The Galilean telescope is often used to increase field of view or magnification in optical systems. However, where there is insufficient space or no collimated light region in which to locate a Galilean, a field lens operating in convergent light may be a solution.
A field lens positioned alternately on either side of an image yields a system with two magnifications, A special arrangement of this type, the “Vari-Mag,” yields parfocal images with two magnifications, m and 1/m. Parfocality, which eliminates focus shift between images in the two magnifications also extends to the entrance and exit pupils in the Vari-Mag concept. Pupils are thereby fixed in location. Fixing the pupil locations gives the optical designer control over the position and direction of travel of off-axis ray bundles through the system, thereby allowing control of vignetting of the field of view.
The diagram below illustrates the principle. A field lens is inserted into the path of an image formed by a preceding objective lens (not shown). The distance between the field lens and the input image, the field lens focal length and the magnification imposed on the input image by the field lens are related by the thin lens equation:
f =s (m/(1-m)
s = the distance between the input image and the field lens
f = field lens focal length
m = magnification of the field lens in the low magnification end
and the field lens image distance s’ is related by :
For high magnification the field lens is repositioned along the axis at a distance equal to the sum of the input and output image distances.
T = s+s’ = ms+s = s(m+1)
This yields parfocality at the two lens positions. The low magnification image is real while the high magnification image is virtual as illustrated in the diagram.
The entrance and exit pupils lie at the points where the chief rays of the off-axis light bundles intersect the optical axis. The pupils will be parfocal if the sum of their object and image distances are the same at the two lens positions. Simultaneous solution of the following equations yields this condition:
1/p’ = 1/p + 1/f
1/(p’-T) = 1/(p-T) + 1/f
where p and p’ are the entrance pupil distance and exit pupil distances in the low magnification setting and T is the distance between the lens positions at the low and high magnification settings.
The entrance pupil distance for the low magnification case is determined by simultaneous solution of the above equations:
p = (-2f +T)/2 +/- sqrt(4f² +T²)/2
The exit pupil distance p’ is then found by solution of the thin lens equation:
p’ = fp/(f+p)
The entrance and exit pupil distances for the lens in the high magnification position are the inverse values of the pupil distances for the low magnification position .
Conveniently, the pupil distances can be expressed in terms of the input image distance s and the magnification m by substituting into the expression for p, the expressions for T and f established above for parfocality between the high and low magnification images. This yields:
p = -m s (1+m)/(1-m)
p’ = s (1+m)/(1-m)
Thus, specifying only the magnification range m² and input image distance s, remarkably, is sufficient information to determine all other factors including field lens focal length, high and low end magnifications, separation of the lens at the low and high end magnification settings, and the entrance and exit pupil distances at each setting.
Further details can be found at Vari-mag Equations Derivation.
The Vari-Mag concept can be implemented by sliding the field lens along the axis from one position to the other. It can also be flipped 180 degrees about a transverse axis, In this case the optical design treats the inverse lens geometry for the lens in the flipped position. By introducing a pair of fold mirrors as illustrated below the field lens can be switched easily and quickly from one position to the next while achieving a compact configuration.