I missed this post when it first appeared. Work has been a bear for quite a while now, and I do get the chance to come visit here as often as I want. That and the title was a little broad.
There is no reason why a fixed power target scope would inherently have a greater dept of field compared to a variable scope. But you astutely added another qualifier in your comparison, and that qualifier is "shorter." Therein lies the rub.
As we all know, a riflescope's magnification is the result of the focal length of the objective lens, divided by focal length of the eyepiece, and then multiplied by the erector lens assembly's magnification. Now all riflescopes have these three parts: objective lens assembly, eyepiece assembly and erector lens assembly. The difference between a fixed power and a variable will be found in the erector lens assembly. That's the piece you twist when you turn the zoom ring.
The objective lens assembly is that part starting from the big lenses in front to where it focuses the image on the first focal plane, at the front of the erector tube. This location will usually be found near or at the adjustment knobs. This picture, at the first focal plane, is where the depth of field of the image is determined and shows up.
In order to determine the depth of field, you have to know several things.
1- Distance to object on which we focus.
2- Focal length of the objective lens assembly.
3- Aperture size of the objective lens assembly.
4- Size of the Circle of Confusion (CoC).
Armed with those data, you can go to an on-line DOF calculator and figure out the DOF and hyperfocal distance of the riflescope.
1- Let's say Distance is 100 yards (300 feet.)
Sope manufacturers do not publish the focal length of their various offerings those are company secrets. But we can "approximate" these values, especially if we're just trying to compare riflescopes. If you measure the distance between the middle of the knobs to about a half inch from the end of the objective lens for the scopes you are trying to compare, you should get a focal length that can be used for this.
2- Measure in a line parallel to the main tube, not angled down. So let's say on your big long fixed target scope you get something like 200mm. Then on your short variable, you get 150mm. You now have the data points for #2.
3- Next up, aperture size, or f-number, or f-stop. Again, that's not something that manufacturers reveal, but we can figure it out using the focal length we just got in 2- and the published size of the objective lens. Let's use 50mm for the objective lens for both riflescopes. So for the fixed power, the f-number will be 200/50 or f/4. For the variable the f-number will be 150/50 or f/3.
The image with its depth of field will be formed on the FFP of each scope, using those first 3 values. Here are some rules.
- The further the focus point is, the greater the depth of field.
- The higher the magnification (focal length) the shallower the depth of field.
- The smaller the f-number, the greater the depth of field.
So if we keep the focus point the same, we can see that the fixed power scope has a longer focal length (200mm) but a smaller f-number (f/4). So the question becomes, is the DOF of the F-P (200mm, f/4) shallower or deeper than the DOF of the V-P (150, f/3)?
That's the optical equation, but there is one more item that determines the actual DOF, and that's the mysterious Circle of Confusion (CoC) that I mentioned at the top. Essentially the CoC is the value at which you deem the object to be in focus, or not in focus. It's somewhat subjective, but it is something quite important to get right. I'll have more to say about that later, but for now, let's just pick a size that is very common, .020mm.
Alright, off to the DOF calculator with those values: 300 feet, CoC .020, F-P (200, f/4), V-P (150,f/3).
I use this calculator:
Depth of field and hyperfocal distance calculator
www.dofmaster.com
Plugging in the value above we get:
F-P: Near limit 254ft, far limit 367ft for a total DOF of 113 ft.
V-P: Near limit 244ft, far limit 389ft for a total DOF of 145 ft. (I used f/2.8 as the calculator did not have f/3. Using f/3.2 made it even larger.
So, that seems to indicate that in this case, the F-P does not have a deeper DOF than the V-P. But wait, there's more.
We have to account for the magnification if we want to compare like for like. Since the focal length of the objective of the V-P is shorter than the one for the F-P, we probably have to increase the magnification on the V-P to get to the same magnification as the F-P. This is where the CoC comes into play. We calculated the DOF values using the same CoC for both scope, essentially the same quality. But when we increase the magnification on the V-P to match the one in the F-P, we have to take into account that we should be using the same equivalent CoC, and that means that we need to reduce the CoC for the V-P at let's say 1.3X to match the .020 CoC of the F-P at the same magnification. So the CoC value for the V-P calculation should be .020/1.3 or 0.15.
Quick, back to the DOF calculator with that CoC for the V-P. Near limit 255, far limit 362, total DOF 106ft.
Now this is all based on guesses about the actual focal length of the objective lens assembly. But it would seem to me that, at equivalent magnification, a short variable with a large objective lens will have a shallower DOF compared to a longer F-P scope with the same size objective lens.
It's important to keep as many things as possible the same, magnification, objective lens size, to be able to reach some measure of conclusion.
Anyway, this was all stream of consciousness, it's entirely possible I got one or more items wrong.
