Page 81 - Modern Optical Engineering The Design of Optical Systems

64 Chapter Four

We can write the first set in matrix notation as
u′
1 y (4.27)
u

y
0 1
The second set becomes
u′
1 0 (4.28)
u
2

1
y
2 d 1 y 1
Substituting the left side of Eq. 4.27 into Eq. 4.28 and multiplying the
two inner matrices, we get
1
u
u
2

1
y
2 d 1 d y 1
which is the matrix form of Eqs. 4.1 and 4.2.
This process can be chained to encompass an entire optical system if
desired, and the final product of all the inner matrices can be interpreted
to yield the cardinal points, focal lengths, etc., of the system.
Note well that there is absolutely no magic in this process. The amount
of computation involved is exactly the same as in the corresponding
paraxial raytrace. To this author it seems far more informative to trace
the ray paths and to have the added benefit of a knowledge of the
paraxial ray heights and slopes. However, for those to whom matrix
manipulation is second nature, this formulation has a definite appeal,
although no advantage.

4.4 The y-ybar Diagram
The y-ybar diagram is a plot of the ray height y of an axial ray versus
the ray height, ybar, of an oblique (i.e., principal or chief) ray. Thus each
point on the plot represents a component (or surface) of the system.
Figure 4.5a shows an erecting telescope and Fig. 4.5b shows the cor-
responding y-ybar diagram. Note that point A in the y-ybar diagram
corresponds to component A, etc. An experienced practitioner can
quickly sketch up a system in y-ybar form in the same way that a system
can be sketched using elements and rays.
The reduction of either a y-ybar diagram or a sketch with rays to a
set of numerical values for the component powers and spacings
involves the same amount of computation in either case. Although the
y-ybar diagram is simpler to draw than a ray sketch, there is obviously
more information in the ray sketch, and an experienced practitioner
can easily draw a ray sketch accurately enough to allow conclusions to
be drawn as to its practicality, size, etc. which the y-ybar diagram does
not readily provide.