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In this paper, core-cladding modal beating in a short piece of all-solid photonic bandgap fiber (AS-PBF) is observed in longitudinal propagation direction. It is demonstrated that at the stopband range of AS-PBF, the power could transfer back and forth between the fiber core and the first layer of high-index rods. Both experimental results and the theoretical analysis from transverse coupled mode theory confirm that the 3-dB width of the sharp stopband could be significantly narrowed by multicycles of such core-cladding modal couplings, which is of great benefit to the high-resolution sensing applications. Based on such a guiding regime, a high-temperature sensor head is also made and its response to temperature is tested to be of 59.9 pm/°C.
©2011 Optical Society of America
1. Introduction
In recent years, solid-core photonic bandgap fibers (SC-PBFs) have received great attentions due to their designable pattern of transmission-inhibited bands and no suffering from surface modes, which can be applied to many devices such as tunable filters, optical modulators, directional optical couplers and laser amplifiers. Different from hollow-core photonic bandgap fibers, SC-PBFs have a solid fiber core and a fiber cladding composed of high-index rods instead of air holes. According to their different fabrication process, they can be divided into two categories, the liquid-cladding SC-PBFs transformed from index-guiding PCFs filled with Cargille liquid or liquid crystal and the all-solid-cladding type of SC-PBFs with index-raised solid rods in cladding [1, 2] which are so called all-solid photonic bandgap fibers.
In SC-PBFs, the propagation of light in the low-index fiber core can be explained by an antiresonant reflecting optical waveguide (ARROW) model, and if the guiding modes of the fiber coincide with the cutoffs of the guiding modes of individual high-index rod, the whole structure becomes transparent substantially and the power escapes from the fiber core which leads to a transmission spectral dip [3, 4]. The position of the spectral dips is shifted by changing the temperature dependent refractive index of high-index rods, which can be exploited for fiber sensors as the detailed theoretical analysis of Ref. 5 and 6. A quality factor Q, defined by Q = (Δλ/Δα)/λ BW, can be introduced to characterize wavelength-encoded fiber sensors, where Δλ is shift of spectral pattern, Δα is a measurand value and λ BW is the 3-dB bandwidth of a spectral dip [7]. However, in practice, sharp spectral dip is difficult to be observed especially for the liquid-filled SC-PBFs [8, 9] and always exhibits a blunt and wide stopband, which leads to low sensing quality and measurement errors. Hence, how to make the stopband narrow and sharp is essential to the ARROW-based fiber sensors. So far, many strategies have been developed to obviate the problem, for example by introducing depressed area around the high-index rods [10] or by filling only some of the air-holes [11, 12] to form SC-PBFs.
In this paper, we present that in a short piece of AS-PBF, the effective 3-dB bandwith could be significantly narrowed by experiencing multicycles of modal coupling between core mode and the rod supermode. Both experimental and numerical results demonstrate that at the stopband wavelength range of AS-PBF, the power transfers back and forth between the fiber core and six high-index rods of first layer of fiber cladding, and very sharp transmission dips exceeding 20dB are observed and its bandwidth could be narrowed with increased fiber length, which is consistent well with transverse coupled-mode theory. A high-temperature sensor head has also been made using such a guidance regime of AS-PBF and its response to high-temperature is investigated.
2. Spectral properties of modal beating in AS-PBFs
The spectral evolution of the stopband in longitudinal propagation direction with different lengths of AS-PBF is examined by the cutback method. Figure 1 (a) shows the schematic of experimental setup. A small piece of AS-PBF with an initial length of around L = 15 mm is used and cut back with a step of around 0.2 mm. In each cut step, both ends are spliced with standard single-mode fibers (SMF28) as lead-in/out fibers and the transmission spectra are recorded. An incandescent white light source is employed as an incident light source and an optical spectrum analyzer (Agilent 86146B) is used to record the transmission spectra.
Fig. 1 (a) Schematic of experimental setup; (b) Cross section of the all-solid photonic bandgap fiber employed; (c) an enlarged unit cell of the high-index rod.
The AS-PBF we used is fabricated by Yangtze Optical Fiber and Cable Company (YOFC co., China). It is a new type photonic bandgap fiber with a similar index profile as Ref. 2 and exhibits very low transmission loss (~0.5 dB/km). As a unit cell shows in Fig. 1(c) in such a fiber, the germanium-doped high-index rod is surrounded by an index-depressed layer doped with fluorine. Figure 1(b) shows the fiber cross section. It is based on a so-called triangular arranged structure and the fiber core is formed by missing a high-index rod. The outer diameter of the fiber is 144 µm and the core diameter is around 14 µm. The other structure parameters are as follows: the rod-to-rod spacing pitch Λ is 10.8 µm. The relative diameters of high-index rods and depressed layers are d Ge/Λ = 0.39 and d F/Λ = 0.7, respectively. The refractive index difference of the germanium-doped and fluorine-doped area are approximately ΔnG = 0.0345 and ΔnF = −0.00723, respectively. The background material is silica with nsilica = 1.45 and the material dispersion is neglected for convenience. Plotted in Fig. 2 are the effective indices of fundamental core modes (red curves), bandgap structure (grey region) and measured transmission spectrum (the blue curve) of a 50 cm AS-PBF, which agree well with each other. The first stopband is from 1198 nm to 1265 nm with a width of around 60 nm, and this transmission trench presents a chaos and blunt feature as we are familiar with.
Fig. 2 Effective index of fundamental core modes (red curve) and transmission spectrum (blue curve) for a 50 cm AS-PBF. The grey region denotes the first and second bandgaps.
However, as the fiber is shortened gradually, the stopband exhibits a quite different transmission profile as Fig. 3(a) shows. A sharp spectral dip is firstly observed at L = 7.98 mm. It has a major transmission dip at 1215 nm with an extinction ratio exceeding 20 dB and a small 3-dB bandwidth of 5.6 nm, and around it are some minor lobes. However, such a transmission dip appears periodically with fiber lengths shortened, and the other special fiber lengths are L = 6.35 mm, 4.83 mm, 2.94 mm and 1.20 mm. It should be noted that a small deviation of fiber length from those values could lead to the rapid disappearance of the spectral dips as exampled by the dot curves of L = 0.93 mm and L = 1.92 mm. Besides that, the amount of minor lobes simultaneously increases with fiber lengthened. Figure 3(b) shows the measured 3-dB bandwidth of the major dips, and it has been greatly reduced to 5.6 nm from 41.9 nm. The fitted line illustrates that the 3-dB bandwidth is in reverse proportion to the fiber length with a relationship of 49.87 nm⋅mm−1, which suggests that we can get a much narrower transmission dip using a longer fiber.
Fig. 3 (a) Spectral evolution of the stopband with different lengths L of AS-PBF. (b) 3-dB bandwidths of the stopband versus the AS-PBF lengths.
3. Theoretical analysis of formation mechanism
According to the ARROW model, the first stopband is formed by the index-matching encounter of the core modes with a series of LP11 rod supermodes which are the coupled photonic states of periodic high-index rods. However, the model is unable to give a deep insight into the spectral evolution of the stopband in longitudinal propagation direction. Since the fiber core of AS-PBF is larger than that of SMF28, the recorded transmission spectra actually reflect the core power at the end stub of AS-PBF adopted by the lead-out fiber. Therefore, the periodically emerged transmission dips at 1215 nm indicate that the power in the fiber core varies periodically along the propagation direction. Figure 4 shows the 1215 nm light propagation and the monitored core power simulated by the beam propagation method. It can be seen that the incident energy transfers back and forth between the fiber core and mainly the first layer of fiber cladding, which suggests that the fundamental core mode is coupled to a rod supermode and the two modes beat with each other along the fiber. The coupled length in which the power is completely transferred to the six rods from the fiber core is around L c=0.72 mm, and the experimental value is 0.84 mm, the deviation mainly results from the measurement error of AS-PBF microgragh.
Fig. 4 Light propagation longitudinally (left panel) and the monitored core power (right panel). The incident light is with a wavelength of 1215 nm, and both ends of the AS-PBF are conjunct with 1mm single-mode fibers, respectively.
Since the energy transferred to the six rods immediately surrounding the core are much higher than that in other rods as shown by the inset of Fig. 4 (right panel), the AS-PBF structure can be simplified as a six-rod system. In such a six-rod system, the total amount of six-rod supermodes is 24 for the four degenerated LP11 modes. However, only six ones among them can produce non-zero overlaps with the fundamental mode due to the symmetric principle of mode distribution. As the up-panel of Fig. 5 shown, when the phase-matching condition is satisfied at a particular wavelength, the energy transfer occurs from the core mode to a six-rod supermode which has the largest overlap integral between them. In such an AS-PBF with a depressed area around high-index rod, the six-rod supermode can still be guided and thus could transfer their power back into the fiber core, and then the intermodal beating with several circles of power exchanges can be observed.
Fig. 5 Phase-matching (up-panel) for LP11 supermode and fundamental core mode, and transmission spectra with L = Lc and L = 9Lc (low-panel). The dash line in the up-panel is the linear fitted curve for the core modes.
According to the transverse coupled-mode theory, the normalized power of the fiber core at the end of AS-PBF stub can be expressed as follows:
Where and and are effective index for the fundamental core mode and a six-rod supermode of cladding, κ = 1.96 mm−1 is the coupling coefficient which is determined by the fields overlap integral between the two modes, and L is the fiber length. Thus, when phase-matching condition of δ = 0 is satisfied, κL = mπ/2 (m is an odd integer) is required for completely coupling the power to the cladding. And the 3-dB bandwidth from the phase-matching wavelength λ 0 is expressed as [13]:where and are the corresponding group indices of core mode and six-rod supermode defined by and , respectively. From the above formula, it can be seen the 3-dB bandwith of transmission dip is in reverse proportion to the fiber length and can be controlled by the fiber length. The low-panel of Fig. 5 shows the simulated transmission spectra with fiber lengths of L = L c and L = 9Lc. The dispersion curves encounter at the phase-matching wavelength of around 1215 nm. With a longer fiber of L = 9Lc, that is with 4 circles mode coupling, the 3-dB bandwidth is narrowed by 9 times theoretically, which is close to experimental results (7.5 times).4. Potential application as high-temperature sensors
Such a simple AS-PBF-based device can find its applications in tunable fiber filters and high-resolution temperature sensors. When the fiber temperature increases, the index difference between Ge-doped area and silica is varied, and correspondingly the phase-matching central wavelength will shift according to the above analysis. To demonstrate the feasibility as a high temperature sensor, an AS-PBF-based sensing head is placed into a tube furnace and tested. The temperature is varied from 25 °C to 600 °C in a step of 50 °C and stay approximately 30 min for each step in both heating and cooling process, and the dip wavelength and transmission spectra are recorded. Figure 6 (a) shows the temperature response with an AS-PBF length of 7.98 mm which has the narrowest 3-dB bandwidth and highest quality factor Q. In an adjustable temperature range of 25 °C - 600 °C, the sensitivity is found to be of the order of 59.9 pm/°C, it is higher than that of interfermetric high-temperature sensors based on index-guiding PCFs and hollow-core PBFs [14, 15] and also comparable to the long period fiber grating inscribed in AS-PBF [16–18]. Figure 6 (b) shows the spectral responses to different temperatures. The shift values are easier to be captured for sharpened dips. However, the degeneration is observed with the temperature rising, which mainly gives rise to the unoptimized fiber length we cut.
Fig. 6 (a) Temperature responses of the transmission dip with and 7.98 mm AS-PBF. The dots are the measured values, and the solid lines are the linear fitting. (b) Spectra shift with temperature increasing.
5. Conclusion
In summary, core-cladding modal beating in a short piece of all-solid photonic bandgap fiber is observed. It is demonstrated that the power of the fundamental mode is firstly couple into a six-rod supermode located in the first layer at index-matching point and then coupled back to the fiber core again, and this process repeats with several cycles. When the fiber length is equal to the odd times of coupling length, the transmission dip appear and its 3-dB bandwidth is narrowed with fiber length increasing, which is consistent with the analysis of coupled mode theory. Although sharp dip output spectra hasn’t been observed with fiber lengths beyond 7.98 mm in our experiment mainly due to the gradual power leakage in propagation, with a redesigned fiber structure, more circles of modal beating and narrower spectral dip can be expected. Its response to high-temperature up to 600°C is investigated and the sensitivity is of 59.9 pm/°C. Such a guidance regime can also find its applications in tunable band-reject fiber filters.
Acknowledgments
This research is supported by Doctoral Project of Guangdong Provincial Natural Science Foundation (No. 10451806001005350) and National Science Foundation of China under Grant (No. 60978041).
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