Transient Absorption

emission is, is related to that of stimulated emission, B, via A= 8πhν³

c³ B. (2.64)

In contrast to absorption, spontaneous fluorescence is usually measured directly with a detector that is sensitive to the number of emitted photons. The recorded photon distribution F(ν) can be converted into (relative) cross section spectra for stimulated emission,σSE, by multiplication with 1/ν².

The formalism presented above is not limited to optical absorption and fluorescence, but is valid also for other processes like stimulated and spontaneous Raman scattering.

2.3 Transient Absorption

In a transient absorption experiment, a pump pulse is applied and the induced absorption of a probe pulse is measured at a specific delay timet_d. A chopper extincts every second pump pulse, and the transient absorption signal is the induced difference spectrum. In an isotropic sample, the lowest-order signal contribution is of third-order,

∆OD(ω₂, t_d) = lg(e)zN 2ω₂

I ImEe^∗₂(ω₂)Pe⁽³⁾(ω₂, t_d). (2.65) The third-order polarization spectrum P⁽³⁾(ω₂, t_d) is taken in the probe direction k₂ =

−k₁+k₁+k₂. It may be expressed by its Fourier transform as

In the dipole approximation the nonlinear polarizationP⁽³⁾(r, t, t_d) is given by P⁽³⁾(r, t, t_d) = The total electric field at timet is given by the sum of the pump and probe pulse fields E₁(r, t+t_d) and E₂(r, t),

E(r, t) =E₁(r, t+t_d) +E₂(r, t). (2.69)

The effective field productE⁽³⁾(k₂, t, t₃, t₂, t₁) ink₂direction is obtained by insertion into equation (2.68) and collecting the terms with exp(ik₂r). It is the sum of six time-ordered

2 Nonlinear Spectroscopy The termE_S⁽³⁾in equation (2.70) describes the sequential contribution, where the sample interacts first twice with the pump and then with the probe field. The termsE_C⁽³⁾andE_D⁽³⁾ correspond to the coherent contribution, where the sample interacts with the interference of pump and probe fields. Note that in the description of ref. 30, the double coherent termE_D⁽³⁾ is omitted. By inserting the previous result into equations (2.65) and (2.66), also a transient absorption spectrum ∆OD(ω2, t_d) may be separated into sequential and coherent spectra,

∆OD(ω₂, t_d) = ∆OD_S(ω₂, t_d) + ∆OD_C(ω₂, t_d) + ∆OD_D(ω₂, t_d). (2.72)

∆ODC(ω2, td) and ∆ODD(ω2, td) contain contributions from stimulated Raman scat-tering, two-photon absorption, and perturbed free induction decay.[35,36,39,54,55] If the electronic coherence time is short compared to the pulse duration, these signals appear only during the cross-correlation time of pump and probe pulses, giving rise to a ’co-herent spike’ around zero delay time. Such co’co-herent contributions will be considered in the context of femtosecond stimulated resonance Raman (FSRR) spectroscopy. Here the discussion is focused on the sequential part ∆ODS(ω2, td), which dominates the signal for temporally well-separated pump and probe pulses. The first two interactions with the pump field generate electronic population, whose evolution is queried by interaction with the delayed probe field. The signal ∆ODS(ω2, td) is determined by the third-order polarization spectraP_S⁽³⁾(ω2, t_d). The relevant nonlinear response functionR⁽³⁾(t3, t2, t1)

2.3 Transient Absorption A pathway [k] is specified by subscripts [mn, kl, pq] and corresponds to a third-order sequence of the general form

|gihg|−−→ |mihn|^µab −−→ |kihl|^µcd −−→ |pihq|.^µef (2.74) In the Bloch approximation the third order response function of the [k]th pathway is given by

R⁽³⁾_S[k](t₃, t₂, t₁) =− i

~³p(g)µ_b[k]exp [−iω_emnt₃−iω_eklt₂−iω_epq]. (2.75) Here the product of the dipole matrix elements,µ_{f e}µ_efµ_cdµ_ab, is collected in the operator µ.b

For well separated pulses of negligible duration, the sequential contribution to transient absorption may be factorized into a term that describes the time-dependent population p(j, td) change of state |ji and the associated first-order spectrum ∆D⁽¹⁾_j (ω2,0), Thus by analyzing the temporal evolution of the induced absorption spectrum, one may draw conclusions about the flow of the intermediately created populations p(j, t_d). The time-dependence of p(j, td) depends on the mechanism of excited-state relaxation. In Section 3.6 it will be shown that the evolution of transient populations can be described in most cases by the sum of exponential terms, weighted by the amplitudesa_i,

p(j, t_d) =^X

n

a_nexp (−t/τ_n)). (2.78)

A typical transient absorption experiment may be modelled by considering a system of three electronic states g, e, and f, each with vibrational structure distinguished by primes (g⁰,e⁰,f⁰). Pathways that contribute to the sequential transient-absorption signal are illustrated as energy-ladder diagrams in figure 2.4. For simplicity only sequences are shown that start from the lowest vibrational level in the ground state. The pathways may be classified according to the way in which system is prepared. In the bleaching process (BL) the pump pulse modifies the population in the ground electronic state. The probe induces the e ← g coherence, resulting in an emissive polarization wave. Other processes involve the transfer of population to the excited state. The probe field may then drive thef ←eor thee→g⁰ transition, corresponding to excited-state absorption (ESA) or stimulated emission (SE).

As an example, the transient-absorption spectrum of riboflavin in DMSO at delay t_d= 0.4 ps is shown in figure 2.4 and compared to the stationary absorption and

emis-2 Nonlinear Spectroscopy

Figure 2.4: The transient absorption signal. Left: Energy ladder diagrams represent-ing the sequential pathways that contribute to the transient absorption signal. Right:

Typical transient absorption signal of riboflavin in DMSO upon excitation with 440 nm at a delay oft_d = 0.4 ps, compared to stationary absorption and emission cross-section spectra.

sion cross-section spectra. The excited-state absorption (ESA) gives rise to positive bands, which are overlayed with negative signals from bleach (BL) and stimulated emis-sion (SE). Note that the contributions monitor evolution of the excited molecule from different viewpoints: ESA and SE report on the population and spectroscopic properties of excited states. On the other hand, BL reflects a lack of population in the fully equili-brated electronic ground state and has the shape of the negative ground-state absorption spectrum.

At early delay times, the pulse duration should be taken into account for a description of the sequential contribution. The pulse envelope is in many cases well described by a Gaussian of widthτ1,2. The pump and probe electric field terms are then

E₁(k₁, t+t_d)=exp^h−(t+t_d)²/2τ₁²−iΩ₁(t+t_d) +ik₁rⁱ, (2.79a) E₂(k₂, t)=exp^ht²/2τ₂²−iΩ1(t) +ik2rⁱ. (2.79b) After performing the Fourier transform, equation (2.73) can be solved analyti-cally.^[35,38,39] If electronic dephasing is fast compared to the pulse durations, i.e. Γ_pq →

∞, the sequential transient absorption signal from population transfer may be written as the convolution of the impulsively generated signal, equation (2.76), with the temporal

24

2.3 Transient Absorption

Figure 2.5: Oscillatory modulation of the transient absorption signal of riboflavin in DMSO by impulsive stimulated Raman scattering (SR). The sample was excited with 400 nm pump pulses. Left: Transient absorption band integral over the region 504–514 nm. Right: Fourier power spectrum of the oscillations (top) and stimulated Raman spectrum of DMSO (bottom).

apparatus function F_cc(t),

∆OD_S⁽³⁾(ω₂, t_d) =^X

j





∞

Z

−∞

dt⁰F_cc(t⁰)p(j, td−t⁰)



∆D⁽¹⁾_j (ω₂,0). (2.80)

Impulsive stimulated Raman scattering (SR) gives additional contributions to the se-quential signal. It arises from the intrinsic spectral width of the femtosecond pump pulses, which allows the first two electric field interactions (mn, kl) to occur with dif-fent transition frequencies. This generates a vibrational coherence in the ground or excited state, off which the probe field is scattered. Motion of the prepared vibrational wavepacket around a potential minimum induces oscillatory modulation of the transient absorption signal with the vibrational frequency ω_vib = ω_kl. The accessible frequency range is limited by the spectral width of the pump pulse. For the time-resolved ex-periments presented in this work, typically vibrations along low-frequency modes up to

∼700 cm⁻¹ are excited. The spectral amplitude of coherent oscillations is determined by a combination of real and imaginary parts of the complex Lorentzians that charac-terize the involved transitions.^[39] Therefore, the oscillation phase typically varies across a transient-absorption band, so that not only the signal amplitude, but also its spectral position is modulated.

Experimentally observed oscillations of the transient absorption signal are shown in figure 2.5 for riboflavin in DMSO. The continuous rise is clearly modulated by oscilla-tions. Fourier transform reveals the spectrum of contributing low-frequency vibraoscilla-tions.

Hence, transient absorption spectroscopy allows to perform vibrational spectroscopy in

2 Nonlinear Spectroscopy

Figure 2.6: Comparison of femtosecond-resolved transient absorption and stimulated Raman spectroscopy (FSRS).

time domain. The previous discussion focussed on resonance transitions, but also un-der non-resonant conditions impulsive stimulated Raman scattering may prepare wave packets in the electronic ground state. Whereas this contribution is negligible for the chromophor, it plays a role for solvent, which is present in large excess. Comparison of the Fourier power spectrum in 2.5 with the Raman spectrum of DMSO identifies bands around 670 cm⁻¹ and 336 cm⁻¹as originating from solvent oscillations. In the discussion of femtosecond stimulated resonance Raman spectroscopy on the flavin chromophor, it will be shown that the Fourier power spectrum of Raman background oscillations is inherently free from solvent contributions.