Abstract

We provide an analysis of the electromagnetic modes of three-dimensional metamaterial resonators in the THz frequency range. The fundamental resonance of the structures is fully described by an analytical circuit model, which not only reproduces the resonant frequencies but also the coupling of the metamaterial with an incident THz radiation. We also demonstrate the contribution of the propagation effects, and show how they can be reduced by design. In the optimized design, the electric field energy is lumped into ultra-subwavelength (λ/100) capacitors, where we insert a semiconductor absorber based on the collective electronic excitation in a two dimensional electron gas. The optimized electric field confinement is exhibited by the observation of the ultra-strong light-matter coupling regime, and opens many possible applications for these structures in detectors, modulators and sources of THz radiation.

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2019 (2)

N.-L. Tran, M. Malerba, A. Talneau, G. Biasiol, O. Ouznali, A. Bousseksou, J.-M. Manceau, and R. Colombelli, “III-V on CaF2: a possible waveguiding platform for mid-IR photonic devices,” Opt. Express 27(2), 1672 (2019).
[Crossref]

M. Jeannin, G. M. Nesurini, S. Suffit, D. Gacemi, A. Vasanelli, L. Li, A. G. Davies, E. Linfield, C. Sirtori, and Y. Todorov, “Ultra-strong light-matter coupling in deeply subwavelength THz LC resonators,” ACS Photonics 6(5), 1207–1215 (2019).
[Crossref]

2018 (1)

T. A. P. Tran and P. H. Bolivar, “Terahertz modulator based on vertically coupled fano metamaterial,” IEEE Trans. Terahertz Sci. Technol. 8(5), 502–508 (2018).
[Crossref]

2017 (3)

2016 (1)

H.-T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: physics and applications,” Rep. Prog. Phys. 79(7), 076401 (2016).
[Crossref]

2015 (1)

2014 (2)

C. Maissen, G. Scalari, F. Valmorra, M. Beck, J. Faist, S. Cibella, R. Leoni, C. Reichl, C. Charpentier, and W. Wegscheider, “Ultrastrong coupling in the near field of complementary split-ring resonators,” Phys. Rev. B 90(20), 205309 (2014).
[Crossref]

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref]

2013 (6)

Y.-J. Chiang and T.-J. Yen, “A composite-metamaterial-based terahertz-wave polarization rotator with an ultrathin thickness, an excellent conversion ratio, and enhanced transmission,” Appl. Phys. Lett. 102(1), 011129 (2013).
[Crossref]

L. Cong, W. Cao, X. Zhang, Z. Tian, J. Gu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103(17), 171107 (2013).
[Crossref]

N. K. Grady, E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H.-T. Chen, “Terahertz metamaterials for linear polarization conversion and anomalous refraction,” Science 340(6138), 1304–1307 (2013).
[Crossref]

D. Dietze, A. M. Andrews, P. Klang, G. Strasser, K. Unterrainer, and J. Darmo, “Ultrastrong coupling of intersubband plasmons and terahertz metamaterials,” Appl. Phys. Lett. 103(20), 201106 (2013).
[Crossref]

Y.-B. Chen and F.-C. Chiu, “Trapping mid-infrared rays in a lossy film with the Berreman mode, epsilon near zero mode, and magnetic polaritons,” Opt. Express 21(18), 20771 (2013).
[Crossref]

C. Feuillet-Palma, Y. Todorov, A. Vasanelli, and C. Sirtori, “Strong near field enhancement in THz nano-antenna arrays,” Sci. Rep. 3(1), 1361 (2013).
[Crossref]

2012 (6)

G. Cataldo, J. A. Beall, H.-M. Cho, B. McAndrew, M. D. Niemack, and E. J. Wollack, “Infrared dielectric properties of low-stress silicon nitride,” Opt. Lett. 37(20), 4200 (2012)..
[Crossref]

F. Costa and A. Monorchio, “Closed-Form Analysis of Reflection Losses in Microstrip Reflectarray Antennas,” IEEE Trans. Antennas Propag. 60(10), 4650–4660 (2012)..
[Crossref]

G. Scalari, C. Maissen, D. Turčinková, D. Hagenmüller, S. De Liberato, C. Ciuti, C. Reichl, D. Schuh, W. Wegscheider, M. Beck, and J. Faist, “Ultrastrong coupling of the cyclotron transition of a 2D electron gas to a THz metamaterial,” Science, 335(6074),1323–1326 (2012).
[Crossref]

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12(9), 4932–4936 (2012).
[Crossref]

C. Feuillet-Palma, Y. Todorov, R. Steed, A. Vasanelli, G. Biasiol, L. Sorba, and C. Sirtori, “Extremely sub-wavelength THz metal-dielectric wire microcavities,” Opt. Express 20(27), 29121 (2012).
[Crossref]

M. Geiser, F. Castellano, G. Scalari, M. Beck, L. Nevou, and J. Faist, “Ultrastrong coupling regime and plasmon polaritons in parabolic semiconductor quantum wells,” Phys. Rev. Lett. 108(10), 106402 (2012).
[Crossref]

2011 (2)

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref]

J. Grant, Y. Ma, S. Saha, A. Khalid, and D. R. S. Cumming, “Polarization insensitive, broadband terahertz metamaterial absorber,” Opt. Lett. 36(17), 3476 (2011).
[Crossref]

2010 (2)

C. Walther, G. Scalari, M. I. Amanti, M. Beck, and J. Faist, “Microcavity laser oscillating in a circuit-based resonator,” Science 327(5972), 1495–1497 (2010).
[Crossref]

Y. Todorov, A. M. Andrews, R. Colombelli, S. De Liberato, C. Ciuti, P. Klang, G. Strasser, and C. Sirtori, “Ultrastrong light-matter coupling regime with polariton dots,” Phys. Rev. Lett. 105(19), 196402 (2010).
[Crossref]

2009 (1)

M. G. Blaber, M. D. Arnold, and M. J. Ford, “Search for the ideal plasmonic nanoshell: the effects of surface scattering and alternatives to gold and silver,” J. Phys. Chem. C 113(8), 3041–3045 (2009).
[Crossref]

2008 (1)

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[Crossref]

2004 (1)

N. K. Grady, N. J. Halas, and P. Nordlander, “Influence of dielectric function properties on the optical response of plasmon resonant metallic nanoparticles,” Chem. Phys. Lett. 399(1-3), 167–171 (2004).
[Crossref]

1985 (1)

1982 (1)

J. R. Sambles, K. C. Elsom, and D. J. Jarvis, “The Electrical Resistivity of Gold Films,” Philos. Trans. Royal Soc. 304(1486), 365–396 (1982).
[Crossref]

1937 (1)

H. B. Palmer, “Capacitance of a parallel-plate capacitor by the Schwartz-Christoffel transformation,” Electr. Eng. 56(3), 363–368 (1937).
[Crossref]

Adler, R. B.

R. B. Adler, L. J. Chu, and R. M. Fano, Electromagnetic energy transmission and radiation (John Wiley & Sons, 1960).

Aieta, F.

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12(9), 4932–4936 (2012).
[Crossref]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref]

Alexander, R. W.

Amanti, M. I.

C. Walther, G. Scalari, M. I. Amanti, M. Beck, and J. Faist, “Microcavity laser oscillating in a circuit-based resonator,” Science 327(5972), 1495–1497 (2010).
[Crossref]

Andrews, A. M.

D. Dietze, A. M. Andrews, P. Klang, G. Strasser, K. Unterrainer, and J. Darmo, “Ultrastrong coupling of intersubband plasmons and terahertz metamaterials,” Appl. Phys. Lett. 103(20), 201106 (2013).
[Crossref]

Y. Todorov, A. M. Andrews, R. Colombelli, S. De Liberato, C. Ciuti, P. Klang, G. Strasser, and C. Sirtori, “Ultrastrong light-matter coupling regime with polariton dots,” Phys. Rev. Lett. 105(19), 196402 (2010).
[Crossref]

Arnold, M. D.

M. G. Blaber, M. D. Arnold, and M. J. Ford, “Search for the ideal plasmonic nanoshell: the effects of surface scattering and alternatives to gold and silver,” J. Phys. Chem. C 113(8), 3041–3045 (2009).
[Crossref]

Azad, A. K.

N. K. Grady, E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H.-T. Chen, “Terahertz metamaterials for linear polarization conversion and anomalous refraction,” Science 340(6138), 1304–1307 (2013).
[Crossref]

Bachmann, D.

Bahl, I. J.

I. J. Bahl, Lumped Elements for RF and Microwave Circuits (Artech House, 2003).

Beall, J. A.

Becerra, L.

Beck, M.

C. Maissen, G. Scalari, F. Valmorra, M. Beck, J. Faist, S. Cibella, R. Leoni, C. Reichl, C. Charpentier, and W. Wegscheider, “Ultrastrong coupling in the near field of complementary split-ring resonators,” Phys. Rev. B 90(20), 205309 (2014).
[Crossref]

G. Scalari, C. Maissen, D. Turčinková, D. Hagenmüller, S. De Liberato, C. Ciuti, C. Reichl, D. Schuh, W. Wegscheider, M. Beck, and J. Faist, “Ultrastrong coupling of the cyclotron transition of a 2D electron gas to a THz metamaterial,” Science, 335(6074),1323–1326 (2012).
[Crossref]

M. Geiser, F. Castellano, G. Scalari, M. Beck, L. Nevou, and J. Faist, “Ultrastrong coupling regime and plasmon polaritons in parabolic semiconductor quantum wells,” Phys. Rev. Lett. 108(10), 106402 (2012).
[Crossref]

C. Walther, G. Scalari, M. I. Amanti, M. Beck, and J. Faist, “Microcavity laser oscillating in a circuit-based resonator,” Science 327(5972), 1495–1497 (2010).
[Crossref]

Belacel, C.

Bell, R. J.

Biasiol, G.

Blaber, M. G.

M. G. Blaber, M. D. Arnold, and M. J. Ford, “Search for the ideal plasmonic nanoshell: the effects of surface scattering and alternatives to gold and silver,” J. Phys. Chem. C 113(8), 3041–3045 (2009).
[Crossref]

Blanchard, R.

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12(9), 4932–4936 (2012).
[Crossref]

Bolivar, P. H.

T. A. P. Tran and P. H. Bolivar, “Terahertz modulator based on vertically coupled fano metamaterial,” IEEE Trans. Terahertz Sci. Technol. 8(5), 502–508 (2018).
[Crossref]

Bousseksou, A.

Cameau, M.

Cao, W.

L. Cong, W. Cao, X. Zhang, Z. Tian, J. Gu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103(17), 171107 (2013).
[Crossref]

Capasso, F.

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref]

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12(9), 4932–4936 (2012).
[Crossref]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref]

Castellano, F.

M. Geiser, F. Castellano, G. Scalari, M. Beck, L. Nevou, and J. Faist, “Ultrastrong coupling regime and plasmon polaritons in parabolic semiconductor quantum wells,” Phys. Rev. Lett. 108(10), 106402 (2012).
[Crossref]

Cataldo, G.

Charpentier, C.

C. Maissen, G. Scalari, F. Valmorra, M. Beck, J. Faist, S. Cibella, R. Leoni, C. Reichl, C. Charpentier, and W. Wegscheider, “Ultrastrong coupling in the near field of complementary split-ring resonators,” Phys. Rev. B 90(20), 205309 (2014).
[Crossref]

Chen, H.-T.

H.-T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: physics and applications,” Rep. Prog. Phys. 79(7), 076401 (2016).
[Crossref]

N. K. Grady, E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H.-T. Chen, “Terahertz metamaterials for linear polarization conversion and anomalous refraction,” Science 340(6138), 1304–1307 (2013).
[Crossref]

Chen, Y.-B.

Chiang, Y.-J.

Y.-J. Chiang and T.-J. Yen, “A composite-metamaterial-based terahertz-wave polarization rotator with an ultrathin thickness, an excellent conversion ratio, and enhanced transmission,” Appl. Phys. Lett. 102(1), 011129 (2013).
[Crossref]

Chiu, F.-C.

Cho, H.-M.

Chowdhury, D. R.

N. K. Grady, E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H.-T. Chen, “Terahertz metamaterials for linear polarization conversion and anomalous refraction,” Science 340(6138), 1304–1307 (2013).
[Crossref]

Chu, L. J.

R. B. Adler, L. J. Chu, and R. M. Fano, Electromagnetic energy transmission and radiation (John Wiley & Sons, 1960).

Cibella, S.

C. Maissen, G. Scalari, F. Valmorra, M. Beck, J. Faist, S. Cibella, R. Leoni, C. Reichl, C. Charpentier, and W. Wegscheider, “Ultrastrong coupling in the near field of complementary split-ring resonators,” Phys. Rev. B 90(20), 205309 (2014).
[Crossref]

Ciuti, C.

G. Scalari, C. Maissen, D. Turčinková, D. Hagenmüller, S. De Liberato, C. Ciuti, C. Reichl, D. Schuh, W. Wegscheider, M. Beck, and J. Faist, “Ultrastrong coupling of the cyclotron transition of a 2D electron gas to a THz metamaterial,” Science, 335(6074),1323–1326 (2012).
[Crossref]

Y. Todorov, A. M. Andrews, R. Colombelli, S. De Liberato, C. Ciuti, P. Klang, G. Strasser, and C. Sirtori, “Ultrastrong light-matter coupling regime with polariton dots,” Phys. Rev. Lett. 105(19), 196402 (2010).
[Crossref]

Colombelli, R.

Cong, L.

L. Cong, W. Cao, X. Zhang, Z. Tian, J. Gu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103(17), 171107 (2013).
[Crossref]

Costa, F.

F. Costa and A. Monorchio, “Closed-Form Analysis of Reflection Losses in Microstrip Reflectarray Antennas,” IEEE Trans. Antennas Propag. 60(10), 4650–4660 (2012)..
[Crossref]

Crozat, P.

Cumming, D. R. S.

Dalvit, D. A. R.

N. K. Grady, E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H.-T. Chen, “Terahertz metamaterials for linear polarization conversion and anomalous refraction,” Science 340(6138), 1304–1307 (2013).
[Crossref]

Darmo, J.

C. G. Derntl, D. Bachmann, K. Unterrainer, and J. Darmo, “Disk patch resonators for cavity quantum electrodynamics at the terahertz frequency,” Opt. Express 25(11), 12311 (2017).
[Crossref]

D. Dietze, A. M. Andrews, P. Klang, G. Strasser, K. Unterrainer, and J. Darmo, “Ultrastrong coupling of intersubband plasmons and terahertz metamaterials,” Appl. Phys. Lett. 103(20), 201106 (2013).
[Crossref]

Davies, A. G.

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Figures (7)

Fig. 1.
Fig. 1. (a) Scanning electron microscope image of a THz LC circuit, showing the top Au inductive loop. The bottom plate is delimited by the dashed rectangle. The capacitor area is shown in thin red lines. The dark grey background corresponds to a 3 µm thick SiN layer on top of a gold mirror, while the light grey rod is the semiconductor (SC) active region. (b) Typical reflectivity spectrum of an array of THz LC circuits for the TM (top) and TE (bottom) polarizations, as depicted in the insets on the right. We see two resonances, indicated as LC and M for the TM polarization, and only one resonance (indicated as W) for the TE polarization. The reflectivity of the layered substrate is shown in by the black dashed lines. (c) Reflectivity spectra for different inductor length Y (indicated in the legend, in µm).
Fig. 2.
Fig. 2. (a) Simulated (solid lines) and experimental (dashed lines) reflectivity spectra as a function of inductor length Y (indicated on each curve, in µm), for the TM (left) and TE (right) polarizations. (b) Sketch of the cut planes used in the electric and magnetic field maps in panel (c). (c) Top: Vertical component of the electric (Ez, left) and magnetic (Hz, right) fields in the x-y plane passing through the center of the capacitors (plane 2 in panel (b)). Bottom: Electric energy density (We) for the LC mode in plane 2 (left) and in a x-z plane passing through the middle of the capacitors (right, vertical plane containing the red dashed line in panel (b)). (d): Vertical component of the electric (Ez, left) and magnetic (Hz, right) fields for the M mode in the plane of the bottom plate [plane 3 in panel (b)]. (e): Vertical component of the electric (Ez, left) and magnetic (Hz, right) fields for the W mode in the plane of the top inductive loop (plane 1 in panel (b)). All colormaps are normalized to the maximum value in the plot plane.
Fig. 3.
Fig. 3. (a) Sketch of the metamaterial showing a unit cell (dashed rectangle) with a transmission line length Y, and a characteristic impedance ZR. The SiN layer is modeled by complex impedance ZSiN, and adjacent unit cells are coupled through a capacitive term CM. (b) Equivalent circuit model showing the arrangement of the unit cell, the capacitor modeling the intercell coupling and the SiN layer. The unit cell coupling to free space is modeled as a transmission line port with free space impedance Zfree = 377 Ω. (c) Equivalent circuit model of the resonator. The two segments A and B are modeled with a resistor and inductance in series. The bottom plate is capacitively coupled to the segment B through Ceq. The radiation resistance is placed where the electric field is confined in the circuit, near the capacitors. The two segments are connected with a transmission line of length Y.
Fig. 4.
Fig. 4. (a) Simulated reflectivity obtained from the circuit model (blue lines) compared with the experimental data (red circles) for the different values of the arm length Y. (b) Resistance R of the inductive segments A and B (see Fig. 3) as a function of frequency.
Fig. 5.
Fig. 5. (a) Electric energy localized inside the capacitors (blue solid lines) and in the transmission lines (purple dashed lines) extracted from the circuit model calculations for the four different geometries. (b) Left axis (blue circles): fraction of peak electric energy stored inside the capacitors as a function of the peak frequency. Right axis (purple triangles): Degree of subwavelength confinement.
Fig. 6.
Fig. 6. Effect of the aspect ratio W/d of the capacitive parts of the meta-atom on the electric field confinement. The results are obtained from FEM simulations by integrating the electric energy stored inside the capacitor volume, normalized by the total electric energy. The black dashed line is a guide for the eye following a logarithmic trend. Two geometries are considered, as shown on the right: straight transmission lines (yellow squares), and broken transmission lines (brown circles). The simulation results show the vertical component of the electric field Ez.
Fig. 7.
Fig. 7. (a) Reflectivity spectra of the LC metamaterial at room temperature (blue line) and at 10 K (red line). The arrows represent the resonant frequencies of the LC resonator and the intersubband transition. (b) Low temperature dispersion relation of the modes showing the crossing between the intersubband and the LC resonance. The stars mark the device for which the results are shown in (a). (c) Reflectivity spectra of the optimized LC metamaterial at room temperature (blue line) and low temperature (red line), showing the frequency splitting of the intersubband and LC resonances into the upper and lower polariton modes. (d) Dispersion relation of the upper and lower polaritons showing the characteristic anticrossing and the vacuum Rabi splitting 2ΩR.