Using terahertz near-field imaging we experimentally investigate the resonant electromagnetic field distributions behind a split-ring resonator and its complementary structure with sub-wavelength spatial resolution. For the out-of-plane components we experimentally verify complementarity of electric and magnetic fields as predicted by Babinet’s principle. This duality of near-fields can be used to indirectly map resonant magnetic fields close to metallic microstructures by measuring the electric fields close to their complementary analogues which is particularly useful since magnetic near-fields are still extremely difficult to access in the THz regime. We find excellent agreement between the results from theory, simulation and two different experimental near-field techniques.
© 2011 OSA
Controlling and manipulating light fields by their interaction with metallic micro- or nano-structures enabled many useful applications, such as improved focusing , enhanced spectroscopic sensitivity  or the implementation of novel optical properties such as negative refractive indices [3, 4]. Whereas most experimental studies to date investigate the light distribution in the far-field of such structures, gaining a comprehensive understanding of the underlying mechanisms requires monitoring their near-fields. Due to typical structure dimensions on the wavelength to sub-wavelength scale, however, near-field studies with the required spatial resolution are highly challenging and experiments in the long-wavelength regime where structure sizes are comparably large can be advantageous. Recently, imaging at terahertz (THz) frequencies proved to be immensely powerful for a detailed investigation of the near-fields around sub-wavelength metal structures and apertures [5–8]. As an example, the study of metamaterials, artificial structures consisting of subwavelength-sized sub-units which can give rise to unprecedented optical properties, has benefited considerably from THz near-field imaging [9–11]. Based on the coherent emission and detection of broadband and single-cycle THz pulses this approach allows measuring time-dependent electric fields with sub-ps temporal and sub-wavelength spatial resolution. Fourier-transformation of the measured time traces provides information on field amplitude and phase in a frequency window between typically 50 GHz and ∼ 4 THz .
Whereas most optical techniques typically measure electric fields, directly accessing the magnetic field component is much more challenging [12, 13]. This is mainly due to the fact that the force on a moving charge exerted by the B-field of an electromagnetic wave is by a factor c/v weaker than the force exerted by the electric field. Here, v is the charge velocity and c the speed of light. Nonetheless, in particular the magnetic near-fields play an extremely important role in many plasmonic systems. For example in split-ring resonators (SRRs), one of the fundamental building blocks of many metamaterials, resonant magnetic fields are formed in response to an incident field leading to a negative magnetic response associated with a negative magnetic permeability, which is one of the requirements for realizing a left-handed medium . Furthermore, microscopic magnetic moments can mediate interaction between meta-atoms through magneto-inductive coupling  or may lead to the formation of magneto-inductive waves . This study aims to investigate the interplay between electric and magnetic near-fields in complementary metamaterial structures, like a split-ring and its inverse analogue. Such inverse elements, like complementary SRRs (CSRRs), have been proposed as an alternative to conventional metallic split-rings for the design of metamaterials or metasurfaces . In principle, complementary metamaterials show similar properties as their inverse analogues. However, according to Babinet’s principle, their transmission and reflection behavior as well as their scattered electric and magnetic fields are interchanged [17–20]. As a result they can provide an effective negative permittivity rather than permeability .
Here we present a detailed characterization of the electromagnetic near-fields of a single SRR and its complementary screen, a CSRR, by THz near-field imaging. Using our approach, we experimentally demonstrate the complementarity of magnetic and electric fields stated by the full vectorial formulation of Babinet’s principle.
2. Sample fabrication and experimental setup
SRR and CSRR samples have been fabricated from a 10 μm copper foil on a 50 μm thick dielectric substrate (nTHz = 1.5) by laser cutting. Both, single structures as shown in Fig. 1 (500 μm side length, ∼ 30 μm line-width), as well as square arrays of 20 × 20 SRRs and CSRRs (700 μm period) have been produced.
Our study comprises three different experimental techniques. Conventional THz time-domain spectroscopy (THz-TDS)  was used in order to obtain far-field transmission spectra of our samples, and two different near-field detection schemes to measure the magnetic as well as the electric out-of-plane components behind a single SRR and CSRR. Using a photoconductive antenna as near-field probe as shown in Fig. 1(a) allows us to determine the out-of-plane magnetic field component Bz from two consecutive measurements of both in-plane electric field components, Ex and Ey, by applying Faraday’s law, as described elsewhere [8, 9]. Briefly, a fs-laser beam is focussed through the sapphire substrate into the photoconductive gap between two electrodes on a silicon-on-sapphire detector chip and the current flow between the electrodes induced by an in-plane THz electric field is measured by lock-in amplification. Due to the H-shaped electrode design the detector is polarization sensitive and can be switched between measurements to either detect the x- or the y-component of the THz electric field by 90° rotation around the laser beam axis. Owing to the planar detector electrode geometry, however, this technique does not permit measuring out-of-plane electric field components. Hence the Ez component was measured using a detection scheme based on electro-optic sampling in a nonlinear crystal. This powerful near-field microscopy approach has been pioneered by the Planken group [5, 6, 23]. For electro-optic detection the photoconductive antenna is replaced by a 100 μm thick (100)-oriented ZnTe crystal in optical contact to an index matching 2.5 mm thick sapphire substrate to temporally delay internal reflections within the detector. The fs-laser beam is focused through the sapphire substrate into the ZnTe layer and is back-reflected from the HR-coated front side of the crystal facing the sample. Polarization rotation of the reflected laser beam induced in the crystal by the THz electric field is measured by balanced photo-detection. For the crystal orientation used only the Ez component of the THz field is measured [23, 24].
For both detection schemes the sample was placed in close proximity to the near-field detectors (∼30 μm distance) and the entire detector unit was raster-scanned together with the probe laser beam in x and y direction relative to the stationary sample and the THz beam in order to map the spatial field distribution. This approach has the advantage that the intrinsic inhomogeneity of ZnTe does not have to be taken into account for spatially resolved measurements. In both cases the spatial resolution was estimated to be on the order of 30 μm which corresponds to λ/20 at 0.5 THz.
3. Theoretical background and near-field simulation
Babinet’s principle relates the fields scattered by two complementary plane structures made of infinitely thin perfectly conducting sheets of arbitrary shape, provided that both are illuminated by complementary waves. If we consider an incident electromagnetic field E⃗0, B⃗0, then its complementary field , is defined as16] Fig. 2. The total fields behind the structure are the superposition of the incident and the scattered fields. In case of a strong resonance associated with high Q-factors, the incident fields are much weaker than scattered fields ( , , B⃗) which also include evanescent field contributions and therefore can be neglected in Eqs. (2) and (3). The total electric field behind the structure (z > 0) can then be considered a duplicate of the total magnetic field behind the complementary screen and vice versa, 16, 25].
In order to confirm this predicted near-field behavior for our structures we have performed numerical simulations based on finite element modeling (FEM). Streamline plots of the simulation results are shown in Fig. 3 as perpendicular cross sections of the E and B-fields close to the SRR and the CSRR for their fundamental (n=1) resonance at 75 GHz. The simulated field patterns in the xy-plane represent a magnetic (a) and an electric (e) dipole character of the resonance (n=1). In the xz-plane (b, f) the field patterns, B⃗x,z and , are identical as predicted by Eq. (4). In this case the incident fields B0 and are oriented normal to the plane and hence do not contribute to the in-plane fields. In contrast, the streamline plots in the yz-plane (c, d) show some deviation, in particular in the region behind the structures. Here, the incident field superimposed on the scattered field has a significant contribution, and therefore is not fully negligible. However, the simulation allows us to extract the scattered fields only, as shown in Fig. 3(d) and 3(h), which again exhibit perfect agreement.
4. Experimental results
From Eq. (3) it follows that the total electric field of a SRR, E⃗, and the total magnetic field of the CSRR, B⃗c, in the region behind the structures (z > 0) are related by . Therefore the transmission coefficient tc for the CSRR illuminated by the complementary wave in Eq. (1) is related to the transmission coefficient t for the SRR by19,20]. Figure 4 shows spectra of SRR and CSRR samples measured by THz-TDS for two orthogonal polarizations of the THz field relative to the structures. In agreement with Eq. (5) we observe an inverse spectral response of the array of CSRRs (red curve) as compared to the SRR (blue curve) sample. The transmission minima/maxima in our spectra indicated by the dashed vertical lines are due to the resonant excitation of the metal structures. For the SRRs these resonances correspond to the formation of charge density standing waves along the metallic ring, which occur whenever the length l of the unfolded SRR corresponds to multiples n of half the wavelength, so that l = n · λ/2. Due to the symmetry of the modes relative to the linearly polarized excitation odd-numbered resonances are excited when the electric field of the THz beam is normal and even numbered when it is polarized parallel to the SRR gap . In our spectra we observe resonances up to the order n=4. Spectral modulations at higher frequencies observed for both samples arise due to the excitation of diffractive modes in the square lattice.
All the resonances observed in the far-field spectra can be correlated with characteristic near-field distributions. In Fig. 5 we show near-field measurements of a single SRR (top row) and of its complementary analogue (bottom row), at their individual resonances. In the upper row the arrows visualize the measured in-plane electric field vectors and the color code the corresponding magnetic field out-of-plane component Bz directly behind the SRR structure. The magnetic field Bz has been determined from the Ex and Ey components according to ∂B⃗/∂t = −∇ × E⃗ (Faraday’s law). For the SRR we find the well-known modal patterns of an oscillating ring current (LC-resonance, n=1), a symmetric depolarization along the vertical axis of the ring (n=2) and the formation of an electric quadrupole (n=3) . The current flow and the charge distribution in the ring associated with each resonance is indicated in the inset below each figure. A closer inspection of the in-plane electric field vectors shows, that they are not perfectly perpendicular to the metal surface as expected for a perfectly metallic conductor. This effect occurs mainly due to the non-negligible separation between detector and sample. Effectively our measurements are performed in a plane 30 μm behind the metal surface, where parallel components of the electric field can occur. In this regime the near-field of the structure also becomes superimposed by the linearly polarized incident electric field giving rise to significant parallel field components. However, we note that since Faraday’s law only considers rotational fields, the superimposed linear incident field does not contribute to the magnetic near-field distribution.
The field maps in the lower row show the electric out-of-plane component of the complementary screen (CSRR) at the corresponding resonances measured by electro-optic sampling. The positions of the cross sections through our simulated data shown in Fig. 3 are indicated by dashed lines in Fig. 5 (n=1 resonance). At this point we want to note that all resonant field patterns (n=1,2,3) are reproduced by our simulations. So, in conclusion a striking correspondence between the magnetic and electric near-field maps of both complementary structures is found in perfect agreement with our simulations and in accordance with the theoretical predictions of Babinet’s principle.
On the example of the fundamental n=1 resonance we can intuitively understand this dual behavior if we consider that in the split-ring this fundamental mode corresponds to a circular ring current inducing a magnetic dipole oscillating inside the ring. At the corresponding resonance of the CSRR the inner section is depolarized by the incident electric field with respect to the outer metal part as sketched below the field map. This results in an oscillating electric dipole perpendicular to the metal surface. At higher orders the near-field modes become more complicated due to the decreased wavelengths and the increased degrees of freedom for the charge distribution relative to the structure size. In this case, the driving fields generally concentrate the charges at the edges of the slit.
The correspondence of magnetic and electric fields close to complementary screens opens up the intriguing possibility to indirectly map the magnetic near-fields of a metallic microstructure simply by measuring the electric fields around its complement. This approach is particularly useful since most near-field imaging techniques rely on measuring electric fields. Directly measuring magnetic field vectors is much more challenging, mainly due to their relative weakness as compared to the electric field components.
Finally, we also compare measurements of the out-of-plane magnetic field component behind the CSRR with the corresponding electric field component of the SRR as shown in Fig. 6. For the SRR-resonances (top row) we find that Ez field maxima occur exactly at the positions where charge density accumulations are expected, underlining the strong correlation between the electric out-of-plane component and the charge density as it is expected for surface plasmon polaritons. This can best be seen on the example of the n=3 resonance where the characteristic electric quadrupole pattern can be clearly identified. In the bottom row we show the measured electric in-plane and magnetic out-of-plane field components of the inverse split-ring. Again, we find significant agreement between the electric and magnetic out-of-plane field maps of the SRR and the CSRR, respectively, which further validates the dual behavior of complementary metallic structures as stated by Babinet’s principle.
In conclusion, we have investigated split-ring resonators and their inverse structures by THz far-field spectroscopy and near-field microscopy in order to validate the duality of their transmission spectra, as well as of their near-fields, as predicted by Babinet’s principle. On the example of the out-of-plane components we experimentally demonstrate for the first time the correspondence between electric and magnetic near-fields of an SRR and its complementary structure, the CSRR. Excellent agreement was found between the results from theory, numerical modeling and two different experimental near-field techniques which validates the consistency of the two detection schemes. As an intriguing implication of this study we propose that Babinet’s principle can be utilized to indirectly map resonant magnetic fields of a structure by measuring the electric field of its complement.
A.O. and M.W. acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG), grant No. WA 2641, and by the Baden-Württemberg Ministry for Science and Arts, Research Seed Capital (RiSC) for young researchers. A.B., H.M. and T.F. thank the LiMat project and the Schweizerischer Nationalfonds (SNF) for financial support, grant No. 200020-119934.
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