## Abstract

We show anti-crossings due to strong in-plane coupling of grating excited propagating plasmon modes in dielectric-metal-dielectric structure with 2D dielectric pattern on top. Grating coupled propagating plasmon modes along with their complete dispersion in the measurement range and all different sample orientations are calculated first. Further a coupled mode theory is presented for the specific geometry presented here. Experimentally measured anti-crossing widths are compared with those calculated by coupled mode theory. It is shown that the coupling strength of the plasmon modes and thus the anti-crossing width can be controlled by the orientation of the sample.

© 2013 OSA

## 1. Introduction

Strong coupling of polariton modes results in anti-crossings or avoided crossings in the dispersion plots and are studied in excitonic systems like exciton-photon coupling in cavities and exciton-exciton coupling in coupled quantum dots [1–3]. In the context of plasmons, coupled plasmons are studied where the coupling is between localized (particle) plasmons and propagating plasmons and coupling induced changes in dispersion in planar and corrugated structures [4–8]. In various sub-wavelength structures, these coupled localized and propagating plasmons are studied for basic physics as well as for different applications [9–16]. Also, in metal-dielectric-metal or dielectric-metal-dielectric structures when the middle layer is thin enough, coupling between plasmons at the top and bottom interfaces results in splitting of the modes to symmetric (short range) and anti-symmetric (long range) plasmon modes [17]. The general case of surface plasmon polariton (SPP) excitation in 1-d system when the grating vector is not contained in the plane of incidence has also been considered [18]. To our knowledge, there is no report on in-plane coupled propagating plasmons. We have recently shown grating coupled plasmon modes in dielectric-metal-dielectric layer structure with 2D dielectric grating on top and conditions under which near dispersionless plasmon modes can be excited in such structures [19]. Such excitation geometry would allow the coupling of in-plane propagating plasmons. In this paper we report on strong coupling of in-plane propagating plasmons at the unpatterned dielectric-metal interface originating due to the 2D grating structure on the top in the non-conical excitation geometry. In addition, we show that the coupling strength and thus the anti-crossing gap (split) width can be controlled. We give a model based on coupled mode theory to calculate the anti-crossing width.

## 2. Sample fabrication and measurement geometry

Samples studied have dielectric-metal-dielectric layer structure with 2D air holes in dielectric pattern on top. On quartz substrates, 50nm Gold was deposited by sputtering followed by spin coating of Shipley’s S1805 photoresist on top. By interference lithography shallow 2D gratings of air holes in resist with depth of air holes of about 60 nm were prepared. There is an unpatterned resist of 400 nm thickness on the Gold layer constituting the top dielectric layer above the metal film. AFM and SEM images showed 2D air hole pattern is of rectangular lattice with periods 610 nm (a_{x}) and 625 nm (a_{y}) in the two orthogonal directions. The fill factors (air to dielectric ratio) are 0.65 and 0.7 in x and y directions, respectively. Figure 1 shows the schematic of the sample along with the measurement geometry and the SEM image. Further details about the structure are in Ref.19 and references therein. When the azimuthal angle is, φ = 0°, for TM polarized incident light launched in the plane perpendicular to the 2D pattern, the electric field component has finite component along a_{x} which excites the SPPs. Similarly, for φ = 90° the finite electric field component along a_{y} excites the SPPs. Angle θ is the launch angle in the plane perpendicular to the XY-plane.

## 3. Experimental results

Angle resolved transmission measurements are performed with the launch angle varied from −25° to + 25° range with 0.3° step. A 100 W halogen lamp is used as white light source (400-1000 nm wavelength range) and the output is collimated using a combination of apertures and lenses to have <0.3° divergence. Transmission spectra are recorded using a fiber spectrometer with 0.1nm spectral resolution.

Figure 2 shows the measured dispersion for three different angles φ. The data is normalized with that measured in sample with unpatterned top dielectric so that the dielectric loaded SPPs and/or waveguide modes are normalized to unity and thus do not contribute to the data presented here. It may, any case, be noted that the slab waveguide modes for this particular structure are expected at 526 nm (1,0), 460 nm (1,1), and 530 nm (0,1) while the (1,2) and (0,2) modes are out of range of the experimental wavelength range.

In addition to the propagating plasmons and the waveguide modes, hybrid photonic-plasmonic modes are also reported in structures with unpatterned dielectric layer or patterned dielectrics (photonic crystals) on metal layers [20, 21]. For the pattern under consideration, there are no photonic bandgap regions in the wavelength range of interest. Like in our recent report, we calculated all possible grating coupled propagating plasmon modes and their dispersion for all measured azimuthal angles [19]. Good match between the measured and calculated dispersion of all measured modes including pure grating modes can be seen in Fig. 2. It may be noted that Fig. 2 shows the SPP dispersion in grey scale plot thus the white regions show dips in the transmission spectra corresponding to plasmon excitation. Dashed lines are the fits based on a model that calculates the grating allowed propagating plasmon modes at the three layer structure [19]. Different SPP modes are labeled by (m,n)_{x}. Use of thin metal layer results in coupling of SPP modes at the top and bottom interfaces resulting in symmetric and antisymmetric modes. Thus, modes with subscript “A” are anti-symmetric and those with “S” are symmetric modes. Modes with subscript “d” are Rayleigh anomalies [19]. While for φ ~0° and φ ~90° orientations (not shown) we see the flat dispersion modes, for all azimuthal angles we see the clear anti-crossing signatures. In Fig. 2 anti-crossings are marked by circles with the zoomed in region shown on the right hand side. In the following we give a model to calculate the anti-crossing width and its dependence on the azimuthal angle.

## 4. Coupled mode theory for calculating anti-crossing width

Coupled mode theory is well known for the orthogonal set of waveguide modes, gratings and other systems [22, 23]. Briefly, when one of the modes locally modifies the dielectric constant then this perturbation induces coupling between different orthogonal modes. In such a picture for the slab waveguide geometry (infinite in x- and y-directions and finite in z-direction) for modes propagating along x-direction, the in-plane and out-of-plane coupling constants are well known. For 1-D metal grating structure, such a model has been applied recently [24]. We apply the coupled mode theory to the three layer structure with 2D pattern on top by the following procedure. We first calculate the field expressions for the plasmon (surface) modes possible in a 3-level structure and get the expressions for coupling constants for non-collinearly coupled modes. In the 2nd step, we consider the effect of 2D dielectric pattern on top by introducing the relevant k components. By this, we consider only the grating coupled plasmon modes and their coupling.

In the first step, we consider the 3-level structure and the relevant expressions for the TM polarization components (E^{x}, H^{y}, and E^{z}) and calculate the coupling constants. For a three layer system with a thin metal film sandwiched between dielectric materials, solutions to Maxwell’s equations with appropriate boundary conditions at the interfaces for TM polarized wave are well known [25]. In the three regions, (in the top dielectric, in the metal and in the bottom dielectric), expressions for the three relevant field components (H^{y}, E^{z} and E^{x}) are in 3 unknown constants. We simplify these equations and write them in terms of a single constant which is solved for by using the power normalization relation,

_{β}governs the evolution of the field. When two of the TM modes are coupled, the in-plane and normal to plane coupling constants are given similar to the overlap integrals [26].

In the 2nd step, we extend the model to the present structure of a three layer system with a 2-D shallow grating on top by considering a periodic perturbative correction to the dielectric constant in the x-y plane. The periodic function can be Fourier expanded as,

*K*= 2π/a

_{x}_{x}and

*K*= 2π/a

_{y}_{y}with a

_{x}and a

_{y}being the periodicities in the x and y directions, respectively. On substituting this in the field propagation equation, it reduces to,

_{α}and φ

_{β}are the angles made by the propagation vector of each of the waves with the x-axis and E

^{x}is the electric field component along the propagation direction of each mode. The total coupling constant is the sum of the transverse and in-plane coupling constants. Next we calculate the specific k values allowed by the sample and excitation geometry to find the coupling between coplanar SPP modes. For the three layer system with appropriate boundary conditions when we invoke continuity of the field and its derivative at each interface, we get the SPP dispersion relation given by [25],

The first term in the square root is the k_{x} component and the second term is the k_{y} component. For given structure (that is, a_{x} and a_{y}) and the measurement geometry (θ, φ), we find the k_{in-plane} values satisfying both Eq. (5) and Eq. (6) simultaneously. We calculate the coupling constant between two such resonant modes A1(x) and A2(x) given by,

We solve simultaneously the coupled differential equations (given by Eq. (7)) for these two modes in order to get the splitting due to the coupling. For perfect phase matching i.e. when k_{α} = k_{β} + mK_{x}, splitting comes out to be Δ = 2K_{αβ} in the k_{x} component. It may be seen from Eq. (4) that the azimuthal angle φ gives control on the coupling strength.

In Fig. 2, we can observe anti-crossing feature between different modes and that the splitting related to the anti-crossing changes with φ. It has been shown earlier that the splitting in the energy and momentum could occur due to conservative and dissipative coupling, respectively [24].

As an example, we consider the case of interaction of (0,2) and (1,1) plasmon modes and how the interaction strength changes with φ. The theoretical values of the splitting have been calculated using Eqs. (4)-(7) which are compared with the measured values in Table 1 as a function of φ.

## 5. Conclusion

In summary, we have shown the anti-crossing of grating (2D patterned top dielectric) coupled propagating plasmon modes at unpatterned dielectric-metal-dielectric structure due to in-plane coupling. Coupled mode theory for in-plane propagating plasmons, with the sample and measurement geometry considered, shows that the coupling strength and thus the anti-crossing width are azimuthal angle dependent. A good match with the measured and calculated split gap width for in-plane coupled plasmon modes for different azimuthal angles shows that the in-plane coupling strength can be controlled by the azimuthal angle.

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