Petzval sum

The Petzval sum or the resulting radius of the Petzval surface describes the field curvature of an optical system. It was developed by Josef Maximilian Petzval and published in 1843. The following applies to a number of thin lenses with the respective focal length and refractive index : ${\displaystyle f_{i}}$ ${\displaystyle n_{i}}$

${\displaystyle {\frac {1}{r_{p}}}=\sum _{i}{\frac {1}{n_{i}f_{i}}}}$

The reciprocal radius of the Petzval area is equal to the Petzval sum. ${\displaystyle r_{p}}$

More generally applies:

${\displaystyle {\frac {1}{r_{p}}}=n_{k+1}\sum _{i}^{k}{\begin{cases}\rho _{i}\left({\frac {1}{n_{i}}}-{\frac {1}{n_{i+1}}}\right),&{\text{refraktive Fl}}{\mathrm {\ddot {a}} }{\text{che}}\\2\rho _{i},&{\text{reflexive Fl}}{\mathrm {\ddot {a}} }{\text{che}}\end{cases}},}$

where is the curvature of the i-th surface ( reciprocal of the radius; 0 for flat surface). is positive for a surface that is convex in the direction of light propagation, negative for a concave one. is the refractive index before the i-th face and the refractive index after it. is the index of refraction after the last surface. ${\displaystyle \rho _{i}}$ ${\displaystyle \rho _{i}}$ ${\displaystyle n_{i}}$ ${\displaystyle n_{i+1}}$ ${\displaystyle n_{k+1}}$

Petzval condition

The Petzval condition says that the curvature of the Petzval surface disappears when the Petzval sum is zero. In addition, if no astigmatism occurs, the image field is flat.

If astigmatism is present, there is the following relationship between the curvature of the Petzval surface and the curvature of the tangential and sagittal image plane: ${\displaystyle r_{t}}$ ${\displaystyle r_{s}}$

${\displaystyle {\frac {2}{r_{p}}}={\frac {3}{r_{s}}}-{\frac {1}{r_{t}}}}$

The mean field curvature is the reciprocal mean of the tangential and sagittal curvature.

Web links

• F. Pedrotti, W. Bausch, H. Schmitt: Optics for engineers
• Image Field Curvature (en)
• H. Zinken called Sommer : On the calculation of the image curvature in optical apparatus , Annalen der Physik , p. 563 ff., 1864