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We show that a periodic array of metal-insulator-metal resonators can be described as a high refractive index metamaterial. This approach permits to obtain analytically the optical properties of the structure and thus to establish conception rules on the quality factor or on total absorption. Furthermore, we extend this formalism to the combination of two independent resonators.
© 2013 Optical Society of America
1. Introduction
Perfect absorbers based on subwavelength metamaterials have attracted much attention for a decade. In particular, plasmonic resonators based on vertical or horizontal Metal-Insulator-Metal (MIM) cavities have been shown to exhibit angularly independent, spectrally tunable, nearly total absorption, in strong subwavelength volumes [1–9]. They have been proposed for various applications such as spectrally signed high performance infrared sensors [1], solar-cells [10], bio-sensors [11] or non reflective coatings [12]. Due to their small size, several resonators can be combined in the same subwavelength period, with almost uncoupled resonances, leading to photon sorting [6, 13], and broadband tailored absorption [7, 8, 12].
Inspired by the work of Shen et al. [14], we describe periodic arrays of MIM cavities as an equivalent metamaterial, consisting of a single homogeneous layer deposited on a perfect conductor. The effective index and effective thickness of this layer are derived analytically from the parameters of the resonators. Based on this analogy, we deduce a simple expression for the spectral response of MIM resonators. These resonators can be individually and quickly designed using this formalism. This analytical model also permits a deeper insight on the parameters acting upon the quality factor of the resonance. Moreover we show that several MIM ribbons combined in the same subwavelength period are almost independent and that their spectral response can thus also be analytically predicted.
2. Metamaterial equivalence
In their seminal work, Shen et al. [14] described a mechanism to design high refractive index metamaterial exploiting Fabry-Perot like resonances in perfect metallic gratings. Based on heuristic considerations about the energy stored in the slits and the energy flux entering the system, they have shown that a grating of perfect metallic slits in transverse magnetic polarization is equivalent to a homogeneous layer of thickness t̄ and of effective index n̄. In this paper we extend Shen’s approach to subwavelength periodic arrays of real (i.e. lossy) MIM absorbers, which is an important step towards the conception and applicability of such integrated structures.
Such a structure, represented in Fig. 1(a), is made of an insulator ribbon of index nI, width w and thickness hI, sandwiched between a gold substrate and an upper gold ribbon of the same width. The thickness of this latter has almost no influence on the optical properties of the structure as soon as it is larger than the skin depth δ; its value is set at 50 nm in the following of this paper (δ ≃ 25 nm for gold at λ = 10 μm). The ribbons are periodically arranged with a subwavelength period d. The computed spectral response of such a structure for a normally incident transverse magnetic (TM) polarized wave, with parameters nI = 4 (corresponding to germanium in the infrared range [15]), w = 1.087 μm, hI = 200 nm, d = 3.8 μm, is given in Fig. 1(c) in blue. It was computed with a B-spline modal method on a non uniform discretization that allows fast and accurate computations [16]. The optical index of gold is described by a Drude function: εAu = 1 − [(λp/λ + iγ)λp/λ]−1, with λp = 159 nm and γ = 0.0048 [6].
Fig. 1 (a) Scheme of a periodic MIM resonator of width w, dielectric thickness hI, period d. Metal is taken as gold, and the upper layer has a thickness hAu = 50 nm. (b) The MIM structure is equivalent to a half MIM resonator with an extremity closed by a perfect mirror and disposed with a period d/2 (c) Equivalent structure made of an effective medium of index n̄ + ik̄, and thickness t̄ on a perfect mirror index (d) Computed absorption spectra of a MIM structure with parameters nI = 4, w = 1.087 μm, hI = 200 nm, d = 3.8 μm in blue straight line and of the equivalent metamaterial with n̄+ik̄ = 33.76+0.508i and t̄ = 74 nm in red dashed line. The incoming light is normally incident and TM polarized.
The response exhibits a nearly total absorption at λr = 10 μm. It is due to a Fabry-Perot like resonance of the guided mode in the insulator layer. Its resonance wavelength is roughly given by λr = 2neffw + λϕ, where λϕ accounts for the phase shifts and neff is the effective index of the guided mode in the MIM cavity approximated by [17]:
The physical mechanism in MIM resonators (i.e. a Fabry-Perot resonance) is similar to the one which occurs in slits. This indicates that the formalism developed by Shen et al. [14] could be extended and adapted to these MIM cavities in order to describe them as an equivalent metamaterial. However, the resonator geometries is not exactly the same in both structures, and to extend the formalism, we have to determine which geometry of slits would give the same behavior as in MIM. To do so, we first take into account the symmetry of our resonators: they present two planes of symmetry α and β represented in dashed line in Fig. 1(a). We propose to limit the problem to an unit cell of width d/2 with a half resonator. Since in MIM structure the in-plane electric field component is zero on α, a perfect conductor can be introduced here without changing the optical response. Thus in a period d, we have two Fabry-Perot resonators, which appear following Shen’s formalism, similar to a grating of grooves having a period d/2 deposited on a perfect conductor, see Fig. 1(b). The electric and magnetic fields plots (data not shown) for this geometry of slits are very similar to those obtained for the initial MIM resonator geometry, which underpins the analogy between these two structures. This slits grating can be described by an equivalent layer (see Fig. 1(c)), and we propose to write its index as:
The period of the new unit cell is d/2. Besides, metal losses have to be taken into account in order to describe nearly perfect absorbers. They are included in our model through the complex effective index of the MIM cavity given by Eq. (1). The metal skin depth δ was added to hI to take into account the penetration of the electromagnetic wave inside the non-perfect metal. Besides, so that the structure has a resonance at the same wavelength λr as the MIM cavity, the thickness of the equivalent layer is:For the geometrical parameters chosen for previously mentioned MIM structure, the equivalent homogeneous layer has an index n̄+ik̄ = 33.76+0.508i and a thickness t̄ = 74 nm. It must be highlighted that n̄ is much higher than natural refractive index in the mid-infrared range. The reflexion coefficient amplitude r of the equivalent structure for a TM-polarized, normally incident plane wave is written as:
where r12 = (1 − n̄ − ik̄)/(1 + n̄ + ik̄) is the classical Fresnel coefficient. The absorption spectrum A of the effective layer computed thanks to Eq. (4) (i.e. A = 1 − |r|2) is plotted in red dashed line in Fig. 1(d) and compared to the numerical calculations made for the grating of MIM ribbons. The perfect agreement between the two curves validates our extension of Shen’s formalism for horizontal MIM resonators with lossy metals.Our model can in fact describe a large variety of structures such as C-shape MIM resonators (e.g. MIM resonators, in which one side is closed as represented in Fig. 2(a)), or vertical MIM cavity (e.g. slits in a gold layer deposited on a perfect mirror as shown on Fig. 2(c)), and even resonators whose insulator presents losses (i.e. imaginary part of nI is positive), which are interesting for optoelectronic devices [1]. As an illustration, Fig. 2 represents the spectra computed numerically (red) and with our model (dashed blue) of (b) a C-shape MIM resonator whose insulator presents losses: nI = 4 + 0.2i and (d) MIM slits.
Fig. 2 (a) Scheme of a periodic C-shape MIM resonator of width w, dielectric thickness hI, and period d. (b) Computed absorption spectra of a C-shape MIM structure with parameters nI = 4 + 0.2i, w = 0.552 μm, hI = 250 nm, d = 0.7 μm in red and of the equivalent metamaterial with n̄+ik̄ = 9.98+0.61i and t̄ = 251 nm in blue dashed line (c) Scheme of a periodic slit MIM resonator of depth w, dielectric width hI, and period d (d) Computed absorption spectra of a slit MIM structure with parameters nI = 4, w = 0.55 μm, hI = 200 nm, d = 1.9 μm in red and of the equivalent metamaterial with n̄ + ik̄ = 34.9 + 0.63i and t̄ = 72 nm in blue dashed line
Our analytical model is much faster than a numerical computation, and it is shown in the next sections for the case of MIM resonators that it can be used for a rapid conception of a nearly-total absorbing structure, including the tuning of its quality factor.
3. Total absorption condition
The analytical model offers the possibility to investigate directly the influence of the various parameters (i.e. nI, hI, d, w) on the spectral response of the MIM resonators and in particular, the conditions necessary to perform nearly total absorption, r ≃ 0. By neglecting the low imaginary part of r12 in Eq. (4), there is nearly total absorption when λr = 4n̄t̄, and r12 = −e−πk̄/n̄. The first condition sets the Fabry-Perot resonance wavelength, while the second condition corresponds to the impedance matching between the MIM resonators and the free space. Using the approximate expression of the Fresnel coefficient r12 and Eq. (2), this gives a condition on the metamaterial index:
As a consequence, for a given choice of wavelength, metal and cavity parameters (i.e. hI and nI), there exists only one period that permits to obtain a nearly total absorption. To illustrate this conception law, Fig. 3 represents the period values computed both analytically (plain lines) and numerically (stars, A > 99.9%) that are needed to obtain a nearly total absorption at 10 μm, as a function of the dielectric thickness hI, for various nI. The agreement is rather fair except when the period is greater than 8 μm. This is explained by the fact that its value goes near the resonance wavelength λr. Indeed surface plasmons excited by the grating are known to couple strongly with cavity resonance [18], and this physical phenomenon is not taken into account in the proposed metamaterial formalism.Fig. 3 Period-insulator thickness abacus for various insulator index nI giving the nearly total absorption condition at 10 μm for a MIM resonator according to our analytical model (plain lines). Comparison with the period that allows to perform numerically 99.9% absorption (stars).
4. Quality factor of MIM resonators
The quality factor Q = λr/FWHM (where FWHM is the full width at half maximum) is an essential characteristic of a resonator, which determines how losses are managed in the system. Thanks to the previous analysis, we show how this quality factor can be engineered in MIM ribbons. The quality factors of MIM structures exhibiting a nearly total absorption at 10 μm for insulator index of nI = 4 (red) and nI = 2 (blue) calculated numerically (stars) and analytically (circles) are represented as a function of hI on Fig. 4(a). It must be emphasized that the quality factor appears to depend linearly on the insulator thickness and to be independent of nI. As an illustration, Fig. 4(b) shows the spectra of three of such resonators for various values of hI, and nI. This is consistent with the expansion at first order of Eq. (4), which gives the absorption spectra. Indeed, from this formula, the quality factor can be fairly approximated by Q ≃ neff/4keff. By an expansion at first order of Eq. (1), it comes that , which explains the linear dependence on hI, the independence on nI and d and the range of the obtained values. To conclude, the quality factor of these MIM resonators, limited by metal losses [19] can be linearly tuned thanks to hI in the range from 10 to 25.
Fig. 4 (a) Quality factor of various MIM structures with nI = 4 (red) and nI = 2 (blue) versus hI, obtained numerically (stars) and analytically (circles). One can note that the quality factor depends linearly and only on hI. (b) Analytically calculated spectra of four MIM structures. Their period d and width w have been optimized to perform nearly total absorption at 10 μm.
5. Combination of resonators
Recently the possibility to combine two MIM resonators in the same period d was proposed [6, 7]. Figure 5(a) represents such a structure, named biMIM, that combines two MIM of respective width w1 and w2 within the same subwavelength period d, and separated by a distance l = (d − w1 − w2)/2. To avoid a slight coupling between the two MIM, the distance l between them must be greater than 300 nm. It is not possible to describe such biMIM structure by a single equivalent layer as previously done.
Fig. 5 (a) Scheme of a structure combining two MIM resonators of width w1 and w2 inserted in the same subwavelength period d. (b–d) Absorption spectra computed numerically (blue plain line) and analytically (blue dashed line) for various values of w1 and w2. The insulator thickness, index, and period are respectively hI = 250 nm, nI = 4, and d = 5.8 μm. Resonators widths are (b) w1 = 0.987 μm, w2 = 1.216 μm, (c) w1 = 1.073 μm, w2 = 1.13 μm, and (d) w1 = 1.079 μm, w2 = 1.125 μm.
Nevertheless it has been shown theoretically as well as experimentally that the absorption spectrum A of such biMIM structures exhibit two peaks at the resonance wavelengths of the single MIM resonators taken alone in the same period d (we call their individual spectra A1 and A2). Indeed thanks to the localized nature of Fabry-Perot resonance, the two resonators behave independently in the biMIM structure. First, we propose to treat them separately thanks to our model (i.e. to compute analytically A1 and A2). Then, to express the probability (1 − A) for a photon to escape from absorption by the biMIM as the product of the independent probabilities (1 − A1) and (1 − A2) to be non absorbed by the single MIM resonators taken separately. This leads to the following law:
The analytically computed spectra of three biMIM structures in which the relative differences of the peak wavelengths Δλr/λr are respectively 0.2, 0.1 and 0.05, are represented in Figs. 5(b–d) for insulator thickness, index, and period of respectively hI = 250 nm, nI = 4, and d = 5.8 μm. We add, on the same figures, the spectra of the biMIM calculated numerically. When the relative difference in resonance wavelength is large (see Figs. 5(b–c), where Δλr/λr is respectively 0.2, and 0.1), our fully analytical model fits perfectly the biMIM behavior and confirms the validity of Eq. (6). Slight discrepancies appear when Δλr/λr decreases (typically Δλr/λr ≤ 0.05, see near the resonance peaks on Fig. 5(d)), which evidences a weak coupling between the two resonators. The analytical model for one MIM resonator used together with Eq. (6), which is based on their independence when they are inserted in the same period, allow the fast conception of dual band absorbers.6. Conclusion
We propose an analytical formalism able to describe a plasmonic MIM absorber as a high index metamaterial layer. It allows to determine the optical properties and to obtain simple conception rules. Nearly total absorbing structures with a given quality factor can thus be designed very fast. Eventually, we have shown that the optical properties of the combination of two independent MIM resonators within the same subwavelength period can also be predicted. We expect these engineering rules to be helpful for the conception of absorbing devices such as infrared photodetectors, solar cells or biosensors.
Acknowledgments
The authors acknowledge Benjamin Portier for fruitful discussions. This work was supported by the ANR project INTREPID, the Carnot project ANTARES, and PRF SONS.
References and links
1. J. Le Perchec, Y. Desieres, and R. E. de Lamaestre, “Plasmon-based photosensors comprising a very thin semiconducting region,” Appl. Phys. Lett. 94, 181104 (2009). [CrossRef]
2. J. Hao, J. Wang, X. Liu, W. J. Padilla, L. Zhou, and M. Qiu, “High performance optical absorber based on a plasmonic metamaterial,” Appl. Phys. Lett. 96, 251104 (2010). [CrossRef]
3. J. Hao, L. Zhou, and M. Qiu, “Nearly total absorption of light and heat generation by plasmonic metamaterials,” Phys. Rev. B 83, 165107 (2011). [CrossRef]
4. P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011). [CrossRef]
5. F. Pardo, P. Bouchon, R. Haïdar, and J. Pelouard, “Light funneling mechanism explained by magnetoelectric interference,” Phys. Rev. Lett. 107, 93902 (2011). [CrossRef]
6. C. Koechlin, P. Bouchon, F. Pardo, J. Jaeck, X. Lafosse, J. Pelouard, and R. Haidar, “Total routing and absorption of photons in dual color plasmonic antennas,” Appl. Phys. Lett. 99, 241104–241104 (2011). [CrossRef]
7. P. Bouchon, C. Koechlin, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Wideband omnidirectional infrared absorber with a patchwork of plasmonic nanoantennas,” Opt. Lett. 37, 1038–1040 (2012). [CrossRef] [PubMed]
8. M. Nielsen, A. Pors, O. Albrektsen, and S. Bozhevolnyi, “Efficient absorption of visible radiation by gap plasmon resonators,” Opt. Express 20, 13311–13319 (2012). [CrossRef] [PubMed]
9. J. Yang, S. Sauvan, A. Jouanin, S. Collin, J.-L. Pelouard, and P. Lalanne, “Quality factor of metal-insulator-metal nanoresonators,” Opt. Express 20, 16880–16891 (2012). [CrossRef]
10. Y. Wang, T. Sun, T. Paudel, Y. Zhang, Z. Ren, and K. Kempa, “Metamaterial-plasmonic absorber structure for high efficiency amorphous silicon solar cells,” Nano Lett. 12, 440–445 (2012). [CrossRef]
11. A. Cattoni, P. Ghenuche, A. Haghiri-Gosnet, D. Decanini, J. Chen, J. Pelouard, and S. Collin, “λ3/1000 plasmonic nanocavities for biosensing fabricated by soft uv nanoimprint lithography,” Nano Lett. 11, 3557–3563 (2011). [CrossRef] [PubMed]
12. Z. Jiang, S. Yun, F. Toor, D. Werner, and T. Mayer, “Conformal dual band near-perfectly absorbing mid-infrared metamaterial coating,” ACS Nano 5, 4641–4647 (2011). [CrossRef] [PubMed]
13. J. Le Perchec, Y. Desieres, N. Rochat, and R. Espiau de Lamaestre, “Subwavelength optical absorber with an integrated photon sorter,” Appl. Phys. Lett. 100, 113305–113305 (2012). [CrossRef]
14. J. Shen, P. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. lett. 94, 197401 (2005). [CrossRef] [PubMed]
15. E. Palik and G. Ghosh, Handbook of optical constants of solids (Academic press, 1985).
16. P. Bouchon, F. Pardo, R. Haïdar, and J. Pelouard, “Fast modal method for subwavelength gratings based on b-spline formulation,” J. Opt. Soc. Am. A 27, 696–702 (2010). [CrossRef]
17. S. Collin, F. Pardo, and J. Pelouard, “Waveguiding in nanoscale metallic apertures,” Opt. Express 15, 4310–4320 (2007). [CrossRef] [PubMed]
18. P. Jouy, A. Vasanelli, Y. Todorov, A. Delteil, G. Biasiol, L. Sorba, and C. Sirtori, “Transition from strong to ultrastrong coupling regime in mid-infrared metal-dielectric-metal cavities,” Appl. Phys. Lett. 98, 231114–231114 (2011). [CrossRef]
19. J. Khurgin and G. Sun, “Scaling of losses with size and wavelength in nanoplasmonics and metamaterials,” Appl. Phys. Lett. 99, 211106–211106 (2011). [CrossRef]
