Abstract

The recent demonstration that an optical vortex could be generated at x-ray wavelengths brings this interesting topological phenomenon into an entirely new regime with several possible applications. We examine the analytic propagation of an optical vortex generated in a synchrotron x-ray beam line. We compare the results obtained with the existing experimental data and further consider the generation and interpretation of mixed vortex-edge discontinuities which might be considered as non-integer charge vortices.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |

  1. J. F. Nye and M. V. Berry, �??Dislocations in wave trains,�?? Proc. R. Soc. London, Ser. A 336, 165 �?? 190 (1974).
    [CrossRef]
  2. N. B. Baranova, B. Ya, Zel�??dovich, A. V. Mamayev, N. F. Pilipetskii, and V. V. Shkukov, �??Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),�?? Pis�??ma Zh. Eks. Teor. Fiz. 33, 206 �?? 210 (1981) [JETP Lett. 33, 195 �?? 199 (1981)].
  3. Z. S. Sacks, D. Rozas, and G. A. Swartzlander, Jr., �??Holographic formation of optical-vortex filaments,�?? J. Opt. Soc. Am. B 15, 2226 �?? 2234 (1998).
    [CrossRef]
  4. V. Yu. Bazhenov, M. S. Soskin and M. V. Vasnetsov, �??Screw dislocations in light wavefronts,�?? J. Mod. Opt. 39, 985 - 990 (1992).
    [CrossRef]
  5. I. Freund, �??Critical point explosions in two-dimensional wave fields,�?? Opt. Commun. 159, 99 - 117(1999).
    [CrossRef]
  6. G. S. Agarwal and J. Banerji, �??Spatial coherence and information entropy in optical vortex fields,�?? Opt. Lett. 27, 800 �?? 802 (2002).
    [CrossRef]
  7. N. B. Simpson, L. Allen and M. J. Padgett, �??Optical tweezers and optical spanners with Laguerre-Gaussian modes,�?? J. Mod. Opt. 43, 2485 �?? 2491 (1996).
    [CrossRef]
  8. E. Wolf, Progress in optics 42, (Elsevier, 2001).
  9. M. Harris, �??Light-field fluctuations in space and time,�?? Contemporary Phys. 36, 215 �?? 233 (1995).
    [CrossRef]
  10. M. V. Vasnetsov, I. G. Marienko and M. S. Soskin, �??Self-reconstruction of an optical vortex�?? JETP Lett. 71, 130 �?? 133 (2000).
    [CrossRef]
  11. Z. Bouchal, �??Resistance of nondiffracting vortex beam against amplitude and phase pertubations,�?? Opt. Commun. 210, 155 �?? 164 (2002).
    [CrossRef]
  12. D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Jr., �??Experimental observation of fluidlike motion of optical vortices,�?? Phys. Rev. Lett. 79, 3399 �?? 3402 (1997).
    [CrossRef]
  13. M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, �??Transverse laser patterns.m I. Phase singularity crystals,�?? Phys. Rev. A 43, 5090 �?? 5117 (1991)
    [CrossRef] [PubMed]
  14. W. Whewell, �??Essay towards a first approximation to a map of cotidal lines,�?? Phil. Trans. R. Soc. Lond. 123, 147 �?? 236 (1833).
    [CrossRef]
  15. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, �??Astigmatic laser mode converters and transfer of orbital angular momentum,�?? Opt. Commun. 96, 123 �?? 132 (1993).
    [CrossRef]
  16. 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 �?? 223 (1992).
    [CrossRef] [PubMed]
  17. 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]
  18. G. A. Turnball, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, �??The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,�?? Opt. Commun. 127, 183 �?? 188 (1996).
    [CrossRef]
  19. G. H. Kim, J. H. Jeon, K. H. Ko, H. J. Moon, J. H. Lee, and J. S. Chang, �??Optical vortices produced with a nonspiral phase plate,�?? Appl. Opt. 36, 8614 �?? 8621 (1997).
    [CrossRef]
  20. A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. C. Harvey, B. Lai, I. McNulty, �??Observation of an x-ray vortex,�?? Opt. Lett. 27, 1752 �?? 1754 (2002).
    [CrossRef]
  21. G. A. Swartzlander, Jr., �??Peering into darkness with a vortex spatial filter,�?? Opt. Lett. 26, 497 �?? 499 (2001).
    [CrossRef]
  22. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, �??Quantitative phase imaging using hard x rays,�?? Phys. Rev. Lett. 77, 2961 �?? 2964 (1996).
    [CrossRef] [PubMed]
  23. L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, �??Phase retrieval from images in the presence of first-order vortices,�?? Phys. Rev. E 63, 037602 (2001).
    [CrossRef]
  24. D. Dragoman, �??Unambiguous coherence retrieval from intensity measurements,�?? J. Opt. Soc. Am. A 20, 290 �?? 295 (2003).
    [CrossRef]
  25. D. Paterson, B. E. Allman, P. J. McMahon, J. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, D. C. Mancini, �??Spatial coherence measurement of X-ray undulator radiation,�?? Opt. Commun. 195, 79 �?? 84 (2001).
    [CrossRef]
  26. G. Indebetouw, �??Optical vortices and their propagation,�?? J. Mod. Opt. 40, 73 �?? 87 (1993)
    [CrossRef]
  27. I. V. Basistiy, V. Yu Bazhenov, M. S. Soskin, and M. V. Vasnetov, �??Optics of light beams with screw dislocations,�?? Opt. Commun. 103, 422 �?? 428 (1993).
    [CrossRef]
  28. D. Rozas, C. T. Law, and G. A. Swartzlander, Jr., �??Propagation dynamics of optical vortices,�?? J. Opt. Soc. Am. B 14, 3054 �?? 3065 (1997).
    [CrossRef]
  29. U. T. Schwarz, S. Sogomonian, M. Maier, �??Propagation dynamics of phase dislocations embedded in a Bessel light beam,�?? Opt. Commun. 208, 255 �?? 262 (2002).
    [CrossRef]
  30. S. Orlov, K.Regelskis, V. Smilgevicius, and A. Stabinis, �??Propagation of Bessel beams carrying optical vortices,�?? Opt. Commun. 209, 155 �?? 165 (2002).
    [CrossRef]
  31. V. Pyragaite and A. Stabinis, �??Free-space propagation of overlapping light vortex beams,�?? Opt. Commun. 213, 187 �?? 191 (2002).
    [CrossRef]
  32. R. P. Singh and S. R. Chowdhury, �??Trajectory of an optical vortex: canonical vs. non-canonical,�?? Opt. Commun. 215, 231 �?? 237 (2003).
    [CrossRef]
  33. J. Masajada, �??Half-plane diffraction in the case of Gaussian beams containing an optical vortex,�?? Opt. Commun. 175, 289 �?? 294 (2000).
    [CrossRef]
  34. D. V. Petrov, �??Vortex-edge dislocation interaction in a linear medium,�?? Opt. Commun. 188, 307 �?? 312 (2001).
    [CrossRef]
  35. A. E. Siegman, An Introduction to Lasers and Masers, (McGraw-Hill, 1971).
  36. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 5th ed., (Academic Press,1994).
  37. I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, �??Optical wavefront dislocations and their properties,�?? Opt. Commun. 119, 604 �?? 612 (1995).
  38. M. S. Naschie, ed., �??Special issue on nonlinear optical structures, patterns, chaos,�?? Chaos Solitons Fractals 4(8/9) (1994).
  39. M. D. Levenson, G. Dai, T. Ebihara, "Vortex Mask: Making 80nm contacts with a twist!" Proc. SPIE 4889, 1293-1303 (2002).
  40. D. Rozas and G. A. Swartzlander, Jr., �??Observed rotational enhancement of nonlinear optical vorticies,�?? Opt. Lett. 25, 126 �?? 128 (2000).

Appl. Opt. (1)

Chaos Soliton Fractals (1)

M. S. Naschie, ed., �??Special issue on nonlinear optical structures, patterns, chaos,�?? Chaos Solitons Fractals 4(8/9) (1994).

Contemp. Phys. (1)

M. Harris, �??Light-field fluctuations in space and time,�?? Contemporary Phys. 36, 215 �?? 233 (1995).
[CrossRef]

J. Mod. Opt. (3)

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

V. Yu. Bazhenov, M. S. Soskin and M. V. Vasnetsov, �??Screw dislocations in light wavefronts,�?? J. Mod. Opt. 39, 985 - 990 (1992).
[CrossRef]

G. Indebetouw, �??Optical vortices and their propagation,�?? J. Mod. Opt. 40, 73 �?? 87 (1993)
[CrossRef]

J. Opt. Soc. Am. A (1)

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

JEPT Lett. (1)

M. V. Vasnetsov, I. G. Marienko and M. S. Soskin, �??Self-reconstruction of an optical vortex�?? JETP Lett. 71, 130 �?? 133 (2000).
[CrossRef]

Opt. Commun. (14)

Z. Bouchal, �??Resistance of nondiffracting vortex beam against amplitude and phase pertubations,�?? Opt. Commun. 210, 155 �?? 164 (2002).
[CrossRef]

I. Freund, �??Critical point explosions in two-dimensional wave fields,�?? Opt. Commun. 159, 99 - 117(1999).
[CrossRef]

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]

G. A. Turnball, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, �??The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,�?? Opt. Commun. 127, 183 �?? 188 (1996).
[CrossRef]

U. T. Schwarz, S. Sogomonian, M. Maier, �??Propagation dynamics of phase dislocations embedded in a Bessel light beam,�?? Opt. Commun. 208, 255 �?? 262 (2002).
[CrossRef]

S. Orlov, K.Regelskis, V. Smilgevicius, and A. Stabinis, �??Propagation of Bessel beams carrying optical vortices,�?? Opt. Commun. 209, 155 �?? 165 (2002).
[CrossRef]

V. Pyragaite and A. Stabinis, �??Free-space propagation of overlapping light vortex beams,�?? Opt. Commun. 213, 187 �?? 191 (2002).
[CrossRef]

R. P. Singh and S. R. Chowdhury, �??Trajectory of an optical vortex: canonical vs. non-canonical,�?? Opt. Commun. 215, 231 �?? 237 (2003).
[CrossRef]

J. Masajada, �??Half-plane diffraction in the case of Gaussian beams containing an optical vortex,�?? Opt. Commun. 175, 289 �?? 294 (2000).
[CrossRef]

D. V. Petrov, �??Vortex-edge dislocation interaction in a linear medium,�?? Opt. Commun. 188, 307 �?? 312 (2001).
[CrossRef]

I. V. Basistiy, V. Yu Bazhenov, M. S. Soskin, and M. V. Vasnetov, �??Optics of light beams with screw dislocations,�?? Opt. Commun. 103, 422 �?? 428 (1993).
[CrossRef]

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

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, �??Optical wavefront dislocations and their properties,�?? Opt. Commun. 119, 604 �?? 612 (1995).

D. Paterson, B. E. Allman, P. J. McMahon, J. Lin, N. Moldovan, K. A. Nugent, I. McNulty, C. T. Chantler, C. C. Retsch, T. H. K. Irving, D. C. Mancini, �??Spatial coherence measurement of X-ray undulator radiation,�?? Opt. Commun. 195, 79 �?? 84 (2001).
[CrossRef]

Opt. Lett. (5)

Phil. Trans. R. Soc. Lond. (1)

W. Whewell, �??Essay towards a first approximation to a map of cotidal lines,�?? Phil. Trans. R. Soc. Lond. 123, 147 �?? 236 (1833).
[CrossRef]

Phys. Rev. A (1)

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, �??Transverse laser patterns.m I. Phase singularity crystals,�?? Phys. Rev. A 43, 5090 �?? 5117 (1991)
[CrossRef] [PubMed]

Phys. Rev. E (1)

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, �??Phase retrieval from images in the presence of first-order vortices,�?? Phys. Rev. E 63, 037602 (2001).
[CrossRef]

Phys. Rev. Lett. (2)

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, �??Quantitative phase imaging using hard x rays,�?? Phys. Rev. Lett. 77, 2961 �?? 2964 (1996).
[CrossRef] [PubMed]

D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Jr., �??Experimental observation of fluidlike motion of optical vortices,�?? Phys. Rev. Lett. 79, 3399 �?? 3402 (1997).
[CrossRef]

Pis???ma Zh. Eks. Teor. Fiz. (1)

N. B. Baranova, B. Ya, Zel�??dovich, A. V. Mamayev, N. F. Pilipetskii, and V. V. Shkukov, �??Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),�?? Pis�??ma Zh. Eks. Teor. Fiz. 33, 206 �?? 210 (1981) [JETP Lett. 33, 195 �?? 199 (1981)].

Proc. R. Soc. London, Ser. A (1)

J. F. Nye and M. V. Berry, �??Dislocations in wave trains,�?? Proc. R. Soc. London, Ser. A 336, 165 �?? 190 (1974).
[CrossRef]

Proc. SPIE (1)

M. D. Levenson, G. Dai, T. Ebihara, "Vortex Mask: Making 80nm contacts with a twist!" Proc. SPIE 4889, 1293-1303 (2002).

Other (3)

E. Wolf, Progress in optics 42, (Elsevier, 2001).

A. E. Siegman, An Introduction to Lasers and Masers, (McGraw-Hill, 1971).

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 5th ed., (Academic Press,1994).

Supplementary Material (1)

» Media 1: MPG (1166 KB)     

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1.

Vortex detector plane intensity from Eq. (9). ω0=239.5 µm, z1=41.4 m, Z=5.8 m, λ=0.13 nm.

Fig. 2.
Fig. 2.

Interferograms of vortex phase structure produced analytically (a and b), numerically (c) and experimentally (d). From left to right, (a) Eq. (12), (b) Eq. (13), (c) Intensity distribution based on Eq. (10) method, (d) experimental result.

Fig. 3.
Fig. 3.

Interferograms of vortex phase structure for non-integer charge and a rotated phaseplate. From left to right; (a) Modified form of Eq. (13) with ν=0.5, (b) Eq. (14) with ν=0.5 and α=π/4, and (c) Eq. (14) with ν=0.5 and α=π/2.

Fig. 4.
Fig. 4.

(1.2 MB) Movie of evolution in the interferogram as the energy varies from 4.5 keV (charge=2.01) to 9 keV (charge=1).

Equations (27)

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

u ( r , θ , 0 ) = 2 π ω 0 exp [ r 2 ω 0 2 ] ,
z 3 π 4 λ ( r r ) max 4 ,
u ( ρ , ϕ , z 1 ) = i λz 2 π 1 ω ( z 1 ) exp [ ρ 2 ω ( z 1 ) 2 ] exp [ i k 2 ρ 2 R ( z 1 ) ] exp [ i ( k z 1 Ψ ( z 1 ) ) ] ,
ω ( z ) = ω 0 [ 1 + ( z z R ) 2 ] 1 2 ; z R = π ω 0 2 λ ;
Ψ ( z ) = atan ( z z R ) ; and R ( z ) = z [ 1 + ( z R z ) 2 ] .
u ( ρ , ϕ , z 1 ) = A m ( ρ , z 1 ) exp [ i Φ ( ρ , z 1 ) ] exp [ imϕ ] ,
A m ( ρ , z 1 ) = i λ z 1 2 π 1 ω ( z 1 ) exp [ ρ 2 ω ( z 1 ) 2 ] ; and
Φ ( ρ , z 1 ) = k 2 ρ 2 R ( z 1 ) + k z 1 Ψ ( z 1 ) .
u ( R , Θ , Z ) = i λZ exp [ λZ R 2 ] 0 0 2 π u ( ρ , ϕ , z 1 ) ρ exp [ λZ ρ 2 ] ×
exp [ i k Z ρ R cos ( ϕ Θ ) ] d ρ d ϕ ,
J m ( α ) = 1 2 π 0 2 π exp [ im ( θ π 2 ) ] exp [ i α cos θ ] d θ ;
u ( R , Θ , Z ) = i λZ exp [ im ( Θ + π 2 ) ] exp [ λZ R 2 ] 0 ρ A m ( ρ , z 1 ) exp [ λZ ρ 2 ] ×
exp [ i Φ ( ρ , z 1 ) ] J m ( Rkρ Z ) d ρ .
u ( R , Θ , Z ) = A ( Z ) exp [ A ( Z ) R 2 ] exp [ im Θ ] exp [ i Φ ( Z ) ] exp [ i Φ ˝ ( Z ) R 2 ] ×
R [ I 1 2 ( m 1 ) ( γ 2 8 β ) I 1 2 ( m + 1 ) ( γ 2 8 β ) ] ,
A ( Z ) = π 2 π λZ 1 λ z 1 2 π 1 ω ( z 1 ) k 8 Z [ 1 ω ( z 1 ) 4 + ( π λ ) 2 { 1 R ( z 1 ) 1 Z } 2 ] 3 4 ;
A ( Z ) = ( k Z ) 2 1 8 1 ω ( z 1 ) 2 1 [ 1 ω ( z 1 ) 4 + ( π λ ) 2 ( 1 R ( z 1 ) 1 Z ) 2 ] ;
Φ ( Z ) = π m π 2 + k z 1 Ψ ( z 1 ) 3 2 atan [ ( π λ ) ω ( z 1 ) 2 ( 1 R ( z 1 ) 1 Z ) ] ;
Φ ˝ ( Z ) = k 2 Z + ( k Z ) 2 1 8 π λ ( 1 R ( z 1 ) 1 Z ) 1 [ 1 ω ( z 1 ) 4 + ( π λ ) 2 ( 1 R ( z 1 ) 1 Z ) 2 ] ;
γ = Rk Z ; and β = 1 ω ( z 1 ) 2 + i π λ ( 1 R ( z 1 ) 1 Z ) .
I ( R , Θ , Z ) = u ( R , Θ , Z ) 2 .
a ( x , y , z ) = F 1 { exp i 2 π z 1 k x 2 k y 2 F { a ( x , y , z = 0 ) } } .
I = A v 2 + A cyl 2 + 2 A v A cyl cos θ i nterf ,
θ interf = { m Θ + Φ ( Z ) + Φ ˝ ( Z ) R 2 + atan ( I 1 2 ( m 1 ) ( γ 2 8 β ) I 1 2 ( m + 1 ) ( γ 2 8 β ) )
+ k ( R cos Θ x offs ) 2 + ( Z z offs ) 2 + m atan ( R sin Θ x offs ) } ,
θ interf = ( m atan ( y x ) + kZ + k ( x x offs ) 2 + ( Z z offs ) 2 + m atan ( y x offs ) ) .
θ interf = ( ν atan ( y cos α x sin α x cos α + y sin α ) + kZ + k ( x x offs ) 2 + ( Z z offs ) 2 + ν atan ( y cos α x sin α x offs ) ) ,

Metrics