Abstract

Three-dimensional microparticle movements induced by laser beams with a funnel- and tubular pod-like structure, in the neighbourhood of the focal plane of an optical trapping setup, are experimentally studied. The funnel and pod beams constructed as coherent superpositions of helical Laguerre-Gaussian modes are synthesized by a computer generated hologram using a phase-only spatial light modulator. Particle tracking is achieved by in-line holography method which allows an accurate position measurement. It is experimentally demonstrated that the trapped particle follows different trajectories depending on the orbital angular momentum density of the beam. In particular applying the proposed pod beam the particle rotates in opposite directions during its movement in the optical trap. Possible applications of these single-beam traps for volumetric optical particle manipulation are discussed.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Ashkin, Optical Trapping and Manipulation of Neutral Particles Using Lasers: A Reprint Volume With Commentaries (World Scientific Publishing Company, 2006).
    [CrossRef] [PubMed]
  2. M. Padgett, and L. Allen, "Light with a twist in its tail," Contemp. Phys. 41, 275-285 (2000).
    [CrossRef]
  3. K. Ladavac, and D. Grier, "Microoptomechanical pumps assembled and driven by holographic optical vortex arrays," Opt. Express 12, 1144-1149 (2004).
    [CrossRef] [PubMed]
  4. A. Jesacher, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, "Holographic optical tweezers for object manipulations at an air-liquid surface," Opt. Express 14, 6342-6352 (2006).
    [CrossRef] [PubMed]
  5. L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
    [CrossRef] [PubMed]
  6. E. G. Abramochkin, and V. G. Volostnikov, "Spiral light beams," Phys. Usp. 47, 1177 (2004).
    [CrossRef]
  7. T. Alieva, E. Abramochkin, A. Asenjo-Garcia, and E. Razueva, "Rotating beams in isotropic optical system," Opt. Express 18, 3568-3573 (2010).
    [CrossRef] [PubMed]
  8. E. Abramochkin, S. Kotova, A. Korobtsov, N. Losevsky, A. Mayorova, M. Rakhmatulin, and V. Volostnikov, "Micro-object manipulations using laser beams with nonzero orbital angular momentum," Laser Phys. 16, 842-848 (2006).
    [CrossRef]
  9. A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, "Size selective trapping with optical cogwheel tweezers," Opt. Express 12, 4129-4135 (2004).
    [CrossRef] [PubMed]
  10. K. Dholakia, M. P. MacDonald, P. Zemanek, and T. Cizmár, Laser manipulation of cells and tissues methods in cell biology (Elsevier, 2007), chap. Cellular and colloidal separation using optical forces, pp. 467-495.
    [CrossRef]
  11. D. G. Grier, and Y. Roichman, "Holographic optical trapping," Appl. Opt. 45, 880-887 (2006).
    [CrossRef] [PubMed]
  12. M. J. Padgett, J. E. Molloy, and D. Mcgloin, eds., Optical Tweezers: Methods and Applications (CRC Press, 2010).
    [CrossRef]
  13. A. E. Siegman, Lasers (University Science Books, 1986).
  14. R. Zambrini, and S. M. Barnett, "Angular momentum of multimode and polarization patterns," Opt. Express 15, 15214-15227 (2007).
    [CrossRef] [PubMed]
  15. A. M. Caravaca-Aguirre, and T. Alieva, "Orbital angular moment density of beam given as a superposition of Hermite-Laguerre-Gauss functions," in "PIERS 2011, Marrakesh," (2011).
  16. R. Ozeri, L. Khaykovich, N. Friedman, and N. Davidson, "Large-volume single-beam dark optical trap for atoms using binary phase elements," J. Opt. Soc. Am. B 17, 1113-1116 (2000).
    [CrossRef]
  17. J. Arlt, and M. J. Padgett, "Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam," Opt. Lett. 25, 191-193 (2000).
    [CrossRef]
  18. N. Bokor, and N. Davidson, "A three dimensional dark focal spot uniformly surrounded by light," Opt. Commun. 279, 229-234 (2007).
    [CrossRef]
  19. B. Sun, Y. Roichman, and D. G. Grier, "Theory of holographic optical trapping," Opt. Express 16, 15765-15776 (2008).
    [CrossRef] [PubMed]
  20. E. Martín-Badosa, M. Montes-Usategui, A. Carnicer, J. Andilla, E. Pleguezuelos, and I. Juvells, "Design strategies for optimizing holographic optical tweezers set-ups," J. Opt. A, Pure Appl. Opt. 9, S267 (2007).
    [CrossRef]
  21. F. C. Cheong, B. J. Krishnatreya, and D. G. Grier, "Strategies for three-dimensional particle tracking with holographic video microscopy," Opt. Express 18, 13563-13573 (2010).
    [CrossRef] [PubMed]
  22. J. P. Kirk, and A. L. Jones, "Phase-Only Complex-Valued Spatial Filter," J. Opt. Soc. Am. 61, 1023-1028 (1971).
    [CrossRef]
  23. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, "Encoding amplitude information onto phase-only filters," Appl. Opt. 38, 5004-5013 (1999).
    [CrossRef]
  24. V. Arrizón, U. Ruiz, R. Carrada, and L. A. González, "Pixelated phase computer holograms for the accurate encoding of scalar complex fields," J. Opt. Soc. Am. A 24, 3500-3507 (2007).
    [CrossRef]
  25. T. Ando, Y. Ohtake, N. Matsumoto, T. Inoue, and N. Fukuchi, "Mode purities of Laguerre-Gaussian beams generated via complex-amplitude modulation using phase-only spatial light modulators," Opt. Lett. 34, 34-36 (2009).
    [CrossRef]
  26. I. Moreno, A. Lizana, A. Márquez, C. Iemmi, E. Fernández, J. Campos, and M. J. Yzuel, "Time fluctuations of the phase modulation in a liquid crystal on silicon display: characterization and effects in diffractive optics," Opt. Express 16, 16711-16722 (2008).
    [CrossRef] [PubMed]
  27. Y. Roichman, A. Waldron, E. Gardel, and D. G. Grier, "Optical traps with geometric aberrations," Appl. Opt. 45, 3425-3429 (2006).
    [CrossRef] [PubMed]
  28. C. López-Quesada, J. Andilla, and E. Martín-Badosa, "Correction of aberration in holographic optical tweezers using a Shack-Hartmann sensor," Appl. Opt. 48, 1084-1090 (2009).
    [CrossRef]
  29. Y. Roichman, I. Cholis, and D. G. Grier, "Volumetric imaging of holographic optical traps," Opt. Express 14, 10907-10912 (2006).
    [CrossRef] [PubMed]

2010 (2)

2009 (2)

2008 (2)

2007 (4)

V. Arrizón, U. Ruiz, R. Carrada, and L. A. González, "Pixelated phase computer holograms for the accurate encoding of scalar complex fields," J. Opt. Soc. Am. A 24, 3500-3507 (2007).
[CrossRef]

R. Zambrini, and S. M. Barnett, "Angular momentum of multimode and polarization patterns," Opt. Express 15, 15214-15227 (2007).
[CrossRef] [PubMed]

N. Bokor, and N. Davidson, "A three dimensional dark focal spot uniformly surrounded by light," Opt. Commun. 279, 229-234 (2007).
[CrossRef]

E. Martín-Badosa, M. Montes-Usategui, A. Carnicer, J. Andilla, E. Pleguezuelos, and I. Juvells, "Design strategies for optimizing holographic optical tweezers set-ups," J. Opt. A, Pure Appl. Opt. 9, S267 (2007).
[CrossRef]

2006 (5)

2004 (3)

2001 (1)

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef] [PubMed]

2000 (3)

1999 (1)

1971 (1)

Abramochkin, E.

T. Alieva, E. Abramochkin, A. Asenjo-Garcia, and E. Razueva, "Rotating beams in isotropic optical system," Opt. Express 18, 3568-3573 (2010).
[CrossRef] [PubMed]

E. Abramochkin, S. Kotova, A. Korobtsov, N. Losevsky, A. Mayorova, M. Rakhmatulin, and V. Volostnikov, "Micro-object manipulations using laser beams with nonzero orbital angular momentum," Laser Phys. 16, 842-848 (2006).
[CrossRef]

Abramochkin, E. G.

E. G. Abramochkin, and V. G. Volostnikov, "Spiral light beams," Phys. Usp. 47, 1177 (2004).
[CrossRef]

Alieva, T.

Allen, L.

M. Padgett, and L. Allen, "Light with a twist in its tail," Contemp. Phys. 41, 275-285 (2000).
[CrossRef]

Andilla, J.

C. López-Quesada, J. Andilla, and E. Martín-Badosa, "Correction of aberration in holographic optical tweezers using a Shack-Hartmann sensor," Appl. Opt. 48, 1084-1090 (2009).
[CrossRef]

E. Martín-Badosa, M. Montes-Usategui, A. Carnicer, J. Andilla, E. Pleguezuelos, and I. Juvells, "Design strategies for optimizing holographic optical tweezers set-ups," J. Opt. A, Pure Appl. Opt. 9, S267 (2007).
[CrossRef]

Ando, T.

Arlt, J.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef] [PubMed]

J. Arlt, and M. J. Padgett, "Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam," Opt. Lett. 25, 191-193 (2000).
[CrossRef]

Arrizón, V.

Asenjo-Garcia, A.

Barnett, S. M.

Bernet, S.

Bokor, N.

N. Bokor, and N. Davidson, "A three dimensional dark focal spot uniformly surrounded by light," Opt. Commun. 279, 229-234 (2007).
[CrossRef]

Bryant, P. E.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Campos, J.

Carnicer, A.

E. Martín-Badosa, M. Montes-Usategui, A. Carnicer, J. Andilla, E. Pleguezuelos, and I. Juvells, "Design strategies for optimizing holographic optical tweezers set-ups," J. Opt. A, Pure Appl. Opt. 9, S267 (2007).
[CrossRef]

Carrada, R.

Cheong, F. C.

Cholis, I.

Cottrell, D. M.

Davidson, N.

Davis, J. A.

Dholakia, K.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Fernández, E.

Friedman, N.

Fukuchi, N.

Fürhapter, S.

Gardel, E.

González, L. A.

Grier, D.

Grier, D. G.

Iemmi, C.

Inoue, T.

Jesacher, A.

Jones, A. L.

Juvells, I.

E. Martín-Badosa, M. Montes-Usategui, A. Carnicer, J. Andilla, E. Pleguezuelos, and I. Juvells, "Design strategies for optimizing holographic optical tweezers set-ups," J. Opt. A, Pure Appl. Opt. 9, S267 (2007).
[CrossRef]

Khaykovich, L.

Kirk, J. P.

Korobtsov, A.

E. Abramochkin, S. Kotova, A. Korobtsov, N. Losevsky, A. Mayorova, M. Rakhmatulin, and V. Volostnikov, "Micro-object manipulations using laser beams with nonzero orbital angular momentum," Laser Phys. 16, 842-848 (2006).
[CrossRef]

Kotova, S.

E. Abramochkin, S. Kotova, A. Korobtsov, N. Losevsky, A. Mayorova, M. Rakhmatulin, and V. Volostnikov, "Micro-object manipulations using laser beams with nonzero orbital angular momentum," Laser Phys. 16, 842-848 (2006).
[CrossRef]

Krishnatreya, B. J.

Ladavac, K.

Lizana, A.

López-Quesada, C.

Losevsky, N.

E. Abramochkin, S. Kotova, A. Korobtsov, N. Losevsky, A. Mayorova, M. Rakhmatulin, and V. Volostnikov, "Micro-object manipulations using laser beams with nonzero orbital angular momentum," Laser Phys. 16, 842-848 (2006).
[CrossRef]

MacDonald, M. P.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Márquez, A.

Martín-Badosa, E.

C. López-Quesada, J. Andilla, and E. Martín-Badosa, "Correction of aberration in holographic optical tweezers using a Shack-Hartmann sensor," Appl. Opt. 48, 1084-1090 (2009).
[CrossRef]

E. Martín-Badosa, M. Montes-Usategui, A. Carnicer, J. Andilla, E. Pleguezuelos, and I. Juvells, "Design strategies for optimizing holographic optical tweezers set-ups," J. Opt. A, Pure Appl. Opt. 9, S267 (2007).
[CrossRef]

Matsumoto, N.

Maurer, C.

Mayorova, A.

E. Abramochkin, S. Kotova, A. Korobtsov, N. Losevsky, A. Mayorova, M. Rakhmatulin, and V. Volostnikov, "Micro-object manipulations using laser beams with nonzero orbital angular momentum," Laser Phys. 16, 842-848 (2006).
[CrossRef]

Montes-Usategui, M.

E. Martín-Badosa, M. Montes-Usategui, A. Carnicer, J. Andilla, E. Pleguezuelos, and I. Juvells, "Design strategies for optimizing holographic optical tweezers set-ups," J. Opt. A, Pure Appl. Opt. 9, S267 (2007).
[CrossRef]

Moreno, I.

Ohtake, Y.

Ozeri, R.

Padgett, M.

M. Padgett, and L. Allen, "Light with a twist in its tail," Contemp. Phys. 41, 275-285 (2000).
[CrossRef]

Padgett, M. J.

Paterson, L.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Pleguezuelos, E.

E. Martín-Badosa, M. Montes-Usategui, A. Carnicer, J. Andilla, E. Pleguezuelos, and I. Juvells, "Design strategies for optimizing holographic optical tweezers set-ups," J. Opt. A, Pure Appl. Opt. 9, S267 (2007).
[CrossRef]

Rakhmatulin, M.

E. Abramochkin, S. Kotova, A. Korobtsov, N. Losevsky, A. Mayorova, M. Rakhmatulin, and V. Volostnikov, "Micro-object manipulations using laser beams with nonzero orbital angular momentum," Laser Phys. 16, 842-848 (2006).
[CrossRef]

Razueva, E.

Ritsch-Marte, M.

Roichman, Y.

Ruiz, U.

Sibbett, W.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Sun, B.

Volostnikov, V.

E. Abramochkin, S. Kotova, A. Korobtsov, N. Losevsky, A. Mayorova, M. Rakhmatulin, and V. Volostnikov, "Micro-object manipulations using laser beams with nonzero orbital angular momentum," Laser Phys. 16, 842-848 (2006).
[CrossRef]

Volostnikov, V. G.

E. G. Abramochkin, and V. G. Volostnikov, "Spiral light beams," Phys. Usp. 47, 1177 (2004).
[CrossRef]

Waldron, A.

Yzuel, M. J.

Zambrini, R.

Appl. Opt. (4)

Contemp. Phys. (1)

M. Padgett, and L. Allen, "Light with a twist in its tail," Contemp. Phys. 41, 275-285 (2000).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

E. Martín-Badosa, M. Montes-Usategui, A. Carnicer, J. Andilla, E. Pleguezuelos, and I. Juvells, "Design strategies for optimizing holographic optical tweezers set-ups," J. Opt. A, Pure Appl. Opt. 9, S267 (2007).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Laser Phys. (1)

E. Abramochkin, S. Kotova, A. Korobtsov, N. Losevsky, A. Mayorova, M. Rakhmatulin, and V. Volostnikov, "Micro-object manipulations using laser beams with nonzero orbital angular momentum," Laser Phys. 16, 842-848 (2006).
[CrossRef]

Opt. Commun. (1)

N. Bokor, and N. Davidson, "A three dimensional dark focal spot uniformly surrounded by light," Opt. Commun. 279, 229-234 (2007).
[CrossRef]

Opt. Express (9)

K. Ladavac, and D. Grier, "Microoptomechanical pumps assembled and driven by holographic optical vortex arrays," Opt. Express 12, 1144-1149 (2004).
[CrossRef] [PubMed]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, "Size selective trapping with optical cogwheel tweezers," Opt. Express 12, 4129-4135 (2004).
[CrossRef] [PubMed]

R. Zambrini, and S. M. Barnett, "Angular momentum of multimode and polarization patterns," Opt. Express 15, 15214-15227 (2007).
[CrossRef] [PubMed]

B. Sun, Y. Roichman, and D. G. Grier, "Theory of holographic optical trapping," Opt. Express 16, 15765-15776 (2008).
[CrossRef] [PubMed]

I. Moreno, A. Lizana, A. Márquez, C. Iemmi, E. Fernández, J. Campos, and M. J. Yzuel, "Time fluctuations of the phase modulation in a liquid crystal on silicon display: characterization and effects in diffractive optics," Opt. Express 16, 16711-16722 (2008).
[CrossRef] [PubMed]

A. Jesacher, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, "Holographic optical tweezers for object manipulations at an air-liquid surface," Opt. Express 14, 6342-6352 (2006).
[CrossRef] [PubMed]

Y. Roichman, I. Cholis, and D. G. Grier, "Volumetric imaging of holographic optical traps," Opt. Express 14, 10907-10912 (2006).
[CrossRef] [PubMed]

T. Alieva, E. Abramochkin, A. Asenjo-Garcia, and E. Razueva, "Rotating beams in isotropic optical system," Opt. Express 18, 3568-3573 (2010).
[CrossRef] [PubMed]

F. C. Cheong, B. J. Krishnatreya, and D. G. Grier, "Strategies for three-dimensional particle tracking with holographic video microscopy," Opt. Express 18, 13563-13573 (2010).
[CrossRef] [PubMed]

Opt. Lett. (2)

Phys. Usp. (1)

E. G. Abramochkin, and V. G. Volostnikov, "Spiral light beams," Phys. Usp. 47, 1177 (2004).
[CrossRef]

Science (1)

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Other (5)

A. Ashkin, Optical Trapping and Manipulation of Neutral Particles Using Lasers: A Reprint Volume With Commentaries (World Scientific Publishing Company, 2006).
[CrossRef] [PubMed]

K. Dholakia, M. P. MacDonald, P. Zemanek, and T. Cizmár, Laser manipulation of cells and tissues methods in cell biology (Elsevier, 2007), chap. Cellular and colloidal separation using optical forces, pp. 467-495.
[CrossRef]

M. J. Padgett, J. E. Molloy, and D. Mcgloin, eds., Optical Tweezers: Methods and Applications (CRC Press, 2010).
[CrossRef]

A. E. Siegman, Lasers (University Science Books, 1986).

A. M. Caravaca-Aguirre, and T. Alieva, "Orbital angular moment density of beam given as a superposition of Hermite-Laguerre-Gauss functions," in "PIERS 2011, Marrakesh," (2011).

Supplementary Material (9)

» Media 1: MOV (1600 KB)     
» Media 2: MOV (790 KB)     
» Media 3: MOV (190 KB)     
» Media 4: MOV (881 KB)     
» Media 5: MOV (763 KB)     
» Media 6: MOV (840 KB)     
» Media 7: MOV (3078 KB)     
» Media 8: MOV (2747 KB)     
» Media 9: MOV (2422 KB)     

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Intensity, phase and OAM density distributions of the beams ℱ1 (a), ℱ2 (b), ℱ3 (c), displayed in the first, second and third row respectively, at the minimum beam waist plane z = 0.

Fig. 2
Fig. 2

Three-dimensional representation for the propagation of beams ℱ1 (a), ℱ2 (b), ℱ3 (c). Minimum beam waist w0 = 3 μm is obtained at focal plane (z = 0) of the 100× microscope’s objective lens.

Fig. 3
Fig. 3

Three-dimensional representation for the propagation of beam ��1,1 (a) and ��2,2 (b). To help visualization a 3D cut-out view of such beams is displayed in the second row.

Fig. 4
Fig. 4

Transversal intensity distributions for propagation distance z = 0 (first row) and z = −11 μm (second row) for the beams: ℱ1 (a), ℱ2 (b), ℱ3 (c), ��1,1 (d) and ��2,2 (e).

Fig. 5
Fig. 5

Schematic representation of the holographic optical trapping system. SLM displays the hologram that is imaged into the back focal plane of the microscope objective (MO) by using two relay lenses, L1 and L2, working as 0.25× telescope. The sample is illuminated by an additional laser beam B2 in order to acquire its image as in-line hologram, which is recorded in real time by a CCD camera.

Fig. 6
Fig. 6

Experimental results ( Media 1): Transversal sections at z = 0 (first row) and z = −11 μm (second row) for the beams: ℱ1 (a), ℱ2 (b), ℱ3 (c), ��1,1 (d) and ��2,2 (e).

Fig. 7
Fig. 7

In-line holograms showing the trapped particle (2 μm diameter) for the beam ℱ1 Media 2 and ℱ2 Media 3, first and second row respectively.

Fig. 8
Fig. 8

3D particle-tracking reconstruction for the beams: ℱ1 (a) Media 4, ℱ2 (b) Media 5, ℱ3 (c). Particle’s trajectory is also projected into x − y, x − z and y − z planes (grey lines). It is experimentally measured from in-line holograms stored as video as the ones showed in (d) Media 6, (e) Media 7, and (f), correspondingly. The trajectories start from the lower coverslip surface at t = 0 s (z = −30 μm) and ends at the focal plane z = 0 μm.

Fig. 9
Fig. 9

3D particle-tracking reconstruction for the trapping beam ��1,1 (a) Media 8 and (b). Particle’s trajectory is also projected into xy, xz and yz planes (grey lines). The in-line holograms (as the ones shown in (c)) used for the reconstruction are stored as video, see Media 9.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

LG p , l ± ( r ; z ) = w 1 ( z ) 2 p ! π ( p + l ) ! ( 2 x ± i y w ( z ) ) l p l ( 2 r 2 w 2 ( z ) ) × exp ( r 2 w 2 ( z ) ) exp ( i π r 2 λ R ( z ) ) exp ( i ( 2 p + l + 1 ) ζ ( z ) ) ,
1 ( r ) = LG 1 , 18 + ( r ; 0 ) ,
j z ( r ) = p , l , p , l ± | a p , l | 2 | LG p , l ± ( r ; 0 ) | 2 l ± Re [ a p , l * a p , l LG p , l ( r ; 0 ) LG p , l ± ( r ; 0 ) ] ( l + l ) .
2 ( r ) = [ LG 1 , 17 + ( r ; 0 ) + LG 1 , 19 + ( r ; 0 ) ] / 2 ,
3 ( r ) = [ LG 1 , 18 + ( r ; 0 ) + LG 1 , 18 ( r ; 0 ) ] / 2 .
g ( d ) = exp ( i 2 π n m d r 2 λ f 2 d ) ,
𝒫 n , m ( r ) = a 1 n * ( r ) + a 2 m ( r ) ,
ψ ( a , ϕ ) = f ( a ) sin ϕ ,

Metrics