Abstract

We report for the first time that an ultra-thin hybrid metamaterial slab can reflect an incident plane wave in −1st diffraction order, giving rise to anomalous reflection in a “negative” way. The functionality is derived from the hybridized surface resonant states of the slab. The retro-directive reflection is demonstrated numerically for a Gaussian beam at oblique incidence and verified experimentally at microwave frequencies.

©2010 Optical Society of America

1. Introduction

One conventional way to modulate reflected waves in free space is to use surface relief grating which diffracts the incident beam into several different directions [1]. The directions of these reflected beams depend on the period of grating, the wavelength and the incident angle of light. A blazed grating can guide most wave energy into one diffraction order at a specified wavelength [2]. The magnitude of the blaze angle, at which the grating will be most efficient, requires the optical path of grating thickness and period to be larger than operational wavelength. Photonic crystal (PC) gratings [35] also provide a route to achieving negative reflection. When the frequency of incident wave falls inside a complete or partial photonic bandgap, the incident wave can be selectively coupled to higher order Bloch modes at a properly designed hybrid [3] or truncated PC-air interface [4,5] and reflected back along the direction of the −1st diffraction order. To fully utilize the bandgap effect for anomalous reflection, the size of photonic crystal gratings must be much larger than operational wavelength in all dimensions. For a planar slab with a thickness much thinner than the operational wavelength, a common notion is that the reflected or transmitted waves in the zeroth order can never be suppressed.

Metamaterial interacts with electromagnetic waves in a resonant manner, affording us new options for wave manipulation as well as subwavelength photonic devices [6]. To the best of our knowledge, most previous studies about metamaterial mainly focus on local resonance in fundamental mode which is diffraction free by operating at long wavelength limit. As a consequence, the applications of metamaterial in diffraction optics have not drawn much attention yet. In this paper, we explore the dispersive functionality of metamaterial by properly harvesting the harmonic modes of local resonance. One simplest model to be investigated is the high impendence surface [7] which can modulate electromagnetic waves within a thickness much smaller than operational wavelength [8]. We show that, by designing a hybrid supercell, such a magnetic “metaface”, with a thickness of about one-twentieth of the operational wavelength, can selectively reflect the incident wave only along the direction of −1st diffraction order in a “negative way”. A specific example of negative reflection is numerically and experimentally demonstrated to verify our theoretical prediction. We attribute this phenomenon to the coincidence of hybridized surface resonant states and the selective Bragg scattering effect of supercell. The simple structure, hybridized with two different magnetic resonance cavities in one dimension, has great application potential in the Littrow Mounting configuration [2].

2. Model description and analytical methodology

The schematic configuration of the hybrid slab is illustrated in Fig. 1(a) . The structure consists of an upper layer of metallic lamellar gratings with a thickness of t, a dielectric spacer layer as a slab waveguide with a thickness of h and a metallic ground plane. We introduce inhomogeneity by employing two adjacent metallic strips with different width sizes ab while keeping the same width g for air gaps between the strips. Each metallic strip together with the metallic ground plane beneath it constitutes a magnetic resonance cavity [7,8]. The hybrid slab is a one-dimensional array composed of two kinds of cavities [labeled as A and B in the schematic picture Fig. 1(b)]. When a = b, it regresses to a homogeneous model, which has been extensively investigated before [912]. In our model, the geometric parameters are as follows: a=13mm+δ,t=0.035mm,h=1.6mm,b=13mmδ and g=1mm, whereδ=7mm as hybrid parameter is adjustable; The period of the slab along y^ direction isp=a+b+2g=28mmgiving rise to a Rayleigh frequency atfR=c0/p10.71GHzunder normal incidence [13], where c0 is the light speed in vacuum. The permittivity of the dielectric layer isεr=2.65. The metallic grating is along thex^ direction so that the air gaps only allow the transverse magnetic (TM) polarized wave (with the electric fieldE in y^z^ plane) to penetrate into the slab waveguide [14]. The field energy is re-distributed through the Bloch channels by the coupled cavity chain and re-emitted out via the air gaps.

 figure: Fig. 1

Fig. 1 (a) Schematic configuration of the hybrid slab. The regions containing perfectly electric conductor (PEC) or dielectric are marked by orange color or blue color. The geometrical parameters are a=20mm,b=6mm,g=1mm,h=1.6mmand t=0.035mm. The permittivity of dielectric slab isεr=2.65. A transverse magnetic (TM) plane wave is incident in y^z^plane with an incident angle θ, the magnetic field is along the y axis. (b) The functionality of the hybrid slab can be understood as the result of the coupling of a chain of alternating cavities A and B.

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For a TM polarized incident wave from the free semi-space at z>h+t (region I), the magnetic field component along x^ direction Hi(i=1,3) in region I, and region III (0<z<h) can be expressed in the form of the Bloch mode expansion [1518],

H1=m=+[δm,0eikm,z1(zht)+rmeikm,z1(zht)]ei(ky+Gm)yH3=m=+{tmeikm,z3(zh)+tmei[km,z3(zh)+φm]}ei(ky+Gm)y,
where δm,0eikm,z1(zht) denotes the incident field with a wave vectork=k//+z^k0,z1,δ being the Kronecker function, and integer m being the order of the Bloch channel. The reciprocal wave vector is Gm=2πm/p, and the corresponding wave vector component along z^ direction iskm,zi=(niω/c0)2(Gm+ky)2(i=1,3), where ni denotes the refractive index of the dielectric medium in region I or III; rm or tm is the coefficient of the mth reflected or guided Bloch waves in region I or III; φm=2hkm,z3 is the phase delay with respect to the thickness h of slab waveguide. In the inhomogeneous layer of region II (hz<h+t),the electromagnetic fields can be expanded, in terms of Rayleigh’s scattering theory, as superpositions of plane-wave harmonics [15,16]. Here we assume perfect conductivity for the metal in our model system so that the electromagnetic field in region II only exists inside the air gaps. The in-plane distribution of magnetic field in region II can be expanded with a series of waveguide modesgl(y)=cos(yl/g), (l=0,1,...,n,...) inside the gaps alternatively. Thus the magnetic field can be simply expressed as,
H2=lgl(y)[aleikl,z2(zht)+bleikl,z2(zh)].
In addition, it is accurate enough to acquire a single mode, the fundamental mode g0(y), to approximate the wave fields inside the air gaps, provided that the air gap width g is much smaller than the period p [17,18]. By applying the boundary continuity conditions for the tangential electric fields and magnetic fields (over the slits) at the interfaces z=h andz=h+t, we obtain the coefficients tm(f,k//)and rm(f,k//) of the mthguided and reflected waves, the dispersion ω(k//) and the wave functions of surface resonances as well. We note that, with perfectly electric conductor (PEC) assumption for metals, the simplified modal expansion method stated above is much faster and more convergent in our case compared to the Rayleigh’s scattering theory.

3. Reflection coefficients derived from hybrid structure

We mainly consider the TM polarized plane wave incident in the yz plane (Hy^). The directions of the incidence and the reflected waves are illustrated by colored arrows in the inset of Fig. 2(a) . Specifically, we calculate the0th,±1st orders of reflectivity spectra Rm=Re(km,z1/k0,z1)|rm|2(m=0,±1) under different incident angles θ=0°,10°,20°and 30°for TM polarized incidence, as shown in Figs. 2(a)~(d). It is noted that within the frequency range fromc0/pk0sin(θ)c0/2πtoc0/p+k0sin(θ)c0/2π, the reflection behaves as the competition between the 0th order and 1storder of reflected waves with the reflection coefficients satisfying toR0+R1=1. The 0th order reflection at some specific frequencies and incident angles is greatly suppressed with extremely small value of R0. For examples, we haveR0=105atf=9.88GHz,θ=10°[Fig. 2(b)], R0=0.08 atf=10.5GHz,θ=20°[Fig. 2(c)], andR0=0.003 at f=10.9GHz,θ=30°[Fig. 2(d)]. We also note that the −1st order of reflection usually occurs at the same side of the surface normal with the incidence, so that the slab appears to reflect the incidence in a “negative” way.

 figure: Fig. 2

Fig. 2 Reflectivity spectra of the0th(red solid lines) −1st (blue dashed lines) and + 1st(dark yellow short-dashed lines) order of reflected waves for TM polarized incident plane waves with different incident angles θ=0°,10°,20°and 30°. The frequencies in white regions [(b)-(d)] confirm that only 0th and −1st reflected waves propagate in the free space with real wave vector component along z axis. The colored arrows in the inset of (a) schematically illustrate the directions of the incidence I0 (black) and the corresponding reflected waves in the 0th order (red), the −1st order (blue), and the + 1st order (dark yellow).

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To understand the anomalous reflection in Fig. 2 more clearly, we employ the modal expansion method to calculate the surface resonance dispersion along y^ direction for a hybrid slab withδ=7mm stated above. This system supports multiple magnetic resonances, which can be identified in the 0th order reflection spectra R0(f,θ) at frequencies satisfying arg(R0(f,θ))=0. Under normal incidence the hybrid metal sandwich structure supports three magnetic resonances at 3.97GHz, 7.95GHz and 9.36GHz no higher than Rayleigh frequency (10.70GHz), corresponding to those Γ points of branches labeled as B2, B3 and B4 in Fig. 3(a) . More calculations show that the two magnetic resonances at 3.97GHz and 7.95GHz are the fundamental mode and its second harmonic mode of localized magnetic resonance derived from cavity A with induced surface current solenoids located on it; The frequencies of these two modes are mainly scaled by the size a of cavity A when ab is satisfied, while the magnetic resonance at 9.36GHz is not scaled by the size b of small cavity B and always appears below the Rayleigh frequency. The magnetic resonance at 9.36GHz is derived from the coupling effect of two different magnetic cavities labeled as A and B in Fig. 1(b) instead.

 figure: Fig. 3

Fig. 3 (a) Dispersion diagram of surface resonance states on the hybrid slab. Reflectivity of the 0th or -1st reflected diffractive waves matching to theky0orky<0 surface resonances on branch B4 and B5 in colored region of reciprocal space are marked with the size of the symbol being proportional the magnitude of the reflectivity (see text for details). The reflectivity is also plotted as a function of incident angles at (b) for f=7.65GHz, (c) forf=9.88GHz and (d) forf=10.88GHz.

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As shown in Fig. 3(a), the surface states come from the hybridization of these localized magnetic resonances with the (folded) light lines. Numerical calculations show that these resonance states are TEM-like guided Bloch modes mainly confined in the dielectric spacer layer. We note that the (folded) light lines L1, L2, L3 and L4, marked by green lines in Fig. 3(a), surround a closed zone (in yellow color) in the reciprocal space. The closed zone defines the region of reciprocal space in which only the 0th and −1st orders of reflected waves of our model system are propagating modes. It means that a state in the closed zone can be converted into free space photons and only along the directions of the 0th and −1st diffraction orders. As such, the 4th and 5th branches B4, B5 of Fig. 3(a) in colored zone are of particular interest for anomalous reflection in the −1st diffraction order. In contrast, we cannot obtain1storder of reflection matching to the same zone of reciprocal space for the homogeneous case withδ=0. Whenδ0, the unit cell is nearly doubled in real space and zone folding due to Bragg scattering occurs at a short wave vectorky=π/p=π/(a+b) instead of ky=π/a in the homogenous slab. We also note that the branch B2 is very flat with a resonant frequency (withk//=0at Γ2point) inversely proportional to the size a of the large resonant cavity A, while the higher order branches from B2 to B5 are very dispersive, revealing hybridized resonance properties. Our prediction from Fig. 2 about anomalous reflection, which happens below Rayleigh frequency, is a unique feature of our hybrid design where the zone folding occurs at a shorter wave vector.

To investigate the Bragg scattering effect from the hybrid supercell, we consider a plane wave incident from the upper left of the slab [depicted in Fig. 1(a)] with the incident angle ranging from 0° to 90°, calculate the spectra R(f) of the 0th and higher order reflection coefficients. We find that at the resonance condition when an incident plane wave is precisely phase-matched to a state on the 4th or 5th branch (ky=k0sinθ), the 0th order reflection coefficient will find its minimum in the spectra R0(f) at a given incident angle, or equivalently, the –1st order reflection coefficient will find its maximum in its spectraR1(f) as shown in Fig. 2.

As shown in Fig. 3(a), we mark all the peaks of the 0th order reflection coefficientR0 on the right side (red circles) and the −1st order reflection coefficientR1 on the left side (blue squares) with the sizes of the red circles and blue squares being proportional to the reflection amplitudes (i.e. the bigger the symbol, the stronger the reflection). We note that 100% conversion efficiency is achieved with R1=1 for the incidence matching to those states at 7.65GHz, 9.88GHz and 10.88GHz with zero group velocity (ω/k//~0). It implies that, with the Bragg folding effect of the supercell, all energy of the incident plane wave is redistributed on the slab surface matching to the −1st order of surface resonant state instead of the 0th order one at on-resonance condition.

To characterize the reflection properties more generally, we also calculated the angle-resolved reflectivity spectra at those frequencies at which the surface states have nearly zero group velocityω/k//~0). As shown in Figs. 3(b)-3(d) R1 reaches unity atθ=44.6°,f=7.65GHz, atθ=10°,65.4°,f=9.88GHz, and atθ=29.4°,f=10.89GHz, where the incident light is phase matched to the surface states with zero group velocity shown in Fig. 3(a). The rather flat plateau of the −1st reflectivity spectrum around R1=1 in Fig. 3(b) means that at 7.65GHz the −1st order reflection has a high efficiency of R1>0.99for40.4°<θ<48.6°. In other words, the “negative” reflection from the hybrid slab is not difficult to implement in practice with an acceptable angular tolerance. We also note that the frequency bandwidth of negative reflection is wide enough for practical application.

4. Numerical simulations and experiments on negative reflection

By adjusting the hybrid parameter δ as well as other structural parameters, we can suppress zero order reflection at prescribed frequencies, incident angles and reflection angles. As a matter of fact, the hybrid parameterδ=7mmalready ensures that the −1st order of reflection withR1=1 at 7.65GHz propagates precisely along the counter-propagating direction of incidence atθ'=44.6°, i.e. the “negative” reflection angle. We employ our own finite-difference-in-time-domain (FDTD) code to numerically verify the phenomenon of negative reflection by adopting a one-way monochromatic Gaussian beam [19]. Perfectly-matched-layer (PML) technique for boundary condition and one-way source technique for Gaussian beam initialization are applied in our simulations. As a proof of principle, Fig. 4(a) presents the numerical results on the specular reflection from a PEC surface. The width of beam waist is 100 mm, and the incident angle isθ=44.6°. Figure 4(b) presents the numerical simulation on the reflection from the hybrid slab under the same condition of incidence. Perfect retro-reflection is verified numerically.

 figure: Fig. 4

Fig. 4 Negative reflection from the hybrid slab at 7.65GHz with an incident angle atθ=44.6°. FDTD simulations with one-way Gaussian beam incident on (a) a PEC slab, and (b) a hybrid slab. (c) The angular spectra of the measured reflection intensity in far-field. (d) Snapshot of local field patterns inside the rectangle box illustrated in Fig. 4(b). The insets in (a) and (b) illustrate the structure of PEC slab and hybrid slab. The positions of one-way Gaussian beam are indicated by horizontal thin lines in (a) and (b). Black arrows refers to the directions of incident Gaussian beams, while red and blue arrows refer to the directions of specular reflection from PEC slab and that of negative reflection from hybrid slab respectively.

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We have fabricated a sample slab with exactly same parameters as our theoretical model. The lateral size of our sample is 1000mm × 1000mm. A horn antenna with a gain factor of 20dB feeds a Gaussian beam at a fixed incident angle θ45°. The inset of Fig. 4(c) schematically illustrates the experimental setup for the angular reflection measurements. Shown as blue dots in Fig. 4(c), a peak intensity of reflection at 7.65GHz is measured along the anti-direction of incidence at angleθ'45°, which is in very good agreement with FDTD numerical simulations. The surface states on B4 and B5 cannot be resolved in eigenmode analysis because they are leaking modes. However the perfect negative reflection shown in Fig. 4(b) explicitly demonstrates that only the surface state in the −1st Bloch order is excited, and we can directly visualize the mode from the spatial distribution of the wave field in the proximity of the hybrid slab. Figure 4(d) shows the wave patterns inside the rectangle box illustrated in Fig. 4(b). The box is at the center of the hybrid slab. The magnetic fields are plotted with strength in color map, and the electric fields are shown as green arrows with field strength proportional to the arrow length. In free space, the reflected plane wave is along the counter-direction of the incident plane wave. A quasi-TEM wave pattern, with field intensity much stronger than that of the incident plane wave in free space, is excited inside the dielectric layer and propagating along the reverse direction of the in-plane wavevector ky=π/pof incidence. This is precisely the wave form of surface state, at 7.65GHzwith an in-plane wavevectorkyG=π/p, inside the supercell.

5. Conclusion

In summary, we have shown that for a hybrid metamaterial slab that is very thin compared with the operational wavelength, both the incident and reflected waves can be at same side of the surface normal at a frequency lower than Rayleigh frequency. The retro-reflection can occur with nearly 100% efficiency, which has great application potentials in radar-tracking systems, long-distance energy transfer and Littrow-mounting configuration. Our hybrid design is a simple layered structure which can be realized in the optical regime.

Acknowledgement

This work was supported by the National 863 Program of China (Grant No. 2006AA03Z407), NSFC (Grant No. 10974144, 60674778, 10574099), CNKBRSF (Grant (No. 2006CB921701), Hong Kong RGC (grant 600308), NCET (07-0621), STCSM and by SHEDF (No. 06SG24).

References and links

1. M. Born, and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon Press, Oxford, Angleterre, 1980).

2. M. Mansuripur, Classical Optics and its Applications (Cambridge University Press, 2002).

3. D. Maystre, “Photonic crystal diffraction gratings,” Opt. Express 8(3), 209–216 (2001). [CrossRef]   [PubMed]  

4. G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gosele, “Diffraction properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 83(4), 614–616 (2003). [CrossRef]  

5. V. Mocella, P. Dardano, L. Moretti, and I. Rendina, “Influence of surface termination on negative reflection by photonic crystals,” Opt. Express 15(11), 6605–6611 (2007). [CrossRef]   [PubMed]  

6. N. Engheta, and R. W. Ziolkowski, Metamaterials: physics and engineering explorations (Wiley & Sons., 2006).

7. D. Sievenpiper, L. J. Zhang, R. F. J. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microw. Theory Tech. 47(11), 2059–2074 (1999). [CrossRef]  

8. M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, “Microwave surface-plasmon-like modes on thin metamaterials,” Phys. Rev. Lett. 102(7), 073901 (2009). [CrossRef]   [PubMed]  

9. A. P. Hibbins, J. R. Sambles, C. R. Lawrence, and J. R. Brown, “Squeezing millimeter waves into microns,” Phys. Rev. Lett. 92(14), 143904 (2004). [CrossRef]   [PubMed]  

10. A. Hibbins, W. Murray, J. Tyler, S. Wedge, W. Barnes, and J. Sambles, “Resonant absorption of electromagnetic fields by surface plasmons buried in a multilayered plasmonic nanostructure,” Phys. Rev. B 74(7), 073408 (2006). [CrossRef]  

11. J. Brown, A. Hibbins, M. Lockyear, C. Lawrence, and J. Sambles, “Angle-independent microwave absorption by ultrathin microcavity arrays,” J. Appl. Phys. 104(4), 043105 (2008). [CrossRef]  

12. M. Diem, T. Koschny, and C. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009). [CrossRef]  

13. R. W. Wood, “Anomalous Diffraction Gratings,” Phys. Rev. 48(12), 928–936 (1935). [CrossRef]  

14. The transverse electric (TE) polarized incident wave with the electric field E//x is blind to the air gaps between the metallic strips`, and treats the whole structure as a homogeneous dielectric slab with PEC ground effectively.

15. P. Rayleigh, “On the Dynamical Theory of Gratings,” R. Soc. London Ser. A 79(532), 399–416 (1907). [CrossRef]  

16. P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: Application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26(6), 2907–2916 (1982). [CrossRef]  

17. P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A-Pure Appl. Opt . 2, 48–51 (2000). [CrossRef]  

18. Z. Wei, J. Fu, Y. Cao, C. Wu, and H. Li, “The impact of local resonance on the enhanced transmission and dispersion of surface resonances,” Photon. Nanostruct.: Fundam. Appl. 8(2), 94–101 (2010). [CrossRef]  

19. A. Taflove, and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Norwood, MA, 2000).

References

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  1. M. Born, and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon Press, Oxford, Angleterre, 1980).
  2. M. Mansuripur, Classical Optics and its Applications (Cambridge University Press, 2002).
  3. D. Maystre, “Photonic crystal diffraction gratings,” Opt. Express 8(3), 209–216 (2001).
    [Crossref] [PubMed]
  4. G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gosele, “Diffraction properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 83(4), 614–616 (2003).
    [Crossref]
  5. V. Mocella, P. Dardano, L. Moretti, and I. Rendina, “Influence of surface termination on negative reflection by photonic crystals,” Opt. Express 15(11), 6605–6611 (2007).
    [Crossref] [PubMed]
  6. N. Engheta, and R. W. Ziolkowski, Metamaterials: physics and engineering explorations (Wiley & Sons., 2006).
  7. D. Sievenpiper, L. J. Zhang, R. F. J. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microw. Theory Tech. 47(11), 2059–2074 (1999).
    [Crossref]
  8. M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, “Microwave surface-plasmon-like modes on thin metamaterials,” Phys. Rev. Lett. 102(7), 073901 (2009).
    [Crossref] [PubMed]
  9. A. P. Hibbins, J. R. Sambles, C. R. Lawrence, and J. R. Brown, “Squeezing millimeter waves into microns,” Phys. Rev. Lett. 92(14), 143904 (2004).
    [Crossref] [PubMed]
  10. A. Hibbins, W. Murray, J. Tyler, S. Wedge, W. Barnes, and J. Sambles, “Resonant absorption of electromagnetic fields by surface plasmons buried in a multilayered plasmonic nanostructure,” Phys. Rev. B 74(7), 073408 (2006).
    [Crossref]
  11. J. Brown, A. Hibbins, M. Lockyear, C. Lawrence, and J. Sambles, “Angle-independent microwave absorption by ultrathin microcavity arrays,” J. Appl. Phys. 104(4), 043105 (2008).
    [Crossref]
  12. M. Diem, T. Koschny, and C. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
    [Crossref]
  13. R. W. Wood, “Anomalous Diffraction Gratings,” Phys. Rev. 48(12), 928–936 (1935).
    [Crossref]
  14. The transverse electric (TE) polarized incident wave with the electric field E//x is blind to the air gaps between the metallic strips`, and treats the whole structure as a homogeneous dielectric slab with PEC ground effectively.
  15. P. Rayleigh, “On the Dynamical Theory of Gratings,” R. Soc. London Ser. A 79(532), 399–416 (1907).
    [Crossref]
  16. P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: Application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26(6), 2907–2916 (1982).
    [Crossref]
  17. P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A-Pure Appl. Opt.  2, 48–51 (2000).
    [Crossref]
  18. Z. Wei, J. Fu, Y. Cao, C. Wu, and H. Li, “The impact of local resonance on the enhanced transmission and dispersion of surface resonances,” Photon. Nanostruct.: Fundam. Appl. 8(2), 94–101 (2010).
    [Crossref]
  19. A. Taflove, and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Norwood, MA, 2000).

2010 (1)

Z. Wei, J. Fu, Y. Cao, C. Wu, and H. Li, “The impact of local resonance on the enhanced transmission and dispersion of surface resonances,” Photon. Nanostruct.: Fundam. Appl. 8(2), 94–101 (2010).
[Crossref]

2009 (2)

M. Diem, T. Koschny, and C. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
[Crossref]

M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, “Microwave surface-plasmon-like modes on thin metamaterials,” Phys. Rev. Lett. 102(7), 073901 (2009).
[Crossref] [PubMed]

2008 (1)

J. Brown, A. Hibbins, M. Lockyear, C. Lawrence, and J. Sambles, “Angle-independent microwave absorption by ultrathin microcavity arrays,” J. Appl. Phys. 104(4), 043105 (2008).
[Crossref]

2007 (1)

2006 (1)

A. Hibbins, W. Murray, J. Tyler, S. Wedge, W. Barnes, and J. Sambles, “Resonant absorption of electromagnetic fields by surface plasmons buried in a multilayered plasmonic nanostructure,” Phys. Rev. B 74(7), 073408 (2006).
[Crossref]

2004 (1)

A. P. Hibbins, J. R. Sambles, C. R. Lawrence, and J. R. Brown, “Squeezing millimeter waves into microns,” Phys. Rev. Lett. 92(14), 143904 (2004).
[Crossref] [PubMed]

2003 (1)

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gosele, “Diffraction properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 83(4), 614–616 (2003).
[Crossref]

2001 (1)

2000 (1)

P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A-Pure Appl. Opt.  2, 48–51 (2000).
[Crossref]

1999 (1)

D. Sievenpiper, L. J. Zhang, R. F. J. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microw. Theory Tech. 47(11), 2059–2074 (1999).
[Crossref]

1982 (1)

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: Application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26(6), 2907–2916 (1982).
[Crossref]

1935 (1)

R. W. Wood, “Anomalous Diffraction Gratings,” Phys. Rev. 48(12), 928–936 (1935).
[Crossref]

1907 (1)

P. Rayleigh, “On the Dynamical Theory of Gratings,” R. Soc. London Ser. A 79(532), 399–416 (1907).
[Crossref]

Alexopolous, N. G.

D. Sievenpiper, L. J. Zhang, R. F. J. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microw. Theory Tech. 47(11), 2059–2074 (1999).
[Crossref]

Astilean, S.

P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A-Pure Appl. Opt.  2, 48–51 (2000).
[Crossref]

Barnes, W.

A. Hibbins, W. Murray, J. Tyler, S. Wedge, W. Barnes, and J. Sambles, “Resonant absorption of electromagnetic fields by surface plasmons buried in a multilayered plasmonic nanostructure,” Phys. Rev. B 74(7), 073408 (2006).
[Crossref]

Broas, R. F. J.

D. Sievenpiper, L. J. Zhang, R. F. J. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microw. Theory Tech. 47(11), 2059–2074 (1999).
[Crossref]

Brown, J.

J. Brown, A. Hibbins, M. Lockyear, C. Lawrence, and J. Sambles, “Angle-independent microwave absorption by ultrathin microcavity arrays,” J. Appl. Phys. 104(4), 043105 (2008).
[Crossref]

Brown, J. R.

A. P. Hibbins, J. R. Sambles, C. R. Lawrence, and J. R. Brown, “Squeezing millimeter waves into microns,” Phys. Rev. Lett. 92(14), 143904 (2004).
[Crossref] [PubMed]

Busch, K.

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gosele, “Diffraction properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 83(4), 614–616 (2003).
[Crossref]

Cao, Y.

Z. Wei, J. Fu, Y. Cao, C. Wu, and H. Li, “The impact of local resonance on the enhanced transmission and dispersion of surface resonances,” Photon. Nanostruct.: Fundam. Appl. 8(2), 94–101 (2010).
[Crossref]

Dardano, P.

Diem, M.

M. Diem, T. Koschny, and C. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
[Crossref]

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gosele, “Diffraction properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 83(4), 614–616 (2003).
[Crossref]

Fu, J.

Z. Wei, J. Fu, Y. Cao, C. Wu, and H. Li, “The impact of local resonance on the enhanced transmission and dispersion of surface resonances,” Photon. Nanostruct.: Fundam. Appl. 8(2), 94–101 (2010).
[Crossref]

Garcia-Martin, A.

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gosele, “Diffraction properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 83(4), 614–616 (2003).
[Crossref]

Gosele, U.

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gosele, “Diffraction properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 83(4), 614–616 (2003).
[Crossref]

Hibbins, A.

J. Brown, A. Hibbins, M. Lockyear, C. Lawrence, and J. Sambles, “Angle-independent microwave absorption by ultrathin microcavity arrays,” J. Appl. Phys. 104(4), 043105 (2008).
[Crossref]

A. Hibbins, W. Murray, J. Tyler, S. Wedge, W. Barnes, and J. Sambles, “Resonant absorption of electromagnetic fields by surface plasmons buried in a multilayered plasmonic nanostructure,” Phys. Rev. B 74(7), 073408 (2006).
[Crossref]

Hibbins, A. P.

M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, “Microwave surface-plasmon-like modes on thin metamaterials,” Phys. Rev. Lett. 102(7), 073901 (2009).
[Crossref] [PubMed]

A. P. Hibbins, J. R. Sambles, C. R. Lawrence, and J. R. Brown, “Squeezing millimeter waves into microns,” Phys. Rev. Lett. 92(14), 143904 (2004).
[Crossref] [PubMed]

Hugonin, J. P.

P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A-Pure Appl. Opt.  2, 48–51 (2000).
[Crossref]

Koch, W.

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gosele, “Diffraction properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 83(4), 614–616 (2003).
[Crossref]

Koschny, T.

M. Diem, T. Koschny, and C. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
[Crossref]

Lalanne, P.

P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A-Pure Appl. Opt.  2, 48–51 (2000).
[Crossref]

Lawrence, C.

J. Brown, A. Hibbins, M. Lockyear, C. Lawrence, and J. Sambles, “Angle-independent microwave absorption by ultrathin microcavity arrays,” J. Appl. Phys. 104(4), 043105 (2008).
[Crossref]

Lawrence, C. R.

A. P. Hibbins, J. R. Sambles, C. R. Lawrence, and J. R. Brown, “Squeezing millimeter waves into microns,” Phys. Rev. Lett. 92(14), 143904 (2004).
[Crossref] [PubMed]

Li, H.

Z. Wei, J. Fu, Y. Cao, C. Wu, and H. Li, “The impact of local resonance on the enhanced transmission and dispersion of surface resonances,” Photon. Nanostruct.: Fundam. Appl. 8(2), 94–101 (2010).
[Crossref]

Lockyear, M.

J. Brown, A. Hibbins, M. Lockyear, C. Lawrence, and J. Sambles, “Angle-independent microwave absorption by ultrathin microcavity arrays,” J. Appl. Phys. 104(4), 043105 (2008).
[Crossref]

Lockyear, M. J.

M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, “Microwave surface-plasmon-like modes on thin metamaterials,” Phys. Rev. Lett. 102(7), 073901 (2009).
[Crossref] [PubMed]

Maystre, D.

Meisel, D. C.

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gosele, “Diffraction properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 83(4), 614–616 (2003).
[Crossref]

Mocella, V.

Moller, K. D.

P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A-Pure Appl. Opt.  2, 48–51 (2000).
[Crossref]

Moretti, L.

Murray, W.

A. Hibbins, W. Murray, J. Tyler, S. Wedge, W. Barnes, and J. Sambles, “Resonant absorption of electromagnetic fields by surface plasmons buried in a multilayered plasmonic nanostructure,” Phys. Rev. B 74(7), 073408 (2006).
[Crossref]

Palamaru, M.

P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A-Pure Appl. Opt.  2, 48–51 (2000).
[Crossref]

Pereira, S.

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gosele, “Diffraction properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 83(4), 614–616 (2003).
[Crossref]

Rayleigh, P.

P. Rayleigh, “On the Dynamical Theory of Gratings,” R. Soc. London Ser. A 79(532), 399–416 (1907).
[Crossref]

Rendina, I.

Sambles, J.

J. Brown, A. Hibbins, M. Lockyear, C. Lawrence, and J. Sambles, “Angle-independent microwave absorption by ultrathin microcavity arrays,” J. Appl. Phys. 104(4), 043105 (2008).
[Crossref]

A. Hibbins, W. Murray, J. Tyler, S. Wedge, W. Barnes, and J. Sambles, “Resonant absorption of electromagnetic fields by surface plasmons buried in a multilayered plasmonic nanostructure,” Phys. Rev. B 74(7), 073408 (2006).
[Crossref]

Sambles, J. R.

M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, “Microwave surface-plasmon-like modes on thin metamaterials,” Phys. Rev. Lett. 102(7), 073901 (2009).
[Crossref] [PubMed]

A. P. Hibbins, J. R. Sambles, C. R. Lawrence, and J. R. Brown, “Squeezing millimeter waves into microns,” Phys. Rev. Lett. 92(14), 143904 (2004).
[Crossref] [PubMed]

Sanda, P. N.

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: Application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26(6), 2907–2916 (1982).
[Crossref]

Schilling, J.

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gosele, “Diffraction properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 83(4), 614–616 (2003).
[Crossref]

Sheng, P.

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: Application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26(6), 2907–2916 (1982).
[Crossref]

Sievenpiper, D.

D. Sievenpiper, L. J. Zhang, R. F. J. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microw. Theory Tech. 47(11), 2059–2074 (1999).
[Crossref]

Soukoulis, C.

M. Diem, T. Koschny, and C. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
[Crossref]

Stepleman, R. S.

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: Application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26(6), 2907–2916 (1982).
[Crossref]

Tyler, J.

A. Hibbins, W. Murray, J. Tyler, S. Wedge, W. Barnes, and J. Sambles, “Resonant absorption of electromagnetic fields by surface plasmons buried in a multilayered plasmonic nanostructure,” Phys. Rev. B 74(7), 073408 (2006).
[Crossref]

von Freymann, G.

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gosele, “Diffraction properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 83(4), 614–616 (2003).
[Crossref]

Wedge, S.

A. Hibbins, W. Murray, J. Tyler, S. Wedge, W. Barnes, and J. Sambles, “Resonant absorption of electromagnetic fields by surface plasmons buried in a multilayered plasmonic nanostructure,” Phys. Rev. B 74(7), 073408 (2006).
[Crossref]

Wegener, M.

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gosele, “Diffraction properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 83(4), 614–616 (2003).
[Crossref]

Wehrspohn, R. B.

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gosele, “Diffraction properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 83(4), 614–616 (2003).
[Crossref]

Wei, Z.

Z. Wei, J. Fu, Y. Cao, C. Wu, and H. Li, “The impact of local resonance on the enhanced transmission and dispersion of surface resonances,” Photon. Nanostruct.: Fundam. Appl. 8(2), 94–101 (2010).
[Crossref]

Wood, R. W.

R. W. Wood, “Anomalous Diffraction Gratings,” Phys. Rev. 48(12), 928–936 (1935).
[Crossref]

Wu, C.

Z. Wei, J. Fu, Y. Cao, C. Wu, and H. Li, “The impact of local resonance on the enhanced transmission and dispersion of surface resonances,” Photon. Nanostruct.: Fundam. Appl. 8(2), 94–101 (2010).
[Crossref]

Yablonovitch, E.

D. Sievenpiper, L. J. Zhang, R. F. J. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microw. Theory Tech. 47(11), 2059–2074 (1999).
[Crossref]

Zhang, L. J.

D. Sievenpiper, L. J. Zhang, R. F. J. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microw. Theory Tech. 47(11), 2059–2074 (1999).
[Crossref]

Appl. Phys. Lett. (1)

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gosele, “Diffraction properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 83(4), 614–616 (2003).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

D. Sievenpiper, L. J. Zhang, R. F. J. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microw. Theory Tech. 47(11), 2059–2074 (1999).
[Crossref]

J. Appl. Phys. (1)

J. Brown, A. Hibbins, M. Lockyear, C. Lawrence, and J. Sambles, “Angle-independent microwave absorption by ultrathin microcavity arrays,” J. Appl. Phys. 104(4), 043105 (2008).
[Crossref]

J. Opt. A-Pure Appl. Opt (1)

P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A-Pure Appl. Opt.  2, 48–51 (2000).
[Crossref]

Opt. Express (2)

Photon. Nanostruct.: Fundam. Appl. (1)

Z. Wei, J. Fu, Y. Cao, C. Wu, and H. Li, “The impact of local resonance on the enhanced transmission and dispersion of surface resonances,” Photon. Nanostruct.: Fundam. Appl. 8(2), 94–101 (2010).
[Crossref]

Phys. Rev. (1)

R. W. Wood, “Anomalous Diffraction Gratings,” Phys. Rev. 48(12), 928–936 (1935).
[Crossref]

Phys. Rev. B (3)

M. Diem, T. Koschny, and C. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
[Crossref]

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: Application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26(6), 2907–2916 (1982).
[Crossref]

A. Hibbins, W. Murray, J. Tyler, S. Wedge, W. Barnes, and J. Sambles, “Resonant absorption of electromagnetic fields by surface plasmons buried in a multilayered plasmonic nanostructure,” Phys. Rev. B 74(7), 073408 (2006).
[Crossref]

Phys. Rev. Lett. (2)

M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, “Microwave surface-plasmon-like modes on thin metamaterials,” Phys. Rev. Lett. 102(7), 073901 (2009).
[Crossref] [PubMed]

A. P. Hibbins, J. R. Sambles, C. R. Lawrence, and J. R. Brown, “Squeezing millimeter waves into microns,” Phys. Rev. Lett. 92(14), 143904 (2004).
[Crossref] [PubMed]

R. Soc. London Ser. A (1)

P. Rayleigh, “On the Dynamical Theory of Gratings,” R. Soc. London Ser. A 79(532), 399–416 (1907).
[Crossref]

Other (5)

A. Taflove, and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Norwood, MA, 2000).

The transverse electric (TE) polarized incident wave with the electric field E//x is blind to the air gaps between the metallic strips`, and treats the whole structure as a homogeneous dielectric slab with PEC ground effectively.

N. Engheta, and R. W. Ziolkowski, Metamaterials: physics and engineering explorations (Wiley & Sons., 2006).

M. Born, and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon Press, Oxford, Angleterre, 1980).

M. Mansuripur, Classical Optics and its Applications (Cambridge University Press, 2002).

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Figures (4)

Fig. 1
Fig. 1 (a) Schematic configuration of the hybrid slab. The regions containing perfectly electric conductor (PEC) or dielectric are marked by orange color or blue color. The geometrical parameters are a = 20 mm , b = 6 mm , g = 1 mm , h = 1.6 mm and t = 0.035 mm . The permittivity of dielectric slab is ε r = 2.65 . A transverse magnetic (TM) plane wave is incident in y ^ z ^ plane with an incident angle θ, the magnetic field is along the y axis. (b) The functionality of the hybrid slab can be understood as the result of the coupling of a chain of alternating cavities A and B.
Fig. 2
Fig. 2 Reflectivity spectra of the 0 th (red solid lines) −1st (blue dashed lines) and + 1st(dark yellow short-dashed lines) order of reflected waves for TM polarized incident plane waves with different incident angles θ = 0 ° , 10 ° , 20 ° and 30 ° . The frequencies in white regions [(b)-(d)] confirm that only 0th and −1st reflected waves propagate in the free space with real wave vector component along z axis. The colored arrows in the inset of (a) schematically illustrate the directions of the incidence I0 (black) and the corresponding reflected waves in the 0th order (red), the −1st order (blue), and the + 1st order (dark yellow).
Fig. 3
Fig. 3 (a) Dispersion diagram of surface resonance states on the hybrid slab. Reflectivity of the 0th or -1st reflected diffractive waves matching to the k y 0 or k y < 0 surface resonances on branch B4 and B5 in colored region of reciprocal space are marked with the size of the symbol being proportional the magnitude of the reflectivity (see text for details). The reflectivity is also plotted as a function of incident angles at (b) for f = 7.65GHz, (c) for f = 9.88GHz and (d) for f = 10.88GHz.
Fig. 4
Fig. 4 Negative reflection from the hybrid slab at 7.65GHz with an incident angle at θ = 44.6 ° . FDTD simulations with one-way Gaussian beam incident on (a) a PEC slab, and (b) a hybrid slab. (c) The angular spectra of the measured reflection intensity in far-field. (d) Snapshot of local field patterns inside the rectangle box illustrated in Fig. 4(b). The insets in (a) and (b) illustrate the structure of PEC slab and hybrid slab. The positions of one-way Gaussian beam are indicated by horizontal thin lines in (a) and (b). Black arrows refers to the directions of incident Gaussian beams, while red and blue arrows refer to the directions of specular reflection from PEC slab and that of negative reflection from hybrid slab respectively.

Equations (2)

Equations on this page are rendered with MathJax. Learn more.

H 1 = m = + [ δ m , 0 e i k m , z 1 ( z h t ) + r m e i k m , z 1 ( z h t ) ] e i ( k y + G m ) y H 3 = m = + { t m e i k m , z 3 ( z h ) + t m e i [ k m , z 3 ( z h ) + φ m ] } e i ( k y + G m ) y ,
H 2 = l g l ( y ) [ a l e i k l , z 2 ( z h t ) + b l e i k l , z 2 ( z h ) ] .

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