What a topsy turvy 2 weeks!
Imagine a Wimbledon where Roger Federer and Serena Williams are ousted like that! And then the first Ashes tests with all its twists and turns….In all this tumult, I need a steadying anchor.
Where else to turn but Science and writing?
I am going to look at some of my own past research in a bid to calm my mind (England have just lost the 5th wicket in their second innings).
Wide angle beam propagation methods (BPM) were the core focus of my PhD thesis (that seems so long ago now!). It may sound like a mouthful and/or gobbledygook but the wide angle BPM is really important for photonics modelling and the logic is straight forward.
The beginning of BPM methods was with algorithms that made the paraxial or Fresnel approximation. This amounted to saying that the light waves would make a small angle with respect to the propagation axis, or in other words light would be almost parallel to the propagation axis (we will take that as the z axis in this post). But with complicated structures that have branches (y-junctions, sharp 90o bends), or high index contrast, this assumption leads to significant error when simulating light propagation.
The light waves (for example in a y-junction) do not travel at small angles to the z axis, so obviously carrying on with the assumption was not practical. In other cases where the refractive index change is large (relative to the propagation variable), backward propagating waves or waves at large angles get generated and again the paraxial approach is not sufficient.
Hence the need for wide angle formulations was felt.
You may ask: why didn’t people incorporate wide angle assumption from the start?
The answer to some extent is due to the mathematical difficulty. If you look at the wave equation it contains second order derivatives with respect to x,y and z.
The x and y derivatives are the transverse derivatives and go to the right hand side of the equation (along with the refractive index variation), while the change with respect to z, stays on the left. Effectively we are trying to see the evolution of the field at different z values. Solving this second order differential equation in z is quite a challenging task. The Fresnel or paraxial approximation allows us to convert it to a first order differential equation in z, thus hugely simplifying things.
Going to wide angle formulations implies, reinstating the second order derivative or finding clever ways to deal with it. There has been a huge body of work on this especially in the 90’s and the last decade. As a result we have wide angle BPM formulations in most popular numerical methods: Finite Difference, Finite element etc. Many of the popular commercial software now incorporate wide angle calculations or offer the option.