Geometrical Optics: Paraxial Approximation, Ray Trace Method

This is the first of about 25 posts I'll be doing over the next couple of months to help me study for the dreaded qualifying exam. It's said that the best way to learn something is to have to explain it to someone else, so I'm hoping to improve my own understanding of these topics by trying to explain them to a broader audience.

The first topic I'm tackling is ray optics, for a couple of reasons. Its the first optical topic that I learned, and it is in a way the first optical physics created historically. Its also the optical topic that I like least. Hate would be an appropriate word, actually, for my feelings towards ray optics.

Why do I hate it so much? Possibly because the book I used to first learn it was poorly written and so poorly laid out that diagrams and text overlapped. Mostly because I find it unbelievably tedious, simplistic, and it is of little to no use to me, since I am the one optics person who isn't into photography.

What I specifically dislike about Ray Optics:
1)It assumes all light travels as a ray (hence the name). No wave properties, no photons, no interacting EM fields, just lines of light. So, kinda ignores my whole subspecialty (no wave properties, no vortices, no research for me).

2) There are two ways to go about it. The 'correct'-er way, which is essentially geometry and repeated use of Snell's Law, over and over and over. This is usually done with a computer these days. The other is to use the paraxial approximation, which on top of assuming that light is just rays also assumes that we are only interested in the light that goes through a very small area at the center of the lens (para- near, beside axial - axis). This method is only good for a narrow bundle of rays near the center and, while it is easier to do by hand than non-paraxial, it still requires pages of tedious and easily messed up algebra

Why Ray Optics is still taught:
1) It can help a person, such as a lens or systems designer, know what is happening with the light passing through the system. Many errors and aberrations can be determined through ray diagrams and optimization takes place using it.

2) It's an intuitive place to start for a lot of people. Every little kid given  a yellow crayon and a piece of paper will draw a sun that looks like this:

Light travels in straight lines. Sundials, shadow puppets and dressing room mirrors that let you see your back all seemingly demonstrate this to us every day. 

3) Realistically, its probably the most practical optical analysis for the real world. Wave optics may be more correct, more precise, show things in greater detail, but it is also computationally intensive. In a commercial setting, the approximation that takes 5 minutes to run and give you an answer you can use immediately is going to trump the precise calculation that takes a  week to run and another week to analyze.

I'm going to focus on the paraxial approximation, and the simplest method of doing for this post, because otherwise it was threatening to become a book.

The paraxial approximation is takes all the assumptions made to get ray optics in the first place and adds the assumption that you are only interested in using the lens right around its center, so all your angles are very tiny, and that the lenses are 'thin' so the curvature is small and you can neglect the distance the light vertically travels in the lens itself. Is it realistic? No, of course light is going to go through the whole lens, not just at the center. But this analysis can be done by hand in about 15 minutes if you aren't prone to sign errors, and can give a person a good idea of what the limiting factors of their system is and what the main aberrations are likely to be. Its kind of like a systematized back of the napkin calculation you do before you do a proper analysis with ray trace software, which is finicky and you want to know what your answer is going to look like before you use it. 

The crux of this method is that we assume $sin \theta = \theta$. That is, that the angles are so small that the sine of the angle is (roughly) equal to the angle itself. When doing sketches of this method the angles often look huge (30 degrees or more) but the sketches have the implicit caveat 'not to scale'. It would be unenlightening (and frustrating) to be drawing angles of a few degrees. For one thing, the lines would end up overlapping.

The simplest method of doing this is ray tracing. It lets you follow individual rays through the system, and see what it is doing at every point. Its also long winded, because it involves tracing each ray at each interface in the system. Not too bad if you only have one lens, not so fun if you have lots of lenses or lenses and stops. It relies on two basic equations:

1) refracting formula: $n'_{i} u'_{i} - n_i u_i = - h_i K_i $

2) transfer formula: $h_{i+1} = h_i + d'_{i} u'_{i}$

$n$ is the refractive index (the subscripts denote which side of the interface it refers to), $h$ is the height above the optical axis of the ray, $u$ is the small angle (no sines or cosines needed), while $K$ refers to the 'focusing power' which for a surface (such as a mirror) is $(n_{i+1} - n_i)c_i$ where $c$ is the curvature or $\frac{1}{f}$ for a lens, where f is the focal length. $d$ refers to the distance between the $ith$ and $(i+1)th$ planes. Once you get the hang of the labeling, its a tedious but simple solution.

For example, two thin lenses in air:

Each dotted blue line represents a 'plane', and the red arrows represent a marginal ray. Number the planes from left to right, starting with the 'zeroth' plane. We will have a total of seven equations, (transfer, refract, refract, transfer, refract, refract, transfer) so we could solve for up to seven unknowns.

1) $ h_1 = h_0 +d'_0 u'_0$
2) $ n'_1 u'_1 - n_1 u_1 = -h_1 K_1$
3) $ n'_2 u'_2 - n_2 u_2 = -h_2 K_2$ {note: $n'_1 = n_2$, $u'_1 = u_2$, assume h1 = h2}
4) $ h_3 = h_2 +d'_2 u'_2$
5) $ n'_3 u'_3 - n_3 u_3 = -h_3 K_3$
6) $ n'_4 u'_4 - n_4 u_4 = -h_4 K_4$
7) $ h_5 = h_4 +d'_4 u'_4$

Note that each equation relies on information from the one before it to proceed. Depending on the kind of ray you are tracing, you can immediately make some assumptions. For example, since we are tracing a marginal ray, we can assume that its start and end height are zero. I've set up the example so that the first lens acts as the ASTOP, so we assume its height at the first lens is the edge of the lens. You can assume all n's that represent air are 1. This plus a little additional information will allow a complete solution.

A faster method is the transfer matrix method, which I will get into in the next post, because this one has already taken long enough to write. Hopefully I can make future posts a little less dry as dust.

~PhysicsGal