Electronic resonances in broadband stimulated Raman spectroscopy: Supplementary Information

Transcription

1 Electronic resonances in broadband stimulated Raman spectroscopy: Supplementary Information G. Batignani,2, E. Pontecorvo, G. Giovannetti, C. Ferrante, G. Fumero, T. Scopigno,3 Dipartimento di Fisica, Universitá di Roma La Sapienza, I-85, Roma, Italy 2 Dipartimento di Scienze Fisiche e Chimiche, Universitá degli Studi dell Aquila, I-67, L Aquila, Italy and 3 Center for Life Nano Istituto Italiano di Tecnologia, 295 Viale Regina Elena, I-6, Roma, Italy (Dated: September 23, 25)

2 Third order nonlinear polarization The complete third order response of the system is given by (t) = dτ 3 dτ 2 dτ E(t τ 3 )E(t τ 2 τ 3 )E(t τ τ 2 τ 3 )S (3) (τ,τ 2,τ 3 ) (S) where S (3) (τ,τ 2,τ 3 ) is S (3) (τ,τ 2,τ 3 ) = ( i ) 3 Tr(µ(τ +τ 2 +τ 3 )[µ(τ +τ 2 ),[µ(τ ),[µ(),ρ()]]]) (S2) and E(t) = i E i(t)e iωit +c.c.. By the use of energy-level diagrams, two contributions can be isolated to reproduce the stimulated Raman signal in the red and blue side of the Raman pulse (see. Fig ). RED ( (t) = i ) 3 µ ab 2 µ bc 2 dτ 3 dτ 2 dτ ER (t τ τ 2 τ 3 )E P (t τ 2 τ 3 )E R (t τ 3 ) e iωr(t τ τ2 τ3) e iωp(t τ2 τ3) e iωr(t τ3) e i ω abτ e i ωacτ2 e i ω bcτ 3 (S3) BLUE ( (t) = i ) 3 µ ab 2 µ bc 2 dτ 3 dτ 2 dτ E P (t τ τ 2 τ 3 )ER(t τ 2 τ 3 )E R (t τ 3 ) e iωp(t τ τ2 τ3) e iωr(t τ2 τ3) e iωr(t τ3) e i ω baτ e i ωcaτ2 e i ω cbτ 3 (S4) where ω ij = ω i ω j iγ ij. The frequency dispersed signal can simply be calculated by the Fourier transform (ω) = P(3) (t)e iωt dt. To reduce the computational effort to calculate S3 and S4 it is useful to write the pulse field in terms of their Fourier transform RED ( (ω) = i ) 3 µ ab 2 µ bc 2 dte iωt dτ 3 dτ 2 dτ dω dω 2 dω 3 E R (ω )e i(ωr+ω)(t τ τ2 τ3) E P (ω 2 )e i(ωp+ω2)(t τ2 τ3) E R (ω 3 )e i(ωr+ω3)(t τ3) e i ω abτ e i ωacτ2 e i ω bcτ 3 (S5) In this way all the temporal integral may be solved analytically. RED (ω) = µ ab 2 µ bc 2 ER dω (ω ) E R (ω 3 )E P (ω ω P +ω ω 3 ) dω 3 ω bc ω ω R +ω + ω ab ω ω 3 ω R ω ac (S6) 2

3 Off-Resonance Resonance Red Side (ω bc ω)(ω R ω ba ) ( iγ bc )( iγ ab ) Blue Side (ω ω ba ) 2 (iγ ba ) 2 TABLE I: Denominator of electronic factors in Eqs. S and S, for the red and the blue side under resonance and off-resonance condition. where we used the conservation of energy ei(ω ωp+ω ω3 ω2)t dt = δ(ω ω P + ω ω 3 ω 2 ) to simplify the ω 2 integral. BLUE (ω) = µ ab 2 µ bc 2 E R (ω ) ER dω dω (ω 3)E P (ω +ω P ω +ω 3 ) 3 ω ω ba ω ω R ω ω ca ω ω ba +ω 3 ω (S7) By considering monochromatic RP and continuous PP, Eqs. S6 and S7 may be recast as and RED (ω) = µ ab 2 µ bc 2 (ω bc ω iγ bc )(ω R ω ba iγ ab )(ω ω R +ω ca +iγ ac ) BLUE (ω,λ R) = µ ab 2 µ bc 2 (ω ω ba +iγ ba ) 2 (ω ω R ω ca +iγ ca ) (S8) (S9) The electronic resonance enhancement is contained in (ω bc ω iγ bc )(ω R ω ba iγ ab ) (S) for the red side, and (ω ω ba +iγ ba ) 2 (S) for the blue side. The denominator of the former contribution can be approximated as shown in table I. Remarkably, while in the red side of the spectrum the electronic resonance factor gives always the same sign to the total, in the blue side it is positive under resonance condition and negative under the off-resonance one. The sign on the off resonance red side can be easily evaluated by the substitution (ω bc ω) (ω bc (ω R ω ca )) = (ω bc ω R +ω ca ) = (ω ba ω R ), from which it follows (ω bc ω)(ω R ω ba ) (ω R ω ba ) 2. The Stimulated Raman Excitation Profile REP(λ R ) = A(λ R) ω R+ω ca ω ba +iγ ba 2 in Eq. 7 is obtained combining Eqs. and S. 3

4 Fit Procedure and Data Treatment To isolate the Raman contributions in Myoglobin SRS spectra a background signal has to be removed. Transient absorption of the Raman pulse and additional nonlinear contributions from both the solute and the solvent generate a broad baseline signal, which can be removed adding a low (second or third) order polynomial to the fit procedure presented previously (Eq. ). In Fig. S we report the polynomial background signal obtained in the fit procedure for the high frequency region around the ν 4 mode, where the filled area represents the SRS isolated contributions. R = 438 nm FIG. S: Baseline Removal example: Polynomial background signal obtained in the fit procedure for the high frequency region around the ν 4 mode, where the filled area represent the SRS isolated contributions. Broadband SRS spectra of Cyclohexane is acquired and used for a calibration of the Raman pulse and Raman probe parameters (see Eq. 4). The precise narrow-band Raman pulse wavelength (ω R ) and its intensity ER are measured by fitting the Cyclohexane Raman modes position and intensity, while the Raman pulse time duration (σ R ) and the delay t P are obtained by the same fit procedure on the more resolved 8 cm Cyclohexane Raman mode, using Eq. S6, as shown in Fig. S2. 4

5 Raman Gain (AU).2. ω ca =82 cm γ =2. ps ca σ R =3.4 ps t P = 3.2 ps Raman Shift (cm ) FIG. S2: Cyclohexane Calibration: The Cyclohexane spectrum is fitted to calibrate the Raman pulse duration σ R and its time delay t P with the probe pulse. In the figure we report these parameters together with the fitted peak position ω ca and the bandwidth γ ca of the reported mode. The small negative side wings indicate that the PP is preceding the RP, ensuring the best condition in terms of spectral resolution and maximum of Raman Gain.. Yoon, S. et al. Dependence of line shapes in femtosecond broadband stimulated raman spectroscopy on pump-probe time delay. J. Chem. Phys 22, 2455 (25). 5