1. Introduction

For the last few decades, scientists and engineers have studied nonlinear optical phenomena in photonic crystal fibers because of the ability to manipulate the core area and total dispersion [1]. The propagation of an optical pulse in a standard fiber has been thoroughly investigated to study the effects of polarization, dispersion, and Kerr nonlinearity [1]. Fibers with a single core have been extensively used in various fields like telecommunication and medical fields [2–5]. However, in recent years, with the demands of high speed and high capacity optical systems, fibers with well-spaced multi-cores have emerged as an essential long-distance transmission medium for optical pulses [6]. Several recent research articles show that multicore fiber is emerging as a strong candidate for ultra-high capacity transmission over a long-haul distance in future optical transmission systems [7,8]. It has been shown that multicore fibers are advantageous for use in couplers, operating at higher output power compared with single-core fiber with sharp switching characteristics and greater sensitivity to the input state [9,10]. They have potential applications in the spatial division multiplexing by providing parallel channels for individual signals and thus improving the system capacity [11]. Each of the generated super-modes in multi-core fiber acts as one spatial channel.

In this letter, we report on linear and nonlinear optics, including phase matched four-wave mixing (FWM) [12], in a 3-core micro-structure fiber for a wavelength of 1.064$\; \mu m$ and for peak powers ranging up to $0.6\; kW$. We find that, by adjusting the pump power and physical length of the fiber, we can observe a transition where FWM happens primarily in a single core to the case where FWM happens in multiple cores with an effective nonlinear coefficient reduced by a factor of 3 (the number of cores). The reduction in the effective nonlinear coefficient leads to a three-fold reduction in the FWM bandwidth. We present measurements of the core-to-core coupling length, including wavelength and polarization-dependence of the coupling and the effective nonlinearity as a function of power. These are the first experimental measurements of FWM in a 3-core microstructure fiber raising the possibility of scaling up the power of microstructure fiber parametric amplifiers and oscillators by distributing the energy among the multiple cores while maintaining the phase matching afforded by tailoring the fiber’s microstructure. This study provides essential guidance for using multi-core fibers as parametric devices.

In a multi-core fiber, there is a complicated interplay between the linear and nonlinear effects as energy couples among the various cores [13]. Linear and non-linear optics in fiber relates to the intensity-independent and intensity-dependent process respectively, and both linear and non-linear effects are affected by the fiber dimensions and pulse parameters [14]. Similarly configured triple-core fibers have been studied in Yan, et al., [15,16] where they characterized linear and nonlinear periodic coupling among the cores. We build upon these previous studies to include phase-matched FWM.

2. Experimental methods

Figure 1(a) shows the fiber under test, which is a microstructure fiber having 3-cores arranged in a line. Each of the 3-cores has a diameter of 3.90 μm surrounded by a triangular lattice of air holes with diameters of 1.08 μm. The hole-to-hole spacing (Λ) is 2.52 μm and the separation between the cores is 4Λ. The fiber length ranged from 35 cm – 52 m, depending on the measurement. Laser light is launched into one of the cores by careful alignment of a 3-axis translation stage and the outputs from each core are characterized. The output from each core can be observed with a CCD camera (see Fig. 1(b)) or individually selected using a standard single-mode fiber aligned with a single core for power or spectral measurements as shown in Fig. 2. For most of the measurements we use a fixed-wavelength mode-locked laser (Time-Bandwidth, Nd-VAN) delivering 8-ps pulses at 1064 nm. Wavelength-dependent measurements are made using a continuous-wave tunable laser (Toptica DL-Pro) having a power less than 40 mW. Figure 1(b) shows, for example, the 1064-nm, 8-ps beams emerging from the three cores of a 52-m long fiber where only a single side core was excited.

 

Fig. 1. (a) The cross-section of the fiber with 3-cores of diameter 3.90 μm in a linear configuration arranged in a triangular lattice of air holes with a diameter of 1.08 μm. The hole to hole spacing (${\Lambda }$) is 2.52 μm and the distance of separation between the cores is 4${\; \Lambda }$. (b) Intensity distribution for a long (52 m) piece where coupling among the cores is evident. (c) Coupling as a function of power for a 4.9-m long fiber. The coupling among the cores is independent of power until spectral broadening becomes large. Insets show spectral broadening due to self-phase modulation and FWM. (d) Mode coupling in a 3-core fiber [17]. κ_ij represents the coupling coefficient between cores i and j.

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Fig. 2. Schematic of experimental setup. $\lambda /2$: half wave plate; ISO: Isolator; M1-M2: mirrors; PBS: polarizing beam splitter; MCF: multi-core fiber; det. fiber: single core fiber for power measurements; PS: power sensor; OSA: optical spectrum analyzer.

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3. Results

In Fig. 1(b) the laser energy is distributed among the 3 cores in a manner consistent with propagation over a distance greater than the fiber’s coupling length, ${L_c}$, for 1064 nm. In this case the energy is roughly equally distributed among the three cores regardless of the core into which the light was initially launched. The mode shape emerging from each core is robust to movements and bending of the fiber although the energy in each core does slightly. Measurements of the core-to-core coupling using a 200 fs-pulsed, 1550 nm laser exhibit the similar stability and coupling behavior.

Figure 1(c) shows plots of the normalized output power as a function of input power for coupling from the center core, C2, to each of the side cores, C1 and C3. In Fig. 1(c), we observe that the coupling coefficients between the central core and each of the side cores (as depicted in Fig. 1(d)) are not equivalent, ${\kappa _{12}} \ne {\kappa _{23}}$. For the 4.9-m long fiber, the coupling between the central core and the two side cores are about 10% and 20% with an average of 15% when measured at low powers. Assuming the coupling fraction to be $T = {\sin ^2}({\kappa L} )$, and the coupling length to be ${L_c} = \frac{\pi }{{2\kappa }}$ we obtain an average coupling length of ${L_c} = 20\; m$ for a wavelength of 1.064 μm. Observations of the coupling between cores for fiber lengths of 35 cm, 4.9 m, and 47 m are consistent with ${L_c} = 20\; m$. Our observations of core-to-core coupling during low-power, linear propagation are consistent with those shown for in a similar triple-core PCF in [14]. In the case of [15], they also observed asymmetric coupling. We do, however, observe a different power-dependence than that observed in [15]. We find in our measurements that the coupling increases with power. The inset in Fig. 1(c) shows that the change in coupling is likely caused by spectral broadening when the power increases beyond 20W. Further discussion of this observation is given later in the paper.

The nonlinear-optical response of the fiber is shown in Fig. 3 where we present the spectrum emerging from the center core when the pump is launched into the center core. The spectra emerging from the side cores under the same launch conditions are similar in width and shape. The spectra for both the short (4.9 m in Fig. 3(a)) and long (47 m in Fig. 3(b)) show the broadening of the spectrum of the pulse train due to self-phase modulation and phase matched FWM. In both cases the overall width of the spectrum increases dramatically as a function of power, but we easily recognize the growth of FWM peaks at about 1049 nm and 1079 nm in the 4.9-m case. Cascaded FWM peaks arise as the peak power exceeds 220 W, and a continuum follows for higher powers. In Fig. 3(b) it is difficult to discern the FWM peaks for the 47-m long fiber, but broadening into a continuum is readily observed.

 

Fig. 3. (a) Spectra showing spontaneous FWM in the 4.9-m long fiber where the phase-matched peaks arise at 1049 nm and 1079 nm. (b) shows the same measurement for a 47-m long fiber. The first-order FWM peaks are located at 1061 nm and 1067 nm.

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The dominance of linear vs. nonlinear effects during pulse propagation through a fiber is determined by comparing the fiber’s physical length, L, dispersion length, ${L_D} = \frac{{2\pi c}}{{{\lambda ^2}}}\frac{{T_0^2}}{{|D |}}$, and nonlinear length, ${L_{NL}} = \frac{1}{{\gamma P}}$, where ${T_0}$ is the input pulse duration, $\lambda $ is the wavelength, and $D \approx 12.5\; ps/({nm\; km} )\; $ quantifies the group velocity dispersion. For our pulses of $\lambda = 1.064\; \mu m$ and ${T_0} = 8\; ps$, the dispersion length is $8\; \pm 1\; km$, and so dispersive pulse broadening plays a negligible role here.

The interplay between nonlinearity and core-to-core coupling over the fiber’s physical length plays an important role for this 3-core fiber. If the launched pump power is sufficiently high, the nonlinear length is less than the coupling length (${L_{NL}} < {L_c}$), and the spectrum is primarily determined by propagation through a single core followed by coupling to the other cores. The effective nonlinear coefficient is given by that for a single core ($\gamma = \frac{{2\pi {n_2}}}{{\lambda {A_{eff}}}}$) where ${n_2}$ is the nonlinear refractive index and ${A_{eff}}$ is the effective cross-sectional area of a single core. If, on the other hand, the launched pump power is low such that the coupling length is less than the nonlinear length, (${L_c} < {L_{NL}}$), the effective nonlinear coefficient is reduced by a factor between 2 and 3 as the energy is split among the 3 cores. The reduction in $\gamma $ could be beneficial in terms of allowing parametric devices to operate at higher power, but it has huge consequences as it reduces the FWM phase matching bandwidth.

Figure 4 shows the measured optical spectrum (for constant ${P_0}L\sim 1\; kW \cdot m$) at the onset of spontaneous FWM plotted along with the calculated gain spectrum [18,19]. In Fig. 4(a) the gain is calculated for the experimental pump power for reasonable choices of the group velocity dispersion, D, and nonlinear coefficient, $\gamma $. The peak of the gain matches with the peak of the FWM data. In Fig. 4(b), we choose D to remain the same because we do not expect the overall dispersion to change with core-to-core coupling. Since the fiber’s physical length is now longer than ${L_c}$, we expect to be able to observe FWM for the case where the nonlinear length extends beyond ${L_c}$. We replace $\gamma $ by $\gamma /3$ and the calculated curve now produces narrow peaks which match the data well. The key hypothesis formed from the previous analysis is that when FWM occurs in multiple cores, $\gamma $ is effectively reduced by a factor of 3. Note that since we have fixed ${P_0}L$ in Fig. 4, we find that changing the power is not sufficient to explain the peak locations in the observed spectra. The effective nonlinearity, $\gamma $, must change as well.

 

Fig. 4. FWM gain. (a) short fiber piece (4.9 m) (b) long fiber piece (47 m). The estimated values of $\gamma $ are 14 and 4.67 ${W^{ - 1}}k{m^{ - 1}}$ for short and long fiber pieces respectively.

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We confirm our hypothesis about the interplay between γ, ${P_0}$ and L by measuring the spectral broadening due to self-phase modulation (SPM) of the pump pulses, as shown in Fig. 5. An estimate of $\gamma $ is obtained [1] from the slope of width vs. power graph according to $\gamma = 1.16\; slope/\mathrm{\Delta }{\lambda _0}L$, where $\mathrm{\Delta }{\lambda _0}$ is the spectral width of the unbroadened pump, and L is the fiber length. Figure 5 shows an evaluation of the width of the central peak as a function of the peak power, wherein the 6-dB width was used to mitigate the effects of noise near the peak of the spectra. Light is launched into the central core and the output of each of the 3 cores is recorded.

 

Fig. 5. Spectral broadening measurement as a function of input power when central core is excited. (a) for short fiber (4.9 m) and (b) for long fiber (47 m). Straight line (red) is the linear fit for linear and nonlinear power regimes in (a) and (b). The slope of the line in nonlinear power regime is 0.04 (for (a)) and 0.12 (for(b)). The value of $\mathrm{\Delta }{\lambda _0}$ is 0.80 . We observed that the spectral broadening is solely due to the fiber SPM parameter and is significant above 100 W.

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In Fig. 5(a) we see that the spectrum broadens in all 3 cores as expected in a linear fashion below about 100 W with the energy splitting to some extent among the cores. The rate of SPM induced broadening (the slope of the curve) increases in the central core when power exceeds about 100 W. Above 100 W, ${L_{NL}} < {L_c}$ and the slope is determined by SPM arising from propagation in a single core leading to $\gamma \approx 14\; \pm 1\; {({W\; km} )^{ - 1}}$. Below 100 W, ${L_c} < {L_{NL}}$ and the result is $\gamma \approx 5\; \pm 1\; {({W\; km} )^{ - 1}}$. Light emerging from the other two side cores also exhibits the relatively small $\gamma $.

Figure 5(b) illustrates the same SPM broadening experiment performed on a 47-m long fiber. In this case the physical fiber length is greater than the coupling length, and ${L_c} < {L_{NL}}$ and we again obtain the result of $\gamma \approx 5\; \pm 1\; {({W\; km} )^{ - 1}}$. As the peak power is increased from 6 W to 30 W, SPM combined with FWM leads to an increasing number of peaks in the spectrum. Specifying broadening due to SPM for powers larger than about 30W is obscured by FWM peaks which arise close to the pump’s spectral envelope.

The observed FWM efficiency fluctuates by ∼20% as the fiber moves. To understand this better we investigated the core-to-core coupling as a function of polarization and wavelength. We launched low-power laser light into a single core and measured the output power emerging from each of the cores. The input peak power is 40 mW and the fiber length is 4.9 m. In Fig. 6(a) we again observe that the core-to-core coupling coefficients between the central core and each of the side cores are not equivalent $({\kappa _{12}} \ne {\kappa _{23}}$) and that the coupling varies by about 50% as a function of the polarization of the launched light. The FWM spectra also fluctuate as the polarization state of the launched light changes. Core-to-core coupling is dependent on wavelength as shown in Fig. 6(b). A wavelength-tunable CW laser is launched into a single core and we measure the power emerging from each core. The coupling is seen to vary from about 15% to as much as 50% depending on wavelength. Consequently, even a moderate amount of spectral broadening will change the resulting core-to-core coupling. For the fiber under test, SPM and FWM followed by wavelength-dependent coupling explains the increase in coupling out of the launched core which is observed above 50 W as seen in Fig. 1(c) and the inset.

 

Fig. 6. (a) Plots of the relative output power emerging from each core as a function of half-wave plate (HWP) angle for 1064 nm pulses launched into a 4.9-m long fiber. (b) Plots of output power from each core as a function of wavelength for a tunable continuous wave laser launched into the central core. Coupling is seen to be wavelength dependent.

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4. Conclusion

To conclude, we study FWM in a 3-core microstructure fiber. Significant FWM is observed despite polarization and wavelength dependence in the coupling coefficient. By adjusting the pump power for a fiber whose physical length is slightly longer than the coupling length we observe the transition between single-core nonlinear effects and multi-core nonlinear effects. The critical design challenge is balancing the effective nonlinearity with the desired phase matching bandwidth. If the coupling length is short compared with the nonlinear length, then the effective nonlinearity, $\gamma $, is reduced by the number of cores present. While this increases the maximum power at which one can operate the FWM device, it reduces the phase matching bandwidth. In our linearly-aligned 3-core fiber, we found that the coupling coefficients were not equal, but microstructure fibers provide large design space. For example, one can fine tune the core-to-core spacing to achieve balanced coupling or change the design to introduce more coupled cores. Studying the limitations and opportunities offered by the fiber linearity and nonlinearity help in optimizing fiber parameters during its fabrication to extract maximum possible gain from a system. We anticipate that our findings help to mitigate the linear and nonlinear effects while implementing a multicore fiber for practical use.