General lossless spatial polarization transformations

In this section, we describe the transformation of polarization by birefringent media in terms of rotations on the Poincaré sphere. Since many mutually inconsistent polarization conventions exist in the literature, we give a brief introduction and review of polarization transformations in appendix A, which sets the conventions and notation used in this paper. Light passing through birefringent media, such as a waveplate, gains a phase between the electric field component along the media's optical axis and its orthogonal component. This phase is known as the 'retardance,' Δ. On the Poincaré sphere, the action of a waveplate corresponds to a rotation of an input polarization $\hat{{\bf{s}}}$ by Δ about a rotation axis $\hat{{\bf{k}}}(2{\rm{\Phi }},0)$, i.e. one lying in the ${\hat{{\bf{s}}}}_{1}$ ${\hat{{\bf{s}}}}_{2}$ plane at $2{\rm{\Phi }}$ from the positive ${\hat{{\bf{s}}}}_{1}$ axis. Here, Φ is the angle in the laboratory between the fast axis and horizontal, $\hat{{\bf{x}}}$, with increasing angle defined to be towards $\hat{{\bf{y}}}$. Accordingly, waveplates, such as half-wave plates (${\rm{\Delta }}=\pi$, HWP) and quarter-wave plates (${\rm{\Delta }}=\pi /2$, QWP), can be described by a rotation matrix. More generally, a birefringent waveplate with arbitrary retardance Δ (e.g. an electro-optic modulator or a nematic-phase parallel-aligned liquid crystal) will have the rotation matrix [50],

$$\begin{eqnarray}{{\bf{R}}}_{{\bf{k}}(2{\rm{\Phi }},0)}({\rm{\Delta }})=\left[\begin{array}{ccc}{\sin }^{2}2{\rm{\Phi }}\cos {\rm{\Delta }}+{\cos }^{2}2{\rm{\Phi }} & {\sin }^{2}\left(\tfrac{{\rm{\Delta }}}{2}\right)\sin 4{\rm{\Phi }} & \sin 2{\rm{\Phi }}\sin {\rm{\Delta }}\\ {\sin }^{2}\left(\tfrac{{\rm{\Delta }}}{2}\right)\sin 4{\rm{\Phi }} & {\cos }^{2}2{\rm{\Phi }}\cos {\rm{\Delta }}+{\sin }^{2}2{\rm{\Phi }} & -\cos 2{\rm{\Phi }}\sin {\rm{\Delta }}\\ -\sin 2{\rm{\Phi }}\sin {\rm{\Delta }} & \cos 2{\rm{\Phi }}\sin {\rm{\Delta }} & \cos {\rm{\Delta }}\end{array}\right].\end{eqnarray} \tag{ 1 }$$

Since we will be using HWP and QWPs frequently, we define ${\bf{HWP}}[\![ {\rm{\Phi }}]\!]$ $\equiv \,{{\bf{R}}}_{{\bf{k}}(2{\rm{\Phi }},0)}(\pi )$ and ${\bf{QWP}}[\![ {\rm{\Phi }}]\!] \,\equiv {{\bf{R}}}_{{\bf{k}}(2{\rm{\Phi }},0)}(\pi /2)$. To avoid any confusion between the reference frames the angles are defined in, an angle expressed in the laboratory frame is written using $[\![ \cdot ]\!]$.

In the Poincaré sphere, an arbitrary general polarization transformation ${{\bf{R}}}_{{\bf{k}}}(\xi )$ consists of a rotation about an arbitrary axis $\hat{{\bf{k}}}$ by angle ξ (i.e. equation (25)). One can implement this with HWP- and QWPs cascaded in the following sequence:

$$\begin{eqnarray}&&{{\bf{R}}}_{{\bf{k}}}(\xi )={\bf{QWP}}[\![ {{\rm{\Phi }}}_{3}]\!] {\bf{HWP}}[\![ {{\rm{\Phi }}}_{2}]\!] {\bf{QWP}}[\![ {{\rm{\Phi }}}_{1}]\!] .\end{eqnarray} \tag{ 2 }$$

Ordered first to last, the waveplates the light passes through respectively correspond to the elements in equation (2) from right to left. By an appropriate rotation of each of the three waveplates in the laboratory, any unitary polarization transformation can be created [2]. However, this mechanical rotation will prohibit fast changes of the unitary polarization transformation.

Now suppose we had a birefringent optical element with retardance Δ and rotation axis ${\hat{{\bf{k}}}}_{A}$, and we wished to convert ${\hat{{\bf{k}}}}_{A}$ to be ${\hat{{\bf{k}}}}_{B}$, as in figure 1(a). This could be done by sandwiching the optical element between two sequences of equation (2),

$$\begin{eqnarray}{{\bf{R}}}_{{{\bf{k}}}_{B}}({\rm{\Delta }}) & = & {\bf{QWP}}[\![ {{\rm{\Phi }}}_{1}+90^\circ ]\!] {\bf{HWP}}[\![ {{\rm{\Phi }}}_{2}+90^\circ ]\!] \\ & \cdot & {\bf{QWP}}[\![ {{\rm{\Phi }}}_{3}+90^\circ ]\!] {{\bf{R}}}_{{{\bf{k}}}_{A}}({\rm{\Delta }}){\bf{QWP}}[\![ {{\rm{\Phi }}}_{3}]\!] \\ & \cdot & {\bf{HWP}}[\![ {{\rm{\Phi }}}_{2}]\!] {\bf{QWP}}[\![ {{\rm{\Phi }}}_{1}]\!] ,\end{eqnarray} \tag{ 3 }$$

where ${{\bf{R}}}_{{{\bf{k}}}_{A}}({\rm{\Delta }})$ is the corresponding rotation matrix of the birefringent optical element, and again, light passes through the elements from right to left. The first sequence of QWP- and HWPs rotates the polarization state ${\hat{{\bf{s}}}}_{B}$ (positioned at ${\hat{{\bf{k}}}}_{B}$ on the sphere) to ${\hat{{\bf{s}}}}_{A}$ (positioned at ${\hat{{\bf{k}}}}_{A}$). The second sequence applies the reverse rotation, rotating it back to ${\hat{{\bf{s}}}}_{B}$. In essence, this is a change of basis such that ${\hat{{\bf{s}}}}_{B}$, a polarization eigenstate of ${{\bf{R}}}_{{{\bf{k}}}_{B}}({\rm{\Delta }})$, is unaffected by the waveplates and optical elements—apart from a global phase—and all other polarization states undergo a rotation by Δ about ${\hat{{\bf{k}}}}_{B}$.

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The most commonly used liquid crystal in optics laboratories is in the nematic phase and is parallel-aligned. It has a variable retardance Δ, but a fixed rotation axis $\hat{{\bf{k}}}(2{\rm{\Phi }},0)$ in the ${\hat{{\bf{s}}}}_{1}$ ${\hat{{\bf{s}}}}_{2}$ plane. Since it has a fixed axis with linear polarization eigenstates, fewer waveplates are necessary in equation (3): only a QWP- and HWP are required to transform any linear polarization to an arbitrary polarization state [1]. The QWP removes the ellipticity from ${\hat{{\bf{k}}}}_{B}(\varphi ,\theta )$ rotating it intermediately to ${\hat{{\bf{s}}}}_{M}(\varphi ,0)$, a linear polarization state. This state is rotated by the HWP to ${\hat{{\bf{k}}}}_{A}=\hat{{\bf{k}}}(2{\rm{\Phi }},0)$, that is, along the axis of the liquid crystal. Consequently, a four waveplate combination is sufficient for a liquid crystal cell to effectively create an arbitrary rotation ${{\bf{R}}}_{{{\bf{k}}}_{B}(\varphi ,\theta )}({\rm{\Delta }})$,

$$\begin{eqnarray}{{\bf{R}}}_{{{\bf{k}}}_{B}(\varphi ,\theta )}({\rm{\Delta }}) & = & {\bf{QWP}}\left[\!\!\left[\displaystyle \frac{\varphi }{2}+90^\circ \right]\!\!\right]{\bf{HWP}}\left[\!\!\left[\displaystyle \frac{\varphi -\theta }{4}+\displaystyle \frac{{\rm{\Phi }}}{2}+90^\circ \right]\!\!\right]\\ & \cdot & {{\bf{R}}}_{{\bf{k}}(2{\rm{\Phi }},0)}({\rm{\Delta }})\\ & \cdot & {\bf{HWP}}\left[\!\!\left[\displaystyle \frac{\varphi -\theta }{4}+\displaystyle \frac{{\rm{\Phi }}}{2}\right]\!\!\right]{\bf{QWP}}\left[\!\!\left[\displaystyle \frac{\varphi }{2}\right]\!\!\right],\end{eqnarray} \tag{ 4 }$$

where φ and θ respectively are the azimuthal and polar angles of the target rotation axis ${\hat{{\bf{k}}}}_{B}(\varphi ,\theta )$ in the Poincaré Sphere (see figure A1(a) for details). The action of the waveplates in equation (4) is shown in figure 1(b) for $\hat{{\bf{k}}}(2{\rm{\Phi }},0)={\hat{{\bf{s}}}}_{1}$. We will repeatedly use this method of converting rotation axes on the Poincaré sphere in the following sections.

If many liquid crystal cells are arranged in an array, we obtain a spatial light modulator (LC-SLM). Here, For simplicity, we take the fast axis of each liquid crystal cell, and thus the whole LC-SLM, to be along the horizontal. i.e. with ${\rm{\Phi }}=0$. If one used uniformly H polarized light, a nematic-phase parallel-aligned LC-SLM would function as it is commonly used: as a 'phase-only' spatial light modulator. Here, we consider other input polarizations. The LC-SLM has a rotation matrix equivalent to equation (28) but with a spatially variable retardance (or 'phase distribution') of ${\rm{\Delta }}(x,y)$. In this way, LC-SLMs provide the spatial degree of freedom to extend the general polarization transformation in equation (4) to be

$$\begin{eqnarray}&&{{\bf{R}}}_{{{\bf{k}}}_{B}(\varphi ,\theta )}({\rm{\Delta }}(x,y))\,={\bf{QWP}}\left[\!\!\left[\displaystyle \frac{\varphi }{2}+90^\circ \right]\!\!\right]{\bf{HWP}}\left[\!\!\left[\displaystyle \frac{\varphi -\theta }{4}+90^\circ \right]\!\!\right]\\ &&\,\cdot \,{\bf{SLM}}({\rm{\Delta }}(x,y))\\ &&\,\cdot \,{\bf{HWP}}\left[\!\!\left[\displaystyle \frac{\varphi -\theta }{4}\right]\!\!\right]{\bf{QWP}}\left[\!\!\left[\displaystyle \frac{\varphi }{2}\right]\!\!\right].\end{eqnarray} \tag{ 5 }$$

The corresponding rotation matrix can then be computed by using equation (1) for the QWP- and HWPs, and equation (28) for the LC-SLM. This configuration gives the possibility for the whole LC-SLM to have an arbitrary fixed rotation axis ${\hat{{\bf{k}}}}_{B}(\varphi ,\theta )$, but with a spatially variable rotation angle ${\rm{\Delta }}(x,y)$. Figure 1(b) traces the path of a state ${\hat{{\bf{s}}}}_{B}={\hat{{\bf{k}}}}_{B}$ on the Poincaré Sphere as it travels through the waveplates in equation (5).

3.1.1. Practical examples and special cases

Let us look at some practical examples of equation (5).

(i) For a rotation axis on the equator (i.e. ${\hat{{\bf{s}}}}_{1}$ ${\hat{{\bf{s}}}}_{2}$ plane), the QWPs are not required. In equation (5), the ${\bf{QWP}}[\![ ]\!]$ terms are eliminated and $\varphi =0$ in the ${\bf{HWP}}[\![ ]\!]$ arguments. For example, for rotations about ${\hat{{\bf{k}}}}_{B}={\hat{{\bf{s}}}}_{2}$, the diagonal polarization axis, the sequence would be ${{\bf{R}}}_{{{\bf{s}}}_{2}}({\rm{\Delta }}(x,y))$ $\,=\,{\bf{HWP}}[\![ 112.5^\circ ]\!]$ ${\bf{SLM}}({\rm{\Delta }}(x,y)){\bf{HWP}}$ $[\![ 22.5^\circ ]\!]$.

(ii) For rotations about a circular polarization axis, i.e. the ${\hat{{\bf{s}}}}_{3}$ axis, we can remove the HWPs, and use the sequence ${{\bf{R}}}_{{{\bf{s}}}_{3}}({\rm{\Delta }}(x,y))$ $=\,{\bf{QWP}}[\![ -45^\circ ]\!]$ ${\bf{SLM}}({\rm{\Delta }}(x,y)){\bf{QWP}}$ $[\![ 45^\circ ]\!] .$ This sequence has been demonstrated to be particularly useful in creating vector beams such as radial and azimuthal polarization distributions [51]. For this, one begins with uniform horizontally polarized light, ${\bf{E}}=\hat{{\bf{x}}}=(\hat{{\bf{l}}}+\hat{{\bf{r}}})/\sqrt{2}$. The right-circular component is retarded with respect to the left-circular component resulting in the following polarization distribution after the waveplates and LC-SLM,

$$\begin{eqnarray}{\bf{E}} & = & \displaystyle \frac{1}{\sqrt{2}}(\hat{{\bf{l}}}+{{\rm{e}}}^{{\rm{i}}{\rm{\Delta }}(x,y)}\hat{{\bf{r}}})\\ & = & \left(\cos \left(\displaystyle \frac{{\rm{\Delta }}(x,y)}{2}\right)\hat{{\bf{x}}}\right.-\left.\sin \left(\displaystyle \frac{{\rm{\Delta }}(x,y)}{2}\right)\hat{{\bf{y}}}\right){{\rm{e}}}^{{\rm{i}}{\rm{\Delta }}(x,y)/2}.\end{eqnarray} \tag{ 6 }$$

The polarization remains linear and is rotated in the laboratory by an angle of ${\rm{\Delta }}(x,y)/2$. Thus, the linear polarization direction can vary spatially in an arbitrary manner. However, on the right side of equation (6) there appears an additional phase term that will not explicitly arise in our analysis using rotations in the Poincaré sphere. This phase is addressed in the appendix B.

In figure 2(a) we demonstrate that the polarization can be aligned radially. In figure 2(b), the polarization follows the local asymptote of the letters 'uO' (i.e. University of Ottawa). The presence of light at the center may seem surprising since radially polarized beams have a null in intensity at their center, as in a Laguerre–Gauss mode beam. However, in figure 2 we are imaging the LC-SLM surface, at which only the phase, rather than intensity is changed. The resulting field distribution is no longer a 'beam' (for more detail on this subject see appendix B).

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In our next step in increasing generality, we introduce a scheme to transform an arbitrary input polarization ${\hat{{\bf{s}}}}_{i}$ to another arbitrary output polarization, ${\hat{{\bf{s}}}}_{o}$. Both polarization distributions can spatially vary independently. Crucially, though, both ${\hat{{\bf{s}}}}_{i}$ and ${\hat{{\bf{s}}}}_{o}$ must be known at every position $(x,y)$ before implementing the transformation, potentially through prior measurements.

To achieve this transformation, a second LC-SLM is added to the compound device of equation (5), thereby giving the ability to perform rotations about two distinct axes on the Poincaré sphere. Through two rotations about two orthogonal axes, say ${\hat{{\bf{s}}}}_{1}$ and ${\hat{{\bf{s}}}}_{2}$, one can transform any point to any other point on the unit sphere [50]. However, this only holds if one can change the order of the orthogonal rotations. The order will depend on the particular input state and output state. In contrast, we consider a fixed ordering defined by the sequence of LC-SLMs and waveplates in the experimental setup.

In this section, it is useful to visualize the Poincaré sphere by projecting it on a plane spanned by our two rotation axes, ${\hat{{\bf{s}}}}_{1}$ and ${\hat{{\bf{s}}}}_{2}$. This is shown in figure 3(a). A rotation about ${\hat{{\bf{s}}}}_{1}$ will take ${\hat{{\bf{s}}}}_{i}$ to ${\hat{{\bf{s}}}}_{m}$. A following rotation about ${\hat{{\bf{s}}}}_{2}$ takes ${\hat{{\bf{s}}}}_{m}$ to ${\hat{{\bf{s}}}}_{o}$. Figure 3(b) shows this sequence in a three-dimensional view of the Poincaré sphere for reference. Reversing the order of the rotations would also work, but would instead pass through the intermediate state ${\hat{{\bf{s}}}}_{m^{\prime} }$. This will not be generally true though. For many pairs of states ${\hat{{\bf{s}}}}_{i}$ and ${\hat{{\bf{s}}}}_{o}$, only one ordering will work. In particular, the region outlined in purple in figure 3(a) contains the subset of output states $\{{\hat{{\bf{s}}}}_{o}\}$ that can be reached from the specific ${\hat{{\bf{s}}}}_{i}$ when rotating about ${\hat{{\bf{s}}}}_{1}$ first, followed by ${\hat{{\bf{s}}}}_{2}$. To understand this, consider moving horizontally in either direction from ${\hat{{\bf{s}}}}_{i}$ along a line at ${s}_{1}^{i}={\hat{{\bf{s}}}}_{1}\cdot {\hat{{\bf{s}}}}_{i}$ (i.e. a rotation about ${\hat{{\bf{s}}}}_{1}$). In doing so, the largest achievable magnitude for ${s}_{2}^{o}={\hat{{\bf{s}}}}_{2}\cdot {\hat{{\bf{s}}}}_{o}$ is set by the intersection of the line with the circle bounding the Poincaré sphere. That is, at $1={({s}_{1}^{i})}^{2}+{({s}_{2}^{o})}^{2}$.

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3.2.1. Required retardances

Keeping this restriction in mind, we now calculate the required retardances, α and β, of the two LC-SLMs. We assume a configuration which rotates first about ${\hat{{\bf{s}}}}_{1}$ (by α) then ${\hat{{\bf{s}}}}_{2}$ (by β):

$$\begin{eqnarray}&&{{\bf{T}}}_{{{\bf{s}}}_{i}\to {{\bf{s}}}_{o}}={{\bf{R}}}_{{{\bf{s}}}_{2}}(\beta ){{\bf{R}}}_{{{\bf{s}}}_{1}}(\alpha )\\ &&\,={\bf{HWP}}\left[\!\!\left[112.5^\circ \right]\!\!\right]{\bf{SLM}}(\beta ){\bf{HWP}}\left[\!\!\left[22.5^\circ \right]\!\!\right]{\bf{SLM}}(\alpha ).\end{eqnarray} \tag{ 7 }$$

While for brevity we omit an explicit spatial dependence in these vectors and retardances, it should be understood to be implicit in what follows. Specifically, all quantities are for the same transverse point in the light field. In order to transform an input polarization ${\hat{{\bf{s}}}}_{i}=[{s}_{1}^{i},{s}_{2}^{i},{s}_{3}^{i}]$ to a target output polarization ${\hat{{\bf{s}}}}_{o}=[{s}_{1}^{o},{s}_{2}^{o},{s}_{3}^{o}]$, one requires the following retardances:

$$\begin{eqnarray}&&\alpha ^{\prime} =\mathrm{atan}2([0,{s}_{2}^{i},{s}_{3}^{i}]\cdot [0,{s}_{2}^{m},{s}_{3}^{m}],\\ &&\,\mathrm{sign}({s}_{2}^{i}{s}_{3}^{m}-{s}_{3}^{i}{s}_{2}^{m})\parallel [0,{s}_{2}^{i},{s}_{3}^{i}]\times [0,{s}_{2}^{m},{s}_{3}^{m}]\parallel ),\,\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&\beta ^{\prime} =\mathrm{atan}2([{s}_{1}^{m},0,{s}_{3}^{m}]\cdot [{s}_{1}^{o},0,{s}_{3}^{o}],\\ &&\quad \quad \,\mathrm{sign}({s}_{3}^{m}{s}_{1}^{o}-{s}_{1}^{m}{s}_{3}^{o})\parallel [{s}_{1}^{m},0,{s}_{3}^{m}]\times [{s}_{1}^{o},0,{s}_{3}^{o}]\parallel ),\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{\hat{{\bf{s}}}}_{m}=[{s}_{1}^{i},\,{s}_{2}^{o},\,\mathrm{sign}({s}_{3}^{o})\sqrt{{({s}_{3}^{i})}^{2}+{({s}_{2}^{i})}^{2}-{({s}_{2}^{o})}^{2}}],\,\,\end{eqnarray} \tag{ 10 }$$

where ${\hat{{\bf{s}}}}_{m}=[{s}_{1}^{m},{s}_{2}^{m},{s}_{3}^{m}]$ is the intermediate point, ·is the dot product, × is the vector cross product, $\parallel \hat{s}\parallel =\sqrt{{s}_{1}^{2}+{s}_{2}^{2}+{s}_{3}^{2}}$ is the vector norm, and $\mathrm{sign}$ is the standard signum function with the convention that $\mathrm{sign}(0)=1$. The function $\mathrm{atan}2(x,y)$ is defined as the angle between the positive x axis and the point (x, y), with angle increasing towards the positive y axis. Additionally, we take $\alpha =\alpha ^{\prime} +2k\pi$ and $\beta =\beta ^{\prime} +2k\pi$, in order to ensure that the two LC-SLMs rotate in the positive sense (the angle is increasing).

3.2.2. Applications of arbitrary to arbitrary polarization transformations

We now present three examples of applications that use this transformation.

Ellipticity (de)magnifier: The ellipticity of an input state can be either magnified to be more circular, or demagnified to be more linear. In terms of the Poincaré sphere, changing the ellipticity of state ${\hat{{\bf{s}}}}_{i}$ corresponds to changing the input state's polar angle θ, while maintaining its azimuthal angle φ. This could be used to change a light field containing a spatial polarization distribution with an assortment of elliptical states to one with only linear states. It could also flip the polarization handedness. An example of the rotation paths is shown in figure 4.

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Beam healer: Passage through birefringent optical media can undesirably transform a uniform polarization into a non-uniform polarization distribution. This effect could potentially degrade the performance of imaging systems. The method previously introduced in this section (i.e. equation (7)) can restore polarization uniformity.

As discussed in the beginning of the subsection 3.2, when the order of the rotation is fixed, it is only possible to reach a particular subset of output states ${\hat{{\bf{s}}}}_{o}$ (the polarization states included in the purple box in figure 3(a)) from a specific input state ${\hat{{\bf{s}}}}_{i}$. However, regardless of the input state ${\hat{{\bf{s}}}}_{i}$, it is always possible to reach the poles of the Poincaré Sphere, the right-handed and left-handed circular polarizations. In short, the transformation a beam healer performs on an non-uniform distribution is ${{\bf{T}}}_{{{\bf{s}}}_{i}\to \pm {{\bf{s}}}_{3}}$. While transforming an arbitrary polarization to another arbitrary polarization might seem completely general, it is not. We clarify this point in appendix B.

The most general polarization transformation is a rotation by an arbitrary angle ξ about an arbitrary axis $\hat{{\bf{k}}}$ in the Poincaré sphere, ${{\bf{R}}}_{{\bf{k}}}(\xi )$. This corresponds to a retardation of an arbitrary polarization state. It is a universal unitary transformation for polarization. The axis $\hat{{\bf{k}}}$ is defined by two free parameters, its spherical coordinates $(\varphi ,\theta )$, and rotation angle ξ adds a third parameter. It follows that one needs at least three control parameters in order to implement this general transformation. One solution is to use three variable liquid crystals and appropriate fixed waveplates. Considering LC-SLMs, this would implement a different general transformation at each and every transverse position in a light field.

As in section 3.2, the LC-SLMs and waveplates effectively implement rotations about orthogonal axes. The addition of the third LC-SLM, and thus third rotation, allows us to draw upon the concept of proper extrinsic Euler angles, in which a general 3D rotation ${{\bf{R}}}_{{\bf{k}}}(\xi )$ can be decomposed into three successive rotations about any two orthogonal axes. Here, we use ${\hat{{\bf{s}}}}_{1}$ and ${\hat{{\bf{s}}}}_{2}$. Accordingly, we compose the general rotation,

$$\begin{eqnarray}{{\bf{R}}}_{{\bf{k}}}(\xi )&=&{{\bf{R}}}_{{{\bf{s}}}_{1}}(\gamma ){{\bf{R}}}_{{{\bf{s}}}_{2}}(\beta ){{\bf{R}}}_{{{\bf{s}}}_{1}}(\alpha )\\ \,&=&\,{\bf{SLM}}(\gamma ){\bf{HWP}}[\![ 112.5^\circ ]\!] {\bf{SLM}}(\beta ){\bf{HWP}}[\![ 22.5^\circ ]\!] {\bf{SLM}}(\alpha )\\ \,&=&\,\left[\begin{array}{ccc}\cos \beta & \sin \alpha \sin \beta & \cos \alpha \sin \beta \\ \sin \beta \sin \gamma & \cos \alpha \cos \gamma -\cos \beta \sin \alpha \sin \gamma & -\cos \gamma \sin \alpha -\cos \alpha \beta \sin \gamma \\ -\cos \gamma \sin \beta & \cos \beta \cos \gamma \sin \alpha +\cos \alpha \sin \gamma & \cos \alpha \cos \beta \cos \gamma -\sin \alpha \sin \gamma \end{array}\right].\end{eqnarray} \tag{ 11 }$$

Figure 5 demonstrates the sequential rotations that ${{\bf{R}}}_{{\bf{k}}}(\xi )$ performs on the principal axes in terms of the three angles α, β, and γ.

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3.3.1. Required retardances

To simplify our notation we write ${\bf{R}}\equiv {{\bf{R}}}_{{\bf{k}}}(\xi )$ in the following and define the ${\bf{R}}$_ij matrix element to be the ith row from the top and jth column from the left. In terms of these elements, the angle of each rotation is,

$$\begin{eqnarray}&&\alpha ^{\prime} =\mathrm{atan}2({{\bf{R}}}_{13},{{\bf{R}}}_{12}),\,\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&\beta ^{\prime} =\mathrm{atan}2({{\bf{R}}}_{11},\sqrt{{{\bf{R}}}_{21}^{2}+{{\bf{R}}}_{31}^{2}}),\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&\gamma ^{\prime} =\mathrm{atan}2(-{{\bf{R}}}_{31},{{\bf{R}}}_{21}).\,\end{eqnarray} \tag{ 14 }$$

These angles, modded by 2π, are the retardances, $\alpha ,$ $\beta ,$ and γ, that are applied at each transverse position in the light field by the three LC-SLMs.

These angles are sufficient if one begins with the actual matrix for ${{\bf{R}}}_{{\bf{k}}}(\xi )$, but it may be more useful if they are expressed in terms of the rotation axis $\hat{{\bf{k}}}=[{k}_{1},\,{k}_{1},\,{k}_{3}]$ and angle ξ,

$$\begin{eqnarray}\alpha ^{\prime} & = & \displaystyle \frac{{k}_{1}{k}_{2}(1-\cos \xi )-{k}_{3}\sin \xi }{{k}_{2}\sin \xi +{k}_{1}{k}_{3}(1-\cos \xi )},\,\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}\beta ^{\prime} & = & \displaystyle \frac{[{({k}_{3}\sin \xi +{k}_{1}{k}_{2}(1-\cos \xi ))}^{2}}{\cos \xi +{k}_{1}^{2}(1-\cos \xi )}\\ & & +\displaystyle \frac{{{({k}_{1}{k}_{3}(1-\cos \xi )-{k}_{2}\sin \xi )}^{2}]}^{1/2}}{\cos \xi +{k}_{1}^{2}(1-\cos \xi )},\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&\gamma ^{\prime} =\displaystyle \frac{{k}_{3}\sin \xi +{k}_{1}{k}_{2}(1-\cos \xi )}{-({k}_{1}{k}_{3}(1-\cos \xi )-{k}_{2}\sin \xi )}.\,\end{eqnarray} \tag{ 17 }$$

These follow from the matrix expression of the Euler–Rodrigues formula, equation (25) and equations (12)–(14). With these retardances, a completely general polarization unitary can be applied at each transverse position $(x,y)$ in a light field.

3.3.2. Applications and examples

However, a residual spatially varying phase $b(x,y)$ can indeed arise in the polarization transformations that we present. This phase is not evident from rotations in the Poincaré sphere but does arise when using Jones Matrices [48]. If not compensated, this residual phase can have physical consequences. As an example, consider the case where we produce a radially polarized field according to equation (6). On the right-hand side, a residual phase of ${{\rm{e}}}^{{\rm{i}}{\rm{\Delta }}(x,y)}$ appears. Here, ${\rm{\Delta }}(x,y)=2\phi$, where ϕ is the azimuthal angle about the center of the radial field. It is impact can be understood by considering the left-hand side of equation (6), ${\bf{E}}\,=\tfrac{1}{\sqrt{2}}(\hat{{\bf{l}}}+{{\rm{e}}}^{{\rm{i}}2\phi }\hat{{\bf{r}}})$. If an incoming optical field had a Gaussian transverse profile, the left-handed component $\hat{{\bf{l}}}$ would be unchanged, whereas the right-handed component $\hat{{\bf{r}}}$ would receive an azimuthally varying phase carrying ${\ell }=2$ units of orbital angular momentum. This phase profile causes the right-handed light-field to no longer be a solution to the paraxial wave equation. Hence, it is no longer a 'beam' in the sense that it does not maintain its spatial and polarization distribution upon propagation (up to a scale factor). We experimentally demonstrated this by allowing the radial field in figure B1(a) to propagate 10 cm further. The polarization and intensity distribution at this point is shown in figure B1(b). We call this the 'far-field'. Physically, the residual phase arises from a variety of causes including, but not limited to, the Pancharatnam–Berry phase. The latter is known to affect propagation, as was in demonstrated in [54].

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In all three types of transformations, a single extra SLM is all that is required to correct this residual phase $b(x,y)$. Our methods control the phase $\delta (x,y)$ between $\hat{{\bf{x}}}$ and $\hat{{\bf{y}}}$ polarizations at any particular point. Therefore, if we phase-shift $\hat{{\bf{x}}}$ by $\mu (x,y)$ with a single extra SLM, we would have ${\bf{E}}(x,y)\,=({a}_{x}\,\hat{{\bf{x}}}+{a}_{y}{{\rm{e}}}^{{\rm{i}}\delta (x,y)-\mu (x,y)}\,\hat{{\bf{y}}}){{\rm{e}}}^{{\rm{i}}{b}(x,y)+\mu (x,y)}{{\rm{e}}}^{{\rm{i}}{c}}$. Setting $\mu (x,y)\,=-b(x,y)$ would cancel the residual phase. One would then also need to pre-compensate in our initial transformation by increasing $\delta (x,y)$ by $\mu (x,y)$. Formulas for $\mu (x,y)$ can be found using the Jones matrix formalism.

Figure B1 shows there are notable differences between the output polarization distribution at the image plane of the LC-SLM (near-field) and after propagation (far-field) for the same input polarization and LC-SLM phase pattern, ${\rm{\Delta }}(x,y)$. However, throughout this paper we assumed that LC-SLMs can be placed in series while acting on the same unchanged optical field. To achieve this, the field at one LC-SLM is imaged onto the next with a 4-f system. The latter acts as a relay imaging system with a one-to-one magnification ratio.

Ideally, in order to create a simple setup with high transmissivity, the LC-SLMs would be transmissive and placed in a line. However, reflective LC-SLMs often have better performance specifications. A reflective setup usually involves picking off a beam that is reflected at a small angle. Since, the pick-off mirror acts as an aperture this can dramatically change the optical field imaged by a 4-f setup. Our setup, shown in figure B2, uses a reflective LC-SLM. In order to separate the reflected beam from incident beam we use a non-polarizing beam-splitter (NPBS) instead of a pick-off mirror. The drawback is that this NPBS introduces loss. Nonetheless, the setup allows us to test the transformations, which are lossless in principle.

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While transforming an arbitrary polarization to another arbitrary polarization might seem completely general, it is not. The most general rotation is ${{\bf{R}}}_{{\bf{k}}}(\xi )$, whereas the transformation that is implemented through this method is ${{\bf{R}}}_{{{\bf{s}}}_{2}}(\beta ){{\bf{R}}}_{{{\bf{s}}}_{1}}(\alpha )$. The latter implements ${{\bf{T}}}_{{{\bf{s}}}_{i}\to {{\bf{s}}}_{o}}$, which takes ${\hat{{\bf{s}}}}_{i}\to {\hat{{\bf{s}}}}_{o}$. It also links the polarization states diametrically opposed to these on the Poincaré sphere, $-{\hat{{\bf{s}}}}_{i}\to -{\hat{{\bf{s}}}}_{o}$. However, ${{\bf{T}}}_{{{\bf{s}}}_{i}\to {{\bf{s}}}_{o}}$ does not fix the retardance ζ between ${\hat{{\bf{s}}}}_{o}$ and $-{\hat{{\bf{s}}}}_{o}$. This retardance is crucial when considering how ${{\bf{T}}}_{{{\bf{s}}}_{i}\to {{\bf{s}}}_{o}}$ transforms any input state other than $\pm {\hat{{\bf{s}}}}_{i}$. Polarizations that are a superposition of $-{\hat{{\bf{s}}}}_{o}$ and $+{\hat{{\bf{s}}}}_{o}$ will transform in an undetermined way.