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Abstract
Geometric phase metasurfaces feature complete phase manipulation of light at the nanoscale. While a majority of prior works assume the structure rotation in a fixed lattice of unit cells as equivalent to the element rotation required by the geometric phase principle, we argue that this assumption is fundamentally challenged for many current schematics which induce phase modulation inaccuracy. Here we take the dielectric nanobar type geometric phase metasurfaces as an example and perform an in-depth analysis about the physical origins of the phase modulation inaccuracy: imperfect structure rotation, resonance, tilted incidence and aperiodic arrays. We clarify the trade-off in phase modulation accuracy, efficiency, broadband property and wide angle acceptance. Furthermore, we present several examples of geometric phase metasurface devices to evaluate the performance degradation under different applications. Finally, based on the research, we provide a set of practical design and optimization guidelines to outperform the present devices of geometric phase metasurface.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Metasurfaces, the 2D arrays of subwavelength structures, provide exceptional prospects for engineering light [1–4]. Through a smart design on each unit-cell and a proper spatial arrangement of the 2D arrays, the metasurfaces can manipulate light planarly as a thin surface [1,4–13]. In the comparison of conventional optical devices such as lenses or prisms that steer light by bulky elements, metasurface devices excel in their flexibility and compactness enabling unprecedented opportunities for integrated nanophotonics on chip.
The metasurfaces based on the geometric phase, i.e. the Pancharatnam and Berry phase, received intense interest in the optics community due to their unique optical phase modulation principle [3,14–19]. The Pancharatnam and Berry phase optical element (PBOE) controls the phase of the orthogonal polarization component of the scattered light with respect to the incident light through the in-plane rotation of the element. Such a phase modulation depends only on the rotation angle. Therefore, the phase modulation by PBOE is accurate, which is highly desirable for metasurface devices [7,8,10,20–23]. In past decades, geometric phase metasurface devices have advanced intensely towards higher efficiency along with versatile functionalities [7,20,24]. The energy efficiency of the geometric phase metasurface devices depends on the polarization conversion efficiency (PCE) of each PBOE unit cell, which is defined as the energy ratio of the converted polarization component to the incident beam. In contrast to the typically lossy plasmonic PBOE with a low PCE, the all-dielectric PBOE nanostructures have been designed and demonstrated to achieve near-perfect PCE [5,20,25,26]. Furthermore, to acquire the ultimate broadband performance, super-cell structures containing different PBOEs are also designed to eliminate structure-induced dispersion [26–28]. Last but not the least, the metasurfaces that enable a variety of combined functionalities, or multitasking, have been proposed by a number of research groups [10,24,26,29]. To summarize, a plethora of geometric phase metasurface devices are emerging towards the superior performance of high efficiency, broadband operation, and ultra-fast switching as well as versatile functionalities, which include beam-steering, focusing, wavefront-twisting, holography, and imaging. Consequently, it gives rise to the following fundamental question: is there any physical challenge to make the “perfect” geometric phase metasurface devices that are simultaneously energy-efficient, accurate, broadband and with wide acceptance angles?
Indeed, due to inevitable physical challenges, the absolutely “perfect” geometric metasurface is challenging for current mainstream geometric phase metasurface schematics. Here, we point out that the geometric phase principle assumes rotation symmetry of the light-PBOE interaction system as an essential prerequisite condition. However, this assumption is not always generally guaranteed which can be attributed to the broken rotational symmetry. This may include structure rotation, resonance, tilted incidence and the aperiod array. The broken rotational symmetry challenges conventional wisdom in many realistic geometric phase metasurfaces, which induces the nontrivial phase modulation inaccuracy (PMI). Here we define the PMI as $\textrm{PMI} = |{{\varphi_{id}} - {\varphi_{re}}} |$, where ${\varphi _{id}}$ is the predicted phase following the PBOE design principle and ${\varphi _{re}}$ is the actual phase produced by a realistic geometric phase metasurface.
In this work, we perform a rigorous theoretical investigation to reveal the fundamental challenges and optimization guidelines of geometric phase metasurfaces with broken rotation symmetry. Firstly, we classify the major factors leading to the rotation symmetry breaking which induces the PMI and use the dielectric nanobar metasurface as an example of PBOE. Then, we quantify the trade-off between PCE and PMI based on the PBOE unit cell, the illumination direction, and the wavelength. Next, we show that the accumulated PMI in each PBOE unit cell eventually becomes critical to the overall far-field performance under different illumination for certain array arrangements of realistic metasurface devices. Lastly, based on our analysis and trade-off above, we give a brief optimization guideline for concrete applications.
2. Result and discussion
2.1 Fundamental challenges induced by PMI of PBOE
The geometric phase principle has a fundamental assumption of rotation symmetry. Here we elaborate on general causes of broken rotation symmetry of realistic geometric phase metasurfaces, which induces the PMI. First, the rotation of only the structure without lattice can lead to broken symmetry. As shown in Fig. 1, metasurfaces are typically built of periodic unit-cells, each of which contains a functional structure and its lattice periodicity. Figure 1(a-c) illustrates 9 arbitrary PBOE unit-cells in a 3×3 square lattice. Without losing generality, we use a rectangle pillar to represent an anisotropic PBOE. We define the periodicity rotation of such elements as the simultaneous rotation of both the functional structure and its lattice, while the structure rotation only rotates the former. The periodicity rotation preserves the ideal rotation symmetry, and the structure rotation can break the rotation symmetry [30,31].
Fig. 1. Schematic of (a) the original geometric phase metasurface containing 9 unit-cells in a 3×3 array (we use a rectangle to represent an arbitrary anisotropic structure), (b) the imperfect structure rotation that only rotates the structure without rotating the periodicity, (c) the periodicity rotation that simultaneously rotates the structure and the periodicity (d) the periodic array (e) the slowly varying array (f) the random array.
Another important and unavoidable source of broken rotation symmetry is the aperiodic array in realistic geometric phase metasurface devices for inhomogeneous phase modulation. Figure 1(d-f) illustrates a periodic array, slowly-varying aperiodic array and random aperiodic array, respectively. It is also straightforward to understand that an element of interest in an aperiodic array generally does not satisfy rotation symmetry.
To elaborate on the PMI induced by the imperfect structure rotation, we employ the dielectric nanobars as an example [25]. Dielectric nanobars are often chosen as the PBOEs due to their low-loss and fabrication convenience that leads to maximum energy efficiency. As illustrated in Fig. 2(a), we set the illuminated light propagating through a TiO2 nanobar (refractive index n≈2.42 with height h=600nm, length L=250nm and width W=90nm) and a glass substrate (SiO2, refractive index n≈1.52 with lattice period P=325nm). We define the sample plane as the x-y plane, and the default illumination is along the z-axis from the structure side to the substrate side. Also, we define the transverse-electric (TE) or transverse-magnetic (TM) polarizations as when the incident electric field E or magnetic field H is parallel to the long side of the nanobar, respectively. The consequence of the imperfect structure rotation is generally too complicated to reach a closed analytical form. However, if the metasurfaces are operating outside the resonance wavelength regions where the multiple internal reflections inside the structure can be ignored, a relatively simple form of expression can be obtained [20,32,33]. Therefore, we consider the nanobar as a simple phase retarder to reach the analytical expression as:
Fig. 2. (a) Schematic diagram of the nano-structure. (b, c) The simulated effective refractive index of the infinite waveguide under TE and TM polarization, respectively. (d) The unit-cell structure of the dielectric nanobar with different structure rotations.
Arg means the argument. When the structure rotates at an arbitrary angle $\theta$, according to the PBOE principle, the total transmitted phase should be the PB phase ${\varphi _{PB}}$ plus the phase shift of the unrotated structure ${\varphi _{l\_retard}}$ (we keep $N{eff}{_{\mathbf{TE\_}\theta }}$ and $N{eff}{_{\mathbf{TM\_}\theta }}$ to indicate periodicity rotation in this case):
Similarly, we can deduce the phase shift (TE polarization at an angle $\theta$) as:
Therefore, the overall phase error, i.e. the phase difference of the rotating structures minus the PB phase, can be expressed as:
Equation (5) shows that the PMI is related to the effective refractive index differences induced by the structure rotation and the height of the nanobar.
Outside the resonance wavelength region, the PMI of a geometric phase element under the structure rotation can be small and even negligible. However, when we consider the resonances of the dielectric nanobar, the combined effect of the structure rotation and the resonance induces a significantly larger PMI. To study its effect, we simulate the transmission phase of the orthogonal polarization component of the transmitted beam versus the wavelength and the in-plane structure rotation angle, as shown in Fig. 3(a). The transmission phase spectra in Fig. 3(a) exhibit abrupt phase changes which indicate the presence of resonances [34–38]. We conclude that these resonances are Mie type resonance of a dense dielectric structure, in which the resonance wavelength is decided by the shape, size, materials of the structure. At the first resonance mode (λ=590.4 nm) shown in Fig. 3(g), we observe that the magnetic field is localized at the center of the nanobar like a dipole while the vortex-like electric field surrounds the magnetic dipole. This indicates the first order magnetic Mie resonance. At the second resonance mode (λ=528.8 nm) shown in Fig. 3(h), the magnetic field is localized into two dipoles surrounded by the vortex electric fields, which indicates a second order magnetic Mie resonance. Therefore, for the dielectric structure such resonances are inevitable at certain wavelength regions. As indicated in Fig. 3(a), while the transmission phase spectra change abruptly near the resonance wavelengths, the overall PMI becomes critical.
Fig. 3. The phase transmission of the orthogonal circular polarization (CP) component versus wavelength under different structure rotation angles and different arrays: (a) Normal (0° tilted) incidence in a periodic array. (b) Normal (0° tilted) incidence in a slowly varying array. (c) Normal (0° tilted) incidence in a random array. (d) 30° tilted incidence in a periodic array. (e) 30° tilted incidence in a slowly varying array. (f) 30° tilted incidence in a random array. (g, h) Simulated magnetic field (color map) and the electric field (arrows) of the first and second magnetic resonance of the dielectric unit under TM illumination, at the x-z cutting plane including the center point of the nanobar. (i) The electric field (color map) and the magnetic field (arrows) of a representative longitudinal resonance under tilt incidence, at the x-y cutting plane including the center point of the nanobar.
We emphasize that the resonances themselves do not break the rotation symmetry; rather, they enhance the PMI induced by the broken symmetry of the structure rotation. Specifically, the resonances of the structure induce an abrupt phase delay in the spectrum. Then, the broken rotation symmetry slightly changes the frequency of the resonance, shown in Fig. 4. So, the abrupt phase change of each rotation angle isn't coincident in the spectrum, and the phase modulation of the PBOEs becomes significantly inaccurate.
Fig. 4. The simulated parametric study of (a, c, e) the standard deviation of PMI, and (b, d, f) the PCE versus tilt angle, height and wavelength, respectively. The height of the nanobar is 600 nm in (a, b) and 100 nm in (c, d), respectively.
Tilted incidence also contributes to broken rotation symmetry of unit-cells. In general, the tilted incident beam can carry s-polarization (electric field normal to the incident plane) and p-polarization (electric field parallel to the incident plane) simultaneously, which presents strong illumination anisotropy. Then, the simultaneous rotation of the incident polarization and the PBOE does not provide a rotational symmetric light-matter interaction.
More importantly, the combination of tilted incidence and resonance further breaks the rotation symmetry of the system. One essential reason is that the tilted incidence can excite certain cavity modes (also Mie resonances) that are not supported by the normal incidence. The normal incident plane wave has no z-component of fields, thus cannot excite the electric or magnetic Mie resonances of the nanobar, or producing an effective magnetic or electric multipole along that direction, in which this phenomenon is presented in Fig. 3(i) [34,37]. However, this restriction does not apply to the tilted incidence case. Since the resonance together with structure rotation can produce critical PMI, the tilted incidence can induce more high-PMI wavelength regions (corresponding to the resonances).
Moreover, under different incidence angles will occur an angular dispersion, which has been indicated in previous works [39,40]. It will deteriorate the performance degradation of the metasurface array.
For the particular dielectric nanobar structure, we show the transmission phase spectra under 30° tilted incidence in Fig. 3(b). The anomaly abrupt phase changes indicate additional resonances at 523 nm. Such resonances are unique for the tilted incidence since normal incidence lacks the longitudinal field component. These new resonances contribute to more high-PMI wavelength regions. Also, the resonance bandwidth gradually broadens via increasing the tilting angle, leading to broader high-PMI wavelength regions. Finally, some exceptionally narrow peaks and dips of the phase modulation lines in Fig. 3(b) are produced by the excitation of the “dark mode” [41].
The aperiodicity also breaks the rotation symmetry of the geometric phase metasurface array. Since practical geometric phase metasurface devices often contain an aperiodic array to achieve inhomogeneous phase modulation, it is important to study its consequence to the phase modulation. For this purpose, we compare the transmission phase spectrum of the converted polarized light from the interested element in the periodic array, the slowly varying array and the random array, respectively, illustrated in Fig. 3.
On one aspect, the neighbor environment in the aperiodic array affects the electromagnetic response of the PBOEs, which breaks the rotation symmetry. Therefore, as shown in Figs. 3(b, c, e, f), the aperiodicity in these arrays induces an additional general PMI in all wavelengths. However, the general PMI is relatively small.
On another aspect, aperiodicity will also affect the resonances of the PBOEs. Here, we emphasize that the resonances origin from the Mie-scattering between the incident beam and the individual dielectric nanobar structure. Therefore, the excitation of such resonances does not require any arrays. However, the array can indeed affect the wavelength and the strength of the resonance. In the periodic array, all elements have the same resonance wavelengths, so the overall resonance Q factor of the interested element is enhanced (sharper resonance peaks/dips) In the aperiodic array, the resonances of the different elements can have slightly different wavelengths, thus the resonance Q factor of the interested element can be reduced (shallower resonance peaks/dips) compared to the periodic array case. The trend of resonance enhancement is positively related to the homogeneousness of the array. As shown in Fig. 3(c), although the random array can have a smaller resonance PMI, it also induces a significant general PMI that could deteriorate the metasurface performance in all wavelengths.
2.2 Trade-off and design benchmark
Then we quantitatively describe the PCE and PMI versus wavelength, nanobar height, and incident angle. For this purpose, we define the standard deviation of the PMI as:
Here, we sample the rotation angle $\theta$ from 0 to 180°. We also define a critical PMI threshold as when its standard deviation is larger than 0.1.
In Fig. 4, we plot the standard deviation of the PMI versus the wavelength and tilting angle and comparing it with the PCE. The critical PMI areas, denoted by the red-colored area in SD plots, always occur near the resonance wavelengths. Also, with the increase of the tilted angle, the critical PMI areas become larger owing to the increase of the resonance number and the broadening of the resonance wavelength bandwidths. Therefore, based on the overlapped area of low PMI and high PCE, we can find the proper combination of wavelength and tilting angle to make an optimized PBOE. In addition, as shown in Figs. 4(c, d), we also calculate the similar PMI and PCE parametric maps for the dielectric nanobar with a smaller height (h=100 nm) while all other structure dimensions remain the same. Comparing to the higher dielectric nanobar, the shorter one gains the desirable smaller PMI with the undesirable smaller PCE. We further study the PCE and PMI versus height and wavelength with a normal incidence as shown in Figs. 4(e, f). The results present an obvious trade-off among PCE, PMI, large angle acceptance and broadband properties. Hence, for different applications that emphasize more on low PMI or high PCE, we can accordingly choose the proper parameters of height, wavelength band and incident angle according to the maps.
2.3 Macroscopic performance of the geometric phase metasurface: examples
In the above, we analyze the fundamental challenges of the PBOE unit, especially the origins of PMI. In this subsection, we will discuss how the phase modulation inaccuracy leads to under-performance revealed by relating the near-fields of a single PBOE to the far-fields of a PBOE-array metasurface. Namely, we take several typical applications including optical vortex generation, planar lens and holography [1,5–8,10,11,13,20,22,26,29,33,42–46] as examples, study their applications, and compare the far-field performance of the metasurface devices with different incidence angles and different wavelengths(in/out resonance).
We create a 2π rotation array of the illustrated dielectric PBOE to generate an optical vortex with orbital angular momentum (OAM) charge-2. We simulate the intensity profiles and the interference pattern of the generated optical vortices shown in Figs. 5(a, b). Firstly, under the normal incidence, the generated optical vortex exhibits the expected symmetric donut shape and a 1-3 fork interference pattern. Then, under 15° incidence angle, the generated optical vortex performs well out of the resonance region as Fig. 5 shows, while the optical vortices exhibit the split singularities in the resonance region, where the interference pattern shows two 1-2 forks. Finally, under 30° incidence angle, the generated optical vortices are no longer symmetric donut shape and the interference patterns are two 1-2 forks. We emphasize that such split singularities are deviant from the original higher order singularity [47], which can be a severe problem for OAM applications [13].
Fig. 5. (a) The simulated intensity profiles and interference pattern of the produced optical vortices under different tilted incidence angle off/on resonance, respectively. (b) The simulated intensity profiles in the focus plane and the x-z plane under different tilted incidence angle off/on resonance, respectively.
Next, we perform a similar comparative simulation for the planar metalens. The simulated intensity profiles of the focused beam at the focal plane and the x-z plane are shown in Figs. 5(c, d). Under the normal incidence and the small tilted incidence, the metalens can focus the light wave at the focus plane. However, under the poor condition not only does the focused beam exhibit an asymmetric profile but also produces a redundant diffractive pattern. What is more, the light focus at different focus plane under different incidence angles, which illustrate the spherical aberration of the metalens.
Here, we discuss the difference of dispersion between the vortex generation and metalens. The dispersion can be divided into material dispersion which origins from the materials or the meta-atoms and structure dispersion which origins from the structure (profile distribution) of the element like lens and gratings. Here, we emphasize that the PBOEs are free of the material dispersion rather than the structure dispersion. So the broadband unit can lead to a broadband vortex generator. However, the broadband metalens not only needs a good performance unit but also need to manage the total dispersion [42,48].
Lastly, we simulate the holography generation using the geometric phase metasurface. We design the holography metasurface based on the Fourier transform in the holography plane and the imaging plane and choose the logo of HUST as the holography image. The simulated holography images under different incidence angles and wavelengths are shown in Fig. 6. The image intensity density is normalized to the incident beam intensity density. Firstly, the generated holography exhibits the highest signal-to-noise ratio (SNR) out of the resonance region under normal incidence. Then, the SNR of the holography decrease with the tilted incidence angle increasing. Next, the SNRs in the resonance region are significantly lower than the SNRs out of the resonance region. Finally, under the 30° tilted incidence, the generated holography image is unrecognizable.
Fig. 6. The simulated holography image under different tilted incidence angle off/on resonance, respectively.
In all, our results show that the critical PMI resulting from the large tilted angle in the resonance condition of a single PBOE unit-cell deteriorates the macroscopic performance of metasurface application. And it demonstrates that the known schemes based on a single metasurface element lack the versatility of simultaneously broadband and broad acceptance angle.
3. Discussion and optimization guidelines
In this section, we discuss some important issues about the PMI, trade-offs and provide practical optimization guidelines. First and most important, we emphasize that although imperfect structure rotation poses a fundamental challenge to the PBOE principle, under certain conditions the structure rotation can be equivalent to the periodicity rotation and preserve the rotation symmetry. This is when the structure rotation angle is the same as the rotation symmetry angle of the unit-cell lattice, i.e. the lattice can overlap itself under the rotation of this angle from its center point. For example, the square lattice has rotation symmetry angles including 0, 90°, 180°, 270°. Therefore, the perfect structure rotation, i.e. when it is equivalent to periodicity rotation, depends on the rotation symmetry of a particular lattice. The group theory has demonstrated that the 2D lattice can have the symmetry angle only at 0° (360°), 60°, 90°, 120°, 180° [49]. Interestingly, recent works demonstrate that the metasurface with a particular quasi-lattice (the lattice that has no translation symmetry but has rotation symmetry) can support the symmetric angle of 36° and its integer times [30,31]. The quasi-lattice can be an important inspiration to create more symmetric angles for geometric phase metasurface. Still, it is a great challenge to realize the perfect structure rotation at any arbitrary angle for any orientation.
Second, we pay special attention to the trade-off of the resonances. According to Fig. 4, the critical PMI region coincides with the resonant condition, whereas out of the resonance the PMI is relatively small. It means we can accomplish a high PCE and low PMI in a relatively narrow band. Additional, a sensible way to decrease the PMI of the unit in broadband is to move the resonances out of the interested wavelength region. Since the higher order resonances often appear in the shorter wavelength side of the first order resonance, it is advisable to move the first order resonance wavelength on the shorter (blue) side of the interested wavelength region. With the numerical details appended in the Supplement 1, in summary, we can either reduce the height or transverse size, of the nanobar to blue-shift the resonance wavelengths. However, scaling down the transverse size is difficult for fabrication and it is hard to reach the π phase difference, and we can achieve low PMI and broadband properties with a relatively low PCE. In all, the resonance results in a trade-off in PCE, PMI and broadband properties.
Third, we focus on the trade-off of the tilted incidence. According to Fig. 4, using a smaller height of the PBOE, preferably as an “optically thin” surface that is much smaller than the operating wavelength, can minimize the PMI. However, these measures will decrease the PCE of the units. So the tilted incidence also leads to the trade-off of the high PCE, low PMI and broadband properties.
Fourth, we remind that aperiodicity is an important source of broken rotation symmetry. Aperiodicity can induce a general PMI in all wavelengths and reduce the Q factor of the resonance. And we emphasize that lower Q factor resonance still induces a significantly large PMI. In reality, most metasurface applications including vortex generators and planar lens, are based on a slowly-varying array which has a very similar property of a periodic array. Additionally, we can use the full-wave simulation and discrete space impulse response technique to continuous design or topological optimize the arrays which can improve the performance, however the deterioration still exists significantly [40].
Finally, we discuss the generalization of the observation results obtained from the dielectric nanobar as a particular example. We emphasize that although the dielectric nanobar is just a typical type of geometric phase element, the analytical results are extendable to all passive and achiral geometric elements including both the plasmonic ones and dielectric ones. Based on the geometric phase principle, any arbitrary subwavelength passive and achiral anisotropic structure (all kinds of shapes and materials) can be a geometric phase element, and we can define the two anisotropic axes of such anisotropic structure similar to the TE and TM polarizations of the dielectric nanobar. Therefore, the discussion about the structure rotation versus the periodicity rotation, aperiodicity, and their quantification of PMI discussed in this manuscript can be reasonably extended to all passive and achiral structures.
For dielectric geometric phase elements, particularly, the Mie resonances excited by normal and oblique incidence exhibited in the dielectric nanobar can be excited in any subwavelength dielectric structure, which makes resonances a generalized key factor of PMI enhancement for arbitrary dielectric geometric phase elements. As a result, the presented physical analysis of PMI in geometric phase metasurfaces and the design guidelines of the geometric phase metasurface devices presented in this manuscript is applicable for arbitrary passive achiral dielectric geometric phase element.
For the plasmonic geometric phase elements, the imperfect rotation and the aperiodic array can also contribute to a small PMI which is similar to the dielectric one. And the resonances also widely exist in the plasmonic metasurfaces. However, the resonance in the plasmonic metasurfaces always has a lower Q than the dielectric metasurfaces, so it will induce a smaller PMI in the resonance area but have a larger critical-PMI area than the dielectric metasurfaces. As for the tiled incidence, the thick plasmonic metasurfaces are saved from the birefringerence [1], but the z-component of the incidence light will also excite the z-direction resonance of the plasmonic unit. Therefore, the performance of the plasmonic geometric phase metasurfaces also suffered from the factors leading to PMI. Moreover, due to the high loss of the plasmonic materials, the transmission type plasmonic geometric phase metasurfaces typically have low energy efficiency, which is unsuitable for the aforementioned applications.
According to all the above discussion, we see that it is fundamentally challenging to design the perfect geometric phase metasurface device, which requires simultaneous wide acceptance angle, broadband, maximum energy efficiency and minimum PMI. These fundamental challenges result in a trade-off in broadband, low PMI, wide angle acceptance, and high efficiency. However, it does not mean we give a negative conclusion. On the contrary, by clarifying these symmetry breaking factors, we can make better choices to trade-off when we design the PBOE devices. Furthermore, we take some widely used applications as examples and provide an application guideline based on the trade-off in Fig. 4 and optimization in Table 1.
Table 1. Guidelines for several applications
The structured light generator and multiplexing need high efficiency. Owing to the self-recovery property, they have a tolerance of the inaccurate phase. According to the simulation, it can cover broadband with high efficiency and good beam quality under normal incidence. Therefore, avoiding the extremely high PMI region such as the large tilted incidence angle is essential for structured light applications.
The planar lens for broadband, large-angle acceptance and the high energy efficiency obviously also suffers from such trade-offs. The planar can work properly under a small tilted angle in the off resonance region at a narrow wavelength. However, compared to the commercial lens which can work in broadband, our simulation indicates that the broadband planar lens can work with a small NA even we can manage the dispersion due to the unavoidable resonance and narrow incidence angle acceptance. Additionally, our simulation illustrates that the high NA planar lens is also extremely difficult due to the spherical aberration and the high PMI under the large tilted incidence angle. It is advisable to use a narrow operation band and near-normal incident angle for the planar lens.
The holography applications also need an accurate design for phase modulation. Therefore, both tilted incidence and resonances can significantly decrease its SNR. When we need high efficiency, the normal incidence at the design wavelength is usually preferable. However, in some applications, the holography still needs to be tilted illuminated sometimes due to the narrow space. We then specifically suggest scaling down the height of the nanobar to get the higher SNR, while sacrificing the PCE to some extent.
Besides the applications above, we can also choose the PBOE unit based on the demands and the trade-off in PCE, PMI, large-angle acceptance and broadband.
4. Conclusion
In summary, we identify the often-overlooked prerequisite assumption of the geometric phase metasurfaces that the light-PBOE interaction needs to be rotationally symmetric and investigate the consequences when this assumption is not satisfied. We have taken the dielectric nanobar as an example and perform an in-depth analysis of the physical origin of its PMI. As a fundamental conclusion, we pinpoint the challenge of achieving exactly accurate phase modulation due to the inevitable imperfect structure rotation in square lattice metasurfaces. We also show the trade-off between maximum PCE and minimum PMI in the PBOEthrough a parametric study. Furthermore, we compare performances of geometric phase metasurface devices under different tilted incidence angles and wavelengths. By pinpointing the physical challenge induced by the PMI of the geometric phase metasurfaces, this work provides a practical design guideline to optimize its optical performance in the broader scope of integrated photonics.
Funding
National Key Research and Development Program of China (2019YFB2203604); National Natural Science Foundation of China (11774116, 11804087); Science, Technology and Innovation Commission of Shenzhen Municipality (JCYJ20200109114018750); Natural Science Foundation of Hubei Province (2020BAB001); Key R&D Program of Guangdong Province (2018B030325002); Fundamental Research Funds for the Central Universities (2019kfyRCPY037).
Acknowledgements
The authors gratefully acknowledge many helpful discussions with Prof. Nautalia Litchinitser from Duke University, Prof. Filippo Capolino from University of California Irvine, Prof. Lei Sun and Dr. Ling Li both from Shenzhen University.
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
Supplemental document
See Supplement 1 for supporting content.
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