Abstract

A simple correspondence between the paraxial propagation formulas along the optical axis of a uniaxial crystal and inside an isotropic medium is found in the case of beams with linearly polarized circularly symmetric boundary distributions. The electric fields of the ordinary and the extraordinary beams are related to the corresponding expressions in a medium with refractive index no and ne2/no, where no and ne are the ordinary and the extraordinary refractive indexes, respectively. Closed-form expressions for Laguerre–Gauss and Bessel–Gauss beams propagating through an anisotropic crystal are given.

© 2002 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
  34. P. L. Greene, D. G. Hall, “Properties and diffraction of vector Bessel–Gauss beam,” J. Opt. Soc. Am. A 15, 3020–3027 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  38. S. Chavez-Cerda, E. Tepichin, M. A. Meneses-Nava, G. Ramirez, “Experimental observation of interfering Bessel beams” Opt. Express 13, 524–529 (1998).
    [CrossRef]
  39. S. Chavez-Cerda, M. A. Meneses-Nava, J. M. Hicknann, “Interference of travelling nondiffracting beams,” Opt. Lett. 23, 1871–1873 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  45. See Ref. 40, p. 51, formula 1.14.1.8.
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    [CrossRef]
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    [CrossRef]

2002 (2)

A. Ciattoni, G. Cincotti, C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19, 792–796 (2002).
[CrossRef]

G. Cincotti, A. Ciattoni, C. Palma, “Hermite–Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 1517–1524 (2002).
[CrossRef]

2001 (1)

J. Arlt, R. Kuhn, K. Dholakia, “Spatial transformation of Laguerre–Gaussian laser modes,” J. Mod. Opt. 48, 783–787 (2001).

2000 (2)

R. Oron, N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182, 205–208 (2000).
[CrossRef]

R. M. Herman, T. A. Wiggins, “Bessel-like beams modulated by arbitrary radial functions,” J. Opt. Soc. Am. A 17, 1021–1032 (2000).
[CrossRef]

1998 (7)

S. Saghafi, C. J. R. Sheppard, “Near field and far field of elegant Hermite–Gaussian and Laguerre–Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
[CrossRef]

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor laser,” Appl. Phys. Lett. 72, 1284–1268 (1998).
[CrossRef]

S. Chavez-Cerda, E. Tepichin, M. A. Meneses-Nava, G. Ramirez, “Experimental observation of interfering Bessel beams” Opt. Express 13, 524–529 (1998).
[CrossRef]

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

P. L. Greene, D. G. Hall, “Properties and diffraction of vector Bessel–Gauss beam,” J. Opt. Soc. Am. A 15, 3020–3027 (1998).
[CrossRef]

A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre–Gaussian laser beams,” J. Opt. Soc. Am. A 15, 2705–2711 (1998).
[CrossRef]

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998).
[CrossRef]

1997 (4)

1996 (8)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[CrossRef]

M. Santarsiero, “Propagation of generalized Bessel–Gauss beams through ABCD optical systems,” Opt. Commun. 132, 1–7 (1996).
[CrossRef]

B. Lu, W. Huang, “Three-dimensional intensity distribution of focused Bessel–Gauss beams,” J. Mod. Opt. 43, 509–515 (1996).
[CrossRef]

N. B. Simpson, L. Allen, M. J. Padgett, “Optical tweezers and optical spanners with Laguerre–Gaussian modes,” J. Mod. Opt. 43, 2485–2491 (1996).
[CrossRef]

R. Borghi, F. Gori, M. Santarsiero, “Optimization of Laguerre–Gauss truncated series,” Opt. Commun. 125, 197–203 (1996).
[CrossRef]

P. L. Greene, D. G. Hall, “Diffraction characteristics of the azimuthal Bessel–Gauss beam,” J. Opt. Soc. Am. A 13, 962–966 (1996).
[CrossRef]

Z. Bouchal, R. Horak, J. Wagner, “Propagation-invariant electromagnetic fields: theory and experiment,” J. Mod. Opt. 43, 1905–1920 (1996).
[CrossRef]

1995 (1)

M. J. Padgett, L. Allen, “The Poynting vector in Laguerre–Gaussian laser modes,” Opt. Commun. 121, 36–40 (1995).
[CrossRef]

1994 (7)

M. Babiker, W. L. Power, L. Allen, “Light-induced torque on moving atoms,” Phys. Rev. Lett. 73, 1239–1242 (1994).
[CrossRef] [PubMed]

L. Allen, M. Babiker, W. L. Power, “Azimuthal doppler-shift in laser beams with orbital angular-momentum,” Opt. Commun. 112, 141–144 (1994).
[CrossRef]

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral-interference-fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

B. Lu, W. Huang, “Focal shift in unaperturated Bessel–Gauss beams,” Opt. Commun. 109, 43–46 (1994).
[CrossRef]

S. Ruschin, “Modified Bessel nondiffracting beams,” J. Opt. Soc. Am. A 11, 3224–3228 (1994).
[CrossRef]

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

R. H. Jordan, G. D. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel–Gauss beam solution,” Opt. Lett. 19, 427–429 (1994).
[CrossRef] [PubMed]

1993 (2)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

S. C. Tidwell, G. H. Kim, W. D. Kimura, “Efficient radially polarized laser beam generation with a double interferometer,” Appl. Opt. 32, 5222–5229 (1993).
[CrossRef] [PubMed]

1992 (1)

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, M. J. Wegener, “Laser-beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

1991 (1)

1990 (2)

1988 (1)

1987 (3)

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

J. E. Durnin, “Exact solutions for nondiffracting beams,” J. Opt. Soc. Am. A 4, 651–653 (1987).
[CrossRef]

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 54, 1499–1501 (1987).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Allen, L.

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998).
[CrossRef]

N. B. Simpson, L. Allen, M. J. Padgett, “Optical tweezers and optical spanners with Laguerre–Gaussian modes,” J. Mod. Opt. 43, 2485–2491 (1996).
[CrossRef]

M. J. Padgett, L. Allen, “The Poynting vector in Laguerre–Gaussian laser modes,” Opt. Commun. 121, 36–40 (1995).
[CrossRef]

M. Babiker, W. L. Power, L. Allen, “Light-induced torque on moving atoms,” Phys. Rev. Lett. 73, 1239–1242 (1994).
[CrossRef] [PubMed]

L. Allen, M. Babiker, W. L. Power, “Azimuthal doppler-shift in laser beams with orbital angular-momentum,” Opt. Commun. 112, 141–144 (1994).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Arlt, J.

J. Arlt, R. Kuhn, K. Dholakia, “Spatial transformation of Laguerre–Gaussian laser modes,” J. Mod. Opt. 48, 783–787 (2001).

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998).
[CrossRef]

Babiker, M.

L. Allen, M. Babiker, W. L. Power, “Azimuthal doppler-shift in laser beams with orbital angular-momentum,” Opt. Commun. 112, 141–144 (1994).
[CrossRef]

M. Babiker, W. L. Power, L. Allen, “Light-induced torque on moving atoms,” Phys. Rev. Lett. 73, 1239–1242 (1994).
[CrossRef] [PubMed]

Bagini, V.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Beijersbergen, M. W.

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

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Borghi, R.

R. Borghi, M. Santarsiero, “M2 factor of Bessel–Gauss beams,” Opt. Lett. 22, 262–264 (1997).
[CrossRef] [PubMed]

R. Borghi, F. Gori, M. Santarsiero, “Optimization of Laguerre–Gauss truncated series,” Opt. Commun. 125, 197–203 (1996).
[CrossRef]

Bouchal, Z.

Z. Bouchal, R. Horak, J. Wagner, “Propagation-invariant electromagnetic fields: theory and experiment,” J. Mod. Opt. 43, 1905–1920 (1996).
[CrossRef]

Brychkov, Yu. A.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon & Breach Science, New York, 1986), Vol. 2, p. 223, formula 2.12.39.3.

Chavez-Cerda, S.

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

S. Chavez-Cerda, E. Tepichin, M. A. Meneses-Nava, G. Ramirez, “Experimental observation of interfering Bessel beams” Opt. Express 13, 524–529 (1998).
[CrossRef]

Ciattoni, A.

G. Cincotti, A. Ciattoni, C. Palma, “Hermite–Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 1517–1524 (2002).
[CrossRef]

A. Ciattoni, G. Cincotti, C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19, 792–796 (2002).
[CrossRef]

Cincotti, G.

A. Ciattoni, G. Cincotti, C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19, 792–796 (2002).
[CrossRef]

G. Cincotti, A. Ciattoni, C. Palma, “Hermite–Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 1517–1524 (2002).
[CrossRef]

C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[CrossRef]

Ciofini, M.

R. Meucci, A. Labate, M. Ciofini, “Polarization properties of low-order Laguerre–Gauss modes in a CO2 laser,” Quantum Semiclassic. Opt. 9, L31–L35 (1997).
[CrossRef]

Clark, G. H.

Coerwinkel, R. P. C.

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

Davidson, N.

R. Oron, N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182, 205–208 (2000).
[CrossRef]

Dholakia, K.

J. Arlt, R. Kuhn, K. Dholakia, “Spatial transformation of Laguerre–Gaussian laser modes,” J. Mod. Opt. 48, 783–787 (2001).

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998).
[CrossRef]

Durnin, J. E.

J. E. Durnin, “Exact solutions for nondiffracting beams,” J. Opt. Soc. Am. A 4, 651–653 (1987).
[CrossRef]

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 54, 1499–1501 (1987).
[CrossRef]

Eberly, J. H.

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 54, 1499–1501 (1987).
[CrossRef]

Ford, D. H.

Frezza, F.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Friberg, A. T.

Friesem, A. A.

R. Oron, N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182, 205–208 (2000).
[CrossRef]

Goodman, J. W.

See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gori, F.

R. Borghi, F. Gori, M. Santarsiero, “Optimization of Laguerre–Gauss truncated series,” Opt. Commun. 125, 197–203 (1996).
[CrossRef]

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, San Diego, Calif., 1994), p. 1001, formula 8.578.1.

Greene, P. L.

P. L. Greene, D. G. Hall, “Properties and diffraction of vector Bessel–Gauss beam,” J. Opt. Soc. Am. A 15, 3020–3027 (1998).
[CrossRef]

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor laser,” Appl. Phys. Lett. 72, 1284–1268 (1998).
[CrossRef]

P. L. Greene, D. G. Hall, “Diffraction characteristics of the azimuthal Bessel–Gauss beam,” J. Opt. Soc. Am. A 13, 962–966 (1996).
[CrossRef]

Guattari, G.

C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[CrossRef]

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Hall, D. G.

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor laser,” Appl. Phys. Lett. 72, 1284–1268 (1998).
[CrossRef]

P. L. Greene, D. G. Hall, “Properties and diffraction of vector Bessel–Gauss beam,” J. Opt. Soc. Am. A 15, 3020–3027 (1998).
[CrossRef]

P. L. Greene, D. G. Hall, “Diffraction characteristics of the azimuthal Bessel–Gauss beam,” J. Opt. Soc. Am. A 13, 962–966 (1996).
[CrossRef]

Hall, G. D.

Harris, M.

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral-interference-fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

Hasman, E.

R. Oron, N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182, 205–208 (2000).
[CrossRef]

Heckenberg, N. R.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, M. J. Wegener, “Laser-beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

Herman, R. M.

Hicknann, J. M.

Hill, C. A.

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral-interference-fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

Horak, R.

Z. Bouchal, R. Horak, J. Wagner, “Propagation-invariant electromagnetic fields: theory and experiment,” J. Mod. Opt. 43, 1905–1920 (1996).
[CrossRef]

Huang, W.

B. Lu, W. Huang, “Three-dimensional intensity distribution of focused Bessel–Gauss beams,” J. Mod. Opt. 43, 509–515 (1996).
[CrossRef]

B. Lu, W. Huang, “Focal shift in unaperturated Bessel–Gauss beams,” Opt. Commun. 109, 43–46 (1994).
[CrossRef]

Jordan, R. H.

Kim, G. H.

Kimura, W. D.

King, O.

Kristensen, M.

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

Kuhn, R.

J. Arlt, R. Kuhn, K. Dholakia, “Spatial transformation of Laguerre–Gaussian laser modes,” J. Mod. Opt. 48, 783–787 (2001).

Labate, A.

R. Meucci, A. Labate, M. Ciofini, “Polarization properties of low-order Laguerre–Gauss modes in a CO2 laser,” Quantum Semiclassic. Opt. 9, L31–L35 (1997).
[CrossRef]

Lu, B.

B. Lu, W. Huang, “Three-dimensional intensity distribution of focused Bessel–Gauss beams,” J. Mod. Opt. 43, 509–515 (1996).
[CrossRef]

B. Lu, W. Huang, “Focal shift in unaperturated Bessel–Gauss beams,” Opt. Commun. 109, 43–46 (1994).
[CrossRef]

Marichev, O. I.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon & Breach Science, New York, 1986), Vol. 2, p. 223, formula 2.12.39.3.

McDuff, R.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, M. J. Wegener, “Laser-beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

Meneses-Nava, M. A.

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

S. Chavez-Cerda, E. Tepichin, M. A. Meneses-Nava, G. Ramirez, “Experimental observation of interfering Bessel beams” Opt. Express 13, 524–529 (1998).
[CrossRef]

Meucci, R.

R. Meucci, A. Labate, M. Ciofini, “Polarization properties of low-order Laguerre–Gauss modes in a CO2 laser,” Quantum Semiclassic. Opt. 9, L31–L35 (1997).
[CrossRef]

Miceli, J. J.

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 54, 1499–1501 (1987).
[CrossRef]

Olson, C.

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor laser,” Appl. Phys. Lett. 72, 1284–1268 (1998).
[CrossRef]

Oron, R.

R. Oron, N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182, 205–208 (2000).
[CrossRef]

Padgett, M. J.

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998).
[CrossRef]

N. B. Simpson, L. Allen, M. J. Padgett, “Optical tweezers and optical spanners with Laguerre–Gaussian modes,” J. Mod. Opt. 43, 2485–2491 (1996).
[CrossRef]

M. J. Padgett, L. Allen, “The Poynting vector in Laguerre–Gaussian laser modes,” Opt. Commun. 121, 36–40 (1995).
[CrossRef]

Padovani, C.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Palma, C.

A. Ciattoni, G. Cincotti, C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19, 792–796 (2002).
[CrossRef]

G. Cincotti, A. Ciattoni, C. Palma, “Hermite–Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 1517–1524 (2002).
[CrossRef]

C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[CrossRef]

Power, W. L.

M. Babiker, W. L. Power, L. Allen, “Light-induced torque on moving atoms,” Phys. Rev. Lett. 73, 1239–1242 (1994).
[CrossRef] [PubMed]

L. Allen, M. Babiker, W. L. Power, “Azimuthal doppler-shift in laser beams with orbital angular-momentum,” Opt. Commun. 112, 141–144 (1994).
[CrossRef]

Prudnikov, A. P.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon & Breach Science, New York, 1986), Vol. 2, p. 223, formula 2.12.39.3.

Ramirez, G.

S. Chavez-Cerda, E. Tepichin, M. A. Meneses-Nava, G. Ramirez, “Experimental observation of interfering Bessel beams” Opt. Express 13, 524–529 (1998).
[CrossRef]

Rishton, S.

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor laser,” Appl. Phys. Lett. 72, 1284–1268 (1998).
[CrossRef]

R. H. Jordan, G. D. Hall, O. King, G. Wicks, S. Rishton, “Lasing behavior of circular grating surface-emitting semiconductor lasers,” J. Opt. Soc. Am. B 14, 449–453 (1997).
[CrossRef]

Rubinsztein-Dunlop, H.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, M. J. Wegener, “Laser-beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

Ruschin, S.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, San Diego, Calif., 1994), p. 1001, formula 8.578.1.

Saghafi, S.

S. Saghafi, C. J. R. Sheppard, “Near field and far field of elegant Hermite–Gaussian and Laguerre–Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
[CrossRef]

Santarsiero, M.

R. Borghi, M. Santarsiero, “M2 factor of Bessel–Gauss beams,” Opt. Lett. 22, 262–264 (1997).
[CrossRef] [PubMed]

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

R. Borghi, F. Gori, M. Santarsiero, “Optimization of Laguerre–Gauss truncated series,” Opt. Commun. 125, 197–203 (1996).
[CrossRef]

C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[CrossRef]

M. Santarsiero, “Propagation of generalized Bessel–Gauss beams through ABCD optical systems,” Opt. Commun. 132, 1–7 (1996).
[CrossRef]

Schettini, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Schirripa Spagnolo, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Sheppard, C. J. R.

S. Saghafi, C. J. R. Sheppard, “Near field and far field of elegant Hermite–Gaussian and Laguerre–Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Simpson, N. B.

N. B. Simpson, L. Allen, M. J. Padgett, “Optical tweezers and optical spanners with Laguerre–Gaussian modes,” J. Mod. Opt. 43, 2485–2491 (1996).
[CrossRef]

Smith, C. P.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, M. J. Wegener, “Laser-beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Tamm, C.

Tepichin, E.

S. Chavez-Cerda, E. Tepichin, M. A. Meneses-Nava, G. Ramirez, “Experimental observation of interfering Bessel beams” Opt. Express 13, 524–529 (1998).
[CrossRef]

Tidwell, S. C.

Tovar, A. A.

Turunen, J.

van der Veen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Vasara, A.

Vaughan, J. M.

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral-interference-fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

Wagner, J.

Z. Bouchal, R. Horak, J. Wagner, “Propagation-invariant electromagnetic fields: theory and experiment,” J. Mod. Opt. 43, 1905–1920 (1996).
[CrossRef]

Wegener, M. J.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, M. J. Wegener, “Laser-beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

Weiss, C. O.

Wicks, G.

Wicks, G. W.

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor laser,” Appl. Phys. Lett. 72, 1284–1268 (1998).
[CrossRef]

Wiggins, T. A.

Woerdman, J. P.

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

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor laser,” Appl. Phys. Lett. 72, 1284–1268 (1998).
[CrossRef]

IEEE J. Quantum Electron. (1)

G. Cincotti, A. Ciattoni, C. Palma, “Hermite–Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 1517–1524 (2002).
[CrossRef]

J. Mod. Opt. (8)

S. Saghafi, C. J. R. Sheppard, “Near field and far field of elegant Hermite–Gaussian and Laguerre–Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
[CrossRef]

Z. Bouchal, R. Horak, J. Wagner, “Propagation-invariant electromagnetic fields: theory and experiment,” J. Mod. Opt. 43, 1905–1920 (1996).
[CrossRef]

B. Lu, W. Huang, “Three-dimensional intensity distribution of focused Bessel–Gauss beams,” J. Mod. Opt. 43, 509–515 (1996).
[CrossRef]

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[CrossRef]

J. Arlt, R. Kuhn, K. Dholakia, “Spatial transformation of Laguerre–Gaussian laser modes,” J. Mod. Opt. 48, 783–787 (2001).

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998).
[CrossRef]

N. B. Simpson, L. Allen, M. J. Padgett, “Optical tweezers and optical spanners with Laguerre–Gaussian modes,” J. Mod. Opt. 43, 2485–2491 (1996).
[CrossRef]

J. Opt. Soc. Am. A (9)

J. Opt. Soc. Am. B (2)

Opt. Commun. (10)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

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

M. J. Padgett, L. Allen, “The Poynting vector in Laguerre–Gaussian laser modes,” Opt. Commun. 121, 36–40 (1995).
[CrossRef]

R. Borghi, F. Gori, M. Santarsiero, “Optimization of Laguerre–Gauss truncated series,” Opt. Commun. 125, 197–203 (1996).
[CrossRef]

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral-interference-fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

R. Oron, N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182, 205–208 (2000).
[CrossRef]

L. Allen, M. Babiker, W. L. Power, “Azimuthal doppler-shift in laser beams with orbital angular-momentum,” Opt. Commun. 112, 141–144 (1994).
[CrossRef]

M. Santarsiero, “Propagation of generalized Bessel–Gauss beams through ABCD optical systems,” Opt. Commun. 132, 1–7 (1996).
[CrossRef]

B. Lu, W. Huang, “Focal shift in unaperturated Bessel–Gauss beams,” Opt. Commun. 109, 43–46 (1994).
[CrossRef]

Opt. Express (1)

S. Chavez-Cerda, E. Tepichin, M. A. Meneses-Nava, G. Ramirez, “Experimental observation of interfering Bessel beams” Opt. Express 13, 524–529 (1998).
[CrossRef]

Opt. Lett. (3)

Opt. Quantum Electron. (1)

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, M. J. Wegener, “Laser-beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

Phys. Rev. Lett. (2)

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 54, 1499–1501 (1987).
[CrossRef]

M. Babiker, W. L. Power, L. Allen, “Light-induced torque on moving atoms,” Phys. Rev. Lett. 73, 1239–1242 (1994).
[CrossRef] [PubMed]

Quantum Semiclassic. Opt. (1)

R. Meucci, A. Labate, M. Ciofini, “Polarization properties of low-order Laguerre–Gauss modes in a CO2 laser,” Quantum Semiclassic. Opt. 9, L31–L35 (1997).
[CrossRef]

Other (7)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon & Breach Science, New York, 1986), Vol. 2, p. 223, formula 2.12.39.3.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

See Ref. 40, p. 51, formula 1.14.1.8.

See Ref. 40, p. 659, formula 5.7.5.1.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, San Diego, Calif., 1994), p. 1001, formula 8.578.1.

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Figures (2)

Fig. 1
Fig. 1

Profiles of the electric field originated by an input LG1 mode. (a) x component at the plane z=0, (b) x component at the plane 2z/k0nov02=2, and (c) y component at the plane 2z/k0nov02=2. The x and y variables are normalized with respect to the waist spotsize v0, and we set ne/no=1.1097 (rutile crystal).

Fig. 2
Fig. 2

Profiles of the electric field originated by an input BG beam. (a) x component at the plane z=0, (b) x component at the plane 2z/k0nov02=2, and (c) y component at the plane 2z/k0nov02=2. The x and y variables are normalized with respect to the waist spotsize v0, and we set βv02/2=2.

Equations (57)

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

E(r, ϕ, 0)=E(r, 0)eˆx.
Ex(r, ϕ, z)=exp(ik0noz){Ao(0)(r, z)+Ae(0)(r, z)+cos 2ϕ[Ao(2)(r, z)-Ae(2)(r, z)]},
Ey(r, ϕ, z)=exp(ik0noz)sin 2ϕ[Ao(2)(r, z)-Ae(2)(r, z)],
Ao(m)(r, z)=π0dk k exp-iz2k0nok2Jm(kr)E˜(k),
Ae(m)(r, z)=π0dk k exp-inoz2k0ne2k2Jm(kr)E˜(k)
m=0, 2.
E˜(k)=12π0drrJ0(kr)E(r, 0).
A(m)(r, z)=π0dkk exp-iz2k0nk2Jm(kr)E˜(k)
m=0, 2,
A(m)(r, z)=120drrE(r, 0)0dkk exp-iz2k0nk2×J0(kr)Jm(kr),m=0, 2.
A(0)(r, z)=120drrE(r, 0)0dkk exp-iz2k0nk2×J0(kr)J0(kr),
0dξ exp(-αξ2)Jν(γξ)Jν(δξ)=12αexp-γ2+δ24αIνγδ2α,
Iν(ξ)=(-i)νJν(iξ).
A(0)(r, z)=-ik0n2z0drrE(r, 0)×exp-ik0nr2+r2zJ0k0nrrz,
A(2)(r, z)=120drrE(r, 0)0dkk×exp-iz2k0nk2J0(kr)J2(kr),
ddξ[ξνJν(ξ)]=ξνJν-1(ξ),
r[r2A(2)(r, z)]=-r2rA(0)(r, z)
A(2)(r, z)=-1r20rdξξ2ξA(0)(ξ, z)=2r20rdξξA(0)(ξ, z)-A(0)(r, z).
E(r, 0)=E¯Lp2r2v02exp-r2v02,
Lp(q)(t)=s=0pp+qp-s(-t)ss!.
Ao(0)(r, z)=E¯2v0vo(z)exp[-i(2p+1)Φo(z)]Lp2r2vo2(z)×exp-r2Qo(z)=E¯2v02p+2vo2p(z)Qo2p+1(z)Lp2r2vo2(z)exp-r2Qo(z),
Ae(0)(r, z)=E¯2v0ve(z)exp[-i(2p+1)Φe(z)]Lp2r2ve2(z)×exp-r2Qe(z)=E¯2v02p+2ve2p(z)Qe2p+1(z)Lp2r2ve2(z)exp-r2Qe(z),
v0(z)=v01+2zk0nov0221/2,
ve(z)=v01+2nozk0ne2v0221/2,
Qo(z)=v02+i2zk0no,Qe(z)=v02+i2nozk0ne2,
Φo(z)=arctan2zk0nov02,
Φe(z)=arctan2nozk0ne2v02.
Ao(2)(r, z)=E¯2r2v02p+2s=0ppp-s(-2)ss!vo2p-2s(z)Qo2p-s(z)×γs+1,r2Qo(z)-Ao(0)(r, z),
Ae(2)(r, z)=E¯2r2v02p+2s=0ppp-s(-2)ss!ve2p-2s(z)Qe2p-s(z)×γs+1,r2Qe(z)-Ae(0)(r, z),
γ(α, β)=0βξα-1 exp(-ξ)dξ.
Ex(r, ϕ, z)=E¯exp(ik0noz)v02p+2sin2 ϕvo2p(z)Qo2p+1(z)Lp2r2vo2(z)×exp-r2Qo(z)+cos2 ϕve2p(z)Qe2p+1(z)Lp2r2ve2(z)×exp-r2Qe(z)+cos 2ϕ2r2s=0ppp-s(-2)ss!×vo2p-2s(z)Qo2p-s(z)γs+1,r2Qo(z)-ve2p-2s(z)Qe2p-s(z)×γs+1,r2Qe(z),
Ey(r, ϕ, z)=-E¯2exp(ik0noz)v02p+2sin 2ϕvo2p(z)Qo2p+1(z)Lp2r2vo2(z)×exp-r2Qo(z)-ve2p(z)Qe2p+1(z)Lp2r2ve2(z)×exp-r2Qe(z)-1r2s=0ppp-s(-2)ss!vo2p-2s(z)Qo2p-s(z)×γs+1,r2Qo(z)-ve2p-2s(z)Qe2p-s(z)γs+1,r2Qe(z).
Ex(r, ϕ, z)=E¯exp(ik0noz)v02sin2 ϕexp-r2Qo(z)Qo(z)+cos2 ϕexp-r2Qe(z)Qe(z)-cos 2ϕexp-r2Qo(z)-exp-r2Qe(z)2r2,
Ey(r, ϕ, z)=-E¯2exp(ik0noz)v02sin 2ϕexp-r2Qo(z)Qo(z)-exp-r2Qe(z)Qe(z)+exp-r2Qo(z)-exp-r2Qe(z)r2,
E(r, 0)=E¯Lpr2v02exp-r2v02.
Ao(0)(r, z)=E¯2v02Qo(z)p+1Lpr2Qo(z)exp-r2Qo(z),
Ae(0)(r, z)=E¯2v02Qe(z)p+1Lpr2Qe(z)exp-r2Qe(z),
ξα exp(-ξ)Lp(α)(ξ)dξ=ξα+1pexp(-ξ)Lp-1(α+1)(ξ),
Ao(2)(r, z)=E¯2v02Qo(z)p+11pLp-1(1)r2Qo(z)-Lpr2Qo(z)exp-r2Qo(z),
Ae(2)(r, z)=E¯2v02Qe(z)p+11pLp-1(1)r2Qe(z)-Lpr2Qe(z)exp-r2Qe(z).
Ex(r, ϕ, z)=E¯exp(ik0noz)v02p+2exp-r2Qo(z)Qop+1(z)×sin2 ϕLpr2Qo(z)+cos 2ϕ2pLp-1(1)r2Qo(z)+exp-r2Qc(z)Qep+1(z)cos2 ϕLpr2Qe(z)-cos 2ϕ2pLp-1(1)r2Qe(z)
Ey(r, ϕ, z)=-E¯2exp(ik0noz)v02p+2sin 2ϕexp-r2Qo(z)Qop+1(z)Lpr2Qo(z)-1pLp-1(1)r2Qo(z)-exp-r2Qe(z)Qep+1(z)Lpr2Qe(z)-1pLp-1(1)r2Qe(z).
E(r, 0)=E¯J0(βr)exp(-r2/v02),
Ao(0)(r, z)=E¯2v0vo(z)exp-iβ2z2k0no+Φo(z)×J0βv02Qo(z)rexp-r2+β2z2k02no2Qo(z)=E¯2v02Qo(z)exp-iβ2v022k0noQo(z)z×J0βv02Qo(z)rexp-r2Qo(z),
Ae(0)(r, z)=E¯2v0ve(z)exp-iβ2noz2k0ne2+Φe(z)×J0βv02Qe(z)rexp-r2+β2no2z2k02ne4Qe(z)=E¯2v02Qe(z)exp-iβ2nov022k0ne2Qe(z)z×J0βv02Qe(z)rexp-r2Qe(z).
0ζdξξν exp-ηξ22ζJν-1(ξ)=iζηνexp-ηζ2×[Uν(-iηζ, ζ)+iUν+1(-iηζ, ζ)]
=ζν exp-ηζ2k=0ηkJk+ν(ζ),
Uν(ξ, ψ)=m=0(-1)mξψ2m+νJ2m+ν(ψ),
Ao(2)(r, z)=E¯2r2v02 exp-iβ2v022k0noQo(z)zexp-r2Qo(z)×k=12rβv02kJkβv02Qo(z)r-Ao(0)(r, z),
Ae(2)(r, z)=E¯2r2v02 exp-iβ2nov022k0ne2Qe(z)zexp-r2Qe(z)×k=12rβv02kJkβv02Qe(z)r-Ae(0)(r, z).
Ex(r, ϕ, z)=E¯exp(ik0noz)v02exp-iβ2v022k0noQo(z)z×exp-r2Qo(z)sin2 ϕQo(z)J0βv02Qo(z)r+cos 2ϕ2r2k=12rβv02kJkβv02Qo(z)r+exp-iβ2nov022k0ne2Qe(z)zexp-r2Qe(z)×cos2 ϕQe(z)J0βv02Qe(z)r-cos 2ϕ2r2k=12rβv02kJkβv02Qe(z)r,
Ey(r, ϕ, z)=-E¯2exp(ik0noz)v02sin 2ϕ×exp-iβ2v022k0noQo(z)zexp-r2Qo(z)×1Qo(z)J0βv02Qo(z)r-1r2k=12rβv02k×Jkβv02Qo(z)r-exp-iβ2nov022k0ne2Qe(z)z×exp-r2Qe(z)1Qe(z)J0βv02Qe(z)r-1r2k=12rβv02kJkβv02Qe(z)r.
Ex(r, ϕ, z)=E¯exp(ik0noz)exp-iβ22k0noz×sin2 ϕJ0(βr)+cos 2ϕrβJ1(βr)+exp-iβ2no2k0ne2zcos2 ϕJ0(βr)-cos 2ϕrβJ1(βr),
Ey(r, ϕ, z)=-E¯2exp(ik0noz)sin 2ϕexp-iβ22k0noz×J0(βr)-2rβJ1(βr)-exp-iβ2no2k0ne2z×J0(βr)-2βrJ1(βr)=E¯2exp(ik0noz)sin 2ϕexp-iβ22k0noz-exp-iβ2no2k0ne2zJ2(βr).
|Ex(r, ϕ, z)|2=|E¯|2cos2πzz¯J02(βr)+cos2 2ϕ sin2πzz¯J22(βr),
|Ey(r, ϕ, z)|2=|E¯|2 sin2πzz¯sin2 2ϕJ02(βr),
z¯=4πk0noβ2ne2no2-ne2.

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