3Stokes Parameters and the Poincaré Sphere

3.1The Description of Polychromatic Light

Unpolarized light and partially polarized light are the dominant forms of light in the universe. Starlight is almost always unpolarized. This unpolarized light is incoherent; it does not form stable interference patterns. Monochromatic light, the topic of Chapter 2, which mostly comes from lasers, is the exception. The polarization vectors and Jones vectors from Chapter 2 do not describe partially polarized light.

The Stokes parameters are a method for characterizing the polarization properties of light beams, suitable for unpolarized, partially polarized, and polarized beams. The beams described by Stokes parameters can be incoherent or coherent light beams.1–3 Incoherent light obeys the same basic principles as coherent light during interference and diffraction; however, because of incoherent light’s polychromatic nature, the details are more complex. Polychromatic light refers to light of multiple wavelengths, light that is not monochromatic. Figure 3.1 (left) shows the electric field amplitude of four monochromatic waves of different frequencies. When these fields are added, the resulting amplitude, shown in Figure 3.1 (right), has a random appearance with peaks of different heights and irregular periods. This is the character of polychromatic light. One difference between monochromatic and polychromatic light is that monochromatic light cannot be unpolarized. Monochromatic light has a single frequency; it is cosinusoidal. Thus, monochromatic light is trapped in a single polarization state for eternity.

Figure 3.1Plot of the electric field amplitude versus time for a polychromatic light waveform (right) formed from the addition of four monochromatic waves of different wavelengths (left).

In this chapter, the Stokes parameters and their mathematical properties are presented, and partially polarized and unpolarized light are discussed. The Stokes parameter properties of linear, circular, and elliptically polarized light are developed. The transformation between Stokes parameters and Jones vectors is presented. The Stokes parameter’s non-orthogonal coordinate system is described; this unusual coordinate system correctly handles the superposition and interference of incoherent light. Then, the Poincaré sphere, a graphical representation of the polarization state related to Stokes parameters, is developed. The addition of incoherent light waves and their interference properties are covered in Chapter 4.

The four Stokes parameters are often referred to as Stokes vectors. The Stokes parameters are not true vectors since they do not transform or rotate as vectors. The Stokes parameters are added as vectors to simulate the addition of incoherent light beams. The Stokes parameters are also operated on by Mueller matrices using matrix–vector multiplication. Thus, it is acceptable and common to refer to the Stokes parameters as Stokes vectors, but one should remember that they are not true vectors.

3.2Phenomenological Definition of the Stokes Parameters

The four Stokes parameters,4S0, S1, S2, and S3, are defined in terms of six polarized flux measurements performed with ideal polarizers as

$$\mathbf{S}=\left(\begin{array}{l}{S}_0\\ S_1\\ S_2\\ S_3\end{array}\right)=\left(\begin{array}{l}{P}_H+P_V\\ P_H-P_V\\ P_{45}-P_{135}\\ P_R-P_L\end{array}\right)=\left(\begin{array}{l}I\\ Q\\ U\\ V\end{array}\right).$$3.1

3.1

The Stokes parameters are usually written in vector-form S with four real-valued elements. S0 is the total irradiance of the beam. S1 is the horizontal (0°) polarized flux component (PH) minus the vertical (90°) flux component (PV). When S1 = 0, the flux measured through a horizontal linear polarizer and that measured through a vertical linear polarizer are equal. Thus, S1 measures the excess of horizontal polarization over vertical polarization and is negative if PV > PH. Similarly, S2 is the 45° flux (P45) minus the 135° flux (P135). Finally, S3 measures the difference of the right (PR) minus left (PL) circularly polarized flux. The Stokes parameters are frequently labeled I, Q, U, and V, particularly in remote sensing and astronomy.

Table 3.1 lists the Stokes parameters for the six basis polarization states; each has been normalized to a flux (S0) of one. The normalized Stokes parameters for unpolarized light are appended to the end.

Table 3.1Basis Polarization States and Their Stokes Parameters

Type of Polarization	Symbol	Stokes Parameters
Horizontal linearly polarized	H	(1, 1, 0, 0)
Vertical linearly polarized	V	(1, −1, 0, 0)
45° Linearly polarized	45	(1, 0, 1, 0)
135° Linearly polarized	135	(1, 0, −1, 0)
Right circularly polarized	R	(1, 0, 0, 1)
Left circularly polarized	L	(1, 0, 0, −1)
Unpolarized light	U	(1, 0, 0, 0)

Example 3.1Stokes Parameters

A beam of light is measured to have the following Stokes parameters, $\mathbf{S}=\left(\begin{array}{c}{S}_0\\ S_1\\ S_2\\ S_3\end{array}\right)=\left(\begin{array}{c}6\\ 4\\ 2\\ -4\end{array}\right).$ Find the six fluxes that define the Stokes parameters in Equation 3.1.

Since S0 = PH + PV and S1 = PH − PV, $P_{\text{H}}=\frac{S_0+S_1}{2}\text{and}{P}_{\text{V}}=\frac{S_0-S_1}{2}$, and similar equations are obtained for the other four fluxes. Thus, the six flux measurements, expressed as a flux vector P, are

$$\mathbf{P}=\left(\begin{array}{l}{P}_H\\ P_V\\ P_{45}\\ \begin{array}{l}{P}_{135}\\ P_R\\ P_L\end{array}\end{array}\right)=\left(\begin{array}{l}5\\ 1\\ 4\\ \begin{array}{l}2\\ 1\\ 5\end{array}\end{array}\right).$$

3.3Unpolarized Light

Unpolarized light has a randomly varying polarization state with no preference for any particular state. Sunlight is unpolarized. The atoms emitting in the sun’s photosphere have no preferred orientation and are constantly buffeted by other atoms moving at high velocity. For unpolarized light, any ideal polarizer will transmit one-half the light. If a linear polarizer is rotated, the exiting light has a constant flux.

Following the electric field of unpolarized light, as in Figure 3.2, the tip of the vector randomly moves around the origin, sometimes clockwise, sometimes counterclockwise, sometimes nearly linearly. Each small segment of the arc is nearly elliptical, but the ellipse parameters are continuously evolving. Unpolarized light is generated when a light source emits photons with random distribution of polarization states. Each photon’s polarization is uncorrelated with the photons previously emitted by the emitting atom and by its neighbors. The polarization ellipse of the beam is constantly evolving.

Figure 3.2An example of electric field patterns for unpolarized light. The tip of the electric field traces patterns with a randomly evolving polarization ellipse. This pattern was generated to simulate sunlight, where the vector field changes its local ellipse parameters in less than an optical period.

3.4Partially Polarized Light and the Degree of Polarization

The degree of polarization (DoP) is a metric for Stokes parameters that characterizes the randomness of a polarization state. The degree of polarization is defined as

$$DoP(\mathbf{S})=\frac{\sqrt{S_1^2+S_2^2+S_3^2}}{S_0},0\le DoP\le 1.$$3.2

3.2

A beam with a DoP of 1 is polarized. The light is in a single polarization state and the light can be completely blocked by the matched polarizer. For such a completely polarized beam, the Stokes vector elements obey the identity

$$S_0^2=S_1^2+S_2^2+S_3^2.$$3.3

3.3

Conversely, when the DoP is 0, the beam is unpolarized. Then, all ideal polarizers will block one-half of the beam, because the light has no preference for any particular polarization state.

Partially polarized light has a randomly changing polarization state but has a tendency toward a particular polarization state; there is an overall average polarization state. In partially polarized light, some polarization ellipses are more likely than others. Figure 3.3 shows example electric field patterns for circularly polarized (left) and partially circularly polarized beams with decreasing degree of polarization.

Figure 3.3Examples of time-varying electric fields with evolving polarization ellipses for partially circularly polarized light with DoCP of 1.0, 0.95, 0.75, and 0.5 (from left to right).

The polarized flux

$$\sqrt{S_1^2+S_2^2+S_3^2}$$3.4

3.4

is the amount of flux that is polarized. Thus, the DoP is the ratio of the polarized flux to the total flux S0. With a beam of linearly polarized light, a linear polarizer can be oriented to transmit the entire beam, and the orthogonal polarizer, rotated by 90°, blocks the entire beam. Partially linearly polarized light has a random distribution but a preference for a particular linearly polarized state. The degree of linear polarization, DoLP,

$$DoLP(\mathbf{S})=\frac{\sqrt{S_1^2+S_2^2}}{S_0},$$3.5

3.5

describes the extent to which the polarization ellipse distribution tends to be linear, or equivalently the extent to which the electric field is confined in one plane. When the DoLP is equal to 1, the light is linearly polarized. A linear polarizer rotated in this beam will transmit all the light when aligned with the plane of polarization. As the DoP or DoLP decreases from 1 to 0, the polarization ellipse becomes more random as shown in Figure 3.4 for 45° linearly polarized light.

Figure 3.4Examples of time-varying electric fields with randomly evolving polarization ellipses for partially linearly polarized light oriented at 45° with DoLP of 1.0, 0.8, 0.6, 0.4, 0.2, and 0.

The DoLP also relates to the flux modulation that occurs when a polarizer rotates in a beam, where PMax and PMin are the maximum and minimum transmitted fluxes,

$$DoLP(\mathbf{S})=\frac{P_{\text{Max}}-P_{\text{Min}}}{P_{\text{Max}}+P_{\text{Min}}}=\frac{\sqrt{S_1^2+S_2^2}}{S_0},$$3.6

3.6

as is shown in Figure 3.5. The orientation ψ of the polarization is the angle of the polarizer for PMax. Both a circularly polarized beam and an unpolarized beam have a DoLP of 0.

Figure 3.5The transmittance through a rotating polarizer in a partially linearly polarized beam results in modulation of the flux proportional to the degree of linear polarization. Unpolarized light is unmodulated. The different lines shown correspond to light with DoLP of 0 (unpolarized), 0.2, 0.4, 0.6, 0.8, and 1 (polarized). The polarized case is described by Malus’ law.

Partially linearly polarized light often arises when unpolarized sunlight scatters from surfaces; typically more light scatters polarized in the plane perpendicular to the plane of incidence (s-polarized light as described in Section 1.7.1 and Figure 1.32) than scatters polarized in the plane of incidence (p-polarized light). The polarization ellipse of the scattered light is still randomly varying in time and space, but scattering has increased the fraction of the s-component. As a result, sunlight or incandescent light scattered from surfaces is usually partially polarized, not unpolarized. When unpolarized light reflects from a smooth water surface, the reflected beam becomes partially polarized. Figure 3.6 shows the DoP for the reflected light, which can be calculated using the Fresnel equations (Section 8.3.3).

Figure 3.6The degree of polarization of unpolarized incident light after specularly reflecting from a smooth water surface becomes increasingly linearly polarized until 57°, the Brewster angle, where the reflected light is completely polarized in the horizontal direction. Beyond the Brewster angle, the degree of polarization decreases to grazing incidence where the reflected light is unpolarized, as it is at normal incidence.

The angle of polarization, AoP, for linearly polarized light is the angle of the electric field oscillation measured counterclockwise from the x-axis in radians,

$$AoP(\mathbf{S})=\frac{1}{2}\text{arctan}\frac{S_2}{S_1}=\frac{1}{2}\text{arctan}2(S_1,S_2).$$3.7

3.7

The function arctan2 is the form of the arctan function that takes the numerator (S2) and the denominator (S1) as two separate arguments, so that it can return values over twice the range, −π to π, of the conventional arctan function by using the signs of the numerator and denominator separately. For elliptically polarized light, AoP is the angle of the major axis of the polarization ellipse. The degree of circular polarization, DoCP,

$$DoCP(\mathbf{S})=\frac{S_3}{S_0},$$3.8

3.8

characterizes the fraction of the polarized flux that is circularly polarized and expresses the sign of the helicity. DoCP = 1 indicates right circularly polarized light; DoCP = −1 indicates left circularly polarized, and DoCP = 0 indicates linearly polarized, partially linearly polarized, or unpolarized light. Natural light in the environment generally has a DoCP near 0.

Example 3.2Stokes Image of a Building

Figure 3.7 shows an intensity image, S0, of the University of Arizona’s Optical Sciences building in 660-nm light taken with an imaging polarimeter, GroundMSPI.5,6 GroundMSPI accurately measures the linearly polarized Stokes parameters (S0, S1, S2). Figure 3.8 shows the Stokes parameter images S1 and S2 in grayscale while Figure 3.9 shows the same data in false color.

Figure 3.7An intensity image of the University of Arizona’s Meinel Optical Sciences building. The entire north face of the building is windows tilted at various angles. The top of the football stadium is seen at the upper right. High intensity is white, and low intensity is black.

Figure 3.8The S1 (first and third panel) and S2 (second and fourth panel) Stokes parameters for the image of Figure 3.7. In panels 1 and 2, gray is zero, and brighter regions have positive S1 or S2 and darker regions have negative S1 and S2. In the third and fourth panels, black is zero, red is positive, and green is negative. The Stokes parameters clearly vary with the window angle.

Figure 3.9The DoP image (left) for the Meinel building image shows large variations. The structures in the image, windows, trees, lamppost, sidewalks, and so on, are easily identified, but the contrast mechanisms are completely different from the intensity image. The AoP image (middle) shows that the polarization orientation for most of the image (red and purple) doesn’t vary much, but some regions (blue) show significantly different orientation. Regions of low DoP tend to be much noisier in AoP. In the quiver plot (right), an array of line segments show DoP as line length and AoP as line orientation. The trees and roof structures are seen to have low DoP. Again, the variation of DoP and AoP from window pane to window pane is striking.

3.5Spectral Bandwidth

Figures 3.2 through 3.4 show unpolarized and partially polarized light as randomly evolving ellipses. This section examines how rapidly the ellipses evolve.

The spectrum of a light beam refers to the range of wavelengths in the beam and, in particular, to plots of the spectral density, flux units per nanometer, of the light. The spectrum of visible light can be made visible with a prism, or measured with a spectrometer. Spectral bandwidth Δλ refers to the range of wavelengths in a spectrum and is frequently specified in terms of the full width at half maximum of the light. The spectral bandwidth of the light determines how rapidly the polarization ellipse can change in partially polarized light. True monochromatic light has a spectral bandwidth of zero, and the ellipse does not change. Laser light is quasi-monochromatic, the spectral bandwidth is close to, but not equal to, zero. Some ultra-long coherence length lasers achieve a spectral bandwidth Δλ/λ < 10−9.

For unpolarized light, different wavelengths can have different polarization states, and as these evolve in and out of phase, the polarization state fluctuates. For sunlight and other broadband visible light, these polarization changes occur so rapidly, with variations on the order of 10−14 s, that light detectors cannot follow the rapid fluctuations of polarization state. In Figure 3.10 (left), a simulation of the electric field of unpolarized sunlight is shown. The electric field vector traces a curve that can be described as being elliptical for short segments, but these local ellipse parameters are continuously changing. For sunlight, the elliptical parameters change significantly in less than an optical period, and the ellipse arcs appear random. The rate of change is related to the spectral bandwidth of the light. For nearly monochromatic light, the ellipse parameters for unpolarized light change rather slowly. Figure 3.10 (center) shows the evolving electric field for a spectral bandwidth Δλ/λ = 0.1 while Figure 3.10 (right) shows the case of Δλ/λ = 0.05. Such small spectral bandwidths produce an evolving ellipse where the change in ellipse is small from period to period; the unpolarized beam’s ellipse parameters are nearly constant for a few periods. The center and right examples look partially polarized (i.e., not randomized over all states), but only because a short portion of the waveform is shown. Over a long period of time, these functions will fully randomize. Figure 3.11 (four figures from the left) shows the state of Figure 3.10 (right) for four consecutive time periods of equal duration, demonstrating the evolution of the polarization ellipse through many different regions of polarization state. Figure 3.11 (right) shows the state over a longer period, demonstrating the overall randomization of the state.

Figure 3.10Unpolarized light has a randomly evolving polarization ellipse. For sunlight (left), the vector field changes its local ellipse parameters in less than an optical period. For a small spectral bandwidth, the polarization ellipse for unpolarized light evolves more slowly, remaining in a similar ellipse over several periods. Two unpolarized examples with the spectral bandwidth Δλ/λ = 0.1 (center) and Δλ/λ = 0.05 (right) show typical polarization ellipse evolution.

Figure 3.11The randomly evolving polarization ellipse for the state of Figure 3.10 (right) in four (left, second, third, and fourth) consecutive time periods of equal duration. (Right) The polarization ellipse graphed over a longer period of time shows it approaching a random unpolarized distribution.

Polarized light (DoP = 1) with a spectral bandwidth also has an electric field with a time-varying polarization ellipse. As the ellipses for different wavelengths add, the ellipse parameters evolve. Figure 3.12 shows examples of the field for circularly polarized light for several Δλ/λ. For monochromatic light, Δλ/λ = 0, the ellipse (circle) is fixed. As the spectral bandwidth increases, the ellipse changes more rapidly, but always remains close to its basic circular form. In these simulations, every wavelength has the same circular state. These figures are generated by adding circular functions of different frequencies. These fully polarized figures could easily be mistaken for partially polarized light such as Figure 3.3. A Fourier transform of the x- and y-components of the functions in Figure 3.12 would reveal that each spectral component had a circularly polarized Jones vector.

Figure 3.12Circularly polarized light (DoP = 1) of spectral bandwidth 0%, 4%, 8%, 12%, 16%, and 20%.

3.6Rotation of the Polarization Ellipse

Next, the mathematical operations commonly performed with Stokes parameters are developed. When a polarization state is rotated relative to the coordinate axes by ψ, as in Figure 3.13, the Stokes parameters transform as

Figure 3.13Rotation of a polarization state counterclockwise through an angle of ψ.

$${\mathbf{S}}_{\text{ER}}={\mathbf{R}}_{\text{M}}(\psi )\cdot \mathbf{S}=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& \text{cos}2\psi & -\text{sin}2\psi & 0\\ 0& \text{sin}2\psi & \text{cos}2\psi & 0\\ 0& 0& 0& 1\end{array}\right)\cdot \left(\begin{array}{l}{S}_0\\ S_1\\ S_2\\ S_3\end{array}\right)=\left(\begin{array}{l}{S}_0\\ S_1\text{cos}2\psi -S_2\text{sin}2\psi \\ S_1\text{sin}2\psi +S_2\text{cos}2\psi \\ S_3\end{array}\right).$$3.9

3.9

SER indicates the Stokes parameters for a rotated ellipse. Hence, Stokes parameters are rotated with respect to their coordinate system by the matrix RM(ψ),

$${\mathbf{R}}_{\text{M}}(\theta )=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& \text{cos}2\psi & -\text{sin}2\psi & 0\\ 0& \text{sin}2\psi & \text{cos}2\psi & 0\\ 0& 0& 0& 1\end{array}\right).$$3.10

3.10

In Section 6.5, this matrix is also shown to be the rotation matrix for Mueller matrices. The 2ψ occurs because the Stokes S1 and S2 axes are only 45° apart (see Section 3.12). A rotation of 180° returns a “Stokes vector” to its original state: 0° horizontally polarized light rotates into 180° horizontally polarized light; thus, RM (180°) must equal the identity matrix. By definition, ψ is positive for rotations that initially move an x-component toward the +y-axis, rotations that are counterclockwise looking into the beam.

Example 3.3Stokes Parameter Rotation by 45°

To rotate the Stokes parameters describing a polarization state by + 45°, the transformation is

$${\mathbf{S}}_{\text{ER}}={\mathbf{R}}_{\text{M}}(45°)\cdot S=\left(\begin{array}{rrrr}\hfill 1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill -1& \hfill 0\\ \hfill 0& \hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 1\end{array}\right)\left(\begin{array}{l}{S}_0\\ S_1\\ S_2\\ S_3\end{array}\right)=\left(\begin{array}{r}\hfill S_0\\ \hfill -S_2\\ \hfill S_1\\ \hfill S_3\end{array}\right).$$3.11

3.11

The S1 and S2 elements are reversed and the S2 element changes sign. The circular polarization component S3 is unchanged by rotation.

3.7Linearly Polarized Stokes Parameters

The normalized Stokes parameters LPS(ψ) for linearly polarized light oriented at an angle ψ, measured counterclockwise from zero, are readily determined using the Mueller rotation matrix (Equation 3.9) operating on horizontally polarized Stokes parameters H as

$$\mathbf{L}{\mathbf{P}}_{\text{S}}(\psi )={\mathbf{R}}_{\text{M}}(\psi )\cdot \mathbf{H}=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& \text{cos}2\psi & -\text{sin}2\psi & 0\\ 0& \text{sin}2\psi & \text{cos}2\psi & 0\\ 0& 0& 0& 1\end{array}\right)\cdot \left(\begin{array}{l}1\\ 1\\ 0\\ 0\end{array}\right)=\left(\begin{array}{l}1\\ \text{cos}2\psi \\ \text{sin}2\psi \\ 0\end{array}\right).$$3.12

3.12

Again, notice that the 2ψ dependence is necessary, so that light polarized at 0° and 180° have the same Stokes parameters. Thus, it is seen that the Stokes parameters do not transform as vectors and are not true vectors.

3.8Elliptical Polarization Parameters

The Stokes parameters S for a partially polarized beam can be mathematically represented as a sum of a completely polarized “Stokes vector” SP and an unpolarized “Stokes vector” SU, which are uniquely related to S as follows:

$$\mathbf{S}={\mathbf{S}}_{\text{P}}+{\mathbf{S}}_{\text{U}}=\left(\begin{array}{l}{S}_0\\ S_1\\ S_2\\ S_3\end{array}\right)=\left(\begin{array}{l}\sqrt{S_1^2+S_2^2+S_3^2}\\ S_1\\ S_2\\ S_3\end{array}\right)+\left(S_0-\sqrt{S_1^2+S_2^2+S_3^2}\right)\left(\begin{array}{l}1\\ 0\\ 0\\ 0\end{array}\right).$$3.13

3.13

The first vector has a DoP of 1 while the second vector is unpolarized (1, 0, 0, 0) with a DoP of 0. Thus, partially polarized light can be treated mathematically as a superposition of polarized and unpolarized light. Although the Stokes parameters are mathematically separated into polarized and unpolarized parts by Equation 3.13, there is not a corresponding polarization element to separate polarized and unpolarized parts of a beam.

The polarization ellipse corresponding to the fully polarized part of the Stokes parameters has the following parameters:

$$\text{Orientation}\text{of}\text{major}\text{axis},\text{azimuth:}\psi =\frac{1}{2}\text{arctan}\left(\frac{S_2}{S_1}\right),$$3.14

3.14

$$\text{Ellipticity:}\epsilon =\frac{b}{a}=\frac{\left|S_3\right|}{\sqrt{S_1^2+S_2^2+S_3^2}+\sqrt{S_1^2+S_2^2}},0\le \epsilon \le 1,$$3.15

3.15

$$\text{Eccentricity:}e=\sqrt{1-{\epsilon }^2},$$3.16

3.16

$$\text{Degree}\text{of}\text{circular}\text{polarization:}DoCP=S_3\text{/}{S}_0,-1\le DoCP\le 1.$$3.17

3.17

The ellipticity ε is the ratio of the minor (b) to the major axis (a) of the corresponding electric field polarization ellipse and varies from 0 for linearly polarized light to 1 for circularly polarized light. The polarization ellipse is alternatively described by its eccentricity, which is zero for circularly polarized light, increases as the ellipse becomes thinner (more cigar-shaped), and becomes one for linearly polarized light. Figure 3.14 shows a series of polarization ellipses (always as electric field amplitudes and never as irradiances) for a set of ellipticities and DoCPs.

Figure 3.14Polarization ellipses with varying ellipticity and degree of circular polarization.

3.9Orthogonal Polarization States

Polarization states are orthogonal when their major axes are 90° apart, their helicities are opposite, and the ellipticities have equal magnitude, as seen in Figure 3.15 and discussed in Section 2.12. For a polarized state (DoP = 1), the states with Stokes parameters S and Sorth,

Figure 3.15An orthogonal pair of polarization ellipses have orthogonal major axes, equal ellipticities, and opposite helicities.

$$\begin{array}{lll}\mathbf{S}=\left(\begin{array}{l}{S}_0\\ S_1\\ S_2\\ S_3\end{array}\right)& \text{and}& {\mathbf{S}}_{\text{orth}}=\left(\begin{array}{l}{S}_0\\ -S_1\\ -S_2\\ -S_3\end{array}\right),\end{array}$$3.18

3.18

are orthogonal polarization states. Orthogonal states are not defined for partially polarized states; for example, unpolarized light does not have an orthogonal polarization state. But the polarized parts of partially polarized beams have orthogonal states.

3.10Stokes Parameter and Jones Vector Sign Conventions

The conversion between Jones vectors and Stokes parameters depends on the sign convention chosen for circular polarization in the two representations. Using the decreasing phase for the electric fields of monochromatic waves (Table 3.2), left circularly polarized light is $(1,i)\text{/}\sqrt{2}$, the y-component is positive complex. This book has adopted the most common sign convention for the Stokes parameters that uses a positive S3 for a right circular polarized component. Thus, right circular polarization is positive in our Stokes parameters and left circular polarization is positive in our Jones vectors. Appropriate minus signs are included in our equations that convert between Jones vectors and Stokes parameters and later in the equations that convert between Jones matrices and Mueller matrices (Section 6.12). For other sign convention choices, checking a few conversions between elliptical and circular states and elliptical and circular retarders can quickly verify the consistency of sign conventions for other choices.

Table 3.2Basis Polarization States and Their Stokes Parameters

Type of Polarization	Symbol	Stokes Parameters	Jones Vector
Horizontal linearly polarized	H	(1, 1, 0, 0)	(1, 0)
Vertical linearly polarized	V	(1, −1, 0, 0)	(0, 1)
45° Linearly polarized	45	(1, 0, 1, 0)	$(1,1)\text{/}\sqrt{2}$
135° Linearly polarized	135	(1, 0, −1, 0)	$(1,-1)\text{/}\sqrt{2}$
Right circularly polarized	R	(1, 0, 0, 1)	$(1,-i)\text{/}\sqrt{2}$
Left circularly polarized	L	(1, 0, 0, −1)	$(1,i)\text{/}\sqrt{2}$
Unpolarized light	U	(1, 0, 0, 0)	Not available

3.11Polarized Fluxes and Conversions between Stokes Parameters and Jones Vectors

For a beam of light with Stokes parameters S, a certain fraction of the flux will be transmitted by a particular polarizer. That flux component is the polarized flux in that particular polarization state. Next, the polarized fluxes for the basis polarization states will be calculated as a step in the conversion between Stokes parameters and Jones vectors. The Jones vector $\mathbf{E}=\left(A_x{e}^{-{\varphi }_x},A_y{e}^{-{\varphi }_y}\right)$ (Section 2.6) describes a monochromatic plane wave propagating along z as

$$\mathbf{E}(\mathbf{r},t)=\mathit{ℛ}\mathit{e}\left\{\widehat{\mathbf{x}}{A}_x\text{exp}\left[i\left(\frac{2\pi z}{\lambda }-\omega t-{\varphi }_x\right)\right]+\widehat{\mathbf{y}}{A}_y\text{exp}\left[i\left(\frac{2\pi z}{\lambda }-\omega t-{\varphi }_y\right)\right]\right\}.$$3.19

3.19

The Jones vector has units of volts per meter while the Stokes parameters have units of watts per meter squared, Sets of Stokes parameters describing fully polarized light are readily converted into equivalent Jones vectors, and vice versa. Equivalent means both the Jones vector and Stokes parameters describe a beam with the same polarization ellipse and flux. The Stokes parameters, however, will not specify the light’s phase. For a partially polarized beam, the polarized part of the flux PP is

$$P_{\text{P}}=\sqrt{S_1^2+S_2^2+S_3^2}.$$3.20

3.20

The horizontal PH (0°) and vertical PV (90°) polarized fluxes of the Jones vector

$$\mathbf{E}=\left(\begin{array}{l}{E}_x\\ E_y\end{array}\right)=\left(\begin{array}{l}{A}_x{\text{e}}^{-i{\varphi }_x}\\ A_y{\text{e}}^{-i{\varphi }_y}\end{array}\right)$$3.21

3.21

are the components of flux transmitted through ideal x- and y-oriented polarizers:

$$P_{\text{H}}=\frac{{\epsilon }_{0}c}{2}{A}_x^2,P_{\text{V}}=\frac{{\epsilon }_{0}c}{2}{A}_y^2.$$3.22

3.22

PH and PV are independent of the phases. Here, c is the speed of light and ε0 is the permittivity of free space. The factor $\frac{{\epsilon }_{0}c}{2}$ is the conversion between amplitude squared, given in (volts per meter)2, and watts/meter2. The 45° and 135° polarized fluxes P45 and P135 expressed in terms of the x and y amplitudes and phases are

$$P_{45}=\frac{{\epsilon }_{0}c}{4}\left(A_x^2+A_y^2+2A_x{A}_y\text{cos}({\varphi }_x-{\varphi }_y)\right),P_{135}=\frac{{\epsilon }_{0}c}{4}\left(A_x^2+A_y^2-2A_x{A}_y\text{cos}({\varphi }_x-{\varphi }_y)\right).$$3.23

3.23

For 45° polarized light, the x- and y-components are in phase or equal ϕx − ϕy = 0, while for 135° polarized light ϕx − ϕy = π. The right and left circularly polarized fluxes PR and PL are associated with the x- and y-phases differing by ±π/2,

$$P_{\text{R}}=\frac{{\epsilon }_{0}c}{4}\left(A_x^2+A_y^2+2A_x{A}_y\text{sin}({\varphi }_x-{\varphi }_y)\right),P_{\text{L}}=\frac{{\epsilon }_{0}c}{4}\left(A_x^2+A_y^2-2A_x{A}_y\text{sin}({\varphi }_x-{\varphi }_y)\right).$$3.24

3.24

To convert a Jones vector into Stokes parameters, these expressions can be applied to the definition of the Stokes parameters in Equation 3.1. A Jones vector E transforms into Stokes parameters as follows:

$$\mathbf{S}(\mathbf{E})=\frac{{\epsilon }_{0}c}{2}\left(\begin{array}{l}{A}_x^2+A_y^2\\ A_x^2-A_y^2\\ 2A_x{A}_y\text{cos}({\varphi }_x-{\varphi }_y)\\ 2A_x{A}_y\text{sin}({\varphi }_x-{\varphi }_y)\end{array}\right).$$3.25

3.25

When the relative phase of the x- and y-components of the electric field is 0 or π, the light is linearly polarized, and the Stokes parameters have non-zero S1 and/or S2 components, but zero S3 components, so sin (ϕx − ϕy) = 0. Similarly, when the relative phase is ±π/2, cos (ϕx − ϕy) = 0, the polarization ellipse has maximum ellipticity for a given set of amplitudes Ax and Ay. The absolute phase of the Jones vector does not change the corresponding Stokes parameters; therefore,

$$\mathbf{S}(e^{-i\varphi }\mathbf{E})=\mathbf{S}(\mathbf{E}).$$3.26

3.26

Example 3.4Finding the Flux Components

Figure 3.16 (left) graphs the polarization ellipse for the polarized state with the Jones vector E,

$$\mathbf{E}=\sqrt{\frac{2}{{\epsilon }_{0}c}}{e}^{0.6i}\left(\begin{array}{l}i\\ 0.8+0.5i\end{array}\right).$$3.27

3.27

The $\sqrt{2\text{/}({\epsilon }_{0}c)}$ is the unit conversion to express the Jones vector E in a normalized form, not in units of volts per meter. The polarized flux component pairs are PH = |i|2 = 1 and PV = |0.8 + 0.5 i|2 = 0.89. According to Equations 3.23 and 3.24, P45 = 1.445 and P135 = 0.445, and PR = 1.745 and PL = 0.145, yielding Stokes parameters S = (1.89, 0.11, 1, 1.6). The corresponding orthogonal pairs of states are graphed to scale in Figure 3.16 (second, third, and last figure).

Figure 3.16An elliptical polarization state represented by its polarization ellipse (left) and its decomposition into horizontal and vertical flux components (second), 45° and 135° components (third), and right and left circular components (right). Note how the arrows on the horizontal and vertical components line up with the elliptical state’s arrow. Similarly, the arrows on the 45° and 135° components add as vectors into the elliptical state’s arrow. Vectors from the origin to the right and to the left circular component’s arrows also add, yielding the location of the elliptical state’s arrow.

A fully polarized set of Stokes parameters, or the polarized part of a set of partially polarized Stokes parameters, is equivalent to the Jones vector E,

$$\mathbf{E}(\mathbf{S})=\sqrt{\frac{2}{{\epsilon }_{0}c}}\left(\begin{array}{l}\sqrt{\frac{S_0+S_1}{2}}\\ \sqrt{\frac{S_0-S_1}{2}}{e}^{-2i\text{arctan}(S_2,S_3)}\end{array}\right).$$3.28

3.28

This Jones vector can be multiplied by an arbitrary phase e−iϕ since the Stokes parameters do not specify an absolute phase.

Example 3.5Converting a Jones Vector into Stokes Parameters

The polarization ellipse of the Jones vector

$$\mathbf{E}=\sqrt{\frac{2}{{\epsilon }_{0}c}}\left(\begin{array}{l}1.5\\ 0.5+0.5i\end{array}\right)=\frac{2}{{\epsilon }_{0}c}\left(\begin{array}{l}1.5\\ 0.707e^{i0.785}\end{array}\right)$$3.29

3.29

is plotted in Figure 3.17 along with its decomposition into the pairs of basis polarization states. Note the positions of the arrows identifying the phases of the polarized E-field components. When two vectors from the origin to these arrows are added, they equal the vector from the origin to the ellipse in Figure 3.17 (left). E has the following polarized flux component pairs: PH = 2.25 and PV = 0.5, P45 = 2.125 and P135 = 0.625, and PR = 0.625 and PL = 2.125. The corresponding Stokes parameters are

$$\mathbf{S}=\left(\begin{array}{l}{1.5}^2+{0.707}^2\\ {1.5}^2-{0.707}^2\\ 2(1.5)(0.707)\text{cos}(-0.785)\\ 2(1.5)(0.707)\text{sin}(-0.785)\end{array}\right)=\left(\begin{array}{l}2.75\\ 1.75\\ 1.5\\ -1.5\end{array}\right).$$3.30

3.30

The degree of polarization must be 1, since this state also has a Jones vector. The other elliptical polarization parameters are as follows: $DoLP=\sqrt{85}\text{/}11\approx 0.838$, orientation of major axis ψ ≈ 20°, DoCP = −6/11, ellipticity ε = −0.30, and eccentricity e ≈ 0.95.

Figure 3.17An elliptical polarization state represented by its polarization ellipse (left) and its decomposition into horizontal and vertical flux components (second), 45° and 135° components (third), and right and left circular components (right).

3.12The Stokes Parameters’ Non-Orthogonal Coordinate System

The Stokes parameters have an unusual coordinate system, because the S1, S2, and S3 axes do not form an orthogonal coordinate system. This non-orthogonal coordinate system is an extremely clever system, which usefully describes the addition of partially polarized light beams.

An orthogonal coordinate system uses basis vectors that are perpendicular to each other, such as the Cartesian basis vector set $\widehat{\mathbf{x}}$, $\widehat{\mathbf{y}}$, and $\widehat{\mathbf{z}}$:

$$\{\begin{array}{l}\widehat{\mathbf{x}}\times \widehat{\mathbf{y}}=0,\\ \widehat{\mathbf{y}}\times \widehat{\mathbf{z}}=0,\text{and}\\ \widehat{\mathbf{z}}\times \widehat{\mathbf{x}}=0.\end{array}$$3.31

3.31

Orthogonal coordinate systems simplify most geometrical calculations and are the principal coordinate systems used in physics and optics. The Jones vector uses the standard x–y orthogonal coordinate system. On the other hand, Stokes parameters use basis vectors where plus and minus values represent orthogonal polarizations, not opposite directions. The S1 and S2 basis states represent electric fields that are only 45° apart; these two states are halfway between orthogonal in the Jones calculus as shown in Figure 3.18.

Figure 3.18The three basis vectors for the Stokes parameters, H, 45°, and R, corresponding to the S1, S2, and S3 basis states, are not orthogonal in Cartesian space.

There are two reasons for the great utility of this non-orthogonal coordinate system. First, the Stokes parameters do not differentiate between light polarized at 0° and 180° because for incoherent light, there is no effective difference. Linearly polarized light is polarized along a line, such as the 0° and 180° line in the transverse plane. The electric field takes equal excursions in both the +x- and –x-directions. A polarizer oriented at 0° has the same optical effect as a polarizer rotated through 180°. This is why polarizers have a plane of polarization, such as “horizontal,” not a single direction such as +x. Since the Stokes parameters do not differentiate between light polarized at 0° and 180°, the Stokes parameters for linearly polarized light $\widehat{\mathbf{L}}(\psi )$ repeat after rotations of 180°,

$$\widehat{\mathbf{L}}(\psi )=\left(\begin{array}{l}1\\ \text{cos}2\psi \\ \text{sin}2\psi \\ 0\end{array}\right)=\widehat{\mathbf{L}}(\psi +\pi )=\left(\begin{array}{l}1\\ \text{cos}(2(\psi +\pi ))\\ \text{sin}(2(\psi +\pi ))\\ 0\end{array}\right).$$3.32

3.32

This explains the 2ψ in the Mueller rotation matrix. Second, the plus and minus signs of the Stokes parameter elements represent orthogonal polarizations. Adding orthogonally polarized incoherent beams reduces the degree of polarization of the light. For example, two beams of horizontally and vertically polarized polychromatic light with equal flux per area add to yield unpolarized light.

3.13The Poincaré Sphere

The Poincaré sphere is a geometric representation for polarization states and the Stokes parameters that simplifies the analysis of many polarization problems, particularly problems involving retarders.2 Henri Poincaré introduced the sphere in 1892 in his text Traité de Lumieré.7 The Poincaré sphere, shown in Figure 3.19, maps all fully polarized states into points covering the surface of a sphere. Three views of a partially transparent (front and back) Poincaré sphere are provided. Figure 3.20 shows additional views of the Poincaré sphere with latitude and longitude lines superposed. Ellipses are drawn looking toward the center of the sphere; hence, the northern (top) hemisphere is right elliptically polarized.

Figure 3.19The Poincaré sphere is a three-dimensional representation of polarization states. The coordinates at a set of points on the surface have been converted into normalized Stokes parameters and the ellipses plotted on the surface. These transparent views from three different angles show the front and back simultaneously showing how the orientation of the polarization rotated through 180° for a full 360° circuit around the sphere.

Figure 3.20The Poincaré sphere, shown here in four views centered above 135° light or −S2 (left), centered just above 0° light or +S1 (second from left), centered just below 45° light or +S2 (second from right), and centered below 90° light or −S1 (right). All fully polarized states are represented on the surface of the sphere. The dark band around the equator represents linearly polarized light, with circularly polarized light at the poles. Polarization states are drawn looking into the sphere, so the top hemisphere is right elliptically polarized.

Consider the normalized Stokes parameters$\widehat{\mathbf{S}}$ obtained by dividing the Stokes parameters S by the flux S0,

$$\widehat{\mathbf{S}}=\left(\begin{array}{l}1\\ s_1\\ \begin{array}{l}{s}_2\\ s_3\end{array}\end{array}\right)=\frac{\mathbf{S}}{S_0}=\left(\begin{array}{l}1\\ S_1/S_0\\ \begin{array}{l}{S}_2/S_0\\ S_3/S_0\end{array}\end{array}\right).$$3.33

3.33

The degree of polarization of $\widehat{\mathbf{S}}$, $DoP(\widehat{\mathbf{S}})$ is

$$DoP(\widehat{\mathbf{S}})=\sqrt{s_1^2+s_2^2+s_3^2}.$$3.34

3.34

For all polarized states, the degree of polarization is one; hence, the normalized Stokes components obey the relation

$$\sqrt{s_1^2+s_2^2+s_3^2}=1.$$3.35

3.35

The Poincaré sphere represents a normalized set of Stokes parameters for each polarization state (s1, s2, s3) as a point in three-dimensional space. The axes of the Poincaré sphere are the Stokes parameter basis states, ±S1, ±S2, ±S3, as shown in Figure 3.21. The fully polarized states lie on the surface of the sphere because their DoP is equal to 1. Figure 3.22 shows views of the circular polarized regions at the top and bottom of the Poincaré sphere. Around each pole, the light is in elliptical states that are nearly circular and the major axis rotates through 180° as we move in a circle of latitude about the pole.

Figure 3.21The axes associated with the Poincaré sphere are the Stokes parameter basis polarization states.

Figure 3.22The polar regions of the Poincaré sphere represent states around right circularly polarized (left, north pole) and left circularly polarized light (right, south pole).

The origin (0, 0, 0) of the Poincaré sphere represents unpolarized light. Linearly polarized states occupy the unit circle in the s1–s2 plane. Right circularly polarized light is at the top of the sphere (0, 0, 1) at the north pole. Left circularly polarized light is at the bottom of the sphere (0, 0, −1) at the south pole. The rest of the surface of the sphere describes elliptically polarized light, with a right circular helicity in the northern hemisphere and a left circular helicity in the southern hemisphere. The ellipticity of the light approaches 0 when approaching the equator; the ellipticity approaches ±1 when approaching the poles. The surface of the Poincaré sphere continuously represents all possible polarized states.

The interior of the Poincaré sphere represents partially polarized light with the distance from the origin indicating the DoP. Thus, a sphere of radius ½ centered at the origin describes all the partially polarized states with a DoP of ½. Figure 3.23 (top) shows a series of spheres; each contains the polarization states with a fixed DoP. A radial line from the center through each sphere describes a state with the same polarization ellipse but varying DoP. Along the line segment (s1, 0, 0), −1 ≤ s1 ≤ 0, shown in Figure 3.23 (top), the polarization state varies from fully vertically polarized, strongly vertically polarized, weakly vertically polarized to unpolarized, with examples of the random polarization ellipses plotted in Figure 3.23 (bottom).

Figure 3.23(Top) Surfaces of constant degree of polarization are a family of spheres inside the Poincaré sphere centered at the origin. The point at the center of these spheres represents unpolarized light. (Bottom) Random polarization ellipses for polarization states along the white line, varying from (left) vertically polarized to (right) unpolarized.

Moving around the Poincaré sphere equator from (1, 0, 0), to (0, 1, 0), (−1, 0, 0), (0, −1, 0), and back to the beginning, (1, 0, 0), the polarization states change from horizontal or 0°, to 45°, 90°, 135°, and back to horizontal, 0°, as shown in Figure 3.24 (left). The polarization state with an orientation of 180° is the same as 0°; polarizers oriented at 0° and 180° generate the same polarization state. Thus, moving around the equator of the Poincaré sphere, the orientation of the polarization axis ψ changes at one-half the rate of the longitude ζ,

Figure 3.24(Left) The linearly polarized states on the equator rotate by 180° as we move 360° around the equator. (Middle) Polarization ellipses along the longitude great circle from left circularly polarized light, through horizontal, to right circularly polarized light all have a horizontal major axis. Continuing onto the back side, all the ellipses have a vertical major axis. (Right) Polarization change shown around two circles of latitude. At each latitude, the ellipse’s major axis orientation changes while the degree of circular polarization remains constant.

$$\psi =\frac{\zeta }{2},$$3.36

3.36

a consequence of the factors of two in Equation 3.7.

Along a longitude line, moving from the south (left circularly polarized) pole, across the equator, to the north (right circularly polarized) pole, the ellipticity of the light changes but the orientation of the major axis remains constant as shown in the front side of the sphere in Figure 3.24 (middle). The lines of latitude on a sphere, the circles in planes perpendicular to the S3 axis through the poles, are labeled from −90° at the south pole, to 0° at the equator, to 90° at the north pole, or as we prefer to work in radians, from –π/2 to 0 to π/2. Moving around a circle of latitude, the normalized Stokes parameter s3 and the ellipticity remain constant, but the orientation of the ellipse’s major axis rotates through 180° as shown in Figure 3.24 (right). Each circle of latitude corresponds to the states of a constant degree of circular polarization. The degree of circular polarization for a normalized set of Stokes parameters is

$$DoCP(\mathbf{S})=\frac{S_3}{S_0}=s_3.$$3.37

3.37

Thus, the latitude η on the Poincaré sphere in radians is proportional to s3,

$$\text{sin}\eta =s_3=DoCP.$$3.38

3.38

Figure 3.25 shows the orientation of the major axis on the Poincaré sphere as positioned at 2ψ in sphere coordinates and at a latitude η.

Figure 3.25Position of a state with major axis orientation ψ and latitude η on the Poincaré sphere.

3.14Flat Mappings of the Poincaré Sphere

Just as it is useful to represent the earth, not just on spherical globes, but on flat maps, it is similarly helpful to use various flat representations for the surface of the Poincaré sphere. First, the Poincaré sphere can be represented within a rectangle where the x-axis represents the orientation of the polarization axis and the y-axis is DoCP, as shown in Figure 3.26a. In this projection, the equi-rectangular projection, the entire line across the top represents right circularly polarized light and the entire line across the bottom represents left circularly polarized light. Other standard map transformations can be used to represent the Poincaré sphere. Figure 3.26b depicts the Mollweide projection of the sphere; Figure 3.26c is the sinusoidal projection and Figure 3.26d is the interrupted sinusoidal projection.

Figure 3.26Several flat representations of the Poincaré sphere: (a) the equi-rectangular projection, (b) the Mollweide projection, (c) the sinusoidal projection, and (d) the interrupted sinusoidal projection.

Before Henri Poincaré introduced the Poincaré sphere, he first developed an interesting representation of polarized light on the complex plane. This parameterization of the Stokes parameters is generated by placing a plane tangent to the bottom of the Poincaré sphere (at left circular), and projecting polarization states from the sphere onto the complex plane along lines as shown in Figure 3.27. If the radius of the Poincaré sphere is set to ½, conveniently, linearly polarized light lies around the unit circle. The resulting parameterization of the normalized Stokes parameters in terms of the complex number z = x + iy is

Figure 3.27The set of all polarized states can be represented on the complex plane by projecting the Poincaré sphere from the right circularly polarized state.

$$\mathbf{S}=\left(\begin{array}{l}1\\ s_1\\ s_2\\ s_3\end{array}\right)=\left(\begin{array}{l}1\\ \frac{2x}{1+x^2+y^2}\\ \frac{2y}{1+x^2+y^2}\\ \frac{2}{1+x^2+y^2}-1\end{array}\right).$$3.39

3.39

Figure 3.28 shows the polarization states near the origin of Poincaré’s complex plane. Centered circles contain all states of common ellipticity. Radial lines represent constant major axis orientation. The parameterization of the complex plane in terms of normalized Stokes vector elements is

Figure 3.28Projection of the Poincaré sphere onto the plane such that left circularly polarized light is at the origin, linearly polarized light is on the unit circle, and right circularly polarized light is at infinity.

$$x+iy=\frac{s_1+i{s}_2}{1+s_3}.$$3.40

3.40

3.15Summary and Conclusion

The Stokes parameters are a standard method in optics for characterizing the polarization states of light beams. Jones vectors are most suited for coherent light beams. Laser light is well approximated by Jones vectors and polarization vectors since the spectral bandwidth is nearly monochromatic (Δλ << 1 nm) and since laser beams can be well collimated. These laser-generated light waves are comparatively simple to describe. On the other hand, light from the sun, light bulbs, LEDs, and most other sources is far from monochromatic. The superposition of all these waves is more challenging to describe mathematically, as treated in Section 4.7.

Stokes parameters can be used for either coherent or incoherent beams. Incoherent light obeys the same basic principles as coherent light during interference and diffraction; however, because of the incoherent light’s polychromatic nature, the details are more complex. For example, monochromatic light cannot be unpolarized. In particular, polarized incoherent beams can add to form unpolarized light whereas mutually coherent beams cannot. Combining the light from one flashlight with a horizontal polarizer and another with a vertical polarizer generates unpolarized light. Combining the light from a horizontally polarized and vertically polarized laser generates a family of polarized states, not unpolarized light; this is treated further in Section 4.4. Thus, incoherent light uses different polarization mathematics compared to the Jones vector, and this mathematics has the remarkable coordinate system of Section 3.12.

Stokes parameters are particularly useful for characterizing the polarization in outdoor and indoor scenes.8 These Stokes parameter images and spectra are measured with Stokes polarimeters (Chapter 7). Jones vectors are not used to describe natural light.

3.16Problem Sets

1. For the following Stokes parameters, determine the degree of polarization, the degree of linear polarization, the orientation of the major axis of the polarization ellipse, the degree of circular polarization, and the ellipticity.

   1. (1, 0, 1, 0)

   2. (1, 0, 0.5, 0)

   3. (2, 1, 0.5, −0.5)

   4. (3, 1, 0, −1)

   5. (1, 0.2, 0.3, 0.6)

   6. (1, −0.8, 0.1, 0.4)

      If all six beams are incoherently combined, what is the resulting Stokes vector?

2. Decompose the following Stokes parameters in Problem 3.1 into completely polarized, SP, and unpolarized, SU, components. Plot the polarization ellipses for the SP all to the same scale.

3. Each row in the table below represents a set of four polarization component measurements chosen from the six measurement types on the heading. Calculate the Stokes vector and fill in the expected measurements for the two missing measurements (marked □).

   	PH	PV	P45	P135	PR	PL
   a.	10	1	3	□	6	□
   b.	5	2	5	□	□	2
   c.	9	□	9	□	0	18
   d.	6	□	□	4	10	7
   e.	□	7	6	6	10	□
   f.	4	□	6	4	□	4

4. Why do Stokes vector elements for linearly polarized light vary with orientation as 2ψ while Jones vector elements have a ψ dependence?

5. Show that for a partially polarized beam with Stokes parameters (S0, S1, S2, S3), the maximum amount of light that can be transmitted through the matched ideal polarizer is more than the polarized flux $\sqrt{S_1^2+S_2^2+S_3^2}$. How much more flux can be transmitted?

6. 1. Where on the Poincaré sphere are all the states with a DoCP = 2/3?

   2. Where are all the states on the Poincaré sphere with the major axis orientation at 30°?

   3. Indicate the states on the Poincaré sphere with a transmittance of 50% through a 45° linear polarizer.

   4. Show the trajectory for 45° linearly polarized light propagating through, first, a quarter wave linear retarder with fast axis at 0°, then a quarter wave linear retarder with fast axis at 45°, and then a quarter wave left circular retarder.

7. Match the Jones vectors with the Stokes parameters that represent the same polarization state:

   a. (1, 1, 0, 0)	i. (1 + i, 0)
   b. $\left(\sqrt{2,}1,1,0\right)$	ii. (0, 1)
   c.  (1, 0, 1, 0)	iii. (i, i)
   d. $\left(\sqrt{2},-1,1,0\right)$	iv. (1, i)
   e.  (1, −1, 0, 0)	v. 2¼ (cos 22.5°, sin 22.5°)
   f.  (1, 0, 0, 1)	vi. (−i, 1)
   g. (1, 0, 0, 0)	vii. 2¼ (cos 67.5°, −sin 67.5°)

8. Determine the completely polarized set of Stokes parameters with the specified irradiance P, orientation θ, and ellipticity ε:

   P (W/m2)	θ	ε
   2	0°	0
   10	22.5°	0.1
   10	45°	0.25
   100	60°	0.4
   30	90°	0.5
   0.1	113°	1

9. Transform the following Jones vectors into Stokes parameters:

   a. (1, 1)

   b. $\left(e^{\frac{i\pi }{4}},0\right)$

   c. (1, eiδ)

   d. (1, i/2)

10. Transform the following Stokes parameters into Jones vectors:

    a. (1, 0, 0, 1)	b. $\left(1,1/\sqrt{2},1/\sqrt{2},0\right)$	c. $\left(1,0,1/\sqrt{2},1/\sqrt{2}\right)$
    d. $\left(\sqrt{14},1,2,3\right)$	e. (1, − cos 2θ, sin 2θ, 0)	f. $\left(\sqrt{3},1,1,1\right)$

11. Light with the Stokes vector S = (11, 6, 6, 7) is transmitted through an adjustable polarization rotator that rotates the major axis of the polarization ellipse.

    1. Determine the resulting Stokes vector for the following rotations: 10°, 22.5°, 30°, 45°, 90°, 135°, and 180°.

    2. Draw a Poincaré sphere and label the basis polarization states.

    3. Show the incident state and all the states from part (a). Describe the trajectory on the Poincaré sphere as the rotation angle increases.

12. Perform the following operations on the Stokes vector below:

    (1, 0, −1, 0)

    (1, −0.5, 0, 0)

    (1, 0.5, 0.5, 0)

    (5, 0, 0, 2)

    1. Determine the degree of polarization.

    2. Identify the polarization state and note whether it is polarized, partially polarized, or unpolarized.

    3. Decompose the Stokes parameters S into a polarized SP and unpolarized SU component.

    4. Determine the polarized Stokes vector S0 (D0 = 1) orthogonal to SP.

13. An elliptically polarized state has a horizontal major axis, has a degree of polarization of one, and rotates clockwise looking into the beam. A horizontal polarizer passes 70% of the flux and a vertical polarizer passes 30%.

    1. Find the normalized Stokes parameters, S.

    2. What is the ellipticity and degree of circular polarization?

    3. Find the orthogonal polarization state.

    4. If the major axis of the ellipse is rotated by 60°, find the resulting Stokes vector.

14. Light with the Stokes parameters S = (27, 22, 14, 7) is transmitted through an adjustable polarization rotator that rotates the major axis of the polarization ellipse.

    1. Determine the resulting Stokes parameters for the following rotations: 10°, 22.5°, 30°, 45°, 90°, 135°, and 180°.

    2. Draw a Poincare sphere and label the six basis polarization states. Show the incident state and all the states from part (a). Describe the trajectory.

    3. For S, determine the degree of polarization and plot the polarization ellipse.

15. Consider the partially polarized Stokes vector S = (1, 0, 0.3, 0).

    1. Decompose S into the sum of two fully polarized linear states L1 and L2 with equal flux.

    2. Are L1 and L2 orthogonal Stokes parameters?

    3. Is this decomposition of S into two the sum of two fully polarized Stokes parameters with equal flux unique? Choose a numerical example if that helps. For an arbitrary Stokes vector, is the decomposition unique? It might help to reason on the Poincaré sphere.

    4. For which state or states is the decomposition not unique?

16. Why does the Stokes vector use a non-orthogonal basis such that the x- and y-components occur as positive and negative values on the same basis state, S1. In typical vectors, x and y occur as two orthogonal vector basis states.

17. Show that if two partially polarized beams with DoP1 > DoP2 are added, the resulting degree of polarization cannot be greater than DoP1, but it can be less than DoP2. When does it equal DoP1?

18. If two polarized Stokes parameters with equal amplitude are added, where in the Poincaré sphere is the result located? If three polarized Stokes parameters with equal amplitude are added, where in the Poincaré sphere is the result located?

References

1R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 2nd edition, Amsterdam: Elsevier (1987).

2W. A. Shurcliff, Polarized Light—Production and Use, Cambridge, MA: Harvard University Press (1962).

3D. Goldstein, Polarized Light, 2nd edition, New York, NY: Marcel Dekker (2003).

4G. G. Stokes, On the composition and resolution of streams of polarized light from different sources, Transactions of the Cambridge Philosophical Society 9 (1851): 3995.

5D. J. Diner, A. Davis, B. Hancock, S. Geier, B. Rheingans, V. Jovanovic, M. Bull, D. M. Rider, R. A. Chipman, A.-B. Mahler, and S. C. McClain, First results from a dual photoelastic-modulator-based polarimetric camera, Applied Optics 49 (2010): 2929–29465.

6D. J. Diner, A. Davis, B. Hancock, G. Gutt, R. A. Chipman, and B. Cairns, Dual-photoelastic-modulator-based polarimetric imaging concept for aerosol remote sensing, Applied Optics, 46 (2007): 8428–84455.

7H. Poincaré, Traite de la Lumiere, Paris 2, 165 (1892).

8G. P. Können, Polarized Light in Nature, CUP Archive (1985).