4. Metamaterials – Negative Index Materials

The term metamaterial was introduced in 1999 by Rodger M. Walser of the University of Texas (Austin). He defined metamaterials as “macroscopic composites having a manmade, three-dimensional, periodic cellular architecture designed to produce an optimized combination, not available in nature, of two or more responses to specific excitation”. By now, this definition is a bit outdated, as the field of metamaterials has evolved tremendously over the past decade. A better definition would be “Man-made artificial materials with a response not readily available in nature which gain their properties from their structural composition rather than their atomic composition”. In that sense, any engineered structure such as an IC could be considered as a metamaterial, but usually the term refers to “optical metamaterials” which are artificial materials with an engineered response to EM-waves. This artificial response is obtained from the sub-wavelength sized building blocks or metamaterial atoms, which can be ordered randomly or in a periodic fashion, in order to obtain the desired effective medium response of the metamaterial.

The birth of the field of metamaterials dates back to a seminal paper by Victor Veselago from 1968 in which the electromagnetic response of a material with simultaneously negative values for the electric permittivity \epsilon and the magnetic permeability \mu was described theoretically. Veselago pointed out that such a material would have a negative refractive index n, and therefore a flat slab of such a material would act as a lens. As such a material was not known to exist at the time, the concept raised a lot of skepticism, until it was picked up in 2000 by John Pendry who pointed out that the flat lens proposed by Victor Veselago would also have non-diffraction-limited resolution. Around the same time the first negative index material (NIM) for microwave frequencies was demonstrated, which soon would lead to many new designs that would allow to obtain NIMs for frequencies up to the visible range.

Negative Index Materials (NIMs)

In this section the theoretical description for NIMs is given based on the papers by Veselago and Pendry. The electric permittivity \epsilon and the magnetic permeability \mu are assumed to be purely real and negative numbers. For the theoretical description of such a material we start from the constitutive Maxwell equations

(1) {\bf D} = \epsilon {\bf E}
(2) \nabla \times \mathbf{{\bf E}} = - \frac{\partial{{\bf B}}}{\partial{t}}
(3) {\bf B} = \mu {\bf H}
(4) \nabla \times \mathbf{{\bf B}} = \frac{\partial{{\bf D}}}{\partial{t}}

We consider a homogeneous plane wave which propagates in an isotropic slab of material with \epsilon_r = \mu_r = -1 (here \epsilon_r and \mu_r are the relative permittivity and permeability with respect to the vacuum values, such that \epsilon = \epsilon_0 \epsilon_r and \mu = \mu_0 \mu_r) for which the propagation behavior is described by

(5)  k^2 = \frac{\omega^2}{c^2} n^2

with n^2 = \epsilon_r \mu_r. We consider a plane wave propagating in the positive z-direction so the field amplitudes show a e^{i(k z - \omega t)}-dependence. By substitution into equations 1-4, these are reduced to

(6) k \times {\bf E} = \omega \mu_0 \mu_r {\bf H}
(7) k \times {\bf H} = - \omega \epsilon_0 \epsilon_r {\bf E}

From these equations we can see that a simultaneous change of the sign of \epsilon_r and \mu_r changes nothing to this generalized solution of Maxwell’s equations. Moreover it shows that for negative values of \epsilon_r and \mu_r, the triplet of vectors {\bf k}, {\bf E} and {\bf H} form a left-handed system of reference, which is why Veselago named these materials left-handed materials (LHM).

From the definition of the refractive index n^2 = \epsilon_r \mu_r we see that there are two possible choices for the sign of n, and the correct solution is dictated by causality. If we assume that a plane wave is propagating in the positive z-direction in a lossy LHM, then the complex values of the permittivity and permeability can be written as

(8) \epsilon_r = | \epsilon_r | e^{i \alpha}
(9) \mu_r = | \mu_r | e^{i \beta}

while the refractive index in given by n = \sqrt{| \epsilon_r \mu_r |} e^{i \gamma} where \gamma can take the value (\alpha + \beta)/2 or (\alpha + \beta)/2 + \pi. As we consider a plane wave propagating in a lossy LHM, the wave should decay in the positive z-direction. This can only be achieved when Im(n) > 0, which fixes the choice of the sign for Re(n) to be negative, as illustrated in figure 1.

Figure 1: Choice of the appropriate sign of the refractive index for a LHM in the complex plane. Given the lossy nature of the material, the wave should decay as it propagates, which implies that Im(n) > 0 and fixes the choice for Re(n).

The direction of the energy flow in a LHM is given by the Poynting vector {\bf S}.

(10) S = \frac{1}{2} {\bf E} \times {\bf H^*}

Due to the presence of the vector product of {\bf E} and {\bf B}, the Poynting vector {\bf S} and the wave vector {\bf k} are aligned anti-parallel, contrary to conventional right-handed materials (RHM). As a consequence, in a LHM the phase velocity \nu_p is aligned anti-parallel to the energy flow and to the group velocity \nu_g. This implies that the direction of the energy flow in LHM and RHM is the same, but the wave fronts travel in opposite directions.

Figure 2: Illustration of phase and group velocity for a wave packet traveling in a LHM. While the pulse and energy propagate to the right, the wavefronts (indicated by a red dot) propagate in the opposite direction.

The propagation behavior of EM-waves in LHMs discussed above was allready descibed by Veselago in 1968, but in 2000 Pendry pointed out that a NIM can also cancel out the exponential decay in amplitude of evanescent waves, allowing to use it as a perfect lens. We consider a slab of LHM in vacuum with thickness d and permittivity \epsilon_r = -1 and permeability \mu_r = -1 (figure 3) and derive the propagation behavior for a S-polarized evanescent wave travelling in the positive (+) z-direction. The incident wave can be descibed as follows

(11) E_{0,+} = E_{y,0} e^{i k_z z + i k_x x - i \omega t}

with k_z = i \sqrt{k_x^2+k_y^2- \omega^2 c^{-2}} and \omega^2 / c^2 < k_x^2+k_y^2[/latex] such that the wave decays exponentially in the propagation direction. At the first interface between the LHM and the surroundings, part of the light will be reflected ([latex]E_{0,-}[/latex]), and part of the light will be transmitted ([latex]E_{1,+}[/latex])  (12) [latex]E_{0,-} = r_s E_{y,0} e^{-i k_z z + i k_x x - i \omega t}[/latex] (13) [latex]E_{1,+} = t_s E_{y,0} e^{i k_z^{'} z + i k_x x - i \omega t}[/latex]  where the choice of [latex]k_z^{'} = i \sqrt{k_x^2+k_y^2- \epsilon \mu \omega^2 / c^2}[/latex] and [latex]\epsilon \mu \omega^2 / c^2 < k_x^2+k_y^2[/latex] is defined by causality as the wave should decay as it propagates away from the source. The transmission and reflection coefficients at the interfaces are given by the Fresnel equations, which for the S-polarized case reduce to  (14) [latex]t_s = \frac{2 \mu_r k_z}{\mu_r k_z + k_z^{'}}[/latex] (15) [latex]r_s = \frac{\mu_r k_z - k_z^{'}}{\mu_r k_z + k_z^{'}}[/latex] (16) [latex]t_s^{'} = \frac{2 k_z^{'}}{k_z^{'}+ \mu_r k_z }[/latex] (17) [latex]r_s^{'} = \frac{k_z^{'} - \mu_r k_z}{k_z^{'} + \mu_r k_z}[/latex]  where [latex]t_s[/latex] and [latex]r_s[/latex] are the coefficients for the vacuum-NIM interface and [latex]t_s^{'}[/latex] and [latex]r_s^{'}[/latex] are the coefficients for the NIM-vacuum interface. The overall transmission and reflection coefficients [latex]T_s[/latex] and [latex]R_s[/latex] can be obtained by calculating the sum of all scattering events  (18) [latex]T_s = t_s t_s{'} e^{i k_z^{'} d} + t_s t_s{'} r_s^{'2} e^{3 i k_z^{'} d} + t_s t_s{'} r_s^{'4} e^{5 i k_z^{'} d} + ... = \frac{t_s t_s{'} e^{i k_z^{'} d}}{1-r_s^{'2} e^{2 i k_z^{'} d}}[/latex]  The actual transmission and reflection is calculated by taking the limit for [latex]\epsilon_r \rightarrow -1[/latex] and [latex]\mu_r \rightarrow -1[/latex] which yields  (19) [latex]\lim_{{\epsilon_r,\mu_r} \to {-1}} T_s = \frac{t_s t_s{'} e^{i k_z^{'} d}}{1-r_s^{'2} e^{2 i k_z^{'} d}} = e^{-i k_z^{'} d} = e^{-i k_z d}[/latex]  Which shows that the wave travels through the medium without experiencing any decay. For the reflection we can make a similar derivation which yields that the overall reflection equals zero.  (20) [latex]\lim_{{\epsilon_r,\mu_r} \to {-1}} R_s = \lim_{{\epsilon_r,\mu_r} \to {-1}} r_s + \frac{t_s t_s{'} r_s^{'} e^{2 i k_z^{'} d}}{1-r_s^{'2} e^{2 i k_z^{'} d}} = 0[/latex]  A similar derivation can be made for P-polarized waves by replacing [latex]\epsilon_r[/latex] for [latex]\mu_r[/latex] in equations 14-17. This means that for a flat slab of a LHM which is perfectly impedance matched to the surroundings, any wave with an arbitrary polarization state will be transmitted without any decay. The amplification of the amplitude of evanescent waves does not violate energy conservation, as evanescent waves do not transport energy. This was the main conclusion from the paper by Pendry as it proves that both low- and high-frequency components of an image can travel through a NIM without any decay, resulting in a perfect lens.   <a href="http://www.kristoflodewijks.be/wp-content/uploads/2013/05/PerfectLens.jpg"><img src="http://www.kristoflodewijks.be/wp-content/uploads/2013/05/PerfectLens.jpg" alt="PerfectLens" width="376" height="594" class="alignnone size-full wp-image-240" srcset="http://www.kristoflodewijks.be/wp-content/uploads/2013/05/PerfectLens.jpg 376w, http://www.kristoflodewijks.be/wp-content/uploads/2013/05/PerfectLens-189x300.jpg 189w" sizes="(max-width: 376px) 100vw, 376px" /></a><br /> <em>Figure 3: Illustration of a NIM as perfect lens. All frequency components can travel through the lens without any decay. (a) Ray-tracing picture of imaging by a NIM lens (red lines) and comparison with a conventional lens (dotted blue lines). The image of the object is focussed once inside the lens and once in the image plane. (b) Illustration of enhancement of evanescent waves in a NIM (red lines) compared to the decay in a conventional lens (blue dotted lines).</em></p> <p>In conventional optical systems, the high-frequency components which make up the smallest details of an object are evanescent in nature and decay exponentially with the distance from the source. Therefore these high-frequency components are lost in the image plane, and the resolution of a conventional lens is \emph{diffraction limited}. The smallest distance between two points in the object plane ([latex]\Delta x) which can be distinguished in the image plane is given by

(21) \Delta x \approx \frac{\lambda}{NA . n}

in which \lambda is the illumination wavelength, NA is the numerical aperture of the optical system and n the refractive index. For a NIM lens all components can travel without any decay, which implies that inside the NIM the evanescent waves are amplified in order to allow for transmission to the image plane without any losses (figure 3(b)).

Effective material parameters

Metamaterials are often described in terms of effective material parameters such as permittivity \epsilon_{eff}, permeability \mu_{eff}, refractive index n_{eff} and impedance Z_{eff}. These parameters describe the propagation properties of EM-waves through metamaterials by treating them as an effective medium. The metamaterial is considered as a homogeneous material (in the propagation direction) which implies that the metamaterial atoms or building blocks should be deep sub-wavelength in order for this approximation to be valid. In fact, this criterion is often used to discriminate between photonic crystals (a \approx \lambda) and metamaterials (a \ll \lambda) in function of the unit cell dimension a.

For optical metamaterials the effective parameters can be deduced from measurements or simulations in which both the amplitude and phase information of reflected and transmitted waves are recorded. The experimental extraction of these parameters is rather cumbersome for VIS and NIR metamaterials, but it is possible to extract both phase and amplitude by interferometric measurements. Often the effective parameter extraction is based on finite element simulations which are compared with the measured transmission and reflection spectra. The homogenization step is performed by averaging out the electric and magnetic field amplitudes over one unit cell of the metamaterial and calculating the complex transmission and reflection coefficients at each side of the metamaterial. In case of bi-anisotropy in the propagation direction, the effective parameters can be extracted through a modified protocol in which the transmission and reflection coefficients are extracted for illumination from both sides of the metamaterial layer. The values for the refractive index n and impedance Z are calculated by inversion of the scattering parameters (S parameters), the complex reflection and transmission coefficients.

We start from a homogeneous 1D slab of metamaterial with a thickness d and consider the transmission and reflection in terms of the transfer function which relates the transmitted and reflected waves to the incident wave according to

(22) {\bf F'} = {\bf TF}


(23) {\bf F} =\begin{Bmatrix}E \\ H_{red} \\ \end{Bmatrix}

with E and H_{red} the electric and magnetic field amplitudes of the incident wave ({\bf F}) and the transmitted wave ({\bf F'}). Here we use the reduced magnetic field which is the normalized magnetic field according to H_{red} = i \omega \mu_0 H. For an isotropic slab of material the transfer matrix can then be written as

(24) {\bf T} = \begin{pmatrix} \cos (n k d) & - \frac{Z}{k} \sin (n k d) \\ \frac{k}{Z} \sin (n k d) & \cos (n k d) \\ \end{pmatrix}

In practical applications we don't have direct access to the components of the transfer matrix {\bf T} but to the scattering parameters (S-parameters) in the scattering matrix {\bf S}

(25) {\bf S} = \begin{pmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \\ \end{pmatrix}

which can be related to the parameters of the T-matrix by

(26) S_{11} = \frac{T_{11}-T_{22}+(i k T_{12} - \frac{T_{21}}{i k})}{T_{11}+T_{22}+(i k T_{12} + \frac{T_{21}}{i k})}
(27) S_{12} = \frac{2 | {\bf T} |}{T_{11}+T_{22}+(i k T_{12} + \frac{T_{21}}{i k})}
(28) S_{21} = \frac{2}{T_{11}+T_{22}+(i k T_{12} + \frac{T_{21}}{i k})}
(29) S_{22} = \frac{T_{22}-T_{11}+(i k T_{12} - \frac{T_{21}}{i k})}{T_{11}+T_{22}+(i k T_{12} + \frac{T_{21}}{i k})}

where S_{11} and S_{21} are the complex reflection and transmission coefficients for incidence from the top and S_{22} and S_{12} are the complex reflection and transmission coefficients for incidence from the bottom, which are extracted from two separate experiments or simulation runs.

Isotropic materials

For an isotropic material T_{11} = T_{22} = T_s and | {\bf T} | = 1 (equation 24), such that the S-matrix is symmetric:

(30) S_{11} = S_{22} = \frac{\frac{1}{2} (\frac{T_{21}}{i k} - i k T_{12})}{T_S + \frac{1}{2}(i k T_{12} + \frac{T_{21}}{ik})}
(31) S_{12} = S_{21} = \frac{1}{T_S + \frac{1}{2}(i k T_{12} + \frac{T_{21}}{ik})}

By substituting the T-matrix elements from equation 24 we obtain the relationship between the S-parameters and n and Z

(32) S_{11} = S_{22} = \frac{i}{2} (\frac{1}{Z}-Z) \sin (n k d)
(33) S_{12} = S_{21} = \frac{1}{\cos (n k d) - \frac{i}{2} (Z - \frac{1}{Z}) \sin (n k d)}

If all of the S-parameters are known, the equations above can be inversed in order to obtain n and Z

(34) n = \frac{1}{k d} \cos^{-1} [ \frac{1}{2 S_{21}} (1 - S_{11}^{2} + S_{21}^{2}) ]
(35) Z = \sqrt{\frac{(1+S_{11})^2 - S_{21}^2}{(1-S_{11})^2 - S_{21}^2}}

In order to obtain physically sound values for the refractive index and impedance, the right branch of the cosine function has to be selected, bearing in mind that for passive materials, both Im(n) > 0 and Re(Z) > 0 should be fulfilled. The relative permittivity and permeability can subsequently be calculated from

(36) \epsilon_r = \frac{n}{Z}
(37) \mu_r = n Z

which are complex functions of the wavelength.

Bi-anisotropic metamaterials

When the metamaterial is not isotropic in the propagation direction, the S-matrix is no longer symmetric as the reflected and transmitted signals are different for illumination from the top or bottom. The S-parameters from both experiments are used in order to construct the overall T-matrix for the anisotropic metamaterial:

(38) T_{11} = \frac{(1+S_{11})(1-S_{22})+S_{21} S_{12}}{2 S_{21}}
(39) T_{12} = \frac{(1+S_{11})(1+S_{22})-S_{21} S_{12}}{2 S_{21}}
(40) T_{21} = \frac{(1-S_{11})(1-S_{22})-S_{21} S_{12}}{2 S_{21}}
(41) T_{22} = \frac{(1-S_{11})(1+S_{22})+S_{21} S_{12}}{2 S_{21}}

For anisotropic metamaterials, the obtained values of the refractive index n are different for both propagation directions and these 2 values are very similar to equation 34:

(42) n_1 = \frac{1}{k d} \cos^{-1} [ \frac{1}{2 S_{12}} (1 - S_{11}^{2} + S_{12}^{2}) ]
(43) n_2 = \frac{1}{k d} \cos^{-1} [ \frac{1}{2 S_{12}} (1 - S_{22}^{2} + S_{12}^{2}) ]

while an overall effective refractive index n_{eff} can be defined by replacing S_{11} or S_{22} in the equations above by an average S-parameter S_{av} = \sqrt{S_{11} S_{22}} such that

(44) n_{eff} = \frac{1}{k d} \cos^{-1} [ \frac{1}{2 S_{21}} (1 - S_{av}^{2} + S_{21}^{2}) ]

The impedance is also different for both propagation directions and can be defined in terms of the T-matrix elements (equation 38-41)

(45) Z_{eff} = \frac{(T_{22} - T_{11}) \pm \sqrt{(T_{22}-T_{11})^2+4 T_{12} T_{21}}}{2 T_{12}}

where the choice of the sign determines the different propagation directions. The extracted effective values for the refractive index and impedance are to be found in between the effective values obtained for top and bottom incidence without bi-anisotropy correction, as defined by equation 36-37.

Plasmonic metamaterial building blocks

Many of the plasmonic metamaterial atoms originated from the original attempts to realize NIMs at lower frequencies, for example in the microwave range. Researchers were trying to realize a material with simultaneously negative values for the electric permittivity \epsilon and the magnetic permeability \mu. The first realization of a NIM consisted of two seperate structures that governed a negative electric and a negative magnetic response at an operational frequency of 10.5 GHz (figure 4). The negative electric response was realized by an array of parallel wires that act as a diluted plasma, while the negative magnetic response was induced by using split-ring resonators (SRRs), in which magnetic resonances can be excited. The SRRs can be seen as subwavelength LC-circuits: the ring behaves like a coil while the slit behaves as a capacitor.

Figure 4: First experimental realization of a NIM. (a) Picture of the NIM structure consisting of metallic wires and split-ring resonators. (b) Experimental setup to verify the negative value of the refractive index of a prism structure by Snell's law. (c) Angle dependent transmission through a NIM prism and a conventional prism strucutre.

In initial attempts to realize NIMs at higher frequencies, the SRRs were shrunk down to the limiting sizes that could be achieved with e-beam lithography, resulting in the first plasmonic NIMs in the NIR. These metamaterials obtained their negative magnetic response from LC-resonances, while the negative electric response is governed by the intrinsic material properties of the metals in the NIR. The first truly 3D metamaterials in the NIR were also realized by means of multilayers of stacked SRRs. Soon it was realized that further downscaling of conventional SRRs would lead to saturation of the magnetic resonance, and new metamaterial atoms were proposed in order to push negative magnetic responses into the visible wavelength range. The saturation frequency can be increased by introducing more splits in a classical SRR, which is one of the paths that was pursued by different research groups simultaneously, resulting in new metamaterial atoms such as double-wire pairs. The next generation of NIMs emerged shortly afterwards and consisted of metal-insulator-metal (MIM) layers, perforated by a periodic array of holes, the so-called double fishnet. The double fishnets are one of the most widely studied classes of NIMs for which negative refraction at NIR wavelengths could be experimentally verified.

Figure 5: Evolution of the scaling of NIMs. The different colors in the plot indicate the different metamaterial atoms used: Orange for double SRRs, Green for U-shaped SRRs, Blue for pairs of wires and Red for double fishnet materials.

In recent years, the focus of the metamaterial community has diverged into several new areas such as chiral metamaterials (allowing to control the polarization state of light) and transition optics (slowing down light, invisibility cloaking), which use many different metamaterial atom designs.