Basic definitions and terminology

Definition of optical properties. Radiation incident on or passing through objects is subject to various processes changing the direction of propagation, attenuating or amplifying the radiant intensity, and changing the spectral distribution or polarization of radiation. With-out going into the details of the complex physical processes governing the interaction of radiation with the molecular structure of objects, the macroscopic optical properties of objects are quantified by the follow-ing dimensionless quantities (Fig.2.11):

Reflectivity Reflectivity or reflectance ρ˜defines the ratio of the re-flected radiative fluxΦr to the incident radiative fluxΦi,

˜ ρ=Φr

Φ_i (2.32)

Absorptivity Absorptivity or absorptance α˜ defines the ratio of the absorbed radiative fluxΦa to the incident radiative fluxΦ_i,

˜ α=Φa

Φi (2.33)

Transmissivity Transmissivity ortransmittanceτ˜defines the ratio of the radiative fluxΦttransmitting the object to the incident radiative fluxΦi,

τ˜=Φt

Φi (2.34)

2.5 Interaction of radiation with matter 33 Emissivity The forementioned quantities ˜ρ, ˜α, and ˜τ define the prop-erty ofpassive receivers in modifying incident radiative flux. The emissivity oremittance˜.quantifies the performance of anactively radiating object compared to a blackbody, which provides the upper limit of the spectral exitance of a source. It is defined by the ratio of the exitances,

˜

.= Ms(T )

Mb(T ) (2.35)

whereMs andM_b denote the exitance of the emitting source, and the exitance of the blackbody at the temperatureT, respectively. A blackbody is defined as an ideal body absorbing all radiation inci-dent on it regardless of wavelength or angle of incidence. No radia-tion is reflected from the surface or passing through the blackbody.

Such a body is a perfect absorber. Kirchhoff demonstrated in 1860 that a good absorber is a good emitter and, consequently, a perfect absorber is a perfect emitter. A blackbody, therefore, would emit the maximum possible radiative flux that any body can radiate at a given kinetic temperature, unless it contains fluorescent or radioac-tive materials. As a blackbody has the maximum possible exitance of an object at the given temperature, ˜.is always smaller than 1.

Spectral and directional dependencies. All of the foregoing intro-duced quantities can have strong variations with direction, wavelength, and polarization state that have to be specified in order to measure the optical properties of an object. The emissivity of surfaces usually only slightly decreases for angles of up to 50° and rapidly falls off for angles larger than 60°; it approaches zero for 90° [8]. The reflectivity shows the inverse behavior.

To account for these dependencies, we define the spectral direc-tionalemissivity ˜.(λ,θ,φ)as ratio of the source spectral radianceLλ,s

to the spectral radiance of a blackbodyLλ,bat the same temperatureT:

˜

.(λ,θ,φ)= Lλ,s(θ,φ,T )

Lλ,b(θ,φ,T ) (2.36) The spectralhemispherical emissivity ˜.(λ)is similarly given by the ra-diant exitance of the source and a blackbody at the same temperature, T:

˜

.(λ)= Mλ,s(T )

Mλ,b(T ) (2.37)

Correspondingly, we can define the spectral directional reflectivity, the spectral directional absorptivity, and the spectral directional trans-missivity as functions of direction and wavelength. In order to simplify

notation, the symbols are restricted to ˜ρ, ˜α, ˜τand ˜.without further in-dices. Spectral and/or directional dependencies will be indicated by the variables and are mentioned in the text.

Terminology conventions. Emission, transmission, reflection, and absorption of radiation either refer to surfaces and interfaces between objects or to the net effect of extended objects of finite thickness. In accordance with Siegel and Howell [9] and McCluney [3] we assign the suffix -ivity to surface-related (intrinsic) material properties and the suf-fix -ance to volume-related (extrinsic) object properties. To reduce the number of equations we exclusively use the symbols ˜., ˜α, ˜ρ and ˜τ for both types. If not further specified, surface- and volume-related prop-erties can be differentiated by the suffixes -ivity and -ance, respectively.

More detailed definitions can be found in theCIE International Lighting Vocabulary[10].

Spectral selectivity. For most applications the spectral optical prop-erties have to be related to the spectral sensitivity of the detector sys-tem or the spectral distribution of the radiation source. Let ˜p(λ)be any of the following material properties: ˜α, ˜ρ, ˜τ, or ˜.. Thespectral selective optical properties ˜ps can be defined by integrating the corresponding spectral optical property ˜p(λ)over the entire spectrum, weighted by a spectral window functionw(λ):

p˜s= ∞ 0

w(λ)p(λ)˜ dλ ∞

0

w(λ)dλ

(2.38)

Examples of spectral selective quantities include thephotopic luminous transmittance or reflectance for w(λ) = V(λ) (Section 2.4.1), the so-lar transmittance, reflectance, or absorptance for w(λ) = Eλ,s (so-lar irradiance), and theemittance of an object at temperatureT for w(λ)=Eλ,b(T )(blackbody irradiance). Thetotal quantities ˜p can be obtained by integrating ˜p(λ)over all wavelengths without weighting.

Kirchhoff’s law. Consider a body that is in thermodynamic equilib-rium with its surrounding environment. Conservation of energy re-quiresΦi=Φa+Φr+Φt and, therefore,

˜

α+ρ˜+τ˜=1 (2.39)

In order to maintain equilibrium, the emitted flux must equal the ab-sorbed flux at each wavelength and in each direction. Thus

˜

α(λ,θ,φ)=˜.(λ,θ,φ) (2.40)

2.5 Interaction of radiation with matter 35

Table 2.3: Basic (idealized) object and surface types Object Properties Description

Cannot be penetrated by radiation. All exi-tant radiation is either reflected or emitted.

AR coating .(λ)˜ +τ(λ)˜ =1,

˜ ρ(λ)=0

No radiation is reflected at the surface. All exitant radiation is transmitted or emitted.

Ideal

All radiation passes without attenuation.

The temperature is not accessible by IR thermography because no thermal emission takes place.

Mirror .(λ)˜ =τ(λ)˜ =0,

˜ ρ(λ)=1

All incident radiation is reflected. The tem-perature is not accessible by IR thermo-graphy because no thermal emission takes place.

Blackbody τ(λ)˜ =ρ(λ)˜ =0,

˜

.(λ)=˜.=1

All incident radiation is absorbed. It has the maximum possible exitance of all objects.

Graybody .(λ)˜ =˜. <1,

˜

ρ(λ)=1−.˜, τ(λ)˜ =0

Opaque object with wavelength independent emissivity. Same spectral radiance as a blackbody but reduced by the factor˜..

This relation is known asKirchhoff’s law [11]. It also holds for the in-tegrated quantities ˜.(λ)and ˜.. Kirchoff’s law does not hold for active optical effects shifting energy between wavelengths, such as fluores-cence, or if thermodynamic equilibrium is not reached. Kirchhoff’s law also does not apply generally for two different components of polar-ization [6,12].

Table2.3 summarizes basic idealized object and surface types in terms of the optical properties defined in this section. Real objects and surfaces can be considered a mixture of these types. Although the ideal cases usually do not exist for the entire spectrum, they can be realized for selective wavelengths. Surface coatings, such as, for example, antireflection (AR) coatings, can be technically produced with high precision for a narrow spectral region.

Figure2.12shows how radiometric measurements are influenced by the optical properties of objects. In order to measure the emitted flux Φ1(e. g., to estimate the temperature of the object), the remaining seven quantities ˜.1, ˜.2, ˜.3, ˜ρ1, ˜τ1, Φ2, andΦ3 have to be known. Only for a blackbody is the total received flux the flux emitted from the object of interest.

object 1

object 2

object 3

ε Φ~1 1

ε Φ3 3

~

ε Φ~2 2

ρ ε Φ~1 3~ 3

τ ε Φ~1 2~ 2

Figure 2.12:Radiometric measurements of object 1 are biased by the radiation of the environment emitted from objects 2 and 3.

Index of refraction. Solving the Maxwell equations for electromag-netic radiation in matter yields thecomplex index of refraction,N:

N(λ)=n(λ)+ik(λ) (2.41)

with the real partnand the imaginary partk.

The real part n constitutes the well-known index of refraction of geometric optics (Section2.5.2; Chapter3). From the complex partk other important optical properties of materials, such asreflection, and absorptioncan be derived (Sections2.5.2and2.5.3).