Open Access
Add to BibSonomyAbstract
We propose an approach for generating a multifocal spot array (MSA) with a high numerical aperture (NA) objective. The MSA is generated by using a special designed phase-only modulation at the back aperture of an objective. Without using any iteration algorithm, the modulated phase pattern is directly obtained by the simple analytical expressions based on the fractional Talbot effect. It is shown that the number of the spots in the focal region depends solely on the fractional Talbot parameter. By engineering the phase pattern with a large fractional Talbot parameter, a large number of focal spots can be created. Furthermore, the intensity distribution of each focal spot can be manipulated by introducing a composite spatially shifted vortex beam (CSSVB) as the incident field, leading to creation of various kinds of specific shaped spots. Consequently, the MSA composed of multiple individual spots with specific shape is created by focusing the CSSVB combined with the multifocal phase-only modulation. These kinds of MSAs may be found applications in parallel optical micromanipulation, multifocal multiphoton microscopic imaging, and parallel laser printing nanofabrication.
© 2014 Optical Society of America
1. Introduction
Recently, the multifocal spots array (MSA) under tight focusing condition in a high NA objective has attracted great attention due to its intrinsic advantage of high processing efficiency. This MSA has been widely used in a range of applications, including multiphoton multifocal microscopic imaging [1–3], optical trapping and manipulation [4–6], real-time molecular imaging [7], and laser micro/nano fabrication [8–10]. There are diverse ways to produce MSAs in the focal region of a high NA objective. These approaches can be summarized in the following two categories. One is obtained by applying optical elements, such as beam splitters [11,12], etalon [13], microlens arrays [7,14,15], and diffractive optical elements (DOEs) [16–19], to divide the incident beam into multiple beams. The other is obtained by modulating the amplitude, phase or polarization of some special vector beams, such as composite vector beams [20–24], vector-vortex Bessel-Gauss beams [25,26], Laguerre-Gauss beams [27]. In order to achieve a high-quality MSA for practical applications, the high uniformity, high diffraction efficiency and superresolved focal spots are necessary. However, such high-quality MSA is difficult to accomplish by the methods previously mentioned. For example, the quality of multi foci reduces with increasing resolution due to interference effects. Hence, it is necessary to use an ultrashort-pulsed laser to eliminate the interference among the split beams caused by splitters or etalon. Furthermore, it is difficult to create two dimensional (2D) spot array by using that beam splitters. Using microlens arrays or DOEs, a 2D focal spot array can be produced, but the manufacturing process of that optical element is usually very complicated, and the structure parameters of a microlens array or a diffractive optical element cannot be changed once the process is completed. Therefore, it is not convenient to produce the MSA with tunable period and spacing (the distance between two adjacent focal spots), and the method of using microlens arrays or DOEs to produce the MSA lacks flexibility. By using the special vector beams, a multifocal spots with high quality can be created. However, the number of the focal spots is limited to small value (typically <10) in the focal region of the high NA objective. Hence, it is hard to produce a 2D MSA with a large number of spots by using the vector beams.
Phase-only modulation implemented by a spatial light modulator (SLM) is a preferable method for producing MSAs, owing to its programmable capability to dynamically update the intensity distributions in the focal plane by varying the incident phase patterns. Based on the method of using SLMs, not only a large number of focal spots with high quality can be obtained, but also the shape of each spot can be engineered. Many approaches have been reported for the purpose in recent years [1,9,20,28–33]. These previous investigations have verified that the MSA can be achieved by spatial phase modulation at the back aperture of the objective. A key issue in creation of a high-quality MSA is to retrieve accurate phase patterns at the back aperture of the objective. Many approaches have been explored and developed for this purpose, such as computer-generated hologram (CGH) [30–32], Gerchberg–Saxton method (GS) [33], weighted Gerchberg-Saxton algorithm (GSW) [1,9,34,35], and other modified Fourier transform methods [36,37]. 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, based on the noniterative method, how to retrieve accurate phase patterns to produce MSA in the focal region of a high NA objective has been an attractive and important task for practical applications.
As is well known, the fractional Talbot effect can generate spot arrays with high uniformity and high compression ratio [38]. Zhou et al. have given a complete set of phase equations for Talbot array illuminators with any compressed ratio [39,40]. And in our previous research [41], some simple analytical equations are also given for designing some typical spot arrays. Without any iterative algorithm, MSAs can be achieved by just using these phase equations. However, the spot arrays are produced in Fresnel plane by scalar diffraction theory. Using these equations to produce MSAs in the focal region of a high NA objective has not been reported.
In this study, we demonstrate a phase-only modulation method based on the fractional Talbot effect for the generation of MSAs under high-NA focusing condition. In section 2, we will introduce the theory and principle of generation of MSAs by phase-only modulation. 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. When the beam at back aperture of the objective is modulated by phase-only distribution, the focus field can be further expressed as a spatial convolution of two terms (phase-only distribution and the focal field distribution without the phase-only modulation) in the Fourier transform domain. Therefore, if the Fourier transform of the phase-only distribution is a lattice function, the MSA will be generated in the focal region of the objective. In section 3, with numerical simulations, the relationship between the fractional Talbot parameter and the focus spacing is discussed in detail. In addition, we describe a method of creating much more split spots in focal region of the high NA objective. By focusing a spatially shifted vortex beam combined with multifocal phase-modulation, multifocal split-spots arrays are created. These properties suggest the potential capability of our method for producing controllable MSAs, which might find applications in micro/nano fabrication, optical tweezers and metamaterials fabrication, etc.
2. Theory and principle of generating multifocal spots array
We starts by considering the incidence of a monochromatic, uniform plane wave beam on the back aperture of an aberration-free high NA objective obeying the sine condition. In the case of left-handed circular polarization, the beam of the incident field can be express as, where A and ϕ are the amplitude and phase distribution, and are the unit vectors along the x and y direction on the back aperture plane. In this study, the phase-only modulation is discussed only. For simplicity, let
where R is the radius of the aperture stop.According to the vectorial Debye integral, the electric field in the neighborhood of the focus is expressed as [42]
where U (θ,φ) = exp(iϕ) is the function of the phase-only modulation on the back aperture plane. is the transmitted field after the objective. r = Rsinθ/sinα and φ are the normalized polar coordinates in back aperture plane. θ and α are the 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. (2) can be rewritten as a Fourier transform [43], which iswhere the symbol “” represents the convolution operator, denotes the Fourier transform, and is the wave vector, which can be given by θ and φ. is the focus field distribution without phase modulation, which isFrom the Eq. (3), we can see that the focus field distribution is a spatial convolution with the Fourier transform of the phase-only pupil function and the Fourier transform of the original focusing field without phase modulation. It means that the MSA will be produced if the Fourier transform of the phase-only distribution is a lattice function. That is the key reason why the MSA can be produced by the phase patterns using the conventional FFT methods, such as the GS iterative algorithm [9,34–37].As we know, the fractional Talbot effect can produce a lattice-structured phase-only distribution, to convert a uniform plane wave into many concentrated spots of light of equal intensity, with little or no energy loss in the conversion [38,41]. Here, we apply it to produce MSAs in the high NA objective. The simple analysis formula of the phase pattern of a square array at the fractional Talbot distance can be expressed as
where m, n are integers, ordinal numbers of the pixels in the SLM. β is the fractional Talbot parameter and γ is a constant depending on the fractional parameter [41]. Figure 1(a) shows the phase pattern of the modulated objective aperture with the fractional Talbot parameter of β = 5. The enlarged phase distribution in one unit cell is shown in Fig. 1(b). It is shown that the phase in each squared sub-area, or called the “pixel” of the modulation is different and the phase distribution in one unit cell varies periodically, and the number of sub-area is equal to the fractional Talbot parameter of β = 5. Because of the periodicity of the phase distribution, the Eq. (5) can be further expressed as a convolution between a unit cell function and a lattice function. So within one period, the complex amplitude distribution of the phase-only modulation in the back aperture plane can be further expressed aswhere rect(x0/Δ, y0/Δ) is the rectangle function with the width of Δ = βd, giving the size of the unit cell; h1 and h2 are integers, and d is the size of a single sub-area. Then, the phase distribution in the back aperture plane can be expressed aswhere n1 and n2 are integers. Then, based on the Fourier transform theorems and the property of the δ function, the Fourier transform of the phase-only distribution in Eq. (3) can be obtained. It is important to note that the Fourier transform is in the k space over (kx,ky)∈R2. Finally, the intensity distribution in the focal region can be expressed aswhere c(n1,n2,β) is the Fourier transform coefficient, Δx and Δy are focus spacings in the focal region along the x and y directions, respectively, which can be written aswhere NA is the numerical aperture. Nx and Ny are the maximum period number inside the objective aperture along the x and y directions, which can be calculated from Nx = Ny = 2R/Δ. From Eq. (8), we can see that the Fourier transform of the phase-only modulation produced by the fractional Talbot effect is a lattice function. Therefore, the MSA can be generated by convolving the lattice function with the focusing field of without the phase modulation.
Fig. 1 (a) Phase distribution inside the back aperture of a MSA system with fractional Talbot parameter β = 5, and period number inside the objective aperture Nx = Ny = 20. (b) Enlarged phase distribution in one unit cell. (c) Intensity distribution on the focal plane. (d) Enlarged intensity distributions of one focal spot. (e) and (f) are the corresponding intensity cross-section profiles in (c) and (d), respectively. (Ix and Iy are the cross-section profiles in x and y direction, respectively).
3. Simulations and discussions
As discussed above, the MSA can be created by modulation of a fractional Talbot phase-only distribution in the back aperture of the high NA objective. 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 1.20 NA water immersion objective. Figure 1(c) shows the corresponding multifocal intensity distribution with the phase modulation in Fig. 1(a). We can see that the intensity distribution is a 5 × 5 multifocal array arranged in a square manner. Figure 1(e) shows the intensity cross-section profile of the MSA. It is shown that the MSA has a good-uniformity. The focus spot spacing is 4.43 μm, which is identical with the theoretical calculation Δx = 20 × 0.532/(2 × 1.2) = 4.43 μm, see Eq. (9). It is clearly seen from Figs. 1(d) and 1(f) that the enlarged intensity distribution of one focal spot in arrays is circularly symmetric, as a result of a circularly polarized beam is used. If a linearly polarized beam is used, the focal spot is ellipsoidal, elongating in the polarization direction. The full width at half maximum (FWHM) of the focal spots in both the x and y directions in the array are found to be identical, see Fig. 1(f), which indicates that the multifocal phase modulation doesn’t affect the spot shape, and the diffraction-limited condition is achieved for each focal spot in the MSA. Hence, the multifocal diffraction-limited spots circular symmetry can be obtained by a circularly polarized beam modulated by fractional Talbot phase-only distribution, which is also suitable for other various polarized beams, such as radially and azimuthally polarized beams. This circular symmetry multifocal spots arrays could be used for high-resolution and high-speed imaging [21].
It should be pointed out that these MSAs are obtained on the focal plane. To obtain more comprehensive properties of the MSA, we must study the intensity distribution when it is out of the focal plane. Study of the three dimensional (3D) distribution near focus is of particular importance in estimating the tolerance in the setting of the receiving plane of an objective. Figure 2(a) shows the 2D intensity distribution on a meridional plane near focus. We can see that the depth of focus is about 1 μm. Within that focal region (depth of focus), the MSA with high-quality and high-energy will still be produced. Once beyond the depth of focus, the size of the focal spot in the MSA will widen. Figure 2(b) shows the out-of-focus image of the MSA at distance of z = 1 μm from the focus. We can clearly see that focal spots diffuse in the MSA. Figure 2(c) shows the 3D iso-intensity surfaces of one focal spot. Of particular interest is the tubular structure of the intensity distribution seen clearly in the Fig. 2(c). It is this structure that is responsible for the tolerance in the setting of a receiving plane in the MSA system.
Fig. 2 (a) 2D intensity distribution (yz plane) in a meridional plane near focus with the same parameters in Fig. 1. (b) Intensity distribution (xy plane) on the plane of z = 1μm. (c) The 3D iso-intensity surfaces of one focal spot (I(x,y,z) = e-1…-4Imax).
3.1 Effects of the fractional Talbot parameter β on the focus spacing and number of arrays
From Eq. (9), we can see that the focus spacing solely depends on the period number inside the objective aperture when the wavelength and the NA are constants. Thus the focus spacing can be engineered by varying the total number Nx = Ny. Figure 3(a) shows the phase pattern of the back aperture with β = 5, which is composed of Nx = Ny = 40 periods inside the aperture. The intensity distribution on the focal plane resulting from this phase pattern, calculated using the Debye theory, is shown in Fig. 3(c). We can see that the number of focal spots is also 5 × 5 arranged in a squared lattice structure, which is similar to Fig. 1(c). The only one difference from Fig. 1(c) is that the focus spacing is different due to different periods (Nx = Ny = 40 in this case). It can be seen from Fig. 3(e) that the focus spot spacing is 8.86 μm, which agrees with the theoretical calculation Δx = 40 × 0.532/(2 × 1.2) = 8.86 μm, see Eq. (9). When the radius of the back aperture is a constant, the focus spacing and focus number can be engineered simultaneously by varying the fractional Talbot parameter β. Figure 3(b) shows the phase pattern with β = 15, with the same radius of the back aperture as shown in Fig. 3(a). Thus the total number of period Nx = Ny is a third of that in Fig. 3(a). The corresponding intensity distribution is shown in Fig. 3(d). We can clearly see that a MSA arranged in a 15 × 15 squared lattice structure is achieved in this case. Furthermore, it is can be seen from Fig. 3(e) that the focus spot spacing for β = 15 is decrease to Δx = 2.95 μm, which is exactly one third of that for the case of β = 5.
Fig. 3 Phase patterns inside the objective aperture of fractional Talbot phase-only modulation with (a) β = 5, (b) β = 15. (c) and (d) Corresponding intensity distributions on the focal plane. (e) Intensity cross-section profiles in (c) and (d), respectively.
The above numerical results show that, MSAs can be well generated by the fractional Talbot phase modulation in tight focusing system with a high NA objective, and the structure of the MSAs can be changed by varying the fractional Talbot parameter β. When the radius of the back aperture of the objective is constant, the larger the β is, the larger number and the smaller focus spacing could be achieved.
3.2 Generation of non-Airy multifocal spot arrays with vortex beams
Based on the above analysis, it is shown that, by engineering the phase distribution in back aperture of the objective, the MSA with multiple diffraction-limited Airy spots can be created. Then in the following subsections, we will show that the MSA of non-Airy patterns can also be produced by combining the fractional Talbot phase and the vortex phase. One important way to achieve non-Airy focal pattern is to use vortex beams, which typically possess helical wave fronts. In this section, a left-hand circular polarized (LCP) vortex beam with topological charge l = 1 is studied. The vortex beam can be expressed as Ei = Aexp(ilφ). The phase pattern of the vortex is shown in Fig. 4(a). According to the above mentioned method for generating Debye diffraction-limited MSA, the multifocal phase is shown in Fig. 4(b), the same as Fig. 3(b). The combination of Figs. 4(a) and 4(b) in the back aperture results in a phase modulation shown in Fig. 4(c). The corresponding intensity distribution of the combined phase modulation is shown in Fig. 4(d). We can see that a 15 × 15 multifocal array of non-Airy spot shape is created in the focal region. Figure 4(e) shows the enlarged image of one focal spot. It is clearly seen that a doughnut-shaped focal spot with central zero intensity point is generated by focused LCP vortex beam with topological charge l = 1, which is different from the focal shape of focused LCP vortex beam with l = −1 [44], and also differ from the focal shape of the linearly vortex beam in a high NA objective. Similarly, the MSA with each spot of doughnut-shaped can be generated by combination of multifocal phase modulation for right-hand polarized (RCP) vortex beam.
Fig. 4 Phase patterns of (a) a vortex beam with topological charge l = −1, (b) the multifocal phase, and (c) the combined phase. (d) Intensity distribution on the focal plane. (e) Enlarged image of one focal spot. (f) Intensity cross-section profiles of single focal spot generated by vortex beam without multifocal phase modulation (Iv), and with array phase in the arrays Ia1 with Δx = 2.95 μm, Ia2 with Δx = 4.43 μm, respectively
The diffraction-limited feature of each spot in the array is checked by plotting the intensity profile Ia1 (marked by red circle line in Fig. 4(f)) along the x direction. The FWHM of the focal spot in the array is slightly larger than that of the focal spot Iv (marked by black square line in Fig. 4(f)) of focused vortex beam without the array phase distribution (as shown in Fig. 4(b)). The reason is that the spot spacing is only Δx = 2.95 μm with β = 15 (as shown in Fig. 3(e)), leading to influence between each other of adjacent spots. A larger spot spacing in the focal array Ia2 (marked by blue triangle line in Fig. 4(f)) can decrease or eliminate the effect, which is found to be identical to Iv. Therefore, when the radius of the back aperture of the objective is a constant, the fractional Talbot parameter β cannot be a big number, limited to the spot spacing. When the spot spacing is too small, the adjacent spots influence each other, resulting in interference effects.
3.3 Generation of multifocal split-spots arrays with composite spatially shifted vortex beams
Recently, a non-Airy spot of split-ring (SR) pattern has been created by using a spatially shifted circularly polarized vortex beam through a high NA objective [45,46]. The SR pattern is generated by a circularly polarized vortex beam with three vortices spatially shifted in the back aperture. Here, we will show that a split-spots pattern will be produced by composite spatially shifted vortices beam (CSSVB). The multifocal split-spots array can be created by combination of a fractional Talbot phase and a CSSVB with more vortices in the back aperture. The split-spots pattern further increase the number of the focal spot in array.
The phase modulation function consisting of multiple vortices at different positions within the back aperture can be expressed as
where ln is the topological charge. N is the total number of vortices. an and bn are the different position parameters.First, we consider a circularly polarized vortex beam with four vortices spatially shifted along the y direction. The resulted phase modulation is shown in Fig. 5(b) with ln = −1, [an = 0, bn = 0.25,0.65,-0.25,-0.65]. By superposing the fractional Talbot phase, see Fig. 5(a) with β = 9, the MSA with split-spots pattern can be generated. The corresponding intensity distribution in focal region is shown in Fig. 5(d). It is shown that a 9 × 18 multifocal array is created, which is two times what is modulated without the CSSVB. It can be seen from the enlarged focal spot in the focal arrays (as shown in Fig. 5(e)) that one focal spot splits into two along the y direction, resulting from vortex beam modulated by spatially shifted with four vortices along the y direction in the back aperture. Similarly, the split-spots pattern along the x direction can be generated by spatially shifted along the x direction.
Fig. 5 Phase patterns of (a) multifocal phase with β = 9, (b) spatially shifted vortex phase along x direction, (c) spatially shifted vortex phase along x and y directions. (d) and (g) are the intensity distributions on the focal plane with the combined multifocal and shifted vortex phase. (e) and (h) are the enlarged images of one spot. (f) and (i) are the iso-intenstiy surfaces given by the intensity at the half of the peak value, corresponding to (e) and (h), respectively.
Additionally, we also study the circularly polarized vortex beam with eight vortices spatially shifted along both the x and y direction. The resulted phase distribution is shown in Fig. 5(c) with ln = −1, [an = 0, bn = 0.25,0.65,-0.25,-0.65; bn = 0, an = 0.25,0.65,-0.25,-0.65]. The corresponding multifocal intensity distribution and the enlarged focal spot in the focal arrays are shown in Fig. 5(g) and Fig. 5(h), respectively. It is shown that the number of focal spot arrays (18 × 18) is four times as many as that one without the CSSVB. We can see from the enlarged focal spot (as shown in Fig. 5(h)) that the one spot splits into four spots. The corresponding 3D isosurface images of the modulated focal spot are shown in Fig. 5(f) and Fig. 5(i). It is shown that the split spots are produced, which is a spiral arranged distribution as a result of the spiral phase appearance from the 3D view. These split-spots microstructure multifocal array can be used for direct laser writing of complex structures, such as metamaterial [47], or controlled rotation of microparticles in micro fluidics and cell biology [48].
4. Conclusion
In summary, we have demonstrated that the MSA can be generated by using the fractional Talbot phase-only modulation at the back aperture of a high NA objective. Without using complicated iteration algorithm, the modulation phase is directly obtained by the simple analytical expressions based on the fractional Talbot effect, which differs from the conventional Fourier transform method. This fractional Talbot phase modulation also provides more controllable parameters. For example, the number and spacing of the MSA in the focal region can be modulated by the fractional Talbot parameter to fit different focusing conditions. Furthermore, a non-Airy MSA can also be achieved by combining the vortex phase with the fractional Talbot phase, in which each focal spot is doughnut-shaped. In particular, by spatially shifting positions of the phase vortices, we have been able to create an array of much more split-spots, which could be suitable for fast fabrication of large area metamaterials. The method developed in this paper also has potentials in a number of other applications, such as parallel optical micromanipulation, multifocal microscopy, and parallel laser printing nanofabrication.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (61205014, 61308077, and 11074105), the Project of Shandong Province Higher Educational Science and Technology Program (J12LJ02), and the Doctoral foundation of Shandong Province (BS2012DX006). It is also partially supported by National Natural Science Foundation of China (61378060) and Dawn Program of Shanghai Education Commission (11SG44).
References and links
1. M. Gu, H. Lin, and X. Li, “Parallel multiphoton microscopy with cylindrically polarized multifocal arrays,” Opt. Lett. 38(18), 3627–3630 (2013). [CrossRef] [PubMed]
2. D. N. Fittinghoff and J. A. Squier, “Time-decorrelated multifocal array for multiphoton microscopy and micromachining,” Opt. Lett. 25(16), 1213–1215 (2000). [CrossRef] [PubMed]
3. K. Bahlmann, P. T. C. So, M. Kirber, R. Reich, B. Kosicki, W. McGonagle, and K. Bellve, “Multifocal multiphoton microscopy (MMM) at a frame rate beyond 600 Hz,” Opt. Express 15(17), 10991–10998 (2007). [CrossRef] [PubMed]
4. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1–6), 169–175 (2002). [CrossRef]
5. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef] [PubMed]
6. R. L. Eriksen, V. R. Daria, and J. Gluckstad, “Fully dynamic multiple-beam optical tweezers,” Opt. Express 10(14), 597–602 (2002). [CrossRef] [PubMed]
7. T. Minamikawa, M. Hashimoto, K. Fujita, S. Kawata, and T. Araki, “Multi-focus excitation coherent anti-Stokes Raman scattering (CARS) microscopy and its applications for real-time imaging,” Opt. Express 17(12), 9526–9536 (2009). [CrossRef] [PubMed]
8. A. Jesacher and M. J. Booth, “Parallel direct laser writing in three dimensions with spatially dependent aberration correction,” Opt. Express 18(20), 21090–21099 (2010). [CrossRef] [PubMed]
9. H. Lin, B. Jia, and M. Gu, “Dynamic generation of Debye diffraction-limited multifocal arrays for direct laser printing nanofabrication,” Opt. Lett. 36(3), 406–408 (2011). [CrossRef] [PubMed]
10. J. Kato, N. Takeyasu, Y. Adachi, H. Sun, and S. Kawata, “Multiple-spot parallel processing for laser micronanofabrication,” Appl. Phys. Lett. 86(4), 044102 (2005). [CrossRef]
11. T. Nielsen, M. Fricke, D. Hellweg, and P. Andresen, “High efficiency beam splitter for multifocal multiphoton microscopy,” J. Microsc. 201(3), 368–376 (2001). [CrossRef] [PubMed]
12. A. Cheng, J. T. Gonçalves, P. Golshani, K. Arisaka, and C. Portera-Cailliau, “Simultaneous two-photon calcium imaging at different depths with spatiotemporal multiplexing,” Nat. Methods 8(2), 139–142 (2011). [CrossRef] [PubMed]
13. D. N. Fittinghoff, P. W. Wiseman, and J. A. Squier, “Widefield multiphoton and temporally decorrelated multifocal multiphoton microscopy,” Opt. Express 7(8), 273–279 (2000). [CrossRef] [PubMed]
14. F. Merenda, J. Rohner, J.-M. Fournier, and R.-P. Salathé, “Miniaturized high-NA focusing-mirror multiple optical tweezers,” Opt. Express 15(10), 6075–6086 (2007). [CrossRef] [PubMed]
15. Y. Shao, J. Qu, H. Li, Y. Wang, J. Qi, G. Xu, and H. Niu, “High-speed spectrally resolved multifocal multiphoton microscopy,” Appl. Phys. B 99(4), 633–637 (2010). [CrossRef]
16. L. Sacconi, E. Froner, R. Antolini, M. R. Taghizadeh, A. Choudhury, and F. S. Pavone, “Multiphoton multifocal microscopy exploiting a diffractive optical element,” Opt. Lett. 28(20), 1918–1920 (2003). [CrossRef] [PubMed]
17. J. Yu, C. Zhou, W. Jia, W. Cao, S. Wang, J. Ma, and H. Cao, “Three-dimensional Dammann array,” Appl. Opt. 51(10), 1619–1630 (2012). [CrossRef] [PubMed]
18. J. Yu, C. Zhou, W. Jia, A. Hu, W. Cao, J. Wu, and S. Wang, “Three-dimensional Dammann vortex array with tunable topological charge,” Appl. Opt. 51(13), 2485–2490 (2012). [CrossRef] [PubMed]
19. J. A. Davis, I. Moreno, J. L. Martínez, T. J. Hernandez, and D. M. Cottrell, “Creating three-dimensional lattice patterns using programmable Dammann gratings,” Appl. Opt. 50(20), 3653–3657 (2011). [CrossRef] [PubMed]
20. M. Cai, C. Tu, H. Zhang, S. Qian, K. Lou, Y. Li, and H. T. Wang, “Subwavelength multiple focal spots produced by tight focusing the patterned vector optical fields,” Opt. Express 21(25), 31469–31482 (2013). [CrossRef] [PubMed]
21. H. Guo, X. Dong, X. Weng, G. Sui, N. Yang, and S. Zhuang, “Multifocus with small size, uniform intensity, and nearly circular symmetry,” Opt. Lett. 36(12), 2200–2202 (2011). [CrossRef] [PubMed]
22. H. Guo, G. Sui, X. Weng, X. Dong, Q. Hu, and S. Zhuang, “Control of the multifocal properties of composite vector beams in tightly focusing systems,” Opt. Express 19(24), 24067–24077 (2011). [CrossRef] [PubMed]
23. M. Sakamoto, K. Oka, R. Morita, and N. Murakami, “Stable and flexible ring-shaped optical-lattice generation by use of axially symmetric polarization elements,” Opt. Lett. 38(18), 3661–3664 (2013). [CrossRef] [PubMed]
24. J. Chen, Q. Xu, and G. Wang, “A four-quadrant phase filter for creating two focusing spots,” Opt. Commun. 285(6), 900–904 (2012). [CrossRef]
25. K. Huang, P. Shi, G. W. Cao, K. Li, X. B. Zhang, and Y. P. Li, “Vector-vortex Bessel-Gauss beams and their tightly focusing properties,” Opt. Lett. 36(6), 888–890 (2011). [CrossRef] [PubMed]
26. J. Chen and Y. Yu, “The focusing property of vector Bessel–Gauss beams by a high numerical aperture objective,” Opt. Commun. 283(9), 1655–1660 (2010). [CrossRef]
27. J. Zhao, B. Li, H. Zhao, Y. Hu, W. Wang, and Y. Wang, “Tight focusing properties of the azimuthal discrete phase modulated radially polarized LG11 beam,” Opt. Commun. 296, 95–100 (2013). [CrossRef]
28. N. J. Jenness, K. D. Wulff, M. S. Johannes, M. J. Padgett, D. G. Cole, and R. L. Clark, “Three-dimensional parallel holographic micropatterning using a spatial light modulator,” Opt. Express 16(20), 15942–15948 (2008). [CrossRef] [PubMed]
29. E. Schonbrun, R. Piestun, P. Jordan, J. Cooper, K. D. Wulff, J. Courtial, and M. Padgett, “3D interferometric optical tweezers using a single spatial light modulator,” Opt. Express 13(10), 3777–3786 (2005). [CrossRef] [PubMed]
30. D. R. Burnham, T. Schneider, and D. T. Chiu, “Effects of aliasing on the fidelity of a two dimensional array of foci generated with a kinoform,” Opt. Express 19(18), 17121–17126 (2011). [CrossRef] [PubMed]
31. G. Lee, S. H. Song, C.-H. Oh, and P.-S. Kim, “Arbitrary structuring of two-dimensional photonic crystals by use of phase-only Fourier gratings,” Opt. Lett. 29(21), 2539–2541 (2004). [CrossRef] [PubMed]
32. K. Obata, J. Koch, U. Hinze, and B. N. Chichkov, “Multi-focus two-photon polymerization technique based on individually controlled phase modulation,” Opt. Express 18(16), 17193–17200 (2010). [CrossRef] [PubMed]
33. R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik (Stuttg.) 34(3), 275–284 (1971).
34. R. Di Leonardo, F. Ianni, and G. Ruocco, “Computer generation of optimal holograms for optical trap arrays,” Opt. Express 15(4), 1913–1922 (2007). [CrossRef] [PubMed]
35. D. Engström, A. Frank, J. Backsten, M. Goksör, and J. Bengtsson, “Grid-free 3D multiple spot generation with an efficient single-plane FFT-based algorithm,” Opt. Express 17(12), 9989–10000 (2009). [CrossRef] [PubMed]
36. H. Duadi and Z. Zalevsky, “Optimized design for realizing a large and uniform 2-D spot array,” J. Opt. Soc. Am. A 27(9), 2027–2032 (2010). [CrossRef] [PubMed]
37. E. H. Waller and G. von Freymann, “Multi foci with diffraction limited resolution,” Opt. Express 21(18), 21708–21713 (2013). [CrossRef] [PubMed]
38. A. W. Lohmann and J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29(29), 4337–4340 (1990). [CrossRef] [PubMed]
39. C. Zhou and L. Liu, “Simple equations for the calculation of a multilevel phase grating for Talbot array illumination,” Opt. Commun. 115(1–2), 40–44 (1995). [CrossRef]
40. C. Zhou, S. Stankovic, and T. Tschudi, “Analytic phase-factor equations for Talbot array illuminations,” Appl. Opt. 38(2), 284–290 (1999). [CrossRef] [PubMed]
41. L.-W. Zhu, X. Yin, Z. Hong, and C.-S. Guo, “Reciprocal vector theory for diffractive self-imaging,” J. Opt. Soc. Am. A 25(1), 203–210 (2008). [CrossRef] [PubMed]
42. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959). [CrossRef]
43. M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14(23), 11277–11291 (2006). [CrossRef] [PubMed]
44. Q. Zhan, “Properties of circularly polarized vortex beams,” Opt. Lett. 31(7), 867–869 (2006). [CrossRef] [PubMed]
45. H. Lin and M. Gu, “Creation of diffraction-limited non-Airy multifocal arrays using a spatially shifted vortex beam,” Appl. Phys. Lett. 102(8), 084103 (2013). [CrossRef]
46. J. Chen, X. Gao, L. Zhu, Q. Xu, and W. Ma, “The generation of a complete spiral spot and multi split rings by focusing three circularly polarized vortex beams,” Opt. Commun. 318, 100–104 (2014). [CrossRef]
47. J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009). [CrossRef] [PubMed]
48. J. Xavier, R. Dasgupta, S. Ahlawat, J. Joseph, and P. K. Gupta, “Three dimensional optical twisters-driven helically stacked multi-layered microrotors,” Appl. Phys. Lett. 100(12), 121101 (2012). [CrossRef]
