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The surface integral equation (SIE) method is used for the computational study of radiation torque on arbitrarily shaped homogeneous particles. The Multilevel Fast Multipole Algorithm (MLFMA) is employed to reduce memory requirements and improve the capability of SIE. The resultant matrix equations are solved iteratively to obtain equivalent electric and magnetic currents. Then, radiation torque is computed using the vector flux of the pseudotensor over a spherical surface tightly enclosing the particle. We use, therefore, the analytical electromagnetic field expression for incident waves in the near region, instead of the far-field approximation. This avoids the error which may be caused when describing the incident beam. The numerical results of three kinds of non-spherical particles are presented to illustrate the validity and capability of the developed method. It is shown that our method can be applied to predict, in the rigorous sense, the torque from a beam of any shape on a particle of complex configuration with a size parameter as large as 650. The radiation torques on large ellipsoids are exemplified to show the performance of the method and to study the influence that different aspect ratios have on the results. Then, the code is used for the calculation of radiation torque on objects of complex shape including a biconcave cell-like particle and a motor with a non-smooth surface.
© 2015 Optical Society of America
1. Introduction
When a particle is illuminated by a beam of light, it experiences a force called the radiation pressure force (RPF) and/or a torque. Under the influence of the RPF generated by tightly focused laser beams, small particles can be trapped and moved to a desired location [1–3 ], while the orientation of the particles can be controlled through torque exerted by the beam [4–7 ]. The computation of RPF and torque is of growing interest due to its importance in practical applications, such as biological cell trapping [8], micromotor design [9] and laser based measurement techniques [10].
Theories for RPF and torque computations have already been developed using different approaches. When a particle is much smaller than the wavelength, the Rayleigh regime is concerned; so, RPF can be calculated using Rayleigh-Debye theory [11]. In order to enlarge the object size range, a suitable rigorous theory had to be developed. Hence, Gouesbet et al. proposed generalized Lorenz-Mie theory (GLMT) [12] for studies of RPF, or torque, exerted on particles of simple shape such as homogeneous spheres [13–15 ], multilayered spheres [16], spheroids [17,18] and infinitely long cylinders of circular cross section [19]. However, rigorous solutions to Maxwell’s equations exist only for a particle whose shape coincides with a specific coordinate system. Furthermore, even for large regularly shaped particles such as spheroids or ellipsoids, the numerical computation of the involved special function is yet another obstacle.
The use of numerical techniques is a possible way to overcome these limitations. Research on the predictions of RPF and torque exerted on non-spherical particles has been performed using the discrete dipole approximation (DDA) [20–29 ], the T -matrix method [30–35 ] and the finite difference time domain method (FDTD) [36–40 ]. The T -matrix method is efficient and applicable to relatively large particles; however, it usually relies on central expansions of the electromagnetic field in vector spherical wave functions (VSWF), and it is suitable for particles with minimal rotational symmetry. Both DDA and FDTD are volume discretized methods. They are flexible and robust for inhomogeneous, anisotropic particles, but the computational demands increase quickly and the calculable size of the particle is severely limited, especially for particles with a high refractive index. Approximate methods, such as ray optics, can provide reasonable results for large particles, but their precision is usually not good enough [5, 41–43 ].
When comparing the theoretical efforts of calculating RPF and torque for spherical particles, less work has been done for their large non-spherical counterparts, yet irregularly shaped objects such as nanotubes and nanorods are quite prevalent in biophysics, microfluidics, microelectronics and photonics [44–46 ]. Moreover, experimental research on non-spherical particles has found interesting and greatly different phenomena in the juxtaposition with the spherical scenario [5, 7, 43, 45, 47, 48]. Trapping and manipulation of large non-spherical particles is far less mastered in both experimental and theoretical research.
On the other hand, the Multilevel Fast Multipole Algorithm (MLFMA) enhanced surface integral equation (SIE) method has shown great potency when solving electromagnetic scattering problems [49–51 ]. Recently, we presented the MLFMA approach for computing light scattering, RPF and stress for large arbitrarily shaped homogeneous particles with a size parameter larger than 600 [52–54 ]. In this paper, MLFMA is applied to the study of radiation torque on arbitrarily shaped particles by integrating the dot product of the outwardly directed normal unit vector and the pseudotensor over a spherical surface tightly enclosing the particle. We use the accurately computed near region electromagnetic fields, since such treatment can help overcome the obstacle of getting the mathematical description of the electromagnetic field components. As an example, we will present the radiation torque exerted by a Gaussian beam on different kinds of non-spherical particles. Therefore, by following this method readers can easily deal with other beam types and/or particle shapes.
The rest of the paper is organized as follows: the formulation of the SIE with MLFMA is presented in Section 2. Then, in Section 3 we will describe how to compute the radiation torque. The numerical performance and capability of MLFMA will be investigated in Section 4, where radiation torques will be calculated on large spheroids, a biconcave cell-like particle as well as a non-smooth motor. Finally, Section 5 is devoted to the conclusions.
2. Combined tangential formulation with MLFMA
The integral equation approach is often preferred for homogeneous objects because it limits the discretization of the unknowns to the surface of the object and the discontinuous interfaces between different materials [50]. In this section, we will give a brief description of the SIE method for a homogeneous dielectric particle illuminated by an arbitrarily shaped beam. The coordinate system (O; xyz) is attached to the particle, while the incident beam is polarized in u and propagates along w within the beam coordinate system (O′; uvw). The origin of the beam, O′, in the particle system, is x 0 , y 0 and z 0. The relation between the axes of the two coordinate systems, (O, xyz) and (O′; uvw), are defined by conventional Euler angles (Fig. 1).
Fig. 1 Schematic of arbitrarily shaped homogeneous particle illuminated by a shaped beam and the definition of the Euler angles.
For a given homogeneous dielectric object, using either the equivalence principle or the vector form of Green’s theorem, we can formulate a set of integral equations to calculate the electric and magnetic fields (E, H) in terms of equivalent electric and magnetic currents (J, M). The boundary S of the dielectric body is taken as the equivalent surface, as shown in Fig. 1, with the incident beam denoted as (E i, H i) and the equivalent sources as (J, M). Four basic integral equations are expressed in the homogeneous medium: the electric and magnetic field integral equations outside the dielectric body, (EFIE-O) and (MFIE-O) respectively, as well as the electric and magnetic field integral equations inside the dielectric body, (EFIE-I) and (MFIE-I) [50]:
where Zl = (μl/εl)1/2 denote the wave impedances in medium l with l = 1, 2 respectively for medium outside and inside the object. The operators L l and K l are defined as: where , kl = ω(μl/εl)1/2, X is either the equivalent electric current J or the equivalent magnetic current M on the surface of the object, and In order to overcome the resonance problem, the combined field integral equation is used. We can do this by combining the electric and magnetic field integral equations, or the internal and external field integral equations. Various forms of this can be constructed, such as the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHW) equations [55], the Combined tangential formulation(CTF), the combined normal formulation (CNF), and the electric and magnetic current combined-field integral equations (JMCFIE) [51]. Among those forms, the nature of CTF equations is close to a first-kind integral, which has the best accuracy, especially when dealing with objects containing sharp edges, corners, or high dielectric constants. CTF can be expressed as follow: where t̂ is the tangential vector at any point on the surface. Eq. (8) can be discretized by first expanding (J, M) as: where Ns denotes the total number of edges on S and g i denotes the Rao, Wilton and Glisson (RWG) vector basis functions [56]. By substituting Eq. (9) into Eq. (8) and applying g i as the trial functions for the tangential field equations, a complete matrix equation system can be obtained. This numerical solution process is well known as the Method of Moments (MoM). The MoM is conventionally limited to dealing with electrically small dielectric objects due to the computational and storage complexity of O(N 2), where N is the number of unknowns. To circumvent this problem, the Multilevel Fast Multipole Algorithm (MLFMA) is employed to speed up the matrix-vector multiplication and reduce the memory requirements. By using the MLFMA, both the time and memory complexity can be reduced to O(NlogN) [49, 50].In MLFMA, interactions between the basis and testing elements can be divided into two types: near-field interactions and far-field interactions. Near-field interactions are computed directly, the same way as those in MoM, then stored as a sparse matrix. However, for far interactions we first construct a tree structure of levels by placing the scatterer in a cubic box and recursively dividing the computational domain into sub-boxes. Then, MLFMA calculates the interactions between the basis and testing elements, which are far from each other in a group-by-group manner consisting of three stages called aggregation, translation and disaggregation. Since both of the operators, L and K, are involved in the matrix equation of CTF, it can be deduced from Eq. (8) that we need to solve two types of multiplication, which can be expressed with the multipole expansion as follows [50]:
where the aggregation terms V 1l and V 2l, the disaggregation term V ′l, and the translation term Tl are explicitly expressed as: where I⃡ denotes the 3×3 unit dyad and the integral is evaluated on the unit sphere, k l = kl k̂. g i and g j are the basis functions at the ith and jth edges, which reside in the groups m or m′ centred at r m and r m′ respectively, and we note that r im = r i − r m, r jm′ = r j − r m′ and r mm′ = r m − r m′. denotes the spherical Hankel function of the second kind, P nl represents the Legendre polynomial of degree nl, and L is the number of multipole expansion terms. Then, MLFMA can be used to accelerate matrix-vector multiplication in the iterative solving process of the resultant equation matrix.3. Radiation Torque
When an arbitrarily shaped particle is illuminated by a given type of beam, the radiation torque exerted on the particle can be determined by integrating the dot product of the surface normal n̂ and the pseudotensor T⃡ × r over a surface enclosing the particle [57, 58]
where is the time averaged Maxwell stress tensor, the star * stands for conjugate, and E(r) and H(r) are total electromagnetic fields: with If we choose a virtual sphere that tightly encloses the particle, has a radius, rs, and its centre located in the scattering object, then Eq. (13) can be written as [57] where the electric and magnetic field components are evaluated on the surface of the virtual sphere. Usually, to be convenient, the radius rs is chosen to be infinite so that the asymptotic forms can be used for the special functions [18,57]. For time-harmonic plane waves, such work is straightforward since the wave field expressions rigorously satisfy the Maxwell equations throughout space [59]. However, for a shaped beam, such as a Gaussian, it is hard or even impossible to get the mathematical description of the electromagnetic fields that satisfy Maxwell’s equations in the same way. Hence, to avoid inaccuracy caused by the description of the incident electromagnetic fields, their analytical expression will be used. Since only the near fields are needed for radiation, the calculation of the incident electromagnetic fields in the far regions is avoided. Theoretically, the virtual sphere can be chosen arbitrarily; however, to avoid special treatment for dealing with the singularity of Green’s function (7) when r = r′, we choose a spherical surface very close to the outer boundary of the particle (minimum 0.1λ away from the particle surface S). The same analytical expressions for the incident wave are used for computing both the equivalent sources in SIE and the Maxwell stress tensor.Now, we need to deal with the spherical integral in Eq. (19). Among different kinds of numerical integral methods, the Gaussian surface integral is simple and with high accuracy. We choose an integral number NL to divide the interval [0, π], such that cosθ satisfies the Gauss Legendre quadrature rule, and 2NL points averagely in [0, 2π] for ϕ. Usually, NL is determined by
To show the precision and capability of our method for the computation of radiation torque, we use a focused Gaussian beam as an example, but other types of beams can be employed in a similar way. When the beam waist radius is much greater than the wavelength, the Davis first-order Gaussian beam description [60] has been found to be good enough. However, for tightly focused beams, one should consider the use of higher-order approximate expressions. In this paper, we adopt the Davis-Barton fifth-order approximation [61].
However, to calculate the equivalent source of the incident wave and the total electromagnetic field on the virtual sphere, we need the incident beam expressed in the particle coordinate system (O; xyz). The relation between the two systems can be obtained by performing one translational and three rotational operations according to the Euler angles, (α, β, γ) [62]. In the following section, we will apply the method presented above to evaluate the radiation torque exerted by a Gaussian beam on particles with various shapes.
4. Numerical results
Based on the algorithm described above, software for computing radiation torque has been realized in Fortran 95. Moreover, parts of the software are parallelized with shared memory multiprocessing programming-OpenMP. All the computations are performed on the render farm, ANTARES, located in “Centre de Ressources Informatiques de HAute-Normandie”(CRIHAN), France. Each node has a dual-processor hexa-core Intel Westmere EP at 2.8 GHz and a maximum of 96 GB DDR3 memory. Mesh density in the numerical realization is set to about 0.08 – 0.1 λ. The generalized minimal residual method (GMRES) iteration solver is employed for solving the final matrix equation system.
The radiation torque on the particle is proportional to the power of the incident beam. The results presented in this section are normalized according to the power of the Gaussian beam, given by [61]:
and the intensity I 0 is related to the amplitude of the electric field at the beam centre E 0 by .We now consider a triaxial ellipsoid of semi-axes a, b and c along the x, y and z directions respectively, as shown in Fig. 2. The shape of the particle is characterized by two aspect ratios, defined as κ 1 = b/a, κ 2 = c/a.
Firstly, we check the validity of our method and our code by comparing the radiation torque calculated using GLMT, by Xu et al. [18]. Consider a spheroidal particle with a refractive index of m = 1.573 + 6.0 × 10−4 i being illuminated by a Gaussian beam. The radiation torque as a function of the incident angle β computed by MLFMA is presented in Fig. 3. A comparison of the results for κ 2 = 1.01 and κ 2 = 1.10 with those of Xu et al. (Fig. 3 in [18]) shows that the agreement is very good.
Fig. 3 Comparison of the radiation torque on two prolate particles (m = 1.573 + 6.0 × 10−4 i) of aspect ratios κ 2 = 1.01 and κ 2 = 1.10 computed using our approach. The wavelength and the beam waist radius of the Gaussian beam are λ = 0.785 μm and w 0 = 1.0 μm respectively. The centre of the particle coincides with the beam centre. The spheroids have the same volume as a sphere of radius r = 1.0 μm (Fig. 3 in [18]).
Now, we focus on the influence that the aspect ratio of ellipsoids has on radiation torque. To this end, we choose polystyrene particles (m = 1.59) located in water (m = 1.33) with different aspect ratios. In the following computations, all the ellipsoids have the same volume as a sphere of radius 4.7622 μm. The wavelength and the beam waist radius of the incident Gaussian beam are λ = 0.5145 μm and 1.3 μm respectively. The beam centre is located at x 0 = y 0 = z 0 = 0. We fix α = γ = 0° and let the Gaussian beam be incident on the particle at different angles β. In such conditions, the axis of the incident beam remains in the (x, z) plane. Since there are no torque components which make the particle rotate along the x or z axes, only the y component will be plotted. The computed torque, as a function of incident angle β for prolate spheroids with different aspect ratios, is shown in Fig. 4. We note first that, as expected, the torque on a spherical particle κ 1 = κ 2 = 1.0, is zero whatever the incident angle, due to its symmetry. Then, from the figure we find that, similar to the spheroids with small aspect ratio, the torque is always positive for a prolate spheroid. Such a phenomenon can easily be understood, because optically trapped objects have been found to align with their major axis along the direction of the laser beam under rotational torque. The maximum torque value appears at an incident angle smaller than 45°. When the aspect ratio is very large, for example κ 2 = 4.0, two maximum torques appear, while for a prolate spheroid with a smaller aspect ratio, (κ 2 = 2.0), only one maximum torque appears.
Fig. 4 Radiation torque as a function of incident angle β on prolate spheroids (κ 1 = 1.0) with different aspect ratios κ 2. The polystyrene particle (m = 1.59) is submerged in water (m = 1.33) and illuminated by a Gaussian beam (λ = 0.5145 μm, w 0 = 1.3 μm). All the prolate spheroids have the same volume as a sphere with a radius of 4.7622 μm.
Now, we study the torques on oblate spheroids. We keep all the parameters of the incident beam the same as in Fig. 4, but k 2 is now less than unity. The computed torque as a function of incident angle, β, with different aspect ratios is shown in Fig. 5. Contrary to the prolate particles, the torque is always negative for an oblate spheroid. The maximum torques appear at an incident angle larger than 45°. We can also observe that two maximum torques appear when the aspect ratio is very small (κ 2 = 0.25).
Fig. 5 Radiation torque as a function of incident angle β on oblate spheroids (κ 1 = 1.0) with different aspect ratios κ 2. The polystyrene particle (m = 1.59) is submerged in water (m = 1.33) and illuminated by a Gaussian beam (λ = 0.5145 μm, w 0 = 1.3 μm). All the oblate spheroids have the same volume as a sphere with a radius of 4.7622 μm.
Next, we show a case corresponding to a potential experiment, i.e. the rotation of a spheroidal particle around its centre, or around one of its tips. The computations are carried out for an ellipsoid of aspect ratios κ 1 = 1.0 and κ 2 = 4.0 illuminated by a Gaussian beam under two different conditions. First, the beam centre coincides with the centre of the particle (x 0 = y 0 = z 0 = 0) and the incident beam rotates around its centre as shown in Fig. 6(a). Second, the incident beam rotates around the bottom tip of particle but the beam centre keeps a distance equal to the semi-axis c away from it, as shown in Fig. 6(b). The computed torques are shown in Fig. 7. We found that when the direction of the incident beam rotates around the centre of the particle (x 0 = y 0 = z 0 = 0), the radiation torque is always positive, which makes the prolate particle in this case rotate clockwise, so as to align its long axis along the incident beam axis, as presented in Fig. 6(a). In the case that the beam rotates around the bottom tip of the prolate spheroid (6(b)), if the incident angle is β = 0°, then the torque is zero because of symmetry of the particle. For all β ≠ 0° the radiation force will push the illuminated part of the particle, so the torque is always negative. This can also be understood by the fact that the lower part of the prolate particle (Fig. 6(b)) experiences larger radiation force than the upper part.
Fig. 7 Radiation torque on a spheroidal polystyrene particle (κ 1 = 1.0, κ 2 = 4.0, m = 1.59) in water (m = 1.33) illuminated by a Gaussian beam (λ = 0.5145 μm, w 0 = 1.3 μm). The incident Gaussian beam rotates in two different manners. The particle has the same volume as a sphere with a radius of 4.7622 μm.
Our approach can also be used to compute the radiation torque exerted on particles of refractive index smaller than the surrounding medium, such as bubbles. We consider a spheroidal air bubble submerged in water with all other parameters being the same as in Fig. 7. The radiation torques computed for the two kinds of rotation of the Gaussian beam are shown in Fig. 8. We find that when the incident Gaussian beam rotates around the centre of the particle, the torque for a bubble, (m < 1), is the opposite to that of a droplet, (m > 1). But in the bottom rotation case, the radiation torque changes in direction at about 57°, i.e. when the incident angle β is small, the particle experiences a positive torque, making it rotate counter-clockwise, while if the incident angle is big, then the particle experiences a negative torque.
Fig. 8 Radiation torque on a spheroidal bubble (κ 1 = 1.0, κ 2 = 4.0, m = 1.0) in water (m = 1.33) illuminated by a Gaussian beam (λ = 0.5145 μm, w 0 = 1.3 μm). All the other parameters are the same as those in Fig. 7.
The next study we present is the more general case of an ellipsoidal polystyrene particle (m = 1.59) of aspect ratios κ 1 = 3.0 and κ 2 = 4.5, illuminated by a Gaussian beam with α = 45°, γ = 0°, and a variable β angle. The beam centre is at x 0 = 0, y 0 = z 0 = 0. The three components of the radiation torque as a function of the incident angle, β, are plotted in Fig. 9. We observe that when β = 0 all three components are zero due to the symmetry, while when β = 90° both the x and y components are zero, but z component is negative. In other general cases the three components of the torque are non-zero.
Fig. 9 Radiation torque as a function of the incident angle, β, on an ellipsoid with aspect ratios κ 1 = 3.0 and κ 2 = 4.5. The particle is made of polystyrene(m = 1.59) and is submerged in water (m = 1.33) whilst being illuminated by a focused Gaussian beam (λ = 0.5145 μm, w 0 = 1.3 μm). The Euler angles, α and γ, of the incident wave are set to 45° and 0° respectively.
To show the applicability of our method, we now examine the radiation torque exerted by a Gaussian beam on two kinds of non-spherical particles: a symmetric, biconcave, red blood cell-like particle and a regular motor. The first one is of smooth surface, while the second one has edges.
The shape of a symmetric, biconcave, red blood cell-like particle (m = 1.4) can be described by a simple function [53]:
In our calculation, we use the same parameters as Yu et al [63], i.e. a = 3.8, b = 0.41, q = 9, and we suppose that the red blood cell is submerged in water (m = 1.33). The centre of a strongly focused Gaussian beam w 0 = 2 μm coincides with the centre of the particle. The corresponding radiation torques for two wavelengths as functions of the incident angle, β, are shown in Fig. 10. It can be observed that the radiation torques on the biconcave cell are always negative and that their profiles are similar to those of an oblate particle.Fig. 10 Comparison of the radiation torque on the biconcave, cell-like particle in water. The waist radius of the Gaussian beam is w 0 = 2 μm. The diameter of the particle is d = 8.419 μm (xy plane), with the maximum and the minimum thickness being hM = 1.765 μm and hm = 0.718 μm.
Lastly, we apply our method to the computation of the radiation torque on a regular motor (Fig. 11) made of polystyrene (m = 1.58) and composed of 12 identical rectangular fans, with a hole at its centre. The hole is 2 μm in diameter. Each fan is 10 μm in length, 1 μm in width and 2 μm in thickness (in the x direction). The motor is submerged in water (m = 1.33) as the Gaussian beam illuminates it along the z axis, with the beam centre at x = 0 and z = 10 μm, but moving along the y axis [9]. The radiation torques exerted on the micro motor versus the position of the beam centre along the y axis are shown in Fig. 12. The torque is found to be symmetrical with respect to the y 0 = 0 axis. Because of the symmetric nature of the structure, the y and z components are always zero.
Fig. 12 Torque versus a varying offset in y evaluated for a motor (m = 1.58) in water (m = 1.33). The incident beam is set to λ = 1.07 μm, w 0 = 3.6 μm and kept at a constant distance z = 10 μm from centre of the motor in its direction of propagation. The motor is 2 μm in height (x – axis), with diameter 10 μm (yz – plane).
5. Conclusions
In this paper, the SIE method is used to compute the torque on arbitrarily shaped homogeneous particles. Triangular patches are used to discretize the outer surface of the particle, which makes this method especially flexible and efficient for modelling irregularly shaped particles. MLFMA is employed to reduce the computational and storage complexity. The RPF and torque are computed by integrating the dot product of the surface normal and Maxwell’s stress tensor over an area enclosing the particle. Furthermore, the analytical electromagnetic field expression in the near-field region is utilised. The presented method is validated and its capability illustrated in several characteristic examples. Torque on an arbitrary particle by other kinds of shaped beam are currently being studied further.
Acknowledgments
The authors acknowledge the support from the China Scholarship Council for the stay of Minglin Yang at CORIA. This work has also been partially supported with substantial computation facilities from CRIHAN (Centre de Ressources Informatiques de Haute-Normandie), the French National Research Agency under the grant ANR-13-BS090008-01 (AMO-COPS), the National Basic Research Program (973) under Grants 2012CB720702, 61320601-1 and 111 Project of China under the grant B14010.
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