OSA's Digital Library

Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 19 — Sep. 22, 2003
  • pp: 2315–2322
« Show journal navigation

X-ray vortex beams: A theoretical analysis

A. G. Peele and K. A. Nugent  »View Author Affiliations


Optics Express, Vol. 11, Issue 19, pp. 2315-2322 (2003)
http://dx.doi.org/10.1364/OE.11.002315


View Full Text Article

Acrobat PDF (366 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

1. Introduction

The ubiquity of wavefront dislocations, including point singularities in the transverse phase commonly referred to as screw-type dislocations or optical vortices, is, following the analysis of Nye and Berry [1

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

], now generally recognized [2

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

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]

]. Likewise, optical vortices have exercised the imagination of the optics community to a considerable degree over the past decade [4

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

]. Potential applications include wave-guides for atom optics [5

5. V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992). [CrossRef]

], optical computing [6

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

], optical communication [7

7. G. S. Agarwal and J. Banerji, “Spatial coherence and information entropy in optical vortex fields,” Opt. Lett. 27, 800–802 (2002). [CrossRef]

], optical tweezers, optical spanners [8

8. 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]

], high-resolution imaging [9

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

], and probing scattering media [10

10. M. Harris, “Light-field fluctuations in space and time,” Contemporary Phys. 36, 215–233 (1995). [CrossRef]

]. Optical vortices also exhibit the ability to regenerate their initial phase distribution after propagation through an obstacle [11

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

,12

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

] which offers promise in micromanipulation and in lithography. Several studies [13

13. 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]

,14

14. D. Rozas and G. A. Swartzlander, Jr., “Observed rotational enhancement of nonlinear optical vorticies,” Opt. Lett. 25, 126–128 (2000). [CrossRef]

] have also been undertaken because of insights that may be gained into vortex systems in media other than light such as fluids, superfluids and superconductors [15

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

]. Indeed, one of the earliest descriptions of a vortex system appears in a study on amphidronic tidal points [16

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

]. Optical vortices will be generated when coherent light is scattered by a rough medium [10

10. M. Harris, “Light-field fluctuations in space and time,” Contemporary Phys. 36, 215–233 (1995). [CrossRef]

], hence their ubiquity. While, for the purposes of investigation, controlled production via laser cavity modes [15

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

], cylindrical lens mode conversion of laser modes [17

17. 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]

], computer generated holograms - including spiral zone plates [18

18. 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]

] and forked diffraction gratings [5

5. V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992). [CrossRef]

] - and phase plates, both spiral [19

19. 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]

,20

20. 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]

] and non-spiral [21

21. 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]

] have all been demonstrated.

A recent publication made the perhaps surprising demonstration that production and detection of optical vortices was possible at x-ray wavelengths [22

22. 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, and I. McNulty, “Observation of an x-ray vortex,” Opt. Lett. 27, 1752–1754 (2002). [CrossRef]

]. This opens up a whole new parameter space of possible application for optical vortices. Possible x-ray applications include using the dark vortex core to mask a bright source, as in an astronomical x-ray source [23

23. G. A. Swartzlander, Jr., “Peering into darkness with a vortex spatial filter,” Opt. Lett. 26, 497–499 (2001). [CrossRef]

], or in a synchrotron-based small angle scattering experiment or, as a component in an x-ray phase-shifting mask, where the central zero in the vortex intensity pattern defines a small dot in the negative photoresist that creates a contact that can be used to connect layers [24

24. M. D. Levenson, G. Dai, and T. Ebihara, “Vortex Mask: Making 80nm contacts with a twist!” Proc. SPIE 4889, 1293–1303 (2002). [CrossRef]

]. Another idea might be to take advantage of elemental characteristic absorption energies to allow the selective transfer of orbital angular momentum thus allowing the manipulation of components of a structure that contain elements matching the tuned beam energy - for instance microfabricated objects carrying certain metal compounds may be moved while leaving their surroundings unaffected.

The stability of optical vortices on propagation makes them particularly problematic in the context of non-interferometric phase measurements as these fail where there are zeros in the intensity or where there is rotational symmetry in the phase [25

25. 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]

,26

26. 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]

]. It appears that singularities also play a significant role in certain methods to recover correlations in the wavefield. Techniques that use only measurements of intensity will fail when the intensity distribution is unchanged after propagation through a rotationally symmetric optical system, which is the case for a vortex beam [27

27. D. Dragoman, “Unambiguous coherence retrieval from intensity measurements,” J. Opt. Soc. Am. A 20, 290–295 (2003). [CrossRef]

]. Both these types of measurements are of particular interest in the x-ray regime [28

28. 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, and D. C. Mancini, “Spatial coherence measurement of X-ray undulator radiation,” Opt. Commun. 195, 79–84 (2001). [CrossRef]

] and in the context of synchrotron experiments where experimental difficulties are greatly alleviated by minimal instrumentation.

2. Vortex beam propagation

Based on typical experimentally observed distributions (for instance as seen by the authors in the work reported in [2225,28]), we model the synchrotron source as a Gaussian beam with constant phase at the source and amplitude given by:

u(r,θ,0)=2πω0exp[r2ω02],
(1)

where r and θ are cylindrical coordinates and ω0 is the 1/e half-waist at z=0. In the following derivations we explicitly retain the physical parameters so that their effect may be readily discerned. Typically, for a synchrotron ω0 is of the order of 10 microns in the vertical direction and order 100’s of microns in the horizontal direction. The Fresnel condition is satisfied if:

z3π4λ(rr)max4,
(2)

where r and r′ are the detector and source positions respectively. For the synchrotron experiment reported in [22

22. 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, and I. McNulty, “Observation of an x-ray vortex,” Opt. Lett. 27, 1752–1754 (2002). [CrossRef]

] where r is of the order of millimeters, r′ of the order of 10’s or 100’s of microns, the wavelength was 0.13 nm and z was approximately 40 m, this condition is satisfied. The Fresnel diffraction integral of Eq. (1) evaluated at z=z 1 gives [39

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

]:

u(ρ,ϕ,z1)=iλz2π1ω(z1)exp[ρ2ω(z1)2]exp[ik2ρ2R(z1)]exp[i(kz1Ψ(z1))],
(3a)

where: ρ and ϕ are the cylindrical coordinates in the z 1 plane, k=2π/λ is the wavenumber;

ω(z)=ω0[1+(zzR)2]12;zR=πω02λ;
Ψ(z)=atan(zzR);andR(z)=z[1+(zRz)2].
(3b)

Our technique for generating a vortex beam is to insert a phase plate that, ideally, is non-absorbing and imparts a helical phase to the beam. It should be noted that material properties at x-ray wavelengths are such that the real part of the refractive index, 1-δ, is less than unity, with δ<10-3. Consequently, for a phase plate that consists of a spiral ramp cut into a substrate, the phase of the beam will be advanced through the thick parts of the ramp compared to the thin parts. This is opposite to the case in visible wavelength optics and results in a vortex with the opposite helicity to that which might be expected. After insertion of the phase plate so that the centre of the plate is coaxial with the beam the amplitude is:

u(ρ,ϕ,z1)=Am(ρ,z1)exp[iΦ(ρ,z1)]exp[imϕ],
(4a)

where m is an integer referred to as the charge of the vortex:

Am(ρ,z1)=iλz12π1ω(z1)exp[ρ2ω(z1)2];and
Φ(ρ,z1)=k2ρ2R(z1)+kz1Ψ(z1).
(4b)

We can then calculate the Fresnel propagation of this vortex beam by evaluating the Fresnel integral at some detector plane Z=Z′-z 1, where Z′ is the distance of the detector plane from the synchrotron source:

u(R,Θ,Z)=iλZexp[λZR2]002πu(ρ,ϕ,z1)ρexp[λZρ2]×
exp[ikZρRcos(ϕΘ)]dρdϕ,
(5)

where R and Θ are the cylindrical coordinates in the Z plane. Recalling the definition of a Bessel function of the first kind of order m:

Jm(α)=12π02πexp[im(θπ2)]exp[iαcosθ]dθ;
(6)

and inserting Eq. (4a) into Eq. (5) we can write:

u(R,Θ,Z)=iλZexp[im(Θ+π2)]exp[λZR2]0ρAm(ρ,z1)exp[λZρ2]×
exp[iΦ(ρ,z1)]Jm(RkρZ)dρ.
(7)

This can be evaluated [40

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

] to give:

u(R,Θ,Z)=A(Z)exp[A(Z)R2]exp[imΘ]exp[iΦ(Z)]exp[iΦ˝(Z)R2]×
R[I12(m1)(γ28β)I12(m+1)(γ28β)],
(8a)

where: In is a Modified Bessel function of the first kind of order n;

A(Z)=π2πλZ1λz12π1ω(z1)k8Z[1ω(z1)4+(πλ)2{1R(z1)1Z}2]34;
A(Z)=(kZ)2181ω(z1)21[1ω(z1)4+(πλ)2(1R(z1)1Z)2];
Φ(Z)=πmπ2+kz1Ψ(z1)32atan[(πλ)ω(z1)2(1R(z1)1Z)];
Φ˝(Z)=k2Z+(kZ)218πλ(1R(z1)1Z)1[1ω(z1)4+(πλ)2(1R(z1)1Z)2];
γ=RkZ;andβ=1ω(z1)2+iπλ(1R(z1)1Z).
(8b)

Typically, at visible wavelengths, Eq. (1) is used as the input to Eq. (5). However, as discussed above, in the synchrotron context we use Eq. (3). The retention of the phase terms in the input beam before implanting the vortex via a phase plate greatly complicates Eq. (8) compared to the corresponding result for visible wavelength.

The detected intensity is given by:

I(R,Θ,Z)=u(R,Θ,Z)2.
(9)

A plot of Eq. (9) is shown in Fig. 1 for the synchrotron parameters described above and with Z=5.8 m, thus satisfying Eq. (2) above. We assume ω0 is the same both vertically and horizontally. We also calculate the detector plane intensity using a standard numerical propagator to calculate the detector plane complex amplitude:

a(x,y,z)=F1{expi2πz1kx2ky2F{a(x,y,z=0)}}.
(10)

Here F and F-1 are the Fourier transform of the argument, kx and ky are the Fourier transform spatial frequencies, a is the complex amplitude at coordinates x,y,z, and a(x,y,0) is given by Eq. (3a). The resultant intensity plotted as for Fig. 1 is indistinguishable from 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.

3. Vortex ‘interferograms’ produced by division of wavefront

In practice the existence of a vortex is not demonstrated merely by producing the type of intensity distribution shown in Fig. 1. Ideally some demonstration of the phase distribution should be made. This can be done interferometrically. In the visible regime division of amplitude interferometric techniques are commonly used, however at x-ray wavelengths this type of experiment is not straightforward. Accordingly, in the x-ray context, we use a division of wavefront technique such as the introduction of a wire to act as a secondary source. The wire is placed close to the phase plate and the Fresnel integral can be calculated for the resultant field. The result in [36

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

] for diffraction of a vortex beam through an aperture with appropriate modification for the limits of the integral could be used to derive a complicated expression for the diffracted field. However, the parameters imposed by the typical synchrotron experiment allow us to analyse the experiment as if it were the interference of the vortex beam in Eq. (8) and a cylindrical wave generated at the wire which has a phase gradient along it due to the incident vortex phase. Strictly, the wave generated by the wire will not be a cylindrical wave due to the curvature in the beam and the vortex phase imparted by the phase plate, however, the approximation is shown to be valid under numerical simulation for the synchrotron parameters used above. The resulting intensity is:

I=Av2+Acyl2+2AvAcylcosθinterf,
(11)

where Av is the real amplitude of the vortex beam given by Eq. (8), Acyl is the real amplitude of the cylindrical wave and there is a modulation to the intensity distribution given by the cosine of the interference term:

θinterf={mΘ+Φ(Z)+Φ˝(Z)R2+atan(I12(m1)(γ28β)I12(m+1)(γ28β))
+k(RcosΘxoffs)2+(Zzoffs)2+matan(RsinΘxoffs)},
(12)

where xoffs and zoffs are the offset coordinates of the wire with respect to the phase plate and where the first atan expression evaluates the phase of its complex argument. If we approximate the vortex beam as a plane wave with a vortex phase then Eq. (12), after switching to Cartesian coordinates, becomes:

θinterf=(matan(yx)+kZ+k(xxoffs)2+(Zzoffs)2+matan(yxoffs)).
(13)

Accordingly, we see that the analytic form for the interferogram for x-rays at a synchrotron becomes similar to the form used in calculating a hologram for visible wavelength vortices [3

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]

,5

5. V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992). [CrossRef]

].

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.

Figure 2(a) shows the modulation of the intensity distribution produced by the cosine of θinterf given by Eq. (12), while Fig. 2(b) shows the result for Eq. (13). Figure 2(c) shows the result for a numerical simulation based on Eq. (10). Figure 2d shows an experimental result as reported in [22

22. 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, and I. McNulty, “Observation of an x-ray vortex,” Opt. Lett. 27, 1752–1754 (2002). [CrossRef]

]. In all cases it can be seen that the phase discontinuity introduces a fork in one of the fringes. The main difference between the first two and the last two images is that the intensity distribution due to the amplitude terms in Eq. (11) are not included in the first two images and that the finite extent of the wire, absorption through the wire and the effects of partial coherence are included in the latter pair.

4. Vortex and an edge discontinuity: non-integer charge

Consider now what happens if the spiral phase plate is not matched to the energy of the incident x-ray beam so that the phase step across the step on the ramp is not an integer multiple of 2π. In general the ‘charge’ of the spiral will be some non-integer, ν. In the derivation above Eq. (6) is modified by replacing m with ν and the Bessel function of the first kind of order m becomes a sum of an Anger and a Weber function of the first kind of order ν. Note that the function is not replaced by a Bessel function of order ν as this has a quite different integral representation [40

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

]. We could proceed via the modified form of Eq. (7), however the success of Eq. (13) in predicting the qualitative form of the downstream ‘interferogram’ suggests that replacing m with ν in that equation will suffice. The approach here becomes similar to that taken elsewhere in calculating the hologram required to generate a vortex [3

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]

,5

5. V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992). [CrossRef]

,37

37. I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995). [CrossRef]

].

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.

Figure 3(a) shows a plot of the cosine of the modified form of θinterf given by Eq. (13) with ν=0.5. The phase indicated is due to an edge discontinuity terminating at the position of the point discontinuity. It is also straightforward to examine the effect of rotating the phaseplate about the beam propagation axis by rotating the vortex phase terms in the modified form of Eq. (13) to give:

θinterf=(νatan(ycosαxsinαxcosα+ysinα)+kZ+k(xxoffs)2+(Zzoffs)2+νatan(ycosαxsinαxoffs)),
(14)

where α is the rotation angle of the phaseplate. Figure 3(b) shows a plot of the cosine of θinterf given by Eq. (14) with ν=0.5 and α=π/4, while in Fig. 3(c) α=π/2.

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).

5. Conclusions

We have developed a useful range of results dealing with vortex phenomena at x-ray energies and for experimental parameters appropriate to a synchrotron source. Analytic expressions that differ from those obtained with visible wavelength assumptions are obtained and care has been taken to retain physical parameters explicitly. It is shown that the analytic forms can usefully predict intensity and phase distributions obtained numerically and experimentally. Finally, we have generated a prediction of vortex phase structure as a function of energy for a given phaseplate. We will compare this with experimental results in future work.

Acknowledgements

The authors acknowledge the support of Australian Research Council Fellowship (AGP: QEII Fellowship; KAN: Federation Fellowship) and grant funding. This work was supported by the Australian Synchrotron Research Program, which is funded by the Commonwealth of Australia under the Major National Research Facilities Program, and by the Australian Research Council. Use of the Advanced Photon Source was supported by the U.S. Department of Energy, Basic Energy Sciences, Office of Science under Contract No. W-31-109-Eng-38. We also acknowledge an anonymous referee for pointing out the application described by Ref. [24

24. M. D. Levenson, G. Dai, and T. Ebihara, “Vortex Mask: Making 80nm contacts with a twist!” Proc. SPIE 4889, 1293–1303 (2002). [CrossRef]

].

References and Links

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.

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

5.

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

6.

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

7.

G. S. Agarwal and J. Banerji, “Spatial coherence and information entropy in optical vortex fields,” Opt. Lett. 27, 800–802 (2002). [CrossRef]

8.

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]

9.

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

10.

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

11.

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

12.

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

13.

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]

14.

D. Rozas and G. A. Swartzlander, Jr., “Observed rotational enhancement of nonlinear optical vorticies,” Opt. Lett. 25, 126–128 (2000). [CrossRef]

15.

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

16.

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

17.

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]

18.

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]

19.

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]

20.

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]

21.

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]

22.

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, and I. McNulty, “Observation of an x-ray vortex,” Opt. Lett. 27, 1752–1754 (2002). [CrossRef]

23.

G. A. Swartzlander, Jr., “Peering into darkness with a vortex spatial filter,” Opt. Lett. 26, 497–499 (2001). [CrossRef]

24.

M. D. Levenson, G. Dai, and T. Ebihara, “Vortex Mask: Making 80nm contacts with a twist!” Proc. SPIE 4889, 1293–1303 (2002). [CrossRef]

25.

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]

26.

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]

27.

D. Dragoman, “Unambiguous coherence retrieval from intensity measurements,” J. Opt. Soc. Am. A 20, 290–295 (2003). [CrossRef]

28.

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, and D. C. Mancini, “Spatial coherence measurement of X-ray undulator radiation,” Opt. Commun. 195, 79–84 (2001). [CrossRef]

29.

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

30.

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]

31.

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]

32.

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

33.

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

34.

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

35.

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

36.

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

37.

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995). [CrossRef]

38.

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

39.

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

40.

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

OCIS Codes
(340.6720) X-ray optics : Synchrotron radiation
(350.7420) Other areas of optics : Waves

ToC Category:
Focus Issue: Coherent x-ray optics

History
Original Manuscript: July 28, 2003
Revised Manuscript: August 23, 2003
Published: September 22, 2003

Citation
A. Peele and K. Nugent, "X-ray vortex beams: A theoretical analysis," Opt. Express 11, 2315-2322 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-19-2315


Sort:  Journal  |  Reset  

References

  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).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1. Fig. 2. Fig. 3.
 
Fig. 4.
 

Supplementary Material


» Media 1: MPG (1166 KB)     

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited