Saturable absorption

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, atoms in the ground state of a saturable absorber material become excited into an upper energy state at such a rate that there is insufficient time for them to decay back to the ground state before the ground state becomes depleted, and the absorption subsequently saturates. Saturable absorbers are useful in laser cavities. The key parameters for a saturable absorber are its wavelength range (where it absorbs), its dynamic response (how fast it recovers), and its saturation intensity and fluence (at what intensity or pulse energy it saturates). They are commonly used for passive Q-switching. 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, atoms in the ground state of a saturable absorber material become excited into an upper energy state at such a rate that there is insufficient time for them to decay back to the ground state before the ground state becomes depleted, and the absorption subsequently saturates. Saturable absorbers are useful in laser cavities. The key parameters for a saturable absorber are its wavelength range (where it absorbs), its dynamic response (how fast it recovers), and its saturation intensity and fluence (at what intensity or pulse energy it saturates). They are commonly used for passive Q-switching. Within the simple model of saturated absorption, the relaxation rate of excitations does not depend on the intensity.Then, for the continuous-wave operation, the absorption rate (or simply absorption) A {displaystyle A} is determined by intensity I {displaystyle I} : 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 au } of the excitations. In the simplest geometry, when the rays of the absorbing light are parallel, the intensity can be described with the Beer–Lambert law, where z {displaystyle z} is coordinate in the direction of propagation.Substitution of (1) into (2) gives the equation 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 The solution can be expressed in terms of the Wright Omega function ω {displaystyle omega } : The solution can be expressed also through the related Lambert W function. Let u = V ( − e t ) {displaystyle u=V{ig (}-mathrm {e} ^{t}{ig )}} . Then With new independent variable p = − e t {displaystyle p=-mathrm {e} ^{t}} ,Equation (6) leads to the equation