Focusing behavior of the fractal vector optical fields designed by fractal lattice growth model

Xu-Zhen Gao; Yue Pan; Meng-Dan Zhao; Guan-Lin Zhang; Yu Zhang; Chenghou Tu; Yongnan Li; Hui-Tian Wang

doi:10.1364/OE.26.001597

1. Introduction

Fractal concept was first introduced in 1967 by Mandelbrot in the famous article entitled “How long is the coast of Britain? Statistical self-similarity and fractional dimension” [1]. A fractal can be constructed based on the incorporation of identical motifs that repeat on differing size scales [2]. Fractal is a universal phenomenon in nature, for instance, clouds, trees, waves on a lake, the human circulatory system, and mountains etc. As is well known, the iteration of similarity maps often produces an invariant fractal with intricate self-similarity structure. Fractal can appear in various branches, such as nature, art, mathematics, chemistry, materials, and physics. In optics issue, the fractal has also attracted wide attention including laser optics [3], light scattering [4–6], light diffraction [7–12], and zone plates [13–15]. Most of these researches investigating fractal applied in the realm of optics only consider one or several specific kinds of fractals at one time, such as Sierpinski triangle [3], Koch curves [4, 9, 10], Vicsek fractals [7, 12], Sierpinski carpet [6, 8, 11], Cantor set [5, 11–15] and so on. How to broaden the using range of fractals in the realm of optics becomes an urgent demand.

Since vector optical fields (VOFs) with space-variant polarization structure hold great potential for a variety of scientific, technical and engineering applications [16–27], we created fractal vector optical fields (F-VOFs) for the first time in a previous work [28], building a bridge between fractals and VOFs. Due to the unique feature of fractals and VOFs, the F-VOF consisting of “lattice” and “base” will become a rather appealing topic. Although we have mentioned that various kinds of fractals can be used to construct F-VOFs, only Sierpinski carpet was used in [28]. No specific scheme of fractal “lattices” is proposed to design the F-VOFs, so the enormous potential properties and applications of F-VOFs are still not tapped enough.

In this paper we introduce a universal fractal lattice growth (FLG) model and apply this model in the realm of optics. In addition, most of the fractals used in the former researches [3–15] can be designed by the FLG model. With the FLG model, we design and generate various kinds of new F-VOFs, and explore their weak and tight focusing behavior. The trapping experimental results demonstrate the feasibility of controlling multiple focal spots by the F-VOFs. We also study the recovery performance of the F-VOFs when the input field or the spatial frequency spectrum is obstructed, which illustrates the F-VOFs can increase the robustness of the information transferring.

2. Fractal lattice growth model

In this section, we will give a detailed principle of the FLG model to describe the standard fractals and to design various kinds of new fractals. If we use the idea introduced in [28], that is to construct the fractals by calculating the convolution of lattices and bases, fractals can be designed. Here the FLG model is introduced for the purpose of flexibly designing the fractal lattices. The fractal lattice in designing both fractal patterns and F-VOFs can be designed with the FLG model as every original set of the base in the lattice grows to next-generation sets in a certain rule in each iteration. The distance between these newly growing sets and the original set can be either equal or unequal. The set in the fractal lattice is defined by the δ function, and the fractal lattice L_n(x, y) is described by the recursion formula L_n₊₁(x, y) = L_n(x, y) ⊗ m_n₊₁ (x, y) with $m_{n} (x, y) = \sum_{j = 1}^{t} δ (x - x_{j} / k^{n}, y - y_{j} / k^{n})$ , where m_n(x, y) is the iteration function of the fractal lattice, n is the hierarchy of the fractal, t stands for the number of the sets derived from one set during the iteration, (x_j/kⁿ, y_j/kⁿ) are the coordinates of the sets, and k is the scaling factor of the iteration. According to the above formula, L_n(x, y) can be rewritten as L_n(x, y) = L₀(x, y) ⊗ m₁(x, y) ⊗ m₂(x, y) ⊗ ⋯ ⊗ m_n(x, y), where L₀(x, y) = δ(x, y) is the lattice of the 0th-generation fractal. Here we give the value of k, t, and (x_j, y_j) of seven exemplary fractals in Table 1, and these lattices are limited in the zone with a size of L × L.

Table 1. The parameters k, t and (x_j, y_j) in the lattices of the FLG model corresponding to 7 kinds of exemplary fractals.

View Table

To show the FLG model in detail, we give some typical examples associated with exemplary fractals as well as arbitrarily designed fractals in Fig. 1, where the blue-green spheres represent the locations of the sets in the lattice and the fuzzy ones are the locations of the former-existing sets before the iteration. The lattice corresponding to the famous Cantor set, as the most typical one-dimensional fractal, is depicted in Fig. 1(a). During each iteration, one set splits into two in opposite directions with the same distance. Another famous fractal is the Sierpinski triangle, which is defined by an equilateral triangle, subdivided recursively into smaller equilateral triangles in each iteration, as shown in Fig. 1(b). Correspondingly, the set in the lattice splits into three along three directions with the same included angle of 120° in each iteration. In order to get insights into the FLG model, we also explore more kinds of lattices which can be arbitrarily designed and controlled to indicate the universality of the FLG model. As two representative examples shown in Figs. 1(c) and 1(d), one set in the lattice can split into several sets in different directions with different distances, which also illustrates that the number and the location of the splitted sets can be flexibly controlled based on needs. One set in the lattice decomposes into three sets, and the original set is removed in Fig. 1(c). Meanwhile, Fig. 1(d) shows the lattice in which one set is subdivided into five sets and the original set is retained. Obviously, the sets in the lattice designed by the FLG model can be flexibly controlled obeying a certain rule.

Fig. 1 The FLG model when one set splits into (a) two (corresponding to Cantor set), (b) three (corresponding to Sierpinski triangle), (c) three and (d) five sets. The blue-green spherules represent the locations of sets and the fuzzy ones just represent the locations of the former-existing sets before the iteration.

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Based on the design method of the FLG model, we can obtain various kinds of fractals and exploit their applications in optics. The method to construct the fractal is to calculate the convolution of the lattice and base, and the lattice designed by the FLG model can determine the location of the base. Here we present several examples demonstrating the fractals and their applications by applying the FLG model. The first example shown in Fig. 2(a) is the schematic of designing Sierpinski carpet with a square as base, which has been used in light scattering [6], light diffraction [8, 11], and wide-area signal transmission [29]. The second example is shown in Fig. 2(b), wherein how a fractal zone plate is constructed by the line segment as base and a lattice designed by the FLG model is shown. In fact, the radial distribution of this fractal zone plate is the Cantor set [13–15]. From Figs. 2(a) and 2(b) we can see that the existing fractal patterns can be constructed with the lattice derived from the FLG model and the base of geometry. If we change the lattices according to the FLG model or change the shape of bases, various kinds of new fractal patterns will appear. For instance, we can use circle, triangle, rectangle or even arbitrary polygon as bases to obtain another kind of “Sierpinski carpet” with the lattice shown in Fig. 2(a). On the other hand, the generalized Cantor set can be gained when changing the distance between the next-generation sets and the original set in the lattice in Fig. 2(b), and far more kinds of fractal zone plate can be designed as a result. We believe these new kinds of fractals can be very useful in the areas of light diffraction, light scattering, focusing and so on.

Fig. 2 Three examples demonstrating the combination of different bases and lattices. (a) Schematic of designing Sierpinski carpet with a square as base and the Sierpinski carpet as lattice. (b) Fractal zone plate constructed by a line segment as base and the Cantor set as lattice. (c) F-VOF designed by the radially polarized VOF as base and an arbitrary lattice.

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Besides the various fractals which can be designed by the FLG model, Fig. 2(c) also shows one example of the new F-VOF, whose lattice is designed by the FLG model and base is radially polarized VOF instead of the geometries in Figs. 2(a) and 2(b). The lattice in this F-VOF does not correspond to any exemplary fractals, showing the variety of the F-VOFs designed by the FLG model. It is worth mentioning that besides the variety of the lattices, the bases of optical fields can be diverse. Seven categories of bases are: three unitary bases of (i) amplitude-only, (ii) phase-only, and (iii) polarization-only; three binary bases of (iv) amplitude-phase, (v) phase-polarization, and (vi) amplitude-polarization; one ternary base of (vii) amplitude-phase-polarization. With the flexible FLG model as well as the various choices of bases, a great number of F-VOFs can be designed in principle.

3. Design and generation of F-VOFs with FLG model

Now we will focus on the design and generation of the F-VOFs to discuss the detailed effect of the FLG model. The mathematical description of the flexible FLG model has been introduced above, but it is not enough to construct a F-VOF only with the lattice, as the base is also necessary. The base of optical field with arbitrary amplitude, phase and polarization distribution (corresponding to seven categories) can be expressed as

E_{n A} (r, φ) = A \exp (i Φ) [\sin σ \exp (- i δ) {\hat{e}}_{r} + \cos σ \exp (i δ) {\hat{e}}_{l}],

where

{{\hat{e}}_{r}, {\hat{e}}_{l}}

is a pair of orthogonal right- and left-handed circularly polarized unit vectors, which can be defined by a pair of unit vectors

{{\hat{e}}_{x}, {\hat{e}}_{y}}

in the x and y directions, as

{\hat{e}}_{r} = \frac{1}{\sqrt{2}} ({\hat{e}}_{x} + j {\hat{e}}_{y})

and

{\hat{e}}_{l} = \frac{1}{\sqrt{2}} ({\hat{e}}_{x} - j {\hat{e}}_{y})

. A is the space-variant amplitude, Φ is the space-variant phase, and σ and δ control the ellipticity and orientation of the long axis for the local polarization state, respectively. Four parameters, A, Φ, σ and δ, are all functions of the polar coordinates (r, φ).

Here we utilize a typical kind of vortex VOFs as bases for simplicity, which are local linearly polarized VOFs with azimuthal variant phase distribution [30]. This kind of vortex VOFs can be expressed as

E_{n A} (r, φ) = U_{n} (r, φ) \exp (i l φ) [\exp (- i m φ) {\hat{e}}_{r} + \exp (i m φ) {\hat{e}}_{l}],

where m is the topological charge of azimuthal variant polarization distribution, while l is the topological charge of azimuthal variant phase distribution. (r, φ) are the polar coordinates. U_n(r, φ) is defined as the shape function as it controls the shape and size of the bases, and U_n(r, φ) is associated with n because the size of the base reduces to k⁻¹ during each iteration.

When these bases are combined with the lattices as mentioned above, the F-VOFs can be described as

E_{n} (r, φ) = E_{n A} (r, φ) \otimes L_{n} (x, y) .

where ⊗ is the convolution operator. We can see from Eq. (3) that the F-VOF can be described as the convolution of the base E_nA(r, φ) and the lattice L_n(x, y). The base E_nA(r, φ) with the cylindrical symmetry is designed in the polar coordinates (r, φ), while the lattice L_n(x, y) is defined in the Cartesian coordinates (x, y) by the FLG model.

The experimental setup for generating the F-VOFs is shown in Fig. 3, which is a common-path interferometric configuration with the aid of a 4f system composed of two lenses L1 (f = 300 mm) and L2 (f = 200 mm), similar to [30, 31]. The input linearly polarized light delivered from a laser (Verdi-5, Coherent Inc.) is incident on the spatial light modulator (SLM, Pluto-Vis, Holoeye System Inc.) with 1920 × 1024 pixels (each pixel has a dimension of 8 × 8 µm²). Its first-order diffraction will produce four beams, which are the ±1st orders in both the x- and y-axes. The two +1st orders are converted into the right- and left-handed circularly-polarized fields by two 1/4 wave plates located at the Fourier plane of the 4f system, respectively. The two orthogonally polarized orders are recombined by the Ronchi phase grating placed in the output plane of the 4f system. To generate the F-VOFs designed by the FLG model, the transmission function of the computer-generated hologram displayed on the SLM can be written as

t_{n} (x, y) = t_{n A} (x, y) \otimes L_{n} (x, y) .

with

t_{n A} (x, y) = \frac{1}{2} + \frac{γ (x, y)}{4} {\cos [2 π f_{0} x + δ_{1} (x, y)] + \cos [2 π f_{0} y + δ_{2} (x, y)]}

where t_nA(x, y) is the transmission function of the bases, δ₁(x, y) and δ₂(x, y) are the phase distributions imposed on the vertical and horizontal holographic gratings, f₀ is the spatial frequency of the grating and γ(x, y) is the gray scale function of the grating. After the two right-and left-handed circularly-polarized orders are collinearly recombined behind the Ronchi grating, the normalized generated F-VOF can be written as

\begin{array}{l} E_{n} (x, y) = \frac{1}{\sqrt{2}} γ (x, y) [\exp (j δ_{1}) {\hat{e}}_{r} + \exp (j δ_{2}) {\hat{e}}_{l}] \otimes L_{n} (x, y) \\ = γ (x, y) \exp (j \frac{δ_{1} + δ_{2}}{2}) [\cos (\frac{δ_{2} - δ_{1}}{2}) {\hat{e}}_{x} + \sin (\frac{δ_{2} - δ_{1}}{2}) {\hat{e}}_{y}] \otimes L_{n} (x, y) . \end{array}

When we set γ(x, y) = A_n(r, φ), δ₁(r, φ) = (l − m)φ, δ₂(φ, r) = (l + m)φ, the F-VOFs with vortex VOFs as bases can be generated experimentally.

Fig. 3 Schematic of the experimental setup for generating the desired F-VOFs designed by FLG model. SLM, spatial light modutor; L1 (f = 300 mm) and L2 (f = 200 mm), a pair of lenses; λ/4, quarter-wave plates; SF, spatial filter; G, Ronchi phase grating; PC, computer. A polarizer may be inserted in the field, then the intensity patterns can be observed by a CCD.

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Firstly, we will demonstrate experimentally the generation of the F-VOFs by the above method. The six columns in Fig. 4 show the total intensity patterns and the x-component intensity patterns of the F-VOFs with different lattices of the Sierpinski triangle, Vicsek fractal II, pentagon, hexaflake, deformed-hexaflake, and arbitrary fractals, which are all designed by the FLG model. The F-VOFs with different hierarchies of n = 1, 2 and 3 are shown in different rows of Fig. 4. We should emphasize that the bases of all the F-VOFs are always the radially polarized VOFs, and we choose U_n(r, φ) = circ(kⁿr/L, kⁿr/L) in Eq. (2) as a typical kind of shape function for circular base in this case. There is always a singularity in the total intensity pattern of each base, meaning that the generated F-VOFs are a kind of VOFs with multi-singularities. In the first column, the F-VOFs are generated using the Sierpinski triangle as lattice and the size of the base shrinks to half in each iteration. In the second column, the F-VOFs have the Vicsek fractal II as lattice and the size of the base reduces to 1/3 in each iteration. The lattices of the F-VOFs in the third and fourth columns correspond to the regular pentagon and hexaflake fractals shown in Table 1, and the size of the base reduces to 1/[2 + 2 sin (π/10)] and 1/3 in each iteration, respectively. The lattices of the F-VOFs in the fifth and sixth columns do not correspond to regular fractals, but they can also be designed and generated by the FLG model, and the size of base reduces to 1/3 in each iteration. The F-VOFs in the fifth column are in fact the shear deformation of that in the fourth column, in which the upper and lower two sets in the lattice move in opposite horizontal directions and the size of the base also slightly shrinks compared with the fourth column. Clearly, the intensity patterns of the F-VOFs exhibit the self-similarity as the fractal hierarchy n increases.

Fig. 4 Experimental intensity patterns of the generated F-VOFs with the same radially polarized VOF as base and different lattices designed by the FLG model. The six columns show the F-VOFs obtained with the following lattices: Siepinski triangle, vicsek fractal II, pentagon, hexaflake, deformed-hexaflake and arbitrary fractals, respectively. The first/second rows, third/fourth rows, and fifth/sixth rows correspond to the F-VOFs with hierarchies of n = 1, n = 2, and n = 3, respectively.

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Secondly, we explore the F-VOFs with the same hexaflake fractal lattice and different bases. The six columns in Fig. 5 show six different shapes of bases corresponding to the hexagon, square, diamond, pentagram, double-ring, and Bessel-Gauss profile, respectively. The first (second), third (fourth) and fifth (sixth) rows correspond to the measured total (x-component) intensity patterns of the F-VOFs with topological charges of m = 1, 2 and 3, respectively. The total intensity patterns clearly show that there is always one singularity in any base, and its size becomes larger as the topological charge m increases, because the SLM with limited resolution cannot distinguish the rapidly changing polarization of the VOF in the center of each base. However, the singularity should theoretically be a geometric point independent of m. The x-component intensity patterns shown in the second, fourth and sixth rows exhibit the fan-like extinction patterns in each base (number of the fans is 2m), which are in good agreement with the simulation results.

Fig. 5 Experimental intensity patterns of the generated F-VOFs with the same hexaflake fractal lattice and different bases. The six columns show the F-VOFs obtained with the following bases: hexagon, square, diamond, pentagram, double-ring, and Bessel-Gauss, respectively. The topological charges of the bases of the F-VOFs are m = 1 in the first and second rows, m = 2 in the third and fourth rows, and m = 3 in the fifth and sixth rows, respectively.

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Now we analyze the limiting factor of this experimental method. The efficiency of the system generating the F-VOFs depends dominantly on the two diffraction elements—the SLM and the Ronchi phase grating. The diffraction efficiency of the two-dimensional holographic grating displayed at the SLM and the Ronchi phase grating are ~3.5% and ~30%, respectively. As a result, the generating efficiency of the vortex VOFs is ~1% in our experiment. The period of the holographic grating displayed at the SLM and the modulation manner of the SLM can have a certain extent influence on the generating efficiency. The different kinds of F-VOFs can also have different intensity under the same input laser power due to the different fractal structures. In addition, the response time and precision of generating the F-VOF are determined by the response time (< 100 ms) and the spatial resolution of the SLM.

4. Controllable multiple focal spots generated by F-VOFs

4.1. Universal focusing formula of F-VOFs

As we know, researches in recent years have shown unique tight focusing properties of the VOFs, such as, far-field focusing beyond the diffraction limit [17, 18], the light needle of a longitudinally polarized field [19], the optical cage [20], and the optical chain [21]. The tight focusing fields are also useful in optical micro-machining and micro-manipulation due to the smaller focal fields with respect to the case of weak focusing. As we have deduced the weak focusing properties of the F-VOFs with the low numerical aperture in our previous report based on the Huygens-Fresnel principle [28], we now discuss the universal calculation theory of the focused F-VOFs to generate various kinds of multiple focal spots. This universal calculation theory can be used to calculate both weak and tight focusing fields, which can cover all the cases of focusing calculation.

Firstly, we deduce the formula of the tightly focused F-VOFs based on the vectorial Richards-Wolf diffraction integral [16, 17, 32]. The vectorial diffraction integral can be written in terms of the Fourier transform [33], which is particularly suited for rapid numerical evaluation, especially, for the tight focusing process. It is very essential to introduce the fast Fourier transform (FFT) to calculate the tight focusing fields of the F-VOFs, as the F-VOFs can be very complicated when n increases and the running time of the calculation program with the traditional theory will be too long. Thus the tight focusing field of the F-VOF can be described by

E_{f} (x^{'}, y^{'}) = ℱ {E_{t} (θ, ϕ) e^{i k_{z} z} / \cos θ},

where E_t(θ, ϕ) is the field in the exit aperture, the phase factor

e^{i k_{z} z}

accounts for the phase accumulation when propagating along the z-axis, k_z is the z component of the wave vector k,

ℱ {}

stands for the Fourier transform.

Due to the refraction of the lens, the relation between the input F-VOF and the exit field can be described as

E_{t} (θ, ϕ) = A (θ) M E_{n} (x, y),

where A(θ) is the pupil plane apodization function, which can be chosen as

A (θ) = \sqrt{\cos θ}

. M is the polarization transformation matrix, which can be written in the cylindrical coordinate system (ρ, ϕ, z) as

M = [\begin{matrix} \cos θ \cos^{2} ϕ + \sin^{2} ϕ & (\cos θ - 1) \sin ϕ \cos ϕ & - \sin θ \cos ϕ \\ (\cos θ - 1) \sin ϕ \cos ϕ & \cos^{2} ϕ + \sin^{2} ϕ \cos θ & - \sin θ \sin ϕ \\ \sin θ \cos ϕ & \sin θ \sin ϕ & \cos θ \end{matrix}] .

With Eq. (7) and the convolution theorem, the tight focusing field of the F-VOF in Eq. (6) is rewritten as

\begin{array}{l} E_{f} (x^{'}, y^{'}) = e^{i k_{z} z} ℱ {M / \sqrt{\cos θ}} \otimes ℱ {E_{n} (x, y)} \\ = e^{i k_{z} z} ℱ {M / \sqrt{\cos θ}} \otimes {ℱ [E_{n A} (φ, r)] ℱ [L_{n} (x, y)]} . \end{array}

From the above equation, we can see that the tight focusing field of the F-VOF is the convolution of the Fourier transforms of the polarization transformation matrix and the F-VOFs. This means, if we know the focusing field calculated by the scalar diffraction theory, we can obtain the tight focusing field directly by calculating the convolution in Eq. (9). For a low number aperture focusing system, θ ≈ 0 and M is a unit matrix, thus, the tight focusing formula degenerates into the weak focusing form, meaning that the focal field of the F-VOF becomes the product of the Fourier transforms of the lattice and the base. Next, we will give the simulation results of the focal fields of the F-VOFs, according the above equations.

4.2. Design and generation of multiple focal spots by manipulating bases

As mentioned above, there are two factors including lattices and bases in designing the F-VOFs, which are also two useful factors in designing and controlling multiple focal spots. Firstly, we discuss the influence of the bases in designing and generating multiple focal spots. Figure 6 shows the focusing properties of six F-VOFs with different bases and the same Sierpinski carpet lattice with fractal hierarchy of n = 2. All the original bases of the F-VOFs are the vortex VOFs [topological charges are shown in the top left of Figs. 6(a1)–6(f1)], and have the same square shapes with the same dimension of 2.7 × 2.7 mm² [the intensity of each base is shown in the top right insets of Figs. 6(a1)–6(f1)].

Fig. 6 Focal fields of the F-VOFs with the same Sierpinski carpet lattice and the different bases are shown in the first to sixth columns. The first row shows the simulated focal field intensity patterns of bases and the inset in the top right of each pattern exhibits the shape of the base. The second row depicts the experimentally generated multiple focal spots. The seventh column shows the simulated focal field of the Sierpinski carpet lattice. In our simulation and experiment, the focal length of the lens is f = 250 mm, and all the pictures have the same dimension of 1.5 × 1.5 mm².

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The first row in Fig. 6 shows the simulated focal field intensity patterns of the six different bases. The focal field of the common Sierpinski carpet lattice, shown in the seventh column, exhibits a 3 × 3 spot array. The weak focusing field of any F-VOF can be calculated as the product of the Fourier transforms of its lattice and base. In Fig. 6(a2) [Fig. 6(b2)], the focal field intensity pattern of the F-VOF with its base shown in Fig. 6(a1) [Fig. 6(b1)] exhibits four focal spots located at four vertices of a diamond [eight focal spots, where four spots locate at four vertices of a square while the other four are at four midpoints of four sides]. To flexibly design and control the focal fields, we also introduce symmetry breaking of the bases with different amplitude-type obstacles [shown by the top right insets of Figs. 6(c1)–6(f1)]. As shown in Figs. 6(c1) and 6(d1), the focal fields of the broken bases (both original bases are in fact the same scalar vortex fields with m = 0 and l = 2) are rotated by an angle of π/2 with respect to the obstacle, similar to [34, 35]. As a result, the focal field of the F-VOF shown in Fig. 6(c1) [Fig. 6(d1)] exhibits three focal spots (like the “Altair” star in night sky) shown in Fig. 6(c2) [five focal spots (arranging a V-shape like the “flying geese queue”) shown in Fig. 6(d2)]. When the original base (vortex VOF with m = l = 1) is broken by four-fold rotation symmetric obstacle, the focal field of the broken base exhibits the “rudder” of four handles shown in Fig. 6(e1). As shown in Fig. 6(e2), the focal field of the F-VOF has five focal spots, where the focal spot at the center is the strongest one while the four vertices of a diamond have the four weak spots. As shown in Fig. 6(f1), the focal field of the broken base [vortex VOFs with (m, l) = (2, 0)] exhibits four large focal spots located at four corners of a square, correspondingly, the focal field of the F-VOF exhibits four focal spots shown in Fig. 6(f2). The experimentally measured focal field patterns of the F-VOFs under the weak focusing condition are in good agreement with the simulated results (the simulated results are not shown here). Obviously, we can find from the above discussion that for one specific kind of lattice, flexibly controllable multiple focal spots can be generated by designing different kinds of bases to obtain different pattern of focal spots.

4.3. Design of multiple focal spots by manipulating lattices and experiment of optical trapping

As we have simulated and generated multiple focal spots by manipulating bases of the F-VOFs under the weak focusing condition in Section 4.2, now we will study the tightly focused F-VOFs and verify the generated focal spots in optical trapping experiment. To explore the influence of the lattice designed by the FLG model on the focal spots of F-VOFs, we carry out a comparative study on the standard hexaflake fractal lattice and the shear-deformed lattice, while the base is always a radially polarized VOF, as shown in Fig. 7.

Fig. 7 Multiple focal spots of the F-VOFs with radially polarized VOF as base and shear-deformed lattices. The first column shows the case of the F-VOF with the hexaflake fractal as lattice. The second to fifth columns show the cases of the F-VOFs with different deformed-hexaflake fractals as lattices. The first row shows the experimental intensity patterns of the F-VOFs, where the white circles demarcate the boundary of the input aperture of the objective. The second row shows the simulated tight focusing fields when n = 2. The third and fourth rows show the snapshots of locations of the trapped particles around the focal spots generated by the F-VOFs for n = 2 and n = 3, respectively. Any focal picture for n = 2 (n = 3) has a dimension of 15 × 15 µm² (45 × 45 µm²).

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For the standard hexaflake fractal lattice listed in Table 1, two upper sets locate at (L/2kⁿ, $\sqrt{3} L / 2 k^{n}$ ) and (−L/2kⁿ, $\sqrt{3} L / 2 k^{n}$ ), while two lower sets are at (L/2kⁿ, $- \sqrt{3} L / 2 k^{n}$ ) and (−L/2kⁿ, $- \sqrt{3} L / 2 k^{n}$ ), and the corresponding F-VOF is shown in Fig. 7(a1). From Figs. 7(b1)–(e1), the two upper sets of the hexaflake fractal lattice are moved toward the right by a step of L/8kⁿ in sequence, while the two lower sets are shifted toward the left by a step of L/8kⁿ in sequence. The first row in Fig. 7 depicts the experimental intensity patterns of the F-VOFs with the standard hexaflake fractal lattice in Fig. 7(a1) and the shear-deformed lattices in Figs. 7(b1)–7(e1) for the same fractal hierarchy of n = 2, and Fig. 7(c1) corresponds to the experimental intensity pattern in the third row of the fifth column of Fig. 4. The white circle in the intensity pattern in the first row of Fig. 7 demarcates the boundary of the input aperture of the objective, so we only consider the portion of F-VOF within this circle for both simulation and experiment. The simulated tight focusing fields are shown in Figs. 7(a2)–7(e2) with n = 2, and the six focal spots are marked by numbers in Fig. 7(a2). After comparing the first row with the second row, we find that when the lattice is sheared along the horizontal direction, its focal field pattern is sheared along the vertical direction. The focal spots of the standard hexaflake lattice exhibit a hexagon [Fig. 7(a2)], while the focal spots of the last shear-deformed lattice become a square lattice [Fig. 7(e2)]. During this evolution, the left two focal spots (1 and 2) gradually rise up, while the right two spots (4 and 5) gradually move down, as the shear increases. As shown in Figs. 7(b2)–7(e2), two new focal spots occur at the upper-right and lower-left corners, respectively. As a result, for the largest shear case in Fig. 7(e2), the focal field of the shear-deformed F-VOF has four strongest focal spots located at four midpoints of four sides of a square, while four weak focal spots locate at four corners of the square. It is also worth mentioning that if we remove the aperture limit of the objective and consider the tight focusing fields of the entire F-VOFs, the simulated focusing fields are only slightly different from those in the second row of Fig. 7. We can find from Fig. 7 that various patterns of multiple focal spots can be generated and controlled by designing different kinds of lattices with the help of the FLG model. The laser source used in our experiment is a green laser at a wavelength of λ = 532 nm. The light wavelength will affect the focusing property of the optical field, which is the reason why the wavelength is often used as the unit charactering the focusing fields [17, 19–21]. Specifically, the size of each focal spot and the distance between focal spots are proportional to the wavelength.

To confirm the simulation results discussed above, we choose the optical tweezers experiment as an indirect way to characterize the tight focusing fields of the F-VOFs. The experimental schematic is similar to those used in [23, 24]. The created F-VOF is introduced into an optical tweezers system composed of an inverted microscope including a 60× objective with NA = 0.75. The neutral isotropic colloidal microspheres with the same diameter of 2.8 µm are dispersed in a layer of sodium dodecyl sulfate solution between a glass coverslip and a microscope slide. To guarantee enough laser power to trap and manipulate particles, we display one-dimensional gratings on the SLM in experiment in Fig. 3, which can generate only the VOFs instead of vortex VOFs [31]. As a result, the overall efficiency generating the traditional VOFs is ~10%, and the efficiency is lower than 10% for the F-VOFs due to the fractal structures. In the experiment, the laser power in the focal region keeps to be ~4.7 mW. The neutral microparticles can be trapped at the locations of the multiple focal spots, as shown in Figs. 7(a3)–7(e3) for n = 2 and Figs. 7(a4)–7(e4) for n = 3. We can see from Fig. 7(a3) that the distance of two adjacent particles is ~3.63 µm, which is approximately equal to the simulated distance 3.67 µm. When the shear-deformation is over, the distance of two adjacent particles is ~3.25 µm [the simulated distance is 3.18 µm, which is 3.67 µm × sin(π/3)]. For the case of n = 3, the distance of the particles expand three times larger than the case of n = 2, which can also be verified by the experimental results of optical trapping in Fig. 7. Clearly, all the experimental results are in good agreement with the simulated intensity patterns of the tight focusing fields, implying that the simulation method proposed in Section 4.1 is correct and powerful.

To further verify the flexibility of the FLG model, we explore the effect of the lattice rotation. Here we choose the hexaflake fractal lattice as an example, as shown in Fig. 8. For the standard hexaflake fractal lattice, its seven sets locate at (0, 0) and (L cos α_q/kⁿ, L sin α_q/kⁿ) with α_q = 0, π/3, 2π/3, π, 4π/3 and 5π/3, as shown in Table 1. When the lattice is rotated anticlockwise by an angle of β, the sets in the new lattice should be written as (0, 0) and (L cos[α_q + β]/kⁿ, L sin[α_q + β]/kⁿ. Here we take β = π/12, π/6, π/4 and π/3 as examples. The first row of Fig. 8 shows the experimental intensity patterns of the rotated F-VOFs, and the second row displays the simulated tight focusing fields. Obviously, the focal field patterns are also rotated anticlockwise by the same angle of β synchronously. To confirm our idea, we also use these rotated multiple focal spots to trap micrometer-sized particles. The third and fourth rows in Fig. 8 give the experimental results of optical trapping, corresponding to the cases of n = 2 and n = 3, respectively. We can see that the experimental results are in good agreement with the simulated tight focusing fields. In addition, the results illustrate that we can use the F-VOFs to move the particles in a circle by changing the holograms with continuously variable β on the SLM. It should be pointed out that the rotation of the lattice cannot be considered as the simple rotation of the F-VOFs, because the bases in this case are not rotated while the lattice is rotated. Specifically, we cannot get the F-VOFs in Fig. 8 by simply rotating the grating on the SLM, as every F-VOF in Fig. 8 corresponds to a unique grating. The rotation of sets is one basic operation for constructing the fractal lattices with the help of the FLG model, and there are also other operations to design the lattices, including the translation of sets, scaling the distance between two sets, skewing of sets, and so on.

Fig. 8 Multiple focal spots of the F-VOFs with the radially polarized VOF as base and the rotated lattices. The first column shows the case of F-VOF with the hexaflake fractal lattice, and the second to fifth columns show the cases of F-VOFs with the rotated hexaflake lattices. The first row shows the experimental intensity patterns of the F-VOFs, and the second row shows the simulated tight focusing fields for n = 2. The third and fourth rows show the snapshots of locations of the trapped particles around the focal spots generated by the F-VOFs for n = 2 and n = 3, respectively. Any focal picture for n = 2 (n = 3) has a dimension of 15 × 15 µm² (45 × 45 µm²).

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5. Recovery performance of F-VOFs in focusing and imaging processes

As we have shown, multiple focal spots can be designed and generated by different kinds of F-VOFs, which shows the flexibility in engineering the focal fields. Recently, the application of vector optical fields in optical communication and transmission has attracted great attention [25–27]. Hence, we find it interesting to study the recovery performance of the F-VOFs in propagation, as there may be various kinds of noises and interferences in focusing and imaging processes [29, 35––37]. We will discuss the recovery performance of the focused F-VOFs when the input field is obstructed and then study the recovery performance of F-VOFs when the spatial frequency spectrum is obscured, to investigate the possibility of recovering information carried by F-VOFs.

5.1. Recovery performance of the focused F-VOFs with the obscured input field

We will now investigate the case when the input F-VOF has radially polarized VOF as base and the hexaflake fractal as lattice, and the sector-shaped obstacle is placed at the pupil plane. The size of the obstacle is defined by the center angle b of the sector, and we choose b = 0, π/3, 2π/3, π, 4π/3 and 5π/3 in experiment. Figure 9 shows the experimental intensity patterns of the input F-VOFs with n = 1 at the pupil plane and the total intensity patterns of the focused F-VOF with n = 1, 2, 3, respectively. The first column of Fig. 9 depicts the case of the unobstructed F-VOF, while the other columns show the results for different sizes of sector-shaped obstacles. Although b is different, the multiple focal spots still locate at the six vertices of the hexagon. For n = 1, 2, 3, the focal spots are elongated as b increases. In the first three columns, this elongation is slight, and the size of each focal spot is similar. When b increases to π or even larger, the focal spots become obviously larger and there occurs a focal spot at the center of the focal plane. It is also noticed that the symmetric axis of the sector-shaped obstacle is also the symmetric axis of the focal spot. Although the locations of the multiple focal spots when n = 1 are the same for different sector-shaped obstacles, the quality of each focal spot is unsatisfactory when b > π. However, as n increases, the multiple focal spots become smaller no matter what size the obstacle is, as shown in the first, second and third rows of Fig. 9. This illustrates that the multiple focal spots can be maintained though the input F-VOF is obstructed, and the hierarchy n of the fractal has the “recovery” function in the focusing process.

Fig. 9 Recovery performance of the focal spots when the input F-VOF is obstructed. The first row depicts the input F-VOFs with radially polarized VOF as base and the hexaflake fractal when n = 1 as lattice. The second, third and fourth rows show the corresponding focal intensity patterns with dimension of 0.6 × 0.6 mm², 1.8 × 1.8 mm² and 5.4 × 5.4 mm² when n = 1, 2 and 3, respectively. The six columns correspond to different sizes of sector-shaped obstacles with b = 0, π/3, 2π/3, π, 4π/3 and 5π/3, respectively. All the patterns are experimental results.

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5.2. Recovery performance of the F-VOFs with the obscured spatial frequency spectrum

Recently, the fractal structure has shown its good property in far field propagation of information [29], and the cylindrical VOFs also behave well when the information is obstructed in the frequency domain [35]. Motivated by these facts, it is necessary to explore the recovery performance of the F-VOF with the obstructed spatial frequency spectrum, as the F-VOF connects the fractal and the VOFs. The experimental setup is similar to [35], as one sector-shaped filter is placed in the spatial frequency domain of the F-VOF, and the output optical fields are observed at the imaging plane. The input F-VOFs in Fig. 10 have the hexaflake fractal lattice with n = 1 and the bases are designed as x-polarized and y-polarized vortex fields with the topological charge l = 1 for the purpose of adding polarization information. The first row of Fig. 10 depicts the total intensity of focal fields for different sizes of sector-shaped filters, and the other three rows show the corresponding total, x-component and y-component intensity patterns observed at the imaging plane, respectively. For the sake of comparison, the first column of Fig. 10 shows the output field at the imaging plane when the spatial frequency spectrum is not obscured, and the other columns give the other cases when different sector-shaped filters are inserted with b = π/3, 2π/3, π, 4π/3 and 5π/3, respectively. It can be clearly seen from Fig. 10 that we can distinguish the bases of the F-VOF with different polarizations at the imaging plane no matter what filter is inserted. The intensity patterns also show that the locations of the bases remain the same for both obstructed and unobstructed cases. Thus the lattice of the F-VOF can remain unchanged even if the spatial frequency spectrum of the F-VOF are obstructed. From these results, we can conclude that the structure of the fractal lattice and the polarization of the bases (one specific polarization for each base) can be recovered when the spatial frequency spectrum is obstructed.

Fig. 10 The recovered F-VOFs at the imaging plane when the spatial frequency spectrum is obscured. The bases are x-polarized and y-polarized vortex fields (the topological charge l = 1), and the lattice corresponds to the hexaflake fractal when n = 1. The first row depicts the total intensity of focal fields for different sizes of sector-shaped filters (b = 0, π/3, 2π/3, π, 4π/3 and 5π/3) when n = 1. The lower rows show the corresponding total, x-component and y-component intensity patterns of the recovered F-VOFs at the imaging plane, respectively. All the patterns are experimental results.

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Now we further investigate the possibility of transmitting and recovering information using the recovery ability with obstructed spatial frequency domain, as the information carried by the optical fields may be disturbed or distorted during transmitting for a long distance. In this case, we can use the focal field as the spatial frequency spectrum of the F-VOF to simulate the far field, and receive the information at the imaging plane [29]. For the experimental results in Fig. 11, we store the information in the x-polarized vortex bases of the F-VOFs with the hexaflake fractal lattice, and the sector-shaped obstacles are still placed in the frequency domain. The first row in Fig. 11 shows the total intensity patterns of the F-VOFs when n = 2 at the imaging plane, and the second row shows the corresponding x-component intensity patterns which contain the information of “N”. The third and fourth rows show the x-component intensity patterns of the recovered F-VOFs with n = 3 at the imaging plane, and the information for these two rows are “N” and “FRACTAL”, respectively. When no obstacle is inserted in the frequency domain, the F-VOF can be recovered with high quality, as shown in the first column in Fig. 11. By increasing the size of the sector-shaped filter, the intensity of each base becomes nonuniform, which is similar to the experimental results in Fig. 10. Meanwhile, the bases at the imaging plane maintain the same locations no matter what size the sector-shaped filter is, meaning that the information stored in the x-polarized bases can be recovered in the imaging plane. Therefore the F-VOFs can be used for information transmission and optical imaging, and the complete information or image can still be received even if the spatial frequency spectrum of the F-VOF is most partially blocked. As a result, the F-VOFs can be used to increase the robustness of the information transmission.

Fig. 11 Robustness of the information transmisson when the frequency domain of F-VOF is inserted with different sector-shapter filters with b = 0, π/3, 2π/3, π, 4π/3 and 5π/3, respectively. The first and second rows show the total and x-component intensity patterns of the output fields at the imaging plane for input F-VOFs with information of “N” when n = 2. The third (fourth) row gives the x-component intensity patterns of the output fields for input F-VOFs with information of “N” (“FRACTAL”) when n = 3. All the patterns are experimental results.

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6. Conclusion

In summary, we introduce a universal FLG model and apply it in the realm of optics, which can significantly expand the application scope of the fractal in optics. With the help of the FLG model, we theoretically and experimentally demonstrate various kinds of F-VOFs. Based on the vectorial Richards-Wolf diffraction theory and with the aid of the Fourier transform, we develop a universal tight focusing formula of the F-VOFs. We can use the Fast Fourier transform to perform the simulation of the tight focusing field patterns of F-VOFs, so it would be high-speed, high-efficiency and high-precision. Multiple focal spots are flexibly designed with the F-VOF and these controllable focal spots can be applied to trap and manipulate micrometer-sized particles in optical tweezers experiment. As the good recovery performance of the F-VOFs is very important in many applications, we explore two representative cases with the obstructed input field and spatial frequency spectrum. The experiments prove that multiple focal spots can be in fact maintained with the obstructed input F-VOF, and the information can be received and recovered when the spatial frequency spectrum of the F-VOF is obstructed. Based on the tight focusing behavior of the F-VOFs, the F-VOFs designed by the FLG model can be applied in a variety of regions, which are not limited to optical trapping, optical machining and information transmission. We hope more applications can be developed by the F-VOFs and the FLG model, in view of the broad prospect.

Funding

National key R&D program of China (2017YFA0303800, 2017YFA0303700)); National Natural Science Foundation of China (NSFC) (11534006, 11674184, 11374166.

Acknowledgments

We acknowledge the support by Collaborative Innovation Center of Extreme Optics.

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Fractal	k	t	Coordinates (x_j, y_j)
Cantor set	3	2	(L, 0), (−L, 0)
Sierpinski triangle	2	3	(0, L/2), $(\sqrt{3} L / 4, - L / 4)$ , $(- \sqrt{3} L / 4, - L / 4)$
Vicsek fractal I	3	5	(0, 0), (L, 0), (−L, 0),(0, L), (0, −L)
Vicsek fractal II	3	5	(0, 0), (L, L), (−L, L),(L, −L), (−L, −L)
Sierpinski carpet	3	8	(0, L), (0, −L), (L, 0), (−L, 0), (L, L), (−L, L), (L, −L) (−L, −L)
Pentagon fractal	$2 + 2 \sin \frac{π}{10}$	5	$(0, L \cos \frac{π}{5})$ , $(L \cos \frac{π}{5} \cos \frac{π}{10}, L \cos \frac{π}{5} \sin \frac{π}{10})$ , $(- L \cos \frac{π}{5} \cos \frac{π}{10}, L \cos \frac{π}{5} \sin \frac{π}{10})$ , $(- L \sin \frac{π}{5} \sin \frac{3 π}{10}, - L \cos^{2} \frac{π}{10} + L \cos \frac{π}{5} \sin \frac{π}{10})$ , $(L \sin \frac{π}{5} \sin \frac{3 π}{10}, - L \cos^{2} \frac{π}{10} + L \cos \frac{π}{5} \sin \frac{π}{10})$
Hexaflake fractal	3	7	(0, 0), $(L / 2, - \sqrt{3} L / 2)$ , $(- L / 2, - \sqrt{3} L / 2)$ , (L, 0), (−L, 0), $(L / 2, \sqrt{3} L / 2)$ , $(- L / 2, \sqrt{3} L / 2)$

Focusing behavior of the fractal vector optical fields designed by fractal lattice growth model

Abstract

1. Introduction

2. Fractal lattice growth model

3. Design and generation of F-VOFs with FLG model

4. Controllable multiple focal spots generated by F-VOFs

4.1. Universal focusing formula of F-VOFs

4.2. Design and generation of multiple focal spots by manipulating bases

4.3. Design of multiple focal spots by manipulating lattices and experiment of optical trapping

5. Recovery performance of F-VOFs in focusing and imaging processes

5.1. Recovery performance of the focused F-VOFs with the obscured input field

5.2. Recovery performance of the F-VOFs with the obscured spatial frequency spectrum

6. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (11)

Tables (1)

Equations (10)

Optics Express