1 - 2Rto cos y + R The reflectivity of the infinite (only one boundary) medium is

and the phase difference as a consequence of two parallel boundaries a finite distance apart is

4nnh

We use the term coherent reflectivity to emphasize that, like a particle, a slab is a coherent object.

The reflection cross section is defined as Wr/F0, and hence the reflection cross section per unit volume v = Ah is

For a sufficiently thin slab (relative to the wavelength), y C 1, and Eq. (3.113) is approximately

This expression resembles what we predicted in Sections 3.1 and 3.4.8 for scattering by a small particle (at wavelengths far from absorption bands), namely volumetric scattering proportional to size (volume) and to wavelength to a power (-4). A thin slab is not a dipole but can be looked upon as a sheet of dipoles, and hence the different (but similar) dependence of the volumetric reflection cross section on size and wavelength, increasing with increasing size and proportional to a power of the wavelength (-2). And note that this power is not exactly -2 because n also depends on wavelength. We note in passing that Eq. (3.114) is the source of the incorrect attribution of the blue sky (Sec. 8.1) to reflection by thin plates. Close but not close enough.

We can imagine the slab thickness to increase by piling one sheet of dipoles onto another. At first, the volumetric reflection cross section increases monotonically with each added sheet (Fig. 3.7) as long as all parts of the slab are excited in phase with all other parts. But once the total thickness reaches the point where appreciable (~n/2) phase differences among the

Slab Thickness / Wavelength

Figure 3.7: Reflection (scattering in the backward direction) cross section per unit volume of a transparent slab (with refractive index 1.33) illuminated at normal incidence as a function of slab thickness relative to the wavelength of the illumination (1 |m).

Slab Thickness / Wavelength

Figure 3.7: Reflection (scattering in the backward direction) cross section per unit volume of a transparent slab (with refractive index 1.33) illuminated at normal incidence as a function of slab thickness relative to the wavelength of the illumination (1 |m).

different parts are possible, the cross section decreases with increasing thickness, then oscillates with decreasing amplitude and decreasing distance between adjacent peaks. Moreover, the wavelength dependence of reflection ceases to resemble that for a dipole sheet. Because of interference, the sum of dipoles is not a dipole, which, in a nutshell, is what makes optics interesting.

Despite appearances to the contrary a plot of Cref/v versus h for fixed frequency is not equivalent to a plot versus 1/A (frequency) for fixed h. Frequency cannot be increased indefinitely and n remain constant. No such material exists, except a perfect vacuum, which also does not exist. Moreover, frequency cannot be varied arbitrarily without encountering non-negligible absorption (see Figure 3.8). If you pretend that absorption is negligible for more than a decade of wavelengths you probably have slipped into the world of fantasy.

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