**2. Principles of the non-linear optical media**

The basic nonlinear optical polarization can be studied at the molecular level. That involves an effort devoted to understanding the relation between the molecular structure and materials properties. Another perspective regards the discussion of the effective arrangement of the nonlinear molecules into macroscopic responses (single crystal, quasi-crystalline thin film, and a polymer matrix, etc.). To describe the nature of nonlinear optical response in materials is necessary to consider the aspects of frequency dependence and symmetry.

A molecular medium, such as an organic crystalline or polymeric solid, is generally non-conducting and non-magnetic and the electrons are regarded as being tightly bound to the nuclei. For such media, the interaction with light can generally be regarded within the framework of a dielectric subjected to an electric field [1–3].

At molecular level, the interaction of molecule with electric field can be described by the polarizability (p):

$$\mathbf{p} = \mathbf{a}\mathbf{E} + \beta \mathbf{E}^2 + \gamma \mathbf{E}^3 + \dots \tag{1}$$

where: α, β, γ are the molecular polarizabilities (1st, 2nd, 3rd order).

The fundamental relationship describing the induced charge in molecular dipole moment (polarization) upon interaction with an oscillating external field (light) can be expressed in a power series:

$$\mathbf{p}\_{\mathbf{i}} = \sum\_{\mathbf{j}} \mathbf{a}\_{\mathbf{i}\mathbf{j}} \mathbf{E}\_{\mathbf{j}} + \sum\_{\mathbf{j} \le k} \emptyset\_{\mathbf{i}\mathbf{j}\mathbf{k}} \mathbf{E}\_{\mathbf{j}} \mathbf{E}\_{\mathbf{k}} + \sum\_{j \le k \le l} \gamma\_{\mathbf{i}\mathbf{j}\mathbf{k}\mathbf{l}} \mathbf{E}\_{\mathbf{j}} \mathbf{E}\_{\mathbf{k}} \mathbf{E}\_{\mathbf{l}} + \dots ; \tag{2}$$

here pi is the electronic polarization induced along the ith molecular axis, Ej is the j th component of the applied electric field, α is the linear polarizability, β is the quadratic hyperpolarizabilitiy and γ is the cubic hyperpolarizability. Starting with the second term of equation are described the nonlinear response to the radiation field and account for terminology "nonlinear optics". Importantly, the even order tensor, β is responsible for doubling frequency of incident light (second harmonic generation, SHG) and γ describe the third order response of material at molecular level (third harmonic generation, THG).

At macroscopic level, nonlinear optics is concerned with the electric polarization P (dipole moment per unit volume) induced in a material by an electric field E.

*Polymer Architectures for Optical and Photonic Applications DOI: http://dx.doi.org/10.5772/intechopen.99695*

Typically, P is proportional to E and oscillates at the same frequency*. So, the response is linear!* As the magnitude of the field increases, P deviates from proportionality and nonlinear phenomena appear:

$$\mathbf{P}\_{\mathbf{i}}/\mathbf{e}\_{0} = \sum\_{\mathbf{j}} \chi\_{\mathbf{i}\mathbf{j}}^{(1)} \mathbf{E}\_{\mathbf{i}} + \sum\_{\mathbf{j} \le \mathbf{k}} \chi\_{\mathbf{i}\mathbf{k}}^{(2)} \mathfrak{R}\_{\mathbf{i}\mathbf{k}} \mathbf{E}\_{\mathbf{j}} \mathbf{E}\_{\mathbf{k}} + \sum\_{\mathbf{j} \le \mathbf{k} \le \mathbf{l}} \chi\_{\mathbf{i}\mathbf{j}}^{(3)} \mathbf{E}\_{\mathbf{j}} \mathbf{E}\_{\mathbf{k}} \mathbf{E}\_{\mathbf{l}} + \dots \tag{3}$$

χ(n) = electric susceptibilities of order n; linear for n = 1, quadratic: n = 2, cubic: n = 3. Considering the dual character of light, electric and magnetic:

$$\mathbf{E} = \mathbf{E}(a) + \mathbf{E}(\Omega) = \mathbf{E}\_0 \cos \left( a \mathbf{t} \right) + \mathbf{E}(\Omega), \tag{4}$$

where E(ω) is the optical component of field and E(Ω) is characteristic for external applied field, can be written the mathematical expression of polarization as function of frequency:

$$\mathbf{P}(\boldsymbol{\alpha}) = \left(\boldsymbol{\chi}^{(1)} + 2\boldsymbol{\chi}^{(2)}\mathbf{E}(\boldsymbol{\Omega}) + \mathbf{3}\boldsymbol{\chi}^{(3)}\mathbf{E}(\boldsymbol{\Omega})^2 - \boldsymbol{\upbeta}\boldsymbol{\chi}^{(3)}\mathbf{E}\_0^2\right)\mathbf{E}\_0\cos(\boldsymbol{\alpha}t),\tag{5}$$

where: the second term describe the Pockels or linear electrooptic effect (1th order) the third is for the Kerr of quadratic electrooptic effect (2th order), and the last is specifically for optical Kerr effect.

$$\mathbf{P}(2\alpha) = \left(\mathbb{1}\langle\mathbf{2}\chi^{(2)} + \mathbb{1}\langle\mathbf{2}'^{(3)}\mathbf{E}(\mathbf{2})\rangle\right)\mathbf{E}\_0^2\cos(2\alpha t),\tag{6}$$

where the first terms specifically for second harmonic generation (SHG) and the second describe the electric-field induced SHG (EFISH),

$$\mathbf{P}(\mathfrak{so}) = \left( \mathbb{W} \mathbf{\hat{x}}^{(3)} \mathbf{E}(\mathfrak{Q}) \right) \circ \mathbb{W}^{(3)} \mathbf{E}\_0^3 \cos \left( \mathfrak{so} t \mathbf{\hat{z}} \right) \mathbf{E} = \mathbf{E}\_0 \cos \left( \mathfrak{so} t \right) + E(\mathfrak{Q}), \tag{7}$$

which is specifically equation for third harmonic generation (THG).

Starting from this fundamental description of interaction light-material, without forgetting about the multicomponent structure of materials (different polar species), can be formulate two fundamental aspects for the researchers interested by the synthesis of such materials:


$$
\chi^{(2)} = \text{Nf}\beta , \text{respectively } \chi^{(3)} = \text{Nf}\gamma , \tag{8}
$$

where N is the number of molecular dipols (polar or polarizable molecules), f is the local field factor.

Synthetic materials of interest in ONL applications are not isotropic. For example:

• Organic crystals consist of molecules with specific orientations of structuring the constituent motif. Thus, the molecules themselves often have electronic structures with a high degree of anisotropy and consequently, the response of the material to the applied electromagnetic field will not always respect the direction of incidence, as is usual in classical optics.

• Similarly, in terms of orientation, polymeric chains comply with the principle of minimum energy. If these materials are generally characterized by the random orientation of chains - isotropic structure - depending on the composition, the chains constituting them may be oriented preferentially, so that their energy distribution will require their response to external stimuli.

For this reason, susceptibilities are tensor quantities related to the polarization response in one of the three directions of the field. The material characteristic that reflects the degree of anisotropy is the dielectric constant εij, defined by the electric field and the dielectric displacement, being a second-degree tensor. The birefringence phenomenon is one of the consequences of this dielectric anisotropy. Visualizing a uniaxial crystal with nx = ny + nz, the refractive index of the materials, n, is given as

$$\mathbf{n} \approx \sqrt{\mathbf{e}} \tag{9}$$

and with a light wave propagating from left to right along the x-axis with its electric field polarized in the z-direction, the propagation constant kz will be determined by nz from:

$$\mathbf{k}\_{\mathbf{z}} = \frac{\mathbf{e}}{\mathbf{c}} \mathbf{n}\_{\mathbf{z}} \tag{10}$$

Thus, while all material structures from the point of view of the optical properties, are characterized by the presence of the both hyperpolarizabilities (β, γ), do not display simultaneous second and third order nonlinear response. That is due to the peculiar structural requirement for the two NLO properties. The high concentration of chromophores with large product of the dipole moment, μ, and the first hyperpolarizability, β, stimulate aggregation tendency, diminishing their contribution to the electrooptic effect of materials. The last decade of the 1990s has established that high electro-optical coefficients for guest-host material systems, with polymeric matrix, are essentially due to electrostatic chromophore-chromophore interactions. Thus, when doping the polymer matrices with chromophe species characterized by high electrostatic interactions, there is a decrease of the electro-optical coefficients in the value with an increase of the chromophore concentration. As well, the poling efficiency is limited because a disrouption in alignment of chromophes appears when the polarization field is stoped, owing to the electrostatic interactions and thermal randomization. This behavior is the result of a major relaxation phenomena of chromophors' in the first moments after stopping the polarization field [4, 5].

The improvement in the value of electro-optical coefficients requires intervention in the composition and structure of the chromophore, without perturbation of of the π-electron structure to avoid changing the molecular hyperpolarizability β. In such cases, the syntheses that favor the perpendicular orientation of constituent segments with distinct polarity (i.e. acceptor, donor) to the chromophore conjugation system, are preferred. Such sterical hidrance limits the distance at which chromophoric molecules can approach each other, thus reducing undesirable dipole–dipole interactions; finally becoming competitive with lithium niobate [6].
