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I suppose that z is the distance at which objects appear sharp to the naked eye at the same time that objects at infinity.
M is the binocular magnification
d is the distance at witch the binocular is perfectly focused.
Without accommodation of the eye, we can see objects perfectly sharp at the distance d'.
By definition, the depth of field is (d – d').
Now, if I suppose that the exit pupil is always larger than the eye pupil, I find the equation :

1/z = M².(1/d'-1/d)

.....So, the only optical parameter which determines depth of field is the magnification. Its influence is huge, because of the 2-power of M in the equation.
Now, why people find that binoculars with equal magnification have quite different depth of field ?
I think that the perceived depth of field in binoculars is determined by other parameters than optical ones.
JCB

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focal length has almost none impact on the DOF. It is the magnification which dominates.

In summary it seems to be that only magnification and effective exit pupil are dominating factors for DOF. Focal length has some influence but not much. However, I am not sure how well the assumptions made for these calculations are satisfied. For example, a binocular is not made of thin lenses. Only professionals may be able to figure out the validity of these assumptions, maybe with the help of ray-tracing software.

With regards,
Holger

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The results can be found with the formula I wrote in a previous post :

1/z = M².(1/d'-1/d)

Here d is infinity, z=b , M=V and d'=G.
We have therefore : b=G/V² which is nearly the same as the formula on your post, in which the negligible terms have been omitted.
(For V=10 and G=100000 mm, we find b=1000 mm)
My formula is not rigorously exact, but is more general because it is also valid when the binoculars are not focused to infinity, but to the distance d.

I think it's worth doing some applications of this formula :
We suppose that the binoculars are focused to infinity, and that with naked eye we can see sharply objects without accommodation if they are 1 m away. Then DOF are :
For a 7x binocular : 49 m to infinity
For a 8x binocular : 64 m to infinity
For a 10x binocular : 100 m to infinity
For a 12x binocular : 144 m to infinity
People more than 60 years old, lacking in eye accommodation, and who have to rapidly focus between two distances (like birders), have to very carefully examine the drawbacks of high power binoculars, considering their poor depth of field.

Jean-Charles

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Often associated with Depth of Field discussion is what is known as the 3D effect. So, here is some discussion about 3D, what is seen, what cannot be seen and the mathematics related to how things seem to have depth of field or appear 3D.
3D or not,3D!

Binoculars with a greater objective lens separation WILL provide more depth perception than a binocular with a lesser separation, regardless of the prism types . . . PERIOD!

the separation between objective lenses is the most important factor: a Porro 8x30 will show more 3D view than a roof 8x30.

Magnification has effect on 3D view, a greater magnification increases the compression of the fields reducing the 3D view: a roof 10x30 will show less 3D view than a roof 8x30.

Also distance has a significant impact on the so-called 3D effect;

For a binocular with objectives 150mm apart, observing two objects placed at

10 meters and 12 meters distant, the angles of view (with reference to centerline) would be 26 arcminutes and 21 arcminutes, a difference of 5 arcminutes, easily perceivably by the human eye.

100 meters and 120 meters, the angles would be 2.6 arcmin and 2.1 arcmin, only one half arcmin difference or 30 arcseconds. Already this narrow angle is beyond the perception of the human eye.

But if the objects were at
1000 meters and 1200 meters, the angles are now 0.26 arcmin and 0.21 arcmin. The difference in these two angles is merely 3 arcseconds, not only beyond the ability of the human eye, but beyond the capabilities of diffraction limited optics smaller than 46mm.

Curvature and Depth of Field
Here's a discussion why some binoculars appear to have a greater depth of field than others, even when the lens equations won't account for any difference in depth of field. As noted above, there is more to perception of depth of field than just mathematics. In this case a strong correlation is shown between the aberration Field Curvature and Depth of Field. A lens that has strong Field Curvature has a shorter focal length in the center and a longer focal length towards the edges. This has the affect of making closer objects apppear in focus if the center is focused on a distant object and the closer objects are viewed further out in the field of view. Hence the binocular has a greater depth of field, but only for nearer objects.







Edited by EdZ (02/02/08 09:59 AM)

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