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Abstract
Multifocal spots in high numerical-aperture (NA) objectives has emerged as a rapid, parallel, and multi-location method in a multitude of applications. However, the typical method used for forming three-dimensional (3D) multifocal spots based on iterative algorithms limits the potential applications. We demonstrate a non-iterative method using annular subzone phases (ASPs) that are composed of many annular subareas in which phase-only distributions with different 3D displacements are filled. The dynamic 3D multifocal spots with controllable position of each focal spot in the focal volume of the objective are created using the ASPs. The experimental results of such dynamic tunable 3D multifocal spots offer the possibility of versatile process in laser 3D fabrication, optical trapping, and fast focusing scanned microscopic imaging.
© 2017 Optical Society of America
1. Introduction
In some applications, such as laser micro/nano fabrication, optical trapping, and high-resolution optical imaging, to achieve the multi-location, parallel, and simultaneous processing, multifocal spots with high-NA (typically NA>0.75) objectives are required. It is straightforward to implement two-dimensional (2D) lateral multifocal spots in the focal plane of an objective, and a number of methods can be exploited to realize it. For example, using optical elements, such as beam splitters, etalons, and microlens arrays, as well as diffractive optical elements, to divide the incident beam into multiple beams, we can generate 2D lateral multifocal spots in the focal plane of a high-NA objective [1–5]. Vector beams can also be used to achieve this by modulating the amplitude, phase, or polarization of the back aperture of the objective [6,7]. Phase-only modulation implemented via a spatial light modulator (SLM) is preferable owing to the SLM’s programmable ability to dynamically update the intensity distributions in the focal region by varying the incident phase patterns [8–12]. The phase-only patterns located at the back aperture of the objective can be obtained using 2D Fourier transform (FT) iterative algorithms, such as the Gerchberg–Saxton (GS) method, weighted Gerchberg-Saxton (WGS) algorithm, or other modified FT methods [13–19].
However, it is not straightforward to achieve a three-dimensional (3D) volumetric multifocal spots with good uniformity over the entire focal region of a high-NA objective, although superposing phase patterns of light fields from multiple discrete focal planes can provide the possibility to create volumetric multifocal spot. This is because 2D FT methods that do not consider the optical field within the entire focal region fail to generate diffraction-limited volumetric multifocal spot with high uniformity because of interlayer crosstalk. Using a Dammann zone plate combined with a conventional 2D Dammann grating located at the back aperture of the objective, J. Yu et al. demonstrated a 3D focused spot array [20]. M. Gu et al. reported 3D multi-layer recording using an aberration-free volumetric multifocal array using the vectorial Debye-based 3D FT method [21]. We have previously shown that a 3D shape-controllable multifocal spots can be generated by combining a 2D pure-phase modulation grating and additional axially shifted pure-phase modulation composed of four-quadrant phase distribution units at the back aperture of a high-NA objective [22]. However, these methods were all based on iterative algorithms, which typically require a large number of iterations and may not yield a unique solution. Furthermore, the phase patterns designed via optimization algorithms are random, which is not relevant to the positional parameters of the spots in the focal region of the objective. If we wish to produce 3D volumetric multifocal spots with various structural parameters (such as different numbers or positions), the multifocal spots must be redesigned with the new optimization parameters. To obtain good uniformity of the intensity distribution in the volumetric multifocal spots, the diffraction efficiency of the phase patterns is not regarded as a criterion in the optimization procedure. Hence, the phase patterns produced via different optimization procedures yield various diffraction efficiencies (i.e., the focal intensity of each spots changes following each iteration of the optimization process), which is not practical for dynamic laser micro/nano fabrication and high-resolution optical imaging. In some situations, dynamic and position-controllable 3D volumetric multifocal spot with good uniformity and high stability are required. For example, we can realize dynamic optical micromanipulation at any position using a position-tunable 3D volumetric multifocal spots. Another potential application of 3D dynamic and controllable multifocal spots is the fabrication of metamaterials with an arbitrary structure, as opposed to a fixed array structure [23]. Therefore, if an analytic expression can be obtained without any iterative optimization algorithm, it may enable us to design 3D volumetric multifocal spots with high quality, good uniformity, high stability, and dynamic control, and would be very useful in extending the applications of the multifocal spots with high-NA objectives.
Here we introduce a non-iterative method for generating 3D dynamic and controllable multifocal spots in the focal volume of the objective. In section 2, the 3D FT expression of Debye vectorial integrals in high-NA objectives is discussed. Based on the theory, a phase-only analytical expression is derived, which can be used for 3D dynamic focus control in the focal region of a high-NA objective. Then, based on the phase-only analytical expression, we design an annular subzone phase (ASP) composed of a large number of annular subareas, which contains phase-only distributions with different displacements. Using the ASP, the 3D dynamic volumetric multifocal spots with good uniformity and stability in the focal volume of the high NA objective can be achieved. In section 3, some experimental results are given and discussed. In section 4, we provide a brief summary of the method, the most significant results, and the potential applications.
2. Principle
2.1 Principle of the 3D displacement control by a phase-only expression
According to the Richards–Wolf vectorial diffraction integral [24,25], the electric field distribution at any point in the focal volume of an aberration-free high-NA objective is
where A is a constant; α is the maximum aperture angle of the objective; k = 2π/λ is the wave number; P(θ) is the apodization function; Et (θ, ϕ) is the transmitted field; θ = arcsin[rNA/(Rnt)] is the converge angle, assuming that the focusing system obeys Abbe’s sine condition, where R is the aperture stop radius, NA is the numerical aperture, and nt is the refractive index of the immersion medium. The coordinates r and ϕ are polar coordinates at the back-aperture plane, and x, y and z are Cartesian coordinates in the focal region, as shown in Fig. 1.
Fig. 1 Schematic showing the geometry for the calculation of the focused field distribution of a high-NA objective.
In the Debye approximation, and neglecting the constant coefficients, Eq. (1) can be rewritten as
where ξ = -cosϕsinθ/λ, η = -sinϕsinθ/λ, ζ = cosθ/λ, are the Cartesian components of the spatial frequency in the x, y, and z directions, respectively. We can see that the electric field distribution of an aberration-free high-NA objective can be further rewritten as a 3D Fourier transform of the weighted field; that is,where FT3D {·} denotes the 3D Fourier transform, and G(ξ, η, ζ) = P(θ)Et(θ, ϕ)/cosθ, in which the constant coefficient is neglected. From Eq. (3), we can see that the focus field distribution of a high-NA objective is now expressed as a 3D Fourier transform (FT) of the field distribution in the incidence aperture plane of the objective.Hence, according to the shift theorem of the FT, the additional phase shift can introduce a linear displacement shift in the spatial domain; i.e., E (x-Δx, y-Δy, z-Δz) = FT3D {e-i2π(ξΔx + ηΔy + ζΔz)·G(ξ, η, ζ)}, which shows that the focused spot of the high-NA objective has a shifted displacement of Δx, Δy and Δz in the focal region of the objective, when a phase distribution is added in the incidence aperture plane. Therefore, by modulating the additional phase distribution, we can control the position of the focal spot in the focal region of the high-NA objective.
Based on the relationship of the converge angle θ and the spatial frequencies ξ, η, ζ, the phase-only analytical equation for controlling the 3D position of the highly focused spot can be finally written as
where x0 and y0 are Cartesian coordinates at the back-aperture plane of the objective. Δx, Δy and Δz are the positions of the focal spot in the 3D focal region of the objective (see Fig. 1). Thus, once the positions of the focal spot and the parameters of the objective are determined, the phase distribution for controlling the 3D position of the focal spot can be obtained using Eq. (4).Based on the phase-only distribution calculated using Eq. (4), the position of the highly focused spots can be controlled in the 3D focal region of a high-NA objective. As an example, we assume a monochromatic, uniform, circularly polarized beam with a wavelength of 532 nm impinges onto the back aperture of a 1.4 NA objective in oil (nt = 1.518). Figure 2 shows an example of the focal field distributions when the objective is modulated by 3D phase-only modulation calculated using Eq. (4), with the displacements of Δx = Δy = Δz = 3λ. Figure 2(a) shows the phase pattern calculated using Eq. (4). The 3D iso-intensity surface distribution of the focal spot in the focal volume is shown in Fig. 2(b). We can see that the highly focused spot shifted to the position (3λ, 3λ, 3λ), and the displacements depended only on the parameters (Δx, Δy, Δz). Without the phase-only modulation of Fig. 2(a), the central position of the focal spot should be at the point (0, 0, 0) in the focal region. Hence, by directly changing the spatial relative displacement parameters Δx, Δy, and Δz in Eq. (4), we can manipulate the focal spot to any location in the 3D focal region of the objective.
Fig. 2 (a) Phase pattern calculated using Eq. (4) with Δx = Δy = Δz = 3λ. (b) The 3D iso-intensity surface in the focal region with the surface intensity of I = 0.5Imax.
2.2 Dynamic 3D control of the focal spot using the phase-only analytical expression
Based on the above discussion, we may expect that the focal spots can be dynamically controlled to any position in the focal region of the high-NA objective, as long as a set of phase patterns can be obtained and the position parameters varied continuously. This provides potential new methods of 3D laser micro/nano fabrication and 3D optical micro-manipulation. Only an SLM rather than a high-precision positioning system is required for controlling the highly focused spot in the 3D focal volume, we can produce the desired artificial micro-structures or manipulate micro-particle in real time.
To realize 3D dynamic control over the focal spot, we just make use of the phase-only expression in Eq. (4). Without any iterative algorithm, the phase patterns calculated by Eq. (4) can be exploited to control the displacement of the focal spot in the 3D focal region of the objective. Figure 3 shows an example of the focal spot with the 3D displacements satisfying a cylindrical spiral equation, i.e., Δx = 5λcosφ; Δy = 5λsinφ; Δz = 5λφ/(2π). It is clear that dynamic control over the 3D displacement of the focal spot can be realized in the focal region of the high-NA objective using the phase-only analytical expression (see Visualization 1).
Fig. 3 Example of simulation results with 3D displacement satisfying the cylindrical spiral equation: Δx = 5λcosφ, Δy = 5λsinφ, Δz = 5λφ/(2π) (see Visualization 1). (a) Phase pattern, (b) 3D iso-intensity surface distribution with I = 0.5Imax.
2.3 Annular subzone phase (ASP) method for producing 3D multifocal spots
We have shown that a focal spot can be manipulated within the focal region, and the displacements can be controlled using the analytical phase expression; however, multifocal spots with tunable position cannot be produced in the focal region of the objective by modulating only the phase distributions using Eq. (4). In this section, we describe a type of annular subzone phase (ASP) method for generating tunable multiple spots in the focal region of the objective.
Firstly, as shown in Fig. 4(a), the aperture function with the radius R of the objective is divided equally into N annular areas (A1, A2, …, AN) with an equal width R/N. Secondly, each annular area is further divided into multiple smaller annular subareas (S1, S2, …, SM), in which the width of each annular subareas is equal to R/(NM), as shown in Fig. 4(b). Lastly, as shown in Fig. 4(c), each of these annular subareas are filled with different phase-only distributions, which are calculated by Eq. (4) with different displacement parameters. That is, if each annular area (A1, A2, …, AN) is divided into M annular subareas, the one annular area (such as AN) will be filled with M phase-only distributions of different displacements. The other N-1 annular areas are filled with the same phase-only distributions of each subarea in order, where the phase is calculated by the same parameters in Eq. (4). To further explain the procedure of generating the desired phase pattern, an example of three focal spots with different axial position is given in Fig. 4(d). In this case, the number of annular areas N = 3, the number of subareas M = 3. In Fig. 4(d), the color (red, yellow and blue) is just used for showing the subareas more clearly. Multifocal spots with M spots can be generated using this ASP to modulate the aperture function of the objective.
Fig. 4 Schematic diagram of the ASP distributions. (a) The aperture stop plane of the objective. (b) A single annulus with M subareas. (c) The phase distribution in one of the M subareas. (d) An example of a procedure to show how to generate the desired phase pattern.
Using the ASPs, here we show some simulated examples of dynamic multifocal spots. If the filled phase is given by the 2D lateral phase, as described by Eq. (4) with Δz = 0, we will obtain a 2D lateral multifocal spots. Figure 5(a) show an ASP pattern which is composed of nine subareas (i.e., M = 9), in which each subarea is filled with different 2D lateral phase-only distributions. The filled phases are calculated using Eq. (4) with nine different displacements, i.e. Δxm = mπ/2 μm, Δym = 2sin(Δxm) μm, m = 1, 2…9, and Δz = 0, which can be used for generating a sine-shaped multifocal spots in the focal plane. Figure 5(b) shows the corresponding intensity distribution. We can see that a sine-shaped distribution with nine spots was created using the ASP shown in Fig. 5(a). For other multi-spots distributions with different numbers and displacements, we only need to change M and the displacement parameters (Δx, Δy, Δz) in Eq. (4). So, it is easy to realize the dynamic control of the multifocal spots by continuously varying the displacement parameters of each spot. Visualization 2 shows the dynamic control of the sine-shaped multifocal spots.
Fig. 5 (a) ASP patterns filled with the 2D lateral phase-only data as described by Eq. (4) with M = 9, and (b) the corresponding intensity distribution in the focal plane (see Visualization 2).
It is necessary to note that the number of N affects the uniformity of the multifocal spots. The uniformity u is defined as
where Im is the mth maximum intensity at the center of the focal spot. Figure 6(a) shows the uniformity of the nine focal spots (shown in Fig. 5) as a function of the number N. It is clear to see that the high uniformity (> 95%) was achieved when the number N > 34. The more the number of N used in the ASP, the better uniformity for the multifocal spots. However, the upper limit on the number of N is limited by the pixel resolution of the SLM. The N and M must satisfy the condition that the width of annular subareas R/(NM) has two pixels at least. Figure 6(b) shows the uniformity of the nine spots in sine-shaped multifocal spots (shown in Fig. 5) in the process of dynamic shift. It is shown that the multifocal spots maintain a high uniformity, even with continuous changes in distance. Such tunable and highly uniform multifocal spots with dynamically controllable position is not easily created using other iterative methods.Fig. 6 The uniformity of the nine spots shown in Fig. 5 as a function of (a) number N of annular areas and (b) dynamic shift.
Figure 7 shows another example of a 3D dynamic multifocal spots with four spots in the volume of a high NA objective. The position parameters of each focal spot are: Δx1 = d1sin(Δφ + π/2), Δx2 = d1sin(Δφ + 7π/6), Δx3 = d1sin(Δφ + 11π/6), Δx4 = 0; Δy1 = d1cos(Δφ + π/2), Δy2 = d1cos(Δφ + 7π/6), Δy3 = d1cos(Δφ + 11π/6), Δy4 = 0; Δz1 = Δz2 = Δz3 = d2, Δz4 = 0. d1 and d2 can be used to control the relative location of each spot in radial and axial direction, respectively. Δφ is used to control the spot position in angular direction. Figure 7(a) shows an ASP pattern filled with the 3D phase distributions of the four different displacements. The corresponding 3D iso-intensity surface distribution in the focal volume is shown in Fig. 7(b). It is clear that 3D volumetric multifocal spots have been achieved in the focal region of a high NA objective. The dynamic control of this 3D multifocal spots is also given in 3D directions with continuous displacements (see Visualization 3).
Fig. 7 (a) An ASP filled with 3D phase-only data as described by Eq. (4) when M = 4, and (b) the 3D iso-intensity surfaces of the intensity distribution in the focal region with I = e−2Imax (see Visualization 3).
3. Experimental results and discussion
We evaluate the proposed phase-only modulation method using the optical setup shown in Fig. 8. In this setup, we use a laser light with a wavelength of 532 nm for incident light (LWGL532, Beijing Laserwave Optoelectronics Technology Co., Ltd, China). A spatial light modulator (LETO, HOLOEYE Photonics AG, Germany) is used to modulate the back-aperture plane of a 100 × NA 1.4 oil immersion objective (UPlanSApo 100 × , Olympus, Japan) with a designated pattern calculated by Eq. (4). A CCD (EO-5012, Edmund Optics, US) is used to record the image of the focal spot reflected from a plane mirror, which is fastened in the focal plane of the objective. The spatial light modulator (SLM) used in this experiment is a reflective phase-only modulator with full HD (1920 x 1080 pixel) resolution, a pixel pitch of 6.4 µm and a linear 256-level phase response for the specified user wavelength, thereby the phase modulated wave front used for controlling the focal spot can be realized easily.
Fig. 8 Schematic of the set-up in experiment, where PHF is a spatial pinhole filter, CL is a collimation lens, BS is a beam splitter, P is a polarizer, SLM is a spatial light modulator, PM is a plane mirror.
Figure 9 shows the experimental results of one focal spot with the 3D displacements satisfying the cylindrical spiral equation. The phase patterns used for the SLM have been shown in Fig. 3(a). Figure 9(a) shows the intensity image of the focal spot recorded by CCD when the back-aperture plane of the objective modulated by the phase-only pattern with φ = 0. In this case, the focal plane of the laser coincides with the fastened image plane of the CCD. So, we can see a clear and highly focused focal spot. Figures 9(b)-9(d) show some other images of the focal spot with different displacement satisfying the cylindrical spiral equation. In these cases, the images of the focal spot become large and diffuse. This is because the focal plane that does not coincide with the fastened image plane of the CCD. As shown in Fig. 9(c), the focal plane has shifted away from Δz = 0 μm to Δz = 1.25λ when φ = 90°. It is clear that dynamic control over the 3D displacement of one focal spot can be realized in the focal volume of the high-NA objective using the phase-only analytical expression.
Fig. 9 The corresponding experimental results of one focal spot with 3D displacement using the phase patterns of Fig. 3. The intensity distributions recorded by the CCD: (a) φ = 0°, (b) φ = 45°, (c) φ = 90°, (d) φ = 135°. (see Visualization 4)
Figure 10 shows the experimental result of the sine-shaped multifocal spots. The phase patterns used for the SLM to produce these multifocal spots have been shown in Fig. 5(a). We can see that a sine-shaped distribution with nine spots was created using the ASPs shown in Fig. 5(a). The intensity image recorded by CCD in the focal plane of the objective is consistent with the simulated results shown in Fig. 5(b). Therefore, without using any iterative algorithm, the multifocal spots with controllable position in each focal spot can be created rapidly using the ASP method.
Fig. 10 Experimental results of nine sine-shaped multifocal spots with 2D phase modulation shown in Fig. 5(a). (see Visualization 5)
Figure 11 shows the experimental result of the 3D voluminal multifocal spots distribution. The phase patterns used for this experiment are composed of four subareas (i.e., M = 4), which have been shown in Fig. 7(a). Figures 11(a)-11(c) show the intensity distribution of four spots with different radial displacements. Figures 11(d)-11(f) show the intensity distributions in the focal region with different angular phases. In this case, the radial interval (the distance between two adjacent spots) is d1 = 10 μm. It is clear that a dynamic rotation multifocal spots can also be realized by using ASPs. The images of 3D volumetric multifocal spots are also given in Figs. 11(g)-11(i). The dynamic control of each focal spot in the 3D multifocal spots is given in Visualization 6. We can see that dynamic 3D volumetric multifocal spots can be easily achieved by directly changing the spatial relative displacement parameters (Δx, Δy, and Δz) in Eq. (4).
Fig. 11 Experimental results of the 3D volumetric multifocal spots. (a) – (c) Radial position control; (d) – (f) Angular rotation control; (g)-(i) axial position control. The dynamic intensity distributions are shown in Visualization 6.
4. Conclusion
We have demonstrated the generation of tunable multifocal spots with high uniformity and stability using ASP distributions and a phase-only analytical expression. Compared to the frequently-used iterative method, the phase-only pattern targeted to dynamically control the multifocal spot can be more rapidly and efficiently obtained using our proposed method. It is Because the displacement parameters are included in the phase-only expression, the position of the focal spots can be controlled within any position of the 3D focal regions of a high-NA objective. We have shown that the highly focused spot can be moved along a 3D cylindrical spiral trace. This phase-only analytical expression has potential applications in 3D laser micro/nano fabrication and optical micro-manipulation using an SLM rather than high-precision positioning systems.
An ASP scheme was designed for generating dynamic 3D multifocal spots. The ASP was composed of numerous annular subareas, in which phase-only distributions with different displacements were filled. By using a lot of ASPs with different displacements, dynamic multifocal spots in the focal volume of the high-NA objective were generated. We have shown that dynamic 3D multifocal spots with high uniformity and high stability can be created in the experiment. This 3D dynamic volumetric multifocal spots have potential applications in the fabrication of metamaterials with special opto-electric properties [26–28], as well as rapid materials processing, parallel optical manipulation, and multi-dimensional excitation and imaging [29–31]. Moreover, this method is also suitable for low NA focusing system, and some potential applications could be found in this field.
Funding
National Natural Science Foundation of China (NSFC) (61675093, 61705096); Shandong Provincial Natural Science Foundation, China (ZR2017MA035).
Acknowledgments
The authors are very thankful to the reviewers for their valuable comments.
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