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## COVER ARTICLE

With the development of intelligent technology, it is essential to develop polarization-conversion devices with adaptable electromagnetic (EM) performance for practical applications. Up to now, most of attempts have relied on PIN diodes and varactor diodes for electrical tuning, typically featuring simplicity and timelineness. However, the shortcomings are also notable, such as less degrees of freedom (DoFs), more complex circuits and more expensive. In view of this, here we propose a kind of spatial-order metasurface for reconfigurable polarization conversion based on kirigami concept. By adjusting the folding angle β, the interaction between neighboring dipoles can be progressively changed and thus the operation frequency of polarization conversion can be shifted. Such a mechanical reconfigurable strategy brings about more DoFs for tuning and is cheaper and extraordinary convenient in practice. To verify the feasibility of our concept, a proof-of-concept spatial-order kirigami metasurface is proposed for the dual-band reconfigurable linear polarization conversion based on asymmetric chiral split ring resonators (SRRs). Experimental results show that the linear polarization operates at 5 and 5.8 GHz when folding angle is β = 10°, these frequencies are shifted to 5.8 and 7.2 GHz when β = 45°: a tuning range is expanded by 18.5%. In addition, the Poisson’s ratio and relative density of proposed kirigami metasurface as a function of β are also theoretically analyzed. The results show that the Poisson’s ratio increases with the value of β increasing. The relative density can be reduced to 1.5% of its unfolded planar counterpart. Our spatial-order kirigami metasurface strategy paves the way for implementing the reconfigurable linear polarization conversion and multifunctional devices. Xu He-Xiu Acta Physica Sinica.2021, 70(15): 154101.

## INVITED REVIEW

2021, 70 (15): 154302. doi: 10.7498/aps.70.20210284
Abstract +
Odontocetes have evolved to own a unique natural sonar system to detect targets. Odontocetes use their sound emission systems in their foreheads to produce echolocation clicking targets. Echoes contain information about the size, material and ranges of the targets. Odontocetes can probe into the echoes in both time domain and frequency domain to realize the target discrimination. More studies are necessary to reveal how odontcoetes collect meaningful information from echoes. In this paper, the target detection by odontocetes is reviewed from three aspects, i.e. detection range, target discrimination and biomimetic target detection system. Odontocetes can actively adjust their biosonar systems to realize optimal detection. Numerical simulation and bioinspired systems can help to shed light on physical mechanism of odontocetes’ target detection process. Multiple theories are needed to deepen our understanding of target detection by odontocetes, which can provide references for designing intelligent biomimetic signal processors.

## EDITOR'S SUGGESTION

2021, 70 (15): 157301. doi: 10.7498/aps.70.20210065
Abstract +
2021, 70 (15): 157501. doi: 10.7498/aps.70.20210097
Abstract +
Hole-doped perovskite-type manganites have received intensive attention due to their intriguing physical phenomena such as giant magnetocaloric effect and magnetic-phase transitions. However, the mechanism of internal ferromagnetic interaction still needs to be further explored due to the complex natures of competing double-exchange (DE) and super-exchange (SE) interaction, Jahn-Teller (JT) polaron localization, charge ordering, and phase separation scenarios. Critical exponent analysis near magnetic phase transition is a powerful tool to investigate the details of the ferromagnetic interactions and has been used frequently in various magnetocaloric materials. In this article, the critical behavior analyses of perovskite manganites in recent years are comprehensively reviewed. A large number of studies have shown that even in single-phase materials with uniform structure and composition, the critical behavior can be affected by multiple factors such as grain boundary density and the degree of disorder, making them difficult to fully describe the intrinsic ferromagnetism. In this review, firstly, the critical behaviors of typical manganites with different bandwidths in single crystal and polycrystalline are discussed. In a double-exchange dominated system such as La-Sr-Mn-O, short-range 3D-Heisenberg model is basically in good accordance with optimally-doped single crystal sample. However, it would be replaced by long-range mean-field critical behavior in polycrystalline sample when the correlation length exceeds the crystallite size. In a typical intermediate bandwidth system such as La-Ca-Mn-O exhibiting a complex phase diagram described by competing SE/DE interactions, JT polaron localization/delocalization, and Griffith phase disorder, the critical exponent can vary from 3D-Heisenberg model to tricritical mean-field model, for the crossover from first to second order phase transition. Secondly, the studies of elements doping and different fabrication methods indicate that the critical behavior of manganites can be effectively modulated, and vary between different theoretical models including even nonuniversal exponent for highly disordered magnetic system. In the following part, the influence of magnetic field on the critical behavior and field induced crossover phenomena of La-Ca-Mn-O system near tricritical point is analyzed and discussed in detail. Furthermore, the magnetocaloric effects of materials near the tricritical point collected in many studies are listed and compared with each other. Excellent magnetocaloric properties with high magnetic entropy change and relative cooling power in plenty of researches indicate that ideal magnetocaloric material would be very likely to be found in the materials near the tricritical point, which lay at the borderline between first-order and second-order phase transition. Consequently, it is suggested that perovskite manganites are still quite promising in the potential magnetic refrigeration applications, and need to be further developed.

## EDITOR'S SUGGESTION

2021, 70 (15): 150201. doi: 10.7498/aps.70.20210411
Abstract +
$O[{\rm{poly}}(p q)]$. We conduct the principle demonstration of the algorithm in the NMR quantum computing system. In the experiment, $^{13}{\rm C}$ labeled cromaric acid is used as a four-bit sample, dissolved in d6-acetone, and $^1 {\rm H }$ is decoupled in the whole process. In the case of a large number of bits, quantum principal component analysis, quantum recommendation algorithm, and other quantum algorithms can achieve the same goal, and their time complexities are basically the same. In this paper, the resonance transition algorithm is used to effectively replace the phase estimation algorithm in this kind of problem, which greatly reduces the need of auxiliary bits. Only one auxiliary bit is used and a singular value is retained to better restore the image, which is currently unable to be achieved by other algorithms based on phase estimation. Firstly, an $8\times8$-dimensional image matrix is selected, and the pseudo-pure state is prepared by using the spatial averaging method. The quantum state reaches the target state by using gradient descent pulse to complete the preparation of the initial state. Then the shape pulse is used to apply the time-evolution operator to the initial state several times to realize the time evolution of the Hamiltonian $\mathcal{H}$ of the resonance transition algorithm. Finally, the quantum state chromatography is used to read out the different components of the density matrix and reconstruct the density matrix. The experimental results are analyzed by quantum state chromatography, and the experimental values are in agreement with the theoretical ones. The fidelity is 99.84%, and the error comes mainly from the experimental equipment and the gradient pulse’s optimization algorithm. This verifies the correctness of the matrix low-rank approximate quantum algorithm proposed in this paper within the error range. For the classical algorithm, it usually takes $O[{\rm{poly}}(p q)]$ to solve the low-rank matrix on the classical computer. Compared with the classical algorithm, the quantum algorithm achieves exponential acceleration.">In the era of big data, efficient data processing is crucial. Quantum computing has the capability of parallel computing, which provides a new solution for convenient data processing. We propose a matrix low-rank approximate quantum algorithm based on singular value decomposition with a complexity of $O[{\rm{poly}}(p q)]$. We conduct the principle demonstration of the algorithm in the NMR quantum computing system. In the experiment, $^{13}{\rm C}$ labeled cromaric acid is used as a four-bit sample, dissolved in d6-acetone, and $^1 {\rm H }$ is decoupled in the whole process. In the case of a large number of bits, quantum principal component analysis, quantum recommendation algorithm, and other quantum algorithms can achieve the same goal, and their time complexities are basically the same. In this paper, the resonance transition algorithm is used to effectively replace the phase estimation algorithm in this kind of problem, which greatly reduces the need of auxiliary bits. Only one auxiliary bit is used and a singular value is retained to better restore the image, which is currently unable to be achieved by other algorithms based on phase estimation. Firstly, an $8\times8$-dimensional image matrix is selected, and the pseudo-pure state is prepared by using the spatial averaging method. The quantum state reaches the target state by using gradient descent pulse to complete the preparation of the initial state. Then the shape pulse is used to apply the time-evolution operator to the initial state several times to realize the time evolution of the Hamiltonian $\mathcal{H}$ of the resonance transition algorithm. Finally, the quantum state chromatography is used to read out the different components of the density matrix and reconstruct the density matrix. The experimental results are analyzed by quantum state chromatography, and the experimental values are in agreement with the theoretical ones. The fidelity is 99.84%, and the error comes mainly from the experimental equipment and the gradient pulse’s optimization algorithm. This verifies the correctness of the matrix low-rank approximate quantum algorithm proposed in this paper within the error range. For the classical algorithm, it usually takes $O[{\rm{poly}}(p q)]$ to solve the low-rank matrix on the classical computer. Compared with the classical algorithm, the quantum algorithm achieves exponential acceleration.
2021, 70 (15): 150701. doi: 10.7498/aps.70.20210151
Abstract +
In this paper we present a miniaturized pre-calibration based forward-viewing Lissajous scanning fiber probe for endoscopic optical coherence tomography (OCT). The probe is based on an asymmetric fiber cantilever driven by the piezoelectric bender to realize the two-dimensional (2D) Lissajous scanning, which can realize a relatively large scanning range under a low driving voltage. A capillary metal tube is mounted at the end of the main fiber to reduce the resonant frequency of the fiber cantilever. The relationship between the filling rate and the side-lobe number of the Lissajous scanning pattern is studied, and a method of selecting the orthogonal resonant frequency of the Lissajous scanning is proposed. Through the numerical simulation by COMSOL software, the structural parameters of the asymmetric fiber cantilever are determined. The orthogonal resonant frequencies of the asymmetric fiber cantilever are 169 Hz and 122 Hz. The lengths of the main imaging fiber, the auxiliary fiber and the metal capillary tube are 15.94 mm, 4.49 mm and 2 mm, respectively. The probe is fully packaged in a metal tube for endoscopic imaging. The focal spot and the working distance are 25 µm and 5 mm, respectively. The field of view is larger than 1.5 mm × 1.5 mm. The total rigid length and the outer diameter of the probe are 35 mm and 3.5 mm, respectively. The stability and repeatability of the Lissajous scanning trajectory, and the imaging stability with the rotation of the probe are investigated and verified. The probe is incorporated into a 50 kHz swept source OCT system. The axial resolution of the endoscopic OCT is 10.3 μm, and the imaging frame rate is 1 FPS (frames per second). The maximum signal-to-noise ratio of the imaging system is 110 dB. The imaging performance of the probe is validated by the 2D en-face and three-dimensional volumetric OCT imaging of the high scattering sample and the biological tissue. The probe can be used for the endoscopic imaging of the human tooth. From the result we can distinguish the dental enamel, dental essence and the dental calculus. The developed forward-viewing Lissajous scanning fiber probe is expected to be used in dental applications such as early calculus detection.
2021, 70 (15): 150702. doi: 10.7498/aps.70.20210312
Abstract +
The quantitative method of iodine monoxide radical (IO) using incoherent broadband cavity enhanced absorption spectroscopy (IBBCEAS) in the 435–465 nm band is described in this paper. In order to obtain the concentration of IO accurately, the parameters such as the mirror reflectivity, effective cavity length and sample loss of the IBBCEAS system are evaluated. Using the difference of Rayleigh scattering between nitrogen and helium, the reflectivity curve of the high-reflection mirror is obtained. The reflectivity R of the mirror at 436.1 nm of the IO absorption peak is about 0.99982, and the effective absorption optical path reaches 3.83 km under vacuum condition. According to the absorption of O4, the effective cavity length of the modified system is 60.7 cm. The Allan deviation is used to evaluate the performance of the system, and the standard deviation is used to analyze the detection sensitivity of the system. When the time resolution is 60 s, the detection sensitivity (2σ) of the system for IO and NO2 are 1.9 pptv and 20 pptv (part per trillion by volume), respectively. The iodine dissolved in potassium iodide (KI) solution is taken out by the bubbling method and react with ozone after photolysis to produce a stable concentration of IO sample gas. The IO loss in the sampling tube is calibrated, and the results show that the sampling tube has no significant effect on the IO loss. The IBBCEAS system is used to determine the linearity of IO, and the correlation coefficient R2 between the measured concentration of IO and the proportioned concentration in a concentration range from 39 to 530 pptv is 0.99. The IO produced by the reaction of iodine released from kelp with ozone is measured.
###### ATOMIC AND MOLECULAR PHYSICS
2021, 70 (15): 153301. doi: 10.7498/aps.70.20210450
Abstract +
The analytical potential energy function (APEF) of SiH+(X1Σ+) is fitted by Aguado-Paniagua function with 112 ab initio energy points, which are calculated by Molpro 2012 Package with the multi-reference configuration interaction including the Davidson correction method using AVX Z and AVX dZ (X = Q, 5, 6) basis sets. Moreover, the calculated ab initio energy points are subsequently extrapolated to complete basis set (CBS) limit to avoid the basis set superposition error. All the fitting parameters of APEFs for AV6Z, CBS(Q, 5), AV6dZ, CBS(Qd, 5d), SA-AV6dZ and SOC-AV6dZ methods are gathered. The potential energy curves (PEC) and the corresponding ab initio points are also shown. As can be seen, the PECs show excellent agreement with the ab initio points and a smooth behavior both in short range and long range, which ensures the high quality of fitting process for the current APEFs. Based on these APEFs of different basis sets and methods including AVQZ, AV5Z, AV6Z, CBS(Q, 5), AVQdZ, AV5dZ, AV6dZ and CBS(Qd, 5d), the spectral constants De, Re, ωe, Be, αe and ωeχe are obtained. In addition, the effects of spin-orbit coupling interaction (SOC) on the system are also investigated. By comparing the spectral constants of SA-AV6dZ with the ones of SOC-AV6dZ, it is found that the effect of SOC on SiH+(X1Σ+) is small and can be ignored. We also compare the spectral constants in this work with the experimental values and other theoretical results. The results of this work accord well with the corresponding experimental and other theoretical results. It is worth noting that the deviation of dissociation energy between the theoretical calculations and experimental values is relatively large. Based on this conclusion, we suggest that the spectral constants including the dissociation energy for SiH+(X1Σ+) should be remeasured. Based on the APEF of SOC-AV6dZ which should be more accurate than others in theory, the top 23 vibrational states (j = 0) of SiH+(X1Σ+) are calculated first by solving the radial Schrödinger equation. All the vibrational energy levels, classical turning points, rotation constants and six centrifugal distortion constants are also provided. The results of this work can provide significant references for the experimental and other theoretical work.

## COVER ARTICLE

2021, 70 (15): 154101. doi: 10.7498/aps.70.20210188
Abstract +
With the development of intelligent technology, it is essential to develop polarization-conversion devices with adaptable electromagnetic (EM) performance for practical applications. Up to now, most of attempts have relied on PIN diodes and varactor diodes for electrical tuning, typically featuring simplicity and timelineness. However, the shortcomings are also notable, such as less degrees of freedom (DoFs), more complex circuits and more expensive. In view of this, here we propose a kind of spatial-order metasurface for reconfigurable polarization conversion based on kirigami concept. By adjusting the folding angle β, the interaction between neighboring dipoles can be progressively changed and thus the operation frequency of polarization conversion can be shifted. Such a mechanical reconfigurable strategy brings about more DoFs for tuning and is cheaper and extraordinary convenient in practice. To verify the feasibility of our concept, a proof-of-concept spatial-order kirigami metasurface is proposed for the dual-band reconfigurable linear polarization conversion based on asymmetric chiral split ring resonators (SRRs). Experimental results show that the linear polarization operates at 5 and 5.8 GHz when folding angle is β = 10°, these frequencies are shifted to 5.8 and 7.2 GHz when β = 45°: a tuning range is expanded by 18.5%. In addition, the Poisson’s ratio and relative density of proposed kirigami metasurface as a function of β are also theoretically analyzed. The results show that the Poisson’s ratio increases with the value of β increasing. The relative density can be reduced to 1.5% of its unfolded planar counterpart. Our spatial-order kirigami metasurface strategy paves the way for implementing the reconfigurable linear polarization conversion and multifunctional devices.
2021, 70 (15): 154201. doi: 10.7498/aps.70.20210261
Abstract +
Corner cube retroreflector (CCR) as a main optical component of laser ranging retroreflector array plays a key role in satellite laser ranging (SLR) to cooperative targets. To accurately estimate the echo energy of SLR, it is necessary to precompute photons’ distribution in the distance of SLR station by calculating the Far field diffraction pattern (FFDP) of CCR under various conditions. In this paper, the analysis of the effective reflection area and optical reflectivity of CCR for arbitrary incidence angle are carried out, in which a method to calculate the CCR effective reflection area with a wider applicability is used, and the difference between optical reflectivity of metal-coated CCR and uncoated (total internal reflection) CCR is also discussed. On this basis, combined with optical diffraction theory, a simulation program for CCR FFDP calculation is established, thereby producing FFDPs of CCR for a variety of incidence angles under different coating conditions. The results show that the FFDP of metal-coated CCR is almost unrelated to the azimuth angle of incident light or polarization, but is determined only by elevation angle of incident light. The pattern is always like Airy spot or its tensile deformation. Relatively, uncoated CCR’s FFDP has a more complex figure, its reflected energy is divided into several lobes whose size, number and position are all influenced by elevation and azimuth angle of incidence, and also by the polarization. Generally, the incidence direction which has a large total intensity of far optical field is to an extent the same as that of large effective reflection area and optical reflectivity. Furthermore, simulation results with uncoated CCR presents a higher relevance of incidence direction and FFDP.To verify the reliability and correctness of these simulation results, a diffraction optical experimental system at a wavelength of 1550 nm is set up to conduct laboratory confirmation, including laser, camera, beam expander, rotating platform and other essential optics. A silver-coated CCR and an uncoated CCR (both made of fuse silica, each has different dihedral angle offset) are chosen to measure their FFDPs with random polarization directions at several random incidence angles. All the measurement results are in good agreement with the simulations of FFDPs.
2021, 70 (15): 154202. doi: 10.7498/aps.70.20210190
Abstract +
In-line digital holography usually employs a phase retrieval algorithm to decouple the phase information but fails to eliminate the unwanted DC and twin image terms when the measured sample does not agree with the sparsity. While the off-axis digital holography can efficiently remove the unwanted image terms but can not reserve the high frequencies of the sample to realize high resolution. The in-line-and-off-axis hybrid digital holography was then developed to provide a relatively high resolution digital holographic imaging without considering the effect of the unwanted terms. In other words, the in-line-and-off-axis hybrid digital holography merges all of the best virtues of the mentioned-above methods in an efficient and elegant way. However, this state-of-the-art method requires prior knowledge about the diffraction distance, which results in time-consuming and low accuracy. In other sense, telecentric technology can realize non-diffractive imaging without the knowledge about the diffraction distance or spherical aberration or defocusing aberration. Therefore, in this paper, a novel in-line-and-off-axis hybrid digital holography is proposed by introducing telecentric imaging architecture, and the corresponding reconstruction method is further proposed by utilizing constrained iterative approach. In this method, telecentric in-line-and-off-axis hybrid digital holography is first used to acquire focused off-axis and in-line holograms, respectively. The low resolution phase information is reconstructed from the off-axis hologram by using Fourier transform method with the help of the sample-free off-axis hologram, and then multiplexed with the amplitude information obtained from the in-line hologram to act as the initial complex amplitude in the iterative recovery process. As a result, constrained iterations are carried out in the spatial domain and frequency domain to realize high resolution and high speed reconstruction. After simulations, we build an experimental setup and demonstrate the operation of the method with USAF resolution target, onion cells and bee wings. Both the simulation and experimental results show that the proposed method can require no prior knowledge to suppress the phase disturbance caused by the unwanted image terms and optical aberrations, resulting in high speed and full utilization of spatial bandwidth product of the digital camera to yield high resolution reconstruction. We hope that the proposed method will have most practical applications in the case where large resolution, high speed and good quality are needed.

## EDITOR'S SUGGESTION

2021, 70 (15): 154203. doi: 10.7498/aps.70.20210318
Abstract +
Generation and manipulation of continuous variable quantum states are the building blocks of quantum communication, quantum key distribution and quantum networks. According to the second-order nonlinear process of the periodically-poled potassium titanyl phosphate (PPKTP) crystal, we design a semi-monolithic optical parametric amplifier (OPA) cavity to generate the bright squeezed light at a wavelength of 1064 nm. With the injection of a seed beam, the squeezed state generated by the OPA has a coherent amplitude, so called bright squeezed state. The squeezing level is directly observed to be –11.6 dB when the pump power is 310 mW at an analysis frequency of 3 MHz. However, with the increase of the pump power, the purity of the squeezed state gets lower and lower due to the increased influence of the anti-squeezing quadrature component on the squeezed quadrature component in the detection process. To obtain a higher purity of the squeezed state for achieving linear optical manipulation and quantum tomography, we choose the pump power of 50 mW, the squeezing level decreases to –6 dB, and the purity of the squeezed state is 98.5% in this case. An electro-optic modulator is adopted to realize the liner manipulation of the squeezed light in the phase space. During the measurement of the bright squeezed state, all the data are taken on condition that the length of the OPA cavity and relative phase between the seed beam and the pump beam are locked by a locking loop. The direct current (DC) signal of the balanced homodyne detection (BHD) is used to accurately determine the phase corresponding to the time domain signal of the squeezed state, while the alternate current (AC) signal of the BHD is mixed with the signal generated by the function generator, after passing through a low-pass filter and a high-pass filter, the signal is then amplified by using a low-noise amplifier. A high-performance oscilloscope is finally used to simultaneously collect the signals, thus obtaining the quantum noise signal of the bright squeezed light after linear manipulation. Together with the maximum likelihood estimation algorithm, the quantum tomography, the density matrix and the Wigner function of the bright squeezed light are obtained, that is, all the information such as the photon number distribution of the quantum state is determined. Multiple iterations are taken in the maximum likelihood estimation algorithm process to eliminate the influence of the low quantum efficiency on the detection system, so that the density matrix is fitted well with the theoretical results.
2021, 70 (15): 154204. doi: 10.7498/aps.70.20210173
Abstract +
$|1, 1\rangle$ and $|1, 0\rangle$ of the 87Rb atom. In the experiment, we observe the relation between the percentage of atoms in the two quantum states and the relative phase between the two pairs of Raman light beams. This method provides a unique control parameter for ultracold atom quantum simulation experiments, which is the laser phase. It is hoped that this technology can be used to manipulate the interaction between light and atoms in the future to achieve more abundant physical phenomena.">In this paper, we develop a new method to adjust the Raman coupling strength by using the relative phase between two pairs of Raman lasers. The stimulated Raman transition process is highly controllable and has the characteristics of multiple degrees of freedom. In experiments on ultracold atoms, the populations of atomic energy levels can be adjusted by taking an appropriate Raman light intensity and interaction time, and by detuning the two-photon frequency. The intensity of the Raman laser is usually changed to adjust the Raman coupling strength. Based on two-level atoms, a new method of accurately controlling the Raman coupling strength by using the relative phase between two pairs of Raman light beams is developed. This technology can achieve coherent manipulation of atomic quantum states, which greatly broadens the ability of ultracold atoms to perform quantum simulations. First, the 87Rb Bose-Einstein condensate is realized by using an optical dipole trap. Then, the two pairs of Raman lasers are designed with a special optical path to keep the relative phase of the two pairs of Raman lasers stable in the transmission process, and can be controlled accurately. Then the two pairs of Raman light beams act on the two ground state hyperfine energy levels $|1, 1\rangle$ and $|1, 0\rangle$ of the 87Rb atom. In the experiment, we observe the relation between the percentage of atoms in the two quantum states and the relative phase between the two pairs of Raman light beams. This method provides a unique control parameter for ultracold atom quantum simulation experiments, which is the laser phase. It is hoped that this technology can be used to manipulate the interaction between light and atoms in the future to achieve more abundant physical phenomena.
2021, 70 (15): 154205. doi: 10.7498/aps.70.20210326
Abstract +
Stimulated Brillouin scattering (SBS) is a typical inelastic scattering effect generated by the interaction between intense incident laser and the acoustic wave field in medium and has always been an active research topic in nonlinear optics. The SBS can be used as a novel LIDAR technology for active optical remote sensing of temperature and sound speed structure in ocean. Although, the threshold value and gain property of SBS at normal temperature are studied, none of the threshold values and gain coefficients of SBS at different temperatures, pressures and attenuation coefficients has been investigated in detail. Further, neither the relation between threshold value and water pressure nor the relation between gain coefficient and water pressure is clear now, and little work has been reported. The theoretical and experimental studies of the influence of water parameters on the threshold value and gain coefficient of SBS are still scanty. In this paper, the effects of temperature, pressure and attenuation coefficient of water on threshold value and gain coefficient of SBS are studied theoretically and experimentally. Theoretically, the variations of threshold value and gain coefficient of SBS with temperature, pressure and attenuation coefficient are analyzed by the average attenuation coefficient method based on the distributed noise model (DNM) and coupled wave equations. The temporal waveforms of Stokes-, pump- and transmission-beam at different water parameters are obtained by using the DNM. Experimentally, a temperature-pressure controlled simulator is designed to obtain the threshold values and gain coefficients of SBS in water at different temperatures, pressures and attenuation coefficients through measuring the change of attenuation coefficient of laser pulses. The results indicate that (i) the threshold value of SBS increases with pressure increasing at the same temperature and decreases with temperature increasing at the same pressure; (ii) the threshold value is positively correlated with the attenuation coefficient at the same temperature and pressure; (iii) the gain coefficient of SBS increases with temperature increasing at the same pressure and decreases with pressure increasing at the same temperature. We also find that the temperature and attenuation coefficient have greater effect on threshold value and gain coefficient of SBS than the water pressure. The studied results are of great significance in realizing the ocean remote sensing by SBS lidar.

## EDITOR'S SUGGESTION

2021, 70 (15): 154206. doi: 10.7498/aps.70.20210332
Abstract +
Subwavelength artificial structures of high refractive index dielectrics provide an effective way to control and manipulate light on a nanoscale by enhancing electric and magnetic fields. This kind of structure usually has low absorption loss, but its performance is also limited by radiation loss, which will reduce the efficiency of its nonlinear response. This problem can be solved by using bound states in the continuum (BICs). The BICs are a kind of unconventional state which is in continuous domain but remains localized. They exist within a light cone and have an infinite quality factor. By combining BICs with nonlinear optics, high-Q resonances from quasi-BICs are used to excite and enhance the nonlinear response. The simulation shows that when the symmetry of the unit cell of the silicon nanoparticle arrays is broken, the BIC become the quasi-BIC, and the transmission spectrum will produce a high-Q narrow resonance valley. The resonance has polarization dependence of electric field. With the change of pump wavelength, the third-harmonic generation (THG) intensity first increases and then decreases gradually. The pump wavelength changed by several nanometers can change THG intensity by at least one order of magnitude. When the pump wavelength is adjusted to the resonance wavelength, the nonlinearity is significantly enhanced as a result of the strong field localization. The THG intensity is highly sensitive to the variation of asymmetric parameters. Only a change of 75 nm will result in a decrease of THG intensity by at least one order of magnitude. There is a third-order relationship between pump power and THG power. For the proposed structure, the factors affecting the conversion efficiency of THG include pump power, pump wavelength, polarization angle of pump light, and asymmetry parameter. When the polarization direction of electric field is along the short axis of the structure and the pump light at resonance wavelength is vertically incident to the structure with an asymmetric parameter of 0.125, the conversion efficiency of THG can be increased to ～2.6 × 10–6 and the intensity of THG is increased by six orders of magnitude. The results are expected to be applied to designing the silicon-based optical nonlinear devices.
2021, 70 (15): 154207. doi: 10.7498/aps.70.20210228
Abstract +
Active illumination is a crucial technology for active imaging, active tracking and aiming system. But the atmosphere turbulence distributed over the entire path causes the intensity to fluctuate, which reduces the illumination uniformity seriously. Therefore, it is desirable to find ways to reduce the intensity fluctuation and improve the uniformity of active illumination. It has been revealed that one can improve illumination uniformity by using multi-beam laser illuminator. Another effective approach is partially coherent beam illumination.In this paper, a novel method is suggested to improve the illumination uniformity. Phase disturbance is induced by a ladder-like phase modulator (LPM) and the transmitting field of narrow spectrum laser is confused, and thus the atmosphere turbulence will be compensated and the illumination uniformity will be improved. The physical models of narrow spectrum laser phase modulation and atmosphere propagation are deduced, and the expression of the facular distribution is obtained. The active-illumination experimental setup with a laser propagation distance of 1.8 km through horizontal atmosphere is established. Based on the facular distribution of illumination laser at 1.8 km, the uniformity and stabilization are achieved. The experimental results indicate that the illumination uniformity and stabilization are both improved. The spatial and central time scintillation indexes are improved from 0.73 to 0.33 and from 0.38 to 0.14, respectively.

## EDITOR'S SUGGESTION

2021, 70 (15): 154208. doi: 10.7498/aps.70.20210118
Abstract +
The red upconversion (UC) emission of Ho3+ ions is located in an “optical window” range of the biological tissue, which has great prospects in the biology application. In this work, the NaYF4:20%Yb3+/2%Ho3+/12%Ce3+@NaYF4:x%Yb3+ and NaYF4:20%Yb3+/2%Ho3+/12%Ce3+@NaYF4:15%Yb3+/x%Nd3+ core-shell (CS) nanoparticles (NPs) are built based on the epitaxial growth technology by the high-temperature co-precipitation method in order to enhance red UC emission. The crystal structure and morphology of NaYF4 CS NPs are characterized by X-ray diffraction and transmission electron microscope. It can be found that the morphology of NaYF4 CS NPs changes from sphere into rod shape when coated with NaYF4 shell, and has a pure hexagonal-phase crystal structure. Under 980 nm excitation, the red UC emission intensity of NaYF4:20%Yb3+/2%Ho3+/12%Ce3+@NaYF4:5%Yb3+ CS NPs is strongest and enhanced about 5.2 times than that of NaYF4:20%Yb3+/2%Ho3+/12%Ce3+ NPs. Under 800 nm excitation, the red emission intensity of NaYF4:20%Yb3+/2%Ho3+/12%Ce3+@NaYF4:15%Yb3+/20%Nd3+ CS NPs is increased about 6.1 times compared with that of the NaYF4:20%Yb3+/2%Ho3+/12%Ce3+@NaYF4:15%Yb3+/5%Nd3+ CS NPs. This is because the constructed CS effectively reduces the non-radiative decay from the surface defects of NPs, and the doped Yb3+ and Nd3+ ions in the NaYF4 shells can transfer more excitation energy to Ho3+ ions in the core. In addition, the NaYF4: 20%Yb3+/2%Ho3+/12%Ce3+@NaYF4:15%Yb3+/20%Nd3+ CS NP is excited by dual-wavelengths co-excitation (800 nm + 980 nm). It is found that the red UC emission intensity under the co-excitation of dual-wavelengths is higher than the sum of the excitation intensities of two single wavelengths (800 nm and 980 nm), which is due to the synergistic effect generated under the co-excitation of 980 nm and 800 nm near infrared laser. Therefore, different CS structures constructed by introducing different energy transfer channels can achieve the enhancement of the red UC emission under different excitation conditions, and the dual-wavelength co-excitation provides a new way to improve the penetration depth and the detection sensitivity for further expanding the applications in the field of biomedicine.
2021, 70 (15): 154209. doi: 10.7498/aps.70.20210355
Abstract +
Prediction of chaotic laser has a wide prospect of applications, such as retrieving lost data, providing assists for data analysis, testing data encryption security in cryptography based on chaotic synchronization of lasers. We propose and demonstrate a new method of using time delayed photonic reservoir computing (RC) to forecast the continuous dynamical evolution of chaotic laser from previous measurements. Specifically, the time delayed photonic RC based on semiconductor laser with optical injection and feedback structure is established as a prediction system. Chaotic laser, as input signal, is generated by semiconductor laser with external disturbance.The time delayed photonic RC used in this stage is a novel implementation, which consists of three parts: the input layer, the reservoir and the output layer. In the input layer, the chaos laser from the semiconductor with an optical feedback needs to preprocess and multiply by a mask signal. The reservoir is the master-slave configuration consisting of a response laser with the optical feedback and light injection. In the feedback loop, there are N virtual nodes at each interval θ with a delay time of τ (N = τ/θ). The reservoir performs the mapping of the input signal onto a high-dimensional state space. In the output layer, the output of the reservoir is a linear combination of the reservoir state and the output weight. The output weight is optimized by minimizing the mean-square error between target value and output value through using the ridge regression algorithm.The results demonstrate that time delayed photonic RC based on semiconductor laser can forecast the trajectory of chaotic laser in about 2 ns. Moreover, we also investigate the influence of critical parameters on prediction result, including the type of the mask, the quantity of the virtual nodes, the length of the training data, the input gain, the feedback strength, the injection strength, the ridge parameter and the leakage rate.The method used here in this work has many attractive advantages, such as simple configuration, low training cost and eminently suitable for hardware implementation. Although the prediction length is limited, the significant innovation using time delayed photonic RC based on semiconductor lasers as the prediction system of chaotic laser presents a new opportunity for further developing a technique for predicting chaotic laser.
2021, 70 (15): 154301. doi: 10.7498/aps.70.20202172
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Owing to the low energy density of sound energy in nature, it is difficult to realize the local enhancement effect of sound field in air. Therefore, it is of great significance to explore new physical mechanisms and methods to achieve sound field enhancement. In recent years, artificial Mie resonance structure as a kind of acoustic metamaterial has attracted considerable attention, which has a variety of resonant modes, such as monopolar, dipolar, quadrupolar and higher multipolar modes. Compared with local resonance, acoustic Mie resonance mode has strong acoustic interaction, which can effectively enhance the acoustic field by the coupling of the Mie resonance. In this paper, we design an acoustic metamaterial composed of multiple-cavity unit cells, which is capable of realizing sound field enhancement. The multiple-cavity unit is circular in external shape and it is composed of a circular central cavity and twelve resonators. The twelve resonators are evenly distributed around the circular central cavity, with three resonators combined into a group. This exotic function arises from the compound monopole Mie resonance introduced by mutual coupling between the system structure and the monopole Mie resonance of each unit cell. Symmetric and asymmetric metamaterials are constructed by arranging several multiple-cavity unit cells in different forms. These two kinds of metamaterials can be used to achieve sound field enhancement with different effects. The results show that due to the symmetry of metamaterial structure, the symmetric metamaterials with square, circle, rectangle and regular hexagon shapes can realize the sound field enhancement, which is independent of the direction of incident wave. However, for the asymmetric metamaterial with equilateral triangle shape, the sound intensity in the center of the system varies with incident direction, which indicates that the designed asymmetric metamaterial has a strong dependence on the direction of incident wave. These two kinds of metamaterials constructed in this research can possess a number of potential applications such as in sound insulation, acoustic sensor, noise location, acoustic communication and asymmetric acoustic device. These two kinds of metamaterials constructed in this research can possess a number of potential applications such as in sound insulation.
2021, 70 (15): 154701. doi: 10.7498/aps.70.20210194
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Ultrasound-induced cavitation in the viscoelastic biological tissues has attracted considerable attention due to its mechanical bio-effects, such as cell sonoporation, hemolysis, vascular disruption and tissue erosion. Cavitation can exert strong mechanical stresses on the surrounding tissues during the rapid bubble growth and collapse. The occurrence of cavitation needs the ultrasound exposure exceeding a certain acoustic pressure threshold, and the cavitation threshold is very high in most tissues, probably causing undesirable side-effects. Introducing artificial cavitation nuclei, e.g., microbubbles and nanodroplets stabilized with a shell such as albumin, lipids or polymers, into the targeted region can effectively reduce the cavitation threshold and significantly enhance the cavitation effects. However, neither the cavitation dynamics of an encapsulated microbubble nor the cavitation-induced stress field around the bubble in a soft tissue is quite clear. In this study, a comprehensive numerical model is developed to describe the dynamics of a lipid-shelled microbubble in vivo and quantify the cavitation-induced mechanical stress in the tissue. Considering the nonlinear changes of both shell viscosity and elasticity, a Gilmore model that has been considered as the most developed and realistic cavitation model is coupled with the Zener viscoelastic model for precisely describing tissue viscoelastic behavior with both creep recovery and stress relaxation of tissue. The developed model has an advantage of accurately describing the bubble behaviors in different biological tissues at high ultrasound intensities, especially for the bubble collapse. Furthermore, the spatiotemporal evolution of mechanical stress in the surrounding tissue generated by the cavitation bubble is investigated. Finally, the effects of encapsulated shell, elasticity modulus and viscosity of tissue as well as ultrasound amplitude are examined. The results show that the viscoelasticity of encapsulated shell and tissue both inhibit the bubble oscillations, and the tissue viscoelasticity has a larger inhibition effect. During the bubble oscillation, the compressive (negative) stress is generated in the tissue with the bubble growing and it continuously increases until it reaches a maximum value at a maximum radius, while the tensile (positive) stress is generated at the stage of bubble collapse and initial stage of bubble rebound due to the restoration of deformed tissue. The stress magnitude is greatest near the bubble wall and decreases rapidly with depth extending into the surrounding tissue. By contrast, the tensile stress decreases at a higher rate than the compressive stress. The encapsulated bubble presents a smaller stress in the tissue, but the decrease of the stress can be ignored at large acoustic pressures. Moreover, the stress decreases with the increase of tissue elasticity modulus, whereas it first increases and then decreases with tissue viscosity increasing, showing a maximum at 15 mPa·s. The increasing of the ultrasound amplitude enhances the bubble oscillations and consequently increases the stress in the tissue. This study is helpful in understanding the bubble dynamics and cavitation-induced mechanical stress of an encapsulated microbubble in soft tissue, which is essential for a safe and precise ultrasound therapy.
2021, 70 (15): 154702. doi: 10.7498/aps.70.20202024
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${M_{\rm{A}}} = 3.33$) on the vorticity and pressure evolution of mixing layer, and also compared with those in the hydrodynamics situation. Moreover, the mechanism of weakening the effect of magnetic field on the KH instability is revealed from the perspectives of the magnetic pressure and the magnetic tension. The results show that the external magnetic field has a great influence on the flow structure of the mixing layer. Specifically, the magnetic pressure has a major effect in the vorticity deposition on the interface, whereas the magnetic tension generates a torque to counter the scrolling of vortex. As a result, the large vortex structure is stretched and destroyed, and finally restrains the vortex rolling-up. In addition, with the development of mixing layer, the interface will separate at the points of maximum curvature under the joint effect of the magnetic pressure, the magnetic tension and the pressure field, and finally form a fishhook-like vortex structure.">The evolution of the Kelvin-Helmholtz (KH) instability in the presence of classical hydrodynamics and magneto-hydro-dynamics is investigated numerically by using the magneto-hydro-dynamic (MHD) equations. The MHD equations are solved with the corner transport upwind plus constrained transport algorithm that guarantees the divergence-free constraint in the magnetic field. The numerical results are used to analyze the effects of magnetic field (${M_{\rm{A}}} = 3.33$) on the vorticity and pressure evolution of mixing layer, and also compared with those in the hydrodynamics situation. Moreover, the mechanism of weakening the effect of magnetic field on the KH instability is revealed from the perspectives of the magnetic pressure and the magnetic tension. The results show that the external magnetic field has a great influence on the flow structure of the mixing layer. Specifically, the magnetic pressure has a major effect in the vorticity deposition on the interface, whereas the magnetic tension generates a torque to counter the scrolling of vortex. As a result, the large vortex structure is stretched and destroyed, and finally restrains the vortex rolling-up. In addition, with the development of mixing layer, the interface will separate at the points of maximum curvature under the joint effect of the magnetic pressure, the magnetic tension and the pressure field, and finally form a fishhook-like vortex structure.
###### PHYSICS OF GASES, PLASMAS, AND ELECTRIC DISCHARGES
2021, 70 (15): 155201. doi: 10.7498/aps.70.20202091
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Atmospheric pressure non-equilibrium low-temperature plasma has been widely used in biomedicine, surface treatment and other fields, which has attracted the attention of researchers extensively. As one of the important methods to generate such a plasma, the plasma jet has become a popular method, which can generate a remote plasma plume at the nozzle through introducing a rare gas flow. However, plasma plume has a small diameter, which results in deficiency for the large-scale surface treatment. A dielectric barrier discharge device with three electrodes is utilized to produce a large brush-shaped plasma plume (50.0 mm × 40.0 mm) downstream of flowing argon under the combined excitation of an alternate current (AC) voltage and a negative bias voltage, thereby increasing the plume scale. The results show that the luminescence intensity of the plasma plume increases with AC peak voltage increasing. By fast photography implemented with an intensified charge coupled device (ICCD), it is found that the plasma plume is composed of temporally superposed branched-streamers. The ICCD images also reveal that the number of branches increases with AC peak voltage increasing. Moreover, the waveforms of AC voltage and light emission signal recorded simultaneously indicate that the plasma plume initiates once per AC voltage cycle, which occurs in the positive half cycle of the applied voltage. With AC peak voltage increasing, the duration and intensity of discharge pulse increase, which results from more branches of the branched streamer. Besides, optical emission spectrum in a range from 300 nm to 850 nm mainly includes OH (A2Σ+–X2Π) peaked at 308.0 nm, the second positive system of N2 (C3Πu–B3Πg), Ar I (4p–4s), and O I (3p3 P–3s3 S) at 844.6 nm. Based on the optical emission spectrum, the plasma parameters such as vibrational temperature and intensity ratio of spectral lines (correlated with electron density and electron temperature) are investigated. Besides, the variation of concentration of oxygen atoms in the plasma plume with experimental parameters is investigated by optical actinometry. The results indicate that the concentration of oxygen atoms first increases and then decreases with the distance increasing along the argon flow direction or with oxygen content of the working gas increasing. In addition, the concentration of oxygen atoms increases with AC peak voltage increasing. All these results are discussed qualitatively. These results are of great importance in modifying the plasma surface on a large scale.
2021, 70 (15): 155202. doi: 10.7498/aps.70.20202135
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Radio frequency (RF) thermal plasma involves abundant and complex physics. The understanding of the physical field distributions of the RF thermal plasma is helpful to its applications in industrial field. In this paper, an electro-thermal-magnetic-flow strong coupling mathematical and physical model of three-dimensional RF thermal plasma is established, the actual solenoid structure of the induction coil is considered, and a C++ code is developed for calculating the complex electromagnetic field in a customized version of the computational fluid dynamics commercial code FLUENT. The physical fields of RF thermal plasma, such as temperature field, flow field and electromagnetic field are studied. The electrical conductivity, thermal conductivity and viscosity distribution of the plasma are investigated. The results show that the physical field distribution of RF thermal plasma has an important non-axisymmetric three-dimensional effect due to the actual shape of the non-axisymmetric induction coil structure. The plasma discharge presents an annular distribution with a certain deflection angle. The distribution of plasma flow field shows a non-axisymmetric electromagnetic pump effect which is different from that of the two-dimensional model. The results have great guiding significance for optimizing and controlling the RF thermal plasma in various application areas.
###### CONDENSED MATTER: STRUCTURAL, MECHANICAL, AND THERMAL PROPERTIES
2021, 70 (15): 156101. doi: 10.7498/aps.70.20210351
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2021, 70 (15): 156201. doi: 10.7498/aps.70.20210253
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Amorphous selenium (Se) can be easily prepared by quenching the melt, which indicates that the Se possesses the good glass-forming ability. However, crystallization occurs after rapidly compressing the melt within about 20 ms. In this work, we investigate the mechanism of rapid compression-induced crystallization from Se melt. Compressing Se melt experiments are carried out at the following temperatures: 513, 523 and 533 K. The melt is rapidly compressed under 2.4 GPa for about 20 ms. Different holding times, i.e. 0, 30, 60 min after solidification are adopted. The samples are quenched to room temperature and then unloaded to ambient pressure. The X-ray diffraction analysis of the recovered sample indicates that the crystallization product is the t-Se. It is found that with the prolongation of holding time, the grain size increases due to the continuous aggregation growth of crystal grains. By comparing with the isothermal crystallization products of amorphous Se and ultrafine Se powder, it is suggested that the rapid compression-induced solidification product should be t-Se crystalline. The speculation that the solidification product is amorphous Se and it crystallizes in the cooling process does not hold true. The amorphous Se cannot be prepared through the rapid compression process on a millisecond scale. It is related to the thermal stability of amorphous Se under high pressure. It is reported that the dependence of crystallization temperature Tx on pressure i.e. dTx/dP for amorphous Se is about 40–50 K/GPa in a range of 0.1 MPa–1 GPa. However, the Tx of amorphous Se is almost constant in a range of 2–6 GPa. It means that the thermal stability of amorphous Se against crystallization does not increase with increasing pressure after 2 GPa. In this work, the temperature of 513–533 K in the experiments is higher than the Tx of amorphous Se. Therefore, the t-Se crystal is the stable phase and amorphous Se is unstable. The Se melt tends to crystallize in the supercooled liquid state after rapid compression. It is interesting to investigate the mechanism of dTx/dP curve discontinuous change at around 2 GPa in the future. Both the Se melt after rapid compression and the amorphous Se before crystallization are in supercooled liquid state. We speculate that high pressure may result in the microstructure transition in supercooled liquid state Se.
2021, 70 (15): 156801. doi: 10.7498/aps.70.20210101
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###### CONDENSED MATTER: ELECTRONIC STRUCTURE, ELECTRICAL, MAGNETIC, AND OPTICAL PROPERTIES
2021, 70 (15): 157101. doi: 10.7498/aps.70.20201952
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Since the successful synthesis of graphene, two-dimensional materials, including hexagonal boron nitride and transition mental dichalcogenides, have attracted wide attention due to their extraordinary properties and extensive applications. Recent researches have revealed that the sensing system based on graphene or MoS2 can efficiently sense various gas molecules. However, the utility of these materials is limited by their inherent weakness, i.e. the zero bandgap in graphene and the relatively low mobility in MoS2, which impede their applications in electronic devices. This further stimulates the motivation of researchers to find more novel 2D materials. Black arsenic phosphide (AsP) monolayer, a novel two-dimensional nanomaterial with the characteristics of model direct bandgap and superhigh carrier mobility, is an ideal material for gas sensor. Here in this work, we investigate the electronic and magnetic properties of monolayer AsP absorbed with small gas molecules by using first-principle calculations based on density functional theory. Four initial absorption sites are selected to explore the optimal absorption positions of CO, CO2, NH3, SO2, NO and NO2 absorbed on the monolayer AsP. The purpose is to calculate the optimal absorption configurations, the absorption energy, absorption distance, and charge transfer, thereby investigating the absorption types. The results revel that the monolayer AsP is sensitive to NO2 gas and SO2 gas via strong physical absorption, and NO gas by chemical absorption, forming a new bond between N atom and O atom. The CO, CO2 and NH3 gas are absorbed on AsP monolayer with weak van Waals force. From the point of view of charge transfer, the CO, CO2, and NH3 molecules are one order of magnitude smaller than SO2, NO and NO2, approximately 0.03e and the charge transfer of NO gas is 0.21e, highest in all gases. Besides, the effects of absorption on the electrons of AsP are investigated. The results show that the absorption of CO, CO2 and NH3 molecules have little effect on band structure, and that the absorption of SO2 molecule increases the bandgap. The absorption of magnetic gas NO and NO2 reduce the bandgap by introducing impurity level near Fermi level, giving rise to their magnetic moments of 0.83μB and 0.78μB and making the whole system magnetic. Theoretical research shows that monolayer AsP is sensitive to NO, NO2 and SO2 gas molecules, which provides theoretical guidance for the experimental preparation of gas sensors band on black arsenic phosphorus.
2021, 70 (15): 157302. doi: 10.7498/aps.70.20202156
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The cylindrical surrounding double-gate metal-oxide-semiconductor field-effect transistor (CSDG MOSFET) is formed by adding an internal control gate to the cylindrical surrounding-gate (CSG) MOSFET. The inner gate of CSDG MOSFET acts as a second gate for enhanced charge control. At present, the research of CSDG MOSFET structure is widely concerned. Compared with double-gate MOSFET, triple-gate MOSFET and CSG MOSFET, the CSDG MOSFET provides good controllability of the gate over the channel. Additionally, the device allows for higher volume inversion than CSG MOSFET, which leads to better output characteristics.In order to study the electrical characteristics of CSDG MOSFET, the potential model of CSDG MOSFET is obtained by solving the two-dimensional Poisson equation in cylindrical coordinates. The effects of gate dielectric, channel length and gate dielectric thickness on the surface potential and electric field of CSDG MOSFET are studied. Besides, the drain-source current model of CSDG MOSFET is established by integrating the inverse charge along the channel. The effects of gate dielectric and gate dielectric thickness on the transconductance of CSDG MOSFET are studied. In addition, the effects of the downscaling of device parameters on the transfer characteristics and transconductance of CSDG MOSFET are studied.The electrical characteristics of CSDG MOSFET are analyzed and discussed. The results show that the minimum surface potential along the channel of CSDG MOSFET decreases with the increase of gate dielectric constant of gate dielectric layer. The electric field along the channel and along the radius, drain-source current and transconductance of CSDG MOSFET increase as the gate dielectric constant increases. The threshold voltage of CSDG MOSFET decreases as the gate dielectric constant increases. Moreover, with the downscaling of device parameters, the transfer characteristics and transconductance of CSDG MOSFET decrease. The performance of CSDG MOSFET can be significantly improved by using high-k gate dielectrics.
2021, 70 (15): 157401. doi: 10.7498/aps.70.20202020
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Lead-free chalcogenide SnTe has a similar crystal structure and energy band structure to high performance thermoelectric material PbTe, which has been widely concerned in recent years. However, due to its low Seebeck coefficient, high intrinsic Sn vacancy concentration and high thermal conductivity, its intrinsic thermoelectric performance is poor. In this study, Mn-In-Cu co-doping SnTe-based thermoelectric materials are prepared by hot pressing sintering at high-temperature and high-pressure. Indium (In) doping brings the resonant level in SnTe and increases the density of states which greatly improves Seebeck coefficient at room temperature; the Seebeck coefficient of Sn1.04In0.01Te(Cu2Te)0.05 reaches 70 μV·K–1 at room temperature. With adding manganese (Mn), the Seebeck coefficient at room temperature is well preserved, indicating that Mn doping has little effect on the resonant level brought by In doping. In addition, due to the band convergence brought by Mn doping, the high temperature Seebeck coefficient of the material is improved, the maximum Seebeck coefficient reaches 215 μV·K–1 for the sample with 17% Mn doping amount at 873 K. Owing to the combination of band convergence and resonant level, the Seebeck coefficient of the whole temperature range of the material increases, the power factor of the material is also greatly optimized, and all samples have a power factor of more than 1.0 mW·m–1·K–2 at room temperature. On the other hand, the point defects brought by Mn alloying and the interstitial defects introduced by copper (Cu) enhance the phonon scattering and effectively reduce the lattice thermal conductivity of the material, the lattice thermal conductivity decreases to 0.68 W·m–1·K–1 at 873 K. The electrical and thermal properties of the materials are optimized simultaneously under the combination of various strategies, the peak zT ≈ 1.45 is obtained at 873 K in the p-type Sn0.89Mn0.15In0.01Te(Cu2Te)0.05 sample and the average zT of 300–873 K reaches 0.76. In the process of multi-strategy coordinated regulation of SnTe-based thermoelectric materials, the excellent properties of single strategy can be well maintained, which provides a possibility for further improving the performance of SnTe-based thermoelectric materials.
2021, 70 (15): 157502. doi: 10.7498/aps.70.20210014
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$\varGamma$ point, which is different from the Dirac state formed by the pz orbital reported previously. In addition, it can still retain its original topological properties even if strongly hybridized with the substrate. The calculated phonon spectrum shows no imaginary frequency in the entire Brillouin zone, indicating that the monolayer BaPb system is dynamically stable. By using Monte Carlo simulation, we determine the Curie temperature of BaPb monolayer toreach up to 378 K. We also calculate the magnetic anisotropy energy of the BaPb cell, defined as $\Delta E={E_{100}}-{E_{001}}$. Here, we consider two magnetization easy-axis directions, [100] and [001]. To our surprise, the MAE of monolayer BaPb is as high as 52.01 meV/cell by considering the spin-orbit coupling effect. Furthermore, the nontrivial band gap is opened with a magnitude of 177.39 meV when the spin-orbit coupling effect is included. The calculations of Berry curvature and edge states further prove that the monolayer BaPb system can realize the quantum anomalous Hall state. This discovery indicates that the monolayer BaPb materials can be used as a candidate for quantum anomalous Hall effect materials, thereby promoting the development of spintronics.">The quantum anomalous Hall effect is an intriguing quantum state that exhibits chiral edge states in the absence of a magnetic field. The chiral edge states are topologically protected and robust against electron scattering, which possesses great potential applications in designing low energy consumption and dissipation less spintronic devices. The experimental conditions are required to be very high, such as extremely low temperature (< 100 mK) due to the small band gap and the greatly accurate control of the extrinsic impurities. These greatly hinder their devices from being put into applications further. Hence, it would be meaningful to search for a new Chern insulator with a large band gap and high Curie temperature. According to the first-principles calculations, we predict the room temperature quantum anomalous Hall effect in the monolayer BaPb. The nontrivial topology of this new type of ferroelectric semi-metal material derives from fully spin-polarized quadratic non-Dirac bands. The quantum anomalous Hall effect can be realized in the monolayer BaPb with fully spin-polarized quadratic px,y non-Dirac bands with the nonzero Chern number (C = 1). Because of the trigonal symmetry of monolayer BaPb material, these bands composed of px,y orbitals are at the $\varGamma$ point, which is different from the Dirac state formed by the pz orbital reported previously. In addition, it can still retain its original topological properties even if strongly hybridized with the substrate. The calculated phonon spectrum shows no imaginary frequency in the entire Brillouin zone, indicating that the monolayer BaPb system is dynamically stable. By using Monte Carlo simulation, we determine the Curie temperature of BaPb monolayer toreach up to 378 K. We also calculate the magnetic anisotropy energy of the BaPb cell, defined as $\Delta E={E_{100}}-{E_{001}}$. Here, we consider two magnetization easy-axis directions, [100] and [001]. To our surprise, the MAE of monolayer BaPb is as high as 52.01 meV/cell by considering the spin-orbit coupling effect. Furthermore, the nontrivial band gap is opened with a magnitude of 177.39 meV when the spin-orbit coupling effect is included. The calculations of Berry curvature and edge states further prove that the monolayer BaPb system can realize the quantum anomalous Hall state. This discovery indicates that the monolayer BaPb materials can be used as a candidate for quantum anomalous Hall effect materials, thereby promoting the development of spintronics.
2021, 70 (15): 158101. doi: 10.7498/aps.70.20202141
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High temperature superconductor has become one of the hotspots of research, because of its high critical temperature, strong trapped flux density, stable suspension characteristics and large magnet levitation force. The single domain REBa2Cu3O7–δ (REBCO) superconductors have the wide and potential applications in the high-tech fields, such as micro-magnet superconducting maglev train, superconducting motor and superconducting magnetic separation system. However, a large number of multi-domain samples are easy to produce in the preparation process, which leads the success rate to decrease significantly and the cost to increase considerably, which restricts its practical application process. Inspired by the top seeded infiltration growth method, we develop a reliable method of recycling failed GdBCO sample by re-supplementing the liquid phase lost in the primary growth process and pretreating the failed sample as solid phase source billets. We recycle a series of GdBCO samples by using this new technique successfully. The growth morphology, superconducting properties, and microstructures of the recycled GdBCO bulk superconductors are investigated in detail in this study. The results show that the magnetic levitation forces of the recycled GdBCO samples are all greater than 30 N, their magnetic flux densities are all above 0.3 T, and their capture efficiencies are above 60%. These results provide the scientific basis and new ideas for developing the low cost and high efficient yield of fabrication of the REBCO bulk superconductors.
2021, 70 (15): 158201. doi: 10.7498/aps.70.20201710
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${k_1}$ in the Lengyel-Epstein model is excited and interacts with the higher-order harmonics $\sqrt 3 {k_1}$ located in the Hopf region in the Brusselator model, and thus giving rise to the synchronous oscillatory hexagon pattern. The harmonic $\sqrt 2 {k_1}$ that can also be excited initially is some parameter domain, but it is unstable and vanishes finally. As the parameter b is increased, this oscillatory hexagon pattern first undergoes period-doubling bifurcation and transits into two-period oscillation, and then into multiple-period oscillation. When the Hopf mode participates in the interaction, the pattern will eventually transit into chaos. The synchronous oscillatory hexagon pattern can only be obtained when the subcritical Turing mode ${k_2}$ in the Brusselator model is weaker than the higher-order harmonics $\sqrt 3 {k_1}$ located in the Hopf region and neither of the two Turing modes satisfies the spatial resonance condition. The system favorites the spatial resonance and selects the super-lattice patterns when these modes interact with each other. The interaction between Hopf mode and Turing mode can only give rise to non-synchronous oscillatory patterns. Moreover, the coupling strength also has an important effect on the oscillatory Turing pattern. These results not only provide a new pattern forming mechanism which can be extended to other nonlinear systems, but also gives an opportunity for more in-depth understanding the nature and their relevance to technological applications.">Pattern formation and self-organization are ubiquitous in nature and commonly observed in spatially extended non-equilibrium systems. As is well known, the origin of spatio-temporal patterns can be traced to the instability of the system, and is always accompanied by a symmetry breaking phenomenon. In reality, most of non-equilibrium systems are constructed by interactions among several different units, each of which has its unique symmetry breaking mechanism. The interaction among different units described by coupled pattern forming system gives rise to a variety of self-organized patterns including stationary and/or oscillatory patterns. In this paper, the dynamics of oscillatory Turing patterns in two-layered coupled non-symmetric reaction diffusion systems are numerically investigated by linearly coupling the Brusselator model and the Lengyel-Epstein model. The interaction among the Turing modes, higher-order harmonics and Hopf mode, and their effects on oscillatory Turing pattern are also analyzed. It is shown that the supercritical Turing mode ${k_1}$ in the Lengyel-Epstein model is excited and interacts with the higher-order harmonics $\sqrt 3 {k_1}$ located in the Hopf region in the Brusselator model, and thus giving rise to the synchronous oscillatory hexagon pattern. The harmonic $\sqrt 2 {k_1}$ that can also be excited initially is some parameter domain, but it is unstable and vanishes finally. As the parameter b is increased, this oscillatory hexagon pattern first undergoes period-doubling bifurcation and transits into two-period oscillation, and then into multiple-period oscillation. When the Hopf mode participates in the interaction, the pattern will eventually transit into chaos. The synchronous oscillatory hexagon pattern can only be obtained when the subcritical Turing mode ${k_2}$ in the Brusselator model is weaker than the higher-order harmonics $\sqrt 3 {k_1}$ located in the Hopf region and neither of the two Turing modes satisfies the spatial resonance condition. The system favorites the spatial resonance and selects the super-lattice patterns when these modes interact with each other. The interaction between Hopf mode and Turing mode can only give rise to non-synchronous oscillatory patterns. Moreover, the coupling strength also has an important effect on the oscillatory Turing pattern. These results not only provide a new pattern forming mechanism which can be extended to other nonlinear systems, but also gives an opportunity for more in-depth understanding the nature and their relevance to technological applications.
2021, 70 (15): 158202. doi: 10.7498/aps.70.20201279
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## EDITOR'S SUGGESTION

2021, 70 (15): 158203. doi: 10.7498/aps.70.20210227
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Doping is one of the most important methods to improve the electronic conductivity and modify its electrochemical performance of LiFePO4. Rare earth elements have become an effective selection for doping modification due to their high electronic charges, large ion radii and strong self-polarization ability. In this work, we study the structural, electronic and ionic diffusion properties of LiFePO4 with rare earth (RE) doping (La, Ce, Pr) by using first-principles calculation based on density functional theory. The calculated results show that the lattice constant and cell volume of LiFePO4 increase to a different degree after RE doping. In the delithiation process, the volume change rate of the material after RE doping is significantly reduced, indicating the cycle performance of the material is improved, on the other hand, the energy density is reduced. The calculated density of states suggests that RE-doped LiFePO4 exhibits metallic characteristics, which is different from the undoped one with semiconductor characteristics. As a result, the RE-doping can increase the electronic conductivity of the material. The calculation of elastic modulus demonstrates the increase of ductility for RE-doped LiFePO4, and it can be predicted that the cycle performance and the rate performance of the RE-doped battery have great improvement. In addition, La and Ce doped LiFePO4 materials exhibit that the complex energy barrier can change during the Li ion migration, and the migration barriers vary considerably, depending on different paths, which is related to the variation of potential energy surface caused by the doping of rare-earth elements. The Li-ions are far from the RE ions, the migration barriers are obviously lower than the undoped one, while the Li-ions are closest to RE ions, the migration barriers increase essentially. Compared with Ce doping, the change of the Li-ion migration barrier caused by La doping is great, indicating that RE ion doping has a greater influence on the local structure of the system.
2021, 70 (15): 158401. doi: 10.7498/aps.70.20210075
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2021, 70 (15): 158501. doi: 10.7498/aps.70.20210168
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Power divider is a useful device that divides the power of signal into different subpowers at a certain ratio. The superconducting power divider plays an important role in various superconducting quantum computing circuits and superconducting microwave photon detectors. Therefore, in this paper we investigate how to design and prepare a typical coplanar waveguide superconducting microwave power divider. The parameters are designed by using the odd-even mode method to analyze the transport features of a three-port microwave network. Specifically, the microwave transport properties of the device with a center frequency of 5 GHz and 3 dB power division ratio are simulated. Then, the designed aluminum coplanar waveguide superconducting power divider on silicon is prepared by micro-processing technology and experimentally tested at low temperature. It is shown that the measurement results are consistent with the design parameters. It is noted that the center frequency of the actually prepared power divider is measured to be about 5.25 GHz, which is slightly different from the result of the design and simulation. This difference is probably due to the following main reasons. Firstly, the limited precision of the micromachining process is caused by the fact that the fabricated quarter-wave impedance matching line is etched incompletely, leading the length of the impedance matching line to be shortened. As a consequence, the frequency of the prepared power divider is slightly higher. Secondly, the simulation software is not designed specially for superconducting device simulations, thereby yielding the design parameters slightly different from those of the fabricated superconducting devices. Additionally, a series of attenuations has been used in the experimental test system of the superconducting microwave power dividers for reducing the various noises. This causes the input test signal to weaken, thus the reflected signal turns significantly small. Therefore, none of the S11 parameters of the device can be effectively measured. Finally, neither of S21 and S31 parameters measured in the experiment is the predicted –3 dB, which is mainly due to the imperfections in the welding between SMA connectors and high-frequency transmission lines, and the spot welding between high-frequency transmission lines and power divider samples, and also due to the discontinuities of the high-frequency transmission line and the power divider and so on. All these factors can yield the tested insertion loss of the device. Hopefully, the method in this work can be extended to designing and preparing other passive superconducting microwave devices.

## EDITOR'S SUGGESTION

2021, 70 (15): 158701. doi: 10.7498/aps.70.20210707
Abstract +
DNA polymerase is essential for DNA replication and repair. As it only performs the 5′-3′ polymerization, there are two kinds of DNA replication. One of them is called strand-displacement synthesis: DNA polymerase opens the double-strand (ds) DNA to attain the 3′-5′strand (leading strand) and copy this template in a continuous way, and the other is extension synthesis: DNA polymerase copies the newly separated 5′-3′ strand (lagging strand) in a discontinuous manner. The replication complex of T7 phage is an optimal model to investigate the mechanism of replication because it is only constituted by 4 terms of protein which are DNA helicase gp4, DNA polymerase gp5 with co-factor thioredoxin (Trx), and single-strand (ss) DNA-binding protein gp2.5. The replication complex of T7 encounters both strand-displacement synthesis and extension synthesis. Previous researches reported that gp5 can have rapid extension synthesis but lacks the ability to attain strand-displacement synthesis. It also reported that gp4 translocates on ssDNA at a rapid speed but unwinds dsDNA at a very low speed. However, gp5 and gp4 together can attain rapid and processive strand-displacement synthesis. Although extensively studied, this mechanism remains unclear. Here in this work, the dynamic of strand-displacement synthesis by gp5 is investigated with single-molecule Förster (fluorescence) resonance energy transfer (smFRET). It is found that gp5, without the help of external tension, can open dsDNA but only attain strand-displacement synthesis about 4 base pairs (bp), because its exonuclease activity excises the nascent nucleotides. Therefore gp5 repeats in the synthesis-excision cycle which results in the less production of strand-displacement synthesis. We conduct another control experiment by nano-tensioner, a high precision smFRET setup which can exert a tension on dsDNA, to change the dsDNA regression pressure on gp5. It is observed that reduced dsDNA regression pressure can increase the length of strand-displacement synthesis and reduce the length of excision which indicates that the dsDNA regression pressure can regulate the strand-displacement synthesis of gp5. The further experiment shows that after gp5 and gp4 are assembled into a replisome, it can have a processive strand-displacement synthesis and barely any excision presented. The speed of replisome is a little higher than gp5 alone but much higher than gp4 alone. Additionally, the length of strand-displacement synthesis by replisome is much longer than gp5 alone. Therefore it is indicated that the gp4 can reduce dsDNA regression pressure to enables gp5 to attain processive strand-displacement synthesis. On the other hand, the gp5 facilitates gp4 to unwind the dsDNA.
2021, 70 (15): 158702. doi: 10.7498/aps.70.20201884
Abstract +
Quantitative phase imaging (QPI), which combines phase imaging with optical microscopy technology, provides a marker-free, fast, non-destructive, and high-resolution imaging method for observing transparent biological samples. It is widely used in life science, biomedicine, etc. As an emerging QPI technology, spiral phase contrast microscopy (SPCM) uses a spiral phase filter to achieve edge enhancement of amplitude or phase objects. Using the multi-step phase-shifting technology, a complex sample can be measured quantitatively, which has the advantages of high stability, high sensitivity and high precision. However, the SPCM requires at least three-step phase-shifted spiral phase filtered images to achieve the quantitative reconstruction of the amplitude and phase of a sample, and the image acquisition process and the reconstruction process are relatively complicated, which require high stability of system, and the SPCM has low temporal resolution. In order to further improve the performance of SPCM and increase the system stability, sensitivity and temporal resolution, in this paper a quantitative phase imaging method and system based on a fractional spiral phase plate is proposed. Through a sample intensity image filtered by a fractional spiral phase plate, the modified Gerchberg-Saxton iterative phase retrieval algorithm is used to quantitatively reconstruct the phase of a pure phase sample, which simplifies the experimental process and phase reconstruction steps of spiral phase contrast microsocopy. In the computer simulation experiments, the phase imaging process and the reconstruction process of spiral phase plates based on different topological charges are studied, the feasibility of which is analyzed. Finally, through imaging and phase reconstruction of the phase grating and biological cell sample, it is verified that the phase contrast microscopy method based on the fractional spiral phase plate can effectively improve the contrast of spiral phase contrast microscopy and can obtain a quantitative reconstruciton of a weak phase object. The phase information of a sample has significance in research and application for developing the spiral phase contrast microscopy.
2021, 70 (15): 158703. doi: 10.7498/aps.70.20210421
Abstract +
Neurons collect information from different parts of the biological body, generate signals and control their functions and activities. There are electromagnetic communication channels between neurons apart from the action potentials. Microtubules are the largest cytoskeletal filaments in neurons, with a diameter of about 25 nm. Microtubule is composed of alpha- and beta-tubulin subunits assembled into hollow cylindrical polymers supporting dynamical growth, and facilitate transport of proteins. In axons, dendrites, growth cones, and migratory neurons, microtubules are generally tightly organized in array and uniformly oriented. Because of the polarity and charge distribution of tubulins, the vibrations of microtubules generate electromagnetic fields. In this paper, electromagnetic fields induced by different vibrational modes of microtubules are studied. The vibrational mode of tubulins calculated using the normal mode analysis shows that there are abundant vibrational modes in the terahertz range. The electric fields of different vibration modes show distinct distribution features. The induced electromagnetic fields of microtubules can be stronger than thermal noise because of reduced permittivity of intracellular fluid for higher frequencies in a nanometric confined region. Since water exhibits layered structuring near all surfaces independent of their hydrophilicity, the permittivity of water surrounding tubulins between microtubules is expected to decrease significantly because of surface-induced alignment of water molecular dipoles. While the permittivity of surrounding medium decreases to 5, the electromagnetic potential energy between two 100-nm-long microtubules can be stronger than the thermal energy within a 30-nm-long distance. As high frequency vibrations are generally localized in the microtubule, terahertz electromagnetic interactions can be present between tubulins and short microtubules. Because the separation between microtubule arrays in neurons is in a range from 20 nm to 100 nm, electromagnetic interactions between microtubules can dominate the thermal motions, and affect the biological functions. Simulation results show that the electromagnetic potential energy increases over one order of magnitude when the vibration amplitude is changed from 0.1 nm to 0.4 nm. The results indicate that the electromagnetic interaction between microtubules is important for a better understanding of neural functions and communication. Terahertz stimulations can be used to detect and modulate the neural signals. The microtubule vibration generated magnetic field can be applied to disease diagnosis and brain-machine interface.
2021, 70 (15): 158704. doi: 10.7498/aps.70.20210297
Abstract +
${{{Z}}'_{\rm{relax}}}$ and imaginary part spectrum ${{{Z}}''_{\rm{relax}}}$ of electrical impedance are extracted from the amplitude spectrum of high frequency band. The quantitative analyses of these three kinds of tissues show that the ${{{Z}}'_{\rm{relax}}}$ and ${{{Z}}''_{\rm{relax}}}$ of tumor tissue are the highest, and those of normal tissue are the lowest. Finally, the cancerous tissue coefficients α and β (the relative change percentage of the real part and imaginary part of the impedance value between the tested tissue and normal tissue) are determined according to these three electrical parameters for tumor tissue identification. The results show that when α ≤ 36.5% and β ≤ 31.2%, the tissue is normal; when α ≥ 36.5% and β ≥ 31.2%, the tissue may be mixed with tumor tissue; when α ≥ 82.7% and β ≥ 73.6%, the tissue is tumor tissue.">A fast and convenient method of recognizing the tongue tumor tissue based on bioelectrical impedance spectroscopy (BIS) is proposed. According to the difference among the electrical characteristics of tongue tissue under different pathological and physiological conditions, we can judge whether it is pathological. This method can help the surgeon in the clinical resection of tongue cancer not only to remove the tumor completely, but also to retain the patient’s tongue function as much as possible. In this paper, a model of human tongue squamous cell carcinoma (HSC3) xenografted in situ is established in mice. The electrical properties of the normal tissue, mixed tumor tissue and tumor tissue are studied by the BIS technology. The amplitude spectrum shows that none of the three tissues can be distinguished well in a low frequency range of 100–8.09 × 105 Hz due to the influence of contact impedance, but they can be distinguished according to their electrical characteristics in a high frequency range of 8.09 × 105–5 × 106 Hz. In the process of the experiment, first of all, the tip, middle and root of the normal tongue are detected, and the results show that the impedance values of these three parts are similar in the high frequency band, so the influence of different positions of tongue on the impedance value can be excluded. Then, the same three regions of the cancerous tongue are detected, and three electrical parameters, namely relaxation frequency frelax, real part spectrum ${{{Z}}'_{\rm{relax}}}$ and imaginary part spectrum ${{{Z}}''_{\rm{relax}}}$ of electrical impedance are extracted from the amplitude spectrum of high frequency band. The quantitative analyses of these three kinds of tissues show that the ${{{Z}}'_{\rm{relax}}}$ and ${{{Z}}''_{\rm{relax}}}$ of tumor tissue are the highest, and those of normal tissue are the lowest. Finally, the cancerous tissue coefficients α and β (the relative change percentage of the real part and imaginary part of the impedance value between the tested tissue and normal tissue) are determined according to these three electrical parameters for tumor tissue identification. The results show that when α ≤ 36.5% and β ≤ 31.2%, the tissue is normal; when α ≥ 36.5% and β ≥ 31.2%, the tissue may be mixed with tumor tissue; when α ≥ 82.7% and β ≥ 73.6%, the tissue is tumor tissue.
2021, 70 (15): 158801. doi: 10.7498/aps.70.20210397
Abstract +
As an important perovskite solar cell (PSC) material, CsSnBr3 has been widely studied. Based on the density functional theory (DFT), the photoelectric properties of CsSnBr3 are studied by using the first-principles at different hydrostatic pressures. It is found that CsSnBr3 has an optimal optical band gap value of 1.34 eV under a pressure of 2.6 GPa, so only the photoelectric properties of CsSnBr3 under the hydrostatic pressure of 0 GPa and 2.6 GPa are studied, respectively. When the pressure is 2.6 GPa, CsSnBr3 has larger values of dielectric, conductivity, absorption coefficient and refractive index, the red-shifted absorption spectrum, and relatively small effective mass of electron and hole and exciton binding energy, indicating that CsSnBr3 is an efficient light absorbing material. According to the triple calculations of Born-Huang stability standard criterion, the tolerance factor T and phonon spectrum with or without virtual frequency, it is found that CsSnBr3 is stable under the pressure of 0 GPa and 2.6 GPa. According to the elastic modulus value of CsSnBr3 before and after pressure, it can be seen that the CsSnBr3 is soft, with good ductility and anisotropy. The Debye temperature and heat capacity of CsSnBr3, soon after it has been pressured, tend to be stable and are independent of temperature. The enthalpy and entropy increase with temperature increasing, and the increased amplitude is larger than those of the unpressured CsSnBr3. Gibbs free energy shows a decreasing trend, and the decrease is slightly faster when unpressured. This study shows that CsSnBr3 is a good photoelectric material after having been pressured hydrostatically, which is suitable for perovskite solar cells.


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Theory Manual Version 3.4
$\newcommand{\lyxlock}{}$
Subsection 6.3.1: Rigid Body Rotation Up Subsection 6.3.1: Rigid Body Rotation Subsubsection 6.3.1.2: Cayley Transform

#### 6.3.1.1 Exponential Map

Conventionally, the rigid body rotation tensor corresponding to a rotation of angle about the unit vector may be expressed in terms of the vector as where is the third-order permutation pseudo-tensor with Cartesian components . Making use of the trigonometric identity, this expression may be rearranged as where we have made use of the identity Letting represent the antisymmetric tensor with axial vector , may now be represented as where is known as the exponential map. Thus, the exponential map provides the rotation tensor for a rotation about the unit vector . Note that , since .
Let be any orthogonal transformation, then where and its corresponding axial vector is , implying that . This property of the exponential map is used in the next derivation.
Consider a vector in the reference configuration of a rigid body. Upon rigid body rotation, this vector is currently at The corresponding axial vector of is . At a subsequent time , we would similarly have where here, is the incremental (finite) rotation from to . Alternatively, we may choose to write such that implying that Note from these relations that . Thus, is the material representation of the incremental rotation from to , while is the corresponding spatial representation.
Subsection 6.3.1: Rigid Body Rotation Up Subsection 6.3.1: Rigid Body Rotation Subsubsection 6.3.1.2: Cayley Transform

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### Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

### Calendar Capers

Choose any three by three square of dates on a calendar page...

### Days and Dates

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

# Palindromic Milometer

##### Age 11 to 14 ShortChallenge Level

Answer: 70$\frac13$

Maximum possible distance: 3$\times$75 = 225

Maximum possible milometer reading: 29792 + 225 = 30017

Largest palindromic number before 30017 will have first digit 3 $\therefore$ last digit 3
second and third digits 0 so fourth digit 0
Distance travelled: 30003 $-$ 29792 = 211     (or 14 less than maximum)
Speed: 211$\div$3 = 70$\frac{1}{3}$                                  (or 14$\div$3 = 4$\frac23$ less than speed limit)


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# Yuvraj purchased a bike on the condition that at the time of purchasing he will pay a down payment of Rs. 15000 and also pay Rs. 13500 and Rs. 17500 respectively at the end of 1st and 2nd year. If the rate of compound interest is 10% per annum compounded annually, then find the approximate price of bike.

1. Rs. 41735.53
2. Rs. 55342.54
3. Rs. 45345.53
4. Rs. 48045.54

Option 1 : Rs. 41735.53
Free
SSC CGL 2021 Tier-I (Held On : 11 April 2022 Shift 1)
1.4Lac Users
100 Questions 200 Marks 60 Mins

## Detailed Solution

Given:

Down payment = Rs. 15000.

He will pay Rs. 13500 and Rs. 17500, at the end of 1st and 2nd year respectively.

The rate of compound interest is 10% per annum compounded annually

Calculation:

Let the price of bike be Rs. x

After down payment the principal of bike will be Rs. (x – 15000)

Amount at the end of 1st year = (x – 15000) × 11/10

Principal for 2nd year = Amount of 1st year – payment by Yuvraj at the end of 1st year

Principal for 2nd year = 11(x – 15000)/10 – 13500

Amount at the end of 2nd year = [11 (x – 15000)/10 – 13500] × 11/10 = [121 (x – 15000)/100 – 14850]

Principal for 3rd year = Amount of 2nd year – payment by Yuvraj at the end of 2nd year

Principal for 3rd year = [121 (x – 15000)/100 – 14850]

Since, principal for 3rd year = amount paid by him at the end of 2nd year

So, [121 (x – 15000)/100 – 14850] = 17500

⇒ 121 (x – 15000)/100 = 17500 + 14850

⇒ x – 15000 = 32350 × 100/121

⇒ x = 26735.53 + 15000

Hence price of bike = Rs. 41735.53


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Chapter 3 - Introduction to Graphing - 3.5 Slope - 3.5 Exercise Set: 19

$\frac{1}{2}$ point per $\$1000$increase in family income. Work Step by Step The graph contains the points$(50, 490)$and$(90, 510)$. To find the rate of change in the SAT critical reading scores with respect to family income, use the formula$\dfrac{\text{change in y}}{\text{change in x}}$. rate of change =$\dfrac{510-490}{90-50}=\dfrac{20}{40}=\dfrac{1}{2}$Thus, the rate of change in SAT critical reading scores with respect to family income is$\frac{1}{2}$point per$\$1000$ increase in family income.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


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# Primitive $6^{th}$ root of unity

Why does any extension of $\mathbb{Q}$ that contains a root of $x^6+3$ have to contain a primitive $6^{th}$ root of unity? The roots of this equation are $\{\sqrt[6]{3}e^{\pi i/2},\sqrt[6]{3}e^{5\pi i/6},\sqrt[6]{3}e^{7\pi i/6},\sqrt[6]{3}e^{3\pi i/2},\sqrt[6]{3}e^{11\pi i/6},e^{13\pi i/6}\}$. A primitive $6^{th}$ root of unity has the form $\frac{1}{2}+\frac{\sqrt{3}}{2}i$...

I don't see how to get $\frac{1}{2}+\frac{\sqrt{3}}{2}i$ from $\sqrt[6]{3}e^{\pi i/2}$, for example.

Let $\alpha$ be such a root, then $\beta=\alpha^3$ satisfies $\beta^2+3=0$, hence ${1+\beta\over 2}={1+\alpha^3\over 2}$ is a $6^{th}$ root of $1$.

You can even verify this directly:

$$(1+\beta)^6 = 1+6\beta+15\beta^2+20\beta^3+15\beta^4+6\beta^5+\beta^6$$

Now use that $\beta^2=-3$ to reduce this to:

$$(1+5\cdot(-3)+15\cdot(-3)^2+(-3)^3)+\beta(6+20\cdot(-3)+6\cdot(-3)^2)$$ $$=(1-45+135-27)+\beta(6-60+54)= 64+0\beta$$ $$= 2^6.$$

And this is primitive as if we only cube things we get

$$(1+\beta)^3 = 1+3\beta + 3\beta^2+\beta^3 = 1-9 + \beta(3-3) = -8=-2^3.$$

• $(1+\beta)^3=-8$ is easier and proves primitivity. – lhf Dec 2 '16 at 17:15
• @lhf true, but if I'm already verifying, I find it's satisfying to see it all the way when it's not extremely cumbersome. Thanks for the feedback! – Adam Hughes Dec 2 '16 at 17:19
• @lhf ah, good point. That is an excellent reason to change it. Much appreciated! – Adam Hughes Dec 2 '16 at 17:21
• I don't understand. Why do we automatically have $\frac{1+\beta}{2}$ a $6^{th}$ root of $1$. And why does cubing show it is primitive? – FTem Dec 2 '16 at 17:24
• @FTem I verified it above, so there's your proof it's a $6^{th}$ root. But it only cubes to $-1$ so the smallest power we can raise it to and get $1$ is 6, that is definition primitive. – Adam Hughes Dec 2 '16 at 17:25