Characterization of anisotropy of the porous anodic alumina by the Mueller matrix imaging method

Chuhui Wang; Chuhui Wang; Peiwu Qin; Peiwu Qin; Donghong Lv; Jiachen Wan; Shuqing Sun; Hui Ma; Hui Ma

doi:10.1364/OE.380070

1. Introduction

Mueller Matrix Imaging (MMI) has been has been a powerful tool in material science and medical diagnosis for quantitative characterization of complex samples [1–6]. The sixteen Mueller matrix elements, which contain very rich information on the microstructural features and the optical properties of the sample, can be used individually as polarization para-meters, and these elements are sensitive to the orientation angles of the sample [7]. New polarization parameters can also be derived from the Mueller matrix by different decomposition [8] or transformation [9] techniques to characterize more explicitly the optical anisotropy such as diattenuation and retardance or the microstructural features such as the alignment of fibrous scatters.

Porous anodic alumina (PAA) is a kind of special nano-material and it is often used as a template to fabricate the nanostructure arrays or microfilters for its controllable aspect ratio, diameter and length of the pores [10–13]. PAA is also a type of photonic crystal [14]. Its optical properties can be altered by varying the geometric features of the nano-porous structure such as pore diameter, porosity and inter pore distance [15]. The common characterization methods of PAA include scanning electron microscope (SEM), X-Ray diffraction, and Rutherford back scattering [16]. However, these methods can’t help to obtain the anisotropy of PAA, which is useful to learn more about the photonic crystal properties of the sample [17]. The PAA films with thickness of hundreds of nanometers to tens of micrometers can be fabricated by two-step anodization of aluminum in oxalic acid [18–21].

In this study, we use the MMI method to examine the anisotropic optical properties of PAA. Since alumina crystals are birefringent, the PAA can be regarded as the combination of many thin wave-plates at different orientations with highly aligned and patterned pores. To separate the different contributions to optical anisotropy by the alumina crystal sheets and the pores, we fabricated a PAA sample with two different pore diameters and take the Mueller matrix images with different sample orientations. It is found that the pores can contribute to the anisotropy (mainly birefringence) of the sample and this effect increases with the inclination angle of the sample. We propose a symmetrical rotation measurements method based on MMI helping to determine the optical axis orientation of the uniaxial crystal as well as the average refractive index (RI). The results show that the pores effect can also be indicated by these two parameters.

2. Material and methods

PAA samples with single pore diameter and two different pore diameters were fabricated in the laboratory using secondary anodization method, which is an electrochemical method and Fig. 1 shows the setup.

Fig. 1. The diagram of the secondary anodization method.

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The scanning electron microscopy (SEM) views of the section and surface of PAA are shown as follows:

Fig. 2. The SEM images of the surface and section view of the PAA sample with single diameter.

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In Fig. 2, we can see that there are many pores that arrange in regular hexagonal pattern on the surface of the sample and the pores are better vertical to the base.

Dual plates rotation method [22] is conducted to measure the Mueller matrix of the sample. The diagram of the optical path and the laboratory coordinate system (LCS) are shown in Fig. 3:

Fig. 3. The diagram of the dual-plates rotation method.

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In the experiments, we use a LED with center wavelength of 633nm as light source. P1 and P2 in Fig. 3 are two linear polarizer (Thorlabs), Q1 and Q2 are two λ/4 waveplates (633nm, Thorlabs), L1 and L2 are two lenses that L1 is for collimation and L2 is for collecting the light and forming an image recorded by the CCD (Q-imaging Retiga Exi, 12-bit). The light passes through P1 and Q1 to generate 6 different polarization states, and then exits from the sample, passing through the analyzer that consists of Q2 and P2, and finally is collected by the CCD camera. The second retarder Q2 is rotated at a rate of five times that of the Q1, which generates a 5:1 rotation ratio to enable the Fourier coefficients to be inverted to give the Mueller matrix elements. The errors due to the retarder and polarizers are compensated by measuring the Mueller matrices of the air. The maximum errors for all the Mueller matrix elements are less than 2%.

3. Polarization measurements of PAA

3.1 Normal incidence experiments of the PAA with single pore parameters

To understand the polarization properties of the PAA, we firstly measured the Mueller matrix of the sample with single pore diameter at normal incidence:

Fig. 4. The Mueller matrix of PAA with single pore diameter under normal incidence. There are 16 elements in the figure. To illustrate the sample signal, we mark it with a red rectangle in M₄₂. It can be seen that the lower right 3*3 elements are stronger than the first row and first column.

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According to the Ref. [23], M₁₁ is invariant under any rotation and retarder transformation, so all the Mueller matrixes in this paper are usually normalized by m11. From Fig. 4, we find the signals of the elements at the bottom right of the matrix are obvious while the elements at the first line and first row are weak. This indicates that the anisotropy of PAA is mainly reflected on birefringence and the dichroism is not capital, for that birefringence is reflected by the bottom right 3*3 elements and the first line and the first row give expression to dichroism. We can see that there are many small blocks on the sample. According to the previous study [7], we concluded that the optical orientations of these blocks in the refraction index ellipsoid are different. Based on the above analysis, the Mueller matrix of PAA can be analyzed from the birefringence effect, including the phase delay caused by birefringence and the orientation angle of the phase delay.

3.2 Rotation experiments of the PAA with two different pore diameters

In order to observe the effect of pores on the PAA, the Mueller matrices of the sample with two different pore diameters were measured at three angles: normal incidence, the incident angle is 9 degrees and 32 degrees respectively (the incident angles were changed by tilting the sample.). The corresponding measured Mueller matrixes are:

Fig. 5. The Mueller matrixes of PAA sample with two different diameters pores at different incidence angles. The order from left to right is: (a) normal incidence, (b) incident angle 9 degrees, (c) incident angle 32 degrees. The Mueller matrixes of the black point denoted in M₂₃ are shown in Table 1.

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To quantitatively check the variation of the Mueller matrix with the changes of the incident angle, we take the Mueller matrix of one random point on the sample presented in Table 1. The position of the point is the black marker shown in the image M₂₄ elements in Fig. 5.

Table 1. The Mueller matrixes under different incident angles of the selected point

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In Fig. 5 and the Table 1, it can be seen that the elements signals in the first row and first column of the three matrices are not obvious which is similar to the Fig. 4. As for the lower right 3*3 elements which reflect the birefringence properties of the sample, the signals of the sample are strong. There is a demarcation line between the two areas with different pore diameters on the surface of the sample. The line is not clear at normal incidence and the inclination angle is small (9 degrees is an example). When the inclination angle is up to 32 degrees, the demarcation line can be clearly seen from some matrix elements especially like M₂₂, M₂₄, M₄₂ and M₄₄ in the lower right, and there are obvious differences on both sides. This phenomenon not only reveal that the anisotropy of the PAA mainly is birefringence other than dichroism, but also show that the pores can affect it. Under normal incidence, the pores and the light are colinear, so their effects on the anisotropy are not obvious. When the PAA is titled, an angle appears between the pores’ orientation and the light so that their anisotropy becomes distinct. The anisotropy of pores becomes stronger with the increase of the angle and it is affected by the pore parameters like diameter at the same time.

Next, we will discuss the birefringence of PAA with the sample of single pore parameters.

4. Study on the birefringence of PAA

4.1 Analysis based on the Mueller matrix of the retarder

From above analyzation, we consider birefringence as the main anisotropy of the PAA sample and the dichroism is not. Next, we will model and analyze the PAA according to crystal optics knowledge and Mueller matrix measurement method.

The Mueller matrix of a pure phase retarder like a wave-plate (WP) is as follows:

(1)$${{\textbf M}_{LR}}({\delta ,\varphi } )= \left( {\begin{array}{cccc} 1&0&0&0\\ 0&{{{\cos }^2}2\varphi + {{\sin }^2}2\varphi \cos \delta }&{\cos 2\varphi \sin 2\varphi ({1 - \cos \delta } )}&{ - \sin 2\varphi \sin \delta }\\ 0&{\cos 2\varphi \sin 2\varphi ({1 - \cos \delta } )}&{{{\sin }^2}2\varphi + {{\cos }^2}2\varphi \cos \delta }&{\cos 2\varphi \sin \delta }\\ 0&{\sin 2\varphi \sin \delta }&{ - \cos 2\varphi \sin \delta }&{\cos \delta } \end{array}} \right).$$

In Eq. (1), φ is the azimuth angle of the anisotropy of linear phase delay, that is, the angle between the fast axis direction and the X axis, δ is the measured phase delay of the linear phase retarder. Among the two parameters, for a uniaxial crystal sheet with the optical axis lying on the surface, δ has its own calculation formula which can be expressed as Eq. (2):

(2)$${\delta _{{\mathop{\rm int}} }} = \frac{{2\pi d({n_e} - {n_o})}}{\lambda }.$$

Here, we use δ_int as the intrinsic retardance of the uniaxial crystal sheet. d is the thickness of the sample, which represents the effective optical path in the sample at normal incidence. n_e and n_o represent the principal axis refractive Indexes in the RI ellipsoid of the Sample, that is, the effective refractive indexes of the ordinary light and extraordinary light which emerge in the sample when light is incident on the surface of a uniaxial crystal sheet. Use δ_mea as the measured retardance, it can be expressed as:

(3)$${\delta _{mea}} = \frac{{2\pi l(n{^{\prime}_e} - {n_o})}}{\lambda }.$$

Among which l represents the effective optical path in the sample, which is not equal to the thickness of the sample for the oblique incidence. Figure 6 describes the geometric relationship between d and l:

Fig. 6. Effective path change in crystal samples caused by oblique incidence. (a). Normal incidence; (b). Oblique incidence.

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n'_e and n_o represent the effective refractive indexes of the ordinary light and extraordinary light. δ_mea can be expressed as [24]:

(4)$${\delta _{\textrm{mea}}} \approx \frac{{2\pi l({n_e} - {n_o})}}{\lambda }{\sin ^2}\theta \approx \frac{{2\pi d({n_e} - {n_o})}}{{\lambda \cos \gamma }}{\sin ^2}\theta = {\delta _{{\mathop{\rm int}} }}\frac{{{{\sin }^2}\theta }}{{\cos \gamma }}.$$

Here, θ represents the angle between the wave normal direction of refracted light in crystal and the optical axis direction of the uniaxial crystal. γ corresponds to the average of γ₁ and γ₂ in Fig. 6(b). Since for the uniaxial crystals, the refracted light of o-ray is in the same direction as its wave normal, while the e-ray is not and the angle between these two directions is called Walk-Off angle α which relates to the principal refractive indexes of the uniaxial crystals and the angle θ is decided by the following formula:

(5)$$\tan \alpha = \frac{{(1 - \frac{{n_o^2}}{{n_e^2}})\tan \theta }}{{(1 + \frac{{n_o^2}}{{n_e^2}}{{\tan }^2}\theta )}}.$$

In the situation that tan θ=n_e/n_o, α has its maximum. Through calculation, the maximum Walk- Off angles of the sample used in this paper are less than 0.5 degrees. Also considering the thickness of the sample (tens of microns and 1mm), in the calculation, we use the directions of the refracted light which can be calculated by the law of refraction as the wave normal direction. At the same time, since the values of n_o and n_e are close to each other, the average of γ₁ and γ₂ is taken as the refraction angle.

Next, we test the errors of Eq. (4) with polarization measurements of two standard true zero order wave-plates (wps), and design experiments to analyze the effect of pores on the anisotropy of PAA according to this formula.

To quantitatively analyze the error between Eq. (4) and the reality, we take a true zero order λ/8 wp (n_o = 1.5350, n_e = 1.5438, λ = 633nm) and a true zero order λ/4 wp (n_o = 1.544, n_e = 1.553, λ = 633nm) as standard samples to conduct a series of polarization measurements. During the process, the incident light always follows the Z-axis direction of the LCS. We rotated the wp along the Y-axis of the LCS and the axis of wp itself to change the orientation of the sample in LCS, and measured the corresponding Mueller matrix. The rotation schematic is shown as Fig. 7. Six incident angles are taken from normal incidence to 50 degrees at intervals of 10 degrees. At each incident angle, 10 azimuths are rotated around the wp's axis at intervals of 10 degrees:

Fig. 7. Rotation schematic in LCS. ω is the incident angle. Incident light is along Z-axis. (a). Change ω through rotating wp along Y-axis; (b). Under every ω, change the orientation of the wp through rotating it along its own axis (the red dotted line).

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Find corresponding parameters in each orientation and substitute them to the Eq. (4) to obtain δ_mea, then further substitute δ_mea and the value of φ into Eq. (1), we can calculate the theoretical Mueller matrix of the samples under different orientations. To quantitatively assess the errors between the experimental and corresponding theoretical data, we calculated the differences and show the variation of the differences with the incident angles of the true zero order λ/8 wp and λ/4 wp in Fig. 8:

Fig. 8. Differences between experimental data and theory of Mueller matrix of wps under different incident angles.

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Wps are non-dichroic in theory as a kind of standard phase retarders, but in practice, dichroism occurs on the surface of the device during experiments. Here, we mainly analyze the lower right 3*3 elements which reflect the birefringence of wp. It can be seen that the modulus of all the errors are under 0.02, which is consistent with the error of the Mueller matrix measurement system.

We conclude that the errors of the theory belong to random errors and there may be three main sources for them: (1) In the process of obtaining the Eq. (4), reasonable approximations are made according to the actual situation; (2) Our polarization measurement system allows errors of less than 2% in calibration. Stated thus, we conclude that the Eq. (4) is reliable to be used to analyze the birefringence of the PAA.

4.2 The symmetric rotation measurements based on the retardance of uniaxial crystals

A symmetrical rotation measurements method was designed to obtain the average RI of the PAA and the orientation of the optic axis of every single area of the sample which characterizes the orientation of the area in the RI ellipsoid based on the Eq. (1) to (4).

Before introducing the method, another important approximation needs to be illustrated. Because the thin sheets of crystals we refer to, such as wps or PAA, are very thin, we can approximately assume that the wave normal direction coincides with the light direction when light propagates in the crystal. This is essential for that on this basis, the fast and slow axis direction of the crystal can be linked with the optical axis direction. Also, the θ in Eq. (4) represents the angle between the wave normal direction and the optical axis direction. The normal direction of light waves here can also be replaced by the direction of refracted light, since the latter is more easily accessible by the law of refraction.

For a uniform uniaxial crystal sheets with known crystal type, that is, positive or negative uniaxial crystals, but unknown RI and optical axis orientation, we can model it in three-dimensional space coordinate system. The arbitrary optical axis orientation can be represented by two angles:

Fig. 9. The optical axis of uniaxial crystal sheet in three-dimensional space coordinate system.

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Figure 9 shows the case of light normally incident on the surface of a uniaxial crystal sheet in LCS. The sample is located on the XOY plane and the incident light always follows the Z-axis direction. The optical axis of the sample can be expressed as:

(6)$$\widehat {{l_0}} = (\sin {\theta _0}\cos {\varphi _0},\sin {\theta _0}\sin {\varphi _0},\cos {\theta _0}).$$

Vectors representing directions are an important tool for our subsequent angle calculation. In the laboratory coordinate system shown in the Fig. 9, the operations of the symmetric rotation measurements include:

(1) Place the sample on the XOY plane, and the light is normally incident on the surface of the sample. Measure the Mueller matrix of the sample with the forward Mueller matrix measuring Device.
(2) Rotate the sample around the Y-axis from Z to X at an angle ω. Now the incident angle is ω. Measure the Mueller matrix of the sample.
(3) Rotate the sample around the Y-axis from X to Z at an angle ω. Now the incident angle is -ω. Measure the Mueller matrix of the sample.

The phase delay of the sample in three cases which are represented by δv, δ₁, δ₂ respectively can be calculated by MMD method (the Mueller matrix are all physical Mueller matrix). According to the Eq. (4), these 3 parameters can also be expressed as:

(7)$$\left\{ {\begin{array}{c} {{\delta_v} = {\delta_0}{{\sin }^2}{\theta_0}}\\ {{\delta_1} = {\delta_0}\frac{{{{\sin }^2}{\theta_1}}}{{\cos \gamma }}}\\ {{\delta_2} = {\delta_0}\frac{{{{\sin }^2}{\theta_2}}}{{\cos \gamma }}} \end{array}} \right..$$

γ is the refraction angle in step (2) and (3). According to the law of refraction, the expression of γ is:

(8)$$\cos \gamma = \sqrt {1 - \frac{{{{\sin }^2}\omega }}{{{n^2}}}} .$$

δ₀ is the intrinsic phase delay of the sample. θ₀, θ₁, θ₂ are the angle between the optical axis and the refraction ray. These angles can be calculated by the vector inner products that are expressions containing the angle parameters ω, θ₀ and φ₀ among which the last 2 are unknowns. Now 2 equations can be listed:

(9)$$\left\{ {\begin{array}{c} {\frac{{{\delta_1}}}{{{\delta_2}}} = \frac{{{{\sin }^2}{\theta_1}}}{{{{\sin }^2}{\theta_2}}}}\\ {\frac{{{\delta_\textrm{v}}}}{{{\delta_1}}} = \frac{{{{\sin }^2}{\theta_0}}}{{{{\sin }^2}{\theta_1}}}\sqrt {1 - \frac{{{{\sin }^2}\omega }}{{{n^2}}}} } \end{array}} \right.$$

Since there is another unknown n that is the average RI of the sample birefringence, another equation is need. The last one is based on the Eq. (1) that the Mueller matrix of the phase retarder. The unknown φ₀ can be obtained using the values of M₂₄, M₃₄, M₄₂ and M₄₃ of the Mueller matrix in Eq. (1):

(10)$$\tan 2\varphi ={-} \frac{{{M_{24}}}}{{{M_{34}}}} ={-} \frac{{{M_{42}}}}{{{M_{43}}}}.$$

From this equation we can calculate the value of the angle between the fast axis and X-axis which is related to the angle between the projection of the optical axis on the XOY plane and X-axis φ₀. The relationship between φ and φ₀ differs with the types of uniaxial crystals. On the basis that the light direction of refracted light coincides with the wave normal direction when light propagates in the sample, it can be inferred that in the case which the sample is located on the XOY plane, the projection of the optical axis on XOY plane coincide with the direction of the fast axis of the sample for the negative uniaxial crystal while for the positive uniaxial crystal these two directions are perpendicular to each other. Combining these analyzations, the model that can be used to get the average RI and optical axis orientation that characterize the anisotropy of the uniaxial crystal sheet is constructed.

For the negative uniaxial crystal:

(11)$$ \left\{\begin{array}{c} {\frac{\delta_{1}}{\delta_{2}}=\frac{\sin ^{2} \theta_{1}}{\sin ^{2} \theta_{2}}} \\ {\frac{\delta_{v}}{\delta_{1}}=\frac{\sin ^{2} \theta_{0}}{\sin ^{2} \theta_{1}} \sqrt{1-\frac{\sin ^{2} \omega}{n^{2}}}} \\ {\varphi_{0}=\frac{1}{2} \arctan \left(-\frac{M_{24}}{M_{34}}\right)} \end{array}\right. $$

For the positive uniaxial crystal:

(12)$$ \left\{\begin{array}{c} {\frac{\delta_{1}}{\delta_{2}}=\frac{\sin ^{2} \theta_{1}}{\sin ^{2} \theta_{2}}} \\ {\frac{\delta_{\mathrm{v}}}{\delta_{1}}=\frac{\sin ^{2} \theta_{0}}{\sin ^{2} \theta_{1}} \sqrt{1-\frac{\sin ^{2} \omega}{n^{2}}}} \\ {\varphi_{0}=\frac{1}{2} \arctan \left(-\frac{M_{24}}{M_{34}}\right) \pm \frac{\pi}{2}} \end{array}\right. $$

Here we should emphasize that this model is also suitable for the characterization of the birefringence of biological tissues and other samples with birefringent properties. Under these circumstances, φ₀ should be calculated by MMD method other than the numerical calculation using Mueller matrix elements M₂₄, M₃₄, M₄₂, M₄₃ directly.

The validation experiments use the zero-order λ/8 wp and the zero-order λ/4 wp mentioned earlier as samples and take ω equals to 10 degrees. In addition, we changed φ₀ 10 times by rotating wp around the axis of itself for each wp and do one group of symmetrical rotation measurements under every orientation and analyzed the data using the model (Eq. (12)). The results of θ₀ for the two wps are shown as follows:

Table 2. The results of θ₀ of the true zero order wps

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For the values of θ₀ calculated using the data under different orientations shown in the table, it can be seen that they all around 90 degrees and the maximum error is less than 2%. On the sources of errors, first, a point-to-point calculation method is conducted to average the final results of every measurement. This operation is based on the assumption that the wp is an absolutely uniform sample. But during the experiments we find that there are slight differences in the results when different regions are taken for calculation, so the sample itself is not absolutely uniform, which is reasonable in reality. This may cause some errors. Second, the error range between 1% and 2% during the calibration process of the optical system for measuring Mueller matrix is permissible. In summary, the error less than 2% of the results calculated by symmetric rotation measurements model is acceptable.

5. Results

Now use the model to analyze the PAA. From part 3 the conclusion that the PAA can be seen as many uniaxial sheets with different optical axis orientations. Take the PAA with nano-pores that have uniform pore parameters as sample and ω equals to 20 degrees and 30 degrees respectively to do two groups of symmetrical rotation measurements. Since the size of the blocks on the sample will change due to the tilt, so we selected and intercepted 11 blocks with relatively more uniform phase delay to analyze them separately. The selected areas are shown below in Fig. 10:

Fig. 10. The 11 selected areas of the PAA.

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The Mueller matrixes of one random point in block 1, 2, 3 are presented in Table 3:

Table 3. The Mueller matrix of the block 1, 2, 3 of the PAA with single pore diameter

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About the analysis results of the sample anisotropy, here we just show the average RI of the birefringence n_ave and the angle between the optical axis of the block and the Z-axis (the direction of the refracted ray at normal incidence) θ₀ since the angle φ₀ can be obtained directly by the Mueller matrix elements or MMD method.

Calculate the n_ave at ω=20 degrees and ω=30 degrees and put the results together to make a comparison. The results of four areas were selected and shown as follow:

From the Fig. 11, the n_ave in the selected regions is relatively uniform and the results of the different ω are similar. This is in line with the actual situation since for the pure alumina, the principal axis refractive indexes of the RI ellipsoid are n_o=1.768, n_e=1.760 that the difference between the two is only 0.008. In general, the average RI of every block is almost equal. To quantitatively check the results of the n_ave of all the selected blocks at different ω, take the average value in every block with uniform calculation results as the final n_ave of the corresponding block, the details are shown in Table 2. To verify the calculation results, we refer to the effective medium approximation (EMA) in which the cylindrical pore geometry is considered [25,26].

Fig. 11. The selected regions 1, 2, 4 and the average RI calculated at 20 degrees and 30 degrees. The left is the M₄₄ at normal incidence representing the selected area; the middle is the n_ave results at 20 degrees and the right is the n_ave results at 30 degrees.

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According to the EMA method, the pore diameter of the sample used in the experiments is about 40nm and corresponding refractive indexes are 1.7007 (paralleling to the sheet surface) and 1.7132 (paralleling to the pores). Their average is 1.7070. The results of n_ave of the selected 11 blocks are shown in Table 4 and make a comparison with 1.7070:

Table 4. The results of the average RI of the birefringence n_ave of the selected areas of the PAA

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According to the above table, we can see the most errors are under 1% except for one which equals to −1.22%. This illustrates that the calculation is accurate.

Finally, the results of the orientation angle of optical axis relative to Z-axis at normal incidence θ₀ are analyzed. Here, we choose the results of 3 blocks (1, 9 and 11) to show and make a detailed analysis:

Fig. 12. The results of θ₀ of the selected blocks 1, 9 and 11. The left one is the M₄₄ at normal incidence which is used to show the original blocks. The middle one is the calculation results of θ₀ at ω=20 degrees. The right one is the calculation results of θ₀ at ω=30 degrees.

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From Fig. 12, we can see that the most calculation results are uniform in the intercepted blocks. Noticing the values of the areas in the red circles in M₄₄ (the 3 images on the left) are large, that is, their colors tend to be more yellow which represents large in the colorbar. In accordance with the formula (1), M₄₄ is expressed as cosδ. The larger M₄₄ means the smaller δ and the less obvious birefringence effect. That is to say, in these areas the angle between the optical axis and the refraction ray θ₀ is small. Observing the corresponding areas in the middle and right images, the values are blue which represents the smaller θ₀ in the colorbar. This is consistent with the analysis and verify the correctness of the calculation method.

It can also be observed that the results calculated by the data at ω=20 degrees are not equal to that calculated by the data at ω=30 degrees. Combining the experiments with the PAA sample with 2 different pore diameters, it can be seen that the pores will exhibit obvious anisotropy when the sample is tilted, that is, when the direction of the pores has an angle with the direction of the incident ray. And the anisotropy is different at different angles. The larger the angle is, the more obvious the anisotropy of the pores is. And this part of anisotropy will affect the calculation of the actual orientation of the optical axis of the PAA. The difference between the results at ω=20 degrees and ω=30 degrees is the effect of pores on the anisotropy of the sample. Since the anisotropy of the crystal itself and the pores are all sensitive to the orientation of PAA and they are coupled together, the suitable method to separate these two kinds of anisotropy has not been found.

During the calculation process, it is found that the results of calculation in some areas are rather messy, such as the block 8 (Fig. 13):

This is mainly because that in the equation set θ₀ is included in the trigonometric functions (sinθ₀ and cosθ₀). The solving process involves finding the inverse function of the trigonometric function in which the calculation of the angle is prone to winding. For example, in the range from 0 to 180 degrees, when θ₀=30 degrees or θ₀=150 degrees, according to the formula of the effective RI of e-ray in the uniaxial crystals:

(13)$$n_e^, = \frac{{n_o^2n_e^2}}{{n_o^2{{\sin }^2}\theta + n_e^2{{\cos }^2}\theta }}.$$

It can be judged that n_e’ has the same value under these two θ₀ and the corresponding δ is same. This may make the results messy. Generally, according to the observation of the calculation results of the selected 11 blocks, these cluttered points mostly appear on the boundary of the blocks or in the region where the original data is uneven.

Fig. 13. The results of θ₀ of the selected block 8. The left one is the M₄₄ at normal incidence which is used to show the original blocks. The middle one is the calculation results of θ₀ at ω=20 degrees. The right one is the calculation results of θ₀ at ω=30 degrees.

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6. Conclusion

In this paper, the Mueller matrix measurements method is applied to characterize the anisotropy of the photonic crystal PAA. It is proved that the birefringence is the main manifestation of the anisotropy of the sample and this property It is proved that the birefringence is the main manifestation of the anisotropy of the sample and this property is affected by the pore parameters such as the pore diameter and the optical orientation of the sample in the RI ellipsoid. To better describe the effects induced by the pores, we study PAA as a phase retarder and propose a symmetrical rotation measurement method based on the Mueller matrix of the standard phase retarder and the crystal optics to calculate the average RI and the orientation angle θ of the optical axis of the sample. This method is verified by the experiments with two true zero order wps. The PAA with single pore parameter experiments show that this method can obtain the average RI accurately with the errors under 1%. θ is a key parameter that we want since it can help to calculate the intrinsic retardance of the sample which is independent of the crystal orientations. To further study the pores’ effects on the anisotropy of the sample, we can fabricate PAA samples with different pore parameters and study the relationship between the intrinsic retardance and these parameters.

The calculation results of θ sometimes are messy because the inverse of trigonometric function is involved in the solving process. These messy points are prone to appear on the boundary of the areas or in the region where the original data are uneven which is related to the real surface morphology of the sample. In summary, this paper provides a new idea for the optical characterization of porous anodic alumina. It also finds a new way to calculate the orientation of the optical axis of the uniaxial crystal sheet based on the MMI.

Funding

National Natural Science Foundation of China (11974206, 61527826); Shenzhen Bureau of Science and Innovation (JCYJ20160818143050110, JCYJ20170412170814624).

Disclosures

The authors declare no conflicts of interest.

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	0 degree				9 degrees				32 degrees
	1	2	3	4	1	2	3	4	1	2	3	4
1	1	−0.01	−0.01	0	1	−0.01	−0.01	0.01	1	0.01	0.03	0.02
2	0	1.01	0	−0.03	0	0.98	0.01	−0.08	0.03	0.82	0.08	0.62
3	0	0	1.02	−0.03	−0.01	−0.01	0.99	−0.1	0.08	0.09	0.98	0.17
4	0	0.03	0.03	0.99	0.01	0.07	0.08	0.99	0.04	0.67	0.18	0.76

λ/8 truth zero order wp		λ/4 truth zero order wp
θ₀ (degrees)	Relative error (\|θ₀-90\|/90)	θ₀ (degrees)	Relative error (\|θ₀-90\|/90)
89.58	0.46%	90.19	0.21%
88.99	1.12%	88.55	1.61%
89.89	0.12%	88.82	1.31%
89.89	0.12%	91.00	1.11%
89.88	0.13%	90.79	0.88%
89.16	0.94%	90.04	0.05%
90.21	0.24%	89.66	0.38%
89.92	0.08%	89.17	0.92%
90.98	1.08%	90.89	0.99%
90.79	0.88%	89.50	0.56%

	Block 1				Block 2				Block 3
	1	2	3	4	1	2	3	4	1	2	3	4
1	1	−0.01	0.01	0	1	−0.02	−0.01	−0.02	1	0	0.02	0.01
2	0	0.98	0.01	−0.16	0.01	0.78	−0.16	0.52	0	0.87	0.02	0.21
3	−0.01	0.03	0.97	0.19	−0.03	−0.16	0.84	0.41	0.04	0.04	0.98	0.15
4	−0.02	0.19	−0.2	0.93	0.01	−0.5	−0.46	0.65	0.01	−0.2	0.12	0.83

Area	n_ave(ω=20°)	Relative error(%)	n_ave(ω=30°)	Relative error (%)
1	1.6959	−0.65	1.6921	−0.87
2	1.6987	−0.49	1.6972	−0.58
3	1.6992	−0.46	1.6961	−0.64
4	1.7077	0.04	1.7087	0.10
5	1.7108	0.22	1.6988	−0.48
6	1.7172	0.60	1.6986	−0.49
7	1.6966	−0.61	1.6862	−1.22
8	1.7036	−0.20	1.7058	−0.07
9	1.7191	0.71	1.6929	−0.80
10	1.7041	−0.17	1.7067	−0.02
11	1.7191	0.71	1.7044	−0.15

	0 degree				9 degrees				32 degrees
	1	2	3	4	1	2	3	4	1	2	3	4
1	1	−0.01	−0.01	0	1	−0.01	−0.01	0.01	1	0.01	0.03	0.02
2	0	1.01	0	−0.03	0	0.98	0.01	−0.08	0.03	0.82	0.08	0.62
3	0	0	1.02	−0.03	−0.01	−0.01	0.99	−0.1	0.08	0.09	0.98	0.17
4	0	0.03	0.03	0.99	0.01	0.07	0.08	0.99	0.04	0.67	0.18	0.76

Characterization of anisotropy of the porous anodic alumina by the Mueller matrix imaging method

Abstract

1. Introduction

2. Material and methods

3. Polarization measurements of PAA

3.1 Normal incidence experiments of the PAA with single pore parameters

3.2 Rotation experiments of the PAA with two different pore diameters

4. Study on the birefringence of PAA

4.1 Analysis based on the Mueller matrix of the retarder

4.2 The symmetric rotation measurements based on the retardance of uniaxial crystals

5. Results

6. Conclusion

Funding

Disclosures

References

Cited By

Figures (13)

Tables (4)

Equations (13)

Optics Express