## Abstract

Light with orbital angular momentum (OAM), or twisted light, is widely investigated in the fields of optical communications, quantum information science, and nonlinear optics by harnessing its unbounded dimension. For nonlinear frequency conversion of twisted light, the efficiency usually decreases exponentially with increasing topological charge, which severely degrades the quality of the output modes. We here conceive and develop a method of eliminating the dependence of the conversion efficiency on the topological charge in second-harmonic generation using a single lens imaging technique. We first give the theoretical model and simulate the evolution of light beams near the image point of the lens, then the independence of the SHG conversion efficiency from topological charge is verified for different integer values of the topological charge; this independence is confirmed for varying pump power. The method can be generalized to other light–matter interactions and is a promising method of achieving higher efficiency and fidelity in the interaction of twisted light with matter.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Light carrying orbital angular momentum (OAM), or twisted light, has been one of the most investigated light fields since it was proposed by L. Allen in 1992 [1]. Investigations have harnessed the unbounded dimension and unique mechanical properties of twisted light in many scientific fields [2–4]; e.g., high-capacity spatial-mode-division multiplexing optical communications [5], high-dimensional quantum information processing such as that of quantum memory [6–8] and the quantum key distribution [9,10], nonlinear frequency conversion of twisted light [11–20], high-precision optical metrology [21–23], and optical trapping and manipulation of micro-particles [24]. The different aspects of twisted light remain a hot topic of research.

In the interaction of matter with twisted light, such as in the case of quantum memory based on atomic ensembles [6–8] and nonlinear frequency conversion in a second- [11,16,25] or third-order nonlinear medium [26], the storage or conversion efficiency strongly depends on the topological charge of the input beam [6,8,11,25], usually decreasing exponentially with increasing topological charge. There are two main drawbacks of the dependence of the conversion efficiency on topological charge in light–matter interactions. On one hand, the decrease in conversion efficiency with increasing topological charge limits the maximum topological charge that can be efficiently stored or converted; on the other hand, for high-dimensional OAM superposition state input, there is severe distortion after storage or frequency conversion. Solving these problems will be an important step towards realizing highly efficient and high-quality light–matter interaction with high-dimensional superposition states. For frequency doubling of OAM-carrying laser beam at femtosecond regime, Ref. [15] reports on the independence of conversion efficiency on topological charges by using perfect vortex beams, in this work, Laguerre-Gaussian modes with different topological charges first been transformed to Bessel Gaussian mode with axicon, then they are converted to perfect vortex beams by using single lens, then they achieved the same intensity distribution at the center of the nonlinear crystal, therefore the conversion efficiency for different topological charges are the same.

In this work, we use a single image technique to achieve the same goal as in Ref. [15], but our experimental setup is simpler than that in Ref. [15]. The second-harmonic generation (SHG) of twisted light in quasi-phase matching periodically poled potassium titanyl phosphate (PPKTP) is presented as an example to verify our method, but we note that the present method can also be generalized to other light–matter interactions (i.e., second- or third-order nonlinear interactions or quantum storage based on electromagnetically induced transparent media) involving twisted light. We first present the general theoretical model of our method for SHG and then verify the principle of the method for SHG of twisted light with different integer topological charge by pumping a PPKTP crystal with a 1560-nm femtosecond fiber laser. Conversion independent of the integer topological charge is achieved for integer topological charge varying from 0 to 3, with pump power as high as 1.1 W. The maximum SHG power for the different pump modes achieved is 150 mW at pump power of 1.1 W, corresponding to a conversion efficiency of 14%. The conservation of topological charge is also verified in the conversion.

## 2. Theoretical model and simulation

A coupling equation can be used to describe the SHG of a twisted-light beam [27,28]. For strong focusing beams or short crystal approximations, the main contribution to the second-harmonic light is from the pump beam waist, as determined by

Where*r*and

*α*are values in cylindrical coordinates. We have assumed that the phase-matching condition is achieved. We use the relationship between power and the electric field:

*n*is the refractive index of the medium,

*c*is the speed of light in a vacuum, and μ

_{0}is the permeability of a vacuum. We obtain

Diffractive optical elements, such as the vortex phase plate (VPP) and spatial light modulator (SLM), are commonly used to generate twisted light [3]. These diffractive elements are often illuminated with collimated Gaussian beams. The fact is that when a beam simply propagates out from the facet of the diffractive optical element, though it is imprinted with a specific topological charge, it still has an intensity distribution similar to that of a standard Gaussian beam. This is central to our method. By imaging the field of the light beam from the output facet of the diffractive elements to the center of the nonlinear crystal using a smaller beam, we can convert the twisted light efficiently without dependence on the topological charge. The ABCD matrix description of an imaging system should meet the conditions that B = 0 and AD − BC = 1 [30]. In practice, there are usually two ways of constructing an imaging system; one is 4-f imaging with two lenses while the other is single-lens imaging. The ABCD matrices for the two systems are

*f*

_{1}and

*f*

_{2}are the focal lengths of the two lenses;

*f*is the focal length of the single lens, and

*u*is the distance between the object and lens. For such imaging systems, the light field on the imaging plane is related to the light field on the object plane as [30]

*k*is the wave vector of the light field,

*z*is the distance between the imaging plane and object plane, and

*r*and

_{i}*α*

_{i}are the coordinates on the image plane. The formula indicates that despite a magnification factor

*A*, the intensity distribution on the image plane is the same as that on the object plane. The difference between two-lens imaging and single-lens imaging is a phase modulation factor; i.e., there is no phase modulation for two-lens imaging. For a collimated Gaussian illuminating beam, the light field at the position of the VPP can be expressed as [31]

*r*and

_{o}*α*

_{o}are coordinates on the object plane and

*w*is the waist of the illuminating Gaussian beam. By combining Eqs. (6) and (7) and using diffraction integration equations as in Refs. [28,29], we can obtain the light field near the center of the crystal:

_{0}*z*is at the image plane; F(p, q, x) is hypergeometric function; $\mbox{a} = 1/{A^2}w_0^2 + ik/2{z_1} + ig$,$\mbox{b} = k{r_c}/2{z_1}$, here $g = k(u - f)/{{\textrm{f}}^2}$. By using Eq. (8), we can simulate the evolution of the light field near the center of the crystal.

_{1}The simulated results of the evolution of the light field for different topological charges by using the experimental parameters are shown in Fig. 1. It is obvious that all the beam with different topological charges have the same beam size and intensity distribution as Gaussian mode at the image point [Figs. 1(a5), 1(b5), 1(c5)], then the beam gradually evolution to ring shapes after propagating away from the image point. The beam with larger topological charge has stronger diffraction ability, and the distance to keep a Gaussian like intensity distribution is shorter, for our experimental parameters, the total propagation distance to keep a Gaussian intensity distribution is less than 10 μm, therefore the present scheme is only effective for thin crystal. Figs. 1(a1), 1(b1), 1(c1) are the intensity distributions at the focus plane of the lens, we can see that, they have nearly ideal intensity distributions as a standard LG modes, the beam size is increasing with topological charges, this the reason for the observation of exponentially decreasing of conversion efficiency with topological charges by put the crystal at the focus point of the lens. The parameters for the simulation are λ=1.56 μm, u = 1050 mm, f = 50 mm, w_{0}=1 mm, A = 1/20. Now we will experimentally verification the predication of the Eq. (8).

## 3. Experimental setups

We conducted a second-harmonic generation experiment from 1560 to 780 nm to illustrate the method described above. The experimental setup is shown in Fig. 2. A femtosecond fiber laser (Calmer Laser, Mendocino 1560 nm) was used as the pump laser for SHG. The central wavelength of the pump laser was 1560 nm, the pulse width was 100 fs, and the repetition rate was 80 MHz. A half-wave plate (HWP1) and polarization beam splitter (PBS) were combined as a power adjustment unit that adjusted the pump power for SHG. The maximum average power that we could obtain with our setup was about 1.1 W. Another half-wave plate (HWP2) was used to change the polarization of the pump light so that it aligned with the z axis of the nonlinear crystal. A vortex phase plate (which had conversion efficiency of around 95% for different integer topological charge) was placed behind HWP2 to imprint different integer OAM on the pump beam. The pump light was then imaged onto the center of the nonlinear crystal.

To simplify the experimental setup, a single lens (L1) was used in imaging the plane at the VPP onto the center of the nonlinear crystal. The focal length of L1 was 50 mm and the distance between the VPP and L1 was 1.05 m, according to the Gaussian imaging formula $1/{s_0} + 1/{s_i} = 1/f$, where s_{o} and s_{i} are the object and image distances from the lens, f is the focal length of the lens, and A =${s_i}/{s_o} = 1/20$ is the magnification factor of the imaging system, with s_{i} = 52.5 mm (See the insert figure in Fig. 2). The beam waist would be about 50 μm at the center of the nonlinear crystal if the beam waist was 1 mm at the VPP position. Moreover, the adoption of this configuration results in the image plane of the VPP almost coinciding with the focal plane of the lens. The image plane nearby therefore has maximum conversion efficiency. We used a type-0 PPKTP crystal for SHG. The crystal had dimensions of 1 mm × 2 mm × 1 mm, both end faces of the crystal had anti-reflection coatings for both pump and SHG wavelengths, and the temperature of the crystal was kept at 45.5°C. The pump beam and SHG beam were then separated by a dichroic mirror (DM, R > 99.5% for 780 nm, T > 99% for 1560 nm). A collimating lens (L2) transformed the second-harmonic light into a collimated beam. We used a power meter to measure the power of the second-harmonic light beam. A camera (BC106-VIS, Thorlabs) was used to capture the intensity distribution of the SHG beam. To distinguish the topological charge of converted light, a pair of cylindrical lenses (CL1 and CL2) was used to transform the second-harmonic beam to a Hermite–Gaussian beam. We then captured the image of transformed light and identified the topological charge of the SHG beam.

## 4. Experimental results

We obtained the conversion efficiency with the topological charge ranging from 0 to 3 at different pump powers. Figure 3 shows the conversion efficiency at different integer topological charge versus pump power. The energy per pulse of the pump beam is also shown on the upper horizontal axis. The conversion efficiency is almost independent of topological charge for a fixed power of pump light. The conversion efficiency of OAM light is almost the same as that of the fundamental Gaussian beam because the object plane of the output facet VPP1 is imaged onto the center of the PPKTP crystal. This illustrates that light with a non-Gaussian spatial distribution can have the same conversion efficiency as fundamental Gaussian light. In reality, however, at the focal point of the lens, the conversion efficiency of the Gaussian beam is higher than that of higher-order modes. Despite the small distance between the image point and the focal point (2.5 mm in free space), the conversion behaviors are rather different, as the beam profile changes rapidly near the focal point. As the interaction length of the nonlinear crystal is only 1 mm, the scaling factor of the SHG beam is affected by the topological charge at the focal point.

This topologically independent phenomenon persisted up to pump power of 1.1 W, the maximum power output of our fiber laser. The conversion efficiency increased with pump power, with maximum SHG power of 150 mW obtained at pump power of 1.1 W, corresponding to a conversion efficiency of 14%. The small deviation of larger topological charge from the Gaussian beam arising from length of the crystal. From the simulation, it is clear that the Gaussian like shape can only hold for a thickness of around 10μm. It can further increase for higher pump power and a smaller pump beam waist.

To verify that the topological charge of the SHG beam is twice that of the pump light, we used a pair of cylindrical lenses to transform the SHG beam to Hermite Gaussian (HG) beam. With proper parameters, the transformed light can form interference-like fringes. The number of dark zones between bright zones matched the topology of the SHG beam.

Figure 4 shows images of both SHG beams and the transformed fringes for the pump with integer topological charge of 1, 2, and 3. The intensity distribution of the second-harmonic light had a torus-like shape and a larger radius at a larger topological charge of the pump beam. The fringes indicate that the integer topological charge of the SHG beams is 2, 4, and 6. This confirms that the topological charge of the SHG beam is twice that of the pump beam and OAM is conserved in SHG. We should point out that because the SHG beam is not pure LG mode, the converted HG mode are not pure HG mode, because of slight misalignment of the cylindrical lens pair, the beam is distorted after transformation.

## 5. Conclusion and discussion

In summary, we developed a method of eliminating the dependence of the conversion efficiency on the integer topological charge in SHG by imaging the VPP output facet onto the center of a nonlinear crystal. We experimentally demonstrated the independence of the SHG conversion efficiency from topological charge ranging from 0 to 3. The relationship held for different pump powers up to 1.1 W. OAM conservation in SHG was verified by identifying the topological charge of the SHG beam with cylindrical lens transformation.

We note that the conversion efficiency of 14% achieved here is moderate, with other demonstrations achieving power conversion efficiencies exceeding 30%. There are limitations to the present scheme: the object distance increases for higher magnification, resulting in the occupation of more space; a higher magnification ratio results in greater beam aberration, the beam profile will be distorted; and only a thin crystal can be used for a smaller beam diameter. The conversion efficiency in the present scheme can be further improved using high-magnification achromatic objectives (which simultaneously reduce the spatial distance and beam size) to achieve a smaller beam waist or using higher pump power (as the SHG power has a squared scaling with the pump power). The present methods can be generalized to other nonlinear processes, such as sum frequency generation, difference frequency generation, four-wave mixing, and electromagnetically induced transparency for quantum storage. In future work, we will attempt to apply the principle to realizing the quantum frequency conversion of high-dimensional OAM entanglement states with higher efficiency and quality. The present findings indicate a promising method for the study of the interaction of matter and twisted light. The method will be important in the fields of laser optical physics and high-dimensional quantum information processing based on twisted light–matter interaction.

## Funding

National Natural Science Foundation of China (NSFC) (61435011, 61525504, 61605194); Anhui Initiative in Quantum Information Technologies (AHY020200); China Postdoctoral Science Foundation (2017M622003, 2018M642517).

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