Electromagnetically induced two-dimensional grating assisted by incoherent pump

Electromagnetically induced two-dimensional grating assisted by incoherent pump

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Electromagnetically induced two-dimensional grating assisted by incoherent pump Yu-Yuan Chen, Zhuan-Zhuan Liu, Ren-Gang Wan ∗ School of Physics and Information Technology, Shaanxi Normal University, Xi’an 710062, China

a r t i c l e

i n f o

Article history: Received 16 December 2016 Received in revised form 27 January 2017 Accepted 9 February 2017 Available online xxxx Communicated by V.A. Markel Keywords: Electromagnetically induced grating Two-dimensional grating Incoherent pump Refractive index enhancement

a b s t r a c t We propose a scheme for realizing electromagnetically induced two-dimensional grating in a double- system driven simultaneously by a coherent field and an incoherent pump field. In such an atomic configuration, the absorption is suppressed owing to the incoherent pumping process and the probe can be even amplified, while the refractivity is mainly attributed to the dynamically induced coherence. With the help of a standing-wave pattern coherent field, we obtain periodically modulated refractive index without or with gain, and therefore phase grating or gain-phase grating which diffracts a probe light into high-order direction efficiently can be formed in the medium via appropriate manipulation of the system parameters. The diffraction efficiency attainable by the present gratings can be controlled by tuning the coherent field intensity or the interaction length. Hence, the two-dimensional grating can be utilized as all-optical splitter or router in optical networking and communication. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Electromagnetically induced transparency (EIT) [1,2], which plays an important role in the manipulation of optical properties of atomic medium, has been extensively investigated in the past two decades due to its enormous promising applications, such as slow light [3], optical storage [4], enhanced nonlinear optical processes at low light levels [5,6] and optical switching [7]. By replacing the traveling-wave field in EIT with an intensity dependent standingwave field, a type of all-optical device called electromagnetically induced grating (EIG) [8–26] is formed as a result of the periodic variation of absorption and refractive index experienced by the probe field. And then, it acts as a Bragg grating or a diffraction grating according to the propagation direction of probe beam relative to the standing-wave. Based on that, EIG has many potential applications in optical communication, such as coherently induced photonic band gaps [12–14], beam splitter [15], optical switching and routing [16], optical bistability [17]. As a diffraction grating, the EIG is usually a hybrid grating with synchronous modulations on amplitude and phase, and then the diffraction efficiency in first-order directions is quite limited. To overcome the absorption, much attention was paid to create a medium that is transparent to the probe field, but can induce a deep phase modulation. Araujo proposed electromagnetically in-

*

Corresponding author. E-mail address: [email protected] (R.-G. Wan).

http://dx.doi.org/10.1016/j.physleta.2017.02.015 0375-9601/© 2017 Elsevier B.V. All rights reserved.

duced phase grating (EIPG) based on nonlinear modulation in a four-level N-type atomic system [18]. Due to the enhancement of nonlinearity accompanied by nearly vanishing linear absorption, the diffracting power of grating is improved. Later, spontaneously generated coherence (SGC) was utilized to realize EIPG and the attainable diffraction efficiency is also enhanced via quantum interference [19,20]. To improve the high-order diffraction efficiencies further, electromagnetically induced gain-phase grating based on active Raman gain (ARG) was presented by Kuang et al. [21]. Moreover, multi-level atomic systems and other quantum systems were utilized to enhance the diffraction efficiency [22–26]. Recently, by using two orthogonal standing-wave fields, electromagnetically induced cross gratings, which can diffract the probe into two-dimensional directions, were proposed [27–29]. The phase modulation, which plays an important role in EIG, corresponds to the refractive index experienced by probe field. Therefore, attention must be paid to obtain enhanced refractive index with vanishing absorption which can significantly improve the diffraction efficiency. The most straightforward way is to tuning the light frequency close to an atomic resonance, however such enhancement is accompanied with inevitable absorption which prevents the usage of obtained refractive index. In order to overcome this disadvantage, several schemes on resonantly enhancing refractive index with zero-absorption have been proposed [30,31]. The refractive index enhancement has many potential applications, such as high sensitivity magnetometer [32], phase shifter [33], enhanced Kerr nonlinearity [34]. Most recently, a scheme for uniform phase modulation without paraxial diffractions via control of re-

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5P 1/2 : |3 = | F = 2, m F = 1, |4 = | F = 1, m F = 1 as the four considering atomic states. Therefore, the light fields should be circular polarized and propagate in the same directions. Fig. 1(b) shows the geometry of beams clearly. In order to form standing-wave, the coupling fields should propagate nearly parallel to the probe field with a small angle, and then they interfere with each other. As a result, we obtain the standing-wave pattern, of which the amplitude varies perpendicular to the propagating direction of probe field and coupling fields. In the framework of the semiclassical theory, using the dipole approximation and the rotating wave approximation, we can obtain the Hamiltonian of the system in the interaction picture as follows



H = h¯ (c −  p )|22| − h¯  p |33| − h¯  p |13| + c |23| Fig. 1. (a) The schematic diagram of a four-level double- systems. The atomic system interacts with the weak probe, coherent and a two-way incoherent pump field. (b) The sketch of the spatial configuration of the probe, coherent and incoherent pump field.

fractive index was proposed in a four-level double- system driven by a coherent field and an incoherent pump field [35]. In such atomic system, amplification can be induced owing to the incoherent pumping, thus compensating the absorption. Meanwhile, a linear constant dispersion appears for the probe field. Therefore it is not difficult to find that such a system is in favor of realizing EIG with high diffracting power. In this paper, we theoretically present a scheme for electromagnetically induced two-dimensional grating which is formed in a four-level double- system driven by a coherent field and an incoherent pump field. With the help of a coherent field with two-dimensional standing-wave pattern, a probe light, which propagates along a direction normal to the standing-wave, can be diffracted into high-order directions. We investigate the effect of interaction length and coherent field intensity on the diffraction efficiency of the grating. The results show that, high diffraction intensity can be obtained via proper manipulation of the system parameters. The grating here is created with the help of an incoherent pump field, and the absorption of the probe beam is eliminated. Then the resulted diffraction efficiency can be improved as comparing with other schemes for two-dimensional gratings. Consequently, the present grating is more suitable for beam splitting and fanning in optical communication and networking. This paper is organized as follows: In Sec. 2 the atomic model and relevant equations are presented. In Sec. 3 we investigate the phase and gain-phase grating assisted by incoherent pump field. In Sec. 4 we show the conclusions of this paper.

 + H.c .

(1)

Considering the relaxation process, the motion equations for the density matrix of the atomic system are given by

ρ˙11 = −i (− p ρ31 +  p ρ13 ) + 31 ρ33 − p (ρ11 − ρ44 ) + 41 ρ44 ,

(2a) p

ρ˙21 = −i (− p ρ21 − c ρ31 +  p ρ23 ) − γ21 ρ21 − ρ21 , 2

ρ˙22 = −i (− p ρ22 − c ρ32 +  p ρ22 + c ρ23 ) + 32 ρ33 + 42 ρ44 ,

(2c)

ρ˙31 = −i (− p ρ11 − c ρ21 −  p ρ31 +  p ρ33 ) − γ31 ρ31 −

p 2

ρ31 ,

(2d)

ρ˙32 = −i (− p ρ12 − c ρ22 −  p ρ32 +  p ρ32 + c ρ33 ) − γ32 ρ32 ,

(2e)

ρ˙33 = −i (− p ρ13 − c ρ23 −  p ρ33 +  p ρ31 + c ρ32 +  p ρ33 ) − 31 ρ33 − 32 ρ33 ,

(2f)

ρ˙41 = −i  p ρ43 − γ41 ρ41 − p ρ41 ,

(2g) p

ρ˙42 = −i ( p ρ42 + c ρ43 ) − γ42 ρ42 − ρ42 ,

(2h)

2

p

2. Atomic model and relevant equations The four-level double- atomic system is shown in Fig. 1(a). A weak probe field E p with Rabi frequency  p and detuning  p = ω p − ω13 couples the transition |1 ↔ |3. The transition |2 ↔ |3 is resonantly driven by a coherent field E c with Rabi frequency c and detuning c = ωc − ω32 . Simultaneously, an incoherent field with pumping rate p is imposed to the transition |1 ↔ |4. The Rabi frequencies of the corresponding laser fields are  p = μ13 E p /2h¯ and c = μ23 E c /2h¯ , respectively. Here, E p ( E c ) represents the amplitude of the probe (coupling) field and μ13 (μ23 ) is the electric-dipole moment of the transition |1 ↔ |3 (|2 ↔ |3). For simplicity, we take these Rabi frequencies as real. The level structure can be realized in the D 1 line of cold Rb87 for the reason that it can effectively reduce the impact of undesired effects such as AC Stark shifts, collisional broadening and power broadening of Raman transition. We choose the magnetic sublevels 5S 1/2 : |1 = | F = 1, m F = 0, |2 = | F = 2, m F = 2 and

(2b)

ρ˙43 = −i ( p ρ41 + c ρ42 +  p ρ43 ) − γ43 ρ43 − ρ43 ,

(2i)

ρ˙44 = −42 ρ44 − 41 ρ44 + p (ρ11 − ρ44 ),

(2j)

2

 j +k

where γ jk = ( j , k = 1, 2, 3, 4) are the dephasing rates of 2 the relevant transitions and mn (m = 3, 4; n = 1, 2) denote the spontaneous decay rates from |m to |n, 3 = 31 + 32 and 4 = 41 + 42 represent the spontaneous decay rates of the upper levels while 1 ≈ 0 and 2 ≈ 0 are those of the lower levels. In the presence of the incoherent pump field, the steady-state solutions of density matrix elements in zeroth order of  p are given by ( 0) ρ11 = 431 ( p + 4 )c2 / w ,

(3a)

( 0) ρ33 = 4p 42 c2 / w ,

(3b)

( 0) ρ23 = −i2p 3 42 c / w ,

(3c)

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where w = p 32 42 + 4c2 [2p (31 + 42 ) + 31 4 ]. Therefore, the steady solution of the element ( 0)

(1 )

ρ31 = −

( 0)

ρ11 − ρ33 + 

 p + i (γ31 + 2p ) −

ρ13 is expressed as

1 E (θ) =

( 0)

ρ23

c2  p +i (γ21 + 2p )

N 0 μ213 2 0 h¯ 31

1

χ,

(5)

where N 0 is the atomic density and

χ =−

( 0) ( 0) ρ11 − ρ33 + 

c p p +i (γ21 + 2 )

 p + i (γ31 + 2p ) −

( 0) ρ23

c2  p +i (γ21 + 2p )

(6)

31 .

Due to the intensity-dependent susceptibility, a coherent field with intensity pattern can lead to absorption and phase modulations for the probe field. Thus, when we employ a coherent field with standing-wave pattern, which is given by c = [sin(π x/x ) + sin(π y / y )] with x and  y being the space frequency of the standing-waves as shown in Fig. 1(b), the probe field propagating through the standing-wave region in z direction experiences periodical variation of absorption and refractive index. Therefore, the atomic medium acts as a two-dimensional grating so that the probe field which propagates perpendicular to the standing-wave can be diffracted into high-order directions. To obtain the diffraction pattern for the probe field which propagates through the medium, we begin with the Maxwell’s equation. Under the slowly varying envelope approximation and in the steady-state regime, the equation for the probe field reduces to [8]

∂Ep π =i P p, ∂z 0 λ

(7)

where λ is the wavelength of the probe field. In the derivation of Eq. (7), a thin vapor cell is preferred for the proposed scheme. Such a situation requires large atomic density, i.e. high optical depth of the medium. For the recent technique of laser cooling, an ensemble of cold atoms with an optical depth exceeding 500 can be obtained [36]. Hence, the assumption of thin grating is valid. Here, we choose x ( y ) as the unit for x ( y ) and ζ = 2λ¯h 0 31 /π N 0 μ213 as the unit for z. ζ is the amplitude absorption distance at e −1 for a resonant probe field in the two-level atomic system. From the polarization of the medium P p = 0 χ p E p , we can obtain the transmission function for an interaction length L of the two-dimensional grating as follows

T (x, y ) = e − Im(χ ) L e i Re(χ ) L = e −α (x, y ) L e i φ(x, y ) L ,

(8)

where the first (second) term in the exponential corresponds to the grating amplitude (phase) modulation, which are given by

      T (x, y ) = exp − Im(χ ) L ,

(9a)

(x, y ) = Re(χ ) L /π .

(9b)

In the case which the probe field is a plane wave, we can obtain the Fraunhofer diffraction equation by taking the Fourier transformation of the transmission function T (x, y )



2 sin2 ( M π x sin θx /λ) sin2 ( N π  y sin θx /λ)

I p (θ) =  E (θ)

exp(−i2π xx sin θx /λ)dx 0

(4)

p .

From the polarization of medium, which is given by P p = 0 χ p E p = N 0 μ13 ρ31 , we can obtain the susceptibility for probe field as follows

χ p ( p ) =

where

(1)

c p p +i (γ21 + 2 )

3

M 2 sin2 (π x sin θx /λ) N 2 sin2 (π  y sin θx /λ)

,

(10)

×

T (x, y ) exp(−i2π y  y sin θ y /λ)dy

(11)

0

represents the Fraunhofer diffraction of a single space period, θx and θ y denote the diffraction angle respect to the z direction, m and n are the spatial periods numbers of the atomic grating illuminated by the probe field. The (m, n)-th order diffraction angle is determined by the grating equation sin θx = mλ/x and sin θ y = nλ/ y . Therefore, we can obtain the diffraction intensities of (0, 0) order, (1, 0) order, (0, 1) order and (1, 1) order as follows

I



θx0 , θ y0



1 1 2     =  dx T (x, y )dy  ,   0

I



θx1 , θ y0



(12a)

0

1 1 2    −i2π x  =  dx T (x, y )e dy  ,   0

0

0

0

(12b)

1 1 2   0 1    −i2π y dy  , I θx , θ y =  dx T (x, y )e   I



θx1 , θ y1



(12c)

1 1 2     −i2π x −i2π y =  dx T (x, y )e e dy  .   0

(12d)

0

To better understand the physical mechanism, we begin with the analysis of such atomic system. As shown in Fig. 1(a), in the absence of incoherent pump field, this configuration reduces to a three-level  system. When the two-photon detuning equals to zero, EIT resonance is present such that not only zero-absorption but also vanishing susceptibility appears for the probe field. Hence we prefer a nonzero two-photon detuning that results in a deviation from EIT resonance so that a nonzero linear dispersion appears for the probe field. However, this leads undesired singlephoton absorption. Namely, when the maximal contribution from the atomic resonance to the dispersion is realized, the absorption is on the same order which prohibits the usage of the enhancement of refractive index. Therefore, we prefer to achieve enhanced refractivity with vanishing absorption. As indicated in Eqs. (3), in the absence of incoherent pump, i.e. p = 0, the system becomes an EIT system with only one coupling field, the population is almost in level |1, and there is no atomic (0) coherence between states |2 and |3, i.e. ρ23 = 0. Then we can only obtain refractivity accompanied by strong absorption. However, when applying the incoherent pump and coupling field at the same time, with the increase of the incoherent pump rate p, the (0) population in the ground state ρ11 decreases while the population (0)

(0)

in the excited state ρ33 and the atomic coherence ρ23 increase as shown in Fig. 2(a). It means that the single-photon absorption due to the non-zero two-photon detuning, which is proportional to the (0) (0) population difference ρ11 − ρ33 , can be overcome via appropriate manipulation of p. As shown in Fig. 1(a), when the incoherent pump field is imposed to the transition |1 ↔ |4, the atoms initially remaining in the level |1 are motivated and then redistributed in all the four levels. With the help of the coherent field, the atomic coherence between states |2 and |3 is generated, and such an atomic coherence also contributes to the refraction experienced by the probe field. As a result, we can obtain enhanced

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3. Numerical calculation and discussion 3.1. Electromagnetically induced two-dimensional phase grating

0 0 0 Fig. 2. (a) The population difference ρ11 − ρ33 and atomic coherence ρ23 as a function of the incoherent pumping rate p. (b) The absorptive part [Im(ρ31 )] and refractive part [Re(ρ31 )] of the probe field versus the probe detuning  p when p = 2.531 . The other parameters are  = 2π × 5.75 MHz, 31 = /4, 32 = /6, 41 = /12, 42 = /2, γ21 = 0.00131 , c = 0 and  = 0.231 .

refractive index with vanishing absorption or even gain under the assistance of incoherent pump field [see Fig. 2(b)]. The incoherent pump field can be obtained from a free-running diode laser, of which the linewidth is broad enough and therefore does not introduce any atomic coherence [37,38].

As pointed out in the above discussions, large refractive index accompanied with vanishing absorption can be realized by proper controlling the system parameters. When we apply a coherent field with two-dimensional standing-wave pattern, a periodic phase modulation across the beam profile of the probe field with low loss can be induced. In the following, we present a numerical study with the parameters p = 2.531 ,  p = 1.831 and L = 58ζ . Fig. 3 displays the amplitude and phase modulations of the two-dimensional grating. As shown in Fig. 3(a1), with the assistance of incoherent pump field, we can obtain a relative small amplitude modulation which oscillates near an average transmissivity of 97%. Namely, the induced absorption modulation and energy loss can be neglected. In order to investigate how the popula(0) (0) (0) tion difference ρ11 − ρ33 and atomic coherence ρ23 affect the absorption experienced by probe field, we display the amplitude of transmission function resulting from the population difference and atomic coherence in Figs. 3(a2) and 3(a3), respectively. In such condition, we find that the population in the excited state is more (0) (0) than that in the ground state (i.e. ρ33 − ρ11 > 0), so that it contributes gain with inversion for the probe. Meanwhile, the atomic (0) coherence ρ23 leads to an absorption to probe field as a result of non-zero two-photon detuning. Therefore, owing to the balance between these two mechanisms, a small amplitude modulation with high average transmissivity is achieved. On the other hand, due to the cross-phase modulation of the coherent field, we obtain a 1.5π phase modulation [see Fig. 3(b1)]. To understand the (0) (0) effects of the population difference ρ11 − ρ33 and atomic coher(0)

ence ρ23 on phase modulation, we show their contributions in Figs. 3(b2) and 3(b3), respectively. It is clear that the atomic coherence provides significant contribution to the phase modulation, whereas the population difference nearly has no effect on phase modulation, thus we can draw a conclusion that the phase mod-

Fig. 3. ((a1) and (b1)) The amplitude modulation | T (x, y )| and phase modulation (x, y )/π of the transmission function. ((a2) and (b2)) The amplitude and phase modulations (0)

(0)

resulted from the population difference ρ11 − ρ33 . ((a3) and (b3)) The amplitude and phase modulations resulted from the atomic coherence p = 2.531 ,  p = 1.831 and L = 58ζ . Other parameters are the same as in Fig. 2.

(0) ρ23 . The parameters are

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Fig. 4. The Fraunhofer diffraction of (a) | T (x, y )|, (b) e i (x, y ) , (c) T (x, y ) as a function of sin θx and sin θ y when M = N = 5. Other parameters are the same as in Fig. 3.

Fig. 5. (a) The diffraction intensities I (θx0 , θ y0 ), I (θx1 , θ y0 ) and I (θx1 , θ y1 ) as a function of the interaction length L. The diffraction patterns of the transmission function for (b) L = 81ζ and (c) L = 93ζ . Other parameters are the same as in Fig. 3.

ulation of the grating is mainly induced by the atomic coherence (0) ρ23 . So far, a two-dimensional phase grating with low energy loss is formed in the atomic medium. In order to illustrate the roles of the amplitude modulation and phase modulation in such phase grating, we display the Fraunhofer diffraction of | T (x, y )|, e i (x, y ) and T (x, y ) in Figs. 4(a), 4(b) and 4(c), respectively. It can be seen that the relative small absorption modulation | T (x, y )| gathers the probe field energy in the center maximum and the intensity is about 0.9. While the pure phase modulation diffracts a small portion of probe energy into (1, 0) and (0, 1) order diffraction peaks, which are located at (sin θx = ±0.25, sin θ y = 0) and (sin θx = 0, sin θ y = ±0.25). The resulting diffraction intensity is about 0.12. When both absorption and phase modulations are considered, the diffraction distribution profile is nearly the same as that for pure modulation except for very little energy loss due to the single-photon absorption. Thus, we can conclude that the transfer of probe energy from (0, 0) order to high-order directions is mainly resulted from the phase modulation. The diffraction efficiency, which is determined by the phase modulation, is strongly dependent on the interaction length. Fig. 5(a) displays the variations of different order diffraction intensities I (θx0 , θ y0 ) [solid line], I (θx1 , θ y0 ) [dashed line] and I (θx1 , θ y1 ) [dashed–dotted line] as a function of the interaction length L. With the increasement of L, the phase modulation depth is enhanced which makes more probe energy be diffracted into high-order directions, thus improving the diffracting power. However, I (θx1 , θ y0 ) and I (θx1 , θ y1 ) get their maximums at optimum values of L, exceed which they decrease gradually owing to the energy transfer to higher order diffraction peaks and the decrease of average transmissivity. In order to see more detail, we present the diffraction patterns for two different interaction lengths in Figs. 5(b) and 5(c), respectively. As shown in Fig. 5(b), a significant portion of probe energy is diffracted into (1, 0), (0, 1) and (1, 1) order diffraction peaks [(sin θx = ±0.25, sin θ y = 0), (sin θx = 0, sin θ y = ±0.25) and (sin θx = ±0.25, sin θ y = ±0.25)] and the resulting diffraction intensities are all equal to 0.07. Whereas for a larger interaction length, the probe energy is mainly shifted into (1, 1) order diffraction peaks and the diffraction efficiency is improved to 0.08 [see Fig. 5(c)]. In addition to the interaction length, the diffraction efficiency can also be tuned by the coherent field intensity [see Fig. 6]. In

Fig. 6. The diffraction intensities I (θx0 , θ y0 ), I (θx1 , θ y0 ) and I (θx1 , θ y1 ) as a function of the coherent field intensity . Other parameters are the same as in Fig. 3.

the absence of coherent field, because the incoherent pump field has optically pumped the atoms from ground state level |1 to |2, there is no interaction between the probe and atoms. Then the probe field propagates as in free space such that the probe energy still gathers in the center maximum. However, as the coherent field intensity increases, such interaction becomes significant so that the phase modulation depth increases, thus improving the high-order diffraction efficiency. 3.2. Electromagnetically induced two-dimensional gain-phase grating As presented in above attention, the largest diffraction efficiencies attainable by the phase grating are not large enough. In order to significantly improve the diffraction efficiency, we prefer to create a medium which can lead to an amplification for the probe field. According to the above discussion, the absorption is attributed to the two-photon detuning, thus we can decrease the absorption by tuning the probe field detuning such that the gain across probe takes place. Fig. 7 displays the amplitude and phase modulations of the two-dimensional grating. Under the assistance of incoherent pump, we can obtain a 0.6π amplified amplitude modulation which oscillates around an average transmissivity of 140%. In other words, the energy loss is eliminated. In order to investigate the effect of (0) (0) (0) population difference ρ11 − ρ33 and atomic coherence ρ23 on the amplitude modulation, we show the amplitude of transmission function resulted from the two parts in Figs. 7(a2) and 7(a3),

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Fig. 7. ((a1) and (b1)) The amplitude modulation | T (x, y )| and phase modulation (x, y )/π of the transmission function. ((a2) and (b2)) The amplitude and phase modulations (0)

(0)

resulted from the population difference ρ11 − ρ33 . ((a3) and (b3)) The amplitude and phase modulations resulted from the atomic coherence same as in Fig. 3 except for  p = 1.631 .

(0) ρ23 . The parameters are the

Fig. 8. The Fraunhofer diffraction of (a) | T (x, y )|, (b) e i (x, y ) and (c) T (x, y ) as a function of sin θx and sin θ y when M = N = 5. Other parameters are the same as in Fig. 7.

respectively. In such a case, it is found that the population in the (0) (0) excited state ρ33 is more than that in the ground state ρ11 , therefore it contributes gain with inversion for the probe field. At the same time, the atomic coherence also results in a small gain to the probe. As a result, a non-ignorable gain modulation is realized due to the superposition of these two mechanisms. On the other hand, owing to the cross-phase modulation of coherent field, a 1.6π phase modulation is achieved [see Fig. 7(b1)]. To illustrate the role of the population difference and atomic coherence in phase modulation, we display their contributions in Figs. 7(b2) (0) (0) and 7(b3), respectively. It is obvious that ρ11 − ρ33 nearly has (0)

no effect on the phase modulation while ρ23 provides significant contribution to the phase modulation. So far, a two-dimensional gain-phase grating is formed in the atomic medium. To better understand the effect of the gain modulation and phase modulation on the grating, we present in Fig. 8 three graphs: the first one is the Fraunhofer diffraction of | T (x, y )|, the second one is the Fraunhofer diffraction of e i (x, y ) and the third one is the Fraunhofer diffraction of T (x, y ). It is not difficult to find that the pure gain modulation, of which the depth is not large enough, leads to an amplification and the probe energy is gathered in center maximum [see Fig. 8(a)], whereas the pure phase modulation tends to diffract a significant portion of probe energy into highorder directions [see Fig. 8(b)]. Therefore we can draw a conclusion

that the diffraction of the grating mainly results from the phase modulation while the population difference just provides gain to the probe beam. As a result, the diffraction efficiency attainable by the grating is significantly improved as indicated from the comparison of Fig. 8(c) with Fig. 4(c). To study the controllability of diffraction efficiency, we display the dependence of different order diffraction intensities I (θx0 , θ y0 ) [solid line], I (θx1 , θ y0 ) [dashed line] and I (θx1 , θ y1 ) [dashed–dotted line] on the interaction length L in Fig. 9(a). It shows that the diffraction intensities in different order can get their maximums at optimum values of L, beyond which they fade away significantly. For large interaction length, the phase modulation becomes dominant, thus the probe energy is shifted from center maximum to high-order directions as shown in Figs. 9(b) and 9(c). The result is the same for the variation of different order diffraction intensities as a function of the coherent field intensity [see Fig. 10]. 4. Conclusions In summary, we have theoretically studied the electromagnetically induced two-dimensional grating in a double- atomic system driven by a coherent standing-wave field and an incoherent pump field. Moreover, the effect of the interaction length and intensity of coherent field on the diffracting power of the presented

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Fig. 9. (a) The diffraction intensities I (θx0 , θ y0 ), I (θx1 , θ y0 ) and I (θx1 , θ y1 ) as a function of the interaction length L. The diffraction patterns of the transmission function for (b) L = 74ζ and (c) L = 92ζ . Other parameters are the same as in Fig. 7. [5] [6] [7] [8] [9] [10] [11]

Fig. 10. The diffraction intensities I (θx0 , θ y0 ), I (θx1 , θ y0 ) and I (θx1 , θ y1 ) as a function of the coherent field intensity . Other parameters are the same as in Fig. 7.

gratings are also investigated. Results show that the incoherent pump field can help in inducing the gain to the probe, which cancels out the absorption, thus a phase grating or a gain-phase grating can be formed in the atomic system. We believe the presented two-dimensional gratings can be used as all-optical splitter or router in optical networking and communication. Additionally, the proposed scheme may allow for further improvement by using other well-investigated atomic configuration and atomic system driven by more than one incoherent pump fields. Acknowledgements This work is supported by the NSFC (Grant Nos. 11204367 and 61475191) and the Fundamental Research Funds for the Central Universities (Grant Nos. GK201503022 and GK201601008). References [1] [2] [3] [4]

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