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Profile function properties with different variables are discussed, the formulae of stimulated absorption, spontaneous and stimulated emission, absorption and emission coefficients, and cross sections are deduced, and some confusing issues are clarified.
Due to defects, strain, disorder, lattice vibration, and so on in solids, an ideal optical absorption or emission transition line of rare earth ions, described by the Dirac function δ, is usually a wider band, which can be described by a profile function ϕ, such as the Voigt function,[1–4] with independent variable wavelength λ, frequency ν, or angle frequency ω, which has the following relationships:
From Eq. (
On the other hand, there are many unit systems of electromagnetics, such as the rationalized MKSA system of units, the absolute electrostatic system of units (e.s.u. or CGSE), the absolute electromagnetic system of units (e.m.u. or CGSM), and the Gauss system of units (CGS). Electromagnetic quantities and formulae usually have a variety of values and expressions. For example, the dielectric constant ε0 and permeability μ0 in vacuum are 8.85× 10−12 F/m and
The present work first discusses the distribution property of a physical quantity with a profile in wavelength, frequency, or angle frequency, and further clarifies the formulae of emission and absorption. The unit system of this work is SI units.
Assume a physical quantity Q, such as the absorption coefficient, emission or absorption cross section, or transition probability, with the variable angle frequency ω and profile
Substituting Eqs. (
Although equations (
Then, integrating Eq. (
The left integral should be equal to the right integral, so
Comparing Eq. (
Integrate Eq. (
In summary, if a physical quantity Q can be expressed as the product of amplitude and a profile function with variable wavelength, frequency, or angle frequency, it has the same distribution expression and amplitude Q0 in different variable spaces, and the profile function value is normalized in the respective variable spaces. This reflects the fact that a physical quantity should be the same no matter how it is described by a profile function in variables of wavelength, frequency, or angle frequency.
Let us consider a plane electromagnetic wave, its electric and magnetic vector can be written as
During a period
The additional energies
With Eq. (
Consider a transition from the initial state
For convenient comparison, we can substitute Eq. (
It can be seen that the right side of either Eq. (
If we want to know the total value W0 of W by integrating (
In fact, this has been proved in Section
Consider that the local electric field correction
Then the average transition rate WIF of
Furthermore, n is separated from
It is usually to call
The ion number variation rate in EI caused by the stimulated absorption, stimulated and spontaneous emission between
Assume a light beam with cross section Λ and intensity
The absorption coefficient is defined as
Substituting Eqs. (
Substituting Eq. (
According to the conclusion in Section
Based on Einstein’s relationship and Eq. (
For laser operation, spontaneous radiation does not need to be considered, so equation (
Like the absorption coefficient, the emission coefficient or gain coefficient β is defined as
Substitute Eqs. (
Substituting Eq. (
Using Einstein’s relationship Eq. (
Substituting Eqs. (
If the emission spectrum
According to the conclusion in Section
A physical quantity Q such as absorption and emission that can be expressed as a product of amplitude and a profile function with variable wavelength, frequency, or angle frequency has the same distribution expression and amplitude Q0 in any of these variable spaces, and the profile function value is normalized in the respective space. Optical transition formulae about emission and absorption were deduced, and the related confusions were clarified.
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