Abstract

We investigate a new class of nondiffracting optical beams carrying fractional orbital angular momentum (FOAM Bessel beams), which can be constructed by modifying the phase step of the spiral phase plate in a setup originally for the generation of traditional Bessel beams. The FOAM Bessel beams have entirely stable vortex structures, and form an infinite number of orthogonal and complete subsets for monochromatic scalar light in free space. With the advantages of structural stability and orthogonality-completeness, the FOAM Bessel beams are expected to be useful for study and application in various fields of modern optics.

© 2018 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2017 (1)

A. Turpin, L. Rego, A. Picon, J. San Roman, and C. Hernandez-Garcia, “Extreme ultraviolet fractional orbital angular momentum beams from high harmonic generation,” Sci. Rep. 7, 43888 (2017).
[Crossref]

2016 (1)

T. Trichili, C. Rosales-Guzman, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,” Sci. Rep. 6, 27674 (2016).
[Crossref]

2015 (1)

2014 (3)

A. Lehmuskero, Y. Li, P. Johansson, and M. Kall, “Plasmonic particles set into fast orbital motion by an optical vortex beam,” Opt. Express 22, 4349–4356 (2014).
[Crossref]

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Diffraction-free asymmetric elegant Bessel beams with fractional orbital angular momentum,” Comput. Opt. 38, 4–10 (2014).
[Crossref]

2013 (2)

M. Agnew, J. Z. Salvail, J. Leach, and R. W. Boyd, “Generation of orbital angular momentum Bell states and their verification via accessible nonlinear witnesses,” Phys. Rev. Lett. 111, 030402 (2013).
[Crossref]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref]

2011 (1)

2009 (2)

S. H. Simpson and S. Hanna, “Optical angular momentum transfer by Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 26, 625–638 (2009).
[Crossref]

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

2008 (1)

2007 (1)

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007).
[Crossref]

2006 (1)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref]

2005 (2)

2004 (2)

2003 (1)

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref]

1997 (1)

1994 (1)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[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 (2)

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

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

1956 (1)

N. Marcuvitz, “On field representations in terms of leaky modes or eigenmodes,” IRE Trans. Antennas Propag. 4, 192–194 (1956).
[Crossref]

1936 (1)

R. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
[Crossref]

1909 (1)

J. Poynting, “The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light,” Proc. R. Soc. London A 82, 560–567 (1909).
[Crossref]

Agnew, M.

M. Agnew, J. Z. Salvail, J. Leach, and R. W. Boyd, “Generation of orbital angular momentum Bell states and their verification via accessible nonlinear witnesses,” Phys. Rev. Lett. 111, 030402 (2013).
[Crossref]

Allen, L.

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997).
[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]

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th ed. (Harcourt, 2005).

Barnett, S. M.

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[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]

Bekshaev, A. Y.

Berry, M. V.

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

Beth, R.

R. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
[Crossref]

Boyd, R. W.

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[Crossref]

M. Agnew, J. Z. Salvail, J. Leach, and R. W. Boyd, “Generation of orbital angular momentum Bell states and their verification via accessible nonlinear witnesses,” Phys. Rev. Lett. 111, 030402 (2013).
[Crossref]

Bozinovic, N.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref]

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

Courtial, J.

Dennis, M. R.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

Dholakia, K.

Dudley, A.

T. Trichili, C. Rosales-Guzman, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,” Sci. Rep. 6, 27674 (2016).
[Crossref]

Durnin, J.

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

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (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]

Flossmann, F.

Forbes, A.

T. Trichili, C. Rosales-Guzman, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,” Sci. Rep. 6, 27674 (2016).
[Crossref]

Franke-Arnold, S.

Gibson, G.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

Götte, J. B.

J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16, 993–1006 (2008).
[Crossref]

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007).
[Crossref]

Hanna, S.

Hernandez-Garcia, C.

A. Turpin, L. Rego, A. Picon, J. San Roman, and C. Hernandez-Garcia, “Extreme ultraviolet fractional orbital angular momentum beams from high harmonic generation,” Sci. Rep. 7, 43888 (2017).
[Crossref]

Huang, H.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref]

Johansson, P.

Kall, M.

Kotlyar, V. V.

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Diffraction-free asymmetric elegant Bessel beams with fractional orbital angular momentum,” Comput. Opt. 38, 4–10 (2014).
[Crossref]

Kovalev, A. A.

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Diffraction-free asymmetric elegant Bessel beams with fractional orbital angular momentum,” Comput. Opt. 38, 4–10 (2014).
[Crossref]

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

Kristensen, P.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref]

Lavery, M. P.

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[Crossref]

Leach, J.

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[Crossref]

M. Agnew, J. Z. Salvail, J. Leach, and R. W. Boyd, “Generation of orbital angular momentum Bell states and their verification via accessible nonlinear witnesses,” Phys. Rev. Lett. 111, 030402 (2013).
[Crossref]

Lehmuskero, A.

Li, Y.

Lin, J.

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref]

Malik, M.

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[Crossref]

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref]

Marcuvitz, N.

N. Marcuvitz, “On field representations in terms of leaky modes or eigenmodes,” IRE Trans. Antennas Propag. 4, 192–194 (1956).
[Crossref]

Marrucci, L.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref]

McGloin, D.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[Crossref]

Miceli, J. J.

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

Mirhosseini, M.

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[Crossref]

Ndagano, B.

T. Trichili, C. Rosales-Guzman, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,” Sci. Rep. 6, 27674 (2016).
[Crossref]

Niu, H. B.

O’Holleran, K.

Padgett, M. J.

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref]

Pasko, V.

Peng, X.

Picon, A.

A. Turpin, L. Rego, A. Picon, J. San Roman, and C. Hernandez-Garcia, “Extreme ultraviolet fractional orbital angular momentum beams from high harmonic generation,” Sci. Rep. 7, 43888 (2017).
[Crossref]

Poynting, J.

J. Poynting, “The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light,” Proc. R. Soc. London A 82, 560–567 (1909).
[Crossref]

Preece, D.

Ramachandran, S.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref]

Rego, L.

A. Turpin, L. Rego, A. Picon, J. San Roman, and C. Hernandez-Garcia, “Extreme ultraviolet fractional orbital angular momentum beams from high harmonic generation,” Sci. Rep. 7, 43888 (2017).
[Crossref]

Ren, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref]

Rosales-Guzman, C.

T. Trichili, C. Rosales-Guzman, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,” Sci. Rep. 6, 27674 (2016).
[Crossref]

Salem, A. B.

T. Trichili, C. Rosales-Guzman, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,” Sci. Rep. 6, 27674 (2016).
[Crossref]

Salvail, J. Z.

M. Agnew, J. Z. Salvail, J. Leach, and R. W. Boyd, “Generation of orbital angular momentum Bell states and their verification via accessible nonlinear witnesses,” Phys. Rev. Lett. 111, 030402 (2013).
[Crossref]

San Roman, J.

A. Turpin, L. Rego, A. Picon, J. San Roman, and C. Hernandez-Garcia, “Extreme ultraviolet fractional orbital angular momentum beams from high harmonic generation,” Sci. Rep. 7, 43888 (2017).
[Crossref]

Simpson, N. B.

Simpson, S. H.

Soifer, V. A.

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Diffraction-free asymmetric elegant Bessel beams with fractional orbital angular momentum,” Comput. Opt. 38, 4–10 (2014).
[Crossref]

Soskin, M. S.

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]

Tao, S. H.

Temme, N. M.

N. M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics, 2nd ed. (Wiley, 1996).

Trichili, T.

T. Trichili, C. Rosales-Guzman, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,” Sci. Rep. 6, 27674 (2016).
[Crossref]

Tur, M.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref]

Turpin, A.

A. Turpin, L. Rego, A. Picon, J. San Roman, and C. Hernandez-Garcia, “Extreme ultraviolet fractional orbital angular momentum beams from high harmonic generation,” Sci. Rep. 7, 43888 (2017).
[Crossref]

Vasnetsov, M.

Vasnetsov, M. V.

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref]

Watson, G. N.

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

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

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th ed. (Harcourt, 2005).

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref]

Willner, A. E.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref]

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[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]

Yang, Z. S.

Yao, A. M.

Yuan, X. C.

Yue, Y.

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[Crossref]

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T. Trichili, C. Rosales-Guzman, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,” Sci. Rep. 6, 27674 (2016).
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[Crossref]

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Figures (6)

Fig. 1.
Fig. 1. Setup for the generation of Bessel beams of integer and fractional orbital angular momentum.
Fig. 2.
Fig. 2. Modulus and phase of JM as functions of ϕ for M=3.3, μ=0.1k, and large ρ values. (a) and (b) ρ=1000λ, (c) and (d) ρ=2500λ, (e) and (f) ρ=5000λ.
Fig. 3.
Fig. 3. Transverse-plane intensity and phase profiles of BM,μ(+) with M=3.3 and μ=0.1k at (a) and (b) z=0; (c) and (d) z=10.3λ.
Fig. 4.
Fig. 4. Intensity profiles of FOAM Bessel beams JM for μ=0.1k and (a) M=3.0, (b) M=3.1, (c) M=3.4, (d) M=3.7, (e) M=3.9, and (f) M=4.0.
Fig. 5.
Fig. 5. Phase profiles of FOAM Bessel beams JM for μ=0.1k and (a) M=3.0, (b) M=3.1, (c) M=3.4, (d) M=3.7, (e) M=3.9, and (f) M=4.0.
Fig. 6.
Fig. 6. Normalized intensity in the central region of the FOAM Bessel beam versus propagation distance L for M=3.4, μ=0.1k, and various aperture radii.

Equations (48)

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(2+k2)U(ρ,ϕ,z)=0,
Bm,μ(±)(ρ,ϕ,z)=Jm(μρ)exp(imϕ)exp(±iβz),0μk,β=k2μ2,m=0,±1,.
12πϕ0ϕ0+2πdϕ0dρρ[Jm(μρ)eimϕ]*[Jm(μρ)eimϕ]=1μδ(μμ)δmm,0μ,μ<,ϕ0:any fixed real value,
12πLL2L2dzϕ0ϕ0+2πdϕ0dρρBm,μ(τ)*(ρ,ϕ,z)Bm,μ(τ)(ρ,ϕ,z)=1μδ(μμ)δmmδττ,
12πm=0dμμ[Jm(μρ)eimϕ][Jm(μρ)eimϕ]*=1ρδ(ρρ)δ(ϕϕ),0ρ,ρ<,ϕ0ϕ,ϕ<ϕ0+2π,
U(ρ,ϕ,z)=m=0kdμμam(+)(μ)Bm,μ(+)(ρ,ϕ,z)+m=0kdμμam()(μ)Bm,μ()(ρ,ϕ,z).
Uim(x,y)=keikfi2πfexp[ik2f(x2+y2)]dξdηUini(ξ,η)eik2f(ξ2+η2)exp[ikf(xξ+yη)].
T(x,y)=exp[ik2f(x2+y2)],
Ufin(x,y)=Uim(x,y)exp[ik2f(x2+y2)]=keikfi2πfdξdηUini(ξ,η)eik2f(ξ2+η2)exp[ikf(xξ+yη)],
Ufin(ρ,ϕ)=keikfi2πf0dμ02πdθμUini(μ,θ)eik2fμ2exp[ikfμρ(cosϕcosθ+sinϕsinθ)],
Uini(μ,θ)=U0δ(μR)exp(imθ),
Ufin(ρ,ϕ)=U0kReik(f+R22f)i2πf02πdθexp(imθ)exp[ikfRρ(cosϕcosθ+sinϕsinθ)]=CJm(kRfρ)exp[im(ϕπ2)].
Jm(ρ)exp(imϕ)=12π02πdθexp[im(θ+π2)]exp[iρ(cosϕcosθ+sinϕsinθ)],
U(ρ,ϕ,z>f)=Cexp[i(mπ2+βRf)]Jm(μRρ)exp(imϕ)exp(iβRz)=Cexp[i(mπ2+βRf)]Bm,μR(+)(ρ,ϕ,z),μR=kRf,βR=k2μR2.
Uini(μ,θ)=U0δ(μR)exp(iMθ),0θ<2π,
exp(iMθ)=m=cm[M]exp(imθ),
cm[M]=12π02πdϕexp[i(Mm)ϕ]=exp(iνπ)sin(νπ)π1Mm.
Uini(μ,θ)=m=cm[M][U0δ(μR)exp(imθ)].
U(ρ,ϕ,z=f+)=Ufin(ρ,ϕ)=Cm=cm[M]Jm(μRρ)exp[im(ϕπ2)]=m=0dμμ[Cμδ(μμR)cm[M]exp(imπ2)]Jm(μRρ)exp(imϕ).
U(ρ,ϕ,z>f)=Cm=cm[M]exp(imπ2)Jm(μRρ)exp(imϕ)exp[iβR(zf)]=Cexp(iβRf)JM(μRρ,ϕ)exp(iβRz)C˜BM,μR(+)(ρ,ϕ,z),
JM(ρ,ϕ)=m=cm[M]Jm(ρ)exp[im(ϕπ2)],
BM,μ(+)(ρ,ϕ,z)=JM(μρ,ϕ)exp(ik2μ2z).
BM,μ(+)(ρ,ϕ,z)=12π02πdϑexp[iM(ϑπ)+i2πνfπ(ϑ)]exp[iμρ(cosϕcosϑ+sinϕsinϑ)]exp(ik2μ2z)=12πdkk02πdϑδ(kμ)μexp[iM(ϑπ)+i2πνfπ(ϑ)]exp[ikρ(cosϕcosϑ+sinϕsinϑ)]exp(ik2k2z),
m=cm[M]exp[im(ϑπ)]=exp[iM(ϑπ)+i2πνfπ(ϑ)],
A(kx,ky)=1μδ(kμ)exp[iM(ϑπ)+i2πνfπ(ϑ)],k=kx2+ky2.
12π02π0dρρJm+ν*(μρ,ϕ)Jm+ν(μρ,ϕ)=1μδ(μμ)δmm,
12πm=0dμμJm+ν(μρ,ϕ)Jm+ν*(μρ,ϕ)=1ρδ(ρρ)δ(ϕϕ),0ρ,ρ<,0ϕ,ϕ<2π.
12π02πdϕ0dρρJm+ν*(μρ,ϕ)Jm+ν(μρ,ϕ)=n,n=cn*[m+ν]cn[m+ν]ei(nn)π2×02πdϕ2π0dρρ[Jn(μρ)einϕ]*[Jn(μρ)einϕ]=1μδ(μμ)n=cn*[m+ν]cn[m+ν],
12π02πdϕ0dρρJm+ν*(μρ,ϕ)Jm+ν(μρ,ϕ)=1μδ(μμ)12π02πdϕ1dϕ2ei[(m+ν)ϕ2(m+ν)ϕ1](12πn=ein(ϕ1ϕ2))=1μδ(μμ)12π02πdϕ1dϕ2ei[(m+ν)ϕ2(m+ν)ϕ1]δ(ϕ1ϕ2)=1μδ(μμ)12π02πdϕ1ei(mm)ϕ1=1μδ(μμ)δmm,
{Bm+ν,μ(±)(ρ,ϕ,z)Jm+ν(μρ,ϕ)exp(±ik2μ2z),m=0,±1,,0μ<k},
Jσ(ρ)2πρcos(ρπ4σπ2)=12πρ[ei(ρπ4σπ2)+ei(ρπ4σπ2)],
JM(ρ,ϕ)12πρm=cm[M][ei(ρπ4mπ2)+ei(ρπ4mπ2)]exp[im(ϕπ2)]=12πρ{ei(ρπ4)m=cm[M]eim(ϕπ)+ei(ρπ4)m=cm[M]eimϕ}.
JM(ρ,ϕ)12πρexp(iMϕ){ei(ρπ4)exp[iMπ+i2πνfπ(ϕ)]+ei(ρπ4)},
JM(ρ,ϕ)12πρexp(iMϕ)[ei(ρπ4)exp(iMπ+i2πν)+ei(ρπ4)]=2πρexp(iMϕi(M2ν)π2)cos(ρπ4(M2ν)π2)exp[iM(ϕπ2)+iνπ]Jmν(ρ);
JM(ρ,ϕ)12πρexp(iMϕ)[ei(ρπ4)exp(iMπ)+ei(ρπ4)]=2πρexp[iM(ϕπ2)]cos(ρπ4Mπ2)exp[iM(ϕπ2)]Jm+ν(ρ).
U(ρ,ϕ,z)=m=0dμμαm(μ,z)Jm(μρ)exp(imϕ).
(2+μ2)[Jm(μρ)exp(imϕ)]=0,2=2x2+2y2,
2αm(μ,z)z2+(k2μ2)αm(μ,z)=0.
αm(μ,z)=αm(+)(μ)exp(iβz)+αm()(μ)exp(iβz),
β={k2μ20μkiμ2k2μ>k.
U(ρ,ϕ,z)=m=0kdμμam(+)(μ)Bm,μ(+)(ρ,ϕ,z)+m=0kdμμam()(μ)Bm,μ()(ρ,ϕ,z),
Bm,μ(±)(ρ,ϕ,z)=Jm(μρ)exp(imϕ)exp(±iβz),β=k2μ2,
αm(μ,z)=αm(μ,z0)exp[iβ(zz0)],
U(ρ,ϕ,z0)=m=0dμμαm(μ,z0)Jm(μρ)exp(imϕ),U(ρ,ϕ,z)=m=0dμμαm(μ,z0)Jm(μρ)exp(imϕ)exp[iβ(zz0)].
U(ρ,ϕ,z0)=χJm(μρ)exp(imϕ),
U(ρ,ϕ,z)=χexp(iβz0)Jm(μρ)exp(imϕ)exp(iβz)=χexp(iβz0)Bm,μ(+)(ρ,ϕ,z).
Jm(ρ)=1π0πdθcos(ρsinθmθ),
Jm(ρ)eimϕ=1π0πdθcos(ρsinθmθ)eimϕ=12π{0πdθei[ρsinθm(θϕ)]+0πdθei[ρsinθm(θ+ϕ)]}=12π{ϕ3π2ϕπ2dθ1ei[ρsin(ϕθ1π2)+m(θ1+π2)]+ϕπ2ϕ+π2dθ2ei[ρsin(ϕθ2π2)+m(θ2+π2)]}=12πϕ3π2ϕ+π2dθexp{i[ρsin(ϕθπ2)+m(θ+π2)]}=12πϕ3π2ϕ+π2dθexp[im(θ+π2)]exp[iρcos(ϕθ)]=12π02πdθexp[im(θ+π2)]exp[iρ(cosϕcosθ+sinϕsinθ)].

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