## Abstract

When two vortex beams with unequal topological charges superpose coherently, orbital angular momentum (OAM) in the two beams would not be cancelled out completely in the interference. The residual OAMs contained by the superposed beam are located at different concentric rings and may have opposite orientations owing to the difference of the charges. The residual OAM can be confirmed by the rotation of microparticles when difference between the charges of two interfering beams is large.

© 2006 Optical Society of America

## 1. Introduction

In recent years, studies on the orbital angular momentum (OAM) of a vortex beam have extensively been reported.^{1–4} A vortex beam with helical phase structure of exp(*ilθ*), where *l* is the topological charge and *θ* is the azimuthal angle, possesses OAM.^{1} Helical phase can exist in a Gaussian beam, namely the Laguerre-Gaussian (LG) beam, modulated vortices,^{5–7} or a high-order Bessel beam.^{8–10} A vortex beam with an integer charge always forms a zero-intensity core encompassed by a bright ring in free-space propagation. As the tightly focused doughnut is associated with optical gradient force and OAM, so the beam can be used for trapping and rotating particles intrinsically or extrinsically.^{3} Moreover, the OAM carried by a vortex beam can also be applied for information encryption/decryption in free-space optical communications.^{11} Recently, generation and characterization of beams interfered by double vortex beams have intensively been implemented, and interferences by two vortex beams with charges of *l*
_{1}=-*l*
_{2} have also been employed for stably trapping microparticles without rotation as the OAMs in each interfering beam have been cancelled out after the coherent superposition.^{12} However, when two vortex beams with charges of ∣*l*
_{1}∣≠∣*l*
_{2}∣ interfere, the OAMs of both the interfering beams are not equal and cannot totally be neutralized; as a result, the interferenced beam still possesses residual OAM, which is not identical to that of either of the vortex beams any more. To our best knowledge, optical rotation induced by a beam interfered by two vortex beams with unequal charges of ∣*l*
_{1}∣≠∣*l*
_{2}∣ has not been experimentally demonstrated.

In this paper, we investigate intensity and phase evolutions of the beams interfered by two vortex beams with unequal charges, and generate the interferenced beams experimentally. Optical rotation induced by the interferenced beam is also demonstrated.

## 2. Simulation of beam propagation

As we know, a vortex beam can be generated by a phase-only hologram with expression of exp(*ilθ*), and for a beam interfered by two vortex beams its complex amplitude can be expressed by,

where *A*
_{1}(*x*, *y*) and *A*
_{2}(*x*, *y*), *l*
_{1} and *l*
_{2} are the corresponding interfering vortex beams’ amplitudes and topological charges, respectively. The interference can be classified into two groups based on the charges: (1) *l*
_{1} ∙ *l*
_{2} > 0 and (2) *l*
_{1} ∙ *l*
_{2} < 0. Obviously, for the first group, the total OAM would be enhanced when two interfering beams have charges of the same sign. However, for the interference by two vortex beams with charges of opposite signs and ∣*l*
_{1}∣≠∣*l*
_{2}∣, the total OAMs may not be neutralized completely. Hence, residual OAM in the interferences of *l*
_{1} ∙ *l*
_{2} < 0 will be of interest for investigation. Without losing generality, three cases of interferences by two vortex beams with charges of (*l*
_{1}=20, *l*
_{2}=-3), (*l*
_{1}=-20, *l*
_{2}=3), and (*l*
_{1}=20, *l*
_{2}=-20) are investigated. We use the angular spectrum of plane waves method ^{13} to simulate the beam’s propagation in free space. In the simulation the grid of sampling points is 256 by 256 with pixel size of 15 μm by 15 μm and the wavelength is 0.6328 μm. On the basis of Eq. (1), where *A*
_{1}(*x*, *y*) and *A*
_{2}(*x*, *y*) is set as 1, the interferenced beams’ intensities and phases in free-space propagation are simulated and displayed in Fig. 1. Videos for phase evolutions of (*l*
_{1}=20, *l*
_{2}=-3), (*l*
_{1}=-20, *l*
_{2}=3), and (*l*
_{1}=20, *l*
_{2}=-20) in propagation are also attached, respectively.

Figures 1(a, b, c) show intensity patterns of the three interferenced beams at propagation distance of 166 mm, which is randomly chosen in the range of the beam reconstruction. The grey scales ranging from the black to the white are corresponding to the beam intensities from zero to maximum. One can observe from Figs. 1(a, b) that both the intensity patterns comprise two parts: one innermost bright ring and one fringed ring with ∣*l*
_{1}-*l*
_{2}∣ fringes. The innermost rings in Figs. 1(a, b) are clearly visible, but in Fig. 1(c) where both the interfering vortex beams have the same charge and opposite signs, the innermost ring disappears and the fringes exhibit higher visibility. Since the positions of the innermost rings and the fringed rings in the interferenced beams are the same as those of the respective doughnuts of the interfering beams, both the rings probably keep some OAM of the original interfering beams. For further verification, phase evolutions of the interferenced beams with (*l*
_{1}=20, *l*
_{2}=-3), (*l*
_{1}=-20, *l*
_{2}=3), and (*l*
_{1}=20, *l*
_{2}=-20) in free-space propagation from 166 mm + λ/3 to 166 mm + λ with a step of λ/3 are shown in Figs. 1(A1-3, B1-3, C1-3), respectively, where the grey scales ranging from the black to the white are corresponding to the phases from 0 to 2π.

From Figs. 1(A1-3, B1-3) one can observe that the innermost and the outer parts of the phases are rotating separately at different speeds in opposite directions. The numbers of the rotating cycles of the innermost and the outer segments are equal to ∣*l*
_{1}∣^{-1} or ∣*l*
_{2}∣^{-1} in one wavelength propagation, respectively. Furthermore, when the signs of the two interfering beams are swapped, the rotating directions are swapped, too. When we further propagate the beams, the innermost segments rotates a full cycle in the propagation distance of λ∙min(∣*l*
_{1}∣, ∣*l*
_{2}∣) while the outer segments in the distance of λ∙max(∣*l*
_{1}∣, ∣*l*
_{2}∣). The rotating phases prove that the interferenced beams still possess OAMs, which are located at separate concentric rings and rotate in opposite directions. The rotations in the inner and the outer parts resemble to the respective interfering beams and confirm that the interferenced beams with unequal charges can possess OAMs. However, as shown in Figs. 1(C1-3), the interferenced beam formed by vortex beams with equal charges but opposite signs has no rotation observed at all. Clearly, in this case the OAMs possessed by two interfering beams have totally been cancelled out in the interference. In terms of topological charge and beam size the innermost rings in the interferences of (*l*
_{1}=20, *l*
_{2}=-3) and (*l*
_{1}=-20, *l*
_{2}=3) are similar to the doughnuts of the corresponding interfering vortex beam with a smaller charge and the fringed rings are related to the interfering beam with a greater charge. Furthermore, from the visibility of the fringed rings shown in Figs. 1(a, b) one can deduce that the interfering beams could have unequal intensities or incomplete superposition. As we know, the maximum visibility for an interferenced pattern occurs when two interfering beams have equal or nearly equal intensities so that the interferenced pattern composes dark and bright fringes. Thus, for the interfering beams with equal charges their interferogram has highest visibility.

For an LG beam its field amplitude in free-space propagation can be written as,

where *L ^{l}_{P}* ( ) is a generalized Laguerre polynomial with radial index

*p*,

*R(z)*is the radius of curvature of the wave front, Ψ(z) is the Guoy phase, and w is the beam’s radius.

^{1}It has been found that an LG beam with

*p*=0 has a bright intensity ring, whose radius varies as its topological charge

*l*as

*R*∞ √

_{l}*l*; however, for a vortex beam reconstructed from a hologram by uniform illumination with a monochromatic light, the relation between the radius of the ring versus the charges was found to be

*R*∞

_{l}*l*.

^{6}Both clearly stated that the doughnut in a vortex beam with a smaller charge would have smaller radius and vice versa. Thus, one can understand that, since the intensity rings of the two interfering vortex beams with unequal charges have not been fully superposed in the interference and only part of them interferes, the superposed part is closely related to the interfering beam with the greater charge while in the non-overlapping area the OAM originated from the interfering beam with the smaller charge will be maintained.

Figure 2(a) illustrates the cross-section intensity plots of individual vortex beams of (*l*
_{1}=20) and (*l*
_{1}=-3) before the interference, and Fig. 2(b) the plot of the interferenced pattern. The simulated plots are obtained by illuminating holograms with a uniform monochromatic light at normal incidence. It can be observed from Fig. 2(a) that both the peaks of the interfering beams are not fully overlapped. The interferenced pattern in Fig. 2(b) has similar peaks of the innermost and the outer, which are respectively corresponding to the peaks of the interfering beams in Fig. 2(a). Note that the sampling cross-section plot in Fig. 2(b) is not symmetrical due to the spiral intensity segments shown in Fig. 1(a). Compare the curves in Figs. 2(a, b) and we can find that both the innermost peaks are almost identical but the outer rings are varying due to the interference occurred. As mentioned above, each interfering vortex beam has a dark core encompassed by the bright rings, and when the difference between the ring sizes of the two doughnuts is considerable, both the bright rings of the two beams would not overlap completely during the interference. Thus, the innermost ring in the interference pattern most probably maintains the original smaller-charge vortex beams’ properties and the fringed ring still maintains some of the properties of the greater-charge one. In this view, we can explain why the phases are localized in different concentric rings in the interferenced beams such as (*l*
_{1}=20, *l*
_{1}=-3) and (*l*
_{1}=-20, *l*
_{1}=3) and rotate oppositely. Fig. 3 shows the energy percentages of the areas inside the fringed ring and in the fringed ring in the interferenced beams with varying charges of (*l*
_{1}=20, l2=-1∼-20) at the propagation distance of 166 mm, respectively. A video for the cross-section intensities of the corresponding interferenced beams is also attached. In Fig. 3, the upper curve represents the intensity percentage of the fringed ring and the lower one the percentage of the area encircled by the fringed ring. The intensity percentage of the fringed ring increasing with ∣*l*
_{2}∣ illustrates that the interference in the ring is becoming stronger while the decreasing curve of the lower plot exhibits the OAM in the innermost part transferring to the fringed ring. The sums of both the percentages are always constant because the total energy in the interferences is unchanged.

## 3. Experimental results

As numerous reports have described the generation of a vortex beam with a hologram in detail, we will not dwell on the method of generation. Using an expanded and collimated laser beam to illuminate phase-only holograms, which are loaded onto a spatial light modulator (SLM) beforehand, we obtain the desired interferenced beams. However, it is worth mentioning that, to avoid interference of the direct-component of the reflected light from the SLM, we add phase of a blazed grating to the hologram to divert the reconstructed on-axis beam to a defined off-axis angle. Mathematically, a displaced vortex beam along *x* axis in (*x*, *y*) coordinates can be written as exp(*iαx*) ∙ exp(*ilθ*), where *α* is a coefficient adjusting the off-axis displacement. Thus, phase distribution of the hologram loaded onto the SLM is written by the following expression,

where *angle*( ) is a function extracting the phase from the enclosed complex expression. Although the amplitude information in Eq. (3) has been ignored in the hologram, the reconstructed beam still much approximates to the desired one, especially for the beam with smaller value of ∣*l*
_{1}+*l*
_{2}∣. In the experiment, the optical tweezers system mainly consists of a diode-pumped solid-state (DPSS) YAG:Yvo4 laser (Verdi 8, Coherent) with wavelength of 532 nm and power up to 8 W, an SLM (Holoeye, LCR-3000) with resolution of 1920 × 1200 at ∼ 9.5 um pixel pitch, and a microscope (Carl-Zeiss, Axiovert 25). First, the green light is emitted from the DPSS laser, expanded and collimated by a telescope system, and then directed onto the SLM screen by the mirrors. It should be noted that, in the setup a half-wave plate is inserted to adjust the polarization orientation of the linearly polarized light incident to the SLM so as to modulate the light in phase only.^{14} After that, the interferenced beam reconstructed by the SLM is condensed by an inverted telescope system and steered into the microscope by the mirrors. Finally, the beam was focused by an oil-immersion objective lens (100×, numerical aperture 1.25) onto the sample stage. In the mean time, a couple-charged device camera is attached to the microscope to view the samples reflected by a dichroic mirror. The detailed configuration can be also found in Refs [10, 14]. As an example, some interferenced patterns formed by the vortex beams with charges of (*l*
_{1}=20, *l*
_{2}=-1), (*l*
_{1}=20, *l*
_{2}=-3), (*l*
_{1}=20, *l*
_{2}=-8), (*l*
_{1}=20, *l*
_{2}=-12), (*l*
_{1}=20, *l*
_{2}=-15), and (*l*
_{1}=20, *l*
_{2}=-20) are shown in Fig. 4, respectively, where diameters of the brightest fringed rings are around 20 μm. It can be seen that diameters of the innermost rings increase with the smaller topological charge of the two vortex beams and when *l*
_{1}=- *l*
_{2} the innermost ring disappears while the outer fringed rings have increasing contrast, fixed radius and ∣*l*
_{1}-*l*
_{2}∣ bright fringes.

In the trapping experiments, the laser power of the DPSS laser is set as 850 mW. Some latex spheres (3.1 um in diameter) are suspended in de-ionized water and enclosed in a chamber, which sandwiches a slide-well film with thickness of ∼50–70 μm by two cover slides. We achieved the optical rotation induced by interferenced beams, and the ring sizes of trapped particles are commensurate with those of the respective interfering vortex beams. Moreover, when we swap the signs of the charges of the two interfering beams, the particles rotate in the reverse direction. Fig. 5(a) shows an example of optical rotation induced by both the residual OAM of the interferenced beams of (*l*
_{1}=10, *l*
_{2}=-2) and (*l*
_{1}=-10, *l*
_{2}=2). In the attached video, particles rotate clockwisely first and then rotate anticlockwisely after the swap of the charges of the two interfering beams. Although in the simulations both the phases of the innermost ring and the fringed one in the interferenced beam with unequal charges are found to possess residual OAM, it is difficult to verify the OAM with optical rotation induced by the fringed ring because the intensity is discontinuous and the intensity variation greatly hinders the particles moving along the ring. Furthermore, the clearer the visibility of the fringes, the harder the rotation is. However, if both the doughnuts of the interfering beams are not overlapped at all, it will be easy to fulfill the rotations both by the smaller doughnut and the larger one simultaneously. In our previous study^{10}, we implemented fractional vortex beams for optical rotation. Although a fractional vortex beam possesses OAM, it also has a low-intensity gap in the bright ring. The low-intensity gap impeded the rotation remarkably, and for the beam with half-integer charge the hindrance was maximized as the beam had the widest gap. Since the fringed ring has so many low-intensity gaps, particles trapped in the bright fringes would be very difficult to overcome the adjacent low-intensity barriers even though the existence of OAM in the fringed ring. The fringed ring can still be utilized to stably trap particles. As shown in Fig. 5(b), some particles are stably trapped in the fringed ring of a beam interfered by two beams of (*l*
_{1}=10, *l*
_{2}=-3) while a full ring of particles trapped by the innermost ring of the interferenced beam rotates. Such a beam has possible application for measurement of an object’s torque by fixing the object’s outer part and rotating the inner part.

The energy distributed in each ring of an interferenced beam can be re-allocated by adjusting the coefficients *A*
_{1} and *A*
_{2} in Eq. (1), so the modified hologram would diffract more energy to the desired ring corresponding to the greater coefficient. Phase of a blazed grating can also be added to this modified hologram to separate the interferenced beam from the direct component reflected by the SLM. It is expected that, with the development in optical encoding/decoding and communications, an interference pattern interfered by double vortex beams with unequal charges could be also useful for quantum communications owing to the quantum nature and coexistence of multiple states of OAM in one beam.

## 4. Conclusion

In conclusion, we have demonstrated that a beam interfered by two vortex beams with unequal charges still possess OAMs, which rotate at opposite orientations and are distributed in different concentric rings. The residual OAM in the interferenced beam can be employed to rotate particles when the difference of the charges is large.

## Acknowledgments

This work is supported by the Agency for Science, Technology and Research (A^{*}STAR) of Singapore under A^{*}STAR SERC Grant No. 032 101 0025.

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