JW2A.76.pdf
CLEO 2018 ยฉ OSA 2018
Efficient Design of Diffractive THz Lenses for Aberration Rectified Focusing via Modified Binary Search Algorithm Sourangsu Banerji, Ashish Chanana, Hugo Condori, Ajay Nahata and Berardi Sensale-Rodriguez Department of Electrical and Computer Engineering, The University of Utah Author e-mail address:
[email protected]
Abstract: We experimentally demonstrate thin, error tolerant, and aberration corrected 3D printed diffractive 1D and 2D THz lenses for focusing THz beams with NA > 0.25 and an average efficiency greater than 75% (for all cases). Our work has relevance to THz lens design, and in general, THz imaging systems.ยฉ2018TheAut h o r ( s ) OCIS codes: (050.1965) Diffractive lenses; (050.1970) Diffractive optics; (110.6795) Terahertz imaging
1. Introduction A wide variety of THz systems rely on focusing of THz radiation onto a single spatial line or a point, thus maximizing the intensity within a special region of interest. However, the problems associated with conventional THz lenses are multifold: they are bulky, thick, expensive and suffer from strong wavefront (geometric as well as chromatic) aberrations which hinders the performance of THz systems [1]. Metamaterials and metasurfaces can provide effective solutions; however, such alternatives are usually polarization dependent, hard to fabricate, and difficult to simulate; in contrast to an all-dielectric based approach [2-3]. In lieu of this, in this work we seek to experimentally demonstrate that the basic principles of scalar diffractive optics, under proper design considerations, can be coupled with a computer-aided optimization based search algorithm, i.e. modified direct binary search technique, to design aberration rectified diffractive THz lenses for both narrowband and broadband focusing.
Fig. 1. (a) Focusing of THz beams with focal length f and its axial focal length shift. Fabricated: (b) 1D lens for narrowband (0.3 THz) focusing and (c) 2D lens for broadband (0.3 to 0.6 THz) focusing. 2. Methods Depicted in Fig. 1(a) is a schematic of a spherical THz lens with its corresponding shift in focus, alongside which is our aberration rectified planar THz lens that consists of multi-height dielectric โpixelsโ. Design and optimization: For aberration rectified focusing, we implemented a computer-aided optimization based search algorithm i.e. modified direct binary search technique, to optimize the distribution of pixel heights. Unlike a perturbation-based iterative method like the direct binary search algorithm employed in previous works [4], our modified direct binary search employs a gradient descent to the search for the optimal solution in the design space, thus intuitively reducing the computation time as well as the search-space complexity. Our results evidence that our algorithm can reduce by >10X the number of iterations required so to achieve a desired convergence. The target function was defined as a diffraction-limited Gaussian with full-width-at-half-maximum (FWHM) determined by the expression: ๐ค๐ค = ๐๐/2๐๐๐๐, where NA denotes the numerical aperture, which in turn is given by ๐๐๐๐ = sin(tanโ1 (2๐๐/๐ฟ๐ฟ)) in where f is the desired focal length, L = Nรฮ is the total length of the lens, and N denotes the total number of pixels. For narrowband focusing at 0.3 THz, N = 260 (1D) and N = 1600 (2D), focal length f = 50 mm, and NA = 0.2516. In order to focus a continuous broadband THz beam across the 0.3 THz to 0.6 THz frequency range, the frequency-sampling was kept to 0.05 THz in the design; a proper FoM was defined so to maximize the lens efficiency across the desired frequency window.
JW2A.76.pdf
CLEO 2018 ยฉ OSA 2018
Fabrication and test: The size of each lens is 26 mm. The aberration corrected 1D and 2D diffractive THz lenses were 3D printed (Cura Ultimaker 2+) with PLA (Poly(lactic) acid) as shown in Figs. 1(b-c), and are composed of linear pixels of width ฮ = 100 ฮผm (1D) and square pixels with a side length ฮ = 650 ฮผm (2D), respectively. The maximum pixel height for these planar structures was fixed at 1400 ฮผm with a minimum height of 100 ฮผm and a layer resolution of 100 ฮผm, which was limited by the resolution of the 3D printer. An important note here is that for 1D lenses, the width is limited by the spool diameter of the 3D printer in contrast to 2D lenses, where the side length dimension is limited by the computational complexity of the design. To experimentally demonstrate aberrationrectified focusing, a THz Transmitter (VDIE synthesizer) was used to generate the THz beam which was spatially collimated using a commercial convex lens. The collimated beam was later passed through a square metal aperture of the same size as our 3D printed lenses to guarantee that the experimental conditions were similar to the theoretical assumptions made in our semi-analytic design algorithm. The lenses were placed after the metal aperture. The transmission spectra were recorded with a THz imager (Terasense) fixed to a motorized one-axis stage with 1 ฮผm steps along the z-axis. The transmitted spectra were collected at locations corresponding to 20 mm, 30 mm, 40 mm, 50 mm and finally at 60 mm. The final results were derived after performing a background normalization with the THz imager. Further details about these methods can be found in [5]. 3. Results and Discussion The semi-analytic and measured efficiency of the narrowband designs was >85% for the 1D lenses (see Fig. 2(a)) and >90% for its 2D counterpart. For the broadband case, the semi-analytic and measured efficiency as a function of frequency for 2D diffractive lenses are plotted in Fig. 2(b). Similar results were observed for 1D cases. Efficiency is defined as the ratio of the power within the zero-order lobe to the total energy contained within the lens aperture area. As expected, the overall efficiency is higher for 2D lenses because of the design-space containing a larger number of pixels thus more degrees of freedom. The observed differences between the semi-analytic and the measured efficiencies as seen in the plots are mainly due to 3D printing errors in individual pixel heights. In order to improve the agreement between practical results and simulations: (i) a different FoM could be defined so to improve the sensitivity of the results to geometrical variations, or (ii) the quality of the overall printing process, i.e. resolution, could be improved. We numerically analyzed this scenario by adopting a standard deviation based model to add random errors to the pixel height distribution. The results are shown in Fig. 2(c) for the particular case of a 1D THz diffractive lens at 0.3 THz, which indicate that these lenses are error tolerant to height errors of up to ~250 ฮผm. The predicted efficiency decreases with increasing the fabrication errors for both the 1D and 2D lens cases, irrespective of narrow or broadband focusing. Our work paves the way for the design and optimization of future efficient THz diffractive optical components that are error tolerant and which could provide any tailored design response.
Fig. 2. (a) Semi-analytic and measured PSFs for a 1D lens with a focal length, f =50 mm under narrowband focusing (0.3 THz). (b) Average efficiency as a function of frequency for a 2D broadband lens (0.3 to 0.6 THz). (b) Effect of fabrication error on the efficiency of a 1D THz lens. 4. References [1] R.A.Lewis, Terahertz physics. Cambridge University Press, 2012. [2] M. Sypek, "Aberrations of the large aperture attenuating THz lenses." In SPIE OPTO. International Society for Optics and Photonics, 2012. [3] J. Neu, "Metamaterial-based gradient index lens with strong focusing in the THz frequency range." Optics express 18 (2010): 27748-27757. [4] P. Wang, "Chromatic-aberration-corrected diffractive lenses for ultra-broadband focusing." Scientific reports 6 (2016). [5] A. Chanana, โColour selective control of terahertz radiation using two-dimensional hybrid organic inorganic lead-trihalide perovskites.โ Nature Communications 8 (2017): 1328.
