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An Accurate Design of Graphene Oxide Ultrathin Flat Lens Based on Rayleigh-Sommerfeld Theory

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An Accurate Design of Graphene Oxide Ultrathin Flat Lens Based on Rayleigh-Sommerfeld Theory Opto-Electronic Original Article Advances 2018, Vol. 1, No. 7 DOI: 10.29026/oea.2018.180012 An accurate design of graphene oxide ultrathin flat lens based on Rayleigh‐Sommerfeld theory Guiyuan Cao, Xiaosong Gan, Han Lin and Baohua Jia* Graphene oxide (GO) ultrathin flat lenses have provided a new and viable solution to achieve high resolution, high effi- ciency, ultra-light weight, integratable and flexible optical systems. Current GO lenses are designed based on the Fresnel diffraction model, which uses a paraxial approximation for low numerical aperture (NA) focusing process. Herein we de- velop a lens design method based on the Rayleigh-Sommerfeld (RS) diffraction theory that is able to unambiguously determine the radii of each ring without the optimization process for the first time. More importantly, the RS design me- thod is able to accurately design GO lenses with arbitrary NA and focal length. Our design is experimentally confirmed by fabricating high NA GO lenses with both short and long focal lengths. Compared with the conventional Fresnel design methods, the differences in ring positions and the resulted focal length are up to 13.9% and 9.1%, respectively. Our me- thod can be further applied to design high performance flat lenses of arbitrary materials given the NA and focal length requirements, including metasurfaces or other two-dimensional materials. Keywords: ultrathin flat lens; graphene oxide; Rayleigh-Sommerfeld diffraction; Fresnel diffraction Cao G Y, Gan X S, Lin H, Jia B H. An accurate design of graphene oxide ultrathin flat lens based on Rayleigh‐Sommerfeld theory. Op‐ to‐Electronic Advances 1, 180012 (2018). ous optical components to change or optimize their func- Introduction tionalities, such as conventional optical lenses, fiber tips, Flat lenses made of ultrathin materials have the advan- and on chip optical systems. tages of astigmatism and coma aberrations free, which are The current GO lenses are designed using the Fresnel otherwise common problems for conventional curved diffraction model16–21, which is only applicable for lenses surface lenses, especially when the numerical aperture with a low NA satisfying the paraxial approximation. For (NA) is high1. In addition, flat lenses offer a compact de- a high NA lens, it is not able to accurately predict the fo- sign for a myriad of rapid development of nanophotonics cusing performance. While the vigorous finite difference and integrated photonic systems, as well as electro-optical time domain (FDTD) method8,22–24 provides accurate applications, such as solar cells and fiber communication simulation on the focusing process, the required amount systems. A number of ultrathin flat lens concepts2–6 have of computational memory and time increase significantly been proposed, such as metamaterials7, metasurfaces8 and as geometry progression. One simulation could take days. planar diffraction lenses9–11. Recently, planar diffraction As a result, the FDTD method is limited to lenses with a lenses have attracted much attention due to the possibility focal length less than 10 μm. Therefore, to promote prac- to achieve wide operational bandwidth12, chromatic ab- tical GO lens applications, it is necessary to develop a erration compensation13 and high focusing performance. theoretical modeling method that is able to accurately Among them, graphene oxide (GO) ultrathin flat calculate the focusing process of GO lenses with arbitrary lenses11,14,15 have demonstrated attractive properties, such NA and focal lengths with high speed and efficiency and as nanometer thickness, high focusing resolution and low computational cost. Such a model can be equally ap- efficiency, high mechanical strength and flexibility, and plicable to other ultrathin lens with high NA and large fast and low-cost fabrication process. In addition, the GO focal length. ultrathin flat lens can potentially be integrated onto vari- In this paper, we develop an accurate method based on Centre for Micro-Photonics, Faculty of Engineering, Science and Technology, Swinburne University of Technology, John Street, Hawthorn, VIC 3122, Australia * Correspondence: B H Jia, E-mail: [email protected] Received 1 July 2018; accepted 26 July 2018; accepted article preview online 21 August 2018 180012‐1 © 2018 Institute of Optics and Electronics, Chinese Academy of Sciences. All rights reserved. Opto-Electronic Advances DOI: 10.29026/oea.2018.180012 the Rayleigh-Sommerfeld (RS) diffraction theory without 112π Urθz22(, 2 ,) Urθ 11 (, 1 )(i k ) the paraxial constrain to design GO ultrathin flat lens. 2π 00 ε The ring radii of a desired GO lens is decided directly exp( ikε ) from the RS diffraction theory without the optimization zrdd r θ , (1) process, such as iterative25, local optimization algorithm26 ε 11 1 and global-search-optimization algorithm27, that is nor- 222 εz r12 r2cos() rr 1212 θ θ, mally used in other design methods for high NA lenses. where k=2π/λ is the wave vector, λ is the wavelength of Most importantly, the RS design method is able to design the incident beam in vacuum; Urθ11(, 1 ) is the E-field GO lenses with arbitrary NA, size and focal length. To immediately behind the GO ultrathin flat lens. When the verify our RS design method, we designed two GO lenses incident wave Urθ11(, 1 ) impinges on the GO lens, the with focal lengths of 3.1 μm (NA=0.82) and 9.1 μm beam is partly absorbed and diffracted by the reduced (NA=0.71), respectively. The designs were verified both graphene oxide (rGO) and GO zones. Urθ11(, 1 ) can be theoretically by FDTD simulation and experimentally by expressed as28: laser fabrication and point spread function characteriza- Urθ(, ) Urθtrθ (, )(, )exp[i kΦrθ (, )], (2) tion. 111 111 11 11 where tr(,11 θ ) is the transmission coefficient, Φr(,11 θ ) Theoretical model is the phase modulation provided by the GO film and air, The schematic of the GO lens focusing is shown in Fig. as shown in Fig. 1(b). The GO ultrathin flat lens is able to 1(a). The incident light wave (Urθ11(, 1 )) is propagating modulate both phase and amplitude simultaneously from along the positive z direction. The amplitude and phase of the conversion of GO to rGO. During the conversion31,32, the incident light that are modulated by the GO lens be- the GO film shows three continuously tunable physical come Urθ11(, 1 ). The GO lens plane is the diffraction property variations: the reduction of film thickness, the 221/2 plane, rxy111(), r1 and θ1 are the polar coor- increase of refractive index and the decrease of transmis- dinates in the GO lens plane. The r2 – θ2 plane is the sion. These three property variations provide the required observation plane, which is usually the focal plane. phase and amplitude modulation in designing a GO lens. 221/2 rxy222(), r2 and θ2 are the polar coordinates in The GO lenses can be one-step fabricated by the direct the focal plane. z is the distance between the diffraction laser writing technology33, which introduces localized plane and the observation plane. rGO region by a tightly focused laser beam. According to the RS diffraction theory28–30, the field at To design the GO lens with the targeted focal length f an arbitrary observation plane at a distance z can be writ- and diameter D based on the RS diffraction theory, we ten as: consider the intensity distribution on the z axis, namely y1 y2 (x1, y1) n a x1 r1 x2 r (x2, y2) r 2 U2(r2, θ2) U1(r1, θ1) U1(r1, θ1) θ2 z θ1 GO zone rGO zone b c Min Max Min Max a1 l a2 a3 0 -50 ) Transmission Phase modulation ) (nm) 1 t 1 θ profile profile θ ∆ -100 , , 1 1 r -6 -4 -2 0 2 4 6 r ( ( T Φ Lateral (μm) Fig. 1 | The design of the GO ultrathin flat lens. (a) Diffraction of GO ultrathin flat lens in polar and Cartesian coordinate systems. (b) Transmission and phase modulations provided by the GO lens. (c) Topographic profile of the GO ultrathin flat lens measured by an atomic force microscope. 180012‐2 © 2018 Institute of Optics and Electronics, Chinese Academy of Sciences. All rights reserved. Opto-Electronic Advances DOI: 10.29026/oea.2018.180012 r2=0, z=f. And based on the Euler’s equation, the expres- zone. sion of intensity distribution on the z axis can be simpli- fied to (Supplementary Section 1): Results and discussion 2πf Ir() ( )(22 ω υ 2 ), (3) We first compare the lens designs using the RS and the 1 λ Fresnel methods. The targeted focal length is 3.1 μm 22 (Lens 1) with a lens thickness of 0.2 μm. The lenses are cos(kf r1 ) composed of three concentric rings with radii of a1, a2 here ωUr 11() rdr 1 1 , 0 22 fr1 and a3. Here we assume the incident beam is a plane wave. 22 The resulted plots of radii of the rings are shown in Fig. 2. sin(kf r ) 1 There is a significant difference (0.215 μm or 13.9%) be- υUr0 11() 22 rdr 1 1 . fr 1 tween the radii of the two models at the first ring, and the The maximal destructive interference positions on the differences decrease to 0.175 μm (6.6%) for the second intensity distribution I(r1) predict the ring radii of the GO ring, and 0.157 μm (4.4%) for the third ring. lens with a focal length f. In the meantime, the diameter is decided by the number of rings. In this way, the lenses RS design can be designed according to different incident fields 3 Fresnel design (Urθ11(, 1 )). 2 On the other hand, for a situation that the observation 4.4% 1 6.6% difference point is not far away from the optical axis, it is assumed m) μ 13.9% difference ( 0 22 2 difference |(rrz12 )/ | 1, which is the paraxial approximation, R Eq.
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