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parfocal

A parfocal lens stays in focus when the magnification / focal length changes.
Some amount of defocussing is inevitable, but the amount is so small that it is considered as insignificant.

In microscopy parfocal lenses stay in focus when the magnification is changed.

The image stays focused when the magnification is changed from 40x to 10x
Most bright field microscoped are parfocal.
In photography, a zoom lens is parfocal, a varifocal lens is not.
Motor-Zoom lenses are actually varifocal. When you change the magnification, also the focus changes. However the focus motor can work in parallel to the focal length motor, so for the user the lens seems to be parfocal.

pericentric

= hypercentric

With pericentric lenses objects at larger distances appear larger(!) and objects at closer distances appear smaller.

Perizentric lenses allow for example to view a can from top and the sides at the same time.

This reverses our normal viewing experience.

Pericentric lenses got to be MUCH larger than the object under inspection.

see “comparison: entocentrictelecentric – pericentric”

Application : A cylinder with a drilling that is centered on one circular side and decentered on the opposite side is to be inspected for foreign parts in the drilling.
The rotation of the cylinder is not known, so we would need a lens that can look from all sides “outside-in” at the correct angle.
Solution: DIY with the help of a Fresnel Lens, a normal M12 lens and the graphic calculator below …

Petzval Sum

In optical lens design the Petzval sum describes the image curvature of an optical system.

The formula was developed by Josef Maximilian Petzval and published 1843.

For a series of thin lenses holds

    \[\frac{1}{r_p} = \sum \limits_{i} \frac{1}{n_i \cdot f_i}\]

or more general

    \[\frac {1}{r_{p}}=n_{k+1}\sum \limits_{i=1}^{k}{\begin{cases}{\frac {1}{r_{i}}}\left({\frac {1}{n_{i}}}-{\frac {1}{n_{i+1}}}\right),&{\text{refractive surface}}\\{\frac {2}{r_{i}}},&{\text{reflective surface}}\end{cases}}\]

where:
r_{i} : radius of the i’th surface,
n_{i} : refraction index before the refraction
n_{i+1} : refraction index after the refraction

The displacement of an image point at height y_i from the paraxial image plane is

    \[ \Delta{x} = \frac{y_{i}^2}{2} \sum \limits_{i=1}^{k}\frac{1}{n_{i} \cdot f_{i}} \]

The Petzval condition says that the curvature of the Petzval area is zero if the Petzval sum is zero.

If in addition there is no astigmatism, then the image surface is flat.

If there is astigmatism however then the folowing relationship holds between the curvature of the petzval surface and the curvature of saggital and tangential surface

    \[ \frac {2}{r_{p}}= \frac {3}{r_{s}}-\frac {1}{r_{t}}\]

The mean image curvature is here the reciproke mean of tangential and saggital curvature.

pixel vignetting

A kind of vignetting which occurs exclusively with digital cameras.

Possible causes are:

  • the pixels are not completely flat due to construction on the sensor surface, but in small cavities. Too shallow light cast shadows on the edges of the pixels, like the evening sun at some point no longer reaches mountain valleys.
  • The sensor uses micro-lenses (small converging lenses) to capture as much light as possible for each pixel. From a certain off axis angle lenses are no longer capable to deflect the light strong enough and the light can’t reach the pixel no more.
  • With image side telecentric lenses such vignetting does not occur because the incident light rays are parallel to the optical axis.
    The latest sensor technologies however try to correct the Pixelvignettierung on-chip (= directly in the sensor) or by micro lenses that have differently shape in the corners than in the center.
    Thus it may happen that the image side telecentric lenses surprisingly show vignetting.

principal plane

The prolongued a light ray parallel to the optical axis entering a the first lens of a  lens system and the corresponding ray leaving the last lens element intersect in a plane called image side principal plane.

The prolongued a light ray parallel to the optical axis entering a the last lens of a  lens system and the corresponding ray leaving the first lens element intersect in a plane called object side  principal plane.

Each (symmetric) lens has two principal planes. These (hypothetical) planes are perpendicular to the optical axis and are the planes on which light beams parallel to the axis coming from infinity seem to bend (and then go through the respective focal points).

NOTE:
This only applies to the paraxial optics, ie very close to the optical axis.
For rays more distant to the optical axis spherical aberration distorts this behaviour.
In single thin lens, the two principal planes merge and can be approximated by the center of the lens.