We began the previous section by asserting that we can ignore the consequences of interference (phase differences) in determining reflection and transmission by a pile of plates if they are separated by distances large compared with the wavelength and if the reflectivities and transmissivities are averages over bands of wavelengths. Because this is such an important yet often misunderstood point we digress briefly to support our assertion.

Consider a single transparent (negligible absorption) plate of uniform thickness h and refractive index n illuminated at normal incidence by light of wavelength A. If we ignore interference, the reflectivity of this plate is t- ,5-23'

where Ris the reflectivity of a single interface of the plate, also the reflectivity of an infinitely thick slab of the same material. The similarity of this equation to Eq. (5.9) is not an accident. A single plate has two identical interfaces, so is formally equivalent to two identical plates. Both equations were derived under the same assumption, namely that we can add irradiances of different beams of radiation without having to concern ourselves with phase differences. Note that the thickness of the plate is neither explicit nor implicit in Eq. (5.23), and wavelength is only implicit through the possible dependence of Ron A.

Now consider the reflectivity of this same plate but taking into account interference (the wave nature of light):

where $ = 4nnh/A. This equation can be derived by adding waves, taking account of phase differences, multiply reflected because of the two interfaces of the plate (see Prob. 7.55). We find the total reflected electromagnetic wave, which has amplitude and phase, then square it to obtain reflected irradiance. We call Eq. (5.24) the coherent reflectivity to emphasize that it was derived taking interference into account and to distinguish it from the incoherent reflectivity [Eq. (5.23)]. These two equations are quite different. For example, the minimum value of R is zero, its maximum value is 4RTO/(1 + RTO)2. Both equations apply to the same plate. Which is correct? Well, they both are correct, although in a strict sense Eq. (5.24) is more correct. To make sense out of this seemingly paradoxical statement consider the average of Eq. (5.24):

This integral can be looked upon as either an average over one period (2n), any integral number of periods, or any non-integral number appreciably greater than 1. With a fair amount of labor, using tables of integrals or attacking Eq. (5.25) with hammer and tongs, we obtain

Note that because $ = 4nnhv/c, where c is the free-space speed of light and v is the frequency, an average over $ is essentially an average over a band of frequencies (if the band is sufficiently narrow that n does not vary appreciably). For a narrow range of frequencies Av the corresponding range of phases $ is nh Av

A v which is many multiples of 2n only if nh/A ^ 1. If this condition is satisfied, the average of R over this frequency interval is R to good approximation. But if the plate is thinner than or comparable with the wavelength, the average of R over the frequency interval is not likely to be R except by accident (see Fig. 5.9).

We could have obtained Eq. (5.23) by first deriving Eq. (5.24). This would have required us to obtain R from electromagnetic theory (a theory of waves with amplitudes and phases), which is not especially difficult but entails considerably more labor than deriving R. Then we would have had to evaluate the integral Eq. (5.25), which is difficult (but not impossible). This laborious procedure would have been like shooting an ant with a machine gun if we could have gone directly to Eq. (5.23) from the outset because the plate of interest is many wavelengths thick. We always are allowed to do more work than necessary, but life is short so we look for shortcuts. And keep in mind this was the easiest problem we could come up with to illustrate our point. A more complicated problem might not have been soluble exactly (i.e., within the framework of electromagnetic theory) or its solution might have been extremely tedious.

We hope that now you have a better understanding of why we could ignore interference when deriving the reflectivity of a pile of plates subject to the requirement that the separation between them is large compared with the wavelength. Note, however, that the plates themselves need not be thick compared with the wavelength. We could use Eq. (5.24) for the individual plates, then find the reflectivity of a pile of them using Eq. (5.12) if the separation between plates is large compared with the wavelength and if by reflectivity we mean a simple average over a range of frequencies.

Figure 5.9: Coherent reflectivity of a single, uniformly thick, nonabsorbing plate versus frequency (solid line). The top curve is for a plate with thickness times refractive index about 25 times the wavelength of the incident illumination; the bottom curve is for a plate with thickness times refractive index of about one wavelength. The horizontal dashed lines are the incoherent reflectivity obtained by ignoring interference. Vertical dotted lines show the range of frequencies over which the coherent reflectivity is averaged. For the plate much thicker than the wavelength, this average is essentially the incoherent reflectivity, whereas this is not true for the plate with thickness comparable with the wavelength.

Figure 5.9: Coherent reflectivity of a single, uniformly thick, nonabsorbing plate versus frequency (solid line). The top curve is for a plate with thickness times refractive index about 25 times the wavelength of the incident illumination; the bottom curve is for a plate with thickness times refractive index of about one wavelength. The horizontal dashed lines are the incoherent reflectivity obtained by ignoring interference. Vertical dotted lines show the range of frequencies over which the coherent reflectivity is averaged. For the plate much thicker than the wavelength, this average is essentially the incoherent reflectivity, whereas this is not true for the plate with thickness comparable with the wavelength.

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