shows sample traces for the spectral re-shaping (a) and the Z-scan technique (b) at different concentrations of the intralipid solution. To calibrate the scattering strength as a function of intralipid concentration, we performed a set of linear transmission measurements. For these measurements we recorded only the unscattered light passing through the cuvette (the ballistic photons). We fit the resulting transmission data with the exponential function T = exp(−a0
C L) (Beer’s law), where C is the concentration (in % vol) and L is the length of the cuvette. We obtained an exponential reduction parameter of b0 = a0
L = 10.4.
Fig. 2 Part (a) shows spectral re-shaping scans at concentrations C0, C1 and C2. Part (b) shows averaged Z-scan traces at concentrations C0 and C1. The red lines in the graphs are the corresponding fit traces. Part (c) shows relative linear transmission of the (more ...)
The traces in show qualitative differences. The spectral re-shaping traces in (a) provide a signal that is proportional to the nonlinear coefficient at every position within the sample [18
]. It is apparent that glass exhibits a larger self-phase modulation coefficient than water and the intralipid solution. We can also see that in a scattering sample, the signal decays exponentially as a function of focus position, because scattering reduces the amount of light reaching the focal volume (the left side of the graph corresponds to the light entrance side). Even in the case of pure water (C0
= 0) we observe a reduction in signal when moving the focus towards the exit surface. We traced this behavior to the combination of two effects: the dispersive broadening of the ultrafast pulse in the solution and the small loss of power due to the refractive index mismatch at the water/glass interface.
The Z-scan traces in exhibit the typical features of self-focusing/defocusing. The transmission changes occur near interfaces (air/glass and glass solution) and are sensitive to the relative change of nonlinear index (this is in contrast to the absolute nature of the spectral re-shaping signal). Here, too, a signal decrease due to scattering, index mismatch and dispersion is apparent in the asymmetry of the peaks at the air/glass and glass/air interfaces.
To quantify the influence of scattering on the two measurement techniques we performed nonlinear least-square fits on the signal trace averages (9 traces were averaged for each concentration). For the spectral re-shaping method, we approximated the trace within the cuvette as the sum of an exponential decay and a Gaussian peak centered at the position of the glass slide. Note that with this model function, the Gaussian amplitude measures the difference in nonlinear coefficients between the glass and the surrounding solution (in analogy to the Z-scan case). The traces in show the best fit curve as thin red lines. We then extracted the values of the Gaussian amplitude and normalized this set to the case of pure water. The resulting data set is shown in (top).
Fig. 3 Relative nonlinear self-phase modulation (SPM) signals in the glass slide as a function of concentration of the intralipid solution. Shown are spectral re-shaping (top) and Z-scan results (bottom). The inset shows a high-resolution scan for low concentrations. (more ...)
The influence of scattering in these measurements can be estimated as follows. Once the light has entered the cuvette it is attenuated by the factor exp(−a0C L / 2) on its way to the focus assuming that the glass is centered in the cuvette. Since the spectral re-shaping scales quadratically with intensity, the signal at the focus is reduced by exp(−a0C L). The resulting spectral signature is further attenuated by exp(−a0C L / 2) between the focus (glass slide) and the exit side of the cuvette. This results in a reduction of exp(−3/2 a0C L), yielding bSpec = 3/2 b0, where b0 is the reduction parameter for linear transmission. An exponential curve with this parameter is indicated in the top panel of and is seen to match the experimental data very well (note that this curve has not been obtained by fitting the SPM data).
We also extracted quantitative data from the Z-scan traces via nonlinear least-square fitting. The model function we used is applicable to “thick” refractive samples (Eq. (29) in ref [20
]). For concentrations above C1
we were are not able to fit Z-scan traces due to extremely low signal to noise ratios. We normalized the obtained value for nonlinear refraction (d0R
in ref [20
]) to the pure water case. The results are displayed in (bottom). The estimation for the influence of scattering can be performed similarly. However, in a Z-scan trace the signal that is analyzed is the power transmitted through the aperture divided by the transmission when the sample is out of focus (i.e. the linear transmission through the aperture). The absolute reduction of aperture transmission also scales quadratically with focal intensity, leading to a reduction of exp(−3/2 a0C L
). However, division by the linear transmission (proportional to exp(−a0C L
) ) results in a reduction parameter of bZ-scan=
. A curve with such a behavior is displayed in (bottom). To further test this model, we replaced the glass sample with a 1 mm thick color glass filter (RG665), which shows strong two-photon absorption (TPA) at our laser wavelength. We then performed an open aperture Z-scan to extract the TPA coefficient for several concentrations of the surrounding scattering solution. The resulting relative TPA values are also plotted in . In contrast to the SPM values, the TPA values fit the expected reduction curve well.
In order to illustrate the difficulty in measuring nonlinear refractive index in scattering media we plotted the ratio of fit value to the statistical fit error obtained by the nonlinear least-squares fit in for both measurement techniques. It is apparent that the reliability of the fit decreases rapidly with increased scattering. For the Z-scan only concentrations below 0.05 percent by volume resulted in reliable fit values.
Fig. 4 Ratio of best fit values for the nonlinear coefficient to the fit error (standard deviation) for the spectral re-shaping and Z-scan technique as a function of concentration of the intralipid solution. The fit was performed on a 9-trace average. Note that (more ...)