Abstract

In this paper, we present a free-standing metallodielectric grating structure that can achieve multiple transmission dips and peaks at normal incidence over the visible spectrum. The amount of dips and peaks can be adjusted by the thickness of dielectric film. In our proposed structure, there are three types of resonance modes supported: Surface plasmon polarition (SPP) at horizontal metal/dielectric interface, vertical cavity mode in the metal slits, and guide mode in the dielectric film. Physically the coupling and resonant interactions among these modes lead to the generation of dips and peaks in the transmission spectrum. The transmission peaks is further interpreted by using Fano resonance. More surprisingly, the simultaneous excitation of three types of resonance modes can enhance the field distribution, which results in unexpected nearly perfect absorption in such simple structure. Moreover, compared to other absorption peaks, this high absorption peak originates from that guide mode resonance in the dielectric film inhibits transmission induced by cavity mode resonance in the metal slits. These results can be used in the design of many photonics components.

© 2014 Optical Society of America

1. Introduction

In recent years, subwavelength metal-dielectric nanostructures have attracted much attention because of its extraordinary optical properties for various applications [1]. A tremendous number of metal-dielectric nanostructures have been proposed with novel optical properties and new functionalities, such as color filters [2, 3], metamaterial absorbers [46], and photodectors [7]. As we know, the design of these metal-dielectric nanostructures is mainly based on the coupling among various EM modes to cause either enhanced transmission or enhanced absorption. In the case of enhanced transmission, recently, a new bandpass filters have been designed experimentally based on the coupling between metallic grating and thin dielectric film with up to 78% transmission at resonance in the mid-IR wavelength range [8]. To overcome the drawbacks of poor angular tolerance, another bi-atom pattern grating-dielectric structure designed can obtain significant improvement of the angular tolerance without modifying the spectral line shape [9]. In the case of enhanced absorption, high absorption efficiency up to 75% were obtained under normal incidence at a freestanding thin metal grating because of the coupling between surface plasmon polariton (SPP) modes and the cavity modes [10]. Compared to that in metal-insulator-metal metamaterrial structure [11], the absorption efficiency is relatively low due to the downward transmission and upward reflection of metallic grating. To solve the issue of low absorption efficiency, the structure of two-dimensional aluminum grating base on guide mode resonance effect is proposed [12]. Though employing the coupling between the quasiguided mode supported by a waveguide film and cavity mode, high absorption (maximum value 99.16%) can be obtained.

In this paper, we report some extraordinary optical properties of a subwavelength metallodielectric free-standing grating structure in the visible wavelength range. In our structure, we substitute the substrate film in traditional metallic grating investigated in previous works [13] with a thin dielectric film that can be used to support guide mode resonance. As a result, multiple transmission dips and peaks are generated at normal incidence for TM polarized light (i.e., magnetic field parallel to the slits). The physical mechanism of dips and peaks is explored by three types of resonances modes: SPP at the interface, guide mode resonance supported in the thin dielectric film and vertical cavity mode in the slits of metallic grating. The transmission peaks is further explained by employing Fano resonance. Furthermore, the nearly perfect absorption peak can be achieved in such simple structure. This is because the simultaneous excitation (at the same wavelength) of horizontal SPP mode, guide mode resonance and vertical cavity mode in the structure.

2. Model construction

Figure 1(a) presents the schematics of proposed metallodielectric free-standing grating. The structure is free-standing and consists of a silver film periodically pierced by narrow slits deposited on a thin SiO2 dielectric film, as represented in Fig. 1(a), where H and h indicate SiO2 film thickness and metal film thickness, w is the slit width, and P is the grating period. The wavelength range chosen in this paper is mainly relative to grating period P and the refractive index of dielectric film nSiO2 (P < λ < nSiO2 P). In this wavelength range chosen, the first diffracted orders are trapped in the SiO2 film by reflection on both sides [8]. This structure can be fabricated by using a combination of electron beam lithography and a subsequent lift-off process [14]. We select the period of metallic grating P = 500 nm and silt width w = 100 nm and keep them constant in the following work. Meanwhile, the thickness of metal film and SiO2 are firstly set to h = 200 nm and H = 800 nm, respectively. H and h are considered variable and serve as parameters suitable for optimizing the structure behavior. The electromagnetic wave is transmitted through this structure for TM polarized light, whereas the TE polarized light are almost totally reflected by the metallic grating.

 

Fig. 1 Structure schematic of the proposed metallodielectric free-standing grating and its representative optical phenomenon. (a) Schematic diagram and its structure parameters. (b) Calculated transmission spectrum in TM polarized light and normal incidence with the thickness of SiO2 film H = 800 nm (red line) and H approaching infinite (blue dashed line).

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We analyze the optical characteristics of this structure by employing a finite-difference time-domain (FDTD) method (Lumerical FDTD Solutions, Inc.) [15]. To achieve high precision in the simulation, we use one period of the grating in the following calculation, the periodic boundary condition is set in x direction and perfectly matched film is used in ± z direction under normal incidence. For oblique incidence, Bloch boundary condition in x direction is used. The numerical calculations are performed with extremely well convergence condition. The structure is assumed to be infinite long in y direction and all simulation results have been normalized to the incident light power.

The frequency-dependent permittivity εm of the metal silver is expressed by Drude–Lorentz model, which is defined as [16]

εm(ω)=εr-ωP02ω(ω+iγ0)-Δε0Ω02ω2-Ω02+iωΓ0
In Eq. (1), the first two items are given by the Drude model, where ω is the angle frequency, ωP0 is the plasma frequency, and γ0 is the damping coefficient. The third term is the Lorentzian term where Ω0 and Γ0, stand for, respectively, the oscillator strength and the spectral width of the Lorentz oscillators, and Δε0 can be interpreted as a weighting factor. But, the Drude model cannot give a detailed description about the permittivity of metal in a wide frequency range. In order to overcome the limitation of the Drude model and be able to consider the interband transitions, one or several Lorentzian terms should be added to make a better fitting effect than only Drude model. In this paper, we use silver as the deposited metal film material because of its low dissipation in visible and near-IR regions compared to other metals and its well-defined plasmonic properties. Therefore, in Eq. (1) εr = 4.6, ωP0 = 1.37 × 1016 rad/s, γ0 = 1.62 × 1014 rad/s, Δε0 = 1.10, Ω0 = 7.43 × 1015 rad/s, Γ0 = 1.82 × 1015 rad/s. The permittivity of SiO2 dielectric film is 2.25.

The SPP resonances (SPPs) are surface waves which propagate at a metal/dielectric interface and are evanescent in the direction normal to the substrate. SPPs excitation condition on both surfaces of the metallic grating is

2πλsinθ-n2πP=-2πλεm(ω)εεm(ω)+ε=ksppn=0,±1,±2,...,±N
where λ is the incident light wavelength, n is diffraction order, θ is the incident angle and ε is the relative permittivity of the medium on the top and bottom surfaces of metallic grating (for the top air medium, ε = 1; for the bottom SiO2 medium, ε = 2.25). When the above resonance condition is satisfied, the energy carried by the incident wave can be transferred to the SPPs, which propagate along the metal/dielectric interface.

The cavity mode resonance is excited inside the slits of metallic grating. It is determined by resonant length hm, and given by the formula [12]:

φ12+φ23+khm=2nπn=0,1,2,...,N
Where φ12 and φ23 are phases obtained by the cavity mode and caused by the reflection at the two terminations, n is resonance order interger, and k is the complex wave vector in the slit.

The guide mode resonance in the dielectric film is excited for TM polarized light when the following condition is satisfied [17]

(k02ε1β2)H=nπ+arctan(ε1ε2β2k02ε2k02ε1β2)1/2+arctan(ε1ε3β2k02ε3k02ε1β2)1/2
where β is the propagation constant, k0 is the free space wave number, H is the waveguide film thickness, and n is an integer. The permittivity of each film is given to be ε1, ε2, and ε3, respectively.

3. Results and discussion

The calculated transmission spectrum of this proposed structure with SiO2 film thickness H = 800 nm is presented in Fig. 1(b) for TM polarized light (red line) with other parameters P = 500 nm, w = 100 nm, and h = 200 nm. As a comparison, the calculated transmission spectrum of the identical metallic grating deposited on the SiO2 substrate (H → infinite) is presented (blue dashed line). It is interesting to see that four transmission dips and three remarkable peaks under normal incidence are obtained with SiO2 film thickness H = 800 nm. However, keeping other parameters constant, when H approaches infinite, there is only two transmission dips with a wide band and low transmission peak in-between in the wavelength range from 500 nm to 800 nm. Remarkably, when the thickness of SiO2 film is reduced to H = 800 nm, a wide band and low transmission peak is transformed to several peaks with narrow and high optical transmission. The appearance of these peaks can make this structure to be designed to act as a multispectral bandpass filter in the visible and near-IR wavelength range. In Fig. 1(b), two dips of SiO2 film thickness H approaching infinite coincides with the leftmost and rightmost dips of SiO2 film thickness H = 800 nm. The transmission peaks are only appeared between the two dips used as two boundary values.

The influence of metal film thickness h on transmission spectrum is shown in Fig. 2. To show clearly transmission minima, transmission spectrum depending on metal film thickness h at normal incidence are depicted in linear color map [Fig. 2(a)] and in logarithmic color map [Fig. 2(b)], respectively. In addition, the four dips with h = 200 nm from left to right in Fig. 2 are denoted by D1, D2, D3, and D4, respectively. It is shown that the wavelength position of four dips keep constant with the increase of metal film thickness h. So it can be inferred that the appearance of dips does not depend on the electromagnetic field distribution of metal silts and it is relative to that of metal/dielectric interface and SiO2 dielectric film. It is well known that cavity mode resonance can be excited inside the metal silts when metal film thickness h satisfies certain condition (Eq. (3)). That is, the appearance of dips has nothing to do with cavity mode inside metal slit. The leftmost and rightmost white dashed line indicate transmission minima appearing at the wavelengths of λD1 = 533 nm and λD4 = 764 nm, corresponding to the wavelength of SPPs resonant excitation at the metal/air and metal/SiO2 surfaces, respectively. This can be verified by Eq. (2). Moreover, as metal film thickness h increases, transmission peaks have a red shift. It is found that transmission peak is divided into three areas by transmission dips. Compared to that of no SiO2 dielectric film H = 0 nm [18], transmission peak of Fig. 2 have a different dependence on metal film thickness h. As metal film thickness h increases, cavity mode inside metal slits with different order is excited. The thicker metal film thickness is, the higher order the cavity mode have. In three areas, transmission peak depending on metal film thickness h has same tendency. These extraordinary optical transmission properties are resulted from the existence of a thin SiO2 dielectric film.

 

Fig. 2 Transmission versus wavelength and metal film thickness h at normal incidence (a) in linear color map and (b) in logarithmic color map with fixed P = 500 nm, w = 100 nm and H = 800 nm. Transmission minima in Fig. 2(a) are shown clearly in Fig. 2(b). The vertical white lines in Fig. 2(a) indicate the position of transmission minima.

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Additionally, we make further investigation of extraordinary optical properties in the subwavelength metallodielectric free-standing grating in Fig. 3. Figure 3 illustrates the effect of SiO2 film thickness H on the transmission spectrum. Similarly, transmission spectrum depending on SiO2 film thickness H at normal incidence is depicted in linear color map [Fig. 3(a)] and in logarithmic color map [Fig. 3(b)], respectively. It is seen that the amount of transmission dips and peaks increases gradually as the SiO2 film thickness H increases. When the SiO2 film thickness H is up to 1400 nm, six transmission dips and five peaks under normal incidence are obtained. The white dashed lines in Fig. 3(a) [blue dashed lines in Fig. 3(b)] is at the wavelength λ = 533 nm and λ = 764 nm, corresponding to the wavelength of SPPs resonant excitation at the metal/air and metal/SiO2 surfaces, respectively. The two wavelengths are regarded as the boundaries of the appearance of all other dips and peaks. As the SiO2 film thickness H increases consistently, the amount of transmission peaks in the transmission spectrum of the structure increases continuously. When SiO2 film thickness H approaches infinite, transmission dips will not be detected and the countless transmission peaks will be connected to form a continue line, as shown in the blue dashed line of Fig. 1(b).

 

Fig. 3 Transmission versus wavelength and SiO2 film thickness H at normal incidence (a) in linear color map and (b) in logarithmic color map with fixed P = 500 nm, w = 100 nm and h = 200 nm. Transmission minima in Fig. 3(a) are shown clearly in Fig. 3(b). The vertical white lines in Fig. 3(a) indicate the position of transmission minima.

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3.1 The explanation of transmission dips

As shown in Fig. 4, to further understand the mechanism of the generation of transmission dips, we calculate the normalized magnetic field distribution of transmission dips indicated by D1, D2, D3, and D4 in Fig. 2(a). Two unit cells of metallic grating in the following areconsidered. As mentioned above, dips D1 and D4 are attributed to SPPs resonant excitation of the top and down surfaces of metallic grating, respectively. As shown in Figs. 4(a) and 4(d), magnetic field of λD1 = 533 nm only is localized on the top metal/air surface of metallic grating. However, magnetic field of λD4 = 764 nm is localized on the top and down surfaces of metallic grating, and at λD4 = 764 nm magnetic field on the metal/SiO2 surface paly a leading role. They demonstrate the clear SPP characteristics: the magnetic field is mostly confined and have its maximum on the surface of metal film. The exponential dependence of field in the vertical direction is shown. This can be also verified by Eq. (2). Moreover, when the wavelength of incident light is longer than nSiO2 P = 1.5 × 500 nm = 750 nm, there is only zero-order transmission and first diffracted orders are disappear in the SiO2 film. So guide mode resonance cannot be excited as wavelength longer than 750 nm because of wave vector mismatch. Compared to that of Figs. 4(a) and 4(d), magnetic field of Figs. 4(b) and 4(c) has a most striking feature: guide mode is excited in SiO2 dielectric film with a perfect standing wave along the x direction generated in the waveguide film. Meanwhile, it is found that SPP mode on the metal/SiO2 surface in Figs. 4(b) and 4(c) is also excited. The SPP mode and guide mode can be simultaneously excited at λD2 = 608 nm and λD3 = 686 nm in the SiO2 film. Different waveguide mode is excited in SiO2 film: that at λD2 = 608 nm corresponding to first order mode, and that λD3 = 686 nm corresponding to zeroth order mode. Here ± 1 diffracted orders in the SiO2 film are involved in the excitation of different guide mode. They are trapped in the SiO2 waveguide. It is concluded that transmission minima at λD2 = 608 nm and λD3 = 686 nm is attribute to the hybrid mode between SPP mode on the metal/SiO2 surface and guide mode resonance in SiO2 film.

 

Fig. 4 Spatial magnetic field (|H|) distribution for (a) λD1 = 533 nm, (b) λD2 = 608 nm, (c) λD3 = 686 nm, and (d) λD4 = 764 nm transmission dips at normal incidence. White lines depict schematically the profile of metallic grating and SiO2 dielectric film.

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3.2 The explanation of transmission peaks

Furthermore, the normalized magnetic field distribution of transmission peaks indicated by P1, P2, and P3 in Fig. 2(a) are shown in Fig. 5. It is seen clearly that zeroth order vertical cavity mode are excited in the slit of metallic grating at the three peaks. The excitation of cavity mode in the metal slits is attributed to the combination of the SPP and FP-like mode [18]. Moreover, strong magnetic field is localized in the SiO2 dielectric film. Compared to that of Fig. 4, magnetic field intensity of Fig. 5 is stronger, which can be indicated by numerical value of color bar. The Stronger magnetic field intensity is mainly attributed to the cavity mode supported inside the slits. The formation of cavity mode results in weak reflection and strong transmission, making most of electromagnetic field focus inside the waveguide film. But magnetic field distribution of Fig. 5 is different from that of Fig. 4 in the waveguide film: a perfect standing wave along the x direction is not generated in the waveguide film, however, magnetic field distribution of Fig. 4 has a perfect standing wave generated. So it is speculated that the physical mechanism at the peaks is different from that of dips in Fig. 4. We find that the transmission spectrum in Fig. 1(b) (solid red line) shows a characteristic Fano resonance: a transmission peak close to a transmission dip. According to the previous publications [1], the fundamental criterion for a Fano resonance in the designed structure is the interference between a broad resonance and a narrow discrete resonance. That is, this Fano resonance is the result of interference between two transmission channels: one corresponding to a broad zeroth order resonance, which is directly transmitted through the structure; and the other corresponding to the narrow discrete resonance, which is transmitted in the zeroth order after coupling and trapping of the ± 1 diffracted orders in the SiO2 waveguide film [8]. The appearance of three different peaks is due to the excitation of different narrow discrete resonances in the waveguide film. The thicker the waveguide film is, the more narrow discrete resonant modes the waveguide film can excite. So high optical transmission peaks mainly result from different Fano resonances and is relative to the excitation of cavity mode in the slits of metallic grating, rather than only guide mode resonance in the SiO2 waveguide film.

 

Fig. 5 Spatial magnetic field (|H|) distribution for (a) λP1 = 584 nm, (b) λP2 = 638 nm, and (c) λP3 = 700nm transmission peaks at normal incidence. White lines depict schematically the profile of metallic grating and SiO2 dielectric film.

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As shown in Figs. 6(a) and 6(b), we also investigate the influence of oblique incident light on transmission spectrum in the designed structure. The structure parameters include P = 500 nm, w = 100 nm, H = 800 nm, h = 200 nm, and incident angles range from 0° to 40°. When the incident angle increases, transmission dips at normal incidence is divided into two transmission dip bands. As mentioned above, transmission dips at normal incidence is relative to the excitation of SPP on the metal/dielectric interfaces. At oblique incidence, two transmission dip bands correspond to the excitation of two SPP with opposite wave vectors in the x direction and opposite direction of propagation on the metal/dielectric interface. In Eq. (2), two SPP correspond to the excitation of SPP waves for n = −1, and 1, respectively. The explanations in details to one transmission dip divided to two ones at the oblique incidence can be found in the final section of ref. 19. In addition, at non-normal incidence, more transmission peaks appear in the transmission spectrum, which further reveal Fano resonance [8]. The effect of oblique light on reflection spectra and absorption is depicted, as shown in Fig. 6(c) and Fig. 6(d). There is some important information provided by comparing four figures in Fig. 6. We observe that regions of high absorption in Fig. 6(d) are in agreement with that of high transmission in Fig. 6(a), but justly correspond to regions of low reflection in Fig. 6(c). Hence, it indicates that high absorption, high transmission and low reflection can be attributed to the resonant excitation of cavity mode in the slits of metallic grating. This high absorption always accompanied by high transmission produces a problem that perfect absorption is difficult to obtain. In the following, we will discuss that how to achieve perfect absorption in our structure by employing the guide mode resonance in the SiO2 film.

 

Fig. 6 Transmission versus wavelength and incident angle (a) in linear color map and (b) in logarithmic color map. Transmission minima in Fig. 6(a) are shown clearly in Fig. 6(b). (c) Reflection and (d) absorption versus wavelength and incident angle in linear color map with fixed P = 500 nm, w = 100 nm, H = 800 nm and h = 200 nm.

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3.3 The explanation of perfect absorption peak

To illustrate clearly perfect absorption, the calculated transmission, reflection and absorption spectra of the proposed metallodielectric free-standing grating at normal incidence are demonstrated in Fig. 7. It is interesting to notice that there is a near perfect absorption peak with wavelength λA1 = 608 nm at normal incidence, as shown in Fig. 7(a), where structure parameters are taken to be h = 332 nm, P = 500 nm, w = 100 nm and H = 800 nm. We observe that the appearance of the high absorption peak is characterized by very low reflection and low transmission with a nearly symmetric dip, which is significantly different from other absorption peaks accompanied with low reflection and high transmission peaks. So the appearance of this absorption peak is mainly ascribed to the formation of low transmission dips. As mentioned above, we know that the hybrid mode between SPP mode on the metal/SiO2 surface and guide mode resonance in SiO2 film at λA1 = 608 nm can be excited. So we can speculate the perfect absorption peak is related to this hybrid mode in SiO2 film. Transmission, reflection and absorption spectra of structure with metal film thickness h = 668 nm is shown in Fig. 7(b). Similarly, a near perfect absorption peak at λA2 = 686 nm is also obtained, which is also characterized by very low reflection and low transmission with a nearly symmetric dip. In the previously report, the perfect absorption peak is usually achieved by adding an opaque metal film under our propose structure [11]. So far, in such simple metallodielectric free-standing grating, the perfect absorption peak obtained has relative few reports.

 

Fig. 7 Transmission, reflection and absorption spectra of the proposed metallodielectric free-standing grating at normal incidence for two specific metal film thickness (a) h = 332 nm and (b) h = 668 nm with fixed P = 500 nm, w = 100 nm and H = 800 nm.

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To reveal the physical mechanism of the perfect absorption peak in the propose structure, we calculate the normalized magnetic field distribution of absorption peak indicated by A1 and A2 in Fig. 7. The magnetic field distribution at λA1 = 608 nm and λA2 = 686 nm is shown in Figs. 8(a) and 8(b), respectively. Firstly, the magnetic field is found to be confined at the metal/SiO2 interface, demonstrating the clear SPP characteristics. Secondly, guide mode resonance in the SiO2 dielectric film is excited. Thirdly, the pronounced distinction between Fig. 8(a) and Fig. 4(b) is that the cavity mode resonance is excited in the metal slits of metallic grating in Fig. 8(a). At absorption peaks, the intensity of magnetic field in Fig. 8 is nearly four times stronger than that of Fig. 4, which is relative to the excitation of cavity mode in the metal slits. The nearly perfect absorption peak is generated because transmission peak is transformed into transmission dip. That is, the presence of the hybrid mode between SPP mode on the metal/SiO2 surface and guide mode in SiO2 film modifies the optical response of structure in the cavity mode vicinity. This modification reinforces the electromagnetic field energy distribution in the structure, which causes the extraordinary optical absorption. Hence, the interaction among SPP mode at the interface, guide mode in the SiO2 film and vertical cavity mode in the metal slit results in this pronounced absorption. This absorption peaks can be generated only when three modes can coexist. The absorption peak of λA1 = 608 nm and λA2 = 686 nm corresponds to different orders guide mode resonance in the SiO2 film and different orders cavity mode resonance inside the slit: that at λA1 = 608 nm corresponding to first order guide mode resonance and first order cavity mode (a node in the middle of the cavity); that at λA2 = 686 nm corresponding to zeroth order guide mode resonance and second order cavity mode.

 

Fig. 8 Spatial magnetic field (|H|) distribution for (a) λA1 = 608 nm, and (b) λA2 = 686 nm high absorption peaks at normal incidence. White lines depict schematically the profile of metallic grating and SiO2 dielectric film.

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4. Conclusions

In conclusion, multiple transmission dips and peaks base on one dimension metallic grating deposited on a thin dielectric film, can be obtained under normal incidence. The amount of dips and peaks can be adjusted by the thickness of dielectric film. The hybrid mode of SPP at the interface and guide mode resonance result in the generation of resonant dips, and transmission peaks mainly results from different Fano resonances and is relative to the cavity mode resonance inside the slits of metallic grating. More importantly, the nearly perfect absorption can be achieved when SPP mode, guide mode resonance and cavity mode resonance is excited simultaneously in the proposed structure. As a result, these extraordinary optical properties could be exploited in numerous photonics applications, such as photodetectors, visible spectral imaging systems, and so on.

Acknowledgments

The authors would like to thank financial supports from the National Nature Science Foundation of China (Grant Nos. 61137005 and 60977055) and the Ministry of Education of China (Grant No. DUT14ZD211 and SRFDP 20120041110040).

References and links

1. B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010). [CrossRef]   [PubMed]  

2. Y.-T. Yoon, C.-H. Park, and S.-S. Lee, “Highly efficient color filter incorporating a thin metal-dielectric resonant structure,” Appl. Phys. Express 5(2), 022501 (2012). [CrossRef]  

3. 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(21), 23769–23777 (2012). [CrossRef]   [PubMed]  

4. N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010). [CrossRef]   [PubMed]  

5. B. Zhang, J. Hendrickson, and J. Guo, “Multispectral near-perfect metamaterial absorbers using spatially multiplexed plasmon resonance metal square structures,” J. Opt. Soc. Am. B 30(3), 656–662 (2013). [CrossRef]  

6. M. Diem, T. Koschny, and C. M. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009). [CrossRef]  

7. Z. Yu, G. Veronis, S. Fan, and M. L. Brongersma, “Design of midinfrared photodetectors enhanced by surface plasmonson grating structures,” Appl. Phys. Lett. 89(15), 151116 (2006). [CrossRef]  

8. 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(16), 3054–3056 (2011). [CrossRef]   [PubMed]  

9. 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(4), 425–427 (2013). [CrossRef]   [PubMed]  

10. A. Roszkiewicz and W. Nasalski, “Reflection suppression and absorption enhancement of optical field at thin metal gratings with narrow slits,” Opt. Lett. 37(18), 3759–3761 (2012). [CrossRef]   [PubMed]  

11. C. W. Cheng, M. N. Abbas, C. W. Chiu, K. T. Lai, M. H. Shih, and Y. C. Chang, “Wide-angle polarization independent infrared broadband absorbers based on metallic multi-sized disk arrays,” Opt. Express 20(9), 10376–10381 (2012). [CrossRef]   [PubMed]  

12. W. Zhou, Y. Wu, M. Yu, P. Hao, G. Liu, and K. Li, “Extraordinary optical absorption based on guided-mode resonance,” Opt. Lett. 38(24), 5393–5396 (2013). [CrossRef]   [PubMed]  

13. T. Ongarello, F. Romanato, P. Zilio, and M. Massari, “Polarization independence of extraordinary transmission trough 1D metallic gratings,” Opt. Express 19(10), 9426–9433 (2011). [CrossRef]   [PubMed]  

14. G. Vincent, S. Collin, N. Bardou, J.-L. Pelouard, and R. Haïdar, “Large-area dielectric and metallic freestanding gratings for midinfrared optical filtering applications,” J. Vac. Sci. Technol. B 26(6), 1852–1855 (2008). [CrossRef]  

15. Lumerical Solutions, http://www.lumerical.com.

16. S. G. Rodrigo, F. J. García-Vidal, and L. Martín-Moreno, “Influence of material properties on extraordinary optical transmission through hole arrays,” Phys. Rev. B 77(7), 075401 (2008). [CrossRef]  

17. A. F. Kaplan, T. Xu, and L. J. Guo, “High efficiency resonance-based spectrum filters with tunable transmission bandwidth fabricated using nanoimprint lithography,” Appl. Phys. Lett. 99(14), 143111 (2011). [CrossRef]  

18. Y. Liang and W. Peng, “Theoretical study of transmission characteristics of sub-wavelength nano-structured metallic grating,” Appl. Spectrosc. 67(1), 49–53 (2013). [CrossRef]   [PubMed]  

19. Y. Liang, W. Peng, R. Hu, and H. Zou, “Extraordinary optical transmission based on subwavelength metallic grating with ellipse walls,” Opt. Express 21(5), 6139–6152 (2013). [CrossRef]   [PubMed]  

References

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  1. B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
    [CrossRef] [PubMed]
  2. Y.-T. Yoon, C.-H. Park, and S.-S. Lee, “Highly efficient color filter incorporating a thin metal-dielectric resonant structure,” Appl. Phys. Express 5(2), 022501 (2012).
    [CrossRef]
  3. 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(21), 23769–23777 (2012).
    [CrossRef] [PubMed]
  4. N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
    [CrossRef] [PubMed]
  5. B. Zhang, J. Hendrickson, and J. Guo, “Multispectral near-perfect metamaterial absorbers using spatially multiplexed plasmon resonance metal square structures,” J. Opt. Soc. Am. B 30(3), 656–662 (2013).
    [CrossRef]
  6. M. Diem, T. Koschny, and C. M. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
    [CrossRef]
  7. Z. Yu, G. Veronis, S. Fan, and M. L. Brongersma, “Design of midinfrared photodetectors enhanced by surface plasmonson grating structures,” Appl. Phys. Lett. 89(15), 151116 (2006).
    [CrossRef]
  8. 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(16), 3054–3056 (2011).
    [CrossRef] [PubMed]
  9. 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(4), 425–427 (2013).
    [CrossRef] [PubMed]
  10. A. Roszkiewicz and W. Nasalski, “Reflection suppression and absorption enhancement of optical field at thin metal gratings with narrow slits,” Opt. Lett. 37(18), 3759–3761 (2012).
    [CrossRef] [PubMed]
  11. C. W. Cheng, M. N. Abbas, C. W. Chiu, K. T. Lai, M. H. Shih, and Y. C. Chang, “Wide-angle polarization independent infrared broadband absorbers based on metallic multi-sized disk arrays,” Opt. Express 20(9), 10376–10381 (2012).
    [CrossRef] [PubMed]
  12. W. Zhou, Y. Wu, M. Yu, P. Hao, G. Liu, and K. Li, “Extraordinary optical absorption based on guided-mode resonance,” Opt. Lett. 38(24), 5393–5396 (2013).
    [CrossRef] [PubMed]
  13. T. Ongarello, F. Romanato, P. Zilio, and M. Massari, “Polarization independence of extraordinary transmission trough 1D metallic gratings,” Opt. Express 19(10), 9426–9433 (2011).
    [CrossRef] [PubMed]
  14. G. Vincent, S. Collin, N. Bardou, J.-L. Pelouard, and R. Haïdar, “Large-area dielectric and metallic freestanding gratings for midinfrared optical filtering applications,” J. Vac. Sci. Technol. B 26(6), 1852–1855 (2008).
    [CrossRef]
  15. Lumerical Solutions, http://www.lumerical.com .
  16. S. G. Rodrigo, F. J. García-Vidal, and L. Martín-Moreno, “Influence of material properties on extraordinary optical transmission through hole arrays,” Phys. Rev. B 77(7), 075401 (2008).
    [CrossRef]
  17. A. F. Kaplan, T. Xu, and L. J. Guo, “High efficiency resonance-based spectrum filters with tunable transmission bandwidth fabricated using nanoimprint lithography,” Appl. Phys. Lett. 99(14), 143111 (2011).
    [CrossRef]
  18. Y. Liang and W. Peng, “Theoretical study of transmission characteristics of sub-wavelength nano-structured metallic grating,” Appl. Spectrosc. 67(1), 49–53 (2013).
    [CrossRef] [PubMed]
  19. Y. Liang, W. Peng, R. Hu, and H. Zou, “Extraordinary optical transmission based on subwavelength metallic grating with ellipse walls,” Opt. Express 21(5), 6139–6152 (2013).
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2013 (5)

2012 (4)

2011 (3)

2010 (2)

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
[CrossRef] [PubMed]

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[CrossRef] [PubMed]

2009 (1)

M. Diem, T. Koschny, and C. M. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
[CrossRef]

2008 (2)

G. Vincent, S. Collin, N. Bardou, J.-L. Pelouard, and R. Haïdar, “Large-area dielectric and metallic freestanding gratings for midinfrared optical filtering applications,” J. Vac. Sci. Technol. B 26(6), 1852–1855 (2008).
[CrossRef]

S. G. Rodrigo, F. J. García-Vidal, and L. Martín-Moreno, “Influence of material properties on extraordinary optical transmission through hole arrays,” Phys. Rev. B 77(7), 075401 (2008).
[CrossRef]

2006 (1)

Z. Yu, G. Veronis, S. Fan, and M. L. Brongersma, “Design of midinfrared photodetectors enhanced by surface plasmonson grating structures,” Appl. Phys. Lett. 89(15), 151116 (2006).
[CrossRef]

Abbas, M. N.

Bardou, N.

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(16), 3054–3056 (2011).
[CrossRef] [PubMed]

G. Vincent, S. Collin, N. Bardou, J.-L. Pelouard, and R. Haïdar, “Large-area dielectric and metallic freestanding gratings for midinfrared optical filtering applications,” J. Vac. Sci. Technol. B 26(6), 1852–1855 (2008).
[CrossRef]

Bouchon, P.

Brongersma, M. L.

Z. Yu, G. Veronis, S. Fan, and M. L. Brongersma, “Design of midinfrared photodetectors enhanced by surface plasmonson grating structures,” Appl. Phys. Lett. 89(15), 151116 (2006).
[CrossRef]

Chang, Y. C.

Cheng, C. W.

Chiu, C. W.

Chong, C. T.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[CrossRef] [PubMed]

Collin, S.

Diem, M.

M. Diem, T. Koschny, and C. M. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
[CrossRef]

Fan, S.

Z. Yu, G. Veronis, S. Fan, and M. L. Brongersma, “Design of midinfrared photodetectors enhanced by surface plasmonson grating structures,” Appl. Phys. Lett. 89(15), 151116 (2006).
[CrossRef]

García-Vidal, F. J.

S. G. Rodrigo, F. J. García-Vidal, and L. Martín-Moreno, “Influence of material properties on extraordinary optical transmission through hole arrays,” Phys. Rev. B 77(7), 075401 (2008).
[CrossRef]

Ghenuche, P.

Giessen, H.

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
[CrossRef] [PubMed]

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[CrossRef] [PubMed]

Guo, J.

Guo, L. J.

A. F. Kaplan, T. Xu, and L. J. Guo, “High efficiency resonance-based spectrum filters with tunable transmission bandwidth fabricated using nanoimprint lithography,” Appl. Phys. Lett. 99(14), 143111 (2011).
[CrossRef]

Haïdar, R.

Halas, N. J.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[CrossRef] [PubMed]

Hao, P.

Hendrickson, J.

Hentschel, M.

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
[CrossRef] [PubMed]

Héron, S.

Hu, R.

Kaplan, A. F.

A. F. Kaplan, T. Xu, and L. J. Guo, “High efficiency resonance-based spectrum filters with tunable transmission bandwidth fabricated using nanoimprint lithography,” Appl. Phys. Lett. 99(14), 143111 (2011).
[CrossRef]

Koschny, T.

M. Diem, T. Koschny, and C. M. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
[CrossRef]

Lai, K. T.

Lee, S.-S.

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(21), 23769–23777 (2012).
[CrossRef] [PubMed]

Y.-T. Yoon, C.-H. Park, and S.-S. Lee, “Highly efficient color filter incorporating a thin metal-dielectric resonant structure,” Appl. Phys. Express 5(2), 022501 (2012).
[CrossRef]

Li, K.

Liang, Y.

Liu, G.

Liu, N.

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
[CrossRef] [PubMed]

Luk’yanchuk, B.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[CrossRef] [PubMed]

Maier, S. A.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[CrossRef] [PubMed]

Martín-Moreno, L.

S. G. Rodrigo, F. J. García-Vidal, and L. Martín-Moreno, “Influence of material properties on extraordinary optical transmission through hole arrays,” Phys. Rev. B 77(7), 075401 (2008).
[CrossRef]

Massari, M.

Mesch, M.

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
[CrossRef] [PubMed]

Nasalski, W.

Nordlander, P.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[CrossRef] [PubMed]

Ongarello, T.

Pardo, F.

Park, C.-H.

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(21), 23769–23777 (2012).
[CrossRef] [PubMed]

Y.-T. Yoon, C.-H. Park, and S.-S. Lee, “Highly efficient color filter incorporating a thin metal-dielectric resonant structure,” Appl. Phys. Express 5(2), 022501 (2012).
[CrossRef]

Pelouard, J.-L.

Peng, W.

Rodrigo, S. G.

S. G. Rodrigo, F. J. García-Vidal, and L. Martín-Moreno, “Influence of material properties on extraordinary optical transmission through hole arrays,” Phys. Rev. B 77(7), 075401 (2008).
[CrossRef]

Romanato, F.

Roszkiewicz, A.

Sakat, E.

Shih, M. H.

Soukoulis, C. M.

M. Diem, T. Koschny, and C. M. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
[CrossRef]

Veronis, G.

Z. Yu, G. Veronis, S. Fan, and M. L. Brongersma, “Design of midinfrared photodetectors enhanced by surface plasmonson grating structures,” Appl. Phys. Lett. 89(15), 151116 (2006).
[CrossRef]

Vincent, G.

Weiss, T.

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
[CrossRef] [PubMed]

Wu, Y.

Xu, T.

A. F. Kaplan, T. Xu, and L. J. Guo, “High efficiency resonance-based spectrum filters with tunable transmission bandwidth fabricated using nanoimprint lithography,” Appl. Phys. Lett. 99(14), 143111 (2011).
[CrossRef]

Yoon, Y.-T.

Y.-T. Yoon, C.-H. Park, and S.-S. Lee, “Highly efficient color filter incorporating a thin metal-dielectric resonant structure,” Appl. Phys. Express 5(2), 022501 (2012).
[CrossRef]

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(21), 23769–23777 (2012).
[CrossRef] [PubMed]

Yu, M.

Yu, Z.

Z. Yu, G. Veronis, S. Fan, and M. L. Brongersma, “Design of midinfrared photodetectors enhanced by surface plasmonson grating structures,” Appl. Phys. Lett. 89(15), 151116 (2006).
[CrossRef]

Zhang, B.

Zheludev, N. I.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[CrossRef] [PubMed]

Zhou, W.

Zilio, P.

Zou, H.

Appl. Phys. Express (1)

Y.-T. Yoon, C.-H. Park, and S.-S. Lee, “Highly efficient color filter incorporating a thin metal-dielectric resonant structure,” Appl. Phys. Express 5(2), 022501 (2012).
[CrossRef]

Appl. Phys. Lett. (2)

Z. Yu, G. Veronis, S. Fan, and M. L. Brongersma, “Design of midinfrared photodetectors enhanced by surface plasmonson grating structures,” Appl. Phys. Lett. 89(15), 151116 (2006).
[CrossRef]

A. F. Kaplan, T. Xu, and L. J. Guo, “High efficiency resonance-based spectrum filters with tunable transmission bandwidth fabricated using nanoimprint lithography,” Appl. Phys. Lett. 99(14), 143111 (2011).
[CrossRef]

Appl. Spectrosc. (1)

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

J. Vac. Sci. Technol. B (1)

G. Vincent, S. Collin, N. Bardou, J.-L. Pelouard, and R. Haïdar, “Large-area dielectric and metallic freestanding gratings for midinfrared optical filtering applications,” J. Vac. Sci. Technol. B 26(6), 1852–1855 (2008).
[CrossRef]

Nano Lett. (1)

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
[CrossRef] [PubMed]

Nat. Mater. (1)

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[CrossRef] [PubMed]

Opt. Express (4)

Opt. Lett. (4)

Phys. Rev. B (2)

S. G. Rodrigo, F. J. García-Vidal, and L. Martín-Moreno, “Influence of material properties on extraordinary optical transmission through hole arrays,” Phys. Rev. B 77(7), 075401 (2008).
[CrossRef]

M. Diem, T. Koschny, and C. M. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
[CrossRef]

Other (1)

Lumerical Solutions, http://www.lumerical.com .

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Figures (8)

Fig. 1
Fig. 1

Structure schematic of the proposed metallodielectric free-standing grating and its representative optical phenomenon. (a) Schematic diagram and its structure parameters. (b) Calculated transmission spectrum in TM polarized light and normal incidence with the thickness of SiO2 film H = 800 nm (red line) and H approaching infinite (blue dashed line).

Fig. 2
Fig. 2

Transmission versus wavelength and metal film thickness h at normal incidence (a) in linear color map and (b) in logarithmic color map with fixed P = 500 nm, w = 100 nm and H = 800 nm. Transmission minima in Fig. 2(a) are shown clearly in Fig. 2(b). The vertical white lines in Fig. 2(a) indicate the position of transmission minima.

Fig. 3
Fig. 3

Transmission versus wavelength and SiO2 film thickness H at normal incidence (a) in linear color map and (b) in logarithmic color map with fixed P = 500 nm, w = 100 nm and h = 200 nm. Transmission minima in Fig. 3(a) are shown clearly in Fig. 3(b). The vertical white lines in Fig. 3(a) indicate the position of transmission minima.

Fig. 4
Fig. 4

Spatial magnetic field (|H|) distribution for (a) λD1 = 533 nm, (b) λD2 = 608 nm, (c) λD3 = 686 nm, and (d) λD4 = 764 nm transmission dips at normal incidence. White lines depict schematically the profile of metallic grating and SiO2 dielectric film.

Fig. 5
Fig. 5

Spatial magnetic field (|H|) distribution for (a) λP1 = 584 nm, (b) λP2 = 638 nm, and (c) λP3 = 700nm transmission peaks at normal incidence. White lines depict schematically the profile of metallic grating and SiO2 dielectric film.

Fig. 6
Fig. 6

Transmission versus wavelength and incident angle (a) in linear color map and (b) in logarithmic color map. Transmission minima in Fig. 6(a) are shown clearly in Fig. 6(b). (c) Reflection and (d) absorption versus wavelength and incident angle in linear color map with fixed P = 500 nm, w = 100 nm, H = 800 nm and h = 200 nm.

Fig. 7
Fig. 7

Transmission, reflection and absorption spectra of the proposed metallodielectric free-standing grating at normal incidence for two specific metal film thickness (a) h = 332 nm and (b) h = 668 nm with fixed P = 500 nm, w = 100 nm and H = 800 nm.

Fig. 8
Fig. 8

Spatial magnetic field (|H|) distribution for (a) λA1 = 608 nm, and (b) λA2 = 686 nm high absorption peaks at normal incidence. White lines depict schematically the profile of metallic grating and SiO2 dielectric film.

Equations (4)

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

ε m ( ω )= ε r - ω P0 2 ω( ω+i γ 0 ) - Δ ε 0 Ω 0 2 ω 2 - Ω 0 2 +iω Γ 0
2π λ sinθ-n 2π P =- 2π λ ε m (ω)ε ε m (ω)+ε = k spp n=0,±1,±2,...,±N
φ 12 + φ 23 +k h m =2nπn=0,1,2,...,N
( k 0 2 ε 1 β 2 )H=nπ+arctan ( ε 1 ε 2 β 2 k 0 2 ε 2 k 0 2 ε 1 β 2 ) 1/2 +arctan ( ε 1 ε 3 β 2 k 0 2 ε 3 k 0 2 ε 1 β 2 ) 1/2

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