Compact devices for the characterization of retarders based on a common-path interferometer

Compact devices for the characterization of retarders based on a common-path interferometer Ines Diaz-Garcia a , Jesus del Hoyo a, ∗ , Joaquin Andres-Porras a , Angela Soria-Garcia a , Mahmoud H. Elshorbagy b,c , Javier Alda b , Luis Miguel Sanchez-Brea a a Applied Optics Complutense Group, Optics Department, Faculty of Physics, Universidad Complutense de Madrid, Plaza de las Ciencias, 1, 28040, Madrid, Spain b Applied Optics Complutense Group, Optics Department, Faculty of Optics and Optometry, Universidad Complutense de Madrid, C/ Arcos de Jalón, 118, 28037, Madrid, Spain c Physics Department, Faculty of Science, Minia University, El Minia, 61519, Egypt Abstract Interferometry has been proven accurate for measuring the absolute retardance and azimuth of optical retarders, with the additional advantage over previous devices of not requiring the use of additional characterized retarders in the measurement process, but only linear polarizers. Retarder parameters are calculated from the fringe pattern displacement produced by the retarder rotation. In this work, we propose a common optical path conguration for removing interferometric instabilities. We have developed two devices, one based on a double slit, each one with a dierent linear polarizer ( 0 ° and 90◦ ) and another based on a Wollaston prism. We also developed a procedure to remove the additional shift produced by tilted or non-plane-parallel retarders on the fringe pattern upon rotation. First, we have analyzed the eect of using imperfect polarizers in the measurement process using numerical simulations. Then, we experimentally characterized several retarders using both interferometers, obtaining t errors below 0.1 ° and repeatabilities down to 0.3 ° . We also characterized a liquid crystal variable retarder using the Wollaston prism interferometer. In both cases, we compared our results with Mueller polarimetry, obtaining good agreement. 1. Introduction An accurate characterization of optical retarders is crucial for numerous experiments and devices such as biological tests, optical communications, optical microscopy, spectral ∗ [email protected] Preprint submitted to Elsevier October 23, 2025 2 analysis and so forth [1,2,3,4]. Also, they are used in polarimetry for both measuring the polarization properties of a sample or the polarization state of a light source [5,6]. Some techniques based on rotating polarizers, spectroscopy, nonlinear optical methods, etc. have been proposed to obtain the retardance and azimuth of the fast axis of the retarders with accuracy [7]. A simple method consists of placing and rotating the optical retarder to be characterized between two xed polarizers with their transmission axis perpendicular to each other. The retardance and fast axis azimuth are determined from the variations of the transmitted intensity [8,9]. However, this procedure does not allow to distinguish between the fast and slow axis. Xie et al. used amplitude division to characterize the retarder [10], but it requires two quarter waveplates and several Wollaston prisms and intensity detectors, and requires a very precise characterization of setup elements. Other procedures are founded on the interferometry principle. Nakadate [11] used a Young interferometer, Vargas et al. [12] a Fabri-Perot interferometer, and Schnoor et al. [13] a Sagnac interferometer to determine the retardance of retarders when the fast axis azimuth is known. Ming et al. used a heterodyne interferometric technique for measuring both the retardance and fast axis azimuth of the retarder [14]. Later, Chen et al. [15] used a Mach-Zehnder interferometer for the same purpose. Nevertheless, the rst procedure requires a very stable laser and the second is very sensitive to vibrations and temperature variations. These mechanical and stability limitations are present in previous methodologies. Moreover, Litwin et al. [16] presented a method of measuring the retardance of waveplates by a continuous phase registration and across the visible spectrum. Nevertheless, to employ this technique, it is necessary to use a variable wavelength light source and to previously know the azimuth of the fast axis. In [17] we proposed an experimental device based on a Michelson interferometer that allows to simultaneously obtain the retardance and azimuth of the retarders. It consists of a common-path interferometer in which the retarder rotation produces a displacement of interferometric fringes. The fringe displacement is related to the the retarder parameters according to Φm(θ) = −arctan sin ∆ cos[2(θ−θ0)] cos ∆ cos2[2(θ−θ0)] + sin2[2(θ−θ0)]!+ Φ0, (1) where Φm is the phase displacement produced by the rotation of the retarder, ∆ is the retardance of the retarder, θ is the rotation angle in the mount reference system, θ0 is the angle in the mount reference system where the fast axis azimuth corresponds to 0 ° in the lab reference system so θ−θ0 is the fast axis azimuth in the lab reference system, and Φ0 represents the arbitrary phase origin. The retardance can be obtained from Eq. 1 as the dierence between the maximum and minimum phase shift divided by two, ∆ = (Φmax −Φmin)/2 , as shown in Fig. 1. In addition, the azimuth is obtained as the angle of the motor where the phase shift is minimum, θ0= argmin [Φm(θ)] . Both parameters can also be obtained by tting Eq. 1with Φ0 , ∆ and θ0 as t parameters. 3 (a) (b) Figure 1: (a) Proposed methodology for obtaining the retardance and azimuth of the sample once the phase displacement is experimentally obtained. 2∆ is the double of the retardance and θ0 is the mount angle where the minimum shift is located. For this example, ∆ = 90 ° and θ0= 90 ° . (b) General scheme of interference phenomenon. However, Michelson interferometer experiment presented some drawbacks. Small variations in air and pressure produce a variation in the refractive index of air. Then, as Michelson interferometer arms are separated spatially, these variations produce shifts in the pattern. These variations are slow in time, so its eect can be minimized by reducing the number of measurements but not completely overcome. Also, when the interferometer is not adjusted so both arms have exactly the same length, tiny variations in the illumination wavelength also produce a shift in the pattern. These variations may be produced by a variation in LED temperature or variation in the dominating longitudinal mode of lasers. In this work, we have developed two dierent common-path interferometers that do not present these drawbacks: one based on a double-slit and another on a Wollaston prism. As both of them are common-path, air refractive index variations aect both arms almost identically, so no shift is produced. Also, common-path interferometers have the same arm length, so wavelength variations aect only the fringes period but do not produce any shift. Double slit interferometers have been commonly used for dierent interferometric applications. Walborn et al. proposed a quantum eraser based on the interference from a double slit [18]. Later, Wu et. al used several double slit interferometers together with imaging sensors to reconstruct non-axisymmetric temperature elds [19]. In particular, Nakadate used a Young interferometer to determine the retardance of retarders [11]. Wollaston prisms have also been used for polarimetry. Williams et al. developed a polarimeter based on a Wollaston beam-splitter polarizer [20]. Later, Xie et al. used three Wollaston prisms and a complex optical system to characterize retarders [10]. Also, Litwin et al. used a device based on a Wollaston prism to spectrally determine the retardance of retarders. Since the measurement process is based on the fringe displacement, we have identied an additional drawback consisting on the existence of an additional fringe pattern displacement 4 (a) (b) Figure 2: Deviation of the beam due to the fact that the retarder (orange element) (a) is not perpendicular to the optical axis or (b) its facets are not plano-parallel. when the optical retarder is not perfectly placed perpendicular to the optical axis or its facets are not plano-parallel, as we show in Fig. 2. This additional displacement, Φ tilt , is periodical over 2π . Then, when performing the experiment, the displacement measured is Φ=Φm+ Φ tilt . As a consequence, this displacement must be independently measured to correctly characterize the retarder. The process is described in Section 3. In the case of the Wollaston prism interferometer, we developed an experimental setup to measure it at the same time as the main experiment, avoiding the need to perform two dierent measurements. Summarizing, we present two new devices for characterizing retarders based on a commonpath interferometer, one of them consisting of a double-slit and the other of a Wollaston prism. In Section 2, we carry out an analysis of the experimental system together with the study of its tolerances by numerical simulations. We present the experimental setups that we have developed and the measurement and data processing process in Section 3. We show in Section 4the experimental results of the characterization of dierent optical retarders and a liquid crystal using both interferometers. These results are compared with Mueller polarimetry for validation [5]. 2. Simulations The mathematical description of the the proposed double-slit and Wollaston devices is the same to that of the Michelson interferometer [17], as they are based on optical path 5 dierences. As a consequence, Eq. 1is still valid to determine the fringe pattern for the ideal case. Nevertheless a complete description of the performance for a real device produce very complex equations. As a consequence, we performed numerical simulations, using the open source Python library Py-pol, to determine the capabilities of the device and its tolerances [21]. We tested the inuence of using real optical elements. Particularly, we varied the value of the maximum and minimum eld transmission coecients (p1, p2) and the retardance of the polarizers ( ∆p ). We also tested the inuence of experimental errors like using dierent light source ellipticity angle. Then, the phase displacement is calculated as described in the previous section. As an example, Fig. 3shows a graphical comparison between an ideal ( p1= 1 , p2= 0 ) and non-ideal case ( p1= 1 , p2= 0.06 ) for a theoretical retarder with ∆ = 120 ° and θ0= 90 ° . The dotted red line represents the simulated displacement while the blue line represents t to Eq. 1. The simulated data perfectly t the theoretical curve when optical elements are ideal. On the other hand, when polarizers and the light source are not perfect, the matching is not perfect. This is better observed in the residuals graph, calculated as the dierence between the modeled and simulated curve. The error for the ideal case is ∆Φ = ±0.05 ° and for the real case is ∆Φ = ±10 ° , the latter being increased due to the remarkable mismatch in the maximums, minimums and inection points. The standard deviation for each is case is σ= 0.05 ° and σ= 5.98 ° , respectively. Considering these interpretations of the simulation, we obtained the t results of ∆ t = 120.01 ° and θ t 0= 90.00 ° for ideal conditions, which is a very good agreement, and ∆ t = 130.39 ° and θ t 0= 91.19 ° for non-ideal conditions, which poses a great deviation from the real values. We carried out an analysis of the tolerances of the system by measuring the dierence between the nominal parameter and the calculated one. For that purpose, we considered two dierent deviations from the ideal case:  Polarizers with p1= 1 , p2∈[0,0.1] and ∆p∈[0,360 ° ] .  A retarder presenting a small diattenuation with p1= 1 and p2∈[0.9,1] . We calculated displacement of the fringes pattern for dierent values of ∆ , ∆p and p2/p1, with p1= 1 and θ0= 90 ° . Then, we calculated the apparent retardance ∆ t and azimuth θ t 0 . Finally, we calculated the errors as the dierence between the apparent and the real values. Fig. 4shows the retardance error for imperfect polarizers distributed throughout all the experimental setup. The error oscillates in a range of almost up to 30◦ depending on the polarizer parameters. This shows that it is important that the polarizers present a very low minimum transmission and it is preferable to employ polarizers of ∆ = 90 ° or 270 ° to accurately determine ∆. From this simulation, we also deduced that the phase displacement maxima and minima position do not vary, so θ t 0≈90.00 ° in all cases We have also analyzed the case of perfect polarizers but introducing diattenuation in the retarder for materials which accompany birefringence with dichroism. The retardance 6 (a) (b) Figure 3: Simulation of the phase displacement of a 120◦ retardance waveplate when considering (a) ideal optical elements and (b) real optical elements. For the latter case, the ellipticity of the light source is χ= 15 ° , polarizers possess p1= 0.74 , p2= 0.06 and ∆p= 180 ° and the sample retarders presents p1=p2= 0.74 . The red dots describe the simulated behavior of the phase and the one that would be expected. The blue line represents the result obtained with Eq. 1. Figure 4: Error in retardance estimation when the ratio p2/p1 of the polarizer is not null and ∆ of the polarizers is in the range from 0 ° to 360 ° . 7 Figure 5: Error in the determination of the retardance when the retarder presents diattenuation and perfect polarizers. error is shown in Fig. 5. We used several retardance values in the range from 1◦ to 179 ° , avoiding 0◦ and 180◦ because these are extreme values dicult to simulate and measure. The retardance error is negligible except for ∆>135 ° , and severely increases for ∆≈180 ° . The error does not reach exactly a null value for p2= 1 due to numerical error in the calculation procedure, as shown in the beginning of this section. The variation of the azimuth is negligible in all cases, lower than 0.003 ° . 3. Experimental setup and procedure We used two experimental setups. The rst one is based on a double slit with polarizers as shown in Fig. 6a. A collimated laser beam, whose wavelength is λ= 639 nm , passes through a polarizer ( P ) to control the amount of power entering the optical system and avoid the saturation of the camera. Then, a polarizer oriented at 45◦ ( P45◦ ) is used to set the polarization to 45 ° so the output intensity of both slits is equal. Afterwards, the Young double slit presents two polarizers, one for each slit, with orthogonal polarization axis ( P0◦ , P90◦ ). Then, the sample retarder ( Q ) is placed on a motorized rotation stage ( M ). Finally, the interference fringes are generated in the focal plane of a lens, using a polarizer oriented at 45◦ ( Pα 45◦ and Pβ 45 ). The camera is placed in the focal plane of the lens. 8 (a) (b) Figure 6: Scheme of the experimental setups. (a) Young double slit interferometer and (b) Wollaston prism interferometer. In this scheme P , P45◦ and P90◦ are polarizers, Q is the sample waveplate placed on the motor M , L is a lens with a focal distance of f= 25 cm, and CL is a cylindrical lens of f= 25 mm. Each dotted region refers to the portion of the image for measuring Φ tilt (green) and Φ tilt + Φ m (blue). When recording the additional shift produced by tiled or non-plane-parallel retarders, Φ tilt , the last polarizer is placed between the double slit and Q . This removes the polarimetric eect of rotating it. The other proposed experimental setup, based on a Wollaston prism, is represented in Fig. 6b. It consists of the same light source of λ= 639 nm and the polarizer oriented at 45◦ ( P45◦ ). Then, the Wollaston prism is followed by a linear polarized cut in half and oriented at 45◦ ( Pα 45◦ ) before the sample retarder ( Q ) mounted at the motor ( M ). After that, we placed a cylindrical lens ( CL ) is used so the two beams at the exit of the Wollaston prism merge spatially. Also, by adjusting the distance from the lens to the camera, the period of the interference fringes can be varied. Finally, another polarizer cut in half and complementary to the rst one, also oriented at 45◦ ( Pβ 45 ), is set before the camera. The cut polarizers are used to record at the same time the fringe displacement due to tilted or non-plane-parallel retarders and the characterization experiment, the top part is used to characterize Φ tilt , while using the bottom part Φm+ Φ tilt is measured. Both setups accomplish the aim of obtaining a more compact and stabilized system. The Young interferometer was the rst approach we analyzed and we checked that uctuations due to pressure, temperature and wavelength changes are eliminated. The Wollaston interferometer also fullls this requirement and allowed to signicantly decrease the size of the setup, leading to a more compact device. Also, Wollaston interferometer allowed us 9 to determine Φ tilt and Φm with a single measurement instead of two like with the Young interferometer. The interference pattern corresponding to each setup is also represented in Fig. 6. The images of these patterns are collected and post-processed as the sum all values in each column to obtain a 1D intensity prole for that retarder rotation angle, I(θ) . In the case of the Young interferometer, the power needs to be controlled using polarizer P to avoid saturating the camera, while this is not necessary for Wollaston interferometer, as the intensity is not focused in the y dimension. In the case of the Wollaston interferometer, this is performed independently for each half of the image avoiding the central region dominated by diraction of Pα 45 ° and Pβ 45 ° polarizer borders. From the obtained intensity proles, it is possible to calculate the spatial displacement of the fringes, ∆x . We used two dierent algorithms to calculate the fringe displacement. The rst method consists on calculating the correlation between I(θ) and I(0) . The position of the central peak is ∆x and the distance between peaks is the period, d . Then, Φ=2π∆x/d . The second consists on calculating the fast Fourier transform (FFT) of I(θ) , ltering the rst order, calculating its inverse FFT and then calculating Φ as the mean complex phase of the retrieved signal. We did this for both measurements, Φ1= Φ tilt and Φ2= Φ tilt + Φm . Then, Φm= Φ2+ Φ1. Finally, we t Φm to Eq. 1with ∆ , θ0 and Φ0 to calculate their values. Fit parameter error is calculated using the procedure described in [22]. Figure 7shows the experimental setups of the the schematics depicted in Fig. 6. From these images it is possible to appreciate a considerable dierent size from the double slit to the Wollaston prism, being the latter much more compact. The light source is a laser diode (MC6320CPWR-004S, Monocrom). Young interferometer setup used two dierent types of linear polarizers: thin lm Thorlabs LPVISE-100A for Pα 45 ° and Pβ 45 ° , and dichroic plastic Itos HN32 for the slits ( P0 and P90 ) as they are easily cut. The double slit is composed of two slits of 160 µm width separated 730 µm . A lens of f= 25 cm is used so the main diraction lobe lls the width of the camera. Wollaston interferometer also used Thorlabs LPVISE-100A for P45 ° and dichroic plastic Itos HN32 polarizers for Pα 45 ° and Pβ 45 ° , a Thorlabs WPQ10 wollaston prism with a separation angle between beams of ≈1◦ and a cylindrical lens of f= 25 mm. In both cases, the intensity images are recorded using a UI1492LE-M µ Eye camera chip. The sample retarder is placed on a motorized rotary stages (X-RSW60A-E03, Zaber). Sample retarders are mounted in the motor without considering the position of the fast axis respect to the motor reference system, so θ0 is random. 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