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FAQs

*Construction:*

- The SAM consists of a mirror and a saturable absorber in front of it.
- The single crystalline layers of the mirror and the absorber are grown on a GaAs wafer.

*Behaviour:*

- The SAM is a nonlinear mirror.
- The reflectance increases with increasing pulse energy.

*Application:*

- Using a saturable absorber mirror (SAM), a self-starting, passively mode-locked diode pumped laser with a very stable pulse repetition rate can be realized.
- SAMs for solid state and fiber lasers in a wide spectral region from 800 nm up to 3 µm wavelength are available.
- Also Q-switched mircochip lasers can be realized using a SAM as nonlinear optical device.

• Absorptance

*Absorptance*

The absorptance of the SAM consists of two parts:

- saturable absorption A
_{s} - non-saturable absorption A
_{ns}.

The non-saturable part A_{ns} 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
A_{ns} ~ 0.2 · A_{0}.

The total absorption is the sum of both parts A_{1} = A_{s} + A_{ns}.

The saturation of the absorption can be described with the following formula:

with

- A
_{1}- Sum of saturable and non-saturable absorption - A
_{s}- saturable absorption - A
_{ns}- non-saturable absorption - F - radial dependent pulse fluence
- F
_{sat}- saturation fluence of the absorber.

The pulse fluence F depends on the beam radius r for a Gaussian pulse as follows:

with

- F(r) radial dependent pulse fluence
- F
_{0}- maximum fluence on the beam axis - r
_{0}- Gaussian beam radius

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 A_{1} and the averaged absorption A with a
certain value of non-saturable absorption A_{ns} = 0.1 · A_{0}.

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 F_{sat} ~ 70 µJ/cm^{2}.

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, F_{sat} ~ 50 µJ/cm^{2}.

*Two photon absorption *

For short pulses the two-photon absorption A_{TPA }increases the total absorption as follows:

with

- β - two-photon absorption coefficient
- I - pulse intensity
- d - absorber layer thickness
- F - pulse fluence
- τ
_{P}- pulse duration.

The integration of the time dependent intensity I(t) over the pulse duration τ_{P} results in the pulse fluence F.
Therefore the pulse fluence can be approximated as F ~ I · τ_{P}.

• Reflectance

*Reflectance*

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

- A
_{s}- saturable absorption - A
_{ns}- non-saturable absorption - F - pulse fluence
- F
_{sat}- saturation fluence - β - two-photon absorption coefficient
- d - absorber layer thickness
- τ
_{P}- pulse duration.

The figures below show the reflectance of a SAM with A_{ns} = 0.1, A_{s} = 0.4, F_{sat} = 0.5 J/m^{2}
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.

*Modulation depth *

The modulation depth ΔR is smaller then the saturable absorption A_{s} 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

- β ~ 25 cm/GW for AlAs rich material,
- β ~ 40 cm/GW for GaAs rich material and
- β ~ 60 cm/GW for InGaAs rich material.

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:

- low-temperature molecular beam epitaxy (LT-MBE)
- ion implantation.

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 F _{sat} *

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

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 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

• Influence of absorber temperature

* Influence of absorber temperature*

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

This value is substantial larger then the linear thermal expansion coefficient α

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

n·d(T) = n·d(T_{0} + ΔT) = n·d(T_{0}) · (1 + α_{ot} · ΔT)

Δ(n·d) = n·d(T_{0}) · α_{ot} · ΔT

with

- n·d - optical thickness
- T
_{0}- reference temperature - ΔT = T - T
_{0}- temperature difference - α
_{ot}- temperature coefficient for optical thickness - Δ(n·d) - optical thickness difference.

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

α_{k} ~ 6·10^{-3}/K.

With this coefficent the temperature dependent value for k can be calculated by

k(T) = k(T_{0})·(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.

• Absorber temperature

* 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 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)

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 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)

- λ - thermal conductivity
- r
_{i}- inner (illuminated) radius of a sphere - r
_{o}- outer radius of a sphere - T
_{0}- heat sink temperature

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. Then the mean absorbed optical power is

eq.(3) with

- P
_{m}- mean absorbed optical power - A - absorptance of the absorber
- r - spot radius on the absorber
- f - optical pulse repetition rate.

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

- ΔT
_{stat}- static temperature rise - F - pulse fluence
- λ - thermal conductivity,(55 W/(mK) for GaAs)

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>t_{p} can be written in this case as

eq. (5) with

- z - coordinate in heat flow direction
- t - time after optical pulse
- a - thermal diffusivity.

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 t_{p} and the carrier relaxation time τ

eq.(6) with

- t
_{P}- pulse duration - τ - absorber relaxation time

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. (7) with

- ΔT - temperature rise

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. (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 t_{p}
the electrons store the absorbed energy for a short time whereas the thermal energy
diffuses already from the absorber layer into the substrate.

• Dispersion

* Dispersion*

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. 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 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.