## 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] 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] or in wider context (as our paper does), or for different objects like electrons [3], neutrons [4], atoms [5], molecules [6] or light but in strange states like entangled photons (see for example [7]). 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]. Since that time hundreds of papers as well as few books have been published in this field [9, 10, 11]. The general papers in Progress in Optics have also been published [12, 13]. 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.

In papers [14, 15] 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]. Some basic measurements with the OVI were reported in Refs. [17, 18, 19]. The extended theory of the OVI was presented in [20] and partially in [21]. The polarization versions of the OVI were built using a Wollastone prism and were presented in Refs. [22 and 23]. 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]. The topology of vortex lines generated by three-, four-and five-plane waves’ interference was discussed in paper [24]. 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]. These relations are necessary when computing the values of physical parameters which are measured with the OVI.

In our experiment the interfering waves are more like spherical waves than plane waves. The vortex lines are not parallel (as in the case of three interfering plane waves), but they form a bundle of diverging vortex lines (see formulas (8) in the text). The geometry of vortex lattice is slightly different and well described by our formulae. In this paper we analyze such a vortex lattice in the plane cross section, which is in accordance with our previous papers and with the character of the classical Young’s experiment.

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].

## 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]. 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:

$$\phantom{\rule{.2em}{0ex}}\left[1+\mathrm{exp}\left\{2\mathrm{\pi i}\left({T}_{\mathrm{Bx}}{f}_{x}+{T}_{{B}_{y}}{f}_{y}\right)\right\}+\phantom{\rule{.2em}{0ex}}\mathrm{exp}\left\{2\pi \phantom{\rule{.2em}{0ex}}i({T}_{\mathrm{Cx}}{f}_{x}+{T}_{\mathrm{By}}{f}_{y})\right\}\right]\phantom{\rule{.2em}{0ex}}.$$

*f*
^{2}
_{p} = *f*
^{2}
_{x} + *f*
^{2}
_{y}, *f _{x}*,

*f*are spatial frequencies,

_{y}**T**

_{B}(

*T*

_{Bx},

*T*

_{By}),

**T**

_{C}(

*T*

_{Cx},

*T*

_{Cy}) are translation vectors indicating the position of B and C holes, respectively:

The function somb is defined as follows:

Here J_{1} is a Bessel function of the first kind and first order and
*R* is the radius of the hole.

Now we are looking for points for which the above expression equals zero. The whole
analysis is performed in a frequency domain. Having a single circular hole A we get
the single somb function as a solution to the far field diffraction integral. By
adding the next hole B shifted by the translation vector **T**
_{B}
we get a sum of the same somb functions. Thus one of them must be multiplied by the
phase term
exp{2*πi*(*T _{Bx}*

*f*+

_{x}*T*

_{By}*f*)} (shift theorem for Fourier transforms). The phase differences between three contributing waves are described by the sum in square parenthesis in Eq. (1). Since we are interested in phase differences between interfering waves we can assume that wave A is our reference wave. This interference field can be represented with the use of phasors. The length of the phasor represents wave amplitude and its angle represents wave phase. Towards zero amplitude the three phasors must form a triangle (must sum up to zero). Since the amplitudes of the three interfering waves are the same at each propagation direction, the resulting triangles must be equilateral. The phases of waves A B, C in vortex points are:

_{y}or

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

or

The solutions to the above two sets of equations are:

or

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].

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.

Figure 5 shows the position of the vortex points for the same plate calculated in far and near field approximations. The difference between both far and near field approximations is more obvious than in the case of the double-slit experiment. This problem will be discussed in the next section. If more accurate results are necessary we have to find the vortex points position using near field approximation. For this purpose we refer the reader to the paraxial approximation, which is well known in the geometrical optics. This is coherent with the near field approximation.

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

Here *q*∊(A, B, C), *x _{q}*,

*y*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.

_{q}The condition for the vortex points position in the observation plane located at
distance *z* from the plate is:

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

where

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.

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, 21, 24]. 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}(*k _{2x}*,

*k*,

_{2y}*k*). The far field solution is

_{2z}Where *f*′^{2}
_{p} = (*f _{x}* -

*f*

_{0x})

^{2}+ (

*f*-

_{y}*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).

## 3. The case of illumination with the spherical wave

In this section we are discussing the illumination of our plate with holes with the spherical wave. Figure 10 shows the way we want to start the analysis of this problem. If the holes are small we can assume that each hole is illuminated by the plane wave of different angle of incidence. Therefore we assume that the incident wavefront can be locally approximated by the tangent plane. In such a case a far field solution to our problem cannot be factorized into two components–one with somb function and the other representing the additional phase shift as it was possible in previous example [Eq. (10)].

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, 21] 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]. Using these formulae we can calculate

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]. 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.

## 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]. 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] or methods presented by other authors [26, 33] will be necessary.

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