Radiometric Measurement of Light

Basic Radiometric Quantities

The whole discipline of optical measurement techniques can be roughly subdivided into photometry and radiometry. Whereas photometry focuses on determining optical quantities that are closely related to the sensitivity of the human eye, radiometry deals with the measurement of energy per time (= power, given in watts) emitted by light sources or impinging on a particular surface. Thus, the units of all radiometric quantities are based on watts (W). According to CIE regulations, symbols for radiometric quantities are denoted with the subscript “e” for “energy”. Similarly, radiometric quantities given as a function of wavelength are labelled with the prefix “spectral” and the subscript “λ” (for example spectral radiant power Φ_λ).

The optical radiation wavelength range

Remark: The definitions of radiometric quantities cannot be understood without a basic comprehension of differential quantities. For an intuitive understanding of these quantities, which is the main objective of this paragraph, the differential quantities dλ, dA and dΩ can be regarded as tiny intervals or elements (Δλ, ΔA and ΔΩ) of the respective quantity. As a consequence of the fact that these intervals or elements are very small, radiometric quantities can be considered constant over the range defined by dλ, dA and / or dΩ. Similarly, dΦ_e, dI_e, dL_e and dE_e can be regarded as small portions which add up to the total value of the respective quantity.

Definitions

The geometric quantity of a solid angle Ω quantifies a part of an observer’s visual field. If we imagine an observer located at point P, his full visual field can be described by a sphere of an arbitrary radius r (see fig. 1). Here, a certain part of this full visual field defines an area A on the sphere’s surface and the solid angle Ω is defined as:

Ω = A / r²

Since the area A is proportional to r², this fraction is independent of the actual choice of r.

If we want to calculate the solid angle determined by a cone (as shown in fig. 1) area A is the area of a spherical calotte. However, area A can have any arbitrary shape on the sphere’s surface because the solid angle is only defined for conical parts of the full visual field.

Although Ω is dimensionless, it is common to use the unit steradian (sr). The observer’s total visual field is described by the whole surface of the sphere, which is given by 4πr², and thus covers the solid angle:

Ω_total = 4π r² / r²  = 4π sr = 12.57 sr

Source (valid as of 2002): http://whatis.techtarget.com/definition/0,,sid9_gci528813,00.html

Radiant power Φ_e is defined as the total power of radiation emitted by a source (lamp, light emitting diode, etc.), transmitted through a surface, or impinging upon a surface. Radiant power is measured in watts (W). The definitions of all other radiometric quantities are based on radiant power. If a light source emits uniformly in all directions, it is called an isotropic light source.

Radiant power characterizes the output of a source of electromagnetic radiation only by a single number and does not contain any information on the spectral distribution or the directional distribution of the lamp output.

Fig. 2: The radiant power (Φ_e) of a light source is given by its total emitted radiation

Radiant intensity I_e describes the radiant power of a source emitted in a certain direction. The source’s (differential) radiant power dΦ_e emitted in the direction of the (differential) solid angle element dΩ is given by

Φ_e =_4π ∫ I_e dΩ

dΦ_e = I_e dΩ

and thus

In general, radiant intensity depends on spatial direction. The unit of radiant intensity is W / sr.

Fig. 3: Typical directional distribution of radiant intensity for an incandescent bulb

The radiance L_e is the intensity of optical radiation emitted or reflected from a certain location on an emitting or reflecting surface in a particular direction (the CIE definition of radiance is more general. This tutorial focuses on the most relevant radiance application describing the spatial emission characteristics of a source).

In this relation, ϑ is the angle between the direction of the solid angle element dΩ and the normal of the emitting or reflecting surface element dA.

whereby ϑ is the angle between the emitting surface element dA and the direction for which I_e is calculated.

The unit of radiance is W/(m²sr).

Generally, the surface element can be oriented at any angle towards the direction of the beam. However, irradiance is maximized when the surface element is perpendicular to the beam:

dΦ_e = E_e,normal dA_normal

The unit of irradiance is W/m².

The integration is performed over the solid angle of 2π steradian corresponding to the directions on one side of the surface and ϑ denotes the angle between the respective direction and the surface’s normal.

The unit of radiant exitance is W/m². In some particular cases, M_e = E_e (see  “Reflectance ρ, Transmittance τ and Absorptance α” ).

The radiometric quantities discussed above are defined without any regard to the wavelength(s) of the quantified optical radiation. In order to not only quantify the absolute amount of these quantities but also the contribution of light from different wavelengths, it is important to also define the respective spectral quantities.

This equation can be visualized geometrically (see Fig. 5). Because dλ is infinitesimally small, spectral radiant power Φ_λ(λ) is approximately constant in the interval between λ and λ+dλ. Thus, the product Φ_λ(λ)dλ equals the area under the graph of Φ_λ(λ) in the interval between λ and λ+dλ. This area describes the contribution of this very wavelength interval to the total value of radiant power Φ_e, which is graphically represented by the total area under the graph of spectral radiant power Φ_λ(λ).

The unit of spectral radiant power is W/nm or W/Å.

The other spectral quantities are defined correspondingly and their units are given by the unit of the respective quantity, divided by nm or Å. Generally, a radiant quantity can be calculated from the respective spectral quantity by integrating over the wavelength from λ = 0 to λ = ∞. However, this integration is often restricted to a certain wavelength range, which is indicated by the respective prefix.

since the UVA range is between λ = 315 nm and λ = 400 nm.

Radiant power emitted in the wavelength interval between λ  and λ+dλ  is given by the area of the shaded rectangle, which amounts to Φ_λ(λ)dλ.  The total amount of radiant power Φ_e  emitted over  the whole spectrum is given by the area under the curve describing Φ_λ(λ), which is mathematically expressed by an integral.

Fig. 5: Relation between spectral radiant power Φ_λ(λ) and radiant power Φ_e, visualized  at a hypothetical example. 

Integral measuring Devices

Measuring instruments that directly provide the integrated, i.e. not spectrally resolved, radiometric unit are often called integral measuring devices. These in turn are specially optimized for the various applications in their spectral sensitivity as well as design to ideally meet the applications.

Spectral measuring Devices

Measuring instruments which output the optical radiation spectrally resolved and in radiometric units are called spectral measuring devices or spectroradiometers. These Spectroradiometers are also optimally adapted to the application in their design and specifications.