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Abstract
We propose and demonstrate nonlinear amplifications of self-accelerating Airy beams by two-wave mixing in photorefractive crystals both numerically and experimentally. By employing a broad Gaussian beam as the pump beam, we show that weak signal Airy beams can be significantly amplified under both diffusion and drift mechanisms. It is revealed that not only higher optical gains but also faster response time can be achieved in the presence of an external electric field, where the drift mechanism dominates. We verify that the self-accelerating and self-healing characteristics of the Airy beams are well preserved during the nonlinear amplification.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Self-accelerating Airy beams has been growing into a research focus in optical community at large in the past ten years since the first introduction of optical Airy wavepackets from quantum mechanics [1–3]. The self-bending and self-healing features in addition to the nondiffracting nature of the beam enables many attracting applications including generating curved plasma channel in air [4], guiding surface plasmons on metallic interfaces [5–7], manipulating micro-particles [8,9], light bullets [10], microscopy [11,12], and laser micromachining [13], just name a few. Moreover, the study has been extended into other wave systems including acoustics [14], electron beams [15], and water waves [16]. In general, self-accelerating nondiffracting beams represent exact solutions of wave equations and possess infinite energies associated with their endless tail structures [1–3,17–20]. In practical applications, certain truncations have to be imposed onto the beam profiles, which will unavoidably lead to severe attenuations of the main lobes after long distance propagations. Especially for those applications involving strong light matter interactions, optical loss due to absorption is another factor of major concern. Thus far, many linear and nonlinear strategies have been developed to alleviate the problem, e.g., constructing attenuation-corrected input beam profiles [21,22], nonlinear solitonary balancing [23,24], as well as compensation with active Raman gain [25]. However, all those aforementioned methods represent internal means, relying either on beam designs or the nonlinear actions of the beams. In this Letter, we propose and demonstrate an external approach for amplifying Airy beams with an optical pump beam through two-wave mixing [26,27]. As well known, such optical amplifications realize the energy transfer through the light-induced phase grating of refractive index, which does not introduce quantum noise in the process of optical signal amplification.
Here we demonstrate the optical amplifications of Airy beams by utilizing a photorefractive strontium barium niobate (SBN) crystal, which has been serving as a versatile ground for exploring different nonlinear effect of Airy beams [23,28]. A broad Gaussian beam is utilized as an optical pump beam, which is coherent with the signal Airy beam. Both drift and diffusion photorefractive mechanism are exploited for the Airy beam amplifications. It is revealed that Airy beams can be amplified under both mechanism. By adjusting the external bias field in the crystal, not only the response time but also the magnitude of the amplification can be controlled at ease. The self-accelerating and self-healing properties are also verified under nonlinear amplifications, which further proves two-wave mixing offers an effective way for amplifying Airy beams.
2. Methods
The experimental setup for optical amplification of Airy beams is shown in Fig. 1. A photorefractive SBN:60 crystal with the dimension of 5mm × 5mm(c) × 10mm is used. A beam splitter splits the incident laser beam at wavelength of 532 nm into two parts. One serves as a pump beam impinges onto the SBN crystal after being reflected by the mirrors M1 and M3. Another beam passes through a pair of orthogonal cylindrical lens providing a 2D cubic phase modulation [29] after being reflected by the mirror M2, serving as the signal beam. After being modulated by the cylindrical lens pair, the signal beam is turned into an Airy beam. The powers of the pump and the signal beams are set to be 0.3W/cm2 and 0.3mW/cm2, respectively. Therefore the pump-to-signal power ratio is 1000:1. The signal Airy beam and the pump beam enter the SBN crystal at an angle 2θ = 15° along the axis perpendicular to the crystal c-axis. To study the photorefractive two-wave-mixing under drift mechanism, an external electric field along the crystal c-axis is applied. Otherwise the diffusion mechanism will be dominant during the optical amplification [27]. After the crystal, the signal Airy beam is detected by a CCD camera through an imaging system with a spatial filter. Note that although the signal and pump beams possess identical wavelength and share the same polarization, the signal beam can be easily separated from the pump by spatial filtering due to the distinct differences between the Fourier spectra of the two beams. Therefore, in practice, one simply launch an Airy beam into the two-wave mixing system, and the exiting beam possesses the same beam structure but with an amplified intensity.
Fig. 1 Experimental setup for optical amplification of Airy beams with photorefractive two-wave-mixing. SBN, strontium barium niobate crystal; M, mirror; BS, beam splitter; LS, lens system with a spatial filter; CL, cylindrical lens. Note that the CLs are composed of two titled orthogonal cylindrical lens, providing the cubic phase modulation for generating Airy beams.
To help us understand the two-wave mixing process of Airy beams in the photorefractive crystal, we perform numerical simulations on the static nonlinear propagations of an Airy beam along with the pump beam, which can be described by the following lossless nonlinear Schrödinger equation [30,31]
where U = U1 + U2 is the wave function of the input Airy signal beam (U1) and the Gaussian pump beam (U1), k0 is the vacuum wave vector, n0 is the unperturbed refractive index, and γ33 and E0 are the crystal electrooptic coefficient and the amplitude of the bias field along the crystal c-axis (x-axis). When E0 = 0, the nonlinearity is solely determined by the last term in Eq. (1), representing a diffusion mechanism. With the increase of E0, the drift mechanism will be gradually taking over. The wave function of a two-dimensional Airy beam can be expressed aswhere Ai denotes the Airy function, X and Y are respectively equivalent to (x + y)/2 and (-x + y)/2, x0 and y0 are constants governing the transverse size of the Airy beam, and α is the decay factor for the truncated beam profile. The simple broad pump beam is described by a Gaussian function along with a tilt factorwhere x1 and y1 determines the transverse profile of the pump Gaussian beam.3. Results and discussion
As shown in Fig. 2, the signal Airy beam depicted in the top panel of Fig. 2(a) is launched into the SBN crystal at E0 = 0 along with a broad Gaussian pump beam (bottom panel) at an angle 2θ = 15°. The nonlinear interaction between the mutually coherent signal and pump beams can be simulated by numerically solving Eq. (1). Figure 2(b) describes the nonlinear evolution of the two beams. It can be seen that the signal Airy beam does not change much over the propagation, while the pump beam transfers energy to the signal beam through the diffraction of the optically induced grating corresponding to the interference pattern of the two beams. The Airy beam components originated from the pump beam, which can be clearly identified from the bottom panel of Fig. 2(c), constructively combine with the signal Airy beam, displaying as an optical gain.
Fig. 2 Numerical simulation of the optical amplifying of an Airy beam with photorefractive two-wave-mixing. (a) The top panel depicts the Airy beam, which is propagating through the SBN crystal and being pumped with a tilted broad Gaussian beam as shown at the bottom panel. (b) The nonlinear evolutions of the Airy (top) and pump(bottom) beams inside the crystal, where the intensity profiles are taken along the vertical direction through the center of the main lobes. (c) Zoom-in plot of the transverse intensity pattern of the beam along the dashed line in (b).
Our static and dynamic experimental results are shown in Fig. 3 and the corresponding quantitative measurements are summarized in Table 1. By setting the external bias field at different levels, we monitor the time evolution of the amplified airy beam by the CCD camera. At E0 = 0, the buildup of the optical induced spatial charge field in the SBN crystal will be dominated by the diffusion mechanism, while the drifting process will be gradually taking over with the increase of E0. In our experiment, three different electric field E0 = 500V/cm, 1000V/cm, and 2000V/cm are applied in addition to the pure diffusion case at E0 = 0. As we can see from Fig. 3(a), at all the bias levels, the power of the amplified Airy beam begins to decline with the removal of the signal light, representing a typical time response of photorefractive two-wave mixing processes [27].
Fig. 3 (a) Time resolved output power of the Airy beam at different bias voltages. (b-f) The intensity pattern of the amplified Airy beam at different bias conditions, where the top and bottom panels are corresponding to experimental and simulation results, respectively. (b) represents the initial linear output intensity pattern, and (c-f) displays the nonlinear ones at E0 = 0V/cm, 500V/cm, 1000V/cm, and 2000V/cm, respectively.
Table 1. Output Power, Gain, and Response Time at different external bias conditions.
Under the diffusion mechanism, when there is no external electric field is applied on the SBN crystal, the optical gain of the Airy beam can be achieved as expected. However, the response time is relatively long. As illustrated in Table 1, by increasing the bias voltage, not only can the response time be significantly shortened, but also higher optical gain can be obtained at a higher bias level. In particular, when the applied electric field is increased to 2000V/cm, the gain of the amplified Airy beam reaches 41cm−1 and the response time decreases to 130s, in comparison with 12cm−1 and 410s for the pure diffusion case, respectively.
The amplified Airy beam intensity patterns at the steady state of different bias conditions are depicted in Figs. 3(b)-3(f), where the gradually saturated intensity reflects the increasing gains at different conditions. It is clear that although the Airy beam has a higher optical gain at a higher external electric field, however the amplified Airy beam is accompanied by a distortion as indicated from Fig. 3(e), which becomes more prominent at higher applied voltages [see Fig. 3(f)]. Apparently, under the action of the externally applied electric field, the migration process of the photo-induced carriers can be accelerated, therefore leading to a shorten response time of the photorefractive effect [27]. With the increase of the bias field as well as the power of the amplified signal Airy beam, an apparent ringing effect shows up, which is consistent with the apparent distortion of the Airy beam profile as shown in Fig. 3(f) [32]. Therefore, there is a compromise of the beam quality, when using an external bias field to achieve faster response time and larger optical gain during the amplification of Airy beams with two-wave mixing.
Now we have shown that optical energies can be transferred from the optical pump beam to the signal Airy beam during the nonlinear two-wave mixing process. It is natural to ask whether if the two most important characteristics of the signal Airy beam, i.e., self-bending [2] and self-healing [33] could be preserved during the nonlinear amplification. First we examine the self-bending propagation of the Airy beam during the two-wave mixing. Our experimental results along with the numerical simulations are presented in Fig. 4. Note that the z = 0 position marks the peak of the parabolic trajectory of the Airy beam [2,33]. It is clear that the nonlinear amplification process does not impact the self-bending property of the Airy beam at all. The amplified Airy beam propagates along the identical trajectory as the original one, except that the beam power becomes much stronger [see Figs. 4(b) and 4(c)]. We mention that at a higher bias level, the self-bending trajectory will be maintained, however the transverse pattern suffers from severe distortions.
Fig. 4 Self-accelerating of an amplified Airy beam. (a) Experimentally observed transverse intensity patterns of the amplified Airy beam at different propagation distances, where the solid line depict the curved trajectory of the main lobe of the original Airy beam without amplification. (b) and (c) depict the propagations of the Airy beam in the absence and presence of the pump beam, respectively, where the intensity profiles are taken along the vertical direction through the center of the main lobe of the Airy beam.
In order to demonstrate the self-healing properties of amplified Airy beam (as they have been perturbed or partially blocked), we monitor its self-reconstruction against propagation. In our experiment, we use an opaque rectangular obstruction to block the main lobe of the Airy beam in its initial intensity distribution, which contains a large percentage of the power of the beam. The resulted intensity distribution is shown in Figs. 5(a) and 5(b). Figures 5(c) and 5(d) depicts the reconstruction of the amplified Airy beam after a distance of z = 10cm. The self-healing evolution of the beam is numerically confirmed with our simulation, as shown in Fig. 5(e). It is clear that the nonlinear amplified Airy beam possesses the same self-healing capability as the original beam.
Fig. 5 Self-healing of an amplified Airy beam when its main lobe is blocked. (a-d) Intensity patterns at the input z = 0 (a, b), and z = 10cm (c, d). (a, c) and (b, d) are experimental and simulation results respectively. (e) describes the numerically simulated evolution of the intensity profile along the dashed line in (b) during the self-healing propagation.
4. Conclusion
In summary, we have numerically and experimentally demonstrated the nonlinear amplifications of Airy beams with photorefractive two-wave mixing. We found that both diffusion and drift mechanisms can be employed for optically amplifying the Airy beam. By changing the external bias level, both response time and the magnitude of the amplification can be adjusted. We proved that the amplified Airy beams possess all the intrinsic properties of the original beams, including self-accelerating and self-healing. Our approach has a far-reaching impact on long-range Airy signal transmission and can be extended to other nonlinear media and wave systems.
Funding
National Natural Science Foundation of China (NSFC) (11304250, 11574389), Ningbo Natural Science Foundation (ZX2015000617) and K.C.Wong Magna Fund in Ningbo University, China.
Acknowledgments
P.Z. acknowledges the support from the One Hundred-Talent Plan of Chinese Academy of Sciences.
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