# Conjugate Planes

Definition: pairs of planes where an optical system images one into the other and vice versa

German: konjugigerte Ebenen

Category: vision, displays and imaging

Author: Dr. Rüdiger Paschotta

An optical imaging system often works such that all points in a certain plane are imaged into points of another plane. At least within geometrical optics, there is then a one-to-one correspondence between points in the two planes: One point in one of the planes is mapped onto a certain point on the other plane, and vice versa, as shown in Figure 1. The two planes are then called *conjugate* to each other.

Conjugate planes usually exist only within the paraxial approximation. For larger ray angles against the optical axis, the image points corresponding to points in one plane are often found to lie on a curved surface. This phenomenon is related to image distortion, as explained in the article on optical aberrations.

Just like conjugate planes, there are also conjugate points – one point in a plane is conjugate to another point in the conjugate plane.

## Conjugate Planes in Imaging Instruments

Conjugate planes do not only come in pairs; there can be subsequent imaging stages e.g. in a microscope, creating multiple planes which are conjugate to each other. One may also distinguish different sets of conjugate plane in a microscope:

- An
*image-forming plane set*has planes at the specimen, an intermediate image plane near the eyepiece, and the retina of the observing eye, and normally also a plane in the illumination beam path. - There is also an
*illumination plane set*with the lamp filament (typically of a halogen lamp), condenser diaphragm, objective back focal plane and eye iris.

Those two plane sets are usually separated such that the structure of the lamp filament is not significantly affecting the image of the specimen (principle of Köhler illumination): the plane of the lamp filament must be far from being conjugate to the specimen plane.

The understanding of conjugate planes is also vital in the context of adaptive optics. Generally, it preferable to locate the wavefront corrector in a plane which is conjugate to that where the phase distortions to be compensated are generated.

## Conjugate Planes at Infinity

In a generalization of the concept, one or both of the planes can lie at infinity (i.e., at infinite distance from the imaging system). This can be the case for a telescope in its basic afocal configuration, where objects in an infinite distance are mapped to an image which is also at infinite distance. In other words, parallel incoming light rays are transformed into parallel rays at the output, just with some linear and angular magnification. In a modified configuration, there can be an image plane at a finite distance on one side, where an image sensor can be placed, for example. Further, a slight increase of the distance between objective and ocular lens can also bring the object plane to a finite distance.

See also: image planes, imaging with a lens, adaptive optics, geometrical optics, paraxial approximation

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