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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 22 — Nov. 4, 2013
  • pp: 27219–27237
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Mode-expansion theory for inhomogeneous meta-surfaces

Shiyi Xiao, Qiong He, Che Qu, Xin Li, Shulin Sun, and Lei Zhou  »View Author Affiliations


Optics Express, Vol. 21, Issue 22, pp. 27219-27237 (2013)
http://dx.doi.org/10.1364/OE.21.027219


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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

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

1. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]

], super and hyper lensing [2

2. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]

5

5. S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt. 50(9), 1419–1430 (2003).

], polarization control [6

6. J. M. Hao, Y. Yuan, L. X. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett. 99(6), 063908 (2007). [CrossRef] [PubMed]

], and so on. Recently, with the help of transformation optics (TO) theory [7

7. U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]

,8

8. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

], inhomogeneous MTMs with slowly changing material properties were widely studied, based on which more fascinating effects were discovered, such as invisibility cloaking [9

9. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006). [CrossRef] [PubMed]

,10

10. W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007). [CrossRef]

], illusion optics [11

11. Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009). [CrossRef] [PubMed]

13

13. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010). [CrossRef] [PubMed]

], lensing [14

14. U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett. 98(24), 243901 (2007). [CrossRef] [PubMed]

17

17. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010). [CrossRef] [PubMed]

] and beam bending [18

18. B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express 18(19), 20321–20333 (2010). [CrossRef] [PubMed]

]. 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.

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

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

32. F. L. Zhang, Q. Zhao, L. Kang, J. Zhou, and D. Lippens, “Experimental verification of isotropic and polarization properties of high permittivity-based metamaterial,” Phys. Rev. B 80(19), 195119 (2009). [CrossRef]

] onto a PEC with laterally different densities to realize this model. Also, it has been proven in [6

6. J. M. Hao, Y. Yuan, L. X. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett. 99(6), 063908 (2007). [CrossRef] [PubMed]

, 33

33. J. M. Hao, L. Zhou, and C. T. Chan, “An effective-medium model for high-impedance surfaces,” Appl. Phys. A Mater. Sci. Process. 87(2), 281–284 (2007). [CrossRef]

, 34

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

] 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
εM(x)=(εMx(x)000εMy(x)000εMz),μM(x)=(μMx(x)000μMy(x)000μMz),
(1)
and the PEC substrate is defined as region III. Note that the parallel components of εM(x),μM(x) 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)
Ein(r)=ei(kxinxkzinz)(010),Hin(r)=1Z0ei(kxinxkzinz)(kzin/k00kxin/k0).
(2)
Here, k0=ω/c is the wave vector in vacuum with ω being the working frequency and c the speed of light, kxin and kzin=(k02(kxin)2)1/2 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[iωt] 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

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

22

22. S. Sun, K.-Y. Yang, C.-M. Wang, T.-K. Juan, W. T. Chen, C. Y. Liao, Q. He, S. Y. Xiao, W.-T. Kung, G.-Y. Guo, L. Zhou, and D.-P. Tsai, “High-efficiency broadband anomalous reflection by gradient meta-surfaces,” Nano Lett. 12(12), 6223–6229 (2012). [CrossRef] [PubMed]

].

For each reflection channel, the corresponding EM fields can be written as
Eref,n(r)=ei(kxr,nx+kzr,nz)(010),Href,n(r)=1Z0ei(kxr,nx+kzr,nz)(kzr,n/k00kxr,n/k0),
(3)
where kxr,n=kxin+2πn/L (n=0,±1,...,±N,...), and kzr,n=(k02(kxr,n)2)1/2 [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,
{EI=Ein+nEref,nρ(kxr,n)HI=Hin+nHref,nρ(kxr,n),
(4)
in which ρ(kxr,n) 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
εM1(x)×[μM1(x)(×E(r))]=ω2E(r),
(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
Ey±(qz,x,z)=G(qz,x)eiqzz,
(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 G(qz,x) function satisfies

μMx(x)μMzd2G(qz,x)dx2+[k02εMy(x)μMx(x)qz2]G(qz,x)=0.
(7)

Once Ey is known by solving Eq. (7), H fields can be derived from Maxwell’s equations. Explicitly, we have
Hx±(qz,x,z)=1iωμ0μMx(x)Eyz=qzk0Z0μMx(x)G(qz,x)eiqzz
(8)
and

Hz±(qz,x,z)=1iωμ0μMzEyx=eiqzzik0Z0μMzG(qz,x)x.
(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

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.

], each with length h = L/(2N + 1), we can rewrite the differential Eq. (7) as the following 2N + 1 linear equations
μM,mxG(qz,m1)γ+[k02εM,myμM,mxμM,mxγ]G(qz,m)+μM,m+1xG(qz,m+1)γ=qz2G(qz,m).
(10)
Here γ(μMz)1h2, m[1,2N+1] labels asub cell with position located at xm = (2m - 1)h/2, and εM,my, μM,my and G(qz,m) are the values of functions εMy(x), μMx(x) and G(qz,x) taken at the position x = xm. It is worth mentioning that we take the periodic boundary condition so thatεM,m+2N+1y=εM,my, μM,m+2N+1x=μM,mx and G(qz,m+2N+1)=G(qz,m). We can further rewrite Eq. (10) as the following matrix form
m'Hmm'Gm'=qz2Gm,
(11)
where Hmm' is a (2N+1)×(2N+1) matrix with elements defined by [37

37. For two boundary indexes, we have the following off-diagonal matrix elementsH1,2N+1=μM,1xγ, H2N+1,1=μM,2N+1xγ according to the periodic boundary condition.

]
Hmm'=(k02μM,mxεM,my2μM,mxγ)δmm'+μM,mxγδm,m'1+μM,mxγδm,m'+1.
(12)
Diagonalizing the H matrix, we can obtain 2N + 1 eigenvalues labeled as qz,j2. The eigen vector corresponding to the j-th eigenvalue is just [G(qz,j,1),…, G(qz,j,i),…, G(qz,j,2N+1)]T, which gives the wave function of G(qz,x) in a discretized manner. The discretized versions of Ey, Hx, Hz can be easily obtained from the G(qz,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
EII(r)=j[C+(qz,j)E+(qz,j,r)+C(qz,j)E(qz,j,r)],HII(r)=j[C+(qz,j)H+(qz,j,r)+C(qz,j)H(qz,j,r)],
(13)
where the summation runs over all (2N + 1) eigenvalues of qz and [C+(qz,j), C(qz,j)] is another set of coefficients to be determined.

The same technique can be easily extended to the case of a transverse-magnetic (TM) plane wave excitation (i.e., Hinc||y^,kin=kxinx^kzinz^). After tedious calculations, we found that the reflection coefficients ρ(kxr,n) can still be calculated by Eq. (18), except that the two overlapping integrals are now defined as
{S(qz,j,kxr,n)=1Lmh(1+ei2qz,jd)G(qz,j,m)eikxr,nmhS'(qz,j,kxr,n)=1Lmhqz,jεM,mxkz0(1ei2qz,jd)G(qz,j,m)eikxr,nmh,
(22)
and the H matrix originally defined in Eq. (12) should now be defined as

Hmm'=(k02εM,mxμM,my2εM,mxγ)δmm'+εM,mxγδm,m'1+εM,mxγδm,m'+1.
(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 εM and μM 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

6. J. M. Hao, Y. Yuan, L. X. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett. 99(6), 063908 (2007). [CrossRef] [PubMed]

, 20

20. S. L. Sun, Q. He, S. Y. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012). [CrossRef] [PubMed]

, 22

22. S. Sun, K.-Y. Yang, C.-M. Wang, T.-K. Juan, W. T. Chen, C. Y. Liao, Q. He, S. Y. Xiao, W.-T. Kung, G.-Y. Guo, L. Zhou, and D.-P. Tsai, “High-efficiency broadband anomalous reflection by gradient meta-surfaces,” Nano Lett. 12(12), 6223–6229 (2012). [CrossRef] [PubMed]

, 33

33. J. M. Hao, L. Zhou, and C. T. Chan, “An effective-medium model for high-impedance surfaces,” Appl. Phys. A Mater. Sci. Process. 87(2), 281–284 (2007). [CrossRef]

, 34

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

], but extensions of the theory to the cases without ground planes (e.g., single-layer meta-surfaces [19

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

,21

21. X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science 335(6067), 427 (2012). [CrossRef] [PubMed]

,23

23. P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012). [CrossRef]

28

28. X. Chen, L. Huang, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, C. W. Qiu, S. Zhang, and T. Zentgraf, “Dual-polarity plasmonic metalens for visible light,” Nat. Commun. 3, 1198 (2012). [CrossRef] [PubMed]

]) 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

20. S. L. Sun, Q. He, S. Y. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012). [CrossRef] [PubMed]

]. 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., εM(x),μM(x)) by letting the whole structure exhibit a linearly varying reflection-phase profile
Φ(x)=Φ0+ξx,
(24)
for normally incident EM wave with polarization E||y^ (instead of E||x^ assumed in [20

20. S. L. Sun, Q. He, S. Y. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012). [CrossRef] [PubMed]

]). We note that there are multiple solutions for εM(x),μM(x) 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 εM(x)=μM(x). A simple calculation shows that
εMy(x)=μMx(x)=1+κx
(25)
with κ ≡ ξ /2k0d. Realizing the difficulties in matching the impedance at every local point, in the second model we set εMy=1 and let μMx vary as a function of x. The μMx(x) distribution can be easily obtained by solving Eq. (24) with local reflection phase determined by the following equation [20

20. S. L. Sun, Q. He, S. Y. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012). [CrossRef] [PubMed]

]
Φ(x)=cos1{[εMy+μMxtan2(εMyμMxk0d)]/[εMy+μMxtan2(εMyμMxk0d)]}.
(26)
All other parameters in both models are simply set as 1, i.e., εMx(x)=μMy(x)=εMz=μMz=1.

As explicit illustrations, we show in Figs. 2(a)
Fig. 2 Material properties of meta-surfaces with different ξ designed based the (a) [εMy=μMx] model and (b) the [εMy=1,μMx] model. (c) and (d): Calculated scattering coefficients |ρ(kxr)|2 versus kxr for different meta-surfaces.
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 |ρ(kxr)|2 [38

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(kxr,n) for simplicity.

] take maximum values at kxr=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)
Fig. 3 (a) Calculated scattering coefficients |ρ(kxr)|2 of the ξ=0.8k0 meta-surface designed with the [εMy=μMx] model, under illuminations of TE-polarized input wave with different parallel wave-vectors. (b) Parallel wave-vector kxr of the reflected beam as functions that of the indent beam kxr, calculated by the mode-expansion theory for two meta-surfaces with different ξ and a PEC (with ξ = 0).
shows the |ρ(kxr)|2 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 kxin. Different spectra are maximized at different values of kxr, but it is interesting to note that the relation
kxr=ξ+kxin
(27)
holds well for all the cases. Equation (27), known as the generalized Snell’s law [19

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

, 20

20. S. L. Sun, Q. He, S. Y. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012). [CrossRef] [PubMed]

], 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.

We now identify the conversion efficiency for such anomalous reflection. Since the anomalous reflection beam (with parallel wave vector kxr) travels along an off-normal direction, its beam width is reduced by a factor of cosθr=(1(kxr/k0)2)1/2 as compared to the incident beam along the normal direction [20

20. S. L. Sun, Q. He, S. Y. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012). [CrossRef] [PubMed]

, 39

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). [CrossRef]

]. Therefore, the final expression for the conversion efficiency should be
Rc=|ρ(kxr)|2cosθr=|ρ(kxr)|21(ξ/k0)2.
(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 cosθr 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 [εMy=1,μMx(x)] model with ξ = 0.8k0, we found that |ρ(0.8k0)|2=1.58 [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

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

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

, 21

21. X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science 335(6067), 427 (2012). [CrossRef] [PubMed]

26

26. L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett. 12(11), 5750–5755 (2012). [CrossRef] [PubMed]

, 30

30. D. Berry, R. Malech, and W. Kennedy, “The reflectarray antenna,” IEEE Trans. Antennas Propag. 11(6), 645–651 (1963). [CrossRef]

, 31

31. S. Larouche and D. R. Smith, “Reconciliation of generalized refraction with diffraction theory,” Opt. Lett. 37(12), 2391–2393 (2012). [CrossRef] [PubMed]

, 41

41. M. Albooyeh, D. Morits, and C. R. Simovski, “Electromagnetic characterization of substrated metasurfaces,” Metamaterials 5(4), 178–205 (2011). [CrossRef]

] 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 εMy=μMx=f(x) 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,
δ(kxrkxin)+ρ(kxr)=0dqzC+(qz)12π+(1ei2qzd)G(qz,x)eikxrxdx,
(32)
δ(kxrkxin)ρ(kxr)=0dqzC+(qz)12π+(1+ei2qzd)G(qz,x)eikxrxdx,
(33)
where a continuous notation has been adopted for convenience. By adding Eqs. (32) and (33), we obtain
δ(kxrkxin)=12π+eikxrxdx0dqzG(qz,x)C+(qz).
(34)
Put δ(kxrkxin)=(2π)1/2+eikxrxeikxinxdx into Eq. (34), we found that
0dqzG(qz,x)C+(qz)=eikxinx.
(35)
Multiply (2π)1+G*(qz,x)dx to both sides of Eq. (35) and using the orthonormality of the G functions, we get the solution for C+(qz) as,
C+(qz)=12π+G*(qz,x)eikxinxdx.
(36)
Inserting Eq. (36) into Eqs. (32) and (33), we finally get the scattering coefficients as
ρ(kxr)=1(2π)2+eikxrx'dx'+eikxinxdx0G(qz,x')ei2qzdG*(qz,x)dqz,
(37)
which can be re-casted into a bra-ket form following [42

42. P. Sheng, “Wave scattering formalism,” in Introduction to Wave Scattering, Localization and Macroscopic Phenomena, R. Hull, R. M. Osgood, eds. (Springer, 2006).

],
ρ(kxr)=1(2π)2+dx'+dx0dqz0dqz×kxr|x'x'|G|qzqz|P|qzqz|G*|xx|kxin.
(38)
Here, we have formally defined that kxr|x'=eikxrx', G*(qz,x)=x|G*|qz, G(qz,x')=x'|G|qz, x|kxin=eikxinx. In particular, we define in a general way that Pqz,qz=ei2qzdδ(qzqz) with its diagonal element describing the propagation of an eigenstate inside the MTM layer. Let us further define two quantities as
{Vqz,kxin=12π+dxqz|G*|xx|kxinVkxr,qz*=12π+dxkxr|xx|G|qz,
(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,

ρ(kxr)=Tkxr,kxin=qz,qzVkxr,qz*Pqz,qzVqz,kxin.
(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 kxin strikes on the meta-surface, it first couples into all possible eigenstates inside the MTM with coupling strength Vqz,kxin. 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 kxr with coupling coefficient Vkxr,qz*. Obviously, Eq. (40) neglected the multiple scattering processes.

Base on Eq. (40), we can obtain an analytical solution of ρ(kxr) if the V matrix (Eq. (39)) is known. However, the eigen wave-function G(qz,x) is difficult to solve analytically from Eq. (7). Fortunately, numerical solutions of G(qz,x) function give us enough hints to “guess” an analytical form. Figure 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.
shows the computed G(qz,x) functions (in a discretized version) with different qz, obtained by the numerical approach described in Sec. 2. The most striking feature of the G(qz,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
qz=k0μMx(x)εMy(x)=k0f(x)
(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
G(qz,x)=δ(xf1(qz/k0)),
(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 qz=k0(μMx(x)εMy(x))1/2=k0f(x). 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.

Put Eq. (42) into Eq. (38), we finally obtain that
ρ(kxr)=12π+dxkxr|xrlocal(x)x|kxin,
(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

Esca(x)=rlocal(x)x|kxin.
(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

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

, 21

21. X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science 335(6067), 427 (2012). [CrossRef] [PubMed]

26

26. L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett. 12(11), 5750–5755 (2012). [CrossRef] [PubMed]

, 30

30. D. Berry, R. Malech, and W. Kennedy, “The reflectarray antenna,” IEEE Trans. Antennas Propag. 11(6), 645–651 (1963). [CrossRef]

, 31

31. S. Larouche and D. R. Smith, “Reconciliation of generalized refraction with diffraction theory,” Opt. Lett. 37(12), 2391–2393 (2012). [CrossRef] [PubMed]

]. 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+κx model to explicitly illustrate this point. Put κ ≡ ξ/2k0d into Eq. (43), we get that

ρLRM(kxr)=e2ik0dδ(kxrkxinξ).
(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
R=|ρ(kx)|2k02kx2dkxk02(kxin)2,
(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 kxin. 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)
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 kxin, calculated by the rigorous mode-expansion theory (black solid lines) and the LRM (red dotted lines). Here we adopted the impedance-matched model εMy=μMx=1+ξx/2k0d for all meta-surfaces studied.
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 kxin 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 kxin<0.2k0. In contrast, our full theory always gives energy-conserved results.

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 εM(x),μM(x) distributions. Instead, typically the designed/fabricated MTM layers exhibit stepwise discontinuous distributions of εM(x),μM(x), depending on how many building blocks adopted in one super cell. To model such realistic situations, we truncate the continuous distributions of the μ(x) profiles in the [εMy=1,μMx] models for both ξ = 0.4k0 and ξ = 0.8k0 cases to stepwise versions as shown in Fig. 7(a)
Fig. 7 (a) Distributions of μMx(x) for meta-surfaces with ξ = 0.4k0 (circles) and ξ = 0.8k0 (triangles), designed based on the stepwise [εMy=1,μMx] models. (b) Scattering coefficients |ρ(kxr)|2 versus kxr for meta-surfaces with properties depicted in (a), calculated by the mode-expansion theory.
. 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 |ρ(kxr)|2 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.

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

6. J. M. Hao, Y. Yuan, L. X. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett. 99(6), 063908 (2007). [CrossRef] [PubMed]

, 33

33. J. M. Hao, L. Zhou, and C. T. Chan, “An effective-medium model for high-impedance surfaces,” Appl. Phys. A Mater. Sci. Process. 87(2), 281–284 (2007). [CrossRef]

, 34

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

], 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

33. J. M. Hao, L. Zhou, and C. T. Chan, “An effective-medium model for high-impedance surfaces,” Appl. Phys. A Mater. Sci. Process. 87(2), 281–284 (2007). [CrossRef]

]. Both propagating-wave and surface-wave properties of a HIS can be accurately reproduced by calculations based on such an effective-medium model [33

33. J. M. Hao, L. Zhou, and C. T. Chan, “An effective-medium model for high-impedance surfaces,” Appl. Phys. A Mater. Sci. Process. 87(2), 281–284 (2007). [CrossRef]

], 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
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.
, 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

20. S. L. Sun, Q. He, S. Y. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012). [CrossRef] [PubMed]

], 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

20. S. L. Sun, Q. He, S. Y. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012). [CrossRef] [PubMed]

].

We first employed finite-difference-time-domain (FDTD) simulations [43

43. EastFDTD v2.0 Beta, DONGJUN Science and Technology Co., China.

] 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

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.

]. 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, |S21|2, 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 θrθi(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 E||y^) microwaves with a double-ridged horn antenna [45

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

], 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 kxr=k0sinθr to retrieve the parallel k vectors of the reflected beams, we find that the measured kxr 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)).

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 kxr for the reflected beam versus kxin 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

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.

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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.

OCIS Codes
(080.2710) Geometric optics : Inhomogeneous optical media
(110.2760) Imaging systems : Gradient-index lenses
(240.0240) Optics at surfaces : Optics at surfaces
(260.2065) Physical optics : Effective medium theory
(160.3918) Materials : Metamaterials
(050.6624) Diffraction and gratings : Subwavelength structures

ToC Category:
Metamaterials

History
Original Manuscript: September 6, 2013
Revised Manuscript: October 27, 2013
Manuscript Accepted: October 27, 2013
Published: November 1, 2013

Citation
Shiyi Xiao, Qiong He, Che Qu, Xin Li, Shulin Sun, and Lei Zhou, "Mode-expansion theory for inhomogeneous meta-surfaces," Opt. Express 21, 27219-27237 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-22-27219


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  35. 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 elementsH1,2N+1=μM,1xγ, H2N+1,1=μM,2N+1xγ 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(kxr,n) for simplicity.
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  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.

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