Focused terahertz waves generated by a phase velocity gradient in a parallel-plate waveguide

We demonstrate the focusing of a free-space THz beam emerging from a leaky parallel-plate waveguide (PPWG). Focusing is accomplished by grading the launch angle of the leaky wave using a PPWG with gradient plate separation. Inside the PPWG, the phase velocity of the guided TE1 mode exceeds the vacuum light speed, allowing the wave to leak into free space from a slit cut along the top plate. Since the leaky wave angle changes as the plate separation decreases, the beam divergence can be controlled by grading the plate separation along the propagation axis. We experimentally demonstrate focusing of the leaky wave at a selected location at frequencies of 100 GHz and 170 GHz, and compare our measurements with numerical simulations. The proposed concept can be valuable for implementing a flat and wide-aperture beam-former for THz communications systems. ©2015 Optical Society of America OCIS codes: (230.7370) Waveguides; (060.5625) Radio frequency photonics. References and links 1. R. Piesiewicz, T. Kleine-Ostmann, N. Krumbholz, D. M. Mittleman, M. Koch, J. Schoebel, and T. Kürner, “Short-range ultra broadband terahertz communications: Concepts and perspectives,” IEEE Antennas Propag. Mag. 49(6), 24–39 (2007). 2. J. Federici and L. Moeller, “Review of terahertz and sub-terahertz wireless communications,” J. Appl. Phys. 107(11), 111101 (2010). 3. T. Kleine-Ostmann and T. Nagatsuma, “A review on terahertz communications research,” J. Infrared Milli. Terahertz Waves 32, 143–171 (2011). 4. H.-T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006). 5. W. Gao, J. Shu, K. Reichel, D. V. Nickel, X. He, G. Shi, R. Vajtai, P. M. Ajayan, J. Kono, D. M. Mittleman, and Q. Xu, “High-contrast terahertz wave modulation by gated graphene enhanced by extraordinary transmission through ring apertures,” Nano Lett. 14(3), 1242–1248 (2014). 6. B. Bowden, J. A. Harrington, and O. Mitrofanov, “Silver/polystyrene-coated hollow glass waveguides for the transmission of terahertz radiation,” Opt. Lett. 32(20), 2945–2947 (2007). 7. M. Mbonye, R. Mendis, and D. M. Mittleman, “Measuring TE1 mode losses in terahertz parallel-plate waveguides,” J. Infrared Milli. Terahertz Waves 34, 416–422 (2013). 8. Y. Monnai, K. Altmann, C. Jansen, H. Hillmer, M. Koch, and H. Shinoda, “Terahertz beam steering and variable focusing using programmable diffraction gratings,” Opt. Express 21(2), 2347–2354 (2013). 9. J. V. Rudd and D. M. Mittleman, “The influence of substrate lens design in terahertz time-domain spectroscopy,” J. Opt. Soc. Am. B 19(2), 319–329 (2002). 10. W. L. Chan, H. T. Chen, A. J. Taylor, I. Brener, M. J. Cich, and D. M. Mittleman, “A spatial light modulator for terahertz beams,” Appl. Phys. Lett. 94(21), 213511 (2009). 11. C. M. Watts, D. Shrekenhamer, J. Montoya, G. Lipworth, J. Hunt, T. Sleasman, S. Krishna, D. R. Smith, and W. J. Padilla, “Terahertz compressive imaging with metamaterial spatial light modulators,” Nat. Photonics 8(8), 605–609 (2014). 12. C. A. Balanis, Modern Antenna Handbook (John Wiley & Sons, 2008). 13. M. Garcia-Vigueras, J. L. Gomez-Tornero, G. Goussetis, A. R. Weily, and Y. J. Guo, “1D-leaky wave antenna employing parallel-plate waveguide loaded with PRS and HIS,” IEEE Trans. Antenn. Propag. 59(10), 3687– 3694 (2011). 14. N. Karl, R. W. McKinney, Y. Monnai, R. Mendis, and D. M. Mittleman, “Frequency-division multiplexing in the terahertz range using a leaky wave antenna,” Nat. Photonics 9, 717–720 (2015). #245988 Received 15 Jul 2015; revised 9 Sep 2015; accepted 29 Sep 2015; published 15 Oct 2015 (C) 2015 OSA 19 Oct 2015 | Vol. 23, No. 21 | DOI:10.1364/OE.23.027947 | OPTICS EXPRESS 27947 15. T. A. Milligan, Modern Antenna Design (John Wiley & Sons, 2005). 16. R. Mendis and D. M. Mittleman, “Comparison of the lowest-order transverse-electric (TE1) and transversemagnetic (TEM) modes of the parallel-plate waveguide for terahertz pulse applications,” Opt. Express 17(17), 14839–14850 (2009). 17. R. Mendis and D. M. Mittleman, “An investigation of the lowest-order transverse-electric (TE1) mode of the parallel-plate waveguide for THz pulse propagation,” J. Opt. Soc. Am. B 26(9), A6–A13 (2009). 18. R. Mendis and D. M. Mittleman, “A 2D artificial dielectric with 0 < n < 1 for the terahertz region,” IEEE Trans. Microw. Theory Tech. 58(7), 1993–1998 (2010). 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,” Appl. Phys. Lett. 101(11), 111108 (2012). 20. J. Liu, R. Mendis, and D. M. Mittleman, “A Maxwell’s fish eye lens for the terahertz region,” Appl. Phys. Lett. 103(3), 031104 (2013). 21. M. Mbonye, R. Mendis, and D. M. Mittleman, “Inhibiting the TE1-mode diffraction losses in terahertz parallelplate waveguides using concave plates,” Opt. Express 20(25), 27800–27809 (2012). 22. J. Deibel, M. Escarra, N. Berndsen, K. Wang, and D. M. Mittleman, “Finite element method simulations of guided wave phenomena at terahertz frequencies,” Proc. IEEE 95(8), 1624–1640 (2007).


Introduction
As the growth of mobile data traffic accelerates, bandwidth availability is becoming an increasing concern [1][2][3].With current wireless technology, operating in the 1-5 GHz range, it is already not possible to meet the existing demand.Even the next-generation wireless protocols at 60 GHz will not provide adequate bandwidth within a few years.As a result, there is growing interest in using the frequency range above a few hundred GHz for shortrange, broadband wireless communications.At such high frequencies, crossing into the THz range, communications systems will require capabilities that are not supported by existing devices.This challenge has motivated much recent research, for example in the development of modulators [4,5], low-loss waveguides [6,7], and beam steering components [8].Despite this growing body of research, numerous difficult problems remain unaddressed.One important example is wavefront engineering -that is, the manipulation of a wavefront for optimizing the throughput of a communication channel.In the THz range, wireless transmission will likely rely on point-to-point connections using highly directional or even focused beams [1,2].
In such applications, wavefront engineering will be a critical need for optimizing system performance.Conventional integrated THz transmitters are often equipped with silicon substrate lenses [9].However, fixed components do not afford the possibility of dynamically changing the THz wave front in response to changing system needs.More recent work has reported the development of metamaterial-based multi-pixel modulators for spatial wavefront control, which can be dynamically reconfigured [10,11].Such bulky components are less favorable in mobile device applications.Alternative techniques are important for fulfilling this compelling need.
Leaky-wave antennas have been used in the microwave and RF spectral ranges as highly directional and versatile antennas since the 1940's and continue to be used as novel transmitters and receivers to this day [12][13][14].In a leaky-wave antenna, a travelling wave is guided along a main waveguiding structure and radiates outward azimuthally from the propagation axis [15].Conventionally, a rectangular waveguide has been used as the main waveguide structure.The travelling wave inside the rectangular waveguide is a fast wave with a phase velocity greater than the vacuum speed of light.This leads to an effective refractive index less than unity [12][13][14][15].By opening a slot along the side wall of the waveguide, one connects the two media inside and outside the waveguide, thus allowing the traveling wave to leak into free space at an angle determined by Snell's Law.Although this mechanism would also work in the THz range, direct scaling of a rectangular waveguide from the microwave to the THz range is impractical due both to increasing metallic losses and significantly more challenging fabrication.

Results and discussion
In this article, we propose the use of a metal parallel-plate waveguide (PPWG) as a leakywave antenna which can also offer a unique method for wavefront engineering.The PPWG, operating in its lowest-order transverse electric (TE 1 ) mode [16,17], offers a practical implementation of a leaky-wave antenna for the THz range, as illustrated in Fig. 1(a).As in the rectangular waveguide, the PPWG's TE 1 mode, where the electric field is polarized parallel to the metal plates, propagates with an effective refractive index smaller than one.Therefore, a slit cut along the top plate can leak the travelling wave into free space.Specifically, the effective refractive index for waves propagating in the TE 1 mode in a PPWG is given by where b is the plate separation, f is the frequency, and c is the light speed [18].Then, the launch angle is determined by the Snell's law as where θ is the angle measured from the broadside orientation as illustrated in Fig. 1b.
Since the angle at which the leaky wave is launched depends on the plate separation b, it is possible to adjust the beam divergence by varying the plate separation along the propagation axis.Ordinarily, a variation in the plate separation could lead to the introduction of higherorder waveguide modes.However, we have previously demonstrated [18][19][20] that a continuous variation of the plate separation can be effected without inducing any higher-order mode conversion, as long as the change in the plate spacing is gradual (on the scale of the wavelength).Thus, one can introduce a smoothly varying effective refractive index for the guided wave inside the waveguide by controlling the plate separation, and therefore create an almost arbitrary spatial profile for the effective refractive index n e (x,y) in the waveguide plane.This degree of freedom proves to be extremely valuable for various applications including the manipulation of terahertz guided wavefronts [19] and for the construction of novel waveguide-based optical components [20], as well as for improving the lateral confinement in an open waveguide structure like the PPWG [7,21].Here, we combine this idea of engineering n e (x,y) with the concept of a leaky-wave waveguide to demonstrate that a leaky-wave can be focused in free space.As seen from Eqs.
(1) and ( 2), the launch angle of the guided wave depends on the plate separation.Thus, the beam emerging from the waveguide can be focused at a predefined location above the leakywave slot when all of the waves arrive at that point with equivalent phase (i.e., exhibiting constructive interference).To compute the necessary plate separation function, we first choose a specific location (x 0 ,z 0 ) for the focal point, referenced to the coordinate system defined in Fig. 1 and Fig. 2. Here, the x coordinate is the waveguide propagation axis (parallel to the slot in the upper waveguide plate), and the z axis points perpendicular to the axis of the slot.(The plane x = 0 corresponds to the waveguide input facet.The z = 0 plane is the upper surface of the upper metal plate.)We can use the fact that the emission angle θ is geometrically related to the values x 0 and z 0 according to ( ) and then solve for the plate separation b as a function of position x along the waveguide to find ( ) It should be mentioned that the plate separation b(x) becomes minimum at x = x 0 , which is the x coordinate of the chosen focal point, and this plate separation corresponds to the cutoff condition for the TE 1 mode.Therefore, it is preferable to define the location of the focal point so that x 0 is no less than L, where L is the length of the waveguide as illustrated in Fig. 2. Using Eq. ( 4), we construct several leaky-wave PPWGs to demonstrate the principle.For each waveguide, the top plate was fabricated using an aluminum sheet of 1 mm thickness, cut to a 50 mm by 50 mm square with a slit of 3 mm width and a length of 42 mm cut along the middle of the square.Thus, during the first 8 mm of guided wave propagation, the guided mode is strictly contained between the two parallel plates, and can only leak out after it has passed this initial propagation length.To fabricate the bottom plate with a curved inner surface, we employ 3D printing.We print a polymer form with the surface height profile defined by Eq. ( 4), and then apply a metal coating using a metal spray paint, with sufficient thickness so that the object behaves as if it were a solid metal [20].We then join the top and bottom plates with plastic spacers, to guarantee that the plate separation is correct at all points inside the waveguide.We demonstrate two particular focusing leaky waveguide geometries, designed to focus frequencies of 100 GHz and 170 GHz respectively.For both designs, we chose the focal point to be at x 0 = 50 mm; that is, the focal point was chosen to be directly above the end of the waveguide (since x 0 = L).For the 100 GHz model, we chose a focal height of z 0 = 50 mm above the top plate, while for the 170 GHz version, we chose a height of z 0 = 30 mm.
The device was experimentally characterized using the setup shown in Fig. 2. It consists of a commercial THz-TDS system for generating and detecting THz radiation, using fibercoupled photoconductive antennas.A broadband THz pulse is delivered through a confocal lens system consisting of 60 mm and 100 mm focal length lenses.This produced a 1/e beam width of 10 mm at the waveguide input facet for all frequencies of interest.The transmitter was oriented to produce radiation polarized in the y-direction, in order to efficiently excite the TE 1 waveguide mode.To map out the electric field of the radiation above the slot, we raster scan the receiver in two dimensions, and measure the time-domain waveform at each spatial location.We extract the spectral amplitude at a given frequency by Fourier transform, and plot the results as the amplitude of the extracted frequency component vs. the receiver position.The rectangular area (white dashed line) in the simulation indicates the area scanned in the measurements.Calculations are the result of finite element simulations using a commercial software package.Note that the color bar is normalized for the peak value inside the measured area, which is indicated by the broken rectangle.Outside the measured area, especially in the vicinity of the waveguide surface, the E-field is larger and the red color is saturated.
In Fig. 3, we compare these experimental results to numerical simulations of the realistic waveguide configuration, performed using a commercial software package [22].In both cases, we clearly observe focusing at the predefined focal points.The cross-sectional field profiles are shown in Fig. 4, compared with the corresponding results from the numerical simulations.We observe that the measured field profiles both peak at x = 50 mm, precisely consistent with the design criteria and with the numerical simulations, at both frequencies.The agreement is not as accurate for the z axis, but the trend is still reproduced.Clearly, the experimentally observed beam width is much wider than in the simulation, for both waveguides.This effect is mostly due to the finite spatial resolution and finite acceptance angle of the receiver used to map the field profile, but could also be a result of small fabrication errors or finite conductivity losses and surface roughness of the metal components which are not accounted for in the simulations.The leaky waveguide performance can be improved in a number of ways.For example, it will be possible to compensate for the field decay along the radiation aperture by gradually increasing the slit width so that more radiation is emitted as the wave travels.Moreover, while we have considered variation of the plate separation along the propagation axis in this paper, we could also implement a gradient or curved plate separation in the direction transverse to the guided wave propagation axis.This would be advantageous for counteracting beam divergence in the unconfined direction inside the PPWG [7,21] and enhancing the efficiency of the device.Also, we note that, while the leaky-wave antenna focuses in the x-z plane, the radiation is freely diffracting in the y direction.To focus the beam also in the y direction, we could extend the aperture geometry, for example by making a parallel slot array.Finally, we can imagine constructing an active device using a MEMS architecture.

Conclusions
In conclusion, we have demonstrated a focusing THz leaky wave antenna using a PPWG with a gradient plate separation.With a phase velocity greater than the vacuum speed of light, the TE 1 mode can be utilized to control the spatial mode of the leaky waves.The proposed structure could replace bulky lens components and offer efficient off-axis quasi-optic coupling of THz radiation from planar guided-wave THz systems.

Fig. 1 .
Fig. 1.(a) An illustration of the parallel-plate waveguide with a slot aperture.The TE 1 mode is a fast mode, and therefore can couple to free-space waves through the slot.(b) The launch angle θ is defined relative to the waveguide normal.

Fig. 2 .
Fig. 2. (left) A schematic of the experimental setup.A THz pulse is emitted from the fibercoupled photoconductive antenna, and then focused using a confocal lens setup with teflon lenses.The focused beam excites the TE 1 mode of the PPWG.The receiver performs a raster scan around the focal point.(right) The launch angle is adjusted along the waveguide axis by variation of the plate separation b(x).Focusing is achieved by adjusting the angle such that all emerging rays converge at (x 0 ,z 0 ).

Fig. 3 .
Fig. 3. Simulated and experimental results of the leaky-wave focusing.The calculated (left) and measured (right) field amplitudes are presented for the two waveguides designed for focusing at (a) (x 0 , z 0 ) = (50 mm, 50 mm) at 100 GHz, and (b) (50 mm, 30 mm) at 170 GHz.The rectangular area (white dashed line) in the simulation indicates the area scanned in the measurements.Calculations are the result of finite element simulations using a commercial software package.Note that the color bar is normalized for the peak value inside the measured area, which is indicated by the broken rectangle.Outside the measured area, especially in the vicinity of the waveguide surface, the E-field is larger and the red color is saturated.

Fig. 4 .
Fig. 4. Cross-sectional field profiles through the focal points, for the two design frequencies along (a) the x (horizontal) axis and (b) the z (vertical) axis.The thick curves are the measured data, while the thin curves show the numerical simulations.