In laser physics, gain or amplification is a process where the medium transfers part of its energy to the emitted electromagnetic radiation, resulting in an increase in optical power. This is the basic principle of all lasers. Quantitatively, gain is a measure of the ability of a laser medium to increase optical power. However, overall a laser consumes energy.
Definition The gain can be defined as the derivative of logarithm of power P {\displaystyle ~P~}
as it passes through the medium. The factor by which an input beam is amplified by a medium is called the gain and is represented by G.
G = d d z ln ( P ) = d P / d z P {\displaystyle G={\frac {\rm {d}}{{\rm {d}}z}}\ln(P)={\frac {{\rm {d}}P/{\rm {d}}z}{P}}}
where z {\displaystyle ~z~} is the coordinate in the direction of propagation. This equation neglects the effects of the transversal profile of the beam. In the quasi-monochromatic paraxial approximation, the gain can be taken into account with the following equation
2 i k ∂ E ∂ z = Δ ⊥ E + 2 ν E + i G E {\displaystyle 2ik{\frac {\partial E}{\partial z}}=\Delta _{\perp }E+2\nu E+iGE} , where
ν {\displaystyle ~\nu ~} is variation of index of refraction (Which is supposed to be small),
E {\displaystyle ~E~} is complex field, related to the physical electric field
E p h y s {\displaystyle ~E_{\rm {phys}}~} with relation
E p h y s = R e ( e → E exp ( i k z − i ω t ) ) {\displaystyle ~E_{\rm {phys}}={\rm {Re}}\left({\vec {e}}E\exp(ikz-i\omega t)\right)~} , where
e → {\displaystyle ~{\vec {e}}~} is vector of polarization,
k {\displaystyle ~k~} is wavenumber,
ω {\displaystyle ~\omega ~} is frequency,
Δ ⊥ = ( ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 ) {\displaystyle ~\Delta _{\rm {\perp }}=\left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}\right)~}
is transversal Laplacian;
R e {\displaystyle ~{\rm {Re~}}} means real part.
Gain in quasi two-level system In the simple quasi two-level system, the gain can be expressed in terms of populations
N 1 {\displaystyle ~N_{1}~} and
N 2 {\displaystyle ~N_{2}~} of lower and excited states:
G = σ e N 2 − σ a N 1 {\displaystyle ~G=\sigma _{\rm {e}}N_{2}-\sigma _{\rm {a}}N_{1}~}
where
σ e {\displaystyle ~\sigma _{\rm {e}}~} and
σ a {\displaystyle ~\sigma _{\rm {a}}~}
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