Читать книгу Optical Engineering Science - Stephen Rolt - Страница 46

The introduction of two distinct sets of ray fans, tangential and sagittal, together with the inclusion of skew rays confirms that sequential ray propagation in an axial geometry is essentially a two-dimensional problem. Hitherto, all discussion and, in particular, the matrix analysis, has been presented in a strictly one-dimensional form. However, the strict description of a ray in two dimensions requires the definition of four parameters, two spatial and two angular. In this more complete description, a ray vector would be written as:

Figure 2.6 (a) Tangential ray fan; (b) Sagittal ray fan.

 hx is the x component of the distance of the ray from the optical axis

 θx is the x component of the angle of the ray to the optical axis

 hy is the y component of the distance of the ray from the optical axis

 θy is the y component of the angle of the ray to the optical axis

In this two dimensional representation, the matrix element representing each optical element would be a 4 × 4 matrix instead of a 2 × 2 matrix. However, the matrix is not fully populated in any realistic scenario. For a rotationally symmetric optical system, as we have been considering thus far, there can only be four elements:

That is to say, the impact of each optical surface is identical in both the x and y directions in this instance. However, there are optical components where the behaviour is different in the x and y directions. An example of this might be a cylindrical lens, whose curvature in just one dimension produces focusing only in one direction. The two dimensional matrix for a cylindrical lens would look as follows:

A component that possesses different paraxial properties in the two dimensions is said to be anamorphic. A more general description of an anamorphic element is illustrated next:

Note there are no non-zero elements connecting ray properties in different dimensions, x and y. This would require the surfaces produce some form of skew behaviour and this is not consistent with ideal paraxial behaviour. Since this is the case, the two orthogonal components, x and y, can be separated out and presented as two sets of 2 × 2 matrices and analysed as previously set out. All relevant optical properties, cardinal points are then calculated separately for x and y components. Even if focal points are identical for the two dimensions, the principal planes may not be co-located. This gives rise to different focal lengths for the x and y dimension and potentially differential image magnification. This differential magnification is referred to as anamorphic magnification. Significantly, in a system possessing anamorphic optical properties, the exit pupil may not be co-located in the two dimensions.