Polarization imaging based on time-integration by a continuous rotating polarizer

Naiting Gu; Naiting Gu; Naiting Gu; Yawei Xiao; Yawei Xiao; Linhai Huang; Linhai Huang; Changhui Rao; Changhui Rao

doi:10.1364/OE.444230

1. Introduction

Polarization imaging is very important because of its ability to acquire the state of polarization (SoP) information of light, which contains object’s special characteristics such as surface features, shapes, roughness and material, etc. [1,2]. The valuable information is hard to be provided by the traditional intensity-based cameras, thus polarization imaging technologies are widely used in many applications for example remote sensing [3], astronomy [4–6], biomedical imaging [7,8], material characterization [9,10] and target detection [11]. Referring the basic principle and configuration, the polarimeter can be categorized simply as four classic types: division of time polarimeter (DoTP) [12–16], division of amplitude polarimeter (DoAmP) [17,18], division of aperture polarimeter (DoAP) [19–21] and division of focal plane polarimeter (DoFP) [22–25]. The DoTP is the most classic polarimeter by adopting temporal polarization modulator and time-sequential data acquisition. Such DoTP is generally treated as slowly or static polarimeter for stationary scene to avoid long time-consuming and temporal blur. Some fast polarimeter based Fourier analysis (such as Thorlabs PAX series) by rotating a retarder can only be used to measure SoP of collimated light [26]. The DoAmP and the DoAP are two kinds of simultaneous polarimeter, which divide light beam under test into several sub-beams (four beams in general) and capture multiple images simultaneously by using independent photodetector or single photodetector. These kinds of design promote the real-time property obviously, but comes with complex structure, poor stability and difficult alignment because of bulky optical components [17]. Thus, the broad application of DoAmP and DoAP is limited. The DoFP, which utilizes micro-grid array to obtain all or part of components of Stokes vector, is the most conventional polarimeter because of its performance of compact structure, light weight and snapshot nature. However, its performance is often limited by the fixed pattern noise (FPN), low and non-uniform extinction ratio, non-uniform photon response, orientation misalignment of micro-polarizer and instantaneous field of view error [27–29]. Which all will contribute to polarimetric inaccuracy due to polarimetric loss and strong edge artifacts. Compared to the other kinds of polarimeter, the DoTP collect the total intensity of each polarization component by a photodetector, ensuring the high and uniform signal-to-noise (SNR) and extinction ratio, which make the DoTP theoretically of higher measuring accuracy [30] and suitable for high-sensitive measuring scenarios [31]. For a typical DoTP, the SoP can be retrieved from optical intensities by rotating polarizer (linear Stokes components) or retarder (full Stokes components) to several certain orientations [11,12]. To improve the speed of DoTP, some authors proposed new kinds of DoTP by using photoelastic modulators [13,32], liquid crystal retarders [15,33], electro-optical modulators [14,34] or binary polarization rotators [16,35], but disadvantages of chromatic dispersion, narrow spectral band and temperature sensitivity still hinder the widespread application of them. So can we avoid these disadvantages while keeping relative high speed?

In this paper, we demonstrate a dynamic imaging polarimeter based on time-integration method by using a continuous rotating polarization modulator. The traditional DoTPs include linear Stokes imaging polarimeter and full Stokes imaging polarimeter, and the main difference between them is that the state of polarization of incident light is modulated by a rotating polarizer or by a rotating retarder and a fixed polarizer, respectively [36]. Due to the fact that there is very little expected circular polarization in most passive imaging scenarios [3], here we use a single polarizer with high extinction ratio as the polarization modulator for simplicity. The polarizer together with an accurate magnetic encoder (ME) are mounted in a brush-less DC motor (BLDCM) with high speed. A photodetector integrated with the BLDCM will be used to collect the polarization intensity after modulation of the polarizer. The SoP of the incoming scenarios can be retrieved once matrix calculation of the sequential optical intensities is made. In this paper, we derived the basic principle and the corresponding mathematical expressions and made the corresponding error analysis. The accuracy and dynamic performance are also validated by two kinds of experiments including illuminated light with different but static and dynamic SoPs. Experimental results show frame-frequency of polarization image is 80fps limited mainly by the speed of the photodetector, but its maximum frame-frequency can achieve over 270fps for some special applications. Compared to the imaging polarimeters listed in above referring papers, the presented polarimeter acquires high measurement speed of polarization imaging while achieving good performances of simplicity, high-accuracy, compact size. That may give this kind of classic polarimeter new attractive prospects and life.

2. Theory

A polarimeter for polarization imaging is generally composed of imaging lens, polarization modulator and photodetector. Figure 1 gives the schematic layout of time-integration polarimeter with a continuously rotating polarizer. Besides the imaging lens and photodetector, the core component of the presented polarimeter is the polarization modulator, which consists of a fast BLDCM, a high-accuracy ME and a high-extinction-ratio polarizer. The polarizer is used to modulate the polarization state of incident light by adjusting its filtering angle, and the ME can record the instantaneous angle position of it. Thus the polarizer and ME been equipped together and mounted on the inner rotating rotor in axial direction. In this case, the polarizer and ME rotate synchronously with BLDCM.

Fig. 1. Schematic layout of time-integration polarimeter with continuously rotating polarizer (BLDCM: brushless DC motor; ME: magnetic encoder; yellow color: inner rotator; pink color: stator; θ: polarization filtering angle relative to horizontal direction; I_in: input intensity of light; I_out:output intensity detected by the photodetector after the polarizer modulation).

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The first three components of Stokes vector are used to describe the SoP of incident light without considering the circular part, i.e. S_in= [I_in, Q_in, U_in]^T. The Mueller matrix of a polarizer with a fixed filtering angle θ can be presented by [2]

(1)$${{\textbf M}_P}(\theta ) = \frac{1}{2}\left[ {\begin{array}{{ccc}} 1&{\cos 2\theta }&{\sin 2\theta }\\ {\cos 2\theta }&{{{\cos }^2}2\theta }&{\sin 2\theta \cos 2\theta }\\ {\sin 2\theta }&{\sin 2\theta \cos 2\theta }&{{{\sin }^2}2\theta } \end{array}} \right]$$

Thus the SoP of output light S_out= [I_out, Q_out, U_out]^T after polarization modulation can be solved by

(2)$$\left[ {\begin{array}{{c}} {{I_{out}}}\\ {{Q_{out}}}\\ {{U_{out}}} \end{array}} \right] = \frac{1}{2}\left[ {\begin{array}{{ccc}} 1&{\cos 2\theta }&{\sin 2\theta }\\ {\cos 2\theta }&{{{\cos }^2}2\theta }&{\sin 2\theta \cos 2\theta }\\ {\sin 2\theta }&{\sin 2\theta \cos 2\theta }&{{{\sin }^2}2\theta } \end{array}} \right] \times \left[ {\begin{array}{{c}} {{I_{in}}}\\ {{Q_{in}}}\\ {{U_{in}}} \end{array}} \right]$$

According to the Eq. (2), the total light intensity can be detected such as

(3)$${I_{out}} = \frac{1}{2}({{I_{in}} + \cos 2\theta \cdot {Q_{in}} + \sin 2\theta \cdot {U_{in}}} )$$

For solving the SoP of incident light, the light intensity should be measured for several times at un-equivalent polarization filtering angles, and that can be presented by

(4)$$\left[ {\begin{array}{{c}} {I_{out}^1}\\ {I_{out}^2}\\ \vdots \\ {I_{out}^N} \end{array}} \right] = \frac{1}{2}\left[ {\begin{array}{{ccc}} 1&{\cos 2{\theta_1}}&{\sin 2{\theta_1}}\\ 1&{\cos 2{\theta_2}}&{\sin 2{\theta_2}}\\ \vdots & \vdots & \vdots \\ 1&{\cos 2{\theta_N}}&{\sin 2{\theta_N}} \end{array}} \right]\left[ {\begin{array}{{c}} {{I_{in}}}\\ {{Q_{in}}}\\ {{U_{in}}} \end{array}} \right]$$

where the variable N ≥3 presents the measuring times in one cycle. Generally N is equal to 4 and the corresponding [θ₁, θ₂, θ₃, θ₄] are fixed at [0, π/4, π/2, 3π/4] respectively, and the choice of those angles makes the calculation easier since they correspond to projections on canonical SoPs. Under this special condition, the Eq. (4) can be simplified as

(5)$$\left[ {\begin{array}{{c}} {I_{out}^1}\\ {I_{out}^2}\\ {I_{out}^3}\\ {I_{out}^4} \end{array}} \right] = \frac{1}{2}\left[ {\begin{array}{{ccc}} 1&1&0\\ 1&0&1\\ 1&{\textrm{ - }1}&0\\ 1&0&{\textrm{ - }1} \end{array}} \right]\left[ {\begin{array}{{c}} {{I_{in}}}\\ {{Q_{in}}}\\ {{U_{in}}} \end{array}} \right]$$

For acquiring the accurate light intensity, the polarizer must stop and stay at the corresponding polarization filtering angles, otherwise the measuring efficiency will decrease. In fact, for a continuously rotating polarizer, the light intensity can be treated as a time-varying function of the SoP of incident light and the polarization filtering angle, and the Eq. (4) can be represented by

(6)$${I_{out}}(t )= \frac{1}{2}[{{I_{in}}(t )+ \cos 2\theta (t )\cdot {Q_{in}}(t )+ \sin 2\theta (t )\cdot {U_{in}}(t )} ]$$

For simplicity, here we suppose that the SoP of incident light is slow changing and can be treated as constant during ultra-short time (several times of exposure time). As a result, the practical light intensity detected by photodetector during exposure time can be presented by

(7)$$\begin{aligned} \int_{{t_1}}^{{t_2}} {{I_{out}}(t)} dt &= \frac{1}{2}\int_{{t_1}}^{{t_2}} {[{{I_{in}} + \cos 2\theta (t) \cdot {Q_{in}} + \sin 2\theta (t) \cdot {U_{in}}} ]dt} \\ &= \begin{array}{{c}} { \frac{1}{2}\left( {\int_{{t_1}}^{{t_2}} {{I_{in}}dt} + {Q_{in}}\int_{{t_1}}^{{t_2}} {\cos 2\theta (t)dt} + {U_{in}}\int_{{t_1}}^{{t_2}} {\sin 2\theta (t)dt} } \right)} \end{array} \end{aligned}$$

If we denote ω₀ as the constant angle velocity of the BLDCM in rad unit, the instantaneous filtering angle of the polarizer θ(t) will be

(8)$$\theta (t)\textrm{ = }{\omega _0}t$$

and the Eq. (7) can be derived further as

(9)$$\begin{aligned} \int_{{t_1}}^{{t_2}} {{I_{out}}(t)} dt &= \frac{1}{2}\left[ { {{I_{in}}t} |_{{t_1}}^{{t_2}} + \frac{{{Q_{in}}}}{{2{\omega_0}}} {\sin ({2{\omega_0}t} )} |_{{t_1}}^{{t_2}} - \frac{{{U_{in}}}}{{2{\omega_0}}} {\cos ({2{\omega_0}t} )} |_{{t_1}}^{{t_2}}} \right]\\ &= \frac{1}{{4{\omega _0}}}[{2{I_{in}}{\omega_0}({{t_2} - {t_1}} )+ {Q_{in}}({\sin 2{\omega_0}{t_2} - \sin 2{\omega_0}{t_1}} )- {U_{in}}({\cos 2{\omega_0}{t_2} - \cos 2{\omega_0}{t_1}} )} ]\end{aligned}$$

Denote t₀= (t₁+ t₂)/2 and Δt = (t₂-t₁) as the center time and exposure time from t₁ to t₂, respectively. Thus, the Eq. (9) can be written as

(10)$$\begin{aligned} \int_{{t_1}}^{{t_2}} {{I_{out}}(t)} dt &= \frac{1}{{4{\omega _0}}}\{{2{I_{in}}{\omega_0}\Delta t + {Q_{in}}[{\sin {\omega_0}({2{t_0}\textrm{ + }\Delta t} )- \sin {\omega_0}({2{t_0} - \Delta t} )} ]- {U_{in}}[{\cos {\omega_0}({2{t_0}\textrm{ + }\Delta t} )- \cos {\omega_0}({2{t_0} - \Delta t} )} ]} \}\\ &= \frac{1}{{2{\omega _0}}}({{I_{in}}{\omega_0}\Delta t + {Q_{in}}\cos 2{\omega_0}{t_0}\sin {\omega_0}\Delta t\textrm{ + }{U_{in}}\sin 2{\omega_0}{t_0}\sin {\omega_0}\Delta t} \end{aligned}$$

Define, for each exposure window i (i = 1,2,…, N), the nominal angle θ₀=θ(t_0,i)= ω₀t_0,i and exposure angle Δθ=ω₀Δt, which represents the polarization filtering angle at time t_0,i and sweeping angle during exposure time, respectively. The Eq. (10) can be rewritten further as

(11)$$\int_{{t_1}}^{{t_2}} {{I_{out}}(t)} dt\textrm{ = }\frac{1}{{2{\omega _0}}}({{I_{in}}\Delta \theta + {Q_{in}}\cos 2{\theta_0}\sin \Delta \theta \textrm{ + }{U_{in}}\sin 2{\theta_0}\sin \Delta \theta } )$$

In a practical imaging polarimeter, the exposure time Δt of the photodetector depends mainly on the brightness of target, and it will not be changed during one polarization measurement cycle. Meanwhile, the angular velocity ω₀ of the BLDCM or the polarizer keeps a constant value. Thus the exposure angle Δθ for each exposure window should be the same, and that is why the triangles in Fig. 2(a) and 2(b) seem to be equal.

Fig. 2. Sketch map of the relationship between the nominal angle, exposure angle and constant angle velocity. (a) Nominal angle locating at the center of the exposure angle; (b) Nominal angle locating at the beginning of the exposure angle; Blue triangles: exposure angle for each different but uniformly spaced nominal angles during half of a round; Green triangles: the exposure angle for each different but uniformly spaced equivalent nominal during the rest half of a round).

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Similar with the Eq. (4), the Eq. (11) can be represented by its matrix form for N times uniformly spaced angles and during the same exposure angle, and that is

(12)$$\left[ {\begin{array}{{c}} {I_{out}^1}\\ {I_{out}^2}\\ \vdots \\ {I_{out}^N} \end{array}} \right] = \frac{1}{{2{\omega _0}}}\left[ {\begin{array}{{ccc}} {\Delta \theta }&{\sin \Delta \theta \cos 2{\theta_{0,1}}}&{\sin \Delta \theta \sin 2{\theta_{0,1}}}\\ {\Delta \theta }&{\sin \Delta \theta \cos 2{\theta_{0,2}}}&{\sin \Delta \theta \sin 2{\theta_{0,2}}}\\ \vdots & \vdots & \vdots \\ {\Delta \theta }&{\sin \Delta \theta \cos 2{\theta_{0,N}}}&{\sin \Delta \theta \sin 2{\theta_{0,N}}} \end{array}} \right]\left[ {\begin{array}{{c}} {{I_{in}}}\\ {{Q_{in}}}\\ {{U_{in}}} \end{array}} \right]$$

where Iⁱ_out, i = 1,2,…,N, represents the time-integration of I_out(t) during the i^th exposure time.

The Fig. 2(a) draws the relationship between the nominal angles, exposure angles and constant rotation velocity of the polarizer. The exposure angle can be calculated by multiplying the rotation velocity of polarizer and the corresponding exposure time. The photodetector should be triggered half of exposure time Δt in advance, i.e. the trigger time can be determined by

(13)$${t_{1,i}} = \frac{{{t_{0,i}}}}{{{\omega _0}}} - \frac{{\Delta t}}{2} = \frac{{{\theta _{0,i}}}}{{{\omega _0}}} - \frac{{\Delta t}}{2},i = 1,2, \cdots N$$

As presented in the Eq. (13), the trigger time must be adjusted for different exposure time. In case of fixed trigger time application, the Eq. (11) can also be represented by substituting t₀= t₁ into the Eq. (9) as

(14)$$\int_{{t_1}}^{{t_2}} {{I_{out}}(t)} dt\textrm{ = }\frac{{2{I_{in}}{\omega _0}\Delta t + {Q_{in}}[{\sin 2{\omega_0}({t_0} + \Delta t) - \sin 2{\omega_0}{t_0}} ]- {U_{in}}[{\cos 2{\omega_0}({t_0} + \Delta t) - \cos 2{\omega_0}{t_0}} ]}}{{4{\omega _0}}}$$

Applying the definitions of nominal angle and exposure angle above, the Eq. (14) can be derived further as

(15)$$\int_{{t_1}}^{{t_2}} {{I_{out}}(t)} dt\textrm{ = }\frac{{2{I_{in}}\Delta \theta + {Q_{in}}[{\sin 2({{\theta_0}\textrm{ + }\Delta \theta } )- \sin 2{\theta_0}} ]- {U_{in}}[{\cos 2({{\theta_0}\textrm{ + }\Delta \theta } )- \cos 2{\theta_0}} ]}}{{4{\omega _0}}}$$

and its matrix form is

(16)$$\left[ {\begin{array}{{c}} {I_{out}^1}\\ {I_{out}^2}\\ \vdots \\ {I_{out}^N} \end{array}} \right] = \frac{1}{{4{\omega _0}}}\left[ {\begin{array}{{ccc}} {2\Delta \theta }&{\sin 2({{\theta_{0,1}}\textrm{ + }\Delta \theta } )- \sin 2{\theta_{0,1}}}&{\cos 2{\theta_{0,1}} - \cos 2({{\theta_{0,1}}\textrm{ + }\Delta \theta } )}\\ {2\Delta \theta }&{\sin 2({{\theta_{0,2}}\textrm{ + }\Delta \theta } )- \sin 2{\theta_{0,2}}}&{\cos 2{\theta_{0,2}} - \cos 2({{\theta_{0,2}}\textrm{ + }\Delta \theta } )}\\ \vdots & \vdots & \vdots \\ {2\Delta \theta }&{\sin 2({{\theta_{0,N}}\textrm{ + }\Delta \theta } )- \sin 2{\theta_{0,N}}}&{\cos 2{\theta_{0,N}} - \cos 2({{\theta_{0,N}}\textrm{ + }\Delta \theta } )} \end{array}} \right]\left[ {\begin{array}{{c}} {{I_{in}}}\\ {{Q_{in}}}\\ {{U_{in}}} \end{array}} \right]$$

The Eq. (15) seems more complex, but the trigger time of the photodetector will not change no matter how long the exposure time is, as shown in Fig. 2(b).

Specially, if four polarization filtering angles of [0, π/4, π/2, 3π/4] are employed as the nominal angles, the Eqs. (12) and (16) will be simplified respectively as

(17)$$\left[ {\begin{array}{{c}} {I_{out}^1}\\ {I_{out}^2}\\ {I_{out}^3}\\ {I_{out}^4} \end{array}} \right] = \frac{1}{{2{\omega _0}}}\left[ {\begin{array}{{ccc}} {\Delta \theta }&{\sin \Delta \theta }&0\\ {\Delta \theta }&0&{\sin \Delta \theta }\\ {\Delta \theta }&{ - \sin \Delta \theta }&0\\ {\Delta \theta }&0&{ - \sin \Delta \theta } \end{array}} \right]\left[ {\begin{array}{{c}} {{I_{in}}}\\ {{Q_{in}}}\\ {{U_{in}}} \end{array}} \right]$$

(18)$$\left[ {\begin{array}{{c}} {I_{out}^1}\\ {I_{out}^2}\\ {I_{out}^3}\\ {I_{out}^4} \end{array}} \right] = \frac{1}{{4{\omega _0}}}\left[ {\begin{array}{{ccc}} {2\Delta \theta }&{\sin 2\Delta \theta }&{1 - \cos 2\Delta \theta }\\ {2\Delta \theta }&{\cos 2\Delta \theta - 1}&{\sin 2\Delta \theta }\\ {2\Delta \theta }&{ - \sin 2\Delta \theta }&{\cos 2\Delta \theta - 1}\\ {2\Delta \theta }&{1 - \cos 2\Delta \theta }&{ - \sin 2\Delta \theta } \end{array}} \right]\left[ {\begin{array}{{c}} {{I_{in}}}\\ {{Q_{in}}}\\ {{U_{in}}} \end{array}} \right]$$

It can be concluded obviously from the Fig. 2 that the exposure angle should not be larger than the uniformly angle space of nominal angles, i.e.

(19)$$\Delta \theta \le {\theta _{0,i + 1}} - {\theta _{0,i}},i = 1,2 \cdots N$$

In practical applications the exposure time Δt depends on the intensity of incident light, thus the Eq. (19) hints the rotation velocity of the polarizer should satisfy

(20)$${\omega _0} \le \frac{{{\theta _{0,i + 1}} - {\theta _{0,i}}}}{{\Delta t}},i = 1,2 \cdots N$$

As shown in Eq. (19), the maximum allowable exposure angle Δθ must less than the angle space, thus the maximum measurement speed of polarization imaging is mainly depends on the exposure time Δt, as shown in Eq. (20). As a result, the presented linear imaging polarimeter has a slower measurement speed for a darker scene.

For four polarization filtering angles of [0, π/4, π/2, 3π/4], the exposure angle and the corresponding rotation velocity should be not larger than π/4 and π/(4Δt) respectively. Besides, the SoP of incident light can be solved by N measurements according to the Eq. (12) or (15). Similarly, m group measurements of the SoP will be solved from the m×N measurements if m periods exist when the polarizer is rotated one circle (0∼2π). In fact, one can reuse the measured light intensities by referring the gliding window method [37]. In fact, the retrieved SoP can be treated as an average result of the SoP of incident light during the neighboring N measurements. When a new light intensity after polarization modulation is detected by the photodetector, a new group of light intensities is formed by reusing the previous N−1 light intensities and replacing the oldest value into the new light intensity. As a result, a new SoP of incident light is measured when a new light intensity is detected, and the data selection and substitution seems a gliding window. Figure 3 shows its basic principle by taking four nominal angles of [0, π/4, π/2, 3π/4] (i.e. m = 2 and N = 4) as an example. As shown in Fig. 3, a linear Stokes vector is calculated by detected light intensity at four different nominal angles. When the polarizer is rotated via the next angle position, a new light intensity is detected and the oldest light intensity is replaced by it. Thus a new SoP is measured when the polarizer is rotated via a new position, and 8 group of measurements of SoP can be measured when the polarization is rotating continuous a round. That will improve in some certain dynamic performance of the presented method, although it is not a real kind of snapshot polarimeter.

Fig. 3. Principle of data processing by using gliding window.

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In practical applications, the rotation velocity of compact BLDCM with inner rotor is over 2000 revolutions per minute (rpm) (∼209 rad/s). Thus the maximum measurement speed of polarization imaging is about 270 frames per second (fps) in case of four nominal polarization angles been used. Certainly, it is not always beneficial for faster rotation velocity because the Eq. (20) must be satisfied for the exposure time of photodetector.

3. Numeric analysis

3.1 Condition number under different exposure angles

For a general purpose polarimeter, the polarimeter modulation matrix should be as far from singular as possible. The condition number (CN) is a typical indicator without considering redundant measurements to evaluate whether the modulation matrix is well-conditioned or not, and the lower its value the better performance the polarimeter is. The theoretical minimum CN of instrument matrix for a linear imaging polarizer is $\sqrt 2$, and that hints the modulation matrix presented in the Eq. (5) is one of the best parameter settings of the Eq. (4). However, modulation matrix of time-integration polarization imaging will deviate from the best optimized CN, and the performance of error propagation will decrease as a result. In this paper, we analyze the change of CN with the exposure angle changing based on two models, which are model1 and model2 presented by the Eqs. (17) and (18), respectively. In fact, these two models are alternative, and the main difference between them is the different location of the nominal measurement angle, as shown in Fig. 2. According to the Eq. (19), the exposure angles for these two models (model1 and model2 hereafter) are ranging from -π/8 to +π/8 and from 0 to π/4 respectively. The analysis results are plotted in Fig. 4.

Fig. 4. The CN with the changing of the exposure angles of two models. Model1: the nominal angle locating the center of the exposure angle (presented by the Eq. (17)); Model2: the nominal angle locating the beginning of the exposure angle(presented by the Eq. (18)).

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As shown in Fig. 4, the CNs of the mentioned two models are the same and equal to the theoretical minimum value of 1.414 when the exposure angle is 0. Under this special condition these two models will return back to the traditional model without consideration of time-integration, as shown in Eq. (5). In case of present of the exposure angle, as discussed above, the CNs deviate from the theoretical minimum value, and the larger deviations will produce worse CNs. We note that the CN curve of the model1 is symmetric, and the maximum value is about 1.451 at the maximum exposure angles of ±π/8, which increases of 2.62% comparing with the theoretical minimum value. The model2 has a single increasing CN curve, and the maximum CN is less than 1.571 at the maximum exposure angle of π/4, which has an increase of 11.1% relative to 1.414. In general, the CNs smaller than 3 can be accepted for most polarimetric applications [38]. Thus the presented polarimeter based on time-integration can provide good performance based on the small deviation of CN from its theoretical value no matter the selection of exposure angle.

3.2 Error analysis

Accurate knowledge of the starting angles and exposure angle is a necessary premise of the linear Stokes vector measurement with high accuracy, no matter which kind of model is selected. In general, the starting angle error depends on the calibration accuracy between the ME reading and the real polarization filtering angle of the rotating polarizer, while the exposure angle error is mainly caused by unstable rotation velocity of the BLDCM. In this section, we will evaluate the influence of those possible error sources based on the model2 since its starting angles is fixed compared to the model1 regardless of exposure angle selection. [0, π/4, π/2, 3π/4] are chosen as the corresponding four nominal angles.

(1). Starting angle error

The Eq. (17) is the simplified mathematical model when the nominal angles are fixed at [0, π/4, π/2, 3π/4]. The nominal angles should be calibrated by building the relationship between the ME reading and the real polarization filtering angle. However, the calibration error always exists, and as a result the starting angle error will be introduced into the presented linear polarimeter and lead to the measurement error of SoP of incident light. In fact, the measurement error of SoP is mainly contributed by the difference between the ideal modulation matrix presented by the Eq. (18) and the practical modulation matrix presented by the origin Eq. (16) when the starting angle error θ_err exists.

Let us assume that the starting angle error θ_err is ranging from −1° to +1°, and the corresponding time integration of light intensities are calculated from the Eq. (16). Then the Eq. (17) is employed to retrieve the SoP of incident light since the value of θ_err is unknown. The assumed ideal normalized SoP of incident light with a full period of polarization angle is drawn in Fig. 5(a), and the corresponding SoP errors of Q component and U component for all polarization angles under different starting angle errors are plotted in Fig. 5(b) and (c) respectively. It is obvious the SoP error will increase with increasing of the absolute value of the starting angle error. The linear relationship between the starting angle errors and its corresponding SoP errors can be found easily from the Fig. 5(d). The SoP error will achieve its maximum value of 3.5×10⁻² when the starting angle error is moves close to −1° or 1°. To achieve a maximum SoP error of smaller than 1×10⁻², the tolerance of the starting angle error must be less than ±0.3°. That means the starting angle must be calibrated carefully before applying in polarization imaging. There are two calibration methods: one is the direct method by using a light source with a known SoP, which builds the relationship between the ME reading and its real polarization filtering angle, and the starting angle error θ_err is minimized; another one is indirect method by calibrating the modulation matrix presented in the Eq. (16) when a set of assumed nominal angles are used to measure a known polarized light source, and the starting angle error θ_err can be calibrated and concealed in the following measurements. In fact, a small angle error after calibration of less than ±0.1° can be achieved easily no matter which kind of calibration method is used, which means the SoP error will be less than 3.5×10⁻³.

Fig. 5. Performance analysis of the presented polarimeter under different starting angle errors. (a) The ideal and normalized SoP of incident light with a full period polarization angles of 2π; (b) The measuring errors of Q component under different starting angle errors ranging from −1° to +1°; (c) The measuring errors of U component under the same conditions of Q; (d) The maximum SoP errors under different starting angle errors (ΔQ&ΔU-SoP errors of Q and U component).

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(2). Unstable rotation velocity

As presentation of the Eq. (18), the exposure angle is another parameter of the modulation matrix. Thus its error will influence the measuring accuracy of the presented polarimeter. The exposure angle equals to the product of the constant rotation velocity ω₀ of the employed polarizer and the exposure time Δt of the used photodetector. The exposure time Δt depends on intensity of incident light and is controlled by photodetector, and its error is generally very small to -µs level. Thus the error of exposure angle mainly depends on the unstable rotation velocity of the polarizer. The error of rotation velocity for a standard BLDCM with high accuracy is generally less than ±5 rpm, and more stable performance will be achieved when it works on higher rotation velocity. Here we assume instability of the rotation velocity has a random uniform distribution with maximum error of ±10 rpm and reuse the curves shown in Fig. 5(a) to present the ideal normalized SoP of incident light. The random errors of the rotation velocity are plotted in Fig. 6(a). The corresponding retrieved SoPs and its errors under a rotation velocity of 600 rpm (the fastest velocity in our experiments) and an exposure angle of π/4 are plotted in Fig. 6(b) and (c) respectively. The SoP errors are almost less than ±2×10⁻² for both of Q and U components, and the corresponding RMS errors for all SoPs is about 9×10⁻³. We also analyze the RMS of SoP errors for different rotation velocities ranging from 100 rpm to 2000rpm, and the results are drawn in Fig. 6(d). The RMS of SoP errors is decreased obviously with the rotation velocity increasing, and the smallest RMS value of 2.7×10⁻³ is achieved at a rotation velocity of 2000rpm. Figure 6(d) also plots the change of RMS of SoP errors with the rotation velocities increasing when the instability of rotation velocity is less than ±5 rpm. It is obvious the SoP error can also be decreased further to 1.3×10⁻³ by improving the stability performance of BLDCM.

Fig. 6. Performance deviation due to unstable rotation velocity of the used BLDCM. (a) The simulated random errors of the rotation velocity with uniform distribution within ±10 rpm; (b) The retrieved Q and U components under the rotation velocity of 600 rpm by considering its instability shown in (a); (c) The measuring errors of Q and U components due to unstable rotation velocity; (d) RMS curves of SoP errors for different rotation velocities ranging from 100 rpm to 2000rpm with unstable rotation velocities of ±10 rpm and ±5 rpm respectively.

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3.3 Dynamic performance

In Eq. (7), the SoP of incident light is supposed as slow changing and can be treated as a steady light during several times of exposure time. However, in practical applications the limitations of its dynamic performance also need to be considered and analyzed. For this purpose, a light source with time-varying SoP is simulated as the beam under test, which is generated numerically from a polarization modulated natural light by rotating a referring polarizer. The rotation velocity of the referring polarizer is 10°/s, and as a result 36s will be taken when the polarizer is rotated a round (0∼360°). The normalized linear Stokes components Q and U is drawn in Fig. 7(a). The measuring polarizer is rotated with the rotation velocity of 100 rpm (i.e. 10.47 rad/s) firstly, and the corresponding exposure time is about 75 ms when the model2 and the maximum exposure angle of π/4 are adopted to analyze the dynamic measuring accuracy of the presented polarimeter. The simulated light intensities at each nominal angle are plotted in Fig. 7(b), which are analyzed by applying the Malus Law and its time-integration value during exposure time. The detected light intensities looks like smaller than the incident one because of time-integration effect and modulation of cosine function. Applying the Eq. (17), the SoP of the dynamic light source can be retrieved easily, and they are plotted in Fig. 7(c). The outputting Q and U components keep good consistence with the ideal SoP of incident light. However, an additional serration wave exist comparing with the ideal SoP, which is more obvious near extreme points. The reason may be the difference of incident light intensities at adjacent four measuring nominal angles, and this kind of difference become worse when the input SoP is near extreme points. The measured errors, as shown in Fig. 7(d), are less than 2.6×10⁻², and the root-mean-square (RMS) is about 1.3×10⁻². It is not very large for the relative slow rotation velocity of 100 rpm. We also study the dynamic performance for different rotation velocities of 500 rpm, 1000 rpm and 2000rpm of the measuring polarizer for the same dynamic light source, and the corresponding normalized SoP errors are plotted in Fig. 8. The maximum errors of the plotted curves are less than 2.6×10⁻², 5.2×10⁻³, 2.6×10⁻³ and 1.3×10⁻³, and the corresponding RMS values of them are less than 1.3×10⁻², 2.5×10⁻³, 1.3×10⁻³ and 6.4×10⁻⁴ respectively. The RMS values of the SoP errors for different rotation velocities of the referring polarizer are plotted in Fig. 9, and they are decreasing along inverse proportional function with increasing rotation velocities. In addition to the 10°/s polarizer rotation velocity, the curves under the rotation velocity of 5°/s, 15°/s, 20°/s, 25°/s and 30°/s of the referring polarizer are also analyzed and plotted in the same figure, as shown in the Fig. 9. It is obvious that higher accuracy of SoP measurement will be obtained for slower changing of light source or faster measuring speed of the presented polarimeter. Extremely, the SoP error will be close to 0 when the light source is close to static or the rotation velocity of the measuring polarizer is close to infinity.

Fig. 7. Numeric validation of dynamic performance of the presented polarizer. (a) Normalized SoP of simulated dynamic light source generated by a rotating polarizer with a speed of 10°/s; (b) The detected light intensity when the dynamic light source is modulated by a fast rotating polarizer with speed of 100 rpm and exposure angle of π/4; (c) The measured normalized SoP calculated from the model2 and the measured light intensity as shown in (b); (d) Errors of the measured SoP.

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Fig. 8. Normalized SoP errors under different rotation velocities of 100 rpm, 500 rpm, 1000 rpm and 2000rpm of the measuring polarizer.

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Fig. 9. Normalized RMS curves of SoP errors for different dynamic light sources under different polarization measuring velocities (The rotation velocities of the polarizer of dynamic light sources are 5°/s, 15°/s, 20°/s, 25°/s and 30°/s, and the rotation velocity of the polarizer of the presented linear polarimeter is ranging from 100 rpm to 2000rpm).

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4. Experiments

4.1 Experimental setup

An experimental setup is built to validate the performance of the presented polarimeter, which is composed of a laser light source, two attenuators, a lens, a referring polarizer mounted in a rotatable motor, a measuring polarizer together with BLDCM and ME, and an industrial CMOS camera, as shown in Fig. 10. The center wavelength of the light source is 650 nm, and light beam with linear SoP can be collimated. The referring polarizer and the measuring polarizer are identical model LPVIS100 manufactured by Thorlabs Inc., with a transmission of ∼75% and extinction ratio of ∼1×10⁶. The referring polarizer will be used to modulate the SoP of the laser light source as incident light, and the resolution and the maximum velocity of its rotatable motor are about 1.8 arc secs and 25°/s. The maximum permitted rotation velocity of the BLDCM is up to 2052rpm with instability of ±5 rpm. The model of ME is eCoder35S-R20B produced by Zeroerr.inc, and its resolution is about 1.2 arc secs. The industrial CMOS camera is employed as the photodetector, which has 1280×1024 resolution and maximum frame frequency of 90fps. Before experiments, we have calibrated the relationship between the ME and the filtering angle of the polarizer by using the direct calibration method, and the initial zero position is 12.77° with uncertainty of ±0.07°. For convenient, the exposure time of the CMOS camera is fixed as 9 ms, and as a result the exposure angle will be different as the changing of rotation velocity of the polarizer. The static and dynamic experiments are also done based on this experimental setup.

Fig. 10. Experimental setup of the presented polarimeter with time integration.

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4.2 Static experiments

The rotatable motor controls the referring polarizer stopping at five fixed filter angles ranging from 0°∼90°, which are 1.1°, 23.1°, 46.1°, 61.1° and 81.1° respectively. Then we rotate the measuring polarizer with the velocities of 100 rpm, 200 rpm, 300 rpm, 400 rpm, 500 rpm and 600 rpm respectively by controlling the BLDCM. The nominal starting angles are 0°, 45°, 90° and 135° and their equivalent angles of 180°, 225°, 270° and 315° during a round. The exposure angles depend on the exposure times and the rotation velocities of the BLDCM. We measure the first referring angle of 1.1° by adjusting the velocity of BLDCM to 100 rpm firstly. A set of time-integration intensity images are obtained by triggering the photodetector at 8 nominal angles, and the corresponding polarization angle is calculated according to the Eq. (17). Time averaging method can be used to alleviate the influence of intensity fluctuations of light source. Similar measurements are also done by adjusting the velocity of BLDCM to the rest of five velocities. Then we adjust the referring polarizer to the next polarization angle and repeat the above measurements under 6 different rotation velocities of the measuring polarizer. As a result, the SoP corresponding to five referring modulation angles was measured with 6 rotation velocities for each referring modulation angle, as drawn in Fig. 11. Five almost horizontal lines hint the stable performance of measurements under different rotation velocities. In fact there exists small difference between the referring polarization angles and the measured ones under all rotation velocities. The corresponding peak and valley deviations of the measured values for different referring polarization angles are also listed in Table 1. As shown in Table 1, all deviations are less than 0.298° for all referring polarization angles and all rotation velocities, and that may cause the largest error of less than 1.04×10⁻² for all possible normalized SoPs.

Fig. 11. The measured results for five different referencing polarization angles under the rotation velocities ranging from 100 rpm to 600 rpm.

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Table 1. The maximum deviations between the referring polarization angles and the measured ones

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4.3 Dynamic experiments

As shown in Fig. 10, the dynamic light source is composed of the linear polarized laser and the rotating referring polarizer, which generates both time-varying light intensity and SoP. Dynamic light intensity can be detected by the employed photodetector, which is modulated obviously by square of cosine function of polarization angle difference between the referring polarizer and laser light source according to the Malus Law. The mentioned polarization modulation process can be presented by

(21)$${I_{in}}(t) = {I_{laser}} \cdot {\cos ^2}[{{\theta_{laser}} - {\theta_{referring}}(t)} ]$$

where I_laser and θ_laser are both constant variables and represent the light intensity and polarization filtering angle respectively. The θ_referring(t) is a time-varying variable and represents the instantaneous polarization filtering angle of the rotating referring polarizer. The I_in(t) represents the instantaneous light intensity outputting after modulating by the referring polarizer.

When the dynamic light source (presented by the Eq. (21)) is modulated further by the rotating measuring polarizer, the total outputting light intensity can be represented by

(22)$$\begin{aligned} {I_{out}}(t) &= {I_{in}}(t) \cdot {\cos ^2}[{{\theta_{referring}}(t) - {\theta_{measuring}}(t)} ]\\ &= {I_{laser}} \cdot {\cos ^2}[{{\theta_{laser}} - {\theta_{referring}}(t)} ]\cdot {\cos ^2}[{{\theta_{referring}}(t) - {\theta_{measuring}}(t)} ]\end{aligned}$$

where θ_measuring(t) represents the instantaneous polarization filtering angle of the rotating measuring polarizer. The I_out(t) represents the instantaneous light intensity outputting after modulating by both the referring polarizer and the measuring polarizer.

In our experiments, the rotation velocities of the rotating referring and measuring polarizer are 10°/s and 600 rpm firstly, respectively. Thus the maximum frame rate of the presented imaging polarimeter is up to 80fps, which near to the extreme performance of the industrial CMOS camera. The exposure angle is about 32.4° (0.57 rad) during exposure time of 9 ms. Four outputting light intensities (i.e. I_out_,1, I_out_,2 I_out_,3 and I_out_,4, normalized to the peak value of I_in) for four nominal angles of [0, π/4, π/2, 3π/4] are detected by the employed photodetector sequentially at four nominal angles, as shown in Fig. 12(a). It is noted that intensity fluctuations exist with small amplitude but high frequency due to fluctuation of laser source and detecting noise. Substituting the system parameters and the detected light intensities into the Eq. (18), one can retrieve the light intensity and SoPs of incident light, as shown in Fig. 12(b) and (c).

Fig. 12. Linear SoP results of incident light. (a) Four group of outputting light intensities Iout,1, Iout,2 Iout,3 and Iout,4, normalized to its peak value; (b) The retrieved light intensities by applying proposed model2; (c) The retrieved SoPs of incident light by applying proposed model2; (d) The experimental linear degree of polarization of incident light.

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The retrieved light intensities are also normalized to its peak value, and the fluctuation is amplified obviously as shown in Fig. 12(b). The retrieved Q and U components, which are normalized to its absolute peak value, seem some different with the standard sine function because of the polarization modulation of incident light intensities, as shown in Fig. 12(c). As a result, the linear degree of polarization (DoP) of incident light can be analyzed and plotted in Fig. 12(d). The average and RMS values of the linear DoPs are about 1.01 and 0.016 respectively, which prove that the incident light is completely linear polarized. The experimental results show some fluctuations around 1.0 due to inescapable detective noise. For convenient comparison, the retrieved Q and U components are also normalized by dividing the quadratic sum of themselves, as plotted in Fig. 13(c). We also plot the theoretical normalized Q and U components of incident light in Fig. 13(a) with time elapse, and the corresponding errors between the retrieved SoPs and theoretical ones can be calculated. As shown in Fig. 13(e), almost all measurement errors of Q and U components are less than ±4×10⁻², and the average value of them is about 5×10⁻⁴. The distribution of the errors is similar in some certain with the simulated results in numeric analysis, although the amplitude is influenced obviously by the high frequency fluctuation of light intensities. In fact, the errors after subtracting the high frequency fluctuation should be less than ±2×10⁻². The polarization angles can be calculated according to the measured SoPs of incident light, and they are plotted in Fig. 13(d). We also draw the theoretical ones with time elapse in Fig. 13(b) for comparison. The errors of polarization angles between them are ranging from −0.94° to +0.97°, and the average value of them is about −0.0061°, as shown in Fig. 13(f). Similarly, the errors of polarization angles should be less than ±0.5° without considering the high frequency of fluctuations.

Fig. 13. Experimental results of SoP measurements. (a) Theoretical normalized Q and U components of incident light; (b) Theoretical polarization angles; (c) Retrieved and normalized Q and U components; (d) Retrieved polarization angles; (e) Measurement errors for the SoPs of incident light; (f) Measurement errors for the polarization angles of incident light.

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Besides, we also repeat the dynamic experiments under different rotation velocities of 5°/s, 15°/s and 20°/s of the referring polarizer. As shown Fig. 14, the average RMS of ΔQ and ΔU are 7.4×10⁻³, 7.4×10⁻³, 7.9×10⁻³, 9.8×10⁻³ and 9.3×10⁻³ respectively, which almost double the numeric analysis results, as shown in Fig. 9. The possible reason is that many error sources exist in experiments besides the dynamic light source.

Fig. 14. Variation tendency of average RMS of the normalized SoPs for different rotation velocities of the referring polarizer.

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5. Conclusion and Discussion

In conclusion, we have demonstrated a time-integration polarimeter by rotating continuously a polarizer. The basic principle and mathematical presentation are also derived. The condition number of the presented polarimeter is positive correlative with the exposure angle, and it is deviated from the theoretical minimum value of 1.414 up to the maximum value of 1.571. The accuracy is limited by the dynamic performance of incident light and rotation velocity of the used polarizer. For a certain application, starting angle error, unstable rotation velocity and detective noise will be the main influence factors of measurement accuracy. To achieve the measurement SoP of incident light with accuracy better than 1×10⁻², the starting angle error should be controlled lower than ±0.3°. In some necessary applications, starting angle error of less than ±0.1° can be achieved easily after calibration, which leads to the SoP error less than 3.5×10⁻³. The unstable rotation velocity will influence the measurement accuracy of SoP, and the error will be decreased with increasing rotation velocity of the polarizer. The RMS errors of SoP are about 9×10⁻³ and 2.7×10⁻³ when the polarizer is working on the rotation velocities of 600 rpm and 2000rpm respectively. The dynamic performance will be better when the polarizer is rotated under higher velocity. The RMS errors of SoP are about 6.3×10⁻³ and 1.9×10⁻³ under the rotation velocities of 600 rpm and 2000rpm respectively when the SoP of light source is changed by a velocity of 30°/s. We also do experiments by using static and dynamic light sources. The maximum errors of polarization angles for 5 static SoP is up to 0.298° for different rotation velocities, which will lead a SoP error of 1.04×10⁻². The measurement accuracy for the dynamic light is worse comparing to static scene. The practical measurement errors of SoPs for different dynamic light sources are all less than ±4×10⁻² with average value of 5×10⁻⁴, and the errors of polarization angles are less than ±0.97° with an average value of 0.0061°. In our experiments, the rotation velocity is up to 600 rpm, which generate the polarization images with speed of 80fps. It is almost the extreme performance by the frame rate of the used industrial CMOS camera. The fastest frame rate of the polarization imaging for the used BLDCM is over 270fps, which should satisfy the requirement of some extreme applications. Besides, the dynamic performance will be improved with the rotation velocity increasing. That may give this kind of classic polarimeter new attractive prospects.

Acknowledgments. The authors would like to thank Prof. Xuejun Zhang and engineer Cheng Su for their benefit discussions and good suggestions. They would also like to thank the reviewers for their constructive comments and helpful suggestions.

Funding

National Natural Science Foundation of China (12073031, 61905252); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2018412); National Science Fund for Distinguished Young Scholars (12022308).

Disclosures

The authors declare no conflicts of interest related to this article.

Data availability

The data that support the findings of this study are available from the first corresponding author upon reasonable request.

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Referring Angle	Valley Deviation	Peak Deviation
1.1°	−0.196°	0.122°
23.1°	−0.190°	0.084°
46.1°	−0.194°	−0.033°
61.1°	−0.039°	0.177°
81.1°	−0.136°	0.298°

Polarization imaging based on time-integration by a continuous rotating polarizer

Abstract

1. Introduction

2. Theory

3. Numeric analysis

3.1 Condition number under different exposure angles

3.2 Error analysis

3.3 Dynamic performance

4. Experiments

4.1 Experimental setup

4.2 Static experiments

4.3 Dynamic experiments

5. Conclusion and Discussion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (14)

Tables (1)

Equations (22)

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