# Luminance

The luminance Lv (English luminance ) provides detailed information about the spatial and directional dependence of light emitted from a light source luminous flux. It is the photometric measure of what the human eye perceives as brightness of a surface. Luminance describes the brightness of extended, planar light sources; for a description of the brightness of the point light sources, the light intensity is more suitable.

- 4.1 General
- 4.2 Light intensity
- 4.3 Specific light emission
- 4.4 Luminous Flux

- 5.1 light emission
- 5.2 light irradiation
- 5.3 conclusion
- 5.4 Example

## Definition

### Introduction

Consider serving as a light source body ( for example, a light bulb, an illuminated sheet of paper ), which (measured in lumens ) emits a light flux into its surroundings. In general, different points of the body are different give off much light, and he will send for many different light in different directions. If these characteristics are described in detail, the concept of luminance is necessary.

It is not possible to specify how many lumens emanating from an infinitely small point on the surface of the body, since the finite number of radiated lumens spread over an infinite number of such points, and therefore accounts for a single surface point zero lumen. Instead, one considers a small neighborhood of the point, sets the outgoing of this environment (finite) light power in relation to its (finite) surface and leaves the environment mentally shrink to zero. Although the emitted luminous flux as well as the radiating surface thereby going against each zero, seeks both ratio to a finite limit, the radiance of the point, measured in lumens per square meter or equivalent lux

Likewise, it is not possible to specify how many lumens are emitted in a certain direction, since the finite number of radiated lumens spread over an infinite number of possible directions and therefore attributable to each individual towards zero lumen. Instead, one considers a small, surrounding the desired direction solid angle sets the cast in this solid angle (finite) luminous flux in relation to the ( finite) size of the solid angle and the solid angle can theoretically shrink to zero. Again, this aim both the solid angle and the emitted light power contained in it each to zero, but their relationship to a finite limit, the light intensity emitted in that direction, measured in lumens per steradian or equivalent Candela.

The concept of a combination of both luminance and described in this manner, both the location and the direction dependence of the output from an infinitely small surface element luminous flux.

For the definition of luminance, it is irrelevant whether it is in the light emitted from the surface element to ( thermal or non-thermal ) Own mission is transmitted or reflected light or a combination thereof.

The luminance is defined at every point of the space is provided on the light. Imagine instead of a light-emitting surface element, where appropriate, a fictional irradiated by light surface element in the room.

### Luminance

Luminance indicates how luminous flux emitted from a given point of the light source in the information given by the polar angle and the azimuth angle direction of the projected area per element and per solid angle:

- Symbols: L, Lv
- SI unit: lumen per square meter per steradian candela per square meter
- Unit symbol: lm · m -2 · sr- 1, cd · m -2

The definition of the luminance has the special feature that the output light power is not, as usual with respect to the radiating planar element but on the projected area in the emission element. The emitted light flux in a particular direction depends namely on the one of the (possibly directional ) radiation physical properties of the surface and the other purely geometrical from the effective projection in the emission of radiant surface element. The second effect will cause the output at the polar angle of luminous flux by a factor lower than the vertically emitted light power. The division by a factor calculated out this geometric effect, so that the luminance of only a possible physical directionality due to the surface properties remains.

### Lambertian Emitter

Surfaces which have after eliminating the factor no directional dependence of the luminance more, called diffuse or Lambertian radiator radiator. A lambertian surface element is in all directions from the same luminance. The luminance is therefore no longer dependent on the angle:

The of a Lambertian radiation emitted in a certain direction luminous flux only varies with the cosine of the emission angle. Such radiators are therefore particularly easy to treat mathematically:

In particular, the integration over the solid angle which is now angularly independent luminance can be considered as constant, out of the integral, which often greatly simplifies the integration ( see below).

An example of a diffuse emitting surface is an illuminated sheet of paper. Looking at it from different directions, then the perceived brightness of the area while remaining constant, while the total amount of light reaching the viewer ( the luminous intensity ) depends on the projected area, and therefore varies with the cosine of the viewing angle.

## Sensitivity of the eyes

The observer takes the luminance of the surrounding surfaces immediately perceive as their surface brightnesses. Due to the adaptability of the eye, the perceived luminance, numerous orders of magnitude sweep. The values given vary from person to person and also depend on the frequency of the light.

## Examples

## Connection with other photometric quantities

### General

Luminance indicates how much light is emitted from a given infinitesimal surface element in a given direction and thus providing the most detailed description of the light emitting characteristics of the surface in question. Switch the defining equation for the luminance provides the infinitesimal light flux irradiated from the at the position lying surface element in the solid angle, which is in the described direction by the angle and:

If the radiance of a finitely large radiating surface are determined in a finitely large solid angle, so is over and integrate:

The representation of the solid angle element was used in spherical coordinates:

Since in general, the place on the light surface and of the cleaned directions and depend arises under certain circumstances a very complicated integral. A major simplification occurs when the light-emitting surface is a Lambertian emitter ( luminance therefore independent of direction ) with constant surface properties ( luminance therefore independent of location ) is. Then, the luminance is a constant number and can be pulled out of the integral:

The remaining integral now depends only on the shape and position of the solid angle and can be independent of be solved. In this way, dependent only on the transmitter and receiver geometry general view factors can be determined and ready tabulated.

For example, the light emission seen in the entire surveyed by the light emitting surface half-space, we obtain for the integral of the value

And the luminous flux of a surface homogeneous Lambertian reflector of area in the entire half-space is simple:

Similarly, the other photometric quantities can be derived by integrating over the entire surface and / or all directions of the half- space from the luminance.

### Luminous intensity

Considering, instead of the radiation of a surface element, the emission of the total surface of a body in a given direction, as is the emission but not integrated over the direction and the intensity of the body is obtained in this direction:

Where the coordinates that describe the position of the surface element on the total area and specify the angle, the observed beam direction with respect to the surface normal of. In particular, is again the angle between the beam direction and considered one of the surface normal. The integral is to extend over that part of the entire surface is available for the.

The light intensity is effectively the sum of all votes cast in a particular direction luminance of the body surface.

### Radiance

Considering, instead of the radiation of the surface element in a certain direction in its emission over the whole looked from the surface element half space, then all directions but can not be integrated over the entire surface and to give the specific radiance of the surface element:

In the special case of a Lambertian emitter is independent of the angles and and can be drawn out of the integral. The remaining integral has, as explained above, the value, and there is the simple relationship

### Luminous flux

Integrating the luminance over all directions in the hemisphere and all surface elements of the radiating surface, or the light intensity over all directions, or the radiance over all surface elements, we obtain the total flux of the luminous body:

Is the special case of a Lambertian emitter and the light flux can be calculated directly from the luminance:

The luminance Furthermore, homogeneous surface (that is the same over the entire surface ), then the integral simplifies to a simple multiplication:

Already shown by direct integration as above.

## Photometric Basic Law

### Light emission

Looking at an area element which illuminates a surface element located within a distance to the luminance as viewed from the space spanned by angle, and follows from the first equation in the previous section:

In this case, and the angle of inclination of the surface elements of the common connecting line.

This is the photometric Basic Law. By integration over the two surfaces results in the flowing of total surface area of 1 by 2 luminous flux.

### Light irradiation

The illumination density is analogous to the luminance, but defined for the radiation case. It specifies which light stream is received from the description given by the polar angle and the azimuth angle direction per projected surface element and per solid angle element. The previously derived equations apply analogously. In particular, for the data received on surface element of the emitted light power:

Except that which occurs from the space spanned angle.

### Conclusion

Of the light emitted according to need and the power of received light to be identical (if not lost at a temperature between the surfaces of the light medium due to absorption or scattering), and by comparing the two equations, the following:

The light emitted from surface element is identical to the luminance of the incident light on surface element density.

Thus, it should be noted that the luminance does not decrease with distance. The total transmitted flux and increases on the other hand, as expected, with the square of the distance from (due to the factor in the denominator of both equation ), this is because the plane defined by the channel surface of the solid angle of view of the receiving surface decreases quadratically with the distance.

If the illumination density integrated over the solid angle from which it is derived, then the illuminance -called single-beam current surface density is on the receiver surface in lm/m2. If the emitted in a certain direction luminance of the station area is known, is now also immediately derived with her identical from the same direction illumination density of the receiving surface is known and the illumination intensity on the receiver surface can be calculated from the luminance distribution of the transmitter surface immediately:

### Example

Comparing a nearby billboard with an identically illuminated further away, both appeared equally bright (they have a distance-independent and, therefore, in both cases identical luminance ). However, the detailed wall takes for the observer a larger solid angle, so that the observer of this larger solid angle of an overall larger luminous flux achieved ( the near wall generated due to their larger solid angle greater illuminance at the observer, it can read in their lighting newspaper ).

## Units of measurement

The SI unit for luminance is candela per square meter ( cd / m²).

In the English-speaking world, especially in the U.S., but also the name Nit ( unit symbol: nt, from Latin nitere = " seem " most nits ) will be used. Specifying the luminance is used for brightness of computer monitors, typically 200 to 300 nits. Also, the brightness of LED video walls, such as are used in the event industry, often indicated with Nit. Values usually in the low to mid four -figure range. The Nit is not a legal entity in Germany and Switzerland.

In addition, in the U.S. and unity Lambert is in use.

Conversion factors to other units of luminance are not limited to:

- Nit: 1 nt = 1 cd / m²
- Stilb: 1 sb = 1 cd / cm ² = 10 000 cd / m²
- Candela per square foot: 1 cd / ft ² ≈ 10,764 cd / m²
- Candelas per square inch: 1 cd / in ² ≈ 1550 cd / m²
- Apostilbes, Blondel: 1 asb = 1 blondel = 1 / π × 10-4 sb = 1 / π cd / m²
- Skot: 1 skot = 0.001 = 0.001 blondel asb = 10-3 / π cd / m²
- Lambert: la 1 = 1 L = 104 / π cd / m ≈ 3183 cd / m²
- Footlambert: 1 fL = 1 / π cd / ft ² ≈ 3.426 cd / m²