No Scattering Medium

For an emitter made of a single crystal, such as the rare earth garnets mentioned in Section 3.5, there are few imperfections that can act as scattering centers. As a result, scattering can be neglected compared to absorption in the extinction coefficient. This approximation should be applicable in the emission bands of all rare earth selective emitters where emission and absorption from the lowest excited states of the rare earth ion dominate. However, polycrystalline materials have many grain boundaries that act as scattering centers. Therefore, outside the emission bands scattering may be important for the sintered and fibrous rare earth emitters discussed in Section 3.3.1.

image390 Подпись: (3.22)

If scattering can be neglected, then a great deal of simplification of the source function equation occurs. For no scattering (Q = 0), the source function [equation (3.15)] reduces to the blackbody intensity.

Thus, if the temperature and index of refraction are constant throughout the medium, S(k) is also a constant. In that case, the к* integrations in equation (3.21) can be completed so that the radiation flux is the following.

Подпись: qdq

image393 Подпись: рф + n2ib ( T)[E3 (к, - к) - E3 (к)] J Подпись: (3.23)

l(K, X) = 2лі ji+(0, p, A)exp

For a medium with n = 1 emitting into a vacuum (no = 1) there is no reflection at the vacuum-medium interface. As a result, i+(0) and i"(Kd) vanish if there is no radiation entering the medium at the boundaries. Therefore, at the boundaries q(Kd) and q(0) become the following.

q(Kd, X) = – q(0,X) = яц (X, T)[1 -2Ed (кл)] (3.24)

Подпись: e(X) Подпись: q (Kd> 4 (XT) Подпись: 1 - 2E3(Kd) image399

The spectral emittance for the medium is the following.

(3.25)

For a cylindrical geometry, replace Kd with kr.