Hi everyone,

the "official" formula to calculate the focal lenght of microscope objectives is: Focal lenght = (Tube lenght / M + 1).

But this formula is good only for a "thin lens" and is scarcely usable for a real lens.

Moreover (and above all), two objectives with the same magnification (for example Olympus PlanApo 10x and Olympus MPlan 10x, both 160mm), but with very different working distances (respectively 0.16mm and 7.10 mm) can't have the same focal lenght.

So, I ask myself if there really is a formula that uses tube lenght, magnification an working distance to give the correct focal lenght.

Ciao

Toni

## Tube lenght, magnification and working distance

**Moderators:** Pau, rjlittlefield, ChrisR, Chris S.

- rjlittlefield
- Site Admin
**Posts:**21197**Joined:**Tue Aug 01, 2006 8:34 am**Location:**Richland, Washington State, USA-
**Contact:**

### Re: Tube lenght, magnification and working distance

No, there is not. Two lenses with the same magnifications and working distances can have different focal lengths. The focal length can be computed from magnification, distances, and positions of the two "principal planes" of the lens. But there is no way to know where those planes are without doing some other measurements.Tonikon wrote:So, I ask myself if there really is a formula that uses tube lenght, magnification an working distance to give the correct focal lenght.

However, you can determine the focal length by measuring magnification at two different extensions and working carefully through the algebra that relates total extension to focal length and magnification.

If I have done the algebra correctly, an intermediate stage is just

FL = (e1-e2)/(m1-m2)

where FL is focal length, e1 and e2 are different extensions, and m1 and m2 are the corresponding magnifications. Of course you don't know what the total extensions are, but you do know the difference between them. So then the formula reduces to

FL = (difference of extensions) / (difference of magnifications)

I have not experimentally checked this result and I don't have a reference for it. Please let me know if it does or doesn't work.

--Rik

You are really right, we have to use two magnifications at two different extensions...simply ingenious!

I have calculated some different magnifications with 165mm and 193mm...only 28mm of difference in tube lenght (because I have already available these values) but tomorrow I'll make other measurations with more difference in tube lenght.

Applying this formula with some objectives (with known focal lenght) I have obtained:

Zeiss Luminar 63mm f/4.5 = Focal lenght resulting: 56mm

Canon 35mm f/2.8 Photomicro = Focal lenght resulting: 31mm

Minolta 12.5mm f/2.8 = Focal lenght resulting: 9mm

These results are in good accord with nominal values, but probably using more difference in tube lenght I'll obtain more correct values).

Using the same formula, it results that my Nikon U10 0.22 (a 10x for metallurgical microscope) is a 20mm and my Olympus MPlan 1.3x 0.03 is a 52mm.

Thank you for your precious suggestion.

Ciao

Toni

I have just tried the formula suggested by Rik using a deeper tube lenght difference (56mm) and results was exciting!

Olympus MPlan 1.3x 0.03 = Focal lenght calculated: 70mm

Luminar 63mm f/4.5 = Focal lenght calculated: 64,5mm

Canon 35mm f/2.8 Photomicro = Focal lenght calculated: 34,5mm

Nikon U10 0.22 = Focal lenght calculated: 21.4 mm

Leitz 10x 0.25 170mm = Focal lenght calculated: 15.7mm

Luminar 63 and Canon 35 was used as "controller" lenses and the focal lenght calculated was very very similar to nominal focal lenght.

This formula (thank you very much, Rik) is perfect to calculate experimentally the focal lenght of microscope objectives!

Nikon U10 and Leitz 10x (the first one has a very long WD, about 15mm) have very different focal lenght , how it's reasonable to expect!

Ciao

Toni

- rjlittlefield
- Site Admin
**Posts:**21197**Joined:**Tue Aug 01, 2006 8:34 am**Location:**Richland, Washington State, USA-
**Contact:**

then measure the magnification at the detector with a ruler or stage micrometer.

fl = distance measured/( m + 1/m + 2)

this neglects the distance between the principal planes. On a side note you could find the distance between the principal planes by combining the methods.

Calulate the focal length by Rik's method and plug it into the fomula with the same magnificatiob and calculate the expected distance, subtract it form the real distance and you have the distance between the principal planes... theoretically, if you have accurate enough measurements