Abstract

We find a two-parameter family of astigmatic elliptical Gaussian (AEG) optical vortices, which are free space modes up to scale and rotation. We calculate total normalized orbital angular momentum of AEG vortices, which can be an integer, fractional and zero, and which is equal to the algebraic sum of two terms reflecting the contribution of the vortex and astigmatic components of the light field. In any transverse plane, such a beam has an isolated n-fold degenerate intensity null on the optical axis (an optical vortex) embedded into an elliptical Gaussian beam. In addition to the quadratic elliptical phase, a beam has the phase of a cylindrical lens rotated by an angle of 45 degrees with respect to the principal axes of the ellipse of the Gaussian beam intensity distribution. The degenerated central intensity null in these beams does not split it into n spatially separated intensity nulls, as is usually assumed for elliptical astigmatic beams.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Astigmatic transforms of an optical vortex for measurement of its topological charge,” Appl. Opt. 56(14), 4095–4104 (2017).
    [Crossref] [PubMed]
  2. A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams,” J. Opt. Soc. Am. A 20(8), 1635–1643 (2003).
    [Crossref] [PubMed]
  3. V. Denisenko, V. Shvedov, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, A. Volyar, M. Soskin, and Y. S. Kivshar, “Determination of topological charges of polychromatic optical vortices,” Opt. Express 17(26), 23374–23379 (2009).
    [Crossref] [PubMed]
  4. S. N. Alperin, R. D. Niederriter, J. T. Gopinath, and M. E. Siemens, “Quantitative measurement of the orbital angular momentum of light with a single, stationary lens,” Opt. Lett. 41(21), 5019–5022 (2016).
    [Crossref] [PubMed]
  5. S. N. Alperin and M. E. Siemens, “Angular momentum of topologically structured darkness,” Phys. Rev. Lett. 119(20), 203902 (2017).
    [Crossref] [PubMed]
  6. E. G. Abramochkin and V. G. Volostnikov, “Beam transformations and transformed beams,” Opt. Commun. 83(1-2), 123–135 (1991).
    [Crossref]
  7. J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian Beams with very high orbital angular momentum,” Opt. Commun. 144(4–6), 210–213 (1997).
    [Crossref]
  8. V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Astigmatic laser beams with a large orbital angular momentum,” Opt. Express 26(1), 141–156 (2018).
    [Crossref] [PubMed]
  9. S. Chávez-Cerda, J. C. Gutiérrez-Vega, and G. H. New, “Elliptic vortices of electromagnetic wave fields,” Opt. Lett. 26(22), 1803–1805 (2001).
    [Crossref] [PubMed]
  10. M. A. Bandres and J. C. Gutiérrez-Vega, “Ince-Gaussian modes of the paraxial wave equation and stable resonators,” J. Opt. Soc. Am. A 21(5), 873–880 (2004).
    [Crossref] [PubMed]
  11. V. V. Kotlyar and A. A. Kovalev, “Hermite-Gaussian modal laser beams with orbital angular momentum,” J. Opt. Soc. Am. A 31(2), 274–282 (2014).
    [Crossref] [PubMed]
  12. V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Vortex Hermite-Gaussian laser beams,” Opt. Lett. 40(5), 701–704 (2015).
    [Crossref] [PubMed]
  13. A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241(4-6), 237–247 (2004).
    [Crossref]
  14. S. Maji and M. M. Brundavanam, “Controlled noncanonical vortices from higher-order fractional screw dislocations,” Opt. Lett. 42(12), 2322–2325 (2017).
    [Crossref] [PubMed]
  15. A. V. Volyar and Y. A. Egorov, “Super pulses of orbital angular momentum in fractional-order spiroid vortex beams,” Opt. Lett. 43(1), 74–77 (2018).
    [Crossref] [PubMed]
  16. L. A. Melo, A. J. Jesus-Silva, S. Chávez-Cerda, P. H. S. Ribeiro, and W. C. Soares, “Direct measurement of the topoloigical charge in elliptical beams using diffraction by a triangular aperture,” Sci. Rep. 8(1), 6370 (2018).
    [Crossref] [PubMed]

2018 (3)

2017 (3)

2016 (1)

2015 (1)

2014 (1)

2009 (1)

2004 (2)

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241(4-6), 237–247 (2004).
[Crossref]

M. A. Bandres and J. C. Gutiérrez-Vega, “Ince-Gaussian modes of the paraxial wave equation and stable resonators,” J. Opt. Soc. Am. A 21(5), 873–880 (2004).
[Crossref] [PubMed]

2003 (1)

2001 (1)

1997 (1)

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian Beams with very high orbital angular momentum,” Opt. Commun. 144(4–6), 210–213 (1997).
[Crossref]

1991 (1)

E. G. Abramochkin and V. G. Volostnikov, “Beam transformations and transformed beams,” Opt. Commun. 83(1-2), 123–135 (1991).
[Crossref]

Abramochkin, E. G.

E. G. Abramochkin and V. G. Volostnikov, “Beam transformations and transformed beams,” Opt. Commun. 83(1-2), 123–135 (1991).
[Crossref]

Allen, L.

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian Beams with very high orbital angular momentum,” Opt. Commun. 144(4–6), 210–213 (1997).
[Crossref]

Alperin, S. N.

Bandres, M. A.

Bekshaev, A. Ya.

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241(4-6), 237–247 (2004).
[Crossref]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams,” J. Opt. Soc. Am. A 20(8), 1635–1643 (2003).
[Crossref] [PubMed]

Brundavanam, M. M.

Chávez-Cerda, S.

L. A. Melo, A. J. Jesus-Silva, S. Chávez-Cerda, P. H. S. Ribeiro, and W. C. Soares, “Direct measurement of the topoloigical charge in elliptical beams using diffraction by a triangular aperture,” Sci. Rep. 8(1), 6370 (2018).
[Crossref] [PubMed]

S. Chávez-Cerda, J. C. Gutiérrez-Vega, and G. H. New, “Elliptic vortices of electromagnetic wave fields,” Opt. Lett. 26(22), 1803–1805 (2001).
[Crossref] [PubMed]

Courtial, J.

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian Beams with very high orbital angular momentum,” Opt. Commun. 144(4–6), 210–213 (1997).
[Crossref]

Denisenko, V.

Desyatnikov, A. S.

Dholakia, K.

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian Beams with very high orbital angular momentum,” Opt. Commun. 144(4–6), 210–213 (1997).
[Crossref]

Egorov, Y. A.

Gopinath, J. T.

Gutiérrez-Vega, J. C.

Jesus-Silva, A. J.

L. A. Melo, A. J. Jesus-Silva, S. Chávez-Cerda, P. H. S. Ribeiro, and W. C. Soares, “Direct measurement of the topoloigical charge in elliptical beams using diffraction by a triangular aperture,” Sci. Rep. 8(1), 6370 (2018).
[Crossref] [PubMed]

Kivshar, Y. S.

Kotlyar, V. V.

Kovalev, A. A.

Krolikowski, W.

Maji, S.

Melo, L. A.

L. A. Melo, A. J. Jesus-Silva, S. Chávez-Cerda, P. H. S. Ribeiro, and W. C. Soares, “Direct measurement of the topoloigical charge in elliptical beams using diffraction by a triangular aperture,” Sci. Rep. 8(1), 6370 (2018).
[Crossref] [PubMed]

Neshev, D. N.

New, G. H.

Niederriter, R. D.

Padgett, M. J.

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian Beams with very high orbital angular momentum,” Opt. Commun. 144(4–6), 210–213 (1997).
[Crossref]

Porfirev, A. P.

Ribeiro, P. H. S.

L. A. Melo, A. J. Jesus-Silva, S. Chávez-Cerda, P. H. S. Ribeiro, and W. C. Soares, “Direct measurement of the topoloigical charge in elliptical beams using diffraction by a triangular aperture,” Sci. Rep. 8(1), 6370 (2018).
[Crossref] [PubMed]

Shvedov, V.

Siemens, M. E.

Soares, W. C.

L. A. Melo, A. J. Jesus-Silva, S. Chávez-Cerda, P. H. S. Ribeiro, and W. C. Soares, “Direct measurement of the topoloigical charge in elliptical beams using diffraction by a triangular aperture,” Sci. Rep. 8(1), 6370 (2018).
[Crossref] [PubMed]

Soskin, M.

Soskin, M. S.

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241(4-6), 237–247 (2004).
[Crossref]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams,” J. Opt. Soc. Am. A 20(8), 1635–1643 (2003).
[Crossref] [PubMed]

Vasnetsov, M. V.

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241(4-6), 237–247 (2004).
[Crossref]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams,” J. Opt. Soc. Am. A 20(8), 1635–1643 (2003).
[Crossref] [PubMed]

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, “Beam transformations and transformed beams,” Opt. Commun. 83(1-2), 123–135 (1991).
[Crossref]

Volyar, A.

Volyar, A. V.

Appl. Opt. (1)

J. Opt. Soc. Am. A (3)

Opt. Commun. (3)

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241(4-6), 237–247 (2004).
[Crossref]

E. G. Abramochkin and V. G. Volostnikov, “Beam transformations and transformed beams,” Opt. Commun. 83(1-2), 123–135 (1991).
[Crossref]

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian Beams with very high orbital angular momentum,” Opt. Commun. 144(4–6), 210–213 (1997).
[Crossref]

Opt. Express (2)

Opt. Lett. (5)

Phys. Rev. Lett. (1)

S. N. Alperin and M. E. Siemens, “Angular momentum of topologically structured darkness,” Phys. Rev. Lett. 119(20), 203902 (2017).
[Crossref] [PubMed]

Sci. Rep. (1)

L. A. Melo, A. J. Jesus-Silva, S. Chávez-Cerda, P. H. S. Ribeiro, and W. C. Soares, “Direct measurement of the topoloigical charge in elliptical beams using diffraction by a triangular aperture,” Sci. Rep. 8(1), 6370 (2018).
[Crossref] [PubMed]

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Figures (4)

Fig. 1
Fig. 1 Intensity (lines 1, 3, 5, 7) and phase (lines 2, 4, 6, 8) distributions calculated by Eq. (18) for AEG vortices with ellipticity parameter 1:3 and for the topological charges n = 0, 1, 3, 5, 7 at different propagation distances: z = 0, z = f = 100 mm, z = 150 mm and z = 2f = 200 mm. The frame size is 2300 × 2300 μm.
Fig. 2
Fig. 2 Optical setup for studying focusing of elliptical optical vortices by a cylindrical lens: SLM is the spatial light modulator HOLOEYE PLUTO-VIS, M1 and M2 are the mirrors, L1 and L2 are the spherical lenses (f1 = 350 mm, f2 = 150 mm), CL1 is the cylindrical lens (f3 = 100 mm), CMOS is a camera ToupCam U3CMOS08500KPA.
Fig. 3
Fig. 3 Obtained intensity distributions of AEG-vortices with the ellipticity parameter 1: 3 and the topological charge n = 0, 1, 3, 5 and 7 at different distances from the plane of the cylindrical lens: z = 0, 100 (focal length), 150 and 200 mm (double focal length). The frame size is 2300 × 2300 μm.
Fig. 4
Fig. 4 Interferograms of AEG vortices obtained at a distance z = 2f [Fig. 3, line 4] for n = 1 (a) and n = 3 (b).

Equations (28)

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E n (x,y)= w n ( x+iy ) n exp( x 2 w x 2 y 2 w y 2 )× ×exp( ik x 2 cos 2 α 2f ik y 2 sin 2 α 2f ikxysin2α 2f ).
E n (u,v,z)= ik 2πz exp( ik u 2 2z + ik v 2 2z ) w n ( x+iy ) n exp( x 2 w x 2 y 2 w y 2 ) × ×exp( ik x 2 cos 2 α 2f ik y 2 sin 2 α 2f ikxysin2α 2f )× ×exp( ik x 2 2z + ik y 2 2z ik(xu+yv) z )dxdy.
E n (u,v,z=2f)= ik 4πf exp( ik u 2 4f + ik v 2 4f )× × w n ( x+iy ) n exp[ x 2 w x 2 y 2 w y 2 ikxy 2f ik(xu+yv) 2f ]dxdy .
E n ( x,y,z=0 )= ( x+iy ) n exp( γ 2 x 2 β 2 y 2 i x 2 i y 2 2ixy ).
E n ( u,v,z=2f )= i π exp( i u 2 +i v 2 ) ( x+iy ) n × ×exp( γ 2 x 2 β 2 y 2 2ixy2ixu2iyv )dxdy.
E n ( u,v,z=2f )=exp( i u 2 +i v 2 ) ( i ) n+1 2 n 1+ γ 2 β 2 ( 2 γ 2 + β 2 1+ γ 2 β 2 ) n/2 × × H n [ ( 1+ β 2 )u+i( γ 2 1 )v 1+ γ 2 β 2 2 γ 2 + β 2 ]exp( β 2 u 2 + γ 2 v 2 2iuv 1+ γ 2 β 2 ).
J z =Im E ¯ (x,y)( x E(x,y) y y E(x,y) x )dxdy ,
W= E ¯ (x,y)E(x,y)dxdy ,
J z W =n+isgn( w y w x ) k w x w y sin2α 4f P n+1 1 ( ξ ) P n 0 ( ξ ) ,
P n m ( ξ )= ( 1 ) m 2 n n! ( 1 ξ 2 ) m/2 d n+m d ξ n+m ( ξ 2 1 ) n .
J z W =n+i sgn( γβ ) βγ P n+1 1 ( ξ ) P n 0 ( ξ ) ,
γ 2 β 2 =2/3 .
E n ( u,v,z=2f )= ( i 1+ β 2 ) n+1 ( u+iv ) n × ×exp[ i u 2 +i v 2 β 2 u 2 +( β 2 +2 ) v 2 2iuv ( 1+ β 2 ) 2 ],
E n ( x,y,z=0 )= ( x+iy ) n exp[ ( β 2 +2 ) x 2 β 2 y 2 i x 2 i y 2 2ixy ].
E n ( u,v,z=2f )= ( i 2 ) n+1 ( u+iv ) n exp( i u 2 +i v 2 u 2 +3 v 2 2iuv 4 ).
( x+iy ) n exp[ 3 x 2 y 2 2ixyi( u ¯ x+ v ¯ y ) ]dxdy = = π 2 ( i 4 ) n ( u ¯ +i v ¯ ) n exp( u ¯ 2 +3 v ¯ 2 2i u ¯ v ¯ 16 ).
J z W =n+ i β β 2 +2 P n+1 1 ( ξ ) P n 0 ( ξ ) ,
E n ( x,y,z )= ( i ) n+1 z 0 ( u+iv ) n z( 1+ b y ) exp[ ik 2z ( x 2 + y 2 ) b y u 2 b x v 2 +2iuv ],
u= kf z( 1+ b y ) x,v= kf z( 1+ b y ) y, b x,y = 4f k ( 1 w x,y 2 + ik 4f ik 2z ).
1 w x 2 = 1 w y 2 + k 2f .
[ 2ik z + 2 x 2 + 2 y 2 ] E n ( x,y,z )=0.
E n ( x,y,z=2f )= ( i ) n+1 ( 1+ 2f z 0y ) n1 ( 1 2 k f ) n ( x+iy ) n × ×exp{ ik 4f ( x 2 + y 2 )[ k x 2 2 z 0y + k y 2 2f ( 1+ f z 0y ) ikxy 2f ] ( 1+ 2f z 0y ) 2 },
E n ( x,y,z= z 0y =2f )= ( i 2 ) n+1 ( 1 2 k f ) n ( x+iy ) n × ×exp[ ik 4f ( x 2 + y 2 ) k 16f ( x 2 +3 y 2 2ixy ) ].
L= w y (z) n ,
w y (z)= (kf w 0y ) 1 [ z 2 ( z 0y +2f) 2 + z 0y 2 (z2f) 2 ] 1/2 .
J z W =n+ 4i 3 P n+1 1 ( 5/3 ) P n 0 ( 5/3 ) .
tgθ=( 2f z y +1 ) ( 2f z 1 ) 1 .
θ=arctan( 2f z y +1 ).

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