Abstract

We present a simple refractive index sensor based on a step-index fiber with a hollow micro-channel running parallel to its core. This channel becomes waveguiding when filled with a liquid of index greater than silica, causing sharp dips to appear in the transmission spectrum at wavelengths where the glass-core mode phase-matches to a mode of the liquid-core. The sensitivity of the dip-wavelengths to changes in liquid refractive index is quantified and the results used to study the dynamic flow characteristics of fluids in narrow channels. Potential applications of this fiber microstructure include measuring the optical properties of liquids, refractive index sensing, biophotonics and studies of fluid dynamics on the nanoscale.

© 2011 OSA

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  1. W. Liang, Y. Huang, Y. Xu, R. K. Lee, and A. Yariv, “Highly sensitive fiber Bragg grating refractive index sensors,” Appl. Phys. Lett. 86(15), 151122 (2005).
    [CrossRef]
  2. V. Bhatia and A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21(9), 692–694 (1996).
    [CrossRef] [PubMed]
  3. X. Fang, C. R. Liao, and D. N. Wang, “Femtosecond laser fabricated fiber Bragg grating in microfiber for refractive index sensing,” Opt. Lett. 35(7), 1007–1009 (2010).
    [CrossRef] [PubMed]
  4. T. Allsop, R. Neal, S. Rehman, D. J. Webb, D. Mapps, and I. Bennion, “Generation of infrared surface plasmon resonances with high refractive index sensitivity utilizing tilted fiber Bragg gratings,” Appl. Opt. 46(22), 5456–5460 (2007).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  6. J. Wu, D. Day, and M. Gu, “A microfluidic refractive index sensor based on an integrated three-dimensional photonic crystal,” Appl. Phys. Lett. 92(7), 071108 (2008).
    [CrossRef]
  7. T. W. Lu, Y. H. Hsiao, W. D. Ho, and P. T. Lee, “Photonic crystal heteroslab-edge microcavity with high quality factor surface mode for index sensing,” Appl. Phys. Lett. 94(14), 141110 (2009).
    [CrossRef]
  8. J. Jágerská, H. Zhang, Z. Diao, N. Le Thomas, and R. Houdré, “Refractive index sensing with an air-slot photonic crystal nanocavity,” Opt. Lett. 35(15), 2523–2525 (2010).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  12. P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24(12), 4729–4749 (2006).
    [CrossRef]
  13. R. Jha, J. Villatoro, G. Badenes, and V. Pruneri, “Refractometry based on a photonic crystal fiber interferometer,” Opt. Lett. 34(5), 617–619 (2009).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  15. L. Rindorf and O. Bang, “Highly sensitive refractometer with a photonic-crystal-fiber long-period grating,” Opt. Lett. 33(6), 563–565 (2008).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  17. T. Han, Y. G. Liu, Z. Wang, B. Zou, B. Tai, and B. Liu, “Avoided-crossing-based ultrasensitive photonic crystal fiber refractive index sensor,” Opt. Lett. 35(12), 2061–2063 (2010).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  23. K. Shiraishi, Y. Aizawa, and S. Kawakawi, “Beam expanding fiber using thermal diffusion of the dopant,” J. Lightwave Technol. 8(8), 1151–1161 (1990).
    [CrossRef]
  24. E. W. Washburn, “The dynamics of capillary flow,” Phys. Rev. 17(3), 273–283 (1921).
    [CrossRef]
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    [CrossRef]
  26. K. Nielsen, D. Noordegraaf, T. Sørensen, A. Bjarklev, and T. P. Hansen, “Selective filling of photonic crystal fibres,” J. Opt. B 7, L13–L20 (2005).
  27. M. Vieweg, T. Gissibl, S. Pricking, B. T. Kuhlmey, D. C. Wu, B. J. Eggleton, and H. Giessen, “Ultrafast nonlinear optofluidics in selectively liquid-filled photonic crystal fibers,” Opt. Express 18(24), 25232–25240 (2010).
    [CrossRef] [PubMed]

2010 (6)

2009 (4)

2008 (3)

2007 (5)

2006 (1)

2005 (2)

W. Liang, Y. Huang, Y. Xu, R. K. Lee, and A. Yariv, “Highly sensitive fiber Bragg grating refractive index sensors,” Appl. Phys. Lett. 86(15), 151122 (2005).
[CrossRef]

K. Nielsen, D. Noordegraaf, T. Sørensen, A. Bjarklev, and T. P. Hansen, “Selective filling of photonic crystal fibres,” J. Opt. B 7, L13–L20 (2005).

1996 (1)

1990 (1)

K. Shiraishi, Y. Aizawa, and S. Kawakawi, “Beam expanding fiber using thermal diffusion of the dopant,” J. Lightwave Technol. 8(8), 1151–1161 (1990).
[CrossRef]

1984 (1)

1921 (1)

E. W. Washburn, “The dynamics of capillary flow,” Phys. Rev. 17(3), 273–283 (1921).
[CrossRef]

Agarwal, A.

Aizawa, Y.

K. Shiraishi, Y. Aizawa, and S. Kawakawi, “Beam expanding fiber using thermal diffusion of the dopant,” J. Lightwave Technol. 8(8), 1151–1161 (1990).
[CrossRef]

Allsop, T.

Auguste, J. L.

Badenes, G.

Bang, O.

Bennion, I.

Bhatia, V.

Bjarklev, A.

K. Nielsen, D. Noordegraaf, T. Sørensen, A. Bjarklev, and T. P. Hansen, “Selective filling of photonic crystal fibres,” J. Opt. B 7, L13–L20 (2005).

Blanc, W.

Carlie, N.

Da, N.

N. Da, L. Wondraczek, M. A. Schmidt, N. Granzow, and P. St. J. Russell, ““High index-contrast all-solid photonic crystal fibers by pressure-assisted melt infiltration of silica matrices,” J. Non-Cryst Solid. 356, 1829–1836 (2010).
[CrossRef]

Day, D.

J. Wu, D. Day, and M. Gu, “A microfluidic refractive index sensor based on an integrated three-dimensional photonic crystal,” Appl. Phys. Lett. 92(7), 071108 (2008).
[CrossRef]

Dellith, J.

J. Kirchhof, S. Unger, B. Knappe, and J. Dellith, “Diffusion in binary GeO2–SiO2 glasses,” Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B 48, 129–133 (2007).

Dewynter, V.

Diao, Z.

Dulashko, Y.

Dussardier, B.

Eggleton, B. J.

Fan, X.

Fang, X.

Feng, N. N.

Ferdinand, P.

Fleming, J. W.

Giessen, H.

Gissibl, T.

Granzow, N.

N. Da, L. Wondraczek, M. A. Schmidt, N. Granzow, and P. St. J. Russell, ““High index-contrast all-solid photonic crystal fibers by pressure-assisted melt infiltration of silica matrices,” J. Non-Cryst Solid. 356, 1829–1836 (2010).
[CrossRef]

Gu, M.

J. Wu, D. Day, and M. Gu, “A microfluidic refractive index sensor based on an integrated three-dimensional photonic crystal,” Appl. Phys. Lett. 92(7), 071108 (2008).
[CrossRef]

Han, T.

Hansen, T. P.

K. Nielsen, D. Noordegraaf, T. Sørensen, A. Bjarklev, and T. P. Hansen, “Selective filling of photonic crystal fibres,” J. Opt. B 7, L13–L20 (2005).

Ho, W. D.

T. W. Lu, Y. H. Hsiao, W. D. Ho, and P. T. Lee, “Photonic crystal heteroslab-edge microcavity with high quality factor surface mode for index sensing,” Appl. Phys. Lett. 94(14), 141110 (2009).
[CrossRef]

Houdré, R.

Hsiao, Y. H.

T. W. Lu, Y. H. Hsiao, W. D. Ho, and P. T. Lee, “Photonic crystal heteroslab-edge microcavity with high quality factor surface mode for index sensing,” Appl. Phys. Lett. 94(14), 141110 (2009).
[CrossRef]

Hu, J.

Huang, Y.

W. Liang, Y. Huang, Y. Xu, R. K. Lee, and A. Yariv, “Highly sensitive fiber Bragg grating refractive index sensors,” Appl. Phys. Lett. 86(15), 151122 (2005).
[CrossRef]

Jágerská, J.

Jha, R.

Joly, N.

Kanie, T.

H. Tazawa, T. Kanie, and M. Katayama, “Fiber-optic coupler based refractive index sensor and its application to biosensing,” Appl. Phys. Lett. 91(11), 113901 (2007).
[CrossRef]

Katagiri, T.

Katayama, M.

H. Tazawa, T. Kanie, and M. Katayama, “Fiber-optic coupler based refractive index sensor and its application to biosensing,” Appl. Phys. Lett. 91(11), 113901 (2007).
[CrossRef]

Kawakawi, S.

K. Shiraishi, Y. Aizawa, and S. Kawakawi, “Beam expanding fiber using thermal diffusion of the dopant,” J. Lightwave Technol. 8(8), 1151–1161 (1990).
[CrossRef]

Kimerling, L.

Kirchhof, J.

J. Kirchhof, S. Unger, B. Knappe, and J. Dellith, “Diffusion in binary GeO2–SiO2 glasses,” Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B 48, 129–133 (2007).

Knappe, B.

J. Kirchhof, S. Unger, B. Knappe, and J. Dellith, “Diffusion in binary GeO2–SiO2 glasses,” Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B 48, 129–133 (2007).

Kuhlmey, B. T.

Laffont, G.

Le Thomas, N.

Lee, H. W.

Lee, P. T.

T. W. Lu, Y. H. Hsiao, W. D. Ho, and P. T. Lee, “Photonic crystal heteroslab-edge microcavity with high quality factor surface mode for index sensing,” Appl. Phys. Lett. 94(14), 141110 (2009).
[CrossRef]

Lee, R. K.

W. Liang, Y. Huang, Y. Xu, R. K. Lee, and A. Yariv, “Highly sensitive fiber Bragg grating refractive index sensors,” Appl. Phys. Lett. 86(15), 151122 (2005).
[CrossRef]

Liang, W.

W. Liang, Y. Huang, Y. Xu, R. K. Lee, and A. Yariv, “Highly sensitive fiber Bragg grating refractive index sensors,” Appl. Phys. Lett. 86(15), 151122 (2005).
[CrossRef]

Liao, C. R.

Liu, B.

Liu, Y. G.

Lu, T. W.

T. W. Lu, Y. H. Hsiao, W. D. Ho, and P. T. Lee, “Photonic crystal heteroslab-edge microcavity with high quality factor surface mode for index sensing,” Appl. Phys. Lett. 94(14), 141110 (2009).
[CrossRef]

Ma, L.

Mapps, D.

Matsuura, Y.

Neal, R.

Nielsen, K.

K. Nielsen, D. Noordegraaf, T. Sørensen, A. Bjarklev, and T. P. Hansen, “Selective filling of photonic crystal fibres,” J. Opt. B 7, L13–L20 (2005).

Noordegraaf, D.

K. Nielsen, D. Noordegraaf, T. Sørensen, A. Bjarklev, and T. P. Hansen, “Selective filling of photonic crystal fibres,” J. Opt. B 7, L13–L20 (2005).

Pagnoux, D.

Petit, L.

Phan Huy, M. C.

Pricking, S.

Pruneri, V.

Rehman, S.

Richardson, K.

Rindorf, L.

Roy, P.

Russell, P. St. J.

H. K. Tyagi, H. W. Lee, P. Uebel, M. A. Schmidt, N. Joly, M. Scharrer, and P. St. J. Russell, “Plasmon resonances on gold nanowires directly drawn in a step-index fiber,” Opt. Lett. 35(15), 2573–2575 (2010).
[CrossRef] [PubMed]

N. Da, L. Wondraczek, M. A. Schmidt, N. Granzow, and P. St. J. Russell, ““High index-contrast all-solid photonic crystal fibers by pressure-assisted melt infiltration of silica matrices,” J. Non-Cryst Solid. 356, 1829–1836 (2010).
[CrossRef]

P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24(12), 4729–4749 (2006).
[CrossRef]

Scharrer, M.

Schmidt, M. A.

H. K. Tyagi, H. W. Lee, P. Uebel, M. A. Schmidt, N. Joly, M. Scharrer, and P. St. J. Russell, “Plasmon resonances on gold nanowires directly drawn in a step-index fiber,” Opt. Lett. 35(15), 2573–2575 (2010).
[CrossRef] [PubMed]

N. Da, L. Wondraczek, M. A. Schmidt, N. Granzow, and P. St. J. Russell, ““High index-contrast all-solid photonic crystal fibers by pressure-assisted melt infiltration of silica matrices,” J. Non-Cryst Solid. 356, 1829–1836 (2010).
[CrossRef]

Shiraishi, K.

K. Shiraishi, Y. Aizawa, and S. Kawakawi, “Beam expanding fiber using thermal diffusion of the dopant,” J. Lightwave Technol. 8(8), 1151–1161 (1990).
[CrossRef]

Sørensen, T.

K. Nielsen, D. Noordegraaf, T. Sørensen, A. Bjarklev, and T. P. Hansen, “Selective filling of photonic crystal fibres,” J. Opt. B 7, L13–L20 (2005).

Sumetsky, M.

Tai, B.

Tazawa, H.

H. Tazawa, T. Kanie, and M. Katayama, “Fiber-optic coupler based refractive index sensor and its application to biosensing,” Appl. Phys. Lett. 91(11), 113901 (2007).
[CrossRef]

Tyagi, H. K.

Uebel, P.

Unger, S.

J. Kirchhof, S. Unger, B. Knappe, and J. Dellith, “Diffusion in binary GeO2–SiO2 glasses,” Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B 48, 129–133 (2007).

Vengsarkar, A. M.

Vieweg, M.

Villatoro, J.

Wang, D. N.

Wang, Z.

Washburn, E. W.

E. W. Washburn, “The dynamics of capillary flow,” Phys. Rev. 17(3), 273–283 (1921).
[CrossRef]

Webb, D. J.

Windeler, R. S.

Wondraczek, L.

N. Da, L. Wondraczek, M. A. Schmidt, N. Granzow, and P. St. J. Russell, ““High index-contrast all-solid photonic crystal fibers by pressure-assisted melt infiltration of silica matrices,” J. Non-Cryst Solid. 356, 1829–1836 (2010).
[CrossRef]

Wu, D. C.

Wu, D. K. C.

Wu, J.

J. Wu, D. Day, and M. Gu, “A microfluidic refractive index sensor based on an integrated three-dimensional photonic crystal,” Appl. Phys. Lett. 92(7), 071108 (2008).
[CrossRef]

Xu, Y.

W. Liang, Y. Huang, Y. Xu, R. K. Lee, and A. Yariv, “Highly sensitive fiber Bragg grating refractive index sensors,” Appl. Phys. Lett. 86(15), 151122 (2005).
[CrossRef]

Yariv, A.

W. Liang, Y. Huang, Y. Xu, R. K. Lee, and A. Yariv, “Highly sensitive fiber Bragg grating refractive index sensors,” Appl. Phys. Lett. 86(15), 151122 (2005).
[CrossRef]

Zhang, H.

Zou, B.

Appl. Opt. (2)

Appl. Phys. Lett. (4)

W. Liang, Y. Huang, Y. Xu, R. K. Lee, and A. Yariv, “Highly sensitive fiber Bragg grating refractive index sensors,” Appl. Phys. Lett. 86(15), 151122 (2005).
[CrossRef]

J. Wu, D. Day, and M. Gu, “A microfluidic refractive index sensor based on an integrated three-dimensional photonic crystal,” Appl. Phys. Lett. 92(7), 071108 (2008).
[CrossRef]

T. W. Lu, Y. H. Hsiao, W. D. Ho, and P. T. Lee, “Photonic crystal heteroslab-edge microcavity with high quality factor surface mode for index sensing,” Appl. Phys. Lett. 94(14), 141110 (2009).
[CrossRef]

H. Tazawa, T. Kanie, and M. Katayama, “Fiber-optic coupler based refractive index sensor and its application to biosensing,” Appl. Phys. Lett. 91(11), 113901 (2007).
[CrossRef]

J. Lightwave Technol. (2)

K. Shiraishi, Y. Aizawa, and S. Kawakawi, “Beam expanding fiber using thermal diffusion of the dopant,” J. Lightwave Technol. 8(8), 1151–1161 (1990).
[CrossRef]

P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24(12), 4729–4749 (2006).
[CrossRef]

J. Non-Cryst Solid. (1)

N. Da, L. Wondraczek, M. A. Schmidt, N. Granzow, and P. St. J. Russell, ““High index-contrast all-solid photonic crystal fibers by pressure-assisted melt infiltration of silica matrices,” J. Non-Cryst Solid. 356, 1829–1836 (2010).
[CrossRef]

J. Opt. B (1)

K. Nielsen, D. Noordegraaf, T. Sørensen, A. Bjarklev, and T. P. Hansen, “Selective filling of photonic crystal fibres,” J. Opt. B 7, L13–L20 (2005).

Opt. Express (2)

Opt. Lett. (11)

M. C. Phan Huy, G. Laffont, V. Dewynter, P. Ferdinand, P. Roy, J. L. Auguste, D. Pagnoux, W. Blanc, and B. Dussardier, “Three-hole microstructured optical fiber for efficient fiber Bragg grating refractometer,” Opt. Lett. 32(16), 2390–2392 (2007).
[CrossRef] [PubMed]

V. Bhatia and A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21(9), 692–694 (1996).
[CrossRef] [PubMed]

L. Rindorf and O. Bang, “Highly sensitive refractometer with a photonic-crystal-fiber long-period grating,” Opt. Lett. 33(6), 563–565 (2008).
[CrossRef] [PubMed]

J. Hu, N. Carlie, N. N. Feng, L. Petit, A. Agarwal, K. Richardson, and L. Kimerling, “Planar waveguide-coupled, high-index-contrast, high-Q resonators in chalcogenide glass for sensing,” Opt. Lett. 33(21), 2500–2502 (2008).
[CrossRef] [PubMed]

D. K. C. Wu, B. T. Kuhlmey, and B. J. Eggleton, “Ultrasensitive photonic crystal fiber refractive index sensor,” Opt. Lett. 34(3), 322–324 (2009).
[CrossRef] [PubMed]

R. Jha, J. Villatoro, G. Badenes, and V. Pruneri, “Refractometry based on a photonic crystal fiber interferometer,” Opt. Lett. 34(5), 617–619 (2009).
[CrossRef] [PubMed]

L. Ma, T. Katagiri, and Y. Matsuura, “Surface-plasmon resonance sensor using silica-core Bragg fiber,” Opt. Lett. 34(7), 1069–1071 (2009).
[CrossRef] [PubMed]

X. Fang, C. R. Liao, and D. N. Wang, “Femtosecond laser fabricated fiber Bragg grating in microfiber for refractive index sensing,” Opt. Lett. 35(7), 1007–1009 (2010).
[CrossRef] [PubMed]

T. Han, Y. G. Liu, Z. Wang, B. Zou, B. Tai, and B. Liu, “Avoided-crossing-based ultrasensitive photonic crystal fiber refractive index sensor,” Opt. Lett. 35(12), 2061–2063 (2010).
[CrossRef] [PubMed]

J. Jágerská, H. Zhang, Z. Diao, N. Le Thomas, and R. Houdré, “Refractive index sensing with an air-slot photonic crystal nanocavity,” Opt. Lett. 35(15), 2523–2525 (2010).
[CrossRef] [PubMed]

H. K. Tyagi, H. W. Lee, P. Uebel, M. A. Schmidt, N. Joly, M. Scharrer, and P. St. J. Russell, “Plasmon resonances on gold nanowires directly drawn in a step-index fiber,” Opt. Lett. 35(15), 2573–2575 (2010).
[CrossRef] [PubMed]

Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B (1)

J. Kirchhof, S. Unger, B. Knappe, and J. Dellith, “Diffusion in binary GeO2–SiO2 glasses,” Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B 48, 129–133 (2007).

Phys. Rev. (1)

E. W. Washburn, “The dynamics of capillary flow,” Phys. Rev. 17(3), 273–283 (1921).
[CrossRef]

Other (2)

A. Yariv, and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford University Press, 6th edition, 2006).

A. W. Snyder, and J. Love, Optical Waveguide Theory (Springer, 1st edition, 1983).

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Figures (5)

Fig. 1
Fig. 1

(a) Sketch of the device. The hollow channel (radius a F) close to the core (radius a C) is filled with fluid (green, filling length L F) and the centre-centre spacing between core and channel is d. (b) Representative modal indices (nn R) in the vicinity of an anti-crossing (the red and blue lines represent the uncoupled glass and liquid core modes and the black curves the supermodes), calculated using coupled mode theory [18]. (c) Typical plot of transmitted power P(L F) (linear scale) plotted versus wavelength at z = z C for Δn g = 0.0878. For a minimum detectable change δP min in P(L F) the minimum measurable wavelength change (caused by a change in liquid refractive index) is δλ min. The inset is a photograph of the near-field distribution at the output of the fluid-filled structure when broad-band supercontinuum light is launched into the glass core from the unfilled fiber end. The formation of a liquid meniscus distorts the image.

Fig. 2
Fig. 2

Plot of the detection function D. It has a minimum value of 1.55 at κL = 0.876 and diverges at integral multiples of twice the coupling length. The yellow shaded region indicates over-coupling cycles. The device is most sensitive in the first coupling cycle. Inset: (red) wavelength dependence of the transmitted power in the vicinity of the HE21 resonance, calculated using Eq. (3) at the filling length of minimum detectable index change; (blue) corresponding quadratic approximation.

Fig. 4
Fig. 4

(a) Measured resonance wavelengths (dip positions) versus the refractive index of the individual liquid for the three different modes (black/grey circles: HE21, red circles: TM01, green circles: TE01). The lines are linear fits to the experimental data using Eq. (1). (b) Comparison between the experimental points and the results calculated using the resonance condition. The inset is a theoretical plot of the sensitivity as a function of liquid core diameter for the HE21 mode at a fixed liquid RI of 1.6. The blue dot indicates the sensitivity in the experiments reported.

Fig. 3
Fig. 3

(a) Measured transmission spectra for a liquid of refractive index n fl = 1.58 at 598 nm for x (green curve) and y (blue curve) input polarization states. The inset shows the coordinate system (fluid core in yellow). Lower three dashed curves: Calculated dispersion of three modes of the fluid waveguide. The black solid line represents the glass core mode. (b) Calculated axial Poynting vector distributions (at the resonant wavelengths) for the HE21, TM01, and TE01 modes of the isolated (i.e., uncoupled) fluid-filled channel. The dashed white circle indicates the edge of the channel and the arrows the instantaneous local electric field.

Fig. 5
Fig. 5

(a) Optical set-up for investigating the filling dynamics of liquids (b) Dip depth (1 − p(2L F)) as a function of filling time for the HE21 resonance (y-polarization, n fl = 1.58). The solid blue line is a fit using Eqs. (3) and 7. The inset depicts the simulated filling speed of the silica hole as function of time and the corresponding experimental data (green dots). The filling time in the experiment was 10 minutes (grey shaded region).

Equations (7)

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

n k ( λ ) n R n g k ( λ λ R ) / λ R = n R n g k δ λ / λ R
ϑ = β F β C Δ n g 2 π δ λ / λ R 2
p ( z ) = P ( z ) / P 0 = 1 sin 2 ( κ z 1 + ( ϑ / 2 κ ) 2 ) 1 + ( ϑ / 2 κ ) 2
δ n fl = δ λ min S = λ R 2 2 L S π Δ n g D ( 2 κ L ) δ P min P 0
D ( 2 κ L ) = 1 / | ( sin c 2 κ L cos 2 κ L ) sin c 2 κ L | ,
FoM = λ R 2 2 L S π Δ n g
L F = R F σ cos θ 2 η t = C F t

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