Abstract

In this work, we demonstrate the possibility of generating and controlling any given kind of structured radially symmetric intensity profile with an embedded optical vortex. This is achieved with the use of Sturm–Liouville theory on a circular domain with Bessel, Laguerre–Gauss, Zernike, and Fourier bases. We show that the core intensity profile can be constructed independently of the topological charge of the vortex.

© 2014 Optical Society of America

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  1. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011).
    [CrossRef]
  2. J. Curtis and D. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
    [CrossRef]
  3. J. P. Treviño, O. López-Cruz, and S. Chávez-Cerda, “Segmented vortex telescope and its tolerance to diffraction effects and primary aberrations,” Opt. Eng. 52, 081605 (2013).
    [CrossRef]
  4. K. Dholakia and W. M. Lee, “Optical trapping takes shape: the use of structured light fields,” Adv. At. Mol. Opt. Phys. 56, 261–337 (2008).
  5. M. Hartrumpf and R. Munser, “Optical three-dimensional measurements by radially symmetric structured light projection,” Appl. Opt. 36, 2923–2928 (1997).
    [CrossRef]
  6. 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]
  7. K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4, S82–S89 (2002).
  8. J. A. Hernández Nolasco, “Wave field families of the Helmholtz equation in eleven orthogonal coordinate systems,” Ph.D. thesis (INAOE, 2011).
  9. H. Feshbach and P. M. Morse, Methods of Theoretical Physics: Part II (Cambridge University, 1953).
  10. K. T. Tang, Mathematical Methods for Engineers and Scientists (Springer, 2007).
  11. H. F. Davis, Fourier Series and Orthogonal Functions (Dover, 1989).
  12. G. E. Andrews, R. Askey, and R. Roy, Special Functions (Cambridge University, 2000).
  13. M. A. Al-Gwaiz, Sturm-Liouville Theory and its Applications (Springer, 2008).
  14. H. F. Harmuth, Transmission of Information by Orthogonal Functions (Springer, 1970).
  15. G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
    [CrossRef]
  16. M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. E. Wilner, and M. J. Padgett, “Efficient measurement of an optical orbital- angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
    [CrossRef]
  17. E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via Hong-Ou-Mandel interference,” Phys. Rev. A 89, 013829 (2014).
    [CrossRef]
  18. A. S. Ostrovsky, C. Rickenstorff-Parrao, and V. Arrizón, “Generation of the ‘perfect’ optical vortex using a liquid-crystal spatial light modulator,” Opt. Lett. 38, 534–536 (2013).
    [CrossRef]
  19. M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Dynamics of microparticles trapped in a perfect vortex beam,” Opt. Lett. 38, 4919–4922 (2013).
    [CrossRef]
  20. A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications,” J. Appl. Phys. 34, 2722–2727 (1963).
    [CrossRef]
  21. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268 (2004).
    [CrossRef]
  22. A. E. Siegman, Lasers (University Science Books, 1986).
  23. S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics (Wiley, 1984).
  24. B. G. Korenev, Bessel Functions and Their Applications (Taylor and Francis, 2002).
  25. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, 1995).
  26. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free Beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef]
  27. S. Chávez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. 46, 923–930 (1999).
  28. S. Chávez-Cerda, M. A. Meneses-Nava, and J. M. Hickmann, “Interference of traveling nondiffracting beams,” Opt. Lett. 23, 1871–1873 (1998).
    [CrossRef]
  29. J. P. Treviño, J. E. Gómez-Correa, D. R. Iskander, and S. Chávez-Cerda, “Zernike vs. Bessel circular functions in visual optics,” Ophthalmic Physiolog. Opt. 33, 394–402 (2013).
    [CrossRef]
  30. B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. thesis (University of Groningen, 1942).
  31. F. Zernike, “Beugungstheorie des schneidenverfahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1, 689–704 (1934).
    [CrossRef]
  32. V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58, 545–561 (2011).
    [CrossRef]
  33. R. Navarro, R. Rivera, and J. Aporta, “Representation of wavefronts in free-form transmission pupils with Complex Zernike Polynomials,” J. Optom. 4, 41–48 (2011).
    [CrossRef]
  34. G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express 18, 13851–13862 (2010).
    [CrossRef]
  35. J. A. Murphy, “Examples of circularly symmetric diffraction using beam modes,” Eur. J. Phys. 14, 268–271 (1993).
    [CrossRef]
  36. J. A. Murphy and A. Egan, “Examples of Fresnel diffraction using Gaussian modes,” Eur. J. Phys. 14, 121–127 (1993).
    [CrossRef]

2014 (1)

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via Hong-Ou-Mandel interference,” Phys. Rev. A 89, 013829 (2014).
[CrossRef]

2013 (5)

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. E. Wilner, and M. J. Padgett, “Efficient measurement of an optical orbital- angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
[CrossRef]

J. P. Treviño, O. López-Cruz, and S. Chávez-Cerda, “Segmented vortex telescope and its tolerance to diffraction effects and primary aberrations,” Opt. Eng. 52, 081605 (2013).
[CrossRef]

J. P. Treviño, J. E. Gómez-Correa, D. R. Iskander, and S. Chávez-Cerda, “Zernike vs. Bessel circular functions in visual optics,” Ophthalmic Physiolog. Opt. 33, 394–402 (2013).
[CrossRef]

A. S. Ostrovsky, C. Rickenstorff-Parrao, and V. Arrizón, “Generation of the ‘perfect’ optical vortex using a liquid-crystal spatial light modulator,” Opt. Lett. 38, 534–536 (2013).
[CrossRef]

M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Dynamics of microparticles trapped in a perfect vortex beam,” Opt. Lett. 38, 4919–4922 (2013).
[CrossRef]

2011 (3)

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011).
[CrossRef]

V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58, 545–561 (2011).
[CrossRef]

R. Navarro, R. Rivera, and J. Aporta, “Representation of wavefronts in free-form transmission pupils with Complex Zernike Polynomials,” J. Optom. 4, 41–48 (2011).
[CrossRef]

2010 (2)

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express 18, 13851–13862 (2010).
[CrossRef]

2008 (1)

K. Dholakia and W. M. Lee, “Optical trapping takes shape: the use of structured light fields,” Adv. At. Mol. Opt. Phys. 56, 261–337 (2008).

2004 (1)

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268 (2004).
[CrossRef]

2003 (1)

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

2002 (1)

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4, S82–S89 (2002).

1999 (1)

S. Chávez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. 46, 923–930 (1999).

1998 (1)

1997 (1)

1993 (2)

J. A. Murphy, “Examples of circularly symmetric diffraction using beam modes,” Eur. J. Phys. 14, 268–271 (1993).
[CrossRef]

J. A. Murphy and A. Egan, “Examples of Fresnel diffraction using Gaussian modes,” Eur. J. Phys. 14, 121–127 (1993).
[CrossRef]

1992 (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,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef]

1987 (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free Beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

1963 (1)

A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications,” J. Appl. Phys. 34, 2722–2727 (1963).
[CrossRef]

1934 (1)

F. Zernike, “Beugungstheorie des schneidenverfahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1, 689–704 (1934).
[CrossRef]

Al-Gwaiz, M. A.

M. A. Al-Gwaiz, Sturm-Liouville Theory and its Applications (Springer, 2008).

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]

Andrews, G. E.

G. E. Andrews, R. Askey, and R. Roy, Special Functions (Cambridge University, 2000).

Aporta, J.

R. Navarro, R. Rivera, and J. Aporta, “Representation of wavefronts in free-form transmission pupils with Complex Zernike Polynomials,” J. Optom. 4, 41–48 (2011).
[CrossRef]

Arita, Y.

Arlt, J.

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4, S82–S89 (2002).

Arrizón, V.

Askey, R.

G. E. Andrews, R. Askey, and R. Roy, Special Functions (Cambridge University, 2000).

Beijersbergen, M. W.

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[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]

Bent, N.

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via Hong-Ou-Mandel interference,” Phys. Rev. A 89, 013829 (2014).
[CrossRef]

Berkhout, G. C. G.

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

Berry, M. V.

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268 (2004).
[CrossRef]

Bolduc, E.

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via Hong-Ou-Mandel interference,” Phys. Rev. A 89, 013829 (2014).
[CrossRef]

Boyd, R. W.

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via Hong-Ou-Mandel interference,” Phys. Rev. A 89, 013829 (2014).
[CrossRef]

Chávez-Cerda, S.

J. P. Treviño, J. E. Gómez-Correa, D. R. Iskander, and S. Chávez-Cerda, “Zernike vs. Bessel circular functions in visual optics,” Ophthalmic Physiolog. Opt. 33, 394–402 (2013).
[CrossRef]

J. P. Treviño, O. López-Cruz, and S. Chávez-Cerda, “Segmented vortex telescope and its tolerance to diffraction effects and primary aberrations,” Opt. Eng. 52, 081605 (2013).
[CrossRef]

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4, S82–S89 (2002).

S. Chávez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. 46, 923–930 (1999).

S. Chávez-Cerda, M. A. Meneses-Nava, and J. M. Hickmann, “Interference of traveling nondiffracting beams,” Opt. Lett. 23, 1871–1873 (1998).
[CrossRef]

Chen, M.

Cormack, A. M.

A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications,” J. Appl. Phys. 34, 2722–2727 (1963).
[CrossRef]

Courtial, J.

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. E. Wilner, and M. J. Padgett, “Efficient measurement of an optical orbital- angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
[CrossRef]

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

Curtis, J.

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

Davis, H. F.

H. F. Davis, Fourier Series and Orthogonal Functions (Dover, 1989).

Dholakia, K.

M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Dynamics of microparticles trapped in a perfect vortex beam,” Opt. Lett. 38, 4919–4922 (2013).
[CrossRef]

K. Dholakia and W. M. Lee, “Optical trapping takes shape: the use of structured light fields,” Adv. At. Mol. Opt. Phys. 56, 261–337 (2008).

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4, S82–S89 (2002).

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free Beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free Beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Egan, A.

J. A. Murphy and A. Egan, “Examples of Fresnel diffraction using Gaussian modes,” Eur. J. Phys. 14, 121–127 (1993).
[CrossRef]

Feshbach, H.

H. Feshbach and P. M. Morse, Methods of Theoretical Physics: Part II (Cambridge University, 1953).

Fleck, A.

V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58, 545–561 (2011).
[CrossRef]

Forbes, G. W.

Garcés-Chávez, V.

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4, S82–S89 (2002).

Giovannini, D.

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via Hong-Ou-Mandel interference,” Phys. Rev. A 89, 013829 (2014).
[CrossRef]

Gómez-Correa, J. E.

J. P. Treviño, J. E. Gómez-Correa, D. R. Iskander, and S. Chávez-Cerda, “Zernike vs. Bessel circular functions in visual optics,” Ophthalmic Physiolog. Opt. 33, 394–402 (2013).
[CrossRef]

Grier, D.

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

Harmuth, H. F.

H. F. Harmuth, Transmission of Information by Orthogonal Functions (Springer, 1970).

Hartrumpf, M.

Hernández Nolasco, J. A.

J. A. Hernández Nolasco, “Wave field families of the Helmholtz equation in eleven orthogonal coordinate systems,” Ph.D. thesis (INAOE, 2011).

Hickmann, J. M.

Iskander, D. R.

J. P. Treviño, J. E. Gómez-Correa, D. R. Iskander, and S. Chávez-Cerda, “Zernike vs. Bessel circular functions in visual optics,” Ophthalmic Physiolog. Opt. 33, 394–402 (2013).
[CrossRef]

Karimi, E.

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via Hong-Ou-Mandel interference,” Phys. Rev. A 89, 013829 (2014).
[CrossRef]

Korenev, B. G.

B. G. Korenev, Bessel Functions and Their Applications (Taylor and Francis, 2002).

Lakshminarayanan, V.

V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58, 545–561 (2011).
[CrossRef]

Lavery, M. P. J.

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. E. Wilner, and M. J. Padgett, “Efficient measurement of an optical orbital- angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
[CrossRef]

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

Lee, W. M.

K. Dholakia and W. M. Lee, “Optical trapping takes shape: the use of structured light fields,” Adv. At. Mol. Opt. Phys. 56, 261–337 (2008).

López-Cruz, O.

J. P. Treviño, O. López-Cruz, and S. Chávez-Cerda, “Segmented vortex telescope and its tolerance to diffraction effects and primary aberrations,” Opt. Eng. 52, 081605 (2013).
[CrossRef]

Mazilu, M.

Meneses-Nava, M. A.

Miatto, F. M.

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via Hong-Ou-Mandel interference,” Phys. Rev. A 89, 013829 (2014).
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free Beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Morse, P. M.

H. Feshbach and P. M. Morse, Methods of Theoretical Physics: Part II (Cambridge University, 1953).

Munser, R.

Murphy, J. A.

J. A. Murphy, “Examples of circularly symmetric diffraction using beam modes,” Eur. J. Phys. 14, 268–271 (1993).
[CrossRef]

J. A. Murphy and A. Egan, “Examples of Fresnel diffraction using Gaussian modes,” Eur. J. Phys. 14, 121–127 (1993).
[CrossRef]

Navarro, R.

R. Navarro, R. Rivera, and J. Aporta, “Representation of wavefronts in free-form transmission pupils with Complex Zernike Polynomials,” J. Optom. 4, 41–48 (2011).
[CrossRef]

Nijboer, B. R. A.

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. thesis (University of Groningen, 1942).

Ostrovsky, A. S.

Padgett, M. J.

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via Hong-Ou-Mandel interference,” Phys. Rev. A 89, 013829 (2014).
[CrossRef]

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. E. Wilner, and M. J. Padgett, “Efficient measurement of an optical orbital- angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
[CrossRef]

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011).
[CrossRef]

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

Ramo, S.

S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics (Wiley, 1984).

Rickenstorff-Parrao, C.

Rivera, R.

R. Navarro, R. Rivera, and J. Aporta, “Representation of wavefronts in free-form transmission pupils with Complex Zernike Polynomials,” J. Optom. 4, 41–48 (2011).
[CrossRef]

Robertson, D. J.

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. E. Wilner, and M. J. Padgett, “Efficient measurement of an optical orbital- angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
[CrossRef]

Roy, R.

G. E. Andrews, R. Askey, and R. Roy, Special Functions (Cambridge University, 2000).

Siegman, A. E.

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

Sponselli, A.

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. E. Wilner, and M. J. Padgett, “Efficient measurement of an optical orbital- angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
[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]

Steinhoff, N. K.

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. E. Wilner, and M. J. Padgett, “Efficient measurement of an optical orbital- angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
[CrossRef]

Tang, K. T.

K. T. Tang, Mathematical Methods for Engineers and Scientists (Springer, 2007).

Treviño, J. P.

J. P. Treviño, O. López-Cruz, and S. Chávez-Cerda, “Segmented vortex telescope and its tolerance to diffraction effects and primary aberrations,” Opt. Eng. 52, 081605 (2013).
[CrossRef]

J. P. Treviño, J. E. Gómez-Correa, D. R. Iskander, and S. Chávez-Cerda, “Zernike vs. Bessel circular functions in visual optics,” Ophthalmic Physiolog. Opt. 33, 394–402 (2013).
[CrossRef]

Tyler, G. A.

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. E. Wilner, and M. J. Padgett, “Efficient measurement of an optical orbital- angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
[CrossRef]

Van Duzer, T.

S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics (Wiley, 1984).

Volke-Sepulveda, K.

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4, S82–S89 (2002).

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, 1995).

Whinnery, J. R.

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Figures (3)

Fig. 1.
Fig. 1.

(a) Laguerre–Gauss functions with azimuthal order m=5 and radial order n=0 showing the radial dependence of the profile on the Gaussian width; w0=0.1, 0.2 and 0.3. (b) Laguerre–Gauss functions with same widths and m=5 but radial order n=2. (c) BFs of the same order m=5 but increasing radial frequency kr corresponding to the first three eigenvalues. (d) ZPs with m=5 and increasing radial order n=1, 2, and 3.

Fig. 2.
Fig. 2.

Composed staircase vortex. The first panel in the top-left corner shows the target profile to be constructed with the proposed scheme, using topological charges m=1, 7, 20. The subsequent panels show the rms approximation error (dB) for Bessel, Zernike, Laguerre–Gauss, and Fourier bases for each topological charge.

Fig. 3.
Fig. 3.

Intensity of a vortex with sawtooth profile with radially growing amplitude as shown in the first top-left panel. The rest of the panels show the rms error (dB) of the approximations for three topological charges m=1, 7, 20 using Bessel, Zernike, Laguerre–Gauss, and Fourier bases. The Fourier basis performs best, while the B, Z, and LG show a similar behavior.

Equations (8)

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

f(x)=n=0un|fun(x),
um|un=abum*(x)un(x)s(x)dx=δmn,
ELG(r,φ,z)=E0w0w(z)(2r2w2(z))|m|2Ln|m|(2r2w2(z))×exp[r2w2(z)ikr22R(z)+i(2n+|m|+1)Φ(z)+imφ],
EB(x,y,z)=E0J|m|(krr)eikzz+imφ.
Znm(r,φ)=Rn|m|(r)eimφ,
01Rn|m|(r)J|m|(ρr)rdr=(1)n|m|2Jn+1(ρ)ρ.
Fnm(r,φ)=(1)ncossin(2nπar)eimφ.
f=Uc+εN,

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