1. Introduction

Tapered chalcogenide-polymer fiber structures composed of an As₂Se₃ core and a polymethyl methacrylate (PMMA) cladding are a promising platform for nonlinear applications because the As₂Se₃ core provides high nonlinearity over the near- and mid-infrared spectral ranges for compact nonlinear devices with low power consumption and the PMMA cladding provides high mechanical robustness for easy handling. Indeed, tapered As₂Se₃-PMMA fibers have been used in a variety of practical applications including super-continuum generation [1], broadband parametric amplification [2], polarization switching [3], and laser pulse generation [4]. Advanced As₂Se₃-PMMA fiber structures such as dual-core and birefringent elliptical-core fibers enable novel all-optical signal processing devices for a broader set of applications. Dual-core As₂Se₃-PMMA fiber tapers composed of two As₂Se₃ cores and a PMMA cladding are especially promising because they support guided propagation of two main modes, an even mode and an odd mode, allowing for advanced nonlinear applications such as modulation instability in both the normal and anomalous dispersion regimes [5], pulse self-switching [6], and phase-matched four-wave mixing [7].

Photosensitivity of As₂Se₃ glass to optical signals at a wavelength of 1550 nm has been demonstrated and utilized for the inscription of fiber Bragg gratings in tapered As₂Se₃-PMMA fibers [8]. Two identical optical pulses propagating in opposite directions in the tapered As₂Se₃-PMMA fiber form a standing wave which inscribes a longitudinal periodic refractive-index variation leading to the formation of a fiber Bragg grating. The resonance wavelength of the resulting fiber Bragg-grating coincides with the central wavelength of the pulse that is used for grating inscription, which allows for changing the resonance wavelength by tuning the pulse central wavelength. Furthermore, these gratings are inscribed over the entire As₂Se₃-PMMA taper waist allowing for the inscription of long fiber Bragg gratings.

In this paper, we report for the first time that the propagation of optical pulses in a dual-core As₂Se₃-PMMA fiber inscribes an antisymmetric long-period grating with resonance at the central wavelength of the propagating pulses. Experimentally measured transmission spectra of the dual-core As₂Se₃-PMMA fiber show that propagation of optical pulses centered at a wavelength of 1550 nm causes the formation of an antisymmetric long-period grating due to photosensitivity of As₂Se₃. A theoretical model is then developed to computationally reproduce experimental measurements confirming the formation of an antisymmetric long-period grating in the dual-core fiber. The experimentally measured transmission spectra are analyzed to deduce the effect of the antisymmetric long-period grating on the difference between the phases of the even and odd modes of the dual-core fiber. The deduced phase-difference graph indicates that the antisymmetric long-period grating induces effective group-velocity matching between the even and odd modes, and can potentially lead to slow light propagation velocities.

2. Experiment and results

Two multimode step-index As₂Se₃ fibers (from Coractive Inc.) with a core diameter of $D_{AsSe}^{co}=96\mathrm{\mu }\mathrm{m}$, a cladding diameter of $D_{AsSe}^{cl}=170\mathrm{\mu }\mathrm{m}$, and a numerical aperture of N A_AsSe = 0.18 are coated by a PMMA layer with an outer diameter of ~ 9.5 mm to obtain a dual-core fiber preform. The preform is then drawn into a fiber with an As₂Se₃ core diameter of 12 µm, an As₂Se₃ cladding diameter of 21.25 µm and an outer PMMA cladding diameter of ~ 1.2 mm. A fiber piece with a length of 7 cm is cut, both ends are polished, the input and the output of core 1 are butt-coupled to standard single-mode silica fibers, and UV-cured epoxy permanently fixes the butt-coupling interfaces between the dual-core fiber and the silica fibers. The dual-core fiber is then tapered using the heat-brush method [9–11] to obtain a microwire with a 1.5 µm diameter for each As₂Se₃ core, an 84.7 µm diameter PMMA cladding, and a 10 cm long waist.

Figure 1(a) presents the setup used for the inscription and characterization of an antisymmetric long-period grating in the tapered dual-core As₂Se₃-PMMA fiber. Optical pulses with a width of 12 ps centered at a wavelength of λ_c = 1550.3 nm are generated using a laser source (Pritel FFL-1550-20). An electro-optic intensity modulator (Photline MXER-LN-10) that is driven with an electrical square pulse with a duration of 100 µs and repetition rate of 1 MHz reduces the average power of the optical pulses by 10 dB to avoid melting the tapered dual-core fiber. The optical pulses pass through a linear polarizer into core-1 of the tapered dual-core fiber to inscribe a permanent refractive-index change by photosensitivity at λ_c. Broadband amplified spontaneous emission noise from an Erbium-doped fiber amplifier (EDFA) is also passed through the linear polarizer to obtain polarized broadband light. The polarized broadband light is launched into core-1 of the dual-core fiber and the output light from core-1 is passed to an optical spectrum analyzer to obtain the transmission spectrum of the tapered fiber. A polarization controller aligns the polarized broadband light and the optical pulses with one principal polarization axis of the tapered dual-core fiber.

 

Fig. 1 a) Schematic of the setup for inscription and characterization of an antisymmetric grating in a tapered dual-core As₂Se₃-PMMA fiber. b) Initial growth of the transmission spectrum as the cumulative exposure time is increased from 0 s to 50 s in steps of 10 s. c) Evolution of the transmission spectrum as the cumulative exposure time is increased from 10 s to 610 s in steps of 20 s. EDFA: Erbium-doped fiber amplifier, LP: linear polarizer, PC: polarization Controller, OSA: optical spectrum analyzer, FUT: fiber under test.

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The EDFA is switched OFF and the pulsed laser is switched ON to induce refractive-index change by photosensitivity in the tapered dual-core fiber. After 10 seconds, the pulsed laser is switched OFF, the EDFA is switched ON, and the transmission spectrum of the tapered dual-core fiber is measured using the optical spectrum analyzer. The procedure of pulse exposure and transmission measurement is repeated for exposure durations of 10 s to obtain the transmission spectra of the tapered dual-core fiber as the cumulative exposure duration increases. Figure 1(b) presents the measured transmission spectra as the cumulative exposure duration is increased from 0 s to 50 s in steps of 10 s. To get further insight into the changes induced on the tapered dual-core fiber, Fig. 1(c) presents the measured transmission spectra as the cumulative exposure duration is increased from 10 s to 610 s in steps of 20 s.

3. Discussion

The changes in the transmission spectrum observed in Figs. 1(b) and 1(c) occur due to the formation of an antisymmetric long-period grating in the tapered dual-core fiber. When an optical pulse is launched into core 1 of the dual-core fiber, the pulse power is split equally between the even and odd modes. These two modes superpose along the tapered dual-core fiber to form an antisymmetric periodic spatial power distribution as illustrated in Fig. 2. The intensity is proportional to cos² [πz/Λ] in core 1 and sin² [πz/Λ] in core 2 where Λ is the spatial period of intensity oscillations given by Λ = 2π/(β_e − β_o), β_e, β_o are the propagation constants of the even, odd modes, respectively, and z is the propagation distance. A refractive-index change is inscribed by the optical pulse in the fiber cores due to photosensitivity of As₂Se₃ at the pulse central wavelength, λ_c [8]. The inscribed refractive-index change is proportional to the pulse power, and hence, the antisymmetric spatial power distribution leads to an antisymmetric grating with a period Λ in the tapered dual-core fiber.

 

Fig. 2 Illustration of spatial power distribution in a dual-core fiber when light is launched into core 1.

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Calculation of the transmission spectrum confirms that an antisymmetric long-period grating is formed in the tapered dual-core fiber. The propagation equations for the field amplitudes of the even and odd modes in a dual-core fiber with an antisymmetric long-period grating are derived using a perturbation analysis by reciprocity theorem [12] leading to

Optimal coupling between the even and odd modes occurrs at a grating resonance-wavelength λ_r defined as the wavelength λ₀ at which the phase-matching condition q2π/Λ = β_e,₀ − β_o,₀ is satisfied. When the pulse utilized for inscribiong the long-period grating has a relatively narrow spectral-width such that β_m = β_m,_c where ${\beta }_{m,c}={\beta }_m{|}_{\lambda ={\lambda }_c}$, then Λ = 2π/(β_e,_c − β_o,_c) and λ_r = λ_c. Using q = 1 and Λ = 2π/(β_e,_r − β_o,_r) with ${\beta }_{m,r}={\beta }_m{|}_{\lambda ={\lambda }_r}$, Eqs. 1 and 2 reduce to a system of ordinary differential equations

A scalar field-correction method [13, 14] is utilized to calculate F_e, F_o, β_e, and β_o for a dual-core As₂Se₃-PMMA fiber with 1.5 µm As₂Se₃ cores, and a core separation of 1.25 µm. A core separation of 1.25 µm is used in the numerical calculations because the As₂Se₃ cores of the tapered dual-core fiber slightly fuse during the fabrication process [14]. The value of κ_e,_o (λ) is calculated using a_q,F_e, F_o, then κ_e,_o (λ), β_e (λ) and β_o (λ) are utilized to calculate the transmission of the dual-core fiber that is inscribed with an antisymmetric long-period grating. The numerically calculated values of β_e (λ), β_o (λ), κ_e,_o (λ)/a_q are closely fitted by β_e (λ) = −7.4439 × 10¹²λ + 2.2609 × 10⁷, β_o (λ) = −7.5341 × 10¹²λ + 2.2623 × 10⁷, and κ_e,_o (λ)/a_q = −2.4489 × 10¹² λ + 7.6594 × 10⁶. The grating period Λ of the grating is determined by λ_c and is given by Λ = 2π/[β_e (λc) − β_o (λ_c)] = 50 µm. Figure 3 presents the calculated transmission spectra of the dual-core fiber with an antisymmetric long-period grating as a_q is increased from 0 to 4.0 × 10⁻⁵ in steps of 1.33 × 10⁻⁶. The calculated transmission spectra in Fig. 3 show similar progression behavior to that of the experimentally observed spectra in Fig 1(c) which confirms the formation of an antisymmetric long-period grating and indicates that the magnitude of the refractive-index change a_q increases as the exposure time t_c increases.

 

Fig. 3 Evolution of the theoretically calculated transmission spectrum of core 1 of a dual-core fiber with an antisymmetric long-period grating as the amplitude of the refractive-index change a_q is increased from 0 to 4.0 × 10⁻⁵ in steps of 1.33 × 10⁻⁶.

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The difference between the phases of the even and odd modes Δϕ = ϕ_e − ϕ_o, where ϕ_m is the phase of the electric field in mode m, is deduced from the transmission spectra in Fig 1(c). The phase-difference Δϕ is a function of both the wavelength λ and the cumulative exposure time t_c, and is given by Δϕ (λ, t_c) = (2p + 1) π at the minima where p is an integer. As the exposure time is increased by Δt_c, the values of Δϕ corresponding to the minima on the transmission spectrum do not change leading to (∂Δϕ/∂λ) Δλ + (∂Δϕ/∂t_c) Δt_c = 0. The minima at λ < λ_r shift towards longer wavelengths as t_c increases corresponding to Δλ > 0, hence, ∂Δϕ/∂λ and ∂Δϕ/∂t_c have opposite signs. Simulations of the dual-core fiber show that ∂Δϕ/∂λ > 0 leading to ∂Δϕ/∂t_c < 0, which indicates that Δϕ decreases as t_c increases for λ < λ_r. Similarly, the minima at λ > λ_r shift towards shorter wavelengths corresponding to Δλ < 0 as t_c increases, which indicates that Δϕ increases as t_c increases for λ > λ_r. Finally, the transmission spectra have slower oscillations near λ_r which indicates that Δϕ has a slower variation with λ. Figure 4 presents Δϕ as a function of λ which is deduced from the information that Δϕ decreases for λ < λ_r, increases for λ > λ_r, and has slow variation near λ_r.

 

Fig. 4 Illustration of the difference between the phases of even and odd modes before and after inscription of the antisymmetric long-period grating.

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The illustration in Fig. 4 shows that Δϕ (λ) has slow variation near λ_r indicating effective group-velocity matching. The dual-core fiber is multimode as it supports an even mode and an odd mode. The antisymmetric long-period grating causes the electric fields to couple back and forth between the even and odd modes, and hence, the fields travel half the propagation distance in the even mode and the other half in the odd mode leading to effective group-velocity matching. Group-velocity matching can be used for enhanced nonlinear processing of subpicosecond pulses due to a short pulse walk-off, and for increased efficiency of phase-matched four-wave mixing.

Furthermore, fast variation of Δϕ at λ_r in Fig. 4 implies that ϕ_e and ϕ_o have fast variations with wavelength indicating the potential for inducing a slow light propagation velocity. This is the first time to our knowledge that long-period gratings are shown to have the potential for achieving slow light. This slow light feature can be utilized for the implementation of highly sensitive devices for the measurement of temperature and refractive-index change of a liquid solution. Moreover, As₂Se₃-PMMA fiber tapers are highly nonlinear [1,3], and the introduction of slow light will further enhance the waveguide nonlinearity parameter [15]. Finally, antisymmetric long-period gratings can be inscribed in long dual-core fibers allowing for long propagation delays.

4. Conclusion

We report the first observation of self-inscribed antisymmetric long-period grating in a tapered dual-core As₂Se₃-PMMA fiber. Propagation of optical pulses centered at wavelength of 1550 nm in a dual-core As₂Se₃-PMMA fiber leads to the inscription of an antisymmetric long-period grating on the tapered fiber due to photosensitivity of As₂Se₃ at 1550 nm. Experimental results show that antisymmetric long-period gratings in dual-core fibers can be used to achieve group-velocity matching for nonlinear processing of sub-picosecond optical pulses. Experimental results also show the potential for achieving slow light velocity in dual-core fibers with antisymmetric long-period gratings.