1. Introduction

Triangular-lattice hollow core photonic crystal fibers (HC-PCF) are able to guide light within an air-core of a few-µm diameter over a distance that can attain a kilometer scale [1]. This exceptional property has underpinned the many recent advances in gas-laser interactions such as ultra-low threshold stimulated Raman scattering in H2-filled HC-PCF [2] and electromagnetically induced transparency in both molecular gases [3, 4] and atomic vapors [5, 6]. The guidance at such a low attenuation is achieved by the appearance of a photonic band gap (PBG) in the mode spectrum of the fiber cladding, arising from the waveguiding properties of identified structural cladding constituents [7]. Whilst this fiber can in principle exhibit an attenuation lower than the Rayleigh scattering limit encountered in conventional fibers, its transmission bandwidth is restricted to an optical frequency span of typically 50THz [8].

Concurrent to the development of the low-loss PBG fiber, it was shown that air-guidance over a length of a few meters can also be achieved in the absence of a PBG using a Kagome cladding lattice [9, 10], with impact on nonlinear applications as profound as those based upon using PBG HC-PCF [9, 11]. In these fibers, the cladding is made up of thin (<300 nm) parallel sheets of silica that intersect to form a tessellated ‘Star-of-David’ pattern with a large pitch (6–20µm), see Fig. 1(a). Among the salient features of these fibers are their large transmission bandwidth - more than 20 times larger than the PBG HC-PCF - and a much lower chromatic dispersion [10].

The guidance was first described by analogy to Bragg fibres [12] but a more general study recognized that the photonic guidance in the Kagome-lattice HC-PCF relies on a mechanism akin to Von Neumann-Wigner quasi-bound states within a continuum [13], in the sense that the fiber-guided modes occupy the same region of frequency - effective index space as modes of the cladding, without appreciably interacting with them [11]. The latter description does not require the struts which connect concentric glass layers to be considered as a perturbation. The inhibited coupling between a core-guided mode and the continuum of cladding-guided modes results from the disparate nature of the modes. The cladding-guided modes are highly confined to the silica strut network forming the cladding (i.e. small overlap with the air-region) and exhibit a fast transverse phase oscillation, whereas the core modes are characterized by slower field variations. The confluence of these properties results in a dramatic decrease in a pertinent overlap integral between the two types of mode. A further favorable factor for this type of guidance is the limited number of these cladding modes at effective index values (n _eff) close to the core guided mode index, i.e. there exists a low density of photonic states (DOPS) close to the air light-line n _eff=1.

Among the conclusions drawn from [11], it was recognized that in order to minimize the transmission loss associated with this type of guidance, it is desirable to have, in the case of a periodic structure, a topology that entails a network of extended silica struts whose thickness is very small relative to the air hole diameters (low glass fraction per unit cell), and in which the air-hole diameters are many times the free space wavelength (the large pitch regime). Furthermore, this silica network should ideally exhibit no additional structural features such as glass-nodes that can appear at the intersection of the struts as a result of surface tension forces which act during the fiber drawing process.

Armed with this insight, broadband guidance has been experimentally demonstrated in silica based large-pitch HC-PCF with cladding structures based on alternative forms to the Kagome-lattice, such as triangular and square lattices [14, 15], as well as in fibers made out of polymer [16].

Here, we investigate theoretically and experimentally the guidance characteristics of silica-based large pitch square lattice HC-PCF (Fig. 1(b)) by studying the effects of the air-filling fraction and core shape on the guidance properties of such fibers. The numerical simulations show that, similarly to the Kagome fiber, away from specific resonance wavelengths, the square-lattice exhibits low DOPS at effective index values close to the air light-line. Furthermore, the modes that do exist are primarily confined in the silica struts and exhibit fast transverse phase-oscillations. However, in contrast to the Kagome-lattice, the theoretical results show that a loss figure of a few 10’s of dB/km is potentially achievable over a broad bandwidth if the glass nodes are kept small, strut thickness variations remain slight and the defect which forms the core region is suitably chosen.

Experimentally, we show fabricated large-pitch square-lattice fibers that exhibit a flat attenuation spectrum below a loss of 4dB/m over an optical wavelength range >800nm wide, and a flat and low group velocity dispersion around 1ps/nm/km which is, to our knowledge, the lowest broadband dispersion reported so far for a HC-PCF. Furthermore, the fiber exhibits a very long birefringence beat length, yet polarization cross-coupling remains at a low level. This enables the maintenance of an arbitrary polarization over several meters of propagation.

2. Large pitch HC-PCF guidance mechanism

The study of the guidance mechanism given below is not to be confused with previous reports on square lattice HC-PCF where the investigation concentrated on the guidance by a photonic band gap. The bandgap fibers, which are characterized by a much smaller cladding pitch and a narrower transmission spectral range, require sizable glass nodes to occur where the glass planes intersect otherwise a bandgap does not form [17].

Comparing the unit cell of a square-lattice (Fig. 1(b)) with that of a Kagome-lattice (Fig. 1(a)) in the context of (1) the broadband optical guidance mechanism mentioned above, and (2) attaining a low power-in-glass fraction, one notices a number of features which favor the square fiber. For a given lattice pitch Λ and glass strut thickness t, the glass filling fraction, 2t/Λ for a square-lattice, is smaller than the corresponding value e 2√3 t/Λ for a Kagome lattice. Furthermore, per primitive unit-cell, the square-lattice has fewer nodes than in the Kagome lattice as well as three times fewer air-holes. Since the area of the Kagome unit cell in terms of the pitch is √3Λ² 2, compared with Λ² for the square unit cell, the number of nodes and holes per unit area at a common value of Λ is a factor 2√3 smaller for the square fiber. The strut length (i.e. node separation) is also twice as long for the square lattice compared to the Kagome form, when expressed in terms of Λ. These key features motivated our choice of this photonic cladding structure.

Figures 1(c) and 1(d) compare the calculated density of photonic states (DOPS) of a square cladding to that of a Kagome lattice cladding for a common glass strut thickness t (taken to be 0.044Λ) and a similar node size. The density of photonic states that exist in the cladding structure of a HC-PCF was computed using a finite element mode solver [11] as a function of normalized wavenumber kΛ (k is the free-space wavenumber) and mode effective index n_eff. The DOPS diagram shows very similar features for both structures. Most notably, both exhibit large regions of low DOPS extending below n_eff~1, with the square lattice showing less pronounced DOPS peaks and a lower average DOPS than the Kagome lattice.

Similarly to the Kagome lattice, the square-lattice shows narrow high DOPS regions around kΛ~70 and kΛ~140 (see Fig. 1). These correspond to strong anti-crossing events between particular modes of the glass struts and of the air holes within the photonic structure, occurring at wavelengths given approximately by the transverse resonance condition $k\sqrt{n_{\mathrm{gl}}^2-1}t=\pi j$, where n_gl is the refractive index of the glass and j is a positive integer. They are characterized by slow variation along each strut and, therefore, around each hole’s perimeter; this enables the strong interaction with hole-related modes when close to phase-matching. The cladding modes which result from the interaction are in turn expected to strongly interact with core guided modes, hence increasing the attenuation of the fiber for frequencies in the vicinity of each anti-crossing [11]. Away from these resonant frequencies the cladding modes near the air lines show intensity profiles typified by the ones shown in Fig. 1(e) and Fig. 1(f) for the Kagome-lattice and square lattice, respectively. In both cases the field distribution exhibits a sharp localization in the silica region and a fast transverse oscillation so that coupling with the air-hole modes is strongly inhibited. This feature underpins the optical guidance in such fibers.

 

Fig. 1. (A, B) Unit cell of a Kagome lattice (the grey lines show the boundary of the unit cell) and of the square lattice, respectively. (C, D) Density of Photonic States (DOPS) diagrams for Kagome lattice and square-lattice structures, respectively, with the same strut thickness and similar node size. Black designates high DOPS and white low DOPS. (E) The cladding mode at kΛ=100 located at the Γ-point of the Brillouin zone, with an effective index n_eff=0.998, calculated for Kagome structure. (F) Idem for the square crystal, but with n_eff=0.999.

Download Full Size | PDF

In comparison to the Kagome lattice, the anti-crossing induced DOPS enhancement spreads over a slightly narrower spectral range for the square lattice. This relative narrowing of the strong anti-crossing ranges in the square lattice is expected from the key geometrical features outlined above. A reduction in the residual coupling between strut modes and air-hole modes is also observed within the low DOPS region of the square lattice, which in turn leads to a more inhibited interaction between square-lattice cladding modes and air core-guided modes of a HC-PCF within the low DOPS frequency region. The core-shape also has a strong effect on the overall loss spectrum of a broadband guiding HC-PCF, as is discussed in section 4 below.

3. Fiber fabrication

Similar to the fabrication of the Kagome fiber described in [10], the square lattice large pitch HC-PCF is made by stacking thin capillaries of circular cross section into the required lattice and drawing the obtained preform into intermediate canes that are later pulled into fiber. During the cane drawing process of PBG fibers, vacuum is usually applied to the preform so that the interstitial holes collapse, creating the required structure which includes the thick interstitial glass nodes required for achieving PBG guidance. However, in fabricating fibers which guide by inhibited coupling, such as Kagome lattice or the present square lattice HCPCF, these interstitials need to be as small as possible, and as a result, the interstitial holes need to be kept open during this phase of fabrication. The resulting canes are composed of 3–4 layers of squares created alternatively by a glass capillary or by an interstitial square hole between four capillaries. Two types of canes, 3mm in diameter, are drawn from the initial 25mm-OD preform. The first type consists of a square-lattice cladding where the core defect is created by removing a single capillary from the center of the structure (Fig. 2(a)). An example of fiber obtained from this cane is shown in Figs. 2(b) and 2(c) and shall be called “single-cell core” fiber in the remainder of this paper. The second type of cane does not contain any defect (Fig. 2(d)). The core defect is created during the fiber drawing process by independently pressurizing the hole at the centre of the square structure. As no capillaries have been removed from the structure to create the core, the fiber shall be called a “zero-cell core” fiber (Figs. 2(e) and 2(f)). The shape of the core is controlled via the pressurization of the core and cladding during the fiber pulling process. As in [11], this pressurization is also used to expand the fiber structure, and hence control the glass strut thickness.

 

Fig. 2. (A) Optical micrograph of the cane and (B,C) scanning electron micrograph of fiber Single Cell #4. (D,E,F) Idem for fiber Zero Cell #3.

Download Full Size | PDF

Table 1 summarizes the physical characteristics of the fabricated fibers as well as the gas pressure applied to the core and cladding during the drawing process. In the single-cell core fiber (Fig. 2(b)), the cladding includes three full square layers of approximately equal thicknesses surrounding the core defect. The maximum core diameter is 46µm for a pitch of 17µm, obtained for a fiber fabricated at the highest manufacturing pressure (Fiber Single cell #4), yielding an air-filling fraction approaching 96%. In the zero-cell core fiber (Fig. 2(e)), the core defect is created by pressurizing the central hole. This leads to a core becoming somewhat circularized compared with the initial square shape. The major axis (largest straight line distance across the core passing through its geometric center is 42µm. The expansion of the core also leads to the slight deformation of the surrounding cladding structure (Fiber Zero cell #3). In this fiber, the cladding pressure was fixed to 50 kPa and the air-filling fraction was about 95% for all the fibers. The outer diameter of all fabricated fibers is 300µm.

Table 1. Drawing parameters and physical characteristics of fabricated square-lattice large pitch HC-PCF.

View Table

4. Optical properties

The transmission performance of large-pitch HC-PCF is sensitive to the thickness t of silica struts and their uniformity. Figure 3 illustrates this for the square lattice by showing the transmission spectra for the single- and zero-cell core fibers under the different drawing parameters listed in table 1. Figure 3(left) shows the evolution of the single-cell core fiber optical transmission spectra as the applied pressure in the cladding increases from 40 kPa to 60 kPa. As expected, all the spectra exhibit two flat transmission bands separated by a low-transmission band [11]. As the cladding pressure is increased, the silica struts become thinner which leads to a blue-shift the low-transmission band, following the transverse resonance formula $k\Lambda =\pi j(\Lambda ⁄t)⁄\sqrt{n_{\mathrm{gl}}^2-1}(j=1,2,3\dots )$. These spectral locations are illustrated in Figs. 1(c) and 1(d) by strong anti-crossings at kΛ~70 and kΛ~140. The potential transmission red-shift that could arise due to the slight increase of the pitch is overpowered by the blue shift that results from the strut thinning. The transverse resonance condition also explains the observed widening of the transmission bands as the cladding pressure is increased, since the bandwidth of a transmission band is given by $\Delta \left(k\Lambda \right)=\pi (\Lambda /t)\left(j+1-j\right)⁄\sqrt{n_{\mathrm{gl}}^2-1}=\pi (\Lambda ⁄t)⁄\sqrt{n_{\mathrm{gl}}^2-1}$. The fiber with the broadest continuous transmission range is therefore the fiber with the largest cladding pressure - i.e. with the thinnest struts (Fiber Single cell #4).

In order to investigate the role played by the core shape in the optical properties of large-pitch square lattice HC-PCF, the spectrum characterization was repeated for the zero-cell defect fiber by varying the pressure in the core whilst keeping the cladding pressure constant. Figure 3(right) shows the transmission spectrum evolution as the core pressure is increased. This increases the core diameter (see table 1) and decreases the core-surround ring thickness, but the cladding strut thickness and pitch are kept almost constant at 400nm and 17µm, respectively. This core-shape inflation, or more precisely the thinning of the core surround ring, dramatically alters the transmission spectrum by narrowing and improving the guidance within the weaker-guiding band of the spectrum around 900nm. The spectrum reaches the limits of our characterization system for the fiber fabricated with the largest differential pressure between cladding and core (Fiber Zero Cell #3), where the resonance associated with the core surround is shifted to a wavelength below the measured range, leaving only the resonance associated with cladding struts. Since less modal field reaches these cladding struts than the core surround ring, a reduction in influence of the resonances at wavelengths determined by the cladding strut thickness is expected. The core inflation needed to create the central defect, however, also impacts the thickness of the struts within surrounding square layers, and is believed to be the cause of dips in the transmission spectrum. This problem can in principle be overcome by using capillaries of incremental thickness during the stacking process to compensate for the inflation.

 

Fig. 3. Transmission spectra through 2m of (left) single-cell core fibers and (right) zero-cell core fibers manufactured at cladding and core pressures indicated in the insets and listed in Table 1. All traces are normalized to the supercontinuum source. The peak at 1064nm is residual from the supercontinuum source used in the setup.

Download Full Size | PDF

Figure 4(left) presents the loss spectrum of both fibers identified above as giving optimum guidance (i.e. Single Cell #4 and Zero Cell #3) within the near-IR high transmission window between 850 nm and 1750 nm. The optical attenuation has been measured using a 7–8m long sample cut back to 1m. The minimum loss of both fibers is well below 2dB/m. The singlecell core fiber shows a generally lower loss figure and a flatter loss spectrum. The latter is explained by the difference in the core-shape. The zero-cell core fiber exhibits several relatively large glass nodes and some shorter struts (Fig. 2(b)), which favors coupling between the cladding and core modes, and hence larger loss and a more structured transmission spectrum is expected.

The confinement loss of a structure which approximates the geometry of the fabricated fiber Zero Cell #3 is given in Fig. 4(right) (red curve labeled A), and was calculated using the finite element mode solver. The modeled geometry, shown as an inset, assumes that the thickness of each strut is constant. The loss is estimated to be of order 1dB/m over broad wavelength ranges, in general accordance to measurements on the fabricated form. Furthermore, there is good agreement between the theoretical and measured wavelength dependence of the attenuation, which exhibits lower optical loss at short wavelengths than at longer wavelengths. This suggests that the experimental results are limited by the confinement loss which is set by the residual coupling between the cladding modes and the core ones. This trend contrasts with the state-of-the-art PBG HC-PCF [1], where the loss increases with decreasing wavelength due to the nature of the air/glass interface roughness scattering that provides the limiting factor.

The black curve in Fig. 4(right) (labeled B) shows the confinement loss calculated for a more ideal structure. This structure shares the same strut thickness and similar node positions as the model of fiber Zero Cell #3, but the nodes where struts cross are smaller and, more importantly, the core is square in shape. The confinement loss is seen to be almost a hundred times lower than in Zero cell #3 fiber for most of the calculated spectrum. The loss figure reaches 10dB/km over a broad wavelength range. Such numerical simulations motivated the development of this type of fiber geometry as a candidate for a low-loss ultra-broadband HC-PCF. More control during the fabrication process is, however, clearly required to experimentally realize the more optimal geometry.

 

Fig. 4. Left: Measured attenuation spectrum for (top) fiber Zero cell #3 and (bottom) fiber Single cell #4. Right: Calculated transmission loss spectrum for a structure which approximates fiber Zero Cell #3 (red curve), and for a more ideal structure (black curve).

Download Full Size | PDF

Apart from the ultra-broadband guidance achieved through these large pitch fibers, another notable feature is the coexistence of guided core modes and cladding modes, as illustrated in Fig. 5. Figures 5(a) and 5(c) present optical micrographs, obtained under a microscope by illuminating 2cm samples from below, of the two types of square-lattice fiber under investigation. As observed in the Kagome-lattice HC-PCF [11], visible light is not only guided through the fiber core but also through the cladding holes and silica struts, hinting at an absence of a photonic bandgap in the guidance mechanism. The near field profile of the fundamental mode of the fiber, taken at 1000nm for a 2m-long sample, is shown in Fig. 5(b) and 5(d1), for fiber Single Cell #4 and fiber Zero Cell #3, respectively. As expected theoretically, very little light is detected outside of the square core boundaries (the detection apparatus has a 30dB dynamic range); despite the large core size, with the optimized launch conditions employed, the majority of the light can be efficiently coupled into the fundamental mode.

 

Fig. 5. Optical micrograph and experimental near-field profile at 1000nm of (A,B) single-cell and (C,D1) and zero-cell core fibers. (D2) Coupling into the silica cladding is possible with similar coupling conditions and operating wavelength as in D1. (D3) At a wavelength of 654nm, light is guided inside the air-holes of the cladding. However, it is still possible to guide in the core defect (sample length: 2m). (E) Fundamental core mode intensity profile calculated at kΛ=100 for a “zero-cell” core fibre, shown on a log scale with 60dB dynamic range. Very little field penetrates into the cladding. (F) Example of HOM calculated at kΛ=100, which is close to phase-matching with cladding modes with a high field component in the air holes. The absolute value of the E-field is shown on a linear scale. This core mode shows strong hybridization with cladding components and consequently an enhanced loss.

Download Full Size | PDF

Experimentally, predominant guidance within the cladding can be achieved at the same wavelength as the fundamental core-guided mode by simply shifting the focused light beam onto part of the cladding (this keeps the numerical aperture of the coupling beam essentially constant). The nature of these cladding modes is illustrated in Fig. 5(d2), for a normalized frequency kΛ=100 (i.e. wavelength ~1000nm) where the recorded near-field shows a mode where the light is dominantly confined in the silica constituents (mainly the strut part), and in Fig. 5(d3), at higher normalized frequencies kΛ=153 (i.e. wavelength ~650nm), where light propagates mainly in the cladding’s air-holes. The latter modes are identified in the DOPS diagram (Fig. 1(c) and 1(d)) by the narrow high DOPS areas below the air-line. For high normalized frequency, the effective indices of these modes follows those of a capillary [18]:

where ρ_hole is the air-hole diameter and u_m1 is the smallest positive root of the Bessel equation J(u)=0 _m. Consequently, since the holes in the square-lattice cladding are larger than the ones in the Kagome-lattice cladding, the effective index n_eff of these modes is closer to the air line and the core-guided mode for the square geometry. This co-existence of core modes with various classes of cladding modes at nearly the same coupling conditions (i.e. similar n_eff) and at the same wavelength (i.e. same kΛ), in conjunction with the low attenuation of the core mode, confirms the predicted inhibited coupling between cladding and core modes constituents.

It is noteworthy that, although the core size of the fibers is such that higher order modes (HOMs) are guided in the core, their loss is significantly higher than for the “fundamental” (HE₁₁-like) core mode (Fig. 5(e)). Indeed, HOM effective indices are close to matching those of compatible cladding modes (Fig. 5(f)) and the resulting hybridization of the HOMs throughout the cladding causes their higher loss. As a result, these broadband fibers exhibit a far better single-modal behavior than bandgap fibers with a similar core size.

 

Fig. 6. (A). Top: Measured group delay after a propagation in a 1 meter long single cell #4 fiber (open circle) and a nonlinear fit of the experimental data (solid line). Bottom: Experimental group velocity dispersion of the single cell #4 fiber (red open circles), dispersion inferred from the fit to the experimental group delay data (black line), and that of a silica capillary with 52 µm diameter. (B). Crossed-polarizers experiment: Near-field profile after propagation through 10m of single cell #4 fiber with output polarizer/analyzer (top) aligned or (bottom) crossed with the input polarization. The residual light in crossed configuration is in a higher-order mode that accounts for a polarization cross-talk figure of -13.4dB.

Download Full Size | PDF

According to a Kramer-Kronig relation, the flat, ultra-wide transmission bandwidth achieved in the square-lattice fibers presented here should yield a group velocity dispersion that is close to an ideal air-waveguide dispersion in the middle of the transmission band; the effects of the resonant features causing high loss - and hence, higher dispersion - act only towards the edges of the guided band. The group velocity dispersion of fiber Single Cell #4 has been measured using an interference method. Figure 6(left) presents the delay incurred by the light as a function of wavelength as it travels through 1m of fiber and the group velocity dispersion calculated as the derivative of this measured delay. The observed dispersion is around 1ps/nm/km over the whole 600nm-wide transmission band and closely matches the dispersion of a silica capillary of similar core size (deduced from Eq. (1)). This ultra-low chromatic dispersion figure for the fundamental core guided mode is attributed to a combination of low light-in-glass fraction, the broad wavelength range of high transmission and the relatively large core size. It is, to our knowledge, the lowest ever reported for a broadband HC-PCF and further confirms the inhibited interaction of the fundamental core mode with silica. One drawback of the dispersion measurement method used here is that it does not take into account weak and narrow resonant features, visible in the attenuation spectrum, that result in small variations of the dispersion. Further work would be needed in order to confirm whether these variations could be problematic for example in future femto-or atto-second pulse applications.

As the fiber core mode is mainly guided in air, its birefringence is also expected to be ultra-low, so that one can expect the polarization of a guided mode to behave as if it is propagating through free-space. Figure 6(b) presents the near-field profile at the end of 10m of square lattice fiber Single cell #4 coupled to a linearly polarized 1064nm laser, when a polarizer/analyzer at the output of the fiber is aligned (top) or crossed (bottom) with regards to the input polarization. The measured polarization extinction ratio between the two analyzer’s positions is 13.4dB, indicating that the polarization maintained mostly linear. Such a degree of polarization conservation is usually associated with a large birefringence of the fiber and with the requirement for the input polarization to be aligned with one of the polarization axes of the fiber. The square-lattice HCPCF, however, does not exhibit a birefringence that is high enough to be detected using typical polarization characterization experiments (wavelength sweep or mechanical perturbation method), indicating a beat length of the order of a meter or more, similar to - or potentially greater then - that of spun SMF.

The polarization properties of the square lattice large pitch fibers are closer to that of low PMD SMF than conventional band-gap HC-PCF [19] in that the fiber will maintain the polarization so long as no undue stress or bending is applied to it. As a matter of fact, we generally found the dependence of optical properties to bending in the square-lattice fiber to be similar to that of the Kagome form for comparable core sizes [11]. The 10m sample used for the experiment above was bent with a 20cm diameter along its length and held with two magnets at the input and output end of the fiber. The observed polarization cross-talk could therefore come from these experimental conditions and could be reduced by using a straight length of fiber or alternative fiber holding solutions. The weak presence of higher-order core modes observed with crossed polarizers may also contribute to the measured polarization cross-talk (PCT) value.

5. Conclusion

Two types of square lattice large pitch HC-PCF have been fabricated and characterized in terms of attenuation, dispersion and polarization properties. The fibers have similar guidance properties to the Kagome-lattice HC-PCF and can therefore be used in applications requiring broadband transmission and an extremely high quality spatial field emanating from a square-or rectangular-shaped core defect. A geometry which shows considerably reduced loss over broad wavelength bands has been identified by numerical modeling, and efforts are under way to realize it experimentally.

The fabricated fiber exhibits an ultra-low dispersion characteristic over a wide wavelength range, as well as beneficial polarization properties, making it the perfect candidate for non-linear applications where both polarization and dispersion need to be accurately controlled.