We present a geometric-analytic introductory treatment of polarization based on the circular polarization basis, which connects directly to the Poincaré sphere. This treatment 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 can reinforce understanding of all states of polarization, and in particular, elliptical polarization. In addition, it exposes students 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.

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. Polarization 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 This description 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, so 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 light on a table-top gives students a setting to explore optical phenomena via simple experiments, allowing 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 dumbbell-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 Plexiglass,17 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 between crossed polarizers, is a challenging but rich topic that has much to offer in teaching not only the vectorial nature of light but also 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 plastics,24 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 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 light fields in the educational context include their generation by passage through birefringent media,34 or by simply crossing optical beams with orthogonal polarizations.35 An early incarnation of Poincaré beams is vector beams,36 which contain spatially variable linear polarization states. These beams can be produced by using a variable-fast-axis half-wave plate made of cellophane tape,37 or its liquid-crystal counterparts, the radial- or z-polarizer, and 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 has 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 properties41 that can be used as a topic of experimentation in the undergraduate laboratory in 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; the results are shown in Sec. IV. In two appendices, we give details on the programming of the SLM and in alignment procedures. A third  Appendix contains a discussion of equipment costs and experiments performed with inexpensive wave plates and polarizers.

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 of elliptical 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 provide 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 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 = E 0 ( cos α e ̂ H + sin α e i 2 δ e ̂ V ) e i ( k z ω t ) = E 0 e ̂ e i ( k z ω t ) ,
(1)
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 e ̂ H and e ̂ V. As can be seen in Eq. (1), we specify the vectorial part of the field—the polarization—by e ̂. Linear polarization states aligned along the H and V (i.e., x and 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 2 δ = ± 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 ( e ̂ R) when the field rotates clockwise when looking into the beam of light (along the negative z-direction), for which 2 δ = + π / 2. Conversely, left-handed circular polarization ( e ̂ L) corresponds to a counter-clockwise rotation of the field, for which 2 δ = π / 2. We formalize these definitions with the relations2 (and ignoring an overall phase)
e ̂ R = 1 2 ( e ̂ H i e ̂ V )
(2)
and
e ̂ L = 1 2 ( e ̂ H + i e ̂ V ) .
(3)
For completeness, we define two other useful basis states, those with diagonal linear polarization (D) and anti-diagonal linear polarization (A)
e ̂ D = 1 2 ( e ̂ H + e ̂ V )
(4)
and
e ̂ A = 1 2 ( e ̂ H e ̂ V ) .
(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 this question is non-trivial, 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 straightforward, as we now demonstrate.

It is useful to describe the Poincaré sphere, shown in Fig. 1(a), before we present the representation in terms 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 right-handed ellipses and, correspondingly, points in the southern hemisphere represent left-handed ellipses. The polar angle is defined as 2 χ and the azimuthal angle is 2 θ.

Fig. 1.

(a) (Color online) Poincaré sphere representing all the states of polarization; (b) dependence of ellipticity and orientation on the half polar and azimuthal angles of the sphere. Right-handed states occur for 0 χ < π / 4 and left-handed states occur for π / 4 < χ π / 2.

Fig. 1.

(a) (Color online) Poincaré sphere representing all the states of polarization; (b) dependence of ellipticity and orientation on the half polar and azimuthal angles of the sphere. Right-handed states occur for 0 χ < π / 4 and left-handed states occur for π / 4 < χ π / 2.

Close modal
The ellipticity of the polarization state is given by45 
ϵ = ± b a = tan ( π / 4 χ ) ,
(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 e ̂ R and e ̂ L, the general equation of the state of polarization is given by
e ̂ = cos χ e ̂ R + sin χ e i 2 θ e ̂ L .
(7)
Note the simplicity of the description: the relative magnitude of the circular components exclusively determines the ellipticity (related to χ) and their relative phase exclusively determines 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.

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 e ̂ H and e ̂ V 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 2 δ is the corresponding azimuthal angle, with 2 δ = 0 being on the equatorial plane of the (R, L) polar sphere (see Fig. 2).

Fig. 2.

Poincaré sphere showing two spherical triangles: WPY of angular sides 2 α (WP, the spherical hypotenuse), π / 2 2 χ (PY), and 2 θ (YW); and UPZ of angular sides 2 χ (UP, the spherical hypotenuse), π / 2 2 α (PZ), and π / 2 2 δ (ZU).

Fig. 2.

Poincaré sphere showing two spherical triangles: WPY of angular sides 2 α (WP, the spherical hypotenuse), π / 2 2 χ (PY), and 2 θ (YW); and UPZ of angular sides 2 χ (UP, the spherical hypotenuse), π / 2 2 α (PZ), and π / 2 2 δ (ZU).

Close modal

Let us look at particular cases. When 2 δ = π / 2 or 3 π / 2, ellipses have semi-major axes that are either horizontal (for 2 α < π / 2) or vertical (for 2 α > π / 2), regardless of ellipticity (see Fig. 2). Similarly, when 2 α = π / 2 the semi-major axes are either diagonal (for 2 δ < π / 2 and 2 δ > 3 π / 2) or anti-diagonal (for π / 2 < 2 δ < 3 π / 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 ( 2 α ) = cos ( 2 θ ) sin ( 2 χ )
(8)
and
cos ( 2 χ ) = sin ( 2 α ) sin ( 2 δ ) .
(9)
We can use these relations to transform the polarization state of Eq. (1), with α and δ given, to the circular basis [Eq. (7)]
cos χ = 1 + sin ( 2 α ) sin ( 2 δ ) 2 ,
(10)
sin χ = 1 sin ( 2 α ) sin ( 2 δ ) 2 ,
(11)
and
2 θ = cos 1 [ cos ( 2 α ) 1 sin 2 ( 2 α ) sin 2 ( 2 δ ) ] .
(12)

For example, suppose α = π / 8 and 2 δ = 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°. Using the above relations, we get χ = 30 °, which gives an ellipticity ϵ = tan 15 ° = 0.27; and 2 θ = 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.

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 RL-axis is
s 3 = cos ( 2 χ ) = cos 2 χ sin 2 χ = ( I R I L ) / I 0 ,
(13)
where IR and IL are the intensities of the light in the right and left circular states, and
I 0 = I R + I L = I H + I V = I D + I A .
(14)
Likewise, IH, IV, ID, and IA are 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 HV polar axis
s 1 = cos ( 2 α ) = cos 2 α sin 2 α = ( I H I V ) / I 0 .
(15)
If the polar angle in the sphere with the DA polar axis is β (angle subtended by points P and T in Fig. 2), then
s 2 = cos ( 2 β ) = cos 2 β sin 2 β = ( I D I A ) / I 0 .
(16)
Once the normalized Stokes parameters are obtained through measurements, we can then derive the polarization ellipse parameters defined earlier (see also Fig. 1) as
χ = 1 2 cos 1 ( s 3 s 1 2 + s 2 2 + s 3 2 )
(17)
and
θ = 1 2 tan 1 ( s 2 s 1 ) .
(18)
These definitions account for the case when the light is not in a pure state, or having an unpolarized component, for which s 1 2 + s 2 2 + s 3 2 < 1.

The presentation given in this section can be incorporated into one or two days of class in 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 

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 that 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.

This laboratory experience can be presented at several levels. We have used it as a semester-long capstone-project level, as a summer experience to introduce the student to more advanced topics, and more recently as an advanced laboratory in our Physical Optics course. The starting point for these experiences involve the student and the faculty member working first on understanding the theory and formalism as presented in Sec. II. 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, this experience can be introduced as a several-week-long advanced lab experiment.

Beyond polarization, this laboratory introduces spatial modes of light. In particular, Laguerre-Gauss modes, which are a requirement for 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 

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) modes, which have been subject of previous articles in this journal.42,43,49 There are numerous formal studies on these modes,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
L G 0 = A r | | e i ϕ G W ,
(19)
parameterized by , known as the topological charge. The variables r and ϕ are the transverse polar coordinates. A is a normalization constant given by
A = ( 2 | | + 1 π | | ! ) 1 2 ( 1 w ) | | + 1 .
(20)
Here, w is a parameter representing the half-width of the mode. The defining characteristic of these modes is the term e i ϕ, which denotes a phase that varies with the angular coordinate. Thus, is the number of times that the phase advances by 2 π in one turn about the center of the mode. The factor r | | gives the mode the distinctive doughnut shape for 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 / w 2 .
(21)
The function W accounts for propagation phase effects, such as the z-dependence of the phase, the radius of curvature of the expanding wavefront, and an evolving phase known as the Gouy phase. This function is given by
W = e i [ k z + k r 2 / ( 2 R ) φ ] ,
(22)
where k is the wave vector, R is the radius of curvature of the wavefront, and φ is the Gouy phase, given by50,
φ = ( | | + 1 ) tan 1 ( z / z R ) ,
(23)
where zR is the Rayleigh range. In general, LG modes are multi-ringed. In their most general form they are labeled as L G p , with the additional parameter p specifying 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 left-circular 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 
U ( r , ϕ ) = 1 2 ( L G 0 e ̂ R + L G 0 0 e i 2 γ e ̂ L ) ,
(24)
where 2 γ 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 ) = U 0 ( r , ϕ ) [ cos χ ( r ) e ̂ R + sin χ ( r ) e i 2 θ ( ϕ ) e ̂ L ] ,
(25)
where U 0 ( r , ϕ ) is a scalar function. We are interested in the factor in square brackets, which describes the vectorial part of the state with
χ ( r ) = tan 1 ( A 0 A r )
(26)
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)–(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 Sec. III B, 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.
Fig. 3.

Patterns showing four notable cases of Poincaré modes. According to Eq. (24), these states correspond to (a) = 1, (b) = 1; (c) = 2 with γ = 0, and (d) = 2 with γ = π / 2. The false color encodes the orientation of the semi-major axis of the ellipse; the saturation encodes the intensity of the mode.

Fig. 3.

Patterns showing four notable cases of Poincaré modes. According to Eq. (24), these states correspond to (a) = 1, (b) = 1; (c) = 2 with γ = 0, and (d) = 2 with γ = π / 2. The false color encodes the orientation of the semi-major axis of the ellipse; the saturation encodes the intensity of the mode.

Close modal

The modes shown in Fig. 3 are important in characterizing modes with spatially variable polarization. These modes 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: 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 Figs. 3(a) and 3(b) are known, respectively, as lemon and star.53–55 The patterns shown 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.

A schematic of the experimental setup is shown in Fig. 4(a), and a photo 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 is linearly polarized along the horizontal 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. 4.

(a) Schematic of the apparatus to generate Poncaré beams. Optical components include mirrors (M), a spatial light modulator (SLM) with inset 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.

Fig. 4.

(a) Schematic of the apparatus to generate Poncaré beams. Optical components include mirrors (M), a spatial light modulator (SLM) with inset 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.

Close modal
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 inset showing the paths of the beams.

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 inset showing the paths of the beams.

Close modal

The zero-order and first-order beams were reflected off two large (>2-in.) rectangular mirrors (M3 and M4) and passed through another horizontal linear polarizer (P2; again, for cleaning up the polarization). The zero-order 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-in. collar of a 1/2-in. 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 significant 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 horizontally polarized first-order beam and reflected the vertically polarized zero-order beam. The SLM, M5, and the PBS formed a polarization interferometer. A critical part of the setup 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 in 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 half-wave 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 2 ft . × 4 ft . optical breadboard. The camera was just outside the breadboard, so we recommend at least a 2 ft . × 5 ft . 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-h lab period.

There are several ways to detect an arbitrary state of polarization. The simplest uses two elements: a quarter-wave 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.

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.5,53 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 ϵ goes through a half-wave plate with fast axis forming h 2 = θ / 2 with the horizontal; this rotates the polarization so that the semi-major 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 h 3 = ( 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 = 2 h 2 ,
(28)
for | h 2 | π / 2, and
ϵ = tan 2 h 3 ,
(29)
for | h 3 | π / 8. Note that Eqs. (28) and (29) do not hold outside the bounds given. For other angles, they must be modified slightly, and this may lead to confusion.

An alternative method of detecting the polarization state of the entire beam involves 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 the imread instruction), and using Eqs. (13) and (15)–(18) we found the state of polarization (θ and ϵ) at 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 states of polarization. Polarizer P4 has two settings: (i) as a filter (to transmit the state) and (ii) as a null (to block the state).

State h2 (deg) q2 (deg) h3 (deg) p4 (filter/null) (deg)
H  90  0/90 
D  22.5  90  0/90 
V  45  90  0/90 
A  67.5  90  0/90 
R  90  22.5  0/90 
L  90  + 22.5  0/90 
State h2 (deg) q2 (deg) h3 (deg) p4 (filter/null) (deg)
H  90  0/90 
D  22.5  90  0/90 
V  45  90  0/90 
A  67.5  90  0/90 
R  90  22.5  0/90 
L  90  + 22.5  0/90 

The experiments presented here can be performed without the SLM. We have done the same experiments using a plastic binary forked grating43 (see Subsection 4 of  Appendix A 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 would be in transmission mode.

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 with 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 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 p 4 = 90 ° for state nulling measurements. The student was asked to map (that is, to draw carefully on 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 and 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 small 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 h 2 = 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 reaches it when h 3 = 22.5 °, which corresponds to ϵ = 1 or left-circular polarization. Indeed, when h 3 = 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 h 3 = 0 the null state has ϵ = 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 to the forefront of the lab an understanding of the concept of elliptical polarization.

Fig. 6.

Case with = 1. 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.

Fig. 6.

Case with = 1. 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.

Close modal

A second part of the experiment entailed finding the exact pattern and comparing it to the nulling measurements. Here, we needed to set p 3 = 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 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 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 of polarization at the point where they were drawn. These ellipses complemented the false color in understanding the image. This program and sample files are available as supplementary material.57 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, and time devoted to the experiment, 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.

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.

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.

Close modal

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.

When we combined the higher-order = 2 (LG) mode 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.

Fig. 8.

Case with = 2. 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. 8.

Case with = 2. 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.

Close modal

As 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 h 3 = 22.5 °, as seen in Fig. 8 for the case of = 2.

Fig. 9.

Image obtained by blocking the state of polarization of an elliptical state of ellipticity ϵ = + 0.58 and orientation θ = 55 °.

Fig. 9.

Image obtained by blocking the state of polarization of an elliptical state of ellipticity ϵ = + 0.58 and orientation θ = 55 °.

Close modal

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 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 students 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.

This work was supported by Colgate University, the Schlichting Fellowship, and NSF Grant No. PHY-1506321.

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. Because it has 8-bit resolution, we created images in gray scale. Its shortcoming is that it does not encode the full 2 π phase (the vendor claims 0 0.8 π 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 are, respectively, the number of columns and rows of pixels of the SLM, and the third dimension specifies each color. Once we calculated the values of each matrix element (see below), the program output the matrix into a file. This program is available as supplementary material.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. Hence, the phase that is encoded along the rows of pixels has the cross 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 L G 0 and a plane wave forming an angle Φ, the phase difference at the SLM is60 
Θ = tan 1 ( y x ) 2 π x sin Φ λ ,
(A1)
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., those forming vertical lines). The fringe separation corresponds to points separated by Δ Θ = 2 π, or
Δ x = λ sin Φ .
(A2)
If we turn the argument around, a grating with fringe separation Δ x will produce a first-order diffraction at an angle Φ away from the zero order. If we set 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 Δ 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 2 π ) / 2 π .
(A3)
Figure 10(a) shows the pattern that is obtained when programming the SLM by combining Eqs. (A1) and (A3). For sake of clarity, we increased the fringe separation by a factor of 10.
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.

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.

Close modal
2. Amplitude modulation
The grating described in Subsection 1 of  Appendix A 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 ( r 2 + r 2 w 2 ) ,
(A4)
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 = g f .
(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.
3. Phase-blaze correction

There is one more correction that we can make. Subsections 1 and 2 of  Appendix A described a method that works well with the ideal SLM, where in changing from h = 0 to h = 1 results in a 2 π phase shift. However, the commercial SLM that we had only went as far as 0.8 π. 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 2 π / fringe. Since we do not have the full phase range, we instead program 40% of each fringe from 0 to 0.8 π 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. 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 Subsection 1 of  Appendix A but instead of defining g as in Eq. (A3), we define it as
g bin = { 0 , if   ( Θ mod 2 π ) < π 1 , if   ( Θ mod 2 π ) π .
(A6)
The resulting binary grating is shown in Fig. 10(d). It is one of the options of the program we have made available.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.

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.

1. 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 min.

2. 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, as shown in 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, as shown in 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 min.

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. This step includes a tilting piece of glass (G) and placing polarizer P5. Insets 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.

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. This step includes a tilting piece of glass (G) and placing polarizer P5. Insets 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.

Close modal
3. 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 min.

4. 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 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 image-inset 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 Δ 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-inset 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 min (depending on experience).

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 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.

Table II.

Prices of components needed for the experiment.

Item Price (each) ($) Quantity
2 × 5 optical breadboard  1900 
SLM (Cambridge Correlators)  1200 
Polarized HeNe  >850 
1 / 2 (12.7 mm) PBS  190 
Prism mount  80 
Quarter-wave plate  270/3 
Half-wave plate  270/3 
Polarizer  90/6 
Rotation Mount for 1   90 
USB CMOS Camera  350 
Mounted mirrors  55–150 
Translation stage  250 
Iris  40 
Item Price (each) ($) Quantity
2 × 5 optical breadboard  1900 
SLM (Cambridge Correlators)  1200 
Polarized HeNe  >850 
1 / 2 (12.7 mm) PBS  190 
Prism mount  80 
Quarter-wave plate  270/3 
Half-wave plate  270/3 
Polarizer  90/6 
Rotation Mount for 1   90 
USB CMOS Camera  350 
Mounted mirrors  55–150 
Translation stage  250 
Iris  40 
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.

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.

Close modal

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 of the camera was disabled.

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Supplementary Material