Magnifiers: a closer look (X)
Focal length measurement for amateurs - the practical procedure!

In the previous issue we had presented the theory behind an advanced focal length measurement method (equivalent to an almost universal micro-optics magnification measurement method). The respective mathematical formula behind was as follows:

Fig. 1: Formula for the calculation of a focal length on the basis of the intermediate microscope images measured at two different microscope tube extensions.
f: focal length to be measured (e.g. focal length of a magnifier)
t₁ and t₂: two different microscope tube extension values
m₁ and m₂: observed magnification at the respective tube lengths t₁ and t₂
Source [Johnson 1960].

Clearly a mathematical formula is nice by itself but its content might even be more respected at the moment when it will be verified by practical checks. In fact we followed this pathway and were quite impressed by the turnout:

Tab. 1: Practical check of the formula shown in Fig. 1. The objects chosen were well-documented Nikon "CF M Plan" achromatic objectives, i.e. professional "MPlan" microscope objectives designed for use with the famous Nikon "Optiphot" microscopes at a tube length of 210 mm.
Grey: Our measurements
Blue: Average result, calculated from our own measurements
Green: Official focal length values, as provided by the Nikon datasheet (see lit.)

Table 1 is indicating an excellent accordance between our measurements and the official data by Nikon, with maximum deviation values far below 1 percent!

One might compare with modern methods on the basis of more expensive equipment. The results of the old-fashioned method appear to be quite competitive with the ones published by [Liao 2012]. In the latter publication the focal length of the Nikon MPlan 5x objective is found to be 37.43 mm, differing not dramatically, but still significantly from the official Nikon value shown in table 1 (37.64 mm). Of course one might argue that Nikon might have published slightly wrong values or that there might have been some variance among the individual objectives but we tend to trust in the official Nikon values as we were able to reproduce them.

Now it is up to you to explore the microscope based focal length measuremenent method by yourself! Just follow the procedure as described below:

Fig. 2: The recommended hardware. We used a vintage "Nachet DXS Grand Microscope" with draw tube and scale markers on the tube. But you might use any other high quality monocular microscope with draw tube as well. For lower demands one might work without a dedicated eyepiece micrometer (marked as "2" in fig. 2) and use a more common eyepiece with traditional scale. But please keep in mind that the latter will only work when the field lens is removed. The measurement precision on the basis of the regular, simple eyepiece with scale is said to be lower by a factor of 10 [Göke 1988].

Meaning of the red numbers on the left side of fig. 2:
(1) The draw tube with scale
(2) A Zeiss (or other company) micrometer eyepiece for precision length measurements
(3) The object - in this case a folding magnifier fitted to the objective revolver.

Fig. 3: It is important to fix the object under investigation in a precise orthogonal manner and to avoid stray-light penetrating the joint. Of course RMS objectives can be measured without any adapter, just by screwing them into the microscope revolver.
For other objects we are using a simple DIY adapter as shown above. It is consisting of two RMS extension tube adapters fitted with a primitive diaphragma. The object can be fixed on the downside of the adapter by means of sticky tape.

In order to measure the two magnifications at two tube extensions we need an object micrometer (a microscope slide with a 1/100 mm ruling). We did prefer an old-fashioned Leitz object micrometer with a vertically oriented 2 mm range but in fact any common object microscope object micrometer might be sufficient, too.
The decisive measurement is performed as follows:

(1) The object micrometer is placed on the stage and properly focused

(2) The draw tube is precisely adjusted to an (arbitrary) minimal tube length value on the tube scale. Typically this will be about 150 mm

(3) The image of the object micrometer scale and the image of the micrometer eyepiece scale are x-y adjusted for direct comparison (see fig. 4 with both scales visible)

(4) Now we can determine a first magnification value m₁. In the situation shown in fig. 4 and fig. 5 we do find that an object micrometer range of 1.5 mm is corresponding to an eye-piece scale value of 6.608 mm indicating a magnification value of 4.4053. This is our value m₁ to be inserted into the formula in fig. 1.

(5) The second magnification value m₂ is determined in the same way at maximum tube extension length, e.g. 200 mm.

(6) Finally we fill the formula with the difference of the two tube lengths (here 200 mm minus 150 mm, resulting in 50 mm) and the two magnification values m₁ and m₂. Life is easier when using a spreadsheet calculation software for this task, e.g. Microscoft^TM Excel^TM or OpenOffice^TM Calc^TM.

Fig. 4: View through the micrometer eyepiece showing the object micrometer scale (the vertical 2 mm range scale with very fine divisions) and the eyepiece scale (appearing more coarse, also vertical, on the right side). The double-line cursor of the micrometer eyepiece is pointing towards the 1.5 mm ruling on the object micrometer scale and, at the same time at an eyepiece micrometer scale value of six-plus-something (we don't need to specify this more exactly at the moment). The exact eyepiece micrometer scale must be read on the outside, by means of the micrometer screw of the micrometer eyepiece (see fig. 5).

Fig. 5: The decimal places of the eyepiece value (tenth, hundreds and thousands of a millimeter) must be read from the micrometer screw on the outside of the microscope. In our example the micrometer screw is helping to determine the missing decimal places between the "6" and the "7" (as found in fig. 4). The final value is a combination of the rough estimate (fig. 4, indicating a value between 6 and 7) and the micrometer screw value which is approximately 608. This adds up to a total value of 6.608 mm. The magnification value is found by divison of those 6.608 mm through 1.5 mm, with a magnification value result of 4.41.

Obviously some practice will be needed in order to reach a good precision similar to the one documented in table 1. Small experimental negligences might result in noticable deviations. Just to give some examples we would like to mention that the overall system must be kept mechanically stable during the tube length variation. The micrometer slide must be properly fixed on the table, the tube should not allowed to sink down unnoticed during measurement etc. For additional safety one should check the procedure with known focal length objects and various tube length intervals (e.g. 200 mm to 150 mmm and 180 mm to 140 mm). The good news is that this focal length measurement method can be used for a wide variety of lenses, lens combinations and microscope objectives and that it can be very precise when used properly.
As a funny aside we would like to mention once more that an old fashioned monocular microscope with draw tube is needed. Typical modern 50,000 US $ microscopes do have no draw tube ... and most scientific institutions will have scrapped this kind of vintage equipment a long time ago ... what a pity!

We will come up with many interesting magnification results of magnifiers in the upcoming issues of our magazine. This will be fascinating, though sometimes a little bit sad and sometimes very funny. See you in September!

Literature

Gerhard Göke: Moderne Methoden der Lichtmikroskopie. p. 280. Stuttgart 1988.

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

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

Nikon Ddata sheet [1993]: Nikon CF Objectives 210 mm Tube Length for industrial Applications. [Annotation: we found thus data file on the webpage of the reknowned microcope photographer Charles Krebs].

© 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.

Main Page