The total diameter of the full-illumination LFOV is approximately three-quarters of a millimeter; however, the half-illumination FOV/2 is ±1.85 mm and the absolute FOV/2 is ±3.33 mm (Table 3).
Figure 5 illustrates the effect of eye pupil diameter D a on the LFOV at 50% vignetting of the 4× Galilean telescope (f o = 100 mm, D o = 20 mm, and D′o = 5 mm) with a 100-mm focal length (+10 D) lens cap and an eye relief of 15 mm. The LFOV is constant (±1.85 mm) for the system as long as the objective lens is the field stop, i.e., D a < D′o. When the eye pupil dilates so that D a > D′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 = ±2.96 mm.
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′o results in a chief ray slope from the fixed center of D′o to the edge of the eye pupil. As the pupil dilates, the chief ray angle (tan θ′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 θ′50 is increased when the eye pupil is the exit port. The enlargement in the 50%-vignetted FOV when D a > D′o is equal to D a M/D o. For example, when D a = 8 mm, the enlargement is 32/20 = 1.6x. 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 Do = 20 mm, Da = 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 Da = 8 mm, the objective lens is the aperture stop, i.e., Da > D′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.
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.
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Derivation of FOV Equations for Galilean Telescopes
In Figure A1, D a < D′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′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′ is the distance of the virtual image of the objective lens, e is the eye relief and (−u′ + 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 −(f o + f e). Image distance u′ 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′o) is given by
The half FOV at 50% vignetting when the objective lens is the field stop (D a > D′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 θ′) 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 θos = tan θ′/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 θos = tan θ′/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 (θcap) of rays incident on the lens cap, we may use the Refraction and a Transfer paraxial ray trace equations11 as follows:
The substitution of Eq. (A9) into Eq. (10) results in
where tan θcap = ray slope incident on lens cap, tan θ′/M = slope of ray after refraction by lens cap, Ycap = Do/2, and dcap = fcap.
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:© 2007 American Academy of Optometry
Galilean telescopes; field of view; pupils; ports; vignetting; eye relief; low vision; telemicroscopes