Bessel method

As Bessel method or Bessel method is a measurement method for determining the focal length of a condenser lens , respectively. It is named after Friedrich Wilhelm Bessel , who published it in 1840. ${\displaystyle f}$

Basics

Structure and names

If an object G is imaged on a screen as image B by means of an optical lens , then a sharp image is obtained in two positions of the lens: In this position the image is enlarged, in the position it is reduced. The distance between the object and the screen must be greater than four times the focal length plus the distance between the two main planes of the lens: ${\displaystyle P_{1}}$ ${\displaystyle P_{2}}$ ${\displaystyle a}$ ${\displaystyle f}$ ${\displaystyle {\overline {HH'}}}$

${\displaystyle a>4f+{\overline {HH'}}}$

In practice, the lens is shifted back and forth several times between these two positions and the respective distances and from an edge of the arrangement are measured. The difference between these two lens positions gives the distance between the two lens positions, from which the focal length with the equations ${\displaystyle x_{1}}$ ${\displaystyle x_{2}}$ ${\displaystyle e}$

${\displaystyle f={\frac {a^{2}-e^{2}}{4a}}}$  for thin lenses ,

${\displaystyle f={\frac {(a-{\overline {HH'}})^{2}-e^{2}}{4(a-{\overline {HH'}})}}}$  for thick lenses

can be calculated.

Compared to the simple calculation from image and object distance using the lens equation , the Bessel method has the advantage that the position of the main planes H and H ′ does not have to be known for thick lenses or lens systems . However, the result for thick lenses will be half to a quarter of too large, depending on . ${\displaystyle {\overline {HH'}}}$ ${\displaystyle a}$

Compared to the more complex Abbe method , which also determines the position of the main planes, the Bessel method has the advantage that many lenses can be measured quickly with a fixed structure (light source, object and screen at a fixed distance).

Derivation

Thin lenses

Designations on the thin lens

1. Derivation:
With thin lenses the distance between the two main planes can be neglected. The following applies:

${\displaystyle b+g=a}$,

where is the image distance and the object distance . Because of the symmetry of the arrangement, the following must also apply: ${\displaystyle b}$ ${\displaystyle g}$

${\displaystyle e=a-2b}$

(the object is supposed to be shown in focus, so the distance between the two points at which it is shown in focus can only be described by this equation). Using the lens equation ${\displaystyle e}$

${\displaystyle {\frac {1}{f}}={\frac {1}{b}}+{\frac {1}{g}}}$

and inserting as well ${\displaystyle g=a-b}$

${\displaystyle b={\frac {a-e}{2}}}$

you get

${\displaystyle {\frac {1}{f}}={\frac {1}{\frac {a-e}{2}}}+{\frac {1}{a-{\frac {a-e}{2}}}}={\frac {2}{a-e}}+{\frac {2}{a+e}}={\frac {4a}{a^{2}-e^{2}}}}$.

The transformation results

${\displaystyle f={\frac {a^{2}-e^{2}}{4a}}}$.

2. Derivation:
Using the lens equation

${\displaystyle {\frac {1}{f}}={\frac {1}{b}}+{\frac {1}{g}}}$

and inserting (the distance between the principal planes of a thin lens is zero) one obtains an equation for : ${\displaystyle b=a-g}$ ${\displaystyle g}$

${\displaystyle {\frac {1}{f}}={\frac {a}{ag-g^{2}}}}$.

When this quadratic equation to solved, we obtain ${\displaystyle g}$

${\displaystyle e=\Delta g={\sqrt {a^{2}-4af}}}$.

where is defined as the difference between the two object widths. The transformation results ${\displaystyle e}$ ${\displaystyle g}$

${\displaystyle f={\frac {a^{2}-e^{2}}{4a}}}$ .

Thick lenses

The distance between the main planes is not negligible. It applies ${\displaystyle {\overline {HH'}}}$

${\displaystyle a=g+b+{\overline {HH'}}}$ and

${\displaystyle e=(a-{\overline {HH'}})-2b}$ .

With the above considerations the formula is obtained

${\displaystyle f={\frac {(a-{\overline {HH'}})^{2}-e^{2}}{4(a-{\overline {HH'}})}}}$ .

literature

• Eugene Hecht: Optik , Oldenbourg Verlag, 4th edition 2005, ISBN 3-486-27359-0

Remarks

1. ↑ FW Bessel: About a means for determining the focal length of the objective glass of a telescope. In: Astronomical News , Volume XVII (1840), No. 403, pp. 289–294 ( digitized version )