1. Introduction

Long period gratings (LPGs) induce couplings between two phase-matched co-propagating modes, usually core and cladding modes, in an optical fiber, and produce discrete attenuation bands in their transmission spectra [1]. LPGs are very useful photonic devices in optical filtering, gain flatterning and fiber sensing. The emergence of microstructured optical fibers (MOFs) and photonic crystal fibers (PCFs) provides a versatile platform for LPG devices [2], because LPGs can be formed in these fibers by various approaches, especially by permanent structural modulation. Furthermore, LPGs fabricated in PCFs have presented some unique characteristics, such as temperature insensitivity [6] and highly polarization-dependent properties [7]. The PCFs, especially the photonic bandgap fibers (PBGFs), present completely different modal and dispersion properties compared with index-guided fibers, thus LPG resonances in these fibers need detailed understanding for the sake of developing novel LPG devices. LPGs have been fabricated in air-core and material-filled PBGFs in the past several years [8, 9], and most recently in all-solid ones by use of acoustic gratings [10]. In this paper, we demonstrate LPGs inscribed in the cladding rods of all-solid bandgap fibers, by means of UV illumination. A series of investigations are carried out to gain a detailed insight over the modal and dispersion properties of the PBGF: With the assistance of measurement of their responses to high-index oil immersion, the “upper” and “lower” supermode couplings can be distinguished and the dispersion curves for supermode bands and bandgap-like modes are reconstructed. The spectral widths of the guided-supermode peaks as a function of wavelength are measured to determine the widths of the supermode bands. We calculated the supermode rensonance strengths, which are mainly determined by the local symmetry in the rods of the fundamental modes and the energy portion of modes in the rods. Moreover, the guided-supermode peaks present maximum temperature sensitivity in the middle of the bandgaps, rather than at the bandgap edges, which are associated with the dispersion slope of the modes.

2. Modal properties of the PBGF

The PBGF was fabricated by using a modified stack-and-draw process. Its microstructure is shown in Fig. 1. Triangularly arrayed germanosilicate rods of six layers (including 126 rods totally) are embedded in pure silica background. The fiber core is formed by omitting a single rod from the array. The outside diameter of the fiber is 240 µm. The pitch of the rod lattice is 13.2 µm and the nominal ratio (the ratio between the rod diameter and the pitch) is 0.4. The index difference between the Ge-doped rods and the silica background is about 1%. 5 cm of the PBGF is spliced with singlemode fibers (SMF) at both ends. The SMF-PBGF-SMF structure presents higher insertion loss towards shorter wavelength because of enhanced mode field mismatch. The insertion loss at 1550 nm is about 3 dB. This structure also causes fringes in the transmission spectrum, which is probably induced by interference of the core mode and the excited cladding-rod modes of the PBGF because of modal mismatch. This fringes can hardly be avoided even the two fibers are well aligned before fusion splicing. The PBGF is loaded in hydrogen atmosphere at 100 atm, 100 °C for 48 hours before LPG inscription to enhance its photosensitivity. The grating growth is monitored by use of an optical spectrum analyzer (OSA) and a supercontinuum light source, which is realized by pumping a section of highly nonlinear PCF with a 1064 nm microchip nanosecond laser.

In this paper, mode profiling for the PBGF is carried out with the commercial FEM software package COMSOL. A quarter model is used to decrease the required finite elements. Proper boundary conditions for the two orthogonal radiuses are set up to obtain both symmetric and antisymmetric modes. PML condition is used at the outer boundary of the fiber to calculate confinement loss. Fig. 1 exhibits the dispersion map and some typical modal profiles of supermodes and bandgap modes. The 126 index-raised rods in the PBGF and the silica background determine 126 index-guided eigen modes, which are known as supermodes or modes of the microstructure. Each order of supermodes in a certain band can be considered as a linear combination of rods modes with specific phase relationships of the electric field distribution in neighboring cylinders. (see LP₀₁ supermodes of different orders at points e and f in Fig. 1). The supermode bands are distinguished by modal profile (such as LP₀₁, LP₁₁, LP₀₂……) in the individual rod, as can be seen from the demonstrated supermode profiles at points e, g and i from three individual bands, respectively. For the guided supermodes, whose effective indices are higher than the index of silica, light is confined well in the rods, and they have almost identical effective indices in a certain band. The guided supermodes become radiative near cutoff, which leads to wider supermode bands and much separated index curves. For the radiative supermodes with lower effective indices than that of pure silica, a significant portion of mode energy lies in the low index region, due to the strong coupling between the rods (compare the modal energy distributions of the guided and radiative LP₁₁ supermodes at points f and g in Fig. 1.). The couplings between the neighboring rods create specific modal profiles over the whole microstructure, which is analogous to the cladding modes in index-guided fibers [8,11]. The guidance in the PBGF can be understood by the anti-resonant reflecting optical waveguides (ARROWs) model [12]: when the modes in the defect core are in resonance with the supermodes, the phase-matched supermodes are excited and the energy in the core decays rapidly (see the bandgap mode near the bandgap edges at points a, b and c. These modes are lossier than those in the middle of the bandgaps due to the resonance coupling). When they are not resonant, guidance in the defect core is established by antiresonant scattering from the high-index cylinders. The transmission spectrum of the PBGF is measured, as shown by the black curve in Fig. 2. The region at the longer wavelength side of 1100 nm and the one between 700 and 1100 nm are identified as the first and second bandgap-guided windows, respectively. The dispersion curves for the higher-order core modes are also presented in Fig. 1. Unlike the description in Ref. 8, these modes lie in the continuum of the radiative supermodes, rather in the bandgaps, because the low index contrast and much narrower bandgaps for this PBGF. Most energy of these bandgap-like modes is localized in the core region. The individual index curves for the supermodes are not shown in Fig. 1, to avoid the ambiguity between the radiative supermodes and the bandgap-like modes. However, the bandgap edges are presented to define the individual supermode bands and the bandgaps, which is calculated based on expansion-plane wave method. Material dispersion of pure and doped silica is not taken into consideration, but the calculated results for transmission windows and index contrasts will not be affected.

 

Fig. 1. (a) Dispersion map for the PBGF. Supermode bands and bandgaps are divided by the black curves. (b) Modal energy distributions for some typical supermodes and fundamental modes of the PBGF. Arrows represent the amplitudes and directions of transverse electric fields.

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3. Inscription and characterization of the LPGs

LPGs are inscribed into all-solid PBGFs through a point-by-point UV side illumination process. The setup for LPG fabrication is shown in Fig. 1. The UV light from a 248 nm KrF excimer laser is reflected by a mirror and then focused by a cylindrical lens onto the PBGF. The PBGF is placed a little away from the focal point of the lens to ensure a uniform illumination over its microstructure. A tunable slit is placed close to the PBGF to control the size of the laser spot on the fiber. Its aperture is adjusted to be a half of the grating pitch Λ _LPG to obtain a high visibility of the index modulation. The mirror, cylindrical lens and the slit are mounted on a high-precision motorized linear stage to determine the position of the laser beam along the fiber length. The PBGF is exposed by 400 laser pulses at a repetition rate of 3 Hz at each point before the laser beam moves to the next point by Λ _LPG. The average pulse energy is 30 mJ and the energy density of the laser spot on the fiber is about 600 mJ/cm²/pulse. The length of the grating is 3.5 cm.

 

Fig. 2. Schematic setup for LPG inscription in all-solid PBGFs. Inset, microscopic image for the cross section of the PBGF.

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In our experiment, UV-induced index modulation is considered as a single-photon absorption process, since the energy density on the fiber is relatively low. Index modulation can only take place in the rods. As can be seen from Fig. 1(b), a small portion of energy of the fundamental mode is distributed in the Ge-doped rods, due to the modification of photonic states when creating the defect core. Consequently, grating resonances can be possibly excited due to the non-zero overlap between the fundamental mode and the grating. Note that LPGs can possibly couple light to both guided supermodes in the “upper” bands and the radiative supermodes and bandgap-like modes in the “lower” bands, as long as the phase-matching condition λ _res=|n _fund-n _res|·Λ is satisfied.

LPGs are inscribed with different pitches into the PBGFs. The grating pitche varies from 145 µm to 435 µm. Fig. 3(a) demonstrates the transmission spectrum of a grating with a pitch of 256 µm, measured with a resolution of 1 nm. Two deep resonant peaks are observed in the fundamental bandgap, which are labeled as peaks A (λ_A: 1339.6 nm, depth: 9.6 dB, 3dB bandwidth: 14.6 nm) and C (λ_C: 1422.8 nm, depth: 6.3 dB, 3dB bandwidth: 5.6 nm). Peak A has a much larger bandwidth than peak C, and presents a complicated spectral profile. Several weaker resonance peaks are located at around peaks A and C, labeled as peaks D-G. In addition, a weak peak B is found at around 820 nm in the secondary bandgap. The short wavelength edges of the bandgaps red shift during the illumination, as exhibited in Fig. 2, due to the index raises of the exposed cladding rods. According to the ARROWs model, the measured 22 nm shift of the fundamental bandgap short-wavelength edge corresponds to a rod index raise of 2.2×10^-4.

We found that peaks A and C red shift when LPG pitch increases. These two peaks are located at around 1550 nm when the pitch is 340 µm so that their polarization-dependant property can be measured with a photonic all-parameter analyzer. The measured PDL for the LPG is given in Fig. 3(b). The maximum PDLs for peaks A’ and C’ are 1.15 dB and 4.11 dB, respectively. The high PDL of peak C’ indicates that an asymmetric index modulation over the rod lattice is formed, which is intrinsically caused by the side illumination of the laser beam.

In order to identify what kind of modes are involved in the formation of peaks A’ and C’, we recorded the near field images at the resonance wavelengths. This is carried out by use of a microscope with its eyepiece lens replaced by an infrared camera. A wavelength-swept laser was used as the light source to illuminate the LPG via the lead-in singlemode fiber. Fig. 3(c) demonstrates the recorded images. The near filed profile for the fundamental mode was recorded at 1570 nm as a reference. As can be seen, a small fraction of energy lie in the inner six rods as depicted above. The image taken at 1532 nm shows that the LPG couples light from the core into LP₀₁ guided supermodes. The image recorded at 1609 nm indicates that light is coupled to a mode with a LP₁₁ profile at peak C’, whose energy is mainly localized in the core region. Peaks A’ and C’ can be spectrally observed because the supermodes and LP₁₁-like mode will suffer great losses at the joint between the two fibers due to modal mismatch.

 

Fig. 3. (a) Transmission spectrum of a section of all-solid PBGF before and after LPG inscription. The grating pitch is 256µm. (b) Measured PDL for an LPG with a period of 340µm. Inset, zoomed measurement result for peak C’. (c) Measured near field profiles of the LPG at different resonant wavelengths. Peaks A’ and C’ corresponds to the resonances to guided LP₀₁ supermodes and a LP₁₁-like mode, respectively. (d) Variation of transmission spectrum before and after the LPG is immersed into a high-index liquid. Peaks A and C are hardly influenced, while peaks D-G decrease in strength.

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Figure 3(d) shows the variation of the LPG spectrum when the grating is immersed in a high-index liquid (n=1.47). Peaks A and C remain unchanged because the guided supermodes and the LP₁₁-like mode which are involved in the couplings are well confined. Peaks E-G, however, are eliminated by the liquid. It can be deduced that these relatively weak peaks arise from couplings to radiative supermodes. Their modal fields spread into the outer silica region and can be influenced by the surrounding refractive index. The discrete distribution of the peaks is a result of the much dispersed index curves of the radiative supermodes. Peak D, which has a larger depth than peaks E-G, decreases in strength but does not entirely disappear. This result indicates that it comes from coupling to a lossy bandgap-like mode, rather than a radiative supermode.

4. Numerical analysis

4.1 Reconstruction of dispersion curves of supermode and bandgap-like modes

The dispersion curves for the supermodes and bandgap-like modes can be experimentally reconstructed, by measuring the transmission spectra with different grating pitches [10]. Before the reconstruction work, the “upper” and “lower” couplings are first distinguished: The guided-supermode peaks present higher depths and wider spectral widths, and remain unchanged in the oil immersion measurement; the radiative-supermode peaks have the lowest depths and will be eliminated by the high-index oil; The bandgap-like mode peaks are deeper than the radiative-supermode peaks and will be weakened by high-index oil. This step is important for the reconstruction because it can avoid the ambiguity in Ref. [10] about whether “upper” or “lower” coupling one peak come from. For the convenience to compare our results with Ref. [10], we plot the evolutions of index difference with wavelength in Fig. 4(a). The data of Δn _eff are taken from the transmission spectra of LPGs through Δn _eff=λ _res/Λ. Simulated dispersion curves are then superimposed. Figure 4(a) shows that the calculated and experimental results for the LP₀₁ and LP₁₁ guided-supermode resonances are in good agreement. The red squares which respresent the radiative supermode resonances below the zero line reflect the highly dispersive nature of the radiative supermodes. Peak D is found to be a result of coupling to a LP₀₃ bandgap-like mode, by comparison with the calculated result. The dispersion curve for peak C presents a very different evolution from any other bandgap-like modes and radiative supermodes. The corresponding index difference becomes smaller with wavelength, which could not be well explained so far.

Guided-supermode peaks A and B are actually composed of many overlapping peaks, because a large number of supermodes are contained in the narrow bands. Consequently, the widths of these peaks are associated with the width of the bands and the density of modes. Fig. 4(b) demonstrates calculated and experimental results of the spectral widths of peaks A and B as an evolution with wavelength. As the bands at around cutoffs become wider, the peaks broaden and the depths of the corresponding peaks will decrease. Furthermore, the radiative-supermode resonance peaks can not overlap with each other and a series of discrete peaks are formed, which reflects the distribution of the much dispersive index curves. Note that the widths of the peaks are probably also relevant with the azimuthal angle of the incident of the laser beam with respect to the rod lattice, which cause the disagreement between the calculated and experimental results.

The composition of the guided-supermode peaks also explains the different PDL profiles between peaks A’ and C’ in Fig. 3(b). Since the envelope of peak A’ contains a large amount of supermode peaks which correspond to the multiple modes in a band, the measured PDL profile is actually a result of compensation of the individual peaks with each other. Therefore, peak A’ presents a much lower maximum PDL amplitude than peak C’.

4.2 Calculation of coupling coefficients

Transmission of an LPG can be defined as T=1-sin²(κL), where coupling coefficient κ is usually used to measure the coupling strength between two modes. The coupling coefficient is determined by the overlap integral over the photosensitive region as

In this subsection, we assume a uniform index modulation over the fiber, so Δn _UV is a constant. Eq. (1) suggests that one can calculate the coupling strength of supermode resonances in all-solid PBGFs by calculating the integral over the rod lattice. However, since the energy portions in the six rods immediately surrounding the core are much higher than other rods, the overlap almost entirely take place over these six rods. As a result, we can calculate the overlap with the six-rod supermode for simplification. The modes of the six-rod system can be considered as a linear combination of six orthonormal supermodes with specific phase relationships [13]. The phase difference between two adjacent rods is Δφ=2πn/6 with n=0,…, 5. The overlap is zero when n=1, 3, and 5 and non-zero overlaps are produced when n=0, 2, and 4, which decided by the symmetry of the phase relationships, regardless of modal profile in the rods. The symmetry of the LP₀₁ supermodes is totally determined by the phase relationship, because of the local symmetry in each rod, as can be seen from the modal fields demonstrated in Fig. 5(a). The in-phase one (n=0) produces the highest overlap among all the supermodes because the directions of the transverse electric fields in the rods are the same. For example, at 1350 nm, the calculated integral amplitudes are 0.048 for the in-phase supermode, and 3.9×10^-3 and 6.7×10^-4 for n=2 and 4, respectively. LP₁₁ supermodes with each n are degenerated by four modes, which can further divided into two groups: two modes with all the rod modes antisymmetrical about the radial line from the center of the microstructure, thus overlap integral between fundamental mode and the mode in each rod is zero with whatever phase relationships because of its local antisymmetry. The antisymmetrical axes of the other two modes are orthogonal to the radial line, which enable non-zero overlaps [as shown in Fig. 5(b)]. As a result, only six ones among the 24 supemodes produce non-zero overlaps.

 

Fig. 4. (a) Simulated and experimental results of the dispersion curves for the supermodes and bandgap-like modes, relative to the effective index of the fundamental mode. Curves: Calculated results; Squares: experimental results. (b) Spectral widths of LP₀₁ and LP₁₁ guided-supermode peaks as a function of wavelength. Curves: calculated result; Squares: experimental result.

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The calculated overlap integral as a function of wavelength for the couplings to LP₀₁ and LP₁₁ guided supermodes is demonstrated in Fig. 5(c). The overlap for LP₀₁ supermode resonance decreases sharply at around the LP₁₁ band cutoff. The LP₁₁ supermode resonance presents lower overlaps at both bandgap edges. The local symmetry of the fundamental mode in the rods largely determines the evolution of the curves. The fundamental in the rods evolves from LP₀₁ to LP₁₁ in the first bandgap and from LP₁₁ to LP₀₂ in the second bandgap, as frequency increases [see Fig. 1(b)] [14]. As a result, low overlaps for LP₀₁ supermodes are obtained due to the integration between a local LP₀₁ mode and an increasing LP₁₁-like mode. Similarly, for the LP₁₁ supermode resonances, integrating a local LP₁₁ mode with an increasing LP₀₂-like rod mode cause the rapid decrease of the overlap integral at the blue edge of the second bandgap. Energy portion in the rods of a mode is another factor to affect the integral value. The weakening of the LP₁₁ supermode resonance at the red edge of the second bandgap arises from the decreasing energy portion of supermodes in the rod lattice.

Figure 5(d) demonstrates the experimental result of coupling strengths of the gratings. The amplitudes of coupling coefficients could be estimated by measuring the depth of each peak. For the distinct comparison, the index modulation, obtained by measuring the bandgap shift, is normalized with 1×10^-4. The calculated and experimental results agree well at the shorter wavelength region in each transmission window. As wavelength increases, the experimental strength becomes lower than the calculated ones because of the broandening of the resonance peaks.

4.3 Influence of the asymmetric index modulation

The PDL and the near field measurement results for peak C’ indicate that the UV laser beam introduces an asymmetric index modulation over the rod lattice. As a result, non-zero overlap integrals are produced between the fundamental mode and supermodes with n=1,3 and 5. We assume the amplitude of index raise is Δn₀ when a perfectly uniform index raise is formed over the microstructure, thus the overlap integral can be expressed by ∬_rodsΔn ₀·e⃑_fund;·e⃑_superdxdy. In order to estimate the influence of the asymmetric index modulation, we consider an ideal linear decrease of index raise over the six rods, which is defined by $\Delta n\left(x\right)=\left(\Delta n_0-\Delta n\text{'}\right)+\frac{\Delta n\text{'}}{R\text{'}}x$, as shown in Fig. 5(a). Index variation along y axis is neglected.

Fig. 5(b) demonstrates calculated coupling coefficient for supermodes with odd n as an evolution of wavelength when Δn’ is 1×10^-4. Meanwhile, the decrease of the coefficient for supermodes with even n, whose amplitude is ∬_rodsΔn′·e⃑_fund;·e⃑_superdxdy, is also given for comparison. The two curves present similar evolutions and the amplitudes are close, which indicates that integral decrease for supermodes with even n can be compensated by the ones with odd n to some extent. In practice, the index modulation over the rods will not be simply linear varying. The determination of actual amplitudes of the coupling coefficients depends on further index measurement for fiber gratings in the PBGF or multi-core fibers.

5. Temperature response

Figure 6(a) plots the measured temperature responses for peaks A and C. The two peaks present maximum sensitivities at 1480nm and 1560nm, respectively. This result indicates that one can obtain highest tunable sensitivity in the middle of a bandgap, rather than at the lossy region at the bandgap edges. This is useful for the design and optimization of tunable LPG devices in solid-core PBGFs. (We believe couplings to guided-supermodes can be realized in the LPG device in liquid crystal PBGF described in Ref. [11].) Transmission spectrum of the LPG with a pitch of 256 µm under room temperature and 100 ⁰C is exhibited in Fig. 6(b). The measured sensitivities for peaks A and C are 19.1 pm/⁰C and 25.2 pm/⁰C, respectively.

 

Fig. 5. (a) Modal profiles of the LP₀₁ supermodes for the six-rod fiber with n=0,…, 5. The amplitudes and directions of electric fields are represented by the arrows. (b) Some modal profiles of the LP₁₁ supermodes. The former three with n=0, 2, and 4 produce non-zero overlaps. The modes with all the rod modes antisymmetrical about the radial line from the center of the microstructure, like the last one, cause zero overlap. (c) Calculated overlap integral as a function of wavelength over the rod lattice between the fundamental mode and the guided supermodes. (d) Curves, calculated variations of coupling constants with wavelength for the guided-supermode resonances. Squares, experimental results.

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Fig. 6. (a) Schematic index distribution over the six rods. We assume an idea linear gradient is established by the laser beam for simplification. (b) Calculated coupling constant for supermodes with odd n, compared with the decrease of that for supermodes with even n.

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Fig. 7. (a) Solid curves, calculated variations of temperature sensitivities for peaks A and C. Squares, experimental measurement. (b) Transmission spectra of the LPG measured at room temperature and 100 °C. Insets, zoomed pictures of spectral variations for peaks A and C.

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The temperature sensitivity of LPGs in solid-core PBGFs depends on both the shift of bandaps and the dispersion characteristics of the modes, as described in [15]. Based on this work, the temperature sensitivity of the LPG in an all-solid PBGF can be expressed by

where n₁₀ and n₂₀ represent the refractive indices of pure and doped silica. ξ₁ and ξ₂ are the thermal-optic coefficients of pure and doped silica, respectively. α is the thermal-expansion coefficient of the fiber glass. The last term is the waveguide factor, which is decided by the dispersive property of the mode pairs. The subscripts “eff” and “g” means effective index and group index, respectively. Although the index contrast between n₁₀ and n₂₀ is very small, rather low temperature sensitivity is obtained because the amplitudes of ξ₁ and ξ₂ are very close.

The calculated evolutions of temperature sensitivity with wavelength are demonstrated in Fig. 6(a). The curves for peaks A and C are obtained based on the polynomial-fit result from the measured wavelengths as a function of grating pitch. The deviation between the calculated and experimental results probably arises from difference in the amplitudes of index modulation for the individual LPGs. We presume the second term on the right hand of Eq. (2) is a constant and the amplitudes of α, ξ₁ and ξ₂ are 5×10^-7, 6×10^-6 and 7.5×10^-6, respectively. The profiles of the curves for guided-supermode couplings are totally different from the result demonstrated in Ref. [15], due to the different evolution of Δn, as shown in Fig. 4(a). The evolution of the red curve is quite similar with the blue one, which also indicates that peak C does not arise from couplings to any radiative supermodes or bandgap-like modes. The formation of this resonance peak needs further investigation. The multiple radiativesupermode peaks red shift with different sensitivities. We did not track the variation for these peaks because we could not distinguish the peaks with each other due to the large amount of radiative supermode. However, the evolutions of temperature sensitivities for these peaks should be similar with the result in Ref. [15], only with much smaller amplitudes and the positive sign.

6. Conclusion

LPG inscription into an optical fiber is considered as an effective method to study the modal and dispersive properties of the fiber. In this paper, UV-induced LPGs are fabricated in all-solid PBGFs and a series of simulations and experimental investigations are carried out, so that we can obtain a detailed insight over the properties of the PBGF. The dispersion curves for the modes are reconstructed by plotting the resonance wavelengths of LPGs with different pitches, with the assistance of the measurement of their oil-immersion responses and temperature sensitivities. The widths of the supermode bands are detected by measuring the spectral widths of the corresponding peaks. The coupling strengths of peaks reflect the local symmetry of the fundamental modes in the rods and the energy portion of modes in the rods. Maximum temperature sensitivity for guide-supermode peaks can be obtained in the middle region of a bandgap, which is determined by the dispersion property of the PBGF.