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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 27 — Dec. 19, 2011
  • pp: 26318–26324
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Phase sensitive sensor on plasmonic nanograting structures.

M. Maisonneuve, O. d’Allivy Kelly, A-P. Blanchard-Dionne, S. Patskovsky, and M. Meunier  »View Author Affiliations


Optics Express, Vol. 19, Issue 27, pp. 26318-26324 (2011)
http://dx.doi.org/10.1364/OE.19.026318


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Abstract

In this paper, a concept of phase sensitive sensor based on plasmonic nanograting structures with normal incidence and transmission detection is presented. Performed theoretical modeling enables fabrication of nanostructures with optimal geometry for polarimetric measurements of the phase difference between s- and p- polarized light. High phase resolution of the optical setup (6*10−3 deg.) allows detection of the bulk refractive index with sensitivity equal to 3.8*10−6 RIU. Proposed technique presents a more efficient alternative to the conventional spectral interrogation method of nanoplasmonic-based sensing and could be used for multisensing or imaging applications.

© 2011 OSA

1. Introduction

In this article, we propose to use a phase-sensitive detection approach to biosensing based on plasmonic nanograting structures. We think that existing phase measurement methodology which was shown as very efficient for SPR optical sensing [5

5. S. Patskovsky, M. Meunier, P. N. Prasad, and A. V. Kabashin, “Self-noise-filtering phase-sensitive surface plasmon resonance biosensing,” Opt. Express 18(14), 14353–14358 (2010). [CrossRef] [PubMed]

, 6

6. W.-C. Law, P. Markowicz, K.-T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H.-P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron. 23(5), 627–632 (2007). [CrossRef] [PubMed]

], could be used in combination with stabilized monochromatic laser sources to further improve the sensitivity, selectivity and resolution of plasmonics-based biosensing up to the detection limit of conventional SPR and to develop novel nanoscale plasmonics complexes and architectures.

2. Basic idea and approach

While the proposed methodology could be applied to any types of gases or liquids biosensing, we concentrated our effort, as a proof of concept, on gas sensing (n = 1) by performing both simulations and experiments on rectangular-slits gratings. The proposed structures (Fig. 1a
Fig. 1 a) Schematic representation of the nanograting with period d, ridge size r, slit size s and thickness h; a) Intensity and phase of the light transmitted through the nanoslit array with 600nm period, 450nm ridge size and 100 nm thickness.
.) consist of a h = 100nm thick gold film deposited on a transparent glass substrate (BK7) with a fixed period of Λ = 600nm and a varying ridge size r as an optimization parameter.

As it was shown in the literature [8

8. M. R. Gadsdon, I. R. Hooper, and J. R. Sambles, “Optical resonances on sub-wavelength silver lamellar gratings,” Opt. Express 16(26), 22003–22028 (2008). [CrossRef] [PubMed]

] the interaction of the incident light with subwavelength metallic structures results in an excitation of normal propagating plasmonic modes and coupled and uncoupled modes that are highly localized resonances. Moreover, these resonances may not propagate between slits, are independent of the grating period and are associated with reflection, extinction and transmission enhancement of the light. Using the rigorous couple wave analysis (RCWA) formalism [9

9. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71(7), 811–818 (1981). [CrossRef]

], we performed numerical simulations to obtain the spectral characteristics for phase and intensity of light transmitted through such plasmonic nanostructures . Experimental spectral dependences (Fig. 1b) for an incident light polarization perpendicular to the grating slits contain distinguishing features characterized by a sharp phase jump and a minimum intensity at about 600nm and 900nm attributed to the SP excitations at respectively the air/metal and metal/substrate interfaces. Since sensing occurs at the medium/gold interface, only the spectral range around 600nm is shown later in this work. Figure 2a
Fig. 2 a) Theoretical spectral phase curves for nanograting structures with different ridge sizes; dashed blue line presents a shifted spectral phase curve for r = 450nm resulting from a change in refractive index equal to 3*10−3RIU; b) Experimental spectral phase characteristics.
presents theoretical spectral phase dependences for gratings having a fixed period of Λ = 600nm and varying ridge sizes of r = 300nm, 350nm, 400nm, 450nm and 500nm. Maximising the sensitivity requires to obtain a sharp phase variation around a predetermined working wavelength. As follow from Fig. 2a, the larger is r, the sharper is the phase change. For our system, the optimum condition was achieved for ridge size equal to 450nm at the working wavelength of 632.8nm (He-Ne laser). Here we propose to apply a phase sensing methodology to follow changes of medium refractive index. As an example, after slightly increasing the medium RI by a value equal to 0.003 RIU for sample with r= 450nm, the spectral phase curve is shifting, as shown by the dashed blue line on Fig. 2a. This shift resulted in a corresponding phase change A->B at the fixed working wavelength. Dynamic range of the sensor can be estimated from the absolute phase variation (Δλ on Fig. 2a). It is interesting to point out that the subsequent increase of r to 500nm has rather different impact on phase characteristic, showing opposite spectral trend. Such behaviour is very similar to the surface plasmon phase dependences on the thin metal film thickness in the Kretschmann SPR geometry.

For experimental tests, nanograting arrays with a fixed period of 600nm, ridge sizes ranging from 300nm to 500nm and overall dimension of 150x150µm were produced by Focused Ion Beam (FIB) on a 100nm thick gold metal film (Platypus) deposited on the transparent substrate (glass BK7). Experimental spectral phase curves were measured on an ellipsometer (J.A. Wollam) and are presented on Fig. 2b. Overall tendencies confirm the theoretical predictions while some differences are observed in spectral phase curves positions and absolute phase variation, which are due to the imperfections of the fabricated samples by FIB method. It is anticipated that surface roughness [10

10. D. L. Mills, “Attenuation of surface polaritons by surface roughness,” Phys. Rev. B 12(10), 4036–4046 (1975). [CrossRef]

], grating lines and period precision, slight convergence of the probe light could affect the surface waves propagation parameters especially for smaller slit sizes. Additional irregularities presented on the experimental curves are probably due to the complex nature of the light transmission through nanoplasmonic structures [11

11. L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of Extraordinary Optical Transmission through Subwavelength Hole Arrays,” Phys. Rev. Lett. 86(6), 1114–1117 (2001). [CrossRef] [PubMed]

].

Sensitivity and dynamic range of the proposed phase-sensitive method depends on the sharpness and absolute variation of the spectral phase curves (Fig. 2). Usually these sensor parameters are tuned by the plasmon coupling grating period. Here we show how important is the optimization of the ridge sizes and thickness for the nanogratings based sensor. Figure 3
Fig. 3 Theoretical (solid curves; scale on the left) and experimental data (points; scale on the right) for phase sensitivity dependence on the ridge sizes.
shows the calculated and measured results for sensor phase sensitivity as a function of the ridge width. The obtained lower experimental sensitivity could be improved by performing E-Beam nanofabrication method on larger samples with smoother metal layers.

Optimal grating thickness that corresponds to the maximum spectral phase variation was evaluated theoretically for different ridge sizes (Fig. 4
Fig. 4 Theoretical results for optimal grating thickness and corresponding working wavelength position for different ridge sizes; Inserts: Typical phase curves for marked region.
. black line). Calculated working wavelengths at the position of the phase maximum are shown as red lines on Fig. 4. It is clear that the minima of transmission and large phase changes follow the SP dispersion line (Eq. (1).). We can apply the concept that the optimal resonance condition of surface plasmon excitation is obtained when internal damping Γi is equal to the radiation damping Γrad [12

12. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Berlin, 1988).

] and that the direction of phase variation depends on the relation of these two parameters [13

13. F. Abeles, “Surface electromagnetic waves ellipsometry,” Surf. Sci. 56, 237–251 (1976). [CrossRef]

]. Radiation damping mechanism in our case contains two components. The first one is backscattering by the grating that increases with the grating thickness. The second is excitation of surface plasmon on the air/gold interface, for which enhanced transmission through nanoslits and coupling to the plasmon on the gold/substrate interface induces reemission to the far field. Γi is usually equal to the imaginary part of the propagating surface plasmon wavevector. But, in the case of asymmetrical gratings, the wave which travels parallel to a surface of a medium with a higher refractive index tends to lose energy into that medium and, similarly, the grating lines tend to collect light which initially travels into the grooves. Thus, this effect and the amounts of energy emerging from the grating lines and grooves will affect the internal damping parameter [14

14. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68(9), 1206–1210 (1978). [CrossRef]

]. Using this concept we can explain the behavior of the curve for optimal grating thickness. First, the grating thickness of about 100nm allows compensating Γi [15

15. I. Pockrand, “Resonance anomalies in the light intensity reflected at silver gratings with dielectric coatings,” J. Phys. D Appl. Phys. 9(17), 2423–2432 (1976). [CrossRef]

]. Increasing of the ridge sizes results in the increase of Γi and requires thicker grating to maintain condition Γi = Γrad. The optimal grating thickness of 98nm is found for ridge width 470nm with a working wavelength of 627nm. This wavelength corresponds exactly to the condition of surface plasmon excitation at gold/air interface on the 600nm grating with normal incidence (Eq. 1). A very interesting effect is observed after this point: splitting of the curve for optimal grating thickness and inversion of the phase curve (Fig. 4, inserts). This effect could be explained by the resonance properties of enhanced light transmission through small subwavelength slits. For example, for nanoslits less than 100nm, different Fabry-Perot modes exist that depend on the slit size and depth [8

8. M. R. Gadsdon, I. R. Hooper, and J. R. Sambles, “Optical resonances on sub-wavelength silver lamellar gratings,” Opt. Express 16(26), 22003–22028 (2008). [CrossRef] [PubMed]

,16

16. G. D’Aguanno, N. Mattiucci, M. J. Bloemer, D. de Ceglia, M. A. Vincenti, and A. Alù, “Transmission resonances in plasmonic metallic gratings,” J. Opt. Soc. Am. B 28(2), 253–264 (2011). [CrossRef]

]. Leaking of evanescent surface plasmon also can contribute to the appearance of several optimal thicknesses [11

11. L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of Extraordinary Optical Transmission through Subwavelength Hole Arrays,” Phys. Rev. Lett. 86(6), 1114–1117 (2001). [CrossRef] [PubMed]

,17

17. F. García de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79(4), 1267–1290 (2007). [CrossRef]

]. A deeper, more physical understanding of these phenomena would be desirable, but it takes substantial experimental work and is a subject of a future work.

Due to the nanofabrication technology limitations and the relatively low dynamic range of the phase-sensitive method, coincidence of probing laser light wavelength with the phase curve’s maximum slope is rather difficult experimentally. It is critical to find another parameter to tune the starting point, which is generally the maximum phase slope point. To maximise the sensor performance, we opted to tilt the structure. As it could be observed in Fig. 5
Fig. 5 Experimental phase dependences for different incident angles;
, angular deviation of the sample enables to tune this point by changing the wavelength position of the maximum phase slope. Therefore, by doing angular measurement at fixed wavelength, it is possible to retrieve the starting point of the experiment.

3. Instrumental methodology

An experimental set-up illustrated in Fig. 6
Fig. 6 AFM sample images and optical set-up for phase measurements.
was developed to measure time variations of phase properties in air or liquid medium of our samples. In the proposed polarimetric set-up, spatially filtered light from a 5 mW stabilized He-Ne laser with 632.8 nm wavelength is passed through a Glan-Taylor polarizer to provide a 45° linearly polarized beam. The light is directed to a Photoelastic Modulator (PEM, HINDS I/FS50) after passing through a Soleil -Babinet compensator, which serves to optimize the initial phase retardation. The PEM is used to sinusoidally modulate the phase of the p-component at a frequency of 50 kHz. The beam is then directed through a diaphragm to the nanoplasmonic structure placed on the high precision rotation stage, thus allowing for incident angle fine tuning. All experiments in air or in gases were performed in a specially designed measurements chamber.

The final periodic signal of light transmitted through the tested nanoplasmonic sample and the 45° oriented analyzer is measured by the detector and then decomposed into harmonics using a lock-in amplifier. Since the time domain signal is periodic and continuous, the harmonics of a frequency spectrum could be modeled using a Fourier transform method. Thus, the first two harmonics are given by [6

6. W.-C. Law, P. Markowicz, K.-T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H.-P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron. 23(5), 627–632 (2007). [CrossRef] [PubMed]

]:

F1=2J1(M)RpRssin(α+(ϕpϕs))
(2)
F2=2J2(M)RpRscos(α+(ϕpϕs))
(3)

Here, α is the initial phase retardance introduced by the Soleil-Babinet compensator. Rp, ϕp and Rs, ϕs are reflection and phase under p- and s- polarization, respectively. Jn indicates a Bessel function of order n and M, the modulation amplitude. Phase retardation ϕ = ϕps can be derived from Eq. (2) and (3) and are given by:

ϕ=tan1(F1J2(M)F2J1(M))
(4)

4. Results and discussion

In order to estimate the efficiency of the polarimetric experimental set-up for the proposed nanoplasmonic phase-sensitive methodology, initial calibration tests were performed. In the absence of the sensing block, we measured responses of the first and second harmonic to the small phase retardation introduced by a Soleil-Babinet compensator. Optimal signal-to-noise ratio for the phase measurements (Eq. 4) was obtained at 154° phase modulation depth (M), which provides equal harmonics intensities and therefore optimal conditions for the noise elimination (Fig. 7a
Fig. 7 (a) Phase sensitivity experiment. b) Determination of optimal incident angle.
). Looking at the noise level, the high phase resolution of 6*10−3 deg. confirms the efficiency of our polarimetric set-up for direct real time phase measurements at fixed incident angle.

To test the proposed methodology, we adapted the PEM-based phase measurement system for nanoplasmonic sensing measurements, as shown in the Fig. 6. Nanoplasmonic samples were placed on the rotation stage and the measured initial phase retardation was compensated by a Soleil -Babinet compensator. Experimental phase-angular dependencies serve as calibration procedure to find the optimal tilt position for highest phase sensitivity For example, as seen in Fig. 7b, for a nanograting array with ridge = 450nm, the maximum phase derivative is obtained at 2.88 deg. angle tilt which is later fixed for real time gas sensing.

We then used a well-established gas methodology for small refractive index variations Δn [5

5. S. Patskovsky, M. Meunier, P. N. Prasad, and A. V. Kabashin, “Self-noise-filtering phase-sensitive surface plasmon resonance biosensing,” Opt. Express 18(14), 14353–14358 (2010). [CrossRef] [PubMed]

]. This method involves comparing the system response, while different inert gases with known refractive indices are brought in contact with the gold film. In the experiments, Ar and N2 were used, for which the refractive indices differ by Δn ≅1.5*10−5 RIU under the normal conditions [18

18. E. R. Peckt and B. N. Khanna, “Dispersion of nitrogen,” J. Opt. Soc. Am. 56, 1059–1063 (1966).

, 19

19. A. Michels and A. Botzen, “Refractive index and Lorentz-Lorenz function of argon up to 2300 atmospheres at 25°C,” Physica 15(8-9), 769–773 (1949). [CrossRef]

]. The gases were brought to the cell through a long spiral copper tube to equilibrate their temperatures with the room temperature. A flow meter is used to equalize the pressure to the flow cell in contact with the SPR-supporting nanoplasmonic gold film. Figure 8
Fig. 8 Experimental results for phase under the replacement of pure N2 by Ar.
demonstrates typical phase signals under the replacement of pure N2 by Ar gas. Taking into account the level of residual noises that is equal to 0.06deg. (equal to three times of the mean standard deviation on each step), we can determine that the detection limit of our system is about 3.8*10−6 RIU, which is significantly better compared to conventional spectral-sensitive nanoplasmonic devices. Further improvement of the detection limit could be achieved by optimizing grating parameters, precision of nanofabrication and application of an active thermostabilisation system.

Conclusion

We proposed a dephasing method of SP signal/noise enhancement to improve overall sensitivity of nanoplasmonic structures in bio- and chemical sensing. The proposed method enables to obtain low noise signals with an anticipated resolution of about 3.8*10−6 RIU. Such phase-sensitive approach makes possible the achievement of a relatively low detection limit and is expected to provide substantial advantages in the development of low-cost, portable gas- or biological sensor devices.

Acknowledgment

The authors acknowledge the financial contribution from the NSERC Strategic Network for Bioplasmonic Systems (Biopsys) and NanoQuébec.

References and Links

1.

F. J. Garcia-Vidal, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010). [CrossRef]

2.

L. Guyot, A. P. Blanchard-Dionne, S. Patskovsky, and M. Meunier, “Integrated silicon-based nanoplasmonic sensor,” Opt. Express 19(10), 9962–9967 (2011). [CrossRef] [PubMed]

3.

K. A. Willets and R. P. Van Duyne, “Localized surface plasmon resonance spectroscopy and sensing,” Annu. Rev. Phys. Chem. 58(1), 267–297 (2007). [CrossRef] [PubMed]

4.

M. Vala, K. Chadt, M. Piliarik, and J. Homola, “High-performance compact SPR sensor for multi-analyte sensing,” Sens. Actuators B Chem. 148(2), 544–549 (2010). [CrossRef]

5.

S. Patskovsky, M. Meunier, P. N. Prasad, and A. V. Kabashin, “Self-noise-filtering phase-sensitive surface plasmon resonance biosensing,” Opt. Express 18(14), 14353–14358 (2010). [CrossRef] [PubMed]

6.

W.-C. Law, P. Markowicz, K.-T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H.-P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron. 23(5), 627–632 (2007). [CrossRef] [PubMed]

7.

P. S. Vukusic, G. P. Bryan-Brown, and J. R. Sambles, “Surface plasmon resonance on gratings as a novel means for gas sensing,” Sens. Actuators B Chem. 8(2), 155–160 (1992). [CrossRef]

8.

M. R. Gadsdon, I. R. Hooper, and J. R. Sambles, “Optical resonances on sub-wavelength silver lamellar gratings,” Opt. Express 16(26), 22003–22028 (2008). [CrossRef] [PubMed]

9.

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71(7), 811–818 (1981). [CrossRef]

10.

D. L. Mills, “Attenuation of surface polaritons by surface roughness,” Phys. Rev. B 12(10), 4036–4046 (1975). [CrossRef]

11.

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of Extraordinary Optical Transmission through Subwavelength Hole Arrays,” Phys. Rev. Lett. 86(6), 1114–1117 (2001). [CrossRef] [PubMed]

12.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Berlin, 1988).

13.

F. Abeles, “Surface electromagnetic waves ellipsometry,” Surf. Sci. 56, 237–251 (1976). [CrossRef]

14.

K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68(9), 1206–1210 (1978). [CrossRef]

15.

I. Pockrand, “Resonance anomalies in the light intensity reflected at silver gratings with dielectric coatings,” J. Phys. D Appl. Phys. 9(17), 2423–2432 (1976). [CrossRef]

16.

G. D’Aguanno, N. Mattiucci, M. J. Bloemer, D. de Ceglia, M. A. Vincenti, and A. Alù, “Transmission resonances in plasmonic metallic gratings,” J. Opt. Soc. Am. B 28(2), 253–264 (2011). [CrossRef]

17.

F. García de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79(4), 1267–1290 (2007). [CrossRef]

18.

E. R. Peckt and B. N. Khanna, “Dispersion of nitrogen,” J. Opt. Soc. Am. 56, 1059–1063 (1966).

19.

A. Michels and A. Botzen, “Refractive index and Lorentz-Lorenz function of argon up to 2300 atmospheres at 25°C,” Physica 15(8-9), 769–773 (1949). [CrossRef]

OCIS Codes
(050.2770) Diffraction and gratings : Gratings
(120.5050) Instrumentation, measurement, and metrology : Phase measurement
(240.6680) Optics at surfaces : Surface plasmons

ToC Category:
Sensors

History
Original Manuscript: July 7, 2011
Revised Manuscript: September 15, 2011
Manuscript Accepted: October 6, 2011
Published: December 9, 2011

Virtual Issues
Vol. 7, Iss. 2 Virtual Journal for Biomedical Optics

Citation
M. Maisonneuve, O. d’Allivy Kelly, A-P. Blanchard-Dionne, S. Patskovsky, and M. Meunier, "Phase sensitive sensor on plasmonic nanograting structures.," Opt. Express 19, 26318-26324 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-27-26318


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References

  1. F. J. Garcia-Vidal, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys.82(1), 729–787 (2010). [CrossRef]
  2. L. Guyot, A. P. Blanchard-Dionne, S. Patskovsky, and M. Meunier, “Integrated silicon-based nanoplasmonic sensor,” Opt. Express19(10), 9962–9967 (2011). [CrossRef] [PubMed]
  3. K. A. Willets and R. P. Van Duyne, “Localized surface plasmon resonance spectroscopy and sensing,” Annu. Rev. Phys. Chem.58(1), 267–297 (2007). [CrossRef] [PubMed]
  4. M. Vala, K. Chadt, M. Piliarik, and J. Homola, “High-performance compact SPR sensor for multi-analyte sensing,” Sens. Actuators B Chem.148(2), 544–549 (2010). [CrossRef]
  5. S. Patskovsky, M. Meunier, P. N. Prasad, and A. V. Kabashin, “Self-noise-filtering phase-sensitive surface plasmon resonance biosensing,” Opt. Express18(14), 14353–14358 (2010). [CrossRef] [PubMed]
  6. W.-C. Law, P. Markowicz, K.-T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H.-P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron.23(5), 627–632 (2007). [CrossRef] [PubMed]
  7. P. S. Vukusic, G. P. Bryan-Brown, and J. R. Sambles, “Surface plasmon resonance on gratings as a novel means for gas sensing,” Sens. Actuators B Chem.8(2), 155–160 (1992). [CrossRef]
  8. M. R. Gadsdon, I. R. Hooper, and J. R. Sambles, “Optical resonances on sub-wavelength silver lamellar gratings,” Opt. Express16(26), 22003–22028 (2008). [CrossRef] [PubMed]
  9. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am.71(7), 811–818 (1981). [CrossRef]
  10. D. L. Mills, “Attenuation of surface polaritons by surface roughness,” Phys. Rev. B12(10), 4036–4046 (1975). [CrossRef]
  11. L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of Extraordinary Optical Transmission through Subwavelength Hole Arrays,” Phys. Rev. Lett.86(6), 1114–1117 (2001). [CrossRef] [PubMed]
  12. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Berlin, 1988).
  13. F. Abeles, “Surface electromagnetic waves ellipsometry,” Surf. Sci.56, 237–251 (1976). [CrossRef]
  14. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am.68(9), 1206–1210 (1978). [CrossRef]
  15. I. Pockrand, “Resonance anomalies in the light intensity reflected at silver gratings with dielectric coatings,” J. Phys. D Appl. Phys.9(17), 2423–2432 (1976). [CrossRef]
  16. G. D’Aguanno, N. Mattiucci, M. J. Bloemer, D. de Ceglia, M. A. Vincenti, and A. Alù, “Transmission resonances in plasmonic metallic gratings,” J. Opt. Soc. Am. B28(2), 253–264 (2011). [CrossRef]
  17. F. García de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys.79(4), 1267–1290 (2007). [CrossRef]
  18. E. R. Peckt and B. N. Khanna, “Dispersion of nitrogen,” J. Opt. Soc. Am.56, 1059–1063 (1966).
  19. A. Michels and A. Botzen, “Refractive index and Lorentz-Lorenz function of argon up to 2300 atmospheres at 25°C,” Physica15(8-9), 769–773 (1949). [CrossRef]

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