## Abstract

We report for the first time that transmission of optical pulses centered at a wavelength of 1550 nm through a tapered dual-core As_{2}Se_{3}-PMMA fiber inscribes an antisymmetric long-period grating. The pulse power is equally divided between even and odd modes that superpose along the dual-core fiber to form an antisymmetric intensity distribution. A permanent refractive-index change that matches the antisymmetric intensity distribution is inscribed due to photosensitivity at the pulse central wavelength. The evolution of the transmission spectrum of the dual-core fiber is experimentally measured as the accumulated time that the fiber is exposed to the pulse is increased. A theoretical model of an antisymmetric long-period grating in a dual-core fiber computationally reproduces the experimentally observed evolution of the transmission spectrum. Experimental results indicate that antisymmetric long-period gratings induce effective group-velocity matching between the even and odd modes of the dual-core fiber, and reveal for the first time that long-period gratings can lead to slow light propagation velocities.

© 2017 Optical Society of America

## 1. Introduction

Tapered chalcogenide-polymer fiber structures composed of an As_{2}Se_{3} core and a polymethyl methacrylate (PMMA) cladding are a promising platform for nonlinear applications because the As_{2}Se_{3} 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 As_{2}Se_{3}-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 As_{2}Se_{3}-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 As_{2}Se_{3}-PMMA fiber tapers composed of two As_{2}Se_{3} 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 As_{2}Se_{3} glass to optical signals at a wavelength of 1550 nm has been demonstrated and utilized for the inscription of fiber Bragg gratings in tapered As_{2}Se_{3}-PMMA fibers [8]. Two identical optical pulses propagating in opposite directions in the tapered As_{2}Se_{3}-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 As_{2}Se_{3}-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 As_{2}Se_{3}-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 As_{2}Se_{3}-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 As_{2}Se_{3}. 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 As_{2}Se_{3} fibers (from Coractive Inc.) with a core diameter of
${D}_{AsSe}^{co}=96\phantom{\rule{0.2em}{0ex}}\mathrm{\mu}\mathrm{m}$, a cladding diameter of
${D}_{AsSe}^{cl}=170\phantom{\rule{0.2em}{0ex}}\mathrm{\mu}\mathrm{m}$, 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 As

_{2}Se

_{3}core diameter of 12 µm, an As

_{2}Se

_{3}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 As

_{2}Se

_{3}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 As_{2}Se_{3}-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*. 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.

_{c}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 cos^{2} [*πz*/Λ] in core 1 and sin^{2} [*πz*/Λ] in core 2 where Λ is the spatial period of intensity oscillations given by Λ = 2*π*/(*β _{e}* −

*β*),

_{o}*β*,

_{e}*β*are the propagation constants of the even, odd modes, respectively, and

_{o}*z*is the propagation distance. A refractive-index change is inscribed by the optical pulse in the fiber cores due to photosensitivity of As

_{2}Se

_{3}at the pulse central wavelength,

*λ*[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.

_{c}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

*β*are respectively the electric field amplitude and the propagation constant of mode

_{m}*m*, ${\beta}_{m,0}={\beta}_{m}{|}_{\lambda ={\lambda}_{0}}$ with

*λ*

_{0}being the carrier wavelength of an optical signal,

*q*is an integer, Λ is the grating period,

*κ*,

_{o}*and*

_{e}*κ*,

_{e}*are the coupling coefficients with ${\kappa}_{e,o}={\kappa}_{o,e}^{\ast}$. The coupling coefficients are calculated using*

_{o}*ε*

_{0}is the permittivity of free space,

*ω*

_{0}= 2

*πc*/

*λ*

_{0}is the optical angular frequency,

*c*is the speed of light in free space, ${N}_{m}=0.5{\displaystyle \iint \left({\overrightarrow{F}}_{m}\times {\overrightarrow{G}}_{m}^{\ast}\right)\cdot \widehat{z}\partial x\partial y}$ is the field normalization factor,

*n*is the refractive-index, ${\overrightarrow{F}}_{m}$, ${\overrightarrow{G}}_{m}$ are respectively the electric and magnetic field distributions of mode

*m*,

*a*is the amplitude of the refractive-index change,

_{q}*u*(

_{j}*x*,

*y*) equals 1 in core

*j*and 0 elsewhere.

Optimal coupling between the even and odd modes occurrs at a grating resonance-wavelength *λ _{r}* defined as the wavelength

*λ*

_{0}at which the phase-matching condition

*q*2

*π*/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}*where ${\beta}_{m,c}={\beta}_{m}{|}_{\lambda ={\lambda}_{c}}$, then Λ = 2*

_{c}*π*/(

*β*,

_{e}*−*

_{c}*β*,

_{o}*) and*

_{c}*λ*=

_{r}*λ*. Using

_{c}*q*= 1 and Λ = 2

*π*/(

*β*,

_{e}*−*

_{r}*β*,

_{o}*) with ${\beta}_{m,r}={\beta}_{m}{|}_{\lambda ={\lambda}_{r}}$, Eqs. 1 and 2 reduce to a system of ordinary differential equations*

_{r}*E*

_{1}(

*L*) = [

*E*(

_{e}*L*) +

*E*(

_{o}*L*)]/2, the output of core 2 is given by

*E*

_{2}(

*L*) = [

*E*(

_{e}*L*) −

*E*(

_{o}*L*)]/2, and the transmission of the dual-core fiber is given by

*T*= |

*E*

_{1}(

*L*)|

^{2}/|

*E*

_{1}(0)|

^{2}, where

*E*(

_{m}*z*) = ${\tilde{A}}_{m}$ (

*z*) exp (

*iβ*,0

_{m}*z*) for mode

*m*and

*L*is the length of the dual-core fiber.

A scalar field-correction method [13, 14] is utilized to calculate *F _{e}*,

*F*,

_{o}*β*, and

_{e}*β*for a dual-core As

_{o}_{2}Se

_{3}-PMMA fiber with 1.5 µm As

_{2}Se

_{3}cores, and a core separation of 1.25 µm. A core separation of 1.25 µm is used in the numerical calculations because the As

_{2}Se

_{3}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*, then

_{o}*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*are closely fitted by

_{q}*β*(

_{e}*λ*) = −7.4439 × 10

^{12}

*λ*+ 2.2609 × 10

^{7},

*β*(

_{o}*λ*) = −7.5341 × 10

^{12}

*λ*+ 2.2623 × 10

^{7}, and

*κ*,

_{e}*(*

_{o}*λ*)/

*a*= −2.4489 × 10

_{q}^{12}

*λ*+ 7.6594 × 10

^{6}. The grating period Λ of the grating is determined by

*λ*and is given by Λ = 2

_{c}*π*/[

*β*(

_{e}*λc*) −

*β*(

_{o}*λ*)] = 50 µm. Figure 3 presents the calculated transmission spectra of the dual-core fiber with an antisymmetric long-period grating as

_{c}*a*is increased from 0 to 4.0 × 10

_{q}^{−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*increases as the exposure time

_{q}*t*increases.

_{c}The difference between the phases of the even and odd modes Δ*ϕ* = *ϕ _{e}* −

*ϕ*, where

_{o}*ϕ*is the phase of the electric field in mode

_{m}*m*, is deduced from the transmission spectra in Fig 1(c). The phase-difference Δ

*ϕ*is a function of both the wavelength

*λ*and the cumulative exposure time

*t*, and is given by Δ

_{c}*ϕ*(

*λ*,

*t*) = (2

_{c}*p*+ 1)

*π*at the minima where

*p*is an integer. As the exposure time is increased by Δ

*t*, the values of Δ

_{c}*ϕ*corresponding to the minima on the transmission spectrum do not change leading to (∂Δ

*ϕ*/∂

*λ*) Δ

*λ*+ (∂Δ

*ϕ*/∂

*t*) Δ

_{c}*t*= 0. The minima at

_{c}*λ*<

*λ*shift towards longer wavelengths as

_{r}*t*increases corresponding to Δ

_{c}*λ*> 0, hence, ∂Δ

*ϕ*/∂

*λ*and ∂Δ

*ϕ*/∂

*t*have opposite signs. Simulations of the dual-core fiber show that ∂Δ

_{c}*ϕ*/∂

*λ*> 0 leading to ∂Δ

*ϕ*/∂

*t*< 0, which indicates that Δ

_{c}*ϕ*decreases as

*t*increases for

_{c}*λ*<

*λ*. Similarly, the minima at

_{r}*λ*>

*λ*shift towards shorter wavelengths corresponding to Δ

_{r}*λ*< 0 as

*t*increases, which indicates that Δ

_{c}*ϕ*increases as

*t*increases for

_{c}*λ*>

*λ*. Finally, the transmission spectra have slower oscillations near

_{r}*λ*which indicates that Δ

_{r}*ϕ*has a slower variation with

*λ*. Figure 4 presents Δ

*ϕ*as a function of

*λ*which is deduced from the information that Δ

*ϕ*decreases for

*λ*<

*λ*, increases for

_{r}*λ*>

*λ*, and has slow variation near

_{r}*λ*.

_{r}The illustration in Fig. 4 shows that Δ*ϕ* (*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 Δ*ϕ* at *λ _{r}* in Fig. 4 implies that

*ϕ*and

_{e}*ϕ*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, As

_{o}_{2}Se

_{3}-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 As_{2}Se_{3}-PMMA fiber. Propagation of optical pulses centered at wavelength of 1550 nm in a dual-core As_{2}Se_{3}-PMMA fiber leads to the inscription of an antisymmetric long-period grating on the tapered fiber due to photosensitivity of As_{2}Se_{3} 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.

## Funding

NSERC Discovery Grant (06071-RGPIN-2015); Canada Research Chair Program (CRC in Fiber Optics and Photonics)(75-67138).

## Acknowledgments

The authors are thankful to Coractive Inc. for providing the As_{2}Se_{3} multimode fiber that is utilized in the fabrication of our dual-core As_{2}Se_{3}-PMMA fibers.

## References and links

**1. **C. Baker and M. Rochette, “Highly nonlinear hybrid AsSe-PMMA microtapers,” Opt. Express **18**, 12391–12398 (2010). [CrossRef] [PubMed]

**2. **R. Ahmad and M. Rochette, “High efficiency and ultra broadband optical parametric four-wave mixing in chalcogenide-pmma hybrid microwires,” Opt. Express **20**, 9572–9580 (2012). [CrossRef] [PubMed]

**3. **C. Baker and M. Rochette, “High nonlinearity and single-mode transmission in tapered multi-mode As2Se3-PMMA fibers,” J. IEEE Photon. **4**, 960–969 (2012). [CrossRef]

**4. **A. Al-Kadry, M. E. Amraoui, Y. Messaddeq, and M. Rochette, “Mode-locked fiber laser based on chalcogenide microwires,” Opt. Lett. **40**, 4309–4312 (2015). [CrossRef] [PubMed]

**5. **J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A **42**, 682–685 (1990). [CrossRef] [PubMed]

**6. **S. Jensen, “The nonlinear coherent coupler,” J. Quantum Electron. **18**, 1580–1583 (1982). [CrossRef]

**7. **F. Setzpfandt, A. S. Solntsev, J. Titchener, C. W. Wu, C. Xiong, R. Schiek, T. Pertsch, D. N. Neshev, and A. A. Sukhorukov, “Tunable generation of entangled photons in a nonlinear directional coupler,” Laser Photon. Rev. **10**, 131–136 (2016). [CrossRef]

**8. **R. Ahmad and M. Rochette, “Photosensitivity at 1550 nm and bragg grating inscription in As_{2}Se_{3} chalcogenide microwires,” App. Phys. Lett. **99**, 061109 (2011). [CrossRef]

**9. **F. Bilodeau, K. O. Hill, D. C. Johnson, and S. Faucher, “Compact, low-loss, fused biconical taper couplers: over-coupled operation and antisymmetric supermode cutoff,” Opt. Lett. **12**, 634–636 (1987). [CrossRef] [PubMed]

**10. **T. A. Birks and Y. W. Li, “The shape of fiber tapers,” J. Lightwave Technol. **10**, 432–438 (1992). [CrossRef]

**11. **C. Baker and M. Rochette, “A generalized heat-brush approach for precise control of the waist profile in fiber tapers,” Opt. Mater. Express **1**, 1065–1076 (2011). [CrossRef]

**12. **A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. **62**, 1267–1277 (1972). [CrossRef]

**13. **F. Gonthier, S. Lacroix, and J. Bures, “Numerical calculations of modes of optical waveguides with two-dimensional refractive index profiles by a field correction method,” Opt. Quantum Electron. **26**, S135–S149 (1994). [CrossRef]

**14. **S. Lacroix, F. Gonthier, and J. Bures, “Modeling of symmetric 2 × 2 fused-fiber couplers,” Appl. Opt. **33**, 8361–8369 (1994). [CrossRef] [PubMed]

**15. **R. W. Boyd, “Material slow light and structural slow light: similarities and differences for nonlinear optics,” J. Opt. Soc. Am. B **28**, A38–A44 (2011). [CrossRef]