Open Access
Add to BibSonomy
Abstract
The line optical tweezers (LOT) has been proven to be an alternative technique to manipulating the biological cells because of the bigger potential compared with traditional optical tweezers with a highly focused spot. We deduce the 4 × 4 optical matrix of the astigmatic LOT to investigate the optical characteristics related to the systematic parameters. The comparison of the initial and scattered electric fields by the cell under the astigmatic and stigmatic LOT is implemented to illustrate that the forward scattered light from the astigmatic LOT is much stronger than that from the stigmatic LOT, so as to the cell deformations. It is demonstrated that the astigmatic LOT could provide a more efficient way to deform the cell not only in the focal plane, but also along the optical axis to screen large biomaterials in biomechanics.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The mechanical properties of biological cells are used to indicate physiological and pathological characteristics in cells as an intrinsic, label-free biomarker. Discrepancy in the cellular deformability (elasticity) of the cell membrane and cytoskeleton is associated with a broad spectrum of cellular functional changes, including differentiation, apoptosis, disease transformation, and drug response. Diseased cells may exhibit different elasticity from normal cells, suggesting that measuring the mechanical properties of cells could provide an efficient way to differentiate abnormal cells from normal cells, which is called pre-diagnostic screening [1]. The past decades have witnessed many applications of lasers in the biomedical field [2,3]. One notable event is laser trapping, which can manipulate biological cells in a non-contact manner by means of optical tweezers exerting a radiation force deforming the cell [4–6]. Furthermore, it has been found that the local force on the surface of the object might be much larger than the total net force under laser irradiation, and this surface stress is always perpendicular to the interface between the two media, towards the low refractive index medium [7,8].
With the feature of trapping and deforming the biological cells by exerting the pico-Newton force on the cell, optical tweezers have been proved to be a significant tool to measure the mechanical properties of the cell. It is, however, difficult to trap the cell (e.g. the biconcave red blood cell) which might be flickering, or be pushed away in the limited volume of the potential well with a single trapping beam. For the reason that the size of the highly focused spot of optical tweezers is much smaller than that of the biological cell, it is thus necessary to create more trapping beams [9,10] or a longer one [11]. It is called the LOT [12] or light-sheet optical tweezer [13], which has a larger volume of the trapping potential well with the simple construction and easy-to-operate feature, and it is of great interest for transporting and sorting biological cells in medical diagnostics and self-propulsion applications in drug delivery [14].
The LOT could be classified into two types according to the focusing conditions: One is the stigmatic LOT, the highly focused Gaussian elliptical beam that is focused only once by the microscopic objective; The other is the astigmatic LOT, the lower numerical-aperture Gaussian elliptical beam that has two focal planes. The stigmatic LOT is mostly created through scanning and navigating the highly focused trapping beam with acousto-optic deflector (AOD), resonant scanning mirror and Galvanometer mirror at a high frequency, which could mimic the steady trapping beam to manipulate the cell [12,15,16]. The astigmatic LOT is generally produced by shaping the laser beam into the elliptical profile with the spatial light modulator (SLM) or cylindrical lens to adequately trap the cell [13,17,18]. Due to the distinctive strategies generating LOT, the manipulated cell presents diverse mechanical responses accordingly. The radiation force on the swollen cell surface manipulated by the astigmatic beam is calculated in the ray-tracing method or in Zemax, and the applied optical stresses and cell deformation by line traps generated from laser diode bar with different microscope objectives are compared [19,20]. Instead of the laser diode bar of the elliptical profile, the laser source of the circle spot combined with the lens system of cylindrical lenses, spherical lenses and objectives could be used to construct the astigmatic LOT [21]. This means the latter LOT are more flexible by selecting appropriate lenses with suitable parameters of the lens system for the specified line trap. Furthermore, the feature that astigmatic beam profile is both divergent and convergent as passing through the first focal plane results in the dense light distribution on the cell, compared to the feature of the stigmatic LOT which is totally divergent after the beam passing by the first focal plane. The distinctive astigmatic LOT might provide a broader way for the detection of the mechanical properties of single cells, adherent cells and even biological tissues.
In this paper, the positions and angles of the astigmatic beam in the astigmatic LOT system is deduced in 4 × 4 matrix optical method [22]. We discuss the tendency and the dependent relationship for the major axis (lx’, ly’) and divergent angles (αx’, αy’) of elliptical beam in both focal planes, as well as the distance df between two focal planes for the purpose of trapping characteristics. The scattered light by the spherical dielectric micro-particle in both astigmatic and stigmatic beams is then calculated in electromagnetic optics by using the finite element method in COMSOL Multiphysics. The simulation results are demonstrated that the astigmatic LOT could provide a more efficient way to deform the cell not only in the focal plane, but also along the optical axis.
2. Methods
2.1 Matrix optical method
The optical system for the astigmatic LOT generally consists of a pair of orthogonal cylindrical lenses, a pair of spherical-convex lenses and an objective as shown in Fig. 1. For example, if the first cylindrical lens (CL1) is used to focus the beam in the x-direction, the tangential rays would be created without varying the divergence angle of the light in the y-direction. Meanwhile the second one (CL2) was used to focus the beam in the y-direction generating the sagittal rays without changing the divergence state in the x-direction. Two plano-convex lenses are placed behind cylindrical lenses and in front of the objective in case of a part of the beam being cut off by the entrance pupil of objective. This is because tangential rays of the astigmatic beam passing by the focal plane of the first cylindrical lens becomes divergent and the optical path from the entrance of microscopic frame to the entrance pupil of microscopic objective is so long that the beam size arriving at the entrance pupil of objective is much larger than the entrance aperture of objective.
We firstly set the input matrix for the light travelling along z axis as [22]
Departing from the lens system of transmission matrix TL, the light enters into the microscopic objective (Obj) of the translation matrix A5 and is imaged at the distance of d5 or ${d_{5^{\prime}}}$ as shown in Fig. 1. The transmission matrix of objective is ${\textrm{A}_5} = \left[ {\begin{array}{{cccc}} {{a_{11}}}&0&{{a_{13}}}&0\\ 0&{{a_{22}}}&0&{{a_{24}}}\\ {{a_{31}}}&0&{{a_{33}}}&0\\ 0&{{a_{42}}}&0&{{a_{44}}} \end{array}} \right]$, and the translation matrices between the image and the objective is ${T_5} = \left[ {\begin{array}{{cccc}} 1&0&{{d_5}}&0\\ 0&1&0&{{d_5}}\\ 0&0&1&0\\ 0&0&0&1 \end{array}} \right]$. The matrix at the image could be written by
The displacements and angles are expressed as follows:
The astigmatic beam is finally imaged by the objective in x- and y-axis in the two focal planes, respectively. The two focal lengths of d5 and ${d_{5^{\prime}}}$ should satisfy the two focusing conditions in which the terms multiplied by ${\alpha _x}$ and ${\alpha _y}$ in Eq. (5a) and (5b) are supposed to be zero, respectively. They are thus deduced to be:
The separation between the primary and the second focal planes is
When the object distance between the laser source and the cylindrical lens is infinity ${d_0} = \infty $, two focal lengths become
Generally, the parameters of the transmission matrix for objective follow the relationship as a11 = a22, a13 = a24, a31 = a42, a33 = a44. Then separation of two foci df is formulated as
According to Eqs. (5a)-(5d), the focal length f1 does not have any effect on the angle αx’ and the length lx’ while ly’ = 0 in the primary focal plane, but is inversely proportional to the divergence angle αy’. Contrarily, the focal length f2 plays an important role in shaping the geometry of the astigmatic LOT:
- 1. f2 is inversely proportional to the length lx’ in the primary focal plane and to the divergence angle αx’;
- 2. f2 is linearly proportional to the divergence angle αy’.
Furthermore, the difference ${d_4} - m{f_3}$ influences not only the length lx’ and the divergence angle αx, but also the separation of two foci df in Eq. (8). The astigmatic LOT are thus formed in the first focal plane as the beam passes through the objective, as shown in Fig. 2(a-d). In the condition of the infinite object distance, Eqs. (8) and (9) illustrate that the focal lengths d5 and ${d_{5^{\prime}}}$ and the separation of two foci df are all independent on the cylindrical lenses parameters f1 and f2, but dependent on the objective parameters a11, ···, a42 and the difference d4 - mf3. One can choose appropriate parameters of optical elements in the astigmatic LOT system to obtain the specific electric fields of the astigmatic LOT in experiment.
Fig. 2. Normalized initial electric fields in transparent contour plot for astigmatic beam in upper panel in the view of spatial coordinate (a), in plane y = 0 (b) plane x = 0 (c) and zoomed plot around focal spot (d); and those fields for stigmatic beam in lower panel in the view of spatial coordinate (e), in plane y = 0 (f) plane x = 0 (g) and zoomed plot around focal spot (h).
2.2 Finite element model
We perform two-way coupled opto-mechanical simulations to investigate the diffused light and cell deformation of both astigmatic and stigmatic LOT system in the finite element method (FEM). For the case of the astigmatic LOT, the vector electric field EBAst traveling along z-axis is the linearly polarized elliptical-Gaussian beam with the wave-front of the plane wave, expressed as [19,23]
For the case of the stigmatic LOT propagating along z-axis, it is modeled by stretching the highly focused laser spots along x-axis in the plane z = 0, as shown in Figs. 2(e–h). The vector electric field EBSt is thus written by [24]
Table 1. Parameters of electric field
Figures 2(d) and 2(h) present the normalized initial electric field distribution surrounding the two LOTs respectively in the region of interest that locates around the concentrated electric field focused by the objective. They are both in the cross section of plane x = 0. The stigmatic LOT with the larger NA = 1.25 makes the beam concentrate along x-axis and spread over a larger volume, but its electric field decreases dramatically away from the focal spot. It is obvious to see that the electric field is symmetric with respect to the plane z = 0, as shown in Fig. 2(h). Nevertheless, the astigmatic LOT with lower NAx = 0.21 and NAy = 0.39 manifesting stronger the asymmetric electric field: The forward (z > 0) electric field is bigger than the backward (z < 0), as shown in Fig. 2(d). The reason is that one part of the light is convergent to be focused in the primary focal plane and then divergent afterwards as shown in Fig. 2(c), while the other part is convergent passing by the primary focal plane until arriving at the second focal plane as shown in Fig. 2(b). It is obvious to observe that the astigmatic LOT makes the beam more intensive than stigmatic LOT as z > 0. The major radius of astigmatic LOT was predicted to be ω0x = 9.6 µm which determines the magnification coefficient of the beam length in x-axis N = 160 ensuring the stigmatic LOT with the same ω0x.
The optical radiation stress as a function of the scattered electric field with respect to the Maxwell stress tensor is [24]
3. Results and discussion
3.1 Electric fields
The light scattering by the cell under manipulations of astigmatic and stigmatic LOT was computed in the same way as the paper [24], in which the model contains cytosol, cell membrane, buffer solution and perfect matched layer (PML) which is used to limit the calculation area in case of the scattered light travelling beyond. The cell is a sphere with a radius of 3.5 µm. The electrical conductivity σ = 0 S/m and relative permeability µr = 1 are adaptable to all the materials in the model. The scattered electric field could be calculated in the condition that the refractive index of the cell n2 = 1.378 is larger than that of the buffer solution n1 = 1.335. ESAst is the scattered electric field of astigmatic LOT, and ESSt is the scattered electric field of stigmatic LOT. Figure 3 illustrates the normalized electric fields distribution on the cell surface (upper row) and in the cross section of the plane y = 0 (lower row). It is obvious to see that the normalized initial electric field |EBAst|/|EBmax| looks like a light-sheet cutting the cell along x-axis, as shown in Fig. 3(a). It is almost evenly distributed across the section plane along the propagation direction in Fig. 3(e). That is why the astigmatic beam is also used for the light-sheet illumination in the fluorescent microscope. The initial electric field of stigmatic LOT is mostly concentrated at the both ends along x axis and it is very weak around the equator of the cell, as shown in Fig. 3(b), (f). The higher numerical aperture of stigmatic LOT makes the initial electric field more concentrated along the focused axis, which is compliance with the 3D electric field distribution in Fig. 2(e-h). The forward scattered light is enhanced compared with the initial LOT, not only on the cell surface, but also in the cross section, as shown in Fig. 3 (a–h). The scattered light of astigmatic LOT on the output surface is much higher than that of stigmatic LOT, so as to the scattered light of astigmatic LOT in the cross section higher than that of stigmatic LOT, as shown in Fig. 3(c), (d), (g), (h). Further illustration of electric field distributions on the cell surfaces is presented in Fig. 4.
Fig. 3. Normalized electric field on the cell surface of for |EBAst|/|EBmax| (a), |EBSt|/|EBmax| (b), |ESAst|/|EBmax| (c), and |ESSt|/|EBmax| (d).
Fig. 4. Normalized modulus of electric fields distribution of |EBAst|/|EBmax| (a), |EBSt|/|EBmax| (b), |ESAst|/|EBmax| (c) and |ESSt|/|EBmax| (d) on the hemispherical surface in the top view and |EBAst|/|EBmax| (e), |EBSt|/|EBmax| (f), |ESAst|/|EBmax| (g), |ESSt|/|EBmax| (h) in the bottom view, and the corresponding electric field versus the arc length with black solid line denoting EBAst, green dash-dot-dotted line denoting EBSt, red dash-dotted line denoting ESAst, orange dashed line denoting ESSt on the outgoing surface (i) and incident surface (j).
The first and second rows of Fig. 4(a–h) exhibit the normalized electric field distributions on the output and incident hemispherical surfaces of the spherical cell, and the third row of Fig. 4(i), (j) depicts the normalized electric field of the corresponding colored line profiles on cell surfaces, respectively. The light-sheet normalized initial electric field |EBAst|/|EBmax| is on the output and incident surfaces as shown in Fig. 4(a), (e), (i), (j), while |EBSt|/|EBmax| distributes evenly on the output and incident surfaces other than the weak region near the equator z = 0 and the extremely intense region around x-axis as shown in Fig. 4(b), (f), (i), (j). |EBAst|/|EBmax| on the output hemispherical surface in Fig. 4(a) is slightly higher than that on the incident hemispherical surface in Fig. 4(e), complying with the initial electric field distribution around the focal plane in Fig. 2(d) and the corresponding black solid curves in Fig. 4(i), (j). Meanwhile, |EBSt|/|EBmax| on the both hemispherical surfaces in Fig. 4(b), (f) is symmetric with respective to the plane z = 0, coinciding with the initial electric field distribution in Fig. 2(h) and 3(b).
The scattered light is redistributed since the refractive index of the cell is bigger than that of the buffer solution, i.e. n2 > n1. The scattered electric fields on the output hemispherical surfaces are enhanced compared with the initial electric fields on the incident hemispherical surfaces, as shown in Fig. 4(c), (d), (g–j). Furthermore, the intense parts of the forward scattered light for astigmatic LOT is about 3∼4 times stronger than that for stigmatic LOT, which means that astigmatic LOT is able to result in the stronger forward scattered light than stigmatic LOT when the same beam power is used in both cases.
3.2 Cell deformation
The cell gets the surface radiation stress as a function of normal and tangent electric fields based on the Maxwell stress tensor and the final deformation of the cell is thus calculated in the two- way coupled RF and Continuum mechanics COMSOL modules as the Ref. [24]. The mechanical parameters of cell membrane and cytosol are listed in Table 2.
Table 2. Parameters of cell
Figure 5(a-d) shows the simulated displacement of the deformed cell in the astigmatic and stigmatic LOT, in which the cell is elongated along the x-axis. It is clear to see that the cell under the astigmatic LOT is not only stretched along the major axis, but also in z-axis, whereas the cell under the stigmatic LOT deforms mainly in x axis without the obvious deformation in z-axis. The reason is that the forward scattering by the cell in the astigmatic LOT is highly enhanced, so as to the optical radiation stress on the output surface as well as the cell deformation. The cell deformation in the astigmatic LOT in z-axis is much bigger than that in the stigmatic LOT. This characteristic may provide an alternative way to manipulate the biological samples more efficiently. Figure 5(e) displays the simulated cell deformations divided by the cell diameter in x-, y-, and z-directions as a function of the laser power. These simulation results of the cell elongation in x-axis in the astigmatic LOT could be well fit to the experimental results [19], which means the simulation method is valid. In the condition of the incompressible material, the cell is stretched in x- and z- directions and must be shrunk in y-direction. The deformation in y-axis is minus. The error bars mean the displacements of iterative calculation until at equilibrium. The percentage of deformations in x-, y-, and z-directions as a function of the laser power are fit to three lines in corresponding colors. The fitting line for the cell deformation in z-direction has larger slope than the other two lines in x- and y-directions, which means the astigmatic LOT facilitates the mechanical responses in z-axis more efficiently than those in other directions when the laser power is bigger.
Fig. 5. Simulation results of cell deformations in astigmatic LOT in spatial view (a) and bottom view (b), and in stigmatic LOT in spatial view (c) and bottom view (d) in the equilibrium state, and cell deformation percentage versus laser power and the fitting lines (e).
4. Conclusion
In this paper, we firstly deduce the 4 × 4 optical matrix of the whole optical system to investigate the effects of the specific parameters (the focal lengths f1, f2, and the difference d4 – mf3) on the astigmatic characteristics. Then, the electric fields of the astigmatic and stigmatic LOT are characterized and compared. The scattered electric fields by the cell under both LOTs are calculated and compared, which is hard to be detected in experiment. The enhancement of the forward scattered light by the cell under the astigmatic LOT facilitates the stronger optical radiation stress on the output surface of the cell due to the positive correlation between the optical radiation stress and the scattered electric fields. The cell deformation is computed in the Structure Mechanics module which is fully coupled to the RF module of COMSOL Multiphysics in the iterative manner. The simulated cell elongations as a function of laser power are well fit to the experimental results in literature, which is valid to demonstrate that the astigmatic LOT could be used to manipulate the cell not only in x-axis, but also in z axis. In addition, the astigmatic LOT facilitates the mechanical responses in z axis more efficiently than those in other directions when the laser power is bigger. The optical manipulation by the astigmatic LOT may not occur in the focal plane, but along optical axis, which could broaden the biological samples from the single suspended cell to adherent cells and even biological tissues.
Funding
National Natural Science Foundation of China (61965008); Natural Science Foundation of Guangxi Province (2022GXNSFAA035643, 2023GXNSFBA026071, AD21220086); Scientific Research Project for Guangxi University (2020KY05022); Guangxi Key Laboratory (GD21103, YQ21109); Innovation Project of GUET Graduate Education (2023YCXS207).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. Z. Yao, C. C. Kwan, and A. W. Poon, “An optofluidic “tweeze-and-drag” cell stretcher in a microfluidic channel,” Lab Chip 20(3), 601–613 (2020). [CrossRef]
2. E. Mester, A. F. Mester, and A. Mester, “The biomedical effects of laser application,” Lasers Surg. Med. 5(1), 31–39 (1985). [CrossRef]
3. J. R. Basford, “Low intensity laser therapy: still not an established clinical tool,” Lasers Surg. Med. 16(4), 331–342 (1995). [CrossRef]
4. A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330(6150), 769–771 (1987). [CrossRef]
5. S. C. Kuo and M. P. Sheetz, “Optical tweezers in cell biology,” Trends Cell Biol. 2(4), 116–118 (1992). [CrossRef]
6. I. A. Favre-Bulle and E. K. Scott, “Optical tweezers across scales in cell biology,” Trends Cell Biol. 32(11), 932–946 (2022). [CrossRef]
7. K. Okamoto and S. Kawata, “Radiation Force Exerted on Subwavelength Particles near a Nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999). [CrossRef]
8. J. Guck, R. Ananthakrishnan, T. J. Moon, et al., “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84(23), 5451–5454 (2000). [CrossRef]
9. R. Liu, M. Shao, Z. Ke, et al., “Measurement of red blood cell deformability during morphological changes using rotating-glass-plate-based scanning optical tweezers,” Biomed. Opt. Express 14(9), 4979–4989 (2023). [CrossRef]
10. E. V. Lyubin, M. D. Khokhlova, M. N. Skryabina, et al., “Cellular viscoelasticity probed by active rheology in optical tweezers,” J. Biomed. Opt. 17(10), 101510 (2012). [CrossRef]
11. B. Ma, B. Yao, F. Peng, et al., “Optical sorting of particles by dual-channel line optical tweezers,” J. Opt. 14(10), 105702 (2012). [CrossRef]
12. R. Nambiar and J. C. Meiners, “Fast position measurements with scanning line optical tweezers,” Opt. Lett. 27(10), 836–838 (2002). [CrossRef]
13. P. P. Mondal, N. Baro, A. Singh, et al., “Lightsheet optical tweezer (LOT) for optical manipulation of microscopic particles and live cells,” Sci. Rep. 12(1), 10229 (2022). [CrossRef]
14. Y. Shen, D. A. Weitz, N. R. Forde, et al., “Line optical tweezers as controllable micromachines: techniques and emerging trends,” Soft Matter 18(29), 5359–5365 (2022). [CrossRef]
15. B. Trankle, M. Speidel, and A. Rohrbach, “Interaction dynamics of two colloids in a single optical potential,” Phys. Rev. E 86(2), 021401 (2012). [CrossRef]
16. L. Yu and Y. Sheng, “Mechanical analysis of the optical tweezers in time-sharing regime,” Opt. Express 22(7), 7953–7961 (2014). [CrossRef]
17. G. T. Tietjen, Y. Kong, and R. Parthasarathy, “An efficient method for the creation of tunable optical line traps via control of gradient and scattering forces,” Opt. Express 16(14), 10341–10348 (2008). [CrossRef]
18. E. Spyratou, E. Cunaj, G. Tsigaridas, et al., “Measurements of liposome biomechanical properties by combining line optical tweezers and dielectrophoresis,” J. Liposome Res. 25(3), 202–210 (2015). [CrossRef]
19. T. Sawetzki, C. D. Eggleton, and D. W. M. Marr, “Cell elongation via intrinsic antipodal stretching forces,” Phys. Rev. E 86(6), 061901 (2012). [CrossRef]
20. K. B. Roth, K. B. Neeves, J. Squier, et al., “Imaging of a linear diode bar for an optical cell stretcher,” Biomed. Opt. Express 6(3), 807–814 (2015). [CrossRef]
21. L. Yu, Y. Jia, X. Hu, et al., “Trapping and revolving micron particles by transformable line traps of optical tweezers,” Chin. Opt. Lett. 20(5), 053801 (2022). [CrossRef]
22. A. E. Attard, “Matrix optical analysis of skew rays in mixed systems of spherical and orthogonal cylindrical lenses,” Appl. Opt. 23(16), 2706 (1984). [CrossRef]
23. S. K. Mohanty, R. S. Verma, and P. K. Gupta, “Trapping and controlled rotation of low-refractive-index particles using dual line optical tweezers,” Appl. Phys. B 87(2), 211–215 (2007). [CrossRef]
24. L. Yu, Y. Sheng, and A. Chiou, “Three-dimensional light-scattering and deformation of individual biconcave human blood cells in optical tweezers,” Opt. Express 21(10), 12174–12184 (2013). [CrossRef]
