Implementation and evaluation of data analysis strategies for time-resolved optical spectroscopy

Transcription

1 Supporting information Implementation and evaluation of data analysis strategies for time-resolved optical spectroscopy Chavdar Slavov, Helvi Hartmann, Josef Wachtveitl Institute of Physical and Theoretical Chemistry, Goethe University, Frankfurt am Main, Germany ABSTRACT: The supporting information (SI) contains details on the implementation of approaches for treating various experimental artifacts observed in ultrafast optical spectroscopy, e.g. deconvolution of the instrument response function, correction of time zero wavelength dependence, coherent artifact treatment. Furthermore, the SI, contains also a number of figures that show in detail the results from the evaluation of the algorithms that were discussed in the main part of the article. S-1

2 MATERIALS AND METHODS General considerations Here we present the detailed implementation procedures of different strategies for dealing with a number of experimental artifacts that typically appear in ultrafast optical spectroscopy experiments. Instrument response function Despite the significant advances in the experimental techniques, the time-resolution of the set-ups is finite and determined by the response of the instrument. In this respect the recorded experimental data, S exp, do not represent the pure molecular response of the studied compound to a Dirac δ- function input, but rather contain also the instrument response function (IRF). For techniques like single-photon counting and streak camera, the timing of the signal is provided by the detection system itself and thus the IRF is determined by the response of the detector and by the length of the excitation pulse. When the timing is done mechanically via a delay stage, as in pump-probe (e.g. transient absorption) or pump-gate (e.g. up-conversion and Kerr-shutter) techniques, the width of the IRF is determined by the width of the cross-correlation between the two pulses. In any of these cases the detected experimental signal, S exp, is a convolution,, between the pure molecular response, eventually approximated by a sum-ofexponents function in GLA, GTA and LDA, and the IRF (Eq. 5). The reconstruction of the pure molecular response involves solving the ill-posed deconvolution operation, which is a tedious task and thus the problem is typically addressed by performing iterative re-convolution of the IRF with the molecular response approximated by sum-of-exponents function. Depending on the properties of the IRF it can either be measured (like in single-photon counting experiments) or approximated by a function (e.g. Gaussian). The re-convolution procedure implemented in OPTIMUS follows 1, where an analytical expression for the convolution of an exponential decay with relaxation rate, k = 1 τ, and a Gaussian function (σ = FWHM 2 2ln2) with center location c is used: S(t) = exp( kt) 2 exp (k (c + kσ2 2 )) {1 + erf (t (c + kσ2 ) )} (S1) σ 2 In future releases other options, like convolution with measured IRF, are planned for implementation. Wavelength-dependent time zero position (chirp) The wavelength dependence of the time zero position, c, which is mainly due to the group velocity dispersion of the light but also to self-phase modulation in certain techniques, is another distortion of the recorded signal that needs to be accounted for. The chirp is pronounced on the sub-picosecond timescale and especially for wavelengths approaching the ultraviolet range. The chirp is modeled in OPTIMUS by a polynomial function of order n. 1 The time zero position of the IRF at the central wavelength, λ c, is given by c 0. n c(λ) = c 0 + c i ( λ λ i c 100 ) (S2) i=1 Coherent artifact Nowadays, ultrafast transient absorption spectroscopy is routinely performed with ultrashort excitation (pump) pulses of ~100 fs or shorter. However, to obtain a reasonable S/N ratio in those experiments the number of excited molecules needs to be sufficiently large and thus a minimum level for the pump energy is required. In effect, the experiments are conducted with relatively high peak intensities which can eventually lead to distortions in the recorded signal. The distortions could be caused for example by stimulated Raman scattering or by temporal changes in the refractive index of the medium induced by the intense pump, which leads to spectral changes in the weak probe via cross-phase modulation 2. Often the CA problem is addressed by subtracting a solvent measurement from the measurement of the studied compound. However, this procedure is tedious and usually does not yield high quality results. Furthermore, it is highly advisable to avoid alteration of the original experimental data unless it is absolutely necessary. The alternative and far more elegant way to treat the CA contributions is to approximate them with a function composed of a Gaussian and/or its first and second derivative (Eqs. 8-10) 2,3. The latter approach is implemented in OPTIMUS. It should be noted here that the time zero position, c, is the same for both the IRF and the CA function. g(t) = 1 (t c)2 exp ( σ 2π 2σ 2 ) (S3) (c g(t) t) = σ 3 2π g(t) = (t2 2ct σ 2 + c 2 ) σ 5 2π exp ( (t c)2 2σ 2 ) (S4) (t c)2 exp ( 2σ 2 ) (S5) A separate module is implemented in OPTIMUS for analysis of the pure CA from a solvent measurement. This analysis allows determination of the cross-correlation width as well as the chirp parameters. Performing such an analysis on a solvent measurement at exactly the same conditions as the sample measurement is typically very helpful for finding the appropriate parameters for the analysis of the experimental data. REFERENCES (1) van Stokkum, I. H. M.; Larsen, D. S.; van Grondelle, R. Biochim. Biophys. Acta, Bioenerg. 2004, 1657, (2) Kovalenko, S. A.; Dobryakov, A. L.; Ruthmann, J.; Ernsting, N. P. Phys. Rev. A 1999, 59, (3) Dobryakov, A. L.; Kovalenko, S. A.; Weigel, A.; Perez- Lustres, J. L.; Lange, J.; Müller, A.; Ernsting, N. P. Rev. Sci. Instrum. 2010, 81, S-2

3 Fig. S1. Results of fitting a CA signal using the CA module of OPTIMUS. The CA signal was induced in an ethanol solution (1 mm path length) by a 15 nj, 340 nm laser beam. S-3

4 Fig. S2. Evaluation of the capabilities of the GLA module of OPTIMUS. Detailed illustration of the quality of the fit for different wavelength channels. S-4

5 Fig. 3. Evaluation of the capabilities of the TGA module of OPTIMUS. Results from the unconstrained fit of the synthetic dataset. A&B) correctly determined SAS of C3 in the nm range; C&D) local minimum solution with inadequately recovered C3 SAS in the nm range. S-5

6 Fig. S4. Evaluation of the capabilities of the TGA module of OPTIMUS. Detailed illustration of the quality of the fit for different wavelength channels. S-6

7 Fig. S5. Evaluation of the capabilities of the LDA module of OPTIMUS. Comparison of the simulated datasets (different noise levels) for LDA analysis and the results from the fitting. S-7

8 Fig. S6. Evaluation of the capabilities of the LDA module of OPTIMUS. Detailed illustration of the quality of the fit for different wavelength channels for a synthetic dataset with 0.14% noise. S-8

9 Fig. S7. Evaluation of the capabilities of the LDA module of OPTIMUS. Detailed illustration of the quality of the fit for different wavelength channels for a synthetic dataset with 1.2% noise. S-9

10 Fig. S8. Evaluation of the capabilities of the LDA module of OPTIMUS. Detailed illustration of the quality of the fit for different wavelength channels for a synthetic dataset with 2.5% noise. S-10

11 Fig. S9. Evaluation of the capabilities of the LDA module of OPTIMUS. Detailed illustration of the quality of the fit for different wavelength channels for a synthetic dataset with 4.6% noise. S-11

12 Fig. S10. Evaluation of the capabilities of the LDA module of OPTIMUS. LDMs obtained using different regularization terms at two noise levels 0.14 and 2.5% S-12

13 Table S1. Testing of OPTIMUS-GLA on a synthetic data set. 95% non-linear regression prediction confidence intervals for the lifetimes. Original Lower limit Predicted Upper limit τ ps 0.15 ps 0.16 ps 0.17 τ ps 5.79 ps 6.49 ps 7.19 τ ps 35.6 ps 41.5 ps 47.4 τ ps ps ps τ 5 1 ns 1 ns fixed 1 ns S-13