Field of View of Telescopes

By Rafael Chamón Cobos

Created: December 2016

From a Post by Glenn LeDrew in CN Binoculars Forum:

In a Keplerian telescope the FOV is determined by two factors:

- Objective focal length.

- Eyepiece field stop diameter.

These two dimensions form a triangle, the angle of the narrower apex at the objective defining the angular FOV. This triangle's narrow base is the field stop aperture, and the height is the distance between the field stop and the center of the objective. The two lines emanating from the objective center and intercepting the field stop at opposite sides are chief rays which delimit the envelope of sight lines to objects which can simultaneously be encompassed by the field stop. See the following figure:

Other things being equal, if you double the objective focal length you halve the FOV. Or if you double the field stop diameter you double the FOV.

From the figure we can obtain the formula giving the FOV angle:

FOV = 2*arctan((FS/2)/FL)

I'm surprised by the number of folks who think FOV increases with objective diameter, as though inducing a 'picture window' effect. This cannot happen in the way imagined for the kind of optics we're considering here. They being of the Keplerian configuration, employing a positive eyepiece.

Now, for a Galilean system (the best known examples being simple opera glasses), the negative eyepiece causes the exit pupil to lie *inside* the instrument. Being thereby inaccessible, the FOV scales directly as the objective aperture; the field edge is the out-of-focus edge of the objective. Other things being equal, if you double the objective diameter you double the FOV.

From a Post by Rich V. in CN Binoculars Forum:

The previous calculation is also valid for binoculars.

Generally, keeping the same focal ratio, as the objective diameter increases, so does the focal length of the lens. Here's the problem; as the focal length increases, the field stop of the eyepiece must also increase proportionally to provide the same TFOV. To keep the bino's eyepiece field stop and prism sizes reasonable a compromise must be made. This explains why a 7x35 binocular has normally a greater FOV than a 7x50. To explain this, let's compare a 7x35 and a 7x50 both having objectives of same focal ratio F/3.5 and same field stop size=22mm.

7x35 binocular:

Magnification=7

Objective diameter=35mm,

Objective focal length=35*3.5=122.5mm

Eyepiece focal length= 122.5/7=17.5mm

Field stop diameter= 22mm

Field of view=2*arctan((22/2)/122.5)=10,26º

Apparent field of view=10.26*7=71.8º (this is a fairly wide angle)

7x50 binocular:

Magnification=7

Objective diameter=50mm,

Objective focal length=50*3.5=175mm

Eyepiece focal length= 175/7=25mm

Field stop diameter= 22mm

Field of view=2*arctan((22/2)/175)=7.19º

Apparent field of view=7.19*7=50.3º (this is not a wide angle)

Let us calculate the field stop diameter of the same 7x50 to have a 10.26° FOV like the 7x35:

Field Stop=2*FL*tan(FOV/2)=2*175*tan(10.26/2)=31.4mm

This wide angle 7x50 binocular would need a whopping 31.4mm field stop. This would require a larger build with much larger prisms to accommodate the much larger eyepieces. Therefore, to keep the bino's eyepiece field stop and prism sizes reasonable a compromise must be made. This means that for practical pourposes the field stop size cannot be so big and therefore, 7x50 binoculars are generally not wide angle.

Another way to achieve a 7x50 with 10.26° FOV would be by making the focal ratio much shorter and using the same sized eyepiece as the 7x35. Edge correction and eye relief could suffer, though.