Multi-beams engineered to increase patterns of vortex lattices by employing zero lines of the coherent non-diffracting field

Kunpeng Liu; Zhigang Liu; Xiaochun Dong

doi:10.1364/OE.27.021652

1. Introduction

An OV is a screw dislocation that occurs in a wavefront phase distribution [1–4]. The screw dislocation wavefront shows itself as a system of l nested helicoids, where l is the TC, or the order of the dislocation [5,6]. Optical vortices (OVs) have unique properties [7,8] to the left-handed or right-handed of its helical wavefront and, specifically, to their ability carrying orbital angular momentum (OAM) [9], which make them extensively used in applications ranging from optical communications [10,11] to optical manipulation [12–14]. Comparing to the isolated OV, OV arrays or OV lattices present some distinct properties and advantages, and in hence, they have broader utilization. As an example, an isolated OV can capture and observe a single particle in particle manipulation. However, the OV lattice consisting of multiple isolated vortex that numerous particles can be manipulated simultaneously, improving the efficiency [15]. Beyond that, the OV lattice has also found interesting applications in the microlithography [16] and microoptomechanical pumps assemble and actuate [17].

The regular array of OVs with first order generated by three plane waves interference which obtained by using six cubic beam splitters, was originally put forward by Masajada et al. [18], in 2001. After almost two decades of research and development, there are abounding approaches to generate the OV lattice, such as holograms [17], helical phase spatial filtering [19,20], the transformation of the Laguerre–Gaussian mode [20], spatial light modulator (SLM) [21–23], liquid crystals [24] and arrangement pinhole plates [25]. But relatively speaking, using SLM is a convenient and flexible way to get the OV lattice just by setting appropriate parameters of the device. In the intervening time, quite a few scholars focus on the theoretical research and analysis about how to create higher order or other patterns OV lattices, most of which are on the basis of multi-waves interference. For example, Kagome pattern is a classic OV lattice with TC 2 that has been proposed [26]. In all these methods, the distribution of interfering waves is relatively fixed for the engineered beams are lying on the surface of a cone. Well then, could we encode other distributions or combination modes of multi-beams to achieve other patterns of OV lattice when the engineered waves interference occurs in free space? Furthermore, how to acquire an intuitive encoding method is worthy of an intensive study.

Mitsui et al. [27] presented a geometric approach for solving algebraic equations employing the map of zero lines of the real and imaginary parts. Naturally, zero lines can also be applied in the optical analysis as well. Freund et al. [8] analyzed and gave the sign principle of the wave-field phase singularities by employing zero crossing diagram. Similarly, other sorts of optical fields also have been studied with the aid of zero lines [28,29]. Nonetheless, the above studies are all aim to explore the relationship between the research objective and the distribution of zero lines. In this letter, the relationship between the TC and zero lines of the real and imaginary parts is explored at first, in the isolate OV condition. Then combined with zero lines diagrams of the multi-waves coherent non-diffracting field, we stack two groups of zero lines of two non-diffracting fields to satisfy the zero lines distribution of an OV lattice. That being said, the two groups of multi-waves propagation vectors and initial phases are encoded actively. Thereafter, the expected OV lattice can be generated by interference. Taking the design process of generating two different three order OV arrays as an example, the theoretical analysis is given and the verification experiment is carried out in the end.

2. Design principle and relative analysis

For a vortex beam traveling along the z direction, the complex amplitude of the field in cylindrical coordinates is

E (r, τ, z) = u (r, z) \exp (i l τ) \exp (- i k z),

where u(r, z) is the real amplitude of the vortex optical field, and exp(ilτ) is the phase factor, kz is the phase change causing by the propagation along axis-z, and τ and k are the polar angle and the wavenumber, respectively. The quasi monochromatic optical field in the transverse plane can be written by means of the real amplitude u(x, y) and the phase ϕ in the form

E (x, y) = u (x, y) \exp [i ϕ (x, y)] .

Further, Eq. (2) can be manipulated in the real and imaginary parts combination form: E (x, y) = Re(x,y) + iIm(x,y). The polar angle τ can be expressed as tan⁻¹(y/x) in Cartesian coordinate system. Therefore, the complex amplitude of the vortex beam in the x-y plane can be rewritten as

E (x, y) = u (x, y) \exp [i l \tan^{- 1} (y / x)] .

Meanwhile, the vortex manifests as a point on the transverse plane where the phase dislocation and zero intensity appear [3,7]. The position can be determined by the set of equations

{\begin{matrix} Re [u (x, y)] = u (x, y) \cos [l \tan^{- 1} (y / x)] = 0 \\ Im [u (x, y)] = u (x, y) \sin [l \tan^{- 1} (y / x)] = 0 \end{matrix},

where Re[u(x,y)] and Im[u(x,y)] are operations of the real and imaginary parts, respectively. If the Eq. (4) is solved, there will appear two kinds of isolines which are boundaries between the positive and the negative values of the real and imaginary parts, portraying the zero lines distribution and intersect at one point. Moreover, the Eq. (5) will be performed, along a closed path surrounding the intersection.

\oint_{L} \nabla ϕ \cdot d L = \pm 2 l π .

The value of variable l represents the TC of an OV and is equal to the number of zero lines of the real (or imaginary) part. The sign of Eq. (5) distinguishes between a clockwise vortex (-) and a counter-clockwise vortex ( + ). To sum up, the topological state of the OV depends on the number of the real and imaginary parts zero lines which intersect at one point, and the number of zero lines of the real part equals that of the imaginary part, as depicted in Fig. 1.

Fig. 1 Visualization the distribution of the real part zero lines (red dashed lines) and the imaginary part zero lines (green dashed lines) of the isolate OV on the transverse plane. The OV with TC l has l zero lines of the real part and l zero lines of the imaginary part: (a) l = 1, (b) l = 2, (c) l = 3. Zero lines intersect at one point where the phase singularity exist. Insets depict the corresponding phase map.

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The coherent non-diffracting field that can be looked upon as the interference field produced by the interference of plane waves whose propagation vectors distribute on the conical surface and form an angle of θ with the central axis-k_z. Also, the arrowheads of wave vectors are located on a circle and the lines of adjacent tips construct a N-regular polygon on the transverse Fourier plane, as can be seen in Fig. 2. The complex amplitude of the non-diffracting field can be expressed in Cartesian coordinates as

E_{n, m} = A \sum_{j = 1}^{n} \exp [i k_{0} (x \sin θ \cos φ_{j} + y \sin θ \sin φ_{j} + z \cos θ) + i m φ_{j}],

where k₀ = 2π/λ₀ is the wave number with wavelength λ₀, φ_j = 2jπ/n is the angle between the transverse component and the k_x-axis, Φ_j = mφ_j is an absolute phase assigned to the j-th plane wave.

Fig. 2 Sketch of the wave vectors distribution of the interfering plane waves. (a) A set of six plane waves on a cone. (b) Two sets of six plane waves on two cones with different opening angles. The arrowheads of those vectors locate on a sphere with a radius of wave number k₀.

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Consequently, there is a constant phase increment 2mπ/n between two neighboring wave components, and all plane waves are encoded by the total phase shift 2mπ. Because of the phase of 2π periodicity, the parameter m in the interval 0 ≤ m ≤ n/2 (m∈N) can describe all patterns of non-diffracting field when the number of beams n is fixed. In addition, the negative m values situations are just equal to that of the positive ones. Hence, the real and imaginary parts of Eq. (6) are given as

Re = A \sum_{j = 1}^{n} \cos [k_{0} (x \sin θ \cos φ_{j} + y \sin θ \sin φ_{j} + z \cos θ) + m φ_{j})],

Im = A \sum_{j = 1}^{n} \sin [k_{0} (x \sin θ \cos φ_{j} + y \sin θ \sin φ_{j} + z \cos θ) + m φ_{j})],

and the phase information could be obtained from ϕ = tan⁻¹ [Im (x, y)/Re (x, y)].

If the real part and imaginary parts zero lines, both arrange periodically, as well as the points created by the intersection of the lines have periodic property correspondingly, as shown in Figs. 3(j) and 3(k). Therefore, the optical field is a vortex array. In accordance with Eq. (6), four completely different kinds of non-diffracting fields created by six plane waves interference are depicted in Fig. 3(first row), which their unit cells are sixfold symmetry. The azimuthal phase related parameter m is chosen as the integer numbers between 0 and 3. In addition, the phase diagrams of the four sixfold field intensities are demonstrated in the middle row. The last row illustrates the distribution of the real part zero lines (black lines) and the imaginary part zero lines (blue lines) of the corresponding field in the same column. Insets in the bottom right corner show the corresponding local amplification of images.

Fig. 3 The discrete beams interference of E_6, _m and m = 0, 1, 2, 3 from left to right. The first row: numerical simulation of intensity in x-y plane. The middle row: numerical simulation of the phase map. The last row: zero lines distribution of the real (blue lines) and imaginary parts in the bottom right corner (black lines). Inserts in the bottom right corner of each simulation display the magnification of the area (red circle) in the host graph. Red points are OVs.

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There will create two classical OV lattices-Honeycomb pattern and Kagome pattern, when m is 1 and 2 respectively. Zero lines of the real part and imaginary parts are distributed periodically and the number of zero lines of the real part at the intersection (red points in the magnifications) is equal to that of the imaginary part. Furthermore, the number also equals the TC. From the insert, we can see that the OVs distribute at six corners and the center of the regular hexagon (red points). It can be simply verified by Mach-Zehnder interference that the central bifurcation number of the fork pattern is equal to the TC (the insert of the first row is a partial zoom-in of the interference pattern). It also can be seen whether the light field exists OVs and the properties of them, through the phase distribution diagram in the second row. Nevertheless, the resultant field does not contain any OV when the parameter m = 0 or 3. Zero lines of the real part coincide with zero lines of the imaginary part, that means the OV cannot be established in this situation. Figs. 3(i) and 3(l) only show the distribution of the real part zero lines, covering the imaginary part zero lines, and the inset presents the corresponding zero lines of the imaginary part distribution in the red circle range.

But it is worth noting that the intersections of zero lines are also arrayed points in the case of m = 3 [see Fig. 3(k)]. However, according to the above-mentioned theory, only zero lines of the real and imaginary parts do not overlap but cut, an OV can be detected at the position of the intersection, that means those array points are not OVs. Then, we attempt to stack the two different interference fields with this situation that zero lines of the real and imaginary parts are both overlapping. It is equivalent to assembling two sets of zero lines with distinct distributions, one of them can be regarded as zero lines of the real part and the other can be considered as zero lines of the imaginary part. Thus, the two sets of zero lines with dissimilar placement states certainly will cross over and the intersection will meet the requirement to be an OV. The two different fields are obtained by the interference of two groups of multi-waves whose propagation vectors distribute on two conical surfaces like Fig. 2(b).

The first chosen group is called as original group Eog n,m and the other is named as introduced group Eig n,m. If wave vectors of the introduced group all rotate δ degree clockwise about the axis-k_z, it will lead to zero lines of the interference field rotate by clockwise around the origin of the x-y plane correspondingly. The superposition of two groups of multi-planar spatial wavefunctions can be defined as

\begin{array}{l} Ψ_{s u m} = E_{n, m}^{o g} + E_{n, m}^{i g} \\ = A \sum_{j = 1}^{n} \exp [i k_{0} (x \sin θ_{o g} \cos φ_{j} + y \sin θ_{o g} \sin φ_{j} + z \cos θ_{o g}) + i m φ_{j}] \\ + A \sum_{j = 1}^{n} \exp [i k_{0} (x \sin θ_{i g} \cos φ_{j}^{*} + y \sin θ_{i g} \sin φ_{j}^{*} + z \cos θ_{i g}) + i m φ_{j}^{*}], \end{array}

where φ* j = 2jπ/n + δ. θ_og and θ_ig are the angle of the original and introduced group respectively.

We assemble two sets of zero lines of two arbitrary E_6,3 but with distinct distributions, and only one intersection of two sets of three zero lines will occur at the origin of transverse plane, as shown in the central region of Fig. 4(a), which can be regarded as three zero lines of the real part crossing with three zero lines of the imaginary part. Thereupon, an OV with third order TC can be detected at that point. Nevertheless, this special intersection does not exist in other locations on the transverse plane, that is to say, the third order OV cannot be formed except at the origin. In other places, a zero line of the real part and a zero line of the imaginary part will cross and generate an OV which the TC is first order [red circle in Figs. 4 (a) and 4(c)]. In Fig. 4(c), there are many homomorphism intersections, which are one-to-one correspondence with the intersections in Fig. 4(a). Despite the fact that the real and imaginary parts of the twelve-plane waves interference field do not equal to the simple addition of the two sets of six-waves interference fields, still, it is intuitive to guide us to design the engineered waves configuration to generate OV lattices.

Fig. 4 (a) Zero lines of the imaginary part plot of the original group waves interference field (blue lines) and the introduced group waves (δ = π/6) interference field (black lines). (b) The transverse intensity map of the field that formed by the two groups of waves interference directly. (c) The real (green) and imaginary (black) parts zero lines crossing diagram of the wave field (b). (d) Phase pattern of (b). TC of the OV at the center is 3. The vortex with TC 1 is marked by a red circle and insert is the magnification of the circular domain.

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In order to make the two sets of three zero lines intersect in other places, but not just the origin of the x-y plane, which means, to construct an OV lattice field with TC 3. Since the distance d between two adjacent zero lines intersections is related to the parameter of θ, the angle between the wave vector and the axis-k_z, as shown in Fig. 2(a), it needs us to predesign the direction of introduced group wave vectors ingeniously, to cater for the demand of the two sets of zero lines intersection points coincide partially.

Before that, the functional relationship between the angle θ and distance d should first be made. Applying Eq. (6) and Eq. (7), we obtain the expression of the real part of E_6,3

\begin{array}{l} Re (x, y) = \sum_{j = 1}^{6} \cos [k_{0} (x \sin θ \cos φ_{j} + y \sin θ \sin φ_{j}) + ς) + m φ_{j}] \\ = \sum_{j = 1}^{6} \cos [κ (x \cos φ_{j} + y \sin φ_{j}) + ς + m φ_{j}], \end{array}

where, κ = k₀sinθ, ς = z₀cosθ, φ₁ = 0, φ₂ = π/3, φ₃ = 2π/3, φ₄ = π, φ₅ = 4π/3, φ₆ = 5π/3. Eq. (10) can be rewritten by employing trigonometric function formula

\begin{array}{l} Re (x, y) = \cos [- κ x + ς] - \cos [κ x + ς] + \cos [- κ (\frac{1}{2} x + \frac{\sqrt{3}}{2} y) - ς] \\ - \cos [κ (\frac{1}{2} x + \frac{\sqrt{3}}{2} y) - ς] + \cos [- κ (\frac{1}{2} x - \frac{\sqrt{3}}{2} y) - ς] - \cos [κ (\frac{1}{2} x - \frac{\sqrt{3}}{2} y) - ς] . \end{array}

In the same way, the imaginary part of E_6,3 is given by

\begin{array}{l} Im (x, y) = \sin [- κ x + ς] - \sin [κ x + ς] + \sin [κ (\frac{1}{2} x + \frac{\sqrt{3}}{2} y) + ς] \\ - \sin [κ (\frac{- 1}{2} x - \frac{\sqrt{3}}{2} y) + ς] + \sin [κ (\frac{1}{2} x - \frac{\sqrt{3}}{2} y) + ς] - \sin [κ (\frac{- 1}{2} x + \frac{\sqrt{3}}{2} y) + ς] . \end{array}

The zero-value coordinate (x, y) of the real part and the imaginary part both can be determined from the following equations:

{\begin{array}{l} - κ x + ς = κ x + ς + 2 p π \\ κ (\frac{1}{2} x + \frac{\sqrt{3}}{2} y) + ς = - κ (\frac{1}{2} x + \frac{\sqrt{3}}{2} y) + ς + 2 p π \\ κ (\frac{1}{2} x - \frac{\sqrt{3}}{2} y) + ς = - κ (\frac{1}{2} x - \frac{\sqrt{3}}{2} y) + ς + 2 p π \end{array}, p \in ℤ .

That is, zero points of the real part coincide with zero lines of the imaginary part and the set of equations has the solution

{\begin{cases} x = p π / κ \\ x + \sqrt{3} y = 2 p π / κ, p \in ℤ \\ x - \sqrt{3} y = 2 p π / κ \end{cases} .

Consequently, the zero contour lines of the real (or imaginary) part is composed by a linear cluster perpendicular and two linear clusters with slopes of ± 1/3^1/2, respectively. Besides, the distances between the adjacent lines of the three types of linear clusters all are the same. Then, the distance d between the adjacent intersections of zero contour lines is

d = \frac{2 π}{\sqrt{3} κ} \propto \frac{1}{\sin θ} .

Fig. 5(a) depicts a kind of two sets of zero lines arrangement that meet the requirement: blue lines delineate the real part zeros of the field generated by the original group waves interference, and the other cluster zero lines of the real part is black, belonging to the introduced group interference field. We consider those two groups of waves as an integrated whole and tab it as the combination mode 1. Now, we know that zero lines of the real part of the two sets of interference fields coincide with that of the real part. As before, we regard the two different sets of zero lines as one set of zero lines of the real part and one set of zero lines of the imaginary part respectively. Hence, OVs can be observed at the points where the two sets of zero lines intersect.

Fig. 5 (a) Zero lines map of combination mode 1 that consisted by two sets of wave groups. (blue lines: Re(Eog 6,3) = Im(Eog 6,3) = 0, black lines: Re(Eig 6,3) = Im(Eig 6,3) = 0). The distance between adjacent zeros are d₁ and d₂ respectively. (b) The intensity field generated by the interference of whole waves of combination mode 1. (c) Zero crossing map of (b) (the real part: green, the imaginary part: black).(d) The phase distribution of (b). (e) Zero crossing map of combination mode 2. (blue lines: Re(Eog 6,3) = Im(Eog 6,3) = 0, black lines: Re(Eig 6,3′) = Im(Eig 6,3′) = 0). The distances between adjacent zeros are d₁ and d₃ respectively. (f) The intensity field generated by the interference of whole waves of combination mode 2. (g) Zero crossing map of (f) (the real part: green, the imaginary part: black). (h) The phase distribution of (f).

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On the basis of Fig. 5(a), joining the adjacent intersections of two groups zero lines can construct an equilateral triangle network. The global rotation angle δ = π/6 of the introduced group wave vectors are available to obtain, and the ratio d₁:d₂ between the two sets of zero distances is 3^1/2:2, which can be got by the triangular relationship (OM = d₁, OA = d₂). The relationship between the opening angles of two cones and zero spacings can be derived through Eq. (15)

\sin θ_{1} : \sin θ_{2} = d_{2} : d_{1} .

If the angle θ₁ which between the original group wave vectors and the axis-k_z is predetermined. After that, the angle θ₂ of the introduced group wave vectors can be calculated via the Eq. (16). Fig. 5(b) presents the intensity distribution of the resultant field that generated by the two groups of waves interference directly. Fig. 5(c) is the real (green) and imaginary (black) parts zero crossing plot. The cross state of zero lines of the red circle region in Figs. 5(a) and 5(c) both perform in the similar way. It can be seen that the points formed by the three real and imaginary part zero lines intersection show up as an array, unlike Fig. 4. Hence, the light field is a third order OV lattice, whose phase profile is shown in Fig. 5(d).

The combination mode 2, another kind of zero lines crossing state, also compliance with the requirements is shown in Fig. 5(e). The original group of mode 2 is the same as that in mode 1. The global rotation angle of the introduced group waves also is π/6, and the ratio d₁: d₃ (OM = d₁, OA = 2d₃) of distances between zero points belonging to the two fields changes to 3^1/2:1. The corresponding angles between the wave vectors and the axis-k_z can also be achieved on the basis of Eq. (16). Fig. 5(f) is the whole groups’ waves interference field. Fig. 5(g) shows the zero lines distributions of the real (green) and imaginary (black) parts. The cross state of zero lines of the red circle region in Figs. 5(e) and 5(g) are similar. The third order OV lattice also can be observed. The phase distribution is shown in Fig. 5(h). In theory, we can develop various kinds of combination modes and obtain different OV lattices with the same order but different patterns.

Fig. 6 profiles the singularity distribution of cells which intercept from Fig. 5. The parameter ρ = N_OV/C_s is used to describe the cell density of OV lattice by analogy to the definition of cell density in Crystallography, where C_s is the cell size and N_OV is the OV number in the cell. The cell center of the mode 1 is an OV with TC −3. It also contains six OVs with TC −1 and six OVs with TC + 1. The OV in the cell center of mode 2 is same with the mode 1 and it surrounded with six OVs with TC + 1. Note that the six OVs with TC + 1 are shared equally with the six adjacent cells. For mode 1 and mode 2, the equivalent number of OVs with TC + 1 is three. Therefore, the sum of the TCs of the cells both are zero, which means the total TC is conserved. Basing on the Eq. (15) and Fig. 5, the cell sizes of the two modes can be computed as follows: C_s₁ = 8π²/[3^1/2 (k₀sinθ*)²], C_s₂ = 2π²/[3^1/2 (k₀sinθ*)²], θ* = max{θ_og, θ_ig}. Consequently, the singularity density of the two modes can be obtained and then the number of singularities in the field can be estimated when the transverse section area is determined.

Fig. 6 (a) Fringe pattern of a cell of the mode 1. (b) Phase map of (a). (c) Fringe pattern of a cell of the mode 2. (d) Phase map of (c). The circles mark the position of the singularities. The sign of TC can be confirmed by observing the fringe bifurcation direction.

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The tips of wave vectors of the combination mode lie on the different transverse Fourier plane, similar to those shown in Fig. 2(b). The resultant fields are periodicity but not non-diffracting, along the propagating direction, for the distinctness of k_z component between the two groups of waves. The transmission period is given by

T = | \frac{2 π}{k_{0} (\cos θ_{o g} - \cos θ_{i g})} | .

It can be seen from Fig. 7(a) that the interference region is a shuttle shaped volume with the longitudinal length L ≈2r/sinθ* in real space, where r is the beam radius. The range depends on the cross-sectional area of the beams and the parameter θ*. If the r is 5mm and the θ* is 0.125°, which the assumed parameters are close to the actual situation. Taking for example of the mode 1, the length that the lattice could be sustained can reach about 4.58m in theory. The range is in the order of meters and it is necessary to explore the propagation properties next.

Fig. 7 (a) The shuttle shape interference region. (b) First row: transverse intensity maps of a cell; middle row: intensity maps of a cell interference with a reference plane wave (fringe pattern); last row: the phase distribution of the cell corresponding to the same column.

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To present propagation characteristics of the generated OV lattices, we take an example of combined mode 1. The maximum cross section [z = z₀, Fig. 7(a)] is set to be the datum, the transverse intensity map and the corresponding phase distribution of the cell of the OV lattice are depicted in Fig. 7(b), with step length T/4 along the z direction. Propagating along the z-axis, the phase rotates gradually, which reveals the OV moving forward in a spiral. The second row of Fig. 7(b) shows the fork directions are opposite when z = 0 and z = T/2, which means the sign of the TC is varied regularly in a period. Moreover, the vortex will disappear when z = (2p-1)T/4 (p∈Z), since the fringe pattern does not bifurcate. Therefore, all these characteristics show the generated OV lattices are different from the existing ones and may result in wider application.

Through the systemic research in the case of n = 6, m = 3, we can design more zero lines crossing ways and get the corresponding combination modes, to generate third order OV lattice with different structures. Furthermore, we attempt to engender higher order OV lattice fields. Based on the theory above, zero lines of the real and imaginary parts of the original group wave interference field should coincide, and the intersections that generated by the zeros crossing are in the form of arrays. Unfortunately, except for the interference field in the situations of n = 4, m = 2 and n = 6, m = 3, other multi-beam interference fields cannot accord with the demands.

Analogously, we present a combination mode for generating second order OV lattice. The superposition of two groups of multi-planar spatial wavefunctions is Eog 4,2 + Eig 4,2. The intensity and phase distribution of the field obtained by the interference of two groups of multi-waves, are shown in Fig. 8. From Fig. 8(c), the ratio d₁:d₂ is 1:2^1/2. Furthermore, the opening angles belonging to the two groups of multi-waves can be obtained, through a process like the situation of Eog 6,3 + Eig 6,3. Furthermore, in the experimental section, Fig. 10(b) shows a new different pattern of second-order OV lattice, which is form by anther combined mode (d₁:d₂ = 2^1/2:3/2). Fig. 9(c) is the corresponding phase map of it.

Fig. 8 (a) Simulated intensity modulations of E4,2. (b) Phase profile of (a). (c) Zero lines map of combination mode that consisted by two sets of wave groups interference fields. (blue lines: Re(Eog 4,2) = Im(Eog 4,2) = 0, black lines: Re(Eig 4,2) = Im(Eig 4,2) = 0). Zero lines of the real and imaginary parts are both coincident in each field. (d) The intensity field generated by the interference of whole waves of combination mode. (e) Phase profile of (d). (f) Zero lines (the real part: green, the imaginary part: black) crossing map of (d). The cross state of zero lines of the red circle region in (c) and (f) both perform in the similar way.

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Fig. 9 (a) Experimental configuration for induction of compound OV lattice. SF: Spatial filter; BE: Beam expander; BS: Beam splitter; M: Mirror; S: shutter; LP: Linear polarizer; SLM: Spatial light modulator; FF: Fourier filter; NA: Neutral attenuator; TS: Translation stage. (b)-(e) Phase maps that will be loaded on the SLM in turn.

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3. Experiment implementation

The experimental configuration is illustrated in Fig. 9 to demonstrate the presented compound OV lattices. He-Ne laser with central wavelength at 632.8 nm is spatially filtered and then collimated by a beam expander. The beam will be divided into two paths which are used as experimental and reference beam respectively, by passing through the Beam splitter. The experimental beam with the appropriate polarization state is directed onto the reflective surface of the SLM (Upolabs-HDSLM80R, China, Resolution 1920 × 1200, Pixel size 8.0µm × 8.0µm) and modulated basing on the loaded phase mask. To ensure leave only the first diffraction order, the 4F spatial filtering system is set up by employing two lenses (f = 300mm) and a particular intensity filter. A CCD camera (AVT-Prosilica GT, Germany, Resolution 5120 × 5120, Pixel size 4.5µm × 4.5µm) is to capture the generated intensity profile of the OV lattice. A reference plane wave is introduced to additionally detect the phase structure of the resultant field when the shutter is unblocking.

For the SLM in our experiment, only the cell phase map is loaded on 30 × 30 pixels at least, can the incident light be modulated to the expected field effectively. Correspondingly, if use the camera to record the resultant field, it will take up about 54 × 54 pixels. To sum up, the resolution and phase modulation capability of the SLM are the main factor limiting OV lattice generation. Different brands of SLMs have different phase modulation capabilities, but the issue is not covered here. The minimal cell area C_min is 5.76 × 10⁴µm². We have made statistics of singularities with different TCs in a cell. Combining with the resolution of the SLM, the maximum number of OVs can be estimated. Moreover, for recording the modulated field sufficiently, the working area of the CCD should be larger than the LCD screen of the SLM, which our devices meet this condition.

In order to make comparisons between the experimental results and the simulation effectively, a cell of OV lattice is taken as the unit to determine appropriate design parameters. The formula for calculating the cell area has been given. The cell area is related to the parameter θ. For the smallest cell area C_min has been determined in the previous part, the range of θ can be derived by using inequality C_min < C_s. Hence, the selected θ should be less than 0.162° to facilitate the observation with the naked eye in the experiment.

Firstly, we concentrate on the analysis of the four new OV lattice patterns in transverse plane. The top row of the Fig. 10 illustrates the experimentally recorded images of the intensity profiles of OV lattices. The inset in the lower right corner is the corresponding simulation of each OV array. When a reference plane wave is superimposed on the resultant field, interferogram will be generated, as shown in the last row of Fig. 10. OV exists at the bifurcation point of the interferogram and the bifurcation fringe number of the fork pattern is equal to the TC. Therefore, the topological properties of the resultant field can be detected by analyzing the bifurcation states of the interferogram. We can see that the TC of Figs. 10(a) and 10(b) are 2 and the TC of Figs. 10(c) and 10(d) are 3. OV lattices of second and third orders different from the previous patterns verify the feasibility of the method.

Fig. 10 Experimental records of transverse intensity profile (top row) and interferogram (last row). (a) and (b) OV lattice with TC 2; (c) and (d) OV lattice with TC 3; Insets show the according simulation. The sequence from left to right corresponds to the experimental results when the SLM is loaded phase maps of Figs. 9 (b)-9(e) respectively.

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What is more, the pattern like Fig. 10(c) with the lattice period of lp = 670μm and the transmission period of T = 1063.5mm is chosen to be studied, to verify the transmission property of the OV lattice generated by the two groups waves interference (θ_og = 0.1250°, θ_ig = 0.1083°). The camera is installed on the slider of a translation stage, moving in the direction of the OV lattices propagation (see Fig. 9). Move the slider back or forth referring to the standard position (z = 0) where the OV array is captured by the camera. Fig. 11(a) shows the OV lattice transmission performance in the interval of −30mm to + 30mm. there are some disturbances in the intensity distribution along the propagation, but it still can be regarded as quasi non-diffraction in the small neighborhood. The sign of TC remains constant during the process of moving the camera to the position of z = 265mm where close to T/4, but it will be opposite on the domain [267, 531]. Through the experimental results and comparative analysis, both the intensity distribution and transmission characteristics tally closely with the simulation patterns.

Fig. 11 (a) and (b) Experimental records of intensity patterns in the x-z plane and y-z plane. (c), (e) and (g) Transverse intensity profile at the position of z = 265mm < T/4, z = 267mm > T/4 and z = 531mm ≈T/2, respectively. (d), (f) and (h) are the corresponding interferogram patterns. The sign of TC can be confirmed by observing the fringe bifurcation direction (red arrow) from the interferogram.

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4. Conclusions and discussions

The OV appears at the intersection of zero lines of real and imaginary parts. The TC of the vortex is equal to the number of zero lines of the real part, which also equal to the number of zero lines of the imaginary part. Accordingly, zero lines distributions at each vortex in the OV lattice also meet this relation. In general, the coherent non-diffracting field can be obtained by the interference of phase engineered multi-waves whose propagation vectors distribute symmetrically on a conical surface. Some of the interference fields contain vortices, and some do not. Zero lines of the real and imaginary parts are coincident when the OV does not exist. We superimpose two non-diffracting fields without any vortex. Due to zero lines of the real and imaginary parts coincidence, there will be two clusters of zero lines in the superimposed field. One cluster can be seen as zero lines of the real part, and the other can be regarded as zero lines of the imaginary part. Since zero lines of non-diffracting fields distribute periodically, array intersections can be generated when the superposition state is designed ingeniously. Moreover, the same number of zero lines of the real and imaginary parts or OVs with the same TC exist at some intersections which still have the array feature. Then, the OV lattice can be obtained when the two groups of multi-waves corresponding to the two non-diffracting fields are interfering together.

The lattices have the periodic characteristics in propagation unlike the former ones and more various patterns can be obtained by changing the combination mode. It can be applied in the region of nanofabrication and optical trapping. At present, the highest order of the resultant OV lattice is only the third. Whether it is possible to encode another form of combination mode, for example, more groups of multi-waves, a higher order OV lattice can be obtained when interference occurs, which will be further studied in the coming part of this paper. Nevertheless, we can anticipate that the present intuitive method is to be an inspiration for encoding waves to generate more patterns of OV lattices.

Funding

National Natural Science Foundation of China (51875447).

Acknowledgments

We thank the editors and the anonymous reviewers for their careful reading and valuable comments that very helpful for revision and improving this paper. Particularly, the authors sincerely appreciate Zengkun Wang, the engineer of UPOLabs (http://www.realic.cn), for the help of the optical experimental setup and test.

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Multi-beams engineered to increase patterns of vortex lattices by employing zero lines of the coherent non-diffracting field

Abstract

1. Introduction

2. Design principle and relative analysis

3. Experiment implementation

4. Conclusions and discussions

Funding

Acknowledgments

References

Cited By

Figures (11)

Equations (17)

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