Where is the orbital angular momentum in vortex superposition states?

Huajie Hu; Hehe Li; Xueyun Qin; Xinzhong Li

doi:10.1364/OE.523305

1. Introduction

Interference is a fundamental physical phenomenon [1,2]. After the overlapping region, each beam maintains the independence property. Many reports address the superposition state of coaxial vortex beams carrying an orbital angular momentum (OAM) [3–5], which tend to rotate along their optical axis upon propagation [6,7]. This type of beams have many applications in optical communication [8,9], as they trap and guide particles [10]. The conservation of the angular momentum in this process and the location of the OAM in the superposition state have not been discussed. The angular momentum includes the spin angular momentum and the OAM [11,12]. The spin angular momentum corresponds to the polarization state, and vortex beams carrying an OAM exhibit helicity patterns in their cross sections [13,14]. The phase profile can be described as $\exp (i l_{n} \varphi )$. $l_{n}\in Z$ denotes the topological charge, Which is equal to the OAM quantum number, and $\varphi$ represents the azimuthal angle. Many novel physical phenomena exist due to the conservation of the angular momentum in the interaction with a medium [15]. For instance, when the fundamental Gaussian mode propagates in an-isotropic medium, the spin-OAM mapping phenomenon exists [16]. This physical phenomenon can be explained by the different phase modulation for two orthogonal circular polarization basis vectors. The generation principle of a vector beam is also based on the conservation of angular momentum [17,18]. Using the conservation of angular momentum in the multi-wave mixing effect in an atomic medium, researchers realized the frequency conversion of structural beams [19,20]. To realize interference, three conditions should be satisfied by the encountering beams: same polarization state, same frequency, and fixed phase difference. The same polarization reflects the spin angular momentum conservation. Here, we only need to demonstrate the OAM conservation in interference [15,21].

In this letter, we selected two coaxial perfect vortex beams with different waist spot size as the input beams to analyze the conservation of the OAM in the interference process. According to the relationship between topological charge and OAM, we analyzed the conservation of the topological charge in the interference process. We found that in this case there exist some specific singular points in the spatial intensity distribution. The first type of singular point is located at the center point of the spatial intensity distribution whose topological charge value is equal to that of a vortex beam with a small waist spot size. The second type of specific singular point exists at the critical location of the overlapping area in angular direction due to the phase difference, and the number of singular points and the topological charge value contained in those singular points are related to the difference of the internal and external topological charge values. Intuitively, from the interferogram, the topological charge value contained in the superposition field is not equal to that of the incident field. To calculate the total topological charge value of the superposition state, we propose a hypothesis: The center point is located at the overlapping area. Therefore, double topological charge value actually exists at the center point. The sum of those two types of topological charge value corresponds to the total topological charge value of the vortex superposition state, which is equal to that of the incident superposition field. This law can also be applied to the superposition of multiple coaxial vortex beams.

2. Simulation

We adopted the superposition states of horizontal polarization perfect vortex beams with different topological charges values as the input field [22]:

(1)$$E(r, \varphi,z) = \sum_{l_{n}}i^{l_{n}-1} \frac{\omega_{g}}{\omega_{0}} \exp (i l_{n} \varphi)\exp (i k z) \exp \left(\frac{-(r-R_{n})^{2}}{\omega_{0}^{2}}\right)$$

where ($r,\varphi,z$) is the cylindrical coordinate vector, $l_{n}$ is the topological charge order of the perfect vortex beams, $\omega _{g}$ is the waist spot, $\omega _{0}=\frac {2f}{k\omega _{g}}$, where $f$ is the lens whose focus is 5 mm, $k$ is the wave vector, and $R_{n}$ is the radial values. Firstly, we selected two coaxial perfect vortex beams with different waist spot size as the input beams to analyze the conservation of the OAM in the interference process. In this case the radial values of the inner and outer circles $R_{n}$ as 40 and 60 $\mu$m, respectively. Here, we consider two cases. In the first case, the topological charge value in the outer circle is bigger than that in the inner circle. In the second case, the topological charge value in the outer circle is smaller than that in the inner circle.

We show the interference diagram of the two-coaxial vortex beams in Fig. 1. In Fig. 1(a)- (a2), the topological charge values in the inner and outer circles are 1 and 3, 2 and 4, and 3 and 5, respectively. Here, the difference in the topological charge value between the inner and outer circles is −2. In this case, the interference diagram has three minimum values in total. The first one is located at the center intensity spot, whereas the other two are evenly distributed in the angular direction. To determine the OAM distribution in this superposition field, we realized the interference process between the reference Gauss sphere beam and the superposition field shown in Fig. 1, second line. Through the interference diagram, we can see that the topological charge value in the angular direction is +1. The number of the singular point in angular direction of the superposition states is 2. The topological charge values in the center intensity spot in Fig. 1(b)-(b2) are 1, 2, 3, respectively, which correspond to the topological charge value in the inner circle. As we know, the total topological charge values of the input beam are 4, 6, and 8, respectively. However, the total topological charge values of the superposed field are 3, 4, and 5, respectively. Where does the residual topological charge exist? It is suggested that the center point is located at the overlapping area. Therefore, a double topological charge actually exists in the center point. Based on this suggestion, the total topological charge values in Fig. 1(a)-(a2) are 2*1+2*1=4, 2*2+2*1=6, and 2*3+2*1=8, and this result is consistent with the conservation of the OAM in superposed states. In Fig. 1(c)- (c2), the topological charge values in the inner and outer circles are 3 and 1, 4 and 2, and 5 and 3, respectively. The difference in the topological charge value between the inner and outer circles is 2. In this case, the interference diagram has also three minimum values in total. The interference diagrams between the reference Gauss sphere beam and the superposed field are shown in Fig. 1(d)-(d2). We can see that the topological charge value in the angular direction is −1. The topological charge values in the center intensity spot in Fig. 1(d)- (d2) are 3, 4, and 5, respectively. Given the above suggestion, the total topological charge values in Fig. 1(c)-(c2) are 2*3+2*(−1)=4, 2*4+2*(−1)=6, and 2*5+2*(−1)=8. This result shows that the OAM is conservative in the interference process.

Fig. 1. Interference diagram of the two-coaxial vortex beams. Figure 1(a)-(a2) shows the intensity distribution of the superposed states where the topological charge values in the inner and outer circles are 1 and 3, 2 and 4, and 3 and 5, respectively. Figure 2(b)-(b2) shows the corresponding interference diagram between the reference Gauss sphere beam and the superposed field. Figure 1(c)-(c2) shows the intensity distribution of the superposed states where the topological charge values in the inner and outer circles are 3 and 1, 4 and 2, and 5 and 3, respectively. Figure 2(d)-(d2) shows the corresponding interference diagram between the reference Gauss sphere beam and the superposed field.

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Compared with the interference diagram shown in Fig. 1, second and fourth column, the topological charge value in the angular direction are opposite of each other. This physical phenomenon can also be explained by the conservation of the OAM. For instance, in Fig. 1(a), the total topological charge value in the center spot is 2, which is smaller than that of the input vortex beams. Therefore, the OAM in the angular direction should be consistent with the rotation in the central position. Meanwhile, in Fig. 1(c), the total topological charge value in the center spot is 6, which is greater than that of the input vortex beams. Therefore, the rotation of the OAM in the angular direction should be opposite to that in the central position.

To demonstrate the applicability of this suggestion, we analyzed the superposition of three coaxial vortex beams with different topological charge values, as shown in Fig. 2. In Fig. 2(a)-(e), the topological charge values from the innermost to outermost circle are 3, 3, 3; 3, 3, −3; 3, −3, 3; −3, 3, 3; and −3, −3, 3, respectively. The radial values from the innermost to outermost circle $R_{n}$ are 40, 70, and 100$\mu$m, respectively. As the vortex order of each perfect vortex beam component is the same in Fig. 2(a), the interference diagram between the reference Gauss sphere beam and the superposed field shown in Fig. 2(a1) has only one minimum value located at the center intensity spot, and the corresponding topological charge value is 3. Based on the above suggestion, the total topological charge value in Fig. 2(a) is equal to 3*3=9. When the topological charge value of the outermost ring is replaced by its opposite number, as shown in Fig. 2(b), the phase singularity exists at the critical location of the overlapping area between the middle and outermost rings. As can be observed from the interference diagram shown in Fig. 2(b1), the topological charge value in the angular direction is −1. The topological charge value in the center intensity spot in Fig. 2(b1) is 3. The total topological charge value in Fig. 2(b) is 3*3+6*(−1)=3. When the OAM of the middle ring is replaced by its opposite number, shown in Fig. 2(c), the phase singularity exists at the critical location of the overlapping area. From the interference diagram shown in Fig. 2(c1), we can see that the topological charge value at the critical location of the overlapping area between the innermost and middle rings is −1, whereas that between the middle and outermost rings is 1. The topological charge value in the center intensity spot in Fig. 2(c1) is 3. The total topological charge value in Fig. 2(c) is 3*3+6*(−1)+6*(1)=9. The value is not equal to the total OAM of the input field. Here, we should consider that the outermost ring also contains the critical location of the overlapping area between the innermost and middle rings, and the topological charge value at this area should be double. Therefore, the total topological charge value in Fig. 2(c) is 3*3+6*(−1)*2+6*1=3. Based on this suggestion, the total topological charge value in Fig. 2(a) is amended to 3*3+0*2+0*1=9. The total topological charge value in Fig. 2(b) is 3*3+0*2+6*(−1)=3. When the OAM of the innermost ring is replaced by its opposite number, as shown in Fig. 2(d), the phase singularity exists at the critical location of the overlapping area between the middle and innermost rings, and the topological charge value in the angular direction is 1. The topological charge value in the center intensity spot in Fig. 2(d1) is −3. The total topological charge value in Fig. 2(d) is (−3)*3+6*1*2+0*1=3. When the OAM of the innermost and middle rings is replaced by the opposite number, as shown in Fig. 2(e), the phase singularity exists at the critical location of the overlapping area between the middle and outermost rings. The total topological charge value in Fig. 2(e) is (−3)*3+0*2+6*1=−3. This law can be applied to the interference of n-layer ring vortex beams with different topological charge values. Supposing that the corresponding vortex order of each layer ring from the innermost to outermost rings is $l_{1}$, $l_{2}$,…$l_{n}$. The total topological charge value is equal to $l_{1}*n+(l_{2}−l_{1})*(n−1)+(l_{3}−l_{2})*(n−2)\cdots +(l_{n}−l_{n−1})=l_{1}+l_{2}\cdots +l_{n}$. It should be noted that the number of the singular point in angular direction located at the critical location of the overlapping area between the $n$th and $(n-1)$th rings of the superposition states is equal to the value of $\left |l_{n}-l_{n-1}\right |$. This result satisfies the conservation of the OAM in interference process.

Fig. 2. Interference diagram of the three-coaxial vortex beams. Figure 2(a)-(e) shows the intensity distribution of the superposed states, and the topological charge values from the inner to outer circles are 3, 3, 3; 3, 3, −3; 3, −3, 3; −3, 3, 3; and −3, −3, 3, respectively. Figure 2(a1)- (e1) shows the corresponding interference diagram between the reference Gauss sphere beam and the superposed field.

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3. Experiment

We performed an experiment to demonstrate the suggestion. Here, we selected the combination of the 3rd vortex beam in the inner circle and the 11th vortex beam in the outer circle as the superposition state component. In the first line of Fig. 3, the interference diagram has two types of minimum values in total. The first type is located at the center intensity spot, whereas the values of the second type are evenly distributed in angular direction. The interference diagram between the reference Gauss sphere beam and the superposed field is shown in the second line. The blue circles show the location of the second type of the singular points, and the blue auxiliary dotted line are added to calculate the topological charge value. We can see that the topological charge value in the center intensity spot in Fig. 3(a1) is 3. The topological charge value in the angular direction is 1. The total topological charge value in Fig. 3(a) is 2*3+8*1=14. Utilizing the N-fold phase doubling technique mentioned in Reference [10], we also analyzed the case with high topological charge value in the angular direction. When N=2, the total topological charge value of the input field is 28. The interference diagram shown in Fig. 3(b) has also two types of minimum values. The interference between the reference Gauss sphere beam and the superposed field is shown in Fig. 3(b1). The topological charge value in the center intensity spot in Fig. 3(b1) is 6, and the topological charge value in the angular direction is +2. The total topological charge value in Fig. 3(b) is 2*6+8*2=28. When N=3, the total topological charge value of the input field is 42. Through the interference diagram between the reference Gauss sphere beam and the superposed field shown in Fig. 3(c1), we can see that the topological charge value in the center intensity spot in Fig. 3(c1) is 9, and the topological charge value in the angular direction is +3. The total topological charge value in Fig. 3(c) is 2*9+8*3=42. Therefore, the OAM is conservative in the case with high topological charge value in the angular direction.

Fig. 3. Interference diagram of the two-coaxial vortex beams in the experiment. In Fig. 3(a)-(c), the waist size $\omega _{g}=10 \mu$m, and N=1,2,3, respectively. Figure 3(a1)-(c1) shows the corresponding interference diagram between the reference Gauss sphere beam and the superposed field. In Fig. 3(a2)-(c2), the waist size is $\omega _{g}=20 \mu$m, and N=1,2,3, respectively. Figure 3(a3)-(c3) shows the corresponding interference diagram between the reference Gauss sphere beam and the superposed field. In the second line the blue circles show the location of the second type of the singular points, and the blue auxiliary dotted line are added to calculate the topological charge value.

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In the experiment, when the waist size increased, we could only observe the OAM distribution in the angular direction in Fig. 3, fourth line. The reason of this physical phenomenon is that when the waist size increases, the input beam is more like a plane wave and the intensity focus is on the ring area, whereas the intensity value at the spatial center area approaches zero. Meanwhile, the radial ring size of the generated perfect vortex beam decreased, and the center area where the intensity is zero became bigger. As we know, we cannot measure the phase at the spot where the electric field value is zero. Therefore, when we did the interference process between the reference Gauss sphere beam and the superposed field, the interference diagram in the center intensity spot fourth line could not be observed. With propagation, the perfect vortex beam evolved into the general vortex beam, and the interference diagram in the center intensity spot could be observed again. To show this physical phenomenon, we used the split-step Fourier algorithm to solve the propagation equation [23,24], Eq. (2), in the free space shown in Fig. 4.

(2)$$\frac{\partial E(r, \varphi,z)}{\partial z}=\frac{i}{2 k} \nabla_{{\perp}}^{2} E(r, \varphi,z)$$

Fig. 4. Evolution of the superposed states at different propagation distances z. Figure 4(a)-(d) shows the intensity distribution of the superposed states at different propagation distances z=0, 1, 4, 7 mm, respectively. Figure 4(a1)-(d1) shows the corresponding interference diagram between the reference Gauss sphere beam and the superposed field.

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The first line in Fig. 4 shows the evolution of the superposed states at different propagation distances z. In Fig. 4(a)-(d) z=0, 1, 4, 7mm, respectively. We can see that with propagation, the waist size became bigger owing to the diffraction effect, and the interference diagram in the center intensity spot appeared again at a distance z=7 mm, as shown in Fig. 4(d1). Adding the effect of the helical phase, the superposed state rotates clockwise during propagation [25].

4. Conclusion

In conclusion, we discussed the location and conservation of the OAM in the coaxial interference process of vortex beams based on the independent propagation principle of light. We found that the interference diagram has two types of minimum values in total. This result is analogous to the superposition of angular momentum of two axially rotating homogeneous disks with different radius in rigid body. At the overlapping area the contribution of the amplitude components of each superposition state should be taken into account. Based on this suggestion, the sum of those two types of topological charge values is exactly equal to that contained in the input superposition state. The conservation of the OAM in the coaxial interference process is demonstrated. The evolution law of the OAM in the off-axial interference process of vortex beams is a complex physical process due to the external OAM. New topological charges may also be generated in the interference process [26].

Funding

National Natural Science Foundation of China (12204155, 11974101, 12274116).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Where is the orbital angular momentum in vortex superposition states?

Abstract

1. Introduction

2. Simulation

3. Experiment

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (4)

Equations (2)

Optics Express