Nonlinear vortex beam array generation by spatially modulated fundamental wave

We experimentally demonstrated the generation of a dynamic nonlinear vortex beam array by utilizing a fundamental wave with a modulated phase structure, which was incident into a homogeneous nonlinear medium. In our experiment, one-dimensional and twodimensional second harmonic vortex beam arrays were investigated, and the topological charge of second harmonic vortex beam of different order was measured. This study presents a method of dynamic control of the nonlinear vortex beam array, which may have applications in multiple-particles optical trapping, optical communication, and so on. © 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement OCIS codes: (190.0190) Nonlinear optics; (050.1970) Diffractive optics; (190.2620) Harmonic generation and mixing; (080.4865) Optical vortices. References and links 1. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. 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Introduction
It is well known that optical angular momentum is composed of spin angular momentum and orbital angular momentum (OAM) [1].The OAM of light was firstly observed in 1930s [2].In 1992, Allen et al. verified that OAM has an azimuthal phase term il e ϕ , where l and ϕ represent the topological charge and azimuth angle, respectively [3].In the last two decades, optical OAM has attracted great interest due to its potential applications in various research fields, such as optical trapping [4][5][6][7][8], quantum computation [9], optical communications [10].In the field of optical trapping, the beam carrying OAM can be used to rotate particles with different angular velocities [4][5][6][7][8].Quantum computation and quantum imaging have been realized by utilizing the entanglement of optical OAM [9].In addition, such optical OAM could also be used to increase the capacity of optical communications [10].The nonlinear generation process enables vortex beams to be obtained at new wavelengths, which opens up new possibilities for all-optical switching and manipulation of vortex beams.Hence, second harmonic generation (SHG) [11][12][13][14], sum-frequency generation [11][12][13], difference frequency generation [12] and even high-harmonic [15][16][17] of the vortex beam has been investigated.To get SHG of the vortex beam, there are two main methods.One is to manipulate the structure of nonlinear photonic crystals (NPCs) [11,13].The other is to pattern the output facet of a nonlinear crystal [18,19], which combines both the nonlinear process and the beam spatial shaping.However, these two methods possess the same drawbacks, including complex fabrication and unchangeable nonlinear wave patterns.To solve these problems, some researchers turn their attention to pre-shaping the fundamental wave (FW) before its incidence into a nonlinear crystal [14,[20][21][22][23].In our previous researches, we have already introduced the structured FW into nonlinear frequency conversion processes.For example, the nonlinear Raman-Nath SHG of different types of FW was achieved [24].
In this paper, we experimentally demonstrated the nonlinear vortex beam array dynamically generated by utilizing the FW with a modulated phase structure via a spatial light modulator (SLM), which was incident into a homogeneous nonlinear medium.In the onedimensional (1D) case, a series of second harmonic (SH) vortex beam was achieved in 1D direction.In the two-dimensional (2D) case, SH vortex beam array in 2D direction was generated.Besides, the topological charge of SH vortex beam of different order located in such SH vortex beam array was also experimentally measured.

Theoretical analysis
The phase structure of the FW was periodically and sharply modulated from 0 to φ into fork-shaped structure.The diffraction of FW is negligible in case of long-period modulation of the FW in a sufficient short medium [24].And supposing the light propagates along the yaxis (longitudinal direction) of the crystal, in 1D case, the FW is given in the form of the Fourier series by: ( ) ( ) where 1 A and 1 k are the amplitude and wave vector of the FW, respectively.m C is the Fourier coefficients ( ( ) is the azimuthal angle, ( ) ( ) is the spiral structure function, Λ is spatial period of the plane wave, 1 l is the topological charge of the FW.The SH can be expressed as ( ) , where 2 A , 2t k  and 2 y k represent the amplitude, transverse and longitudinal wavevector, respectively.Under the non-depletion assumption, the evolution of the SH wave can be written as [15]: where , is the Fourier coefficient of the nonlinear polarization.In Eq. ( 2), the first exponential term denotes the longitudinal phase mismatch, the middle exponential term is the nonlinear Raman-Nath diffraction, and the last exponential term represents the topological charge of the generated SH vortex beam.As can be seen, the topological charge of the generated SH vortex beam obeys the law of When the phase is modulated in 2D, the FW can be expressed as where x l and z l are the topological charges of the FW along x and z direction, respectively.Then the coupled wave equation of SH can be written as:

Experimental results and discussion
In our experiments, a Nd:YAG nanosecond laser with a wavelength and a repetition rate of 1064 nm and 20 Hz, respectively, was applied.The sample was a homogenous 5-mol% MgO:LiNbO 3 crystal (10 × 0.5 × 10 mm 3 in x × y × z dimensions).The FW was an ordinary wave and incident into the crystal along the y direction.The nonlinear process was Type-I (oo-e) phase matching.The FW was modulated by a SLM, whose resolution is 512 × 512 pixels and each with a square area of 19.5 × 19.5 2 m μ .A 4-f system was used to imprint the phase structure on the nonlinear crystal and the beam waist was 2 mm.A shortpass filter was placed after the crystal to obstruct the FW.At last, the generated SH vortex beam array was projected on a screen and then recorded by a camera.In order to illuminate the relationship between the generated SH vortex beam array and the structure of the FW, we firstly imprinted different fork-shaped structured FW in 1D case onto the nonlinear crystal.The structure of the FW is shown in the left column of Fig. 1, where the fork-shaped phase structure is 0-π/2 sharply modulated.The duty cycle is 0.5.The corresponding experimental results of the SH vortex beams are shown in the right column of Fig. 1, where u represents the Raman-Nath order.The phase singularity (dark core) always exists as long as SH l is nonzero.The radius of the SH vortex ring gets larger as the topological charge increases.According to Eq. ( 2), the topological charge of the SH vortex beam obeys the law of for the uth order SH.In order to determine the topological charge of the vortex beam in different orders, we measured it by utilizing a cylindrical lens [25,26].The results are displayed in Fig. 2, which corresponds to the 1, 2, 3 SH l = ± ± ± orders of the SH vortex beam in the case of 1 1 l = .The number of dark stripes equals to the topological charge of the SH, which agrees well with the theoretical prediction.
) if u is odd.
In addition to the generation of SH vortex beam with integer topological charges, the case of fraction topological charge was also investigated.The FW structure is shown in the left column of Fig. 4 and the corresponding experimental results are shown in the right column.It is interesting that the SH vortex beam was not a closed ring when SH l was not an integer.In this case, a dark stripe exists on the SH vortex beam according to Berry's theory in 2004 [27].As the topological charge of the SH vortex beam was further tuned away from the integer, the gap becomes larger.For example, the gap is the largest when SH l is a half-integer ( 2.5 SH l = ).When SH l is close to an integer (such as 2.9 SH l = shown the + 1 order in the right column of the last row of Fig. 4), the SH vortex beam returned to a ring.Similar to the integer cases of SH vortex beam ( 0 SH l ≠ ), the phase singularity still exists and the radius of the ring is determined by the topological charge SH l .To further reveal the flexibility of our method, the 2D SH vortex beam array generation was studied.The structure of the FW is modulated along the x and z direction, as shown in the top row of Fig. 6, where x l and z l denote the topological charge of the fork-shaped FW along the x-axis and z-axis, respectively.The 2D SH vortex beam array was achieved, as shown in the bottom row of Fig. 6.The topological charge of the SH vortex beam located in the SH vortex beam array can be written as SH  Compared with the method of modulating NPCs, our method is much more convenient, since it is dynamically controlled.Different from [14] where the authors conducted the experiment making use of diffraction of FW, we paid much attention to the nonlinear diffraction process.In ref [14], two FWs with different topological charge were incident into the nonlinear crystal.In our experiment, the whole phase structure of the FW loaded on the SLM was printed onto the nonlinear crystal.Hence, the completely different experimental phenomena were obtained.SH vortex array in 1D and 2D cases are also systematically studied in our work.

Conclusion
In summary, we experimentally demonstrated the nonlinear vortex beam array dynamically generated by utilizing the FW, in a homogeneous nonlinear medium.In 1D cases, the topological charge obeys the law of in such SH vortex beam array was also experimentally measured.This study presents a method of dynamic control of the nonlinear vortex beam array, which may have applications in multiple-particles optical trapping, optical communication, and so on.

Fig. 1 .
Fig. 1.Left column: The different phase structures of the FW with 0-π/2 phase modulation.Right column: The generated SH vortex beams in different orders of nonlinear Raman-Nath diffraction. 1 SH l ul = for the uth order SH.

Fig. 3 .
Fig. 3. Left column: different phase structures of the FW with 0-π phase modulation.Right column: The generated SH vortex beam in different orders of nonlinear Raman-Nath diffraction. 1 SH l ul = for the uth order SH.When the FW phase was sharply modulated from 0 to π , the odd orders of SH vortex beam vanishes, as shown in Fig. 3.Because the Fourier coefficients m C are zero when m is even, as a result 0 u b = (

Figure 5
Figure 5 shows the process after the vortex beams passing through the cylindrical lens with the topological charge changing from 2 to 3. The original two dark stripes correspond to 2 SH l = .With the fractional topological charge of SH vortex beams increasing, another dark stripe comes into being and becomes more and more obvious.At last, the right bright stripe splits into two stripes completely when SH l is close to 3.

Fig. 5 .
Fig. 5. Evolution of dark stripes with the topological charge changing from 2 to 3.
ul and z vl are the topological charge along x and z-axis, respectively.The topological charges of SH vortex beams of different orders are measured to be identical to the analysis (

Fig. 6 .
Fig. 6.Top row: The different phase structures of the FW with 0-π/2 phase modulation.x l and . In 2D case, SH vortex beam array in 2D direction can be generated and the topological charge obeys the law of , the topological charge of SH vortex beam of different orders located