We report on the experimental visualization of the cladding Bloch-modes of a hollow-core photonic crystal fiber. Both spectral and spatial field information is extracted using the approach, which is based on measurement of the near-field and Fresnel-zone that results after propagation over a short length of fiber. A detailed study of the modes near the edges of the band gap shows that it is formed by the influence of three types of resonator: the glass interstitial apex, the silica strut which joins the neighboring apexes, and the air hole. The cladding electromagnetic field which survives the propagation is found to be spatially coherent and to contain contributions from just a few types of cladding mode.
© 2007 Optical Society of America
Hollow Core Photonic Crystal Fibers (HC-PCF) have been the object of an ever growing interest over the past decade, with applications spanning from telecoms  and metrology , to gas-laser systems [3-6]. In these fibers, the guidance is not achieved via total internal reflection but through the trapping of the light in a core defect by an out-of plane photonic band gap (PBG) created by the 2-dimensional photonic crystal cladding . Both theoretical and experimental work has previously been carried out in order to study and understand the guidance properties of these fibers. These studies include dispersion characteristics, modal attributes, anti-crossings between core-guided and core-interface modes and the scaling laws for the PBG location (see e.g. references [8-13]). These studies were mostly related to the defect modes, i.e. core modes or interface modes located within the cladding band gap, and little work has been reported on the nature of the cladding states which exist outside the band gap spectral region. An understanding of how light propagates within different constituent features of the photonic crystal cladding should provide insight on how out-of-plane PBGs form.
An interesting approach to understanding optical guidance in waveguides with micro-structured media as a cladding was provided by Yariv and Yeh in analyzing their proposal of Bragg fiber (i.e. a fiber with 1-D cladding made of concentric solid rings with alternating indices) in the 1970s [14,15]. In that proposal, they pointed out that the allowed bands (i.e. cladding modes) which sandwich the bandgaps in the β-ω diagram relating mode propagation constant (β) to the optical frequency (ω), are composed of several constituent modes mainly confined in the high-index layer and whose number is given by the number of these layers. As the frequency and the effective indices of the modes in a particular allowed band become higher, this band shrinks to a set of degenerate modes confined in the high-index layers. This observation clearly indicates the role of the resonance properties of the high-index layer in the formation of the bandgap .
The work of Yeh and Yariv gave rise to the Anti-Resonant Reflective Optical Waveguides (ARROW) model , which has since been successfully applied to some classes of photonic crystal fibers [18,19]. In these studies [18,19], the authors consider PCFs with a structure made of a low-index background material with isolated high-index cylindrical inclusions. They show, using the ARROW model, that the frequencies of transmission bands are primarily determined by the geometry and the modal properties of the individual high-index inclusions irrespective of the lattice constant value. Consequently, one can consider the high-index inclusions as optical resonators whose modal properties determine the properties of the formed bandgaps. More recently, Birks et al  proposed a semi-analytical approach to understanding the formation of bandgaps in all-solid bandgap PCFs with the same type of cladding structure as in references [18, 19]. The authors modeled the out-of-plane mode structure of the fiber cladding by employing an approximate technique related to the cellular method used in solid-state physics . This method explicitly takes into account electromagnetic coupling between each high index cylinder within the cladding and, in contrast to the ARROW model, provides information on the width of the cladding pass-bands. The model also elucidates the hybridization which occurs between resonances associated with the localized high index regions and the low index regions which fit between them.
Numerical solutions to Maxwell’s equations have also confirmed that the cladding pass-bands in all-solid band gap fibers are formed from modes of the individual high index inclusions  based on plane wave scattering of a single cylinder. The connection between the scattered field and cylinder modes was also described . These solutions verified that the width of a particular pass-band is related to the confinement of the field within each inclusion . The light confining properties of a core introduced within an arrangement of high index inclusions has also been linked to the plane-wave scattering properties of the individual inclusions . In particular, a high degree of confinement has been linked to a Fano resonance effect caused by the interference of different multipole components within the scattered field. Such resonances are, however, only observed for well separated scatterers.
The above approaches provide important and useful insights into the guidance mechanism of Bragg and all-solid photonic crystal fibers which comprise arrangements of separated high index regions within a lower index background, and on the nature of bandgap formation within the claddings of these fibers. However, the complex topological structure of the silica-air PCF cladding, such as possessed by the HC-PCF shown in Fig. 1, makes it very challenging to identify the relevant optical resonators responsible for the PBG formation. Indeed, the singly-connected nature of the high index component of such fibers calls into question the applicability of the picture of guidance and band gap formation based on the influence of a few constituent elementary resonator features. Consequently, in order to assess the applicability of a resonator picture as an aid to designing realistic HC-PCF, it is useful to develop experimental or theoretical tools aimed at identifying any relevant cladding features which may be present.
In this article, we report on experimental and theoretical results related to the modal properties of the cladding modes of fabricated HC-PCFs with a triangular lattice arrangement defining the cladding geometry (Fig. 1) . Two experimental procedures have been used in the study, both of which involve optical transmission over short lengths of fiber. The first entails optical imaging of the near-field and Fresnel zone close to the air light-line (i.e. β=ω/c, with c the speed of light in vacuum). Since the cladding electromagnetic field which survives the propagation is found to possess a high degree of spatial coherence, the field diffracting from the end of the fiber into free-space shows a complicated but stable self-interference pattern, rather than a speckle-form which would result from a superposition of randomly phased contributions from a number of different cladding modes. By comparing the observed Fresnel evolution with modeled results, the dominant cladding mode contributions can be extracted. The second experiment is based on the simultaneous measurement of the fiber near field pattern and the transmission spectrum recorded at particular features within its core and cladding regions using a Scanning Near-field Optical Microscope (SNOM). The results from the experiments show that the upper-edge of the band-gap is determined by a single resonator
The experiments, in conjunction with numerical simulations, also establish which cladding modes are best transmitted over a few millimeters of HC-PCF for different wavelength ranges. Cladding modes are of interest in their own right since they can probe the holes within the cladding as well as within the core, thus making them suitable for some sensing applications where preferential channeling of the sample into the central core is not possible .
2. Numerical modeling of the HC-PCF cladding modes near the photonic bandgap
In order to understand the nature of the cladding states on either side of the band gap, the coupled resonator picture  will be reviewed and applied to a HC-PCF. A HC-PCF is composed of an air core defect surrounded by a periodic lattice of air holes in glass. The triangular lattice fiber (Fig. 1), now widely available commercially, has been identified as supporting a low attenuation HE11-like core mode [27,28]. For our current purposes, sufficient information is encapsulated by the cladding mode density-of-states (DOS) [29,30] plotted as a function of wavenumber k = 2π/λ and effective index neff =β/k. Here λ is the vacuum optical wavelength.
The DOS diagram, calculated for a crystalline approximation to the cladding of the fiber shown in Fig. 1, is presented in Fig. 2(A). The modeled cladding structure comprises a triangular lattice arrangement of hexagonal holes with rounded corners , characterized by an air filling fraction of 91.9% and a meniscus radius at the corners of 0.24Λ, with Λ being the lattice pitch. A plane wave approach, similar to the one presented in , was used for the calculation.
The band gap creates a finger-shaped region near the air light-line (neff=1) where the DOS falls to zero and propagation of light through the cladding is forbidden [shown in black within Fig. 2(a)]. Either sides of this region are allowed bands where the cladding can support propagating modes [regions 1 and 2 in Fig. 2(a)]. In the case of an infinite cladding structure, these cladding modes form a continuum (the cladding modes are also known as Bloch modes of the periodic medium). For higher normalized wavenumbers kΛ, the upper allowed-bands (within region 1) shrink to an extremely narrow region in the k-neff plane and the light is well localized within the glass apexes which exist between three nearest-neighbor holes. This behavior is amenable to a tight-binding description of band formation familiar from solid state physics, whereby electronic bands in a solid (e.g. a crystal) are considered to be the result of bringing together isolated and identical atomic-like orbitals  so that a degree of spatial overlap is incurred. In the context of the fiber cladding, the allowed bands result from an overlap of a mode associated with a particular structural feature, in this case a glass apex, with equivalent modes of neighboring apexes. The overlap increases with increasing wavelength (due to the consequent reduction in confinement) or with a decrease in separation of neighboring apexes (which is driven by the lattice constant Λ). The effect of the overlap is to cause the mode-line associated with individual apex states to spread and form a band of finite width in the k-neff plane, and also to introduce a delocalized character to the cladding states . This picture is well illustrated by the shape of region 1 in Fig. 2(a), and in Fig. 2(e) which shows the DOS over an extended frequency range. Figure 2(a) also shows a selected set of modes (colored lines) in both regions. The solid lines correspond to modes found at the Γ-point of the Brillouin zone and dotted lines to J-point modes [see Fig. 2(f)]. As there are two glass apexes within a primitive unit cell of the cladding crystal, the tight binding model indicates that, at the Γ-point of the reciprocal lattice Brillouin Zone, the modes in region 1 will either be “symmetric” or “anti-symmetric”. The symmetric form is characterized by the fields in all the apexes being in-phase with one another, whereas the field changes sign between nearest neighbor apexes for anti-symmetric modes. The symmetric mode is known as the fundamental space filling mode  and forms the upper boundary to region 1, i.e. its mode-line [shown by the purple line in Fig. 2(a)] designates the cladding cut-off index variation with wavenumber k. Of particular interest with regards to guidance within HC-PCF is the antisymmetric mode since it forms the upper edge of the out-of-plane photonic bandgap of the cladding. In the remainder of this paper we shall term this mode as the “apex” mode. The dispersion of the apex mode is represented in red in Fig. 2(a) and its near-field, which is shown in Fig. 2(b) at a representative normalized wavenumber kΛ, confirms that the light is predominantly guided in the interstitial apexes. The apexes are thus identified as the most important optical resonators associated with the upper bandgap edge .
The nature of the modes at the lower edge of the PBG is more complicated than those at the upper edge. Indeed, the lower edge is formed from the trajectories of two cladding modes of different symmetry. At frequencies (i.e. kΛ) below kΛ=16.9, the edge is due to modes associated with the J-point within the Brillouin zone, and above this frequency, the edge is due to a mode located at the Γ-point. The modes at the J and Γ symmetry points which form part of the lower band gap edge are represented by the dotted green and continuous blue traces in Fig. 2(A), respectively. The J-point mode [Fig. 2(d)] guides predominantly in the air-holes of the cladding lattice and hence we shall call it an “airy” mode, whereas the Γ-point mode [Fig. 2(c)] guides predominantly within and close to the silica struts which join neighboring apexes with little of the field penetrating into the apexes. It is now clear what effect the silica struts have on the performance of the HC-PCF: they directly limit the upper-frequency of the fiber transmission band. Compared to an all-solid bandgap fiber form (assuming the same index contrast) which has no struts connecting high index inclusions, the connected HC-PCF structure will therefore show narrower guiding wavebands. Moreover, the curve of the J-point airy mode, which forms the lower bandgap edge below kΛ=16.9, suffers from anti-crossing events with modes associated with glass features, thus limiting the PBG depth. One such event, occurring around kΛ=18, is responsible for the point of inflection in the mode trajectory of the airy mode (green line) observed near this frequency value. The influence of this strong anti-crossing is long-ranged in frequency and introduces a field component to the J-point mode at the bandgap edge which lies within and near the glass struts. Similar anti-crossing events are in fact commonplace within region 2 and result in cladding modes which typically show a high degree of hybridization between air-guided and silica-guided components. Such mode interactions also cause higher-order cladding band gaps to close up [see Fig. 2(e)] at kΛ=26).
Well below the lower bandgap edge, the interactions are numerous and involve a variety of higher-order resonances associated with the various cladding features, so that the intuitive resonator-based picture of cladding states becomes of less use. Near the bandgap, which is the region of most relevance to HC-PCF guidance, sufficiently few resonators are involved in forming the cladding modes for the resonator picture to be a useful conceptual framework.
3. Optical imaging of HC-PCF cladding modes
3.1 Propagation and diffraction properties of HC-PCF cladding modes
In order to find out whether it is possible to measure and identify the cladding modes surviving propagation through a short length of fiber using a simple optical imaging technique, we shall discuss some relevant properties of these modes. The influence of the core and of structural disorder on the cladding modes will be neglected in the analysis. The cladding field at input can then be decomposed into contributions of the various cladding modes throughout the Brillouin Zone.
The cladding field components which best survive propagation along the fiber will be characterized by a small transverse group velocity vg⊥ component within the crystal plane, since the leakage rate from a finite cladding region is approximately proportional to ∣vg⊥∣. For states within the n’th band, vg⊥ is determined from the band dispersion by
where kn(k ⊥,β) is the wavenumber of the n’th band at the point k ⊥ within the first Brillouin zone for propagation constant β.
The transverse group velocity and the DOS are in fact inextricably linked. The number of states per unit fiber length and per unit cell within infinitesimal wavenumber and propagation constant intervals δk and δβ, respectively, is given by ρ(k,β)δkδβ where
with the contour Cn in the first Brillouin Zone defined by kn(k⊥,β) = k and Ac the unit cell area. Only bands which cross the wavenumber value of k contribute in the summation. The density of states ρ(k,n eff) plotted in Fig. 2(a) is related to ρ(k,β) by ρ(k,n eff)=kρ(k,β). A high DOS is thus associated with a low transverse group velocity and consequently a slow leakage from the region of cladding near the core. Stationary points, where vg⊥ tends to zero, can occur in the band dispersion surfaces. Generally these occur at positions of high-symmetry such as the Γ- and J-points. The DOS in the vicinity of these stationary points is determined by the band curvature, with low curvature implying a high DOS.
If the optics at input excites cladding modes at wavenumber k over a spread of β-values Δβ, initially the evolution of the field within the cladding is complex, due to the various cladding mode components acquiring different phases due to propagation. Further along the fiber, most components have leaked out and only modes with low vg⊥ and preferentially associated with a high DOS within the spread Δβ remain. The field then has a much simpler structure, typically comprising states from one or two bands within small regions of the Brillouin zone centered at high-symmetry points. Often the main part of the field is ascribable to a single (or perhaps doubly degenerate) band close to a single high-symmetry point. The field will then show the same spatial dependence as a single (perhaps degenerate) cladding mode. Thus the cladding field, after adequate propagation, will regain a high degree of spatial coherence, since only very small regions of (k ⊥,β) space contribute. This implies that the field escaping from the output end of the fiber will show a self-interference pattern devoid of speckle. An illustration of such a diffraction pattern is shown in the animations of Fig. 3.
The diffraction patterns shown here were calculated using the same methodology as has been used for a core guided mode . The field is obtained by applying the electromagnetic boundary conditions at the fiber end to match the cladding mode field to the fixed-frequency plane-wave eigenmodes appropriate to free-space. The field within the fiber is assumed to comprise an incident field in the cladding mode and a component which is reflected back into this mode at the fiber end. The matching procedure gives expressions for the plane-wave components which can be expressed in terms of a 2-d FFT. The FFT is taken over a supercell which contains an integer number of primitive unit cells. A supercell is used rather than a primitive cell to enable a higher density of k ⊥-space sampling. The supercell was taken to be a 6Λ×4√3Λ rectangle. For cladding modes not located at the Γ-point of the Brillouin zone, the origin of k ⊥-space is appropriately shifted so that the periodic boundary conditions implicit with the FFT remain appropriate. The field is propagated a distance z from the fiber end by ikzz), where kz = √k 2 - k 2 ⊥. Evanescent as well as propagating components are retained in the field expansion.
Figures 3(a) and 3(b) show the evolution of the Fresnel zone patterns of the “apex” mode and the “strut” mode, respectively, as they propagate along a distance of 2Λ from the output end of the fiber. The strut mode was calculated at normalized wavenumber kΛ=18.2 at the bandgap edge where n eff=0.995. The apex mode was calculated at kΛ=14.5 and n eff=0.995, corresponding to the long wavelength bandgap edge. In comparison to the strut mode, the apex mode exhibits a much slower and simpler evolution of the diffracted field from the fiber end. This is in part due to the lower frequency of the apex mode, which implies a lower spatial resolution of the glass features close to the air light-line. The evolution of the field within the Fresnel zone contains more information than just the near-field, and measurements within this region can be used to infer which cladding mode components are present in experimental observations, as is done in the following section.
3.2 HC-PCF cladding-mode imaging using CCD camera
To optically image the HC-PCF cladding modes of interest, i.e. those close to the band gap edges, we excite a short length of HC-PCF at an effective index close to the air light-line. This is achieved by splicing a 3mm long HC-PCF, which guides with low loss (within its air core) around 800nm, to one end of an SMF-980 fiber (using the technique described in Ref. ). The pitch of the fiber is measured to be Λ=2.15μm. The SMF fiber is aligned such that the coupling of light (i.e. mode field matching) into the HE11-like mode of the PBG fiber is optimized for the frequencies lying in the HC-PCF transmission band. This ensures, via directional coupling, the excitation of cladding modes over a narrow Δβ range centered around an effective index close to the value neff =0.995 representative of the HE11-like guided mode within the band gap range [shown by a white dotted line in Fig. 2(a)].
The other end of the SMF-980 fiber is coupled to a super-continuum light source. This is generated by a commercially available ns-pulsed 1062nm microchip laser and 5m of highly nonlinear PCF (solid-core PCF with 5μm core diameter). The generated super-continuum is filtered to the desired wavelength using 3nm-bandwidth interference filters. The transmitted HC-PCF cladding modes are then imaged onto a CCD camera using a microscope objective.
The filter bandwidth of 3nm corresponds to a typical spread in normalized wavenumber of ΔkΔ∼0.05. This is a sufficiently narrow range for the cladding eigenmode field distributions not to vary significantly. Each frequency component gives rise to a different narrow range of β-values contributing at output. The diffracted intensity pattern from the fiber end shows some variation in spatial dependence with k, due mainly to an associated change in the phase factors exp(ikzz) in the free-space Fourier expansion. Over the propagation distances for which results are presented below, however, this variation is found to be small. Although the overall phase of the contributing frequency components varies over multiples of 2π, each component shows a closely similar distribution of phase variation within the image (x-y) plane. Since the frequency contributions show similar intensity and phase-difference distributions, the frequency spread does not cause a speckle-averaging in the collected Fresnel-zone patterns: similar results would be obtained if a narrow laser source were used instead.
A key issue with optically imaging the HC-PCF cladding modes with use of this technique at short wavelengths is associated with the rapid evolution of the field pattern from the end of the fiber: the pattern can change appreciably after propagation of distances less than 1μm. This often precludes a direct observation of the near-field. In order to overcome this limitation, we record the field evolution through the Fresnel zone and match it with the calculated evolution of candidate mode components. From the discussion in the previous section, one should expect identification of cladding modes can be accomplished by imaging the Fresnel zone from the fiber end. This is experimentally achieved by recording the image evolution as the fiber output end is moved away from the imaging lens, and along the optical axis of propagation, over a travel distance of ∼100μm. The initial position corresponds to the fiber end being in contact with the lens. Figure 4 shows an animated evolution of the imaged field over a selected length interval of ∼2-20μm. The input light is centered at 950nm, which, near the air light-line corresponds to excitation of cladding mode components associated with the apex resonator. The animation clearly shows a single mode propagating in a self-interference manner, free from components due to other cladding-mode types and matches very well with the calculated sequence [Fig. 3(a)].
A noteworthy feature inferred from the experimentally measured diffraction pattern is that the output cladding field is spatially phase coherent (i.e. free of speckle) and extended over many lattice periods, despite the input field being spatially confined to the core of the fiber. This is perhaps rather surprising, particularly in view of the structural disorder which necessarily occurs within the cladding of a fabricated HC-PCF. For very narrow bands associated with resonators with a small overlap, the disorder-induced phase mismatch between the individual resonator modes will halt the hybridization associated with Bloch state formation (for a periodic arrangement), resulting in cladding modes essentially confined to individual resonators. The band will then effectively break up into a point spectrum covering a larger β-range, Λβ say. Phase coherence (and negligible diffraction) associated with the contributions from the entire band will remain at the output end-face if the fiber length satisfies 
with λ the wavelength and Δneff the spread in effective index values. For the fiber length of 3 millimeters used in the experiments and λ∼1.0μm, an index spread of less than 3×10-4 is needed to maintain the phase coherence. This is somewhat narrower than the bands associated with the HC-PCF cladding over the wavelength range probed [c.f. Fig. 2(a)], and considerably narrower than the spread in β arising from the input coupling optics in the experiments. The maintenance of the spatial coherence is interpreted as being due to the selection of a narrow range of Δβ associated with a high DOS and low vg⊥, and preferentially a high DOS, as described in subsection 3.1. Moreover, the phase matching between individual resonators is sufficiently good and the overlap between neighboring resonators sufficiently high to cause the hybridization. In other words, over the range Δβ that contributes at output, the band dispersion is sufficiently high for the range Δk ⊥ of cladding modes at output to satisfy ∣Δk ⊥∣Λ<<1, so that transverse spreading due to diffraction is expected.
Using the above-mentioned imaging technique, we recorded the Fresnel zones at different wavelengths on both sides of the transmission band of the HC-PCF. Figure 5 summarizes the results for the relevant cladding modes.
The column (A) represents the calculated near-field patterns for the apex mode (kΛ=14.5), the strut mode (kΛ=18.2) and the airy mode (kΛ=19.2), respectively. The airy mode, located at the J-point, in fact shows a high degree of hybridization with glassy field components, as discussed in section 2. The column (B) in Fig. 5 shows the calculated Fresnel zone patterns of the three modes after they have propagated a distance of 1.8Λ, 1.5Λ and 2Λ respectively, from the fiber end. The column (C) shows the measured Fresnel zone patterns of the modes corresponding to wavelengths centered at 950nm, 750nm and 700nm respectively. The patterns correspond to an imaged cladding section covering ∼9 air-holes and located at 2-3 rings away for the fiber-core. The distance from the fiber end at which the pattern is taken was deduced from the position of the pattern’s frame in the animation collected as the fiber-end is moved away from the imaging lens to a final distance of 100μm. The deduced position of the diffracted pattern relative to the lens was found to be 4μm or 5μm, which is in agreement with the corresponding calculated distance since the fiber pitch is estimated as Λ=2.15 μm. The general matching between the experimental and theoretical results is excellent and makes this simple technique a rather powerful and accurate tool for imaging and identifying HC-PCF cladding modes.
From the images presented in Fig. 5, it becomes clear that the PBG is indeed caused by the interactions of the three identified resonators.
3.3 Mode Imaging and spectral analysis using SNOM
In order to corroborate the above findings, we have taken spectrally-resolved near-field images using a scanning near-field optical microscope (SNOM). Combined with the high power of the super-continuum source, the SNOM enables the detection of a spatially resolved optical frequency spectrum with a sub-wavelength near-field profile resolution. The input part of the experimental setup is similar to the one used in the optical imaging experiment. The 3mm-long HC-PCF sample is placed onto the SNOM sample vertical holder and an Au/Cr coated, 150nm core diameter fiber tip is lowered onto the HC-PCF end. The tip is maintained at a constant height just above the silica part of the cladding. This prevents the tip from moving uncontrollably into the air holes. The collected light is then detected with an avalanche photo-detector (APD) for near-field profiling. Alternatively, the input wavelength filter can be removed and an optical spectrum recorded using an optical spectrum analyzer (OSA) with a resolution of 1nm. Once a near field profile is acquired and the position of the core and cladding marked, it is possible to align the SNOM tip to a selected region of the fiber end face and record its optical spectra.
Figure 6(b) shows near-field profiles measured using the SNOM when the tip is scanned over a fiber region of 20 × 20μm2 area which includes the HC-PCF core. The scan step, which contributes to the final spatial resolution of the near-field, is set to 0.15μm. The fiber is a HC-PCF whose transmission band is centered at 1550nm and has a pitch of Λ=3.8μm. Such a long-range scan is done for each spliced fiber sample in order to check that we are exciting the fiber near the air-line. The near-field profiles shown in Fig. 6 are displayed as a function of the filtered input wavelength. The near-field images and the transmission spectrum recorded when the tip is aligned with the fiber core show that our lunching conditions are optimized for HE11-like mode guidance. Despite the relatively low resolution due to the large scan step, a close look at the near-field profiles both in the cladding and the core provide us with the observations described below.
Cladding modes with a large field component in the air holes [green line in Fig. 2(a)] are observed at a wavelength λ=600nm, corresponding to kΛ∼39 i.e. well beyond the PBG region at the air core guided mode index (neff ∼0.995). At such a high frequency, the cladding band containing modes with most of the light concentrated in the air holes attains an index value close to 0.995 so that it is close to phase-matched with the field incident at the input end of the fiber. As the input wavelength is increased, the NF of the supported cladding modes does evolve: between λ=800nm and about 1200nm, corresponding to a kΛ range of 27-20 which extends close to the upper-frequency edge of the PBG at neff∼0.995, cladding modes are observed with fields localized in and around the cladding struts [At the Γ-point, these are represented by the blue line in Fig. 2(a)]. As expected, for wavelengths corresponding to kΛ values satisfying the PBG conditions, there are no detectable cladding modes and most of the propagated light is situated in the core. These observations are corroborated by the transmission spectra collected by the SNOM tip when aligned with a cladding region (grey line in Fig. 6) and with the core (black line). The PBG operates as an optical pass-band filter with low levels of guidance outside the PBG.
A few additional remarks can be made about the results shown in Fig. 6: (i) Windows of core guidance are observed at 1100nm and 600nm, although these are very lossy. The short wavelength peak in core transmission is due to an antiresonance of the core wall : sections of the core wall have a thickness, t, of approximately 140 nm (estimated from the fiber SEM) and therefore show a lowest-order antiresonance at a wavelength of approximately 4t(n 2 gl - 1)1/2 ≈600nm, where n gl is the refractive index of silica. The peak at 1100nm is probably due to phase-matching between the leaky core guided mode and narrow high-DOS band of cladding states with low vg⊥ and a significant portion of the light within air. This band of states has a trajectory at the Γ-point given by the uppermost continuous yellow line in Fig. 2(a). The narrow high-DOS band close to this yellow line is seen to intersect the neff=0.995 core mode line at about kΛ=22, which indeed corresponds to a wavelength close to 1100nm. The ring of holes immediately surrounding the core is deformed compared to the rest of the cladding structure and therefore has its own short wavelength cut-off at around 1400nm, which is somewhat higher than the cut-off for the remainder of the cladding.
In order to selectively image and spectrally analyze the three cladding modes of interest, we used two different fibers whose transmission bandwidth lies within the OSA wavelength detection range. We reduced the scan area to a desired cladding feature and enhanced the spatial resolution by setting the scan step to 20nm. Figure 7 summarizes the results collected using HC-PCFs, one showing core-guidance around 800nm [Fig. 7(b)] and the other around 1064nm [Fig. 7(c)].
Figure 7(a) shows the typical NF profiles when the fiber sample is excited by light near the lower-frequency band gap edge and the upper-frequency edge [Fig. 7(A1) and 7(A2), respectively]. Figure 7(A1) clearly shows that the imaged mode corresponds to that of the apex resonator. Moreover, when the tip is aligned with an interstitial apex, the transmission spectrum of both fibers shows a cut-off [solid black line on the lower graph of Fig. 7(C)]. This corresponds to the apex mode frequency cut-off near the air-line in accordance with the numerical simulation [see Fig. 2(A)]. Similarly, for strut and airy modes, the transmission spectra show a clear cut-off at the short wavelength side of the HC-PCF transmission bandwidth. However, due to the limited spatial resolution of the SNOM and to the hybridization between the two constituent resonators, the transmission spectra collected when the tip was aligned on top of a strut or in an air hole do not show a measurable difference in their frequency cut-off. Nevertheless, the above results using the SNOM do confirm that the PBG is formed by the interplay of three distinct resonators.
We developed a simple and powerful experimental technique for measuring and identifying HC-PCF cladding Bloch modes. This technique mainly relies on optically imaging the Fresnel zone associated with the cladding modes which survive propagation along short lengths of fiber (∼3mm). The matching of the experimental patterns with corresponding calculated forms is excellent and enabled us to identify unambiguously individual cladding mode contributions. The findings of this new technique were corroborated with SNOM measurements. The results obtained using a fabricated HC-PCF suggest that the fiber PBG is formed by three cladding resonators. These resonators are the interstitial silica apexes, the silica struts and the air holes. Finally, these results confirm that the coupled resonator picture remains a useful conceptual framework for the design and analysis of PCF with high index contrast and a complex topological structure, such as the studied HC-PCF.
This work was supported by the EPSRC. The authors would like to thank T. A. Birks and D. M. Bird for insightful discussions. F. Benabid is an EPSRC Advanced Research Fellow.
References and links
1. B. J. Mangan, L. Farr, and A. Langford, et al., “Low loss (1.7 dB/km) hollow core photonic bandgap fiber,” presented at the OFC 2004, 2004.
2. F. Benabid, F. Couny, and J. C. Knight, et al.., “Compact, stable and efficient all-fibre gas cells using hollow-core photonic crystal fibres,” Nature434,488–491 (2005).
6. F. Couny, P.S. Light, and F. Benabid, et al., “Electromagnetically induced transparency and saturable absorption in all-fiber devices based on 12C2H2-filled hollow-core photonic crystal fiber,” Opt. Commun. 263,28–31 (2006). [CrossRef]
7. T. A. Birks, P. J. Roberts, and P. S. J. Russell, et al., “Full 2-D photonic bandgaps in silica/air structures,” Electron. Lett. 31,1941–1943 (1995). [CrossRef]
11. T. A. Birks, D. M. Bird, and T. Hedley, et al., “Scaling laws and vector effects in bandgap-guiding fibres,” Opt. Express 12,69–74 (2003). [CrossRef]
12. F. Benabid and P. St. J. Russell, “Hollow-core PCF; progress and prospects,” Proc. SPIE 5733,176–190 (2005). [CrossRef]
13. G. Antonopoulos, F. Benabid, and T. A. Birks, et al., “Experimental demonstration of the frequency shift of bandgaps in photonic crystal fibers due to refractive index scaling,” Opt. Express 14,3000–3006 (2006). [CrossRef]
14. P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19,427–430 (1976). [CrossRef]
15. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am 68,1196–1201 (1978). [CrossRef]
16. P. Yeh, Optical waves in layered media (John Wiley and Sons, New York, 1988).
17. M. A. Duguay, Y. Kukubun, and T. L. Koch, et al., “Antiresonant reflecting optical waveguides in Sio2-Si multilayer strucutures,” Appl. Phys. Lett. 49,13–15 (1986). [CrossRef]
19. N. M. Litchinister, A. K. Abeeluck, and C. Headley, et al., “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27,1320–1323 (2002).
21. N. W. Aschcroft and N. D. Mermin, Solid State Physics (Saunders College, Philadelphia, PA 19105, 1976).
22. T. P. White, R. C. McPhedran, and C. M. De Sterke, et al., “Resonance and scattering in microstructured optical fibers,” Opt. Lett. 27,1977–1979 (2002). [CrossRef]
23. J. Lægsgaard, “Gap formation and guided modes in photonic bandgap fibres with high-index rods,” J. Opt. A: Pure Appl. Opt. 6,798–804 (2004). [CrossRef]
25. Here the triangular lattice labeling refers to the placement of the air holes. Though the cladding structure of the fiber could be seen as a set of silica rods aligned in a honeycomb lattice, we keep the triangular labeling for historical reasons.
26. J. B. Jensen, L. H. Pedersen, and P. E. Hoiby, et al., “Photonic crystal fiber based evanescent-wave sensor for detection of biomolecules in aqueous solutions,” Opt. Lett. 29,1974–1976 (2004). [CrossRef]
29. T. D. Hedley, D. M. Bird, and F. Benabid, et al., “Modelling of a novel hollow-core photonic crystal fibre,” presented at the Quantum Electronics and Laser Science, 2003. QELS. Postconference Digest , pp. 2, (2003).
34. J. D. Shephard, P. J. Roberts, and J. D. C. Jones, et al., “Measuring beam quality of Hollow Core Photonic Crystal Fibers,” J. Lightwave Technol. 24,3761–3769 (2006). [CrossRef]