How is the “negative dispersion” derived?
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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
fkjNjgkgj(1−NkgjNjgk)fkjNjgkgj(1−NkgjNjgk)
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
fkjgj(Njgk−Nkgj).fkjgj(Njgk−Nkgj).
This can be obtained only if Nkgj=NjgkNkgj=Njgkleading to a formula
fkjNj(gk−gjgj),fkjNj(gk−gjgj),
equivalent to the first one presented here. Now, let me not forget to mention that NjNjand NkNk are the number of atoms per cm33 in the states jj and kk, gjgj is the statistical weight in the state jj and fkjfkj is the probability coefficient for the transition j→kj→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 NkgjNjgkNkgjNjgk might be of a certain value to achieve "negative dispersion" and then test that hypothesis experimentally?
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
fkjNjgkgj(1−NkgjNjgk)fkjNjgkgj(1−NkgjNjgk)
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
fkjgj(Njgk−Nkgj).fkjgj(Njgk−Nkgj).
This can be obtained only if Nkgj=NjgkNkgj=Njgkleading to a formula
fkjNj(gk−gjgj),fkjNj(gk−gjgj),
equivalent to the first one presented here. Now, let me not forget to mention that NjNjand NkNk are the number of atoms per cm33 in the states jj and kk, gjgj is the statistical weight in the state jj and fkjfkj is the probability coefficient for the transition j→kj→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 NkgjNjgkNkgjNjgk might be of a certain value to achieve "negative dispersion" and then test that hypothesis experimentally?
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