Arbitrary amplitude and phase control in visible by dielectric metasurface

Qiang Jiang; Leyong Hu; Guangzhou Geng; Junjie Li; Yongtian Wang; Lingling Huang

doi:10.1364/OE.454967

1. Introduction

Manipulating the amplitude, phase, and polarization of an optical field is essential to numerous engineering applications. Conventionally, the modulation of amplitude and phase can be achieved using traditional optical elements, such as lenses, gratings, and prism. By removing the redundant phase, diffractive optical elements (DOEs) have been proposed to shrink the size of optical system. Using DOEs, optical field manipulation with flexible control at pixel level can be realized. Spatial light modulators (SLMs), a device consists of many repeated units and can modulates the amplitude or phase of the light wave independently in space and time at pixel-level, such as liquid crystal on silicon (LCoS) devices and digital micro-mirror devices (DMD), were excellent candidates for controlling light fields. However, these two types of SLMs cannot control the complex amplitude of the light field directly. Nevertheless, in the area of holography, only by using the complex hologram can the object be reconstructed without missing any information, especially for high resolution displays that make full use of the bandwidth product. Therefore, independently controlling the spatial amplitude and phase distributions of an electromagnetic wave is essential to forming the desired accurate optical field. Methods such as multi-SLM [1], single SLM with space-division modulation [2] and super pixel [3] methods have been proposed to arbitrarily modulate the complex amplitude. However, challenges such as the precise alignment, low resolution, low refresh rate and complex setup have restricted the applications of these methods. Moreover, the principle of phase accumulation adopted by traditional optical elements leads to physically large systems.

To meet the trend of miniaturization and integration of optoelectronic devices without sacrificing performance or even improving it, great efforts have been made in reducing the size, weight and cost. Metasurface, one type of the so-called third generation optical element which consist of numerous artificial nanostructures with meticulous feature [4], possessing the powerful ability in arbitrarily modulating the amplitude, phase and polarization of a local field in subwavelength scale, have been proposed in recent years. The subwavelength feature leads to growing potential for the design of ultra-thin electromagnetic flat optical components and systems, such as achromatic metalenses [5], holograms with high resolution and enlarged viewing angle [6], vortex beam generators [7], compact optical spectrometers [8] and so on.

Through subtle design, metasurface can modulate either amplitude or phase. Binary amplitude modulation realized by scatters such as photon sieves and dielectric nanostructures is widely used for high resolution amplitude hologram and optical switcher [9–11]. Phase modulation is often achieved by geometric metasurfaces, Huygens’metasurfaces and high-contrast metasurfaces by rotating the antennas, spectrally overlapping of electric and magnetic dipole resonances and treating each nanoantenna as a nanoscale optical waveguide, respectively [12,13]. For arbitrary complex amplitude modulation, different designs of metasurface from microwave to visible region have been proposed [14–26]. From the perspective of metasurface structure, the existing metasurfaces with complex amplitude modulation in visible region can be classified into two kinds, single-layered metasurface [14–23], and bi-layered metasurface [24–26]. The latter often makes high requirements on processing technology and alignment accuracy, while the former reduces these requirements, but spatially multiplexed metasurface sacrifices the resolution [19–22]. Only single-atom units in single-layered metasurface maintains high resolution along with lower process requirements. In the visible region, the X-shaped meta-atom achieves complex amplitude modulation by varying the angles of two rods. However, rods will cross each other when the angle difference is too small, resulting in a maximin polarization conversion efficiency limited to 49% [14]. Besides, pillars with different heights and rotations can realize complex amplitude modulation but with unwanted high order diffraction [18]. In a word, transmissive complex-amplitude modulation metasurface in the visible region with high resolution, high efficiency, easy processing, arbitrary complex-amplitude modulation, and large modulation depths are rarely reported. Here, we present an ultrathin dielectric metasurface with arbitrary and simultaneous control of the amplitude and phase at visible frequencies in transmission mode. The amplitude is controlled by varying the width of one rod, while the phase is controlled by the in-plane orientation of the meta-atoms. The orthogonal structure introduces new freedom of adjustment, and also avoids the cross-overlap problem like the X-shape meta-atom. With a thickness of ∼λ/5, the fabrication of these dielectric metasurface not only CMOS compatible, but also has the potential for mass-fabricated via nanoimprinting. To demonstrate the arbitrary complex amplitude modulation, complex amplitude fields for generating holographic images and structure light are encoded on metasurface, and the reconstructed results are analyzed by simulation and experiment.

2. Principle and design

In this work, an ultra-thin dielectric cross-shaped metasurface have been proposed for independent and simultaneous control of arbitrary amplitude and phase in the visible region. Titanium dioxide (TiO₂) rods embedded in polydimethylsiloxanes (PDMS) that support multi-mode resonances have been delicately designed to meet the first Kerker condition for efficient complex amplitude modulation. The amplitude is modulated by changing the geometric size of each selected rod to deviate from the Kerker condition, and the phase is controlled by the rotation angle based on PB phase principle. With two parameters independently control the amplitude and phase, the mapping relationship between complex amplitude and the modulation parameters is simpler than other structures, making it easier to realize arbitrary complex amplitude control. It should be mentioned that while ensuring the required refractive index difference, the PDMS also prevents the microstructure from being contaminated by dust. The inset of Fig. 1(a) shows the proposed structure. The parameters can be adjusted according to the required operating wavelength. Figure 1(a) shows the distribution of amplitude and phase of the RCP light when W₁ and θ vary. The phase changes with θ, but does not change much when W₁ changes. The phase response is composed of dynamic phase and geometric phase. While the amplitude varies with both W₁ and θ.

Fig. 1. Half-wave plate feature of the proposed meta-atom. (a) the propose orthogonal rods above fuse silica is embedded in PDMS. Arbitrary amplitude and phase can be modulated by varying W₁ and rotating angle θ respectively. (b) Transmission coefficients of two branches. (c) Polarization conversion efficiency under circular polarized light.

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With this sophisticated design scheme, the light-matter interaction can be greatly enhanced to realize high efficiency scattering. According to Jones matrix, the scattering property of this proposed meta-atom can be expressed as

(1)$$J = \left[ {\begin{array}{{cc}} {({{\tilde{t}}_1} + {{\tilde{t}}_2})/2}&{({{\tilde{t}}_1} - {{\tilde{t}}_2}){e^{2i\theta }}/2}\\ {({{\tilde{t}}_1} - {{\tilde{t}}_2}){e^{ - 2i\theta }}/2}&{({{\tilde{t}}_1} + {{\tilde{t}}_2})/2} \end{array}} \right].$$

where ${\tilde{t}_1}$=t₁exp(-iφ₁) and ${\tilde{t}_2}$=t₂exp(-iφ₂) are complex transmission coefficients along two rods. As shown in Fig. 1(b), the amplitudes of ${\tilde{t}_1}$ and ${\tilde{t}_2}$ along two rods are close to unity at the target working wavelength of 606nm, and the phase difference equals to π for the largest polarization conversion. Taking the high polarization conversion as an example, which is often used in pure phase modulation cases, we set the period of antenna to 390 nm, the lengths of two rods to L₁= 300 nm and L₂= 150 nm, while the widths to W₁= 110 nm, W₂= 80 nm, the thickness H to 140 nm. Hence the anisotropic meta-atom behaves as a half-wave plate at this specific wavelength, leading to a high polarization conversion efficiency (PCE) under the illumination of left/right circular polarized (LCP/RCP) light. Figure 1(c) shows a PCE as higher as 95.7% at wavelength of 606nm.

The dominant modes can be analyzed and determined by expanding the enhanced scattering field into a series of multipole emitters according to the Cartesian multipole expansion technique [27]

(2)$${\textrm{P}_{sca}} \approx \frac{{{c^2}k_0^4{Z_0}}}{{12\mathrm{\pi }}}{\left| {\mathrm p} \right|^2} + \frac{{k_0^4{Z_0}}}{{12\mathrm{\pi }}}{\left| {\mathrm m} \right|^2} + \frac{{{c^2}k_0^6{Z_0}}}{{1440\mathrm{\pi }}}{\left| {\mathop {Q}\limits^ \wedge} \right|^2} + \frac{{k_0^6{Z_0}}}{{160\mathrm{\pi }}}{\left| {\mathop {\mathrm M}\limits^ \wedge} \right|^2}\textrm{ + } \ldots .$$

where Z₀ is the vacuum wave impedance, c is the speed of light, k₀ is the wavenumber in vacuum and p, m, $\mathop {Q}\limits^ \wedge $ and $\mathop {M}\limits^ \wedge $ are the multipole moments which can be calculated based on the light-induced polarization P(r)=ɛ₀(ɛ_d-1)E(r), where ɛ₀ and ɛ_d are the vacuum permittivity and relative dielectric permittivity, respectively. E(r) is the total electric field inside the nanoparticle. The first four terms in Eq. (2) correspond to the electric dipole (ED), magnetic dipole (MD), electric quadrupole (EQ) and magnetic quadrupole (MQ) modes, respectively. By exquisitely constructing the supported multipole resonances, the desired amplitude response of the meta-atom can be easily engineered. Figure 2 shows the multipolar decomposition results of scattering field of the proposed structure under LCP incident light, from which we can see that the total contributions T_f of the first four terms are consistent with the total scattering field T_s. These form terms dominate the three obvious peaks in the total scattered field. These peaks are contributed by the combination of multipole resonances. The weaker scattering peak at 576 nm is mainly contributed by MQ and ED modes, while peak at 587nm is dominated by EQ and MD modes. For the highest peak at 606nm, all these four modes play import roles. At this specific wavelength, EQ and MQ modes, ED and MD modes intersect with each other, namely, the amplitudes for dipoles are equal, so are quadrupoles, as shown in the insets of Fig. 2. The phase differences between two dipoles and quadrupoles are analyzed respectively. The inset in the left of Fig. 2 shows a phase difference of zero between ED and MD, means that the first KerKer condition is fully met for dipoles, with ED and MD in phase (Φ=0) and having equal amplitude. Meanwhile, we notice that EQ and MQ are also in phase and have the same amplitude. Thus, the backscattering can be totally suppressed and an enhanced forward scattering can be achieved.

Fig. 2. Multipole expansion of the total scattering field. Only the first four terms (ED, MD, EQ, MQ) are analyzed. The highest peak at 606 nm is dominated by both dipoles and quadrupoles. The left inset shows the amplitudes of ED and MD, and the phase difference between these two. While the right inset shows that for EQ and MQ.

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By varying the geometric parameters L or W slightly around the delicately designed parameters mentioned above, the KerKer condition will slowly become unsatisfied, leading to a slow change in amplitude of the scattering light. In our work, W₁ is changed in the range of 60-110nm for amplitude modulation, while the phase of scattering light is controlled based on PB phase principle. Thus, by searching the intersection in these two distribution maps, arbitrary complex wavefront modulation can be flexibly implemented by the proposed meta-atom.

3. Result and discussion

To demonstrate the arbitrary amplitude and phase control of the proposed metasurface and the application of complex amplitude modulation, we encode two kinds of complex amplitude fields that generate holographic images and structure light on metasurface, respectively, and analyze the reconstructed results.

The advantages of different holograms can be evaluated by comparing the quality of the reconstructed images. Holograms generated by the complex-amplitude angular spectrum method (CA-ASM) and phase-only ASM (PO-ASM) were utilized for the comparison. In addition, as the Gerchberg–Saxton algorithm (GSA) is often employed for the generation of high-quality holographic image, hologram calculated by the GSA was also used for comparison. These three holograms, as shown in Fig. 3(a) (e) and (i), are encoded on the proposed metasurface. For the convenience of description, these three holograms are abbreviated as HCA-ASM, HPO-ASM, and HGSA, respectively. The particle swarm optimization (PSO) was utilized to search the suitable parameters of meta-atoms from library built by FDTD simulation for each hologram with 16 amplitude levels or 16 phase levels. As is shown in Fig. 3 (b), (f) and (j), the complex amplitude values mapped to the searched parameters A_sexp(iφ_s) are very close to the theoretical values A_texp(iφ_t), with most of the errors between searched amplitude (phase) values and real amplitude (phase) values around 0. The tiny errors may affect the quality of the reconstructed images. Slight unwanted noise can be seen in the four corners of the logo reconstructed from searched HCA-ASM by simulation, as shown in Fig. 3(c). While the tiny error seems have little effect on the image reconstructed from HPO-ASM since the quality of this image is too low. However, for phase-only GS hologram, the errors in amplitude disturb the image reconstructed from the searched HGSA. When taking a closer look in Fig. 3(k), it reveals a low-intensity inverted image superimposed on the reconstructed image. This unwanted low-intensity inverted image is reconstructed from the searched amplitude which is modulated by the designed phase since we used the figure of merit (FOM) |A_texp(iφ_t)-A_sexp(iφ_s)| in the PSO. We have verified in simulation that the superimposed image can be depressed either by adjusting the weight of A_s of the FOM, or replacing the searched A_s by unity value with keeping the searched phase φ_s unchanged.

Fig. 3. Hologram and holographic images. (a), (e), (i): holograms generated by CA-ASM, PO-ASM and GSA, respectively. (b), (f), (j): amplitude and phase errors for these three holograms between searched values from PSO and calculated values. (c), (g), (k): images reconstructed by searched values. (d), (h), (l): holographic images captured in experiment. (m), (n): the comparison of details in (c) and (k), (d) and (l), respectively.

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Three samples, each has an area of 700 µm×700 µm, were applied to obtain the desired holographic images in the experiment. A supercontinuum source (SuperK EVO, NKT Photonics) and multiwavelength filter (SuperK SELECT, NKT Photonics, 400-650nm) were used as the light source. The reconstructed images at the plane of nearly 1.25mm were then collected and magnified by an objective (Nikon, NA = 0.65, 40×) and a tube lens. A color CCD camera (Thorlabs 8051C-USB) was then used to capture the reconstructed images after the analyzer. The captured reconstructed images of these three samples are shown in Fig. 3(d), (h) and (l), and are very consistent with the simulated result. The reconstruction results from HCA-ASM is very clear, better than that from HGSA, and significantly better than that from HPO-ASM. To intuitively demonstrate the quality improvement of the reconstructed image from HCA-ASM, we compare the details along the white dash line in Fig. 3(c) and (k), Fig. 3(d) and (l), and exhibit them in Fig. 3(m) and (n) respectively. Apparently, the intensity and SNR of image reconstructed from HCA-ASM is better than that from HGSA either in the simulated results or experiment results. MSE, defined as [28]

(3)$$(1/M \times N){\sum\nolimits_{i,j} {|I(i,j) - {I_0}(i,j)|} ^2},$$

is used to estimate the SNR. The MSE values for the experimental results in Fig. 3(d),(h) and (l) are 0.1495, 0.174 and 0.1554, respectively. The experimental result in Fig. 3(l) presents a more distinct inverted image overlaps with the reconstructed logo, from which we can see the dove wings at the top superimposing on the tail of the eagle at the bottom and the eagle tail at the bottom overlap to the body of dove at the top. The dotted box in Fig. 3(m) and (n) indicate the positions of the inner ring and tail of the inverted image, it exhibits the intensity of undesired overlapped image in experiment has a higher intensity than that in simulation. This may be caused by fabrication error. The shortest protruding part of the designed short arm, which can be express as (L₂-W₁)/2, is only 40nm, which poses a challenge to the processing ability. The samples, each containing 1794×1794 pixels, were fabricated by using standard EBL exposure and ALD processes. Figure 4(a) shows the processing flow of metasurfaces. As can be seen from Fig. 4(b) which displays the SEM images of the samples before spin-coating PDMS, the protruding part of the short arm is no longer a regular rectangle due to fabrication error, which enlarges the error in amplitude modulation.

It should be mentioned that with the geometric phase dominating the whole phase modulation, the designed metasurface was confirmed to work over a wide range. As shown in Fig. 5, the quality of the images reconstructed from HCA-ASM, HPO-ASM and HGSA does not change much when the wavelength increases from 600nm to 650nm. In addition, when taking a closer look at the “1940” at the bottom of images from HCA-ASM in Fig. 5(a), it can be seen that as the wavelength increases, the image quality improves, but the intensity decreases. From the perspective of image quality, due to the thickness error of TiO₂, the actual working wavelength seems to be shifted to around 640 nm instead of the design value of 606 nm. The efficiency obtained by directly measuring the holographic image is the result of multiplying the polarization conversion efficiency and the transmittance. The measured efficiency decreases from 18.2% to 3.4% when the illumination wavelength changes from 600 nm to 650 nm for the HCA-ASM structure, and from 24.1% to 7.9% for HPO-ASM structure. However, for HGSA structure, the efficiency improves to 45.8% at 600 nm. Though the proposed complex-amplitude metasurface theoretically exhibits high polarization conversion efficiency, Fig. 3(a) exhibits that most of the amplitude value is zero, not all unity as in phase-only hologram, thus resulting in a low efficiency. For the images reconstructed from HGSA, as shown in Fig. 5(c), if we set wavelength around 640 nm as the actual wavelength, as the wavelength deviates from 640 nm, the overlapped image becomes more obvious dues to the amplitude deviation, while for images reconstructed from HPO-ASM, in Fig. 5(b), the background noise decreases when wavelength gets longer.

Fig. 4. (a) processing flow. (b) SEM images of the fabricated metasurface before spin-coating PDMS.

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Fig. 5. The proposed complex-amplitude metasurface exhibits broadband feature in the range of 600-650 nm in the experiment. (a),(b) and (c) are images reconstructed from HCA-ASM, HPO-ASM and HGSA respectively.

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The quality of optical reconstruction of the hologram depends on many factors, including the algorithm design, encoding technique, and optical systems. To further verify the reliability of the proposed structure for arbitrary complex amplitude modulation, complex amplitude fields that generate a complex structured trapping beam are also studied [29]. We generate the complex amplitude by focusing the polymorphic beam and encode it on metasurface. Figure 6 (a) and (d) show the desired structured trapping beam, which is a ring of uniform intensity and varying phase gradient. The complex amplitude in Fig. 6(b) and (e) are encoded on the proposed metasurface with an error of less than 10%. By using FDTD simulation and far field projection, we obtain the field distribution of this structured trapping beam in the focal plane, as exhibited in Fig. 6 (c) and (f). It shows that the amplitude is uniformly distributed along the ring and the phase varies periodically along the ring, that is, the complex fields generated by the proposed complex-amplitude metasurface are in good agreement with the desired structured trapping beam.

Fig. 6. structured trapping beam generated by the proposed structure. (a) and (d) are the amplitude and phase distribution of the desired structured trapping beam; (b) and (e) are the complex amplitude for generating the structured trapping beam; (c) and (f) are the results by simulation.

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4. Conclusion

In summary, we have demonstrated an ultrathin single-cell metasurface with complex-amplitude modulation for the CP incidences in the visible region. The proposed metasurface is composed of two orthogonal TiO₂ rods with different lengths and widths. By adjusting one of the geometric parameters and rotating the whole meta-atom, the amplitude and phase can be simultaneously and independently controlled, realizing the arbitrary complex amplitude modulation. By utilizing the proposed metasurface, it is experimentally confirmed that the quality of holographic image reconstructed from complex-amplitude hologram obtained by the angular spectrum method is much better than that reconstructed from phase-only hologram obtained by angular spectrum method, and better than holographic image from phase-only GS algorithm. The reliability of the proposed structure for the modulation of arbitrary complex amplitude is further verified by generating a complex structured trapping beam in simulation. The PDMS background layer can prevent external dust from polluting the nanostructure of metasurface. In addition, the structure can reconstruct a clear holographic image at wide wavelength region of 600-650nm. The thickness of the meta-atom is almost one fifth of the wavelength, making it possible to be fabricated by nanoimprinting, which has the advantage of low-cost and large-area manufacturing.

Funding

National Natural Science Foundation of China (62005017, 92050117, U21A20140); Open Research Fund Program of State Key Laboratory of Precision Measurement Technology and Instruments (TH20-02); Beijing Outstanding Young Scientist Program of Ministry of Science and Technology of China (BJJWZYJH01201910007022); National Key Research and Development Program of China (2021YFA1401200).

Disclosures

The authors declare no conflict of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Arbitrary amplitude and phase control in visible by dielectric metasurface

Abstract

1. Introduction

2. Principle and design

3. Result and discussion

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (3)

Optics Express