**A.1 Theory: the classical approach**

The wave equations for pump and Stokes laser pulses with electromagnetic field amplitude *<sup>E</sup>* → *<sup>P</sup>* and *<sup>E</sup>* → *<sup>S</sup>* at frequencies *ωP* and *ωS* (<sup>ω</sup>*<sup>P</sup> <sup>&</sup>gt;* <sup>ω</sup>*S*), are:

$$\begin{array}{rcl} \text{at frequencies } \omega\_P \text{ and } \omega\_S \text{ (}\omega\_P \text{ > }\omega\_S\text{), are:}\\ \nabla & \times \left(\nabla \quad \times \vec{E}\_P\right) - \frac{\alpha\_P^2}{c^2} \varepsilon\_P \vec{E}\_P = \frac{4\pi\alpha\_P^2}{c^2} \vec{\vec{P}}^{(3)} \text{ (}\mathbf{0}\nu\text{)}\\ \nabla & \times \left(\nabla \quad \times \vec{E}\_S\right) - \frac{\alpha\_S^2}{c^2} \varepsilon\_S \vec{E}\_S = \frac{4\pi\alpha\_S^2}{c^2} \vec{\vec{P}}^{(3)} \text{ (}\mathbf{0}\nu\text{)} \end{array} \tag{1}$$

where *<sup>P</sup>* →(3) is the nonlinear polarizations, is the dielectric constants and *c* is the light velocity.

In the case of SRS, the material interaction is classically treated through a thirdorder nonlinear susceptibility tensor *χ*(3) given by:

$$\mathcal{X}\_{\text{(G)}} = \mathcal{X}\_{\text{(G)}\text{NR}} + \mathcal{X}\_{\text{(G)}\text{R}} \tag{2}$$

**137**

*Stimulated Raman Scattering in Micro- and Nanophotonics*

*P*(3)

(1) can be solved with Eq. (3) by knowing *χ*(3)*R*.

<sup>α</sup>(*t*) <sup>=</sup> <sup>α</sup><sup>0</sup> <sup>+</sup> (

Starting from Eqs. (4) and (5), we obtain

**−**ω<sup>2</sup>*q*(*Ω*) − 2*iq*(*Ω*) + ωυ

*g* = −4*π*ω<sup>2</sup>

When the depletion of the pump field ⌈*EP*⌉

is held at its equilibrium value.

gain by the following relation:

⌈*ES*⌉

*i*(ω*Pt*−*kPz*)

(ω*P*) = [χ*<sup>P</sup>*

(ω*S*) = [χ*<sup>S</sup>*

depends on the internuclear distance according to the equation:

where *m* represents the reduced nuclear mass and *<sup>Ω</sup>* <sup>=</sup> <sup>ω</sup>*<sup>P</sup>* <sup>−</sup> <sup>ω</sup>*<sup>s</sup>*

of Stokes field is given by the exponentially growing solution of

<sup>2</sup> = ⌈*ES*(0)⌉

satisfied. In other words, Raman amplification is a pure gain process.

explicity *q*(*Ω*) as a material excitation resonantly driven by optical mixing *EPES*

phonons can therefore be considered a result of coupling three waves *EP*, *ES* and *q*(*Ω*) governated by the wave equations (3) and (6). This system is essentially the wave equation coupled to an oscillator equation. Starting from Eq. (6) it is possible to calculate the resonant Raman susceptibility, which, for the steady-state case, is related to the Raman

> 2 \_\_\_\_ *<sup>c</sup>*<sup>2</sup>*k*<sup>2</sup> (*Im*(χ*<sup>S</sup>*

susceptibility *χ*(3)*R* a negative imaginary, we find that the evolution of the intensity

2

<sup>2</sup> exp(*g* ∗ ⌈*EP*⌉

The Stokes wave is amplified if the gain exceeds the losses. We note that Raman amplification is a process for which the phase matching condition is automatically

+ *ES e i*(ω*St*−*kSz*)

In order to simplify, we study the special case of an isotropic medium with *EP* and *ES* with the same polarization direction and propagation along *z*. The whole

(3)*NR* |*EP*|

(3)*<sup>R</sup>* |*EP*|

and <sup>ϵ</sup> *<sup>S</sup>* in Eq. (1). They are responsible for the field induced birefringence, selffocusing, etc., but have no direct effect on SRS. Therefore, in the following discussion, we neglect them. The *χ*(3)*R* terms in *P*(3), instead, effectively couple *EP* and *ES* in Eq. (1) and is the reason of energy transfer between the two fields. They are the cause of the stimulated Raman process and are called Raman susceptibilities. Eq.

A molecular vibration or optical phonon is the most common case of SRS. The optical radiation is assumed interacting with a vibrational mode of a molecule and this vibrational mode can be defined as a simple harmonic oscillator of resonance frequency *ω*υ, damping constant *γ*. The analysis is one-dimensional, thus, each oscillator can be distinguished by its position *z* and normal vibrational coordinate *q* [2]. The key assumption of the theory is that the optical polarizability of the molecule (which is typically predominantly electronic in origin) is not constant, but

> \_\_\_ ∂α <sup>∂</sup>*q*)<sup>0</sup>

<sup>2</sup> *q*(*Ω*) = \_\_1

(**3**)*R*

*m* ( \_\_\_ ∂α <sup>∂</sup>*t*)<sup>0</sup>

This quantity is a tensor, but to simplify the discussion we will consider it as a scalar. Here *α*0 is the polarizability of a molecule in which the internuclear distance

<sup>2</sup> + χ*<sup>P</sup>*

<sup>2</sup> + χ*<sup>S</sup>*

(3)*<sup>R</sup>* |*ES*| 2 ]*EP* 

(3)*NR* |*ES*| 2 ]*ES*

(3)*NR* terms in *P*(3) only act to modify the dielectric constant <sup>ϵ</sup> *<sup>P</sup>*

. According to Eq. (2), the nonlinear

*q*(*t*) (4)

*EPES*

)) (6)

<sup>2</sup> ∗ *z* − α ∗ *z*) (7)

is negligible, being the Raman

<sup>∗</sup> (5)

∗ . SRS by

. This equation shows

(3)

*DOI: http://dx.doi.org/10.5772/intechopen.80814*

field amplitude is: *<sup>E</sup>*(*z*,*t*) <sup>=</sup> *EP <sup>e</sup>*

polarizations take the form:

*<sup>P</sup>*(3)

(3)*NR* and <sup>χ</sup> *<sup>S</sup>*

The <sup>χ</sup> *<sup>P</sup>*

which defines both electronic (*χ*(3)*NR*, 'non-resonant') and vibrational (*χ*(3)*R*, 'resonant') responses. When input laser pulse frequencies are different from electronic resonances, the first term *χ*(3)*NR* does not depend on frequency, i.e. it is linked to a flat spectral background that changes immediately with the excitation change; thus, it is a real quantity. The second term, the complex quantity *χ*(3)*R*, characterizes the nuclear response of the molecules and yields the intrinsic vibrational mechanism of SRS [2].

*Stimulated Raman Scattering in Micro- and Nanophotonics DOI: http://dx.doi.org/10.5772/intechopen.80814*

*Nonlinear Optics - Novel Results in Theory and Applications*

overcome in low dimensional materials [30].

**4. Conclusion(s)**

**A.Appendix**

amplitude *<sup>E</sup>*

where *<sup>P</sup>* →(3)

light velocity.

→ *<sup>P</sup>* and *<sup>E</sup>* →

and matter based on SRS.

electronic and photonic devices.

effects are out of the scope of this chapter [44–46].

∇ × (∇ × *E*

∇ × (∇ × *E*

order nonlinear susceptibility tensor *χ*(3) given by:

**A.1 Theory: the classical approach**

silicon-based materials. By combining our earlier results on the broadening of the Raman gain spectra [33–35] with the observation of higher Raman gain [2, 36–42], bring us to state that the traditional trade-off between gain and bandwidth is

In this book chapter, some of the most significant experimental investigations of SRS in micro- and nano-photonics are reported. The focuses are microstructures and nanostructures, which are able to enhance nonlinear interaction between light

We try to highlight how the nonlinear interaction based on SRS can take advantage of micro- and nanostructure with respect to bulk structure in order to improve SRS efficiency. In addition, we try to discuss new perspectives for the realization of Raman lasers with ultra small sizes, which would increase the synergy between

We note that pulsed lasers are often used in SRS experiment; therefore, we have to consider the time dependence of the output. If the pulsewidth is much longer than the relation time of the Raman excitation and the time required for light toi traverse the medium, we can expect from physical argument that the output pulse will follow the temporal variation of the input pulse. This is the quasi-steady-state case. Otherwise, the output should exhibit a transient behavior. The transient

The wave equations for pump and Stokes laser pulses with electromagnetic field

*<sup>P</sup>* <sup>=</sup> <sup>4</sup>*<sup>P</sup>*

*<sup>S</sup>*<sup>=</sup> <sup>4</sup>*<sup>S</sup>*

is the nonlinear polarizations, is the dielectric constants and *c* is the

In the case of SRS, the material interaction is classically treated through a third-

χ(3) = χ(3)*NR* + χ(3)*<sup>R</sup>* (2)

which defines both electronic (*χ*(3)*NR*, 'non-resonant') and vibrational (*χ*(3)*R*,

'resonant') responses. When input laser pulse frequencies are different from electronic resonances, the first term *χ*(3)*NR* does not depend on frequency, i.e. it is linked to a flat spectral background that changes immediately with the excitation change; thus, it is a real quantity. The second term, the complex quantity *χ*(3)*R*, characterizes the nuclear response of the molecules and yields the intrinsic vibra-

2 \_\_\_\_ *<sup>c</sup>*<sup>2</sup> *<sup>P</sup>* <sup>→</sup>(3) (ω*P*) 

2 \_\_\_\_ *<sup>c</sup>*<sup>2</sup> *<sup>P</sup>* <sup>→</sup>(3) (ω*S*) (1)

*<sup>S</sup>* at frequencies *ωP* and *ωS* (<sup>ω</sup>*<sup>P</sup> <sup>&</sup>gt;* <sup>ω</sup>*S*), are:

→ *<sup>P</sup>*) − <sup>ω</sup>*<sup>P</sup>* 2 \_\_ *<sup>c</sup>*<sup>2</sup> <sup>ϵ</sup>*PE* →

→ *<sup>S</sup>*) <sup>−</sup> <sup>ω</sup>*<sup>S</sup>* 2 \_\_ *<sup>c</sup>*<sup>2</sup> <sup>ϵ</sup>*SE* →

**136**

tional mechanism of SRS [2].

In order to simplify, we study the special case of an isotropic medium with *EP* and *ES* with the same polarization direction and propagation along *z*. The whole field amplitude is: *<sup>E</sup>*(*z*,*t*) <sup>=</sup> *EP <sup>e</sup> i*(ω*Pt*−*kPz*) + *ES e i*(ω*St*−*kSz*) . According to Eq. (2), the nonlinear polarizations take the form:

$$\begin{array}{ll}\text{minimize} & \cdots(\text{") } - \text{"} & \cdots & \text{"} \\ \text{ variations take the form:} & & & \\ & & & \end{array}$$

$$\begin{array}{ll}P^{\{3\}}\{\mathbf{u}\_{P}\} &=& \left[\chi\_{P}^{\{3\} \text{NR}} \left|E\_{P}\right|^{2} + \chi\_{P}^{\{3\} \text{R}} \left|E\_{S}\right|^{2}\right]E\_{P} \\ & P^{\{3\}}\{\mathbf{u}\_{S}\} &=& \left[\chi\_{S}^{\{3\} \text{R}} \left|E\_{P}\right|^{2} + \chi\_{S}^{\{3\} \text{NR}} \left|E\_{S}\right|^{2}\right]E\_{S} \end{array} \tag{3}$$

The <sup>χ</sup> *<sup>P</sup>* (3)*NR* and <sup>χ</sup> *<sup>S</sup>* (3)*NR* terms in *P*(3) only act to modify the dielectric constant <sup>ϵ</sup> *<sup>P</sup>* and <sup>ϵ</sup> *<sup>S</sup>* in Eq. (1). They are responsible for the field induced birefringence, selffocusing, etc., but have no direct effect on SRS. Therefore, in the following discussion, we neglect them. The *χ*(3)*R* terms in *P*(3), instead, effectively couple *EP* and *ES* in Eq. (1) and is the reason of energy transfer between the two fields. They are the cause of the stimulated Raman process and are called Raman susceptibilities. Eq. (1) can be solved with Eq. (3) by knowing *χ*(3)*R*.

A molecular vibration or optical phonon is the most common case of SRS. The optical radiation is assumed interacting with a vibrational mode of a molecule and this vibrational mode can be defined as a simple harmonic oscillator of resonance frequency *ω*υ, damping constant *γ*. The analysis is one-dimensional, thus, each oscillator can be distinguished by its position *z* and normal vibrational coordinate *q* [2].

The key assumption of the theory is that the optical polarizability of the molecule (which is typically predominantly electronic in origin) is not constant, but depends on the internuclear distance according to the equation:

$$\mathbf{a}(t) \,\, = \,\, \mathbf{a}\_0 \,\, \star \left(\frac{\partial \mathbf{a}}{\partial q}\right)\_0 \,\, \mathbf{q}(t) \tag{4}$$

This quantity is a tensor, but to simplify the discussion we will consider it as a scalar. Here *α*0 is the polarizability of a molecule in which the internuclear distance is held at its equilibrium value.

Starting from Eqs. (4) and (5), we obtain

$$-\alpha^2 q(\mathfrak{Q}\mathfrak{Q}) - 2i\alpha \eta q(\mathfrak{Q}\mathfrak{Q}) + \alpha^2\_\alpha q(\mathfrak{Q}\mathfrak{Q}) \ = \frac{1}{m} \left(\frac{\partial \mathfrak{q}}{\partial t}\right)\_0 E\_P E\_\mathcal{S}^\* \tag{5}$$

where *m* represents the reduced nuclear mass and *<sup>Ω</sup>* <sup>=</sup> <sup>ω</sup>*<sup>P</sup>* <sup>−</sup> <sup>ω</sup>*<sup>s</sup>* . This equation shows explicity *q*(*Ω*) as a material excitation resonantly driven by optical mixing *EPES* ∗ . SRS by phonons can therefore be considered a result of coupling three waves *EP*, *ES* and *q*(*Ω*) governated by the wave equations (3) and (6). This system is essentially the wave equation coupled to an oscillator equation. Starting from Eq. (6) it is possible to calculate the resonant Raman susceptibility, which, for the steady-state case, is related to the Raman gain by the following relation:

 *g* = −4*π*ω<sup>2</sup> 2 \_\_\_\_ *<sup>c</sup>*<sup>2</sup>*k*<sup>2</sup> (*Im*(χ*<sup>S</sup>* (**3**)*R* )) (6)

When the depletion of the pump field ⌈*EP*⌉ 2 is negligible, being the Raman susceptibility *χ*(3)*R* a negative imaginary, we find that the evolution of the intensity of Stokes field is given by the exponentially growing solution of

$$\left[E\_{\mathbb{S}}\right]^2 = \left[E\_{\mathbb{S}}(\mathbb{O})\right]^2 \exp\left(\mathbb{g} \ast \left[E\_P\right]^2 \ast z - \mathfrak{a} \ast z\right) \tag{7}$$

The Stokes wave is amplified if the gain exceeds the losses. We note that Raman amplification is a process for which the phase matching condition is automatically satisfied. In other words, Raman amplification is a pure gain process.
