Blackbody Radiation and Complex Refractive Index

[elad]

TL;DR

Emitted blackbody radiation into a medium with complex refractive index

Hi.

If the refractive index of a medium equals one, the total emitted blackbody intensity inside a medium is sigma*T^4.
In general, if the refractive index of a is a real number, the total emitted blackbody intensity inside a medium is n^2*sigma*T^4.
Now, when the refractive index of the medium is complex (n + ik), what will be the expression of the total emitted blackbody intensity inside the medium?
Will it be n^2*sigma*T^4 as before, or maybe (n^2+k^2)*sigma*T^4?

Thanks.

 

[tech99]

If the medium has an imaginary component, this will cause the electric and magnetic fields to be out of phase, so the real component of intensity will be reduced. I suppose that n^2*sigma*T^4 would have a cos theta factor applied. In addition, an imaginary component of refractive index means that the medium has attenuation, so the intensity will reduce exponentially with distance.

 

[pines-demon]

TL;DR Summary: Emitted blackbody radiation into a medium with complex refractive index

Hi.

If the refractive index of a medium equals one, the total emitted blackbody intensity inside a medium is sigma*T^4.
In general, if the refractive index of a is a real number, the total emitted blackbody intensity inside a medium is n^2*sigma*T^4.
Now, when the refractive index of the medium is complex (n + ik), what will be the expression of the total emitted blackbody intensity inside the medium?
Will it be n^2*sigma*T^4 as before, or maybe (n^2+k^2)*sigma*T^4?

Thanks.

Emissivity of thermal radiation is indeed tied to the (imaginary part of the) refractive index but it is not a simple polynomial factor.

Edit: I do not have the right book at hand but check for example this: https://neutrium.net/heat-transfer/calculation-of-emissivity-for-metals/

Edit2:

In general, if the refractive index of a is a real number, the total emitted blackbody intensity inside a medium is n^2*sigma*T^4.

actually where did you get this from?

 

[elad]

If the medium has an imaginary component, this will cause the electric and magnetic fields to be out of phase, so the real component of intensity will be reduced. I suppose that n^2*sigma*T^4 would have a cos theta factor applied. In addition, an imaginary component of refractive index means that the medium has attenuation, so the intensity will reduce exponentially with distance.

tech99 - what do you mean by theta and why cos?

actually where did you get this from?

pines-demon - I got the form n^2*sigma*T^4 from the book thermal radiation heat transfer (Howell and Siegel). According to the book:
"If the refractive index is constant with frequency, integrating equation 17.44 over all ni yields the local total emitted blackbody intensity inside a medium, Ib,m=(n^2)*Ib, where Ib in this relation is for n=1. "
Equation 17.44 is the blackbody spectral intensity emitted locally inside a medium with n~=1 and with n a function of frequency.
Actually, you get n^2*sigma*T^4 from integration of Planck's law over frequency, when n~=1.

 

[pines-demon]

pines-demon - I got the form n^2*sigma*T^4 from the book thermal radiation heat transfer (Howell and Siegel). According to the book:
"If the refractive index is constant with frequency, integrating equation 17.44 over all ni yields the local total emitted blackbody intensity inside a medium, Ib,m=(n^2)*Ib, where Ib in this relation is for n=1. "
Equation 17.44 is the blackbody spectral intensity emitted locally inside a medium with n~=1 and with n a function of frequency.
Actually, you get n^2*sigma*T^4 from integration of Planck's law over frequency, when n~=1.

Oh I see this is within the media! Never seen this, how is it derived that would give you a clue. As intensities are involved there might be some $|\tilde{n}|^2=n^2+\kappa^2$ instead of a simple square.

 

[Philip Koeck]

pines-demon - I got the form n^2*sigma*T^4 from the book thermal radiation heat transfer (Howell and Siegel). According to the book:
"If the refractive index is constant with frequency, integrating equation 17.44 over all ni yields the local total emitted blackbody intensity inside a medium, Ib,m=(n^2)*Ib, where Ib in this relation is for n=1. "

Is this expression only valid when there is direct contact between the radiation source and the medium or does it even hold if the source is surrounded by a layer of vacuum and the medium forms a shell around this layer?

 

[pines-demon]

Is this expression only valid when there is direct contact between the radiation source and the medium or does it even hold if the source is surrounded by a layer of vacuum and the medium forms a shell around this layer?

It is only valid if the source (and detector) are within the media. Imagine a sort of hot object in glass, the object emits $n^2 \sigma T^4$ where $n$ is the index of glass. If there object has a different emissivity $\epsilon$ then it is $n^2\epsilon \sigma T^4$. If there is vacuum somewhere, then that has to be taken into account and would depend on the geometry.

Edit: vacuum not glass