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Abstract
We investigate the propagation properties of finite-energy Airy beams in nonlocal nonlinear atomic vapor. The nonlocality is caused by the transportation of optically excited atoms and described with a Green function. By conducting numerical simulation on Airy beams in the case of one-dimension, two-dimensions, and ring geometry, respectively, we find that the beams have a more stable shape and a longer propagation distance in the presence of nonlocality. The results show that the nonlocality can slow down the beams’ distortion induced by the self-focusing effect and contributes to a more stable propagation of Airy beams in the nonlinear media.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Airy Beams were first predicted by Berry and Balazs in 1979 [1] and were experimentally confirmed by Siviloglou et al. in 2007 [2,3]. Due to their properties of self-healing, non-diffraction, and lateral self-acceleration, Airy Beams have extensive potential applications, including manipulation of particles [4], self-bending plasma channels [5], autofocusing waves [6] and light bullets [7]. Now the propagation dynamics of Airy beams have been extensively studied in the linear and nonlinear regime. The effect of the Kerr nonlinearity on an Airy beam [8,9], and the interaction of Airy beams in the nonlinear media [10] have been investigated. These studies mainly focused on the local conditions in which the refractive index of the media at a given point only depends on the light intensity at that position, that is, $n({\textbf{r}}) = n(I({\textbf{r}}))$. In fact, in many other situations, the nonlinearity of one medium is not only determined by the local conditions, but also depends on nonlocal contributions. Nonlocal nonlinearity refers to the spatial correlation of the refractive index distribution in the media, implying that the change of refractive index at a given point is not only related to the light intensity of the same point, but also related to the light intensity of all nearby points, that is, $n({\textbf{r}}) = \int d{\textbf{r}_{\textbf{0}}}R({\textbf{r}}, {\textbf{r}_{\textbf{0}}})I({\textbf{r}_{\textbf{0}}})$, where $R({\textbf{r}}, {\textbf{r}_{\textbf{0}}})$ is the nonlocal response function of the media[11]. Nonlocality is often the consequence of transport processes or long range interactions [12], so it mainly exists in atomic vapors [13], thermal media [14], and nematic liquid crystals [15].
In recent years, studies on the propagation of Airy beams in nonlocal nonlinear media have also been demonstrated, but these works mainly focus on special models for the nonlocal response function, considering the nonlocal response function as an angular type function [16], a Gaussian-type function [17], or a hyperbolic function [18]. Atomic vapor is a good platform for studying the generation, propagation, and manipulation of Airy beams due to the atomic favorable quantum coherence effect. The atomic susceptibilities can be easily controlled, in comparison with that in solid materials. In the past, our group demonstrated the generation and self-healing properties of Airy beams in atomic vapor assisted by four wave mixing and electromagnetically-induced-transparency [19,20]. Here we investigate the propagation properties of finite-energy Airy beams in atomic vapor. From the perspective of the susceptibilities, we use a Green function to describe the nonlocal nonlinearity of atomic vapor. The interaction between any two atoms in the media is described by the Green function which can be derived from the Maxwell-Boltzmann velocity distribution. This model can give us a clearer picture about the origin of the nonlocality. Under this theoretical model, our numerical simulations show that the beams can propagate with a more stable shape and longer distance in the nonlocal nonlinear media compared with the results in the local case. The results may provide valuable reference in applications of Airy beams.
2. Theoretical model
Considering an Airy beam propagates in Rubidium atomic vapor where the atoms can be viewed as a two-level system. The density matrix equations of motion for a closed two-level atomic system in the presence of a monochromatic, steady-state field are [21]
where ${\omega }_{ba}$ is the frequency corresponding to the energy separation between the ground level and the excited level, $V_{ba}=-{\mu }_{ba}Ee^{-i{\omega }t}$ is the matrix elements of the interaction Hamiltonian in the rotating-wave approximation, $T_1$ and $T_2$ are the ground state recovery time and the dipole moment dephasing time respectively. The population inversion is given by $W={\rho }_{bb}-{\rho }_{aa}$, and $W^{eq}$ represents the population inversion in thermal equilibrium. The susceptibility for a homogeneously broadened two-level atom can expressed as the form [21]The propagation process of Airy beams in the atomic vapor follows the paraxial wave equation
where $\psi$ is the wave function of the Airy beam and $k = n{\omega }/c=2{\pi }n/\lambda$ is the wave number. ${\chi }(I)$ is the total susceptibility which can be divided into real and imaginary parts and expressed as ${\chi }(I) = {\chi }'(I) + i{\chi }''(I)$. ${\chi }'(I)$ and ${\chi }''$ determine the refractive index and absorption coefficient of the medium, respectively. Now we can divide them into the linear part that is independent of light intensity and the nonlinear part that depends on light intensity, that is, ${\chi }'(I) = {\chi }'_l + {\chi }'_{nl}(I)$ and ${\chi }''(I) = {\chi }''_l + {\chi }''_{nl}(I)$. Usually, ${\chi }'_l$ is very small and negligible for atomic medium, so the total susceptibility can be expressed as ${\chi }(I) = {\chi }'_{nl}(I) + i[{\chi }''_l + {\chi }''_{nl}(I)]$. The absorption coefficient of the atomic medium is given by ${\alpha }(I) = k[{\chi }''_l+{\chi }''_{nl}(I)]/2$.Considering that the thermal motion of atoms follows the Maxwell-Boltzmann velocity distribution, these results can be extended to the case of Doppler broadened two-level system. Under local condition, the susceptibilities are given by [22]
where ${\chi _0} = 6{\pi }N qc^3/{\omega _a}^3$ is the linear susceptibility without input beams, $q_I = q\sqrt {1+I/I_S}$ is power broadened hole size with $q=\sqrt {{\bf {ln}}2}\gamma /\omega _D$ and the normalized detuning frequency $p=2q\Delta _0/\gamma$. $\gamma$ is the full width at half maximum natural linewidth, $\Delta _0 = \omega -\omega _a$ is the detuning between the laser frequency $\omega$ and the atomic transition frequency $\omega _a = 2{\pi }c/{\lambda _a}$, and $\omega _D = k\sqrt {8ln2{k_B}T/m}$ represents the full width at half maximum of the Doppler profile for the atom. The function $\xi (z)=i\sqrt {\pi }exp(-z^2)erfc(-iz)$ is the plasma dispersion function where $erfc(z)=1-\frac {2}{\sqrt {\pi }}\int _0^z exp(-t^2)dt$ is the complementary error function.Under the nonlocal condition, we need to consider the complex influence of the motion of atoms and the inhomogeneous distribution of light intensity in the media on the susceptibilities. The shorter the distance between two atoms, the more likely they are to interact. Therefore, the contribution of the neighbor atoms to the susceptibilities of a given point in the medium is greater than that of distant atoms. We use the Green function $g({\textbf{r}}, {\textbf{r}_{\textbf{0}}})$ to describe the nonlocality of the media, and modify the nonlinear susceptibilities to
Through the above analysis, we use the Green function to express the nonlinear susceptibilities in the nonlocal case. Combining the Eqs. (4), (5), the Eq. (3) can be solved numerically, and then we can learn about the phenomena when the beams propagate in the nonlocal nonlinear media. At the temperature $T=155^{\circ }C$, the atom number density $N = 10^{20} m^{-3}$. If we set the incident wavelength $\lambda = 780nm$, the detuning frequency ${\Delta _0}/2{\pi }=1.46 GHz$, and saturated intensity $I_S = 16.7 W/m^2$, we can obtain the parameters of the susceptibilities is $p=4.0$, $q=8.3{\times }10^{-3}$, ${\chi _0}=0.03$. We will use this set of parameters in the whole numerical simulation.
3. Numerical results
3.1. Variation of susceptibilities with the intensity
Because the intensity of the beams as well as the susceptibilities will changed in the nonlinear propagation process, we firstly investigate the effect of intensity on the nonlinear susceptibilities though the Eqs. (4a),(4b) and (4c) before numerically solving the nonlinear propagation Eq. (3). Figure 1 shows the relationship between the susceptibilities and the intensity. In Fig. 1(a) and Fig. 1(b), the abscissa shows the ratio of the intensity to the saturation intensity, and the ordinates are susceptibilities ${\chi }'$ and ${\chi }''$, respectively. We see that the absolute values of ${\chi }'$ and ${\chi }''$ increases with the intensity, while the absolute value of the slope gradually decreases. The behavior of susceptibilities with the change of intensity dominates the propagation properties of the beams in the nonlinear media. The local susceptibilities of a given point on the facula depend on the intensity of the same point, while the nonlocal susceptibilities of a given point depend on the distribution of the entire facula.
Fig. 1. The relationship between the susceptibilities and the intensity with parameters $p=4.0$, $q=0.0083$, ${\chi _0}=0.03$.
In order to elucidate the significance of the Green function and clarify the difference between local and nonlocal susceptibilities, we use the simplest Gaussian light field as an example for numerical simulation. For a given Gaussian field, the distributions of local and nonlocal susceptibilities corresponding to the spot diameter are calculated as shown in Fig. 2. The Gaussian field is expressed as $E(x,y)={A_0}exp(-r^2/{w_0}^2)$, where the initial amplitude is set as ${A_0}=3{\times }10^{5}V/m$, and the beam size ${w_0}=10{\mu }m$. The real and imaginary parts of the susceptibilities are shown in Fig. 2(a) and Fig. 2(b), respectively. The blue and solid lines show the susceptibilities in the local case, while the red and dashed lines show the susceptibilities in the nonlocal case. By comparison, we can find that the absolute values of the extreme susceptibilities in the nonlocal case are less than those in the local case, while the widths of susceptibilities at half extremum in the nonlocal case are greater than those in the local case. The results in the Fig. 2 indicate that the role of the Green function in the Eqs. (5a) and (5b) is actually to broaden the local susceptibility distribution, forming the nonlocal susceptibility distribution. In essence, this conclusion is also applicable to the susceptibility distribution for other types of beams, including Airy beams.
Fig. 2. The local and nonlocal susceptibilities on the spot diameter for Gaussian field, with parameters ${A_0}=3{\times }10^{5}V/m$, ${w_0}=10{\mu }m$, $p=4.0$, $q=0.0083$, ${\chi _0}=0.03$.
3.2. Propagation of one-dimensional (1D) Airy beams in nonlocal nonlinear media(NNM)
Considering the simplest case, one-dimensional finite-energy Airy beams, we investigate the propagation process of the beams in Rb atomic vapor. We study the propagation of the beam in local and nonlocal cases respectively. The envelope function of the input is expressed in the form of
where ${A_0}$ is initial amplitude and Airy function is defined as $Ai(x)=\frac {1}{2{\pi }i}\int _{-i\infty }^{i\infty } exp(xt-t^{3}/3)dt$. The decay constant $\alpha$ satisfies ${\alpha }>0$ and $w_0$ denotes the central lobe. The nonlinear propagation Eq. (3) was solved by utilizing the split-step fast Fourier transform method in our numerical simulation. If the size of the beam is close to the atomic characteristic length, the nonlocality of the atomic vapor will be prominent. The characteristic length of the atomic vapor is $7.5{\mu }m$ when $T=155^{\circ }C$, so the size of the beam can’t exceed $7.5{\mu }m$ too much and we set ${w_0}=15{\mu }m$. The intensity distribution at different propagation distances is shown in the Fig. 3. In the simulation, we set ${A_0}=3.2{\times }10^{4}V/m$ and ${\alpha }=0.1$.Fig. 3. Nonlocal and local propagation of a one-dimensional Airy beams. The first column shows the propagation in local case and the second column shows the propagation in nonlocal case. The propagation distances are (a0)(b0)z=0; (a1)(b1)z=0.5mm; (a2)(b2)z=1mm; (a3)(b3)z=2mm. The parameters are $p=4.0$, $q=0.0083$, ${\chi _0}=0.03$, ${A_0}=3.2{\times }10^{4}V/m$, ${w_0}=15{\mu }m$, ${\alpha }=0.1$.
Due to the nonlinearity, the main lobe of the Airy beams is self-focusing in both local and nonlocal cases. After the beam focuses to the minimum size, it will diverge outwards and the intensity distribution in the form of Airy function will be destroyed by the nonlinearity. By comparing the results in the first column in Fig. 3 with those in the second column, we find that the self-focusing effect and collapse of the beams are obviously suppressed under the nonlocal condition. At the same distance, the peak in the second column of Fig. 3 is not as sharp as the first column, which shows that the interaction of atoms has an inhibitory effect on the self-focusing process. In addition, a ”toe” appears beside the main lobe in the first column (Fig. 3(a3)), while it is not obvious in the second column (Fig. 3(b3)). The above results indicate that nonlocality contributes to the stable propagation in NNM for one-dimensional Airy beam.
Figure 4 shows the peak intensity of the one-dimensional Airy beams during their propagation. From the Fig. 4, we can clearly see that the peak intensity under nonlocal condition is lower than that in the local case, and the beam propagates a longer distance before reaching the maximum intensity under nonlocal condition. The result shows that nonlocality suppresses the self-focusing process, which is consistent with that in Fig. 3.
Fig. 4. The maximum intensity of the one-dimensional Airy beam at different propagation distances in the local case (the blue line) and nonlocal case (the red dashed line). The parameters are $p=4.0$, $q=0.0083$, ${\chi _0}=0.03$, ${A_0}=3.2{\times }10^{4}V/m$, ${w_0}=15{\mu }m$, ${\alpha }=0.1$.
3.3. Propagation of two-dimensional (2D) Airy beams in NNM
In two-dimensional situation, the electric field of the incident beam is expressed by
The propagation dynamics of the 2D Airy beams in Rb atomic vapor are showed in Fig. 5. In both local and nonlocal cases, the main lobe of the Airy beams experience the process of self-focusing and diffusion. Whether the beams will focus or diffuse depends on the competition between linear and nonlinear effects. In the initial stage, the nonlinear effect of the media is stronger than the spatial linear effect, causing the energy to flow inward, which leads to the decrease of the beam size and a much larger peak intensity in the main lobe. With the self-focusing process of the beams, the spot become very small, then the linear diffraction effect becomes very significant. The limitation of diffraction effect leads to the beam diffusing during the next propagation process. Eventually, the distribution of the beams will collapse under the combined action of linear and nonlinear effects.Fig. 5. Nonlocal and local propagation of a 2D Airy beam with the input power ${P_0}=0.51mW$. The first column shows the propagation in local case and the second column shows the propagation in nonlocal case. The propagation distances are (a0)(b0)z=0; (a1)(b1)z=2mm; (a2)(b2)z=4mm; (a3)(b3)z=6mm. The parameters are $p=4.0$, $q=0.0083$, ${\chi _0}=0.03$, ${A_0}=3.43{\times }10^{4}V/m$, ${w_0}=30{\mu }m$, ${\alpha }=0.1$.
We found that nonlocality has a significant effect on the propagation of the 2D Airy beams. Under the nonlocal condition (the second column in Fig. 5), the self-focusing process and collapse of the beam are obviously suppressed in comparison with the results in the first column of Fig. 5. During the self-focusing process, the size of the main lode in the nonlocal case is larger than that in the local case at the same distance, as shown in Fig. 5(a1) and (b1) or Fig. 5(a2) and (b2). At the distance $z=6mm$, the shape of the main lobe in the local case is fiercely distorted due to the intense transverse flow of energy, while the intensity distribution of the nonlocal case is still regular. The results indicate that nonlocality delays the collapse of the main beam.
Figure 6 shows the peak intensity of the 2D Airy beam at different propagation distances. we can see that the peak intensity in the local case is about $6{\times }10^{5}W/m^2$, which is less than a half of that in the local case (about $1.6{\times }10^{6}W/m^2$), and the nonlocal focusing distance is also obviously shorter than that in the local situation. The results are consistent with those of 1D Airy beams in Fig. 4.
Fig. 6. The maximum intensity of the 2D Airy beam at different propagation distances in the local case (the blue line) and nonlocal case (the red dashed line). The parameters are $p=4.0$, $q=0.0083$, ${\chi _0}=0.03$, ${A_0}=3.43{\times }10^{4}V/m$, ${w_0}=30{\mu }m$, ${\alpha }=0.1$.
3.4. Propagation of ring-Airy beams in NNM
We also investigated the propagation dynamics of ring-Airy beams in Rb atomic vapor. The ring-Airy beams [24–26] possess an abrupt autofocus behavior during their propagation, that is, the beams will shrink in size in the free space and becomes into Bessel beams gradually after the focal point. This unique property is determined by their cylindrically symmetric phase and amplitude distribution. The electric field of the input beam is expressed by
Here we consider the propagation process of ring Airy beams in different cases. Figure 7(a0)-(a3) show the propagation process in the free space. We can see that the beams get smaller in the beginning process and then the beams diverge outward. As for the nonlinear situations, the beam will shrink faster in both local and nonlocal cases, as shown in the second column and the third column in Fig. 7, respectively. The beam will focus on a very small area, and the peak intensity will increase several times or even tens of times in the atomic vapor, as shown in Fig. 8. After the beam focuses into a very small size, the diffraction effect will be very prominent, leading the beams to diverge around.Fig. 7. Propagation of ring-Airy beams with the input power ${P_0}=0.58mW$ and width ${w_0}=15{\mu }m$. The three column pictures show the propagation in free space, local and nonlocal nonlinear conditions, respectively. The distances are (a0)(b0)(c0)z=0; (a1)(b1)(c1)z=3mm; (a2)(b2)(a2)z=3.8mm; (a3)(b3)(a3)z=6.3mm. The parameters are $p=4.0$, $q=0.0083$, ${\chi _0}=0.03$, ${A_0}=1.2{\times }10^{4}V/m$, ${r_0}=15{\mu }m$, ${\alpha }=0.1$.
Fig. 8. The maximum intensity of the ring-Airy beam at different propagation distances in the case of the free space (the black dashed line), local conditions (the blue line) and nonlocal conditions (the red line marked with dots), respectively. The parameters are $p=4.0$, $q=0.0083$, ${\chi _0}=0.03$, ${A_0}=1.2{\times }10^{4}V/m$, ${w_0}=15{\mu }m$, ${\alpha }=0.1$.
By comparing the nonlinear propagation of ring-Airy beams under nonlocal condition with that under the local condition, we find that the nonlocality has a significant effect on light field regulation. Under the nonlocal condition (Fig. 7(c0)-(c3)), the self-focusing process and collapse of the beam are suppressed to certain extent, compared with the results under the local condition (Fig. 7(b0)-(b3)). During the self-focusing process, the size of the main lode in the nonlocal case is larger than that in the local case at the same distance, as shown in Fig. 7(b2). At the distance $z=6.3mm$, the shape of the beam in the local case begins to distort, while the shape of the beam in nonlocal case is obviously more stable. From the results of Fig. 8, as the blue line and the red dashed line show, nonlocality delays the process of self-focusing and diffusion.
4. Conclusion
In summary, we have numerically investigated the propagation of Airy beams in Rb atomic vapor and find nonlocality delays the process of self-focusing and diffusion. we use the Green function to describe the interaction between two atoms in media, and then obtain the nonlocal nonlinear susceptibilities. By solving the propagation equation through the split-step fast Fourier transform method, we have simulated the propagation process of Airy beams. Different kinds of input beams, including one-dimensional Airy beams, two-dimensional Airy beams and ring-Airy beams, have been considered in our study. In all of these cases, the beam will propagate with a more stable shape and longer distance in nonlocal condition than that in the local case. The phenomena indicate that nonlocality contributes to the stable propagation of light beams. This result provides a possible method for nonlinear control of the beam, which is conducive to the application of particle manipulation. The media with nonlocal propagation may be considered to control the focusing distance and size of the beams.
Funding
National Natural Science Foundation of China (NSFC) (11534008, 11574247, 11774286, 11874296); Natural Science Foundation of Shaanxi Province (2015JQ6233).
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