Fundamental precision limits of full Stokes polarimeters based on DoFP polarization cameras for an arbitrary number of acquisitions

Xiaobo Li; Xiaobo Li; Haofeng Hu; Haofeng Hu; Haofeng Hu; François Goudail; François Goudail; Tiegen Liu; Tiegen Liu

doi:10.1364/OE.27.031261

1. Introduction

Polarimetry is a powerful tool with various applications, such as object identification, thin-film characterization and image recovery [1–5]. The complete polarization state can be expressed by the Stokes vector and up to now, various Stokes polarimeter architectures have been proposed [3]. In particular, as an emerging technology, the polarimeters based on linear division-of-focal-plane (DoFP) polarization cameras, whose pixels are arranged periodically with different orientations of linear micro-polarizers, have recently raised attention and have been widely used due to their ability to perform real-time polarimetric measurements [6–10]. These “linear” DoFP camera can directly measure the linear Stokes parameters in a single image acquisition, however, they cannot measure its circular properties since the micro-polarizer array does not include any retarder element [8–10]. Therefore, to measure the full Stokes vector, one must place an adjustable retarder in front of the DoFP camera and perform at least two image acquisitions with different settings of the retarder [9,11].

With such a setup, one can perform an arbitrary number N of image acquisitions with different settings of the retarder. In the presence of measurement noise, increasing N of course increases the signal to noise ratio, and one obtains a versatile polarization imager where the tradeoff between acquisition time and precision of Stokes vector estimation can be optimized depending on the application needs and timescale of the scene variations [12,13]. The question thus arises of best way to perform these N acquisitions, i.e., the optimal sequence of N retarder settings that maximizes the estimation precision of the Stokes vector. This question has long been investigated in the case of unconstrained polarimeters where the measurement vectors of the N acquisitions can lie anywhere on the Poincaré sphere [14–20]. Different optimization criteria have been considered, such as the determinant of the measurement matrix [14,15], its condition number [16,17], or the variances of the Stokes vector estimated in the presence of additive noise [12,18] and Poisson noise [19,20]. Globally, the solution to this problem can be synthetized has follows: optimizations of these three different criteria lead to the same optimal solutions, which consist in measurement vectors forming a “spherical design” on the Poincaré sphere [13,15]. However, the structure of DoFP-based full Stokes polarimeters is constrained since each image acquisition consists of a group of four intensity measurements done with the same retarder setting and four fixed orientations of the polarizers. Optimization of the sequence of retarder settings for this type of polarization imagers is thus a specific problem. The purposes of this paper are to solve this problem, and to determine the fundamental precision limits of DOFP polarization imagers as has been done for unconstrained ones. We will show that for a number of acquisitions greater than three, the performance of constrained polarimeters can be reached with some retarder architectures, but not all of them.

This paper is organized as follows. In Section 2, we define the addressed problem in precise mathematical terms and review the state of the art on this issue. Then we perform optimization of several retarder architectures in the presence of additive white noise (AWN) and Poisson shot noise (PSN): we consider a rotating quarter-waveplate (QWP) in Section 3, a rotating retarder with optimized retardance in Section 4, and a variable retarder with fixed orientation in Section 5. We summarize the obtained results and propose the conclusion in Section 6, besides, the perspectives of this work are also drawn in Section 6.

2. Definition and model of the problem

We consider an imaging system composed of a linear DoFP camera based on a micro-polarizer array with four different orientations of 0°, 90°, 45° and 135° [ Fig. 1(a)] and a variable retarder in front of it [Fig. 1(b)]. We assume that the orientation angle $\theta $ and the retardance $\delta $ of the retarder is variable and can be controlled.

Fig. 1. (a) Structure of the linear DOFP polarization camera. (b) Setup of the full Stokes polarimeter based on a linear DoFP camera and a controllable retarder.

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Let ${\bf S} = {({{s_0},{s_1},{s_2},{s_3}} )^T}$ denote the input Stokes vector to be measured, where s₀ refers to the intensity of input light. One performs N image acquisitions by changing the configuration of the retarder (the orientation and/or the retardance), and one thus obtains 4N intensity measurements that can be expressed as follows:

(1)$${\bf I} = W \cdot {\bf S},$$

where ${\bf I}$ is a 4N-element intensity vector, each element ${I_{4({j - 1} )+ i}}$ of which denotes the intensity measured under retarder orientation angle ${\theta _j}$ and retardance ${\delta _j}, j \in [{1,N} ]$ by the pixel with micro-polarizer orientation ${\varphi _i}, i \in [{1,4} ]$. Each value of ${\varphi _i}$ corresponds respectively to orientation 0°, 45°, 90° or 135°. The Mueller matrix of a retarder with retardance ${\delta _j}$ and orientation ${\theta _j}$ is expressed as [1]:

(2)$${M_R}({{\delta_j},{\theta_j}} )= \left[ {\begin{array}{{cccc}} 1&0&0&0\\ 0&{{{\cos }^2}2{\theta_j} + \cos {\delta_j}{{\sin }^2}2{\theta_j}}&{(1 - \cos {\delta_j})\sin 2{\theta_j}\cos 2{\theta_j}}&{ - \sin {\delta_j}\sin 2{\theta_j}}\\ 0&{(1 - \cos {\delta_j})\sin 2{\theta_j}\cos 2{\theta_j}}&{{{\sin }^2}2{\theta_j} + \cos {\delta_j}{{\cos }^2}2{\theta_j}}&{\sin {\delta_j}\cos 2{\theta_j}}\\ 0&{\sin {\delta_j}\sin 2{\theta_j}}&{ - \sin {\delta_j}\cos 2{\theta_j}}&{\cos {\delta_j}} \end{array}} \right],$$

and the Mueller matrix of the polarizer with orientation ${\varphi _i}$ is:

(3)$${M_P}({\varphi _i}) = \frac{1}{2}\left( {\begin{array}{{cccc}} 1&{\cos 2{\varphi_i}}&{\sin 2{\varphi_i}}&0\\ {\cos 2{\varphi_i}}&{{{\cos }^2}2{\varphi_i}}&{\sin 2{\varphi_i}\cos 2{\varphi_i}}&0\\ {\sin 2{\varphi_i}}&{\sin 2{\varphi_i}\cos 2{\varphi_i}}&{{{\sin }^2}2{\varphi_i}}&0\\ 0&0&0&0 \end{array}} \right),$$

The matrix W defined in Eq. (1) is a 4N×4 measurement matrix since there is a total of 4N intensity measurements when we perform N image acquisitions by DoFP camera, and the [4(j-1)+i]^th row vector ${{\bf W}_{4({j - 1} )+ i}}$ of matrix W is equal to $[{1,0,0,0} ]{M_P}({\varphi _i}){M_R}({{\delta_j},{\theta_j}} )$. By taking into account the four possible values of ${\varphi _i}$, one obtains:

(4)$${{\bf W}_{4({j - 1} )+ i}} = \left\{ {\begin{array}{{c}} \begin{array}{l} \frac{1}{2}[{1,({{{\cos }^2}2{\theta_j} + \cos {\delta_j}{{\sin }^2}2{\theta_j}} ),} \\ {(1 - \cos {\delta_j})\sin 2{\theta_j}\cos 2{\theta_j}, - \sin {\delta_j}\sin 2{\theta_j}} ],\textrm{ if }i = \textrm{1} \end{array}\\ \begin{array}{l} \frac{1}{2}[{1, - ({{{\cos }^2}2{\theta_j} + \cos {\delta_j}{{\sin }^2}2{\theta_j}} ),} \\ { - (1 - \cos {\delta_j})\sin 2{\theta_j}\cos 2{\theta_j},\sin {\delta_j}\sin 2{\theta_j}} ],\textrm{ if }i = \textrm{2} \end{array}\\ \begin{array}{l} \frac{1}{2}[{1,({{{\cos }^2}2{\theta_j} + \cos {\delta_j}{{\sin }^2}2{\theta_j}} ),} \\ {(1 - \cos {\delta_j})\sin 2{\theta_j}\cos 2{\theta_j},\sin {\delta_j}\sin 2{\theta_j}} ],\textrm{ if }i = \textrm{3} \end{array}\\ \begin{array}{l} \frac{1}{2}[{1, - ({{{\cos }^2}2{\theta_j} + \cos {\delta_j}{{\sin }^2}2{\theta_j}} ),} \\ { - (1 - \cos {\delta_j})\sin 2{\theta_j}\cos 2{\theta_j}, - \sin {\delta_j}\sin 2{\theta_j}} ],\textrm{ if }i = \textrm{4} \end{array} \end{array}} \right.,\textrm{with }j \in [{1,N} ].$$

In practice, intensity measurements are always corrupted by noise, so that the vector I defined in Eq. (1) is a random vector. We will consider in this paper the two fundamental sources of noise that affect measurements. The first one is AWN, in which case I is a random vector with variance ${\sigma ^2}$ and a diagonal covariance matrix. The second one is PSN, in which case the covariance matrix is still diagonal, but the variance of the element ${I_i}, i \in [{1,N} ]$ is equal to ${I_i}$ if ${I_i}$ is expressed in terms of a number of photoelectrons [15]. The full Stokes vector can be estimated by inverting Eq. (1) as follows:

(5)$$\hat{{\bf S}} = {W^ + }{\bf I},$$

where

(6)$${W^ + } = {[{{W^T}W} ]^{ - 1}}{W^T}$$

is the Moore-Penrose pseudo-inverse matrix. Since I is random, so is $\hat{{\bf S}}$. Our goal is to find the measurement parameters, that is, the configuration parameters $({{\theta_j},{\delta_j}} )$ that minimize the variances of the elements of the vector $\hat{{\bf S}}$. A relevant criterion to quantify the performance of a polarimeter is the sum of the variances for the four elements of the estimated Stokes vector. This sum is referred to as “Equally Weighted Variance (EWV)” [15,19,20].

A recent article [11] studied the optimal strategies for full Stokes measurement with two image acquisitions with the setup in Fig. 1(b). It showed that by optimizing the retardance and the orientations of the retarder, the EWV of the estimated full Stokes vector can be reduced to $5.5{\sigma ^2}$ in the presence of AWN with variance of ${\sigma ^2}$. On the other hand, previous works have demonstrated that for any polarimeter involving M intensity measurements, the minimal possible value of the EWV is ${{\textrm{40}{\sigma ^2}} \mathord{\left/ {\vphantom {{\textrm{40}{\sigma^2}} M}} \right.} M}$ in the presence of AWN [13,19]. Since one acquisition by the DoFP camera is equivalent to four intensity measurements corresponding to the four different polarizer orientations, N acquisitions are thus equivalent to M = 4N intensity measurements. Hence the theoretical lower bound on the EWV is equal to ${{\textrm{10}{\sigma ^2}} \mathord{\left/ {\vphantom {{\textrm{10}{\sigma^2}} N}} \right.} N}$, and thus for N = 2 acquisitions, it is equal to $5{\sigma ^2}$. Obviously, the value of $5.5{\sigma ^2}$ obtained in Ref. [11] is larger than this lower bound. This means that for N = 2 acquisitions, the constraints of DoFP architecture are such that the lower bound cannot be reached. Therefore, the question we shall address in the following is whether this lower bound can be reached with a larger number N > 2 of acquisitions. Furthermore, in the presence of PSN, even if the EWV is independent of the input Stokes vector, the individual estimation variances of each Stokes parameters may depend on it [19–21]. We will thus identify the configurations that both minimize the EWV and equalize the individual estimation variances.

3. Full Stokes estimation with a rotatable QWP

We assume in this section that the retarder is a rotatable QWP, since this type of retarder is widely available [1,3,12]. In Eq. (2), ${\delta _j}$ is thus constant and equal to ${\pi \mathord{\left/ {\vphantom {\pi 2}} \right.} 2}$, and only the orientation angle ${\theta _j}$ varies between different acquisitions. Our goal is to determine the sets of orientation angles that minimize EWV in the presence of AWN and PSN. In the presence of AWN, it is known that the EWV has the following expression [19]:

(7)$$\textrm{EW}{\textrm{V}_{\textrm{AWN}}} = {\sigma ^2} \cdot \textrm{trace}\{{{{[{{W^T}W} ]}^{ - 1}}} \},$$

where $\textrm{trace}\{{\cdot} \}$ denotes the trace of a matrix and ${\sigma ^2}$ is the variance of AWN. The QWP is rotated through N steps with evenly-spaced orientation angles of:

(8)$${\theta _i} = {\theta _0} + \frac{{({i - 1} )\cdot \textrm{18}{\textrm{0}^ \circ }}}{N},\forall i \in [{1,N} ]$$

in the range of 0^°∼180^° [22,23]. Without loss of generality, the starting-angle ${\theta _\textrm{0}}$ is set to 0^°. Substituting these angle values into Eq. (4), setting $\delta = {90^ \circ }$ and assuming that N > 2, one obtains after simple calculations:

(9)$${[{{W^T}W} ]^{ - 1}} = \textrm{diag}\left\{ {\frac{1}{N},\frac{4}{N},\frac{4}{N},\frac{2}{N}} \right\},$$

where $\textrm{diag}\{{\cdot} \}$ denotes a diagonal matrix. Consequently, the individual estimation variances of each Stokes parameter and the EWV have the following values:

(10)$${\gamma _0} = \frac{{{\sigma ^2}}}{N},{\gamma _1} = {\gamma _2} = \frac{{4{\sigma ^2}}}{N},{\gamma _3} = \frac{{2{\sigma ^2}}}{N},\textrm{EW}{\textrm{V}_{\textrm{AWN}}} = \frac{{11{\sigma ^2}}}{N}.$$

It is interesting to note that this result is also valid for N = 2 acquisitions if the orientation angles of the QWP are ${\theta _0}$ and ${\theta _0} + {45^ \circ }$ [11].

Let us now assume that the noise source is PSN. The variances of each intensity measurement are equal to its mean $\sum\nolimits_{j = 0}^3 {{W_{ij}}{s_j}} $, and therefore, the individual estimation variances of each Stokes parameter are given by [19]:

(11)$$\forall i \in [{0,3} ],{\gamma _i} = \sum\limits_{j = 0}^3 {{s_j}\sum\limits_{n = 0}^{4N - 1} {{{[{W_{in}^\dagger } ]}^2}{W_{nj}}} } = \sum\limits_{j = 0}^3 {{Q_{ij}}{s_j}} ,$$

where the matrix Q is defined as:

(12)$${Q_{ij}} = \sum\limits_{n = 0}^{4N - 1} {{{[{W_{in}^ + } ]}^2}{W_{nj}}} .$$

It is seen in Eq. (11) that the estimation variance depends on all the elements of the input Stokes vector [11,13]. It has been shown in Ref. [19] that an appropriate optimization criterion, derived from a minmax approach, is:

(13)$$\textrm{EW}{\textrm{V}_{\textrm{PSN}}} = {s_0} \cdot \left( {\frac{1}{{2{\sigma^2}}}\textrm{EW}{\textrm{V}_{\textrm{AWN}}} + ||{\bf v} ||} \right),$$

where $||{\bf v} ||= {\left( {\sum\nolimits_{j = 1}^3 {q_j^2} } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}}},$ and ${q_j} = {\sum\nolimits_{i = 0}^3 Q _{ij}}$. When the orientation angles are those defined in Eq. (8), the matrix Q defined in Eq. (12) has the following structure:

(14)$$Q = \frac{1}{{2N}}\left[ {\begin{array}{{cccc}} 1&0&0&0\\ 4&0&0&0\\ 4&0&0&0\\ 2&0&0&0 \end{array}} \right].$$

Consequently, the individual estimation variances of the four Stokes parameters and the EWV are calculated by substituting Eq. (14) into Eqs. (11) and (13):

(15)$${\gamma _0} = \frac{{{s_o}}}{{2N}},{\gamma _1} = {\gamma _2} = \frac{{2{s_o}}}{N},{\gamma _3} = \frac{{{s_o}}}{N},\textrm{ EW}{\textrm{V}_{\textrm{PSN}}} = \frac{{11{s_o}}}{{2N}}.$$

It is seen in this equation that both the individual estimation variances and the EWV depend only on the intensity s₀ and not on the other Stokes parameters.

The closed-form analysis above was based on the hypothesis of evenly-spaced orientation angles [see Eq. (8)]. This is the most common rotation strategy, but it may not be optimal. We thus performed a global optimization considering arbitrary sets of orientation angles. For that purpose, we used the shuffled complex evolution (SCE) method [24,25], which is robust to the presence of local maxima when multiple parameters have to be optimized. The results show that even with arbitrary sets of orientations, the minimal values of EWV and individual variances are identical to those given in Eqs. (10) and (15). Analyzing in more detail the optimal set of orientations found in this case, it is observed that they are identical in the presence of AWN and PSN, and have the following characteristics:

1. $\forall i \in [{1,N} ],$ the i^th optimal rotation angle has to be chosen as one of the two angles: $(16)$${A_i} = {\theta _0} + \left( {\frac{{({i - 1} )\cdot \textrm{9}{0^ \circ }}}{N},{{90}^ \circ } + \frac{{({i - 1} )\cdot \textrm{9}{0^ \circ }}}{N}} \right),$$$ with an arbitrary starting starting-angle ${\theta _\textrm{0}}$.
2. With the assumption of ${\theta _2} \le {\theta _3}$, the number of optimal solution sets is equal to $\underbrace{{C_2^1 \times C_2^1 \times \cdots \times C_2^1}}_{N} = {2^N}$, where $C_n^k$ denotes the number of k-combinations from a given set of n elements.

For example, we have represented in Fig. 2 the value of 1/EWV_AWN as a function of ${\theta _2}$ and ${\theta _3}$ for N = 3 acquisitions, under the assumptions of ${\theta _1} = {0^ \circ }$ and ${\theta _2} \le {\theta _3}$ . The points associated to the minima of EWV_AWN are marked with black crosses, which correspond to four sets of optimal solutions $({{\theta_2},{\theta_3}} )$:

(17)$$({{\theta_2},{\theta_3}} )= \{{({{{30}^ \circ },{{60}^ \circ }} ),({{{30}^ \circ },{{150}^ \circ }} ),({{{60}^ \circ },{{120}^ \circ }} ),({{{120}^ \circ },{{150}^ \circ }} )} \}.$$

Besides, it should be noted that there are other four sets consisting of the same values of $({{\theta_2},{\theta_3}} )$ as in Eq. (17) but with ${\theta _1} = {90^ \circ }$.

Fig. 2. For the setup based on a QWP, the value of 1/EWV_AWN as a function of the ${\theta _2}$ and ${\theta _3}$ under the assumptions of ${\theta _1} = {0^ \circ }$ and ${\theta _2} \le {\theta _3}$ . The black crosses mark the minima of EWV_AWN.

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As a conclusion, for an arbitrary number N of acquisitions, the values of EWV given in Eqs. (10) and (15) are the lower bounds on estimation variance of a full Stokes polarimeter with a linear DoFP camera and a rotatable QWP in the presence of AWN and PSN. This means that the lower bounds of an unconstrained polarimeter with 4N intensity measurements, namely, $10{{{\sigma ^2}} \mathord{\left/ {\vphantom {{{\sigma^2}} N}} \right.} N}$ in the presence of AWN and $5{{{s_0}} \mathord{\left/ {\vphantom {{{s_0}} N}} \right.} N}$ in the presence of PSN, cannot be reached with a rotatable QWP. However, what if we change the retardance? In order to answer this question, we will assume in the following section that the retardance $\delta $ of the rotating retarder can have an arbitrary value.

4. Full Stokes estimation with optimal retardance

Let us now assume that the retardance $\delta $ of the rotating retarder is arbitrary. Using Eq. (4), assuming that the angles are evenly-spaced and the number of acquisitions is N > 2, it can be shown that the matrix ${({{W^T}W} )^{ - 1}}$ has the following expression:

(18)$$\textrm{diag}\left\{ {\frac{1}{N},\frac{4}{{N(1 + {{\cos }^2}\delta )}},\frac{4}{{N(1 + {{\cos }^2}\delta )}},\frac{2}{{N{{\sin }^2}\delta }}} \right\}.$$

Consequently, the EWV in the presence of AWN is equal to:

(19)$$\textrm{EW}{\textrm{V}_{\textrm{AWN}}} = \frac{{({11 - 6{{\cos }^2}\delta - {{\cos }^4}\delta } )}}{{N({1 - {{\cos }^4}\delta } )}}{\sigma ^2}.$$

The value of EWV_AWN as a function of the retardance is plotted in Fig. 3. Interestingly, a retardance equal to 90° corresponds to a local maximum, but EWV_AWN reaches a global minimum equal to ${{10{\sigma ^2}} \mathord{\left/ {\vphantom {{10{\sigma^2}} N}} \right.} N}$ for two other values of $\delta $. Annulling the derivative of Eq. (19) with respect to $\delta $, we find that the exact values of these optimal retardances are such that $\cos \delta = {{ \pm \sqrt 3 } \mathord{\left/ {\vphantom {{ \pm \sqrt 3 } 3}} \right.} 3}$, which means that ${\delta _{\textrm{opt}}} = {125.26^ \circ }$ or ${54.74^ \circ }$. It is also interesting to remember that, for the full Stokes polarimeter featuring a rotatable retarder and a fixed polarizer, there is only one optimal retardance equal to 131.80° [21]. This difference results from the fact that the micro-polarizers in DoFP-based polarimeters have four different fixed directions.

Fig. 3. The value of EWV_AWN as a function of the retardance of the rotatable retarder.

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For these optimal retardance values, Eq. (18) yields:

(20)$$\textrm{diag}\left\{ {\begin{array}{{cccc}} {\frac{1}{N},}&{\frac{3}{N},}&{\frac{3}{N},}&{\frac{3}{N}} \end{array}} \right\}.$$

Consequently, the individual estimation variances of each Stokes parameter and the EWV have the following values:

(21)$$\begin{array}{{ccc}} {{\gamma _0} = \frac{{{\sigma ^2}}}{N},}&{{\gamma _1} = {\gamma _2} = {\gamma _3} = \frac{{3{\sigma ^2}}}{N},}&{\textrm{EW}{\textrm{V}_{\textrm{AWN}}} = \frac{{10{\sigma ^2}}}{N}.} \end{array}$$

Since the value of EWV_AWN is $10{{{\sigma ^2}} \mathord{\left/ {\vphantom {{{\sigma^2}} N}} \right.} N}$, the lower bound of unconstrained polarimeters is reached.

Furthermore, in the presence of PSN, it is found that with the optimal retardances, one has:

(22)$$\begin{array}{{cc}} {\forall j \in [{1,3} ],}&{{q_j} = \sum\limits_{i = 0}^3 {{Q_{ij}}} = \frac{9}{{{N^2}}}\sum\limits_{n = 0}^{4N - 1} {{W_{nj}}\sum\limits_{i = 0}^3 {{{[{{W_{ni}}} ]}^2}} = 0} .} \end{array}$$

Consequently, the second term $||{\bf v} ||$ in Eq. (13) is null and the values of the individual estimation variances and of the EWV are given by:

(23)$$\begin{array}{{ccc}} {{\gamma _0} = \frac{{{s_0}}}{{2N}},}&{{\gamma _1} = {\gamma _2} = {\gamma _3} = \frac{{3{s_0}}}{{2N}},}&{\textrm{EW}{\textrm{V}_{\textrm{PSN}}} = \frac{{5{s_0}}}{N}} \end{array}.$$

Thus in the presence of PSN, the lower bound on EWV is also reached. Moreover, the estimation variances of the last three Stokes parameters are “equalized” as in optimal unconstrained polarimeters based on spherical designs. This performance is indeed same as that obtained with unconstrained full Stokes polarimeters in previous works [15,19,21,26].

In addition, we performed a global optimization considering arbitrary sets of retarder orientation angles by SCE method. In contrast with the setup based on a QWP, the results show that only evenly-spaced orientation angles in the range of 0°∼180° (with arbitrary starting-angle ${\theta _0}$) lead to the minimal values of EWV given in Eqs. (21) and (23). The value of 1/EWV_AWN as a function of ${\theta _2}$ and ${\theta _3}$ is displayed in Fig. 4 under the assumptions of ${\theta _1} = {0^ \circ }$ and ${\theta _2} \le {\theta _3}$. The point corresponds to the minimum value of EWV_AWN is also marked with black cross. It is seen that only the set (0°, 60°, 120°) leads to the optimal value of EWV.

Fig. 4. For the setup based on a retarder with retardance of 125.26^° or 54.74^°, the value of 1/EWV_AWN as a function of ${\theta _2}$ and ${\theta _3}$ under the assumptions of ${\theta _1} = {0^ \circ }$ and ${\theta _2} \le {\theta _3}$. The black cross marks the minima of EWV_AWN.

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As a conclusion, we have shown that by performing a number N > 2 of acquisitions, a polarimeter based on DoFP camera and rotating retarder with optimal retardance can reach the same performance as an unconstrained polarimeter, namely, the theoretical lower bounds on EWV of $10{{{\sigma ^2}} \mathord{\left/ {\vphantom {{{\sigma^2}} N}} \right.} N}$ in the presence of AWN and of $5{{{s_0}} \mathord{\left/ {\vphantom {{{s_0}} N}} \right.} N}$ in the presence of PSN. Moreover, since it has been shown in Ref. [11] that no retarder, be it elliptical, can reach this lower bound with N = 2 acquisitions, the least number of acquisitions to reach this bound is N = 3.

As a remark concerning practical implementation, it is noticed that retarder of retardance 125.26° is not common and may be difficult to realize. An alternative strategy is to replace it by two rotatable QWPs. For a number N = 3 of acquisitions, optimization results show that the following set of orientation angles of the two QWPs $\{{({{0^ \circ },{{45}^ \circ }} ), ({{{45}^ \circ },{{45}^ \circ }} ),({{{45}^ \circ },{0^ \circ }} )} \}$ (among others) reaches the minimal EWV and individual estimation variances given in Eqs. (21) and (23).

5. Full Stokes estimation based on a retarder with variable retardance

The previous results have been obtained with a rotating retarder having fixed retardance and varying orientation angle. Let us now consider a linear retarder with fixed orientation angle $\theta $ and variable retardances ${\delta _i},\, i \in [{1,N} ]$. Such a variable retarder strategy is easily implementable with a liquid crystal variable retarder (LCVR) [8,27]. Compared to the rotating retarder, it has the advantage of not having any moving parts. Let us first assume that the retardances are evenly-spaced:

(24)$${\delta _i} = {\delta _\textrm{0}}\textrm{ + }\frac{{({i - 1} )\cdot \textrm{18}{\textrm{0}^ \circ }}}{N}.$$

Without loss of generality, the starting retardance is set to 0°. As in the previous sections, the estimation variance can be obtained by substituting the retardance values defined in Eq. (24) into Eq (4) in order to compute the matrix W, and then into Eqs. (7) and (13) to compute the values of EWV in the presence of AWN and PSN. After cumbersome but elementary calculations, one obtains the values of the individual estimation variances and of the EWV that are shown in Table 1. It is seen in this table that both the individual estimation variances and the EWVs are independent of the input Stokes vector in the presence of AWN and PSN. Moreover, the values of EWV_AWN and EWV_PSN are the same as those obtained with a rotating QWP. However, the individual estimation variances of the Stokes parameters s₁ and s₂ vary with the orientation angle $\theta $ of the variable retarder. We have represented the variations of the four individual variances as a function of $\theta $ in the presence of AWN and PSN in Fig. 5. It is seen in particular that with the variable retarder, the variance of s₃ is larger than those of s₁ and s₂, while it was the opposite with the rotating QWP [see Eqs. (10) and (15)].

Fig. 5. Individual estimation variances of four Stokes parameters as a function of the orientation angle of the variable retarder in the presence of AWN and PSN with number of N acquisitions.

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Table 1. Individual estimation variances and the corresponding EWV in the presence of AWN and PSN.

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Besides, we also performed a global search among arbitrary sets of retardances instead of evenly-spaced ones. The results show that only the evenly-spaced strategy can lead to the minimal values of EWV shown in Table 1. This means that with a single variable linear retarder, for any number N of acquisitions, the minimum of EWV is equal to $11{{{\sigma ^2}} \mathord{\left/ {\vphantom {{{\sigma^2}} N}} \right.} N}$ in the presence of AWN and to ${{5.5{s_0}} \mathord{\left/ {\vphantom {{5.5{s_0}} N}} \right.} N}$ in the presence of PSN. This type of setup thus cannot reach the theoretical lower bound whatever the value of N.

6. Summary and conclusion

As a summary, we have listed the minimal EWVs and the corresponding individual estimation variances of all the investigated setups in Table 2. Since the individual estimation variances of s₁ and s₂ vary with the orientation angle $\theta $ in the variable retarder setup, we only show the case $\theta = \textrm{22}\textrm{.}{\textrm{5}^ \circ }$, for which the individual estimation variances are best balanced.

Table 2. Minimal EWVs and corresponding individual estimation variances of the investigated setups with number of N > 2 acquisition.

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As a final remark, let us note that in practice, AWN and PSN are often simultaneously present in images. Since these noise sources are statistically independent, their variances simply add up if both the AWN standard deviation $\sigma $ and the intensity s₀ are expressed in terms of numbers of electrons. According to the results obtained in the previous sections, the optimal measurement configurations are identical in the presence of AWN and PSN, therefore, they are also optimal when AWN and PSN are simultaneously present. Hence only the setups with one rotatable retarder with fixed retardance of 125.26° or two rotatable QWPs reach the lower bound in the presence of mixed noise. We have given the expressions of the optimal EWV and individual variances in the presence of mixed noise in the last row of Table 2. It is noted that since the optimal configurations are identical, only the multiplicative coefficient changes: it is ${{({{\sigma^\textrm{2}}\textrm{ + }{{{s_0}} \mathord{\left/ {\vphantom {{{s_0}} 2}} \right.} 2}} )} \mathord{\left/ {\vphantom {{({{\sigma^\textrm{2}}\textrm{ + }{{{s_0}} \mathord{\left/ {\vphantom {{{s_0}} 2}} \right.} 2}} )} N}} \right.} N}$ for mixed noise.

In conclusion, for the first time to our knowledge, we have systematically investigated the optimization of full Stokes polarimeters based on a linear DoFP camera and a retarder for an arbitrary number N of acquisitions, and determined their fundamental precision limits in the presence of AWN and PSN. We have demonstrated that it is possible to reach the theoretical lower bound on EWV of unconstrained polarimeters with rotating retarder if N is greater than or equal to three, the measurement angles are evenly spaced, and the retardance is equal to 125.26°. With this setup, estimations variances are exactly the same as those obtained with unconstrained polarimeters based on spherical designs [13,15]. Moreover, in practice, all the properties above can also be obtained by using a system with two rotatable QWPs in pair. In contrast, with a rotating QWP or a variable retarder, the EWV cannot reach this bound. Moreover, in the latter case, the individual estimation variances of s₁ and s₂ vary with the orientation angle of the variable retarder both in the presence of AWN and PSN. This means that with these setups, one has not enough degrees of freedom to reach the performance of unconstrained polarimeters whatever the value of N. These results are important to get the most out of DoFP polarization imagers in real applications. However, it is also seen that the performance loss among the different considered setups is slight, since it amounts to a 10% increase of EWV. Thus in practice, simpler or handier solutions may be preferred if ultimate precision is not needed.

An important perspective of this work is to investigate how optimized full Stokes imagers based on DoFP camera can improve spatially resolved polarimetric measurements in different applications. Investigation of speckle pattern polarization properties is particularly interesting since it requires high precision measurements. [28,29]. In addition, applications of the optimization approach described in this paper to systems with other sources of perturbations such as instrumental errors [30–32] and to Mueller polarimeters [33,34] based on DoFP camera with an arbitrary number of acquisitions are two other interesting perspectives to the present work.

Funding

National Natural Science Foundation of China (61775163); Young Elite Scientists Sponsorship Program by CAST (2017QNRC001); Director Fund of Qingdao National Laboratory for Marine Science and Technology (QNLM201717).

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Value	AWN	PSN
$γ_{0}$	$σ^{2} / N$	$s_{0} / 2 N$
$γ_{1}$	$3 σ^{2} / N - σ^{2} / N \cos (4 θ)$	$3 s_{0} / 2 N - s_{0} / 2 N \cos (4 θ)$
$γ_{2}$	$3 σ^{2} / N + σ^{2} / N \cos (4 θ)$	$3 s_{0} / 2 N + s_{0} / 2 N \cos (4 θ)$
$γ_{3}$	$4 σ^{2} / N$	$2 s_{0} / N$
EWV	$11 σ^{2} / N$	$11 s_{0} / 2 N$

		Rotating retarder		Variable retarder
		QWP	$δ = 125 .2 6^{\circ}$ /two QWPs	$θ = 22 . 5^{\circ}$
AWN	EWV $\times σ^{2} / N$	11	10 (Lower bound)	11
AWN	Individual variance $\times σ^{2} / N$	1:4:4:2	1:3:3:3 (Equalized)	1:3:3:4
PSN	EWV $\times s_{0} / 2 N$	11	10 (Lower bound)	11
PSN	Individual variance $\times s_{0} / 2 N$	1:4:4:2	1:3:3:3 (Equalized)	1:3:3:4
Mixed noise	EWV $\times (σ^{2} + s_{0} / 2) / N$	11	10 (Lower bound)	11
Mixed noise	Individual variance $\times (σ^{2} + s_{0} / 2) / N$	1:4:4:2	1:3:3:3 (Equalized)	1:3:3:4

Value	AWN	PSN
$γ_{0}$	$σ^{2} / N$	$s_{0} / 2 N$
$γ_{1}$	$3 σ^{2} / N - σ^{2} / N \cos (4 θ)$	$3 s_{0} / 2 N - s_{0} / 2 N \cos (4 θ)$
$γ_{2}$	$3 σ^{2} / N + σ^{2} / N \cos (4 θ)$	$3 s_{0} / 2 N + s_{0} / 2 N \cos (4 θ)$
$γ_{3}$	$4 σ^{2} / N$	$2 s_{0} / N$
EWV	$11 σ^{2} / N$	$11 s_{0} / 2 N$

		Rotating retarder		Variable retarder
		QWP	$δ = 125 .2 6^{\circ}$ /two QWPs	$θ = 22 . 5^{\circ}$
AWN	EWV $\times σ^{2} / N$	11	10 (Lower bound)	11
AWN	Individual variance $\times σ^{2} / N$	1:4:4:2	1:3:3:3 (Equalized)	1:3:3:4
PSN	EWV $\times s_{0} / 2 N$	11	10 (Lower bound)	11
PSN	Individual variance $\times s_{0} / 2 N$	1:4:4:2	1:3:3:3 (Equalized)	1:3:3:4
Mixed noise	EWV $\times (σ^{2} + s_{0} / 2) / N$	11	10 (Lower bound)	11
Mixed noise	Individual variance $\times (σ^{2} + s_{0} / 2) / N$	1:4:4:2	1:3:3:3 (Equalized)	1:3:3:4

Fundamental precision limits of full Stokes polarimeters based on DoFP polarization cameras for an arbitrary number of acquisitions

Abstract

1. Introduction

2. Definition and model of the problem

3. Full Stokes estimation with a rotatable QWP

4. Full Stokes estimation with optimal retardance

5. Full Stokes estimation based on a retarder with variable retardance

6. Summary and conclusion

Funding

References

Cited By

Figures (5)

Tables (2)

Equations (24)

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