Artificial dielectric stepped-refractive-index lens for the terahertz region

In this paper we theoretically and experimentally demonstrate a steppedrefractiveindex convergent lens made of a parallel stack of metallic plates for terahertz frequencies based on artificial dielectrics. The lens consist of a non-uniformly spaced stack of metallic plates, forming a mirror-symmetric array of parallel-plate waveguides (PPWGs). The operation of the device is based on the TE1 mode of the PPWG. The e ective refractive index of the TE1 mode is a function of the frequency of operation and the spacing between the plates of the PPWG. By varying the spacing between the plates, we can modify the local refractive index of the structure in every individual PPWG that constitutes the lens producing a stepped refractive index profile across the multi stack structure. The theoretical and experimental results show that this structure is capable of focusing a 1 cm diameter beam to a line focus of less than 4 mm for the design frequency of 0.18 THz. This structure shows that this artificial-dielectric concept is an important technology for the fabrication of next generation terahertz devices. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement OCIS codes: (230.7370) Waveguides; (220.3630) Lenses; (300.6495) Spectroscopy, terahertz. References and links 1. S. Dhillon, M. Vitiello, E. Linfield, A. Davies, M. C. Ho mann, J. Booske, C. Paoloni, M. Gensch, P. Weightman, G. Williams et al., “The 2017 terahertz science and technology roadmap,” Journal of Physics D: Applied Physics 50, 043001 (2017). 2. A. Squires, E. Constable, and R. A. Lewis, “3d printed terahertz di raction gratings and lenses,” Journal of Infrared, Millimeter, and Terahertz Waves 36, 72–80 (2015). 3. S. Busch, M. Weidenbach, M. Fey, F. Schäfer, T. Probst, and M. Koch, “Optical properties of 3d printable plastics in the thz regime and their application for 3d printed thz optics,” Journal of Infrared, Millimeter, and Terahertz Waves 35, 993–997 (2014). 4. A. Hernandez-Serrano, M. Weidenbach, S. Busch, M. Koch, and E. Castro-Camus, “Fabrication of gradient-refractiveindex lenses for terahertz applications by three-dimensional printing,” JOSA B 33, 928–931 (2016). 5. S. F. Busch, J. C. Balzer, G. Bastian, G. E. Town, and M. Koch, “Extending the alvarez-lens concept to arbitrary optical devices: Tunable gratings, lenses, and spiral phase plates,” IEEE Transactions on Terahertz Science and Technology 7, 320–325 (2017). 6. B. Scherger, M. Scheller, C. Jansen, M. Koch, and K. Wiesauer, “Terahertz lenses made by compression molding of micropowders,” Applied optics 50, 2256–2262 (2011). 7. J. Neu, B. Krolla, O. Paul, B. Reinhard, R. Beigang, and M. Rahm, “Metamaterial-based gradient index lens with strong focusing in the thz frequency range,” Optics express 18, 27748–27757 (2010). 8. D. Smith, J. Mock, A. Starr, and D. Schurig, “Gradient index metamaterials,” Physical Review E 71, 036609 (2005). 9. R. Liu, Q. Cheng, J. Y. Chin, J. J. Mock, T. J. Cui, and D. R. Smith, “Broadband gradient index microwave quasi-optical elements based on non-resonant metamaterials,” Optics express 17, 21030–21041 (2009). 10. O. Paul, B. Reinhard, B. Krolla, R. Beigang, and M. Rahm, “Gradient index metamaterial based on slot elements,” Applied Physics Letters 96, 241110 (2010). 11. R. Mendis and D. M. Mittleman, “A 2-d artificial dielectric with 0n<1 for the terahertz region,” IEEE Transactions on Microwave Theory and Techniques 58, 1993–1998 (2010). 12. R. Mendis, M. Nagai, Y. Wang, N. Karl, and D. M. Mittleman, “Terahertz artificial dielectric lens,” Scientific reports 6 (2016). 13. S. Jones and J. Brown, “Metallic delay lenses,” Nature 163, 324–325 (1949). 14. W. E. Kock, “Metal-lens antennas,” Proceedings of the IRE 34, 828–836 (1946). 15. J. Brown, “Artificial dielectrics having refractive indices less than unity,” Proceedings of the IEE-Part IV: Institution Monographs 100, 51–62 (1953). 16. V. Torres, V. Pacheco-Peña, P. Rodríguez-Ulibarri, M. Navarro-Cía, M. Beruete, M. Sorolla, and N. Engheta, “Terahertz epsilon-near-zero graded-index lens,” Optics express 21, 9156–9166 (2013). 17. R. Mendis and D. M. Mittleman, “Comparison of the lowest-order transverse-electric (TE1) and transverse-magnetic (TEM) modes of the parallel-plate waveguide for terahertz pulse applications,” Optics express 17, 14839–14850 (2009). 18. R. Mendis, M. Nagai, W. Zhang, and D. M. Mittleman, “Artificial dielectric polarizing-beamsplitter and isolator for the terahertz region,” Scientific Reports 7, 5909 (2017). 19. R. Mendis, J. Liu, and D. M. Mittleman, “Terahertz mirage: Deflecting terahertz beams in an inhomogeneous artificial dielectric based on a parallel-plate waveguide,” Applied Physics Letters 101, 111108 (2012). 20. C. A. Balanis, Advanced engineering electromagnetics (John Wiley & Sons, 1999). 21. D. G. Voelz, Computational fourier optics: a MATLAB tutorial (SPIE press, 2011).


Introduction
In order to expand the applications of terahertz (THz) radiation [1], the generation of new devices and components for the manipulation of THz radiation are necessary.There has been an e ort to fabricate lenses using 3D printing techniques [2][3][4][5], compressing powders [6], using ring-resonators-based metamaterials [7][8][9][10] and using artificial dielectrics made of parallel plate waveguides (PPWGs) [11][12][13][14][15][16].Artificial dielectrics are man-made media that mimic the properties of naturally occurring dielectric media, or even manifest properties that cannot generally occur in nature.For example, the well-known dielectric property, the refractive index, which usually has a value greater than unity, can have a value less than unity in an artificial dielectric.In this work we design and experimentally demonstrate that a modulation of the refractive index of an artificial dielectric made of PPWGs is possible by varying the plate spacing of a stack of waveguides.In our device, every PPWG has the same propagation length, resulting in a flat lens.These plate spacings, compared to the input beam size, ensure that only the TE 1 mode is dominantly excited in the PPWGs [17].We demonstrate that this artificial dielectric stepped-index lens is capable of focusing a 10 mm diameter beam into line focus of less than 4 mm at the design frequency of 0.18 THz.

Design and Fabrication
A simplified schematic diagram of the artificial dielectric device is shown in Fig. 1 (a) and Fig. 1 (b).It consists of an assembly of non-uniformly spaced identical parallel plates made of 100 µm thick titanium with spacings from 0.8 mm to 1.5 mm, which gives a ratio between 8 and 15 of the spacing compared to the thickness.The plates were fabricated by chemical etching in order to prevent any burring [18].We used three types of spacers with di erent thickness, 0.3 mm, 0.5 mm and 1.0 mm.Using linear combinations of these three thicknesses, we realized plate spacings varying from 0.8 mm to 1.5 mm in steps of 0.1 mm.Fig. 1 (b) shows a front view of the device, where the variation of the spacing between plates is illustrated.The plot in Fig. 1 (c) is the theoretical variation of the e ective refractive index as a function of the plate spacing for di erent frequencies [19,20].This plot shows that the index is highly dependent on the plate spacing.The e ective-refractive-index function of the device is given by where f is the frequency, c is the speed of light in vacuum and h is the plate spacing.Fig. 1 (d) shows a photograph of the fabricated device, looking on axis, which illustrates a clear aperture of 20 mm by 18 mm.Fig. 1 (e) shows the refractive-index profile of the structure at 0.18 THz along the red dotted line in (d).In the same figure, we plot a parabolic fit in order to emphasize the quadratic variation of the refractive index profile .The parabolic fit of the refractive index is represented by n = 5.2x10 3 RIU mm 2 x 2 + 0.83 RIU where x is the position in mm.The operating principle of our device relies on the propagation of the TE 1 mode in every individual PPWG constituting the artificial dielectric structure.The refractive index of each PPWG is a function of the plate spacing as shown in Fig. 1 (c) .When the spacing between the plates decreases, the refractive index also decreases, for a given frequency.
In order to verify the operation of this device, we used a commercial finite-element method (FEM) software, COMSOL Multiphysics, to perform numerical simulations for frequencies from 0.17 THz to 0.20 THz in steps of 0.01 THz.The metal plates of the device are considered as perfect electric conductors (PECs).The device is illuminated by a collimated Gaussian beam with an amplitude of 1 V/m and a 1/e diameter of 10 mm.In order to ensure the accuracy, the length scale of the mesh is set to be less than or equal to /8 throughout the simulation domain, where is the wavelength of the incident radiation.For the boundaries we used scattering boundary conditions.These results are shown in Fig. 2. The focusing of the THz radiation by the device is clearly observed in Fig. 2 (a) and Fig. 2 (b).Fig. 2 (a) and Fig. 2 (b) show the instantaneous electric field and the normalized intensity respectively, at a frequency of 0.18 THz.The input electric-field vector is parallel to the plates to excite the TE 1 mode.In Fig. 2 (a) the formation of the wavefront curvature inside the structure is clearly seen as the wave propagates, which causes to generate a focus.In the same figure the polarization of the input electric field is indicated.In Fig. 2 (b) the intensity shows a strong focus approximately 10 mm after the front face of the lens.In Fig. 2 (c) we present simulation results for 0.17 -0.21 THz in the in order to find the maximum electric-field amplitude.This figure shows that the strongest focus is at 0.18 THz (the design frequency).These simulation results predict a focus with an approximately 2 mm beam waist.The focal length clearly exhibits chromatic aberration, shifting to larger distances for larger frequencies.In Fig. 2 (d) the predicted wavefront position after propagating 4.3 ps is shown in red lines superimposed on the simulation.The position of the wavefront was found theoretically using where != ck 0 n, where c is the speed of light in vaccum, k 0 the wavenumber in vaccum, n the refractive index of the device as given in Eq. ( 1), and t the time.Fig. 2 (d) shows excellent agreement between the simulation and the analytical result; the wavefront engineering due to the particular index profile of the structure.

Experimental characterization
The experimental characterization was carried out using a fiber-coupled commercial time-domain spectrometer which generates and detects radiation in the THz band [18].The schematic of the experimental setup is illustrated in Fig. 3.The input beam to the lens was formed to a 1/e-amplitude diameter of 10 mm as used in the numerical simulations.For the detection we used an e ective aperture of 0.5 mm diameter placed in front of the receiver (silicon lens) in order to improve the spatial resolution.The data were collected by scanning the aperture-integrated receiver along a 20 mm line with steps of 0.5 mm in the direction transverse to the input beam axis, as shown in Fig 3 .We recorded the broadband pulses passing through the device which were subsequently Fourier transformed in order to extract their spectral content.In Fig. 4 we show the measured beam profiles for 0.15 THz, 0.18 THz and 0.20 THz.
From these results we see that the maximum amplitude is achieved at 0.18 THz, in agreement with the simulations.The major di erences between the simulations and the experimental results are in the focal distance and the size of the focus.They were respectively, 10 mm and 2 mm for the simulation and 18 mm and 3.34 mm for the experiment.These discrepancies are most likely due to the imperfections in the device; in other words, the plates were not perfectly flat.In addition, secondary lobes are seen on either side of the main lobe.The secondary lobes are due the truncation of the input beam, which becomes more severe as the frequency goes down, due to the TE 1 cuto .This can be understood from Fig. 2 (a).For example, at 0.18 THz the clear aperture is determined by the location of PPWGs with spacing of 0.9 mm, since smaller spacings are below cuto at this frequency.As a result, the beam is clipped for plate spacings smaller than 0.9 mm, causing this "ringing" e ect in the focal plane.
In order to understand the device further, we also carried out an analytical examination of the device.In the model, we used the Rayleigh-Sommerfeld Integral given by [21].
where U 2 is the electric field at the observation position, the wavelength, z the perpendicular distance from the incident plane to the observation plane, U 1 the electric field at the incident plane, ⌘ and ⇠ are the coordinates at the object plane, k = k 0 n the wavenumber in vacuum multiplied by the fitted refractive index of the device given by the red dashed curve in Fig. 1 (e) and r 12 the distance from the incident plane to a point on the observation plane.
In Fig. 5 we compare the experimental beam profile with our numerical and theoretical result at 0.18 THz.This comparison shows very good agreement between the analytical and simulation results of the size of the focus.The slightly larger experimental result is due to device imperfections as noted earlier.

Conclusion
In conclusion, we showed that it is possible to realize a stepped-index lens for the THz region using artificial dielectrics.In this work, we linearly increase the spacing of the plates from 0.8 mm at the outer periphery to 1.5 mm in the center of the device in steps of 0.1 mm.We experimentally demonstrated that this novel device is capable of focusing a 10 mm diameter beam to a size of 3.3 mm at 0.18 THz.The wavefront engineering of this device is achieved via the spatial index variation as in a gradient-index (GRIN) lens, and not via a geometric curvature, making this a flat lens.This device made of metal plates of all the same size has advantages in comparison to other artificial dielectric lenses [8] since the faces of the device are planar, and therefore easier to integrate into a composite assembly.This kind of planar convergent lens with a frequency-dependent focus may be valuable in future applications in THz communications and imaging.

Fig. 1 .
Fig. 1.Schematic diagram of the device.(a) lateral view, (b) front view (the aspect ratio of the device was exaggerated for clarity).(c) refractive index of a typical PPWG operating in the TE 1 mode as function of the plate spacing for frequencies from 0.15 THz to 0.20 THz (d) Photograph of the fabricated device.The dotted circle denotes the incident beam.e) Index profile of the structure at 0.18 THz along the red dotted line in (d).In (e) the red dashed line is a parabolic fit to the refractive index profile.

Fig. 2 .
Fig. 2. Numerical simulation results.(a) Instantaneous electric field and (b) normalized electric field at 0.18 THz.(c) longitudinal cross-section along the dotted line in Fig (b) for the same simulation but for frequencies from 0.17 THz to 0.21 THz in steps of 0.01 THz.In (a) and (b) the focus is clearly identified at approximately 10 mm from the front face of the lens.This distance is confirmed in (c) where the maximum amplitude is at approximately 10 mm for f = 0.18 THz. Figure (d) shows the predicted wavefront position after 4.3 ps (red lines) using eq.(2).

Fig. 3 .
Fig. 3. Schematic of the experimental setup for the experimental characterization of the focusing properties for the artificial dielectric stepped-index device.

Fig. 4 .
Fig. 4. Cross sectional profile for the input and output beams for a) 0.15 THz, b) 0.18 THz and c) 0.20 THz.The data were recorded at 18 mm from the front surface of the device.In the same figures Gaussian fits are shown.

Fig. 5 .
Fig.5.Comparison between cross section of the beam for the numerical simulation (blue line), experimental result (red line) and theoretical result (yellow line) at 180 GHz.It is observed that the beam waist for the experimental result is larger than the theoretical and the simulation one.This is most likely due the imperfections on the device.