PHYSICAL REVIEW A 100, 043819 (2019) Optical mode conversion through nonlinear two-wave mixing D. G. Pires, J. C. A. Rocha, A. J. Jesus-Silva, and E. J. S. Fonseca * Optics and Nanoscopy Group, Universidade Federal de Alagoas, 57061-670, Maceió, AL, Brazil (Received 3 May 2019; published 14 October 2019) Hermite-Gaussian (HG), Laguerre-Gaussian (LG), and Ince-Gaussian (IG) modes have been of considerable importance for photonics. They have revealed a unique way to optical manipulation and are particularly promising for optical communications. This paper combines HG modes, as a basis, inside the nonlinear crystal and the generated second harmonic field turns to LG and IG modes. Here we present a way to use second order nonlinear media to convert optical beam modes as we wish. DOI: 10.1103/PhysRevA.100.043819 I. INTRODUCTION Undoubtedly, photonics achieved a great position in the present scientific picture. Among the many applications it has led us to, the enhancements in the communication system and optical micromanipulation are the most highlighted ones. Spatially structured light, which can be defined as a bidimen- sional design of the transverse higher-order light modes, is one of the ways to code information or be used in optical tweezers and trapping. The potential of higher order spatial light modes with defined, complex propagation properties, such as the Hermite-Gaussian (HG), Laguerre-Gaussian (LG), and, recently, Ince-Gaussian (IG) modes, has attracted steady attention of researchers over the last decades. Particularly, LG modes are of highest interest in optical micromanipulation [1] and communication [2]. Since its discovery [3], many studies on fundamental properties [4,5], optical tweezers [6], spin- orbit coupling [7], teleportation schemes [8,9], imaging [10], manipulation of ultracold atoms [11], and quantum protocols [12,13] to cite a few, were realized. On other hand, nonlinear responses can be useful in order to mediate some photonic processes. Nonlinear media can bring very interesting effects like second harmonic generation [14] and sum-difference frequency generation [15], optical parametric oscillation [16], parametric fluorescence [17], non- linear mixing [18,19], and four-wave mixing in atomic media [20,21]. The beam coupling due to two-wave mixing has been studied [22–24], showing how optical vortices behave in the second harmonic generation (SHG). The nonlinear media acts like a mode selector, leading to mode superposition in the second harmonic field. This effect happens when the wave mixing is under longitudinal and transversal phase match, and this role is played by the overlap integral within the nonlinear media. Optical mode conversion is mainly realized by using diffractive and linear optics [25–27]. Here we present a way to use second order nonlinear media to convert optical beam modes as we wish. It is well known that HG and LG are transversal eigenmodes of typical laser resonators and can * efonseca@fis.ufal.br easily be created with high efficiency [28,29]. These two modes separately form two complete families of exact and orthogonal solutions of the paraxial wave equation (PWE) in rectangular and cylindrical coordinates, respectively. More recently, IG modes have been proposed as a third complete family of PWE solutions in elliptic coordinates [30]. By using HG modes as a basis, we combine multiple beams inside the nonlinear crystal and the generated second harmonic field turns as a combination of HG modes, leading to the conversion into LG and IG modes. In order to better visualize the mode decomposition, Fig. 1(a) shows a Poincaré sphere for a set of mode conversions in a particular HG basis. Modal conver- sion and decomposition of HG modes in the spatial-temporal degree of freedom were considered before for tailoring its temporal-mode structures [31–34]. The results presented here, together with previous works on spatial-temporal control of structured beams [35,36], are important for communication systems. II. NONLINEARTWO-WAVE MIXING OF HERMITE-GAUSSIAN BEAMS In the paraxial approximation, the second harmonic gen- eration under longitudinal and transversal phase match con- ditions is described by a coupled evolution for the incoming fields U h and U v , orthogonally polarized in respect of each other and frequency ω, generating a field U 2ω [24,37]. The expansion of these fields in an orthonormal basis is [24] U j =  ω j n j  m,n A j mn u j mn (r, z ), (1) where n j is the refractive index for the U j field, u j mn is the mode basis, and A j mn is the amplitude with j = h, v, 2ω. Although the basis of the expansion is arbitrary, it must be chosen carefully. In our case, the bases used are the HG modes. Figure 1(c) shows a sketch of the nonlinear two-wave mixing process considered here. Notice that the generated field U 2ω is a sum-frequency field but, since its frequency is 2ω and for convention reasons, we will call U 2ω as a second harmonic generated field. 2469-9926/2019/100(4)/043819(7) 043819-1 ©2019 American Physical Society