**2. Basic principles of NLO**

In the linear optical domain, photons interact with the glass structure leading to various optical effects, such as dispersion, refraction, reflection, absorption, diffraction, and scattering. For example, the linear refractive index of a material, *n*, describes how light propagates through it, and the index defines how much light is bent, or refracted when it across the material. However, these properties may become nonlinear if the intensity is high enough to modify the glass optical properties, resulting in the creation of new beam lights of different wavelengths.

A nonlinear optical behavior is a deviation from the linear interaction between a material's polarization response and the electric component of an applied electromagnetic field [10]. This phenomenon involves various optical exchanges such as frequency doubling, conversion, data transformation, etc. Because the magnetic component of light can be ignored in a glass (photons and magnetic fields usually do not interact), the electric component (E) becomes the main field that interacts with the medium. The polarization (P) induced by this interaction produces nonlinear responses that can be explained due to the distortion/deflection of the electronic structure of any atom or molecule (deformation of the electron cloud) due to the

application of the electric field, thus producing a resulting dipole moment (vector that separates the positive and negative charges).

Once an external E field is applied to the material the positive charges tend to move in the opposite direction of the electrons. This interaction causes a charge separation that gives rise to microscopic dipole moments within the material. Under the influence of an electric field, these dipoles oscillate at the same frequency (ω) of the incident light. The sum of all the microscopic dipoles of the medium oscillating with time gives rise to material polarization. At low light intensities, Hook's law is valid and the deformation of the electrons cloud is proportional to the applied field strength of the incident light: the light waves and excited electrons oscillate sinusoidally. The induced polarization is also oscillatory and is directly proportional to the incident electric field, as described by:

$$P = \varepsilon\_0 \chi^{(1)} E \tag{1}$$

where ε<sup>0</sup> is the vacuum permittivity and *χ*ð Þ<sup>1</sup> (or *χ*) is the linear susceptibility, which, in this case, depends on the frequency, and thus is directly linked to the linear refractive index, but does not depend on the amplitude of the electric field, which implies that the frequency of light does not change as it passes through matter. However, at high intensities, the electrons are extremely deflected from their orbit, and their movements become distorted giving rise to important deviation from harmonic oscillation. As a result, the amplitude of dipoles oscillation increases, and they emit light not only at the wavelength that excites them but in other frequencies (new color!!!) (**Figure 2**) [11]. At large intensities, P is a nonlinear function of E whereas, at low intensities, the interaction is a linear function. So, for materials with nonlinear characteristics, in which the polarization given by the Eq. (1) is no longer valid, P must be written in a more general form, as a power series of E:

$$P = \varepsilon\_0 \left( \chi^{(1)} E^1 + \chi^{(2)} E^2 + \chi^{(3)} E^3 + \dots \right) \tag{2}$$

where the values of χ(2) and χ(3), are, respectively, the second-order and thirdorder susceptibilities which appear due to the nonlinear response of charged particles and are determined by the symmetry properties of the medium. Consequently, nonlinear refractive index (*n*2), second (χ (2)), and third-order (χ (3)) nonlinear

#### **Figure 2.**

*(a) Linear optics, a light wave acts on the material constituents, which vibrates and then emits its own light wave that interferes with the original light wave, (b) nonlinear optics. Adapted from [11].*

*Optical Nonlinearities in Glasses DOI: http://dx.doi.org/10.5772/intechopen.101774*

susceptibilities can be measured. In isotropic, nondispersive, and homogeneous mediums, the material susceptibilities can be considered constants. However, in anisotropic media where properties are directionally dependent, the susceptibilities of the material are tensor quantities and therefore, depend on the microscopic structure (electronic e nuclear) of the material [12].

Considering the relations *<sup>n</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> *<sup>χ</sup>*ð Þ<sup>1</sup> , *<sup>c</sup>* <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffi *ε*0*μ*<sup>0</sup> <sup>p</sup> , and *<sup>n</sup>* <sup>¼</sup> *<sup>c</sup> c*0 , where *n* represents the linear refractive index, *c* is the speed of light in vacuum, c0 the speed of light in the material, and μ<sup>0</sup> the vacuum permeability; Maxwell's equations can be used to obtain the wave equations in a nonlinear material:

$$
\nabla^2 E - (\mathbf{1} + \chi) \varepsilon\_0 \mu\_0 \partial\_t^2 E = \frac{\mathbf{1}}{\varepsilon\_0 c^2} \partial\_t^2 P \tag{3}
$$

where the term *∂*<sup>2</sup> *<sup>t</sup> P*, represents a measure of the acceleration of the charges that constitute the material, which plays a fundamental role in the theory of nonlinear optics. This term acts as a source in the generation of new radiation field components, producing oscillating electric fields within a linear medium of refractive index n.

Assuming an external electric field of the type E(t) = Eexp(�*i*ωt) + c.c, where c.c. denotes "complex conjugate", the term related to second order polarization is given by:

$$P^{(2)} = \varepsilon\_0 \chi^{(2)} E^2 = 2\varepsilon\_0 \chi^{(2)} |E|^2 + \varepsilon\_0 \chi^{(2)} E^2 \exp\left(-2i\alpha t\right) + c.c.\tag{4}$$

being responsible for the generation of a field with twice the frequency of the incident radiation (2ω), taking the designation of the second harmonic generation process. However, in centrosymmetric materials, or isotropic materials like glass, which have macroscopic inversion symmetry, the polarization must reverse when the optical electric field is reversed, which implies that χ(2) must be zero, i.e., all second-order components of the susceptibility tensor are null and GSH does not manifest unless the glass has been poled. It is possible to induce GSH in glasses to break its centrosymmetry, using heat treatments or high energy excitation in the UV [13]. But, without the use of this strategy to eliminate glass's isotropy, only a χ(3) is 6¼0 and may lead to NLO character in glass [10] and the dominant term in (2) is then the third order:

$$P^{(3)} = \varepsilon\_0 \chi^{(3)} E^3 \tag{5}$$

which will give rise to frequency tripled light, called third-harmonic generation (THG). According to (5) this nonlinear polarization contains a component of frequency ω and an additional one at 3ω:

$$P^{(3)} = \underbrace{\mathfrak{Jac}\_{0} \chi^{(3)} |E|^2 E}\_{P(o)} + \underbrace{\mathfrak{e}\_{0} \chi^{(3)} E}\_{P(3o)} \tag{6}$$

The term P (3ω) shows that the THG of light is produced while the term P (ω) denotes an incremental change of the susceptibility (Δχ) at the frequency ω, given by:

$$
\varepsilon\_0 \Delta \chi = \frac{P(\alpha)}{E} = 3\chi^{(3)}|E|^2 = \frac{6}{n\varepsilon\_0 c} \chi^{(3)} I \tag{7}
$$

Where *I* is the intensity of the incident light that become significantly the value of χ(3). The χ(3), which gives the dependence of refraction on the intensity of the

propagated optical beam, is responsible for the lowest order nonlinear effects in the glass as self-phase modulation and other parametric effects.

Since n<sup>2</sup> = 1+ χ, Δχ is equivalent to an incremental change in the refractive index, Δn is an increase (or decrease) of the total refractive index due to nonlinear effects:

$$
\Delta n = \left(\frac{\partial \chi}{\partial n}\right)^{-1} \Delta \chi = \frac{\Delta \chi}{2n} = \frac{3}{n^2 e\_0^2 c} \chi^{(3)} I = n\_2 I \tag{8}
$$

where n2 is the nonlinear refractive. This change of the linear refractive index, n, is proportional to the light intensity, and therefore it becomes a linear function of I:

$$n(\mathbf{I}) = n + n\_2 \mathbf{I} \tag{9}$$

$$m\_2 = \frac{\mathfrak{Z}}{n^2 \varepsilon\_0^2 c} \chi^{(3)} \tag{10}$$

The intensity-dependent refractive index is generally given as:

$$n(\mathbf{I}) = n + n\_1 \mathbf{E} + n\_2 \mathbf{E}^2 \tag{11}$$

where n1 is the Pockel's coefficient (insignificant for isotropic materials as glasses) and n2 is known as the Kerr coefficient (from the optical Kerr effect) [10]. However, the classical wave theory says that the intensity of the electric field of the light is equal to the square of its amplitude, and thus one can also write n(I) in the form of Eq. (9). The optical Kerr effect is very sensitive to the operating wavelength and polarization dependence and so the prevalent non-linearity occurs at a frequency well below the glass band gap and this effect is called non-resonant [10].

Typical values of the Kerr coefficient (in cm2 /W) are 10�<sup>16</sup> to 10�<sup>14</sup> in transparent crystals and glasses. Silica glass (e.g. silica fibers), has an n2 index of 2.7 �10�16cm2 /W at the wavelength of 1500 nm, whereas most of the chalcogenide glasses exhibit higher values, about several orders of magnitude larger than silica [14]. Since the values of the nonlinear refractive index in glasses are very small, resulting in a slight change of Δn=n2I, the effect is measurable only for very intense light beams (lasers) of the order of 1GWcm�<sup>2</sup> . From **Figure 3**, it can be noted that n and n2 are usually directly correlated, such that high index (n) glasses, like chalcogenides, have also high n2 [16] and exhibit ultrahigh n2 greater than silica, as plotted in **Figure 3**.

For all-optical signal processing and switching devices, glasses with large n (hence a large n2) are very attractive. **Figure 4** shows the relationship between the linear refractive index (n), and the third-order nonlinear optical susceptibility χ(3) of various types of glass. High index (n) glasses, like chalcogenide ones, have also high n2, which seem to have the largest non-resonant third-order optical nonlinearities related so far. As previously mentioned, χ(3) arises from light-induced changes in the refraction index that result in the Kerr effect or in parametric interactions (mixing of optical beams). In a glass fiber, the third-order susceptibility is related to n2 by Eq. (10) and the magnitude of the corresponding nonlinear effect is given by:

$$
\gamma = \frac{2\pi}{\lambda A\_{\text{eff}}} n\_2 \tag{12}
$$

where λ is the free-space wavelength and Aeff is the efficient core area [6]. Since 1999, single-mode silica fibers with γ of 20 W�<sup>1</sup> km�<sup>1</sup> were fabricated [18] with a

*Optical Nonlinearities in Glasses DOI: http://dx.doi.org/10.5772/intechopen.101774*

**Figure 3.**

*Nonlinear refractive index, n2, versus refractive index,* n*, for various glasses, and silica glasses. Adapted from [15].*

core that was only 10.7 μm<sup>2</sup> , but typical Aeff values in silica fibers can reach 50 μm<sup>2</sup> for 1.5 μm wavelengths. The self-phase modulation is a phenomenon arising from the dependence between the refractive index of a nonlinear medium and the strength of the electric field, which induces a phase shift of the propagating light, *φ*NL(z):

$$
\phi\_{\rm NL}(\mathbf{z}) = \mathbf{y}P\_0 \mathbf{z} = \frac{\mathbf{z}}{L\_{\rm NL}} \tag{13}
$$

where P0 is the input power and LNL is the non-linear length that corresponds to the propagation distance at which the phase modulation becomes relevant, being defined by:

$$L\_{\rm NL} = \left(\eta P\_0\right)^{-1} \tag{14}$$

If the input power is only 1 mW at λ =1.55 μm, and the Aeff = 50 μm<sup>2</sup> , the LNL is �500 m [6]. As the refractive index in silica is weakly dependent on power, nonlinearities are introduced into the signal propagation and significantly increase in optical networks over relevant distances.

The various nonlinearities can be expressed in terms of the real and imaginary parts of each of the nonlinear susceptibilities χ(1), χ(2), χ(3), … that appear in (2). The real part is associated with the refractive index and the imaginary part with a time or phase delay in the reply of the material, giving rise to loss or gain. **Table 1** exhibits the principal third-order NLO effects usually showed by dielectric materials like most glasses. For example, the nuclear contribution to stimulated Raman scattering (resulting in loss or gain) can be expressed in terms of the imaginary part of a χ(3) susceptibility, while the four-wave mixing, which is only of electronic nature and almost an instantaneous effect, result in frequency conversion and in related to the real part of the χ(3) susceptibility [6]. The imaginary part of χ(3) provides a change in the absorption coefficient, α, as a function of light intensity:

$$a(I) = a\_0 + \mathfrak{J}I \tag{15}$$

where α is the linear absorption, and β is the non-linear absorption coefficient. As a result, occurs a prevalence of non-linearities at frequencies above the electronic absorption edge is known as resonant. The third-order non-linearity may be analyzed in phase conjugate mirrors, like in Mach-Zehnder interferometer pulse selectors or in Fabry-Perot interferometers filled with a nonlinear medium.

The χ(3) susceptibility is often measured by degenerate four-wave mixing, by the maker fringe method (THG method), or by the Z-scan method. The latter is by far the most used and meticulous method involving the analysis of third-order nonlinear optical properties arising from pulsed laser or CW irradiation at a given wavelength [20].
