4Interference of Polarized Light

4.1Introduction

Interference is the ability of light waves when combined or interfered to produce interference fringes. Monochromatic light easily produces interference fringes and speckle patterns over a wide variety of situations. Thus, laser light is used for most holographic recording and as the light source for many interferometers. Incoherent light, such as sunlight and light from light-emitting diodes, only produces interference fringes in very restricted conditions where the optical path lengths (OPLs) of the combined beams differ by less than a few wavelengths. This chapter first treats the interference of coherent monochromatic polarized light. Then, the interference of polychromatic light is studied to understand why this beam combination process is well modeled by the Stokes parameters.

The performance of an interferometer is critically dependent on the flux, spectrum, wavefront quality, and polarization of the beams; the latter is our focus. The fundamental metric for coherence is the quality of the interferograms it can produce, the fringe visibility, which is the contrast of the fringes. The visibility of interference fringes is highest when the polarization states of the interfering beams are aligned and decreases as the angle between polarization vectors increases. The fringe visibility becomes zero when orthogonal polarizations interfere. There may still be polarization fringes present, a modulation of the polarization state. Even though the fringe visibility may be zero, an interference pattern can be recovered by introducing a polarizer oriented at an intermediate angle.

The interference of light has provided essential clues into the nature of light since Thomas Young published the results of his early interferometry experiments in 1803.1 The fact that two equal beams of light could cancel in some areas and generate twice the flux in other areas provided compelling evidence for the wave nature of light. Later, Fresnel and Arago 2 discovered that interference fringes were not formed if the two beams had orthogonal polarization states. Therefore, the control of polarization is an important factor in optimizing the performance of interferometers and holographic recording setups.

4.2Combining Light Waves

Interference refers to the measurable effects that occur when two or more light waves are combined. When the two beams are coherent, they can constructively and destructively interfere, creating interference patterns that reveal important information of the two beams.

When beams of light are combined in a linear optical medium, their electric fields add. Labeling individual beams with index q, the electric field throughout the region of space where the beams overlap becomes

$$\mathbf{E}(\mathbf{r},t)={\mathbf{E}}_1(\mathbf{r},t)+{\mathbf{E}}_2(\mathbf{r},t)+{\mathbf{E}}_3(\mathbf{r},t)+\dots ={\displaystyle \sum _q{\mathbf{E}}_q(\mathbf{r},t)}.$$4.1

4.1

When the beams are monochromatic, such as if the beams all derive from the same laser, all beams have a common angular frequency ω and a fixed phase relationship with respect to each other. The interference of the beams is expressed by summing their fields, where each field has an x, y, and z amplitude distribution A(r) and a phase distribution ϕ(r),

$$\begin{array}{l}\mathbf{E}(\mathbf{r},t)=\mathit{ℛ}\mathit{e}\left(e^{-i\omega t}{\displaystyle \sum _q{\mathbf{E}}_q(\mathbf{r})e^{-i\frac{2\pi }{\lambda }{\mathbf{k}}_q\cdot \mathbf{r}}}\right)=\mathit{ℛ}\mathit{e}\left(e^{-i\omega t}{\displaystyle \sum _q\left(\begin{array}{l}{E}_{x,q}(\mathbf{r})\\ E_{y,q}(\mathbf{r})\\ E_{z,q}(\mathbf{r})\end{array}\right)}\right)\\ \phantom{E(r,t)}=\mathit{ℛ}\mathit{e}\left(e^{-i\omega t}{\displaystyle \sum _q\left(\begin{array}{l}{A}_{x,q}(\mathbf{r})e^{-i{\varphi }_x(\mathbf{r})}\\ A_{y,q}(\mathbf{r})e^{-i{\varphi }_y(\mathbf{r})}\\ A_{z,q}(\mathbf{r})e^{-i{\varphi }_z(\mathbf{r})}\end{array}\right)}\right).\end{array}$$4.2

4.2

Young’s double slit interferometer, shown in Figure 4.1, creates an interference pattern by division of wavefront, transmitting two beams of light from an incident wavefront through two slits of nominally equal width. The light from each slit spreads out in angle due to diffraction, and interference fringes form on the observation screen. When light is normally incident on the two slits, the phase of the light emerging from both slits is equal. At the screen, the phase of the two beams differs due to the different path length ΔL to the screen as a function of angle. Where ΔL is an integral number of wavelengths, constructive interference occurs, the interference pattern is bright, and the two beams are in phase. Where the two beams are π radians out of phase, the interference pattern is dark. Since the slits are assumed to not change the polarization, the two beams have the same polarization, and polarization is not a consideration in the calculation of the interference pattern.

Figure 4.1Schematic of Young’s double slit. The corresponding interference pattern is shown on the right.

Young’s double slit generates interference by division of wavefront, taking two separate pieces from a wavefront and combining them. Most interferometers operate by division of amplitude, splitting the amplitude of a wave with a beam splitter or diffractive optical element such as a grating to create two or more beams. Figure 4.2 shows a non-polarizing beam splitter dividing an incident beam into two beams (left) and combining two beams (right). Beam splitters can be designed to divide a beam equally into two beams of equal amplitude, but in general, the division is unequal. In practice, beam splitters also change the polarization of the light; the first beam will have one polarization state and the second beam will have a different polarization state. Beam splitters can be designed to be non-polarizing, where the polarization changes are minimal. Alternatively, polarizing beam splitters are common where one polarization component, the p-polarized component, is transmitted and the s-polarized component is reflected.

Figure 4.2(Left) Non-polarizing beam splitter dividing incident light E0 into two beams αE0 and βE0. (Right) Beam splitter combining two beams. The beam incident from the left exits with Jones vector E1. The beam incident from the bottom exits with E2. For coherent monochromatic beams, the combined beam has Jones vector E1 + E2.

4.3Interferometers

Interferometers divide and combine waves and measure the resulting constructive and destructive interference, often to obtain information about the shape of wavefronts or the shape of optical surfaces. Hundreds of configurations for interferometers have been described and interferometers are employed for a large variety of applications, including optical testing, optical communications, and holography.3–10 Two representative interferometers are the Mach–Zehnder interferometer, shown in Figure 4.3, and the Twyman–Green interferometer, shown in Figure 4.4. Both of these interferometers split an incident wavefront into two waves, operate on the two beams separately, and then recombine the wavefronts with a beam splitter to generate an interference pattern. In Figure 4.3, the wavefront quality of a transmitting sample is shown being tested. A laser beam is collimated by a beam expander illuminating beam splitter 1. The transmitted beam, the test beam, is transmitted through a sample, reflecting from mirror 1, and then reflecting out of beam splitter 2 toward the camera. The beam reflected from beam splitter 1, the reference beam, reflects from mirror 2 and transmits through beam splitter 2. Upon exiting beam splitter 2, a lens images the sample onto a screen, a camera, or a detector where the interference can be observed. If the reference beam is a high-quality plane wave, the aberrations of the sample can be measured with the interference pattern. The Twyman–Green interferometer in Figure 4.4 is similar, except that the single beam splitter is used twice. The Twyman–Green interferometer is common in optical shops for testing optical components.3 The collimated transmitted beam is shown focusing after a lens, retro reflecting from a concave mirror, and double passing the lens before reflecting from the beam splitter. This configuration can be used to test either the lens, the shape of the mirror, or the combination of the two elements.

Figure 4.3A schematic of a Mach–Zehnder interferometer, which utilizes two beam splitters to interfere beams taking two different paths. A transmitting sample under test is shown, but many different configurations of samples are used.

Figure 4.4A schematic of a Twyman–Green interferometer, which uses a single beam splitter and interferes wavefronts returning from the interferometer’s two arms.

Figure 4.5 shows a Mach–Zehnder interferometer configured such that the polarization states and the relative amplitude of the two exiting beams can be adjusted arbitrarily. Linearly polarized light from a laser is collimated and passes through a half waveplate. A polarizing beam splitter transmits p-polarized light to mirror 1 and s-polarized light to mirror 2. Both paths have a two-retarder polarization controller. First, the orientation of the quarter wave linear retarder adjusts the ellipticity of the light to any desired value from left circular through linear to right circular. Then, the orientation of the half wave linear retarder adjusts the orientation of the major axis of each polarization state’s ellipse. The two beams are combined with a non-polarizing beam splitter and their interference pattern is measured by a camera, film, or some other detector. The angle between the beams is adjusted by tilting the mirrors; this controls the spatial frequency of the fringes. The phase is adjusted by translating one of the mirrors; this moves the fringes. The relative amplitude of the two beams is adjusted by rotating the initial half wave linear retarder to send more light into one channel and less into the other; this changes the fringe visibility. With this interferometer, interference between two arbitrary polarization states and arbitrary amplitudes can be created. Section 4.4 examines some of the resulting interference patterns.

Figure 4.5Mach–Zehnder interferometer configured to study the interference of polarized light. The rotatable quarter wave (λ/4) and half wave (λ/2) linear retarders in each path allow any pair of polarization ellipses to be generated. The initial rotatable half wave retarder adjusts the ratio of the flux in each arm. When this polarization beam splitter (PBS) outputs p-polarized light, the beam splitter transmits all the light to mirror 1. When the PBS outputs s-polarized light, the polarizing beam splitter transmits all the flux to mirror 2.

4.4Interference of Nearly Parallel Monochromatic Plane Waves

Consider the interference of two plane waves propagating in nearly parallel directions as shown in Figure 4.6. The first beam has polarization vector E1 and the second beam has polarization vector E2. Where the beams overlap, they interfere and the electric fields add,

Figure 4.6Interference of two plane waves, blue and pink, propagating at a small angle to each other and the z-axis.

$$\mathbf{E}(\mathbf{r},t)=\mathit{ℛ}\mathit{e}\left({\mathbf{E}}_1{e}^{i\left(\frac{2\pi }{\lambda }{\mathbf{k}}_1\cdot \mathbf{r}-\omega t-{\varphi }_1\right)}+{\mathbf{E}}_2{e}^{i\left(\frac{2\pi }{\lambda }{\mathbf{k}}_2\cdot \mathbf{r}-\omega t-{\varphi }_2\right)}\right).$$4.3

4.3

Let the normalized propagation vectorsk1 and k2 lie very near the z-axis in the x–z plane, so the z-components of the E-field are very close to zero. The propagation angles measured from the z-axis are ζ1 and ζ2. The propagators in the phase become

$${\mathbf{k}}_1\cdot \mathbf{r}=\left(\begin{array}{l}\text{sin}{\zeta }_1\\ 0\\ \text{cos}{\zeta }_1\end{array}\right)\cdot \left(\begin{array}{l}x\\ y\\ z\end{array}\right)=x\text{sin}{\zeta }_1+z\text{cos}{\zeta }_1,{\mathbf{k}}_2\cdot \mathbf{r}=\left(\begin{array}{l}\text{sin}{\zeta }_2\\ 0\\ \text{cos}{\zeta }_2\end{array}\right)\cdot \left(\begin{array}{l}x\\ y\\ z\end{array}\right)=x\text{sin}{\zeta }_2+z\text{cos}{\zeta }_2.$$4.4

4.4

Thus, the E-field is

$$\mathbf{E}(z,t)=\mathit{ℛ}\mathit{e}\left[e^{-i\omega t}\left(\left(\begin{array}{l}{E}_{1,x}\\ E_{1,y}\\ E_{1,z}\end{array}\right)e^{i\frac{2\pi }{\lambda }(x\text{sin}{\zeta }_1+z\text{cos}{\zeta }_1)-i{\varphi }_1}+\left(\begin{array}{l}{E}_{2,x}\\ E_{2,y}\\ E_{2,z}\end{array}\right)e^{i\frac{2\pi }{\lambda }(x\text{sin}{\zeta }_2+z\text{cos}{\zeta }_2)-i{\varphi }_2}\right)\right].$$4.5

4.5

The interference pattern can be viewed in any plane. For simplicity, let z = 0. As the propagation vectors approach the z-axis, the z-components approach zero. Then, the z-components can be dropped and the E-vectors can be replaced with 2 × 1 Jones vectors. The E-field at the observation plane becomes

$$\mathbf{E}(0,t)=\mathit{ℛ}\mathit{e}\left[e^{-i\omega t}\left(\left(\begin{array}{l}{E}_{1,x}\\ E_{1,y}\end{array}\right)e^{i\left(\frac{2\pi }{\lambda }x\text{sin}{\zeta }_1-{\varphi }_1\right)}+\left(\begin{array}{l}{E}_{2,x}\\ E_{2,y}\end{array}\right)e^{i\left(\frac{2\pi }{\lambda }x\text{sin}{\zeta }_2-{\varphi }_2\right)}\right)\right].$$4.6

4.6

First, consider the case where the two interfering beams are in the same polarization state but may have different amplitudes A1 and A2. The phase of the x-component can be used as the reference phase (ϕ1 = 0) and the y-phase becomes the relative phase (ϕ2 = ϕ). The normalized Jones vector, now labeled as (F1,x, F2,x), is being factored out, so Equation 4.6 becomes

$$\mathbf{E}(0,t)=\mathit{ℛ}\mathit{e}\left[e^{-i\omega t}\left(\begin{array}{l}{F}_{1,x}\\ F_{1,y}.\end{array}\right)\left(A_1{\text{e}}^{i\left(\frac{2\pi }{\lambda }x\sin {\zeta }_1\right)}+A_2{\text{e}}^{i\left(\frac{2\pi }{\lambda }x\sin {\zeta }_2-\varphi \right)}\right)\right]$$4.7

4.7

The intensity distribution P(x) in the interference pattern is

$$\begin{array}{l}P(x)=\frac{{\epsilon }_{0}c}{2}{\mathbf{E}}^{†}\mathbf{E}\\ \phantom{P(x)}=\frac{{\epsilon }_{0}c}{2}\left(A_1{\text{e}}^{-i\left(\frac{2\pi }{\lambda }x\text{sin}{\zeta }_1\right)}+A_2{\text{e}}^{-i\left(\frac{2\pi }{\lambda }x\text{sin}{\zeta }_2-\varphi \right)}\right)\left(A_1{\text{e}}^{i\left(\frac{2\pi }{\lambda }x\text{sin}{\zeta }_1\right)}+A_2{\text{e}}^{i\left(\frac{2\pi }{\lambda }x\text{sin}{\zeta }_2-\varphi \right)}\right)\\ \phantom{P(x)}=\frac{{\epsilon }_{0}c}{2}\left(A_1^2+A_2^2+A_1{A}_2{\text{e}}^{i\frac{2\pi x}{\lambda }(\text{sin}{\zeta }_1-\text{sin}{\zeta }_2)+i\varphi }+A_1{A}_2{\text{e}}^{-i\frac{2\pi x}{\lambda }(\text{sin}{\zeta }_1-\text{sin}{\zeta }_2)-i\varphi }\right)\\ \phantom{P(x)}=\frac{{\epsilon }_{0}c}{2}\left\{A_1^2+A_2^2+2A_1{A}_2\text{cos}\left[\frac{2\pi x}{\lambda }(\text{sin}{\zeta }_1-\text{sin}{\zeta }_2)+\varphi \right]\right\}.\end{array}$$4.8

4.8

This yields a cosinusoidal flux variation, the interference pattern, with a period of Λ = λ / (sin ζ1 − sin ζ2). The minimum flux (Pmin) and the maximum flux (Pmax) are

$$P_{\text{min}}=\frac{{\epsilon }_{0}c}{2}\left(A_1^2+A_2^2-2A_1{A}_2\right)\text{and}{P}_{\text{max}}=\frac{{\epsilon }_{0}c}{2}\left(A_1^2+A_2^2+2A_1{A}_2\right).$$4.9

4.9

The quality of the fringes is described by the metric fringe visibility V,

$$V=\frac{P_{\text{max}}-P_{\text{min}}}{P_{\text{max}}+P_{\text{min}}}=\frac{2A_1{A}_2}{A_1^2+A_2^2},$$4.10

4.10

which depends on the relative flux between the two beams. The fringe visibility is a maximum, V = 1, when the amplitudes are equal, A1 = A2, and thus the minimum flux is 0. These are the easiest fringes to detect. As the ratio of the amplitudes A1/A2 increases or decreases, Pmin increases, the fringe visibility decreases, and the fringes become harder to detect. As the ratio A1/A2 approaches zero, Pmin approaches Pmax, and the fringes become difficult to measure due to the small intensity modulation. Figure 4.7 shows interference fringes of different visibility resulting from different ratios of the two fluxes. The higher the fringe visibility, the easier the fringes are to measure and the better the signal to noise of the interferometric measurement will be. For the remainder of the chapter, the term $\frac{{\epsilon }_{0}c}{2}$ will be dropped from flux equations and the equations are presented in normalized flux units.

Figure 4.7Fringe patterns between two beams with identical polarization states obtained by interfering two beams of equal flux, P1/P2 = 1, (top) yielding a fringe visibility of one, (middle) P1/P2 = 1/3 with a visibility of 0.5, and (bottom) P1/P2 = 9/11 with a visibility of 0.1.

Next, interference between beams with different polarization states is examined in Examples 4.1 and 4.2.

Example 4.1Interference of Horizontally and Vertically Polarized Light

Consider the interference between a horizontally polarized, 0°, and a vertically polarized, 90°, plane wave propagating in the x–z plane close to and symmetrically about the z-axis, ζ1 = −ζ2 = ζ. Assume the beams have equal amplitudes; hence, the Jones vectors are E1 = (1, 0) and E2 = (0, 1). The E-field in the z = 0 plane, Equation 4.7, becomes

$$\mathbf{E}(0,t)=\mathit{ℛ}\mathit{e}\left[e^{-i\omega t}\left(\left(\begin{array}{l}1\\ 0\end{array}\right){\text{e}}^{i\left(\frac{2\pi }{\lambda }x\text{sin}\zeta \right)}+\left(\begin{array}{l}0\\ 1\end{array}\right){\text{e}}^{-i\left(\frac{2\pi }{\lambda }x\text{sin}\zeta -\varphi \right)}\right)\right].$$4.11

4.11

This produces a Jones vector varying in the x-direction. The propagators (exponents) describe a linearly varying relative phase between the two beams. The resulting interference pattern is plotted in Figure 4.8. When the x- and y-polarized beams are in phase, the light is polarized at 45°. As the relative phase varies, polarization state is modulating along a great circle on the Poincaré sphere in the S2 and S3 plane from 45°, to right circular, to 135°, to left circular, and so on. These are polarization fringes. The intensity distribution in the polarization fringe pattern is constant,

$$P(x)={\mathbf{E}}^{†}\mathbf{E}=1,$$4.12

4.12

so the fringe visibility is zero. A camera will register no intensity fluctuations and thus no information beyond a constant flux. Thus, many sources stated that orthogonal polarization states don’t interfere. These orthogonal states do interfere, the polarization state is modulating, but there are no intensity fringes. Thus, it is more precise to state that orthogonal polarization states do not produce interference (intensity) fringes, only polarization fringes. When a linear polarizer oriented at 45° is inserted into the polarization fringes, as in Figure 4.9, intensity fringes are recovered. A circular polarizer recovers intensity fringes at a different phase, as shown in Figure 4.10. Many interferometers interfere orthogonally polarized beams and then use a polarizer to generate the fringes.

Figure 4.8The interference of a horizontally polarized and vertically polarized beam of equal amplitude produces a periodic polarization state. The flux in these polarization fringes is constant. The maximum flux (Pmax) and the minimum flux (Pmin) are equal and the fringe visibility is zero.

Figure 4.9Insert a linear polarizer into the polarization fringes of Figure 4.8 and intensity fringes (bright and dark bands) are recovered and can be measured.

Figure 4.10Insert a circular polarizer into the fringes of Figure 4.8 and the intensity fringes are recovered at a different phase.

A similar pattern of polarization fringes is formed from the interference of two plane waves with 45° and 135° polarization and equal amplitudes as shown in Figure 4.11. Now, the polarization state evolves on the Poincaré sphere along the great circle through the S1- and S3-axes.

Figure 4.11The interference of 45° and 135° tilted plane waves yields a pattern similar to the fringes of Figure 4.8. The polarization evolves along a different great circle around the Poincaré sphere in the S1 and S3 plane.

Example 4.2Interference of Right and Left Circularly Polarized Light

A particularly useful set of polarization fringes is formed from the interference of right and left circularly polarized light. The E-field, from Equation 4.7, becomes

$$\begin{array}{l}\mathbf{E}(0,t)=\frac{1}{\sqrt{2}}\mathit{ℛ}\mathit{e}\left[e^{-i\omega t}\left(\left(\begin{array}{l}1\\ i\end{array}\right)e^{i\left(\frac{2\pi }{\lambda }x\sin \zeta +\varphi \right)}+\left(\begin{array}{r}\hfill 1\\ \hfill -i\end{array}\right)e^{-i\left(\frac{2\pi }{\lambda }x\sin \zeta +\varphi \right)}\right)\right]\\ \phantom{E(0,t)}=\sqrt{2}\mathit{ℛ}\mathit{e}\left[e^{-i\omega t}\left(\begin{array}{l}\text{cos}\left(\frac{2\pi }{\lambda }x\text{sin}\zeta +\varphi \right)\\ \text{sin}\left(\frac{2\pi }{\lambda }x\text{sin}\zeta +\varphi \right)\end{array}\right)\right],\end{array}$$4.13

4.13

which is a rotating linearly polarized state with period Λ = λ/(2 sin ζ), as seen in Figure 4.12. When a linear polarizer is inserted, intensity fringes are recovered. When the polarizer is rotated, as in Figure 4.13, the fringes move with the polarizer. This provides a simple method of moving an interference pattern, such as the moving or stepped fringes used in phase-stepping interferometry.

Figure 4.12Addition of equal amplitude right and left circularly polarized beams with a 5% frequency difference yields a rotating linearly polarized state.

Figure 4.13Inserting a polarizer at 0° (top) into the polarization fringes of Figure 4.12 yields intensity fringes. Rotating the polarizer (remaining rows) causes the fringe pattern to move, providing a convenient method to shift an interference pattern.

4.5Interference of Plane Waves at Large Angles

In interferometers, the two beams are typically propagating very close to each other, within milliradians. For hologram writing, the angles are often much larger. The principles of Section 4.4 apply for arbitrary propagation directions as well, but the fringe visibility depends on the orientation of the polarization relative to the plane containing the propagation vectors k1 and k2. Figure 4.14 shows two cases of monochromatic plane waves propagating at 90° to each other. In the case on the left, the linear polarization of both beams is along the z-direction, k1 × k2, and the fields E1 and E2 are parallel. In this case, the beams can constructively and destructively interfere, producing interference fringes with good visibility. In the case on the right, E1 and E2 are in the plane defined by k1 and k2 but are perpendicular to each other, E1 · E2 = 0. These fields cannot produce intensity fringes.

Figure 4.14Two plane waves propagating at 90° from each other with aligned polarizations (left) and orthogonal polarizations (right).

Similarly, consider the case of Figure 4.15 where two circularly polarized beams, E1 and E2,

Figure 4.15In the interference of two circularly polarized beams propagating at 90° to each other, the components along k1 × k2 (vertical) can produce interference fringes while the components in the k1 and k2 plane cannot.

$${\mathbf{E}}_1=\frac{1}{\sqrt{2}}\left(\begin{array}{l}1\\ i\\ 0\end{array}\right),{\mathbf{E}}_2=\frac{1}{\sqrt{2}}\left(\begin{array}{l}1\\ 0\\ i\end{array}\right)$$4.14

4.14

are propagating at a relative angle of 90°. In this case, the z-components of each beam will interfere, but the orthogonal components will not. Assuming both beams have a flux of 1, the average flux over a period of the interference pattern is 2. The flux of the x- and y-components would add separately while the amplitudes of the two z-components would add or subtract. The maximum and minimum fluxes, Pmax and Pmin, and the fringe visibility are

$$\begin{array}{l}{P}_{\text{max}}={\left(\frac{1}{\sqrt{2}}\left(\begin{array}{l}1\\ i\\ 0\end{array}\right)+\frac{1}{\sqrt{2}}\left(\begin{array}{l}1\\ 0\\ i\end{array}\right)\right)}^{†}\left(\frac{1}{\sqrt{2}}\left(\begin{array}{l}1\\ i\\ 0\end{array}\right)+\frac{1}{\sqrt{2}}\left(\begin{array}{l}1\\ 0\\ i\end{array}\right)\right)=3,\\ P_{\text{min}}={\left(\frac{1}{\sqrt{2}}\left(\begin{array}{l}1\\ i\\ 0\end{array}\right)+\frac{1}{\sqrt{2}}\left(\begin{array}{l}-1\\ 0\\ -i\end{array}\right)\right)}^{†}\left(\frac{1}{\sqrt{2}}\left(\begin{array}{l}1\\ i\\ 0\end{array}\right)+\frac{1}{\sqrt{2}}\left(\begin{array}{l}-1\\ 0\\ -i\end{array}\right)\right)=1,\\ V=\frac{P_{\text{max}}-P_{\text{min}}}{P_{\text{max}}+P_{\text{min}}}=\frac{3-1}{3+1}=\frac{1}{2}.\end{array}$$4.15

4.15

In this case, fringes of visibility V = ½ would result.

4.6Polarization Considerations in Holography

The previous section described how, when beams are propagating at angles near 90° to each other, the polarization components in the plane of the propagation vectors have reduced fringe forming capability, because they are not parallel. The fringe visibility can be high for one polarization state and low for the other. This has important implications for setting the polarization in holographic setups.

Holography is a method for recording optical waves and recreating the waves at a later time. A hologram is an optical element, such as a transparency, containing a coded record of the optical wave. This coded record is often an intensity interference pattern. Since the recorded wave can be a complex wavefront, holograms are capable of creating three-dimensional views of complex objects. Large numbers of holographic setups have been described for a wide variety of tasks. One of the most common configurations will be considered in this section to highlight the polarization issues that are common to most holographic recording setups.

Figure 4.16 shows a common configuration for recording holograms. A laser is split at a variable beam splitter. Each beam is spatially filtered. The test beam illuminates the sample and the scattered light illuminates the hologram. The reference beam is a clean spherical wave that illuminates the hologram. The interference between the test and sample beams is recorded in photographic film or a similar holographic media and developed to form an amplitude or phase pattern. Later, when the hologram is illuminated with a copy of the reference beam, several diffraction orders are generated at and propagate beyond the hologram, including one order that will be a continuation of the test beam. This holographic wavefront contains a three-dimensional view of the object.

Figure 4.16A typical optical setup for recording holograms. A laser beam with a good coherence length is split at beam splitters. Part of the light transmitted through the beam splitters is diverging, and its spherical wave illuminates the holographic recording medium. The larger fraction of the light is spatially filtered and illuminates a scattering object. The scattered light, which can be considered as a large ensemble of spherical waves emanating from the object, is incident on the holographic recording medium. The interference pattern between the object’s wave and reference wave is recorded as the hologram. Later when the hologram is illuminated with a spherical wave (right), several diffracted orders result from interaction with the fine fringes in the hologram. The 0th-order beam is the continuation of the incident spherical wave. The more interesting beams are the ±1st-order diffracted beams. The first order beam is the continuation of the object wave, which, when viewed, projects a three-dimensional image of the object. The other order contains a distorted view of the object.

To produce holographic fringes with good visibility, the polarization states of the reference beam and the test beam should be nearly the same. In Figure 4.16, the best hologram is obtained when the polarization state is perpendicular to the plane of the page, as in Figure 4.14 (left). If the polarization is in the plane of the page, similar to Figure 4.14 (right), the fringe visibility will be poor. At some parts of the hologram, the reference and sample beam polarizations might be orthogonal, and no fringes would occur in that area.

4.7The Addition of Polarized Beams

4.7.1Addition of Polarized Light of Two Different Frequencies

The addition of monochromatic beams of two different frequencies does not produce a polarization ellipse. The tip of the electric field vector traces a more general shape that can be considered as a time-varying polarization state that appears as an evolving ellipse. For example, consider two plane waves propagating along the z-axis. Let E1(r, t) be an x-polarized beam of angular frequency ω1 combined with E2(r, t), which is a y-polarized beam of equal amplitude but different angular frequency ω2. The resulting field is the sum of the individual electric fields,

$$\mathbf{E}(\mathbf{r},t)=\mathit{ℛ}\mathit{e}\left(\widehat{\mathbf{x}}{\mathbf{E}}_1{e}^{i(k_{1}z-{\omega }_{1}t-{\varphi }_x)}+\widehat{\mathbf{y}}{\mathbf{E}}_2{e}^{i(k_{2}z-{\omega }_{2}t-{\varphi }_y)}\right).$$4.16

4.16

For a 10% frequency difference, ω1 = 1.1ω2, the electric field traces the pattern shown in Figure 4.17. For each short instant, the electric field is tracing a figure that is nearly an ellipse, but the ellipticity is steadily changing. This particular shape is an example of a Lissajous figure, the type of curve formed by the parametric equation

Figure 4.17Addition of equal horizontally and vertically polarized beams with a 10% frequency difference yields a time-varying polarization state that, when averaged, behaves as unpolarized light.

$$\left(\begin{array}{l}x(t)\\ y(t)\end{array}\right)=\left(\begin{array}{l}a\cos (2\pi ct)\\ b\text{cos}(2\pi dt+\varphi )\end{array}\right),$$4.17

4.17

where c and d are two integers, and a and b are arbitrary amplitudes. This wave has a constantly varying polarization ellipse because the phase difference between the E1 and E2 components is changing, ω1t − ω2t = 1.1ω2t − ω2t = 0.1ω2t. For these two different frequencies, the phase difference increases linearly in time. This beam starts as 45° linearly polarized with the two beams in phase. The light becomes elliptically polarized as the x- and y-components become out of phase and steadily increases in its ellipticity until it is right circularly polarized. The polarization ellipse evolves into 135° polarized light when the two beams are 180° out of phase and continues changing, becoming left circularly polarized. After 10 periods of ω1 and 11 periods of ω2, the beams are in phase again and the ellipse is 45° linearly polarized. The time-varying Stokes parameters for this evolving shape are

$$\mathbf{S}(t)=\left(\begin{array}{l}2\\ 0\\ 2\text{cos}(0.1{\omega }_{2}t)\\ 2\text{sin}(0.1{\omega }_{2}t)\end{array}\right),$$4.18

4.18

which is rapidly fluctuating, completing a cycle every 10 periods of ω1, about 2 × 10−14 s for visible light. S1 is zero because the wave always has equal x- and y-components.

To measure the Stokes parameters of this beam, a series of polarized flux measurements are performed. Each measurement requires much more than 10 optical periods; hence, the time-averaged Stokes parameters $\langle \mathbf{S}(t)\rangle$,

$$\mathbf{S}=\langle \mathbf{S}(t)\rangle =\left(\begin{array}{l}2\\ 0\\ \langle 2\text{cos}(0.1{\omega }_{2}t)\rangle \\ \langle 2\text{sin}(0.1{\omega }_{2}t)\rangle \end{array}\right)=\left(\begin{array}{l}2\\ 0\\ 0\\ 0\end{array}\right),$$4.19

4.19

are measured, yielding the Stokes parameters for unpolarized light. Although the measured Stokes parameters indicate unpolarized light, the beam has horizontal and vertical linearly polarized components at different frequencies. To a Stokes polarimeter, this beam is indistinguishable from unpolarized light.

In conclusion, the polarization state measured by a Stokes polarimeter when combining two laser beams of different frequencies is the sum of the Stokes parameters of the two individual laser beams. For the previous example, the Stokes parameter equation becomes

$$\mathbf{S}=\mathbf{H}+\mathbf{V}=\left(\begin{array}{l}1\\ 1\\ 0\\ 0\end{array}\right)+\left(\begin{array}{r}\hfill 1\\ \hfill -1\\ \hfill 0\\ \hfill 0\end{array}\right)=\left(\begin{array}{l}2\\ 0\\ 0\\ 0\end{array}\right).$$4.20

4.20

Figure 4.18 shows the addition of equal beams of right and left circularly polarized light with a 5% frequency difference. This flower petal-like pattern describes a state that is nearly linear but steadily rotating in orientation, moving through 360° in about 20 optical periods. Optical detectors are too slow to follow this rapid evolution of the polarization state. For this wave, the component fluxes PH and PV are equal. Similarly, P45 and P135 are present in equal amounts, so S2 = 0. Finally, careful analysis shows that the electric field vector spends equal time moving in the clockwise and counterclockwise directions, so S3 = PR − PL = 0. These example waves are coherent, not incoherent, because the beam is periodic. The interference that occurs is too fast to observe, but the interference can be inferred and understood from the equations. Similarly, incoherent waves also interfere; however, the waves are random; the variations that occur in any particular circumstance cannot be known through measurement.

Figure 4.18Addition of equal right and left circularly polarized beams with a 5% frequency difference yields a rotating nearly linearly polarized state.

Figure 4.19 shows the results of adding several other pairs of monochromatic beams with a 10% frequency difference.

Figure 4.19Three examples of the sum of non-orthogonally polarized beams with different frequencies. (Top) Interference of horizontal and 45° linearly polarized beams with 10% frequency difference yields a partially polarized beam at 22.5°. (Middle) Interference of horizontal and right circularly polarized beams with 10% frequency difference yields a partially elliptically polarized beam with a horizontal fast axis. (Bottom) Summation of horizontal and 20° linearly polarized beams with 10% frequency difference yields a nearly polarized beam oriented at 10°.

4.7.2Addition of Polychromatic Beams

In this section, the measurement of a polychromatic unpolarized beam will be examined in detail and numerically modeled in time. A horizontally polarized polychromatic wave is added to a vertically polarized polychromatic wave, and the measurement of the resulting unpolarized beam by a Stokes polarimeter is simulated. Section 4.7.1 showed how two plane wave components at different frequencies generate rapidly evolving polarization states, reducing the degree of polarization. Similarly, in the next example, since most of the pairs of frequency components in two white light beams are different, the degree of polarization is greatly reduced when polychromatic beams are added.

The wavelengths in white light span the visual response of the eye, roughly from 400 to 700 nm, and the frequencies span the range from 750 to 430 terahertz (THz, 1012 Hz). The light’s electric field is a superposition of the electric fields at each of the constituent frequencies. Thus, a polychromatic beam can be written as an integral over frequency of monochromatic beams.

A collimated white light beam propagating in the z-direction that has transmitted through a horizontal polarizer has an electric field

$${\mathbf{E}}_x(\mathbf{r},t)=\mathit{ℛ}\mathit{e}\left(\widehat{\mathbf{x}}{\displaystyle \underset{0}{\overset{\infty }{\int }}{A}_x(\omega )e^{i(kz-\omega t-{\varphi }_x(\omega ))}d\omega }\right),$$4.21

4.21

where Ax(ω) is the real amplitude of this white light beam as a function of angular frequency and ϕx(ω) is the phase of each frequency component at t = 0. Any component in the y-direction has been removed by the polarizer. The resulting polychromatic electric field is determined by integrating the electric field spectrum, Ex(ω), with respect to frequency. Similarly, another white light beam transmitted through a y-polarizer has the electric field

$${\mathbf{E}}_y(\mathbf{r},t)=\mathit{ℛ}\mathit{e}\left(\widehat{\mathbf{y}}{\displaystyle \underset{0}{\overset{\infty }{\int }}{A}_y(\omega )e^{i(kz-\omega t-{\varphi }_y(\omega ))}d\omega }\right).$$4.22

4.22

When these two beams are combined, the polarization state and its Stokes parameters are

$$\mathbf{E}(\mathbf{r},t)={\mathbf{E}}_x(\mathbf{r},t)+{\mathbf{E}}_y(\mathbf{r},t).$$4.23

4.23

When performing simulations, the integral is usually replaced by the sum of discrete frequency components of the form

$${\mathbf{E}}_1(z,t)=\widehat{\mathbf{x}}\mathit{ℛ}\mathit{e}\left[{\displaystyle \sum _{q=1}^Q\left(A_{x,q}{e}^{i(k_{q}z-{\omega }_{q}t-{\varphi }_{x,q})}\right)}\right],$$4.24

4.24

$${\mathbf{E}}_2(z,t)=\widehat{\mathbf{y}}\mathit{ℛ}\mathit{e}\left[{\displaystyle \sum _{r=1}^R\left(A_{y,r}{e}^{i\left(k_{r}z-{\omega }_{r}t-{\varphi }_{y,r}\right)}\right)}\right].$$4.25

4.25

For this simulation, Q = R = 8 randomly generated frequencies, amplitudes, and phases are used as represented in Figure 4.20. Figure 4.21 shows the x-polarized (left first panel) and y-polarized (left second panel) electric fields for the first 3 × 10−14 s. Note that these waves are non-periodic because of their polychromatic nature. The widths and heights of individual oscillations vary between pairs of peaks. These two waves combine to form the unpolarized state as seen in Figure 4.21 (right).

Figure 4.20Randomly generated frequencies and phases used in Equation 4.25 for the example shown in Figure 4.21.

Figure 4.21A short duration (10−13 s) showing the oscillations of an example polychromatic E1(0, t), x-polarized and E2(0, t), y-polarized beams (left). The resulting electric field vector traces a random figure (middle), an evolving polarization ellipse (right).

The instantaneous fluxes in the electric field components, P(r, t), in watts per meter squared, conveyed by this wave are proportional to the square of the component electric fields (the Poynting vector),

$$P(\mathbf{r},t)=\frac{{\epsilon }_{0}c}{2}{\left|\mathbf{E}(\mathbf{r},t)\right|}^2.$$4.26

4.26

Figure 4.22 shows the x and y instantaneous fluxes (in units of ε0c/2), which are always positive. For flux measurements through x-polarizers (green) and y-polarizers (orange), these signals are integrated by an optical detector producing the photocurrent. The S0 Stokes parameter is equal to the integral of the sum of the green and orange curves. Similarly, S1 is equal to their difference, shown in Figure 4.22 in purple, a signal that goes positive and negative and has an integral that approaches zero for long integration times.

Figure 4.22The instantaneous flux through a horizontal (green) and a vertical (orange) polarizer. The purple curve is the difference of the horizontal flux minus the vertical flux. Integrating this purple function yields S1.

The components of the instantaneous flux through 45° and 135° polarizers is

$${\left|{\widehat{\mathbf{E}}}_{45}^{†}\cdot \mathbf{E}\right|}^2=\frac{{(A_x+A_y)}^2}{2}\text{and}{\left|{\widehat{\mathbf{E}}}_{135}^{†}\cdot \mathbf{E}\right|}^2=\frac{{(A_x-A_y)}^2}{2}.$$4.27

4.27

The circularly polarized fluxes can be simulated by delaying the y-polarized component by a quarter of a wavelength, Ey(t, π/2), before calculating the flux through 45° and 135° linear polarizers. The quarter wavelength delay is achromatic; it is not a delay in time; every frequency is delayed by one quarter of a period, as shown in Figure 4.23. The polarized fluxes in the four basis states, 45°, 135°, right and left circularly polarized, are plotted in Figure 4.24, and the instantaneous differences are plotted in blue. The integrals of the blue curves in Figure 4.22 and Figure 4.24 are the Stokes parameters S1, S2, and S3.

Figure 4.23Ey(t), orange, and Ey(t, π/2), green, where every frequency component has been delayed by a quarter of a wavelength, which involves a longer time delay for longer wavelengths and a shorter delay for shorter wavelengths.

Figure 4.24The instantaneous flux through (left) 45° (green) and 135° (orange) polarizers, and (right) (green) and left (orange) circular polarizers over 3 × 10−14 s. The blue curves show the difference as a function of time. Integrals of the blue curves yield S2 (left) and S3 (right). These integrals tend to become zero over times for this wave.

$${\mathbf{E}}_y(t,\pi /2)=\mathit{ℛ}\mathit{e}\left[\widehat{\mathbf{y}}{\displaystyle \sum _{r=1}^R\left(A_{y,r}{e}^{i\left(k_{r}z-{\omega }_{r}t+\pi /2\right)}\right)}\right].$$4.28

4.28

This simulation demonstrated how the addition of horizontal and vertical polychromatic polarized beams of equal irradiance yield an approximately unpolarized beam. The answer is only “approximately unpolarized” because the result of this calculation is random, depending on the initial parameters and integration time. When the simulation is rerun with many different initial conditions, the answers are distributed with unpolarized light as the mean, or the most likely result. By this method, the addition of beams of arbitrary polarization states can be simulated and the resulting polarization states are well approximated by the sum of the Stokes parameters of the two separate beams. Thus, it is seen how the addition of Stokes parameters describes the addition of polychromatic waves.

4.7.3A Gaussian Wave Packet Example

Another example of the addition of polychromatic waves will be given, which produces mode locked pulses with rapidly varying polarization states. Pulses with time-varying polarization states are used in quantum optics and spectroscopy to get fine control over quantum states and atomic transitions. Complex pulses with rapidly changing frequencies and polarizations can put atoms or molecules into particular quantum states with unique density matrices.

An approximately Gaussian wave packet is formed from the combination of Q = 21 parallel monochromatic plane waves with different frequencies, each linearly polarized at a different orientation. The center frequency of ω0 = 3 × 1015 radians/s corresponds to λ0 = 627.9 nm. The 21 frequencies ωq are separated by Δω = 3 × 1013 radians/s with q = −10, −9, …, 10, where

$${\omega }_q=3\times {10}^{15}+3\times {10}^{13}q.$$4.29

4.29

The real amplitudes Aq vary as a Gaussian function (Figure 4.25),

$$A_q=\frac{\text{exp}\left(\frac{-q^2}{18}\right)}{\sqrt{2\pi }}.$$4.30

4.30

Figure 4.25The 21 real amplitudes for the wave packet form a Gaussian envelope.

These choices are arbitrary, but the conclusions to be drawn are quite general in nature.

The phases are all set equal to zero at t = 0. Each frequency component is linearly polarized. The center frequency ω0 is polarized at 0°. Then, there is an 18° (π/10 radians) rotation of the linear polarization angle χ between each frequency,

$${\chi }_q=\frac{\pi q}{10}.$$4.31

4.31

Hence, the corresponding time-dependent expression for the wave’s electric field is

$$\mathbf{E}(t)={\displaystyle \sum _{q=-10}^{10}\mathit{ℛ}\mathit{e}\left[e^{-i{\omega }_{q}t}{A}_q\left(\begin{array}{l}\cos (\pi q/10)\\ \sin (\pi q/10)\end{array}\right)\right]}.$$4.32

4.32

Figure 4.26 shows the x- and y-electric fields as a function of time. Since the amplitude distribution is approximately Gaussian, the waveform is also nearly Gaussian. Since the frequencies are discrete and periodic, the waveform is also periodic (a Fourier series) with a period of 21 × 10−12 s as seen in Figure 4.27. A Stokes polarimeter would measure the polarization state S as the sum of the Stokes parameters for each frequency.

Figure 4.26The x- and y-electric fields within one wave packet.

Figure 4.27The x-component of the electric field for two pulses from the sequence of pulses.

$$\mathbf{S}={\displaystyle \sum _{q=-10}^{10}{\mathbf{S}}_q={\displaystyle \sum _{q=-10}^{10}{A}_q^2\left(\begin{array}{l}1\\ \text{cos}(\pi q/5)\\ \text{sin}(\pi q/5)\\ 0\end{array}\right)=\left(\begin{array}{l}1\\ \text{cos}(\pi q/5)\\ \text{sin}(\pi q/5)\\ 0\end{array}\right)=\left(\begin{array}{l}3\\ 0.5\\ 0\\ 0\end{array}\right).}}$$4.33

4.33

Figure 4.28 provides a 3D view of the time-varying polarization ellipse for one Gaussian pulse width. The field is linearly polarized at 0° at the center of the pulse and then begins spiraling in ellipses of increasing ellipticity, before the flux of the pulse fades out. Figure 4.29 shows the instantaneous fluxes that constitute the Stokes parameters. Each of the frequency components has a set of Stokes parameters Sq listed in Table 4.1. The Stokes parameters for this beam is the sum of the component Stokes parameters, S = (3, 0.5, 0, 0), with DoP = 1/6.

Figure 4.28The x- and y-components versus time spiral around the axis in a steadily evolving polarization ellipse shown in a 3D view. The color cycles from red, green, blue, to red every 2.1 × 10−15 s, the approximate period. The first half of the pulse has left helicity (counterclockwise). After passing through horizontal polarization, the second half of the pulse has right helicity. Time is marked along the z-axis in units of 10−14 s.

Figure 4.29The time-dependent contributions to the Stokes parameters: (upper left) S0, (upper right) S1, (lower left) S2, and (lower right) S3. The Stokes parameters for this beam are proportional to the integrals of these functions.

Table 4.1Stokes Parameters for Gaussian Wave Packet Spectral Components

Index	S0	S1	S2	S3
−10	0.00000238	0.00000238	0	0
−9	0.0000196	0.0000159	0.0000115	0
−8	0.000129868	0.0000401316	0.000123512	0
−7	0.000687587	−0.000212476	0.000653935	0
−6	0.00291502	−0.0023583	0.00171341	0
−5	0.0098957	−0.0098957	0	0
−4	0.0268993	−0.021762	−0.015811	0
−3	0.0585498	−0.0180929	−0.0556842	0
−2	0.102047	0.0315343	−0.0970525	0
−1	0.142418	0.115219	−0.0837113	0
0	0.159155	0.159155	0	0
1	0.142418	0.115219	0.0837113	0
2	0.102047	0.0315343	0.0970525	0
3	0.0585498	−0.0180929	0.0556842	0
4	0.0268993	−0.021762	0.015811	0
5	0.0098957	−0.0098957	0	0
6	0.00291502	−0.0023583	−0.00171341	0
7	0.000687587	−0.000212476	−0.000653935	0
8	0.000129868	0.0000401316	-0.000123512	0
9	0.0000196413	0.0000158901	-0.0000115449	0
10	0.00000238	0.00000238	0	0

Mode locked lasers produce pulse trains containing a comb of frequencies, such as Figure 4.25. This pulse begins by rotating clockwise, which will tend to cause a transition from state A into state B, changing the angular momentum of a molecule. A horizontally oscillating field can now drive the molecule to state C, and finally the counterclockwise field may drive it into another state D.

Mode locked lasers typically produce linearly polarized light. One method to produce this complex pulse is to utilize a thick optically active plate, such as C-cut quartz, with its dispersion of optical activity. This will rotate the plane of polarization of different frequencies by different amounts, to produce pulses with polarizations like this example. Placing the optically active plate inside the laser cavity can increase the polarization rotation by the cavity Q-factor. Other combinations of retarders can produce a wide array of pulse polarization properties.

The relative phases of the comb of frequencies are very important for shaping the pulse. If the 21 phases ϕq are chosen randomly over the interval (−π, π), as in Figure 4.30 (left), the pulse is no longer Gaussian but has an irregular shape as in Figure 4.30 (right).

Figure 4.30When the 21 Gaussian pulse amplitudes of Figure 4.25 are given random phases (left), the pulse shape is corrupted and irregular (right). Two periods are shown.

$$\mathbf{E}(t)={\displaystyle \sum _{q=-10}^{10}\mathit{ℛ}\mathit{e}\left[e^{-i({\omega }_{q}t+{\varphi }_q)}{A}_q\left(\begin{array}{l}\text{cos}(\pi q/10)\\ \text{sin}(\pi q/10)\end{array}\right)\right]}$$4.34

4.34

4.8Conclusion

In conclusion, when adding two monochromatic beams with different wavelengths and different polarization states, the resulting polarization state evolves rapidly in time. The smaller the frequency difference, the longer the pattern takes to repeat. A polarimeter averages over many of these cycles in making its measurements, and the degree of polarization measured is reduced by averaging over polarization states. The Stokes parameters of the combined beam are equal to the sum of the Stokes parameters of the two individual laser beams.

A monochromatic beam resembles a single note from a piano, while an incoherent beam is similar to the sound played by leaning on many piano keys with your arm. Adding Jones vectors is similar to playing the exact same note on several different pianos. These sound waves will constructively and destructively interfere, and interference fringes of sound can be set up in the room. Adding Stokes parameters resembles having several small children banging on pianos in the same room.

4.9Problem Sets

1. Find the degree of polarization of the three beams in Figure 4.19.

2. What polarization states result from adding the following Jones vectors for right $\mathbf{R}=(1,-i)\text{/}\sqrt{2}$ and left $\mathbf{L}=(1,i)\text{/}\sqrt{2}$ circularly polarized light with the following relative phases ϕ:

   1. In phase, ϕ = 0.

   2. Out of phase, ϕ = π.

   3. In quadrature, ϕ = π/2.

   Assume the phase of R is fixed and the phase of L changes.

3. Write a program to plot the polarization ellipse for $\mathbf{E}=\left(\begin{array}{c}{a}_x+i{b}_x\\ a_y+i{b}_y\end{array}\right)$ by drawing a series of lines from (xj, yj) to (xj+1, yj+1) where $\left(\begin{array}{c}{x}_j\\ y_j\end{array}\right)=\mathit{ℛ}\mathit{e}\left[e^{-i2\pi t_j}\left(\begin{array}{c}{a}_x+i{b}_x\\ a_y+i{b}_y\end{array}\right)\right]$ for t = (0, Δt, 2Δt, …, 1).

4. When monochromatic 0° linearly polarized light is interfered with 90° linearly polarized light, the Jones vector function for the polarization fringes is of the form $\mathbf{E}(x)={\text{e}}^{-i2\pi x/\Lambda }\left(\begin{array}{c}1\\ 0\end{array}\right)+{\text{e}}^{i2\pi x/\Lambda }\left(\begin{array}{c}0\\ 1\end{array}\right)$, where Λ is the period of the fringes.

   1. Plot the fringes over two periods.

   2. What is the resulting flux P(x)?

   3. Plot the path of the fringes on the Poincaré sphere.

5. Plot the polarization fringes for the interference of monochromatic right and left circularly polarized beams of equal amplitude A0 over two periods.

   1. Plot the path of the fringes on the Poincaré sphere.

   2. Plot the path of the fringes on the Poincaré sphere when the amplitudes change to AR = 2A0/3 and AL = 4A0/3. What is the resulting flux P(x)?

   3. Repeat (c.) for AR = A0/3 and AL = 5A0/3.

6. Consider the polarization fringe pattern $\mathbf{E}(x)=\left(\begin{array}{c}\text{cos}(2\pi x/\Lambda )\\ i\text{sin}(2\pi x/\Lambda )\end{array}\right)$. This pattern can only result from the interference E (x) = e−i2πx/ΛE1 + ei2πx/ΛE2 of two particular polarization states E1 and E2 in a particular amplitude ratio |E1|/|E2|.

   1. Plot the polarization fringes E(x).

   2. Find the unique states E1 and E2.

7. Estimate the two unique Jones vectors that interfere to yield the following interference patterns. Estimate the ratios of their amplitudes and fluxes.

   1. 

   2. 

   3. 

   4. 

   5. 

8. Consider the interference between the monochromatic linear ${\mathbf{E}}_1=\left(\begin{array}{c}1\\ 0\end{array}\right)$ and elliptical ${\mathbf{E}}_2=\left(\begin{array}{c}\text{cos}\zeta \\ i\text{sin}\zeta \end{array}\right)$ beams.

   1. Find the ellipticity of E2 as a function of ζ.

   2. Find the fringe visibility V of the polarization fringes as a function of ζ for 0 ≤ ζ ≤ π.

   3. Plot the polarization fringes for ζ = 0, π/6, π/3, π/2, 3π/4, and π.

9. Three lasers with frequencies υ1 = 480 THz, υ2 = 500 THz, and υ3 = 520 THz are combined with dichroic filters and are propagating with the same propagation vector. The associated Jones vectors are E1 = (1, 0), E2 = (cos(π/3), sin(π/3)), and E3 = (cos(2π/3), sin(2π/3)).

   1. What is the resulting polarization state Eα? Use Jones vectors or Stokes parameters as appropriate.

   2. What is the degree of polarization of Eα?

10. A fourth beam E4 is added at a frequency υ4 = 480 THz with the Jones vector E4 = (0, i).

    1. What is the resulting polarization state Eβ?

    2. What is the degree of polarization of Eβ?

    3. Now, beam E1 is removed. What is the degree of polarization of the combination of beams E2, E3, and E4?

11. Two polarization states, E1 = (2, 0) and E2 = (−1, i), are interfered. What relative phase yields the brightest beam? What relative phase yields the faintest beam? If two plane waves propagate at an angle, what will the fringe visibility be?

12. Young’s double slit with polarizers. Young’s double slit is configured with a right circular polarizer over one slit and a left circular polarizer over the other slit. The slits are very narrow and have equal width. The slits are separated by 10 wavelengths. The slits are equally illuminated with nearly monochromatic linearly polarized light at normal incidence.

    1. Describe the interference pattern formed on a screen in the far field as a function of the angle θ from the centerline between the slits.

    2. Draw the polarization fringes.

13. This problem simulates the addition of two incoherent polarized beams, one horizontally polarized, one vertically polarized. This example provides an example of why the superposition of polychromatic beams should be treated by the sum of Stokes parameters. Both beams are described by the polychromatic plane wave equations:

    $$\begin{array}{l}{\mathbf{E}}_1(z,t)=\mathit{ℛ}\mathit{e}\left[\widehat{\mathbf{x}}{\displaystyle \sum _{q=1}^Q{A}_{x,q}{e}^{i\left(k_{q}z-2\pi {\nu }_{q}t+{\varphi }_{x,q}\right)}}\right]\\ {\mathbf{E}}_2(z,t)=\mathit{ℛ}\mathit{e}\left[\widehat{\text{y}}{\displaystyle \sum _{q=1}^Q{A}_{y,q}{e}^{i\left(k_{q}z-2\pi {\nu }_{q}t+{\varphi }_{y,q}\right)}}\right]\end{array}$$

    

    Frequencies for visible light span the range from 430 to 750 THz. For simplicity, we can drop the THz and consider eight frequencies distributed between 430 and 750 Hz. For both x- and y-components separately, generate a set of eight frequencies in our spectral band. Choose eight phases in the range from 0 to 2π. Let the eight amplitudes all equal 1.

    1. Tabulate all your values in two tables. First, present Table 1 containing your parameters for E1(r, t) with 1 ≤ q ≤ 8 and Table 2 containing parameters for E2(r, t) with 1 ≤ r ≤ 8.

    2. Plot the instantaneous values of Ex(0, t, ϕx = 0), Ey(0, t, ϕy = 0), and Ey(0, t, ϕy = π/2), which is the Ey signal where each component is shifted by a quarter of a wavelength, as a function of time for the x-polarized beam and separately for the y-polarized beam. Use enough points to resolve the oscillations and plot a list of at least 400 values.

    3. Plot the flux as a function of time transmitted through the six basis polarizers, PH, PV, P45, P135, PR, and PL.

    4. Perform numerical integrals to calculate the Stokes parameters. As the integration time is increased, how rapidly do the integrals converge? Your Stokes parameters will likely converge to unpolarized light, but this is not guaranteed.

    5. To understand this better, set your first y-frequency equal to your first x-frequency, and leave all the rest of the frequencies as before. Recalculate the Stokes parameters.

    6. Finally, set all eight y-frequencies equal to the eight x-frequencies. Then, set all the y-phases equal to the corresponding x-phases plus π/2. Recalculate the Stokes parameters.

    7. Draw some conclusions from parts a to part e.

14. 1. Consider the phase shifting Twyman–Green interferometer (as shown below) which uses a PBS in conjunction with two linear quarter wave retarders to get the light through the system with minimal loss. Once the beams are recombined, an analyzer is used to get the fringes. What are the Jones matrices for the two paths before 45° linear polarizer? What is the Jones matrix for the entire system? Be sure to account for unmatched phase between the arms.

       

    2. Suppose one of the quarter waveplates is rotated (from 45°) by some small angle δ. What is the new Jones matrix for that arm? For the system? How will this affect the measured phase?

    3. Suppose the retardance of one of the waveplates is off by a small amount δ. What is the new Jones matrix for that arm? For the overall system? How will this affect the measured phase?

    4. When L45 is incident, plot the polarization (before the LP45) as a function of the phase shift.

    5. Plot fringe contrast as a function of input polarization.

    6. For what polarization is fringe contrast maximized?

    7. For what polarization state is fringe contrast minimized?

References

1O. S. Heavens and R. W. Ditchburn, Insight into Optics, New York: John Wiley & Sons (1991).

2E. Hecht, Optics, 4th edition, Addison Wesley (2002), pp. 386–387.

3D. Malacara, (ed.), Optical Shop Testing, Vol. 59, New York: John Wiley & Sons (2007).

4E. P. Goodwin and J. C. Wyant, Field guide to interferometric optical testing, SPIE (2006).

5C. M. Vest, Holographic Interferometry, Vol. 476, New York: John Wiley & Sons (1979), p. 1.

6P. K. Rastogi, Holographic Interferometry, Vol. 1, Heidelberg/Berlin: Springer (1994).

7S. Tolansky, An Introduction to Interferometry, New York: John Wiley & Sons (1973).

8P. L. Polavarapu, (ed.), Principles and Applications of Polarization-Division Interferometry, New York: John Wiley & Sons (1998).

9A. Ya Karasik and V. A. Zubov, Laser Interferometry Principles, Boca Raton, FL: CRC Press (1995).

10M. P. Rimmer and J. C. Wyant, Evaluation of large aberrations using a lateral-shear interferometer having variable shear, Applied Optics 14.1 (1975): 142–150.