Optical Engineering Science. Stephen Rolt

from the object by rotation alone. Conversely, if the optical system were to contain an even number of reflective surfaces, then the parity between the object and image geometries would be conserved.

      Another way in which a plane mirror is different from a plane refractive surface is that a plane mirror is the one (and perhaps only) example of a perfect imaging system. Regardless of any approximation with regard to small angles discussed previously, following reflection at a planar surface, all rays diverging from a single image point would, when projected as in Figure 1.15, be seen to emerge exactly from a single object point.

      1.4.6 Reflection from a Curved (Spherical) Surface

Figure 1.16 illustrates the reflection of a ray from a curved surface.

      The incident ray is at an angle, θ, with respect to the optical axis and the reflected ray is at an angle, ϕ to the optical axis. If we designate the incident angle as θ₁ and the reflected angle as θ₂ (with respect to the local surface normal), then the following apply, assuming all relevant angles are small:

Figure 1.16 Reflection from a curved surface.

We now need to calculate the angle, ϕ, the refracted ray makes to the optical axis:

(1.17)

In form, Eq. (1.17) is similar to Eq. (1.14) with a linear dependence of the reflected ray angle on both incident ray angle and height. The two equations may be made to correspond exactly if we make the substitution, n₁ = 1, n₂ = −1. This runs in accord with the empirical observation made previously that a reflective surface acts like a medium with a refractive index of −1. Once more, the sign convention observed dictates that positive axial displacement, z, is in the direction from left to right and positive height is vertically upwards. A ray with a positive angle, θ, has a positive gradient in h with respect to z.

      As with the curved refractive surface, a curved mirror is image forming. It is therefore possible to set out the Cardinal Points, as before: Cardinal points for a spherical mirror

Both Principal Points: At vertex
Both Nodal Points: At centre of sphere

      The focal length of a curved mirror is half the base radius, with both focal points co-located. In fact, the two focal lengths are of opposite sign. Again, this fits in with the notion that reflective surfaces act as media with a refractive index of −1. Both nodal points are co-located at the centre of curvature and the principal points are also co-located at the surface vertex.

1.5 Paraxial Approximation and Gaussian Optics

      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.

1.6 Matrix Ray Tracing

      1.6.1