Application: Which resolution formula?

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Resolution formulae: Do they work?

A popular formula for the net resolution of combined optical systems is

Rnet = 1/(1/R1 + 1/R2), where Rnet is the resolution of the combination and R1 and R2 are the individual resolutions of the two sub-systems. For example, if a particular film has a resolution of 100 lp/mm and a lens has a resolution of 45 lp/mm at the selected aperture, the predicted net resolution when using them together is 1/(1/100 + 1/45) = 31 lp/mm. This is based on a simple idea, namely that the diameters of the "blur circles" of each subsystem simply add together. But is this correct? I decided to find out, and this is a problem ideally suited to the optical simulator.

For this study, I chose MTF50 resolution to be the criterion, using single-point objects. Then MTF50 occurs when the separation between the single-point images is equal to the 1/4-power diameter of the point spread function. That is, resolution in lp/mm is 1/(1/4-power dia. in mm).

The specific approach becomes: Run the simulator to combine a diffraction response with a lens point-spread response, then compare the 1/4-power diameter of the result with the 1/4-power diameters of the inputs. For example, the f/8 diffraction curve has a 1/4-power diameter of 6.1 microns, so in the presence of diffraction alone, MTF50 occurs when these are placed 6.1 microns apart; in other words, a resolution of 164 lp/mm. If we combine this diffraction with various lens point-spread functions, how does the 1/4-power diameter of the result relate to those of the inputs? Here are a few sample numbers from my study:

Diffraction 6.1 microns + lens PSF 1.4 micron -> 1/4-pwr dia. 6.2 microns (net res 161lp/mm).

Diffraction 6.1 microns + lens PSF 4.2 micron -> 1/4-pwr dia. 7.2 microns (net res 138lp/mm)

Diffraction 6.1 microns + lens PSF 7.1 micron -> 1/4-pwr dia. 9.2 microns (net res 108lp/mm)

Diffraction 6.1 microns + lens PSF 11.3 micron -> 1/4-pwr dia. 13 microns (net res 77lp/mm)

We see immediately that the diameters do not directly add. For most of the cases I tried, the formula quoted above returns values that are up to 30% off - not a very accurate formula at all! However, an alternative rule found in some references does provide good accuracy (within 3% in the cases I tried):

Rnet = 1/sqrt(1/R1^2 + 1/R2^2), which is the formula which I would recommend adopting.

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- Marianne

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