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New class of highly nonlinear fibers possessing dispersive characteristics invariant to transverse geometry fluctuations is described. The sensitivity to stochastic core fluctuations is reduced by order of magnitude while maintaining the fiber nonlinear coefficient. The effectiveness of the new highly nonlinear fiber type is demonstrated on stochastically perturbed distant-band mixer that could not be previously constructed with high-confinement fiber. The new fiber design offers a unique platform for ideally phase matched parametric exchange with significantly increased Brillouin threshold.
©2012 Optical Society of America
1. Introduction
Three general trends in signal processing have recently emerged in communication and sensing applications. The first is recognized in increased signal bandwidths that are no longer compatible with conventional electronics. The second trend is driven by large dynamic range expected from a mixed (analog/digital) signal processor. Finally, the standard (incoherent) processing is being eliminated with the introduction of fully-coherent optical links. Consequently, advanced optical signal processors are coherent, able to manipulate THz-wide signals in real time, and possess dynamic range compatible with analog processing.
Parametric mixers are recognized as a core processing technology in this regard and have been investigated in silica, silicon and semiconductor platforms [1–3]. With octave-wide bandwidth and high figure of merit (FoM) driven by a long interaction length [4], fiber-optic mixers represent particularly important class of processing engines. They allow for power-efficient access to ultrafast signals, with applications ranging from Tb/s channel transport [5], propagation impairment reversal [6,7], packet manipulation [8], and, more recently, complex analog signal acquisition [9] and wavelength conversion to non-conventional bands [10,11]. Highly-nonlinear fiber (HNLF) has been the most important mixer platform to date, and is responsible for majority of advances made in low-power, high-bandwidth parametric devices [4].
Regardless of specific device functionality, its power efficiency, noise and bandwidth metrics are uniquely determined by the phase-matching condition of the underlying four-photon mixing (FPM) processes. Phase matching in fiber parametric mixers is defined by the combination of material and waveguide dispersion characteristics. The latter is recognized as a powerful tool in tailoring the mixer response over large spectral range: a conventional HNLF index profile allows for significant dispersion change even with small transverse fiber geometry alterations [12]. While this sensitivity is clearly desirable for phase-matched design, it also introduces the fundamental phase-matching limit [13]. Indeed, in addition to deterministic core variation, HNLF fabrication is accompanied by inherently stochastic microscopic fluctuation that leads to considerable dispersion variation [12]. As a result, phase-matching variance along the fiber length then severely impacts the performance of a mixer relying on long interaction length [14], denying the advantage of fiber-based mixer construction. Specifically, distant-band parametric mixer relying on negative fourth-order dispersion [15,16] is recognized as the most sensitive to such dispersive fluctuations. For this reason, recent demonstrations have exclusively relied on standard dispersion-shifted fiber (DSF), rather than existing HNLF type [17,18]. To maintain the FoM of a DSF mixer, pump powers had to be scaled by nearly an order of magnitude, as required by the ratio between HNLF and DSF nonlinear coefficients [17].
Recognizing this limitation, various post-fabrication correction schemes have been demonstrated. Invariably, they require longitudinal mapping of local dispersion fluctuations [13], and must be followed by either selection/concatenation of useful sections [19] or localized dispersion equalization [20]. In contrast, pre-fabrication measures for HNLF design with inherent resilience to local geometry fluctuation are highly desirable yet remain largely unexplored. Rare attempts to rectify the dispersion fluctuation via unconstrained optimization were driven by purely mathematical formulation and have resulted in index profiles that are either not manufacturable or possessing nonlinear coefficients similar to that of the standard fibers [21].
This paper reports a new fiber design inherently resilient to transverse geometry fluctuation that is compatible with practical HNLF manufacturing process. The new approach relies on physical optimization of waveguiding in multi-layered fiber geometry. The effectiveness of the new HNLF type is assessed by the efficiency of the distant-band parametric mixer. In this regime, the phase-matching requires negative fourth-order dispersion to achieve gain projection to a pair of GHz-wide windows separated by 100-THz. This regime defines the most sensitive FPM process to localized dispersion fluctuation, and is the singular reason for the absence of HNLF-based device.
This report first introduces the tolerance analysis for dispersion fluctuation of narrow- and distant-band parametric amplification process. Subsequently, the origin of the large dispersion fluctuation in conventional HNLF is described, and is followed by the new design strategy used to minimize dispersion fluctuation via multi-layered cladding design. The linear and nonlinear characteristics of an exemplary design are presented in Section 3. Finally, the new HNLF type is compared with the conventional HNLF used in narrow- and distant-band mixer.
2. Dispersion fluctuation tolerance for narrow-band parametric amplification
In case when optical pump centered at frequency ωP is launched to silica fiber, an efficient parametric amplification (conversion) occurs within the phase-matching window, defined by:
where γ and PP denote the nonlinear coefficient of the fiber and peak pump power. The spectral dependency of the parametric gain is defined by the linear phase-mismatch term ∆β. In silica, the phase mismatch is described with sufficient accuracy by the second- and the fourth-order dispersion β2 and β4 terms:In case when phase matching is dominated by strong negative β4, it is possible to achieve strong spectral localization of the parametric process: instead of producing a contiguous gain band in immediate pump vicinity, parametric amplification (conversion) only occurs in a pair of narrow spectral windows [15,16] centered away from the pump frequency. The frequency offset ∆ωS (defined relative to the pump frequency) and the bandwidth δω of the gain (conversion) window are approximated by the following expressions [16]:
where β3 and ω0 represent the third-order dispersion and the zero-dispersion frequency (ZDF) of the fiber.In addition to highly selective spectral response inherent to reduced gain bandwidth, this operational mode for a fiber-optical parametric amplifier (FOPA) is used to access distant bands without paying the penalty of excessive out-of-band amplified quantum noise (AQN), inherent to wide-band FOPAs [22]. This feature is not easily accomplished in practice: the strong gain (conversion) spectral dependence on the fiber dispersion requires stringent control of the dispersion uniformity along the entire length of the mixing fiber. The importance of the distributed dispersion uniformity can be demonstrated by a gain model incorporating random dispersion fluctuation [23]. The model predicts the mean gain attainable in a FOPA perturbed by the influence of random dispersion variation along the fiber span, in form of a perturbation δβ2 to the mean second-order dispersion:
where p is a zero-mean Gaussian random variable with standard deviation of σb. The perturbation in Eq. (5) corresponds to a random process with ensemble-wise Gaussian statistics of N(0,σb) and an auto-correlation σb2exp(-∆z/Lc), with Lc corresponding to the correlation length of the random fluctuation. Physically, this means that the main effect of the random fluctuations along the fiber length is ZDF shift, rather than change in dispersive characteristic curvature. Indeed, it is recognized that the influence of second-order dispersion perturbation to a particular FOPA configuration depends on the mean dispersion β2 and β4, and nonlinear interaction strength γPP.In order to study the influence of dispersion fluctuation independent of the particular FOPA configuration, we introduce the normalized fluctuation parameter f, defined as the ratio of the fluctuation of the gain peak frequency and the mean bandwidth of the gain window. Using Eq. (3) and (4) with the assumption that the perturbation δβ2 is a small fraction of the mean second-order dispersion, the parameter f can be expressed as follows:
Figure 1 shows the impact of the dispersion fluctuation in a narrow-band FOPA, obtained by geometrically-averaging the gain in 100 realizations of the Gaussian-perturbed FOPA. It is readily recognized that the existence of dispersion perturbation in non-ideal fibers imposes significant penalty on the attainable gain. Statistically, a fluctuation factor f below 3.38(γPPL)-0.454 is required at a correlation length of 1m in order to attain half of the unperturbed gain level. A practical illustration can be defined by γPP = (0.01/Wm) × (10W); β4 = −10−4 ps4/km; λP = 1550 nm; λS = 2000nm, dictating that the HNLF dispersion (D) be maintained within 5 × 10−3 ps/nm/km range. Unfortunately, this stringent requirement also poses an insurmountable challenge with HNLF transverse geometry control during the fabrication process.
Fig. 1 Attainable mean gain of a narrow-band FOPA in the presence of fiber dispersion fluctuation. Fiber length: 100m; fluctuation correlation length: 1m. Dotted line denotes the half-gain level with respect to the ideal case.
In spite of this limitation, HNLF remain indispensible in practical parametric device construction since its high nonlinear coefficient alleviates the need for very high pump powers. Furthermore, the strong mode confinement within conventional HNLFs provides means for achieving small chromatic dispersion by imposing strong waveguide-dispersion effect [12]. The dispersion reduction from the intrinsic (material) silica characteristic facilitates phase-matching for pumps situated in conventional telecom band (1530 – 1610 nm) where mature technologies allow for economical generation of high-quality pump beams. However, the large dispersion shift that originates in down-sizing of the fiber mode also means that the dispersion becomes highly sensitive to random fluctuations in the fiber transversal geometry. As an example, a typical HNLF profile shown in Fig. 2(a) [12] exhibits 0.35 ps/nm/km dispersion shifts when subjected to a 10-nm variation in core radius, as illustrated in Fig. 2(b). Consequently, enforcing the dispersion tolerance of 5 × 10−3 ps/nm/km to a typical high-confinement fiber requires core radius control accuracy better than 196 pm. Unfortunately, this scale is also comparable to the radius of a silicon atom (111 pm), and much smaller than the silica Si-O molecular ring (600 pm) that forms the glass matrix. Even though contemporary fiber fabrication techniques have allowed dispersion stability for approaching 0.05 ps/nm/km levels in special cases [12], it is not realistic to expect that any future process will close this order-of-magnitude disparity with the stability requirement for a narrow-band FOPA.
Fig. 2 (a) Refractive index profile of a conventional HNLF; (b) Dispersion profiles of (a) when subjected to 10-nm deviation in core radius..
3. Geometric-variation desensitized fiber design
Rather than addressing the unphysical requirement imposed on typical HNLF geometry control accuracy, we consider a new class of fiber profile capable of high modal confinement and inherent resilience to the transverse perturbations. Fiber dispersion control can be achieved by variety of core/cladding geometries to synthesize waveguide dispersion and offset the material dispersion of silica glass. The total dispersion of a step-index fiber waveguide is expressed as the combination of the material and waveguide contributions:
where DM indicates material, DW is waveguide and DMW is the cross-term contribution. When expanded using standard normalized notation, the dispersion terms are expressed as [24]: where a, n1 and nc are the core diameter, core index and cladding index of a fiber respectively. The effective index experienced by the mode field is denoted as neff., while parameters V and b indicate normalized frequency and propagation constant, respectively. Since the index contrast is limited to few percents by a Rayleigh-scattering limit, variation in material dispersion is considered insignificant [25]. In contrast, the wave-guiding contribution changes dramatically in response to subtle deviation from the designed geometry.In conventional high-confinement fibers such as HNLFs, a single delta-like core provides highly negative waveguide dispersion to negate the material dispersion of silica [13]. The wave-guiding characteristics of conventional designs is best illustrated by the normalized waveguide dispersion (Fig. 3 ), computed by solving the scalar field equation using finite difference method [26]. In order to provide high nonlinearity and low dispersion spread across the operational band, conventional HNLFs are designed to operate in the vicinity of the cut-off frequency where the mode field is tightly confined in the core, and the waveguide dispersion has an inverted slope with respect to the material dispersion. However, the dispersion profile in this region is highly sensitive to the geometry fluctuations: a mere 1% shift in core radius results in 3% change in dispersion (Fig. 3(b)). In practice, this is equivalent to 0.5 ps/nm/km deviation in total dispersion for a fiber designed to have zero dispersion wavelength (ZDW) at 1550 nm. Consequently, this also represents two orders of magnitude larger dispersive fluctuation than that allowed by a test parametric device described earlier.
Fig. 3 (a) Normalized waveguide dispersion of a HNLF with refractive index profile shown in Fig. 2(a). Right column are zoom-in views of the dispersion curves at various positions: (b) V = 3.67 (1550 nm) and (c) V = 1.28. Green line represents dispersion of unperturbed waveguide, whereas red and blue lines depict dispersion of waveguides with + 1% and −1% geometry deviation.
By departing from the conventional, single-core design strategy, the required dispersion stability can be provided by a multi-layered waveguide design (Fig. 4 ) without compromising confinement capability. The design consists of four concentric layers: an inner core (a1, ∆n1) defining the field-confinement characteristics, an outer core (a2, ∆n2) serving to stabilize the dispersion profile, an intermediate cladding layer (a3, ∆n3) which supports guiding for both cores, and the outer cladding providing mainly the mechanical support. The underlying principle of the new design can be understood by noting that the beam waist of the confined optical field depends on its wavelength. Furthermore, the direction of dispersion shift is reversed in the region with V-number below the dispersion minimum point (Fig. 3(c)). The double-core geometry then serves to impose different guiding regime to the short- and long-wavelength components of the optical field by utilizing the dependence of beam waist on wavelength: optical field with shorter wavelength (higher V-number) will experience guiding by the inner core with the outer core acting as the cladding, whereas the long-wavelength beam is confined collectively by the core layers, and bounded by the cladding layers. The disparity between the guiding characteristics in these two regimes results in a double-dip dispersion profile, as shown in Fig. 5 . When the profile is stretched radially, the saddle region sandwiched between two depressions will see an opposite dispersion shift being exerted by the two guiding regimes, thereby reducing the net dispersion shift. As demonstrated in Fig. 5 (b) and (c), the outer core radius and index contrast provide a comprehensive control mechanism that can be used to center the saddle dispersive region and minimize the dispersion fluctuation. Indeed, the desired waveguide dispersion can be obtained by adjusting the index contrast, while radial trimming will control the extent of dispersion fluctuation and shift the frequency range where the saddle region is situated.
Fig. 5 Dispersion profiles of (a) the optimal double-core profile, (b) various a2, and (c) various ∆n2. Dashed and dotted lines indicate the V-parameters corresponding to wavelengths of 1.2 and 2.0 μm.
In comparison to conventional high-confinement fiber, the dispersion profile of the new fiber geometry is significantly less sensitive to core size variation. For instance, the exemplary fiber depicted in Fig. 6 has a dispersion variation within the telecom band (1550 nm) reduced by 99%, compared to the single-core design depicted in Fig. 2. More importantly, this reduction is achieved while attaining field-confinement metric similar to the standard HNLF, as characterized by an effective mode area of 16.2 μm2 (24.7 μm2) at 1550-nm (2000-nm) wavelength, versus 11.3 μm2 (16.5 μm2) for the conventional HNLF design. The nonlinear coefficients (γ) at 1550 nm, calculated with consideration of the glass composition [27], were 11.4 and 18.8 W−1km−1 for the new and conventional design examples respectively. The new design maintained single-mode operation within the telecom band, characterized by a cut-off wavelength at 1480nm. While the example design allows higher-order mode (LP11) to propagate at short wavelength, the large index contrast (> 0.01) between the fundamental (LP01) and higher-order mode over the band from 1100 nm to 1480 nm guarantee negligible mode coupling within the operating band (1200 to 2000 nm) of a parametric mixer pumped by telecom-band lasers
Fig. 6 Dispersion profiles of a double-core fiber subject to 10-nm core radius perturbation. Refractive index profile parameters: a1 = 1.9μm, ∆n1 = 2.4%, a2 = 3.8μm, ∆n2 = 0.7%, a3 = 20μm, ∆n3 = −0.5%.
Finally, the benefit of the new fiber design was quantified using a numerical model of the narrow-band parametric amplification process, taking into account of dispersion fluctuations due to stochastic transversal geometry variation along the fiber span. The model evaluated the gain attainable with a 50-m fiber section, where the core-radius variation of the fiber was characterized by a standard deviation of 1 nm and a correlation length of 1 m. Figure 7 shows the gain profiles produced by conventional and newly-designed fibers, aggregated over 100 random realizations of longitudinal core-radius variation profiles. The pump power PP and wavelengths λP (listed in Fig. 7) were chosen to produce 20-dB of gain centered at 2-μm wavelength in the absence of geometry fluctuations. The benefit of the new design is evident: the new fiber attained 17-dB higher average gain, despite under a more stringent condition due to the fact that the gain window produced by the new fiber was 3.6 times narrower than that by the conventional fiber. When compared in terms of yield, all realizations of the new fiber design produced gain above the half-ideal-gain benchmark, whereas no fiber designed using the conventional approach attained the comparable level of performance.
Fig. 7 FOPA gain profiles of (a) conventional HNLF type and (b) new HNLF type incorporating geometry-variation desensitized design. Colored traces correspond to the gain produced by each member of the geometry perturbation ensemble, whereas the grey trace indicates the average gain profile. The black traces show the gain profile of the ideal (unperturbed) fibers for comparison. Inset shows a zoom-in view of the peak region in (b).
4. Conclusion
We described the new highly-nonlinear fiber design with inherent dispersion resilience to transverse waveguide fluctuation. The new fiber design relies on an additional guiding layer surrounding the central core. This layer cancels the waveguide dispersion shift due to core diameter fluctuations while preserving confinement capability. The efficiency of the new HNLF type was explored using parametric amplification phase-matched by negative fourth-order dispersion. This mixer type was specifically chosen as no known HNLF type has been successfully used in its construction. While the conventional HNLF design requires subatomic level of precision in core diameter, the new design relieved the tolerance by more than an order of magnitude, bringing it well within the range of practical fabrication process. In terms of parametric gain synthesis, the new fiber provided near-ideal gain level associated with uniform fiber, in sharp contrast to non-existent gain generated by the conventional HNLF type. Consequently, the new fiber design principle should render mass-scale manufacturing of parametric devices considerably more economical, by easing the need for post-fabrication characterization and correction of nonlinear fiber dispersion used to date.
More importantly, the new HNLF type allows for longitudinal strain to be introduced for stimulated Brillouin scattering suppression [28] with near impunity with respect to dispersive fluctuations. As a result, the new HNLF type should lead to ideally phase-matched mixers with qualitatively increased Brillouin threshold, thus removing the impairments from standard Brillouin mitigation techniques [29]. Characterization of manufactured HNLF with the proposed design will be presented in future reports.
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