Refractive index tensor

This type of material was the subject of the example calculation in section 1.2.1. The refractive index tensor is anisotropic with principal values ( , n. n ), defined in... 

This as been written in a reference frame coincident with the principal axes of n. The case of noncoaxial real and imaginary parts of the refractive index tensor requires a much more complex calculation and is treated in section 2.4.6. 

In an important series of papers [6,7], Jones established an approach for the treatment of materials where the refractive index tensor, n (z), varies along the propagation direction of the transmitted light. This procedure also lays the foundation for the analysis of complex systems possessing any combination of optical anisotropies. 

A material with an axial variation in its refractive index tensor is shown in Figure 2.6. Light propagating through a differential element of thickness Az is transformed according to... 

If the principal axis of the real part of the refractive index tensor is oriented at an angle 0 relative to the laboratory frame, equation (2.11) must be used to rotate the Jones matrices used in the calculation of N in the rotated frame to obtain ... 

Figure 2.8 is a vector diagram describing the principal axes of the refractive index tensor of this material. Using equation (2.47), the Jones matrix is given by (2.46) with parameters ... 

Figure 2.8 Vector diagram of the refractive index tensor for a material with noncoaxial...

The procedure to extract the refractive index tensor is to first determine the principal... 

Once these eigenvalues and known, the principal values of the refractive index tensor are... 

The experimental arrangement analyzed in this section is shown in Figure 2.11. Here a shear flow is applied in the (x, y) plane, with the flow in the x direction. Due to the symmetry of the flow, the principal axes of the refractive index tensor will have 3 oriented... 

Equation (4.12) connects the polarizability of a particle with its orientation and can be used to derive the following equations for the particle contribution to the refractive index tensor [16] ... 

The Lorentz-Lorenz equation can be used to express the components of the refractive index tensor in terms of the polarizability tensor. Recognizing that the birefringence normalized by the mean refractive index is normally very small, ( A/i / 1), it is assumed that Aa /a 1, where the mean polarizability is a = (al + 2oc2)/3 and the polarizability anisotropy is Aa = a1-a2. It is expected that the macroscopic refractive... 

This expression can be combined with equation (4.12) to obtain the following expression for the refractive index tensor as a function of the orientation distribution of the rods making up the bulk sample ... 

Using this model, the birefringence and orientation angle, %, of the principal directions of the refractive index tensor can be determined. If light is propagating along the y axis, we find that... 

Equation (7.24) predicts that the refractive index tensor is proportional to the second-moment tensor of the orientation distribution of the end-to-end vector. This expression was developed using a number of assumptions, however, and is strictly valid only for small... 

In principle, once the probability distribution function is available, bulk solution properties can be evaluated by averaging appropriate functions of conformation space and time. From the Kuhn and Grun analysis leading to equation (7.24) for the refractive index tensor, we are particularly interested in the second moment tensor,... 

Bulk material properties can be determined quite simply using this model. For example, consider the calculation of the second-moment tensor, Q = (u u ), which is required for the stress and refractive index tensors. Using the independent alignment approximation, we have... 

Since the polarizers discussed above involve light reflection combined with the real part of the refractive index tensor, they can be used effectively over a broad spectral range about a central wavelength. Calcite Glan-Thompson polarizers, for example, operate successfully over the entire visible spectrum. When fabricated of crystalline quartz, these polarizers can be used to polarize ultraviolet light as well as visible light. 

The 3x3 complex refractive index tensor N = n — ik is related to the dielectric tensor e and the magnetic tensor p by the Maxwell relation ... 

So far, however, one still needs an expression for the reflection matrix that shows how to extract from it the tensor elements for the refractive index tensor of the biaxial medium. We seek the reflection matrix R for the semi-infinite anisotropic biaxial medium. Using Eq. (2.15.8) and Eq. (2.15.21), we can relate the 4x4 differential propagation matrix A to the dielectric tensor e from Eqs. (2.15.21) and (2.15.24). Then it can be shown that... 

Birefringence and dichroism represent two optical methods which can be applied to materials under flow conditions, forming the basis of Optical Rheometry [3,4]. The aim of these two techniques is to measure the anisotropy of the complex refractive index tensor n = n - i n". Birefringence is related to the anisotropy of the real part, whereas dichroism deals with the imaginary part. Recent applications of birefringence measurements to polymer melts can be foimd in Chapter III.l of the present book. 

Flow birefringence is due to the optical anisotropy created by the orientation of the macromolecules within the flow field. In a plane flow, if we denote by I and II the principal axes of the refractive index tensor (principal axes are those for which the tensor is diagonal they are defined by the eigenvectors of the tensor), the local birefringence A is defined as ... 

In solid photoelasticimetry, birefringence is related to local stresses through the stress optical law, which expresses that the principal axes of stress and refractive index tensors are parallel and that the deviatoric parts of the refractive index and stress tensors are proportional ... 

In optically active substances, according to equations (12a) and (17a), the non-diagonal components of the refractive index tensor differ from zero and describe the variation in optical rotation due to a static electric field, an effect studied by Tinoco. ... 

The birefiingence of a material is expressed by the anisotropic dielectric tensor at optical frequency, or the refractive index tensor which is the square root of i.e.. 

Below, taking an electric field as a typical example of an external field, we will discuss the electro-optic effects that result from the effect of this field on the optical properties. Electro-optic effects are effects such as changes to the deformation and rotation of the index ellipsoid caused by the action of an electric field (E). When the field is applied, the change in the refractive index tensor A(l/np, which is an optical parameter, can be expressed as follows by expanding the electric field terms ... 

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