Abstract

This paper describes theoretical and experimental study of the fundamentals of using surface plasmon resonance (SPR) for label-free detection of voltage. Plasmonic voltage sensing relies on the capacitive properties of metal-electrolyte interface that are governed by electrostatic interactions between charge carriers in both phases. Externally-applied voltage leads to changes in the free electron density in the surface of the metal, shifting the SPR position. The study shows the effects of the applied voltage on the shape of the SPR curve. It also provides a comparison between the theoretical and experimental response to the applied voltage. The response is presented in a universal term that can be used to assess the voltage sensitivity of different SPR instruments. Finally, it demonstrates the capacity of the SPR system in resolving dynamic voltage signals; a detection limit of 10mV with a temporal resolution of 5ms is achievable. These findings pave the way for the use of SPR systems in the detection of electrical activity of biological cells.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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References

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2016 (2)

S. A. Abayzeed, R. J. Smith, K. F. Webb, M. G. Somekh, and C. W. See, “Responsivity of the differential-intensity surface plasmon resonance instrument,” Sens. Actuators, B 235, 627–635 (2016).
[Crossref]

H.-M. Tan, S. Pechprasarn, J. Zhang, M. C. Pitter, and M. G. Somekh, “High resolution quantitative angle-scanning widefield surface plasmon microscopy,” Sci. Rep. 6, 20195 (2016).
[Crossref] [PubMed]

2013 (1)

Y. Huang, M. Pitter, M. Somekh, W. Zhang, W. Xie, H. Zhang, H. Wang, and S. Fang, “Plasmonic response of gold film to potential perturbation”, Sci. China. Phys. Mech. Astron. 56, 1495–1503 (2013).
[Crossref]

2012 (4)

T. H. Grandy, S. A. Greenfield, and I. M. Devonshire, “An evaluation of in vivo voltage-sensitive dyes: pharmacological side effects and signal-to-noise ratios after effective removal of brain-pulsation artifacts,” J. Neurophysiol. 108, 2931–2945 (2012).
[Crossref] [PubMed]

A. Dahlin, B. Dielacher, P. Rajendran, K. Sugihara, T. Sannomiya, M. Zenobi-Wong, and J. Vörös, “Electrochemical plasmonic sensors,” Anal. Bioanal. Chem. 402, 1773–1784 (2012).
[Crossref]

Y. Huang, M. Pitter, and M. Somekh, “Time-dependent scattering of ultrathin gold film under potential perturbation,” ACS Appl. Mater. Interfaces 4, 3829–3836 (2012).
[Crossref] [PubMed]

S. Kim, S. Kim, H. Moon, and S. Jun, “In vivo optical neural recording using fiber-based surface plasmon resonance,” Opt. Lett. 37, 614–616 (2012).
[Crossref] [PubMed]

2011 (3)

W. Wang, K. Foley, X. Shan, S. Wang, S. Eaton, V. J. Nagaraj, P. Wiktor, U. Patel, and N. Tao, “Single cells and intracellular processes studied by a plasmonic-based electrochemical impedance microscopy,” Nat. Chem. 3, 249–255 (2011).
[Crossref] [PubMed]

Y. Huang, M. Pitter, and M. Somekh, “Morphology-dependent voltage sensitivity of a gold nanostructure,” Langmuir 27, 13950–13961 (2011).
[Crossref] [PubMed]

X. Shan, S. Wang, W. Wang, and N. Tao, “Plasmonic-based imaging of local square wave,” Anal. Chem. 83, 7394–7399 (2011).
[Crossref] [PubMed]

2010 (1)

X. Shan, U. Patel, S. Wang, R. Iglesias, and N. Tao, “Imaging local electrochemical current via surface plasmon resonance,” Science 327, 1363–1366 (2010).
[Crossref] [PubMed]

2009 (2)

O. Bolduc, L. Live, and J.-F. Masson, “High-resolution surface plasmon resonance sensors based on a dove prism,” Talanta 77, 1680–1687 (2009).
[Crossref] [PubMed]

J. Zhang, T. Atay, and A. Nurmikko, “Optical detection of brain cell activity using plasmonic gold nanoparticles”, Nano Lett. 9, 519–524 (2009).
[Crossref] [PubMed]

2008 (4)

J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. 108, 462–493 (2008).
[Crossref] [PubMed]

A. M. Lopatynskyi, O. G. Lopatynska, M. D. Guiver, L. V. Poperenko, and V. I. Chegel, “Factor of interfacial potential for the surface plasmon-polariton resonance sensor response,” Semicond. Phys. Quantum Electron. Optoelectron 11, 329–336 (2008).

K. Foley, X. Shan, and N. J. Tao, “Surface impedance imaging technique,” Anal. Chem. 80, 5146–5151 (2008).
[Crossref] [PubMed]

S. Ae Kim, K. Min Byun, J. Lee, J. Hoon Kim, D. Albert Kim, H. Baac, M. Shuler, and S. June Kim, “Optical measurement of neural activity using surface plasmon resonance,” Opt. Lett. 33, 914–916 (2008).
[Crossref] [PubMed]

2007 (1)

K. Kato, T. Ishimuro, Y. Arima, I. Hirata, and H. Iwata, “High-throughput immunophenotyping by surface plasmon resonance imaging,” Anal. Chem. 79, 8616–8623 (2007).
[Crossref] [PubMed]

2006 (2)

C. Celedón, M. Flores, P. Häberle, and J. Valdés, “Surface roughness of thin gold films and its effects on the proton energy loss straggling,” Braz. J. Phys. 36, 956–959 (2006).
[Crossref]

K. Chu and M. Bazant, “Nonlinear electrochemical relaxation around conductors,” Phys. Rev. E 74, 011501 (2006).
[Crossref]

2005 (1)

T. Pajkossy, “Impedance spectroscopy at interfaces of metals and aqueous solutions : surface roughness, CPE and related issues,” Solid State Ion. 176, 1997–2003 (2005).
[Crossref]

2004 (2)

2003 (2)

A. N. Bashkatov and E. Genina, “Water refractive index in dependence on temperature and wavelength: a simple approximation,” Proc. SPIE 5068, 393–395 (2003).
[Crossref]

J. E. Garland, K. A. Assiongbon, C. M. Pettit, and D. Roy, “Surface plasmon resonance transients at an electrochemical interface: time resolved measurements using a bicell photodiode,” Anal. Chim. Acta 475, 47–58 (2003).
[Crossref]

2000 (3)

Z. Kerner and T. Pajkossy, “On the origin of capacitance dispersion of rough electrodes,” Electrochim. Acta 46, 207–211 (2000).
[Crossref]

R. J. Green, R. A. Frazier, K. M. Shakesheff, M. C. Davies, C. J. Roberts, and S. J. Tendler, “Surface plasmon resonance analysis of dynamic biological interactions with biomaterials,” Biomaterials 21, 1823–1835 (2000).
[Crossref] [PubMed]

R. L. Rich and D. G. Myszka, “Advances in surface plasmon resonance biosensor analysis,” Curr Opin Biotechnol. 11, 54–61 (2000).
[Crossref] [PubMed]

1999 (2)

D. Kröger, F. Hucho, and H. Vogel, “Ligand binding to nicotinic acetylcholine receptor investigated by surface plasmon resonance,” Anal. Chem. 71, 3157–3165 (1999).
[Crossref] [PubMed]

N. Tao, S. Boussaad, W.L. Huang, R. A. Arechabaleta, and J. D’Agnese, “High resolution surface plasmon resonance spectroscopy,” Rev. Sci. Instrum. 70, 4656–4660 (1999).
[Crossref]

1998 (2)

A. Kabashin and P. Nikitin, “Surface plasmon resonance interferometer for bio-and chemical-sensors,” Opt. Commun. 150, 5–8 (1998).
[Crossref]

A. D. Rakić, A. B. Djurišić, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for verticalcavity optoelectronic devices,” Appl. Opt. 37, 5271–5283 (1998).
[Crossref]

1997 (2)

A. Kabashin and P. I. Nikitin, “Interferometer based on a surface-plasmon resonance for sensor applications,” Quant. Electron. 27, 653–654 (1997).
[Crossref]

L. Daikhin, A. Kornyshev, and M. Urbakh, “Double layer capacitance on a rough metal surface: surface roughness measured by “Debye ruler”,” electrochim. acta 42, 2853–2860 (1997).
[Crossref]

1996 (1)

L. Daikhin, A. Kornyshev, and M. Urbakh, “Double-layer capacitance on a rough metal surface,” Phys. Rev. E 53, 6192 (1996).
[Crossref]

1990 (1)

P. Schiebener, J. Straub, J. L. Sengers, and J. Gallagher, “Refractive index of water and steam as function of wavelength, temperature and density,” J. Phys. Chem. Ref. Data 19, 677–717 (1990).
[Crossref]

1977 (1)

R. Katz, D. Kolb, and J. Sass, “Electron density effects in surface plasmon excitation on silver and gold electrodes,” Surf. Sci. 69, 359–364 (1977).
[Crossref]

1975 (1)

F. Abeles, T. Lopez-Rios, and A. Tadjeddine, “Investigation of the metal-electrolyte interface using surface plasma waves with ellipsometric detection,” Solid State Commun. 16, 843–847 (1975).
[Crossref]

1973 (2)

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

1955 (1)

J. Macdonald and M. Brachman, “The charging and discharging of nonlinear capacitors,” Proceedings of the IRE 43, 71–78 (1955).
[Crossref]

Abayzeed, S. A.

S. A. Abayzeed, R. J. Smith, K. F. Webb, M. G. Somekh, and C. W. See, “Responsivity of the differential-intensity surface plasmon resonance instrument,” Sens. Actuators, B 235, 627–635 (2016).
[Crossref]

Abeles, F.

F. Abeles, T. Lopez-Rios, and A. Tadjeddine, “Investigation of the metal-electrolyte interface using surface plasma waves with ellipsometric detection,” Solid State Commun. 16, 843–847 (1975).
[Crossref]

Ae Kim, S.

Albert Kim, D.

Arechabaleta, R. A.

N. Tao, S. Boussaad, W.L. Huang, R. A. Arechabaleta, and J. D’Agnese, “High resolution surface plasmon resonance spectroscopy,” Rev. Sci. Instrum. 70, 4656–4660 (1999).
[Crossref]

Arima, Y.

K. Kato, T. Ishimuro, Y. Arima, I. Hirata, and H. Iwata, “High-throughput immunophenotyping by surface plasmon resonance imaging,” Anal. Chem. 79, 8616–8623 (2007).
[Crossref] [PubMed]

Assiongbon, K. A.

J. E. Garland, K. A. Assiongbon, C. M. Pettit, and D. Roy, “Surface plasmon resonance transients at an electrochemical interface: time resolved measurements using a bicell photodiode,” Anal. Chim. Acta 475, 47–58 (2003).
[Crossref]

Atay, T.

J. Zhang, T. Atay, and A. Nurmikko, “Optical detection of brain cell activity using plasmonic gold nanoparticles”, Nano Lett. 9, 519–524 (2009).
[Crossref] [PubMed]

Azzam, R.

R. Azzam and N. Bashara, Ellipsometry and polarized light (Elsevier science, 1987).

Baac, H.

Bard, A. J.

A. J. Bard and L. R. Faulkner, Electrochemical Methods: Fundamentals and Applications (WileyNew York, 2001).

Bashara, N.

R. Azzam and N. Bashara, Ellipsometry and polarized light (Elsevier science, 1987).

Bashkatov, A. N.

A. N. Bashkatov and E. Genina, “Water refractive index in dependence on temperature and wavelength: a simple approximation,” Proc. SPIE 5068, 393–395 (2003).
[Crossref]

Bazant, M.

K. Chu and M. Bazant, “Nonlinear electrochemical relaxation around conductors,” Phys. Rev. E 74, 011501 (2006).
[Crossref]

Bolduc, O.

O. Bolduc, L. Live, and J.-F. Masson, “High-resolution surface plasmon resonance sensors based on a dove prism,” Talanta 77, 1680–1687 (2009).
[Crossref] [PubMed]

Boussaad, S.

N. Tao, S. Boussaad, W.L. Huang, R. A. Arechabaleta, and J. D’Agnese, “High resolution surface plasmon resonance spectroscopy,” Rev. Sci. Instrum. 70, 4656–4660 (1999).
[Crossref]

Brachman, M.

J. Macdonald and M. Brachman, “The charging and discharging of nonlinear capacitors,” Proceedings of the IRE 43, 71–78 (1955).
[Crossref]

Britland, S.

M. A. Jamil, F. Sefat, S. Khaghani, S. Lobo, F. A. Javid, M. Youseffi, S. Britland, S. Liu, C. See, M. Somekh, and M. Denyer, “Cell imaging with the widefield surface plasmon microscope,” in “4th Kuala Lumpur International Conference on Biomedical Engineering 2008,” (Springer, 2008), pp. 528–531.

Celedón, C.

C. Celedón, M. Flores, P. Häberle, and J. Valdés, “Surface roughness of thin gold films and its effects on the proton energy loss straggling,” Braz. J. Phys. 36, 956–959 (2006).
[Crossref]

Chegel, V. I.

A. M. Lopatynskyi, O. G. Lopatynska, M. D. Guiver, L. V. Poperenko, and V. I. Chegel, “Factor of interfacial potential for the surface plasmon-polariton resonance sensor response,” Semicond. Phys. Quantum Electron. Optoelectron 11, 329–336 (2008).

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Chu, K.

K. Chu and M. Bazant, “Nonlinear electrochemical relaxation around conductors,” Phys. Rev. E 74, 011501 (2006).
[Crossref]

Conway, B.

B. Conway, Electrochemical supercapacitors: scientific fundamentals and technological applications (Springer Science & Business Media, 2013).

D’Agnese, J.

N. Tao, S. Boussaad, W.L. Huang, R. A. Arechabaleta, and J. D’Agnese, “High resolution surface plasmon resonance spectroscopy,” Rev. Sci. Instrum. 70, 4656–4660 (1999).
[Crossref]

Dahlin, A.

A. Dahlin, B. Dielacher, P. Rajendran, K. Sugihara, T. Sannomiya, M. Zenobi-Wong, and J. Vörös, “Electrochemical plasmonic sensors,” Anal. Bioanal. Chem. 402, 1773–1784 (2012).
[Crossref]

Daikhin, L.

L. Daikhin, A. Kornyshev, and M. Urbakh, “Double layer capacitance on a rough metal surface: surface roughness measured by “Debye ruler”,” electrochim. acta 42, 2853–2860 (1997).
[Crossref]

L. Daikhin, A. Kornyshev, and M. Urbakh, “Double-layer capacitance on a rough metal surface,” Phys. Rev. E 53, 6192 (1996).
[Crossref]

Davies, M. C.

R. J. Green, R. A. Frazier, K. M. Shakesheff, M. C. Davies, C. J. Roberts, and S. J. Tendler, “Surface plasmon resonance analysis of dynamic biological interactions with biomaterials,” Biomaterials 21, 1823–1835 (2000).
[Crossref] [PubMed]

Denyer, M.

M. A. Jamil, F. Sefat, S. Khaghani, S. Lobo, F. A. Javid, M. Youseffi, S. Britland, S. Liu, C. See, M. Somekh, and M. Denyer, “Cell imaging with the widefield surface plasmon microscope,” in “4th Kuala Lumpur International Conference on Biomedical Engineering 2008,” (Springer, 2008), pp. 528–531.

Devonshire, I. M.

T. H. Grandy, S. A. Greenfield, and I. M. Devonshire, “An evaluation of in vivo voltage-sensitive dyes: pharmacological side effects and signal-to-noise ratios after effective removal of brain-pulsation artifacts,” J. Neurophysiol. 108, 2931–2945 (2012).
[Crossref] [PubMed]

Dielacher, B.

A. Dahlin, B. Dielacher, P. Rajendran, K. Sugihara, T. Sannomiya, M. Zenobi-Wong, and J. Vörös, “Electrochemical plasmonic sensors,” Anal. Bioanal. Chem. 402, 1773–1784 (2012).
[Crossref]

Djurišic, A. B.

Eaton, S.

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X. Shan, S. Wang, W. Wang, and N. Tao, “Plasmonic-based imaging of local square wave,” Anal. Chem. 83, 7394–7399 (2011).
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W. Wang, K. Foley, X. Shan, S. Wang, S. Eaton, V. J. Nagaraj, P. Wiktor, U. Patel, and N. Tao, “Single cells and intracellular processes studied by a plasmonic-based electrochemical impedance microscopy,” Nat. Chem. 3, 249–255 (2011).
[Crossref] [PubMed]

X. Shan, U. Patel, S. Wang, R. Iglesias, and N. Tao, “Imaging local electrochemical current via surface plasmon resonance,” Science 327, 1363–1366 (2010).
[Crossref] [PubMed]

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Y. Huang, M. Pitter, M. Somekh, W. Zhang, W. Xie, H. Zhang, H. Wang, and S. Fang, “Plasmonic response of gold film to potential perturbation”, Sci. China. Phys. Mech. Astron. 56, 1495–1503 (2013).
[Crossref]

Y. Huang, M. Pitter, and M. Somekh, “Time-dependent scattering of ultrathin gold film under potential perturbation,” ACS Appl. Mater. Interfaces 4, 3829–3836 (2012).
[Crossref] [PubMed]

Y. Huang, M. Pitter, and M. Somekh, “Morphology-dependent voltage sensitivity of a gold nanostructure,” Langmuir 27, 13950–13961 (2011).
[Crossref] [PubMed]

G. Stabler, M. Somekh, and C. See, “High-resolution wide-field surface plasmon microscopy,” J. Microsc. 214, 328–333 (2004).
[Crossref] [PubMed]

M. A. Jamil, F. Sefat, S. Khaghani, S. Lobo, F. A. Javid, M. Youseffi, S. Britland, S. Liu, C. See, M. Somekh, and M. Denyer, “Cell imaging with the widefield surface plasmon microscope,” in “4th Kuala Lumpur International Conference on Biomedical Engineering 2008,” (Springer, 2008), pp. 528–531.

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H.-M. Tan, S. Pechprasarn, J. Zhang, M. C. Pitter, and M. G. Somekh, “High resolution quantitative angle-scanning widefield surface plasmon microscopy,” Sci. Rep. 6, 20195 (2016).
[Crossref] [PubMed]

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[Crossref]

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[Crossref] [PubMed]

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[Crossref]

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A. Dahlin, B. Dielacher, P. Rajendran, K. Sugihara, T. Sannomiya, M. Zenobi-Wong, and J. Vörös, “Electrochemical plasmonic sensors,” Anal. Bioanal. Chem. 402, 1773–1784 (2012).
[Crossref]

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F. Abeles, T. Lopez-Rios, and A. Tadjeddine, “Investigation of the metal-electrolyte interface using surface plasma waves with ellipsometric detection,” Solid State Commun. 16, 843–847 (1975).
[Crossref]

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H.-M. Tan, S. Pechprasarn, J. Zhang, M. C. Pitter, and M. G. Somekh, “High resolution quantitative angle-scanning widefield surface plasmon microscopy,” Sci. Rep. 6, 20195 (2016).
[Crossref] [PubMed]

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X. Shan, S. Wang, W. Wang, and N. Tao, “Plasmonic-based imaging of local square wave,” Anal. Chem. 83, 7394–7399 (2011).
[Crossref] [PubMed]

W. Wang, K. Foley, X. Shan, S. Wang, S. Eaton, V. J. Nagaraj, P. Wiktor, U. Patel, and N. Tao, “Single cells and intracellular processes studied by a plasmonic-based electrochemical impedance microscopy,” Nat. Chem. 3, 249–255 (2011).
[Crossref] [PubMed]

X. Shan, U. Patel, S. Wang, R. Iglesias, and N. Tao, “Imaging local electrochemical current via surface plasmon resonance,” Science 327, 1363–1366 (2010).
[Crossref] [PubMed]

N. Tao, S. Boussaad, W.L. Huang, R. A. Arechabaleta, and J. D’Agnese, “High resolution surface plasmon resonance spectroscopy,” Rev. Sci. Instrum. 70, 4656–4660 (1999).
[Crossref]

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K. Foley, X. Shan, and N. J. Tao, “Surface impedance imaging technique,” Anal. Chem. 80, 5146–5151 (2008).
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R. J. Green, R. A. Frazier, K. M. Shakesheff, M. C. Davies, C. J. Roberts, and S. J. Tendler, “Surface plasmon resonance analysis of dynamic biological interactions with biomaterials,” Biomaterials 21, 1823–1835 (2000).
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L. Daikhin, A. Kornyshev, and M. Urbakh, “Double layer capacitance on a rough metal surface: surface roughness measured by “Debye ruler”,” electrochim. acta 42, 2853–2860 (1997).
[Crossref]

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[Crossref]

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C. Celedón, M. Flores, P. Häberle, and J. Valdés, “Surface roughness of thin gold films and its effects on the proton energy loss straggling,” Braz. J. Phys. 36, 956–959 (2006).
[Crossref]

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Figures (11)

Fig. 1
Fig. 1 The concept of voltage sensing using surface plasmon resonance sensors. Surface plasmons are sensitive to changes in charge density at the gold-electrolyte interface.
Fig. 2
Fig. 2 One dimensional model of the sensing structure showing the glass substrate coated with 50 nm of gold while the electrolyte is placed on the top of the gold surface. Debye and Thomas Fermi layers are highlighted in electrolyte and gold respectively.
Fig. 3
Fig. 3 The graph shows the key regions in a metal-electrolyte interface that are formed due to the extended electrostatic interactions between charge carriers in the metal and the electrolyte. The Stern layer is formed by electrostatic interaction similar to parallel-plate capacitor while diffuse layer is formed due to competitive effect of the thermal energy and electrostatic potential. The Debye layer is the depth at which the potential drops to 1/e of its initial value. The profile is simulated for a 0.15M NaCl.
Fig. 4
Fig. 4 Electrochemical Surface Plasmon Resonance System: a prism based Kretchmann-Raether configuration, that is used to excite surface plasmons at the metal-electrolyte interface. A three electrode, electrochemical cell is mounted on the prism, which is used to modulate the potential at the metal-electrolyte interface.
Fig. 5
Fig. 5 The effect of the applied voltage on the SPR curve. Theoretical SPR curves were generated for a sensing structure formed of BK-7 glass coated with 50 nm of gold. The wavelength of the incident light is 632.8nm. Voltage series were simulated for 0.9% sodium chloride as an electrolyte solution.
Fig. 6
Fig. 6 The effects of the applied voltage on the properties of the SPR simulated using the procedure described in Sec.3.1. Change to the minimum reflectivity is shown in (a) while changes to the full-width half-maximum (FWHM) and the resonance angle are presented in (b) and (c) respectively.
Fig. 7
Fig. 7 (a) The input voltage waveform. Voltage is applied versus an Ag/AgCl reference electrode at a rate of 10V/s (b) The response of the DI-SPR system to the input waveform for 0.15M NaCl.
Fig. 8
Fig. 8 Experimental resonance shifts (Δθ0) for a series of applied voltages compared to theoretical resonance angle shift.
Fig. 9
Fig. 9 Experimental voltage-equivalent refractive index units compared to theoretically calculated figures.
Fig. 10
Fig. 10 The DI-SPR response to voltage pulses with a duration of 5ms. In (a) the applied 100mV voltage signals and corresponding charging and discharging currents are shown in (b). The simultaneous SPR response is presented in (c). (d–f) shows the a window of the three signals on a magnified scale.
Fig. 11
Fig. 11 (a) Dynamic response of the SPR system to the applied voltage (100mV) fitted to a second-order exponential saturation function. (b) A comparison between time constants and onset times of the fast and the slow responses of the system to the applied voltage. (c) Expected response of the system for pulses with duration less than 0.2s(5τ of the fast response).

Equations (7)

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

1 C = x s 0 + 1 2 0 n o z 2 e 2 k T cosh ( z e ψ s 2 k T )
ψ ( x ) = ψ 0 x 8 k T n 0 0 sinh ( z e ψ s 2 k T )
Δ N = C ψ 0 e d T F
ω p = ( N + Δ N ) e 2 m 0
( ω ) = 1 f 0 ω p 2 ω ( ω + i γ )
y ( t ) = a 1 ( 1 e t / τ 1 ) + a 2 ( 1 e t / τ 2 ) + c
y ( t ) = a 1 τ 1 e t / τ 1 + a 2 τ 2 e t / τ 2

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