26Multi-Order Retarders and the Mystery of Discontinuities

26.1Introduction

Throughout this book, retardance has proven to be one of the more difficult concepts to define. This chapter examines the spectra of high-order retarders and how the retardance varies with wavelength. The retardance spectra of these compound multi-order retarders appear to “turn around” and avoid integer numbers of waves of retardance, when it may be obvious that the retardance must be continuously increasing. Such issues that occur in measurement and modeling of retarders are addressed in this chapter. The operation of retardance unwrapping of the principal retardance is developed to explain the behavior of retardance of the compound retarders. This allows us to generalize the concept of retarder order for what at first appears to be mysterious jumps in the retardance.

Two approaches to retardance discontinuities are developed. One approach uses a dispersion model to describe the retardance behavior. Another approach considers multiple wavefronts exiting the compound retarder system, that is, a multivalued optical path length (OPL) view.

The common definition of a retarder is a device that divides a beam into two orthogonal modes and introduces a relative phase difference δ.1 Another view of retarders is provided by the Mueller calculus and Poincaré sphere. A retarder rotates polarization states on the Poincaré sphere about an axis by the retardance δ; as light propagates through a retarder, the incident state on the Poincaré sphere is rotated about a retardance axis to another state. In this Mueller/Poincaré picture, cascading retarders is equivalent to cascading rotations on the Poincaré sphere. This view of retarders and the associated Mueller matrices have ambiguities of retardance order; all retarders with retardance 2nπ ± δ, with integer n, have the same Mueller matrix. In this Mueller retardance picture, the final polarization state always ends up in the right place but the retardance is only known to modulo 2π and the retardance fast and slow states may be either of two orthogonal states along an axis through the Poincaré sphere.

A multi-order retarder is a retarder with more than a half wave of retardance. A compound retarder is a combination of two or more retarders in sequence. A misaligned retarder is defined here as a compound retarder where the fast axes of the components are not at 0° or 90° to each other as shown in Figure 26.1. For misaligned compound multi-order retarders, more than two beams are involved and the common definition of retarders is inadequate. Using theretarder space and algorithms first introduced in Chapter 14 (Jones Matrix Data Reduction with Pauli Matrices), a retardance unwrapping algorithm is derived. An example compound retarder system is studied to demonstrate discontinuities in the retardance space. These discontinuities correspond to the retardance spectra missing the identity matrix as the retardance transitions from less than an integer number of waves of retardance to greater than that integer number of waves. The explanation for this breakdown to the conventional definition of retardance arises from the interference of the four or more beams as shown in Figure 26.1 such that the simple definition of a retarder does not apply.

Figure 26.1A misaligned two-element compound linear retarder has an arbitrary angle between the two fast axes, different from 0° or 90°. Four output beams exit a misaligned retarder with four different combinations of the fast and slow OPLs. For an aligned retarder, only the FF and SS beams exit.

In this chapter, the retardance spectrum of an example Quartz and Sapphire misaligned retarder is modeled and compared to Mueller matrix imaging polarimeter measurements.

26.2Mystery of Retardance Discontinuity

Polarimeters measure principal retardance, which varies between 0 and π. As shown in Figure 26.2, the orientation of the retarder’s fast axis flips by 90° at a retardance of 0° and 180° while the retardance abruptly increases or decreases. Similar behavior can be observed in interferometers. Most interferometers measure phase between 0 and 2π and often use phase unwrapping to extend the range of phase beyond one wave.2–4 For non-compound retarders (Figure 26.2, left), the principal retardance can be unwrapped to a smooth curve with no discontinuity as shown in the red curve of Figure 26.3. On the other hand, the principal retardance of a compound retarder does not reach zero, as shown in Figure 26.2 (right), and thus the unwrapped retardance has discontinuity as the blue curve shown in Figure 26.3. In this chapter, this mystery of retardance discontinuity will be explained.

Figure 26.2(Left) Principal retardance (solid line) and fast axis orientation (dashed line) of a non-compound retarder as a function of wavelength. (Right) Principal retardance (solid line) and fast axis orientation (dashed line) of a misaligned retarder with non-parallel or non-perpendicular fast axes as a function of wavelength. The principal retardance has a tendency to avoid zero retardance (green ovals).

Figure 26.3The unwrapped retardance of a single retarder of Figure 26.2 (left) in red and the compound retarder of Figure 26.2 (right) in blue showing retardance discontinuities.

In Section 17.2.1, the effect of rotating the polarization state analyzer (PSA) in a Jones matrix polarimeter is analyzed. The resultant family of Jones matrices forms a trajectory in the retarder space. The concept of the retarder space will be used to understand the unwrapped retardance in later sections. Figure 26.4 shows a trajectory of a half wave horizontal fast axis linear retarder (HLR) as the PSA rotates from 0 to π; fast axis orientation θ rotates from 0 to π. Figure 26.5 shows two different views of the trajectories of HLR with retardance δ0 (written in blue) as the PSA rotates.

Figure 26.4A trajectory of HLR in the retarder space as the PSA rotates from 0 to π. The trajectory starts at (π, 0, 0), at where the letter “S” is, moves to (0, π, 0), and to (0, −π, 0), and comes back to the starting point.

Figure 26.5Trajectories of HLR (δ = 0, π/3, 2π/3, and π) in the retarder space as the PSA rotates from 0 to π are shown from the side views (left two figures) and the δ45 − δR view (right). Each trajectory starts on a point along δH and then follows the same color scheme as Figure 26.4.

Since the Mueller matrices for retarders repeat every nπ, the retardance space trajectory repeats every π rotation of θ. For the half wave retarder (δ0 = π), the trajectory stays in δR − δ45 plane and moves around the π retardance sphere in a semicircle. When δ0 = 0, the retardance trajectory remains along the δR-axis; the Jones matrix for an empty sample compartment appears as a circular retarder as the PSA rotates. As δ0 increases from 0, the trajectory starts to curve and form a spiral. All the initial points for each δ0 start from the δH-axis. When the trajectory reaches a distance π from the origin, the retardance components jump to the opposite point with respect to the origin and continue to move until it returns to the starting point as θ reaches π. Unwrapping retardance is equivalent to understanding the trajectory of retarders in the retarder space.

26.3Retardance Unwrapping for Homogeneous Retarder Systems Using a Simple Dispersion Model

A dispersion model of retardance and the use of the model for unwrapping the continuous retardance of a homogeneous retarder system are shown in this section.

26.3.1Dispersion Model

The Jones matrices for monochromatic beams with different absolute phase terms cannot be distinguished; for example,

$$J_1=\left(\begin{array}{ll}1& 0\\ 0& 1\end{array}\right)=J_2=\left(\begin{array}{ll}1& 0\\ 0& e^{-2\pi i}\end{array}\right)=J_3=\left(\begin{array}{ll}1& 0\\ 0& e^{-4\pi i}\end{array}\right),$$26.1

26.1

where J1, J2, and J3 are Jones matrices for 0, 1, and 2 waves of retardance. Thus, the absolute phase term is often ignored for monochromatic light sources, and all three matrices are treated as an identity matrix. In the Michelson interferometer with polychromatic light, the location of the zero optical path difference (OPD) can be found, the so-called white light or zero order fringe. Thus, with polychromatic light, it may be possible to distinguish absolute phases in cases such as Equation 26.1.

A simple model for the wavelength-dependent retardance of a waveplate assumes the retardance varies as 1/λ,

$$\delta (\lambda )=\frac{{\delta }_0{\lambda }_0}{\lambda }$$26.2

26.2

where δ0 is the retardance for a reference wavelength λ0. This model follows from the assumption that the ordinary and extraordinary refractive indices are not dependent on the wavelength. Equation 26.2 is the dispersion model of the retardance. This model is used for the retardance unwrapping of the principal retardance for homogeneous and inhomogeneous retarder systems.

26.3.2Retardance of the Homogeneous Retarder System

In Section 26.3.3, a retardance unwrapping algorithm using the dispersion model equation is introduced. The Quartz and the Sapphire retarder models and their principal retardance and fast axes orientation for 0.45 μm < λ < 0.74 μm are shown in Figure 26.6. These models are created to compare with experimental results. The principal retardance and the fast axis orientation are calculated by the algorithms in Chapter 14 (Jones Matrix Data Reduction with Pauli Matrices). For this example, both retarders’ ordinary axes are along the horizontal axis. For Quartz, a positive uniaxial material, horizontal is the fast axis and its thickness is 0.5831 mm. For Sapphire, a negative uniaxial material, horizontal is the slow axis and its thickness is 0.37427 mm. The fast axis orientations measured by a Mueller matrix polarimeter as a function of wavelength switches between 0° and 90° because the value of the principal retardance is bounded between 0 and π although the retardance is steadily increasing from longer to shorter wavelengths; hence, the measured fast axis orientation changes from 0° to 90° each time the principal retardance reaches 0 or π. The Mueller matrix of the retarders varies continuously as the wavelength changes from 0.74 to 0.45 μm.

Figure 26.6The principal retardance spectra measured by Mueller matrix polarimetry (solid lines) and fast axis orientation (dashed lines). (Left) Quartz retarder and (right) Sapphire retarder oscillate between 0 and π. Each time the retardance changes direction, the fast axis changes by 90°.

From a Jones or Mueller matrix spectrum, the retarder order at each λ is determined using the dispersion model by rearranging sections of the principal retardance spectrum. Retardance unwrapping is an algorithm to arrange the pieces of the principal retardance to generate a plausible overall retardance spectrum. Our example of a compound retarder, formed from a Sapphire retarder followed by a Quartz retarder, is used to understand the retardance unwrapping algorithm. When two horizontal linear retarders made from Sapphire and Quartz are aligned so that ordinary axes of each crystal are aligned or perpendicular to each other, the combination forms another linear retarder system since the eigenpolarizations are parallel or orthogonal to each other. Here, the two ordinary axes are aligned and the retarder’s fast axes are perpendicular to each other. Thus, the total retardance is the difference of the individual retarders, the Quartz retardance subtracted from the Sapphire retardance,

$${\delta }_{Total}(\lambda )={\delta }_{Quartz}(\lambda )-{\delta }_{Sapphire}(\lambda ),$$26.3

26.3

where δQuartz(λ) > 0 and δSapphire(λ) > 0.

Figure 26.7 shows the continuous principal retardance of the Quartz retarder and the Sapphire retarder and the total retardance δTotal(λ) of the homogeneous retarder system.

Figure 26.7The principal retardance is plotted as a function of wavelength for the Quartz and Sapphire (green and blue) retarders and the combination with the fast axes crossed so the retardances subtract (red).

Figure 26.8 compares the simulation and an actual Mueller matrix polarimeter measurement of the principal retardance magnitude and orientation of the compound retarder. The measurement is taken every 0.01 μm between 0.45 and 0.74 μm. The simulation closely matches the measured retardance data.

Figure 26.8Principal retardance (red) and orientation (dark red) of the Sapphire and Quartz compound retarder as a function of wavelength for the simulation (lines) and measurement (dots).

Examining the curves from longer toward shorter wavelengths, the principal retardance oscillates between 0 and π more rapidly as the wavelength becomes shorter. Coming from the right side of Figure 26.8, the retardance increases to π with a horizontal fast axis and then decreases with a vertical fast axis. When the retardance reaches zero, it increases again with a horizontal fast axis, and so on. To unwrap the principal retardance, a mode number q = γ is assigned to each segment of the principal retardance with different fast axis orientation; odd q’s correspond to a horizontal fast axis, and even q’s correspond to a vertical fast axis. Figure 26.9 shows the mode numbers with an odd value γ, odd mode numbers in blue, and even in red.

Figure 26.9Each segment of the principal retardance is assigned a mode number q in units of half wavelengths. Starting from the right side of the graph, blue segments have odd mode numbers and the red segments have even mode numbers.

26.3.3Homogeneous Retarder’s Trajectory and Retardance Unwrapping in Retarder Space

Figure 26.10 shows the retarder space trajectory (see Section 14.5 for more details) for the principal retardance, which remains within a sphere of radius π, jumping from one side of the sphere to the other at a radius δH = π. The left figure corresponds to the q = γ, and the red color indicates the principal retardance vector (δH, δ45, δR) for the longest wavelength. In each plot, arrow color changes from red → orange → green → blue → purple as the wavelength becomes shorter. The δR-axis points out of the page, and each segment shows the points along the same fast axis. The left figure has the horizontal fast axis (along +δH-axis) and the retardance is increasing. Once the retardance reaches the π sphere at (π, 0, 0), its fast axis changes to the vertical direction (along the −δH-axis) and jumps to the symmetric point (−π, 0, 0). The next trajectory is continued in the middle figure, and so on. The origin (0, 0, 0) is equivalent to the identity matrix, which represents no retardance, one wave retarder (2π), or a 2nπ retarder for integer n. As the wavelength reduces, the fast axis orientation changes four times alternating along the horizontal and vertical directions.

Figure 26.10The principal retardance vector trajectories are shown in the retarder space as the wavelength changes. Each figure corresponds to a different mode number starting from the longest wavelength to the shortest wavelength: (left) q = γ, (middle) q = γ + 1, γ + 2, (right) q = γ + 3, γ + 4.

The retardance unwrapping algorithm maintains the fast axis orientation as the wavelength reduces and estimates the true retardance at each wavelength. Thus, there should be no more jumps from one side to the other of the sphere. One assumption is that if one starts measuring a multi-order retarder at a long enough wavelength, the retardance will be less than a half wave of retardance. This gives a known starting point for the unwrapping, which would be the true retardance of that retarder. When this is the case, then when q = 1, the principal retardance is the true retardance. For even q’s, the true retardance is qπ − δprincipal and the fast axis orientation remains along the horizontal axis. For odd q’s, the true retardance is (q − 1)π + δprincipal with the fast axis along the horizontal axis, that is,

$${\delta }_{unwrapped}=\{\begin{array}{l}{\delta }_{principal}\text{ when }q=1\\ q\pi -{\delta }_{principal}\text{ when }q\in even\\ (q-1)\pi +{\delta }_{principal}\text{ when }q\in odd\end{array}.$$26.4

26.4

Thus, the range of the unwrapped retardance has no upper limit.

Figure 26.11 (left) shows the principal retardance trajectory in the retarder space for the aligned example system as the wavelength reduces; when the principal retardance reaches the boundary value π, the trajectory moves to the origin symmetric point on the sphere and changes its fast axis to the orthogonal direction. Figure 26.11 (right) shows the retardance trajectory of the same system after the retardance is unwrapped; the horizontal retardance increases continuously, keeping the fast axis orientation along the +δH horizontal direction. As shown in Section 14.5, the distance from the origin to a point in the retarder space is the total retardance.

Figure 26.11(Left) Principal retardance trajectory of the aligned retarder in the retarder space as the wavelength decreases. (Right) Retardance trajectory after retardance unwrapping.

Retardance plots in Figure 26.7 are unwrapped in Figure 26.12. In this figure, the total retardance (red) value is always δQuartz(λ) − δSapphire(λ) (Equation 26.3). There is no discontinuity in the unwrapped retardance for homogeneous retarder systems that consist of two or more linear retarders with parallel or perpendicular fast axes.

Figure 26.12The unwrapped retardance as a function of wavelength for the two linear retarders individually (green and blue) and the combined system with their fast axes aligned (red).

26.4Discontinuities in Unwrapped Retardance Values for Compound Retarder Systems with Arbitrary Alignment

Things become interesting with misaligned retarders, compound linear retarders whose fast axes are neither parallel nor perpendicular. Such retarders may result from misalignment or be intentionally aligned at arbitrary angles, as with Pancharatnam-design retarders.5–7 Sequences of linear retarders whose axes are neither parallel nor perpendicular are in general elliptical retarders as is seen in the next section and have discontinuous retardance spectra. In practice, the axes of compound retarders would always be expected to have a small misalignment, within some alignment tolerance.

26.4.1Compound Retarder Jones Matrix Decomposition

If the fast axis of the second retarder is slightly misaligned from that of the first retarder (it always will be in practice), the net retardance shows a different behavior from the aligned system. For example, if the two horizontal fast axis linear retarders with retardance δ1(λ) and δ2(λ) are misaligned by θ, the Jones matrix of the system is

$$\begin{array}{l}{J}_{\mathrm{Total}}=J_1{J}_2\\ \phantom{J_{Total}}=e^{\frac{-i({\delta }_1+{\delta }_2)}{2}}\left(\begin{array}{ll}{\cos }^2(\theta )+e^{i{\delta }_2}{\sin }^2(\theta )& (1-e^{i{\delta }_2})\cos (\theta )\sin (\theta )\\ e^{i{\delta }_1}(1-e^{i{\delta }_2})\cos (\theta )\sin (\theta )& e^{i{\delta }_1}(e^{i{\delta }_2}{\cos }^2(\theta )+{\sin }^2(\theta ))\end{array}\right).\end{array}$$26.5

26.5

Using Equation 14.77, the principal retardance is

$$\delta =2\arccos \left[\cos \left(\frac{{\delta }_1+{\delta }_2}{2}\right){\cos }^2\left(\theta \right)+\cos \left(\frac{{\delta }_1-{\delta }_2}{2}\right)si{n}^2(\theta )\right],$$26.6

26.6

which reduces to δ = δ1 + δ2 if θ → 0 and δ = δ1 − δ2 if θ → π/2. Equation 26.6 shows that the retardance of the system depends on θ. If both retarders are half wave retarders with horizontal fast axes, the total retardance will be 2π (one wave) when the two fast axes are aligned, and the total retardance will be 0 when the fast axes are orthogonal. Figure 26.13 shows how the total retardance of the system changes as the second retarder’s fast axis orientation (θ) varies.

Figure 26.13The principal retardance for two combined half wave linear retarders as a function of the fast axis orientation (θ) of the second retarder with respect to the first retarder.

Note that in Figure 26.13, zero principal retardance occurs three times but they imply different total retardance; at θ = π/2, zero principal retardance means zero total retardance but at θ = 0 and π, the total retardance is 2π.

J1 and J2 in Equation 26.5 can be written as the sum of Pauli matrices,

$$\begin{array}{l}{J}_1=L{R}_1({\delta }_1,0)=\cos \left(\frac{{\delta }_1}{2}\right){\mathbf{\sigma }}_0-i\sin \left(\frac{{\delta }_1}{2}\right){\mathbf{\sigma }}_1\text{and}\\ J_2=L{R}_2({\delta }_2,\theta )=\cos \left(\frac{{\delta }_2}{2}\right){\mathbf{\sigma }}_0-i\cos (2\theta )\sin \left(\frac{{\delta }_2}{2}\right){\mathbf{\sigma }}_1-i\sin (2\theta )\sin \left(\frac{{\delta }_2}{2}\right){\mathbf{\sigma }}_2,\end{array}$$26.7

26.7

where LR(δ, θ) is a linear retarder with a principal retardance δ and its fast axis along θ. The apparent retardance discontinuities (Figure 26.3) are explained by separating the JTotal in Equation 26.5 into two parts, a θ-independent Jones matrix JMajor and θ-dependent Jones matrix JMinor,

$$J_{\mathrm{Total}}=L{R}_1({\delta }_1,0)L{R}_2({\delta }_2,\theta )=J_{\mathrm{Major}}{J}_{\mathrm{Minor}}.$$26.8

26.8

JMajor matrix is a θ-independent part, a horizontal fast axis linear retarder with retardance δMajor = δ1 + δ2,

$$J_{\mathrm{Major}}=\mathrm{LR}({\delta }_1+{\delta }_2,0)$$26.9

26.9

and JMinor is a θ-dependent part,

$$\begin{array}{l}{J}_{\mathrm{Minor}}=c_0\left[{\mathbf{\sigma }}_0+d_1{\mathbf{\sigma }}_1+d_2{\mathbf{\sigma }}_2+d_3{\mathbf{\sigma }}_3\right]\\ \phantom{J_{Minor}}=\left[{\cos }^2(\theta )+\cos ({\delta }_2){\sin }^2(\theta )\right][{\mathbf{\sigma }}_0+\frac{i}{\cot ({\delta }_2)+{\cot }^2(\theta )\csc ({\delta }_2)}{\mathbf{\sigma }}_1\\ -\frac{i\cot (\theta )}{\cot ({\delta }_2)+{\cot }^2(\theta )\csc ({\delta }_2)}{\mathbf{\sigma }}_2+\frac{i{\sin }^2({\delta }_2\text{/}2)sin(2\theta )}{{\cos }^2(\theta )+{\sin }^2(\theta )\cos ({\delta }_2)}{\mathbf{\sigma }}_3].\end{array}$$26.10

26.10

JMinor is a retarder since the coefficients d1, d2, and d3 are pure imaginary (see Chapter 14 [Jones Matrix Data Reduction with Pauli Matrices]). Using Equation 14.77, the retardance of JMinor is

$${\delta }_{\mathrm{Minor}}=2\arctan \left[\sqrt{{\csc }^2\left(\frac{{\delta }_1}{2}\right){\csc }^2(\theta )-1}\right].$$26.11

26.11

Note that JMajor contains the phase difference between the FF and SS modes in Figure 26.1 while JMinor contains the phase difference between the FS and SF modes. As shown in the previous section, JMajor always has continuous unwrapped retardance. Thus, the discontinuities in the unwrapped retardance of JTotal come from JMinor. Note that the amplitude of JMinor has θ dependence; the discontinuity becomes maximum at θ = π/4. Figure 26.14 shows the principal retardance of JMajor in red and JMinor in blue when θ = 10°.

Figure 26.14The principal retardance of JMajor (red) and JMinor (blue) is plotted for 10° fast axis misalignment.

As an example, consider our two laboratory linear retarders in sequence—a Sapphire retarder with retardance δ2 at 10° followed by a Quartz HLR with retardance δ1 at 0°. The compound retarder system has a Jones matrix

$$\begin{array}{l}{J}_{\mathrm{Total}}=L{R}_1({\delta }_1,0)L{R}_2({\delta }_2,{10}^{\circ })\\ \phantom{J_{Total}}=e^{\frac{-i({\delta }_1+{\delta }_2)}{2}}\left(\begin{array}{ll}{\cos }^2({10}^{\circ })+e^{i{\delta }_2}{\sin }^2({10}^{\circ })& (1-e^{i{\delta }_2})\cos ({10}^{\circ })\sin ({10}^{\circ })\\ e^{i{\delta }_1}(1-e^{i{\delta }_2})\cos ({10}^{\circ })\sin ({10}^{\circ })& e^{i{\delta }_1}\left[e^{i{\delta }_2}{\cos }^2({10}^{\circ })+{\sin }^2({10}^{\circ })\right]\end{array}\right).\end{array}$$26.12

26.12

Using the simple dispersion model for retardance, the simulated principal retardance and the fast axis orientation of the combination (JTotal) are plotted in Figure 26.15, which also shows the measured principal retardance and the fast axis orientation yielding a close match between simulated values and measurements. Coming from the right side, the principal retardance increases to π (half wave), and then the fast axis changes to vertical and the principal retardance decreases. The principal retardance only decreases to 0.7 before turning around and increases to π a second time, the second maximum from the right. At this point, the retardance (δ) is expected to be 3π, corresponding to the second maxima in the top figure in Figure 26.8; however, the retardance never reaches 0, which would correspond to δ = 2π. Thus, it appears as if the retardance has passed from π to 3π without passing through 2π!

Figure 26.15(Lines) Simulated principal retardance (blue) and the fast axis orientation (dark blue) of a system of JTotal, a Sapphire retarder with fast axis at 10° followed by a Quartz retarder with horizontal fast axis are plotted as a function of wavelength. Green circles indicate regions where the principal retardance changes its slope without going to zero, thus avoiding one wave, two waves, and so on, retardance values. (Dots) Measured principal retardance (blue) and the fast axis orientation (dark blue) of the system are also plotted as a function of wavelength.

26.4.2Compound Retarder’s Trajectory in Retarder Space

Full wave retarders (2nπ retardance) have Jones matrices, which are the identity matrix, corresponding to the origin (0, 0, 0) of the retarder space. The trajectory of the principal retardance of the example compound system repeatedly misses the origin of the retarder space as the wavelength scans, as shown in Figure 26.16. The trajectories’ segments from π to π leaving one side of the π sphere jump to the other side as the wavelength decreases. Each segment shows the trajectory as it approaches the π sphere boundary. The left figure’s trajectory starts at λ = 0.74 μm and moves as λ becomes shorter, and the right figure’s trajectory ends at λ = 0.45 μm. Unlike the aligned system’s retarder space trajectory in Figure 26.10, the compound system’s retarder space trajectory repeatedly misses the origin, which indicates discontinuities in unwrapped retardance; points closest to the origin in Figure 26.16 middle and right are where the discontinuity happens, although the retarder space trajectory is not discontinuous. When unwrapped in retarder space, the discontinuities at integer numbers of waves become obvious.

Figure 26.16The retarder space inside the π sphere (purple sphere) and the principal retardance component trajectories are plotted as the wavelength decreases (left) λ = 0.74 to 0.64 μm, (middle) λ = 0.64 to 0.515 μm, and (right) λ = 0.515 to 0.45 μm. All figures have the same color scheme as Figure 26.10; red → yellow → green → blue → purple. The left figure’s ending point is where the retardance is π. The middle figure starts from the symmetric point of the ending point of the left figure, and so on. Note that the trajectories for the second and third figures miss the origin.

Using the principal retardance of JTotal and the fast axis orientation, a retardance can be assigned by using the retardance unwrapping algorithm in Equation 26.4. Discontinuities are clearly visible when the unwrapped retardance for the misaligned (blue) and aligned (red) systems are plotted together in Figure 26.17. The blue plot has similar values as the red plot since the misalignment is small. However, the blue plot has discontinuities whenever the retardance value crosses 2nπ boundaries; that is, the unwrapped retardance of the compound system increases from π to 3π without passing through 2π, the point of origin. The discontinuities come from non-zero amplitude in JMinor, that is, the existence of multiple modes exiting the compound system, which will be discussed in the next section.

Figure 26.17The unwrapped retardance plotted as a function of wavelength for the aligned (red) and misaligned (blue) two-retarder system.

26.4.3Multiple Modes Exit the Compound Retarder System

Another view of the multi-order retarders considers each of the four ray paths as a different polarizer, with the output beams interfering upon exit. Thus, a ray propagating through a retarder can be modeled as coherent addition of two rays exiting from two polarizers with corresponding OPLs due to birefringence. A retarder has two modes, the fast and slow modes. For a misaligned system of two retarders, four modes exit, F1F2, F1S2, S1F2, and S1S2, where F and S stand for the fast and slow modes, and 1 and 2 stand for the first and the second retarders. At normal incidence, F1F2, F1S2, S1F2, and S1S2, lie on top of each other. When θ is close to zero, two of the modes (F1F2 and S1S2) have most of the intensity and the other two modes (F1S2 and S1F2) are weak. The F1F2 and S1S2 modes for JTotal in Equation 26.12, that is, θ = 10°, together have cos2(10°) ≈ 97% of the intensity and the other two modes have sin2(10°) ≈ 3% of the intensity. Therefore, the unwrapped retardance is similar to the aligned system. However, whenever the phase difference between F1F2 and S1S2 modes is a multiple of 2π (full waves of retardance), effects from the other two modes become substantial, creating the discontinuities as shown in Figure 26.17.

The Jones matrix for an HLR can be represented as a sum of horizontal and vertical polarizers with OPLs along the fast and slow axes, OPLf1 and OPLs1, respectively,

$$J_{HLR}\text{ = }{e}^{+i\frac{2\pi }{\lambda }OP{L}_{f1}}\left(\begin{array}{ll}1& 0\\ 0& 0\end{array}\right)+e^{+i\frac{2\pi }{\lambda }OP{L}_{s1}}\left(\begin{array}{ll}0& 0\\ 0& 1\end{array}\right).$$26.13

26.13

Thus, a linear retarder at θ is equal to a sum of linear polarizers at θ and θ + π/2 with absolute phases equal to the OPLs along the fast and slow axes, respectively,

$$\begin{array}{l}LR(\theta ,\mathrm{OP}{L}_{f2},OP{L}_{s2})\\ \phantom{}=\frac{e^{+i\frac{2\pi }{\lambda }OP{L}_{f2}}}{2}\left(\begin{array}{ll}1+\cos (2\theta )& \sin (2\theta )\\ \sin (2\theta )& 1-\cos (2\theta )\end{array}\right)+\frac{e^{+i\frac{2\pi }{\lambda }OP{L}_{s2}}}{2}\left(\begin{array}{ll}1-\cos (2\theta )& -\sin (2\theta )\\ -\sin (2\theta )& 1+\cos (2\theta )\end{array}\right).\end{array}$$26.14

26.14

Therefore, the Jones vector for each mode can be calculated from the matrix multiplication of two linear polarizers with associated OPLs as absolute phases. For example, the first mode (F1F2) is a horizontal linear polarizer followed by a linear polarizer at θ with associated absolute phases, yielding the Jones vector

$${\mathbf{\text{X}}}_1=F_1{F}_2=e^{+i(OP{L}_{f1}+OP{L}_{f2})}\left(\begin{array}{l}\cos (\theta )\\ \sin (\theta )\end{array}\right).$$26.15

26.15

Similarly, the other three modes’ Jones vectors are

$$\begin{array}{l}{\mathbf{\text{X}}}_2=S_1{F}_2=e^{+i(OP{L}_{s1}+OP{L}_{f2})}\left(\begin{array}{l}\cos (\theta )\\ \sin (\theta )\end{array}\right)\\ {\mathbf{\text{X}}}_3=F_1{S}_2=e^{+i(OP{L}_{f1}+OP{L}_{s2})}\left(\begin{array}{l}-\sin (\theta )\\ \cos (\theta )\end{array}\right)\\ {\mathbf{\text{X}}}_4=S_1{S}_2=e^{+i(OP{L}_{s1}+OP{L}_{s2})}\left(\begin{array}{l}-\sin (\theta )\\ \cos (\theta )\end{array}\right).\end{array}$$26.16

26.16

For simplicity, one OPL or absolute phase can be set to zero; here, OPLs is chosen. This does not affect the retardance calculation. Using Equation 26.2, each mode’s phase is

$$\begin{array}{l}\arg ({\mathbf{\text{X}}}_1)=OP{L}_{f1}+OP{L}_{f2}=\frac{{\delta }_1{\lambda }_0}{\lambda }+\frac{{\delta }_2{\lambda }_0}{\lambda }\\ \arg ({\mathbf{\text{X}}}_2)=OP{L}_{s1}+OP{L}_{f2}=0+\frac{{\delta }_2{\lambda }_0}{\lambda }\\ \arg ({\mathbf{\text{X}}}_3)=OP{L}_{f1}+OP{L}_{s2}=\frac{{\delta }_1{\lambda }_0}{\lambda }+0\\ \arg ({\mathbf{\text{X}}}_4)=OP{L}_{s1}+OP{L}_{s2}=0+0=0,\end{array}$$26.17

26.17

where δi is the retardance of the ith retarder at λ0.

The retardance is a function of the amplitude and OPD between the four modes. When the misalignment is small, most of the intensity of the exiting light has the phases in X1 and X4. Therefore, the retardance of the compound system as a function of wavelength follows the curve of the OPD between X1 and X4,

$$\begin{array}{l}{\delta }_{\mathrm{major}}=\arg ({\mathbf{\text{X}}}_1)-\arg ({\mathbf{\text{X}}}_4)\\ \phantom{{\delta }_{major}}=OP{L}_{f1}+OP{L}_{f2}-OP{L}_{s1}-OP{L}_{s2}=\frac{{\lambda }_0({\delta }_1+{\delta }_2)}{\lambda }.\end{array}$$26.18

26.18

However, whenever δmajor becomes 2nπ, effects from the OPLs of the other two modes (X2 and X3) increase, and the retardance of the compound system deviates from δmajor.

26.4.4Compound Retarder Example at 45°

In this section, the properties of a Sapphire retarder with the fast axis at 45° followed by a Quartz retarder with the fast axis at 0° is simulated and measured. Section 26.6 (Appendix) shows the principal retardance for a Sapphire retarder with the slow axis at θ followed by a Quartz retarder with the fast axis at 0° as θ varies from 0° to 90°. Figure 26.18 compares a simulation and measured principal retardance and fast axis orientation as a function of wavelength of the compound retarder. Both simulated and measured principal retardance plots have minima far from zero, which indicates that this system will have large discontinuities in unwrapped retardance values near 2nπ.

Figure 26.18Simulated (lines) and measured (dots) principal retardance and the fast axis orientation of a system of a Sapphire retarder with fast axis at 45° followed by a Quartz retarder with horizontal fast axis are plotted as a function of wavelength.

To better understand the behavior of the fast axis orientation of the compound system, Figure 26.19 shows the Sapphire (blue) and Quartz (green) retarders’ retardance as a function of wavelength along with the fast axis orientation of the compound system, θfast. When the retardance of the Sapphire retarder becomes 2nπ, the fast axis of the system is aligned with the Quartz retarder, that is, θfast = 0. When the retardance of the Quartz retarder becomes 2nπ, the system’s fast axis is aligned with the Sapphire retarder, that is, θfast = π/4.

Figure 26.19The unwrapped retardance of the Quartz (green) and Sapphire (blue) retarders with the fast axis orientation (θfast) of the compound system are plotted. The × symbols mark wavelengths where individual plates have integer waves of retardance and don’t contribute to the axis of the retarder.

Using Equation 26.4, the principal retardance can be unwrapped assuming that the overall behavior of the retardance is 1/λ. Figure 26.20 shows the unwrapped retardance of the Quartz HLR (green), a 45° fast axis Sapphire retarder (blue), and the combined system (orange). Each mode in this system has 25% of the total intensity, and thus the discontinuities in the unwrapped retardance are much larger than Figure 26.15.

Figure 26.20The green line shows the retardance of the Quartz retarder, δ1(λ) as a function of wavelength. The blue line is the retardance of the 45° fast axis Sapphire retarder, δ2(λ). The orange line shows the retardance of the combined system, δTotal(λ).

Note that the discontinuities in the orange line occur whenever the retardance of one of the retarders is 2nπ; when one of the retarders has multiple waves of retardance, interferences occur between the two modes in the same polarization, that is, interference between S1F2 and F1F2, as well as interference between F1S2 and S1S2.

The retardance space trajectory for the principal retardance of the example system is shown in Figure 26.21. For clarity, the top and side views reveal the 3D character of the space curve. The trajectory starts at point A (0.675, 1.463, 0.675) and crosses the inside of the π sphere; once the trajectory reaches the opposite boundary of the sphere (point B), it jumps to the opposite point (point B′) on the π sphere and the fast axis changes to the orthogonal state. Thus, points B and B′ correspond to the same Jones matrix and retarder components. The trajectory follows A → B → B′ → C → C′ → D → D′ → E → E ′→ F → F′ → G → G′ → H→ H′ → I → I′ → J. Point A corresponds to the retardance for the longest wavelength and point J corresponds to the retardance for the shortest wavelength.

Figure 26.21A principal retardance trajectory of the compound retarder. The two left columns show the trajectories in one viewpoint and the two right columns show the trajectories viewed from δR. When this trajectory reaches the boundary of π, the trajectory jumps to the opposite point on the π sphere and the fast axis changes to the orthogonal state.

26.5Conclusion

The mysterious behavior of compound retarders and their discontinuity is explained by the interference of the multiple modes exiting the system. Using the wavelength dependence of the retarders and the fast axis orientation, an algorithm to unwrap the principal retardance is derived. Discontinuities occur when the unwrapped retardance of compound retarders avoids the values around 2nπ. This is because the trajectory of the principal retardance in the retardance space can easily avoid the origin, which is just a single point representing the identity matrix. But for the unwrapped retardance, this point at the origin becomes the entire 2π sphere, 4π sphere, and so on. As the retardance increases, the trajectory appears to jump discontinuously across the 2nπ sphere in the retarder space.

For small misalignment of fast axis orientation in a compound retarder, the trajectory passes close to the origin; thus, the discontinuities are small. These compound multi-order retarders have a single Jones and Mueller matrix and perform a single rotation on the Poincaré sphere. However, because of the multiple exiting beams, they don’t exist in a single state of “retardance.”

Although this chapter only showed examples of two waveplates, systems with multiple waveplates can be understood in a similar logic; for N waveplates, there will be 2N different OPLs and the mathematics is analogous. Also, understanding the multi-order nature of the retardance provides greater insights on understanding liquid crystals, biaxial films, and optical fiber.

26.6Appendix

This section shows Figure 26.22, the principal retardance for a set of compound retarder systems.

Figure 26.22The principal retardance is plotted as a function of wavelength for Quartz and Sapphire retarders (green and blue) and a compound retarder (red), a Sapphire retarder with the slow axis at θ followed by a Quartz retarder with the fast axis at 0°, as the angle between the two fast axes θ varies.

Whenever the principal retardance of the misaligned HLR system has minima other than zero, the unwrapped retardance has discontinuity. At θ = 90°, the fast axes of two HLRs are aligned to each other since Quartz is a positive uniaxial material and Sapphire is a negative uniaxial material. Thus, the total retardance at θ = 90° is

$${\delta }_{\mathrm{Total}}(\lambda )={\delta }_{\mathrm{Quartz}}(\lambda )+{\delta }_{\mathrm{Sapphire}}(\lambda ),$$26.19

26.19

where δQuartz(λ) > 0 and δSapphire(λ) > 0.

26.7Problem Sets

1. Consider a compound retarder system consisting of two retarders with slight misalignment. What happens to the unwrapped retardance when one of the retarders is an integer number of retardance?

2. What if we have two birefringent wedges instead of two retarders? Now you are using the thickness instead of a wavelength to unwrap the retardance. How does the retardance unwrap?

3. Given the Mueller matrix spectrum in Figure 26.23, find the retardance magnitude and orientation through wavelength and then unwrap its retardance.

4. Given the Mueller matrix spectrum in Figure 26.24, plot the retardance trajectory in retarder space.

5. What are the retardance and fast axis orientation when two one-wave retarders are aligned at θ = 0°, 45°, 90°?

6. Plot the trajectory of a horizontal fast axis one-wave retarder in retarder space as the fast axis of the retarder rotates.

7. How does the gap, shown in Figure 26.17, change as you rotate the second retarder?

8. Why do multi-wave retarders avoid 2nπ?

Figure 26.23Mueller matrix spectrum.

Figure 26.24Mueller matrix spectrum.

References

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