Examples

Lens Combinations

The following examples demonstrate how to determine the parameters of simple two-element optical systems using the equations found in the Lens Combination Formulae section of this guide. In the examples,
f =  combination focal length (EFL), positive if combination final focal point falls to right of the combination secondary principal point; negative otherwise. f1 =  focal length (EFL) of first element f2 =  focal length (EFL) of second element d =  distance from secondary principal point of first element to primary principal point of second element.; positive if primary principal point is t right of the secondary principal point, negative otherwise. s2" =  distance from secondary principal point of second element to final combination focal point (location of the final image for object at infinity to left); positive If the focal point is to the right of the second element secondary principal point. z =  distance to combinationsecondary principal point measured from secondary principal point of second element; positive if combination secondary principal point is to right of secondary principal point of second element.
Positive lenses separated by distance greater than f1 + f2
f is negative, while both s2 and z are positive. Lens symmetry is not required.   Achromatic combinations
Air-spaced lens combinations can be made nearly achromatic, even though both elements are made of the same material. Achieving achromatism requires that, in the thin-lens approximation,
	.
This is the basis for Huygens and Ramsden eyepieces. This approximation is adequate for most thick-lens situations. The signs of f1, f2, and d are unrestricted, but d must have a value that guarantees the existence of an air space. Element shapes are unrestricted and can be chosen to compensate for other aberrations. Telephoto combination
The most important characteristic of the telephoto is that the EFL, and hence the image size, can be made much larger than the distance from the first lens surface to the image would suggest by using a positive lens followed by a negative lens (but not necessarily the lens shapes shown in the figure). For example,  f1 is positive and f2 = 4 f1/ 2. Then f is negative for d less than f1/ 2, infinite for d = f1/ 2 (Galilean telescope or beam expander), and positive for d larger than f1/ 2. To make the example even more specific, catalog lenses 01 LDX 189 and 01 LDK 021, with d = 78.2 mm, will yield s2 = 2.0 m, f = 5.2 m, and z = 43.2 m. Condenser configuration
A pair of identical plano-convex lenses have their convex vertices in contact. (The lenses could also be plano aspheres.) Because d = 0, f = f1/ 2 = f2/ 2, f1/ 2 = s2", and z = 0. The secondary principal point of the second element and the secondary principal point of the combination coincide at H", at depth tc/n beneath the vertex of the plano surface of the second element, where tc is the element center thickness and n is the refractive index of the element. By symmetry, the primary principal point of the combination is similarly located in the first element. Combination conjugate distances must be measured from these points.

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Optics Guide Copyright 2002 Melles Griot Inc.