# How is the “negative dispersion” derived?

I'm looking at Kopfermann H., Ladenburg R., Nature, 122, 338-339 (1928) and it appears Ladenburg in Ladenburg R., Z.Physik, 4, 451-468 (1921) was the first to discover the phenomenon of "negative dispersion" and/or "negative absorption" which is at the basis of the laser theory. That idea is present in as early as Planck's 1901 paper but apparently Ladenburg was the first to see it. Unfortunately, I can't read German, so the 1921 paper is off-limits for me and therefore I'm missing most of the derivation. I'm, however, seeing the formula $$f_{kj} N_j \frac{g_k}{g_j} \left( 1 - \frac{N_kg_j}{N_j g_k} \right)$$ in Kopfermann H., Ladenburg R., Nature, 122, 338-339 (1928), which appears later in Fabrikant's dissertation but because only two pages from that dissertation are available I can't tell if Fabrikant has cited Ladenburg in it. Anyway, Ladenburg claims that the refractive index should be proportional to the above formula which isn't at all obvious to me. Further, it should be said that the above formula is arrived at from $$\frac{f_{kj}}{g_j} \left( N_{jgk} - N_kg_j \right) .$$ This can be obtained only if $N_{kgj} = N_j g_k$ leading to a formula $$f_{kj} N_j \left( \frac{g_k - g_j} {g_j} \right) ,$$ equivalent to the first one presented here. Now, let me not forget to mention that $N_j$ and $N_k$ are the number of atoms per cm$^3$ in the states $j$ and $k$, $g_j$ is the statistical weight in the state $j$ and $f_{kj}$ is the probability coefficient for the transition $j \rightarrow k$. Can anyone tell how the second equation is derived and what the physical meaning of the third formula might be? Also, how it would follow from the first formula naturally that there may be "negative dispersion" other than speculating that somehow the term $\frac{N_kg_j}{N_j g_k}$ might be of a certain value to achieve "negative dispersion" and then test that hypothesis experimentally?

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I think this all goes back to Einstein's original paper on stimulated emission. Incidentally, the wikipedia article on stimulated emission appears to go through this derivation (using nearly identical notation!) – sevenofdiamonds Sep 20 '11 at 19:56
@sevenodiamonds, I guess you mean this paper Einstein A., Zur Quantentheorie der Strahlung, Phys.ZS., 18, 121-137 (1917). Can you please explain where in it Einstein talks about negative dispersion"? – ganzewoort Sep 20 '11 at 23:51
I don't think he explicitly mentions negative dispersion, I meant the derivation you're looking at is rooted in Einstein's. – sevenofdiamonds Sep 26 '11 at 21:38