Effect of composite vortex beam on a two-dimensional gain assisted atomic grating

We propose an atomic grating based on an electromagnetically induced transparency phenomenon that switches between zeroth-order diffraction to a distinct higher-order diffraction pattern by driving a planar gaseous medium of a four-level tripod ( ⋔ ) atoms with three laser beams: modulation of standing wave control beam propagating nearly perpendicular to the planar medium, while vortex and weak plane probe beams directed perpendicular to the medium. We numerically investigate the behavior of the amplitude, phase modulations, and probe field diffraction intensities of different orders by the variation of the field detunings and orbital angular momentum number of the composite vortex light beam. Specifically, in the off-resonant case, the interplay between a square lattice of the control and an additional spatial variation of the vortex beam allows the emergence of higher diffraction orders and variable gain due to double transparency windows in this complex optical system. We believe that our proposed scheme might be useful in optical memory devices via the storage of information to diffraction orders of the atomic grating.


Introduction
The topic of an electromagnetically induced grating (EIG) [1] is intriguing as a probe beam can thereby diffract into high-orders by employing a standing-wave (SW) control field, to construct a spatial absorption (amplitude) and dispersion (phase) grating in a medium, which provides tunability advantages over conventional optical gratings. EIG offers a wide range of applications, for example, the grating created by an optically induced lattice may change the structure of photonic band gaps [2,3], and also produce the electromagnetically induced Talbot effect [4], which is extremely beneficial for imaging of two-dimensional (2D) cold atoms. Undoubtedly, controllable diffraction gratings have more potential applications [5][6][7][8].
Our aim is to create EIG through the interaction between an atomic four-level tripod (⋔) atom and the following electromagnetic fields: a probe, a 2D SW control, and a superposition of two composite vortex light beams. Unlike previous EIG schemes, our scheme has the advantage of allowing the probe light beam to diffract to several higher-order directions with amplification and the intensity of the diffracted beam may become greater than the input beam.
The concept of EIG was observed experimentally in hot [6,9] and cold [10,11] atomic systems. The basic Λ EIG scheme has been expanded to other multi-level schemes such as ∨ [12] and Ξ excitations [13]. Phase-controlled EIG can be realized in symmetric and asymmetric double and triple quantum wells and quantum dot systems [14][15][16], with efficiency, was enhanced via microwave modulation [17], and parity-time symmetry [18,19]. Theoretical schemes about EIG with PT symmetry were proposed in different optical systems [20,21]. A number of experimental studies on 1D and 2D EIGs have recently been performed in atomic gas medium [22][23][24].
An optical vortex refers to a beam of photons with non-zero orbital angular momentum (OAM) whose phase changes along the beam's propagation direction in a corkscrew-like fashion that circulates around the core in a helical wavefront [25][26][27]. When interacting with atoms such optical vortex beams reveal a number of interesting effects, including light-induced torque [28,29], entanglement of OAM states of photon pairs [30], spatially dependent optical transparency [31], exchange of OAM modes in multilevel quantum systems [32], atom vortex beams [33] and the vortex slow light [34]. Vortex beams have also found fascinating applications in optical information processing during the storage and retrieval of slow light [35].
Conventional EIG systems lack phase sensitivity, limiting their ability to encode photons for quantum information processing. Vortex beam with OAM has emerged as a powerful toolbox and gives an extra degree of freedom, which makes it a higher-dimensional system for high-capacity information transfer. Recently, the use of special optical beams in an EIG, particularly optical vortices, has drawn increasing interest [36]. Specifically, by adjusting the OAM number of vortex beam, the diffraction intensity distribution of the 2D EIG can be modified in various regions [37], which provides direct control on the grating performance. Moreover in a conventional EIG system number of grating lines is fixed, while in our proposed setup we can increase or decrease the number of grating by increasing or decreasing the control SW field strength and with the addition of control over the transfer of energy to diffraction orders of grating by tuning the OAM of structure light without changing the medium properties.
Inspired by the ability of electromagnetic OAM to give extra degrees of freedom to numerous control schemes, we present a theoretical framework for realizing controlled symmetric 2D EIG. Analytical and numerical analyses show that depending on whether the probe, SW control, and vortex light fields are resonant or off-resonant, the amplitude, phase modulations, and Fraunhofer diffraction intensity profile display very different behavior. Furthermore, by simply changing the OAM of the optical vortex light, the resulting 2D EIG takes on symmetric nature, shifting the probe energy from zeroth to higher-order directions. In particular, our proposed system introduced double EIT (DEIT) [38,39] effect with the gain occurs at the position of the second transparency window, which is responsible for 2D gain phase grating [40], with amplification and such exclusivity may provide a substantial advantage over prior works [5,16,[41][42][43].
In this paper, we consider the ⋔ atomic system with one upper state and three lower states. We use three simultaneous fields: a traveling wave probe beam, a spatially dependent control beam, and a composite vortex field. We use the semi-classical model to investigate an atomic four-level EIG scheme that interacts with a probe, SW control, and vortex light fields. In order to see the modulation of OAM of structure light beam we consider two scenarios in our system. First, we consider an all-fields resonance scenario that leads to a single EIT window. Then we look at the scenario where the SW control field is resonant, while the probe and vortex fields are off-resonant, which leads to DEIT windows with the amplification in the second EIT window. We see the effects of OAM l on 2D EIG diffraction orders in the amplification window. Furthermore, we consider the case where we see the effect of interaction length on intensities of 2D EIG. Finally, we discuss the effect of radial diffraction intensity distributions on intensities of 2D EIG. Our analysis reveals that, in the first scenario, we can only get an amplitude grating. However, both amplitude and phase grating are feasible in the second scenario. In the third and fourth scenarios, the strength of 2D EIG diffraction relies on interaction length and radial distance.
The important feature of our suggested scheme is that we offer a different, intriguing, and straightforward way to modify the phase grating process using vortex light in a ⋔ configuration with amplification. We can control the probe field energy in different-order diffractions by simply altering the vortex light OAM number and azimuthal angle, which provides us a better understanding of the possibilities for excellent EIG control. Because of its simple controllability, we anticipate that the suggested 2D EIG model will be experimentally achievable in a typical EIG setup [10,11,44], with potential applications in EIT-based quantum devices.
Our research is structured as follows. In section 2, we present the background of our system. In section 3, we present our model of ⋔ atomic system for 2D EIG. In section 4 we present our mathematics. In section 5 we present our results and discuss these results in section 6. Finally, in section 7 we present our conclusions.

Background
In this section, we briefly explain the background for our work on 2D EIG via vortex light, which is needed for the further development of our approach and results. More details related to the background of EIG and DEIG can be found in section appendix. , F = 1, m = 1⟩) same as in [45]. Transitions are controlled by the probe, control, and vortex fields. The Ωp, Ωc and Ωv shows the Rabi frequencies, whereas ∆p, ∆c and ∆v are the detunings of the respective fields. The γ j for j ∈ {2, 3, 4} represents the decay rates of the level |j⟩. (b) Schematic of the diffraction grating when a probe field diffracts through an SW (like slits). Where L shows the interaction length, θ ∈ {θx, θy} is the diffraction angle, and Λ ∈ {Λx, Λy} is the spatial period.
Various models consisting of multi-level systems have been used to examine the optical characteristics of EIGs [43]. Here, we briefly discuss about the ⋔ atomic system. Specifically, we consider hyperfine four-level ultra-cold 87 Rb D1 line [45][46][47][48][49] The detailed schematic of the ⋔ system is described in figure 1(a). A ⋔ system involves three ground states coupled to an excited state by probe, control, and vortex beams respectively. The weak probe beam is applied for the transition The control and vortex beams with Rabi frequencies Ω c and Ω v coupled with atomic transitions and respectively. The parameters ∆ p , ∆ c and ∆ v are the detunings of the probe, control, and vortex fields respective. The γ j for j ∈ {2, 3, 4} represents the decay rates of the level |j⟩.
The semi-classical Hamiltonian of the ⋔ type atomic system by considering the rotating-wave and dipole approximation is [50] in which H.c indicates the Hermitian conjugate, where ∆ xy := ∆ x − ∆ y are the two-photon detuning. The ⋔ atomic system response to the applied fields can be determined by the time evolution quantum master equation, which is [50] ρ : Equation (6) comprises both spontaneous emissions as well as dephasing. Here γ j i shows the decay rate between the state |j⟩ → |i⟩ and γ ϕ j is the dephasing of the state |j⟩. The complete solution of density matrix ρ under steady-state conditions can be found in [51,52].

Model
In order to quantify the effect of composite vortex beam on 2D EIG, we consider experimentally realizable parametric values in our numerical simulations. For the 2D EIG system, our measurements were done on the D1 line of ultra-cold 87 Rb (wavelength 795 nm) atoms in a vapor cell magneto-optical trap (MOT), which is the same as in [45,53,54] to construct the ⋔ type structure. The atoms with a temperature of T ∼ 2.8 µK [54], atomic number density N ∼ 1.6 × 10 13 cm −3 [54] and having Gaussian distribution are populated between states |1⟩, and |3⟩, i.e. ρ 11 ≈ ρ 33 ≈ 0.5 [46,52]. The schematic of the possible experimental realization scheme is displayed in figure 2.
The weak probe laser beam having σ − polarization is generated such that its direction is directed perpendicular to the atomic medium (z-direction) to produce the 2D EIG. The appropriate vortex beams Ω v are created by passing a Gaussian plane laser beam through a spatial light modulator with a fork-shaped grating device before interacting with atoms. This composite vortex beam has σ + polarization that varies with respect to the azimuthal angle of the plane and propagates directly perpendicular to the medium along the z direction.
A strong control beam with π transition making a SW field is displaced symmetrically with respect to z, and they are incident on MOT consisting of ⋔ atoms at such an angle that they intersect and create an SW with a space period along the transverse x and y-direction inside the medium perpendicular to propagation direction z [55].

Mathematics
In this section, we use the ⋔ atomic model of section 2 and develop mathematics. First, we determine the atomic-optical susceptibility of the ⋔ system. Next, we consider SW control and vortex beams and use these fields to derive the transmission function equations for probe light. Finally, the Fraunhofer diffraction intensity is calculated by using the transmission function equations for the probe light.

Atomic-optical susceptibility
Let us now start with equation (6) and determine the off-diagonal (ρ 14 ) equation for the probe light in the steady-state regime. Here, we consider constant populations such as ρ 11 ≈ ρ 33 ≈ 0.5 in our suggested model [46,52]. This assumption makes it possible to solve the equations analytically. Also, we assume that Consequently, the off-diagonal term (ρ 14 ) can be computed as where with Γ kl = γ k + γ l . Therefore, the optical response of the probe light is defined as the steady-state linear susceptibility (χ p ), which belongs to ρ 14 , where the monitoring probe field is applied and written as Here, in equation (10) N , µ 14 , ϵ 0 and ℏ represents the atomic number density, dipole moments, permittivity of the free space, and Planck constant respectively. The final probe susceptibility equation is

Control SW and composite vortex field
As is well known, control fields with SW patterns result in spatially varied absorption and dispersion of the probe field due to its intensity-dependent susceptibility. As a response, the atoms serve as gratings, allowing the probe to diffract in various directions.
To do this, we can modify the control field Ω c to a 2D SW Ω c (x, y) in our proposed model, i.e.
with Λ x and Λ y being the spatial period of SW. This 2D control SW is responsible for 2D EIG.
Let us now replace the vortex field Rabi frequency (Ω v = Ω v (r, φ)) with a composite vortex beam and can be defined as which is essentially a combination of different vortex fields. Here, r = x 2 + y 2 is the radius from the core vortex, with l, w and φ are the vorticity, beam waist, and azimuthal angle of the LG field respectively. If we consider that l 1 = −l 2 = l, so one can write as Ω v = Ω s1 cos(lφ), where Ω s1 = 2Ω v1 e −r 2 /w 2 (r/w) |l| . So in equation (11) we clearly see the direct dependence of the OAM number of the composite beam to the susceptibility of the medium. As a result, we can arrive at the conclusion that the vortex field's azimuthal angle has a significant impact in equation (11), which increases the ability to change the EIG pattern on demand.

2D transmission function and Fraunhofer diffraction intensity equations for probe light
Here, if we assume that the control field is position-dependent in 2D (x, y), so according to equation (A2), the probe beam 2D transmission function takes the form, The amplitude of the grating is indicated by the first term in the exponent, while the phase modulations are shown by the second term, which are rewritten as Equation (15) are used to derive the results for amplitude and phase modulation of 2D EIG. The interaction length L is defined in units of ζ = λpγ4 ηp . As in our model subsection, we treat the probe field as a plane wave and we can get the Fraunhofer diffraction intensity equation by using the Fourier transform of the transmission function T(x, y) that is written as It should be noted that our calculations are based on equal values of P and Q (P and Q shows the spatial periods of the atomic grating). Here we havẽ with θ x and θ y represent the diffraction angles with reference to z-direction. The grating constant sin θ x = mλ p/Λx and sin θ y = nλ p/Λy are used to find the (m, n)th order diffraction angles, where m and n shows the order of diffraction angles of the grating. The diffraction intensities of the (m, n)th orders can be obtained as

Results
In this section, we go through our results for 2D EIG via a vortex field by considering four cases i.e. in section 5.1 we present the case, when all fields are resonant. In section 5.2 we present our results when the control field is resonant while probe and vortex fields are off-resonant. In section 5.3 we see the effect of interaction length on 2D EIG intensities. Finally in section 5.4, we discuss our results for radial diffraction intensity distributions for 2D EIG with respect to various azimuthal angle (φ) of the vortex beam.

All fields are resonant
In this subsection, we see the scenario, when all fields are in resonance and see how different system parameters influence the amplitude and phase modulation of the transmission profile of the probe beam. In addition, we also investigate the effect of various system parameters on the diffraction intensity pattern of the probe light beam. As shown in figures 3(a) and (b), we depict the 2D amplitude and phase modulation of the transmitted probe beam versus position x and y over four spatial intervals. We consider the OAM l = 1 for vortex light and within a single space region, i.e. −0.5Λ x ⩽ x ⩽ 0.5Λ x , −0.5Λ y ⩽ y ⩽ 0.5Λ y , the amplitude modulation intensity is maximum with amplitude of 3.5 (see red in color bars). Here we see that for the resonant case, the composite vortex beam effect is very weak enough and DEIT windows merge into a single EIT window. Our ⋔ system is converted to three level EIT system with a plane probe light beam propagating along a SW control field, which forms the grating and we see the transmission spectrum. At every antinode where the SW control field has maximum amplitude, we have maximum transmitted amplitude. However, for nodes where the SW control field has minimum amplitude, we get minimum transmitted amplitude (which is indicated by amplitude 1, see blue in the color bar). The remaining variables are mentioned in the caption of figure 3. From the above discussion we analyze, our scheme at resonance field gives periodic modulation in amplitude transmission which leads to the 2D grating with maximum probe light passing through where SW control field intensity is maximum (antinodes), see figure 3(a).
Also, we would like to mention that phase modulation of the transmission profile is zero at resonance, see figure 3(b) because the real part of the susceptibility vanishes (ϕ(x, y) = β(x, y)L = 0) so no phase modulation takes place. As a result, a phenomenon similar to pure amplitude-type grating is realized. This phenomenon coincides with our expectations.
Next, we see the interference of the probe and vortex light beam passing through the slits which are made by the SW control field and see the diffraction intensity pattern orders in the near field regime called Fraunhofer diffraction versus diffraction angle and azimuthal angle of vortex beam see figures 3(c) and (d).
In figure 3(c) we observe that due to the pure 2D amplitude grating, with zero phase modulation, the entire energy of the probe field predominantly occupies the center maximum, resulting in a minimal absorption to zeroth order of probe beam, while the higher orders of diffraction are completely lost. The corresponding intensity orders of diffraction I(θ m x , θ n y ), with respect to vortex light azimuthal angle are shown in figure 3(d). The figure verifies our prior finding that the entire probe energy is in the zeroth order, with essentially no higher-order diffraction intensity.

Off resonant probe and vortex fields
In this subsection, we describe our results for off-resonant probe and vortex fields while the detuning of strong SW control field is taken as zero. First, we discuss our results for equal probe and vortex field detuning that differs from control field detuning which is taken as zero, and study their impact on 2D EIG. Next, we apply the same field detunings and analyze the influence of vortex beam OAM on the diffraction intensity profile of the 2D EIG.
We consider the scenario where the control field has a high intensity and ∆ c = 0 in order to maintain a high level of transparency all over the beam profile, whereas the probe and vortex fields are off-resonance (∆ p = ∆ v = 1γ 4 ). In this case we have DEIT windows at ∆ p = 0 and ∆ p = 1γ 4 . However, when we switch the vortex beam OAM l = 1, the off-resonant window converts amplification. Thus we have amplification in amplitude and phase modulation that leads to an increase in the performance of 2D EIG. This is due to the fact that the coherence induces in a system due to a structure light beam leads to amplification in phase modulation. When we alter the OAM number (l), the transfer of probe energy to higher-order diffraction occurs. Based on the suitable choice of l, the diffraction of probe light can occur into four distinct regions, i.e. region- We see that phase modulation is zero for all fields to be resonant. In that case, the diffraction efficiency is low. Here, we expect some non-zero phase modulation that can diffract a part of the probe energy into high-order diffractions.
To verify these predictions, in figures 4-6 we show the results of amplitude, phase modulation of transmitted probe light beam versus position x and y, Fraunhofer diffraction pattern and different diffraction intensities orders for various values of OAM number l.
In figures 4(a) and (b) for l = 1, we show the amplitude and phase of transmission function T(x, y). In this scenario, we notice the off-resonant phase modulation. The reason is that at the equal probe and vortex field detuning we observe two EIT windows. The first window appears at equal detunings of all the applied fields, due to the destructive interference between the indirect channels [56]. The second window is shown at the position where the probe and vortex field detunings are the same but distinct from the control-field detuning. The second window is very narrow with amplification, which is due to the Raman gain effect. Mathematically, this gain is understood as coherence in equation (8), which is a combination of two terms: one proportional to ρ 11 − ρ 44 and the other is proportional to ρ 43 [50].
Due to the gain in the second window, the amplification of light takes place, which enhanced the amplitude modulations resulting in a large transmission. And we clearly see in figure 4(a) the enhanced amplitude modulations.
Previously for resonant case, the phase modulation is zero while in figure 4(b) the phase modulation is definitely non-zero because of the higher refractive index in the second window, as a result, we may conclude that amplitude and phase modulation has a considerable influence on the probe field transmission profile.
In figure 4(c) we clearly see the amplified diffraction intensity, which corresponds to the gain in the second transparency window, with the maximum amount of energy gathered in the zeroth order. Furthermore, it can be shown that the insertion of phase modulation transfers probe light beams from zeroth to higher diffraction patterns.
In figure 4(d) we get to the conclusion that as the azimuthal angle changes, the various orders of diffraction are changed. For any value of l, some part of the probe light beam remains non-diffracted and lies in zeroth order I(θ 0 x , θ 0 y ) and some probe beam energy shifted to high-orders. In figures 5(a) and (b) at l = 2 we notice a considerable change in the modulation of amplitude and phase of the transmitted probe light beam. The amplitude modulation is decreased while the phase modulation increases. This is again due to coherence induced in a system by a structure light beam that leads to enhanced phase modulation that is directly related to enhancing the diffraction intensity to higher orders. So the overall effect of the transmission profile on 2D grating increases. The decrease in amplitude of probe field transmission is due to the fact that amplification decreases at the second EIT window at equal probe and vortex field detuning. The vortex field mainly affects the second EIT window, however, it has no influence on the first EIT window. However, we see the phase modulation of the transmission profile is increase with the increase of the OAM number. This is because phase modulation is dependent on the dispersion behavior of the medium's response, which gets steeper as OAM increases, allowing phase transmission to increase.
As the amplitude and phase transmission modulation influence the diffraction intensity pattern. Now we see the effect of the OAM number on the Fraunhofer diffraction intensity pattern. In figures 5(c) and (d) we  figure 5(c) we would like to mention that the OAM of the structure light beam is taken as l = 2 for which we have enhanced amplitude modulation and weak phase modulation at certain detuning (∆ p = ∆ v = 1γ 4 , ∆ c = 0). So, in this case, we have most of the probe energy lies in the zeroth order positions at (sin θ x = sin θ y = 0), with a peak intensity value is 0.035. However, a very little amount of probe energy shifts towards higher orders, which may be observed at the position (sin θ x = sin θ y = ±0. 25). In order to comprehend the scenario more clearly, we can observe in figure 5(d) that most of the probe light beam is still gathered in zero-order with a small amount of energy shifting towards higher orders.
In figure 6 as l increases to 3 (see figures 6(a) and (b)) we see the change in amplitude and phase modulations. In this case, the amplitude modulation further decreases as seen in figure 6(a), and phase modulation increases as shown in figure 6(b). The decrease in amplitude modulation with OAM number increases is due to the fact that amplification of transmission profile further decreases and there is the absorption of probe light beam at that certain frequency. So for certain OAM l = 3, we have a decrease in coherence. However, we have a steep dispersion profile at that certain frequency, which has a direct relation with phase modulation enhancement.
In figures 6(c) and (d) we notice that the transfer of probe energy takes place to higher orders; however, the zeroth order intensity still remains higher than other diffracted orders. In figure 6(c) the OAM of the vortex beam is taken as l = 3 for which the amplitude modulation becomes weak but the phase modulation is amplified. As the phase modulation enhancement leads to the transfer of energy to diffraction orders of grating with asymmetric behavior. We see that very less probe energy lies at (sin θ y = 0) position, with a peak intensity value is 0.001 and slightly more energy lies in (sin θ x = 0) position, with peak intensity value is 0.009. However, there is more energy transfer to the first order of sin θ y as compared to sin θ x in x and y directions, which will have asymmetric behavior.
The key finding in this subsection is that by varying the detuning of the fields, a phase grating or a gain-phase grating may be generated. Furthermore, increasing the OAM number of the vortex field shifts probe field energy to higher diffraction orders. So one may conclude that phase modulation is primarily responsible for transferring probe field energy from zeroth to higher diffraction orders.

Effect of interaction length on diffraction intensities
In this subsection, we study the effect of the interaction length of the atomic medium through which the probe field propagates and analyze the diffraction intensities of the 2D EIG. Furthermore, we also see how the interaction length influences the diffraction efficiency profile of the 2D EIG. We plot the diffraction intensities of different orders I(θ 1 x , θ 0 y ) (blue sold line), I(θ 1 x , θ 1 y ) (red dotted line), I(θ 2 x , θ 1 y ) (green dashed line) and I(θ 2 x , θ 2 y ) (black dotted line) as shown in figure 7(a). At l = 1, the diffraction intensities of I(θ 1 x , θ 0 y ), I(θ 1 x , θ 1 y ), I(θ 2 x , θ 1 y ) and I(θ 2 x , θ 2 y ) orders will start to increase, as the medium interaction length (L) increases at some optimum values (say L = 1 ζ). The increase of the diffraction intensity of order I(θ 1 x , θ 0 y ) is greater then I(θ 1 x , θ 1 y ), I(θ 2 x , θ 1 y ) and I(θ 2 x , θ 2 y ) orders. This can be shown in the inset of figure 7(a). However, at exact L = 10 ζ, the diffraction intensities of I(θ 1 x , θ 0 y ), and I(θ 1 x , θ 1 y ) reach to 0.45, whereas the diffraction intensities of I(θ 2 x , θ 1 y ) and I(θ 2 x , θ 2 y ) reach to 0.2, see figure 7(a). These results indicate that, although the diffraction intensities of the first and second orders increase, however at L = 10 ζ, the increase in efficiency of the first order is greater than that of the second order. Now when the interaction length (L) is greater than 10 ζ, the diffraction intensity of I(θ 1 x , θ 0 y ) is increase with the small increment increase in interaction length L = 12 ζ and attain its maximum value (0.55), and then start decreasing as shown in figure 7(a). However, the diffraction intensities of I(θ 1 x , θ 1 y ), I(θ 2 x , θ 1 y ) and I(θ 2 x , θ 2 y ) orders have same increasing behavior. This is due to the fact that as the interaction length L increases, the phase modulation is increased, which increases the diffraction efficiency of the 2D EIG. In another sense, the probe energy can be transferred to the high-order directions as the interaction length increases.
To illustrate in more detail for l = 2 and l = 3, we show the diffraction intensities of different orders, we clearly see in figures 7(b) and (c) that the diffraction intensity of I(θ 1 x , θ 0 y ) increase rapidly and then decrease so there is an exchange of probe energy to the diffraction orders. However, the magnitude of these diffraction intensities is less than one.
In the next step, we present the 2D surface plots of diffraction intensity of transmitted probe beam I(θ x , θ y ) by incorporating different interaction lengths i.e. L = 10 ζ, L = 15 ζ and L = 20 ζ respectively. As depicted in figures 8(a)-(c), at L = 10 ζ, the probe field energy is primarily distributed to the I(θ 1 x , θ 0 y ) order directions, being positioned at (sin θ x = 0, sin θ y = 0). However, very small energy distributes to other orders.  The central peak diffraction efficiency is about 1.15, while the efficiencies of the remaining orders are around 0.24. By increasing the interaction length to L = 15 ζ, there is a small increment transfer of probe energy to other orders as in figure 8(b).
On further changing the interaction length i.e. L = 20 ζ, we clearly see that there is an almost complete probe energy transfer from zeroth to first order. Therefore, as demonstrated in figure 8(c), the central peak of the probe beam is suppressed and the probe energy shifts to (sin θ x = ±0.25, sin θ y = ±0. 25). Without a doubt, extending the interaction length at a specific optimal value can greatly enhance the diffraction intensity pattern. The gain phase grating together with periodic phase change is the reason for such a result, which may assist to boost the efficiency of higher-order diffraction.
From the above discussion, we clearly see that we can transfer the probe energy from the zeroth order to higher diffraction orders by varying certain parameters, which can be used for the storage and retrieval of information.

Radial diffraction intensity distributions with respect to various φ
In this subsection, we investigate the radial diffraction intensity distributions and 2D transverse diffraction intensity pattern of 2D EIG with respect to various azimuthal angles φ of vortex beam. Let us now examine the effect of azimuthal angle φ on radial diffraction intensity distributions. For this, we plot the diffraction intensities of different orders versus radial distance r/w as shown in figure 9. When (φ = π/6 radian, π radian) in figures 9(a) and (c), under certain situations, the probe field is not only being amplified but extensively distributed into the high-order directions at a certain radial distance where the vortex beam has maximum amplitude. As a result, their intensity may be significantly more than the probe field intensity impacting on the grating. In other words, high-order diffraction intensities are also increased. This is due to the fact that at a certain point, we get gain in the second transparency window. The enhanced high-order diffraction appears at a sufficiently high Raman gain. With increasing amplification, the intensity of the high-order diffraction increases. Now, when φ = π/4 radian in figure 9(b), our system is still in DEIT, but there is no gain in the second transparency window, hence there is no intensity amplification of higher orders.
The 2D diffraction intensity pattern for the transmitted profile of probe beam I(θ x , θ y ) has been analyzed in figure 10 by incorporating various azimuthal angle φ of vortex beam i.e (φ = π/6 radian, π/4 radian, π radian). At (φ = π/6 radian, π radian) in figures 10(a) and (c) the probe field transmission intensity is obviously enhanced, which is owing to the gain in the second window as a result of the nonlinear process. Due to the gain and high diffraction intensity, most of the light intensity is distributed in the I(θ 1 x , θ 1 y ) order direction, however, only a small amount of energy is transferred to the other orders. At φ = π/4 radian, in figure 10(b), the gain in the second window is suppressed, and the peak intensity are just 0.15. Most of the probe energy is confined inside the I(θ 1 x , θ 0 y ) order, with only a small amount transferring to the other orders. As a conclusion, we may conclude that the probe energy can be shifted to different orders by altering the azimuthal angle (φ) of the vortex beam.

Discussion
In this section, we briefly give an overview of our results and analysis. We have developed an approach to achieving the 2D EIG by considering four different cases: First, we have discussed the case where all fields are resonant with their respective atomic transitions. Next, we see the case, when the control field is resonant while the probe and composite vortex lights are off-resonant and see the effects of field detuning and OAM l on 2D EIG. Furthermore, we consider the case where we see the effect of interaction length on intensities of 2D EIG. Finally, we discuss the effect of radial diffraction intensity distributions on intensities of 2D EIG.
Our approach involves calculating the density matrix equations in the steady-state regime to determine the probe field optical susceptibility of the proposed atomic system. Furthermore, we solve the probe light transmission function equations for 2D EIG and Fraunhofer intensity diffraction equations by using SW control and optical vortex fields.
We have neglected the Doppler broadening effect in the calculation of the response of the medium using density matrix formalism. The MOT is the most often used technology for trapping and cooling atoms in this environment. In hot atomic medium investigations, the counter-propagating beam approach may be employed to reduce the Doppler broadening effect [57]. The easiest technique to avoid the Doppler broadening effect, however, is to use cold atoms.
In all fields' resonant cases, we have seen that we only get the amplitude grating. And most of our probe field energy lies within the zeroth order. In the off-resonant case, as our system has DEIT we obtain a gain in the second EIT window. Due to the gain in the present system, light amplification could be achieved, which should lead to a larger transmission function. Most importantly, we have seen that by changing the OAM of the composite vortex beam the probe energy shifts to higher orders.
Furthermore, we investigate the impact of medium interaction length on the 2D EIG diffraction intensities pattern. We see that at a certain value of interaction length, most of the probe energy gathers at the center. On increasing the interaction length first-order energy completely vanished and probe energy shifted to the higher orders. Additionally, the gain in the diffraction intensities of higher order may be achieved by appropriately altering the azimuthal angle φ with respect to the radial distance.
These findings suggest that optimal system and laser settings are crucial for producing high-efficiency gratings. As a result, properly adjusting the probe, vortex, and control field detuning, as well as the vortex light beam azimuthal angle, is a good way to achieve high diffraction efficiencies in all-optical absorption and gain-phase gratings.

Conclusions
In conclusion, we have presented a comprehensive analysis of a 2D EIG in a coherently prepared ⋔ type ultra-cold rubidium 87 Rb atomic system that could be trapped and cooled using the MOT approach to suppress the Doppler broadening effect. Here, we present four different cases, i.e. all fields being resonant and secondly the case of control field resonant, and off-resonant vortex and probe fields. Furthermore, we investigate the influence of radial distance and interaction length on 2D EIG diffraction intensities.
Our ⋔ excitation scheme is driven by a probe field, a 2D (x, y) SW control field, and a vortex composite beam. The numerical analysis of our proposed scheme revealed that by selecting appropriate system parameters, the behavior of the transmission function's amplitude and phase, as well as the Fraunhofer transverse diffraction intensity pattern, shows the symmetric grating, for field detuning and OAM of vortex beam.
Additionally, we can see that changing the azimuthal number and interaction length serves as a straightforward control knob for directly controlling the diffraction intensity distribution as well as the efficiency of the suggested grating. This additional level of flexibility provided by the vortex field allows us to go beyond the possibilities afforded by previously investigated EIGs.
Our proposed model has an advantage over previous research in that it offers a simple method for controlling the amplitude and phase grating processes by just altering the system parameters of the vortex light to transfer from zeroth to the first order of probe energy. We anticipate a feasible practical application of the suggested grating approach in conventional atomic EIG configurations due to its easy controllability.
In our analysis here, we establish a framework for innovative applications in quantum information and all-optical information processing technologies in order to improve the flexibility of EIG-based quantum memory devices.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.
Considering the input probe beam to be a plane wave, the diffraction intensity distribution equation may be represented as I p (θ x ) = |E(θ x )| 2 sin 2 (Nπ Λ x sin(θ x )/λ p ) N 2 sin 2 (πΛ x sin(θ x )/λ p ) , where N denotes how many spatial periods there are in an atomic grating, θ x is the angle of diffraction, whereas E(θ x ) is the Fourier transform of transmission function T(x), which can be express as ORCID iDs