- Photonics Research
- Vol. 10, Issue 4, 1031 (2022)
Abstract
1. INTRODUCTION
As a technique to fully engineer the wavefronts of light, holography has demonstrated remarkable modulation abilities for both free-space beams [1–4] and surface waves [5–7], and facilitated applications of wavefront shaping, data storage, three-dimensional display, and so on. However, for optical holographic video display and complex optical encryption, the bandwidth is still limited, and undesired diffraction orders exist in using traditional approaches. The metasurface, proposed as a novel artificial planar element with subwavelength units, has become a powerful platform for hologram recording in recent years. It can overcome the above limitations and also provide unprecedented spatial resolution and a large field of view (FOV). Over the past decade, delicately designed meta-atoms have shown flexible light manipulation properties, such as amplitude [6,8–13], phase [8–15], polarization [14,16–21], orbital angular momentum (OAM) [22–26], and frequency [12,25,27–31]. Through coding the meta-atoms with diverse holograms in different optical channels, holographic information capacity can be further augmented, and multiplexed functions are available [8,10–14,17–27,31–33].
Among the fundamental light properties, polarization records the vectorial nature of light containing rich invisible information to human eyes. Due to the unique advantages of metasurfaces in designing anisotropic optical response, polarization multiplexed holography based on various metasurface design strategies has emerged [8–12,14,18–21,25,34–37], and greatly promotes potential applications such as optical document security and optical switching devices. However, most schemes are dependent on orthogonal polarization states [14,19,21,24], rather than using all states on the full Poincaré sphere, which decreases the versality and information capacity of polarization multiplexed holography. As one of the most widely used strategies, the spatial multiplexing method [8,20,34,35] relying on supercells suffers from low conversion efficiency and undesired diffraction orders. Especially for supercell metasurfaces tailored by the detour phase [8,14,36], oblique incidence becomes necessary and does not adapt to applications working under normal illumination. So far, progress has put forward challenging requirements for an advanced encoding method and more delicate metasurface design to achieve greater polarization modulation possibilities, more information capacity, and high conversion efficiency.
Here, we propose a novel metasurface encoding method to achieve polarization multiplexed holography, which can realize diverse holographic mappings from one full-Stokes space to another efficiently with subwavelength units. As shown in Fig. 1, based on metasurfaces encoded with vectorial holograms with unbounded possibilities, we can selectively address the intensity distributions in the transmitted field according to input and output polarization states, which greatly improves the availability of polarization modes and information capacity. The delicately designed metasurfaces have high polarization conversion in the circular polarization channels, and can achieve independent phase modulations that combine dynamic phase and geometric phase. With the help of a hybrid genetic algorithm, we generated phase-only holograms to synthesize multiple vectorial fields based on the designed meta-atoms. By loading the time-varying polarization channels in sequence, a holographic video and complex optical encryption with double secret keys have been experimentally demonstrated in real time. Such a scheme opens new avenues for multi customizable polarization modulations, and is expected to be used in dynamic display, dynamic optical manipulation, optical encryption, and anticounterfeiting.
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Figure 1.Principle of polarization multiplexed holography for time sequence holographic video and optical encryption based on metasurfaces. (a) Time-dependent image frames encoded in diverse linear polarization states ranging from zero to
2. RESULTS AND DISCUSSION
A. Principle of Polarization Multiplexed Holography
Mathematically, at a given input polarization , the polarization multiplexed hologram can be expressed as the superposition of sub-holograms with diverse polarization states: , where , , and represent the amplitude, phase, and polarization state of the th hologram. Here, we utilize the interaction of the input polarized plane wave and metasurface to construct , which satisfies the relationship , and is the Jones matrix of metasurfaces at each pixel. As we build the vectorial images in the far field, the reconstructed field can be regarded as the Fourier transform of , namely, , where is the Fourier transform operator. Polarization state connected with the th hologram is independent of the spatial coordinate and not involved in the process of Fourier transform, so we can recast the reconstructed field as according to the linear property of Fourier transform. By imposing polarization selection to the whole field with a spatial inhomogeneous vectorial nature, we can modulate the optical field and intensity distribution to and , respectively. Clearly, the intensity of each vectorial component is governed by the polarization correlation, and the polarization of the sub-image with maximum intensity distribution is consistent with . Thus, a holographic mapping from customized to diverse is available for optical data storage and information processing. By simplifying the Jones matrix of nanostructures into two off-diagonal components, we can further build the holographic connections between diverse input polarization states and specific output polarization selections clearly.
Next, we describe the recording rules of vectorial holography to achieve dynamic modulations by diverse input/output polarization channels. Since arbitrary is described by two orthogonally polarized vectors, at least two polarization channels are required to digitize the vectorial holography. We decomposed the vectorial holography into two circularly polarized holograms, which can be recorded with metasurfaces in subwavelength pixels commendably. The complex-amplitude holograms of right circular polarization (RCP) and left circular polarization (LCP) channels can be expressed as and , respectively, where and are the left vector of RCP and LCP, respectively. On the other hand, the phase-only holograms of RCP and LCP channels can be described as and , respectively, where and in some cases. To efficiently reconstruct the desired field, the final phase-only holograms and are usually provided by suitable optimization algorithms. Obviously, the reconstructed field in the far field decomposed as satisfies the conditions as and , where and are the complex amplitude fields of RCP and LCP channels in the far field, respectively. We purposely designed the metasurface as a transmitted half-wave plate with sufficient phase modulation capacity, whose Jones matrix can be described as the following equation in the circular polarization basis:
When we use another polarized input to interact with metasurfaces, the output RCP and LCP holograms become and approximately, and the synthetic multiplexed hologram at input can be expressed as
B. Metasurface Design
Next, we introduce the design strategy of metasurface digitalization. First, we developed an efficient approach to design meta-atoms with separated and sufficient phase modulations in and channels. As the most popular method of metasurface designs, geometric phase, also known as Pancharatnam–Berry (PB) phase, utilizes polarization evolution to realize continuous phase modulation ability related to azimuthal rotations. For meta-atoms with rotation angle , a phase response can be introduced to and , respectively. To make the phase modulation of and independent, we integrated geometric phase together with dynamic phase. We chose amorphous silicon (α-Si) nanofins with rectangular cross sections as modulated units, which have excellent optical response in the near infrared band and the same dynamic phase response in and channels resulting from geometric symmetry. That is, the and of each modulated unit are equal to and , respectively, where can be tailored by the change of geometric cross sections. We conducted a parameter sweep of -Si nanofins with diverse cross sections by the rigorous coupled wave analysis (RCWA) method to determine suitable structures. The period and height are fixed at 400 nm and 600 nm, respectively, the refractive index we used for calculation is at a wavelength of 800 nm, and the sweeping range of the width and length is from 80 nm to 240 nm. The amplitude response and the calculated dynamic phase of the channel are shown in Fig. 2(b). Among the nanofins with amplitude response above 0.83, 56 structures were selected to cover a full phase range at the same time. Afterwards, by rotating the selected nanofins, we got an effective database to encode each separate hologram.
Figure 2.Metasurface design and digitalization based on hybrid genetic algorithm. (a) Schematic illustration of phase modulation by combining dynamic phase and geometric phase. (b) Simulated results for the amplitude and dynamic phase of
Then we designed a hybrid genetic algorithm to calculate the phase-only holograms of RCP and LCP channels based on the structural database. As shown in Fig. 2(c), we first generated the initial metasurface by randomly setting the lengths, widths, and rotations based on the above nanofins. The constructed fields of both circularly polarized channels and can be calculated based on diffraction theory. To reconstruct the vectorial images, we have three target parameters: intensity distribution , polarization components and of all images, and the design degrees of freedom (DOFs) containing only two phase distributions and . In this case, we nested an alternative selection in the iterative loops to manipulate more parameters by a few ones. The idea is that if the number of target parameters is larger than the number of DOFs , at least (a nature number rounded up) alternative selections are nested in the iterative loop. Here, we need to nest the selections only twice. The complete loop begins with the odd iteration number (iterative number ; is positive integer), and in this case, we selected the absolute phase of as the imaging phase . With the later replacement of target intensities and polarization distributions and , the next holograms can be generated by inverse Fourier transform and nanofin digitalization. The iteration number becomes even (), and in this selection process, we chose the absolute phase of calculated as , which has the characteristic of previous and target fields. The replacement process was kept the same as in the previous step. Through the hybrid iteration, both characteristics of the two polarization channels can be preserved based on the above database of nanofins. Therefore, holography recording based on metasurfaces is algorithmically realized.
C. Experimental Characterizations
We designed and fabricated two metasurfaces to demonstrate our method. As described in Fig. 1, the first is the time sequence holographic display of a prose poem, and the second is an optical encryption composed of a metasurface and two separate keys. Nano fabrication depends on the standard electron beam lithography and inductively coupled plasma reactive ion etching process. Scanning electron microscopy images of sample 1 (dynamic display) and sample 2 (optical encryption) are shown in Fig. 3(a) and Fig. 3(b). Both metasurfaces have a footprint of . By using the experimental setup shown in Fig. 3(c), we collected holographic images at diverse input/output polarization selections.
Figure 3.Scanning electron microscopy images of the fabricated samples and the experimental setup. (a), (b) Scanning electron microscopy images of sample 1 (time sequence holographic video) and sample 2 (optical encryption), composed of diverse cross sections and rotations. (c) Experimental setup for vectorial holography, which works at various input/output polarization channels.
1. Time Sequence Holographic Video
To facilitate practical application, we encoded the time-dependent frames with linearly polarized input that orbits around the equator of the Poincaré sphere, and the analyzer in the output space was fixed without any adjustment. For the th word of the poem, we designed various phase delays into the RCP and LCP holograms, which helps to create the multiplexed sequence. At the initial -polarized incidence, the output poem is composed of diverse linearly polarized words before modulation of the analyzer, where is the intensity distribution of the th part, is the total number of words and equals 16, and in the circularly polarized basis . We changed the input linear polarization angle and described it with (that is, ), and the linear polarization angle of the analyzer was set as (). The collected intensity of the th vectorial component at this input/output polarization channel can be calculated based on the above principles, which can be expressed as
Here, we make be zero, and in the poem ranges from 0° to 180°. Thus, when the incident changes from 0° to 180°, the word with maximum intensity can be selectively addressed in sequence. The designed metasurface is composed of nanofins, with different cross sections and rotations optimized by the hybrid genetic algorithm. As shown in Fig. 3(c), we used and a half-wave plate to generate varied linearly polarized light, and in the output, we removed the quarter-wave plate and fixed as polarized. The simulated and experimental results shown in Fig. 4(c) are consistent with the theoretical design; the poem is dynamically displayed by varying the incident polarization angle , following the order of designed . In addition, the collected profile of the input / output channel shows uniform distribution of each word, and the detected signals of and channels are relatively weak (see Fig. 6 in Appendix A).
Figure 4.Design principle of the time sequence holographic video of Tagore’s poem. (a) Time-dependent frames encoded in the linearly polarized input, which round the equator of the Poincaré sphere. (b) The preset phase modulation
Above all, the dynamic display of holographic videos with high performance has been experimentally realized. It does not require adjustment of the analyzer at the output end, which is more practical for system integration. By increasing the multiplexed number, the movie can be encoded with more information. Furthermore, different display requirements are also available by the polarization multiplexed strategy. For example, for videos played in linearly polarized input/output channels, the display time of sub-images can be precisely customized according to the ellipticity, and real-time editing, rewinding, and inserting of the movie can be easily realized based on diverse polarization selections. Compared with other multiplexed schemes, such as utilizing incident wavelength, OAM, and nonlinear frequencies as multiplexing parameters, the polarization multiplexed strategy shows the advantage of simple manipulation. Various functionalities can be realized by modulating the input/output polarizations. In contrast, nonlinear frequency multiplexing needs high incident intensity, large nonlinear susceptibility, and a complex design to meet phase matching conditions. OAM multiplexing relies on a spatially distributed phase profile generated from spiral plates or spatial light modulators. Furthermore, polarization multiplexing based on our method can facilitate applications of compact devices with the integration of liquid crystal.
2. Optical Encryption with High Complexity
The first design mainly depends on linear polarization multiplexing of input/output channels. In the second design, we extended the polarization modulation range to the full-Stokes space, and a demonstration of optical encryption with high security is provided as follows. The secret information is hidden in multiple polarization multiplexed images, which has many possibilities due to distinct input/output polarization selections. As shown in Fig. 5(b), the designed pattern of a dove with five wings and leaves has seven polarization states, which can convert an arbitrary incident polarization state (except for circular polarization states) into different polarization states in the output full-Stokes space. Upon -polarized incidence, we describe the vectorial dove as , where , and and imply the changes of amplitude and phase in circularly polarized channels, respectively. Next, we change input polarization to another state , and the constructed vectorial field becomes . That is, the generated vectorial field and the output intensity of each sub-image change upon different illuminance conditions except for linearly polarized input. With further decoding with an analyzer that can be expressed as , we can get a reliable tool to encode the secret information into polarization multiplexed images. The intensity distribution of the th vectorial component at the input / output polarization channel can be expressed as follows:
Figure 5.Design principle of the optical encryption with double secret keys. (a) First secret key provided by input polarization states. The blue dots represent some deceptive keys, and the blue star marked at
Figure 6.Experimental results of sample 1 at diverse input and output polarization selections. The blue arrows and orange arrows are polarization states of input and output, respectively.
For the use of optical encryption, we set the first secret key as -polarized incidence. When using the first secret key to interact with the metasurface, all vectorial patterns are displayed in Fig. 5(c). The second secret key is set as polarized, which is expected to show brighter third and fourth wings and the absence of second wings; the experimental result in the third column of Fig. 5(e) is consistent with the theoretical design. Furthermore, much deceptive information can be realized at other input/output polarization channels (results at more input/output polarization selections are provided in Figs. 7 and 8, Appendix A). Since the proposed encoding rules are based on the linear additivity of Fourier transformation, it is more suitable for large capacity information multiplexing compared with other schemes. The polarization states of input and output as secret keys have innumerable possibilities in the full-Stokes space, and the complexity and security of such optical encryption are extremely high, especially when the three elements (metasurfaces and two secret keys) are preserved and transferred separately.
Figure 7.Experimental results of sample 2 at six diverse input and output polarization selections. The blue arrows and orange arrows are polarization states of input and output, respectively.
Figure 8.Theoretical results of sample 2 at six diverse input and output polarization selections. The blue arrows and orange arrows are polarization states of input and output, respectively.
3. CONCLUSION
In conclusion, we proposed and realized polarization multiplexed holography based on metasurfaces optimized by a hybrid genetic algorithm, which demonstrated the viability and versatility of time sequence dynamic display and optical encryption. Through full-Stokes polarization transformations and delicate metasurface design, multiple and independent input/output polarization modulations can be customized flexibly in two full-Stokes spaces. The subwavelength feature of the nanofin without coherent pixels or any spatial multiplexing not only decreases fabrication difficulties, but also ensures a larger FOV. Based on the efficient and independent phase encoding of a circular polarization basis, the output vectorial wavefront can be freely customized within an observable numerical aperture that reaches 0.80 (see Appendix A, Note 2) and conversion efficiency reaches 40.24%. The designed holographic videos with large information capacity have more displayed possibilities by using different polarization combinations. The optical encryption shows significantly improved performance compared with other polarization encryption schemes in security and data capacity. This method is expected to be used in dynamic display and beam shaping, polarization detection, holographic tweezers, optical encryption/anticounterfeiting, and so on.
4. METHODS
APPENDIX A
This section includes the following.
Figure
Figure
Figure
Note 1. Conversion efficiency of metasurfaces (including Figs.
Figure 9.Conversion efficiency and broadband behavior of sample 1. (a) Data processing of holographic construction efficiency, using ROIs generated by the algorithm to extract effective information. (b) Polarization conversion efficiencies of four circularly polarized channels. (c) Conversion efficiencies
Figure 10.Conversion efficiency and broadband behavior of sample 2. (a) Data processing of holographic construction efficiency, using ROIs generated by the algorithm to extract effective information. (b) Polarization conversion efficiencies of four circularly polarized channels. (c) Conversion efficiencies
Note 2. Numerical aperture of vectorial holography.
Note 3. Dynamic display of the designed vectorial patterns.
Other materials for this paper include Visualizations 1 and 2.
The conversion efficiency of polarization multiplexed holography is defined as the ratio of effective diffracted power with circular polarization conversion to incident power. It equals the product of polarization conversion efficiency and holographic reconstruction efficiency. That is, , , where the subscripts represent incident/transmitted polarization channels, and superscripts and represent holography and polarization conversion, respectively.
First, we measured and calculated the polarization conversion efficiency of circularly polarized channels based on the optical setup shown in Fig.
Furthermore, to get accurate efficiency for each vectorial component of the holographic image, we calculated the holographic reconstruction efficiency of circularly polarized channels based on an image recognition algorithm. Through such data processing, we defined the region of interest (ROI) of collected images and extracted effective information. ROIs at different wavelengths were generated according to comparisons between collected images with the designed patterns, which serve as a mask to extract holographic imaging. By calculating the ratio of extracted information to the initial images as shown in Figs.
The method we used ensures that each subwavelength unit is connected with all vectorial fields, which is totally different from the spatial multiplexing scheme and guarantees a larger FOV. The numerical aperture (NA) of metasurfaces is defined as
Here, we used varied linearly polarized input and -polarized output selections to test the dynamic property and large information capacity of our devices. Two movies include:
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