# Angular and Linear Fields of View of Galilean Telescopes and Telemicroscopes

Purpose. The calculation of the angular fields of view (FOVs) of Galilean telescopes generally necessitates the calculation of the pupils and ports. This, in turn, requires knowledge of the optical design of the telescope, in particular, the focal lengths or powers of the objective and ocular lenses. Equations for finding the FOV that obviate the need to calculate pupils and ports, or even to know the lens powers of the telescope, are presented in this article. The equations can be used to find the FOVs in image space of real Galilean telescopes of known magnification, merely by measuring the distance between the objective and ocular lenses and the diameter of the objective lens. The equations include the effects of eye pupil diameter and eye relief. Linear FOVs (LFOVs) of Galilean telemicroscopes are similarly determined.

Methods. Two image space angular FOV equations were derived: (1) an equation to determine the angular FOVs of a telescope with various amounts of vignetting and eye relief; and (2) an equivalent equation for the LFOVs of telescopes fitted with lens caps for near vision.

Results. The FOV increases linearly with increasing vignetting. Increasing the eye relief results in a nonlinear decrease in the FOV, shown as a fraction of the normalized value for zero eye relief. Decrements in the FOVs with increasing eye relief as a fraction of the normalized field angle when the eye relief = 0 are shown to be constant regardless of the vignetting level. A transition of the objective lens from field stop to aperture stop occurs when the eye pupil diameter exceeds the diameter of the objective lens divided by the magnification.

Conclusions. Equations have been derived for Galilean telescopes and telemicroscopes that make it unnecessary to find pupils and ports, or to know the powers of the lenses. They provide a direct and simple evaluation of angular and LFOVs as functions of magnification, objective lens diameter, eye pupil diameter, eye relief, and vignetting, and enable comparisons of actual telescopes.

State University of New York, State College of Optometry, New York, New York

Received May 12, 2006; accepted February 9, 2007.

The Galilean telescope is widely used in low-vision prescriptions. As is common in compound optical instruments, its field of view (FOV) is greatly limited. Consequently, for telescopes of a given magnification (all other things being equal) it is desirable to prefer the one with the widest FOV.

Descriptions of the Galilean telescope in the literature^{1â€“6} are cursory, and generally limited to noting that it comprises a positive objective and a negative ocular lens, provides erect images that are angularly magnified, and is limited to low magnification. The objective lens is a field stop, and when coupled to the eye, the pupil is the aperture stop of the telescope and commonly shown coincident with the ocular lens. Smith and Atchison^{7} describe how to calculate the FOV in object space (real FOV) by calculating the position and size of the entrance pupil and determining the field angles given by the directions between the edge of the objective lens (the entrance port) and the center and edges of the entrance pupil, that is, with full illumination or zero vignetting, half illumination or 50% vignetting, and the absolute or 100%-vignetted field. Keating^{8} also details the use of the entrance port and pupil to calculate the FOV of the Galilean telescope. Westheimer^{9} extensively treats the angular and linear FOVs (LFOVs) of various visual aids that range from magnifiers to telescopes. His equation for the LFOV of a telescope adjusted for near vision requires the calculation of the image of the entrance pupil of the eye produced by the ocular lens. Reich^{10} treats the LFOV and equivalent viewing power of Galilean and Keplerian telescopes fitted with lens caps or adjusted in length to focus for near vision. His analysis also requires the calculation of the entrance pupil.

Two equations that obviate the need to calculate pupils and ports or the powers of the lenses are presented. The first is an equation to calculate the FOV in image space or the apparent angular FOV of Galilean telescopes. The equation requires knowledge of the telescope's angular magnification (*M*), the diameter of the objective lens (*D*_{o}), and the distance (*d*) between the objective and ocular lenses. The latter two items may be measured on actual telescopes. In addition, by specifying the diameter of the pupil of the eye (*D*_{a}) and the eye relief (*e*), the equation provides the FOVs at 0, 50, and 100% vignetting (full illumination, half illumination, and absolute field) and illustrate the effect of eye relief and eye pupil diameter on the FOV.

The diameter of the pupil of the eye determines whether the objective lens is a field stop or an aperture stop. When the pupil diameter is less than the diameter of the virtual image of the objective lens produced by the ocular lens, the pupil is the exit pupil. However, if the pupil diameter exceeds the diameter of the virtual image of the objective lens, the pupil becomes the exit port of the telescope.

The determination of the more relevant LFOVs when Galilean telescopes are converted with lens caps into telemicroscopes for near vision is provided by the second equation.

## METHODS

The geometry for finding FOVs of a Galilean telescope in which all components are centered on an optical axis is illustrated in Figure 1(a). Because the diameter of the pupil of the eye *D*_{a} is smaller than the diameter of the virtual image of the objective lens *D*â€²_{o}, (*D*_{a} < *D*â€²_{o}), the objective lens is the field stop and entrance port. The diameter of the objective lens determines the FOV and its virtual image is the exit port located at a distance *u*â€² from the ocular lens [Fig. 1(b)]. The pupil of the eye is the aperture stop and exit pupil of the system. In most treatments, the eye is coincident with the ocular lens, that is, the eye relief *e* = 0. In the derived equation, the exit pupil may be at any distance *e* behind the telescope. The exit port and exit pupil replace the telescope for finding the FOVs in image space. The exit port is a window through which the eye (exit pupil) views the magnified FOV.

### Zero Vignetting

The angular FOV for which there is full illumination or zero vignetting is indicated by the path of a real ray in object space with a slope Î¸_{0} that strikes the upper edge of the field stop and emerges from the telescope to pass through the upper edge of the aperture stop (exit pupil). Real ray paths are shown as unbroken lines. The forward projections of the emerging rays in image space are shown as broken lines. They correspond to the apparent field angles.

The emergent ray 1 in Figure 1(b) appears to graze the upper edge of the exit port and grazes the upper edge of the exit pupil. Its slope Î¸â€²_{0} is the apparent field angle within which there is zero vignetting. Parallel ray 2 in image space grazes the lower edge of the exit pupil; thus, the pupil is filled with light from this direction. Vignetting is zero for all lesser field angles.

### 50% Vignetting

As shown in Figure 1(c), the image space field of half illumination or apparent field angle Î¸â€²_{50} where vignetting is 50% is delimited by apparent image space ray 1 (the chief ray) through the upper edge of the exit port and the center of the exit pupil. The corresponding real ray (not shown) enters the telescope at the upper edge of the objective lens and emerges toward the center of the pupil of the eye. Parallel image space ray 2 enters the lower edge of the pupil so that only half the pupil receives light from this direction. When unqualified, the FOV of optical systems is denoted by this chief ray angle, because as the pupil diameter becomes vanishingly small, the chief ray is the last ray to be transmitted.

### 100% Vignetting

The absolute field angle, at which vignetting reaches 100%, is given by the ray through opposite edges of the exit port and exit pupil. The object space portion of this ray strikes the upper edge of the objective lens. In image space the ray shown in Figure 1(d) has a slope Î¸â€²_{100} = *M* Î¸_{100.} Object space rays that graze the upper edge of the field stop with greater field angles will not enter the exit pupil and will not be visible to the observer [Fig. 1(d)].

### LFOV of Telemicroscopes

Figure 2 illustrates the half-linear FOVs *Y*_{j} with indicated vignetting of an object at the anterior focal plane of a lens cap. The slopes of the rays from each *Y* point are such that the rays are incident at the lens cap with a height equal to the semidiameter of the objective lens. It is assumed that the distance between the cap and the objective is zero. After refracting through the telescope the zero-vignetted ray from height *Y*_{0} enters the upper edge of the eye (exit pupil); the 50%-vignetted ray from height *Y*_{50} enters the center of the exit pupil; and the 100%-vignetted ray enters the lower edge of the pupil. Projected back, these rays appear to come from the upper edge of the exit port.

## RESULTS

### The Angular FOV

As derived in the Appendix (available online at www.optvissci.com), based on Figure A1, the apparent field angle Î¸â€²_{j} is given by Eq. (A8), where *j* is 0, 50, or 100% vignetting.

where, *M* = angular magnification, is given, *d* = distance between objective and ocular lenses (this may be the measured tube length or the sum of their focal lengths f_{o} + *f*_{e}, if they are known.), *D*_{o} = diameter of objective lens, *D*_{a} = diameter of pupil of eye (aperture stop), and *e* = eye relief (the distance between the ocular lens and the pupil). *D*_{a} and *e* are specified.

Figure A1 shows the negative slopes of the emergent rays that correspond to the three degrees of vignetting when *D*_{a} < *D*â€²_{o}. The *D*_{o} term in Eq. (1) is negative for zero vignetting, thus yielding a minimum slope (or tan Î¸â€²_{0}). The ray grazes corresponding edges of port and pupil. *D*_{a}*M* equals zero for 50% vignetting and results in the intermediate slope (or tan Î¸â€²_{50}) of the chief ray through the center of the exit pupil. *D*_{o} is positive for the ray with the greatest slope that corresponds to 100% vignetting. It grazes opposite edges of the port and pupil (or tan Î¸â€²_{100}).

*Example:* Consider as an example a 4Ã— telescope in which *d* = 75 mm (e.g., *f*_{o} = 100 mm, f_{e} = âˆ’25 mm), *D*_{o} = 20 mm, *D*â€²_{o} = *D*_{o}/*M* = 5 mm, *e* = 15 mm, 2(d + eM) = 270 mm) and *D*_{a} = 4 mm. Note: *D*_{a} < *D*â€²_{o}, and therefore, the objective lens is the field stop. The total FOVs are:

Table 1 summarizes the FOVs with vignetting and eye relief for this telescope in which the objective lens is the field stop.

Figure A2 shows the negative slopes of the emergent rays that correspond to the three degrees of vignetting when the objective lens is the aperture stop, i.e., *D*_{a} > *D*â€²_{o}. Equation (1) still applies. The *D*_{a}*M* term is negative for zero vignetting and positive for 100% vignetting. However, the *D*_{o} term is zero for 50% vignetting. If the diameter of the pupil *D*_{a} is increased to 8 mm and other parameters remain the same, the total FOVs are increased, as follows:

Table 2 summarizes the FOVs with vignetting and eye relief for this telescope in which the objective lens is the aperture stop.

Tables 1 and 2 also show identical normalized tan Î¸â€² values. They are decrements in the FOVs with increasing eye relief as a fraction of the normalized FOV at zero eye relief, that is, the normalized Î¸â€² = e_{n}/e_{0}, where e_{n} > 0. These normalized values are constant for any given eye relief, regardless of the vignetting level and whether the objective lens acts as a field or aperture stop. For example, the values of tan Î¸â€² for eye relief e = 15 mm are 56% of the zero eye relief values. As seen in Figure 3, the FOVs, at each eye relief, increase linearly with vignetting. Figure 4 also shows the normalized tan Î¸â€²values. The tables also show the apparent and real field angles in degrees.

### LFOV of Telemicroscopes

With the attachment of a plus lens or cap, a telescope is converted into a telemicroscope and can be used to magnify near objects. The cap acts as a magnifier with a nominal magnification of M_{mag} = 250/f_{cap}, where f is in millimeters. The magnification of the telemicroscope is equal to the product of the magnification of the magnifier M_{mag} and the magnification of the telescope M, i.e., M_{tms} = M_{mag} Ã— M.

If it is assumed that the separation between the cap and the objective lens is zero, the semidiameter of the FOV is given by (f_{cap}/M) Ã— tan Î¸â€² [see Appendix 2, Equation (A13)].

Given a +10 D or 100-mm focal length cap and the telescope parameters of the prior examples, the half-linear FOVs as functions of vignetting and eye relief are shown in Table 3. For example: Let *e* = 15 mm, find the LFOV at zero vignetting.

The total diameter of the full-illumination LFOV is approximately three-quarters of a millimeter; however, the half-illumination FOV/2 is Â±1.85 mm and the absolute FOV/2 is Â±3.33 mm (Table 3).

Figure 5 illustrates the effect of eye pupil diameter *D*_{a} on the LFOV at 50% vignetting of the 4Ã— Galilean telescope (*f*_{o} = 100 mm, *D*_{o} = 20 mm, and *D*â€²_{o} = 5 mm) with a 100-mm focal length (+10 D) lens cap and an eye relief of 15 mm. The LFOV is constant (Â±1.85 mm) for the system as long as the objective lens is the field stop, i.e., *D*_{a} < *D*â€²_{o}. When the eye pupil dilates so that *D*_{a} > *D*â€²_{o}, the objective lens becomes the aperture stop of the system and the LFOV increases. As indicated by the figure and Eq. (2), when *D*_{a} = 8 mm, the LFOV = Â±2.96 mm.

## DISCUSSION

The slope of the chief ray is through the edge of the port and the center of the pupil. It corresponds to the field angle where vignetting is 50%. This is the unqualified FOV of the telescope. It is the FOV that remains when the exit pupil is reduced to a pinhole and so it is independent of pupil size. When the objective lens is the field stop and the eye pupil is the exit pupil, the slope of the chief ray for a given eye relief is fixed because the diameter of the exit port is fixed. However, the transition of the objective lens from field stop to aperture stop that occurs when the eye pupil diameter *D*_{a} > *D*â€²_{o} results in a chief ray slope from the fixed center of *D*â€²_{o} to the edge of the eye pupil. As the pupil dilates, the chief ray angle (tan Î¸â€²_{50}) increases. For the prior example, this occurs when *D*_{a} > 5 mm. Figure 5 illustrates this effect for LFOV.

The numerator in Eq. (1) is *D*_{a}*M* or *D*_{o} depending upon whether the pupil of the eye is the exit port or exit pupil. Tan Î¸â€²_{50} is increased when the eye pupil is the exit port. The enlargement in the 50%-vignetted FOV when *D*_{a} > *D*â€²_{o} is equal to *D*_{a}*M*/*D*_{o}. For example, when *D*_{a} = 8 mm, the enlargement is 32/20 = 1.6*x*. This may be seen in Tables 1 and 2 where, for *e* = 15 mm, the 50%- vignetted FOVs are 0.119/0.0741 = 1.6. Similarly, the 50%- vignetted LFOVs are in the ratio of 2.98/1.85 = 1.6 (Table 3).

For the 4-mm pupil, the 100%-vignetted FOV is 1.44 times larger than the zero-vignetted FOV. This factor is increased to three times with the 8-mm pupil.

The advantage of dilated pupils is clearly indicated by comparing the enlargement factor EF for the absolute vs. full-illumination FOVs at a given *D*_{a}. EF equals the ratio of the corresponding numerators of Eq. (1), namely,

Given D_{o} = 20 mm, D_{a} = 4 mm, and M = 4, the absolute field is 144/16 = 9 times larger than the fully illuminated field. Table 1 shows that this ratio is constant regardless of eye relief. If D_{a} = 8 mm, the objective lens is the aperture stop, i.e., D_{a} > *D*â€²_{o}. The enlargement factor becomes 4.333.

The reduction in field that occurs as the eye relief increases can be readily visualized by simplifying the telescope to an exit port or window through which the eye (exit pupil) looks. Thus, Table 1 shows that the maximum FOV is obtained when *e* = 0, but is about a quarter as large when *e* = 60 mm. This reduction in angular field is constant regardless of vignetting. Thus, from Table 1, the tangents for *e*_{0}/*e*_{60} at 0, 50, and 100% vignetting = 0.0063/0.0267 = 0.0317/0.1333 = 0.0571/0.240 = 0.24.

The reduction in field angle as a function of eye relief may be found with

where *e*_{1} and *e*_{2} are the two eye reliefs to be compared. For example, the substitution *e*_{1} = 0 mm and *e*_{2} = 60 mm results in 75/315 = 0.24.

Table 1 also shows the effect of vignetting on the angular FOV. For example, when *e* = 0, the factors of increase in FOV are:

These factors are independent of eye relief.

The effect of vignetting on the LFOVs of the telemicroscope is identical to the angular FOVs of the telescope. Thus, the 100%-vignetted FOV is nine times that of the zero-vignetted FOV, and 1.8 times that of the 50%-vignetted FOV regardless of the eye relief.

## CONCLUSIONS

Equations (1) and (2) provide a direct and simple evaluation of the angular and LFOVs of Galilean telescopes and telemicroscopes and enable comparisons of telescopes. Reasonably accurate FOVs will be obtained, given the angular magnification and simple measurements on actual telescopes of the separation between the objective and ocular lens, and the diameter of the clear aperture of the objective lens, assuming the telescope lenses are relatively thin and mounted in a tube. More accurate results will be obtained if the focal length of the objective lens is known. The transition of the objective lens from field to aperture stop results in an increased FOV at 50% vignetting; however, it has no effect on the 0 and 100%-vignetted FOV.

Milton Katz

SUNY College of Optometry

33 West 42nd Street

New York, NY 10036

e-mail: mkatz@sunyopt.edu

## REFERENCES

## APPENDIX

### Derivation of FOV Equations for Galilean Telescopes

In Figure A1, *D*_{a} < *D*â€²_{o}. The objective lens is the field stop and all rays have negative slopes between the exit port and exit pupil. In Figure A2, *D*_{a} > *D*â€²_{o}. The objective lens is the aperture stop and all rays have negative slopes between the exit pupil and exit port. Let *y*_{1} be the height of a ray at the port, and *y*_{2} be its height at the pupil. Then, the FOVs in image space as functions of percent vignetting (*j*) are given by the inverse tangent

where, *u*â€² is the distance of the virtual image of the objective lens, *e* is the eye relief and (âˆ’*u*â€² + *e*) is the positive distance between the port and pupil.

The values of *y*_{1} and *y*_{2} for the indicated vignettings are shown in Table A1.

As shown in Figures A1 and A2 object distance *u* in the afocal telescope is equal to âˆ’(*f*_{o} + *f*_{e}). Image distance *u*â€² can be found with the thin lens equation.

According to the angular magnification of the telescope

Substitution of Eqs. (A2) and (A3) into Eq. (A1) results in the half FOV at zero vignetting

The half FOV at 50% vignetting when the objective lens is the field stop (*D*_{a} < *D*â€²_{o}) is given by

The half FOV at 50% vignetting when the objective lens is the field stop (*D*_{a} > *D*â€²_{o}) is given by

The half FOV at 100% vignetting is given by

The FOV equations for 0 and 100% vignetting apply regardless of whether the eye is the pupil or the port.

The general equation for the FOV/2 is

where, *D*_{o} is negative for zero vignetting and positive for 100% vignetting. Depending on whether the pupil of the eye is the exit pupil or exit port, *D*_{a}*M* = 0 or *D*_{o} is 0.

### Derivation of LFOV Equations for Galilean Telemicroscopes

Equation (1) solves for the image space slopes (tan Î¸â€²) of the variously vignetted rays that are incident at the edge of the objective lens (*D*_{o}/2) of the telescope. The slopes of these rays in object space of the telescope are tan Î¸_{os} = tan Î¸â€²/*M*.

To find the LFOVs of the telemicroscopes, it is merely necessary to find the paths of rays from points in the object plane (*Y*_{obj}) that are incident on the lens cap at a height of *Y*_{cap} = *D*_{o}/2 and have slopes after refraction equal to tan Î¸_{os} = tan Î¸â€²/*M*. It is assumed that the separation between the lens cap and the objective lens is zero (Fig. 2).

To find the ray height at the object plane (*Y*_{obj} = LFOV/2) and object space slopes (Î¸_{cap}) of rays incident on the lens cap, we may use the Refraction and a Transfer paraxial ray trace equations^{11} as follows:

and

The substitution of Eq. (A9) into Eq. (10) results in

where tan Î¸_{cap} = ray slope incident on lens cap, tan Î¸â€²/M = slope of ray after refraction by lens cap, Y_{cap} = D_{o}/2, and d_{cap} = f_{cap}.

Substitution of these terms into Eq. (A11) results in

The LFOVs are simply equal to *f*_{cap} times the angular FOVs in the object space of the telescope.

**Keywords:**

Galilean telescopes; field of view; pupils; ports; vignetting; eye relief; low vision; telemicroscopes