# Cos 4 law

Loss of brightness for different focal lengths, based on the 35mm format (43.2 mm diagonal)

The Cos 4 law (read: Cosine to the power of 4 law) describes the natural fall-off in marginal light . It says that the image brightness when imaging a uniformly bright subject through a lens decreases by a factor of the brightness in the center of the image. The image brightness in the angle outside the image center is ${\ displaystyle \ cos ^ {4} \ alpha}$${\ displaystyle B (\ alpha)}$${\ displaystyle \ alpha}$

${\ displaystyle B (\ alpha) = B_ {0} \ cdot \ cos ^ {4} \ alpha}$.

The law applies under the prerequisites that the image is free of distortion and the lens does not vignette and does not show any pupillary aberration (i.e. spherical aberration that occurs when imaging by the main rays) and other light losses, e.g. due to reflection on the lens surfaces, are neglected. In practice, however, these requirements are often not met. Most lenses vignette when the aperture is wide open , which increases the drop in brightness, and the pupillary aberration of many wide-angle lenses reduces the drop in brightness considerably.

## causes

Illustration of the solid angle ω relevant for the image brightness

Various factors are summarized in the Cos4 law. To illustrate the individual causes for the natural drop in edge light, one assumes a two-dimensional motif, e.g. B. a uniformly illuminated wall with the luminance L . This requirement does not apply to all lenses. See section Lens Construction .

The imaging should be done with an ideal lens . If one considers an infinitesimal surface element of the wall of the size that is supposed to be on the optical axis , then the photometric light intensity I is perpendicular to the surface (i.e. in the direction of the lens) ${\ displaystyle dA = dx \ cdot dy}$

${\ displaystyle I = dA \ cdot L}$.

The luminous flux Φ penetrating the lens depends on the solid angle ω of the light cone , which is formed by the lens (cone base) and the object point (cone tip):

${\ displaystyle \ qquad \ qquad \ Phi = \ omega \ cdot I}$(Equation 1)

If the surface element lies on the optical axis , the solid angle ω results from the lens diameter d and the distance g to the wall (approximately for ): ${\ displaystyle d \ ll g}$

${\ displaystyle \ omega = {\ frac {d ^ {2}} {4 \ cdot g ^ {2}}} \ cdot 1sr}$

If the surface element is now shifted laterally by an amount s so that it appears at the angle α to the optical axis when viewed from the lens , then both the photometric light intensity I and the solid angle ω change :

Illustration of the Cos4 law
Reduction of the photometric light intensity
Viewed from the lens, the surface element now appears to be shortened by the factor because it is tilted by the angle α to the viewing direction ( shortened perspective ) and therefore emits correspondingly less light in the direction of the lens ( Lambert's law ).${\ displaystyle \ cos (\ alpha)}$
This gives the first cosine factor:
${\ displaystyle \ qquad \ qquad I '= I \ cdot \ cos \ left (\ alpha \ right)}$(Equation 2)
Reduction of the solid angle ω
Moving the subject increased the distance g as follows:
${\ displaystyle g '= {\ frac {g} {\ cos \ left (\ alpha \ right)}}}$
The lens now also no longer appears circular as seen from the object point, but as an ellipse , the short axis of which has been shortened too .${\ displaystyle q = d \ cdot \ cos (\ alpha)}$
Here the freedom from vignetting and pupillary aberration must be assumed. In the case of the single lens, the lens edge is also the diaphragm and thus the entry pupil (EP) of the system. In a more complex system, the diaphragm is imaged onto the EP by the lenses in front of the diaphragm, whereby it is more or less distorted (pupillary aberration). The EP then limits the incident beam and it can let in more or less light than in the case of an undistorted image. In addition, the incident bundle of rays can still be cut by fixed apertures or lens edges (vignetting) so that it no longer fills the entire EP.
This finally reduces the solid angle to:
${\ displaystyle \ omega '= {\ frac {d \ cdot q} {4 \ cdot g' ^ {2}}} \ cdot 1sr = {\ frac {\ cos ^ {3} \ alpha \ cdot d ^ {2 }} {4 \ cdot g ^ {2}}} \ cdot 1sr}$,
and this is:
${\ displaystyle \ qquad \ qquad \ omega '= \ omega \ cdot \ cos ^ {3} \ alpha}$(Equation 3)

Analogous to equation 1 , the resulting luminous flux Φ results in

${\ displaystyle \ Phi '= \ omega' \ cdot I '}$.

Substituting and I from equations 2 and 3 results in: ${\ displaystyle \ omega '}$

${\ displaystyle \ Phi '= \ Phi \ cdot \ cos ^ {4} \ alpha}$

The angle β corresponds to the double of α , as it passes over the same angle on the α opposite side of the optical axis, therefore the following applies:

${\ displaystyle \ Phi '= \ Phi \ cdot \ cos ^ {4} \ left ({\ frac {\ beta} {2}} \ right)}$

This expression indicates how large the luminous flux is from a small surface element at an angle to the axis through a circular opening (here the lens edge). It applies universally, even if there is no imaging optics at all. ${\ displaystyle \ alpha = \ beta / 2}$

If now the surface element of the lens or the lens distortion without further loss of light to the imaging scale displayed on the image plane, it assumes independent of always the same area a in the picture. The illuminance of the image plane at this point is then ${\ displaystyle dA}$ ${\ displaystyle m}$${\ displaystyle \ alpha}$${\ displaystyle dA '= m ^ {2} \ cdot dA}$${\ displaystyle E}$

${\ displaystyle E = {\ frac {\ Phi '} {dA'}} = {\ frac {\ Phi '} {m ^ {2} \ cdot dA}} = {\ frac {\ Phi} {m ^ { 2} \ cdot dA}} \ cos ^ {4} \ left ({\ frac {\ beta} {2}} \ right)}$

Thus the illuminance of the image decreases with the factor under the conditions described (Cos4 law). ${\ displaystyle \ cos ^ {4} \ left ({\ frac {\ beta} {2}} \ right)}$

On the other hand, if the lens is distorting, then the area of ​​the picture element depends on the angle of view. This has a considerable influence on fisheye lenses , which have a strong barrel shape. As the angle of view increases , it becomes smaller and smaller, and the light is concentrated on a smaller area, which results in a higher illuminance and thus a reduction in the peripheral light drop. ${\ displaystyle dA '}$${\ displaystyle \ alpha}$${\ displaystyle dA '}$

In the event of light losses within the optical system - e.g. B. by filters, diaphragms, reflection and absorption losses - which evenly darken the image field , the measured brightness distribution still corresponds to the Cos4 law, because this does not give any specific brightness information, but describes the ratio of the brightnesses in the center or at the edge of the Field of view. However, this does not apply to light losses depending on the angle of view .

## Countermeasures

Illustration with a ball lens

The apparent reduction in size of the surface element due to the shortening in perspective and the increase in its distance from the objective cannot be influenced. But the "squeezing" of the light cone by shortening the perspective of the lens (or the entrance pupil as an image of the aperture in a lens ) to form an ellipse can be reduced with technical measures. ${\ displaystyle dA}$

One could replace a lens with a glass ball (or a cobbler ball ) - the projection of which appears as a circle at every angle and is therefore not shortened in perspective.

In real lenses, a pupil aberration is often used for this purpose: the lenses in front of the diaphragm, which images it onto the entrance pupil (EP), are designed in such a way that the EP is distorted. With an increasing angle of incidence of the light, the area of ​​the EP increases. The cross section of the beam let in by the objective is thereby reduced by less than the factor , and with a correspondingly strong pupillary aberration it can even become larger. With a reasonable effort, the decrease in brightness can be reduced by a maximum of around two cosine factors: ${\ displaystyle \ beta / 2}$${\ displaystyle \ cos (\ beta / 2)}$

${\ displaystyle B (\ beta) \, \ approx \, B (0) \ cdot \ cos \ left ({\ frac {\ beta} {2}} \ right) ^ {2}}$

However, such a severe pupillary aberration is associated with considerable effort. The resulting aberrations must be corrected again by the lenses after the aperture. The gain in brightness is also noticeable at large angles of view , which is why a correction based on this principle is only used in photography for wide-angle lenses with large angles of view . ${\ displaystyle \ beta}$

### Stitching

The drop in edge light can be counteracted by stitching when capturing still motifs . Individual images are combined into one image with a larger angle of view. The camera is swiveled horizontally or vertically through a small angle between the individual recordings. This also swivels the focal plane (similar to a fish-eye lens - see next section) and the image plane. The edge light drop can in principle be completely eliminated. While the drop in peripheral light can be counteracted by stitching, conversely the difference in brightness caused by the peripheral light decrease has a particularly disruptive effect when combining images, especially when the individual images themselves already cover a larger angle of view.

### Lens constructions

Some lens constructions show a peripheral light falloff that differs from the Cos4 law:

Object-side telecentric lens
Systems that are telecentric on the object side show a (in principle correctable) natural edge light decrease of
${\ displaystyle B '= B \ cdot {\ frac {f} {\ sqrt {{\ left ({\ frac {d} {2}} \ right)} ^ {2} + f ^ {2}}}} }$
at the edge of the image circle . Since the angle of view is always 0 °, the same amount of light always passes through the front lens. Systems that are telecentric on the object side, however, have a diaphragm in the focal point on the image side , through which the edge rays - depending on the front lens diameter and the focal length - are penetrated at an angle. The bundle of rays is therefore reduced to an elliptical cross section, and this causes the above-mentioned edge light drop.
Fisheye lens
With fisheye lenses , the plane of focus is not flat, but has the shape of a spherical shell . This makes it possible to display image angles of 180 ° and even beyond. This also means, however, that the patch can no longer simply be shifted perpendicular to the optical axis , as shown in the Causes section ; Instead, you have to move it on the spherical shell or rotate it around the center of the sphere.
If the center of the sphere is close to the objective, the distance of the surface element does not change during the shift. There is also no longer any perspective distortion of the surface element, because this has also changed its orientation in space, so that its surface normal still points to the lens. Only the (in principle correctable) perspective distortion of the diaphragm to an ellipse leads to a peripheral light drop of
${\ displaystyle B '= B \ cdot \ cos \ left ({\ frac {\ beta} {2}} \ right)}$

### Relative decrease in brightness

While the absolute decrease in brightness can only be influenced to an insignificant extent, the relative decrease in brightness, on the other hand, can simply be corrected by simply darkening the image in the center of the image. For example, in extreme wide-angle lenses are graduated filter (center filter) are used which darken the picture center and brighter outward.

Darkening with a center filter makes the image darker overall (the absolute decrease in brightness is therefore even greater), so that longer exposure times are required for photography . But the brightness gradient is less steep and the image illumination is more even.

### Mathematical correction

In digital cameras which can firmware of the camera while recording or later, the image processing software on the basis of metadata about the recording parameters in the digital image data , a compensation calculating the peripheral light waste. In the corners of the image, however, an increase in the image noise is to be expected due to the amplification of the brightness values.