1. Introduction

Metamaterials constructed from subwavelength resonator structures exhibit unprecedented electromagnetic properties [1, 2, 3] which are based on the microscopic electric and magnetic response of their constituting elements to an incident light wave. As a result artificially engineered metamaterials make possible a range of novel applications such as perfect lensing [4] or invisibility cloaking [5]. Since their first demonstration at microwave frequencies [6] progress in structure miniaturization allowed to build metamaterials for higher frequencies, ranging from the terahertz [7] over the mid-infrared [8, 9] to the near-infrared regime [10, 11, 12]. Typically, they are made of metallic structures, e.g. split-ring resonators (SRRs) [13, 7], specifically designed to respond resonantly to the electric and the magnetic fields of an incident light wave. Near resonance the induced fields can exceed the driving fields by orders of magnitude and generate a counteracting response of the medium. This can be expressed in terms of a negative effective permittivity ε _eff and permeability μ _eff of the metamaterial and can lead to a negative refractive index [14]. So far, photonic metamaterials are mostly characterized by far-field measurements. While this provides important information about the characteristic resonances and allows to extract the macroscopic quantities ε _eff and μ _eff [15, 10], such far-field investigations do not provide direct insight into the formation and dynamics of the microscopic internal fields responsible for the material’s negative response. An experimental characterization of the involved electric and in particular the magnetic near-fields which are localized to the microscopic building blocks remains highly challenging and current studies rely mostly on numerical simulations.

Only recently scanning near-field optical microscopy (SNOM) methods have been developed to map optical near-fields of metallic structure on the micro- to nanometer scale [16, 17] and were also applied to the investigation of near-fields in metamaterial structures [18, 19]. Although being extremely powerful, SNOM techniques are mainly sensitive to only the longitudinal electric field component (E_z) and can not directly provide the magnetic information. Experimental studies of the entire electromagnetic near-fields in metamaterials are still lacking, mainly due to the difficulties involved in measuring electric and in particular magnetic field components with the required subwavelength spatial resolution.

Here, we employ terahertz time-domain imaging [20, 21, 22] to investigate the complex resonant activity of a planar metamaterial and trace their localized electric and magnetic near-field response. Our approach provides spatially resolved measurements of the amplitude, phase and polarization of the electric field from which we extract the microscopic magnetic near-field signatures in a metamaterial at terahertz frequencies.

2. Experimental setup and sample fabrication

The experimental setup is illustrated in Fig. 1. Terahertz pulses were emitted and detected by photoconductive antennas optically gated by the output of a mode-locked Ti:sapphire laser (< 20 fs, 800 nm, 75 MHz repetition rate). The pulsed THz beam was focused by a pair of off-axis parabolic mirrors to a frequency dependent spot of about 1 mm diameter (at 0.5 THz) at the position of the sample. The transmitted electric field was detected at the backside of the sample in the 10 μm wide photoconductive gap between the H-shaped electrodes on an ion implanted silicon-on-sapphire substrate [23]. The detector chip was mounted with its electrodes and photo-active silicon layer facing the backside of the sample and was optically gated by focusing the probe laser beam through the sapphire substrate (Fig. 1(b)). This allows us to perform measurements in direct proximity of the sample, typically at a distance of 30 μm from its surface. Spatial scans were performed by synchronous translation of the probe laser and the detector chip. This is achieved by a pair of mirrors mounted on separate translation stages arranged in a periscope configuration as described elsewhere [24] (Fig. 2). The spatial resolution of the experiment is mainly limited by the finite dimensions of the probe beam focus and of the photoconductive gap and has been determined to be on the order of 20 μm. Finally, by scanning a variable delay line in the detector beam path the temporal waveform of the electric field was recorded in each spatial pixel.

 

Fig. 1. THz near-field microscope. (a) Overview of the setup. The output of a mode-locked Ti:sapphire laser is split by a beam splitter (BS) into excitation and probe beam which are focused by lenses (L) onto a photoconductive THz emitter and a detector, respectively. The emitted THz pulses are focused by a pair of off-axis paraboloidal mirrors M₁ and M₂ onto the sample. The THz electric near-field is spatially resolved at the backside of the sample by raster-scanning the detector unit along the x- and y-axis. (b) Cross section of the detector chip with the sample for a measurement of the x-component of the electric field.

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Fig. 2. Photograph of the detector unit. Two dimensional raster scans are performed by moving the detector mount with detector chip (D), the laser focusing lens (L) and the probe laser (red line) via the periscope mirrors (P) in x- and y-directions relative to the stationary sample (S) and THz beam (green line).

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The transmitted electric field is detected at the backside of our sample which is positioned in the focus of a pulsed THz beam. We perform measurements at a typical distance of 30 μm away from the sample surface, corresponding to λ/25 at 400 GHz, the highest frequency considered in this study. The incident terahertz beam has a fixed polarization along the x-axis. The linearly polarized detector is oriented to be sensitive to both, the x- and the y-component of the transmitted electric field by rotating the entire detector chip by 90 degrees. Spatially resolved measurements are performed by moving the detector (D) together with the probe laser beam and the focusing optics (L) in x- and y-directions relative to the stationary sample (S) by the movable detector mount shown in Fig. 2. At each spatial position the time dependent electric field is measured by scanning an optical delay. From two separate measurements of the electric field polarizations E_x(x,y,t) and E_y(x,y,t) the in-plane vector field E_xy(x,y,t) can be reconstructed. By Fourier transforming the time traces we obtain the spectral amplitude and phase distributions. In order to study the oscillation of the vector field at an individual frequency ν ₀, the amplitude and phase at the selected frequency, E_xy (x,y, ν ₀) and ϕ_xy (x,y, ν ₀), are transformed back to the time-domain. By this frequency filtering we obtain the oscillation signature of the field in a narrow frequency window (Δv=15 GHz) around ν ₀ without background from other superimposed frequencies, which allows us to unambiguously assign the specific resonances observed in far-field transmission measurements to near-field patterns.

 

Fig. 3. Microscope images of the split-ring resonator (SRR) samples. The structures are made from 9 μm thick copper on a 120 μm thick PTFE substrate. (a) Single SRR. (b) Periodic array of SRRs. o= 500 μm; i= 300 μm; g= 100 μm; t= 30 μm; p= 600 μm.

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The samples were fabricated by conventional photolithography and etching techniques and were made from 9 μm thick copper on a 120 μm thick polytetrafluoroethene (PTFE) substrate (ε_{r PTFE} = 2.28) from Rogers (RT/duroid 5880 ®). A thickness of the substrate smaller than the terahertz wavelength was chosen to avoid multiple reflections within the substrate. We investigate samples consisting of one individual double SRR and of a square array of this structure as shown in Fig. 3(a) and (b) in order to analyze the resonant behavior of an individual element and its interaction with neighboring resonators.

3. Experimental results and simulation

First, the far-field transmission of the SRR array sample has been characterized under normal incidence using a conventional THz time-domain spectrometer [25]. Figure 4 shows the transmitted power spectra for two perpendicular orientations of the structure relative to the polarization of the incident wave. Four dominant resonances (A₁-A₄) are observed for orthogonal and two (B₁ and B₂) for parallel polarization of the electric field relative to the slits in the split-rings. All modes can be interpreted in terms of fundamental excitations of the constituting SRR elements leading to strong reflection peaks and the observed transmission minima [8]. In our case dissipative losses are negligible since in the THz regime metals are almost perfect conductors and the teflon substrate is non-absorbing. Current investigations of the underlying mechanisms leading to these sharp and strong resonances rely almost entirely on numerical simulations which normally assume idealized geometries, material properties, excitation and boundary conditions.

 

Fig. 4. Power transmission spectrum of the SRR array sample. (a) Spectrum showing four dominant resonances (A₁-A₄) for orthogonal polarization of the incident electric field relative to the slits. (b) Spectrum for parallel polarization showing only two resonances (B₁ and B₂).

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Fig. 5. Measured electromagnetic near-field distribution and simulation of the current density. The vector plots show the electric in-plane field vectors in the x-y-plane at the backside of the sample at the resonances A ₁ - A ₄ for (a) the isolated SRR sample (Media 1) and (c) the array sample (Media 2). The color code indicates the time derivative of the out-of-plane component of the corresponding magnetic field, ∂H_z/∂_t, derived from the electric field vectors. All plots show the fields at an individually chosen phase within one oscillation cycle. (b) Simulated surface current density. Polarization and propagation direction of the incident wave is indicated in (b).

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Our THz near-field microscopy approach permits to characterize the electric and magnetic near-field distributions of these structures at their far-field resonances. Figures 5(a) and (c) show the measured near-field distributions of a single SRR and a periodic array of identical SRRs. The electric field vectors are plotted, all at a fixed phase within an oscillation cycle, at the frequencies of the four dominant resonances A₁-A₄. The complete oscillation cycles for each resonance are available online as movies (Media 1) and (Media 2). For resonances A₁ and A₂ of the isolated SRR the vector plots indicate the formation of rotational fields oscillating around the center of the structure. These modes correspond to the LC-resonances of the outer and inner split-ring, associated with circulating currents in the metal structures leading to the formation of a magnetic moment normal to the SRR plane [26]. On the other hand, the modes at 225 and 405 GHz (A₃ and A₄) show a fundamentally different character. For both modes the field vectors point toward and away from the corners of the resonators indicative of electric quadrupole excitation.

 

Fig. 6. Rotated orientation of the structure. (a) Measured electric and magnetic near-fields at resonances B₁ and B₂ for the isolated SRR (Media 3) and (c) for the SRR array sample (Media 4)). (b) Simulated current densities.

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According to Maxwell’s equations a curl of the electric field induces a change in the magnetic field by ∂H⃗/∂t = − μ ⁻¹ ∇ × E⃗. This relation allows us to determine the change in the out-of-plane magnetic field component, ∂H_z/∂t, from the measured in-plane electric field vectors, E_xy, at each spatial pixel, indicated in the plots in Fig. 5 and 6 by the additional color code. Since the field is harmonically oscillating the magnetic field distribution H_z(x,y,t) simply corresponds to its time derivative phase-shifted by π/2. The magnetic field distributions shown in Fig. 5 reveal that the LC-resonances, A₁ and A₂, are associated with the formation of magnetic dipoles normal to the sample plane whereas the higher order modes, A₃ and A₄, exhibit a fourfold magnetic field pattern.

Complementary numerical simulations based on finite element method (FEM) modeling [27] reproduce all four fundamental resonances, albeit slightly shifted (< 10%) to higher frequencies. In the simulation an individual double SRR structure was positioned in the center of a 3-dimensional simulation box of refractive index n=1. Drude conductivity was assumed for the rings using the literature values of copper for the plasma frequency and the damping rate [28]. A plane harmonic wave normally incident onto the SRR plane was excited on one boundary. Scattering boundary conditions have been imposed on the remaining sides of the simulation volume. This simplified model does not account for the frequency-dependent Gaussian THz beam focus nor the changing position of the detector close to the sample surface. Therefore, instead of the fields we consider the surface currents in the rings which are extracted from the simulation, since they proved to be less dependent on excitation conditions and external influences. The current densities were determined from the simulated electric field in the rings by j⃗ = σE⃗, where E⃗ is the field in the metal 50 nm below the surface.

 

Fig. 7. Illustration of the fundamental modes. For the three fundamental modes (A₁, A₃, B₁) of the outer ring the currents j⃗ (red arrows), the induced magnetic field H⃗ (green arrows) and the resulting charge density are sketched, each at its maximum. The orientation of the magnetic out-of-plane component H_z is indicated by the colored areas.

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The simulated surface current densities at each resonance are shown in Fig. 5(b). We find that the measured magnetic field distributions are fully consistent with the directions of the driving currents predicted by the simulation for each resonance, the magnetic dipole mode (A₁ and A₂) and the fourfold magnetic field pattern of an electric quadrupole (A₃ and A₄). An illustration of the relative orientation of the current flux and associated magnetic fields is shown in Fig. 7 for the three characteristic resonances of the outer ring.

The lower panel of Fig. 5 shows the respective fields observed for the SRR-array sample. Since the resonances significantly extend into the surrounding area of the SRRs field interference between neighbors has to be considered for the array sample. Destructive field interference between neighboring SRRs for the LC-resonance A₁ is observed considerably attenuating the field rotations. In contrast, for the higher order mode A₃ constructive interference significantly enhances the rotation patterns predominantly in-between the structures. We attribute the remarkably stronger resonance at A₃ observed in the far field transmission spectrum in Fig. 4 to this mechanism. The modes of the inner ring (A₂ and A ₄) remain relatively unperturbed by influence from neighboring structures due to their larger separation and the screening by the outer rings.

Figure 6 shows the field patterns and simulated currents when the SRRs are rotated by 90°. Both modes B₁ and B₂ arise from currents symmetrically induced in the horizontal sides of the outer and the inner ring, respectively, leading to charge accumulations on the vertical sides and the formation of a depolarization field. The measured magnetic fields are again consistent with the orientation of the simulated current flux in the rings as illustrated for the resonance of the outer ring in Fig. 7. For the periodic array sample constructive interference between neighbors leads to enhanced field patterns, albeit slightly distorted due to the inhomogeneous illumination in particular at higher frequencies [24].

4. Conclusion

In summary we have utilized THz near-field microscopy to investigate a planar metamaterial based on double split-ring resonator structures. By our approach we are able to trace the in-plane electric and out-of-plane magnetic fields at the metamaterial’s characteristic resonances which are responsible for its extraordinary optical properties. We find that for our structure three fundamental resonances occur for the outer and the inner ring, respectively, leading to the six dominant resonances observed in the far-field transmission spectra. Using our imaging technique we map the electromagnetic near-field signature of each fundamental mode for the first time. The ability to visualize near-field distributions of microstructured surfaces opens a unique way to experimentally investigate and optimize metamaterials and their complex interaction with light.