#
The Young Interferometer as an Optical System for a Variable Depolarizer Characterization^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Liquid Crystal Variable Depolarizer

_{1}, S

_{2}, S

_{3}, S

_{0}are Stokes parameters. In our experiment, a polarimeter (PAX5710VIS, Thorlabs) was employed to measure the DOP of the transmitted light. Figure 2 shows the measured DOP as a function of the applied voltage calculated from the experimentally measured Stokes parameters. Therefore, the DOP for VAN can be determined with high accuracy, as shown in Figure 2, for different inputs of SOPs [17]. These results show that with increasing applied voltage, the DOP of transmitted light decreases because the cell generates a disordered birefringent medium related to the undefined switching direction of molecules. This phenomenon produces mixed SOPs in the cross-section of the transmitted light. A series of the minima and maxima of the DOP are related to the cell’s retardation changes. For a linearly polarized beam, which passes through a depolarizer cell composed of a disordered birefringence medium, the DOP becomes zero when the optical retardation induced by the cell matches that of a half-wavelength of the incident beam. However, the incident circular polarization state DOP is zero when the phase retardation is tuned to the quarter-wave condition of incident circular polarization [19]. The proposed VAN cell transmits the polarized component of incident light with a minimum DOP (about 16%) for a horizontally polarized input light obtained at 2.65 V. In such situations, minimum intensity loss and scattering can be observed. It is also worth mentioning that the depolarizer cell’s orientation does not have an impact on the obtained measurements.

## 3. Calibration and Essential Parameters of the Applied Interferometric Measurement Setup

_{1}). Next, the light splits into two beams by the beam splitter (BS). The probe beam is reflected by the mirror (M

_{1}), and a reference beam is reflected by the second mirror (M

_{2}), which is mounted on the mirror mount with the piezoelectric adjuster. Changing the optical path in one arm of the interferometer causes the phase shift change. In both paths, telescopes (T

_{1}, T

_{2}) consisting of two positive lenses were inserted. To adjust the reference equality and the probe beams’ diameter, two collimators with a circular pinhole were situated after the telescopes to obtain 400 μm diameter collimated beams (d = 400 µm). The interferometer includes a half-wave plate (HP) and a quarter-wave plate (QP) in the probe arm to change the beam incident polarization, while the reference beam is kept linearly polarized. Then, a right-angled gold coated prism bends the light coming from the collimators by 90°. Thus, the reflected beams become parallel to each other and are focused by the Fourier lens (L

_{2}, f = 170 mm) on a two-channel photodetector (PhD). The PhD consists of the right (PhD

_{R}) and left (PhD

_{L}) photodetectors separated by a gap (q ≈ 50 μm). The prism is mounted on a translation and rotation stage. The possibility of setting the prism position enables us to change the distance between two beams (2b) and the distribution of the fringe pattern, i.e., the number of fringes. These two parameters, 2b and q, significantly affect our setup performance. Due to the fact that q is fixed, the calibration process requires the adjustment of 2b, which is affected by the prism position. If the prism position is changed, the distribution of the field intensity in the Fourier plane collected by PhDs also changes. These intensities are collected in an acquisition card (DAC). Software for data recording was created in LabView.

_{i}is the Bessel functions; $u=\frac{x}{\lambda f}$, $v=\frac{y}{\lambda f}$ are the components of the spatial frequencies in the x and y directions in the Fourier plane; $\rho =\sqrt{{u}^{2}+{v}^{2}}$; and $\mu =\frac{2\mathrm{cos}\left(\theta \right)}{1+{\mathrm{cos}}^{2}\left(\theta \right)}$. Equation (3) is the basic dependency on the basis of which a number of simulations have been made. The simulations of the cross-section of power spectral density (PSD) and the corresponding interferograms of the fringe patterns for three different values of θ (θ = 0°, θ = 70°, and θ = 90°) are presented in Figure 4. As can be seen, the PSD envelope is the same for all cases because it depends on the beam’s aperture d. Along with the θ change, the PSD distribution is averaged until the disappearance of the fringes at θ = 90°.

_{R}(S

_{R}) and PhD

_{L}(S

_{L}) can be described as follows:

_{1}, a

_{2}, a

_{3}are the Fourier coefficients. Then, using the relationship describing signals from the left and right side, the sum Sum = (S

_{R}+ S

_{L}) and the difference Diff = (S

_{R}− S

_{L}) can be expressed as:

_{1}is the amplitude of the sum function, a

_{2}is the average value of the sum function, and a

_{3}is the average value of the difference function. From simple trigonometry, $\mathrm{tan}\left(\Delta \phi \right)=\mathrm{sin}\left(\Delta \phi \right)/\mathrm{cos}\left(\Delta \phi \right)$, the phase difference can be expressed as:

_{R}and S

_{L}depend on the combination of cosine and sine of the measured phase shift. The coefficients a

_{1}, a

_{2}, a

_{3}are constants depending on the wavelength λ, the distance between two beams 2b, the focal length f of the Fourier lens, the polarization, and the size of the beam d. The validation of a

_{1}, a

_{2}, a

_{3}is essential to optimize the system operation. Constants a

_{1}, a

_{2}, a

_{3}can be determined starting from a dependence describing the intensity:

^{2}(ρ) is the spatial distribution of amplitude in the Fourier plane and ${J}_{1}$ is the Bessel function of the first-order. The above formulas are components which take into account the form of the intensity distribution after Fourier transform with the polarization orientation difference θ between the interfering beams. Then, expressing the above components of complex functions in the form of trigonometric functions, the intensity distribution can be described as:

_{2}and a

_{3}is influential. The experimental investigations of coefficients a

_{2}, a

_{3}, and the calibration of the setup were performed using a mirror (M

_{2}) mounted on a moving table with a piezoelectric adjuster. Thus, the phase shift was induced in our system. During the calibration process, the mount of the mirror was moved by the external electric field. Results of the phase demodulation as a function of time for three different positions of the prism are presented in Figure 6a. As can be seen, the variation of the distance between two beams had a significant impact on the values of coefficients a

_{2}and a

_{3}as well as on the performance quality of our device. If one of the coefficients was much larger, the resulting phase shift recorded by our system was not linear. This error had to be eliminated during the measurement. In the case where a

_{2}≈ a

_{3}, the experimental results were in good agreement with the theoretical model presented in Figure 6b. The theoretical model was calculated from Equation (8), assuming the optical path shift induced by the mirror displacement.

## 4. Analysis of Dynamic Polarization Transmitted from the VAN Cell

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Measurement results of the DOP for VAN as a function of applied voltage for different input SOP; Linear horizontal (H) and vertical (V), linear at an angle of +45°(/) and −45°(\), circular right (R) and left (L).

**Figure 3.**The scheme of the interferometer setup. Light source: He-Ne laser with λ = 633 nm; P—linear polarizer; SF—spatial filter; L

_{1}, L

_{2}—lenses, BS—beam splitter; M

_{1}, M

_{2}—mirrors; T

_{1}, T

_{2}—telescopes, VAN—vertically aligned nematic liquid crystal cell; PhD—photodetector, Gen—generator, HP—half-wave plate, QP—quarter-wave plate. Adapted from [16,17].

**Figure 4.**Simulation results of the distribution of the power spectral density (PSD) and the corresponding interferograms of the fringe patterns for a different polarization orientation difference (θ) between the interfering beams: (

**a**) θ = 0°, (

**b**) θ = 70°, and (

**c**) θ = 90°.

**Figure 5.**The fringe pattern photos captured by the CCD camera placed in the Fourier plane. The VAN cell was inserted in the interferometer’s probe arm. Linear horizontal polarization of the incident beam was used and the reference beam was linearly polarized. By applying different voltages to the cell, the DOP of the probe beam was: (

**a**) 100%, (

**b**) 50%, and (

**c**) 2%.

**Figure 6.**System calibration. (

**a**) Dynamic phase shift measurements at three different values of the distance between two beams d. (

**b**) Comparison of the theoretical model (dotted line) and measured phase shift when coefficients a

_{2}and a

_{3}are equal. Adapted from [16].

**Figure 7.**Real time phase shift measurements when the modulated waveform varied between zero and 6 V was applied to the vertically aligned nematic (VAN) cell. (

**a**) Arbitrary polarization state in the input of the depolarizer while the reference beam is kept linearly polarized. (

**b**) Horizontal polarization input while the reference beam polarization has been changed to horizontal (H), vertical (V), linear ±45°(+45 and −45), circular left (L), and circular right (R).

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**MDPI and ACS Style**

Kalbarczyk, A.; Jaroszewicz, L.R.; Bennis, N.; Chrusciel, M.; Marc, P.
The Young Interferometer as an Optical System for a Variable Depolarizer Characterization. *Sensors* **2019**, *19*, 3037.
https://doi.org/10.3390/s19143037

**AMA Style**

Kalbarczyk A, Jaroszewicz LR, Bennis N, Chrusciel M, Marc P.
The Young Interferometer as an Optical System for a Variable Depolarizer Characterization. *Sensors*. 2019; 19(14):3037.
https://doi.org/10.3390/s19143037

**Chicago/Turabian Style**

Kalbarczyk, Aleksandra, Leszek R. Jaroszewicz, Noureddine Bennis, Monika Chrusciel, and Pawel Marc.
2019. "The Young Interferometer as an Optical System for a Variable Depolarizer Characterization" *Sensors* 19, no. 14: 3037.
https://doi.org/10.3390/s19143037