Dielectric metasurfaces have been widely developed as ultra-compact photonic elements based on which prominent miniaturized devices of general interest, such as spectrometers, achromatic lens, and polarization cameras, have been implemented. With metasurface applications taking off, realizing versatile manipulation of light waves is becoming crucial. Here, by detailedly analyzing the light wave modulation principles raising from an individual meta-atom, we discuss the minimalist design strategy of dielectric metasurfaces for multi-dimensionally manipulating light waves, including parameter and spatial dimensions. As proof-of-concepts, those on-demand manipulations in different dimensions and their application potentials are exemplified by metasurfaces composed of polycrystalline silicon rectangle nanopillars. This framework provides basic guidelines for the flexible design of functionalized metasurfaces and the expansion of their applications as well as implementation approaches of more abundant light wave manipulations and applications using hybrid structures.

The available functionalities of optical elements come from the effective manipulation of the light wave’s fundamental parameters, e.g., amplitude, phase, and polarization. Therefore, structuring materials with capabilities to manipulate light waves has been a long-concerned issue and attracted significant interest. To overcome the physical limitations imposed by conventional natural materials and traditional optical devices, the emerging metamaterials1,2 exhibit unprecedented properties and lead to various novel optical effects.3–9 However, challenging problems, e.g., high losses and costly fabrication associated with bulky structures, especially hinder them from practical applications. Until recently, the advent of metasurfaces10–17 that are characterized as reduced dimensionality of metamaterials makes the breakthrough to dramatically reduce the fabrication complexity and increase the design flexibility,18–27 providing an elegant solution to those problems aroused in metamaterial-based optical devices.

In the past decade, metasurfaces have been extensively studied for engineering the fundamental parameters of light waves.28–35 A considerable amount of metasurfaces have been developed with impressive applications in realms of holographic imaging,36,37 polarization conversion,38,39 functional devices,16,17 multiplexing,26,27 and nonlinear optics.40–42 Nowadays, particular initiatives have been taken to enable multifunctional metasurfaces, which are based on the multi-dimensional manipulation of light waves. Recent progress has made some achievements, for instance, the introduction of unique structures (few-layer,43 diatomic,44 and folding45) and compositional materials (liquid crystal,46 phase change material,47 and two-dimensional material48) provides additional degree of freedoms (DoFs) for manipulating light waves with metasurfaces in both parameter and space dimensions. In contrast, the use of a simpler structure to achieve multi-dimensional light wave control has greater advantages in practical applications and device fabrications. Although some relevant studies have been reported,20,21 the characteristics of multi-dimensional light wave control that can be realized by a minimalist structure have not been systematically analyzed, and the relationship between the manipulation of parameter dimension with spatial DoFs has not been well discussed.

Here, by detailedly analyzing the structural birefringence of an individual meta-atom in single-layer dielectric metasurfaces, the light wave modulation principle for different parameter dimensions and spatial DoFs is discussed, based on which the minimalist design strategy of dielectric metasurfaces for modulation requirement of multiple dimensions and DoFs is demonstrated. According to diverse control principles, we design metasurfaces composed of polycrystalline silicon rectangle nanopillars and then demonstrate multifunctional applications of such minimalist metasurfaces, including phase-only holography, complex-amplitude holography, 3D holographic scene, axial modulation of light field, and polarization-encrypted holography. Meanwhile, the applicable principles of manipulating light waves in broadband and 3D space are analyzed.

To construct the modulation principles for different parameter dimensions, we first investigate the light wave modulation effect of an individual meta-atom in a single-layer dielectric metasurface. Figure 1 presents the schematic illustration of the wavefront modulation mechanism of the single-layer dielectric metasurface. According to the effective medium theory,49 the meta-atom is an effective anisotropic structure that supports large refractive index contrast between orthogonal polarizations of light. Therefore, the complex transmission property of such a birefringent meta-atom can be expressed as

J=RθToeiφo00ToeiφeRθ,
(1)

where R(θ) is the rotation matrix, and the middle matrix accounts for the transmission amplitudes (To, Te) and phases (φo, φe) along the ordinary and extraordinary axes, respectively, as shown in Fig. 1(a). Assuming that two orthogonal polarizations have uniform transmission amplitude, i.e., To = Te = T, one can further simplify the Jones matrix according to the incident polarization. It is well known that the light–matter interaction is generally described as the response of two kinds of polarization states, that is, the linear polarization (LP) and circular polarization (CP). Thus, we take these two typical polarizations as examples, and the corresponding Jones matrices in the CP basis [EREL]T and LP basis [EHEV]T subsequently can be written as (the subscript R/L denotes the right/left CP state, and H/V denotes the horizontal/vertical LP state, respectively)

JCP=Teiφ0cosδ/2isinδ/2ei2θisinδ/2ei2θcosδ/2,
(2)
JLP=Teiφ1eiφ2eiφ2eiφ3,eiφ1=cos2θeiφo+sin2eiφe,eiφ2=cosθsinθeiφecosθsinθeiφo,eiφ3=sin2θeiφo+cos2θeiφe,
(3)

where δ = (φoφe) and φ0 = (φo + φe)/2 depict the phase retardation and propagation phase based on ordinary and extraordinary components, respectively.

FIG. 1.

Wavefront modulation mechanism of the single-layer dielectric metasurface. (a) Schematic illustration of the dielectric metasurface. Inset: transmission property of a meta-atom. (b) and (c) Transmission properties of a meta-atom corresponding to two kinds of bases. (d) Conversion of polarization states on the Poincaré sphere.

FIG. 1.

Wavefront modulation mechanism of the single-layer dielectric metasurface. (a) Schematic illustration of the dielectric metasurface. Inset: transmission property of a meta-atom. (b) and (c) Transmission properties of a meta-atom corresponding to two kinds of bases. (d) Conversion of polarization states on the Poincaré sphere.

Close modal

The above Jones matrices cannot be directly connected with the modulable parameter dimensions. Therefore, to address legible modulation principles, we further consider the whole output fields, which are the composition of different polarizations, as schematically shown in Figs. 1(b) and 1(c). For the incidence of the |R⟩ state, the output field naturally consists of two components with orthogonal polarizations, namely, the co-polarized and cross-polarized components; thus, the output vector field is expressed as

Eoutcp=Teiφocosδ/2R+iTeiφosinδ/2ei2θL.
(4)

As for the incidence of linear polarized light field with |H⟩ or |V⟩ state, the output field consequently presents a [eiφ1 eiφ2]T or [eiφ2 eiφ3]T state. Here, we consider a superposition state of the |H⟩ and |V⟩ states, namely, the |D⟩ state, as a generalized model, and thus, the output vector field can be further expressed as

Eoutlp=Teiφ1H+Teiφ2D+Teiφ3V.
(5)

The modulation process introduced by the conversion of polarization states on the Poincaré sphere is shown in Fig. 1(d). Clearly, the above equations provide intuitionally controllable parameter dimensions, including amplitude, phase, and polarization. According to Eq. (1), meta-atoms can be regarded as waveplates with arbitrary phase retardation (δ) achieved by structuring the birefringence. Meanwhile, this phase retardation results in that the two components with orthogonal CPs have complementary intensities of T2 cos2(δ/2) and T2 sin2(δ/2), as shown in Eq. (4). In this principle, some polarization transformers50 and ultrathin energy tailorable splitters16,51 have been designed. Obviously, this modulation on amplitude or polarization only refers to single DoF control.

To showcase the modulation capabilities of different parameter dimensions and spatial DoFs, we categorize the corresponding controllable wavefront into different cases, which are shown in Table I. It is worth noting that, for the CP basis, this inherent intensity relationship disables the independent control of the wavefront amplitudes corresponding to these two components; therefore, the wavefront modulation has been focused on the cross-polarized component, and given that the co-polarized component is a background noise.52 For pure phase modulation, as Eq. (4) shows, these two components have a communal propagation phase exp(iφ0), and the cross-polarized component experiences an abrupt phase change of ±2θ, i.e., the well-known geometric phase.53 These two types of phases are determined by the geometric size and azimuthal angle of the meta-atom, respectively, which can be directly modulated by the φ0 (case 1) and θ (case 2), corresponding to two DoF modulations.

TABLE I.

Categorized modulation capabilities of a single meta-atom in a single-layer dielectric metasurface. PD: parameter dimension, SDoF: spatial degree of freedom.

Master variableControllable wavefrontPD × SDoF
Case 1 φ0 Eout = exp(iφ01 × 2 
Case 2 θ Eout = exp(±i2θ1 × 2 
Case 3 δ, θ Eout = sin(δ/2)exp(i2θ2 × 3 
Case 4 T, δ, θ Eout = T sin(δ/2)exp(i2θ2 × 3 
Case 5 T, φ0, θ Eout = T exp[i(φ0 + 2θ)] 2 × 3, 1 × 1 
Case 6 φ0, θ Eout = exp[i(φ0 + 2θ)]|L⟩ + exp[i(φ0 − 2θ)]|R⟩ 2 × 2 
Case 7 φo, φe Eout = exp(iφo)|H⟩ + exp(iφe)|V⟩ 2 × 2 
Case 8 φo, φe, θ Eout = exp(iφ1)|H⟩ + exp(iφ2)|D⟩ + exp(iφ3)|V⟩ 2 × 2 
Master variableControllable wavefrontPD × SDoF
Case 1 φ0 Eout = exp(iφ01 × 2 
Case 2 θ Eout = exp(±i2θ1 × 2 
Case 3 δ, θ Eout = sin(δ/2)exp(i2θ2 × 3 
Case 4 T, δ, θ Eout = T sin(δ/2)exp(i2θ2 × 3 
Case 5 T, φ0, θ Eout = T exp[i(φ0 + 2θ)] 2 × 3, 1 × 1 
Case 6 φ0, θ Eout = exp[i(φ0 + 2θ)]|L⟩ + exp[i(φ0 − 2θ)]|R⟩ 2 × 2 
Case 7 φo, φe Eout = exp(iφo)|H⟩ + exp(iφe)|V⟩ 2 × 2 
Case 8 φo, φe, θ Eout = exp(iφ1)|H⟩ + exp(iφ2)|D⟩ + exp(iφ3)|V⟩ 2 × 2 

The phase modulation has been widely utilized for two-dimensional holographic imaging and reproducing special phase pattern. While by contrast, the manipulation capability with respect to three DoFs greatly improves the performance of metasurfaces in integrated multifunctional optical devices. In scalar optics, the complete information of a light field requires both amplitude and phase, namely, complex amplitude. Here, from the complex amplitude distribution of the cross-polarized component in the CP basis, i.e., T sin(δ/2)exp[i(φ0 + 2θ)], one can recognize that the amplitude and phase are determined by T, δ and φ0, 2θ, respectively (case 3 and case 4). Thus, both the amplitude and phase can be completely and independently controlled, and benefiting from this, the complex amplitude modulation has advantages in 3D space imaging over amplitude- or phase-only modulation schemes.52 It is important to point out that these two methods have an unavoidable directly transmitted component, which especially affects the axial modulation. Therefore, a complex amplitude modulation method with extra axial DoF control is introduced here (case 5).

The above discussions focus on the scalar field, while the possibility in simultaneous control of polarization and phase provides huge prospect to develop polarization-dependent optical devices and introduces extra polarization channels to increase the DoFs. For this reason, the optical responses to each component should be taken into account. For instance, in the case of CP basis, the combined effect of propagation phase and opposite geometric phases endows independent modulation phases φ0 ± 2θ onto two orthogonal bases54,55 (case 6). While for the case of LP basis, as Eq. (3) shows, one can obtain φ1 = φo and φ3 = φe when the meta-atoms are arranged without rotation (θ = 0), that is, two independent phase patterns can be implemented on two orthogonal linear polarization states (case 7). Then, taking rotation into consideration, as shown in Eq. (5), three phase patterns, φ1, φ2, and φ3, which are dependent on the geometric parameters φo, φe, and θ, can be implemented on three linear polarization states56 (case 8).

For modulation with more parameter dimensions, amplitude, phase, and polarization response are inevitably associated with each other, when adjusting the geometric parameters of individual meta-atom, that is, the number of controllable parameters is limited to two, as shown in Table I. To break this limitation, two orthogonal polarization bases whose amplitude and phase can be precisely and independently modulated are primarily required. For the CP basis, this expectation cannot be achieved due to their correlated amplitudes, while for the LP basis, the background noise arising from phase-only modulation leads to the inaccuracy of superposition state. Therefore, a capable implementation is using hybrid structures based on exploring the inherent relationship between meta-atoms and associating each parameter dimension with a certain structural parameter of meta-atom.

As a proof-of-concept, we design and fabricate metasurfaces corresponding to each case. Here, we choose the poly-Si meta-atoms on a fused silica substrate, which have rectangular cross sections with square lattice arrangement, to design and fabricate metasurfaces by using COMS compatible processes (details can be found in the  Appendix). The geometric size (height H, length L, width W, and period P) of the meta-atom is variable for different cases, depending on master variables. For simplicity and generality, computer-generated holograms (CGHs)57 are chosen to implement most of the following experiments, which are succinct to testify the capability of manipulating the light wave. All experiments were performed at the wavelength of visible light band.

To testify the performance of pure phase modulation, that is, cases 1 and 2, two-dimensional holographic imaging is implemented experimentally. In practice, phase-only CGHs are generated by use of the typical Gerchberg–Saxton (GS) algorithm.58 Crucially, the pure phase modulation based on propagation phase and geometric phase have different modulation precisions and distinct requirements for the selection of meta-atoms. In the case of propagation phase modulation, the phase-only CGH needs to be discretized, and higher nanopillars are required to ensure sufficient phase modulation depth. In contrast, geometric phase modulation only requires a single geometry and has higher modulation accuracy and efficiency via rotating nanopillars. Figures 2(b) and 2(d) show the scanning electron microscope (SEM) images of fabricated metasurfaces corresponding to cases 1 and 2, where the metasurfaces both have 1000 × 1000 meta-atoms, but different heights (case 1: 610 nm and case 2: 350 nm) and periods (case 1: 450 nm and case 2: 300 nm).

FIG. 2.

Holographic imaging based on the phase-only modulations of metasurfaces. (a) Target images and experimental results of case 1 at a wavelength of 633 nm. (b) and (d) SEM images of fabricated metasurfaces corresponding to cases 1 and 2, respectively. Scale bars are 1 µm. (c) Experimental results of case 2 at the wavelengths of 473, 488, 532, 633, and 670 nm, respectively. (e) Simulated transmittance and sinusoidal term spectra of the selected meta-atom in case 2. The geometric parameters are L = 174 nm and W = 104 nm.

FIG. 2.

Holographic imaging based on the phase-only modulations of metasurfaces. (a) Target images and experimental results of case 1 at a wavelength of 633 nm. (b) and (d) SEM images of fabricated metasurfaces corresponding to cases 1 and 2, respectively. Scale bars are 1 µm. (c) Experimental results of case 2 at the wavelengths of 473, 488, 532, 633, and 670 nm, respectively. (e) Simulated transmittance and sinusoidal term spectra of the selected meta-atom in case 2. The geometric parameters are L = 174 nm and W = 104 nm.

Close modal

Figure 2(a) shows two target images (grayscale and binary images, respectively) and experimentally reconstructed results in case 1 at a wavelength of 633 nm. It can be seen that these target images are reconstructed with high performance. As a contrast, the binary image is also reconstructed by means of case 2 under the same experimental condition, and the corresponding result is shown in the fourth column of Fig. 2(c). Clearly, the reconstructed image in this case exhibits a higher fidelity, which results from the high accuracy of geometric phase modulation. In addition, the characteristic of independence of wavelength enables us to operate geometric phase modulation in a broad bandwidth. Figures 2(c) and 2(e) show the reconstructed results and response spectrum of the selected meta-atom at multiple wavelengths. As shown, although the geometric phase modulation has lower transmittance at short wavelengths, it still exhibits good broadband characteristics as the experimental results present clear reconstructed holographic images at all operating wavelengths.

Further testifications of complex amplitude modulation are shown in Figs. 3 and 4, where complex amplitude hologram and 3D holographic scenes are demonstrated. Figures 3(a) and 3(b) show the simulated transmittance, propagation phase, and sinusoidal term of selected meta-atoms in case 3. The complex amplitude holograms are calculated by the Fourier transform of the target image. Figures 3(d) and 3(e) show the fabricated metasurface (H = 610 nm, P = 450 nm) and reconstructed result of case 3. The experiment is carried out with the setup shown in Fig. 3(c) at a wavelength of 670 nm, and the image reconstructed on the screen is photoed by a camera. Compared with these two previous phase-only modulation methods [Figs. 2(a) and 2(c)], the complex amplitude modulation method intuitively improves the imaging quality and reduces the background noise since both the amplitude and phase are faithfully reproduced.

FIG. 3.

Holographic imaging of the metasurface with complex amplitude modulation. (a) and (b) Simulated transmission amplitudes, propagation phases, and sinusoidal term of these selected meta-atoms in case 3. (c) Schematic illustration of the experimental setup. HWP: half-wave plate and QWP: quarter-wave plate. (d) SEM image of the fabricated metasurface corresponding to case 3. The scale bar is 1 µm. (e) Reconstructed results of case 3 at a wavelength of 670 nm.

FIG. 3.

Holographic imaging of the metasurface with complex amplitude modulation. (a) and (b) Simulated transmission amplitudes, propagation phases, and sinusoidal term of these selected meta-atoms in case 3. (c) Schematic illustration of the experimental setup. HWP: half-wave plate and QWP: quarter-wave plate. (d) SEM image of the fabricated metasurface corresponding to case 3. The scale bar is 1 µm. (e) Reconstructed results of case 3 at a wavelength of 670 nm.

Close modal
FIG. 4.

3D imaging of the metasurface with complex amplitude modulation. (a) Simulated amplitudes and propagation phase of the selected meta-atoms in case 4. (b) Schematic illustration of the 3D holographic scene. (c) Schematic illustration of the experimental setup. Inset: SEM image of the fabricated metasurface. The scale bar is 1 µm. (d) Simulated and experiment results at a wavelength of 633 nm.

FIG. 4.

3D imaging of the metasurface with complex amplitude modulation. (a) Simulated amplitudes and propagation phase of the selected meta-atoms in case 4. (b) Schematic illustration of the 3D holographic scene. (c) Schematic illustration of the experimental setup. Inset: SEM image of the fabricated metasurface. The scale bar is 1 µm. (d) Simulated and experiment results at a wavelength of 633 nm.

Close modal

Figure 4(a) shows the combined amplitudes T sin(δ/2) and propagation phases of selected meta-atoms in case 4. In case 3, the precondition that transmittances of these meta-atoms are constant limits the geometry selectivity. While in case 4, the control of parameter T does not directly affect the final modulation effect but supplies a greater tolerance to the selection of the meta-atom in case 4. Consequently, under the same height of the meta-atom, the operating wavelength is reduced to 633 nm. In order to fully demonstrate the advantages of complex amplitude modulation, a 3D holographic scene, which consists of letters “N,” “P,” and “U” localized at three lateral planes, is performed. Figures 4(b) and 4(c) illustrate the operation principle and experimental setup. For the calculation of CGH, each letter image at certain diffraction distances is back-propagated to the metasurface plane by the beam-propagation method. Figure 4(d) shows the simulated and experimentally observed results at three lateral planes, respectively. In this experiment, we introduce an optical microscopy setup with the cross-polarized analyzer in order to avoid the influence of the co-polarized component. It is noteworthy that the experimentally reconstructed images have almost identical profiles with simulated ones, which powerfully demonstrates the great capability of the complex amplitude modulation of the metasurface for reconstructing target images in 3D space.

The full control of the amplitude and phase significantly improves the quality and capability of the reconstructed image. Nevertheless, in the above two cases, the unavoidable co-polarized component arising from the incomplete spin conversion, i.e., the non-zero T cos(δ/2), leads to a drawback that prevents the above methods from axial modulation without polarization filtering. However, eliminating the co-polarized component is difficult to implement in some special situations, such as focusing. Therefore, in case 5, the phase retardation δ is fixed as π, making sure that the incident spin polarization is totally transformed into the orthogonal one; hence, the amplitude and phase modulations are dependent on T and φ0 + 2θ, respectively, as shown in Fig. 5(a). Obviously, the amplitude is only dependent on the transmittance of meta-atom [Fig. 5(b)], but the phase term is related to both the propagation and geometric phases. Unfortunately, the transmittance and propagation phase are jointly related to the geometry of meta-atoms. To break this relationship, an opposite rotation angle φ0/2 should be added onto θ, i.e., θ' = θφ0/2, and then the amplitude and phase are independently and completely controllable.

FIG. 5.

Longitudinal modulation enabled by the metasurface. (a) Schematic of the modulation effect in case 5. (b) Transmittances of 17 selected meta-atoms. (c) Schematic illustration of the experimental setup. Inset: SEM image of the metasurface. The scale bar is 100 µm. (d) Measured intensity distribution in the yz plane. (e) Simulated and measured on-axis intensity distributions.

FIG. 5.

Longitudinal modulation enabled by the metasurface. (a) Schematic of the modulation effect in case 5. (b) Transmittances of 17 selected meta-atoms. (c) Schematic illustration of the experimental setup. Inset: SEM image of the metasurface. The scale bar is 100 µm. (d) Measured intensity distribution in the yz plane. (e) Simulated and measured on-axis intensity distributions.

Close modal

As an example, an axially structured light field with sinc-functional intensity distribution (calculated by the spatial spectrum optimization method based on the Durnin ring59,60) is demonstrated to assess this axial tailoring capability. Figure 5(d) shows the measured intensity distribution (normalized) in the yz plane, which is observed through the setup shown in Fig. 5(c). The microscope system is localized on a linear translation stage with a scanning interval of 10 μm. The simulated and measured on-axis intensity distributions (normalized) are displayed in Fig. 5(e). As shown, this method can sustain the construction of axial light field.

The above discussions all refer to the wavefront manipulation of scalar light field, namely, the cross-polarized component. In addition to the enhancement of multiplexing capability, numerous intriguing phenomena related to vector fields, such as their construction, enhanced longitudinally polarized component, and super-resolution focusing, are based on the combined modulation of two spin states.61,62 To address polarization-dependent light field modulation, more parameters should be taken into account. As is known, the geometric phase is always accompanied by a “twin field,” which originates from the phase accumulation of opposite CP state transition. Therefore, by combining the propagation phase, two CPs can obtain independent phase modulation of φ0 ± 2θ, as shown in Fig. 6(a). However, in this case, the inherent amplitude correlation disables the independent amplitude modulation of two CPs. Thus two CPs are commonly considered to have unitary amplitude, i.e., sin(δ/2) = 1. Here, a holographic reconstruction of two complementary images [Figs. 6(c) and 6(d)] is employed to showcase the potential in polarization-encrypted application. Figures 6(e)6(g) display the experimental results under the incidences of light fields with different polarizations. As shown, when a linear polarized light field illuminates, the metasurface outputs a uniform spot without pattern, but the polarization-dependent patterns show up for the CP incident light fields.

FIG. 6.

Polarization-dependent holographic imaging of the metasurface based on the combined modulations of two CPs. (a) Schematic of the combined modulation of two CPs. (b) SEM image of the metasurface. The scale bar is 1 µm. (c) and (d) Target images encoded on two CPs; reconstructed results for the incidence of a (e) linearly polarized, (f) right-handed CP, and (g) left-handed CP light field.

FIG. 6.

Polarization-dependent holographic imaging of the metasurface based on the combined modulations of two CPs. (a) Schematic of the combined modulation of two CPs. (b) SEM image of the metasurface. The scale bar is 1 µm. (c) and (d) Target images encoded on two CPs; reconstructed results for the incidence of a (e) linearly polarized, (f) right-handed CP, and (g) left-handed CP light field.

Close modal

In comparison, LP-based methods provide more optional channels. Thanks to the “structural birefringence” of meta-atoms, arbitrary manipulation can be implemented on a certain polarized component modulated along ordinary or extraordinary axis theoretically, as described in Eq. (1). Here, the transmittance of each meta-atom is set to be unitary. In Eq. (3), when θ = 0, one obtains φ1 = φo and φ3 = φe, namely, two independent phase modulations can be implemented on two orthogonal linear polarization channels. While taking rotation into account, three independent phase modulations can be implemented on three linear polarization channels. The experimental setup and SEM images of two LP-based metasurfaces without and with rotation are shown in Figs 7(a)7(c).

FIG. 7.

Polarization-encrypted imaging of the metasurface. (a) Schematic illustration of the experimental setup. (b) and (c) SEM images of the metasurface corresponding to cases 7 and 8. Scale bars are 0.5 µm. (d) and (e) Experimental results of polarization-encrypted imaging based on modulation mechanisms of cases 7 and 8. The red and blue arrows depict the incident and detected polarization directions, respectively.

FIG. 7.

Polarization-encrypted imaging of the metasurface. (a) Schematic illustration of the experimental setup. (b) and (c) SEM images of the metasurface corresponding to cases 7 and 8. Scale bars are 0.5 µm. (d) and (e) Experimental results of polarization-encrypted imaging based on modulation mechanisms of cases 7 and 8. The red and blue arrows depict the incident and detected polarization directions, respectively.

Close modal

For the metasurface without rotation, four letters are encoded into the horizontal and vertical polarizations by phases φ1 and φ3, respectively. The experimental results are depicted in Fig. 7(d). Under the illumination of a diagonal polarized light field, patterns encoded in two polarization channels are simultaneously reconstructed. By rotating the polarization analyzer, the reconstructed pattern is switched from “AB” to “CD.” For the second metasurface, an additional polarization channel is available due to the introduction of geometry rotation. Three letter patterns are encoded into the horizontal, vertical, and diagonal linear polarizations by phases φ1, φ2, and φ3, respectively. As shown in Fig. 7(e), for the incidence of a diagonal polarized light field, three patterns in three polarization channels are simultaneously reconstructed without the analyzer. While for the cases of H- or V-polarization incidence, the pattern in the orthogonal polarization channel disappears, respectively. Furthermore, individual polarization channels can be switched by changing the incident and analyzed polarization directions. Thereinto, the “XYZ” pattern, namely, diagonal polarization channel, can be obtained with the orthogonal polarizer and analyzer.

The on-demand modulation principles of single-layer dielectric metasurfaces for multiple dimension control have been theoretically and experimentally exhibited, but more details merit discussions. Notably, arbitrary modulation can be implemented through manipulating the “structural birefringence” of meta-atoms and various applications can be realized according to the above principle. However, limited by the properties of natural materials, the modulation depth and width are restricted to meet some particular applications, which also results in confined operating wavelengths and represents a daunting exploratory and computational problem. Therefore, invariant parameters and varying thicknesses are used in different cases to obtain an enough modulation range.

Second, the systematic strategy for on-demand light wave manipulation demonstrated here avoids unnecessary complexity in both the design process and experimental operation, which presents the full potential of single-layer dielectric metasurfaces, and leads to a series of applications. The pure phase modulation can be realized in two ways, among which the geometric phase modulation has been widely used in device design due to its convenience, high precision, and broad bandwidth. By contrast, the complex amplitude modulation has an advantage of information density over phase-only hologram, which leads to holographic images with higher quality, higher fidelity, and the reconstruction in 3D space. Moreover, for applications involving holographic data encryption or storage, the complex amplitude hologram can greatly increase the storage capacity. Furthermore, the axial modulation method in case 5 enriches the functionalities of complex amplitude modulation and provides an additional DoF in 3D light wave manipulation, as well as an approach for constructing tightly focusing fields with longitudinally oscillating polarization. In addition, the polarization-encrypted holography exhibited in cases 6–8 effectively enlarges the design space of polarization-dependent devices, and further applications, such as polarization-multiplexing and information encryption, can be expected. It is no doubt that such a strategy provides a basic guideline for the flexible design of optical metasurfaces and an effective way for the expansion of their applications.

Finally, besides the functionality, the modulation efficiency is another concerned issue. Notably, the pure phase modulation methods have advantages of efficiency because of the excellent encoding techniques. On the other hand, the modulation efficiency is closely related to the amplitude coefficient of the Fourier transform CGHs, i.e., T sin(δ/2) ≠ 1. In our experiment, taking the absorption of material into account, the diffraction efficiency in case 2 is 62.5% (with polarization conversion efficiency exceeding 95%), while in cases 3 and 4, it is about 10%. For axial modulation, the optimized spatial spectrum can significantly enhance the generation efficiency to about 20%, but it is still strongly dependent on the pre-established distribution. In polarization-dependent modulations, the response of the meta-atoms and the quality of fabrication are the main factors affecting the efficiency; here, the diffraction efficiency of cases 6–8 is all about 50%. As a whole, the demonstrated design principles and devices can be characterized as low loss.

In summary, we have systematically discussed multi-dimensional light wave manipulation via single-layer dielectric metasurfaces. To showcase such a strategy, the “structural birefringence” of meta-atoms on different polarization bases is considered, and the modulation capabilities from single to multiple parameter dimensions are categorized. Based on the proposed mechanism, complete manipulation of the wavefront amplitude, phase, and polarization state has been achieved, and the poly-Si meta-atoms and holographic method are employed to experimentally demonstrate how various functionalities are achieved. The results show that single-layer dielectric metasurfaces exhibit strong modulation capability in various light wave manipulation, and the design principle is simple but has powerful extension for the flexible design of optical metasurfaces. This work offers a systematic and generalizable method toward manipulating light waves at will with meta-devices, and provides a possible approach for achieving more abundant manipulation and applications through hybrid structures.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 91850118, 11774289, 11634010, 61675168, 12074313, and 11804277), the National Key Research and Development Program of China (Grant No. 2017YFA0303800), the Natural Science Basic Research Program of Shaanxi (Grant No. 2020JM-104), the Fundamental Research Funds for the Central Universities (Grant Nos. 3102019JC008 and 310201911cx022), and the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (Grant Nos. CX202046, CX202047, and CX202048). We thank the Zhiwei Song of National Center for Nanoscience and Technology for supplying the materials as well as the Analytical and Testing Center of Northwestern Polytechnical University.

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

The metasurfaces were fabricated based on the process of deposition, patterning, lift off, and etching. At first, a 350 nm (610 nm)-thick poly-Si film was deposited on a 500 μm-thick fused silica substrate by inductively coupled plasma enhanced chemical vapor deposition (ICPECVD), and then a 100 nm-thick hydrogen silsesquioxane electron beam spin-on resist (HSQ, XR-1541) was spin-coated onto the poly-Si film and baked on a hot plate at 100 °C for 2 min. Next, the desired structures were imprinted by using standard electron beam lithography (EBL, Nanobeam Limited, NB5) and subsequently developed in NMD-3 solution (concentration 2.38%) for 2 min. Finally, by using inductively coupled plasma etching (ICP, Oxford Instruments, Oxford Plasma Pro 100 Cobra300), the desired structures were transferred from resistance to the poly-Si film.

1.
W.
Cai
and
V. M.
Shalaev
,
Optical Metamaterials
(
Springer
,
2010
).
2.
Y.
Liu
and
X.
Zhang
, “
Metamaterials: A new frontier of science and technology
,”
Chem. Soc. Rev.
40
,
2494
(
2011
).
3.
W.
Cai
,
U. K.
Chettiar
,
A. V.
Kildishev
, and
V. M.
Shalaev
, “
Optical cloaking with metamaterials
,”
Nat. Photonics
1
,
224
(
2007
).
4.
J. B.
Pendry
, “
Negative refraction makes a perfect lens
,”
Phys. Rev. Lett.
85
,
3966
(
2000
).
5.
D. R.
Smith
,
W. J.
Padilla
,
D. C.
Vier
,
S. C.
Nemat-Nasser
, and
S.
Schultz
, “
Composite medium with simultaneously negative permeability and permittivity
,”
Phys. Rev. Lett.
84
,
4184
(
2000
).
6.
D.
Schurig
,
J. J.
Mock
,
B. J.
Justice
,
S. A.
Cummer
,
J. B.
Pendry
,
A. F.
Starr
, and
D. R.
Smith
, “
Metamaterial electromagnetic cloak at microwave frequencies
,”
Science
314
,
977
(
2006
).
7.
R. A.
Shelby
,
D. R.
Smith
, and
S.
Schultz
, “
Experimental verification of a negative index of refraction
,”
Science
292
,
77
(
2001
).
8.
P. V.
Kapitanova
,
P.
Ginzburg
,
F. J.
Rodríguez-Fortuño
,
D. S.
Filonov
,
P. M.
Voroshilov
,
P. A.
Belov
,
A. N.
Poddubny
,
Y. S.
Kivshar
,
G. A.
Wurtz
, and
A. V.
Zayats
, “
Photonic spin Hall effect in hyperbolic metamaterials for polarization-controlled routing of subwavelength modes
,”
Nat. Commun.
5
,
3226
(
2014
).
9.
G. A.
Wurtz
,
R.
Pollard
,
W.
Hendren
,
G. P.
Wiederrecht
,
D. J.
Gosztola
,
V. A.
Podolskiy
, and
A. V.
Zayats
, “
Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality
,”
Nat. Nanotechnol.
6
,
107
(
2011
).
10.
T.
Cai
,
G.
Wang
,
S.
Tang
,
H.
Xu
,
J.
Duan
,
H.
Guo
,
F.
Guan
,
S.
Sun
,
Q.
He
, and
L.
Zhou
, “
High-efficiency and full-space manipulation of electromagnetic wave fronts with metasurfaces
,”
Phys. Rev. Appl.
8
,
034033
(
2017
).
11.
L.
Huang
,
X.
Chen
,
H.
Mühlenbernd
,
H.
Zhang
,
S.
Chen
,
B.
Bai
,
Q.
Tan
,
G.
Jin
,
K.-W.
Cheah
,
C.-W.
Qiu
,
J.
Li
,
T.
Zentgraf
, and
S.
Zhang
, “
Three-dimensional optical holography using a plasmonic metasurface
,”
Nat. Commun.
4
,
2808
(
2013
).
12.
L.
Jin
,
Z.
Dong
,
S.
Mei
,
Y. F.
Yu
,
Z.
Wei
,
Z.
Pan
,
S. D.
Rezaei
,
X.
Li
,
A. I.
Kuznetsov
,
Y. S.
Kivshar
,
J. K. W.
Yang
, and
C.-W.
Qiu
, “
Noninterleaved metasurface for (26-1) spin-and wavelength-encoded holograms
,”
Nano Lett.
18
,
8016
(
2018
).
13.
J.
Li
,
S.
Chen
,
H.
Yang
,
J.
Li
,
P.
Yu
,
H.
Cheng
,
C.
Gu
,
H.-T.
Chen
, and
J.
Tian
, “
Simultaneous control of light polarization and phase distributions using plasmonic metasurfaces
,”
Adv. Funct. Mater.
25
,
704
(
2015
).
14.
S.
Liu
,
T. J.
Cui
,
Q.
Xu
,
D.
Bao
,
L.
Du
,
X.
Wan
,
W. X.
Tang
,
C.
Ouyang
,
X. Y.
Zhou
,
H.
Yuan
,
H. F.
Ma
,
W. X.
Jiang
,
J.
Han
,
W.
Zhang
, and
Q.
Cheng
, “
Anisotropic coding metamaterials and their powerful manipulation of differently polarized terahertz waves
,”
Light: Sci. Appl.
5
,
e16076
(
2016
).
15.
D.
Wen
,
F.
Yue
,
G.
Li
,
G.
Zheng
,
K.
Chan
,
S.
Chen
,
M.
Chen
,
K. F.
Li
,
P. W. H.
Wong
,
K. W.
Cheah
,
E.
Yue Bun Pun
,
S.
Zhang
, and
X.
Chen
, “
Helicity multiplexed broadband metasurface holograms
,”
Nat. Commun.
6
,
8241
(
2015
).
16.
W.
Liu
,
Z.
Li
,
Z.
Li
,
H.
Cheng
,
C.
Tang
,
J.
Li
,
S.
Chen
, and
J.
Tian
, “
Energy-tailorable spin-selective multifunctional metasurfaces with full Fourier components
,”
Adv. Mater.
31
,
1901729
(
2019
).
17.
J.
Li
,
P.
Yu
,
S.
Zhang
, and
N.
Liu
, “
Electrically-controlled digital metasurface device for light projection displays
,”
Nat. Commun.
11
,
3574
(
2020
).
18.
T.
Xu
,
Y.-K.
Wu
,
X.
Luo
, and
L. J.
Guo
, “
Plasmonic nanoresonators for high-resolution colour filtering and spectral imaging
,”
Nat. Commun.
1
,
59
(
2010
).
19.
D. J.
Roth
,
M.
Jin
,
A. E.
Minovich
,
S.
Liu
,
G.
Li
, and
A. V.
Zayats
, “
3D full-color image projection based on reflective metasurfaces under incoherent illumination
,”
Nano Lett.
20
,
4481
(
2020
).
20.
S.
Chen
,
Z.
Li
,
W.
Liu
,
H.
Cheng
, and
J.
Tian
, “
From single-dimensional to multidimensional manipulation of optical waves with metasurfaces
,”
Adv. Mater.
31
,
1802458
(
2019
).
21.
Z. L.
Deng
,
M.
Jin
,
X.
Ye
,
S.
Wang
,
T.
Shi
,
J.
Deng
,
N.
Mao
,
Y.
Cao
,
B. O.
Guan
,
A.
Alù
,
G.
Li
, and
X.
Li
, “
Full-color complex-amplitude vectorial holograms based on multi-freedom metasurfaces
,”
Adv. Funct. Mater.
30
,
1910610
(
2020
).
22.
S.
Wang
,
Z.
Deng
,
Y.
Wang
,
Q.
Zhou
,
X.
Wang
,
Y.
Cao
,
B.
Guan
,
S.
Xiao
, and
X.
Li
, “
Arbitrary polarization conversion dichroism metasurfaces for all-in-one full Poincaré sphere polarizers
,”
Light: Sci. Appl.
10
,
24
(
2021
).
23.
S.
Chen
,
K.
Li
,
J.
Deng
,
G.
Li
, and
S.
Zhang
, “
High-order nonlinear spin–orbit interaction on plasmonic metasurfaces
,”
Nano Lett.
20
,
8549
(
2020
).
24.
L.
Huang
,
S.
Zhang
, and
T.
Zentgraf
, “
Metasurface holography: From fundamentals to applications
,”
Nanophotonics
7
,
1169
(
2018
).
25.
B.
Xu
,
H.
Li
,
S.
Gao
,
X.
Hua
,
C.
Yang
,
C.
Chen
,
F.
Yan
,
S.
Zhu
, and
T.
Li
, “
Metalens-integrated compact imaging devices for wide-field microscopy
,”
Adv. Photonics
2
,
066004
(
2020
).
26.
S.
Chen
,
W.
Liu
,
Z.
Li
,
H.
Cheng
, and
J.
Tian
, “
Metasurface-empowered optical multiplexing and multifunction
,”
Adv. Mater.
32
,
1805912
(
2020
).
27.
L.
Deng
,
J.
Deng
,
Z.
Guan
,
J.
Tao
,
Y.
Chen
,
Y.
Yang
,
D.
Zhang
,
J.
Tang
,
Z.
Li
,
Z.
Li
,
S.
Yu
,
G.
Zheng
,
H.
Xu
,
C. W.
Qiu
, and
S.
Zhang
, “
Malus-metasurface-assisted polarization multiplexing
,”
Light: Sci. Appl.
9
,
101
(
2020
).
28.
G.
Li
,
G.
Sartorello
,
S.
Chen
,
L. H.
Nicholls
,
K. F.
Li
,
T.
Zentgraf
,
S.
Zhang
, and
A. V.
Zayats
, “
Spin and geometric phase control four-wave mixing from metasurfaces
,”
Laser Photonics Rev.
12
,
1800034
(
2018
).
29.
G.
Qu
,
W.
Yang
,
Q.
Song
,
Y.
Liu
,
C.-W.
Qiu
,
J.
Han
,
D.-P.
Tsai
, and
S.
Xiao
, “
Reprogrammable meta-hologram for optical encryption
,”
Nat. Commun.
11
,
5484
(
2020
).
30.
C.
Zhang
,
S.
Xiao
,
Y.
Wang
,
Y.
Gao
,
Y.
Fan
,
C.
Huang
,
N.
Zhang
,
W.
Yang
, and
Q.
Song
, “
Dynamic perovskite metasurfaces: Lead halide perovskite-based dynamic metasurfaces
,”
Laser Photonics Rev.
13
,
1970030
(
2019
).
31.
Z.
Li
,
C.
Chen
,
Z.
Guan
,
J.
Tao
,
S.
Chang
,
Q.
Dai
,
Y.
Xiao
,
Y.
Cui
,
Y.
Wang
,
S.
Yu
,
G.
Zheng
, and
S.
Zhang
, “
Three-channel metasurfaces for simultaneous meta-holography and meta-nanoprinting: A single-cell design approach
,”
Laser Photonics Rev.
14
,
2000032
(
2020
).
32.
T.
Li
,
X.
Li
,
S.
Yan
,
X.
Xu
, and
S.
Zhu
, “
Generation and conversion dynamics of dual Bessel beams with a photonic spin-dependent dielectric metasurface
,”
Phys. Rev. Appl.
15
,
014059
(
2021
).
33.
X.
Fan
,
P.
Li
,
X.
Guo
,
B.
Li
,
Y.
Li
,
S.
Liu
,
Y.
Zhang
, and
J.
Zhao
, “
Axially tailored light field by means of a dielectric metalens
,”
Phys. Rev. Appl.
14
,
024035
(
2020
).
34.
X.
Guo
,
P.
Li
,
J.
Zhong
,
S.
Liu
,
B.
Wei
,
W.
Zhu
,
S.
Qi
,
H.
Cheng
, and
J.
Zhao
, “
Tying polarization-switchable optical vortex knots and links via holographic all-dielectric metasurfaces
,”
Laser Photonics Rev.
14
,
1900366
(
2020
).
35.
P.
Li
,
X.
Guo
,
J.
Zhong
,
S.
Liu
,
Y.
Zhang
,
B.
Wei
, and
J.
Zhao
, “
Optical vortex knots and links via holographic metasurfaces
,”
Adv. Phys.: X
6
,
1843535
(
2021
).
36.
R. J.
Lin
,
V.-C.
Su
,
S.
Wang
,
M. K.
Chen
,
T. L.
Chung
,
Y. H.
Chen
,
H. Y.
Kuo
,
J.-W.
Chen
,
J.
Chen
,
Y.-T.
Huang
,
J.-H.
Wang
,
C. H.
Chu
,
P. C.
Wu
,
T.
Li
,
Z.
Wang
,
S.
Zhu
, and
D. P.
Tsai
, “
Achromatic metalens array for full-colour light-field imaging
,”
Nat. Nanotechnol.
14
,
227
(
2019
).
37.
X.
Guo
,
P.
Li
,
B.
Li
,
S.
Liu
,
B.
Wei
,
W.
Zhu
,
J.
Zhong
,
S.
Qi
, and
J.
Zhao
, “
Visible-frequency broadband dielectric metahologram by random Fourier phase-only encoding
,”
Sci. China: Phys., Mech. Astron.
64
,
214211
(
2021
).
38.
X.
Gao
,
X.
Han
,
W.-P.
Cao
,
H. O.
Li
,
H. F.
Ma
, and
T. J.
Cui
, “
Ultrawideband and high-efficiency linear polarization converter based on double V-shaped metasurface
,”
IEEE Trans. Antennas Propag.
63
,
3522
(
2015
).
39.
P. C.
Wu
,
W.
Zhu
,
Z. X.
Shen
,
P. H. J.
Chong
,
W.
Ser
,
D. P.
Tsai
, and
A.-Q.
Liu
, “
Broadband wide-angle multifunctional polarization converter via liquid-metal-based metasurface
,”
Adv. Opt. Mater.
5
,
1600938
(
2017
).
40.
W.
Ye
,
F.
Zeuner
,
X.
Li
,
B.
Reineke
,
S.
He
,
C. W.
Qiu
,
J.
Liu
,
Y.
Wang
,
S.
Zhang
, and
T.
Zentgraf
, “
Spin and wavelength multiplexed nonlinear metasurface holography
,”
Nat. Commun.
7
,
11930
(
2016
).
41.
S.
Chen
,
B.
Reineke
,
G.
Li
,
T.
Zentgraf
, and
S.
Zhang
, “
Strong nonlinear optical activity induced by lattice surface modes on plasmonic metasurface
,”
Nano Lett.
19
,
6278
(
2019
).
42.
Y.
Tang
,
Y.
Intaravanne
,
J.
Deng
,
K. F.
Li
, and
G.
Li
, “
Nonlinear vectorial metasurface for optical encryption
,”
Phys. Rev. Appl.
12
,
024028
(
2019
).
43.
H.
Cheng
,
Z.
Liu
,
S.
Chen
, and
J.
Tian
, “
Emergent functionality and controllability in few-layer metasurfaces
,”
Adv. Mater.
27
,
5410
(
2015
).
44.
Z.-L.
Deng
,
J.
Deng
,
X.
Zhuang
,
S.
Wang
,
K.
Li
,
Y.
Wang
,
Y.
Chi
,
X.
Ye
,
J.
Xu
,
G. P.
Wang
,
R.
Zhao
,
X.
Wang
,
Y.
Cao
,
X.
Cheng
,
G.
Li
, and
X.
Li
, “
Diatomic metasurface for vectorial holography
,”
Nano Lett.
18
,
2885
(
2018
).
45.
A.
Cui
,
Z.
Liu
,
J.
Li
,
T. H.
Shen
,
X.
Xia
,
Z.
Li
,
Z.
Gong
,
H.
Li
,
B.
Wang
,
J.
Li
,
H.
Yang
,
W.
Li
, and
C.
Gu
, “
Directly patterned substrate-free plasmonic ‘nanograter’ structures with unusual Fano resonances
,”
Light: Sci. Appl.
4
,
e308
(
2015
).
46.
S.-Q.
Li
,
X.
Xu
,
R.
Maruthiyodan Veetil
,
V.
Valuckas
,
R.
Paniagua-Domínguez
, and
A. I.
Kuznetsov
, “
Phase-only transmissive spatial light modulator based on tunable dielectric metasurface
,”
Science
364
,
1087
(
2019
).
47.
Y.
Qu
,
Q.
Li
,
K.
Du
,
L.
Cai
,
J.
Lu
, and
M.
Qiu
, “
Dynamic thermal emission control based on ultrathin plasmonic metamaterials including phase-changing material GST
,”
Laser Photonics Rev.
11
,
1700091
(
2017
).
48.
J.
Li
,
P.
Yu
,
H.
Cheng
,
W.
Liu
,
Z.
Li
,
B.
Xie
,
S.
Chen
, and
J.
Tian
, “
Optical polarization encoding using graphene-loaded plasmonic metasurfaces
,”
Adv. Opt. Mater.
4
,
91
(
2016
).
49.
M.
Khorasaninejad
and
F.
Capasso
, “
Metalenses: Versatile multifunctional photonic components
,”
Science
358
,
eaam8100
(
2017
).
50.
F.
Ding
,
B.
Chang
,
Q.
Wei
,
L.
Huang
,
X.
Guan
, and
S. I.
Bozhevolnyi
, “
Versatile polarization generation and manipulation using dielectric metasurfaces
,”
Laser Photonics Rev.
14
,
2000116
(
2020
).
51.
B.
Wang
,
F.
Dong
,
H.
Feng
,
D.
Yang
,
Z.
Song
,
L.
Xu
,
W.
Chu
,
Q.
Gong
, and
Y.
Li
, “
Rochon-prism-like planar circularly polarized beam splitters based on dielectric metasurfaces
,”
ACS Photonics
5
,
1660
(
2017
).
52.
A. C.
Overvig
,
S.
Shrestha
,
S. C.
Malek
,
M.
Lu
,
A.
Stein
,
C.
Zheng
, and
N.
Yu
, “
Dielectric metasurfaces for complete and independent control of the optical amplitude and phase
,”
Light: Sci. Appl.
8
,
92
(
2019
).
53.
Z.
Bomzon
,
V.
Kleiner
, and
E.
Hasman
, “
Pancharatnam–Berry phase in space-variant polarization-state manipulations with subwavelength gratings
,”
Opt. Lett.
26
,
1424
(
2001
).
54.
R. C.
Devlin
,
A.
Ambrosio
,
N. A.
Rubin
,
J. P. B.
Mueller
, and
F.
Capasso
, “
Arbitrary spin-to-orbital angular momentum conversion of light
,”
Science
358
,
896
(
2017
).
55.
J. P.
Balthasar Mueller
,
N. A.
Rubin
,
R. C.
Devlin
,
B.
Groever
, and
F.
Capasso
, “
Metasurface polarization optics: Independent phase control of arbitrary orthogonal states of polarization
,”
Phys. Rev. Lett.
118
,
113901
(
2017
).
56.
R.
Zhao
,
B.
Sain
,
Q.
Wei
,
C.
Tang
,
X.
Li
,
T.
Weiss
,
L.
Huang
,
Y.
Wang
, and
T.
Zentgraf
, “
Multichannel vectorial holographic display and encryption
,”
Light: Sci. Appl.
7
,
95
(
2018
).
57.
J. W.
Cooley
and
J. W.
Tukey
, “
An algorithm for the machine calculation of complex Fourier series
,”
Math. Comput.
19
,
297
(
1965
).
58.
R. W.
Gerchberg
and
W. O.
Saxton
, “
A practical algorithm for the determination of phase from image and diffraction plane pictures
,”
Optik
35
,
237
(
1972
).
59.
P.
Li
,
Y.
Zhang
,
S.
Liu
,
L.
Han
,
H.
Cheng
,
F.
Yu
, and
J.
Zhao
, “
Quasi-Bessel beams with longitudinally varying polarization state generated by employing spectrum engineering
,”
Opt. Lett.
41
,
4811
(
2016
).
60.
T.
Čižmár
and
K.
Dholakia
, “
Tunable Bessel light modes: Engineering the axial propagation
,”
Opt. Express
17
,
15558
(
2009
).
61.
T.
Li
,
H.
Liu
,
S.
Wang
,
X.
Yin
,
F.
Wang
,
S.
Zhu
, and
X.
Zhang
, “
Manipulating optical rotation in extraordinary transmission by hybrid plasmonic excitations
,”
Appl. Phys. Lett.
93
,
021110
(
2008
).
62.
Y.
Zhao
and
A.
Alù
, “
Tailoring the dispersion of plasmonic nanorods to realize broadband optical meta-waveplates
,”
Nano Lett.
13
,
1086
(
2013
).