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Abstract
A liquid crystal variable retarder (LCVR) enables fast, automated control of retardance that can be used as a variable waveplate in polarimetric instruments. However, precise control of the polarization state requires calibration of the LCVR. A manufacturer calibration curve is typically supplied for a single specific wavelength and temperature, but for applications under different conditions, additional calibration is needed. Calibration is typically performed with crossed polarizers to generate an intensity curve that is converted to retardance, but this method is prone to noise when retardance is close to zero. Here, we demonstrate a simple common-path Sagnac interferometer to measure retardance and provide open source software for automated generation of calibration curves for retardance as a function of wavelength and voltage. We also provide a curve fitting method and closed-form functional representation that outputs the voltage needed to achieve a desired retardance given a specified wavelength.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
The ability to manipulate polarization is essential in many optical instruments, and automated control via a computer interface can improve speed and accuracy of changes to the polarization state. A liquid crystal variable retarder (LCVR) is often used for polarization state generating and polarization state analyzing, but requires prior calibration of the retardance as a function of applied voltage [1–3]. A Stokes vector is one way of describing polarized light and takes the normalized form of $\hat S$ in Eq. (1), such that ${I^2} = {Q^2} + {U^2} + {V^2} = 1$ [4]. This leaves $Q$, $U$, and $V$ to represent the three independent polarization states of vertical/horizontal linear, ${\pm}{45^ \circ}$ linear, and right/left handed circular, respectively. $Q$, $U$, and $V$ can all vary between ${-}1$ and one dependently. When one of these vector components is ${\pm}1$, the others are zero, and a pure polarization state is described:
LCVRs cause a variable retardance of the phase of one polarization component relative to the other by a voltage induced change in the alignment of birefringent molecules [6,7]. The rotation of these liquid crystal molecules provides a variable optical path length for one polarization component while the other remains fixed. The relationship between applied voltage and retardance is nonlinear and also depends on temperatures and wavelength.
Calibration of retardance as a function of input voltage is necessary for open loop operation. The wavelength of the incoming light must be taken into account due to the slightly dispersive crystals [6,8]. This calibration is critical in applications using several wavelengths of a laser.
Additionally, documentation suggests the temperature of the liquid crystal cavity affects the liquid’s viscosity and can cause an additional transient time delay for the crystals to align as well as slightly limiting the amount of attainable retardance by hindering the amount the crystals can tilt [1,6,7].
The retardance is maximum at zero volts and decreases as voltage increases, approaching but not reaching zero retardance. Manufacturers apply a compensating film with fast and slow axes opposite to the LCVR to reduce the total retardance and allow zero retardance to be reached [7]. This creates a particular voltage where the compensator’s retardance cancels the crystals’ retardance, and there is a true zero phase regardless of orientation angle and wavelength used. However, the combination can result in wavelength dependence of the zero retardance voltage due to the differences in dispersion of the materials and the wavelength dependence of the compensator.
Calibration of the LCVR and other polarization components has been performed previously, most notably using the crossed polarizers method [1–3,6,9–13], variations of Mueller matrix polarimetry [1–3,14,15], and Mach–Zender interferometry [13]. Previous attempts at mathematically describing the retardance curve consist mainly of look-up tables of calibration data [3,13], many parameter polynomial fits [13,15], exponential fitting over only a portion of the curve [15], and support vector machine learning [10]. In this paper, an alternative interferometric method is demonstrated that is insensitive to vibration, does not suffer from noise near intensity minima, lends itself to automation, and simplifies unwrapping of the retardance curve. A fit function is described that provides a closed-form equation with only five parameters for a single wavelength and residuals within $\lambda /60$ over the entire voltage range. Two additional parameters may be used to describe spectral dependence. The equation allows calculation of the voltage required to attain a desired retardance for simple open loop operation of the LCVR.
2. METHODS
A. Crossed Polarizers Method
The standard method for measuring retardance is the crossed polarizers method [6]. Two perpendicularly aligned linear polarizers (LPs) (LPVIS100, Thorlabs, Newton, New Jersey) are placed in the beam path such that they extinguish the beam. A power meter (Newport 1919-R, 918D-SL-OD3R) records the transmitted intensity. The device to be measured (LVR-200, Meadowlark Optics, Frederick, CO) is placed between the two polarizers, with fast and slow axes oriented at 45° from the two linear crossed polarizers. Figure 1 depicts the setup, with the LCVR’s fast axis at the labeled $\theta {= 45^ \circ}$ above horizontal. An electronic controller (D5020, Meadowlark Optics) is used to modulate the LCVR’s input voltage and temperature, changing the resulting intensity through the setup. The intensity is normalized using ${I_{\text{norm}}} = (I - {I_{\text{min}}})/({I_{\text{max}}} - {I_{\text{min}}})$ to account for the background ${I_{\text{min}}}$ and the maximum transmitted intensity ${I_{\text{max}}}$ acquired during the voltage scan (corresponding to LCVR retardance of half and whole waves, respectively) [9]. The normalized intensity is in the range of [0,1], as seen in Fig. 1(a), and has a simple relationship with retardance $\delta$ in Eq. (3) [1,6,9]. This equation calculates all retardances to be in a range of $[0,\pi]$ [Fig. 1(b)], which requires uncoiling: inverting and vertically translating sections of the curve between extrema to create a continuous curve [Fig. 1(c)]. Here, we use the term coiled/uncoiled to distinguish it from the simpler case of wrapped/unwrapped shown later. This can be done in a few lines of code, but is still slightly more complicated than a simple built-in unwrap() function [3,9,12,13,15]. Finally, there is ambiguity as to which maximum corresponds to zero retardance, so one must be chosen, and the curve is then vertically shifted or aligned to that value:
Fig. 1. Crossed polarizer configuration and illustration of data analysis steps. The (a) intensity is recorded as a function of voltage, normalized and converted to (b) retardance using Eq. (3) and then uncoiled and vertically aligned to produce the (c) calibration curve.
One challenge with this common method is that noise increases near the intensity minima ${I_{\text{min}}}$ leading to errors in the uncoiling procedure. Equation (3) indicates that the intensity will drop to zero for retardances $\delta = \pi$ and odd integer multiples of $\pi$. Weak intensities are both challenging to detect as well as prone to noise from low SNR. To avoid the detection issue, a powerful source should be used to ensure that as much of the intensity curve falls within the detector’s dynamic range as possible. To improve SNR, a dark environment is required to minimize background. Even with these optimizations, there is still difficulty obtaining continuity after uncoiling the retardance curves.
B. Co-polarizers Method
A similar process occurs when both LPs are oriented in the same direction [2]. By rotating the second LP in Fig. 1 to match the first LP at 0° and keeping the LCVR at 45°, this method can be easily adapted from the crossed polarizers method setup. Despite being so similar to the crossed polarizers method, this method is much less commonly used. Analysis for this method is similar to the crossed polarizers method, replacing the cosine in Eq. (3) with sine to give Eq. (4):
The uncoiling and noise issues are identical to the crossed polarizers method, but occur at retardance values that are even integer multiples of $\pi$. These two methods could be combined to reduce the noise around intensity minima.
C. Interferometric Method
A new method based on the polarization sensitive Sagnac interferometer in Fig. 2 was also used to characterize the LCVR. This method avoids the difficulties of detecting weak intensities as in the crossed and co-polarizers methods.
Fig. 2. Common-path interferometer schematic and illustration of the data analysis steps. The (a) fringe pattern is averaged over the pixel rows, and the FFT is calculated with the (b) absolute value exhibiting a peak at the fringe frequency. This is repeated for each voltage, and the phase of the fringes is calculated to produce a (c) wrapped retardance curve, which is (d) unwrapped and vertically aligned.
In this configuration, a laser source was coupled into a single-mode optical fiber (Thorlabs S405-XP) and almost collimated using a 10x objective (10x HC PL APO, Leica Microsystems, Wetzlar, Germany) placed beyond focus to diverge the beam and sufficiently fill the camera sensor. The output beam was polarized by a LP (LPVIS100, Thorlabs, Newton, New Jersey), LP1, rotated at 45° relative to the vertical axis. This polarization state is equally split in intensity by the polarizing beam splitter (PBS) (PBS251, Thorlabs) transmitting horizontal polarization and reflecting vertical. Two mirrors angled at 22.5° create a common-path Sagnac interferometer with an even number of reflections. The even number of reflections ensures that one beam is not inverted relative to the other on the camera [4]. The LCVR is placed in the beam path so both polarizations pass antiparallel through its aperture. The beams recombine again at the PBS. A LP, LP2, at 45° passes a component of each, allowing interference to be observed between the beams.
The tilts of M1 and M2 are carefully adjusted to produce vertical fringes on the camera’s sensor array. The common-path geometry ensures no defocus between the diverging beams, resulting in only straight tilt fringes. The resulting fringes of the interferogram are recorded by a CMOS camera (Thorlabs DC3240N), as in Fig. 2(a). The number of fringes on the camera is not critical: consistent and accurate data were acquired with anywhere between six and 150 fringes across the 1280 pixel wide camera image. Voltage induced retardance in the LCVR changes the phase delay between the two counter-propagating beams, resulting in a lateral shift of the fringe pattern. A phase difference of one wave will shift the fringe pattern by one complete fringe period, thus allowing the retardance to be recorded as a relative shift in fringe pattern. Aligning the fast axis of the LCVR head to the horizontal axis ensures each beam encounters only one of the LCVR’s birefringent axes and simplifies the matrix from Eq. (2) to Eq. (5) by asserting $\theta = 0$:
The retardance $\delta$ is calculated from the fringe shift. Averaging the columns of the camera frame helps to reduce noise and dust artifacts and results in an approximate sinusoid. A fast Fourier transform (FFT) is applied to the mean intensity vector. The absolute value of the FFT exhibits a strong peak corresponding to the number of fringes in the frame, as seen in Fig. 2(b). Note that an FFT’s first value is the zero frequency contribution, so if the peak occurs in the 10th element of the array, this corresponds to $(n - 1)$ or nine fringes across the frame. The phase of the lateral shift of the interference fringes is then calculated by taking the phase of the Fourier space peak: the arctangent of the ratio of the imaginary and real components of the FFT at the peak, as in Eq. (6) and as seen in Fig. 2(c). The retardance values are phase-unwrapped using MATLAB’s built in unwrap() function, resulting in a plot similar to Fig. 2(d). This mathematical interpretation is most similar to the Mach–Zender interferometry method [13] and is nearly identical to methods used for birefringence imaging techniques [16]:
Applying this calculation for camera frames over a range of input voltages produces the signature LCVR retardance curve. The voltage sweep can then be performed for different wavelengths, cavity temperatures, and LCVR orientation angles to provide a more comprehensive characterization. The discontinuity in the wrapped retardance curve occurs at $\delta = \lambda /2$, and this can be used to align the vertical axis. As can be seen in Fig. 3, the discontinuity is sharpest and easiest to determine programmatically when the LCVR is rotated 45° by finding the maximum of the absolute value of the phase derivative with respect to voltage. This allows unambiguous determination of the voltage that gives a half-wave retardance for a given wavelength.
This setup can also be adapted for the co-polarizers method by simply rotating the LCVR so that its fast axis is 45° above horizontal and replacing the camera with a power meter. A camera may be used, but the dynamic range is far lower than a power meter, resulting in limited SNR. The PBS then acts as both of the aligned LPs in the co-polarizer configuration, and there are two counter-propagating beams that pass through the setup. The LP before the beam splitter in Fig. 2 must be kept to nullify any existing retardance from the source polarization and can be rotated to control the intensity balance of the clockwise and counterclockwise propagating beams. The LP after the beam splitter is unnecessary, as the two beams need not be recombined—this method uses intensity rather than interference to quantify retardance. Equation (4) as well as the same uncoiling as the crossed polarizers method must also be applied for this setup of the co-polarizers method.
3. EXPERIMENTAL RESULTS
LCVR calibration data were collected for different temperatures, angles of orientation, and input wavelengths. For all data shown, voltage was swept from 0 V to 10 V. Unless otherwise indicated, the interferometric method was used. Data were acquired and analyzed with MATLAB.
A. Temperature Dependence
The LCVR device used here has active heating, but relies on passive cooling, so temperatures were set in an increasing sequence to minimize acquisition time [7]. Data were collected for temperatures from 22.5°C to 50°C in 0.5°C increments using a He–Ne laser (Thorlabs HNLS008L) at 632.8 nm for the source and with LCVR angle at $\theta {= 0^ \circ}$.
The resulting curves for this range of temperatures were very similar, with lower temperatures achieving a slightly larger range in retardance across the 10 V. The difference between the maximum and minimum retardances for each temperature is plotted in Fig. 4 and shows a decrease in retardance range as a function of temperature. The slope of the fit (${-}{0.0022} {\pm} {0.001}\;{\rm rad/ ^\circ {\rm C}}$) corresponds to a ${-}{0.27}\% {/^\circ {\rm C}}$ change with temperature at 30°C and is consistent with the reported specification in the manufacturer’s information sheet (approximately ${-}{0.4}\% { /^ \circ {\rm C}}$) for this LCVR model [7].
Fig. 3. Wrapped retardance curves for different LCVR orientation angles converge at even $\pi$ multiples of retardance at 1.60 V and 4.45 V. The 45° curve has a sharp discontinuity at 2.35 V, marking the half-wave retardance.
Fig. 4. Maximum range of retardance decreases with temperature. A linear fit with slope ${-}{0.022}\;{\pm}\;{0.001}\;{\rm rad/^\circ{\rm C}}$ corresponds to ${-}{0.27}\% {/^\circ{\rm C}}$ change at 30°C.
The Sagnac interferometer configuration minimizes sensitivity to vibration, and much of these data were even collected on a wooden basement workbench with no vibration isolation, yet initial measurements exhibited noise greater than 10 nm standard deviation for repeated measurements at the same voltage. This was eventually attributed to turbulent air currents and was greatly reduced to less than 2 nm by shielding the beam path with foil. However, as the temperature is increased, the noise increases, as can be seen in Fig. 4. This is due likely to increased convection and suggests that a temperature slightly above ambient may be optimal for most applications.
B. Angle Dependence
LCVRs with compensators are able to achieve zero retardance, at which point rotation of the device about the propagation axis has no effect, and the curves for different fast axis orientations converge to the same value. To find this voltage, the He–Ne laser was used with the LCVR at ambient temperature (a consistent ${21.9^ \circ}{\rm C}$). Retardance curves were obtained as a function of fast axis orientation angle, $\theta$, between ${-}{5}^\circ$ and 50° above the horizontal and plotted such that the mean of the flat 45° curve is set as zero retardance in Fig. 3. The crossing point could be used to determine the voltage for zero retardance, but this relies on the intersection of multiple curves acquired separately and can suffer from instability. At 45°, the curve should be independent of voltage since the delayed polarization component is equally split between horizontal and vertical. However, since the polarizers and beam splitter are imperfect polarization elements, some variation is observed with a sharp discontinuity near $\lambda /2$. This voltage is easy to identify automatically, as described in Section 2.C, and is used to set the retardance axis for all subsequent data.
C. Spectral Dependence
The LCVR was tested with a range of different laser wavelengths: 500 nm to 680 nm in increments of 10 nm using a tunable white-light laser (SuperK EXR-15 with AOTF, NKT Photonics, Birkerød, Denmark). The temperature was set to 25°C, and all wavelength data were taken for LCVR angles of 0° and 45°. The 45° data were used only to determine the half-wave retardance value to vertically align the 0° data. Figure 5 shows spectral dependence of retardance in both radians and nanometers of optical path difference. The retardance in radians decreased as a function of wavelength as expected, but absolute path difference in nanometers also decreased with wavelength, due likely to dispersion in the LCVR and accompanying compensator.
Fig. 5. LCVR retardance exhibits a decrease as a function of wavelength from 550 nm to 680 nm even when expressed in nanometers, due likely to dispersion of the LCVR and compensator. Also shown in each plot in black is the manufacturer’s calibration data.
In previous experiments, large (${\gt}{100}\;{\rm nm}$) deviations were observed for data at 0 V. These deviations resulted from residual molecular tilt when returning to 0 V from the 10 V of the previous acquisition. A pause of 0.2 s between setting the voltage to zero and data acquisition was found to be sufficient to eliminate the effect. It was found to be necessary to pause only when two subsequent voltages had expected retardances that were considerably different (${\gt}{100}\;{\rm nm}$).
4. MODELING THE RETARDANCE CURVE
A closed-form expression for the LCVR calibration curve is desirable since by inverting the expression, the voltage needed to achieve a desired retardance can be determined for a given set of conditions (e.g., wavelength and temperature) using only a small number of fit coefficients. Previous efforts to fit the LCVR curve are described in Section 1. Our approach attempts to minimize the number of coefficients while maintaining a fit over the entire range of voltages that can be applied. The equation $\phi = a + b/(1 + {(V/c)^d}{)^e}$ was found to match the shape of the retardance curves in linear and log–log space. Each parameter controls a different characteristic of the curve: $a$ is the lower bound at the limit $V \to \infty$, $b$ is the maximum retardance at 0 V, $c$ controls the shoulder where the retardance fall off begins, $d$ controls the sharpness of the shoulder, and $e$ controls rate of decay (slope in log–log space) of the asymptotic tail. The performance of this expression is shown in Fig. 6 where it is applied to the manufacturer supplied retardance data with ${R^2} = 0.99996$ and exhibits residuals less than ${\pm}9\;{\rm nm} $ (2.1 nm standard deviation).
Fig. 6. Using Eq. (7) to fit the manufacturer’s supplied data for 633 nm provides a concise, accurate analytic model for retardance curves with residuals all within ${\pm}9\;{\rm nm} $.
Fig. 7. Expanding Eq. (7) to account for the spectral dependence of parameter $b$ allows the model to be robust across a range of wavelengths. This surface fit for wavelengths from 550 nm to 680 nm provides a maximum residual of ${\pm}20.0\;{\rm nm} $ and standard deviation of 6.61 nm.
To describe spectral dependence, only parameter $b$ was expanded, as shown in Eq. (7). The wavelength dependence of parameters $a$, $c$, $d$, and $e$ was slight, and these were left constant to minimize over-fitting. Retardance is related to the change in refractive index, and so a two-term equation of the form of the Cauchy model of refractive index was used. This gives the parameters ${b_1}$ and ${b_2}$ such that $b(\lambda) = {b_1} + {b_2}/{\lambda ^2}$. Linear and quadratic models (not shown) provide a similar but slightly worse spectral fit. If the spectral bandwidth were increased sufficiently to warrant more coefficients, a Sellmeier equation model could be employed. Least squares fitting was used, and the parameters calculated for this particular LCVR along with the inverse equation are shown in Eq. (7). These parameters will vary depending on what wavelength the LCVR is designed for as well as manufacturer and model:
The surface fit in Fig. 7 was then created over the ranges of voltages and wavelengths tested with residuals within $\pm {19.6}\;{\rm nm}$ and standard deviation 5.61 nm from the collected data. The MATLAB scripts used to acquire, analyze, and generate fit coefficients is available in the Rogers Lab GIT repository [17].
5. COMPARISON OF METHODS
Due to their configurational and mathematical similarities, the crossed polarizers and co-polarizers methods are grouped together in this comparison. To illustrate some of the differences between the interferometric method and the crossed/co-polarizers methods, data were collected across many wavelengths for the crossed polarizers configuration with a wavelength sensitive power meter. The same source, LCVR, and LPs were used. The data from one acquisition (without averaging) are plotted in Fig. 8 for both methods.
Fig. 8. Wavelengths from 560 nm to 680 nm in 20 nm steps are used to compare data (without averaging) from two methods of LCVR calibration.
The interferometric method was introduced to avoid the limited sensitivity of the crossed polarizers method for low intensities around odd multiples of $\pi$ retardance. Noise in these regions cause data irregularities, as seen in Fig. 8. Uncoiling the data is also challenging because the noise is maximum around these points, and even the manufacturer’s data in Fig. 6 omit values around these points. By averaging intensities over a longer period of time, the noise in this method can be reduced. Similar averaging could also be employed for the interferometric data to reduce noise levels if necessary.
The interferometric method also exhibits noise, but more uniformly over the range of retardance. To quantify noise, Eq. (7) was fit for each method individually, and residuals were calculated for $n = 14$ wavelengths (every 10 nm from 550 nm to 680 nm). The comparison of these residuals is illustrated in Fig. 9 for only every 20 nm to reduce clutter. These residual plots show that in general, the crossed polarizers method has slightly lower noise away from the intensity minima, but greater absolute residuals across the entire data range. In addition to the differences in magnitude and distribution of noise, the unwrapping of the interferometric method is simpler. The crossed polarizers method involves identifying the maxima and minima (where noise is greatest), inverting, and shifting the data at these points. The interferometric method produces a simpler jump discontinuity that is easily unwrapped using a simple built-in MATLAB command.
Fig. 9. Comparison of the residuals between the two methods’ data and smooth curve fittings shows how the interferometric method is more stable at $\pi$ multiple retardances, but the crossed polarizers method is less noisy overall.
The interferometric method also has disadvantages. Physically, the setup requires more parts and alignment. It is also sensitive to air currents such as the convection currents caused by a heated LCVR. For the analysis, the largest drawback lies in the arbitrary initial fringe phase. To determine the zero retardance point requires a second measurement with the LCVR rotated to 45°. Ultimately, it is beneficial to use both methods in tandem to calibrate the LCVR and to average over many measurements.
6. CONCLUSION
The common method for determining calibration of LCVRs uses crossed polarizers and suffers from increased noise around intensity minima and an error prone unwrapping method. A common-path Sagnac interferometer can be used as an alternative and provides some advantages with more uniformly distributed noise and requires simpler unwrapping. Each method has advantages and disadvantages, making the additional method a useful option. The interferometric method uses simple components and a low-cost camera, which are often available in an optics laboratory. In addition, a new model equation is presented that accurately describes the calibration curve over the entire operating voltage range. This equation can be inverted to calculate the voltage needed to obtain a desired retardance given parameters such as wavelength. The open source code to acquire and analyze data and fit this model equation are provided via GIT.
Funding
Retina Research Foundation (Edwin and Dorothy Gamewell Professorship); National Institute of Biomedical Imaging and Bioengineering (R21EB025513); National Science Foundation (1845801).
Disclosures
The authors declare no conflicts of interest.
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