Optical resonators and Gaussian beams

Transfer matrices and ABCD law

Introduction

Even if obtaining an efficient laser with unstable cavities is possible in some special cases (see later), it is generally more favourable to have a stable resonator. The theoretical study of the stability is the first step before designing any laser cavity (choice of the curvature radii, size of the cavity...).

The transfer matrices

The study of the resonator stability is made through the use of so-called “transfer matrices” (or ABCD matrices)

The principle of this method is the following : each optical element (lens, mirror, or even simple propagation into any kind of media, including air) is associated with a specific 2x2 matrix. The propagation characteristics could then be obtained very simply by multiplying the basic matrices.

Let's consider a beam propagating in the yOz plane, Oz being the cavity axis. In this plane, a given ray is characterized by its y-coordinate for z=0 (h) and by the slope between the direction of the ray and the z axis (see figure 5).

Figure 5 : definition of the parameters used in the text
Figure 5 : definition of the parameters used in the text [zoom...]Info

In the Gauss conditions, the relations between before a given optical system and after passing through this system are linear, and could be expressed in a matrix form :

where the diagonal elements have no dimension, B and C being respectively a length and the inverse of a length.

The matrix fully characterise the optical system

We will then give the basic ABCD matrices for some very useful elements, and show how to get the ABCD matrix of the full system from these elements.

  • Propagation over a distance d

The demonstration is straightforward (see figure 5) :

(because the rays are paraxial, we can assume tan x =x)

If we develop the matrix relation given above, we obtain :

We can then deduce by identification  : A =1, B=d, C=0 et D=1.

We will not demonstrate the other relations (the argument is exactly the same) : this is a good exercise !

  • Propagation over a distance d in a medium (index of refraction n)

  • flat diopter between two media (refractive indexes n1 and n2 )

  • Thin lens (focal length f)

  • Mirror (radius of curvature R)

We find again here the equivalence between a mirror and a lens with R=2f (see above).

Generally speaking, for an optical system S made of N systems in a row ( Si (i=1,2,,,,N)) , each of them being characterized by a matrix Ti, the matrix T corresponding to the whole system S is the product of the matrices of each element, in the reverse order : T= TN...Ti...T3T2T1

The matrices do not commute, so the order is really important !

We will see in the C paragraph that this method could be used not only with geometric optics, but also with Gaussian beams.

Some laser cavities use astigmatic components (off-axis spherical mirrors, Brewster plates, cylindrical lenses, prisms...) : in this case, the calculation has to be done for each orthogonal direction (x and y), because the matrices are not the same for x and y !

ABCD law

The ABCD law describes the propagation of a spherical wave through an optical system.

Let's consider a spherical wave with its origin at O1, and a radius of curvature R1 at the entrance of a given optical system. This wave converges toward the point O2 after the system, with a radius R2 . We will take R>0 for diverging waves and R<0 for converging ones.

With this convention, et (see figure 6)

Figure 6 : Parameters used to demonstrate the ABCD law
Figure 6 : Parameters used to demonstrate the ABCD law [zoom...]Info

We can thus deduce from the definition of the ABCD matrix :

math  :   (ABCD law)

This relation is very important. We will see that it can be also applied to complex radii of curvature (see the chapter dealing with Gaussian beams)

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