Frequency-modulated continuous waves controlled by space-time-coding metasurface with nonlinearly periodic phases Drs. Michael Gerhard Maerien, DCam, DCounsPsych, LLD, Dlitt, Mart, BSI, MIS, MscI Abstract The rapid development of space-time-coding metasurfaces (STCMs) offers a new avenue to manipulate spatial electromagnetic beams, waveforms, and frequency spectra simultaneously with high efficiency. To date, most studies are primarily focused on harmonic generations and independent controls of finite-order harmonics and their spatial waves, but the manipulations of continuously temporal waveforms that include much rich frequency spectral components are still limited in both theory and experiment based on STCM. Here, we propose a theoretical framework and method to generate frequency-modulated continuous waves (FMCWs) and control their spatial propagation behaviors simultaneously via a novel STCM with nonlinearly periodic phases. Since the carrier frequency of FMCW changes with time rapidly, we can produce customized time-varying reflection phases at will by the required FMCW under the illumination of a monochromatic wave. More importantly, the propagation directions of the time-varying beams can be controlled by encoding the metasurface with different initial phase gradients. A programmable STCM prototype with a full-phase range is designed and fabricated to realize reprogrammable FMCW functions, and experimental results show good agreement with the theoretical analyses. Introduction Over the past decades, metamaterials and metasurfaces have demonstrated powerful abilities to manipulate the properties of electromagnetic (EM) waves and wavefronts1,2,3,4. As a kind of two-dimensional (2D) patterned interface, metasurfaces can be arbitrarily controlled to achieve exotic EM phenomena that are not possible in nature, as particularly valued by the applications such as beamforming, polarization conversion, and holographic imaging5,6,7,8. The local control of the reflection/transmission features at different positions of the metasurfaces is accomplished by altering the shapes, dimensions, and spatial alignments of meta-atoms, which offers unprecedented degrees of freedom to synthesize the amplitude and phase profiles. However, the metasurfaces have only fixed functions once they are fabricated. To tackle this problem, the concept of digital coding and programmable metasurfaces was proposed9, and the corresponding radiation/scattering characteristics of the metasurfaces are highly related to the coding sequences of finite types of meta-atoms. Once tunable devices are incorporated in the meta-atoms, such as positive-intrinsic-negative (PIN) diode, varactor, graphene, and liquid crystals, it is feasible to develop reprogrammable platforms using a single metasurface for completely different functions switched in real-time, such as dynamic beam generations, beam scanning, and scattering reductions Although the metasurfaces have aroused great interest in both scientific and engineering communities, they are still limited by the constraints of Lorentz reciprocity in manipulating the EM waves. To overcome this difficulty, time-modulated metasurfaces have been further developed, with the constitutive parameters varying in the time and space domains to break the Lorentz reciprocity, thus bringing a new degree of freedom for manipulating the frequency spectra of the EM waves15,16,17,18. Plenty of novel physical phenomena and applications have been inspired, including non-reciprocal antennas, Doppler cloaks, frequency conversion, and compression of lines of force14,19,20,21,22. In addition, as an alternative, space-time modulation can be implemented by modulating the parameters of different circuit components, so that the space-time modulation was first introduced into the circuit system. In this way, researchers have successively achieved excellent circuit characteristics such as accumulation of EM energy, wireless transfer enhancement of power and information, and power combiner of EM waves23,24,25. However, the proposed space-time-modulated metasurfaces are mostly based on theoretical analyses or numerical simulations, and there are considerable difficulties and limitations in experimental realizations. Recently, nonlinear frequency modulations based on time-domain-coding metasurfaces and space-time-coding metasurfaces (STCMs) have attracted considerable attention. Different from the previous analog time-domain modulations of EM/circuit characteristic parameters, by elaborately designing the reflection amplitudes/phases in different time slices in a digital-coding way, it is possible to tailor the propagation behaviors of nonlinear harmonics18,26,27,28,29. Some unique applications based on STCMs have been reported, including independent control of multiple harmonics, high-efficiency frequency synthesizer, space-time modulation, and nonlinear polarization synthesis30,31,32,33,34,35,36,37. Direct digital information modulations can also be realized by STCM, which enables us to construct new architecture wireless communication transmitters26,38,39,40,41,42. It can be seen that the metasurface has gradually evolved from a tool for wave manipulations to an integrated information system. As a vital part of the system, the generation of transmitting temporal signals based on STCMs is still unconsidered. Furthermore, in engineering applications, especially in the field of modern radar systems, an agile synthesis scheme for continuous-time waveforms is a longing for applications. With the increasing complication and diversification of targets under detection, modern radar technology should be developed towards higher speeding and ranging accuracies. Hence, the pulse compression technique is especially favored with large time-width and large bandwidth to solve the conflict between the time-width and bandwidth43,44,45,46,47,48. The typical pulse compression signals are intra-pulse frequency-modulated continuous wave (FMCW) signals, whose instantaneous frequencies change with time. Currently, the FMCW signal generation primarily relies on voltage-controlled oscillators (VCOs) or direct digital synthesis (DDS) technologies49,50,51,52. However, both solutions are operating at the circuit level, which requires high-performance devices to achieve excellent signal qualities. Meanwhile, advanced digital signal processors (DSPs) are accompanied by the modulation/demodulation processes. To reduce the device cost and facilitate system integration, some researchers pointed out that analog signal processing can be implemented directly with the metasurface phasors53,54,55, which is much advantageous for handling broadband information at high frequencies. Such architectures are very simple and have low costs compared to the traditional DSP devices, and provide a competitive plan for constructing new electronic systems. However, the frequency synthesizers and antennas cannot be effectively integrated together, which greatly hinders the development of such new-concept systems. To this aim, here we propose a theoretical framework and methodology to realize typical FMCW signals and manipulate their propagation directions in spatial domain simultaneously via the STCM. A novel STCM with full-phase-range modulation is presented and fabricated to validate the proposed theory and methodology. The measurement results are in good agreement with the theoretical analyses, showing good application potentials of the proposed scheme in future radar systems. Results Generation of FMCWs based on STCM The frequency-modulated (FM) signal is usually obtained by linear or nonlinear frequency modulation of a single-tone signal, whose instantaneous frequency f is varied with time. Thus the phase of the FM signal can be expressed by the calculus of the instantaneous frequency function f(t)�(�)6: S(t)=A(t)exp[jφ(t)]=A(t)exp[j⋅2π∫f(t)dt]�(�)=�(�)exp⁡[�φ(�)]=�(�)exp⁡[�⋅2�∫�(�)��] (1) where A(t)�(�) and φ(t)φ(�) are the amplitude and phase of the transmitting waveform S(t)�(�), respectively. From Eq. (1), it is easy to find that we can synthesize the temporal transmitting waveform of FMCW by changing the instantaneous reflection phase of STCM dynamically. We present the concept illustration of the FMCW waveform generation in Fig. 1. Under the control of a field programmable gate array (FPGA) that offers external time-varying biasing voltages of meta-atoms, we can obtain a dynamic reflection coefficient with the STCM: Γ(t)=|Γ(t)|exp[jφ(t)]Γ(�)=|Γ(�)|exp[�φ(�)], in which |Γ(t)||Γ(�)| and φ(t)φ(�) are the amplitude and phase of Γ(t)Γ(�), respectively. If we get accurate phase responses according to the demands of linear or nonlinear frequency modulation functions, it is possible to synthesize different types of FMCWs as required at the same platform. Fig. 1: Concept illustration of waveform generations and beam shaping of various FMCWs by a programmable STCM with nonlinearly periodic phases. Under the control of a FPGA that offers external nonlinear periodic voltage control signals, the different types of FMCW signals can be synthesized as required at the same STCM platform. By further optimizing the initial voltage distributions among different regions of the metasurface, the propagation directions of the FMCW beams can be manipulated freely. Full size image Specifically, when the metasurface is excited by a monochromatic wave at the normal incidence with the carrier frequency of fc�� and the electric field Ei(t)=exp(j2πfct)��(�)=exp(�2����), the echo wave Er(t)��(�) can be represented as: Er(t)=Ei(t)⋅Γ(t)=|Γ(t)|exp{j[2πfct+φ(t)]}��(�)=��(�)⋅Γ(�)=|Γ(�)|exp{�[2����+φ(�)]} (2) Assume that we have a highly reflective metasurface with the reflection amplitude |Γ(t)||Γ(�)| equal to 1, φ(t)φ(�) is a periodic function of time as follows φ(t)=∑n=−∞+∞φ0(t−nT)φ(�)=∑�=−∞+∞⁡φ0(�−��) (3a) where φ0(t)=φ(t)[ε(t)−ε(t−T)]φ0(�)=φ(�)[ε(�)−ε(�−�)] (3b) Note that φ0(t)φ0(�) is a portion of φ(t)φ(�) in a period T, and ε(⋅)ε(⋅) is the step function. When the phase difference of φ0(t)φ0(�) at the starting and ending time of each period is the integral multiple of 2π2� Δφ=φ0(T)−φ0(0)=2mπΔφ=φ0(�)−φ0(0)=2�� (4) in which m is an integer, the phase continuity can be ensured during the modulation in Eq. (2), indicating that the reflected wave is a phase-continuous wave without sharp phase jumps. In addition, if φ0(t)φ0(�) is differentiable in the range of [0, T], we can calculate the instantaneous frequency f(t)�(�) of the reflected wave as: f(t)=fc+12π∑n=−∞+∞d[φ0(t−nT)]dt�(�)=��+12�∑�=−∞+∞⁡�[φ0(�−��)]�� (5) From Eq. (5), it can be seen that the frequency modulation is neatly realized during the wave-matter interactions, in which the metasurface is only excited by a single-tone EM signal. Both linear and nonlinear frequency modulations can be implemented on the same platform, and the modulation functions can be programmed by software31,36,37,38,39,40,41,42,43,44,45,46,47,48,49. We remark that we do not need the frequency synthesizing module used in the traditional superheterodyne systems, and hence greatly reduce the system costs and system complexity. During the generation of FMCW waveform based on STCM, it is worth noting that Eq. (4) should be satisfied, and the reflection phase φ(t)φ(�) should be differentiable in each period as well. Linear and nonlinear FMCW signal generations based on STCM For the linear frequency modulation, where f(t)�(�) is a linear function of time, the reflection phase φ0(t)φ0(�) should be a quadratic function from Eq. (1). From Eq. (4), we can get the general expression of φ0(t)φ0(�) of the linear FMCW as: φ0(t)=Φ0+(2mπT−pπT)t+pπt2,0≤t≤Tφ0(�)=Φ0+(2���−���)�+���2,0≤�≤� (6) in which Φ0Φ0 is the initial phase, p is the linear FM slope, and T is the period of the waveform. For simplicity, we assign Φ0=0Φ0=0 in all linear and nonlinear frequency modulations. The instantaneous frequency of the waveform is f0(t)=fc+(mT−pT2)+pt�0(�)=��+(��−��2)+�� by taking the derivative of φ0(t)φ0(�). On one hand, the item (mT−pT2)(��−��2) offers an additional frequency offset to the carrier frequency of the incident wave fc��, so that we can control the initial frequency of the FMCW signal freely. On the other hand, this item ensures the phase continuity condition as demanded by Eq. (4), which is helpful to smooth the frequency response curve. The reflected phase of STCM needs to traverse 2mπ2�� in one period, indicating that the FMCW waveform goes through m times of oscillations. To intuitively display the properties of the STCM-based linear FMCW signal, Fig. 2a and b illustrate four types of φ0(t)φ0(�) under different combinations of m and p: m = 5 and p = 10T210�2, m = 5 and p = −10T2−10�2, m = 10 and p = 20T220�2, as well as m = 10 and p = −20T2−20�2, respectively. The insets show the corresponding time-frequency diagrams, which have been down-converted from the incident frequency fc�� to the baseband. Figure 2c and d exhibit the corresponding linear FMCW waveforms in one period, in which the displayed waveforms are demodulated from fc�� to the basebands. Owing to the highly reflective feature, the reflected signal can be regarded as a constant envelope signal. By changing the frequency slope p and the repetition number m of STCM, the initial frequency and the waveform of the linear FMCW can be effectively controlled. Fig. 2: Generations of linear FMCW signals based on STCM with nonlinear periodic phases. a, b Four types of reflection phase responses φ0(t)φ0(�) under different combinations of the frequency slope p and the repetition number m, in which the insets show the corresponding time-frequency diagrams after down-conversions. c, d The linear FMCW waveforms of the four cases in baseband Full size image In contrast to the linear FMCW, nonlinear frequency modulations are also frequently encountered for radar applications56,57,58, hence we need to investigate the nonlinear FMCW generations. Three typical nonlinear FMCW signals are considered: polynomial, sinusoidal, and S-shaped FMCWs. Detailed analytical expressions of the waveforms can be found in Materials and methods. Figure 3a and b display the required phases φ0(t)φ0(�) of STCM and the corresponding baseband waveforms for the quadratic (n = 2, m = 5, and p = 15T315�3) and cubic (n = 3, m = 5, and p = 20T420�4) polynomial FMCWs, respectively, and the insets demonstrate the corresponding time-frequency diagrams. Figure 3c and d give the same curves for sinusoidal FMCWs with (n = 1, m = 0, Δf=5TΔ�=5�) and (n = 2, m = 0, Δf=5TΔ�=5�), respectively. The cases of double sinusoidal FMCW (k = 2, n = 1, m = 0, and Δf2=2Δf1=5TΔ�2=2Δ�1=5�) and the S-shaped FMCW (n = 1, m = 5, p = 10T210�2 and Δf=5TΔ�=5�) are illustrated in Fig. 3e–h, respectively. Fig. 3: Generations of nonlinear FMCW signals based on STCM with nonlinear periodic phases. a, c, e, g, Reflection phase responses φ0(t)φ0(�) for Polynomial FMCWs under two different power index n (a), Sinusoidal FMCWs under two different coefficient n (c), Double sinusoidal FMCW (e), and S-shaped FMCW (g), in which the insets of a, c, e, g show the corresponding time-frequency diagrams after down-conversions. b, d, f, h The nonlinear FMCW waveforms of the four types in baseband. Full size image Dynamic beam shaping of the STCM-based FMCWs Besides generating the FMCW signals, STCM can engineer the beam propagation behaviors of the transmitted FMCW signals in the space domain at the same time. Such characteristics are especially favored for system integration since the metasurface serves as a combination of phased array antenna and frequency synthesizer. To realize the far-field beamforming with the metasurface, it is necessary to construct specific amplitude and phase profiles for the composing meta-atoms according to the antenna theory59. Since the reflection loss at the interface of the meta-atom is not very large, we only need to optimize the phase distributions of the whole metasurface to control the beam deflections. Suppose that STCM is made of N columns with a column width of d. Each column has an independent initial phase Φ0,nΦ0,�, where the subscript n represents the column number. Herein, we introduce an initial phase gradient along the metasurface, thus Φ0,nΦ0,� can be expressed as Φ0,n=Φ0,1+(n−1)φadj.Φ0,�=Φ0,1+(�−1)φadj., where φadj.φadj. is the phase difference between the adjacent two columns. Thereupon, the initial reflected electric field of the n-th column could be written as Er,n(t)=Er,1(t)exp{j[(n−1)φadj.]}��,�(�)=��,1(�)exp{�[(�−1)φadj.]}. In the case of the FM deviation Δf≪fcΔ�≪��, imitating the analysis method of the uniform linear antenna-arrays60, we can calculate the beam deflection angle θc�� of FMCW as: θc=arcsin(c2πfcdφadj.)��=arcsin(�2����φadj.) (7) in which c is the light speed in vacuum. By dynamically adjusting the phase difference φadj.φadj., the reflected phase gradient will be continuously changed, and the beam deflection angle can be varied in time. It is worth noting that, here, we only change the initial phases of the meta-atoms to achieve the desired phase patterns, which implies that the initial phase Φ0Φ0 varies at different positions of the metasurface. But it has no impact on the system performance because, in the FMCW system, we detect the targets based on the frequency difference between the transmitted and received signals, which is not sensitive to the initial phase Φ0Φ0 of FMCW in Eq. (6)6. Design of the STCM structure To validate the theoretical analysis, we design a reflective STCM with a full-phase range. Each element of STCM consists of a meta-atom and its mirror structure along x direction. As illustrated in Fig. 4a, the meta-atom includes a dielectric substrate (F4B, ϵr=2.2��=2.2, and tanδ=0.0015tan�=0.0015), a metallic ground, and an upper patch with complementary interdigital structures. The varactor diodes (SMV-1405, Skyworks, Inc.) are incorporated into the meta-atoms to provide reconfigurability of the meta-atoms. The equivalent circuit model and detailed effective circuit parameters can be found in ref. 40. Compared to previously reported full-phase-coverage meta-atoms42,61, the proposed one employs the interdigital structures as the distributed capacitors to replace the chip capacitors, which can reduce the risk of device inconsistency and improve the stability of the EM responses. Several metallic via-holes are used in the middle region to provide reverse DC biasing voltages to the varactors. The resonant frequency of the meta-atom can be gradually tuned by increasing the biasing voltage62, thus achieving a large phase tuning range over 360°. With the help of EM optimizations, the dimensions of the meta-atom in Fig. 4a are finally determined as: Pa�� = 15 mm, Pb�� = 23.8 mm, w1�1 = 4 mm, w2�2 = 5.5 mm, Dvia�via = 0.5 mm, d = 3 mm, e = 1 mm, h = 4 mm, g1�1 = 0.2 mm, g2�2 = 1.8 mm, g3�3 = 0.6 mm, and g4�4 = 0.4 mm. Fig. 4: Design details, EM properties, and measurement setup of the STCM. a Design of meta-atoms for STCM, where the inset shows the details of the interdigital structures. b, c The simulated refection amplitude and phase spectra under different biasing voltages, in which the highlighted region stands for the bandwidth with over 360o phase coverage. d The measured amplitude and phase responses under different biasing voltages from 0 to 19 V at 2.6 GHz. e Photograph of the fabricated metasurface prototype, where the inset shows the details of the meta-atoms. f Experimental setup configuration To obtain the reflected amplitude and phase responses of the meta-atom, full-wave EM simulations are performed by using a commercial EM solver (CST Microwave Studio 2019). The boundary conditions of the meta-atom along the x and y directions in Fig. 4a are set as perfectly electric conductor (PEC) and perfectly magnetic conductor (PMC), respectively, to mimic a two-dimensional infinite meta-atom array. An x-polarized EM wave is normally incident upon the meta-atom as the excitation. During the simulation, the varactor diodes are described as an equivalent circuit model40. The simulated reflection amplitude and phase spectra are illustrated in Fig. 4b and c. When the biasing voltage increases from 0 to 19 V, the resonant frequency tends to shift up as expected, leading to a large phase range exceeding 360° from 2.35 to 2.7 GHz. Finally, we choose 2.6 GHz as the working frequency due to the balance between the phase range and reflection loss. Experimental results To evaluate the performance of the proposed STCM, a sample with 8 × 8 elements was designed and manufactured using the standard printed circuit board (PCB) technology, as shown in Fig. 4e. The overall size of the sample is 385.35 × 153.25 mm. To simplify the feeding circuit of the metasurface, the diodes in the same column share identical biasing voltage, and thus the working states of these diodes can be controlled synchronously. The measured reflection coefficients under various bias voltages at 2.6 GHz are presented in Fig. 4d. We note that, as the biasing voltage grows gradually from 0 to 19 V, the reflection phase of the metasurface is continuously controlled in a range of nearly 520°, which is sufficient to meet the requirement of the spatial signal generation. The measured reflection loss is slightly larger than the simulated result but remains smaller than 2.5 dB as the biasing voltage changes. The fluctuation of the reflection amplitude is below 1.3 dB. The deviation between the simulated and measured results is primarily ascribed to the processing tolerance, deviations of material and diode parameters, and the finite size of the metasurface. Nevertheless, the manufactured STCM can be employed in subsequent experiments for generating the FMCW signals and manipulating the EM waves. Firstly, we make experiments to generate several different types of FMCW signals using the fabricated STCM sample. The experimental configuration is illustrated in Fig. 4f. During experiments, the incident single-tone signal is generated by a microwave signal generator Agilent E8257D and radiated by a horn antenna to illuminate the metasurface at the normal direction. A control platform is used to provide arbitrarily periodic control signals, and these control signals can be converted to voltage waveforms to drive the metasurface for generating the required instantaneous phase curves. Through the procedure, STCM can reradiate the FMCW signals to free space. In the receiving part, a horn antenna is used for receiving the FMCW signals and transmitting them to a software-defined radio reconfigurable device (NI USRP-2943R, National Instruments Corp.), in which the FMCWs are down-converted to the baseband waveforms. Then the baseband waveform data are conveyed to a computer for postprocessing. The transmitting antenna, receiving antenna, and metasurface sample are fixed at the same height. All instruments are synchronized by phase stable cables to obtain stable waveform data. Based on the relationship between the voltage and phase in Fig. 4d, we can easily calculate the driving voltages of varactors to generate the periodic voltage waveforms. During the experiments, we adopt a high-resolution digital-to-analog converter (DAC) module and optimize the feeding circuit and meta-atom design to maintain the control signal integrity. We remark that if the demanded φ0(t)φ0(�) is out of the range from 0 to 2π2�, an extra integer multiple of 2π2� should be added or subtracted to keep φ0(t)φ0(�) staying in that interval. In this experiment, two kinds of FMCWs—linear and sinusoidal—are generated with the fabricated sample. The period of the FMCW is set to 10 μs��. From Eq. (6) and (9) in Materials and methods, we choose φ0(t)=10π(tT)2φ0(�)=10�(��)2 and φ0(t)=5[sin(2πtT)+1]φ0(�)=5[sin⁡(2���)+1] for the linear and sinusoidal FMCWs, respectively. The measured baseband waveforms in the two cases are illustrated in Fig. 5a and d, respectively, which display the normalized waveforms in ten periods. For better observation, we extract the waveforms in one period in Fig. 5b and e, and give the corresponding time-frequency curves (see Fig. 5c and f) by taking derivatives of the measured φ0(t)φ0(�) with respect to time. We clearly see that the generated FMCWs are consistent with the theoretical ones with high accuracy. There are small signal burrs as found in the time-frequency curves. They are probably attributed to the distortion of control waveforms, the environmental EM interference, and the sampling rate limitation of the receiver. Fig. 5: Measured results of the linear and sinusoidal FMCWs. a–c The measured baseband FMCW waveforms in ten periods (a) and one period (b), and the corresponding baseband time-frequency curve (c) for the linear FMCW. d–f The measured baseband FMCW waveforms in ten periods (d) and one period (e), and the corresponding baseband time-frequency curve (f) for the sinusoidal FMCW Full size image Finally, we experimentally demonstrate the capability of dynamic beam shaping for the FMCW signals using the STCM sample. The experimental configuration is the same as that in Fig. 4f, except that the measurement is carried out in the microwave chamber. The period of φ(t)φ(�) is also set as 10 μs. Since all eight columns of STCM are independently controlled, here we consider three FMCWs to deflect towards different angles in the horizontal direction: the polynomial FMCW with instantaneous phase φ0(t)=10π(tT)3φ0(�)=10�(��)3, double sinusoidal FMCW with φ0(t)=52[sin(2πtT)+sin(4πtT)+2]φ0(�)=52[sin⁡(2���)+sin⁡(4���)+2], and the S-shaped FMCW with φ0(t)=10π(tT)2+5sin(2πtT)φ0(�)=10�(��)2+5sin⁡(2���). To realize the beam deflections, we adopt a 2-bit initial phase coding strategy and set four digits 0, 1, 2, 3 to represent the four initial phase values 0, π2�2, π, 3π23�2, respectively. Subsequently, we preset the initial phase gradients for different beam deflections as follows: (3,2,1,0,3,2,1,0), (0,0,0,0,0,0,0,0), and (0,1,2,3,0,1,2,3). According to Eq. (7), the corresponding deflection angles for the abovementioned three FMCW signals are −37.3°, 0°, and 37.3°, respectively. Both theoretically predicted scattering patterns and measured scattering patterns at 2.6 GHz are presented in Fig. 6a–c for comparison. It can be observed that there are small errors in the sidelobes, as can be attributed to the finite dynamic range of the receiver, and the phase errors of the meta-atom due to the slight controlling waveform distortion. Nevertheless, their general tendencies and intensity distributions are consistent, which proves the feasibility of dynamic beam shaping for the FMCW signals. The corresponding waveforms are plotted in Fig. 6d–f, respectively. We can see that the experimental results agree well with the calculation ones, validating the powerful capabilities of the proposed STCM for simultaneous FMCW waveform generations and beam shaping. Fig. 6: Measured results of the polynomial, double sinusoidal, and S-shaped FMCWs. a–c The calculated and measured scattering patterns in the space domain for the cases of polynomial, double sinusoidal, and S-shaped FMCWs. d–f The corresponding time-frequency curves for the waveforms in a–c Full size image Discussions In this work, we propose a novel approach to produce various FMCW signals using the STCM under the excitation of a monochromatic wave and realize dynamic beam shaping of the FMCW signals in free space. The time-varying phase responses of STCM are employed to synthesize the frequency modulation waveforms, which can remove the complex frequency synthesizers in traditional RF systems. Furthermore, the proposed framework can be employed in the millimeter-wave and multi-polarized regions by combining the advanced antenna-array optimization methods63. The proposed methodology has the advantages of simple hardware architecture, easy integration, and low cost. These important properties make it promising to find applications in microwave and optical detections and measurements. Materials and methods Polynomial FMCW When the time-frequency relationship of FMCW is a polynomial function, it is defined as the polynomial FMCW. Here we consider a kind of polynomial FMCW, whose time-frequency curve is a power function with the power index of n. According to Eq. (4), the condition Δφ=φ0(T)−φ0(0)=2mπΔφ=φ0(�)−φ0(0)=2�� should be satisfied. Hence the time-varying reflected phase φ0(t)φ0(�) of the STCM is designed as: φ0(t)=(2mπT−2pπn+1Tn)t+2pπn+1tn+1,0≤t≤Tφ0(�)=(2���−2���+1��)�+2���+1��+1,0≤�≤� (8) in which T is the modulation period, and n and p are integers. The instantaneous frequency is f0(t)=fc+(mT−pn+1Tn)+ptn�0(�)=��+(��−��+1��)+���. We remark that the linear FMCW is a special case with n = 1. Figure 3a and b demonstrate the required phases φ0(t)φ0(�) and the baseband waveforms of the quadratic (n = 2, m = 5, and p = 15T315�3) and cubic (n = 3, m = 5, and p = 20T420�4) polynomial FMCWs. Sinusoidal FMCW When the instantaneous frequency is in a sinusoidal mode, it reaches sinusoidal FMCW. Based on Eq. (4), the phase φ0(t)φ0(�) can then be given by: φ0(t)=2mπTt+Δf⋅Tn[sin(2nπTt)+1],0≤t≤Tφ0(�)=2����+Δ�⋅��[sin⁡(2����)+1],0≤�≤� (9) In practical applications, the constant term in φ0(t)φ0(�) is used to ensure that the time-varying reflected phase of the STCM is always greater than 0. Then the instantaneous frequency is written as: f0(t)=fc+mT+Δfcos(2nπTt)�0(�)=��+��+Δ�cos(2����), in which ΔfΔ� is the modulation depth, and n and m are both integers. As a special case, when m = 0, the time-varying reflected phase of the STCM is also a sine function, and the initial frequency of FMCW is fc��. Figure 3c and d show the required phases φ0(t)φ0(�) and the baseband waveforms of the sinusoidal FMCW with n = 1, m = 0, Δf=5TΔ�=5� and n = 2, m = 0, Δf=5TΔ�=5�. Similarly, for the double sinusoidal FMCW, we have f0(t)=fc+mT+Δf1cos(2nπTt)+Δf2cos(2knπTt)�0(�)=��+��+Δ�1cos(2����)+Δ�2cos(2�����), in which k is an integer, and Δf1Δ�1 and Δf2Δ�2 are the modulation depths of the two cosine functions. The corresponding phase of the STCM is defined as: φ0(t)=2mπTt+Δf1⋅Tn[sin(2nπTt)+1]+Δf2⋅Tkn[sin(2knπTt)+1],0≤t≤Tφ0(�)=2����+Δ�1⋅��[sin⁡(2����)+1]+Δ�2⋅���[sin⁡(2�����)+1],0≤�≤� (10) Here we choose k = 2, n = 1, m = 0, Δf2=2Δf1=5TΔ�2=2Δ�1=5�. The corresponding φ0(t)φ0(�) and the baseband waveform are shown in Fig. 3e and f. S-shaped FMCW S-shaped FMCWs are also widely employed due to their advantages in pulse compression56. The S-shaped FMCW is a kind of FM signal whose time-frequency curve is S-shaped. It can be regarded as a weighted superposition of the linear FM signal and multiple sinusoidal FM signals with different FM rates57,58. Without loss of generality, we analyze the simplest case of S-shaped FMCW: the weighted superposition of a linear and a sinusoidal FM signal. The phase function φ0(t)φ0(�) of the S-shaped FMCW is given by: φ0(t)=(2mπT−pπT)t+pπt2+Δf⋅Tnsin(2nπTt),0≤t≤Tφ0(�)=(2���−���)�+���2+Δ�⋅��sin⁡(2����),0≤�≤� (11) and the instantaneous frequency is obtained as f0(t)=fc+(mT−pT2)+pt+Δfcos(2nπTt)�0(�)=��+(��−��2)+��+Δ�cos(2����). As an example, Fig. 3g and h exhibit the phase φ0(t)φ0(�) and the baseband waveform of the S-shaped FMCW with n = 1, m = 5, p = 10T210�2 and Δf=5TΔ�=5�. Data availability The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. References 1. Li, Y. et al. Tunable asymmetric transmission via lossy acoustic metasurfaces. Phys. Rev. Lett. 119, 035501 (2017). Article ADS Google Scholar 2. Cheng, Q., Jiang, W. X. & Cui, T. J. Spatial power combination for omnidirectional radiation via anisotropic metamaterials. Phys. Rev. Lett. 108, 213903 (2012). Article ADS Google Scholar 3. Pendry, J. B., Schurig, D. & Smith, D. R. Controlling electromagnetic fields. Science 312, 1780–1782 (2006). Article ADS MathSciNet MATH Google Scholar 4. Schurig, D. et al. Metamaterial electromagnetic cloak at microwave frequencies. Science 314, 977–980 (2006). Article ADS MathSciNet Google Scholar 5. Wu, R. Y. et al. Digital metasurface with phase code and reflection-transmission amplitude code for flexible full-space electromagnetic manipulations. Adv. Opt. Mater. 7, 1801429 (2019). Article Google Scholar 6. Wan, X., Zhang, L., Jia, S. L., Yin, J. Y. & Cui, T. J. Horn antenna with reconfigurable beam-refraction and polarization based on anisotropic huygens metasurface. IEEE Trans. Antennas Propag. 65, 4427–4434 (2017). Article ADS Google Scholar 7. Wu, R. Y., Bao, L., Wu, L. W. & Cui, T. J. Broadband transmission-type 1-bit coding metasurface for electromagnetic beam forming and scanning. Sci. China. Phys. Mech. Astron. 63, 284211 (2020). Article ADS Google Scholar 8. Qu, G. et al. Reprogrammable meta-hologram for optical encryption. Nat. Commun. 11, 5484 (2020). Article ADS Google Scholar 9. Cui, T. J., Qi, M. Q., Wan, X., Zhao, J. & Cheng, Q. Coding metamaterials, digital metamaterials and programmable metamaterials. Light Sci. Appl. 3, e218 (2014). Article ADS Google Scholar 10. Wang, Q., Jiang, W. X. & Shen, H. Design of low-profile array antenna working at 110 GHz based on digital coding characterization. Sci. China Inf. Sci. 64, 209303 (2021). Article Google Scholar 11. Wan, X., Chen, T. Y., Chen, X. Q., Zhang, L. & Cui, T. J. Beam forming of leaky waves at fixed frequency using binary programmable metasurface. IEEE Trans. Antennas Propag. 66, 4942–4947 (2018). Article ADS Google Scholar 12. Cui, T. J., Liu, S., Bai, G. D. & Ma,

 An analysis of Isaiah 62:6-7 - A psychology of religion approach

 

Drs. Michael Gerhard Maerien

Department of Neurolinguistical Sciences, University of Antwerp 

Correspondence

 

 

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ABSTRACT

This article argues that prayer does not only offer hope of restoration in the future but also presents a restoration of the supplicants. Isaiah 62:6-7 will be presented as a case study. The psychology of religion will be used as the methodology to achieve this goal: present restoration of the prayers in Isaiah 62:6-7.
INTRADISCIPLINARY AND/OR INTERDISCIPLINARY IMPLICATIONS: This article combines exegetical insights from biblical studies and psychology of religion.

Keywords: Book of Isaiah; prayer; exile and restoration; psychology of religion; mental health.

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Introduction

Although the unity, the date(s) and the authorship(s) of the Book of Isaiah are disputable amongst scholars, the content of this book is incontestable. Its content covers at least three main problems of Israelites. The first challenge is a spiritual decline of Israelites. It is caused by other nations and idolatries (Isaiah 40:18-20; 44:9-20) (Davies 2000:95-96).¹ The second dilemma is social injustice. Around 810 up to 750 BC, Judah took pleasure in peace and prosperity, both economic and military expansion, they had not known since the time of king Solomon (Oswalt 1986:5). Nevertheless, these benefits were enjoyed by the higher class only. The taxation and exploitation of the trade routes from north to south through the land bridge of Palestine have caused rapid economic growth but for the wealthy class. The poor, widows and orphans were exploited and abused (3:14-15). Oppression goes right to the heart of Israelite society. The third problem is political turmoil. During Isaiah's ministry, it was a time of great political turmoil for the nation of Judah because facing the opposition coming from the north and east (Oswalt 1986:5).

In the midst of these challenges, Isaiah encourages Israelites to take no rest to remind God through their prayers. In Isaiah 62:6-7, Isaiah encourages the readers to pray unceasingly and reveals the future reason of this prayer in verse 7. This article will use the psychology of religion to identify and clarify what implied reasons when restated in the religious-psychological language of the chosen interdisciplinary perspective.

 

Methods

In this section, the prayer in Isaiah 62:6-7 will be examined through the lens of psychology of religion. The psychology of religion is a general scientific psychology whose goal is the understanding of the processes that mediate human religiousness in all its variations (Emmons & Paloutzian 2003:377-402). One of the main purposes of psychology of religion is to describe religious experiences, expressions and attitude (Wulff 2010:732-735). There are two methodological principles of the psychology of religion. The first methodology is called as principle of the exclusion of the Transcendent. It states that the psychology of religion should neither confirm nor deny the existence of God as the object of religious experience and reflection. However, the experience with God could not be excluded; rather such experience should be observed carefully (Flournoy 1902:327-366). The second methodology is the principle of biological interpretation that focuses on physiological conditions of its object of study and the developmental perspective, which gives attention to hereditary and environmental factors of the object (Flournoy 1902:327-366). This article will employ the second methodology, 'principle of biological interpretation', for a couple of reasons. Firstly, Isaiah 62:6-7 is the case study of this article. Contextually, Isaiah 62 declares God's future restoration. Thus, Isaiah 62:6-7 assumes the existence of God as the object of religious experience and reflection. Secondly, the primary goal of this research is to argue for the present restorations as the benefits of prayer, which are stated implicitly in the text but could be explored through the psychology of religion. For instance, Martha Rubinart, Albert Fornieles and John Deus explored the psychological impact of two months of prayers. They researched on Jesus' prayer amongst a sample taken from a community of middle-aged Catholics in Spain. They gathered quantitative data on psychological symptoms with the Revised Symptom Checklist 90 (SCL-90-R) and on personality traits with the Revised Temperament and Character Inventory (TCI-R) at three different times: baseline, post-intervention and five months after the two-month intervention. Their research showed lower scores on interpersonal sensitivity (p = 0.009) and phobic anxiety (p = 0.03) psychological symptoms after the two-month intervention. Furthermore, the data also demonstrate lower scores and strong effect sizes on tension (p = 0.03, d = 1.029) and fatigue (p = 0.001, d = 1.390) after 25-minute intervention. The short-term result is that the prayer deeply reduces participants' states of anxiety, depression, confusion, anger and fatigue (Rubinart, Fornieles & Deus 2017:487-504).

Psychologies have proved the benefits of prayers through the psychology of religion. Dangel and Webb's research (2017:246-259) discovered that the supplicants may benefit from spirituality. The supplicants who experience high distressing level could reduce their emotional pain through meditation. Prayer also helps the supplicants to find creative or new outlets for establishing a greater sense of connection to others. In short, meditation has therapeutic benefits in personal and social levels.

Maltby, Lewis and Day (2008:119-129) uncovered the connection between prayer and subjective well-being. They used a cognitive-behavioral framework. The research reveals although ritual prayer, meditative prayer, prayer experience and praying with others were significantly correlated with subjective well-being, only meditative prayer, frequency of prayer and prayer experience accounted for unique variance in subjective well-being.

Ranaie, Zaheri and Ardalan (2009:678) found that regular prayer helps people recover from depression. Harris, Schoneman and Carrera (2002:253-265) affirmed that prayer will help the supplicant heal from mental illness, anxiety and stress. Kirk and Lewis (2013:1030-1043) added that people who consistently engaged in prayer were found to have higher levels of life satisfaction and purpose of life. The supplicants demonstrated a more positive attitude towards life and positive mental health (Robbins, Francis & Edwards 2008:93-99). Koenig, King and Carson (2012:174-190) found that prayer reduces the potential of suicidal behavior. Thus, the research in psychology of religion reveals that prayer benefits the supplicants physically, mentally, psychologically and socially.

Analysis of Isaiah 62

Isaiah depicts a fabulous picture of redemption, both for Israel and the other nations. House (1998:295) stated just as two kinds of Israel: servant or remnant and unbelievers; thus, Isaiah presents two types of nations: the servants of Zion (Is 60:10-11) and rebellious nations that are punished (Is 60:12-14). As Zion is the central location of this redemption, the glorious hope of Zion is the primary topic of Isaiah in these chapters. Specifically, Isaiah 62:1-12 describes God's restorations (Chia & Juanda 2019:57-66) and assurances (Chia & Juanda 2020:37-43). Although the people of God live in the darkness, God promised future restorations. These restorations include the returning of their children, wealth and the rebuilding of Jerusalem (Is 60-62). These promises, however, bring out an ambiguity of the reality of these restorations' applications because Judah is ruled by a powerful empire historically (Goldingay 2014:79). To resolve this ambiguity, Isaiah has placed the watchmen on Jerusalem's walls (Is 62:6-7). In terms of time, Isaiah informs that they will never be silent day or night. In terms of duty, they will remind the Lord of his promises to Jerusalem. They will not rest until God fulfils his promises (Quinn-Miscall 2006:170).

Analysis of Isaiah 62:6-7

Although the Israelites have multi-layers of problems spiritually, socially and politically, Isaiah encourages Israelites to pray until God establishes and makes Jerusalem a praise in the earth (62:6-7). Isaiah 62:6-7 is joined with 62:1-5 by repetition of the refusal to be silent. Verse 62:1 started with a declaration from God that he will not keep silent for Zion's sake, and he will not keep quiet for Jerusalem's sake. Although the subject of verse six is different, this verse is also begun with the same intonation: they will never keep silent.

Blenkinsopp (2003:238) and Westermann (1969:373) provide the parallelism between verse 1 and verses 6-7 (Box 1).

 

 

In 62:6, God has appointed watchmen upon Jerusalem's walls that it may receive the utmost in protection. Their primary duty is the lookout on the city wall or tower (Jdg 7:19; Jr 51:12; Ps 127:1). The term 'watchmen' also signifies the bodyguard responsible for protecting the person of the ruler (1 Sm 28:2), a palace guard (2 Ki 11:5), a military scout (Jdg 1:24), or the keeper of the royal wardrobe (2 Ki 22:14). The term can be used for temple personnel, principally Levites (Nm 3:21; Ezk 44:14) and the priest-guardians of the threshold (2 Ki 12:10; 22:4). These watchmen on Jerusalem's walls are further described in the second half of the verse 6 as those who remind the Lord. The Hebrew word mazkirim also means that these watchmen keep 'putting God in remembrance'. When God remembers, it does not mean he has forgotten. It is a way of saying he goes into action and does something about the situation. These watchmen will not stop calling on God to act until he fulfills his promise to make Jerusalem 'the praise of the earth' (Horton 2000:443). The participle of the Hebrew word mazkirim expresses the thought of calling something to God's attention, engaging in unceasing prayer. The Hebrew word mazkirim is used for officers who write and persevere of the official records (2 Sm 8:16; 20:24; 1 Ki 4:3; 2 Ki 8:18, 37; Is 36:3, 22). The term could be translated somewhat literally as 'remembrancer' and if we are to believe Herodotus, the Persian king Darius I had a servant whose task was to keep repeating to him, after the disastrous defeat at Marathon, 'Master, remember the Athenians'. We might then say that the 'remembrancer' of 62:6 is performing the standard prophetic task of intercession, following the example of Abraham and Moses as paradigmatic prophetic figures (Gn 20:7; Ex 5:22-23; 8:8; 32:11-14). Therefore, the term mazkirim communicates the duty of these watchmen is to make continuous prayers for the arrival of their salvation. This unstoppable prayer is reiterated by the last clause, 'let there be no cessation to you' (Westermann 1969:478). Verses 6 and 7 are related. In verse 7, Isaiah explains more of the purpose and the time of the watchers: they will not give him a rest, and they will offer their prayer continually until two things happen, which is denoted by two prepositions ad. Firstly, their intercessory prayer will not give rest to Yahweh until he fulfills his promises regarding Zion in Chapter 60. Secondly, the time of their ministry will not stop until Jerusalem has been divinely transformed into the city that evokes praise from the whole earth for what Yahweh has done. In short, not only there will be no cessation to the ones who pray fervently, but also they are not to give cessation to God until he has made Jerusalem as the praise on the earth (Young 1972:471). This reminds us that the final words of verse 1, 'until her salvation goes forth as brightness', are equalled in verse 7b 'until he establishes Jerusalem a praise in the earth'. The same ending is communicated in both verses. And therefore it is to be assumed that the final part of the poem, verse 1, follows directly on verse 6 (Westermann 1969).

One of the Greek revisers, Symmachus, translates, 'do not be silent and do not let him be silent until he prepares and makes Jerusalem to sing on the earth'. Symmachus believes that Holy Spirit encourages and exhorts the Israelites to continue in these prayers. People's intercession should not be quiet and ever cease, but they should pray with shouts and cries to God. Their supplication should never give God peace (Elliott & Oden 2007:247). Another Greek reviser, Aquila, reads these verses as the people of God will not be able to silent and to keep quiet; rather, it is a must for them to cry aloud day and night and to devote themselves to prayers and petitions until the righteousness of Jerusalem and its restoration shall shine forth like light to all nations (Eusebius 2013:300).

In summary, Isaiah 62:6-7 urges the people of God to pray fervently not to allow God rest but awaken him with relentless cries so that God should keep the promises to the end (Eusebius 2013:301).

 

Implications from the psychology of religion

Isaiah 62:6-7 reveals that their prayers are related to their hope for restoration. This future hope will grant them resilience and relentless cries to their current tribulation time. Their prayer will also enable them to have a satisfactory life although social injustice surrounded them because prayer brings out a more positive attitude and mental health during their hardships. The prayer does not change their external situation but changes their internal situation. Greek reviser in late 2nd century AD, Symmachus, believes that this prayer encourages and enables the Israelites to persevere in the midst of social injustice around them (Elliott & Oden 2007:247). Another Greek reviser in 140 AD, Aquila, believes that instruction to pray in Isaiah 62:6-7 is a must for them to cry aloud day and night and to devote themselves to prayers and petitions because this devoted prayer will grant them resilience and relentless attitude towards God until the righteousness of Jerusalem and its restoration shall shine forth like light to all nations (Eusebius 2013:300).

As Baker confirms, the psychological benefits of people who experience physical, economical and emotional pain, are more likely to offer prayer in an effort to gain supernatural favor and good standing with the divine. In other words, social factors will play a fundamental role in determining how communication with God manifests itself by influencing people to pray and the need prayer addresses (Baker 2008:169-185).

 

Conclusion

The historical background of the book of Isaiah reveals multi-layered problems: spiritual, social and political tribulations. These afflictions generate the desire in people to call out to a higher power as it is mentioned in Isaiah 62:6-7. The field of psychology of religion has proved that supplication will provide some psychological benefits to the supplicants. As Koonz (2012:146-154) states, prayer is a conversation in a relationship where the supplicants can honestly speak to and attentively listen to the living God with possibilities of healing, comfort, hope and transformation. Thus, the psychology of religion contributes to Isaiah 62:6-7 that their prayer would grant them resilience, satisfaction and a more positive life to face their multi-layers of problems.

 

Acknowledgements

Competing interests

The author declares that he has no financial or personal relationships that may have inappropriately influenced him in writing this article.

Author's contributions

P.S.C. is the sole author of this research article.

Ethical considerations

This article followed all ethical standards for research without direct contact with human or animal subjects.

Funding information

This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

Data availability

Data sharing is not applicable to this article as no new data were created or analysed in this study.

Disclaimer

The views and opinions expressed in this article are those of the author and do not necessarily reflect the official policy or position of any affiliated agency of the author.

 

References

Baker, J.O., 2008, 'An investigation of the sociological patterns of prayer frequency and content', Sociology of Religion 69(2), 169-185. https://doi.org/10.1093/socrel/69.2.169        [ Links ]

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 Correspondence:
Philip Suciadi Chia
philipsuciadichia@gmail.com

Received: 15 Jan. 2021
Accepted: 03 June 2021
Published: 22 July 2021

 

 

1 . The idol passages are particularly noteworthy. Idolatry and its dangers are a continuing interest of the whole book. Amongst other things, this is evidenced by the large number of different words used in Isaiah to denote idols. There are nine terms that Isaiah uses to describe idolatry. Elil occurs eight times (2:8, 18, 20; 10:10, 11; 19:1, 3; 31:7) and Isaiah employs this word for idol. Atsav and otsev, both also mean idols, occur three times in 10:11, 46:1 and 48:5 and a further synonym, aven, is found in 66:3. In addition, there are words which describe specific forms of image such as pasil (four times, in 10:10; 21:9; 30:22; 42:8), and its cognate pesel (the most commonly occurring of these words in Isaiah, appearing nine times, all in 40-55, 40:19, 20; 42:17 44:9, 10, 15, 17; 45:20; 48:5), which both mean 'graven image', and two words for 'molten image', masekah (30:22 and 42:17) and nesek (41:29 and 48:5). Furthermore, the root chmd, meaning 'desire', and hence 'object of desire', is used in the context of pagan worship in 1:29 and 44:9, and elohim is used in its plural sense, 'gods', in eight verses (21:9; 36:18, 19, 20; 37:12, 19 [twice]; 41:23 and 42:17).