The effective dielectric function out of thermal equilibrium

4.2.1. The effective dielectric function in the LTE approximation

From Eq. (4.17) we see that the effects of the nonlinear susceptibility can be taken into account with an effective electromagnetic potential or dielectric function, just like in Chapter 3. Substituting the field correlator from Eq. (4.17) into Eq. (2.127), we can

obtain a closed expression for the effective dielectric function,

Notice that the equilibrium expression from Eq. (3.17) is recovered if all temperatures Tmare equal sincePN

m=0Im GV_mG†

=ImG˜.

As discussed in Section3.2, Eq. (4.21) can be written equivalently as an integral over positive frequencies as well,

4.2.2. The non-equilibrium part to the effective dielectric function

In order to distinguish and highlight the non-equilibrium properties ofV˜^LTEandε˜^LTE, it is useful to look at the difference of these quantities to their equilibrium counterparts, V˜^eqandε˜^eq. We will denote this difference with the superscript “∆T” to signify that it is the contribution due to temperature differences,

N^∆T r,r^′;ω

The last expression can be evaluated directly using Eqs. (4.21) and (3.17). Practically, however, it is easier to use the expression forε˜^LTEand set the temperatures equal. It is important, however, to choose a particular temperature forε˜^eq. Our choice is to consider

the “reference temperature” to be the temperature of the object that the position vector One can see now directly thatN^∆T becomes zero in objectnif the temperature of all other objects is the same.

We can also see from Eq. (4.27) that the effective dielectric function of a particular object starts to directly depend on the temperatures of other objects. While local temperature-dependence of epsilon is well known (for example, ordinary thermal expansion leads to changes in the dielectric function), the dependence on the temperature at other lo-cations and even other objects is a novel prediction.

Furthermore, Eq. (4.27) is neat since it shows that really only objects which have differ-ent temperatures contribute, while the terms for objectsnandmwithTn=Tmwill be zero. Most importantly, that includes the contribution from the same bodym=n. That means in all non-zero terms of Eq. (4.27) the two Green’s functions will only connect points in different objects, therefore avoiding the divergenceImG(r,r) as discussed in the previous chapter in Section3.2. The divergence is still there, inε˜^eq, but we can proceed to study the physically relevant partN^∆T without problems, exactly like the distance dependent part of the effective potential in Section3.2.3.

4.2.3. One object in vacuum – passive gain media

A special case to consider is a single object at temperatureT_objin vacuum at temperature T_env. In that case Eq. (4.27) can be written explicitly as

N_single^∆T (r, ω)_ij =−18π

where we used Eq. (3.8) for the environment dust. Remember thatV0 is the volume of all space except for (absorbing) objects. This volume is infinite, but the integral

1 V0

´

V0dr^′is nevertheless finite.

As an illustration, the space and temperature dependence ofN^∆T for the case of a single planar interface with homogeneous (and temperature independent) bare coefficients is shown in Figures4.3aand4.3b. It shows clearly how the effective dielectric function depends on the temperature of external objects (in this case, the environment). As mentioned above, this is a purely nonlinear effect, absent for linear materials.

Kirchhoff’s law of radiation

From Eq. (4.28) and also Figure4.3b, we can make an interesting observation regarding the Kirchhoff’s law of radiation. While this law applies strictly only in equilibrium, it is valid in the LTE approximation as well. However, care must be taken if nonlinear ma-terials are present, since the effective dielectric properties depend on the temperatures of all the objects. For example, comparing the absorption in equilibrium and emissiv-ity after lowering the temperature of the environment to zero will seemingly violate the Kirchhoff’s law. All measurements must be performed at equal conditions (same temperatures and locations of objects).

Passive gain media

Eq. (4.28) is interesting, because even though it is hard to predict the sign ofN_single^∆T due to the integration and the unknown sign ofχ⁽³⁾, it is clear that theN_single^∆T changes sign ifT_env andT_obj are switched (because b(ω)is monotonic in temperature). This has some very pertinent consequences for the imaginary part ofε, which is generally˜ positive for passive media, which can only dissipate passing light but not amplify it.

Materials with negative Im˜εare called gain media and normally require some energy source in order to achieve this.

We can see from Eq. (4.28), however, that passive (meaning Imε,Im˜ε > 0,Imχ⁽³⁾ <

0) but nonlinear materials can become gain media in some cases. Specifically, this could happen in scenario where for some frequencies and temperatures|Im˜ε^eq(ω)|<

ImN^∆T (ω)

. Since ImN^∆T (ω)depends on the sign of T_env−T_obj, this must happen either whenT_env > T_objorT_env< T_obj. This change in sign will also have a further very interesting effect on the heat radiation, which we will discuss below.

(a)

(b)

Figure 4.3.: Dependence of the effective dielectric function of an isotropic dielectric slab at a fixed temperatureT_obj = 300K on the temperatureTenvof the environ-ment, for the non-absorbing [ε(ω^′) = 4, top] and absorbing [ε(ω^′) = 4 +i, bottom] case. These plots are calculated forω^′ = ₅₀^2πc_µm, which sets the decay length into the material (skin depth). Compare with Figures3.1and3.6.