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Prior to launching into a discussion of optical specification, fabrication, and testing, a coordinate system and a consistent sign convention need to be defined. These definitions will enable easy viewing of the figures and diagrams and interpretation of the physical quantities that are being illustrated. In general, the axis for an optical system will be taken as the z axis. Typically, light will travel from left (more negative values of z) to right (more positive values of z) with regard to this axis, unless the light is reflected from a surface. The vertical axis will typically be taken as the positive y axis. The coordinate system will be considered right-handed. In the following derivations, there are several signed quantities: distances, radii of curvature, and angles. Distances are measured from a reference point and are signed to be consistent with the coordinate system. For example, if one optical surface is located at z = 0 and a second optical surface is located at z = +10, then the distance from the first surface to the second surface is positive. Similarly, distances measured in the -z direction are negative. The same convention holds for the x and y directions as well. An object with its base on the z axis and its top at y = 5 would have a positive height, while its image may be upside down with its base still on the z axis but its "top" now at y = -3. In this latter case, the image height is negative.
Angles are measured with regard to a reference line such as the z axis or a normal to an optical surface. Counterclockwise angles are positive and clockwise angles are negative. For example, a ray that starts on the z axis (y = 0) at the first optical surface described above and that has a height of y = 2 at the second optical surface would form a positive angle with regard to the z axis. Finally, the sign convention for radii of curvature is that positive radii have their center of curvature to the right of the surface, while negative radii have their center of curvatures to the left of the surface. In general, arrowheads will be used in the figures to help illustrate the sign of the quantities depicted. Figure 1.1 illustrates the various conventions outlined above.
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