A double-layered metallic grating (metal-dielectric-metal) with a complementary capacitive (isolated discs) / inductive (connected film with apertures) structure exhibits multiple infrared transmission resonances peaks with up to 70% at wavelength ranges corresponding to local modes for geometric dimensions less than a wavelength. The period, dielectric thickness, refractive index and unit cell size of the periodic structure modulate the local mode positions and amplitudes. The electromagnetic field distribution and energy flow in the structure explain the relation of transmission resonance, local modes, and distributed surface plasma wave modes.
© 2007 Optical Society of America
Periodic structures fabricated on dielectric or semiconductor substrates with unit cell dimensions less than the wavelength of incident light have been the subject of intense investigation both theoretically and experimentally as a result of increasing interest in metamaterials and photonic crystals. These structures demonstrate many remarkable optical properties, such as large photonic crystal band gaps , negative refractive indices [2–4], extraordinary optical transmission properties [5–7], nonlinear optical properties  and strong coupling effects between electronic and photonic resonances . Surface plasma waves (SPWs), electromagnetic surface excitations localized near the metal-dielectric interfaces originating from the free electrons in the metal, coupled with modes due to the local structure are the major physical mechanisms controlling these effects.
Christ et al. proposed one-dimensional periodic gold nanowire arrays on top of a dielectric waveguide supporting the coupling of plasmon resonances to optical waveguide modes . Fan et al. present a coaxial metallic array with high transmission (>50%) due to the coupling between localized modes and SPWs [6,7]. In this paper we present experimental and theoretical results for the complementary double layered 2D metallic grating shown in Fig.1. Each disc in the bottom gold layer is exactly under each hole in the top layer; so that at normal incidence, there is no direct line of sight past the two metal layers. This structure is much different from metal structures with apertures that can let light pass directly through. Even though the structure when viewed at normal incidence appears opaque, a transmission efficiency of as high as 70% is achieved at resonant incident wavelengths. The SPWs bounded on the top and bottom metallic gratings and the resonant localized modes confined within the complementary unit are the main factors controlling the transmission properties. In this paper, experimental and simulation results for five samples with varying pitch and dielectric thickness are reported. Transmission spectra, measured by FTIR, are compared at normal incidence and as a function of the angle of incidence for TM polarization. Rigorous coupled wave analysis (RCWA) simulations fit the experimental transmission spectra very well, and elucidate the relation between the local resonant wavelengths and the structure parameters. Increases of pitch, of the diameter of the holes, of the dielectric thickness and of the refractive index of the matching liquid, red shift the localized modes confirming the existence of local fields confined in the unit structure. The electromagnetic field distribution on the top-hole layer and the bottom-disc layer and the energy flow through this structure are simulated.
The process for fabricating large-area double layered metallic gratings with a complementary structure starts with a glass substrate (BK7), coated with spin-on glass (SOG). After baking at 300°C for one hour on a hot plate, the SOG is solidified into a soft glass. A bottom anti-reflection coating (BARC) and positive photoresist (PR) SPR 510 are spun on top of the SOG layer. Two dimensional interferometric lithography  using a 355-nm UV source is applied to pattern a large area (2×2 cm2) periodic array of discs in the PR. After developing the PR, Cr is deposited on the wafer by e-beam evaporation, followed by a lift-off process leaving a 2D array of holes in the Cr. Reactive ion etching (RIE) is used to remove the BARC (O2) and SOG (CHF3 and O2) in the holes. Then, an O2 plasma is used to etch away the remaining BARC, thus removing the sacrificial Cr mask. A 5-nm layer of Ti and 95-nm layer of Au are deposited by e-beam evaporation. A scanning electron micrograph (SEM) and a schematic diagram of the structure of one sample are shown in Fig. 1 (a, b). The pitch of grating is controlled by the incident angle between the two optical beams during interferometric lithography and the size of the opening is controlled by the exposure time. The SOG thickness is determined by coating times and spin speed. Finally, a refractive index matching liquid with n = 1.4 at 633 nm is used to cover the sample which is then clamped with another BK7 glass plate as shown in Fig.1(c). This is done in order to ensure degeneracy for all of the single-surface (glass-metal) SPWs at the interfaces to simplify the structure and the analysis. In this paper, we only report experiments and simulations on the sandwich structure shown in Fig.1(c).
The parameters of five samples fabricated with varying pitch and SOG thickness are listed in Table I. The thickness of each Au layer was fixed at 100 nm. The relevant parameters in determining the behavior of the structure are the period of the grating (Λ), the SOG thickness between Au layers (T), the diameter of each hole (D) and the refractive index of the dielectric (n). The average hole diameter is approximately 0.68 μm which is very large compared with the scale of the pitch, thus all peaks due to SPWs and order local modes are observed in transmission. The transmission spectra are measured with a Fourier transform infrared spectrometer (FTIR) using unpolarized normally incident light from an incoherent source. The transmission spectra are shown in Fig. 2, and are normalized to the transmission (approximately 90%) of two bare glass substrates with matching liquid between.
Where Λ is the pitch of the grating, εm(λ), εd are the dielectric constants of the metal and dielectric, respectively. εm(λ) is expressed by the Drude formula, Eq. (2),
where λ p, λ γ are electron plasma wavelength and scattering wavelength. The pitch, the permittivity of incident medium and substrate and the polarization angle of incident light influence the resonance wavelength.
The resonant wavelength λ due to SPWs is a monotonically increasing function of the pitch and is related to the incident angle in Eq (1) which can be used to identify the SPW peaks. This resonance is at 1.176 μm for Λ = 0.8 μm, 1.464 μm for Λ = 1.0 μm and 1.75 μm for Λ = 1.2 μm with n = 1.47 at normal incidence. In experimental spectrum of sample B as shown in Fig. 2(a), three dominant transmission peaks are observed: 20% transmission at 1.19 μm, 14% transmission at 1.47 μm and 67% at 2.40 μm. When the SOG layer thickness (approximately 0.68 μm) is fixed but the pitch is varied (0.8-, 1.0- and 1.2-μm), the peaks red shift: (0.923-μm, 1.22-μm, and 2.2-μm), (1.19-μm, 1.47-μm, 2.40-μm) and (1.49-μm, 1.84- μm, 2.54-μm). Additional short wavelength peaks in the sample C spectrum compared with three peaks in A and B spectra are due to the second order SPW coupling. Only the middle series of peaks are close to the calculated wavelength. The longer wavelength peaks which move in much smaller steps than the short and middle wavelength peaks with the increase of pitch are localized mode transmission peaks which are affected, but not induced, by SPWs. The left and middle peaks show weak transmission around 20-30%. The right peaks have the strongest transmission varying from 32.5%, 66.7% to 70% with increasing pitch. The transmission is enhanced because the SPW resonance is very close to the local mode.
The experimental spectra for samples D, B and E with a fixed 1-μm pitch but varying dielectric slab thickness 0.5-, 0.7- and 1.5-μm are shown in Fig. 2(b). The spectrum of an single Au layer hole array with the same pitch only has a single ~30% transmission peak at 2.25 μm due to the coupling of SPW to localized a mode confined in the hole . While the double layered complementary structure exhibits multiple transmission peaks as high as 70%. The two shorter wavelength peaks of the three spectra around 1.15- and 1.47-μm are independent of the dielectric thickness, while the long wavelength peaks strongly red shift: 1.8-μm, 2.40-μm and 3.9-μm, also another peak appears at 2.29 μm in the spectrum of sample E. The absorption of atmospheric gases and of the refractive index matching liquid causes the noise at 3.32- and 4.4-μm.
To understand the relations between transmission peaks and the SPW and localized modes, the angular dependence of the transmission peaks is examined. A linear polarizer is set in front of the FTIR. The sample filled by matching liquid is fixed on an angular rotator. The measurements in Fig. 2 are repeated as a function of the sample tilt from the normal for TM polarization. The relation between the photon energy (in eV) corresponding to the transmission peaks and the incident angles are shown in Fig. 3 (a), (b) and (c) which correspond to samples D, B and E. The transmittance associated with the coupled SPWs will show a strong dependence on θ as in Eq. (1) while the local modes are relatively independent of θ.
In Fig. 3, the colored dots correspond to measured transmission peaks, while the black lines represent the calculated SPW resonance wavelengths as a function of incident angle. A stack matrix method  is used to simulate the modes for the multiple interfaces (ignoring the periodic structure). For a very thin layer Au slab (less than 30nm), the degenerate electromagnetic waves on the top and bottom surfaces interact, and split . Four modes form with two Au layer slabs sandwiching a dielectric slab . When the thickness of Au and dielectric slabs increases, the four modes are all degenerate. Since both thick Au (100nm) and dielectric (>500nm) slabs are used in this work, the coupling of the four metal-dielectric surface modes is very weak and is largely negligible. The simulated SPW lines are almost the same as that of a single interface. The red and yellow dots follow the K+ and K- SPWs lines at large incident angle, corresponding to the ± signs in Eq. (1). However, they are not degenerate but separate at θ = 0. This is attributed to a repulsive interaction between the top and bottom SPWs with a π phase shift. The modes marked by blue dots are almost independent of θ and are localized modes resulting from vertical Fabry-Perot resonances, sequenced in series number. The red dots and the blue dots below SPW converge from 0 degree to 45 degree and then diverge at larger angles, due to the coupling between the SPW modes and the localized modes . There are some smaller marks in additional colors representing observed transmission peaks with low efficiency that are not assigned.
The simulation is based on a rigorous coupled wave analysis (RCWA) , which is commonly used for the analysis of scattering of electromagnetic waves from periodic structures. The round apertures/discs were replaced in the simulation with square apertures/discs of the same area for numerical convergence reasons . In Fig. 4, the simulated spectrum fits the experimental spectrum of sample E very well with resonant wavelengths at 0.94-, 1.45-, 2.29- and 3.9-μm. We have previously found that using a scattering frequency three times higher than literature values provides a good fit to our experiments, accounting for both fabrication inhomogeneity across the large area and additional scattering in these very thin non-optimized films .
Varying the main parameters in the RCWA simulation, such as the dielectric thickness, the refractive index of the matching liquid and the hole diameter, tracing the localized modes wavelengths in transmission spectra, we can get the relations between the localized mode positions and geometric parameters. In Fig. 5(a), with the increase of the SOG slab thickness, the localized modes wavelengths shift to longer wavelengths and more orders of localized modes appear. For different thickness samples (D, B, E), the first order localized modes are at (0.67 eV, 0.52 eV, 0.323 eV), the second order at (0.84 ev, 0.82 eV, 0.54 eV), a third order peak at 0.88 eV was only observed for sample E [Fig. 3(c)]. Plotting a trend line for each order simulation marks, we find that the resonance wavelength of localized modes is a linear function of the dielectric slab thickness. The slope of the second order trend line (1.1) is about a half of that of the first order (2.223). The slope of the third order line (0.7) is about one third that of the first order. The slight experimental variations of hole sizes result in the displacement between simulation and experiment. In Fig. 5(b), the experimental and simulation results of the first order of sample B are marked with varied refractive indices of matching liquid. The effect of hole diameter is only checked by simulation. The localized mode position is linear with the refractive index of the filled materials and hole diameter.
From the above analysis, the relation between the local mode positions and geometric parameters can be roughly expressed in Eq. (3).
Where n is the refractive index of the matching liquid, T is the dielectric thickness between the two Au slabs, D is the diameter of the hole, m represents the integer order, and Δ is the wavelength shift due to other factors.
The electrical (E) and magnetic (H) fields at each point in one unit cell of sample E are extracted by RCWA. The propagation factor exp(±γL) in the transfer matrix becomes large/small for thick and/or lossy dielectric layers which results in numerical difficulties. Using a scattering matrix method from the front medium to the back medium can solve this numerical issue . The magnitudes and phases of E and H in the mid-plane of the upper Au hole and in the mid-plane of the lower Au disc at λ = 3.9 μm are plotted in Fig. 6. We define the incident electromagnetic fields Ex = 1 and Hy= 1 at normal incidence. The electric and magnetic fields are confined in the Au hole and outside of the Au disc. The field distributions are similar to the fields propagating through a single-layer hole and a single-layer disc structure, respectively.
Compared with the magnitude of the field distributions in the Au layers, the fields in the dielectric slab are roughly constant with phase oscillating in the propagation direction. The forward and backward waves interact, forming a standing wave as shown in Fig.7. The incident electromagnetic waves illuminate the top Au hole, oscillate between the two Au layers, go around the Au discs and radiate from the substrate. The average energy flow is constant in the incident medium and transmitted medium as required, but decay very rapidly in the vicinity of the lossy Au layers. The small spikes at the interface of Au and dielectric are due to the limited number of diffraction orders retained in the RCWA simulation. The phase increases in the propagation direction, confirm that this structure exhibits positive refraction. The phase propagation in round trip of the first order localized mode within the dielectric slab is close to, but larger than 2π, that of the second order localized mode is close to 4π (not show here). That means that the resonant wavelength is shorter than T/2n, where T is the thickness of the dielectric slab. The high absorption and reflection of Au at longer wavelength makes the transmission peak happen at shorter wavelength. Compared with the standing wave in a single dielectric slab with transmission peaks relate to 2nπ, the Au material and structure will affect the fields and the resonant wavelengths. Using this simulation, we also investigated the effective phase propagation with varying hole size for a fixed pitch. The phase of a larger hole structure with 0.8 μm diameter is 11 which is larger than that of 9.8 (circled in pink in Fig. 7) for a 0.675 μm diameter sample at the output surface Z = 3.7μm in simulation. The light propagates faster for large holes than for small holes with the same pitch. This property can be adopted to make a nano-lens by annularly arranging different hole diameters in the same substrate with a fixed pitch. In our simulation, a 1.4-m focal length would be achieved by a coaxial arrangement with varying diameters.
In conclusion, compared with single-layer Au aperture arrays with ~30% transmission peaks at infrared wavelengths, double-layered Au arrays with a complementary aperture/disc structure with geometrical scales less than the incident wavelength show transmission resonances > 70% due to excitation of surface electromagnetic modes and localized resonant modes even though there is no line-of-sight for direct transmission. The strong coupling between SPWs and localized modes give rise to multiple resonances in this structure. The metallic grating pitch, the thickness between metal slabs, refractive index inside local structure and critical dimension are the main parameters that control the peak positions and transmission amplitudes. The electromagnetic propagation of localized modes within the structure is analyzed. This structure can be used as multiple bandpass filter to select wavelengths with strong transmission efficiency. This structure also modulates the phase propagation by changing the structural parameters which can be applied to fabricate nano-lenses and other nano-optical elements.
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