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Phase-assisted angular-multiplexing nanoprinting based on the Jacobi-Anger expansion

Kuixian Chen, Jiaxin Li, Guodong Zhu, Weiguo Zhang, Zhixue He, Guoxing Zheng, and Zile Li

Open Access Open Access

Abstract

Featuring with ultracompactness and subwavelength resolution, metasurface-assisted nanoprinting has been widely researched as an optical device for image display. It also provides a platform for information multiplexing, and a series of multiplexed works based on incident polarizations, operating wavelengths and observation angles have emerged. However, the angular-multiplexing nanoprinting is realized at the cost of image resolution reduction or the increase of fabrication difficulty, hindering its practical applications. Here, inspired by the Jacobi-Anger expansion, a phase-assisted design paradigm, called Bessel metasurface, was proposed for angular multiplexing nanoprinting. By elaborately designing the phase distribution of the Bessel metasurface, the target images can be encoded into the desired observation angles, reaching angular multiplexing. With the merits of ultracompactness and easy fabrication, we believe that our design strategy would be attractive in the real-world applications, including optical information storage, encryption/concealment, multifunctional switchable optical devices, and 3D stereoscopic displays, etc.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Metasurface, made of elaborately designed sub-wavelength nanostructures periodically arranged on a planar substrate, has been widely researched as an optical material for manipulating the electro-magnetic waves [1–11], leading to various optical applications of holography [12–16], imaging [17,18], nanoprinting [19–21], and etc. Along with the continuous advances in nano-fabrication [22], commercially viable metasurfaces are going to become a reality. Thanks to the exciting characteristics of ultrathin nature and mature fabrication, metasurface provides a novel way for image display. For instance, a metasurface-assisted nanoprinting is formed by spatially varied intensity or/and color distributions at the interface of fabricated samples, and the experimental result has demonstrated its competitive edges of ultracompactness, subwavelength resolution near the optical diffraction limit, non-fading, etc [23,24]. Recently, there sprung up a series of multiplexed nanoprinting works based on incident polarizations [25–33], operating wavelengths [34,35] or observation angles [36–38]. For instance, as for the angular multiplexing nanoprinting, Bao et al. proposed a design strategy, called coherent pixel, for encoding multiple nanoprinting images at the cost of image resolution reduction [37]. Wang et al. optimized and fabricated the asymmetric plasmonic nanostructures for reaching the angular multiplexing, but these inclined nanopillars would inevitably raise the difficulty level of fabrication, causing in an unwanted increase in the cost [38]. In conclusion, it is still challenging to achieve the angular multiplexing nanoprinting with the characteristics of ultracompactness and easy fabrication.

In this study, taking inspiration from the Jacobi-Anger expansion and the phase-only image hologram based on the interference recording [39], a phase-assisted design paradigm, called Bessel metasurface, was proposed for angular multiplexing nanoprinting, as indicated in Fig. 1. Here, we theoretically and experimentally demonstrate the feasibility and practical performance of the Bessel metasurface in realizing angular multiplexing. By elaborately designing the phase distribution of the Bessel metasurface, the target images can be encoded into the desired observation angles. Compared with the aforementioned design strategies, our method breaks the limitation of image resolution reduction while does not increase the burden of the nanostructure fabrication, which makes the Bessel metasurface attractive for practical applications. Therefore, our research paves a convenient and efficient way for achieving angular multiplexing nanoprinting, which enriches the study of multifunctional metasurfaces, and possesses promising applications in optical information encryption, multifunctional switchable optical devices, and advanced virtual reality, etc.

 figure: Fig. 1.

Fig. 1. Schematic illustration of angular multiplexing nanoprinting based on a phase-assisted Bessel metasurface that simultaneously encodes independent nanoprinting images into different observation angles of θ1 and θ2. The incident light is a left circularly polarized light beam.

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2. Working principle of intensity manipulation based on the phase-assisted angular-multiplexing nanoprinting

Firstly, for illustrating the intensity manipulation of a phase-assisted Bessel metasurface, we start with a simple case of a phase-only grating with cosine distribution (${e^{ia\cos \theta }}$). According to the Jacobi-Anger expansion, this expression can be expanded as a sum of Bessel functions:

$$E(x,y) = {e^{ia(x,y)\cos \theta (x,y)}} = \sum\limits_{n ={-} \infty }^{ + \infty } {{i^n}{J_n}[a(x,y)]{e^{in\theta (x,y)}}}, $$
where Jn is the nth order of the Bessel function of the first kind, the amplitude of each order depends on the distribution of a(x, y), θ(x, y) is an initial phase with periodic distribution, and its periodicity would influence the spatial frequency of each order. It is obviously that the wavefront of the phase-only grating can be divided into many parts, which corresponds to different diffraction orders of the phase-only grating during the reconstruction. When the incident light illuminates the phase-only grating, the complex amplitude distribution of nth diffraction order of output light can be expressed as ${i^n}{J_n}[a(x,y)]{e^{in\theta (x,y)}}$. Therefore, by elaborately designing the distribution of a(x, y), the intensity manipulation can be achieved on the surface of the grating. This unique intensity modulation gifted by the phase-only grating paves an original and minimalist way for nanoprinting and the target information can be directly obtained from the encoded phase distribution $a(x,y)\cos [\theta (x,y)]$. To go a further step, owing to the outstanding performance of phase grating in beam shaping/deflection, the observation angle of the encoded meta-image can be manipulated by simply attaching periodic phase, thus the phase-assisted intensity modulation provides a new platform for meta-images multiplexing. Here, we take the 0th order of the Bessel function as the information channel, and the periodicity of the initial phase θ is 600 nm.

3. Design of angular multiplexed nanoprinting

Thanks to the capability of blazed gratings in beam deflection, we can attach a periodic phase distribution of a blazed grating to the aforementioned phase-only grating for deflecting the information channel (i.e. 0th order of the Bessel function) to the designed observation angle. Thus the phase-only grating can be re-expressed as

$${e^{i[a(x,y)\cos \theta (x,y) + \beta (x,y)]}} = \sum\limits_{n ={-} \infty }^{ + \infty } {{i^n}{J_n}[a(x,y)]{e^{in\theta (x,y)}}{e^{i\beta (x,y)}}} , $$
where β(x, y) is a periodic phase distribution for beam deflection. Owing to the additional phase β(x, y), the complex amplitude distribution of nth diffraction order changes to ${i^n}{J_n}[a(x,y)]{e^{in\theta (x,y)}}{e^{i\beta (x,y)}}$. It is clearly observed that the amplitude of nth diffraction order keeps unchanged even with the attachment of the periodic phase distribution, which means the intensity manipulation of phase-assisted Bessel metasurface would not be affected by the additional periodic phase. Therefore, we can arbitrarily vary the observation angle via changing the periodicity of the periodic phase, which is hardly achieved in traditional nanoprinting. Based on this design flexibility of observation angle, we can easily reach angular multiplexing nanoprinting by encoding different target meta-images into different observation angles.

As a proof of concept, we take dual meta-images multiplexing to demonstrate the design paradigm of the angular multiplexed nanoprinting. The target meta-images and the flowchart of the angular multiplexing process are indicated in Fig. 2. In step 1, based on the 0th order of the Bessel function of the first kind, the information of the meta-images would be encoded into the phase distributions of the holograms, which can be derived as

$$\left\{ {\begin{array}{*{20}{c}} {{\varphi_1}(x,y) = J_0^{ - 1}[\sqrt {{I_1}(x,y)} ]\cos \theta (x,y)}\\ {{\varphi_2}(x,y) = J_0^{ - 1}[\sqrt {{I_2}(x,y)} ]\cos \theta (x,y)} \end{array}} \right., $$
where I­1(x, y) and I­2(x, y) represent the intensity distributions of the two multiplexed meta-images respectively, and $J_0^{ - 1}$ is the zero-order anti-Bessel function of the first kind. In step 2, according to the obtained phase distributions (φ1(x, y) and φ­2(x, y)) and selected observation angles, the final phase φf(x, y) can be calculated by
$${\varphi _f}(x,y) = \arg \{ {e^{i[{\varphi _1}(x,y) + {\beta _1}(x,y)]}} + {e^{i[{\varphi _2}(x,y) + {\beta _2}(x,y)]}}\} , $$
where β1(x, y) and β2(x, y) are two periodic phase distributions with different periodicities, which are designed for deflecting the information channels to their given observation angles. Here, the periodicities of β1(x, y) and β2(x, y) are 0 nm and 1200 nm, thus the corresponding observation angles of the two multiplexed meta-images are 0° and 26.7° at the operating wavelength of 540 nm (i.e. the included angle between the direction of the 0th diffraction order and the normal direction of the metasurface), respectively. Moreover, for simplifying the fabrication of the Bessel metasurface, we chose to only keep the phase distribution of the complex amplitude instead of considering the complex amplitude distribution, i.e. the fabricated sample, consisting of silicon-on-insulator (SOI) material, is only composed of single-sized nanostructures. The designed phase can be realized by manipulate the orientations of the single-sized nanostructures according to the working principle of geometric phase [13]. With the aforementioned selected parameters, we further utilize silicon nanobrick arrays sitting on a bi-layer substrate of silica and silicon to form the Bessel metasurface. After meticulously designing the geometric size of these nanostructures (CS = 300 nm, L = 200 nm, W = 100 nm and H = 220 nm), each working unit can function as a phase retarder at the operating wavelength of 540 nm as shown in Fig. 3(a).

 figure: Fig. 2.

Fig. 2. Design procedure of angular multiplexing nanoprinting based on a phase-assisted Bessel metasurface. Two independent target greyscale images with 500 × 500 pixels are selected for multiplexing design. First, based on the Bessel-function-assisted encoding algorithm, the target information can be encoded into two phase gratings (with different phase distribution φ1 and φ2). Then, the final phase distribution can be obtained by only keep the phase of the superposition of complex amplitudes.

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 figure: Fig. 3.

Fig. 3. (a) Schematic diagram and simulated reflectivity of a unit-cell of the phase-assisted Bessel metasurface consisted of SOI material. The thickness of the silica layer is 2 μm. R­co and Rcross represent the reflectivities of co- and cross-polarized light, respectively. (b) Simulated intensity profiles for different information channels with (top) and without (bottom) considering the amplitude distribution at the operating wavelength of 540 nm.

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4. Experiment and discussion

Next, Kirchhoff diffraction was employed for calculating the effect of not considering the amplitude distribution, i.e., only keep the phase distribution of the complex amplitude. To explore the performance of the phase-assisted angular multiplexing nanoprinting, the simulation results (with or without considering the amplitude information of the complex amplitude distribution in step 2) are shown in Fig. 3(b). It is clearly observed that there are nearly no image blurring and information loss, and the encoded meta-images can be intactly obtained in simulation results, which verifies the feasibility of realizing angular multiplexed nanoprinting based on the phase-assisted Bessel metasurface. However, when we zoomed in on the obtained images of simulations, it can be found that there existed a little crosstalk in both information channels. Although the phase-only simulation results look slightly different from the simulation results of considering the amplitude information, considering the amplitude information would bring the burden to the metasurface fabrication since the nanostructures with varied geometric dimensions are required. Considering the trade-off between the little crosstalk of information channels and convenience in fabrication, we chose to employ a phase-only metasurface, which would bring the phase-assisted Bessel metasurface one step closer to real-world applications.

To investigate the practical performance of angular multiplexed nanoprinting, a single-sized Bessel metasurface was fabricated with electron beam lithography, and a CMOS digital camera (Moticam X) was employed for observation at the wavelength of 540 nm. As indicated in Fig. 4(a), the nanoprinting images are collected by using a 10× objective with a numerical aperture of NA = 0.25 (aperture angle 14.5°) and the aforementioned CMOS camera. Owing to the filtering of the numerical aperture, the effect of non-0th diffraction orders would be eliminated. By changing the incident angle from 0° to 26.7°, the experimental results of different observation channels can be obtained, as shown in Fig. 4(b). We can see that the experimental results with incident angles of 0° and 26.7° have a good visual effect, and there only exists little crosstalk in the experiment. Comparing with the simulated phase-only results in Fig. 3(b), it is clearly observed that the crosstalk of the two information channels is consistent with the simulation results, and the little blurring of encoded information is mainly originated from not considering the amplitude of the complex amplitude distribution in the design flowchart. Moreover, we further took the scanning electron microscope (SEM) image, which demonstrates the specific fabrication errors including size mismatch and round errors of these fabricated nanostructures as illustrated in Fig. 4(c). In general, the great agreement between simulation and experimental results at the operating wavelength of 540 nm has shown that our design strategy of Bessel metasurface is robust against the fabrication errors, which makes the proposed Bessel metasurface suitable for practical applications.

 figure: Fig. 4.

Fig. 4. Experimental illustration of the angular multiplexing nanoprinting based on the phase-assisted Bessel metasurface. (a) Optical setups for observing the angle-multiplexed metasurface. By varying the incident angle, the multiplexed information can be decoded. The incident light is left circularly polarized. (b) Experimentally captured images of different incident angles with the incident wavelength of 540 nm. The inset angles of the experimental images are the incident angles of the fabricated sample. The sizes of the captured images are both 150 × 150 μm2. (c) The scanning electron microscope (SEM) image of the fabricated Bessel metasurface. The scale bar is 2 μm.

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5. Conclusion

In summary, we theoretically and experimentally demonstrated angular multiplexing nanoprinting based on the phase-assisted Bessel metasurface. According to the Jacobi-Anger expansion, we can easily reach information multiplexing and encode the multiplexed information into different observation channels via deliberately designing the orientation arrangement of the single-sized nanostructures. Moreover, the observation angles of the multiplexed information channels can be further encrypted by selecting the operating wavelength. In our demonstrated sample, the good agreement between the target images and the experimental results further verifies the feasibility and the outstanding performance of the phase-assisted Bessel metasurface. Besides, the fabricated sample is composed of the widely available SOI material, and owing to the simple fabrication and compatibility of SOI, this Bessel metasurface can be mass manufactured via the complementary metal-oxide-semiconductor (CMOS) processing platform, gifting it great opportunity for large-scale commercial applications. Overall, the proposed phase-assisted Bessel metasurface for angular multiplexing can further enrich metasurface multiplexing schemes, and potentially facilitate the state-of-the-art optical information storage, encryption/concealment, multifunctional switchable optical devices, and 3D stereoscopic displays, etc.

Appendix A. Sample fabrication

The Bessel metasurface was fabricated with a standard EBL process based on SOI substrate which possesses a mid-layer silica of 2 µm and a top-layer crystalline silicon of 220 nm. First of all, a mask of polymethyl methacrylate (PMMA) was patterned on the aforementioned SOI substrate via the EBL process. Then a thermal evaporator was employed to deposite a 30 nm Cr film on the PMMA layer, and Cr played the role of etch mask here. After that, the sample was cleaned with ultrasonic waves by immersing it into hot acetone at 75 °C. Subsequently, a solution mixed with 250 sccm (standard-state cubic centimeter per minute) SF6, 95 sccm O2 (at 200 WRF power) and 300 sccm CHF3 was employed for removing the part without Cr by reactive ion etching. Finally, by removing Cr with the Cr etchant, only silicon nanobricks remained on the substrate.

Funding

National Key Research and Development Program of China (2021YFE0205800); National Natural Science Foundation of China (11904267, 12174292, 91950110); Natural Science Foundation of Hubei Province (ZRMS2021000211); Fundamental Research Funds for the Central Universities (2042021kf0018).

Disclosures

The authors declare no conflicts 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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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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Figures (4)
Fig. 1.
Fig. 1. Schematic illustration of angular multiplexing nanoprinting based on a phase-assisted Bessel metasurface that simultaneously encodes independent nanoprinting images into different observation angles of θ1 and θ2. The incident light is a left circularly polarized light beam.

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Fig. 2.
Fig. 2. Design procedure of angular multiplexing nanoprinting based on a phase-assisted Bessel metasurface. Two independent target greyscale images with 500 × 500 pixels are selected for multiplexing design. First, based on the Bessel-function-assisted encoding algorithm, the target information can be encoded into two phase gratings (with different phase distribution φ1 and φ2). Then, the final phase distribution can be obtained by only keep the phase of the superposition of complex amplitudes.

View in Article | Download Full Size | PDF

Fig. 3.
Fig. 3. (a) Schematic diagram and simulated reflectivity of a unit-cell of the phase-assisted Bessel metasurface consisted of SOI material. The thickness of the silica layer is 2 μm. R­co and Rcross represent the reflectivities of co- and cross-polarized light, respectively. (b) Simulated intensity profiles for different information channels with (top) and without (bottom) considering the amplitude distribution at the operating wavelength of 540 nm.

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. Experimental illustration of the angular multiplexing nanoprinting based on the phase-assisted Bessel metasurface. (a) Optical setups for observing the angle-multiplexed metasurface. By varying the incident angle, the multiplexed information can be decoded. The incident light is left circularly polarized. (b) Experimentally captured images of different incident angles with the incident wavelength of 540 nm. The inset angles of the experimental images are the incident angles of the fabricated sample. The sizes of the captured images are both 150 × 150 μm2. (c) The scanning electron microscope (SEM) image of the fabricated Bessel metasurface. The scale bar is 2 μm.

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Equations (4)

Equations on this page are rendered with MathJax. Learn more.

E ( x , y ) = e i a ( x , y ) cos ⁡ θ ( x , y ) = ∑ n = − ∞ + ∞ i n J n [ a ( x , y ) ] e i n θ ( x , y ) ,
e i [ a ( x , y ) cos ⁡ θ ( x , y ) + β ( x , y ) ] = ∑ n = − ∞ + ∞ i n J n [ a ( x , y ) ] e i n θ ( x , y ) e i β ( x , y ) ,
{ φ 1 ( x , y ) = J 0 − 1 [ I 1 ( x , y ) ] cos ⁡ θ ( x , y ) φ 2 ( x , y ) = J 0 − 1 [ I 2 ( x , y ) ] cos ⁡ θ ( x , y ) ,
φ f ( x , y ) = arg ⁡ { e i [ φ 1 ( x , y ) + β 1 ( x , y ) ] + e i [ φ 2 ( x , y ) + β 2 ( x , y ) ] } ,
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