Abstract

The concept of photonic bandgap fiber-based surface plasmon resonance sensor operating with low refractive index analytes is developed. Plasmon wave on the surface of a thin metal film embedded into a fiber microstructure is excited by a leaky Gaussian-like core mode of a fiber. We demonstrate that by judicious design of the photonic crystal reflector, the effective refractive index of the core mode can be made considerably smaller than that of the core material, thus enabling efficient phase matching with a plasmon, high sensitivity, and high coupling efficiency from an external Gaussian source, at any wavelength of choice from the visible to near-IR. To our knowledge, this is not achievable by any other traditional sensor design. Moreover, unlike the case of total internal reflection waveguide-based sensors, there is no limitation on the upper value of the waveguide core refractive index, therefore, any optical materials can be used in fabrication of photonic bandgap fiber-based sensors. Based on numerical simulations, we finally present designs using various types of photonic bandgap fibers, including solid and hollow core Bragg fibers, as well as honeycomb photonic crystal fibers. Amplitude and spectrum based methodologies for the detection of changes in the analyte refractive index are discussed. Furthermore, sensitivity enhancement of a degenerate double plasmon peak excitation is demonstrated for the case of a honeycomb fiber. Sensor resolutions in the range 7·10-6-5·10-5 RIU were demonstrated for an aqueous analyte.

© 2007 Optical Society of America

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References

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  1. V. M. Agranovich and D. L. Mills, Surface Polaritons - ElectromagneticWaves at Surfaces and Interfaces, (North- Holland, Amsterdam, 1982).
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    [CrossRef]
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  5. L.M. Zhang and D. Uttamchandani, "Optical chemical sensing employing surface plasmon resonance," Electron. Lett. 23, 1469 (1988).
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  15. D. Monzon-Hernandez, J. Villatoro, D. Talavera, D. Luna-Moreno, "Optical-fiber surface-plasmon resonance sensor with multiple resonance peaks," Appl. Opt. 43, 1216 (2004).
    [CrossRef] [PubMed]
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    [CrossRef]
  32. M. N. Weiss, R. Srivastava, and H. Grogner, "Experimental investigation of a surface plasmon-based integratedoptic humidity sensor," Electron. Lett. 32, 842 (1996).
    [CrossRef]
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  34. J. Dostalek, J. Ctyroky, J. Homola, E. Brynda, M. Skalsky, P. Nekvindova, J. Spirkova, J. Skvor, and J. Schrofel, "Surface plasmon resonance biosensor based on integrated optical waveguide," Sens. Actuators B 76, 8 (2001).
    [CrossRef]
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    [CrossRef]
  36. Y. Gao, N. Guo, B. Gauvreau, M. Rajabian, O. Skorobogata, E. Pone, O. Zabeida, L. Martinu, C. Dubois, and M. Skorobogatiy, "Consecutive Solvent Evaporation and Co-Rolling Techniques for Polymer Multilayer Hollow Fiber Preform Fabrication," J. Mat. Res. 21, 2246-2254 (2006)
    [CrossRef]
  37. M. Skorobogatiy, "Efficient anti-guiding of TE and TM polarizations in low index core waveguides without the need of omnidirectional reflector," Opt. Lett. 30, 2991 (2005)
    [CrossRef] [PubMed]
  38. T. D. Engeness, M. Ibanescu, S. G. Johnson, O. Weisberg, M. Skorobogatiy, S. Jacobs, and Y. Fink, "Dispersion tailoring and compensation by modal interactions in OmniGuide fibers," Opt. Express 11, 1175-1198 (2003)
    [CrossRef] [PubMed]
  39. T. Murao, K. Saitoh, and M. Koshiba, "Design of air-guiding modified honeycomb photonic band-gap fibers for effectively single mode operation," Opt. Express 14, 2404-2412 (2006).
    [CrossRef] [PubMed]
  40. S. E. Barkou, J. Broeng, and A. Bjarklev, "Silica-air photonic crystal fiber design that permits waveguiding by a true photonic bandgap effect," Opt. Lett. 24, 46-49 (1999).
    [CrossRef]
  41. J. Homola, S. S. Yee and G. Gauglitz, "Surface plasmon resonance sensors: review," Sens. Act. B 54, 3-15 (1999).
    [CrossRef]

2006 (8)

D. Monzon-Hernandez and J. Villatoro, "High-resolution refractive index sensing by means of a multiple-peak surface plasmon resonance optical fiber sensor," Sens. Actuators B 115, 227 (2006).
[CrossRef]

H. Suzuki, M. Sugimoto, Y. Matsuiand, J. Kondoh, "Fundamental characteristics of a dual-colour fibre optic SPR sensor," Meas. Sci. Technol. 17, 1547 (2006).
[CrossRef]

B. T. Kuhlmey, K. Pathmanandavel, and R. C. McPhedran, "Multipole analysis of photonic crystal fibers with coated inclusions," Opt. Express 14, 10851-10864 (2006)
[CrossRef] [PubMed]

A. Hassani and M. Skorobogatiy, "Design of the microstructured optical fiber-based surface plasmon resonance sensors with enhanced microfluidics," Opt. Express 14, 11616-11621 (2006)
[CrossRef] [PubMed]

M. Skorobogatiy and A. Kabashin, "Plasmon excitation by the Gaussian-like core mode of a photonic crystal waveguide," Opt. Express 14, 8419 (2006)
[CrossRef] [PubMed]

M. Skorobogatiy, A. Kabashin, "Photon crystal waveguide-based surface plasmon resonance biosensor," Appl. Phys. Lett. 89, 211641 (2006)
[CrossRef]

Y. Gao, N. Guo, B. Gauvreau, M. Rajabian, O. Skorobogata, E. Pone, O. Zabeida, L. Martinu, C. Dubois, and M. Skorobogatiy, "Consecutive Solvent Evaporation and Co-Rolling Techniques for Polymer Multilayer Hollow Fiber Preform Fabrication," J. Mat. Res. 21, 2246-2254 (2006)
[CrossRef]

T. Murao, K. Saitoh, and M. Koshiba, "Design of air-guiding modified honeycomb photonic band-gap fibers for effectively single mode operation," Opt. Express 14, 2404-2412 (2006).
[CrossRef] [PubMed]

2005 (2)

M. Skorobogatiy, "Efficient anti-guiding of TE and TM polarizations in low index core waveguides without the need of omnidirectional reflector," Opt. Lett. 30, 2991 (2005)
[CrossRef] [PubMed]

B. D. Gupta and A. K. Sharma, "Sensitivity evaluation of a multi-layered surface plasmon resonance-based fiber optic sensor: a theoretical study," Sens. Actuators B 107, 40 (2005).
[CrossRef]

2004 (2)

D. Monzon-Hernandez, J. Villatoro, D. Talavera, D. Luna-Moreno, "Optical-fiber surface-plasmon resonance sensor with multiple resonance peaks," Appl. Opt. 43, 1216 (2004).
[CrossRef] [PubMed]

A. K. Sheridan, R. D. Harris, P. N. Bartlett, and J. S. Wilkinson, "Phase interrogation of an integrated optical SPR sensor," Sens. Actuators B 97, 114 (2004).
[CrossRef]

2003 (2)

2001 (2)

J. Dostalek, J. Ctyroky, J. Homola, E. Brynda, M. Skalsky, P. Nekvindova, J. Spirkova, J. Skvor, and J. Schrofel, "Surface plasmon resonance biosensor based on integrated optical waveguide," Sens. Actuators B 76, 8 (2001).
[CrossRef]

A. Diez, M. V. Andres, J. L. Cruz, "In-line fiber-optic sensors based on the excitation of surface plasma modes in metal-coated tapered fibers," Sens. Actuators B 73, 95 (2001).
[CrossRef]

1999 (6)

J. Ctyroky, F. Abdelmalek, W. Ecke, K. Usbeck, "Modelling of the surface plasmon resonance waveguide sensor with Bragg grating," Opt. Quantum Electron. 31, 927 (1999).
[CrossRef]

A. N. Grigorenko, P. Nikitin, and A. V. Kabashin, "Phase jumps and interferometric surface plasmon resonance imaging," Appl. Phys. Lett. 75, 3917 (1999).
[CrossRef]

J. Ctyroky, J. Homola, P. V. Lambeck, S. Musa, H. J. W. M. Hoekstra, R. D. Harris, J. S. Wilkinson, B. Usievich, and N. M. Lyndin "Theory and modelling of optical waveguide sensors utilising surface plasmon resonance," Sens. Actuators B 54, 66 (1999).
[CrossRef]

M. Weisser, B. Menges, and S. Mittler-Neher, "Refractive index and thickness determination of monolayers by plasmons," Sens. Actuators B 56, 189 (1999).
[CrossRef]

S. E. Barkou, J. Broeng, and A. Bjarklev, "Silica-air photonic crystal fiber design that permits waveguiding by a true photonic bandgap effect," Opt. Lett. 24, 46-49 (1999).
[CrossRef]

J. Homola, S. S. Yee and G. Gauglitz, "Surface plasmon resonance sensors: review," Sens. Act. B 54, 3-15 (1999).
[CrossRef]

1998 (1)

A. V. Kabashin and P. Nikitin, "Surface plasmon resonance interferometer for bio- and chemical-sensors," Opt. Commun. 150, 5 (1998).
[CrossRef]

1997 (3)

A. J. C. Tubb, F. P. Payne, R. B. Millington, and C. R. Lowe, "Single-mode optical fibre surface plasma wave chemical sensor," Sens. Actuators B 41, 71 (1997).
[CrossRef]

J. Homola, R. Slavik, J. Ctyroky, "Intreaction between fiber modes and surface plasmon wave: spectral properties," Opt. Lett. 22, 1403 (1997).
[CrossRef]

J. Homola, J. Ctyroky, M. Skalky, J. Hradiliva, and P. Kolarova, "A surface plasmon resonance based integrated optical sensor," Sens. Actuators B 39, 286 (1997).
[CrossRef]

1996 (3)

A. Trouillet, C. Ronot-Trioli, C. Veillas, H. Gagnaire, "Chemical sensing by surface plasmon resonance in a multimode optical fibre," Pure Appl. Opt. 5, 227 (1996).
[CrossRef]

M. N. Weiss, R. Srivastava, and H. Grogner, "Experimental investigation of a surface plasmon-based integratedoptic humidity sensor," Electron. Lett. 32, 842 (1996).
[CrossRef]

J. L. Melendez, R. Carr, D. U. Bartholomew, K. A. Kukanskis, J. Elkind, S. S. Yee, C. E. Furlong, R. G. Woodbury, "A commercial solution for surface plasmon sensing," Sens. Actuators B 35, 212 (1996).
[CrossRef]

1995 (2)

J. Homola, "Optical fiber sensor based on surface plasmon resonance excitation," Sens. Actuators B 29, 401 (1995).
[CrossRef]

R. Harris and J. S. Wilkinson, "Waveguide surface plasmon resonance sensors," Sens. Actuators B 29, 261 (1995).
[CrossRef]

1994 (2)

1993 (3)

M. B. Vidal, R. Lopez, S. Aleggret, J. Alonso-Chamarro, I. Garces and J. Mateo, "Determination of probable alcohol yield in musts by means of an SPR optical sensor," Sens. Actuators B 11, 455 (1993).
[CrossRef]

S.J. Al-Bader andM. Imtaar, "Optical fiber hybrid-surface plasmon polaritons," J. Opt. Soc. Am. B 10, 83 (1993).
[CrossRef]

R. C. Jorgenson and S. S. Yee, "A fiber-optic chemical sensor based on surface plasmon resonance," Sens. Actuators B 12, 213 (1993).
[CrossRef]

1988 (1)

L.M. Zhang and D. Uttamchandani, "Optical chemical sensing employing surface plasmon resonance," Electron. Lett. 23, 1469 (1988).
[CrossRef]

1983 (1)

B. Liedberg, C. Nylander, I. Lundstrom, "Surface plasmon resonance for gas detection and biosensing," Sens. Actuators B 4, 299 (1983).
[CrossRef]

1968 (1)

E. Kretschmann and H. Raether, "Radiative decay of non radiative surface plasmons excited by light," Naturforschung A 23, 2135 (1968).

Appl. Opt. (2)

Appl. Phys. Lett. (2)

M. Skorobogatiy, A. Kabashin, "Photon crystal waveguide-based surface plasmon resonance biosensor," Appl. Phys. Lett. 89, 211641 (2006)
[CrossRef]

A. N. Grigorenko, P. Nikitin, and A. V. Kabashin, "Phase jumps and interferometric surface plasmon resonance imaging," Appl. Phys. Lett. 75, 3917 (1999).
[CrossRef]

Electron. Lett. (2)

L.M. Zhang and D. Uttamchandani, "Optical chemical sensing employing surface plasmon resonance," Electron. Lett. 23, 1469 (1988).
[CrossRef]

M. N. Weiss, R. Srivastava, and H. Grogner, "Experimental investigation of a surface plasmon-based integratedoptic humidity sensor," Electron. Lett. 32, 842 (1996).
[CrossRef]

J. Mat. Res. (1)

Y. Gao, N. Guo, B. Gauvreau, M. Rajabian, O. Skorobogata, E. Pone, O. Zabeida, L. Martinu, C. Dubois, and M. Skorobogatiy, "Consecutive Solvent Evaporation and Co-Rolling Techniques for Polymer Multilayer Hollow Fiber Preform Fabrication," J. Mat. Res. 21, 2246-2254 (2006)
[CrossRef]

J. Opt. Soc. Am. B (1)

Meas. Sci. Technol. (1)

H. Suzuki, M. Sugimoto, Y. Matsuiand, J. Kondoh, "Fundamental characteristics of a dual-colour fibre optic SPR sensor," Meas. Sci. Technol. 17, 1547 (2006).
[CrossRef]

Naturforschung A (1)

E. Kretschmann and H. Raether, "Radiative decay of non radiative surface plasmons excited by light," Naturforschung A 23, 2135 (1968).

Opt. Commun. (1)

A. V. Kabashin and P. Nikitin, "Surface plasmon resonance interferometer for bio- and chemical-sensors," Opt. Commun. 150, 5 (1998).
[CrossRef]

Opt. Express (5)

Opt. Lett. (3)

Opt. Quantum Electron. (1)

J. Ctyroky, F. Abdelmalek, W. Ecke, K. Usbeck, "Modelling of the surface plasmon resonance waveguide sensor with Bragg grating," Opt. Quantum Electron. 31, 927 (1999).
[CrossRef]

Pure Appl. Opt. (1)

A. Trouillet, C. Ronot-Trioli, C. Veillas, H. Gagnaire, "Chemical sensing by surface plasmon resonance in a multimode optical fibre," Pure Appl. Opt. 5, 227 (1996).
[CrossRef]

Sens. Act. B (1)

J. Homola, S. S. Yee and G. Gauglitz, "Surface plasmon resonance sensors: review," Sens. Act. B 54, 3-15 (1999).
[CrossRef]

Sens. Actuators B (17)

C. P. Lavers and J.S. Wilkinson, "A waveguide-coupled surface-plasmon sensor for an aqueous environment," Sens. Actuators B 22, 75 (1994).
[CrossRef]

R. Harris and J. S. Wilkinson, "Waveguide surface plasmon resonance sensors," Sens. Actuators B 29, 261 (1995).
[CrossRef]

J. Homola, J. Ctyroky, M. Skalky, J. Hradiliva, and P. Kolarova, "A surface plasmon resonance based integrated optical sensor," Sens. Actuators B 39, 286 (1997).
[CrossRef]

J. Dostalek, J. Ctyroky, J. Homola, E. Brynda, M. Skalsky, P. Nekvindova, J. Spirkova, J. Skvor, and J. Schrofel, "Surface plasmon resonance biosensor based on integrated optical waveguide," Sens. Actuators B 76, 8 (2001).
[CrossRef]

A. K. Sheridan, R. D. Harris, P. N. Bartlett, and J. S. Wilkinson, "Phase interrogation of an integrated optical SPR sensor," Sens. Actuators B 97, 114 (2004).
[CrossRef]

J. Ctyroky, J. Homola, P. V. Lambeck, S. Musa, H. J. W. M. Hoekstra, R. D. Harris, J. S. Wilkinson, B. Usievich, and N. M. Lyndin "Theory and modelling of optical waveguide sensors utilising surface plasmon resonance," Sens. Actuators B 54, 66 (1999).
[CrossRef]

M. Weisser, B. Menges, and S. Mittler-Neher, "Refractive index and thickness determination of monolayers by plasmons," Sens. Actuators B 56, 189 (1999).
[CrossRef]

B. D. Gupta and A. K. Sharma, "Sensitivity evaluation of a multi-layered surface plasmon resonance-based fiber optic sensor: a theoretical study," Sens. Actuators B 107, 40 (2005).
[CrossRef]

R. C. Jorgenson and S. S. Yee, "A fiber-optic chemical sensor based on surface plasmon resonance," Sens. Actuators B 12, 213 (1993).
[CrossRef]

B. Liedberg, C. Nylander, I. Lundstrom, "Surface plasmon resonance for gas detection and biosensing," Sens. Actuators B 4, 299 (1983).
[CrossRef]

J. L. Melendez, R. Carr, D. U. Bartholomew, K. A. Kukanskis, J. Elkind, S. S. Yee, C. E. Furlong, R. G. Woodbury, "A commercial solution for surface plasmon sensing," Sens. Actuators B 35, 212 (1996).
[CrossRef]

M. B. Vidal, R. Lopez, S. Aleggret, J. Alonso-Chamarro, I. Garces and J. Mateo, "Determination of probable alcohol yield in musts by means of an SPR optical sensor," Sens. Actuators B 11, 455 (1993).
[CrossRef]

J. Homola, "Optical fiber sensor based on surface plasmon resonance excitation," Sens. Actuators B 29, 401 (1995).
[CrossRef]

A. J. C. Tubb, F. P. Payne, R. B. Millington, and C. R. Lowe, "Single-mode optical fibre surface plasma wave chemical sensor," Sens. Actuators B 41, 71 (1997).
[CrossRef]

A. Diez, M. V. Andres, J. L. Cruz, "In-line fiber-optic sensors based on the excitation of surface plasma modes in metal-coated tapered fibers," Sens. Actuators B 73, 95 (2001).
[CrossRef]

M. Piliarik, J. Homola, Z. Manikova, J. Ctyroky, "Surface plasmon resonance based on a polarization-maintaining optical fiber," Sens. Actuators B 90, 236 (2003).
[CrossRef]

D. Monzon-Hernandez and J. Villatoro, "High-resolution refractive index sensing by means of a multiple-peak surface plasmon resonance optical fiber sensor," Sens. Actuators B 115, 227 (2006).
[CrossRef]

Other (2)

A. Hassani, M. Skorobogatiy, "Design criteria for the Microstructured Optical Fiber-based Surface Plasmon Resonance sensors," accepted for publication in the J. Opt. Soc. Am. B, February (2007).

V. M. Agranovich and D. L. Mills, Surface Polaritons - ElectromagneticWaves at Surfaces and Interfaces, (North- Holland, Amsterdam, 1982).

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Figures (6)

Fig. 1.
Fig. 1.

Schematics of various photonic crystal waveguide-based SPR sensor implementations. a) Single mode planar photonic crystal waveguide-based SPR sensor. The dispersion relation of the core guided mode is in solid blue, that of the plasmon is in thick dashed red. Inset - coupler schematic; |S z | of a plasmon (left) and a core mode (right). b) Solid core Bragg fiber-based SPR sensor. c) Microstructured core, honeycomb photonic crystal fiber-based SPR sensor.

Fig. 2.
Fig. 2.

Large solid core photonic crystal Bragg fiber-based SPR sensor. a) Schematic of the sensor. The low refractive index core is surrounded by a concentric photonic crystal reflector. Outside, the reflector is goldplated for plasmon excitation. The gold layer is bordered by an aqueous analyte. The energy flux distribution across the fiber cross-section is shown with a solid curve for the fundamental core mode, and with a dashed curve for the plasmonic mode. b) Band diagram of the sensor modes showing the dispersion relation of the fundamental core mode (thick solid curve), and plasmonic mode (dashed curve). Common part of the TE, TM bandgaps of a periodic planar reflector is shown as a clear region, while gray regions correspond to the continuum of bulk reflector states. In a large core Bragg fiber, the effective refractive index of the fundamental core mode is close to the refractive index of the core material. c) Upper part: solid curve shows loss of the fundamental core mode near the phase matching point with the plasmon. The modal loss reaches its maximum at the phase matching wavelength. The dashed line shows a shift of the modal loss curve when the refractive index of the analyte is varied. Lower part: computed dependence of the sensor amplitude sensitivity over wavelength.

Fig. 3.
Fig. 3.

Small solid core photonic crystal Bragg fiber-based SPR sensor. a) Schematic of the sensor. A low refractive index core is surrounded by a concentric photonic crystal reflector. Outside, the reflector is goldplated for plasmon excitation. The gold layer is bordered by an aqueous analyte. The energy flux distribution across the fiber cross-section is shown with a solid curve for the fundamental core mode, and with a dashed curve for the plasmonic mode. b) Band diagram of sensor modes. Dispersion relation of the fundamental core mode (thick solid curve), and plasmonic mode (dashed curve). The common part of the TE, TM bandgaps of a periodic planar reflector is shown as a clear region, while gray regions correspond to the continuum of bulk reflector states. In a small core Bragg fiber, the effective refractive index of the fundamental core mode can be much smaller than the refractive index of the core material. c) Upper part: the solid curve shows loss of the fundamental core mode near the phase matching point with the plasmon. The modal loss reaches its maximum at the phase matching wavelength. The dashed line shows a shift of the modal loss curve when the refractive index of the analyte is varied. Lower part: computed dependence of the sensor amplitude sensitivity on wavelength.

Fig. 4.
Fig. 4.

Analyte-filled large hollow core photonic crystal Bragg fiber-based SPR sensor. a) Schematic of the sensor. The analyte-filled hollow core is surrounded by a concentric photonic crystal reflector. Outside, the reflector is goldplated for plasmon excitation. The gold layer is bordered by an aqueous analyte. The energy flux distribution across the fiber crosssection is shown in solid curve for the fundamental core mode, and in dashed curve for the plasmonic mode. b) Band diagram of sensor modes. Dispersion relation of the fundamental core mode crossing over the analyte light line (thick solid curve), and plasmonic mode (dashed curve). Common part of the TE, TM bandgaps of a periodic planar reflector is shown as a clear region, while gray regions correspond to the continuum of bulk reflector states. In a large analyte-filled hollow core Bragg fiber, the effective refractive index of the fundamental core mode is close to the refractive index of the analyte. By introducing a defect into the multilayer structure, one can force a core mode to cross over the analyte radiation line. c) Upper part: solid curve shows loss of the crossed-over core mode near the phase matching point with the plasmon. The modal loss reaches its maximum at the phase matching wavelength. The dashed line shows a shift of the modal loss curve when the refractive index of the analyte is varied. Lower part: dependence of the sensor amplitude sensitivity on wavelength.

Fig. 5.
Fig. 5.

Solid core honeycomb photonic crystal fiber-based SPR sensor. a) Schematic of the sensor. The solid core having a small central hole is surrounded with a honeycomb photonic crystal reflector. Two large channels are integrated to implement analyte access to the fiber reflector region. The channels are goldplated for plasmon excitation. The gold layer is bordered by an aqueous analyte. b) Band diagram of sensor modes. Dispersion relation of the fundamental core mode (thick solid curve), analyte bound plasmonic mode (dashed curve with circles), and cladding bound plasmonic mode (dashed curve). The bandgap of an infinitely periodic reflector is shown as a clear region. c) The energy flux distributions across the fiber cross-section are shown for the fundamental core mode (II) as well as the analyte and cladding bound plasmon modes (I,III) outside of the phase matching region. The energy flux distribution is also shown for the fundamental core mode at the phase matching point (IV) showing strong mixing of the fundamental core mode with plasmonic modes.

Fig. 6.
Fig. 6.

Sensitivity of the honeycomb photonic crystal fiber-based SPR sensor. a) The solid curve shows loss of the fundamental core mode near the degenerate phase matching point with two plasmonic modes and nanalyte =1.32. Due to degeneracy, only one peak is distinguishable in the loss curve. Dashed line shows splitting of the degeneracy in plasmonic modes when the analyte refractive index is changed to nanalyte =1.322. b) Dependence of the sensor amplitude sensitivity on wavelength.

Equations (6)

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d l , h = λ c 4 n l , h 2 n eff 2 ( λ c ) ,
n eff 2 ( λ c ) ε gold ( λ c ) ε a ( λ c ) ε gold ( λ c ) + ε a ( λ c ) ,
ε gold = ε ( λ λ p ) 2 1 + i ( λ λ t ) ,
S A ( λ ) [ RIU 1 ] = 1 P ( L , λ , n a ) P ( L , λ , n a ) n a = 1 α ( λ , n a ) α ( λ , n a ) n a .
S λ [ nm · RIU 1 ] = d λ peak ( n a ) d n a .
S λ = λ peak 1 λ peak 2 d n analyte ,

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