One of my tasks related to metamaterials is to prepare a dielectric metamaterial whose building blocks are made of TiO2 (rutile) microspheres of diameters around 30 to 100 um. This idea was proposed in several articles by other groups, namely vendik2009.
A high-permittivity sphere behaves as a spherical resonator thanks to internal reflections (Mie resonances). As a rule of thumb, a TiO2 sphere with index of refraction of ca. 10 and diameter of 100 um is expected to resonate near to the frequency of an electromagnetic wave with free-space wavelength of 1 mm, which corresponds to the frequency of roughly 300 GHz.
The behaviour of TiO2 microspheres had been studied by our group experimentally in the TDTS lab. The TiO2 samples were provided by P. Mounaix and U.-C. Chung and a clever echo deconvolution technique was used to obtain both reflection and transmission. The first results were published in Němec2012 and shown quite a good match to a simulation data obtained by H. Němec and R. Yahiaoui using Ansys HFSS.
Numerical simulation is helpful to verify experiments, understand the physical nature of observations and also to propose further improvements of the structure. Fortunately, the microsphere is one of the simplest structures to be simulated. More details on my experimenting with FDTD simulations are at my MEEP page, where also the whole simulation script can be downloaded.
In brief, the simulation uses a 70x70x300 um volume containing one TiO2 sphere in its center, with diameter of 39 um. The shorter axes x, y are set to be periodic, as if the spheres were in a 2-D square periodic array. A broadband pulse (500-2500 GHz) is then sent along the longer z axis, scatters on the sphere and both the transmitted and reflected fields are recorded and processed.
The figures above show incomprehensible superposition of several modes excited by one pulse. The modes of a spherical resonator were described long time ago in microwave and laser technology. Thanks to a very high permittivity contrast, the lowest Mie resonances observed in a TiO2 sphere have quite a high quality and occur at distinct frequencies.
By applying Fourier transform on the reflected and transmitted field, the modes can be easily identified. Knowing the frequency of modes, one can rerun the simulation with a narrower frequency source to observe each mode separately:
As the microsphere is regarded to as a single unit cell in a metamaterial, it is desirable to calculate its effective material parameters. For some more complicated metamaterials, this task may be very tricky and it is not even guaranteed to always work. Several approaches were suggested by different groups (optimization [Chen2004], "Wave Propagation Retrieval Method" [Andryieuski2010], etc.). For the dielectric spheres, fortunately it was possible to get realistic results with the most common method based on S-parameters [Smith2002], which uses the reflected and transmitted waves and gives approximate effective refractive index N(omega), impedance Z(omega) and therefore also permittivity epsilon(omega) and permeability µ(omega) (all as complex functions of frequency). The result from this method is on the second graph on the above figure.
Two problems had to be resolved to precisely match my results to those obtained by R. Yahiaoui. First I constantly obtained unrealistic parameters breaking the Kramers-Kronig relations -- until I found out that my simulation software uses negative frequency convention which requires me to use complex conjugate of the impedance during the calculation. After that, choosing the correct branch of solution was easyhere (see Smith2002 etc.). The second problem was to determine the complex permittivity of the TiO2 the microsphere consists of. Basically, the real part gives the frequency of resonance and the imaginary part gives the losses and therefore the width of resonance. A plausible value of epsilon = 85+2i exactly matched the R. Yahiaoui's data. See Characterisation of the samples below for further info.
The reflection/transmission spectra show that there are several resonant modes, where a reflection of the structure has a maximum. Understanding the spatial E- and H-field patterns of these different modes may be useful e. g. when the resonator is to be surrounded by some other structures. Let us use Mayavi2 to make a short excursion into the mode field patterns.
It is easy to excite one separate resonance by a narrowband pulse once we know the resonant frequencies. Here we used 800 GHz, 1150 GHz and 1550 GHz for the first three modes. The source bandwidth was set to 200 GHz.
In the first resonant mode, the electric field is circular and confined within the volume of the sphere, whereas the magnetic field reaches out of the sphere. Therefore the first mode couples strongly to magnetic field only. The circular electric field of the first mode is illustrated on the 3-D view of the X-Z cut:
A general rule is that odd modes (1st, 3rd...) couple to magnetic field while even modes couple more to the electric field.
Note: The reflection amplitude easily reaches 0.75 for the first resonance in a 40 um sphere in 70x70 um cell. This is obviously in contrast to the fact that its cross-section is only 1250 um2, whereas the crossection of the cell is 4900 um2. How is it possible? I think the explanation lies in that it is the sphere's near field that covers most of the cell's cross section. This also means that in the aforementioned geometry the resonant fields have to couple to fields in neighbouring spheres in the virtual periodic array. In the detailed view in Mayavi one can easily see that this indeed happens, but fortunately not too much to distort the spectra. (However, for a periodic array the mode splitting cannot be observed because also the fields are periodic in the cells.)
The experimental measurements based on the terahertz time-domain spectroscopy has been described many times elsewhere, so only a brief explanation is needed.
Our setup was in principle similar to the simulated one. A broadband terahertz pulse passed through a single layer of microspheres inserted between two sapphire slabs and randomly arranged. The transmitted electric field was optically sampled, providing spectral data ranging from 100 GHz to 2500 GHz. In order to obtain not only transmission, but also reflection spectra in one experiment, we employed two sapphire slabs 3 mm and 6 mm thick, which enabled us to retrieve the reflected wave from the echoes introduced by the sapphire-vacuum and sapphire-sample boundaries. The experimental data were processed in a manner similar to those given by simulation .
The resonant frequency of the microsphere in metamaterial have to be very similar in order to obtain negative permeability. The samples our group obtained for measurement in [Němec2009] had quite a broad distribution of sizes which manifested in broadened and weak resonance curves when whole sample was measured.
The size distribution of the microspheres in the samples was obtained by automated numerical processing of many microscopic photographs. They were taken by a common laboratory microscope and by a compact "Infinity X" microscopic camera. I attached an aplanate objective from an old scanner to the camera objective, which approximately doubled the resolution and also shifted the camera focal plane to few millimeters under the objective (which does not bring any problem for transmission measurements). It was proven that the camera still gives decent results and is easier to use. The resolution of camera with objective was ca. 0.6 pixel/um and it was always calibrated using a ruler.
The program processing the photographs uses Python and ImageJ and is documented on the page: Calculating size distribution of powder particles using ImageJ. (Note: There is probably a bug in the watershed function in Scikits-image, which forced me to launch ImageJ instead of python module.) The next figures only illustrate the method used:
Most of the samples were prepared in Bordeaux by spray-dry technique, sintering, washing and finally triple sieving on commercial sieves weaved from stainless steel wire. However none of the samples seems to have distribution of sizes narrow enough. The sieving done in Bordeaux was the best described by U-Chan Chung:
(It shall be noted that there are many other techniques possible. For instance centrifuging in viscous liquid, free fall with initial horizontal speed, bouncing the spheres in vacuum, using optical tweezer and automated individual microscopic nulometry etc. One interesting option is to use the pressure of monochromatic terahertz radiation with radial gradient high enough to attract the particles resonating at one frequency and repulse others, but this is rather a sci-fi.)
The simulations prove that all dimensions of the TiO2 microsphere, x, y and z, more or less influence the resonant frequencies of modes. In any case, we can not control the orientation of the nearly-spherical particle. Thus the only well defined sample is only such that consists of spheres of one radius and we have to sort the particles not only by one of dimensions, but by shape as well. I tried to sieve the samples again with this in mind.
The key idea I tried to employ was in using a circular-elliptical pair of sieves. The circular sieve should first remove all oversized particles. The elliptical sieve should have holes with major axis slightly bigger and minor axis slightly smaller than the radius of holes in the upper sieve. Therefore, only the particles whose cross-section is nearly circular and very narrow in size should be trapped between the sieves. Theoretically the particles in the intermediate fraction, remaining on the second sieve, should be either perfect spheres or oblong ellipsoids (i. e. rugby balls).
In the first setup, nylon sieves with ca. 65 um holes were used; therefore it was very easy to stretch the second sieve diagonally to obtain the "circular-elliptical" (or rather a "square-rhombic") sieves pair. A tiny motor vibrated the 4 cm long glass tube with two sieves stacked. An illustration of such setup along with the first sieving device is shown below:
In few seconds, the sieves stuck with the spheres and the sieving virtually stopped. Therefore, a circular spring with diameter of 4 mm was added to jump in the layer between a coarse auxiliary sieve and each fine sieve (see figure above). This helped partially to clean the nylon sieves from particles. Optionally a little plastic sphere jumped on top of each fine sieve with similar purpose.
The results of sieving could be checked both by granulometry and by THz spectroscopy. On the spectral dependence of permeability (from 1st Mie mode), the not sieved sample had central frequency of 475 GHz and spectral peak-to-notch width of 26 % (126 GHz). The sieved sample had slightly higher resonant frequency of 510 GHz and spectral width of 18 % (96 GHz). So we obviously separated some oversized particles, but the improvement was quite modest. Other samples were too small to be tested with the pair of nylon sieves we had.
One issue of nylon sieves was that the sphere distribution below the sieves contained a lot of spheres bigger than the nominal sieve hole was! The obvious cause was in the elasticity of the nylon mesh; such sieving was rather a statistical process. Nonetheless, the sieving provided promising results. After a week of operation, the size distribution of few micrograms of spheres narrowed at least twice. We measured the size statistics, the resonant THz spectra and tried to match the results.
With experience from the first setup, I made the following changes in sieving:
The quality of sieves is crucial for proper results and to my knowledge no commercial sieves match our demanding requirements. To prepare the sieves (and also electromagnetical fishnets discussed later), we have built a simple femtosecond micromachining robot from the Spitfire titanium-sapphire laser, a precise X-Y translational stage and electronics+software borrowed from the FDCNC project.
The holes, with diameter from 50 to 250 um and spacing 300 um, could be easily viewed and measured by microscope. I tried also to obtain some statistics of their sizes. I consider the drilling process to be perfectly repeatable, except for the longitudinal position of the metal sheet, which may move forth and back due to thermal stress. The precision of laser micromachining will surely require further experimenting. The results from granulometry applied to the sieve photographs are worse than I expected, but maybe the photographs themselves were a bit imprecise:
For comfortable experiments with sieving, I built a simple device with a little acoustic transducer tightly attached to a little glass tube, which has a sieve on its base as depicted below. It turned out that the sound frequency has to be tuned to acoustic resonance, whose frequency depends on the sieve being used, but generally ranges from 800 to 1200 Hz. An ideal amplitude is such that the particles form a continuously churned cloud of dust ca. 2-5 mm above the sieve. The sieving would be quite loud at such amplitude, so the apparatus is usually covered with a massive bell jar.
When sieving a sample with mean diameter under 50 um, another issue had to be overcome. The lighter particles tend to stick to glass surface and virtually in any other place where the acoustic vibrations are not strong enough. After turning the sound on, the particles danced on the sieve, but in few seconds they mostly hid in such chill-out positions. Fortunately, tapping the apparatus with a screwdriver forced the "tired" particles to leave their chill-out and return to the dancing crowd for a while again. To avoid having to spend hours and days sitting by the noisy sieving apparatus, I built a tiny motorized hammer that tapped on the apparatus roughly twice a second. This finally enabled a reliable around-the-clock sieving. A slightly slanted position seems to better spread the sample on the sieve area.
The results from THz spectroscopy seem to be a bit better than with the first sieving setup. Whereas the not sieved sample "40-50" had a resonance width of 42 % (260 GHz) at central frequency 605 GHz, the sample sieved for few days on the 10 and 11 sieves had a resonance width "only" 18 % at frequency 640 GHz. Here it seems that we removed not only the oversized, but also a substantial fraction of undersized particles, which corresponds to observation from sieving.
We may also compare the second sieving result to the first one, where sample "100" (red) has frequency axis multiplied by 1.25, amplitude difference from 1.0 divided by approx. two. This comparison proves that the resonance width has a similar ratio to the central frequency as we obtained in May. Although this may not look as any improvement, we should not forget that this time the sieved particles were approx. 1.25-times smaller and that the "40-50" sample had much broader initial distribution. Last but not least, it was a first experiment with metallic sieves and it is likely that this result will be gradually improved at least a bit.
We have two sources of spectra for various parameters, which in an ideal case should match: the experiment and the simulation. The spectral response from experiment is generally much broader than from simulation, it is also burdened by additive noise and some complicated offset error. We believe that the largest source of discrepancy is the broad size distribution of sphere sizes, which is however known from granulometry. We can therefore try to broaden the simulation data in a similar manner and compare the curves to the experiment.
The results show a good match, however, it shall be simultaneously noted that this is thanks to X and Y axis fitting. First the resonance frequency had to be fitted, because we did not know the real part of the microsphere permittivity precisely. Second, also the height of resonance was fitted as we did not know either the actual filling fraction in the experiment. (On the other hand, the unknown imaginary part of permittivity, causing losses, has only a little influence on the matching.) To put it into nutshell, the graph above proves two statements only: 1. the values needed to match the curves are very realistic, 2. the shapes of the fitted curves are quite similar.
Although the dielectric resonators have the advantage of confining most of the resonant energy within their volume, the coupling of neighboring resonators is not negligible. This obviously results in some additional broadening of resonance when a sample is randomly arranged.
While so far we performed no experiments on a periodic array of spheres, we have assessed the influence of spheres arrangement by numerical means. As in figures above, the wave propagates along the Z axis; its E vector is parallel to the X axis and its magnetic vector thus has to be parallel to the Y axis. The following figures depict the transmission as a function of frequency and one parameter describing the spacing of spheres with 20 um radius in a 2-D rectangular lattice:
Coupling between resonant modes begins to manifest when the distance along the Y axis (i. e. along the magnetic field of the incoming wave) is lower than 10 um. The resonant frequency of the first mode drops by roughly 10 % at Yspacing of 40 um, when the spheres are nearly touching. Further compression of the lattice means that the spheres would connect to a rough TiO2 cylinder with lower resonant frequency, but still behaving very similarly to a insulated sphere.
Compressing the lattice along the electric field makes the modes get more pronounced. The second (electric) mode slightly shifts frequency down due to coupling. Interestingly, the first (magnetic) mode shifts frequency up. When the spheres touch, they start to behave similarly to a continuous high-permittivity layer with high reflectivity.
When the lattice is compressed along both axes simultaneously, all the effects sum up. The first mode therefore retains its original frequency quite exactly, which may be useful for building variable-density lattices!
Any periodic arrangement might introduce anisotropy of the metamaterial, as the near-fields in the vicinity of a resonator are located in volume much smaller than the wavelength. Therefore, we tried also to rotate the square lattice by 45 degrees (see above) and to build a dense hexagonal lattice, which can also have two symmetric orientations. Either it is "electric friendly" in that the spheres are arranged along the X axis, or it is "magnetic friendly" where the lattice is rotated by 30 degrees:
The diagonal lattice shows little mode splitting, which may be due to a numerical artifact or to an (unclear but observed) formation of symmetric and antisymmetric magnetic modes between interlaced layers along X axis.
The hexagonal lattice looks better: We can see little differences between the lattices when the spheres are nearly touching (spacing between 40-50 um); however for bigger spacing the near field does not resolve the position of adjacent particles, thus the lattice is isotropic and has a highest possible filling fraction (115 % higher density than square lattice for the same spacing).
It was proven that the TiO2 microspheres exhibit effective permeability around 0.5 and numerical simulations suggest that they may provide negative permeability when properly sorted. They key advantages are in that they isotropic even in a lattice and easy to manufacture. The following issues have to be resolved in particular to build a NIM:
Ensuring simultaneous negative permittivity and permeability may be a complicated task. One suggested way is to mix bigger and smaller spheres, which would introduce magnetic and electric resonances at the same frequency [vendik2009], but the simulations suggest that the second mode is not strong enough to provide negative permittivity. Other options involve embedding the resonators in some more complicated, possibly metallic structures. In this case one has to be very careful not to damage the resonance in the microsphere. I have made few simulations of this and it might be the way to go.
If a bulk metamaterial is to be built, the spatial arrangement of the resonators is to be thought up as well, both from electromagnetic and technological point of view. Otherwise the microspheres will naturally stick together and their resonances would couple. The spheres might be interlaced by a dielectric sheet with sub-diameter holes. This may also define the spatial shape of the metamaterial object. Maybe they might even self-assemble into a plane- or space-filling "crystal" if mixed with triangles or octohedra of proper size.
While the transmission and reflection spectra of dielectric structures exhibited distinct resonances with characteristic shapes, interpreting the transmission spectra of continuous metallic fishnets becomes more tricky. The key phenomenon (which still puzzles me a bit) is the extraordinary optical transmission, or EOT. Let us study it on a simpler, onedimensionally periodic structure first.
On the following animation, a broadband pulse was sent against a metal sheet (60 um thick) with narrow slits perpendicular to the electric field polarization. In the first simulation, the slits were 60 um wide and their spacing was 300 um.
As usually, I performed series of parametric sweeps to get some feel on how EOT behaves. Before changing the geometry, I verified that doubling nor halving metal conductivity has no significant effect. (The reason for checking this parameter is the unrealistically low value of metal conductivity used for FDTD simulation, for numerical stability reasons. If the conductivity was reduced by more than 10 times, the metal's plasma frequency gets in the THz range and complicated oscillations emerge at higher frequencies.)
The first parameter to scan was the slit width, with the spacing of 300 um and constant metal thickness of 60 um unchanged.
It is obviously the slit spacing that predominantly determines the resonance frequency, which is always between 960-1000 GHz for the given geometry, just below the onset of first diffraction order. The slit width ratio (AKA filling fraction) has almost no impact on the frequency. This means that whatever effect causes the EOT, it is not confined in the volume between the metallic bars. Everything suggests that the EOT is caused by standing oscillation of surface plasmons, as we will see later. The width ratio, however, has a major impact on the amplitude of transmission at non-resonant frequencies.
The second figure shows what happens when such slit width scan is calculated on thinner sheet of metal. At first glance, reducing the metal thickness seems to suppress the EOT, but when a better frequency resolution was used, the transmission peak could be found. Only it is much narrower than before, e. g. 1 GHz for 20 um metal thickness.
Both figures confirm a known fact that thin wires perpendicular to the polarisation do not reflect much. For instance, 20 um thick and 20 um wide metallic wires have transmission > 99 % for the perpendicularly polarized radiation from 0 to 2 THz.
At the right hand side, we are approaching the high-frequency limit, where the transmission should grow with square root of slit width ratio.
As can be seen from the previous figures, the metal thickness has a big impact at least on the EOT spectral width. Let us use constant slit width of 60 um and scan the metal thickness now.
If we assume that the EOT is mediated by coupled oscillation of surface plasmons at both sides of the slit array, we may test this by another numerical experiment. Let us add a dielectric layer on one side of metal to introduce a mismatch of the plasmon resonances. Second, we may add identical dielectric layers on both sides as a reference.
When a dielectric of permittivity 4.0+0i and thickness higher than 20-30 um is added on one side of the metal, the EOT is suppressed. This is obviously by frequency mismatch between plasmons!
On the other hand, when the structure is kept symmetric, the plasmons shift their resonance to lower frequency, but the EOT still works. We may guess that the frequency of first mode shifts from 980 GHz (no plastic) to 840 GHz (25 um of plastic) and converges to 700 GHz (infinite plastic). Therefore, we may roughly estimate that half the energy of the plasmons is localised up to 25 um above the metal surface.
The situation with circular holes is similar as above, except for the metallic sheet is conductive also along the electric field of the incident wave. Therefore, it is predicted that the low frequency limit will have zero transmission.