Technical Note: A universal method for measuring the thickness of microscopic calcite crystals, based on Bidirectional Circular Polarization

Abstract. The coccoliths are major contributors to the particulate inorganic carbon in the ocean that is a key part of the carbon cycle. The coccoliths are few microns in length and weigh few picograms. Their birefringence characteristics in polarized optical microscopy has been used to estimate their mass. This method is rapid and precise because camera sensors produce excellent measurement of light. However, current method is limited because it requires a precise and replicable set up and calibration of the light in the optical apparatus. Precisely, the light intensity, the diaphragm opening, the position of the condenser, and the exposure time of the camera have to be strictly identical during the calibration and the analysis of calcite crystal. Here we present a new method that is universal in the sense that the thickness estimations are independent from a calibration but results from a simple equation. It can be used with different cameras and microscope brand. Moreover, the light intensity used in the microscope does not have to be strictly and precisely controlled. This method permits to measure crystal thickness up to 1.7&htinsp;μm. It is based on the use of one left circular polarizer and one right circular polarizer with a monochromatic light source using the following equation: d = λ/πΔn arctan (√(ILR/ILL)) where d is the thickness, λ the wavelength of the light used, Δn the birefringence, ILR and ILL are the light intensity measured with a right and a left circular polarizer. Because of the alternative and rotative motion of the quarter-wave plate of the circular polarizer, we coined the name of this method Bidirectional Circular Polarization (BCP).



Introduction
Coccolithophores are abundant oceanic single cell algae that produce calcite plate called coccoliths. The coccoliths are major contributors to the particulate inorganic carbon (i.e., PIC) in the ocean that is a key part of the carbon cycle. The coccoliths are 25 so minute (few microns in length) and light (few picograms) that they can be weighed individually only with extreme labor and expensive apparatus (Hassenkam et al., 2011;Beuvier et al., 2019). Alternatively, the birefringence characteristic of coccolith in polarized optical microscopy has been used to estimate their mass (Beaufort, 2005;Beaufort et al., 2014;Bollmann, https://doi.org/10.5194/bg-2020-28 Preprint. Discussion started: 4 February 2020 c Author(s) 2020. CC BY 4.0 License. 2 2014; Fuertes et al., 2014). This method is rapid and precise. The camera sensor produces excellent measurement of light. The camera sensor measures the light that had come through the polarizers and a calcite crystals to convert into a thickness value. 30 The estimation made by this method has been recently positively evaluated by the independent measurements made by X-ray tomography at the European Synchrotron Radiation Facility (ESRF) (Beuvier et al., 2019). One of its limitations is that it needs a precise calibration of the lightness of the microscope. The light intensity, the diaphragm opening, the position of the condenser, and the exposure time of the camera, have to be strictly identical between the calibration and the analysis of the calcite crystal. Slight change on one of those parameters have important consequence on the results. Another limitation is that 35 the measured light intensity is not linearly proportional to the thickness but follow a sigmoid (Beaufort et al., 2014;Bollmann, 2014) making difficult to estimate the thickness precisely at the two ends of the calibration. The use of standard polychromatic « white » light induce a small imprecision, because the temperature of light that depends on the microscope -some have a bluish light other have it more yellowish -will change slightly the result if not calibrated. There is a theoretical limit of the thickness estimation to about 1.56 µm when using a black and white camera. The estimation of calcite particles thicker than 40 1.56 µm needs to be done with a color camera with several calibration equations (Beaufort et al., 2014;González-Lemos et al., 2018). Here we propose a new method that solve those problems: the estimations are not the results of a calibration, they can be applied to crystals as thick as 1.7 µm, and are not dependent on the precise tuning of the light of the microscope.

Principles
The representation of the polarized light is based on Jones's calculus (Jones, 1941). The microscope is composed of two 45 circular polarizers -one left oriented and the other right oriented -used alternatively and one circular analyzer.

Jones Matrices
For an anisotropic material having its ordinary neutral axis horizontal, Jones matrix is given by 8 = : where T is the (complex) transmission coefficient, is the diattenuation, and is the retardation, with = F$ # (where λ 50 is the wavelength, is the birefringence, is the thickness).
If the neutral axis is rotated by an angle , the Jones matrix becomes

Proposed measurement scheme
Assuming that = 0 (no diattenuation), the input field is left-circularly polarized

Retrieving thickness
One can see that 55 and 56 do not depend on the orientation of the neutral axes.
Moreover, the ratio

75
does not depend on the transmission coefficient .
In the case that we can assume that then there is only one solution, , to Eq (1) : ,- Therefore the thickness can be estimated by grabbing two images of a thin calcite crystals, taken one through a right circular polarizer ( 56 ) and a second through a left circular polarizer ( 55 ). 55 has a dark background and calcite crystals appear lighter. 56 has a light background and calcite particles appear darker. They are negative images of each other (Fig. 1a). The ratio The methodology presented here was developed on a Leica DM6000 microscope, with a x100 objective having a numerical aperture of 1.47, and a condenser lens having a 1.2 numerical aperture. Three circular polarizers made by Chroma Technology Corp. are integrated in the microscope. One right circular polarizer is positioned as analyzer. It consists of a linear polarizer oriented at +90° placed below a quarter-wave plate oriented at +45° mounted in a Leica cube and placed in the upper automatic 90 turret of the microscope. This is a convenient place when one wants to automatically remove this analyzer to use other filters.
If this not necessarily the analyzer can be placed in its regular position.
One (and one) left (right) circular polarizer, LCP (RCP), made of a quarter-wave plate oriented at 45° (-45°) followed by a linear polarizer oriented at 0°. These two polarizers are used alternatively when taking images of the same crystal. LCP and RCP are placed in the revolving filter chamber of the automated condenser block. For a manual use, a quarter-wave plate could 95 be placed under a linear polarizer, and rotated manually from -45° (LCP) to 45° (RCP).
One of five monochromatic bandpass filters centered at 435, 460, 560, 655, and 700 nm (AT435/20X, AT460/50M, ZET561/10X, AT655/30M and ET700/50M; all from Chroma Technology Corp.) is positioned in the light trajectory after the light bulb. The 561 nm filter is used in routine work because of its versatility (see below). The other filters are used to test the method and in special occasions when study of relatively thick calcite particles in the range of 1.4-1.9 µm in the case of 100 700 nm; or for detail measurements of thin particles in the range of 0.2-0.4 nm for the 435 nm filter.
Two black and white numerical cameras are set up. A SpotFlex from Diagnostic Instrument, with a CCD image sensor of 2048x2048 pixels that are 7.4 µm large. It is a 14-bit camera (16383 grey levels in depth). And an Orca Flash 4.0 V2 from Hamamatsu, with a CMOS image sensor of 2048x2048 pixels that are 6.3 µm wide. It is a 16-bit camera (65548 grey levels in depth). The tests of this method presented in results have been made with (i) surface sediment retrieved in the Southern 105 Pacific and spread onto a slide, and (ii) calcium carbonate crystals precipitated onto a slide.

Results
To test the quality of the thickness estimations with the BCP method, the same field of view has been studied in different light conditions (lightness, opening, and wavelength) and with different cameras. In each condition, the two images 55 and 56 are captured and used to compute the thickness , with Equation 2. In some cases, in order to illustrate , an image frame B in 8-110 bits, was computed using the following equation: where abc represent the maximum measurable thickness at a given wavelength. It is calculated using the following equation: For calcite crystals, abc ranges between 1.17 µm at 405 nm and 2.03 µm at 700 nm. 115 https://doi.org/10.5194/bg-2020-28 Preprint. Discussion started: 4 February 2020 c Author(s) 2020. CC BY 4.0 License.

Lightness
The same field of view was captured at different time exposures with the SpotFlex camera. Exposure time is the simplest way to change the lightness of an image. Figure 2 shows that the fields of view captured at short exposure time (e.g., 5 ms) are extremely dark and conversely those captured at long exposure time (e.g., 320 ms) are light with many saturated areas (maximum Grey Level (GL) values). Except for those two extreme expositions (i.e., 5 ms and 320 ms) the GL values are 120 identical. In Fig. 3 the histograms of 55 , 56 and are shown. At 320 ms the images are too light, and many areas are saturated both in 55 and 56 and thus have the same GL values. Knowing that the solution of Equation 2 is 0.81 µm when 55 = 56 and = 561 nm, a spurious density peak appears in the histograms at a thickness of 0.81 µm with an exposure time longer than 320 ms (Fig. 3). In areas where 55 is saturated but not 56 , the estimations are shifted toward thicker values, explaining the thicker density pick found at 0.7 µm in the histogram of 320 ms (Fig. 3). The image background, materialized in the histograms 125 by the first peak, is around 0.1 µm for all exposures but is shifted toward higher thickness up to 0.2 µm at 320 ms.
At 5 ms, the images are too dark to provide correct estimation of the background level ( Fig. 3) which, in turn, increases noise in the results. Therefore, in order to get correct thickness values, it is important to avoid too low or too high lightness. Between those extremes light conditions, the estimates of thicknesses are independent from the lightness. To get the maximum depth details, it is suggested to use the maximum light before saturation in 55 , providing the highest range of grey levels in both 130 images. In the example given in Fig. 2, this maximum detail would be achieved between 80 ms and 160 ms.
The optical setting used in this experiment was not able to produce the darkest values (close to 1) and lightest value (equivalent to 255 in 8-bit). The reason why those extreme values are not reached is largely due to the imperfections of the circular polarizers that are composed of two layers. Those imperfections are amplified at the extremes of the light ranges because of the sigmoid shape of the thickness function (Fig. 1). In practice, the ratio 56 / 55 is reached in the flattest part of the sigmoids 135 ( Fig. 1b), for example between 0.10 µm and 1.41 µm with 561 nm light wavelength. In consequence, the thickness measured in absence of particle was 0.10 µm at 561 nm when it should be 0. Also, the maximum measurable thickness is lower than the maximum theoretical thickness: using a wavelength of 561 nm, we obtain a maximum of 1.45 µm of thickness instead 1.62 µm (Fig. 3).

Aperture 140
The illumination tuning of the microscope is also important. The range of measurable thickness is largest when the condenser is focused and centered following the Köhler illumination (Köhler, 1894). More the field diaphragm is close, wider is the range of measurable thickness (Fig. 4). Hence, both diaphragms (i.e., field and aperture) should be closed at their maximum in order to maximize the range of measurable thickness.

4.3
Camera Type 145 The two tested camera types (CMOS vs CCD; 14-bit vs 16-bit; different brand) produced the same measure. The same view field was captured with two different camera type without measurable difference between the two resulting thickness images (Fig. 5).
The theoretical maximum measurable thickness ( fabc ) depends on the number of grey levels ( ) achieved by the camera: fabc = ij 1 k 150 At = 561 , fabc is 1.565 µm with an 8-bit camera, 1.622 µm with a 14-bit camera and 1.626 µm with a 16-bit camera.
These fabc are far above the maximum measurable thickness of 1.45 µm described in section 5.a. However, the low depth resolution of an 8-bit camera should further limits the range of measurable thickness, although this was not tested here. Hence, both 14-bit and 16-bit can be used but we don't recommend to use 8-bit camera.

Accuracy and Precision 155
It is extremely difficult to estimate the measurement error in the present case because there is no standard material for thickness comparison in the range of few nanometers. The thickness of the wedge used to estimate the accuracy in González-Lemos et al. (2018) is measured at 250 nm intervals which is not enough in our case. Also, its measurements are based on a birefringence principle that is not strictly independent from our methodology. However, González-Lemos et al. (2018) clearly validate the accuracy of birefringence method at 250 nm. The measurement of coccoliths made by coherent X-ray diffraction (CXDI) at 160 ESRF (Beuvier et al., 2019) require the use of silicon nitride (Si3N4) TEM windows influencing birefringence. Hence, those coccoliths cannot be used later as standard. However, in this study, coccoliths mass and size measurements from the same culture using both birefringence and CXDI provide a comparison on statistically similar results. The validity of the birefringence method is also demonstrated, although without giving a value to the accuracy. The use of cylindric rods such as rhabdoliths (Beaufort et al., 2014;Fuertes et al., 2014) is limited by the precision of the microscope used to produce the 165 measurement of their diameter, around 0.2 µm in our microscope. The BCP method does not use any calibration, it is therefore theoretically absolute. It is accurate in the range given by the inflection points in Fig. 1.
We determine the precision of the BCP method at the five different wavelengths by using the two cameras on the same 7.74 µm transect of a Pontosphaera japonica (Fig. 6), producing 10 series of measurements. At the difference with Fig. 5, we voluntarily did not produce identical focus and use different wavelengths in order to produce generalized values. The root-170 mean-square error (RMSE) between two series is used to determine the precision of the method. When the wavelength are separated, the 5 RMSE range between 14 nm and 47 nm. The large RMSE values result from different focuses and/or red colors (635 nm and 700 nm). Best results were obtained at 561 nm and 435 nm with similar focus. When one series of measurements was compared to the average of all the other series, the RMSE = 32 nm. When it is limited to 435 nm to 561 nm, the RMSE = 12 nm. As we explain in detail in the next section, longer wavelengths in red lower the precision. This is an order 7 of magnitude smaller than the spatial optical resolution which ranges between 150 nm and 240 nm in the present microscopic setting at the 5 different wavelengths. The precision of the BCP method is expected to be smaller in many cases. For example, the RMSE in the transect of Fig. 5 is 5 nm. The difference of RMSE between Figs. 5 and 6 is related essentially to the focus that was well reproduced in Fig. 5. The estimated masses of P. japonica in Fig. 6, is ranging from 65.3 pg to 69.9 pg with a standard deviation of 1.28 pg (N=10) and depends again, on the wavelength and the focus. 180

Wavelength and range of mesurable thickness
The comparisons of the same transects captured at different wavelengths along an image frame containing thick CaCO3 particles emphasize the advantages and limits of each light wavelength. The range of thickness measurable at a given wavelength is presented in Fig. 7. In the transects, a plateau is reached at the maximum practical thickness (MPT) ; and when the particle thickness is about 0.5 µm above the MPT, the thickness values decrease. It is not entirely clear why MPT is about 185 84% lower than the maximum measurable thickness ( abc ). This difference has been described earlier (Bollmann, 2014). This discrepancy could be resulting from the quality of circular polarizers used. The circular polarizers are made with polaroid filters that are not perfect and are composed of two filters -a quarter-wave plate and a polarizer -creating some imperfections.
As an example, linear polarizers exhibit generally larger range of grey levels with darker background than circular polarizers.
For the study coccoliths thicker than 1 µm like those of the Eocene, we recommend to use a light with long wavelengths (e.g., 190 red at 700 nm). On the contrary, for the study of thin coccoliths such as most extant and Pleistocene species, we recommend to use shorter wavelengths (e.g., green or blue). Short wavelengths reached a MPT at lower thickness but offer higher precision in the measurement of the thickness and higher optical resolution permitting higher precision in the measurement of the area.
The distal shield of Emiliania huxleyi coccoliths illustrate well an extreme measurement cases where the lower wavelength has to be used to get a precise thickness and mass measurements. The distal shield of E. huxleyi is constructed with thin -195 ~100 nm -elements that do not touch each other (Fig. 8). The detection of those elements above the background is extremely difficult using wavelength at 700 nm but is possible using wavelength at 435 nm. In consequence, mass measurements are underestimated at 700 nm because the distal shield is not completely detected and producing a total area smaller than it is really (Table 1). Finally, this new method cannot give accurate results for calcareous nannofossils) thickness above 1.7 µm like Cretaceous Nannoconus species. For such material, we recommend to be critical with results close to MPT and to use a 200 color camera (Beaufort et al., 2014;González-Lemos et al., 2018) as in Fig. 7, although less precise than the BCP method related to some calibration issues (González-Lemos et al., 2018).

Conclusions
The alternative use of left and right circular polarization permits to measure the thickness of calcite crystals in a universal manner without precise calibration of light. The BCP method has a great advantage from previous methods for which it is 205 difficult to maintain stable light (i) in time (i.e., bulb aging, condenser vertical position,…) and (ii) in space since the field of 8 view may not be uniformly illuminated (i.e., low quality lens, uncentered condenser, ...). In all situations mentioned above, the previously published linear or circular polarizer methods will provide different thicknesses measurements whereas the BCP method described here will provide the same values. The choice of the wavelength of the light used for the measurements is specific to a targeted thickness. Thicker crystal will require longer wavelengths. Shorter wavelengths are recommended for 210 precise measurement of thin crystals. In practice, upper and lower limits of measurements depend on the quality of polarizers and on the tuning of the microscope (Kohler illumination and close diaphragms). With our microscope, the practical range of measurements is 84% of the theoretical range. For example,at 561 nm, the lower measurable thickness is 0. 10 µm and the largest is 1.45 µm when theoretically the range should be 0 to 1.61 µm. It could be interesting to test if other type of circular polarizers such as mineral ones could provide larger practical ranges. The precision of the thickness measurements are an order 215 of magnitude smaller -0.012 µm to 0.030 µm -than that measurements of the length related to the resolution of an optical microscope that is approximatively 0.20 µm using natural light.