## The inherent optical properties

There are only two things that can happen to photons within water: they can be absorbed or they can be scattered. Thus if we are to understand what happens to solar radiation as it passes into any given water body, we need some measure of the extent to which that water absorbs and scatters light. The absorption and scattering properties of the aquatic medium for light of any given wavelength are specified in terms of the absorption coefficient, the scattering coefficient and the volume scattering function. These have been referred to by Preisendorfer (1961) as inherent optical properties (IOP), because their magnitudes depend only on the substances comprising the aquatic medium and not on the geometric structure of the light fields that may pervade it. They are defined with the help of an imaginary, infinitesimally thin, plane parallel layer of medium, illuminated at right angles by a parallel beam of monochromatic light (Fig. 1.4). Some of the incident light is absorbed by the thin layer. Some is

Fig. 1.4 Interaction of a beam of light with a thin layer of aquatic medium. Of the light that is not absorbed, most is transmitted without deviation from its original path: some light is scattered, mainly in a forward direction.

Fig. 1.4 Interaction of a beam of light with a thin layer of aquatic medium. Of the light that is not absorbed, most is transmitted without deviation from its original path: some light is scattered, mainly in a forward direction.

scattered - that is, caused to diverge from its original path. The fraction of the incident flux that is absorbed, divided by the thickness of the layer, is the absorption coefficient, a. The fraction of the incident flux that is scattered, divided by the thickness of the layer, is the scattering coefficient, b.

To express the definitions quantitatively we make use of the quantities absorptance, A, and scatterance, B. If F0 is the radiant flux (energy or quanta per unit time) incident in the form of a parallel beam on some physical system, Fa is the radiant flux absorbed by the system, and Fb is the radiant flux scattered by the system. Then i.e. absorptance and scatterance are the fractions of the radiant flux lost from the incident beam, by absorption and scattering, respectively. The sum of absorptance and scatterance is referred to as attenuance, C: it is the fraction of the radiant flux lost from the incident beam by absorption and scattering combined. In the case of the infinitesimally thin layer, thickness Dr, we represent the very small fractions of the incident flux that are lost by absorption and scattering as DA and DB, respectively. Then

An additional inherent optical property that we may now define is the beam attenuation coefficient, c. It is given by c — a + b (1.30)

and is the fraction of the incident flux that is absorbed and scattered, divided by the thickness of the layer. If the very small fraction of the incident flux that is lost by absorption and scattering combined is given the symbol DC (where DC — DA + DB) then c — DC=Dr (1.31)

The absorption, scattering and beam attenuation coefficients all have units of 1/length, and are normally expressed in m-1.

In the real world we cannot carry out measurements on infinitesimally thin layers, and so if we are to determine the values of a, b and c we need expressions that relate these coefficients to the absorptance, scatterance and beam attenuance of layers of finite thickness. Consider a medium illuminated perpendicularly with a thin parallel beam of radiant flux, F0. As the beam passes through, it loses intensity by absorption and scattering. Consider now an infinitesimally thin layer, thickness Dr, within the medium at a depth, r, where the radiant flux in the beam has diminished to F. The change in radiant flux in passing through Dr is DF. The attenuance of the thin layer is

(the negative sign is necessary since DF must be negative)

DF|F — —cDr Integrating between 0 and r we obtain ln — —-cr (1.32) $0

indicating that the radiant flux diminishes exponentially with distance along the path of the beam. Equation 1.32 may be rewritten c = ^ln — (1.34)

The value of the beam attenuation coefficient, c, can therefore, using eqn 1.34 or 1.35, be obtained from measurements of the diminution in intensity of a parallel beam passing through a known pathlength of medium, r.

The theoretical basis for the measurement of the absorption and scattering coefficients is less simple. In a medium with absorption but negligible scattering, the relation a =-^ln(1 - A) (1.36)

r holds, and in a medium with scattering but negligible absorption, the relation b =-^ln(1 - B) (1.37)

r holds, but in any medium that both absorbs and scatters light to a significant extent, neither relation is true. This can readily be seen by considering the application of these equations to such a medium.

In the case of eqn 1.37 some of the measuring beam will be removed by absorption within the pathlength r before it has had the opportunity to be scattered, and so the amount of light scattered, B, will be lower than that required to satisfy the equation. Similarly, A will have a value lower than that required to satisfy eqn 1.36 since some of the light will be removed from the measuring beam by scattering before it has had the chance to be absorbed.

In order to actually measure a or b these problems must be circumvented. In the case of the absorption coefficient, it is possible to arrange that most of the light scattered from the measuring beam still passes through approximately the same pathlength of medium and is collected by the detection system. Thus the contribution of scattering to total attenuation is made very small and eqn 1.36 may be used. In the case of the scattering coefficient there is no instrumental way of avoiding the losses due to absorption and so the absorption must be determined separately and appropriate corrections made to the scattering data. We shall consider ways of measuring a and b in more detail later (§§ 3.2 and 4.2).

The way in which scattering affects the penetration of light into the medium depends not only on the value of the scattering coefficient but also on the angular distribution of the scattered flux resulting from the primary scattering process. This angular distribution has a characteristic shape for any given medium and is specified in terms of the volume scattering function, b(0). This is defined as the radiant intensity in a given direction from a volume element, dV illuminated by a parallel beam of light, per unit of irradiance on the cross-section of the volume, and per unit volume (Fig. 1.5a). The definition is usually expressed mathematically in the form

Since, from the definitions in §1.3

where dF(0) is the radiant flux in the element of solid angle do, oriented at angle 0 to the beam, and F0 is the flux incident on the cross-sectional area, dS, and since dV = dS.dr where dr is the thickness of the volume element, then we may write

The volume scattering function has the units m_1 sr_1.

Light scattering from a parallel light beam passing through a thin layer of medium is radially symmetrical around the direction of the beam. Thus, the light scattered at angle 0 should be thought of as a cone with half-angle 0, rather than as a pencil of light (Fig. 1.5b).

From eqn 1.39 we see that b(0) is the radiant flux per unit solid angle scattered in the direction 0, per unit pathlength in the medium, expressed as a proportion of the incident flux. The angular interval 0 to 0 +D0 corresponds to an element of solid angle equal to 2p sin 0 D0 (Fig. 1.5b) and so the proportion of the incident radiant flux scattered (per unit pathlength) in this angular interval is b(0) 2p sin 0 D0. To obtain the proportion of the incident flux that is scattered in dV,

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