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2.4. General curvilinear axis
2.4 General curvilinear axis
For many systems, it is convenient to formulate the optics in terms
of transverse coordinates in a plane which is locally perpendicular
to a central optic axis. As this axis need not be a straight line,
we designate it a general curvilinear axis. We designate an axial
coordinate z, and transverse Cartesian coordinates x j = (x, y) for
j = (1, 2) in a plane locally perpendicular to the axis. We further
designate ray slope components x j
� = (x
� , y
� ) = dx j /dz. A ray is
completely specified at any plane z by its two-vector transverse
position x and its two-vector slope x
� .
The central problem in this formulation may be stated as follows: given the transverse position x a and slope x
�
a at an arbitrary
starting axial coordinate z a , find the transverse position x b and
slope x
�
b at an arbitrary ending axial coordinate z b . This is shown
schematically in Figure 2.6. It is implicit here and in the following
Figure 2.6: General curvilinear axis.

that the slope x b
� be finite. This excludes the case of a particle
mirror, for which the slope is infinite where the ray turns around.