Spectrometric Imaging of Polarization Colors and Its Application in Forensic Fiber Analysis

Abstract

A hyperspectral imaging instrument analyzing fibers between crossed polarizers spectrometrically is presented. The principle of operation is introduced and illustrated briefly by the theory of polarization and birefingence and calculations based on Stokes parameters and Müller matrices. Afterward, the developed instrumentation and results are detailed. Results obtained by the calculations and developed instrumentation indicate that the retardance of birefringent materials can be calculated over a high range and with a high accuracy. In addition, the spatial resolution of the instrument suffices to analyze different regions within a fiber individually. This allows the selection of a region at the center of an analyzed fiber that is shown to provide an optimal signal. The developed software enables analysis of the polarization color and the “normal”, i.e., absorptive color of the sample individually. These features make possible a preliminary identification of fibers, without isolation of the fiber from a microscope slide. The instrument forms a powerful new approach to automated analyses in forensic fiber evidence, as it can establish analyses of morphology, color, and identity of a set of samples in a high-throughput, automated, and objective way.

Keywords

Polarization microscopy Polarization color Hyperspectral imaging Forensic fiber examinations Fiber finder

INTRODUCTION

Many textiles shed fibers. Only a mild contact with a shedding textile suffices to cause transfer of textile material to a receptor. If transferred material (“traces”) can be attributed to a certain donor, it may establish a relation between a suspect and a crime scene, or between a suspect and a victim. The easy transfer of textile material thereby makes investigation of transferred fibers a powerful forensic tool.

Fiber traces present on a receptor can be collected using tapes. A main challenge in forensic fiber analyses is the sheer quantity of fibers that are present on such tapes in many cases. Tapes may contain hundreds or thousands of fibers, and scanning the tapes for relevant traces is often laborious and time-consuming.

Current Automated Microscopy Systems. Different automated microscopy systems, also called fiber finders, have been introduced to facilitate fiber scanning.¹ These systems are designed to scan a large surface and locate the relevant traces. Many forensic laboratories have carried out tests to introduce the automated microscopy systems into their case work; however, it was generally found that the automated microscopes did not discriminate fibers well enough, i.e., they found too many “false positives” and hence did not improve efficiency.

The poor discrimination achieved by the automated microscopy systems can be explained by their restricted optical capabilities. Most systems discriminate on a single parameter, namely, fiber color as observed by a standard color camera. Moreover, the magnification of these systems is low, and the illumination systems are limited to white light sources. The restrictions of the optical capabilities of these automated systems become apparent when they are compared to the full set of features available to microscopists and spectrometrists; these features include morphology, fluorescence, optical spectrometry, infrared spectrometry, and polarization microscopy. The discrimination that can be reached by the combination of these sophisticated and powerful techniques exceeds the discrimination of current automated microscopy systems and explains the false positive errors of fiber finder systems.

Optimization of the optical capabilities of automated microscopes would improve automated microscopy systems. In addition, it would enable the objective discrimination of fibers and thereby facilitate a more scientific and numeric basis for forensic fiber investigation.

Hyperspectral Imaging. An optical capability that has already been explored is spectrometric imaging of fiber color. Spectrometric detection was used on the Foster and Freeman FX5 fiber finder.¹ The spectrometric resolution of this system is limited to ten wavelength regions, and no numeric data have been found on the improvement of discrimination. A newer approach to spectrometric detection is the use of a liquid crystal tunable filter (LCTF), as presented by Markstrom and Mabbott.² This system has been proposed for the analysis of fiber evidence. Currently, no technical implementation of LCTF filters for use in an automated microscope system is available.

Spectrometric Imaging of Polarization Colors. In the current study, spectrometric imaging of polarization colors is explored for the first time. Polarization colors can be observed when a suitable sample is placed between crossed polarizers. An image obtained with crossed polarizers is characterized by a dark background on which birefringent samples are visible, often in bright and beautiful colors. Polarization colors can be used for a preliminary identification of, e.g., textile fibers, pigments, and minerals. An identification based on polarization colors is considered less reliable than identification by chemical methods or infrared spectrometry.³ Nevertheless, experienced microscopists frequently use polarization microscopy to visually obtain a preliminary identification of textile fibers, as the procedure is simple, cheap, nondestructive, and does not require isolation of a sample from a microscopic slide.

The possibility for a preliminary identification would be very advantageous for an automated microscopy system. Nevertheless, acquisition of polarization colors using a standard color camera is not considered promising. This will be illustrated in the next section, where the principles of spectrometric analysis of polarization colors are described. This section contains information provided by Yang et al.,⁴ who proposed a method to determine the retardance of aramide fibers, and by Valaskovic,⁵ who used matrix algebra to explain a complex microscopic observation.

Next, the instrumentation developed to acquire polarization spectra with a high spatial resolution is presented. The high spatial resolution is achieved by the implementation of a spectrometric imaging system, based on the LCTF described by Markstrom and Mabbott,² and it enables the individual analysis of regions that are in proximity. The results detailed below show that the different spectra over the width of a single fiber show marked differences. The hyphenation of an imaging approach thus allows derivation of the morphology, spectrometric color description, and (preliminary determination of the) identity of a sample independently. The developed instrumentation is thereby very promising, both as an approach for a purely objective description of fiber characteristics and as an extension to automated microscopy systems. It is shown that preliminary identification of fibers can be made effectively and efficiently by a spectrometric analysis of the polarization colors.

Spectrometric Analysis Of Polarization Colors

A combination of two crossed polarizers, i.e., polarizers placed at right angles, does not transmit light. However, a change of the polarization state of light between the polarizers may lead to a partial or even full transmission by the second polarizer (usually called the “analyzer”). This effect is illustrated in Fig. 1. Figure 1a shows the transmission image of three nylon fibers with different colors. The fibers have a thickness of about 22 μm. The background in this image is white as it transmits all incoming light. Figure 1b shows the same fibers placed between two crossed polarizers. The polarization color of the fibers is different from the “normal” absorptive color of the sample, shown in Fig. 1a. The background in Fig. 1b appears black as the light polarized by the first polarizer is not transmitted by the second polarizer. However, the fibers are visible in Fig. 1b, indicating that the fibers alter the polarization state of the incoming beam in such a way that part of the light is transmitted by the analyzer.

Fig. 1.

Microscopic images of three nylon (polyamide) fibers, (a) Transmission microscopy and (b) polarization microscopy, where the fibers are positioned between two crossed polarizers. The inset shows the definitions of the two refractive indices of fibers, namely, n|| and n⊥.

Fibers alter the polarization state of light due to their birefringence, which means that the refractive index of a sample cannot be stated as a single value. For most fibers, the refractive index experienced by light polarized parallel to the main axis of a fiber (n||) is different from the refractive index experienced by light polarized perpendicular to the main axis of a fiber (n⊥), as shown in the inset of Fig. 1b.

The refractive index of a material is proportional to the speed of light through that material. A polarized beam entering a birefringent sample placed at 45° with respect to the polarization of the incoming beam will experience both refractive indices (n|| and n⊥) and therefore “split” into parts that travel at different speeds. During travel through the sample, a path difference, or “retardance” between the split parts develops, and the interference between these parts alters the polarization state of the light beam.

The retardance R of a sample is related to the thickness of the sample d and its birefringence Δn by

R = d \cdot Δ n

(1)

If the final retardance, i.e., the retardance after the light beam left the sample, equals the wavelength of the incoming beam (or a multiplex of this wavelength), the polarization state is not effectively changed and all light is blocked by the analyzer. In other cases, however, the retardance alters the polarization state and light is transmitted by the analyzer. This principle is illustrated in Fig. 2 and is based on an illustration by Winchell and Emmons.⁶ The different plots in Fig. 2a show the light transmission as a function of wavelength of the incoming beam (vertical axis) and the retardance (horizontal axis). The gray scale patterns in Fig. 2a show the absorptions (black parts) and transmissions (white parts) and show that the light transmission of the sample depends not only on the retardance but also on the wavelength of the incoming beam. For example, a sample with a retardance of 1000 nm induces full transmission of light at 400 nm by the analyzer and hence visibility in the resulting image. Light at 500 nm, however, is not changed effectively by this sample and thus is not visible in the resulting image.

Fig. 2.

Explanation of the Michel-Lévy chart of polarization colors, (a) Transmission intensity of light through a birefringent sample placed between crossed polarizers and illuminated by monochromatic light with a wavelength indicated on the vertical axis, The retardance of the sample is indicated on the horizontal axis, Transmission patterns for different monochromatic beams are provided on the vertical axis, (b) Transmission color of a birefringent sample placed between crossed polarizers and illuminated by white light, [Adapted from Ref. 6. Copyright John Wiley & Sons, 1931.]

Figure 2a shows the transmission patterns for a restricted number of wavelengths; it can be inferred that patterns can be derived for all wavelengths in the same manner. A birefringent sample illuminated with white light thus gives rise to a complex set of light transmissions and blockages. This can be observed as polarization colors as provided in a Michel-Lévy chart,⁷ such as the (simplified) chart shown in Fig. 2b.

Comparison of the polarization color of an unknown sample to the colors on a Michel–Lévy chart can reveal the retardance of the sample. Once the retardance and the thickness of a sample are determined, their ratio equals the sample birefringence (see Eq. 1) and can be used for identification purposes. Tables for the identification of textile fibers based on their birefringence can be found in the literature.^8,9

This procedure, used in many laboratories, includes a visual comparison of polarization colors. This step is subjective and the colors that need to be discriminated can be very similar. For example, retardances of 500, 1000, and 1500 nm all lead to a reddish color (see Fig. 2b). Moreover, polarization colors become less and less distinct at higher retardances (data not shown). It is for these reasons that observation of polarization colors by a standard color camera is not considered promising.

Alternatively, retardance can be obtained using a device with a tunable, calibrated retardance, usually called a compensator. The compensator is placed in the light path to cancel (“compensate”) the retardance of the sample. On full compensation, the sample appears black in the microscopic image, and the retardance can be read from the compensator scale. The compensator yields rather accurate results, but it has a limited range (often four to six “orders,” amounting to about 2000–3000 nm). The retardance of many fibers, including most polyester fibers, cannot be accurately analyzed by a compensator due to the limited range.

It can be anticipated that spectrometric analysis yields a more accurate description and discrimination of polarization colors. Results of a few calculations are presented in Fig. 3. A first advantage of spectrometric analysis is that samples with a similar polarization color, e.g., samples with a retardance of 500, 1000, and 1500 nm, show markedly different polarization spectra (black curves in Fig. 3).

Fig. 3.

Theoretical polarization spectra, based on the Stokes parameters and Müller matrices presented in the Appendix. All spectra shown lead to a reddish polarization color. The gray spectra shown are shifted to higher retardance by 20 nm.

A second advantage of spectrometric analysis is the high accuracy with which samples with a slightly different retardance can be discriminated. To illustrate this advantage, every plot in Fig. 3 presents calculated polarization spectra of samples that differ by only 20 nm. The black curves (having retardances of 500, 1000, and 1500 nm) can easily be distinguished from the gray curves (having retardances of 520, 1020, and 1520 nm), and the spectral resolution of even basic current spectrometers suffices to resolve these small changes. By contrast, it is hardly possible to reliably observe the color differences caused by such small retardance differences by eye.

A third advantage of spectrometric observation is the high range of retardances that can be analyzed. in fact, spectrometric observation was first proposed by Yang et al.⁴, who determined the retardance of an aramide fiber to be 8529 nm. This retardance is far higher than the 2000–3000 nm that can be achieved using a compensator.

In summary, spectrometric analyses of polarization colors (or polarization spectra) allow the unambiguous and accurate determination of birefringence over a very high range.

EXPERIMENTAL

Calculations. Calculations on the transmission of our optical system were carried out using Stokes parameters and Muller matrices. A less complex description would suffice for the current purposes, but future work is expected to require the higher flexibility offered by the chosen description. The optical setup can be summarized as follows: light source → polarizer → birefringent sample → analyzer → detector. The vector and matrices used to simulate this setup are provided in the Appendix. Retardance was calculated from experimental spectra by comparing them to calculated spectra calculated using the formulae in the Appendix. All calculations were carried out in Matlab release 2010B (The Mathworks, Inc.).

Instrumentation. A scheme of the developed instrument is provided in Fig. 4. The design is aimed at a maximum stability and therefore minimizes the number of moving parts. In fact, the only moving part needed in the optical setup is a shutter. The instrument is built of commercially available components. The illumination path consists of a 100 W halogen light source (Carl Zeiss B.V.), two mirrors, two polarizing beam splitters, an achromatic condenser lens (focal length = 40 mm), and a variable aperture (Edmund Optics). The two polarizing beam splitters separate and recombine the illumination. Polarized illumination is obtained by blocking one of the pathways using the shutter.

Fig. 4.

Setup of the developed instrument. See Experimental section for details. The light incident on the sample can either be polarized or depolarized by closing and opening the shutter. Patent pending.

The image path consists of a Mitutuyo 5X objective (Edmund Optics), a Varispec LCTF (Cambridge Research Instruments), and a Pike F100 black-and-white camera (Allied Vision Technologies). The Pike F100 camera has a pixel size of 7.4 μm; combined with a 5X magnifying objective, this leads to a theoretical spatial resolution of about 1.5 μm. The Varispec filter acts both as a spectrometer and as a polarizing filter; no additional polarizing optics were introduced in the image path.

The halogen light source was driven by a direct current source (SPS 1540 PFC, Voltcraft) to optimize the stability of the light intensity and prevent interference between the mains voltage and camera shutter times. The light source was switched using a relais board (EasyDAQ USB8PR2).

The camera was controlled from Matlab, using the Image Acquisition Toolbox and the 64-bit camera driver (AVT). The LCTF and the relay board were controlled from Matlab by direct access of the serial ports. The wavelength accuracy of the LCTF was checked using a mercury–argon calibration source (HG-1, Ocean Optics).

Measurements and Data Processing. Procedures to drive the measurements were written in Matlab. In our procedure, the shutter time is optimized at every wavelength in such a way that the signal intensity of the reference spectra (see below) has an optimal intensity (set to approximately 50 000 counts for the used 16-bit camera). Variable shutter times have been implemented as the transmission of the LCTF spectrometer depends on the selected wavelength: transmission is relatively low for wavelengths near 400 nm. Analyses with a fixed shutter time therefore lead to high noise levels near 400 nm, or to saturation of the camera in higher wavelength ranges.

After the shutter times are optimized, three sets of spectra are acquired:

Reference spectra, acquired with an unpolarized excitation beam (no shutter in light path; see Fig. 4); no sample in light path.

Visible light absorption spectra, acquired with an unpolarized excitation beam (no shutter in light path; see Fig. 4); sample positioned in light path.

Polarization spectra, acquired with a polarized excitation beam (shutter in light path; see Fig. 4); sample positioned in light path.

Polarization spectra are divided by the visible spectrum, to correct for both the sensitivity of the optical system and the absorptive color of the sample; visible spectra are divided by the reference spectrum to correct for the sensitivity of the optical system. The acquisition time of full dataset is about 20 s. Several of measurements in the current study did not require the spatial resolution supplied by the imaging system. In those cases, the acquired spectra were averaged before further processing.

Contrast ratios are calculated by dividing the reference spectra (unpolarized light) by the polarization spectra, measured without a sample in the light path.

Samples. Fiber samples were obtained from Spinnerij Geldrop. The analyzed fibers have a thickness of 18-25 μm. Samples were prepared in glass slides, using glycerin as the mounting medium. Synthetic fibers (polyester, polyamide, acrylic) were selected, as natural fibers such as cotton can be discriminated based on their morphology.

RESULTS AND DISCUSSION

Contrast Ratio. Initial tests showed transmission of light through crossed polarizers even without insertion of a sample. This is against expectations and indicates that the performance of the polarization optics is not optimal. The performance of the polarization optics was therefore evaluated by measuring the contrast ratio. The results presented in Fig. 5, show that the contrast ratio is poor (<50) at wavelengths below 450 nm. At higher wavelengths the contrast ratio is better. The low-wavelength spectral region (<450) was excluded from further consideration because of the poor contrast.

Fig. 5.

Contrast ratio as a function of wavelength.

Standard Wave Plates. The instrument was further tested using two standard first-order wave plates provided with a comparison microscope (Leica Microsystems) used in our laboratory. These wave plates have a first-order birefringence, i.e., 550 nm. However, the color of the wave plates, when observed between crossed polarizers, is slightly different; hence, differences between the retardances of the wave plates are expected. Figure 6 shows the obtained polarization spectra. Comparison of the experimental and theoretical spectra shows that the obtained value is close to the expected 550 nm. The two wave plates, having a retardance differing by about 10 nm, can be well discriminated by the instrument.

Fig. 6.

Polarization spectra of two first-order wave plates, showing a small but clear difference.

Fiber Analyses. In a next set of experiments, the birefringence of a set of fibers was studied. Textile fibers have a diameter of typically 10–25 μm and cover only a small part of the field of view of the microscope system. Individual fibers can however be analyzed in the developed instrument, as the spatial resolution of our imaging system is about 1.5 μm. In fact, it is possible to derive information from different spots within a single fiber. This effect is shown in Fig. 7, where several polarization spectra are provided. These spectra were obtained from different positions along the width of a polyester fiber, as illustrated by the drawn fiber in this figure. A detailed polarization spectrum is obtained from the center of the fiber. The retardance of the fiber is determined to be 3510 nm. This fiber has a thickness of 19 μm and the birefringence is calculated (see Eq. 1) at 0.18, a value that is in agreement with literature values for polyester fibers.^8,9

Fig. 7.

Polarization spectra of different positions on a white polyester fiber. The experimental spectra are superimposed on a longitudinal drawing of the fiber.

Figure 7 shows that the polarization signal from the center of the fiber is clear but that the contrast decreases with increasing distance from the center. The low-contrast spectra do not show a decreasing retardance, as could be expected by the lower sample thickness. This effect is attributed to the differences in refractive indices of the fiber and that of the surrounding fluid (glycerin). Due to these differences, the fiber acts as a cylinder lens, and the path length of light through the fiber is not defined accurately. In this case, optimal information is obtained from the center region of the fiber, whereas the contrast in adjacent regions is less clear. The high spatial resolution of the system enables selection of the optimal polarization spectra to calculate the retardance.

Preliminary Identification. In a next experiment, a set containing 12 types of fibers was analyzed. The set contained polyester, acrylic, and nylon fibers in the colors white (i.e., colorless), red, green, and blue. The visible and polarization spectra of these fibers are provided in Fig. 8. Figures 8a and 8b show data obtained from white (colorless) fibers. As expected, the visible spectra of these fibers show high transmission values. The polarization spectra are however widely different for the different materials. The transmission observed in the polarization spectrum of the acrylic fiber (black solid curve, fiber diameter 23 μm) is very low, indicating a low birefringence. The polyester fiber yields a far more detailed spectrum (gray solid curve), containing different bands. The corresponding retardance was determined to be about 3554 nm. The diameter of this fiber is about 19 μm, and the calculated birefringence is thus about 0.19. The retardance of the polyamide spectrum (black dotted curve) is determined to be about 1044 nm. The diameter of this fiber is about 22 μm, and the calculated birefringence is thus about 0.047. These values are in good agreement with values determined by other techniques⁸ and indicate that polarization spectra can be used for a preliminary identification of fibers.

Fig. 8.

Visible (left column) and polarization (right column) spectra of white (a) and (b), red (c) and (d), green (e) and (f), and blue (g) and (h) fibers. Each plot contains the spectra of an acrylic (black), a polyester (gray), and a polyamide (black dotted) fiber. Calculated retardances for the polyester and polyamide fibers are indicated near the polarization spectra.

The fibers shown in Fig. 1b indicate that the apparent color of different nylon fibers is different. These color differences can be explained by random changes between the birefringence of the displayed fibers, by convolution of the absorptive color with the polarization color, or by a combination of these factors. The data processing procedure proposed in the current study implies division of acquired polarization spectra by the visible spectrum of the same fiber. The polarization spectra are thus corrected for the absorptive color and solely carry the information about the birefringence of the analyzed sample. To test the data processing procedure, several colored fibers (red, green, and blue) were analyzed. The acquired data are presented in Figs. 8c–8h. The color of the various spectra is represented by the visible spectra on the left. The color information is not observed in the polarization spectra presented on the right. Instead, the similarity of polarization spectra of white fibers of equal identity (see Fig. 8b) is clear. Small differences are observed, especially for the blue polyester and nylon fiber. These changes are explained by a higher retardance of the involved fibers. The variation in values for the retardances of fibers of the same type is indicated in Fig. 8.

The data in Fig. 8 show that that the current instrumentation introduces some deviations in the obtained spectra. In some cases, an absorption band around 660 nm is visible. This band was not visible in spectra acquired with routine instrumentation (data not shown). In addition, some of the acquired visible and polarization spectra are noisy. Current work in our laboratory is aimed at solving these issues. Nevertheless, even with the current instrumentation, the differences between observed retardances indicate that the calculated retardances provide a preliminary identification. In addition, it becomes possible to discriminate fibers of the same type based on the exact value of the retardance.

CONCLUSIONS AND OUTLOOK

The current contribution presents an instrument that enables the spectrometric imaging of polarization colors. It combines a spectrometric imaging system based on an LCTF filter, with observation of samples positioned between crossed polarizers.

A theoretical framework is presented to predict polarization spectra as a function of the retardance of a sample and shows that the accuracy of the method can be very high. This theoretical framework extends the Michel-Lévy chart, used by many microscopists to analyze polarization colors, to a spectrometric analysis. Advantages of spectrometric analysis of polarization colors are that a polarization spectrum can be related to a retardance value very accurately and unambiguously. In addition, the range of retardances that can be analyzed is much higher than can be obtained using a compensator.

The results obtained with our instrumentation confirm the high accuracy with which retardances can be determined using spectrometric imaging of polarization colors. In addition, they show that the imaging approach allows the independent analysis of adjacent areas within a single fiber. This feature is very useful, as the quality of the results obtained from areas that are in proximity has been shown to vary, e.g., due to the refraction of light beams at the surface of a sample.

The results indicate that polarization spectra of acrylic, nylon, and polyester samples are clearly different. The system can thus be used for a preliminary identification of samples. Polarization spectra are shown to be independent of the absorptive color of the fiber. This independence is facilitated by the presented approach for data processing. In this approach, the polarization spectrum is corrected for the absorptive color by division through a visible light spectrum rather than though the reference spectrum. Independent observation of the polarization color and the absorptive color is an advantage over normal microscopic examination, where a convolution of both colors is observed. The reliability of the identification based on the retardance is not expected to be as high as that with identification using chemical methods or infrared spectrometry. However, the identification is fast, can be carried out in a high-throughput approach, and does not require sample isolation or preparation.

The presented instrument enables the acquisition of visible light spectra and polarization spectra. The instrument is robust, as it operates with a minimum number of moving parts. In our current work, implementation of additional illumination modes and automated image processing techniques is addressed. The developed instrumentation should warrant the acquisition of objective data and allow good discrimination between different types of fibers. This approach provides a means to numerically determine the frequency or rarity of textile materials.

Future work will aim at the development of instrumentation capable of analyzing textile fiber traces, i.e., the collection of fibers recovered from other (textile or non-textile) items. This will require implementation of a moving stage into the automated microscope to extend the field of view. In addition, fiber recovery methods will have to be optimized. Currently, recovery often makes use of tape lifts. The image quality that can be obtained through these tapes is poor and probably limits the value of the acquired data. New tape systems, such as Easylift,¹⁰ may prove important to optimize the image quality of fiber traces. Acquisition of data from fiber traces will make the developed instrumentation applicable as a fiber finder system and provide more efficient methodology in forensic laboratories. In addition, it will allow calculations on the background population of different fibers and provide a next level in the objective, numerical evaluation of fiber evidence.

Footnotes

ACKNOWLEDGMENTS

This research has been made possible by kind support of the Netherlands Forensic Institute (NFI) R&D funds and the dedication and hard work of Dimitry Lamers, Kevin Ferreira da Silva, and Brigitte de Haas, students of the Haagse Hogeschool. H. Habraken is acknowledged for technical assistance. Valuable comments to the manuscript were provided by Drs. Peter Zoon and Karla de Bruin. The authors worked on this project while employed by the Netherlands Forensic Institute. Patent pending on the instrumentation described in this work.

APPENDIX

Transmission of a birefringent, nonabsorbing sample between crossed polarizers is calculated by the Stokes-Müller description of the optical setup used in the current study. General info on this approach is described by Chipman¹¹ and Hecht.¹² The Müller matrix of a linear retarder, such as a birefringent sample, is provided by Chipman:¹¹

[\begin{array}{l} 1 & 0 & 0 & 0 \\ 0 & {cos}^{2} 2 θ + {sin}^{2} 2 θ \cdot cos δ & sin 2 θ \cdot cos 2 θ \cdot (1 - cos δ) & - sin 2 θ \cdot sin δ \\ 0 & sin 2 θ \cdot cos 2 θ \cdot (1 - cos δ) & {sin}^{2} 2 θ + {cos}^{2} 2 θ \cdot cos δ & cos 2 θ \cdot sin δ \\ 0 & sin 2 θ sin δ & - cos 2 θ \cdot sin δ & cos δ \end{array}]

TABLE A1.

Calculated transmittance values for the presented instrument.

	Wavelength (nm)

Retardance (nm)	400	500	600	700	800
0	0.000	0.000	0.000	0.000	0.000
100	0.250	0.173	0.125	0.094	0.073
200	0.500	0.452	0.375	0.306	0.250
300	0.250	0.452	0.500	0.475	0.427
400	0.000	0.173	0.375	0.475	0.500
500	0.250	0.000	0.125	0.306	0.427
600	0.500	0.173	0.000	0.094	0.250
700	0.250	0.452	0.125	0.000	0.073
800	0.000	0.452	0.375	0.094	0.000

The matrix of the described setup (light source → polarizer → birefringent → sample → analyzer → detector) now becomes

\begin{matrix} \frac{1}{2} \cdot [\begin{matrix} 1 & - 1 & 0 & 0 \\ - 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}] \cdot [\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & cos δ & 0 & - sin δ \\ 0 & 0 & 0 & 0 \\ 0 & sin δ & 0 & cos δ \end{array}] \\ \cdot \frac{1}{2} [\begin{matrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}] \cdot [\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}] \end{matrix}

In this description, the last (fourth) term represents a light source providing an unpolarized beam with normalized intensity. The first and the third matrices represent the crossed polarizers. The second term represents the sample, which is placed between the polarisers. The result of the matrix calculations is a vector

\frac{1}{4} [\begin{matrix} 1 - cos δ \\ - 1 + cos δ \\ 0 \\ 0 \end{matrix}]

that contains the intensity of the resulting light beam as the top element, 1/4 (1 - cosδ) or 1/4 · (1 - cos(2πR/λ)). As the intensity of the light source was set to unity, the calculated values equal the transmittance op the setup. A number of calculated transmittance values is provided in

. The maximum transmittance is 0.5, which is intuitively correct, as the first polarizer rejects half the incoming intensity.

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