Saturable absorption is a property of materials where the absorption of light decreases with increasing light intensity. Most materials show some saturable absorption, but often only at very high optical intensities (close to the optical damage). At sufficiently high incident light intensity, the ground state of a saturable absorber material is excited into an upper energy state at such a rate that there is insufficient time for it to decay back to the ground state before the ground state becomes depleted, causing the absorption to saturate. The key parameters for a saturable absorber are its wavelength range (where in the electromagnetic spectrum it absorbs), its dynamic response (how fast it recovers), and its saturation intensity and fluence (at what intensity or pulse energy it saturates). Saturable absorber materials are useful in laser cavities. For instance, they are commonly used for passive Q-switching.
Phenomenology Within the simple model of saturated absorption, the relaxation rate of excitations does not depend on the intensity. Then, for the continuous-wave (cw) operation, the absorption rate (or simply absorption) A {\displaystyle A} is determined by intensity I {\displaystyle I} :
( 1 ) A = α 1 + I / I 0 {\displaystyle (1)~~~~A={\frac {\alpha }{1+I/I_{0}}}}
where α {\displaystyle \alpha } is linear absorption, and
I 0 {\displaystyle I_{0}} is saturation intensity. These parameters are related with the concentration N {\displaystyle N} of the active centers in the medium, the effective cross-sections σ {\displaystyle \sigma } and the lifetime τ {\displaystyle \tau } of the excitations.
Relation with Wright omega function In the simplest geometry, when the rays of the absorbing light are parallel, the intensity can be described with the Beer–Lambert law,
( 2 ) d I d z = − A I {\displaystyle (2)~~~~{\frac {\mathrm {d} I}{\mathrm {d} z}}=-AI}
where z {\displaystyle z} is coordinate in the direction of propagation. Substitution of (1) into (2) gives the equation
( 3 ) d I d z = − α I 1 + I / I 0 {\displaystyle (3)~~~~{\frac {\mathrm {d} I}{\mathrm {d} z}}=-{\frac {\alpha ~I}{1+I/I_{0}}}}
With the dimensionless variables u = I / I 0 {\displaystyle u=I/I_{0}} , t = α z {\displaystyle t=\alpha z} , equation (3) can be rewritten as
( 4 ) d u d t = − u 1 + u {\displaystyle (4)~~~~{\frac {\mathrm {d} u}{\mathrm {d} t}}={\frac {-u}{1+u}}}
The solution can be expressed in terms of the Wright omega function ω {\displaystyle \omega } :
( 5 ) u = ω ( − t ) {\displaystyle (5)~~~~u=\omega (-t)}
Relation with Lambert W function The solution can be expressed also through the related Lambert W function. Let u = V ( − e t ) {\displaystyle u=V{\big (}-\mathrm {e} ^{t}{\big )}} . Then
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