Anomalous reflection from hybrid metamaterial slab

Zeyong Wei; Hongqiang Li; Chao Wu; Yang Cao; Jinzhi Ren; Zhihong Hang; Hong Chen; Daozhong Zhang; C. T. Chan

doi:10.1364/OE.18.012119

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 [3–5] 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 a≠b 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 [9–12]. In our model, the geometric parameters are as follows: $a = 13 mm + δ$ , $t = 0.035 mm$ , $h = 1.6 mm$ , $b = 13 mm - δ$ and $g = 1 mm$ , where $δ = 7 mm$ as hybrid parameter is adjustable; The period of the slab along $\hat{y}$ direction is $p = a + b + 2 g = 28 mm$ giving rise to a Rayleigh frequency at $f_{R} = c_{0} / p \approx 10.71 GHz$ under normal incidence [13], where $c_{0}$ is the light speed in vacuum. The permittivity of the dielectric layer is $ε_{r} = 2.65$ . The metallic grating is along the $\hat{x}$ direction so that the air gaps only allow the transverse magnetic ( $T M$ ) polarized wave (with the electric field $\vec{E}$ in $\hat{y} \hat{z}$ 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.

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 $\hat{y} \hat{z}$ 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 $\hat{x}$ direction $H_{i} (i = 1, 3)$ in region I, and region III ( $0 < z < h$ ) can be expressed in the form of the Bloch mode expansion [15–18],

\begin{array}{l} H_{1} = \sum_{m = - \infty}^{+ \infty} [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} = \sum_{m = - \infty}^{+ \infty} {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}, \end{array}

where

δ_{m, 0} e^{i k_{m, z_{1}} (z - h - t)}

denotes the incident field with a wave vector

\vec{k} = {\vec{k}}_{/ /} + \hat{z} k_{0, z_{1}}

,δ being the Kronecker function, and integer m being the order of the Bloch channel. The reciprocal wave vector is

G_{m} = 2 π m / p

, and the corresponding wave vector component along

\hat{z}

direction is

k_{m, z_{i}} = \sqrt{{(n_{i} ω / c_{0})}^{2} - {(G_{m} + k_{y})}^{2}}

(

i = 1, 3

), where

n_{i}

denotes the refractive index of the dielectric medium in region I or III;

r_{m}

or

t_{m}

is the coefficient of the

m^{t h}

reflected or guided Bloch waves in region I or III;

φ_{m} = 2 h k_{m, z_{3}}

is the phase delay with respect to the thickness h of slab waveguide. In the inhomogeneous layer of region II (

h \leq z < 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 modes

g_{l} (y) = \cos (y l / g)

, (

l = 0, 1, ..., n, ...

) inside the gaps alternatively. Thus the magnetic field can be simply expressed as,

H_{2} = \sum_{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)}] .

In addition, it is accurate enough to acquire a single mode, the fundamental mode

g_{0} (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

and

z = h + t

, we obtain the coefficients

t_{m} (f, {\vec{k}}_{/ /})

and

r_{m} (f, {\vec{k}}_{/ /})

of the

m^{t h}

guided and reflected waves, the dispersion

ω ({\vec{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 ( $\vec{H} ∥ \hat{y}$ ). The directions of the incidence and the reflected waves are illustrated by colored arrows in the inset of Fig. 2(a) . Specifically, we calculate the $0^{th}$ , $\pm 1^{st}$ orders of reflectivity spectra $R_{m} = Re (k_{m, z_{1}} / k_{0, z_{1}}) | r_{m} |^{2}$ ( $m = 0, \pm 1$ ) 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 from $c_{0} / p - k_{0} \sin (θ) c_{0} / 2 π$ to $c_{0} / p + k_{0} \sin (θ) c_{0} / 2 π$ , the reflection behaves as the competition between the $0^{th}$ order and $- 1^{st}$ order of reflected waves with the reflection coefficients satisfying to $R_{0} + R_{- 1} = 1$ . The 0th order reflection at some specific frequencies and incident angles is greatly suppressed with extremely small value of $R_{0}$ . For examples, we have $R_{0} = 10^{- 5}$ at $f = 9.88 G H z$ , $θ = 10^{°}$ [Fig. 2(b)], $R_{0} = 0.08$ at $f = 10.5 G H z$ , $θ = 20^{°}$ [Fig. 2(c)], and $R_{0} = 0.003$ at $f = 10.9 G H z$ , $θ = 30^{°}$ [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.

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).

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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 $\hat{y}$ direction for a hybrid slab with $δ = 7 m m$ stated above. This system supports multiple magnetic resonances, which can be identified in the 0th order reflection spectra $R_{0} (f, θ)$ at frequencies satisfying $\arg (R_{0} (f, θ)) = 0$ . 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 $a ≫ b$ 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.

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 $k_{y} \geq 0$ 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.

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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 obtain $- 1^{st}$ order of reflection matching to the same zone of reciprocal space for the homogeneous case with $δ = 0$ . When $δ \neq 0$ , the unit cell is nearly doubled in real space and zone folding due to Bragg scattering occurs at a short wave vector $k_{y} = π / p = π / (a + b)$ instead of $k_{y} = π / a$ in the homogenous slab. We also note that the branch B2 is very flat with a resonant frequency (with $k_{/ /} = 0$ at $Γ_{2}$ point) 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 ( $k_{y} = k_{0} \sin θ$ ), the 0th order reflection coefficient will find its minimum in the spectra $R_{0} (f)$ at a given incident angle, or equivalently, the –1st order reflection coefficient will find its maximum in its spectra $R_{- 1} (f)$ as shown in Fig. 2.

As shown in Fig. 3(a), we mark all the peaks of the 0th order reflection coefficient $R_{0}$ on the right side (red circles) and the −1st order reflection coefficient $R_{- 1}$ 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 $R_{- 1} = 1$ for the incidence matching to those states at 7.65GHz, 9.88GHz and 10.88GHz with zero group velocity ( $\partial ω / \partial k_{/ /} ~ 0$ ). 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 velocity $\partial ω / \partial k_{/ /} ~ 0$ ). As shown in Figs. 3(b)-3(d) $R_{- 1}$ reaches unity at $θ = {44.6}^{°}, f = 7.65 G H z$ , at $θ = 10^{°}, {65.4}^{°}, f = 9.88 G H z$ , and at $θ = {29.4}^{°}, f = 10.89 G H z$ , 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 $R_{- 1} = 1$ in Fig. 3(b) means that at 7.65GHz the −1st order reflection has a high efficiency of $R_{- 1} > 0.99$ for ${40.4}^{°} < θ < {48.6}^{°}$ . 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 parameter $δ = 7 m m$ already ensures that the −1st order of reflection with $R_{- 1} = 1$ at 7.65GHz propagates precisely along the counter-propagating direction of incidence at $θ^{'} = - {44.6}^{°}$ , 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 is $θ = {44.6}^{°}$ . 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.

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.

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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 $θ \approx 45^{°}$ . 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 $7.65 G H z$ is measured along the anti-direction of incidence at angle $θ^{'} \approx - 45^{°}$ , 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 $k_{y} = π / p$ of incidence. This is precisely the wave form of surface state, at $7.65 G H z$ with an in-plane wavevector $k_{y} - G = - π / p$ , 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).

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Anomalous reflection from hybrid metamaterial slab

Abstract

1. Introduction

2. Model description and analytical methodology

3. Reflection coefficients derived from hybrid structure

4. Numerical simulations and experiments on negative reflection

5. Conclusion

Acknowledgement

References and links

Cited By

Figures (4)

Equations (2)

Optics Express