Abstract

We investigate the properties of angular momentum carrying vortex beams, reflected by a phase-conjugating mirror. It is shown that a self-pumped photorefractive phase-conjugating mirror is suitable to produce stable, high-fidelity phase conjugation of vortex beams. We prove that the topological charge of the vortex beam is maintained, and thus the angular momentum in the laboratory frame of reference is reversed, as it is expected by the time reversal property of the phase-conjugating mirror. The three dimensional interference pattern in front of the phase-conjugating mirror is studied and applications in optical traps are suggested.

© 2009 Optical Society of America

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  1. I. Basistiy, M. Soskin, and M. Vasnetsov, "Optical wave-front dislocations and their properties," Opt. Commun. 119, 604-612 (1995).
    [CrossRef]
  2. K. Ladavac and D. G. Grier, "Microoptomechanical pumps assembled and driven by holographic optical vortex arrays," Opt. Express 12, 1144-1149 (2004).
    [CrossRef] [PubMed]
  3. 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," Phys. Rev. A 45, 8185 - 8189 (1992).
    [CrossRef] [PubMed]
  4. A. Y. Okulov, "Angular momentum of photons and phase conjugation," J. Phys. B: At. Mol. Opt. Phys. 41(10), 101001 (2008).
    [CrossRef]
  5. R. Hellwarth, "Generation Of Time-Reversed Wave Fronts By Nonlinear Refraction," J. Opt. Soc. Am. 67(1), 1-3 (1977).
    [CrossRef]
  6. G. S. He, "Optical phase conjugation: principles, techniques, and applications," Prog. Quantum Electron. 26(3), 131-191 (2002).
    [CrossRef]
  7. J. Feinberg, "Self-Pumped, Continuous-Wave Phase Conjugator Using Internal-Reflection," Opt. Lett. 7(10), 486-488 (1982).
    [CrossRef]
  8. A. Ashkin, "History of optical trapping and manipulation of small-neutral particle, atoms, and molecules," IEEE J. Sel. Top. Quantum Electron. 6(6), 841-856 (2000).
  9. R. Beth, "Mechanical Detection and Measurement of the Angular Momentum of Light," Phys. Rev. 50, 115-125 (1936).
    [CrossRef]
  10. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the orbital angular momentum of a single photon," Phys. Rev. Lett. 88(25), 257901 (2002).
    [CrossRef]
  11. 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(5), 826-829 (1995).
    [CrossRef]
  12. 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(22), 5448-5456 (2004).
    [CrossRef]
  13. R. A. Fisher, ed., Optical Phase Conjugation (Academic Press, Inc., 1983).
  14. P. V. Polyanskii and K. V. Fel’de, "Static holographic phase conjugation of vortex beams," Opt. Spectrosc. 98(6), 913-918 (2005).
    [CrossRef]
  15. I. G. Marienko, M. S. Soskin, and M. V. Vasnetsov, "Phase conjugation of wavefronts containing phase singularities," in International Conference on Singular Optics, M. S. Soskin, ed., 3487, pp. 39-41 (Proceedings of SPIE, 1998).
  16. J. Feinberg, "Continuous-Wave Self-Pumped Phase Conjugator With Wide Field Of View," Opt. Lett. 8(9), 480-482 (1983).
    [CrossRef]
  17. F. C. Jahoda, P. G. Weber, and J. Feinberg, "Optical Feedback, Wavelength Response, And Interference Effects Of Self-Pumped Phase Conjugation In BaTiO3," Opt. Lett. 9(8), 362-364 (1984).
    [CrossRef]
  18. V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, "Screw Dislocations In Light Wave-Fronts," J. Mod. Opt. 39(5), 985-990 (1992).
    [CrossRef]
  19. T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Angular momentum of a strongly focused Gaussian beam," J. Opt. A: Pure Appl. Opt. 10(11), 115005 (2008).
    [CrossRef]
  20. M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296(5570), 1101-1103 (2002).
    [CrossRef]
  21. A. Jesacher, S. F¨urhapter, S. Bernet, and M. Ritsch-Marte, "Size selective trapping with optical "cogwheel" tweezers," Opt. Express 12, 4129-4135 (2004).
    [CrossRef] [PubMed]
  22. J. P. Gordon, "Radiation Forces And Momenta In Dielectric Media," Phys. Rev. A 8(1), 14-21 (1973).
    [CrossRef]
  23. V. G. Shvedov, A. S. Desyatnikov, A. V. Rode,W. Krolikowski, and Y. S. Kivshar, "Optical guiding of absorbing nanoclusters in air." Opt. Express 17(7), 5743-57 (2009).
    [CrossRef]
  24. M. Bhattacharya, "Lattice with a twist: Helical waveguides for ultracold matter," Opt. Commun. 279(1), 219-222 (2007).
    [CrossRef]

2009

2008

T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Angular momentum of a strongly focused Gaussian beam," J. Opt. A: Pure Appl. Opt. 10(11), 115005 (2008).
[CrossRef]

A. Y. Okulov, "Angular momentum of photons and phase conjugation," J. Phys. B: At. Mol. Opt. Phys. 41(10), 101001 (2008).
[CrossRef]

2007

M. Bhattacharya, "Lattice with a twist: Helical waveguides for ultracold matter," Opt. Commun. 279(1), 219-222 (2007).
[CrossRef]

2005

P. V. Polyanskii and K. V. Fel’de, "Static holographic phase conjugation of vortex beams," Opt. Spectrosc. 98(6), 913-918 (2005).
[CrossRef]

2004

2002

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296(5570), 1101-1103 (2002).
[CrossRef]

G. S. He, "Optical phase conjugation: principles, techniques, and applications," Prog. Quantum Electron. 26(3), 131-191 (2002).
[CrossRef]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the orbital angular momentum of a single photon," Phys. Rev. Lett. 88(25), 257901 (2002).
[CrossRef]

2000

A. Ashkin, "History of optical trapping and manipulation of small-neutral particle, atoms, and molecules," IEEE J. Sel. Top. Quantum Electron. 6(6), 841-856 (2000).

1995

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(5), 826-829 (1995).
[CrossRef]

I. Basistiy, M. Soskin, and M. Vasnetsov, "Optical wave-front dislocations and their properties," Opt. Commun. 119, 604-612 (1995).
[CrossRef]

1992

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, "Screw Dislocations In Light Wave-Fronts," J. Mod. Opt. 39(5), 985-990 (1992).
[CrossRef]

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," Phys. Rev. A 45, 8185 - 8189 (1992).
[CrossRef] [PubMed]

1984

1983

1982

1977

1973

J. P. Gordon, "Radiation Forces And Momenta In Dielectric Media," Phys. Rev. A 8(1), 14-21 (1973).
[CrossRef]

1936

R. Beth, "Mechanical Detection and Measurement of the Angular Momentum of Light," Phys. Rev. 50, 115-125 (1936).
[CrossRef]

Allen, L.

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," Phys. Rev. A 45, 8185 - 8189 (1992).
[CrossRef] [PubMed]

Arlt, J.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296(5570), 1101-1103 (2002).
[CrossRef]

Ashkin, A.

A. Ashkin, "History of optical trapping and manipulation of small-neutral particle, atoms, and molecules," IEEE J. Sel. Top. Quantum Electron. 6(6), 841-856 (2000).

Barnett, S. M.

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(22), 5448-5456 (2004).
[CrossRef]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the orbital angular momentum of a single photon," Phys. Rev. Lett. 88(25), 257901 (2002).
[CrossRef]

Basistiy, I.

I. Basistiy, M. Soskin, and M. Vasnetsov, "Optical wave-front dislocations and their properties," Opt. Commun. 119, 604-612 (1995).
[CrossRef]

Bazhenov, V. Y.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, "Screw Dislocations In Light Wave-Fronts," J. Mod. Opt. 39(5), 985-990 (1992).
[CrossRef]

Beijersbergen, M. W.

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," Phys. Rev. A 45, 8185 - 8189 (1992).
[CrossRef] [PubMed]

Bernet, S.

Beth, R.

R. Beth, "Mechanical Detection and Measurement of the Angular Momentum of Light," Phys. Rev. 50, 115-125 (1936).
[CrossRef]

Bhattacharya, M.

M. Bhattacharya, "Lattice with a twist: Helical waveguides for ultracold matter," Opt. Commun. 279(1), 219-222 (2007).
[CrossRef]

Courtial, J.

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(22), 5448-5456 (2004).
[CrossRef]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the orbital angular momentum of a single photon," Phys. Rev. Lett. 88(25), 257901 (2002).
[CrossRef]

Desyatnikov, A. S.

Dholakia, K.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296(5570), 1101-1103 (2002).
[CrossRef]

F¨urhapter, S.

Feinberg, J.

Fel’de, K. V.

P. V. Polyanskii and K. V. Fel’de, "Static holographic phase conjugation of vortex beams," Opt. Spectrosc. 98(6), 913-918 (2005).
[CrossRef]

Franke-Arnold, S.

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(22), 5448-5456 (2004).
[CrossRef]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the orbital angular momentum of a single photon," Phys. Rev. Lett. 88(25), 257901 (2002).
[CrossRef]

Friese, M. E. J.

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(5), 826-829 (1995).
[CrossRef]

Gibson, G.

Gordon, J. P.

J. P. Gordon, "Radiation Forces And Momenta In Dielectric Media," Phys. Rev. A 8(1), 14-21 (1973).
[CrossRef]

Grier, D. G.

He, G. S.

G. S. He, "Optical phase conjugation: principles, techniques, and applications," Prog. Quantum Electron. 26(3), 131-191 (2002).
[CrossRef]

He, H.

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(5), 826-829 (1995).
[CrossRef]

Heckenberg, N. R.

T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Angular momentum of a strongly focused Gaussian beam," J. Opt. A: Pure Appl. Opt. 10(11), 115005 (2008).
[CrossRef]

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(5), 826-829 (1995).
[CrossRef]

Hellwarth, R.

Jahoda, F. C.

Jesacher, A.

Kivshar, Y. S.

Krolikowski, W.

Ladavac, K.

Leach, J.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the orbital angular momentum of a single photon," Phys. Rev. Lett. 88(25), 257901 (2002).
[CrossRef]

MacDonald, M. P.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296(5570), 1101-1103 (2002).
[CrossRef]

Nieminen, T. A.

T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Angular momentum of a strongly focused Gaussian beam," J. Opt. A: Pure Appl. Opt. 10(11), 115005 (2008).
[CrossRef]

Okulov, A. Y.

A. Y. Okulov, "Angular momentum of photons and phase conjugation," J. Phys. B: At. Mol. Opt. Phys. 41(10), 101001 (2008).
[CrossRef]

Padgett, M. J.

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(22), 5448-5456 (2004).
[CrossRef]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the orbital angular momentum of a single photon," Phys. Rev. Lett. 88(25), 257901 (2002).
[CrossRef]

Pas’ko, V.

Paterson, L.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296(5570), 1101-1103 (2002).
[CrossRef]

Polyanskii, P. V.

P. V. Polyanskii and K. V. Fel’de, "Static holographic phase conjugation of vortex beams," Opt. Spectrosc. 98(6), 913-918 (2005).
[CrossRef]

Ritsch-Marte, M.

Rode, A. V.

Rubinsztein-Dunlop, H.

T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Angular momentum of a strongly focused Gaussian beam," J. Opt. A: Pure Appl. Opt. 10(11), 115005 (2008).
[CrossRef]

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(5), 826-829 (1995).
[CrossRef]

Shvedov, V. G.

Sibbett, W.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296(5570), 1101-1103 (2002).
[CrossRef]

Soskin, M.

I. Basistiy, M. Soskin, and M. Vasnetsov, "Optical wave-front dislocations and their properties," Opt. Commun. 119, 604-612 (1995).
[CrossRef]

Soskin, M. S.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, "Screw Dislocations In Light Wave-Fronts," J. Mod. Opt. 39(5), 985-990 (1992).
[CrossRef]

Spreeuw, R. J. C.

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," Phys. Rev. A 45, 8185 - 8189 (1992).
[CrossRef] [PubMed]

Stilgoe, A. B.

T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Angular momentum of a strongly focused Gaussian beam," J. Opt. A: Pure Appl. Opt. 10(11), 115005 (2008).
[CrossRef]

Vasnetsov, M.

Vasnetsov, M. V.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, "Screw Dislocations In Light Wave-Fronts," J. Mod. Opt. 39(5), 985-990 (1992).
[CrossRef]

Volke-Sepulveda, K.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296(5570), 1101-1103 (2002).
[CrossRef]

Weber, P. G.

Woerdman, J. P.

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," Phys. Rev. A 45, 8185 - 8189 (1992).
[CrossRef] [PubMed]

IEEE J. Sel. Top. Quantum Electron.

A. Ashkin, "History of optical trapping and manipulation of small-neutral particle, atoms, and molecules," IEEE J. Sel. Top. Quantum Electron. 6(6), 841-856 (2000).

J. Mod. Opt.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, "Screw Dislocations In Light Wave-Fronts," J. Mod. Opt. 39(5), 985-990 (1992).
[CrossRef]

J. Opt. A: Pure Appl. Opt.

T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Angular momentum of a strongly focused Gaussian beam," J. Opt. A: Pure Appl. Opt. 10(11), 115005 (2008).
[CrossRef]

J. Opt. Soc. Am.

J. Phys. B: At. Mol. Opt. Phys.

A. Y. Okulov, "Angular momentum of photons and phase conjugation," J. Phys. B: At. Mol. Opt. Phys. 41(10), 101001 (2008).
[CrossRef]

Opt. Commun.

I. Basistiy, M. Soskin, and M. Vasnetsov, "Optical wave-front dislocations and their properties," Opt. Commun. 119, 604-612 (1995).
[CrossRef]

M. Bhattacharya, "Lattice with a twist: Helical waveguides for ultracold matter," Opt. Commun. 279(1), 219-222 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Spectrosc.

P. V. Polyanskii and K. V. Fel’de, "Static holographic phase conjugation of vortex beams," Opt. Spectrosc. 98(6), 913-918 (2005).
[CrossRef]

Phys. Rev.

R. Beth, "Mechanical Detection and Measurement of the Angular Momentum of Light," Phys. Rev. 50, 115-125 (1936).
[CrossRef]

Phys. Rev. A

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," Phys. Rev. A 45, 8185 - 8189 (1992).
[CrossRef] [PubMed]

J. P. Gordon, "Radiation Forces And Momenta In Dielectric Media," Phys. Rev. A 8(1), 14-21 (1973).
[CrossRef]

Phys. Rev. Lett.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the orbital angular momentum of a single photon," Phys. Rev. Lett. 88(25), 257901 (2002).
[CrossRef]

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(5), 826-829 (1995).
[CrossRef]

Prog. Quantum Electron.

G. S. He, "Optical phase conjugation: principles, techniques, and applications," Prog. Quantum Electron. 26(3), 131-191 (2002).
[CrossRef]

Science

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296(5570), 1101-1103 (2002).
[CrossRef]

Other

I. G. Marienko, M. S. Soskin, and M. V. Vasnetsov, "Phase conjugation of wavefronts containing phase singularities," in International Conference on Singular Optics, M. S. Soskin, ed., 3487, pp. 39-41 (Proceedings of SPIE, 1998).

R. A. Fisher, ed., Optical Phase Conjugation (Academic Press, Inc., 1983).

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Figures (5)

Fig. 1.
Fig. 1.

Interference pattern of a first order LG beam and its reflection. The reflection is performed by a conventional (a) and a phase-conjugating mirror (b), respectively. A wavelength of λ=532nm is assumed. The drawn isosurfaces correspond to a fraction a of the peak intensity where a=0.82 is chosen for best visibility of the structures.

Fig. 2.
Fig. 2.

Experimental setup for investigation of reflection properties of phase-conjugating mirrors. PCM, phase-conjugating mirror; VH, vortex hologram plate; (P)BS, (polarizing) beams splitters; HWP, half wave plates; L, lenses; M, mirrors; VA, variable attenuator.

Fig. 3.
Fig. 3.

Interference pattern of a plane wave and a vortex reflected by a phase-conjugating mirror (left) and a conventional mirror (right), respectively. It is clearly seen that the topological charge of the vortex is reversed. The circle indicates the center of the phase dislocation.

Fig. 4.
Fig. 4.

Interference pattern of two vortices, reflected by a phase-conjugating mirror and a conventional mirror, respectively (a). Interpretation of the dipole pattern as one plane of the three dimensional helical interference pattern in front of the phase-conjugating mirror (b). The isosurface plot is adopted from Fig. 1.

Fig. 5.
Fig. 5.

Interference pattern of |m|=2 vortices (a,b) and |m|=3 vortices (c,d). The left column shows a comparison between vortices reflected by a phase-conjugating (left) and a conventional mirror (right). At the right column, both vortices are interfering, yielding a 2|m| multipole interference pattern.

Equations (3)

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

E ( z , r , φ , t ) E 0 exp ( i ( k z ω t ) + i m φ ) ( 1 + i z ( k D 2 ) ) 2 r m exp ( r 2 D 2 ( 1 + i z ( k D 2 ) ) )
I 2 E 0 2 ( 1 + cos ( 2 k z ) ) r m exp ( 2 r 2 D 2 ( 1 + z 2 ( k D 2 ) 2 ) ) .
I 2 E 0 2 ( 1 + cos ( 2 k z + 2 m φ ) ) r 2 m exp ( 2 r 2 D 2 ( 1 + z 2 ( k D 2 ) 2 ) ) .

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