Sumfrequency Generation Spectroscopy

General method

Sumfrequency spectroscopy (SFS) is a powerfull sensitive tool to study vibrations of molecules at interfaces. It is based on the generation of a light beam with the sum of the frequencies (SFG) of two incoming powerfull laser beams in a non-linear optical process. One of the lasers is tuned such that its light is resonant with a vibrational transition in a molecule.

Light is an electro-magnetic wave. The electric field associated with that wave induces a certain polarization when interacting with matter. The polarization P depends on the susceptibility χ of the material and the electrical field strength E of the incoming wave. The light induced polarization for a material without a static dipole moment can be written in the following series expansion

\(P(\vec{E}) = \epsilon_0 \chi \vec{E} + \epsilon_0 \chi^{(2)} \vec{E}^2 + \epsilon_0 \chi^{(3)} \vec{E}^3\) .

In approximation we have only to consider the second term when discussing sum frequency generation from two waves with different frequencies ω₁ and ω₂ (as well as second harmonic generation with ω₁= ω₂). χ is a third rank tensors with respect to the three dimensions in space. Then the intensity of the sum frequency signal generated is given by

\(I_{SFG} \propto \left| P^{(2)} \right|^2 = \left| \chi^{(2)} : E_{IR}E_{vis} \right|^2\)

Frequency mixing only takes place in non-centrosymmetric media as in a centrosymmetric environment different components of χ⁽²⁾ cancel each other. Naturally, all gases and liquids are centrosymmetric, as are most solids.[1] At interfaces, the symmetry is broken. Therefore, the sum frequency generation is inherently surface-sensitive. Since the SFG signal is resonantly enhanced when the frequency of one of the incident photons matches a vibrational transition of a molecule this technique is a powerful tool for the analysis of adsorbates. In this case the overall process of photon mixing can be interpreted as a simultaneous infrared and Raman-transition (figure 1a). Therefore, the selection rules for both IR and Raman transitions must be fullfilled for a molecular vibrations to be SFG-active.

The overall nonlinear surface susceptibility χ⁽²⁾ is often composed of two contributions. A resonant part χ_R⁽²⁾ which reflects the contribution of the molecular vibration and a nonresonant background χ_NR⁽²⁾ which is almost frequency independent when observing small spectral windows.

\(\chi^{(2)} = \chi_{NR}^{(2)} + \chi_{R}^{(2)}\)

The nonresonant background is very prominent on transition metal surfaces since the d-electrons are easily excited by the visible upconversion pulse. Within the dipole approximation the overall SFG signal is given by

\(I_{SFG} \propto \left| \chi_{NR}^{(2)} \cdot e^{i \varphi} + \sum_n {\frac{A_n}{\omega_{IR} \, - \, \omega_{vib,n} \, + \, i\Gamma}} \right| ^2\)

The non-resonant background is not simply additiv, as SFG is a coherent process, but rather its amplitude is to be added with a certain phase relation φ to the resonant vibrational parts. Mathematically, both - the resonant and non-resonant susceptibility - are complex entities between which a phase shift exists. Hence, the two contributions cannot be simply added. From this coherent superposition results that the resonance can have a quite distorted line shape deviating from the Lorentzian form. The phase shift φ is represented in the e^iφ term. If the phase shift is close to π the line appears as dip in the non-resonant background.

The field strength of the incoming waves must be quite large to generate a considerable nonlinear polarization effect, as χ⁽²⁾ is typically several orders of magnitude smaller than the linear susceptibility. Therefore, two techniques are commonly used in SF spectroscopy: Broadband SFG setups with femtosecond pulse durations or tunable laser sources in the picosecond regime. Experimentally the laser pulses are overlapped spatially and temporally on the surface and the generated SFG signal is detected in transmission or reflection geometry after spectral separation (figure 1b). Thus, SF spectroscopy is not restricted to solid surfaces. It can be applied to any interface provided at least one of the media is transparent to both the incoming beams and the generated SF beam.

Time dependent measurements

The vibrational lifetimes of adsorbates can be determined in a pump-probe approach in which an IR pump pulse for vibrational excitation is followed by an IR-Vis pulse-pair for probing. We bear in mind that A_res in equation 3 is the SFG transition strength. A_res depends on the population difference between the vibrational ground state and the excited state ∆N = N₁ – N₀. The pump pulse transfers population from the ground state to the excited state. Thus the population difference is decreased accompanied by a decrease of A_res reducing the SFG signal generated by the probe-pulse pair. As the excited vibrations have a finite lifetime the population difference recovers with time. Therefore, the SFG signal S(t) is restored with increasing delay time t between the pump and the probe pulse pair. Assuming that the population transfer of the probe-pulse is small when compared to the pump-pulse the bleach decays exponentially with characteristic time constant τ.

This time constant contains both the vibrational lifetime T₁ and the dephasing time T₂. Dephasing means a loss of temporal coherency of the oscillators after a coherent excitation due to different damping environments on the surface. The contribution of T₂ to τ can safely be neglected if the vibrational lifetime T₁ is much longer than the dephasing time, which is typically found for adsorbates on semiconductors or insulators. In this case τ represents the vibrational lifetime of the adsorbate.

\(1 -\frac{S(t)}{S_0} \propto \exp[-\frac{t}{\tau}]\)

Essentially, it is also possible to extract information on the vibrational relaxation from the line shape of the vibrational resonance like in any other spectroscopical technique. However, various effects including inhomogeneous broadening or the linewidth of the laser system may govern the lineshape and make it impossible to determine the vibrational lifetime correctly.

Exceptions of technological importance, for example, are β–Bariumborate (BBO) or Kaliumdiphosphate (KDP) crystals which are commonly used for frequency doubling in laser systems.