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Abstract
Instantaneous phase shifting interferometry technology, the core component of which is the pixel micropolarizer camera, has been widely used in commercial interferometers. This technology has the superiority of single-frame acquisition, vibration insensitivity, and no need for phase shifting devices. However, due to manufacturing defects and accuracy limitations, the extinction ratios (ER) of the micropolarizer array are different and fairly small, directly affecting the phase calculation accuracy. This paper initially derives a theoretical expression for the phase calculation error introduced by the extinction ratio (ER) and proposes the error correction model to reduce phase calculation errors caused by the extinction ratio. The theoretical analysis can serve as an important basis for accurately assessing the polarization characteristics of a pixel micropolarizer camera. Quantifying the impact of the extinction ratios provides significant support for the selection of polarization equipment. In addition, the paper proposes a calibration model to improve measurement accuracy, which can serve as an effective means to reduce the impact of the extinction ratio (ER). The innovative research content revealed the influence of extinction ratio (ER), serving as a valuable complement to the existing analysis and research on extinction ratio (ER).
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1. Introduction
As a method that can effectively acquire and analyze phase data, phase-shifting interferometry technology has driven the development of interferometry technology [1]. Compared with mechanical phase shifting, polarization phase shifting has many advantages, such as single frame acquisition, no requirement for moving components, and strong resistance to environmental interference [2,3]. The polarization phase shifting method utilizes polarization elements to generate the desired phase shift signal to determine the phase distribution of the interferogram [4]. Smythe and Moore proposed using multiple image sensors to obtain a series of interferograms with fixed phase differences in the multiple-camera system [5]. In other work, Miller et al. adopted the holographic element to divide the beam into four independent beams. They added a polarizer mask to introduce a phase shift to acquire interferograms in the four polarization directions [6–8]. In contrast, Miller et al. used micropolarizers to introduce four phase shifts to four adjacent pixels to obtain interferograms in the four polarization directions [9]. The method for capturing interferograms using multiple image sensors requires fixing, image registration, strict installation accuracy, and high manufacturing costs [10]. The beam-splitting technique by holographic elements has the shortcomings of reduced resolution, limited measurement accuracy, and only operating in a limited wavelength range. Therefore, the two polarization phase-shifting methods limit the commercialization of their interferometers [11]. Pixel micropolarizer cameras are widely used in commercial instruments due to advantages such as single frame acquisition, quantitative phase, and high integration [12].
The traditional assembly method of pixel micropolarizer cameras is to manufacture detector and micropolarizer arrays separately, then mutually align them and fix them in appropriate positions [13–16]. Ideally, the pixel micropolarizer camera would accurately reflect the polarization information of the incident polarized light. However, the fine-spacing line grid of the micropolarizer is difficult to manufacture, which means it is challenging for the micropolarizer to achieve the polarization purity that single-piece polarizing devices can easily attain. Generally, we use the extinction ratio parameter to indicate the purity of a linear polarizer. It reflects the performance of the polarizing element in transmitting and blocking light with different polarization directions [17]. A higher extinction ratio indicates the improved performance of the polarizing element in the blocking non-target polarization states, thus enhancing the purity and performance of the polarizing device. The polarization application with a high extinction ratio corresponds to more accurate and reliable measurement results since it can effectively eliminate interference from non-target polarized light. Therefore, considering the extinction ratio is crucial when selecting and using polarizing elements.
Pixel micropolarizer cameras serve as phase-shifting measurement components in polarization interferometry devices, and the quality of the polarization elements directly affects the measurement accuracy of the interferometric setup. Hence, clarifying the relationship between the polarization parameters and the measurement accuracy is important. This paper derives the calculational formula for the phase calculation error caused by the extinction ratio through theoretical analysis, which complements and enriches the current error analysis of polarization cameras. Based on this scenario, we propose a mathematical model for maximum error, which can be directly used to evaluate the performance of a pixel micropolarizer camera. In addition, our proposed correction model effectively solves the phase calculation error caused by the extinction ratio, which is of practical significance. The experimental results intuitively demonstrate the impact of extinction ratio on phase calculation errors and verify that the correction model can effectively alleviate extinction ratio errors. In conclusion, the thesis conducts thorough research on the fundamental nature of extinction ratio and quantifies the influence of extinction ratio. Furthermore, by modifying the model, the error caused by the extinction ratio can be reduced, effectively addressing the systematic error source from the polarization camera.
2. Principle of polarization phase shifting interference
2.1 Polarization interference theory
In a polarization phase-shifting interferometer system, the polarization data of the outgoing light intensity can determine the phase difference between the mutually orthogonal reference and test beams. This principle is described as follows [18,19].
There is a pair of orthogonal linearly polarized light beams in the horizontal x and the vertical y directions, which assumes that they are the reference beam Er and the test beam Et in a polarization phase-shifting interferometer, respectively. The phase difference between them is φ. Using the Jones matrix, the pair of orthogonal linearly polarized beams can be represented by:
The Jones matrix of the orthogonally polarized reference beam and the test beam after passing through the quarter-wave plate (QWP) and linear polarizer can then be expressed as:
According to Eq. (4), it can be known that the transmitted light is linearly polarized, with its direction aligned with the transmission axis θ of the linear polarizer. The intensity I of this linearly polarized light is written as:
Based on this expression, it can be understood that the intensity distribution of interference fringes in an interferogram relates to the phase difference φ between two orthogonally polarized beams and the azimuth angle θ of the linear polarizer. A rotation of the linear polarizer changes its transmission direction, which affects the brightness of the interference fringes. In this case, the linear polarizer is equivalent to a phase shifter, and by rotating it to a certain azimuth angle, we can obtain a corresponding phase shift, which provides a theoretical basis for the polarization phase-shifting interferometry.
2.2 Phase shifting interferometry theory of the pixel micropolarizer camera
From Eq. (5) in the previous section, it is known that changing the angle of the linear polarizer can achieve a phase-shifting function. The pixel micropolarizer camera is designed based on this principle [20].
The principle of the pixel micropolarizer camera is shown in Fig. 1. A polarizing interferometer generates two orthogonal linearly polarized wavefronts, the reference wavefront R and the test wavefront T. The quarter-wave plate (QWP) converts the linearly polarized wavefronts R and T into left-handed and right-handed circularly polarized lights, respectively [21]. These circularly polarized wavefronts then interfere after passing through the micropolarizer arrays. The pixel micropolarizer camera consists of a finite number of superpixels, each containing four subpixels corresponding to the four polarization directions. The light intensity data captured by the detector array can be resolved into four subarrays, each corresponding to a different phase-shifting interferogram. The directional angles θ of the four sub-arrays are 0, π/4, π/2, and 3π/4, respectively. According to the Eq. (5), the corresponding introduced phase shifts are 2θ, which are 0, π/2, π, and 3π/2, respectively.
The pixel micropolarizer camera utilizes a four-step phase shifting method. The intensity of the four phase-shifting interferograms incident on the detector can be derived from Eq. (5), that is, I0, I45, I90, I135,
The detector array captures the interference intensity distribution, corresponding to the phase difference φ between the R and T wavefronts. However, in practical polarization interference measurement systems, the extinction ratio factors of the micropolarizer lead to a certain deviation between the detected intensity values by the pixel micropolarizer camera and the theoretical intensity values obtained from the formula. This deviation is directly incorporated into the phase calculation formula, which creates inaccurate measurement results. Thus, testing the pixel micropolarizer camera’s extinction ratio and analyzing its impact on phase calculation is necessary. The following section provides a theoretical analysis of the impact of the extinction ratio on the phase calculation errors.
3. Analysis of errors introduced by extinction ratios
3.1 Definition of polarization extinction ratio
Light is a transverse wave whose energy is distributed in the cross-section perpendicular to the direction of propagation. The characterization of this energy distribution is described by the polarization of the light. The latter refers to the bias of its energy distribution during its propagation in a certain direction. The polarization extinction ratio refers to the ratio between the two orthogonal polarized components resolved along the polarization state direction, which is an important parameter used to measure the performance of a linear polarizer. The higher the extinction ratio, the stronger the ability to convert input light into linearly polarized light. The extinction ratio of the linear polarizer can be defined as a fixed intensity and an adjustable direction of linearly polarized light passing through the linear polarizer. When the polarization direction of the incident light is parallel or perpendicular to the polarization axis of the polarizer, the transmitted intensity is q or r, respectively [22]. The extinction ratio, denoted as ρ, is the ratio between q and r,
Theoretically, when the incoming polarized light passes through the ideal linear polarizer, the transmittance of the incoming polarized light parallel to the direction of the ideal linear polarizer is 100%. In contrast, the transmittance of the incoming polarized light perpendicular to the direction of the ideal linear polarizer is zero, which means that the extinction ratio of the ideal linear polarizer should be infinite. However, due to manufacturing defects, the extinction ratio of the polarizer reached this ideal state. The extinction ratio of a monolithic polarizer often exceeds 103, but it is difficult to achieve this level for pixelated micropolarizers due to manufacturing accuracy limitations. Below is an analysis of the impact of the extinction ratio on phase calculation.
3.2 Theoretical derivation of the errors introduced by extinction ratios
When the pixel micropolarizer camera is used for phase-shifting interferometry, the errors generated by the pixel micropolarizer camera itself belong to system errors. The extinction ratio of the pixelated micropolarizer is one of the systematic error sources of the pixel micropolarizer camera. The paper analyzes and deduces the formula for the influence of extinction ratio on phase calculation. Based on this, the error influence model for the extinction ratio is established.
When a beam of linearly polarized light I with phase φ is incident on the 0° pixel of the superpixel, the intensity decomposed in the 0° direction is I0, which is the theoretical value of light intensity in the 0° direction. Since the extinction ratio exists and the extinction ratio value is relatively small, its effect is not negligible. This means part of the light decomposed into the vertical direction I90 also enters the 0° pixel, which forms a crosstalk light intensity. A crosstalk intensity can cause a certain deviation between the actual intensity g0 and the theoretical intensity I0, introducing errors in the phase calculation. In a superpixel, the baseline crosstalk intensity of the 0° pixel is coincidentally equal to the theoretical intensity of the 90° pixel, which makes the analysis simpler. The following expressions are applied:
Next, according to the four-step polarization phase shifting Eq. (7) and actual light intensity (9), the actual phase φ’ is:
Substituting Eq. (6) and Eq. (9) into Eq. (10), the influence of the extinction ratio on the four-step phase shifting can be explained by the following expression:
3.3 Numerical calculation of the errors and simplified model
3.3.1 Numerical calculation of the errors introduced by extinction ratios
In the previous section, a mathematical model describing the effect of the extinction ratio on phase calculation was established. Next, this mathematical model is utilized to quantify the error value due to the extinction ratio.
First, as shown in Fig. 3(a), a single variable is analyzed; this means that the extinction ratio exists in only one direction. Subsequently, the simultaneous presence of extinction ratios in three directions is analyzed. There are four ways to choose three directions in four directions, as shown in Fig. 3(b). Then, the simultaneous presence of extinction ratios in four directions is examined. The error distribution of the simultaneous presence of extinction ratios in four directions is shown in Fig. 3(c). Finally, there are six ways to combine four directions in pairs, which is shown in Fig. 3(d). Table 1 shows the range of the error in different combinations.
Fig. 3. The error distributions of different combinations: (a) single ER variable; (b) the combinations of three ER variables; (c) the combinations of two ER variables; (d) the combinations of foue ER variables.
Table 1. Range of the error in different combinations
3.3.2 Simplified model of the maximum error introduced by extinction ratios
The previous section used a traversal method to obtain the range for the phase calculation errors. Considering the impact of analysis error sources, the focus is centered more on the maximum error value. This section conducts in-depth analysis and research on the maximum error. According to Eq. (11), the difference between the phase value of the measured light intensity and the phase value for the ideal light intensity is that the numerator 2sin(φ) adds one more term 2a, and the denominator 2cos(φ) adds one more term 2b,
The general model is shown in Fig. 4; the model discusses the impact of different values of 2a and 2b on the phase calculation in the cases of φ $\in$ (0, π/2), φ $\in$ (–π/2, 0), φ $\in$ (–π, –π/2) and φ $\in$ (π/2, π), respectively.
Fig. 4. Simplified model of maximum error: (a) the geometric model in φ $\in$ (–π, –π/2) and φ $\in$ (π/2, π); (b) the geometric model in φ $\in$ (0, π/2) and φ $\in$ (–π/2, 0).
As shown in Fig. 5, assuming the length of the hypotenuse of a right triangle to be one unit, the length of the vertical leg is sin(φ), and the length of the horizontal leg is cos(φ). Here, φ represents the desired phase. In the geometric model diagram, the red rectangular box depicts the possible values for the deviation quantities a and b. The angle formed by the yellow dashed lines represents the possible maximum value for the angular deviation, denoted as φ’. Since the values of a and b can be positive or negative, they are divided into four categories in the following, which correspond to the geometric models (a), (b), (c), and (d) in Fig. 5.
From the model in Fig. 6, it is easy to observe that the maximum possible values of the error are shown in Table 2. Hence, a conclusion can be drawn that the maximum error may be Δφ1max’ or Δφ2max’. Namely, the maximum value of the phase calculation error is obtained in φ $\in$ (0, π/2) for a = amin, b = bmax, or a = amax, b = bmin. Similar to the analysis method in the previous sub-section, the maximum value of the phase calculation error is obtained in φ $\in$ (–π/2,0) in a = amax, b = bmax, and a = amin, b = bmin.
Table 2. Error extremum corresponding to the values of a and b
In addition, we also analyze the max errors in φ $\in$ (π/2, π) and φ $\in$ (-π, -π/2). The results are consistent with the above analysis in φ $\in$ (0, π/2) and φ $\in$ (-π/2, 0).
From the above two sections, it can be observed that the maximum value of the error Δφmax of the extinction ratio on the phase calculation may occur for the four cases. As we know, the extinction ratios of the pixel micro polarizer cameras vary within a certain range, namely ρ0 $\in$ [ ρ0min, ρ0max], ρ45 $\in$ [ ρ45min, ρ45max], ρ90 $\in$ [ ρ90min, ρ90max], and ρ135 $\in$ [ ρ135min, ρ135max]. So, the results of the maximum value of a and b can be calculated as follows.
3.3.3 Simplified model simulation verification
The previous section categorized and analyzed the values of a and b when the error is maximum. Subsequently, the accuracy of the theoretical analysis is then validated through simulation.
From the previous experimental data, the extinction ratio data for different directions in the pixel micropolarizer cameras fluctuates within a certain range. The combination of different extinction ratios in the four directions corresponds to various errors in the phase calculation.
Taking a set of measured extinction as an example, as shown in Fig. 7, the blue curve shows the all phase calculation error of all pixels, while the four dashed lines correspond to the maximum error value obtained by the maximum error model. It is found that the errors of all phases vary within the maximum error range.
From the analysis in section 3.3.2, the maximum or minimum values of a and b occur at the combinations of the maximum or minimum values of the four extinction ratios. As a result, the calculation of errors is simplified from traversing all possible cases within a certain range to combining extreme values. Next, it can combine 16 groups, which form 16 curves, as shown in Fig. 8. All the errors fluctuating within the four curves are displayed in Fig. 9.
Comparing Table 3 and Table 4, the combinations of extinction ratios corresponding to the maximum or minimum errors are consistent.
Table 3. Error extremum corresponding to the combinations of extinction ratios
Table 4. Error extremum corresponding to the combinations of extinction ratios
4. Experiment and verification
4.1 Measurement of extinction ratio
In order to validate the impact of extinction ratio, we construct an optical system to measure the extinction ratios of a commercial polarization camera, outlined in Fig. 10. The experimental arrangement comprises an LED light source, an integrating sphere, a rotatable linear polarizer, polarization cameras, and a data processing server. During the experiment, the linear polarizer is controlled to rotate while the polarization camera captures intensity images simultaneously. So, we can acquire the curves of the variation of light intensity over time for each pixel, accordingly capturing the maximum intensity and the minimum intensity for each pixel, thereby obtaining the extinction ratios of all pixels.
The measurement results of the extinction ratio are presented in Fig. 11, where (a), (b), (c), and (d) correspond to the distribution in four directions, respectively. According to the test data, the distribution of the extinction ratio varies in each direction, which indicates that there are certain differences in the manufacturing of polarization camera micropolarizers.
Fig. 11. The extinction ratio of polarization camera: (a) the extinction ratio distribution in 0° direction; (b) the extinction ratio distribution in 45° direction; (c) the extinction ratio distribution in 90° direction; (d) the extinction ratio distribution in 135° direction.
4.2 Correction model of extinction ratio error
Based on the above research content, the paper proposes an error correction model based on actual light intensity. In the Eq. (13), G0 is the corrected light intensity value, g0 is the measured light intensity, and I0 is the theoretical light intensity. The above-modified model indicates that the measuring error of light intensity is from I90/ρ0 drops to I90/(ρ0*ρ90). According to experience, the range of extinction ratio is 50-400, which means that the error in light intensity decreases by tens or even hundreds of times. Similarly, the corrected G45, G90, and G135 can also be obtained.
By correcting the light intensity introduced by the extinction ratio, the accuracy of phase calculation can be improved. The phase errors before and after correction corresponding to the four maximum error extinction ratio combinations Δφ1max’’, Δφ2max’’, Δφ1max’ and Δφ2max’’ are shown in Fig. 12. Then it can be seen that the correction mode can significantly reduce the impact of extinction ratio.
Fig. 12. The simulation of phase calculation error before correction and after correction: (a) the changes in errors before and after correction correction corresponding extinction ratio combination of [ρ0min ρ45min ρ90max ρ135max]; (b) the changes in errors before and after correction correction corresponding extinction ratio combination of [ρ0max ρ45max ρ90min ρ135min]; (c) the changes in errors before and after correction corresponding extinction ratio combination of [ρ0max ρ45min ρ90min ρ135max]; (d) the changes in errors before and after correction corresponding extinction ratio combination of [ρ0min ρ45max ρ90max ρ135min].
4.3 Simulation of extinction ratio error and correction model
In addition, we carry out the phase calculation error simulation on the phase distribution of interferogram from the interferometer. Figure 13 illustrates the results, where (a) displays the original phase of the known 1# interferogram, (b) depicts the calculated phase with extinction ratio, (c) represents the phase calculation error caused by extinction ratio, and (d) shows the error distribution after correction. The simulation experiment confirms that the presence of extinction ratio has an adverse impact on the accuracy of phase calculation. It is worth mentioning that by utilizing a correction model, the influence of extinction ratio can be mitigated, thereby improving the measurement accuracy of the entire optical system. Likewise, the paper conducts a simulation analysis and verification on another set of 2# interferogram in Fig. 14, demonstrating the effect of extinction ratio and the importance of applying the correction model.
Fig. 13. The simulation of the 1# interferogram: (a) the original phase ; (b) the calculated phase with extinction ratio; (c) the phase calculation error introduced by extinction ratio; (d) the error distribution after correction.
Fig. 14. The simulation of the 2# interferogram: (a) the original phase ; (b) the calculated phase with extinction ratio; (c) the phase calculation error introduced by extinction ratio; (d) the error distribution after correction.
5. Conclusion
This article derives a theoretical expression for the error introduced by the extinction ratio, providing a theoretical basis for the quantitative analysis of the impact of extinction ratio. On above foundation, the paper derives an error analysis model for the extinction ratio, providing an effective means for the evaluation of the pixel micropolarizer camera. In addition, the text proposes correction model of extinction ratio on phase calculation, offering the effective methods to reduce phase calculation errors caused by extinction ratio. The research content of the thesis is novel which can complement and enrich the field of polarization. Meanwhile the work in the paper is practical, supplying a quantitative basis for the evaluation of polarization cameras. It can also be used as a practical method for evaluating system errors induced by the extinction ratio in commercial pixel micropolarizer cameras. This provides a good basis for the selection of pixel micropolarizer cameras.
Funding
National Key Research and Development Program of China (2021YFC2202403-2); National Natural Science Foundation of China (62375019).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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