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The absorptance of the SAM consists of two parts:
The non-saturable part Ans of the absorption can be caused by macroscopic crystal defects with very short relaxation time
of the excited carriers.
The non-saturable absorption decreases with increasing relaxation time of the excited carriers in the absorber material.
In case of absorbers with a short relaxation time of ~ 1 ps the non-saturable part of the absorption can be Ans ~ 0.2 · A0.
The total absorption is the sum of both parts A1 = As + Ans.
The saturation of the absorption can be described with the following formula:
The pulse fluence F depends on the beam radius r for a Gaussian pulse as follows:
The fluence and consequently also the saturation depend on the radial distance r from beam axis. To get the effective absorption A for the pulse an averaging over the whole illuminated area on the absorber is needed.
The figures below show the saturation behaviour according to the formulas above for the absorption A1 and the averaged absorption A with a
certain value of non-saturable absorption Ans = 0.1 · A0.
The averaging results in a lower saturation simulating an additional virtual non-saturable absorption.
Typical values of a saturable absorber mirror for mode-locking a solid state laser are A ~ 0.02 and Fsat ~ 70 µJ/cm2.
A fiber laser has typically a higher gain. In this case for mode-locking a SAM with the following parameters are appropriate: A ~ 0.3, Fsat ~ 50 µJ/cm2.
Two photon absorption
For short pulses the two-photon absorption ATPA increases the total absorption as follows:
The integration of the time dependent intensity I(t) over the pulse duration τP results in the pulse fluence F. Therefore the pulse fluence can be approximated as F ~ I · τP.
The saturable absorber mirror has no transmission in the stop band of the Bragg mirror. Therefore the reflectance is R = 1 - A.
If the two photon absorption (TPA) is included then the SAM reflectance R can be calculated as
The figures below show the reflectance of a SAM with Ans = 0.1, As = 0.4, Fsat = 0.5 J/m2 and an absorber thickness d = 2 µm as a function of the pulse fluence F for three different pulse durations in a linear and a logarithmic scale.
The modulation depth ΔR is smaller then the saturable absorption As and depends on the
pulse duration because of TPA (two photon absorption) in the absorber and the Bragg mirror.
The TPA depends on the material. The SAM consists of different materials. The TPA coefficient β increases with decreasing energy gap of the semiconductor material.
The TPA coefficient can be therefore only estimated to
For these values the modulation depth is shown in the graph using the formula for R(F) above.
• Relaxation time
Relaxation time τ
The saturable absorber layer consists of a semiconductor material with a direct
band gap slightly smaller than the photon energy of the laser beam. As a result of photon absorption
electron-hole pairs are created in the absorber film.
For laser mode-lockig the relaxation time τ of the excited carriers shall be somewhat longer than the pulse duration. In this case the back side of the pulse is still free of absorption, but during the whole period between two consecutive pulses the absorber is non saturated and serves for a high absorption.
Because the relaxation time due to the spontaneous photon emission in a direct semiconductor is about 1 ns, some precautions has to be done to shorten it for mode-locking application.
Two technologies are used to introduce lattice defects in the absorber layer for fast non-radiative relaxation of excited carriers:
The parameters to adjust the relaxation time in both technologies are the growth temperature in case of LT-MBE and the ion dose in case of implantation. Typical values of the relaxation time τ of SAMs are between 500 fs and 30 ps.
An example of a pump-probe measurement to determine the relaxation time τ is shown right.
• Saturation fluence
Saturation fluence Fsat
The saturation fluence Fsat 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
To decrease 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 Fsat ~ 30 µJ/cm² = 0.3 J/m² whereas the saturation fluence of an anti-resonant SAM is Fsat ~ 120 µJ/cm² = 1.2 J/m²
• Influence of absorber temperature
Influence of absorber temperature
The SAM parameters depend on the temperature mainly because of two effects:
Temperature dependent optical thickness ot
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
αot ~ 7·10-5/K.
This value is substantial larger then the linear thermal expansion coefficient αt = 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
n·d(T) = n·d(T0 + ΔT) = n·d(T0) · (1 + αot · ΔT)or
Δ(n·d) = n·d(T0) · αot · ΔT
A temperature rise of 100 K shifts the SAM spectral curve towards longer wavelengths by 7 nm at 1 µm wavelength and by 21 nm at 3 µm. Such a shift is important in case of a resonant SAM.
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
αk ~ 6·10-3/K.
With this coefficent the temperature dependent value for k can be calculated by
k(T) = k(T0)·(1+αk·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.
• Absorber temperature
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 ΔTstat.
The heat diffusion into the GaAs substrate and the resulting temperature distribution T(r,t) can be described by the heat diffusion equation (1)
eq. (1) with a - thermal diffusivity (3.1·10-5 m²/s for GaAs)
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 Tstat of the absorber layer caused by the mean absorbed optical power Pm, then we can use the following relations for a centre symmetric three dimensional heat flow (eq. 2)
The approximation used is ri « ro when we consider the GaAs wafer thickness as ro substantial larger then the laser spot radius ri=r. Then the mean absorbed optical power is
To take into account that the heat flow is only into a half space (not to air) and that in the vicinity of the absorber layer the flow is nearly one dimensional (not three dimensional as calculated) the real temperature rise is about by a factor of 4 higher then in the above equation (2), so that
eq. (4) with
If the laser spot radius r is larger then the substrate thickness d then the heat flow is nearly one dimensional and then
r must be replaced by d in the above equation (4).
The time dependent solution of the one dimensional heat equation within the absorber layer can be estimated using the response of the system on a Dirac delta function, which simulates the heating at z=0 during a short laser pulse. The temperature evaluation near the surface (z² < 4at) after the laser pulse at t>tp can be written in this case as
eq. (5) with
Here the heated volume after the laser pulse is approximated by the product of the illuminated area
and the diffusion length (4at)1/2.
The maximum temperature of the absorber layer at z=0 can be expected after the absorbed optical energy is converted into heat. This is roughly after the sum of the pulse duration tp and the carrier relaxation time τ
The maximum total absorber temperature increase ΔTmax can be described by the sum of the time dependent dynamical part ΔTdyn(τ+tp) and the static part ΔTstat.
eq. (7) with
The two figures below show the static temperature rise ΔTstat as a function of the optical spot radius r and the dynamic temperature rise ΔTdyn as a function of the absorber relaxation time for typical parameters in solid state (absorption A = 0.03) and fiber lasers (absorption A = 0.3) according to eqs. (5) and (7).
The maximum temperature rise decreases with increasing absorber relaxation time τ because the absorbed energy is released from the excited electrons into the crystal lattice after the relaxation time. If the relaxation time τ is longer then the pulse duration tp the electrons store the absorbed energy for a short time whereas the thermal energy diffuses already from the absorber layer into the substrate.
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:
If the SAM is anti reflection (AR) coated ( Rs = 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.
If the absorber layer is enclosed in a cavity determined by the Bragg mirror and the surface mirror with Rs > 0 then the light is partial reflected on both mirrors. Two effects determine the net reflectance and the dispersion:
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 resoance 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.