It is now commonly accepted that, in large pitch hollow-core ‘kagomé’ lattice fibers, the loss spectrum is related to resonances of the thin silica webs in the photonic crystal cladding. Moreover, coherent scattering from successive holes’ layers cannot be obtained and adding holes’ layers does not decrease the loss level. In this communication, cross-comparison of experimental data and accurate numerical modeling is presented that helps demonstrate that waveguiding in large pitch hollow-core fibers arises from the antiresonance of the core surround only and does not originate from the photonic crystal cladding. The glass webs only mechanically support the core surround and are sources of extra leakage. Large pitch hollow-core fibers exhibit features of thin walled and thick walled tubular waveguides, the first one tailoring the transmission spectrum while the second one is responsible for the increased loss figure. As a consequence, an approximate calculus, based on specific features of both types of waveguides, gives the loss spectrum, in very good agreement with experimental data. Finally, a minimalist hollow-core microstructured fiber, the cladding of which consists of six thin bridges suspending the core surround, is proposed for the first time.
©2010 Optical Society of America
One of the most outstanding breakthrough in fiber optics has been the possibility to guide light in air, over kilometric distances, by means of an all-dielectric hollow-core photonic bandgap fiber (PBGF) . The guiding mechanism in PBGF is well understood. Light confinement in the air-core arises from antiresonances of the resonators formed at the apices of the air-silica honeycomb lattice. The struts joining the resonators are made small enough not to support parasitic modes that could detrimentally couple to those of the resonators. In other words, the photonic crystal exhibits a full photonic bandgap which forbids propagation through the cladding at the operation wavelength. Very low level of confinement loss may be reached by increasing the number of cladding layers. Furthermore, low (1 dB/m) loss air-guiding has been obtained in so-called “kagomé-latticed” photonic crystal fibers (K-PCF)  for which an explanation of the guidance mechanism has been proposed earlier [3,4]. Contrary to PBGF which exhibits a full photonic bandgap, the photonic crystal of K-PCF does not exhibit full bandgaps around the air-line but spectral regions with low “density of optical states (DOS)” separated from each other by high-DOS regions (see DOS plots in ). Most optical states are highly dispersive, high-order transverse modes guided in the silica bridges from which the cladding is made up. In low-DOS regions, opposite parities of core and bridges’ modes is expected to inhibit their coupling [3,4], thereby implying a relative low-loss transmission through the fiber core. In high-DOS regions, coupling to even bridges’ modes imply a high loss level. On the one hand, this model predicts the location of the transmission windows with great accuracy. The model proposed, however, is qualitative and does not give quantitative information about the loss figure which is of primary importance for any waveguiding application. Later, it has been proposed that K-PCF could be modeled by the Bragg fiber (BF) . The opto-geometrical parameters of the high-index rings of the BF (i.e. their thickness t and their refractive index n) are chosen to be equal to those of the bridges of the K-PCF of interest. Again, the location of the transmission windows is very accurately predicted by the cut-off wavelengths of the rings’ even modes. It is worth noting that Pearce et al., in , numerically demonstrate that adding hole-layers (from 2 to 4) does not decrease the loss figure. Furthermore they infer, from the numerical results, that the connecting struts are responsible for the increase of the loss figure when compared to the ‘unconnected’ Bragg fiber model. It is worth noting that the loss figure, either computed  or measured  is always higher than approximately 1 dB/m.
Low-index contrast photonic bandgap fibers, such as the Bragg fiber , guiding light in a solid-core, are a convenient platform for understanding bandgap guiding mechanism. In stricly unperturbed fiber, coupling from the core to the cladding is allowed for modes with the same parity that is, as far as the fundamental HE11 mode is concerned, around cut-off wavelengths of cladding even modes. These coupling regions separate low-loss regions from each other. When the fiber is perturbed by an antisymmetric perturbation, such as a gentle bend, although very weak, the coupling between even core and odd cladding modes is no longer inhibited, thereby leading to the appearance of discrete loss peaks in the low-loss regions , reaching ~1dB/m. The apparent likeness of behavior in solid-core BF and in K-PCF suggests that the inhibited coupling assumption may not be valid.
In this communication, we further improve the knowledge of the waveguiding mechanism by identifying the physical causes responsible for the increase of the loss figure in the realistic fiber. To do so, experimental data obtained by characterizing an in-house fabricated fiber are depicted and confronted to those given by very accurate two dimensional modeling of K-PCF. Perturbations giving rise to mode coupling are then identified. Conclusions about the loss limits are drawn from this cross-comparison and a very simple model for the accurate determination of the loss is proposed. Finally we propose, for the first time to our knowledge, a minimalist design of hollow-core microstructured fiber, which performances are as good as those of more conventional and complex designs.
2. Characterization of the manufactured fiber
A K-PCF allowing for air-guiding in several transmission windows was designed, manufactured by the stack-and-draw process and characterized . The bridges’ thickness t and the pitch Λ were measured, by means of scanning electron microscope, to be 600 nm and 12 µm, respectively. The diameter of the seven-cell core was measured to approximately 32 µm. According to [4,5], the transmission windows are located in between cut-off wavelengths of spatial modes guided in the silica bridges from which the cladding is made up and given by where n and m stand for the glass index and the radial order of the guided mode, respectively. The loss spectra were measured and found in excellent agreement with that computed by the full-vector finite element method (FEM) . It is worth noting that the accuracy of the loss computation was tested beforehand. To do so, the waveguide consisting of a hollow core surrounded by a pure silica glass cladding, for which the HE11 mode loss can be analytically computed as a function of n and RC , was modeled. The departure from the analytical result is a low as ± 0.2% over the broad wavelength range scanned.
When the fabricated fiber input is illuminated by means of a white light source, without any selective launching, the output beam pattern, shown in Fig. 2 , reveals the various guiding regions. The seven-cell core is the lowest loss and broadband guiding region. The antiresonant nature of the waveguiding is evidenced when the cladding holes are considered. Each of them is a poorly efficient waveguide, guiding light is a narrow range of wavelengths, the selective spectral transmission of each giving the characteristic colors observed. The innermost cladding holes guide within the blue region of the spectrum. This means that the antiresonant condition is collegially satisfied for the blue wavelengths and that the bridges’ thickness is rather uniform there. On the contrary, the cladding bridges’ thickness varies for outermost holes, and so do their transmission windows. On the other hand, the fiber modeling assumes a uniform thickness across the whole cross-section. Therefore the very good agreement observed between experimental and numerical results supports the assumption that the waveguiding occurs thanks to the antiresonance of the core surround only.
3. Thorough modeling of the manufactured fiber
The manufactured fiber was then modeled by the Bragg fiber. The parameters of the BF are the number of rings N, their thickness t = 600 nm, their index n = 1.45. The rings are located according to the geometry of the K-PCF after Pearce et al. . Figure 3 compares the spectrum computed by the FEM for N = 2 and N = 1 (see models in inset) and the experimental data.
As far as the Bragg fiber model is concerned, the coherent nature of the waves reflected and transmitted through the various cladding layers implies a drastic reduction of the confinement loss as the number of layers is increased. For N = 2, the imaginary part of the modal effective index is so low that artifacts related to the numerical accuracy appear in the spectrum. Second, the resonant wavelengths coincide with the high-loss peaks measured in the K-PCF transmission spectrum. Finally, it is worth noting that whatever the number of high-index rings, the Bragg fiber model underestimates the loss throughout the spectrum. Two conclusions can be drawn from these observations: (i) the waveguiding is indeed obtained by the sole innermost silica ring, i.e. the core surround, the resonances of which tailor the spectrum, and (ii) the hole-layers do not play a positive role.
To check this assumption the actual cross-section was modeled. The influence of N was evaluated by varying N from 0 to 4 and the theoretical results were plotted in Fig. 4a . Examples of models used are shown in inset.
As shown in Fig. 4a, the number of hole-layers only weakly influences the loss figure, confirming the fact that the periodic cladding plays a minor role in the waveguiding mechanism. This is in total contradiction with the behavior of Bragg fibers exemplified in Fig. 3. The electric field distribution was recorded at λ = 0.5 µm and plotted in inset to Fig. 4 on a logarithmic scale. It is worth noting that air-guided mode couples to high-order bridges’ modes. Along the smallest dimension of the bridge, the mode exhibits three oscillations. The order m = 3 is consistent with the fact that λ = 0.5 µm belongs to the third transmission window. Finally, the whole microstructured cladding was removed in an attempt to prove that air-core waveguiding is obtained by destructive interferences in the core surround only and perturbed by coupling to bridges’ modes. As shown in Fig. 4b, the loss figure obtained with the isolated core surround is by far lower than the experimental data (or the value computed for N = 4). We thus infer that the microstructured cladding does not play a positive role in the waveguiding process which originates from antiresonances in a single high-index material layer as shown in [12,13]. The coupling between the air-guided mode and the high-order bridges’ modes is not inhibited and accounts for the extra leakage. Moreover, it is worth noting that the loss level for N = 0 in Fig. 4b does not reach that in Fig. 3, evidencing the fact that the shape of the core surround affects the loss level. This easily can be understood remembering the fact that the core mode consists of spherical waves starting from the center of the fiber and impinging on the core – cladding interface, bouncing back and forth to build up a stationary wave. Obviously a perfectly circular interface is more likely to produce a constant dephasing and therefore a lower loss mode.
Plotted on a logarithmic scale, the spectrum obtained for the dodecagonal shape of the actual core surround exhibits some features (see solid line in Fig. 4b) whereas that of the single-layer BF is perfectly smooth. Then, as an example, the features located around λ = 0.725 µm were thoroughly analyzed for the dodecagonal core surround. The numerical calculations were carried out in 0.2 nm steps from 0.71 µm to 0.75 µm. The real part of the effective index and the associated loss have been plotted in Fig. 5 .
Strong couplings between the core mode and some highly dispersive high-order modes of the core surround can be observed at, e.g., 0.717, 0.725, 0.734 and 0.742 µm, that is far from the transmission window edges, each of which being associated with high loss since the modes of the core surround approach their cut-off. The distribution of the electric field has been recorded at 0.726 µm and 0.729 µm and plotted in inset to Fig. 5. At λ = 0.726 µm the core mode couples to m = 2 mode supported by the core surround. The associated loss is about 11 dB/m. On the other hand, at λ = 0.729 µm, the mode is confined to the air core and the loss reaches 0.9 dB/m. Remembering the fact that the spectrum obtained for the circular core surround does not exhibit such features, it is concluded that the coupling between the air-guided mode and the bridges’ modes is primarily due to the shape of the core surround that breaks the symmetry of the modes. Obviously this must occur in the actual structure with at least one hole-layer. Then, a single hole-layer was added and the same computations were carried out. Results have been plotted in Fig. 5c where no discrete couplings can be discerned. However, the overall loss level is increased by one order of magnitude compared to Fig. 5b, evidencing the fact that the light spreads into the cladding through the core surround and the silica bridges. As a consequence, the wavelets composing the air-guided mode dephase differently when passing through the silica bridges or through the air holes. This explains why adding reflectors (silica layers) does not improve the loss figure as shown in Fig. 4a.
4. Proposition for a simplified model
Manenkov derived that, in a very thin walled tubular waveguide as that shown in inset to Fig. 3, the air-guided mode loss is proportional to λ3/RC 4 where RC stands for the air-core radius . On the other hand, Marcatili and Schmeltzer showed that, in the thick walled tubular waveguide with core radius RC and t → ∞, loss of the HE11 depends on λ2/RC 3 .
In K-PCF, the antiresonance from the core surround allows for waveguiding whereas the coupling to the continuum of cladding radiative modes implies an increase in the loss figure. Such fibers are therefore at the border between thin walled and thick walled tubular waveguides, the first one fixing the transmission windows and the second one being responsible for the increased loss figure.
We therefore anticipate that the loss spectrum of K-PCF may be inferred from that of the circular core surround divided by a factor of λ/RC. In Fig. 6 , the loss spectrum of the manufactured fiber is compared to that computed for the one-layer Bragg fiber divided by a factor of λ/RC. An excellent quantitative agreement is obtained.
Therefore, the loss spectrum of highly complex structures such as ‘kagomé’ lattice hollow-core PCFs can be calculated in a straightforward manner, from the transfer matrix method for instance.
5. Minimalist hollow-core microstructured fiber
In the previous section, we have demonstrated that the waveguidance in such kagomé-lattice HC-PCF originates from the core surround. In the following, we propose a minimalist hollow-core microstructured fiber composed of a hollow core surrounded by a single layer of holes. The six large air holes merge and, provided the adequate overpressures are used during the fabrication process, should deform and create the hollow core surrounded by a thin web of silica. The proposed design is shown in Fig. 7a while the computed intensity distribution is shown in Fig. 7b.
The various holes have been deformed as expected from our previous drawing experience . The core surround allows for efficiently guiding a HE11-like mode as shown in Fig. 7b. Then, a very first fiber was fabricated in house by the stack-and-draw process . An optical microscope photograph of the fiber end facet is shown in inset to Fig. 8 . It is worth noting that the first silica layer is actually very thin (t was measured to be of the order of 430 nm). The core diameter is about 30 µm along the shortest dimension, rather similar to that of the fiber depicted in the first part. The loss spectrum was measured by the cut-back technique and plotted in Fig. 8.
The fiber exhibits one transmission window in the visible. The UV transmission windows were not investigated. The loss level is rather low (~1 dB/m). The straightforward quasi-analytical model developed in the previous section was applied to the newly fabricated design, assuming t = 0.43 µm and RC = 15 µm. The confinement loss spectrum was computed and the results overlaid in Fig. 8. Despite the discrepancies between the modeled and the actual fiber, a quantitative agreement is obtained, showing that this kind of fiber is intrinsically resistant to morphology deformations. The slight discrepancies can be explained by the presence of one or two high-order modes that increase the overall measured loss figure.
In summary, hollow-core kagomé-lattice photonic crystal fibers have been studied both theoretically and experimentally in order to identify the causes responsible for the loss as well as determining the minimum loss level achievable in a realistic design. It was confirmed that the waveguiding originates from the core surround only and not from the microstructured cladding which acts as a mechanical support for the core surround. The non-circular shape of the core surround breaks the cylindrical symmetry of the air mode and implies coupling to the antisymmetric core surround modes. Then, the bridges couple core light to the cladding radiative modes. Consequently, the loss spectrum can be inferred from quasi-analytical computations of the HE11 mode of the lone core surround, and divided by a factor of λ/RC, accounting for the increase in loss. Then, a minimalist design has been proposed in which the photonic crystal cladding was removed. Low (~1 dB/m) loss was measured as well as computed by the straightforward model developed.
Thanks to the straightforward model developed, it is now possible to predict with good accuracy the confinement loss in a hollow-core antiresonant fiber operated in the short wavelength domain. The main conclusion is that, at a fixed wavelength, the loss level decreases with the increasing core diameter. Therefore, the loss can be made arbitrarily low provided (i) the singlemodedness is not a requirement and (ii) the ratio t/RC can be decreased during the fabrication process.
This work was funded by the French National Research Agency through grant ACI JC 9142 and French Délégation Générale à l’Armement through grant n°05 34 060. The computations were carried out using Cali, the cluster of the University of Limoges. The authors thank the staff of Xlim – Platinom for the fabrication of the fibers.
References and links
1. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic bandgap guidance of light in air,” Science 285(5433), 1537–1539 (1999). [CrossRef] [PubMed]
2. F. Benabid, J. C. Knight, G. Antonopoulos, and P. St. J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298(5592), 399–402 (2002). [CrossRef] [PubMed]
4. F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multi-octave optical-frequency combs,” Science 318(5853), 1118–1121 (2007). [CrossRef] [PubMed]
5. G. J. Pearce, G. S. Wiederhecker, C. G. Poulton, S. Burger, and P. St J Russell, “Models for guidance in kagome-structured hollow-core photonic crystal fibres,” Opt. Express 15(20), 12680–12685 (2007). [CrossRef] [PubMed]
7. S. Février, R. Jamier, J.-M. Blondy, S. L. Semjonov, M. E. Likhachev, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Y. Salganskii, and A. N. Guryanov, “Low-loss singlemode large mode area all-silica photonic bandgap fiber,” Opt. Express 14(2), 562–569 (2006). [CrossRef] [PubMed]
8. F. Gérôme, S. Février, A. D. Pryamikov, J.-L. Auguste, R. Jamier, J.-M. Blondy, M. E. Likhachev, M. M. Bubnov, S. L. Semjonov, and E. M. Dianov, “Highly dispersive large mode area photonic bandgap fiber,” Opt. Lett. 32(10), 1208–1210 (2007). [CrossRef] [PubMed]
10. A. Peyrilloux, S. Février, J. Marcou, L. Berthelot, D. Pagnoux, and P. Sansonetti, “Comparison between the finite element method, the localized function method and a novel equivalent averaged index method for modelling photonic crystal fibres,” J. Opt. A, Pure Appl. Opt. 4, 257–262 (2002). [CrossRef]
11. E. A. J. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
12. A. B. Manenkov, “Quasioptics of waveguides with selective reflecting dielectric walls”, in Proc. of the fifth colloquium on microwave communications, Budapest, Hungary, 24–30 June, 1974.
13. M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49(1), 13–15 (1986). [CrossRef]
14. P. Viale, “Management of nonlinear effects in photonic bandgap fibers,” Ph. D. thesis, University of Limoges, n°42–2006 (2006)
15. F. Gérôme, R. Jamier, J.-L. Auguste, G. Humbert, and J.-M. Blondy, “Simplified hollow-core photonic crystal fiber,” submitted to Conference on Lasers and Electro-Optics (2010)