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Abbe-Method

Measuring method (named after Ernst Abbe) used to determine the focal length and the position of the principal planes of a lens singlet or a lens system (=objective) on the optical axis.

How to determine the focal length:
The position of the lens is fixed and the camera (or the screen ) is moved depending on the object position, that you get a foczussed image (in the image center). Different object positions result in different camera- or screen distances
FocalLength = {\frac  {DistanceOfObjectpositions}{{\frac  {ObjectSize}{ImageSize2}}-{\frac  {ObjectSize}{ImageSize1}}}}

How to determine the focal length of an objective (= (= lens system)):
The Position of a lens (and the lens singlets in it) are fixed and an arbitrary Point O on the optical axis is marked as reference point, for example the center of the lens or the center of the first lens element).
Now we measure the distance x from the reference point to the object, the distance x’ to the image and the image size B.
You get a list of Magnifications
\gamma = \frac {ImageSize}{ObjectSize} = \frac{B}{G},
and equations from refererence Point to object
x=f\left(1+{\frac  {1}{\gamma }}\right)+h
and reference point to image:
x'=f'\left(1+\gamma \right)+h'
Where h und h’ are the distances from object side resp. image side principal planw to the reference point.

Abbe-number

(also known as v-number)
a measure of the materials dispersion (=variation of refractive index with wavelength),
with high values of V indicating low dispersion (low chromatic aberration).

The value Vd is given by

V_d = \frac{n_d-1}{ n_F - n_C }

which defines the Abbe number with respect to the yellow Fraunhofer-Line d (or D3) helium line at 587.5618 nm wavelength.

It can also be defined using the green mercury E-line at 546.073 nm:

V_e = \frac{n_e-1}{ n_{F'} - n_{C'}}

where F’ and C’ are the blue and red cadmium lines at 480.0 nm and 643.8 nm, respectively.