Background Correction

The signal S⁽ⁱ⁾ comprises the transient Raman signal R⁽ⁱ⁾ superimposed on a slowly varying background B⁽ⁱ⁾ from further nonlinear interactions. In previous publications the background has been modelled by splines or polynomials.^[16,19] The performance of this correction depends on the actual background that is described. Whereas such an approach is well suited for stationary Raman spectroscopy, in transient Raman spec-troscopy it is important to remove the background at all delay times consistently. Note that during an FSRS measurement the background may vary significantly, since it repre-sents mainly a bleach of the transient absorption signal induced by the Raman pump. In particular, wave packet motion in the ground or excited state causes an oscillatory mod-ulation of the Raman background. Any correction method that relies on a description of the background shape translates deviations from the model function directly into fluc-tuations of the Raman baseline. Since the background is large compared to the signal, this may render an analysis of the temporal evolution of the Raman spectra impossible.

In this view it appears advantageous to derive the baseline position from a description of the Raman signal itself. In this work a correction procedure was developed, in which global analysis is used to obtain a spectral basis set to the Raman signal.

Generation of a Spectral Basis to the Raman Signal

In contrast to the Raman signal itself, the background is rather insensitive to small changes of the Raman center frequency ν_e⁽ⁱ⁾. For an ideal measurement one expects a linear shift response ofR⁽ⁱ⁾ while the backgroundB⁽ⁱ⁾ stays constant,

S⁽¹⁾(ν) =_e R⁽¹⁾(ν) +_e B⁽¹⁾(ν),_e (3.8)

S⁽²⁾(ν) =_e R⁽²⁾(ν) +_e B⁽²⁾(ν)_e

=R⁽¹⁾(ν_e+ν_e⁽²⁾−ν_e⁽¹⁾) +B⁽¹⁾(ν)._e

(3.9) Hence, recording spectra with two frequency-shifted pulses can help to identify the Ra-man features R⁽ⁱ⁾. In the setup presented above, a chopper switches between the two Raman frequencies on a shot-to-shot basis, thus maximizing the correlation of back-ground fluctuations. Both spectra are compared in 3.8, middle, as a function of Stokes

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3.5 The Femtosecond Stimulated Raman Spectrometer

Figure 3.9: Difference Spectra of trans-stilbene in n-hexane at 0.5 ps delay, given in the Stokes-shift scale (ν_e⁽¹⁾−ν) of Raman1._e

detuning ν. The similarity of the estimated background spectra (dashed black) is evi-dent. If they were exactly equal, a formation of the difference spectrum ∆Swould cancel the two contributions, leaving only the Raman features,

∆S(ν) =_e S⁽¹⁾(_eν)−S⁽²⁾(ν)_e

=R⁽¹⁾(ν)_e −R⁽²⁾(ν_e). (3.10)

As shown in 3.9, top, the experimentally obtained difference spectrum ∆S still bears a vanishing background. The residual may result either from changes in the non-linear interaction or from small deviations in the experimental conditions between the two Raman experiments. As a consequence, the equality of B⁽¹⁾ and B⁽²⁾ was not assumed for background correction, and the B⁽ⁱ⁾ were determined individually. The strong correlation between R⁽¹⁾ and R⁽²⁾, however, was used to identify the Raman features.

The temporal evolution of a Raman band may be reproduced reliably only with a consistent data treatment over all delay times. In previous publications the background has been modeled by splines.^{[16] [19]} To extract Raman peaks, which evolve gradually on a fluctuating background, it seems advantageous to describe the Raman signal itself.

Here the transient Raman spectra are approximated by a linear combination of spectral basis functions. This allows an initial background guess which is then refined. As an exampletrans-stilbene measured in n-hexane with 6 fs steps is discussed (cf. 3.8).

Starting from S⁽ⁱ⁾ we search for an approximation to R⁽ⁱ⁾ which takes into account

3 Experimental Section

Figure 3.10:Modified Raman signalS_mod(ν) =S⁽¹⁾(ν) +S⁽²⁾(ν)/2 oftrans-stilbene inn-hexane at different delay times.

the temporal evolution of the Raman signal. An operation is required that allows to extract the time-dependence of the Raman features R⁽ⁱ⁾(t). One choice would be the frequency derivative ^dS_dν⁽ⁱ⁾, since the background may be assumed to change only slowly with frequency. This would, however, enhance the baseline noise, and therefore a dif-ferent approach was chosen: a 30-point (60-80 cm⁻¹) moving average A₃₀ is applied to S⁽ⁱ⁾ and subtracted, to obtain

S⁽ⁱ⁾_mod=S⁽ⁱ⁾−A₃₀S⁽ⁱ⁾. (3.11)

In a summation the moving average works distributively, hence

A₃₀S⁽ⁱ⁾ =A₃₀R⁽ⁱ⁾+A₃₀B⁽ⁱ⁾. (3.12)

As the background may be assumed to vary only slowly with frequency,A₃₀B⁽ⁱ⁾≈B⁽ⁱ⁾ and

S⁽ⁱ⁾_mod=R⁽ⁱ⁾−A₃₀R⁽ⁱ⁾. (3.13)

3.10 demonstrates that the modified spectra reflect the Raman spectral evolution, al-though the signal shape is distorted.

A global multiexponential analysis is now performed onS_mod⁽ⁱ⁾ , as described in Section 3.6.2. In the current example (3.10), S_mod⁽ⁱ⁾ is fitted with two exponential functions (τ₁

= 0.35 ps,τ2 = 1.1 ps) and an offset (τ3 =∞); the exponential sum is convoluted with a Gaussian system response of 0.15 ps full width at half maximum. Next, a three-state model is introduced assuming that the states interconvert according to the formal kinetic

52

3.5 The Femtosecond Stimulated Raman Spectrometer

Figure 3.11: Basis spectra s⁽ⁱ⁾_k (for Raman1 and Raman2) as derived from a global fit of the modified Raman signal S_mod⁽ⁱ⁾ of trans-stilbene in n-hexane with 0.35 and 1.1 ps decay times. Right: Fit ofs_k exemplified fors₁ (top, Raman1 and Raman2 averaged), and reconstruction of the Raman spectra r₁ and r₂.

sequence 1→2→3. The exponential description of S_mod⁽ⁱ⁾ is then transformed^[76] into S_mod⁽ⁱ⁾ (ν, t) =a₁(t)s⁽ⁱ⁾₁ (ν) +a₂(t)s⁽ⁱ⁾₂ (ν)

+a3(t)s⁽ⁱ⁾₃ (ν).

(3.14)

Here the basis spectra s⁽ⁱ⁾_k (ν) (commonly refered to as evolution associated spectra) are time-independent; the time-dependence is captured by the (relative) populations ak(t), which apply to both Raman1- and Raman2-generated transient spectra. Finally, the spectral basis is reduced to those spectra which describe the dominant spectral changes clearest, in the present example s1 and s2.¹ The spectra s1 and s2 are compared in 3.11,left (note that s⁽¹⁾_k ands⁽²⁾_k are shown as a function of Stokes shiftν). The Raman features are clearly visible and highly reproduced for both excitation wavelengths.

The spectraS_mod⁽ⁱ⁾ are now expressed in the reduced basis by fitting their coefficients a⁰_k(t) at each delaytime,

S_mod⁽ⁱ⁾ (ν, t) =a⁰₁(t)s⁽ⁱ⁾₁ (ν) +a⁰₂(t)s⁽ⁱ⁾₂ (ν). (3.15)

1For the correction of thetrans-stilbene measurement discussed here,s1 ands2 form a sufficient basis.

The offsets3 could be included as well but was dismissed due to the lower signal/noise level.

3 Experimental Section

Similarly, the Raman spectra can be expanded in a basisr⁽ⁱ⁾_k as

R⁽ⁱ⁾(ν, t) =a⁰₁(t)r⁽ⁱ⁾₁ (ν) +a⁰₂(t)r⁽ⁱ⁾₂ (ν). (3.16) The basis spectra are related via

s⁽ⁱ⁾_k (ν) =r_k⁽ⁱ⁾(ν)−A₃₀r_k⁽ⁱ⁾(ν). (3.17) For reconstruction (3.11, right), every r⁽ⁱ⁾_k is described by a sum of Lorentzians L_j and their first derivatives,

The coefficients c_j and d_j, and the spectral positions and widths of the Lorentzians are determined by fitting the basis spectra s⁽ⁱ⁾_j for both Raman wavelengths (i= 1,2) simultaneously. In the fit the equality ofs⁽¹⁾_k (ν) ands⁽²⁾_k (ν) on the Stokes detuning scale is introduced,

The shifted Raman spectra are used to identify the Raman features. For example, the oscillatory features at frequencies>1700 cm⁻¹ can be assigned to background noise. As positive and negative peaks contribute to the signal, this reconstruction is not unique and has to be guided by comparison to the raw data. The information available for the fitting procedure depends on the length of the applied moving average. If the moving average window is smaller, background rejection is stronger and the shape in spectrally congested regions becomes clearer. On the other hand, also the amplitude of broad Raman bands will be reduced, and their reconstruction becomes more difficult.

Generation of the Background Spectrum

An approximation to the Raman spectrum R⁽ⁱ⁾ at each delay time (here denoted as R⁽ⁱ⁾appr) is now constructed by inserting r⁽ⁱ⁾_k into equation 3.16. Subtraction from the initial signalS⁽ⁱ⁾ then gives an approximation to the background spectrum.

B_appr⁽ⁱ⁾ =S⁽ⁱ⁾−R_appr⁽ⁱ⁾ . (3.20)

By smoothing this spectrum with a 30-point moving average, one arrives at the final estimate for the background

B⁽ⁱ⁾=A₃₀B⁽ⁱ⁾_appr. (3.21)

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3.5 The Femtosecond Stimulated Raman Spectrometer

Figure 3.12: Transient Raman spectra of trans-stilbene in n-hexane at 60 ps delay (black), and spectra after subtracting a noise guess S_noise which is common for both signals, Raman1 and Raman2.

The Raman spectrum R⁽ⁱ⁾ is obtained as

R⁽ⁱ⁾=S⁽ⁱ⁾−B⁽ⁱ⁾. (3.22)

Spectrally sharp features which were not reproduced by the base spectra, enter the signal during this step due to the previous smoothing of the background. The background and the retrieved signals are also shown in 3.8.

The dependence of the Raman peak positions on the excitation wavelength was used here to improve the signal recognition during the fitting of the basis spectra s⁽ⁱ⁾_k . In an alternative approach, the Raman spectra can be reconstructed from the difference spec-tra ∆S (eq. 3.10), thus profiting from the inherent reduction of the background.^[77,78]

In stationary Raman spectroscopy this technique has been widely applied for fluores-cence rejection (Shifted Excitation Raman Difference Spectroscopy, SERDS).^[79,80] In the present work, however, difference formation does not completely cancel the back-ground. The residual background and the complicated signal shape hamper spectral reconstruction. In 3.9 the difference spectra at 0.5 ps delay before and after background correction are compared. Clearly all spectral features are preserved, and the shape as well as the amplitudes are reproduced. In future, an inclusion of the difference spectrum into the correction algorithm may further reduce background artefacts.

3 Experimental Section

Noise Reduction for Small Signals

For small signals at long delay times, the identitity ofR⁽¹⁾(ν) andR⁽²⁾(ν) on the Stokes-shift scale allows to guess the pixel-dependent correlated noise (3.12). The denoised spectrumR_dn is initially approximated as the average over R⁽¹⁾ and R⁽²⁾, smoothed by a 4-point moving average.

R_dn(ν) =A₄(R⁽¹⁾(ν) +R⁽²⁾(ν))/2. (3.23)

It is subtracted from the individual Raman signalsR1 and R2. The residuals are then averaged on the absolute frequency scale to achieve an estimate for the noise which is common to both Raman spectra,

S_noise(ν) =_e R¹(ν)_e −R_dn(ν+ν_e⁽¹⁾) +R⁽²⁾(ν)_e −R_dn(ν+ν_e⁽²⁾)/2.

(3.24) On this basis, a new approximation for the denoised spectrum is obtained, and the procedure iterated:

R_dn,new(ν) = (R⁽¹⁾(ν)−S_noise(ν_e⁽¹⁾−ν_e) +R⁽²⁾(ν)−S_noise(_eν⁽²⁾−ν)_e /2.

(3.25) The algorithm slightly improves the signal/noise level while the difference spectrum stays unchanged. It is only applicable in the case of small signal/noise ratios (for example at long delay times) and if the Raman signal is equally reproduced in both R1 andR2. Characteristics of the Background Correction Algorithm

In conclusion, the use of a spectral basis set to correct time-dependent Raman data has several advantages:

(i) The evolution is treated in a uniform way over all delay times. This allows to process large data sets (here typically 660 delay times) without manual adjustment.

(ii) Base spectra obtained from global analysis profit from a signal/noise improvement which allows to identify the Raman bands better.

(iii) The signalR⁽ⁱ⁾can be extracted from a changing background. Transient absorption artefacts like oscillations from wavepacket motion are removed.

(iv) Within the limitation of the moving average length, all sharp spectral features are retained, even if they are not included in the intermediate description by base spectra. This is demonstrated by the oscillations of individual bands and the ultrafast spectral shift of the 200 cm⁻¹ mode.

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3.6 Data Analysis

Im Dokument Femtosecond stimulated resonance Raman spectroscopy (Seite 64-71)