### Retrieval

#### 1. What is retrieval

Metamaterial unit cells are usually made of complex metallic inclusions inside a dielectric matrix, but it is possible to represent them using homogeneous effective optical properties. While you might find this hard to accept, this is not much different from any material composed of electrons more or less bounded to nuclei. The movement of those charged particles (mainly the electrons) under the effect of electric and magnetic fields creates a polarization and a magnetization in the material which gives it its specific electromagnetic response. When using materials, we do not usually consider all the details of their structure, but only the macroscopic response of the material, its refractive index *n* and its impedance *z*, or its permittivity *ε* = *n*/*z* and its permeability *μ* = *nz*. It is well known that the microscopic properties of the material can be related to its homogenized macroscopic properties through the Clausius-Mossotti relation.

Similarly, when electric and magnetic fields are applied on metamaterials, they induce currents and polarizations in the elements constituting every unit cell, creating an electromagnetic response. If the unit cell is small compared to the wavelength of the electromagnetic wave, it is possible to describe the interaction between the wave and the metamaterials as if the material was homogeneous. A rule of thumb is that the unit cell should be at most a tenth of the wavelength.

The retrieval method is a practical approach to determine the effective properties of metamaterials. In this method, we first measure or simulate the reflection and transmission of a slab of metamaterial. Then, we figure out the properties of a homogeneous material with identical thickness that would give the same reflection and transmission. We say that the properties of this equivalent slab of homogeneous material are the effective properties of the metamaterial.

In this tutorial, we first present equations to calculate the transmission and reflection of a slab of homogeneous material. Then, we reverse those equations to obtain the material properties corresponding to a given reflection and transmission.

#### 2. Forward calculation

**SIGN CONVENTION**

When solving the electromagnetic wave equation using the complex exponential form, one can arbitrarily choose plus or minus sign inside the exponential. In this tutorial, we use the *+iωt* convention, like most physicists do. This has an effect on the sign of the imaginary part of the permittivity and the permeability which, for passive homogeneous materials, must be positive. If we apply the opposite convention, like most engineers do, the imaginary part of the permittivity and the permeability must be negative.

You should be careful when using the reflection and transmission simulated by a software, or obtained from experiment, to make sure that it uses the same sign convention, or to convert the data. For example, COMSOL, which is widely used to simulate metamaterials, uses the -*iωt* convention. To account for this, you simply need to replace the reflection and the transmission by their complex conjugate before feeding them to the retrieval algorithm.

Let us suppose that we have an homogeneous slab of material of thickness *d*, refractive index *n*, and impedance *z* in vacuum. Using one of the many ways to calculate the reflection and transmission of a slab of material, such as a transfer matrix approach, it is possible to demonstrate that the reflection of the slab at normal incidence is

$$S_{11} = \frac{R(1-e^{i2nk_{0}d})}{1- R^{2}e^{i2nk_{0}d}}$$

while its transmission is

$$S_{21} = \frac{(1-R)e^{ink_{0}d}}{1- R^{2}e^{i2nk_{0}d}}$$

where

$$R = {\frac{z-1}{z+1}}$$

is the reflection at a single interface, *k _{0 }= 2*

*π / λ*is the wavenumber in free space and

_{0}*λ*is the wavelength in free space.

_{0}#### 3. Retrieval

To retrieve the optical properties of a slab with known reflection and transmission, we need to isolate *z* and *n* in the previous equations. The impedance of the slab is

$$z =\pm \sqrt \frac{(1+S_{11})^2 - S_{21}^2 }{(1-S_{11})^{2} - S_{21}^2}\text{.}$$

Here, the ± sign indicates that we face a first branch selection. This one is easy to resolve since, for a passive material, the real part of the impedance (which in a circuit corresponds to the resistance) must be positive. So for every point in the spectrum, we choose the appropriate sign. Next, we can show that

$$e^{ink_{0}d} = \frac{S_{21}}{1- S_{11}\frac{z-1}{z+1}}\text{.}$$

The magnitude of $e^{ink_0d}$ obtained from this equation is always smaller or equal to 1, which guarantees that the sign of the imaginary part of the refractive index will be correct. Finally, the refractive index is

$$n = \frac{1}{k_0d}(-i\ln{}e^{ink_0d}+2m\pi)$$

where *m* is an integer. Here we face a trickier branch selection: choosing the correct value of *m*. This difficulty is related to the fact that is usually impossible to measure the absolute phase shift occurring when a wave traverses the slab of material. Indeed, a plane wave repeats every period and it is impossible to determine in what period a measurement is done. The phase can therefore only be determined modulo 2π. The branch selection only affects the real part of the refractive index. The imaginary part, which corresponds to absorption, can be determined unambiguously.

Another way of looking at this problem is that trigonometric functions give the same output for many values of their argument. The inverse trigonometric functions, such as the natural logarithm, must therefore have many solutions for the same argument. We say that these functions have many branches. In computers, the inverse trigonometric functions are designed to output values on one specific branch. It is the job of the programmer to adjust the output value correctly according to the nature of the problem.

It should be noted that we often need to select different branches for different parts of the spectrum. Indeed, for shorter *λ _{0}*, or in regions when

*n*is high, the phase shift inside the slab might get higher than 2π and a

*branch jump*occurs.

One approach to select the right branch is to use the fact that the materials are passive. With the sign convention we use, for a homogeneous material, this means that the imaginary part of the permittivity and the permeability must be positive. A little bit of math shows that this condition is equivalent to

$$|\Re(n) \Im (z)|\le \Im(n)\Re(z)\text{.}$$

For a homogeneous material, there is always one and only one branch that respects this inequality. However, for metamaterials, because of a phenomenon called spatial dispersion, it might happen that none of the branches respect the inequality. While the retrieval is not strictly valid in that case, it might still be useful to have effective electromagnetic property values. For numerical reasons, it is also possible that more than one branch respect the inequality. So, while this approach is useful, we have found that it is not the most reliable to determine the right branch.

Another approach that we prefer is to select the right branch at low frequency (high wavelength, low wavenumber) and use the continuity of the refractive index to keep the right branch over the whole spectrum. Indeed, as the frequency tends toward 0, the phase shift caused by the metamaterial must tend toward a finite value. Looking at the equation for the refractive index, we see that when *k _{0 }→ 0* , the refractive index will tend toward infinity, except for

*m*= 0. Therefore, the right branch for small frequency must be branch 0. As the frequency increases, the refractive index should be continuous, so when a discontinuity is observed we know that we should change the value of

*m*.

To demonstrate the retrieval process, let’s take a look at a canonical example of a metamaterial unit cell, that of a negative refractive index metamaterial. The unit cell, shown in the figure above consists of a continuous wire, providing an electric response, and 2 concentric split ring resonators, providing a magnetic response (for more details, see the article "Electromagnetic parameter retrieval from inhomogeneous metamaterials" mentioned in the reference section). The reflection and transmission of the unit cell, surrounded by 1.5 mm of vacuum padding on both side, was simulated in Comsol and is shown below. The phase of the scattering parameters outputted by Comsol, has been adjusted according to the *+i**ωt* sign convention by taking the complex conjugate of the *S* parameters outputted by Comsol.

The first step in the retrieval process is to remove the effect of the padding. This padding is necessary in the simulation process to make sure that the near fields of the metamaterial elements do not interact with the ports, but adds a phase shift to both the reflection and the transmission. It is trivial to remove this phase shift by multiplying *S*_{11} and *S*_{21} by exp(*-i*2*k _{0}d*

_{pad}) where

*d*

_{pad}is the thickness of the padding. The results are shown in Figure 3.

The next step consists in finding the impedance. Applying the equation above and choosing the right branch, we get the impedance shown below.

Then, we can calculate the real part of the refractive index for many branches, as shown in the figure below. One thing you should note is that at 37.2 GHz, the value of the refractive index for all branches suddenly jump. To preserve the continuity of the refractive index, we will need to jump from one branch to another at that point. Also, the figure indicates the points of every branch that respect the above inequality based on the passivity of the material. Only one branch has many points respecting this inequality, and is the one that must be chosen. Furthermore, all branches, except for *m* = 0, diverge for small frequencies. Therefore, at low frequency, the branch *m* = 0 must be chosen and for frequencies larger than 37.2 GHz, the branch *m* = 1 must be chosen.

Once the right branch has been selected, one can easily calculate the complex value of the refractive index, shown below.

Finally, it is trivial to calculate the permittivity and the permeability.

#### 5. Code and data

The Matlab code for the retrieval, as well as the data for the above example can be downloaded here.