radiometry - photometry

Radiometry basics

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 :

.

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 :

.

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 :

.

.

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 :

.

.

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. :

.

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 :

.

The etendue can also be defined as follows :

.

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 :

.

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 :

.

Light sources

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 :

.

E₀ is the irradiance for α = 0 :

.

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 :

.

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 :

.

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 :

.

The intensity is :

.

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

.

The intensity in the direction α is :

.

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 :

.

Radiometry and optical systems

Etendue conservation

Let consider a small light source of area dS_o conjugated by a perfect optical system in Gauss conditions.

The etendue in the object space is :

.

n_o is the refraction index in the object space, dΩ_o is the solid angle defined by the aperture angle i_o in the object space.

.

The etendue in the image space is :

.

n_o is the refraction index in the image space, dΩ_i is the solid angle defined by the aperture angle i_i in the image space and dS_i is the area of the source conjucgated.

.

The paraxial formulas give the following relations :

and .

m is the magnification of the optical system. Therfore :

.

The etendue is thus conserved through the optical system.

Considering the transmission T of the optical system and the flux in the object space d²F_o, the flux in the image space is :

.

Therefore, as , the radiance of the secondary source (conjugated of the initial source by the optical system) is :

.

If the initial and final media have the same refractive index, the ratio between the radiance of the conjugated source and the radiance of the initial source is equal to the transmission.

Irradiance produced by a source and an optical system

The irradiance on a screen can be anaytically calculated in some cases where light go through a perfect optical system ( in Gauss conditions).

In the case where a punctual isotropic light source ( intensity I ) is in the focal plane of a system ( focal length f_i ), the irradiance on a screen, whatever its position, is :

.

D is the aperture diameter and T is the system transmission.

In the case of a small Lambertian source ( area S and radiance L ) located in the front focal plane of the slightly opened system, the beam is slightly diverging after being transmitted trough the optical system. Therefore, the irradiance depends on the position z_i of the screen. Indeed, the flux after the system is :

.

Considering the beam area S_i on the screen : .

Therefore, the irradiance E on the screen is :

.

In the case where the screen is conjugated with a small Lambertian source ( area S_o and radiance L ) and considering that the system has a small aperture, the flux on the screen is :

.

x_o is the algebric distance of the source.

where x_i is the algebric distance of the source conjugate.

The magnification is given by :

.

If S_i is the area on the screen and m is the magnification :

.

Then, the irradiance on the screen is :

.

In the case where the light source is at infinity and the screen is located in the back focal plane of the system :

.

In this case, when defining the aperture with the aperture numbre N, the ratio between the irradiance on the screen with and without the optical system is ( the angular size α of the source is supposed to be small ) :

.

Diffusion

The diffusion is a deflection of light in various directions as a result of interaction with matter. It is caused by the granularity of the material when it is comparable to the wavelength of the incident wave.

Diffusion can be in transmission (rays diffused in the same sense as the incident beam ) or in reflexion (rays diffused in the opposite sense ).

Diffusers that are diffusing uniformly on the whole half space are called Lambertian diffusers.

The radiance of a perfect lambertian diffuser illuminated with a given irradiance E is :

.

The intensity diffused by a lambertian diffuser illuminated with an optical power F in the direction making an angle α with the normal to the diffuser is :

.

Diffusers may also scatter light inside a given cone, which means that a collimated incident beam at a given incidence angle is only diffused inside the cone. This cone can have for instance a circular basis or a rectangular basis.

Some diffusers have a gaussian shaped radiance regarding the incident light direction ( diffuser in transmission ) or regarding the specular reflexion direction ( diffuser in reflexion ). The radiance in a direction making the angle α with the incident (or specular reflexion) direction is :

.

L_max is the maximum radiance and α₀ is the angle corresponding to the directions for which the radiance is L_max / 2.

L_max can be calculated from the irradiance E on the diffuser :

.

Ω_{Half space} is the solid angle corresponding to the half space delimited by the diffuser surface.

Integrating spheres

Integrating spheres are spherical diffusers coated with a lambertian diffusing paint.

They are used to measure the total power of light sources that is normally difficult to realize with sources emitting on wide angles.

They include an aperture through which the light injected (surface area S_I). Alternatively, the light source can be put inside the integrating sphere . They contain also a detector (surface area S_d). In any case, the surface for the light injection (or the surface of the light source) and the detector surface are not diffusing.

Also, a baffle may be needed when the light source is inside the integrated sphere in order to protect the detector from direct illumination by the source.

Because the light is uniformly diffused, the irradiance is the same anywhere in the sphere. After one reflexion its value is :

.

ρ is the sphere reflectivity, S_S the sphere area, F the luminous flux and .

After N reflexions, its value is :

.

Therefore, the total irradiance on the sphere wall is uniform and its value is :

.

Finally :

.

The detected power is then :

.

According to the formula above, the injected power is proportionnal to the detected power and can be calculated from the reflectivity, the sphere area, the detector area and the eventual aperture area.

Black bodies

A black body is a theoretical object absorbing all the electromagnetic energy it receives, which creates thermal agitation. At equilibrium, ie at constant temperature, all the energy absorbed is re-emitted in electromagnetic radiation also called thermal radiation. Indeed, the black body absorption coefficient as well as its emissivity are equal to "1". The emission depends then only on the temperature. A black body is a Lambertian source radiating therefore identically in all directions.

The emittance of a black body is given by the Stefan-Boltzman law:

.

T is the temperature.

Its radiance is :

.

.

The radiance is also calculated as follows :

.

is the spectral radiance at T. It is given by the following formula :

.

The spectral radiance as a function of the wavelength is shown for different temperature on the left curves.

,

is maximum for a wavelength λ_M satisfying :

.

.

Thus, λ_M decreases with temperature.

For "real" bodies, the spectral emissivity ε_λ is not equal to unity. It is calculated as follows, and being the spectral radiances respectively of the "real" body and of the black body at the same temperature :

.

The total emissivity of the "real" body is :

.

It is smaller than 1.

Grey bodies have a spectral emissivity ε which is not dependant on the wavelength. Therefore, their spectral radiance is:

.

The wavelength for the maximum spectral radiance remains the same than for a black body at the same temperature :

.

The radiance and emittance are :

.

.

Photometric units

The human eye a a certain sensitivity to radiations.

The visibility curve V ( λ ) represents the normalized sensitivity of the human eye to radiations depending on the wavelength for daytime vision. The curve is not flat. It has a bell shape with a maximum at 555 nm. This means that for instance, a flux at 555 nm is seen brighter than the same flux at 650 nm. Therefore, a higher flux at 650 nm is needed to be seen at the same brightness than at 555 nm.

V ( λ ) is calculated in the visible range ( from ~ 400 nm to ~ 800 nm).

Whatever the considered parameter, if P_r is the value in SI radiometric and P_p, its value in SI photometric unit, for a monochromatic radiation :

.

K₅₅₅ is a constant defined with the intensity. It is the intensity in SI photometric unit ( which is the candela - abbreviation : Cd) corresponding to an intensity of 1 W/Sr at 555 nm. Therefore, K₅₅₅ is expressed in Cd .Sr / W.

K₅₅₅ = 683 Cd .Sr / W.

At any wavelength λ :

.

The different photometric units and their corrsepondance in radiometric units for monochromatic radiations are therefore :

- Intensity : Cd (Candela) ( ),

- flux or power : lumen ( ),

- irradiance : lux ( ),

- emittance : lux ( ),

- luminance or radiance (in radiometry ) : Cd /m2 ( ).

For polychromatic radiations, whatever the parameter P ( intensity, radiance or luminance, flux or power, irradiance, emittance) Pr can be expressed as follows :

.

Therefore :

.

For instance, it is possible to calculate the luminance of a balck body at a cetain temperature in Cd/m2 ( photometric unit ) , using the precedent formula.

References

"Cours de photométrie radiométrie" - Ecole Supérieure doptique - 1987 - author : F.Desvignes.

"Optique géométrique Imagerie et instruments" - 2007 - author : Bernard Balland.

"cours de radiométrie et détection optique" - author : Jean Louis Meyzonette.