In the past several decades, the study of optical vortices carrying an orbital angular momentum (OAM) of lℏ per photon has been an active area in the field of optics since it was first proposed and experimentally observed by Allen et al. in 1992 [1]. As a structured light beam, the vortex beam contains a helical phase term e ^ilϕ , where l is the topological charge (TC) [2]. The exchange and manipulation of optical vortices have been extensively investigated in a variety of structures and materials, such as liquid crystal films [3], metamaterials [4, 5], Dammann vortex grating [6], cold atomic ensembles [7–11], quantum dots [12, 13], molecular magnets [14], and graphene [15]. Meanwhile, optical vortices have been widely used in optical tweezers [16, 17], optical communication [18, 19], microscopic imaging [20, 21], quantum entanglement [22, 23], quantum teleportation [24, 25], and quantum information processing [26]. For most of these applications, it is of great importance to accurately measure the TCs of optical vortices. Until now, numerous approaches have been proposed to realize the TC measurement, such as using the torque measurement [27, 28], rotational Doppler effect [29], the diffractive optical elements including apertures [30, 31] and gratings [32, 33], the interference mechanisms of oblique plane waves [34], spherical waves [35, 36], Young’s double-slit [37, 38] and Mach–Zehnder [39, 40]. However, these approaches mostly require some specialized optical components or a good number of optical elements with fine alignment. Therefore, realizing the TC measurement of a vortex beam in a simple and efficient optical system still remains to be explored.

On the other hand, as a solid-state material working in the mid-infrared band, semiconductor quantum wells (SQWs) provide a promising platform for the coherent control of mid-infrared light transmission due to their advantages of high nonlinear optical coefficients, large electric dipole moments, and a flexible structure design. Until now, a host of breakthroughs have been made such as electromagnetically induced transparency (EIT) [41, 42], electromagnetically induced grating [43, 44], all-optical switching [45], optical solutions [46, 47], Goos–Hänchen shift [48, 49], and four-wave mixing (FWM) [50–52]. Recently, the transfer and modulation of mid-infrared optical vortices have been realized via the high-efficient FWM process in SQWs [53–55]. Thus, it reminds us of one question: Can we realize the TC measurement of a mid-infrared vortex beam in a SQW system?

To answer this question, a scheme is proposed in this article for measuring the TC of a mid-infrared vortex beam via observing the intensity distribution of the generated FWM field in an asymmetric semiconductor double quantum well (SDQW). In this SDQW, Fano-type interference exists, which arises from the absorption paths of two states coupled to an electronic continuum [52, 56, 57]. Different from previous studies, the distinguishing features of this scheme are given as follows: First and foremost, with the help of the Fano-type interference, the special inherent interference leads to the interference-type phase and intensity patterns for the generated FWM field. This is a significant advantage of our proposed scheme compared with Refs. [53, 54]. Second, by adjusting the intensity and detuning of the control field, the interference-type intensity pattern of the FWM field can be drastically manipulated. In particular, by an appropriate choice of the intensity Ω _c and detuning Δ _c , a high-visibility interference-type pattern accompanied by an appropriate intensity can be achieved. Third, by monitoring the number of light spots of the FWM field, the measurable TC value can reach up to 120 in our scheme, which is a great improvement compared with previous schemes [38, 58].

As shown in Figure 1A, we consider an asymmetric SDQW with four-subband configurations, which can be grown by molecular-beam epitaxy [56]. In this designed SDQW, an 8.3-nm thick Al_0.07Ga_0.93As layer and a 6.9-nm GaAs layer are separated by an Al_0.32Ga_0.68As potential barrier with the thickness of 4.75 nm. On the right side of the right well is a thin barrier with a thickness of 3.8 nm, which is followed by a thick Al_0.16Ga_0.84As layer [52, 59, 60]. The eigenenergies and wave functions for the four conduction subbands can be obtained by solving the effective mass Schrödinger equation [61]. In the proposed SDQW, the eigenenergies of the four subbands |1⟩, |2⟩, |3⟩, and |4⟩ are ω ₁ = 51.53 meV, ω ₂ = 97.78 meV, ω ₃ = 191.3 meV, and ω ₄ = 233.23 meV, respectively [52]. Owing to the existence of resonant tunneling, the two closely spaced delocalized subbands |3⟩ and |4⟩ can be represented by a coherent superposition of the first excited subband in the shallow well |se⟩ and deep well |de⟩, that is, 3 = ( s e − d e ) / 2 and 4 = ( s e + d e ) / 2 . A pulse probe field Ω _p = Ω _p0 exp (−t ²/τ ²) (Ω _p0 and τ are the initial Rabi frequency and pulse width) is applied to the transition |3⟩ ↔|1⟩, while the transitions |3⟩ ↔|2⟩ and |4⟩ ↔|2⟩ are driven by a continuous-wave (cw) control field Ω _c and a vortex driving field Ω _d . Subsequently, a pulse FWM field Ω _m can be efficiently generated via the FWM process 1 → 3 → 2 → 4 → 1 (Figures 1B,C). In our scheme, the vortex driving field Ω _d is a Laguerre–Gaussian (LG) mode with the form [12] Ω d r , ϕ = Ω d 0 1 l ! 2 r w 0 l L p l 2 r 2 / w 0 2 e − r 2 / w 0 2 e i l ϕ , (1) where r and ϕ are the radial radius and azimuthal angle, respectively. Ω _d0 and w ₀ represent the initial Rabi frequency and beam waist, respectively. The radial index and TC are labeled by p and l, respectively. Here, L p l is the Laguerre polynomial. L p l x = e x x − l p ! d p d x p x l + p e − x . (2)

It can be seen from Eqs 1, 2 that the Laguerre polynomial L p l determines the radial distribution of the intensity of the LG mode. For simplicity, it is assumed that all the four subbands have the same effective mass. Furthermore, this SDQW is designed to have a low electron sheet density so that the electron–electron interactions can be reasonably neglected [62]. Under the rotating-wave and electric-dipole approximations, the interaction Hamiltonian for this system in the interaction picture can be written as (ℏ = 1). H int I = Δ p − Δ c 2 2 + Δ p 3 3 + Δ p − Δ c + Δ d 4 4 − Ω p e i k ⃗ p ⋅ r ⃗ 3 1 + Ω c e i k ⃗ c ⋅ r ⃗ 3 2 + Ω d e i k ⃗ d ⋅ r ⃗ 4 2 + Ω m e i k ⃗ m ⋅ r ⃗ 4 1 + H . c . , (3) where Δ _p = (ω ₃ − ω ₁) − ω _p , Δ _c = (ω ₃ − ω ₂) − ω _c , and Δ _d = (ω ₄ − ω ₂) − ω _d are the probe field, control field, and vortex field detunings, respectively. k j ⃗ (j = p, c, d, and m) is the wave vector of the corresponding applied field. The Rabi frequencies of the corresponding applied fields are Ω _p = μ ₃₁ E _p /2ℏ, Ω _c = μ ₃₂ E _c /2ℏ, Ω _d = μ ₂₄ E _d (r, ϕ)/2ℏ and Ω _m = μ ₄₁ E _m (r, ϕ)/2ℏ with μ _ij (i, j = 1–4; i ≠ j) being the transition dipole moment between subbands i ↔ j and E _p,c,d,m being the slowly varying electric field amplitude of the applied field. According to Ref. [63], the light intensity I _i (j = p, c, d, and m) of the applied field is proportional to the square of the electric field amplitude E _i , that is, I i ∝ E i 2 . In other words, I i ∝ Ω i 2 , which means that the Rabi frequency of an applied field can be used to represent its light intensity. Meanwhile, the electron wave function can be written as Ψ = A 1 1 + A 2 e i k ⃗ p − k ⃗ c ⋅ r ⃗ 2 + A 3 e i k p ⃗ ⋅ r ⃗ 3 + A 4 e i k p ⃗ − k ⃗ c + k ⃗ d ⋅ r ⃗ 4 , (4) where A _j (j = 1, 2, 3, 4) stands for the time-dependent probability amplitude for finding particles in the corresponding subband. Substituting Eqs 3, 4 into the Schrödinger equation i ∂ Ψ / ∂ t = H int I Ψ , the equations of motion for the probability amplitudes can be obtained as [52]. i ∂ A 1 ∂ t = − Ω p ∗ A 3 − Ω m ∗ e i δ k ⃗ ⋅ r ⃗ A 4 , (5) i ∂ A 2 ∂ t = Δ p − Δ c A 2 − i γ 2 A 2 − Ω c ∗ A 3 − Ω d ∗ A 4 , (6) i ∂ A 3 ∂ t = Δ p A 3 − i γ 3 A 3 − Ω p A 1 − Ω c A 2 + i ζ A 4 , (7) i ∂ A 4 ∂ t = Δ d − Δ c + Δ p A 4 − i γ 4 A 4 − Ω d A 2 − Ω m e − i δ k ⃗ ⋅ r ⃗ A 1 + i ζ A 3 , (8) in which δ k ⃗ = k ⃗ p − k ⃗ c + k ⃗ d − k ⃗ m denotes a phase mismatching factor. In Eqs 6–8, the decay rate γ _j (j = 2, 3, 4) is introduced phenomenologically. The total decay rate γ _j (j = 2–4) = γ _jl + γ _jd includes the population decay rate γ _jl and the pure dipole dephasing rate γ _jd . The population decay rate γ _jl is induced by the longitudinal optical phonon emission events at low temperatures, which can be calculated in [56]. The pure dipole dephasing rate γ _jd is due to a combination of quasi-elastic interface roughness scattering and acoustic phonon scattering. In the presence of the electronic continuum, the population decay rates γ _3l and γ _4l represent the decay rates from the subbands |3⟩ and |4⟩ to the continuum by tunneling with γ _3l = 1.58 meV and γ _4l = 1.5 meV. In the absence of the electronic continuum, γ _3l and γ _4l stand for the decay rates from the subbands |3⟩ and |4⟩ to the ground subband with γ _3l ≈ γ _4l = 1 meV. For temperatures up to 10 K, the electron density can be kept to 10²⁴m⁻³ [64]. In this sense, the dephasing rates can be estimated as γ _3d = 0.32 meV and γ _4d = 0.3 meV. It is worth noting that a cross coupling term between the two excited states 3 and 4 is introduced as ζ = γ 3 l ⋅ γ 4 l when the electronic continuum exists [56, 57, 64]. In this case, the strength of the Fano-type interference can be denoted by P = ζ / γ 3 ⋅ γ 4 , where p = 0 and p = 1 correspond to no interference and perfect interference, respectively. In the limit of slowly varying amplitude approximation, both the input probe field Ω _p and the generated FWM field Ω _m , which propagate in the z-direction, obey one-dimensional Maxwell wave equations. ∂ Ω p ∂ z + 1 c ∂ Ω p ∂ t = i c 2 ω p ∇ ⊥ 2 Ω p + i κ p A 3 A 1 ∗ , (9) ∂ Ω m ∂ z + 1 c ∂ Ω m ∂ t = i c 2 ω m ∇ ⊥ 2 Ω m + i κ m A 4 A 1 ∗ , (10) where κ p = 2 π N ω p μ 31 2 / ℏ c and κ m = 2 π N ω m μ 41 2 / ℏ c are the propagation constants with N being the electron density. The first terms on the right-hand sides of Eqs 9, 10 account for light diffraction. When the propagation distance L is much smaller than the Rayleigh length (i.e., L ≪ π w 0 2 / λ ), the diffraction term can be ignored. In our scheme, L = 1μm, w ₀ ≈ 500 μm, and λ _m ≈ 6.8 μm are selected so that π w 0 2 / λ m ≈ 1.154 × 1 0 5 μ m ≫ 1 μm, Therefore, we can neglect the diffraction terms in Eqs 9, 10. In the following, we perform a time-dependent analysis for FWM in the asymmetric SDQW, which requires both the input probe field and the generated FWM field as laser pulses, not cw lasers [65]. Then, we can perform the Fourier transformation for Eqs 5–10 by defining A j t = 1 2 π ∫ − ∞ ∞ a j ω exp − i ω t d ω , j = 2,3,4 , (11) Ω n t = 1 2 π ∫ − ∞ ∞ Λ n ω exp − i ω t d ω , n = p , m , (12) where ω is the Fourier transform variable.

In the limit of weak probe and FWM fields, most electrons remain in the ground subband |1⟩, that is, |A ₁|² ≈ 1. Therefore, we can obtain ω + Δ c − Δ p + i γ 2 a 2 + Ω c ∗ a 3 + Ω d ∗ a 4 = 0 , (13) ω − Δ p + i γ 3 a 3 + Ω c a 2 − i ζ a 4 = − Λ p , (14) ω − Δ d + Δ c − Δ p + i γ 4 a 3 + Ω d a 2 − i ζ a 3 = − Λ m , (15) ∂ Λ p ∂ z − i ω c Λ p = i κ p a 3 a 1 ∗ , (16) ∂ Λ m ∂ z − i ω c Λ m = i κ m a 4 a 1 ∗ . (17)

By analytically solving Eqs 13–15, one can obtain a _j (j = 2, 3, and 4) as a 2 = i ζ Ω c ∗ Λ m + ω − Δ d + Δ c − Δ p + i γ 4 Ω c ∗ + i ζ Ω d ∗ Λ p + ω − Δ p + i γ 3 Ω d ∗ Λ m D ω , (18) a 3 = − D p ω D ω Λ p + D 1 ω D ω Λ m , (19) a 4 = − D m ω D ω Λ m + D 2 ω D ω Λ p , (20) where D 1 ( ω ) = i ζ ( ω + Δ c − Δ p + i γ 2 ) + Ω c Ω d ∗ , D 2 ( ω ) = i ζ ( ω + Δ c − Δ p + i γ 2 ) + Ω d Ω c ∗ , D p ( ω ) = Ω d 2 − ( ω + Δ c − Δ p + i γ 2 ) ( ω − Δ d + Δ c − Δ p + i γ 4 ) , D m ( ω ) = Ω c 2 − ( ω + Δ c − Δ p + i γ 2 ) ( ω − Δ p + i γ 3 ) and D ( ω ) = − ζ 2 ( ω + Δ c − Δ p + i γ 2 ) − ( ω + Δ c − Δ p + i γ 2 ) ( ω − Δ p + i γ 3 ) ( ω − Δ d + Δ c − Δ p + i γ 4 ) + i ζ Ω d Ω c ∗ + ( ω − Δ d + Δ c − Δ p + i γ 4 ) Ω c 2 + i ζ Ω c Ω d ∗ + ( ω − Δ p + i γ 3 ) Ω d 2 .

Considering the initial conditions for the pulse probe and FWM fields, that is, Λ _p (0, ω) ≠ 0, and Λ _m (0, ω) = 0, we obtain the analytical solution of the FWM field as follows: Λ m z , ω = Λ p 0 , ω S ω e i z K + ω − e i z K − ω , (21) where K ± ω = ω c − D m ω κ m + D p ω κ p ± G ω 2 D ω = K ± 0 + K ± 1 ω + O ω 2 , (22) S ω = κ m D 2 ω G ω = S 0 + O ω , (23) with G ( ω ) = [ D p ( ω ) κ p − D m ( ω ) κ m ] 2 + 4 D 1 ( ω ) D 2 ( ω ) κ m κ p , by seeking the approximated inverse Fourier transform with the approximation of neglecting the O(ω) term in S(ω) and the O (ω ²) term in K _±(ω) [52]. Then it is straightforward to obtain Ω m z , t = S 0 Ω p 0 , t − z / V g − e i z K − 0 − Ω p 0 , t − z / V g + e i z K + 0 , (24) where the group velocities V _g± are determined by 1 / V g ± = R e [ K ± 1 ( 0 ) ] . Note that there exist two modes described by the dispersion relations K ₊(0) and K ₋(0). Re [K _±(0)] and Im [K _±(0)] represent the phase shifts per unit length and absorption coefficients at the center frequency ω = 0, respectively. A previous study [52] has demonstrated that the absorption of the K ₊(0) mode is much greater than that of the K ₋(0) mode. Therefore, the rapid decay K ₊(0) mode can be reasonably ignored after a short propagation distance L. Therefore, Eq. 24 can be simplified to Ω m L , t = Ω p 0 , t − L / V g S 0 e i K L , (25) where V _g = V _g− and K = K ₋(0). By using S ( 0 ) = κ m [ − ζ ( i Δ p − i Δ c + γ 2 ) + Ω d Ω c ∗ ] / G ( 0 ) and K = Re(K) + iIm(K), Eq. 25 can be rewritten as Ω m L , t = κ m Ω p 0 , t − L / V g − ζ i Δ p − i Δ c + γ 2 + Ω d Ω c ∗ G 0 e − I m K L + i R e K L , (26) where the intensity of the FWM field is ∝ | κ m Ω p ( 0 , t − z / V g ) [ − ζ ( i Δ p − i Δ c + γ 2 ) + Ω d Ω c ∗ ] e − Im ( K ) L / G ( 0 ) | 2 . The factor e ^i Re(K)L reflects the phase distribution of the FWM field. Obviously, both the phase and intensity distribution of the FWM field are modulated by dispersion relation K.

We first explore the influence of the Fano-type interference on the phase and intensity distribution of the generated FWM field in Figure 2. Here, the mode of the vortex driving field Ω _d is L G p = 0 l = 4 . In the absence of the electronic continuum, the Fano-type interference does not exist (i.e., p = 0). In this situation, the system is simplified as a common double-Λ SDQW [55]. One can find from Figures 2A,C that the phase wavefront twists in the anticlockwise direction and the intensity distribution exhibits a single-ring pattern at the radial position r = 0.71 mm. A phase singularity exists at the center of zero intensity, around which the helical phase changes from 0 to 8π. As explained, the OAM of the vortex driving field can be transferred to the generated FWM field via the FWM process [55]. In the presence of the electronic continuum, the Fano-type interference exists (i.e., p = 0.83) [52]. Different from Figures 2A,C, the FWM field displays four phase singularities on the inner side of the twisted phase wavefront in the phase profiles and the helical phase changes from 0 to 2π around every phase singularity (Figure 2B). Meanwhile, a petal-like intensity pattern with four light spots on the circle with radius r = 0.71 mm can be observed (Figure 2D). As a matter of fact, the cross coupling term ζ in Eq. 26 acts as a plane wave and then makes inherent interference with the vortex driving field. Therefore, we can observe an interference-type intensity pattern with four light spots and an interference-type phase pattern with four phase singularities, which satisfies the conservation of OAM [66, 67].

In order to have a deeper understanding for the effect of the Fano-type interference, we plot the spatial distribution of the real and imaginary parts of the dispersion relation K in Figure 3. Note that the phase shift Re(K) per unit length determines the phase wavefront distribution of the FWM field, while the absorption coefficient Im(K) determines the distribution of the intensity [55]. Without the Fano-type interference, the phase shift Re(K) per unit length displays an inverted crater-like pattern with a radius r = 0.71 mm and leads to the helical phase twisting in the anticlockwise direction (Figure 3A). Meanwhile, the absorption coefficient Im(K) also exhibits an inverted crater-like pattern at r = 0.71mm, where the low absorption ring results in the appearance of an intensity ring (Figure 3C). With the Fano-type interference, the inverted crater-like pattern of Re(K) also makes the phase twist in the anticlockwise direction (Figure 3B), while the existence of four independent low-absorption regions along the angular direction leads to a discrete intensity distribution with four light spots (Figure 3D).

A previous study [36] has demonstrated that the number of light spots in the interference spectrum is determined by the TC of the involved optical vortex. Therefore, the inherent interference mechanism in the proposed SDQW allows us to measure the TC of the vortex driving field via directly monitoring the number of light spots of the FWM field. It is worth noting that the precision of the TC measurement would be limited by the intensity and visibility of the inherent interference. To achieve a high-quality interference-type pattern, we explore the influence of the intensity and detuning of the control field based on Eq. 26. Figures 4A–C show the intensity patterns of the FWM field for different values of the control intensity Ω _c . When Ω _c = 10 meV, the FWM field shows a clear four petal-like intensity pattern, but the intensities of the four light spots are very small (Figure 4A). As we adjust Ω _c to 25 meV and then to 40meV, as shown in Figures 4B,C, the intensity of the FWM field becomes more and more stronger, while the visibility of the light spots becomes more and more worse. In order to evaluate the quality of the interference-type intensity pattern, we defined interference visibility η as [63]. η = I max − I min I max + I min , (27) where I _max and I _min represent the maximal and minimal light intensities along the angular direction in the interference-type intensity pattern of the FWM field, respectively. We can set I max = I 0 e − 2 α min L and I min = I 0 e − 2 α max L with α _min and α _max being the corresponding minimal and maximal absorption coefficients, respectively, of the FWM field along the angular direction. Then, Eq. 27 can be rewritten as η = e − 2 α min L − e − 2 α max L e − 2 α min L + e − 2 α max L = 1 − 2 e 2 Δ α L + 1 , (28) where Δα = α _max − α _min stands for the difference between the maximal and minimal absorption coefficients. It can be seen that Δα determines the interference visibility of the output FWM field. Saying concretely, the increase (decrease) of Δα would lead to the increase (decrease) of interference visibility η. Here, the absorption coefficient difference Δα and interference visibility η versus the azimuthal angle ϕ at radius r = 0.71 mm are plotted in Figure 4D. It is found that Δα decreases from 32,720.0 μm⁻¹ to 350.6 μm⁻¹ when Ω _c increases from 10 to 40 meV. Thus, interference visibility η shows a decreasing trend. It is worth noting that η is almost kept to 1 for 10 meV ≤ Ω c ≤ 20 m e V because Δα has an extremely high value in this region so that term e ^2ΔαL in Eq. 28 is close to infinity.

We also investigate the influence of control detuning Δ _c on the intensity pattern of the FWM field in Figures 5A–C. It can be seen that the intensity of the FWM field decreases monotonically as Δ _c increases from 2 to 14 meV (Figures 5A–C). Different from the results shown in Figure 4, Δα increases from 735.8 μm⁻¹ to 42,871 μ m⁻¹ as Δ _c increases from 2 to 14 meV (Figure 5D). In this sense, interference visibility η increases from 0.63 to 1. As Δ _c exceeds 5.2 meV, the value of Δα is extremely high so that interference visibility η ≈ 1. According to the aforementioned discussions, one can conclude that the intensity and detuning of the control field play different roles in modifying the intensity and visibility of the interference-type pattern. Therefore, a high-visibility interference-type pattern accompanied by an appropriate intensity can be achieved via choosing the suitable values of the intensity and detuning of the control field.

Based on the aforementioned discussions, we perform the TC measurement of the vortex driving field via monitoring the number of light spots in the intensity pattern of the generated FWM field. The intensity patterns of the FWM field for different TCs of the vortex driving field are displayed in Figure 6, and the corresponding curves for absorption coefficient difference Δα and interference visibility η are shown in Figure 7. It is worth noting that the waist radius of the vortex driving field is always kept to 500 μm for the selected TCs. In the measurement of the low-order TC (i.e., l ≤ 10), Ω _c = 25 meV and Δ _c = 8 meV are selected. We can clearly observe two, four, and six light spots in the intensity patterns of the FWM field when l = 2, 4, and 6 (Figures 6A–C). As shown in Figure 7A, Δα increases from 2,263.6 μm⁻¹ to 2,436.1 μm⁻¹ as l increases from 2 to 6. Therefore, interference visibility η would increase from 0.9786 to 0.9848 for the low-order TC case. Although the overall output intensity decreases in some degree due to the dependence of the Laguerre polynomial on the TC, the visibility of the interference-type pattern always exceeds 0.97. That is to say, we can realize a high-precision measurement for the low-order TC of a mid-infrared optical vortex. In the measurement of the high-order TC, Ω _c = 13 meV and Δ _c = 2 meV are chosen. When l = 100, 110, and 120, the same number of light spots can be observed in the interference-type pattern of the FWM field (Figures 6D–F). The overall output intensity of the FWM field only decreases slightly with the increase of l from 100 to 120. Δα first slightly increases from 2,126.7 μm⁻¹ to 2,127.1 μm⁻¹ as l increases from 100 to 105 and then decreases to 2,125.6 μm⁻¹ as l increases to 120 (see the blue line in Figure 7B). Therefore, interference visibility η would slightly increase from 0.97197 to 0.97207 and then decrease to 0.97191 with the increase of l from 100 to 120 (see the red line in Figure 7B). The change of Δα is so small that interference visibility η is kept at the level of 0.9719. Therefore, our scheme is suitable for measuring both the low-order and high-order TC. More importantly, the measurable TC value is greatly improved compared with previous schemes [38, 58].

Before concluding, we explore the influence of the TC of the vortex driving field on the conversion efficiency of FWM. FWM efficiency ρ is defined as ρ = E m ( o u t ) / E p ( i n ) 2 [65], where E p ( o u t ) is the is the electric field amplitude E m ( E m 2 = 4 ℏ 2 Ω m 2 / μ 31 2 ) of the generated FWM field at the exit z = L and E p ( i n ) is the electric field amplitude E p ( E p 2 = 4 ℏ 2 Ω p 2 / μ 41 2 ) of the probe field at entrance z = 0. Combined with Eq. 26, the FWM efficiency can be rewritten as ρ = μ 31 2 μ 41 2 Ω m 2 Ω p 2 = μ 31 2 μ 41 2 κ m − ζ i Δ p − i Δ c + γ 2 + Ω d r , ϕ Ω c ∗ e − I m K L G 0 2 , (29) where μ 31 2 / μ 41 2 = κ p ω m / κ m ω p . It can be seen from Eq. 29 that the spatial distribution of the FWM efficiency depends on the spatially dependent vortex driving field. In Figure 8, we plot FWM efficiency ρ versus radius r for different values of the TC by setting ϕ = 0. In this situation, these curves can reflect the radial distribution of the FWM efficiency cross the center of the light spot. As shown in Figure 8, the FWM efficiency is spatially dependent for a certain vortex driving field. In the low-order TC case, the peak value of the FWM efficiency decreases from 9.86 to 3.66% as l increases from 2 to 6 (Figure 8A). In the high-order TC case, the optimal FWM efficiency decreases from 4.79 to 4.04% with the increase of l from 100 to 120 (Figure 8B). In the two cases, the peak of the FWM efficiency moves toward the larger radius with the increase of TC owing to the change of the ring radius of the vortex driving field. Thus, one can conclude that the FWM efficiency would decrease when the ring radius of the vortex driving field increases with the TC.

In conclusion, we have theoretically suggested a scheme to measure the TC of a mid-infrared vortex beam via observing the intensity distribution of the FWM field in an asymmetric SDQW with a four-subband configuration. In this SDQW, the effect of the Fano-type interference exists because the absorption paths of two states are coupled to an electronic continuum [56, 57]. With the help of the Fano-type interference, the cross coupling term acting as a plane wave makes inherent interference with the vortex beam and generates the interference-type patterns for the phase and intensity of the FWM field. It is demonstrated that a high-contrast interference-type pattern accompanied by appropriate intensity can be achieved via adjusting the intensity and detuning the control field. Furthermore, the TC measurement of the vortex beam is performed by monitoring the number of light spots of the FWM field. By properly choosing the parameters of the system, the detectable TC value can reach to 120, which is greatly improved compared with previous schemes [38, 58].

Mid-infrared optical vortices can open up new avenues toward super-resolution microscopy of arbitrarily oriented single molecules [68], fabrication of three-dimensional chiral microstructures [69], and OAM-multiplexing-based free-space optical communication [70]. Our scheme may provide the possibility for the realization of mid-infrared OAM detectors in a compact solid-state system.