1. Introduction

Fiber Bragg grating (FBG) is a mode coupling device which involves two phase-matched counter-propagating modes, by forming a longitudinal index modulation over the photosensitive core along the fiber length with a period of several hundred nanometers [1]. Its performance is largely determined by the modal distribution of an optical fiber and the index perturbation region over fiber cross section. Photonic crystal fibers (PCFs), with of a two-dimensional periodic array of air holes or high-index rods, have presented unique modal properties and thereby provided a versatile platform for the implementation of novel fiber grating devices [2–4]. Index-guided PCFs have a highly controllable cladding index profile, and so FBGs inscribed in these fibers can exhibit very strong or very weak cladding mode resonances, depending on the fiber design [4, 5]. Moreover, by infusing active materials into air holes of PCFs, enhanced tunability of fiber gratings can be achieved [4].

Solid-core PCFs with high-index inclusions confines light to fiber core by a photonic bandgap effect. They transmit light in discrete frequency bands [6, 7]. The anti-resonant reflecting optical waveguiding (ARROW) model explains the guidance in these fibers [8–10]: when the high-index rod array and the modes in the low-index region are in resonance, the phase-matched supermodes in the rods are excited. When they are not resonant, guidance in the defect core is established by antiresonant scattering from the high-index cylinders. Inherent loss in these fibers is attributed to the coupling to radiative supermodes, which is severe at the edges of bandgaps [11, 12]. Grating resonance between core mode and supermodes can also be realized. Mechanically and electrically induced long period gratings in the ARROW fibers, which couple light to forward radiative supermodes, have been demonstrated [13–15]. Both of them are highly tunable devices. High thermal sensitivities are obtained due to the bandgap shift, which is induced by the inclusion index change, and the strong waveguide dispersion of the modes in ARROW fibers. In addition, their attenuation strengths can be adjusted by applied lateral stress and electric field, respectively.

ARROW fibers can be straightforward drawn to form an all-solid structure as well, by including high-index rods (usually Ge-doped ones) into silica background [11,12,16,17]. The all-solid PBGFs have attracted great interest because of their easiness to fabricate and splice, all-solid structure and promising potential application. By proper design, ultra-low bend loss, decoupling between the twin core, and dispersion tailoring in this kind of fiber can be achieved [18–21]. We have recently reported the FBGs inscribed into the Ge-doped cladding rods in all-solid PBGF and observed the grating resonance to the guided LP₀₁ supermodes, which is much stronger than the coupling between the core modes [22]. The corresponding resonance peak is located at the long wavelength side of the Bragg wavelength. In this paper, spectral characteristics and bend response of FBGs in all-solid PBGFs are investigated in detail. FBGs are inscribed into the high-index rods of the fiber within the secondary bandgap by UV side illumination. The resonances to guided LP₀₁ and LP₁₁ supermodes and radiative LP₀₂ supermodes are obtained. The coupling mechanism in the FBG is described, based on analyzing the phase relationship in neighboring rods and the symmetry of the spatial field of the guided supermodes. Moreover, when applying bend to the FBG, the LP₀₁ and LP₁₁ supermode resonances become chirped due to the strain differences between the individual cladding rods. The core mode resonance is enhanced, since energy of the core mode is not well constrained in bent PBGFs at the bandgap edges. The bend response is found to be direction dependant because of the nonuniform average index raises and index modulation over the cladding rod lattice.

2. Spectral characteristics of the FBG

Figure 1 exhibits the transverse microstructure of the all-solid PBGF. The fiber was fabricated by University of Bath, by using a modified stack-and-draw process [16, 17]. In the fiber, a triangular periodic array of germanosilicate rods (represented by the bright spots) is embedded in pure silica background. Fiber core is formed by omitting a single rod from the matrix. The outside diameter D of the fiber is 118 μm. The raised-index rod pitch Λ is 8.5 μm and the nominal ratio d/Λ is 0.44.

 

Fig. 1. Microscopic photograph for the cross section of the all-solid bandgap fiber. The bright spots represent Ge-doped rods.

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Fig. 2. (a). Simulated effective indices of the core modes versus wavelength and the bandgap map for the PBGF. Supermodes are discretely distributed in between the bandgap cutoffs. Due to the large quantity of supermodes in our experiment, they are not given in this figure. (b) Intensity distributions of the core modes in the middle of the bandgap (upper, at 1050nm) and at the edge of the bandgap (lower, at 950nm). Core modes near the bandgap edges have larger mode field areas, and more energy fractions of these modes are distributed over the high-index rods. (c) Some typical phase relationships for the eigen supermodes in our PBGF. The supermodes are distinguished by phase relationship of rod mode fields in adjacent high-index rods, which are represented by different color combinations. (d) The calculated overlap integrals between each LP₀₁ (upper) and LP₁₁ (lower) supermode and the core mode over the rod lattice. The calculation involves considering both the symmetric profiles of the phase relationships and the specific LP₀₁ and LP₁₁ rod modes. As shown in these two figures, only the 60°-symmetric LP₀₁ supermodes are responsible for the LP₀₁ supermode resonances, while the LP₁₁ supermodes with 180°-antisymmetric phase relationships for the LP₁₁ supermode resonances. Mode numbers are ordered with decreased propagation constants.

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There are two kinds of modes in ARROW fibers: core modes and supermodes. The effective indices and modal field profiles for the core modes, and phase relationships among the rods for the eigen supermodes (without considering specific rod mode profiles) in the all-solid bandgap fiber, are plotted in Fig. 2. Core modes are confined to the defect core by bandgap effect, whose effective indices are lower than the index of pure silica n _si. The core modes are represented by discrete curves in Fig. 2(a), which corresponds to the multiple transmission bands. As can be seen from its modal profile in Fig. 2(b), the core mode has a non-zero overlap with the high index rods, though most of its energy is confined in the low-index region, because the presence of the defect modifies the photonic density of states. Supermodes are modes of the microstructure rod lattice, which are the “cladding modes” in the PBGF. However, unlike the conventional index-guided fibers, the PBGF cladding modes can be guided (supermodes above the silica line) or radiative (supermodes below the silica line). For the guided supermodes, the energy is almost entirely confined in the high-index rods. The radiative supermodes, whose energy spread out into the low-index region, are lossy modes in the microstructure. In the limit of an infinite array of cylinders, these supermodes form continuous bands. However, for finite systems, the supermodes of the experimental PBGF are in discrete states.

An N-core waveguide array determines N eigen supermodes. The individual supermodes, whether guided or radiative, are distinguished by phase relationship of the field distribution in neighboring cylinders. For the guided supermodes, the coupling between adjacent rods can be considered as the first-order perturbation. The electric field of the ith supermode can be expressed by

where e⇀_sup and e⇀_rod are the electric fields of the guided supermode and rod mode, respectively. (x_j, y_j) represents the position of the center of each rod. We assume that all the rods are identical in size and refractive index, which ensures identical rod mode profile in each cladding rod. Eigenvector for each supermode is consisted of the values of m_j, which reflects the phase relationship between the rods and determines the propagation constant for the supermode. We calculate the eigenvalues for all the supermodes, based on the method described in [23]. The phase relationships for all the supermodes are represented by color combinations and some typical ones are demonstrated in Fig. 2(c). They present four kinds of symmetrical profiles: 60°-symmetry (D_6h), 60°-antisymmetry (D_3h), 180°-symmetry (D_2h) and 180°-antisymmetry (C_2h). The mode profiles of the guided supermodes can be obtained by arranging the LP₀₁ and LP₁₁ rod modes in each rod with these determined phase relationships. The symmetric profiles of the transverse fields for the supermodes are determined by both the phase relationships and the specific rod mode field profiles.

Grating resonances between core modes and supermodes can probably be obtained by index modulation that covers the high-index lattice, since these modes have nonzero overlap over it. We intended to fabricate FBGs within the secondary bandgap around 1050 nm. In this wavelength range, the modal indices of LP₀₁ and LP₁₁ supermodes are higher than that of the guided mode at around 1050 nm, the possible corresponding resonance peaks will be at the long wavelength side of Bragg wavelength and thereby can be distinguished with the radiative supermode resonance peaks, which are always at the short wavelength side.

The Bragg gratings are fabricated by the phase mask method, by means of a 248nm KrF excimer laser. 5cm of PBGF is first spliced to single mode fibers with minimized splice loss. In order to further enhance its photosensitivity, the fiber is loaded in hydrogen atmosphere at 100 atm, 100 °C for 48 hours. The laser beam is focused by a cylindrical lens with a focal length of 100mm and then diffracted by the phase mask before launching onto the PBGF. In order to avoid damage to the phase mask by huge pulse energy and achieve more uniform illumination of the microstructure, the PBGF is placed a little away from the right focal point towards the cylindrical lens. The repetition rate is 5 Hz and the average pulse energy is 40 mJ. The energy density onto the fiber is about 800mJ/cm²•pulse. The period of the grating is 362.4 nm and the grating length is 1.0 cm. The transmission spectrum is monitored by an ASE light source based on Yb-doped fiber pumped by a 980nm laser diode and an ANDO Q8383 optical spectrum analyzer.

 

Fig. 3 Transmission (upper, solid curve) and reflection (dashed curve) spectra of the FBG in straight all-solid PBGF. Peaks A-C correspond to the resonant couplings to backward LP₀₁ supermodes, LP₁₁ supermodes, and core mode respectively. The green, blue, and red curves represent simulated resonant wavelengths for them, respectively. Note that each of the green and blue ones is actually made up of many closely spaced lines, because of the discretion of the guided supermodes. Peak D and its adjacent peaks at the short wavelength side of peak C arise from couplings to discrete, radiative LP₀₂ supermodes. The lower solid curve represents a typical transmission spectrum of the FBG when it is bent against the UV launch direction. The core mode resonance is enhanced and the two guided supermode resonances are chirped.

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Figure 3 demonstrates the transmission and reflection spectra for the FBG, which are measured with a resolution of 0.1nm. Four obvious attenuation peaks A (1057.5 nm, 2.6 dB), B (1052.62 nm, 3.5 dB), C (1050.7 nm, 0.8 dB), and D (1049.18 nm, 0.6 dB) are observed. Peak C corresponds to the strongest reflection peak and can be identified to arise from coupling between the core modes, since most energy of the backward core mode can couple into singlemode fiber. Peaks A and B, at the long wavelength side of peak C, correspond to guided supermode resonances. They find no reflection peaks because the energy of supermodes distributes in cladding region and can hardly couple into singlemode fiber. Peak D and the other peaks at the shorter wavelength side of peak C are produced by couplings to radiative LP₀₂ supermodes. The separation of these peaks reflects that the radiative supermodes are dispersed in index due to the strong coupling between the adjacent rods. We calculated the theoretical amplitudes of these resonant wavelengths based on phase-matching condition and demonstrate the result in Fig. 3 by color lines. The small deviations between theoretical and experimental results arise from uncertainties of the geometric parameters of the microstructure.

Resonance strength for each peak is measured by coupling coefficient, which is determined by the overlap integral of the coupling mode pairs over the index-modulation region (the high-index rod lattice) by

The value of κ is largely determined by the symmetry of the coupling modes. We assume that UV-induced average index raises and index modulations are identical over the individual rods. The coupling behaviors in the FBG are described as follows:

Figure 4(a) shows the UV-induced shift of the short wavelength edge of the first transmission window. As the PBGF is illuminated by the UV light, the average index of the cladding rods over the grating region is raised. The bandgaps move towards the long wavelength side because of the index contrast variation over the microstructure. As a result, the transmission bands are narrowed due to the different transmissive ranges between the grating region and the unperturbed fiber. The long wavelength edges of the transmission bands are fixed (not given here) while the short wavelength edges red shift. The index raise can be estimated based on the ARROW modal by [9]

where λ_m is the cutoff wavelength of the mth supermode band. For the LP₁₁ supermode bandgap, m = 1. n₀ and Δn represent the index of silica and the index contrast, respectively. For the measured shift of 80nm in our experiment, a corresponding average index raise of 3.2×10^-3 is calculated based on Eq. (1). Figure 4(b) shows the shifts of peaks A, B and C during exposure. They shift by 0.88, 0.52 and 0.08nm, respectively, from which the effective indices of the core mode, the LP₀₁ and LP₁₁ supermodes can be calculated as 1.1×10^-4, 2.32×10^-3, and 1.33×10^-3. The effective indices of the guided supermodes change because they are confined to the region where the index change occurs. The shift of peak C depends on the dispersion curve of the core mode, as previously described in [25]. We simulated the sensitivities of effective indices of LP₀₁, LP₁₁ supermodes and core modes to the raise of rod index over the secondary bandgap, by means of plane-wave expansion method and full-vector finite element method, respectively (For the supermodes, this simulation is simplified by calculating the response of each band cutoff). As can be seen from Fig. 4(c), core mode has a larger sensitivity to rod index variation at the edge of the bandgap, due to a stronger dispersion. However, it is still much smaller than those for the guided supermodes. The supermodes present higher sensitivities at short wavelengths because the energy is better constrained in the rod lattice.

 

Fig. 4. (a). Shift of the short wavelength edge of the fundamental bandgap during exposure (resolution: 1nm); (b). UV-induced shifts for peaks A, B, and C. The result indicates that the effective indices of the LP₀₁, LP₁₁ supermodes and the core mode present different sensitivities to average index raise of the high-index rods; (c). Simulated sensitivities of wavelength variation of peaks A, B, and C to the average index raise of high index rods over the secondary bandgap. The LP₀₁ and LP₁₁ supermodes present much higher sensitivities than that of the core mode.

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The analysis above is based on the assumption of uniform index modulation. However, the side UV illumination causes inherently nonuniform index modulation profiles [25], so that the overlaps between the antisymmetrical supermodes and the core mode are no longer strictly zero. The influence of the nonuniform average index raise to the phase relationship can be ignored because the UV-induced index differences are rather small compared to the refractive index of the rods. We inscribed FBGs in PBGFs with different geometrical orientations with respect to the laser beam and found that it hardly influences the grating spectral profile. The index modulation pattern in our proposed FBG plays an important role in exciting grating resonances to guided supermodes. In previous reported long period fiber gratings mechanically or arc-induced in high-index fluid-infused ARROW fibers, only radiative supermode resonance peaks were observed, because equivalent index modulation or periodic structural variation is imposed on the low-index pure silica region [13, 26]. The strong intercylinder coupling of the radiation supermodes means a lot of the mode energy is localized in the low index regions, and so these modes are perturbed by the presence of the defect core and can have strong overlap with the bandgap guided modes of the defect core. Coupling to guided supermodes can probably be observed in the electrically induced LPGs, since the liquid crystal in the holes are periodically perturbed [15]. The reason they did not observe any guided mode might be the large index contrast of the microstructure, which requires a very small grating period. Fiber gratings in all-solid PBGFs with a low index contrast provide a coupling method between modes of a fiber core and a waveguide array, which would find applications in coherent beam propagation, optical signal routing, mode conversion, etc.

3. Bend response of the FBG

We examined the spectral response of the FBG to directional bends. Figure 5 demonstrates the depths of peaks B and C with curvatures along the UV launch direction. A typical transmission spectrum when the FBG in all-solid PBGF is bent against UV launch direction is shown in Fig. 3. When the PBGF is bent with a radius of R, as demonstrated in Fig. 6, a strain gradient is established over the fiber cross section. The fiber core is at the neutral plane, on which the strain is zero. The upper cladding rods are stretched while the lower ones are compressed. The amplitude of strain is determined by the distance between the individual rods and the neutral plane x and the bending radius R by ε = x/R. Due to the nonuniform strain distribution over the cladding rods, the guided mode resonances will become chirped, with the resonant strength decreasing, as demonstrated in Figs. 3 and 5. Assume that the index modulation over the rods are uniform and the rod lattice covers almost the whole fiber cladding, the resonance peak will broaden uniformly and the bandwidth of the chirp can be estimated by Δλ = (1 - p_e)D/R, where p_e = 0.22 is the effective photoelastic coefficient of fiber glass and D notes the diameter of the fiber. However, we found that the chirped spectral profile of peaks A and B vary with bending direction and the induced spectra are quite complicated, which arises from the nonuniform average index raises and index modulation over the cladding rods, inherently induced by UV side illumination. Each peak splits to several peaks when further increasing the curvature.

 

Fig. 5. Evolution for depths of peaks B and C with directional curvature along the UV launch direction. When applying bendings, peak B will broadens and splits, which causes decreased resonant strength. The core mode resonance is enhanced, because of the energy extension to the high-index rods of the core mode, induced by the narrowed bandgap.

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The core mode resonance is enhanced when the FBG is bent, as shown in Fig. 3. As described in [12], bending of all-solid PBGF results in narrower bandgaps. For the bent PBGF in our experiment, the short wavelength edge moves towards the Bragg wavelength of the FBG. As a result, energy can not be well constrained in the core and more energy of the core mode distributes in the cladding rods (as demonstrated in Fig. 2 (b), lower), which causes a higher overlap integral between core mode and the high-index rods and produces a strengthened resonance. The amplitudes of core mode overlap integral over the rods for these two demonstrated mode fields are calculated as 2.45×10^-2 and 6.15×10^-2, respectively. The bend response of the core mode resonance is direction dependant as well. The resonance presents the largest increment in strength against the UV launch direction, which indicates that strains induced by this bend undoes the nonuniform index modulation to a certain extend. The bend response of radiative supermode resonances is not given because the resonances are relatively weak and can hardly be distinguished from the complicated fluctuations in the spectrum background.

 

Fig. 6. Scheme for strain gradient over the cross section of the bent PBGF. The neutral layer experiences a zero strain. The outside half of the fiber is stretched while the inner half is compressed. The amplitude of strain of the local cladding rod is determined by its distance x to the neutral layer and the bent radius R by ε = x/R.

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It is noted that the all-solid bandgap fiber is sensitive to bend itself [11, 12]. Core mode leaks out as the form of radiative supermode in bent PBGFs and hence causes loss. When the light reach the splice point between the PBGF and singlemode fiber, the core mode and supermode interfere with each other and causes periodic fluctuation in the spectrum background, as can be seen from Fig. 3. We examined several FBGs with different PBGF lengths and found almost the same evolution of these resonance peaks, which indicates that the interference fringings do not obscure the changes in grating spectrum.

4. Conclusion

In this paper, Bragg grating is inscribed into the high-index rod lattice of all-solid bandgap fiber within the secondary bandgap. The FBG comprehensively reflects the modal distribution in the PBGF by exhibiting resonance peaks corresponding to backward core mode, guided and radiative supermodes. We found that only those supermodes with certain phase relationships and symmetric mode field profiles are responsible for the supermode resonances, with an ideal uniform index modulation. During the exposure, bandgaps of the PBGF are narrowed and the resonant wavelengths shift with different rates, which provide approaches to independently monitor the index change in the Ge-doped rods. When the FBG is bent, guided supermode resonances are chirped and the core mode resonance is enhanced. The bend response of the FBG is direction dependent due to the side-illumination induced nonuniform average index raises and index modulation over the cladding rods.