Optical Engineering Science. Stephen Rolt

General

      In Section 1.4 we looked at the behaviour of some very simple components, mirrors and lenses, deriving the locations of the Cardinal Points. As discussed previously, the Cardinal Points provide a complete description of the first order properties of an optical system, no matter how complex.

The question then is how do we calculate the properties of a more complex optical system, such as the camera lens depicted in Figure 1.17? It is not immediately obvious where the Cardinal points lie or what the focal length is. However, we can combine the generalised description of an optical system with the treatment of Gaussian optics to produce a model that describes the entire system as a black box acting on rays with a simple linear transformation. The black box may be visualised as below in Figure 1.18.

Following the basic premise of Gaussian optics, we can relate the input and output rays using a set of linear equations:

(1.23)

(1.24)

Figure 1.17 Complex optical system.

Figure 1.18 Modelling of complex systems.

Equations (1.23) and (1.24) may be combined in a matrix representation:

(1.25)

Equation (1.25) sets out the Matrix Ray Tracing convention used in this book. The reader should be aware that other conventions are used, but this is the most widely used. Equation (1.25) can be used to describe the overall system matrix or that of individual components. The question is how to build up a complex system from a large number of optical elements. The camera lens shown in Figure 1.17 has six lenses and we might represent each lens as a single matrix, i.e. M₁, M₂,…..,M₆. Each matrix describes the relationship between rays incident upon the lens and those leaving. The impact of successive optical elements is determined by successive matrix multiplication. So the system matrix for the lens as a whole is given by the matrix product of all elements:

      (1.26)

      Note the order of the multiplication; this is important. M₁ represents the first optical element seen by rays incident upon the system and the multiplication procedure then works through elements 2–6 successively. For purposes of illustration, each lens has been treated as being represented by a single matrix element. In practice, it is likely that the lens would be reduced to its basic building blocks, namely the two curved surfaces plus the propagation (thickness) between the two surfaces. We also must not forget the propagation through the air between the lens elements.

Representation of the key optical surfaces can be determined by casting Eqs. (1.18)–(1.22) in matrix format.

(1.27a)

      (1.27b)

      (1.27c)

      (1.27d)

      (1.27e)

(1.27f)

      n₁ and n₂ represent the refractive index of first and second media respectively.

      1.6.2 Determination of Cardinal Points

It is very straightforward to calculate the Cardinal Points of a system from the system matrix:

      The matrix above represents the system matrix after propagating through all optical elements as shown in Figure 1.17. However, the convention adopted here is that an additional transformation is added after the final surface. This additional transformation is free space propagation to the original starting point. It must be emphasised that, this is merely a convention, and that the final step traces a dummy ray as opposed to a real ray. That is to say, in reality, the light does not propagate backwards to this point. In fact, this step is a virtual back-projection of the real ray which preserves the original ray geometry. The logic of this, as will be seen, is that in any subsequent analysis, the location of all cardinal points is referenced with respect to a common starting point. If this step were dispensed with, then the three first Cardinal Points would be referenced to the start point and the three second Cardinal Points to the end point. With this in mind, the Cardinal Points, as referenced to the common start point are set out below; the reader might wish to confirm this.

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