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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 9 — May. 5, 2014
  • pp: 11011–11020
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Dielectric multilayer beam splitter with differential phase shift on transmission and reflection for division-of-amplitude photopolarimeter

Wenjia Yuan, Weidong Shen, Yueguang Zhang, and Xu Liu  »View Author Affiliations


Optics Express, Vol. 22, Issue 9, pp. 11011-11020 (2014)
http://dx.doi.org/10.1364/OE.22.011011


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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 Tp = 78.9%, Ts = 21.1% and Δr − Δt = 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 (ψr, Δr) = (26.5°, 135.1°) and (28.2°, 133.5°), as well as in transmission (ψt, Δt) = (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.

© 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

1. H. G. Berry, G. Gabrielse, and A. E. Livingston, “Measurement of the Stokes parameters of light,” Appl. Opt. 16(12), 3200–3205 (1977). [CrossRef] [PubMed]

5

5. L. Weller, T. Dalton, P. Siddons, C. S. Adams, and I. G. Hughes, “Measuring the Stokes parameters for light transmitted by a high-density rubidium vapour in large magnetic fields,” J. Phys. At. Mol. Opt. Phys. 45(5), 055001 (2012). [CrossRef]

]. The division-of-amplitude photopolarimeter (DOAP) introduced by R. M. A. Azzam [6

6. R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” Opt. Acta (Lond.) 29(5), 685–689 (1982). [CrossRef]

] 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

7. R. M. A. Azzam and A. De, “Optimal beam splitters for the division-of-amplitude photopolarimeter,” J. Opt. Soc. Am. A 20(5), 955–958 (2003). [CrossRef] [PubMed]

], 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

8. R. M. A. Azzam and F. F. Sudradjat, “Single-layer-coated beam splitters for the division-of-amplitude photopolarimeter,” Appl. Opt. 44(2), 190–196 (2005). [CrossRef] [PubMed]

]. 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

Fig. 1 Schematic diagram of DOAP.
A typical configuration of DOAP [6

6. R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” Opt. Acta (Lond.) 29(5), 685–689 (1982). [CrossRef]

] 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 D0, D1, D2, D3 and the intensity signals i1, i2, i3, i4 were recorded. It can be written as vector I=[i1,i2,i3,i4]T. So, the four-dimensional Stokes vector S=[S1,S2,S3,S4] can be calculated by S=A1I. Here, A is instrument matrix, usually determined by calibration [9

9. R. M. A. Azzam and A. G. Lopez, “Accurate calibration of the four-detector photopolarimeter with imperfect polarizing optical elements,” J. Opt. Soc. Am. A 6(10), 1513–1521 (1989). [CrossRef]

]. 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

7. R. M. A. Azzam and A. De, “Optimal beam splitters for the division-of-amplitude photopolarimeter,” J. Opt. Soc. Am. A 20(5), 955–958 (2003). [CrossRef] [PubMed]

]:
detA=(RT)2sin2ψrsin2ψt(cos2ψrcos2ψt)sin(ΔrΔt)
(1)
Here, R and T are the reflectance and transmittance of beam splitter. (ψr, Δr) and (ψt, Δt) 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 tan2ψt=Tp/Ts,tan2ψr=Rp/Rs. Δ is the phase difference between p- and s-polarized light, and in this paper, Δ=ϕpϕs is defined. By a mathematical calculation, the maximum absolute value of determinant|detA|max=3/36=0.0481can be obtained when the beam splitter meets the following requirements:
R=T=0.5,ΔrΔt=±π/2(ψr,ψt)=(27.368°,62.632°)or(62.632°,27.368°)
(2)
At oblique incidence, p-polarized light always transmits more than s-polarized light, i.e.,ψt>45°, then (ψr,ψt)=(27.368°,62.632°) is adopted. Therefore, for non-absorbing all dielectric coatings, the optimal parameters of beam splitter can be determined: Tp=78.9%, Ts=21.1% and ΔrΔt=±π/2. It is a beam splitter with fixed transmittances and differential phase shift on transmission and reflection. To evaluate the performance of a beam splitter in DOAP, the normalized determinant can be used:
|detA|norm=|detA|/|detA|max
(3)
The normalized determinant should be as close to 1 as possible.

Fig. 2 The structure of the designed beam splitter.
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.

Nb2O5 and SiO2 are employed as high and low refractive index materials, respectively, and the substrate is BK7. The optical constants Nb2O5 and SiO2 at different wavelengths are listed in Table 1.

Table 1. Optical constants of Nb2O5 and SiO2 at different wavelengths

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The refractive index of Nb2O5 is 2.317 and SiO2 is 1.488 at 532nm, the extinction coefficient of Nb2O5 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 Δr and Δt should be π/2 and the fixed values for Δr and Δt are not required. In this case, there are two design strategies to optimize the phase shift: I) define Δr and Δt separately as the constant values for the optimization targets, II) directly define Δr − Δt as the optimization target in a spectral region.

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 Ts, Tp, Δr, and Δt by design method I, and the corresponding merit function can be expressed as:
MF1=1Mp, sm=1M(T(λm)T(m)δT(m))2+1Mr, tm=1M(Δ(λm)Δ(m)δΔ(m))2
(4)
Here λ is wavelength in the target region and m = 1, 2,…M are a set of wavelengths. T(λm) is the theoretical transmittance for both p and s polarization light at a specified wavelength, while Δ(λm) is the theoretical phase shift difference on p- and s-polarized light for both reflection and transmission at a specified wavelength. T(m) and Δ(m) are the corresponding target values and δT(m), δΔ(m) are the corresponding tolerances.

We use the needle optimization method implemented in OptiLayer [10

10. A. V. Tikhonravov and M. K. Trubetskov, OptiLayer Software, http://www.optilayer.com.

,11

11. A. V. Tikhonravov, M. K. Trubetskov, and G. W. Debell, “Application of the needle optimization technique to the design of optical coatings,” Appl. Opt. 35(28), 5493–5508 (1996). [CrossRef] [PubMed]

] 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 (Nb2O5) and low index material (SiO2) respectively. Sub means a glass substrate which is BK7 glass in this paper.

The values of Δr and Δt, have influence on physical thickness and merit function of design results. In order to find a good solution with proper values of Δr and Δt, we designed beam splitter with different values of Δt and set the difference ΔrΔt=π/2. The different designs with values of Δt = 15, 30, 45, 60 degrees are shown in Table 2.

Table 2. Design results with different values of Δt

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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 Tp and Ts are 0.1%, and the tolerances of Δr, Δt are 0.1°. For Δt = 15, 30, 45, 60 degrees, the corresponding merit function 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 Δt = 45 degrees has the lowest merit function. Therefore, we set Δt = 45 degrees and correspondingly Δr = 135 degrees as the targets.

Fig. 3 Structure and optical characteristics of the designed beam splitter (straight line) and the calculated deviations with a 1nm random thickness error (dash line) (a) structure (b) transmittance (c) Δ.
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 Δt is 44.7° at 532nm and Δr is 135.1° and the value of Δr − Δt is 90.4° as seen in Fig. 3(c).

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 Ts, Tp, and T at 532nm are 4.8%, 5.4%, and 4.9%. For the phase difference shift Δr and Δt, the deviations are 4.2° and 3.8°, and the deviation of Δr − Δt is 6.5° at 532nm.

In the design method I, Δr and Δt are separately set as a constant value. In fact, the value of Δr − Δt is the final target we require. Therefore it is possible to directly define Δr − Δt as an optimization target, not considering the detailed values of Δr and Δt. As a result, the merit function contains only three parameters: Tp, Ts, and Δr – Δt. It can be described as Eq. (5):
MF2=1Mp, sm=1M(T(λm)T(m)δT(m))2+1Mm=1M(Δrt(λm)Δrt(m)δΔrt(m))2
(5)
WhereΔrtmeans Δr – Δt in the equation and other parameters are the same as in Eq. (4)

Fig. 4 Structure and optical characteristics of designed beam splitter (straight line) and the calculated deviations with a 1nm random thickness error (dash line) (a) structure (b) transmittance (c) Δ.
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 Δt is 45.8° at 532nm and on reflectance Δr is 135.9°. The value of Δr – Δt 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.

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 Ts, Tp, and T at 532nm are 5.7%, 5.9%, and 5.4%. For the phase difference shift Δr and Δt, the deviations are 5.4° and 5.8°, and the estimated value of Δr – Δt is 81.2°~98.8°, with a deviation of 8.8° at 532nm.

Compared with the two designs, design method II presents better results of Δr – Δt 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.

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 Δt is minimized as well.

Fig. 5 Structure and optical characteristics of the designed anti-reflectance coating (a) structure (b) reflectance (c) Δt.
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 Δt 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 Δt is under 0.2° at 532nm.

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 O2 were introduced into the system during the process and the working pressure was 5 × 10−2Pa. The deposition rates of Nb2O5 and SiO2 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.

Fig. 6 The measured transmittance and reflectance curves (a) Sample I (b) Sample II (c) Anti-reflection coating.
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 (ψr, Δr) and (ψt, Δt) 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 ψr and ψt of samples are shown in Fig. 7.
Fig. 7 The measured ψr and ψt at different wavelengths (a) Sample I (b) Sample II.
The measured ψr and ψt 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°.

Fig. 8 The measured Δr and Δt at different wavelengths (a) Sample I (b) Sample II.
Figure 8 presents the measured Δr and Δt of the samples. For sample I, Δr and Δt 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.

Fig. 11 Normalized determinant of matrix (A) of the beam splitter at different wavelengths.
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 Δr and Δt as a constant in the targeted spectral range and the other is to define Δr – Δt directly as an optimized target.

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).

References and links

1.

H. G. Berry, G. Gabrielse, and A. E. Livingston, “Measurement of the Stokes parameters of light,” Appl. Opt. 16(12), 3200–3205 (1977). [CrossRef] [PubMed]

2.

F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. 24(9), 584–586 (1999). [CrossRef] [PubMed]

3.

B. Kanseri, S. Rath, and H. C. Kandpal, “Direct determination of the generalized Stokes parameters from the usual Stokes parameters,” Opt. Lett. 34(6), 719–721 (2009). [CrossRef] [PubMed]

4.

T. Kihara, “Measurement method of Stokes parameters using a quarter-wave plate with phase difference errors,” Appl. Opt. 50(17), 2582–2587 (2011). [CrossRef] [PubMed]

5.

L. Weller, T. Dalton, P. Siddons, C. S. Adams, and I. G. Hughes, “Measuring the Stokes parameters for light transmitted by a high-density rubidium vapour in large magnetic fields,” J. Phys. At. Mol. Opt. Phys. 45(5), 055001 (2012). [CrossRef]

6.

R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” Opt. Acta (Lond.) 29(5), 685–689 (1982). [CrossRef]

7.

R. M. A. Azzam and A. De, “Optimal beam splitters for the division-of-amplitude photopolarimeter,” J. Opt. Soc. Am. A 20(5), 955–958 (2003). [CrossRef] [PubMed]

8.

R. M. A. Azzam and F. F. Sudradjat, “Single-layer-coated beam splitters for the division-of-amplitude photopolarimeter,” Appl. Opt. 44(2), 190–196 (2005). [CrossRef] [PubMed]

9.

R. M. A. Azzam and A. G. Lopez, “Accurate calibration of the four-detector photopolarimeter with imperfect polarizing optical elements,” J. Opt. Soc. Am. A 6(10), 1513–1521 (1989). [CrossRef]

10.

A. V. Tikhonravov and M. K. Trubetskov, OptiLayer Software, http://www.optilayer.com.

11.

A. V. Tikhonravov, M. K. Trubetskov, and G. W. Debell, “Application of the needle optimization technique to the design of optical coatings,” Appl. Opt. 35(28), 5493–5508 (1996). [CrossRef] [PubMed]

OCIS Codes
(120.2130) Instrumentation, measurement, and metrology : Ellipsometry and polarimetry
(230.1360) Optical devices : Beam splitters
(310.6860) Thin films : Thin films, optical properties
(310.4165) Thin films : Multilayer design
(310.5696) Thin films : Refinement and synthesis methods

ToC Category:
Optical Devices

History
Original Manuscript: March 31, 2014
Revised Manuscript: April 16, 2014
Manuscript Accepted: April 20, 2014
Published: April 30, 2014

Citation
Wenjia Yuan, Weidong Shen, Yueguang Zhang, and Xu Liu, "Dielectric multilayer beam splitter with differential phase shift on transmission and reflection for division-of-amplitude photopolarimeter," Opt. Express 22, 11011-11020 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-9-11011


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References

  1. H. G. Berry, G. Gabrielse, A. E. Livingston, “Measurement of the Stokes parameters of light,” Appl. Opt. 16(12), 3200–3205 (1977). [CrossRef] [PubMed]
  2. F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. 24(9), 584–586 (1999). [CrossRef] [PubMed]
  3. B. Kanseri, S. Rath, H. C. Kandpal, “Direct determination of the generalized Stokes parameters from the usual Stokes parameters,” Opt. Lett. 34(6), 719–721 (2009). [CrossRef] [PubMed]
  4. T. Kihara, “Measurement method of Stokes parameters using a quarter-wave plate with phase difference errors,” Appl. Opt. 50(17), 2582–2587 (2011). [CrossRef] [PubMed]
  5. L. Weller, T. Dalton, P. Siddons, C. S. Adams, I. G. Hughes, “Measuring the Stokes parameters for light transmitted by a high-density rubidium vapour in large magnetic fields,” J. Phys. At. Mol. Opt. Phys. 45(5), 055001 (2012). [CrossRef]
  6. R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” Opt. Acta (Lond.) 29(5), 685–689 (1982). [CrossRef]
  7. R. M. A. Azzam, A. De, “Optimal beam splitters for the division-of-amplitude photopolarimeter,” J. Opt. Soc. Am. A 20(5), 955–958 (2003). [CrossRef] [PubMed]
  8. R. M. A. Azzam, F. F. Sudradjat, “Single-layer-coated beam splitters for the division-of-amplitude photopolarimeter,” Appl. Opt. 44(2), 190–196 (2005). [CrossRef] [PubMed]
  9. R. M. A. Azzam, A. G. Lopez, “Accurate calibration of the four-detector photopolarimeter with imperfect polarizing optical elements,” J. Opt. Soc. Am. A 6(10), 1513–1521 (1989). [CrossRef]
  10. A. V. Tikhonravov, M. K. Trubetskov, OptiLayer Software, http://www.optilayer.com .
  11. A. V. Tikhonravov, M. K. Trubetskov, G. W. Debell, “Application of the needle optimization technique to the design of optical coatings,” Appl. Opt. 35(28), 5493–5508 (1996). [CrossRef] [PubMed]

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