**2. Nonlinear light-matter interactions**

The interaction of light with matter can be broadly classified in two categories (a) linear interaction (weak light regime) and (b) nonlinear interaction (intense light regime). When the intensity of the incident beam is so high that it is nearly equal to the internal electric field of the atom then the absorption coefficient and the real and imaginary part of the nonlinear refractive index (*n*) and absorption coefficient (*β*) of the material become a function of the incident intensity or electric field [27]. Such that, polarization of the materials tends to behave nonlinearly with the incident electric field (Eq. (1)).

$$P\_i = \varepsilon\_0 \chi\_{ijk}^{(1)} E\_j + \varepsilon\_0 \chi\_{ijk}^{(2)} E\_j E\_k + \varepsilon\_0 \chi\_{ijk}^{(3)} E\_j E\_k E\_l + \dots \tag{1}$$

where, *ε*<sup>0</sup> denotes permittivity of free space, *P* is electric polarization, *χ* symbolizes electric susceptibility tensor, *ε*0*χ* ð Þ1 *ijk Ej* is the linear polarization, *ε*0*χ* ð Þ2 *ijk EjEk* depicts second order nonlinearity, *ε*0*χ* ð Þ3 *ijk EjEkEl* is the third order nonlinearity and so on.

Hence, in comparison with the linear interaction, during nonlinear interaction superposition principle gets violated, light can alter its frequency as it passes through a *Types of Nonlinear Interactions between Plasmonic-Excitonic Hybrids DOI: http://dx.doi.org/10.5772/intechopen.105833*

#### **Figure 1.**

*Schematic representation of nonlinear interaction of light with matter [27].*

NLO material and photons start interacting with each other within the confines of a NLO medium [27, 28]. The nonlinear interactions of light are further classified as resonant and nonresonant interactions (**Figure 1**).

The nonresonant or elastic interactions are the ones in which the incident light is not absorbed by the material (sample). Therefore, these interactions can be described using nonlinear polarization in terms of the Maxwell equations, whereas, in case of resonant interactions or coherent interactions emission/absorption of light inside the material takes place. The density matrix formalism is hence used to evaluate such interactions as discrete structure of the energy levels inside matter and their phase dependent occupation during the light wave period are important to be considered in these interactions [27, 28].

When high intensity light is incident on a medium, based on its inherent response scattering, refraction or absorption takes place. This in turn alters the transmittance of the medium as a function of the input light intensity and is termed as nonlinear light absorption or nonlinear light transmission. However, at very high intensities, the probability of the medium absorbing more than one photon before relaxing to the ground state becomes prominent.

A few phenomena that control nonlinear light transmission are listed below.

#### **2.1 Saturable absorption**

Saturable absorption (SA) is the mechanism in which the absorption of light is inversely proportional to light intensity. That is at sufficiently high intensities of the incident beam, if the rate of excitation of atoms is greater than their decay rate then the ground state gets depleted resulting in a saturable absorption. Further, this phenomenon critically depends on the absorption range of the material, its dynamic response and saturation intensity.

The steady state rate equation for saturable absorption is denoted by Eq. (2).

$$\frac{dN}{dt} = \frac{\sigma I}{h\nu} \left(N\_{\text{g}} - N\right) - \frac{N}{\pi} \tag{2}$$

where, *N* is the number of excited state molecules, *Ng* is the undepleted ground state concentration, *σ* is termed as absorption cross section, *hν* is the photon energy, and *τ* is the lifetime of the excited state population.

Assuming that the absorption coefficient and *α* is proportional to the ground state population. Saturable absorption finds application in passive mode locking and Q-switching of lasers and optical signal processing [27, 28].

#### **2.2 Two-photon absorption**

When simultaneous absorption of two photons from an incident beam causes a transition of atoms/molecules of a material from ground state to a higher energy level, this phenomenon is termed as two-photon absorption (TPA). The intermediate level in this case is a virtual energy level therefore; two photons should be simultaneously absorbed for this process to occur. Also TPA is directly proportional to the square of the input fluence. The propagation of laser light through the system describing the TPA is given by Eq. (3).

$$\frac{dI}{dZ} = -aI - \beta I^2 \tag{3}$$

where, *α* is the linear absorption coefficient and *β* is the two-photon absorption coefficient and *σ* is the individual molecular two-photon absorption cross section, also

$$
\sigma = \frac{\alpha \beta h}{2\pi N} \tag{4}
$$

where *N* is the number of the molecules in the system and *ω* is the incident radiation frequency. It is the imaginary part of the third-order nonlinear susceptibility of the system that determines the strength of the two-photon absorption. The relation between the TPA coefficient and the third-order susceptibility of a centrosymmetric system for linearly polarized incident light is given as,

$$\beta = \frac{3\pi}{\varepsilon\_0 n^2 c\lambda} \operatorname{Im} \left[ \chi^3\_{\text{xxx}}(-a; a, a, -a) \right] \tag{5}$$

here, *c* is the speed of light, *λ* is the wavelength of the incident beam and *n* is the linear refractive index [27].

### **2.3 Three-photon absorption**

The transition of a ground state molecule to higher excited state by simultaneous absorption of three photons from the incident radiation is termed as three-photon absorption (3PA) [28]. It is a fifth-order nonlinear process, and the propagation equation for a medium having significant three-photon absorption is given as,

$$\frac{dI}{dZ} = -aI - \chi I^3 \tag{6}$$

where, *α* is the linear absorption coefficient and *γ* is the 3PA coefficient. For acentrosymmetric system and linearly polarized light, *γ* is related to the imaginary part of the fifth-order susceptibility through the following equation,

*Types of Nonlinear Interactions between Plasmonic-Excitonic Hybrids DOI: http://dx.doi.org/10.5772/intechopen.105833*

$$\chi = \frac{5\pi}{c\_0 \, ^2 \mathfrak{n}^3 c^2 \lambda} \operatorname{Im} \left[ \chi^5\_{\, \text{xxxxx}} (-\alpha; \alpha, \alpha, \alpha, -\alpha, -\alpha) \right] \tag{7}$$

#### **2.4 Reverse saturable absorption**

Reverse saturable absorption (RSA) is the property of materials where the absorption of light increases with increasing incident light intensity. It is basically a two-step, sequential single-photon absorption process where, the excited atoms/molecules make a subsequent transition to higher energy levels by absorbing another single photon [28]. For steady-state condition, the intensity change of the laser beam in the nonlinear medium along its propagation direction for RSA can be expressed as,

$$\frac{dI}{dz} = -\sigma\_{12}(N\_1 - N\_2)I - \sigma\_{23}N\_2I \tag{8}$$

where, *σ*<sup>12</sup> is the transition crossection from the ground state to first excited state, *N*<sup>1</sup> is the no. of molecules in the first excited state, *N*<sup>2</sup> is the no. of molecules in the second excited state and *σ*<sup>23</sup> is the transition crossection from first excited state to second excited state.
