We propose a scheme to generate optical vortices through exciting exciton polariton vortices by a Gaussian beam in a pillar microcavity. With coupled Gross-Piteavskii equations we find that the structure of the exciton polariton vortices and antivortices shows a strong dependence on the microcavity radius, pump geometry, and nonlinear exciton-exciton interaction. Due to the nonlinear exciton-exciton interaction the strong Gaussian beam cannot excite more exciton polariton vortices or antivortices with respect to the weak one. The calculation demonstrates that the weak Gaussian beam can excite vortex-antivortex pairs, vortices with high total orbital angular momentum, and superposition states of vortex and antivortex with high total opposite orbital angular momentum. The pump geometry for the Gaussian beam to excite these vortex structures is analyzed in detail, which shows a potential application for generating optical vortex beams.
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Since optical vortices were demonstrated a few decades ago , researchers have paid much attention to their generation and application [2–4]. Optical vortices are topological excitations characterized by the vanishing of the field density at a given point (the vortex core) and by the quantized winding of the field phase from 0 to 2πl (l is an integer) . They have been extensively studied and observed in cold atoms [5–7], optical systems [8–10], and matter waves [11–14]. Exciton polariton condensates [15–17] as a type of matter waves in semiconductor microcavities provide a platform for generating optical vortex beams. The semiconductor microcavities, consisting of two distributed Bragg reflectors, can exhibit spontaneous coherence for exciton polaritons that are bosonic quasiparticles — a superposition state of excitons in quantum wells and photons in cavities . The polaritons, due to the photonic part, can be coherently excited by an incident laser and detected by their emitted light. The optical beams emitted from the exciton polariton vortices inherit the exciton polariton orbital angular momentum.
Various vortex states in exciton polariton condensates continue gaining much attention on disorder effects [18, 19], vortex-antivortex pairs [11, 20], and vortex-vortex control . These vortices show a strong dependence on the potential landscape designed by fabrication techniques  or using optical potentials induced by exciton-exciton interactions [23,24]. The vortex properties of the non-equilibrium exciton polariton condensates have been diagnosed from experiments  and theories [13,26] in last decades, such as lattices of vortices  and superposition of vortex-antivortex states . To create exciton polariton vortices one can use Laguerre-Gauss optical beams that carry a well-defined external orbital angular momentum [29–31]. The vortex-antivortex superposition states are of potential interest to Sagnac interferometry , being a gyroscope which has been archived in atomic systems , and to quantum information . The exciton polariton vortices also show a potential application for generating optical vortex beams.
The application of the exciton polariton vortices for generating optical vortex beams strongly depends on their excitation in a semiconductor microcavity, so that we will study how to effectively excite them in the present work. As a rational expectation researchers hope that the exciton polariton vortices could be directly generated by Gaussian beams rather than Laguerre-Gauss beams. This can be achieved by using the splitting between transverse-electric and transverse-magnetic polariton modes [34–36] and polariton optical-parametric oscillators [37,38]. In the present work, we will show that Gaussian beams can also excite the exciton polariton vortices in a finite microcavity without these conditions. In the finite microcavity the geometry of the Gaussian beam and microcavity boundaries play major roles. In addition, the polariton-polariton interaction needs to be considered in the strong pump regime, though it can be neglected in the weak pump regime. Therefore, we will focus on their effects on the excitation of the polariton vortices and antivortices in the present work. Fig. 1 shows two possible excitation processes by Gaussian beams. When the polaritons arrive at the microcavity boundary they will change their motion direction and thus can form the vortices. We will find that this boundary-based scheme is a very efficient method for generating optical vortex beams.
The present work is organized as follows. In Sec. II, we first introduce the coupled dynamic equations for the quantum well excitons and cavity photons from quantum field theory and then give the system parameters adopted in numerical calculation. Numerical results and discussion are shown in Sec. III which is separated into two subsections according to the pump strength. Finally, a brief conclusion is summarized in Sec. IV.
2. Gross-Pitaevskii equations
The polariton field in a planar microcavity can be described as the coupling of the quantum well exciton field, Ψ̂X(r, t), and the cavity photon field, Ψ̂C(r, t). They are Bose fields and meet the Bose commutation rules. The mean-field approximation, namely, ψX/C(r) = 〈Ψ̂X/C(r)〉, has proven to be an efficient way to describe the fluid properties of the polariton condensate. The motion equation of ψX/C(r), known as the Gross-Pitaevskii equations [39,40], can be written asEq. (2) and the eigenvalues for the two-branch (upper and lower) polaritons are . VX(r) and VC(r) in Eq. (1) are the single particle potentials acting on the exciton and photon fields, respectively. They can break the translational symmetry of the microcavity along the two in-plane directions. The exciton potential generally dates from natural interface or alloy disorder in the quantum wells, while the photon potential is mainly determined by the cavity height or transversal size. Therefore, it is much easier to design the photon potential than to design the exciton potential [22, 41]. The quantities γX and γC are the exciton and photon decay rates, respectively. At last, gX and Fp(r) measure the exciton-exciton interaction and the external pump field, respectively. For convenience, ħ is set to 1 in the following part if there is no ambiguity.
The Gaussian pump beam considered in the present work is defined asFig. 1. For a pillar microcavity VC(r) is infinite outside the cavity region due to the total internal reflection on the boundaries . In calculation VC(r) = 100 meV for r > R and 0 for others, which cuts out the required pillar microcavity with radius R. Note that we mainly focus on the effects of the pump geometry in the present work and therefore, VX(r) is set to zero to avoid the disorder influence.
In the following numerical calculation the parameters of a typical GaAs-based microcavity are adopted. The energy of the excitons is taken as the zero point, i.e., , and other parameters are mC = 1 × 10−5me where me is the free electron mass, γX = γC = 0.01 meV, gX = 0.015 meV · μm2 , ΩR = 2.5 meV. In addition, two pump cases, namely, weak pump regime with fp = 0.1 meV · μm−1 and another strong with fp = 10 meV · μm−1, are considered to show the influence of the nonlinear interaction on the polariton vortices. As is well known the exciton part concerns the nonlinear interaction, while the photon part in the polaritons relates to the pump efficiency. Consequently, the polaritons with k = kp had better have a suitable ratio between them, for example, half over half. This requires , always maintained in the following calculation. Besides, we take the pump detuning to be δp = ωp − ωLP(kp) = −0.2 meV and the Gaussian beam size to be w = 4 μm.
3. Numerical results and discussion
The polariton fluid has been generated by several types of pump fields [24,43]. In the present work we use the “resonant injection” scheme that the pump frequency ωp is set to be near the lower-branch polariton energy at the pump wave vector, ωLP(kp). The Gaussian beam creates the polariton condensate and determines its properties (such as momentum, energy, density, phase). This controllable scheme allows to study the excitation of vortices and antivortices [20,44]. For clear we divide the present section into two subsections according to the pump field strength: (A) weak pump regime and (B) strong pump regime. For the former the exciton density is low and thus the nonlinear exciton-exciton interaction can be neglected, while for the later the exciton density is so high that the nonlinear interaction must be considered.
The main results are obtained by numerically solving Eq. (1) on a two-dimensional grid 320×320 for a square 32 μm×32 μm microcavity region. The discretization area is 0.1 μm×0.1 μm, smaller than the requirement of the maximum pump wave vector 4.0 μm−1 adopted in calculation. The fourth-order Runge-Kutta algorithm is used to evaluate the photon and exciton fields ψC/X(r, t).
3.1. Weak pump regime
In the weak pump regime the energy due to the exciton-exciton interaction, gX |ψX(r)|2, is far less than the polariton kinetic energy and therefore, its effect can be neglected and the polariton evolution can reach a steady state. Figure 2 shows the steady density distributions of the photon field, |ψC(r)|2, in the first column and corresponding field phases in the second column for four microcavities with radii R = 2 μm, 4 μm, 8 μm, and 16 μm. The exciton field has a similar distribution and so is not shown. The typical velocity of the polariton is and therefore, the characteristic length . When kp = 2 μm−1, ξ ∼ 150 μm and subsequently the polaritons in the four microcavities shown in Fig. 2 can reach the boundary. Due to R less than half of 150 μm the boundary exerts manifest influence on the polariton condensate in four cases. The width of the Gaussian pump beam with center at the original point is w = 4 μm, thus with increasing R the microcavity boundary is increasingly away from the pump beam. The pump beam covers all the microcavity when R ≲ w [see Figs. 2(a–b)], while only a central part when R > w [see Fig. 2(d)]. Accounting for the loss of the polaritons in traveling, the boundary plays a major role on the forming of the polariton states for the small microcavities, that is, the boundary effect decreases with increasing R. This can be seen from the variation of the photon density distribution from Figs. 2(a) to 2(d).
As is known the Gaussian beam has no orbital angular momentum and so cannot excite the vortex by itself. For the present circular and no disorder microcavities to excite the vortices requires two conditions: (i) the microcavity boundary can influence the movement of the polaritons and (ii) the Gaussian pump beam has a nonzero in-plane wave vector. As illustrated in Fig. 1, the polaritons change their moving direction once they arrive at the boundary, accompanied by a complex polariton interference. The polariton interference leads to different spatial structures for the vortices and antivortices, see Figs. 2 and 3. In Fig. 2 the Gaussian beam has a wave vector kp = (2, 0) μm−1 and therefore, can induce the vortices and antivortices, referred to the vortices and antivortices in Fig. 2(b) denoted by dots and stars, respectively. The numbers of the dots and stars are same due to the mirror symmetry of the pump beam along direction x. From Fig. 2(b) the distance between two adjacent vortices can be estimated to be ∼ 1.8 μm (about half of the polariton wavelength) for kp = 2 μm−1. Therefore, it is impossible to generate the vortex excitation for small enough microcavities, also called as photonic dots . Since R = 2 μm ∼ 1.8 μm in Fig. 2(a), no vortex or antivortex is generated in the microcavity. With increasing R more complex polariton vortices can be excited [see Fig. 2(b)], even the vortices with high orbital angular momentum [see Figs. 2(c–d)]. The photon density distributions in Figs. 2(c–d) represent the superposition state of the vortex and antivortex with l = ±3, which is important for application of polaritons to the Sagnac interferometry . The Sagnac interferometry requires large l whose value is mainly determined by kp in the present pump geometry.
The variation of the vortex excitation with kp is shown in Fig. 3 where the pillar radii are set to R = 8 μm. When kp is small [Fig. 3(a)] no vortex or antivortex is excited, while with increasing kp the pattern of the photon density shows more and more complexity and subsequently the vortex and antivortex structures are generated [Figs. 3(b–h)]. In other words, the argument that the distance between two adjacent vortices decreases with increasing kp is responsible for the complicated density patterns of the high total orbital angular momentum states in the large-kp cases. Due to the boundary reflection the high total orbital angular momentum states are composed of several vortex-antivortex pairs with l = ±1, as shown in Figs. 3(b–d). The superposition states of the vortex and antivortex with high orbital angular momentums can also be excited. For examples, the superposition states in Figs. 2(c–d) with l = ±3 and in Fig. 3(e) with l = ±6. Since kp is along the x axis in Fig. 3, there is a mirror symmetry for the vortices and antivortices along the x axis, referred to Fig. 3(c). This mirror symmetry can be broken up by changing the pump geometry from Fig. 1(a) to Fig. 1(b).
We take the pump geometry of Fig. 1(b) in Fig. 4 where the pump position is set to rp = (4, 0) μm and the wave vector kp is along the y axis. Since this pump beam has a non-zero angular momentum with respect to the center of the pillar microcavity, the number of the vortices is different from that of the antivortices. The angular momentum of the pump field isFig. 3. On the contrary, for a nonzero rp the net angular momentum of the polariton condensates should be proportional to kp, see Fig. 4 where the total angular momentum is 0, 1ħ, 4ħ, 6ħ, 3ħ, 6ħ, 11ħ, 15ħ, and 20ħ from Figs. 4(a) to 4(i), respectively. Because the angular momentum is not conserved, these values are not exactly equal to Lpump, but maintain the approximate proportional relation with kp.
For convenient analysis we expand the exciton and photon fields, i.e., ψX(r) and ψC(r), as followEq. (7), the angular momentum of the photon field is mainly determined by the pump coefficient fn,l and the corresponding energy ωn,l. In other words, the excitation of the high angular momentum states requires that the pump beam should have a large angular momentum about the pillar center, see Fig. 4. For example, the states of ϕ1,0, ϕ1,1, ϕ1,4, and ϕ4,3 dominate in Figs. 4(a), 4(b), 4(c), and 4(e), respectively. Even if the photon fields have the same total orbital angular momentum, their density distribution could be much different from each other, as shown in Figs. 4(d) and 4(f). Both of their total orbital angular momentums are 6. When kp is large the photon density distribution appears more complex, see Figs. 4(g–i) where more than one states of ϕn,l are excited. If one wants to excite only one angular momentum state ϕn,l [see Figs. 2(c), 3(e), and 4(e)], equation (7) provides a guidance: (i) increase fn,l by controlling the pump geometry and (ii) achieve a resonant excitation for ϕn,l by tuning ωp. To summarize, by designing the pump geometry the Gaussian beam can efficiently excite the polariton vortices and antivortices in the pillar microcavity, which holds potential applications for Sagnac interferometry and generating optical vortex beams.
3.2. Strong pump regime
When the pump field is strong enough the nonlinear exciton-exciton interaction plays an important role in exciting the polariton vortices and antivortices and can further, make the steady state of the polaritons unreachable. Figure 5 shows the density and phase distributions for the photon and exciton fields under a strong pump field fp = 10 meV · μm−1, where the exciton density is always smaller than its saturation density (∼ 500 μm−2 for GaAs-based quantum wells ). In the strong pump regime the exciton energy is blueshifted due to the nonlinear exciton-exciton interaction. The maximum values of gX |ψX(r)|2 are about 0.49 meV, 1.1 meV, 3.7 meV, and 4.0 meV from Figs. 5(e) to 5(h), which approximate to the Rabi splitting 2 = 5.0 meV gradually. Considered that ωp = ωLP(kp) − 0.2 meV is set, the photon density is larger than the exciton density, comparing Figs. 5(a–d) with Figs. 5(e–h), respectively.
This can also be seen from Eqs. (7) and (8): Cn,l ≫ Xn,l if the exciton energy is much larger than ωp. Note that in the weak pump regime the densities for the photon and exciton fields are in the same order of magnitude. For small kp [see Figs. 5(a–c) and 5(e–g)] the polariton system can reach a steady state, while for large kp the polariton system is unstable [see Figs. 5(d) and 5(h)]. This is owed to that the case with kp = 1.5 μm−1 displayed by Figs. 5(d) and 5(h) has the largest exciton density, as well as the strongest nonlinear effect. Since gX |ψX(r)|2 depends on the coordinate, the exciton field has different oscillation frequencies at different positions, which is not negligible in the strong pump regime. Subsequently, the oscillation of the exciton field at different positions is out of step and appears a random density/phase distribution, referred to Fig. 5(h). This kind of randomness cannot be seen from the photon field, because it is stronger than the exciton field.
For the steady cases in Figs. 5(a–c) the total orbital angular momentums, respectively, are 0, 0, and 3ħ. Compared with Figs. 4(b–c) the Figs. 5(b–c) hold less total orbital angular momentum, indicating that the strong pump cannot excite more steady vortices. This can be argued as follow: the nonlinear exciton-exciton interaction prefers to spread the polariton density equally, while the vortices and antivortices have zero-density points. Consequently, the nonlinear exciton-exciton interaction is harmful to the excitation of the polariton vortices. In addition, since it is not easy to observe the vortices or antivortices in an unstable state, a not-too-strong pump beam is the better choice for generating the polariton vortices.
We have studied the excitation of exciton polariton vortices and antivortices in the pillar microcavities by Gaussian pump beams. The structure of vortices and antivortices strongly depends on the microcavity radius, pump geometry, and nonlinear exciton-exciton interactions. We found that it is hard to observe the excited polariton vortices in the strong pump regime for a pillar microcavity, because the nonlinear exciton-exciton interaction prefers to spread the polariton density equally and can cause the system to be unstable. On the contrary, the polariton system can reach a steady state in the weak pump regime. Though the Gaussian pump beams do not carry angular momentums, they can also excite many kinds of the vortex and antivortex structures in the pillar microcavities, such as vortices with high total orbital angular momentum, and superposition states of vortex and antivortex with high total opposite orbital angular momentums. Our results demonstrated that exciting vortices and antivortices by Gaussian beams are possible for experimental observation, which holds potential applications for Sagnac interferometry and generating optical vortex beams with high angular momentum.
National Natural Science Foundation of China (NSFC) (11304015); Beijing Higher Education Young Elite Teacher Project (YETP1228).
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