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Abstract
Multi-functional metasurface devices are of great importance for miniaturization and integration. Here we propose a novel approach to design dual-functional dielectric metasurface doublets based on the principle of Pancharatnam-Berry phase. The functionalities for the two operating wavelengths can be independently designed by adjusting the orientation angle of the nanofins in the upper layer and that in the lower layer, respectively. The functionality can be switched by simply changing the incident wavelength. The simulated results of the examples demonstrate that the designed dual-functional doublet works well as expected.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent years, metasurfaces have aroused considerable interest for their ability of manipulating light by altering the amplitude, phase, and polarization states through sub-wavelength structures [1–13]. It has been demonstrated that metasurfaces can be utilized in various applications such as wave retarders [14], beam splitters [15], beam generators [16–23], metalenses [24–35], holograms [36–38], and nonlinear devices [39,40]. In addition, because metasurfaces can be much thinner than the conventional bulky optical components, they are of interest for the miniaturization and integration.
Since metasurfaces can be used to realize various types of single-functional devices, a natural question is, can a metasurface device be multi-functional? The answer is yes. In fact, driven by their fantastic features and the important application in integration, in recent years much effort has been put to search for the multi-functional metasurface structures. It has been revealed that the multi-functional metasurface devices can be constructed through different approaches, such as the vertical stacking of multiple metallic metasurfaces [41], the in-plane spatial multiplexing in a single-layer metasurface [42–45], the dispersion engineering of phase shifters [46], and the hybrid design of propagation phase and geometric phase [47,48].
On the other hand, the metasurface doublet, which is composed of a substrate and two metasurface layers, has been proposed to eliminate the monochromatic aberrations or the chromatic aberrations in the past years [30,32,49]. The doublet can be a metalens with significantly reduced monochromatic aberrations, provided that the phase profiles of the two metasurfaces are optimized so that the doublet has an aperture metalens in front of a focusing metalens [32,49]. If the two metasurface layers are designed to act as the operation metalens and correction metalens, respectively, the chromatic aberrations of the doublet metalens is eliminated [30]. These results are of great interest in applications such as diffraction-limited monochromatic imaging and multi-spectral imaging. However, to our best knowledge, the application of the doublet in realizing multi-functional devices is still remain unexplored.
In this paper, we apply the concept of the metasurface doublet to develop a new type of dual-functional metasurface device, on the basis of the principle of Pancharatnam-Berry (PB) phase. The three-dimensional finite-difference time-domain (FDTD) method is utilized to optimize the geometric parameters of the nanofins in the two metasurface layers of the doublet. We also illustrate two examples to show the performance of the designed dual-functional doublet.
2. Design and structure
Figure 1 shows the schematic diagram of the designed dual-functional metasurface doublet, of which the two working wavelengths are $\lambda _{1}$ and $\lambda _{2}$. As shown in Fig. 1(a), the metasurface doublet comprises two layers of elliptical amorphous silicon nanofins patterned on both sides of a glass substrate. The upper (lower) layer of the doublet consists of amorphous silicon elliptical nanofins with the same major axis $a_1$ ($a_2$), minor axis $b_1$ ($b_2$), and height $h_1$ ($h_2$) but different rotation angles $\theta _1$ ($\theta _2$). Figures 1(b)–1(d) present a unit cell which is composed of two amorphous silicon elliptical nanofins with high refractive index and low loss sitting on the upper and lower sides of a fused-silica substrate.
Fig. 1. (a) Schematic diagram of the dual-functional metasurface doublet, which comprises two layers of elliptical amorphous silicon nanofins patterned on both sides of a glass substrate. (b) Side view of the unit cell. (c) Top view of the unit cell for the top layer of metasurface. (d) Bottom view of the unit cell for the lower layer of metasurface. Geometric parameters of each unit cell are shown in (b)-(d).
When transmitting through a unit cell in the metasurface layer, the light polarized along the major axis and that along the minor axis both experience phase shifts (the two phase shifts are defined as $\varphi _x$ and $\varphi _y$, respectively). Because of the chromatic dispersion produced by a combination of the material dispersion and the waveguide dispersion [50], the refractive index for the light polarized along the major axis (defined as $n_x(\omega )$, where $\omega$ is the optical frequency) and that for the light polarized along the minor axis (defined as $n_y(\omega )$) both vary with the wavelength of the incident light field. Therefore, the difference between the phase shift of the light linearly polarized along the major axis and that along the minor axis, i.e.,
is wavelength-dependent. In Eq. (1), $c$ represents the light velocity in vacumm, and $L$ is the propagation distance. Therefore, it is possible for one to search for a unit structure, for which $\triangle \varphi =\pi$ for one operating wavelength and $\triangle \varphi =2\pi$ for the other operating wavelength. In this case, the unit structure works as a half-wave plate for one operating wavelength meanwhile works as a full-wave plate for another wavelength.It is well-known that, for a circularly polarized incident light with a specific wavelength, the nanofins which function as full-wave plates have no effect on the phase distribution of the light field [51]. However, the nanofins which function as half-wave plates would convert a circularly polarized incident light into the transmitted light with opposite helicity and produce the so-called Pancharatnam-Berry phase [16]: $\varphi (x,\;y)= 2\theta (x,\;y)$. Therefore, one can impart the required phase distribution $\varphi (x,\;y)$ by rotating the orientation angle of the nanofins, i.e., $\theta (x,\;y)$.
Based on above analysis, we can construct a dual-functional metasurface doublet. The mechanism is as follows. If the geometric parameters of each unit structure are properly selected, for the incident wavelength $\lambda _{1}$ ($\lambda _{2}$), each unit structure in the upper layer of metasurface functions as a half-wave plate (full-wave plate), and each unit structure in the lower layer of metasurface functions as a full-wave plate (half-wave plate). In this case, the phase distributions of $\lambda _{1}$ and $\lambda _{2}$ can be independently controlled by the upper and lower layers of metasurface, respectively. In this way, the metasurface doublet becomes dual-functional, and the functionality can be switched by simply changing the incident wavelength.
3. Parameter optimization
In order to find the expected meta-structure, we utilize the three-dimensional FDTD method to optimize the geometric parameters of the nanofins. For the example shown in Figs. 2(a) and 2(b), the size of each unit cell is $s_x\times s_y =300nm\times 300nm$, the nanofins are assumed to be elliptic cylindrical, the operating wavelengths are assumed to be $\lambda _{1}=780nm$ and $\lambda _{2}=660nm$, and the heights of the nanofins in the two layers are $h_{1}=600nm$ and $h_{2}=800nm$. To construct the nanofins which function as half-wave plates for one operating wavelength and function as full-wave plates for the other, the major and minor axes of the elliptic nanofins should be optimized first.
Fig. 2. (a) The phase difference ($\triangle \varphi ^{(1)}$) of each unit structure in the upper layer for different lengths of the major and minor axes. The wavelength is $\lambda _1=780nm$. (b) The phase difference ($\triangle \varphi ^{(2)}$) of each unit structure in the lower layer for different lengths of the major and minor axes. The wavelength is $\lambda _2=660nm$. (c) Simulated transmission coefficients for x- and y-polarized light for $U_{1}$ and $U_{2}$. (d) The phase difference as a function of the wavelength for $U_{1}$ and $U_{2}$.
In Figs. 2(a) and 2(b), we change the lengths of the major and minor axes and get the phase difference, i.e., the difference between the phase shift of a light linearly polarized along the major axis ($\varphi _{x}^{(1,2)}$) and that along the minor axis ($\varphi _{y}^{(1,2)}$),
In the simulation, the spatial mesh grids are set as $\triangle x =\triangle y =\triangle z = 10nm$. The perfectly matched layer (PML) boundary condition is used along the $z$ axis and the periodic boundary condition is applied along the $x$ and $y$ axes.By filtering the data shown in Figs. 2(a) and 2(b), we found that, at the point A ($a_{1}=238nm$, $b_{1}=70nm$, $h_{1}=600nm$, the corresponding unit cell is defined as $U_1$), the phase difference $\triangle \varphi ^{(1)}$ approximates well to $\pi$ at $\lambda _{1}=780nm$, and approximates well to zero at $\lambda _{2}=660nm$. Meanwhile, at the point B ($a_{2}=248nm$, $b_{2}=94nm$, $h_{2}=800nm$, the corresponding unit cell is defined as $U_2$), the phase difference $\triangle \varphi ^{(2)}$ approximates well to $\pi$ at $\lambda _{2}=660nm$, and approximates well to zero at $\lambda _{1}=780nm$. In addition, in Fig. 2(d) we illustrate the phase differences for $U_{1}$ and $U_{2}$ at a broadband spectrum of $650nm-900nm$. The results further verify that $U_1$ ($U_2$) can be regarded as a half-wave plate (full-wave plate) for $\lambda _{1}=780nm$ and a full-wave plate (half-wave plate) for $\lambda _{2}=660nm$. Therefore, we can construct a dual-functional doublet (of which the unit cell is defined as $U_1$+$U_2$) by patterning the two metasurface layers (of which the unit cells are $U_1$ and $U_2$, respectively) on both sides of the substrate.
Figure 3(a) shows the polarization conversion efficiency (defined as the ratio of the energy of the cross-polarized light wave to that of the total transmitted light wave) for the single layer cases ($U_1$ and $U_2$) and the doublet case ($U_1+U_2$). For the doublet case, the polarization conversion efficiency arrives at two peaks ($99.5\%$ and $85\%$) around the two designed operating wavelengths $\lambda _1=780nm$ and $\lambda _2=660nm$. The polarization conversion efficiency is closely related to the difference between the transmitted amplitude of a light polarized along the major axis and that along the minor axis [18]. As shown in Fig. 2(c), for $U_1$ and $U_2$, there is an obvious difference between the two transmitted amplitudes at the wavelength of $\lambda _2=660nm$. Therefore, as shown in Fig. 3(a), the polarization conversion efficiency at the wavelength of $\lambda _2=660nm$ is obviously lower than that at the wavelength of $\lambda _1=780nm$.
Fig. 3. (a) Simulated polarization conversion efficiency for the single layer cases ($U_1$, $U_2$) and the doublet case ($U_1+U_2$). (b) Simulated transmission for the single layer cases ($U_1$, $U_2$) and the doublet case ($U_1+U_2$). (c) Comparison between the phase shift as a function of the rotation angle $\theta$ for the single-layer unit cell $U_{1}$ and that for the double-layer unit cell $U_{1}+U_{2}$. The wavelength is $\lambda _1=780nm$. (d) The same as (c) except that the single-layer unit cell is $U_{2}$ and the wavelength is $\lambda _2=660nm$.
Figure 3(b) illustrates the transmission (defined as the ratio of the energy of the cross-polarized light wave to that of the total incident light wave) for the single layer cases ($U_1$, $U_2$) and the doublet case ($U_1+U_2$). It can be observed that the unit cell $U_{1}$ has the highest transmission (about 0.9) around the wavelength of $780 nm$, and the transmission at the wavelength of $660 nm$ is close to $0$. On the other hand, the transmission of the unit cell $U_{2}$ is the highest (about 0.88) around the wavelength of $660 nm$ and approaches $0$ at $780 nm$. In the doublet case $U_{1}+U_{2}$, although the transmission is lower than that in the single-layer cases, the transmission peaks also exist around the wavelengths $\lambda _1=780 nm$ and $\lambda _2=660 nm$.
Figures 3(c) and 3(d) illustrate the comparison between the simulated phase shift as a function of the rotation angle $\theta$ for the single layer case ($U_1$ or $U_2$) and that for the doublet case ($U_1+U_2$). The result shows that, at the designed operating wavelengths $\lambda _1=780 nm$ and $\lambda _2=660 nm$, the phase shift for the doublet case is in good agreement with that for the single layer case. In fact, the relation between the PB phase and the orientation angle of the nanofins, $\varphi (x,\;y)= 2\theta (x,\;y)$, is also satisfied for the doublet case. Therefore, we can reasonably estimate that the meta-structure doublet ($U_{1}+U_{2}$) would work well at the designed operating wavelengths.
4. Performance verification
We introduce two examples to verify the performance of the proposed dual-functional doublet in the following. First, as shown in Fig. 4, we construct a dual-functional device which acts as a meta-axicon and a metalens at the wavelengths $\lambda _1=780nm$ and $\lambda _2=660nm$, respectively. For a planar meta-axicon to generate a zero-order Bessel beam, the phase distribution satisfies [17]
Fig. 4. Design of a dual-functional device which acts as a meta-axicon and a metalens at the wavelengths $\lambda _1=780nm$ and $\lambda _2=660nm$, respectively. (a) Schematic diagram of a meta-axicon for generating a zero-order Bessel beam. (b) The required continuous phase distribution of the meta-axicon for NA=0.7. (c) The arrangement of elliptical amorphous silicon nanofins in the upper layer of metasurface in FDTD simulation. (d) Schematic diagram of a metalens for focusing. (e) The required continuous phase distribution of the metalens for $f=15\mu m$. (f) the arrangement of elliptical silicon nanofins in the lower layer of metasurface in FDTD simulation.
where $\textrm{NA}=\sin \alpha$ represents the numerical aperture. And a metalens for focusing requires that the phase distribution is [24]
where $f$ is the preset focal length. The phase distributions $\varphi _{\lambda _{1}}$ and $\varphi _{\lambda _{2}}$ are obtained by rotating elliptical nanofins on both sides of the doublet according to the relation $\theta _{\lambda _{1},\lambda _{2}} (x,\;y)=(1/2)\varphi _{\lambda _{1},\lambda _{2}} (x,\;y)$.When the incident plane wave is at the wavelength $\lambda _{1}=780nm$, as shown in Fig. 5, the doublet acts as a meta-axicon and generates a zero-order Bessel beam (the efficiency is $71\%$). As expected, the intensity of the main lobe at the beam center is much stronger than that of the side lobes (Figs. 5(a) and 5(b)). Moreover, the size of the main lobe remains almost the same within the maximum non-diffraction distance $Z_{max}={R}/{\tan [\arcsin (\textrm{NA})]}=15.3\mu m$ (Figs. 5(b) and 5(c)), where $R$ is the radius of the aperture. The measured full width at half maximum (FWHM) of the main lobe is about 440nm (Fig. 5(c)), which is very close to its theoretical limit ($=0.358\lambda /\textrm{NA}=400nm$).
Fig. 5. Performance verification for the dual-functional device shown in Fig. 4 when the wavelength is $\lambda _{1}=780nm$, at which the device acts as a meta-axicon with NA = 0.7. (a) Simulated normalized intensity profile in x-z plane. (b) The beam patterns at different propagation distances. (c) The transverse normalized intensity distributions at different propagation distances.
However, as shown in Fig. 6, the doublet acts as a metalens when the incident plane wave is at the wavelength $\lambda _{2}=660nm$ (the focusing efficiency is $27\%$). It provides strong focusing capability and focuses the beam at $z=15.64 \mu m$, which approximates well to the preset focusing location $z=15.4 \mu m$. The FWHM of the focal spot (480nm) is very close to the diffraction limit ($=0.5\lambda /\textrm{NA} = 467nm$). In addition, the FWHM of the beam increases quickly with the distance from the focus. These results demonstrate a good focusing performance of the doublet for $\lambda _{2}=660nm$.
Fig. 6. Performance verification for the dual-functional device shown in Fig. 4 when the wavelength is $\lambda _{2}=660nm$, at which the device acts as a meta-lens. (a) Simulated normalized intensity profile in x-z plane. (b) The beam patterns at different propagation distances. (c) The transverse normalized intensity distributions at different propagation distances.
The second example is a dual-pattern generator, which generates a (0, 1) and a (1, 0) mode Hermite-Gaussian focused pattern when the incident light is composed of two operating wavelengths: $\lambda _1=780nm$ and $\lambda _2=660nm$ (Fig. 7(a)). The phase distribution for $\lambda _1=780nm$ and $\lambda _2=660nm$ are [24]
Fig. 7. (a) Schematic diagram of the dual-pattern generator which generates a (0, 1) and a (1,0) mode Hermite-Gaussian focused pattern when the incident light is composed of two wavelengths: $\lambda _{1}=780nm$ and $\lambda _{2}=660nm$. The values of other parameters are: $f_1=f_2=15\mu m$, $x_1=2\mu m$, $x_2=-2\mu m$, $y_1=y_2=0$, $NA=0.7$. (b) Simulated intensity distribution of the dual-pattern generated by the designed device.
5. Conclusion
In conclusion, the designed dielectric dual-functional doublet, which is composed of a substrate and two metasurface layers, performs two different functionalities at two different operating wavelengths. For one operating wavelength, each unit structure in the upper (lower) layer acts as a half-wave plate (full-wave plate), therefore the PB phase and thereby the functionality can be designed by adjusting the orientation angle of the nanofins in the upper layer. In a similar way, the functionality for the other operating wavelength can be designed by adjusting the orientation angle of the nanofins in the lower layer. It is verified by the examples that the designed dual-functional doublet works well as expected. The proposed approach can be used to design dielectric meta-structures with more layers of metasurface and thereby with more functionalities, and would be of interest for miniaturization and integration.
Funding
National Natural Science Foundation of China (11174090, 11174091, 61575068).
Acknowledgments
The authors are very grateful to the reviewers for valuable comments.
Disclosures
The authors declare no conflicts of interest.
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