Abstract

An anti-resonant reflecting guidance coated with a few layers of graphene has been proposed and experimentally demonstrated for the simultaneous measurement of the refractive index and liquid flow rate. A few layers of graphene were coated on the surface of a hollow core photonic crystal fibre. The refractive index and liquid flow rate can be detected by using the resonant condition of the Fabry–Perot resonator and the effective refractive index of the graphene layers heated by a visible laser beam, which are interrogated through the wavelength shift and visibility of the lossy dip in the transmission spectrum. The experimental results show that the sensitivity of up to 1328 nm/RIU and −2.99 dB/(µL/s) are achieved for the refractive index and flow rate measurement in the refractive index range from 1.345 to 1.363 RIU, respectively. The proposed sensor appears to have potential applications for precise measurement in chemistry, medicine, and biology.

© 2017 Optical Society of America

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2016 (5)

2015 (4)

2013 (4)

T. Han, Y. G. Liu, Z. Wang, J. Guo, Z. Wu, S. Wang, Z. Li, and W. Zhou, “Unique characteristics of a selective-filling photonic crystal fiber Sagnac interferometer and its application as high sensitivity sensor,” Opt. Express 21(1), 122–128 (2013).
[PubMed]

S. Liu, Y. Wang, M. Hou, J. Guo, Z. Li, and P. Lu, “Anti-resonant reflecting guidance in alcohol-filled hollow core photonic crystal fiber for sensing applications,” Opt. Express 21(25), 31690–31697 (2013).
[PubMed]

X. H. Wang, X. Y. Dong, Y. Zhou, K. Ni, J. Cheng, and Z. M. Chen, “Hot-wire anemometer based on silver coated fiber Bragg grating assisted by no-core fiber,” IEEE Photonics Technol. Lett. 25(24), 2458–2461 (2013).

R. Gao, Y. Jiang, W. H. Ding, Z. Wang, and D. Liu, “Filmed extrinsic Fabry–Perot interferometric sensors for the measurement ofarbitrary refractive index of liquid,” Sens. Actuators B Chem. 177, 924–928 (2013).

2012 (2)

Y.-F. Chen, L. Jiang, M. Mancuso, A. Jain, V. Oncescu, and D. Erickson, “Optofluidic opportunities in global health, food, water and energy,” Nanoscale 4(16), 4839–4857 (2012).
[PubMed]

T. Chen, Q. Wang, B. Zhang, R. Chen, and K. P. Chen, “Distributed flow sensing using optical hot -wire grid,” Opt. Express 20(8), 8240–8249 (2012).
[PubMed]

2011 (3)

2010 (3)

S. Bae, H. Kim, Y. Lee, X. Xu, J.-S. Park, Y. Zheng, J. Balakrishnan, T. Lei, H. R. Kim, Y. I. Song, Y. J. Kim, K. S. Kim, B. Ozyilmaz, J. H. Ahn, B. H. Hong, and S. Iijima, “Roll-to-roll production of 30-inch graphene films for transparent electrodes,” Nat. Nanotechnol. 5(8), 574–578 (2010).
[PubMed]

S. L. Stott, C. H. Hsu, D. I. Tsukrov, M. Yu, D. T. Miyamoto, B. A. Waltman, S. M. Rothenberg, A. M. Shah, M. E. Smas, G. K. Korir, F. P. Floyd, A. J. Gilman, J. B. Lord, D. Winokur, S. Springer, D. Irimia, S. Nagrath, L. V. Sequist, R. J. Lee, K. J. Isselbacher, S. Maheswaran, D. A. Haber, and M. Toner, “Isolation of circulating tumor cells using a microvortex-generating herringbone-chip,” Proc. Natl. Acad. Sci. U.S.A. 107(43), 18392–18397 (2010).
[PubMed]

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[PubMed]

2009 (1)

L. M. Malard, M. A. Pimenta, G. Dresselhaus, and M. S. Dresselhaus, “Raman spectroscopy in graphene,” Phys. Rep. 473(5–6), 51–87 (2009).

2008 (1)

G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103(6), 064302 (2008).

2006 (3)

G. J. M. Krijnen, M. Dijkstra, J. J. van Baar, S. S. Shankar, W. J. Kuipers, R. J. de Boer, D. Altpeter, T. S. J. Lammerink, and R. Wiegerink, “MEMS based hair flow-sensors as model systems for acoustic perception studies,” Nanotechnology 17(4), S84–S89 (2006).
[PubMed]

A. Quist, A. Chand, S. Ramachandran, D. Cohen, and R. Lal, “Piezoresistive cantilever based nanoflow and viscosity sensor for microchannels,” Lab Chip 6(11), 1450–1454 (2006).
[PubMed]

D. Psaltis, S. R. Quake, and C. Yang, “Developing optofluidic technology through the fusion of microfluidics and optics,” Nature 442(7101), 381–386 (2006).
[PubMed]

2005 (1)

M. Dijkstra, J. J. Baar, R. J. Wiegerink, T. S. J. Lammerink, J. H. Boer, and G. J. M. Krijnen, “Artificial sensory hairs based on the flow sensitive receptor hairs of crickets,” J. Micromech. Microeng. 15(7), 132–138 (2005).

2004 (1)

J. Collins and A. P. Lee, “Microfluidic flow transducer based on the measurement of electrical admittance,” Lab Chip 4(1), 7–10 (2004).
[PubMed]

2002 (1)

1999 (1)

M. Ashauer, H. Glosch, F. Hedrich, N. Hey, H. Sandmaier, and W. Lang, “Thermal flow sensor for liquids and gases based on combinations of two principles,” Sens. Actuators A Phys. 73(12), 7–13 (1999).

Abeeluck, A. K.

Ahn, J. H.

S. Bae, H. Kim, Y. Lee, X. Xu, J.-S. Park, Y. Zheng, J. Balakrishnan, T. Lei, H. R. Kim, Y. I. Song, Y. J. Kim, K. S. Kim, B. Ozyilmaz, J. H. Ahn, B. H. Hong, and S. Iijima, “Roll-to-roll production of 30-inch graphene films for transparent electrodes,” Nat. Nanotechnol. 5(8), 574–578 (2010).
[PubMed]

Akowuah, E. K.

Altpeter, D.

G. J. M. Krijnen, M. Dijkstra, J. J. van Baar, S. S. Shankar, W. J. Kuipers, R. J. de Boer, D. Altpeter, T. S. J. Lammerink, and R. Wiegerink, “MEMS based hair flow-sensors as model systems for acoustic perception studies,” Nanotechnology 17(4), S84–S89 (2006).
[PubMed]

Andrews, N. L. P.

Ashauer, M.

M. Ashauer, H. Glosch, F. Hedrich, N. Hey, H. Sandmaier, and W. Lang, “Thermal flow sensor for liquids and gases based on combinations of two principles,” Sens. Actuators A Phys. 73(12), 7–13 (1999).

Auguste, J. L.

Baar, J. J.

M. Dijkstra, J. J. Baar, R. J. Wiegerink, T. S. J. Lammerink, J. H. Boer, and G. J. M. Krijnen, “Artificial sensory hairs based on the flow sensitive receptor hairs of crickets,” J. Micromech. Microeng. 15(7), 132–138 (2005).

Bae, S.

S. Bae, H. Kim, Y. Lee, X. Xu, J.-S. Park, Y. Zheng, J. Balakrishnan, T. Lei, H. R. Kim, Y. I. Song, Y. J. Kim, K. S. Kim, B. Ozyilmaz, J. H. Ahn, B. H. Hong, and S. Iijima, “Roll-to-roll production of 30-inch graphene films for transparent electrodes,” Nat. Nanotechnol. 5(8), 574–578 (2010).
[PubMed]

Balakrishnan, J.

S. Bae, H. Kim, Y. Lee, X. Xu, J.-S. Park, Y. Zheng, J. Balakrishnan, T. Lei, H. R. Kim, Y. I. Song, Y. J. Kim, K. S. Kim, B. Ozyilmaz, J. H. Ahn, B. H. Hong, and S. Iijima, “Roll-to-roll production of 30-inch graphene films for transparent electrodes,” Nat. Nanotechnol. 5(8), 574–578 (2010).
[PubMed]

Barnes, J. A.

Benítez, J. L.

Boer, J. H.

M. Dijkstra, J. J. Baar, R. J. Wiegerink, T. S. J. Lammerink, J. H. Boer, and G. J. M. Krijnen, “Artificial sensory hairs based on the flow sensitive receptor hairs of crickets,” J. Micromech. Microeng. 15(7), 132–138 (2005).

Brzezinski, A.

Chand, A.

A. Quist, A. Chand, S. Ramachandran, D. Cohen, and R. Lal, “Piezoresistive cantilever based nanoflow and viscosity sensor for microchannels,” Lab Chip 6(11), 1450–1454 (2006).
[PubMed]

Chang, H. C.

Chen, K. P.

Chen, R.

Chen, T.

Chen, Y.-F.

Y.-F. Chen, L. Jiang, M. Mancuso, A. Jain, V. Oncescu, and D. Erickson, “Optofluidic opportunities in global health, food, water and energy,” Nanoscale 4(16), 4839–4857 (2012).
[PubMed]

Chen, Z. M.

X. H. Wang, X. Y. Dong, Y. Zhou, K. Ni, J. Cheng, and Z. M. Chen, “Hot-wire anemometer based on silver coated fiber Bragg grating assisted by no-core fiber,” IEEE Photonics Technol. Lett. 25(24), 2458–2461 (2013).

Cheng, J.

X. H. Wang, X. Y. Dong, Y. Zhou, K. Ni, J. Cheng, and Z. M. Chen, “Hot-wire anemometer based on silver coated fiber Bragg grating assisted by no-core fiber,” IEEE Photonics Technol. Lett. 25(24), 2458–2461 (2013).

Cho, L. H.

Cohen, D.

A. Quist, A. Chand, S. Ramachandran, D. Cohen, and R. Lal, “Piezoresistive cantilever based nanoflow and viscosity sensor for microchannels,” Lab Chip 6(11), 1450–1454 (2006).
[PubMed]

Collins, J.

J. Collins and A. P. Lee, “Microfluidic flow transducer based on the measurement of electrical admittance,” Lab Chip 4(1), 7–10 (2004).
[PubMed]

cordero, J. H.

Cui, Y.

de Boer, R. J.

G. J. M. Krijnen, M. Dijkstra, J. J. van Baar, S. S. Shankar, W. J. Kuipers, R. J. de Boer, D. Altpeter, T. S. J. Lammerink, and R. Wiegerink, “MEMS based hair flow-sensors as model systems for acoustic perception studies,” Nanotechnology 17(4), S84–S89 (2006).
[PubMed]

Dijkstra, M.

G. J. M. Krijnen, M. Dijkstra, J. J. van Baar, S. S. Shankar, W. J. Kuipers, R. J. de Boer, D. Altpeter, T. S. J. Lammerink, and R. Wiegerink, “MEMS based hair flow-sensors as model systems for acoustic perception studies,” Nanotechnology 17(4), S84–S89 (2006).
[PubMed]

M. Dijkstra, J. J. Baar, R. J. Wiegerink, T. S. J. Lammerink, J. H. Boer, and G. J. M. Krijnen, “Artificial sensory hairs based on the flow sensitive receptor hairs of crickets,” J. Micromech. Microeng. 15(7), 132–138 (2005).

Ding, W. H.

R. Gao, Y. Jiang, W. H. Ding, Z. Wang, and D. Liu, “Filmed extrinsic Fabry–Perot interferometric sensors for the measurement ofarbitrary refractive index of liquid,” Sens. Actuators B Chem. 177, 924–928 (2013).

Dinh, X. Q.

Dong, X. Y.

X. H. Wang, X. Y. Dong, Y. Zhou, K. Ni, J. Cheng, and Z. M. Chen, “Hot-wire anemometer based on silver coated fiber Bragg grating assisted by no-core fiber,” IEEE Photonics Technol. Lett. 25(24), 2458–2461 (2013).

Dresselhaus, G.

L. M. Malard, M. A. Pimenta, G. Dresselhaus, and M. S. Dresselhaus, “Raman spectroscopy in graphene,” Phys. Rep. 473(5–6), 51–87 (2009).

Dresselhaus, M. S.

L. M. Malard, M. A. Pimenta, G. Dresselhaus, and M. S. Dresselhaus, “Raman spectroscopy in graphene,” Phys. Rep. 473(5–6), 51–87 (2009).

Eggleton, B. J.

Engheta, N.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[PubMed]

Erickson, D.

Y.-F. Chen, L. Jiang, M. Mancuso, A. Jain, V. Oncescu, and D. Erickson, “Optofluidic opportunities in global health, food, water and energy,” Nanoscale 4(16), 4839–4857 (2012).
[PubMed]

Floyd, F. P.

S. L. Stott, C. H. Hsu, D. I. Tsukrov, M. Yu, D. T. Miyamoto, B. A. Waltman, S. M. Rothenberg, A. M. Shah, M. E. Smas, G. K. Korir, F. P. Floyd, A. J. Gilman, J. B. Lord, D. Winokur, S. Springer, D. Irimia, S. Nagrath, L. V. Sequist, R. J. Lee, K. J. Isselbacher, S. Maheswaran, D. A. Haber, and M. Toner, “Isolation of circulating tumor cells using a microvortex-generating herringbone-chip,” Proc. Natl. Acad. Sci. U.S.A. 107(43), 18392–18397 (2010).
[PubMed]

Gao, R.

R. Gao, Y. Jiang, W. H. Ding, Z. Wang, and D. Liu, “Filmed extrinsic Fabry–Perot interferometric sensors for the measurement ofarbitrary refractive index of liquid,” Sens. Actuators B Chem. 177, 924–928 (2013).

Gao, S.

Ghasemi, A. H.

J. Sadeghi, A. H. Ghasemi, and H. Latifi, “A label-free infrared opto-fluidic method for real-time determination of flow rate and concentration with temperature cross-sensitivity compensation,” Lab Chip 16(20), 3957–3968 (2016).
[PubMed]

Gilman, A. J.

S. L. Stott, C. H. Hsu, D. I. Tsukrov, M. Yu, D. T. Miyamoto, B. A. Waltman, S. M. Rothenberg, A. M. Shah, M. E. Smas, G. K. Korir, F. P. Floyd, A. J. Gilman, J. B. Lord, D. Winokur, S. Springer, D. Irimia, S. Nagrath, L. V. Sequist, R. J. Lee, K. J. Isselbacher, S. Maheswaran, D. A. Haber, and M. Toner, “Isolation of circulating tumor cells using a microvortex-generating herringbone-chip,” Proc. Natl. Acad. Sci. U.S.A. 107(43), 18392–18397 (2010).
[PubMed]

Glosch, H.

M. Ashauer, H. Glosch, F. Hedrich, N. Hey, H. Sandmaier, and W. Lang, “Thermal flow sensor for liquids and gases based on combinations of two principles,” Sens. Actuators A Phys. 73(12), 7–13 (1999).

Guo, J.

Haber, D. A.

S. L. Stott, C. H. Hsu, D. I. Tsukrov, M. Yu, D. T. Miyamoto, B. A. Waltman, S. M. Rothenberg, A. M. Shah, M. E. Smas, G. K. Korir, F. P. Floyd, A. J. Gilman, J. B. Lord, D. Winokur, S. Springer, D. Irimia, S. Nagrath, L. V. Sequist, R. J. Lee, K. J. Isselbacher, S. Maheswaran, D. A. Haber, and M. Toner, “Isolation of circulating tumor cells using a microvortex-generating herringbone-chip,” Proc. Natl. Acad. Sci. U.S.A. 107(43), 18392–18397 (2010).
[PubMed]

Han, M.

Han, T.

Hanson, G. W.

G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103(6), 064302 (2008).

Haxha, S.

He, S.

Headley, C.

Hedrich, F.

M. Ashauer, H. Glosch, F. Hedrich, N. Hey, H. Sandmaier, and W. Lang, “Thermal flow sensor for liquids and gases based on combinations of two principles,” Sens. Actuators A Phys. 73(12), 7–13 (1999).

Hey, N.

M. Ashauer, H. Glosch, F. Hedrich, N. Hey, H. Sandmaier, and W. Lang, “Thermal flow sensor for liquids and gases based on combinations of two principles,” Sens. Actuators A Phys. 73(12), 7–13 (1999).

Hong, B. H.

S. Bae, H. Kim, Y. Lee, X. Xu, J.-S. Park, Y. Zheng, J. Balakrishnan, T. Lei, H. R. Kim, Y. I. Song, Y. J. Kim, K. S. Kim, B. Ozyilmaz, J. H. Ahn, B. H. Hong, and S. Iijima, “Roll-to-roll production of 30-inch graphene films for transparent electrodes,” Nat. Nanotechnol. 5(8), 574–578 (2010).
[PubMed]

Hou, M.

Hou, W.

Hsu, C. H.

S. L. Stott, C. H. Hsu, D. I. Tsukrov, M. Yu, D. T. Miyamoto, B. A. Waltman, S. M. Rothenberg, A. M. Shah, M. E. Smas, G. K. Korir, F. P. Floyd, A. J. Gilman, J. B. Lord, D. Winokur, S. Springer, D. Irimia, S. Nagrath, L. V. Sequist, R. J. Lee, K. J. Isselbacher, S. Maheswaran, D. A. Haber, and M. Toner, “Isolation of circulating tumor cells using a microvortex-generating herringbone-chip,” Proc. Natl. Acad. Sci. U.S.A. 107(43), 18392–18397 (2010).
[PubMed]

Hu, L.

Humbert, G.

Ianno, N.

Iijima, S.

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S. L. Stott, C. H. Hsu, D. I. Tsukrov, M. Yu, D. T. Miyamoto, B. A. Waltman, S. M. Rothenberg, A. M. Shah, M. E. Smas, G. K. Korir, F. P. Floyd, A. J. Gilman, J. B. Lord, D. Winokur, S. Springer, D. Irimia, S. Nagrath, L. V. Sequist, R. J. Lee, K. J. Isselbacher, S. Maheswaran, D. A. Haber, and M. Toner, “Isolation of circulating tumor cells using a microvortex-generating herringbone-chip,” Proc. Natl. Acad. Sci. U.S.A. 107(43), 18392–18397 (2010).
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S. Bae, H. Kim, Y. Lee, X. Xu, J.-S. Park, Y. Zheng, J. Balakrishnan, T. Lei, H. R. Kim, Y. I. Song, Y. J. Kim, K. S. Kim, B. Ozyilmaz, J. H. Ahn, B. H. Hong, and S. Iijima, “Roll-to-roll production of 30-inch graphene films for transparent electrodes,” Nat. Nanotechnol. 5(8), 574–578 (2010).
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Yan, G.

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Yao, Y.

You, B.

Yu, M.

S. L. Stott, C. H. Hsu, D. I. Tsukrov, M. Yu, D. T. Miyamoto, B. A. Waltman, S. M. Rothenberg, A. M. Shah, M. E. Smas, G. K. Korir, F. P. Floyd, A. J. Gilman, J. B. Lord, D. Winokur, S. Springer, D. Irimia, S. Nagrath, L. V. Sequist, R. J. Lee, K. J. Isselbacher, S. Maheswaran, D. A. Haber, and M. Toner, “Isolation of circulating tumor cells using a microvortex-generating herringbone-chip,” Proc. Natl. Acad. Sci. U.S.A. 107(43), 18392–18397 (2010).
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Zhang, A. P.

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S. Bae, H. Kim, Y. Lee, X. Xu, J.-S. Park, Y. Zheng, J. Balakrishnan, T. Lei, H. R. Kim, Y. I. Song, Y. J. Kim, K. S. Kim, B. Ozyilmaz, J. H. Ahn, B. H. Hong, and S. Iijima, “Roll-to-roll production of 30-inch graphene films for transparent electrodes,” Nat. Nanotechnol. 5(8), 574–578 (2010).
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IEEE Photonics Technol. Lett. (1)

X. H. Wang, X. Y. Dong, Y. Zhou, K. Ni, J. Cheng, and Z. M. Chen, “Hot-wire anemometer based on silver coated fiber Bragg grating assisted by no-core fiber,” IEEE Photonics Technol. Lett. 25(24), 2458–2461 (2013).

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Nanotechnology (1)

G. J. M. Krijnen, M. Dijkstra, J. J. van Baar, S. S. Shankar, W. J. Kuipers, R. J. de Boer, D. Altpeter, T. S. J. Lammerink, and R. Wiegerink, “MEMS based hair flow-sensors as model systems for acoustic perception studies,” Nanotechnology 17(4), S84–S89 (2006).
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Nat. Nanotechnol. (1)

S. Bae, H. Kim, Y. Lee, X. Xu, J.-S. Park, Y. Zheng, J. Balakrishnan, T. Lei, H. R. Kim, Y. I. Song, Y. J. Kim, K. S. Kim, B. Ozyilmaz, J. H. Ahn, B. H. Hong, and S. Iijima, “Roll-to-roll production of 30-inch graphene films for transparent electrodes,” Nat. Nanotechnol. 5(8), 574–578 (2010).
[PubMed]

Nature (1)

D. Psaltis, S. R. Quake, and C. Yang, “Developing optofluidic technology through the fusion of microfluidics and optics,” Nature 442(7101), 381–386 (2006).
[PubMed]

Opt. Express (13)

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S. Gao, A. P. Zhang, H. Y. Tam, L. H. Cho, and C. Lu, “All-optical fiber anemometer based on laser heated fiber Bragg gratings,” Opt. Express 19(11), 10124–10130 (2011).
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Z. Wu, Z. Wang, Y. G. Liu, T. Han, S. Li, and H. Wei, “Mechanism and characteristics of long period fiber gratings in simplified hollow-core photonic crystal fibers,” Opt. Express 19(18), 17344–17349 (2011).
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T. Chen, Q. Wang, B. Zhang, R. Chen, and K. P. Chen, “Distributed flow sensing using optical hot -wire grid,” Opt. Express 20(8), 8240–8249 (2012).
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T. Han, Y. G. Liu, Z. Wang, J. Guo, Z. Wu, S. Wang, Z. Li, and W. Zhou, “Unique characteristics of a selective-filling photonic crystal fiber Sagnac interferometer and its application as high sensitivity sensor,” Opt. Express 21(1), 122–128 (2013).
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S. Liu, Y. Wang, M. Hou, J. Guo, Z. Li, and P. Lu, “Anti-resonant reflecting guidance in alcohol-filled hollow core photonic crystal fiber for sensing applications,” Opt. Express 21(25), 31690–31697 (2013).
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G. Liu, M. Han, and W. Hou, “High-resolution and fast-response fiber-optic temperature sensor using silicon Fabry-Pérot cavity,” Opt. Express 23(6), 7237–7247 (2015).
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Figures (7)

Fig. 1
Fig. 1

(a) Cross-section of the HCPCF. (b) Square microchannel for the inlet and outlet. (c) The schematic diagram of the proposed sensor. (d) FLG-coated HCPCF.

Fig. 2
Fig. 2

(a) Schematic diagram of the cross-section of the liquid-infiltrated HCPCF. (b) Guiding mechanism of the liquid -infiltrated HCPCF. Numerical simulation of the HCPCF filled with liquid(c) at the resonant anti-wavelength (1530.00 nm), (d) at the resonant wavelength (1535.26 nm). (e) Raman spectra of FLG using an excitation wavelength of 532 nm. (f) Transmission spectrum of the FLG.

Fig. 3
Fig. 3

(a) The experimental setup of the proposed sensor. (b) The metal plate. (c) Thermal image obtained with the thermographic camera. The close-up view of the (d) HCPCF, (e) fluorescent image of the HCPCF filled with anti-mouse IgG -FITC in all hollow holes, (f) and only one hollow hole.

Fig. 4
Fig. 4

The transmission spectrum of the in-line optofluidic filled with the ethanol solution.

Fig. 5
Fig. 5

(a) The transmission spectra of the in-line optofluidic filled with different RI of ethanol solutions. (b) Relationship between the RI and wavelength of the lossy dip.

Fig. 6
Fig. 6

(a) The transmission spectra with different flow rates. (b) Relationship between the flow rate and visibility of the lossy dip. (c) Visibility of the lossy dip with different flow rates. (d) Time response of the fibre sensor. (e) The relationship between the temperature, visibility and 532 nm heating laser intensity. (f) The visibility at different heating light intensities.

Fig. 7
Fig. 7

(a) The temperature response of (a) wavelength shift and (b) visibility.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

λ r = 2( d LS n LS 2 n air 2 + d cl n silica 2 n air 2 ) m .
σ(ω,T)=j e 2 k B T π 2 (ωj2Γ) [ u c k B T +2ln( e ( u c / k B T) )+1] +j e 2 4π ln[ 2| u c |(ω+j2Γ) 2| u c |+(ω+j2Γ) ].
Re( n eff )= ( 1 2ωΔ ε 0 ) 1/2 [ σ i + 4 σ r 2 + σ i 2 ] 1/2 .
H power =Δ T h (A+B υ ).
T R = (1rr')(r+r') 1+r ' 4 2r ' 2 I R .

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