## Abstract

We proposed a simple method to realize optical Fourier transform during the nonlinear wave shaping processes. In this method, an integrated optical superlattice is designed to realize multiple optical functions, which plays important roles in both the nonlinear harmonic generation process and the optical Fourier Transform process. We demonstrated our method by the nonlinear generation of Airy beams as an example. It is a universal method for beam shaping and is of practical importance for designing compact nonlinear optical devices.

© 2015 Optical Society of America

## 1. Introduction

Nonlinear beam shaping (NBS) is a hot topic in nonlinear optics in the past several years [1–4], harmonic waves with shaped wave-fronts can be achieved with this technique during nonlinear optical processes. NBS can be realized in the microstructure material with nonlinear susceptibility ${\chi}^{(2)}$ being artificially modulated [5] (hereafter we call it an optical superlattice (OSL) [6]). By changing the interaction length [7] or introducing a transverse modulation to the OSL [1, 8], both the phase and the amplitude of the harmonic wave can be manipulated. NBS is effective for generating harmonic waves with complicated wave-fronts. Special beams such as Airy beams, Laguerre-Gaussian beams and vortex beams have been successfully generated with this technique recently [1, 4, 9–11].

To date, various methods have been developed for NBS. In 2009, T. Ellenbogen *et al.* proposed that Airy beams can be generated in an asymmetric nonlinear photonic crystal, which induces a wave-front with cubic phase (the Fourier Transform (FT) of the Airy function) [1]. This is a reciprocal-space method and in the shaping process optical FT is needed. Recently, we put forward a real-space method to generate nonlinear Airy beams by introducing the concept of nonlinear volume holography [4]. In this way no optical FT is needed [8].

It is noticed that optical FT is generally required for the reciprocal-space nonlinear shaping method and is usually performed with a lens in a linear optical manner. That is, if a light beam propagates through a lens, the FT image can be obtained at the back focal plane, and the OSL structure makes no contribution to the optical FT process. In this paper, we proposed another method to realize NBS. It is well known that the key of optical FT is to provide a quadratic phase-factor [12]. In our method, this phase-factor is provided by the OSL rather than a lens, thus the OSL makes contributions to both the nonlinear harmonic generation process and the optical FT process. Although a lens is still needed but its function is quite different from the previous approach. We have designed an OSL structure for realizing optical FT based on the above method by taking the Airy beams as an example. Numerical simulations have been performed to verify the theory, and the result coincides well with the theoretical predictions.

## 2. Theory

We consider a two-dimensional second-harmonic generation (SHG) process, where the fundamental wave is propagating in the *x*-direction and can be expressed as ${E}_{1}(x,y)={A}_{1}(x,y)\mathrm{exp}(i{k}_{1}x)$, here ${A}_{1}$and ${k}_{1}$are the amplitude and wave vector of the fundamental wave. The corresponding second-harmonic (SH) wave can be expressed as ${E}_{2}(x,y)={A}_{2}(x,y)\mathrm{exp}(i{k}_{2}x)$, where${A}_{2}$and ${k}_{2}$are the amplitude and wave vector of the SH wave. Assuming that the required FT image of the SH wave at *x* plane is$\mathrm{exp}(i\phi )/2\pi $, the corresponding OSL structure which is able to satisfy the quasi-phase-matching condition and provide the quadratic phase for the SH wave can be designed as:

The OSL designed by Eq. (2) can provide multiple optical functions. As a result, the phase item in the structure function can be considered as the sum of two parts. The first part is used to generate the required SH wavefront and the other is used to generate the additional quadratic phase factor. Thus we call it a “x+2” structure.

During the SHG process, the item $\Delta kx$ in Eq. (2) is used to satisfy the quasi-phase-matching condition for the nonlinear process. The generated SH wave in this structure will take a suitable quadratic phase, which is resulting from the transverse components in the structure function, the amplitude of the SH wave at the exit surface of the OSL (plane $x=0$) could be expressed as:

where $A$is a constant related to the conversion efficiency. According to the Fresnel diffraction theory, the amplitude of the SH wave after a propagation distance $x$ can be written as:Taking the Airy beams as an example, to provide the cubic phase (required by the Airy beams) and the quadratic phase (required by the Optical FT) simultaneously, the structure function of the OSL should be designed by:

From Eq. (6) we can see that the obtained structure function is similar to the cubic structure used in [1] but with an extra phase factor of ${k}_{2}{y}^{2}/2f$, and the resulting structure will be much like but not a strict cubic structure.## 3. Results and discussion of numerical simulation

For simplicity, we only consider the ee-e nonlinear process for SHG in the nonlinear OSL. The wavelengths of the fundamental wave and SH wave are 1064nm and 532nm, respectively. LiNiO_{3} is used as the nonlinear material. The size of the OSL is 500$\mu m$ × 1000$\mu m$. The OSL structure is periodic in the x direction and the perioid is equal to $2\pi /\Delta k$, which can be deduced directly from Eq. (6). Part of the structure is shown in Fig. 1(a). The focal length *f* is set to be 5mm. A Gaussian beam with 300$\mu m$ waist radius is chosen as the fundamental wave. And ${C}_{c}$ is set to be 5 × 10^{-4$\mu {m}^{-3}$}.

It is well known that Airy beams have the properties of non-diffracting and self-acceleration [13]. The intensity distribution of SH waves has been calculated numerically with a finite difference method [14]. From the numerical results shown in Fig. 1(b), we can see that the SH wave generated by the “x+2” OSL is still diffractive. However, it becomes non-diffractive completely if a lens is properly placed, which is shown in Fig. 1(c) (the translucent white line marks the location of the lens for phase counteracting). This phenomenon agrees well with the theoretical analysis presented in Part 1. It is obvious that the SH wave also propagates with a transverse self-acceleration after the lens. Figure 2 shows the intensity patterns after the lens at *x* = 5mm, 7.5mm, 10mm and 12.5mm, respectively. Along with the propagation, the intensity distribution of the SH wave keeps the profile of the Airy beams, and the distance between the lobes almost remains unchanged. Through the analysis of self-accelerating and intensity distribution of the profile we can see that it coincides well with the properties of the Airy beams.

As is well known that there are two different optical FT methods that commonly used in linear optics [15, 16]. One method only uses a single lens, and the other method uses two lenses. Both of these two methods can obtain strict optical FT at the conjugate plane. It is interesting to see that the realization of optical FT discussed in [1] just corresponds to the single lens case (as shown in Fig. 3(a)), while the FT process studied in this paper is quite similar to the two-lens method, except that the first lens is no needed now and its function is integrated in the OSL structure (as shown in Fig. 3(b)).

It is worth mentioning that though a lens is still needed in our method, but its optical function is quite different from the one-lens system. We can see that the amplitude of the SH wave has been formed without the lens in Eq. (5), the lens is used to provide a phase counteracting for the generated wave at the focal plane instead of performing a full optical FT. Furthermore, to make the optical device more compact, it is possible to replace this lens by any other optical components which have similar property, such as an OSL with quadratics phase structure in a cascaded nonlinear processes.

## 4. Conclusion

In conclusion, we have proposed a new method to realize NBS. In this method, the required optical FT process is mainly realized by the OSL structure rather than in a linear manner. The realization of optical FT in this method is similar to the two-lens system in linear optics, except that only one lens is needed and the optical function of the other lens is integrated in the OSL structure. The nonlinear generation of Airy beams has been taken as an example for demonstration. This method is universal for beam shaping and can be applied to the generation of other special beams without difficulties. The only change needed is replacing the $\phi $ in Eq. (2) by any other wave-fronts that required. And owing to the participation of the OSL in optical FT, it is possible to design a more compact optical system with this method.

## Acknowledgment

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11274163, 11274164 and 11374150). The authors also thank Prof. J. P. Ding for useful discussions.

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