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Abstract
In this paper, coding Huygens’ metasurface (CHM) is proposed for holographic imaging with enhanced quality. A weighted holographic algorithm is used to calculate the phase distribution at the interface and to design the CHM. Experimental demonstration performed in the microwave region validates holographic imaging with the ability to modulate energy distribution among focal points and improve image quality. By judiciously engineering both electric and magnetic dipolar resonators, the proposed digital Huygens’ meta-atom is able to provide a full transmission–phase covering the whole range of 2π together with a near-unity transmission efficiency. The proof-of-concept experiments show that holographic imaging quality can be indeed improved by using digital meta-atoms with several bits. Furthermore, the modulation of intensity distribution among focal points is experimentally realized by using the 3-bits CHM. The proposed CHM hologram shows great potential in a variety of application fields, such as programmable high-resolution imaging lenses, microscopy, data storage, information processing, and computer-generated holograms.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Metasurfaces, the two-dimensional (2D) version of metamaterials, show great ability in manipulating wavefronts in arbitrary manner by introducing corresponding phase discontinuities at the interface. A metasurface is typically composed of an array of subwavelength artificial meta-atoms with specially designed geometries [1] and orientations [2,3], and several devices have been implemented based on the construction of metasurfaces, including couplers [4,5], beam shapers [6,7], invisibility cloaks [8,9], imaging systems [10–14], and other functional devices [15–19]. A main challenge for most metasurfaces operating in transmission mode is to fully control the transmission phase, while at the same time being able to guarantee high transmission efficiency [20–24]. It can also be solved by using the so called Huygens’ metasurface [25] that enables to fully control both the phase and the amplitude of co-polarized transmitted wave without polarization conversion losses. Due to the outstanding ability in manipulating electromagnetic waves, applications of the Huygens’ metasurface have been reported in beam-refracting [26], focusing [27,28], beam-shaping [29], and hologram imaging [30].
Recently, the concept of coding metasurfaces has been proposed [31] and applied in beam-editing [32], diffuse scattering [33] and energy radiations controlling [34]. Coding metasurface presenting the ability to describe information in a digital way is usually composed of digital meta-atoms with different out-of-phase responses, allowing to manipulate electromagnetic waves. The meta-atoms with different designed functions can be represented by binary codes, which are deemed to be coding elements. For example, based on the out-of-phase responses, binary coding elements for the simplest 1-bit coding metasurface are “0” and “1”, which correspond to “0” and “π” phase response, respectively. Similarly, a 2-bits coding metasurface is composed of four coding elements “00”, “01”, “10” and “11”, which have “0”, “π/2”, “π” and “3π/2” phase response, respectively. The generation of n-bit coding metasurface obeys the same rule. In this way, the information of the metasurface can be recorded in a digital way, which creates the connection between the physical meta-atom and digital codes. The digital description of metasurfaces has changed the way of describing, analyzing, and designing metasurface in the following aspects. Firstly, as scattering properties are uniquely determined by the coding sequence applied on the metasurface, the specific structures of coding particles are no longer needed to be considered, which can significantly simplify the design process. Secondly, as the real-time reconfigurability is a very appealing character for metasurface [35,36], a digital description allows to implement a metasurface with digital logic devices such as PIN diodes and to be controlled by a field-programmable gate array (FPGA). By changing the input coding sequence, functionality of the metasurface can be switched in real time, leading to a programmable metasurface [37,38]. Thirdly, the digital description of metasurface allows to apply theorems in digital signal processing and information science to analysis and design processes, such as application of the Fourier relation between the coding pattern and far-field radiation pattern [39].
Here in this work, we propose the concept of coding Huygens’ metasurface (CHM) for microwave holographic imaging. To construct the CHM, we use Huygens’ meta-atoms as the basic coding elements. By correctly designing geometrical parameters of the electric and magnetic dipoles, meta-atoms able to cover full 2π-phase range with near-unity transmission amplitude can be obtained. A weighted holographic algorithm is proposed to calculate the phase distribution along the metasurface. Then, arbitrary n-bit CHMs can be realized by discretizing the calculated phase distribution by 2n over 2π. For validation of the proposed concept, 1-, 2-, and 3-bits CHMs are designed, fabricated and measured to find the relationship between imaging quality of the CHM holograms and phase-quantization level. Moreover, we modulate the intensity distribution of focal points using the proposed 3-bits CHMs to verify the feasibility of our proposed weighted holographic algorithm. Experimental verifications performed at microwave frequencies agree qualitatively with theoretical calculations and numerical simulations, indicating the feasibility and high imaging quality of the proposed CHM holograms.
2. CHM holograms
In order to achieve the desired holographic imaging, the phase profile at the interface of the metasurface is designed from the proposed holographic algorithm [40–42]. Different from algorithm for manipulating the radiation pattern of the antenna [43], we propose holographic algorithm to converge the incident wave to four specific focal points in the near field, at a distance about 3.3λ away from the metasurface. The schematic diagram for the theoretical analysis is illustrated in Fig. 1. One fundamental method is that we are able to select ideal point sources as virtual sources and place them at pre-designed focal points positions. We consider N focal points located at (i = 1 to N). Then, by superposing the electromagnetic field generated by all the virtual sources, which can be described utilizing Green function, the phase delay at the position of each meta-atom (j = 1 to M) can be retrieved. Accordingly, the reconstructed electric field is converged to the pre-designed focal points. However, the assumption of in-phase radiation can only ensure the convergence of the incident wave to the preset focal points, but is not sufficient to modulate the intensity distribution of the focal points. This problem is solved by introducing a weight factor so as to modify the phase distribution on the metasurface and hence to modulate energy distribution. An iterative procedure is adopted to obtain the phase distribution on the CHM, as follows:
Where is the distance between the mth meta-atom and the nth focal point, k is the phase constant, wn presents the intensity ratio of nth focus to the first one, and the superscript p represents the pth iteration. denotes the electric field intensity of the nth focal point in the (p-1)th iterative step. Once the phase value of each meta-atom is obtained, the reconstructed field can be calculated by superposing the field component emitted by each meta-atom. Since the radiation of the meta-atom also obeys the Greens’ function, the reconstructed electromagnetic field at the position of each focal point can be retrieved as: where sn is the preset intensity ratio of the nth focal point. According to the above details about the algorithm, wn is adjusted to eliminate the deviations of from the desired intensity. The initial condition is set as:
Fig. 1 Schematic illustration of the holographic imaging process. A weighted Huygens’ metasurface incorporating coding electric and magnetic dipolar resonators is illuminated by an incident plane wave to produce a hologram. Photography and structural details of a fabricated CHM sample (image lower right).
3. Design of the coding elements
The proposed CHMs consist of a two-dimensional square array of Huygens’ meta-atoms as shown in Fig. 2(a). A split-ring resonator (SRR) is placed on one side of the substrate playing the role of the magnetic dipole and an electric-LC resonator acting as the electric dipole is arranged on the other side of the substrate. With the combination of magnetic and electric dipole sources, the meta-atom can be considered as a small Huygens’ source. To reveal the mechanism of the proposed Huygens’ meta-atom, the surface current flowing along the metallic patterns of the meta-atom is plotted in Fig. 2(b). A photography of a fabricated CHM sample is shown in the inset of Fig. 1. The proposed CHMs, which are fabricated using conventional PCB photolithography, consist of coding elements covering an area of 205 × 201.2 mm2. A CHM is assembled by stacking 41 identical circuit board strips with 3.5 mm air gap between each strip, and each board strip consists of 57 meta-atoms. The top and the bottom sides of the strips provide the required electric and magnetic currents, respectively.
Fig. 2 Design of the coding elements. (a) Schematic view of the Huygens’ meta-atom composed of electric and magnetic dipoles. The thickness of the substrate (εr = 3 and tan δ = 0.002) is 1.5 mm and the periodicity of the elementary meta-atom is respectively 5 mm and 3.53 mm along x- and y-directions. (b) Simulated electric and magnetic currents along the metallic electric and magnetic dipolar resonators excited by a y-polarized incident wave. (c) Simulated amplitude of the transmission coefficient and (d) simulated phase of the transmission coefficient of the Huygens’ meta-atom with fixed parameters le and lm. (e) Simulated transmission amplitude and (f) transmission phase of the Huygens’ meta-atom with different parameters le and lm.
When a y-polarized incident plane wave illuminates the proposed Huygens’ meta-atoms, currents are induced along the metallic patterns on the two sides of the substrate, resulting in the equivalent transverse magnetic currents parallel to x-direction and electric currents along y-direction, respectively. According to Huygens’ principle, the electromagnetic field distribution can be effectively described by the electric and magnetic sheet impedances. The local complex transmission and reflection coefficients can be obtained by [44,45]:
where Zms is the magnetic sheet impedance, Yes is the electric sheet admittance, and η is the wave impedance of free-space. In our proposed Huygens’ meta-atom, by changing the length le and lm of the electric and magnetic resonators, electric and magnetic sheet impedances can be adjusted to achieve the desired amplitude and phase responses. Therefore, a flexible and efficient method to engineer electromagnetic wavefronts can be anticipated. The proposed meta-atom with fixed parameters (le = 3.5 mm, lm = 2.7 mm) is simulated using the commercial software CST Microwave Studio by applying unit cell boundary conditions in x- and y-directions under y-polarized plane wave incidence. The simulation results are shown in Figs. 2(c) and 2(d). It can be observed that the resonance frequency of the meta-atom is around 10GHz, so the operating frequency of our proposed CHM is designed to be 10 GHz. By modulating the parameters le and lm, the amplitude of the transmission coefficient ranges from 0 to 0.99, and the phase of the transmission coefficient covers the whole 2π range at 10 GHz, as shown in Figs. 2(e) and 2(f). It is important to observe that the transmission phase can be freely manipulated by changing le and lm, while keeping the transmission amplitude at a high level. The designs of 1-, 2-, and 3-bits coding elements with different phase shift of π, π/2 and π/4 are depicted in Fig. 3. The transmission amplitude of all the coding elements can be kept above 0.9 so as to achieve a high transmittance efficiency.Fig. 3 Coding elements for 1-, 2-, and 3-bits CHMs extracted from the simulation results of the Huygens’ meta-atom.
A series of CHMs are fabricated to generate holograms and to modulate focal intensity ratio of the focal points. For the uniform holograms, all the intensity ratio sn are set to the same value and are equal to 1, such that the incident energy can be transformed equally into each focal point. For the modulation of intensity distribution among focal points, the value of sn can be changed to achieve any desired intensity distribution. In our demonstrations, we set the intensity ratio sn to be 1:1:1:1, 4:5:4:5 and 1:2:1:2 respectively to verify the feasibility of the weighted algorithm. To evaluate the imaging quality, two parameters are adopted here: the imaging efficiency and the root-mean-square error (RMSE). The imaging efficiency is defined as the ratio of the energy converged to designated points to the energy of the incident wave. The RMSE describing the deviations between the measured intensity ratios and the theoretical values, is used to evaluate the manipulation ability of energy allocation of focal points. Particularly, the RMSE can be calculated as:
Based on the proposed weighted holographic algorithm, the phase profile at the interface of the CHM can be obtained. The focal distance is chosen to be 100 mm because of the convenience of measurement. To replace continuous phase distribution in proof-of-concept prototypes, approximated discrete phase values are used. Then arbitrary n-bit CHM can be built through discretization of the phase profile over 2π by 2n. Taking for instance 1-bit CHM as example, the calculated phase profile at the interface can be divided into two intervals over 2π. The phases for the two intervals are quantified to be 0 and π, which are represented by code “0” and “1”, respectively. In this way, the phase information can be described by codes distribution on CHM. Figure 4 shows the code distribution of the proposed 1-, 2-, and 3-bits CHMs for uniform holograms, and code distribution of the 3-bits CHMs for modulation of focal intensity distribution. Utilizing the calculated code distributions, the CHMs can be easily built up by arranging coding elements at corresponding positions.
Fig. 4 Code distribution of the proposed CHMs for (a) 1-bit, (b) 2-bit, (c) 3-bit uniform holograms, and code distribution for modulation of focal energy, (d) Intensity distribution ratio of 1:1:1:1, (e) intensity distribution ratio of 4:5:4:5 and (f) intensity distribution ratio of 1:2:1:2.
4. Experimental results and discussion
The fabricated samples of the CHMs are experimentally validated using a near field scanning system. A feeding horn antenna is placed 65 cm away from the metasurface to ensure the incident quasi-plane wave at 10 GHz. A fibre optic active antenna is used as field probe to measure the electric field distribution in the transmission region. The vector network analyzer (Agilent 8722ES) is connected to the feeding horn antenna and field receiving probe to measure the transmission coefficients. The holographic image of a rhombus is realized with uniform intensity distribution as shown in Fig. 5. Figure 5(a) shows the theoretical result obtained from the Huygens’ metasurface with the calculated continuous phase profile. Figures 5(b)-5(d) show the measured images at the frequency of 10.2 GHz of the 1-, 2- and 3-bits CHMs at a distance of 100 mm above the CHMs. The parameters used to evaluate the imaging quality of the CHM holograms are listed in Table 1, where we can clearly observe that imaging efficiency is significantly improved from 27.1% for 1-bit hologram to 51.5% for 3-bits hologram, which is superior compared with previous metasurface holograms [26,29]. It can also be noted that the total image efficiency is improved by 57.2% from 1-bit to 2-bits, and 20.9% from 2-bits to 3-bits hologram, meaning that the imaging quality improvement effect will be slight when the phase-quantization increases to a relatively high level. Therefore, we need to take both design complexity and imaging quality into consideration and choose coding elements with suitable bits for real-life practical applications. Table 1 also depicts the RMSE of the uniform holograms. For the 3-bits hologram, the measured RMSE can be as low as 0.77%, indicating a low deviation between calculation and measurement when the incident energy is transformed to the desired focus points equally. The signal to noise ratio (SNR), which is defined as the ratio of the peak intensity in the image to the standard deviation of the background noise [29], is found to be above 10 for all the measured CHM holograms as shown in Table 1. Figures 5(e)-5(g) show the measured images at 9 GHz, 10 GHz and 11 GHz, respectively. The image quality will decrease when the operating frequency deviates from the designed one. Since the phase responses of the meta-atoms are extracted from the simulated results at 10 GHz, and the phase distribution at the interface is calculated using the operating wavelength at 10 GHz, which means the phase delay at the focal plane will be affected when the operating frequency deviates. The simulation results of 3-bits CHM at different focal distance of 90 mm, 100 mm and 110 mm are also given in Figs. 5(h)-5(j). When the focal distance deviates from the designed value, the image quality will decrease owing to the change of phase delay on the focal plane.
Fig. 5 Holographic images of CHMs with different bits. (a) Theoretical results with calculated continuous phase profile. (b) Experimental image of 1-bit CHM at 10.2 GHz. (c) Experimental image of 2-bits CHM at 10.2 GHz. (d) Experimental image of 3-bits CHM at 10.2 GHz. (e), (f) and (g) Experimental images of 3-bits CHM at 9 GHz, 10 GHz and 11 GHz respectively. (h), (i) and (j) Simulation images of 3-bits CHM at different focal distances of 90 mm, 100 mm and 110 mm respectively.
Table 1. Parameters to Evaluate Imaging Quality of Uniform CHM Holograms
Intensity modulation of the four focal points performed with our proposed 3-bits CHMs is illustrated in Fig. 6. For the three demonstrated CHMs, the focal intensity ratio sn of the four focal points is designed to be 1:1:1:1, 4:5:4:5 and 1:2:1:2, respectively. Figures 6(a), 6(e), and 6(i) show the theoretical results with the different intensity ratios. Figures 6(b), 6(f), and 6(j) depict the simulated results. Figures 6(c), 6(g), and 6(k) present the measured images at the distance z = 100 mm above the CHMs at 10.2 GHz. The parameters to evaluate the manipulation ability are summarized from the experimental results and listed in Table 2. The RMSE verify the outstanding intensity modulation ability of the proposed CHMs. For further verification, the measured intensity distribution along the central vertical and horizontal lines in Figs. 6(c), 6(g), and 6(k) are depicted in Figs. 6(d), 6(h), and 6(l). The measured peak intensity ratios show good agreement with the theoretical calculations. As shown in Fig. 6, there are large diffracted fields near the four focal points. These higher harmonic components in the images are induced by the microwave which is propagating through the Huygens metasurface. During illumination, high harmonics of the fundamental diffracted light patterns are generated within the fields near the four focal points. To restrain the emerging unnecessary information and improve the hologram, the threshold-value filtering indicator can be applied during the post processing operations, owing to this large intensity difference. Furthermore, to more improve the holographic imaging and achieve a better image quality, the spatial denoising operation algorithm which can compromise the spatial resolution and remains a Haze like background, may be included in this post-processing model to reduce the random noise thoroughly.
Fig. 6 Holographic images of CHMs with different focal intensity ratio. (a) Theoretical, (b) simulated and (c) experimental results for the imaging with intensity distribution ratio of 1:1:1:1. (e) Theoretical, (f) simulated and (g) experimental results for the imaging with intensity distribution ratio of 4:5:4:5. (i) Theoretical, (j) simulated and (k) experimental results for the imaging with intensity distribution ratio of 1:2:1:2, and normalized intensity profiles along the central vertical and horizontal lines of measured images, (d) Results for the intensity distribution ratio of 1:1:1:1, (h) Results for the intensity distribution ratio of 4:5:4:5. (l) Results for the intensity distribution ratio of 1:2:1:2.
Table 2. Parameters to Evaluate Intensity Modulation Ability of 3-bit CHMs
To summarize, in order to verify the proposed theoretical concept, the proposed proof-of-concept CHMs only realize a simple hologram with four focal points and are able to modulate the intensity distribution of the latter focal points with three different ratios. However, arbitrary number of focal points and intensity ratio of each focal point can also be further realized utilizing our CHMs with accordingly weighted holographic algorithm, which means that more freedom can be added to the manipulation of electromagnetic waves. Moreover, the code description of phase information allows the proposed CHMs to be extended to some other applications. For example, by incorporating controllable electronic components, such as varactor diodes or micro-electro-mechanical systems, into the Huygens’ meta-atom design, the scattering state of each individual meta-atom can be dynamically controlled by applying different biased voltages to the components. Then the spatial distribution of voltages applied to each meta-atom can be controlled by a field-programmable gate array (FPGA), resulting in a real-time dynamic and efficient manipulation of electromagnetic wave propagation. The code description also allows us to apply the Fourier relation between coding pattern and scattering pattern to the CHM, which provides new freedom to the holograms, such as imaging rotation and imaging spin. Finally, although the proposed CHMs in this paper are only based on phase-coding, the concept can be readily extended to multiple bits or both phase and amplitude coding, which can lead to more advanced, efficient and feasible functionalities.
5. Conclusion
In conclusion, CHM holograms are realized using Huygens’ meta-atoms as coding elements and are experimentally validated in microwave regime. By adjusting the geometrical parameters of the electric and magnetic dipoles, 1-, 2-, and 3-bits coding elements are built based on the transmission phase. A weighted holographic algorithm is proposed to achieve the phase profile along the metasurface, and 1-bit, 2-bits and 3-bits CHM holograms are designed, fabricated and measured. The imaging efficiency of the 3-bits CHM hologram is measured to be as high as 51.5%, showing outstanding imaging quality. Based on the measured results, the influence of the phase-quantization level to the imaging quality is discussed to improve imaging quality. Moreover, several 3-bits CHMs designed to modulate intensity distribution among focal points are experimentally demonstrated. The RMSE of the measured results verify the great intensity modulation ability of our proposed CHMs. The proposed CHM holograms provide more freedom to electromagnetic wave manipulation, which may have many practical applications in computer-generated holograms, imaging lenses and microscopy. Furthermore, the concept shows great potentials in constructing programmable CHMs by loading controllable elements to realize dynamic holograms and wave focusing.
Funding
National Natural Science Foundation of China (61701141); Fundamental Research Funds for the Central Universities (HIT.NSRIF.2019029).
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