Sum frequency generation spectroscopy

For a given nonlinear optical process, the polarization ${\displaystyle {\overrightarrow {P}}}$ which generates the output is given by

${\displaystyle {\overrightarrow {P}}=\epsilon _{0}\left(\chi ^{(1)}{\overrightarrow {E}}+\chi ^{(2)}{\overrightarrow {E}}^{2}+\chi ^{(3)}{\overrightarrow {E}}^{3}+\dots +\chi ^{(n)}{\overrightarrow {E}}^{n}\right)=\epsilon _{0}\sum _{i=1}^{n}\chi ^{(i)}{\overrightarrow {E}}^{i}}$

where ${\displaystyle \chi ^{(i)}}$ is the ${\displaystyle i}$th order nonlinear susceptibility, for ${\displaystyle i\in [1,2,3,\dots ,n]}$.

It is worth noting that all the even order susceptibilities become zero in centrosymmetric media. A proof of this is as follows.

Let ${\displaystyle I_{inv}}$ be the inversion operator, defined by ${\displaystyle I_{inv}{\overrightarrow {L}}=-{\overrightarrow {L}}}$ for some arbitrary vector ${\displaystyle {\overrightarrow {L}}}$. Then applying ${\displaystyle I_{inv}}$ to the left and right hand side of the polarization equation above gives

${\displaystyle I_{inv}{\overrightarrow {P}}=-{\overrightarrow {P}}=I_{inv}\left(\epsilon _{0}\sum _{i=1}^{n}\chi ^{(i)}{\overrightarrow {E}}^{i}\right)=\epsilon _{0}\sum _{i=1}^{n}\chi ^{(i)}\left(I_{inv}{\overrightarrow {E}}\right)^{i}=\epsilon _{0}\sum _{i=1}^{n}(-1)^{i}\chi ^{(i)}{\overrightarrow {E}}^{i}.}$

Adding together this equation with the original polarization equation then gives

${\displaystyle {\overrightarrow {P}}-{\overrightarrow {P}}={\overrightarrow {0}}=\epsilon _{0}\sum _{i=1}^{n}\left(1+(-1)^{i}\right)\chi ^{(i)}{\overrightarrow {E}}^{i}=2\epsilon _{0}\sum _{i=1}^{n/2}\chi ^{(2i)}{\overrightarrow {E}}^{(2i)}}$

which implies ${\displaystyle \chi ^{(2i)}=0}$ for ${\displaystyle i\in [1,2,3,\dots ,n/2]}$ in centrosymmetric media. Q.E.D.

[Note 1: The final equality can be proven by mathematical induction, by considering two cases in the inductive step; where ${\displaystyle k}$ is odd and ${\displaystyle k}$ is even.]

[Note 2: This proof holds for the case where ${\displaystyle n}$ is even. Setting ${\displaystyle m=n-1}$ gives the odd case and the remainder of the proof is the same.]

As a second-order nonlinear process, SFG is dependent on the 2nd order susceptibility ${\displaystyle \chi ^{(2)}}$, which is a third rank tensor. This limits what samples are accessible for SFG. Centrosymmetric media include the bulk of gases, liquids, and most solids under the assumption of the electric-dipole approximation, which neglects the signal generated by multipoles and magnetic moments.[7] At an interface between two different materials or two centrosymmetric media, the inversion symmetry is broken and an SFG signal can be generated. This suggests that the resulting spectra represent a thin layer of molecules. A signal is found when there is a net polar orientation.[7][9]

The output beam is collected by a detector and its intensity ${\displaystyle I}$ is calculated using[7][10]

${\displaystyle I(\omega _{3};\omega _{1},\omega _{2})\propto |\chi ^{(2)}|^{2}I_{1}(\omega _{1})I_{2}(\omega _{2})}$

where ${\displaystyle \omega _{1}}$ is the visible frequency, ${\displaystyle \omega _{2}}$ is the IR frequency and ${\displaystyle \omega _{3}=\omega _{1}+\omega _{2}}$ is the SFG frequency. The constant of proportionality varies across literature, many of them including the product of the square of the output frequency, ${\displaystyle \omega _{2}}$ and the squared secant of the reflection angle, ${\displaystyle \sec ^{2}\beta }$. Other factors include index of refractions for the three beams.[6]

The second order susceptibility has two contributions

${\displaystyle \chi =\chi _{nr}+\chi _{r}}$

where ${\displaystyle \chi _{nr}}$ is the non-resonating contribution and ${\displaystyle \chi _{r}}$ is the resonating contribution. The non-resonating contribution is assumed to be from electronic responses. Although this contribution has often been considered to be constant over the spectrum, because it is generated simultaneously with the resonant response, the two responses must compete for intensity. This competition shapes the nonresonant contribution in the presence of resonant features by resonant attenuation.[11] Because it is not currently known how to adequately correct for nonresonant interferences, it is very important to experimentally isolate the resonant contributions from any nonresonant interference, often done using the technique of nonresonant suppression.[12]

The resonating contribution is from the vibrational modes and shows changes in resonance. It can be expressed as a sum of a series of Lorentz oscillators

${\displaystyle \sum _{q}{\frac {A_{q}}{\omega _{2}-\omega _{0_{q}}+i\Gamma _{q}}}}$

where ${\displaystyle A}$ is the strength or amplitude, ${\displaystyle \omega _{0}}$ is the resonant frequency, ${\displaystyle \Gamma }$ is the damping or linewidth coefficient (FWHM), and each ${\displaystyle q>1}$ indexes the normal (resonant vibrational) mode. The amplitude is a product of ${\displaystyle \mu }$, the induced dipole moment, and ${\displaystyle \alpha }$, the polarizability.[7][9] Together, this indicates that the transition must be both IR and Raman active.[6]

The above equations can be combined to form

${\displaystyle \chi =|\chi _{nr}|e^{i\phi }+\sum _{q}{\frac {A_{q}}{\omega _{2}-\omega _{0_{q}}+i\Gamma _{q}}}}$

which is used to model the SFG output over a range of wavenumbers. When the SFG system scans over a vibrational mode of the surface molecule, the output intensity is resonantly enhanced.[6][9] In a graphical analysis of the output intensity versus wavenumber, this is represented by Lorentzian peaks. Depending on the system, inhomogeneous broadening and interference between peaks may occur. The Lorentz profile can be convoluted with a Gaussian intensity distribution to better fit the intensity distribution.[13]

From the second order susceptibility, it is possible to ascertain information about the orientation of molecules at the surface. ${\displaystyle \chi ^{(2)}}$ describes how the molecules at the interface respond to the input beam. A change in the net orientation of the polar molecules results in a change of sign of ${\displaystyle \chi ^{(2)}}$. As a rank 3 tensor, the individual elements provide information about the orientation. For a surface that has azimuthal symmetry, i.e. assuming ${\displaystyle C_{\infty }}$ rod symmetry, only seven of the twenty seven tensor elements are nonzero (with four being linearly independent), which are

${\displaystyle \chi _{zzz}^{(2)},}$

${\displaystyle \chi _{xxz}^{(2)}=\chi _{yyz}^{(2)},}$

${\displaystyle \chi _{xzx}^{(2)}=\chi _{yzy}^{(2)},}$ and

${\displaystyle \chi _{zxx}^{(2)}=\chi _{zyy}^{(2)}.}$

The tensor elements can be determined by using two different polarizers, one for the electric field vector perpendicular to the plane of incidence, labeled S, and one for the electric field vector parallel to the plane of incidence, labeled P. Four combinations are sufficient: PPP, SSP, SPS, PSS, with the letters listed in decreasing frequency, so the first is for the sum frequency, the second is for the visible beam, and the last is for the infrared beam. The four combinations give rise to four different intensities given by

${\displaystyle I_{PPP}=|f'_{z}f_{z}f_{z}\chi _{zzz}^{(2)}+f'_{z}f_{i}f_{i}\chi _{zii}^{(2)}+f'_{i}f_{z}f_{i}\chi _{zii}^{(2)}+f'_{i}f_{i}f_{z}\chi _{iiz}^{(2)}|^{2},}$

${\displaystyle I_{SSP}=|f'_{i}f_{i}f_{z}\chi _{iiz}^{(2)}|^{2},}$

${\displaystyle I_{SPS}=|f'_{i}f_{z}f_{i}\chi _{zii}^{(2)}|^{2},}$ and

${\displaystyle I_{PSS}=|f'_{z}f_{i}f_{i}\chi _{zii}^{(2)}|^{2}}$

where index ${\displaystyle i}$ is of the interfacial ${\displaystyle xy}$-plane, and ${\displaystyle f}$ and ${\displaystyle f'}$ are the linear and nonlinear Fresnel factors.

By taking the tensor elements and applying the correct transformations, the orientation of the molecules on the surface can be found.[6][9][13]