Читать книгу Optical Engineering Science - Stephen Rolt - Страница 30

Earlier, in order to make our lens and mirror calculations simple and tractable, we introduced the following approximation:

That is to say, all rays make a sufficiently small angle to the optical axis to make the above approximation acceptable in practice. When this approximation is applied more generally to an entire optical system, it is referred to as the paraxial approximation (i.e. ‘almost axial’). If the same consideration is applied to ray heights as well as angles, the paraxial approximations lead to a series of equations describing the transformation of ray heights and angles that are linear in both ray height and angle. This first order theory is generally referred to as Gaussian optics, named after Carl Friedrich Gauss.

If we now assume that all rays are confined to a single plane containing the optical axis, then we can describe all rays by two parameters: θ – the angle the ray make to the optical axis and h – the height above the optical axis. If, after transformation by an optical surface, these parameters change to θ′ and h′, it is possible to write down a series of linear equations describing all transformations. These are set out in Eqs. 1.18–1.21:

(1.18)

(1.19)

(1.20)

(1.21)

Even the most complex optical system may be described as a combination of all the above elements. At first sight, therefore, it would seem that this provides a complete description of the first order behaviour of an optical system. However, there is one important, but seemingly trivial, aspect that is not considered here. This is the case of ray propagation through space. The equations are, of course simple and obvious, but we include them for completeness.

(1.22)

Equation (1.8) introduced the Helmholtz equation, a necessary condition for perfect image formation for an ideal system. It is clear that Gaussian optics represents a mere approximation to the ideal of the Helmholtz equation. The contradiction between the two suggests that there may be imperfections in the ideal treatment of Gaussian optics. This will be considered later when we will look at optical imperfections or aberrations. In the meantime, we will consider a very powerful realisation of Gaussian optics that takes the basic linear equations previously set out and expresses them in terms of matrix algebra. This is the so-called Matrix Ray Tracing technique.