## Abstract

We investigate the effects of geometrical and structural disorders on perfectly asymmetric diffraction (PAD) in Raman-Nath regime. The two types of disorders are realized by introducing random fluctuations in the position and width of one-dimensional (1D) driven atomic lattices. Raman-Nath diffraction is modified differently with respect to the geometrical and structural disorders. It is shown that the PAD is observed with a certain strength range of geometrical disorder, exceeding which it can be destroyed, while the PAD is rather robust against structural disorder. The different behaviors originate from the disorder-induced random variations of the spatial phase shifts of the standing-wave (SW) coupling field and atomic lattices with Gaussian profile. Furthermore, we find that, in the presence of geometrical disorder, the PAD is more susceptible to correlated disorder than to uncorrelated disorder. Our scheme may be useful for understanding the effects of disorder on the diffraction of light and matter waves in disordered potentials..

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In the past few years, the study of grating diffraction has been one of the hot spots in optics due to their significant applications in the fields of natural science and industrial production[1, 2]. As a periodic optical device, the grating can diffract the incident light beam into different diffraction directions. Rapidly developing manufacturing technologies allow fabricating different kinds of diffraction gratings, such as ruled gratings [3], holographic grating [4, 5], silicon gratings [6, 7] and metal gratings [8]. In recently years, atomic gratings have been widely studied in multilevel atomic vapors [9–21] because of their uncommon advantage of real-time all-optical tunable capability. It is worth noting that the plane diffraction gratings generally operate in two different diffraction regime: Bragg regime [2, 4, 5, 14, 15, 22–24] and Raman-Nath regime [2, 9–13, 17–21, 25, 26]. In Bragg regime, essentially only two waves, i.e., one transmitted wave and one diffracted wave, exist when the light wave is incident at or near the Bragg angle, while multiple diffraction waves are generated in Raman-Nath regime [2]. Among these investigations, perfectly asymmetric diffraction (PAD) phenomena in Raman-Nath regime can be achieved by introducing periodic parity-time $\left(\mathcal{P}\mathcal{T}\right)$-symmetric or $\mathcal{P}\mathcal{T}$-antisymmetric refractive index [20, 25, 26] (or susceptibility [19, 21]).

On the other hand, disorder inevitably exists in most of the fabricated diffraction structures due to the imperfect manufacturing process. Consequently, The studies of the influence of disorder on the diffraction of light have attracted considerable interest. Note that disorder caused by the imperfection could diffuse the diffraction light over a broader spectral range [27, 28] with respect to ideal periodic diffraction gratings [9,13], which leads to the weakening of diffraction performance. On the contrast, it has been proven that, in the disordered gratings, the diffused diffraction pattern induced by artificial disorder imposes a positive impact on improving of light harvesting [29, 30], enhancing light absorption [31] and light extraction efficiency of organic light-emitting diodes [32], etc. As far as we know, no reports have been proposed for the impact analysis of disorder on perfectly asymmetric Raman-Nath diffraction. Thus, it reminds us of one question: how robust is the PAD phenomenon against disorder?

In this paper, we investigate the effects of geometrical and structural disorders on the PAD of one-dimensional (1D) atomic grating. Such a disordered atomic grating consists of 1D disordered cold atomic lattices, which have been exploited to investigate the robustness of unidirectional reflectionless and coherent perfect absorption against different types of disorders [33] due to their flexible optical tunability and easy reconfiguration. In our scheme, these two types of disorders are expressed in the random fluctuations of the position and width of each cold atomic lattice. We find that the PAD is destroyed by geometrical disorder when the strengths of disorder exceed the critical values, while it is well maintained in the presence of structural disorder. The different effects of geometrical and structural disorders on the PAD are attributed to the changes of the spatial phase shift between the spatial modulations of coupling field and atomic lattices with the same periodicity. Furthermore, it is demonstrated that, in the presence of geometrical disorder, the disappearance of PAD phenomenon is much more sensitive to correlated disorder than to uncorrelated disorder. Our scheme may provide a feasible method to predict the diffraction features of light (or matter) wave in disordered potential.

## 2. Models and equations

The schematic of a weak probe field diffracted by 1D atomic grating is presented in Fig. 1(a). Such an atomic grating consists of 1D driven atomic lattices, in which cold atoms are distributed into the optical lattices of dipole traps formed by the red-detuned laser beams of wavelength *λ _{o}* [not indicated in Fig. 1(a)]. In the absence of disorder, 1D atomic lattices have a period Λ = 0.5

*λ*/cos

_{o}*θ*, where

_{o}*θ*is the angle between the dipole-trap laser beams and the lattice axis along $\overrightarrow{x}$. Meanwhile, the atomic density

_{o}*N*(

_{j}*x*) for the trapped atoms in the

*j*th unit cell, i.e., (−1 + 2

*j*)Λ/2 ≤

*x*≤(1 + 2

*j*)Λ/2, can be well approximated by a Gaussian function [34, 35]

*N*

_{0}is the average atomic density. For the ideal atomic lattices,

*d*=

_{j}*d*and

*x*=

_{j}*j*×Λ are the

*e*

^{−1}half width and position of the

*j*th atomic lattice, respectively. It is worth noting that the position

*x*of each atomic lattice is located at the bottom of each optical lattice [see Fig. 1(a)].

_{j}However, disordered cold atomic lattices can be achieved by superposing the random perturbation of a speckle potential to the ideal optical lattices [36], which have been used for studying disorder related phenomena, such as Anderson localization [37] and perfect absorption and no reflection [33]. Note that disorder can be introduced to either position or width of the cold atomic lattices [33]. In the following, we would introduce two types of disorders, i.e., geometrical (position) disorder and structural (width) disorder.

In the case of geometrical disorder, the position of each atomic lattice is perturbed while its width remains constant. By introducing random variations of lattice position *x _{j}* in Eq. (1), uncorrelated geometrical disorder is modeled as

*ζ*}represent sequences of uncorrelated random numbers uniformly distributed in the interval [−Δ

_{j}*Δ*

_{g},*] with Δ*

_{g}*the strength of the geometrical disorder.*

_{g}If the position *x _{j}* of the

*j*th atomic lattice is affected by the variations of the positions of all previous atomic lattices, we consider a model of correlated geometrical disorder, which is given by

In the case of structural disorder, the width of each atomic lattice is varied while their position remains unchanged. Uncorrelated structural disorder is obtained by introducing random variations of width *d _{j}* in Eq. (1), which can be written as

*η*}represent sequences of uncorrelated random numbers uniformly distributed in the interval [−Δ

_{j}*Δ*

_{s},*]. Here, Δ*

_{s}*is the strength of the structural disorder.*

_{s}If the width *d _{j}* of the

*j*th atomic lattice depends on the variations of the widths of all previous lattices, the structural disorder becomes correlated. Then we obtain

To ensure the validity of two types of disorders existing in 1D cold atomic lattices, we require the maxima of sequences |{*ζ _{j}*}|, $\left|\left\{{\displaystyle {\sum}_{i=1}^{j}{\zeta}_{i}}\right\}\right|$, |{

*η*}| and $\left|\left\{{\displaystyle {\sum}_{i=1}^{j}{\eta}_{i}}\right\}\right|$ being less than 0.5. Thus, the strengths of two types of uncorrelated disorders satisfy Δ

_{j}*, Δ*

_{g}*≤0.5, while, for two types of correlated disorders, the disorder strengths satisfy Δ*

_{s}*, ${\mathrm{\Delta}}_{s}\le \sqrt{{0.5}^{2}/12}\simeq 0.14$ [27].*

_{g}Next, we consider a four-level *N*-type atomic system shown in Fig. 1(b). The experimental system for this scheme is realized by ^{87}Rb atoms with |5^{2}*S*_{1/2}, *F* = 1〉, |5^{2}*S*_{1/2}, *F* = 2〉, |5^{2}*P*_{1/2}, *F* = 1〉 and |5^{2}*P*_{1/2}, *F* = 2〉 behaving the |1〉, |2〉, |3〉 and |4〉, respectively. A weak probe field with Rabi frequency Ω* _{p}* =

*µ*

_{31}

*E*/2

_{p}*ħ*and a strong driving field with Rabi frequency Ω

*=*

_{d}*µ*

_{32}

*E*/2

_{d}*ħ*are applied to the transitions |3〉→|1〉 and |3〉→|2〉, respectively, while the transition |4〉→ |1〉 is driven by a position-dependent coupling field with Rabi frequency Ω

*(*

_{c}*x*) =

*µ*

_{41}

*E*(

_{c}*x*)/2

*ħ*. Here,

*µ*

_{31},

*µ*

_{41}and

*µ*

_{32}are the corresponding electric-dipole matrix moments. Note that the position-dependent coupling field Ω

*(*

_{c}*x*) is a superposition of a traveling-wave (TW) field and a standing-wave (SW) field, which can be written as

*φ*is the initial phase of the SW field and period Λ can be adjusted by tuning the angle between the forward and backward components of the coupling fields [not shown in Fig. 1(a)].

_{c}Under the electric-dipole and rotating-wave approximations, the interaction Hamiltonian for the *N*-type atomic system is given by

*=*

_{p}*ω*−

_{p}*ω*

_{31}, Δ

*=*

_{c}*ω*−

_{c}*ω*

_{41}and Δ

*=*

_{d}*ω*−

_{d}*ω*

_{32}are the detunings of the probe, coupling and driving fields, respectively. The dynamics of the atomic system can be described by using the density matrix approach as

Here, the Liouvillian matrix *L*[*ρ*(*t*)]indicating the relaxation by spontaneous decay can be written as

*σ*

_{44}= (Γ

_{41}+ Γ

_{42})

*ρ*

_{44},

*σ*

_{33}= (Γ

_{31}+ Γ

_{32})

*ρ*

_{33},

*σ*

_{22}= Γ

_{42}

*ρ*

_{44}+ Γ

_{32}

*ρ*

_{33}and

*σ*

_{11}= Γ

_{41}

*ρ*

_{44}+ Γ

_{31}

*ρ*

_{33}. Γ

*is the spontaneous-emission decay rate from the state |*

_{ij}*i*〉 to the state |

*j*〉 and

*γ*is the decay rate of the coherence between the states |

_{ij}*i*〉 and |

*j*〉 (

*i, j*= 1, 2, 3, 4;

*i > j*). For the cold

^{87}

*Rb*atoms, Γ

_{41}= Γ

_{42}= Γ

_{31}= Γ

_{32}= Γ = 5.9 MHz,

*γ*

_{41}=

*γ*

_{42}=

*γ*

_{31}=

*γ*

_{32}=

*γ*

_{43}/2 = 3 MHz and

*γ*

_{21}= 10kHz [19, 33].

In the limit of weak probe field, the local steady-state probe susceptibility, i.e., ${\chi}_{j}\left(x\right)={\chi}_{j}^{\prime}\left(x\right)+i{\chi}_{j}^{\u2033}\left(x\right)$, in the *j*th unit cell can be written as

*d*

_{1}=

*γ*

_{21}−

*i*(Δ

*−Δ*

_{p}*),*

_{d}*d*

_{2}=

*γ*

_{42}+

*i*(Δ

*+ Δ*

_{c}*−Δ*

_{d}*),*

_{p}*d*

_{3}=

*γ*

_{31}−

*i*Δ

*and*

_{p}*d*

_{4}= 2

*γ*

_{43}−

*i*(Δ

*−Δ*

_{p}*). Then the total probe susceptibility*

_{c}*χ*(

*x*)=

*χ*

^{′}(

*x*)+

*i χ*

^{″}(

*x*)= Σ

*(*

_{j}χ_{j}*x*) can be obtained.

We note that the SW component in Eq. (6) is a sine function of *x* for *φ _{c}* = 0. In the absence of disorder, the position

*x*of 1D atomic lattices, i.e., the bottom position of the formed optical lattices, is situated right at the node of the SW component of the coupling field. Therefore, the optical lattices and the coupling field can be modulated in phase when

_{j}*φ*=

_{c}*π*/2. That is to say, there is a

*π*/2 spatial phase shift between the position-dependent coupling field and the atomic lattices with Gaussian atomic density distribution. In this case, the double modulations in Ω

*(*

_{c}*x*) and

*N*(

_{j}*x*) may result in $\mathcal{P}\mathcal{T}$ symmetric susceptibility [19]. However, probe susceptibility

*χ*(

*x*) will no longer be $\mathcal{P}\mathcal{T}$-symmetric when

*φ*≠ 0, ±

_{c}*π*.

In this work, to focus on the diffraction features of 1D atomic grating operating in Raman-Nath regime. We assume that the incident probe field is a plane wave having an amplitude *E*_{0} uniform across a beam of a width of *M*Λ (*M* is an integer). According to the amplitude transmittance approach in Raman-Nath theory [2], 1D transmission function *T*(*x*) bounded by the width of the incident light beam can expressed as

*β*(

_{I}*x*) =

*πχ*

^{″}(

*x*)/

*λ*and

_{p}*β*(

_{R}*x*) =

*πχ*

^{′}(

*x*)/

*λ*,

_{p}*L*is the grating thickness. ${e}^{-{\beta}_{I}\left(x\right)L}$ and

*β*(

_{R}*x*)

*L*are the amplitude and phase of the transmission function, respectively. Therefore, the diffraction-field distribution of Fraunhofer diffraction can be directly obtained by the Fourier transform of

*T*(

*x*) [9]:

*C*is the proportionality. We define

*I*(

_{p}*θ*) as |

*E*(

_{p}*θ*)|

^{2}normalized to (

*CE*

_{0}

*M*Λ)

^{2}[9], then the intensity distribution of Fraunhofer diffraction patterns can be written as

It is worth noting that, in comparison with the diffraction formulas in periodic diffraction gratings [9–21], Eq. (13) is a more basic diffraction formula to calculate the intensity of Fraunhofer diffraction [38], which can be used to simulate the diffraction-intensity patterns when disorder is introduced into the atomic grating.

## 3. Results and discussions

Before embarking a detailed analysis for the influence of disorder on the PAD. We first explore in Fig. 2 how to realize the PAD in 1D driven atomic lattices without disorder. Figs. 2(a) and 2(b) illustrate that the PAD, where all diffraction fields only appear in the non-positive diffraction orders, can be achieved under different conditions of the phase *φ _{c}* of the SW coupling field and the detuning Δ

*of the driving field. In the former case, i.e.,*

_{d}*φ*= 0 and Δ

_{c}*= 0MHz, the formation of PT-symmetric susceptibility leads to the generation of the PAD [21]. In the latter case, i.e.,*

_{d}*φ*= 0.15

_{c}*π*and Δ

*= 2MHz, the PAD can also be observed although the system is no longer PT-symmetric. That is to say, we can realize the PAD when the atomic grating has a $\mathcal{P}\mathcal{T}$-symmetric or more general susceptibility. Note that the driven atomic lattices with the atomic density of Gaussian profile provide a useful but not only platform to realize the PAD. Previous work has demonstrated the PAD can be achieved in uniformly distributed cold atomic gas via double modulations of the control and Stark fields [20]. In the following, by utilizing the disorder algorithm in [33, 39], we focuson analyzing the effects of two types of disorders on the PAD of 1D atomic grating.*

_{d}We then examine in Fig. 3 the influence of geometrical disorder in the position of the atomic lattices on the PAD. In the case of uncorrelated geometrical disorder [see Fig. 3(a)], it can be found from Fig. 3(c) that the PAD is well maintained in the presence of 7 percent of uncorrelated geometrical disorder. However, in comparison with ideal case shown in Fig. 2(b), the uncorrelated geometrical disorder leads to a redistribution of the diffraction intensities in the diffraction orders and side-lobes. In the case of correlated geometrical disorder, the random configurations of correlated geometrical disorder exhibit maximal absolute values ∼0.28, which is much larger than the corresponding disorder strength, i.e., Δ* _{g}* = 0.07 [see Fig. 3(b)]. It is clear in Fig. 3(d) that already a small percent level (7%) in the position fluctuations can destroy the PAD, where some diffraction fields appear in the positive diffraction orders. Direct comparison of Figs. 3(a) and 3(b) implies that the diffraction intensities induced by correlated geometric disorder can be much larger than that induced by uncorrelated geometric disorder in some random configurations. In the same strength of disorder, the larger variation of the lattice position induced by correlated geometric disorder is easier to decrease the absorption and increase gain, thereby leading to the appearance of diffraction spectrum with high diffraction intensities. In these two cases, we also compute two spatial correlation functions, i.e.,

*C*(Δ

_{R}*x*)= 〈

*χ*

^{′}(

*x*)

*χ*

^{′}(

*x*+ Δ

*x*)〉 and

*C*(Δ

_{I}*x*)= 〈

*χ*

^{″}(

*x*)

*χ*

^{″}(

*x*+ Δ

*x*)〉, where

*χ*

^{′}(

*x*)(

*χ*

^{″}(

*x*)) indicates the dispersion (absorption) at the position

*x*and the brackets 〈

*…*〉 represents averaging over many random configurations with the same disorder strength. As shown in Fig. 3(e), both

*C*and

_{R}*C*are independent of Δ

_{I}*x*when Δ

*x*≠ 0 in the case of uncorrelated geometrical disorder. Thus, the correlation of the system is long-ranged. In the case of correlated geometrical disorder,

*C*and

_{R}*C*decreases with the increase of Δ

_{I}*x*[see Fig. 3(f)], which indicates that the system has a short-range order. Therefore, they play different roles on the modifications of Raman-Nath diffraction.

To address this issue of Raman-Nath diffraction variations with the strength of disorder, we define the average diffraction 〈*I _{p}*(

*θ*)〉 as the mean value of the diffractions for

*N*different disorder realizations with the same disorder type and strength, for a given angle

_{R}*θ*:

*I*(

_{p}*θ, m*) represents the diffraction spectrum induced by the

*m*th disorder realization.

Accordingly, the average diffraction spectra 〈*I _{p}*(

*θ*)〉 for different strengths Δ

*of uncorrelated and correlated geometrical disorders are plotted in Figs. 4(a) and 4(b), respectively. It can be seen that both uncorrelated and correlated geometrical disorders can destroy the PAD as the disorder strength Δ*

_{g}*increases [see the insets in Figs. 4(a) and 4(b)]. However, the diffraction spectra of the atomic grating are in general much more sensitive to correlated disorder than to uncorrelated disorder. In order to gain overall view of the effect of geometrical disorder, we present the corresponding evolutions of the average diffraction spectra 〈*

_{g}*Ip*(

*θ*)〉 with the increase of the disorder strength in Figs. 4(c) and 4(d). To distinguish the PAD from asymmetric diffraction spectra, we define a ”diffraction contrast

*η*”, which is the intensity ratio of the +1st order diffraction to the 0th order diffraction, i.e.,

*η*=

*I*(

_{p}*θ*+1)/

*I*(

_{p}*θ*

_{0}). Asymmetric Raman-Nath diffraction can be treated as the PAD when the diffraction contrast

*η*is very small (i.e.,

*η<*0.02). Therefore, we can identify two critical disorder strengths, i.e., ${\mathrm{\Delta}}_{g}^{c}\simeq 0.17$ for uncorrelated disorder and ${\mathrm{\Delta}}_{g}^{c}\simeq 0.05$ for correlated disorder. As shown in in Figs. 4(c) and 4(d), the PAD can be observed below the critical disorder strengths, beyond which the PAD is degraded to asymmetric diffraction. Meanwhile, it is found that the uncorrelated geometrical disorder diffracts a considerable amount of probe energy into the diffraction side-lobes in strong disorder region, while the correlation in the geometrical disorder leads to probe energy being concentrated into the relevant diffraction orders. However, we should note that the PAD can also be observed in the ideal atomic grating for

*φ*= 0.15

_{c}*π*and Δ

*= 2MHz. In this case, our system will no longer be $\mathcal{P}\mathcal{T}$-symmetric. We examine in Figs. 5(a) and 5(b) the effects of uncorrelated and correlated geometrical disorders on the PAD of the atomic grating with more general susceptibility. Similar to the results in Figs. 4(c) and 4(d), the PAD can also be degraded to asymmetric diffraction with the increase of the disorder strength. The destructive effect of geometrical disorder on the PAD can readily be explained physically. In the ideal atomic grating, there is a fixed spatial phase shift between the double modulations of the SW coupling field and Gaussian atomic density distribution. The disorder in the position*

_{d}*x*of each atomic lattice could induce a series of additional random phase shifts between Ω

_{j}*(*

_{c}*x*) and

*N*(

_{j}*x*). The additional phase shifts become larger with the increase of Δ

*, thereby leading to the decrease of the correlation of disorder and the disappearance of the PAD.*

_{g}We further examine the effect of structural disorder in the width of the atomic lattices on the PAD. In this step, we repeat the above calculation for uncorrelated and correlated structural disorders in Figs. 6. Figs. 6(a) and 6(b) illustrate that, under different strengths of uncorrelated and correlated structural disorders, the random perturbations in the width of Gaussian density profile have few influences on the average diffraction efficiencies in the diffraction orders. More importantly, the PAD phenomenon can also be observed even in the strong disorder effect. To see more details, the evolutions of the average diffraction spectra 〈*I _{p}*(

*θ*)〉 with the strengths of uncorrelated and correlated structural disorders are plotted in Figs. 6(c) and 6(d), respectively. We can find that these results shown in Figs. 6(c) and 6(d) differ dramatically from the corresponding geometrical disorder cases [see Figs. 4(c) and 4(d)]. The structural disorder has less impact on the grating diffraction with a certain range of disorder strength. The similar evolution trends are observed in Figs. 7(a) and 7(b) where the atomic grating has a non-$\mathcal{P}\mathcal{T}$-symmetric susceptibility. Apparently, the PAD is rather robust against the structural disorder whenever the system is $\mathcal{P}\mathcal{T}$-symmetric or not. As a matter of fact, disorder in the width of the atomic lattices would not induce additional random spatial phase shift between the position-dependent coupling field Ω

*(*

_{c}*x*) and the Gaussian density distribution

*N*(

_{j}*x*) of each atomic lattice. That is to say, the spatial phase shift is fixed. In this case, the correlation functions

*C*and

_{R}*C*are almost unchanged. Therefore, the PAD has good robustness against the structural disorder.

_{I}In the above discussions, we focus on investigating the influence of disorder in the position and width of the cold atomic lattices on Raman-Nath diffraction. As a matter of fact, disorder can also be introduced to the number of cold atoms trapped in the optical lattices or the intensity of the SW coupling field. Thus, we check in Figs. 8(a) and 8(b) what will happen when considering random fluctuation *δN _{j}* in the average atomic density

*N*0 and random fluctuation

*δ*Ω

_{c}_{1}in the SW Rabi frequency Ω

_{c}_{1}. Here, we only focus on the uncorrelated cases. In the presence of disorder in the average atomic density of the cold atomic lattices, as shown in Fig. 8(a), it is found that the PAD is well maintained even though the disorder strength Δ

*increases, which is similar to the case shown in Fig. 6(c). This is because the random variation*

_{N}*δN*, similar to the random variation

_{j}*δd*, would not induce additional random phase shifts between Ω

_{j}*(*

_{c}*x*) and

*N*(

_{j}*x*). When the random fluctuation in the intensity of the standing-wave coupling field is considered, although no additional phase shifts between Ω

*(*

_{c}*x*) and

*N*(

_{j}*x*)are generated, the fluctuation of standing-wave intensity can greatly affect the atom-field interaction. In this case, we can find from Fig. 8(b) that the PAD is destroyed when the disorder strength Δ

_{Ω}has a large value. Therefore, it can be concluded that the PAD maybe broken when disorder is introduced to other light parameters, such as the detunings of the applied fields.

## 4. Conclusions

In summary, we have theoretically investigated the effects of geometric and structural disorders on the PAD in Raman-Nath regime. These two types of disorders can be easily realized by introducing random variations of the position and width of each cold atomic lattice and artificially controlled via using optical speckle potential. Our simulation results show distinct Raman-Nath diffraction behaviors in the presence of geometric and structural disorders. In the case of geometric disorder, the PAD exists below the critical strengths of uncorrelated and correlated disorders, beyond which the PAD disappears. However, the PAD remains unchanged in the case of uncorrelated or correlated structural disorder, at least for the disorder strengths considered here. These different diffraction phenomena are associated with the disorder-induced random variation of the spatial phase shift between the modulated coupling field and atomic density distribution. It is found that the disappearance of the PAD is more sensitive to correlated geometric disorder than to uncorrelated geometric disorder. Furthermore, the spectrum spread phenomenon of diffracted light can be observed with increasing the strength of uncorrelated geometric disorder. It is worth noting that our results can be easily extended to the diffraction of atoms and electrons because they obey similar diffraction equations [40–43]. Thus, our scheme provides a possibility to understand the effect of disorder on the Raman-Nath diffraction properties of matter waves in disordered potential.

## Funding

National Natural Science Foundation of China (11374050, 11774054); Natural Science Foundation of Jiangsu Province (BK20161410); Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX18_0055).

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