## Abstract

Nonlinear processes of laser beams carrying orbital angular momentum (OAM) offer a means to generate new wavelengths and to manipulate OAM charge numbers. We demonstrate the second-harmonic generation (SHG) of asymmetric Bessel-Gaussian (BG) beams carrying OAM of both integer and fractional charge numbers. Experimental results show a good one-to-one correspondence of the charge numbers and compliance with the OAM conservation law. The SHG conversion process and efficiency with different combined charge numbers are also discussed.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

OAM beams, a kind of structured light, have recently attracted attention due to their specific properties such as a myriad of transverse amplitude distributions and spiral phase. Laguerre Gaussian (LG) modes are one type of well-studied OAM beam and these modes were first demonstrated by Allen in 1992 [1]. Since then, beams with OAM have been widely utilized in optical communications [2,3], particle manipulation [4,5], remote sensing [6] and quantum entanglement [7,8]. The complex amplitude of LG modes is described as carrying a spiral phase term, $\exp (i\ell \theta )$, where $\ell $ is the so-called topological charge of the beam and $\theta $ is the azimuthal coordinate. As a result, LG modes have a symmetric, donut-like intensity distribution. Typically, an OAM beam profile is azimuthally symmetric and the topological charge number $\ell $, or vortex strength, is an integer and will not evolve through propagation. Illumination from a Gaussian mode on a spiral phase plate with fractional phase step ($2\pi \alpha $, $\alpha $ is fractional) will generate a beam with a fractional OAM charge number. In general, fractionally charged OAM beams do not retain their value with propagation because the structure degrades due to their inherent branch-cut [9]. In addition, adjacent fractional charge numbers generated in this way can become inseparable. Fractional OAMs with better performance will be of great benefit to scientific research. For example, separable fractional OAM beams with good propagation characteristics could serve to increase channel capacity and reduce mode crosstalk in a free-space optical communication (FSO) link [10].

Nonlinear optical processes offer a means to generate new wavelengths of light and to manipulate OAM charge numbers. The OAM charge number is shown to be conserved in nonlinear processes due to the OAM conservation law [11]. Recently, nonlinear processes of OAM such as second-harmonic generation (SHG) [11–17], sum-frequency generation (SFG) [18], difference-frequency generation (DFG) [19], high-harmonic generation (HHG) [20,21], spontaneous parametric down-conversion [22], four-wave mixing [23–25] and vortex-pumped optical parametric oscillator (OPO) [26] have been studied. In this paper, we focus on the development of SHG of OAM beams carrying integer and fractional charge numbers. SHG of OAM using LG modes was first studied by Padgett in which the authors discussed the frequency-doubling process of LG modes [11,12]. In 2011, Bovino *et al.* studied the behavior of light with OAM in a non-collinear second-harmonic generation process [13] in which the pump light beams carrying opposite fractional charge numbers generate zero angular momentum. In 2013, Si-Min Li *et al.* used a non-collinear setup to study the SHG of arbitrary combinations of OAM and confirmed the OAM conservation law experimentally [14]. In 2016, R. Ni *et al.* studied the second-harmonic process of OAM beams carrying fractional charge numbers generated through a fractional phase-step spiral phase plate (SPP) [15]. In 2017, Paulius Stanislovaitis *et al.* studied collinear SHG using a beam with an embedded initial phase singularity of strength $|\ell |= m/2$, where *m* is an integer, and examined the law of OAM conservation on such charge numbers [16]. In 2018, Sabir Ul Alam *et al.* studied the SHG of fractional OAM by generating a beam with a transverse shifted SPP, the measurement of the charge number and efficiency with different shift distances [17]. Utilizing a phase plate with fractional phase steps and transverse shifting of integer phase plates are two usual methods to generate fractional charge numbers. However, the generated charge number is generally different from the order of the phase plate, which is troublesome when determining the second-harmonic generated fractional charge number using these beams. Most research focuses on proving the OAM conservation law on integer and half-integer charge numbers; studies on fractional OAM (non-half-integer) only measure some specific generated charge numbers based on OAM spectra theory.

In this paper, we demonstrate the second-harmonic process of asymmetric BG beams carrying OAM of integer and fractional charge numbers. The OAM SHG theory is given. Charge number measurements are presented based on the cylindrical lens method [27,28]. Both OAM charge numbers of the fundamental frequency and the doubled frequency are measured on a large scale with small charge number steps. The simulation and experimental results show that asymmetric BG beams are able to retain their shape and topological charge with propagation. The measured charge number carried by the asymmetric BG beams has a better one-to-one correspondence to the programmed value than conventional beams carrying fractional charge numbers. This is the first time fractional charge numbers generated through a nonlinear process have been shown in a wide range with small fractional steps. The SHG conversion process and efficiency of beam carrying combined charge numbers are also discussed.

## 2. Method

The generation of asymmetric BG beams was described in our former work [29]. We are able to generate asymmetric BG beams with the Higher Order Bessel-gaussian Beams Integrated in Time (HOBBIT) setup using log-polar elements. The output of the HOBBIT system in the near-field is a ring-shaped beam of charge number $\ell $ given by

Figure 1 gives the basic generation concept of the asymmetric BG beams with the main parameters shown. The log-polar elements wrap an elliptical Gaussian beam with a linear phase in the horizontal direction, Fig. 1(a), into a ring structure, Fig. 1(b), which carries a helical phase distribution known as OAM. The linear phase on the beam is generated by an AOD. By varying the Bragg angle with the AOD, different linear phases can be generated resulting in different charge numbers. As shown in Fib. 1(b), $\beta $ controls the asymmetry, ${\rho _0}$ is the ring radius and ${w_{ring}}$ is the ring half-width of the asymmetric BG beams. By controlling these parameters, the properties of the asymmetric BG beams can be manipulated, for example, controlling the Gaussian envelope and Bessel rings with ${\rho _0}$ and ${w_{ring}}$. Typically, the far-field of Fig. 1(b) is considered the complex amplitude distribution of the asymmetric BG beam shown in Fig. 1(c). The far-field Gaussian envelope is controlled by ${w_{ring}}$. By simply changing ${w_{ring}}$, the transverse distribution (ring shape) can be generated arbitrarily.

We apply the theory of second-harmonic generation of OAM beams mentioned in [12] to do the analysis. In this scenario, both the complex amplitude and the frequency doubling process are simplified. We assume the light field carrying OAM has a simple form as Eq. (2),

Where, $U(\omega )$ is the field, $\omega$ represents the fundamental frequency, and*U*is the arbitrary complex amplitude.

Based on the OAM conservation law, the second-harmonic generation process of a single OAM charge number is shown in Eq. (3). Through the frequency doubling process, the complex amplitude of the field will have a squared distribution and the charge number of the doubled frequency will be twice as large as the one carried by the fundamental frequency. Usually, the nonlinear process of SHG of beams carrying fractional OAM is very complicated and it is difficult to analyze the generated charge number, because of the intensity distribution changing and phase discontinuity existing in the beam profile. However, due to the asymmetric structure of these asymmetric BG beams, the results have good agreement with Eq. (3) which will be shown in the simulation and experimental results.

Now assume the incident beam is a combination of two charge numbers (${\ell _1} < {\ell _2}$). In the nonlinear process of multi-charge-number case, the situation is complicated because the interaction between different charge numbers needs to be considered [30,31, and 32]. Mathematically, Eq. (3) becomes Eq. (4),The far-field of the asymmetric BG beams is a superposition of infinite integer OAM charge numbers which are centered at charge number $\ell$. Here we assume the approximate form as Eq. (6),

*m*stands for the infinite integer OAM charge numbers, ${C_m} = {c_m}{A_m}G{B_m}$ is the complex scaling factor, ${c_m}$ is the weighting factor, ${A_m}$ is the asymmetric amplitude, $G{B_m}$ is the Gaussian envelope, ${J_m}\left( {\frac{{{k_{tm}}r}}{{{\mu_m}}}} \right)$ is the Bessel term, ${k_{tm}}$ is the transverse wave number, ${\mu _m}$ is the scaling factor [33].

Substituting Eq. (6) into Eq. (3), we can get a similar result as Eq. (5),

*p*and

*q*are two interacting charge numbers in the dimension of

*m*, ${C_{pq}} = {c_{pq}}{\gamma _{pq}}B(\varDelta {k_{pq}}){A_{pq}}G{B_{pq}}$ is the complex scaling factor of the generated charge number, ${c_{pq}}$ is the weighting factor based on the power density of the two interacting charge number, ${\gamma _{pq}}$ is the overlapping coefficient of the two charge numbers [30], $B(\varDelta {k_{pq}})$ is the phase-matching coefficient, ${A_{pq}}$ is asymmetric amplitude and $G{B_{pq}}$ is the Gaussian envelope, ${J_p}{J_q}$ is the Bessel term. Similar to Eq. (5), power scaling, interaction and phase-matching condition should be considered in the square of a summation of OAM charge numbers as these will affect the generated OAM spectrum because of the nonlinear process. The nonlinear process changes the OAM spectrum and the weighting for each OAM charge number causing a shift from the expected average OAM charge number.

We use the relationship between the OAM projection along the *z* direction, ${J_z}$, and the light field energy, *W*, to determine the average charge number [28].

By describing the complex amplitude $U{(x,y,z)_\ell }$ as a 1D Fourier transform separately for each coordinate in Eq. (8), we get the cylindrical lens method [27] which will be used in simulation as well as experimentally to measure the OAM charge number. This method is given by Eq. (9).

*f*is the focal length of the cylindrical lens.

## 3. Simulation and experiment realization

#### 3.1 Simulation

The advantage of the asymmetric BG beams is first demonstrated by simulation. By choosing different values of $\beta$, the asymmetry of the beam can be changed. Figure 2(a) (inset) shows the intensity distributions of charge −3 when $\beta =$ 1, 0.7, and 0.5. The design of $\beta$ is discussed in [29] and in our simulation and experimental setup $\beta$ is 0.7. Figure 2(a) also provides the simulation results of charge number measurement curves for different beam models and $\beta$. The curve of OAM-carrying beam generated through plane wave is based on the theory in [9] which shows no measured fractional charge numbers. The range of charge numbers is from −1.5 to 1.5 with steps of 0.1. The curves with three different $\beta$ values are quite linear but with slightly different amplitudes in their undulations. The linear one-to-one correspondence shown here is very different from the fractional OAM charge number results in [27,34]. This is because the branch-cut point is avoided due to the asymmetric properties of the beam structure. The phase discontinuity in conventional fractional OAM charge number will develop singularities during propagation, generating off-axis angular momentum components. As a result, the average OAM measured value will shift away from the designed charge number. Compared to conventional fractional OAM charge numbers, the asymmetry in our BG beams decreases the evolution of phase discontinuities and the generated cascaded phase singularities are eliminated.

Using Eq. (3), the 532 nm charge number measurement curve for these asymmetric BG beams is shown in Fig. 2(b). Based on the OAM conservation law, the charge numbers of the second harmonic generated beams have a range of −3 to 3 with steps of 0.2. The 532 nm measured charge number curve has more undulations than the 1064 nm curve, but again, has little deviation from the programmed value. This result has a good agreement with the prediction of charge number shifting by using Eq. (3). In the simulation, the maximum deviation happens at charge 0.5 with a shift of around 0.15. OAM is still conserved in the SHG process, considering converted and non-converted charge numbers. Figure 2(c) gives the simulated beam profiles of 1064 nm and 532 nm based on Eq. (3). The marked charge number on each beam is based on the OAM conservation law, ignoring that the generated average OAM charge number has a shift from the marked charge number. Different from conventional fractional OAM beams whose intensity distribution is not stable from near-field to far-field, the simulation results show similar intensity distribution to the near-field which indicates that the asymmetric BG beams can retain their shape and charge number through propagation.

#### 3.2 Experiment

We conducted an experiment to examine the SHG of the asymmetric BG beams. An illustration of the experimental setup is shown in Fig. 3. A 1064 nm laser source (Nuphoton 1064 nm Dual Amplifier Assembly) with vertical polarization is incident on the HOBBIT setup. The polarization is required for both a good diffraction efficiency of the AOD and the nonlinear process in the Magnesium doped Periodically Poled Lithium Niobate (MgO:PPLN) nonlinear crystal. The AOD (Gooch & Housego Model #3080-197) adds a linear phase on the 1064 nm Gaussian beam in the horizontal direction. The AOD is controlled by an arbitrary waveform generator (AWG) (Tektronix AWG5208). By generating sine waves with specific frequencies, we can generate asymmetric BG beams carrying arbitrary charge numbers, integer or fractional. A pair of lenses containing a cylindrical lens ($f = 40mm$) and a spherical lens ($f = 200mm$) converts the Gaussian profile to an elliptical Gaussian distribution. A beam block is used to block the 0th order diffraction. The log-polar elements wrap the incident beam into an asymmetric ring shape. This generation procedure is shown in Fig. 1. A lens ($f = 200mm$) is used to focus the beam at the center of a 20 mm long MgO:PPLN nonlinear crystal (Covesion MSHG1064-1.0-20) with a grating period of 6.93 µm. With quasi-phase matching, the 1064 nm laser is frequency doubled to generate 532 nm. An imaging lens ($f = 75mm$) is placed after the PPLN to image the beam profile at the focus inside the crystal. Two dichroic mirrors are utilized to separate the generated 532 nm beam from the non-converted 1064 nm pump. A pick-off optic is placed after the dichroic mirrors to reflect a small amount (10%) of the generated beam to the imaging camera (Spiricon SP300). The cylindrical lens ($f = 100mm$) before the camera is used for the charge number measurement.

The experimental results are shown in Fig. 4. Figures 4(a) and 4(c) show the 1064 nm asymmetric BG beams carrying both integer and fractional charge numbers, respectively. Integer charge numbers from −3 to + 3, fractional charge numbers of −2.5:1:2.5 and two extra random charge numbers of −0.26 and + 0.6 are produced by defining the waveforms generated by the AWG. Figures 4(b) and 4(d) show the second-harmonic generated beams at 532 nm. Relative charge numbers based on the OAM conservation law are marked on each beam profile. The beam profiles of the nonlinear generated beams have a good agreement with the simulation in Fig. 2(c).

The charge number measurement curves of the experimental results are shown in Fig. 5 along with the simulated results in Fig. 2(b). The error bars are based on the standard deviation of the measurement over 9 frames captured by the camera at each charge number. By comparing simulation and experiment results, the 1064 nm curve has a very good linear correspondence between the programmed and measured charge number, and the 532 nm experimental curve has more undulations than the simulation which could be accounted for as suggested in Eq. (5). The simulations and experiment demonstrate that the asymmetric BG beams can carry continuous OAM charge numbers with a good one-to-one correspondence to the programmed value across a wide range of charge numbers with fractional steps. This supports the idea that asymmetric BG beams can provide fractional charge numbers after being converted with a nonlinear process. This result shows that tailoring the transverse distribution, one can generate arbitrary OAM charge numbers that have a better propagation property than those with a generated branch cut as part of the amplitude envelope.

We also measured the single-pass nonlinear conversion efficiency. Shown in Fig. 1, the beam profile of asymmetric BG beams rotate 90 degrees as they propagate from the near-field to the far-field. In order to control the rotation of the beam and keep the PPLN within the Rayleigh range, a Fourier lens with a focal length of 200 mm was used. Because the AOD is driven by the electrical waveforms from the AWG, theoretically any waveform can be used to generate single OAM and combinations of OAM charge numbers. Combinations of OAM charge numbers are generated for 0, ±1, ±2 and ± 3 with the results shown in Fig. 6. Both charge numbers and SHG efficiencies are included. Note that the charge numbers are given as combinations in the field, not as an average value over the coherent combinations. The relative 532 nm charge numbers are provided based on the simple approximation. It is also important to note that the individual beams have different frequencies based on their Doppler frequency shift from the AOD. These different Doppler frequencies will introduce an interference between the different modes at specific beat frequencies. These are not observable in the images, since the camera is averaging over a relatively long interaction time. Also, the AOD’s diffraction efficiency changes depending on the generated Bragg angle. Therefore, different charge number combinations have different maximum power output from the HOBBIT system at 1064 nm which affected the SHG efficiency shown in Fig. 6. Charge 0 had a maximum pump power of 4960 mW. This resulted in the SHG output of 149 mW at the phase-matching temperature of 36.7°C. The charge combination of ± 1 had a maximum pump power of 3170 mW. This resulted in the SHG output of 41 mW at the phase-matching temperature of 38.8°C. The difference in the phase matching condition for charge 0 and charge ± 1 is not only because of the laser power, but also because of the different k vectors for different charge numbers and combinations of the beams can be further optimized as suggested in [32]. The maximum conversion efficiency is 3%, 1.29%, 0.72%, and 0.4% for charge 0, ±2, ±4 and ± 6, respectively. Some of these artifacts based on the Doppler frequencies and SHG conversion efficiency can be eliminated and improved using a high peak power source.

## 4. Conclusion

In summary, we have investigated the second-harmonic generation process of asymmetric BG beams with integer and fractional charges. The main parameters controlling the beam properties were discussed and experimentally verified. The numerical simulations and experiments were carried out to illustrate the nonlinear process including both integer and fractional OAM charge numbers. Using a cylindrical lens measurement technique, measurements on the average charge number were completed for the fundamental and the frequency doubled beams, which show good agreement and demonstrate a one to one correspondence with the programmed fundamental and SHG beams. The results also show a specific advantage of asymmetric BG beams as our experiments demonstrated the ability to maintain their charge number and beam shape with propagation. The SHG conversion process and efficiency of combined charge numbers are also discussed. This is the first time such a relationship between generated and programmed fractional OAM charge numbers is given, both in the fundamental frequency and the nonlinear generated frequency. Our results indicate that it is possible to generate arbitrary OAM charge numbers by carefully tailoring the transverse field distribution. Given the current optical design and bandwidth of the acousto-optic deflector, the present system can generate OAM charge numbers from −3 to + 3. Future work includes a new design to generate higher charge numbers, while still maintaining the linear relationship between the programmed and measured charge numbers. Additionally, theoretical modeling on the SHG process with a superposition of asymmetrical Bessel-Gaussian beams in pulsed or continuous waves is also a subject of interest. With the specific characteristics of fractional OAM charge numbers, the asymmetric BG beams show great potential in applications like optical communication [10], particle manipulation [5] and quantum entanglement [8].

## Funding

Office of Naval Research (N00014-16-1-3090, N00014-17-1-2779, N00014-18-1-2225, N00014-18-1-2377).

## Disclosures

The authors declare no conflicts of interest.

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