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This question was asked in another forum: "How can I determine the focal length of my binoculars"? This individual wanted to change the magnification of his binocular by replacing the eyepieces, but didn't know what focal length eyepieces to use as replacements. I once owned a pair of 15x70s that had been converted to 25x70s by doing the exact same thing.
I wrote this a year or two ago, embedded in an article published in the CN Reviews (4 Way Binocular Comparison, published Jan 2003), but I've corrected it here to account for refractive index and I've cleaned it up a bit and now repost it here.
FOCAL LENGTH IN BINOCULARS
Most Binoculars are Fast Achromats
Although this is not a particularly useful measurement in a fixed system, knowing how to calculate the focal length of binoculars may help you to understand a little more about how fast the optics are operating. Most common porro prism binoculars are fast Achromats. The short f-numbers of these systems certainly help explain why chromatic aberration can be present. Some binoculars are triplet apochromats, the Oberwerk BT100 for example. That would be relevant to the discussion if we were performing this calculation because we were concerned about chromatic aberration. This topic is focal length.
In binoculars, the same as in telescopes, magnification is focal length of objective (Fo) divided by focal length of eyepiece (Fe), or mag = Fo / Fe. Also, Fo + Fe is the total length of the light path. So, we know two formulas with the same terms and we know magnification. We need to find the length of the light path, Fo + Fe. Then we can solve the problem.
Measure the outside lens-to-lens distance of the binoculars. Focal length is generally accepted as taken from the centerline of a lens. So, the measurement should truly be from the center of the objective lens to the center of the eye lens. When I record this measurement, I correct for the thickness of the objective lens. I ignore the thickness of the eye lens. I have checked the thickness of the objective in several binoculars. For a Swift 8x42 it is 11m, the Nikon 10x50 and an Orion 10x50 is 13mm and the Oberwerk 20x80 Deluxe is closer to 16-18mm. Subtract one half the thickness for the glass out-to-out measurement.
Also, closely estimate the length of the light path through the prisms, then divide the length of the path thru the prisms by the glass index, and add this to the measured lens-to-lens distance. This will give you the total Fo + Fe.
If we use a common porro prism binocular, without any reducing lens in the light path, we can use these formulas. This probably refers to about 90% of the common binoculars on the market. It helps to see a diagram. Several of the books referenced in the credits at the end of this article (in the main article originally published in CN Reviews, see note above) show a cut-a-way diagram of a porro prism binocular. I have not included a diagram here. The formula can also be used for roof prism binoculars, but the light path through the prisms is quite a bit more complex to measure. Whether or not the binocular is a doublet achromat, or a triplet apochromat, has no affect on the outcome of this focal length calculation.
When light travels though glass it’s angle changes. The convergence of the light cone from the objective, when it passes thru the prisms, is slowed by a factor determined by the refractive index of the glass. While this occurs in all glass, this does not need to be addressed through thin lenses, but it has a significant impact in binocular prisms. Assuming Bak4 glass, the glass has a refractive index of 1.57. If the physical path through the prism is 100mm, the effective focal length through the prism is 100/1.57 = 64mm.
I have had several binoculars opened up and had the opportunity to measure the light path through the prisms. The prism light path in my smaller binoculars, those with mag. of 7x, 8x, 10x and 12x, measured 90mm to 100mm. The light path in my giant binoculars, 16x70, 16x80 and 20x80, with very large prisms measured 110mm to 130mm. I wouldn’t venture a guess at the length of the light path in the prisms of the BT100, but it’s longer. So the “effective length of the light path” thru the small 10x50 prisms is about 65mm, and ranges from 65mm to 80mm, depending on the size of the binocular and the size of the prisms. A 10mm error in this actual size will produce a difference in the final outcome of only 0.1 in the f# for a 10x binocular and less for a higher power, not a significant error in the final outcome.
These two measurements added together, lens-to-lens + prism effective path, is the total effective light path, the total of effective Fo + Fe. For a Nikon (Earth and Sky) 10x50 the Fo + Fe = 218mm.
Fun With Math
Now we can use our two formulas. We will use the Nikon for our example. We know Fo + Fe = 218 and we have Fo / Fe = 10. Since magnification is given as 10x, it’s easy to understand this second formula will result in a Fo = 10x longer than Fe. If we solve this 2nd formula Fo / Fe = 10, then Fo = 10 Fe. Now substitute this answer for Fo into the first formula, so Fo + Fe = 218 can be written as 10 Fe + Fe = 218. Therefore 11 Fe = 218. So Fe = 218 / 11 = 19.8. Now you know the focal length of the eyepiece = 19.8mm. Now then 218 – Fe = Fo, or Fo = 198. A check of the results is Fo / Fe = 10. Our check, 198 / 19.8 = 10 shows the answers are correct.
All of this just to get the f number of the binoculars? Yes, since now that we know the Fo, we can divide Focal length of objective by the Diameter of the objective and we get f, the speed of the binocular system. In this example it is Fo / Do = f, or 198 / 50 = 3.96. These Nikon 10x50 binoculars are operating at f4.0. Seven out of twelve binoculars I tested are operating within the narrow range of f3.9 to f4.3.
Other checks (exit pupil) will prove out these answers. Exit Pupil = Fe / f, so 19.8 / 3.96 = 5.0. Compared to Exit Pupil = Do / mag, = 50 / 10 = 5, it checks.
Two Things You Need to Know
Having gone through this whole exercise, you need to know only two things to help you determine the f of binoculars. First, the total length of the effective light path, which you must accurately measure, and second, that the shortest form of the formula above, for any binocular, will always be FL eyepiece = Length of Light Path / (magnification + 1). The rest is simple subtraction and division. To get the FL of the binocular it’s Effective Length of the Light Path minus FL eyepiece. To get f of the binocular it’s FL of the binocular divided by aperture.