Arbitrary complex retarders using a sequence of spatial light modulators as the basis for adaptive polarisation compensation

As a starting point, we consider a four pass SLM system that was previously shown to be sufficient to convert between two fixed arbitrary vectorial states [23]. Alternatively, the same effect could be achieved using a sequence of three SLMs and a DM. The sequence is shown in figure 1. For brevity, we will refer to these devices just as 'SLMs'. The axes of the SLMs are oriented in sequence at 0°, 45° and 0°. All of the SLMs are assumed to be in planes optically conjugate to the system pupil, so that their effects can be modelled in terms of the 'Jones pupil', onto which all spatially variant retardance effects in the system can be mapped. The Jones matrix ${{\mathbf{J}}_{\text{T}}}$ of the combination represented in this pupil is given by the product of the four Jones matrices of each of the individual elements.

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Note that all retardances and related variables used here are functions of position in the Jones pupil. For notational simplicity, this spatial variation is not written explicitly

$$\begin{equation}{{\mathbf{J}}_{\text{T}}} = {{\mathbf{J}}_{{\text{DM}}}}{{\mathbf{J}}_{{\text{SLM}}3}}{{\mathbf{J}}_{{\text{SLM}}2}}{{\mathbf{J}}_{{\text{SLM}}1}}.\end{equation} \tag{ 1 }$$

The individual Jones matrices are presented here, where the offset phase corresponding to the fixed phase shift at the non-modulating axis has been neglected

$$\begin{align} {{\mathbf{J}}_{{\text{SLM}}1}} & = \left( {\begin{array}{*{20}{c}} {{{\text{e}}^{{\text{i}}{\psi _1}}}}&0 \\ 0&1 \end{array}} \right) \nonumber\\ {{\mathbf{J}}_{{\text{SLM}}2}}& = \displaystyle\frac{1}{2}\left( {\begin{array}{*{20}{c}} {{{\text{e}}^{{\text{i}}{\psi _2}}} + 1}&{{{\text{e}}^{{\text{i}}{\psi _2}}} - 1} \\ {{{\text{e}}^{{\text{i}}{\psi _2}}} - 1}&{{{\text{e}}^{{\text{i}}{\psi _2}}} + 1} \end{array}} \right) \nonumber\\ {{\mathbf{J}}_{{\text{SLM}}3}}& = \left( {\begin{array}{*{20}{c}} {{{\text{e}}^{{\text{i}}{\psi _3}}}}&0 \\ 0&1 \end{array}} \right) \nonumber\\ {{\mathbf{J}}_{{\text{DM}}}}& = {{\text{e}}^{{\text{i}}{\psi _{{\text{DM}}}}}}\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right) \end{align} \tag{ 2 }$$

${\psi _1}$, ${\psi _2}$, and ${\psi _3}$ are the retardances of SLM1, SLM2 and SLM3, respectively. ${\psi _{{\text{DM}}}}$ is the phase introduced by the DM. As the DM is polarisation insensitive, the Jones matrix is simply defined here as a complex scalar variable multiplied by the identity matrix. We note that this Jones matrix strictly speaking describes a transmissive phase element, rather than a reflective element, but we choose this representation for mathematical clarity. As the retardance of these devices is spatially variable, each of the phase values is a function of the spatial coordinates across the device, although this dependence is omitted from the expressions for the sake of clarity.

The Jones matrix ${{\mathbf{J}}_{\text{E}}}$ of a generalised elliptical retarder can be expressed as

$$\begin{equation}\small{{\mathbf{J}}_{\text{E}}} = {{\text{e}}^{ - {\text{i}}\phi /2}}\left[ {\begin{array}{*{20}{c}} {{{\cos }^2}\theta + {{\text{e}}^{{\text{i}}\phi }}{{\sin }^2}\theta }&{\left( {1 - {{\text{e}}^{{\text{i}}\phi }}} \right){{\text{e}}^{ - {\text{i}}\alpha }}\cos \theta \sin \theta } \\ {\left( {1 - {{\text{e}}^{{\text{i}}\phi }}} \right){{\text{e}}^{{\text{i}}\alpha }}\cos \theta \sin \theta }&{{{\sin }^2}\theta + {{\text{e}}^{{\text{i}}\phi }}{{\cos }^2}\theta } \end{array}} \right]\end{equation} \tag{ 3 }$$

where $\theta$ is the ellipse orientation, $\phi$ is the retardance, and $\alpha$ is the parameter that determines the ellipticity of the eigenstate. Note that this retardance $\phi$ needs to be equal to the negative of the retardance of the retarder for which we need to compensate, so that the Jones matrix ${{\mathbf{J}}_{\text{E}}}$ conjugate to the perturbation retarder. In order for the SLM system to perfectly mimic the elliptical retarder, it is necessary for the Jones matrices ${{\mathbf{J}}_{\text{T}}}$ and ${{\mathbf{J}}_{\text{E}}}$ to be equal. It is a non-trivial task to determine the required retardances directly from these expressions. Hence, we have developed a geometrical approach involving transitions on the PS in order to obtain the necessary SLM settings.

For the two retarders to be equivalent, their eigensystems must also be equivalent. Hence, the task can be reduced to the problem of setting the SLM retardances so that the eigenvectors and eigenvalues in each case are equal. The eigenvectors are equivalent to the waveplate axes and the retardance is given by the difference between the phases of the eigenvalues.

In order to determine the eigenvalues ${\mu _1}$ and ${\mu _2}$, let us define the Jones matrix of the three SLMs as

$$\begin{eqnarray} &&{{\mathbf{J}}_{\text{S}}} = {{\mathbf{J}}_{{\text{SLM}}3}}{{\mathbf{J}}_{{\text{SLM}}2}}{{\mathbf{J}}_{{\text{SLM}}1}} \end{eqnarray}$$

$$\begin{equation} = \frac{1}{2}\left[ {\begin{array}{*{20}{c}} {\left( {{{\text{e}}^{{\text{i}}{\psi _2}}} + 1} \right){{\text{e}}^{{\text{i}}({\psi _1} + {\psi _3})}}}& {\left( {{{\text{e}}^{{\text{i}}{\psi _2}}} - 1} \right){{\text{e}}^{{\text{i}}{\psi _3}}}} \\ {\left( {{{\text{e}}^{{\text{i}}{\psi _2}}} - 1} \right){{\text{e}}^{{\text{i}}{\psi _1}}}}& {{{\text{e}}^{{\text{i}}{\psi _2}}} + 1} \end{array}} \right].\end{equation} \tag{ 4 }$$

The dynamic phase offset ${{\Theta }}$ is given by the argument of the determinant of the Jones matrix, which is readily obtained as

$$\begin{equation}{{\Theta }} = \frac{1}{2}\arg \left( {\det {{\mathbf{J}}_{\text{S}}}} \right) = \frac{1}{2}\left( {{\psi _1} + {\psi _2} + {\psi _3}} \right).\end{equation} \tag{ 5 }$$

We adjust equation (4) by removing this phase factor to give

$$\begin{equation}{{\mathbf{J}}_{\text{S}}} = {{\text{e}}^{{{\text{i}\Theta }}}}{{\mathbf{J{^{^{\prime}}}}}_{\text{S}}}\end{equation} \tag{ 6 }$$

where

$$\begin{equation}{{\mathbf{J{^{^{\prime}}}}}_{\text{S}}} = \frac{1}{2}\left[ {\begin{array}{*{20}{c}} {\left( {{{\text{e}}^{{\text{i}}{\psi _2}/2}} + {{\text{e}}^{ - {\text{i}}{\psi _2}/2}}} \right){{\text{e}}^{{\text{i}}\left( {{\psi _1} + {\psi _3}} \right)/2}}}&{\left( {{{\text{e}}^{{\text{i}}{\psi _2}/2}} - {{\text{e}}^{ - {\text{i}}{\psi _2}/2}}} \right){{\text{e}}^{{\text{i}}\left( {{\psi _3} - {\psi _1}} \right)/2}}} \\ {\left( {{{\text{e}}^{{\text{i}}{\psi _2}/2}} - {{\text{e}}^{ - {\text{i}}{\psi _2}/2}}} \right){{\text{e}}^{{\text{i}}\left( {{\psi _1} - {\psi _3}} \right)/2}}}&{\left( {{{\text{e}}^{{\text{i}}{\psi _2}/2}} + {{\text{e}}^{ - {\text{i}}{\psi _2}/2}}} \right){{\text{e}}^{ - {\text{i}}\left( {{\psi _1} + {\psi _3}} \right)/2}}} \end{array}} \right]\end{equation} \tag{ 7 }$$

has zero dynamic phase. The eigenvectors and eigenvalues of ${{\mathbf{J{^{^{\prime}}}}}_{\text{S}}}$ can thus be found as the solution to the equation

$$\begin{equation}\small{\mu ^2} - \frac{1}{2}\left( {{{\text{e}}^{{\text{i}}{\psi _2}/2}} + {{\text{e}}^{ - {\text{i}}{\psi _2}/2}}} \right)\left( {{{\text{e}}^{{\text{i}}({\psi _1} + {\psi _3})/2}} + {{\text{e}}^{ - {\text{i}}\left( {{\psi _1} + {\psi _3}} \right)/2}}} \right)\mu + 1 = 0\end{equation} \tag{ 8 }$$

or alternatively

$$\begin{equation}{\mu ^2} - 2{\text{cos}}\left( {\frac{{{\psi _2}}}{2}} \right){\text{cos}}\left( {\frac{{{\psi _1} + {\psi _3}}}{2}} \right)\mu + 1 = 0.\end{equation} \tag{ 9 }$$

Solving this quadratic equation gives the two eigenvalues as

$$\begin{equation}{\mu _{1,2}} = Y \pm {\text{i}}\sqrt {1 - {Y^{\,2}}} \end{equation} \tag{ 10 }$$

with

$$\begin{equation}Y = {\text{cos}}\left( {\frac{{{\psi _2}}}{2}} \right){\text{ cos}}\left( {\frac{{{\psi _1} + {\psi _3}}}{2}} \right).\end{equation} \tag{ 11 }$$

The retardance ${{\Psi }}$ of the equivalent retarder described by ${{\mathbf{J{^{^{\prime}}}}}_{\text{S}}}$ is given by

$$\begin{align} {{\Psi }} & = {\text{arg}}\left( {{\mu _1}\mu _2^*} \right) = \arg \left( {{\mu _1}} \right) - \arg \left( {{\mu _2}} \right)\nonumber\\ & = 2{\text{atan}}\left( {\frac{{\sqrt {1 - {Y\,^{2}}} }}{Y}{ }} \right) = 2{\text{acos}}Y. \end{align} \tag{ 12 }$$

This provides a relationship between the retardance of the SLM combination and the settings of the individual SLMs. However further conditions must be placed on the eigenvectors for the SLM combination to be equivalent to the chosen retarder.

There is insufficient information in the expressions shown above to make the Jones matrix of the three-SLM system equivalent to a chosen arbitrary retarder. However, the necessary relationships can be obtained through geometrical considerations concerning the eigenvectors.

We need to note that if the input to the system is an eigenmode, then the output will be the same eigenmode. When an input polarisation state is transformed by a sequence of retarders, the state will evolve along a path on the PS. If the path is closed—i.e. when the output state is the same as the input state—then the input state must be aligned with an eigenvector of the system. For our system to be equivalent to the chosen retarder, we must therefore ensure that the sequence of SLMs causes a closed path to be traced out on the PS that starts from and returns to the point represented by the eigenvector of the retarder. As the two eigenvectors of a retarder are diametrically opposed on the PS, by symmetry, a sequence of SLMs that achieves a closed path for one eigenvector must also achieve the same for the other eigenvector.

Following progression through the sequence of SLMs, the eigenmode acquires a phase shift, which is given by the argument of the corresponding eigenvalue. These phase shifts will generally not be the same for the two eigenmodes, but the difference between the phases is equivalent to the retardance of the system.

We illustrate this principle using the three-SLM system to act as a linear retarder of retardance $\varphi$, whose extraordinary axis is oriented at an angle $\theta$. The primary eigenvector of this retarder is represented by a point on the equator (${S_3} = 0$ plane) of the PS at a longitude of $2\theta$, shown at point A.

Figure 2 shows how a closed path ABCA can be constructed using three arcs: AB is due to the action of the first SLM, whose primary eigenvector is aligned with the ${S_1}$ axis; BC is due to the second SLM with eigenvector along the ${S_2}$ axis; CA is from the third SLM with eigenvector along the ${S_1}$ axis. Note that point C is the reflection of point B in the ${S_3} = 0$ plane. The angles subtended by the arcs at their respective eigenmode axes are equal to the retardances of the SLMs, respectively ${\psi _1}$, ${\psi _2}$, and ${\psi _3}$. The enclosed area is equivalent to the intersection of two spherical caps, one centred on the ${S_1}$ axis and the other on the ${S_2}$ axis. The symmetry of the path about the equator shows that ${\psi _1} = {\psi _3}$. The retardance of the second SLM can found from geometry (figure 3) as

$$\begin{equation}{\psi _2} = 2{\text{atan}}\left[ {\tan \left( {2\theta } \right)\sin \left( \beta \right)} \right].\end{equation} \tag{ 13 }$$

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Where we have set $\beta = {\psi _1} = {\psi _3}$.

The choice of closed path is not unique, as the point B can be chosen to lie on any point on the circle that passes through A and is in a plane perpendicular to the axis ${S_1}$. This degree of freedom permits control of the equivalent retardance of the system. As the retardance is the difference in phase between the two eigenvalues, the position of B can be chosen to ensure that the system retardance is equivalent to that of the desired waveplate.

We show in this section how one can obtain the value of $\beta$ necessary to implement a chosen retardance. Using the condition $\beta = {\psi _1} = {\psi _3}$, from equation (12), we can obtain the following relationship between the retardance ${{\Psi }}$ of the three-SLM system and other parameters:

$$\begin{equation}\cos \left( {\frac{{{\Psi }}}{2}} \right) = {\text{cos}}\left( {\frac{{{\psi _2}}}{2}} \right){\text{ cos}}\left( \beta \right).\end{equation} \tag{ 14 }$$

From equation (13) we have

$$\begin{equation}\tan \left( {\frac{{{\psi _2}}}{2}} \right) = {\text{tan}}\left( {2\theta } \right)\sin \left( \beta \right).\end{equation} \tag{ 15 }$$

Through a sequence of trigonometric manipulations (see appendix A) one can eliminate ${\psi _2}$ to obtain the relationship:

$$\begin{equation}\tan \left( \beta \right) = {\text{ tan}}\left( {\frac{{{\Psi }}}{2}} \right){\text{cos}}\left( {2\theta } \right).\end{equation} \tag{ 16 }$$

We hence find that

$$\begin{equation}\beta = {\psi _1} = {\psi _3} = {\text{atan}}\left[ {{\text{tan}}\left( {\frac{{{\Psi }}}{2}} \right){\text{cos}}\left( {2\theta } \right)} \right].\end{equation} \tag{ 17 }$$

We can also find an explicit expression relating ${\psi _2}$ directly to the retardance and waveplate angle as (see appendix B):

$$\begin{equation}\sin \left( {\frac{{{\psi _2}}}{2}} \right) = \sin \left( {\frac{{{\Psi }}}{2}} \right)\sin \left( {2\theta } \right).\end{equation} \tag{ 18 }$$

So that

$$\begin{equation}{\psi _2} = 2{\text{asin}}\left[ {\sin \left( {\frac{{{\Psi }}}{2}} \right)\sin \left( {2\theta } \right)} \right].\end{equation} \tag{ 19 }$$

We therefore have explicit expressions for the SLM settings in terms of the desired retarder's properties.

Aside from the special cases where $\theta = \pm \pi /4$, which are explained below, there is a one-to-one mapping between ${{\Psi }}$ and $\beta$, in the range $- \frac{\pi }{2} < \left\{ {{{\Psi }},\beta } \right\} \leqslant \frac{\pi }{2}{ }$. Hence, it is clear that any retardance in the range $- \frac{\pi }{2} < {{\Psi }} \leqslant \frac{\pi }{2}\,$ can be achieved with an appropriate choice of $\beta$. Furthermore, retardances in the range $- \pi < {{\Psi }} \leqslant \pi$ are possible when one considers the full range of parameter values that are accessible.

Figure 4 (along with supplementary video 1 (available online at stacks.iop.org/JOPT/23/065602/mmedia)) shows how the retardance ${{\Psi }}$ varies with the choice of the system parameters $\beta$ and $\theta$, as plotted on the surface of the PS. The explicit expression for the retardance in terms of$\,\beta$ and $\theta$ is, from equation (16),

$$\begin{equation}{{\Psi }} = 2{\text{atan}}\left[ {\frac{{\tan \beta }}{{\cos 2\theta }}} \right].\end{equation} \tag{ 20 }$$

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The relationship between the retardance ${{\Psi }}$ and the settings of the three SLMs ${\psi _1}$, ${\psi _2}$ and ${\psi _3}$ can also be expressed geometrically by calculating the total area of spherical triangles formed by points A, B, C and the optical axes of the three SLMs on a PS (see appendix C). Numerical calculations (not included here) based on the Jones matrices of section 2.3 and of the expressions in appendix C have confirmed that these analytical expressions provide equivalent results.

The derivation above shows that the three SLM system can mimic the retardance of an arbitrary linear waveplate. However, the dynamic phase introduced by the system—given by $\frac{1}{2}{\text{arg det}}\left( {{{\mathbf{J}}_{\text{S}}}} \right)$—cannot be independently chosen. For single retarders that modulate the whole beam uniformly, this phase offset is not important. However, for an adaptive system that modulates the beam differently across its profile, phase offsets that differ between pixels correspond to wavefront aberration. Consequently, for full vectorial control we require an additional adaptive element to adjust the overall phase. The DM is included in the system for this purpose.

As the determinant of a matrix is given by the product of its eigenvalues, the dynamic phase can be calculated from the Jones matrix ${{\mathbf{J}}_{\text{S}}}$ based upon the SLM retardances using ${\psi _1} = {\psi _3} = \beta$ and ${\psi _2} = 2{\text{atan}}\left[ {\tan \left( {2\theta } \right)\sin \left( \beta \right)} \right]$ to give the expression

$$\begin{align} {{\Theta }} & = \frac{1}{2}\arg \left[ {{\mu _1}{\mu _2}} \right] = \frac{1}{2}\left( {{\psi _1} + {\psi _2} + {\psi _3}} \right)\nonumber\\ & = \beta + {\text{atan}}\left[ {\tan \left( {2\theta } \right)\sin \left( \beta \right)} \right]. \end{align} \tag{ 21 }$$

This phase would need to be compensated by the DM in order that the combined system behave as the modelled retarder. Figure 5 (along with supplementary video 2) shows how ${{\Theta }}$ varies with the choice of the system parameters $\beta$ and $\theta$, as plotted on the surface of the PS.

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There are two special cases of linear retarders that need to be addressed: one with an x-oriented axis, which would be co-aligned with the primary axis of SLM1 (and SLM3); another where the axis is at 45° and hence aligned with the axis of SLM2. For the x-oriented retarder, we can set ${\psi _2} = 0$ and the retardance would be ${\psi _1} + {\psi _3}$. In this case, no path is traced out on the PS, but the settings of SLM1 (and/or SLM3) can be used to access all retardances.

For the 45° retarder, we set ${\psi _1} = {\psi _3} = 0$ and have freedom to control the retardance purely using ${\psi _2}$. Other SLM settings are possible for ${\psi _1} = {\psi _3} \ne 0$, but in this case the retardance is always limited to $\pi$ radians.

For a general elliptical retarder, we have to consider eigenvectors corresponding to any arbitrary point A on the PS. We can use a geometric construction for the path ABCA that is similar that introduced above for the linear retarders. In this case though, the point A does not lie on the equator. There are two possibilities for the path shape, dependent upon whether the action of SLM1 is to move the state towards or away from the equator. These two cases are illustrated in figure 6. In case 1, the path shape is the same as for the linear retarder in the previous section, but with point A not lying on the equator. In case 2, the path AB due to SLM1 is retraced as part of the path CA due to SLM3.

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In this configuration ${\psi _1} \ne {\psi _3}$. We define D to be the point at which the arc AC crosses the equator, then the angle $\beta$ subtended by DB at the ${S_1}$ axis is given by

$$\begin{equation}\beta = \frac{{\left( {{\psi _1} + {\psi _3}} \right)}}{2} = {\text{atan}}\left( {\frac{{{B_3}}}{{{B_2}}}} \right)\end{equation} \tag{ 22 }$$

where ${B_2}$ and ${B_3}$ are the elements of the Stokes vector at B. We also define $\alpha$ to be the angle subtended by DA at the ${S_1}$axis, such that

$$\begin{equation}\alpha = \beta - {\psi _1} = \frac{{\left( {{\psi _3} - {\psi _1}} \right)}}{2} = {\text{atan}}\left( {\frac{{{A_3}}}{{{A_2}}}} \right)\end{equation} \tag{ 23 }$$

The arc ADC lies in a plane perpendicular to the ${S_1}$ axis, so the point D would represent a linear polarisation at orientation $\theta = \frac{1}{2}{\text{acos}}{A_1}$, where ${A_1}$ is the first element of the normalised Stokes vector at A. Note that ${A_1} = {B_1} = {C_1} = {D_1}\,\,$. We can therefore determine that the retardance of SLM2 should again be given by

$$\begin{equation}{\psi _2} = 2{\text{atan}}\left[ {\tan \left( {2\theta } \right)\sin \left( \beta \right)} \right].\end{equation} \tag{ 24 }$$

As for the linear retarder above, we have one degree of freedom—equivalent to the choice of the position of point B—that enables us to control the retardance of the equivalent waveplate.

When using the three-SLM system to simulate a single elliptical retarder, the same relationships between the desired retardance and $\theta$ and $\beta$ as derived in section 3 are again valid:

$$\begin{equation}\beta = {\text{atan}}\left[ {{\text{tan}}\left( {\frac{{{\Psi }}}{2}} \right){\text{cos}}\left( {2\theta } \right)} \right].\end{equation} \tag{ 25 }$$

We can therefore use this to derive the settings for the three SLMs. Using $\beta \, = \alpha + {\psi _1}$ we obtain the SLM settings as

$$\begin{equation}{\psi _1} = \beta - \alpha \end{equation} \tag{ 26 }$$

and

$$\begin{equation}\begin{array}{*{20}{l}} {{\psi _2} = 2{\text{atan}}\left[ {\tan \left( {2\theta } \right)\sin \left( \beta \right)} \right]} \\ {{\psi _3} = 2\alpha + {\psi _1} = \beta + \alpha } \end{array}.\end{equation} \tag{ 27 }$$

As the retardance is dependent only on the choice of point B and hence on $\theta$ and $\beta$, the plot of figure 4 is also valid for elliptical retarders.

Following the derivation in section 3, we calculate the dynamic phase from the Jones matrices based upon the SLM retardances using ${\psi _1} = \beta - \alpha$, ${\psi _2} = 2{\text{atan}}\left[ {\tan \left( {2\theta } \right)\sin \left( \beta \right)} \right]$, and ${\psi _3} = \beta + \alpha$ to give the equivalent expression

$$\begin{equation}{{\Theta }} = \beta + {\text{atan}}\left[ {\tan \left( {2\theta } \right)\sin \left( \beta \right)} \right].\end{equation} \tag{ 28 }$$

As the expression for dynamic phase is identical to that obtained for linear retarders, the plot of figure 5 is also valid for elliptical retarders.

For circular polarisation eigenmodes, where the point A lies on the ${S_3}$ axis, we can set ${\psi _1} = \pi /2$ and ${\psi _3} = - \pi /2$. The action of SLM1 would be to transform an eigenmode input to a state on the ${S_2}$ axis. SLM2 would then apply the arbitrarily chosen phase shift ${\psi _2}$. SLM3 would transform this state back to the original state at point A. The retardance of the overall system would be equal to ${\psi _2}$.

The total phase introduced by the three-SLM system can be calculated using following expressions [24]

$$\begin{equation}{\zeta _1} = {\phi _1} + {\phi _2} + {\phi _3} + {\phi _4}\end{equation} \tag{ 40 }$$

where ${\phi _1}$, ${\phi _2}$ and ${\phi _3}$ are the phases introduced by each SLM and ${\phi _4}$ is a Pancharatnam phase term. These can be calculated using the areas of spherical triangles as

$$\begin{align}\begin{array}{l} {\phi _1} = \frac{{{\Omega _{{\textrm{AQ}}\prime {\textrm{B}}}}}}{2} = \arctan \left[ {\frac{{{\textbf{A}} \cdot \left( {{\textbf{Q}}\prime \times {\textbf{B}}} \right)}}{{1 + {\textbf{A}} \cdot {\textbf{B}} + {\textbf{B}} \cdot {\textbf{Q}}\prime + {\textbf{Q}}\prime \cdot {\textbf{A}}}}} \right]\\ {\phi _2} = \frac{{{\Omega _{{\textrm{BR}}\prime {\textrm{C}}}}}}{2} = \arctan \left[ {\frac{{{\textbf{B}} \cdot \left( {{\textbf{R}}\prime \times {\textbf{C}}} \right)}}{{1 + {\textbf{B}} \cdot {\textbf{C}} + {\textbf{C}} \cdot {\textbf{R}}\prime + {\textbf{R}}\prime \cdot {\textbf{B}}}}} \right]\\ {\phi _3} = \frac{{{\Omega _{{\textrm{CQ}}\prime {\textrm{A}}}}}}{2} = \arctan \left[ {\frac{{{\textbf{C}} \cdot \left( {{\textbf{Q}}\prime \times {\textbf{A}}} \right)}}{{1 + {\textbf{C}} \cdot {\textbf{A}} + {\textbf{A}} \cdot {\textbf{Q}}\prime + {\textbf{Q}}\prime \cdot {\textbf{C}}}}} \right]\\ {\phi _4} = \frac{{{\Omega _{{\textrm{ACB}}}}}}{2} = \arctan \left[ {\frac{{{\textbf{A}} \cdot \left( {{\textbf{B}} \times {\textbf{C}}} \right)}}{{1 + {\textbf{C}} \cdot {\textbf{B}} + {\textbf{B}} \cdot {\textbf{A}} + {\textbf{A}} \cdot {\textbf{C}}}}} \right] \end{array}\end{align} \tag{ 41 }$$

where ${{{\Omega }}_{XYZ}}$ represents the area of the spherical triangle formed between the points X, Y and Z; ${\mathbf{A}}$, ${\mathbf{B}}$ and ${\mathbf{C}}$ are the normalised Stokes vectors representing points A, B and C; ${\mathbf{Q{^{^{\prime}}}}}$ is the vector representing the non-modulating eigenmode of SLM1 (or equivalently of SLM3), which points along the ${S_1}$ axis in the negative direction; ${\mathbf{R{^{^{\prime}}}}}$ is the vector representing the non-modulating eigenmode of SLM2, which points along the ${S_2}$ axis in the negative direction.

Using the shorthand notation ${\mathcal{C}_x} = \cos x$ and ${\mathcal{S}_x} = \sin x$, we can define the normalised Stokes vectors as

$$\begin{equation*}{\mathbf{A}} = \left( {\begin{array}{*{20}{c}} {{\mathcal{C}_{2\theta }}} \\ {{\mathcal{S}_{2\theta }}} \\ 0 \end{array}} \right),\,\,\,\,{\mathbf{B}} = \left( {\begin{array}{*{20}{c}} {{\mathcal{C}_{2\theta }}} \\ {{\mathcal{S}_{2\theta }}{\mathcal{C}_\beta }} \\ {{\mathcal{S}_{2\theta }}{\mathcal{S}_\beta }} \end{array}} \right),\,\,\,\,{\mathbf{C}} = \left( {\begin{array}{*{20}{c}} {{\mathcal{C}_{2\theta }}} \\ {{\mathcal{S}_{2\theta }}{\mathcal{C}_\beta }} \\ { - {\mathcal{S}_{2\theta }}{\mathcal{S}_\beta }} \end{array}} \right),\end{equation*}$$

$$\begin{equation}{\mathbf{Q{^{^{\prime}}}}} = \left( {\begin{array}{*{20}{c}} { - 1} \\ 0 \\ 0 \end{array}} \right),\,\,\,{\mathbf{R{^{^{\prime}}}}} = \left( {\begin{array}{*{20}{c}} 0 \\ { - 1} \\ 0 \end{array}} \right).\end{equation} \tag{ 42 }$$

Then we find that

$$\begin{equation}{\phi _1} = {\phi _3} = \arctan \left[ {\frac{{{\mathcal{S}_\beta }\mathcal{S}_{2\theta }^2}}{{1 - 2{\mathcal{C}_{2\theta }} + \mathcal{C}_{2\theta }^2 + {\mathcal{C}_\beta }\mathcal{S}_{2\theta }^2}}} \right].\end{equation} \tag{ 43 }$$

Where we have used symmetry to find that the phases from SLM1 and SLM3 must be identical. We find further that

$$\begin{align} \displaystyle {\phi _2}& = \arctan \left[ {\frac{{{\mathcal{S}_\beta }{\mathcal{S}_{4\theta }}}}{{1 + \mathcal{C}_{2\theta }^2 - 2{\mathcal{C}_\beta }{\mathcal{S}_{2\theta }} + {\mathcal{C}_{2\beta }}\mathcal{S}_{2\theta }^2}}} \right] \nonumber\\[3pt] \displaystyle {\phi _4}& = \arctan \left[ {\frac{{2\left( {1 - {\mathcal{C}_\beta }} \right){\mathcal{S}_\beta }{\mathcal{C}_{2\theta }}\mathcal{S}_{2\theta }^2}}{{1 + 3\mathcal{C}_{2\theta }^2 + \left( {2{\mathcal{C}_\beta } + {\mathcal{C}_{2\beta }}} \right)\mathcal{S}_{2\theta }^2}}} \right] .\end{align} \tag{ 44 }$$

As ${\mathbf{A}}$ was chosen to be an eigenvector of the system, then the phase ${\zeta _1}$ must be the phase of the corresponding eigenvalue ${\mu _1} = {\text{exp}}\left( {{\text{i}}{\zeta _1}} \right)$.

To obtain the equivalent phases for the opposite eigenvector passing through the same three-SLM system, we replace ${\mathbf{A}}$, ${\mathbf{B}}$, and ${\mathbf{C}}$ by the diametrically opposed vectors. The resulting phases are

$$\begin{align} \displaystyle {{\phi {^{^{\prime}}}}_1} & = {{\phi {^{^{\prime}}}}_3} = \arctan \left[ {\frac{{{\mathcal{S}_\beta }\mathcal{S}_{2\theta }^2}}{{1 + 2{\mathcal{C}_{2\theta }} + \mathcal{C}_{2\theta }^2 + {\mathcal{C}_\beta }\mathcal{S}_{2\theta }^2}}} \right] \nonumber\\[3pt] \displaystyle {{\phi {^{^{\prime}}}}_2} & = \arctan \left[ {\frac{{{\mathcal{S}_\beta }{\mathcal{S}_{4\theta }}}}{{1 + \mathcal{C}_{2\theta }^2 + 2{\mathcal{C}_\beta }{\mathcal{S}_{2\theta }} + {\mathcal{C}_{2\beta }}\mathcal{S}_{2\theta }^2}}} \right] \nonumber\\[3pt] {{\phi {^{^{\prime}}}}_4}& = - {\phi _4} .\end{align} \tag{ 45 }$$

The eigenvalue corresponding to this eigenvector is therefore ${\mu _2} = {\text{exp}}\left( {{\text{i}}{\zeta _2}} \right)$, where

$$\begin{equation}{\zeta _2} = {\phi {^{^{\prime}}}_1} + {\phi {^{^{\prime}}}_2} + {\phi {^{^{\prime}}}_3} + {\phi {^{^{\prime}}}_4}.\end{equation} \tag{ 46 }$$

The retardance of the three-SLM system is thus given by

$$\begin{align} {{\Psi }} & = \arg \left[ {{ }{\mu _1}\mu _2^*} \right] = \arg \left[ {{{\text{e}}^{{\text{i}}\left( {{\zeta _1} - {\zeta _2}} \right)}}} \right]\nonumber\\ & = 2{\phi _1} + {\phi _2} + 2{\phi _4} - 2{\phi {^{^{\prime}}}_1} - {\phi {^{^{\prime}}}_2} \end{align} \tag{ 47 }$$

where the asterisk represents the complex conjugate.

As the determinant of a matrix is given by the product of its eigenvalues, we can use the geometrical model to calculate the dynamic phase as

$$\begin{equation}{{\Theta }} = \frac{1}{2}\arg \left[ {{\mu _1}{\mu _2}} \right] = \frac{1}{2}\left( {2{\phi _1} + {\phi _2} + 2{{\phi {^{^{\prime}}}}_1} + {{\phi {^{^{\prime}}}}_2}} \right).\end{equation} \tag{ 48 }$$

Here, the normalised Stokes vectors are given by

$$\begin{equation*}{\mathbf{A}} = \left( {\begin{array}{*{20}{c}} {{\mathcal{C}_{2\theta }}} \\ {{\mathcal{S}_{2\theta }}{\mathcal{C}_\alpha }} \\ {{\mathcal{S}_{2\theta }}{\mathcal{S}_\alpha }} \end{array}} \right),\,\,\,\,{\mathbf{B}} = \left( {\begin{array}{*{20}{c}} {{\mathcal{C}_{2\theta }}} \\ {{\mathcal{S}_{2\theta }}{\mathcal{C}_\beta }} \\ {{\mathcal{S}_{2\theta }}{\mathcal{S}_\beta }} \end{array}} \right),\,\,\,{\mathbf{C}} = \left( {\begin{array}{*{20}{c}} {{\mathcal{C}_{2\theta }}} \\ {{\mathcal{S}_{2\theta }}{\mathcal{C}_\beta }} \\ { - {\mathcal{S}_{2\theta }}{\mathcal{S}_\beta }} \end{array}} \right),\end{equation*}$$

$$\begin{equation}{\mathbf{Q{^{^{\prime}}}}} = \left( {\begin{array}{*{20}{c}} { - 1} \\ 0 \\ 0 \end{array}} \right),\,\,\,{\mathbf{R{^{^{\prime}}}}} = \left( {\begin{array}{*{20}{c}} 0 \\ { - 1} \\ 0 \end{array}} \right)\end{equation} \tag{ 49 }$$

where only the definition of ${\mathbf{A}}$ has changed

$$\begin{equation}\begin{gathered} {\phi _1} = \arctan \left[ {\frac{{ - {\mathcal{S}_{\alpha - \beta }}\mathcal{S}_{2\theta }^2}}{{1 - 2{\mathcal{C}_{2\theta }} + \mathcal{C}_{2\theta }^2 + {\mathcal{C}_{\alpha - \beta }}\mathcal{S}_{2\theta }^2}}} \right] \hfill \\ {\phi _2} = \arctan \left[ {\frac{{{\mathcal{S}_\beta }{\mathcal{S}_{4\theta }}}}{{1 + \mathcal{C}_{2\theta }^2 - 2{\mathcal{C}_\beta }{\mathcal{S}_{2\theta }} + {\mathcal{C}_{2\beta }}\mathcal{S}_{2\theta }^2}}} \right] \hfill \\ {\phi _3} = \arctan \left[ {\frac{{{\mathcal{S}_{\alpha + \beta }}\mathcal{S}_{2\theta }^2}}{{1 - 2{\mathcal{C}_{2\theta }} + \mathcal{C}_{2\theta }^2 + {\mathcal{C}_{\alpha + \beta }}\mathcal{S}_{2\theta }^2}}} \right] \hfill \\ {\phi _4} = \arctan \left[ {\frac{{2\left( {{\mathcal{C}_\alpha } - {\mathcal{C}_\beta }} \right){\mathcal{S}_\beta }{\mathcal{C}_{2\theta }}\mathcal{S}_{2\theta }^2}}{{1 + 3\mathcal{C}_{2\theta }^2 + \left( {2{\mathcal{C}_\alpha }{\mathcal{C}_\beta } + {\mathcal{C}_{2\beta }}} \right)\mathcal{S}_{2\theta }^2}}} \right] \hfill \\ {{\phi {^{^{\prime}}}}_1} = \arctan \left[ {\frac{{ - {\mathcal{S}_{\alpha - \beta }}\mathcal{S}_{2\theta }^2}}{{1 + 2{\mathcal{C}_{2\theta }} + \mathcal{C}_{2\theta }^2 + {\mathcal{C}_{\alpha - \beta }}\mathcal{S}_{2\theta }^2}}} \right] \hfill \\ {{\phi {^{^{\prime}}}}_2} = \arctan \left[ {\frac{{{\mathcal{S}_\beta }{\mathcal{S}_{4\theta }}}}{{1 + \mathcal{C}_{2\theta }^2 + 2{\mathcal{C}_\beta }{\mathcal{S}_{2\theta }} + {\mathcal{C}_{2\beta }}\mathcal{S}_{2\theta }^2}}} \right] \hfill \\ {{\phi {^{^{\prime}}}}_3} = \arctan \left[ {\frac{{{\mathcal{S}_{\alpha + \beta }}\mathcal{S}_{2\theta }^2}}{{1 + 2{\mathcal{C}_{2\theta }} + \mathcal{C}_{2\theta }^2 + {\mathcal{C}_{\alpha + \beta }}\mathcal{S}_{2\theta }^2}}} \right] \hfill \\ {{\phi {^{^{\prime}}}}_4} = - {\phi _4}. \hfill \\ \end{gathered} \end{equation} \tag{ 50 }$$

Using these definitions for the individual phase terms, we can obtain the retardance of the three-SLM system as

$$\begin{equation}{{\Psi }} = {\phi _1} + {\phi _2} + {\phi _3} + 2{\phi _4} - {\phi {^{^{\prime}}}_1} - {\phi {^{^{\prime}}}_2} - {\phi {^{^{\prime}}}_3}.\end{equation} \tag{ 51 }$$

Following the derivation in the section for linear retarders, we calculate the dynamic phase introduced by the three-SLM system as

$$\begin{equation}{{\Theta }} = \frac{1}{2}\left( {{\phi _1} + {\phi _2} + {\phi _3} + {{\phi {^{^{\prime}}}}_1} + {{\phi {^{^{\prime}}}}_2} + {{\phi {^{^{\prime}}}}_3}} \right).\end{equation} \tag{ 52 }$$