## Abstract

We propose a method for creating a three-dimensional (3D) shape-controllable focal spot array by combination of a two-dimensional (2D) pure-phase modulation grating and an additional axial shifting pure-phase modulation composed of four-quadrant phase distribution unit at the back aperture of a high numerical aperture (NA) objective. It is demonstrated that the one-dimensional (1D) grating designed by optimized algorithm of selected number of equally spaced arbitrary phase value in a single period could produce desired number of equally spaced diffraction spot with identical intensity. It is also shown that the 2D pure-phase grating designed with this method could generate 2D diffraction spot array. The number of the spots in the array along each of two dimensions depends solely on the number of divided area with different phase values of the dimension. We also show that, by combining the axial translation phase modulation at the back aperture, we can create 3D focal spot array at the focal volume of the high NA objective. Furthermore, the shape or intensity distribution of each focal spot in the 3D focal array can be manipulated by introducing spatially shifted multi vortex beams as the incident beam. These kinds of 3D shape-controllable focal spot array could be utilized in the fabrication of artificial metamaterials, in parallel optical micromanipulation and multifocal multiphoton microscopic imaging.

© 2014 Optical Society of America

## 1. Introduction

Tightly focused far-field light spot of sub-diffraction size with a high numerical aperture (NA) objective have been extensively investigated due to its various applications such as optical data storage, material laser processing, high-resolution optical imaging, optical trapping and manipulation, optical lithography and metamaterial fabrication [1–19]. In some situations, the 2D equally spaced focal spot arrays are demanded because of their intrinsic properties or advantages of fast, multi-location, parallel and simultaneous processing. Currently, there have been diverse ways to generate 2D equally spaced focal spot array. The reported ways could be classified roughly into two categories: One category is applying optical elements, such as beam splitters, etalon, microlens arrays, holography and diffractive optical elements (DOEs) [20–25], to divide the incident beam into multiple beams. The spot sizes created by such approach are usually larger than the wavelength; the interference effect between beams will degrade the quality of multifocal spots with increasing resolution in using methods of multiple beams. Furthermore, the manufacturing process of the optical element, such as microlens arrays is usually very complicated, and the structure parameters of the elements cannot be changed once the process is completed. Therefore, it is not possible to produce spot arrays with tunable period and spacing, and thus lacks flexibility; the other category is by modulating the amplitude, phase or polarization of some special vector beams using the dynamic spatial light modulators at the back aperture of the focusing objective [26–31]. For practical applications, the high uniformity, high diffraction efficiency and superresolved focal spots are required. Among these methods, pure-phase modulation implemented by a spatial light modulator (SLM) is preferable owing to its programmable capability to dynamically update the intensity distributions in the focal plane by varying the incident phase patterns. Furthermore, the shape of each spot can be engineered [30–34]. The pure-phase modulations at the back aperture of the objective can be calculated by Gerchberg–Saxton method (GS), weighted Gerchberg-Saxton algorithm (WGS) [35,36] and other modified Fourier transform methods [37,38]. The most commonly used methods are perhaps those based on the iterative Fourier transform algorithms. These iterative methods usually require a large number of iterations and may not lead to a unique solution. Therefore, how to retrieve accurate phase patterns without iterative calculations to produce multifocal spots with high quality, high uniform and high energy efficiency should be an attractive and significant problem. In our previous research, it has been shown that the fractional Talbot effect can be used to generate spot arrays with high uniformity and high compression ratio both in free space and in far field focusing of a high numerical aperture objective without any iterative algorithm [31, 39–41].

As is well known, the pure-phase Dammann grating is a kind of grating in which each period is composed of several binary phase distribution with different widths [42–46]. With optimized widths, it can produce desired number of diffracting spots with equal intensity. Two-dimensional pure-phase optimized Dammann grating are capable of creating arbitrary N × M diffracting spot array. In 2012, J. Yu and his associates reported that, by combining a Dammann zone plate (DZP) with a conventional 2D Dammann grating(2D DG) at the back aperture of the objective, they successfully demonstrated the generation of a 3D focusing spot array [47]. However, this 3D spot array is only circular symmetrical spot array. The main potential application of a 3D focusing spot array is the fabrication of metamaterials with unique opto-electronic properties. Mostly, for this application, it requires that the single spot in 3D array has a specific shape. For example, a metamaterial composed of 3D gold helix array acts as a broadband circular polarizer [48]. If it is fabricated using dot-by-dot scanning laser direct writing method, it is an extremely time-consuming process. Thus, in the fabrication of metamaterials, one-time exposure fabrication with 3D spot array of a desired shape is more practical and time-saving. For this reason, the generation of 3D spot array with controllable shape is an attractive issue. On the other hand, the ever-increasing demand for ultra-high density data storage has always been pushing the volumetric information recording technical development. Recently, Min Gu and his group members reported the 3D multi-layer recording with aberration-free volumetric multifocal array by using the vectorial Debye-based 3D Fourier-transform method [49].

It has been shown that the vectorial Debye integral can be rewritten as a fast Fourier transform (FFT) for calculating the focus field of high NA objectives [50]. When the beam at back aperture of the objective is modulated by a pure-phase distribution, the focus field can be further expressed as a spatial convolution of two terms in the Fourier transform domain. Therefore, if the Fourier transform of the pure-phase distribution could be viewed as a dot array, the multifocal spot array will be generated in the focal region of the objective.

In this investigation, we first demonstrate that, by varying and optimizing the equally spaced phase values in a single period of the pure-phase grating, it is also possible to create desired number of equally spaced diffracting spot with almost identical intensities. Accordingly, a 2D such designed pure-phase grating can produce 2D diffracting spot array. By combining the different additional phase shift to the discrete but uniformly distributed sub-areas in the back aperture of the high numerical aperture objective, a 3D focusing spot array will also be created. Further, by spatially shifting the vortices of several vortex beams with different topological charge, a shape controllable 3D focusing spot array can also be created.

The paper is organized as follows: In section 2, the theory of pure-phase modulation in a single period of 1D grating for generating 1D diffracting spot array with equal intensities is first presented. The generalization of 1D grating to 2D such designed pure-phase grating is given and illustrated; then in section 3, the fast Fourier transform and convolution calculation method of the focusing field of the incident beam combined with the pure-phase modulation is presented. In section 4, a scheme of combining an additional phase value for splitting the focusing spot into multiple spots along optical axis are designed and shown; the main result which is the creation of 3D sub-diffraction shape-controllable spot array are demonstrated in section 5; Finally, the main method, the significant result and potential applications are concisely summarized in section 6.

## 2. Theory of pure-phase modulation in a single period of grating

For generating a 3D focusing spot array, we first design and optimize a pure-phase grating for generating 1D spot array. Our design method is inspired by the Dammann grating. The pure-phase Dammann grating is a kind of gratings in which each one period is composed of several binary phase distributions with different widths. However, the different widths cannot match the unit cell phase distribution designed for shifting the focusing spots array along the optical axis (discussed later in section 4). Compared with Dammann grating, an ordinary binary pure-phase grating is composed of several periodic two phase values. We assume that, similar to the conception of Dammann grating, our multi-value pure-phase grating(MVPPG) is composed of several different phase values in a single period in which each phase value has the same distribution width. The several phase values are optimized to generate several focal spots. Figure 1 shows the contrast schematic between an ordinary phase grating and the MVPPG which has the same width but has different phase values in a single period. In the case of Fig. 1(b), one period normalized to 1 is divided into 5 segments with identical width of 0.20. Generally, if the period is divided into *N* segments, the transmission function of multi-value 1D MVPPG can be written as

*t*

_{n}can be expressed as

*N*is the total numbers of the unit cell in one period of the grating. Then, the Fourier transform of the Eq. (2) can be written as

*ζ*is the spatial frequency. Then, the corresponding Fourier series coefficients

*A*(that is, the amplitude of the each diffraction order) can be expressed as

_{m}*β*is the total numbers of the splitting spots, defined as splitting parameter.

*I*

_{m}is the intensity of the

*m*

_{th}diffraction order. The symbol “$\ast $” denotes the complex conjugate. The uniformity coefficient of the splitting spots is defined as

Using an optimization algorithm (such as Simulated Annealing) for minimizing the uniformity coefficient, we may obtain the values of *φ*_{n} which determines the phase distribution in a single period of the MVPPG. For example, Table 1 lists the optimization results with *β* = 5 and *β* = 9. From Table 1, we can see that the uniformity coefficient can reach above 99.99%, while the diffraction efficiency exceeds 70% in a 1D MVPPG, and by adjusting the total numbers *N* of the unit cell, the efficiency can be reach up to 84.25% (when *β* = 5 and *N* = 4). For a 2D MVPPG, the efficiency can also reach above 50%. In this article, we mainly discuss the focal spots array with examples of *β* equivalent to *N.* Fig. 2(a) shows an example of the simulated 1D phase distribution of optimized 1D MVPPG when *β* is equal to 9. Figure 2(b) is the stairstep graph of 1D phase distribution of the corresponding MVPPG in Fig. 2(a). Figure 2(c) is the 1D diffracting intensity distribution. Figure 2(d) is the plot of the 1D diffracting intensity distribution. It is obvious that the 1D equal-intensity spot array with high uniformity and high efficiency can be generated with the optimized MVPPG.

One-dimensional optimized MVPPG can be reasonably extended to the 2D optimized MVPPG without any new additional theory or principle. It is expected that the properties of 1D optimized MVPPG can be transferred to 2D optimized MVPPG generated by overlapping two orthogonal 1D MVPPGs. Figure 3(a) shows the phase distribution of a 2D MVPPG with *β* = 9. Figure 3(b) shows the corresponding diffraction intensity distribution of optimized 2D MVPPG. It clearly shows that the 2D diffracting intensity distribution spot array is similar to the 2D lattice structure. The 2D equal-intensity spot array with high uniformity and high efficiency can also be generated by the optimized MVPPG. Then, based on the theory of the optimized MVPPG, a focal spot array could be created by using the pure-phase distribution at the back aperture of a high NA objective. In the following section, we will discuss the reason why a focal spot array can be created by using the optimized MVPPG modulation in a high NA objective.

## 3. Focusing of the incident beam combining the pure-phase modulation

According to the Debye vectorial integral, the electric field in the focus volume is expressed as

*U*(

*θ*,

*φ*) is the function of the pure-phase modulation on the back aperture.${\overrightarrow{E}}_{t}\left(\theta ,\phi \right)$ is the transmitted field after the objective (without phase modulation).

*r*=

*R*sin

*θ*/sin

*α*and

*φ*are the normalized polar coordinates in back aperture plane.

*R*is the maximum radius of the back aperture.

*θ*and

*α*are the half of convergence angle and the maximum aperture angle of the objective, respectively.

*k =*2π/

*λ*is the wave number and

*λ*is the vacuum wavelength. The Debye diffraction integral of Eq. (7) can be rewritten as a Fourier transform [50]. It is as follows

*k*,

_{x}*k*,

_{y}*k*) is the wave vector which can be given by

_{z}*θ*and

*φ*. From the Eq. (8), we can see that the focus field distribution is a spatial convolution with the Fourier transform of the pure-phase modulation function and the Fourier transform of the original incident focusing field (without phase modulation). It means that the tightly focusing 2D spot array will be produced if the Fourier transform of the pure-phase modulation is a 2D lattice function. Hence, based on the pure-phase distribution discussed above in section 2, a 1D or 2D tightly focusing spot array could be created in a high NA objective.

## 4. The scheme for axial shift and splitting of focal spots

In order to generate a 3D focusing spot array, we have to discuss the method for creating a focal array in axial direction. In this section, we propose an axial shifting pure-phase modulation composed of four-quadrant phase distribution unit for generating the axial split focal spot array.

In the Eq. (7), the phase factor exp(i*kz*cos*θ*) tell us that, if there is another additional phase factor exp(i*k*Δ*z*cos*θ*) imposed on the incident beam, the focusing spot will be shifted a distance Δ*z* along the optical axis of the focusing objective. For the 2D focusing spot array, this shifting phase factor will also move it to another focal plane. On the basis of this principle, we divided the optimized 2D MVPPG into sub-areas each of which is composed of four adjacent small squares illustrated in Fig. 4. The four squares construct a big square with the area of 1/*N*^{2}. The small square has the area of 1/(4*N*^{2}). Here *N* is the total number of unit cells in a single period of the MVPPG. The width of a single period is normalized to 1 as described in section 2. Δ*z*_{1}, Δ*z*_{2}, Δ*z*_{3}, Δ*z*_{4}, is set for axial translation of corresponding four 2D focusing spot arrays. The *θ* value in each of four small squares could be evaluated with center location of the each small square because the side length of the small square or the period of the MVPPG is usually small enough comparing to the diameter of the back aperture of the objective as shown in Fig. 4. The phase value of each sub-area with shifting distance Δ*z* is given as

*n*is the refraction index of the immersion,

_{t}*NA*is the numerical aperture of the objective.

According to this scheme for shifting focusing spot along the optical axis of the objective, we simulated the phase distribution on the whole back aperture of the objective for splitting original one focusing spot into four focusing spots which are equally spaced along the optical axis. As an example in this study, we assume that a circularly polarized beam with wavelength of 532 nm impinges onto the back aperture of a 0.95 NA objective in vacuum. Figure 5 is the phase distribution and the split intensity distribution along optical axis of the objective. Figure 5(a) shows the phase distribution with four shifting axial distance Δ*z*_{1} = −6μm, Δ*z*_{2} = −3μm, Δ*z*_{3} = 3μm, Δ*z*_{4} = 6μm. Figure 5(b) shows the enlarged sub-areas of four-quadrant phase distribution unit. Figure 5(c) shows intensity distribution of one focusing spot viewed in *xy* plane. We can see that the focal spot is circularly symmetric, as result of a circularly polarized beam is used. It is clear from Figs. 5(c) and 5(d) that the axial intensity distribution is a 1 × 4 multifocal array with high uniformity. If the one focusing spot has been changed into the 2D focusing spot array distributed transversely by imposing the optimized 2D MVPPG as shown in the Fig. 3(a), then the 3D focusing spot array will be created.

Furthermore, the interval spacing of the axial shifting array spots can be tuned easily using the phase distribution calculated by Eq. (9). Figure 6(a) shows the axial intensity distribution which is a 1 × 4 multifocal array with interval spacing of Δ*z* = 2 μm. It is shown that the axial shifting spots array with different interval spacing can be adjusted flexibly by changing the distance Δ*z* in the phase equation of Eq. (9). Obviously, each spot of the multifocal array is a 3D intensity distribution, which has a limited depth of focus. Once the interval spacing is within the range of the limited depth of focus, the interference will appear between the adjacent focal spots. Figures 6(b) and 6(c) are the axial intensity distributions of multifocal arrays with a smaller axial shifting distance Δ*z* = 1μm and Δ*z* = 0.5μm, respectively. We can see that the intensity of each focal spots overlap together.

## 5. Shape-controllable focal spot array

#### 5.1 Lateral 1D doughnut-shaped focusing spot array

We consider the incidence of a monochromatic, uniform, left-handed circularly polarized vortex beam on the back aperture of an aberration-free high NA objective obeying the sine condition. The beam of the incident field can be expressed as

*A*and

*ϕ*are the amplitude and phase distribution, ${\overrightarrow{a}}_{x}$and ${\overrightarrow{a}}_{y}$ are the unit vectors along the

*x*and

*y*direction on the back aperture plane.

*l*is the topological charge of the vortex beam which typically possess helical wave fronts. When

*l*is equal to zero, the vortex beam will reduce to the plane wave beam. For uniform wave beam, we let

*R*is the radius of the aperture stop. The phase pattern of the vortex beam with topological charge of 1 is shown in Fig. 7(a). Figure 7(b) shows the status of beam polarization. Figures 7(c) and 7(e) are the phase distributions of two optimized one-dimensional MVPPG with

*β*= 5 and

*β*= 9, respectively. Figures 7(d) and 7(f) are the combined phase distribution with the phase pattern in Fig. 7(a). Figures 7(g) and 7(h) shows the 1D focusing spot array when the incident field are modulated at the back aperture by two optimized one-dimensional MVPPG with

*β*= 5 and

*β*= 9, respectively. The single spot in the 1D focusing spot array is a doughnut-shaped focal spot due to the vortex beam with topological charge of 1.

#### 5.2 Lateral 2D doughnut-shaped focusing spot array

Figure 8 shows a 2D focal spot array generated by a circularly polarized vortex beam modulated by 2D MVPPG on the back aperture of a high NA objective. Figures 8(a) is the phase distributions of the optimized 2D MVPPG with *β* = 9. The phase pattern of the vortex beam with topological charge of 1 is shown in Fig. 8(b). Figure 8(c) shows the combined phase distribution with the phase patterns in Figs. 8(a) and 8(b). Figures 8(d) shows the 2D focusing spot array when the incident field are modulated at the back aperture by the combined 2D MVPPG shown in Fig. 8(c). We can see that the intensity distribution is a 9 × 9 multifocal array arranged in a square manner. The single spot in the 2D focusing spot array is also a doughnut-shaped focal spot due to the vortex beam with topological charge *l* = 1.

It should be pointed out that these results are obtained on the focal plane. The focus spot spacing (the distance between two adjacent focal spots) in focal array depends on the incident light wavelength *λ*, the numerical aperture *NA*, and the maximum period number *M* inside the objective aperture, that is the spacing Δ*x* = Δ*y* = *Mλ*/(2*NA*). The detailed discussion can be seen in our previous work [31]. In this simulation, the parameters are *λ* = 532 nm, *NA* = 0.95, *M* = 10, so the spacing Δ*x* = Δ*y* = 10 × 0.532μm/(2 × 0.95) = 2.8 μm, which can be seen from Fig. 8(d).

#### 5.3 The 3D shape-controllable focusing spot array

Based on the above analysis, it is shown that, by engineering the phase distribution in back aperture of the objective, the 1D and 2D focal array with doughnut-shaped spots can be created. Then in the following subsections, we will show that the 3D focal spot array can also be produced by combining the optimized 2D MVPPG and the axial shifting phase modulation. The axial shifting phase modulation imposed on the incident beam at the back aperture is for shifting the focusing spot along the optical axis (discussed in section 4). The incident beam is a plane wave beam which could be taken as a special vortex beam with topological charge of 0. On the optical axis, the shifting distances of focusing spots are Δ*z*_{1} = −12μm, Δ*z*_{2} = −4μm, Δ*z*_{3} = 4μm and Δ*z*_{4} = 12μm. Figure 9 shows a 3D focusing spot array generated by the 2D MVPPG with *β* = 5 and axial shifting phase with four shifting distance. Figures 9(a) is the phase distributions of the optimized 2D MVPPG with *β* = 5. The phase pattern of the axial shifting modulation is shown in Fig. 9(b). Figure 9(c) shows the combined phase distribution with the phase patterns in Figs. 9(a) and 9(b). Figure 9(d) shows the 3D iso-intensity surface (*I* = *I*_{max}/2) of the focusing spot array when the incident field are modulated at the back aperture by the combined phase shown in Fig. 9(c). Figures 9(e) and 9(f) are the cross-section views of the focusing spot array in *xy* and *xz* plane, respectively. We can see that the 3D focusing spot array is a 4 × 5 × 5 array.

In order to manipulate the shape of the 3D focusing spot array, one approach is to use multiple vortex beams. When a vortex beam with topological charge of 1 is as an incident beam, the focusing spot will be a doughnut-shaped spot. However, if using three circularly polarized vortex beams with three vortices spatially shifted in the back aperture of a high NA objective, a spiral-shaped spot can be created.

The phase modulation function consisting of multiple vortices at different positions within the back aperture can be expressed as

*a*and

*b*are the translation coefficient along the

*x*coordinate.

*l*

_{1},

*l*

_{2}, and

*l*

_{3}are the topological charges of three spatially shifted vortex beams.

*R*

_{max}is the maximum radius of the back aperture of the objective. The magnitude of

*a*and

*b*and the three topological charges have effect on the shape of the focal spot. We let that

*l*

_{1}= −1,

*l*

_{2}= −1,

*l*

_{3}= −1,

*a*= 0.25, and

*b*= 0.65. Figure 10(c) shows the phase distribution of the spatial shifting vortex beam with topological charge of

*l*

_{1}=

*l*

_{2}=

*l*

_{3}= −1. The spatial shifting vortex beams combining the optimized 2D MVPPG and the axial shifting phase modulation can create a 3D shape-controllable focusing spot array. Figure 10(e) shows a 2 × 5 × 5 3D shape-controllable focusing spot array. This 3D shape-controllable array can be used for direct laser writing of complex structures, such as metamaterial, or controlled rotation of microparticles in micro fluidics and cell biology.

## 6. Conclusion

We have proposed a one-dimensional multi-value pure phase grating (MVPPG) in which each period is composed of several equally spaced optimized phase values for generating desired number of diffracting order with equal intensities and presented respective theory. This optimized one-dimensional MVPPG can be reasonably generalized into an optimized two-dimensional MVPPG with identical design principle and similar properties in each of two dimensions. With this kind of optimized 2D MVPPG, a two-dimensional diffracting spot array with high uniformity and high diffracting efficiency can be created. Then we theoretically show that the calculation of focusing field of a high NA objective is a spatial convolution between the Fourier transform of the pure phase modulation function and the Fourier transform of the original incident focusing field. It means that the tightly focusing two-dimensional spot array will be produced if the Fourier transform of the pure phase modulation is a two-dimensional lattice function. We also, on the basis of Debye vectorial integral formulae, proposed a constructing approach of an additional pure phase modulation imposed on the optimized 2D MVPPG for shifting and splitting a single focusing spot or transversely distributed 2D focusing spot array into several spots or several layers of 2D spot array aligned along the optical axis of the focusing objective. With the combination of these two pure phase modulation applied on the incident beam, the three-dimensional focusing spot array will be created. Furthermore, using three vortex beams with different topological charges the vortices of which are spatially shifted along one direction, three-dimensional shape-controllable focusing spot arrays are demonstrated. This 3D shape-controllable focal spot array potentially may be utilized in the fabrication of metamaterials with some special opto-electric properties. Compared with dot-by-dot scanning direct laser writing or recording, it is extremely time-saving. It also opens up a possibility for fast material processing, parallel optical manipulation and multi dimensional excitation and imaging.

## Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (61205014, 61378060, and 61378035), the Doctoral foundation of Shandong Province (BS2012DX006). It is also partially supported by Dawn Program of Shanghai Education Commission (11SG44), Special-funded Program on National Scientific Instruments and Equipment Development(2012YQ17000408), and 151 Talent Project of Zhejiang Province of China (12-2-008).

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