We applied the proposed calibration method to a standard sample of air (under stable measuring conditions with a temperature of 23.1 °C and humidity of 51%), based on the configuration illustrated in
Figure 1, to demonstrate the influence of birefringent parasitic polarization on the Mueller matrix elements of the objective lens in the imaging section and to validate the adaptability of the proposed calibration procedure when using the improved error-propagation model. To examine the general performance, we visually examined and quantified the measurement results of the elements
,
,
, and
, together with the polarization parameter
derived from these elements, as expressed in Equation (10) [
24] (which characterizes the amplitude value of birefringence), for an air sample and a quarter-wave plate sample as the standard samples and a Daphnia organism sample with complex internal structure.
3.1. Calibration Application on a Standard Sample of Air
As mentioned in the introduction section, for the polarizing section, it is sufficient to use the conventional calibration method based on the discrete Fourier coefficients [
13,
15,
16] illustrated in
Appendix A. However, in the case of important polarization parameters such as birefringence, the imaging section should be considered for accurate measurement results. We performed computer simulations and experimental measurements on a standard sample of air to reveal the cause of the errors arising from the polarizing section and the imaging section and their propagating effects on the Mueller matrix measurement results. The robustness of the proposed calibration methods was investigated in comparison to the conventional calibration method. It can be seen from Equation (9) that, assuming an optimized combination of controllable small errors when measuring under stable system conditions, the errors of the measurement results caused by different error sources are independent of each other, i.e.,
is independent of
(
); therefore, we can separately calibrate the polarizing section and imaging section.
When calibrated using the conventional calibration method, the maximum error for the absolute values of the Mueller matrix elements of the measurement configuration without an objective lens could be reduced to 0.01. With the same predefined values of the PSG and PSA (
,
,
,
, and
set to 0°, 0°, 0°, 90°, and 90°, respectively), we first analyzed the parasitic polarization effects in the objective lenses of the imaging section using the conventional method. The Mueller matrix of the objective lens is discussed as follows. This matrix was set to
, identical to the Mueller matrix of air, and this indicated that there were no effects caused by the objective lens. In this case, we obtained the error magnification coefficient matrices of the five system parameters
,
,
,
, and
, as expressed in Equation (11):
Equation (11) can be used with Equation (9) to obtain the errors in the Mueller matrix for air, which are caused by the polarizing elements. For example, the orientation error of the quarter-wave plate in the PSG () influences the Mueller matrix elements , , and , and the orientation error of the quarter-wave plate in the PSA () influences the Mueller matrix elements , , and . However, it also follows from Equation (11) that the error magnification coefficients that may influence the Mueller matrix elements , , , , , , , and are zero, implying that the current system parameters in the polarizing section have no effect on these matrix elements.
Generally, the Mueller matrix should be analyzed in normalized form, and the influence on the matrix element
is therefore ignored. The error sources for the other matrix elements need to be investigated. For the conventional calibration method, as described in
Table A1, the Fourier coefficients
,
,
,
,
,
,
,
,
,
,
, and
are combinations of one or more terms from the fourth row or column of the Mueller matrix. However, the system parameters in the polarizing section are calculated in terms of
,
,
,
,
,
,
, and
, while the system parameter
is calculated in terms of
. Therefore, the fourth row and column of the Mueller matrix have been regarded as redundant and ignored. This, however, results in information loss and calibration inaccuracy. From Equation (11), it can be observed that the matrix elements in the fourth row and fourth column of the error magnification coefficient matrices of the polarizing section are all zeros. To discuss whether these results were the outcome of some particular default values, we also assigned values other than the default values to the system parameters in the polarizing section. The results show that the calculated error magnification coefficients are stable, and the elements in the fourth row and column are always near zero when the five system parameters simultaneously deviate from the default; this indicates that the errors of the measurement system that propagate to these elements cannot be calibrated using the conventional method. One simulation result for the error transform coefficient matrix is shown in
Figure 3, where the horizontal axis of each subfigure corresponds to the five parameters in the polarizing section, the vertical axis corresponds to the simulation times, and different colors correspond to the values of the matrix elements. As the maximum absolute value of any element of the error transform coefficient matrices in Equation (11) is 4, random values in the range of −0.05 to 0.05 were assigned to every system parameter (this range was used because it results in a maximum absolute error of 0.2 before the calibration and is available for the visualization of the results, according to the calculated values determined using Equation (9)). Thus, the error magnification coefficient matrix of the imaging section is supplemented, thereby reflecting the error induced by the objective lens, and the calculated results are presented by Equation (12):
The calculated results for rotation angle
and ellipsometric angles
and
are shown in Equation (12). Because all the elements in
are zero, Equation (12) indicates that the rotation angle
has no effect at any point in time. To test whether the zero results were caused by particular default values, deviating values were once again assigned to the parameters, and the Mueller matrix of the objective lens was reset to
, although 0.01 is the recommended value according to our experimental results. The error magnification coefficients of the polarizing section were recalculated to ensure accurate calibration. The results, as presented in
Table 1(a),
Table 1(b), and
Figure 4a, show that the maximum error could usually be reduced to 0.01 or less. Elements
,
,
, and
, even after the polarizing section was calibrated, constitute an exception; therefore, these elements must have been mainly influenced by the imaging section, instead of the polarizing one.
Next, the proposed calibration method was applied. Considering the imaging section as well as the polarizing section, the Mueller matrix of air was recalculated; the results are presented in
Table 1(c) and
Figure 4b, which show that the maximum errors were reduced to less than 0.01 for all the elements. The calibrated system parameters were
,
,
,
,
,
,
, and
.
To test the robustness of the proposed method, the system parameters and were also set to several different values around their default values. These two system parameters were singled out because they can be precisely controlled and constrained in a particular range to avoid a large condition number of the instrument matrix. Thus, the values of the system parameters and were set to −15°, −10°, −5°, 0°, 5°, 10°, and 15°.
Figure 5 and
Figure 6 show the calibration results for the air sample when
and
deviate from their default values;
Figure 5a and
Figure 6a correspond to
, while
Figure 5b and
Figure 6b correspond to
. The vertical axis shows the difference between the calibration and assumed values. The experimental results clearly show that the calibration result of
determined using the conventional calibration method is unstable (
Figure 5b), but the calibration result with the proposed method (
Figure 6b) fluctuates only slightly, and the standard deviations of
for the conventional and proposed methods are 1.6772 and 0.0971, respectively. In addition, the calibration results for all the other parameters in the polarizing section have good consistency, indicating that the proposed method is highly robust.
3.2. General Applicability
Measured samples often have highly variable content and quality in consideration of the polarization properties. To examine the general applicability of the proposed calibration method, the measurement results of samples of air were first examined; then, a quarter-wave plate with a distinct birefringence effect was selected; and finally, Daphnia, a kind of plankton that demonstrates hierarchical structure details in its dying process, was investigated. The polarization parameter , which characterizes the amplitude value of birefringence and is derived from elements , , , and , was considered to discuss the calibration results of the measurement. In addition, the values of the system parameters and were set to −15°, −10°, −5°, 0°, 5°, 10°, and 15° in the measurement of the air and the quarter-wave plate samples, while and were both set to 0° for the daphnia sample to identify the improvement on its hierarchical structure details.
For the air sample, the measurement results of
(with a theoretical value of 0) shown in
Figure 7a are stable when
or
deviates from the default and when the average values are 0.0461 and 0.0455, respectively, meaning that
is independent of
and
and that the errors caused by the imaging section are constant. Thus, after calibrating the imaging section, the order of magnitude of the error of
is reduced to 0.001, as depicted in
Figure 7b.
Figure 7 depicts not only the difference in the measurement results of
for the conventional and proposed calibration methods but also the stability of the errors caused by the imaging section. In addition, the measurement results for the quarter-wave plate sample (
Figure 8b) are closer to the theoretical value of
(with an average error less than 0.001) and have a stable trend, indicating more stable and accurate test results than those illustrated in
Figure 8a.
The calibration result of parameter
in the polarizing section fluctuates considerably (shown in
Figure 5b), while the results in
Figure 7a are constantly stable, which suggests that the birefringence of air is mainly influenced by the imaging section. For a measurement sample that itself exhibits birefringence, such as the quarter-wave plate (with a theoretical value of 0.5 for
), the measurement tendencies of
as shown in
Figure 8a,b are similar to those of
in
Figure 5 and
Figure 6, respectively. In other words, the calibration results of parameter
affect the calculation of the birefringence parameter,
, of materials with strong birefringence.
Finally, we compared the calculation accuracy and the clarity of the hierarchical details of the conventional method and the proposed method when imaging plankton samples for the
parameter. A micrograph of the Daphnia sample is shown in
Figure 9, and the measurement results of this sample are shown in
Figure 10.
The Daphnia occupies the middle area, so the pixels at the edge correspond to the air. The
values of the pixels at the edge in
Figure 10a are larger than those in
Figure 10b; on the other hand, the pixels corresponding to the Daphnia also have large errors, and the birefringence features at these pixels are smaller than the errors caused by the imaging section, thereby leading to an erroneous judgment of the birefringence of the sample. For instance, the values in the red dotted box approximately range from 0.15 to 0.23 in
Figure 10c and from 0.29 to 0.47 in
Figure 10d, and the values of the orange dotted box area approximately range from 0.15 to 0.36 in
Figure 10e but from 0.03 to 0.21 in
Figure 10f. The comparison results shown in
Figure 10 are similar to those obtained for the air and quarter-wave plates, i.e., when the birefringence of the sample is weak, the calculated value of
is larger than its theoretical value, but
is smaller than its theoretical value for the sample with strong birefringence. Then, the polarization features of Daphnia were recovered, as shown in
Figure 10b; this is unlike the results presented in
Figure 10a, where the experimental results were severely distorted by the birefringence of the imaging section. Thus, the proposed method possesses considerable advantages in terms of improving the estimation accuracy of the birefringence caused by the internal structures of the Daphnia and retaining more details of the hierarchical structure of the tissue.