1. Introduction

In recent years, there has been an increasing interest in the research of Airy beams. Airy wavepackets were initially expected as a solution of the Schrodinger equation for a free particle within the context of quantum mechanics [1]. Ideal Airy beamsare not square integrable and carry infinite energy, thus it is difficult to demonstrate them in experiment. In 2007, Siviloglou et al. have firstly implemented optical Airy beams of finite energy by introducing an exponential aperture function [2],and these beams still exhibit the key characteristics of ideal Airy beams. To date, Airy beams have attracted much attention because of their remarkable feathers such as self-accelerating, diffraction-free and self-healing [2–4]. The researches on Airy beams have grown quickly in their generation and control. Many potential applications involving Airy beams have been demonstrated, including optical micromanipulation of small particles [5], curved plasma channel [6], light induced all-optical routing [7], and super-resolution imaging [8]. Airy light pulses of the time domain have been used to form light bullets [9, 10]. Beyond light waves, the concept of Airy beams has been extended to the subsequent realization of Airy surface Plasmon beams [11], electron Airy beams [12], and Airy surface gravity water waves [13].

It is well known that electromagnetically induced transparency (EIT) is an effective method to manipulate light beam [14]. EIT means that the weak light signal passes through an opaque medium without absorption when an additional coupling field is used to produce quantum interference. EIT allows for a large range of light-matter interaction phenomena. In particular, EIT can be used to store a weak probe light [15–17]. This EIT-based light storage is interpreted as dark-state polariton [18], and has important applications in classical and quantum information processing. Recently, there have been some interests in using EIT technique to manipulate the propagation of Airy beams. Some related theoretical and experimental works have been investigated [19–25]. The propagation properties of an Airy beam have been theoretically investigated in a four-level EIT system [19]. Li et al. have examined the selective reflection spectrum of an Airy beam in an EIT-driven medium [20]. Wei et al. have experimentally obtained the generation of an Airy beam by four-wave mixing in EIT-driven atomic vapor [21, 22], and investigated the self-healing property of an optical Airy beam [23]. Hang et al. have theoretically investigated the possibility of slow and storage of Airy wavepackets by using an EIT atomic gas [24, 25]. To our knowledge, the storage of Airy wavepackets has not been reported in experiment until now.

In this paper, we experimentally demonstrate the storage and retrieval of 2D Airy wavepackets in a doped solid driven by EIT. Under EIT condition, the weak probe and strong control fields are applied to the experimental medium with an EIT three-level lambda system. The transverse profile of the probe light carries 2D Airy wavepackets and its temporal shape is a Gaussian pulse. By switching off the control field, the probe Airy wavepackets are stored into atomic spin coherence between two ground levels. After a storage time, by switching back on the control field, the stored Airy wavepackets are retrieved. This storage of Airy wavepackets is based on coherent conversion between light component and atomic spin coherence component. The intensity distribution and similarity of the retrieved Airy pattern are further analyzed. In addition, we investigate the self-healing property of the retrieved Airy wavepackets. The above demonstration develops the manipulation method of Airy wavepackets, and will be useful for light beam manipulation and information processing.

 

Fig. 1 (a)The levels of ${ }^3{H}_4{\to }^1{D}_2$ optical transition of Pr:YSO and the applied light fields. (b)The experimental setup of the storage of Airy wavepackets. AOM, acousto-optic modulator; SLM, spatial light modulator; BS, beam splitter; PD,photodiode; L1 and L2, lens.

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2. Experimental configuration

The optical transition ${ }^3{H}_4{\to }^1{D}_2$ of Pr³⁺:Y₂SiO₅ (Pr:YSO) crystal is used to demonstrate the storage of Airy wavepackets, as shown in Fig. 1(a). Pr:YSO crystal provides spatially-fixed interaction units, narrow spectrum line and long decoherence time. Thus, Pr:YSO can be used to perform EIT-related atomic coherence effects [26, 27]. Two hyperfine states ($\pm 1/2$ and $\pm 3/2$) of ${ }^3{H}_4$ are used as two lower levels of an EIT lambda system, and $\pm 3/2$ of ${ }^1{D}_2$ is used as the upper level. The strong control field ω_c couples the transition of ${ }^3{H}_4\left(\pm 1/2\right){\to }^1{D}_2\left(\pm 3/2\right)$. The weak probe field ω_p couples the transition of ${ }^3{H}_4\left(\pm 3/2\right){\to }^1{D}_2\left(\pm 3/2\right)$, and its transverse profile exhibits 2D Airy wavepackets. The above three levels and two light fields form a typical EIT lambda system. Under EIT condition, the probe Airy wavepackets are expected to be stored into atomic spin coherence between two ground levels. An additional repump field ω_r drives the transition of ${ }^3{H}_4\left(\pm 5/2\right){\to }^1{D}_2\left(\pm 5/2\right)$, and pumps the populations to two lower levels of the EIT lambda system. For our crystal, the optical inhomogeneous broadening is about several GHZ, and spin inhomogeneous broadening of two ground states is about $30\text{ KHz}$ [28]. In experiment, the technique of spectral hole-burning is used to choose single ensembles of Pr ions, thus EIT-related effect can be effectively performed using lower light intensity [29]. The linewidth of our laser is about 1MHz, which dominates the absorption linewidth of the prepared Pr ions [29]. Spin inhomogeneous broadening causes the dephasing of stored atomic coherence, and determines the largest storage time.

The experimental setup is shown in Fig. 1(b). The Pr:YSO crystal with dopant concentration of 0.05% is used as the experimental medium, which is placed inside a cryostat of the temperature of $3.5\text{ K}$. The sizes of the crystal are 4*4*3mm, and the light beams propagate along the optical B axis of $3\text{ mm}$. The initial light source is a continuous single-mode R6G dye laser. The laser wavelength is about $606\text{ nm}$, which corresponds to the optical transition of ${ }^3{H}_4{\to }^1{D}_2$. By beam splitters, the laser output is divided into the experimental light beams. Each light beam is guided to an acousto-optic modulator (AOM), and the experimental parameters of light beam can be controlled independently. It is well known that Airy wavepackets can be generated via the Fourier transformation of a Gaussian beam modulated with a cubic phase [2]. A computer-controlled spatial light modulator (SLM) is used to impose a 2D cubic-phase modulation, and is placed at the front focalplane of the lens L1. After SLM, 2D probe Airy wavepackets of finite energy are obtained at the back focal plane of the lens L1. The size of the probe Airy wavepackets is further decreased in the EIT interaction region by using an additional imaging lens L2. The wavefront of 2D finite-energy Airy wavepackets [2] can be written as $\varphi =Ai\left(x/x_0\right)exp\left(ax/x_0\right)Ai\left(y/y_0\right)exp\left(ay/y_0\right)$, where $Ai()$ represents the Airy function, a represents the exponential truncation factor, x₀, y₀ is anarbitrary transverse scale. The control field and the probe field overlap in the crystal with a small angle of 50mrad. These two fields have the opposite propagation direction, which can reduce the scattering of the control field. The powers of the input control and probe fields are 16mW and 0.4mW, respectively. The probe field passing through the experimental medium is divided into two portions by beam splitter. One portion is sent to a CCD camera to record the transverse profile of the probe Airy pattern. The other portion is sent to a photodiode to record the pulse profile of the time domain.

 

Fig. 2 The storage and retrieval of the probe Airy wavepackets by using the EIT technique. The transverse profile of the probe field is modulated by 2D Airy wavepackets. (a) The time sequence of the storage experiment recorded by photodiode. The storage time is $15\mu \mathrm{s}$. (b) The transverse profile of the probe Airy pattern recorded by CCD camera before and after the storage. (c) The 3D intensity distribution of the Airy pattern before and after the storage.

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3. Experimental results and discussion

In order to demonstrate the EIT-based light storage, the population preparation of an EIT three-level lambda system is firstly done. Through the preparation pulse sequence of [30], the populations are pumped to the level ${ }^3{H}_4\left(\pm 3/2\right)$. After the population preparation, the control and probe fields are applied to the experimental medium in order to perform the storage of Airy wavepackets. The control field has a much stronger power than the probe field, which satisfies the EIT condition. Figure 2(a) shows the time sequence of the storage of Airy wavepackets, recorded by the photodiode. The input probe field is shaped to a Gaussian pulse of $40\mu \mathrm{s}$ width by using AOM, and its transverse profile is modulated by 2D Airy wavepackets. The CCD camera is synchronized with the AOM driver, and is selectively triggered to record the transverse profile of the probe Airy pattern before and after the storage. Once the probe pulse completely enters the experimental medium, the control field is immediately switched off. Then the probe light is converted into atomic spin coherence (a coherent superposition of the two ground levels), and is stored in the crystal. After a storage time of $15\mu \mathrm{s}$, the control field is switched on again, and the stored atomic spin coherence is converted back to a propagating probe pulse. Under our experimental condition, the probe pulse is partially stored and retrieved. This reversible storage is based on the interconversion between atomic spin coherence and light signal.

The retrieved pulse intensity decreases as the storage time increases. In our experiment, the measured largest storage time is about $30\mu \mathrm{s}$, which is the same as our previous work [31]. All Pr ions in the spin inhomogeneous bandwidth contribute to the preparation of stored atomic coherence. Spin inhomogeneous broadening causes the dephasing of stored coherence and limits the largest storage time. The storage efficiency is defined as the ratio between the output pulse energy and the input pulse energy. Under our experimental condition, we have obtained the storage efficiency of about 3%. The low efficiency mainly comes from imperfect EIT interaction and the low optical depth of the crystal. By using a ring-type multipass configuration and optimizing the experimental parameters, high storage efficiency can be achieved in solid memory [32].

By using CCD camera to record the transverse Airy pattern of the probe pulse, the storage of Airy wavepackets is investigated. The EIT-based light storage has been used to store the transverse image and orbital angular momentum of light [33–37]. In order to develop the manipulation of Airy beams, the storage of Airy wavepackets is expected to be realized by using the EIT technique. The storage of Airy wavepackets has been theoretically investigated in the EIT-driven atomic gas [25]. Here, we experimentally demonstrate the storage of 2D Airy wavepackets in an EIT-driven solid. Figure 2(b) shows the transverse Airy pattern of the probe pulse before and after the storage recorded by CCD camera. The input probe field exhibits typical 2D Airy wavepackets in its transverse profile. To efficiently record the transverse profile of the retrieved probe pulse, the CCD camera is triggered after the control field is switched on again in the retrieval process. The exposure time of CCD camera is set to match the width of the retrieved pulse to reduce the background scattering lights. It is clearly seen that the retrieved probe field preserves the same Airy pattern as the unstored light field. Through the above demonstration, the storage andretrieval of a 2D Airy wavepackets are realized in the EIT-driven solid. Unlike atomic gases, the solid-state medium does not have atomic diffusion caused by atomic motion, and provides the spatially-fixed interaction units. Thus the experimental demonstrations in solid-state mediums have more practical applications for further information processing. The storage of Airy wavepackets further proves that the intensity and phase distribution of the transverse profile of light field can be effectively preserved by the technique of EIT-based storage.

We further quantitatively analyze the Airy patterns before and after the sorage. To obtain their 3D intensity distribution, we record Fig. 2(b) under the condition that the main lobe of Airy pattern is not saturated. In this case, due to the limitation of the grey scale of CCD camera, the other lobes of Airy pattern have low intensities. Figure 2(c) shows the 3D intensity distribution of 2D Airy pattern before and after the storage. It is clearly seen that the intensity distribution of the retrieved Airy pattern almost remain invariant compared with that before the storage, and the Airy wavepackets are well preserved through EIT-based storage. The quality of the retrieved Airy pattern is further analyzed by calculating the similarity R of the retrieved Airy pattern compared with the unstored Airy pattern. The similarity R is calculated by using the formula of [35], $R=\frac{{{\displaystyle \sum }}_m{{\displaystyle \sum }}_n{A}_{mn}{B}_{mn}}{\sqrt{{{\displaystyle \sum }}_m{{\displaystyle \sum }}_n{A}_{mn}^2{{\displaystyle \sum }}_m{{\displaystyle \sum }}_n{B}_{mn}^2}}$, where A_mn and B_mn are the gray-scale intensities recorded for pixels m and n of the two images to be compared. Due to the vibration noise and background scattering light of the experimental environments, Airy patterns of each measurement have a slight difference. In experiment, we have recorded a series of retrieved Airy patterns, and the calculated average similarity is about $\left(97.8\pm 0.7\right)\%$. The high similarity means that the storage of Airy wavepackets has a high fidelity.

 

Fig. 3 The self-healing process of the retrieved probe Airy wavepackets for different propagation distances z when its main lobe is blocked. (a), (b) and (c) correspond to the distance $z=0.2\text{ cm}$, $z=0.6\text{ cm}$ and $z=1.0\text{ cm}$, respectively.

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To check the property of the retrieved Airy wavepackets, we further investigate the self-healing process of the retrieved probe Airy wavepackets by monitoring its self-reconstruction during the propagation. The propagation dynamics of the self-healing property can be recorded as a function of the propagation distance by translating the CCD camera. An opaque rectangular obstacle is employed to block the Airy pattern by using a micro-positioner. The main lobe of an Airy pattern contains a large percentage of the total beam power, and is the most prominent intensity characteristic of the light field. Thus, we demonstrate the self-healing of the retrieved Airy wavepackets by blocking the main lobe of the Airy pattern. Figure 3 shows the experimental results of the self-healing process of the retrieved Airy wavepackets for different z, where z is the distance between the opaque obstacle and CCD camera. As the propagation distance increases, the field power flows from the side lobes towards the blocked main lobe to facilitate self-healing. Figure 3(a) is recorded at $z=0.2\text{ cm}$. It is seen that the main lobe is still absent and the self-healing does not happen. After a distance of $z=0.6\text{ cm}$, as shown in Fig. 3(b), the main lobe appears and the reconstruction of the retrieved Airy pattern happens. Figure 3(c) is recorded at $z=1.0\text{ cm}$. It is clearly shown that the main lobe is almost reborn, and the self-healing of this field becomes very apparent. From the above results, it is seen that the retrieved Airy pattern with the blocked main lobe can reconstruct itself when it propagates a certain distance. It is proved that the retrieved probe field preserves the self-healing property of Airy wavepackets.

4. Conclusion

We have experimentally investigated the storage of 2D Airy wavepackets in an EIT-driven solid-sate medium. The weak probe light is modulated by Airy wavepackets in the transverse profile. In an EIT three-level experimental system, the probe Airy wavepackets are stored into atomic spin coherence by switching off the control field. After a certain storage time, the stored Airy wavepackets are retrieved from the experimental medium by the opposite process. The retrieved Airy wavepackets keep a high fidelity in EIT-based storage. The self-healing property of the retrieved Airy wavepackets is further investigated. Such an experimental demonstration extends the manipulation method of Airy wavepackets, and is expected to be useful for the manipulation of light beams and image information processing.