# 1.4 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}).

**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. Find here an brief explanation of the concept of differential quantities and integral calculus for spectral radiometric quantities.

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## Definition of solid angle

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 = 4π sr = 12.57 sr r²

*Fig. 1: The solid angle Ω quantifies a certain part of the visual field seen by an observer located at P*

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

## Radiometric Quantities

### Radiant power or radiant flux Φ_{e}

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}

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

dΦ

_{e }= I_{e}dΩ

and thus

Φ _{e}=∫ I _{e }dΩ4π

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*

### Radiance L_{e}

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). The radiant power dΦ_{e} emitted by a (differential) surface element dA in the direction of the (differential) solid angle element dΩ is given by

Equation 1:

dΦ_{e}= L_{e}× cos(ϑ) dA dΩ

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.

From the definition of radiant intensity I_{e}, it follows that the differential radiant intensity emitted by the differential area element dA in a certain direction is given by

dI

_{e}= L_{e}cos(ϑ) dA

Thus,

Equation 2:

I=_{e}∫ L_{e }× cos(ϑ) dAemitting surface

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/(m ^{2}sr)**.

### Irradiance E_{e}

The irradiance E_{e} is the amount of radiant power impinging upon a surface per unit area. In detail, the (differential) radiant power dΦ_{e} upon the (differential) surface element dA is given by

dΦ

_{e }= E_{e}dA

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}

*Fig. 4: Irradiance is defined as incident radiant power dΦ _{e} per surface area element dA*

Note that the corresponding area element dA_{normal}, which is oriented perpendicular to the incident beam, is given by

dA

_{normal }= cos(ϑ) dA

with ϑ denoting the angle between the beam and the normal of dA, we get

Equation 3:

E_{e }= E_{e,normal}× cos(ϑ)

The unit of irradiance is **W/m ^{2}**.

### Radiant exitance M_{e}

Radiant exitance M_{e} quantifies the radiant power that is emitted or reflected from a certain location on a surface per area. In detail, the (differential) radiant power dΦ_{e} emitted or reflected by the surface element dA is given by

dΦ

_{e}= M_{e }dA

Based on the definition of radiance, the (differential) radiant exitance dMe emitted or reflected by a certain location on a surface in the direction of the (differential) solid angle element dΩ is therefore given by

dM

_{e}= L_{e}cos(ϑ) dΩ

and consequently

Equation 4:

M=_{e}∫ L_{e }× cos(ϑ) dΩ2πsr

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” ).

## Spectroradiometric Quantities

### Spectral radiant power Φ_{λ}(λ), spectral radiant intensity I_{λ}(λ), spectral radiance L_{λ}(λ), spectral irradiance E_{λ}(λ) and spectral radiant exitance M_{λ}(λ)

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.

Spectral radiant power is defined as a source’s radiant power per wavelength interval as a function of wavelength. In detail, the source’s (differential) radiant power dΦ_{e} emitted in the (differential) wavelength interval between λ and λ+dλ is given by

dΦ

_{e}= Φ_{ λ }(λ) dλ

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).

Mathematically, this can be expressed by the integral

Φ _{e}=∞ ∫ Φ λ(λ) dλ 0

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. For instance, UV-A irradiance is defined as

E _{e,UV-A}=400 nm ∫ E _{λ}(λ) d315 nm

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

*Fig. 5: **Relation between spectral radiant **power Φ _{λ}(λ) and *

*radiant power*

*Φ*, visualized_{e}*at a hypothetical example.*

*Radiant power*

*emitted in the wavelength interval between*

*λ**and*

*λ+dλ**is given by the area of the shaded*

*rectangle,*

which amounts to

which amounts to

*Φ*._{λ}(λ)dλ*The*

*total amount of radiant power*

*Φ*emitted over_{e}*the whole spectrum*

*is given*

by the area under

by the area under

*the curve describing*

*Φ*,_{λ}(λ)*which is*

*mathematically expressed by an integral.*

## 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.