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Abstract
In this work, we proposed an optical trapping and manipulation technology based on spatial diffraction of 45° tilted fiber Bragg grating (TFBG). The length of the line-shape-facula of the TFBG diffraction light can be as large as tens of millimeters, which enables the TFBG trapping system control massive dielectric particles. We analyze the light distribution of the spatial diffraction by using the volume current method (VCM) and established a theoretical model to analyze the optical trapping force of TFBG based on the ray tracing method (RTM). Then, we designed several optical trapping schemes, with two-, three- and four-TFBGs respectively. Numeral simulation indicates that only the scheme with axisymmetric layout of TFBGs can achieve stable particle trapping. We comprehensively analyze the trapping force distribution of four- TFBG scheme with different influence factors. In addition, the rotation manipulation based on the two- and four- TFBGs schemes are also demonstrated. The proposed optical trapping technology open a new route for massive particles trapping and manipulation.
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1. Introduction
The massive particle trapping and manipulation is an important technology and has potential usages in the biological and medical areas, such as cell separation and extraction, medical diagnosis, drug delivery, biomedical and chemical analysis, fundamental studies of cell–cell interaction and communication [1–6]. Various approaches have been proposed, including electrophoresis [7], dielectrophoresis [8], magnetophoresis [9], and acoustophoresis [10]. However, these methods often require complicated microfluidic structures and external systems to generate electric fields, magnetic fields or acoustic fields [11]. Although the large space-optic trapping and manipulation technique does not require specific designed microfluidic chip, the bulky structure creates a lack of flexibility in operations and makes it hard to work in small-scale environments like blood vessels [12–14].
Recently, the optical fiber trapping technique has drawn extensive attentions, thanks to its merits of miniaturization, flexibility of manipulation, capability of working in cramped space and remote control, and ease of integration with fluidic chip [15–18]. Many optical fiber schemes for massive particle trapping have been proposed, such as the optical fiber photothermal-phoresis probe [19], the micro/nanofiber evanescent field trapping [20,21], and the abrupt fiber taper probe for cell-patterning [22,23]. The fiber photothermal-phoresis probe employs the output light to heat the surrounding liquid, so as to generate the thermal gradient. Then, the particles can be trapped on the surface at the fiber end. As the output light becomes divergent soon in the free space, the trapping region and the amount of the trapped particles are constrained. Some efforts have been made by using the fiber taper probe or the fiber knot to improve the trapping region and capacity [24,25], even though the structures of these probe are fragile, which affects the trapping stability. The micro/nanofiber trapping technique uses the evanescent field to generate the gradient force and trap the particle onto the fiber surface. The particles move along the propagation direction of the light in the micro/nanofiber, which makes it attractive in the long-distance particle transportation [26]. To excite the evanescent field with sufficient intensity, the radius of the fiber should be reduced to micrometers and even sub-micrometers. In this case, the strength of the taper fiber is deteriorated. Thus, the structural fragility is the main shortcoming for this scheme. The abrupt fiber taper probe is a robust trapping scheme. The fiber taper with a large cone angle can focus light around the tip and trap the particle. However, this trapping scheme can only trap a few particles in general. By using specific designed fiber tapers or involving varieties of particles, many particles can be self-assembled into chain driven by the output light, which is rather attractive in researches of cell-to-cell interactions [27]. However, the fabrication of this fiber taper is complicated and the consistency is hard to guarantee. Hence, an optical fiber trapping technology with robust structure and capability of massive particle trapping is still to be proposed.
Recently, a tilted fiber Bragg grating (TFBG) with 45° tilted angle has been extensively investigated [28–32]. The light propagating in fiber can be diffracted into the free space from one side of 45° TFBG. This spatial property has been utilized to design the sensing demodulation scheme [30], the line-scan imaging [31], and the miniaturized spectrometer [32]. As the size of the side-diffractive facula can be tens of millimeters, the 45° TFBG becomes an ideal fiber diffractive component in the design of massive particle trapping and manipulation technologies. In this work, we propose and theoretically demonstrate an optical massive particle trapping and manipulation technology based on the 45° TFBG. We firstly analyze the light distribution of the spatial diffraction by using the volume current method (VCM) and then established a theoretical model to analyze the optical trapping force of TFBG based on the ray tracing method (RTM). Based on this, we designed several optical trapping schemes, with two, three and four TFBGs respectively. Then, we comprehensively analyze the trapping force distribution of the four- TFBGs scheme with different influence factors. In addition, the rotation manipulation based on the two- and four- TFBGs schemes are also demonstrated.
2. Theoretical model
2.1 Spatial diffraction of the 45° TFBG
In order to analyze the optical force of the proposed 45° TFBG tweezers, we first analyze the light field diffracted by the TFBG. By using the volume current method (VCM) [33,34], the radiation distribution of the TFBG can be achieved. The 45° TFBG and the coordinate system used in VCM analysis are illustrated in Fig. 1, where $\overrightarrow r$ and $\overrightarrow {{r^{\prime}}}$ are respectively the detection vector and source vector. ξ and Λ represent the tilted angle and period of the TFBG, respectively. According to Li’s work, the vector potential of the light field in the TFBG can be expressed as [33]:
In Eq. (2), ω is the angular frequency of the light, κ is the coupling coefficient of the TFBG, and E0 is the electric field intensity for the fundamental mode of fiber. J0() is the 0-order Bessel function of first kind with $u = \sqrt {{k_0}^2{n_0}^2 - {\beta ^2}}$. Δ=β-Kg with β to be the propagation constant in optical fiber. Kt is the transverse wavevector of the grating. r and φ are the coordinate of the detection point defined in Fig. 1. Then, the Poynting vector of the light field radiated by the TFBG can be expressed as:
In Eq. (3), δ is the polarization direction of light, which is equal to 0° for S polarization and 90° for P polarization. J1() is the 1-order Bessel function of first kind. r0 is the radius of the fiber core. ${k_t} = \sqrt {{k_0}^2{n_0}^2 - {\Delta ^2}} = \sqrt {{k_0}^2{n_0}^2 - {\beta ^2} - {K_g}^2 + 2\beta {K_g}}$. c and k0 are respectively the light speed and wavevector in vacuum. Knew is the vector addition of Kt and kt, where:
For the above model, given a tilted angle of 45°, ${K_g} = \beta$ and $|{\overrightarrow {{K_t}} } |= {k_t} = {k_0}{n_0}$, we have ${K_{new}} = \sqrt {{k_t}^2 + {K_t}^2 + 2{K_t}{k_t}\sin \phi } = {k_0}{n_0}\sqrt {2 + 2\sin \phi }$. If we further have the wavelength of light source $\lambda = \sqrt 2 {n_0}\Lambda $ and the phase matching achieved, Eq. (3) can be simplified as:
From Eq. (6), we can obtain a simulated radiation distribution of the TFBG, where the parameters in the simulation are shown in Tab. 1. As we select a proper refractive index (RI) modulation, a line shape facula can be achieved with uniform intensity along the fiber axis. In this case, we can choose a cross section in the grating region to analyze the spatial diffraction of the TFBG in detail. For an optical fiber shown in the coordinate system in Fig. 2(a), where the z axis is perpendicular to the fiber axis and 45° to the grating plane, the energy density flow outside the TFBG is shown in Fig. 2(b) with the white region representing the cross section of optical fiber. From Fig. 2(b), we can clearly see that the light is diffracted from one side of the TFBG, propagates with wavevector along z axis and disperses within a small angle. Moreover, in Fig. 2(c), we observe that the light along the z axis is the base ray of the diffractive beam and its energy density flow dramatically decreases as it propagates away from the center of the TFBG. Note that the basic parameters used in calculation can be found in Table 1. As the length of the TFBG can be tens of millimeters, such facula can be utilized in design of optical tweezers for massive particles trapping.
Table 1. Parameters used in spatial diffraction property analysis
Fig. 2. (a) Coordinate system of the plane for analyzing the spatial diffraction property of TFBG; (b) Energy density flow in the free space outside the TFBG (c) Energy density flow along the z axis.
2.2 Optical force analysis based on the ray-tracing method
In order to design the TFBG based optical tweezers, we first analyze the optical trapping force that apply to the particle. To begin with, we introduce an optical trapping force model based on the ray tracing method (RTM), which has been demonstrated to be valid for trapping particles with radius more than ten times of λ [35].
Figure 3 briefly describes the RTM in the coordinate system for TFBG optical force analysis. The coordinate of the particle is shown in Fig. 3, while the gray semicircle is the collision region between the TFBG and particle. As a dielectric microparticle is placed in an optical field, it is simultaneously driven by scattering and gradient forces. Considering a single ray (labeled by i), the two forces can be then expressed as:
If the particle is in water, the refractivity T and reflectivity R can be determined according to Fresnel formula. Then, the total optical force can be obtained by vector sum of the scattering forces and gradient forces of the light rays, i.e.:
In Fig. 4, the optical forces are illustrated as functions of particle positions for the radius of the dielectric particle radius rp of 7.5 µm, 10 µm, 15 µm and 17.5 µm, respectively, where the RI np and the ambient RI nliq of the particle take the values of 1.573 and 1.333, respectively. The parameters in Table 1 are also used for the analysis in Fig. 4. The color represents the magnitude of optical force, with the reference of the color bar in the figure. The arrows in the figures simultaneously indicate the magnitude (length of arrow) and direction of optical force (orientation of arrow). We can clearly see that the microparticle is applied to the optical force only if it is irradiated by the side-diffractive light of TFBG. As shown in Fig. 4(f), the optical force is attenuated if the particle is away from the TFBG. The zoomed view in Fig. 4(b) illustrates the optical force near the TFBG with particle size of 10 µm, where the component force along the z-axis is positive. With such force, the particle is pushed away from the TFBG. On the other hand, the component force along the x-axis points to the z-axis, which drags the particle toward the base ray of the beam. The results agree with schemes with similar intensity distributions of output light field [36]. In addition, the particles with larger sizes are more tolerant to the horizontal offsets. For the same distance on z-axis, the optical force also increases with the increase of the particle size. This is because a larger microparticle leads to larger cross section, so that more light rays irradiate onto these particles and contribute to the total optical force. As the above optical force analysis indicates that the particle is dragged to the base ray of side-diffractive beam and pushed away from the TFBG. Thus, stable trapping of particle cannot be achieved by using a single TFBG.
Fig. 4. Optical force as functions of particle position for different particle radius (a) 7.5 µm, (b)10 µm, (c)15 µm and (d)17.5 µm (See high resolution images in section S1 of Supplement 1); (e) Optical force as a function of particle position in z axis.
3. Optical trapping and manipulation
3.1 Design of trapping system
In order to achieve particle trapping, we can use multiple 45° TFBGs to generate the force equilibrium region, where the particles can be trapped. In this section, we render two-, three- and four-TFBG trapping system and discuss the trapping performances of these schemes.
Figure 5 illustrates the proposed multiple-TFBG trapping systems and the optical force as a function of the particle position at the cross-section of the grating region. The parameters used in analysis are the same to the above section. The radius of particle rp is 10 µm. As shown in Fig. 5(a), the two TFBGs are axisymmetric to each other and the base rays of the side-diffractive beams for these TFBGs are opposite. Then, the force equilibrium region can be obtained at the central line of the two TFBGs, while the optical force in the two-TFBG scheme can be modeled as a function of the particle position. The cross-section map of the force is shown in Fig. 5(b), where the distance between the two TFBGs δz is equal to 80 times of r0. It’s obvious that there is a force equilibrium point (labeled with ob) between the two TFBGs and the particles would be driven into this region and trapped inside as they are irradiated by the spatial diffractive beam of the TFBGs.
Fig. 5. Arrangement of the TFBGs respectively for (a) two-, (d) three- and (g) four-TFBG scheme; (b)(e)(h) the corresponding optical force as a function of particle position (See high resolution images in section S2 of Supplement 1). (c)(f)(i) the corresponding optical force at the selected axes.
To analyze the trapping performance more explicitly, we plot the force magnitude as a function of particle position along the base ray of the diffractive beam (zb axis) and perpendicular to the base ray (xb axis). The plot is shown in Fig. 5(c), where the equilibrium point lies on the coordinate origin. The red arrows indicate the orientation and magnitude of the force. It’s obvious that no matter for zb axis or xb axis, the optical force would drive the particles back to the equilibrium point, if they deviate from the equilibrium point. In addition, as the optical force increases with the increase of the offset at the zb axis, the particles can be stably trapped in this axis. We refer to the trapping in the zb axis as “strong trapping”. On the other hand, as the optical force increases as the offset at the xb axis increases to 10.5 µm and then rapidly decreases afterwards, the optical force would also disappear if the offset is larger than 41 µm. In this case, the particle would escape. If the escaping force generated by the Brown motion is larger than 0.15 pN [37] (i.e. the peak force in the xb axis), the particle would escape from the xb axis. Hence, to achieve stable trapping and avoid particle escaping, the optical force should be large enough. Due to the risk of escaping, the trapping in the xb axis is defined as “weak trapping”.
The three-TFBG trapping scheme is illustrated in Fig. 5(d), where the TFBGs are placed at three apexes of an equilateral triangle. The base rays of the side-diffractive beam point to the center of the equilateral triangle. Hence, as shown in Fig. 5 (e), the optical force equilibrium point of this scheme is the center of the equilateral triangle, where the distance between the adjacent TFBGs is 80 times of r0. To analyze the trapping performance of this scheme, the optical force as a function of particle position along the base ray of TFBG1 is plotted in Fig. 5(f). From Fig. 5(f), we can clearly see that if the particle deviates from the equilibrium point, the optical force would push the particle away from TFBG1. Although there still exists an equilibrium point, the particle cannot be stably trapped. Hence, such non-axisymmetric multiple-TFBG scheme is not an ideal optical trapping system.
The four-TFBG scheme is shown in Fig. 5(g). Similar to the above designs, the TFBGs are placed in the apexes of a square and the base rays of the side-diffractive beam point to the center of the square. Then, the equilibrium point can be generated at the center, as shown in Fig. 5(h). As a particle is irradiated by the side diffractive beam, it is driven to the equilibrium point ob and stably trapped. If it deviates from the equilibrium point, the optical force would drag it back. The strong trapping and weak trapping conditions also exist. Different from the two-TFBG design, the two axes for the strong trapping and weak trapping are not perpendicular but at an angle of 45°. The optical force as a function of particle position along these axes are illustrated in Fig. 5 (i), where the performances of the strong and weak trappings are similar to those of the two-TFBG scheme. In Fig. 5(i), the dips in the curve for strong trapping are much sharper than those of two-TFBG scheme, and the peak force for weak trapping is about 0.46 pN, which is about 3 times of that in two-TFBG scheme. Thus, compared with the two-TFBG scheme, the four-TFBG scheme outperforms in the trapping stability.
The above analysis indicates that only the axisymmetric scheme can achieve stable trapping. Because the number of TFBGs that are involved in the system design and their arrangement are different, these schemes exhibit different trapping performance in different direction. In order to further investigate their trapping performances in different directions, we plot the optical force around the equilibrium point for the offset as 10 µm, 20 µm, 30 µm, 40 µm and 50 µm, respectively.
As shown in Fig. 6(a), for the two-TFBG scheme, if the offset is 10 µm, the optical forces around the angles 0° and 180° are larger than those around 90° and 270°, which agrees with our above analysis. If offset is larger than 10 µm, there would be no optical force around 90° and 270°. In other words, the angular bondage capability in 90° and 270° is much lower than that in 0° and 180°.
Fig. 6. Angular trapping performance (a) two-TFBG scheme (b) four-TFBG scheme with offset of 10 µm (black), 20 µm (red), 30 µm (green), 40 µm (blue) and 50 µm (cyan), respectively.
For the four-TFBG scheme, when the offset is larger than 30 µm, the optical force only exists around 4 direction, i.e. 0°, 90°, 180°and 270°. When the offset is smaller than 20 µm, the optical force exists in any direction. In particular, if the offset is 10 µm, the optical force is close to uniformly distributed from all directions, indicating that the four-TFBGs scheme exhibits better angular trapping performance in this case.
3.2 Discussion on stable trapping
According to the above analysis, the trapping performance of the four-TFBG scheme would be better than the two-TFBG design. Thus, in this section, we take this scheme as an example to analyze the factors that affect the trapping performance of TFBG-based system, including the arrangement parameter δz, the electric field amplitude of the input light E0, the particle RI np and the particle radius rp.
Figure 7 illustrates the trapping performance by varying the distance δz between the two opposite TFBGs.
Fig. 7. (a) Optical force as a function of particle position in weak and strong condition under different distance δz; (b) optical force versus normalized distance when offset is 10 µm.
As shown in Fig. 7 (a), the peak of the curve for weak trapping does not shift, but the its magnitude decreases dramatically as δz increases from 60r0 to 120r0. In contrast, for strong trapping, the trend of the force versus particle position becomes slower as δz increases. In addition, the working distance become broader due to the increment of the distance in this case.
As discussed in the above section, the angular trapping performance is uniform in the case that the offset is 10 µm for the four-TFBG scheme. As shown in Fig. 7 (b), we utilize the optical force with the offset of 10 µm for the four-TFBG scheme to evaluate how the distance affect the trapping performance. From that figure, we observe that the optical force decreases dramatically from about 1.048 pN to 0.125 pN for weak trapping and from 0.944 pN to 0.138 pN for strong trapping. The trapping capacity of the four-TFBG system become weaker as the distance increases, which may lead to the escape of the trapped particle. Hence, to design a four-TFBG trapping system, the TFBGs should be placed close enough to enhance the trapping performance.
In addition, we increase the amplitude of electric field E0 from 1 to 5 V/m and calculate the optical force as a function of particle position in the strong and the weak trapping situations as shown in Fig. 8. We find that no matter for the strong trapping or the weak trapping, the shape of the curve does not change as E0 changes. However, the magnitude increases significantly. We also use the optical force with the offset of 10 µm to evaluate the influence of E0, where the optical force increases from 0.470 pN to 11.751 pN for the strong trapping and from 0.444 pN to 11.089 pN for the weak trapping. According to Eq. (6), (7), the optical force obeys $F \propto E_0^2$. Particularly, using the second-order allometric formula to fit the data, we get F = 0.47003E02 and F = 0.44356E02, respectively, for strong and weak trapping with R2 = 1 in good allometric behaviors. Thus, the increase of E0 is an effective way to improve the trapping performance.
Moreover, the impact of the particle RI is also an important factor to be considered, as shown in Fig. 9(a). Similar to the influence of E0, the shape of the curve for the optical force versus particle position does not change, but its magnitude increases as the particle RI increases. In Fig. 9 (b), The optical forces with the offset of 10 µm as functions of np for the strong trapping and the weak trapping are both plotted. In this paper, three kinds of particles including the cell (with RI increasing from 1.38 to 1.42), the silica (RI = 1.444) and the polymethyl methacrylate (PMMA, RI = 1.573). As the particle RI increases, the optical force also increases from 0.168 pN to 0.470 pN for the strong trapping and from 0.182 pN to 0.444 pN for the weak trapping. According to Eq. (6)-(8), for small RI variation, the refractivity T and reflectivity R can be regarded as constant values. Then, the optical force obeys $F \propto {n_p}$. The linear vibration behavior of optical force can be found for both the strong and the weak trapping situations, where particle RI varies from 1.38 to1.42. In contrast, for a large RI variation, T and R can no longer be treated as constant values and could be affected by the particle RI, and nonlinear behavior can be observed, as shown in Fig. 9 (b). From Fig. 9(b), it is indicated that by increasing the particle RI, the trapping performance can be improved, which can be utilized in the design of particle sorting systems.
Fig. 8. (a) Optical force as a function of particle position in weak and strong condition under different E0; (b) optical force versus E0 when offset is 10 μm.
Fig. 9. (a) Optical force as a function of particle position in weak and strong condition under different RI of particle np; (b) optical force versus np when offset is 10 µm.
In Fig. 10, the optical force is plotted as a function of particle position with varieties of particle radius rp, where we observe that the peak of the curve for the weak trapping shifts towards the longer distance with its magnitude increased dramatically with the increase of rp. In contrast, in the strong trapping case, the optical force increases with the increase of rp for the offset smaller than 25 µm. However, this increase becomes mild for offset larger than 25 µm. The optical force with offset of 10 µm indicates that the force increases with the increase of rp, which results in more significant influence to the weak trapping than the strong trapping. As the trapping forces are different for different particle sizes, we can also design particle sorting systems.
Fig. 10. (a) Optical force as a function of particle position in weak and strong condition under different particle radius rp; (b) optical force versus rp when offset is 10 µm.
3.3 Displacement manipulation
As demonstrated above, the equilibrium point is generated when the two optical forces are canceled out. Thus, in the case that the input powers of two TFBGs are equal, the equilibrium point is at the midpoint between the two fiber centers. In contrast, as their input powers are different, the equilibrium point would deviate from the midpoint. Figure 11(a) illustrates the optical force as a function of the particle position for the two-TFBG trapping scheme. The input power of the TFBG2 is 1.5 times of that for the TFBG1. We can clearly see from Fig. 11(a) that the equilibrium point moves toward TFBG1. Therefore, the precise linear motion manipulation along the z-axis can be conducted through the adjustment of the input power difference.
Fig. 11. Optical force as a function of particle position for (a) two-TFBG scheme and (b) four-TFBG scheme with unbalanced input power (See high resolution images in section S3 of Supplement 1).
As shown in Fig. 11(b), the equilibrium point deviation can also be observed in four-TFBG scheme with unbalanced input power. In this case, the input powers of TFBG3 and TFBG4 are 1.2 times of those from TFBG1 and TFBG2. As the four-TFBG scheme can be treated as two two-TFBG schemes that are perpendicular to each other. Then, the deviation gets two degrees of freedom, which would be more useful in particle manipulation.
In order to investigate how the unbalanced input power affects the equilibrium point, we model the equilibrium displacement of particle position as a function of the unbalanced power input. As the displacement of four-TFBG scheme can be analyzed through the decomposition into two orthogonal components, we only need to discuss about the equilibrium point displacement in two-TFBG layout. Let us denote the balanced input power as P0 and unbalanced power as ΔP. Then, the input powers of TFBG1 and TFBG2 become P0+ΔP and P0 -ΔP, respectively. By tracking the position of the equilibrium point, a curve of its displacement versus the unbalanced power ΔP can be obtained. As shown in Fig. 12, as the normalized unbalanced power ΔP/ P0 increases from -0.4 to 0.4, the displacement of equilibrium point increases linearly from -42.5 µm to 42.5 µm. Therefore, the fitted relationship between the equilibrium displacement and the unbalanced power input can be expressed as y = 104.6667x, with the coefficient of determination as R2 = 0.9996. These results indicate that we can achieve linear motion manipulation of the particles without mechanical actuators, which would be very useful in particle manipulations in small-scale environments.
3.4 Rotation manipulation
According to the discussion in Section 3.1, only the schemes with axisymmetric structures can achieve stable trapping. As the two base rays of the opposite TFBGs is offset, the optical forces along the base rays would no longer cancel out to generate force equilibrium point, but drive the particle to rotate around a point. This rotation manipulation has been demonstrated in D. D. Coster’s work [35].
In this paper, we also present the particle rotation manipulation based on the base ray misalignment of TFBGs. The detailed arrangements of the TFBGs for rotation manipulation with two- and four- TFBG trapping systems are shown in Fig. 13, where d represents the misalignment of the two base rays and the red dots are the center of the rotation motion.
Fig. 13. Arrangement of the TFBGs of rotation manipulation for two- and four- TFBGs scheme; left: levorotation, right: dextrorotation.
We model the optical forces as functions of particle positions for the rotation manipulation in the two- and four- TFBG schemes. Both the levorotation and the dextrorotation are demonstrated.
Figure 14 (a) and (b) show the optical forces versus particle positions of the two-TFBG scheme for the levorotation and the dextrorotation, respectively. In these designs, the original distance δz between the two TFBGs is 44r0 and the misalignment d is 5 µm. We can see that there still exists a force equilibrium region for each design. But different from stable trapping, as the particle deviates from the equilibrium point, the optical force would drive it to move round in an elliptical trajectory either in lefthanded or righthanded direction, instead of drag it back to the equilibrium point. To clearly observe the influence of optical force, a zoomed view of the red boxed region in Fig. 14 (a) is presented in Fig. 14 (c). The direction of the optical force around the equilibrium point indicates that although the rotation manipulation can be achieved, the particle would still move back to the rotation center at last. Therefore, the particle would still be trapped finally.
Fig. 14. optical force as a function of particle position for rotation manipulation; (a)(d) levorotation, (b)(e) Dextrorotation for two- and four- TFBG scheme (See high resolution images in section S4 of Supplement 1); (c)(f) Zoomed view of the red region in (a) and (d) respectively.
The optical forces versus particle positions of four-TFBG scheme for the levorotation and the dextrorotation are plotted in Fig. 14(d) and (e), respectively, where δz = 60r0 and the d = 10 µm. Similar to the two-TFBG scheme, there also exists force equilibrium regions in these cases. As the particle deviates from the equilibrium point, it would be driven to move round in a circular trajectory in either lefthanded or righthanded direction. Different from the two-TFBG scheme, the optical force round the equilibrium region does not point to the equilibrium point. Therefore, the particle would never be dragged back and trapped. In this case, the levorotation or the dextrorotation can be achieved.
From the above analysis, we can achieve rotation manipulation by using the misalignment of TFBGs, where the four-TFBG scheme exhibits better rotation manipulation performance than two-TFBG scheme.
4. Conclusion
In this paper, an optical trapping and manipulation technology was proposed based on spatial diffraction of 45° TFBG. The line-shape-facula of the TFBG diffraction light enabled the TFBG trapping system to control massive dielectric particles. The light distribution of the spatial diffraction was analyzed by using the VCM. Meanwhile, a theoretical model to analyze the optical trapping force of TFBG based on the RTM was also established. Several optical trapping schemes was proposed and discussed in detail. The simulation results indicated that only the scheme with axisymmetric layout could achieve stable particle trapping. The trapping performances of the two- and four- TFBG schemes were comprehensively discussed. The trapping performance could be effectively improved by reducing the distance δz and increasing the input light amplitude. Furthermore, the optical force would increase with the increase of particle RI and radius, which could be adopted to design the particle sorting systems. Finally, the rotation manipulation, including the levorotation and the dextrorotation, based on these two schemes were demonstrated. Due to the robustness, capability of massive trapping and ease of fabrication, the proposed trapping and manipulation technology can be used in microplastic sensing, medical diagnosis, cellular surgery, drug delivery, biological research, etc.
Funding
National Natural Science Foundation of China (11774181, 11904180, 61875091, 61901233, 62105164); Natural Science Foundation of Tianjin City (19JCYBJC16700, 20JCQNJC01480, 21JCQNJC00210, 21JCYBJC00080); Fundamental Research Funds for the Central Universities (Nankai University); Shandong Postdoctoral Innovation Talent Support Plan (SDBX2020017); China Postdoctoral Science Foundation (2021M693004).
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.
Supplemental document
See Supplement 1 for supporting content.
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