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Abstract
In this study, we demonstrate a polarization imaging camera with a waveplate array of a silica glass fabricated by femtosecond (fs) laser direct writing. To use a waveplate array of silica glass for polarization imaging, non-uniformity of the transmittance and retardance in the waveplates must be considered. Therefore, we used a general method of polarization analysis with system matrices determined experimentally for all the units in the waveplate array. We found that a figure of merit based on the determinant of the system matrix could be applied to improve the accuracy of analysis and the robustness to the retardance dispersion for both the simulated and the fabricated waveplate array.
© 2017 Optical Society of America
1. Introduction
The polarization state of light is one of its important properties for identification of objects. For example, the polarization state provides us with the orientation of the surface of an object, because the reflectance depends on the polarization, i.e., p- and s-polarization [1]. Moreover, the polarization state reflects strain in transparent materials via photoelastic effects [2]. Therefore, detection of the polarization state of light can be used in the evaluation of the thickness or strain in optical films [3], classifications of pollutants in the atmosphere [4], and detection of cancer [5,6].
Real-time acquisition of the distributions of polarization using a single imaging sensor is challenging, because the analysis of a polarization state requires at least three light intensities detected through a waveplate or a polarizer of different orientations [7–9]. In fact, real-time polarization cameras have been reported [10,11]. There are two main types of polarization cameras: one is equipped with micro-arrayed waveplates of at least three different orientations (a waveplates array), and the other is equipped with micro-arrayed polarizers of at least three different orientations (a polarizer array). The waveplate array type has an advantage over the polarizer array type in polarization analysis, because the former can distinguish between clockwise and counter-clockwise elliptical polarizations [8] while the latter cannot. Because determining the rotation of elliptical polarization is indispensable for obtaining the birefringence, an analysis, for example, of photoelastic effects requires a polarization imaging camera with a waveplate array.
The method of analysis used by a polarization imaging camera equipped with a waveplate array is based on a system that consists of a rotatable waveplate (or retarder) and a fixed linear polarizer [8]. Several methods of analysis have been proposed. In the simplest method, the orientation and retardance at each waveplate cell are set to specific values to simplify the analysis [12]. For example, in this method, the retardances must be half or quarter of the wavelength of the detected light, and the orientations should be 22.5°, 45°, and 90°. To satisfy this requirement, the waveplate array must be aligned accurately, the distribution of retardance in a single cell must be uniform, and the transmittance of the waveplate array must be uniform. In more flexible methods of analysis, the Stokes vector is obtained by solving a system of linear equations [7–9]. In that case, the system matrix must be determined based on the orientation and retardance of the waveplates in the optical system. In both methods, the analysis requires accurate alignment of the waveplate array relative to the fixed linear polarizer, a uniform distribution of retardance in the waveplate cell, and uniform light transmittance by the waveplate array.
Satisfying these criteria requires a highly precise technique for fabricating the waveplate array that has a uniform birefringence and transmittance. One method for fabricating a waveplate array is femtosecond (fs) laser direct writing [13]. Shimotsuma et al. have found that focused fs laser pulses create microstructures of nanogratings inside silica glass [14–16]. Because the nanogratings have birefringence and their orientation can be controlled by the polarization of the laser pulses, a waveplate array can be fabricated inside silica glass by fs laser direct writing [17]. Recently, Gecevičius has demonstrated a polarization-sensitive camera with a waveplate array of silica glass fabricated by fs laser direct writing [12]. However, fs laser processing has a critical problem with fabrication of a waveplate array as designed; laser-induced birefringence is sensitive to the laser writing condition, so the sensitivity makes it difficult to fabricate a waveplate array with uniform birefringence and transmittance [18–22]. Non-uniformity of birefringence and transmittance could cause the increase of polarization analytical error and its sensitivity to image noise. Therefore, it is essential for a polarization imaging camera with a waveplate array fabricated via fs laser writing to find a new analysis method that enables accurate polarization analysis regardless of non-uniformity of birefringence and transmittance in the waveplates.
In this paper, we used generalized polarization analysis to assess the performance of a polarization imaging camera with a waveplate array of silica glass fabricated via fs laser direct writing. We found that polarization analysis was possible, even though the transmittance and birefringence of the waveplate were not uniform. In section 2, we explain the basic principles and methods of analysis of a polarization imaging camera and how to determine the parameters needed for the analysis. In section 3, we explain methods for simulation, an experimental method for fabricating a waveplate array, the configuration of the polarization imaging camera, and the flow of the analysis in a computer. In section 4, we simulated the polarization analysis and investigate the influence of the characteristics of the waveplate array to the analytical errors. Although the parameters for the simulation were based on the typical values obtained in our fs laser writing experiments, we obtained a generally useful conclusion: the parameters of a waveplate array should be chosen according to the figure of merit based on the determinant of the system matrix [23–25]. In section 5, we show the calibration process and polarization analysis of a real polarization camera with fabricated waveplate arrays. We showed that the analytical errors in a real polarization imaging camera are comparable to those by the simulation. Finally, we demonstrate real-time capturing of polarization distributions using a polarization imaging camera.
2. Principle
2.1 The configuration of the polarization imaging camera
Figure 1(a) shows the configuration of the polarization imaging camera with a waveplate array. In a waveplate array, small waveplates are placed on a square grid. The light to be detected is transmitted through the waveplate array and a linear polarizer, and the transmitted light intensity is detected by an image sensor. Because the orientations of four adjacent waveplates in 2 × 2 blocks [Fig. 1(b)] differ from each other, four different intensities (I0, I1, I2, and I3) are obtained by 2 × 2 blocks in the image sensor. Here, we define the four waveplates of the 2 × 2 blocks as a “unit” and a waveplate array of one block as a “cell” [Fig. 1(b)]. The four waveplate cells are designated 0, 1, 2, and 3. For polarization imaging, the polarization state of each unit is calculated from the four different intensities. In this study, we defined 0° of this system parallel to x-axis.
Fig. 1 Detection of polarized light by an image sensor (a) Configuration of the polarization imaging camera with a waveplate array. (b) Detection of polarized light in a single unit.
2.2 Principle of polarization analysis
For the polarization analysis, we needed to derive the relation between the polarization state, the intensity (Ik), and the characteristics of the waveplate in the k-th cell. The characteristics of the k-th waveplate cell are transmittance (tk), orientation (χk) and retardance (Δk). We derived the relation using Jones calculus for polarization analysis [26]. The polarization state of incident light is expressed by Jones vector,
where |Ein| is the amplitude of the electric field of the incident light, and θ and δ are the direction and phase of the incident light, respectively. The goal of polarization analysis is to obtain θ and δ, which determine the polarization state. The change of polarization passing through the k-th waveplate cell and the linear polarizer can be obtained by multiplying Ein by the Jones matrices of the waveplate and polarizer:where the first matrix on the right side is the linear polarizer of which transmission orientation is parallel to the x-axis, and the second, third, and fourth matrices are the k-th waveplate cell. From Eq. (2), the light intensity, Ik, can be calculated fromwhere Iin is the intensity of the incident light. To simplify Eq. (3), we introduced the following coefficients:Using these coefficients, Eq. (3) can be rewritten asUsing these coefficients, the light intensities detected at the four cells in a single unit can be expressed bywhere the Sk(θ, δ) are the Stokes parameters of the Poincare sphere, and A is a 4 × 4 matrix of the four waveplate cells, which is called “system matrix” [23–25]. Once the system matrix A is determined and the four light intensities (Ik) are obtained, the four Stokes parameters can be obtained by solving the system of four equations. Therefore, where Bk’,k is an element of the inverse of the matrix A. From Eqs. (8), θ and δ can be obtained to identify the polarization state.2.3 Calculation with non-uniform waveplates
The calculation in Section 2.2 is based on the assumption that the birefringence and the transmittance were uniform in each waveplate cell. In this section, we show that an equation identical to Eq. (6) can be obtained, even if the birefringence and transmittance in each waveplate cell are not uniform.
First, the distributions of the transmittance, orientation, and retardance in the k-th waveplate cell are expressed by tk(x, y), χk(x, y), and Δk(x, y), respectively. In this case, the coefficients αk, βk, γk, and ξk can be calculated by integrating the distributions in the k-th waveplate cell:
where Sk is the area of the k-th waveplate cell. These equations mean that the system of four linear equations in Eq. (6) is the same, even if there are distributions of the transmission and the birefringence in each waveplate cell. The important point is that the polarization analysis needs only the values of αk, βk, γk, and ξk, not tk(x,y), χk(x,y), and Δk(x,y). In other words, we need not measure the distributions of transmittance and birefringence for polarization analysis. Therefore, if the coefficients αk, βk, γk, and ξk can be determined experimentally, the polarization state can be calculated by using Eqs. (6)–(8). The determination of these coefficients is described in the next section.2.4 Experimental determination of αk, βk, γk, and ξk
Figure 2 shows the optical system for the experimental determination of αk, βk, γk, and ξk. In the system, polarized light is produced using one linear polarizer and one waveplate with the retardance of δ0. The polarizer and waveplate are designated the “calibration polarizer” and “calibration waveplate”, respectively. The light intensity is detected by an image sensor in the polarization imaging camera as a function of the angle of the calibration polarizer (θp).
Fig. 2 The optical system for determination of the coefficients for the polarization analysis (calibration of a polarization camera).
The determination of four coefficients (calibration) includes light measurements of three steps with different settings. First, the light intensities are measured without the calibration waveplate as a function of θp. By substituting θ = θp and δ = 0 to Eq. (5), the relation between the detected light intensity and θp can be expressed by
Iinαk, βk and γk can be determined if the Ik-θp curve is fitted with Eq. (10) using these values as the fitting parameters.Secondly, the light intensities are measured with the calibration waveplate as a function of θp. In this case, the slow axis of the calibration waveplate is set to be parallel to the x-axis of the system (i.e., 0°). By substituting θ = θp and δ = δ0 to Eq. (5), the detected light intensity can be expressed by
Fitting of the Ik-θp curve using Eq. (11) yields I'inαk, βk, and γkcosδ0 + ξksinδ0 as the fitting parameters.Finally, the light intensities are measured as a function of θp with the slow axis of the calibration waveplate set to be perpendicular to the x-axis. In that case, by substituting θ = θp and δ = -δ0 to Eq. (5), the detected light intensity is given by
Similarly, fitting of the Ik-θp curve using Eq. (12) yields I'inαk, βk, and γkcosδ0 –ξksinδ0. Because γk has been determined by the first measurement and fitting, δ0 and ξk can be determined from γkcosδ0 + ξksinδ0 and γkcosδ0-ξksinδ0, which are obtained in the second and final fittings, respectively. For a polarization imaging camera, the same procedure are performed at all the cells in the camera. The determination of the coefficients for all the cells in a polarization imaging camera corresponds to the calibration of the camera.3. Method
3.1 Simulation of polarization analysis and error evaluation
Polarization analysis with the polarization imaging camera was simulated by various combinations of four birefringences for 2 × 2 cells in a single unit of a waveplate array. First, the orientation and retardance of the birefringence (χk, Δk) of 2 × 2 cells were chosen, and the coefficients αk, βk, γk, and ξk, (k = 0, 1, 2, 3) were calculated with Eqs. (4) to obtain the system matrix A for Eqs. (6)-(8). Among the four cells, we set the orientation and retardance of 0th-cell to χ0 = 0ᴼ and Δ0 = 0, because the intensity at the cell for which χ0 = 0ᴼ does not depend on Δ0. Next, four light intensities (I0, I1, I2, and I3) at the 2 × 2 cells were calculated via Eq. (5) with one polarization, which was expressed by Eq. (1) with θ = θin and δ = δin. To examine the sensitivity of the analysis to image noise, random intensity fluctuations up to 1% were added to the four light intensities; i.e., Ik = |Ik + noise × Iin| (–0.01 < noise < 0.01). Finally, the Stokes vector was calculated with the system matrix A and the noise-corrupted intensities via Eqs. (8) to obtain θ and δ. The calculated θ and δ were denoted by θcalc and δcalc, respectively.
The analytical error was defined as the distance between the positions on the Poincare sphere of the incident polarization (θin, δin) and the calculated position (θcalc, δcalc):
where S(θ, δ) is a vector, the elements of which are the Stokes parameter, S1(θ, δ), S2(θ, δ), and S3(θ, δ), of the polarization state determined by θ and δ. Because this error depends on the polarization state of the incident light, the mean error of the polarization analysis was defined by the average value of the errors for all possible polarizations (0 ≤ θ ≤ π, 0 ≤ δ ≤ π):where i, j and N are integers to express all possible polarizations (θin = iπ/N, δin = jπ/N) that are used for the error evaluation. In this study, we found that N = 36 was enough to evaluate the mean analytical error.3.2 Fabrication of a waveplate array
Figure 3 shows a direct fs laser-writing system for fabrication of a waveplate array inside a silica glass plate. A femtosecond laser (IFRIT, Cyber Laser Co., Japan) operating at λ = 780 nm, a repetition rate of 1 kHz, and a pulse duration of 220 fs was used as the laser pulse source. The energy of the laser pulse was controlled by a half-waveplate and a polarizer. The energy-controlled laser pulses were reflected on a spatial light modulator (LCOS-SLM, Hamamatsu Photonics K.K., Japan). The spatial light modulator (SLM) was used to perform parallel laser irradiation at multiple spots inside a silica glass for reducing fabrication time of a large waveplate array. The spatial phase distribution of the laser pulse was modulated on the reflection on the SLM. The laser pulse was then passed through a half waveplate and two lenses and focused inside a silica glass plate (VIOSIL-SQ, Shin-Etsu Quartz Products Co., Ltd., Japan) with an objective lens (LU Plan FLUOR, × 20, NA = 0.45, Nikon Corp., Japan). Birefringent structures were generated via nonlinear light absorption at the focal region inside the glass and monitored by a camera above the objective lens. During the irradiation with fs laser pulses, the glass plate was translated perpendicular to the optical axis of the laser beam by using a linear-motor-driven XYZ stage (AI-LM-XY& AI-VC-Z, ALIO Industries, USA) to write straight birefringent lines. Each cell of the waveplate array was made by one birefringent layer and the birefringent layer was fabricated by writing birefringent lines with a spacing of 1.5 µm. The retardance of each sample was controlled by the pulse energy and scanning speed. The polarization of the laser pulse was changed by a half-wave plate after L1 to write birefringent lines with different slow axes.
Fig. 3 Experimental setup for fabrication of a waveplate array of silica glass by fs laser direct writing with an SLM. M1, M2: mirrors; L1: a lens of f = 150 mm; L2: a lens of f = 90 mm; DM: a dichroic mirror which reflects light of 750-850 nm; OL: an objective lens.
We fabricated waveplate arrays with two different cell sizes. For the error evaluation, the cell size was 22.0 µm, which corresponds to the 4 × 4 pixels of the image sensor (pixel size of which was 5.5 µm). For the demonstration of polarization imaging, the cell size was 16.5 µm, which corresponds to the 3 × 3 pixels of the image sensor. The smaller pixel size for the demonstration of polarization imaging was for increasing the spatial resolution.
3.3 Configuration of the polarization imaging camera
Figure 4 shows the configuration of the polarization imaging camera. A light image transmitted through the waveplate array and linear polarizer was imaged onto an image sensor (UI-3360CP-M-GL, IDS Imaging Development Systems GmbH, Germany) by the relay lens, the magnification of which was 1.0 (High Performance Relay Lens #45-760, Edmond Optics Corp., USA). The waveplate arrays were mounted on a XY linear stage with linear polarizer so that the relative positions between the waveplate array and the image sensor could be adjusted. The observed object was focused on the image sensor by adjusting the camera lens. The intensity distribution detected by the image sensor was transferred to a personal computer (PC) via a USB cable and processed for the polarization analysis with application software originally developed by C + + Builder XE.
3.4 Experimental polarization image analysis method
Figure 5 shows the cell and unit in a waveplate array [Figs. 5(a) and 5(b)] and the flow of the polarization analysis in a polarization imaging camera [Fig. 5(c)]. A single unit of a waveplate array is defined by one 2 × 2 cells. The cells are referred by the indices (l, m), and the unit including the cell (l, m) at the top-left is denoted by “Unit (l, m)”. The coefficients of the cell (l, m) are expressed by αl,m, βl,m, γl,m and ξl,m..
Fig. 5 The analysis for a polarization imaging camera. (a) The structure of a waveplate array. (b) Single unit in the waveplate array. (c) The flow of calibration, calculation of the system matrices and polarization analysis for a polarization imaging camera.
The flow of the analysis in a computer with a polarization imaging camera is summarized in Fig. 5(c). First, the coefficients αl,m, βl,m, γl,m and ξl,m of all the cells were determined experimentally by the method described in Section 2.4 (Step 1). Next, the system matrices A of all the units were constructed using the coefficients determined according to Eq. (6) (Step 2). The system matrix of unit (l, m), A(l,m), were constructed by
The inverses of the A(l,m) for all the units, B(l,m), were calculated, and the elements of B(l,m) were stored in the memory of a PC.Once the matrices B(l,m) were determined, the polarization at each unit could be calculated via Eq. (8) using the light intensities {Il, m} captured by the image sensor (Step 3 and Step 4). If the light intensities in the unit (l, m), Il, m, Il + 1, m, Il, m + 1, and Il + 1, m + 1, are designated I0(l, m), I1(l, m), I2(l, m), and I3(l, m), the equations for the Stokes vectors at the position (l, m) are given by
After the Stokes vectors were calculated, the direction θl,m and phase δl,m of the polarization state at the unit (l, m) could be obtained.4. Simulation results
As described in the introduction, it is difficult to make a waveplate array of silica glass with uniform retardance by fs laser direct writing. Therefore, it is necessary to examine how the dispersion of the retardance influences the analytical errors and find the combination of retardances of a waveplate array with smeller analytical errors. In this section, we simulated polarization analysis and examined how the analytical errors depend on the combination of the waveplates and dispersion of the retardances.
4.1 Simulation of polarization analysis
Simulation of polarization imaging was performed by all possible polarizations. In the simulation, a polarization imaging camera with a waveplate array of 36 × 36 units was used, and all possible polarizations were input to the camera. The polarization distribution for the simulation is shown in Fig. 6(a), in which the orientation (long axis) and ellipticity angle are expressed by color and brightness, respectively. In the polarization distribution, θin and δin are changed in the transversal and the longitudinal directions, respectively, i.e. (θin, δin) = (iπ/36, jπ/36) (i, j = 0-35) at the unit (i, j). The long axis (ψ) and ellipticity angle (X) of the polarized light of θ and δ were defined by Eqs. (17) and (18), respectively:
Fig. 6 Simulation of polarization imaging and analysis. (a) Input polarization distribution. The color and brightness indicate the slow axis and ellipticity angle of the polarization state, respectively. CW and CCW mean clockwise and counter-clockwise ellipsoidal polarizations, respectively. (b) Simulated intensity distributions detected by a polarization imaging camera. In this case, the combination of orientation and retardance is (Δ22deg, Δ45deg, Δ67deg) = (0.8π, 0.8π, 0.8π). (c)–(e) Calculated polarization distributions with three different waveplate arrays.
For the parameters of a waveplate array, we assumed that the slow axes of the four waveplates in a single unit were χ0 = 0°, χ1 = 22°, χ2 = 45°, and χ3 = 67°. To facilitate understanding of these results, we designated the retardances Δ0, Δ1, Δ2, and Δ3 by Δ0deg, Δ22deg, Δ45deg, and Δ67deg, respectively. Because Eq. (3) shows that the light intensity through the waveplate of χ0 = 0°does not depend on Δ0, we assumed that Δ0deg = 0.
Figure 6(b) shows the intensity distributions simulated by the polarization imaging camera for Δ22deg = Δ45deg = Δ67deg = 0.8π as an example. The value of 0.8π was chosen just because the maximum retardance inside a silica glass by fs laser writing in our previous report [27]. The intensity mosaic pattern reflects the polarization distribution of Fig. 6(a). In the following polarization analysis in this section, an image noise (Section 3.1) of 1% of the maximum intensity was applied to the simulated intensities for evaluation of the analytical error.
The polarization analysis was performed for three different waveplate arrays, i.e. three different combination of retardance values for Δ22deg, Δ45deg, and Δ67deg. To investigate the influence of magnitude relation between three retardances to polarization analysis, simulation was performed with several retardance combinations of 0.8π and 0.4π. Figures 6(c)–6(e) show the polarization distributions calculated for three waveplate arrays. These polarization distributions show that the analytical errors differ greatly for three waveplate arrays. In the case of (Δ22deg, Δ45deg, Δ67deg) = (0.8π, 0.8π, 0.8π), small analytical errors were found in the calculated polarization distributions [Fig. 6(c)]. The polarization error given by Eq. (13), <εpol>, was 0.08 rad. In the case of (Δ22deg, Δ45deg, Δ67deg) = (0.8π, 0.4π, 0.4π), the calculated polarization distributions were much noisier [Fig. 6(d)] (<εpol> = 0.25 rad). Interestingly, the error of the polarization analysis for (Δ22deg, Δ45deg, Δ67deg) = (0.4π, 0.8π, 0.4π) was much smaller [Fig. 6(e)] (<εpol> = 0.03 rad), although the sum of the retardances (Δ22deg + Δ45deg + Δ67deg = 1.6π) was the same as that of Fig. 6(d). The different analytical errors mean different sensitivities to image noise, and the origin of the difference comes from the system matrix A [Eqs. (7) and (15)], which will be shown in the next section.
In summary, the comparison of the polarization analyses [Figs. 6(c)-6(e)], indicates that the accuracy of the polarization analysis is greatly affected by the combination of four waveplates in a waveplate array.
4.2 The factor that affects the accuracy of the polarization analysis
To find the factor that affects the accuracy of the polarization analysis, we performed the same simulation as the previous section for 1,000 different waveplate arrays 36 × 36 units and evaluated the mean errors of the polarization analyses (<εpol>). The values of retardance for (Δ22deg, Δ45deg, Δ67deg) of the 1,000 waveplate arrays were chosen randomly in the range between π/180 and π.
Figure 7(a) shows the <εpol> for 1,000 waveplate arrays plotted against the sum of the retardances (Δ22deg + Δ45deg + Δ67deg). Clearly, the correlation between the <εpol> and the sum of the retardances was very weak. This weak correlation means that larger retardance of a waveplate array does not always increase the accuracy of the polarization analysis.
Fig. 7 Mean analytical errors <εpol> of 1000 different waveplate arrays. (a) <εpol> plotted against the sum of retardances, Δ22deg + Δ45deg + Δ67deg. (b) <εpol> plotted against |det(A)|−1. (A) is the system matrix of a waveplate array given by Eq. (6).
According to Sabatke et al. and other researchers, the reciprocal of the absolute value of the determinant of the system matrix A (|det(A)|–1) is a good figure of merit for increasing the signal-to-noise ratio of a polarization measurement [23–25]. Based on this figure of merit, we plotted the <εpol> of Fig. 7(a) against |det(A)|–1 in Fig. 7(b). The fact that there was clearly a positive correlation between <εpol> and |det(A)|–1 is consistent with the studies of Sabatke et al. and other researchers [23–25]. Therefore, to increase the accuracy of the polarization analysis, the retardances of a waveplate array should be selected so that |det(A)|–1 becomes smaller.
4.3 Influence of retardance dispersion to analytical errors
In this subsection, we investigated how the dispersion of retardance in a waveplate array influences to the error of polarization analysis. For the simulation, we chose the retardance values for (Δ22deg, Δ45deg, Δ67deg) randomly in the specific ranges listed in Table 1. In each range, the retardance values have a dispersion of ± 0.1π. The mean analytical errors were simulated for 100 randomly chosen combinations of Δ22deg, Δ45deg, and Δ67deg for each range. Figure 8 shows the plot of the mean analytical errors <εpol> against |det(A)|–1. There were two patterns among the ranges of retardances: for Ranges 1, 2, 4, and 8, the dispersion of the analytical errors exceeded one order of magnitude [Fig. 8(a)]. For Ranges 3, 5, 6, and 7, the dispersions were as small as 0.05 rad [Fig. 8(b)]. Interestingly, the dispersions in |det(A)|–1 were also completely different between the two types; in Fig. 8(a), the largest |det(A)|–1 was more than 100 times the smallest one, whereas in Fig. 8(b), the |det(A)|–1 varied from 0.6 to 2.5. The difference means that the influence of retardance dispersion to analytical error strongly depends on the mean values of Δ22deg, Δ45deg, and Δ67deg.
Table 1. Ranges of Δ22deg, Δ45deg and Δ67deg for evaluation of analytical errors.
Fig. 8 Mean analytical errors plotted against |det(A)|−1. (a) For the Ranges 1, 2, 4 and 8, and (b) for the Ranges 3, 5, 6 and 7. The mean analytical errors were calculated under an image noise of 1%.
Although we used typical values, 0.8π and 0.4π, for the simulation, we obtained an important general conclusion: before fabricating a waveplate array, we must choose the combination of retardances in a waveplate array that minimizes the influence of retardance dispersion to analytical error as possible. For example, among the ranges in Table 1, we should choose Range 3 that gives smallest analytical errors with a dispersion of ± 0.1π. In addition, we should choose a combination of orientation and retardance in a waveplate array to make |det(A)|–1 smaller as possible for more accurate polarization.
5. Experimental results
5.1 Fabrication and calibration of laser written waveplate arrays
To confirm the polarization analyses in Sections 4.2 and 4.3 experimentally, we fabricated four kinds of waveplate arrays (Samples 1–4) of silica glass by fs laser direct writing. The laser processing conditions for the waveplate arrays are shown in Table 2. Figure 9(a) shows the retardance images of Samples 1–4, which were obtained with a polarization microscope (LC-Polscope, CRI, Inc., USA). The waveplate array consisted of 8 × 8 cells. The width and height of a single cell were about 22.0 μm. One cell in a unit for χ = 0° had no birefringence, i.e., no laser writing, and the orientations of the birefringence of the other three cells were about 22°, 45°, and 67°. These waveplate arrays were mounted in our polarization imaging camera (Fig. 4) with a bandpass filter of 500 nm.
Table 2. Laser processing conditions for fabricating Sample 1-4. Δ22deg, Δ45deg, and Δ67deg correspond to the cells of k = 1, 2 and 3, respectively.
Fig. 9 Retardance distribution images. (a)Retardance distributions of Sample 1-4, which were captured by a polarization microscope. (b) Histogram of the retardance in a single cell.
The retardances for these waveplate arrays, measured via polarization microscope images [Fig. 9(a)], are shown in Table 3. Compared with Table 1, Sample 1 was categorized in Range 3, because Δ45deg was always larger than Δ22deg and Δ67deg. Based on the relationship between Δ22deg, Δ45deg, and Δ67deg, Samples 2, 3, and 4 were roughly categorized in Ranges 1, 2, and 8, respectively. According to the analysis shown in Fig. 8, it was expected that the analytical errors of Sample 1 would be relatively small, whereas those of Samples 2, 3, and 4 would be relatively large.
Table 3. The retardance of Δ22deg, Δ45deg and Δ67deg at 500 nm wavelength in each sample. The retardances were calculated by the retardance images shown in Fig. 9 under the assumption that the birefringence were the same at the wavelength in a polarization imaging (500) nm and that in polarization microscope (546 nm).
Figure 9(b) shows a histogram of the retardances in a single cell. The histogram shows that the retardances in a single cell are multi-valued, so the coefficients must be determined experimentally. The non-uniformity should have come from the stress between adjacent laser written lines, spatial distribution of birefringence in a single laser written line and so on. Therefore, before the polarization measurement and the analysis, the coefficients at each cell (αk, βk, γk, ξk) were determined by the calibration method described in Section 2.4. Figure 10 show examples of the measurements and fitting that gave the four coefficients of the four cells in a single unit of Sample 1. The light intensity at each cell was measured as a function of the angle of a linear polarizer (θp) with three settings, without calibration waveplate and with calibration waveplate of orthogonal orientations [//x and ⊥x, respectively]. The three intensity data were fitted with Eqs. (10)–(12) to obtain αk, βk, γk and ξk. Figure 10 shows that all the plots of intensity-θp were fitted well. Table 4 shows the coefficients of the four cells obtained via the fitting. One remarkable point was the differences of αk values. Because αk is proportional to the transmittance of the cell according to Eqs. (4), the difference in the coefficients means that the transmittances in laser written cells (k = 1–3) were only 70% of that in the cell with no laser writing (k = 0). As shown in the next subsection, the polarization analysis is possible regardless of such large differences in transmittance.
Fig. 10 Examples of fitting for obtaining the coefficients (αk, βk, χk, ξk) at four cells in a single unit of Sample 1. Red open circles are intensities measured without a calibration waveplate, blue squares and green crosses are those measured with a calibration waveplate of orthogonal orientations (//x and ⊥x). The solid lines are the fitting curves by Eq. (10)-(12).
Table 4. The coefficients of four cells of Sample 1.
5.2 Polarization analysis with laser written waveplate arrays
The polarization analyses with the four waveplate arrays were performed by detecting linearly polarized light with directions that ranged from 0° to 180°. Figure 11(a) shows examples of the polarizations obtained with the four waveplate arrays for linearly polarized lights. The polarization states obtained with 7 × 7 units for each waveplate array were expressed by ellipses. Clearly, we could obtain correct polarizations (linearly polarized lights of almost the same orientations to the input light) with Sample 1, whereas we obtained incorrect results (ellipsoidally polarized lights) with the other waveplate arrays because of the analytical errors. This means that Sample 1 was the best of the four waveplate arrays. Consequently, we confirmed experimentally that the choice of appropriate combination of retardance values is dispensable for accurate polarization analysis, even though the calibration was performed before the measurement.
Fig. 11 Polarization analysis for Sample 1-4. (a) Polarizations obtained by four different waveplate arrays. Ellipsoids of different colors represent the calculated polarizations at individual units. (b), (c) Plot of analytical errors (errors in calculated polarization angle) against |det(A)|−1. (b) For all the samples. (c) Magnified graph for Sample 1.
According to the analysis in Section 4.3, there were two groups of retardance combinations with relatively small and large analytical errors. Among the four samples, only Sample 1 was fabricated to be in the group of relatively small errors (Range 3 in Table 1), whereas Samples 2, 3, and 4 were in the group of relatively large errors (Ranges 1, 2, and 8 in Table 1, respectively). Therefore the analytical errors of the fabricated waveplate arrays were consistent with the errors simulated in Section 4.3. In addition, the analysis of Sample 1 indicated that the polarization distributions could be obtained with our method regardless of non-uniform retardance distributions and difference in transmittance in a waveplate array.
For a more quantitative comparison, the errors of the calculated polarization directions were plotted against |det(A)|–1 in Figs. 11(b). Here, the analytical errors were evaluated by the angle errors for linearly polarized lights, because it was difficult to obtain all ellipsoidal polarizations experimentally. The relations between the analytical errors and |det(A)|–1 were comparable to those in Fig. 8; the analytical errors have a positive correlation with |det(A)|–1. For Sample 3 and 4, the largest values of |det(A)|–1 were more than 100 times the smallest value. This trend is the same as that of Ranges 2 and 8 in Fig. 8(a). In contrast, the value and dispersion of |det(A)|–1 of Sample 1 were much smaller; |det(A)|–1 = 20 ± 5 [Fig. 11(c)]. This smaller dispersion of |det(A)|–1 and smaller error were consistent with those of Range 3 in Fig. 8(b). However, the relation between the angle error and |det(A)|–1 of Sample 2 was different from that of Range 1 in Fig. 8(a), although we expected that Sample 2 was similar to Range 1. This difference might be because Sample 2 is also similar to Range 6 because of the relation of Δ22deg, Δ67deg>Δ45deg. Although there is a slight difference between simulation and experiment, we could experimentally confirm that the figure of merit for the polarization analysis, |det(A)|–1, was applicable to our polarization imaging camera with a waveplate array fabricated by fs laser direct writing.
5.3 Demonstration of the polarization imaging camera
Finally, we demonstrated our polarization-imaging camera with a waveplate array of silica glass fabricated with fs laser direct writing. The fabrication condition for the waveplate array was the same as that for Sample 1, for which we have confirmed accurate polarization analysis. For the demonstration, three polarization films with different directions were imaged on the polarization imaging camera and the distributions of the polarization were analyzed. Figure 12(a) shows the light distributions captured by the camera. The direction of polarization of the backlight was 90°, and the transmission orientation of the polarization films were 65° (upper), 130° (lower left), and 40° (lower right). The magnified images in Figs. 12(b) and 12(c) show the intensity distributions in the regions of different polarization films. The intensities at four adjacent cells clearly depended on the directions of the polarization films. Figure 12(d) shows the polarization distributions calculated from the intensity distributions of Fig. 12(a). The light green color of the background indicates the polarization direction of the backlight (90°). The directions of all three polarization films could be imaged clearly. The calculated directions of the light polarized through the polarization films were 65.8° ± 0.5ᴼ, 132.5° ± 0.6°, and 39.1° ± 1.2°. These directions were roughly the same as the orientation of the films. More demonstrations are available as supplementary materials (see Visualization 1).
Fig. 12 Demonstration of polarization imaging. (a) The light distributions captured by the polarization imaging camera. (b) and (c) The magnified images in the areas of the polarization films. (d) Calculated distribution of the polarization direction from (a).
6. Conclusions
We applied a generalized polarization analysis method to a polarization imaging camera of a waveplate array fabricated by fs laser direct writing. Although the transmittance and birefringence in a cell of the waveplate array were not uniform, polarization analysis was possible with system matrices with four coefficients for each waveplate cell determined experimentally for all the units in the waveplate array. In addition, the influence of the characteristics of a waveplate array on the polarization analysis was investigated by both simulations and experiments. As a result, we confirmed that the figure of merit based on the determinant of the system matrix could be applied to improve the accuracy of the polarization analysis. Further improvement of the polarization analysis can be expected by optimizing the orientation, retardance, and transmittance of each cell in a waveplate array.
Funding
Cross-Ministerial Strategic Innovation Promotion (SIP) Program; JSPS KAKENHI (16K13929 and 17H03040).
Acknowledgments
The authors would like to thank Mr. K. Hayashi and Ms. E. Ueno from Hitachi Zosen Co. for their good suggestions and helpful discussions.
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