Abstract

Symmetric metal-dielectric guided-mode resonators (GMR) can operate as infrared band-pass filters, thanks to high-transmission resonant peaks and good rejection ratio. Starting from matrix formalism, we show that the behavior of the system can be described by a two-mode model. This model reduces to a scalar formula and the GMR is described as the combination of two independent Fabry-Perot resonators. The formalism has then been applied to the case of asymmetric GMR, in order to restore the properties of the symmetric system. This result allows designing GMR-on-substrate as efficient as free-standing systems, the same high transmission maximum value and high quality factor being conserved.

© 2015 Optical Society of America

1. Introduction

Metal-based nanostructured thin films have received a growing interest since the first theoretical and experimental evidences of extraordinary transmission [1–3]. This phenomenon originates from the coupling of the incident light to surface plasma waves and has been applied to various fields of research like microfluidic, photovoltaic devices [4–7]. Metallic gratings with narrow slits have been shown to exhibit high transmission [8] and can be used as an alternative to thin films for infrared band-pass filters, especially in the case of multispectral imaging [9]. More recently, metal-dielectric guided mode resonances (GMR) filters have been introduced [10]. They permit to design unpolarized filters [11, 12] or high angular tolerance filters [13] with an improved rejection ratio away from resonance. Nonetheless, one of the main drawback of these devices is that they are based on membranes, which complicates their fabrication and decreases their mechanical robustness. One solution consists in using a substrate [14]. However, in most of the cases, the introduction of a dissymetry between the top and the bottom interfaces of these filters is to the detriment of its performances. Likewise, for lossless Fabry-Perot filters, 100% transmission are only reached for symmetric filters.

In this article, we develop a two modes model that accurately accounts for the various resonance peaks of metal-dielectric GMR filters, highlighting the paramount influence of the 0th and ±1st orders in the resonance mechanism. This model is further simplified to give a scalar expression of the transmitted intensity, and the GMR can be described by two coupled resonators. By analogy with a Fabry-Perot resonator, a symmetrisation criterion is formulated thanks to the scalar expression for either one of the coupled states. We show in the case of a GMR on substrate filter, that it is possible to satisfy this criterion for the first resonator and to restore the same performances than the symmetric filter.

First, we introduce the matrix formalism to compute the transmission spectra of the metal-dielectric GMR filters, and show that it can be reduced to a two modes model. It can be further simplified to a two terms scalar expression for the transmission. In the Sec. 3, we use this expression to formulate a symmetrisation criterion and apply it to various cases of asymmetric GMR filters.

2. Matrix formalism

GMRs are based on the coupling of the incident light to a waveguide via diffracted orders created by a periodic grating [15, 16]. In particular ref. [11] shows that the GMR is due to the coupling of the ±1st orders diffracted by a metallic grating to the dielectric layer guided-modes. The metal-dielectric GMR structure we consider consists in two identical one-dimensional gold gratings with a silicon carbide (SiC) layer in-between, as depicted on Fig. 1(a). The choice of the geometrical parameters stems from the procedure that was used in refs. [11] and [13] to obtain a transmission peak at 4.65μm. With the following values (grating period d = 2 μm, SiC layer thickness td = 700 nm, metal thickness tm = 100 nm and slit width a = 250 nm), we expect a high internal reflectivity of the metallic interfaces for TM-polarized light (with the magnetic field Hy parallel to the slits) [17,18], and no diffracted order in the 3 to 6 μm spectral range. The thickness td is large enough to prevent any coupling of the evanescent waves from the two sides, including when the surface plasmon-polaritons are excited at wavelength close to dnd ≈ 5.7μm, where nd the refractive index of the waveguide (2.84 for SiC).

 figure: Fig. 1

Fig. 1 Optical response of a metal-dielectric GMR filter. (a) Schematic of the freestanding GMR filter (surrounded by air). The geometrical parameters are: grating period d = 2 μm, SiC layer thickness td = 700 nm, metal thickness tm = 100 nm and slit width a = 250 nm. (b) Transmission spectra at normal incidence of the metal-dielectric structure considering all the diffracted orders (solid line) and considering only the three first diffracted orders (dashed line). (c) Definition of the S-matrices used for the transmission spectrum calculations. Contour plots of the magnetic field distributions of the overall field Hysum and the field contributions Hy0, Hy±1, Hy0,±1 and Hyevan.: (d) at λa = 4.65 μm and (e) at λb = 4.04 μm.

Download Full Size | PPT Slide | PDF

2.1. Resonance analysis

A first transmission spectrum is calculated with the Rigorous Maxwell Constitutive Approximation (RMCA) method [19] considering all the diffracted orders using N mesh points (N ≈ 100), a SiC permittivity fixed at εd = 2.842 and a dielectric function of gold given by a Drude model : εm(λ) = 1 − [(λp + )λp]−1 with λp = 159 nm and γ = 0.0048 [20, 21]. The full-calculation spectrum on Fig. 1(b) exhibits two transmission peaks at λa = 4.65 μm and λb = 4.04 μm. The resonance ”b” has a very low intensity compared to the resonance ”a” with a transmission up to 80%.

A second transmission spectrum is calculated by considering a smaller number of diffracted orders m, therefore the size of the S-matrices depicted on Fig. 1(c) is reduced. By using the scattering matrix formalism, the transmission amplitude S matrix of the all structure is written

S31=S32P1/2(k=0P1/2S212PS232P1/2)k)P1/2S21,
or assuming that the geometrical serie converges
S31=S32P1/2(IP1/2S212PS232P1/2)1P1/2S21,
with I being the m × m identity matrix. The matrices S32 and S21 describe the transmission of the diffracted orders through the metallic gratings. The order 0 is the only one considered in this calculation, as the only order propagating in the incoming medium 1 and the outcoming medium 3. Thus, the matrice S32 is indeed a 1 × m row vector, and the matrice S21 a m × 1 column vector. The matrices S232 and S212 are m × m square matrices describing the internal scattering of the orders propagating in the medium 2 and reflecting onto the metallic mirrors. P = (pij) is diagonal with its components pjj=eikz(j)td and 1 ≤ jm; for example kz(0), kz(1) et kz(2) are the propagation constants of the orders 0, +1 and −1. Finally, the transmission for the all structure is T = |S31|2.

From now on, we consider the case of normal incidence, for which orders +1 and −1 are excited symmetrically. The truncated response considering only the orders 0, +1 and −1 corresponds to a 2 × 2 matrix (m = 2), and is plotted (dotted line) on Fig. 1(b). It shows a very good agreement with the full calculation. In this approximation, S21 and S32 are considered as column and row vectors with two components, and P, S232 and S212 are 2 × 2 matrices.

On Fig. 1(b), the very good agreement between the exact response and the two-mode model spectrum on the 3.5 – 6 μm wavelength range demonstrates that only the propagating orders play a significant role in the optical response, while evanescent orders have a low influence. In fact, since only the three orders 0, +1 and −1 are propagating, all the propagation factors for j ≥ 3 tend to 0. Accordingly, the discrepancy below 3.5 μm between the resonance peaks can be corrected by taking the ±2nd diffracted orders into account in our simplified model. Figures 1(d) and 1(e) represent the magnetic field distribution over the GMR structure at λa and λb. The overall magnetic field Hysum is compared to the magnetic field contributions Hy0, Hy±1 and Hyevan. of the 0th order, the ±1st order and the evanescent orders. We also plot the magnetic field contribution Hy0,±1 which corresponds to the interference between the 0th and ±1st orders ( Hy0,±1 is given by the cross-coupling term appearing when developing the sum |Hy0+Hy±1|). At λb, the prevalence of the 0th order is due to the Fabry-Perot resonance and therefore explains the intensity distribution of Hysum: the x–invariant Hy0 field is slightly perturbed by the d-periodic Hy±1 field. At λa, the ±1st diffracted orders dominate. This is consistent with the observation of a d/2-periodic field Hy±1 of the standing-wave resulting from the interference of the d-periodic orders +1 and −1. Nonetheless, for the resonance ”a” we observe that evanescent orders are quite intense. We can still neglect them because we do not consider their propagation in the SiC layer but take them into account inside the transmission and reflection matrix coefficients. This might yet explain the small difference we see between the two spectra on Fig. 1(a).

2.2. Scalar expression for the transmission

The loop matrix M = P1/2S212PS232P1/2 is introduced. It refers to all the multiple reflections onto the metallic gratings and the propagation inside the SiC layer of the 0th and ±1st diffracted orders. Diagonalizing M allows to write M = ΠDΠ−1 with D=(da00db) (Π is a transfer matrix). Finally the Eq, (2) is equivalent to

S31=S32P1/2Π(ID)1Π1P1/2S21.
Then setting down Π=(Π0aΠ0bΠ1aΠ1b), Π1=(Πa0Πa1Πb0Πb1), S32=(s0βs1β), S21=(s0αs1α) and P1/2=(p01/200p11/2) (with p0=eikz(0)td, p1=eikz(±1)td) and developing the Eq. (3), we obtain a scalar expression for the transmission amplitude
S31=(Π0ap01/2s0β+Π1ap11/2s1β)11da(Πa0p01/2s0α+Πa1p11/2s1α)+(Π0bp01/2s0β+Π1bp11/2s1β)11db(Πb0p01/2s0α+Πb1p11/2s1α)
Figure 2(a) shows that the transmission spectrum can be divided in two parts in agreement with the two-term scalar expression found for the transmission intensity
T=|S31|2Ta+Tb,
where Ta=|Π0ap01/2s0β+Π1ap11/2s1β(Πa0p01/2s0α+Πa1p11/2s1α)/(1da)|2 and Tb=|Π0bp01/2s0β+Π1bp11/2s1β(Πb0p01/2s0α+Πb1p11/2s1α)/(1db)|2. Ta approximates well the intensity spectrum around λa and Tb around λb. We also notice that the transmission peaks appear at λa when arg(da) ≡ 0[2π] and at λb when arg(db) ≡ 0[2π] (cf Fig. 2(b)). Therefore the 0th and ±1st orders can be viewed as coupled resonators that interfere constructively in the GMR structure when arg(da) ≡ 0[2π] or arg(db) ≡ 0[2π].

 figure: Fig. 2

Fig. 2 Scalar expression for the transmitted intensity. (a) Plot of the scalar expression using Eq. (4) (black dashed line) separated in two terms: term proportional to 1/1 − da (light blue line) and term proportional to 1/1 − db (red line). (b) Plot of the two complex eigenvalues da and db of the loop matrix M.

Download Full Size | PPT Slide | PDF

3. Symmetrisation of guided-mode resonance filters

3.1. Symmetry criterion

We want to study the effect of symmetry on the resonance ”a”. Although the Fabry-Perot peak corresponds to a single-mode resonance associated to a wavevector kz, we note two analogies between the Fabry-Perot resonance and the GMR resonance. (i) Considering the expression Ta we note that it is analogous to the classical expression |t32t21eikztd/(1 − r212r232e2ikztd)|2 for the intensity transmitted through a Fabry-Perot cavity of thickness td as a function of the transmission and reflection Fresnel coefficients t32, t21, r212 and r232. (ii) Fabry-Perot resonances occur similarly when arg(r212r232e2ikztd) ≡ 0[2π]. For a Fabry-Perot cavity, this condition defines similarly a resonance round trip.

The eigenvalue da of the matrix M also turns to be the eigenvalue of (P1/2S212P1/2)(P21/2S232P21/2), P1/2 and P21/2 being seen as the propagation matrices related to two dielectric layers of thicknesses t and t2, given that td = t + t2. Finally, as a symmetric Fabry-Perot cavity is defined by r212eikztd = r232eikztd, we introduce the eigenvalues d212 and d232 of the matrices P1/2S212P1/2 and P21/2S232P21/2. A symmetry criterion is formulated, the necessary condition to obtain a symmetric filter being: d212 = d232.

3.2. Symmetrisation results and discussions

Figure 3(a) presents three different freestanding GMR structures A, B and C. The structure A is the one originally shown on Fig. 1(a): it is made of two identical periodic gold gratings with a SiC layer in-between. The structure B only differs from A because the slits of the lower grating are filled with SiC. The structure B is therefore not symmetric, and we observe that d212d232 (on Fig. 3(b), solid lines refer to d212 and dotted lines to d232). To restore the equality, three degrees of freedom are available: the thickness t2 of the dielectric waveguide lower part and the slit width a2 and metal thickness tm2 of the lower grating. In theory, three parameters are more than enough to make the complex d232 equal to d212 at the resonance wavelength. For the structure C, the equalization is even obtained on a large wavelength range. As a consequence, the slopes of the arguments of d212 and d232 also coincide, which allows to restore the same quality factor.

 figure: Fig. 3

Fig. 3 Validation of the symmetry criterion. (a) Schematic of three freestanding GMR structures A, B and C. For the structures A and B, the gold grating period is d = 2 μm, the SiC layer thickness is td = 700 nm, the gold grating slit width is a = 250 nm and the gold grating thickness is tm = 100 nm. For the structure C, the gold grating period is d = 2 μm; the SiC layer is divided in two parts of thickness t = 350 nm and t2 = 320 nm; the gold gratings are different, the upper one having a thickness tm = 100 nm and a slit width a = 250 nm, and the lower one a thickness tm2 = 60 nm and slit width a2 = 350 nm. (b) Modules and arguments of coefficients d212 and d232 corresponding to the structures A, B and C. (c) Transmission spectra at normal incidence of the structures A, B and C calculated using Eq. (2).

Download Full Size | PPT Slide | PDF

To derive an expression for the quality factor, we consider the denominator of the scalar expression Ta. We introduce the length α by setting down: da = |da|eiδkaα where δka = k0ka with k0 = 2π/λ and ka = 2π/λa. Consecutively around the resonance wavelength λa, |1da|2(1|da|)2+|da|δka2α2. The quality factor Q being defined as Q = k0/2ka,1/2, the Lorentzian shape of Ta allows to write |da|ka,1/22α2=(1|da|)2. Finally we obtain the following expression for the quality factor

Q=k02|da|1/21|da|α
The length α (resp. β defined similarly by db = |db|eiδkbβ) can be retrieved numerically, as α = δda/δka (we will choose δka = k0/1000). To understand the origin of phase variations, we derive da, considering that da=Πa1(P1/2S212P1/2)(P21/2S232P21/2)Πa, with Πa1=(Πa0Πa1) and Πa=(Π0aΠ1a), and we obtain the first order differential
δda=(δΠa1)P1/2S212P1/2P21/2S232P21/2Πa+Πa1(δP1/2)S212P1/2P21/2S232P21/2Πa+Πa1P1/2(δS212)P1/2P21/2S232P21/2Πa+Πa1P1/2S212(δP1/2)P21/2S232P21/2Πa+Πa1P1/2S212P1/2(δP21/2)S232P21/2Πa+Πa1P1/2S212P1/2P21/2(δS232)P21/2Πa+Πa1P1/2S212P1/2P21/2S232(δP1/2)Πa+Πa1P1/2S212P1/2P21/2S232P21/2(δΠa)+o(δda).

These 8 terms have been evaluated numerically. It appears that, contrary to the case of a simple Fabry-Perot filter where the phase variations at reflection onto the mirrors are negligible, the terms Πa1P1/2(δS212)P1/2P21/2S232P21/2Πa and Πa1P1/2S212P1/2P21/2(δS232)P21/2Πa must be taken into account. Furthermore, we notice that the first and last terms of this sum (δΠa1)P1/2S212P1/2P21/2S232P21/2Πa and Πa1P1/2S212P1/2P21/2S232P21/2(δΠa) are negligible.

The eigenmodes complex frequencies ωa and ωb are solutions of the implicit equation

det(IM)=0.
These complex values permit to fully characterize the non-conservative open system around a peak wavelength λ = 2πc/ω: the decay rate of the mode ”a” is γa = ℑ(ωa) and the quality factor is defined by Qa = ℜ(ωa)/2γa [22, 23]. The expression (6) for the quality factor Q is analogous to the typical quality factor of a Fabry-Perot resonator of thickness td filled with a dielectric of refractive index nd [24]:
Q=k02|r|1/21|r|(2ndtd),
given that r = r212r232 for highly reflecting interfaces. α is therefore similar to 2ndtd, and can therefore be considered as the cavity optical length.

The assumption that the system is separated in two independent resonators (indexed as a and b) is confirmed by Fig. 4, which shows the exact value of det(IM) and the approximated value obtained as the product of terms (1 − da) and (1 − db). The two curves are in good agreement over a wide spectral range around the resonance peaks λa and λb.

 figure: Fig. 4

Fig. 4 Determinant as a function of wavelength of the 2 × 2 matrix IM compared to the numerically-calculated expression (1 − |da|eiδkaα)(1 − |db|eiδkbβ).

Download Full Size | PPT Slide | PDF

Figure 5 shows the result of the symmetrisation approach for a GMR filter on a sapphire substrate. We manage to make d212 equal to d232 at the resonance wavelength λa. This allows to obtain the same transmission maxima for the original freestanding structure A and the structure on substrate C′, but not the same quality factor (on Fig. 5 (c) the transmission peak of the structure C′ appears larger than the one related to the original structure A). The equalization is obtained by enlarging the slits and reducing the thickness of the lower grating. Under these conditions, the hypothesis of a highly reflecting interface and the approximation 1ln|r||r|1/21|r| at first order may no longer be valid, especially as the index contrast between the dielectric waveguide and the sapphire substrate is low. Therefore, second order terms should be considered. Finally, this explains why the the quality factor is not restored in addition to the transmission maximum value at the original resonance wavelength λa, even though the symmetry criterion is fulfilled: in this case, the influence of second order terms would require in addition to suppress the small discrepancy we observe on Fig. 5(b) between the slopes of the coefficients d212 and d232 (both for the modules and arguments), responsible for the slightly lower quality factor we observed for the structure C′.

 figure: Fig. 5

Fig. 5 Symmetrisation of GMR filters on a sapphire substrate of refractive 1.7. (a) Schematic of three GMR structures A, B′ and C′. The structure B′ has the same geometrical parameters than the structure B. For the structure C′, the gold grating period is d = 2 μm; the SiC layer is divided in two parts of thickness t = 350 nm and t2 = 230 nm; the gold gratings are different, the upper one having a thickness tm = 100 nm and a slit width a = 250 nm, and the lower one a thickness tm2 = 50 nm and slit width a2 = 290 nm. (b) Modules and arguments of coefficients d212 and d232 corresponding to the structures A, B′ and C′. (c) Transmission spectra at normal incidence of the structures A, B′ and C′ calculated using equation (2).

Download Full Size | PPT Slide | PDF

4. Conclusion

We have proposed a mathematical framework to study the symmetry of metal-dielectric GMR filters. The analysis of the GMR resonance mechanism shows that only the 0th and ±1st diffracted orders play a significant role. As a consequence, we can reduce the size of the scattering matrices involved in the RMCA method computation. This reduction allows to write a scalar expression for the intensity transmitted through the GMR structures. A symmetry criterion is then formulated by noticing strong analogies between this expression and the one for standard Fabry-Perot filters. Satisfying this symmetry criterion by adapting the geometrical parameters at the resonance wavelength allows to restore the same transmission maximum value and quality factor than the original symmetric freestanding structure, at the same original resonance wavelength.

This simplification could also be applied to design GMR filters with a large angular acceptance [13], GMR filters exhibiting perfect extinction [25] or for example resonators to be integrated in VCSELs [26, 27], providing the minimum number of electromagnetic modes describing the system is small.

References and links

1. T. W. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998). [CrossRef]  

2. H. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779 (1998). [CrossRef]  

3. T. J. Kim, T. Thio, T. W. Ebbesen, D. E. Grupp, and H. J. Lezec, “Control of optical transmission through metals perforated with subwavelength hole arrays,” Opt. Lett. 24, 256–258 (1999). [CrossRef]  

4. S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics 1, 641–648 (2007). [CrossRef]  

5. J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008). [CrossRef]   [PubMed]  

6. J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010). [CrossRef]   [PubMed]  

7. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010). [CrossRef]   [PubMed]  

8. J. Porto, F. García-Vidal, and J. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845 (1999). [CrossRef]  

9. R. Haïdar, G. Vincent, S. Collin, N. Bardou, N. Guérineau, J. Deschamps, and J.-L. Pelouard, “Free-standing subwavelength metallic gratings for snapshot multispectral imaging,” Appl. Phys. Lett. 96, 221104 (2010). [CrossRef]  

10. E. Sakat, G. Vincent, P. Ghenuche, N. Bardou, S. Collin, F. Pardo, J.-L. Pelouard, and R. Haïdar, “Guided mode resonance in subwavelength metallodielectric free-standing grating for bandpass filtering,” Opt. Lett. 36, 3054–3056 (2011). [CrossRef]   [PubMed]  

11. E. Sakat, G. Vincent, P. Ghenuche, N. Bardou, C. Dupuis, S. Collin, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Free-standing guided-mode resonance band-pass filters: from 1d to 2d structures,” Opt. Express 20, 13082–13090 (2012). [CrossRef]   [PubMed]  

12. C.-H. Park, Y.-T. Yoon, and S.-S. Lee, “Polarization-independent visible wavelength filter incorporating a symmetric metal-dielectric resonant structure,” Opt. Express 20, 23769–23777 (2012). [CrossRef]   [PubMed]  

13. E. Sakat, S. Héron, P. Bouchon, G. Vincent, F. Pardo, S. Collin, J.-L. Pelouard, and R. Haïdar, “Metal-dielectric bi-atomic structure for angular-tolerant spectral filtering,” Opt. Lett. 38, 425–427 (2013). [CrossRef]   [PubMed]  

14. J. Le Perchec, R. E. de Lamaestre, M. Brun, N. Rochat, O. Gravrand, G. Badano, J. Hazart, and S. Nicoletti, “High rejection bandpass optical filters based on sub-wavelength metal patch arrays,” Opt. Express 19, 15720–15731 (2011). [CrossRef]   [PubMed]  

15. S. S. Wang, M. G. Moharam, R. Magnusson, and J. S. Bagby, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 7, 1470–1474 (1990). [CrossRef]  

16. R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992). [CrossRef]  

17. P. Lalanne, J.-P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, “One-mode model and airy-like formulae for one-dimensional metallic gratings,” J. Opt. A: Pure Appl. Opt. 2, 48 (2000). [CrossRef]  

18. S. Collin, F. Pardo, and J.-L. Pelouard, “Waveguiding in nanoscale metallic apertures,” Opt. Express 15, 4310–4320 (2007). [CrossRef]   [PubMed]  

19. B. Portier, F. Pardo, P. Bouchon, R. Haïdar, and J.-L. Pelouard, “Fast modal method for crossed grating computation, combining finite formulation of maxwell equations with polynomial approximated constitutive relations,” J. Opt. Soc. Am. A 30, 573–581 (2013). [CrossRef]  

20. C. Koechlin, P. Bouchon, F. Pardo, J. Jaeck, X. Lafosse, J.-L. Pelouard, and R. Haïdar, “Total routing and absorption of photons in dual color plasmonic antennas,” Appl. Phys. Lett. 99, 241104 (2011). [CrossRef]  

21. R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S.-H. Oh, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of gold,” Phys. Rev. B 86, 235147 (2012). [CrossRef]  

22. P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057–3067 (1994). [CrossRef]   [PubMed]  

23. R. K. Chang and A. J. Campillo, Optical Processes in Microcavities (World Scientific, 1996).

24. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Inc., 2001).

25. T. Estruch, J. Jaeck, F. Pardo, S. Derelle, J. Primot, J.-L. Pelouard, and R. Haïdar, “Perfect extinction in subwave-length dual metallic transmitting gratings,” Opt. Lett. 36, 3160–3162 (2011). [CrossRef]   [PubMed]  

26. V. Karagodsky, C. Chase, and C. J. Chang-Hasnain, “Matrix fabry-perot resonance mechanism in high-contrast gratings,” Opt. Lett. 36, 1704–1706 (2011). [CrossRef]   [PubMed]  

27. C.J. Chang-Hasnain and W. Yang, “High-contrast gratings for integrated optoelectronics,” Adv. Opt. Photonics 4, 379–440 (2012). [CrossRef]  

References

  • View by:

  1. T. W. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
    [Crossref]
  2. H. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779 (1998).
    [Crossref]
  3. T. J. Kim, T. Thio, T. W. Ebbesen, D. E. Grupp, and H. J. Lezec, “Control of optical transmission through metals perforated with subwavelength hole arrays,” Opt. Lett. 24, 256–258 (1999).
    [Crossref]
  4. S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics 1, 641–648 (2007).
    [Crossref]
  5. J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
    [Crossref] [PubMed]
  6. J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
    [Crossref] [PubMed]
  7. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
    [Crossref] [PubMed]
  8. J. Porto, F. García-Vidal, and J. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845 (1999).
    [Crossref]
  9. R. Haïdar, G. Vincent, S. Collin, N. Bardou, N. Guérineau, J. Deschamps, and J.-L. Pelouard, “Free-standing subwavelength metallic gratings for snapshot multispectral imaging,” Appl. Phys. Lett. 96, 221104 (2010).
    [Crossref]
  10. E. Sakat, G. Vincent, P. Ghenuche, N. Bardou, S. Collin, F. Pardo, J.-L. Pelouard, and R. Haïdar, “Guided mode resonance in subwavelength metallodielectric free-standing grating for bandpass filtering,” Opt. Lett. 36, 3054–3056 (2011).
    [Crossref] [PubMed]
  11. E. Sakat, G. Vincent, P. Ghenuche, N. Bardou, C. Dupuis, S. Collin, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Free-standing guided-mode resonance band-pass filters: from 1d to 2d structures,” Opt. Express 20, 13082–13090 (2012).
    [Crossref] [PubMed]
  12. C.-H. Park, Y.-T. Yoon, and S.-S. Lee, “Polarization-independent visible wavelength filter incorporating a symmetric metal-dielectric resonant structure,” Opt. Express 20, 23769–23777 (2012).
    [Crossref] [PubMed]
  13. E. Sakat, S. Héron, P. Bouchon, G. Vincent, F. Pardo, S. Collin, J.-L. Pelouard, and R. Haïdar, “Metal-dielectric bi-atomic structure for angular-tolerant spectral filtering,” Opt. Lett. 38, 425–427 (2013).
    [Crossref] [PubMed]
  14. J. Le Perchec, R. E. de Lamaestre, M. Brun, N. Rochat, O. Gravrand, G. Badano, J. Hazart, and S. Nicoletti, “High rejection bandpass optical filters based on sub-wavelength metal patch arrays,” Opt. Express 19, 15720–15731 (2011).
    [Crossref] [PubMed]
  15. S. S. Wang, M. G. Moharam, R. Magnusson, and J. S. Bagby, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 7, 1470–1474 (1990).
    [Crossref]
  16. R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
    [Crossref]
  17. P. Lalanne, J.-P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, “One-mode model and airy-like formulae for one-dimensional metallic gratings,” J. Opt. A: Pure Appl. Opt. 2, 48 (2000).
    [Crossref]
  18. S. Collin, F. Pardo, and J.-L. Pelouard, “Waveguiding in nanoscale metallic apertures,” Opt. Express 15, 4310–4320 (2007).
    [Crossref] [PubMed]
  19. B. Portier, F. Pardo, P. Bouchon, R. Haïdar, and J.-L. Pelouard, “Fast modal method for crossed grating computation, combining finite formulation of maxwell equations with polynomial approximated constitutive relations,” J. Opt. Soc. Am. A 30, 573–581 (2013).
    [Crossref]
  20. C. Koechlin, P. Bouchon, F. Pardo, J. Jaeck, X. Lafosse, J.-L. Pelouard, and R. Haïdar, “Total routing and absorption of photons in dual color plasmonic antennas,” Appl. Phys. Lett. 99, 241104 (2011).
    [Crossref]
  21. R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S.-H. Oh, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of gold,” Phys. Rev. B 86, 235147 (2012).
    [Crossref]
  22. P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057–3067 (1994).
    [Crossref] [PubMed]
  23. R. K. Chang and A. J. Campillo, Optical Processes in Microcavities (World Scientific, 1996).
  24. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Inc., 2001).
  25. T. Estruch, J. Jaeck, F. Pardo, S. Derelle, J. Primot, J.-L. Pelouard, and R. Haïdar, “Perfect extinction in subwave-length dual metallic transmitting gratings,” Opt. Lett. 36, 3160–3162 (2011).
    [Crossref] [PubMed]
  26. V. Karagodsky, C. Chase, and C. J. Chang-Hasnain, “Matrix fabry-perot resonance mechanism in high-contrast gratings,” Opt. Lett. 36, 1704–1706 (2011).
    [Crossref] [PubMed]
  27. C.J. Chang-Hasnain and W. Yang, “High-contrast gratings for integrated optoelectronics,” Adv. Opt. Photonics 4, 379–440 (2012).
    [Crossref]

2013 (2)

2012 (4)

2011 (5)

2010 (3)

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[Crossref] [PubMed]

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[Crossref] [PubMed]

R. Haïdar, G. Vincent, S. Collin, N. Bardou, N. Guérineau, J. Deschamps, and J.-L. Pelouard, “Free-standing subwavelength metallic gratings for snapshot multispectral imaging,” Appl. Phys. Lett. 96, 221104 (2010).
[Crossref]

2008 (1)

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
[Crossref] [PubMed]

2007 (2)

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics 1, 641–648 (2007).
[Crossref]

S. Collin, F. Pardo, and J.-L. Pelouard, “Waveguiding in nanoscale metallic apertures,” Opt. Express 15, 4310–4320 (2007).
[Crossref] [PubMed]

2000 (1)

P. Lalanne, J.-P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, “One-mode model and airy-like formulae for one-dimensional metallic gratings,” J. Opt. A: Pure Appl. Opt. 2, 48 (2000).
[Crossref]

1999 (2)

T. J. Kim, T. Thio, T. W. Ebbesen, D. E. Grupp, and H. J. Lezec, “Control of optical transmission through metals perforated with subwavelength hole arrays,” Opt. Lett. 24, 256–258 (1999).
[Crossref]

J. Porto, F. García-Vidal, and J. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845 (1999).
[Crossref]

1998 (2)

T. W. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

H. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779 (1998).
[Crossref]

1994 (1)

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057–3067 (1994).
[Crossref] [PubMed]

1992 (1)

R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[Crossref]

1990 (1)

Anker, J. N.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
[Crossref] [PubMed]

Astilean, S.

P. Lalanne, J.-P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, “One-mode model and airy-like formulae for one-dimensional metallic gratings,” J. Opt. A: Pure Appl. Opt. 2, 48 (2000).
[Crossref]

Atwater, H. A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[Crossref] [PubMed]

Badano, G.

Bagby, J. S.

Bardou, N.

Barnard, E. S.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[Crossref] [PubMed]

Boreman, G. D.

R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S.-H. Oh, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of gold,” Phys. Rev. B 86, 235147 (2012).
[Crossref]

Bouchon, P.

Brongersma, M. L.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[Crossref] [PubMed]

Brun, M.

Cai, W.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[Crossref] [PubMed]

Campillo, A. J.

R. K. Chang and A. J. Campillo, Optical Processes in Microcavities (World Scientific, 1996).

Chang, R. K.

R. K. Chang and A. J. Campillo, Optical Processes in Microcavities (World Scientific, 1996).

Chang-Hasnain, C. J.

Chang-Hasnain, C.J.

C.J. Chang-Hasnain and W. Yang, “High-contrast gratings for integrated optoelectronics,” Adv. Opt. Photonics 4, 379–440 (2012).
[Crossref]

Chase, C.

Collin, S.

de Lamaestre, R. E.

Derelle, S.

Deschamps, J.

R. Haïdar, G. Vincent, S. Collin, N. Bardou, N. Guérineau, J. Deschamps, and J.-L. Pelouard, “Free-standing subwavelength metallic gratings for snapshot multispectral imaging,” Appl. Phys. Lett. 96, 221104 (2010).
[Crossref]

Dupuis, C.

Ebbesen, T. W.

T. J. Kim, T. Thio, T. W. Ebbesen, D. E. Grupp, and H. J. Lezec, “Control of optical transmission through metals perforated with subwavelength hole arrays,” Opt. Lett. 24, 256–258 (1999).
[Crossref]

T. W. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

H. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779 (1998).
[Crossref]

Estruch, T.

García-Vidal, F.

J. Porto, F. García-Vidal, and J. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845 (1999).
[Crossref]

Ghaemi, H.

H. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779 (1998).
[Crossref]

T. W. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

Ghenuche, P.

Gravrand, O.

Grupp, D. E.

T. J. Kim, T. Thio, T. W. Ebbesen, D. E. Grupp, and H. J. Lezec, “Control of optical transmission through metals perforated with subwavelength hole arrays,” Opt. Lett. 24, 256–258 (1999).
[Crossref]

H. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779 (1998).
[Crossref]

Guérineau, N.

R. Haïdar, G. Vincent, S. Collin, N. Bardou, N. Guérineau, J. Deschamps, and J.-L. Pelouard, “Free-standing subwavelength metallic gratings for snapshot multispectral imaging,” Appl. Phys. Lett. 96, 221104 (2010).
[Crossref]

Haïdar, R.

E. Sakat, S. Héron, P. Bouchon, G. Vincent, F. Pardo, S. Collin, J.-L. Pelouard, and R. Haïdar, “Metal-dielectric bi-atomic structure for angular-tolerant spectral filtering,” Opt. Lett. 38, 425–427 (2013).
[Crossref] [PubMed]

B. Portier, F. Pardo, P. Bouchon, R. Haïdar, and J.-L. Pelouard, “Fast modal method for crossed grating computation, combining finite formulation of maxwell equations with polynomial approximated constitutive relations,” J. Opt. Soc. Am. A 30, 573–581 (2013).
[Crossref]

E. Sakat, G. Vincent, P. Ghenuche, N. Bardou, C. Dupuis, S. Collin, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Free-standing guided-mode resonance band-pass filters: from 1d to 2d structures,” Opt. Express 20, 13082–13090 (2012).
[Crossref] [PubMed]

E. Sakat, G. Vincent, P. Ghenuche, N. Bardou, S. Collin, F. Pardo, J.-L. Pelouard, and R. Haïdar, “Guided mode resonance in subwavelength metallodielectric free-standing grating for bandpass filtering,” Opt. Lett. 36, 3054–3056 (2011).
[Crossref] [PubMed]

C. Koechlin, P. Bouchon, F. Pardo, J. Jaeck, X. Lafosse, J.-L. Pelouard, and R. Haïdar, “Total routing and absorption of photons in dual color plasmonic antennas,” Appl. Phys. Lett. 99, 241104 (2011).
[Crossref]

T. Estruch, J. Jaeck, F. Pardo, S. Derelle, J. Primot, J.-L. Pelouard, and R. Haïdar, “Perfect extinction in subwave-length dual metallic transmitting gratings,” Opt. Lett. 36, 3160–3162 (2011).
[Crossref] [PubMed]

R. Haïdar, G. Vincent, S. Collin, N. Bardou, N. Guérineau, J. Deschamps, and J.-L. Pelouard, “Free-standing subwavelength metallic gratings for snapshot multispectral imaging,” Appl. Phys. Lett. 96, 221104 (2010).
[Crossref]

Halas, N. J.

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics 1, 641–648 (2007).
[Crossref]

Hall, W. P.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
[Crossref] [PubMed]

Hazart, J.

Héron, S.

Hugonin, J.-P.

P. Lalanne, J.-P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, “One-mode model and airy-like formulae for one-dimensional metallic gratings,” J. Opt. A: Pure Appl. Opt. 2, 48 (2000).
[Crossref]

Jaeck, J.

C. Koechlin, P. Bouchon, F. Pardo, J. Jaeck, X. Lafosse, J.-L. Pelouard, and R. Haïdar, “Total routing and absorption of photons in dual color plasmonic antennas,” Appl. Phys. Lett. 99, 241104 (2011).
[Crossref]

T. Estruch, J. Jaeck, F. Pardo, S. Derelle, J. Primot, J.-L. Pelouard, and R. Haïdar, “Perfect extinction in subwave-length dual metallic transmitting gratings,” Opt. Lett. 36, 3160–3162 (2011).
[Crossref] [PubMed]

Johnson, T. W.

R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S.-H. Oh, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of gold,” Phys. Rev. B 86, 235147 (2012).
[Crossref]

Jun, Y. C.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[Crossref] [PubMed]

Karagodsky, V.

Kim, T. J.

Koechlin, C.

C. Koechlin, P. Bouchon, F. Pardo, J. Jaeck, X. Lafosse, J.-L. Pelouard, and R. Haïdar, “Total routing and absorption of photons in dual color plasmonic antennas,” Appl. Phys. Lett. 99, 241104 (2011).
[Crossref]

Lafosse, X.

C. Koechlin, P. Bouchon, F. Pardo, J. Jaeck, X. Lafosse, J.-L. Pelouard, and R. Haïdar, “Total routing and absorption of photons in dual color plasmonic antennas,” Appl. Phys. Lett. 99, 241104 (2011).
[Crossref]

Lal, S.

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics 1, 641–648 (2007).
[Crossref]

Lalanne, P.

P. Lalanne, J.-P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, “One-mode model and airy-like formulae for one-dimensional metallic gratings,” J. Opt. A: Pure Appl. Opt. 2, 48 (2000).
[Crossref]

Le Perchec, J.

Lee, S.-S.

Leung, P. T.

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057–3067 (1994).
[Crossref] [PubMed]

Lezec, H.

T. W. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

H. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779 (1998).
[Crossref]

Lezec, H. J.

Link, S.

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics 1, 641–648 (2007).
[Crossref]

Liu, S. Y.

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057–3067 (1994).
[Crossref] [PubMed]

Lyandres, O.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
[Crossref] [PubMed]

Magnusson, R.

Moharam, M. G.

Möller, K. D.

P. Lalanne, J.-P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, “One-mode model and airy-like formulae for one-dimensional metallic gratings,” J. Opt. A: Pure Appl. Opt. 2, 48 (2000).
[Crossref]

Nicoletti, S.

Oh, S.-H.

R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S.-H. Oh, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of gold,” Phys. Rev. B 86, 235147 (2012).
[Crossref]

Olmon, R. L.

R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S.-H. Oh, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of gold,” Phys. Rev. B 86, 235147 (2012).
[Crossref]

Palamaru, M.

P. Lalanne, J.-P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, “One-mode model and airy-like formulae for one-dimensional metallic gratings,” J. Opt. A: Pure Appl. Opt. 2, 48 (2000).
[Crossref]

Pardo, F.

B. Portier, F. Pardo, P. Bouchon, R. Haïdar, and J.-L. Pelouard, “Fast modal method for crossed grating computation, combining finite formulation of maxwell equations with polynomial approximated constitutive relations,” J. Opt. Soc. Am. A 30, 573–581 (2013).
[Crossref]

E. Sakat, S. Héron, P. Bouchon, G. Vincent, F. Pardo, S. Collin, J.-L. Pelouard, and R. Haïdar, “Metal-dielectric bi-atomic structure for angular-tolerant spectral filtering,” Opt. Lett. 38, 425–427 (2013).
[Crossref] [PubMed]

E. Sakat, G. Vincent, P. Ghenuche, N. Bardou, C. Dupuis, S. Collin, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Free-standing guided-mode resonance band-pass filters: from 1d to 2d structures,” Opt. Express 20, 13082–13090 (2012).
[Crossref] [PubMed]

E. Sakat, G. Vincent, P. Ghenuche, N. Bardou, S. Collin, F. Pardo, J.-L. Pelouard, and R. Haïdar, “Guided mode resonance in subwavelength metallodielectric free-standing grating for bandpass filtering,” Opt. Lett. 36, 3054–3056 (2011).
[Crossref] [PubMed]

C. Koechlin, P. Bouchon, F. Pardo, J. Jaeck, X. Lafosse, J.-L. Pelouard, and R. Haïdar, “Total routing and absorption of photons in dual color plasmonic antennas,” Appl. Phys. Lett. 99, 241104 (2011).
[Crossref]

T. Estruch, J. Jaeck, F. Pardo, S. Derelle, J. Primot, J.-L. Pelouard, and R. Haïdar, “Perfect extinction in subwave-length dual metallic transmitting gratings,” Opt. Lett. 36, 3160–3162 (2011).
[Crossref] [PubMed]

S. Collin, F. Pardo, and J.-L. Pelouard, “Waveguiding in nanoscale metallic apertures,” Opt. Express 15, 4310–4320 (2007).
[Crossref] [PubMed]

Park, C.-H.

Pelouard, J.-L.

E. Sakat, S. Héron, P. Bouchon, G. Vincent, F. Pardo, S. Collin, J.-L. Pelouard, and R. Haïdar, “Metal-dielectric bi-atomic structure for angular-tolerant spectral filtering,” Opt. Lett. 38, 425–427 (2013).
[Crossref] [PubMed]

B. Portier, F. Pardo, P. Bouchon, R. Haïdar, and J.-L. Pelouard, “Fast modal method for crossed grating computation, combining finite formulation of maxwell equations with polynomial approximated constitutive relations,” J. Opt. Soc. Am. A 30, 573–581 (2013).
[Crossref]

E. Sakat, G. Vincent, P. Ghenuche, N. Bardou, C. Dupuis, S. Collin, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Free-standing guided-mode resonance band-pass filters: from 1d to 2d structures,” Opt. Express 20, 13082–13090 (2012).
[Crossref] [PubMed]

E. Sakat, G. Vincent, P. Ghenuche, N. Bardou, S. Collin, F. Pardo, J.-L. Pelouard, and R. Haïdar, “Guided mode resonance in subwavelength metallodielectric free-standing grating for bandpass filtering,” Opt. Lett. 36, 3054–3056 (2011).
[Crossref] [PubMed]

C. Koechlin, P. Bouchon, F. Pardo, J. Jaeck, X. Lafosse, J.-L. Pelouard, and R. Haïdar, “Total routing and absorption of photons in dual color plasmonic antennas,” Appl. Phys. Lett. 99, 241104 (2011).
[Crossref]

T. Estruch, J. Jaeck, F. Pardo, S. Derelle, J. Primot, J.-L. Pelouard, and R. Haïdar, “Perfect extinction in subwave-length dual metallic transmitting gratings,” Opt. Lett. 36, 3160–3162 (2011).
[Crossref] [PubMed]

R. Haïdar, G. Vincent, S. Collin, N. Bardou, N. Guérineau, J. Deschamps, and J.-L. Pelouard, “Free-standing subwavelength metallic gratings for snapshot multispectral imaging,” Appl. Phys. Lett. 96, 221104 (2010).
[Crossref]

S. Collin, F. Pardo, and J.-L. Pelouard, “Waveguiding in nanoscale metallic apertures,” Opt. Express 15, 4310–4320 (2007).
[Crossref] [PubMed]

Pendry, J.

J. Porto, F. García-Vidal, and J. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845 (1999).
[Crossref]

Polman, A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[Crossref] [PubMed]

Portier, B.

Porto, J.

J. Porto, F. García-Vidal, and J. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845 (1999).
[Crossref]

Primot, J.

Raschke, M. B.

R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S.-H. Oh, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of gold,” Phys. Rev. B 86, 235147 (2012).
[Crossref]

Rochat, N.

Sakat, E.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Inc., 2001).

Schuller, J. A.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[Crossref] [PubMed]

Shah, N. C.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
[Crossref] [PubMed]

Shelton, D.

R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S.-H. Oh, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of gold,” Phys. Rev. B 86, 235147 (2012).
[Crossref]

Slovick, B.

R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S.-H. Oh, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of gold,” Phys. Rev. B 86, 235147 (2012).
[Crossref]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Inc., 2001).

Thio, T.

T. J. Kim, T. Thio, T. W. Ebbesen, D. E. Grupp, and H. J. Lezec, “Control of optical transmission through metals perforated with subwavelength hole arrays,” Opt. Lett. 24, 256–258 (1999).
[Crossref]

H. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779 (1998).
[Crossref]

T. W. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

Van Duyne, R. P.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
[Crossref] [PubMed]

Vincent, G.

Wang, S. S.

White, J. S.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[Crossref] [PubMed]

Wolff, P.

T. W. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

Yang, W.

C.J. Chang-Hasnain and W. Yang, “High-contrast gratings for integrated optoelectronics,” Adv. Opt. Photonics 4, 379–440 (2012).
[Crossref]

Yoon, Y.-T.

Young, K.

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057–3067 (1994).
[Crossref] [PubMed]

Zhao, J.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
[Crossref] [PubMed]

Adv. Opt. Photonics (1)

C.J. Chang-Hasnain and W. Yang, “High-contrast gratings for integrated optoelectronics,” Adv. Opt. Photonics 4, 379–440 (2012).
[Crossref]

Appl. Phys. Lett. (3)

C. Koechlin, P. Bouchon, F. Pardo, J. Jaeck, X. Lafosse, J.-L. Pelouard, and R. Haïdar, “Total routing and absorption of photons in dual color plasmonic antennas,” Appl. Phys. Lett. 99, 241104 (2011).
[Crossref]

R. Haïdar, G. Vincent, S. Collin, N. Bardou, N. Guérineau, J. Deschamps, and J.-L. Pelouard, “Free-standing subwavelength metallic gratings for snapshot multispectral imaging,” Appl. Phys. Lett. 96, 221104 (2010).
[Crossref]

R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[Crossref]

J. Opt. A: Pure Appl. Opt. (1)

P. Lalanne, J.-P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, “One-mode model and airy-like formulae for one-dimensional metallic gratings,” J. Opt. A: Pure Appl. Opt. 2, 48 (2000).
[Crossref]

J. Opt. Soc. Am. A (2)

Nat. Mater. (3)

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
[Crossref] [PubMed]

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[Crossref] [PubMed]

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[Crossref] [PubMed]

Nat. Photonics (1)

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics 1, 641–648 (2007).
[Crossref]

Nature (1)

T. W. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

Opt. Express (4)

Opt. Lett. (5)

Phys. Rev. A (1)

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057–3067 (1994).
[Crossref] [PubMed]

Phys. Rev. B (2)

R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S.-H. Oh, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of gold,” Phys. Rev. B 86, 235147 (2012).
[Crossref]

H. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779 (1998).
[Crossref]

Phys. Rev. Lett. (1)

J. Porto, F. García-Vidal, and J. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845 (1999).
[Crossref]

Other (2)

R. K. Chang and A. J. Campillo, Optical Processes in Microcavities (World Scientific, 1996).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Inc., 2001).

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1 Optical response of a metal-dielectric GMR filter. (a) Schematic of the freestanding GMR filter (surrounded by air). The geometrical parameters are: grating period d = 2 μm, SiC layer thickness td = 700 nm, metal thickness tm = 100 nm and slit width a = 250 nm. (b) Transmission spectra at normal incidence of the metal-dielectric structure considering all the diffracted orders (solid line) and considering only the three first diffracted orders (dashed line). (c) Definition of the S-matrices used for the transmission spectrum calculations. Contour plots of the magnetic field distributions of the overall field H y sum and the field contributions H y 0, H y ± 1, H y 0 , ± 1 and H y evan .: (d) at λa = 4.65 μm and (e) at λb = 4.04 μm.
Fig. 2
Fig. 2 Scalar expression for the transmitted intensity. (a) Plot of the scalar expression using Eq. (4) (black dashed line) separated in two terms: term proportional to 1/1 − da (light blue line) and term proportional to 1/1 − db (red line). (b) Plot of the two complex eigenvalues da and db of the loop matrix M.
Fig. 3
Fig. 3 Validation of the symmetry criterion. (a) Schematic of three freestanding GMR structures A, B and C. For the structures A and B, the gold grating period is d = 2 μm, the SiC layer thickness is td = 700 nm, the gold grating slit width is a = 250 nm and the gold grating thickness is tm = 100 nm. For the structure C, the gold grating period is d = 2 μm; the SiC layer is divided in two parts of thickness t = 350 nm and t2 = 320 nm; the gold gratings are different, the upper one having a thickness tm = 100 nm and a slit width a = 250 nm, and the lower one a thickness tm2 = 60 nm and slit width a2 = 350 nm. (b) Modules and arguments of coefficients d212 and d232 corresponding to the structures A, B and C. (c) Transmission spectra at normal incidence of the structures A, B and C calculated using Eq. (2).
Fig. 4
Fig. 4 Determinant as a function of wavelength of the 2 × 2 matrix IM compared to the numerically-calculated expression (1 − |da|eiδkaα)(1 − |db|eiδkbβ).
Fig. 5
Fig. 5 Symmetrisation of GMR filters on a sapphire substrate of refractive 1.7. (a) Schematic of three GMR structures A, B′ and C′. The structure B′ has the same geometrical parameters than the structure B. For the structure C′, the gold grating period is d = 2 μm; the SiC layer is divided in two parts of thickness t = 350 nm and t2 = 230 nm; the gold gratings are different, the upper one having a thickness tm = 100 nm and a slit width a = 250 nm, and the lower one a thickness tm2 = 50 nm and slit width a2 = 290 nm. (b) Modules and arguments of coefficients d212 and d232 corresponding to the structures A, B′ and C′. (c) Transmission spectra at normal incidence of the structures A, B′ and C′ calculated using equation (2).

Equations (9)

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

S 31 = S 32 P 1 / 2 ( k = 0 P 1 / 2 S 212 P S 232 P 1 / 2 ) k ) P 1 / 2 S 21 ,
S 31 = S 32 P 1 / 2 ( I P 1 / 2 S 212 P S 232 P 1 / 2 ) 1 P 1 / 2 S 21 ,
S 31 = S 32 P 1 / 2 Π ( I D ) 1 Π 1 P 1 / 2 S 21 .
S 31 = ( Π 0 a p 0 1 / 2 s 0 β + Π 1 a p 1 1 / 2 s 1 β ) 1 1 d a ( Π a 0 p 0 1 / 2 s 0 α + Π a 1 p 1 1 / 2 s 1 α ) + ( Π 0 b p 0 1 / 2 s 0 β + Π 1 b p 1 1 / 2 s 1 β ) 1 1 d b ( Π b 0 p 0 1 / 2 s 0 α + Π b 1 p 1 1 / 2 s 1 α )
T = | S 31 | 2 T a + T b ,
Q = k 0 2 | d a | 1 / 2 1 | d a | α
δ d a = ( δ Π a 1 ) P 1 / 2 S 212 P 1 / 2 P 2 1 / 2 S 232 P 2 1 / 2 Π a + Π a 1 ( δ P 1 / 2 ) S 212 P 1 / 2 P 2 1 / 2 S 232 P 2 1 / 2 Π a + Π a 1 P 1 / 2 ( δ S 212 ) P 1 / 2 P 2 1 / 2 S 232 P 2 1 / 2 Π a + Π a 1 P 1 / 2 S 212 ( δ P 1 / 2 ) P 2 1 / 2 S 232 P 2 1 / 2 Π a + Π a 1 P 1 / 2 S 212 P 1 / 2 ( δ P 2 1 / 2 ) S 232 P 2 1 / 2 Π a + Π a 1 P 1 / 2 S 212 P 1 / 2 P 2 1 / 2 ( δ S 232 ) P 2 1 / 2 Π a + Π a 1 P 1 / 2 S 212 P 1 / 2 P 2 1 / 2 S 232 ( δ P 1 / 2 ) Π a + Π a 1 P 1 / 2 S 212 P 1 / 2 P 2 1 / 2 S 232 P 2 1 / 2 ( δ Π a ) + o ( δ d a ) .
det ( I M ) = 0 .
Q = k 0 2 | r | 1 / 2 1 | r | ( 2 n d t d ) ,

Metrics