McCumber 理论

The McCumber relation (or McCumber theory) refers to the effective

cross-sections of absorption and emission of light in the physics of

solid-state lasers [1][2].

Definition

Let σa(ω) be the effective absorption cross-section σe(ω) be effective

emission cross-sections at frequency ω, and let be the effective

temperature of the medium. The McCumbner relation is

(1)

where is thermal steady-state ratio of populations; frequency ωz

is called "zero-line" frequency [3][4]; is the Planck constant and kB is

the Boltzmann

constant. Note that the right-hand side of Equation (1) does

not depend on .

Gain

It is typical that the lasing properties of a medium are determined by

the temperature and the population at the excited laser level, and are

not sensitive to the method of excitation used to achieve it. In this case,

the absorption cross-section σa(ω) and the emission cross-section σe(ω)

at frequency can be related to the lasers gain in such a way, that the

gain at this frequency can be determined as follows:

(2)

D.E.McCumber had postulated these properties and found that the emisison

and absorption cross-sections are not independent [1][2]; they are related

with Equation (1).

Idealized atoms

In the case of an idealized two-level atom the detailed balance for the

emission and absorption which preserves the Max Planck formula for the

black body

radiation leads to equality of cross-section of absorption and emission. In the solid-state lasers the splitting of each of laser levels

leads to the broadening which greatly exceeds the natural spectral

linewidth. In the case of an ideal two-level atom, the product of the

linewidth and the lifetime is of order of unity, which obeys the

Heisenberg

uncertainty principle. In solid-state laser materials, the linewidth is

several orders of magnitude larger so the spectra of emission and

absorption are determined by distribution of excitation among sublevels

rather than by the shape of the spectral line of each individual transition

between sublevels. This distribution is determined by the effective

temperature within each of laser levels. The McCumber hypothesis is that

the distribution of excitation among sublevels is thermal. The effective

temperature determines the spectra of emission and absorption ( The

effective temperature is called a temperature by scientists even if the

excited medium as whole is pretty far from the thermal state )

Deduction of the McCumber relation

Fig.1. Sketch of sublevels

Consider the set of active centers (fig.1.). Assume fast transition

between sublevels within each level, and slow transition between levels.

According to the McCumber hypothesis, the cross-sections σa and σe do

not depend on the populations N1 and N2. Therefore, we can deduce the

relation, assuming the thermal state.

Let

be group velocity of light in the medium, the product

is spectral rate of

stimulated emission, and is that of absorption; a(ω)n2 is spectral rate of

spontaneous emission. (Note that in this approximation, there is no such

thing as a spontaneous absorption) The balance of photons gives:

(3)

Which can be rewritten as

(4)

The thermal distribution of density of photons follows from blackbody

radiation [5]

(5)

Both (4) and (5) hold for all frequencies . For the case of idealized

two-level active centers, , and ,

which leads to the relation between the spectral rate of spontaneous

emission a(ω) and the emission cross-section [5]. (We keep the term

probability of emission for the quantity , which is probability

of emission of a photon within small spectral interval during

a short time interval , assuming that at time the atom is

excited.) The relation (D2) is a fundamental property of spontaneous and

stimulated emission, and perhaps the only way to prohibit a spontaneous

break of the thermal equilibrium in the thermal state of excitations and

photons. For each site number , for each sublevel number j, the partial spectral

emission probability can be expressed from consideration of

idealized two-level atoms [5]:

(6)

Neglecting the cooperative coherent effects, the emission is additive:

for any concentration of sites and for any partial population of

sublevels, the same proportionality between and holds for the

effective cross-sections:

(7)

Then, the comparison of (D1) and (D2) gives the relation

(8)

This relation is equivalent of the McCumber relation (mc), if we define

the zero-line frequency ωZ as solution of equation

(9)

the subscript indicates that the ratio of populations in evaluated in

the thermal state. The zero-line frequency can be expressed as

(10)

Then (n1n2) becomes equivalent of the McCumber relation (mc).

No specific property of sublevels of active medium is required to keep

the McCumber relation. It follows from the assumption about quick transfer

of energy among excited laser levels and among lower laser levels. The