Abstract

We apply the Collins-Huygens integral to analytically describe propagation of a doughnut beam generated by a spiral phase plate. Measured beam profiles in free space and through an ABCD-lens system illustrate excellent agreement with theory. Applications range from the creation of optical beams with angular momentum to microscopy to trapping neutral atoms. The method extends to other beam shaping components, too.

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References

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  1. T. A. Klar and S. Hell, “Subdiffraction resolution in far-field fluorescence microscopy,” Opt. Lett. 24, 954 (1999)
    [CrossRef]
  2. T. Kuga, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. 78, 4713 (1997)
    [CrossRef]
  3. A. Kaplan, N. Friedman, and N. Davidson, “Optimized single-beam dark optical trap,” J. Opt. Soc. Am. B 19, 1233 (2002)
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  7. M.R. Dennis, K. O’Holleran, and M.J. Padgettin “Progress in Optics,”  53, 293–363 (2009)
  8. V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts”, J. Mod. Opt. 39, 985 (1992)
    [CrossRef]
  9. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221 (1992)
    [CrossRef] [PubMed]
  10. E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123 (1991)
    [CrossRef]
  11. M. Padgett, J. Arlt, N. Simpson, and L. Allen, “Optical tweezers and optical spanners with Laguerre-Gaussian modes,”, Am. J. Phys. 64, 77 (1996)
    [CrossRef]
  12. S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147 (1992)
    [CrossRef]
  13. M.W. Beijersbergen and et al., “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321 (1994)
    [CrossRef]
  14. J. H. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968)
  15. D. Rozas, C. T. Law, and G. A. Swartzlander, “Propagation Dynamics in Optical Vortices,” J. Opt. Soc. Am. B 14, 3054 (1997)
    [CrossRef]
  16. Z. S. Sacks, D. Rozas, and G. A. Swartzlander, “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am. 15, (1998)
  17. S. A. Collins, “Lens-System Diffraction Integral Written in Terms of Matrix Optics,” J. Opt. Soc. Am.60, 1168 (1970); A. Siegman, Lasers, University Science Books (Mill Valley 1986), p. 781
    [CrossRef]
  18. I. S. Gradshteyn and I. M. Rhyzhik, Table of Integrals, Series, and Products, Academic Press (San Diego1979)

2010 (1)

2009 (1)

M.R. Dennis, K. O’Holleran, and M.J. Padgettin “Progress in Optics,”  53, 293–363 (2009)

2003 (1)

2002 (1)

1999 (1)

1998 (1)

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am. 15, (1998)

1997 (2)

T. Kuga, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. 78, 4713 (1997)
[CrossRef]

D. Rozas, C. T. Law, and G. A. Swartzlander, “Propagation Dynamics in Optical Vortices,” J. Opt. Soc. Am. B 14, 3054 (1997)
[CrossRef]

1996 (1)

M. Padgett, J. Arlt, N. Simpson, and L. Allen, “Optical tweezers and optical spanners with Laguerre-Gaussian modes,”, Am. J. Phys. 64, 77 (1996)
[CrossRef]

1994 (1)

M.W. Beijersbergen and et al., “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321 (1994)
[CrossRef]

1992 (3)

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147 (1992)
[CrossRef]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221 (1992)
[CrossRef] [PubMed]

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts”, J. Mod. Opt. 39, 985 (1992)
[CrossRef]

1991 (1)

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123 (1991)
[CrossRef]

Abramochkin, E.

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123 (1991)
[CrossRef]

Allen, L.

M. Padgett, J. Arlt, N. Simpson, and L. Allen, “Optical tweezers and optical spanners with Laguerre-Gaussian modes,”, Am. J. Phys. 64, 77 (1996)
[CrossRef]

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum, Taylor and Francis, (2003)
[CrossRef]

Arlt, J.

M. Padgett, J. Arlt, N. Simpson, and L. Allen, “Optical tweezers and optical spanners with Laguerre-Gaussian modes,”, Am. J. Phys. 64, 77 (1996)
[CrossRef]

Barnett, S. M.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum, Taylor and Francis, (2003)
[CrossRef]

Bazhenov, V. Y.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts”, J. Mod. Opt. 39, 985 (1992)
[CrossRef]

Beijersbergen, M.W.

M.W. Beijersbergen and et al., “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321 (1994)
[CrossRef]

Collins, S. A.

S. A. Collins, “Lens-System Diffraction Integral Written in Terms of Matrix Optics,” J. Opt. Soc. Am.60, 1168 (1970); A. Siegman, Lasers, University Science Books (Mill Valley 1986), p. 781
[CrossRef]

Davidson, N.

Dennis, M.R.

M.R. Dennis, K. O’Holleran, and M.J. Padgettin “Progress in Optics,”  53, 293–363 (2009)

Friedman, N.

Goodman, J. H.

J. H. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968)

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Rhyzhik, Table of Integrals, Series, and Products, Academic Press (San Diego1979)

He, X.

Heckenberg, N. R.

Hell, S.

Kaplan, A.

Khonina, S. N.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147 (1992)
[CrossRef]

Klar, T. A.

Kotlyar, V. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147 (1992)
[CrossRef]

Kuga, T.

T. Kuga, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. 78, 4713 (1997)
[CrossRef]

Law, C. T.

McDuff, R.

O’Holleran, K.

M.R. Dennis, K. O’Holleran, and M.J. Padgettin “Progress in Optics,”  53, 293–363 (2009)

Padgett, M.

M. Padgett, J. Arlt, N. Simpson, and L. Allen, “Optical tweezers and optical spanners with Laguerre-Gaussian modes,”, Am. J. Phys. 64, 77 (1996)
[CrossRef]

Padgett, M. J.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum, Taylor and Francis, (2003)
[CrossRef]

Padgett, M.J.

M.R. Dennis, K. O’Holleran, and M.J. Padgettin “Progress in Optics,”  53, 293–363 (2009)

Rhyzhik, I. M.

I. S. Gradshteyn and I. M. Rhyzhik, Table of Integrals, Series, and Products, Academic Press (San Diego1979)

Rozas, D.

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am. 15, (1998)

D. Rozas, C. T. Law, and G. A. Swartzlander, “Propagation Dynamics in Optical Vortices,” J. Opt. Soc. Am. B 14, 3054 (1997)
[CrossRef]

Sacks, Z. S.

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am. 15, (1998)

Shinkaryev, M. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147 (1992)
[CrossRef]

Simpson, N.

M. Padgett, J. Arlt, N. Simpson, and L. Allen, “Optical tweezers and optical spanners with Laguerre-Gaussian modes,”, Am. J. Phys. 64, 77 (1996)
[CrossRef]

Smith, C. P.

Soifer, V. A.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147 (1992)
[CrossRef]

Soskin, M. S.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts”, J. Mod. Opt. 39, 985 (1992)
[CrossRef]

Swartzlander, G. A.

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am. 15, (1998)

D. Rozas, C. T. Law, and G. A. Swartzlander, “Propagation Dynamics in Optical Vortices,” J. Opt. Soc. Am. B 14, 3054 (1997)
[CrossRef]

Uspleniev, G. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147 (1992)
[CrossRef]

Vasnetsov, M. V.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts”, J. Mod. Opt. 39, 985 (1992)
[CrossRef]

Volostnikov, V.

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123 (1991)
[CrossRef]

Wang, J.

White, A. G.

Xu, P.

Yuan, X.-C.

Zhan, M.

Zhang, D. W.

Am. J. Phys. (1)

M. Padgett, J. Arlt, N. Simpson, and L. Allen, “Optical tweezers and optical spanners with Laguerre-Gaussian modes,”, Am. J. Phys. 64, 77 (1996)
[CrossRef]

J. Mod. Opt. (2)

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147 (1992)
[CrossRef]

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts”, J. Mod. Opt. 39, 985 (1992)
[CrossRef]

J. Opt. Soc. Am. (1)

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am. 15, (1998)

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

Opt. Commun. (2)

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123 (1991)
[CrossRef]

M.W. Beijersbergen and et al., “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321 (1994)
[CrossRef]

Opt. Lett. (4)

Phys. Rev. Lett. (1)

T. Kuga, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. 78, 4713 (1997)
[CrossRef]

Other (5)

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum, Taylor and Francis, (2003)
[CrossRef]

M.R. Dennis, K. O’Holleran, and M.J. Padgettin “Progress in Optics,”  53, 293–363 (2009)

J. H. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968)

S. A. Collins, “Lens-System Diffraction Integral Written in Terms of Matrix Optics,” J. Opt. Soc. Am.60, 1168 (1970); A. Siegman, Lasers, University Science Books (Mill Valley 1986), p. 781
[CrossRef]

I. S. Gradshteyn and I. M. Rhyzhik, Table of Integrals, Series, and Products, Academic Press (San Diego1979)

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Figures (5)

Fig. 1
Fig. 1

A Gaussian beam is converted at position z′ by a spiral phase plate (SPP). We are seeking the field distribution of the optical beam at position z in terms of the ABCD-coefficients of its trajectory.

Fig. 2
Fig. 2

(a): Setup for measuring the beam profile of the propagating doughnut beam in free space. (b): Details of the Galileian telesope used for the test measurement. Abbreviations: AT: attenuator; BS: beam splitter; CCD: beam profile camera; ECDL: external cavity diode laser; FC: fiber coupler; LS: lens system; OI: optical isolator; SMF: single-mode fiber; SPP: spiral phase plate. Parametres: d1–3= 300, 80, 50 mm; f1–3= −100, 175, 250 mm.

Fig. 3
Fig. 3

(a): Beam profile at z = 250 mm for free space propagation. The 1D radial intensity distribution of the doughnut beam is taken along a straight line passing through the beam center (red line). (b): Radial intensity distribution extracted from the measurement. The theoretical curve (shaded area, shifted) shows all the details of the measured profile.

Fig. 4
Fig. 4

(a) Intensity distributions of the SPP generated doughnut beam propagating in free space. The distribution at 250 mm is identical with Fig. 3. (b) One-dimensional radial intensity distributions with measured (black) and calculated (red) intensity distributions.

Fig. 5
Fig. 5

(a) Intensity distributions of the SPP generated propagating doughnut beam transformed by the lens system at different propagation distances. (b) One-dimensional radial intensity distributions with measured (black) and calculated (red) intensity distributions.

Equations (12)

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E ( ρ , ϕ , z ) = 0 w 0 w ( z ) exp [ ρ 2 w 2 ( z ) ] exp [ ik z i k ρ 2 2 R ( z ) + i η ( z ) ] exp ( i ϕ ) .
E ( ρ , ϕ , z ) = i λ B exp ( ikz ) 0 0 2 π E ( ρ , ϕ , z ) exp [ ik 2 B ( A ρ 2 + D ρ 2 ) ] exp [ ik ρ ρ cos ( ϕ ϕ ) B ] ρ d ρ d ϕ .
E ( ρ , ϕ , z ) = i λ B 0 w 0 w ( z ) exp ( ikD ρ 2 2 B ) exp [ ik ( z z ) ] exp [ i η ( z ) ] × 0 0 2 π exp [ ρ 2 w 2 ( z ) i k ρ 2 2 R ( z ) + i kA ρ 2 2 B ] exp [ ik ρ ρ cos ( ϕ ϕ ) B ] exp ( i ϕ ) d ϕ ρ d ρ
E 00 ( ρ , z ) = 0 w 0 w ( z ) exp ( ik ρ 2 2 B / D ) exp ( ik z ) exp [ i η ( z ) ] ,
1 R C 2 ( z ) = [ 1 w 2 ( z ) + ik 2 R ( z ) iAk 2 B ] and 1 / ρ C = k ρ / B ,
E ( ρ , ϕ , z ) = i λ B exp ( ikz ) E 00 ( ρ , z ) × 0 0 2 π exp [ ρ 2 R C 2 ( z ) ] exp [ i ρ ρ C cos ( ϕ ϕ ) ] exp ( i ϕ ) d ϕ ρ d ρ .
E ( ρ , ϕ , z ) = 2 π ( i ) | l | + 1 λ B E 00 ( ρ , z ) exp ( ikz ) exp ( i ϕ ) 0 exp [ ρ 2 / R C 2 ( z ) ] J ( ρ / ρ C ) ρ d ρ .
0 x exp ( α x 2 ) J ν ( β x ) dx = π β 8 α 3 / 2 exp ( β 2 8 α ) [ I 1 2 | ν | 1 2 ( β 2 8 α ) I 1 2 | ν | + 1 2 ( β 2 8 α ) ] .
E ( ρ , ϕ , z ) = 2 π 3 / 2 ( i ) | | + 1 E 00 ( ρ , z ) R C 3 ( z ) 8 ρ C λ B × exp ( ikz ) exp ( i ϕ ) exp ( R C 2 8 ρ C 2 ) × [ I 1 2 | | 1 2 ( R C 2 8 ρ C 2 ) I 1 2 | | + 1 2 ( R C 2 8 ρ C 2 ) ] .
( A B C D ) = ( 1 z 0 1 )
1 R C 2 ( z ) = [ 1 w 2 ( z ) + ik 2 R ( z ) ik 2 z ] and 1 / ρ C = k ρ / z .
E ( ρ , ϕ , z ) = 2 π 3 / 2 E 00 ( ρ , z ) R C 3 ( z ) 8 ρ C λ z exp ( ikz ) exp ( i ϕ ) exp ( R C 2 8 ρ C 2 ) [ I 0 ( R C 2 8 ρ C 2 ) I 1 ( R C 2 8 ρ C 2 ) ] .

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