Switching in multicore fibers using flexural acoustic waves

Gil M. Fernandes; Nelson J. Muga; Ana M. Rocha; Armando N. Pinto

doi:10.1364/OE.23.026313

1. Introduction

Space-division multiplexing (SDM) over multicore fibers (MCF) has been proposed as an alternative to current optical communications systems, based on single-mode fibers (SMF). This disruptive technology can increase drastically the transmission capacity per fiber [1]. However, optical signal processing techniques like switching are mandatory to make multicore fibers transmission systems cost attractive.

Although MCFs with several similar cores, i.e. homogeneous MCF (HoMCF), was been proposed to increase the capacity of a single fiber, the tightly packing of the cores leads to crosstalk [1]. In order to suppress such crosstalk, MCFs were proposed with index trenches, airholes [1] or heterogeneous core structures, i.e. heterogeneous MCF (HeMCF), in which cores have different effective indices [2]. Signal amplification in multicore transmission systems have been demonstrated using Erbium doped [3] and Raman amplification [4,5] achieving promising results. Regarding the signal switching at reconfigurable optical add/drop multiplexers in SDM systems, few solutions have been proposed [6–8]. In [6], it was proposed the use of an array of wavelength-selective switches (WSSs) to implement a ROADM suitable to MCFs transmission systems. More flexible architectures have been achieved using free-space switching based on Micro Electro Mechanical Systems (MEMS) mirrors or on Liquid Crystal on Silicon (LCOS) pixel arrays [6, 7]. The all-optical nonlinear switching in MCFs was also demonstrated using high-power ultrashort laser pulses [8]. Nevertheless, all these solutions are quite complex and tend to have high insertion losses. On the other hand, in standard SMFs the acousto-optic (A-O) effect has been widely used to produce tunable filters and modulators [9,10]. The A-O effect is a specific case of photoelasticity in which the diffraction of light is induced by a sonic grating [9]. The transversal symmetry of this grating depends on the kind of acoustic waves excited in the fiber (i.e. longitudinal, torsional or flexural) [11]. The A-O effect was recently proposed to perform mode switching in few-mode fibers [12,13]. In [12], the mode switching was achieved using a single flexural or a single longitudinal acoustic waves. Furthermore, the double resonant coupling induced by two flexural waves was experimentally demonstrated in few-mode fibers, and used to promote the mode coupling between three distinct spatial modes [13]. Despite the work already developed in the A-O effect for SMFs and few-mode fibers, the interaction between the optical signal and acoustic waves in MCFs was not yet deeply investigated.

We develop a theoretical model describing the interaction between acoustic waves and the optical field in MCFs. Using such model, we show that the antisymmetric perturbation induced by flexural acoustic waves can be used to switch light between the cores of a MCF. We demonstrate the switching between any two cores, in HeMCFs by using the double resonant coupling, and the possibility to distribute the power uniformly among all the cores in HoMCFs. Moreover, we propose an in-line wavelength selective core switch and an in-line wavelength selective core attenuator based on the A-O effect.

This paper is organized in six sections. In section 2, we present the operation principle of the technique proposed to signal core switching. The model addressing the interaction between acoustic and optical waves in MCFs is developed in section 3. In section 4, we discuss the A-O core switching in HoMCFs. In section 5, the signal core switching is enhanced using a double resonant coupling in HeMCFs. Finally, the conclusions are presented in section 6.

2. Core switching operation principle

By means of photoelasticity, the dielectric permittivity in optical fibers can be changed by the strain induced by an acoustic wave [9]. Fibers can support three different kinds of acoustic waves, usually called longitudinal, torsional and flexural [11]. Torsional and longitudinal acoustic waves induce axially symmetric displacements; in the fist case, displacements are purely circular, whereas in the second case displacements are no-circunferential [11]. On the other hand, flexural waves are characterized by a bending motion in which one side of the fiber is stretched and the opposite side compressed [11]. Such waves produce a Bragg structure in the optical fiber with period defined by the acoustic frequency [9] and a transversal antisymmetric perturbation that can be exploited to change the refractive index in the fiber cores. By using two stacked piezoelectric transducers (PZTs) with orthogonal vibration directions and applying in both PZTs equal in-phase radio-frequency (RF) signals, an arbitrary flexural acoustic wave can be generated [14], see right inset in Fig. 1(a). When both RF signal are properly synchronized (i.e., in-phase), the vibration plane of the acoustic wave can be adjusted by changing the amplitude ratio between both applied signals, see Fig. 1(b). In addition, the peak-to-peak voltage (V_p-p) of the RF signals applied on each PZT allow to control the acoustic wave amplitude, see Fig. 1(b). Then, the acoustic wave is transmitted to the fiber through a horn [15]. We are assuming a setup similar to the used in [14] to generate acoustic vortex in optical fiber with controllable amount of orbital angular momentum.

Fig. 1 a) Schematic of the proposed tunable in-line core switch based on A-O effect. In the right-hand side of the figure is showed the configuration with two PZTs used to tune the spatial orientation of the flexural wave. In the left-hand side of the figure is showed the profile of the four cores MCF considered. b) The frequency and amplitudes of the RF signals applied over both PZTs can be tuned to adjust the spatial orientation, φ, of the resultant flexural wave.

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In homogeneous MCF, the index perturbation induced by the flexural wave in the cores depends on the geometry of the HoMCF and in the spatial orientation of the flexural wave (i.e., vibration plane). By adjusting this plane, the refractive index perturbation induced in a given core and, consequently the coupling coefficient between such core and the cladding mode can be adjusted. If the flexural wave enables the phase-matching, the sonic grating leads to the power transfer from the initial core mode to a cladding mode. Then, the coupling coefficient between the cladding mode and the remaining cores define the subsequent energy transfer from the cladding mode to the other cores of the MCF.

Unlike homogeneous MCF, in heterogeneous MCF the resonant coupling induced by a single flexural wave occurs only between a given core and a cladding mode because each core has a different propagation constant. This wave induces the energy transfer between a given core and a particular cladding mode. A second flexural wave, with the proper acoustic frequency, is also applied to the optical fiber in order to induce the resonant coupling between the previous excited cladding mode and the desired output core. Both acoustic waves are propagated jointly in the optical fiber inducing two distinct Bragg structures and enabling the signal switching between any two cores by means of an “intermediary” cladding mode. Note that the core conversion using the double resonant coupling in HeMCF is independent of any relative time delay between both acoustic waves and on the fiber geometry considered.

3. Theoretical model

3.1. Acoustic flexural waves in MCFs

The A-O effect allows the power transfer between two optical modes, named l and p, in an optical fiber. In such effect the momentum and energy are preserved [9],

β_{l} (ω_{l}) - β_{p} (ω_{p}) = k,

and

ω_{l} - ω_{p} = Ω,

where β_i(ω_i) represents the propagation constant and ω_i is the angular frequency for the optical signal, with i = l or p specifying the optical mode. The parameters k and Ω represent the propagation constant and the angular frequency of the acoustic wave, respectively. The A-O effect induces a frequency shift in the optical signal. Nevertheless, this frequency shift is extremely small (Ω ≪ ω_i) [9].

A flexural wave induces a perturbation in the dielectric permittivity of the fiber, given by [9]

ε (r, t) = ℜ {ε_{u} [1 + 2 (1 - χ) S_{z z} (r, t)]},

where t is the temporal coordinate, r = (r,θ,z) denotes the spatial coordinates, with r, θ and z being the radial, the angular and the longitudinal coordinate, respectively. The ε_u is the unperturbed dielectric permittivity, χ is the elasto-optic coefficient (for silica χ = 0.22 [12]) and S_zz(r,t) is the strain distribution along the fiber [12],

S_{z z} (r, t) = \frac{\partial}{\partial z} u_{z} (r, t) .

being u_z(r,t) the longitudinal displacement induced by the flexural wave in the MCF. Such wave induces additional displacements in the radial, u_r(r,t), and azimuthal, u_ϕ(r,t) axis, and the three displacement components can be written as the real part of, [11]

u_{z} (r, t) = W (r) Φ_{z} (ϕ) e^{- i (k z - Ω t)},

u_{r} (r, t) = U (r) Φ_{r} (ϕ) e^{- i (k z - Ω t)},

u_{ϕ} (r, t) = V (r) Φ_{ϕ} (ϕ) e^{- i (k z - Ω t)},

where the U(r), V(r) and W(r) parameters describe the axial variation of displacement distribution,

U (r) = A k_{d} J_{m}^{'} (k_{d} r) + B k J_{m}^{'} (k_{t} r) + C \frac{m}{r} J_{m} (k_{t} r),

V (r) = A \frac{m}{r} J_{m} (k_{d} r) + B \frac{k m}{k_{t} r} J_{m} (k_{t} r) + C k_{t} J_{m}^{'} (k_{t} r),

W (r) = - i [A k J_{m} (k_{d} r) + B k_{t} J_{m} (k_{t} r)],

with the integer m=1 for flexural acoustic waves. The J_m represents a Bessel function of the first kind and order m,

J_{m}^{'}

represent the derivative of the Bessel function with respect to r, and

k_{d} = \frac{Ω^{2}}{c_{d}^{2}} - k,

k_{t} = \frac{Ω^{2}}{c_{t}^{2}} - k,

where

c_{d}^{2} = µ / ρ

and

c_{t}^{2} = (λ_{M} + 2 µ) / ρ

represents the square of the bulk dilatational wave velocity and the transverse wave velocity, respectively. The λ_M = 1.6×10¹⁰ Nm⁻², µ = 3.1×10¹⁰ Nm⁻², and ρ = 2.2 × 10³ kg/m³ are the Lame’s constants and the material density for silica, respectively [11]. The Φ(ϕ) = [Φ_r,Φ_ϕ,Φ_z] represents the circumferential displacement distribution, where Φ_r, Φ_ϕ and Φ_z are the radial, azimuthal and longitudinal components, respectively. The circumferential distribution for flexural acoustic modes can assume two solutions: [sin(mϕ),cos(mϕ),sin(mϕ)] and [cos(mϕ),−sin(mϕ),cos(mϕ)] where the integer m determines the circumferential field variation and consequently the symmetry of the index perturbation induced in the MCF. The constants A, B, and C are calculated considering the maximum longitudinal displacement,

u_{z}^{\max}

, induced by the acoustic wave in the MCF. The dispersion equation for flexural acoustic waves is obtained requiring a stress-free condition at the MCF surface, [11]

| \begin{array}{l} m^{2} - 1 - q^{2} (x - 1) & m^{2} - 1 - q^{2} (2 x - 1) & 2 (m^{2} - 1) (γ_{m} (q_{t}) - m)) - q^{2} (2 x - 1) \\ γ_{m} (q_{d}) - m - 1 & γ_{m} (q_{t}) - m - 1 & 2 m^{2} - 2 (γ_{m} (q_{t}) - m) - q^{2} (2 x - 1) \\ γ_{m} (q_{d}) - m & - (x - 1) (γ_{m} (q_{t}) - m) & m^{2} \end{array} | = 0,

where the parameter a denotes the radius of the MCF, γ_m(k) = kJ_m₋₁(k)/J_m(k), x = Ω²/(kc_t)², q_t = k_ta, q_d = k_da and q = ka [11]. By solving Eq. (7), we calculate the acoustic frequency, Ω. Note that we only consider the fundamental flexural mode because higher modes have larger acoustic frequencies and, consequently, are more difficult to excite from the experimental viewpoint. In order to generalize the A-O theory to MCF, we need to take into account the geometry of the fiber and the spatial orientation of the flexural wave. The spatial distribution of the cores in the MCF is considered in the ε_u, where the dielectric permittivity of the fiber depends on the axial and radial coordinates in order to describe the regions of the core and the cladding. The dielectric permittivity for the core/cladding is calculated considering

ε_{i} = n_{i}^{2} ε_{0}

, where ε₀, n_i and ε_i are the vacuum permittivity, the refractive index and the dielectric permittivity for the region i (denoting i the core or cladding regions), respectively. The refractive index for the region i can be defined by a three term Sellmeier equation,

n {(λ)}_{i}^{2} - 1 = \sum_{j = 1}^{3} \frac{[{SA}_{j} + X ({GA}_{j} - {SA}_{j})] λ^{2}}{λ^{2} - {[{SL}_{j} + X ({GL}_{j} - {SL}_{j})]}^{2}}

where λ represents the signal wavelength and SA, Sl and GA, Gl are the Sellmeier coefficients for the SiO₂ and GeO₂ glasses, respectively [16]. The X parameter represents the mole fraction of GeO₂ [16]. We are assuming SiO₂ MCFs with GeO₂ used as dopant to create the cores, however the model can be applied for other MCF knowing the refractive index of each region. After specified the refractive index profile of the MCF, we define a referential and the fiber is assumed to be fixed relatively to such referential, see Fig. 1(a). Then, the vibration plane of the flexural wave can be rotated relatively to the referential. Such rotation must be considered in the circumferential displacement as an additional phase induced in Φ_z(θ). In that way, the circumferential displacement along the MCF can be rewritten as: Φ_z(θ,φ) = sin(θ + φ) or Φ_z(θ,φ) = cos(θ + φ), where φ is the angle between the vibration plane of the flexural wave and the YZ plane. Therefore, in the following section we assume that the circumferential displacement is described by

u_{z} (r, t) = W (r) \cos (θ + φ) e^{- i (k z - Ω t)},

in which the relative orientation between the vibration plane of the flexural wave and the cores spatial distribution of the MCF can be adjusted changing φ, and consequently the index perturbation induced in a given core. Note that, the amplitude of

u_{z}^{\max}

is defined by the acoustic power applied in the MCF.

3.2. Mode coupling theory

The signal propagation considering the A-O effect can be described through the coupled mode theory [12]. The electromagnetic field is assumed as

E (r, t, ω) = \sum_{p} \frac{F_{p} (x, ω)}{N_{p} (z, ω)} A_{p} (z, ω) e^{i [β_{p} (ω) z - ω t]},

H (r, t, ω) = \sum_{p} \frac{G_{p} (x, ω)}{N_{p} (z, ω)} ℬ_{p} (z, ω) e^{- i [β_{p} (ω) z - ω t]},

where x = (r,θ) are the transversal coordinates,

A_{p} (z, ω)

and ℬ_p(z,ω) are the electric and magnetic complex amplitudes, and F_p(x,ω) and G_p(x,ω) are the transverse distributions of the electric and magnetic field for the p mode at ω. The N_p(z,ω) is a normalization coefficient defined in order to make A_p(z,ω)A_p(z,ω)^∗ equal to the optical power in the p mode,

\iint [F_{p}^{*} (x, ω) \times G_{l} (x, ω) + F_{p} (x, ω) \times G_{l}^{*} (x, ω)] \cdot \hat{z} r d r d θ = 4 δ_{p l} N_{p}^{2} (z, ω),

with δ_pl denoting the Kronecker delta. Assuming a small index perturbation induced by the A-O effect (which allows to neglect the longitudinal variation of N_p, F_p and G_p) the following set of coupled equations is obtained replacing Eq. (10) into the propagation Eq. (11) of [12],

\frac{d}{d z} A_{p} (z) = \sum_{j = 0}^{k} \sum_{l = 1}^{n} A_{l} (z) ϑ_{l p}^{j} e^{i Ω_{j} t} e^{i [β_{p} (ω) - β_{l} (ω) - k_{j}] z},

where

A

is the complex amplitude of the electric field and the subindex l and p denotes the core and cladding mode considered. Note that, Eq. (12) comprises p coupled equations for the four cores and the p−4 cladding modes considered, and

ϑ_{l p}^{j}

represents the coupling coefficient between the l and p modes (i.e. between a given core/cladding mode and the remaining cores and cladding modes considered),

ϑ_{p l}^{j} = \frac{ω}{4} \iint \frac{F_{p} (x, ω) \cdot F_{l}^{*} (x, ω)}{N_{p} (ω) N_{l} (ω)} \frac{\partial ε_{j}}{\partial z} r d r d θ,

being ε_j the permittivity induced by the j acoustic wave. Furthermore, Eq. (12) also allows to describe the mode coupling dynamics considering several index perturbations with distinct spatial distributions and periods.

4. Homogeneous multicore fibers (HoMCFs)

In this section the A-O effect is analyzed in HoMCF with cladding diameter of 125 µm and four single-mode cores. The cores have a step-index profile with core diameter (d) of 7.2 µm and “square” arrangement with pitch (D) of 36.25 µm, see the left-hand inset in Fig. 1. The refractive index of the core/cladding is calculated considering in Eq. (8) the Sellmeier coefficients given in [16]. The mole fraction of GeO₂ considered for all the cores was X = 0.0516 [16]. The fiber considered is similar to the commercial fiber, SM 4C1500(8.0/125) from the company Fibercore [17]. Using the previous parameters in the Comsol Multiphysics^© software package, β_l(ω) and F_l(x,ω) were obtained for the four core modes and the first 22 vectorial cladding modes with the highest effective refractive index, in order to solve the analytical model presented in section 3. In the numerical tool, it was considered an extra layer of air surrounding the fiber two times larger than the silica cladding.

In single core fibers, a flexural wave only enables the energy transfer between symmetric/antisymmetric and antisymmetric/symmetric modes as a result of the antisymmetric perturbation induced in the fiber core [12]. Because of the large spatial overlap between modes, such perturbation can induce the mode coupling between distinct modes supported in the fiber core, or even between the core and cladding modes [12]. In contrast, MCFs have spatial distinct cores and therefore the spatial overlap between core modes tends to be negligible. In this case, the A-O effect can only induce the energy transfer between core and cladding modes.

In MCFs, the magnitude of the refractive index perturbation induced in a specific core depends on the spatial orientation of the flexural wave and, consequently, the magnitude of the coupling coefficient between a given core and a given cladding mode. Moreover, the index perturbation induced by the flexural wave only presents the antisymmetric index profile when the vibration angle of the flexural wave is orthogonal to the given core, for the remaining values of φ the index perturbation does not show the antisymmetric index profile characteristic because the cores are shifted from the fiber center. In that way, flexural waves can induce the mode coupling between a core mode and symmetry and antisymmetric cladding modes. In Fig. 2 are showed the normalized coupling coefficients between the four core modes and a single vectorial mode of a set of symmetric, see Fig. 2(a), and antisymmetric, see Fig. 2(b), cladding modes as function of φ. For the symmetric cladding mode considered the maximum/minimum value of the coupling coefficient is achieved when the vibration plane is aligned with one pair of cores, see Fig. 2(a). In this case, the relative shift between curves is 90° because the maximum overlap is achieved over the core region. For the antisymmetric cladding mode considered in Fig. 2(b), the opposite cores have the same coupling coefficient because the antisymmetric refractive index perturbation induced by the flexural wave. Here, the optical power is mainly distributed between the cores and, thus the maximum overlap is achieved over the cladding region leading to a relative shift between curves of ∼ 60°, see Fig. 2(b). If opposite cores have antisymmetric field distribution over the core region for a given antisymmetric cladding mode, the coupling coefficients between this antisymmetric cladding mode and the cores shows four distinct curves like symmetric cladding modes.

Fig. 2 Normalized coupling coefficient between the cores and a symmetric cladding mode, (a), and an antisymmetric cladding mode, (b), at 1550 nm, as function of the angle between the vibration plane of the flexural wave and the YZ plane, φ. Inset figures show the real part of the x component of the electric field, black and white regions represent the positive and the negative values, respectively.

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In order to induced the resonant coupling between a core mode and a given cladding mode, the acoustic propagation constant of the flexural wave is calculated considering in Eq. (1a) the propagation constant of the core and the cladding mode. When the phase-matching condition between the modes is achieved, the exponent in Eq. (12) becomes null and the resonant coupling occurs efficiently. Then, the acoustic frequency for the fundamental flexural mode is obtained using the acoustic propagation constant, k, in Eq. (7) and the index perturbation induced in the MCF was calculated using Eq. (2) in which the maximum longitudinal displacement was taken as $u_{z}^{\max} = 25 nm$ [12]. In Eq. (12), we consider a flexural wave with acoustic frequency of 1282.95 kHz and an optical signal in the core-1 of the HoMCF, whereas the other considered cores and the set of degenerated cladding modes were assumed to have no optical power. The power evolution for the cores and the cladding mode as function of the fiber length and φ are displayed in Fig. 3. Figure 3(a)–3(c) show the optical power in the core-1, core-2 and core-3, respectively. Note that, the power evolution in core-4 and core-2 are similar and therefore only the power evolution in the core-2 is showed. In Fig. 3(d) we show the sum of the optical power of the two degenerated cladding modes with the second highest refractive index. In the calculations, we assume an almost perfect phase-matching condition between the core and a selected cladding mode at 1550 nm, whereas the resonant coupling only occurs between the cores and the set of degenerated cladding modes and consequently the interaction with the remaining cladding modes can be neglected. Nevertheless, in a real devices such condition may not be completely achieved due to index variations and diameter fluctuations and therefore some penalty in the extinction ratio is expected.

Fig. 3 Normalized power evolution as function of the fiber length and the angle of the acoustic wave, ϕ, assuming an optical signal launched in the core-1 of the HoMCF at 1550 nm and considering a flexural wave with a frequency of 1282.95 kHz and peak deflection of 25 nm. In Figs. 3(a)–3(c), we show the normalized power evolution for core-1, core-2 and core-3, respectively. Note that, the power evolution in core-4 and core-2 are similar and in that way only the power in the core-2 was showed. The sum of the power evolution in the two nearly degenerated cladding modes considered is shown in Fig. 3(d).

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4.1. Switching between two opposite cores

Considering in Fig. 3 an angle of φ = 45°, we can see the power conversion between the core-1 and the core-3. The switch between opposite cores can also be achieved by rotating the vibration plane of the flexural wave by 180° as a result of the antisymmetric index perturbation induced, see Figs. 3(a) and 3(c). The power evolution as function of the fiber length is showed in detail in Fig. 4(a). The power is transferred from the core-1 to the core-3 by means of the selected “intermediary” cladding mode without meaningful power transfer for the remaining cores. In order to analyze the spectral response of the core switch, we obtained the transmission spectrum at the maximum power transfer achieved between opposite cores considering all the cladding modes previously found. The transmission spectrum reflects the interaction between the core-1, core-3 and the cladding mode in the range from 1530 to 1570 nm, see Fig. 4(b). At 1550 nm, we note an attenuation band in core-1 and a transmission window in the core-3 with a full width at half maximum (FWHM) or bandwidth of 5.2 nm. In addition, the transmission bandwidth can be spectrally shifted by fine tuning the acoustic frequency applied in the HoMCF.

Fig. 4 a) and c) Power evolution in the four cores and the cladding mode of the HoMCF as function of the fiber length considering a flexural wave with peak deflection of 25 nm and acoustic frequency of 1282.95 kHz and φ = 45°and φ = 0°, respectively. b) and d) Transmission spectrum considering all the cladding modes and the parameters used in a) and b), respectively.

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4.2. Uniform power distribution

Considering in Fig. 3 an angle of φ = 0°, we can see the uniform distribution of the signal power over all the cores. This can also be achieved for more three angles by rotating the vibration plane of the flexural wave of 90°, see Fig. 3. In Fig. 4(c), we show in detail the power transfer from the core-1 to the cladding mode and then the uniform power distribution by the remaining cores as function of the fiber length. The transmission spectrum considering the uniform power distribution is showed in Fig. 4(d), where the equal power distribution is achieved at 1550 nm with a bandwidth of 5.2 nm. In this case, the spectra show several peaks at both sides of 1550 nm. However, the optical bandwidth and the additional peaks in the transmission spectra are determined by the dispersion profile of the core and the cladding mode considered and, in principle can be customized by engineering the dispersion profile of the HoMCF [10]. This technique can distribute a single signal over all the cores of an HoMCF and, therefore can be used to power optical amplifiers for MCF transmission systems using a single pump source [5].

In order to achieve the mentioned uniform power distribution in HoMCF with ring geometry (i.e. all the cores have the same distance to the center of the fiber), the coupling coefficients between all the cores and the cladding mode should take the same value [5]. Depending on the number of cores and their relative position this can be achieved with a single acoustic wave or eventually more acoustic waves could be required. For instance, in a two core fiber the uniform power distribution can be achieved by properly adjusting the vibration plane of a single acoustic wave. In a eight cores MCF with ring geometry two orthogonal flexural waves are required to uniform distributed the optical power over all the cores.

The results above presented have been obtained considering a given degenerated cladding mode, although similar results were achieved for the remaining symmetric and antisymmetric cladding modes with coupling coefficients described by Fig. 2(a). Note that, different beat length are observed because the acoustic wavelength required to induce the resonant coupling also change. We also investigate the interaction between flexural waves and cladding modes with coupling coefficients described by Fig. 2(b). Such modes leads to a more complex power evolution in which the switch between two opposite cores it is still possible by properly adjusting the φ and $u_{z}^{\max}$ . However, we can not uniform power distribute the optical power using such modes.

5. Heterogeneous multicore fibers (HeMCF)

In this section the A-O effect is investigated in heterogeneous MCFs. We assume the same geometrical parameters as stated in section 4, only the mole fraction of GeO₂ in each core is changed. The refractive index of the core-1, 2, 3 and 4, can be calculated by replacing the X parameter in Eq. (8) by X₁=0.0516, X₂=0.0525, X₃=0.0514 and X₄=0.0530, respectively. In that way, the degeneracy of the core modes can be lifted in terms of propagations constants.

5.1. Wavelength selective core attenuator

Using a single flexural wave in an HeMCF, we can induce a resonant coupling between any core and a cladding mode without energy transfer to the remaining cores. In the inset Fig. 5(a), we show the power transfer from the core-1 to a cladding mode as function of the fiber length without additional coupling to the remaining cores. Likewise in SMFs [9], an inline selective core attenuator can be produced by coupling the optical power from a given core to a cladding mode. By adjusting $u_{z}^{\max}$ , we can additionally control the optical power transfer from the core-1 to the cladding mode and, consequently, the magnitude of the attenuation band induced in such core. In Fig. 5(a), we show the energy transfer between the core-1 and the two degenerated modes of the fourth cladding mode as function of $u_{z}^{\max}$ for a fiber length of 60 mm. In Fig. 5(b), we show the transmission spectrum of the proposed attenuator considering the maximum power transfer between the selected core and the cladding mode. In the given example, the resonant coupling is achieved at 1550 nm with an attenuation bandwidth of 1.47 nm. Nevertheless, the central wavelength of the attenuation bandwidth can be spectrally shifted by fine tuning the acoustic frequency applied in the HeMCF.

Fig. 5 a) Output optical power as function of the peak deflection considering a fiber length of 60 mm. Inset figure show the power evolution as function of the fiber length for the four cores and the selected cladding mode of the HeMCF. A flexural wave with a peak deflection of 25 nm, acoustic frequency of 1283.4 kHz and φ = 180°applied over the fiber. b) Transmission spectra considering a fiber length of 0.06 m and a peak deflection of 25 nm.

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5.2. Wavelength selective core switch

The process behind the light switching between any pair of cores in HeMCF is based in a double resonant coupling induced by two flexural waves. The signal in the input core is transferred to a cladding mode by a flexural wave with frequency, Ω₁. Applying another flexural wave with proper acoustic frequency, Ω₂, the optical power in the excited cladding mode is switched to the desired output core. This resonant process can be described by a system of coupled equations considering k = 2 in Eq. (12). Such interaction is analogous to the lambda-type laser-driven atomic states with no dephasing terms for pure initial states. However, in this process the effective index (n_eff) of each core mode corresponds to the energy level of the lambda-type configuration [13].

The acoustic frequencies of the flexural waves are calculated considering the propagation constant of the cores and the selected cladding mode in Eq. (1a). In Fig. 6(a), we show the power evolution as function of the length between the core-1, the cladding mode and the core-2. The power is progressively transferred from the core-1 to the cladding mode and then to the core-2 as function of the fiber length. The optical power carried in the core-1, core-2 and the cladding mode show a small beating induced by an additional coupling between the cladding mode and the others two remaining cores, see Fig. 6(a). The switching between the core-1 and the core-3/core-4 can also be achieved by properly adjusting the Ω, $u_{z}^{\max}$ , and φ of both flexural waves. Note that, such parameters can be adjusted to achieve the same interaction length for the switching process between all the cores considered. From an analytical viewpoint, the complete power switch between two different cores is achieved when ϑ₁₋_cl ≈ ϑ_cl₋_j, with j = 2,3 and 4 [13].

Fig. 6 a) Power evolution in a four core HeMCF considering two flexural waves with $u_{z}^{\max} = 25 nm$ , φ = 0° and Ω₁ = 1282.1 and Ω₂ = 1283.4 kHz. b) Transmission spectra considering a fiber length of 85 mm.

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In Fig. 6(b) is showed the transmission spectrum for the core switch between the core-1 and core-2. A transmission window at 1550 nm width a bandwidth of 3 nm was obtained. In this case, the energy transference between the core-1 and cladding mode also occurs at others wavelengths (i.e. at 1535 and 1561 nm). However, the transmission spectra are determined by the dispersion profile of the core and the cladding mode considered and, in this sense can be improved by engineering the dispersion profile of the HeMCF [10]. The transmission spectrum considering the core switch between the core-1 and core-3/core-4 was also obtained. The acoustic frequencies required to induce the core switch between two distinct cores are presented in Table 1.

Table 1. Acoustic frequencies (kHz)

View Table

Such spectra show a bandwidth of 2.22 nm for the core switch between the core-1 and core-3, and 3.34 nm for the core switch between the core-1 and core-4 at 1550 nm. Note that, the different values obtained for the transmission bandwidth can be explained by the different dispersion profiles owned by each core.

6. Conclusions

A theoretical model addressing the signal propagation in MCFs considering the index perturbation induced by flexural acoustic waves was developed. Such model takes into account the spatial orientation of the flexural waves. The spatial orientation of the flexural waves determines the coupling coefficient between each core and a cladding mode. In a four cores homogeneous MCF with ring geometry, we show the core switch between two cores diametrically opposed. We also demonstrate that the power launched in one core can be distributed to all the cores by properly adjusting the spatial orientation of the flexural wave. The peak deflection can be used to additionally control the efficiency of power transferred between the cores. Moreover, the acoustic wave frequency can be used to selected the central wavelength of the switching band. In heterogeneous MCF, we demonstrate the in-line wavelength selective core switch between any two cores using two flexural waves. The conversion efficiency between two cores can also be tuned by adjusting the peak deflection. In addition, we proposed an in-line wavelength selective core attenuator using a single flexural wave in HeMCFs. The transmission spectrum of the core switch and the tunable core attenuator was also characterized.

Acknowledgments

This work was supported in part by the FCT – Fundação para a Ciência e a Tecnologia, by the FCT and European Union FEDERPT2020, through DiNEq project, PTDC/EEITEL/3283/2012, UID/EEA/50008/2013, Ph.D. grant SFRH/BD/102631/2014 and the postdoctoral grant SFRH/BPD/77286/2011 and SFRH/BPD/84055/2012.

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Switching in multicore fibers using flexural acoustic waves

Abstract

1. Introduction

2. Core switching operation principle

3. Theoretical model

3.1. Acoustic flexural waves in MCFs

3.2. Mode coupling theory

4. Homogeneous multicore fibers (HoMCFs)

4.1. Switching between two opposite cores

4.2. Uniform power distribution

5. Heterogeneous multicore fibers (HeMCF)

5.1. Wavelength selective core attenuator

5.2. Wavelength selective core switch

6. Conclusions

Acknowledgments

References and links

Cited By

Figures (6)

Tables (1)

Equations (20)

Optics Express