1. Introduction

Photonic bandgap fibers (PBGFs) are a class of photonic crystal fibers (PCFs) [1] for which light propagates in a low-index defect [2], as opposed to high-index core PCFs, which guide light through the modified total internal reflection mechanism (MTIR) [3]. Low-index core PCFs are generally classified in two categories, namely, hollow-core structures (HC) for which light propagates mainly in a gas-filled core [2], and solid-core (SC) structures [4]. In such low index core fibers, the cladding contains 1- or 2-dimensional periodic arrangements of dielectrics, giving rise to photonic bandgaps (PBG) providing guidance of light in the core.

Two dimensional SC-PBGFs with high refractive index inclusions in a low index background (see Fig. 1 ) can guide several core modes in discrete spectral windows, separated by spectral regions with high confinement losses (CL), which depend mainly on the opto-geometrical parameters of the micro-structured cladding [5]. Within each high-transmission band strong waveguide dispersion leads to high normal and anomalous group velocity dispersion (GVD), in particular near the photonic bandgap edges. The transmission, confinement and dispersion properties of SC-PBGF can be adjusted by design of the micro-structure (rod-to-rod spacing, high index inclusion diameter, refractive index contrast). Unlike in Hollow-core-PBGFs (HC-PBGF’s), the solid core of these structures can be doped, enabling laser operation [6], Bragg grating inscription [7], and solid-based non-linear effects [8]. SC-PBGFs are also easier to fabricate and splice compared with HC-PBGFs, and do not suffer from surface modes [9].

 

Fig. 1 Schematic cross section of a 2D-SC-PBGF. The bright gray region represent the low refractive index background (n_low,), whereas dark gray circles are high index inclusions (n_high). A typical transversal refractive index profile of the cladding is shown on the right.

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Owing to their unusual properties, SC-PBGFs have been used as temperature-tunable spectral filters [10–12], in laser systems [6,13], for demonstration or suppression of non-linear effects [8,14] and also as GVD compensators [15,16]. For this last application, these fibers are generally used close to the photonic bandgap edges, where confinement losses and GVD are generally very high [6,15]. When the applications require very small confinement losses or GVD (like generation of nonlinear effects), these fibers are usually used in the centre of the photonic bandgap.

In most of these applications, the fibers are generally treated as being effectively single-mode. However, bandgap guidance is not limited to the fundamental, LP₀₁-type mode, and indeed bandgaps can give rise to interesting behavior of higher order modes: Digonnet et al. [17] pointed out that in hollow-core PBGFs, the number of guided core modes can depend on the wavelength within the photonic bandgap of interest, so that the number of core guided modes at the center of a photonic bandgap and close to the photonic band edges can be different. Digonnet pointed out that spectral regions can exist in which only higher order modes are guided. Euser et al. [18] have demonstrated selective higher-order mode excitation in hollow core PCFs using a complex dynamic holographic technique. This permits launching an input beam shape corresponding to the different core guided modes into the fiber core, leading to their observation at the fiber end one by one. The number of modes has also been studied extensively theoretically in 1D-PBGFs (Bragg fibers) [19,20].

In contrast, in 2D-SC-PBGFs higher-order modes have been succinctly observed both experimentally and numerically [12,21], but to our knowledge no more detailed study of the number of modes in SC-PBGFs and their propagation has been published.

Here, we investigate the existence of higher order modes in 2D-SC-PBGFs. In particular, we observe that as for hollow core PBGFs [17] there are wavelength ranges in which the second mode is the only one guided in an all-solid SC-PBGF. This allowed us to excite the second mode without the fundamental mode, without the requirement for a complex experimental setup, for fiber lengths up to 20 m. These observations, obtained in transmission, occur close to the red edges in wavelength of at least three different photonic bandgaps. We explain the existence and low relative confinement losses of higher order modes near these band edges in terms of band diagrams along the lines of what has been proposed for hollow core PBGFs, and we provide full numerical simulations of confinement losses in good agreement with experimental observations.

2. Experimental observations of second order guided modes

A 2D all-solid PBGF was fabricated using the stack and draw technique. The cladding is made of 7 Germanium-doped rings (circular inclusions) with a step refractive index profile embedded in a pure silica background (doping of 2% ≈3 × 10⁻²). The centre-to-centre distance between two consecutives inclusions, or pitch Λ of the micro-structure is 8.2 μm with a d/Λ ratio of 0.62 (see inset of Fig. 2 ), where d is the rod’s diameter. The external diameter of the fiber is 174 µm. These parameters lead to a low loss operation in the wavelength range of 1-1.1 μm in the 3rd PBG, commonly used in this kind of fiber due to a good compromise between small confinement and bend losses [15].

 

Fig. 2 Schematic representation of the experimental set-up used for the transmission measurements. The near field images of the guided modes are observed thanks to an IR camera by moving the mobile mirror. An optical image of the SC-PBGF is also shown in inset.

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The experimental set-up shown in Fig. 2 is used to measure the different transmission spectra of the SC-PBGF. A super-continuum fiber (≈10m) is used to generate white light directly launched into the SC-PBGF core by butt-coupling. The super-continuum fiber is not endlessly single-mode, but can be considered as effectively single-mode in the wavelength range of interest due to the very high CL of higher order guided modes. At the SC-PBGF end, a microscope objective (MO) images the output beam on a pinhole in order to collect light solely from the core modes. Light is then collimated with another objective onto a multi-mode fiber directly coupled to an optical spectrum analyzer. A mobile mirror allows imaging of the modes using an infrared camera.

Figure 3 represents the transmission spectrum of a 20 m long piece of SC-PBGF (red curve). All the 20 m are wrapped on a 7.9 cm radius spool. We clearly observe three transmission windows due to the PBG effect, labeled from #3 to #5. Moreover, on the right-hand side of each PBG (the red edge), we observe slight discontinuities of the transmitted power (right sides of the vertical dashed lines). In order to highlight their origin, the transmission through a 20 cm long piece of the same SC-PBGF is shown on the same figure (blue curve). The injection conditions are kept constant (the fiber is cut in situ) and the piece of fiber is straight. Whereas transmitted power is a few dB lower for the 20 m long fiber in the middle of each PBG due to confinement losses, its value changes dramatically in the wavelength range where the discontinuities were observed for the 20 m fiber: the transmission drops by 7 dB at 670 nm (PBG #5), 10 dB at 855 nm (PBG #4) and 26 dB at 1175 nm (PBG #3). These high values suggest the existence of two different core guided modes in the SC-PBGF having different confinement losses: in the centre of the bandgap two modes should coexist, one of which has very low confinement losses. At the red end of these three PBGs, beyond the discontinuity, the low-loss mode is no longer confined, but another mode with smaller confinement losses continues to exist.

 

Fig. 3 Transmission spectra of a 20 m (red curve) and a 20 cm (blue curve) piece of the 2D SC-PBGF. The 20 m long piece is wrapped on a 7.9 cm bend radius spool whereas the smallest one is kept straight. Inset: near field image of the fundamental mode observed at 1000 nm (PBG #3).

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In order to verify this interpretation, we took images of the modal field distribution for several wavelengths and light injection conditions using dichroic bandpass filters (with 3dB width of 10 nm, see set up on Fig. 2). For wavelengths in the centre of these three bandgaps, light is predominantly confined in the fundamental (LP₀₁-like) mode. However, at the red edge of the bandgaps (beyond the discontinuity) we were unable to find injection conditions that would result in light at the output of the fiber being in the fundamental mode. Instead, we consistently obtained LP₁₁-like field distributions (ring shape), regardless of injection conditions.

Figure 4 shows selected field distributions, all recorded at the red band edges for a fiber length of 20 m (with a bend radius of 7.9 cm). In each series a-c, picture 1 is the field distribution recorded without polarizer, and clearly shows the presence of a ring shape mode (without the fundamental mode), guided for wavelengths where the different discontinuities in the transmission spectra are observed (right sides of each vertical dashed lines of Fig. 3). Pictures 2 and 3 are recorded with a polarizer and by changing slightly the launching conditions, and show the two polarization states of the ring-shaped mode (doughnut mode). These modes are also clearly observed for a shorter piece of fiber (20 cm), without the presence of the fundamental mode. For the modes presented in Fig. 4, the light is mainly confined in the fiber core. The resonances in the first ring of doped rods (depending on the PBG order) are in agreement with bandgap guidance at the edge of each band. For wavelengths on the left-hand sides of each vertical dashed lines in Fig. 3, light is predominantly in the fundamental core mode, for both fiber lengths. Attempts to obtain evidence of the existence of higher order guided modes in the second PBG (wavelengths between roughly 1.27 −2 μm, beyond the range shown in Fig. 3) were unsuccessful, and only the fundamental mode was observed in the second PBG. However, our experimental setup (and in particular OSA) was limited to wavelengths shorter than 1750 nm, well below the red edge of the second bandgap.

 

Fig. 4 Near field images of the guided modes observed at 675 nm (PBG #5) (a), 850 nm (PBG #4) (b) and 1200 nm (PBG #3) (c). Pictures 2 and 3 are recorded with a polarizer (see Fig. 2). The fiber length is 20 m.

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Note that, while not rigorous, due to the similarity of modal profiles it is common practice to label core guided modes of a PCF using the conventional step-index fiber LP_lm notation, l and m being the azimuthal and radial numbers respectively [22]. We follow this convention by calling the fundamental and second order guided modes of the SC-PBGF LP₀₁ (two fold degenerated: HE₁₁) and LP₁₁ (four fold degenerated: HE₂₁, TE₀₁ and TM₀₁) respectively. In a different context we also use the LP_lm notation for the modes of the individual high-index circular inclusions of the cladding.

3. Interpretation of the observation of LP₁₁-like modes: band diagram and tunneling effect

To understand the modes’ propagation behavior at the photonic band edges, we calculate the frequency-dependent band diagram of an infinite triangular lattice of high-index circular inclusions embedded in a pure silica background, related to the cladding of our structure (Fig. 5 ). This diagram is computed using the analytic method developed in Ref [23], without material dispersion (n_high = 1.4807, n_low = 1.45, d/Λ = 0.62). Note that the use of this simple analytic method is justified by the low index contrast of the structure (≈2%) and by the fact that the d/Λ ratio is not too big (weak coupling between rods).

 

Fig. 5 Plot of the band structure of an infinite triangular lattice constituted of high-index inclusions embedded in pure silica background, related the cladding of our 2D SC-PBGF (without material dispersion). The solid (dotted) black curve shows the effective index of the fundamental (LP₁₁-like) guided core mode of the associated PBGF. Blue horizontal line represents the refractive index of pure silica (n_low).

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For a mode to be guided in the core, its electrical field must be propagative in the core and evanescent in the cladding. The first condition imposes for the mode’s effective index to be smaller than the refractive index of the core (below the horizontal line of Fig. 5), in our case pure silica, and the second condition requires that the effective index lie in a PBG of the cladding (white region of Fig. 5). In Fig. 5 black lines represent the effective index (n_{eff mode}) of the fundamental (solid lines) and second (dotted lines) guided core-modes calculated with the multipole method [24]. Effective indices are simulated for the associated 2D SC-PBGF structure with a core consisting of one missing rod (and 7 rings in the cladding). The bands can be seen as originating from the coupling of the LP_lm modes of individual high-index inclusions [25]: the labels at the top of Fig. 5 indicate which high-index inclusion mode the bands are derived from.

On Fig. 5, we also define the two quantities Δn_eff ^- and Δn_eff ⁺ [26], which are the effective index differences between the core-guided mode and the lower and upper PBG edges, respectively. Following arguments in Refs [23,26], when either of these quantities approaches zero (in particular at the sides of each PBG), evanescent coupling (or tunneling) of light from the core to the external silica jacket increases, and consequently so do the confinement losses. Because Δn_eff ⁺ is very large over most of each PBG (see Fig. 5 and Ref [26].), the minimum CL in each PBG (which typically is away from the band edges) depends mostly on Δn_eff ^- and on cladding size (number of rings), relative to the wavelength.

On the band diagram of Fig. 5 we observe that n_{eff mode} increases with frequency in each PBG for both kinds of modes, as for index-guiding fibers, and generally speaking rises from a PBG to another with their order. The effective indices of higher-order guided modes are below the effective index of the fundamental mode in each PBG. Consequently, the value Δn_eff ^- decreases with increasing mode order and the tunneling rate linked to the confinement loss increase with mode order. Very close to the band edges, the effective index of the guided modes approaches the index of the lower/upper PBG edges and coupling with cladding modes occurs. At the band edge the core mode is no longer guided, a phenomenon equivalent to modal cutoff in index-guiding fibers.

Because the lower edge of bands (green curves) decreases with decreasing frequency, and higher-order modes have lower effective indices than the fundamental mode, the ‘cutoff’ of the fundamental mode occurs at slightly higher frequency than for higher-order modes. Even before this happens, Δn_eff ⁺ for the fundamental mode can become much smaller than both Δn_eff ⁺ and Δn_eff ^- for higher-order modes so that the fundamental mode will suffer higher CL (and bend losses) than higher-order modes. As a consequence, in each bandgap only higher order modes can propagate at low enough frequencies, and even before the ‘cutoff’ of the fundamental mode, higher order modes may have losses below those of the fundamental mode. In particular, there is a wavelength λ_cut for which the confinement losses of both guided modes are equal. Above λ_cut, the CL of the fundamental LP₀₁-like core-mode must be higher than those of the LP₁₁-like mode.

These observations are in good agreement with the experimental results of Section 2. Indeed, we only observe the propagation of the LP₁₁-like core-mode on the red side of several PBGs, not on their blue side, even for a short fiber length of 20 cm. Furthermore Fig. 5 shows that Δn_eff ^- is very small in the second bandgap, indicating the LP₁₁-like is expected to suffer considerable loss. This could well explain why we could only observe the fundamental mode in this bandgap.

4. Numerical modeling of the SC-PBGF: Confinement Losses and Effective Area

In order to confirm our previous experimental observations (Section 2) and physical interpretations (Section 3), the CL of both LP₀₁ and LP₁₁ core modes as well as their effective area are simulated using the multipole method [24]. Figure 6 shows their spectral evolution from the 3rd to the 5th PBG using 7 rings of Ge-doped rods in the cladding (the material dispersion is now taken into account). The dashed vertical lines in Fig. 6 correspond to the cut-off wavelengths of the LP_lm modes of an isolated high-index inclusion of the cladding, approximately delimiting the spectral transmission windows of 2D SC-PBGFs [25].

 

Fig. 6 Simulated spectral evolution of confinement losses (top) in dB/km and of the normalized effective area (bottom) for the LP₀₁ (red curves) and LP₁₁ (blue curves) core modes (logarithmic scale). The opto-geometrical parameters are identical to the fiber presented in Section 2. 7 rings of Ge-doped inclusions are used in the cladding (with material dispersion).

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Figure 6 shows clearly that CL of the LP₀₁-like core modes become higher than those of the second guided mode above the wavelength λ_cut in the three considered PBG. The three simulated λ_cut values of 1.17 µm, 846 nm and 661 nm (from the 3rd to 5th PBG respectively) are very close to those observed experimentally in Section 2: 1.15 µm, 846 nm and 666 nm (dashed vertical lines in Fig. 3). The computed CL at these three wavelengths are 5.5 dB/m, 740 dB/km and 80 dB/km respectively and for both kind of modes. This very good agreement in wavelength confirms our discussion in Sections 2 and 3: the discontinuities observed in the transmission spectrum (Fig. 3) of our 2D SC-PBGF are due to LP₁₁-like mode CL becoming smaller than the fundamental core mode on the red sides of these three PBG. In each of these three PBG and above λ_cut, light in the fundamental mode is lost more rapidly than for the LP₁₁ core modes, and the majority of the output light is in this second mode. This behavior is still valid even for short fiber lengths (20 cm) and we believe that both kinds of modes coexist just below λ_cut in our transmission setup, as observed in Ref [21].

Figure 6 also shows that the normalized effective area (A_eff/Λ²) of the fundamental mode, which is typically smaller than that of LP₁₁-like modes becomes larger than that of LP₁₁ core modes on the red side of these three PBGs and for wavelengths just above λ_cut. To the best of our knowledge, this is the first time that the observation of LP₁₁ core modes with effective area smaller than the fundamental is highlighted. This is of importance for the understanding and simulation of nonlinear propagation near band edges in these fibers.

Finally, Lægsgaard [27] observed that the LP_lm cutoff wavelengths of an isolated high-index rod correspond to a large stop-band of the 2D-SC-PBGF when their azimuthal number l is ≤ 1 (delimiting then the different PBG), and narrow stop-band if l > 1 (see vertical dashed lines in Fig. 6). Figure 6 shows then that the PBG edges are well delimited by these cut off wavelengths (for l ≤ 1) for the fundamental core mode, but not for the LP₁₁-like core mode, which still exist at the cut off wavelength of the LP_0m and LP_1m rod modes. Indeed, Renversez et al. [28] already pointed out that the transmission bands of a 2D-SC-PBGF are limited to modes coupling between the core and one or several high-index rods below their cut off, and then differ from a core guided mode order to another. Consequently, the spectral delimitation of the different PBG of 2D-SC-PBGFs becomes complex if we take into account the multi-mode behavior of the PCF, in particular on the red side of each PBG. This delimitation depends then on the band diagram of the infinite structure associated to the cladding of the 2D-SC-PBGF, and so on the coupling strength between high index inclusion, increasing with the d/Λ ratio and when the refractive index contrast in the cladding decreases.

5. Conclusion

We investigated both experimentally and numerically the propagation of higher order modes in 2D-SC-PBGFs. We showed by a very simple experimental setup in transmission the guidance of only the LP₁₁-like modes, without the fundamental mode, on the red side of several photonic bandgaps for fiber lengths from 20 cm to 20 m. As for Hollow-Core PBGFs, we interpret this behavior in terms of bands, effective index diagrams and confinement losses which are smaller for the LP₁₁-like modes than for the fundamental mode on these wavelength ranges. Good agreements with numerical simulations also confirm our experimental observations. The guidance of ring modes in the absence of the fundamental mode in PBG fibers can then be useful for modal filtering to generate for example pure vortex beams.