## 1. Introduction

Nanoplasmonic waveguides offer unprecedented tight confinement of electromagnetic waves in subwavelength structures. In principle, nanoplasmonic waveguides comprise several metal‐dielectric layers. The propagating light is coupled to the free electrons in the metal, which acts like plasma at the optical frequency. It then follows that the propagating light can be strongly confined even to 100 times smaller than its wavelength. Such tight confinement has the potential to open a new world of scalability and integration. For instance, nanoplasmonic‐based devices can be utilized as matching interconnectors between traditional micro‐photonics and nanoelectronic devices [1–5]. This paves the way for the development of efficient, compatible and ultrafast chips [6–8]. However, the achievable tight confinement in the nanoplasmonic waveguides can be functioned to boost the propagating waves’ intensities. It then follows that the nonlinear effects can be significantly enhanced, and thus, desired functionalities can be achieved. To date, several advances in plasmonic technology have been reported. These include all optical nanoplasmonic logic gates [9], electrically controlled plasmonic devices [10], plasmonic amplification and lasing [11], efficient second harmonic generation using plasmonic waveguides [12], silicon‐on‐insulator‐compatible plasmonic devices [13], focus and enhancement of terahertz (THz) radiation using plasmonic waveguides [14], efficient and ultra‐compact hybrid plasmonic devices [15], solitonic plasmonic waveguides [16] and mode conversion in plasmonic waveguides [17], to mention a few examples.

In this chapter, nanoplasmonic waveguides filled with electro‐optical materials are proposed and discussed. Specifically, metal‐insulator‐metal (MIM) plasmonic waveguides filled with lithium niobate (LiNbO_{3}) are considered. The theoretical models that describe the nonlinear interaction of the confined electromagnetic waves in nanoplasmonic waveguides are developed. Numerical evaluations based on experimentally reported parameters are provided. Two different configurations are studied thoroughly. The first configuration is a MIM nanoplasmonic waveguide designed for compact THz generation [18]. The optical pumps are guided by means of surface plasmon polaritons (SPPs), whereas the generated THz waves are designed to diffract and propagate outside the MIM waveguide. The design of this work attains the following desired properties: first, the THz absorption is minimized. Secondly, the proposed structure is compact and relevant for nanoapplications. Thirdly, the generation efficiency is maximized by reducing losses rather than introducing resonant conditions (i.e. phase matching), and thus, wide THz generation is achieved. We note that this proposal introduces a new modality from the standpoint of applications. For instance, the proposed MIM nanostructure can be immersed inside a target (such as a cell or a biological entity), and the generated THz waves can interact with the (under‐test) surrounding medium of the MIM nanostructure, while the optical modes are tightly confined inside the MIM nanostructure. Potential future applications include nanocommunication networks and body‐centric systems. The second configuration considered in this work includes MIM plasmonic waveguide filled with doped LiNbO_{3} [19]. The main interest is to study the interaction of the interfering SPP modes with the doping impurities. To model the interaction analytically, small intensity modulation interference depth is considered and a perturbation approach is employed. The evolution of the interfering modes is obtained to quantify the interaction. On considering strong pump and weak signal scenario, it is found that the two SPP modes are coupled by means of the photorefractive effect. The modal gain is calculated to characterize the photorefractive effect. First, an ideal case of lossless waveguides is considered, and the gain versus the waveguide length, the doping concentration and the input amplitudes are characterized. Secondly, the modal losses are taken into account. It is found that a weak antisymmetric mode can experience an effect gain up to certain waveguide lengths in the presence of a strong symmetric mode, considering proper doping concentration and input amplitudes. The coupling effect can be conceived either as an amplification or mode‐conversion process, promising novel future application.

The remaining part of this chapter is organized as follows. The propagation modes and the dispersion relation of MIM plasmonic waveguides are introduced in Section 2. The nonlinear and the electro‐optical coefficients of the LiNbO_{3} are introduced in Section 3. THz generation in nanoplasmonic waveguides is discussed in Section 4. The photorefractive effect in nanoplasmonic waveguides is discussed in Section 5. Finally, concluding remarks are presented in Section 6.

## 2. MIM Plasmonic waveguides

Consider an MIM plasmonic waveguide comprising two metallic layers sandwiching a LiNbO_{3} of thickness **Figure 1**. The waveguide is two dimensional and independent of the

The waveguide has two fundamental propagating SPP modes of symmetric and antisymmetric transverse field distribution at specific frequency

(1) |

where

(2) |

Here, _{3} relative permittivity for

On applying the boundary conditions, the dispersion relation for the symmetric and antisymmetric modes can be, respectively, given by the following [20]:

where

One may rewrite the dispersion relation in the following form [18]:

The dispersion form in Eq. (4) can be solved numerically for *β* can then be calculated by

## 3. LiNbO_{3} coefficients

LiNbO_{3} is chosen as the electro‐optical material for this work. Consider the indices of the LiNbO_{3} (1, 2, 3) along the axes (_{3} is given by the following [21]:

where

The electro‐optic coefficients are related to the nonlinear coefficients by the following:

where

The effective permittivity of the LiNbO_{3} depends on the electric field components, by means of electro‐optic effect, through the following relation [21]:

where _{3} and _{3}.

On considering a small induced change in the permittivity, so that

In Section 5, an example of photorefractive effect in MIM plasmonic waveguide based on the electro‐optic effect is discussed.

## 4. THz generation in nanoplasmonic waveguides

In this section, an MIM plasmonic waveguide filled with LiNbO_{3} and with two propagating SPP modes is considered. The frequency difference between the two SPP modes is properly designed for THz generation. Theoretical modelling and numerical evaluations are presented.

### 4.1. Field expressions

As the symmetric mode is a fundamental mode of the MIM waveguides, two symmetric modes with distinct frequencies

Here

The THz wave is generated by means of difference‐frequency generation of the two SPP modes enabled by the LiNbO_{3} nonlinearity.

The MIM structure is designed to guide the SPP pumps, whereas the generated THz waves are designed to diffract and propagate outside the MIM waveguide, as shown in **Figure 2**. The thickness of the metallic layer is equal to

where

### 4.2. Nonlinear polarization

The induced nonlinear polarization components are at the frequencies

(11) |

the nonlinear polarization components at

(12) |

and the nonlinear polarization component at

The

### 4.3. THz generation

The evolution of the generated THz wave and the SPP modes are governed by the nonlinear wave equation, given by:

On the substitution of the fields of Eqs. (9) and (10) into Eq. (14), and using the nonlinear polarization expressions of Eqs. (11–13), the corresponding slowly varying envelope approximation (SVEA) equations can be formed, given by the following [18]:

(17) |

where

The evolution of the THz and the SPP fields can be obtained by numerically solving the SVEA Eqs. (15–17). However, various losses must first be precisely evaluated. These include SPP and THz losses. The decay factor of the SPP linear losses can be calculated from the imaginary part of the propagation constant, i.e. _{3} crystal, and

where _{3} THz absorption coefficient and _{3} absorption coefficients can be found in the literature. For example, in [27] the absorption coefficient of gold is given by _{3} is given by

### 4.4. Numerical evaluations

In the following analysis, we consider gold for the two metallic layers, a slot thickness of **Figure 3**, the power of a 4 THz generated wave is shown versus the waveguide length. Here,

As can be seen in **Figure 3**, the effective waveguide length is limited to

The optics‐to‐THz conversion efficiency can be defined as

In **Figure 4**, the conversion efficiency for a 4 THz generated wave is presented versus the input SPP power. Here, we consider a waveguide length of

The frequency of the generated THz wave can be tuned by controlling the frequency difference between the two SPP modes, i.e. **Figure 5**, the wavelength 1 is fixed at **Figure 5**, THz waves can be generated over the entire range from 1 to 10 THz simply by tuning the frequency **Figure 5**. These include

## 5. Photorefractive effect in nanoplasmonic waveguides

In this section, the LiNbO_{3} filling the MIM waveguide is doped with

### 5.1. Intensity expression

The electric fields of the two symmetric and antisymmetric propagating modes are described by Eq. (1). The optical field intensity inside of the waveguide,

where

On adapting a strong pump and weak signal scenario, i.e.

### 5.2. Band transport model

The interaction of the interfering SPP modes with LiNbO_{3} impurities is governed by the standard band transport model that encompasses the electron continuity equation, the current density equation and the Poisson's equation, as in the following [29]:

where _{3} electron mobility, _{3} permittivity. The three coupled equations of the band transport model shown above can be solved analytically by obeying a perturbation approach for a small depth intensity modulation limit i.e.

where

The space charge electric field

Consequently, on substituting Eq. (23) into Eq. (22), and using the electrostatic condition, one can obtain the following expression [19]:

Where

By substituting Eq. (23) into Eqs. (20) and (21), and considering a steady state so that the time rates of the free electrons and the ionized donor densities are zeros (i.e.

Secondly, by substituting Eq. (23) into Eqs. (20) and (21) and equating the modulation terms (i.e. those with

where

The spatial electric field

### 5.3. Photorefractive effect

The evolution of the SPP modes are governed by nonlinear wave equation, given by:

where

The effective permittivity can be obtained by substituting the spatial electric field, i.e.

where

On the substitution of symmetric and antisymmetric mode expressions of Eq. (1) into Eq. (29) and using Eq. (30), the fields equations (for slowly varying amplitudes) can be obtained, given by the following [19]:

Where

As can be seen in Eqs. (31) and (32), the two SPP modes are coupled in a phase‐matched fashion. The coupling is conducted by the spatial space charge electric field generated by the interfering SPP modes in the doped LiNbO_{3}. This is known as the photorefractive effect. The photorefractive effect has been known since the early 1960s [30] in bulk materials and micro‐meter dielectric waveguides. In this work, the photorefractive effect is investigated in nanoplasmonic waveguides. Numerical evaluations are presented in the following section.

### 5.4. Numerical evaluations

To illustrate the potential of the photorefractive effect in plasmonic waveguides, typical realistic values are considered. Consider _{3} permittivity is

First, the waveguide losses are ignored and the gain versus the doping concentration and the input amplitudes is characterized. Different slot thicknesses are considered. Secondly, the modal losses are taken into account, considering proper doping concentration and input amplitudes.

Our numerical investigations show that gain can be realized only for weak antisymmetric mode (the signal) co‐propagating with strong symmetric mode (the pump). This is because the space charge electric field, that couples the two modes, has an antisymmetric transverse distribution, as can be inferred from Eqs. (2), (19) and (24).

In **Figure 6**, the antisymmetric gain, defined by

As can be seen from the simulations in **Figure 6**, the photorefractive response, characterized by the gain, is qualitatively the same for different waveguide thicknesses, given losses are ignored. However, it can be seen from part (b) that the gain crucially depends on the doping concentration; the stronger the effect, the larger the doping concentration. Furthermore, it can be seen from part (c) that the gain does not dramatically depend on the input ratio, given that the perturbation condition is satisfied, i.e.

Losses can be taken into account by incorporating the effective decay factor, given by

In **Figure 7**, the antisymmetric gain is presented against the waveguide length. Here, losses are taken into account. Three different thicknesses of **Figure 7**, a significant gain can be experienced despite losses with the maximum gain attainable at the same waveguide length

**Figure 8** depicts the results of **Figure 7** in dB units. As can be seen, given the assumed parameters, the maximum achieved gain can be

## 6. Conclusion

Nanoplasmonic waveguides filled with electro‐optical material were proposed and discussed. The aim of this work is to incorporate the unique properties of plasmonic waveguides, ultra‐compact structures and high internal intensities, with electro‐optical material properties, thus achieving novel functionalities. Two configurations were chosen and investigated. First, an Au‐LiNbO_{3}‐Au nanostructure waveguide is considered for THz generation by means of difference‐frequency generation of SPP modes. The SPP modes are designed to be totally confined inside the waveguide, whereas the generated THz waves are concentric with the SSP modes and contained mainly in the surrounding medium of the waveguide. Several advantages are achieved through this design. First, THz losses are minimized. Secondly, an off‐resonance operation is conducted. Finally, a nanoscale and yet simple THz generation is offered. The evolution of the THz wave and the SPP modes are evaluated and found to be mainly limited by losses. Nevertheless, THz generation is shown to be viable over the entire range from 1 to 10 THz by properly designing the SPP wavelengths and waveguide dimension. Possible future applications include nanocommunication systems and body‐centric networks. Secondly, an AL‐LiNbO_{3}‐AL nanostructure is considered with the LiNbO_{3} being doped with donor and acceptor atom impurities. Two SPP modes with symmetric and antisymmetric spatial distribution are considered. The interaction of the interfering SPP modes with the atom impurities is modelled. It was found that the SPP modes are coupled by the means of the photorefractive effect. A net gain was shown viable for weak antisymmetric mode co‐propagating with strong symmetric mode. The antisymmetric gain was studied against the doping concentration and the input amplitudes. This work opens up new opportunities to apply known photorefractive applications to nanoplasmonic devices. The two configurations discussed in this chapter demonstrate the potential of utilizing electro‐optical materials in nanoplasmonic waveguides to achieve novel, efficient and ultra‐compact devices.