"the optics of it"

This week I continued to venture into the realm of nonlinearities by learning about the Lorentz model followed by the Drude model for metals.

It was fun employing the whole "statistical ensemble" trick to relate the microscopic dipole moments of individual atoms being subjected to an applied Electric Field to the macroscopic property of Material Polarization and further towards the Electric Flux Response of a material.

Once again, complex numbers rear their heads and there's growing pains in understanding the physical significance behind "complex dielectric" with the "real permittivity" all while trying to connect it with notions of conductivity in terms of both free space responses and material responses contributing to the total Electric Flux.

It is kinda neat how the complex refractive indexes extinction coefficient can be used to ??? Let us see where a materials absorption is highest near points of resonance?

And furthermore, a interesting connection to metamaterials was established when I realized that if the resonances of our doped elements are in frequencies way below our materials resonances, we do some cool stuff with the maths.

Something something we can use the dielectric permittivity at DC and at very high (infinity) to calculate the plasma frequency of a metal.

Now my plan is to deep dive into crystal anisotropy, do some numerical examples of tensor rotation, and hopefully that will help me better understand optical components like quarter wave plates and all that stuff.

5 posts!

I started to learn about the Index Ellipsoid, and I am beginning to understand how we can quantify a materials allowed polarizations using the "indicatrix" (not going to lie this is a hard word to spell lol)

I now understand that there are two possible electric flux densities (basically the quadratic equations +/- gives two roots, so when you solve for two possible refractive indices, you also get two D solutions) and that these two fluxes are orthogonal.

The index Ellipsoid let's us characterize anisotropic crystals into biaxial (two defined optical axes) and uniaxial (two axes are equal)

But honestly, I'm hitting a bit of a wall here. I just want to be familiar with the material before grad school starts. Since I'm self studying now, I decided to venture off into a related topic: Computational electromagnetics!!

I took numerical methods this winter so when I heard about Finite Difference Method I didn't look like a 🦌 staring at 🚗 lol (honestly I find Runge-Kutta methods scarier haha)

So I learned a bit about collocated grids, recalled Dirichlet boundary conditions, periodic boundary conditions as well as Neuman boundary conditions.

My plan is to use COMSOL multi physics to do a little simulation for my masters thesis, so I'm betting that having a grasp on the numerical side of emag is something I need to accomplish to bear fruitful results.

And of course I get hit with the realization exactly *why* plane waves are used to teach us optics: because they are simply a stepping stone towards us learning about Gaussian Beams!

As someone who never had to use Gaussian form of Maxwell's equations, never used Tensors, and had never used Einsteins notation for summation; let me just say the algebra of the book proved to be a hard nut to crack!

What am I learning about? Anisotropy! Who knew that the direction at which you observe/propagate through a material such as a crystal plays a role in the material properties you will experience!!

I still don't believe you can determine eigenvalues and eigenvectors for an arbitrary 3x3 permittivity matrix (I need to use numerical examples to really see it) and that there exists rotational matricies that let us make the math all neat and proper.

The first step is to recognize that the differential length d_l of each vector pointing from each slit to the observation point P is given as slit separation times the sine of the angle formed by the triangle formed by the slit, the observation point, and a point directly across from the slit.

Now if we assume that the observation point is closest to the topmost slit, we can say that l_2 = l_1 + d_l and likewise for l_3.

So the path difference for slits 2 and 3 are d_l and 2*d_l respectively.

Knowing that the phase difference is defined as the wave number times path difference, we obtain expressions for phi_2 and phi _3 (no phase difference on the topmost slit it's the reference!)

Then, begins the algebraic slog... Using superposition theorem, we generate an expression for the net electric field at the observation point.

A big equation with sine terms whose arguments are sums? Sum of sines formula!

Then it's time to factor out the time dependent sin(wt) and cos(wt) terms.

Then comes the part I don't *really* understand...

1 + cos(phi) + cos(2*phi) is defined as Rcos(v), so we combine it all into one resulting amplitude and phase shift??? Same goes for sin(phi) + sin(2*phi) = Rsin(v)???

And then the big kicker: the whole equation is supposed to be just R*sin(wt + v) ?! How does that work?? (I guess it's time to review trig....)

At this point I'm lost in the algebra. We square R ( I'm beginning to suspect that R here is the net electric field amplitude) and since intensity is proportional to the square of the amplitude, and the incident intensities are I_0, we get I/I_0 = 1 + 4cos(phi) + 2cos(2*phi)

Did anyone else try this problem themselves??? What is this black magic going on in the algebra here?? It's the same answer as in the back of the book but that trick with the R*R really confused me

I thought we were supposed to use the |E|^2 = E times E* the complex number method to generate the interference terms ???

So I am studying optics right now, and the topic of interferometry keeps coming up!!

Because light can be modelled like a wave using maths, we know through the Double Slit experiment that two point sources can exhibit constructive and destructive interference.

So far, I have learned about the Double Slit interferometer ( the first one), the Michelson interferometer (book says it can be used to measure refractive index of gases) and the Fabry Perot interferometer (wavelength measurement).

I don't know *exactly* how they work, but I got excited when I found out that the Fourier Transform is used to show the frequency domain (spectral resolution) of optical signals much like in EE!!

How can we extend the autocorrelation of two points sources derivation to a broader "extended source" using the autocorrelation function and geometry?

What about transverse coherence? Does it also exhibit a periodic spatial dependence?

Big Q: all of the derivations were done using a quasi monochromatic source, what changes when we consider a optical signal with a non-negligible bandwidth?

Even bigger Q: temporal vs spatial coherence, what are the applications of these coherences? What can do with this information in an optical drive train?

Up next is to learn about the Van Cittert-Zernike Theorem. Man, it's been a while since I had to review Bessel Functions!!

Hopefully by the end of this month, I will have a general idea of coherence and interference to be able to grasp the fundamentals of some interferometry systems (I just know Michelson and Fabry-Perot)