#DevOptical Part 2: Paraxial Raytracing and the ABCD Matrix

While the [»] part #1 of the series gave rules for some particular rays, it lacked a general description of how we could determine the exit angle and position for a given arbitrary input ray. While it is essential to be able to think qualitatively about a system, the foundation of science is precisely in its nature to offer quantitative data as well.

Let us take an arbitrary ray refracted by a thin lens, such as shown in Figure 1.

The ray starts at position y₀ with deviation u₀. It travels through air a distance d₁ until it reaches the lens of focal length f at height y₁ (its orientation remains unchanged so u₁=u₀). There it refracts to an output deviation u₂ (its height remains unchanged so y₂=y₁). From there, it travels a new distance d₂ such that it ends at height y₃ (its orientation remains unchanged so u₃=u₂).

By deviation, we are actually talking about the tangent of the angle the rays make to the optical axis. In the paraxial region where the ray departure to the optical axis is considered low enough, we can further simplify into:

The corresponding vector is

Indeed, after normalization we find

which is the vector oriented at angle ϴ relative to the optical axis if we restrict ourselves to the case ϴ>0.

This notation can be extended in 3D using two deviations dx and dy but we will not do it here because we will always consider that the systems are symmetric around the optical axis when we are in the sketching phase. Formal (non-paraxial) raytracing will handle the 3D case later, which will be useful when studying effects such as positioning tolerances.

We will discuss the limit of the paraxial region later when discussing aberrations. Note however that as we already mentioned in [»] part #1 of the series, we will often abuse the paraxial region as we sketch-up the system and refine the design later with more sophisticated methods.

Let’s now focus on paraxial raytracing. In the example of Figure 1 we faced two different operators: the propagation in air and the refraction.

Propagation in air of a distance d is the simplest because it follows simple trigonometric rules (you can use the vector notation as well to obtain the same result)

Refraction by a thin lens of focal length f is the symmetrical of the above operator

I will leave this formula without its derivation but you can check our former assertion that a ray parallel to the optical axis will focus to a point located at a distance f since

When you would like to trace the ray to a more complex system, you just apply the formula sequentially through each interface of the system. Here, an interface can be a thickness of air or a thin-lens.

The two formula required for a thickness of air are

And the two formula required for a thin lens are

The previous sets of equations can be written in the form of matrices which brings some advantages at the system-study level.

The equivalent matrix form of a thickness or air of distance L is

and the matrix form of the thin lens is

The matrix formulation is interesting when you would like to concatenate systems. Indeed, without any loss of generality we can write

where

is the cumulative matrix up to interface #i in the system.

You can therefore express your complete system as a single matrix M which concatenate all the matrices of the system. Let us assume that the content of the matrix is represented by the letters ABCD such that

From this representation, we can extract valuable information from the system.

For a input ray (y,u) we therefore have

Note that when talking about system, it is convenient to use ‘ instead of the interface number notation. The term with the ‘ simply means it lies at the exit of the system.

The distance L at which a collimated ray focus is given by

And therefore

This distance is known as the back focal distance (BFL) of the system.

Similar to the back focal distance, it is possible to derive a formula for the front focal distance (FFL), the distance to the system at which an object will exit collimated. We obtain it by solving the problem

Which gives

Finally, the equivalent focal length (EFFL) of the system is obtained as the inverse reciprocal of C

It is worth noting that the values ABCD are constrained and only have three degrees of freedom because of the general relation

and since both the air thickness operator and the thin lens operators have det(M)=1 it means that

so we can express one of the ABCD parameter as a function of the others

Any system made of thin lenses and air spacing is then completely described by its equivalent focal length, its front focal length and its back focal length

This ends today’s post on paraxial raytracing and the ABCD matrix. It was rather heavy in maths but we will need it for the next post which will be about STOP and pupils. I will come back later to the ABCD notation when we will start talking about system solving and the air-spaced doublet.

I would like to give a big thanks to James, Daniel, Naif, Lilith, Cam and Samuel who have supported this post through [∞] Patreon. I also take the occasion to invite you to donate through Patreon, even as little as $1. I cannot stress it more, you can really help me to post more content and make more experiments!