Thanks for providing the link to John Hallmén's article. It has been a long time since I read that article. The reminder is good. I am also amused to see that my own name now appears late in the comments, suggested as someone who might be able to answer questions similar to yours.
Unfortunately I cannot answer your questions exactly, because exact answers would depend on details of the setup that I don't know and that probably could be determined only by experiment. But perhaps I can give you some ideas of how to think about the calculations.
For starters, let me describe a little more about how these setups work.
At first blush, you can divide the optical system into two parts: a front lens section that forms a real aerial image, and a rear lens section that focuses on that first aerial image and in turn forms a second real image back on the sensor.
The magnification of the rear section is determined by lens characteristics and extension distances. That will be m = totalExtension/FL - 1, where FL is the focal length and totalExtension is the distance from the sensor to the appropriate principal plane of the lens. Unfortunately we don't know where the principal plane of the lens is located, so we cannot compute this exactly. It would be far simpler to just remove the front lens, place a ruler in focus by the rear lens, take a picture, then do the usual division: m = mmOnSensor/mmOnRuler.
Once the magnification of the rear section is known, it's a pretty good approximation that the effective aperture of the rear section by itself is just equal to (m+1)*nominalAperture. For example if you're running at 5X and nominal f/8, then the effective aperture is (5+1)*8 = f/48. An exact calculation would depend on pupil ratio (as described
HERE), but the simple calculation is pretty close when a reversed lens is used at magnifications that are far above 1X.
Now, the great simplification in thinking about these setups is that if the front lens has an f-number that is smaller than the rear lens, and you're not getting any vignetting, then the "limiting aperture" is provided entirely by the rear lens. That, in turn, implies that the effective aperture of the whole system (as seen by the sensor) is simply the effective aperture of the rear section by itself. In this situation, you get to ignore the front lens!
In John's setup, the conditions are satisfied because the front lens is f/1.6 and the rear lens is f2.8 at widest, and typically stopped down quite a bit. So if John's rear lens section is running at 5X and nominal f/8, then the whole setup is still effective f/48 (=(5+1)*8
) even when the front lens is installed for taking pictures.
Overall magnification of the whole setup is just the product of front lens magnification and rear lens magnification.
Note, however, that front lens magnification is very dependent on spacing between the front and rear lenses. With a front lens having FL 2.1 mm, the difference between 0.1X and 0.05X is only 0.105 mm of separation! With a 5X rear section, that 0.105 mm of separation makes the difference between 0.5X and 0.25X overall magnification.
So again, trying to calculate the front section magnification with any accuracy is quite difficult; it's far simpler to just focus on a ruler and get the magnification from that.
Once you have the magnification and the effective f-number, you can calculate DOF using something like the classic circle-of-confusion method:
totalDepthOfField = 2*C*Feff/(m*m), where Feff is the effective aperture, m is the overall magnification for both lens sections combined, and C is the acceptable circle of confusion based on sensor size.
It may be puzzling that the depth of field of the relay system, when calculated this way, comes out to be just the same as any other lens system that provides the same overall magnification and effective aperture. The reason for this equivalence is that the DOF of the wide-angle relay system in fact
IS just the same as any other lens system, when we're talking only about parts of the image that are close enough in focus to be considered sharp. The obvious improvement in DOF comes entirely from areas that are significantly blurred. Compared to a conventional macro lens, what the wide-angle setups do is to render the background more sharply (because the entrance pupil is smaller) and to pack a lot more background into the same image area.
You asked also about effective focal length (EFL) and FOV. I'm not sure exactly what you mean by either of those terms.
For an optical designer, EFL is equivalent to "How much additional extension would I need to add, to add 1X of overall magnification?". But I suspect that's not a number that would be very helpful to you.
So, what do
you mean by EFL?
As for FOV, taken literally that means "field of view" and it's a linear measurement taken across the in-focus plane. But of course that depends on overall magnification, which depends strongly on the separation between front and rear lenses.
I'm wondering instead if you mean "angle of view" (AOV), the angle between lines joining foreground and background points at opposite edges of the frame.
We can estimate that by calculating the width of the aerial image as imageWidth = sensorWidth/magnificationOfRearLensSection, then doing the usual trig calculation 2*arctan((imageWidth/2)/FL), where FL is the front lens FL. That won't account for shifts due to focusing, or for fisheye-like lens distortions typical of very short lenses, but it should get you in the ballbark.
All this is only from thought experiments, no physical experiments to confirm anything, so higher than normal chances of being wrong.
One last comment... If you try building simple models of these relay lens systems, say using just two "thin lenses" with appropriate spacing, I expect you'll find that the models fail rather badly. For example they predict severe vignetting due to the two lens apertures "fighting" with each other. In the simple model, off-axis rays that get through the front lens tend to miss the aperture of the second lens, and that effect gets worse as you stop down the second lens. A simple model of John's setup, with a 1.5 mm diameter aperture on the rear lens at 24mm f/16, predicts hopeless vignetting. What makes the real setups work well, I think, is that these short lenses tend to be telecentric toward the aerial image in the middle of the setup. That is, the front lens is designed so that its image is formed by cones of light whose centers are nearly parallel to the optical axis, and likewise the rear lens is designed so that, when reversed, it accepts cones of light in that same orientation. Given this situation, stopping down the rear lens can make all the cones of light narrower at the same time, without blocking the corners of the frame to cause vignetting. At least, that's my speculation. Again, no experimental confirmation.
I hope this helps.
--Rik
