Holographic metasurfaces simulations applied to realization of non-diffracting waves in the microwave regime
HHolographic metasurfaces simulations applied to realization ofnon-diffracting waves in the microwave regime
Santiago R. C. Fernandez , and Marcos R. R. Gesualdi Universidade Federal do ABC, Av. dos Estados 5001, CEP 09210-580, Santo Andr´e, SP, Brazil.
Abstract –
In this work, we present the computational realization of holographicmetasurfaces to generation of the non-diffracting waves. These holographic metasurfaces(HMS) are simulated by modeling a periodic lattice of metallic patches on dielectricsubstrates with sub-wavelength dimensions, where each one of those unit cells alter thephase of the incoming wave. We use the surface impedance (Z) to control the phase ofthe electromagnetic wave through the metasurface in each unit cell. The sub-wavelengthdimensions guarantees that the effective medium theory is fulfilled. The metasurfaces aredesigned by the holographic technique and the computer-generated holograms (CGHs)of non-diffracting waves are generated and reproduced using such HMS in the microwaveregime. The results is according to the theoretically predicted by non-diffractingwave theory. These results are important given the possibilities of applications of thesetypes of electromagnetic waves in several areas of telecommunications and bioengineering.
1. Holographic Metasurfaces and Surface Impedance — The artificiallystructured materials such as photonic crystals and metamaterials have attracted greatinterest for their remarkable properties to control and manipulate light and electro-magnetic waves [1, 2]. Metamaterials are artificial materials composed by a periodicarray of sub-wavelength unit cells, they have been very explored in a wide range ofapplications due to fact of their properties depends on the geometry and materials oftheir unit cells [3, 4, 5, 6, 7]. The requirement of sub-wavelength dimensions is importantfor approximation of effective media be fulfilled, it means, the incoming wave can’tdistinguish the discontinuities of the metamaterial and therefore, it can be consideredas an homogeneous media and characterized by effective parameters of permittivity andpermeability. The novelty in those metamaterials is the capacity of having permittivityand permeability simultaneously negatives in a same interval of frequencies. Someimportant applications of metamaterials are the obtaining of negative refractive indexwith resonant character [5, 8], superlensing [9, 10], the phenomena of negative refrac-tion [11, 12, 13] and the possibility of cloaking light around certain physical spaces [14, 15].The applications of three-dimensional metamateriais can be also applied to theirtwo-dimensional versions: the metafilms or metasurfaces. Artificial surfaces composedby a periodic array of sub-wavelength unit cells or resonators [16, 17]. Metasurfaces offergreat advantages over their analogous three-dimensional metamaterials, as for example,less losses, more comfortability, occupying less physical space and more connectivity toconventional equipments in laboratories.Metasurfaces are fundamental devices to control or modify wavefront, phase orpolarization state. The resonators introduce abrupt changes of phase in the interface1 a r X i v : . [ phy s i c s . c l a ss - ph ] A ug ue to the discontinuities on the surface. The result is a generalization of the laws of thereflection and refraction, being possible the control of a refracted wave by modulatingthe gradient of phase imposed by the resonators [18, 19]. Thus, metasurfaces provide usto shape the wavefront in shapes designed at will only by building and by ordering thesuitable resonators [20]. In the designing such metasurfaces is very important to havea complete control of the phase of the wave scattered by resonators (from 0 to 2 π ), forachieving this complete interval, much type of resonators have been proposed dependingon the required functionality. As for example, arrays of plasmonic nanoantennas withvariation of angles and orientation for achieving tunable amplitude and polarization stateof the refracted wave [21, 22, 23].On other hand, the holography was developed as a method for reconstruct wavefronts and producing three-dimensional images. Through process of registration, theinformation of phase and amplitude of a wave scattered for the surface of an object(object wave) is stored in determinate photosensitive materials by interference with thereference wave. Thus, the hologram captures the interference pattern between those twooptical waves. The reconstruction occurs when the hologram is illuminated by referencewave, the diffraction pattern reproduces the wavefront from the original object [26, 27, 28].In this work, we use the surface impedance ( Z ) to control the phase of a wave throughthe metasurface by controlling each unit cell, such metasurface is called holographic meta-surface (HMS) [24, 25]. That name comes from holographic principle, where the inter-ference happens between the surface wave ψ surf, the incoming wave passing through thesurface (reference wave), and the radiation wave ψ rad, the transmitted wave from thesurface (object wave). For reconstructed the radiation wave, we use the surface wave toexcite the interference pattern: ( ψ ∗ surf ψ rad) ψ surf = ψ rad | ψ surf | . Thus, to realize theradiation pattern, we need a distribution for a surface impedance on the metasurface, thetheoretical equation for the surface impedance is given by interference of ψ surf and ψ rad[25] Z = i (cid:104) X + M Re ( ψ rad ψ ∗ surf) (cid:105) (1)where X and M are modulation values.The impedance surface ( Z ) is defined as the ratio between the component of electricfield parallel to a current along the surface and the current per unit length of surface: E t = Z ( x t ) J . For the case of a metasurface, we should average that equation over theunit cell, the result for the TM modes (transverse magnetic waves) is given by [25]: Z = iZ (cid:16) k z k (cid:17) (2)where Z is the vacuum impedance, k the wave vector and k z the decay constant, consid-ering the surface wave as Ae − i ( k t · x t ) − k z z e iωt . We can obtain another result for the surfaceimpedance (2) by finding a value for the transverse wave vector, k t . Using the softwareCST Microwave Studio, we can simulate a unit cell with lattice parameter d and to findthe phase ( φ ) through such unit cell for the frequency ω , according to φ = k t d :2 k z k (cid:19) = (cid:18) k t k (cid:19) − (cid:18) φ/dω/c (cid:19) − c the light speed at vacuum. Then, we can calculate the expression that relates thesurface impedance and the phase through a unit cell: Z = Z (cid:112) − φ c /ω d (4)In this way, we design the holographic metasurface (HMS) consisting on a set of unitcells, each one of them formed by a metallic patch on a dielectric substrate, both elementshave square shape being the side of metallic patch always less than the side of substrate,thus, a gap ( g ) is formed in each unit cell (see Figure 1) [41]. We can choose a determinatenumbers of gaps for making the simulations, usually nine or ten values equally spacedare chosen, from a g min till g max. For every value of gap, we design in CST MicrowaveStudio software the corresponding unit cell, applying the function Eingenmode solver toobtain the dispersion curve, i.e. the variation of frequency with the phase. We repeatthis procedure for each value of gap previously chosen, obtaining a graph for each one ofthem. When all those graphs are superimposed we choose the operating frequency ( f ),it is defined so that we be able to find a unique value of phase for every value of gapat such frequency. For waves in the microwave regime, the wavelengths varies from 1mm to 1 m, therefore the dimensions of our holographic metasurfaces must be in orderof millimeters (mm) regarding d (cid:46) λ/
10 for is in according with the effective media theory.We can to obtain for a frequency f the value of Z corresponding to a value of phase φ , according to the equation 4, and at same time, corresponding to a value of gap g . Thisresult is very useful because it allows us to control the surface impedance Z just varyingthe values of gaps. Thus, we can obtain an interval for surface impedance [ Z min , Z max]starting from the chosen values of gaps [ g min , g max]. For having continuous valuesof Z for any value of g , we can make an adjustment of the discrete points Z vs g byinterpolations and, finally to get the relation g = g ( Z ).Then, if we have a map of values of surface impedance Z for every unit cell of theholographic metasurface, we can find the corresponding value of gap and design the wholemetasurface. Those values of Z are given by the expression 1, and therefore, they will de-pend on type of wave that we want obtain through the metasurface. In this work, we wantto obtain the holographic metasurfaces (HMS) of computer-generated holograms (CGH)of object waves previously calculated. Thus, the interference pattern in the expression 1would be in the own CGH, i.e. the information of phase is given in each pixel of CGH.Every pixel of the computer-generated hologram has a gray level between 0 (black) and255 (white), we associate each one of these values to a value of phase between 0 and 2 π forobtaining a matrix of phase (Φ) for the selected CGH. Thus, according to the expression1, we have: Z = i [ X + M Φ] (5)and now, X and M would be adjustment values for making Z to be inside theinterval previously calculated for surface impedance [ Z min , Z max], making the cal-3ulations, we found: X = Z min and M is the maximum integer value satisfying: M ≤ ( Z max − Z min) / π .Thus, we have a matrix of surface impedance for every pixel of the CGH of the non-diffracting wave and and the relation g = g ( Z ). Therefore, we have a unique value of gapfor each pixel of the CGH and thus, we can build an array of such unit cells forming theholographic metasurface working in the microwave regime. Following are shown the resultsobtained for some CGH of known wavefronts. Particularly, in this work this wavefrontsis the non-diffracting waves.Figure 1: (a) Unit cell of the metasurface. (b)
Boundary conditions in the unit celldesigned in CST.
2. Non-diffracting waves — Non-diffracting waves are beams and pulses that keeptheir intensity spatial shape during propagation [29, 30, 31, 32, 33, 34, 35, 36, 37,38, 39, 40]. Pure non-diffracting waves include Bessel beams, Mathieus beams andParabolic beams; as well as the superposition of these waves can produce very specialdiffraction-resistant beams, such as the Frozen Waves. These waves could be applied inmany fields in photonics.In optics, the experimental generation of non-diffracting beams using conventionaldiffractive optical components presents several difficulties, as co-propagating beamsuperposition for instance, and in some cases, is not feasible. Thus, a type of holographyvery important and relevant lately, is the computer-generated holography (CGH). Inthis case, the hologram is created from computational numerical methods. The physicalprocess that allows the reconstruction of the image in far-field is expressed by the theoryof diffraction of Fresnel-Kirchhoff [27]. The computational holography technique with theuse of numerical holograms and spatial light modulators, has efficiently reproduced thebeams cited above. In this case, the construction of the non-diffracting beam hologramis done numerically by a computer generated hologram (CGH) and its reconstruction isperformed optically with its implementation in a spatial light modulator (SLM).In this work, we will focus on computer-generated holograms of phase [28] of sometypes of non-diffracting waves. The non-diffracting waves are solutions to the (linear)wave equation which travel well confined or localized , in a single direction without toexperiment effects of dispersion caused by diffraction. The types of non-diffracting4aves studied are Airy beams [29, 37], Bessel beams [30, 31] and Frozen waves (FW)[32, 33, 34, 35, 36, 37, 38, 39, 40].
3. Simulations and Results — We built two sets of holographic metasurfaces workingin different frequencies in the microwave regime by simulations using CST MicrowaveStudio software [41].Figure 2: (a) Variations of frequency with phase for each v alue of gap ( g ), the operatingfrequency was defined to 24.34 GHz. (b) Variation of values of gaps according to surfaceimpedance and its corresponding adjustment curve.The first case, at operating frequency of 24.34 GHz, the metal used was copper (Cu)on a substrate Rogers RT5880 with permittivity (cid:15) = 2 . t = 1 .
57 mm. Thelattice parameter was set at d = 3 mm, and gaps were set from g = 1 mm till g = 2 . g vs Z , we obtained theinterval for surface impedance: [ Z min , Z max] = [235 .
05 Ω , .
39 Ω], and we also foundthe following modulating values: X = 235 .
05 Ω, M = 56.For this first HMS, we reproduce the CGH of a Bessel wave of zero order, with aresolution of 128x128 pixels, transverse number wave k ρ = 16 mm − , size of centralspot of 0.28 mm and generated at wavelength of λ = 12 .
33 mm, corresponding toour operating frequency of 24.34 GHz (see Figure 3), the corresponding holographicmetasurface was built applying the theory presented above and it is shown in the Figure4. We can note the small unit cells with variations of gap, the darkest zones correspondto cells with smaller gaps or high surface impedance whilst the clearest zones correspond5o cells with larger gaps or low surface impedance.Figure 3: Image of the computer-generated hologram of a Bessel beam with resolution of128x128 pixels at wavelength λ = 12 .
33 mm.Figure 4: Holographic metasurface of the CGH of Bessel beam implemented using unitcells with gaps variation. 6igure 5: Image of the computer-generated hologram of an Airy beam with resolution of128x128 pixels at wavelength λ = 12 .
33 mm.Figure 6: Holographic metasurface of the CGH of an Airy beam implemented using unitcells with gaps variation. 7igure 7: Image of the computer-generated hologram of a Frozen Wave (FW) beam withresolution of 128x128 pixels at wavelength λ = 12 .
33 mm.Figure 8: Holographic metasurface of the CGH of FW beam implemented using unit cellswith gaps variation. 8e also reproduce the CGH of an Airy wave, with a resolution of 128x128 pixels,parameter of decay a = 0 . ? ] and generated at wavelength of λ = 12 .
33 mm, corre-sponding to our operating frequency of 24.34 GHz (see Figure 5), the correspondingHMS is shown in the Figure 6.And, we also reproduce the CGH of a Frozen Wave (FW), with a resolutionof 128x128 pixels, number of Bessel beams superposed N = 6, size of central spot∆ ρ = 7 . Q = 407 .
67 [32] and generated at wavelength of λ = 12 .
33 mm,corresponding to our operating frequency of 24.34 GHz (see Figure 7), the correspondingHMS is shown in the Figure 8.A second holographic metasurface was built by simulation using CST MicrowaveStudio. This case, the operating frequency is 2.4 GHz, the metal used was copper (Cu)on a substrate Rogers TM6 with permittivity (cid:15) = 6 and thickness t = 7 .
85 mm. The latticeparameter was set at d = 15 mm, and gaps were set from g = 1 mm till g = 5 mm with aninterval of 0.5 mm. The range of frequencies in the simulation was defined from 0 till 5.5GHz. In the four sides of the unit cell (see Figure 1) were set periodic boundary conditionsand Perfect Electrical Conductor (PEC) for the propagation direction. The graph of thedispersion curves for each value of gap and variation of gaps with surface impedanceare shown in the Figure 9. After making the adjustment of the curve of gaps ( g ) vssurface impedance ( Z ), we obtained the interval for surface impedance: [ Z min , Z max] =[188 . , . X = 188 . M = 46.Figure 9: (a) Variations of frequency with phase for each value of gap ( g ), the operatingfrequency was defined to 2.4 GHz. (b) Variation of values of gaps according to surfaceimpedances and its corresponding adjustment curve.9igure 10: Image of the computer-generated hologram of a Bessel beam with resolutionof 128x128 pixels at wavelength 125 mm.Figure 11: Holographic metasurface of the CGH of Bessel beam implemented using unitcells with gaps variation. 10igure 12: Image of the computer-generated hologram of an Airy beam with resolutionof 128x128 pixels at wavelength 125 mm.Figure 13: Holographic metasurface of the CGH of Airy beam implemented using unitcells with gaps variation. 11igure 14: Image of the computer-generated hologram of a frozen wave beam with reso-lution of 128x128 pixels at wavelength 125 mm.Figure 15: Holographic metasurface of the CGH of FW beam implemented using unitcells with gaps variation. 12or this second HMS, we reproduce the CGH of a Bessel wave of zero order, with aresolution of 128x128 pixels, transverse number wave k ρ = 16 mm − , size of central spotof 0.28 mm and generated at wavelength of λ = 125 mm, corresponding to our operatingfrequency of 2.4 GHz (see Figure 10), the corresponding holographic metasurface isshown in the Figure 11.Figure 12 shows the CGH of an Airy wave, with a resolution of 128x128 pixels,parameter of decay a = 0 . ? ], the corresponding HMS is shown in the Figure 13.Finally, we reproduce the CGH of a Frozen Wave (FW), with a resolution of 128x128pixels, number of Bessel beams superposed N = 6, size of central spot ∆ ρ = 47 . Q = 4 .
52 [32], in Figure 14), and the corresponding HMS is shown in the Figure15.
4. Conclusions — This work presents a way for controlling and manipulating theelectromagnetic radiation through the computational realization of holographic metasur-faces to generation of the non-diffracting waves. Holographic metasurfaces (HMS) aresimulated by modeling a periodic lattice of metallic patches on dielectric substrates withsub-wavelength dimensions, where each one of those unit cells alter the phase of theincoming wave. The surface impedance (Z) allows to control the phase of a wave throughthe metasurface in each unit cell. The sub-wavelength dimensions guarantees that theeffective medium theory is fulfilled. The metasurfaces are designed by the computer-generated hologram (CGH) of non-diffracting waves are generated and reproduced usingsuch HMS in the microwave regime. Two sets of holographic metasurfaces was built forworking in the microwave regime, the first at 24.34 GHz and the second one at 2.4 GHz.The results is according to the theoretically predicted and allows applications of thesetypes of electromagnetic waves in several areas of telecommunications and bioengineering.
Acknowledgments
The authors acknowledge partial support from UFABC, CAPES,FAPESP (UNDER GRANTS 16/19131-6) and CNPq (UNDER GRANTS 302070/2017-6).