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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 8 — Apr. 16, 2007
  • pp: 5196–5207
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Creation of vortex lattices by a wavefront division

J. Masajada, A. Popiolek-Masajada, and M. Leniec  »View Author Affiliations


Optics Express, Vol. 15, Issue 8, pp. 5196-5207 (2007)
http://dx.doi.org/10.1364/OE.15.005196


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Abstract

The Young’s double-slit experiment is one of the most popular stories in the history of physics. This paper, like many others, has emerged from the Young’s idea. It investigates the diffraction of the plane or spherical wave produced by three or four small holes in an opaque screen. It was noticed that the interference field contained a lattice of optical vortices which were equivalent to those produced in optical vortex interferometer. The vortex lattice generated by the three holes possessed some unique properties from which the analytical formulae for vortex points position were derived. We also pointed out the differences between our case and the double-slit experiment. Finally, some remarks on possible applications of our arrangement are discussed briefly. These theoretical considerations are illustrated with the use of experimental results.

© 2007 Optical Society of America

1. Introduction

The Young double-slit experiment has had a great impact on the optics and physics and is simple enough to be explained in physics text books. No wonder the Young’s experiment is one of the most widely recognized by the general public. After Young’s famous papers [1

1. T. Young, “Experimental demonstration of the General Law of the Interference of Light,” Philos. Trans. R. Soc. London94, (1804).

] a number of subsequent works were published on this subject by different authors. The present paper is yet another contribution to this famous topic. Of course, none can compete with the Young’s work. The contributions following the original Young’s paper have dealt with the subject in more details [2

2. R. Welti, “Light transmission through two slits: the Young experiment revisited,” J. Opt. A: Pure Appl. Opt. 8, 606–609 (2006). [CrossRef]

] or in wider context (as our paper does), or for different objects like electrons [3

3. C. Jönsson, “Electron diffraction at multiple slits,” Am. J. Phys. 42, 4–11 (1974). [CrossRef]

], neutrons [4

4. A. Zeilinger, R. Gähler, C.G. Shull, W. Treimer, and W. Mampe, “Single and double-slit diffraction of neutrons,” Rev. Mod. Phys. 60, 1067–1073 (1988). [CrossRef]

], atoms [5

5. O. Carnal and J. Mlynek, “Young’s double-slit experiment with atoms: A simple atom interferometer,” Phys. Rev. Lett. 66, 2689–2692 (1991). [CrossRef] [PubMed]

], molecules [6

6. W. Schöllkopf and J. Toennies, “The nondestructive mass selection of small van der Waals clusters,” Science 226, 1345–1348 (1994). [CrossRef]

] or light but in strange states like entangled photons (see for example [7

7. C. K. Hong and T. G. Noh, “Two-photon double slit interference experiment,” J. Opt. Soc. Am. B 15, 1192–1197 (1997). [CrossRef]

]). In this paper we explore the three and four hole experiment and focus our attention on the optical vortices which are generated in such an interference field.

The interest in phase singularities (optical vortices in particular) started with the paper by Nye and Berry published in 1974 [8

8. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. Roy. Soc. Lond. A 336, 165–189 (1974). [CrossRef]

]. Since that time hundreds of papers as well as few books have been published in this field [9

9. J. F. Nye, Natural focusing and fine structure of light (IoP, Bristol and Philadelphia, 1999).

, 10

10. M. Vasnetsov and K. Staliunas, eds., Optical vortices (Nova Science Publishers, 1999).

, 11

11. L. Allen, S. M. Barnett, and M. J. Padgett, Optical angular momentum (IoP, Bristol, Philadelphia, 2003). [CrossRef]

]. The general papers in Progress in Optics have also been published [12

12. L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt.39, chapter IV (1999). [CrossRef]

, 13

13. M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” Prog. Opt.42, chapter IV (2001).

]. The interested reader may refer to this rich literature. Here, to give a general view on optical vortices, we note that in a three dimensional picture the wavefront containing a single optical vortex has a characteristic helical geometry. The phase along the helice axis is undetermined. In the plane cross section this vortex line becomes a point. The optical vortices possess a number of interesting properties. Due to the two possible handedness for example of the helice the optical vortices can be classified as either left or right oriented (we say that they have negative or positive topological charge) and the wavefront carrying optical vortices possesses non-zero angular momentum.

Fig. 1. The basic optical setup of the optical vortex interferometer.
Fig. 2. The example of the equiphase line plot (for the vortex lattice generated by three plane waves' interference). The lattice was computer generated.

In papers [14

14. J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198, 21–27 (2001). [CrossRef]

, 15

15. J. Masajada, A. Popiołek-Masajada, and D. Wieliczka, “The interferometric system using optical vortices as a phase markers,” Opt. Commun. 207, 85–93 (2002). [CrossRef]

] the concept of the new interferometer based on the regular lattice of optical vortices was introduced. Such a lattice was generated by the interference of three plane waves. The basic Optical Vortex Interferometer (OVI) configuration used by the authors is given in Fig. 1. Figure 2 shows an example of the phase-field plot produced by the OVI. The problem of vortex points (i.e. points in which the phase is undetermined) localization was discussed in paper [16

16. J. Masajada, A. Popiołek Masajada, E. Frączek, and W. Frączek, “Vortex points localization problem in optical vortices interferometr,” Opt., Commun. 234, 23–28 (2004). [CrossRef]

]. Some basic measurements with the OVI were reported in Refs. [17

17. J. Masajada, “Small rotation-angle measurement with optical vortex interferometer,” Opt. Commun. 234, 373–381 (2004). [CrossRef]

, 18

18. A. PopioВek-Masajada, M Borwińska, and W Frączek, “Testing a new method for small-angle rotation measurements with Optical Vortices Interferometer,” Meas. Sci. Technol. 17, 653–658 (2006). [CrossRef]

, 19

19. M. Borwińska, A. Popiołek-Masajada, and B. Dubik, “Reconstruction of the plane wave tilt and its orientation using Optical Vortex Interferometer,” Opt. Eng. (to be published).

]. The extended theory of the OVI was presented in [20

20. J. Masajada, “The interferometry based on regular net of optical vortices,” Opt. Appl. to be published in vol.37, (2007).

] and partially in [21

21. J. Masajada, “The optical vortex interferometer, theory, technology and applications,” Proc. SPIE 6254, 62540C1–10 (2006).

]. The polarization versions of the OVI were built using a Wollastone prism and were presented in Refs. [22

22. P. Kurzynowski, A. WoŴniak, and E. Frączek, “Optical vortices generation using the Wollaston prism,” Appl. Opt. 45, 7898–7903 (2006). [CrossRef] [PubMed]

and 23

23. P. Kurzynowski and M. Borwińska, “Generation of the vortex type markers in a one wave setup,” Appl. Opt. 46, 676–679 (2007). [CrossRef] [PubMed]

]. The three-hole version of the Young’s experiment can be treated as a new version of the OVI – the OVI with a wavefront division.

The physics of vortex lattice produced by the wave interference or scattering has attracted attention of other researchers [24–26

24. K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four and five plane waves,” Opt. Express 14, 3039 – 3044 (2006). [CrossRef] [PubMed]

]. The topology of vortex lines generated by three-, four-and five-plane waves’ interference was discussed in paper [24

24. K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four and five plane waves,” Opt. Express 14, 3039 – 3044 (2006). [CrossRef] [PubMed]

]. Such a three dimensional picture seems to be more general than our view. However, the authors of that paper were focused on general topological properties of the vortex lines in the interference field and such a description is not sufficient when using the OVI. Our approach – limited to the plane cross-section – gives the analytical relations between vortex lattice geometry and the physical parameters of interfering waves (wavevector coordinates, waves amplitude and phases) [20

20. J. Masajada, “The interferometry based on regular net of optical vortices,” Opt. Appl. to be published in vol.37, (2007).

]. These relations are necessary when computing the values of physical parameters which are measured with the OVI.

It should be noticed that some authors have studied the three or more plane waves’ interferometer, but focused their attention on the bright spots in the interference pattern. No remarks on the vortex lattice appearance in such a pattern have been made [27–31

27. J. Primot and L. Sogno, “Achromatic three-wave (or more) lateral shearing interferometer,” J. Opt. Soc. Am. A 12, 2679–2685 (1995). [CrossRef]

].

2. The plane wave illumination

As the first step we are considering the opaque screen with the three circular holes marked as A, B, C, illuminated by the plane wave under normal incidence. This screen is further in the text called “plate”. The centers of the holes can not be collinear. We have adopted the coordinate system in which the z-axis is perpendicular to the plate. In such a coordinate system the wavevector of the incident wave is k(0,0,k). We have assumed that the center of the hole A overlaps with the center of coordinate system. The waves emerging from the holes A, B, C, will be marked by the same letters A, B, C.

The analytical description of the interference field is rather complicated. Therefore we will limit our consideration to the points where the light amplitude is zero – they are the vortex points. At first the interference field will be analyzed in a far field approximation (Fraunhofer approximation) [32

32. W. Singer, M. Totzeck, and H. Gross, Handbook of Optical System, (Wiley-VCH, Berlin, 2005) Vol. 2

]. Applying the Fraunhofer diffraction integral, which is just a Fourier transform of the transmittance function of our plate multiplied by complex amplitude of the incident wave, we get:

U(fx,fy)=Ωexp{iπfp2z}somb(Rfp)×
[1+exp{2πi(TBxfx+TByfy)}+exp{2πi(TCxfx+TByfy)}].
(1)

f 2 p = f 2 x + f 2 y, fx, fy are spatial frequencies, T B(T Bx, T By), T C(T Cx, T Cy) are translation vectors indicating the position of B and C holes, respectively:

Ω=exp{ikz}2iλz.
(2)

The function somb is defined as follows:

somb(x)=2J1(x)x.
(3)

Here J1 is a Bessel function of the first kind and first order and R is the radius of the hole.

ψA=0;ψB=23π+2πm;ψc=43π+2πn,
(3a)

or

ψA=0;ψB=43π+2πm;ψc=23π+2πn.
(3b)

Here m and n are integers. The relations (3) lead to the following conditions:

2π(TBxfx+TByfy)=23π+2πm,
(4a)
2π(TCxfx+TCyfy)=43π+2πn,
(4b)

or

2π(TBxfx+TByfy)=43π+2πm,
(4c)
2π(TCxfx+TCyfy)=23π+2πn.
(4d)

The solutions to the above two sets of equations are:

fx=13TBy(2+3n)TCy(1+3m)TByTCxTBxTCy,
(5a)
fy=13TBx(2+3n)TCx(1+3m)TByTCxTBxTCy,
(5b)

or

fx=13TCy(2+3m)TBy(1+3n)TByTCxTBxTCy,
(5c)
fy=13TBx(1+3n)TCx(2+3m)TByTCxTBxTCy.
(5d)
Fig 3. The comparison between the experiment and the results of far field computations. The circles correspond to the computed positions of the vortex points. The crosses mark the vortex points localized in the interferogram. Both figures differ in the arrangement of the three holes as shown under each interferogram. The plates were inspected under microscopy while applying to the calculations.

We have found the positions of vortex points in the frequency domain. They can be easily recomputed to normal coordinates by multiplying the formulae (5) by factor λz; where z is the distance between the plate and the observation plane. The two sets of solutions correspond to two topological charges of the optical vortices. Figure 3 shows two experimental examples as well as comparison to our theory. Figure 4 shows the interference pattern obtained as a result of adding the light emerging from the plate together with an additional plane wave. The fork like fringe pattern in Fig. 4 is an experimental evidence that our plate generates the vortex lattice.

For the experiment we have manufactured the plate with holes using a piece of thin metal sheet and needles of different thickness. After some tests our plates were manufactured with enough precision to perform the experiments we needed for this paper. The diameters of the holes and the distances between them are given in Fig. 3. The plate was illuminated with collimated He-Ne laser (632.8 nm) and observed in CCD camera. The distance between the plate and the observation plane was 900 mm. The vortex points were localized using the adopted minima method [16

16. J. Masajada, A. Popiołek Masajada, E. Frączek, and W. Frączek, “Vortex points localization problem in optical vortices interferometr,” Opt., Commun. 234, 23–28 (2004). [CrossRef]

].

Obviously, within the far field approximation the result does not depend on the shape of the holes. It should be mentioned that by “result” we mean the position of the vortex points. On the other hand the details of the whole diffraction pattern do depend on the shape of the holes. We should also remember that the holes cannot be too big since the far field approximation may fail. That is why long slits couldn’t be used in our experiment. This is an important difference between our experiment and a classical double-slit experiment. In a double-slit experiment the interference field is analyzed in one direction which is perpendicular to the direction of the slit elongation. In our case we had to make our analysis in the two dimensional areas of the observation plane, so our “slit” had to be small in both directions.

Fig. 4. The interference pattern obtained by adding the light emerging from the plate with three holes together with an additional plane wave. The fork like fringes indicates the presence of optical vortices. The vortex points localized on this interferogram are marked with crosses.
Fig. 5 The position of vortex points calculated for the plate shown in Fig. 3(a) using far (pluses) and near (circles) field diffractions integral (numerical integration). The positions of the three vortex points were adjusted manually to show the difference between the lattice determined by both methods.

The distance between the q-hole and (x,y) the point on the screen can be approximated as (Fig. 6)

rq=z+12(xqx)2+(yqy)2z.
(6)
Fig. 6. The scheme for the parabolic approximation.

Here q∊(A, B, C), xq, yq are coordinates of the center of the q-hole. We can still assume that the center of the hole A lies at the origin of a coordinate system. We can also assume that the phase of the incident wave equals zero in the center of this hole.

The condition for the vortex points position in the observation plane located at distance z from the plate is:

k(rbra)+δb=ψb+2πm,
(7a)
k(rcra)+δc=ψc+2πn.
(7b)

Where δb δc are the phase differences between the center of the holes B, C and the hole A for the given incident wave. The value of δb δc is important when the incident plane wave is inclined in respect to the plate; in the case of normal illumination δb = δc = 0. Using the parabolic approximation we get:

x=12yc(Hb+2mλz)yb(Hc+2nλz)xbycybxc,
(8a)
y=12xc(Hb+2mλz)xb(Hc+2nλz)xbycybxc,
(8b)

where

Hq=2z(ψqδqk)xq2yq2.
(9)

Figure 7(a) shows the difference between the vortex points position calculated using Eqs. (5) and (8). Figure 7(b) shows the comparison to the experimental results.

Fig. 7. (a). The position of the vortex points for the plate shown in Fig. 3(a), calculated using parabolic (crosses) and near (circles) field approximation, (b) comparison between computations using parabolic field approximation and experimental results.
Fig. 8. The interference pattern produced by the four holes in the arrangement shown below each interferogram. The vortex points (marked as crosses) were localized by using minima method. Contrary to the three-holes case both lattice shown in these interferograms are irregular.

The case of the four holes is much more difficult. The four phasors which sum up to zero, form a quadrilateral. The quadrilateral angles are not defined uniquely by the length of its sides. Thus, we cannot write conditions like in (3). The situation is similar to the four-plane waves interference which form a lattice of optical vortices but of much more complicated structures. The difficulties with such lattices are discussed in [20

20. J. Masajada, “The interferometry based on regular net of optical vortices,” Opt. Appl. to be published in vol.37, (2007).

, 21

21. J. Masajada, “The optical vortex interferometer, theory, technology and applications,” Proc. SPIE 6254, 62540C1–10 (2006).

, 24

24. K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four and five plane waves,” Opt. Express 14, 3039 – 3044 (2006). [CrossRef] [PubMed]

]. As a result the vortex points' distribution can be irregular and their internal geometry may change from one vortex point to another. The internal geometry of the vortex points is, by definition, the geometry of figures formed by the phasors representing contributing waves. Figure 8 shows an example of the lattice produced by the four holes (experiment). The problem gets even more complicated with more than four holes.

Now we are considering the case when the incident wave is inclined. Let its wavevector coordinates be k 2(k2x, k2y, k2z). The far field solution is

U(fx,fy)=Ωexp{iπf′p2z}somb(f′p)(1+exp{2πi(TBxfx+TByfy)}+exp{2πi(TCxfx+TByfy)}).
(10)

Where f2 p = (fx - f 0x)2 + (fy - f 0y)2 , f 0x = k 2x/2π, f 0y = k 2y/2π . We can read from this solution that although nothing changes the whole pattern is shifted in the frequency domain, which means that the vortex lattice moves like a rigid body. The same conclusion can be drawn from formula (8). There we have a very simple method for measuring the small tilt angles of the incident wave. Instead of tracing the shift of the whole spot the vortex lattice shift can be traced. The small shift of vortex points can be determined with higher accuracy than the whole light spot. Moreover, the determination of the tilt axis orientation in a single measurement is possible. Figure 9 shows an example of vortex lattice reaction to the wave tilt (experiment). The wave was tilted by introducing 90arcsec wedge in front of the plate. The wave tilt is half of the wedge angle – 45arcsec in our case. Knowing the distance between the plate and the screen the tilt angle can be determined. In the experiment the vortex lattice shift equal to 193 μm (30 pixels) was measured; the z distance was 890mm. If we recalculate these data to the tilt angle we get the value of 44.8 ±1 arcsec. All this procedures are correct if the incidence angle is small enough (we still work in far or near field approximations).

Fig. 9. The vortex lattice without (circles) and with (pluses) wedge. The plate shown in Fig. 3(a) was used. Due to wave tilt the lattice was shifted as a rigid body (experiment).

3. The case of illumination with the spherical wave

The analysis of this solution is difficult since each somb function is shifted in the frequency domain by a different translation vector. As a result the waves propagate with different amplitude in the same direction (Fig. 11). In this case we do not know what the triangle angles at vortex points will be. We can write the appropriate conditions for both amplitudes and phases but due to trigonometric function there will be no closed solution to it. The same problem arises when we follow the way in which Eq. (8) were derived. Nevertheless, we can still use such a plate to determine the wavefront geometry.

As was shown in [20

20. J. Masajada, “The interferometry based on regular net of optical vortices,” Opt. Appl. to be published in vol.37, (2007).

, 21

21. J. Masajada, “The optical vortex interferometer, theory, technology and applications,” Proc. SPIE 6254, 62540C1–10 (2006).

] the lattice generated by the three waves possesses some useful properties. One is that the geometry of vortex sublattice (i.e. the lattice containing vortices of the same topological charge) does not depend on the interfering waves amplitudes. This property can be used if we assume that the ratios between waves amplitudes in the direction of the neighboring vortex points of the same topological charge differ by a negligible amount. In such a case we can refer to the relations describing the mutual shift between positive and negative vortex sublattice which were derived in Ref. [20

20. J. Masajada, “The interferometry based on regular net of optical vortices,” Opt. Appl. to be published in vol.37, (2007).

]. Using these formulae we can calculate

Fig. 10. The spherical wave incident on the plate; here drawn in cross section. Since the holes are small we can assume that each hole is illuminated by the plane wave tangent to the given spherical wave at the hole center.
Fig. 11. When the plate is illuminated by the spherical waves the somb functions representing the holes are mutually shifted in frequency domain. Here, for simplicity somb function representing wave A and one of the other two waves are plotted. For this reason in a given direction each hole contributes to a different amplitude, which complicates remarkably the conditions for vortex points positions.

the difference in the wave vector direction in the area of holes B, C and A. Moving the plate perpendicularly to the z-axis we can scan larger area of the wavefront. With the use of this method the changes in the wavevector direction the can be measured. This is an important advantage of our method. In the standard interferometry we measure the phase differences which are recomputed to wavefront slope on the basis of some additional assumptions. Figure 12(a) shows the comparison between two vortex lattices. Both were obtained using spherical illumination. After registering the first interferogram the plate was slightly shifted (in the direction perpendicular to the z-axis), so the wavevectors in the center of the holes changed their orientation. As a result the geometry of the positive and negative sublattices was the same but the distance between these two sublattices changed. The results of our calculations are in qualitative agreement with the estimation of the wavefront geometry. However, more precise experiments are necessary to judge the value of this method.

In the classical double-slit experiment the spherical illumination does not make such a difference. The reason is that interference minima do not have to be true zeros. In fact, for spherical illumination the minima usually are not intensity zeros since the propagating amplitudes in the same direction have different values. The interference minima occur in the direction where the phasors representing contributions from both holes are in antiphase. As a result, small amplitude variations change the positions of intensity minima at a smaller rate than in our case. The shift between somb functions shown in Fig. 11 plays a negligible role, therefore we can still focus on the expression in parenthesis [10

10. M. Vasnetsov and K. Staliunas, eds., Optical vortices (Nova Science Publishers, 1999).

]. This is yet another difference between the classical double slit and our experiment.

Figure 13 shows the results of a similar experiment but with the plate with four holes (experiment). In this case the analysis is much more complicated, but the system is definitely more sensitive.

Fig. 12 The vortex lattice shift due to small plate shift in the direction perpendicular to the optical axis. Figure 12(a) shows two vortex lattices (circles-no shift, x-after the test was shifted). Each lattice can be divided into two sublattices which differ in the topological charge (the red signs plus and minus are marked next to the vortices). The plus sublattices overlaps – they have the same geometry. The minus sublattices are mutually shifted. This is in agreement with our conclusions drawn from the theory. Figures 12(b) and 12(c) show the corresponding interferograms (before and after the plate shift).
Fig. 13 The same experiment as in Fig. 13, but with four holes. This setup is much more sensitive than the previous one, but the analysis is much more complicated. The relevant methods are not ready yet.

4. Conclusions and remarks

In this paper we have discussed the creation of the vortex lattice generated by the plate with three or four holes illuminated by the plane or spherical coherent wave. To find the intensity minima the whole two dimensional area of the screen must be analyzed. Nevertheless, in the case of three holes illuminated by a plane wave the position of the intensity minima, which are vortex points, can be easily computed. It is worth noticing that the intensity maxima can also be determined. Since the phasors have the same angle at points of maximum intensity, the set of Eqs. (3) and (7) can be rewritten and solved for such a case. For four and more holes the problem is more difficult. The difficulties are of the same character as in the case of four or more waves OVI identified in Ref. [20

20. J. Masajada, “The interferometry based on regular net of optical vortices,” Opt. Appl. to be published in vol.37, (2007).

]. Also in the case of intensity maxima the system of equations requires some additional information to be solved uniquely.

We have discussed the differences between the classical double-slit experiment and our own. In the two-hole case the minima are dependant on the phase differences between interfering waves. The amplitude ratio is of lesser importance. In the three (or more) hole case the amplitude ratio is an important factor and we have to be more careful when applying standard analytical methods such as far and near field approximations. In particular, the three wave system is sensitive to the differences between the incidence angle of the wavefront in the center of the holes, which is not the case in the two-hole system. On the other hand this fact may be one of the advantages of the more than two-hole cases.

The analyzed system is sensitive to wavelength, which results from Eqs. (5) and (8). For each wavelength the vortex lattice changes its position. The question of applications of the OVI with wavefront division is open. Here we have suggested two of them i.e. wave tilt measurement and the wavefront geometry reconstruction. Certainly more possibilities exist. However, a lot of work must be done to improve technology and theory of the OVI with wavefront division before it becomes a fully competitive device in particular measurements. Specifically more accurate and dedicated localization procedures developed from our methods [16

16. J. Masajada, A. Popiołek Masajada, E. Frączek, and W. Frączek, “Vortex points localization problem in optical vortices interferometr,” Opt., Commun. 234, 23–28 (2004). [CrossRef]

] or methods presented by other authors [26

26. W. Wang, N. Ishii, S. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005). [CrossRef]

, 33

33. W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. Hanson, “Optical vortex metrology for nanometric speckle displacement measurement,” Opt. Express 14, 120–127 (2006). [CrossRef] [PubMed]

] will be necessary.

References and links

1.

T. Young, “Experimental demonstration of the General Law of the Interference of Light,” Philos. Trans. R. Soc. London94, (1804).

2.

R. Welti, “Light transmission through two slits: the Young experiment revisited,” J. Opt. A: Pure Appl. Opt. 8, 606–609 (2006). [CrossRef]

3.

C. Jönsson, “Electron diffraction at multiple slits,” Am. J. Phys. 42, 4–11 (1974). [CrossRef]

4.

A. Zeilinger, R. Gähler, C.G. Shull, W. Treimer, and W. Mampe, “Single and double-slit diffraction of neutrons,” Rev. Mod. Phys. 60, 1067–1073 (1988). [CrossRef]

5.

O. Carnal and J. Mlynek, “Young’s double-slit experiment with atoms: A simple atom interferometer,” Phys. Rev. Lett. 66, 2689–2692 (1991). [CrossRef] [PubMed]

6.

W. Schöllkopf and J. Toennies, “The nondestructive mass selection of small van der Waals clusters,” Science 226, 1345–1348 (1994). [CrossRef]

7.

C. K. Hong and T. G. Noh, “Two-photon double slit interference experiment,” J. Opt. Soc. Am. B 15, 1192–1197 (1997). [CrossRef]

8.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. Roy. Soc. Lond. A 336, 165–189 (1974). [CrossRef]

9.

J. F. Nye, Natural focusing and fine structure of light (IoP, Bristol and Philadelphia, 1999).

10.

M. Vasnetsov and K. Staliunas, eds., Optical vortices (Nova Science Publishers, 1999).

11.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical angular momentum (IoP, Bristol, Philadelphia, 2003). [CrossRef]

12.

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt.39, chapter IV (1999). [CrossRef]

13.

M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” Prog. Opt.42, chapter IV (2001).

14.

J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198, 21–27 (2001). [CrossRef]

15.

J. Masajada, A. Popiołek-Masajada, and D. Wieliczka, “The interferometric system using optical vortices as a phase markers,” Opt. Commun. 207, 85–93 (2002). [CrossRef]

16.

J. Masajada, A. Popiołek Masajada, E. Frączek, and W. Frączek, “Vortex points localization problem in optical vortices interferometr,” Opt., Commun. 234, 23–28 (2004). [CrossRef]

17.

J. Masajada, “Small rotation-angle measurement with optical vortex interferometer,” Opt. Commun. 234, 373–381 (2004). [CrossRef]

18.

A. PopioВek-Masajada, M Borwińska, and W Frączek, “Testing a new method for small-angle rotation measurements with Optical Vortices Interferometer,” Meas. Sci. Technol. 17, 653–658 (2006). [CrossRef]

19.

M. Borwińska, A. Popiołek-Masajada, and B. Dubik, “Reconstruction of the plane wave tilt and its orientation using Optical Vortex Interferometer,” Opt. Eng. (to be published).

20.

J. Masajada, “The interferometry based on regular net of optical vortices,” Opt. Appl. to be published in vol.37, (2007).

21.

J. Masajada, “The optical vortex interferometer, theory, technology and applications,” Proc. SPIE 6254, 62540C1–10 (2006).

22.

P. Kurzynowski, A. WoŴniak, and E. Frączek, “Optical vortices generation using the Wollaston prism,” Appl. Opt. 45, 7898–7903 (2006). [CrossRef] [PubMed]

23.

P. Kurzynowski and M. Borwińska, “Generation of the vortex type markers in a one wave setup,” Appl. Opt. 46, 676–679 (2007). [CrossRef] [PubMed]

24.

K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four and five plane waves,” Opt. Express 14, 3039 – 3044 (2006). [CrossRef] [PubMed]

25.

J. Courtial, R. Zambrini, M. Dennis, and M. Vasnetsov, “Angular momentum of optical vortex arrays”, Opt. Express 14, 938–949 (2006) [CrossRef] [PubMed]

26.

W. Wang, N. Ishii, S. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005). [CrossRef]

27.

J. Primot and L. Sogno, “Achromatic three-wave (or more) lateral shearing interferometer,” J. Opt. Soc. Am. A 12, 2679–2685 (1995). [CrossRef]

28.

J. S. Darlin, P. Senthilkumaran, S. Bhattacharaya, M. P. Kothiyal, and R. S. Sirohi, “Fabrication of an array illuminator using tandem Michelson interferometers,” Opt. Commun. 123, 1–4 (1996). [CrossRef]

29.

N. Guerineau and J. Primot, “Nondiffracting array generation using an N-wave interferometer,” J. Opt. Soc. Am. A 16, 293–298 (1999). [CrossRef]

30.

S. Velghe, J. Primot, N. Guerineau, M. Cohen, and B. Wattellier, “Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers,” Opt. Lett. 30, 245–247 (2005). [CrossRef] [PubMed]

31.

A. S. Patra and A. Khare, “Interferometric array generation,” Opt. Laser Techn. 38, 37–45 (2006). [CrossRef]

32.

W. Singer, M. Totzeck, and H. Gross, Handbook of Optical System, (Wiley-VCH, Berlin, 2005) Vol. 2

33.

W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. Hanson, “Optical vortex metrology for nanometric speckle displacement measurement,” Opt. Express 14, 120–127 (2006). [CrossRef] [PubMed]

OCIS Codes
(070.2580) Fourier optics and signal processing : Paraxial wave optics
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(260.3160) Physical optics : Interference

ToC Category:
Physical Optics

History
Original Manuscript: February 21, 2007
Revised Manuscript: March 19, 2007
Manuscript Accepted: March 28, 2007
Published: April 13, 2007

Citation
J. Masajada, A. Popiolek-Masajada, and M. Leniec, "Creation of vortex lattices by a wavefront division," Opt. Express 15, 5196-5207 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-5196


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References

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  9. J. F. Nye, Natural focusing and fine structure of light (IoP, Bristol and Philadelphia, 1999).
  10. M. Vasnetsov, K. Staliunas, eds., Optical vortices (Nova Science Publishers, 1999).
  11. L. Allen, S. M. Barnett, and M. J. Padgett, Optical angular momentum (IoP, Bristol, Philadelphia, 2003). [CrossRef]
  12. L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Prog. Opt. 39, chapter IV (1999). [CrossRef]
  13. M. S. Soskin and M. V. Vasnetsov, "Singular Optics," Prog. Opt. 42, chapter IV (2001).
  14. J. Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001). [CrossRef]
  15. J. Masajada, A. Popiołek-Masajada, and D. Wieliczka, "The interferometric system using optical vortices as a phase markers," Opt. Commun. 207, 85-93 (2002). [CrossRef]
  16. J. Masajada, A. Popiołek Masajada, E. Frączek and W. Frączek, "Vortex points localization problem in optical vortices interferometr," Opt. Commun. 234, 23-28 (2004). [CrossRef]
  17. J. Masajada, "Small rotation-angle measurement with optical vortex interferometer," Opt. Commun. 234, 373-381 (2004). [CrossRef]
  18. A. Popiołek-Masajada, M Borwińska and W Frączek, "Testing a new method for small-angle rotation measurements with Optical Vortices Interferometer," Meas. Sci. Technol. 17, 653-658 (2006). [CrossRef]
  19. M. Borwińska, A. Popiołek-Masajada and B. Dubik, "Reconstruction of the plane wave tilt and its orientation using Optical Vortex Interferometer," Opt. Eng. (to be published).
  20. J. Masajada, "The interferometry based on regular net of optical vortices," Opt. Appl. to be published in vol. 37, (2007).
  21. J. Masajada, "The optical vortex interferometer, theory, technology, and applications," Proc. SPIE 6254, 62540C1-10 (2006).
  22. P. Kurzynowski, A. Woźniak and E. Frączek, "Optical vortices generation using the Wollaston prism," Appl. Opt. 45, 7898-7903 (2006). [CrossRef] [PubMed]
  23. P. Kurzynowski and M. Borwińska, "Generation of the vortex type markers in a one wave setup," Appl. Opt. 46, 676-679 (2007). [CrossRef] [PubMed]
  24. K. O’Holleran, M. J. Padgett, and M. R. Dennis, "Topology of optical vortex lines formed by the interference of three, four and five plane waves," Opt. Express 14, 3039 - 3044 (2006). [CrossRef] [PubMed]
  25. J. Courtial, R. Zambrini, M. Dennis, and M. Vasnetsov, "Angular momentum of optical vortex arrays", Opt. Express 14, 938-949 (2006) [CrossRef] [PubMed]
  26. W. Wang, N. Ishii, S. Hanson, Y. Miyamoto, and M. Takeda, "Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement," Opt. Commun. 248, 59-68 (2005). [CrossRef]
  27. J. Primot and L. Sogno, "Achromatic three-wave (or more) lateral shearing interferometer," J. Opt. Soc. Am. A 12, 2679-2685 (1995). [CrossRef]
  28. J. S. Darlin, P. Senthilkumaran, S. Bhattacharaya, M. P. Kothiyal and R. S. Sirohi, "Fabrication of an array illuminator using tandem Michelson interferometers," Opt. Commun. 123, 1-4 (1996). [CrossRef]
  29. N. Guerineau and J. Primot, "Nondiffracting array generation using an N-wave interferometer," J. Opt. Soc. Am. A 16, 293-298 (1999). [CrossRef]
  30. S. Velghe, J. Primot, N. Guerineau, M. Cohen and B. Wattellier, "Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers," Opt. Lett. 30, 245-247 (2005). [CrossRef] [PubMed]
  31. A. S. Patra and A. Khare, "Interferometric array generation," Opt. Laser Techn. 38, 37-45 (2006). [CrossRef]
  32. W. Singer, M. Totzeck and H. Gross, Handbook of Optical System, (Wiley-VCH, Berlin, 2005) Vol. 2
  33. W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda and S. Hanson, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express 14, 120-127 (2006). [CrossRef] [PubMed]

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