Up to this point we have assumed that absorption is negligible and taken k to be identically zero. But an absolutely nonabsorbing medium does not exist. What about free space? It, too, does not exist. Even the vast reaches of interstellar space are populated sparsely with molecules and particles. If you know how to make an absolute vacuum, run, do not walk, to the Patent Office. Moreover absorption is rarely if ever absolutely negligible: it may be negligible for some purposes but not others. So the term "nonabsorbing" should be read as shorthand for "negligibly absorbing for our particular purposes." In this section we do not neglect absorption.

If we assume that w and g are independent of t and differentiate Eq. (5.47) with respect to t, then substitute Eq. (5.48) in the resulting equation, and vice versa, we obtain two second-

order differential equations of the form d2 F

where

and F is either the sum or difference of F^ and F^. As a rule, differentiating differential equations to obtain ones of higher order is jumping out of the frying pan into the fire. But here the result was an equation [Eq. (5.69)] the solutions to which are simple exponential functions A exp(±Kr), which can be verified by differentiating them twice; A is a constant of integration determined by boundary conditions. For simplicity, take the medium to be infinite, which means that for the irradiances to be finite we have to exclude solutions with positive exponent. Subject to the boundary condition that the downward irradiance at t = 0 is F0 and the requirement that the sum and differences of the two irradiances be linked by Eq. (5.47) or, equivalently, by Eq. (5.48), the solution to Eq. (5.69) for an infinite medium is

As a check on the correctness of this solution, note that R= 0 when g =1, which is what we expect on physical grounds: if scattering is entirely downward incident photons cannot contribute to the reflected (upward) irradiance.

Unlike in a nonabsorbing (infinite) medium, the radiation field in an absorbing medium is not isotropic, the upward irradiance being less than the downward irradiance, although the radiation field is approximately isotropic if Ris close to 1. Where the really striking differences occur, however, is in attenuation. According to Eq. (5.71) both irradiances are attenuated exponentially with (optical) depth. For ease of physical interpretation it will be more convenient to transform to physical depth z. If we take k, 3, and g to be independent of 2 we can write

where we call a the attenuation coefficient. Note that K is dimensionless whereas a has the dimensions of inverse length. A necessary and sufficient condition for attenuation is that k be nonzero, and hence attenuation is a consequence of absorption but with a twist: scattering amplifies this absorption. If ¡3(1 — g) = 0, a = k, as we expect. But suppose that 13(1 — g) is not zero, indeed, suppose that it is much larger than k. In this instance a can be considerably larger than k:

,M , /?(i -g) ^ . i3(i-g) „ i-g (,1A, a = KXl 1 + (5.74)

Another way of looking at this is that attenuation at physical depth z in a medium for which /3(1 — #) ^ k is the same at depth z > z in a medium for which a = k:

Multiple scattering increases photon path lengths, and the longer the path, the greater the chance of absorption.

Perhaps the most striking difference between a nonabsorbing and an absorbing medium is in the evolution of the transmitted spectrum with depth. We showed previously that the radiation field in an infinite nonabsorbing medium is isotropic and constant with depth. From Eq. (5.52) the spectrum of the radiation transmitted by a finite nonabsorbing medium of physical thickness z is (for sufficiently large optical thicknesses)

3(1 - g)z if ¡3 is independent of z, whereas the spectrum of the downward irradiance at depth z in the infinite absorbing medium is

Although the magnitude of the irradiance in Eq. (5.76) decreases with depth, the shape of the spectrum is invariant: F0 modulated by the wavelength variation of 1 /{3(1 - g)} at all depths. But in the absorbing medium the shape of the transmitted spectrum can change markedly with depth. For simplicity take F0 to be constant. It follows from Eq. (5.77) that the ratio of the irradiance at the wavelength for which attenuation is a minimum (over the range of wavelengths of interest) to the irradiance at any other wavelength is exp{(a - amin)z}, (5.78)

where amin is the minimum value of the attenuation coefficient. Because a > amin the limit of this ratio as z becomes indefinitely large is infinity. Thus with increasing depth the transmitted spectrum is dominated more and more by light at and around the wavelength of minimum attenuation (of course, at large depths the magnitude of the irradiance may be such that the light is imperceptible). This makes sense given that the light surviving to a depth z is that which has not been absorbed. The wavelength dependence of attenuation can be a consequence of the wavelength dependence of k, of 3, or of g, of any two of these quantities, or of all three [Eq. (5.74)], although in the examples we consider in following sections the wavelength dependence of a is essentially that of k.

Now let us contrast the spectra of reflected and transmitted light. In the spectrum of transmitted light one restriction on the medium is that k is not zero. With the additional restriction that k C 3(1 - g) (implying 3 > k and w close to 1), the reflectivity R [Eq. (5.72)] is approximately i^l-2,/^—^ (5.79)

Figure 5.14: Reflectivity of an absorbing, plane-parallel medium of infinite depth for various values of the asymmetry parameter g. The horizontal axis is the negative logarithm of the ratio of absorption to absorption plus scattering.

Figure 5.14: Reflectivity of an absorbing, plane-parallel medium of infinite depth for various values of the asymmetry parameter g. The horizontal axis is the negative logarithm of the ratio of absorption to absorption plus scattering.

It follows that Ro is close to 1, and hence the spectrum of the reflected light is essentially that of the incident light. This is much different from our result in the previous paragraph that the spectrum of light transmitted sufficiently deep in the medium can be markedly different from the incident spectrum. Why the difference?

To resolve this we turn to Eq. (5.51) for the reflectivity of a finite, nonabsorbing medium of physical thickness h, according to which the reflectivity is within 10% of its asymptotic value (1) when 3(1 - g)h is about 20. This in turns implies that the reflectivity of the infinite absorbing medium [with k C 3(1 —g)] is a consequence mostly of scattering within a distance h of the upper boundary, where h =

How much attenuation occurs in this layer? According to Eqs. (5.71) and (5.74) attenuation over a distance h is determined by the negative exponential of vVa -g)h = 20^

Thus if k/3(1 - g) is sufficiently small, so is the argument [Eq. (5.81)] of the exponential function, and hence attenuation is negligible. As far as reflection is concerned most of the action occurs in a layer near the surface sufficiently thin that few incident photons traversing this layer are absorbed, whereas the deeper they penetrate into the medium the fewer escape absorption.

Before turning to particular observations you can make for yourselves that will breathe life and meaning into these mathematical results we consider some general conclusions that

follow from Eq. (5.72). The relative change of Rx with respect to w (g = 1) 1 dRx 1

at w =1 (Rto = 1) is infinite. Thus the addition of a small amount of absorption (relative to scattering) to a nonabsorbing medium results in a relatively much greater decrease in its reflectivity: the purer you are, the more easily your corruption is evident for all to see. Equation (5.79) suggests that we should plot Rversus the logarithm of 1 - w (Fig. 5.14). We may interpret 1 - w as the probability that a photon is absorbed in a single interaction with a molecule or particle, whereas 1 - Ris the probability of absorption in many such interactions. From Fig. 5.14 it follows that even when the former is quite small, the latter can be many times greater. Once again we see that, as with the pile of absorbing plates (Sec. 5.1), a little bit of absorption goes a long way in a multiple-scattering medium.

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