## Mode-expansion theory for inhomogeneous meta-surfaces |

Optics Express, Vol. 21, Issue 22, pp. 27219-27237 (2013)

http://dx.doi.org/10.1364/OE.21.027219

Acrobat PDF (1890 KB)

### 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

*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]

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

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]

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

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. 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]

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

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

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]

*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.

*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

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]

29. X. Li, S. Y. Xiao, B. G. Cai, Q. He, T. J. Cui, and L. Zhou, “Flat metasurfaces to focus electromagnetic waves in reflection geometry,” Opt. Lett. **37**(23), 4940–4942 (2012). [CrossRef] [PubMed]

*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

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. 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]

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]

27. F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. **12**(9), 4932–4936 (2012). [CrossRef] [PubMed]

29. X. Li, S. Y. Xiao, B. G. Cai, Q. He, T. J. Cui, and L. Zhou, “Flat metasurfaces to focus electromagnetic waves in reflection geometry,” Opt. Lett. **37**(23), 4940–4942 (2012). [CrossRef] [PubMed]

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]

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]

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

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

*slowly varying*properties, is also not suitable for present inhomogeneous meta-surfaces exhibiting

*abruptly*changing material properties [19

**334**(6054), 333–337 (2011). [CrossRef] [PubMed]

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]

**334**(6054), 333–337 (2011). [CrossRef] [PubMed]

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]

29. X. Li, S. Y. Xiao, B. G. Cai, Q. He, T. J. Cui, and L. Zhou, “Flat metasurfaces to focus electromagnetic waves in reflection geometry,” Opt. Lett. **37**(23), 4940–4942 (2012). [CrossRef] [PubMed]

**334**(6054), 333–337 (2011). [CrossRef] [PubMed]

*locally*to the incident wave, was established previously for such systems [30

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

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

*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.

**11**(5), 426–431 (2012). [CrossRef] [PubMed]

## 2. Developments of the mode-expansion theory

*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.

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]

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. 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. 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]

*x*. We are mostly concerned on the scattering properties of such an inhomogeneous system under arbitrary light illuminations.

*ω*being the working frequency and

*c*the speed of light,

**k**vector for the incident wave, and

*Z*

_{0}= (

*μ*

_{0}/

*ε*

_{0})

^{1/2}is the impedance of vacuum. Here, we use the physics convention and omit the common time oscillation term

*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

*k*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

_{x}*L*), i.e., (

**E**,

**H**)

_{x}_{= 0}= (

**E**,

**H**)

_{x}_{=}

*. Introducing a super periodicity makes computations more tractable and will not affect the final results when*

_{L}*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

**334**(6054), 333–337 (2011). [CrossRef] [PubMed]

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]

*N*for

*n*so that the final number of reflection channels is 2

*N*+ 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,in which

*E*field component. Since the MTM is homogenous along

_{y}*z*direction, we can perform variable separation to assume thatwhere superscript

*q*being a positive number. The parameter

_{z}*q*is used to label the eigenmodes inside region II and will be determined later. Inserting Eq. (6) to (5), we find that the

_{z}*E*is known by solving Eq. (7),

_{y}**H**fields can be derived from Maxwell’s equations. Explicitly, we haveand

*N*+ 1 sub-cells [36], each with length

*h*=

*L*/(2

*N*+ 1), we can rewrite the differential Eq. (7) as the following 2

*N*+ 1 linear equationsHere

*x*= (2

_{m}*m*- 1)

*h*/2, and

*x*=

*x*. It is worth mentioning that we take the periodic boundary condition so that

_{m}**H**matrix, we can obtain 2

*N*+ 1 eigenvalues labeled as

*j-*th eigenvalue is just [

*, which gives the wave function of*

^{T}*E*,

_{y}*H*,

_{x}*H*can be easily obtained from the

_{z}*N*+ 1) eigenvalues of

*q*and [

_{z}*z*= −

*d*, EM fields should follow the PEC boundary condition (

*z*= 0. The tangential EM fields (

*E*and

_{y}*H*) should be continuous crossing the interface, i.e.,Put the explicit forms of fields (Eqs. (4) and (13)) into Eq. (16), we get that

_{x}*N*+ 1) linear equations with (2

*N*+ 1) unknowns

*N*+ 1) unknowns

*n*th-order plane wave (in region I) and the eigenmodes in region II, and

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

*iq*

_{z}_{,}

*] 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*

_{j}d*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]

**11**(5), 426–431 (2012). [CrossRef] [PubMed]

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. 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. 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]

*without*ground planes (e.g., single-layer meta-surfaces [19

**334**(6054), 333–337 (2011). [CrossRef] [PubMed]

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. 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. 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]

## 3. Applications of the theory

### 1. Meta-surfaces with linear reflection-phase profiles

**11**(5), 426–431 (2012). [CrossRef] [PubMed]

**11**(5), 426–431 (2012). [CrossRef] [PubMed]

*κ*≡

*ξ*/

*2k*

_{0}

*d*. Realizing the difficulties in matching the impedance at every local point, in the second model we set

*x*. The

**11**(5), 426–431 (2012). [CrossRef] [PubMed]

*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

*ξ*= 0.8

*k*

_{0}meta-surface (based on the impedance-matched model) under illuminations with different oblique angles specified by the values of

**334**(6054), 333–337 (2011). [CrossRef] [PubMed]

**11**(5), 426–431 (2012). [CrossRef] [PubMed]

*ξ*is always provided by the meta-surface. We performed a systematic study on three different meta-surfaces with

*ξ*= 0, 0.4

*k*

_{0}, 0.8

*k*

_{0}, 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.

*off-normal*direction, its beam width is reduced by a factor of

*normal*direction [20

**11**(5), 426–431 (2012). [CrossRef] [PubMed]

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]

*anomalous*reflections. One may easily verify that the conversion efficiency

*R*→ 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 [

_{c}*ξ*= 0.8

*k*

_{0}, we found that

*R*~0.95 based on Eq. (28). We note that a small peak appears at

_{c}*k*= −0.8

_{x}*k*

_{0}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

27. F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. **12**(9), 4932–4936 (2012). [CrossRef] [PubMed]

**37**(23), 4940–4942 (2012). [CrossRef] [PubMed]

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). [CrossRef] [PubMed]

*ultra-thin flat*MTM lens exhibiting a parabolic reflection-phase distributioncan achieve the desired functionality. Here,

*l*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

_{focus}27. F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. **12**(9), 4932–4936 (2012). [CrossRef] [PubMed]

**37**(23), 4940–4942 (2012). [CrossRef] [PubMed]

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). [CrossRef] [PubMed]

*l*= 5

_{focus}*λ*and took a super periodicity

*L*= 10

*λ*to truncate the otherwise divergent parameter profile. We studied the scattering properties of such a meta-surface, and Fig. 4(c) depicts the calculated

*k*and does not show a delta-like peak at some particular

_{x}*k*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

_{x}**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

**37**(23), 4940–4942 (2012). [CrossRef] [PubMed]

## 4. Comparisons with the local response model

**334**(6054), 333–337 (2011). [CrossRef] [PubMed]

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

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

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

*first approximation*is to set

*k*/

_{z}*k*

_{0}= 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, where a continuous notation has been adopted for convenience. By adding Eqs. (32) and (33), we obtainPut

*G*functions, we get the solution for

*k*) and the inner eigen wave-function specified by the perpendicular wave-vector

_{x}*q*. With Eq. (39), we can finally rewrite Eq. (38) as a standard T matrix form,

_{z}*under the adopted approximation*. When a plane wave with parallel wave-vector

*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

*V*matrix (Eq. (39)) is known. However, the eigen wave-function

*in a discretized version*) with different

*q*, obtained by the numerical approach described in Sec. 2. The most striking feature of the

_{z}*discretized version*, with peak appearing at

*q*values. Excellent agreement between dashed lines and the peak positions is noted.

_{z}*second approximation*. We assume that the eigen wave function inside the inhomogeneous MTM is given bywhere 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

*q*and therefore do not talk to each other. Then, the final eigen wave-function associated with a particular

_{z}*q*will be highly localized at the very sub-cell that supports this wave-vector, as shown in Fig. 5.

_{z}*r*

_{local}(x) = −exp[2

*if*(

*x*)

*k*

_{0}

*d*] 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

**334**(6054), 333–337 (2011). [CrossRef] [PubMed]

**335**(6067), 427 (2012). [CrossRef] [PubMed]

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

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

*f*(

*x*) =

*κ*≡

*ξ*/2

*k*

_{0}

*d*into Eq. (43), we get that

*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 aswhich 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

*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

*ξ*= 0.4

*k*

_{0}. Again, we found that the LRM cannot yield energy-conserved results in general, and can even yield reflection efficiency exceeding 1 when

*k*/

_{z}*k*

_{0}= (1 − (

*ξ*/

*k*

_{0})

^{2})

^{1/2}≈1. When

*ξ*is large, such a simplification is no longer valid. In fact, the term

*k*/

_{z}*k*

_{0}represents the impedance mismatch between the scattered and incident waves. When the scattered wave is not along the specular channel, the term

*k*/

_{z}*k*

_{0}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

*continuous*

*discontinuous*distributions of

**]**models for both

*ξ*= 0.4

*k*

_{0}and

*ξ*= 0.8

*k*

_{0}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

*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

**99**(6), 063908 (2007). [CrossRef] [PubMed]

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. 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]

**87**(2), 281–284 (2007). [CrossRef]

*accurately*reproduced by calculations based on such an effective-medium model [33

**87**(2), 281–284 (2007). [CrossRef]

*μ*(

*x*) profiles depicted in Fig. 7(a).

*ε*= 3.9) of thickness

_{r}*d*. We note that although the building block is essentially the same as that adopted in meta-surfaces designed previously [20

**11**(5), 426–431 (2012). [CrossRef] [PubMed]

**11**(5), 426–431 (2012). [CrossRef] [PubMed]

*μ*parameter of a building block is “tuned” by varying its geometrical parameters. By changing the central bar length

_{eff}*L*

_{1}of the “H”, the magnetic resonance frequency of a building block can be efficiently changed, so that the

*μ*parameter can be dramatically modified at a given frequency. Choosing the working frequency as 15 GHz, we performed FDTD simulations to retrieve

_{eff}*μ*parameters of such structures, and depicted the retrieved

_{eff}*μ*parameters as functions of

_{eff}*L*

_{1}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*= 0.4

*k*

_{0}and 0.8

*k*

_{0}meta-surfaces [44

44. For the *ξ* = 0.4*k*_{0} sample, a super cell contains 10 pairs of “H” (altogether 20 ones) in one supercell, with *L*_{1} 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* = 0.8*k*_{0} sample, a super cell contains 10 “H” in one super cell with *L*_{1} parameters the same as the case of *ξ* = 0.4*k*_{0}.

*ξ*= 0.4

*k*

_{0}sample is shown in Fig. 9(a).

*ξ*= 0.4

*k*

_{0}and

*ξ*= 0.8

*k*

_{0}, respectively. The experimental results are in excellent agreement with FDTD simulations on realistic structures.

**k**vectors of the reflected beams, we find that the measured

*k*

_{0}and 0.79

*k*

_{0}, 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*= 0.4

*k*

_{0}and

*ξ*= 0.8

*k*

_{0}cases, which are again in good agreement with the

*ρ*(

*k*) spectra calculated by the mode-expansion theory (Fig. 7(b)).

_{x}**334**(6054), 333–337 (2011). [CrossRef] [PubMed]

**335**(6067), 427 (2012). [CrossRef] [PubMed]

**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

## References and links

1. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science |

2. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

3. | N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science |

4. | D. O. S. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express |

5. | S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt. |

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. |

7. | U. Leonhardt, “Optical conformal mapping,” Science |

8. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science |

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 |

10. | W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics |

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. |

12. | Y. Lai, H. Y. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett. |

13. | H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. |

14. | U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett. |

15. | A. O. Pinchuk and G. C. Schatz, “Metamaterials with gradient negative index of refraction,” J. Opt. Soc. Am. A |

16. | O. Paul, B. Reinhard, B. Krolla, R. Beigang, and M. Rahm, “Gradient index metamaterial based on slot elements,” Appl. Phys. Lett. |

17. | N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. |

18. | B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express |

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 |

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. |

21. | X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science |

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. |

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. |

24. | F. Aieta, P. Genevet, N. Yu, M. A. Kats, Z. Gaburro, and F. Capasso, “Out-of-plane reflection and refraction of light by anisotropic optical antenna metasurfaces with phase discontinuities,” Nano Lett. |

25. | N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. |

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. |

27. | F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. |

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. |

29. | X. Li, S. Y. Xiao, B. G. Cai, Q. He, T. J. Cui, and L. Zhou, “Flat metasurfaces to focus electromagnetic waves in reflection geometry,” Opt. Lett. |

30. | D. Berry, R. Malech, and W. Kennedy, “The reflectarray antenna,” IEEE Trans. Antennas Propag. |

31. | S. Larouche and D. R. Smith, “Reconciliation of generalized refraction with diffraction theory,” Opt. Lett. |

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 |

33. | J. M. Hao, L. Zhou, and C. T. Chan, “An effective-medium model for high-impedance surfaces,” Appl. Phys. A Mater. Sci. Process. |

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. |

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 2 |

37. | For two boundary indexes, we have the following off-diagonal matrix elements |

38. | Since the super-cell length ρ(k) to represent _{x}^{r}ρ(k) for simplicity._{x}^{r,n} |

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. |

40. | A. Pors, M. G. Nielsen, R. L. Eriksen, and S. I. Bozhevolnyi, “Broadband focusing flat mirrors based on plasmonic gradient metasurfaces,” Nano Lett. |

41. | M. Albooyeh, D. Morits, and C. R. Simovski, “Electromagnetic characterization of substrated metasurfaces,” Metamaterials |

42. | P. Sheng, “Wave scattering formalism,” in |

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

44. | For the |

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

Sort: Year | Journal | Reset

### References

- R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science292(5514), 77–79 (2001). [CrossRef] [PubMed]
- J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85(18), 3966–3969 (2000). [CrossRef] [PubMed]
- N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science308(5721), 534–537 (2005). [CrossRef] [PubMed]
- D. O. S. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express13(6), 2127–2134 (2005). [CrossRef] [PubMed]
- 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).
- 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]
- U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
- J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
- 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,” Science314(5801), 977–980 (2006). [CrossRef] [PubMed]
- W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics1(4), 224–227 (2007). [CrossRef]
- 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]
- Y. Lai, H. Y. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett.102(9), 093901 (2009). [CrossRef] [PubMed]
- H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater.9(5), 387–396 (2010). [CrossRef] [PubMed]
- 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]
- A. O. Pinchuk and G. C. Schatz, “Metamaterials with gradient negative index of refraction,” J. Opt. Soc. Am. A24(10), A39–A44 (2007). [CrossRef] [PubMed]
- O. Paul, B. Reinhard, B. Krolla, R. Beigang, and M. Rahm, “Gradient index metamaterial based on slot elements,” Appl. Phys. Lett.96(24), 241110 (2010). [CrossRef]
- N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater.9(2), 129–132 (2010). [CrossRef] [PubMed]
- B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express18(19), 20321–20333 (2010). [CrossRef] [PubMed]
- 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,” Science334(6054), 333–337 (2011). [CrossRef] [PubMed]
- 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]
- X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science335(6067), 427 (2012). [CrossRef] [PubMed]
- 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]
- 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]
- F. Aieta, P. Genevet, N. Yu, M. A. Kats, Z. Gaburro, and F. Capasso, “Out-of-plane reflection and refraction of light by anisotropic optical antenna metasurfaces with phase discontinuities,” Nano Lett.12(3), 1702–1706 (2012). [CrossRef] [PubMed]
- N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett.12(12), 6328–6333 (2012). [CrossRef] [PubMed]
- 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]
- F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett.12(9), 4932–4936 (2012). [CrossRef] [PubMed]
- 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]
- X. Li, S. Y. Xiao, B. G. Cai, Q. He, T. J. Cui, and L. Zhou, “Flat metasurfaces to focus electromagnetic waves in reflection geometry,” Opt. Lett.37(23), 4940–4942 (2012). [CrossRef] [PubMed]
- D. Berry, R. Malech, and W. Kennedy, “The reflectarray antenna,” IEEE Trans. Antennas Propag.11(6), 645–651 (1963). [CrossRef]
- S. Larouche and D. R. Smith, “Reconciliation of generalized refraction with diffraction theory,” Opt. Lett.37(12), 2391–2393 (2012). [CrossRef] [PubMed]
- 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. B80(19), 195119 (2009). [CrossRef]
- 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]
- 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]
- 35. These reflection channels could also be understood as the Floquet modes diffracted by our super-periodic system.
- 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.
- 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.
- 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.
- 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]
- 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). [CrossRef] [PubMed]
- M. Albooyeh, D. Morits, and C. R. Simovski, “Electromagnetic characterization of substrated metasurfaces,” Metamaterials5(4), 178–205 (2011). [CrossRef]
- P. Sheng, “Wave scattering formalism,” in Introduction to Wave Scattering, Localization and Macroscopic Phenomena, R. Hull, R. M. Osgood, eds. (Springer, 2006).
- EastFDTD v2.0 Beta, DONGJUN Science and Technology Co., China.
- 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.
- The gain of the employed double-ridged horn antenna is about 14dB~15dB in this frequency region.

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.