Mode-expansion theory for inhomogeneous meta-surfaces · 38. Since the super-cell length L is very...

Mode-expansion theory for inhomogeneous meta-surfaces

Shiyi Xiao,1 Qiong He,1 Che Qu,1 Xin Li,1 Shulin Sun,2 and Lei Zhou1,* 1State Key Laboratory of Surface Physics, Key Laboratory of Micro and Nano Photonic Structures (Ministry of

Education) and Physics Department, Fudan University, Shanghai 200433, China 2Key Laboratory for Micro and Nanophotonic Structures (Ministry of Education), Department of Optical Science and

Engineering, School of Information Science and Engineering, Fudan University, Shanghai 200433, China *[email protected]

Abstract: Modeling meta-surfaces as thin metamaterial layers with continuously varying bulk parameters, we employed a rigorous mode-expansion theory to study the scattering properties of such systems. We found that a meta-surface with a linear reflection-phase profile could redirect an impinging light to a non-specular channel with nearly 100% efficiency, and a meta-surface with a parabolic reflection-phase profile could focus incident plane wave to a point image. Under certain approximations, our theory reduces to the local response model (LRM) established for such problems previously, but our full theory has overcome the energy non-conservation problems suffered by the LRM. Microwave experiments were performed on realistic samples to verify the key theoretical predictions, which match well with full-wave simulations.

©2013 Optical Society of America

OCIS codes: (160.3918) Metamaterials; (260.2065) Effective medium theory; (080.2710) Inhomogeneous optical media; (110.2760) Gradient-index lenses; (240.0240) Optics at surfaces; (050.6624) Subwavelength structures.

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#196945 - $15.00 USD Received 6 Sep 2013; revised 27 Oct 2013; accepted 27 Oct 2013; published 1 Nov 2013(C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.027219 | OPTICS EXPRESS 27219

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35. These reflection channels could also be understood as the Floquet modes diffracted by our super-periodic system. 36. In our computational approach, we have to set the number of sub-cells divided identical to the number plane

waves adopted in region (both are 2N + 1), in order to ensure that the number of restraints equals to that of variables.

37. For two boundary indexes, we have the following off-diagonal matrix elements1, 2 1 ,1

x

N MH μ γ

+= ,

2 1,1 , 2 1

x

N M NH μ γ

+ += according to the periodic boundary condition.

38. Since the super-cell length L is very large, the distribution of those discretized kxr,n is almost continuous. Thus, in

what follows, we use ρ(kxr) to represent ρ(kx

r,n) for simplicity. 39. C. Qu, S. Y. Xiao, S. L. Sun, Q. He, and L. Zhou, “A theoretical study on the conversion efficiencies of gradient

meta-surfaces,” Europhys. Lett. 101(5), 54002 (2013). 40. A. Pors, M. G. Nielsen, R. L. Eriksen, and S. I. Bozhevolnyi, “Broadband focusing flat mirrors based on

plasmonic gradient metasurfaces,” Nano Lett. 13(2), 829–834 (2013). 41. M. Albooyeh, D. Morits, and C. R. Simovski, “Electromagnetic characterization of substrated metasurfaces,”

Metamaterials 5(4), 178–205 (2011). 42. P. Sheng, “Wave scattering formalism,” in Introduction to Wave Scattering, Localization and Macroscopic

Phenomena, R. Hull, R. M. Osgood, eds. (Springer, 2006). 43. EastFDTD v2.0 Beta, DONGJUN Science and Technology Co., China. 44. For the ξ = 0.4k0 sample, a super cell contains 10 pairs of “H” (altogether 20 ones) in one supercell, with L1

values of those 10 pairs set as 1.3 mm, 2.68 mm, 2.98 mm, 3.14 mm, 3.24 mm, 3.36 mm, 3.48 mm, 3.66 mm, 4.08 mm, and 5.7 mm. For the ξ = 0.8k0 sample, a super cell contains 10 “H” in one super cell with L1 parameters the same as the case of ξ = 0.4k0.

45. The gain of the employed double-ridged horn antenna is about 14dB~15dB in this frequency region.

1. Introduction

Last decade has witnessed tremendous progresses in using metamaterials (MTMs) to manipulate light on a subwavelength scale. MTMs are artificial materials composed by manmade functional electromagnetic (EM) microstructures, typically in deep-subwavelength


sizes and exhibiting tailored electric and/or magnetic responses. Early studies were largely conducted on homogenous MTMs, with discovered light-manipulation phenomena including negative refraction [1], super and hyper lensing [2–5], polarization control [6], and so on. Recently, with the help of transformation optics (TO) theory [7,8], inhomogeneous MTMs with slowly changing material properties were widely studied, based on which more fascinating effects were discovered, such as invisibility cloaking [9,10], illusion optics [11–13], lensing [14–17] and beam bending [18]. These works revealed that a carefully designed “inhomogeneity” could manipulate the local propagation phase inside the medium so as to adiabatically guide light travelling along a desired trajectory dictated by the TO theory, leading to new physics and phenomena.

Very recently, much attention were paid to inhomogeneous systems with abruptly changing materials properties, in particular, ultra-thin MTMs (also called meta-surfaces) constructed by carefully designed planar subwavelength components with tailored EM responses [19–29]. Rather than modulating the propagation phase inside a bulk medium, these planar systems explored another degree of freedom to modulate the lateral distribution of the abrupt phase change of reflected/transmitted light across the systems. It was shown that the transmission/reflection of light follow a generalized Snell’s law at the interface between air and a carefully designed gradient meta-surface, in which the parallel momentum of light is not conserved at the interface but rather gain an additional term contributed by the lateral gradient of the transmission/reflection phase change [19,20]. In particular, the meta-surface can perfectly convert an impinging propagating wave to a bounded surface wave under certain conditions [20]. Other fascinating wave-manipulation effects include focusing by planar lens [27–29], generalization of optical vortex [19,23], and so forth.

In sharp contrast to the exciting achievements on experimental side, the theoretical developments for studying such complex systems are far behind. Apparently, standard approaches for homogeneous MTMs are not applicable here. Furthermore, the TO theory [7,8], being a powerful tool for inhomogeneous MTMs with slowly varying properties, is also not suitable for present inhomogeneous meta-surfaces exhibiting abruptly changing material properties [19,20]. In previous works, theoretical understandings were either based on heuristic Fermat-Huygens wave interference arguments or based on full-wave simulations [19,21–29]. The former is very intuitive but is far from rigorous and thus cannot fully explain the rich physical phenomena observed (say, the multi-channel reflections/refractions discovered in [19]). The latter is basically a computational experiment, which is rigorous enough but suffers the limitation of physically less illuminating. We notice that a local response model (LRM), which assumed each local part of the inhomogeneous meta-surface to response locally to the incident wave, was established previously for such systems [30, 31]. However, we will show in the following sections that the LRM suffers severe energy non-conservation problems in many cases. Therefore, a general theoretical approach to study light scatterings by such inhomogeneous meta-surfaces, which can yield energy-conserved results, is still lacking and is highly desired.

Here, we develop such a theoretical framework. In contrast to previous theories assuming zero thicknesses for the studied systems, we model the meta-surfaces as thin MTM layers with finite thickness and with laterally varying bulk EM parameters. Although some limited results of the theory (obtained under restricted conditions) have been reported in [20], here we present all details of the theoretical developments under general conditions (Sec. 2) and then apply the theory to study two specific examples, one exhibiting a linear reflection-phase distribution and another a parabolic one (Sec. 3). Section 4 is devoted to highlighting the key advantage of our theoretical approach. Specifically, we show that our theory can reduce to the LRM under several approximations, but the full theory (without taking these approximations) has overcome the energy non-conservation problems faced by the LRM. Finally, we design and fabricate a series of realistic samples and perform microwave experiments to verify the theoretical predictions (Sec. 5). We conclude our paper in Section 6.


2. Developments of the mode-expansion theory

We chose a particular system as a concrete example to illustrate the developments of our theory. The model system is schematically shown in Fig. 1(a), which is an inhomogeneous MTM layer of thickness d (much smaller than wavelength λ) put on top of a perfect electric conductor (PEC). For simplicity, we assume that the MTM layer is inhomogeneous only along x direction, but is invariant along both y and z directions. Adding a PEC substrate makes the entire system totally reflecting for EM waves so that we do not need to worry about the transmitted signals, which significantly simplifies the theoretical developments.

Fig. 1. (a) Geometry of the system under study. (b) Discretized model for the inhomogeneous structure.

Although the model looks ideal, we emphasize that actually it can be realized by various means in practice. For example, one can put powders of high-index materials [32] onto a PEC with laterally different densities to realize this model. Also, it has been proven in [6, 33, 34] that the usual high-impedance structures (HIS) can be well represented by such a double-layer model (in a homogeneous version), and thus it is straightforward to design an (laterally) inhomogeneous HIS to practically realize the model presented in Fig. 1.

As shown in Fig. 1, the entire space is divided into three regions, where region I denotes the vacuum, region II denotes the inhomogeneous MTM with permittivity and permeability matrices given by

( ) 0 0 ( ) 0 0

( ) 0 ( ) 0 , ( ) 0 ( ) 0 ,

0 0 0 0

x xM M

y yM M M M

z zM M

x x

x x x x

ε με ε μ μ

ε μ

= =

(1)

and the PEC substrate is defined as region III. Note that the parallel components of ( ), ( )M Mx xε μ

can vary as function of lateral position x. We are mostly concerned on the scattering properties of such an inhomogeneous system under arbitrary light illuminations.

Consider first the case that the excitation is a transverse-electric (TE) polarized incident plane wave, with field components explicitly given by (in region I)


0

( ( ))

00

0 /1

( ) 1 , ( ) 0 .

0 /

in inin inx xz z

inz

i k x i k xk z k zin in

inx

k k

E r e H r eZ

k k

−−

= = −

(2)

Here, 0 /k cω= is the wave vector in vacuum with ω being the working frequency and c the

speed of light, inxk and 2 2 1/2

0( ( ) )in inz xk k k= − are the parallel and vertical components of the k

vector for the incident wave, and Z0 = (μ0/ε0)1/2 is the impedance of vacuum. Here, we use the

physics convention and omit the common time oscillation term exp[ ]i tω− throughout this paper. Due to lacking of translational invariance on the xy-plane, the reflected beam does not necessarily exhibit a single k vector, but must in principle be a linear combination of plane waves with all allowed k vectors (each defined as a reflection channel). In general, there is no restriction on choosing the value of kx for each channel. However, it is more convenient in practical computations to introduce periodic boundary conditions at the two ends of considered system (with length denoted by L), i.e., (E, H)x = 0 = (E, H)x = L. Introducing a super periodicity makes computations more tractable and will not affect the final results when L is large enough. In addition, we note that some of the designed/fabricated meta-surfaces already exhibited super periodicities to which our method is naturally applicable [19–22].

For each reflection channel, the corresponding EM fields can be written as

, ,, ,

,0

( () ), ,

0 ,0

0 /1

( ) 1 , ( ) 0 ,

0 /

r n r nr n r nx xz z

r nz

i k x i k xk z k zref n ref n

r nx

k k

E r e H r eZ

k k

+ +

= =

−

(3)

where , 2 /r n inx xk k n Lπ= + ( 0, 1,..., ,...n N= ± ± ), and , 2 , 2 1/2

0( ( ) )r n r nz xk k k= − [35]. In practical

calculations, we typically adopt a large cutoff value N for n so that the final number of reflection channels is 2N + 1. To obtain reasonable results, convergence tests against both N and L (in case of a finite system without super periodicity) should be carefully performed. With these notations, EM fields in region I can be formally expressed as a sum of the incident plane wave and the reflected waves,

, ,

, ,

( ),

( )

I in ref n r nxn

I in ref n r nxn

E E E k

H H H k

ρ

ρ

= +

= +

(4)

in which ,( )r nxkρ are a set of coefficients to be determined.

Let us turn to consider the EM fields in region II. We need to first calculate the eigen wave functions of EM waves travelling inside this region, which are governed by the following equation

1 1 2( ) [ ( )( ( ))] ( ),M Mx x E r E rε μ ω− −∇ × ∇ × =

(5)

derived from the original Maxwell’s equations. For the present TE polarization, we only need to consider the Ey field component. Since the MTM is homogenous along z direction, we can perform variable separation to assume that

( , , ) ( , ) ,ziq zy z zE q x z G q x e± = (6)

where superscript ± stands for forward ( + ) and backward ( − ) propagating waves with wave-vector qz being a positive number. The parameter qz is used to label the eigenmodes inside region II and will be determined later. Inserting Eq. (6) to (5), we find that the ( , )zG q x function satisfies


2

2 202

( ) ( , )( ) ( ) ( , ) 0.

xy xM zM M z zz

M

x d G q xk x x q G q x

dx

μ ε μμ

+ − = (7)

Once Ey is known by solving Eq. (7), H fields can be derived from Maxwell’s equations. Explicitly, we have

0 0 0

1( , , ) ( , )

( ) ( )zy iq zz

x z zx xM M

E qH q x z G q x e

i x z k Z xωμ μ μ± ∂

= =∂

(8)

and

0 0 0

( , )1( , , ) .

ziq zy z

z z z zM M

E G q xeH q x z

i x ik Z xωμ μ μ± ∂ ∂= − = −

∂ ∂

(9)

It is hard to solve Eq. (7) analytically, so that we now develop a numerical approach. As shown in Fig. 1(b), discretizing a super cell of the inhomogeneous MTM slab into 2N + 1 sub-cells [36], each with length h = L/(2N + 1), we can rewrite the differential Eq. (7) as the following 2N + 1 linear equations

2

, 0 , , ,

2, 1

( , 1) [ ] ( , )

( , 1) ( , ).

x y x xM m z M m M m M m z

xM m z z z

G q m k G q m

G q m q G q m

μ γ ε μ μ γ

μ γ+

− + −

+ + = (10)

Here 1 2( )zM hγ μ − −≡ , [1,2 1]m N∈ + labels a sub cell with position located at xm = (2m - 1)h/2,

and ,yM mε , ,

yM mμ and ( , )zG q m are the values of functions ( )y

M xε , ( )xM xμ and ( , )zG q x taken

at the position x = xm. It is worth mentioning that we take the periodic boundary condition so that , 2 1 ,

y yM m N M mε ε+ + = , , 2 1 ,

x xM m N M mμ μ+ + = and ( , 2 1) ( , ).z zG q m N G q m+ + = We can further

rewrite Eq. (10) as the following matrix form

2' '

'

,mm m z mm

H G q G= (11)

where 'mmH is a (2 1) (2 1)N N+ × + matrix with elements defined by [37]

2' 0 , , , ' , , ' 1 , , ' 1( 2 ) .x y x x x

mm M m M m M m mm M m m m M m m mH k μ ε μ γ δ μ γδ μ γδ− += − + + (12)

Diagonalizing the H matrix, we can obtain 2N + 1 eigenvalues labeled as 2,z jq . The eigen

vector corresponding to the j-th eigenvalue is just [ ,( ,1)z jG q ,…, ,( , )z jG q i ,…,

,( , 2 1)z jG q N + ]T, which gives the wave function of ( , )zG q x in a discretized manner. The

discretized versions of Ey, Hx, Hz can be easily obtained from the ( , )zG q x function based on Eqs. (6)–(9).

Knowing all non-vanishing field components for every eigenmode, we can then formally write the EM fields in region II as linear combinations of these eigenmodes. Thus, in general we have

, , , ,

, , , ,

( ) ( ) ( , ) ( ) ( , ) ,

( ) ( ) ( , ) ( ) ( , ) ,

IIz j z j z j z j

j

IIz j z j z j z j

j

E r C q E q r C q E q r

H r C q H q r C q H q r

+ + − −

+ + − −

= +

= +

(13)

where the summation runs over all (2N + 1) eigenvalues of qz and [ ,( )z jC q+ , ,( )z jC q− ] is

another set of coefficients to be determined.


We now match the boundary conditions at two interfaces. On the MTM/PEC interface

located at z = −d, EM fields should follow the PEC boundary condition ( 0n E× ≡

) so that

, , ,( , , ) ( ) ( ) ( , ) 0.z ziq d iq dIIy z j z j z j

j

E x y z d C q e C q e G q x−+ − = − = + ≡ (14)

Since all eigenmodes are independent with each other, we get that

2, ,( ) ( ) .zi q d

z j z jC q C q e− += − (15)

Consider next the air/MTM interface located at z = 0. The tangential EM fields (Ey and Hx) should be continuous crossing the interface, i.e.,

, , at 0.I II I IIy y x xE E H H z= = = (16)

Put the explicit forms of fields (Eqs. (4) and (13)) into Eq. (16), we get that

,

,

,

, , ,

,,

0 0

,, , ,

, 0

( )

( , ) ( ) ( )

.( )

( , ) ( ) ( )

in r nx m x m

in r nx m x m

ik x ik xr nx

n

z j z j z jj

in r nik x ik xr nz z

xn

z jz j z j z jx

j M m

e k e

G q m C q C q

k ke k e

k k

qG q m C q C q

k

ρ

ρ

μ

+ −

+ −

+

= +

−

= −

(17)

Put Eq. (15) to Eq. (17), we have a set of 2(2N + 1) linear equations with (2N + 1) unknowns

{ },( )r nxkρ and (2N + 1) unknowns { },( )z jC q+ . Solve these equations by standard linear

algebra, we get the final form of reflection coefficients as

, 1 ,, ,( ) ( , ) ( , ),r n in r n

x x z j z j xj

k A k q B q kρ −= (18)

where we have introduced two matrixes defined as

,, ,

, , ,, ,

, ,, , ,, ,

( , ) ( , ) '( , )

,

( , ) ( , ) '( , )

r n inin r n r nz z

z j x z j x z j xr n in r n inz z z z

in inr n n r nz z

z j x z j x z j xr n in r n inz z z z

k kA q k S q k S q k

k k k k

k kB q k S q k S q k

k k k k

= + + +

= − + +

(19)

with

,,

,,

2,, ,

2,,, ,

,

1( , ) (1 ) ( , )

1'( , ) (1 ) ( , )

r nz j x

r nz j x

i q d ik mhr nz j x z j

m

i q dz j ik mhr nz j x z jx in

m M m z

S q k h e G q m eL

qS q k h e G q m e

L kμ

−

−

= − = − +

(20)

being the overlap integrals between the nth-order plane wave (in region I) and the eigenmodes in region II, and 1

,( , )inx z jA k q− is the inverse matrix of ,( , )in

z j xA q k . Therefore, all reflection

coefficients of the EM waves scattered to different channels can be calculated from the above equations.


The same technique can be easily extended to the case of a transverse-magnetic (TM)

plane wave excitation (i.e., ˆ ˆ ˆ|| ,inc in in inx zH y k k x k z= −

). After tedious calculations, we found

that the reflection coefficients ,( )r nxkρ can still be calculated by Eq. (18), except that the two

overlapping integrals are now defined as

,,

,,

2,, ,

2,,, ,0

,

1( , ) (1 ) ( , )

,1

'( , ) (1 ) ( , )

r nz j x

r nz j x

i q d ik mhr nz j x z j

m

i q dz j ik mhr nz j x z jx

m M m z

S q k h e G q m eL

qS q k h e G q m e

L kε

−

−

= + = − −

(22)

and the H matrix originally defined in Eq. (12) should now be defined as

2' 0 , , , ' , , ' 1 , , ' 1( 2 ) .x y x x x

mm M m M m M m mm M m m m M m m mH k ε μ ε γ δ ε γδ ε γδ− += − + + (23)

We note that the coefficients in front of exp[2iqz,jd] exhibit different signs in Eqs. (20) and (22), due to different boundary condition requirements for TE and TM cases. Besides this, in fact we can interchange Mε and Mμ to derive the TM formulas from the TE case, thanks to the excellent symmetry properties of EM fields.

We mention three points before concluding this section. First, after knowing the scattering properties of the system for both TE and TM excitations, we can in principle obtain all information of the scattered field under an arbitrary excitation (not necessarily a plane wave). Second, the developed technique is so general that there are no difficulties to extend it to more complicated cases, say, the inhomogeneous MTMs with materials properties depending on both x and y. Third, so far the developed formulas and the model adopted are directly applicable only to those meta-surfaces with ground planes [6, 20, 22, 33, 34], but extensions of the theory to the cases without ground planes (e.g., single-layer meta-surfaces [19,21,23–28]) are straightforward.

3. Applications of the theory

The developed theoretical approach can be applied to many inhomogeneous meta-surfaces. Below we present two explicit examples.

1. Meta-surfaces with linear reflection-phase profiles

In this subsection, we design a gradient system (based on the model depicted in Fig. (1)) working for the TE polarization, and then employ the newly developed mode-expansion theory to study its scattering properties. To determine the model parameters of the system, we adopt a local-phase argument similar to that taken in [20]. Although obviously such a designing scheme neglected the diffraction effects, our mode-expansion theory will take all such effects into account, and therefore can serve as a serious justification on such a designing scheme. Specifically, we fix the model parameters (i.e., ( ), ( )M Mx xε μ

) by letting the whole structure exhibit a linearly varying reflection-phase profile

0( ) ,x xξΦ = Φ + (24)

for normally incident EM wave with polarization ˆ||E y

(instead of ˆ||E x

assumed in [20]).

We note that there are multiple solutions for ( ), ( )M Mx xε μ making Eq. (24) satisfied, and

here we take two specific solutions to illustrate the applications of our theory. The first model is an ideal impedance-matched model, where we assume that ( ) ( )M Mx xε μ=

. A simple calculation shows that

( ) ( ) 1y xM Mx x xε μ κ= = + (25)


with κ ≡ ξ /2k0d. Realizing the difficulties in matching the impedance at every local point, in the second model we set 1y

Mε = and let xMμ vary as a function of x. The ( )x

M xμ distribution can be easily obtained by solving Eq. (24) with local reflection phase determined by the following equation [20]

{ }1 2 20 0( ) cos tan ( ) / tan ( ) .y x y x y x y x

M M M M M M M Mx k d k dε μ ε μ ε μ ε μ− Φ = − + + (26)

All other parameters in both models are simply set as 1, i.e., ( ) ( ) 1x y z zM M M Mx xε μ ε μ= = = = .

Fig. 2. Material properties of meta-surfaces with different ξ designed based the (a) [y x

M Mε μ= ]

model and (b) the [ 1,y x

M Mε μ= ] model. (c) and (d): Calculated scattering coefficients

2| ( ) |

r

xkρ versus

rxk for different meta-surfaces.

As explicit illustrations, we show in Figs. 2(a) and 2(b) the distributions of material properties for two models with different values of ξ. We employed the mode-expansion theory to study the scattering coefficients of meta-surfaces constructed by two different models with different ξ under normal excitations with TE polarizations. The spectra depicted in Figs. 2(c) and 2(d) show that 2| ( ) |r

xkρ [38] take maximum values at rxk ξ= for all the

cases studied, indicating that the incident wave is indeed redirected to the desired anomalous channel after reflections.

We also employed the mode-expansion theory to study the cases of oblique incident excitations. Figure 3(a) shows the 2| ( ) |r

xkρ spectra for a ξ = 0.8k0 meta-surface (based on the impedance-matched model) under illuminations with different oblique angles specified by the values of in

xk . Different spectra are maximized at different values of rxk , but it is interesting to

note that the relation

r inx xk kξ= + (27)

holds well for all the cases. Equation (27), known as the generalized Snell’s law [19, 20], is remarkable since it points out that an additional wave-vector ξ is always provided by the meta-surface. We performed a systematic study on three different meta-surfaces with ξ = 0,


0.4k0, 0.8k0, respectively, under TE excitations with different incident angles. The results depicted in Fig. 3(b) show that Eq. (27) holds perfectly for all the cases studied.

Fig. 3. (a) Calculated scattering coefficients 2

| ( ) |r

xkρ of the

00.8kξ = meta-surface designed

with the [y x

M Mε μ= ] model, under illuminations of TE-polarized input wave with different

parallel wave-vectors. (b) Parallel wave-vector r

xk of the reflected beam as functions that of

the indent beam r

xk , calculated by the mode-expansion theory for two meta-surfaces with

different ξ and a PEC (with ξ = 0).

We now identify the conversion efficiency for such anomalous reflection. Since the anomalous reflection beam (with parallel wave vector r

xk ) travels along an off-normal

direction, its beam width is reduced by a factor of 2 1/20cos (1 ( / ) )r

r xk kθ = − as compared to the incident beam along the normal direction [20, 39]. Therefore, the final expression for the conversion efficiency should be

( )2 2 2

0( ) cos ( ) 1 / .r rc x r xR k k kρ θ ρ ξ= ⋅ = ⋅ − (28)

In fact, since the reflectance should be defined as the ratio between normal Poynting-vector components of the reflected and incident lights, a factor cos rθ naturally appears in the expression of Eq. (28) for such anomalous reflections. One may easily verify that the conversion efficiency Rc → 1 for most cases studied, indicating that almost all incoming energies are converted to these non-specular channels after reflections by the meta-surfaces. However, in the case of the [ 1, ( )y x

M M xε μ= ] model with ξ = 0.8k0, we found that 2

0| (0.8 ) | 1.58kρ = [see Fig. 2(d)] so that the corresponding conversion efficiency can be easily calculated as Rc ~0.95 based on Eq. (28). We note that a small peak appears at kx = −0.8k0 in the spectrum [see blue line in Fig. 2(d)], which means that some of the incoming energy is converted to other channels due to the Bragg scatterings, so that the conversion efficiency to the desired anomalous reflection channel is not 100%.


2. Meta-surfaces with parabolic reflection-phase distributions

Fig. 4. (a) Working scheme of the flat meta-surface lens. (b) Distributions of parameter values

and reflection-phases Φ for the meta-surface. (c) Calculated 2

| ( ) |r

xkρ spectrum for the

designed lens under TE normal incident excitation. (d) Calculated E-field distribution for the waves scattered by the meta-surface.

Focusing a plane wave to a point image is always fascinating. Conventional lenses are much thicker than wavelength and exhibit certain curved shapes. Several previous works [27, 29, 40] have shown that an ultra-thin flat MTM lens exhibiting a parabolic reflection-phase distribution

2 20 0 0( ) focus focusx k x l k lΦ = Φ − + + ⋅ (29)

can achieve the desired functionality. Here, lfocus is the focal length. The key idea is to use the reflection phase gained at the meta-surface to compensate the propagation phase difference for waves radiated from different local positions at the meta-surface [see Fig. 4(a)]. However, previous works [27, 29, 40] only contained simulation and experimental results, without analytical calculations on model systems. In this subsection, we employ the newly developed mode-expansion theory to study the scattering properties of a meta-surface satisfying Eq. (29), as another application of our theory. Still based on the geometry shown in Fig. 1, we assume that the material parameters for the capping layer are given by

2 2( ) ( ) ( ) / 2 ,y xM M focus focusx x v l x l dε μ= = + − + (30)

where v is an arbitrary value to keep all parameters positive (here we set it as 30). Figure

4(b) depicts the profiles of ( )yM xε as well as ( )xΦ for the system under study, where we

have assumed lfocus = 5λ and took a super periodicity L = 10λ to truncate the otherwise divergent parameter profile. We studied the scattering properties of such a meta-surface, and


Fig. 4(c) depicts the calculated 2| ( ) |rxkρ spectrum. We note that the spectrum is symmetrical

for ± kx and does not show a delta-like peak at some particular kx position, which is different from the cases studied in last subsection. This is reasonable since we do not expect the reflected beam to be a plane wave. However, it is difficult to see the focusing effect from the

2| ( ) |rxkρ spectrum. Therefore, we computed the real-space field distribution from the

2| ( ) |rxkρ spectrum based on Eq. (4). Figure 4(d) presents the calculated result (with incoming

plane wave deducted), from which we can clearly identify a focal point at the pre-decided position. The E field at the focus is enhanced roughly 6 times with respect to that of the incident wave. However, the focusing effect suffers some distortions, which is due to the finite size of the flat lens (i.e., the super-periodicity L here), as already discussed in [29].

4. Comparisons with the local response model

Although the theory developed in last section is rigorous, it looks quite complicated and how it is connected with previously established theories (say, the LRM) is unclear. In this section, we show that our theory can recover the LRM [19, 21–26, 30, 31, 41] under several approximations. However, we note that the LRM inevitably face energy non-conservation problems in non-specular reflection cases, while our full theory (without taking approximations) always yields correct (energy-conserved) results. For simplicity, we study the impedance-matched meta-surfaces with ( )y x

M M f xε μ= = throughout this section. The first approximation is to set kz /k0 = 1 in the second equation of Eq. (17). The physics

and limitation of this approximation will be discussed later. Under this approximation, Eq. (17) can be rewritten as,

2

0

1( ) ( ) ( ) (1 ) ( , ) ,

2

rxz ik xi q dr in r

x x x z z zk k k dq C q e G q x e dxδ ρπ

∞ +∞ −+

−∞− + = − (32)

2

0

1( ) ( ) ( ) (1 ) ( , ) ,

2

rxz ik xi q dr in r

x x x z z zk k k dq C q e G q x e dxδ ρπ

∞ +∞ −+

−∞− − = + (33)

where a continuous notation has been adopted for convenience. By adding Eqs. (32) and (33), we obtain

0

1( ) ( , ) ( ).

2

rxik xr in

x x z z zk k e dx dq G q x C qδπ

+∞ ∞− +

−∞− = (34)

Put ( ) 1/2( ) 2

r inx xik x ik xr in

x xk k e e dxδ π+∞− −

−∞− = into Eq. (34), we found that

0

( , ) ( ) .inxik x

z z zdq G q x C q e∞ + = (35)

Multiply 1 *(2 ) ( , )zG q x dxπ+∞−

−∞′ to both sides of Eq. (35) and using the orthonormality of the

G functions, we get the solution for ( )zC q+ as,

*1( ) ( , ) .

2

inxik x

z zC q G q x e dxπ

+∞+

−∞= (36)

Inserting Eq. (36) into Eqs. (32) and (33), we finally get the scattering coefficients as

' 2 *2 0

1( ) ' ( , ') ( , ) ,

(2 )

r inx x zik x ik x i q dr

x z z zk e dx e dx G q x e G q x dqρπ

+∞ +∞ ∞−

−∞ −∞= − (37)

which can be re-casted into a bra-ket form following [42],


2 0 0

*

1( ) '

(2 )

' ' .

rx z z

r inx z z z z x

k dx dx dq dq

k x x G q q P q q G x x k

ρπ

+∞ +∞ ∞ ∞

−∞ −∞′= −

′ ′×

(38)

Here, we have formally defined that ''rxik xr

xk x e−= , * *( , )z zG q x x G q= ,

( , ') 'z zG q x x G q′ ′= , inxik xin

xx k e= . In particular, we define in a general way that 2

, ( )z

z z

i q dq q z zP e q qδ′ ′= − − with its diagonal element describing the propagation of an eigenstate

inside the MTM layer. Let us further define two quantities as

*

,

*

,

1

2 ,1

2

inz x

rx z

inz xq k

rx zk q

V dx q G x x k

V dx k x x G q

π

π

+∞

−∞

+∞

−∞

= =

(39)

which have clear physical explanations of the couplings between an external plane wave (with a parallel wave-vector kx) and the inner eigen wave-function specified by the perpendicular wave-vector qz. With Eq. (39), we can finally rewrite Eq. (38) as a standard T matrix form,

*,, , ,

,

( ) .r in r inz zx x x z z x

z z

rx q qk k k q q k

q q

k T V P Vρ ′′′

= = (40)

Equation (40) clearly shows that the considered problem is a second-order scattering process under the adopted approximation. When a plane wave with parallel wave-vector in

xk strikes on the meta-surface, it first couples into all possible eigenstates inside the MTM with coupling strength

, inz xq k

V . These eigenstates propagate inside the MTM (along z direction)

without interacting each other, and after reflection by the MTM/PEC interface, they propagate along -z direction and couple out of the MTM to the out-side plane wave mode r

xk with

coupling coefficient *

,rx zk q

V ′ . Obviously, Eq. (40) neglected the multiple scattering processes.

Base on Eq. (40), we can obtain an analytical solution of ( )rxkρ if the V matrix (Eq. (39))

is known. However, the eigen wave-function ( , )zG q x is difficult to solve analytically from

Eq. (7). Fortunately, numerical solutions of ( , )zG q x function give us enough hints to “guess”

an analytical form. Figure 5 shows the computed ( , )zG q x functions (in a discretized version) with different qz, obtained by the numerical approach described in Sec. 2. The most striking feature of the ( , )zG q x function is that it resembles very much as a δ function in a discretized version, with peak appearing at x which makes the condition

0 0( ) ( ) ( )x yz M Mq k x x k f xμ ε= = (41)

satisfied. Dashed lines represent the positions calculated by Eq. (41) for the adopted qz values. Excellent agreement between dashed lines and the peak positions is noted.

Such an important observation motives us to take the second approximation. We assume that the eigen wave function inside the inhomogeneous MTM is given by

( )10( , ) ( / ) ,z zG q x x f q kδ −= − (42)

where we have again used a continuous representation in consistency with our analytical solution. In addition to the obvious supports from the numerical solutions (see Fig. 5), we can also interpret Eq. (42) from a different aspect. As shown in Fig. 1(b), as we divide the


inhomogeneous MTM into many sub-cells, each sub-cell represents an independent open-end waveguide with 1/2

0 0( ( ) ( )) ( )x yz M Mq k x x k f xμ ε= = . Since the MTM is inhomogeneous, those

waveguides possess different qz and therefore do not talk to each other. Then, the final eigen wave-function associated with a particular qz will be highly localized at the very sub-cell that supports this wave-vector, as shown in Fig. 5.

Fig. 5. Calculated G functions in discretized versions for different eigenvalues qz for the model with f(x) = 1 + ξx/2k0d. Dashed lines represent the x positions satisfying Eq. (41). Here, ξ = 0.4k0, d = λ /20, L = 200Ls, Ls = 2π / ξ with λ being the working wavelength.

Put Eq. (42) into Eq. (38), we finally obtain that

local

1( ) ( ) ,

2r r inx x xk dx k x r x x kρ

π+∞

−∞= (43)

where rlocal(x) = −exp[2if(x)k0d] is the local reflection coefficient of the system at the position x. An inverse Fourier transform of Eq. (43) gives the distribution of scattered field (measured at the z = 0 plane) as

sca local( ) ( ) .inxE x r x x k= (44)

Equations (43) and (44), derived from our full theory with two approximations, are exactly the same as the LRM widely used for such systems [19, 21–26, 30, 31]. However, we must point out that Eq. (43) and thus the LRM actually suffer severe energy non-conservation problems. We take the f (x) = 1 xκ+ model to explicitly illustrate this point. Put κ ≡ ξ/2k0d into Eq. (43), we get that

02LRM ( ) ( ).ik dr r in

x x xk e k kρ δ ξ= − − − (45) Equation (45) tells us that the LRM predicts that the reflected wave always carries an

additional wave vector ξ with amplitude 1. However, energy-conservation law requires the normal components of energy fluxes to be conserved after the EM wave is reflected by a flat surface. Let us define a function as

2 2 2

0

2 20

| ( ) |,

( )

x x x

inx

k k k dkR

k k

ρ ⋅ − ⋅=

− (46)

which is the reflection efficiency of the whole device summing up all reflection channels. Energy-conservation law requires R ≡ 1 for such a system since there is no loss and transmission here. Unfortunately, the LRM cannot yield the energy-conserved results in many cases, and the total energy of reflected wave can be either larger or smaller than that of the


incident beam, depending on the values of ξ and inxk . As an illustration, we employed both our

full theory (without taking the two approximations) and the LRM to calculate the reflection efficiency R for meta-surfaces with different ξ under normal-incidence excitations, and Fig. 6(a) compares the R ~ξ relations calculated by two theories. Obviously, the energy non-conservation issue is more severe in large ξ cases for the LRM while our full theory always yields energy-conserved results. As another example, we show in Fig. 6(b) how R depends on the incidence parallel wave-vector in

xk for a meta-surface with a fixed ξ = 0.4k0. Again, we found that the LRM cannot yield energy-conserved results in general, and can even yield reflection efficiency exceeding 1 when 00.2in

xk k< − . In contrast, our full theory always gives energy-conserved results.

Fig. 6. The reflection efficiency R for (a) meta-surfaces with different ξ /k0 under normal-incidence excitations and for (b) a ξ = 0.4k0 meta-surface illuminated by plane waves with

different inxk , calculated by the rigorous mode-expansion theory (black solid lines) and the

LRM (red dotted lines). Here we adopted the impedance-matched model

01 / 2

y x

M Mx k dε μ ξ= = + for all meta-surfaces studied.

To understand the inherent physics accounting for the failure of LRM, we re-examined the two approximations adopted. For the first one, we found it can be justified only in small-ξ cases, since in such cases for the most relevant channel (the anomalous reflection channel) we have kz / k0 = (1 − (ξ / k0)

2)1/2 ≈1. When ξ is large, such a simplification is no longer valid. In fact, the term kz / k0 represents the impedance mismatch between the scattered and incident waves. When the scattered wave is not along the specular channel, the term kz /k0 generates important local-field corrections which cannot be neglected. However, the LRM assumed that the response of each part of the system is solely dependent on the incident field on that very point, but is independent on the direction of outgoing wave. Obviously, this approximation is too rough to completely neglect the local-field corrections for the non-specular reflections. The second approximation neglected the couplings between adjacent units, which is also questionable in general cases. Therefore, we conclude that the failure of the LRM in certain cases is due to its neglecting the local field corrections and the coupling effects.

5. Experimental and simulation verifications

In this section, we design and fabricate realistic gradient meta-surfaces to experimentally verify the theoretical predictions presented in Fig. 2(b). In practice, it is difficult to construct a MTM system representing the model depicted in Fig. 1(a) with continuous ( ), ( )M Mx xε μ

distributions. Instead, typically the designed/fabricated MTM layers exhibit stepwise discontinuous distributions of ( ), ( )M Mx xε μ

, depending on how many building blocks adopted in one super cell. To model such realistic situations, we truncate the continuous distributions of the ( )xμ profiles in the [ 1,y x

M Mε μ= ] models for both ξ = 0.4k0 and ξ = 0.8k0

cases to stepwise versions as shown in Fig. 7(a). We then employed the mode-expansion


theory to calculate the scattering properties of such systems. Figure 7(b) shows that, for such stepwise models which can better represent the realistic situations, the 2| ( ) |r

xkρ spectra remain essentially the same as those of their continuous counterparts, indicating that such models still work very well to achieve the desired anomalous reflection effect.

Fig. 7. (a) Distributions of ( )x

M xμ for meta-surfaces with ξ = 0.4k0 (circles) and ξ = 0.8k0

(triangles), designed based on the stepwise [ 1,y x

M Mε μ= ] models. (b) Scattering coefficients

2| ( ) |

r

xkρ versus

r

xk for meta-surfaces with properties depicted in (a), calculated by the mode-

expansion theory.

We can therefore design realistic samples according to the stepwise μ(x) profiles shown in Fig. 7(a). As already discussed in Sec. 3, there are multiple ways to realize such model practically, and here we choose one of them. In [6, 33, 34], it had already been proved that the HIS can be very well represented by a double-layer model with a thin homogeneous magnetic MTM layer put on a PEC (e.g., homogeneous version of Fig. 1(a)). The physics is that near-field interaction between the top metallic layer and the bottom PEC ground plane in a HIS can generate a magnetic resonance with anti-parallel currents induced on two layers [33]. Both propagating-wave and surface-wave properties of a HIS can be accurately reproduced by calculations based on such an effective-medium model [33], demonstrating the validity of the model. Thus, we can use the HIS as a building block to design our gradient meta-surfaces according to the μ(x) profiles depicted in Fig. 7(a).

A building block is shown in the inset to Fig. 8, which is a sandwich system consisting of a metallic “H” and a metallic ground plane, separated by a dielectric spacer (with εr = 3.9) of thickness d. We note that although the building block is essentially the same as that adopted in meta-surfaces designed previously [20], here we have to carefully re-perform the designs since the present meta-surface works for TE-polarized incidence wave rather than for TM case considered in [20].

Fig. 8. FDTD-retrieved μeff parameter (line) for HIS’ consisting of periodic arrangements of unit cells depicted in the inset, with different values of central bar length L1. Scatters represent those units adopted in designing the ξ = 0.4k0 model. Other parameters Px, Py, d, w and L2 are fixed as 2.5 mm, 6 mm, 1 mm, 0.5 mm, and 2 mm. The working frequency is 15 GHz.


We first employed finite-difference-time-domain (FDTD) simulations [43] to study how the μeff parameter of a building block is “tuned” by varying its geometrical parameters. By changing the central bar length L1 of the “H”, the magnetic resonance frequency of a building block can be efficiently changed, so that the μeff parameter can be dramatically modified at a given frequency. Choosing the working frequency as 15 GHz, we performed FDTD simulations to retrieve μeff parameters of such structures, and depicted the retrieved μeff parameters as functions of L1 in Fig. 8. Such calculations greatly facilitate us to select appropriate building blocks to construct the desired stepwise μ(x) profiles as shown in Fig. 7, for both ξ = 0.4k0 and 0.8k0 meta-surfaces [44]. We then fabricated the two samples and a picture of the ξ = 0.4k0 sample is shown in Fig. 9(a).

Fig. 9. (a) Picture of part of the fabricated ξ = 0.4k0 sample. (b) Schematics of the FF

characterization. Measured (scatters) and simulated (lines) scattering patterns, 2

21| |S , for the

samples with (c) ξ = 0.4k0 and (d) ξ = 0.8k0. In our experiments, we cannot measure the

reflection signals within the angle region of r iθ θ≈ (grey area) where the two antennas touch

each other.

We performed microwave experiments to characterize the functionalities of the fabricated samples. As schematically shown in Fig. 9(b), we illuminated these meta-surfaces by

normally incident TE-polarized (with ˆ||E y

) microwaves with a double-ridged horn antenna [45], and then measured the far-field (FF) scattering patterns using another identical double-ridged horn antenna. Both emitting and receiving horn antennas were connected to a vector-field analyzer (Agilent E8362C). The measured signals were normalized against a reference single, which was obtained through replacing the meta-surface by a metal plate of the same size. Figures 9(c) and 9(d) depict the normalized scattering patterns for the samples with ξ = 0.4k0 and ξ = 0.8k0, respectively. The experimental results are in excellent agreement with FDTD simulations on realistic structures.

It is difficult to make direct comparisons between experimental/simulation results and the mode-expansion model calculations, since the former are obtained with finite-sized samples while the latter are with infinite systems. Nevertheless, meaningful comparisons can still be made in terms of reflection angle and reflection efficiency. We can easily identify from Figs. 9(c) and 9(d) that the peaks of two scattering patterns appear at 22.5° and 52.5°, respectively. Using the formula 0 sinr

x rk k θ= to retrieve the parallel k vectors of the reflected beams, we

find that the measured rxk are about 0.38k0 and 0.79k0, respectively, which are in good

agreement with the rigorous mode expansion calculations in Fig. 7(b). Meanwhile, both


measured and simulated radiation patterns exhibit only a single main peak at the anomalous reflection angle in both ξ = 0.4k0 and ξ = 0.8k0 cases, which are again in good agreement with the ρ(kx) spectra calculated by the mode-expansion theory (Fig. 7(b)).

Fig. 10. Measured (blue line) and simulated (green circles) scattering patterns for (a) the ξ = 0.4k0 meta-surface illuminated by an TE wave with incident angle 30° and (b) the ξ = 0.8k0 meta-surface for incident angle 45°. Red lines denote the incident angle, and the grey area

denotes the angle region where we cannot measure the reflected signals. (c) r in

x xk k relations

obtained by experiments (crosses), simulations (triangles) and model (lines) for meta-surfaces with ξ = 0.4k0 (green) ξ = 0.8k0 (blue) and a flat PEC surface(black). The shadowed area denotes the region where negative reflection happens.

We also studied the scattering patterns of the fabricated meta-surfaces under oblique-angle excitations. Figures 10(a) and 10(b) present the normalized FF patterns of the two meta-surfaces under external illuminations at different incident angles. Again, the measured patterns are in excellent agreement with FDTD simulations, both showing that the reflected beams have been redirected to non-specular channels with very high efficiencies. Two points are worthy being emphasized. First, we note that in the case of Fig. 10(b) the reflection beam and the incident one appear at the same side with respect to the surface normal, indicating that the reflection is the so-called “negative” reflection. Second, in the measured scattering patterns, the normal (specular) reflection modes are nearly completely suppressed, which are also consistent with the theoretical calculations depicted in Fig. 2. This later property is an important advantage of present system over previous structures suffering the multi-mode conversion shortcomings [19, 21], and explains why the present system can have such high conversion efficiencies (nearly ~100%) for the anomalous reflection.

Finally, we systematically measured the scattering patterns for two samples under illuminations of input waves with incident angles varying within the whole angle region allowed. Figure 10(c) depicts the obtained r

xk for the reflected beam versus inxk which is the

parallel k vector of the incident wave for the two samples, which are again in excellent agreement with corresponding FDTD simulation results. We note that all measured/simulated data fall into two separate lines defined by Eq. (27) - the general Snell’s law, which are in turn, agreeing perfectly with the mode-expansion results recorded in Fig. 3(b).

6. Conclusions

In summary, based on a new model for inhomogeneous meta-surfaces, we established a general theoretical framework to study the scattering properties of such systems, and applied it to two particular examples. Our theory recovers previously established local response model


under several simplifications, but the full theory has overcome the energy non-conservation problems encountered by previous theory. We designed and fabricated realistic structures according to theoretical calculations, and performed microwave experiments and full-wave simulations to verify the key theoretical predictions of our theory.

Acknowledgments

This work was supported by NSFC (60990321, 11174055), the Program of Shanghai Subject Chief Scientist (12XD1400700) and MOE of China (B06011). QH acknowledges financial supports from NSFC (11204040) and China Postdoctoral Science Foundation.


Mode-expansion theory for inhomogeneous meta-surfaces · 38. Since the super-cell length L is very...

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