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. 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, its dynamic response, and its saturation intensity and fluence.
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 operation, the absorption rate is determined by intensity :
where is linear absorption, and
is saturation intensity.
These parameters are related with the concentration of the active centers in the medium,
the effective cross-sections and the lifetime 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 theBeer–Lambert law,
where is coordinate in the direction of propagation.
Substitution of into gives the equation
With the dimensionless variables,,
equation can be rewritten as
The solution can be expressed in terms of the Wright omega function :
Relation with Lambert W function
The solution can be expressed also through the related Lambert W function.Let. Then
With new independent variable,
Equation leads to the equation
The formal solution can be written
where is constant, but the equation may correspond to the non-physical value of intensity
or to the unusual branch of the Lambert W function.
Saturation fluence
For pulsed operation, in the limiting case of short pulses, absorption can be expressed through the fluencewhere time should be small compared to the relaxation time of the medium; it is assumed that the intensity is zero at.
Then, the saturable absorption can be written as follows:
where saturation fluence is constant.
In the intermediate case, the rate equations for excitation and relaxation in the optical medium must be considered together.
Saturation fluence is one of the factors that determine threshold in the gain media and limits the storage of energy in a pulsed disk laser.
Mechanisms and examples
Absorption saturation, which results in decreased absorption at high incident light intensity, competes with other mechanisms, which result in increased absorption.In particular, saturable absorption is only one of several mechanisms that produce self-pulsation in lasers, especially in semiconductor lasers.
One atom thick layer of carbon, graphene, can be seen with the naked eye because it absorbs approximately 2.3% of white light, which is π times fine-structure constant. The saturable absorption response of graphene is wavelength independent from UV to IR, mid-IR and even to THz frequencies. In rolled-up graphene sheets, saturable absorption is dependent on diameter and chirality.