Research ArticleAPPLIED PHYSICS

From the generalized reflection law to the realization of perfect anomalous reflectors

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Science Advances  11 Aug 2017:
Vol. 3, no. 8, e1602714
DOI: 10.1126/sciadv.1602714
  • Fig. 1 Anomalous reflective metasurface.

    (A) Illustration of the performance of a reflective metasurface. Propagating waves in the system when θi = 0° and θr > 30° are represented in the scheme. (B) Comparative overview of the efficiency of anomalous reflection metasurfaces and optical gratings. Blue dots represent previous results found in the literature (6, 8, 13, 16, 17, 20, 21); the black line represents the numerical estimation of the designs based on a linear 2π phase gradient calculated according to Eq. 1; the red star represents the results obtained in this paper.

  • Fig. 2 Comparison between the different reflective metasurface proposals when θi = 0° and θr = 70°.

    Conventional design: Real part of the scattered electric field (A) and total power density distribution (D). Lossy design: Real part of the scattered electric field (B) and total power density distribution (E). Active design: Real part of the scattered electric field (C) and total power density distribution (F).

  • Fig. 3 Local design of a lossless perfect anomalous reflector.

    (A) Schematic representation of the proposed design methodology for the perfect reflecting metasurface based on the leaky-wave antenna behavior with a modulation in the impedance profile. Tangential (B) and vertical (C) wave numbers introduced in the system due to a periodic perturbation with period Dx = λ/|sin θi − sin θr| as a function of the surface impedance Zs = jXs. (D) Surface impedance of the metasurface when θr = 70° and θi = 0°. Blue line represents the initial estimation of the surface reactance (Eq. 11), and orange dots represent the optimized values of the surface reactance on the discretized surface. Dashed line represents the impedance of conventional design described by Eq. 1. Numerical results of a perfect reflectarray based on the linearly modulated leaky-wave antenna structure: Real part of the scattered electric field (E) and total power density flow distribution (F).

  • Fig. 4 Bandwidth analysis.

    Bandwidth comparison between anomalous reflectors based on (A and C) conventional design (Eq. 1) and on (B and D) inhomogeneous leaky-wave antenna (see Fig. 3D). Both metasurfaces are designed to produce anomalous reflection from 0° to 70° at 8 GHz and modeled as an inhomogeneous impedance boundary. (A and B) Comparison between wavelength bandwidths. (C and D) Comparison between angular bandwidths.

  • Fig. 5 Nonlocal implementation.

    (A) Proposed inhomogeneous nonlocal leaky-wave reflector. (B) Frequency bandwidth of the proposed design. Simulated real part of the total electric field (C) on the xz plane when y = 0 and (D) on the xy plane when z = 0. Real part of the total Poynting vector on (E) the xz plane when y = Dy/2 and real part of the Poynting vector (F) on the xy plane when z = 0.

  • Fig. 6 Experimental verification.

    (A) Fabricated metasurface. (B) Experimental setup in an anechoic chamber. Signals measured by the receiving antenna for different orientation angles ϕ of (C) the metasurface and (D) the metal plate of equal size. (E) Amplitude efficiency of the metasurface illuminated normally.

  • Table 1 Numerical results and comparison between the different design possibilities for reflectarrays.

    Amplitude/power of waves sent into the respective directions, absorption coefficient, and power efficiency.

    θiθr−θrAbsorptionEfficiency
    Generalized reflection law (Eq. 1)0.24/0.061.50/0.760.73/0.180.0075.7%
    Lossy design dictated by Eq. 20.00/0.001.00/0.340.00/0.000.6634.0%
    Active-lossy design dictated by Eq. 30.03/0.001.77/1.040.11/0.000.00104.4%
    Inhomogeneous leaky-wave antennas introduced in this paper0.04/0.001.70/0.990.03/0.000.0099.7%
    Implementation with metal patches (lossy materials)0.04/0.001.66/0.940.03/0.000.0694.0%

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/8/e1602714/DC1

    section S1. Power modulation produced by reflective metasurfaces

    section S2. Nonlocal perfect anomalous reflectors for different reflection angles

    section S3. Analysis of the unit cell using two-dimensional evanescent fields

    section S4. Scattered fields from a reference metal plate illuminated obliquely

    section S5. Scattered fields from the normally illuminated metasurface

    section S6. Correction factor for the signal amplitudes measured in the experiment

    fig. S1. Total power density distribution for the conventional, lossy, and active designs (from left to right).

    fig. S2. Initial approximation for nonlocal designs.

    fig. S3. Real part of the scattered electric field when the designed metasurfaces are illuminated normally at f = 8 GHz.

    fig. S4. Illustration of the desired performance of an ideally reflecting metasurface.

    fig. S5. Real part of the total electric field.

    fig. S6. Geometry of the problem with a metal plate.

    fig. S7. Geometry of the problem with a metasurface.

    table S1. Length of the strips, l, for different designs.

    movie S1. Bandwidth analysis of conventional designs.

    movie S2. Bandwidth analysis of nonlocal designs.

  • Supplementary Materials

    This PDF file includes:

    • section S1. Power modulation produced by reflective metasurfaces
    • section S2. Nonlocal perfect anomalous reflectors for different reflection angles
    • section S3. Analysis of the unit cell using two-dimensional evanescent fields
    • section S4. Scattered fields from a reference metal plate illuminated obliquely
    • section S5. Scattered fields from the normally illuminated metasurface
    • section S6. Correction factor for the signal amplitudes measured in the experiment
    • fig. S1. Total power density distribution for the conventional, lossy, and active designs (from left to right).
    • fig. S2. Initial approximation for nonlocal designs.
    • fig. S3. Real part of the scattered electric field when the designed metasurfaces are illuminated normally at f = 8 GHz.
    • fig. S4. Illustration of the desired performance of an ideally reflecting metasurface.
    • fig. S5. Real part of the total electric field.
    • fig. S6. Geometry of the problem with a metal plate.
    • fig. S7. Geometry of the problem with a metasurface.
    • table S1. Length of the strips, l, for different designs.

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    Other Supplementary Material for this manuscript includes the following:

    • movie S1 (.avi format). Bandwidth analysis of conventional designs.
    • movie S2 (.avi format). Bandwidth analysis of nonlocal designs.

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