Magnifiers: a closer look (IX)
Professional focal length measurement for amateurs!

Naturalists tend to belittle multiple lens magnifiers as gadgets, or even worse: as toys. So you might simply stop reading here.

But we have to admit that we do like the Octoscop^TM multiple magnifier as presented here in January 2022. Its symmetrical design is simply beautiful. Moreover, it can serve as an interesting object in order to check optical theories and practical focal length measurement methods.

So let's repeat the basic properties of the Octoscop magnifier (fig. 1 and fig. 2): it is providing eight (!) different magnifications in a very compact housing. Those magnifications are achieved by crosswise, parallel or antiparallel orientation of its optical components.

The Octosctop is boasting with the following lens combinations: 2x, 4x, 6x (overlay of 2x plus 4x), 10x, 14x (overlay of 4x plus 10x), 18x, 20x (overlay of 2x plus 18x) and 28x (overlay of 10x plus 18x).

Fig. 1: The Octoscop in its unfolded (crosswise) orientation. In this orientation the following four magnifications are provided without any lens overlay: 2x, 4x, 10x and 18x.

The most extreme (28x) magnification is reached by means of an overlay of the 10x and 18x optical element - easy:

Fig. 2: Maximum magnification by an overlay of the 10x and 18x optical elements (smaller lens combination, on the left side of the image), resulting in an assumed 28x magnification, signalized in yellow by the respective indicator window. But see our warning and annotation below.

And yes, 10 plus 18 is definitely summing up to 28? Well, not quite in case. There exists an old "Seibert" branded magnifier claiming to provide an 28x magnification by a combination of a 10x and a 20x optical element! Hmm - is it possible that the reknowned Seibert company managers Wilhelm and Arthur Seibert were not able to calculate the sum of 10 plus 20? No, in fact they were perfectly right: when looking into optical theory textbooks it becomes clear that the sum magnification of an overlay system is by no means equivalent to the sum of the magnification values of its individual components. There is a mathematical formula for this:

Fig. 3: The calculation of the focal length of combined lens systems [Fowles 1975]. f is the resulting focal length of the overlay, f₁ and f₂ are representing the focal lengths of the two subsystems. Furthermore the formula is taking into account the distance between the two optical elements (the third term has a negative sign, thus indicating that the resulting inverted focal length value - i.e., the magnification - of the combined system will decrease with increasing lens distance).

Here is a practical example how to make use of the formula: just insert a focal length value f₁ of 25mm (10x lens) and f₂ of 13.9mm (18x lens). For a lens distance of 3mm the formula is delivering a combined focal length value f of 9.68mm which is equivalent to a magnification value V = 250mm / f = 26. This is corroborating the fact that the combined magnification power is markedly lower than the simple mathematical sum of its component magnification values.
For a 5mm lens distance the actual magnification outcome is even lower, ca. 24x!

It is good practice to check those theoretical forumulas by means of experimental systems. For this endeavour we need a focal length measurement method. And there is an ingenious way to perform this task. Let's have a look at the theory first. The line of thought is said to be based on ideas by the famous mathematician Carl Friedrich Gauß [Johnson 1960]. The mathematics are so simple that we are going to explain them even here, in our popular e-zine:

Fig. 4: The schematic two-lens model as presented by [Johnson 1960]. Please note that there is no need to investigate the light path between the two lenses! Instead the diagram is representing the characteristic rays forming images g' and r' of one and the same object g = r at two different magnifications. An easy going object is e.g. a given distance on an object micrometer. t₁ und t₂ are representing two different microscope tube lengths (which are are causing different magnification effects). One big advantage of the method is that we needn't know the absolute t₁ and t₂ values, which might be difficult to measure. Instead the difference between the two (at two different tube extensions) will suffice. Again, f is the focal length of the combined system and the overall magnification V is calculated as V = 250mm/f.

Fig. 5: The magnification of the system can be calculated as m₁ = g'/g (for the green case)

Fig. 6: and in the same way for the red case ...

Fig. 7: ... as m₂ = r'/r.

Fig. 8: We can combine the formulas in fig. 5 und fig. 7 by subtraction (ending up as common, single formula). After a few further steps we will arrive at the fascinating final formula (fig. 9).

Fig. 9: This formula is indicating that we can find out the focal length of a given optical system just by measuring the two magnification values which can be observed at two different microscope tube lengths!
All this is based on the publication by [Johnson 1960]. The figures 4 and 6 were slightly simplified. Potential errors are our fault, not the fault of our publication source!

It is good news for us amateurs that all this can be transferred to practice by means of an old-fashioned horse-shoe microscope. We will explain the exact procedure in our next magazine. The hardware needed is fairly simple:

Fig. 10: Setup for focal length precision measurements to be used in order to check the focal length values of microscope objective lenses and magnifier lenses:
(1) Microscope tube with variable extension and millimeter scale reading
(2) Micrometer eyepiece for precise measurement of the magnification values
(3) The object under investigation - in this case a chrome-colored folding magnifier
(4) An object micrometer (with 1/100 mm spacing) is positioned on the microscope table, waiting to be measured at two different tube extensions.

Fascinating - there it is, a highly interesting focal length measurement method, tremendously useful for a wide palette of optical components! We will come up with a bunch of results in the following issues of our magazine.

Literature

Grant R. Fowles: Introduction to Modern Optics. 2nd ed., New York 1975. p. 297.

B.K. Johnson: Optics and optical Instruments. 2n ed., London 1960. p. 31-32.

The following citation might be helpful in order to understand that the above method obviously has been sinking into oblivion and appears to have been replaced by other methods on the basis of a much more expensive equipment:
Lin-Yao Liao, Bráulio Fonseca Carneiro de Albuquerque, Robert E. Parks, and José Sasián: Precision focal-length measurement using imaging conjugates, Optical Engineering 51(11), 113604 (2 November 2012). https://doi.org/10.1117/1.OE.51.11.113604

© Text, images and video clips by  Martin Mach  (webmaster@baertierchen.de).
The Water Bear web base is a licensed and revised version of the German language monthly magazine  Bärtierchen-Journal . Style and grammar amendments by native speakers are warmly welcomed.

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