The second order ($ \propto \chi ^{(2)} E^2$) nonlinear spectroscopies of SHG, SFG, DFG, and optical rectification are the simplest examples of nonlinear molecular spectroscopies and will serve as our example of diagrams that describe nonlinear spectroscopy. They are **three wave mixing (TWM)** methods. Second order spectroscopies vanish for isotropic samples and are therefore surface selective. Figure 1a shows the flow of coherences that describes all the processes. The second order nonlinear spectroscopies involve two transitions from the initial ground state population, $gg$, to $cg$ for SFG, $ba$ for DFG, or back to $gg$ for optical rectification. SFG is a parametric process where the final emitting coherence involves the initial state, *g*, so the molecule returns to the ground state after the cg coherence emits. It has only a single coherence pathway, $gg \rightarrow bg \rightarrow cg$. SHG is a special case of SFG where the two excitation frequencies are identical. DFG is a nonparametric process where the final emitting coherence, $ba$, does not involve the ground state so the molecule is left in an $aa$ population after the *ba* coherence emits. It has two coherence pathways that interfere, $gg \rightarrow bg \rightarrow ba$ and $gg \rightarrow ga \rightarrow ba$. Optical rectification is a parametric process with one pathway, $gg \rightarrow bg \rightarrow gg$. If the excitation beam frequencies are labeled $\omega_1$ and $\omega_2$, SHG, SFG, DFG, and optical rectification have output frequencies at $2\omega_1$ (or $2\omega_2$), $\omega_1 + \omega_2$, $\omega_1-\omega_2$, and dc, respectively.

SFG and DFG spectra result from scanning one of the excitation frequencies while monitoring the intensity of the output signal from the final coherence. The final coherence emits light at a different frequency than the excitation beams so it is simple to spectrally discriminate between the excitation beams and the output signal. The output intensity increases when the excitation frequencies match molecular resonances with vibrational and/or electronic states. Figure 1b shows the WMEL diagrams that describe the possible resonances for the three processes. Figure 1c shows Feynman diagrams that contain the same information more explicitly. Here, the two vertical lines represent the ket (left) and bra (right) states as time advances vertically from the bottom. Interactions are represented by sloped lines. The quantum states before and after an interaction are indicated by letters. Lines sloped downward from a ket or bra vertex are absorption events $e^{i(kz-\omega t)}$ for ket side interactions and $e^{-i(kz-\omega t)}$ for bra side interactions) and lines sloped upward from a vertex are emission events $e^{-i(kz-\omega t)}$and $e^{i(kz-\omega t)}$ for ket and bra side interactions, respectively.) The resonance enhancements occur when the arrows or combination of arrows match the frequencies of the coherences. The coherences are defined by identifying the ket and bra states at a given time (eg. after the second interaction, the ket-bra states are *cg* for SFG and *ba* for DFG in figure 1c). Resonant enhancements occur when a combination of excitation frequencies matches the coherence frequency.

The relative intensity of the different TWM methods depends on the phase matching conditions. The polarization induced in any given molecule will have all the possible frequency components formed by linear combinations of the excitation frequencies but the observed emission at these frequencies will depend on how the phases of the emission from the polarizations of all the molecules add together. Phase matching requires $\vec{k}_3=2 \vec{k}$, $\vec{k}_3=\vec{k}_1+\vec{k}_2$, $\vec{k}_3 = \vec{k}_1 - \vec{k}_2$, and $\vec{k}_3=0$ for SHG, SFG, DFG, and optical rectification, respectively. On a surface, the perpendicular component of $\vec{k}$ is unimportant and phase matching requires matching the component parallel to the surface.