Imaging Properties of a Lens System

Imaging Formulae for Lenses in Arbitrary Media

These formulae allow for the possibility of distinct and completely arbitrary refractive indices for the object space medium (refractive index n), lens (refractive index n'), and image space medium (refractive index n"). In this situation, the effective focal length assumes two distinct values, namely f in object space and f " in image space, as shown in the figure below. It is also necessary to distinguish the principal points from the nodal points. The lens serves both as a lens and as a window separating the object space and image space media.
Symmetric lens with disparate object and image space indices
The nodal points are indicated by N and N". A ray directed at the primary nodal point N of a lens appears to emerge from the secondary nodal point N" without change of direction. Conversely, a ray directed at N" appears to emerge from N without change of direction. At the infinite conjugate ratio, if a lens is rotated about a rotational axis orthogonal to the optical axis at the secondary nodal point (i.e., if N" is the center of rotation), the image remains stationary during the rotation. This fact is the basis for the nodal slide method for measuring nodal-point location. The nodal points coincide with their corresponding principal points when the image space and object space refractive indices are equal ( n = n"). This makes the nodal slide method the most precise method of principal-point location. The situation of a lens immersed in a homogenous fluid is included as a special case ( n = n" ). This case is of considerable practical importance. The two values f and f " are again equal, so that the lens-combination formulas are applicable to systems immersed in a common fluid. The general case (two different fluids) is more difficult, and it must be approached by ray tracing on a surface-by-surface basis.
Lens Constant (k)
This number appears frequently in the following formulas. It is an explicit function of the complete lens prescription (both radii, tc and n' ) and both media indices (n and n" ). This dependence is implicit anywhere that k appears.
Effective Focal Lengths
Lens Formula (Gaussian form)
Lens Formula (Newtonian form)
where x = s - f and x" = s" - f "
Principal-Point Locations
Object-to-First-Principal-Point distance
Second-Principal-Point-to-Image
Magnification
Lens Maker's Formula
Nodal-Point Locations
Separation of Nodal Point from Corresponding Principal Point
Back Focal Length
Front Focal Length
Focal Ratios   The focal ratios are f /f and f /f, where f is the diameter of the clear aperture of the lens.
Numerical Apertures
where
and
where
Solid Angles
where
and
where
To convert from steradians to spheres, simply divide by 4p

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Optics Guide Copyright 2002 Melles Griot Inc.