Based on electromagnetically induced transparency (EIT),^[1–3] electromagnetically induced grating (EIG) is formed and has been investigated widely.^[4–10] By replacing the coupling field with a standing wave, alternating regions of high transmission and absorption can be created, acting as a grating on which the probe beam can be diffracted. As shown in Ref. [4], when the probe field is resonant with the related states, EIG is a kind of amplitude grating which tends to diffract light into the central maximum, and the high-order diffraction efficiency is very low. If a high efficiency is desired for the high-order diffraction directions, detuning should be introduced. The detuning can cause a phase modulation to the probe beam and the formed phase grating then disperses energy into the high-order directions. However, the detuning gives rise to large absorption and the whole diffraction efficiency is lowered accordingly. Alternatively, electromagnetically induced phase grating (EIPG) based on the giant Kerr nonlinearity has been proposed to improve the first-order diffraction efficiency.^[11,12] In the EIPG, a π cross-phase modulation across the probe beam with low energy loss has been accomplished. Furthermore, the effects of microwave modulation,^[13] spontaneous generated coherence,^[14,15] and static magnetic field^[16] have also been considered to achieve improved diffraction efficiency in the first-order direction.

On the other hand, in the active Raman gain (ARG) systems, the probe field can be amplified and some related studies have been reported, such as superluminal light propagation,^[17–19] ultraslow propagation,^[20] rapidly responding nonlinear effects,^[21–23] etc. A kind of gain grating that is based on the spatial modulation of ARG has also been proposed.^[24–26] This kind of gain grating not only amplifies the zero-order diffraction intensity but it also effectively diffracts light into the high-order directions.

In our previous work, we have proposed a scheme for a two-dimensional electromagnetically induced cross-grating (EICG) based on EIT in a four-level tripod-type atomic system.^[27] As two standing-wave fields along the x and y directions drive their corresponding atomic transitions, a two-dimensional EICG is created and diffracts light into the high-order directions. In order to improve the diffraction efficiency of the two-dimensional grating, in this paper we theoretically investigate a two-dimensional gain cross-grating (GCG) in a cold atomic medium, taking advantage of ARG. As the probe field propagates along the z direction and passes through the intersectant region of the two orthogonal standing-wave fields in the x–y plane, it can be significantly diffracted into the first-order direction and the zero-order diffraction intensity is also amplified. In contrast with the two-dimensional EICG, the first-order diffraction intensity of the two-dimensional GCG is greatly enhanced.

We consider a four-level N-type cold atomic system as shown in Fig. 1(a). A Raman field with Rabi frequency Ω_r drives the transition |3〉 ↔ |1〉. A weak probe field with Rabi frequency Ω_p(z) couples to the transition |3〉 ↔ |2〉. The transition |4〉 ↔ |2〉 is driven by a standing-wave field with position-dependent Rabi frequency Ω_s(x, y). As shown in Fig. 1(b), the standing-wave field Ω_s(x, y) is the combination of two orthogonal standing-wave fields with the same frequency, that is Ω_s(x, y) = Ω_s0[sin(πx/Λ_x) + sin(πy/Λ_y)], where Ω_s0 is assumed to be real for simplicity, and Λ_x and Λ_y are the standing-wave field’s spatial periods in the x and y directions, respectively. The probe field propagates along the z direction and passes through the intersectant region of the two orthogonal standing-wave fields in the x–y plane, while the Raman field may propagate along an arbitrary direction.

Fig. 1.

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	Fig. 1. (a) Schematic diagram of the four-level N-type atomic system. The possible levels correspond to the hyperfine levels of 85Rb atoms: |5S1/2, F = 3, m = −3〉 = |1〉, |5S1/2, F = 2, m = −1〉 = |2〉, |5P1/2, F = 2, m = −2〉 = |3〉, and |5P3/2, F = 1, m = 0〉 = |4〉. (b) Sketch of a prototype.

In the interaction picture, with the rotating-wave approximation and the electric-dipole approximation, the Hamiltonian is given by

In order to obtain the diffraction pattern for the probe field propagating through the medium, we begin with the Maxwell’s equation. Under the slowly varying envelope approximation and in the steady-state regime, the propagation of the probe field driven by the atomic polarization can be described as

In this section, we demonstrate the two-dimensional gain cross-grating and analyze the controllability of the diffraction efficiency. We assume γ₃₁ = γ₃₂ = γ₄₁ = γ₄₂ = γ, γ₁₂ = 10⁻³ γ with γ being the natural linewidth of state |3〉. L is given in units of Figure 2(a) displays the amplitude of the transmission function as a function of x and y. This shows that the probe field is amplified and the probe gain oscillates along the x and y directions in the periods of Λ_x and Λ_y, respectively. As shown in Ref. [24], the amplification of the probe field is attributed to the effect of ARG induced by the Raman field, and the probe gain is periodically changed due to the periodic intensity pattern of the standing wave in the x–y plane. In Fig. 2(b), we describe the normalized Fraunhofer diffraction intensity according to the transmission function of Fig. 2(a). An investigation of Fig. 2(b) shows that the probe field is not only amplified in the zero-order direction but also distributes significantly into the high-order directions. The first-order diffraction efficiency can approach 22%, which is much higher than that (2%) of the two-dimensional EICG.^[27] Meanwhile, the dispersion of the probe field also changes periodically, and then the phase modulation of the probe field occurs along the x and y directions, as shown in Fig. 2(c). When the phase modulation is considered, a fraction of the zero-order diffraction intensity will deflect into the high-order directions and the first-order diffraction efficiency approaches to 24%, as shown in Fig. 2(d). In comparison with the two-dimensional EICG,^[27] due to the ARG, the two-dimensional GCG has a much high diffraction intensity in the first-order direction.

Fig. 2.

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Fig. 2. The amplitude |T(x, y)|(a) and the phase Φ/π (c) of the transmission function as a function of x (in units of Λx) and y (in units of Λy). (b) The normalized diffraction intensity Ip(θ1,θ2) as a function of sin(θ1) and sin(θ2) for the corresponding transmission function shown in panel (a). (d) The two-dimensional gain cross-grating considers the gain and phase modulation. The insets in panels (b) and (d) show the residual parts of the diffraction patterns along sin(θ1). The parameters are Ωr = 0.2γ, Ωs0 = 0.2γ, Δr = Δp = 80γ, Δs = 100γ, Λx/λp = Λy/λp = 4, L = 200ξ, and M = N = 5.

In order to meet the requirement of ARG that most atoms populate in the ground state |1〉, we ensure that Δ_r and Δ_p are much larger than the Rabi frequencies and the atomic decay rates. Under this condition, we investigate the controllability of the first-order diffraction intensity of the two-dimensional GCG. Figure 3(a) depicts the effects of the Raman field’s Rabi frequency on the first-order diffraction intensity. It is found that the first-order diffraction intensity increases with the increase of Ω_r, which is consistent with the one-dimensional gain-phase grating.^[24] This can be explained by the gain and phase modulation as shown in Figs. 3(b) and 3(c). When the Raman field’s Rabi frequency is small, the gain and phase modulation induced by the standing-wave field are not obvious due to the small probe gain and dispersion. With the increase of Ω_r, the probe gain and dispersion increase, which enlarges the gain and phase modulation depth and makes more light available for diffraction into the first-order direction. However, in our scheme, the Raman field’s Rabi frequency cannot be set too large since the weak-field approximation for the probe field is applied in our numerical simulation. If the Raman field’s Rabi frequency is too large, then the probe field will be amplified greatly and even exceed the other two fields. So we chose Ω_r = 0.2γ in the following simulations.

Fig. 3.

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	Fig. 3. (a) The first-order diffraction intensity or as a function of Ωr. (b) The amplitude and (c) the phase of the transmission function in the x direction for different Ωr at y = 0. The other parameters are the same as those in Fig. 2.

Figure 4(a) displays the first-order diffraction intensity as a function of Ω_s0. At small Ω_s0, the first-order diffraction intensity increases with Ω_s0. This is within our expectation. As shown in Figs. 4(b) and 4(c), with the increase of Ω_s0, the gain and dispersion are gradually enhanced at the individual antinodes, which increases the gain and phase modulation depth. As a result, more light can be diffracted into the first-order direction. However, the first-order diffraction intensity peaks at a certain Ω_s0 (Ω_s0 = 0.2γ) beyond which it fades away. As Ω_s0 increases further and exceeds 0.2γ, the phase modulation is always increased (see Fig. 4(c)) but the gain is gradually decreased at the individual antinodes and the gain modulation depth is also decreased, as shown in Fig. 4(b). So, light available for diffraction is reduced, and the first-order diffraction intensity decreases as Ω_s0 > 0.2γ (see Fig. 4(a)). At the same time, we also investigate the effect of the interaction length L on the first-order diffraction intensity. It is obvious that the first-order diffraction intensity will increase with the increase of the interaction length L. In order to meet the actual experiment condition and the weak-field approximation of the probe field, the appropriate interaction length we chosen is L = 230ξ, which leads to an amplification of the probe field by a factor of four.^[24] Therefore, due to the ARG, the two-dimensional GCG has a much high diffraction intensity in the first-order direction, and it has a great potential to be used as all-optical switching and routing^[28] in optical networking and communication.

Fig. 4.

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	Fig. 4. (a) The first-order diffraction intensity or as a function of Ωs0. (b) The amplitude and (c) the phase of the transmission function in the x direction for different Ωs0 at y = 0. The other parameters are the same as those in Fig. 2.

We theoretically demonstrate a two-dimensional GCG based on the spatial modulation of active Raman gain and dispersion in a four-level N-type cold atomic system. It is shown that not only the zero-order diffraction intensity is amplified but the first-order diffraction efficiency is also improved. As can be seen, it is mainly the probe gain that contributes to the gain modulation, while the phase modulation further improves the first-order and the high-order diffraction intensities. In comparison with the two-dimensional EICG, the two-dimensional GCG has much high diffraction intensities in the first-order and the high-order directions. Thus it may be utilized as all-optical switching and routing, which are important optical devices in optical networking and communication.