Light beams possessing an azimuthal phase front e i l φ are called vortex beams [1], where φ is the azimuthal angle and l is an integer which indicates the topological charge of the optical vortex. Unlike the spin angular momentum of photons, which only has two possible states ± 1 , theoretically achievable orbital angular momentum (OAM) states are infinite, that is, l is unbounded and can take any integer value. Vortex beams have unique characteristics such as a helical wave-front and a donut-shaped intensity profile. Such beams carry an OAM of l ℏ per photon. Since l   is unbounded, photons carrying OAM can carry multidimensional information. Due to the aforementioned characteristics of vortex beams, they have been extensively used in many applications, including super-resolution microscopy [2], optical tweezers [3, 4], and quantum information technology [5, 6]. The generation, manipulation, and detection of optical vortices are the focus of this research area. To generate vortex beams, the conventional ways are based on linear optics, such as vortex phase plate, spatial phase modulator based on computational holograph, and q-plate. However, these methods can only generate vortex beams with limited wavelength, since the devices usually have a limited operating wavelength bandwidth. In addition to the linear methods, nonlinear frequency conversion has been proved to be an important scheme to extend the operating wavelength of vortex beams [7–19]. For instance, the operating wavelength of vortex beams can be extended to the visible and ultraviolet band through frequency up-conversion processes such as third-harmonic generation (THG), sum-frequency generation (SFG), and high harmonic generation (HHG) [9–11]. In addition, vortex beams with tunable wavelength ranging from near-infrared to the mid-infrared can be obtained by parametric down-conversion processes such as difference-frequency generation (DFG) [12, 13] and optical parametric oscillation (OPO) [14–17]. Vortex beams in the mid-infrared region have the potential to be used in the areas of molecular spectroscopy and creating chiral nanostructures [20, 21]. For the OPOs pumped by vortex beams, the researchers reported that these are commonly in a forward-propagating configuration, which means the interacting waves are propagating in the same direction. As we know, in addition to the forward-propagating OPO, counter-propagating OPO [22, 23] is another type of configuration in which the interacting waves are propagating in the opposite directions. In backward OPO, the pump beam is down-converted into a counter-propagating signal and idler waves, and the oscillation is established by the distributed feedback due to the presence of two counter-propagating parametric waves. For the operation of backward OPOs, it does not require external mirrors or surface coating to form the optical cavity and operates in a simple single-pass geometry. Compared with conventional OPO, backward OPO has a simpler optical setup, a narrower linewidth, and less temperature sensitivity [24, 25]. Up to now, there is no research report on backward OPO pumped by vortex beams. In this work, we investigate in theory the backward OPO pumped by vortex beams using the nonlinear coupled-wave equations. The OAM conservation transfer law, the threshold, and the conversion efficiency of the vortex beam pumped backward OPO are studied.

Theoretically, the backward OPO pumped by vortex beams can be described by the coupled-wave equation. We assume that both the pump and the signal waves travel along the + X direction, while the idler wave travels along the − X direction. The polarization direction of the three beams is along the Z direction, corresponding to the Z axis of the KTP crystal, as shown in Figure 1.

The electric fields of the interacting waves are given by the following equation: E j = ω j n j A j u j ( ρ ) e i ( k j z + l j φ ) δ j − i ω j t + c . c , (1) where j = p , i , s represents the pump, idler, and the signal wave, respectively. ρ  and  φ are the radius and azimuthal angle in the Z-Y plane, respectively. δ j = 1 for j = p , s , while δ j = − 1 for j = i . ω j , n j , A j , l j , and u j are the frequency, refractive index, amplitude, topological charge, and the normalized intensity profile of the three interacting waves, respectively. k j = 2 π / λ j is the wave vector of the corresponding light field, where λ j is the wavelength. Substituting Eq. 1 into the nonlinear coupled-wave equation [26] and considering the normalization condition ∫ u p , s , i ∗ u p , s , i r d r d φ = 1 , we can obtain d A s dz = + iκ A p A i * exp ( iΔkz ) , d A i dz = − iκ A p A s * exp ( iΔkz ) , d A p dz = + i κ * A s A i exp ( − iΔkz ) , (2) where κ = 2 c d e f f S ω s ω i ω p n s n i n p is the nonlinear coupling constant, with c being the velocity of light, d e f f being the effective nonlinear coefficient, and S = ∫ u s u i u p e i ( l p + l i − l s ) r d r d φ being the overlapping integral of the nonlinear process. Δ k = k p + k i − k s is the wave vector mismatch of the backward OPO process. In order to achieve a high conversion efficiency of the counter-propagating nonlinear process, the quasi-phase-matching technique is desirable. Here in this work, we choose periodically poled KTP(PPKTP) as the quasi-phase-matched nonlinear crystal. For the KTP crystal, the difference of the lattice constant along the X and Y principal axes leads to the highly anisotropic domain growth velocity; thus, KTP is favorable for the fabrication of short-pitched periodically poled structures. Canaltas et al. demonstrated in an experiment the first backward OPO with a short-period PPKTP [27]. The first-order reciprocal vector provided by the PPKTP is G = 2 π / Λ , with Λ being the poling period. Thus, the wave vector mismatch can be compensated by the reciprocal vector, that is, Δ k ' = Δ k − G = 0 . At this time, Eq. (2) has the similar form as that under the plane wave approximation [28]. The difference is that the overlapping integral S between the interacting light fields should be considered in the nonlinear coupling constant of Eq. (2) in the backward OPO process involved with vortex light. In order to ensure that the overlapping integral is non-zero, the topological charges of the pump, signal, and idler waves should satisfy the following relationship: l p = l s − l i . (3)

Equation 3 is the OAM conservation condition in the backward OPO, which is different from that in conventional OPO due to the reversed propagation direction of the idler wave.

Equation 3 presents the OAM conservation law of the backward OPO, and in principle, there are infinite combinations of the topological charge of the signal and idler waves [ l s ,   l i ] satisfying the OAM conservation. To determine the OAM transfer in an optical parametric oscillator, one should consider the threshold of each combination, which is the basic knowledge of mode selection in the laser techniques. In previous studies, OAM transfer in the OPO process can be controlled by adjusting the cavity losses or the spatial overlap integral of the interaction waves [29–31]. The purpose is to change the OPO threshold of different combinations since the OAM preferentially transfers to the combination with the lower threshold. Therefore, we use the method in Ref. [28] to solve Eq. 2 to obtain the threshold of the backward OPO pumped by vortex beams: P th = c ε 0 n s n i n p λ s λ i 32 L 2 d eff 2 S 2 , (4) where ε 0 is the permittivity of vacuum and L is the nonlinear interaction length. The Rayleigh distance of light waves is set to be much longer than the sample length, thus the overlapping integral is approximately constant and equal to that in the beam waist.

According to Eq. 4, the threshold P t h for the backward OPO is inversely proportional to the square of overlapping integral. When the topological charge of [ l s ,   l i ] the pump wave is fixed, processes with different combinations correspond to different overlapping integrals. Therefore, in order to control the OAM conversion and ensure the purity of output beams, we can control the pump power to only excite the process with the largest overlap integral.

In the study, we assume that the pump, signal light, and idler light are in the form of the Laguerre–Gaussian mode [1], and have the same confocal parameters [32]: n p w p 2 λ p = n s w s 2 λ s = n i w i 2 λ i , (5) where w p , w s , w i are the beam waist radius of the three interacting waves, respectively. By substituting the normalized light intensity of a Laguerre–Gaussian mode, the OAM conservation law | l p | = | l s | + | l i | , and Eq. 5 into the expression of the overlapping integral, the overlapping integral of light field at the beam waist is as follows: S l p , l s , l i = 2 π ⋅ | l p | ! | l s | ! ⋅ | l i | ! ⋅ γ s | l s | + 1 γ i | l i | + 1 w p × ( 2 1 + γ s 2 + γ i 2 ) | l p | + 1 , (6) where γ s = w p / w s ,   γ i = w p / w i . We should be aware that the combination with | l p | = | l s | − | l i | may also satisfy the OAM conservation condition, but the overlapping integral of the corresponding process is smaller than that with | l p | = | l s | + | l i | , and thus the discussions are not included in this article.

When OAM conservation in Eq. 3 is satisfied, we use Eq. 6 to calculate the overlapping integrals of all the combinations of l s and l i when the pump light carries three different OAMs, as shown in Figure 2. For the numerical calculations, the temperature was set at 100°C, the beam waist was set to be w p = 150   μ m , and the pump light wavelength was 1.064   μ m . We systematically studied the backward OPO process when the wavelength of signal wave is in the range of 1.5 − 2.128   μ m .

As shown in Figure 2A, when the signal wavelength of the output signal is near the degenerate point, the overlapping integral of S 1,0 , − 1 and S 1,1,0 is close; hence, both processes may oscillate. As the signal wavelength decreased from the degeneracy point, the S 1,1,0 increases while the S 1,0 , − 1 decreases gradually, which means S 1,1,0 is more preferentially to oscillate than S 1,0 , − 1 . Thus, the OAM of the pump wave tends to transfer to the signal wave. Figures 2B,C have the same trends. When the signal light wavelength is in the range of 1.7–1.9 μ m , S 1,1,0 , S 2,1 , − 1 , and S 3,2 , − 1 are obviously larger than the other [ l s ,   l i ] combinations when l P is 1,2, and 3, respectively. From Figure 2C, we can see that the overlap integral of S 3,1 , − 2 and S 3 , − 1,2 is higher than that of S 3,0 , − 3 and S 3,3,0 , and this can be explained as follows. At the degenerated point, the beam waist of the signal and the idler waves are the same. For S 3,1 , − 2 and S 3 , − 1,2 , the signal and the idler waves are both vortex beams, while for S 3,0 , − 3 and S 3,3,0 , the parametric waves, respectively, possess a Gaussian profile and a vortex one. According to the calculation, the overlap integral between the two vortex beams is larger than that between a Gaussian beam and a vortex beam; hence, the overlap integral for S 3,1 , − 2 and S 3 , − 1,2 is larger than the situation for S 3,0 , − 3 and S 3,3,0 . Next, we will make a detailed analysis of the threshold of the backward OPO when the wavelength of the signal wave is set to be λ s = 1.8   μ m . Eq. 4 is used to obtain the relationship between the OPO threshold and the sample length when the topological charge of the pump wave is   l p = 3 , and the relationship is plotted in Figure 3A.

As can be seen from Figure 3A, the oscillation threshold decreases with the increase in the length of the nonlinear crystal. Meanwhile, the difference in the threshold value among the different [ l s ,   l i ] combinations becomes smaller. The results imply that the threshold difference between different combinations [ l s ,   l i ] of the backward OPOs can be controlled by varying the length of the nonlinear crystal. Figure 3B shows the pump threshold of the processes with the largest overlap integrals with different interaction lengths and different topological charges of the pump wave, which follows the same trend as that in Figure 3A. Moreover, the oscillation threshold increases with the increase in the topological charges of the pump. In Figure 3B, we list the threshold of the combination [ l s ,   l i ] when l p = 0,1,2,3 , respectively, and then we can control the pump power for more precise excitation of the required backward OPO.

The conversion efficiency η is a key parameter, and under a stable oscillation state, it is determined by the following formula, which can be obtained by solving Eq. 2: ∫ 0 π 2 dθ 1 − ηsi n 2 θ = π 2 ⋅ P p P th , (7) where P p is the power of the pump wave and θ is the incident angle of the pump wave [28]. Here, we assume that the length of PPKTP is 1 cm, and the relationship between the conversion efficiency and the peak power of the pump wave with different topological charge of the pump wave is shown in Figure 4.

According to Figure 4, we can see that when the input power of the pump light exceeds the threshold, the conversion efficiency increases rapidly. Since only the [ l s ,   l i ] combination with the largest overlapping integral is considered in the calculation of Eq. 7, and when the pump power is too high, processes with a high threshold may also oscillate. Thus, the purity of the output vortex light will be reduced, and the efficiency curve may not be accurate (dotted lines). In the experiment, it would be better if we controlled the input power in the solid line area; thus, we can ensure a high purity OAM state of the signal wave.

For practical realization of the backward OPO pumped by vortex beams, a high-energy pulsed laser light source together with a sub-micrometer periodically poled KTP is required. In our theoretical investigations, we choose a long-pulsed laser as the pump rather than ultrashort pulses because oscillation may not be observed due to the appearance of stimulated Raman scattering at pulses shorter than 20 pico-seconds [27]. The main difficulty of the practical demonstration is the nonlinear crystal. For backward OPO, a sub-micrometer QPM structure is required; however, fabrication of such a structure is still a big challenge. Up to now, there are only a few reports on the experimental demonstration of backward OPOs with Gaussian beams as the pump.

In this study, we have investigated the backward OPO pumped by vortex beams using the nonlinear coupled-wave equations. The OAM conservation law was determined to be l p = l s − l i , which is different from that of the conventional forward-propagating OPO. To figure out the OAM transfer during the backward OPO process, we studied the oscillation threshold for different topological charge combinations of the signal and the idler waves, which is closely related to the overlapping integral, the length of the nonlinear crystal, and the topological charge of the pump wave. In addition, the conversion efficiency of the oscillation was numerically calculated. Our results can help understand the OAM transfer in the backward OPO, and obtain the output vortex beam with a pure OAM component. Further work will be focused on the experimental demonstration of the backward OPO pumped by vortex beams.