Calculating, NOT measuring, aperture of compound lenses

**Emmanuel BIGLER** · 6-Apr-2015, 02:35

Struan, we're all working in a vacuum.

Yes, Dan, and this situation greatly simplifies our calculations wih pencil and paper, since the refractive index between glasses is exactly 1.0000000 and not 1.000293 in air.

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And going back to purely academic exercises, as usual, it is shortly after posting my long message that I remembered from the series of lectures I attended a long time ago, that determining the pupils of a system made of 2 thin groups is best achieved by considering the intermediate optical space between the two thin groups. Simply because the apertures limiting the sizes of the 2 groups are self-imaged in this intermediate space.
Hence the quickest procedure to find the pupils for the Galilean telescope, (e.g. (4 / 3 / -1) for a 4x instrument) the classical refractor telescope (2 positive groups e.g. a 8x pair of binoculars (8 / 9 / 1) ), the Peztval lens represented by 2 equivalent singlets in terms of focal lengths and pupils (and of course NOT in terms of aberrations!), and the academic telephoto analysed above, is to consider the angular diameter of both groups as seen from a point in the intermediate optical space view which is simply the focal point F'1 of group#1.

In the case of the academic telephoto (1.5 / 1 / -1), it is easy to see without any calculation that the smallest aperture seen from F'1 is the entrance aperture as long as the 2nd aperture is bigger in diameter than one-third of the 1st aperture.
And this approach proves for the same (1.5 / 1 / -1) design that any section of a cylindrical barrel connecting two singlets of the same diameter will not act as the pupil.
At least in infinity-focus setting.
See the attached sketch.

**Struan Gray** · 7-Apr-2015, 00:23

Emmanuel, I will admit to being confused a while by an intermediate space which wasn't actually between the elements. But I think I've got it now.

For my purposes, I just treated the elements as loupes, each being used to look at the disc of the other. The positive element gives you a magnified image, the negative element a diminished one. Job done, albeit not on such a general level.

**Emmanuel BIGLER** · 7-Apr-2015, 08:01

I will admit to being confused a while by an intermediate space which wasn't actually between the elements.

Yes Struan, this is one of the most difficult things in geometrical optics, to visualize all those optical spaces that are actually stacked one upon each other like in a those kinds of multilayer structures used in digital image post-processing.

For example in the Petzval lens, the distance between doublets (in some Petzval designs the rear group is made of an un-cemented doublet, both elements are very close to each other with a tiny air gap, or with 2 curvatures slightly different) is shorter than the focal length of the first (positive) doublet.
In means that in the intermediate space, the image of an object on-axis at infinity is located at F'1, which is outside the physical space between the two doublets, behind the 2nd doublet.
The diagram required to locate the pupils in the Petzval design is the same as for the academic telephoto, and same rule applies: in a Petzval lens with no iris inside, and 2 doublets of same diameter as a cylindrical barrel, the first doublet's mount is the entrance pupil, and the 2nd doublet is the output window through which you can look a the pupil from the rear of the lens, and see if the cat's eye appears or not when looking from the edges of the image.

I just ran Oslo-edu on a Petzval design available in the demo lens files.
In the Petzval demo file, the aperture stop (AST in oslo-jargon) is located in between both doublets and has the same diameter as both doublets; the paraxial calculation, as expected, locates the entrance pupil at the front doublet when the iris is wide open to a diameter equal to the barrel saize.
If you stop down the Petzval lens with an iris located in between both groups, at full aperture the entrance pupil will be located at the 1st doublet, but as soon as the iris is stopped down beyond a certain reduced diameter, the pupil will "jump" to the iris, the iris acting then as the pupil in the intermediate space, the image of the iris viewed through the 1st doublet wil be the entrance pupil.