## Abstract

Porous anodic alumina (PAA) is a photonic crystal with a hexagonal porous structure. To learn more about the effects brought by pores on the anisotropy of the PAA, we use the orientation sensitive Mueller matrix imaging (MMI) method to study it. We fabricated the PAA samples with uniform pores and two different pore diameters. By the MMI experiments with these samples, we found that the birefringence is the major anisotropy of the PAA and that there are many small areas with different orientations that formed spontaneously in the process of production on the surface of the PAA. By the MMI experiments at different orientations of the sample with two different pore diameters, we found that the pores affect the birefringence of the sample and the effect increases with the increased inclination of the sample. To further analyze the PAA, we present a symmetrical rotation measurement method according to the Mueller matrix of the retarder. With this method, we can calculate the average refractive index (RI) of birefringence and the orientation of the optical axis of uniaxial crystal. The results also show the effect of the pores on the anisotropy of PAA.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 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.

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

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:

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:

According to the Ref. [23], M_{11} 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:

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 M_{24} elements in Fig. 5.

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 M_{22}, M_{24}, M_{42} and M_{44} 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:

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):

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:

n'_{e} and n_{o} represent the effective refractive indexes of the ordinary light and extraordinary light. δ_{mea} can be expressed as [24]:

_{1}and γ

_{2}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:

_{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 γ

_{1}and γ

_{2}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:

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:

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:

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:

- (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, δ_{1}, δ_{2} 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:

_{0}is the intrinsic phase delay of the sample. θ

_{0}, θ

_{1}, θ

_{2}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, θ

_{0}and φ

_{0}among which the last 2 are unknowns. Now 2 equations can be listed:

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 φ_{0} can be obtained using the values of M_{24}, M_{34}, M_{42} and M_{43} of the Mueller matrix in Eq. (1):

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 φ_{0}. The relationship between φ and φ_{0} 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:

For the positive uniaxial crystal:

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, φ_{0} should be calculated by MMD method other than the numerical calculation using Mueller matrix elements M_{24}, M_{34}, M_{42}, M_{43} 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 φ_{0} 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 θ_{0} for the two wps are shown as follows:

For the values of θ_{0} 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:

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

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) θ_{0} since the angle φ_{0} 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].

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:

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 θ_{0} are analyzed. Here, we choose the results of 3 blocks (1, 9 and 11) to show and make a detailed analysis:

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 M_{44} (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), M_{44} is expressed as cosδ. The larger M_{44} 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 θ_{0} is small. Observing the corresponding areas in the middle and right images, the values are blue which represents the smaller θ_{0} 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 θ_{0} is included in the trigonometric functions (sinθ_{0} and cosθ_{0}). 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 θ_{0}=30 degrees or θ_{0}=150 degrees, according to the formula of the effective RI of e-ray in the uniaxial crystals:

_{e}’ has the same value under these two θ

_{0}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.

## 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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