The dispersion essentially comes from how much phase the wave accumulates through the plate. Take a look at Slides 61 and 62 in Lecture 5 here:

http://emlab.utep.edu/ee5390cem.htm

Essentially, you calculate the scattering matrix, or transfer matrix, of a wave through your plate. From this matrix you calculate the eigen-values from which you extract the dispersion information. The matrix comes from a matrix wave equation where the eigen-values are the propagation constants of the waves. It is from these that you calculate dispersion.

For a discussion on stability, see Slides 33 to 38 in Lecture 4 at this same link. The short answer is that the transfer matrix method treats everything as forward propagating. There are certainly cases where waves will be cutoff and will decay exponentially. If everything is treated as forward propagating, then backward decaying waves will grow exponentially instead of decay. This is the numerical instability. There are ways to fix this, but they all involve treating forward and backward waves separately to avoid the situation altogether.

Does this help?

In the paper i am able to go upto eq 4 and then lose the author

I think this paper may be confusing you because they adopt so many closed form solutions. I teach the transfer matrix method in my Computational Electromagnetics class that you may find a little more straightforward. See Lectures 4 and 5 here:

http://emlab.utep.edu/ee5390cem.htm

You can download the lecture notes and watch recorded lectures on YouTube. Unfortunately, I don't discuss in too much detail the case for anisotropic media, but you can see the final formulation on Slide 16 of Lecture 4. I teach in this class the 2x2 method for isotropic materials. When generalized to anisotropic media, the main matrix equation becomes 4x4 and it becomes necessary to sort your modes to separate the backward and forward propagating waves. If you don't do this, your code will be numerically unstable. After that it is the same.

Once you get through this, you will see that the paper you referenced is just another variant of TMM.

Hope this helps!!

Thanks

my eventual objective is to understand how another author used 4x4 method to come up with expressions for dispersion in birefringent plate. and i am also wondering if i can use this method not just for plane wave but a guassian wave.

'Dispersion of linearly polarized light propagating in a thin birefringent plate'

http://www.opticsinfobase.org/josaa/abstract.cfm?uri=josaa-22-4-752

Can you please elaborate what you mean by this

' it becomes necessary to sort your modes to separate the backward and forward propagating waves. If you don't do this, your code will be numerically unstable.'

The dispersion essentially comes from how much phase the wave accumulates through the plate. Take a look at Slides 61 and 62 in Lecture 5 here:

http://emlab.utep.edu/ee5390cem.htm

Essentially, you calculate the scattering matrix, or transfer matrix, of a wave through your plate. From this matrix you calculate the eigen-values from which you extract the dispersion information. The matrix comes from a matrix wave equation where the eigen-values are the propagation constants of the waves. It is from these that you calculate dispersion.

For a discussion on stability, see Slides 33 to 38 in Lecture 4 at this same link. The short answer is that the transfer matrix method treats everything as forward propagating. There are certainly cases where waves will be cutoff and will decay exponentially. If everything is treated as forward propagating, then backward decaying waves will grow exponentially instead of decay. This is the numerical instability. There are ways to fix this, but they all involve treating forward and backward waves separately to avoid the situation altogether.

Does this help?

Thanks