4 Experimental methods for SESAM characterization
4.4 Data analysis and interpretation
4.4.2 Matrix formalism for calculating the reflectivity of a multilayer stackmultilayer stack
The reflectivity spectrum of a multilayer stack consisting of two materials with dif-ferent refractive indexes and a substrate can be calculated taking into account the transmission and reflection at every single layer interface. A widely used method for the calculation is the so called transfer matrix method as it can be found for example in [Tr¨a07]. Therefore, one determines a matrix Mi representing the i-th layer of the stack. Due to boundary conditions the electric field strength Ei-1 and magnetic field strengthHi-1 of thei-th layer compared to the field strengths at the rear of thei+ 1-th layerEi and Hi is index of the material and λ the wavelength of the incoming light and the traveling distance through thei-th layer: di =di,0cosθ wheredi,0 is the layer thickness and θthe angle of incidence. Multiplying the transfer matrices of each of theN layers results in the transfer matrix MS of the entire layer stack: MS =MNMn-1. . . M2M1. If the layer stack is built on a substrate theN-th layer represents the substrate.
The reflectivity R(λ) for a wavelength λ is given by R(λ) = wherekout = 2πnout/λ is the wave number outside the structure, so that for air atmo-sphere the refractive index nout = 1.
For the calculation of a reflectivity spectrum over a wide wavelength range it has to be kept in mind that the reflectivity for every single wavelengthλ has to be calculated according to Eq. (4.7) and that the refractive indexes of the materials do depend on the wavelength, ni =ni(λ).
4.4.3 Single pulse spectroscopy
Based on the four-state signal detected by a photo-diode the absolute reflectivity of the SESAM is calculated by the ratio of the SESAM signal to the signal reflected from the HR mirror, each with the dark signal subtracted and averaged over many measurements for high precision. The left panel of Fig. 4.4 shows the four-state signal.
The absolute reflectivity R for a given fluence F is: R(F) = B(F)/A(F). Due to a fluence range of several orders of magnitude the linearity of the signal processing
4.4 Data analysis and interpretation
1 2
3
4
Time Signhoo DoalPtide
d kar
A B
100 101 102 103 104 60
70 80 90 100
Fluence (µJ/cm²)
Reflectivity (%) SESAM
HR mirror
Figure 4.4: Left: Four-state signal of the photo-diode in SPS setup: Depending on the chopper position four signal states occur: Arm with HR mirror is open (1), both arms are open (2), arm with SESAM is open (3), both arms are blocked, respectively dark measurement (4). The absolute reflectivity of the SESAM for a predefined fluence is the ratio B/A. The red dots mark one period, respectively one full chopper rotation.
The figure is taken from [Maa08]. Right: Example for nonlinear reflectivity curve ex-perimentally measured with SPS. For reasons of reference and accuracy measurements the experimentally obtained reflectivity of the HR mirror at 100 % is also plotted. The fit curve of the SESAM reflectivity is based on Eq. (3.2).
is demanding. However, an accuracy level of 0.1 % could be reached [Fle13]. The photo-diode detects the reflected light and is connected to a signal amplifier that sends the signals to an USB oscilloscope. The calculation of the absolute reflectivity of the SESAM, even taking into account signal amplification at low fluences, is done by a Matlab program developed in our working group.
The resulting reflectivity curves based on experimental data obtained by single pulse spectroscopy are shown in Fig. 4.4 (right). The experimental data of the reflectivity of the HR mirror are plotted as well as the nonlinear reflectivity of a SESAM. Based on the constant reflectivity of the mirror at 100 % the accuracy level of 0.1 % was extracted [Fle13]. The obtained nonlinear reflectivity curve of the SESAM is fitted with Eq. (3.2). The fit routine delivers directly the following parameters: Saturation fluence Fsat, roll-over parameter F2, the linear reflectivity Rlin, and the nonsaturable reflectivityRns, as they are introduced in Section 3.2.2. The effective modulation depth
∆Reff is the maximum change in reflectivity:
∆Reff = Maximum(R−Rlin) = Maximum(∆R). (4.8)
A detailed theoretical analysis of the reflectivity measured by means of SPS is published in [Hai04, Maa08, Fle13].
4.4.4 Degenerate pump-probe spectroscopy
Pump-probe measurements deliver reflectivity transients, i.e. the change in reflectivity versus time. For a quantitative analysis the change in reflectivity ∆R is related to the linear reflectivityRlin, respectively the reflectivity without any excitation of the pump pulse. Fig. 4.5(a) shows the signal detected with the photo-diode where Rlin can be extracted. The time dependent transient ∆R/Rlin is plotted in Fig. 4.5(b). Zero time delay is set by a correlation of the pump and the probe pulse in a TPA or SHG crystal.
Based on this time resolved measurement one can mainly extract information about carrier dynamics in the SESAM. However, a correct analysis also allows to extract the fluence dependent nonlinear reflectivity curve of the SESAM [Fle13]. The following paragraphs will expand on both issues.
By measuring reflectivity transients for different pump fluences important character-istics of the dynamics of a SESAM can be investigated. The left panel of Fig. 4.6 shows transients for low fluences. The transients exhibit a sharp peak around zero time delay. For low fluences the reflectivity peak increases linearly with the incident pump fluence. This peak results from the pump pulse that excites electrons into the valence band of the quantum well absorber leading to absorber bleaching. A stronger absorber bleaching leads to a higher reflectivity.
This first maximum is followed by an exponential decay determined by the recombi-nation processes of the excited electron-hole pairs. The absorber recovery time corre-sponds to the time constant ∆τslow related to this exponential decay. The reflectivity of a normalized transient reaches the reflectivity value 1/e of its maximum for a time delay ∆τslow. Since this time constant is in the ps regime and much slower than the fast time constant corresponding to intraband relaxation processes (see Section 3.3 for carrier dynamics after optical excitation), this recombination time is often called slow time constant [Kel03]. As it is summarized in Section 3.4, this slow time constant can be influenced by a variety of fabrication and post-growth processes. Since the fast time constant, respectively the time constant related to intraband relaxation processes, is in the order of hundreds of fs, its experimental observation requires a laser with short enough laser pulses. The Yb:YAG laser presented in Section 7.3 emits ps-pulses, so that fast time constants<1 ps cannot be resolved in the transients.
For higher pump fluences the shape of the transients changes as it can be seen in the right panel of Fig. 4.6: The first reflectivity peak saturates or even decreases again whereas after a few ps a second increase of the reflectivity occurs. The saturation of the first reflectivity peak comes from the saturation of the absorber, i.e. all states in the valence band of the quantum well absorber are occupied. A further increase of the pump fluence leads to induced absorption. As it is discussed in Section 3.3.2, due to two-photon absorption (TPA) and free-carrier absorption (FCA) electrons are excited in high states reducing the absorber bleaching. Hence, the first reflectivity peak decreases again. Due to a delayed relaxation of the highly excited carriers into the minimum of the valence band the absorber is bleached time-delayed leading to a second increase of the reflectivity after a few ps [Lan99, Jos00, Gop01]. For very high
4.4 Data analysis and interpretation
Rlin
∆ R (a)
Voltage Photo Diode
Time delay (ps)
0 (b)
∆R/R lin (%)
Time delay (ps)
−5 0 5 10
0 (c)
∆R/R lin (%)
Time delay (ps)
101 102 103
0 (d)
∆R 0/R lin (%)
Fluence (µJ/cm2)
exp. data fit
Figure 4.5: Steps how to extract nonlinear reflectivity curves based on PPS data. (a) Signal of photo-diode, i.e. voltage versus time delay. The linear reflectivityRlin and the change in reflectivity ∆R are marked. Provided that stray light of the pump and other disturbances on the diode signal are small the dark current of the photo-diode can be identified as the reflectivity of the probe without influences of the pump, respectively asRlin. It is clear that these values are voltage values. For quantitative analysis of the reflectivity change the ratio ∆R/Rlin is built and plotted versus time. The resulting transient is plotted in (b). Measuring transients for different pump fluences allows fluence dependent analysis. (c) Transients for different fluence values (arrow points in direction of increasing pump fluence). For better visibility of the zero time delay it is zoomed in. In the transients the reflectivity change at zero time delay ∆R0/Rlin is marked with red dots. Plotting ∆R0/Rlin versus pump fluence results in a nonlinear reflectivity curve relative to Rlin, as it is plotted in (d). The experimental data points are fitted with Eq. (4.10). Since the curve shows the change in reflectivity relative to Rlin, the curve starts at 0 for small pump fluences.
0
∆R/R lin (%)
Time delay (ps)
0
∆R/R lin (%)
Time delay (ps)
Figure 4.6: Typical reflectivity transients of a SESAM for different pump fluences measured with PPS. The arrow points in direction of increasing fluence. Left: Typical transients for low fluence values. With increasing fluence the reflectivity at zero time delay increases due to the stronger absorber bleaching. Right: Typical transients for high fluence values. With increasing pump fluence effects based on induced absorption occur. Due to TPA and FCA the absorber bleaches delayed leading to a second increase of the reflectivity after the first peak at zero time delay. Since the transients are exemplary, axes ticks are omitted. The change in reflectivity can vary between a few percent and tens of percent. Typical absorber recovery time constants are in the range of a few ps up to hundreds of ps.
fluences TPA and FCA can even be so strong that the probe pulse is fully absorbed leading to a dip of the reflectivity at zero time delay.
Information such as the absorber recovery time and the appearance of TPA and FCA are very important for SESAM mode-locking. Hence, time resolved reflectivity mea-surements are a crucial part of SESAM characterization.
Apart from carrier dynamics also the nonlinear reflectivity curve of a SESAM is impor-tant, since it allows to characterize the modulation depth, the saturation fluence, and the roll-over parameter. The nonlinear reflectivity curve can also be extracted from the time resolved transients. Fleischhaker et al. showed the consistency of measuring the nonlinear reflectivity curve of a SESAM based on SPS data and extracted from PPS data [Fle13]. Only for a perfect overlap in space and time of the pump and the probe beam the reflectivity of the probe pulse is exactly the same as the reflectivity of a single pulse in the SPS setup. Thus, extracting the reflectivity of the probe pulse at zero time delay of transients of different pump fluences allows for conclusions regard-ing the nonlinear reflectivity curve. With particular care the zero time delay has to be determined. Due to the time dependent reflectivity of the SESAM resulting from the temporal shape of the pump pulse the reflectivity transients, respectively the time resolved reflectivity of the probe pulse, do not show the maximum at zero time delay [Fle13]. Hence, zero time delay has to be fixed by means of correlation of the pump and probe pulse.
4.4 Data analysis and interpretation
As already mentioned above to extract the nonlinear reflectivity curve special atten-tion has to be turned on the reflectivity at zero time delay. For a fluence dependent reflectivity curve transients at different pump fluences have to be measured. Taking the relative reflectivity change at zero time delay ∆R0/Rlin for different pump fluences gives a nonlinear reflectivity curve relative to Rlin. The different steps how to extract fluence dependent reflectivity curves of a SESAM based on pump-probe measurements are illustrated in Fig. 4.5. Based on the voltage signal detected by the photo-diode (see Fig. 4.5(a)) the transient (Fig. 4.5(b)) is obtained. For quantitative analysis the ratio ∆R/Rlin is taken giving relative values. For fluence dependent analysis transients for different pump fluences have to be measured (see Fig. 4.5(c)). Plotting the relative reflectivity change at zero time delay ∆R0/Rlin versus incoming pump fluence leads to a nonlinear reflectivity curve in relation to Rlin, as it is shown in Fig. 4.5(d).
It is obvious that this is no absolute reflectivity curve, but a nonlinear reflectivity rel-ative to Rlin. This results in two consequences. Firstly, the equation for fitting this relative nonlinear reflectivity curve is a modified version of Eq. (3.2). The fluence dependent reflectivity change is
∆R0(FP)
Rlin = R0(FP)−Rlin
Rlin = R0(FP)
Rlin −1. (4.9)
Since the reflectivity at zero time delayR0(FP) for a given pump fluenceFPis the same as the reflectivityR(FP) of a single pulse with pulse fluence FP [Fle13], Eq. (4.9) can be rewritten inserting Eq. (3.2) with a new fit parameter ρ = Rlin/Rns. The relative nonlinear reflectivity curve of SESAMs extracted from pump-probe transients can then be fitted with Eq. (4.10).
Secondly, the parameter that can be extracted from this fitting routine are the
saturation fluenceFsat and the roll-over parameterF2. Instead of the linear reflectivity Rlin and the nonsaturable reflectivity Rns that can be directly obtained by SPS data the fitting routine based on PPS data just allows the extraction of the ratio ρ = Rlin/Rns. For nonsaturable reflectivity closed to 100 % the linear reflectivity is similar to ρ. Furthermore, based on the relative nonlinear reflectivity curve only the maximum of
∆R0/Rlin can be obtained. Consequently, no absolute value for ∆Reff can be specified, but only the ratio ∆Reff/Rns.
The derivation of ∆Reff/Rns based on the fit parameters is explained in the following.
It begins with Eq. (4.8)
∆Reff = Maximum(∆R) = Maximum(∆R0) (4.11) with ∆R0 being the reflectivity change at zero time delay obtained by pump-probe measurements.
The relative nonlinear reflectivity curve only gives the maximum of the relative reflec-tivity change, i.e. Maximum(∆R0/Rlin).
To maintain the equality of Eq. (4.11) ∆Reff is:
∆Reff = Maximum
∆R0 Rlin
·Rlin. (4.12)
However, the fit routine of the relative nonlinear reflectivity curve does not give Rlin
but only
ρ= Rlin
Rns. (4.13)
Inserting Eq. (4.13) into Eq. (4.12) without changing the equality ∆Reff = Maximum(∆R0) gives
Hence, ∆Reff/Rns is obtained by multiplying the maximum value of the relative non-linear reflectivity curve, Maximum
∆R0
Rlin
, with the fit parameter ρ.
Thus, PPS measurements only deliver relative nonlinear reflectivity curves and relative SESAM parameters. For an absolute nonlinear reflectivity curve as well as absolute values for ∆Reff and Rns the linear reflectivity of the SESAM has to be determined.
Rlin can be obtained for example by FTIR measurements.
4.4.5 High-speed asynchronous optical sampling
The measured signal of the ASOPS system is a typical transient showing the time de-pendent reflectivity change, as it is shown in the left panel of Fig. 4.7. This transient exhibit a sharp increase of the reflectivity followed by a multiexponential decay. The oscillations on the transient result from the generated phonon modes. For a better analysis of these phonon oscillations background of the time resolved signal is numer-ically subtracted leading to oscillations as they are shown in the right panel of Fig.
4.7. The frequency spectrum of the active modes is obtained by Fourier transform of the extracted oscillations in the time domain signal. Further information about the data analysis of ASOPS transients can be found in [Het13]. An introduction to phonon generation and detection is given in Section 6.1 of this thesis.
The experimentally obtained frequencies of the phonon modes are measured with ac-curacy of 1 GHz at center frequencies around∼300 GHz. If the phonon modes are used for structural analysis this frequency resolution results in an error of the thickness extraction of approximately 1 nm for periods of the Bragg mirror of∼150 nm.
4.4 Data analysis and interpretation
0 200 400 600 800 1000 1200 0
0.05 0.1
Time delay (ps)
∆ R/R 0
300 500 700 900
0 1 2x 10−3
200 400 600 800 1000 1200
−2
−1 0 1 2x 10−4
∆R/R 0
Time delay (ps)
Figure 4.7: Left: Typical reflectivity transient measured with the ASOPS system.
In the zoom oscillations originating from coherent acoustic phonons are visible. The numerically extracted oscillations are shown in the right panel.