Phase sensitivity of fundamental mode of hollow-core photonic bandgap fiber to internal gas pressure

Yingchun Cao; Wei Jin; Fan Yang; Hoi Lut Ho

doi:10.1364/OE.22.013190

1. Introduction

Hollow-core photonic bandgap fibers (HC-PBFs) have unique optical properties and can be used to create novel sensing devices or improve the performance of existing sensors [1, 2]. For example, HC-PBFs can confine an optical beam in the low-index fiber core, and the hollow-core and cladding-holes can be selectively filled with liquid or gaseous materials. These features make HC-PBFs ideal platforms for investigation light-material interaction [3–6], as well as high sensitivity chemical and biological sensing [7–9]. Air-silica HC-PBFs have low temperature sensitivity and ultra-low Kerr and Faraday coefficients, thus could be used to improve the performance of fiber-optic gyroscopes [10].

Several research groups have studied the effects of pressure on the transmission characteristics of HC-PBFs. These include the influence of air pressure on soliton formation [11], sensing based on pressure-induced transmission loss in non-conventional air-guided transmission windows [12], and phase sensitivity of the fundamental mode to external pressure [13] and acoustic wave [14, 15]. Compared with solid silica optical fibers, HC-PBFs have much lower effective Young’s modulus, and hence enhanced phase sensitivity to external pressure [13–15]. Recent experimental with modified HC-1550-02 fiber demonstrated (acoustic) pressure sensitivity of 1.72 × 10⁻³ rad/(Pa∙m) [15]. This is ~25 dB higher than the conventional single mode fibers (SMFs) [14] and could significantly enhance the performance of optical fiber acoustic sensor systems.

In this paper, we report the results of our recent experimental investigation on the phase sensitivity of fundamental mode to pressure applied internally to the hollow-core of a HC-PBF. It was found that, for the same fiber length, the phase sensitivity to internal pressure is over two orders of magnitude more sensitive than to external pressure. This would have important applications in high sensitivity static and dynamic pressure detection, phase manipulation of guided mode, and other processes that involve the change of gas pressure. To better understand the physics behind the pressure sensitivity we also developed an analytical model and carried out numerical simulation to compare with the experimental results.

2. Experimental procedures and results

The HC-PBF used in our experiments is the HC-1550-02 fiber from NKT Photonics and a microscopic image of the cross-section of the fiber is shown in Fig. 1. The air-hole pitch and core diameter were measured to be ~3.8 µm and 10.8 µm, respectively.

Fig. 1 Microscopic image of the cross section of a HC-PBF (HC-1550-02) with polymer coating removed. I and II denote the regions of honeycomb inner-cladding and pure silica outer-cladding. r₁, r₂ and r₃ are the radius of the fiber core, honeycomb inner-cladding and silica outer-cladding, respectively.

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2.1 Preparation of HC-PBF sample

As our intention was to study the response of the HC-PBF to pressure variation inside the hollow-core, we need to block the cladding holes but leave the central hollow-core open for applying gas pressure to the core. This was done by use of an arc discharge technique similar to that described in [16]. A cleaved HC-PBF was firstly fusion spliced to a SMF with a fusion current of 14.5 mA, time duration of 0.3 s and offset of 50 µm. The arc discharge was offset from the splice joint so that, in a section adjacent to the splicing joint, the cladding air holes are collapsed completely while the central hollow-core is stilled open, as shown in Fig. 2(a). The HC-PBF was then cleaved near the edge of the collapsed region, as indicated by the green wedge in Fig. 2(a). This was achieved by scanning a focused femtosecond infrared laser beam across the fiber transversely to induce a damage at which the fiber broke upon bending. Figures 2(b) and 2(c) show respectively the end and side views of the cleaved end. It is clear that the cladding holes are fully blocked at the fiber end while the hollow-core remains open. The other end of the HC-PBF was processed identically to ensure the air-core is open while all the cladding holes are completely blocked.

Fig. 2 Procedures for HC-PBF end processing. (a) The cladding holes of HC-PBF are blocked by fusion splicing to a SMF; (b) end view and (c) side view of the HC-PBF end cleaved at a position indicated by the green wedge in (a).

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The so treated HC-PBF was then butt connected to SMFs at both ends by following the procedures shown in Figs. 3(a)–3(c): (i) two identical fiber ferrules with inner diameter of 125 µm were plugged into a mechanical splicer with a side slot as shown in Fig. 3(a); the splicer aligns the two ferrules and holds them together tightly with a small gap left between the ferrules. The side-slot allows easy visual observation and also facilities gas diffusion into the HC-PBF; (ii) with the aid of a microscope and two translation stages, SMF and HC-PBF were inserted carefully into the fiber ferrules from opposite sides as depicted in Fig. 3(b), and a gap of ~20 µm was left between the two fiber ends via inspection from the microscope; (iii) the fibers, ferrules and mechanical splicer were then fixed together with glue as shown in Fig. 3(c). The other end of the HC-PBF was similarly connected to a SMF. The two connection joints were then sealed into two separate gas chambers for applying pressure into the hollow-core. The length of the HC-PBF used is ~95 cm. The total loss of the SMF/HC-PBF/SMF sample around 1550 nm was measured to be ~20 dB, which is believed to be mainly due to the loss at the connection joints. This HC-PBF sample was used to study the response of HC-PBF to internal gas pressure.

Fig. 3 Procedures for connecting HC-PBF with SMF. (a) Two identical fiber ferrules were plugged into a mechanic splicer to ensure the ferrules be aligned to each other and a distance of a few hundreds of µm was left between the ferrules; (b) SMF and HC-PBF were inserted into the ferrules from opposite sides and a small gap of ~20 µm was left between the two fiber ends; (c) The fibers, ferrules and mechanical splicer were fixed together with glue (marked in yellow).

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2.2 Effect of internal pressure on transmission spectrum

The HC-PBF sample described in section 2.1 was connected to an optical broadband source (BBS) and optical spectrum analyzer (OSA) as shown in Fig. 4. A bottle of high pressure N₂ gas was connected to the two gas chambers and the gas pressure level inside the chambers (and hence in the hollow-core) was adjusted and monitored with a standard pressure gauge.

Fig. 4 Experimental setup for studying the effect of varying gas pressure inside the hollow-core on the transmission spectrum of the HC-1550-02 fiber. BBS: broadband source, OSA: optical spectrum analyzer.

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The transmission spectrums of the HC-PBF at different applied pressure levels were measured and shown in Fig. 5. The fiber sample shows a wide transmission window from 1420 to 1660 nm. The loss peaks located in the 1440-1490 nm spectrum range correspond to coupling from the guiding mode to lossy surface modes [17, 18]. The transmission window is hardly shifted when the pressure level is varied from 0 to 4 bar, while the transmission loss increases slightly with increasing internal pressure, which agrees with the results in [12].

Fig. 5 Transmission spectrum of HC-1550-02 for different applied internal pressure levels.

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2.3 Phase sensitivity of the fundamental mode to internal pressure

The setup used to investigate the phase sensitivity of the fundamental mode to internal pressure is shown in Fig. 6. The pressurization process is similar to that of section 2.2 but the fiber sample is now placed in one arm of an optical fiber Mach-Zehnder interferometer (MZI). The other interferometer arm is a section of SMF with its optical-path-length approximately matched to that of the HC-PBF arm. Light from a 1.53-µm distributed feedback (DFB) laser was split equally by a 50/50 coupler and recombined via a second 70/30 fiber coupler, to partly compensate the large loss in HC-PBF arm, and fed into a photodetector (PD). The coupling ratios of the couplers were not optimized but the interference fringes were found to have acceptable contrast and are sufficient for the current experiment. The signal from the PD was recorded by an oscilloscope for further data processing.

Fig. 6 Experimental setup for studying the phase response of HC-1550-02 to internal pressure.

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When the gas pressure inside the hollow-core of HC-PBF was varied gradually, the output intensity of the MZI changed sensitively and periodically, indicating a fast optical phase variation has happened to the HC-PBF. Figure 7 shows the recorded intensity variation of the MZI output when the gas pressure in the hollow-core was increased from 0 to 0.5 bar. The pressurization process started at ~23 s and ended at ~40 s. It reveals that the light intensity changed periodically with a quasi-sinusoidal waveform. The non-uniformity of the fringe spacing indicates that the speed of pressure increase in the fiber core is not constant. At a constant applied pressure, the output intensity was found varying randomly and slowly due to environmental disturbance on the two arms of the fiber interferometers [19].

Fig. 7 Evolution of the output intensity of the MZI when the gas pressure inside the hollow-core was increased from 0 to 0.5 bar.

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To quantitatively investigate the relationship between the internal pressure in the hollow-core and the phase variation, we increased the gas pressure step by step from 0 to 4 bar and recorded the number of induced interference fringe change accordingly. Measurement over a larger pressure range is possible but was not conducted due to the limitation of facilities available in our lab. The accumulated phase variation in terms of number of interference fringes for varying applied pressure is plotted in Fig. 8. The number of interference fringes shows a linear dependence on the applied pressure, with a slope of 157.8 fringes per bar. The variation of fringe numbers for decreasing pressure from 4 to 0 bar is also provided in Fig. 8. In this case, the optical phase variation is reversed but the slope is approximately the same with that of the pressure increasing case. The normalized phase sensitivity to pressure is calculated to be 166.1 fringes per bar per meter, or 1.044 × 10⁻² rad/(Pa∙m). This sensitivity is over two orders of magnitude higher than that to external pressure [13].

Fig. 8 Number of induced interference fringes as function of applied internal pressures.

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3. Theoretical analysis and discussion

The accumulated phase of the fundamental mode in HC-PBF at wavelength $λ$ after propagating over a length L may be expressed as

ϕ = \frac{2 π}{λ} n_{e f f} L,

where

n_{e f f}

is the effective refractive index of the fiber mode. The phase sensitivity to pressure, normalized to the fiber length, may be expressed as

S = \frac{1}{L} \frac{d ϕ}{d P} = \frac{2 π}{λ} (\frac{n_{e f f}}{L} \frac{d L}{d P} + \frac{d n_{e f f}}{d P}) = S_{L} + S_{n} .

The two terms on the right-hand side of Eq. (2) are respectively due to the pressure-induced changes of the fiber length and the effective index of the fundamental mode. Since the HC-PBF is fixed near its ends, the pressure applied from the two ends would have little or no effect on its length. Furthermore, because that the Poisson’s ratios ( $ν_{r - z}$ and $ν_{θ - z}$ ) of the honeycomb cladding are approximately zero [14] and the pressure is applied internally from the hollow-core, the pressure-induced longitudinal strain $ε_{z}$ is also negligible. Hence, the length term S_L in Eq. (2) may be regarded to be negligible, i.e., S_L = 0. We only need to consider the refractive index term S_n.

The change of the effective refractive index due to internal pressure may be attributed to three factors: (i) refractive index change of gas (air) within the hollow-core S_air, (ii) structural deformation of the honeycomb inner cladding S_structure, and (iii) refractive index change of silica material due to strain-optic effect S_silica. The normalized phase sensitivity to internal pressure may then be rewritten as

S = S_{n} = S_{a i r} + S_{s t r u c t u r e} + S_{s i l i c a} = \frac{2 π}{λ} [{(\frac{d n_{e f f}}{d P})}_{a i r} + {(\frac{d n_{e f f}}{d P})}_{s t r c t u r e} + {(\frac{d n_{e f f}}{d P})}_{s i l i c a}] .

Firstly, we concern the contribution from the refractive index change of air, i.e., S_air. From an updated Edlén equation, the dependence of the refractive index of air on temperature and pressure at wavelength λ (in µm) is expressed as [20–22]:

\begin{array}{l} n_{a i r} (λ) = 1 + \frac{10^{- 8} P}{96095.43} \cdot (8342.54 + \frac{2406147}{130 - {(1 / λ)}^{2}} + \frac{15998}{38.9 - {(1 / λ)}^{2}}) \\ \cdot (\frac{1 + 10^{- 10} (60.1 - 0.972 t) P}{1 + 0.003661 t}), \end{array}

where P is air pressure in Pa and t is temperature in °C. At room temperature of 25 °C and wavelength of 1.53 µm, the refractive indices of air were calculated for pressure from 0 to 4 bar and are listed as the second column in Table 1. With these refractive index values of air and assuming the pressure is applied only to the hollow-core, we calculated the effective refractive indexes of the fundamental mode by a numerical model with parameters of HC-1550-02 given in [18], and the results are listed in the third column in Table 1. The changes of the effective refractive index of the fundamental mode at different applied pressure are also shown in the last column. From these results, the normalized phase sensitivity due to pressure induced change in air refractive index is calculated to be 1.031 × 10⁻² rad/(Pa∙m). This value is very close to that obtained experimentally in section 2.3, indicating that the air-index change played a major role in the observed pressure sensitivity.

Table 1. Calculated refractive indices of air and the effective refractive index of the fundamental mode of the HC-1550-02 fiber at different pressures.

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To estimate the magnitude of S_structure and S_silica, the mechanical deformation and strain distributions over the honeycomb region, i.e., region I indicated in Fig. 1, need to be evaluated. This was done by using the elasticity model of the HC-PBF as described in [14, 23] with the Young’s modulus and Poisson’s ratio of the honeycomb region expressed as [14, 23, 24]:

{\begin{cases} E_{r} = E_{θ} = \frac{3}{2} (^{1 - η) 3} E_{s i} = E_{t} \\ E_{z} = (1 - η) E_{s i} \end{cases}

and

{\begin{cases} ν_{r - θ} = ν_{θ - r} = 1 \\ ν_{z - θ} = ν_{z - r} = ν_{s i} \\ ν_{r - z} = ν_{θ - z} \approx 0 \end{cases},

where

η

is air-filling ratio of the honeycomb cladding,

E_{s i}

and

ν_{s i}

are the Young’s modulus and Poisson’s ratio of silica material. The air-filling ratio of the honeycomb cladding in HC-1550-02 fiber is well over 90%, which means that E_r and E_z are very small and the honeycomb region is highly compressive. The approaching zero values of

ν_{r - z}

and

ν_{θ - z}

show that strain in the transverse plane would induce little or no strain in the longitudinal direction, which supports our earlier assumption of S_L = 0.

The stress expression in different regions can be written as:

{\begin{cases} σ_{r}^{i} = A_{i} / r^{2} + B_{i} \\ σ_{θ}^{i} = - A_{i} / r^{2} + B_{i} \\ σ_{z}^{i} = C_{i} \end{cases}, i = I, I I,

where

A_{i}

,

B_{i}

and

C_{i}

are constants,

I

and

I I

represent the honeycomb inner-cladding and silica outer-cladding regions, respectively. By substituting the above equations into the Hooke’s law, we obtain the strain tensor for the honeycomb layer as:

{\begin{cases} ε_{r}^{I} = \frac{σ_{r}^{I}}{E_{t}} - \frac{ν_{θ - r}^{I} σ_{θ}^{I}}{E_{t}} - \frac{ν_{z - r}^{I} σ_{z}^{I}}{E_{z}} = \frac{2 A_{I}}{E_{t} r^{2}} - \frac{ν_{s i} C_{I}}{E_{z}} \\ ε_{θ}^{I} = \frac{σ_{θ}^{I}}{E_{t}} - \frac{ν_{r - θ}^{I} σ_{r}^{I}}{E_{t}} - \frac{ν_{z - θ}^{I} σ_{z}^{I}}{E_{z}} = - \frac{2 A_{I}}{E_{t} r^{2}} - \frac{ν_{s i} C_{I}}{E_{z}} \\ ε_{z}^{I} = \frac{σ_{z}^{I}}{E_{z}} - \frac{ν_{r - z}^{I} σ_{r}^{I}}{E_{t}} - \frac{ν_{θ - z}^{I} σ_{θ}^{I}}{E_{t}} = \frac{C_{I}}{E_{z}} \end{cases}

and for the silica cladding layer as

{\begin{cases} ε_{r}^{I I} = \frac{1}{E_{s i}} [σ_{r}^{I I} - ν_{s i} (σ_{θ}^{I I} + σ_{z}^{I I})] = \frac{1}{E_{s i}} [(1 + ν_{s i}) \frac{A_{I I}}{r^{2}} + (1 - ν_{s i}) B_{I I} - ν_{s i} C_{I I}] \\ ε_{θ}^{I I} = \frac{1}{E_{s i}} [σ_{θ}^{I I} - ν_{s i} (σ_{r}^{I I} + σ_{z}^{I I})] = \frac{1}{E_{s i}} [- (1 + ν_{s i}) \frac{A_{I I}}{r^{2}} + (1 - ν_{s i}) B_{I I} - ν_{s i} C_{I I}] \\ ε_{z}^{I I} = \frac{1}{E_{s i}} [σ_{z}^{I I} - ν_{s i} (σ_{r}^{I I} + σ_{θ}^{I I})] = \frac{1}{E_{s i}} (C_{I I} - 2 ν_{s i} B_{I I}) \end{cases} .

Since the HC-PBF is fixed at both ends, the force applied to the fiber along the longitudinal direction may be regarded as zero. Therefore, the boundary and continuity conditions may be written as:

{\begin{cases} σ_{r}^{I} |_{r_{1}} = - P \\ σ_{r}^{II} |_{r_{3}} = 0 \\ σ_{r}^{I} |_{r_{2}} = σ_{r}^{II} |_{r_{2}} \\ ε_{θ}^{I} |_{r_{2}} = ε_{θ}^{II} |_{r_{2}} \\ σ_{z}^{I} π (r_{2}^{2} - r_{1}^{2}) + σ_{z}^{II} π (r_{3}^{2} - r_{2}^{2}) = 0 \end{cases} .

With Eqs. (7)–(10), $A_{i}$ , $B_{i}$ and $C_{i}$ , and hence the stress and strain fields over the honeycomb and silica regions can be obtained. For different applied pressures in the hollow-core, the radial and azimuthal strains ( $ε_{r}$ and $ε_{θ}$ ), and the radial displacement ( $u_{r} = ε_{θ} r$ ) distribution for the entire cladding region are plotted in Fig. 9. As expected, the maximum strain and displacement happen near the wall of hollow-core, and the maximum displacement is less than 3 nm for an applied pressure of 4 bar, which is much smaller than the thinnest strut of fiber structure.

Fig. 9 Distribution of (a) radial strain, (b) azimuthal strain and (c) radial displacement of HC-PBF for different pressures applied to the air-core. The honeycomb inner-cladding is 5<r<35 μm, while the silica outer-cladding corresponds to 35<r<60 μm.

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So far, we have obtained the strain distribution in the cross section of the HC-PBF. It should be noted that the actual strain can only exist in the silica region, and the refractive index of the silica will then be modified via strain-optic effect. The refractive index changes may be estimated by using [13, 25]:

Δ {(\frac{1}{n^{2}})}_{i} = \sum_{j = 1}^{6} p_{i j} {[\begin{matrix} ε_{r} & ε_{θ} & 0 & 0 & 0 & 0 \end{matrix}]}^{T},

where

p_{i j}

is strain-optic tensor and

ε_{i}

(i = r, θ) are strain components obtained from Eq. (8). Here

ε_{z}

equals to zero considering the negligible longitudinal strain. Silica is an isotropic material and the strain tensor has only two nonzero values,

p_{11}

and

p_{12}

. Thus, the changes of the three components of silica refractive index may be written as

{\begin{cases} Δ n_{r} = - \frac{1}{2} n_{0}^{3} (p_{11} ε_{r} + p_{12} ε_{θ}) \\ Δ n_{θ} = - \frac{1}{2} n_{0}^{3} (p_{12} ε_{r} + p_{11} ε_{θ}) \\ Δ n_{z} = - \frac{1}{2} n_{0}^{3} (p_{12} ε_{r} + p_{12} ε_{θ}) \end{cases},

where

n_{0}

is the refractive index of silica under strain-free condition. With n₀ = 1.444, p₁₁ = 0.121 and p₁₂ = 0.27 for bulk silica material, we calculated the changes of refractive index in silica for 4 bar internal pressure and the results are shown in Fig. 10. It is shown that the change of the longitudinal component Δn_z is zero while transverse components reduce quickly with increasing radius. The maximum refractive index changes occur in the wall surrounding the hollow-core and have value of ~1 × 10⁻⁴.

Fig. 10 The changes for individual refractive index component of silica in the honeycomb cladding region for a 4 bar pressure applied in the hollow-core.

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By importing the updated displacement and refractive index distribution of silica into the numerical model of HC-PBF, we calculated the effective refractive index of the fundamental mode under different pressure conditions. The effective mode index remains unchanged for applied internal pressure level up to 4 bar. When the pressure level is increased to 30 bar, the change of the mode index is ~10⁻⁶, corresponding to a phase sensitivity to pressure of ~1.4 × 10⁻⁶ rad/(Pa∙m), by far less than the contribution of the pressure-induced refractive index change of air within the hollow-core.

We also numerically simulated the pressure-induced deformation and the refractive index distribution of silica webs with the Solid Mechanics model in COMSOL, and then imported these values in the Electromagnetic Waves model to calculate the effective refractive index of the fundamental mode. Figures 11(a) and 11(b) show the structural deformation and refractive index change Δn_x of silica near the core region, respectively, for an applied internal pressure of 4 bar. It is found that the fiber structure is hardly modified by a 4-bar pressure and the mostevident deformation occurs at the innermost silica ring surrounding the fiber core with a maximum displacement of ~10 nm. The largest refractive index change happens at the “T-junctions” at the innermost silica wall and has a value of ~1 × 10⁻⁴. The effective refractive index change of the fundamental mode under 4 bar pressure was found to be beyond the numerical accuracy of 10⁻⁶, indicating again that the effect of structure deformation and silica refractive index change due to internal pressure on the effective refractive index is negligible.

Fig. 11 (a) Total displacement of the HC-PBF structure in the cross section (rainbow color map), and (b) change of the refractive index (x-component) of silica webs due to strain-optic effect (rainbow color map) for an applied pressure of 4 bar. An amplification factor of 20 is applied to the structural deformation in Fig. 11(a) for better visibility. The electric field (red arrows) and intensity (thermal color map with color bar shown under the panels) distributions of the fundamental mode are also shown in Figs. 11(a) and 11(b).

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From the above numerical analysis, we may conclude that, for a pressure level of several bar applied to the hollow-core, the structural deformation and the perturbation of silica refractive index are very small and would cause negligible effect on the transmission window of the HC-PBF and the effective refractive index of its fundamental mode. The effective refractive index of the fundamental mode is affected dominantly by the pressure-induced refractive index change of air inside the hollow-core, which determines the phase sensitivity of the fundamental mode to internal pressure. These results agree with the experimental results in section 2.

Under the same experimental conditions, we carried out further experiments with the same HC-1550-02 fiber but without sealing any of the cladding holes. In this case, the pressure is simultaneously applied to all the cladding holes as well as the hollow-core, thus the structural deformation may be neglected in this case. We found that the transmission spectrum was not affected at all for a pressure level up to 4 bar, and the normalized phase sensitivity to pressure is 1.055 × 10⁻² rad/(Pa∙m). We also numerically calculated the phase sensitivity to pressure by assuming that pressure is applied to all the air-holes and the result is 1.072 × 10⁻² rad/(Pa∙m). These values are very close to the ones for the sealed cladding holes, as shown in Table 2, and further confirm that the pressure-induced change of air refractive index is the dominant factor that determines the effective index of the fiber mode and hence the phase sensitivity to internal pressure.

Table 2. Phase sensitivities of the fundamental mode of HC-1550-02 to internal pressure with cladding holes of the fiber sealed/unsealed.

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4. Conclusion

The effects of internal gas pressure on the transmission properties of hollow-core photonic bandgap fibers were investigated experimentally and numerically. The transmission spectrum was found hardly affected for pressure level up to 4 bar, while the phase sensitivity of the fundamental mode to pressure was measured to be ~1.044 × 10⁻² rad/(Pa∙m). This phase sensitivity is over two orders of magnitude higher than that to external pressure, showing that a much shorter fiber is required to achieve the same target detection sensitivity. Alternatively, for the same fiber length, more sensitive pressure sensors could be developed with the internal pressurization configuration. It was found that the pressure-induced refractive index change of air inside the hollow-core is the dominant factor that determines the phase sensitivity, while the effects of pressure-induced fiber structural change and index change of silica web are negligible for a pressure level up to several bar. This research would promise several useful applications such as high sensitivity static and dynamic pressure measurement, optical phase manipulation, and monitoring of pressure related biochemical processes.

Acknowledgments

We acknowledge the support of the National Natural Science Foundation of China (NSFC) through Grant No. 61290313 and The Hong Kong Polytechnic University through grant G-YK62.

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Pressure (bar)	n_air-1	n_eff	Δn_eff
0	0	0.99455	0
1	2.6063 × 10⁻⁴	0.994801	2.51 × 10⁻⁴
2	5.2145 × 10⁻⁴	0.995052	5.02 × 10⁻⁴
3	7.8246 × 10⁻⁴	0.995303	7.53 × 10⁻⁴
4	10.4365 × 10⁻⁴	0.995554	10.04 × 10⁻⁴

Pressure (bar)	n_air-1	n_eff	Δn_eff
0	0	0.99455	0
1	2.6063 × 10⁻⁴	0.994801	2.51 × 10⁻⁴
2	5.2145 × 10⁻⁴	0.995052	5.02 × 10⁻⁴
3	7.8246 × 10⁻⁴	0.995303	7.53 × 10⁻⁴
4	10.4365 × 10⁻⁴	0.995554	10.04 × 10⁻⁴

Phase sensitivity of fundamental mode of hollow-core photonic bandgap fiber to internal gas pressure

Abstract

1. Introduction

2. Experimental procedures and results

2.1 Preparation of HC-PBF sample

2.2 Effect of internal pressure on transmission spectrum

2.3 Phase sensitivity of the fundamental mode to internal pressure

3. Theoretical analysis and discussion

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (11)

Tables (2)

Equations (12)

Optics Express

Cladding holes	Phase sensitivity (rad/(Pa∙m))
Cladding holes	theoretical	experimental
sealed	1.031 × 10⁻²	1.044 × 10⁻²
unsealed	1.072 × 10⁻²	1.055 × 10⁻²