## Abstract

Dielectric multilayer beam splitter with differential phase shift on transmission and reflection for division-of-amplitude photopolarimeter (DOAP) was presented for the first time to our knowledge. The optimal parameters for the beam splitter are *T _{p}* = 78.9%,

*T*= 21.1% and Δ

_{s}*− Δ*

_{r}*= π/2 at 532nm at an angle of incidence of 45°. Multilayer anti-reflection coating with low phase shift was applied to reduce the backside reflection. Different design strategies that can achieve all optimal targets at the wavelength were tested. Two design methods were presented to optimize the differential phase shift. The samples were prepared by ion beam sputtering (IBS). The experimental results show good agreement with those of the design. The ellipsometric parameters of samples were measured in reflection (ψ*

_{t}

_{r}_{,}Δ

*) = (26.5°, 135.1°) and (28.2°, 133.5°), as well as in transmission (ψ*

_{r}*, Δ*

_{t}*) = (62.5°, 46.1°) and (63.5°, 46°) at 532.6nm. The normalized determinant of instrument matrix to evaluate the performance of samples is respectively 0.998 and 0.991 at 532.6nm.*

_{t}© 2014 Optical Society of America

## 1. Introduction

Stokes parameters can describe the general states of polarization of a light beam. Various types of instruments have been applied to measure Stokes vector [1–5]. The division-of-amplitude photopolarimeter (DOAP) introduced by R. M. A. Azzam [6] is capable of measuring all four stokes parameters simultaneously by four independent detectors. It is a rapid real-time method without any moving parts or modulators and is now widely used for many applications to measure the light polarization or scattering matrices under dynamic conditions. DOAP employs a beam splitter and two Wollaston prisms. In a typical DOAP, the beam splitter serves as a key component, dividing light into two beams as well as providing phase shift on reflection and transmission. In reference [7], Azzam presented the optimal parameters (transmittance, reflectance and phase shift) for beam splitter to obtain the best performance in DOAP. Some simple examples such as single and bi-layer coated beam splitters with high index substrate or film were also provided in later works [8]. However, the performances of these beam splitters are limited and can only satisfy the near optimal conditions. Moreover, only theoretical results are provided and no experimental results are reported so far.

It is a direct and effective way to improve the performance of DOAP by using a well designed and manufactured multilayer beam splitter. The thickness and quantity of the layers constituting a beam splitter can be precisely tailored to provide all optimal parameters in a spectral and incident angle range. So, a compact system can be constructed without additional quarter wave plates. In this paper, a dielectric multilayer beam splitter with differential phase shift on transmission and reflection at 532nm was proposed for the first time to our knowledge. Different design strategies that can achieve all optimal targets at the wavelength were tested and compared. The samples were prepared by ion beam sputtering (IBS), and the results show good agreement with the theoretical design. The normalized determinant of instrument matrix to evaluate the performance of samples is respectively 0.998 and 0.991 at 532.6nm.

## 2. Design of multilayer beam splitter

A typical configuration of DOAP [6] is shown in
Fig. 1. The incident light beam whose stokes parameters
to be measured is divided into four separate beams by a beam splitter and two Wollaston prisms
WP1, WP2. The light fluxes of four split beams can be detected by photodetectors D_{0},
D_{1}, D_{2}, D_{3} and the intensity signals i_{1},
i_{2}, i_{3}, i_{4} were recorded. It can be written as vector
$I={\left[{i}_{1},{i}_{2},{i}_{3},{i}_{4}\right]}^{T}$. So, the four-dimensional Stokes vector $S=\left[{S}_{1},{S}_{2},{S}_{3},{S}_{4}\right]$ can be calculated by $S={A}^{-1}I$. Here, **A** is instrument matrix, usually determined by
calibration [9]. As matrix **A** is nonsingular,
its inverse **A**^{−1} exists. By maximizing the absolute value of the
determinant of matrix **A**, the measurement accuracy can be increased. The determinant
of matrix **A** is determined by the optical parameters of the beam splitter and can be
expressed by [7]:

*R*and

*T*are the reflectance and transmittance of beam splitter. (ψ

*, Δ*

_{r}*) and (ψ*

_{r}*, Δ*

_{t}*) are the ellipsometric parameters written in reflection and transmission, respectively. ψ is the intensity ratio of p-polarized and s-polarized light in reflection or transmission, and ${\mathrm{tan}}^{2}{\psi}_{t}={T}_{p}/{T}_{s}$,${\mathrm{tan}}^{2}{\psi}_{r}={R}_{p}/{R}_{s}$. Δ is the phase difference between p- and s-polarized light, and in this paper, $\Delta ={\varphi}_{p}-{\varphi}_{s}$ is defined. By a mathematical calculation, the maximum absolute value of determinant${\left|\mathrm{det}A\right|}_{\mathrm{max}}=\sqrt{3}/36=0.0481$can be obtained when the beam splitter meets the following requirements:*

_{t}The designed beam splitter contains two sides and is placed an angle of incidence of 45°. The structure of beam splitter is shown in Fig. 2. The front side is beam splitter and an anti-reflection coating is applied on the back side of the substrate to reduce the reflection from the rear face.

Nb_{2}O_{5} and SiO_{2} are employed as high and low refractive index
materials, respectively, and the substrate is BK7. The optical constants
Nb_{2}O_{5} and SiO_{2} at different wavelengths are listed in Table 1.The refractive index of Nb_{2}O_{5} is 2.317 and SiO_{2} is
1.488 at 532nm, the extinction coefficient of Nb_{2}O_{5} is 2.2 ×
10^{−4}, which can be regarded as non-absorption.

#### 2.1 Design of the beam splitter

According to Eq. (2), the optimal parameters of the beam splitter are determined. The transmittances of p-polarized, s-polarized light are equal to 78.9% and 21.1% respectively. The difference between Δ* _{r}* and Δ

*should be π/2 and the fixed values for Δ*

_{t}*and Δ*

_{r}*are not required. In this case, there are two design strategies to optimize the phase shift: I) define Δ*

_{t}*and Δ*

_{r}*separately as the constant values for the optimization targets, II) directly define Δ*

_{t}*− Δ*

_{r}*as the optimization target in a spectral region.*

_{t}The design problem of beam splitter by the two optimization methods is discussed in the following section. A merit function *MF* is introduced to evaluate the deviation between the design results and the targets, which allows to convert the design problem to a minimization of merit function. Therefore, there are four optimization targets *T _{s}*,

*T*, Δ

_{p}*, and Δ*

_{r}*by design method I, and the corresponding merit function can be expressed as:*

_{t}*λ*is wavelength in the target region and m = 1, 2,…M are a set of wavelengths.

*T*(

*λ*) is the theoretical transmittance for both p and s polarization light at a specified wavelength, while Δ(

_{m}*λ*) is the theoretical phase shift difference on p- and s-polarized light for both reflection and transmission at a specified wavelength.

_{m}*T*

^{(}

^{m}^{)}and Δ

^{(}

^{m}^{)}are the corresponding target values and

*δT*

^{(}

^{m}^{)},

*δ*Δ

^{(}

^{m}^{)}are the corresponding tolerances.

We use the needle optimization method implemented in OptiLayer [10,11] software to design and analyze the design results of the beam splitter. Needle optimization is a process in which new layers are automatically inserted into the design during the optimization procedure. The optimization begins with a starting design stack Sub/ (HL)^{10} /Glass on the front surface with air and glass as the incident and exit medium. H and L are quarterwave layer of high index material (Nb_{2}O_{5}) and low index material (SiO_{2}) respectively. Sub means a glass substrate which is BK7 glass in this paper.

The values of Δ* _{r}* and Δ

*, have influence on physical thickness and merit function of design results. In order to find a good solution with proper values of Δ*

_{t}*and Δ*

_{r}*, we designed beam splitter with different values of Δ*

_{t}*and set the difference ${\Delta}_{r}-{\Delta}_{t}=\pi /2$. The different designs with values of Δ*

_{t}*= 15, 30, 45, 60 degrees are shown in Table 2.The spectral region is from 517nm to 547nm, and the transmittances of p-polarized, s-polarized light are 78.9% and 21.1% respectively. The tolerances of*

_{t}*T*and

_{p}*T*are 0.1%, and the tolerances of Δ

_{s}*, Δ*

_{r}*are 0.1°. For Δ*

_{t}*= 15, 30, 45, 60 degrees, the corresponding merit function*

_{t}*MF1*are 7.7, 6.07, 5.81, and 11.04, and the total physical thickness are 2353nm, 2430nm, 2420nm, and 2675nm. The design with Δ

*= 45 degrees has the lowest merit function. Therefore, we set Δ*

_{t}*= 45 degrees and correspondingly Δ*

_{t}*= 135 degrees as the targets.*

_{r}The structure of the splitter was further refined by needle method with the adjusted
parameter tolerances. The final results are shown in Fig.
3(a), which consists of 36 layers with a total thickness of about 2273nm. The thinnest
layer has a thickness of 29nm. The optical characteristics of the designed splitter including
transmittance and Δ are shown in Fig. 3(b) and
3(c) (straight line). The transmittance of p-,
s-polarized are 78.95% and 20.91% at 532nm, and the average transmittance is 49.93%. The phase
shift on transmittance Δ* _{t}* is 44.7° at 532nm and
Δ

*is 135.1° and the value of Δ*

_{r}*− Δ*

_{r}*is 90.4° as seen in Fig. 3(c).*

_{t}Usually, the manufacturing of non-quarterwave multilayer filters requires precise control of layer thickness. However, the manufacturing errors inevitably exist and make the experimental results deviate from the theoretical design. Therefore, it is necessary to perform an analysis of error sensitivity before fabrication. When small errors in layer thickness or refractive index take place, error analysis of design can describe the influences to the optical performance. In Fig. 3(b) and 3(c), the area enclosed by two dash lines neighboring the design curve represents the calculated results with a probability of 68.3% when a random deviation of layer thickness is 1nm for each layer. The deviations of *T _{s}*,

*T*, and

_{p}*T*at 532nm are 4.8%, 5.4%, and 4.9%. For the phase difference shift Δ

*and Δ*

_{r}*, the deviations are 4.2° and 3.8°, and the deviation of Δ*

_{t}*− Δ*

_{r}*is 6.5° at 532nm.*

_{t}In the design method I, Δ* _{r}* and Δ

*are separately set as a constant value. In fact, the value of Δ*

_{t}*− Δ*

_{r}*is the final target we require. Therefore it is possible to directly define Δ*

_{t}*− Δ*

_{r}*as an optimization target, not considering the detailed values of Δ*

_{t}*and Δ*

_{r}*. As a result, the merit function contains only three parameters:*

_{t}*T*,

_{p}*T*, and Δ

_{s}*– Δ*

_{r}*. It can be described as Eq. (5):*

_{t}*– Δ*

_{r}*in the equation and other parameters are the same as in Eq. (4)*

_{t}The optimization was also performed with needle method and the final results are shown in
Fig. 4. The design consists of 34 layers with a total
thickness of about 2374nm. The thinnest layer has a thickness of 19nm. The optical
characteristic of the splitter including transmittance and Δ are shown in Fig. 4(b) and 4(c).
The transmittances of p-, s-polarized light are 78.93% and 20.92% at 532nm, and the average
transmittance is 49.93%. The phase difference shift on transmittance
Δ* _{t}* is 45.8° at 532nm and on reflectance
Δ

*is 135.9°. The value of Δ*

_{r}*– Δ*

_{r}*is 90.05° with a deviation of under ± 0.2° at wavelengths from 517nm to 547nm, and is close to the target in the desired wavelength region.*

_{t}The Fig. 4(b) and 4(c) show optical characteristics performed by error analysis with a random thickness deviation of 1nm, plotted in dash line. The deviations of *T _{s}*,

*T*, and

_{p}*T*at 532nm are 5.7%, 5.9%, and 5.4%. For the phase difference shift Δ

*and Δ*

_{r}*, the deviations are 5.4° and 5.8°, and the estimated value of Δ*

_{t}*– Δ*

_{r}*is 81.2°~98.8°, with a deviation of 8.8° at 532nm.*

_{t}Compared with the two designs, design method II presents better results of Δ* _{r}* – Δ

*by direct optimization. However, according to error analysis with a random thickness deviation of 1nm, the deviations of the transmittance and phase difference shift in design I are lower.*

_{t}#### 2.2 Design of anti-reflection coating on the backside

The reflection on the backside surface of substrate will cause a loss of light and also change the parameters of the beam splitter. It is necessary to add anti-reflection coating on the back side. To avoid the extra phase shift, the anti-reflection multilayer coating must have a low phase shift on transmission. Therefore, the reflectance of p-polarized and s-polarized light should be close to zero as possible, and the value of phase shift on transmittance Δ* _{t}* is minimized as well.

The design structure of the anti-reflectance coating is shown in Fig. 5. It consists of 7 layers with a total thickness about 470nm. The
refined reflectance and Δ* _{t}* are shown in Fig. 5(b) and 5(c) respectively. P-,
s-polarized and the average reflectance are lower than 0.5%, meanwhile the absolute value of
Δ

*is under 0.2° at 532nm.*

_{t}## 3. Experimental results and analysis

The beam splitter and anti-reflection coating were prepared by ion beam sputtering. The ion beam sputtering deposition is a highly stable process and considered as one of the best optical thin film deposition techniques. The coating plant is our home-made dual ion beam sputtering system equipped with two RF ion sources (16cm and 12cm, Veeco Inc.). The background pressure was 2 × 10^{−4} Pa. Ar and O_{2} were introduced into the system during the process and the working pressure was 5 × 10^{−2}Pa. The deposition rates of Nb_{2}O_{5} and SiO_{2} were respectively 0.12nm/s and 0.1nm/s. Quartz crystal monitoring was used to control the thickness of thin films during the deposition process.

The transmittance curves were measured by a spectrophotometer (Perkin-Elmer Lambda 900). The beam splitter sample I and II designed by method I and II were both prepared. The measured results of the average, p- and s-polarized transmittance are shown in Fig. 6. All the results were measured with an angle of incidence of 45°. In Fig. 6(a), the transmittance of p-polarized light of sample I is 81.8%, s-polarized light is 20.6% and the average transmittance is 51.2%. In Fig. 6(b), the transmittance of p-polarized light of sample II is 75.7%, s-polarized light is 18.9%, and the average transmittance is 47.3%. The reflectance curves of anti-reflection coating were measured and shown in Fig. 6(c). For comparison, all the theoretical curves are also shown in the Figs with dash lines. It can be seen that all the experimental results agree well with those of the design.

Ellipsometric parameters (ψ* _{r}*,
Δ

*) and (ψ*

_{r}*, Δ*

_{t}*) of the beam splitters were measured by spectroscopic ellipsometer (J. A. Woollam M-2000). The J. A. Woollam M-2000 can measure ellipsometric parameters in reflectance and transmittance, and cover a wavelength range from 193nm to 1690nm with a resolution of 1.6nm in the visible spectral range. So, the values at wavelength of 532.6nm are obtained in fact. The measured ψ*

_{t}*and ψ*

_{r}*of samples are shown in Fig. 7.The measured ψ*

_{t}*and ψ*

_{r}*of sample I are 26.5° and 62.5° at 532.6nm, while 28.2° and 63.5° for sample II. Both their values are very close to the target values of 27.4° and 62.6°, with a deviation of less than 1°.*

_{t}Figure 8 presents the measured
Δ* _{r}* and Δ

*of the samples. For sample I, Δ*

_{t}*and Δ*

_{r}*are respectively 135.1° and 46.1°, while 133.5° and 46° for sample II. The two beam splitters designed by different methods has the similar measured results at 532.6nm.*

_{t}We can obtain Δ* _{r}* –
Δ

*of samples from previous results, as presented in Fig. 9.For sample I, Δ*

_{t}*– Δ*

_{r}*is 89.0° at 532.6nm and the value is 89.0°~91.3° from 526.2nm to 532.6nm. For sample II, Δ*

_{t}*– Δ*

_{r}*is 87.5° at 532.6nm and it is 88.7°~90.1° in 537.4-548.5nm. Phase shift on transmittance of the anti-reflection coating was measured, as shown in Fig. 10.The value of Δ*

_{t}*is −0.3° at 532.6nm while the design value is −0.1°. The results match the theoretical data very well at the interested spectral range. So, the anti-reflection coating can significantly reduce reflection caused by the backside of substrate and very low phase shift is introduced.*

_{t}The normalized determinant of matrix **A** in DOAP can be used to evaluate the
performance of a beam splitter. For a perfect beam splitter, normalized determinant should be
equal to 1. Normalized determinants of sample I and II at different wavelengths are shown in
Fig. 11. The value of sample I is 0.998 at wavelength of
532.6nm and is larger than 0.98 from 521.4nm to 542.1nm. The value of sample II is 0.991 at
532.6nm and is larger than 0.98 from 519.8nm to 548.5nm.

The samples present a very good performance in the target spectral region. Compared with samples of two designs, the normalized determinant of sample I is higher than sample II from 523.0nm to 539.0nm, because design I has a relative lower deviation caused by thickness errors, according to the error analysis. Meanwhile, sample II has a broader wavelength region where the normalized determinant is above 0.98. Furthermore, design method II with less target parameters has a quicker optimizing process to find the optimal solution. And it is more effective than design method I, especially for beam splitters with larger total thickness applied in infrared spectral range.

## 4. Conclusions

In this paper, we presented a dielectric multilayer beam splitter with differential phase shift on transmission and reflection which serves as a key component in DOAP. The splitter contains two stacks on both sides of substrate. Multilayer anti-reflection coating with low phase shift was applied to reduce the backside reflection. Two design methods to optimize the phase difference shift are performed. One is to define Δ* _{r}* and Δ

*as a constant in the targeted spectral range and the other is to define Δ*

_{t}*– Δ*

_{r}*directly as an optimized target.*

_{t}The samples designed by the two optimization method were both prepared by ion beam sputtering (IBS). The measured results show an excellent agreement with the design curves. The normalized determinant of matrix **A** of sample I is 0.998, and the value of sample II is 0.991 at 532.6nm The results demonstrated that the normalized determinant of matrix **A** of sample II is lower at 532.6nm than sample I, but has a broader wavelength region where the normalized determinant of matrix **A** is above 0.98. The samples present a very good performance and satisfy the optimal parameters for DOAP. Furthermore, the extensions to different spectral ranges or angles and a broad bandwidth application are also possible.

## Acknowledgments

It is a pleasure for authors to acknowledge the funding support from the National High Technology Research and Development Program 863 (No. 2012AA040401), the National Natural Science Foundation of China (No. 61275161), the Zhejiang Provincial Natural Science Foundation (No. LY13F050001), and the Fundamental Research Funds for the Central Universities (No. 2014FZA5004).

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