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How does a SAM™ work ?


Contents  
>  1. Aim of SAM  
Passive modelocking techniques for the generation of ultrashort pulse trains are preferred over active techniques due to the ease of incorporation of passive devices into various laser cavities. 

>  2. Parameters  
A SAM consists of a Braggmirror on a semiconductor wafer like GaAs, covered by an absorber layer and a more or less sophisticated top film system, determining the absorption. 

>  3. Absorption  
A SAM is a nonlinear optical device. Therefore the absorption A_{1} depends on the pulse fluence F. If the pulse duration τ_{p} is shorter than the relaxation time τ of the absorber material, then the fluence dependent absorption is given by  
with  eq. (1)  
A_{0}  small signal saturable absorption  
F(r)  radial dependent fluence of a Gaussian pulse  
F_{sat}  saturation fluence of the absorber material  
F_{0}  average value of the pulse fluence  
r  radius, distance from the beam axis  
r_{0}  Gaussian beam radius  
The effective absorption A of a Gaussian beam is the result of an averaging over the radial dependent fluence F(r) of the pulse:  
eq. (2)  
In the figure below the saturation of the absorption according to equations (1) and (2) (blue curve) is shown. For small fluence F < F_{sat} the absorption drops down linear with increasing fluence  
The small signal absorption A_{0} is proportional to the square of the electric field strength of the standing wave at the position of the absorber layer. Therefore the saturable absorption of the SAM can be adjusted by the design. A typical value for the saturation fluence F_{sat} is 50 µJ/cm^{2}. 

For short pulses the twophoton absorption A_{TPA }increases the total absorption as follows:  
eq. (3)  
with  β  twophoton absorption coefficient  
I  pulse intensity  
d  thickness of the absorber layer  
F  pulse fluence  
τ_{P}  pulse duration  
>  4. Modulation depth ΔR  
The reflectance R of a saturable absorber mirror (SAM) is in the region of the stopband with zero transmittance determined by the absorption A according to R = 1A. The modulation depth ΔR is smaller than the absorption A_{0} because of nonsaturable losses A_{ns}: ΔR = A_{0} A_{ns}. The main reason for the nonsaturable losses are crystal defects, which are needed for fast relaxation of the excited carriers. The modulation depth increases with increasing relaxation time t. Typical values for ΔR are
The pulse fluence dependent reflection R(F) of a saturable absorber mirror is governed by the effective absorption according to eq.(2). For short pulses and high pulse intensities I the two photon absorption decreases the reflection and therefore also the effective modulation depth according to eq.(3). Therefore the reflectance R of a SAM can be written as 

eq. (4)  
The calculated SAM reflectance R as a function of pulse fluence F according to eq. (4) is shown for three different pulse durations in the figure below.  
>  5. Relaxation time  
The saturable absorber layer consists of a semiconductor material with a direct band gap slightly lower than the photon energy. During the absorption electronhole pairs are created in the film. The relaxation time t of the carriers has to be a little bit 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 prevents Qswitched modelocking of the laser.
The parameters to adjust the relaxation time in both technologies are the growth temperature in case of LTMBE and the ion dose in case of implantation. Typical values of the relaxation time t of SAMs are between 500 fs and 10 ps. An example of a pumpprobe measurement to determine the relaxation time τ is shown below. 

>  6. Saturation fluence F_{sat}  
The saturation fluence depends on the semiconductor material parameters and on the optical design of the SAM. To prevent the SAM from unwanted degradation and destruction due to high pulse fluences, the saturation fluence must be low. 

>  7. 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. 

eq. (5)  
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. 

eq. (6)  
Here λ is the thermal conductivity, r_{i} is the inner (illuminated) radius of a sphere and r_{o} an outer radius of a sphere with the heat sink temperature T_{0}. The approximation used is r_{i} « r_{o} when we consider the GaAs wafer thickness as r_{o} substantial larger then the laser spot radius r_{i}=r. The mean absorbed optical power is 

eq. (7)  
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 formula (6), so that 

eq. (8)  
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 (8). 

eq. (9)  
Here the heated volume after the laser pulse is approximated by the product of the illuminated area and the diffusion length (4at)^{1/2}. 

eq. (10)  
The maximum total absorber temperature increase ΔT_{max} can be described by the sum of the time dependent dynamical part ΔT_{dyn}(τ+t_{p}) and the static part ΔT_{stat}. 

eq. (11)  
with  ΔT  temperature rise  
A  absorption  
F  pulse fluence  
λ_{th}  thermal conductivity of the absorber material (55 W/(mK) for GaAs)  
a  thermal diffusivity of the absorber material (3.1 10^{5} m^{2}/s for GaAs)  
t_{p}  optical pulse duration  
τ  carrier relaxation time in the absorber  
r  spot radius on the absorber  
f  optical pulse repetition rate  
The two figures below show the static temperature rise ΔT_{stat} as a function of the optical spot radius r and the dynamic temperature rise ΔT_{dyn} 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. (9) and (11).  
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 t_{p} the electrons store the absorbed energy for a short time whereas the thermal energy diffuse already from the absorber layer into the substrate.  
>  8. Reflection and absorption bandwidth  
7.1 Timebandwidth product (TBWP)  
From Heisenberg's uncertainty principle for the conjugated variables pulse width Δt and photon energy E = hν the TBWP of a laser pulse is limited to about Δt^{.}Δν ≥ 1/(2π).
The minimum TBWP for a Sech^{2}pulse is Δt^{.}Δν = 0.32 . Most people do not work with frequency ν but prefer wavelength λ. Using the relation c=λ^{.}ν the frequency interval Δν is related to the wavelength interval Δλ by Δν =  c^{.} Δλ/λ^{2}. c = 2.988 ^{.} 10^{8} m/s is the speed of light in the vacuum. 

Numerical values for the minimum bandwidth Δν as a function of pulse duration Δt  


Numerical values for the minimum bandwidth in nm as a function of pulse duration Δt  


7.2 Reflection bandwidth  
The reflection bandwidth of the SAM has to be larger than the pulse bandwidth. In case of a SAM with an underlying Braggmirror the reflection bandwidth is determined by the ratio of the refractive indices n_{H}/n_{L} of the layers in the thin film stack. More about Braggmirrors ... 

7.3 Absorption bandwidth  
An ideal SAM has a constant saturable absorption for all wavelengths of the pulse spectrum. Because of the wavelength dependence of the absorption in a semiconductor material above the band gap, the absorption increases with decreasing wavelength. In case of a resonant SAM this dependency may be changed by the standing waves inside the cavity in such a way, that the maximum absorption is at the resonance wavelength of the SAM. 
