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 (PT)-symmetric or PT-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 λo [not indicated in Fig. 1(a)]. In the absence of disorder, 1D atomic lattices have a period Λ = 0.5λo/cos θo, where θo is the angle between the dipole-trap laser beams and the lattice axis along x. Meanwhile, the atomic density Nj (x) for the trapped atoms in the jth unit cell, i.e., (−1 + 2j)Λ/2 ≤ x ≤(1 + 2j)Λ/2, can be well approximated by a Gaussian function [34, 35]

Nj(x)=N02πdexp[(xxj)2dj2],
where N0 is the average atomic density. For the ideal atomic lattices, dj = d and xj = j ×Λ are the e−1 half width and position of the jth atomic lattice, respectively. It is worth noting that the position xj of each atomic lattice is located at the bottom of each optical lattice [see Fig. 1(a)].

 

Fig. 1 (a) Sketch of 1D atomic grating, which consists of 1D optical lattices of driven cold 87Rb atoms. The area shown in purple is the spatial distribution of the atomic density. (b) Schematic of diagram of a four-level N-type atomic system interacting with three applied laser fields.

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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 xj in Eq. (1), uncorrelated geometrical disorder is modeled as

xj=jΛ+δxj=jΛ+Λζj,
where {ζj}represent sequences of uncorrelated random numbers uniformly distributed in the interval [−Δg, Δg] with Δg the strength of the geometrical disorder.

If the position xj of the jth 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

xj(c)=jΛ+i=1jδxi=jΛ+Λi=1jζi,

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 dj in Eq. (1), which can be written as

dj=d+δdj=d+dηj,
in which {ηj}represent sequences of uncorrelated random numbers uniformly distributed in the interval [−Δs, Δs]. Here, Δs is the strength of the structural disorder.

If the width dj of the jth atomic lattice depends on the variations of the widths of all previous lattices, the structural disorder becomes correlated. Then we obtain

dj(c)=d+i=1jδdi=d+di=1jηi.

To ensure the validity of two types of disorders existing in 1D cold atomic lattices, we require the maxima of sequences |{ζj}|, |{i=1jζi}|, |{ηj}| and |{i=1jηi}| being less than 0.5. Thus, the strengths of two types of uncorrelated disorders satisfy Δg, Δs ≤0.5, while, for two types of correlated disorders, the disorder strengths satisfy Δg, Δs0.52/120.14 [27].

Next, we consider a four-level N-type atomic system shown in Fig. 1(b). The experimental system for this scheme is realized by 87Rb atoms with |52S1/2, F = 1〉, |52S1/2, F = 2〉, |52P1/2, F = 1〉 and |52P1/2, F = 2〉 behaving the |1〉, |2〉, |3〉 and |4〉, respectively. A weak probe field with Rabi frequency Ωp = µ31Ep/2ħ and a strong driving field with Rabi frequency Ωd = µ32Ed/2ħ 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 Ωc(x) = µ41Ec(x)/2ħ. Here, µ31, µ41 and µ32 are the corresponding electric-dipole matrix moments. Note that the position-dependent coupling field Ωc(x) is a superposition of a traveling-wave (TW) field and a standing-wave (SW) field, which can be written as

Ωc(x)=Ωc0+Ωc1sin(2πxΛφc),
where φc 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)].

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

HI=(Δc00Ωc(x)0ΔpΩdΩp0Ωd*(ΔpΔd)0Ωc*(x)Ωp*00),
where Δp = ωpω31, Δc = ωcω41 and Δd = ωdω32 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
dρdt=i[HI,ρ]+L[ρ(t)].

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

L[ρ(t)]=(σ44γ43ρ43γ42ρ42γ41ρ41γ43ρ34σ33γ32ρ32γ31ρ31γ42ρ24γ32ρ23σ22γ21ρ21γ41ρ14γ31ρ13γ21ρ12σ11),
where σ44 = (Γ41 + Γ42)ρ44, σ33 = (Γ31 + Γ32)ρ33, σ22 = Γ42 ρ44 + Γ32 ρ33 and σ11 = Γ41 ρ44 + Γ31 ρ33. Γij is the spontaneous-emission decay rate from the state |i〉 to the state |j〉 and γij is the decay rate of the coherence between the states |i〉 and |j〉 (i, j = 1, 2, 3, 4;i > j). For the cold 87Rb atoms, Γ41 = Γ42 = Γ31 = Γ32= Γ = 5.9 MHz, γ41 = γ42 = γ31 = γ32 = γ43/2 = 3 MHz and γ21 = 10kHz [19, 33].

In the limit of weak probe field, the local steady-state probe susceptibility, i.e., χj(x)=χj(x)+iχj(x), in the jth unit cell can be written as

χj(x)=Nj(x)|μ31|2ε0Ωpρ31=Nj(x)|μ31|2ε0[iκ3(κ4+κ5)Ωc2(x)]Ωd2κ1κ2,
where κ1=d1d2d3d4+[Ωc2(x)Ωd2+d1d2+d3d4]Ωc2(x)+[Ωd2Ωc2(x)+d1d3+d2d4]Ωd2, κ2=[2Ωd2+Δd2+Γ2]Ωc2(x)+[2Ωc2(x)+Δc2+Γ2]Ωd2, κ3=(Δc2+Γ2)[d1Ωd2+d4Ωc2(x)+d1d2d4], κ4=[Ωc2(x)Ωd2+d1d2](Δc+iΓ), κ5=[Ωd2Ωc2(x)+d2d4](Δd+iΓ), d1 = γ21ip −Δd), d2 = γ42 + ic + Δd −Δp), d3 = γ31iΔp and d4 = 2γ43ip −Δc). Then the total probe susceptibility χ(x)= χ(x)+ i χ(x)= Σj χj(x) can be obtained.

We note that the SW component in Eq. (6) is a sine function of x for φc = 0. In the absence of disorder, the position xj 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 φc = 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 Ωc(x) and Nj (x) may result in PT symmetric susceptibility [19]. However, probe susceptibility χ(x) will no longer be PT-symmetric when φc ≠ 0, ±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 E0 uniform across a beam of a width of MΛ (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

T(x)={eβI(x)LeiβR(x)L,ifx[Λ/2,(2M+1)Λ/2]0,otherwise,
where βI (x) = πχ(x)/λp and βR(x) = πχ(x)/λp, L is the grating thickness. eβI(x)L and βR(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]:
Ep(θ)=C+E0T(x)ei2πxsinθ/λpdx,
where C is the proportionality. We define Ip(θ) as |Ep(θ)|2 normalized to (CE0MΛ)2 [9], then the intensity distribution of Fraunhofer diffraction patterns can be written as
Ip(θ)=1(MΛ)2|+T(x)ei2πxsinθ/λpdx|2.

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 φc of the SW coupling field and the detuning Δd of the driving field. In the former case, i.e., φc = 0 and Δd = 0MHz, the formation of PT-symmetric susceptibility leads to the generation of the PAD [21]. In the latter case, i.e., φc = 0.15π and Δd = 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 PT-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.

 

Fig. 2 The Raman-Nath diffraction intensities as a function of sin θ for (a) φc = 0, Δd = 0MHz and (b) φc = 0.15π, Δd = 2MHz. Other parameters are N0 = 5 ×1012cm−3, µ31 = 2.5377 ×10−29C m, Ωc0 = 6MHz, Ωc1 = 1.53MHz, Ωd = 2MHz, Δp = 7MHz, Δc = 0MHz, λp = 795.5nm, Λ/λp = 4, d = 0.2Λ, M = 11 and L = 30µm.

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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., Δg = 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., CRx)= 〈χ(x)χ(x + Δx)〉 and CIx)= 〈χ(x)χ(x + Δx)〉, 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 CR and CI are independent of Δx when Δx ≠ 0 in the case of uncorrelated geometrical disorder. Thus, the correlation of the system is long-ranged. In the case of correlated geometrical disorder, CR and CI decreases with the increase of Δx [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.

 

Fig. 3 Two examples of 20 random configurations of (a) uncorrelated geometric disorder [Eq. (2)] and (b) correlated geometric disorder [Eq. (3)]. The dependence of Raman-Nath diffraction spectra on the (c) uncorrelated and (d) correlated geometric disorders. Each curve represents a diffraction profile induced by a random configuration of geometric disorder. The spatial correlation functions CRx) and CIx) for the real part χ(x) and imaginary part χ(x) of the spatial susceptibility with the (e) uncorrelated and (f) correlated geometric disorders. The strength of geometric disorder Δg = 0.07 and other parameters are the same as in Fig. 2(a).

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To address this issue of Raman-Nath diffraction variations with the strength of disorder, we define the average diffraction 〈Ip(θ)〉 as the mean value of the diffractions for NR different disorder realizations with the same disorder type and strength, for a given angle θ:

Ip(θ)=1NRm=1NRIp(θ,m),
where Ip (θ, m) represents the diffraction spectrum induced by the mth disorder realization.

Accordingly, the average diffraction spectra 〈Ip(θ)〉 for different strengths Δg 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 Δg 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 〈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 = Ip(θ+1)/Ip(θ0). 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., Δgc0.17 for uncorrelated disorder and Δgc0.05 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 φc = 0.15π and Δd = 2MHz. In this case, our system will no longer be PT-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 xj of each atomic lattice could induce a series of additional random phase shifts between Ωc (x) and Nj (x). The additional phase shifts become larger with the increase of Δg, thereby leading to the decrease of the correlation of disorder and the disappearance of the PAD.

 

Fig. 4 The average Raman-Nath diffraction spectra 〈Ip(θ)〉 of the atomic grating for different strengths of (a) uncorrelated and (b) correlated geometric disorders. The corresponding evolutions of the average Raman-Nath diffraction 〈Ip(θ)〉 with the strengths of (c)uncorrelated and (d) correlated disorders. The average diffraction spectra are attained through averaging over 50 different random configurations of disorder (NR = 50). Other parameters are the same as in Fig. 2(a).

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Fig. 5 The evolutions of the average Raman-Nath diffraction 〈Ip(θ)〉 with the strengths Δg of (a) uncorrelated (b) correlated geometric disorders. Other parameters are the same as in Fig. 2(b).

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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 〈Ip(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-PT-symmetric susceptibility. Apparently, the PAD is rather robust against the structural disorder whenever the system is PT-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 Ωc(x) and the Gaussian density distribution Nj (x) of each atomic lattice. That is to say, the spatial phase shift is fixed. In this case, the correlation functions CR and CI are almost unchanged. Therefore, the PAD has good robustness against the structural disorder.

 

Fig. 6 The average Raman-Nath diffraction spectra 〈Ip(θ)〉 of the atomic grating for different strengths of (a) uncorrelated and (b) correlated structural disorders. The corresponding evolutions of the average Raman-Nath diffraction 〈Ip(θ)〉 with the strengths of (c)uncorrelated and (d) correlated disorders. Other parameters are the same as in Fig. 2(a).

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Fig. 7 The evolutions of the average Raman-Nath diffraction 〈Ip(θ)〉 with the strengths Δs of (a) uncorrelated (b) correlated structural disorders. Other parameters are the same as in Fig. 2(b).

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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 δNj in the average atomic density N0 and random fluctuation δΩc1 in the SW Rabi frequency Ωc1. 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 ΔN increases, which is similar to the case shown in Fig. 6(c). This is because the random variation δNj, similar to the random variation δdj, would not induce additional random phase shifts between Ωc(x) and Nj (x). When the random fluctuation in the intensity of the standing-wave coupling field is considered, although no additional phase shifts between Ωc (x) and Nj (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 ΔΩ 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.

 

Fig. 8 The evolutions of the average Raman-Nath diffraction 〈Ip(θ)〉 with the strengths ΔN and ΔΩ of uncorrelated disorders in (a) the average atomic density N0 and (b) the standing-wave Rabi frequency Ωc1. Other parameters are the same as in Fig. 2(a).

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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).

References

1. C. A. Palmer and E. G. Loewen, Diffraction Grating Handbook (Newport Corporation, 2005).

2. T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985). [CrossRef]  

3. R. B. Witmer and J. M. Cork, “The measurement of x-ray emission wave-lengths by means of the ruled grating,” Phys. Rev. 42, 743–748 (1932). [CrossRef]  

4. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” The Bell Syst. Tech. J. 48, 2909–2947 (1969). [CrossRef]  

5. T. Kubota, “Characteristics of thick hologram grating recorded in absorptive medium,” Opt. Acta: Int. J. Opt. 25, 1035–1053 (1978). [CrossRef]  

6. B. Wang, C. Zhou, S. Wang, and J. Feng, “Polarizing beam splitter of a deep-etched fused-silica grating,” Opt. Lett. 32, 1299–1301 (2007). [CrossRef]   [PubMed]  

7. D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4, 466–470 (2010). [CrossRef]  

8. M. B. Sobnack, W. Tan, N. Wanstall, T. Preist, and J. Sambles, “Stationary surface plasmons on a zero-order metal grating,” Phys. Rev. Lett. 80, 5667–5670 (1998). [CrossRef]  

9. H. Y. Ling, Y.-Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57, 1334–1338 (1998). [CrossRef]  

10. L. E. de Araujo, “Electromagnetically induced phase grating,” Opt. Lett. 35, 977–979 (2010). [CrossRef]   [PubMed]  

11. S. Q. Kuang, C. S. Jin, and C. Li, “Gain-phase grating based on spatial modulation of active raman gain in cold atoms,” Phys. Rev. A 84, 033831 (2011). [CrossRef]  

12. R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J. Y. Gao, “Electromagnetically induced grating via enhanced nonlinear modulation by spontaneously generated coherence,” Phys. Rev. A 83, 033824 (2011). [CrossRef]  

13. L. Wang, F. Zhou, P. Hu, Y. Niu, and S. Gong, “Two-dimensional electromagnetically induced cross-grating in a four-level tripod-type atomic system,” J. Phys. B: At. Mol. Opt. Phys. 47, 225501 (2014). [CrossRef]  

14. L. Zhao, W. Duan, and S. Yelin, “Generation of tunable-volume transmission-holographic gratings at low light levels,” Phys. Rev. A 84, 033806 (2011). [CrossRef]  

15. S. A. Carvalho and L. E. de Araujo, “Electromagnetically-induced phase grating: A coupled-wave theory analysis,” Opt. Express 19, 1936–1944 (2011). [CrossRef]   [PubMed]  

16. M. Mitsunaga and N. Imoto, “Observation of an electromagnetically induced grating in cold sodium atoms,” Phys. Rev. A 59, 4773–4776 (1999). [CrossRef]  

17. A. W. Brown and M. Xiao, “All-optical switching and routing based on an electromagnetically induced absorption grating,” Opt. Lett. 30, 699–701 (2005). [CrossRef]   [PubMed]  

18. L. Zhao, W. Duan, and S. Yelin, “All-optical beam control with high speed using image-induced blazed gratings in coherent media,” Phys. Rev. A 82, 013809 (2010). [CrossRef]  

19. Y. M. Liu, F. Gao, C. H. Fan, and J. H. Wu, “Asymmetric light diffraction of an atomic grating with PT symmetry,” Opt. Lett. 42, 4283–4286 (2017). [CrossRef]   [PubMed]  

20. T. Shui, W. X. Yang, S. P. Liu, L. Li, and Z. H. Zhu, “Asymmetric diffraction by atomic gratings with optical PT symmetry in the Raman-Nath regime,” Phys. Rev. A 97, 033819 (2018). [CrossRef]  

21. T. Shui, W. X. Yang, L. Li, and X. Wang, “Lop-sided Raman-Nath diffraction in PT-antisymmetric atomic lattices,” Opt. Lett. 44, 2089–2092 (2019). [CrossRef]   [PubMed]  

22. M. Kulishov, H. Jones, and B. Kress, “Analysis of PT -symmetric volume gratings beyond the paraxial approximation,” Opt. Express 23, 9347–9362 (2015). [CrossRef]   [PubMed]  

23. X. Y. Zhu, Y. L. Xu, Y. Zou, X. C. Sun, C. He, M. H. Lu, X. P. Liu, and Y. F. Chen, “Asymmetric diffraction based on a passive parity-time grating,” Appl. Phys. Lett. 109, 111101 (2016). [CrossRef]  

24. V. Bushuev, L. Dergacheva, and B. Mantsyzov, “Asymmetric pendulum effect and transparency change of PT-symmetric photonic crystals under dynamical bragg diffraction beyond the paraxial approximation,” Phys. Rev. A 95, 033843 (2017). [CrossRef]  

25. M. Kulishov and B. Kress, “Free space diffraction on active gratings with balanced phase and gain/loss modulations,” Opt. Express 20, 29319–29328 (2012). [CrossRef]  

26. S. C. Tian, R. G. Wan, L. J. Wang, S. L. Shu, H. Y. Lu, X. Zhang, C. Z. Tong, J. L. Feng, M. Xiao, and L. J. Wang, “Asymmetric light diffraction of two-dimensional electromagnetically induced grating with PT symmetry in asymmetric double quantum wells,” Opt. Express 26, 32918–32930 (2018). [CrossRef]  

27. P. Licinio, M. Lerotic, and M. Dantas, “Diffraction by disordered gratings and the Debye–Waller effect,” Am. J. Phys. 67, 1013–1016 (1999). [CrossRef]  

28. A. D. Sinelnik, M. V. Rybin, S. Y. Lukashenko, M. F. Limonov, and K. B. Samusev, “Evolution of optical diffraction patterns on disordered woodpile photonic structures,” Phys. Solid State 60, 1387–1393 (2018). [CrossRef]  

29. B. M. DeVetter, B. E. Bernacki, and K. J. Alvine, “Disordered spontaneously buckled optical gratings for improved lighting applications,” Opt. Lett. 43, 4895–4898 (2018). [CrossRef]   [PubMed]  

30. S. Schauer, R. Schmager, R. Hünig, K. Ding, U. W. Paetzold, U. Lemmer, M. Worgull, H. Hölscher, and G. Gomard, “Disordered diffraction gratings tailored by shape-memory based wrinkling and their application to photovoltaics,” Opt. Mater. Express 8, 184–198 (2018). [CrossRef]  

31. J. B. Kim, P. Kim, N. C. Pégard, S. J. Oh, C. R. Kagan, J. W. Fleischer, H. A. Stone, and Y.-L. Loo, “Wrinkles and deep folds as photonic structures in photovoltaics,” Nat. Photonics 6, 327–332 (2012). [CrossRef]  

32. T.-B. Lim, K. H. Cho, Y.-H. Kim, and Y.-C. Jeong, “Enhanced light extraction efficiency of oleds with quasiperiodic diffraction grating layer,” Opt. Express 24, 17950–17959 (2016). [CrossRef]   [PubMed]  

33. J. H. Wu, M. Artoni, and G. C. La Rocca, “Perfect absorption and no reflection in disordered photonic crystals,” Phys. Rev. A 95, 053862 (2017). [CrossRef]  

34. A. Schilke, C. Zimmermann, and W. Guerin, “Photonic properties of one-dimensionally-ordered cold atomic vapors under conditions of electromagnetically induced transparency,” Phys. Rev. A 86, 023809 (2012). [CrossRef]  

35. J. H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014). [CrossRef]   [PubMed]  

36. M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56, 243–379 (2007). [CrossRef]  

37. G. Modugno, “Anderson localization in Bose–Einstein condensates,” Reports on Prog. Phys. 73, 102401 (2010). [CrossRef]  

38. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999). [CrossRef]  

39. S. F. Liew and H. Cao, “Optical properties of 1D photonic crystals with correlated and uncorrelated disorder,” J. Opt. 12, 024011 (2010). [CrossRef]  

40. M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Atom waves in crystals of light,” Phys. Rev. Lett. 77, 4980–4983 (1996). [CrossRef]   [PubMed]  

41. M. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. A: Math. Gen. 31, 3493–3502 (1998). [CrossRef]  

42. H. Batelaan, “Colloquium: Illuminating the Kapitza-Dirac effect with electron matter optics,” Rev. Mod. Phys. 79, 929–941 (2007). [CrossRef]  

43. P. Vidil and B. Chalopin, “Controllable blazed grating for electrons using Kapitza-Dirac diffraction with multiple-harmonic standing waves,” Phys. Rev. A 92, 062117 (2015). [CrossRef]  

References

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  1. C. A. Palmer and E. G. Loewen, Diffraction Grating Handbook (Newport Corporation, 2005).
  2. T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [Crossref]
  3. R. B. Witmer and J. M. Cork, “The measurement of x-ray emission wave-lengths by means of the ruled grating,” Phys. Rev. 42, 743–748 (1932).
    [Crossref]
  4. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” The Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [Crossref]
  5. T. Kubota, “Characteristics of thick hologram grating recorded in absorptive medium,” Opt. Acta: Int. J. Opt. 25, 1035–1053 (1978).
    [Crossref]
  6. B. Wang, C. Zhou, S. Wang, and J. Feng, “Polarizing beam splitter of a deep-etched fused-silica grating,” Opt. Lett. 32, 1299–1301 (2007).
    [Crossref] [PubMed]
  7. D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4, 466–470 (2010).
    [Crossref]
  8. M. B. Sobnack, W. Tan, N. Wanstall, T. Preist, and J. Sambles, “Stationary surface plasmons on a zero-order metal grating,” Phys. Rev. Lett. 80, 5667–5670 (1998).
    [Crossref]
  9. H. Y. Ling, Y.-Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57, 1334–1338 (1998).
    [Crossref]
  10. L. E. de Araujo, “Electromagnetically induced phase grating,” Opt. Lett. 35, 977–979 (2010).
    [Crossref] [PubMed]
  11. S. Q. Kuang, C. S. Jin, and C. Li, “Gain-phase grating based on spatial modulation of active raman gain in cold atoms,” Phys. Rev. A 84, 033831 (2011).
    [Crossref]
  12. R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J. Y. Gao, “Electromagnetically induced grating via enhanced nonlinear modulation by spontaneously generated coherence,” Phys. Rev. A 83, 033824 (2011).
    [Crossref]
  13. L. Wang, F. Zhou, P. Hu, Y. Niu, and S. Gong, “Two-dimensional electromagnetically induced cross-grating in a four-level tripod-type atomic system,” J. Phys. B: At. Mol. Opt. Phys. 47, 225501 (2014).
    [Crossref]
  14. L. Zhao, W. Duan, and S. Yelin, “Generation of tunable-volume transmission-holographic gratings at low light levels,” Phys. Rev. A 84, 033806 (2011).
    [Crossref]
  15. S. A. Carvalho and L. E. de Araujo, “Electromagnetically-induced phase grating: A coupled-wave theory analysis,” Opt. Express 19, 1936–1944 (2011).
    [Crossref] [PubMed]
  16. M. Mitsunaga and N. Imoto, “Observation of an electromagnetically induced grating in cold sodium atoms,” Phys. Rev. A 59, 4773–4776 (1999).
    [Crossref]
  17. A. W. Brown and M. Xiao, “All-optical switching and routing based on an electromagnetically induced absorption grating,” Opt. Lett. 30, 699–701 (2005).
    [Crossref] [PubMed]
  18. L. Zhao, W. Duan, and S. Yelin, “All-optical beam control with high speed using image-induced blazed gratings in coherent media,” Phys. Rev. A 82, 013809 (2010).
    [Crossref]
  19. Y. M. Liu, F. Gao, C. H. Fan, and J. H. Wu, “Asymmetric light diffraction of an atomic grating with PT symmetry,” Opt. Lett. 42, 4283–4286 (2017).
    [Crossref] [PubMed]
  20. T. Shui, W. X. Yang, S. P. Liu, L. Li, and Z. H. Zhu, “Asymmetric diffraction by atomic gratings with optical PT symmetry in the Raman-Nath regime,” Phys. Rev. A 97, 033819 (2018).
    [Crossref]
  21. T. Shui, W. X. Yang, L. Li, and X. Wang, “Lop-sided Raman-Nath diffraction in PT-antisymmetric atomic lattices,” Opt. Lett. 44, 2089–2092 (2019).
    [Crossref] [PubMed]
  22. M. Kulishov, H. Jones, and B. Kress, “Analysis of PT -symmetric volume gratings beyond the paraxial approximation,” Opt. Express 23, 9347–9362 (2015).
    [Crossref] [PubMed]
  23. X. Y. Zhu, Y. L. Xu, Y. Zou, X. C. Sun, C. He, M. H. Lu, X. P. Liu, and Y. F. Chen, “Asymmetric diffraction based on a passive parity-time grating,” Appl. Phys. Lett. 109, 111101 (2016).
    [Crossref]
  24. V. Bushuev, L. Dergacheva, and B. Mantsyzov, “Asymmetric pendulum effect and transparency change of PT-symmetric photonic crystals under dynamical bragg diffraction beyond the paraxial approximation,” Phys. Rev. A 95, 033843 (2017).
    [Crossref]
  25. M. Kulishov and B. Kress, “Free space diffraction on active gratings with balanced phase and gain/loss modulations,” Opt. Express 20, 29319–29328 (2012).
    [Crossref]
  26. S. C. Tian, R. G. Wan, L. J. Wang, S. L. Shu, H. Y. Lu, X. Zhang, C. Z. Tong, J. L. Feng, M. Xiao, and L. J. Wang, “Asymmetric light diffraction of two-dimensional electromagnetically induced grating with PT symmetry in asymmetric double quantum wells,” Opt. Express 26, 32918–32930 (2018).
    [Crossref]
  27. P. Licinio, M. Lerotic, and M. Dantas, “Diffraction by disordered gratings and the Debye–Waller effect,” Am. J. Phys. 67, 1013–1016 (1999).
    [Crossref]
  28. A. D. Sinelnik, M. V. Rybin, S. Y. Lukashenko, M. F. Limonov, and K. B. Samusev, “Evolution of optical diffraction patterns on disordered woodpile photonic structures,” Phys. Solid State 60, 1387–1393 (2018).
    [Crossref]
  29. B. M. DeVetter, B. E. Bernacki, and K. J. Alvine, “Disordered spontaneously buckled optical gratings for improved lighting applications,” Opt. Lett. 43, 4895–4898 (2018).
    [Crossref] [PubMed]
  30. S. Schauer, R. Schmager, R. Hünig, K. Ding, U. W. Paetzold, U. Lemmer, M. Worgull, H. Hölscher, and G. Gomard, “Disordered diffraction gratings tailored by shape-memory based wrinkling and their application to photovoltaics,” Opt. Mater. Express 8, 184–198 (2018).
    [Crossref]
  31. J. B. Kim, P. Kim, N. C. Pégard, S. J. Oh, C. R. Kagan, J. W. Fleischer, H. A. Stone, and Y.-L. Loo, “Wrinkles and deep folds as photonic structures in photovoltaics,” Nat. Photonics 6, 327–332 (2012).
    [Crossref]
  32. T.-B. Lim, K. H. Cho, Y.-H. Kim, and Y.-C. Jeong, “Enhanced light extraction efficiency of oleds with quasiperiodic diffraction grating layer,” Opt. Express 24, 17950–17959 (2016).
    [Crossref] [PubMed]
  33. J. H. Wu, M. Artoni, and G. C. La Rocca, “Perfect absorption and no reflection in disordered photonic crystals,” Phys. Rev. A 95, 053862 (2017).
    [Crossref]
  34. A. Schilke, C. Zimmermann, and W. Guerin, “Photonic properties of one-dimensionally-ordered cold atomic vapors under conditions of electromagnetically induced transparency,” Phys. Rev. A 86, 023809 (2012).
    [Crossref]
  35. J. H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
    [Crossref] [PubMed]
  36. M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56, 243–379 (2007).
    [Crossref]
  37. G. Modugno, “Anderson localization in Bose–Einstein condensates,” Reports on Prog. Phys. 73, 102401 (2010).
    [Crossref]
  38. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
    [Crossref]
  39. S. F. Liew and H. Cao, “Optical properties of 1D photonic crystals with correlated and uncorrelated disorder,” J. Opt. 12, 024011 (2010).
    [Crossref]
  40. M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Atom waves in crystals of light,” Phys. Rev. Lett. 77, 4980–4983 (1996).
    [Crossref] [PubMed]
  41. M. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. A: Math. Gen. 31, 3493–3502 (1998).
    [Crossref]
  42. H. Batelaan, “Colloquium: Illuminating the Kapitza-Dirac effect with electron matter optics,” Rev. Mod. Phys. 79, 929–941 (2007).
    [Crossref]
  43. P. Vidil and B. Chalopin, “Controllable blazed grating for electrons using Kapitza-Dirac diffraction with multiple-harmonic standing waves,” Phys. Rev. A 92, 062117 (2015).
    [Crossref]

2019 (1)

2018 (5)

2017 (3)

J. H. Wu, M. Artoni, and G. C. La Rocca, “Perfect absorption and no reflection in disordered photonic crystals,” Phys. Rev. A 95, 053862 (2017).
[Crossref]

V. Bushuev, L. Dergacheva, and B. Mantsyzov, “Asymmetric pendulum effect and transparency change of PT-symmetric photonic crystals under dynamical bragg diffraction beyond the paraxial approximation,” Phys. Rev. A 95, 033843 (2017).
[Crossref]

Y. M. Liu, F. Gao, C. H. Fan, and J. H. Wu, “Asymmetric light diffraction of an atomic grating with PT symmetry,” Opt. Lett. 42, 4283–4286 (2017).
[Crossref] [PubMed]

2016 (2)

X. Y. Zhu, Y. L. Xu, Y. Zou, X. C. Sun, C. He, M. H. Lu, X. P. Liu, and Y. F. Chen, “Asymmetric diffraction based on a passive parity-time grating,” Appl. Phys. Lett. 109, 111101 (2016).
[Crossref]

T.-B. Lim, K. H. Cho, Y.-H. Kim, and Y.-C. Jeong, “Enhanced light extraction efficiency of oleds with quasiperiodic diffraction grating layer,” Opt. Express 24, 17950–17959 (2016).
[Crossref] [PubMed]

2015 (2)

M. Kulishov, H. Jones, and B. Kress, “Analysis of PT -symmetric volume gratings beyond the paraxial approximation,” Opt. Express 23, 9347–9362 (2015).
[Crossref] [PubMed]

P. Vidil and B. Chalopin, “Controllable blazed grating for electrons using Kapitza-Dirac diffraction with multiple-harmonic standing waves,” Phys. Rev. A 92, 062117 (2015).
[Crossref]

2014 (2)

J. H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[Crossref] [PubMed]

L. Wang, F. Zhou, P. Hu, Y. Niu, and S. Gong, “Two-dimensional electromagnetically induced cross-grating in a four-level tripod-type atomic system,” J. Phys. B: At. Mol. Opt. Phys. 47, 225501 (2014).
[Crossref]

2012 (3)

M. Kulishov and B. Kress, “Free space diffraction on active gratings with balanced phase and gain/loss modulations,” Opt. Express 20, 29319–29328 (2012).
[Crossref]

A. Schilke, C. Zimmermann, and W. Guerin, “Photonic properties of one-dimensionally-ordered cold atomic vapors under conditions of electromagnetically induced transparency,” Phys. Rev. A 86, 023809 (2012).
[Crossref]

J. B. Kim, P. Kim, N. C. Pégard, S. J. Oh, C. R. Kagan, J. W. Fleischer, H. A. Stone, and Y.-L. Loo, “Wrinkles and deep folds as photonic structures in photovoltaics,” Nat. Photonics 6, 327–332 (2012).
[Crossref]

2011 (4)

L. Zhao, W. Duan, and S. Yelin, “Generation of tunable-volume transmission-holographic gratings at low light levels,” Phys. Rev. A 84, 033806 (2011).
[Crossref]

S. A. Carvalho and L. E. de Araujo, “Electromagnetically-induced phase grating: A coupled-wave theory analysis,” Opt. Express 19, 1936–1944 (2011).
[Crossref] [PubMed]

S. Q. Kuang, C. S. Jin, and C. Li, “Gain-phase grating based on spatial modulation of active raman gain in cold atoms,” Phys. Rev. A 84, 033831 (2011).
[Crossref]

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J. Y. Gao, “Electromagnetically induced grating via enhanced nonlinear modulation by spontaneously generated coherence,” Phys. Rev. A 83, 033824 (2011).
[Crossref]

2010 (5)

L. Zhao, W. Duan, and S. Yelin, “All-optical beam control with high speed using image-induced blazed gratings in coherent media,” Phys. Rev. A 82, 013809 (2010).
[Crossref]

D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4, 466–470 (2010).
[Crossref]

L. E. de Araujo, “Electromagnetically induced phase grating,” Opt. Lett. 35, 977–979 (2010).
[Crossref] [PubMed]

G. Modugno, “Anderson localization in Bose–Einstein condensates,” Reports on Prog. Phys. 73, 102401 (2010).
[Crossref]

S. F. Liew and H. Cao, “Optical properties of 1D photonic crystals with correlated and uncorrelated disorder,” J. Opt. 12, 024011 (2010).
[Crossref]

2007 (3)

M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56, 243–379 (2007).
[Crossref]

H. Batelaan, “Colloquium: Illuminating the Kapitza-Dirac effect with electron matter optics,” Rev. Mod. Phys. 79, 929–941 (2007).
[Crossref]

B. Wang, C. Zhou, S. Wang, and J. Feng, “Polarizing beam splitter of a deep-etched fused-silica grating,” Opt. Lett. 32, 1299–1301 (2007).
[Crossref] [PubMed]

2005 (1)

1999 (2)

M. Mitsunaga and N. Imoto, “Observation of an electromagnetically induced grating in cold sodium atoms,” Phys. Rev. A 59, 4773–4776 (1999).
[Crossref]

P. Licinio, M. Lerotic, and M. Dantas, “Diffraction by disordered gratings and the Debye–Waller effect,” Am. J. Phys. 67, 1013–1016 (1999).
[Crossref]

1998 (3)

M. B. Sobnack, W. Tan, N. Wanstall, T. Preist, and J. Sambles, “Stationary surface plasmons on a zero-order metal grating,” Phys. Rev. Lett. 80, 5667–5670 (1998).
[Crossref]

H. Y. Ling, Y.-Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57, 1334–1338 (1998).
[Crossref]

M. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. A: Math. Gen. 31, 3493–3502 (1998).
[Crossref]

1996 (1)

M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Atom waves in crystals of light,” Phys. Rev. Lett. 77, 4980–4983 (1996).
[Crossref] [PubMed]

1985 (1)

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[Crossref]

1978 (1)

T. Kubota, “Characteristics of thick hologram grating recorded in absorptive medium,” Opt. Acta: Int. J. Opt. 25, 1035–1053 (1978).
[Crossref]

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” The Bell Syst. Tech. J. 48, 2909–2947 (1969).
[Crossref]

1932 (1)

R. B. Witmer and J. M. Cork, “The measurement of x-ray emission wave-lengths by means of the ruled grating,” Phys. Rev. 42, 743–748 (1932).
[Crossref]

Abfalterer, R.

M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Atom waves in crystals of light,” Phys. Rev. Lett. 77, 4980–4983 (1996).
[Crossref] [PubMed]

Ahufinger, V.

M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56, 243–379 (2007).
[Crossref]

Alvine, K. J.

Artoni, M.

J. H. Wu, M. Artoni, and G. C. La Rocca, “Perfect absorption and no reflection in disordered photonic crystals,” Phys. Rev. A 95, 053862 (2017).
[Crossref]

J. H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[Crossref] [PubMed]

Batelaan, H.

H. Batelaan, “Colloquium: Illuminating the Kapitza-Dirac effect with electron matter optics,” Rev. Mod. Phys. 79, 929–941 (2007).
[Crossref]

Beausoleil, R. G.

D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4, 466–470 (2010).
[Crossref]

Bernacki, B. E.

Bernet, S.

M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Atom waves in crystals of light,” Phys. Rev. Lett. 77, 4980–4983 (1996).
[Crossref] [PubMed]

Berry, M.

M. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. A: Math. Gen. 31, 3493–3502 (1998).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
[Crossref]

Brown, A. W.

Bushuev, V.

V. Bushuev, L. Dergacheva, and B. Mantsyzov, “Asymmetric pendulum effect and transparency change of PT-symmetric photonic crystals under dynamical bragg diffraction beyond the paraxial approximation,” Phys. Rev. A 95, 033843 (2017).
[Crossref]

Cao, H.

S. F. Liew and H. Cao, “Optical properties of 1D photonic crystals with correlated and uncorrelated disorder,” J. Opt. 12, 024011 (2010).
[Crossref]

Carvalho, S. A.

Chalopin, B.

P. Vidil and B. Chalopin, “Controllable blazed grating for electrons using Kapitza-Dirac diffraction with multiple-harmonic standing waves,” Phys. Rev. A 92, 062117 (2015).
[Crossref]

Chen, Y. F.

X. Y. Zhu, Y. L. Xu, Y. Zou, X. C. Sun, C. He, M. H. Lu, X. P. Liu, and Y. F. Chen, “Asymmetric diffraction based on a passive parity-time grating,” Appl. Phys. Lett. 109, 111101 (2016).
[Crossref]

Cho, K. H.

Cork, J. M.

R. B. Witmer and J. M. Cork, “The measurement of x-ray emission wave-lengths by means of the ruled grating,” Phys. Rev. 42, 743–748 (1932).
[Crossref]

Damski, B.

M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56, 243–379 (2007).
[Crossref]

Dantas, M.

P. Licinio, M. Lerotic, and M. Dantas, “Diffraction by disordered gratings and the Debye–Waller effect,” Am. J. Phys. 67, 1013–1016 (1999).
[Crossref]

de Araujo, L. E.

Dergacheva, L.

V. Bushuev, L. Dergacheva, and B. Mantsyzov, “Asymmetric pendulum effect and transparency change of PT-symmetric photonic crystals under dynamical bragg diffraction beyond the paraxial approximation,” Phys. Rev. A 95, 033843 (2017).
[Crossref]

DeVetter, B. M.

Ding, K.

Duan, W.

L. Zhao, W. Duan, and S. Yelin, “Generation of tunable-volume transmission-holographic gratings at low light levels,” Phys. Rev. A 84, 033806 (2011).
[Crossref]

L. Zhao, W. Duan, and S. Yelin, “All-optical beam control with high speed using image-induced blazed gratings in coherent media,” Phys. Rev. A 82, 013809 (2010).
[Crossref]

Fan, C. H.

Fattal, D.

D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4, 466–470 (2010).
[Crossref]

Feng, J.

Feng, J. L.

Fiorentino, M.

D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4, 466–470 (2010).
[Crossref]

Fleischer, J. W.

J. B. Kim, P. Kim, N. C. Pégard, S. J. Oh, C. R. Kagan, J. W. Fleischer, H. A. Stone, and Y.-L. Loo, “Wrinkles and deep folds as photonic structures in photovoltaics,” Nat. Photonics 6, 327–332 (2012).
[Crossref]

Gao, F.

Gao, J. Y.

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J. Y. Gao, “Electromagnetically induced grating via enhanced nonlinear modulation by spontaneously generated coherence,” Phys. Rev. A 83, 033824 (2011).
[Crossref]

Gaylord, T. K.

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[Crossref]

Gomard, G.

Gong, S.

L. Wang, F. Zhou, P. Hu, Y. Niu, and S. Gong, “Two-dimensional electromagnetically induced cross-grating in a four-level tripod-type atomic system,” J. Phys. B: At. Mol. Opt. Phys. 47, 225501 (2014).
[Crossref]

Guerin, W.

A. Schilke, C. Zimmermann, and W. Guerin, “Photonic properties of one-dimensionally-ordered cold atomic vapors under conditions of electromagnetically induced transparency,” Phys. Rev. A 86, 023809 (2012).
[Crossref]

He, C.

X. Y. Zhu, Y. L. Xu, Y. Zou, X. C. Sun, C. He, M. H. Lu, X. P. Liu, and Y. F. Chen, “Asymmetric diffraction based on a passive parity-time grating,” Appl. Phys. Lett. 109, 111101 (2016).
[Crossref]

Hölscher, H.

Hu, P.

L. Wang, F. Zhou, P. Hu, Y. Niu, and S. Gong, “Two-dimensional electromagnetically induced cross-grating in a four-level tripod-type atomic system,” J. Phys. B: At. Mol. Opt. Phys. 47, 225501 (2014).
[Crossref]

Hünig, R.

Imoto, N.

M. Mitsunaga and N. Imoto, “Observation of an electromagnetically induced grating in cold sodium atoms,” Phys. Rev. A 59, 4773–4776 (1999).
[Crossref]

Jeong, Y.-C.

Jiang, L.

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J. Y. Gao, “Electromagnetically induced grating via enhanced nonlinear modulation by spontaneously generated coherence,” Phys. Rev. A 83, 033824 (2011).
[Crossref]

Jiang, Y.

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J. Y. Gao, “Electromagnetically induced grating via enhanced nonlinear modulation by spontaneously generated coherence,” Phys. Rev. A 83, 033824 (2011).
[Crossref]

Jin, C. S.

S. Q. Kuang, C. S. Jin, and C. Li, “Gain-phase grating based on spatial modulation of active raman gain in cold atoms,” Phys. Rev. A 84, 033831 (2011).
[Crossref]

Jones, H.

Kagan, C. R.

J. B. Kim, P. Kim, N. C. Pégard, S. J. Oh, C. R. Kagan, J. W. Fleischer, H. A. Stone, and Y.-L. Loo, “Wrinkles and deep folds as photonic structures in photovoltaics,” Nat. Photonics 6, 327–332 (2012).
[Crossref]

Kim, J. B.

J. B. Kim, P. Kim, N. C. Pégard, S. J. Oh, C. R. Kagan, J. W. Fleischer, H. A. Stone, and Y.-L. Loo, “Wrinkles and deep folds as photonic structures in photovoltaics,” Nat. Photonics 6, 327–332 (2012).
[Crossref]

Kim, P.

J. B. Kim, P. Kim, N. C. Pégard, S. J. Oh, C. R. Kagan, J. W. Fleischer, H. A. Stone, and Y.-L. Loo, “Wrinkles and deep folds as photonic structures in photovoltaics,” Nat. Photonics 6, 327–332 (2012).
[Crossref]

Kim, Y.-H.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” The Bell Syst. Tech. J. 48, 2909–2947 (1969).
[Crossref]

Kou, J.

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J. Y. Gao, “Electromagnetically induced grating via enhanced nonlinear modulation by spontaneously generated coherence,” Phys. Rev. A 83, 033824 (2011).
[Crossref]

Kress, B.

Kuang, S. Q.

S. Q. Kuang, C. S. Jin, and C. Li, “Gain-phase grating based on spatial modulation of active raman gain in cold atoms,” Phys. Rev. A 84, 033831 (2011).
[Crossref]

Kubota, T.

T. Kubota, “Characteristics of thick hologram grating recorded in absorptive medium,” Opt. Acta: Int. J. Opt. 25, 1035–1053 (1978).
[Crossref]

Kulishov, M.

Lemmer, U.

Lerotic, M.

P. Licinio, M. Lerotic, and M. Dantas, “Diffraction by disordered gratings and the Debye–Waller effect,” Am. J. Phys. 67, 1013–1016 (1999).
[Crossref]

Lewenstein, M.

M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56, 243–379 (2007).
[Crossref]

Li, C.

S. Q. Kuang, C. S. Jin, and C. Li, “Gain-phase grating based on spatial modulation of active raman gain in cold atoms,” Phys. Rev. A 84, 033831 (2011).
[Crossref]

Li, J.

D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4, 466–470 (2010).
[Crossref]

Li, L.

T. Shui, W. X. Yang, L. Li, and X. Wang, “Lop-sided Raman-Nath diffraction in PT-antisymmetric atomic lattices,” Opt. Lett. 44, 2089–2092 (2019).
[Crossref] [PubMed]

T. Shui, W. X. Yang, S. P. Liu, L. Li, and Z. H. Zhu, “Asymmetric diffraction by atomic gratings with optical PT symmetry in the Raman-Nath regime,” Phys. Rev. A 97, 033819 (2018).
[Crossref]

Li, Y.-Q.

H. Y. Ling, Y.-Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57, 1334–1338 (1998).
[Crossref]

Licinio, P.

P. Licinio, M. Lerotic, and M. Dantas, “Diffraction by disordered gratings and the Debye–Waller effect,” Am. J. Phys. 67, 1013–1016 (1999).
[Crossref]

Liew, S. F.

S. F. Liew and H. Cao, “Optical properties of 1D photonic crystals with correlated and uncorrelated disorder,” J. Opt. 12, 024011 (2010).
[Crossref]

Lim, T.-B.

Limonov, M. F.

A. D. Sinelnik, M. V. Rybin, S. Y. Lukashenko, M. F. Limonov, and K. B. Samusev, “Evolution of optical diffraction patterns on disordered woodpile photonic structures,” Phys. Solid State 60, 1387–1393 (2018).
[Crossref]

Ling, H. Y.

H. Y. Ling, Y.-Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57, 1334–1338 (1998).
[Crossref]

Liu, S. P.

T. Shui, W. X. Yang, S. P. Liu, L. Li, and Z. H. Zhu, “Asymmetric diffraction by atomic gratings with optical PT symmetry in the Raman-Nath regime,” Phys. Rev. A 97, 033819 (2018).
[Crossref]

Liu, X. P.

X. Y. Zhu, Y. L. Xu, Y. Zou, X. C. Sun, C. He, M. H. Lu, X. P. Liu, and Y. F. Chen, “Asymmetric diffraction based on a passive parity-time grating,” Appl. Phys. Lett. 109, 111101 (2016).
[Crossref]

Liu, Y. M.

Loewen, E. G.

C. A. Palmer and E. G. Loewen, Diffraction Grating Handbook (Newport Corporation, 2005).

Loo, Y.-L.

J. B. Kim, P. Kim, N. C. Pégard, S. J. Oh, C. R. Kagan, J. W. Fleischer, H. A. Stone, and Y.-L. Loo, “Wrinkles and deep folds as photonic structures in photovoltaics,” Nat. Photonics 6, 327–332 (2012).
[Crossref]

Lu, H. Y.

Lu, M. H.

X. Y. Zhu, Y. L. Xu, Y. Zou, X. C. Sun, C. He, M. H. Lu, X. P. Liu, and Y. F. Chen, “Asymmetric diffraction based on a passive parity-time grating,” Appl. Phys. Lett. 109, 111101 (2016).
[Crossref]

Lukashenko, S. Y.

A. D. Sinelnik, M. V. Rybin, S. Y. Lukashenko, M. F. Limonov, and K. B. Samusev, “Evolution of optical diffraction patterns on disordered woodpile photonic structures,” Phys. Solid State 60, 1387–1393 (2018).
[Crossref]

Mantsyzov, B.

V. Bushuev, L. Dergacheva, and B. Mantsyzov, “Asymmetric pendulum effect and transparency change of PT-symmetric photonic crystals under dynamical bragg diffraction beyond the paraxial approximation,” Phys. Rev. A 95, 033843 (2017).
[Crossref]

Mitsunaga, M.

M. Mitsunaga and N. Imoto, “Observation of an electromagnetically induced grating in cold sodium atoms,” Phys. Rev. A 59, 4773–4776 (1999).
[Crossref]

Modugno, G.

G. Modugno, “Anderson localization in Bose–Einstein condensates,” Reports on Prog. Phys. 73, 102401 (2010).
[Crossref]

Moharam, M. G.

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[Crossref]

Niu, Y.

L. Wang, F. Zhou, P. Hu, Y. Niu, and S. Gong, “Two-dimensional electromagnetically induced cross-grating in a four-level tripod-type atomic system,” J. Phys. B: At. Mol. Opt. Phys. 47, 225501 (2014).
[Crossref]

Oberthaler, M. K.

M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Atom waves in crystals of light,” Phys. Rev. Lett. 77, 4980–4983 (1996).
[Crossref] [PubMed]

Oh, S. J.

J. B. Kim, P. Kim, N. C. Pégard, S. J. Oh, C. R. Kagan, J. W. Fleischer, H. A. Stone, and Y.-L. Loo, “Wrinkles and deep folds as photonic structures in photovoltaics,” Nat. Photonics 6, 327–332 (2012).
[Crossref]

Paetzold, U. W.

Palmer, C. A.

C. A. Palmer and E. G. Loewen, Diffraction Grating Handbook (Newport Corporation, 2005).

Pégard, N. C.

J. B. Kim, P. Kim, N. C. Pégard, S. J. Oh, C. R. Kagan, J. W. Fleischer, H. A. Stone, and Y.-L. Loo, “Wrinkles and deep folds as photonic structures in photovoltaics,” Nat. Photonics 6, 327–332 (2012).
[Crossref]

Peng, Z.

D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4, 466–470 (2010).
[Crossref]

Preist, T.

M. B. Sobnack, W. Tan, N. Wanstall, T. Preist, and J. Sambles, “Stationary surface plasmons on a zero-order metal grating,” Phys. Rev. Lett. 80, 5667–5670 (1998).
[Crossref]

Rocca, G. C. La

J. H. Wu, M. Artoni, and G. C. La Rocca, “Perfect absorption and no reflection in disordered photonic crystals,” Phys. Rev. A 95, 053862 (2017).
[Crossref]

J. H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[Crossref] [PubMed]

Rybin, M. V.

A. D. Sinelnik, M. V. Rybin, S. Y. Lukashenko, M. F. Limonov, and K. B. Samusev, “Evolution of optical diffraction patterns on disordered woodpile photonic structures,” Phys. Solid State 60, 1387–1393 (2018).
[Crossref]

Sambles, J.

M. B. Sobnack, W. Tan, N. Wanstall, T. Preist, and J. Sambles, “Stationary surface plasmons on a zero-order metal grating,” Phys. Rev. Lett. 80, 5667–5670 (1998).
[Crossref]

Samusev, K. B.

A. D. Sinelnik, M. V. Rybin, S. Y. Lukashenko, M. F. Limonov, and K. B. Samusev, “Evolution of optical diffraction patterns on disordered woodpile photonic structures,” Phys. Solid State 60, 1387–1393 (2018).
[Crossref]

Sanpera, A.

M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56, 243–379 (2007).
[Crossref]

Schauer, S.

Schilke, A.

A. Schilke, C. Zimmermann, and W. Guerin, “Photonic properties of one-dimensionally-ordered cold atomic vapors under conditions of electromagnetically induced transparency,” Phys. Rev. A 86, 023809 (2012).
[Crossref]

Schmager, R.

Schmiedmayer, J.

M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Atom waves in crystals of light,” Phys. Rev. Lett. 77, 4980–4983 (1996).
[Crossref] [PubMed]

Sen, A.

M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56, 243–379 (2007).
[Crossref]

Sen, U.

M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56, 243–379 (2007).
[Crossref]

Shu, S. L.

Shui, T.

T. Shui, W. X. Yang, L. Li, and X. Wang, “Lop-sided Raman-Nath diffraction in PT-antisymmetric atomic lattices,” Opt. Lett. 44, 2089–2092 (2019).
[Crossref] [PubMed]

T. Shui, W. X. Yang, S. P. Liu, L. Li, and Z. H. Zhu, “Asymmetric diffraction by atomic gratings with optical PT symmetry in the Raman-Nath regime,” Phys. Rev. A 97, 033819 (2018).
[Crossref]

Sinelnik, A. D.

A. D. Sinelnik, M. V. Rybin, S. Y. Lukashenko, M. F. Limonov, and K. B. Samusev, “Evolution of optical diffraction patterns on disordered woodpile photonic structures,” Phys. Solid State 60, 1387–1393 (2018).
[Crossref]

Sobnack, M. B.

M. B. Sobnack, W. Tan, N. Wanstall, T. Preist, and J. Sambles, “Stationary surface plasmons on a zero-order metal grating,” Phys. Rev. Lett. 80, 5667–5670 (1998).
[Crossref]

Stone, H. A.

J. B. Kim, P. Kim, N. C. Pégard, S. J. Oh, C. R. Kagan, J. W. Fleischer, H. A. Stone, and Y.-L. Loo, “Wrinkles and deep folds as photonic structures in photovoltaics,” Nat. Photonics 6, 327–332 (2012).
[Crossref]

Sun, X. C.

X. Y. Zhu, Y. L. Xu, Y. Zou, X. C. Sun, C. He, M. H. Lu, X. P. Liu, and Y. F. Chen, “Asymmetric diffraction based on a passive parity-time grating,” Appl. Phys. Lett. 109, 111101 (2016).
[Crossref]

Tan, W.

M. B. Sobnack, W. Tan, N. Wanstall, T. Preist, and J. Sambles, “Stationary surface plasmons on a zero-order metal grating,” Phys. Rev. Lett. 80, 5667–5670 (1998).
[Crossref]

Tian, S. C.

Tong, C. Z.

Vidil, P.

P. Vidil and B. Chalopin, “Controllable blazed grating for electrons using Kapitza-Dirac diffraction with multiple-harmonic standing waves,” Phys. Rev. A 92, 062117 (2015).
[Crossref]

Wan, R. G.

Wan, R.-G.

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J. Y. Gao, “Electromagnetically induced grating via enhanced nonlinear modulation by spontaneously generated coherence,” Phys. Rev. A 83, 033824 (2011).
[Crossref]

Wang, B.

Wang, L.

L. Wang, F. Zhou, P. Hu, Y. Niu, and S. Gong, “Two-dimensional electromagnetically induced cross-grating in a four-level tripod-type atomic system,” J. Phys. B: At. Mol. Opt. Phys. 47, 225501 (2014).
[Crossref]

Wang, L. J.

Wang, S.

Wang, X.

Wanstall, N.

M. B. Sobnack, W. Tan, N. Wanstall, T. Preist, and J. Sambles, “Stationary surface plasmons on a zero-order metal grating,” Phys. Rev. Lett. 80, 5667–5670 (1998).
[Crossref]

Witmer, R. B.

R. B. Witmer and J. M. Cork, “The measurement of x-ray emission wave-lengths by means of the ruled grating,” Phys. Rev. 42, 743–748 (1932).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
[Crossref]

Worgull, M.

Wu, J. H.

J. H. Wu, M. Artoni, and G. C. La Rocca, “Perfect absorption and no reflection in disordered photonic crystals,” Phys. Rev. A 95, 053862 (2017).
[Crossref]

Y. M. Liu, F. Gao, C. H. Fan, and J. H. Wu, “Asymmetric light diffraction of an atomic grating with PT symmetry,” Opt. Lett. 42, 4283–4286 (2017).
[Crossref] [PubMed]

J. H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[Crossref] [PubMed]

Xiao, M.

Xu, Y. L.

X. Y. Zhu, Y. L. Xu, Y. Zou, X. C. Sun, C. He, M. H. Lu, X. P. Liu, and Y. F. Chen, “Asymmetric diffraction based on a passive parity-time grating,” Appl. Phys. Lett. 109, 111101 (2016).
[Crossref]

Yang, W. X.

T. Shui, W. X. Yang, L. Li, and X. Wang, “Lop-sided Raman-Nath diffraction in PT-antisymmetric atomic lattices,” Opt. Lett. 44, 2089–2092 (2019).
[Crossref] [PubMed]

T. Shui, W. X. Yang, S. P. Liu, L. Li, and Z. H. Zhu, “Asymmetric diffraction by atomic gratings with optical PT symmetry in the Raman-Nath regime,” Phys. Rev. A 97, 033819 (2018).
[Crossref]

Yelin, S.

L. Zhao, W. Duan, and S. Yelin, “Generation of tunable-volume transmission-holographic gratings at low light levels,” Phys. Rev. A 84, 033806 (2011).
[Crossref]

L. Zhao, W. Duan, and S. Yelin, “All-optical beam control with high speed using image-induced blazed gratings in coherent media,” Phys. Rev. A 82, 013809 (2010).
[Crossref]

Zeilinger, A.

M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Atom waves in crystals of light,” Phys. Rev. Lett. 77, 4980–4983 (1996).
[Crossref] [PubMed]

Zhang, X.

Zhao, L.

L. Zhao, W. Duan, and S. Yelin, “Generation of tunable-volume transmission-holographic gratings at low light levels,” Phys. Rev. A 84, 033806 (2011).
[Crossref]

L. Zhao, W. Duan, and S. Yelin, “All-optical beam control with high speed using image-induced blazed gratings in coherent media,” Phys. Rev. A 82, 013809 (2010).
[Crossref]

Zhou, C.

Zhou, F.

L. Wang, F. Zhou, P. Hu, Y. Niu, and S. Gong, “Two-dimensional electromagnetically induced cross-grating in a four-level tripod-type atomic system,” J. Phys. B: At. Mol. Opt. Phys. 47, 225501 (2014).
[Crossref]

Zhu, X. Y.

X. Y. Zhu, Y. L. Xu, Y. Zou, X. C. Sun, C. He, M. H. Lu, X. P. Liu, and Y. F. Chen, “Asymmetric diffraction based on a passive parity-time grating,” Appl. Phys. Lett. 109, 111101 (2016).
[Crossref]

Zhu, Z. H.

T. Shui, W. X. Yang, S. P. Liu, L. Li, and Z. H. Zhu, “Asymmetric diffraction by atomic gratings with optical PT symmetry in the Raman-Nath regime,” Phys. Rev. A 97, 033819 (2018).
[Crossref]

Zimmermann, C.

A. Schilke, C. Zimmermann, and W. Guerin, “Photonic properties of one-dimensionally-ordered cold atomic vapors under conditions of electromagnetically induced transparency,” Phys. Rev. A 86, 023809 (2012).
[Crossref]

Zou, Y.

X. Y. Zhu, Y. L. Xu, Y. Zou, X. C. Sun, C. He, M. H. Lu, X. P. Liu, and Y. F. Chen, “Asymmetric diffraction based on a passive parity-time grating,” Appl. Phys. Lett. 109, 111101 (2016).
[Crossref]

Adv. Phys. (1)

M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56, 243–379 (2007).
[Crossref]

Am. J. Phys. (1)

P. Licinio, M. Lerotic, and M. Dantas, “Diffraction by disordered gratings and the Debye–Waller effect,” Am. J. Phys. 67, 1013–1016 (1999).
[Crossref]

Appl. Phys. Lett. (1)

X. Y. Zhu, Y. L. Xu, Y. Zou, X. C. Sun, C. He, M. H. Lu, X. P. Liu, and Y. F. Chen, “Asymmetric diffraction based on a passive parity-time grating,” Appl. Phys. Lett. 109, 111101 (2016).
[Crossref]

J. Opt. (1)

S. F. Liew and H. Cao, “Optical properties of 1D photonic crystals with correlated and uncorrelated disorder,” J. Opt. 12, 024011 (2010).
[Crossref]

J. Phys. A: Math. Gen. (1)

M. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. A: Math. Gen. 31, 3493–3502 (1998).
[Crossref]

J. Phys. B: At. Mol. Opt. Phys. (1)

L. Wang, F. Zhou, P. Hu, Y. Niu, and S. Gong, “Two-dimensional electromagnetically induced cross-grating in a four-level tripod-type atomic system,” J. Phys. B: At. Mol. Opt. Phys. 47, 225501 (2014).
[Crossref]

Nat. Photonics (2)

D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4, 466–470 (2010).
[Crossref]

J. B. Kim, P. Kim, N. C. Pégard, S. J. Oh, C. R. Kagan, J. W. Fleischer, H. A. Stone, and Y.-L. Loo, “Wrinkles and deep folds as photonic structures in photovoltaics,” Nat. Photonics 6, 327–332 (2012).
[Crossref]

Opt. Acta: Int. J. Opt. (1)

T. Kubota, “Characteristics of thick hologram grating recorded in absorptive medium,” Opt. Acta: Int. J. Opt. 25, 1035–1053 (1978).
[Crossref]

Opt. Express (5)

Opt. Lett. (6)

Opt. Mater. Express (1)

Phys. Rev. (1)

R. B. Witmer and J. M. Cork, “The measurement of x-ray emission wave-lengths by means of the ruled grating,” Phys. Rev. 42, 743–748 (1932).
[Crossref]

Phys. Rev. A (11)

S. Q. Kuang, C. S. Jin, and C. Li, “Gain-phase grating based on spatial modulation of active raman gain in cold atoms,” Phys. Rev. A 84, 033831 (2011).
[Crossref]

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J. Y. Gao, “Electromagnetically induced grating via enhanced nonlinear modulation by spontaneously generated coherence,” Phys. Rev. A 83, 033824 (2011).
[Crossref]

L. Zhao, W. Duan, and S. Yelin, “Generation of tunable-volume transmission-holographic gratings at low light levels,” Phys. Rev. A 84, 033806 (2011).
[Crossref]

L. Zhao, W. Duan, and S. Yelin, “All-optical beam control with high speed using image-induced blazed gratings in coherent media,” Phys. Rev. A 82, 013809 (2010).
[Crossref]

M. Mitsunaga and N. Imoto, “Observation of an electromagnetically induced grating in cold sodium atoms,” Phys. Rev. A 59, 4773–4776 (1999).
[Crossref]

J. H. Wu, M. Artoni, and G. C. La Rocca, “Perfect absorption and no reflection in disordered photonic crystals,” Phys. Rev. A 95, 053862 (2017).
[Crossref]

A. Schilke, C. Zimmermann, and W. Guerin, “Photonic properties of one-dimensionally-ordered cold atomic vapors under conditions of electromagnetically induced transparency,” Phys. Rev. A 86, 023809 (2012).
[Crossref]

H. Y. Ling, Y.-Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57, 1334–1338 (1998).
[Crossref]

T. Shui, W. X. Yang, S. P. Liu, L. Li, and Z. H. Zhu, “Asymmetric diffraction by atomic gratings with optical PT symmetry in the Raman-Nath regime,” Phys. Rev. A 97, 033819 (2018).
[Crossref]

V. Bushuev, L. Dergacheva, and B. Mantsyzov, “Asymmetric pendulum effect and transparency change of PT-symmetric photonic crystals under dynamical bragg diffraction beyond the paraxial approximation,” Phys. Rev. A 95, 033843 (2017).
[Crossref]

P. Vidil and B. Chalopin, “Controllable blazed grating for electrons using Kapitza-Dirac diffraction with multiple-harmonic standing waves,” Phys. Rev. A 92, 062117 (2015).
[Crossref]

Phys. Rev. Lett. (3)

M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, “Atom waves in crystals of light,” Phys. Rev. Lett. 77, 4980–4983 (1996).
[Crossref] [PubMed]

J. H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[Crossref] [PubMed]

M. B. Sobnack, W. Tan, N. Wanstall, T. Preist, and J. Sambles, “Stationary surface plasmons on a zero-order metal grating,” Phys. Rev. Lett. 80, 5667–5670 (1998).
[Crossref]

Phys. Solid State (1)

A. D. Sinelnik, M. V. Rybin, S. Y. Lukashenko, M. F. Limonov, and K. B. Samusev, “Evolution of optical diffraction patterns on disordered woodpile photonic structures,” Phys. Solid State 60, 1387–1393 (2018).
[Crossref]

Proc. IEEE (1)

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[Crossref]

Reports on Prog. Phys. (1)

G. Modugno, “Anderson localization in Bose–Einstein condensates,” Reports on Prog. Phys. 73, 102401 (2010).
[Crossref]

Rev. Mod. Phys. (1)

H. Batelaan, “Colloquium: Illuminating the Kapitza-Dirac effect with electron matter optics,” Rev. Mod. Phys. 79, 929–941 (2007).
[Crossref]

The Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” The Bell Syst. Tech. J. 48, 2909–2947 (1969).
[Crossref]

Other (2)

C. A. Palmer and E. G. Loewen, Diffraction Grating Handbook (Newport Corporation, 2005).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 (a) Sketch of 1D atomic grating, which consists of 1D optical lattices of driven cold 87Rb atoms. The area shown in purple is the spatial distribution of the atomic density. (b) Schematic of diagram of a four-level N-type atomic system interacting with three applied laser fields.
Fig. 2
Fig. 2 The Raman-Nath diffraction intensities as a function of sin θ for (a) φc = 0, Δ d = 0MHz and (b) φc = 0.15π, Δ d = 2MHz. Other parameters are N0 = 5 ×1012cm−3, µ31 = 2.5377 ×10−29C m, Ω c 0 = 6MHz, Ω c 1 = 1.53MHz, Ω d = 2MHz, Δ p = 7MHz, Δ c = 0MHz, λp = 795.5nm, Λ/λp = 4, d = 0.2Λ, M = 11 and L = 30µm.
Fig. 3
Fig. 3 Two examples of 20 random configurations of (a) uncorrelated geometric disorder [Eq. (2)] and (b) correlated geometric disorder [Eq. (3)]. The dependence of Raman-Nath diffraction spectra on the (c) uncorrelated and (d) correlated geometric disorders. Each curve represents a diffraction profile induced by a random configuration of geometric disorder. The spatial correlation functions CRx) and CIx) for the real part χ(x) and imaginary part χ(x) of the spatial susceptibility with the (e) uncorrelated and (f) correlated geometric disorders. The strength of geometric disorder Δ g = 0.07 and other parameters are the same as in Fig. 2(a).
Fig. 4
Fig. 4 The average Raman-Nath diffraction spectra 〈Ip(θ)〉 of the atomic grating for different strengths of (a) uncorrelated and (b) correlated geometric disorders. The corresponding evolutions of the average Raman-Nath diffraction 〈Ip(θ)〉 with the strengths of (c)uncorrelated and (d) correlated disorders. The average diffraction spectra are attained through averaging over 50 different random configurations of disorder (NR = 50). Other parameters are the same as in Fig. 2(a).
Fig. 5
Fig. 5 The evolutions of the average Raman-Nath diffraction 〈Ip(θ)〉 with the strengths Δ g of (a) uncorrelated (b) correlated geometric disorders. Other parameters are the same as in Fig. 2(b).
Fig. 6
Fig. 6 The average Raman-Nath diffraction spectra 〈Ip(θ)〉 of the atomic grating for different strengths of (a) uncorrelated and (b) correlated structural disorders. The corresponding evolutions of the average Raman-Nath diffraction 〈Ip(θ)〉 with the strengths of (c)uncorrelated and (d) correlated disorders. Other parameters are the same as in Fig. 2(a).
Fig. 7
Fig. 7 The evolutions of the average Raman-Nath diffraction 〈Ip(θ)〉 with the strengths Δ s of (a) uncorrelated (b) correlated structural disorders. Other parameters are the same as in Fig. 2(b).
Fig. 8
Fig. 8 The evolutions of the average Raman-Nath diffraction 〈Ip(θ)〉 with the strengths Δ N and ΔΩ of uncorrelated disorders in (a) the average atomic density N0 and (b) the standing-wave Rabi frequency Ω c 1. Other parameters are the same as in Fig. 2(a).

Equations (14)

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N j ( x ) = N 0 2 π d exp [ ( x x j ) 2 d j 2 ] ,
x j = j Λ + δ x j = j Λ + Λ ζ j ,
x j ( c ) = j Λ + i = 1 j δ x i = j Λ + Λ i = 1 j ζ i ,
d j = d + δ d j = d + d η j ,
d j ( c ) = d + i = 1 j δ d i = d + d i = 1 j η i .
Ω c ( x ) = Ω c 0 + Ω c 1 sin ( 2 π x Λ φ c ) ,
H I = ( Δ c 0 0 Ω c ( x ) 0 Δ p Ω d Ω p 0 Ω d * ( Δ p Δ d ) 0 Ω c * ( x ) Ω p * 0 0 ) ,
d ρ d t = i [ H I , ρ ] + L [ ρ ( t ) ] .
L [ ρ ( t ) ] = ( σ 44 γ 43 ρ 43 γ 42 ρ 42 γ 41 ρ 41 γ 43 ρ 34 σ 33 γ 32 ρ 32 γ 31 ρ 31 γ 42 ρ 24 γ 32 ρ 23 σ 22 γ 21 ρ 21 γ 41 ρ 14 γ 31 ρ 13 γ 21 ρ 12 σ 11 ) ,
χ j ( x ) = N j ( x ) | μ 31 | 2 ε 0 Ω p ρ 31 = N j ( x ) | μ 31 | 2 ε 0 [ i κ 3 ( κ 4 + κ 5 ) Ω c 2 ( x ) ] Ω d 2 κ 1 κ 2 ,
T ( x ) = { e β I ( x ) L e i β R ( x ) L , if x [ Λ / 2 , ( 2 M + 1 ) Λ / 2 ] 0 , otherwise ,
E p ( θ ) = C + E 0 T ( x ) e i 2 π x sin θ / λ p d x ,
I p ( θ ) = 1 ( M Λ ) 2 | + T ( x ) e i 2 π x sin θ / λ p d x | 2 .
I p ( θ ) = 1 N R m = 1 N R I p ( θ , m ) ,

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