# Galilean and Inverse Galilean Afocal Lenses

This is a continuation from the previous tutorial - keplerian afocal lenses.

The combination of a positive objective and a negative eyepiece forms a galilean telescope. If the objective is negative and the eyepiece positive, it is referred to as an inverse galilean telescope.

The galilean telescope has the advantage that it forms an erect image. It is the oldest form of visual telescope, but it has been largely replaced by terrestrial telescopes for magnified viewing of distant objects, because of field of view limitations.

In terms of number of viewing devices manufactured, there are far more inverse galilean than galilean telescopes. Both are used frequently as power-changing attachments to change the effective focal length of focusing lenses.

## Thin-Lens Model of a Galilean Afocal Lens

Figure 12 shows a thin-lens model of a galilean afocal lens.

The properties of galilean lenses can be derived from Eqs. (9) [refer to the Gaussian analysis of afocal lenses tutorial], (12), and (13) [refer to the keplerian afocal lenses tutorial].

Given that $$f_e$$ is negative and $$f_o$$ is positive, $$M$$ is positive, indicating an erect image. If $$RO$$ is placed at the front focal point of the objective, $$RE$$ is a virtual pupil buried inside the lens system. In fact, galilean lenses cannot form real images of real objects under any conditions, and at least one pupil will always be virtual.

## Field of View in Galilean Telescopes

The fact that only one pupil can be real places strong limitations on the use of galilean telescopes as visual instruments when $$M\gg1x$$. Given the relationship $$\Delta{z_o}=M^2\Delta{z_e}$$, moving $$RE$$ far enough outside the negative eyepiece to provide adequate eye relief moves $$RO$$ far enough into virtual object space to cause drastic vignetting at even small field angles.

Placing $$RE$$ a distance $$ER$$ behind the negative lens moves $$RO$$ to the position shown in Fig. 13, $$SF'$$ units behind $$RE$$, where

$\tag{26}SF'=(M^2-1)ER-(M-1)^2f_e$

In effect, the objective is both field stop and limiting aperture, and vignetting defines the maximum usable field of view. The maximum acceptable object space angle $$u_{po}$$ is taken to be that for the principal ray which passes just inside $$D_o$$, the entrance pupil at the objective.

If the F-number of the objective is $$\text{FN}_{ob}=f_o/D_o$$, then

$\tag{27}\tan{u_{po}}=\frac{-f_e}{2\text{FN}_{ob}(M\cdot{ER}+f_e-Mf_e)}$

For convenience, assume $$ER=-f_e$$. In this case, Eq. (27) reduces to

$\tag{28}\tan{u_{po}}=\frac{1}{2\text{FN}_{ob}(2M-1)}$

For normal achromatic doublets, $$\text{FN}_{ob}\ge4.0$$. For $$M=3x$$, in this case, Eq. (28) indicates that $$u_{po}\le1.43°$$ ($$\text{FOV}\le150$$ feet at 1000 yard). For $$M=7x$$, $$u_{po}\le0.55°$$ ($$\text{FOV}\le57.7$$ feet at 1000 yard).

The effective field of view can be increased by making the objective faster and more complex, as can be seen in early patents by von Rohr and Erfle. In current practice, galilean telescopes for direct viewing are seldom made with $$M$$ larger than $$1.5x–3.0x$$. They are more typically used as power changers in viewing instruments, or to increase the effective focal length of camera lenses.

## Field of View in Inverse Galilean Telescopes

For inverse galilean telescopes, where $$M\ll1x$$, adequate eye relief can be achieved without moving $$RO$$ far inside the first surface of the objective.

Inverse galilean telescopes for which $$u_{po}\rightarrow90°$$ are very common in the form of security viewers of the sort shown in Fig. 14, which are built into doors in hotel rooms, apartments, and many houses.

These may be the most common of all optical systems more complex than eyeglasses. The negative objective lens is designed with enough distortion to allow viewing of all or most of the forward hemisphere, as shown by the principal ray in Fig. 14.

Inverse galilean telescopes are often used in camera view finders. These present reduced scale images of the scene to be photographed, and often have built in arrangements to project a frame of lines representing the field of view into the image.

Inverse galilean power changers are also used to increase the field of view of submarine periscopes and other complex viewing instruments, and to reduce the effective focal length of camera lenses.

## Anamorphic Afocal Attachments

Afocal attachments can compress or expand the scale of an image in one axis . Such devices are called anamorphosers, or anamorphic afocal attachments. One class of anamorphoser is the cylindrical galilean telescope, shown schematically in Fig. 15a.

Cox and Harris have patented representative examples. The keplerian form is seldom if ever used, since a cylindrical keplerian telescope would introduce image inversion in one direction.

Anamorphic compression can also be obtained using two prisms, as shown in Fig. 15b. The adjustable magnification anamorphoser patented by Luboshez is a good example of prismatic anamorphosers. Many anamorphic attachments were developed in the 1950s for the movie industry for use in wide-field cameras and projectors. An extensive list of both types will be found in Wetherell.

Equation (9) [refer to the Gaussian analysis of afocal lenses tutorial] can be modified to describe anamorphic afocal lenses by specifying separate afocal magnifications $$M_x$$ and $$M_y$$ for the two axes.

One important qualification is that separate equations are needed for object and image distances for the $$x$$ and $$y$$ planes. In general, anamorphic galilean attachments work best when used for distant objects, where any difference in $$x$$-axis and $$y$$-axis focus falls within the depth of focus of the associated camera lens.

If it is necessary to use a galilean anamorphoser over a wide range of object distances, it may be necessary to add focus adjustment capabilities within the anamorphoser.

The next tutorial introduces relay trains and periscopes.