxima and minima when G is tilted.
Appendix B: Alignment
Here we focus on the alignment of the optical components after the SLM. A critical pair of components, M5
and the PBS, were aligned so that they superimposed
collinearly the zero- and first-order beams with orthogonal linear polarizations. We note that these optical elements must be adjusted with the working SLM pattern in
place, because the angular spread of the orders depends
on the line spacing of the grating encoded onto the SLM.
a.
Alignment of large mirror M4
The preparation of the beam required best practices
in the alignment of optical beams: whenever possible we
aligned the beams to be parallel to the rows of holes of the
optical breadboard. We find this alignment, described in
an earlier publication,43 indispensable for the alignment
of interferometers. Using this method we aligned the
first-order beam, after mirror M4 , to be parallel to the
rows of holes of the breadboard. Estimated assembly
time: 30 minutes.
c.
Aligning rectangular mirror M5
After aligning the PBS we blocked the first-order beam
and placed mirror M5 . We tilted it so that its reflection,
going through the PBS, was also parallel to the row of
holes of the breadboard. By alternatively blocking the
zero and first-order beams we then adjusted the translation stage so that the two beams merged collinearly into
one beam, as shown in Fig. 11(a). Estimated assembly
time: 30-60 minutes.
d.
Fine adjustment of the collinearity
The final adjustment of the beam was done by looking at the superposition of the two beams. We returned
the axis of polarizer P2 to horizontal and adjusted the
half-wave plate H1 for the two beams to have the same
intensity. After the PBS we placed polarizer P5 with
transmission axis at 45◦ with the horizontal. Because
13
the polarizer projects equally the horizontal and vertical
polarizations, the beam after the polarizer showed the interference of the zero- and first-order modes.43 The alignment just described was too crude to make the beams
exactly collinear, and so the beam projected on a screen
(expanded by a lens) displayed a fringe interference pattern with a fork in its center, as shown in the left imageinsert to Fig. 11(b). It only takes an angle ϕ of a few
minutes of arc between the two modes for the pattern
to show several fringes, of separation ∆x ∼ λ/ϕ (see
Eq. A2).
The next step in the alignment was to make the beams
fully collinear. This was done by looking at the dynamical aspect of the pattern. That is, by varying the relative
phase between the two modes. We did this by placing a
thin glass plate, such as a microscope slide, in the path
of the zero-order beam, and tilting it. The optical path
length of the light increases with the tilt of the plate.
Thus, we saw the fringes move as a function of the tilt,
as shown by the white arrow in Fig. 11(b). Then we
adjusted the tilt of the PBS to increase the fringe separation until there was one dark interference minimum.
Collinearity was achieved when the intensity minimum
rotated symmetrically about the center of the beam as
a function of the tilt of the glass plate, as shown in the
right-insert in Fig. 11(b).
This exercise already provides an excellent setting
to study the phase properties of LG beams with
an interferometer.43 The final step for obtaining the
Poincaré mode was to remove the polarizer P5 and add
a quarter-wave plate with fast axis forming an angle of
45◦ with the horizontal, as shown in Fig. 4(a). The glass
plate can still be used for setting the correct phase γ = 0
between the modes. This was accomplished by adjusting
the tilt until the interference minimum was at φ = 0 with
the detection optics set to nulling horizontal polarization.
Estimated assembly time: 30-120 minutes (depending on
experience).
TABLE II: Prices of components needed for the experiment.
Item
20 × 50 optical breadboard
SLM (Cambridge Correlators)
Polarized HeNe
1/200 (12.7 mm) PBS
Prism mount
Quarter-wave plate
Half-wave plate
Polarizer
Rotation Mount for 100
USB CMOS Camera
Mounted mirrors
Translation stage
Iris
Price (each)
$1900
$1200
>$850
$190
$80
$270/$3
$270/$3
$90/$6
$90
$350
$55–150
$250
$40
Quantity
1
1
1
1
1
2
3
5
9
1
5
1
2
simple plastic sheet polarizers and polymer wave plates
worked well. Only with careful handling and alignment
of the low-cost sheet optics were we able to generate acceptable polarimetry results, as seen in Fig. 12, which
displays a star pattern (see also Fig. 3) by programming
the SLM for ` = −1. Therefore, for the polarimetry analysis we recommend the higher-quality optics.
Appendix C: Equipment Costs
In Table II we list the prices of the required optical
components. We list typical prices of new items obtained
from commercial vendors. However, because many of the
items are standard optical hardware, they do not need to
be new. We only note that the holders of M5 and PBS
have to be stable, preferably mounted on pillars, to eliminate vibration-induced instabilities. The expensive wave
plates were commercial zero-order wave plates made of
quartz and designed for our operating wavelength of 633
nm. To alleviate some of the costs involved with this
lab, we also tested low-cost wave plates and sheet polarizers. The quarter- and half-wave plates were commercial
achromatic polymer sheets obtained from Edmund Optics. They were cut from square sheets, effectively costing
a few dollars each, as listed in the table. For mapping out
the polarization of the beam via the nulling method, the
FIG. 12: A star pattern created using the ` = −1 beam. The
polarimetry measurements were performed using the low-cost
optics. The ellipses represent the polarization, the false color
denotes orientation, and the saturation is the intensity.
Additionally, we have found it possible to perform the
polarimetry analysis using pictures taken of each pattern projected onto a screen with a consumer-electronics
camera. We found that a number of conditions had to
be met for this procedure to be viable: (1) the camera
was mounted to optics hardware so that it did not move
between pictures; (2) stray light was blocked from the
view of the camera, or the lab room was darkened; (3)
the intensity of the light was lowered to prevent pixel saturation; and (4) the automatic intensity control or gain
14
of the camera was disabled.
the Schlichting Fellowship and NSF grant PHY-1506321.
Acknowledgments
We thank G. Millione for help with the polarimetry
program. This work was supported Colgate University,
∗
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Electronic address: [email protected]
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