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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 2 — Jan. 23, 2006
  • pp: 535–541
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Residue orbital angular momentum in interferenced double vortex beams with unequal topological charges

S. H. Tao, X.-C. Yuan, J. Lin, and R. E. Burge  »View Author Affiliations


Optics Express, Vol. 14, Issue 2, pp. 535-541 (2006)
http://dx.doi.org/10.1364/OPEX.14.000535


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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,

E(x,y)=A1(x,y)exp[il1θ(x,y)]+A2⋅exp[il2θ(x,y)],
(1)

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 1l 2 > 0 and (2) l 1l 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 1l 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.

Fig. 1. [1 MB, 1 MB, 1.3 MB, respective videos for phase evolutions of interferenced beams of (l 1=20, l 2=-3), (l 1=-20, l 2=3), and (l 1=20, l 2=-20)]. (a, b, c) Simulated intensities of interferenced beams of (l 1=20, l 2=-3), (l 1=-20, l 2=3), and (l 1=20, l 2=-20), respectively. (A1-3, B1-3, C1-3) Simulated phase distributions of the corresponding interferenced beams at distance of 166 mm+λ/3, 166 mm+λ/3, 166 mm+λ, respectively. [Media 1] [Media 2] [Media 3]

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

uPl(r,θ,z)[2rw(z)]lLPl[2r2w(z)2]exp[r2w(z)2]exp[ikr22R(z)]exp[iΨ(z)]exp(ilθ),
(2)

Fig. 2. (a) Intensity profiles of vortex beams with charge of 20 (the dotted line) and -3 (the solid line), respectively, (b) intensity profile of the interferenced beam of (l 1=20, l 2=-3).
Fig. 3. [226 KB, video for intensity evolution with varying charges of (l 1=20, l 2=-1 to -20)] The intensity percentages of the fringed ring (upper curve) and the innermost part (lower curve) of the interferenced beams of (l 1=20, l 2=-1 to -20). [Media 4]

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,

φ(x,y)=angle{exp[il1θ(x,y)]+exp[il2θ(x,y)]}+αx,
(3)

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

10. S. H. Tao, X.-C. Yuan, and J. Lin, “Fractional optical vortex beam induced rotation of particles,” Opt. Express 13, 7726 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-20-7726. [CrossRef] [PubMed]

, 14

14. S. H. Tao, X.-C. Yuan, H. B. Niu, and X. Peng, “Dynamic optical manipulation using intensity patterns directly projected by a reflective spatial light modulator,” Rev. Sci. Instrum. 76, 056103 (2005). [CrossRef]

]. 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.

Fig. 4. Interferenced patterns recorded from the sample stage of a microscope. The beams are interfered by two vortex beams with (a) (l 1=20, l 2=-1), (b) (l 1=20, l 2=-3), (c) (l 1=20, l 2=-8), (d) (l 1=20, l 2=-12), (e) (l 1=20, l 2=-15), and (f) (l 1=20, l 2=-20), respectively.
Fig. 5. (a) (1.2 MB, video for rotation) Sequent rotations induced by the residual OAM of the interferenced beams of (l 1=10, l 2=-2) and (l 1=-10, l 2=2) [Media 5]. (b) Rotation induced by the innermost ring and static trapping by the fringed ring of the interferenced beam of (l 1=10, l 2=-3).

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.

References and links

01.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phy. Rev. A 45, 8185 (1992). [CrossRef]

02.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826 (1995). [CrossRef] [PubMed]

03.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phy. Rev. Lett. 88, 053601 (2002). [CrossRef]

04.

K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particles,” Opt. Lett. 21, 827 (1996). [CrossRef] [PubMed]

05.

I. D. Maleev and G. A. Swartzlander Jr., “Composite optical vortices,” J. Opt. Soc. Am. B 20, 1169 (2003). [CrossRef]

06.

J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phy. Rev. Lett. 90, 133901 (2003). [CrossRef]

07.

J. Lin, X.-C. Yuan, S. H. Tao, X. Peng, and H. B. Niu, “Deterministic approach to the generation of modified helical beams for optical manipulation,” Opt. Express 13, 3862 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-10-3862. [CrossRef] [PubMed]

08.

A. Vasara, J. Turunen, and A. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748 (1989). [CrossRef] [PubMed]

09.

S. H. Tao, W. M. Lee, and X.-C. Yuan, “Experimental study of holographic generation of fractional Bessel beams,” Appl. Opt. 943, 102 (2004).

10.

S. H. Tao, X.-C. Yuan, and J. Lin, “Fractional optical vortex beam induced rotation of particles,” Opt. Express 13, 7726 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-20-7726. [CrossRef] [PubMed]

11.

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5448. [CrossRef] [PubMed]

12.

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Size selective trapping with optical ”cogwheel” tweezers,” Opt. Express 12, 4129 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-17-4129. [CrossRef] [PubMed]

13.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed (McGraw-Hill, New York, 1996).

14.

S. H. Tao, X.-C. Yuan, H. B. Niu, and X. Peng, “Dynamic optical manipulation using intensity patterns directly projected by a reflective spatial light modulator,” Rev. Sci. Instrum. 76, 056103 (2005). [CrossRef]

OCIS Codes
(050.1970) Diffraction and gratings : Diffractive optics
(090.0090) Holography : Holography
(140.7010) Lasers and laser optics : Laser trapping

ToC Category:
Diffraction and Gratings

Virtual Issues
Vol. 1, Iss. 2 Virtual Journal for Biomedical Optics

Citation
S. H. Tao, X.-C. Yuan, J. Lin, and R. E. Burge, "Residue orbital angular momentum in interferenced double vortex beams with unequal topological charges," Opt. Express 14, 535-541 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-2-535


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References

  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phy. Rev. A 45, 8185 (1992). [CrossRef]
  2. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826 (1995). [CrossRef] [PubMed]
  3. A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phy. Rev. Lett. 88, 053601 (2002). [CrossRef]
  4. K. T. Gahagan, and G. A. Swartzlander Jr., "Optical vortex trapping of particles," Opt. Lett. 21, 827 (1996). [CrossRef] [PubMed]
  5. I. D. Maleev, and G. A. Swartzlander Jr., "Composite optical vortices," J. Opt. Soc. Am. B 20, 1169 (2003). [CrossRef]
  6. J. E. Curtis, and D. G. Grier, "Structure of optical vortices," Phy. Rev. Lett. 90, 133901 (2003). [CrossRef]
  7. J. Lin, X.-C.Yuan, S. H. Tao, X. Peng, and H. B. Niu, "Deterministic approach to the generation of modified helical beams for optical manipulation," Opt. Express 13, 3862 (2005), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-10-3862.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-10-3862.</a> [CrossRef] [PubMed]
  8. A. Vasara, J. Turunen, and A. Friberg, "Realization of general nondiffracting beams with computer-generated holograms," J. Opt. Soc. Am. A 6, 1748 (1989). [CrossRef] [PubMed]
  9. S. H. Tao, W. M. Lee and X.-C. Yuan, "Experimental study of holographic generation of fractional Bessel beams," Appl. Opt. 43, 122 (2004).
  10. S. H. Tao, X.-C. Yuan, and J. Lin, "Fractional optical vortex beam induced rotation of particles," Opt. Express 13, 7726 (2005), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-20-7726.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-20-7726.</a> [CrossRef] [PubMed]
  11. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas'ko, S. M. Barnett, and S. Franke-Arnold, "Freespace information transfer using light beams carrying orbital angular momentum," Opt. Express 12, 5448 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5448.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5448.</a> [CrossRef] [PubMed]
  12. A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, "Size selective trapping with optical "cogwheel" tweezers," Opt. Express 12, 4129 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-17-4129.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-17-4129.</a> [CrossRef] [PubMed]
  13. J. W. Goodman, Introduction to Fourier Optics, 2nd ed (McGraw-Hill, New York, 1996).
  14. S. H. Tao, X.-C. Yuan, H. B. Niu, and X. Peng, "Dynamic optical manipulation using intensity patterns directly projected by a reflective spatial light modulator," Rev. Sci. Instrum. 76, 056103 (2005). [CrossRef]

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