Sensitivity enhancement of plasmonic grating in the local field

L. L. Frumin; D. A. Shapiro

doi:10.1364/OE.400382

1. Introduction

Among devices developed during the last two decades, the sensors based on the surface plasmon resonance are distinguished (see [1,2], and references therein). A plane wave falls to the boundary between the first medium (the glass) from the glass side. Its incident angle is close to the total internal reflection (TIR). A metallic layer or a plasmon-supporting sub-wavelength grating at the interface makes the reflection coefficient’s dependence on the angle (or the wavelength) extremely sensitive to the refractive index of the second medium (analyte). It is a consequence of an interaction between the electric field of plasmon (an evanescent wave) with the analyte. Its high sensitivity is confirmed by theoretical calculations and experiments (see, e.g., [3,4]).

Many plasmonic sensor practical applications concern the measurement of chemical and biological species in a gas or liquid [5–7]. At the same time, there is an essential problem with device improvement, especially the increase in their sensitivity. The progress of a sensor based on the tilted fiber Bragg grating occurs after coating with chemically synthesized silver. The authors obtained a 3.5-fold increase in sensitivity relative to the uncoated fiber [8]. A thin film of graphene oxide deposited over a 45 nm gold film is applied to enhance the sensitivity of the surface plasmon sensor by 2.5 times [9]. Dielectric subwavelength gratings improve the sensitivity of silicon photonics sensors [10–12], due to stronger interaction of the waveguide mode with the analyte. The singularities of open systems, known as exceptional points, are also exploited as an improvement method [13].

The purpose of the present paper is to treat another way to increase sensitivity. We propose to measure the local field in points where the field enhancement coefficient of a grating being extremely high. For this purpose, we consider the subwavelength periodic sequence of metallic cylinders placed above the dielectric half-space. A grating of this type was studied earlier [14], but with no analysis of sensitivity to the neighbor medium’s refractive index.

We study the dependence of local field enhancement factor on the refractive index. The grating, a periodic set of metallic cylinders, is described in Section 2 along with the boundary element method of computing. Section 3 includes the interpretation of the cusp-like resonance structure appearing in the angular dependence for longer waves. Section 4 contains the results of modeling and their discussion. The angular dependence is presented for gold and silver in gas and liquid, respectively. The last section includes a summary of the study and conclusions.

2. Boundary integral equations

We consider the scattering of a plane wave by a periodic grating consisting of equal circular cylinders with dielectric constant $\varepsilon _3$, Fig. 1. The grating is located in half-space 2 near its plane interface with half-space 1. The dielectric constants of the half-spaces are $\varepsilon _2$, $\varepsilon _1$, respectively. The plane wave incidents from half-space 1 by angle $\theta$ to the negative part of the $y$-axis. We denote the wave vectors in half-space $k_{1,2}=k_0\sqrt {\varepsilon _{1,2}}$, where $k_0$ is the wave vector in free space.

Fig. 1. The sketch of medium 2, above the dielectric subspace 1. The circular cylinders of radius $a$ form the grating with period $d$. The circles mark borders $\Gamma _0,\Gamma _{\pm 1}$. Inset illustrates the polarization state of $p$-wave.

Download Full Size | PDF

Choose axis $z$ of the Cartesian coordinates along the axis of a cylinder. In the case of incident $H$-wave, the solution is determined by the $z$-component of the magnetic field $H_z$. We add together the incident field $H^{0}$ and the scattered field $H'$ to get the total field $H_z=H^{0}_z+H'_z$.

The BEM for a cylinder on a substrate with a special Green function $G_1(p,q),G_2(p,q)$ for two dielectric layers had been introduced in Ref. [14,15]. One integral equation is obtained from Green’s theorem when the point $p$ tends to the border contour from its external side, another is the limit while $p$ comes from the internal domain:

(1)$$\begin{aligned} \frac{u(p)}{2}+\int_{\Gamma}\left[\frac{\partial G_3(p,q)}{\partial n} u(q)-G_3(p,q)f(q)\right]\, dq =\\ =\frac{u_0(p)}{2}+\int_{\Gamma}\left[\frac{\partial G_3(p,q)}{\partial n}u_0(q)-G_3(p,q)\frac{\varepsilon_3}{\varepsilon_2}f_0(q)\right] d q, \end{aligned}$$

where $\Gamma$ is the border contour enclosing the corresponding domain, index $0$ denotes the incident field $u_0=H^{0}_z$ and its normal derivative $f_0={\partial H^{0}_z}/{\partial n}$. There are two unknown functions: the scattered field $u=H'_z$ and its normal derivative $f= {\partial H'_z}/{\partial n}$ in boundary point $p=(x,y)$. In the internal domain 3 we use the Green function as a divergent cylindrical wave

G_3(\mathbf{r},\mathbf{r}')=\frac 1{4i}\mathscr{H}_0^{(1)}(k_3|\mathbf{r}-\mathbf{r}'|),

where $\mathscr {H}_0^{(1)}$ is the Hankel function of the first kind and zero order.

There is a consequence of Floquet — Bloch theory of equations with periodic coefficients:

(2)$$u(x_m,y_m)=u(x_0+m d,y_0)=u(x_0,y_0)e^{i k_t m d},$$

where $k_t=k_2 \sin \theta$ is the tangential component of vector $\mathbf {k}_2$, the wave vector in half-space 2, $m=0,\pm 1,\dots$ is the number of cylinder, $p=(x_0,y_0)$ is a point at contour $\Gamma _0$. This property is valid also for the normal derivative $f$ of the magnetic field. When we translate the point by a period, the solution acquires a phase shift.

In the infinite periodic system for calculating the field in arbitrary point, it is sufficient to know functions $u(q),f(q)$ along only one contour, e.g., $m=0$. We find other contributions from the periodicity. Taking into account property (2) we write the external integral equation in the form

(3)$$-\frac{u(p)}{2}+\int_{\Gamma_0}\sum_{m=-\infty}^{+\infty} \left[\frac{\partial G_2(q)}{\partial n} u(q)-G_2(q)f(q)\right]e^{i k_t m d} d q=0,$$

where the first argument $p$ of the Green function $G_2(p,q)$ is omitted for the brevity’s sake.

Let us consider sum

(4)$${G}_2(p,q)=\sum_{m=-\infty}^{+\infty}G_2(p,q_m)e^{i k_t m d}=\sum_{m=-\infty}^{+\infty}G_2(x,y,x_0+m d,y_0)e^{i k_t m d}.$$

Changing the order of summation and integration we get:

(5)$$\begin{aligned} {G}_2(p,q)=-\int_{-\infty}^{\infty}\frac{e^{-\mu_2|y-y_0|+ik(x-x_0) }}{4\pi\mu_2}d k \sum_{m=-\infty}^{+\infty}e^{-i k m d +i k_t m d}\\ -\int_{-\infty}^{\infty}\rho(k) \frac{e^{-\mu_2(y+y_0)+ik(x-x_0) }}{4\pi\mu_2}d k \sum_{m=-\infty}^{+\infty}e^{-i k m d +i k_t m d},\\ \rho(k)=\frac{\varepsilon_1\mu_{2}-\varepsilon_2\mu_{1}}{\varepsilon_1\mu_{2}+\varepsilon_2\mu_{1}},\quad \mu_{1,2}=\sqrt{k^{2}-k^{2}_{1,2}}. \end{aligned}$$

The sum over $m$ gives the periodic $\delta$-function

(6)$$\sum_{m=-\infty}^{+\infty}e^{-i k m d +i k_t m d}=\frac{2\pi}{d}\sum_{n=-\infty}^{\infty}\delta\left( k-k_t-n k_d\right),$$

were $k_d=2\pi /d$ is the reciprocal lattice vector. Substituting (6) into Eq. (5) we denote

(7)$$\begin{aligned} \tilde{G}_2(p,q)=-\sum_{n=-\infty}^{+\infty}\frac{1}{2d \mu_{2n}}\bigl[e^{-\mu_{2n}|y-y_0|+i(k_t+k_d n)(x-x_0)}\\ +\rho_n e^{-\mu_{2n}(y+y_0)+i(k_t+k_d n)(x-x_0)}\bigr].\end{aligned}$$

Here $\rho _n,\mu _{jn}$ are the mirror image source $\rho (k)$ and eigenvalue $\mu _j(k), j=1,2$ at $k=k_t+nk_d$.

Function $\tilde {G}_2(p,q)$ is the effective Green function for the periodic lattice. Expression (7) is valid in half-space 2. A similar Green function exists for the half-space 1 as well. With the help of (7), Eq. (3) can be rewritten in the form

(8)$$-\frac{u(p)}{2}+\int_{\Gamma_0}\left(\frac{\partial \tilde{G}_2(p,q)}{\partial n} u(q)-\tilde{G}_2(p,q)f(q)\right)\,d q.$$

In the course of numerical solution, a pair of integral Eqs. (8) and (1) are replaced by a system of linear algebraic equations whose solution determines the grid functions of the scattered field $u$ and its normal derivative $f$ along selected path $\Gamma _0$. The grating-scattered field in arbitrary point $r$ of the half-space 2 can be calculated with using Green’s integral formula

(9)$$u(r)=\int_{\Gamma_0}\left(\frac{\partial \tilde{G}_2(r,q)}{\partial n} u(q)-\tilde{G}_2(r,q)f(q)\right)\,dq.$$

Thus, the periodicity effectively reduces the scattering problem to an elementary cell with one cylinder. Calculating $\tilde {G}_2$, we sum a finite number of terms (7) and asymptotically estimate the remainder. High efficiency of calculations is provided by the second-order approximation accuracy of both Green’s integrals and the contours.

3. Plasmon resonance near TIR

We find specific plasmon resonances in the grating of metallic cylinders located near the interface between a dielectric and free space [14]. The resonances are excited in the gaps by the evanescent wave. The height of resonance depends on the incidence angle. The maximum value of field enhancement factor (FEF) appears in the spectral intervals corresponding to minimal absorption in the metal, i.e., the small values of permittivity imaginary part $\varepsilon ''$. Maximal FEF takes place at the incidence angle $\theta$ exceeding the TIR angle $\theta _0=\arcsin (n_a/n)$ at the boundary between dielectric $\varepsilon _1$ and free space $\varepsilon _2=1$, where $n=\sqrt {\varepsilon _1}$ is the refractive index of the dielectric. In this domain, the transmitted wave occurs evanescent. Its Pointing vector directs along $x$, and then the surface wave could excite the resonance modes of the grating.

We study the resonances in the near IR spectrum. Figure 2 shows the angular dependence of plasmon intensity in the middle of neighbor cylinders within interval $\theta =0.72-0.74$. The calculation has been performed for gold cylinders with period $d=0.11~\mu$m. We use optical constants from the handbook by Palik [16]. As indicated in the picture, for shorter waves, the peak is relatively broadened and smooth. For longer waves, the peak shifts towards the smaller angles, and its maximum tends to the TIR angle. We see the break of a curve, the jump of the first derivative at this angle. This cusp becomes sharper with increasing wavelength. For $\lambda =0.7749, 0.8266~\mu$m, and for higher wavelength, the maximum is located at the cusp. The sensitivity of curves to the refractive index is a consequence of this very sharp cusp.

We interpret the cusp in terms of the electric field in half-space 2. When the grating is absent, the field is described by the Fresnel formulas [17]. Figure 3 shows the angular dependence of FEF in medium 2 and obtained by the Fresnel formulas. Hereafter we call it the Fresnel field. An evanescent wave induces dipoles in the cylinders. The resonance in the gap arises from the $x$-components created by all the induced dipoles. The losses because of radiation and dissipation processes in cylinders determine the quality factor. We emphasize that in plasmon resonance, only the $x$-component of the field is amplified. The question is, what is the origin of the $x$-component of the field arising in the gap, whereas only $y$-projection of the Fresnel field exists under the TIR conditions. While we consider the field of induced dipoles, the $x$-component in the center turns to zero at the TIR condition, because the induced dipoles have the only $y$-part and they do not create fields along the $x$ axis. To interpret the $x$ component, we have to consider not only the dipoles induced in the cylinders but also the dipoles of the mirror image source. They are oriented along $y$, but their field has both $x$- and $y$- components in the gap. The $x$-component of neighboring image dipoles compensate each other due to symmetry, of course, if we do not take into account the phase difference of these dipoles. Therefore, with increasing wavelength, when the phase difference decreases, the resonance intensity falls under the TIR conditions. Figure 4 confirms the falling dependence.

Fig. 2. FEF $F_1=|E_x/E_0|^{2}$ as a function of $\theta$ (rad) near TIR angle $\theta _0=0.7297$ for gold cylinders $a=0.05~\mu$m at the wavelength $\lambda =0.688$ (dashed line), $0.7293$ (dot-dashed), $0.7749$ (solid), $0.8266~\mu$m (long dashes). The gap between neighbor cylinders and between a cylinder and the interface is $0.1~\mu$m. The tangential component of the electric field is normalized by the amplitude of the incident waves.

Download Full Size | PDF

Fig. 3. Angular dependence of Fresnel field near the plane normalized by the incident field: $F_1=|E_x/E_0|^{2}$ (solid), $F_2=|E_y/E_0|^{2}$ (dashed line).

Download Full Size | PDF

Fig. 4. The same as Fig. 2 but for silver at the wavelength $\lambda =0.8266$ (dashed line), $0.8856$ (dot-dashed), $0.9537$ (solid), $1.033~\mu$m (long dashes).

Download Full Size | PDF

4. Results and discussions

Consider an analyte in half-space 2 with $\varepsilon _2=n_a^{2}$, where $n_a$ is its refractive index. The sharp resonance allows one to increase the sensitivity of refractive index measurement, registering the local field near the central point between the adjacent cylinders. The extensive numerical study has been performed, including calculating the local field angular dependence for gold and silver gratings near the interface with glass for a gaseous or liquid analyte. As the study show, the cusps are very sensitive to tiny variations of the analyte refractive index. Some typical results of numerical modeling are shown in Fig. 5 and Fig. 6.

Fig. 5. Angular dependence of the local field in the gap for a gold lattice above Crown glass ($n=1.5$) at $\lambda =0.7749$ (solid line), $\lambda =0.8266$ (dashed), $\lambda =0.8856~\mu$m (dot-dashed): $n_a=1.00$ at the left, $n_a=1.01$ at the right.

Download Full Size | PDF

Fig. 6. The same as in Fig. 5, but for silver lattice in water at $\lambda =0.9537$ (solid line), $\lambda =1.033$ (dashed), $\lambda =1.127~\mu$m (dot-dashed): $n_a=1.3270, 1.3265, 1.3251$, respectively, for the left plots, $n$ is greater by $\delta n = 0.001n$ for the right curves.

Download Full Size | PDF

Figure 5 demonstrates the comparison of angular dependence of gas when the refractive index changes by 1% ($\delta n=0.01n$) for different wavelengths. We see a significant shift of resonances after changing the refractive index of medium 2 by 1% only. The location of peaks correspond to angles $\theta _0=0.7297$ ($n=1$) or $\theta _0=0.7342$ ($n=1.01$). Figure 6 occurs more indicative. It shows the variation of the angular dependence of the local plasmon field for silver lattice while the index changes by $\delta n=0.001n$. We take the data from the Polyanskiy database [18] for different wavelengths. The refractive index is $n_a=1.3270, 1.3265, 1.3251$, TIR angles are $1.0858,1.0851,1.0831$, respectively. For refractive index greater by $0.1$% they are $1.0877,1.0870,1.0850$. The positions of peaks coincide with the listed values.

To calculate fields of nanowires lying on the glass, we use the Floquet — Bloch theory. We reduce the problem to a calculation for one cylinder in an elementary cell. A modified BEM scheme was used based on the Green function that satisfies the boundary conditions between the dielectric and analyte. It automatically takes into account the dielectric-analyte interface, without considering its discretizing. The calculations cover a wide range of geometric lattice, dielectric, and analyte parameters. In the course of numerical calculations, we study the geometry and location effects. Geometry effects as a whole are entirely predictable. There is no considerable effect of the slit between the glass and lattice. However, the peak grows with decreasing the gap between wires.

There are two essential elements in the model: the Floquet theorem and two-dimensional geometry. Applicability of the Floquet theorem consists of linearity and periodicity. For high-intensity laser radiation, the nonlinear effects influence plasmon resonances. The finite number of wires also breaks down the validity of the theory. Using the cylinders of a limited length transforms the two-dimensional approach to an approximation.

We choose larger sizes for calculations since nanosized lattices are technologically much harder to manufacture. At the same time, note that the researchers are continuously improving the manufacturing technology [19]. The optical properties of nanosized metal grains differ from that of individual particles even in dimers [20,21]. An external field in narrow gaps between grains excites gap plasmon resonance, which is especially powerful in the case of a nano-grain lattice when the gap modes interact with plasmon grain modes [22]. The waveguide modes in an array of closely spaced metal cylinders and their dimers are comprehensively studied [23–25]. The enhanced local field of this resonance contains much more information about the environment than transmitted or reflected waves [26]. High FEF values achieved in gaps in nanoscale gratings are of significant interest for modern sensors. The near-field optical microscopy [27] should help to realize the detection of local fields.

Using the total internal reflection in the Kretschman scheme makes it possible to control the gap plasmons’ excitation by changing the angle $\theta$. The plasmons are efficiently excited by the evanescent wave penetrating the analyte. The enhancement factor reaches its maximum at $\theta$ close to the TIR angle $\theta _0$. For longer waves, especially in near IR, the peak tends to $\theta _0$. The shape of resonance changes dramatically at $\theta =\theta _0$ since the field undergoes realignment: the amplitude of $x$-component turns to zero, and $y$-part, on the contrary, reaches its maximum. The maximal value several times exceeds the amplitude of the incident wave. The angular dependence of the $y$-part satisfying the Fresnel formulas and includes a sharp peak. The position of the cusp is sensitive to the analyte properties and allows one to reveal the distinction in its refractive index at the level of 0.1%.

5. Conclusions

We suggest applying the lattice of parallel metallic cylinders on the glass substrate as a sensitive element for optical refractive index measurements. It is helpful to measure the field between neighbor cylinders. This signal is strong owing to the gap plasmon amplification. Moreover, as shown in the present paper, its angular dependence has a very sharp cusp, which is a consequence of the total internal reflection. We have found a significant shift of resonance at a small variation in the refractive index of an analyte. A small-angle tail of the cusp occurs especially steep, and its derivative tends to the infinity. Both factors are essential for further potential exploiting subwavelength metallic gratings in chemo- and bio-sensing techniques.

Funding

Russian Foundation for Basic Research (20-02-00211).

Acknowledgment

We are grateful to V. P. Drachev and S. V. Perminov for helpful discussions.

Disclosures

The authors declare no conflicts of interest.

References

1. J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. 108(2), 462–493 (2008). [CrossRef]

2. A. Shalabney and I. Abdulhalim, “Sensitivity-enhancement methods for surface plasmon sensors,” Laser Photonics Rev. 5(4), 571–606 (2011). [CrossRef]

3. K. Kim, S. J. Yoon, and D. Kim, “Nanowire-based enhancement of localized surface plasmon resonance for highly sensitive detection: a theoretical study,” Opt. Express 14(25), 12419–12431 (2006). [CrossRef]

4. M. Piliarik, H. Šípová, P. Kvasnička, N. Galler, J. R. Krenn, and J. Homola, “High-resolution biosensor based on localized surface plasmons,” Opt. Express 20(1), 672–680 (2012). [CrossRef]

5. Y. Xu, P. Bai, X. Zhou, Y. Akimov, C. E. Png, L.-K. Ang, W. Knoll, and L. Wu, “Optical refractive index sensors with plasmonic and photonic structures: Promising and inconvenient truth,” Adv. Opt. Mater. 7(9), 1801433 (2019). [CrossRef]

6. Y. Zhao, R. Tong, F. Xia, and Y. Peng, “Current status of optical fiber biosensor based on surface plasmon resonance,” Biosens. Bioelectron. 142, 111505 (2019). [CrossRef]

7. T. Allsop and R. Neal, “A review: Evolution and diversity of optical fibre plasmonic sensors,” Sensors 19(22), 4874 (2019). [CrossRef]

8. A. Bialiayeu, A. Bottomley, D. Prezgot, A. Ianoul, and J. Albert, “Plasmon-enhanced refractometry using silver nanowire coatings on tilted fibre Bragg gratings,” Nanotechnology 23(44), 444012 (2012). [CrossRef]

9. P. T. Arasu, A. S. M. Noor, A. A. Shabaneh, M. H. Yaacob, H. N. Lim, and M. A. Mahdi, “Fiber Bragg grating assisted surface plasmon resonance sensor with graphene oxide sensing layer,” Opt. Commun. 380, 260–266 (2016). [CrossRef]

10. J. G. Wangüemert-Pérez, A. Hadij ElHouati, A. Sánchez-Postigo, J. Leuermann, D.-X. Xu, P. Cheben, A. Ortega-Moñux, R. Halir, and Í. Molina-Fernández, “Subwavelength structures for silicon photonics biosensing,” Opt. Laser Technol. 109, 437–448 (2019). [CrossRef]

11. M. Odeh, K. Twayana, K. Sloyan, J. E. Villegas, S. Chandran, and M. S. Dahlem, “Mode sensitivity analysis of subwavelength grating slot waveguides,” IEEE Photonics J. 11(5), 1–10 (2019). [CrossRef]

12. E. Luan, K. M. Awan, K. C. Cheung, and L. Chrostowski, “High-performance sub-wavelength grating-based resonator sensors with substrate overetch,” Opt. Lett. 44(24), 5981–5984 (2019). [CrossRef]

13. J. Park, A. Ndao, W. Cai, L. Hsu, K. A. T. Lepetit, Y.-H. Lo, and B. Kanté, “Symmetry-breaking-induced plasmonic exceptional points and nanoscale sensing,” Nat. Phys. 16(4), 462–468 (2020). [CrossRef]

14. L. L. Frumin, A. V. Nemykin, S. V. Perminov, and D. A. Shapiro, “Plasmons excited by an evanescent wave in a periodic array of nanowires,” J. Opt. 15(8), 085002 (2013). [CrossRef]

15. O. V. Belai, L. L. Frumin, S. V. Perminov, and D. A. Shapiro, “Scattering of evanescent electromagnetic waves by cylinder near flat boundary: the Green function and fast numerical method,” Opt. Lett. 36(6), 954–956 (2011). [CrossRef]

16. E. D. Palik, ed., Handbook of optical constants of solids, vol. 3 (Academic, 1998).

17. L. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media (Pergamon, 1984).

18. M. N. Polyanskiy, “Refractive index database,” https://refractiveindex.info. Accessed on 2020-05-19.

19. Y.-F. Chou Chau, K.-H. Chen, H.-P. Chiang, C. M. Lim, H. J. Huang, C.-H. Lai, and N. Kumara, “Fabrication and characterization of a metallic–dielectric nanorod array by nanosphere lithography for plasmonic sensing application,” Nanomaterials 9(12), 1691 (2019). [CrossRef]

20. P. E. Vorobev, “Electric field enhancement between two parallel cylinders due to plasmonic resonance,” J. Exp. Theor. Phys. 110(2), 193–198 (2010). [CrossRef]

21. V. E. Babicheva, S. S. Vergeles, P. E. Vorobev, and S. Burger, “Localized surface plasmon modes in a system of two interacting metallic cylinders,” J. Opt. Soc. Am. B 29(6), 1263–1269 (2012). [CrossRef]

22. A. Manjavacas and F. J. Garciá de Abajo, “Robust plasmon waveguides in strongly interacting nanowire arrays,” Nano Lett. 9(4), 1285–1289 (2009). [CrossRef]

23. A. Ivanov, A. Shalygin, V. Lebedev, P. Vorobev, S. Vergiles, and A. K. Sarychev, “Plasmonic extraordinary transmittance in array of metal nanorods,” Appl. Phys. A 107(1), 17–21 (2012). [CrossRef]

24. S. Belan and S. Vergeles, “Plasmon mode propagation in array of closely spaced metallic cylinders,” Opt. Mater. Express 5(1), 130–141 (2015). [CrossRef]

25. S. Belan, V. Parfenyev, and S. S. Vergeles, “Negative-angle refraction and reflection of visible light with a planar array of silver dimers,” Opt. Mater. Express 5(12), 2843–2848 (2015). [CrossRef]

26. C. J. Alleyne, A. G. Kirk, R. C. McPhedran, N.-A. P. Nicorovici, and D. Maystre, “Enhanced SPR sensitivity using periodic metallic structures,” Opt. Express 15(13), 8163–8169 (2007). [CrossRef]

27. B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: Fundamentals and applications,” J. Chem. Phys. 112(18), 7761–7774 (2000). [CrossRef]

Sensitivity enhancement of plasmonic grating in the local field

Abstract

1. Introduction

2. Boundary integral equations

3. Plasmon resonance near TIR

4. Results and discussions

5. Conclusions

Funding

Acknowledgment

Disclosures

References

Cited By

Figures (6)

Equations (10)

Optics Express