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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 focused image (in the image center). Different object positions result in different camera- or screen distances
FocalLength = {\frac  {DifferenceOfObjectpositions}{{\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.

Abbe’s Invariant

In paraxial optics each single refracting surface satisfies the Abbe’s Invariant Q in the paraxial Area, that relates the front focal distance s of an axial object point with the back focal distance s’ of it’s conjugated point behind the surface

Q = n (\frac{1}{r}-\frac{1}{s}) = n' (\frac{1}{r}-\frac{1}{s'})