Fractal Light Vortices Fractal Light Vortices

Vortex lenses produce special wavefronts with zero-axial intensity, and helical phase structure. The variations of the phase and amplitude of the vortex produce a circular flow of energy that allows transmitting orbital angular momentum. This property is especially in optical trapping, because due to the orbital angular momentum of light, they have the ability to set the trapped particles into rotation. Vortex lenses engraved in diffractive optical elements have been proposed in the last few years. These lenses can be described mathematically as a two-dimensional (2D) function, which expressed in polar coordinates are the product of two different separable one-dimensional (1D) functions: One, depends only on the square of radial coordinate, and the other one depends linearly on the azimuthal coordinate and includes the topological charge. The 1D function that depends on the radial coordinate is known as a zone plate. Here, vortex lenses, constructed using different aperiodic zone plates, are reviewed. Their optical properties are studied numerically by computing the intensity distribution along the optical axis and the transverse diffraction patterns along the propagation direction. It is shown that these elements are able to create a chain of optical traps with a tunable separation, strength and transverse section.


Introduction
Vortex lenses (VLs) produce special wavefronts with zero-axial intensity, and helical structure with undefined phase in the vortex centre. These structures, called optical vortices [1][2][3], constitute an intriguing and growing area of research that combines fundamental theoretical aspects and novel applied technologies. A fundamental property of vortices is the conservation of the topological charge which is defined as the 2π module of the total change of the phase along a closed curve surrounding the vortex centre. The variations of the phase and amplitude of the vortex produce a circular flow of energy that allows to carry orbital angular momentum [4,5]. This property is useful for several applications, such as in astrophysics [6,7], transmission of information [8,9], microscopy [10], laser engraving [11] and especially in optical traps [12][13][14][15][16], because due to the orbital angular momentum of light, they have the ability to set the trapped particles into rotation.
Several methods have been proposed to obtain optical vortices [3,4,[17][18][19][20] being spiral phase plates (i.e. lenses with a linear phase dependence on the azimuthal angle) and diffractive optical elements (DOEs) probably the most frequent approach. In fact, due to its simplicity, they can be employed in multiple applications. In particular, zone plates (ZPs) have found a great number of new applications in the last few years [21]. A standard amplitude ZP consists of a series of concentric circular rings of equal area, with alternating transmitting and absorbing zones. This means that along the square of the radial coordinate, a ZP can be thought as a periodic structure. The focusing effect is created by the constructive interference of waves passing through this structure.
FrZPs are characterized by its fractal structure along the square of the radial coordinate that produce multiple foci along the optical axis which are defined by the self-similar Fourier spectrum of the fractal pupil function [23]. These lenses produce a main focus surrounded by numerous secondary foci, which together result in a compound focal volume. It has been demonstrated that these self-similar foci produce reduction of the chromatic aberration under wideband illumination and increase of depth of field [24].
FiZPs are bifocal ZPs with their foci located at certain axial positions given by the Fibonacci numbers, being the 'golden mean', the ratio of the two focal distances [25]. As the name indicates, these lenses are designed following an aperiodic structure generated by the Fibonacci sequence. The focusing and imaging capabilities of Fibonacci lenses have been experimentally demonstrated under monochromatic illumination [26]; however, these lenses are affected by the same limitations of conventional ZPs when broadband illumination is considered, since the twin foci are not self-similar.
TMZPs are based on the deterministic Thue-Morse sequence, which results in a combination of the FrZP and FiZP. For this reason, this new family of aperiodic ZPs combines the advantages of fractal ZPs (reduction of the chromatic aberration) and Fibonacci ZPs (bifocusing along the optical axis) [27].
In this chapter, the combination of a vortex lens and ZPs, constructed using different aperiodic sequences, in a single element is considered. Their optical properties are investigated numerically and experimentally. The focusing properties of different combinations of FZPs and vortex lenses are studied by computing the intensity distribution along the optical axis and the transverse diffraction patterns along the propagation direction. The diffracted field of these vortex lenses was obtained numerically within the Fresnel approximation.
We emphasize that these elements are able to create a chain of optical traps along the optical axis with a tunable separation, strength and transverse section. We discuss the influence of the topological charge on the irradiance propagation and also we investigate the variation of the angular momentum provided by the doughnut-shaped foci.
An optical set-up was implemented to obtain experimental results in which the VLs were registered in a spatial light modulator (SLM).
We have shown that our VLs are able to generate multiple-plane optical trappings with a volumetric extension. In this sense, they are superior to conventional vortices, whose extension is limited to the depth of focus of the beam.

Basic theory
The transmittance of a VL can be expressed as the product of two factors being the first one associated to a given ZP, which has only a radial dependence, and the other one corresponding to a vortex lens with a linear phase dependence on the azimuthal angle θ.
A ZP can be realized from a one-dimensional (1D) compact-supported periodic function qðζÞ as shown in Figure 1, where ζ ¼ r a À Á 2 is the normalized square radial coordinate and a is the external radius of the outermost ring. Therefore, in a binary ZP, every pair of opaque and transparent zones conforms a period. The area of each period is constant over all the ZPs.
In a similar way, the aperiodic zone plates that we consider from now on can be constructed by replacing the periodic function qðζÞ by either the Cantor function, the Fibonacci sequence or the Thue-Morse sequence. In fact, when designing VL, each of these sequences can be used to define the transmission generating function qðζÞ with compact support on the interval ζ∈½0; 1. This interval is partitioned in 2S sub-intervals of length dS ¼ 1 2S , and the transmittance value t S, j , which takes at the jth sub-interval, is associated to the value of the element D S, j , being t S, j ¼ 1 for the transparent zones and t S, j ¼ 0 for the opaque zones.
To study the focusing properties of VLs, we compute the irradiance provided by the transmittance of each lens described in general terms as where the azimuthal dependence is characterized by the topological charge m We consider monochromatic plane wave illumination of wavelength λ. Thus, within the Fresnel approximation the irradiance function is given by where u ¼ a 2 2λz is the dimensionless reduced axial coordinate and v ¼ r a is the normalized transverse coordinate [23]. By using Eq. (1) and taking into account that

Equation 2 is reduced to
J m ðÞ being the Bessel function of the first kind of order m.

Fractal vortex lenses
Two different results were independently obtained by combining spiral phase plates with FrZP to produce a sequence of focused optical vortices along the propagation direction. In Ref. [31], the spiral fractal zone plate is generated as a phase-only FrZP modulated by helical phase structure. Another design of spiral phase plate based on a blazed FrZP, the devil's lens [32], was proposed to improve diffraction efficiency of a spiral FrZP. A devil's lens has a characteristic surface relief along the radial coordinate which is obtained using the devil's staircase function [33]. It is because of its blazed profile that devil vortex lens has improved diffraction efficiency with respect to the spiral fractal zone.
The focal volume generated by Fresnel vortex lenses (FrVLs) results in a chain of vortices that could be used as versatile and very efficient optical tweezers because, in addition to rotating the trapped particles with high refractive index, other particles with a lower refractive index can be trapped in the vortex centre. The relative angular velocity of the particles at the different traps can be changed with different topological charges; the distances between the links of the chain can be modified with different values of S-parameter.
Following the same approach employed in Figure 1, an FrVL is mathematically obtained by replacing the periodic sequence by a Cantor structure developed up to certain stage. Let us consider, for example, the triadic regular Cantor sequence. The construction procedure is shown in Figure 1a. In the first stage, S ¼ 1 , the segment is divided into three parts, and the middle one is removed. In the second stage, this slicing-and-removing process is repeated in each one of the remaining two segments from the first stage. This process is repeated in the following stages. In mathematical terms, the FrVL transmittance function, for a given stage S, can be expressed by replacing the following generating function of the FrZP in Eq. (1) [23]: In which the transparent and opaque zones are replaced by pure phase zones differing in π for the design wavelength. In this equation, the function modðx, yÞ gives the remainder on division of x by y.
Typical results are shown in Figure 2b for a fractal vortex lens with topological charge m ¼ 2 and in Figure 2c for an FVL with topological chargem ¼ 3. By using expression (4), we have computed the irradiance provided by these FrVLs at transverse planes along the optical axis. The result is shown in Figure 3.
As we mentioned, the diffraction efficiency of these FVLs can be improved by imposing a blazed profile to the radial coordinate. This can be done by using the Cantor function, or devil's staircase, as a generating qðζÞ function [32,34]. The focusing properties of these FVL were experimentally analysed in [35]. It has been demonstrated that for multiple-plane optical trappings, they can generate a light beam with axially distributed optical vortices. The transverse patterns appearing along the propagation distance present several concatenated doughnut modes as represented in Figure 3.
Besides, the generation of multiple vortex distributions is of interest as demonstrated by the new methods that have been recently proposed to generate different two-dimensional (2D) and three-dimensional (3D) arrays of vortices. Different methods have been employed, such as interferometric techniques using Michelson or Mach-Zehnder interferometers [36] and Dammann gratings [37]. A simple method to obtain special arrays of vortices is possible by means of a reconfigurable spatial light modulator (SLM). In fact, the use of an SLM allowed the possibility to change in a simple way the characteristics of diffractive lenses, such as their focal length or their topological charge. The implementation of compound 3D optical vortex structures by means of an array of DVLs was reported in [38] with numerical simulations and experimental results.

Fibonacci vortex lenses
Fibonacci vortex lenses (FiVLs) are constructed using the Fibonacci sequence [22]. This sequence has been also employed in the development of different photonic devices and applications [39], such as multilayers and gratings [40], cryptography [41] and photonic crystals [42]. The Fibonacci sequence is a set of ordered numbers, F j ¼ {0; 1; 1; 2; 3; 5; 8; 13; 21;…}, that are obtained following a simple rule: Starting with two elements (called seeds) F 0 ¼ 0 and F 1 ¼ 1, the following numbers of the sequence are obtained as F j þ 1 ¼ F j þ F j−1 , ðj ¼ 0; 1; 2;…Þ. The ratio of two consecutive elements of the Fibonacci sequence approaches asymptotically an irrational number known as the golden mean: Fibonacci series and the golden mean have been ubiquitously observed in nature and on different scientific areas [43,44].
Based on the Fibonacci numbers, a binary Fibonacci sequence can also be generated with two seed elements, S 1 ¼ {A} and S 0 ¼ {B} , as shown in Figure 4a. Then, the next order of sequence is obtained simply as the concatenation of the two previous ones: S jþ1 ¼ {S j S j−1 } for j≥1. Consequently, S 2 ¼ fABg, S 3 ¼ fABAg, S 4 ¼ fABAABg, S 5 ¼ {ABAABABA}, and so on. It should be noted that the total number of elements of the order j sequence is F jþ1 and that, for each S, two consecutive 'B' are separated by either one or two 'A'. Each sequence can be used to define the binary generating function for the radial phase distribution of the FiVL.
In our case, the function, Φ j ðζÞ, is defined in the domain ½0; 1, which is partitioned in F jþ1 subintervals of length d ¼ 1=F jþ1 . Therefore, the function Φ j ðζÞ at the k th sub-interval is either 0 or π if the value of the k th element of the S j sequence, S jk , is 'A' or 'B', respectively. Finally, after performing the coordinate transformation, ζ ¼ r a À Á 2 , the radial part of the transmittance is obtained as qðζÞ ¼ exp½iΦ j ðζÞ as shown in Figure 1.
An FiVL is defined as a pure phase diffractive element whose phase distribution is given by Φ FVL ðζ, θ 0 Þ ¼ mod2π½Φ j ðζÞ þ mθ 0 . Thus, it combines the azimuthal phase variation that characterizes a vortex lens, with the radial phase distribution that is generated through the Fibonacci sequence. Figure 4b and c shows the results for S ¼ 7 and S ¼ 8 for the topological charge m ¼ 2.
Compared with a Fresnel zone plates (which, we recall, can be considered periodic structures along the square radial coordinate ζ), it can be verified that if both have the same number of elements, Fj, but the position of some zones with different phase has been interchanged, the FiZP produces a focal splitting of the main focus of the Fresnel zone plate along the axis. Thus, an FVL with m ¼ 0 can be understood as two Fresnel zone plates interlaced [45].
The irradiances provided by the lenses are shown in Figure 4. The result is shown in Figure 5.
The integrals were computed using Eq. (4) applying Simpson's rule with a step length 1/2000. Note that FiVLs produce twin foci whose locations are coincident with the Fibonacci numbers. In fact, for S 8 FVLs, the first focus is located at u 1 ¼ 13 ¼ F j−1 and the other one at u 2 ¼ 21 ¼ F j : Moreover, the ratio of the focal distances satisfies u2 u1 ≈ϕ. The diameter of these 'twin' vortices provided by FiVLs is related by the golden mean [46], and the diameter of the rings increases proportionally with the topological charge.

Thue-Morse vortex lens
The Thue-Morse sequence [22] is also a binary sequence in which each element is obtained with the previous one by appending to it its Boolean complement. This sequence has been applied in several branches of Physics, as, for example, in the context of photonic crystals [47], quantum wells [48], metamaterials [49] and graphene superlattices [50].
The characteristic function q(ζ) corresponding to the Thue-Morse sequence is constructed defining a seed D 0 ¼ A from which the following elements in the sequence are obtained by replacing A by AB and B by BA. In this way, D 1 ¼ AB, D 2 ¼ ABB, D 3 ¼ BBABAAB, D4 ¼ ABBABAABBAABABBA, and so on. Figure 6a shows the geometrical construction of the TM sequence up to order S ¼ 4. When designing Thue-Morse vortex lens (TMVL), each D i can be used to define the transmission function qðζÞ with compact support on the interval ζ∈½0; 1. This interval is divided in 2S sub-intervals of length dS ¼ 1=2S, where the transmittance, t S, j , of the jth sub-interval is associated to the element D S, j , as t S, j ¼ 1 when D S, jis 'A', and t S, j ¼ 0 when D S, j is 0 B 0 . Figure 1(b) shows the transmittance pupil function of a TMZP of order S ¼ 6 and its equivalent periodic ZP. Note that like a conventional ZP the period of a TMZP is pS ¼ 2dS, where the position of transparent/opaque zones has been interchanged. In mathematical terms, the transmittance function, qðζÞ, can be written as Figure 7 shows the axial irradiance provided by TMVLs of orders S ¼ 4 and S ¼ 5. Note that the Thue-Morse ZP produces a symmetrical splitting of the first-order focus. This zero irradiance in the middle is due to the destructive interference generated by the two conjugated parts of the lens. In this way, like FiVLs, TMVLs are bifocals, but they produce a sequence of secondary foci around each main focus that have a fractal structure. In fact, it has been shown that irradiance provided by these lenses with topological charge m ¼ 0 is self-similar [51], that is, the irradiance distribution corresponding to a TMVL of order S is a  modulated version of the irradiance distribution corresponding to the previous stage, S−1, magnified by a factor 2.
In Figure 7, we represent the axial irradiances provided by the TMVLs represented in Figure 6.
It can be seen that TMVLs produce a bifocal structure with fractal characteristic. It can also be observed that the diameters of these 'twin' vortices provided are proportional to the topological charge.

Experimental results
For the experimental study of the properties of FVLs, we implemented the experimental set-up shown in Figure 8, where the aperiodic VLs were experimentally implemented in a programmable spatial light modulator. The static aberrations caused by the SLM display were characterized with a Shack-Hartmann wavefront sensor, and then compensated as detailed elsewhere [35]. The vortex-lens performance was studied computing the diffraction patterns along different planes along the optical axis.
The proposed ZPs were recorded on a spatial light modulator (Holoeye PLUTO, eight-bit grey level, pixel size 8 μm and resolution equal to 1920 × 1080 pixels), calibrated for a 2π phase shift at λ = 633 nm operating in phase-only modulation mode. A linear phase grating was superposed to the diffractive lenses on the SLM to avoid the specular reflection and the pixelated structure of the SLM. This linear phase was compensated by an appropriate tilt of the SLM. A pin hole (PH) was used to filter high diffraction orders. A scaled image of the lens was achieved at the L3 lens focal plane (exit pupil). A collimated He-Ne laser beam (λ = 633 nm) was sent to the SLM and the diffracted field was registered with an eight-bit grey-level, chargecoupled device (CCD) camera (EO-1312M 1/2" CCD Monochrome USB Camera, pixel pitch of 4.65 μm, 1280 × 1024 pixels) through a microscope objective (10× Zeiss Plan-Apochromat). The microscope and the CCD were mounted on a translation stage (Thorlabs LTS 300, range: 300 mm and 5 μm precision). The computed and experimental irradiances produced by an S8 FiVL with m ¼ 6 are shown in Figure 9.
As predicted by the theoretical analysis (Figure 9a), the axial localization of the focal rings depends on the Fibonacci numbers F j and F j−1 , and such focal distances satisfy the following The diameter of the rings also satisfies Δ1 Δ2 ≈Φ:

Conclusions
A new family of fractal aperiodic VLs with interesting focusing and imaging capabilities has been presented. The transverse patterns appearing along the propagation distance present several concatenated doughnut modes. The ability of these VLs to produce multiple vortex tweezers has been demonstrated. In fact, it was found that contrary to conventional spiral zone plate, which produces a single vortex, each member of this family generates a delimited chain of vortices that are axially distributed. The distances between the links of the chain depend on the level S of the generating function and the radii of the doughnuts increase with the topological charge. Additionally, the multifocal nature of the lens, resulting from its fractal structure along the radial coordinate, could provide a high depth of field, especially with wideband sources.
In the case of FrVL (i.e. Cantor and Devil's VLs), the evolution of the axial irradiance replicates the fractality of the pupil. The orbital angular momentum in each link on the chain also depends on the topological charge but it is nearly independent of its axial location.
Our analysis demonstrated the possibility of simple design procedure arrays of VLs with any desired range of topological charge. As each individual VL can be understood as a light gear capable of driving microstructures around its circumference, applications involving particle transfer and manipulation could be foreseen.
On the other hand, FiVLs and TMVLs are intrinsically bifocal vortex lenses. It was shown that they produce twin optical vortices along the axial coordinate. The positions of both foci depend on the two incommensurable periods of the Fibonacci sequence in which the lenses are based. Moreover, the evolution of the irradiance along the propagation axis reproduces the fractality of the lens [34]. The diameter of these chains of vortices is proportional to the topological charge, and their ratio is close to the golden mean. The volumetric distribution of the diffracted field generated by FiVLs was assessed experimentally using an SLM. An excellent agreement was found between the experimental results and the theoretical predictions.
The peculiar focal volume obtained with DVLs could be exploited as versatile and efficient optical tweezer, because it can also trap low-index particles in the zero-intensity axial zone of the doughnut and at the same time can exert a torque on small objects having a high refraction index. The distances between the links of the vortex chain can be modified with the level S of the generating function and relative angular velocity of the particles at the trap depends on the topological charge of the vortex.
Another potential application of these optical vortices arises in X-ray microscopy where its azimuthal component can be used for detecting the phase component of objects with complex index of refraction since it acts as a Hilbert phase filter. This feature is especially useful in the case of biological specimens to provide enhanced contrast. TMVL produces a couple of self-similar vortex located symmetrically one at each side of the focus of the equivalent periodic VL. Therefore, under broadband illumination, a TMVL produces a pair of images with an extended depth of field and a strong reduction in the chromatic aberration. In this way, TMVLs combine the characteristics of FrVLs and FiVLs and could be used in multiple applications including spectral domain optical coherence tomography (OCT) and X-ray microscopy [35].
Finally, it should be mentioned that all the aperiodic VLs here presented admit fractional topological charges that break down the symmetry of the foci and produce chains of anisotropic vortex foci.