Content

The theory of ideal optical systems (paraxial or Gaussian optics)

Key

In the paraxial region (infinitely close to the optical axis ), any real system behaves as ideal:

Each point in the space of objects can be associated with its conjugate point in the space of images .
Each straight line has a straight line conjugate to it in the image space.
Each plane of the space of objects has a conjugate plane in the space of images.

From these provisions it follows that:

The meridional plane has a meridional plane conjugated to it in the image space.
A plane in the space of objects perpendicular to the optical axis has a conjugate plane perpendicular to the optical axis in the space of images.

Linear, angular, longitudinal magnification

A linear (transverse) increase in the optical system is the ratio of the linear size of the image in the direction perpendicular to the optical axis to the corresponding size of the object in the direction perpendicular to the optical axis (Fig. 1).

$V=\beta ={\frac {y'}{y}}$ ${\ displaystyle V = \ beta = {\ frac {y '} {y}}}$ , (one)

If V > 0, then the segments y and y 'are directed in the same direction, if V <0, then the segments y and y' are directed in different directions, that is, the image is wrapped.

If | V | > 1, then the magnitude of the image is greater than the magnitude of the subject, if | V | <1, then the image size is less than the size of the subject.

For an ideal optical system, the linear magnification for any size of the subject and image in the same planes is the same.

The angular increase in the optical system is the ratio of the tangent of the angle between the beam and the optical axis in the image space to the tangent of the angle between the beam conjugate with it in the space of objects and the axis (Fig. 2).

$W={\frac {\tan \alpha '}{\tan \alpha }}$ ${\ displaystyle W = {\ frac {\ tan \ alpha '} {\ tan \ alpha}}}$ , (2)

In the paraxial region, the angles are small, and therefore, the angular increase is the ratio of any of the following angular values:

W={\frac {\tan \alpha '}{\tan \alpha }}={\frac {\sin \alpha '}{\sin \alpha }}={\frac {\alpha '}{\alpha }}

{\ displaystyle W = {\ frac {\ tan \ alpha '} {\ tan \ alpha}} = {\ frac {\ sin \ alpha'} {\ sin \ alpha}} = {\ frac {\ alpha '} { \ alpha}}}

, (3)

A longitudinal increase in the optical system is the ratio of an infinitesimal segment taken along the optical axis in the space of images to the segment mated with it in the space of objects (Fig. 3).

$Q={\frac {l'}{l}}$ ${\ displaystyle Q = {\ frac {l '} {l}}}$ , (four)

Cardinal points and line segments

Consider the planes in the space of objects and their associated planes in the space of images. We find a pair of planes in which the linear increase is equal to unity. In the general case, such a pair of planes exists, and only one (the exception is afocal or telescopic systems for which such planes may not exist or there can be an infinite number of them).

The main planes of the system are called a pair of conjugate planes in which the linear increase is equal to unity ( V = 1).
The principal points H and H 'are the intersection points of the principal planes with the optical axis.

Consider the case where the linear increase is zero, or infinity. Move the plane of objects infinitely far from the optical system. The plane conjugated to it is called the back focal plane , and the point of intersection of this plane with the optical axis is the back focus F '(Fig. 4).

The distance from the rear main point to the back focus is called the rear focal length f '.
The distance from the last surface to the back focus is called the back focal segment S ' _F.
The front focus is a point on the optical axis in the space of objects, conjugated to an infinitely distant point located on the optical axis in the space of images

If the rays come out of the front focus, then they go in the image space in parallel.

The front focal length f is the distance from the front main point to the front focus.
The front focal length S _F is the distance from the first surface to the front focus.

If f '> 0, then the system is called collecting or positive . If f '<0, then the system is scattering or negative .

The front and rear focal lengths are not completely independent, they are interconnected by the ratio:

{\frac {f'}{f}}=-{\frac {n'}{n}}

{\ displaystyle {\ frac {f '} {f}} = - {\ frac {n'} {n}}}

, (five)

Expression (5) can be rewritten in the form:

{\frac {f'}{n'}}=-{\frac {f}{n}}

{\ displaystyle {\ frac {f '} {n'}} = - {\ frac {f} {n}}}

, (6)

Where ${\frac {f'}{n'}}$ ${\ displaystyle {\ frac {f '} {n'}}}$ - reduced or equivalent focal length .

If the optical system is in a homogeneous medium (for example, in air) n = n ', therefore, the front and rear focal lengths are equal in absolute value | f | = | f '|.

The optical power of the optical system:

$\Phi ={\frac {n'}{f'}}=-{\frac {n}{f}}$ ${\ displaystyle \ Phi = {\ frac {n '} {f'}} = - {\ frac {n} {f}}}$ , (7)

The greater the optical power, the stronger the optical system changes the course of the rays. If Φ = 0 then $f'=\inf$ ${\ displaystyle f '= \ inf}$ .

Image Building

Find the image A 'of point A. For this, it is necessary to construct at least two auxiliary rays, at the intersection of which the point A 'will be located (Fig. 5). The auxiliary beam 1 can be drawn through point A parallel to the optical axis. Then, in the image space, beam 1 'will pass through the back focus of the optical system. The auxiliary beam 2 can be drawn through point A and the front focus of the optical system. Then, in the image space, the beam 2 'will go parallel to the optical axis. At the intersection of rays 1 'and 2' there will be an image of point A. Now at point A 'all the rays (1-2-3) intersecting from point A intersect.

We now construct the ray path r (Fig. 6).

1 way . It is possible to construct an auxiliary beam parallel to this one and passing through the front focus (beam 1). In the image space, beam 1 'will run parallel to the optical axis. Since the rays r and 1 are parallel in the plane of objects, then in the space of images they must intersect in the rear focal plane. Therefore, the ray r 'will pass through the intersection of the ray 1' and the rear focal plane. 2 way . It is possible to construct an auxiliary beam parallel to the optical axis and passing through the point of intersection of the beam r and the front focal plane (beam 2). The corresponding beam in the image space (beam 2 ') will pass through the back focus. Since the rays r and 2 intersect in the front focal plane, they must be parallel in the image space. Therefore, the beam r 'will go parallel to the beam 2'.

Literature

Mikhelson N.N. Optics of astronomical telescopes and methods for its calculation. - M .: Fizmatlit, 1995 .-- 333 p.
Rodionov S.A., Voznesensky N.B., Ivanova T.V. Electronic textbook on discipline: "Fundamentals of optics." https://de.ifmo.ru/bk_netra/page.php?tutindex=201

Gaussian optics