1.2 Assess need for Berreman calculus

[Jashcraf]

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[Jashcraf]

Birefringent Thin Films and Polarizing Elements

Section 3.3: Propagation in Layered Biaxial Media

A plane wave incident on a parallel layer of biaxial medium will excite four plane waves in the medium. These waves are linearly polarized in directions specified by the D fields and share the same value $\beta$ for snells law

$$\beta = nsin(\theta)$$

Where the angles change based on the effective refractive index experienced. We also know that there are four parameters $\alpha$ defined by

$$\alpha = ncos\theta$$

amounts to knowledge of the four indices and angles, because

$$ n = \sqrt{\alpha^2 + \beta^2} $$ $$ \theta = arcsin(\beta/n) $$

These are used to solve Fresnel's quartic equation, which is kind of a beast

But there are simplifications for Uniaxial and Isotropic media later in the chapter. To determine the basis vectors you need to solve for the Auxilliary Matrix L

Which I think is just a function of the layer thickness and electric permitivity. Computing the field matrix is done by solving for the eigenvectors of the auxilliary matrix.

[Jashcraf]

Berreman Calculus

Section 4.4: Berreman Calculus

Armed with the field matrix you have the basis vectors for your four principal directions. Note that the eigenequation can't be used for the isotropic case (but we would just call the characteristic matrix in this case). Beginning with the field matrix and the field coeficients, we can compute the total field vector.

The four waves also change phase linearly with displacement but at different rates. We account for this with a diagonal phase matrix.

Computing the characteristic matrix is done like a rotation of the phase matrix by the field matrix

Through which the m vector can be propagated

For an arbitrary set of thin films this is done with

Repeating this procedure for the full characteristic matrix with the field matrix for the cover medium and the substrate medium to derive the system matrix A.