## Abstract

Rapid development of the integrated optics and photonics makes it necessary to create cheap and simple technology of optical waveguide systems formation. Photolithography methods, widely used for these tasks recently, require the production of a number of precision amplitude and phase masks. This fact makes this technology expensive and the formation process long. On another side there is a cheap and one-step holographic recording method in photopolymer compositions. Parameters of the waveguide system formed by this method are determined by recording geometry and material’s properties. Besides, compositions may contain liquid crystals that make it possible to create elements, controllable by external electric field. In this chapter, the theoretical model of the holographic formation of controllable waveguide channels system in photopolymer liquid crystalline composition is developed. Special attention is paid to localization of waveguides in the media caused by light field attenuation during the formation process.

### Keywords

- photopolymer
- liquid crystal
- waveguide
- holography

## 1. Introduction

The ability to form waveguide systems for optical and terahertz radiation in photopolymerizable compositions recently is of great interest among researchers: [1, 2, 3, 4]. Formed holographically or by photolithography methods, such waveguides are widely used in the integrated optics and photonics devices. Besides, it seems urgent to create the manageable light guides, in which the light propagation conditions can be controlled by external influences, such as an electric field. One of the possible solutions of this problem is a holographic recording of the waveguide channels in a photopolymer composition containing liquid crystals.

The aim of this chapter is to develop the theoretical model of holographic formation of controllable waveguide channels system in photopolymer liquid crystalline composition with dye sensitizer, also known as polymer-dispersed (PDLC) or polymer-stabilized (PSLC) liquid crystals.

## 2. Theoretical model

We consider the incidence of two plane monochromatic waves

Thus, recording waves in the general case can be described as:

where

Investigated material is characterized by optical anisotropy properties; thus, in the material, each of recording wave (Eq. (1)) will be divided into two mutually orthogonal ones called ordinary and extraordinary. So, in the sample, Eq. (1) should be rewritten as:

where * m* =

*corresponds to ordinary waves and*o

*=*m

*corresponds to extraordinary waves, respectively.*e

For two recording geometries (see Figure 1), the spatial distributions of the forming field intensities are determined by the following expression:

where

Under the influence of light field in photopolymer liquid crystalline composition with dye sensitizer, the dye molecule absorbs a light radiation quantum with the dye radical and the primary radical initiator formation. The radical of dye is not involved in further chemical reactions and turns to colorless leuco form.

Thus, during the waveguide channel system formation, dye concentration decreases. This fact causes the light-induced decreasing of light absorption. The absorption coefficient dependence of a mount of absorbed radiation can be written as [5]:

where

where

Then, the solution of Eq. (3) for light-induced absorption change for transmission and reflection geometries is obtained in [5] as follows:

where

The process of waveguide channels’ holographic formation is described by the kinetic equations system (KES), written for monomer concentration and refraction index [6, 7, 8]:

where

Diffusion coefficients can be defined from the following equations:

where

Weight coefficients

where

where

Because of periodical character of the forming fields’ intensities’ spatial distributions, solution of KES can be found as a sum of * H* spatial harmonics [8]:

where

By substituting Eq. (13) to the KES (Eqs. (7 and 8)) and using the orthogonality of spatial harmonics, a system of coupled kinetic differential equations for the amplitudes of monomer concentration harmonics can be obtained for

and also a system of differential equations for the amplitudes of refraction index harmonics:

In equation systems Eqs. (14) and (15), a coefficient matrix is introduced [6]:

where

Coefficients

For solution of the coupled differential equations system (14), the initial conditions should be introduced:

The solution can be found using the operator method [6]. The general solution for the spatial amplitude profiles of monomer concentration harmonics will be:

where functional dependencies of coefficients

where

Then, by substituting (20) to (15) and by integrating the resulting differential equations with initial conditions, we get.

The general solution for the amplitude of * j*-harmonic of the refractive index can be found:

where
* j* = 0,…,

*.*H

Thus, Eqs. (13), (20), and (23) are the general solutions of nonlinear photopolymerization diffusion holographic recording of waveguide channels system in PDLCs (PSLCs) in the case of stable diffusion coefficients and stable absorption. They define kinetics of spatial profiles of monomer concentration—

In case of high nonlinearity, it is possible to form the specific spatial profile of refractive index. The nonlinearity of recording is achieved by changing the ratio of the photopolymerization and diffusion mechanism contributions to the process of structure?s formation.

## 3. Numerical simulations

To investigate the formation processes, numerical simulations of first four refractive index harmonics kinetics for transmission and reflection geometries were made with the following parameters:

As can be seen from Figure 2, in the case of predomination of polymerization (

Corresponding distributions

In Figure 3, a new spatial coordinate is introduced:

According to Eqs. (17) and (18), parameter

It should be also noted that the amplitude of refractive index change when

To investigate the impact of absorption on the spatial profile of structure along the thickness of the sample, numerical simulations were made with the following parameters:
* Np*. Simulations were made by Eq. (23) for the amplitude of the first harmonic of refractive index and for two formation geometries. Results are shown in Figure 4.

As shown in Figure 4, absorption of the formation field causes dependence of the first harmonic on * z*-coordinate (see Figure 1) during the recording process. Thus, the spatial profile of refractive index changes its character in time and space.

In Figure 4, some characteristic cases of localization of maximum of refractive index change in the thickness can be seen. In particular, in the case of transmission geometry (Figure 4a) at
* z*-coordinate and spatial distributions of amplitude, and the phase of recording field (Eqs. (1) and (2)) causes the dependence on

*and*x

*coordinates. In more general cases, there is also a temporal dependence*y

To illustrate the localization of waveguide channels into the material, in Figure 5, there are some more two-dimensional spatial distributions of refractive index, obtained by Eq. (23) for transmission and reflection geometries—

It follows from the abovementioned that by controlling the spatial distribution of light field and taking the absorption effects into account, the waveguide systems localized into the material can be holographically formed. Localization is important because of the necessity to create a predetermined refractive index profile independent from properties of possible substrates (electrodes, glass, polymer layers, etc.). In particular, to create the waveguide system localized near the bottom substrate, formation geometry should be transmissive and formation should be stopped before
_{,} waveguides will be localized near the top substrate for transmission geometry and not localized for reflection geometry. The given values of

In our recent works [9, 10, 11, 12, 13, 14, 15, 16, 17, 18], we have seen that the impact of the external electric field on the holographic structure formed in PDLC (PSLC) leads to the refractive index change due to electro-optical orientation mechanisms, that is typical to liquid crystals, and the recorded structure can be “erased”. So, it can be supposed that the impact of external electric field on individual waveguides will lead to its “erasing”, and it is possible to “switch off” some waveguides or change the system’s period, and so on.

## 4. Conclusion

Thus, in this chapter, the theoretical model of holographic formation of controllable waveguide channels system in photopolymer liquid crystalline composition is developed. The most general cases are described by the developed model, and numerical simulations were made for plane recording waves and stable absorption cases. Special attention is paid to localization of waveguides in the media caused by light field attenuation during the formation process.

It is shown that parameters of the waveguide system formed by holography’s methods are determined by recording geometry and material’s properties. Also, by control of these parameters, waveguides can be localized into the sample that makes them independent from substrates. Also, introduced compositions contain liquid crystals that make it possible to create elements, controllable by external electric field.

Obtained results can be used for new photonics devices based on photopolymer liquid crystalline composition development.

## Acknowledgments

The work is performed as a project that is part of Government Task of Russian Ministry of Education (project No. 3.1110.2017/4.6).

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