Singular optics has recently become a subject of numerous investigations, since Nye and Berry [1] first found dislocations in optical fields in 1974. While phase singularities (wave dislocated, or optical vortices) are frequently encountered in interference of scalar waves [2, 3], they resolve into polarization singularities (PSs) when the vector nature of light is retained [4-19]. A large number of papers have studied polarized properties of vector fields [4-11]. Measuring polarization states is the crucial step in experimental studies of polarization singularities [12]. Experiments have been successful in generating PSs [13-16]. However fields with crowded PSs present challenges in measuring the polarization states [17-19]. The polarization state of a light beam is completely described by the Stokes parameters [18-20]. Indeed, Stokes measurements require wave plates (at least one) and a polarizer to be inserted and rotated in front of the imaging detector, which inevitably modifies the optical wavefront and polarization state of the vector field [17]. The modification obviously leads to erroneous experimental results, especially in the case of measurement of field with crowded PSs, such as the case in [18]. This experimental difficulty is also encountered in polarization imaging of speckle in [17] which a method has been proposed to overcome the mismatches by holding the wave plate and polarizer together. In our previous work [19], we also encountered the mismatches, and we resolved the difficulty well through matching of characteristic points. In this paper, a checkerboard calibration board is used to construct characteristic points in the measured field, which makes it possible to calibrate the mismatches caused by rotation of quarter-wave plate (or polarizer) and to retrieve the original image through matching of characteristic points. Using this method, we measure the polarization state of a field with array of PSs. The result shows that values of shifts between images obtained in the experiment could be big enough to break the distribution of PSs. The comparison and analysis emphasize the necessity and feasibility of this method to measure the PSs.
The traditional setup used to measure the polarization state of the field is shown in Fig. 1(a). By setting the quarter-wave plate Q and polarizer P angle as δ_{i} and θ_{i}, respectively, one can obtain n intensity images I(δ_{i}, θ_{i}) (i=1,2,…n) through the imaging detector CCD. For instance, we got four images (at least) by setting the angles of Q and P as (0°, 0°), (0°, 45°), (0°, 90°) and (45°, 45°). With the help of Stokes formulas [20-22], we can acquire the polarization state of light, as follows:
Where S_{0} is the overall intensity. I(δ, θ ) denotes the intensity image obtained by CCD when the angles of Q and P are set at δ and θ, respectively. According to [20], we can obtain the distribution of polarization states (the azimuthal angle and the ellipticity). However, in practical experiment, the defects of devices (quarter-wave plate, polarizer and rotation mounts) and assembly factors cannot be ignored. Figure 2 illustrates the influence of these factors. The wave-plate (or polarizer) is fixed on a rotation mount whose gyration axis is the red line shown in Fig. 2(a). The blue line is perpendicular to the wave-plate, and has an angle β with the gyration axis. The black dashed line is the incident direction (optical axis) of beams, and have angles α and γ with the gyration axis and the normal, respectively. In Fig. 2(b), the three axes are in a Cartesian coordinate system, which makes it easier to understand Fig. 2(a). Ideally the gyration axis, the normal of wave-plate and the optical axis coincide with each other, namely α, β and γ are all equal to zero. However, because of assembly factors and defects of devices, the three angles are not zero. Because the quarter-wave plate is fixed on the rotation mount which rotates around an invariant gyration axis (the red line in Fig. 2(b)), the values of α and β hold steady, while the value of γ keeps changing with the rotation of devices. Actually this variable angle γ is the incident angle of light irradiating the wave-plate (polarizer). Considering the thickness of the wave-plate (polarizer), different incident angles will cause different degrees of modifications on the wavefront of the vector field. Further, small wedge angles between the front and back planes of wave-plate (or polarizer), namely the polarizer and wave-plate used to measure polarization states are not perfect optical plates, also cause unpredictable modifications on the wavefront. In the process of measurement, two rotation mounts, fixed with a polarizer and a wave-plate respectively, are used in the form of serial transmission, which can make the modifications on the wavefront stronger.
Without considering the distortion, factors analyzed above result in the images’ shift away from the original positions. A hypothetical case shown in Fig. 3 is used to illustrate this phenomenon. A, B, C are three points in image I obtained when the Q and P are absent, and constitute an original coordinate system without modification caused by devices. By inserting the quarter-wave plate and polarizer in front of the CCD and rotating their angles to arbitrary values, points A, B, C may move to A_{1}, B_{1}, C_{1}, respectively, which constitute a new coordinate system. Keep rotating the quarter-plate and polarizer to other angles, the points A_{1}, B_{1}, C_{1} may move to A_{2}, B_{2}, C_{2}, respectively. In Fig. 3, the rotation angle and distance between I and I_{1} are denoted as φ_{1}, Dx_{1} and Dy_{1}, respectively. Generally, the three points move back after the wave-plate and polarizer rotate one circle. The analysis above points out that images obtained with the wave-plate and polarizer set at different angles have mismatches with each other. Because of the huge number of arrangements of δ and θ, it is impossible to calibrate all shifts of images obtained with the wave-plate and polarizer set at arbitrary angles. Fortunately, only four images I corresponding to special angles (δ_{i}, θ_{i}) (i=1, 2, 3, 4) are needed to calculate Stokes parameters. So we just need to calibrate the shifts of images at angles (δ_{i}, θ_{i}), which is easier to implement.
[FIG. 3.] Scheme used to illustrate the shift of field passing through the quarter-wave plate and polarizer. A, B, C are three points in a original image (obtained by CCD without the quarter-wave plate and polarizer in front of the CCD), while A1, B1, C1 and, A2, B2, C2 denote positions of A, B, C, respectively, after the light passing through the quarter-wave plate and polarizer set at different angles. (φ1, Δx1, Δy1) is the mismatch between coordinate systems ABC and A1B1C1.
Assuming that we have rotated an angle φ_{i} and panned a distance (Δx_{1}, Δy_{1}), I(δ_{i}, θ_{i}) can be retrieved to the original image I. Having the following relationship
Where (x, y) is the position of a point in the original image I, while (x_{i}, y_{i}) is the position of the same point in the shifted image I(δ_{i}, θ_{i}. There are three unknown parameters φ_{i}, Δx_{i} and Δy_{i} in Eq. (2). At least two pairs of points are needed to resolve the three parameters, such as pairs of {A(x, y), A_{1}(x_{1}, y_{1})} and {B(x, y), B_{1}(x_{1}, y_{1})} in Fig. 3. When φ_{i}, Δx_{i} and Δy_{i} are acquired, we can retrieve the position of images I(δ_{i}, θ_{i}). Actually the assembly factors and defects of devices do not cause rotation of images (φ_{i} = 0), which is also proved by our experimental results. In our previous work [19], we also encountered mismatches caused by the rotating quarter-wave plate and polarizer, and we resolved it well through characteristic points matching.
However, for an unknown vector field, there are no characteristic points that can used to match images. In order to calibrate values of the shifts (φ_{i}, Δx_{i}, Δy_{i}) caused by rotating quarter-wave and polarizer, we proposed a setup which can construct characteristic points in the field, shown in Fig. 1(b). The light is focused onto the rotating ground glass by Lens L1, and collimated by Lens L2. Then the collimated light gets through a checkerboard calibration board whose size is 9 mm × 9 mm (shown in Fig. 1(b)). The transmittance of white squares of the checkerboard is almost equal to 1, while the transmittance of black squares is 0. We regarded the intersections of the checkerboard as characteristic points. After passing through the checkerboard, the light successively gets through quarter-wave plate Q and polarizer P. Here we emphasized that the combination of rotating ground glass and lens L1, L2 is used to reduce the coherence and ensure uniform intensity distribution of the field, which make extracting characteristic points of checkerboard possible. Figure 4(a) is obtained without the combination of rotating ground glass, L1 and L2, while Fig. 4(b) is obtained with this combination. In Fig. 4(a), information of the calibration sustained serious losses, which makes it impossible to extract the characteristic points from Fig. 4(a). Comparing with Fig. 4(a) the edge diffraction of Fig. 4(b) is well depressed, and the intensity is uniform.
We first implemented binary conversion of Fig. 4(b) by setting a threshold value. We selected a 6 mm × 6 mm window in the acquired images. The reason of selecting a 6 mm × 6 mm window is the size of small lattices in the checkerboard is 3 mm × 3 mm. This means nearly half of the 6 mm × 6 mm window has transmittance 1, while the other half has transmittance 0. So the average gray value in the selected window can be used to implement binary conversion of Fig. 4(b). This value is proved to be very effective in our experiments. The threshold value μ is given as follow
Where m = n = [6 mm /σ ] are pixel numbers of the selected window, σ is the size of pixel of CCD, and brackets indicate rounding. f(i, j) denotes image gray value at point (i, j) in the selected window, and range of the value is 0:255. Figure 4(c) is the digital binary image of Fig. 4(b). In Fig. 4(c) we see that the corner information sustained serious losses. Fortunately, the edge information was maintained perfectly. The edge of the binary image is extracted through an edge detection operator [23]. Then four straight lines are obtained through fitting the edges by using the least squares method, shown in Fig. 4(c). The four straight lines reconstruct the lattices of the checkerboard. According to equations of the four lines, we obtained coordinates of the characteristic points A and B. In one experiment, one image without the quarter-wave and polarizer and four images with the quarter-wave and polarizer set at angles (δ_{1}, θ_{1}) = (0°, 0°), (δ_{2}, θ_{2}) = (0°, 45°), (δ_{3}, θ_{3}) = (0°, 90°), (δ_{4}, θ_{4}) = (45°, 45°) are needed to be obtained, so that extracting characteristic points needs to be implemented five times. According to Eq. (2), we can calibrate the shifts (φ_{i}, Δx_{i}, Δx_{i}) (i=1, 2, 3, 4) caused by the measuring devices.
In this section, we used the setup shown in Fig. 1(b) to measure a field with an array of PSs produced by superposition of three sources of linearly polarized light [19]. The polarization directions of two beams were perpendicular to one another, and polarization direction of the third beam had an angle 45° with polarization directions of the other two beams. The quarter-wave plate (AHWP10M) was offered by Thorlabs, and its thickness was 1.07 mm. The polarizer (LPVISB100) was offered by Thorlabs, and its thickness was 2.0 ± 0.2 mm. According to the detailed calibration procedure illustrated in the above section, we obtained the values (φ_{i}, Δx_{i}, Δy_{i}) (i=1, 2, 3, 4). Then the rotating ground glass and checkerboard calibration board were moved away. By rotating the angle of quarter-wave plate and polarizer to (0°, 0°), (0°, 45°), (0°, 90°), (45°, 45°), respectively, we got four images I'(δ_{i}, θ_{i}) (i=1, 2, 3, 4). Rotating α_{i} angle and panning (Δx_{i}, Δy_{i}) distance, I'(δ_{i}, θ_{i}) were returned to the original positions. Substituting the retrieved intensities into Eq. (1), we acquired the distribution of polarization states.
In our experimental data, the values of φ_{i} were close to zero. This is because the assembly factors and defects of devices discussed above do not cause rotation of images. The most serious mismatch occurred between I(0°, 0°) and I(0°, 90°), and the value of this mismatch was . Figure 5(a) was the result without eliminating the mismatches between the four intensities I'(δ_{i}, θ_{i}) (i=1, 2, 3, 4). The background represents the light intensity. The green ellipses show the distribution of the polarization states and the yellow lines indicated where linear polarization occurs, i.e. L-lines. The blue circles (acquired when S_{1} = 0 and S_{2} = 0) in Fig. 5(a) were pure circular polarized points. These points were regarded as circle polarization singularities [7, 11], i.e. C-points. Ignoring what types these C-points belong to, we compared Fig. 5(a) with the numerical simulation shown in Fig. 5(b) (simulated according the polarization directions of the three linearly polarized beams, with the help of MATLAB). We found that Fig. 5(a) has a completely different distribution of polarization state (including the distribution of L-lines and C-points) compared to Fig. 5(b). This comparison intuitively showed the serious effect of shifts caused by the rotating quarter-wave plate and polarizer, and the necessity to eliminate the mismatches.
The Fig. 5(c) was the same area shown in Fig. 5(a), while the distribution of polarization state in Fig. 5(c) was obtained after the procedure of eliminating the shifts. The vector field is divided into band structure with periodic distribution of -1/2 Stars [yellow squares in Fig. 5(c)] and +1/2 Lemons [red circles in Fig. 5(c)]. The polarization may be either right handed or left handed, but C points of opposite handedness are always separated by an L-line [4]. Compared with the numerical simulation, Fig. 5(c) had basically the same distribution of polarization states, apart from sharp teeth on the L-lines. These sharp teeth were caused by noise. The comparison of Fig. 5(a), (b) and (c) illustrated that the method we proposed can calibrate and eliminate the mismatches caused by the rotating devices, and this method is feasible in practical experiments.
Many papers acquired good experimental results [13-16] without eliminating the mismatches caused by the measuring system. But we noticed that most of these successful experiments were measuring the polarization state of fields with sparse PSs. Mismatches were too small to disturb the distribution of polarization states, and they could be ignored. For example, the distance is smaller than 0.2 mm in our experiment. However when measuring the field with rapidly varying distribution of polarization state (such as a field with intensive PSs), the mismatches may be big enough to destroy the distribution of polarization states, for example, the cases in [17], [18] and [19]. Analyzing from the data, we noticed that the size of field shown in Fig. 5(c) was about 1735 μm × 1735 μm and the density of PSs was about 5.6 / mm^{2}. The distance of the nearest two PSs in Fig. 5(c) was about 100 μm, which was smaller than the mismatch that occurred between I(0°, 0°) and I(0°, 90°). So the wrong result (shown as Fig. 5(a)) caused by this mismatch l = 182 μm can be explained.
It is worthwhile to notice that the mismatches cause more serious effects in fields with higher density of PSs. As shown in Fig 5(a) and (c), the isolated Lemons in Fig. 5(c) maintained their type in Fig. 5(a), while the dense Stars in Fig. 5(c) transformed into several C-points in Fig. 5(a). In other words, for a field with sparse PSs, someone may obtain the right experimental results without calibrating the shifts. However when used to measuring field with crowded PSs, the calibration technique is necessary. However what kind of density of PSs can be regarded as crowded PSs (or sparse PSs) has not been definite. But one thing is certain: if the biggest mismatch between images is close to (or bigger than) the distance of the nearest PSs in the measured field, the calibration is necessary.
In addition, the method requires that the polarization components be removed from the optical path to measure the original image, so the calibration procedure should be repeated when the system is used to measure a new field.
In this paper, we propose a measuring system used to measure the polarization state of a field. This method can calibrate the shifts caused by rotating quarter-wave plate and polarizer through constructing characteristic points in the measured light with a checkerboard calibration board. The more serious obstruction of the mismatches occurs in a field with more intensive PSs. In the setup, a combination of a rotating ground glass and two lenses is used to reduce the coherence of light, and it also ensures uniform distribution of intensities obtained by the CCD. The effect of the combination makes it possible to extract the characteristic points of the checkerboard. According to coordinate values of the characteristic points, we acquire the rotated angle and panned distance when light passes through the quarterwave plate and the polarizer. By eliminating the shifts, we retrieve the original position of images, which ensured the correctness of calculation of the polarization state. Then a field with an array of PSs is measured. The data shows that the mismatches were big enough to result in wrong distribution of polarization state, and thus it is necessary to eliminate the mismatches. Compared with the numerical simulation, the experimental result has the same distribution of polarization states with the simulation, which illustrated the feasibility of this method. Though our results are obtained from measuring the specific three-wave interference, it is clear that our experimental method can be readily generalized to more general cases.