Advances in Optical Technologies

Volume 2016, Article ID 3696094, 10 pages

http://dx.doi.org/10.1155/2016/3696094

## Modeling Nonlinear Acoustooptic Coupling in Fiber Optics Based on Refractive Index Variation due to Local Bending

Facultad de Ciencias, Escuela de Física, Universidad Nacional de Colombia, Medellín, A.A. 3840, Medellín 20036, Colombia

Received 4 April 2016; Revised 22 August 2016; Accepted 23 August 2016

Academic Editor: José Luís Santos

Copyright © 2016 Catalina Hurtado Castano et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A detailed procedure is presented to compute analytically the acoustooptic coupling coefficient between copropagating core and lowest-order cladding modes in tapered fiber optics. Based on the effect of the local bending, the linear and nonlinear variations in the refractive index are modeled. A set of equations and parameters are presented in order to calculate the influence of acoustooptic effect in nonlinear pulse propagation. We will show that as the tapered fiber diameter decreases more energy can be transferred to the cladding and the nonlinear phenomena can compensate the coupling coefficients effects.

#### 1. Introduction

The use of light to solve specific problems has led to the development of new knowledge in materials and optical systems which have found applications in technologies such as photonics [1]. The development of fiber optics (FO) has increased the capacity of processing light by offering systems with ultra high speed such as telecommunication networks and fiber laser [2, 3]. Furthermore, FO can be used as a multiplexers, modulators, and tunable filters based on acoustooptic (AO) coupling effect.

Although the AO coupling mechanism is well known, the study and design of AO devices remain as a topic that deserves attention, especially if one wants to achieve high frequency processing and reproducibility. Studies of AO effect in fibers have been carried out both theoretically and experimentally, where acoustic waves are considered long period fiber grating (LPFG) which are capable of coupling coherent copropagating modes traveling in neighboring wave guide that interact through the evanescent optical field [4]. The phase matching condition between the modes needs the overlapping among the interacting modes, allowing energy transfer between them. LPFGs offer the possibility of nonlinear coupling interaction between copropagating guided modes [5]. LPFGs have been manufactured with UV lasers to study the propagation of nonlinear pulse in coupled modes [6].

Herein, using a local bending model, variations in the refractive index due to the presence of the acoustic wave are calculated. A detailed theory and methodology are presented in order to calculate analytically the nonlinear coupling pulse propagation between copropagating core and linearly polarized (LP) cladding modes in tapered fibers. This paper is structured in the following way: Section 2 presents the fundamental theory of local bending model for linear acoustooptic coupling. Section 3 describes the expressions to calculate field profiles and propagation constants of cladding modes. In Section 4, based on local bending, novel equations for the nonlinear coupling between copropagating core-cladding modes are presented. Section 5 presents the results of numerical simulations. Finally, the conclusions are presented in last section.

#### 2. Local Bending Model for AO Coupling

In this section, the AO coupling interactions between the fundamental core mode and copropagating cladding modes are analyzed. We see that acoustooptic coupling equations described changes in the amplitude of the mode propagated through an optical fiber, either in core region or in cladding region.

The transference of energy between fiber modes can be achieved by the presence of acoustic waves [7]; we assume that a single mode fiber is modulated in the plane and the modulation locally bends it with radius of . The effect of the local bending, as a consequence of acoustic wave, can be modeled as a variation in the refractive index [8] of the fiber which is given by the expressionwhere is the core refractive index for the fundamental fiber mode without bending, is the local coordinate on the straight fiber, and is the silica elastooptic coefficient whose value is . Assuming that the amplitude of oscillation is much smaller than the acoustic wavelength, one can approximate the local curvature aswhere is acoustic amplitude, is the acoustic wave number, and is acoustic wave period. The coupled-mode equations that describe linear coupling interactions in a long-period grating arewhere is the core mode amplitude, the cladding -mode amplitude, and the detuning parameter. In (4) the longitudinal part of the coupling coefficient is neglected since it is substantially much more smaller than the transverse part [9]. The maximum transference of energy is given at the phase condition , where is the resonant wavelength. Then, we can write

The simplified expression of the coupling coefficient between the fundamental core mode and any cladding mode is given by and described in [8] for local bending model; an extensive expression can be seen in [10, Eq. ]. In this paper, we consider the interaction between the fundamental core mode () and the first three lowest cladding modes (, ) since is only nonzero for modes that differ by an odd integer in angular momentum [8]. Using the local bending model, is written aswhere subscripts and reference the core and cladding mode, respectively; is the transverse mode profile, normalized according to . The normalized power carried by the and modes can be written as a function of the propagation distance aswhere , , and . The initial power is pumped through the fundamental core region , and the initial power is pumped through the cladding region .

#### 3. LP Cladding Modes

This section is devoted to describe the expressions to calculate field profiles and propagation constants of cladding modes using exact analytical equations for a step index fiber optics.

The cladding is considered a multimode wave guide; for this to exist, it is necessary to remove the polymer coating to satisfy the condition of total internal reflection at the external boundary [11]; therefore these modes are very sensitive to change because of the surrounding medium. Cladding modes can be excited by using LPFGs which can couple light from core to cladding modes [11]. This happens due to the fact that the condition of total internal reflection is violated when the optical fiber is bending.

We are interested mainly in the interaction between the fundamental core mode () and the lowest () cladding modes in a tapered fiber optics. In the case of the mode guided by the fiber core, field profile and propagation constant are calculated using the expressions given in [12]. For cladding modes, we used the dispersion expression given in [10, 13] for the three-layer fiber model in order to compute the constant propagations. The cladding mode dispersion equation is given by the equationswhere

The variables and functions given in (10) are the following:where and are the core and cladding radius, respectively. is the azimuthal number. denoted the Bessel functions of the first kind, are the Bessel functions of the second kind, and are the modified Bessel functions of the second kind; also, , , and indicate differentiation of Bessel functions with respect to the total argument . Based on the three-layer model, the electric field components for cladding modes are [10, 14]

The conventional mode is equivalent to linearly polarized mode [12], which is our mode of interest. The cladding modes occur in a strict alternating sequence of and , except from the fundamental mode [13]. In our case, we set and in conventional modes which are the first three lowest cladding modes which are the ones we are interested in. Then, Figure 1 shows the -profile for these modes in a standard tapered single mode fiber optics.