Photonic-crystal fibers (PCFs) [1, 2] play an increasingly significant role in modern optical technologies. Dispersion properties of such fibers can be modified and tailored by changing the core--cladding design, allowing dispersion optimization [3] and enabling novel functionalities of fiber-optic components and devices. [4] PCFs with tailored dispersion are intensely used for the frequency conversion of ultrashort pulses and supercontinuum generation [5–7] for metrological, [8–10] spectroscopic, [11] and biomedical [12] applications, as well as for carrier╍envelope phase stabilization in ultrafast optics [13] and pump╍seed synchronization in optical parametric chirped-pulse amplification of few-cycle pulses. [14] In fiber-laser technologies, large-mode-area single-mode PCFs allow the creation of high-power laser sources, [15–19] while unique dispersion properties of PCFs find growing applications for dispersion and chirp compensation in short-pulse fiber-laser oscillators and amplifiers, [20–23] including high-peak power fiber-laser [20,21] and thin-disk-laser-based [24] systems.

Due to the commonly used stack-and-draw fabrication technology, PCFs can come in with a remarkable variety of cross-section geometries (see Refs. [1, 2] for a review). For solid-core PCFs, the required dispersion profile is typically engineered [3, 25–27] by choosing the appropriate geometry of the cladding, as well as by varying the size and the shape of the fiber core. As shown by the recent work, [28–32] an interesting alternative to dispersion engineering by modifying the structure of the fiber cladding involves a careful design of air-hole nanosize defects in the fiber core. Since the field intensity reaches its maximum in the fiber core, small defects in the core region may have a very strong impact on dispersion properties of guided modes. This approach has been recently shown to offer much promise for the design of ultraflattened group-velocity dispersion (GVD) profiles in PCFs, [28] including large-mode-area microstructure fibers, [29] as well as for the control of the fiber nonlinearity and gain. [30, 31] PCF nanomanagement strategies for the optimization of supercontinuum generation have been discussed by Frosz et al. [33] PCFs with a solid core modified with a ring-shaped array of nanosize air-hole defects (NAHDs) have been shown [34] to allow a precise management of dispersion profiles of guided modes for an efficient nonlinear-optical frequency conversion of femtosecond Cr: forsterite laser pulses, resulting in the generation of signals at the desired central wavelengths at the fiber output.

In this work, we present a systematic quantitative fully vectorial finite-difference analysis of PCFs with solid cores modified by arrays of nanosize air-hole defects (NAHDs). Based on this analysis, we identify important regimes of field-profile and dispersion management using NAHD arrays, which reflect physically significant properties of guided optical fields in nanostructured materials and suggest promising ways for the management of dispersion, field localization, and nonlinearity in waveguide structures.

The generic PCF structure considered in this work is shown in Fig. 1. The solid core of this fiber is modified by a ring of six nanosize air-hole defects. Numerical simulations presented here are intended to analyze the field-intensity profiles, as well as wavelength dependences of the effective mode area and dispersion parameters of the fundamental mode in this type of PCF as functions of the NAHD diameter d _a and the distance R of these holes from the center of the PCF core (Fig. 1). Numerical simulations were performed by using a fully vectorial finite-difference scheme, described in detail, e.g., in Ref. [35]. A five-point finite-difference scheme has been used to simulate the electric field. The accuracy of this method for PCF modeling has been checked through a comparison with the finite-difference time-domain analysis, [36] as well as with other numerical techniques, [37] demonstrating the reliability of the approach within the range of the parameters studied in this paper. We emphasize we do not claim here the novelty of the numerical method of PCF analysis. Instead, we are interested in the influence of additional air-hole defects on the mode properties, the physics behind a radical mode modification observed with certain NAHD parameters, and related new opportunities for dispersion and mode-area engineering in PCFs.

In Figs. 2–4, we present the results of simulations performed for nanohole-modified PCFs with a NAHD diameter d _a varying from 200 to 600 nm and two different radii of NAHD rings, R = 1.3 μm [Figs. 2(b)–2(f)] and R = 1.5 μm [Figs. 2(g)–2(k)]. The field-intensity profiles and the wavelength dependences of the GVD for the fundamental mode of the considered type of PCFs are shown in Figs. 3(a) and 3(b), respectively. In Figs. 4(a) and 4(b), we plot the wavelength dependences of the effective mode area for the fundamental PCF mode, defined as [38] A_eff =({∫∫_∑ F (x,y) dxdy)²/∫∫(F (x,y) dxdy, where F (x,y) is the transverse field profile and the integration is performed over the entire fiber cross section.

 

Fig. 1. Sketch of a photonic-crystal fiber with a solid core modified by a ring of six nanosize hole defects: d is the diameter of air holes in the PCF cladding, Λ is the distance between the centers of air holes in the PCF cladding, d _a is the diameter of nanohole defects, and R is the distance of the center of nanoholes from the center of the fiber core. The circle shows the effective size of the modified fiber core.

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Based on the results of simulations presented in Figs. 2–4, we identify three important regimes of field-profile and dispersion management of PCFs using NAHD arrays. In the first regime, the size of nanohole defects is so small that NAHDs can be treated as small perturbations giving rise to a weak distortion of mode field profiles [Figs. 2(b) and 2(g)]. As small NAHDs slightly reduce the effective refractive index of the PCF core, [30, 31] the zero GVD wavelength of a NAHD-modified PCF is slightly red-shifted [the curves corresponding to d _a ranging from 200 to 350 nm in Fig. 3(a) and d _a ranging from 200 to 450 nm in Fig. 3(b)].

In the second regime, the diameter of air holes in the six-NAHD ring in the fiber core is large enough for the NAHD ring to bound a new, modified core of the fiber (shown by a circle in Fig. 1). The diameter of this modified fiber core is effectively reduced as the diameter of NAHDs increases. Correspondingly, the fiber modes become more compact as the parameter d _a is increased in this regime [Figs. 2(c), 2(d), 2(h), 2(i)], the effective mode area A _eff decreases relative to A _eff of the fundamental mode in the original PCF [Figs. 4(a) and 4(b)], while the zero-GVD is blue-shifted [the curves corresponding to da ranging from 400 to 550 nm in Fig. 3(a) and d _a ranging from 500 to 550 nm in Fig. 3(b)].

 

Fig. 2. Field-intensity profiles of 1.5-μm radiation in (a) a PCF without air-hole defects in the fiber core and NAHD-modified PCFs with (b - f) R = 1.3 μm and (g - k) and R = 1.5 μm. The diameter of NAHDs is (b, g) 200 nm, (c, h) 400 nm, (d, i) 500 nm, (e, j) 550 nm, (f, k) 600 nm. The pitch of the PCF cladding is Λ = 2.3 μm. The diameter of air holes in the cladding is d = 2.3 μm.

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Fig. 3. Wavelength dependences of the group-velocity dispersion for the fundamental guided mode of 1.5-μm radiation in NAHD-modified PCFs with (a) R = 1.3 μm and (b) R = 1.5 μm and the diameter of NAHDs ranging from 0 to 550 nm. The pitch of the PCF cladding is Λ = 2.3 μm.

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Fig. 4. Wavelength dependences of the effective mode area for the fundamental guided mode of 1.5-μm radiation in NAHD-modified PCFs with (a) R = 1.3 μm and (b) R = 1.5 μm and the diameter of NAHDs ranging from 0 to 550 nm. The pitch of the PCF cladding is Λ = 2.3 μm.

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To demonstrate that the NAHD-induced fiber modification in this intermediate regime is mainly caused by geometric factors related to the reduction of the effective size of the modified fiber core, we plot in Figs. 5(a) and 5(b) the material, D _m, and waveguide, D _w, components of the fiber GVD along with the total GVD D = D _m + D _w for the considered type of NAHD-modified PCFs As can be seen from these figures, in the short-wavelength range, the waveguide GVD D _w becomes shifted upward as the NAHD diameter d _a is increased (curves 2 – 4). Notably, as can also be seen from the comparison of curves 2–4 in Figs. 5(a) and 5(b), the change in D _w in this wavelength range scales linearly with the variation in d _a. This scaling law can be readily explained by representing the effective mode index for a fiber mode as n _eff ≈n ₁⌊1-u ²/($k_0^2$ $n_1^2$ a ²)⌋, where n ₁ is the refractive index of the fiber material, k ₀ = 2π, λ, λ is the radiation wavelength, u is the guided-mode eugenvalue, and a is the fiber core diameter (which is, of course, a fuzzily defined parameter in the case of a PCF). With the total GVD defined in a standard fashion, D = -(λ/c)(d ² n_eff /dλ ²), with c being the speed of light, the variation of the a-dependent part of D _w scales as (δD _w)_a ∝ - δa/a ³ in the regime where the mode eigenvalue u only weakly depends on a (i.e., in the regime of short wavelengths). Since δa = -δ d_a in the considered case, we recover the δ D_w ∝ δ d_a scaling law observed in Figs. 5(a) and 5(b).

 

Fig. 5. Wavelength dependences of (1) the material GVD of fused silica, D _m, and (2–4) the waveguide GVD component D _w, and (5–7) the total GVD D = D _m + D _w for an NAHD-modified PCF with (a) R = 1.3 μm and (b) R = 1.5 μm and d _a = 300 nm (2, 5), 400 nm (3, 6), and 500 nm (4, 7).

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We now proceed with the analysis of the third regime of PCF modification with a ring-shaped array of NAHDs. It can be seen from Figs. 2(e) and 2(f) that, with a specifically chosen relation between the radiation wavelength and the distance of the nanohole defects from the fiber core center, the increase in the NAHD diameter d _a may give rise to phase-transition-type changes in transverse field profiles in fiber modes, accompanied by dramatic changes in the wavelength dependences of the effective mode area and the GVD, especially in the long-wavelength range (Figs. 3 and 4). The mode profile in Fig. 2(f) displays multiple maxima, reminiscent of those typically observed for higher order waveguide modes. It should be emphasized, however, that the mode profile shown in Fig. 2(f) is not a higher order mode of the fiber core, but rather is a superposition of the core and cladding modes. This type of behavior of PCF modes has been earlier discussed for PCFs with air holes of varying diameters in the cladding by Wilcox et al., [39] who have very instructively interpreted this effect in terms of the effective mode indices of the PCF core and PCF cladding. Unlike the geometry examined in Ref. [39], a kind of a phase transition in the behavior of guided modes in the PCFs considered in this work is induced by nanohole defects in the fiber core. However, the physical origin of this effect is essentially the same as revealed by Wilcox et al. In Fig. 6, we illustrate this argument by comparing the effective mode indices of the guided modes in NAHD-modified PCFs with the refractive index of fused silica and the effective index of the space-filling mode in the microstructure cladding of the considered type of PCF. As can be seen from Fig. 6(b), for the PCF with R = 1.5 μm, no index matching is possible between the mode guided in the PCF core and the space-filling mode of the PCF cladding within the studied range of d _a, prohibiting a phase transition in the mode field profile. For the PCF with R = 1.3 μm, however, the mode index of the core-confined mode in the fiber modified by NAHDs with d _a = 500 nm, as can be seen from Fig. 6(a), asymptotically tends to the effective mode index of the cladding, inducing a localization╍delocalization phase-transition-type switching of mode field profiles in the long-wavelength region. This phase transition is manifested as a dramatic increase in the effective mode area [Figs. 2(e), 2(f)] and a radical modification of the wavelength dependence of the GVD [Figs. 3(a), 4(a)]. As the radiation wavelength changes from 1.20 to 1.33 μm, the effective area A _eff of the mode presented in Fig. 2(f) increases by approximately a factor of two [Fig. 4(a)]. Correspondingly, the nonlinearity is reduced by the same factor. Waveguide modes that spread outside the core area often exhibit much higher bending losses compared to the modes that are well-localized in the fiber core. In the PCF geometry, however, the bending losses of delocalized modes similar to that shown in Fig. 2(f) can be kept under control by using an additional outer microstructure cladding with a high air-filling fraction, preventing the leakage of radiation from the inner part of the PCF. With just a few rings of air holes in such a high-air-filling-fraction outer cladding, as suggested by numerical simulations, no noticeable increase in bending losses for the mode of Fig. 2(f) should be expected.

Some of the applications, such as high-power fiber amplifiers, for example, are quite restrictive on the beam quality, requiring a smooth bell-shaped transverse intensity profiles, typically provided by fibers operating in the single-mode regime. There exist, however, a broad class of important applications where a spatially nonuniform, but regular beam profiles can be tolerated or can be even preferable. One example of such an application is a fiber-optic sensor where an analyte fills the fiber holes. [40, 41] A combination of core and cladding modes shown in Fig. 2(f) would then help to increase the spatial overlap between the light field and the analyte. Another possible application is a fiber-optic probe where the core serves to deliver laser radiation to the sample, while the cladding is used to transmit the fluorescent or Raman signal from the sample in the backward direction toward a detector. [42] While a shorter wavelength excitation light can be guided in a well-localized PCF mode, the longer wavelength fluorescent or Raman signal would be transmitted in a mode with larger A _eff, providing a higher numerical aperture for a more efficient collection of the fluorescent or Raman signal. Both A _eff and D engineering in this regime suggests attractive solutions for the design of novel types of PCFs for applications in ultrafast science and nonlinear optics. In particular, a steep profile of the wavelength dependence of the effective mode area A _eff observed in Fig. 4(a) suggests an interesting way to reduce a timing jitter and wavelength uncertainties in soliton self-frequency shift induced by fluctuations in input laser intensity. Novel dispersion profiles attainable with NAHD-modified PCFs can be used to reduce pulse distortions caused by high-order dispersion effects in fiber stretchers and fiber compressors, thus offering new PCF-based solutions for all-fiber short-pulse laser--amplifier systems.

 

Fig. 6. Wavelength dependences of the refractive index of fused silica (PCF material), the fundamental space-filling mode in the cladding of the considered type of PCF, and effective mode indices in the NAHD-modified PCF with (a) R = 1.3 μm and (b) R = 1.5 μm and the air-hole diameter d _a ranging from 200 to 500 nm.

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To summarize, a fully vectorial finite-difference analysis of PCFs with an NAHD-modified solid core allows us to identify important regimes of field-profile and dispersion management using NAHD arrays. Very small air holes typically play the role of weak perturbations, slightly modifying field profiles in fiber modes and red-shifting the zero-GVD wavelength. Larger holes can, however, effectively reduce the mode area, increasing the confinement of the light field in the fiber core and blue-shifting the zero-GVD wavelength. Air nanoholes with diameters above a certain critical value can induce a phase-transition-type behavior of mode field profiles, dramatically reducing the localization of the field in the fiber core and increasing the radiation power in the fiber cladding. This phase transition in mode field profiles qualitatively modifies the wavelength dependence of the effective mode area and dispersion parameters of fiber modes, especially in the long-wavelength range, suggesting an attractive strategy for fiber dispersion and mode area engineering.