Radiance

The radiance ( specific intensity also, English radiance ) L provides detailed information about the spatial and directional dependence of the votes from a transmitting surface radiation.

• 2.1 Spectral radiance of a black body radiator
• 2.2 overall radiance of a black body radiator

• 4.1 radiance
• 4.2 irradiation
• 4.3 conclusion
• 4.4 Examples

Definition

Introduction and limit viewing

Consider a body (such as an incandescent lamp, a light emitting diode ), which radiation ( for example, measured in watts) gives in his surroundings. In general, different points of the body are different give off much radiation, and he will send out different much radiation in different directions. If these characteristics are described in detail, the concept of radiance is necessary.

It is not possible to specify how many watts emanating from an infinitely small point on the surface of the body, since the finite number of radiated watts spread over an infinite number of such points, and therefore accounts for a single surface point zero watts. Instead, one considers a small neighborhood of the point, is the threat posed by this environment (finite) radiation power in relation to its (finite) surface and leaves the environment mentally shrink to zero. Although the radiated power as well as the radiating surface while each go to zero, seeks both relative to a finite limit, area coverage or emittance of the point, measured in watts per square meters.

Likewise, it is not possible to specify how many watts are delivered in a particular way because the finite number of radiated watts spread over an infinite number of possible directions and therefore account for each direction of zero watts. Instead, one considers a small, surrounding the desired direction solid angle is the angle given in this space (finite) performance 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 radiated power contained in it each to zero, but their relationship to a finite limit, the radiant intensity emitted in that direction, measured in watts per steradian.

Radiance describes both, and combined in this manner, both the location and the direction dependence of the output from an infinitesimal surface element radiation.

Radiance

The radiant flux indicates which radiation power is emitted from a given point of the radiation source in the given by the polar angle and the azimuth angle direction of the projected area per element and per unit solid angle.

• Symbols: L, LΩ
• SI unit: watts per square meter per steradian
• Unit symbol: W · m -2 · sr -1

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

Surfaces which have after eliminating the factor no directional dependence of radiance more, is called diffuse or Lambertian radiator radiator. A lambert cal surface element is in all directions the same radiance from:

The declaration made by him in a certain direction radiation power 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, the angle-independent now be considered as a constant radiance before the integral ( see below).

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

, The beam density is defined at every point of the space is provided on the radiation. Imagine instead a radiating surface element, where appropriate, by a fictional irradiated surface element in the room.

Photometric equivalent radiance is the luminance, which can thus serve for illustrative purposes: The luminance is a measure of the brightness with which a face is detected. If one considers a diffuse emitting surface, eg a sheet of paper from different directions, then the perceived brightness of the area while remaining constant, while the total amount of light reaching the viewer depends on the projected area, and therefore varies with the cosine of the viewing angle. The radiance of a diffuse reflector is analogous in all directions the same, the radiation power emitted in a particular direction depends but additionally from the projected beam area in that direction from.

Spectral radiance

The spectral radiance ( engl. spectral radiance ) (unit: W · m -2 · Hz -1 sr -1 ) of a body indicates which radiation power of the body at the frequency in the given by the polar angle and the azimuth angle direction per projected unit area per unit solid angle and per unit frequency interval emits.

The spectral radiance is also expressed as (unit: W · m-3 sr -1) based on the unit wavelength interval.

The spectral radiance provides the most detailed illustration of the characteristics of a radiation emitter. It specifically describes the direction dependence and the frequency ( or wavelength ) depending on the emitted radiation. From the spectral radiance of the other radiation quantities can by integration over the directions and / or frequencies derived. Integration over the relevant frequency or wavelength interval provides in particular again the beam density, which therefore, when they must be distinguished from the spectral radiance and total radiance is called.

Black body

A black body is an idealized body that completely absorbs all electromagnetic radiation impinging on it. For thermodynamic reasons, the of such a body, in turn emitted thermal radiation is a universal spectrum, and he is bound to be a Lambertian emitter. Reliable radiator achieve these ideal characteristics never complete, but can they come close. The radiation properties of a black body can be used as a good approximation to a real radiator so often.

The deviation of a real radiator of blacks ideal can be detected by an emissivity. As a real spotlight on a given wavelength can not emit more than a black body of the same temperature, the emissivity must always be less than 1. The emissivity is wavelength dependent, and if the real spotlight is not a Lambertian reflector, also be directional. The emission levels are determined by comparing the radiances or the spectral radiances of real and black radiator.

Spectral radiance of a black body radiator

For the spectral radiance of a black body radiator of the absolute temperature T is considered by Planck

In the frequency representation:

With

And in the wave -length representation:

With

Is the radiated power, the dv from the surface element dA in the frequency range between ν and ν in the φ between the azimuth angles and φ d.phi and the polar angles β and β dβ spanned solid angle D? radiated. The direction dependence of this radiation power only comes through the geometrical factor; the spectral radiance itself is independent of direction.

When converting between frequency and wavelength representation is important to note that because

Applies:

The ratio of the votes in a certain direction and looked at a particular wavelength spectral radiance of a surface element of a given radiator to the observed at the same wavelength spectral radiance of a black body of the same temperature is the directional spectral emissivity of the surface element.

Integrating the spectral radiance of a blackbody over all directions of the half- space into which irradiates the surface element, we obtain the spectral emittance of the blackbody radiator. The integral provides an additional factor. For the formula, see the article " Planck's radiation law."

Total radiance of a black body radiator

Integrating the spectral radiance over all frequencies or wavelengths, one obtains the total radiance:

The evaluation of the integral yields due to:

With

Is the radiant power that is radiated from the surface element dA at all frequencies in the β located in the direction of solid angle element d Ⓜ.

The ratio of the votes in a particular direction total radiance of a surface element of a given radiator to the total radiance of a blackbody of the same temperature is the directional total emissivity of the surface element.

Integrating the total radiance of the blackbody over all directions of the half- space into which irradiates the surface element, we obtain the emittance of the blackbody radiator. The integral provides an additional factor. For the formula, see the article " Stefan -Boltzmann law."

Application

Change the definition equation for radiance provides the radiation power which is irradiated from the surface element to the element of solid angle, which is in the described direction by the angle and:

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

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

Since in general may depend on the location on the beam surface and of the cleaned directions arises under circumstances a very complicated integral. A major simplification occurs when the beam area a Lambertian emitter ( ie the radiance independent of direction ) with constant surface properties ( ie the radiance regardless of location ) is. Then the beam density is a constant number and can be pulled out of the integral:

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

For example, the broadcast in the entire surveyed area of the beam half-space is considered, then the result for the integral of the value and the emission of a Lambertian reflector of area in the entire half-space is simple:

If the reflecting surface is a black body temperature, so can the radiance required immediately after the Planck's radiation law, calculate ( formulas see above). Is it a gray radiator, the Planck's radiance is to be reduced by the emissivity. A possible spatial and directional dependence of the emissivity as well as possible reflections can complicate integrations.

Photometric Basic Law

Broadcast

Regarding a surface element, which irradiates an area located within a distance element of the radiance so urged by the solid angle of view of, and results from the first equation in the previous section:

In this case, and the inclination angle of the face elements relative to the common connection line.

This is the photometric Basic Law. By integration of the two surfaces, in turn, results in the current flowing from one surface by surface 2 the radiation performance.

Irradiation

The irradiance is analogous to the beam density, but defined for the radiation case. It indicates that the radiation power 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 received on surface element of emitted radiation output:

Except that which occurs from the space spanned angle.

Conclusion

The after emitted and ( will not be lost if in a lying between the surfaces medium radiation power by absorption or scattering) of radiation received on must be the same performance, and from the comparison of the two equations follows from:

The light emitted from one surface of the beam density is the same as the incident irradiance on surface 2.

Note that the beam density does not decrease with distance. The total transmitted radiation power 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. Photometric equivalent radiance is the luminance which appears for surface light sources is also known, irrespective of their distance ( namely, a near billboard appears to be larger but not identical lit brighter than a more distant ).

If the irradiance integrated over the solid angle from which it is derived, then the irradiance -called single-beam power density results on the receiver surface in W/m2. If the radiance of the transmitter area is known, it is thus now also the irradiance of the receiver surface is known:

Examples

The sun is a good approximation a black body temperature of 5777 K. It appears from Earth, seen under a solid angle of 0.000068 steradian. Compute the resulting irradiance at the surface (perpendicular to the solar radiation and excluding the absorbent atmosphere).

According to the Planck's radiation law is the radiance of the sun's surface. The irradiance at the Earth's surface has the same numerical value. If the arising of the sun irradiance as across the solar disk viewed constant, this reduces the integration over the space occupied by the solar disk solid angle to a multiplication of the irradiance with the solid angle.

This results in an irradiance of 20.10 × 106 W · m -2 · sr -1 sr = 0.000068 × 1367 W · m -2, the solar constant.

A green laser pointer emitting a light beam with a power of one milliwatt. At the outlet of the annular beam has a radius of one millimeter. The radius increases along the beam axis by 0.2 mm per meter. Calculate the beam parameter product of this laser, as well as the solid angle of the beam occupies the cone and the resulting radiance of the laser.

The given increase of the radius along the steel axis by 0.2 mm per meter corresponds to a flat angle of 0.2 mrad. The beam parameter product of this laser is the product of this half- opening angle and the radius of the beam to 0.2 mm mrad ·.

From the flat angle of 0.2 mrad, the solid angle steradian calculated to 3.14 × 10-8 sr.

To determine the radiance of the area of ​​the beam is also calculated at the outlet opening to 3.14 mm2 of the given radius. For the beam density, this results in a value of 1 mW / (3.14 × 3.14 × 10-8 mm2 sr ) = 10 GW · m -2 · sr -1. The density of the laser beam is so by a factor of 500 greater than that of the sun (20 MW · m 2 · sr- 1, see above)!