Abstract

We discuss a three-parameter family of paraxial coherent light fields that originate from a complex amplitude composed of the following four cofactors: the Gaussian beam, a logarithmic axicon, a spiral phase plate (angular harmonic), and an amplitude power function with a possible singularity at the origin of coordinates. For such types of beams, the near-field complex amplitude is proportional to the degenerate hypergeometric function, prompting the beams’ name—hypergeometric (HyG) beams. When the Gaussian beam is replaced by a plane wave, the above beams change to generalized HyG modes that preserve their structure up to scale upon propagation. The intensity profile of the HyG beams is similar to that of the Bessel modes, forming a set of alternating bright and dark rings. However, the thickness of the rings of the HyG beams decreases with increasing ring number.

© 2008 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
  8. V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, "Rotation of laser beams with zero of the orbital angular momentum," Opt. Commun. 274, 8-14 (2007).
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    [CrossRef]
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  13. R. K. Singh, P. Senthilkumaran, and K. Singh, "The effect of astigmatism on the diffraction of a vortex carrying beam with a Gaussian background," J. Opt. A, Pure Appl. Opt. 9, 543-554 (2007).
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    [CrossRef] [PubMed]
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    [CrossRef]
  26. V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Soifer, H. Elfstrom, and J. Turunen, "Generation of phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate," J. Opt. Soc. Am. A 22, 849-861 (2005).
    [CrossRef]

2007

Z. Mei, J. Gu, and D. Zhao, "The elliptical Laguerre-Gaussian beam and its propagation," Optik (Stuttgart) 118, 9-12 (2007).

A. Burvall, K. Kolacz, A. V. Goncharov, Z. Jaroszewicz, and C. Dainty, "Lens axicons in oblique illumination," Appl. Opt. 46, 312-318 (2007).
[CrossRef] [PubMed]

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, "Rotation of laser beams with zero of the orbital angular momentum," Opt. Commun. 274, 8-14 (2007).
[CrossRef]

V. Kotlyar, A. Kovalev, V. Soifer, C. Tuvey, and J. Devis, "Sidelobes contrast reduction for optical vortex beams using a helical axicon," Opt. Lett. 32, 921-923 (2007).
[CrossRef] [PubMed]

R. K. Singh, P. Senthilkumaran, and K. Singh, "Effect of coma on the focusing of an apertured singular beam," Opt. Lasers Eng. 45, 488-494 (2007).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, "Focusing of a vortex carrying beam with Gaussian background by a lens in the presence of spherical aberration and defocusing," Opt. Lasers Eng. 45, 773-782 (2007).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, "Influence of astigmatism and defocusing on the focusing of a singular beam," Opt. Commun. 270, 128-138 (2007).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, "The effect of astigmatism on the diffraction of a vortex carrying beam with a Gaussian background," J. Opt. A, Pure Appl. Opt. 9, 543-554 (2007).
[CrossRef]

V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, "Hypergeometric modes," Opt. Lett. 32, 742-744 (2007).
[CrossRef] [PubMed]

M. Gao, C. Gao, and Z. Lin, "Generation and application of the twisted beam with orbital angular momentum," Chin. Opt. Lett. 5, 89-92 (2007).

K. Duan and B. Lu, "Application of the Wiener distribution function to complex-argument Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation," Opt. Laser Technol. 39, 110-115 (2007).
[CrossRef]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, "Hypergeometric-Gaussian modes," Opt. Lett. 32, 3053-3055 (2007).
[CrossRef] [PubMed]

2006

2005

2004

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, "Rotation of microparticles with Bessel beams generated by diffractive elements," J. Mod. Opt. 51, 2167-2184 (2004).
[CrossRef]

2001

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Pääkkönen, J. Simonen, and J. Turunen, "An analysis of the angular momentum of a light field in terms of angular harmonics," J. Mod. Opt. 48, 1543-1557 (2001).
[CrossRef]

1993

1987

J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1-1501 (1987).
[CrossRef]

1986

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

1983

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marychev, Integrals and Series. Special Functions (Nauka, 1983) (in Russian).

1965

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Applied Mathematics Series (National Bureau of Standards, 1965).

Abramovitz, M.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Applied Mathematics Series (National Bureau of Standards, 1965).

Almazov, A. A.

Balalayev, S. A.

V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and S. A. Balalayev, "Hyper-geometric modes," Comput. Opt. 30, 16-22 (2006) (in Russian).

Brychkov, Yu. A.

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marychev, Integrals and Series. Special Functions (Nauka, 1983) (in Russian).

Burvall, A.

Dainty, C.

Devis, J.

Duan, K.

K. Duan and B. Lu, "Application of the Wiener distribution function to complex-argument Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation," Opt. Laser Technol. 39, 110-115 (2007).
[CrossRef]

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1-1501 (1987).
[CrossRef]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1-1501 (1987).
[CrossRef]

Elfstrom, H.

Gao, C.

Gao, M.

Goncharov, A. V.

Gu, J.

Z. Mei, J. Gu, and D. Zhao, "The elliptical Laguerre-Gaussian beam and its propagation," Optik (Stuttgart) 118, 9-12 (2007).

Jaroszewicz, Z.

Jefimovs, K.

V. V. Kotlyar, S. N. Khonina, A. A. Almazov, V. A. Soifer, K. Jefimovs, and J. Turunen, "Elliptic Laguerre-Gaussian beams," J. Opt. Soc. Am. A 23, 43-56 (2006).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, "Rotation of microparticles with Bessel beams generated by diffractive elements," J. Mod. Opt. 51, 2167-2184 (2004).
[CrossRef]

Karimi, E.

Khonina, S. N.

V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, "Hypergeometric modes," Opt. Lett. 32, 742-744 (2007).
[CrossRef] [PubMed]

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, "Rotation of laser beams with zero of the orbital angular momentum," Opt. Commun. 274, 8-14 (2007).
[CrossRef]

V. V. Kotlyar, S. N. Khonina, A. A. Almazov, V. A. Soifer, K. Jefimovs, and J. Turunen, "Elliptic Laguerre-Gaussian beams," J. Opt. Soc. Am. A 23, 43-56 (2006).
[CrossRef]

V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and S. A. Balalayev, "Hyper-geometric modes," Comput. Opt. 30, 16-22 (2006) (in Russian).

V. V. Kotlyar, A. A. Kovalev, S. N. Khonina, R. V. Skidanov, V. A. Soifer, H. Elfstrom, N. Tossavainen, and J. Turunen, "Diffraction of conic and Gaussian beams by a spiral phase plate," Appl. Opt. 45, 2656-2665 (2006).
[CrossRef] [PubMed]

V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Soifer, H. Elfstrom, and J. Turunen, "Generation of phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate," J. Opt. Soc. Am. A 22, 849-861 (2005).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, "Rotation of microparticles with Bessel beams generated by diffractive elements," J. Mod. Opt. 51, 2167-2184 (2004).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Pääkkönen, J. Simonen, and J. Turunen, "An analysis of the angular momentum of a light field in terms of angular harmonics," J. Mod. Opt. 48, 1543-1557 (2001).
[CrossRef]

Kolacz, K.

Kolodziejczyk, A.

Kostenbauder, A.

Kotlyar, V.

Kotlyar, V. V.

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, "Rotation of laser beams with zero of the orbital angular momentum," Opt. Commun. 274, 8-14 (2007).
[CrossRef]

V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, "Hypergeometric modes," Opt. Lett. 32, 742-744 (2007).
[CrossRef] [PubMed]

V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and S. A. Balalayev, "Hyper-geometric modes," Comput. Opt. 30, 16-22 (2006) (in Russian).

V. V. Kotlyar, S. N. Khonina, A. A. Almazov, V. A. Soifer, K. Jefimovs, and J. Turunen, "Elliptic Laguerre-Gaussian beams," J. Opt. Soc. Am. A 23, 43-56 (2006).
[CrossRef]

V. V. Kotlyar, A. A. Kovalev, S. N. Khonina, R. V. Skidanov, V. A. Soifer, H. Elfstrom, N. Tossavainen, and J. Turunen, "Diffraction of conic and Gaussian beams by a spiral phase plate," Appl. Opt. 45, 2656-2665 (2006).
[CrossRef] [PubMed]

V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Soifer, H. Elfstrom, and J. Turunen, "Generation of phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate," J. Opt. Soc. Am. A 22, 849-861 (2005).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, "Rotation of microparticles with Bessel beams generated by diffractive elements," J. Mod. Opt. 51, 2167-2184 (2004).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Pääkkönen, J. Simonen, and J. Turunen, "An analysis of the angular momentum of a light field in terms of angular harmonics," J. Mod. Opt. 48, 1543-1557 (2001).
[CrossRef]

Kovalev, A.

Kovalev, A. A.

Lin, Z.

Lu, B.

K. Duan and B. Lu, "Application of the Wiener distribution function to complex-argument Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation," Opt. Laser Technol. 39, 110-115 (2007).
[CrossRef]

Marrucci, L.

Marychev, O. I.

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marychev, Integrals and Series. Special Functions (Nauka, 1983) (in Russian).

Mei, Z.

Z. Mei, J. Gu, and D. Zhao, "The elliptical Laguerre-Gaussian beam and its propagation," Optik (Stuttgart) 118, 9-12 (2007).

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1-1501 (1987).
[CrossRef]

Pääkkönen, P.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Pääkkönen, J. Simonen, and J. Turunen, "An analysis of the angular momentum of a light field in terms of angular harmonics," J. Mod. Opt. 48, 1543-1557 (2001).
[CrossRef]

Piccirillo, B.

Prudnikov, A. P.

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marychev, Integrals and Series. Special Functions (Nauka, 1983) (in Russian).

Santamato, E.

Senthilkumaran, P.

R. K. Singh, P. Senthilkumaran, and K. Singh, "The effect of astigmatism on the diffraction of a vortex carrying beam with a Gaussian background," J. Opt. A, Pure Appl. Opt. 9, 543-554 (2007).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, "Focusing of a vortex carrying beam with Gaussian background by a lens in the presence of spherical aberration and defocusing," Opt. Lasers Eng. 45, 773-782 (2007).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, "Influence of astigmatism and defocusing on the focusing of a singular beam," Opt. Commun. 270, 128-138 (2007).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, "Effect of coma on the focusing of an apertured singular beam," Opt. Lasers Eng. 45, 488-494 (2007).
[CrossRef]

Seshardi, S. R.

Siegman, A. E.

Simonen, J.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, "Rotation of microparticles with Bessel beams generated by diffractive elements," J. Mod. Opt. 51, 2167-2184 (2004).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Pääkkönen, J. Simonen, and J. Turunen, "An analysis of the angular momentum of a light field in terms of angular harmonics," J. Mod. Opt. 48, 1543-1557 (2001).
[CrossRef]

Singh, K.

R. K. Singh, P. Senthilkumaran, and K. Singh, "Influence of astigmatism and defocusing on the focusing of a singular beam," Opt. Commun. 270, 128-138 (2007).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, "Focusing of a vortex carrying beam with Gaussian background by a lens in the presence of spherical aberration and defocusing," Opt. Lasers Eng. 45, 773-782 (2007).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, "The effect of astigmatism on the diffraction of a vortex carrying beam with a Gaussian background," J. Opt. A, Pure Appl. Opt. 9, 543-554 (2007).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, "Effect of coma on the focusing of an apertured singular beam," Opt. Lasers Eng. 45, 488-494 (2007).
[CrossRef]

Singh, R. K.

R. K. Singh, P. Senthilkumaran, and K. Singh, "Focusing of a vortex carrying beam with Gaussian background by a lens in the presence of spherical aberration and defocusing," Opt. Lasers Eng. 45, 773-782 (2007).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, "Effect of coma on the focusing of an apertured singular beam," Opt. Lasers Eng. 45, 488-494 (2007).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, "Influence of astigmatism and defocusing on the focusing of a singular beam," Opt. Commun. 270, 128-138 (2007).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, "The effect of astigmatism on the diffraction of a vortex carrying beam with a Gaussian background," J. Opt. A, Pure Appl. Opt. 9, 543-554 (2007).
[CrossRef]

Skidanov, R. V.

V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, "Hypergeometric modes," Opt. Lett. 32, 742-744 (2007).
[CrossRef] [PubMed]

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, "Rotation of laser beams with zero of the orbital angular momentum," Opt. Commun. 274, 8-14 (2007).
[CrossRef]

V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and S. A. Balalayev, "Hyper-geometric modes," Comput. Opt. 30, 16-22 (2006) (in Russian).

V. V. Kotlyar, A. A. Kovalev, S. N. Khonina, R. V. Skidanov, V. A. Soifer, H. Elfstrom, N. Tossavainen, and J. Turunen, "Diffraction of conic and Gaussian beams by a spiral phase plate," Appl. Opt. 45, 2656-2665 (2006).
[CrossRef] [PubMed]

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, "Rotation of microparticles with Bessel beams generated by diffractive elements," J. Mod. Opt. 51, 2167-2184 (2004).
[CrossRef]

Sochacki, J.

Soifer, V.

Soifer, V. A.

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, "Rotation of laser beams with zero of the orbital angular momentum," Opt. Commun. 274, 8-14 (2007).
[CrossRef]

V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, "Hypergeometric modes," Opt. Lett. 32, 742-744 (2007).
[CrossRef] [PubMed]

V. V. Kotlyar, S. N. Khonina, A. A. Almazov, V. A. Soifer, K. Jefimovs, and J. Turunen, "Elliptic Laguerre-Gaussian beams," J. Opt. Soc. Am. A 23, 43-56 (2006).
[CrossRef]

V. V. Kotlyar, A. A. Kovalev, S. N. Khonina, R. V. Skidanov, V. A. Soifer, H. Elfstrom, N. Tossavainen, and J. Turunen, "Diffraction of conic and Gaussian beams by a spiral phase plate," Appl. Opt. 45, 2656-2665 (2006).
[CrossRef] [PubMed]

V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Soifer, H. Elfstrom, and J. Turunen, "Generation of phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate," J. Opt. Soc. Am. A 22, 849-861 (2005).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, "Rotation of microparticles with Bessel beams generated by diffractive elements," J. Mod. Opt. 51, 2167-2184 (2004).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Pääkkönen, J. Simonen, and J. Turunen, "An analysis of the angular momentum of a light field in terms of angular harmonics," J. Mod. Opt. 48, 1543-1557 (2001).
[CrossRef]

Staronski, L. R.

Stegun, I. A.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Applied Mathematics Series (National Bureau of Standards, 1965).

Sun, Y.

Tossavainen, N.

Turunen, J.

Tuvey, C.

Zhao, D.

Z. Mei, J. Gu, and D. Zhao, "The elliptical Laguerre-Gaussian beam and its propagation," Optik (Stuttgart) 118, 9-12 (2007).

Zito, G.

Appl. Opt.

Chin. Opt. Lett.

Comput. Opt.

V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and S. A. Balalayev, "Hyper-geometric modes," Comput. Opt. 30, 16-22 (2006) (in Russian).

J. Mod. Opt.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, K. Jefimovs, J. Simonen, and J. Turunen, "Rotation of microparticles with Bessel beams generated by diffractive elements," J. Mod. Opt. 51, 2167-2184 (2004).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Pääkkönen, J. Simonen, and J. Turunen, "An analysis of the angular momentum of a light field in terms of angular harmonics," J. Mod. Opt. 48, 1543-1557 (2001).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

R. K. Singh, P. Senthilkumaran, and K. Singh, "The effect of astigmatism on the diffraction of a vortex carrying beam with a Gaussian background," J. Opt. A, Pure Appl. Opt. 9, 543-554 (2007).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

R. K. Singh, P. Senthilkumaran, and K. Singh, "Influence of astigmatism and defocusing on the focusing of a singular beam," Opt. Commun. 270, 128-138 (2007).
[CrossRef]

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, "Rotation of laser beams with zero of the orbital angular momentum," Opt. Commun. 274, 8-14 (2007).
[CrossRef]

Opt. Laser Technol.

K. Duan and B. Lu, "Application of the Wiener distribution function to complex-argument Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation," Opt. Laser Technol. 39, 110-115 (2007).
[CrossRef]

Opt. Lasers Eng.

R. K. Singh, P. Senthilkumaran, and K. Singh, "Effect of coma on the focusing of an apertured singular beam," Opt. Lasers Eng. 45, 488-494 (2007).
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R. K. Singh, P. Senthilkumaran, and K. Singh, "Focusing of a vortex carrying beam with Gaussian background by a lens in the presence of spherical aberration and defocusing," Opt. Lasers Eng. 45, 773-782 (2007).
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Opt. Lett.

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A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marychev, Integrals and Series. Special Functions (Nauka, 1983) (in Russian).

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Figures (7)

Fig. 1
Fig. 1

Amplitude distribution E γ n m ( ρ ) (in relative units) versus the radial variable at distance z = 2000 mm for different n: (a) 10 and (b) 200.

Fig. 2
Fig. 2

Radius (a) and the magnitude (b) of the diffraction pattern principal maximum versus the singularity number.

Fig. 3
Fig. 3

Radial amplitude distribution at distance z = 2000 mm at (a) n = 10 , m = 0 , γ = 150 and at (b) γ = 350 .

Fig. 4
Fig. 4

Radius (a) and principal maximum of on-radius amplitude E γ n m ( ρ 0 ) (b) for the diffraction pattern at distance z = 2000 mm as a function of the logarithmic axicon parameter γ.

Fig. 5
Fig. 5

Radial amplitude distribution for a conic axicon at z = 2000 mm for (a) γ = 12.5 and (b) γ = 12.5 .

Fig. 6
Fig. 6

Radial distribution of the complex amplitude modulus in the planes z = 0 mm (a,b) and z = 2000 mm (c,d) for the light field in Eq. (14) with infinite aperture (a,c) and limited by the annular diaphragm of radii R 1 = 0.2 mm and R 2 = 10 mm (b,d).

Fig. 7
Fig. 7

Radial distribution of the complex amplitude modulus for the initial field in Eq. (14) ( n = 10 ) in the plane z = 2000 mm at (a) γ = 25 and (b) γ = 50 .

Equations (57)

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E γ n m ( r , φ ) = 1 2 π ( r w ) m exp ( r 2 2 σ 2 + i γ ln r w + i n φ ) ,
E ( ρ , θ , z ) = i k 2 π z R 2 E ( r , φ , 0 ) exp { i k 2 z [ ρ 2 + r 2 2 ρ r cos ( φ θ ) ] } r d r d φ ,
E ( ρ , θ , z ) = ( i ) n + 1 k z exp ( i k ρ 2 2 z + i n θ ) 0 A ( r ) exp ( i k r 2 2 z ) J n ( k ρ r z ) r d r ,
0 x α 1 exp ( p x 2 ) J ν ( c x ) d x
= c ν p ( ν + α ) 2 2 ν 1 Γ ( ν + α 2 ) Γ 1 ( ν + 1 ) F 1 1 ( ν + α 2 , ν + 1 , c 2 4 p ) ,
E γ n m ( ρ , θ , z ) = ( i ) n + 1 2 π n ! ( z 0 z q 2 ) ( 2 σ w q ) m + i γ ( k σ ρ 2 q z ) n exp ( i k ρ 2 2 z + i n θ ) × Γ ( n + m + 2 + i γ 2 ) F 1 1 ( n + m + 2 + i γ 2 , n + 1 , ( k σ ρ 2 q z ) 2 ) ,
E γ n m ( ρ , θ , z ) x n 2 F 1 1 ( a , b , x ) ,
x = ( k σ ρ 2 q z ) 2 .
F 1 1 ( a , b , x ) = l = 0 C l ( 1 ) l x l ,
C l = Γ ( a + l ) Γ ( a ) Γ ( b ) Γ ( b + l ) l ! .
F 1 1 ( a , b , x ) = l = 0 ( i ) l C l [ k σ ρ 2 z ( 1 + z 0 2 z 2 ) 1 4 ] 2 l exp [ i l tan 1 ( z z 0 ) ] .
1 2 σ 2 i k 2 z i k 2 z , q 2 i z 0 z .
E γ n m ( ρ , θ , z z 0 ) = ( i ) ( n m i γ ) 2 2 π n ! ( k w 2 2 z ) ( m + i γ ) 2 ( k ρ 2 2 z ) n 2 exp ( i k ρ 2 2 z + i n θ ) × Γ ( n + m + 2 + i γ 2 ) F 1 1 ( n + m + 2 + i γ 2 , n + 1 , i k ρ 2 2 z ) .
E γ n m ( ρ , θ , z z 0 ) = ( i ) n + 1 2 π n ! ( z 0 z ) ( 2 σ w ) m + i γ ( k σ ρ 2 z ) n exp ( i k ρ 2 2 z + i n θ ) × Γ ( n + m + 2 + i γ 2 ) F 1 1 ( n + m + 2 + i γ 2 , n + 1 , ( k σ ρ 2 z ) 2 ) .
E γ , n , 1 ( r , φ ) = 1 2 π ( w r ) exp ( r 2 2 σ 2 + i γ ln r w + i n φ ) ,
E γ , n , 1 ( ρ , θ , z ) = ( i ) n + 1 2 π n ! ( z 0 z q 2 ) ( 2 σ w q ) 1 + i γ ( k σ ρ 2 q z ) n exp ( i k ρ 2 2 z + i n θ ) Γ ( n + 1 + i γ 2 ) F 1 1 ( n + 1 + i γ 2 , n + 1 , ( k σ ρ 2 q z ) 2 ) .
E γ , n , 1 ( ρ , θ , z ) = ( i ) ( n + 1 i γ ) 2 2 π n ! ( k w 2 2 z ) ( 1 i γ ) 2 ( k ρ 2 2 z ) n 2 exp ( i k ρ 2 2 z + i n θ ) Γ ( n + 1 + i γ 2 ) F 1 1 ( n + 1 + i γ 2 , n + 1 , i k ρ 2 2 z ) .
F 1 1 ( a , b , z ) = exp ( z ) F 1 1 ( b a , b , z ) ,
E γ , n , 1 ( ρ , θ , z ) = w n ρ n 2 π n ! ( i k w 2 2 z ) ( n + 1 i γ ) 2 × exp ( i n θ ) Γ ( n + 1 + i γ 2 ) F 1 1 ( n + 1 i γ 2 , n + 1 , i k ρ 2 2 z ) .
( 2 i k z + 2 ρ 2 + 1 ρ ρ + 1 ρ 2 θ 2 ) U ( ρ , θ , z ) = 0 ,
[ χ 2 χ 2 + ( b χ ) χ a ] F 1 1 ( a , b , χ ) = 0 ,
a = ( n + 1 i γ ) 2 , b = n + 1 , χ = i k ρ 2 ( 2 z ) .
F 1 1 ( n 2 + 1 2 , 2 n 2 + 1 , i k ρ 2 2 z ) = Γ ( 1 + n 2 ) exp ( i k ρ 2 4 z ) ( k ρ 2 8 z ) ( n 2 ) J n 2 ( k ρ 2 4 z ) .
E 0 , n , 1 ( ρ , θ , z ) = w 2 ( k 2 π z ) 1 2 exp [ i π 4 ( n + 1 ) ] exp ( i k ρ 2 4 z + i n θ ) J n 2 ( k ρ 2 4 z ) .
E 0 , 1 , 1 ( ρ , θ , z ) = i w π ρ exp ( i k ρ 2 4 z + i θ ) sin ( k ρ 2 4 z ) .
E ( ρ , θ , 0 ) = ρ p z q exp ( i n θ ) F ( s ρ m z l ) ,
s 2 m 2 ρ 2 m 2 z 2 l F + [ ( 2 p + m ) s m ρ m 2 z l + 2 i k s l ρ m z l 1 ] F + [ p ( p 1 ) ρ 2 + p ρ 2 n 2 ρ 2 + 2 i k q z 1 ] F = 0 .
s 2 m 2 ρ 2 m 2 z 2 l F + [ s m ( 2 n + m ) ρ m 2 z l + 2 i k s l ρ m z l 1 ] F + 2 i k q z 1 F = 0 .
s ρ m z l F + ( 2 n + m m + 2 i k l ρ 2 z 1 m 2 ) F + 2 i k q ρ 2 m z l 1 s m 2 F = 0 .
m = 2 , l = 1 , s = 2 i k l m 2 = i k 2 .
i k 2 ρ 2 z 1 F + ( n + 1 i k 2 ρ 2 z 1 ) F + q F = 0 .
a = q , b = n + 1 .
F 1 1 ( a , n + 1 , i k ρ 2 2 z ) ;
E a , + n ( ρ , θ , z ) = ρ n z a exp ( i n θ ) F 1 1 ( a , n + 1 , i k ρ 2 2 z ) .
E a , n ( ρ , θ , z ) = ρ n z a exp ( i n θ ) F 1 1 ( a , 1 n , i k ρ 2 2 z ) .
E a n ( ρ , θ , z ) = ρ n z a exp ( ± i n θ ) F 1 1 ( a , n + 1 , i k ρ 2 2 z ) .
E γ n 0 ( r , φ ) = 1 2 π exp ( r 2 2 σ 2 + i γ ln r w + i n φ ) .
E γ n 0 ( ρ , θ , z ) = ( i ) n + 1 2 π n ! ( z 0 z q 2 ) ( 2 σ w q ) i γ ( k σ ρ 2 q z ) n exp ( i k ρ 2 2 z + i n θ ) Γ ( n + 2 + i γ 2 ) F 1 1 [ n + 2 + i γ 2 , n + 1 , ( k σ ρ 2 q z ) 2 ] .
E γ n 0 ( ρ , θ , z z 0 ) = ( i ) ( n i γ ) 2 2 π n ! ( k w 2 2 z ) ( i γ 2 ) ( k ρ 2 2 z ) n 2 exp ( i k ρ 2 2 z + i n θ ) Γ ( n + 2 + i γ 2 ) F 1 1 ( n + 2 + i γ 2 , n + 1 , i k ρ 2 2 z ) .
E γ n 0 ( ρ , θ , z z 0 ) = ( i ) n + 1 2 π n ! ( z 0 z ) ( 2 σ w ) i γ y n 2 × exp ( i k ρ 2 2 z + i n θ ) Γ ( n + 2 + i γ 2 ) F 1 1 ( n + 2 + i γ 2 , n + 1 , y ) ,
E 0 n 0 ( ρ , θ , z z 0 ) = ( i ) n + 1 2 π n ! ( z 0 z ) Γ ( n + 2 2 ) exp ( i k ρ 2 2 z + i n θ ) y n 2 F 1 1 ( n + 2 2 , n + 1 , y ) .
F 1 1 ( n + 2 2 , n + 1 , y ) = exp ( y ) F 1 1 ( n 2 , n + 1 , y ) ,
F 1 1 ( n 2 , n + 1 , y ) = n 2 ( n + 1 ) F 1 1 ( n + 2 2 , n + 2 , y ) ,
F 1 1 ( n 2 , n + 1 , y ) = 1 + n 2 ( n + 1 ) 0 y F 1 1 ( n + 2 2 , n + 2 , y ) d y = 1 + n 2 ( n + 1 ) Γ ( n + 3 2 ) 0 y exp ( y 2 ) ( y 4 ) ( n + 1 ) 2 I ( n + 1 ) 2 ( y 2 ) d y .
0 x x ν exp ( x ) I ν ( x ) d x = x ν + 1 2 ν 1 exp ( x ) [ I ν ( x ) I ν 1 ( x ) ] 2 1 ν ( 2 ν 1 ) Γ ( ν ) ,
F 1 1 ( n 2 , n + 1 , y ) = Γ ( n + 1 2 ) 2 ( n 1 ) 2 ( y 2 ) ( n 1 ) 2 exp ( y 2 ) [ I ( n 1 ) 2 ( y 2 ) I ( n + 1 ) 2 ( y 2 ) ] .
E 0 n 0 ( ρ , θ , z z 0 ) = ( i ) n + 1 2 π ( k σ 2 2 z ) ξ exp ( i k ρ 2 2 z + i n θ ) exp ( ξ ) [ I ( n 1 ) 2 ( ξ ) I ( n + 1 ) 2 ( ξ ) ] ,
Γ ( n + 1 2 ) Γ ( n + 2 2 ) = π n ! 2 n .
E 0 n n ( ρ , θ , z ) = ( i ) n + 1 2 π ( z 0 z q 2 ) ( k σ 2 ρ w q 2 z ) n exp ( i k ρ 2 2 z + i n θ ) F 1 1 ( n + 1 , n + 1 , ( k σ ρ 2 q z ) 2 ) = ( i ) n + 1 2 π ( z 0 z q 2 ) ( k σ 2 ρ w q 2 z ) n exp ( i n θ ) exp [ ρ 2 2 ( σ 2 + i z k ) ] .
E 0 n n ( ρ , θ , z ) = i 2 π ( z 0 z q 2 ) [ ρ w ( z ) ] n exp ( i n θ ) exp [ ρ 2 2 τ 2 ( z ) ] ,
E 0 n n ( ρ , θ , z ) = 1 2 π ( ρ w ) n ( 1 + z 2 z 0 2 ) ( n + 1 ) 2 exp [ i n θ i ( n + 1 ) tan 1 ( z z 0 ) ρ 2 2 σ 2 ( z ) + i k ρ 2 2 R ( z ) ] ,
R ( z ) = R 1 + ( z z 0 ) 2 , z 0 = k R 2 2 ,
ρ 0 ( z ) = 4 α n z k , α n 0.54 n + 1 .
R ( z ) = ρ 0 ( z ) .
n 0 k R 2 2 z + 2 z k R 2 2 250 .
ρ 1 = 2 β n z k ,
β n = k R 2 2 z .

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