Magnifiers: a closer look (IX) 
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, f_{1} and f_{2} 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 f_{1} of 25mm (10x lens) and f_{2} 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. 
Fig. 4: The schematic twolens 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. t_{1} und t_{2} 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 t_{1} and t_{2} 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 m_{1} = g'/g (for the green case) 
Fig. 6: and in the same way for the red case ... 
Fig. 7: ... as m_{2} = 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!

It is good news for us amateurs that all this can be transferred to practice by means of an oldfashioned horseshoe 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:

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 
© Text, images and video clips by
Martin Mach (webmaster@baertierchen.de). 