The excited electrons in the conduction band relax either by spontaneous photon emission or by non-radiative energy loss with a mean relaxation time constant τ.
The sum of states in the valence and conduction band gives the total number of electrons N_T which take place on the absorption process: N_T = N_V + N_C.
Without illumination of the absorber all electrons are in the ground state N_C = NT. In case of very high density of incident photons the maximum value for the excited states N_C=N_V=N_T/2 is approached and the absorber is transparent.
The absorption A of the saturable absorber is proportional to the number of electrons N_C in the ground state. If we consider that the incident photon density is proportional to the time dependent optical beam intensity I(t) then we can rewrite the rate equation for the number of states into a rate equation for the absorption A(t) as

The arbitrary solution of this differential equation for saturable absorption A(t) is

For a certain time dependency of the optical pulse intensity I(t) the integration in the exponential function can be carried out.

In case of a Gaussian pulse the intensity I(t) can be written as

The integration in the exponential function can be executed resulting in the following time dependent absorption A(t)

Approximation for long relaxation time   τ >> t_P

Approximation for short relaxation time   τ << t_P

In this case the dependency dA/dt in the above rate equation for the absorption A can be neglected because at each time the absorption is determined solely by the actual optical intensity I(t) and not on its history.

The integration of the time dependent intensity I(t) over the pulse duration t_P results in the pulse fluence F. Therefore the pulse fluence can be approximated as F ~ I · t_P.
The two-photon absorption coefficient β is a material dependent parameter. For GaAs the value is β = 2.5·10^-10 m/W.

The saturable absorber mirror (SAM) has no transmission in the stop band of the Bragg mirror. Therefore the reflectance is R = 1 - A.
If two photon absorption (TPA) is included and the absorber relaxtion time τ is short in comparison to the pulse duration t_P then the laterally averaged saturation of the SAM reflectance R can be calculated by

The saturation fluence F_sat depends on the semiconductor material and the optical design of the SAM. Low saturation fluence has the advantage that the laser mode-locking can be started at low power level. This avoids a fast absorber degradation
For decreasing the saturation fluence the thickness of the semiconductor absorber layer is reduced below ~ 10 nm. In this case a quantization of the electron energy and the momentum in the direction perpendicular to the absorber layer takes place and as a consequence the density of states decreases below the value of a compact semiconductor. Therefore the absorber layers in a SAM are thin quantum wells with a smaller band gap than the barriers on both sides. If a larger absorption value of the SAM is needed, the number of the quantum wells is increased instead of using a single thick absorber layer.
The electric field intensity in front of the Bragg mirror of a SAM is a periodic function with nodes and antinodes. The absorbing quantum wells are positioned in the antinodes to get a low saturation fluence. Together with the Fresnel reflectance at the semiconductor-air boundary the Bragg mirror builds a Fabry-Perot like resonator, which contains the quantum wells. The optical thickness of the semiconductor material between the reflectors determines the cavity to be resonant or anti-resonant. The saturation fluence of a resonant SAM is lower than that of an anti-resonant SAM because of the field enhancement inside the cavity.
Typical saturation fluence of a resonant SAM is F_sat ~ 30 µJ/cm² = 0.3 J/m² whereas the saturation fluence of an anti-resonant SAM is F_sat ~ 120 µJ/cm² = 1.2 J/m²

The SAM parameters depend on the temperature mainly because of two effects:

• Thermal expansion of the absorber material
• Decrease of the semiconductor band gap with increasing temperature.

Temperature dependent optical thickness o_t
The second effect results in an increase of komplex refractive index n+i·k with increasing temperature. Together with the thermal expansion this leads to an increase of optical thickness n·d (n - real part of refractive index, d - film thickness) with temperature. Therefore with increasing temperature shifts the spectral reflection curve of the SAM towards longer wavelengths. This thermal shift can be measured for a typical SAM consisting of AlAs, GaAs and InGaAs layers. The temperature coefficient for the optical thickness can be determined to
    α_ot ~ 7·10^-5/K.
This value is substantial larger then the linear thermal expansion coefficient α_t = 5.39·10^-6/K of GaAs and therefore mainly determined by the change of the band gap and the refractive index.

A temperature shift of the SAM spectral curve can be calculated using the relation

Temperature dependent absorption
The imaginary part k of the refractive index and consequently the absorption of a direct semiconductor increases with temperature. This is also a consequence of decreasing band gap with increasing temperature.
For a typical SAM the temperature coefficient of the imaginary part k of the refractive index is
    α_k ~ 6·10^-3/K.
With this coefficent the temperature dependent value for k can be calculated by

    k(T) = k(T₀)·(1+α_k·ΔT)

Therefore k and also the SAM absorption increase by 50 % if the absorber temperature increases by 83 K. This temperature dependence has a significant influence on laser mode-locking.

The saturable absorber converts a part of the incoming photon energy into heat. This thermal energy increases the absorber layer temperature during and shortly after an optical pulse. After that the heat is transported through the substrate to the heat sink on the rear substrate side. In case of a substrate like GaAs with a high thermal conductivity only a negligible amount of the dissipated heat goes trough the front surface of the absorber into air.
In case of a pulsed laser beam the absorber temperature T varies periodically with the pulse repetition rate f. A continuous heat flow from the absorber layer to the heat sink leads to a constant absorber temperature rise ΔT_stat.

The heat diffusion into the GaAs substrate and the resulting temperature distribution T(r,t) can be described by the heat diffusion equation (1)

If the laser spot radius r on the absorber surface is small in comparison to the substrate thickness (typically 0.5 mm) then the heat flow in the absorber layer is nearly one dimensional but in the substrate three dimensional (with a point source). Then the heat is distributed into a half space of the substrate resulting in a temperature field with nearly concentric interfaces of equal temperature.
If we consider at first the time average temperature increase delta T_stat of the absorber layer caused by the mean absorbed optical power P_m, then we can use the following relations for a centre symmetric three dimensional heat flow (eq. 2)

The SAM consists of a Bragg mirror and an absorber layer with a certain optical thickness n·d in front of it. The interface between the semiconductor absorber and air builds a second Fresnel reflector. The absorption and the dispersion of the SAM depend on the following parameters:

• Surface reflectance R_s
• Optical absorber thickness n·d
• Absorption coefficient α.

Anti-resonant SAM

If the SAM is anti reflection (AR) coated ( R_s = 0 ) then the optical beam is solely reflected on the Bragg mirror. This results in a small dispersion similar as of a dielectric mirror.

Group velocity dispersion (GVD) of an anti resonant SAM-1040-2-1ps.

The GVD is calculated using the second order derivative of the reflected phase.
For an anti resonant SAM the GVD is small in the spectral region of the stop band.

Resonant SAM

If the absorber layer is enclosed in a cavity determined by the Bragg mirror and the surface mirror with R_s > 0 then the light is partial reflected on both mirrors. The following two effects determine the net reflectance and the dispersion:

• The optical thickness n·d of the absorber layer.
• The absorption coefficient α of the absorber material.

Group velocity dispersion (GVD) of a resonant SAM-1030-32-3ps.

The GVD is calculated using the second order derivative of the reflected phase.
The largest absorption and the lowest reflectance is at the resonant wavelength λ = 2·n·d. Around this resonance the GVD changes rapidly whereas outside the resonance the GVD varies slowly.
The amplitude of the GVD variation near the resonance increases with increasing optical thickness n·d and decreasing absorption coefficient α of the absorber layer.