Loading [MathJax]/extensions/TeX/boldsymbol.js

⇦Chinese Physics Letters, 2020, Vol. 37, No. 10, Article code 106801⇨ Diffraction-Limited Imaging with a Graphene Metalens Xueyan Li (李雪岩)1,2, Han Lin (林瀚)2*, Yuejin Zhao (赵跃进)1*, and Baohua Jia (贾宝华)2,3* Affiliations 1Beijing Key Laboratory for Precision Optoelectronic Measurement Instrument and Technology, School of Optics and Photonics, Beijing Institute of Technology, Beijing 100081, China 2Centre for Translational Atomaterials, Swinburne University of Technology. P.O. Box 218, Hawthorn, VIC 3122, Australia 3The Australian Research Council (ARC) Industrial Transformation Training Centre in Surface Engineering for Advanced Materials (SEAM), Swinburne University of Technology, P.O. Box 218, Hawthorn, VIC 3122, Australia Received 12 July 2020; accepted 8 August 2020; published online 29 September 2020 Supported by the Scholarship of China Scholarship Council (Grant No. 201706030189), the Industrial Transformation Training Centres Scheme (Grant No. IC180100005), and the National Natural Science Foundation of China (Grant No. 61935001).
^*Corresponding authors. Email: hanlin@swin.edu.au; yjzhao@bit.edu.cn; bjia@swin.edu.au Citation Text: Li X Y, Lin H, Zhao Y J and Jia B H 2020 Chin. Phys. Lett. 37 106801 Abstract Planar graphene metalens has demonstrated advantages of ultrathin thickness (200 nm), high focusing resolution (343 nm) and efficiency (

$>$ 32%) and robust mechanical strength and flexibility. However, diffraction-limited imaging with such a graphene metalens has not been realized, which holds the key to designing practical integrated imaging systems. In this work, the imaging rule for graphene metalenses is first derived and theoretically verified by using the Rayleigh-Sommerfeld diffraction theory to simulate the imaging performance of the 200 nm ultrathin graphene metalens. The imaging rule is applicable to graphene metalenses in different immersion media, including water or oil. Based on the theoretical prediction, high-resolution imaging using the graphene metalens with diffraction-limited resolution (500 nm) is demonstrated for the first time. This work opens the possibility for graphene metalenses to be applied in particle tracking, microfluidic chips and biomedical devices. DOI:10.1088/0256-307X/37/10/106801 PACS:68.65.Pq, 78.67.Wj, 78.68.+m, 78.67.Pt © 2020 Chinese Physics Society Article Text Conventional lenses relying on the refraction principle are bulky and not suitable for on-chip integration.[1–3] In contrast, flat lenses offer a compact design for a myriad of rapid development of nanophotonics and integrated photonic systems, as well as electro-optical applications in microfluidic devices, compact optical microscope and cell phone camera to further reduce the volume of the devices. More importantly, in principle flat lenses are able to achieve high performance imaging by eliminating optical aberrations, astigmatism, and coma.[4,5] A number of ultrathin flat lens concepts[2,3,6–8] have been proposed, such as metamaterials,[9] metasurfaces[10] and planar diffraction lenses.[4,5,11] Among them, graphene metalenses[11–13] have demonstrated attractive properties,[6–10] such as nanometer thickness, high focusing resolution and efficiency, high mechanical strength and flexibility, and fast and low-cost fabrication process, which allow the integration onto various optical components to change or optimize their functionalities, such as conventional optical lenses, fiber tips, on chip optical systems. Furthermore, it has been experimentally demonstrated that graphene ultrathin flat lenses can be applied in harsh environments for different applications,[14,15] including low Earth orbit space environment, strong corrosive chemical environment (PH = 0 and PH = 14), and biochemical environment, without compromising the high focusing performance, for broad applications. Although high focusing performance of graphene metalenses has been demonstrated,[16,17] the diffraction-limited imaging capability, which is essential to transform an optical lens to an imaging system, has not been demonstrated yet. In this Letter, to achieve high resolution diffraction limited imaging, we first analyze and derive the imaging rule for graphene metalenses based on geometric optics. The derived imaging rule is theoretically verified by the Rayleigh–Sommerfeld diffraction theory[11,18] to simulate the imaging performance. More importantly, based on the imaging rule, we experimentally demonstrated diffraction-limited imaging with the graphene metalens. The experimental results agree well with theoretical derivation. The imaging rule can be applied to design imaging systems with graphene metalenses for broad applications in particle tracking,[12,13,19–21] lab-on-a-chip microfluidic devices[22–26] and biophotonic devices.[3,19,24,27] Results and Discussion—Derivation of imaging rule of graphene metalenses based on geometric optics. The schematic of the imaging process using a graphene metalens is shown in Fig. 1(a). The graphene metalens is composed of concentric rings with different radii and a focal length

$f$ . For an object at a distance

$l_{1}$ from the lens, the image can be found at a certain distance

$l_{2}$ from the other side of the lens. Imaging rule describes the relationship between the object distance

$l_{1}$ , image distance

$l_{2}$ and the focal length

$f$ . In the meantime, the magnification ratio can be calculated as

$\alpha ={l_{2} } / {l_{1}}$ . We first derive the analytical imaging rule based on geometric optics. Geometric optics can be considered as the case of Maxwell's equation when the wavelength is considered to be infinitely small.[28] The key to describing the geometric optical phenomenon is to determine the optical path.

Fig. 1. (a) Schematic of imaging using a graphene metalens; (b) the light tracing schematic of a graphene metalens.

$L$ : direction of optical path;

$R_{m}$ : radius of a spherical surface;

$l_{1}$ : object distance;

$l_{2}$ : image distance;

$n_{1}$ and

$n_{2}$ : refractive indices.

A graphene metalens bends light based on diffraction. The ray tracing schematic of a graphene metalens is shown in Fig. 1(b). An object can be considered as composing of point light sources at different spatial positions. The light from the point sources is converged by the lens and constructively interferes in the image plane to form the image. Considering a point source located on the optical axis with a distance

$l_{1}$ from the lens, the light from the point source is focused by the lens on the other side of the lens to form a real image of the source at a distance

$l_{2}$ from the lens. The coordinate of any point in the space is

$(x, y, z)$ . The differential equation of the actual light propagation is[28]

$\frac{{d}}{{d}s}\Big({n\frac{{d}\boldsymbol{r}}{{d}s}} \Big)=\nabla n ,~~ \tag {1}$

$\frac{{d}\boldsymbol{r}}{{d}s}=\frac{{d}x}{{d}s}\boldsymbol{i}+\frac{{d}y}{{d}s}\boldsymbol{j}+\frac{{d}z}{{d}s}\boldsymbol{k},~~ \tag {2}$ where

${\boldsymbol r}$ is the position vector of any point on the light beam, and

${\boldsymbol i}$ ,

${\boldsymbol j}$ ,

${\boldsymbol k}$ are the unit vectors along the

$x$ ,

$y$ ,

$z$ directions, respectively;

${d}s$ is the arc element of the extended light and

$n$ is the refractive index of the medium. In geometric optics, for paraxial region, the Eikonal equation can be regarded as[28]

$({\nabla L})^{2}=n^{2},~~ \tag {3}$ where

$L$ is the optical path. The direction of

$\nabla L$ is along the optical path direction. Take the square root of both sides of the formula, we can obtain

$\nabla L=n\frac{{d}\boldsymbol{r}}{{d}s}.~~ \tag {4}$ The component equation of

$x$ can be expressed as

$\frac{\partial L}{\partial x}=n\frac{{d}x}{{d}s}.~~ \tag {5}$ As shown in Fig. 1(b),

$l_{1} =\overline {OS} =[ {x^{2}+({z-z_{1} })^{2}}]^{1 / 2}$ ,

$l_{2} =\overline {IS} =[ {x^{2}+({z_{2} -z})^{2}}]^{1 / 2}$ . To constructively interfere all the light at the imaging point

$I$ , we can consider all the light coming from a spherical surface (the radius is

$R_{m}$ ) with the same phase as indicated by the gray dashed line. Such a surface is created by the light diffraction from the graphene metalens in this case, which is different from the refraction from conventional lenses. We can obtain the optical path

$L$ as follows:

$\begin{alignat}{1} L={}&n_{1} \overline {OS} +n_{2} \overline {IS}\\ ={}&n_{1}[ {x^{2}+({z-z_{1} })^{2}}]^{1 / 2}+n_{2}[ {x^{2}+({z_{2} -z})^{2}}]^{1 / 2},~~ \tag {6} \end{alignat}$ where

$n_{1}$ and

$n_{2}$ are the refractive indices. In the case of a paraxial approximation,

$z={x^{2}} / {2R_{m}}$ ,

${{d}x} / {{d}s}=[ {1-({{{d}z} / {{d}s}})^{2}}]^{1 / 2}\approx 0$ . The formula is taken into

$\begin{align} n_{1} \frac{x}{l_{1} }&+n_{1} \frac{z}{l_{1} }\frac{x}{R_{m} }-n_{1} \frac{z_{1} }{l_{1} }\frac{x}{R_{m} }+n_{2} \frac{x}{l_{2} }\\ &+n_{2} \frac{z}{l_{2} }\frac{x}{R_{m} }-n_{2} \frac{z_{1} }{l_{2} }\frac{x}{RR_{m} }=0.~~ \tag {7} \end{align}$ According to the paraxial approximation,

$l_{1} =z_{1}$ ,

$l_{2} =z_{2}$ , and finally we obtain the equation

$\frac{n_{2} }{z_{2} }+\frac{n_{1} }{z_{1} }=\frac{n_{2} +n_{1} }{R_{m} }.~~ \tag {8}$ For the geometric optics theory model:

$({R_{m} +\Delta \mathit{\varPhi} })-f={m\lambda } / {2\pi}$ ,

$\Delta \varPhi =x+z={m\lambda } / {2\pi}$ , where

$\varPhi$ is the optical path from lens to the focal plane. The imaging rule of a graphene metalens can thus be defined as

$\frac{n_{1} }{l_{1} }+\frac{n_{2} }{l_{2} }=\frac{n_{2} +n_{1} }{f}.~~ \tag {9}$ This imaging rule can be used for graphene metalenses immersed in different media with different

$n_{1}$ , such as water (

$n_{1}=1.33$ ) and immersion oil (

$n_{1}=1.51$ ), and the image can be in free space, such as

$n_{2}=1$ . According to the imaging rule of the system, we can calculate the magnification ratio as

$\alpha ={l_{2} } / {l_{1}}$ . Theoretical Simulation Based on Rayleigh–Sommerfeld Diffraction Theory. As the imaging rule is derived using geometric optics, without considering the wave nature of light, we further verify the accuracy of the imaging rule using the rigorous diffraction theory without the paraxial approximation. Here the Rayleigh–Sommerfeld diffraction theory is used. The focusing schematic of a graphene metalens is shown in Fig. 2(a). An incident laser beam [

$E_{0} ({r_{1},\theta_{1} })]$ is diffracted by the lens and constructively interferes at the focal point. As the lens can produce both amplitude and phase modulations of the incident laser beam due to the transmission and refractive index changes introduced by the fabrication process,[17] the modulated amplitude and phase of the incident beam by the graphene metalens can be written as [

$E({r_{1},\theta_{1} })]$ . The focal point appears at a far-field position. Here

$r_{1}$ and

$\theta_{1}$ are the polar coordinates in the graphene metalens plane,

$r_{2}$ and

$\theta_{2}$ are the polar coordinates in the focal plane.[16]

Fig. 2. (a) The schematic for calculating the imaging process of a graphene metalens based on the Rayleigh–Sommerfeld diffraction theory, and (b) the schematic of light wave from an off-axis point

$S$ .

According to the diffraction theory, the field distribution in the focal region of the graphene metalens at a distance

$z$ can be calculated by[29]

$\begin{align} E'({r_{2},\theta_{2},z})={}&\frac{1}{2\pi }\int_0^{2\pi } {\int_0^\infty E } ({r_{1},\theta_{1} })\Big({jk-\frac{1}{r'}} \Big)\\ &\cdot\frac{\exp ({jkr'})}{r'^{2}}r_{1} zdr_{1} d\theta ,~~ \tag {10} \end{align}$

$\begin{alignat}{1} r'^{2}={}&({x_{2} -x_{1} })^{2}+({y_{2} -y_{1} })^{2}{+z}^{2}\\ ={}&z^{2}+r_{1}^{2} +r_{2}^{2} -2r_{1} r_{2} {\cos}({\theta_{1} -\theta_{2} }),~~ \tag {11} \end{alignat}$ where

$k={2\pi } / {n\lambda}$ is the wave number,

$\lambda$ is the wavelength of the incident beam in vacuum,

$n$ is the refractive index. When the light from an off-axis point resource [

$S$ , the coordinate is

$({x_{0},y_{0},z_{0} })]$ impinges on the graphene metalens located at a plane of

$z=0$ (for simplicity the schematic of the lens is not drawn), the wave front can be regarded as spherical wave as shown in Fig. 2(b), which can be written as

$E_{0} '({r,\theta })=\frac{E_{0} }{r}\exp (jkr),~~ \tag {12}$

$r^{2}=(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2},~~ \tag {13}$ where

$r$ is the distance between the point resource and the graphene metalens. For a complex object, the incident

$E$ -field can be expressed as the linear combination of the

$E$ -fields from different point sources as

$\begin{alignat}{1} E_{\rm obj} =\sum\nolimits_1^M {\frac{E_{0m} }{r_{m} }} \exp ({jkr}),~~m=1,2,3,\dots, M,~~ \tag {14} \end{alignat}$ where

$m$ is an integer, and

$M$ is the total number of point sources from the object.

$E_{0m}$ and

$r_{m}$ are the

$E$ -field and the distance from the

$m$ th point source. The amplitude and phase of the beam are both modulated by a graphene metalens. The modulated

$E$ -field becomes

$E({r_{1},\theta_{1} })=\sum\limits_{m=1}^M {\frac{E_{0m} }{r_{m} }} \exp ({jkr_{m} })t({r_{1},\theta })\exp [{j\mathit{\varPhi} ({r_{1},\theta_{1} })}],~~ \tag {15}$ where

$t({r_{1},\theta })$ and

$\varPhi ({r_{1},\theta_{1} })$ are the transmission and phase modulations provided by the graphene metalens. Substituting Eq. (15) into Eq. (10), we can obtain the intensity distributions of the image as follows:

$I_{2} ({r_{2},\theta_{2},z})=| {E'({r_{2},\theta_{2},z})} |^{2}.~~ \tag {16}$ Different from the geometric optics derivation, there is no analytical expression of the

$E$ -field in the focal region. We must use software to numerically calculate the maximal interference positions directly. In this way, we are able to calculate image of arbitrary objects at different spatial positions produced by a graphene metalens with a focal length

$f$ and diameter

$D$ . Simulation of the Imaging System Using a Graphene Metalens. Graphene metalenses have been experimentally demonstrated before,[14–17,30–38] which were fabricated by laser reduction of graphene oxide (GO). Upon laser reduction, GO is converted to graphene. During the conversion,[16,17] the GO film shows three continuously tunable physical property variations: the reduction of film thickness, the increase of refractive index and the decrease of transmission. These three property variations provide the required phase and amplitude modulations for designing a graphene metalens. The graphene metalenses can be one-step fabricated by direct laser writing technology,[39,40] which introduces localized graphene/GO regions by a tightly focused laser beam. The graphene metalens can be designed by an accurate method based on the Rayleigh–Sommerfeld diffraction theory,[16] which calculates the intensity distribution in the focal region. The radius of each ring is determined by maximizing the focusing intensity and minimizing the full width at half maximum (FWHM) in the

$x$ –

$y$ plane of the focal spot. In this way, we are able to achieve subwavelength focusing resolution, which is very important in diffraction-limited imaging. The designed graphene metalens is composed of concentric rings of GO (the white regions) and graphene (the dark regions) [Fig. 3(a)],[14,17,34] which has a maximum ring radius of

$D = 100$ µm and a focal length of

$f = 200$ µm. The transmission difference contributing to the amplitude modulation and the refractive index difference contributing to the phase modulations provided by the graphene metalens are shown in Fig. 3(c), where a simplified graphene metalens with only three rings is presented to clearly demonstrate the modulations. Here the design of the graphene metalens is first theoretically confirmed by calculating the intensity distribution in the focal region by assuming a plane-wave incidence at the wavelength of

$\lambda = 500$ nm. The intensity distributions in the

$x$ –

$z$ and

$x$ –

$y$ planes are shown in Figs. 3(d) and 3(e). The FWHM of the focal spot along the

$x$ and

$z$ directions are

$\sim$ 372 nm (0.74

$\lambda$ ) and

$\sim$ 4.093 µm (8.2

$\lambda$ ), respectively, confirming the three-dimensional (3D) high focusing performance of the graphene metalens. The effective numerical aperture (NA) of the graphene metalens can be calculated by

${\rm NA}=\frac{0.61\lambda }{\rm FWHM}$ , which is

$\sim$ 0.82, and it is considered as a high NA (

${\rm NA} > 0.7$ ) graphene metalens. It is worth noting that the conventional definition of the NA based on the radius and the focal length of the lens cannot be directly applied to diffractive lenses due to the different working principles. Although the imaging rule is analytically derived based on geometric optics and the paraxial approximation, we would like to test the accuracy in predicting the imaging performance of high NA lenses, which are able to achieve high spatial resolution and more widely used in different applications.

Fig. 3. (a) Design of a graphene metalens, (b) optical microscopic image of a fabricated graphene metalens according to the design. (c) Transmission and phase modulations provided by the graphene metalens. Simulated intensity distributions in the

$x$ –

$z$ (d) and

$x$ –

$y$ (e) planes. Experimentally measured intensity distributions in the

$x$ –

$z$ (f) and

$x$ –

$y$ planes (g). Cross-sectional intensity distribution along the

$z$ direction (h) and

$y$ direction (i), which are marked by the white dashed lines. Scale bars in (a) and (b) are 20 µm.

The designed graphene metalens was fabricated using direct laser writing with a femtosecond laser (Coherent

$^{\circledR}$ , Libra,

$\lambda = 800$ nm, pulse width 100 fs, repetition rate 10 kHz, the setup information can be found from Fig. S1 in the Supplementary Material).[41] The optical microscopic image of the fabricated graphene metalens is shown in Fig. 3(b). The focusing performance of the graphene metalens is measured by a characterization setup with a collimated laser beam at 500 nm as the incident light (the measurement setup is shown in Fig. S2 in the Supplementary Material). The measured intensity distributions in the

$x$ –

$z$ and

$x$ –

$y$ planes are shown in Figs. 3(f) and 3(g). The FWHMs of the focal spot along the

$x$ and

$z$ directions are

$\sim$ 401.46 nm (0.8

$\lambda$ ) and

$\sim$ 5.6568 µm (11.3

$\lambda$ ), respectively. The good agreement between the simulation and experimental results is shown in the intensity cross-sectional plots along the

$z$ and

$y$ directions [Figs. 3(h) and 3(i)]. To theoretically verify the imaging performance of the graphene lens, we used a point light source

$S({x_{0},y_{0},z_{0} })$ as an object [Fig. 2(b)] in the simulation model, which has an incident

$E$ -field expressed by Eq. (12). As only one point source is considered here,

$m$ in Eq. (14) is 1 with

$r=z^{2}+({x-r_{1} \cos \theta })^{2}+({y-r_{1} \sin \theta })^{2}$ . The intensity distribution is then calculated using Eqs. (15) and (16). Here we first consider the immersion media of the lens as air (

$n_{1}=n_{2} = 1$ ). Thus, the imaging rule of a diffractive lens working in air becomes

$\frac{1}{l_{1} }+\frac{1}{l_{2} }=\frac{2}{f}.~~ \tag {17}$ This is different from the imaging rule of a refractive lens, which is

$l_{1}^{-1}+l_{2}^{-1} =f^{-1}$ . To verify the imaging rule in Eq. (17), we first place the point light source on the optical axis (

$r_{1}=0$ ), and change the distance

$l_{1}$ between the point light source (the object) and the lens plane from 110 µm to 410 µm. The corresponding curve from Eq. (17) is shown in Fig. 4 (red solid line). The resulted curve calculated using the Rayleigh–Sommerfeld diffraction theory is illustrated by the blue curve in Fig. 4 for comparison, showing a good agreement with the derived imaging rule, especially in the paraxial region where the distance between the object and the lens is much larger than the diameter of the lens. We note that there is a significant difference between the two curves when the object is very close to the lens (

$l_{1} < 0.8f$ ), in which the paraxial approximation is not valid anymore. In addition, although the lens is designed as a high NA one without using the paraxial approximation, the imaging rule is accurate (only 1% deviation from the diffraction theory) when the object distance is larger than the focal length (

$l_{1}>f$ ). Therefore, we can confirm that the imaging rule is applicable to graphene metalenses, even for high NA cases. According to the imaging rule predicted in Eq. (17), if the object is placed in the focal plane of the lens (

$l_{1}=f$ ), the image will be found in the other focal plane (

$l_{2}=f$ ). In this case, the magnification ratio is

$\alpha =-1$ .

Fig. 4. Relationship between the image and object distances from the derived imaging rule (red solid line) and diffraction theory (blue solid line).

To verify the accuracy of the magnification ratio, we further place the object at different positions in the focal plane. Thus, in this case,

$x_{0}$ and

$y_{0}$ are varied to calculate the corresponding

$x_{i}$ and

$y_{i}$ . The intensity distributions are shown in Figs. 5(a)–5(d). There are two characteristics that can be observed from the simulation: (1) the

$x_{i} =-x_{0}$ and

$y_{i} =-y_{0}$ relation, confirming that the magnification of

$\alpha =-1$ is consistent with the imaging rule, i.e.,

$x_{i} =\alpha x_{0}$ ,

$y_{i} =\alpha y_{0}$ ; (2) slight distortion of the imaged points due to the spatial shift variation, which can be further removed by including sine condition in the design.[29]

Fig. 5. Result from the simulation of the focusing intensity distributions with the target at different positions. The image positions (

$x_{i}$ ,

$y_{i}$ ) from different object positions (

$x_{0}$ ,

$y_{0}$ ): (a)

$x_{i} =-x_{0} =0$ ,

$y_{i} =-y_{0} =0$ ; (b)

$x_{i} =-x_{0} =-5\,µ$ m,

$y_{i} =-y_{0} =-5\,µ$ m; (c)

$x_{i} =-x_{0} =10\,µ$ m,

$y_{i} =-y_{0} =0$ ; (d)

$x_{i} =-x_{0} =-10\,µ$ m,

$y_{i} =-y_{0} =10\,µ$ m.

Imaging Experiment with the Graphene Metalens. Finally, we further experimentally verify the theoretical results using the fabricated graphene metalens to image an object. The optical setup is shown in Fig. 6(a), in which the image from the graphene metalens is magnified by

$100\times$ with a 4

$f$ microscopic imaging system and collected by a CCD camera. We fabricated a letter F with a size of 5 µm on a gold film [Fig. 6(b)] using laser writing as the object and place it at the center of the optical axis in one of the focal planes of the lens (

$l_{1 }=f = 200$ µm). As a result, an inverted image can be found at the center of the CCD field of view in the other focal plane (

$l_{2 }=f = 200$ µm) [Fig. 6(c)], validating the derived imaging rule predicted by Eq. (17). By comparing the object with the invert image, we also confirm that the magnification ratio is

$\alpha =-1$ . Here we intentionally rotate the image by 180

$^{\circ}$ for easy comparison. In this way, the imaging rule of a graphene metalens is experimentally confirmed using a high NA graphene metalens. To further confirm that diffraction-limited imaging capability of the graphene metalens, we used a USAF standard board as the object [Fig. 6(d)], and the corresponding image is shown in Fig. 6(e). As can be seen, although the sine condition has not been implemented in the current design, within our current field of view, there is no significant distortion of the image. In order to quantify the resolving capability of the graphene metalens, the cross-sectional intensity distributions of a set of line patterns in the object and the image are plotted in Figs. 6(f) and 6(g). The interline spacing between the lines is 1 µm, and the FWHM of the linewidth is around 500 nm. It is shown that all the lines are well separated and imaged, confirming the diffraction-limited imaging performance.

Fig. 6. (a) Experimental setup to collect the image from a graphene metalens; (b) optical microscopic image of the object letter F; (c) optical microscopic image of the imaged letter F from the graphene lens. Scale bars in (b) and (c) are 2 µm. (d) Optical microscopic image of the object USAF standard board; (e) optical microscopic image of the imaged USAF standard board from the graphene lens. [(f),(g)] Cross-sectional plots of the object and image along the

$x$ and

$y$ directions, marked by the blue and red dashed lines.

In summary, we have derived the imaging process and revealed the imaging rule for a graphene metalens based on geometric optics for the first time. The prediction has been verified by rigorous diffraction theory. Based on the theoretical principle, we experimentally demonstrate diffraction-limited imaging performance of a graphene metalens. Furthermore, despite the fact that imaging rule is derived based on geometric optics and the paraxial approximation, it can accurately predict the image performance of a high NA graphene metalens in the immersion medium of air. According to the theoretical prediction, this imaging rule should be equally applicable to other immersion media. Thus, the research finding in this work can be applied to other diffractive flat lens, providing guidance in designing imaging system and optimizing imaging quality. Thus it can find broad applications in particle tracing, nanophotonics, integrated optics and biomedical applications. Diffraction-limited imaging can be achieved by other dielectric materials, such as dielectric metasurfaces.[42] The working principle of dielectric metasurfaces requires the thickness

$d$ to be at least the effective wavelength of the incident light

$\lambda$ , which can be expressed as

$d=\lambda /n$ , where

$n$ is the refractive index of the material. In comparison, graphene family materials (including graphene, graphene oxide and reduced graphene oxide) are able to simultaneously provide phase and amplitude modulations. Therefore, the thickness of graphene metalenses can be further reduced. In this study a 200-nm-thick graphene metalens is presented. In addition, the amplitude and phase modulations can be easily introduced by laser reduction of graphene oxide. Thus, graphene metalenses can be fabricated in one-step laser nanofabrication as demonstrated in this study. In contrast, metasurface lenses, including those made by either metal or dielectric materials, need to be fabricated by sophisticated vacuum assisted nanofabrication techniques with multiple steps, such as patterning, etching and washing. Furthermore, graphene metalenses are based on diffraction principle, which allows to achieve high NA focusing with a small size (a few microns)[17] that is suitable for chip integration. In contrast, metasurface lenses (both the metallic and dielectric ones) require a large size to achieve high NA focusing because the size of each nanoelement cannot be too small, which is in the range of a few hundreds of nanometers. It requires a collection of thousands of those nanoelements to achieve a high NA metasurface lens, which results in a large size of a few hundreds of microns to a few millimeters. Method—Simulation. The imaging and focusing simulations are carried out using a homemade Matlab program. Experiment. The graphene oxide (GO) film is prepared using a layer-by-layer self-assembly method[36,37] with a commercial coating equipment (Innofocus, GM-AC1). The graphene lens is fabricated by a commercial laser 3D nanoprinting setup (Special Edition, Innofocus Nanoprint

$^{\rm 3D}$ , design shown in the Supplementary Material). The imaging and focusing performance are characterized by a homemade imaging setup (Fig. S2 in the Supplementary Material) with a commercial supercontinuum laser source (NKT Photonics, SuperK Flanium) and a CCD camera (Watec 902H). The results are analyzed using a homemade Matlab program. References Optical imaging applied to microelectronic chip-to-chip interconnectionsPerformance of a negative index of refraction lensOn-chip optical components and microfluidic systemsRefractive and diffractive properties of planar micro-optical elementsAberrations and adaptive optics in super-resolution microscopySubwavelength-thick lenses with high numerical apertures and large efficiency based on high-contrast transmitarraysFlat optics with designer metasurfacesFlat metasurfaces to focus electromagnetic waves in reflection geometryFast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula3D Particle Trajectories Observed by Orthogonal Tracking Microscopy3D Particle Tracking on a Two-Photon MicroscopeGraphene-Based Multilayered Metamaterials with Phototunable Architecture for on-Chip Photonic DevicesResilient Graphene Ultrathin Flat Lens in Aerospace, Chemical, and Biological Harsh EnvironmentsAn accurate design of graphene oxide ultrathin flat lens based on Rayleigh-Sommerfeld theoryHighly efficient and ultra-broadband graphene oxide ultrathin lenses with three-dimensional subwavelength focusingMethods in EnzymologyGraphene metalens for particle nanotrackingDiffractive photonic applications mediated by laser reduced graphene oxidesPaper-Based Microfluidic Devices: Emerging Themes and ApplicationsRecent Developments in Paper-Based Microfluidic DevicesOn-chip transformation optics for multimode waveguide bendsAchromatic Fresnel optics for wideband extreme-ultraviolet and X-ray imagingPolymer microfluidic devicesFeature point tracking and trajectory analysis for video imaging in cell biologyFormal Quantum Theory of Light RaysA 90-nm-thick graphene metamaterial for strong and extremely broadband absorption of unpolarized lightInvited Article: Enhanced four-wave mixing in waveguides integrated with graphene oxideLaser trimming of graphene oxide for functional photonic applicationsGraphene-Based Ultrathin Flat LensesIn Situ Third-Order Non-linear Responses During Laser Reduction of Graphene Oxide Thin Films Towards On-Chip Non-linear Photonic DevicesBroadband Subwavelength Imaging Using a Tunable Graphene-LensGraphene-Based Electronic Spin LensesGraphene oxide as a chemically tunable platform for optical applicationsThe Focusing of Electron Flow and a Veselago Lens in Graphene p-n JunctionsAdaptive optics enhanced direct laser writing of high refractive index gyroid photonic crystals in chalcogenide glassTwo-Photon Polymerization for Three-Dimensional Photonic Devices in Polymers and NanocompositesFabrication of 3D microoptical lenses in photosensitive glass using femtosecond laser micromachiningMetalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging

[1]	Kostuk R K, Goodman J W and Hesselink L 1985 Appl. Opt. 24 2851
[2]	Parazzoli C G, Greegor R B, Nielsen J A, Thompson M A et al. 2004 Appl. Phys. Lett. 84 3232
[3]	Kou Q, Yesilyurt I, Studer V, Belotti M, Cambril E and Chen Y 2004 Microelectron. Eng. 73 876
[4]	Rossi M, Kunz R E and Herzig H P 1995 Appl. Opt. 34 5996
[5]	Booth M, Andrade D, Burke D, Patton B and Zurauskas M 2015 Microscopy 64 251
[6]	Arbabi A, Horie Y, Ball A J, Bagheri M and Faraon A 2015 Nat. Commun. 6 7069
[7]	O'Shea D C, Suleski T J, Kathman A D and Prather D W 2004 Diffractive Optics: Design, Fabrication and Test (London: SPIE Press)
[8]	Turunen J and Wyrowski F 1998 Diffractive Optics for Industrial and Commercial Applications (New York: Wiley-VCH)
[9]	Yu N and Capasso F 2014 Nat. Mater. 13 139
[10]	Li X, Xiao S, Cai B, He Q, Cui T J and Zhou L 2012 Opt. Lett. 37 4940
[11]	Shen F and Wang A 2006 Appl. Opt. 45 1102
[12]	McMahon M D, Berglund A J, Carmichael P, McClelland J J and Liddle J A 2009 ACS Nano 3 609
[13]	Ragan T, Huang H and So P 2006 J. Fluoresc. 16 325
[14]	Yang Y, Lin H, Zhang B Y, Zhang Y, Zheng X, Yu A et al. 2019 ACS Photon. 6 1033
[15]	Cao G, Lin H, Fraser S, Zheng X, Del Rosal B, Gan Z et al. 2019 ACS Appl. Mater. & Interfaces 11 20298
[16]	Cao G, Gan X, Lin H and Jia B 2018 Opto-Electron. Adv. 1 180012
[17]	Zheng X, Jia B, Lin H, Qiu L, Li D and Gu M 2015 Nat. Commun. 6 8433
[18]	Hecht E 2002 Optics (San Francisco: Addison Wesley)
[19]	Meijering E, Dzyubachyk O and Smal I 2012 Methods Enzymology 504 183
[20]	Li X, Wei S, Cao G, Lin H, Zhao Y and Jia B 2020 Photon. Res. 8 1316
[21]	Wang S, Ouyang X, Feng Z, Cao Y, Gu M and Li X 2018 Opto-Electron. Adv. 1 170002
[22]	Yang Y, Noviana E, Nguyen M P, Geiss B J, Dandy D S and Henry C S 2017 Anal. Chem. 89 71
[23]	Cate D M, Adkins J A, Mettakoonpitak J and Henry C S 2015 Anal. Chem. 87 19
[24]	Gabrielli L H, Liu D, Johnson S G and Lipson M 2012 Nat. Commun. 3 1217
[25]	Wang Y, Yun W and Jacobsen C 2003 Nature 424 50
[26]	Becker H and Locascio L E 2002 Talanta 56 267
[27]	Sbalzarini I F and Koumoutsakos P 2005 J. Struct. Biol. 151 182
[28]	Gloge D and Marcuse D 1969 J. Opt. Soc. Am. 59 1629
[29]	Gu M 2000 Advanced Optical Imaging Theory (Berlin: Springer)
[30]	Lin H, Sturmberg B C, Lin K T, Yang Y, Zheng X, Chong T K et al. 2019 Nat. Photon. 13 270
[31]	Yang Y, Wu J, Xu X, Liang Y, Chu S T, Little B E et al. 2018 APL Photon. 3 120803
[32]	Zheng X, Lin H, Yang T and Jia B 2017 J. Phys. D 50 074003
[33]	Kong X T, Khan A A, Kidambi P R, Deng S, Yetisen A K, Dlubak B et al. 2015 ACS Photon. 2 200
[34]	Zheng X, Jia B, Chen X and Gu M 2014 Adv. Mater. 26 2699
[35]	Li P and Taubner T 2012 ACS Nano 6 10107
[36]	Moghaddam A G and Zareyan M 2010 Phys. Rev. Lett. 105 146803
[37]	Loh K P, Bao Q, Eda G and Chhowalla M 2010 Nat. Chem. 2 1015
[38]	Cheianov V V, Fal'ko V and Altshuler B 2007 Science 315 1252
[39]	Cumming B P, Turner M D, Schröder-Turk G E, Debbarma S, Luther-Davies B and Gu M 2014 Opt. Express 22 689
[40]	Jia B, Li J and Gu M 2007 Aust. J. Chem. 60 484
[41]	Cheng Y, Tsai H L, Sugioka K and Midorikawa K 2006 Appl. Phys. A 85 11
[42]	Khorasaninejad M, Chen W T, Devlin R C, Oh J, Zhu A Y and Capasso F 2016 Science 352 1190