Intense harmonics generation with customized photon frequency and optical vortex

An optical vortex with orbital angular momentum (OAM) enriches the light and matter interaction process, and helps reveal unexpected information in relativistic nonlinear optics. A scheme is proposed for the first time to explore the origin of photons in the generated harmonics, and produce relativistic intense harmonics with expected frequency and an optical vortex. When two counter-propagating Laguerre–Gaussian laser pulses impinge on a solid thin foil and interact with each other, the contribution of each input pulse in producing harmonics can be distinguished with the help of angular momentum conservation of photons, which is almost impossible for harmonic generation without an optical vortex. The generation of tunable, intense vortex harmonics with different photon topological charge is predicted based on the theoretical analysis and three-dimensional particle-in-cell simulations. Inheriting the properties of OAM and harmonics, the obtained intense vortex beam can be applied in a wide range of fields, including atom or molecule control and manipulation.


Introduction
The vortex beam exhibiting a helical wavefront can be applied as a powerful probing tool in studying cold atoms, atomic transition [1][2][3][4][5][6][7], design and operation of micromachines [8], optical communications and quantum space [9][10][11] and even astrophysics [12]. Such a beam can even transfer orbital angular momentum (OAM) to material particles, rotate them, or manipulate their vortex characteristics when the intensity is sufficiently high. Recent researches have proved its possible usage in particle acceleration [13], and the vortex beams can be amplified and generated to petawatt intensities in plasmas through stimulated Raman backscattering [14]. The helical characteristic of a vortex beam is described by the phase component of exp(ilf), where f is the azimuthal coordinate and the integer number l is the topological charge [15][16][17]. Different values of l mean that the photon has different vortex characteristics and carries different OAMs. Owing to the limitation of the etching resolution, it is difficult to obtain a vortex beam of high topological charge through conventional methods such as forked diffraction gratings [18,19] or spiral phase plates [15]. There have been some efforts, such as the use of a helical undulator [20][21][22], Compton backscattering effects [23,24], and high-order harmonics generation (HHG) [25][26][27][28], devoted to generating a light beam with OAM. Among them, the HHG scheme has an extraordinarily promising perspective because of the confluence of OAM and HHG, and experimental results have been demonstrated [26,28]. In the previous HHG scheme, one incident pulse is used, and only certain topological charges of the HHG are generated and cannot be tuned. Thus, the application will be inevitably limited to some extent when a light beam with a specific OAM and frequency is required.
In this paper, we present a new way to generate tunable intense vortex beams of high topological charge from laser-plasma interactions, in which the frequency and mode of the radiation harmonics can be tuned to meet different requirements. More importantly, this is the first attempt to determine the origin of photons of the generated harmonics through the coupling process of high harmonics. In this scheme, two counter-propagating intense driving vortex beams with low topological charges impinge on a thin foil. Due to their interaction in the foil, high-order harmonics are radiated on both sides. With the help of a photon vortex, the contribution of input pulses in producing harmonics can be distinguished. The topological charge and order of harmonics can be designed by adjusting the parameters of the input pulses, which promises more possibilities with an additional pulse compared with the case of a single driving pulse.
Compared with the well-known gas HHG sources, the present scheme is in the relativistic (>10 18 W cm −2 ) laser plasma interaction range. Using plasma as the mode-converter, the intensity of the vortex beam can be increased by several orders of magnitude compared with that of the gas HHG scheme, in which it is impossible to generate such an intense vortex beam because of the limitations (often below 10 15-16 W cm −2 ) of the laser intensity to avoid strong ionization. The generation mechanism in the present scheme is independent of the input laser polarization state, which is different from that in our previous study, in which the high order vortex beam is obtained from the oscillating foil surface by using a linearly polarized laser pulse [27]. Here, the high harmonics with large OAM are radiated from the oscillating electrons, which are well confined by the two pulses, and most importantly, the two laser pulses allow better flexibility in the sense of generating OAM. The mode can be tuned by adjusting the driving pulses individually, thereby satisfying different requirements.

Theoretical analysis
The proposed scheme is shown in figure 1. Two counter-propagating circularly polarized (CP) Laguerre-Gaussian (LG) laser pulses a 1 and a 2 with the same rotation directions (both left handed or right handed CP) with frequencies and topological charges of (n 1 ω, l 1 ) and (n 2 ω, l 2 ), respectively, irradiate on a thin foil. For this scheme, electrons are pushed inside from both sides of the foil and are well confined within the foil by the radiation pressure. High harmonics are generated from the oscillating electrons. The foil is thinner than the skin depth to ensure full interaction of the two pulses inside the target. This is not the first time HHG have been produced with two pulses [29,30], however, LG pulses bring the high topological charge of harmonics, which is our main focus. To overcome the drawback of the limited multiplication factor with a single frequency conversion, the use of beam-echo effect induced by an additional laser beam has been found more efficiently in the HHG generation [31,32]. Here with the similar motivation, the main focus is that the high topological charge concerning to the large OAM can be tunable. The normalized vector potential (a=eA/m e c 2 , where A is the vector potential, c is the light speed in vacuum, m e is the electron mass, and e is the electron charge) of the laser amplitude inside of the electron layer can be written as a a a a n n t l l x n n t l l y n n t l l 2 cos 2 sin 2 sin 2 , where a 1 and a 2 are used as a a n t l x n t l y a a n t l x n t l y sin cos , sin cos for simplicity, since only phase terms are concerned, where a 0 is the normalized vector potential. The electron transverse velocity is ν=a/γ, which is the source term and contributes to the nonlinearity in Maxwell's equation Figure 1. Schematic of the proposed scheme. Two counter-propagating CPLG laser pulses a 1 and a 2 with frequencies and orders of (n 1 ω, l 1 ) and (n 2 ω, l 2 ), respectively, irradiate on a thin foil (n 1 =1, l 1 =1, n 2 =2, and l 2 =2).
where γ=(1+a 2 ) 1/2 is the relativistic factor, ν is normalized to the light speed c and n is the electron density. We only analyze the source term to interpret the dominant source of harmonics generation and the corresponding phase terms which originates from the electron movement under the field inside the foil, rather than an exact solution for all physical variables. From the Fourier expansion of this source term, we can see that it contains the following components: where n m1 (2) and l m1 (2) are the order and topological charge of generated harmonics and and m=0, 1, 2, K. From equation (1), two kinds of harmonics carrying the frequency and OAM information of the input pulses are generated, i.e., the n m1 order harmonics with a topological charge of l m1 and the n m2 order harmonics with a topological charge of l m2 can be obtained with this regime. Based on this, it can be concluded that a vortex beam with a specific topological charge and frequency can be obtained by combining the four given variables n 1 , n 2 , l 1 , l 2 to meet different requirements. Considering photons in the nonlinear process of the present scheme, for the n m1 order harmonic, each photon carries an energy n m1 ħω. For the input pulses without helicity, it is difficult to distinguish where the generated harmonic photons originate from or how many photons of pulse a 1 and a 2 are consumed in the nonlinear process. This is because the photons of two pulses cannot be distinguished completely just according to the energy conservation depending solely on the photon frequency. That is, one cannot determine the exact values of X, Y just from Xn 1 ħω+Yn 2 ħω=n m1 ħω, where X, Y are the photon numbers from pulse a 1 and pulse a 2 . However, when the photons of the input pulses have vortex properties, i.e., the OAM in the present scheme, there is an additional freedom degree l. According to the OAM conservation Xl 1 ħ+Yl 2 ħ=n m1 ħ, we can easily deduce that one photon of the n m1 order harmonic is transformed from (m+1) photons of pulse a 1 and (m) photons of pulse a 2 . Similarly, one photon of the n m2 order harmonic is transformed from (m) photons of pulse a 1 and (m+1) photons of pulse a 2 . It is same with the sum frequency generation in nonlinear optics. From equation (1), it can be seen that the harmonics of n m1 and n m2 order maintain the same rotation direction as pulse a 1 and a 2 , respectively, which means all helicities of photons including spin (polarization) [33] and orbit are conserved in this nonlinear HHG process. Using the helicity of the optical vortex in the present scheme, apart from the generation of the tunable, intense high-order optical vortex beam, we can determine the process that the photons undergo during the generation of different order harmonics, which is helpful to understand this nonlinear process in plasma.

Three-dimensional PIC simulations
To confirm the above analysis, three-dimensional (3D) particle in cell (PIC) simulations have been performed. We carried out all simulations using EPOCH code [34]. The driving LG beam is described as The left pulse a 1 has a frequency of ω corresponding to the wavelength of λ=1 μm and the right pulse a 2 has frequency of 2ω corresponding to the wavelength of λ/2=0.5 μm. r 0 =6 μm, and t 0 =5T are used for the two pulses, where T is the left driving laser period. The topological charges of the a 1 and a 2 pulses are l 1 and l 2 , respectively. For the left pulse, laser wavelength of λ=1 μm, a 0 =5 corresponds to a peak electric field intensity of 9.6×10 12 V m −1 . The same peak electric field for the right laser pulse is used. The thin foil with the thickness of 0.1 μm as the nonlinear converter occupies the region 9.95 μm<x<10.05 μm in the propagation direction of the driving beam and −13.5 μm<y(z)<13.5 μm in the transverse direction with a density of n 0 =20n c , where n c =1.1×10 21 cm −3 is the critical density for the left laser pulse. The simulation box is 20 μm (x)×30 μm (y)×30 μm (z), and corresponds to a window with 1000×400×400 cells and 100 particles per cell for the foil. At t=0, the laser pulses enter the simulation box from both boundaries.
The isosurface distribution of the laser electric field and the thin foil density in this simulation are shown in figure 1. Figure 2 shows the frequency spectrum of the whole laser field including the reflected and transmission parts, which shows that each third harmonic (3th, 6th, K) is missing. Here we note the amplitude of twin harmonics (1st & 2nd, 4th & 5th, and 7th & 8th) are almost equal because of the same intensity of the both pulses a 1 and a 2 . According to equation (2), for n 1 =1, n 2 =2, the frequency spectrum of the generated harmonics only contains (3m+1)ω and (3m+2)ω, which well explains the spectrum in the simulation. Most importantly, the corresponding topological charges of these harmonics are (m+1)l 1 +ml 2 and ml 1 +(m+1)l 2 , which means that the topological charges of the generated harmonics 1, 2, 4, 5, 7, 8, K, Figure 2. Frequency spectrum of the laser field after interaction between the input pulses and thin foil. The field signal at y=5 μm and z=0 μm is used. The peak electric field intensity is 9.6×10 12 V m −1 for both input pulses. (3m+1), (3m+2) should be l=1, 2, 4, 5, 7, 8, K, (3m+1), (3m+2) for l 1 =1, l 2 =2, and l=−1, 2, 0, 3, 1, 4, K, (m−1), (m+2) for l 1 =−1, l 2 =2. Figures 3 and 4 show the electric field isosurface of different order harmonics within one wavelength of the left pulse (1 μm) for the cases of l 1 =1, l 2 =2 and l 1 =−1, l 2 =2, and the corresponding transverse electric field distribution in the same y-z plane after the laser foil interaction. From these structures in the figures, the wavelength, helicity, and topological charge of the harmonics can be seen clearly, and are in good agreement with the theoretical analysis results. For example, the clear LG 80 -like mode is obtained, in which each photon has an OAM of 8ħ and the peak electric field intensity is of 10 11 V cm −1 in the case of l 1 =1, l 2 =2. It should be noted that in the case of l 1 =−1, l 2 =2, the field in the center of the fourth harmonics is close to zero due to the similar intensity distribution of the both input source fields, although its topological charge is l=0.
We should note that the resolution in the x direction is not enough for the high order harmonics (>10th) with the present resolution, but it is enough for the lower order harmonics. The results are in good agreement with the theoretical analysis. Actually, when the resolution is doubled, even the higher order harmonics (10th and 11th) are clearer, besides the low order ones.

Discussion
The normal incidence of two beams has been analyzed to show the main idea clearly. In the perspective of the experimental setup, the influence of the incident angle on the vortex beam generation should be considered.

Figures 5(a) and (b)
show the laser electric fields before and after the interaction with the thin foil when the right laser pulse irradiates obliquely with the incident angle α=π/64. In this case, the electric field E y after the interaction carries on the influence of the incident angle. Different wave fronts of a 2 cross the y-axis at different places, and this forms a sinusoidal profile with the wavelength of λ/(2 sinα) along y that represents the phase, where λ/2 is the wavelength of the laser a 2 . Therefore, the k y component (wave vector in the y direction) of a 2 in the case of oblique incidence results in the distortion of the wave fronts of the reflected beam. And the distortion grows strong with the incident angle. Considering the small influence brought by the k y component of the incident beam, in which the field of k y component of a 2 is weak and does not interrupt the information of harmonics along x-axis, it is better to drive the beam at a small angle. In this case, the harmonics and their corresponding topological charge keep almost the same with the results of the normal incidence case as shown in figures 5(c) and (d), except for little influence on the phase change within one loop shown in figures 5(e) and (f).
In the example above, the two driving CP pulses have the same rotation directions. In the case of the counterrotation directions, the topological charge and order of harmonics can also be determined with the same analytical method, that is, for the (m+1)n 1(2) −mn 2(1) order harmonic, the topological charge is (m+1)l 1(2) −ml 2(1) . For n 1 =1, l 1 =−1 and n 2 =2, l 2 =2, the harmonic order and its corresponding topological charge are 1, 2, 3, 4, 5, K, (m+1) and −1, 2, 5, 8, 11, K, (3m−1), which is confirmed by our simulations. Interestingly, for the counter-rotation case, one photon of the (m+1)n 1(2) −mn 2(1) harmonic is transformed from (m+1) photons of a 1 (2) , and at the same time m photons of a 2(1) are generated, which may be annihilated during the other order harmonics process. Different from the sum frequency generation process in the samerotation case, this is a difference frequency generation process. In addition, CP laser pulses are used as the example to distinguish the HHG in terms of the oscillating mirror model, in which the neither of the CP lasers would produce harmonics by itself, Actually, the present scheme is effective even for linearly polarized lasers, and high harmonics with different topological charges are generated.
The vortex beam with high topological charge opens a wide range of applications, since a new freedom degree, i.e., OAM, is included. According to equation (2), the topological charge of the high order harmonics can be further increased when the vortex beams with high charge acquired with this scheme or others [14,27,35] are input. Large OAM can make a huge mechanical torque when it interacts with matter, and help to control and manipulate micro-particles. The circular or elliptical polarization laser pulse as a chiroptical characterization tool has been applied in probing chiral molecules during the gas HHG process [36,37]. The generation of a much stronger and tunable infrared vortex laser provides an alternative tool with a new freedom degree. The new freedom degree will also help to understand the nonlinear physical process, such as the transformation of photons with different topological charge in the process of multiple frequency generation and sum or difference frequency generation, especially when two or more pulses are involved. Inspired by the amplification and generation of the intense twisted laser pulses via stimulated Roman scattering [14], researchers expect to find more optical-parametric-like process when such beams interact with some chiral materials. Figure 5. Electric field distribution in the y-x plane at z=0 μm before (a) and after (b) the interaction between the laser pulses and thin foil in the case of l 1 =1, l 2 =2. Electric field isosurface of the (c) 2nd and (d) 4th harmonics within a 1 μm distance in the propagating direction after the interaction. Electric field distribution of the (e) 2nd and (f) 4th harmonics in the z-y plane at x=4 μm after the interaction. The field is normalized to m e ωc/e (4×10 12 V m −1 ) and the right laser pulse propagates at an angle of α=π/64.

Conclusion
In conclusion, a new scheme to obtain tunable intense high-order vortex beams in the high frequency region from laser plasma interaction is proposed, and it provides an approach to produce the vortex beam with a specific OAM and frequency as expected. For the first time the additional freedom degree gives us insight into the photon process and helps determine the origin of harmonics photon when two pulses are involved. Both theoretical analysis and 3D PIC simulations have confirmed the generation of high harmonics carrying large OAM when two counter-propagating intense driving vortex beams interact in a thin foil. The topological charge and order of harmonics can be predicted by adjusting the input pulses. Corresponding experimental research is expected based on the well-developed solid HHG diagnostics [38][39][40][41][42][43][44].