For an anisotropic material having its ordinary neutral axis horizontally, the Jones matrix is given by W0=T100(1-n)eiϕ, where T is the (complex) transmission coefficient, η is the diattenuation and ϕ is the retardation, with ϕ=2πλΔnd (where λ is the wavelength, Δn is the birefringence and d is the thickness).

If the neutral axis is rotated by an angle θ, the Jones matrix becomes Wθ=R(-θ)W0R(θ), where R(θ) is the rotation matrix. R(θ)=cos⁡θsin⁡θ-sin⁡θcos⁡θ

Assuming that η=0 (no diattenuation), the input field is left-circularly polarized PL=121i and the polarization analysis involved either a left circular polarizer made of a quarter-wave plate at 45∘ followed by a horizontal polarizer AL=1+i1-i00 or a right circular polarizer (made of a quarter-wave plate at -45∘ followed by a horizontal polarizer) AR=1+i-1+i00 so that the measured intensities are written ILL=|ALWθPL|2=|T|2sin⁡2ϕ2 and ILR=|ARWθPL|2=|T|2cos⁡2ϕ2.

One can see that ILL and ILR do not depend on the orientation θ of the neutral axes.

Moreover, the ratio 1ILLILR=tan⁡2ϕ2=tan⁡2πλΔnd does not depend on the transmission coefficient T.

In the case that we can assume that πλΔnd<π2, implying that d<λ2Δn, then there is only one solution, d, to Eq (1): 2d=λπΔnarctanILRILL. Therefore the thickness can be estimated by grabbing two images of a thin calcite crystal, one taken through a right circular polarizer (ILR) and a second through a left circular polarizer (ILL). ILL has a dark background and calcite crystals appear lighter. ILR has a light background and calcite particles appear darker. They are negative images of each other (Fig. 1a). The ratio ILRILL increases with thickness (Fig. 1b). Applying Eq. 2 to those two images gives the thickness, and this depends on the wavelength (λ) of the light used and the birefringence of calcite (Δn=0.172).

The same field of view was captured at different exposure times with the SPOT Flex camera. Exposure time is the simplest way to change the brightness 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 and 320 ms), the GL values, in the resulting images in the bottom row of Fig. 2, are identical. In Fig. 3 the histograms of ILL, ILR and d are shown. At 320 ms the images are too light, and many areas are saturated in both ILL and ILR and thus have the same GL values. Knowing that the solution of Eq. (2) is 0.81 µm when ILL=ILR and λ=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 ILL is saturated but not ILR, 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 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 brightness. Between those extremes light conditions, the estimates of thicknesses are independent of brightness. To get the maximum depth details, it is suggested to use the maximum light before saturation in ILL, providing the largest range of grey levels in both images and therefore a larger signal-to-noise ratio in the thickness estimates. In the example given in Fig. 2, this maximum detail would be achieved between 80 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 ILR/ILL is reached in the flattest part of the sigmoids (Fig. 1b), for example between 0.10 and 1.41 µm with a 561 nm light wavelength. As a consequence, the thickness measured in an empty part of the field of view 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 of 1.62 µm (Fig. 3).

The illumination tuning of the microscope is also important. The range of measurable thickness is largest when the condenser is focused and centred following the Köhler illumination (Köhler, 1894). The more closed the field diaphragm, the wider 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.

The two tested camera types (CMOS vs CCD; 14 bit vs. 16 bit; different brand) produced the same results. The same view field was captured with two different camera types without measurable difference between the two resulting thickness images (Fig. 5).

The theoretical maximum measurable thickness (dcmax) depends on the number of grey levels (nGL) achieved by the camera: 5dcmax=λπΔnarctannGL1. At λ=561 nm, dcmax 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 dcmax values are far above the maximum measurable thickness of 1.45 µm described in Sect. 5.1. However, the low depth resolution of an 8 bit camera should further limit the range of measurable thickness, although this was not tested here. Hence, both 14 and 16 bit can be used but we do not recommend using 8 bit camera.

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 a few nanometres. 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 ESRF (Beuvier et al., 2019) requires the use of silicon nitride (Si3N4) TEM windows influencing birefringence. Hence, those coccoliths cannot be used later as a standard. However, in this study, coccolith 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 measurement of their diameter, around 0.2 µm in our microscope, and likely due to issues with natural variations in rhabdoliths (parts of which may be hollow). 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, and to produce feasible “user noise”, we have slightly shifted the focus and use different wavelengths. The root-mean-square error (RMSE) between two series is used to determine the precision of the method. The RMSE ranges between 14 and 47 nm. The largest RMSE values result from the largest focus differences and/or red colours (635 and 700 nm). The best results were obtained at 561 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 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 of magnitude smaller than the spatial optical resolution which ranges between 150 and 240 nm in the present microscopic setting at the five 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 essentially related to the focus that was well reproduced in Fig. 5. The measurable masses of P. japonica in Fig. 6 range from 65.3 to 69.9 pg with a standard deviation of 1.28 pg (N=10) and depend again on the wavelength and the focus.

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); 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 84 % lower than the maximum measurable thickness (dmax). 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 a 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 using a light with long wavelengths (e.g. red at 700 nm). On the contrary, for the study of thin coccoliths such as the most extant and Pleistocene species, we recommend using shorter wavelengths (e.g. green or blue). Short wavelengths reached a MPT at a 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. Plate 1a shows an Emiliania huxleyi coccolith, in which the slits, which are present in the distal shield, appear only in blue light. This illustrates an extreme case, for which the low wavelength has to be used to get a most precise thickness and mass measurements. The distal shield of E. huxleyi is constructed with thin – ∼ 100 nm – elements that do not touch each other (Plate 1a). The detection of those elements above the background is extremely difficult using wavelengths at 700 nm but is possible using wavelengths at 435 nm. As a 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 2). Finally, this new method cannot give accurate results for calcareous nannofossils with a thickness above 1.7 µm like Cretaceous Nannoconus species. For such material, we recommend being critical with results close to MPT and using a colour camera (Beaufort et al., 2014; González-Lemos et al., 2018) as in Fig. 7, although less precise than the BCP method related to colour calibration issues (González-Lemos et al., 2018).