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DOI: 10.1039/D0NA00167H
(Paper)
Nanoscale Adv., 2020, Advance Article

Xionggui Tang*^{ac},
Fan Nan^{b} and
Zijie Yan*^{bc}
^{a}Department of Physics, Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education, Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha, 410081, P. R. China. E-mail: tangxg@hunnu.edu.cn
^{b}Department of Applied Physical Sciences, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599, USA. E-mail: zijieyan@unc.edu
^{c}Department of Chemical and Biomolecular Engineering, Clarkson University, Potsdam, New York 13699, USA

Received
28th February 2020
, Accepted 28th April 2020

First published on 29th April 2020

Holographic optical tweezers can be applied to manipulate microscopic particles in various optical patterns, which classical optical tweezers cannot do. This ability relies on accurate computer-generated holography (CGH), yet most CGH techniques can only shape the intensity profiles while the phase distributions remain poor. Here, we introduce a new method for fast generation of holograms that allows for accurately shaping both the intensity and phase distributions of light. The method uses a discrete inverse Fourier transform formula to directly calculate a hologram in one step, in which a random phase factor is introduced into the formula to enable complete control of intensity and phase. Various optical patterns can be created, as demonstrated by the experimentally measured intensity and phase profiles projected from the holograms. The high-quality shaping of intensity and phase of light provides new opportunities for optical trapping and manipulation, such as controllable transportation of nanoparticles in optical trap networks with variable phase profiles.

A phase-only spatial light modulator (SLM), which can be easily used to generate a desired light field with CGH, has been one of the most attractive devices for building holographic optical tweezers.^{17,18} A phase-only SLM consists of a 2D matrix of liquid crystal cells individually controlled by electronic signals. It can retard the optical phase from 0 to 2π, which is determined by the corresponding gray level of the calculated CGH. In the past several years, various methods have been proposed for the design of phase-only holograms, which generally can be categorized as the iterative algorithms, non-iterative algorithms, and integral methods. For most iterative algorithms, such as the iterative Fourier transform algorithm,^{19} direct binary search,^{20} Gerchburg–Saxton algorithm,^{21–24} and adaptive-additive algorithm,^{25} the intensity distribution with relatively good uniformity at the output plane can be obtained by iterative searching and optimization. However, the phase distribution is generally poor, which is obviously unfavorable for nanoparticle manipulation when using optical forces arising from phase gradients.^{26} In addition, the quality of CGH highly depends on the steps and searching algorithm in the optimization process, which largely affects the computation time of hologram generation.

For non-iterative algorithms, the computation time can be reduced, and the speckle noise can be decreased to some extent.^{27,28} However, the quality of the generated hologram still needs to be improved. In order to achieve simultaneous intensity and phase control, Jesacher et al. proposed a method of using two cascaded phase diffractive elements to separately modulate the amplitude and phase of a laser beam,^{29} but it is difficult to precisely control the alignment between two phase elements, which easily results in poor quality of the optical field and low efficiency. Later, Bolduc et al. established a similar method of encoding the amplitude and phase of the optical field by using two SLMs, which can be applied in the quantum encryption.^{30} Besides, Shanblatt et al. presented a method of directly using analytical formula to generate holograms in ring traps with different orientations, which have advantages such as high accuracy and low computation cost.^{31} Recently, Rodrigo et al. demonstrated a strategy for CGH design of optical traps by using the integral method, which can effectively achieve simultaneous control of intensity and phase distributions.^{32–37} However, this method is limited to the realization of smooth optical patterns consisting of special curves, termed as superformula curves. If an optical pattern cannot be expressed by a well-defined formula, the integral method cannot be used directly, and some approximations must be made. Particularly, the method is not suitable for designing even ordinary optical profiles, such as straight lines and point arrays. Therefore, it is highly desirable to generate optical fields with complete control of intensity and phase, especially for optical manipulation of nanoparticles.

In this work, we introduce a new approach for generating phase-only holograms that can produce high-quality intensity and phase profiles at the output plane. The approach is a direct computation method for CGH. Its simple calculation leads to low computational cost yet high accuracy. Through numerical simulation and experimental investigation, we show that the intensity and phase distribution can be simultaneously obtained, which is very useful for exploring new functions in optical nano-manipulation.

(1) |

U(x_{o},y_{o}) = a_{u}(x_{o},y_{o})exp[jφ_{u}(x_{o},y_{o})],
| (2) |

H(x_{i},y_{i}) = a_{h}(x_{i},y_{i})exp[jφ_{h}(x_{i},y_{i})],
| (3) |

Obviously, there exists an inverse Fourier transform relationship between H(x_{i},y_{i}) and U(x_{o},y_{o}), as shown in eqn (1). Theoretically, H(x_{i},y_{i}) can be easily obtained by using a fast inverse Fourier transform (IFFT) if the SLM can modulate both the amplitude and phase. However, commercial SLMs are either amplitude-only or phase-only devices. For example, if the calculated optical field H(x_{i},y_{i}) is projected on a phase-only SLM, all amplitude information of a_{h}(x_{i},y_{i}) will be lost, leading to a large discrepancy between the generated optical field U′(x_{o},y_{o}) and the desired optical field U(x_{o},y_{o}). Usually, the discrepancy will further increase as the uniformity of a_{h}(x_{i},y_{i}) gets worse.

From a different standpoint, the optical field H(x_{i},y_{i}) can be viewed as the superposition of individual plane waves with a weight factor, U(x_{o},y_{o}). Hence, the eqn (1) can be given using discrete expression,

(4) |

(5) |

In this case, a random phase generated by Matlab library function is directly added into each plane wave used as an eigenfunction in the superposition process; therefore eqn (5) is substantially different from eqn (4). Usually, the superposition among different plane waves at the input plane leads to the degradation of uniformity of a_{h}(x_{i},y_{i}). The random phase is employed to minimize their interference, and reshape the profiles of amplitude a_{h}(x_{i},y_{i}), so the uniformity of amplitude a_{h}(x_{i},y_{i}) can be effectively improved, which upgrades the accuracy of intensity and phase profiles at output plane (see Note 1 in ESI†). Accordingly, the phase profiles of holograms can be obtained by,

P(m,n) = arg[H(m,n)], | (6) |

It is worth noting that the holograms in our proposed method are directly calculated in single step, which are different from the conventional iterative methods. The Gerchburg–Saxton algorithm may also use random phases to preset the initial phase of a hologram in the first step of the search process, but the purpose is only to converge to the solution faster.^{21} Random superposition algorithm is an iterative algorithm, in which random values uniformly distributed in [0, 2π] are added while calculating the phase of linear superposition of single trap hologram. It is helpful for improving its efficiency and reducing its computation cost, which is preferred when requiring frequent calculation of large dynamical optical spot arrays with low symmetry geometries.^{39} Additionally, a recent report adopted a similar concept of the random phase factor.^{40} The report shows that Fresnel holograms can produce large volume, high-density, dynamic 3D images for display at different depths by adding random phases into each desired 3-D image, to decrease crosstalk among these different images at output plane. However, the iterative Fourier transform algorithm has been used for generating a set of kinoforms, and phase profiles of each image at different output positions cannot be fully controlled. In our method, the random phase factor is directly introduced in each plane wave of eqn (5) to largely weaken their interference among different plane waves at input plane. The related results reveal that it is very helpful for generating the desired optical traps with accurate control of intensity and phase profiles, which is highly preferred in the optical manipulation of nanoparticles.

Fig. 1 Point and peanut-shaped spot arrays. (a and d) Calculated holograms, (b and e) reconstructed intensity profiles, and (c and f) measured intensity profiles. Note that scale bar is 5 μm. |

Particularly, 3D optical traps with various closed curves at different regions along light propagation direction can also be easily realized by using our method. The position of each trap can be conveniently controlled by adding a spherical wave phase with πz(x_{o}^{2} + y_{o}^{2})/(λf^{2}) in the desired hologram, according to on-demand shifting distance z along optical axis direction. The reconstructed intensity distributions, reconstructed phase profiles and measured intensity distributions at different positions are shown in Fig. 4. The ring and star traps are focused on the focal plane (z = 0), as presented in Fig. 4(b), while the other ring and star traps are axially shifted to the plane of z = −5 μm and z = 5 μm, as shown in Fig. 4(a) and (c), respectively. It finds that the measured intensity distributions are highly consistent with the reconstructed ones. These multiple traps at different axial positions have important applications for simultaneously implementing multitask optical manipulation. Particularly, it needs to note that variation range of grayscale in reconstructed phase is from 0 to 2π, and reconstructed intensity distribution is normalized, whose grayscale range is from 0 to 1.

The computation cost is low in our method, which is very helpful for dynamical optical manipulation. As a comparison, we also calculate the hologram of a typical ring trap by using the integral method,^{33} ant it reveals that the reconstructed intensity and phase are nearly identical to those obtained by our method. Its computation time in integral method is about 90 second, but it is about 13 second in our method.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0na00167h |

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