Rigorous treatment of SFG theory can be found in excellent reviews [44–46] and books [47,48]. Here, we focus on describing the theory to illustrate how SFG can be used as a surface-specific vibrational spectroscopy. SFG uses two pulsed laser beams, one at infrared (IR) frequency ω_IR and the other at visible frequency ω_VIS. When these two beams spatially and temporally overlap at surfaces, a second-order polarization, P⁽²⁾, at the sum frequency (ω_IR + ω_Vis) can be induced to generate an SFG signal. The intensity of the SFG signal is related to the surface second order susceptibility, χ⁽²⁾, and the electric fields of the IR and visible beams as below,

Under the dipole approximation, χ⁽²⁾ is nonzero only if the medium lacks centrosymmetry (Figure 1). This is the case for interfaces because molecules align due to asymmetric physical and chemical properties of two media. The second-order susceptibility of an interface consists of a non-resonant term, χ_NR⁽²⁾, and a sum of vibrational resonant terms, χ_q⁽²⁾, which is considered to be in a Lorentzian lineshape as shown below,

where A_q is the amplitude, Γ_q is the damping coefficient, and ω_q is the vibrational frequency of the q^th vibrational mode, and ω_IR is the incident IR frequency. The SFG signal is enhanced when ω_IR is in resonance with ω_q. Therefore, SFG is a second-order surface-specific vibrational spectroscopy.

Furthermore, one can use SFG to selectively probe the chirality of interfaces. The selectivity of chirality comes from the coherent nature of the SFG signals, which is directly related to a third-rank tensor of 2^nd-order nonlinear optical susceptibility, χ⁽²⁾, as shown in Equation 1. SFG experiments are often performed using linearly s- or p-polarized IR and visible beams, and measuring s- or p-polarized SFG signals. Hence, there are eight possible combinations for the polarization settings: ssp (s-polarized SFG, s-polarized visible, and p-polarized IR), sps, pss, ppp, sss, psp, spp, and pps. For a particular polarization setting, the SFG intensity (I_SFG) can be expressed by an effective second-order susceptibility, χ_eff⁽²⁾, e.g., χ_ssp⁽²⁾, χ_ppp⁽²⁾, χ_psp⁽²⁾, etc.

The effective second-order susceptibility, χ_eff⁽²⁾, is a linear combination of the χ⁽²⁾ tensor elements, χ_IJK⁽²⁾, where I, J, K are laboratory coordinates (x, y, z). For an isotropic achiral surface with the C_∞V symmetry, there are only seven nonzero χ_IJK⁽²⁾ elements: χ_xxz⁽²⁾ = χ_yyz⁽²⁾, χ_xzx⁽²⁾ = χ_yzy⁽²⁾, χ_zxx⁽²⁾ = χ_zyy^(2), and χ_zzz⁽²⁾. For a chiral surface with the C_∞ symmetry, there are six additional nonzero elements: χ_xyz⁽²⁾, χ_yxz^(2), χ_zxy⁽²⁾, χ_zyx^(2), χ_xzy⁽²⁾ and χ_yzx⁽²⁾. These orthogonal elements, χ_IJK⁽²⁾ (I ≠ J ≠ K), are characteristic of chiral surfaces. These chiral and achiral elements can be used to express eight effective χ_eff⁽²⁾, of which five are achiral as shown in Equation 4 and three are chiral as shown in Equation 5 [34,49]:

where α_i is the incident or reflected angle of the i^th laser beam and L(ω_i) is the Fresnel factor. Equation 5 shows three chiral effective susceptibilities, χ_psp⁽²⁾, χ_spp⁽²⁾, and χ_pps⁽²⁾, which contain only the orthogonal chiral tensor elements χ_IJK⁽²⁾ (I ≠ J ≠ K, ), including χ_xyz⁽²⁾, χ_yxz^(2), χ_zxy⁽²⁾, χ_zyx^(2), χ_xzy⁽²⁾ and χ_yzx⁽²⁾. Thus, surface chirality can be probed by using the psp, spp, and pps polarizations without interference from the background of achiral solute and solvent molecules at the interface.

Among these polarization configurations, the psp polarization is commonly used to obtain chiral SFG spectra. As shown in Figure 2, when the psp polarization setting (p-polarized SFG, s-polarized visible, and p-polarized IR) is used, the p-polarized SFG and the p-polarized IR light have electric fields in the x and z directions such that the first (I) and third index (K) of χ_IJK⁽²⁾ denoting the direction of the electric fields of the SFG signal and the IR beam, respectively, can only be either z or x. The s-polarized visible beam has an electric field only in the y direction such that the second index (J) of χ_IJK⁽²⁾ denoting the direction of an electric field, can only be y. Thus, the psp polarization setting can be used to measure two chiral elements χ_xyz⁽²⁾ and χ_zyx⁽²⁾. A combination of Equations 3 and 5 describes how the experimental observable (I_SFG) relates to the two chiral second-order susceptibility tensor elements χ_xyz⁽²⁾ and χ_zyx⁽²⁾.

The experimental observable I_SFG can be further related to microscopic hyperpolarizability (β) of molecules at interfaces because macroscopic χ_IJK⁽²⁾ elements are ensemble averages of microscopic β tensor elements, β_ijk. The χ_IJK⁽²⁾ elements can be expressed by β_ijk using the Euler transformation as shown in Figure 3 and Equation 6, where I, J, K are the laboratory coordinates (x, y, z) and i, j, k are molecular coordinates (a, b, c):

where N_s is the number density of the interface moiety under study and R_Ii, R_Jj, and R_Kk are elements of the rotational transformation matrix from the molecular coordinates to the laboratory coordinates (Figure 3). The microscopic β tensor determines the SFG response of a molecule, which is related to the electric polarizability (α_ij) and electric dipole moment (μ_k),

where Q_q is the q^th normal mode coordinate. Equation 7 shows that the vibrational mode that is SFG active must be both IR and Raman active.

The intensity of the chiral SFG signal measured using the psp polarization is related to χ_psp⁽²⁾ according to Equation 3, which is further related to two chiral elements χ_xyz⁽²⁾ and χ_zyx⁽²⁾ according to Equation 5. The expression of these two chiral elements, as shown in Equations 8 and 9, can be obtained using Equation 6, which introduces the molecular orientation (θ, ψ) into the equation by averaging over the in-plane orientation angle (Φ) from 0 to 2π[50].

Equations 8 and 9 can be further simplified by eliminating the zero hyperpolarizability β elements using Equation 7 and the symmetry of the vibrational modes. Hence, a combination of Equations 3, 5, 6, 8, and 9 provides an expression of the experimental observable I_SFG as a function of β_ijk and molecular orientation (θ, ψ) at interfaces.

Therefore, chiral SFG spectra can provide information about molecular orientations and structures of chiral interfaces. For orientation information, if the microscopic molecular hyperpolarizability (β) tensor is known, the orientation (θ, ψ) can be obtained from the experimentally measured macroscopic second-order susceptibility elements, χ_IJK⁽²⁾ (I ≠ J ≠ K), as indicated by Equations 8 and 9. The microscopic molecular hyperpolarizability (β) tensor can be obtained either by linear IR [51] and Raman [52] measurements or by ab initio quantum chemistry calculations [53]. For structural information, vibrational peaks in the chiral SFG spectra can be assigned to particular vibrational modes of surface structures. This assignment is generally achieved by considering the selection rules, symmetries of vibrational modes, and normal modes analyses of the molecules under study. This approach for peak assignment can also be aided by ab initio quantum chemistry calculations. Such vibrational analyses and peak assignments can reveal molecular details of chiral interfaces.

In the past decade, several research groups have developed theory to describe chiral SFG phenomena. Shen and coworkers applied first-order perturbation theory to describe the chiral signal detected from bulk chiral liquids and found that the anti-Stokes Raman tensor is responsible for the bulk chiral SFG signal [54]. Liu and coworkers also performed calculations to confirm that the chiral nonlinear susceptibility originates from the anti-symmetric component of the anti-Stokes Raman tensor [55]. The anti-symmetric part of the Raman tensor is usually very weak without electronic resonance. Thus, in order to detect this chiral SFG signal from bulk media, the visible beam is required to be in resonance with the electronic transition of the chiral molecules.

On the other hand, Simpson and coworkers also used perturbation theory with two-photon absorption to treat the molecular hyperpolarizability near and off resonance [56]. By analyzing the symmetry of vibrational modes, they determined the nonzero β tensor elements, from which they calculated the macroscopic second-order susceptibility χ⁽²⁾ of interfaces with C_∞ symmetry [50]. They further illustrated that achiral molecules, when arranged in macromolecular chiral architectures, can generate surface-specific chiral SFG signals [53,56–58]. This surface-specific chiral signal is comparable to conventional surface-specific achiral SFG without electronic resonance. Hence, the origin of chiral SFG signal generated at interfaces is different from the origin of chiral SFG signal generated in bulk solution. While the bulk chiral SFG signal is due to the asymmetric part of the Raman tensor associated with the “intrinsic chirality” of molecules, the surface-specific chiral SFG is due to chiral arrangements of chiral or even achiral molecular entities, whose microscopic hyperpolarizabilities are summed up to generate macroscopic chiral susceptibility elements at interfaces. Hence, when chiral SFG is employed to study chiral macromolecular structures, such as protein secondary structure, it can provide vibrational information about the macroscopic chiral structures that is surface-specific.

Furthermore, the origin of the chiral sensitivity of SFG is also different from that of linear spectroscopies, such as CD and ORD. For chiral SFG, the chiroptical response comes from the orthogonal elements of a third-rank susceptibility tensor, χ_IJK⁽²⁾ (I ≠ J ≠ K). Each of these third-rank tensor elements is described by three independent coordinates (I, J, K), which are sufficient to specify the symmetry of the interaction between optical fields and chiral molecular entities. For linear spectroscopy, however, neither the vector of electric dipole moments (μ_k) nor the second-order tensor of Raman polarization (α_ij) alone can reflect the chiral symmetry. Hence, in order to describe the chiroptical responses, the vector of electric dipole moments (μ_ij) or the second-order tensor of Raman polarization (α_ij) need to couple to magnetic dipoles, as in the case for CD or ORD. Such magnetic dipole coupling is generally weak. Hence, the sensitivity of CD or ORD is expected to be lower than chiral SFG.

In 2000, Shen and coworkers detected chiral SFG signal from bulk chiral liquids using transmission optical geometry with electronic resonance enhancement [32]. They demonstrated, for the first time, the chiroptical response in an SFG process. They showed that chiral SFG spectral are useful for determining absolute configurations and conformations of chiral molecules [32,71–73]. Moreover, Ishibashi and coworkers probed porphyrin aggregates by double-resonance chiral SFG [74], and Busson and coworkers studied the anisotropy of a helicene bisquinone sample [75]. Both studies made use of electronic resonance to enhance the vibrational SFG signals. These chiral SFG signals, originating from “intrinsic chirality” of molecules, are due to the asymmetry of the Raman tensor and are usually weak. In order to detect these signals, the “interference” or “mix” methods are often applied, which use pmp (the polarization of the visible beam is in the “m” configuration, where “m” means 50% p-polarized and 50% s-polarized) rather than psp or spp polarizations settings [32].

Without electronic resonance enhancement, the chiral SFG signals have also been observed in various biological systems. Chen and coworkers detected chiral amide I signals using the psp and spp polarization configurations from a peptide, tachyplesin I, which forms an anti-parallel β-sheet structure on a polystyrene surface [33]. They made the first observation of the vibrational chiroptical signal from a protein using chiral SFG and identified three major peaks at 1633, 1685, and 1695 cm⁻¹ in the amide I vibrational regions (Figure 5a and b). The amide I peaks at high frequency (>1680 cm⁻¹) are characteristic of anti-parallel β-sheets. Although it was controversial whether this chiral SFG signal was due to surface chirality [71], the interpretation of surface chirality is supported by a theory developed by Simpson et al. [53,56–58]. According to the theory, the chiral SFG signals observed from proteins were attributed to the macroscopic chiral arrangement of the amide groups along the peptide backbone adopting anti-parallel β-sheet structures at interfaces. Chen’s pioneering work has demonstrated the potential of developing chiral SFG into a characterization method to probe protein structures at interfaces.

Moreover, Knoesen and coworkers also applied chiral vibrational SFG to study the molecular origin of the second-order optical nonlinearity of collagen [76]. They obtained chiral and achiral SFG spectra of type I collagen in both the C-H stretch and amide I vibrational regions (Figure 6). They observed the chiral C-H stretch and amide I signals of type I collagen using the spp polarization configuration.

In addition, chiral SFG can also be used to determine orientations of proteins at interfaces. Using Simpson’s theoretical treatment, Chen and coworkers derived an analytical expression to describe the chiral amide I signals detected from anti-parallel β-sheet at interfaces, and successfully obtained its interfacial orientation at the polystyrene surface [38]. They analyzed both ssp and psp SFG spectra and expressed the effective second-order susceptibility, χ_ssp⁽²⁾ and χ_spp⁽²⁾, as functions of nonzero hyperpolarizability tensor elements, β_ijk, and molecular orientations (θ, ψ), similar to Equations (8) and (9) presented above. They considered the D₂ symmetry of anti-parallel β-sheet and determined the non-zero derived β_ijk, from which they derived the expression connecting χ_ssp⁽²⁾ and χ_spp ⁽²⁾ with nonzero β_ijk and orientation (θ, ψ) for three amide I modes. Combining these results with the ATR-FTIR data and using the previously reported values for IR dipole and Raman polarizability, they successfully determined the orientation of the anti-parallel strands at the air/polystyrene surface.

In addition to proteins, it is worth mentioning that Geiger and coworkers also used chiral SFG to study DNA at interfaces by detecting the chiral C-H stretch. Their work suggests that the applications of chiral SFG to biological systems is not limited to proteins but can also be extended to other native or even synthetic chiral polymers [36,77].

The experimental results summarized in this section represent the current status of SFG applications in molecular chirality. The results demonstrate that chiral SFG is an effective method for detecting chirality at interfaces, which has inspired us to further develop chiral SFG as an analytical tool for in situ and real-time characterization of proteins at interfaces. In the following section, we will present our recent work on using chiral SFG for identifying secondary structures of proteins at interfaces [35] and for probing kinetics of conformational changes of amyloid proteins at membrane surfaces [37].

We recently reported both chiral amide I and N-H stretch signals of peptide backbones from various secondary structures [35]. We obtained chiral (psp polarization) and achiral (ssp polarization) N-H stretch and amide I spectra of model peptides and proteins. The chiral SFG spectra of the model peptides and proteins have shown vibrational signatures unique to their secondary structures, supporting that chiral SFG can be used to distinguish protein secondary structures at interfaces. The model peptides and proteins that we studied include (1) human islet amyloid polypeptide (hIAPP), which forms parallel β-sheets at the air-water interface in the presence of negatively charged lipids [37,78–80]; (2) tachyplesin I, which has two intra-strand disulfide bonds stabilizing an anti-parallel β-sheet structure [81–83]; (3) bovine rhodopsin, which has a 7-α-helical transmembrane structure [84,85]; (4) pH-low insertion peptide (pHLIP), derived from helix 3 of bacteriorhodopsin, which adopts α-helical structures in amphiphilic environments [86,87]; (5) de novo designed LKα14, which forms α-helices at the air/water interface [88–90]; and (6) rat islet amyloid polypeptide (rIAPP) used as a control, which is largely disordered [91]. These proteins and peptides were studied at the plain air/water (phosphate buffer, 10 mM, pH = 7.4) interface, except for hIAPP and rhodopsin. For hIAPP, we added dipalmitoylphospho–glyc–erol (DPPG) to induce formation of parallel β-sheet structures. For rhodopsin, we made a monolayer of rhodopsin in the presence of detergent in a Langmuir trough by controlling the surface pressure as described [92,93].

Figure 7 shows the chiral SFG spectra of the model peptides and proteins. The parallel β-sheet in hIAPP aggregates exhibits amide I peaks at 1622 and 1660 cm⁻¹, but no chiral N-H signal. In contrast, α-helices display N-H signals at 3280 cm⁻¹ (rhodopsin and pHLIP) and 3300 cm⁻¹ (LKα14), but no amide I signal. Antiparallel β-sheets of tachyplesin I displays chiral N-H at 3274 cm⁻¹ and 3175 cm⁻¹, as well as chiral amide I signals at 1634 cm⁻¹. Figure 8 shows the corresponding achiral SFG spectra taken using the ssp polarization setting. The achiral SFG spectra show both amide I and N-H stretch achiral signals at the interface regardless of their secondary structures. Compared to chiral SFG results, the peaks in achiral SFG spectra are generally broad and complex, and need spectral deconvolution to extract structural information. Taken together, chiral SFG shows an advantage over conventional (achiral) SFG by providing background-free and optically-clean signals for robust characterization of protein secondary structures at interfaces.

Figure 7 shows that the N-H stretches of peptide backbones can be used for characterizing protein structures at aqueous interfaces, which is unprecedented. Because the N-H stretch overlaps with the O-H stretch, the N-H stretch of proteins is often masked by water background in conventional vibrational spectra. However, the N-H stretch along chiral peptide backbones can be detected in chiral SFG spectra, but the O-H stretch of water, which lacks any chiral macroscopic structure, is muted in the chiral SFG spectra. Hence, the chiral N-H stretch of peptide backbones can be detected without interference from the water O-H stretch background. Moreover, specific hydrogen bonds between peptide amide (N-H) and carbonyl (C=O) moieties stabilize secondary structures, where the N-H stretch frequency is sensitive to the local hydrogen-bond environment. Hence, the chiral N-H frequency is expected to be characteristic of protein secondary structures, providing additional vibrational signatures to characterize protein secondary structures at interfaces. Nonetheless, the chiral N-H stretch for different secondary structures may overlap. Figure 7 shows similar peak positions for anti-parallel β-sheet tachyplesin I (3274 cm⁻¹) and α-helical pHLIP (3278 cm⁻¹). Hence, the chiral N-H stretch frequency by itself does not distinguish these two structures. However, β-sheet tachyplesin I shows a chiral amide I SFG signal, while α-helical pHLIP does not display any chiral amide I SFG signal. Thus, our results show that the chiral N-H stretch signal combined with the chiral amide I signal of peptide backbones can be potentially developed as robust optical probes for characterizing protein secondary structures at interfaces.

Here, we describe our recent application of chiral SFG to a biomedical problem. We used the SFG vibrational signatures of various secondary structures to study the kinetics of the misfolding of amyloid proteins at membrane surfaces. The misfolding is implicated in various amyloid diseases, including Parkinson’s, Alzheimer’s, and type II diabetes [94,95]. We focused on human islet amyloid polypeptide (hIAPP), which is associated with type II diabetes [96]. In the normal state, hIAPP is secreted by pancreatic β-cells in human body with random-coil structures [91]. In the disease state, hIAPP converts to aggregates in parallel β-sheet structures. Previous biochemical and biophysical studies showed that the aggregation of hIAPP into parallel β-sheet structures is catalyzed by its interactions with membrane surface [95–97]. We probed the aggregation process of hIAPP upon interactions with the negatively charged lipid, DPPG, at the air/water interface. We monitored the chiral N-H stretch and amide I SFG spectra over time. Upon addition of lipids (t = 0), the SFG intensity of the chiral N-H stretch gradually increased to its maximum in 3 hours and then disappeared by 10 h (Figure 9A). In contrast, the intensity of the chiral amide I signal appeared at approximately 4 hours after addition of lipid and increased to maximum by 10 h (Figure 9A). Each measurement was performed in triplicate and the N-H stretch and amide I intensities were plotted as a function of time in Figure 9B. The results show that the N-H stretch signal consistently disappeared prior to accumulation of the amide I signal. The transient N-H signal at 3285 cm⁻¹ corresponds to an α-helical intermediate, while the amide I signal at 1620 cm⁻¹ corresponds to a parallel β-sheet structure. The initial absence of amide I signal and N-H stretch signals reveal that hIAPP is likely unstructured upon adsorption onto the lipid surface. The gradual buildup of the amide I signal and a simultaneous decrease of the N-H chiral signal suggest the conversion of α-helices to parallel β-sheets. We conclude that hIAPP adsorbs at the lipid/aqueous interface as unstructured protein, which then folds into α-helical intermediates, and subsequently converts to parallel β-sheets on the time scale of 10 hours.

This application of chiral SFG to probe aggregation of amyloid proteins at interfaces has illustrated the potential of chiral SFG to address biomedical questions at the molecular level. It is expected that further application of chiral SFG for characterizing protein secondary structures at interfaces can solve problems that have biomedical relevance.