Saturable absorption
Encyclopedia
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
.
Then, for the continuous-wave operation, the absorption rate (or simply absorption)
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.
In the simplest geometry, when the rays of the absorbing light are parallel, the intensity can be described with the
Bouguer law,
where
is coordinate in the direction of propagation.
Substitution of (1) into (2) gives the equation
With the dimensionless variables
, 
,
equation (3) can be rewritten as
The solution can be expressed in terms of the Wright Omega function
:
Let
. Then
With new independent variable
,
Equation (6) 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
(intensity zero) or to the unusual branch of the Lambert W function.
where 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 (neither cw, nor short pulse operation), 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
.
, etc.), 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.
was irradiated with soft X-ray
laser
radiation (wavelength
13.5 nanometre (5.31496062992126E-07 in)). The short laser pulse knocked out core L-shell
electrons without breaking the crystal
line structure of the metal, making it transparent to soft X-rays of the same wavelength for about 40 femtoseconds.
Intensity (physics)
In physics, intensity is a measure of the energy flux, averaged over the period of the wave. The word "intensity" here is not synonymous with "strength", "amplitude", or "level", as it sometimes is in colloquial speech...
. 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
Optical cavity
An optical cavity or optical resonator is an arrangement of mirrors that forms a standing wave cavity resonator for light waves. Optical cavities are a major component of lasers, surrounding the gain medium and providing feedback of the laser light. They are also used in optical parametric...
. The key parameters for a saturable absorber are its wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
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
Q-switching
Q-switching, sometimes known as giant pulse formation, is a technique by which a laser can be made to produce a pulsed output beam. The technique allows the production of light pulses with extremely high peak power, much higher than would be produced by the same laser if it were operating in a...
.
Phenomenology of saturable absorption
Within the simple model of saturated absorption, the relaxation rate of excitations does not depend on the intensityThen, for the continuous-wave operation, the absorption rate (or simply absorption)
is determined by intensity :
where
is linear absorption, and is saturation intensity.These parameters are related with the concentration
Concentration
In chemistry, concentration is defined as the abundance of a constituent divided by the total volume of a mixture. Four types can be distinguished: mass concentration, molar concentration, number concentration, and volume concentration...
of the active centers in the medium,the effective cross-sections
and the lifetime of the excitations.Relation with Wright Omega function
Bouguer law,
where
is coordinate in the direction of propagation.Substitution of (1) into (2) gives the equation
With the dimensionless variables
, ,equation (3) can be rewritten as
The solution can be expressed in terms of the Wright Omega function
Wright Omega function
In mathematics, the Wright omega function, denoted ω, is defined in terms of the Lambert W function as:-Uses:One of the main applications of this function is in the resolution of the equation z = ln, as the only solution is given by z = e−ω.y = ω is the unique solution,...
:Relation with Lambert W function
The solution can be expressed also through the related Lambert W function.Let
. ThenWith new independent variable
,Equation (6) 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(intensity zero) 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 (neither cw, nor short pulse operation), the rate equations for excitation
Excited state
Excitation is an elevation in energy level above an arbitrary baseline energy state. In physics there is a specific technical definition for energy level which is often associated with an atom being excited to an excited state....
and relaxation in the optical medium must be considered together.
Saturation fluence is one of the factors that determine threshold
Lasing threshold
The lasing threshold is the lowest excitation level at which a laser's output is dominated by stimulated emission rather than by spontaneous emission. Below the threshold, the laser's output power rises slowly with increasing excitation. Above threshold, the slope of power vs. excitation is orders...
in the gain media and limits the storage of energy in a pulsed disk laser
Disk laser
A disk laser or active mirror is a type of solid-state laser characterized by a heat sink and laser output that are realized on opposite sides of a thin layer of active gain medium...
.
Mechanisms and examples of saturable absorption
Absorption saturation, which results in decreased absorption at high incident light intensity, competes with other mechanisms (for example, increase in temperature, formation of color centersF-Center
An F-Center or Farbe center is a type of crystallographic defect in which an anionic vacancy in a crystal is filled by one or more electrons, depending on the charge of the missing ion in the crystal. Electrons in such a vacancy tend to absorb light in the visible spectrum such that a material...
, etc.), which result in increased absorption.
In particular, saturable absorption is only one of several mechanisms that produce self-pulsation
Self-pulsation
Self-pulsation takes place at the beginning of laser action.As the pump is switched on, the gainin the active medium rises and exceeds the steady-state value...
in lasers, especially in semiconductor lasers.
One atom thick layer of carbon, graphene
Graphene
Graphene is an allotrope of carbon, whose structure is one-atom-thick planar sheets of sp2-bonded carbon atoms that are densely packed in a honeycomb crystal lattice. The term graphene was coined as a combination of graphite and the suffix -ene by Hanns-Peter Boehm, who described single-layer...
, can be seen with the naked eye because it absorbs approximately 2.3% of white light, which is π times fine-structure constant
Fine-structure constant
In physics, the fine-structure constant is a fundamental physical constant, namely the coupling constant characterizing the strength of the electromagnetic interaction. Being a dimensionless quantity, it has constant numerical value in all systems of units...
. The saturable absorption response of graphene is wavelength independent from UV to IR, mid-IR and even to THz frequencies.
Saturable X-ray absorption
Saturable absorption has been demonstrated for X-rays. In one study, a thin 50 nanometre (1.96850393700787E-06 in) foil of aluminiumAluminium
Aluminium or aluminum is a silvery white member of the boron group of chemical elements. It has the symbol Al, and its atomic number is 13. It is not soluble in water under normal circumstances....
was irradiated with soft X-ray
X-ray
X-radiation is a form of electromagnetic radiation. X-rays have a wavelength in the range of 0.01 to 10 nanometers, corresponding to frequencies in the range 30 petahertz to 30 exahertz and energies in the range 120 eV to 120 keV. They are shorter in wavelength than UV rays and longer than gamma...
laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...
radiation (wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
13.5 nanometre (5.31496062992126E-07 in)). The short laser pulse knocked out core L-shell
Electron shell
An electron shell may be thought of as an orbit followed by electrons around an atom's nucleus. The closest shell to the nucleus is called the "1 shell" , followed by the "2 shell" , then the "3 shell" , and so on further and further from the nucleus. The shell letters K,L,M,.....
electrons without breaking the crystal
Crystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...
line structure of the metal, making it transparent to soft X-rays of the same wavelength for about 40 femtoseconds.