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.

FIGURE A1. Emergent ... Image Tools |
Equation 12 Image Tools |

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:

Equation (Uncited) Image Tools |
Equation (Uncited) Image Tools |
Equation (Uncited) Image Tools |

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:

FIGURE A2. Emergent ... Image Tools |
Equation (Uncited) Image Tools |
Equation (Uncited) Image Tools |

Equation (Uncited) Image Tools |

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.

Figure 3 Image Tools |
Figure 4 Image Tools |

##### 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.

Table 3 Image Tools |
Equation (Uncited) Image Tools |

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.

Equation 2 Image Tools |
Figure 5 Image Tools |

#### 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:

Equation (Uncited) Image Tools |
Equation (Uncited) Image Tools |
Equation (Uncited) Image Tools |

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