Flat lenses made of ultrathin materials have the advantages of astigmatism and coma aberrations free, which are otherwise common problems for conventional curved surface lenses, especially when the numerical aperture (NA) is high^{1}. In addition, flat lenses offer a compact design for a myriad of rapid development of nanophotonics and integrated photonic systems, as well as electrooptical applications, such as solar cells and fiber communication systems. A number of ultrathin flat lens concepts^{26} have been proposed, such as metamaterials^{7}, metasurfaces^{8} and planar diffraction lenses^{911}. Recently, planar diffraction lenses have attracted much attention due to the possibility to achieve wide operational bandwidth^{12}, chromatic aberration compensation^{13} and high focusing performance. Among them, graphene oxide (GO) ultrathin flat lenses^{11, 14, 15} have demonstrated attractive properties, such as nanometer thickness, high focusing resolution and efficiency, high mechanical strength and flexibility, and fast and lowcost fabrication process. In addition, the GO ultrathin flat lens can potentially be integrated onto various optical components to change or optimize their functionalities, such as conventional optical lenses, fiber tips, and on chip optical systems.
The current GO lenses are designed using the Fresnel diffraction model^{1621}, which is only applicable for lenses with a low NA satisfying the paraxial approximation. For a high NA lens, it is not able to accurately predict the focusing performance. While the vigorous finite difference time domain (FDTD) method^{8, 2224} provides accurate simulation on the focusing process, the required amount of computational memory and time increase significantly as geometry progression. One simulation could take days. As a result, the FDTD method is limited to lenses with a focal length less than 10 µm. Therefore, to promote practical GO lens applications, it is necessary to develop a theoretical modeling method that is able to accurately calculate the focusing process of GO lenses with arbitrary NA and focal lengths with high speed and efficiency and low computational cost. Such a model can be equally applicable to other ultrathin lens with high NA and large focal length.
In this paper, we develop an accurate method based on the RayleighSommerfeld (RS) diffraction theory without the paraxial constrain to design GO ultrathin flat lens. The ring radii of a desired GO lens is decided directly from the RS diffraction theory without the optimization process, such as iterative^{25}, local optimization algorithm^{26} and globalsearchoptimization algorithm^{27}, that is normally used in other design methods for high NA lenses. Most importantly, the RS design method is able to design GO lenses with arbitrary NA, size and focal length. To verify our RS design method, we designed two GO lenses with focal lengths of 3.1 μm (NA=0.82) and 9.1 μm (NA=0.71), respectively. The designs were verified both theoretically by FDTD simulation and experimentally by laser fabrication and point spread function characterization.
Theoretical modelThe schematic of the GO lens focusing is shown in Fig. 1(a). The incident light wave (
According to the RS diffraction theory^{2830}, the field at an arbitrary observation plane at a distance z can be written as:
$ \begin{array}{l} {U_2}({r_2}, \;{\theta _2}, z) = \frac{1}{{2{\rm{ \mathsf{ π} }}}}\int_0^{{\rm{2 \mathsf{ π} }}} {\int_0^\infty {{{U'}_1}({r_1}, \;{\theta _1})} } (  {\rm{i}}k  \frac{1}{{\sqrt \varepsilon }})\\ \cdot \frac{{\exp (  {\rm{i}}k\sqrt \varepsilon )}}{\varepsilon }z{r_1}{\rm{d}}{r_1}{\rm{d}}{\theta _1},\\ \varepsilon = {z^2} + r_1^2 + r_2^2  2{r_1}{r_2}\cos ({\theta _1}  {\theta _2}), \end{array} $  (1) 
where k=2π/λ is the wave vector, λ is the wavelength of the incident beam in vacuum;
$ {U'_1}({r_1}, \;{\theta _1}) = {U_1}({r_1}, \;{\theta _1})\;t({r_1}, \;{\theta _1}){\rm{exp}}[  {\rm{i}}k \cdot \Phi ({r_1}, \;{\theta _1})], $  (2) 
where
To design the GO lens with the targeted focal length f and diameter D based on the RS diffraction theory, we consider the intensity distribution on the z axis, namely r_{2}=0, z=f. And based on the Euler's equation, the expression of intensity distribution on the z axis can be simplified to (Supplementary Section 1):
$ I({r_1}) = {(\frac{{2{\rm{ \mathsf{ π} }}f}}{\lambda })^2}({\omega ^2} + {\upsilon ^2}), $  (3) 
here
$ \begin{array}{l} \omega = \int_0^\infty {{U_1}({r_1})\frac{{\cos (  k\sqrt {{f^2} + r_1^2} )}}{{{f^2} + r_1^2}}\;} {r_1}d{r_1},\\ \upsilon = \int_0^\infty {{U_1}({r_1})\frac{{\sin (  k\sqrt {{f^2} + r_1^2} )}}{{{f^2} + r_1^2}}\;} {r_1}d{r_1}. \end{array} $ 
The maximal destructive interference positions on the intensity distribution I(r_{1}) predict the ring radii of the GO lens with a focal length f. In the meantime, the diameter is decided by the number of rings. In this way, the lenses can be designed according to different incident fields (
On the other hand, for a situation that the observation point is not far away from the optical axis, it is assumed
$ \begin{array}{l} {U_2}({r_2}, \;z) = \frac{{{\rm{i2 \mathsf{ π} }}}}{{\lambda z}}{\rm{\exp}} (  {\rm{i}}kz){\rm{\exp}}(  \frac{{{\rm{i}}kr_2^2}}{{2z}})\\ \cdot \int_0^\infty {{{U'}_1}({r_1}){\rm{\exp}}} (  \frac{{{\rm{i}}kr_1^2}}{{2z}}){J_0}(\frac{{k{r_1}{r_2}}}{z}){r_1}{\rm{d}}{r_1}. \end{array} $  (4) 
According to the Fresnel diffraction theory, the radii of a Fresnel zone plate can be defined as follows^{18}:
$ {r_m} = \sqrt {m\lambda f + {{(\frac{{m\lambda }}{2})}^2}} , $  (5) 
where f is the focal length, r_{m} is the outer radius of the m^{th} zone.
Results and discussionWe first compare the lens designs using the RS and the Fresnel methods. The targeted focal length is 3.1 μm (Lens 1) with a lens thickness of 0.2 μm. The lenses are composed of three concentric rings with radii of a_{1}, a_{2} and a_{3}. Here we assume the incident beam is a plane wave. The resulted plots of radii of the rings are shown in Fig. 2. There is a significant difference (0.215 μm or 13.9%) between the radii of the two models at the first ring, and the differences decrease to 0.175 μm (6.6%) for the second ring, and 0.157 μm (4.4%) for the third ring.
To verify the two designs, we numerically simulate the intensity distributions based on the RS diffraction theory and compared to the Fresnel design. The results are shown in Fig. 3. The transverse full widths at a half maximum (FWHMs) of the focal spots are similar (1.0% difference, Fig. 3(c)) for the two designs. The effective NA of the GO lens can be calculated as NA=0.61λ/FWHM, which is 0.82. However, on the z axis (Fig. 3(b)), we can see clearly the difference (9.3%) of the focal lengths between the two designs. The RS model design provides a focal length of 3.11 μm as the design specified, whereas the Fresnel model design gives a 3.40 μm focal length which has a 9.3% deviation. Therefore, for high NA GO lenses, the RS method is able to design lenses according to the requirements with a high accuracy, in contrast to the Fresnel model.
In addition, to demonstrate the flexibility of the RS method, another lens (Lens 2) with 9.1 µm focal length is designed by the RS method. To verify the accuracy of the RS method, these two lenses are analyzed using the RS and Fresnel diffraction theories, and compared with the FDTD model and the experimental results. The parameters of the two lenses are shown in Table 1. The lenses are fabricated using direct laser writing with a femtosecond laser (Coherent^{®}, Libra, λ=800 nm, pulse width is 100 fs, repetition rate is 10 kHz, the setup information can be found in Supplementary Section 2). We use Gaussian function to express the modulation of the complex refractive index and the depth profiles (Fig. 1(b)), which is confirmed by the atomic force microscope (AFM) measurement of the fabricated lens (Fig. 1(c)). The thickness of the GO region is ~200 nm, and the thickness of the thinnest rGO region is ~100 nm, as shown in the topographic profile of Lens 2 in Fig. 1(c). The FWHM of the Gaussian profiled lines is ~300 nm.
The focal intensity distributions of the lenses are measured using a microscopic imaging setup (Supplementary Section 3). The experimentally measured focal intensity distributions of Lens 1 in the xy and xz planes and the corresponding theoretical results from the FDTD, RS and Fresnel theoretical calculations are shown in Fig. 4(a).
To see the effect of the incident polarizations, we assume the incident light polarizing along the xdirection in the FDTD simulation. The intensity distributions along the directions parallel (xdirection) and vertical (ydirection) to the polarization of the incident beam are shown in Fig. 4(c) with dash lines. The FWHM of the focal spot along the x and y directions of the FDTD results are ~0.50 μm (0.79 λ) and ~0.46 μm (0.73 λ), respectively (Table 2). The FWHM of the focal spot predicted by the RS diffraction theory is ~0.47 μm (0.74 λ) (Table 2). No remarkable difference (< 6% in the x direction and 2.1% in the y direction) can be found between the FDTD and RS theoretical results, which is expected considering the effective NA of the lens is approximately 0.82. Based on this result, we can conclude that the scalar RS diffraction theory is able to predict the focusing intensity distributions of the GO lenses accurately.
Lens 1  Lens 2  
f (μm) 
FWHM (μm) 
f (μm) 
FWHM (μm) 

RS theory  3.11  0.47  9.14  0.54 
FDTD (xdirection) 
3.07  0.5     
FDTD (ydirection) 
3.07  0.46     
Fresnel theory  3.48  0.24  10.28  0.5 
Experiment  3.09  0.51  9.15  0.56 
One can see that the match between the FDTD simulation, RS theoretical calculation and the experimental results are reasonably good. In comparison, significant difference can be identified in the Fresnel theoretical results, which gives a much smaller focal spot both in the xy and xz planes. For further detailed comparisons, the intensity distribution along the x and z directions (marked by the black dash lines) of all results are plotted in the same figures (Figs. 4(b) and 4(c)). The designed focal length of Lens 1 is 3.1 µm. The resulted focal lengths of the experimental measurement, the FDTD simulation, the RS theoretical results are 3.09 µm, 3.07 μm and 3.11 μm, respectively (Table 2). However, the Fresnel theoretical result is 3.48 μm. In addition, the experimental result of the FWHM is 0.51 μm (0.81 λ), which is only 2% difference from the FDTD result (xdirection) and 7.8% difference from the RS theoretical calculation. In comparison, the FWHM prediction using the Fresnel diffraction model is only 0.24 μm (0.38 λ), which shows 52.1% and 52.9% differences compared with the FDTD model and experimental result, respectively. It shows that, the RS diffraction theory can predict the focusing process accurately for a high NA lens, in contrast to the Fresnel diffraction theory, which shows nonnegligible difference.
For the Lens 2 case, the focal length and diameter of the lens are 23 times of those used in Lens 1, the FDTD requires nearly 20 times of simulation resources and time. Under such a circumstance, only the results from experiment, RS and Fresnel theoretical calculations are shown in Fig. 5. The crosssectional intensity plots along the axial and lateral directions are shown in Figs. 5(b) and 5(c), respectively. It is noticed that the experimental results and the simulations using the RS diffraction theory match well, while the focal length calculated by the Fresnel diffraction theory is significantly different from the others.
The FWHMs and focal lengths are listed in Table 2. The FWHMs of the experimental result, the RS and the Fresnel theoretical calculations are 0.56 μm (0.89 λ), 0.54 μm (0.86 λ) and 0.50 µm (0.79 λ), respectively. The difference between the experimental result and RS theoretical calculation is 3.7%, confirming the high accuracy of the RS diffraction theory as expected. Meanwhile, the difference between the experimental result and the Fresnel theoretical result is 11.4%. Although the difference for the Fresnel result is still larger than the RS result, it is much smaller compared with the previous case because the effective NA of Lens 2 is smaller (0.71). Whereas, the focal length calculated by the Fresnel theory (10.28 µm) is much larger than the experimental results (9.14 µm) and that from the RS theory (9.15 µm). In addition, there is a false focal spot at around 5 μm in the Fresnel diffraction model, which is not observed in the experimental result. Therefore, under such a circumstance, the Fresnel diffraction theory still fails to give an accurate description of the performance of the GO lens.
ConclusionIn conclusion, we have developed a design method based on the RS diffraction theory, which is able to accurately design GO lenses with arbitrary focal length and diameter without the optimization process for the first time. Compared with the Fresnel model design, it is only applicable for low NA lenses satisfying the paraxial condition. The differences in the ring positions and the resulted focal length are approximately 13.9% and 9.1%, respectively, for the high NA GO lens we targeted. Furthermore, the design flexibility of the RS method has been demonstrated by two GO lenses with desired focal lengths and NAs. The accuracy of our design methods has been verified both experimentally through laser assisted fabrication and point spread function characterized as well as the FDTD simulation. The theoretical and experimental results show that the RS diffraction theory is able to accurately calculate the focusing process of GO lenses with arbitrary NA and focal lengths with a high speed and efficiency. Therefore, the demonstrated RS method is expected to find broad applications in designing and analyzing other ultrathin flat lenses, including metasurface lenses and lenses made of other 2D materials.
AcknowledgementsB H Jia acknowledges the support from Australian Research Council and Defence Institute Australia.
Competing interestsThe authors declare no competing financial interests.
Supplementary informationSupplementary information is available for this paper at https://doi.org/10.29026/oea.2018.180012
Section 1: Lens design method
Section 2: Experimental setup of the laser fabrication system (Fig. S1)
Section 3: Experimental setup of the GO lens characterization (Fig. S2)
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