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We present the first characterisation of cladding modes of a low-index contrast all-solid photonic bandgap fiber using an acousto-optic long-period grating. We experimentally measure the relative band diagrams of the cladding, visualise the fields of the cladding modes in the near field, and find both to be in good agreement with simulations. Our measurements and simulations show that the bands of the cladding are very sensitive to actual details of the structure.
©2008 Optical Society of America
1. Introduction
Solid-core photonic bandgap fibers (SC-PBGF) are optical fibers containing high-index inclusions in a low-index background, typically arranged in a hexagonal lattice around a core consisting of a missing high-index rod. Light guidance in the core of these fibres has been explained in terms of backscattering of single high-index rods, the antiresonant reflecting optical waveguide (ARROW) effect [1–3], and also in terms of photonic bandgaps [4, 5]. While the interpretation in terms of ARROW and scattering properties uses characteristics of the single high-index inclusions to predict light guidance properties, and thus works best when the high-index inclusions are weakly coupled [3], the interpretation in terms of bandgaps puts the accent on the collective effects and consequently works best when coupling between high-index regions are strong [4]. Both interpretations nevertheless rely on the fact that light can only leak out of the core if it can couple to cladding modes, and are thus in essence equivalent. SCPBGFs have been used to demonstrate tunable band pass filters and other devices [6], and also exhibit tunable dispersion properties with several zero-dispersion wavelengths. Combined with the strong confinement SC-PBGFs can provide, this has led to demonstrations of soliton in the near-visible infrared [7, 8]. While most SC-PBGFs use conventional photonic crystal fibres [9] in which holes are filled with high-index fluids, all-solid SC-PBGFs have been demonstrated, in which guidance can be achieved with index contrasts as low as 1% [10, 11].
Much of the properties of SC-PBGFs are determined by the modes of the cladding: the bandgaps in which light is guided in the core corresponds to the absence of cladding modes, and even within a bandgap the losses of the core-mode are largely determined by the separation (in terms of propagation constants) between the core mode and the bands of the cladding [10, 12]. It has been suggested that SC-PBGFs containing long period gratings could be used for measuring temperatures or other physical quantities with extreme sensitivities [13]. As long period gratings work by phase matching the core mode to cladding modes, the exact dispersion of cladding modes and hence the detailed structure of the cladding bands need to be known very precisely.
While the bands of SC-PBGFs have been extensively studied numerically as well as theoretically [2–5, 10, 14], only few methods have been reported for experimentally determining cladding mode properties. Couny et al used angular and spectral measurements of the light tunneling through the cladding to measure the bands of hollow core fibres, but to the best of our knowledge this method has not been applied to SC-PBGFs [15, 16].
Here, we demonstrate the experimental determination of the bands of an all-solid SC-PBGF using an acoustic grating. We carefully simulate these SC-PBGF cladding bands using both the plane wave [17] and multipole methods [18–20]. While the experimentally determined bandgap structure agrees well with simulation results, we find that minute differences in the numerical representation of the structure can lead to significant differences in simulations results, highlighting the need for determining the band structures experimentally.
2. Principle
Long period gratings (LPG), gratings with periodicity of the order of a hundred micrometers, can provide phase matching between the fundamental core mode and higher order core or cladding modes. This has found a number of applications in optical fiber technology, in particular for filtering [21] and gain flattening [22]. LPGs have also been used to probe the existence and properties of cladding modes, and were a very useful tool in the early understanding of guidance properties in index guiding photonic crystal fibres [23]: by applying an LPG onto a photonic crystal fiber, the transmission spectrum shows minima every time the LPG allows coupling between the core mode and cladding modes, allowing to measure the difference in propagation constants between the core and cladding modes. Acoustic gratings -LPGs based on an acoustic wave- have recently been demonstrated in fluid filled SC-PBGFs to show narrowband rejection in spectrum due to the highly dispersive nature of this fibre [24]. Here we apply acoustic gratings to all-solid SC-PBGFs, and by scanning the acoustic frequency of the grating we reconstruct the band diagram of the SC-PBGF’s cladding.
A long period grating can couple two modes of appropriate symmetry satisfying the phase matching condition
where Δn eff is the difference between effective indices of the two modes, λ is the optical wavelength and Λ is the period of the LPG, which in our case is the acoustic wavelength. We inject light into the fundamental core mode of the SC-PBGF, and use an acoustic LPG to couple light to cladding modes (see Section 3.1). At a given acoustic frequency (and hence acoustic pitch), if the phase-matching condition is satisfied at a given optical wavelength, light is coupled from the core into the cladding, resulting in a dip in the core’s transmission.
By scanning the acoustic frequency and measuring the optical transmission spectrum through the core of the SC-PBGF we obtain dips as a function of Δn eff and λ, which allow us to reconstruct the relative dispersion of the cladding modes, and thus the band diagrams (relative to the fundamental core mode’s effective index) of the SC-PBGF’s cladding.
Note that cladding modes of the microstructure are in fact anti-guided, and hence lossy [25–27]; it can be considered that whenever coupling to a cladding mode is possible, light will effectively get lost to the cladding, and beating between core and cladding modes is not observed [28]. The depth of the dip primarily depends on the overlap integrals between acoustic mode, cladding mode and core mode. However, in situations where a large number of cladding modes exist within a narrow range of effective indices, the core mode will couple to several cladding modes simultaneously. In that case the depth of the transmission dip will to some extent reflect the density of modes. Note that the acoustic grating being a flexural wave and thus an antisymmetric perturbation, in theory it can only couple the (symmetric) fundamental mode to antisymmetric modes of the cladding. However, the exact symmetry conditions are somewhat complicated by the acoustic and optical birefringence of SC-PBGFs as discussed in detail in Ref. 29.
3. Experimental determination of bands
3.1. Experimental setup
Figure 1 shows a schematic diagram of the acoustic grating setup, along with a cross-section of the SC-PBGF used in our experiments. The fibre was fabricated using the procedure outlined in Ref. [12], and incorporates high-index rods of Ge doped silica in a hexagonal arrangement. The centre-to-centre spacing between rods is 6.7µm. Each rod has a graded-index distribution given by
where r is the distance from the rod’s center, α≃4.7, Δn GI≃0.0203, r 0≃1.6 and n silica is the index of fused silica.
Fig. 1. A schematic diagram of the acoustic grating setup for spectral measurements (a) and for mode imaging (b). Pol: polariser, Obj: microscope objective, Mono: Monochromator, SC source: supercontinuum source, OSA: optical spectrum analyser, CCD: CCD camera. For spectral measurements the SC source is butt-coupled to the SC-PBGF. (c): Optical microscope image of the SC-PBGF.
The acoustic grating is generated using a piezoelectric transducer driven by an amplified wave-generator and coupled to the fibre through an acoustic horn. The acoustic excitation is transverse to the fibre, and we place an acoustic damper 12cm after the tip of the horn.
To measure the transmission of the acoustic grating (Fig. 1(a)), we spliced the input end of the SC-PBGF onto a short piece of SMF28 pigtail and then butt coupled the fibre to a PCF-based supercontinuum broadband source [30]. The output of the PBGF is butt-coupled to a piece of SMF28, which is connected to an OSA. As SMF28 is multi-mode over the wavelength range we explore (visible and near infrared), higher order core modes of the SC-PBGF will couple to the SMF28 and also contribute to the intensity measured at the OSA. Coupling to higher-order core-modes will hence not result in transmission dips: observed transmission dips can solely result from coupling from the core-mode to the cladding modes.
3.2. Results: transmission measurements
Figure 2 shows the transmission of the fibre with and without acoustic grating. In the absence of acoustic grating, high transmission bands are separated by low transmission bands, as expected for SC-PBGFs [31]. When the acoustic gratings is switched on, dips in transmission appear within the high transmission bands. We measured the transmission of the acoustic grating in the SC-PBGF over a range of acoustic frequencies, and normalized this data by subtracting the transmission of the SC-PBGF without the grating. Figure 3 shows the normalized transmission as a density plot, in which the resonant loss due to coupling between the core and cladding modes now clearly appears as a function of acoustic frequency and optical wavelength. Using Eq. (1) this can be translated into a plot of the strength of the coupling as a function of Δn eff and λ.
Fig. 2. Spectral power density at the output of the fibre when the acoustic grating is turned off (blue) and on (red, acoustic frequency is 750kHz).
Fig. 3. Density plot of the acoustic grating’s transmission normalized to the SC-PBGF’s transmission in the absence of grating, as a function of acoustic frequency and optical wavelength. Blue crosses indicate wavelength and acoustic frequency of the mode images shown in Fig. 8.
4. Simulations
We calculate the modes of the infinite periodic photonic crystal of which the cladding of our PBGF is a subset using a commercial plane wave method [17]. The simulated structure is a hexagonal array of pitch 6.7 µm of graded-index cylinders, with index profile within the unit cell centered on a high-index rod defined by Eq. (2), except that we use r 0=1.5927 (our best estimate for the actual value of r 0) and nsilica=1.458 for all wavelengths. The core modes of the fibre with a defect were calculated using a plane wave method, with a supercell of 5×5 periods with one missing rod at the center. Note that, due to limitations in the software (which solves for frequency at fixed propagation constant), in both these simulations material dispersion of silica and germanium doped silica are neglected. We also simulated the structure using a multipole method, [18–20] where the actual graded-index profile was approximated by a layered cylinder, with profile
where r1=0.65r 0 and n silica is the refractive index of fused silica, obtained using a Sellmeier expansion [32]. Multipole simulations used 5 rings of inclusions around the core, and Bessel series were truncated to the 5th order ensuring excellent numerical convergence over the domain of effective indices considered.
Figure 5(a) shows the effective index of modes obtained through the simulations: results from plane wave simulations are in blue (cladding modes) and green (core modes), while the black crosses are obtained from multipole simulations. Agreement between plane wave methods and multipole method is good at short wavelength, where the index of silica is closest to 1.458. At longer wavelength, the slight disagreement between methods of simulation is partly, but not entirely, due to the index of silica differing markedly (by 0.008 at 1000nm) from the background index used in plane wave simulations.
Fig. 4. Index profiles within the unit cell used for simulations, normalized to the background refractive index, as a distance from the rod’s center. blue: Profile used for plane wave expansion simulations; red: profile used for multipole method simulations.
The bands of the cladding shown in Fig. 5(a) can be subdivided into the “upper bands” and the“lower bands”. The “upper bands” are those with effective index greater than the background index, and can in essence be seen as the supermodes resulting from the coupling of guided modes of the inclusions. The field distribution of such modes is essentially concentrated in the high-index inclusions (see Fig. 5(b) upper inset).
The lower bands are those with effective index below the refractive index of the background, and result from the coupling of the leaky modes of the high-index inclusions [4]. The fields of the modes of the lower band are correspondingly mostly concentrated between high-index inclusions, in the background (Fig. 5(b) lower inset). A long period grating can in principle couple light from the core mode to modes of the upper and lower bands.
Many properties of SC-PBGFs, such as losses [10, 12] or resonant wavelengths of LPGs are determined by the difference between the effective index of the core mode and that of cladding modes, rather than by the absolute value of the effective index of the cladding modes. We therefore use relative band diagrams, normalized with respect to the fundamental core-mode’s effective index n core eff, as shown in Fig. 5(b). Note that this diagram is only meaningful at wavelengths for which the fundamental core mode is clearly confined in the core, explaining the grayed areas in Fig. 5(b).
Fig. 5. (a): calculated effective index of core and cladding modes as a function of wavelength, relative to the background index. Blue lines: modes of the infinite periodic photonic crystal (plane wave simulations). Green curves: core modes of a 5×5 supercell with a missing rod at its center, as obtained using the plane wave method. For both plane wave simulations no material dispersion is taken into account, but the graded index profile of each cylinder is accurately described. Black +symbols: modes of a 5 ring finite structure as calculated by the multipole method using an approximate radial index profile, but with material dispersion taken into account. Insets show examples of field distributions of modes of the upper and lower bands. (b): same bands, but relative to the core mode’s refractive index. (c): Experimentally reconstructed bands of the PBGF’s cladding: Map of acoustic grating resonances as a function of wavelength and difference between effective index of higher order and fundamental modes. -see Figs. 6 and 7 for details.
5. Experiment and simulations: comparison
Figure 5(c) shows the experimentally determined relative band structure on the same scale as the calculated relative band structure of Fig. 5(b). Because the phase matching condition Eq. (1) contains an absolute value, the experimental results contain information on both, Δn eff=λ/Λ and Δn eff=-λ/Λ. In Fig. 5(c) we have hence reproduced the experimental data with both, Δn eff=λ/Λ (upper half of the graph) and Δn eff=-λ/Λ (lower part of the graph).
Over the range covered by our experiment, agreement with numerical calculations seems good at first sight. We now proceed to a detailed comparison between numerical and experimental results over several wavelength and effective index ranges, and in particular distinguish between bands attributed to positive and negative values of Δn eff:
Figure 6 shows the experimentally and numerically determined relative band structures Δn eff(λ) for the band around 600nm: Fig. 6(a) and 6(b) are simulated results, with blue curves from plane wave simulations and black crosses from multipole simulations (blow up of same data as Fig. 5(b)); Fig. 6(c) and 6(d) show the corresponding experimental measurements, where red indicates strong coupling and yellow no coupling. Simulation and experiment are in apparent good agreement, with discrepancies in terms of wavelength or Δn eff of order 10%. In particular, differences between the experimental and simulated results are of the same order as the differences between the two different approaches for simulating the cladding modes.
Fig. 6. Band of the cladding around 600nm wavelength: Experimentally determined (bottom) and simulated (top): lines using plane wave method, crosses using multipole method. Left: using Δn eff=-λ/Λ, right: using Δn eff=λ/Λ. Blue crosses indicate wavelength and acoustic frequency of the mode images shown in Fig. 8.
Figure 7 shows the experimentally determined relative band structure around a wavelength of 900nm, along with the equivalent simulation results. Here discrepancies between simulations and experiment are larger, of the order of 20%. Furthermore, aroundλ=1µm, the experiments indicate a cladding mode with Δn eff increasing steeply with wavelength, which is not present at first sight in the simulations (ellipse in Fig. 7). The right hand side of Fig. 7 shows the same result, but where we have used Δn eff=-λ/Λ for the experimentally determined bands, for which the phase matching is also satisfied. Simulations show a dense band of cladding modes with similar slope. These correspond to the core-mode coupling to the upper band described in Section 4. The same phenomenon, albeit less clearly visible, seems to occur around 650nm in Fig. 6 (marked by an ellipse). Data outside the ellipse in Figs. 6(d) and 7(d) correspond to points with negative Δn eff, and thus do not have corresponding points in Figs. 6(b) and 7(b), but rather in Figs. 6(a) and 7(a).
Fig. 7. Band of the cladding around 900nm wavelength: Experimentally determined (bottom) and simulated (top): lines using plane wave method, crosses using multipole method. Left: using Δn eff=-λ/Λ, right: using Δn eff =λ/Λ.
6. Modal fields
To verify our interpretation, we measure the modal field distribution of modes within the marked region, since modes of the upper band have field distributions concentrated within the high-index inclusions. We used free-space coupling of light into the SC-PBGF instead of butt-coupling or splicing, to allow us to add a monochromator and polariser before the input of the fibre, and another polariser at the output of the fibre. The modes were then imaged on a colour CCD camera (Fig. 1(b)).
Fig. 8. Field distribution of modes in the upper (top) and lower (bottom) bands. Left: simulated (multipole), right: experiment. The experimental mode images link to an animation showing switching between the acoustic grating being on and off. Video file sizes are 1.6MB for (b)(Media 1), 1.2MB for (d) (Media 2). Simulated field distributions show the field intensity (norm of the electric field squared).
Figure 8b shows the field image taken at λ=645nm and an acoustic frequency of 835kHz (acoustic wavelength 960µm), corresponding to a resonance which is part of the steep positive slope indicated by an ellipse in Fig. 6. The fields are clearly concentrated inside the high-index inclusions, showing that the excited cladding mode is indeed on the upper band (blue cross in Fig. 6). Figure 8a shows a simulated mode profile (multipole simulations) from the upper band at wavelength 560nm (n eff-n core eff=-7.0·10-4) showing a very similar modal field distribution. Figure 8d shows the modal field distribution imaged at λ=569nm and an acoustic frequency of 717kHz (acoustic pitch 1.04mm), corresponding to a point on a curve with negative slope (indicated by the blue cross in Fig. 6(c): the field is most intense between inclusions. Figure 8c shows a similar mode from the lower bands obtained with multipole simulations (λ=610nm, n eff-n core eff=3.7·10-4).
7. Discussion and conclusion
We demonstrated that acoustic gratings can be used to experimentally reconstruct the bands of SC-PBGFs: we have obtained the curves of the effective index of the cladding modes (relative to that of the core mode) and were able to image the field distribution of these modes. Comparison of numerical models and experimental results show that the band structure is very sensitive to minute variations in the exact geometry of the SC-PBGF’s high-index inclusions, and that modelling alone cannot be relied upon to accurately predict LPG resonances between the core and cladding modes. The extreme sensitivity of LPGs to the actual refractive index profile of high-index inclusion could be the key to ultra-sensitive SC-PBGF LPG based sensors and widely tunable filters [13]; for the study and development of such devices, the ability to experimentally determine the band structures with accuracy will be essential. While our method of characterisation can only be applied when a core mode is guided, that is within the high transmission bands of PBGF, it is a complement to the method of Couny et al [15, 16] which works best for modes strongly leaking out of the cladding. Indeed LPGs can also couple light to strongly confined modes (including higher order core modes or surface modes when these exist).
Acknowledgments
This research was supported under the Australian Research Council’s (ARC) Discovery Project and Centre of Excellence funding schemes. CUDOS is an ARC Centre of Excellence.
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