Babinet's principle - OSA Publishing

Terahertz near-field microscopy of complementary planar metamaterials: Babinet’s principle Andreas Bitzer,1,3 Alex Ortner,2 Hannes Merbold,1 Thomas Feurer,1 and Markus Walther2,4 1 Institute

of Applied Physics, University of Bern, Sidlerstr. 5, CH-3012 Bern, Switzerland Materials Research Center, University of Freiburg, Stefan-Meier-Strasse 21, D-79104 Freiburg, Germany 3 [email protected] 4 [email protected]

2 Freiburg

Abstract: 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 Optical Society of America OCIS codes: (180.4243) Near-field microscopy; (300.6495) Spectroscopy, terahertz; (160.3918) Metamaterials; (240.6680) Surface plasmons.

References and links 1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). 2. M. Osawa, “Dynamic processes in electrochemical reactions studied by surface-enhanced infrared absorption spectroscopy (seiras),” Bull. Chem. Soc. Jpn. 70, 2861–2880 (1997). 3. V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41–48 (2007). 4. C. M. Soukoulis, S. Linden, and M. Wegener, “Negative refractive index at optical wavelengths,” Science 315, 47–49 (2007). 5. M. A. Seo, A. J. L. Adam, J. H. Kang, J. W. Lee, S. C. Jeoung, Q. H. Park, P. C. M. Planken, and D. S. Kim, “Fourier-transform terahertz near-field imaging of one-dimensional slit arrays: mapping of electric-field-, magnetic-field-, and Poynting vectors,” Opt. Express 15, 11781–11789 (2007). 6. A. J. L. Adam, J. M. Brok, M. A. Seo, K. J. Ahn, D. S. Kim, J. H. Kang, Q. H. Park, M. Nagel, and P. C. M. Planken, “Advanced terahertz electric near-field measurements at sub-wavelength diameter metallic apertures,” Opt. Express 16, 7407–7417 (2008). 7. A. Bitzer and M. Walther, “Terahertz near-field imaging of metallic subwavelength holes and hole arrays,” Appl. Phys. Lett. 92, 231101 (2008). 8. A. Bitzer, A. Ortner, and M. Walther, “Terahertz near-field microscopy with subwavelength spatial resolution based on photoconductive antennas,” Appl. Opt. 49, E1–E6 (2010). 9. A. Bitzer, H. Merbold, A. Thoman, T. Feurer, H. Helm, and M. Walther, “Terahertz near-field imaging of electric and magnetic resonances of a planar metamaterial,” Opt. Express 17, 3826–3834 (2009). 10. A. Bitzer, J. Wallauer, H. Helm, H. Merbold, T. Feurer, and M. Walther, “Lattice modes mediate radiative coupling in metamaterial arrays,” Opt. Express 17, 22108–22113 (2009). 11. V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104, 223901 (2010).

#138230 - $15.00 USD Received 16 Nov 2010; revised 17 Dec 2010; accepted 20 Dec 2010; published 26 Jan 2011

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12. C. C. Chen and J. F. Whitaker, “An optically-interrogated microwave-Poynting-vector sensor using cadmium manganese telluride,” Opt. Express 18, 12239–12248 (2010). 13. M. Burresi, D. van Oosten, T. Kampfrath, H. Schoenmaker, R. Heideman, A. Leinse, and L. Kuipers, “Probing the magnetic field of light at optical frequencies,” Science 326, 550–553 (2009). 14. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999). 15. E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magnetoinductive waves in one, two, and three dimensions,” J. Appl. Phys. 92, 6252–6261 (2002). 16. F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, “Babinet principle applied to the design of metasurfaces and metamaterials,” Phys. Rev. Lett. 93, 197401 (2004). 17. T. Zentgraf, T. P. Meyrath, A. Seidel, S. Kaiser, H. Giessen, C. Rockstuhl, and F. Lederer, “Babinet’s principle for optical frequency metamaterials and nanoantennas,” Phys. Rev. B 76, 033407 (2007). 18. C. Rockstuhl, T. Zentgraf, T. P. Meyrath, H. Giessen, and F. Lederer, “Resonances in complementary metamaterials and nanoapertures,” Opt. Express 16, 2080–2090 (2008). 19. H. T. Chen, J. F. O’Hara, A. J. Taylor, R. D. Averitt, C. Highstrete, M. Lee, and W. J. Padilla, “Complementary planar terahertz metamaterials,” Opt. Express 15, 1084–1095 (2007). 20. I. A. I. Al-Naib, C. Jansen, and M. Koch, “Applying the Babinet principle to asymmetric resonators,” Electron. Lett. 44, 1228–1229 (2008). 21. F. Falcone, T. Lopetegi, J. D. Baena, R. Marques, F. Martin, and M. Sorolla, “Effective negative-epsilon stopband microstrip lines based on complementary split ring resonators,” IEEE Trans. Microwave Theory Tech. 14, 280– 282 (2004). 22. D. Grischkowsky, S. Keiding, M. Vanexter, and C. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006–2015 (1990). 23. N. C. J. van der Valk and P. C. M. Planken, “Electro-optic detection of subwavelength terahertz spot sizes in the near field of a metal tip,” Appl. Phys. Lett. 81, 1558–1560 (2002). 24. P. C. M. Planken, C. E. W. M. van Rijmenam, and R. N. Schouten, “Opto-electronic pulsed THz systems,” Semicond. Sci. Technol. 20, S121–S127 (2005). 25. J. D. Baena, J. Bonache, F. Martin, R. M. Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. Garcia-Garcia, I. Gil, M. F. Portillo, and M. Sorolla, “Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines,” IEEE Trans. Microwave Theory Tech. 53, 1451–1461 (2005).

1.

Introduction

Controlling and manipulating light fields by their interaction with metallic micro- or nanostructures enabled many useful applications, such as improved focusing [1], enhanced spectroscopic sensitivity [2] 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 [8]. 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 #138230 - $15.00 USD Received 16 Nov 2010; revised 17 Dec 2010; accepted 20 Dec 2010; published 26 Jan 2011

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(a)

(b)

(c)

fs laser beam (800 nm)

focussing lens (f=7.5 mm) detector electrodes ion-implanted silicon layer (~ 500 nm)

ZnTe 100-cut (100 µm)

sapphire substrate

500 µm

dielectric HR coating (800nm)

sample

incident THz beam

incident THz beam

(d)

Fig. 1. Schematic illustration of the two different near-field detection techniques used in this study. (a) Photoconductive antenna as polarization sensitive near-field probe. By rotating the antenna by 90◦ both in-plane THz electric field components can be measured. (b) Nonlinear crystal as near-field probe. A (100)-cut ZnTe crystal allows measuring the electric out-of-plane component. In both configurations the detector is scanned together with the laser beam relative to the stationary sample in order to map the electric fields. (c) and (d) show microscope images of the investigated samples, a SRR and a CSRR, respectively.

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 [14]. Furthermore, microscopic magnetic moments can mediate interaction between meta-atoms through magneto-inductive coupling [11] or may lead to the formation of magnetoinductive waves [15]. 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 [16]. 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 [21]. 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-


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domain spectroscopy (THz-TDS) [22] 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 fslaser 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 HRcoated 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 E0 , B0 , then its complementary field E0c , Bc0 is defined as E0c = −cB0

and Bc0 = E0 /c,

(1)

which corresponds to a 90◦ rotation around the propagation axis. Derived from classical diffraction theory the full vectorial formulation of Babinet’s principle states that the field E c , Bc behind a complementary screen (e.g. a CSRR) illuminated by a complementary field is given by [16] E c = E0c + cB

(2)

Bc = Bc0 − E/c,

(3)

where E and B are the scattered fields behind the positive structure (a SRR) as illustrated in 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 (E0c , Bc0 E, B) which also include evanescent field contributions and therefore can be neglected in Eqs. (2) and (3). The total electric field behind the structure #138230 - $15.00 USD Received 16 Nov 2010; revised 17 Dec 2010; accepted 20 Dec 2010; published 26 Jan 2011

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(a)

(b)

y

y

B0 E0

x k

z

c

B0

E B scattered field (z > 0)

c

x k

z

c

E B scattered field (z > 0)

c E0

Fig. 2. Illustration of Babinet’s principle which relates the scattered fields E, B and E c , Bc behind complementary structures. SRR (a) and its complementary screen (b) illuminated by complementary incident fields.

(z > 0) can then be considered a duplicate of the total magnetic field behind the complementary screen and vice versa, E c = cB and Bc = −E/c. (4) Note that on the left-hand side of the screen (z < 0) the scattered fields have opposite sign to ensure, that both the total magnetic polarization perpendicular to the screen and the total electric polarization parallel to the screen vanish, as required for a perfectly conducting surface at (z = 0) [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 simulation n=1 (75 GHz) y E k

yz-plane

yz-plane scat. fields

B x,z

B y,z

B y,z

(a)

(b)

(c)

(d)

x

SRR

B

xz-plane

xy-plane Bz

z SRR

k inc.

y E x

E x,z

E y,z

E y,z

(e)

(f)

(g)

(h)

k

CSRR

B

Ez

z CSRR

Fig. 3. FEM simulation of the field distributions around a SRR and a CSRR under complementary illumination. (a) Density plot of the in-plane magnetic near-field behind a SRR. (b,c) Streamline representation of the magnetic near-field of a SRR plotted along perpendicular cross sections through the center of the structure. (d) yz-cross section of the scattered magnetic field only, shown without superimposed driving fields. (e-h) Cross sections of the corresponding electric field distribution behind the CSRR. The color intensity scales with the amplitude of the corresponding magnetic (upper row) and electric (lower row) fields.


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c , are identical as predicted resonance (n=1). In the xz-plane (b, f) the field patterns, Bx,z and Ex,z by Eq. (4). In this case the incident fields B0 and E0c 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, E, and the total magnetic field of the CSRR, Bc , in the region behind the structures (z > 0) are related by E + cBc = cBc0 . Therefore the transmission coefficient t c for the CSRR illuminated by the complementary wave in Eq. (1) is related to the transmission coefficient t for the SRR by t + t c = 1.

(5)

This complementary behavior of the field transmission can be verified by far-field transmission spectroscopy on both structures [19,20]. 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 [9]. 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. n1

n3

n2

SRR Array

1

n4

SRR Array

1

E E

k

0

CSRR Array

1

E

transmission

transmission

B B

k

0

CSRR Array

1

B

B

E k

0 0.0

0.2

0.4 0.6 frequency THz

0.8

1.0

k

0 0.0

0.2

0.4 0.6 frequency THz

0.8

1.0

Fig. 4. Far-field transmission spectra obtained from a 20x20 array of SRRs for two different polarizations of the incident beam relative to the sample as indicated by the insets (top curves) and corresponding spectra of the complementary sample (bottom curves). Characteristic resonances of the structures are indicated by vertical dashed lines.


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n1 75 GHz

n2 150 GHz

n3 225 GHz Bz

SRR

1

xzplane

0

1

E

B

k

E

k

n1 75 GHz

B

E

yzplane

B

k

n2 150 GHz

n3 225 GHz Ez

CSRR

1

xzplane

0

E k

E

k

B

B

1 E

yzplane

B k

Fig. 5. Near-field distributions measured directly behind our samples at their lowest order resonances n=1-3. Top row: Electric in-plane (arrows) and magnetic out-of-plane (colors) field distributions behind the SRR-sample. Bottom row: Density plots of the electric out-ofplane component close to the surface of the complementary screen (CSRR). Dashed lines are indicating the positions at which the image plane intersects with the simulated xz- and yz-cross sections shown in Fig 3.

All the resonances observed in the far-field spectra can be correlated with characteristic nearfield 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 ∂ B/∂ t = −∇ × E (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) [9]. 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 μ m behind the metal surface, where parallel components of the electric field can occur. In this regime the near-field of the structure also


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n1 75 GHz

n2 150 GHz

n3 225 GHz Ez

SRR

1

0

1 B

E

E

B

k

k

n1 75 GHz

B

E

k

n2 150 GHz

n3 225 GHz Bz

CSRR

1

0

k

E k

E

B

B

1 E

B k

Fig. 6. Near-field distributions of our structures, however, measured with interchanged detection schemes as in Fig. 5. Top row: electric near-field distribution of the out-of-plane component close to the SRR. Bottom row: electric in-plane and magnetic out-of-plane nearfield distributions behind the CSRR.

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


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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. 5.

Conclusion

In conclusion, we have investigated split-ring resonators and their inverse structures by THz farfield 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. Acknowledgment A.O. and M.W. acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG), grant No. WA 2641, and by the Baden-W¨urttemberg 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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