The article information
 Li Fu, Shunli Chen, Wei Gan, Hongfei Wang
 CrossPropagation SumFrequency Generation Vibrational Spectroscopy
 Chinese Journal of Chemical Physics, 2016, 29(1): 7080
 http://dx.doi.org/10.1063/16740068/29/cjcp1512248

Article history
 Received on December 2, 2015
 Accepted on December 28, 2015
b. Laboratory of Environmental Science and Technology, Xinjiang Technical Institute of Physics Chemistry, Key Laboratory of Functional Materials and Devices for Special Environments, Chinese Academy of Sciences, Urumqi 830011, China
Surface sumfrequency generation vibrational spectroscopy (SFGVS),first developed three decades ago ^{[1]},has shown great promises in study of all kinds of surfaces and interfaces as a unique spectroscopic tool with its surface specificity and submonolayer sensitivity ^{[2, 3, 4, 5, 6, 7, 8]}. Many recent developments aiming to quantitative measurement and interpretation of SFGVS data from complex molecular interfaces have focused on the aspects of the spectral phase,polarization and spectral resolution ^{[9, 10, 11, 12]}. Among these developments,the issues on polarization ^{[10]} and spectral resolution ^{[12, 13, 14]} have been less controversial,and the controversies regarding the issues on the consistency ^{[9, 15, 16]} between the phaseresolved and intensity SFGVS spectra measurements are also converging recently ^{[14, 17, 18]}.
In the SFGVS experiment,a visible laser beam ($\omega_1$),and a tunable or broadband infrared (IR) laser beam ($\omega_2$) in the midIR (3$$8 μm) or longIR (8$$15 μm) frequency range are focused simultaneously onto the surface or interface of interest,and a weak signal at the sum of the two frequencies ($\omega$=$\omega_1$+$\omega_2$) is generated. By fixing the visible frequency,usually at 532 or 800 nm,and tuning the IR frequency,the vibrational spectroscopy of the surface or interfacial molecules can be detected,to provide the spectroscopy,structure,conformation and dynamics information of the molecular species of interest. The unique surface selectivity of SFGVS comes from the intrinsic symmetry requirement for the secondorder nonlinear process which is forbidden for the centrosymmetric medium and the fact that the centrosymmetry of any surface or interface is always broken ^{[19, 20]}. The practical considerations in the actual SFGVS experiment are usually on what is the best way for the visible and IR light to be accessible to the surface or interface of interest,and on whether enough SFG signal can be generated and detected under specific polarization and spectral resolution,etc.
In the SFGVS literature,so far almost all SFGVS experiment has been performed using the coplanar geometry (Fig. 1),either in the reflective or transmission modes with copropagation,counterpropagation or twophase propagation arrangements of the visible and IR beams ^{[5, 7, 21, 22, 23, 24]}. In the coplanar geometry,the visible and IR beam are all within the same incident plane that is perpendicular to the surface or interface. It is clear that for the exposed surfaces,there is no problem for the visible and IR beams to access the surface,and the detection of SFG signal can be either reflective or refractive. For the refractive detection,the only requirement is that the lower phase is transmissible for the SFG signal. For the buried interface,the issues,on whether either of the upper or lower phase is transmissible for the visible or IR light to allow them to access the interface and to allow the SFG signal in the visible to be detected,always have to be evaluated in the SFGVS experiment. So,all three geometries for the buried interfaces work only when the upper phase is IR transmissible,either transparent to the IR frequencies of interest or thin enough to allow insignificant absorption for the IR frequencies. This is why even though there have been successful cases for SFGVS measurement of molecules and nanoparticles at buried interfaces ^{[24, 25, 26, 27, 28]},the application of SFGVS to buried interfaces are still challenging and limited ^{[29]}.
In this work,we present the formulation and experiment validation of the SFGVS in a noncoplanar geometry,i.e. the crosspropagation (XP) geometry,in which the incident planes of the visible and IR beams are perpendicular to each other,and they are both perpendicular to the surface. As far as we have known,SFGVS measurement under such geometry has not been reported in the literature. In the XP geometry,the visible and IR are delivered to the surface sharing no common optics. Therefore,the advantage of the XP geometry is that there is no requirement of the optical element to be transparent for both the visible and IR light. The XP geometry is not really useful for the application to buried interface measurement,but it can be useful for the applications to the exposed surface enclosed in a vacuum or pressure chamber for surface science and fundamental catalysis studies ^{[4, 5, 30, 31, 32, 33]},where transparent optical windows for the visible and IR lights are needed. In the coplanar geometry,either copropagation or counterpropagation,at least one window needs to be transparent for both the visible and IR beams. Such requirement significantly narrows the choices of suitable optical window materials that are both robust and stable. For example,CaF$_2$ window allows transmission of the visible frequency and IR frequency above 1000 cm$^{1}$,and BaF$_2$ windows allow transmission of visible frequencies and IR frequency above $\sim$650 cm$^{1}$. But both materials are known to be brittle and not quite robust or stable for vacuum and pressure chambers. While KBr or CsI windows allow transmission of the visible frequencies and the IR frequency down to 300 cm$^{1}$(KBr) or even 200 cm$^{1}$(CsI),these windows are known to be easily degradable under moisture. Since with the XPgeometry the visible and IR beams no longer pass through the same windows,the abovementioned limitations on the choice of the window materials are removed.
In the following section,as the XPSFGVS has not been reported before,we first present the formulation of the XPSFGVS and the derivation of the experimental measurable properties similar to the formulations as in the most commonly used copropagation geometry as in the SFGVS literatures ^{[10, 11, 34, 35]}; then we present experimental results using the XPSFGVS measurement on the air/DMSO (dimethyl sulfoxide) interface,cyclohexanol on the vapor/$z$cut $\alpha$quartz surface in the C$$H stretching vibration region,i.e. in 2800$$3050 cm$^{1}$,and the air/$z$cut $\alpha$quartz surface in the 1000$$1280 cm$^{1}$ phonon mode region. These experimental results provided verification on the effectiveness of the formulations. With these established,vacuum and pressure chambers for XPSFGVS can be designed and constructed accordingly for surface science and fundamental catalysis studies with extended IR frequency range.
Ⅱ. THEORY FORMULATION OF CROSSPROPAGATION GEOMETRYHere we present the formulations of the effective susceptibility terms for the crosspropagation SFGVS,i.e. XPSFGVS of the rotationally isotropic surface with the C$_\infty$ symmetry,and the $z$cut $\alpha$quartz surfaces with the D$_3$ symmetry. Other important crystal surfaces with different symmetries,such as the anatase or rutile TiO$_2$ surfaces,can be formulated similarly. In these formulation,the optical field needs to be projected to the principal axes in the laboratory coordinates system and proper products of the optical fields and the macroscopic susceptibility tensors of the surface need to be obtained.
A. The SFG detection angles in the XPSFGVSFigure 2 shows the wave vector and field directions of the visible,the IR and the SFG signal fields in the XPSFGVS. Here,the incoming visible beam ($\omega_1$) is in the $XZ$ plane with an incident angle of $\beta_1$ from the surface normal $z$; the incoming IR beam ($\omega_2$) is in the $YZ$ plane with an incident angle $\beta_2$; and the outgoing SFG beam ($\omega$) is in a plane that is with an dihedral angle of $\gamma$ from the $XZ$ plane with the outgoing angle of $\beta$ from the surface normal $z$. In coplanar geometry,as in Fig. 1,the visible,IR and SFG signal beams are all in the same incident plane. However,in the XPSFGVS shown in Fig. 2,neither two of the three beams are in the same incident plane. Accordingly,the wave vector projections of the visible,IR and SFG beams are listed in the Table I.
Since in the SFG process the wave vectors of the visible,IR and SFG beams should satisfy both the energy conservation condition and the momentum conservation condition in both the $X$ and $Y$ directions ^{[36, 37]},one has,for energy conservation:
$\omega = {\omega _1} + {\omega _2}$  (1) 
for momentum conservation:
${\omega _1}\sin {\beta _1} = \omega \sin \beta \cos \gamma $  (2) 
${\omega _2}\sin {\beta _2} = \omega \sin \beta \sin \gamma $  (3) 
The momentum conservation condition in the XPSFGVS is related to two angles,i.e. $\beta$ and $\gamma$. In comparison,the momentum conservation condition in the coplanar geometry is only related to one outgoing angle $\beta$ ^{[10, 11, 35]}.Thus,when $\beta _1$=$\beta _2$,
$\begin{array}{l} \tan \gamma = \frac{{{\omega _2}\sin {\beta _2}}}{{{\omega _1}\sin {\beta _1}}} = \frac{{{\omega _2}}}{{{\omega _1}}}\\ \sin \beta = \frac{{{\omega _1}\sin {\beta _1}}}{{\omega \cos \left( {\arctan \frac{{{\omega _2}\sin {\beta _2}}}{{{\omega _1}\sin {\beta _1}}}} \right)}} \end{array}$  (4) 
$ = \frac{{{\omega _1}\sin {\beta _1}}}{{\omega \cos \left( {\arctan \frac{{{\omega _2}}}{{{\omega _1}}}} \right)}}$  (5) 
Eq.(4) and Eq.(5) defines the unique outgoing direction of the SFG signal from the values of the frequencies and incident angles of the visible and IR beams. One can see that the relationships in Eq.(4) and Eq.(5) are simplified when the visible and IR incident angles are equal,i.e. $\beta_1$=$\beta_2$. Since now the visible and IR beams are in $XZ$ and $YZ$ planes,respectively,$\beta_1$=$\beta_2$ is apparently different from the case of collinear propagation condition in the copropagation case with the coplanar geometry.
Using Eq.(4) and Eq.(5),one can calculate the $\beta$ and the $\gamma$ values for the typical SFG experiment with visible beam wavelength at 800 or 532.1 nm,and different IR frequencies. These results for the typical $\beta$ and $\gamma$ values of the simplified case with $\beta_1$=$\beta_2$ are listed in the Table Ⅱ. $\beta$ and $\gamma$ values for the more general cases with $\beta_1$$\neq$$\beta_2$ can also be directly calculated from the Eq.(4) and Eq.(5). One can see from Table Ⅱ that the dihedral angle $\gamma$ values change in the range of 15° with 800 nm visible and IR in the full range of 300$$ 4000 cm$^{1}$,and are about 11° with 532.1 nm visible and IR in the full range of 300$$4000 cm$^{1}$. Such results also indicate that in a typical SFG experiment with IR frequency changing in the range of 300 cm$^{1}$,the $\gamma$ angle for SFG signal detection is usually within 1° or so. The same is true for the outgoing angle $\beta$. For such a small change of the $\beta$ and $\gamma$ values,there is generally no need to vary the detection angle in a typical XPSFGVS experiment. However,if the IR range is much larger than 300 cm$^{1}$,some adjustment of the $\beta$ and $\gamma$ angles is necessary.
B. The field projections in the XPSFGVSIn order to know the SFG signal in different polarizations,the polarizations and the projections of the visible,the IR and the SFG signal fields in the laboratory coordinates system ($X,Y,Z$) need to be known (Table Ⅲ). In Table Ⅲ,the polarization s is defined as the field direction that is perpendicular to the incident (for visible and IR beams) or outgoing (for SFG signal beam) plane,and the polarization p is defined as the field direction is in the incident or outgoing plane. One can see that the s and p polarizations for the visible,the IR and the SFG signal beams are all different. For example,the s polarization of the visible beam is along the $Y$ direction in the laboratory frame,while the s polarization of the IR beam is along the $X$ direction. This cross relationship between the visible and the IR beams is the direct result of the crosspropagation geometry. If the incident plane of the visible and the IR beams are not perpendicular to each other,the vector projection and the filed projection relationships can also be derived,but they are to be more complicated. In this study,we stick to the simplest case with these two incident planes perpendicular to each other as defined in Fig. 1.
Using the polarization definition and the projection of the three laser optical beams,one can directly derive the SFG tensor element expressions by knowing the macroscopic susceptibility tensors of the surfaces. The relationships between the nonzero elements of the macroscopic susceptibility tensor elements of the surface are different for surfaces with different symmetry properties. Below the results for the rotationally isotopic surface (C$_\infty$ symmetry) and the $z$cut $\alpha$quartz surface (D$_3$ symmetry) are presented.
C. Expressions for the rotationally isotropic surface (C∞symmetry)The rotationally isotropic surface (C$_\infty$),has 11 nonzero elements (7 achiral and 4 chiral) as the followings ^{[7, 11]},for Achiral terms:
$ZZZ,XXZ{\rm{ }} = {\rm{ }}YYZ,{\rm{ }}XZX{\rm{ }} = {\rm{ }}YZY,{\rm{ }}ZXX{\rm{ }} = {\rm{ }}ZYY$  (6) 
$XYZ =  YXZ,\quad XZY =  YZX$  (7) 
For an achiral rotationally isotropic surface (C$_\infty$ symmetry),only the 7 achiral terms are nonzero. The complete non zero elements in different polarization combinations are listed in the Table Ⅳ. The terms in red are the achiral susceptibility tensors and the terms in black are the chiral susceptibility tensors.
Therefore,using the definition of the s and p polarization for the visible,the IR and the SFG signal beams in Table Ⅲ,the effective susceptibility terms that can be directly measured in the XPSFGVS experiment with eight polarization combinations are derived and listed in Eq.(8)(14) below.
${\chi _{{\rm{sss}}}} = 0$  (8) 
$\begin{array}{l} {\chi _{{\rm{ssp}}}} = {L_{YY}}(\omega ){L_{YY}}({\omega _1}){L_{ZZ}}({\omega _2})\cos \gamma \sin {\beta _2} \cdot {\chi _{YYZ}}  \\ \;\;\;\;\;\;\;\;\;{L_{XX}}(\omega ){L_{YY}}({\omega _1}){L_{ZZ}}({\omega _2})\sin \gamma \sin {\beta _2} \cdot {\chi _{XYZ}}\\ {\chi _{{\rm{sps}}}} =  {L_{XX}}(\omega ){L_{ZZ}}({\omega _1}){L_{YY}}({\omega _2})\sin \gamma \sin {\beta _1} \cdot {\chi _{XZX}} + \\ \;\;\;\;\;\;\;\;\;{L_{YY}}(\omega ){L_{ZZ}}({\omega _1}){L_{YY}}({\omega _2})\cos \gamma \sin {\beta _1} \cdot {\chi _{YZX}} \end{array}$  (9) 
${\chi _{{\rm{pss}}}} = {L_{ZZ}}(\omega ){L_{YY}}({\omega _1}){L_{YY}}({\omega _2})\sin \beta \cdot {\chi _{ZYX}}$  (10) 
$\begin{array}{l} {\chi _{{\rm{pps}}}} =  {L_{XX}}(\omega ){L_{ZZ}}({\omega _1}){L_{YY}}({\omega _2})\cos \gamma \cos \beta \sin {\beta _1} \cdot \\ \;\;\;\;\;\;\;\;\;{\chi _{XZX}}  {L_{YY}}(\omega ){L_{ZZ}}({\omega _1}){L_{YY}}({\omega _2})\sin \gamma \cos \beta \cdot \\ \;\;\;\;\;\;\;\;\;\sin {\beta _1}{\chi _{YZX}} + {L_{ZZ}}(\omega ){L_{XX}}({\omega _1}){L_{YY}}({\omega _2}) \cdot \\ \;\;\;\;\;\;\;\;\;\;\sin \beta \cos {\beta _1} \cdot {\chi _{ZXX}} \end{array}$  (11) 
$\begin{array}{l} {\chi _{{\rm{psp}}}} =  {L_{XX}}(\omega ){L_{YY}}({\omega _1}){L_{ZZ}}({\omega _2})\cos \gamma \cos \beta \sin {\beta _2} \cdot \\ \;\;\;\;\;\;\;\;\;{\chi _{XYZ}}  {L_{YY}}(\omega ){L_{YY}}({\omega _1}){L_{ZZ}}({\omega _2})\sin \gamma \cos \beta \cdot \\ \;\;\;\;\;\;\;\;\;\sin {\beta _2}{\chi _{YYZ}} + {L_{ZZ}}(\omega ){L_{YY}}({\omega _1}){L_{XX}}({\omega _2})\\ \;\;\;\;\;\;\;\;\;\;\sin \beta \cos {\beta _2} \cdot {\chi _{ZYY}} \end{array}$  (12) 
$\begin{array}{l} {\chi _{{\rm{spp}}}} =  {L_{XX}}(\omega ){L_{ZZ}}({\omega _1}){L_{XX}}({\omega _2})\sin \gamma \sin {\beta _1}\cos {\beta _2} \cdot \\ \;\;\;\;\;\;\;\;\;\;{\chi _{XZY}}  {L_{XX}}(\omega ){L_{XX}}({\omega _1}){L_{ZZ}}({\omega _2})\sin \gamma \cdot \\ \;\;\;\;\;\;\;\;\;\;\cos {\beta _1}\sin {\beta _2} \cdot {\chi _{XXZ}} + {L_{YY}}(\omega ){L_{ZZ}}({\omega _1}) \cdot \\ \;\;\;\;\;\;\;\;\;\;\;{L_{XX}}({\omega _2})\cos \gamma \sin {\beta _1}\cos {\beta _2} \cdot {\chi _{YZY}} + \\ \;\;\;\;\;\;\;\;\;\;\;{L_{YY}}(\omega ){L_{XX}}({\omega _1}){L_{ZZ}}({\omega _2})\cos \gamma \cdot \\ \;\;\;\;\;\;\;\;\;\;\cos {\beta _1}\sin {\beta _2} \cdot {\chi _{YXZ}} \end{array}$  (13) 
$ \begin{array}{l} {\chi _{{\rm{ppp}}}} =  {L_{XX}}(\omega ){L_{XX}}({\omega _1}){L_{ZZ}}({\omega _2})\cos \gamma \cos \beta \cos {\beta _1} \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\sin {\beta _2} \cdot {\chi _{XXZ}}  {L_{XX}}(\omega ){L_{ZZ}}({\omega _1}){L_{XX}}({\omega _2}) \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\cos \gamma \cos \beta \sin {\beta _1}\cos {\beta _2} \cdot {\chi _{XZY}}  {L_{YY}}(\omega ) \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\;{L_{XX}}({\omega _1}){L_{ZZ}}({\omega _2})\sin \gamma \cos \beta \cos {\beta _1}\sin {\beta _2} \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\;\;{\chi _{YXZ}}  {L_{YY}}(\omega ){L_{ZZ}}({\omega _1}){L_{XX}}({\omega _2})\sin \gamma \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\cos \beta \cdot \sin {\beta _1}\cos {\beta _2} \cdot {\chi _{YZY}} + {L_{ZZ}}(\omega ) \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{L_{XX}}({\omega _1}){L_{XX}}({\omega _2})\sin \beta \cos {\beta _1}\cos {\beta _2} \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\chi _{ZXY}} + {L_{ZZ}}(\omega ){L_{ZZ}}({\omega _1}){L_{ZZ}}({\omega _2}) \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\sin \beta sin{\beta _1}\sin {\beta _2} \cdot {\chi _{ZZZ}} \end{array}$  (14) 
In these expressions,the red terms are the achiral terms and the black terms are the chiral terms,and the $L_{II}(\omega_i)$ are the SFG Fresnel factors defined in the SFG literature ^{[11, 35]}. It is to be noted that for SFG Fresnel factors,since the IR beam is in the cross geometry from the visible incident plane,the expressions of $L_{XX}(\omega_2)$ and $L_{YY}(\omega_2)$ terms for the IR beam should exchange their positions with each other,as the $XZ$ plane is defined as the incident plane in obtaining the Fresnel factors,while in the crossgeometry case the $YZ$ plane is the incident plane for the IR beam. For VIS and SF beam,the definition is still the same as in the literature ^{[11, 35]}.
It is interesting to see that for the C$_\infty$ surface,all the 7 nonzero polarization combinationterms,i.e. ssp,sps,pss,pps,psp,spp,and ppp,are not purely achiral. In contrast,in the coplanar geometry,the ssp,sps,pss and ppp terms are always purely achiral ^{[7, 11]}. In the coplanar geometry,the ssp,sps,pss terms are all function of a single achiral macroscopic susceptibility tensors,i.e. $YYZ$,$YZY$,and $ZYY$,respectively ^{[7, 11]}. However,in the XPSFGVS,only the pss term is a function of a single chiral term ZYX (Eq.(10)). Therefore,the polarization combination terms in the XPSFGVS are generally more complicated than the corresponding polarization combination terms. This is probably the principal reason that as far as we have known,there has been no study with XPSFGVS geometry in the SFGVS literature.
However,a closer examination of all the polarization combination terms in Eq.(8}) to Eq.(14) shows that these terms are not so difficult to understand or measure,and they can be simplified or reduced to simple terms with good approximation. For example,for a given molecular surface,the achiral $YYZ$ term is usually larger than the chiral $XYZ$ term,and also in the XPSFGVS experiment $\cos\gamma$ is usually much larger than $\sin\gamma$,because $\gamma$ is usually a small angle (see Table Ⅱ). This is particularly true when at lower IR frequencies such as 300 or 600 cm$^{1}$,the range that the XPSFGVS geometry is most useful in avoiding the limitation of the selection for window materials. Consequently,the contribution from the chiral $XYZ$ term in the ssp polarization combination in XPSFGVS (Eq.(9)) is usually much smaller or even negligible in comparison with the contribution of the achiral $YYZ$ term. Thus,the ssp term can basically be approximated by the single achiral $YYZ$ term,making the ssp polarization term in the XPSFGVS similar to the ssp polarization term in the coplanar geometry. The rest of the terms in the Eq.(8) to Eq.(14) can also be evaluated in the same way. And it is easy to show that despite the differences in their expressions,qualitatively speaking,the polarization terms in the XPSFGVS are usually dominated by the same polarization terms in the coplanar geometry measurement. Of course,detailed quantitative analysis of the XPSFGVS data will show the differences from the coplanar geometry measurements. Nevertheless,qualitative understanding of the XPSFGVS data is not as difficult as the above expressions seemingly imply.
D. Expressions for $z$cut $\alpha$quartz surface (D$_\mathbf{3}$ symmetry)The nonzero macroscopic susceptibility tensors of the $z$cut $\alpha$quartz are listed in the Table Ⅳ. There are eight achiral elements and 6 chiral elements. In order to get the complete expressions of XPSFGVS polarization terms for the $z$cut $\alpha$quartz surface,one needs to consider the azimuthal angle dependences,i.e. rotational dependence along the $z$ axis of the $z$cut $\alpha$quartz crystal. In order to do this,one has to consider the nonzero polarizability tensors of the unit cell of the quartz crystal with D$_3$ group symmetry. There are 8 nonzero polarizability elements (4 achiral and 4 chiral) for the unit cell of the quartz crystal,with the unit cell coordinates system defined as ($x,y,z$) ^{[38]},i.e.,
$ xxx{\rm{ }} = {\rm{ }}  {\rm{ }}xyy{\rm{ }} = {\rm{ }}  {\rm{ }}yzy{\rm{ }} = {\rm{ }}  {\rm{ }}yyz$  (15) 
$xyz{\rm{ }} = {\rm{ }}xzy{\rm{ }} = {\rm{ }}  {\rm{ }}yzx{\rm{ }} = {\rm{ }}  {\rm{ }}yxz$  (16) 
Also it is known that for the lefthanded $\alpha$quartz,$xxx/xyz$$>$0,while for the righthanded $\alpha$quartz $xxx/xyz$$ < $0 ^{[38]}. For $z$cut quartz,the transformation matrix for rotation around the $z$axis with the azimuthal angle $\phi$ placed in the $Z$ direction of the laboratory coordinates system ($X,Y,Z$) (Fig. 3) is,
$R = \left( {\begin{array}{*{20}{c}} {{R_{xx}}}&{{R_{xy}}}&{{R_{xz}}}\\ {{R_{yx}}}&{{R_{yy}}}&{{R_{yz}}}\\ {{R_{zx}}}&{{R_{zy}}}&{{R_{zz}}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\cos \phi }&{  \sin \phi }&0\\ {\sin \phi }&{\cos \phi }&0\\ 0&0&1 \end{array}} \right)$  (17) 
The relationship between the macroscopic susceptibility tensor elements and the microscopic polarizability tensor terms is,
${\chi _{IJK}} = \sum\limits_{ijk} {\left\langle {{R_{Ii}}{R_{Jj}}{R_{Kk}}} \right\rangle {\beta _{ijk}}} $  (18) 
Accordingly,the 14 nonzero elements in Table Ⅳ for $z$cut aquartz crystal are derived and listed in Table Ⅴ.
Using the field vector projections in Table Ⅲ,the nonzero macroscopic susceptibility terms,and the expressions in the Table V,the following effective susceptibility terms in the 8 polarization combinations are derived for the XPSFGVS with $z$cut $\alpha$quartz (Eq.(19)$$Eq.(28)). In these expressions,the red terms are contributions from the achiral terms of the quartz unit cell,and the black are contribution from the chiral terms.
$\begin{array}{l} {\chi _{{\rm{sss}}}} =  {L_{XX}}(\omega ){L_{YY}}({\omega _1}){L_{YY}}({\omega _2})\sin \gamma \cdot {\chi _{XYX}} + \\ \;\;\;\;\;\;\;\;{L_{YY}}(\omega ){L_{YY}}({\omega _1}){L_{YY}}({\omega _2})\cos \gamma \cdot {\chi _{YYX}}\\ \;\;\;\;\;\; =  {L_{XX}}(\omega ){L_{YY}}({\omega _1}){L_{YY}}({\omega _2})\sin \gamma \cdot {\beta _{xxx}}\\ \;\;\;\;\;\;\sin 3\phi  {L_{YY}}(\omega ){L_{YY}}({\omega _1}){L_{YY}}({\omega _2})\cos \gamma \cdot \\ \;\;\;\;\;\;{\beta _{xxx}}\cos 3\phi \end{array}$  (19) 
$\begin{array}{l} {\chi _{{\rm{ssp}}}} =  {L_{XX}}(\omega ){L_{YY}}({\omega _1}){L_{XX}}({\omega _2})\sin \gamma \cos {\beta _2} \cdot \\ \;\;\;\;\;\;\;\;\;{\chi _{XYY}}  {L_{XX}}(\omega ){L_{YY}}({\omega _1}){L_{ZZ}}({\omega _2})\sin \gamma \sin {\beta _2} \cdot \\ \;\;\;\;\;\;\;\;\;{\chi _{XYZ}} + {L_{YY}}(\omega ){L_{YY}}({\omega _1}){L_{XX}}({\omega _2})\cos \gamma \cos {\beta _2} \cdot \\ \;\;\;\;\;\;\;\;\;\;\;{\chi _{YYY}}\\ = + {L_{XX}}(\omega ){L_{YY}}({\omega _1}){L_{XX}}({\omega _2})\sin \gamma \cos {\beta _2} \cdot \\ \;\;\;\;\;\;{\beta _{xxx}}\cos 3\phi  {L_{XX}}(\omega ){L_{YY}}({\omega _1}){L_{ZZ}}({\omega _2}) \cdot \\ \;\;\;\;\;\;\;\sin \gamma \sin {\beta _2}{\beta _{xyz}}  {L_{YY}}(\omega ){L_{YY}}({\omega _1}){L_{XX}}({\omega _2}) \cdot \\ \;\;\;\;\;\;\;\cos \gamma \cos {\beta _2} \cdot {\beta _{xxx}}\sin 3\phi \end{array}$  (20) 
$\begin{array}{l} {\chi _{{\rm{sps}}}} =  {L_{XX}}(\omega ){L_{XX}}({\omega _1}){L_{YY}}({\omega _2})\sin \gamma \cos {\beta _1} \cdot \\ \;\;\;\;\;\;\;\;\;{\chi _{XXX}} + {L_{YY}}(\omega ){L_{XX}}({\omega _1}){L_{YY}}({\omega _2}) \cdot \\ \;\;\;\;\;\;\;\;\;\;\cos \gamma \cos {\beta _1}{\chi _{YXX}} + {L_{YY}}(\omega ){L_{ZZ}}({\omega _1}){L_{YY}}({\omega _2}) \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\cos \gamma \sin {\beta _1} \cdot {\chi _{YZX}} \end{array}$  (21) 
$\begin{array}{l} =  {L_{XX}}(\omega ){L_{XX}}({\omega _1}){L_{YY}}({\omega _2})\sin \gamma \cos {\beta _1} \cdot \\ \;\;\;\;{\beta _{xxx}}\cos 3\phi \cdot {\beta _{xxx}}\cos 3\phi + {L_{YY}}(\omega ){L_{XX}}({\omega _1}) \cdot \\ \;\;\;\;\;{L_{YY}}({\omega _2})\cos \gamma \cos {\beta _1} \cdot {\beta _{xxx}}\sin 3\phi  {L_{YY}}(\omega )\\ \;\;\;\;\;{L_{ZZ}}({\omega _1}){L_{YY}}({\omega _2})\cos \gamma \sin {\beta _1} \cdot {\beta _{xyz}} \end{array}$  (22) 
$\begin{array}{l} {\chi _{{\rm{pss}}}} =  {L_{XX}}(\omega ){L_{YY}}({\omega _1}){L_{YY}}({\omega _2})\cos \gamma \cos \beta \cdot \\ \;\;\;\;\;\;\;\;\;{\chi _{XYX}}  {L_{YY}}(\omega ){L_{YY}}({\omega _1}){L_{YY}}({\omega _2}) \cdot \\ \;\;\;\;\;\;\;\;\;\sin \gamma \cos \beta \cdot {\chi _{YYX}} \end{array}$  (23) 
$\begin{array}{l} =  {L_{XX}}(\omega ){L_{YY}}({\omega _1}){L_{YY}}({\omega _2})\cos \gamma \cos \beta \cdot \\ \;\;\;\;{\beta _{xxx}}\sin 3\phi + {L_{YY}}(\omega ){L_{YY}}({\omega _1}){L_{YY}}({\omega _2}) \cdot \\ \;\;\;\;\;\sin \gamma \cos \beta \cdot {\beta _{xxx}}\cos 3\phi \end{array}$  (24) 
$\begin{array}{l} {\chi _{{\rm{pps}}}} =  {L_{XX}}(\omega ){L_{XX}}({\omega _1}){L_{YY}}({\omega _2})\cos \gamma \cos \beta \cdot \\ \;\;\;\;\;\;\;\;\;\cos {\beta _1} \cdot {\chi _{XXX}}  {L_{YY}}(\omega ){L_{XX}}({\omega _1}){L_{YY}}({\omega _2}) \cdot \\ \;\;\;\;\;\;\;\;\;\;\sin \gamma \cos \beta \cos {\beta _1} \cdot {\chi _{YXX}}  {L_{YY}}(\omega ){L_{ZZ}}({\omega _1}) \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\;{L_{YY}}({\omega _2})\sin \gamma \cos \beta \sin {\beta _1} \cdot {\chi _{YZX}}\\ \;\;\;\;\;\;\;\;\;\; =  {L_{XX}}(\omega ){L_{XX}}({\omega _1}){L_{YY}}({\omega _2})\cos \gamma \cos \beta \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\cos {\beta _1} \cdot {\beta _{xxx}}\cos 3\phi  {L_{YY}}(\omega ){L_{XX}}({\omega _1}) \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{L_{YY}}({\omega _2})\sin \gamma \cos \beta \cos {\beta _1} \cdot {\beta _{xxx}}\sin 3\phi + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{L_{YY}}(\omega ){L_{ZZ}}({\omega _1}){L_{YY}}({\omega _2}) \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\sin \gamma \cos \beta \sin {\beta _1} \cdot {\beta _{xyz}} \end{array}$  (25) 
$\begin{array}{l} {\chi _{{\rm{psp}}}} =  {L_{XX}}(\omega ){L_{YY}}({\omega _1}){L_{XX}}({\omega _2})\cos \gamma \cos \beta \cdot \\ \;\;\;\;\;\;\;\;\;\;\cos {\beta _2} \cdot {\chi _{XYY}}  {L_{XX}}(\omega ){L_{YY}}({\omega _1}){L_{ZZ}}({\omega _2}) \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\cos \gamma \cos \beta \sin {\beta _2} \cdot {\chi _{XYZ}}  {L_{YY}}(\omega ){L_{YY}}({\omega _1}) \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\;{L_{XX}}({\omega _2})\sin \gamma cos\beta \cos {\beta _2} \cdot {\chi _{YYY}}\\ \;\;\;\;\;\;\;\;\;\; = + {L_{XX}}(\omega ){L_{YY}}({\omega _1}){L_{XX}}({\omega _2})\cos \gamma \cos \beta \cos {\beta _2} \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{\beta _{xxx}}\cos 3\phi  {L_{XX}}(\omega ){L_{YY}}({\omega _1}){L_{ZZ}}({\omega _2}) \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\cos \gamma \cos \beta \sin {\beta _2} \cdot {\beta _{xyz}} + {L_{YY}}(\omega ){L_{YY}}({\omega _1}) \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{L_{XX}}({\omega _2})\sin \gamma \cos \beta \cos {\beta _2} \cdot {\beta _{xxx}}\sin 3\phi \end{array}$  (26) 
$\begin{array}{l} {\chi _{{\rm{spp}}}} =  {L_{XX}}(\omega ){L_{XX}}({\omega _1}){L_{XX}}({\omega _2})\sin \gamma \cos {\beta _1} \cdot \\ \;\;\;\;\;\;\;\;\;\;\cos {\beta _2} \cdot {\chi _{XXY}} + {L_{YY}}(\omega ){L_{XX}}({\omega _1}) \cdot \\ \;\;\;\;\;\;\;\;\;{L_{XX}}({\omega _2})\cos \gamma \cos {\beta _1}\cos {\beta _2} \cdot {\chi _{YXY}} + \\ \;\;\;\;\;\;\;\;\;\;{L_{YY}}(\omega ){L_{XX}}({\omega _1}){L_{ZZ}}({\omega _2}) \cdot \\ \;\;\;\;\;\;\;\;\;\;\cos \gamma \cos {\beta _1}\sin {\beta _2} \cdot {\chi _{YXZ}}\\ \;\;\;\; =  {L_{XX}}(\omega ){L_{XX}}({\omega _1}){L_{XX}}({\omega _2})\sin \gamma \cos {\beta _1} \cdot \\ \;\;\;\;\;\;\;\cos {\beta _2} \cdot {\beta _{xxx}}\sin 3\phi  {L_{YY}}(\omega ){L_{XX}}({\omega _1}) \cdot \\ \;\;\;\;\;\;\;\;{L_{XX}}({\omega _2})\cos \gamma \cos {\beta _1}\cos {\beta _2} \cdot {\beta _{xxx}}\cos 3\phi  \\ \;\;\;\;\;\;\;\;{L_{YY}}(\omega ){L_{XX}}({\omega _1}){L_{ZZ}}({\omega _2}) \cdot \\ \;\;\;\;\;\;\;\;\;\cos \gamma \cos {\beta _1}\sin {\beta _2} \cdot {\beta _{xyz}} \end{array}$  (27) 
$\begin{array}{l} {\chi _{{\rm{ppp}}}} =  {L_{XX}}(\omega ){L_{XX}}({\omega _1}){L_{XX}}({\omega _2})\cos \gamma \cos \beta \cdot \\ \;\;\;\;\;\;\;\;\;\cos {\beta _1}\cos {\beta _2} \cdot {\chi _{XXY}}  {L_{XX}}(\omega ){L_{ZZ}}({\omega _1}) \cdot \\ \;\;\;\;\;\;\;\;\;{L_{XX}}({\omega _2})\cos \gamma \cos \beta \sin {\beta _1}\cos {\beta _2} \cdot {\chi _{XZY}}  \\ \;\;\;\;\;\;\;\;\;\;{L_{YY}}(\omega ){L_{XX}}({\omega _1}){L_{XX}}({\omega _2})\sin \gamma \cos \beta \cdot \\ \;\;\;\;\;\;\;\;\;\;\cos {\beta _1}\cos {\beta _2} \cdot {\chi _{YXY}}  {L_{YY}}(\omega ){L_{XX}}({\omega _1}) \cdot \\ \;\;\;\;\;\;\;\;\;\;{L_{ZZ}}({\omega _2})\sin \gamma \cos \beta \cos {\beta _1}\sin {\beta _2} \cdot {\chi _{YXZ}}\\ \;\;\;\;\; =  {L_{XX}}(\omega ){L_{XX}}({\omega _1}){L_{XX}}({\omega _2})\cos \gamma \cos \beta \cdot \\ \;\;\;\;\;\;\;\;\;\cos {\beta _1}\cos {\beta _2} \cdot {\beta _{xxx}}\sin 3\phi  {L_{XX}}(\omega ) \cdot \\ \;\;\;\;\;\;\;\;\;{L_{ZZ}}({\omega _1}){L_{XX}}({\omega _2})\cos \gamma \cos \beta \sin {\beta _1}\cos {\beta _2} \cdot \\ \;\;\;\;\;\;\;\;\;\;\;{\beta _{xyz}} + {L_{YY}}(\omega ){L_{XX}}({\omega _1}){L_{XX}}({\omega _2}) \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\sin \gamma \cos \beta \cos {\beta _1}\cos {\beta _2} \cdot {\beta _{xxx}}\cos 3\phi + \\ \;\;\;\;\;\;\;\;\;\;\;{L_{YY}}(\omega ){L_{XX}}({\omega _1}){L_{ZZ}}({\omega _2})\sin \gamma \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\cos \beta \cos {\beta _1}\sin {\beta _2} \cdot {\beta _{xyz}} \end{array}$  (28) 
Here,Eq.(19)$$Eq.(28) for the XPSFGVS on $z$cut $\alpha$quartz can also be compared with the corresponding terms in the copropagation case in the coplanar geometry in the literature ^{[14, 22, 23, 39]}. It is similar to the C$_\infty$ surface discussed above,there is no longer a pure achiral term in the XPSFGVS expressions. However,the case for the D$_3$ symmetry is apparently much more complicated than the case for the C$_\infty$ symmetry. For example,in Eq.(20),there are contributions from both the $\beta_{xxx}\sin3\phi$ and $\beta_{xxx}\cos3\phi$ terms to the ssp susceptibility term,instead of only the $\beta_{xxx}\cos3\phi$ term in the case for copropagation case. To evaluate the relative contribution and overall azimuthal angle dependence from the sum of these two mutually excluding terms as well as the chiral term can be quite complicated,especially for the phonon modes or quartz in the 750$$1300 cm$^{1}$ region,as shown in previous studies by Liu and Shen ^{[40, 41]}. By applying the small $\sin\gamma$ approximation,one can find that the ssp term in Eq.(20) is dominated by the $\beta_{xxx}\sin3\phi$ azimuthal angle dependence,instead of the $\beta_{xxx}\cos3\phi$ term as in the ssp term in the copropagation case in the coplanar geometry ^{[14]}. However,how good this approximation is depends also on the Fresnel factors at specific frequencies that are resonant with specific vibrational modes ^{[40, 41]}. Nevertheless,the full expressions as presented in the Eq.(19)$$Eq.(28) provide the frameworks to quantitatively evaluate these factors.
One final note to the Eq.(19) to Eq.(28) is that the Fresnel factors in the SFG literature are usually defined by assuming the substrate is without birefringence. Actually this is not always true and may result complications in the evaluation of the overall effective susceptibility in these equations. For example,for $z$cut $\alpha$quartz,since its $z$ axis is the optical axis,any incident or outgoing beam with polarization in the incident or outgoing plane,i.e. the p polarization,is the ordinary ray,while the beam with polarization perpendicular to the incident or outgoing plane,i.e. the s polarization,is the extraordinary ray. When there is birefringence,the refractive indexes of the ordinary and extraordinary rays are different. Thus the Fresnel factors needs to be calculated accordingly for the s and p polarization,respectively. Even though the birefringence of $z$cut $\alpha$quartz is known to be small ($\Delta n$=$n_e$$$$n_o$$\approx$0.009 in the visible region) ^{[42]},its birefringence can still be observed if the SFG measurement is done accurately. The effect is to be much larger for materials such as rutile TiO$_2$ ($\Delta n$=$n_e$$$$n_o$$\approx$0.28 in the visible region) or anatase TiO$_2$ ($\Delta n$=$n_e$$$$n_o$$\approx$0.17 in the visible region) ^{[42]},whose surfaces are important for studying photocatalytic reactions and have been extensively studies using SFGVS ^{[31, 43, 44]}. These effects can be more significant when dealing with their low frequency modes. Further work regarding these effects in the SFGVS in both the XPSFGVS and the coplanar geometry SFGVS measurements is warranted.
Ⅲ. EXPERIMENTSTo demonstrate the XPSFGVS does work,the XPSFGVS spectra of a few model surfaces are measured using both recently developed sub 1cm$^{1}$ highresolution broadband SFGVS (HRBBSFGVS) with a narrow 800 nm visible beam (resolution about 0.6 cm$^{1}$). Data for the $z$cut $\alpha$quartz in the 1000$$1280 cm$^{1}$ region from the highresolution XPSFGVS are also compared with the copropagation geometry data from a commercially available picosecond scanning SFGVS spectrometer with a 532.1 nm visible beam (EKSPLA,Inc.,resolution about 6 cm$^{1}$). The detailed descriptions of the HRBBSFGVS spectrometer ^{[13, 14, 45]} and the picosecond scanning SFGVS spectrometer ^{[27, 46, 47]} can be found in our previous publications.
To achieve XPSFGVS measurement,the IR beam is redirected with IR optical mirrors into the crosspolarization. In all the experiments,we choose the simplest XPSFGVS case with the same incident angles for the visible and the IR beams in their respective incident plane,i.e. $\beta_1$=$\beta_2$ as defined in Fig. 1. The SFG detection angles followed the calculated $\beta$ and $\gamma$ values as in the Table Ⅱ.
Neat dimethyl sulfoxide (SigmaAldrich 99.9%) and cyclohexanol (SigmaAldrich 99.9%) liquids wereused as received. The air/DMSO interface was prepared by pouring~8 mL into ateflon Petry dish which was carefully cleaned with a Nochromix (Godax Laboratories,Inc) and sulfuric acidmixture,rinsed with Millipore water (18 M$\Omega\cdot$cm,Millipore Corp.),and followed by 12 min of UV/ozone cleaning. When making the cyclohexanol on vapor/$z$cut $\alpha$quartz,arighthanded $z$cut $\alpha$quartz with square shape (size 12.7 mm$\times$ 12.7 mm$\times$ 5 mm,Conex System Technology) was placed in a Teflon dish about 8 mm deep,and a drop of cyclohexanol aqueous solution (0.32 mol/L,with 18 M$\Omega\cdot$cm Millipore water) was placed at the bottom of the Teflon dish,and the dish was covered with an ozone plasma cleaned CaF$_2$ window (5 cm diameter$\times$4 mm thickness). The cyclohexanol evaporated and was adsorbed on the quartz surface.
The SFG measurement in the 2800$$3050 cm$^{1}$ range usually takes 5 min acquisition time,and the signal was normalized to the SFG profile of a thick $z$cut $\alpha$quartz (size 12.7 mm$\times$12.7 mm$\times$5 mm) at azimuthal angle of $\phi$=0°,as in all our previous HRBBSFGVS measurement ^{[13, 14]}. The SFG measurement in the 1000$$1280 cm$^{1}$ region on the $z$cut $\alpha$quartz surface usually take 60 s acquisition time,and the signal is normalized to the SFG profile of a ZnS plate (Cleatran,ISP Optics,Inc.). This is because in this lower frequency IR region,$z$cut $\alpha$quartz is no longer spectroscopically flat. So in the broadband SFG measurement the IR spectral profile was measured by the spectroscopically flat ZnS in this region for normalization. While for the scanning SFG measurement,the signal intensity was normalized directly to the visible and IR power. So the absolute value of the two sets of data cannot be directly compared without additional intensity calibration.
All experiments were performed with a freshly poured sample. All experiments were performed at room temperature of 22$\pm$2 °C.
Ⅳ. RESULTS AND DISCUSSIONHere we present the realization of the reflective XPSFGVS measurement on the air/DMSO interface and cyclohexanol on the vapor/$z$cut $\alpha$quartz surface in the C$$H stretching vibration region,i.e. in 2800$$3050 cm$^{1}$,and the air/$z$cut $\alpha$quartz surface in the 1000$$1280 cm$^{1}$ phonon mode region. These results show the feasibility of the XPSFGVS and also provided validation of the formulation presented above.
The air/DMSO interface is one of the benchmark achiral interfaces for HRBBSFGVS measurement ^{[13, 48]}. Figure 4 shows the ssp and ppp spectra of the air/DMSO interface in the XPSFGVS with $\beta_1$=$\beta_2$=45°. In the measurement,the dihedral angle between the SFG and IR planes is fixed at $\gamma$=13°,showing the validity of the values calculated in the Table Ⅱ. One can see that the spectral lineshapes in the ssp and ppp polarization combinations are similar to the lineshapes of the reported SFGVS HRBBSFGVS spectra in the copropagation coplanar geometry with $\beta_1$=45° and $\beta_2$=55° ^{[13, 49]},except for that the relative ssp and ppp intensity of the sharp $\sim$2920 cm$^{1}$ peak is slightly different. These results can be readily understood by comparing the ssp and ppp effective susceptibility expressions,i.e. Eq.(9) and Eq.(14),respectively,for the XPSFGVS,and the ssp and ppp expressions for the copropagation case in the literature ^{[11]}. The achiral ssp terms of the rotationally isotropic liquid surface in the two geometries are both from the contribution from a single the $YYZ$ macroscopic susceptibility term. However,the ppp term in the XPSFGVS in this case are with contributions from the three achiral $XXZ$,$YZY$ and $ZZZ$ terms; while in the copropagation coplanar case the ppp term are with contributions from the four achiral $XXZ$,$XZX$,$ZXX$ and $ZZZ$ terms. Quantitative analysis of the XPSFGVS from the rotationally isotropic chiral and achiral surfaces should follow the Eq.(8) to Eq.(14).
In the experiment we noticed that the ssp signal intensities in the XPSFGVS and in the copropagation coplanar measurement are at similar signal levels for the air/DMSO interface. This fact can be readily comprehended as in the XPSFGVS the ssp term is $L_{YY}(\omega)L_{YY}(\omega_1)L_{ZZ}(\omega_2)\cos\gamma\sin\beta_2$$\times$$\chi_{YYZ}$ (Eq.(9)),while in the copropagation coplanar geometry,the ssp term is $L_{YY}(\omega)L_{YY}(\omega_1)L_{ZZ}(\omega_2)\sin\beta_2$$\times$$\chi_{YYZ}$ ^{[11]}. The only difference between them is the factor of $\cos\gamma$,which is close to unity as the value of $\gamma$ is usually small (Table Ⅱ). The same is also true when comparing the SFGVS signal from the thick $z$cut $\alpha$quartz crystal surface in both geometries,where a single reflection from the quartz crystal region that satisfies phase match condition was responsible for the generated SFG signal. However,the XPSFGVS signal from the thin $z$cut $\alpha$quartz crystal (about 200 μm thickness) is usually 30 times smaller than that with the copropagation coplanar geometry. The reason for this is that there are interferences of SFGVS signals generated from multiple reflections of the visible and IR beams in the thin quartz crystal in the copropagation coplanar geometry; while in the crosspropagation geometry such multiple reflection interference phenomenon is significantly reduced as the visible and IR beams are going different directions after the initial overlapping point. Therefore,one advantage for the XPSFGVS is to avoid the complication from such multiple interferences from molecular layers adsorbed on thin substrate surfaces.
It has been shown recently that the effective susceptibility from the quartz crystal surface contains the bulk contribution of the nonlinear quartz crystal and the surface contribution of the adsorbed molecular layer. Because the bulk SFG response has an intrinsic shift of optical phase by $\pi/2$ from that of the surface response,the overall SFG signal from this interface provided a direct measurement of the phaseresolved spectra of the adsorbed molecular layer,as long as the bulk response is spectroscopically flat in the frequency region ^{[14]}. The interference pattern depends on the relative phase between the quartz bulk term and the orientation of the molecular groups on the interface. It has been known that if the CH$_3$ or CH$_2$ dipole direction is pointing away from the surface,the absolute phase of the imaginary part of their symmetric stretching modes should be positive ^{[14]}. Therefore,in the Fig. 5,the two positive peaks ($\sim$2860 and $\sim$2945 cm$^{1}$) measured with the quartz crystal azimuthal angle the $\phi$=0° suggests that the quartz term at $\phi$=0° is positive in phase; and the quartz term with $\phi$=180° is negative in phase,and the interference pattern are with negative peaks. This results put the ssp response of the $z$cut $\alpha$quartz in the XPSFGVS to be opposite in phase (sign) to that of the copropagation planar geometry ^{[14]}. Nevertheless,these results can be easily understood with the respective expressions of the ssp terms for the $z$cut $\alpha$quartz crystal surface. In the XPSFGVS case with Eq.(\ref{10.2}),the quartz ssp term at $\phi$=0° is dominated by the term $L_{XX}(\omega)L_{YY}(\omega_1)L_{XX}(\omega_2)\sin\gamma\cos\beta_2$$\cdot$$\beta_{xxx}$ and it changes sign at $\phi$=180°. It is therefore positive at $\phi$=0° and negative at $\phi$=180°. While in the copropagation planar geometry case,it has been known that the quartz ssp term at $\phi$=0° is dominated by the term $L_{YY}(\omega)L_{YY}(\omega_1)L_{XX}(\omega_2)\cos\beta_2$$\cdot$$\beta_{xxx}$ and it changes sign at $\phi$=180°. Therefore,it is negative at $\phi$=0° and positive at $\phi$=180° for the copropagation coplanar case. Therefore,the phaseresolved XPSFGVS results in Fig. 5 confirmed the validity of the formulations for $z$cut $\alpha$quartz in the XPSFGVS. Such differences between the XPSFGVS and copropagation coplanar SFGVS needs to be carefully examined when making phase resolved measurements using the two SFGVS geometries.
Figure 6 shows the highresolution broadband XPSFGVS measurement of $z$cut $\alpha$quartz crystal surface in the 1000$$1280 cm$^{1}$ region in the ssp,ppp and spp polarization combinations. These spectra can also be compared with the ssp and ppp spectra in the same region measured with the copropagation coplanar SFGVS using a picosecond scanning SFG spectrometer. One can see that the main spectral features in the XPSFGVS and the copropagation coplanar SFGVS spectra are consistent with each other. The two apparent peaks at about 1065 and 1160 cm$^{1}$ are known from the bulk phonon modes and have been studied extensively by Shen and coworkers previously ^{[40, 41]}. Here,we only presented the proofofprinciple data with azimuthal angle $\phi$=0° with XPSFGVS. The spp spectral intensity at $\phi$=0° is significantly stronger than that of the ssp and ppp spectra. This can only happen in the XPSFGVS but not in the copropagation coplanar geometry ^{[14, 40, 41]}. These differences can be quantitatively understood with the formulations for the XPSFGVS presented in this work and for the copropagation coplanar SFGVS previously ^{[14, 39, 40, 41]}.
Ⅴ. CONCLUSIONIn this work,we presented the theoretical formulation of the crosspropagation nonplanar SFGVS (XPSFGVS) with surface or interface with both C$_\infty$ and D$_3$ symmetries. We also presented experimental results from XPSFGVS measurement on the air/DMSO (dimethyl sulfoxide) interface,adsorbed cyclohexanol layer on the vapor/$z$cut $\alpha$quartz surface in the C$$H stretching vibration region,i.e. in 2800$$3050 cm$^{1}$,and the air/$z$cut $\alpha$quartz surface in the 1000$$1280 cm$^{1}$ phonon mode region. The similarities and differences in the XPSFGVS and the commonly used copropagation coplanar geometry SFGVS are also compared and examined.
Because of the momentum conservation condition,the visible,IR and SFG beams can no longer be in the same incident plane in XPSFGVS. As the result,the expressions for the effective susceptibility in different polarization combinations in XPSFGVS are more complicated than their counterparts in the coplanar SFGVS. However,in this work,we showed how these expression can be derived in detail and how experimental data can be understood using the formulations. It turned out that the XPSFGVS formulation with perpendicular visible and IR incident planes and equal incident angles is the simplest case in the non coplanar SFGVS. Using such conditions,qualitative understanding and quantitative analysis of XPSFGVS data can be achieved in a relatively less complicated fashion.
The ultimate purpose of this work on the XPSFGVS is to provide a practical solution for the limitations on the window material choices in the coplanar geometry SFGVS to study the surfaces in the enclosed vacuum or pressure chamber. SFGVS measurement on these surfaces is most useful for the surface sciences and fundamental catalysis studies,where both the vibrational spectra of the adsorbed molecular layers and the lower IR frequency vibrations of the substrate surface are of interest. We hope the theoretical and experimental exercises presented in this work shall be helpful in designing vacuum and pressure chambers for broader applications of SFGVS measurements in the surface sciences,material sciences,catalytic sciences,as well as lowtemperature molecular sciences,etc.
Ⅵ. AcknowledgementsHongFei Wang thanks Professor Zefeng Ren and Weitao Liu for discussions that later led him to consider the idea of XPSFGVS. Li Fu and Shunli Chen thank Zizwe Chase for experimental assistance. Wei Gan thanks for the support from the "1000 Talent Program" (The Recruitment Program of Global Experts). Part of this work was conducted at the William R. Wiley Environmental Molecular Sciences Laboratory (EMSL),a national scientific user facility located at the Pacific Northwest National Laboratory (PNNL) and sponsored by the Department of Energy's Office of Biological and Environmental Research (BER). Li Fu is the William R. Wiley postdoc fellow at EMSL. Shunli Chen,who is an assistant professor at the Xinjiang Technical Institute of Physics & Chemistry of the Chinese Academy of Sciences,is an Alternate Sponsored Fellow at PNNL working in EMSL during this work.
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