Tutorial

The radiometry is the field of photonics describing and quantifying the energy of light radiations. For instance, it may characterize the amount of light emitted by a source, received by a surface, transmitted through a medium or through an optical system....

When applied to the human vision, the radiometry is called photometry. The light energy is then considered as perceived by the human eye and is therefore expressed with dedicated units (photometric units - see "photometry section").

For example, 1 watt in the green-yellow at 555 nm ( maximum diurnal eye sensitivity ) is seen brighter than 1 Watt in the red at 650 nm. It is then useful to take into account this characteristic. The photometry is the part of photonic science describing and quantifying light energy as seen by the human eye.

The photometry uses its own units like the "lumen" which is a power unit depending on wavelength. The curve giving the ratio Lumen/Watt vs wavelength is homotetic to the diurnal sensitivity of the human eye. When normalized, this curve represents the luminous efficiency of the human eye at each wavelength for the diurnal vision. Lumen is said to be a "photometric" power unit. Derived parameters like intensity, radiance ( or luminance ), irradiance ( or illuminance), emittance can also be expressed in photometric units. Usually, the words luminance and illuminance are used in photometry and are expressed with photometric units while the words radiance and irradiance are used in radiometry and are expressed in radiometric units.

Photometric units are detailed in the "photometric units" section. Up to this section, only radiometry is discussed and therefore radiometric units are considered.

Basic parameters

Among all the parameters used in radiometry, the most important is the "solid angle" representing an angle in three dimensions. Considering a cone C formed by a surface B viewed from a point A, the solid angle defined by C is the area of the surface defined as the intersection of C by the sphere centered on A and wich radius is 1 meter.

The element of solid angle corresponding to the cone formed by the surface element dS viewed from the point A is :

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d is the distance from A to the surface and θ is the angle between the normal to the surface and the direction of incidence.

The solid angle corresponding a cone with a circular basis with a half angle α is :

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Therefore, the solid angle corresponding to the full space and half space are respectively 4 π and 2 π.


There are a some very useful parameters currently used in radiometry.

The flux F represents an optical power. It can be emitted by a light source, transmitted by an optical system, detected by a photodetector, etc. It is expressed in Watts.

The intensity I of a light source represents the flux emitted in a given direction. It is the derivate of the flux by solid angle :

. It is expressed in W/sr.

The radiance L of a light source is the intensity emitted per surface area. It is therefore the derivate of the intensity by surface area :

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It is a local parameter expressed in W/sr/m².

The emittance M of a light source is the total flux emitted per surface area. It is the derivative of the emitted flux by surface area of the source :

.

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It is a local parameter expressed in W/m².

The illuminance or irradiance E is the flux received by surface area. It is the derivative of the received flux by surface area of the receiving surface. :

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It is a local parameter expressed in W/m².

The etendue is a geometric parameter that characterizes the flux emitted by a source and received by a receiver. Let consider two surface elements dS and dS' which normals make respectively an angle teta and teta' with the segment connecting them, the etendue is the value d2G defined by the following relation :

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The etendue can also be defined as follows :

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Let consider dS as a source, the etendue characterizes a light brush emitted by dS and received by dS'. The flux emitted by dS and received by dS' is :

.

L is the radiance of the source.

Reciprocally, if dS' ( radiance L') is as a source, the etendue characterizes a light brush emitted by dS' and received by dS. The flux emitted by dS' and received by dS is :

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For a system made of a source S emitting through an aperture of diameter D, the etendue of the system is the sum of all the etendue elements defined by a source element dS and a solid angle element dΩ :

.

Isotropic sources

An isotropic source is a source which intensity I is constant whatever the direction. Therefore, its total flux is :

.

Let consider a punctual isotropic source, the flux received by a small surface area dS at a distance d is :

. α is the angle between the direction of incidence and the normal to the surface.

The irradiance on the surface is : .

It varies in (Bouguer law).

The flux emitted by a punctual isotropic source in a solid angle limited by a circle is :

.

Let consider a screen at a distance d from the source. The irradiance at a location where the direction of incidence makes an angle α with the normal is :

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E₀ is the irradiance for α = 0 :

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Lambertian sources

Lambertian sources have a constant radiance whatever the location on the source and whatever the considered direction of observation.

The flux emitted by a source element dS in a given direction α and a given solid angle element dΩ is :

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Therefore, the flux emitted by the full source S in a given aperture is Ω :

.

The flux emitted by a surface element in a cone making an angle α with the normal to the surface is :

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The total flux (in the half space correponding to a 2 π solid angle) emitted by the surface element is therefore :

.

Let consider a screen at a distance d from the source. The irradiance at a location where the direction of incidence makes an angle α with the normal is :

.

E₀ is the irradiance for α = 0 :

.


The intensity and flux of a lambertian source can be analytically calculated in some particular cases.

For a flat source which surface area is S, the total flux is :

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The intensity is :

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For a cylindrical source which length and radius are respectively r and d, the total flux is :

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The intensity in the direction α is :

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For a spherical source which radius is r;, the intensity is constant and its value is :

.

The total flux is :

.

Let consider another case where the source is circular, Lambertian and located at infinity. Its size is defined by its angular radius α which is considered small. The irradiance on a surface with an inclination i is then :

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