Spatially coherent surface resonance states derived from magnetic resonances

A thin metamaterial slab comprising a dielectric spacer sandwiched between a metallic grating and a ground plane is shown to possess spatially coherent surface resonance states that span a large frequency range and can be tuned by structural and material parameters. They give rise to nearly perfect angle-selective absorption and thus exhibit directional thermal emissivity. Direct numerical simulations show that the metamaterial slab supports spatially coherent thermal emission in a wide frequency range that is robust against structural disorder.


I. INTRODUCTION
Surface plasmon polaritons (SPPs) can modulate light waves at the metal-dielectric interface with wavelength much smaller than that in free space [1],which enables the control of light in a subwavelength scale for nanophotonic devices [2]. SPPs with large coherent length are useful in many areas, including optical processing, quantum information [3] and novel light-matter interactions [4]. The enhancement of local fields by SPPs is particularly important as it opens a new route to absorption enhancement [5], nonlinear optical amplification [6,7] as well as weak signal probing [8,9]. As the properties of SPP are pretty much determined by the natural (plasmon) resonance frequency, there is not much room for us to adjust the SPP response for practical applications. With induced surface current oscillations on an array of metallic building blocks [10][11][12][13][14][15][16], metamaterial surfaces can manipulate electromagnetic waves in a similar way as SPPs. Such SPPs or surface resonance states on structured metallic surfaces are tunable by geometric parameters.
In this paper, we examine the properties of surface resonance states at a dielectricmetamaterial interface that exhibit magnetic response to the incident waves and strong local field enhancement. We will see that these surface resonance states can give highly directional absorptivity and emissivity, and may thus help to realize interesting effects such as spatially coherent thermal emission. As the structure is very simple, it can be fabricated down to the IR and optical regime [17][18][19].
We will show that a thin metamaterial slab, with a thickness much smaller than the operational wavelength, supports delocalized magnetic surface resonance states with a long coherent length in a wide range of frequencies. Operating in a broad frequency range, these spatially coherent SPPs are surface resonance states with quasi-TEM modes guided in the dielectric layer that are weakly coupled to free space, and the coupling strength can be controlled by tuning structural parameters while the frequency can be controlled by varying structural and material parameters. The high fidelity of these surface resonance states results in directional absorptivity or emissivity, which is angle-dependent with respect to frequency. Finite-difference-in-time-domain (FDTD) simulations verify that the highly directional emissivity from the slab persists in the presence of structural disorder in the grating layer.
Such metal-dielectric-metal (MDM) structures were recognized as artificial magnetic sur-faces with high impedance by the end of last century [12] , the magnetic response were described with an effective permeability in Lorentz type [12,20]. After the concept of metamaterial being proposed [21], P. Alastair and his co-workers numerically and experimentally proved that the ultra-thin MDM structures can resonantly absorb or transmit radiations at low frequency limit [22]. They addressed that the central frequencies of absorption peaks are independent from the incident angle with an interpretation of Farby-Perrot resonant mode (EQ. 1 in Ref. 22). The same group further explored the angle-independent absorption, as the main scenario of the incremental work, by measuring the flat bands of surface wave dispersion in the visible [23] as well as the microwave region [24]. In contrast, we find that the structure with proper design also supports very narrow absorption peaks which are sensitive to the incident angle and obviously do not satisfy to the Fabry-Perot resonance condition suggested in the previous studies.
It is worth noting that the physics origin of an angle-independent peak is quite different from that of angle-dependent ones. The former, investigated in Refs. 22-24, mainly comes from the localized surface resonance states, while the latter, found by us, comes from collective surface resonance states. An intuitive picture is as follows: high order quasi-TEM modes induced inside the dielectric layer can assign phase correlation to the outgoing waves emitted from the air slits of grating, thus are very crucial to the formation of collective response.
Weak enough both the leakage from dielectric layer to air slits and the material absorption, the spatial coherence of surface resonance states will survive. As the interaction between the structure and the incident waves will excite quasi-TEM modes inside the dielectric layer, the magnetic induction must be parallel to the MDM surfaces if it exists. Thus a surface resonance state on a MDM structure is usually magnetic in nature. Our findings about spatially coherent surface resonance states are original compared to the common knowledge, and have great potentials in coherent control of SPPs as well as thermal emission radiations.

II. MODEL DESCRIPTION AND MODE EXPANSION METHOD
Our model system is schematically illustrated in Fig. 1. Lying on thexŷ plane, the slab comprises an upper layer of a metallic lamellar grating with thickness t, a dielectric spacer layer as a slab waveguide with thickness h and a metallic ground plane. The metallic strips are separated by a small air gap g, giving rise to a period of p = a+g for the lamellar grating. Each metallic strip together with the ground plane beneath it constitutes a planar resonant cavity as the building block that gives magnetic responses at cavity resonances [12,16] . The metallic grating is along thex direction so that the guided waves in the dielectric layer (at 0 < z < h in region III) can only couple with the transverse magnetic (T M) polarized wave (with the electric field Einxẑ plane) in the free semi-space (at z > h + t in region I). The geometric parameters of our model are t = 0.2µm, h = 0.8µm, a = 3.8µm, g = 0.2µm and p = a + g = 4.0µm. For a TM incident plane wave with an in-plane wave vector k 0 = k xêx + k yêy , the EM field in region I and in region III can be written in terms of the reflection coefficients r m and the guided Bloch wave coefficients t m [25][26][27][28], as where the term δ m,0 e −ik I zm z r| k 0 denotes the incident plane wave with r| k 0 = are the z components of wave vector for the m th order Bloch eigenmode in region I and region III respectively. ε 0 and ε III are the permittivity of the vacuum and the dielectric slab, µ 0 is the vacuum permeability.
We shall mainly consider infrared frequencies, at which the metals can be well approxi-mated as perfectly electric conductors (PEC). The EM fields at h ≤ z ≤ h + t in region II are squeezed inside the air gaps, in which the magnetic fields can be expressed in terms of the expansion coefficients a l and b l of forward and backward guided waves, as: where r|α l = cos[lπ/g(x + g/2)], (l = 0, 1, ..., n, ...) is the in-plane distribution of guided mode |α l running over all air gaps defined by [29]. q l = ε 0 µ 0 ω 2 − (lπ/g) 2 − k 2 y is the z component of wave vector for the l th guided mode |α l .
We can obtain the coefficients t m (f, k 0 // ) and r m (f, k 0 // ) of the m th guided and reflected waves by applying the boundary continuity conditions for the tangential components of electromagnetic wave fields (over the slits) at the interfaces z = h and z = h + t. Given that surface resonance modes are intrinsic response, we can also assign zero to the incident plane wave and apply the boundary continuity conditions for the tangential components of wave fields to derive the eigen-value equations. A surface resonance state can be determined by searching a zero value/minimum of eigen-equation determinant in the reciprocal space provided that it is non-radiative/radiative with infinite/finite life time below/above light line in free space.

III. ABSORPTION SPECTRA PROPERTIES AND SPACIAL COHERENCE OF MAGNETIC SURFACE RESONACE STATES
We derived the absorption spectra of the slabA includes the contributions from all Bloch orders of reflected waves. As a consequence, A( k 0 , ω) gives information about the surface resonance states as well as the emissivity properties as governed by Kirchhoff's law [30] . We shall assume that the dielectric spacer layer is slightly dissipative by assigning a complex permittivityε III = ε r ε 0 + iσ/ω with ε r = 2.2 and σ = 66.93S/m [Im(ε III ) ≈ 10 −2 ε r ε 0 ] in the calculated frequency regime. In Fig. 2(a), we present the absorption spectra at various incident angles. The spectra exhibit a low and broad peak at 13.2THz which is almost independent of the incident angle, while the other absorption peaks at higher frequencies are narrow and sensitive to the incident angle with a maximum absorption approaching 100%. The slab thus acts as an all-angle absorber at 13.2THz (a similar result can be found in Ref. 31), but exhibits sharp angle-selective  is reduced if the gap size is smaller, as shown with the dashed and dotted lines in Fig. 2(b) for g = 0.2µm and g = 0.1µm at 40THz, which means that the coherent length of the surface resonant modes can be controlled by the gap-period ratio g/p.
To quantitatively characterize the formation of these spatially coherent surface resonance states, we employ the eigenmode expansion method to calculate the surface resonance dispersion (in the limit of no material loss) as shown in Fig. 3(a). The  Figure 5 also indicates that the enhancement of local field of an excited B 3 or B 4 state can be ten times larger than that of an excited B 2 state; the enhancement factor at 50.22THz is about 100, while it is only 10 at 13.2THz.
We see from Fig. 3(a) that the surface resonance dispersion of the slab comes from the interaction between the magnetic resonances and the (folded) light lines L 1 (for dielectrics) and L 2 (for air) grazing on the interfaces. In the limit of a small gap-period ratio (g/p = 0.05 for example), our system is weakly Bragg-scattered, and as such, when a surface resonance state on branches B 3 or B 4 is excited, the induced wave fields inside the dielectric of region III are guided quasi-TEM modes dominated by ±1 st Bloch orders. For that reason, the when the air gap width g ≪ p is satisfied. For the B 2 states, the major Fourier component of the wavefunction is | k i 0 in zero order, and as k III z0 is generally not small, C III 0 is usually very large according to Eq. 3, and the B 2 states leak out easily. The states on branches B 3 and B 4 have major Fourier components in m=±1 order, and as they are asymptotic to the (folded) dielectric light lines L 1 , the absolute value of k III zm (m = +1 for k x < 0 or m = −1 for k x > 0) is very small, resulting in the small coupling coefficients C III −1 or C III +1 . The B 3 and B 4 modes have to travel a long distance before they leak out. They have a long life time and a good spatial coherence. It also explains why the state Γ 3 , a state precisely superposing on folded light line L ′ 1 in dielectric layer, is dark to the incident plane wave as k III zm = 0. Different from B 3 and B 4 states, the B 2 states have a major Fourier component in m = 0 order which directly couples to the free space photons. As a consequence, the B 2 states, forming a flat band far away from the light line L 2 when k 0 is small, are localized with resonant frequency scaled by local geometry of unit cell. The high model fidelity of a B 3 or B 4 state also gives rise to much more intense local field compared to the B 2 states. As shown in Fig. 5, the induced local field is 100 times stronger than the incident field for the state on B 3 ; while it is only 10 times stronger for Γ 2 , and this is consistent with the absorptivity shown in Fig. 2(a). In addition, the coherent length can be adjusted by the gap width as the kernel α 0 | r|k i m is proportional to the gap-period ratio g/p. More calculations demonstrate that the angular FWHM of the absorption peak is reduced from 0.26 o to 0.16 o when the gap is decreased from 0.2µm to 0.1µm, corresponding to a coherent length of 358λ.
We note that most of the attentions in previous studies have been devoted on the localized B 2 states [22][23][24]31] . While the spatially coherent surface resonance states will lead us into a new vision about coherent control of emission radiations. J.-J. Greffet and co-workers showed that highly directional and spatially coherent thermal emission can be obtained by etching a periodic grating structure into a SiC surface [32][33][34][35]. The magnetic resonant modes in our system can do the same, as will be demonstrated below. Our system has the advantage that the operational frequency is tunable by changing the structural parameters, and the operational bandwidth is wide. In addition, our structure supports all-angle functionality for some specific range of frequencies as shown in Fig. 6, although it is periodic only in one direction.

IV. COHERENT THERMAL EMISSION
We performed finite-difference-in-time-domain (FDTD) simulations to emulate the emissions from a slab containing point sources with random phases using the same configuration parameters aforementioned. We purposely put disorder in structure to test the robustness of the phenomena. We assigned two Gaussian distributions(they can be uniform distributions or other types as well) independently to the width of metallic strips and the center positions of air gaps to introduce a 4% (standard deviation) structural disorder. The slab has a lateral size of 60 periods along thex direction. A total of 1200 point sources with random phases are placed at the mesh points inside the dielectric layer. Directional emissions of a wide range of frequencies above 34THz are confirmed by the simulation. The 4% structural disorder has little impact on the directional emissivity. Figs. 6 (a), 6(c) and 6(e) show the far-field emission patterns in thexẑ plane (H-plane) at 40.0THz, 54.3THz and 58.0THz.
The inset in Fig. 6(c) is a control calculation in which the top metal gratings are removed, so that there is just a dielectric layer with random phase sources above a metal ground plane. The directivity of emission from the random sources is lost. Figs. 6(b), 6(d) and 6(f) present the absorptivity (under plane wave incidence) as a function of in-plane wave-vector that there is just a dielectric layer with random phase sources above a metal ground plane. The directivity of emission from the random sources is lost.
(solid angle) at these frequencies. The strong angle selectivity of the absorption is evident, and by Kirchhoff's law, the thermal emission should also be highly directional, which is a direct consequence of the good spatial coherence of the surface resonance states. As shown in Figs. 6(b), 6(d) and 6(e), the absorption/emission peaks generally trace out an arc in the k x ∼ k y plane, but near 54.3THz[ Fig. 6(d)], the dominant emission beam is restricted to a small region near the zone center. This is because the Γ 4 state is a minimum point if we consider the band structure in the k x − k y plane. That means that at 54.3THz, we can obtain a directional emission beam not just in the H-plane, but in all directions, although the structure is periodic in only one direction.
We note that there are other schemes to realize coherent thermal radiations, such as by utilizing three-dimensional photonic crystals [36] or one-dimensional photonic crystal cavities [37]. Our metamaterial slab presents a route to achieve linearly polarized coherent thermal emission radiations in a wide frequency range which can be tuned by adjusting structural parameters and material parameters.

V. CONCLUSION
In summary, we proposed a simple metamaterial slab structure that possesses spatially coherent magnetic surface resonance states in a broad range of frequencies. These states facilitate nearly perfect absorption in a thin metamaterial slab containing slightly absorptive materials. As the absorption spectrum is highly angle-selective, the slab should give directional thermal emission. Direct FDTD simulation with random-phase sources corroborates the existence of strong angular emissivity even in the presence of structural disorder. As the surface resonances originate from artificial resonators, the operational frequency and the response can be tuned by varying the structural configurations. The simple metamaterial structure may be a useful platform to realize the coherent control of thermal emissions, optical antennas, infrared or THz spectroscopy as well as photon detector.