Experiment to find the circle of least confusion (optics)

I want to conduct an experiment where the circle of least confusion (wiki ref) occurs for a lens. Suppose that I shine white light onto the lens and I know the lens EFL is lets say 100 mm, in order to balance the chromatic aberrations between the short wavelength (blue) and long wavelength (red) I will have to move the lens by certain amount (+/-). That is to defocus it.

Is there a method by which I can compute how much to move exactly to reach the circle of least confusion?

The idea I have is, use a reference spectrum (Ne lamp) whose spectral lines are known in the visible region, based on the line width of the red, green and blue lines; calculate how much to defocus the lens to make the red/blur sharper. But, I have no idea how this can be calculated. Any ideas ?

(This can also be related to control of DOF)

• Just to clarify - you are using "circle of confusion" to mean that point near the focal plane of the lens where a point source of white light results in the smallest image, taking into account the chromatic aberration of the lens. Are you assuming a source at infinity? This should be the "focal length" of the lens - if it is given for white light. If it's given for a specific wavelength, you have to determine the chromatic aberration somehow. Am I understanding your question correctly? Dec 15, 2015 at 17:46
• Yes you are almost right, assuming the source is at infinite; but not a white light, lets say red, green and blue wavelength. The critical point here is, the angle at which the light enters the lens is not with respect to optical axis (with an angle instead), hence due to different path lengths there occurs a chromatic aberration. Dec 18, 2015 at 19:39

For anything but the simplest system, you are likely to get the best result by carefully determining the focal length at a number of different wavelengths. Simulation is quite hard - see this question and associated answers for some ideas for (free) software you could use. But it's hard to do it right.

Instead, if you want to go the experimental route, you could do this in a couple of different ways.

First method: use a white point source at a known distance (it could be "very far away", but that doesn't actually matter). Of course a real source is likely to be extended, but for this experiment to work well you really want to focus it through a small pinhole that is on the optical axis of the lens. Then put a second pinhole at the other side of the lens, and scan the intensity of the light at a few different wavelengths as the pinhole steps through the nominal focal distance. This is most simply done by sending the light through a grating and capturing the resulting spectrum with a camera; but you could also use monochromatic filters at a few different wavelengths, and you can filter the beam at any point. Note, however, that you don't want to place filters between the pinholes and the lenses, as they may affect the optical path length. You have to put the filter either before the first pinhole, or after the second.

If I am not mistaken, you would expect to see the intensity of the light coming through the pinhole peak quite sharply as you go through the focal point; but since the focal point will be different for each wavelength, the peaks will not lie exactly on top of each other. Instead, you will get a plot like this (numbers and shape of curves are made up):

The peaks should be fairly sharp, as long as everything is well aligned and the pinholes are small: when the focal blur is equal to the pinhole size, the intensity will drop by a factor 4x. Therefore it should be easy to determine the focal length for each wavelength. You should be able to make a plot of focal length as a function of wavelength. For a well compensated lens, that plot will be (nearly) flat; the example graph I gave above corresponds to a lens with a lot of uncorrected dispersion.

Once you have that relationship, you determine the point of least confusion based on your needs.

For example, if you have all wavelengths evenly represented in your source, you will want to put the POLC at the unweighted mean of all focal lengths; but if you have a predominantly red image, for example, you will want to use a heavier weight for the red focal length than for the blue.

I am reminded of a camera I had many years ago that had manual focus (using a TTL microprism). The lens had an "IR" marking on it - so that if you wanted to do IR photography, you would first focus on the visible light, then make a small adjustment to move the focus setting to the IR dot. Of course IR is somewhat more extreme - but that's an example of what needs to be done to get focus right "for the wavelength of interest" (in this image, the camera is focused at 10 m in the IR, 6 m in the visible).

Note - chromatic aberration occurs both on the optical axis, and off it. You notice it more strongly off axis, because it results in characteristic colored fringes at points of sharp contrast. What you are really seeing with such fringes is the fact that with different focal lengths, images are of slightly different size; and the further to the edge you look, the greater the difference in size will be.

Incidentally, the diagram you use in the question seems to relate to spherical aberration; what I described above relates to chromatic aberration. Both of these things matter if you really want the best focus; and both are diminished by using a smaller aperture lens (bigger f-number).

• chromatic aberration is more prominent off axis, which can also add a factor of spherical aberration. Will conduct a few more experiments and update. Dec 23, 2015 at 13:57

Do you know the material of the lens? If you do then you can find out what how the index of refraction changes with wavelength (dispersion). Using this information you can calculate the focal length for each wavelength using the so-called lens make formula: 1/f = (n-1) (1/R1 - 1/R2)

Where R1 and R2 are the radii of each surface of your lens, and n is the refractive index. f is the focal length corresponding to the index n. Note this formula is only applicable for "thin" lenses.

So, once you have computed the focal length for each wavelength you'll be able to compute the COLC.

• Not exactly the right approach, its not a thin single element lens, for example if its a doublet; we have to consider the Vd (dispersion coefficients) of both flint and crown and calculate the change in focal length and from there the COLC. I have already considered your suggestion. Dec 18, 2015 at 19:42
• The more information you can provide the better. Your questions asks if there's any way you can compute where the COLC is. This cannot be done unless the optical system is defined, materials, surface figures, thicknesses, etc.
– JQK
Dec 18, 2015 at 23:17