SFG is a second-order nonlinear optical process during which two photons of certain frequencies interact simultaneously with a surface molecule to instantaneously emit a new photon at the sum of the two frequencies. The unique advantage of SFG, distinct from other surface-sensitive techniques, is attributed to its interface selectivity. This originates from the fact that coherent second-order optical processes are forbidden in media with inversion symmetry, while they are allowed for an interface layer, where this centrosymmetry is naturally broken. The SFG intensity
\(\left( {{I_{SFG}}} \right)\) is proportional to the two incident laser intensities (
\({I_{Vis}}\) and
\({I_{IR}}\)) and the absolute square of second-order nonlinear susceptibility (
\(\chi _{{{\text{eff}}}}^{{{\text{(2)}}}}\)), as shown in Eq.
1.
$${I_{SFG}} \propto {\left| {{\varvec{\upchi}}_{{{\text{eff}}}}^{{{\text{(2)}}}}} \right|^{\text{2}}}{I_{Vis}}{I_{IR}}$$
(1)
\({\chi _0}\) is the magnitude of the non-resonant susceptibility
\(\chi _{{NR}}^{{(2)}}\) due to electronic excitations of the substrate and the adsorbate, and
\(\phi\) is its phase relative to the resonant term.
\({\chi _q}\),
\({\omega _q}\) and
\({\Gamma _q}\) represent the resonance amplitude, frequency and damping constant of the
qth vibrational mode, respectively.
\({\omega _{IR}}\) is the frequency of the IR laser beam. In general,
\(\chi _{{NR}}^{{(2)}}\) should be small and real when the substrate is not resonant with either
\({\omega _i}\) of incident visible, IR and output SFG beams. For dielectric interfaces, it is negligible. However, for metal or semiconductor substrate interfaces,
\(\chi _{{NR}}^{{(2)}}\) generally becomes complex and can no longer be ignored.
\(\chi _{{{\text{eff}}}}^{{{\text{(2)}}}}\) depends on the experimental polarization and geometry, and there are an infinite number of combinations of experimental configurations that can give different
\(\chi _{{{\text{eff}}}}^{{{\text{(2)}}}}\). In this paper, we mainly study
\(\chi _{{{\text{eff}}}}^{{{\text{(2)}}}}\) with a linear combination of independent experimental polarization combinations, namely, SSP (S-polarized sum frequency, S-polarized visible and P-polarized infrared) and PPP as shown in the following:
$$\begin{gathered} \chi _{{{\text{eff}},SSP}}^{{{\text{(2)}}}}={L_{yy}}\left( {{\omega _{SFG}}} \right){L_{yy}}\left( {{\omega _{Vis}}} \right){L_{zz}}\left( {{\omega _{IR}}} \right)\sin {\alpha _{IR}}{\chi _{yyz}} \\ \chi _{{{\text{eff}},PPP}}^{{{\text{(2)}}}}= - {L_{xx}}\left( {{\omega _{SFG}}} \right){L_{xx}}\left( {{\omega _{Vis}}} \right){L_{zz}}\left( {{\omega _{IR}}} \right)\cos {\alpha _{SFG}}\cos {\alpha _{Vis}}\sin {\alpha _{IR}}{\chi _{xxz}} \\ \;\; - {L_{xx}}\left( {{\omega _{SFG}}} \right){L_{zz}}\left( {{\omega _{Vis}}} \right){L_{xx}}\left( {{\omega _{IR}}} \right)\cos {\alpha _{SFG}}\sin {\alpha _{Vis}}\cos {\alpha _{IR}}{\chi _{xzx}} \\ \;\;+{L_{zz}}\left( {{\omega _{SFG}}} \right){L_{xx}}\left( {{\omega _{Vis}}} \right){L_{xx}}\left( {{\omega _{IR}}} \right)\sin {\alpha _{SFG}}\cos {\alpha _{Vis}}\cos {\alpha _{IR}}{\chi _{zxx}} \\ \;\;+{L_{zz}}\left( {{\omega _{SFG}}} \right){L_{zz}}\left( {{\omega _{Vis}}} \right){L_{zz}}\left( {{\omega _{IR}}} \right)\sin {\alpha _{SFG}}\sin {\alpha _{Vis}}\sin {\alpha _{IR}}{\chi _{zzz}} \\ \end{gathered}$$
(3)
Here
\({\omega _{SFG}}\),
\({\omega _{Vis}}\) and
\({\omega _{IR}}\) are the frequencies;
\({\alpha _{SFG}}\),
\({\alpha _{Vis}}\) and
\({\alpha _{IR}}\) are the angles (with respect to the surface normal), of the SFG signal, visible and IR laser beams, respectively.
\({L_{ii}}({\omega _i})\) denotes the Fresnel factor at frequency
\({\omega _i}\) for the local field corrections, which can be calculated with the knowledge of
\({\alpha _i}\) and refractive indices
n1 (medium 1, the air in which incident and SFG photons propagate),
n2 (medium 2, the single crystal phase) and
n′ (interfacial layer). In this paper, the
n′ values were estimated by the modified Lorentz model, and the expression is
\(n^{\prime}=n{}_{1}{n_2}\sqrt {\frac{{6+{n_2}^{2} - {n_1}^{2}}}{{4{n_2}^{2}+2{n_1}^{2}}}}\) [
26] ; when
n1 = 1, then
\(n^{\prime}={n_2}\sqrt {\frac{{{n_2}^{2}+5}}{{4{n_2}^{2}+2}}}\) [
27].
\(\chi _{{ijk}}^{{(2)}}\) is the macroscopic sum frequency susceptibility, which is related to the microscopic hyperpolarizability tensor elements
\(\beta _{{i^{\prime}\,j^{\prime}\,k^{\prime}}}^{{(2)}}\) in the molecular coordinates system.
\(\chi _{{ijk}}^{{(2)}}\) can be deduced from SFG measurement with three different input/output polarization combinations, for example, SSP, SPS, PSS and PPP.
For
\({C_{\infty v}}\) symmetry group, such as CO, OH, –CH, there are only two independent nonvanishing components in
\(\beta _{{i^{\prime}\,j^{\prime}\,k^{\prime}}}^{{(2)}}\),
\({\beta _{ccc}}\) and
\({\beta _{aac}}={\beta _{bbc}}\). Then, the non-zero macroscopic elements of
\(\chi _{{ijk}}^{{(2)}}\) for a rotationally isotropic interface which are obtained through integration over the Euler angles (
\(\phi\), azimuth angle;
\(\psi\), twist angle) can be expressed as follows [
27,
28].
$$\begin{gathered} \chi _{{xxz}}^{{(2)}}=\chi _{{yyz}}^{{(2)}}=\frac{1}{2}{N_s}{\beta _{ccc}}\left[ {(1+R)\left\langle {\cos \theta } \right\rangle - (1 - R){{\left\langle {\cos \theta } \right\rangle }^3}} \right] \hfill \\ \chi _{{xzx}}^{{(2)}}=\chi _{{zxx}}^{{(2)}}=\chi _{{yzy}}^{{(2)}}=\chi _{{zyy}}^{{(2)}}=\frac{1}{2}{N_s}{\beta _{ccc}}\left[ {(1 - R)\left\langle {\cos \theta } \right\rangle - (1 - R){{\left\langle {\cos \theta } \right\rangle }^3}} \right] \hfill \\ \chi _{{zzz}}^{{(2)}}={N_s}{\beta _{ccc}}\left[ {R\left\langle {\cos \theta } \right\rangle +(1 - R){{\left\langle {\cos \theta } \right\rangle }^3}} \right] \hfill \\ \end{gathered}$$
(4)
Here
\(\beta _{{i^{\prime}\,j^{\prime}\,k^{\prime}}}^{{(2)}}\) is the molecular hyperpolarizability tensor. The hyperpolarizability ratio is
\(R={\beta _{aac}}/{\beta _{ccc}}={\beta _{bbc}}/{\beta _{ccc}}\). For a single bond with
\({C_{\infty v}}\) symmetry, the
R-value equals to the bond polarizability derivative ratio
r. Different
R-values will lead to different results in the SFG orientational analysis [
21,
27,
28].
\(\theta\) is the orientation angle of the moiety of the symmetry axis with respect to the surface normal.
\({N_s}\) is the
effective surface number density of molecules contributing to the SF signal. Then, based on Eqs.
1–
4, we can determine the orientation (
\(\theta\)) and/or molecular hyperpolarizability ratio (
R) of the moiety by the measurements of the ratio of independent nonvanishing
\(\chi _{{ijk}}^{{(2)}}\) components assuming a
δ-function distribution for
\(\theta\). The latter assumption is frequently applied because only a small angular distribution is expected.