Multifractal vector optical fields

Meng-Dan Zhao; Xu-Zhen Gao; Qiang Wang; Guan-Lin Zhang; Ke Wang; Fan Dai; Dan Wang; Yongnan Li; Chenghou Tu; Hui-Tian Wang

doi:10.1364/OE.27.020608

1. Introduction

The concept of fractal was first introduced by Mandelbrot in 1967 [1]. A fractal is a self-similar geometric pattern with a scaling rule to reproduce itself, such as Sierpiński carpet [2] and Koch curve [3]. The fractal exists in various systems, such as biology, medicine, materials, mathematics and physics. In the realm of optics, the fractal has also attracted extensive attention including optical antennas [4], optical scattering [5, 6], and zone plates [7–9]. Due to various novel characteristics, vector optical fields (VOFs) have many potentional applications [10–14]. We introduced the concept of fractral into VOFs and generate the fractal VOFs [15, 16], in which the fractal structure and the VOF act as “lattice” and “base”, respectively. It has been found that the fractal VOFs have the self-healing and information recovering properties [16]. However, the lattice used in the fractal VOFs [15, 16] is a single-fractal (SF) structure, so this kind of fractal VOFs can be called SF-VOFs, which limits the flexibility for designing the fractal VOFs.

The concept of multifractal spectrum [17–19] is proposed to study the complexity and inhomogeneity of fractals. The multifractal (MF) has also attracted much attention, such as optical lattice [20], image processing [21, 22] and optical turbulence [23–25]. Besides the random MFs mentioned above, regular MFs have also attracted attention. Moran fractals, as a kind of regular MF, have been explored [26, 27]. Giménez et al. [28] designed the regular MF zone plates, which improve the axial resolution and also give better performance under polychromatic illumination. A regular MF can be decomposed into multiple fractal geometries, which have different scaling rules. If the MF structure is introduced as the lattice to design the fractal VOFs, various MF-VOFs can be created and expected to have more novel properties.

In this article, we introduce the MF concept to design MF-VOFs, in which the MF structure and the conventional VOF act as the lattice and the base, respectively. We generate two kinds of MF-VOFs experimentally and study their focusing behaviors. In addition, we explore the self-healing and information recovering abilities of MF-VOFs. The results reveal that MF-VOFs have better self-healing and information recovering abilities than SF-VOFs.

2. Generation of MF-VOFs

2.1. MF structures

To describe a fractal structure, a fractal dimension D and a fractal hierarchy n are needed. A fractal dimension is an index for characterizing a fractal pattern or set, by quantifying its complexity as a ratio of the change in detail to the change in scale. For the fractal structure, if one pattern can be divided into K patterns with the size being $1 / ε$ of the original size, D is defined as $D = \ln K / \ln ε$ . In order to clearly describe the procedure of generating the MF structure, we will elaborate from two different perspectives: Intuitive physical picture and Mathematical description.

2.1.1. Intuitive physical picture

Figure 1 illustrates the schematic diagrams of generating the SF and MF structures. We assume that the initial structure be a filled circle with a radius of R₀. Firstly, we introduce the SF structure, as shown in Fig. 1(a). For the SF structure, any hierarchy of fractal has the same fractal geometry but different scale. Here the fractal geometry in the nth hierarchy of fractal is a (3)-circle triangular structure consisting of close-packed 3 identical circles, which is internally tangent with the circles of the $(n - 1)$ th hierarchy of fractal. (i) The initial filled circle is divided by the (3)-circle triangular 1st fractal to generate a ${(3)}_{1}$ -circle triangular SF structure; (ii) any circle of ${(3)}_{1}$ -circle triangular SF structure is divided by the(3)-circle triangular 2nd fractal to generate a ${(3)}_{2}$ -circle triangular SF structure; (iii) any circle of ${(3)}_{2}$ -circle triangular SF structure is divided by the (3)-circle triangular 3rd fractal to generate a ${(3)}_{3}$ -circle triangular SF structure. And so on, we easily construct the ${(3)}_{n}$ -circle triangular SF structure. The subscript n indicates the hierarchy of SF structure. Any circle in the SF structure is divided into three identical circles in the internally tangent form (i.e., K = 3) and the radius of circle in the nth SF structure is 1/2.15 of that in the $(n - 1)$ th SF structure (i.e., $ε = 2.15$ ), implying that the fractal dimension of the ${(3)}_{n}$ -circle triangular SF structure is $D = \ln 3 / \ln (2.15) = 1.44$ . Similarly, when the initial structure of a SF structure is a hexagonal structure composed of 7 [or 6] identical circles, and we call this SF structure a ${(7)}_{n}$ -circle [or ${(6)}_{n}$ -circle] hexagonal SF structure, with a fractal dimension of $D = \ln 7 / \ln 3 = 1.77$ [or $D = \ln 6 / \ln 3 = 1.63$ ].

Secondly, we introduce the MF structure. The generation of the MF structure is very similar to that of the SF structure. However, the SF structure has only one fractal geometry, while the MF structure has more than one fractal geometries. If the MF structure has two or three fractal geometries, we call the MF structure a binary-fractal (BF) or ternary-fractal (TF) structure. As shown in Fig. 1(b), (i) The initial filled circle is divided by the (6)-circle hexagonal 1st fractal to generate a ${(6)}_{1}$ -circle hexagonal SF structure; (ii) any circle of ${(6)}_{1}$ -circle hexagonal SF structure is divided by the (7)-circle hexagonal 2nd fractal to generate a ${(67)}_{1}$ -circle hexagonal BF structure; (iii) any circle of ${(67)}_{1}$ -circle hexagonal BF structure is divided by the (3)-circle triangular 3rd fractal to generate a ${(673)}_{1}$ -circle hexagonal TF structure. And so on, we easily construct the ${(673)}_{n}$ -circle hexagonal TF structure.

Fig. 1 Procedures of generating the SF and MF structures. (a) (3)_n-circle triangular SF structure. (b) (673)_n-circle hexagonal TF structure.

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2.1.2. Mathematical description

In fact, the procedure of constructing the MF structure can be expressed mathematically by convolution operations. Any structure can be abstracted into convolutions of lattice and base. The lattice can be defined by the Dirac delta functions.

The lattice of the (3)-circle triangular fractal geometry with a radius b of the maximum circumcircle is expressed as

T (x, y) = \sum_{p = 1}^{3} δ [x - \frac{2}{\sqrt{3}} r cos (\frac{2 π}{3} p - \frac{π}{6}), y - \frac{2}{\sqrt{3}} r sin (\frac{2 π}{3} p - \frac{π}{6})],

where r is the radius of each circle (or 2r indicates also the distance between adjacent lattice points) in the (3)-circle triangular fractal geometry, as

r = (2 \sqrt{3} - 3) b

.

The lattice of the (7)-circle hexagonal fractal geometry with a radius a of the maximum circumcircle is expressed as

H (x, y) = δ (x, y) + \sum_{p = 1}^{6} δ [x - 2 R cos (\frac{π}{3} p - \frac{π}{6}), y - 2 R sin (\frac{π}{3} p - \frac{π}{6})],

where the radius R of each circle (or 2R indicates also the distance between adjacent lattice points) in the (7)-circle hexagonal fractal geometry, as

R = a / 3

.

The lattice of the (6)-circle hexagonal fractal geometry with a radius $\tilde{a}$ of the maximum circumcircle is expressed as

\tilde{H} (x, y) = \sum_{p = 1}^{6} δ [x - 2 \tilde{R} cos (\frac{π}{3} p - \frac{π}{3}), y - 2 \tilde{R} sin (\frac{π}{3} p - \frac{π}{3})],

where the radius

\tilde{R}

of each circle (or

2 \tilde{R}

indicates also the distance between adjacent lattice points) in the

(6)

-circle hexagonal fractal geometry, as

\tilde{R} = \tilde{a} / 3

.

Generating the nth SF structure

All the cases, we define that the initial structure is a filled circle with a radius of R₀. Based on the above intuitive physical picture, when the initial structure is performed nth $(3)$ -circle triangular fractal, any circle of the (3) $_{n}$ -circletriangular SF structure should have a radius of $r_{n} = {(2 \sqrt{3} - 3)}^{n} b = {(2 \sqrt{3} - 3)}^{n} R_{0}$ . With Eq. (1), the lattice of (3) $_{n}$ -circle triangular SF structure can be written as

{ΓT}_{n} (x, y) = T_{1} (x, y) \otimes T_{2} (x, y) \otimes \dots \otimes T_{n} (x, y),

with

T_{n} (x, y) = \sum_{p = 1}^{3} δ [x - \frac{2}{\sqrt{3}} r_{n} cos (\frac{2 π}{3} p - \frac{π}{6}), y - \frac{2}{\sqrt{3}} r_{n} sin (\frac{2 π}{3} p - \frac{π}{6})] .

Similarly, with Eq. (2), the lattice of (7) $_{n}$ -circle hexagonal SF structure can be written as

{ΓH}_{n} (x, y) = H_{1} (x, y) \otimes H_{2} (x, y) \otimes \dots \otimes H_{n} (x, y),

with

H_{n} (x, y) = δ (x, y) + \sum_{p = 1}^{6} δ [x - 2 R_{n} cos (\frac{π}{3} p - \frac{π}{6}), y - 2 R_{n} sin (\frac{π}{3} p - \frac{π}{6})],

where

R_{n} = a / 3^{n} = R_{0} / 3^{n}

. With Eq. (3), the lattice of (6)

_{n}

-circle hexagonal SF structure can be written as

{Γ \tilde{H}}_{n} (x, y) = {\tilde{H}}_{1} (x, y) \otimes {\tilde{H}}_{2} (x, y) \otimes \dots \otimes {\tilde{H}}_{n} (x, y),

with

{\tilde{H}}_{n} (x, y) = \sum_{p = 1}^{6} δ [x - 2 {\tilde{R}}_{n} cos (\frac{π}{3} p - \frac{π}{3}), y - 2 {\tilde{R}}_{n} sin (\frac{π}{3} p - \frac{π}{3})],

where

{\tilde{R}}_{n} = \tilde{a} / 3^{n} = R_{0} / 3^{n}

.

Generating the nth BF structure

We still define that the initial structure is a filled circle with a radius of R₀. We take the (73) $_{n}$ -circle hexagonal BF structure as an example. Based on the above intuitive physical picture, the initial structure is performed a $(7)$ -circle hexagonal fractal, to generate the ${(7)}_{1}$ -circle hexagonal SF structure, where any circle has a radius of $R_{1} = a / 3 = R_{0} / 3$ . And then any circle of ${(7)}_{1}$ -circle hexagonal SF structure is divided by the (3)-circle triangular fractal to generate a ${(73)}_{1}$ -circle hexagonal BF structure, where any circle has a radius of $r_{1} = (2 \sqrt{3} - 3) R_{1} = (2 \sqrt{3} - 3) R_{0} / 3$ . Repeating the above procedure, we easily generate ${(73)}_{n}$ -circle hexagonal BF structure. With Eqs. (1) and (2), the lattice of (73) $_{n}$ -circle hexagonal BF structure can be written as

{ΓHT}_{n} (x, y) = [H_{1} (x, y) \otimes T_{1} (x, y)] \dots \otimes [H_{n} (x, y) \otimes T_{n} (x, y)],

with

\begin{array}{l} H_{n} (x, y) = = δ (x, y) + \sum_{p = 1}^{6} δ [x - 2 R_{n} \cos (\frac{π}{3} p - \frac{π}{6}), y - 2 R_{n} \sin (\frac{π}{3} p - \frac{π}{6})], \\ T_{n} (x, y) = \sum_{p = 1}^{3} δ [x - \frac{2}{\sqrt{3}} r_{n} \cos (\frac{2 π}{3} p - \frac{π}{6}), y - \frac{2}{\sqrt{3}} r_{n} \sin (\frac{2 π}{3} p - \frac{π}{6})], \end{array}

where

R_{n} = R_{0} / 3^{n}

and

r_{n} = {(2 \sqrt{3} - 3)}^{n} R_{0} / 3^{n}

.

Generating the nth TF structure

We still define that the initial structure is a filled circle with a radius of R₀. We take the (673) $_{n}$ -circle hexagonal TF structure as an example. Based on the above intuitive physical picture, the initial structure is performed a $(6)$ -circle hexagonal fractal, to generate the ${(6)}_{1}$ -circle hexagonal SF structure, where any circle has a radius of ${\tilde{R}}_{1} = \tilde{a} / 3 = R_{0} / 3$ . And then any circle of ${(6)}_{1}$ -circle hexagonal SF structure is divided by a (7)-circle hexagonal fractal to generate a ${(67)}_{1}$ -circle hexagonal BF structure, where any circle has a radius of $R_{1} = a / 3 = {\tilde{R}}_{1} / 3 = R_{0} / 3^{2}$ . Any circle of the ${(67)}_{1}$ -circle hexagonal BF structure is further divided by a (3)-circle triangular fractal to generate a (673) $_{1}$ -circle hexagonal TF structure, where any circle has a radius of $r_{1} = (2 \sqrt{3} - 3) b = (2 \sqrt{3} - 3) R_{1} = (2 \sqrt{3} - 3) R_{0} / 3^{2}$ . Repeating the above procedure, we easily generate ${(673)}_{n}$ -circle hexagonal TF structure. With Eqs. (1)-(3), the lattice of (673) $_{n}$ -circle hexagonal TF structure can be written as

{Γ \tilde{H} HT}_{n} (x, y) = [{\tilde{H}}_{1} (x, y) \otimes H_{1} (x, y) \otimes T_{1} (x, y)] \dots \otimes [{\tilde{H}}_{n} (x, y) \otimes H_{n} (x, y) \otimes T_{n} (x, y)],

with

\begin{array}{l} {\tilde{H}}_{n} (x, y) = \sum_{p = 1}^{6} δ [x - 2 {\tilde{R}}_{n} \cos (\frac{π}{3} p - \frac{π}{3}), y - 2 {\tilde{R}}_{n} \sin (\frac{π}{3} p - \frac{π}{3})], \\ H (x, y) = = δ (x, y) + \sum_{p = 1}^{6} δ [x - 2 R_{n} \cos (\frac{π}{3} p - \frac{π}{6}), y - 2 R_{n} \sin (\frac{π}{3} p - \frac{π}{6})], \\ T_{n} (x, y) = \sum_{p = 1}^{3} δ [x - \frac{2}{\sqrt{3}} r_{n} \cos (\frac{2 π}{3} p - \frac{π}{6}), y - \frac{2}{\sqrt{3}} r_{n} \sin (\frac{2 π}{3} p - \frac{π}{6})], \end{array}

where

{\tilde{R}}_{n} = R_{0} / 3^{n}

,

R_{n} = R_{0} / 3^{2 n}

and

r_{n} = {(2 \sqrt{3} - 3)}^{n} R_{0} / 3^{2 n}

.

Fig. 2 Measured intensity patterns of four kinds of BF-VOFs with the same base. These four columns show (36)_n-circle triangular, (63)_n-circle hexagonal, (37)_n-circle triangular and (73)_n-circle hexagonal BF-VOFs with n = 1 and n = 2, respectively. The first and third rows show the total intensity patterns of BF-VOFs we generated experimentally. The second and fourth rows correspond to the y-component intensity patterns of BF-VOFs.

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2.2. The generation of MF-VOFs

The base with arbitrary amplitude, phase and polarization distribution is designed by the method in [16]. We focus mainly on the generation of MF-VOFs by using multi-fractal structures as lattices. Here we choose the typical radially polarized VOF as the base. We use the MF structure as the lattice, and the radially polarized VOF as the base to generate MF-VOF. The radially polarized VOF with a topological charge m can be expressed as $E_{VOF} = E_{0} [\cos (m ϕ) e_{x} + \sin (m ϕ) e_{y}] S_{n} (x, y)$ , where ${e_{x}, e_{y}}$ is a pair of unit vectors along the x and y directions in the Cartesian coordinate system and ϕ is the azimuthal angle in the corresponding polar coordinate system. $S_{n} (x, y)$ is a shape function of defining the shape of the base, here $S_{n} (x, y) = circ (\sqrt{x^{2} + y^{2}} / w_{n})$ with $circ (\cdot)$ being the well-known circular function and w_n being the radius of the circle. Therefore, the nth iteration MF-VOF can be expressed as $E_{n} (x, y) = Γ_{n} (x, y) \otimes E_{VOF} (x, y)$ , where $Γ_{n} (x, y)$ is the expression for the lattice of MF structure. We use the 4f system and wavefront reconstruction method [29, 30] to generate the MF-VOFs in our experiment, and use the CCD camera to detect their intensity patterns. The experimental setup is the same as that used in [16].

The MF-VOF with a BF structure as the lattice is called BF-VOF. We experimentally generate four kinds of BF-VOFs, as shown in Fig. 2, where the base is the radially polarized VOF. The first and third rows are the total intensity patterns of BF-VOFs with n = 1 and n = 2, respectively. The second and fourth rows are the corresponding y-component intensity patterns, respectively. The initial structures of the lattices in the first and second columns are composed of $(3)$ -circle triangular and $(6)$ -circle hexagonal fractal geometries. We generate two kinds of BF-VOFs: (36) $_{n}$ -circle triangular and (63) $_{n}$ -circle hexagonal BF-VOFs. The initial structures of the lattices in the third and fourth columns are composed of $(3)$ -circle triangular and $(7)$ -circle hexagonal fractal geometries, and two kinds of BF-VOFs are also generated: (37) $_{n}$ -circle triangular and (73) $_{n}$ -circle hexagonal BF-VOFs.

The MF-VOF with a TF structure as the lattice is called TF-VOF. We experimentally generate three kinds of TF-VOFs with the radially polarized VOF as the base, which are shown in Fig. 3. The first and second rows show the total and y-component intensity patterns of TF-VOFs with n = 1, respectively. The initial structures of the lattices in Fig. 3 are composed of $(3)$ -circle triangular, $(7)$ -circle hexagonal and $(6)$ -circle hexagonal fractal geometries. We generate three kinds of TF-VOFs: (376) $_{1}$ -circle triangular, (763) $_{1}$ -circle hexagonal, and (673) $_{1}$ -circle hexagonal TF-VOFs, as shown in Fig. 3.

Fig. 3 Measured intensity patterns of three kinds of TF-VOFs with the same base (radially polarized VOF). The three columns show (376)₁-circle triangular, (763)₁-circle hexagonal, and (673)₁-circle hexagonal TF-VOFs, respectively. The first rowshows the total intensity patterns of TF-VOFs. The second row corresponds to the y-component intensity patterns of TF-VOFs.

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3. The focusing behavior of MF-VOFs

As is well known, the focal field can be expressed by the Fourier transformation of the input optical field under the paraxial approximation. When the input field is expressed by the convolution of the lattice and the base, the focal field can be represented as the product of their Fourier transformations. We take the focal field ${E^{'}}_{n}$ of (73) $_{n}$ -circle hexagonal BF-VOF as an example:

{E^{'}}_{n} (x^{'}, y^{'}) = {ΓHT}^{'}_{n} (x^{'}, y^{'}) {E^{'}}_{VOF} (x^{'}, y^{'}),

with

\begin{array}{l} ΓHT'_{n} (x^{'}, y^{'}) = \prod_{i = 1}^{n} [{H^{'}}_{i} (x^{'}, y^{'}) {T^{'}}_{i} (x^{'}, y^{'})], \\ {H^{'}}_{i} (x^{'}, y^{'}) = 1 + \sum_{p = 1}^{6} \exp {- \frac{j 4 π}{λ f} R_{i} [x^{'} \cos (\frac{π}{3} p - \frac{π}{2}) + y^{'} \sin (\frac{π}{3} p - \frac{π}{2})]}, \\ {T^{'}}_{i} (x^{'}, y^{'}) = \sum_{q = 1}^{3} \exp {- \frac{j 4 π}{\sqrt{3} λ f} r_{i} [x^{'} \cos (\frac{2 π}{3} q - \frac{5 π}{6}) + y^{'} \sin (\frac{2 π}{3} q - \frac{5 π}{6})]}, \end{array}

where x^′ and y^′ are the coordinates in the focal plane.

{ΓHT}^{'}_{n} (x^{'}, y^{'})

and

{E^{'}}_{VOF} (x^{'}, y^{'})

are the Fourier transformations of the lattice and the base, respectively.

{H^{'}}_{i} (x^{'}, y^{'})

and

{T^{'}}_{i} (x^{'}, y^{'})

are the Fourier transformations of

(7)

-circle hexagonal and

(3)

-circle triangular fractal geometries in the ith hierarchy. λ is the light wavelength and f is the focal length.

Fig. 4 Simulated and measured focal fields of different MF-VOFs. (a) corresponds to (3)₁-circle triangular SF-VOF, (73)₁-circle hexagonal BF-VOF and (673)₁-circle hexagonal TF-VOF. (b) shows (73)₂-circle hexagonal BF-VOF. The first and second rows correspond to the simulated and measured results, respectively. The patterns in the first, second, and third columns of (a) have sizes of 2.15×2.15 mm², 6.45×6.45 mm², 19.35×19.35 mm², respectively. The patterns of (b) have sizes of 41.6×41.6 mm².

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As mentioned above, the MF-VOF depends on the lattice and the base, and its focal field size is related to the radius of the minimum circle. Figure 4(a) shows the focal fields of three kinds of MF-VOFs with n = 1. It can be seen that their focal fields all exist six strongest spots, which is mainly determined by (3)-circle triangular fractal. Obviously, the focal fields exhibit a sixfold rotation-symmetry, because the focal fields of their lattices and the base have the sixfold rotation-symmetry and the cylindrical symmetry, respectively. For ${(3)}_{n}$ -circle triangular SF-VOF, (73) $_{n}$ -circle hexagonal BF-VOF and (673) $_{n}$ -circle hexagonal TF-VOF, the radius of their smallest circle in the nth hierarchy are ${(2 \sqrt{3} - 3)}^{n} a$ , ${(2 \sqrt{3} - 3)}^{n} a / 3^{n}$ and ${(2 \sqrt{3} - 3)}^{n} a / 9^{n}$ , respectively, where a is the radius of the maximum circumcircles of their lattice structures. Therefore, in the nth hierarchy, the ratio of the distance between the two adjacent strongest focal spots of the three MF-VOFs is $1 : 3^{n} : 9^{n}$ . If n = 1, then the distance radio is $1 : 3 : 9$ . Figure 4(b) shows the focal field of (73) $_{2}$ -circle hexagonal BF-VOF, and the distance between the two adjacent strongest spots is $3 / (2 \sqrt{3} - 3)$ times of that of (73) $_{1}$ -circle hexagonal BF-VOF. In addition, we find that the measured results are in good agreement with the simulations.

Fig. 5 Measured input and focal fields of MF-VOFs with their lattices composed of two (or three)-hierarchy fractal geometries. (a) fractal VOFs with their lattices composed of two-hierarchy fractal geometries, (3)₂-circle triangular SF-VOF and (73)₁-circle hexagonal BF-VOF. (b) fractal VOFs with their lattices composed of three-hierarchy fractal geometries, (3)₃-circle triangular SF-VOF and (673)₁-circle hexagonal TF-VOF. In the second row, the sizes of focal fields from left to right are 4.62×4.62 mm², 6.45×6.45 mm², 9.94×9.94 mm² and 19.35×19.35 mm², respectively.

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Fig. 6 The self-healing ability of MF-VOF and SF-VOF with their lattices composed of two(or three)-hierarchy fractal geometries. (a) Measured intensity patterns of the focal field of (73)₁-circle hexagonal BF-VOF blocked by fan-shaped obstacles. (b) The γ curves for SF-VOF and BF-VOF with their lattices composed of two-hierarchy fractal geometries blocked by fan-shaped obstacles. (c) Measured intensity patterns of the focal field of (673)₁-circle hexagonal TF-VOF blocked by fan-shaped obstacles. (d) The γ curves for SF-VOF and TF-VOF with their lattices composed of three-hierarchy fractal geometries blocked by fan-shaped obstacles. The sizes of focal fields in (a) and (c) are 6.45×6.45 mm² and 19.35×19.35 mm²,respectively.

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4. Self-healing and information recovery abilities of MF-VOFs

Recently, the application of VOF in optical communication has attracted attention [31, 32]. As demonstrated in [16], the SF-VOF has the self-healing and information recovery abilities. As mentioned above, the SF structure has only one fractal geometry, while the MF structure has more than one fractal geometries. The introduction of MF structures can enrich the design of VOFs and may consequently bring advantages over SF-VOFs. Here we compare the self-healing and information recovery abilities of MF-VOFs with SF-VOFs whose lattices are composed of two(or three)-hierarchy fractal geometries under two cases: (i) the input field of MF-VOF is partially blocked, and the self-healing ability of the focal field is studied; (ii) the spatial frequency spectrum of MF-VOF is partially blocked, and the information recovery ability at the image plane is studied. To quantitatively compare the self-healing and information recovery abilities of MF-VOF with that of SF-VOF, a mean structural similarity index γ [33–35] is introduced to evaluate the whole frame quality from luminance, contrast and structure. γ = 0 implies that the recovered information has no similarity to the original information, while γ = 1 means that the recovered and original information is completely identical.

4.1. Self-healing ability of MF-VOFs

We compare the self-healing ability of MF-VOFs with SF-VOFs whose lattices are composed of two(or three)-hierarchy fractal geometries. As shown in Fig. 5(a), the lattices of MF-VOFs are composed of two-hierarchy fractal geometries. The first column shows ${(3)}_{2}$ -circle triangular SF-VOF, whose lattice is composed of two ${(3)}_{1}$ -circle triangular fractal geometries, and the second column shows (73) $_{1}$ -circle hexagonal BF-VOF, whose lattice is composed of $(7)$ -circle hexagonal and $(3)$ -circle triangular fractal geometries. In Fig. 5(b), the lattices of MF-VOFs are composed of three-hierarchy fractal geometries. The first column depicts ${(3)}_{3}$ -circle triangular TF-VOF, whose lattice is formed by three $(3)$ -circle triangular fractal geometries, and the second column depicts (673) $_{1}$ -circle hexagonal TF-VOF, whose lattice consists of $(6)$ -circle hexagonal, $(7)$ -circle hexagonal and $(3)$ -circle triangular fractal geometries.

Fig. 7 Measured input fields of fractal VOFs with their lattices composed of two(or three)-hierarchy fractal geometries. (a) fractal VOFs with lattices composed of two-hierarchy fractal geometries, (3)₂-circle triangular SF-VOF and (73)₁-circle hexagonal BF-VOF. (b) fractal VOFs with lattices composed of three-hierarchy fractal geometries, (3)₃-circle triangular SF-VOF and (673)₁-circle hexagonal TF-VOF.

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Figures 6(a) and 6(c) show the measured intensity patterns of the focal fields when (73) $_{1}$ -circle hexagonal BF-VOF and (673) $_{1}$ -circle hexagonal TF-VOF are blocked by different fan-shaped obstacles. In the experiment, the blocked angles θ of the fan-shaped obstacles are $θ = 0^{\circ}$ , 60 $^{\circ}$ , 120 $^{\circ}$ , 180 $^{\circ}$ , 240 $^{\circ}$ and 300 $^{\circ}$ , respectively. Clearly, although the focal spots become larger and their qualities degrade as θ increases, their locations have no change. In particular, the focal spots are still clearly visible although five-sixth of the input fractal VOF is blocked, implying that fractal VOFs have the self-healing ability. Figure 6(b) plots relationships of γ and θ for two fractal VOFs with their lattices composed of two-hierarchy fractal geometries, where ${(3)}_{2}$ -circle triangular SF-VOF and (73) $_{1}$ -circle hexagonal BF-VOF are shown by the blue and red curves, respectively. Clearly, the self-healing ability of (73) $_{1}$ -circle hexagonal BF-VOF is better than that of ${(3)}_{2}$ -circle triangular SF-VOF; for instance, the γ value for (73) $_{1}$ -circle hexagonal BF-VOF is ∼2.6 times of that for ${(3)}_{2}$ -circle triangular SF-VOF, when $θ = 300^{\circ}$ . Figure 6(d) plots relationships of γ and θ for two kinds of fractal VOFs with their lattices composed of three-hierarchy fractal geometries, where the blue line represents ${(3)}_{3}$ -circle triangular SF-VOF and the red line represents (673) $_{1}$ -circle hexagonal TF-VOF. It can be seen from the $γ \sim θ$ curves that the self-healing ability of (673) $_{1}$ -circle hexagonal TF-VOF is better than that of ${(3)}_{3}$ -circle triangular SF-VOF; for example, the γ value for (673) $_{1}$ -circle hexagonal TF-VOF is ∼4 times of that for ${(3)}_{3}$ -circle triangular SF-VOF, when the blocked angle is 300 $^{\circ}$ . As a result, when the lattices of the fractal VOFs are composed of the same hierarchy of fractal geometries, the self-healing ability of the MF-VOF is better than that of the traditional SF-VOF.

Fig. 8 The information recovery abilities of SF-VOF and MF-VOF with their lattices composed of two(or three)-hierarchy fractal geometries. (a) Measured total and y-component intensity pattern of (73)₁-circle hexagonal BF-VOF in the image plane when its spatial frequency spectrum is blocked by fan-shaped obstacle. (b) The γ ∼ θ curves for two kinds of fractal VOFs with their lattices composed of two-hierarchy fractal geometries. (c) Measured total and y-component intensity patterns of (673)₁-circle hexagonal TF-VOF in the image plane when its spatial frequency spectrum is blocked by fan-shaped obstacle. (d) The γ ∼ θ curves for two kinds of fractal VOFs with their lattices composed of three-hierarchy fractal geometries.

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4.2. Information recovery ability of MF-VOFs

Now we use two kinds of bases to design MF-VOFs. Base-1 and Base-2 are x- and y-polarized vortices, respectively. Part of the whole F-VOF is filled by Base-1, while the rest part is loaded by Base-2. We compare the information recovery ability of MF-VOFs with SF-VOFs, both have two(or three)-hierarchy structures. As shown in Fig. 7, the first row shows the total intensity patterns, and the second row gives the corresponding x-polarized intensity patterns. Figure 7(a) shows the fractal VOFs whose lattices are composed of two-hierarchy fractal geometries, where the first and second columns are ${(3)}_{2}$ -circle triangular SF-VOF and (73) $_{1}$ -circle hexagonal BF-VOF, respectively. Figure 7(b) depicts the fractal VOFs whose lattices are composed of three-hierarchy fractal geometries, where the first and second columns show ${(3)}_{3}$ -circle triangular SF-VOF and (673) $_{1}$ -circle hexagonal TF-VOF, respectively. In particular, in polarization-encoded (73) $_{1}$ -circle hexagonal BF-VOF and (673) $_{1}$ -circle hexagonal TF-VOF, the letters “N” and “NKU” hidden in the total intensity patterns can be decoded by using the x-polarizer, respectively.

Figures 8(a) and 8(c) show the detected total and x-component intensity patterns of (73) $_{1}$ -circle hexagonal BF-VOF and (673) $_{1}$ -circle hexagonal TF-VOF in the image plane when the spatial frequency spectrum is blocked by three fan-shaped obstacles of θ = 0, 90 $^{\circ}$ and 180 $^{\circ}$ , respectively. We can see that MF-VOF has very good information recovery ability. Particularly, when the spatial frequency spectrum is half blocked, the whole lattices and the letters “N” and “NKU” are still very clear. Figure 8(b) plots the $γ \sim θ$ curves of the two kinds of fractal VOFs whose lattices are composed of two-hierarchy fractal geometries, where the blue and red curves represent ${(3)}_{2}$ -circle triangular SF-VOF and (73) $_{1}$ -circle hexagonal BF-VOF, respectively. It can be seen that (73) $_{1}$ -circle hexagonal BF-VOF has better information recovery ability than that of ${(3)}_{2}$ -circle triangular SF-VOF. Figure 8(d) plots the $γ \sim θ$ curves of the two kinds of fractal VOFs whose lattices are composed of three-hierarchy fractal geometries, where the blue and red curves represent ${(3)}_{3}$ -circle triangular SF-VOF and (673) $_{1}$ -circle hexagonal TF-VOF, respectively. It can be seen that the information recoveryability of (673) $_{1}$ -circle hexagonal TF-VOF is better than that of ${(3)}_{3}$ -circle triangular SF-VOF. As a result, when the fractal VOFs are composed of the same hierarchy of fractal geometries, MF-VOF has the better information recovery ability than that SF-VOF.

5. Conclusion

In conclusion, we introduce the concept of multifractal into the optical fields. We use multifractal structure as the lattice to design multifractal VOFs, which enrich the family of VOFs and have advantages over SF-VOF in information transmission. We experimentally generate two kinds of MF-VOFs (BF-VOFs and TF-VOFs) and study their focusing behaviors. The experimental results prove that MF-VOFs have better self-healing and information recovery abilities than that of SF-VOFs, when their lattices are composed of the same hierarchy of fractal geometries. For the radially and azimuthally polarized VOFs as the base, the self-healing and information recovering abilities of MF-VOF are almost the same, and both are superior to that of the multifractal scalar optical fields. In addition, we also confirm that under the tight focusing condition, the longitudinally polarized components of MF-VOFs also show the self-healing property. All the results imply that MF-VOFs have better ability to resist information missing during the focusing and imaging process. These properties of MF-VOFs may find potential applications in information transmission, optical communication, etc.

Funding

National Key R&D Program of China (2017YFA0303800, 2017YFA0303700); National Natural Science Foundation of China (11534006, 11774183, 11674184); Natural Science Foundation of Tianjin (16JCZDJC31300); 111 Project (B07013).

Acknowledgments

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

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Multifractal vector optical fields

Abstract

1. Introduction

2. Generation of MF-VOFs

2.1. MF structures

2.1.1. Intuitive physical picture

2.1.2. Mathematical description

Generating the nth SF structure

Generating the nth BF structure

Generating the nth TF structure

2.2. The generation of MF-VOFs

3. The focusing behavior of MF-VOFs

4. Self-healing and information recovery abilities of MF-VOFs

4.1. Self-healing ability of MF-VOFs

4.2. Information recovery ability of MF-VOFs

5. Conclusion

Funding

Acknowledgments

References

Cited By

Figures (8)

Equations (15)

Optics Express