# A Unified Mathematical Formalism for First to Third Order Dielectric Response of Matter: Application to Surface-Specific Two-Colour Vibrational Optical Spectroscopy

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## Abstract

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## 1. Introduction

**P**, so-called polarization, whose amplitude depends on the $n\mathrm{th}$ power of the light electric field amplitude: $\left|\mathbf{P}\right|\sim {\left|\mathbf{E}\right|}^{n}$. The proportionality coefficient is then characteristic of the inner properties of the material: crystal structure, molecular vibrations, electronic density, chemical composition, internal symmetries and so on. This response factor is denoted ${\chi}^{\left(n\right)}$. It must be handled with care: the relation between the polarization $\mathbf{P}$ of the material and the electric field $\mathbf{E}$ of the light is not as simple as $\mathbf{P}={\chi}^{\left(n\right)}{\mathbf{E}}^{n}$: first, the $n\mathrm{th}$ power of $\mathbf{E}$ is not necessarily a vector (e.g., ${\mathbf{E}}^{2}$ is a number), while $\mathbf{P}$ is a vector; second, each component ${P}_{i}$ of $\mathbf{P}$ may depend on all the components ${E}_{x}$, ${E}_{y}$ and ${E}_{z}$ of the electric field, so that the response factor is actually a tensor [1]; and third, the frequencies of $\mathbf{P}$ and of the different spectral contributions to $\mathbf{E}$ must be explicitly written. In order to properly describe optical spectroscopies, the response functions ${\chi}^{\left(n\right)}$ must be well defined on a mathematical point of view: dimension of the tensor, number of frequency arguments, relation with the polarization and the electric field. Given that it is not the case in many articles, we dwell on that point through this review.

**P**of a material and that of the emitted/scattered light power $\langle |\mathbf{P}{|}^{2}\rangle $. As explained in Section 4.4, this comes from quantum mechanics.

## 2. Linear Response Theory

#### 2.1. Polarization of Matter and Optical Response Function

**d**denotes the vector connecting the barycentre of the negative charges (of the electronic cloud) to the barycentre of the positive charges (of the nucleus). In this context, reducing the behaviour of matter to that of the dipole moment alone is an approximation. When this is necessary (which will not be our case), we may have to consider quadrupole and octupole moments. It is then possible to define for any macroscopic material system the local polarization $\mathbf{P}$ as the volume density of dipole moments:

#### 2.2. Refraction and Absorption

#### 2.3. Scattering and Extinction

#### 2.4. Extinction Spectroscopies

## 3. Non-Linear Response of Anisotropic Media

#### 3.1. Second Order Non-Linear Optical Processes

- $\omega ={\omega}_{1}+{\omega}_{2}$, via ${\mathbf{\chi}}^{\left(2\right)}({\omega}_{1},\phantom{\rule{0.166667em}{0ex}}{\omega}_{2})$: for SFG, sum-frequency generation;
- $\omega ={\omega}_{1}-{\omega}_{2}$, via ${\mathbf{\chi}}^{\left(2\right)}({\omega}_{1},-{\omega}_{2})$: for DFG, difference-frequency generation;
- $\omega =2{\omega}_{i}$, via ${\mathbf{\chi}}^{\left(2\right)}({\omega}_{i},\phantom{\rule{0.166667em}{0ex}}{\omega}_{i})$, for SHG, second harmonic generation;
- $\omega =0$, via ${\mathbf{\chi}}^{\left(2\right)}({\omega}_{i},-{\omega}_{i})$, corresponding to optical rectification.

#### 3.2. Symmetry Rules

#### 3.3. Sum-Frequency Generation at Interfaces

#### 3.4. Local Field Correction Factors: Light Intensity Modulation by Interface Symmetry

#### 3.5. Third Order Non-Linear Optical Processes

## 4. Vibrational Spectroscopies

#### 4.1. Molecular Vibrations

#### 4.2. Transition Dipole Moments and Susceptibility

#### 4.3. Infrared Spectroscopy

#### 4.4. Raman Spectroscopy

#### 4.5. SFG Spectroscopy

#### 4.6. Prospects in SFG Spectroscopy and Microscopy

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

SFG | Sum-frequency generation |

DFG | Difference-frequency generation |

SHG | Second harmonic generation |

VOA | Vibrational optical activity |

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**Figure 1.**

**Mono- and polychromatic picture of second order response functions.**(

**a**) Common description of sum-frequency generation (SFG) from a monochromatic point of view, considered to derive from a 3-argument function. (

**b**) Description of second order processes (SFG, DFG, SHG) from a polychromatic point of view, considered to derive from the 2-argument second order susceptibility ${\chi}^{\left(2\right)}({\omega}_{1},{\omega}_{2})$ combining any input frequencies ${\omega}_{1}$ and ${\omega}_{2}$.

**Figure 2.**

**Wave propagation in a dielectric medium.**Illustration of refraction and absorption phenomena within a dielectric system of susceptibility ${\chi}^{\left(1\right)}\left(\omega \right)$. The incident electric field $\mathbf{E}\left(\omega \right)$ generates a polarization $\mathbf{P}\left(\omega \right)={\epsilon}_{0}{\chi}^{\prime}\left(\omega \right)\mathbf{E}\left(\omega \right)+{\epsilon}_{0}i{\chi}^{\u2033}\left(\omega \right)\mathbf{E}\left(\omega \right)$. The real part of $\chi \left(\omega \right)$ results in the appearance of a field ${\mathbf{E}}_{r}\left(\omega \right)$ in phase with the incident electric field, while the imaginary part leads to the generation of an electric field ${\mathbf{E}}_{a}\left(\omega \right)$ in phase opposition. The destructive interference thus occurring gives rise to a transmitted electric field ${\mathbf{E}}_{t}\left(\omega \right)$ of weaker amplitude (damped by a factor ${e}^{-{q}^{\u2033}\left(\omega \right)x}$).

**Figure 3.**

**Geometric and wave description of light scattering.**Sketch of the scattering process, (

**a**) from geometric optics and, (

**b**) from wave optics. The dotted arrows represent the intensity of the scattered light according to the scattering angle.

**Figure 4.**

**Second order non-linear optical processes.**Sketch of the processes of second harmonic generation (SHG), sum-frequency generation (SFG) and difference-frequency generation (DFG). These are driven by the second order dielectric susceptibility ${\mathbf{\chi}}^{\left(2\right)}({\omega}_{1},\phantom{\rule{0.166667em}{0ex}}{\omega}_{2})$ of the material, which couples two input frequencies ${\omega}_{1}$ and ${\omega}_{2}$ to generate new signals.

**Figure 5.**

**Sum-frequency generation at nanostructured interfaces.**Schematic representation of the SFG process in the case of functionalized nanoparticles grafted on a solid substrate. The two input frequencies belong to the visible (${\omega}_{1}={\omega}_{\mathrm{vis}}$) and the infrared (${\omega}_{2}={\omega}_{\mathrm{IR}}$) spectral ranges. This configuration is characterized by (

**a**) a breaking of the global centrosymmetry at the macroscopic scale of the substrate and (

**b**) a breaking of the local centrosymmetry at the surface of nanoparticles. (

**c**) These two interfaces can be modelled as a 3-layer surface characterized by three refractive indices ${n}_{1}$, ${n}_{2}$ and ${n}_{\mathrm{lay}}$.

**Figure 6.**

**SFG process at flat interfaces.**(

**a**) Schematic representation of SFG at the surface of a nanostructured sample. The visible, IR and SFG beams belong to the same plane of incidence $(x,z)$. (

**b**) Definition of the $\mathcal{P}$ and $\mathcal{S}$ polarizations, with respect to the plane of incidence $(x,z)$. (

**c**) Directions and polarizations of the beams in $[\mathcal{P}:\mathcal{PP}]$ configuration. (

**d**) Directions and polarizations of the beams in $[\mathcal{S}:\mathcal{SP}]$ configuration.

**Figure 7.**

**Fluorescence and Raman scattering.**Illustration of third order non-linear processes: (

**a**) fluorescence, (

**b**) Stokes Raman scattering and (

**c**) anti-Stokes Raman scattering. We note $|g\rangle $ the ground state, $|v\rangle $ a vibrational state within the ground state, and $|e\rangle $ an electronic excited state. The definition of normal coordinates is given in Section 4.1 in the case of molecules.

**Figure 8.**

**Normal vibration modes.**Schematic representation of the different vibration modes observed within molecules. Here we take the example of CH${}_{2}$ chemical group.

**Figure 9.**

**Infrared spectroscopy of methylene.**Typical shape of the IR spectrum associated with C−H vibration modes within CH${}_{2}$ groups.

**Figure 10.**

**Vibrational spectroscopies.**Comparison between the three techniques of vibrational spectroscopy: infrared absorption, anti-Stokes Raman scattering and sum-frequency generation. We represent three quantum states $|g\rangle $, $|v\rangle $ and $|e\rangle $ for the system, with the same meaning than Figure 7.

**Figure 11.**

**Principle of 2-dimension sum-frequency generation.**(

**a**) Conventional use of SFG spectroscopy on a nanostructured sample functionalized by organic molecules. The SFG spectrum consists in measuring the SFG intensity as a function of the IR wavenumber for a fixed visible wavelength (${\lambda}_{\mathrm{vis}}=550$ nm). (

**b**) Comparison of five vibrational SFG spectra obtained for five different visible wavelengths. For each vibration mode, the variation of the intensity from a visible wavelength to another is characteristic of electronic structure of the system (nanoparticles). (

**c**) Unconventional use of SFG spectroscopy at variable visible wavelength. The spectrum is acquired at a fixed IR wavenumber that coincides with the vibration mode indicated on the spectra of Figure 11b.

**Figure 12.**

**Interference profiles of SFG spectra.**Typical profiles of the vibrational SFG spectra: (

**a**) in the destructive case, (

**b**) in an intermediate case, (

**c**) in the constructive case. The quantity ${A}^{2}$ corresponds to the non-resonant background. It is not specific to the molecular species but to the inorganic components of the system.

**Table 1.**

**Comparison of vibrational spectroscopies.**Comparative table of the three vibrational spectroscopies: infrared absorption, Raman scattering and SFG. ${}^{\u2020}$ Raman scattering can be resonant in the visible spectral range when the frequency of the pump beam coincides with an electronic transition (the excited state is therefore not virtual but indeed real, as illustrated in Figure 10). ${}^{\u2021}$ SFG can be doubly resonant, with respect to the infrared and visible beams, as shown in Figure 10.

IR Absorption | Raman Scattering | SFG |
---|---|---|

${\mathbf{\chi}}^{\left(1\right)}\left({\omega}_{\mathrm{ir}}\right)$ | ${\mathbf{\chi}}^{\left(3\right)}({\omega}_{\mathrm{vis}},-{\omega}_{\mathrm{vis}},\phantom{\rule{0.166667em}{0ex}}\omega )$ | ${\mathbf{\chi}}^{\left(2\right)}({\omega}_{\mathrm{vis}},\phantom{\rule{0.166667em}{0ex}}{\omega}_{\mathrm{ir}})$ |

${\left(\frac{\partial \mathsf{\mu}}{\partial {Q}_{v}}\right)}_{0}\ne \mathbf{0}$ | ${\left(\frac{\partial \mathbf{\alpha}}{\partial {Q}_{v}}\right)}_{0}\ne \mathbf{0}$ | ${\left(\frac{\partial \mathbf{\alpha}}{\partial {Q}_{v}}\right)}_{0}\otimes {\left(\frac{\partial \mathsf{\mu}}{\partial {Q}_{v}}\right)}_{0}\ne \mathbf{0}$ |

${\omega}_{\mathrm{ir}}$-resonant | non ${\omega}_{\mathrm{ir}}$-resonant | ${\omega}_{\mathrm{ir}}$-resonant |

non ${\omega}_{\mathrm{vis}}$-resonant | (${\omega}_{\mathrm{vis}}$-resonant) ${}^{\u2020}$ | (${\omega}_{\mathrm{vis}}$-resonant) ${}^{\u2021}$ |

coherent | incoherent | coherent |

directional | diffused | directional |

non surface-specific | non surface-specific | surface specific |

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**MDPI and ACS Style**

Humbert, C.; Noblet, T.
A Unified Mathematical Formalism for First to Third Order Dielectric Response of Matter: Application to Surface-Specific Two-Colour Vibrational Optical Spectroscopy. *Symmetry* **2021**, *13*, 153.
https://doi.org/10.3390/sym13010153

**AMA Style**

Humbert C, Noblet T.
A Unified Mathematical Formalism for First to Third Order Dielectric Response of Matter: Application to Surface-Specific Two-Colour Vibrational Optical Spectroscopy. *Symmetry*. 2021; 13(1):153.
https://doi.org/10.3390/sym13010153

**Chicago/Turabian Style**

Humbert, Christophe, and Thomas Noblet.
2021. "A Unified Mathematical Formalism for First to Third Order Dielectric Response of Matter: Application to Surface-Specific Two-Colour Vibrational Optical Spectroscopy" *Symmetry* 13, no. 1: 153.
https://doi.org/10.3390/sym13010153