1. Introduction

Following the remarkable discovery of the Laguerre-Gaussian (LG) beam with orbital angular momentum (OAM) by Allen and co-workers [1], understanding the spatial distribution of the intensity and phase diagrams resulting from helical phase fronts in various systems have attracted much attentions [2–6]. Recently, it has been revealed that a dipolar chiral nanostructure was capable of distinguishing the sign of the phase vortex of the incoming LG beam [7]. Moreover, the LG beams with nonzero OAM can be used to investigate phase matching for high-order harmonic generation and which provides a route to enhance the conversion efficiency [8].

Recent researches have been focused on the ability to manipulate the spatial characteristics of the fields during the four-wave mixing (FWM) process [9–12]. The coherent transfer of OAM has been demonstrated using the optical FWM process in semiconductor and atomic systems [13,14]. The vortex solitons of FWM have been experimentally demonstrated in multi-level atomic media, which are generated from the interference patterns by utilizing three or more waves [15,16]. It has been reported that the experimental implementation of OAM multiplexed continuous variables entanglement in a FWM process and $13$ pairs of entangled LG modes are simultaneously created in hot atomic vapor [17]. Later, Rahmatullah and co-workers proposed a scheme to generate structured light and realize the transfer of optical vortices via FWM in a semiconductor quantum-dot nanostructure [18]. Wang et al. [19–22] studied the generation of spatially dependent FWM and its control in atomic and semiconductor quantum well systems. Moreover, Kerr-nonlinearity-modulated dressed vortex FWM and its propagation behaviors have been experimentally investigated in a four-level rubidium atomic system with a photonic band gap [23], which would facilitate the development of optical computing and information processing science. Although, the generation of vortex FWM and their coherent control have been investigated in atomic gases and semiconductor quantum wells [13–23]. Also, the bichromatic field generation from a double-FWM system has been experimentally demonstrated and theoretically simulated in a cold atomic ensemble [24]. However, the vortex beams generation and its spatial characteristics from double-FWM process have not been reported yet in graphene systems.

Motivated by these findings and open questions, here, we explore the azimuthal and radial modulation of double-FWM process by means of two higher-order LG beams in a Landau quantized graphene ensemble. The graphene system serves as an alternative platform for probing the properties of vortex FWM beams for a number of reasons. First, graphene is a two-dimensional material with carbon atoms arranging in a hexagonal honeycomb lattice, which has a giant infrared optical nonlinearity under an external magnetic field [25–27]. Second, graphene has unusual linearly dispersing electronic excitations and many allowed intersubband transitions via the application of a magnetic field [28–30]. In particular, graphene holds special optical selection rule [31], i.e., $|\Delta n|=\pm 1$ with $n$ being the energy quantum number, which provides an extra degree of freedom for choosing intra- and inter-Landau levels transitions. This is to be contrasted with atomic and semiconductor systems with nonlinear dispersion relations [32–36]. Last but not the least, two different kinds of FWM processes could be realized simultaneously, which signifies that the phase matching condition and energy conservation should be satisfied for both the processes.

In this paper, we provide a scheme to study the spatially dependent double-FWM process in a Landau quantized graphene ensemble driven by two higher-order LG beams. We suppose that the carriers are initially prepared in a coherent superposition state of the two lower levels. By adjusting the azimuthal and radial mode orders of the two LG beams, i.e., the azimuthal index $l_j$ and radial index $p_j$ with $j=1, 2$, the two generated FWM fields can be flexibly modulated. We find that there will always be $l_1+l_2$ periods of phase variation in the phase profiles regardless of whether the two LG beams have the same mode orders or not. Moreover, the phase jumps from $-\pi$ to $\pi$ for the first FWM-generated field, while the phase jumps of the second FWM field is just the opposite. We show that when the two LG beams have the same radial orders (i.e., $p_1=p_2=p$), $p+1$ concentric rings can be observed in the intensity profile and the strength is mainly concentrated on the inner ring. When the two LG beams have the same radial orders ($p_1=p_2=p$) and different azimuthal orders ($l_1\neq l_2$), there are $p$ raised narrow rings in the phase profile. However, these raised narrow rings will disappear when the two LG beams have the same mode orders, i.e., $p_1=p_2$ and $l_1=l_2$. On the other hand, when the two LG beams have different radial orders ($p_1\neq p_2$) regardless of whether the two LG beams have the same azimuthal orders or not, we take $p_{max}=max(p_1, p_2)$ and $p_{min}=min(p_1, p_2)$, there appear $p_{max}+1$ concentric rings in the intensity profile. Simultaneously, $|p_1-p_2|$ convex discs and $p_{min}$ raised narrow rings will emerge in the phase diagram. It should be noted that the two generated FWM fields have the same results, the difference is that the phase variations are completely opposite.

This paper is outlined as follows. In Sec. 2, we introduce the model under consideration and provide a theoretical description of the system by means of the density-matrix method. Then, quite accurate analysis of the generation and manipulation of two vortex FWM beams with opposite vortices based on numerical simulations is displayed in Sec. 3. Besides, in Sec. 4 we present the possible experimental realization techniques of double-FWM in a coherently driven graphene ensemble and discuss the feasibility of its implementation. Finally, the conclusions are summarized in Sec. 5.

2. Physical model and governing master equations

The scheme under consideration is shown in Fig. 1 where light-matter interaction in an ensemble consisted by Landau-quantized graphene. The band structure of graphene is comprised of cones in the vicinity of two inequivalent Brillouin zone corners where the conduction and valence bands appear. In the proximity of these points the electron energy relies linearly on its momentum, that is, $E(\vec {p})=\pm v_F|\vec {p}|$, which means that free charge carriers in graphene are controlled by Dirac’s equation for massless particles, with the band parameter (i.e., Fermi velocity) $v_F=1.03\times 10^6$m/s replacing the speed of light $c$. Under the action of an external magnetic field, the Dirac energy spectrum changes into Landau levels (LLs) with energies given by $\varepsilon _n=sgn(n)\hbar \omega _c\sqrt {|n|}$ for electrons (positive) or holes (negative) in the vicinity of the Dirac point, $n=0, \pm 1, \pm 2$, $\omega _c=\sqrt {2}v_F/l_c$ with $l_c=\sqrt {\hbar c/(eB)}$ being the magnetic length [37,38]. The energy of the LLs depends on the magnetic field $B$ and the index $n$. Besides, the optical transitions between the neighboring LLs in graphene fall into the infrared or terahertz (THz) range for an external magnetic field in the region of $0.01-10$ T: $\hbar \omega _c\simeq 36\sqrt {B(\mbox {Tesla})}$ meV. The states $|1\rangle$, $|2\rangle$, $|3\rangle$, $|4\rangle$, and $|5\rangle$ are corresponding to the LLs with quantum numbers $n=-2, -1, 0, +1, +2$, respectively. Taking into account two incident bichromatic electric fields $\vec {E}_p=\hat {e}_{RHS}E_{p1}e^{-i\omega _{p1}t}+\hat {e}_{LHS}E_{p2}e^{-i\omega _{p2}t}+c.c.$ and $\vec {E}_c=\hat {e}_{RHS}E_{c1}e^{-i\omega _{c1}t}+\hat {e}_{LHS}E_{c2}e^{-i\omega _{c2}t}+c.c.$ polarized in the $x$-$y$ plane along vector $\hat {e}$. The unit vectors of the right- and left-hand circularly polarized components can be expressed as $\hat {e}_{RHS}=[\hat {x}+i\hat {y}]/\sqrt {2}$ and $\hat {e}_{LHS}=[\hat {x}-i\hat {y}]/\sqrt {2}$. Based on the unusual selection rules of graphene, $\Delta |n|=\pm 1$, that is, the left-hand circularly polarized light is absorbed when $|n_f|=|n_i|+1$, on the contrary, the right-hand circularly polarized light is absorbed when $|n_f|=|n_i|-1$ with $n_f$ and $n_i$ denoting the final and initial energy quantum numbers [31], we utilize the right- and left-hand circularly polarized pulsed probe fields $E_{p1}$ and $E_{p2}$ with carrier frequencies $\omega _{p1}$ and $\omega _{p2}$ to couple the transitions $|1\rangle \leftrightarrow |2\rangle$ and $|2\rangle \leftrightarrow |5\rangle$, respectively. The transitions $|2\rangle \leftrightarrow |3\rangle$ and $|3\rangle \leftrightarrow |4\rangle$ are respectively driven by the right- and left-hand circularly polarized control vortex beams $E_{c1}$ and $E_{c2}$ with respective carrier frequencies $\omega _{c1}$ and $\omega _{c2}$. The two probe fields and two control fields can make up two possible FWM pathways: $|1\rangle \leftrightarrow |2\rangle \leftrightarrow |3\rangle \leftrightarrow |4\rangle \leftrightarrow |1\rangle$ and $|2\rangle \leftrightarrow |5\rangle \leftrightarrow |4\rangle \leftrightarrow |3\rangle \leftrightarrow |2\rangle$. As a result, $E_{m1}$ with right-hand circular polarization and $E_{m2}$ with left-hand circular polarization are the generated FWM signal fields. Correspondingly, there exist two possible phase-matching conditions in Landau-quantized graphene, i.e., the conservation of energy conditions: $\omega _{p1}+\omega _{c1}+\omega _{c2}=\omega _{m1}$ and $\omega _{m2}+\omega _{c1}+\omega _{c2}=\omega _{p2}$, where $\omega _{m1}$ and $\omega _{m2}$ are the carrier frequencies of two generated FWM fields.

Under the rotating wave approximation (RWA) and electric-dipole approximation (EDA), according to the standard techniques of quantum optics [39], we obtain the following off-diagonal density-matrix elements equations for describing the interaction between laser beams and graphene

 

Fig. 1. Schematic diagram of two four-wave mixing (FWM) signals generation in Landau-quantized graphene. (a) The states $|1\rangle$, $|2\rangle$, $|3\rangle$, $|4\rangle$, and $|5\rangle$ correspond to the Landau levels (LLs) with quantum numbers $n=-2, -1, 0, +1, +2$, respectively. The transitions $|1\rangle \leftrightarrow |2\rangle$ and $|2\rangle \leftrightarrow |5\rangle$ are respectively coupled by the pulsed probe fields $E_{p1}$ and $E_{p2}$. The right- and left-hand circularly polarized laser beams $E_{c1}$ and $E_{c2}$ are used to drive the transitions $|2\rangle \leftrightarrow |3\rangle$ and $|3\rangle \leftrightarrow |4\rangle$, respectively. $E_{m1}$ and $E_{m2}$ are the resulting generated FWM fields. Polarization of laser beam corresponds to the allowed transitions. (b) Geometric configuration of the laser fields for the double-FWM. The magnetic field $B$ is perpendicular to the graphene plane.

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The incoming probe fields and the generated FWM fields satisfy Maxwell’s wave equation, and thus we can obtain

We assume that the atoms are initially prepared in a superposition state of two lower levels, i.e., $|\psi (0)\rangle =c_1|1\rangle +c_2|2\rangle$. Therefore, we can achieve $\rho _{11}=|c_1|^2$ and $\rho _{22}=|c_2|^2$. Let us introduce the method of perturbation expansion $\rho _{ij}=\sum _{k}\rho _{ij}^{(k)}$, here $\rho _{ij}^{(k)}$ is the $k$th-order term of $\rho _{ij}$, we can obtain that $\rho _{ij}^{(0)}=0$ $(i\neq j)$ and $\rho _{33}^{(k)}=\rho _{44}^{(k)}=\rho _{55}^{(k)}=0$. These assumptions are justified by the fact that the control vortex field $\Omega _{cq}$ is strong compared with the pulsed probe field $\Omega _{pq}$ and the generated FWM field $\Omega _{mq}$ $(q=1, 2)$. Then, to the first order of the pulsed probe and FWM fields, in combination with the time Fourier transformation, the Eqs. (1)–(10) and Eqs. (11)–(14) can be written as

Solving the Eqs. (15)–(24), we can get

Substituting Eqs. (29) and (30) into Eqs. (25) and (26), one arrives at the following equations

Under the initial condition of the generated FWM field, i.e., $\Lambda _{m1}(0,\omega )=0$, the solutions to Eqs. (33) and (34) can be given by

Since we are concerned here with the influence of vortex control fields, the analytical expressions of the pulsed probe and FWM-generated fields are so tedious that it’s not easy to carry out the inverse Fourier transformation. Hence, the $\mathcal {O}(\omega )$ terms in $g_{1,2}(\omega )$, $f_{1,2}(\omega )$ and $\mathcal {O}(\omega ^2)$ terms in $K_{\pm }(\omega )$ can be completely neglected under these circumstances, and we still obtain the needed underlying physical mechanisms. By means of the approximated inverse Fourier transformation, the expressions for the probe and FWM fields can be written in the form

Similar to the derivation of Eqs. (37) and (38) described above, we can obtain the results

3. Numerical results and discussion

We consider in this paper the relaxation rates between different levels with the same value, i.e., $\gamma _{mn}=\gamma =3\times 10^{13}$ s$^{-1}$ [42]. Also, in the following discussions, let us suppose that the atoms are initially prepared in a superposition state of the two lower levels $|1\rangle$ and $|2\rangle$. In general, the sign convention for angles is that counterclockwise rotation is positive and clockwise rotation is negative when facing the direction of light beam propagation.

We first consider the case that only the control field $\Omega _{c1}$ is an optical vortex beam with $l_1=p_1=2$ and the control field $\Omega _{c2}$ is a nonvortex beam, we plot the evolution of phase patterns and the corresponding normalized intensity distributions of the generated vortex FWM beams in Fig. 2. When the two probe fields are on resonance (i.e., $\Delta _1=\Delta _4=0$), the phase profile of the first FWM field displays two periods and the phase of each period is $2\pi$, therefore, we can see that the phase jumps from $-\pi$ to $\pi$ around the singularity point and the boundary between the two periods is $x=0$ as shown in Fig. 2(a). Also, there is a raised ring on the phase profile and the boundary between the two periods on the ring is $y=0$ [see Fig. 2(a)]. While, when the first probe field is off resonance, the boundary between the two periods is $y=x$ in the phase profile and the boundary is $y=-x$ of the raised ring as can be seen from Fig. 2(b). At the same time, we can observe three concentric rings with a zero intensity at the vortex core as illustrated in Fig. 2(c). On the other hand, we also plot the phase and the intensity profiles of the second FWM field in Figs. 2(d)–2(f) on the resonance and nonresonance conditions. In the case of resonance, the phase jump of the second FWM field [see Fig. 2(d)] is exactly in contrast to that of the first FWM field [see Fig. 2(a)]. For the case of $\Delta _1=0$ and $\Delta _4=20$ fs$^{-1}$, the boundary between the two periods is $y=-x$ in the phase profile and the boundary is $y=x$ of the raised ring as can be seen from Fig. 2(e). Similarly, there are three concentric rings in the corresponding intensity distribution [see Fig. 2(f)]. Moreover, the maximum intensity is distributed in the inner ring and the strength of the outer ring is relatively weak for the two generated FWM fields.

In order to obtain the conclusion that the generated FWM fields are sensitively dependent on the mode orders of the LG beams, we will numerically simulate the profiles of phase and intensity distribution under different azimuthal indices and different radial indices. Figure 3 shows the phase patterns and their corresponding intensity distributions of the first FWM field in the case of $l_1\neq l_2$ and $p_1\neq p_2$. Under the condition of $(l_1, p_{1}, l_2, p_{2})=(2, 0, 1, 1)$, we observe three periods and the phase jumps from $-\pi$ to $\pi$ in one period, as well as a convex disk with three cycles in the phase profile as shown in Fig. 3(a). Moreover, its corresponding intensity distribution is plotted in Fig. 3(b), which contains two concentric rings with a zero intensity in the central spot. As can be seen from Fig. 3(b) that the strength of the inner ring is greater than that of the outer ring. When the mode orders are adjusted to $(l_1, p_{1}, l_2, p_{2})=(3, 1, 1, 2)$, it can be observed from Fig. 3(c) that four phase jumps from $-\pi$ to $\pi$ appear in the phase pattern, there exists a convex disk and a raised narrow ring, which can be ascribed to the phase vortex of the coherent LG beams. Three concentric rings appear in the corresponding intensity profile illustrated in Fig. 3(d), it can be seen that the intensity is concentrated in the middle ring which is sandwiched by two weak rings. While, when the parameters are tuned to $(l_1, p_{1}, l_2, p_{2})=(2, 2, 3, 3)$, we can achieve five periods of phase jumps from $-\pi$ to $\pi$, as well as a convex disk and two raised narrow rings, as shown in Fig. 3(e). In this case, we can obtain four concentric rings in the intensity distribution [see Fig. 3(f)], the intensity of the inner ring is maximal, and the strength value is decreased from the inside to the outside. Therefore, one can conclude that when the mode orders $l_1\neq l_2$ and $p_1\neq p_2$, one can obtain $p_{2}+1$ concentric rings with a zero intensity at the center of the intensity distribution for the first FWM field. Moreover, we can achieve $l_1+l_2$ periods of phase variation from $-\pi$ to $\pi$ and $p_{1}$ raised rings as well as $p_{2}-p_{1}$ convex disks in the phase profile.

We now consider the effect of the mode orders, which have different azimuthal indices ($l_1\neq l_2$) and the same radial orders ($p_1=p_2$), on the phase profile and intensity pattern of the second FWM field. In this paragraph, for the convenience of discussion, we assume that $p_1=p_2=p$. When $l_1=4$, $l_2=3$, and $p=2$, we observe that seven periods of phase jumps from $\pi$ to $-\pi$ in the phase profile with two raised narrow rings as illustrated in Fig. 4(a). In this case, there exist three concentric rings with zero intensity in its center in the intensity distribution as shown in Fig. 4(b), where the intensity of the inner ring has the maximal value. When the combined value of the mode orders is adjusted to $l_1=3$, $l_2=2$, and $p=3$, five periods and three raised narrow rings can be observed in the phase profile [see Fig. 4(c)]. While, we can obtain four concentric rings and the intensity is mainly concentrated on the inner ring in the corresponding intensity distribution as can be seen from Fig. 4(d). Moreover, when the mode parameters are tuned to $l_1=2$, $l_2=1$, and $p=4$, we can notice that three periods of phase jumps occur in the phase pattern and four raised narrow rings can be achieved as shown in Fig. 4(e). Under the same conditions, the corresponding intensity distribution presented in Fig. 4(f), five concentric rings can be observed and the intensity of the inner ring is the maximal, and the intensity value decreases from the inside ring to the outside ring. A similar evolution happens for the first FWM field (not shown here), the difference is that the phase variation is completely opposite. By summarizing the above discussions, it can be deduced that $l_1+l_2$ periods of phase jumps from $\pi$ to $-\pi$ and $p$ raised narrow rings can be achieved in the phase profile when the two control vortex beams have different azimuthal indices ($l_1\neq l_2$) and the same radial orders ($p_1=p_2=p$). Moreover, there exist $p+1$ concentric rings with zero intensity at the center in the intensity distribution pattern.

Considering the condition that the two control vortex beams have the same azimuthal orders and the same radial orders, i.e., $l_1=l_2=l$ and $p_1=p_2=p$. Additionally, the waist size of the Gaussian beam is adjusted from $\omega _0=0.3$ mm to $\omega _0=0.03$ mm. In the case of $l=2$ and $p=0$, we can observe a windmill-like phase profile with four cycles of phase jumps from $-\pi$ to $\pi$ as shown in Fig. 5(a). While, only one ring appears in the corresponding intensity distribution as plotted in Fig. 5(b). There are six periods of phase jumps from $-\pi$ to $\pi$ appearing in the windmill-like phase profile under the condition of $l=3$ and $p=1$ as shown in Fig. 5(c). Two concentric rings appear in the corresponding intensity pattern, where the strength of the inner ring is stronger than that of the outer ring as displayed in Fig. 5(d). Finally, when the mode orders are increased to $l=4$ and $p=2$, we can observe eight periods of phase jumps from $-\pi$ to $\pi$ in the phase profile [see Fig. 5(e)], the corresponding intensity pattern shows three concentric rings and the intensity is mainly concentrated on the inner ring [see Fig. 5(f)]. This allows to draw conclusions that there are $p+1$ concentric rings in the intensity distribution profile and $2l$ (i.e., $l_1+l_2$) periods of phase jumps from $-\pi$ to $\pi$ in the phase pattern under the condition of $l_1=l_2=l$ and $p_1=p_2=p$.

In the following, we shall discuss the influence of the mode orders on the phase profile and intensity distributions of the second FWM field under the condition of $l_1=l_2$ and $p_1\neq p_2$. For the sake of simplicity, we suppose that $l_1=l_2=l$. When the combination of azimuthal order and radial order is set to $l=3$, $p_1=1$ and $p_2=0$, we can observe a windmill-like phase pattern which has six periods of phase jumps from $\pi$ to $-\pi$ and a convex disk with the same period as shown in Fig. 6(a). Meanwhile, a double-ring structure appears in the intensity profile and it can be seen from Fig. 6(b) that the intensity of the inner ring is bigger than that of the outer ring. For the case that $l=2$, $p_1=2$ and $p_2=1$, there exist five periods of phase jumps and a convex disk as well as a raised narrow ring in the windmill-like phase profile as exhibited in Fig. 6(c). Also, three concentric rings occur in the corresponding intensity distribution as shown in Fig. 6(d). When the combination of mode orders is tuned to $l=1$, $p_1=3$ and $p_2=2$, there are two periods of phase jumps as well as a convex disk and two raised narrow rings in the phase profile as displayed in Fig. 6(e). Moreover, we can obtain four concentric rings with zero intensity at the center in the intensity pattern as plotted in Fig. 6(f), and the strength value gradually decreases from inner ring to outer ring. As a result, when the two control vortex beams have the same azimuthal orders ($l_1=l_2$) and different radial orders ($p_1\neq p_2$), we can achieve $p+1$ concentric rings in the intensity distribution profile and $2l$ (i.e., $l_1+l_2$) periods of phase jumps from $\pi$ to $-\pi$ in the phase pattern, besides, there are $p_1-p_2$ convex discs and $p_2$ raised rings in the phase profile.

 

Fig. 2. The phase patterns in (a), (b) and the corresponding normalized intensity distribution in (c) of the first FWM field. (d), (e), and (f) are respectively the phase pattern and the corresponding normalized intensity of the second FWM field. The parameters are $\Omega _{p10}=\Omega _{p20}=2$ fs$^{-1}$, $\Omega _{c2}=30$ fs$^{-1}$, $\Delta _2=\Delta _3=0$, $\gamma =0.03$ fs$^{-1}$, $\kappa _{12}=10\kappa _{14}=0.0015$ $\mu$m$^{-1}$fs$^{-1}$, $\kappa _{45}=10\kappa _{25}=0.0015$ $\mu$m$^{-1}$fs$^{-1}$, $\tau =33.3$ fs, $t=100$ fs, $\omega _0=0.3$ mm, $c_1=\sqrt {3}/2$, $c_2=1/2$, and $l_1=p_1=2$. (a) $\Delta _1=\Delta _4=0$; (b), (c) $\Delta _1=20$ fs$^{-1}$, $\Delta _4=0$; (d) $\Delta _1=\Delta _4=0$; (e), (f) $\Delta _1=0$, $\Delta _4=20$ fs$^{-1}$.

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Fig. 3. The phase patterns in (a), (c), (e) and the corresponding normalized intensity distributions in (b), (d), (f) of the first FWM field with different OAM numbers. The parameters are $\Omega _{p10}=2$ fs$^{-1}$, $\Delta _1=\Delta _2=\Delta _3=0$, $\gamma =0.03$ fs$^{-1}$, $\kappa _{12}=10\kappa _{14}=0.0015$ $\mu$m$^{-1}$fs$^{-1}$, $\tau =33.3$ fs, $t=100$ fs, $\omega _0=0.3$ mm, $c_1=\sqrt {3}/2$, and $c_2=1/2$. (a), (b) $(l_1, p_1, l_2, p_2)=(2, 0, 1, 1)$; (c), (d) $(l_1, p_1, l_2, p_2)=(3, 1, 1, 2)$; (e), (f) $(l_1, p_1, l_2, p_2)=(2, 2, 3, 3)$.

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Fig. 4. The phase patterns in (a), (c), (e) and the corresponding normalized intensity distributions in (b), (d), (f) of the second FWM field with different OAM numbers. The parameters are $\Omega _{p20}=2$ fs$^{-1}$, $\Delta _1=\Delta _2=\Delta _3=\Delta _4=0$, $\gamma =0.03$ fs$^{-1}$, $\kappa _{45}=10\kappa _{25}=0.0015$ $\mu$m$^{-1}$fs$^{-1}$, $\tau =33.3$ fs, $t=100$ fs, $\omega _0=0.3$ mm, and $c_2=1/2$. (a), (b) $(l_1, p_1, l_2, p_2)=(4, 2, 3, 2)$; (c), (d) $(l_1, p_1, l_2, p_2)=(3, 3, 2, 3)$; (e), (f) $(l_1, p_1, l_2, p_2)=(2, 4, 1, 4)$.

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Fig. 5. The phase patterns in (a), (c), (e) and the corresponding normalized intensity distributions in (b), (d), (f) of the first FWM field with different OAM numbers. The parameters are the same as that shown in Fig. 3 except that $\Delta _1=20$ fs$^{-1}$ and $\omega _0=0.03$ mm. (a), (b) $(l_1, p_1, l_2, p_2)=(2, 0, 2, 0)$; (c), (d) $(l_1, p_1, l_2, p_2)=(3, 1, 3, 1)$; (e), (f) $(l_1, p_1, l_2, p_2)=(4, 2, 4, 2)$.

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Fig. 6. The phase patterns in (a), (c), (e) and the corresponding normalized intensity distributions in (b), (d), (f) of the second FWM field with different OAM numbers. The parameters are the same as that shown in Fig. 4 except that $\Delta _4=20$ fs$^{-1}$ and $\omega _0=0.03$ mm. (a), (b) $(l_1, p_1, l_2, p_2)=(3, 1, 3, 0)$; (c), (d) $(l_1, p_1, l_2, p_2)=(2, 2, 2, 1)$; (e), (f) $(l_1, p_1, l_2, p_2)=(1, 3, 1, 2)$.

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4. Feasibility of experimentally implementing

In this section, let us briefly discuss the possible experimental realization for our proposed scheme by using the quantized graphene nanostructures, quantum cascade laser [43], high-quality spatial light modulator (SLM) loaded with computational hologram [44], and Spiral Phase Plates (SPPs) [40]. When the SLM with hologram is illuminated with a Gaussian laser beam, the first-order diffracted beams can be obtained. Then, the SPP converts the first-order diffracted beams into the desired LG beams with adjustable OAM. On the other hand, we choose the graphene with a cascade of allowed intersubband transitions as our sample. Specifically, $|1\rangle =|n=-2\rangle$, $|2\rangle =|n=-1\rangle$, $|3\rangle =|n=0\rangle$, $|4\rangle =|n=+1\rangle$, and $|5\rangle =|n=+2\rangle$, the dephasing decay rate is $\gamma =3\times 10^{13}$ $s^{-1}$ [42]. All the four laser beams (i.e., two probe fields and two LG beams) are collimated as shown in Fig. 1(b) in order to satisfy the phase-matching conditions simultaneously. Moreover, the carrier frequencies of driving fields and the FWM fields are located in the infrared region, for instance, $\omega _{p1}\sim (\omega _{n=-1}-\omega _{n=-2})/\hbar$ and $\omega _{p2}\sim (\omega _{n=+2}-\omega _{n=-1})/\hbar$ are of the order of $10^{13}$ s$^{-1}$ based on the numerical estimate in the case of $\omega _c\simeq 10^{14}$ s$^{-1}$ at $B=3$ T [37,38,42]. Nonlinear cyclotron resonance in graphene has been studied theoretically in Ref. [45], which can be employed only to electrons in a low magnetic field that occupy highly excited LLs (i.e., $n\gg 1$), when energy and momentum quantization are ignored. In our proposed scheme, we gave a quantum-mechanical density-matrix description of the double-FWM processes of graphene, which is valid for the lower LLs. Owing to the unique optical selection rules for zero rest mass electrons in the vicinity of the Dirac point, we can realize this kind of nonlinear interaction where the optical fields are resonant or near-resonant to the allowed transitions. The third-order susceptibility is proved to be extremely large, the mid-infrared wavelength light would be absorbed in the magnetic field of several Tesla. With the rapid development of a bulk graphene manufacturing technique, the formation and manipulation of vortex FWM beams in graphene ensemble should be accessible under LG beams excitation with suitable mode parameters and moderate laser intensities in experiments in the near future.

5. Conclusions

In conclusion, we have theoretically investigated the spatial distribution of phase and intensity of two FWM-generated fields in a double-channel FWM scheme driven by two weak nonvortex probe fields as well as two strong control laser fields with LG modes. Due to the fact that the two different types of FWM processes can be realized simultaneously, the phase and intensity profiles of FWM-generated fields depend on the mode orders of the control vortex beams and quantum interference between the two transition pathways. Based on the above analysis, we obtained the following conclusions: (i) $l_1+l_2$ periods of phase jumps from $-\pi$ to $\pi$ can be observed in the phase profile of the FWM fields and the two FWM-generated fields have opposite phase. (ii) $p+1$ concentric rings with zero intensity at the center can be achieved in the intensity distribution pattern and simultaneously there exist $p$ raised narrow rings in the phase profile when the two control vortex beams have the same radial orders (i.e., $p_1=p_2=p$) and different azimuthal orders (i.e., $l_1\neq l_2$). Also, the raised rings in the phase profile would disappear in the case of $p_1=p_2$ and $l_1=l_2$. (iii) when the two control vortex beams have different radial orders (i.e., $p_1\neq p_2$), assuming that $p_{max}=max(p_1, p_2)$ and $p_{min}=min(p_1, p_2)$, $p_{max}+1$ concentric rings with zero intensity at the vortex core can be obtained in the intensity profile, $|p_1-p_2|$ raised discs and $p_{min}$ rased narrow rings appear in the phase profile. It has been demonstrated that the phase jumps are opposite for the two FWM-generated fields and the intensity is mainly concentrated on the inner ring due to the quantum interference between the two FWM pathways. This work provides an opportunity to better understand the spatial distribution of the intensity and phase of double-FWM in graphene ensemble, and thus offers a valuable guidance for their spatial coherence modulation.