Abstract

The response of the commercial HC-1550-02 hollow-core photonic bandgap fiber (HC-PBF) to gas pressure applied internally to the hollow-core was experimentally investigated. The transmission spectrum of the HC-PBF was hardly affected by the pressure, while the accumulated phase of the fundamental optical mode showed a normalized pressure sensitivity of 1.044 × 10−2 rad/(Pa∙m), which is over two orders of magnitude higher than that to the external pressure. Numerical simulation showed that the observed high sensitivity to pressure is due to the pressure-induced refractive index change of air inside the hollow-core. This research could find potential applications in high sensitivity static and dynamic pressure measurement and optical phase manipulation.

© 2014 Optical Society of America

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2013 (4)

2011 (1)

2010 (2)

M. Pang, H. F. Xuan, J. Ju, W. Jin, “Influence of strain and pressure to the effective refractive index of the fundamental mode of hollow-core photonic bandgap fibers,” Opt. Express 18(13), 14041–14055 (2010).
[CrossRef] [PubMed]

Y. L. Hoo, S. J. Liu, H. L. Ho, W. Jin, “Fast response microstructured optical fiber methane sensor with multiple side-openings,” IEEE Photonics Technol. Lett. 22(5), 296–298 (2010).
[CrossRef]

2009 (3)

2008 (1)

2007 (1)

C. M. B. Cordeiro, C. J. S. de Matos, E. M. dos Santos, A. Bozolan, J. S. K. Ong, T. Facincani, G. Chesini, A. R. Vaz, C. H. B. Cruz, “Towards practical liquid and gas sensing with photonic crystal fibres: side access to the fibre microstructure and single-mode liquid-core fibre,” Meas. Sci. Technol. 18(10), 3075–3081 (2007).
[CrossRef]

2006 (2)

2005 (3)

2004 (2)

J. M. Fini, “Microstructure fibres for optical sensing in gases and liquids,” Meas. Sci. Technol. 15(6), 1120–1128 (2004).
[CrossRef]

J. A. West, C. M. Smith, N. F. Borrelli, D. C. Allan, K. W. Koch, “Surface modes in air-core photonic band-gap fibers,” Opt. Express 12(8), 1485–1496 (2004).
[CrossRef] [PubMed]

2003 (1)

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424(6949), 657–659 (2003).
[CrossRef] [PubMed]

2002 (1)

F. Benabid, J. C. Knight, G. Antonopoulos, P. S. J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298(5592), 399–402 (2002).
[CrossRef] [PubMed]

1999 (1)

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. Russell, P. J. Roberts, D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285(5433), 1537–1539 (1999).
[CrossRef] [PubMed]

1994 (1)

K. P. Birch, M. J. Downs, “Correction to the updated Edlén equation for the refractive-index of air,” Metrologia 31(4), 315–316 (1994).
[CrossRef]

1993 (1)

K. P. Birch, M. J. Downs, “An updated Edlén equation for the refractive-index of air,” Metrologia 30(3), 155–162 (1993).
[CrossRef]

1978 (1)

Aghaie, K. Z.

Allan, D. C.

J. A. West, C. M. Smith, N. F. Borrelli, D. C. Allan, K. W. Koch, “Surface modes in air-core photonic band-gap fibers,” Opt. Express 12(8), 1485–1496 (2004).
[CrossRef] [PubMed]

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424(6949), 657–659 (2003).
[CrossRef] [PubMed]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. Russell, P. J. Roberts, D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285(5433), 1537–1539 (1999).
[CrossRef] [PubMed]

Antonopoulos, G.

F. Benabid, J. C. Knight, G. Antonopoulos, P. S. J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298(5592), 399–402 (2002).
[CrossRef] [PubMed]

Benabid, F.

F. Benabid, J. C. Knight, G. Antonopoulos, P. S. J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298(5592), 399–402 (2002).
[CrossRef] [PubMed]

Bhagwat, A. R.

Birch, K. P.

K. P. Birch, M. J. Downs, “Correction to the updated Edlén equation for the refractive-index of air,” Metrologia 31(4), 315–316 (1994).
[CrossRef]

K. P. Birch, M. J. Downs, “An updated Edlén equation for the refractive-index of air,” Metrologia 30(3), 155–162 (1993).
[CrossRef]

Birks, T. A.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. Russell, P. J. Roberts, D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285(5433), 1537–1539 (1999).
[CrossRef] [PubMed]

Borrelli, N. F.

J. A. West, C. M. Smith, N. F. Borrelli, D. C. Allan, K. W. Koch, “Surface modes in air-core photonic band-gap fibers,” Opt. Express 12(8), 1485–1496 (2004).
[CrossRef] [PubMed]

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424(6949), 657–659 (2003).
[CrossRef] [PubMed]

Bozolan, A.

C. M. B. Cordeiro, C. J. S. de Matos, E. M. dos Santos, A. Bozolan, J. S. K. Ong, T. Facincani, G. Chesini, A. R. Vaz, C. H. B. Cruz, “Towards practical liquid and gas sensing with photonic crystal fibres: side access to the fibre microstructure and single-mode liquid-core fibre,” Meas. Sci. Technol. 18(10), 3075–3081 (2007).
[CrossRef]

Butter, C. D.

Chang, W. K.

Chesini, G.

C. M. B. Cordeiro, C. J. S. de Matos, E. M. dos Santos, A. Bozolan, J. S. K. Ong, T. Facincani, G. Chesini, A. R. Vaz, C. H. B. Cruz, “Towards practical liquid and gas sensing with photonic crystal fibres: side access to the fibre microstructure and single-mode liquid-core fibre,” Meas. Sci. Technol. 18(10), 3075–3081 (2007).
[CrossRef]

Cordeiro, C. M. B.

R. E. P. de Oliveira, C. J. S. de Matos, J. G. Hayashi, C. M. B. Cordeiro, “Pressure sensing based on nonconventional air-guiding transmission windows in hollow-core photonic crystal fibers,” J. Lightwave Technol. 27(11), 1605–1609 (2009).
[CrossRef]

C. M. B. Cordeiro, C. J. S. de Matos, E. M. dos Santos, A. Bozolan, J. S. K. Ong, T. Facincani, G. Chesini, A. R. Vaz, C. H. B. Cruz, “Towards practical liquid and gas sensing with photonic crystal fibres: side access to the fibre microstructure and single-mode liquid-core fibre,” Meas. Sci. Technol. 18(10), 3075–3081 (2007).
[CrossRef]

Corwin, K. L.

Cregan, R. F.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. Russell, P. J. Roberts, D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285(5433), 1537–1539 (1999).
[CrossRef] [PubMed]

Cruz, C. H. B.

C. M. B. Cordeiro, C. J. S. de Matos, E. M. dos Santos, A. Bozolan, J. S. K. Ong, T. Facincani, G. Chesini, A. R. Vaz, C. H. B. Cruz, “Towards practical liquid and gas sensing with photonic crystal fibres: side access to the fibre microstructure and single-mode liquid-core fibre,” Meas. Sci. Technol. 18(10), 3075–3081 (2007).
[CrossRef]

Dangui, V.

de Matos, C. J. S.

R. E. P. de Oliveira, C. J. S. de Matos, J. G. Hayashi, C. M. B. Cordeiro, “Pressure sensing based on nonconventional air-guiding transmission windows in hollow-core photonic crystal fibers,” J. Lightwave Technol. 27(11), 1605–1609 (2009).
[CrossRef]

C. M. B. Cordeiro, C. J. S. de Matos, E. M. dos Santos, A. Bozolan, J. S. K. Ong, T. Facincani, G. Chesini, A. R. Vaz, C. H. B. Cruz, “Towards practical liquid and gas sensing with photonic crystal fibres: side access to the fibre microstructure and single-mode liquid-core fibre,” Meas. Sci. Technol. 18(10), 3075–3081 (2007).
[CrossRef]

de Oliveira, R. E. P.

Demokan, M. S.

Digonnet, M. J. F.

dos Santos, E. M.

C. M. B. Cordeiro, C. J. S. de Matos, E. M. dos Santos, A. Bozolan, J. S. K. Ong, T. Facincani, G. Chesini, A. R. Vaz, C. H. B. Cruz, “Towards practical liquid and gas sensing with photonic crystal fibres: side access to the fibre microstructure and single-mode liquid-core fibre,” Meas. Sci. Technol. 18(10), 3075–3081 (2007).
[CrossRef]

Downs, M. J.

K. P. Birch, M. J. Downs, “Correction to the updated Edlén equation for the refractive-index of air,” Metrologia 31(4), 315–316 (1994).
[CrossRef]

K. P. Birch, M. J. Downs, “An updated Edlén equation for the refractive-index of air,” Metrologia 30(3), 155–162 (1993).
[CrossRef]

Ducournau, G.

G. Ducournau, O. Latry, M. Kétata, “Fiber-based Mach-Zehnder interferometric structures: principles and required characteristics for efficient modulation format conversion,” Proc. SPIE 6019, 60190A (2005).
[CrossRef]

Facincani, T.

C. M. B. Cordeiro, C. J. S. de Matos, E. M. dos Santos, A. Bozolan, J. S. K. Ong, T. Facincani, G. Chesini, A. R. Vaz, C. H. B. Cruz, “Towards practical liquid and gas sensing with photonic crystal fibres: side access to the fibre microstructure and single-mode liquid-core fibre,” Meas. Sci. Technol. 18(10), 3075–3081 (2007).
[CrossRef]

Faheem, M.

Fan, S. H.

Fini, J. M.

J. M. Fini, “Microstructure fibres for optical sensing in gases and liquids,” Meas. Sci. Technol. 15(6), 1120–1128 (2004).
[CrossRef]

Gaeta, A. L.

Gallagher, M. T.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424(6949), 657–659 (2003).
[CrossRef] [PubMed]

Guan, B.-O.

Hayashi, J. G.

Ho, H. L.

Hocker, G. B.

Hoo, Y. L.

Y. L. Hoo, S. J. Liu, H. L. Ho, W. Jin, “Fast response microstructured optical fiber methane sensor with multiple side-openings,” IEEE Photonics Technol. Lett. 22(5), 296–298 (2010).
[CrossRef]

L. Xiao, W. Jin, M. S. Demokan, H. L. Ho, Y. L. Hoo, C. L. Zhao, “Fabrication of selective injection microstructured optical fibers with a conventional fusion splicer,” Opt. Express 13(22), 9014–9022 (2005).
[CrossRef] [PubMed]

Hu, Y. M.

Jin, L.

Jin, W.

Joly, N. Y.

Ju, J.

Kétata, M.

G. Ducournau, O. Latry, M. Kétata, “Fiber-based Mach-Zehnder interferometric structures: principles and required characteristics for efficient modulation format conversion,” Proc. SPIE 6019, 60190A (2005).
[CrossRef]

Kim, H. K.

Kino, G. S.

Knabe, K.

Knight, J. C.

F. Benabid, J. C. Knight, G. Antonopoulos, P. S. J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298(5592), 399–402 (2002).
[CrossRef] [PubMed]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. Russell, P. J. Roberts, D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285(5433), 1537–1539 (1999).
[CrossRef] [PubMed]

Koch, K. W.

J. A. West, C. M. Smith, N. F. Borrelli, D. C. Allan, K. W. Koch, “Surface modes in air-core photonic band-gap fibers,” Opt. Express 12(8), 1485–1496 (2004).
[CrossRef] [PubMed]

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424(6949), 657–659 (2003).
[CrossRef] [PubMed]

Lægsgaard, J.

Latry, O.

G. Ducournau, O. Latry, M. Kétata, “Fiber-based Mach-Zehnder interferometric structures: principles and required characteristics for efficient modulation format conversion,” Proc. SPIE 6019, 60190A (2005).
[CrossRef]

Limsuwan, P.

M. Ranusawud, P. Limsuwan, T. Somthong, K. Vacharanukul, “Effects of the environment on refractive index of air in long gauge block interferometer,” Precis. Eng. 37(3), 782–786 (2013).
[CrossRef]

Liu, S. J.

Y. L. Hoo, S. J. Liu, H. L. Ho, W. Jin, “Fast response microstructured optical fiber methane sensor with multiple side-openings,” IEEE Photonics Technol. Lett. 22(5), 296–298 (2010).
[CrossRef]

Liu, W.

Ma, L. N.

Mangan, B. J.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. Russell, P. J. Roberts, D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285(5433), 1537–1539 (1999).
[CrossRef] [PubMed]

Müller, D.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424(6949), 657–659 (2003).
[CrossRef] [PubMed]

Naweed, A.

Nold, J.

Ong, J. S. K.

C. M. B. Cordeiro, C. J. S. de Matos, E. M. dos Santos, A. Bozolan, J. S. K. Ong, T. Facincani, G. Chesini, A. R. Vaz, C. H. B. Cruz, “Towards practical liquid and gas sensing with photonic crystal fibres: side access to the fibre microstructure and single-mode liquid-core fibre,” Meas. Sci. Technol. 18(10), 3075–3081 (2007).
[CrossRef]

Pang, M.

Ranusawud, M.

M. Ranusawud, P. Limsuwan, T. Somthong, K. Vacharanukul, “Effects of the environment on refractive index of air in long gauge block interferometer,” Precis. Eng. 37(3), 782–786 (2013).
[CrossRef]

Roberts, P. J.

J. Lægsgaard, P. J. Roberts, “Influence of air pressure on soliton formation in hollow-core photonic bandgap fibers,” J. Opt. Soc. Am. B 26(9), 1795–1800 (2009).
[CrossRef]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. Russell, P. J. Roberts, D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285(5433), 1537–1539 (1999).
[CrossRef] [PubMed]

Russell, P. S.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. Russell, P. J. Roberts, D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285(5433), 1537–1539 (1999).
[CrossRef] [PubMed]

Russell, P. S. J.

J. C. Travers, W. K. Chang, J. Nold, N. Y. Joly, P. S. J. Russell, “Ultrafast nonlinear optics in gas-filled hollow-core photonic crystal fibers [Invited],” J. Opt. Soc. Am. B 28, A11–A26 (2011).
[CrossRef]

F. Benabid, J. C. Knight, G. Antonopoulos, P. S. J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298(5592), 399–402 (2002).
[CrossRef] [PubMed]

Smith, C. M.

J. A. West, C. M. Smith, N. F. Borrelli, D. C. Allan, K. W. Koch, “Surface modes in air-core photonic band-gap fibers,” Opt. Express 12(8), 1485–1496 (2004).
[CrossRef] [PubMed]

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424(6949), 657–659 (2003).
[CrossRef] [PubMed]

Somthong, T.

M. Ranusawud, P. Limsuwan, T. Somthong, K. Vacharanukul, “Effects of the environment on refractive index of air in long gauge block interferometer,” Precis. Eng. 37(3), 782–786 (2013).
[CrossRef]

Thapa, R.

Travers, J. C.

Vacharanukul, K.

M. Ranusawud, P. Limsuwan, T. Somthong, K. Vacharanukul, “Effects of the environment on refractive index of air in long gauge block interferometer,” Precis. Eng. 37(3), 782–786 (2013).
[CrossRef]

Vaz, A. R.

C. M. B. Cordeiro, C. J. S. de Matos, E. M. dos Santos, A. Bozolan, J. S. K. Ong, T. Facincani, G. Chesini, A. R. Vaz, C. H. B. Cruz, “Towards practical liquid and gas sensing with photonic crystal fibres: side access to the fibre microstructure and single-mode liquid-core fibre,” Meas. Sci. Technol. 18(10), 3075–3081 (2007).
[CrossRef]

Venkataraman, N.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424(6949), 657–659 (2003).
[CrossRef] [PubMed]

Wang, F. Y.

Weaver, O. L.

Wei, H.

West, J. A.

J. A. West, C. M. Smith, N. F. Borrelli, D. C. Allan, K. W. Koch, “Surface modes in air-core photonic band-gap fibers,” Opt. Express 12(8), 1485–1496 (2004).
[CrossRef] [PubMed]

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424(6949), 657–659 (2003).
[CrossRef] [PubMed]

Xiao, L.

Xuan, H. F.

Yang, F.

Zhao, C. L.

Appl. Opt. (1)

IEEE Photonics Technol. Lett. (1)

Y. L. Hoo, S. J. Liu, H. L. Ho, W. Jin, “Fast response microstructured optical fiber methane sensor with multiple side-openings,” IEEE Photonics Technol. Lett. 22(5), 296–298 (2010).
[CrossRef]

J. Lightwave Technol. (4)

J. Opt. Soc. Am. B (2)

Meas. Sci. Technol. (2)

J. M. Fini, “Microstructure fibres for optical sensing in gases and liquids,” Meas. Sci. Technol. 15(6), 1120–1128 (2004).
[CrossRef]

C. M. B. Cordeiro, C. J. S. de Matos, E. M. dos Santos, A. Bozolan, J. S. K. Ong, T. Facincani, G. Chesini, A. R. Vaz, C. H. B. Cruz, “Towards practical liquid and gas sensing with photonic crystal fibres: side access to the fibre microstructure and single-mode liquid-core fibre,” Meas. Sci. Technol. 18(10), 3075–3081 (2007).
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Metrologia (2)

K. P. Birch, M. J. Downs, “An updated Edlén equation for the refractive-index of air,” Metrologia 30(3), 155–162 (1993).
[CrossRef]

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[CrossRef]

Nature (1)

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424(6949), 657–659 (2003).
[CrossRef] [PubMed]

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Precis. Eng. (1)

M. Ranusawud, P. Limsuwan, T. Somthong, K. Vacharanukul, “Effects of the environment on refractive index of air in long gauge block interferometer,” Precis. Eng. 37(3), 782–786 (2013).
[CrossRef]

Proc. SPIE (1)

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F. Benabid, J. C. Knight, G. Antonopoulos, P. S. J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298(5592), 399–402 (2002).
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[CrossRef] [PubMed]

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Figures (11)

Fig. 1
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.

Fig. 2
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).

Fig. 3
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).

Fig. 4
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.

Fig. 5
Fig. 5

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

Fig. 6
Fig. 6

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

Fig. 7
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.

Fig. 8
Fig. 8

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

Fig. 9
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.

Fig. 10
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.

Fig. 11
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).

Tables (2)

Tables Icon

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.

Tables Icon

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

Equations (12)

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ϕ= 2π λ n eff L,
S= 1 L dϕ dP = 2π λ ( n eff L dL dP + d n eff dP )= S L + S n .
S= S n = S air + S structure + S silica = 2π λ [ ( d n eff dP ) air + ( d n eff dP ) strcture + ( d n eff dP ) silica ].
n air (λ)=1+ 10 8 P 96095.43 ( 8342.54+ 2406147 130 (1/λ) 2 + 15998 38.9 (1/λ) 2 ) ( 1+ 10 10 (60.10.972t)P 1+0.003661t ),
{ E r = E θ = 3 2 (1η) 3 E si = E t E z =(1η) E si
{ ν rθ = ν θr =1 ν zθ = ν zr = ν si ν rz = ν θz 0 ,
{ σ r i = A i / r 2 + B i σ θ i = A i / r 2 + B i σ z i = C i ,i=I, II,
{ ε r I = σ r I E t ν θr I σ θ I E t ν zr I σ z I E z = 2 A I E t r 2 ν si C I E z ε θ I = σ θ I E t ν rθ I σ r I E t ν zθ I σ z I E z = 2 A I E t r 2 ν si C I E z ε z I = σ z I E z ν rz I σ r I E t ν θz I σ θ I E t = C I E z
{ ε r II = 1 E si [ σ r II ν si ( σ θ II + σ z II ) ]= 1 E si [ ( 1+ ν si ) A II r 2 +( 1 ν si ) B II ν si C II ] ε θ II = 1 E si [ σ θ II ν si ( σ r II + σ z II ) ]= 1 E si [ ( 1+ ν si ) A II r 2 +( 1 ν si ) B II ν si C II ] ε z II = 1 E si [ σ z II ν si ( σ r II + σ θ II ) ]= 1 E si ( C II 2 ν si B II ) .
{ 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 .
Δ ( 1 n 2 ) i = j=1 6 p ij [ ε r ε θ 0 0 0 0 ] T ,
{ Δ n r = 1 2 n 0 3 ( p 11 ε r + p 12 ε θ ) Δ n θ = 1 2 n 0 3 ( p 12 ε r + p 11 ε θ ) Δ n z = 1 2 n 0 3 ( p 12 ε r + p 12 ε θ ) ,

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