Bulk-related properties of objects

ρ.

2.5.3 Bulk-related properties of objects

This section deals with the various processes influencing the propa-gation of radiation within optical materials. The basic processes are attenuation by absorption or scattering, changes in polarization, and frequency shifts. For active emitters, radiation emitted from partially transparent sources can originate from subsurface volumes, which changes the radiance compared to plain surface emission. The most important processes for practical applications are attenuation of radi-ation by absorption or scattering and luminescence. A more detailed treatment of bulk-related properties can be found in CVA1 [Chapter 3].

Attenuation of radiation. Only a few optical materials have a trans-missivity of unity, which allows radiation to penetrate without atten-uation. The best example is ideal crystals with homogeneous regular grid structure. Most materials are either opaque or attenuate transmit-ted radiation to a certain degree. Letzbe the direction of propagation along the optical path. Consider the medium being made up from a number of infinitesimal layers of thickness dz(Fig.2.17). The fraction of radiance dLλ=Lλ(z)−Lλ(z+dz)removed within the layer will be

2.5 Interaction of radiation with matter 41

Figure 2.17: Depth dependence of the volumetric absorption and emission of radiation.

proportional to both the thickness dzand the radianceLλ(z)incident on the layer atz:

dLλ(z)= −κ(λ,z)Lλ(z)dz (2.49) with theextinction coefficientorattenuation coefficientκof the material (in environmental sciences, κ is sometimes referred to as turbidity).

The unit ofκis a reciprocal length, such as m⁻¹. Solving Eq. (2.49) for Land integrating overzyields:

L_λ(z)=L_λ(0)exp

If the medium shows homogeneous attenuation, that is,κ(λ,z)=κ(λ), Eq. (2.50) reduces to

Lλ(z)=Lλ(0)exp(−κ(λ)z) (2.51) which is known asLambert Beer’s orBouguer’s law of attenuation. It has to be pointed out that Bouguer’s law holds only for first-order (lin-ear) processes, Eq. (2.49), where dLis proportional toL. This is true for a wide range of practical applications, but breaks down for very high intensities, such as laser radiation, or if multiscatter processes play a dominant role.

So far there has not been a discussion as to which processes are responsible for attenuation of radiation. The two basic processes are absorption and scattering. Separating the total amount dL of radia-tion that is lost into the parts dLa (absorption) and dLs (scattering), dL= dLa+dLs, the attenuation coefficientκsplits into theabsorption coefficient αand thescattering coefficient β:

κ= −1

Both coefficients have the dimension of a reciprocal length (m⁻¹) and are intrinsic material properties.

In order to separate the effect of absorption and scattering on at-tenuation, both the transmitted as well as the scattered radiation in all directions has to be measured. For the transmitted beam, only the net effect of both processes can be measured if no further knowledge on the material properties is available.

Thetransmittance²of a layer of thicknesszcan be computed from Eq. (2.51) as

τ(λ)˜ =Lλ(z)

Lλ(0)⁼exp(−κ(λ)z) (2.53) Therefore, a layer of thicknessκ⁻¹(λ)has a transmittance of e⁻¹. This distance is calledpenetration depthof the radiation at the specific wave-length. A variety of materials do not exhibit scattering. In these cases κ=α.

Another frequently used term (mainly in spectroscopy) is theoptical depthτ(z1, z2)of a medium. It is defined as integral over the attenu-ation coefficient:

τ(z1, z2)=

z2

z1

κ(z)dz (2.54)

Taking the logarithm of the radiance, Lambert Beer’s law (see Eq. (2.50)) reduces to a sum over the optical depths of allM layers of material:

lnL_λ(z)−lnL_λ(0)= M m=0

τ(zm, zm+1) (2.55) Again, for nonscattering mediaκhas to be replaced byα.

Absorption. The absorption coefficient α of a material can be com-puted from the imaginary part k of the complex index of refraction (Eq. (2.41)):

α(λ)=4πk(λ)

λ (2.56)

Tabulated values of absorption coefficients for a variety of optical ma-terials can be found in [6,15,16,17].

The absorption coefficient of a medium is the basis for quantita-tive spectroscopy. With an imaging spectrometer, the distribution of

2As mentioned in Section2.5.1, thetransmittanceof a layer of finite thickness must not be confused with thetransmissivityof an interface.

2.5 Interaction of radiation with matter 43

dz

L z( )

θ

L z( +d )z

Figure 2.18: Single and multiple scatter of radiation in materials with local inhomogeneities.

a substance can be quantitatively measured, provided there is appro-priate illumination (SectionA23). The measured spectral absorption coefficient of a substance depends on the amount of material along the optical path and, therefore, is proportional to the concentration of the substance:

α=.c (2.57)

wherecis the concentration in units mol l⁻¹ and.denotes the molar absorption coefficient with unit l mol⁻¹m⁻¹).

Scattering. Scatter of radiation is caused by variations of the refrac-tive index as light passes through a material [16]. Causes include for-eign particles or voids, gradual changes of composition, second phases at grain boundaries, and strains in the material. If radiation traverses a perfectly homogeneous medium, it is not scattered. Although any material medium has inhomogeneities as it consists of molecules, each of which can act as a scattering center, whether the scattering will be effective depends on the size and arrangement of these molecules. In a perfect crystal at zero temperature the molecules are arranged in a very regular way and the waves scattered by each molecule interfere in such a way as to cause no scattering at all but just a change in the velocity of propagation, given by the index of refraction (Section2.5.2).

The net effect of scattering on incident radiation can be described in analogy to absorption Eq. (2.49) with thescattering coefficient β(λ,z) defining the proportionality between incident radianceLλ(z)and the amount dLλ removed by scattering along the layer of thickness dz (Fig.2.18).

The basic assumption for applying Eq. (2.49) to scattering is that the effect of a volume containingMscattering particles isMtimes that scat-tered by a single particle. This simple proportionality to the number of particles holds only if the radiation to which each particle is exposed is essentially radiation of the initial beam. For high particle densities

θ

dS dz

L_i L_t

Ls( )θ dΩ

Figure 2.19:Geometry for the definition of the volume scattering functionfV SF.

and, correspondingly, high scattering coefficients, multiple scattering occurs (Fig.2.18) and the simple proportionality does not exist. In this case the theory becomes very complex. A means of testing the propor-tionality is to measure the optical depthτ(Eq. (2.54)) of the sample. As a rule of thumb, single scattering prevails forτ <0.1. For 0.1< τ <0.3 a correction for double scatter may become necessary. For values of τ > 0.3 the full complexity of multiple scattering becomes a factor [18]. Examples of multiple scatter media are white clouds. Although each droplet may be considered an independent scatterer, no direct solar radiation can penetrate the cloud. All droplets only diffuse light that has been scattered by other drops.

So far only the net attenuation of the transmitted beam due to scat-tering has been considered. A quantity accounting for the angular dis-tribution of scattered radiation is thespectral volume scattering func-tion,fV SF:

fV SF(θ)= d²Φs(θ) E_idΩdV ⁼

d²Ls(θ)

L_idΩdz (2.58)

where dV= dSdzdefines a volume element with a cross section of dS and an extension of dzalong the optical path (Fig.2.19). The indices iandsdenote incident and scattered quantities, respectively. The vol-ume scattering function considers scatter to depend only on the angle θ with axial symmetry and defines the fraction of incident radiance being scattered into a ring-shaped element of solid angle (Fig.2.19).

From the volume scattering function, the total scattering coefficient βcan be obtained by integratingfV SF over a full spherical solid angle:

β(λ)=

2π

0

π 0

fV SF(λ,θ)dθ dΦ=2π π 0

sinθfV SF(λ,θ)dθ (2.59)

2.5 Interaction of radiation with matter 45 Calculations offV SF require explicit solutions of Maxwell’s equations in matter. A detailed theoretical derivation of scattering is given in [18].

Luminescence. Luminescencedescribes the emission of radiation from materials by radiative transition between an excited state and a lower state. In a complex molecule, a variety of possible transitions between states exist and not all are optically active. Some have longer lifetimes than others, leading to a delayed energy transfer. Two main cases of luminescence are classified by the time constant of the process:

1. Fluorescence, by definition, constitutes the emission of electromag-netic radiation, especially of visible light, stimulated in a substance by the absorption of incident radiation and persisting only as long as the stimulating radiation is continued. It has short lifetimes, that is, the radiative emission occurs within 1–200 ns after the excitation.

2. Phosphorescencedefines a delayed luminescence, occurring millisec-onds to minutes after the excitation. Prominent examples of such materials are watch displays or light switches that glow in the dark.

The intensity decreases as the time from the last exposure to light increases.

There are a variety of physical and chemical processes leading to a transition between molecular states. A further classification of lumi-nescence accounts for the processes that lead to excitation:

• Photoluminescence: Excitation by absorption of radiation (photons);

• Electroluminescence: Excitation by electric current (in solids and so-lutions) or electrical discharge (in gases);

• Thermoluminescence: Thermal stimulation of the emission of al-ready excited states;

• Radioluminescence: Excitation by absorption of ionizing radiation or particle radiation;

• Chemoluminescence: Excitation by chemical reactions; and

• Bioluminescence: Chemoluminescence in living organisms; promi-nent examples include fireflies and marine organisms.

For practical usage in computer vision applications, we have to con-sider how luminescence can be used to visualize the processes or ob-jects of interest. It is important to note that fluorescent intensity de-pends on both the concentration of the fluorescent material as well as on the mechanism that leads to excitation. Thus, fluorescence allows us to visualizeconcentrationsandprocesses quantitatively.

The most straightforward application can be found in biology. Many biological processes are subject to low-level bioluminescence. Using appropriate cameras, such as amplified intensity cameras (Section4), these processes can be directly visualized (ChapterA25, [CVA1,

Chap-ter 12]). An application example is the imaging ofCa²⁺ concentration in muscle fibers, as will be outlined in CVA3 [Chapter 34].

Other biochemical applications make use of fluorescent markers.

They use different types of fluorescent dyes to mark individual parts of chromosomes or gene sequences. The resulting image data are mul-tispectral confocal microscopic images (SectionA26, [CVA2, Chapter 41]) encoding different territories within the chromosomes).

Fluorescent dyes can also be used as tracers in fluid dynamics to visualize flow patterns. In combination with appropriate chemical trac-ers, the fluorescence intensity can be changed according to the relative concentration of the tracer. Some types of molecules, such as oxygen, are very efficient in deactivating excited states during collision with-out radiative transfer—a process referred to asfluorescence quench-ing. Thus, fluorescence is reduced proportional to the concentration of the quenching molecules. In addition to the flow field, a quantitative analysis of the fluorescence intensity within such images allows direct measurement of trace gas concentrations (SectionA18).