Abstract

The properties of the high order cladding modes of standard optical fibers are measured in real-time during the deposition of gold nanoparticle layers by chemical vapor deposition (CVD). Using a tilted fiber Bragg grating (TFBG), the resonance wavelength and peak-to-peak amplitude of a radially polarized cladding mode resonance located 51 nm away from the core mode reflection resonance shift by 0.17 nm and 13.54 dB respectively during the formation of a ~200 nm thick layer. For the spectrally adjacent azimuthally polarized resonance, the corresponding shifts are 0.45 nm and 16.34 dB. In both cases, the amplitudes of the resonance go through a pronounced minimum of about 5 dB for thickness between 80 and 100 nm and at the same time the wavelengths shift discontinuously. These effects are discussed in terms of the evolving metallic boundary conditions perceived by the cladding modes as the nanoparticles grow. Scanning Electron Micrographs and observations of cladding mode light scattering by nanoparticle layers of various thicknesses reveal a strong correlation between the TFBG polarized transmission spectra, the grain size and fill factor of the nanoparticles, and the scattering efficiency. This allows the preparation of gold nanoparticle layers that strongly discriminate between radially and azimuthally polarized cladding mode evanescent fields, with important consequences in the plasmonic properties of these layers.

© 2013 OSA

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2012 (4)

2011 (6)

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics5(7), 411–415 (2011).
[CrossRef]

C. Caucheteur, Y. Shevchenko, L.-Y. Shao, M. Wuilpart, and J. Albert, “High resolution interrogation of tilted fiber grating SPR sensors from polarization properties measurement,” Opt. Express19(2), 1656–1664 (2011).
[CrossRef] [PubMed]

C. Caucheteur, C. Chen, V. Voisin, P. Berini, and J. Albert, “A thin metal sheath lifts the EH to HE degeneracy in the cladding mode refractometric sensitivity of optical fiber sensors,” Appl. Phys. Lett.99(4), 041118 (2011).
[CrossRef]

A. Bialiayeu, C. Caucheteur, N. Ahamad, A. Ianoul, and J. Albert, “Self-optimized metal coatings for fiber plasmonics by electroless deposition,” Opt. Express19(20), 18742–18753 (2011).
[CrossRef] [PubMed]

L.-Y. Shao, J. P. Coyle, S. T. Barry, and J. Albert, “Anomalous permittivity and plasmon resonances of copper nanoparticle conformal coating on optical fibres,” Opt. Mater. Express1(2), 128–137 (2011).
[CrossRef]

T. J. J. Whitehorne, J. P. Coyle, A. Mahmood, W. H. Monillas, G. P. A. Yap, and S. T. Barry, “Group 11 amidinates and guanidinates: potential precursors for vapour deposition,” Eur. J. Inorg. Chem.2011(21), 3240–3247 (2011).
[CrossRef]

2010 (3)

2009 (1)

J. Pollet, F. Delport, K. P. F. Janssen, K. Jans, G. Maes, H. Pfeiffer, M. Wevers, and J. Lammertyn, “Fiber optic SPR biosensing of DNA hybridization and DNA-protein interactions,” Biosens. Bioelectron.25(4), 864–869 (2009).
[CrossRef] [PubMed]

2008 (1)

W.-J. Lee, J.-E. Kim, H. Y. Park, S. Park, M.-S. Kim, J. T. Kim, and J. J. Ju, “Optical constants of evaporated gold films measured by surface plasmon resonance at telecommunication wavelengths,” J. Appl. Phys.103(7), 073713 (2008).
[CrossRef]

2007 (3)

I. Horcas, R. Fernández, J. M. Gómez-Rodríguez, J. Colchero, J. Gómez-Herrero, and A. M. Baro, “WSXM: A software for scanning probe microscopy and a tool for nanotechnology,” Rev. Sci. Instrum.78(1), 013705 (2007).
[CrossRef] [PubMed]

M. Chen and R. G. Horn, “Refractive index of sparse layers of adsorbed gold nanoparticles,” J. Colloid Interface Sci.315(2), 814–817 (2007).
[CrossRef] [PubMed]

C.-F. Chan, C. Chen, A. Jafari, A. Laronche, D. J. Thomson, and J. Albert, “Optical fiber refractometer using narrowband cladding-mode resonance shifts,” Appl. Opt.46(7), 1142–1149 (2007).
[CrossRef] [PubMed]

2005 (1)

2000 (1)

P. S. Westbrook, T. A. Strasser, and T. Erdogan, “In-line polarimeter using blazed fiber gratings,” Photonics Technol. Lett.12(10), 1352–1354 (2000).
[CrossRef]

Achaerandio, M.

Ahamad, N.

Albert, J.

Arregui, F. J.

Bao, Q.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics5(7), 411–415 (2011).
[CrossRef]

Baro, A. M.

I. Horcas, R. Fernández, J. M. Gómez-Rodríguez, J. Colchero, J. Gómez-Herrero, and A. M. Baro, “WSXM: A software for scanning probe microscopy and a tool for nanotechnology,” Rev. Sci. Instrum.78(1), 013705 (2007).
[CrossRef] [PubMed]

Barry, S. T.

L.-Y. Shao, J. P. Coyle, S. T. Barry, and J. Albert, “Anomalous permittivity and plasmon resonances of copper nanoparticle conformal coating on optical fibres,” Opt. Mater. Express1(2), 128–137 (2011).
[CrossRef]

T. J. J. Whitehorne, J. P. Coyle, A. Mahmood, W. H. Monillas, G. P. A. Yap, and S. T. Barry, “Group 11 amidinates and guanidinates: potential precursors for vapour deposition,” Eur. J. Inorg. Chem.2011(21), 3240–3247 (2011).
[CrossRef]

Berini, P.

C. Caucheteur, C. Chen, V. Voisin, P. Berini, and J. Albert, “A thin metal sheath lifts the EH to HE degeneracy in the cladding mode refractometric sensitivity of optical fiber sensors,” Appl. Phys. Lett.99(4), 041118 (2011).
[CrossRef]

Bialiayeu, A.

Carminati, R.

Castanié, E.

Caucheteur, C.

Cazé, A.

Chan, C.-F.

Chen, C.

Chen, M.

M. Chen and R. G. Horn, “Refractive index of sparse layers of adsorbed gold nanoparticles,” J. Colloid Interface Sci.315(2), 814–817 (2007).
[CrossRef] [PubMed]

Chilkoti, A.

J. J. Mock, R. T. Hill, Y.-J. Tsai, A. Chilkoti, and D. R. Smith, “Probing dynamically tunable localized surface plasmon resonances of film-coupled nanoparticles by evanescent wave excitation,” Nano Lett.12(4), 1757–1764 (2012).
[CrossRef] [PubMed]

Choi, S. H.

Colchero, J.

I. Horcas, R. Fernández, J. M. Gómez-Rodríguez, J. Colchero, J. Gómez-Herrero, and A. M. Baro, “WSXM: A software for scanning probe microscopy and a tool for nanotechnology,” Rev. Sci. Instrum.78(1), 013705 (2007).
[CrossRef] [PubMed]

Coyle, J. P.

L.-Y. Shao, J. P. Coyle, S. T. Barry, and J. Albert, “Anomalous permittivity and plasmon resonances of copper nanoparticle conformal coating on optical fibres,” Opt. Mater. Express1(2), 128–137 (2011).
[CrossRef]

T. J. J. Whitehorne, J. P. Coyle, A. Mahmood, W. H. Monillas, G. P. A. Yap, and S. T. Barry, “Group 11 amidinates and guanidinates: potential precursors for vapour deposition,” Eur. J. Inorg. Chem.2011(21), 3240–3247 (2011).
[CrossRef]

Dakka, M. A.

De Wilde, Y.

Del Villar, I.

Delport, F.

J. Pollet, F. Delport, K. P. F. Janssen, K. Jans, G. Maes, H. Pfeiffer, M. Wevers, and J. Lammertyn, “Fiber optic SPR biosensing of DNA hybridization and DNA-protein interactions,” Biosens. Bioelectron.25(4), 864–869 (2009).
[CrossRef] [PubMed]

Erdogan, T.

P. S. Westbrook, T. A. Strasser, and T. Erdogan, “In-line polarimeter using blazed fiber gratings,” Photonics Technol. Lett.12(10), 1352–1354 (2000).
[CrossRef]

Fernández, R.

I. Horcas, R. Fernández, J. M. Gómez-Rodríguez, J. Colchero, J. Gómez-Herrero, and A. M. Baro, “WSXM: A software for scanning probe microscopy and a tool for nanotechnology,” Rev. Sci. Instrum.78(1), 013705 (2007).
[CrossRef] [PubMed]

Geng, R.

Gómez-Herrero, J.

I. Horcas, R. Fernández, J. M. Gómez-Rodríguez, J. Colchero, J. Gómez-Herrero, and A. M. Baro, “WSXM: A software for scanning probe microscopy and a tool for nanotechnology,” Rev. Sci. Instrum.78(1), 013705 (2007).
[CrossRef] [PubMed]

Gómez-Rodríguez, J. M.

I. Horcas, R. Fernández, J. M. Gómez-Rodríguez, J. Colchero, J. Gómez-Herrero, and A. M. Baro, “WSXM: A software for scanning probe microscopy and a tool for nanotechnology,” Rev. Sci. Instrum.78(1), 013705 (2007).
[CrossRef] [PubMed]

Han, B.

Herschel, R.

Hill, R. T.

J. J. Mock, R. T. Hill, Y.-J. Tsai, A. Chilkoti, and D. R. Smith, “Probing dynamically tunable localized surface plasmon resonances of film-coupled nanoparticles by evanescent wave excitation,” Nano Lett.12(4), 1757–1764 (2012).
[CrossRef] [PubMed]

Horcas, I.

I. Horcas, R. Fernández, J. M. Gómez-Rodríguez, J. Colchero, J. Gómez-Herrero, and A. M. Baro, “WSXM: A software for scanning probe microscopy and a tool for nanotechnology,” Rev. Sci. Instrum.78(1), 013705 (2007).
[CrossRef] [PubMed]

Horn, R. G.

M. Chen and R. G. Horn, “Refractive index of sparse layers of adsorbed gold nanoparticles,” J. Colloid Interface Sci.315(2), 814–817 (2007).
[CrossRef] [PubMed]

Ianoul, A.

Jafari, A.

Jans, K.

J. Pollet, F. Delport, K. P. F. Janssen, K. Jans, G. Maes, H. Pfeiffer, M. Wevers, and J. Lammertyn, “Fiber optic SPR biosensing of DNA hybridization and DNA-protein interactions,” Biosens. Bioelectron.25(4), 864–869 (2009).
[CrossRef] [PubMed]

Janssen, K. P. F.

J. Pollet, F. Delport, K. P. F. Janssen, K. Jans, G. Maes, H. Pfeiffer, M. Wevers, and J. Lammertyn, “Fiber optic SPR biosensing of DNA hybridization and DNA-protein interactions,” Biosens. Bioelectron.25(4), 864–869 (2009).
[CrossRef] [PubMed]

Jian, S.

Ju, J. J.

W.-J. Lee, J.-E. Kim, H. Y. Park, S. Park, M.-S. Kim, J. T. Kim, and J. J. Ju, “Optical constants of evaporated gold films measured by surface plasmon resonance at telecommunication wavelengths,” J. Appl. Phys.103(7), 073713 (2008).
[CrossRef]

Käll, M.

L. Tong, V. D. Miljković, and M. Käll, “Alignment, rotation, and spinning of single plasmonic nanoparticles and nanowires using polarization dependent optical forces,” Nano Lett.10(1), 268–273 (2010).
[CrossRef] [PubMed]

Kim, J. T.

W.-J. Lee, J.-E. Kim, H. Y. Park, S. Park, M.-S. Kim, J. T. Kim, and J. J. Ju, “Optical constants of evaporated gold films measured by surface plasmon resonance at telecommunication wavelengths,” J. Appl. Phys.103(7), 073713 (2008).
[CrossRef]

Kim, J.-E.

W.-J. Lee, J.-E. Kim, H. Y. Park, S. Park, M.-S. Kim, J. T. Kim, and J. J. Ju, “Optical constants of evaporated gold films measured by surface plasmon resonance at telecommunication wavelengths,” J. Appl. Phys.103(7), 073713 (2008).
[CrossRef]

Kim, M.-S.

W.-J. Lee, J.-E. Kim, H. Y. Park, S. Park, M.-S. Kim, J. T. Kim, and J. J. Ju, “Optical constants of evaporated gold films measured by surface plasmon resonance at telecommunication wavelengths,” J. Appl. Phys.103(7), 073713 (2008).
[CrossRef]

Kim, Y. L.

Krachmalnicoff, V.

Kwak, B.

Lammertyn, J.

J. Pollet, F. Delport, K. P. F. Janssen, K. Jans, G. Maes, H. Pfeiffer, M. Wevers, and J. Lammertyn, “Fiber optic SPR biosensing of DNA hybridization and DNA-protein interactions,” Biosens. Bioelectron.25(4), 864–869 (2009).
[CrossRef] [PubMed]

Laronche, A.

Lee, W.-J.

W.-J. Lee, J.-E. Kim, H. Y. Park, S. Park, M.-S. Kim, J. T. Kim, and J. J. Ju, “Optical constants of evaporated gold films measured by surface plasmon resonance at telecommunication wavelengths,” J. Appl. Phys.103(7), 073713 (2008).
[CrossRef]

Lim, C. H. Y. X.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics5(7), 411–415 (2011).
[CrossRef]

Liu, C.

Loh, K. P.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics5(7), 411–415 (2011).
[CrossRef]

Lu, Y.-C.

Maes, G.

J. Pollet, F. Delport, K. P. F. Janssen, K. Jans, G. Maes, H. Pfeiffer, M. Wevers, and J. Lammertyn, “Fiber optic SPR biosensing of DNA hybridization and DNA-protein interactions,” Biosens. Bioelectron.25(4), 864–869 (2009).
[CrossRef] [PubMed]

Mahmood, A.

T. J. J. Whitehorne, J. P. Coyle, A. Mahmood, W. H. Monillas, G. P. A. Yap, and S. T. Barry, “Group 11 amidinates and guanidinates: potential precursors for vapour deposition,” Eur. J. Inorg. Chem.2011(21), 3240–3247 (2011).
[CrossRef]

Matias, I. R.

Miljkovic, V. D.

L. Tong, V. D. Miljković, and M. Käll, “Alignment, rotation, and spinning of single plasmonic nanoparticles and nanowires using polarization dependent optical forces,” Nano Lett.10(1), 268–273 (2010).
[CrossRef] [PubMed]

Mock, J. J.

J. J. Mock, R. T. Hill, Y.-J. Tsai, A. Chilkoti, and D. R. Smith, “Probing dynamically tunable localized surface plasmon resonances of film-coupled nanoparticles by evanescent wave excitation,” Nano Lett.12(4), 1757–1764 (2012).
[CrossRef] [PubMed]

Monillas, W. H.

T. J. J. Whitehorne, J. P. Coyle, A. Mahmood, W. H. Monillas, G. P. A. Yap, and S. T. Barry, “Group 11 amidinates and guanidinates: potential precursors for vapour deposition,” Eur. J. Inorg. Chem.2011(21), 3240–3247 (2011).
[CrossRef]

Neumann, N.

Ni, Z.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics5(7), 411–415 (2011).
[CrossRef]

Ning, T.

Park, H. Y.

W.-J. Lee, J.-E. Kim, H. Y. Park, S. Park, M.-S. Kim, J. T. Kim, and J. J. Ju, “Optical constants of evaporated gold films measured by surface plasmon resonance at telecommunication wavelengths,” J. Appl. Phys.103(7), 073713 (2008).
[CrossRef]

Park, S.

W.-J. Lee, J.-E. Kim, H. Y. Park, S. Park, M.-S. Kim, J. T. Kim, and J. J. Ju, “Optical constants of evaporated gold films measured by surface plasmon resonance at telecommunication wavelengths,” J. Appl. Phys.103(7), 073713 (2008).
[CrossRef]

Pfeiffer, H.

J. Pollet, F. Delport, K. P. F. Janssen, K. Jans, G. Maes, H. Pfeiffer, M. Wevers, and J. Lammertyn, “Fiber optic SPR biosensing of DNA hybridization and DNA-protein interactions,” Biosens. Bioelectron.25(4), 864–869 (2009).
[CrossRef] [PubMed]

Pierrat, R.

Pollet, J.

J. Pollet, F. Delport, K. P. F. Janssen, K. Jans, G. Maes, H. Pfeiffer, M. Wevers, and J. Lammertyn, “Fiber optic SPR biosensing of DNA hybridization and DNA-protein interactions,” Biosens. Bioelectron.25(4), 864–869 (2009).
[CrossRef] [PubMed]

Schaffer, C. G.

Schuster, T.

Shao, L.-Y.

Shevchenko, Y.

Smith, D. R.

J. J. Mock, R. T. Hill, Y.-J. Tsai, A. Chilkoti, and D. R. Smith, “Probing dynamically tunable localized surface plasmon resonances of film-coupled nanoparticles by evanescent wave excitation,” Nano Lett.12(4), 1757–1764 (2012).
[CrossRef] [PubMed]

Strasser, T. A.

P. S. Westbrook, T. A. Strasser, and T. Erdogan, “In-line polarimeter using blazed fiber gratings,” Photonics Technol. Lett.12(10), 1352–1354 (2000).
[CrossRef]

Tang, D. Y.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics5(7), 411–415 (2011).
[CrossRef]

Thomson, D. J.

Tong, L.

L. Tong, V. D. Miljković, and M. Käll, “Alignment, rotation, and spinning of single plasmonic nanoparticles and nanowires using polarization dependent optical forces,” Nano Lett.10(1), 268–273 (2010).
[CrossRef] [PubMed]

Tsai, Y.-J.

J. J. Mock, R. T. Hill, Y.-J. Tsai, A. Chilkoti, and D. R. Smith, “Probing dynamically tunable localized surface plasmon resonances of film-coupled nanoparticles by evanescent wave excitation,” Nano Lett.12(4), 1757–1764 (2012).
[CrossRef] [PubMed]

Voisin, V.

C. Caucheteur, C. Chen, V. Voisin, P. Berini, and J. Albert, “A thin metal sheath lifts the EH to HE degeneracy in the cladding mode refractometric sensitivity of optical fiber sensors,” Appl. Phys. Lett.99(4), 041118 (2011).
[CrossRef]

Wang, B.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics5(7), 411–415 (2011).
[CrossRef]

Wang, C.

Wang, Y.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics5(7), 411–415 (2011).
[CrossRef]

Westbrook, P. S.

P. S. Westbrook, T. A. Strasser, and T. Erdogan, “In-line polarimeter using blazed fiber gratings,” Photonics Technol. Lett.12(10), 1352–1354 (2000).
[CrossRef]

Wevers, M.

J. Pollet, F. Delport, K. P. F. Janssen, K. Jans, G. Maes, H. Pfeiffer, M. Wevers, and J. Lammertyn, “Fiber optic SPR biosensing of DNA hybridization and DNA-protein interactions,” Biosens. Bioelectron.25(4), 864–869 (2009).
[CrossRef] [PubMed]

Whitehorne, T. J. J.

T. J. J. Whitehorne, J. P. Coyle, A. Mahmood, W. H. Monillas, G. P. A. Yap, and S. T. Barry, “Group 11 amidinates and guanidinates: potential precursors for vapour deposition,” Eur. J. Inorg. Chem.2011(21), 3240–3247 (2011).
[CrossRef]

Wuilpart, M.

Yap, G. P. A.

T. J. J. Whitehorne, J. P. Coyle, A. Mahmood, W. H. Monillas, G. P. A. Yap, and S. T. Barry, “Group 11 amidinates and guanidinates: potential precursors for vapour deposition,” Eur. J. Inorg. Chem.2011(21), 3240–3247 (2011).
[CrossRef]

Zhang, F.

Zhang, H.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics5(7), 411–415 (2011).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

C. Caucheteur, C. Chen, V. Voisin, P. Berini, and J. Albert, “A thin metal sheath lifts the EH to HE degeneracy in the cladding mode refractometric sensitivity of optical fiber sensors,” Appl. Phys. Lett.99(4), 041118 (2011).
[CrossRef]

Biosens. Bioelectron. (1)

J. Pollet, F. Delport, K. P. F. Janssen, K. Jans, G. Maes, H. Pfeiffer, M. Wevers, and J. Lammertyn, “Fiber optic SPR biosensing of DNA hybridization and DNA-protein interactions,” Biosens. Bioelectron.25(4), 864–869 (2009).
[CrossRef] [PubMed]

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The cylindrical Finite Difference Fiber Mode Solver included in FIMMWAVE by Photon Design, http://www.photond.com/products/fimmwave/fimmwave_features_25.htm

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of TFBG coated by gold nanoparticles (the arrows show how the incident core guided light (red) goes through the grating twice, each time coupling light to cladding modes). The light remaining in the core (straight green arrow) returns to the interrogation system.

Fig. 2
Fig. 2

Simulated electrical fields of a high-order cladding mode excited by P- (left) and S-(right) polarized light. The figures represent a small area of 4 by 4 μm2 close to cladding boundary (indicated by a horizontal line at Y = 142.5 μm). All results were calculated with FIMMWAVE.

Fig. 3
Fig. 3

Experimental transmission spectra of 10° TFBG with S- and P-polarized input light in air, inset: detailed spectra from 1550 to 1556 nm.

Fig. 4
Fig. 4

Schematic diagram of gold CVD system and spectral monitoring setup. The optical sensing analyzer includes a scanning laser source with a wavelength range from 1520 to 1570 nm and a synchronized photodetector.

Fig. 5
Fig. 5

TFBG spectral evolutions of during gold nanoparticles deposition under S- (a) and P-polarized (b) light (the color scale represent the amplitude of the resonances). Spectra are acquired every second and temperature-corrected with reference to the Bragg mode shifts.

Fig. 6
Fig. 6

Normalized amplitude (a) and wavelength (b) evolutions versus time of gold deposition under S- and P-polarized light. The vertical dashed lines bound the minimum amplitudes for each polarization, highlighting the simultaneous occurrence of the corresponding wavelength change discontinuities.

Fig. 7
Fig. 7

SEM images of gold nanoparticles deposited on the TFBG surface with different deposition times (with a scale of 500 nm). (a) Beginning, (b) middle, and (c) end of TFBG spectral evolution. (d) Sectional image of gold film in a small area with an inset of coated TFBG sectional view.

Fig. 8
Fig. 8

Reflective transmission spectra and IR-scattering images under S- and P-polarized lights of the coated TFBG with spectral response times of ~20 s (a), (b) and 40 s (c), (d).

Fig. 9
Fig. 9

Simulated intensity distributions of electric fields on single layer gold nanoparticles interacted by S- (a) and P-polarized (b) light in free space. The incident light propagates along Y-axis in S case, while along X-axis in P case.

Equations (2)

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λ Bragg =2Nef f Bragg Λ/cosθ
λ cladding i =(Nef f Bragg i +Nef f cladding i )Λ/cosθ

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