## Abstract

We present a new mathematical technique which can be used to determine the main refractive indices and the birefringence of an anisotropic layer by using a polarizing microscope in conoscopic illumination. The values of the birefringence for the yellow radiation of a Na lamp are determined here for a Carpathian quartz sample, but the technique can also be applied to the study of other uniaxial substances such as liquid crystals, model membranes or biological tissues. The validity of the proposed method was tested by comparing the results with those obtained with a Rayleigh interferometer and by using the technology of channeled spectra.

© 2008 Optical Society of America

## 1. Introduction

Measurement of optical linear birefringence has been one of the standard tools in the study of anisotropic properties of materials for nearly two centuries [1, 2]. The polarized light is commonly used to determine the double refraction of an optically anisotropic material such as a crystal, organic tissues, strained glasses, polymers, liquid crystals, etc [3].

The polarizing microscope offers the possibility to analyze an anisotropic layer in divergent polarized light beam [3, 4]. Radiations crossing the plan parallel plate through the same geometrical way emerge from the plate with the same phase difference between the extraordinary and ordinary radiation and determine the same illumination in the focal plane of the microscope objective. Radiations with identical values of pathway between the ordinary and extraordinary components determine the appearance of isochromates [4, 5].

Classical crystal optics has recently undergone a renaissance as developments in optical microscopy and polarimetry, enabled in part by sensitive imaging CCD cameras and personal computers now permit the analytical separation of various optical effects that are otherwise convolved in polarized light micrographs [4]. The new microscopic (Metripol Technique, for example [5]) and polarimetric techniques are applied to problems of crystallographic twinning, phase transformations, stress birefringence, symmetry reduction, and the design of new crystalline materials [6].

The mathematical techniques devoted to the polarizing microscopy are usually based on simplified equations obtained by making certain approximations resulting from different acceptable hypothesis [1–5].

In this paper we develop a new mathematical technique in order to determine with a good precision the main refractive indices and the birefringence of the uniaxial anisotropic plate cut perpendicular to optical axis by using a polarizing microscope in conoscopic illumination. We are interested in this technique because we intend to use it for studying the optical properties of some nematic liquid crystals with a homeotropic orientation of the molecules.

## 2. Theoretical background

Let us consider the perpendicular directions Ox and Oy (which can coincide with the principal directions of the anisotropic layer) and the transmission directions P and A of polarizer and analyzer, respectively. The angles *α* and *β* orient the transmission directions of the polarizers relatively to the Ox direction (Fig. 1).

Let be E_{P} the amplitude of the wave emerging from the polarizer P. Its components E_{x} and E_{y} on the axes Ox and Oy of the Oxyz reference system of coordinates, attached to the anisotropic plate, are [7–11]:

Only the components E_{x,A} and E_{y,A} can pass after the analyzer A:

Let us suppose that the substance placed between P and A is a transparent layer of thickness h. It introduces a phase difference ΔΨ between the two components of the electric field intensity E_{x} and E_{y}. The two components recompose at the analyzer exit and give for the flux density [7, 9–10] the following expression:

In Eq. (3), *φ*
_{P} signifies the flux density after the polarizer P. When one uses crossed polarizers, one can consider: β=π/4 and α=3π/4. In these conditions one obtains a simplified equation for the flux density after the analyzer [9–10]:

From Eq. (4) it results that one can obtain maxima and minima of flux density in the conditions expressed by Eqs. (5) and (6), respectively:

Let also suppose that the anisotropic substance placed between the crossed polarizers is an anisotropic uniax plate cut perpendicular to optical axis. The interference figure obtained in the focal plane of the polarizing microscope depends on the phase difference introduced by the anisotropic plate between the ordinary and extraordinary components.

In Fig. 2(a) the ordinary and the extraordinary rays rising from the incident unpolarized ray at incidence i_{k} are shown. The pathway difference introduced by the anisotropic plate between the ordinary and extraordinary components is the same for all directions having the same incidence angle on the plate surface.

The pathway difference between the ordinary and extraordinary rays, introduced by the anisotropic plate, having the thickness h, can be expressed as:

From Fig. 2(a) one can write:

The angles r_{efk} and r_{ok} can be expressed by using the refraction law in the incidence point I from the separation surface ∑_{1} between air and anisotropic layer:

By using the Eqs. (7), (8) and (9) it results the pathway difference between the ordinary and extraordinary rays:

It is important to notice here that the refractive index for the extraordinary ray of the anisotropic plate, n_{ef}, depends on the incidence angle i_{k} [10–12]:

By using the Eq. (10), the Eq. (11) and the condition (6) it results:

The objective of the polarizing microscope and its image focal plane are shown in Fig. 2(b). For two incidence angles i _{k} and i_{k+1}, corresponding to two consecutive maxima, the radii of the two rings are R_{k} and R_{k+1}which are measurable in our experiment. The angle i _{k} can be determined by the radius of the ring of order k, R_{k}, and the focal distance of the objective (Fig. 2(c)).

The Eq. (13) permits the computation of main refractive indices (n_{e}, n_{o}) and the birefringence by using the experimental results for two rings, when the thickness of the anisotropic plate is known or for three rings, when the thickness of the anisotropic plate is unknown.

If the thickness of the anisotropic plate is known, then the main refractive indices (n_{e} and n_{o}) can be determined by solving the following system of equations:

If the radii of the rings are measured with enough precision, the values of the refractive indices can be estimated with four decimals.

The birefringence of the uniaxial crystal can be estimated by the difference between its two main refractive indices.

## 3. Experimental set-up

Figure 3 shows a schematic illustration of a polarizing light microscope with the minimal number of components. The condenser C makes the image of the source S on the anisotropic plate and the rays leaving the substage lens form an inverted cone whose point (focus) is at the sample. In this way a large angular field is incident on the anisotropic plate (conoscopic illumination). The anisotropic plate A.L. is placed on the microscope stage, between the polarizer P and the analyzer A, and it may be rotated so polarized light will vibrate along different directions within the crystal. The interference fringes are observed near the focal plane of the objective Ob by using an ocular Oc. The observations can be made in black field, when the polarizer and the analyzer are crossed, or in illuminated field, when the polarizer and the analyzer have parallel transmission directions.

The isochromates, characterized by a constant phase difference between the two components, are concentric circles and their centers coincide with the intersection point of the optical axis of the anisotropic plate and the image focal plane of the objective. When the polarizers are crossed, a black cross is superposed on the rings (Figs. 4). The image of the black cross does not change its aspect when the microscope table is rotated around its symmetry axis because the optical axis of the anisotropic plate has the same direction with the symmetry axis of the microscope tube.

The obtained images can be captured and transferred to the computer by using a webcam (Fig. 3), in our case a Creative Web camera LIVE 6L (sensor CMOS, video 800x600px, photo 1.3mp, interface USB [13]). The camera objective was fixed at the microscope ocular, coaxially with the ocular.

The radius of the k-ring was measured, after the calibration of the webcam, by using the GrabIt! application.

## 4. Results and discussions

The measurements were made in yellow radiation (λ_{o}=589.3 nm) emitted by a Na lamp. The obtained results for an angle e of 45o between the transmission directions of the polarizer and the principal axes of the anisotropic layer are given in Fig. 4 for various thickness of Carpathians quartz.

From the Fig. 4 one can see that the number of the isochromates increases with the thickness of the anisotropic layer.

The data from Table 2 were computed by solving the equation system (14) by using the Scientific WorkPlace v.5.5 application [14].

From the Table 2 it results that: n̄_{e}≅1.5549, n̄_{o}≅1.5459 and $\overline{\Delta n}={n}_{e}-{n}_{o}\cong 0.009$.

Comparative data regarding the main refractive indices and birefringence obtained in divergent beam by this new method and by two methods in parallel beams (using a Rayleigh interferometer and by channeled spectra technology) are listed in Table 3.

The method using the channeled spectra [15, 16] of the anisotropic plate permits only the birefringence determination.

From our personal data, the values obtained by this method are in good accordance with those obtained by using a Rayleigh interferometer in polarized light [17, 18] or by using the channeled spectra obtained with special devices attached to a spectrophotometer and with the data existing in literature [19, 20].

## 5. Conclusions

In this paper we have shown the possibility of using the equation system (14) in order to determine, with a good precision, the main refractive indices and the birefringence of a uniaxial layer when the thickness of the anisotropic plate is known by using the conoscopic images. The sign of birefringence permits to obtain information about the optical sign of the anisotropic layer.

The proposed technique can also be used to the study of the optical properties of the uniaxial liquid crystals or other uniaxial substances or systems.

## References and links

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