Under the framework of generalized Lorenz-Mie theory, we calculate the radiation force and torque exerted on a chiral sphere by a Gaussian beam. The theory and codes for axial radiation force are verified when the chiral sphere degenerates into an isotropic sphere. We discuss the influence of a chirality parameter on the radiation force and torque. Linearly and circularly polarized incident Gaussian beams are considered, and the corresponding radiation forces and torques are compared and analyzed. The polarization of the incident beam considerably influences radiation force of a chiral sphere. In trapping a chiral sphere, therefore, the polarization of incident beams should be chosen in accordance with the chirality. Unlike polarization, variation of chirality slightly affects radiation torque, except when the imaginary part of the chirality parameter is considered.
©2013 Optical Society of America
Chiral media were first discovered as optical active media in the 19th century for the rotation of the polarization plane of a linearly polarized light that travels through them. Subsequent studies reveal that chiral microstructures are present in these media. The solutions that contain chiral molecules, such as grape sugar and tartaric acid, are optical active media. By imbedding micro chiral objects, such as metallic helices, randomly in an ordinary dielectric, researchers manufactured chiral material in microwave region. Accordingly, many biological particles may be modeled by chiral particles as they contain chiral microstructures inside.
Since Ashkin reported optical levitation  and optical trapping [2, 3], the techniques have been extensively applied in many fields. Especially in biology, optical tweezers are widely used for manipulating and classifying biological particles as they handle the particles without mechanical contact. Considerable effort has been devoted to studying the theory of interactions between lasers and particles. Researchers have developed several theoretical models for calculating the radiation force and torque exerted on a particle by a laser beam. The geometrical optics (GO) model, first used by Roosen  and later by many other researchers [5–10], conveniently calculates radiation force as it treats a beam as a bundle of rays. However, GO model adapts only to large particles with sizes much larger than the incident wavelength. For particles with sizes much smaller than the wavelength, the Rayleigh regime is appropriate . Although primarily proposed as beam scattering theory , the generalized Lorenz Mie theory (GLMT) was applied to calculate radiation force by Gouesbet , Ren [14, 15] and Lock [16, 17]. Kim [18, 19] and Barton  studied the radiation force by using a similar method in the 1980s. As analytical solutions, GLMT-based models generate more accurate numerical results than GO model and Rayleigh model and can be used for spheres of arbitrary sizes, though calculating a very large sphere may be time consuming. The numerical methods, such as Extended Boundary Condition Method (EBCM, i.e. T-matrix method) [21, 22], finite difference time-domain method , and discrete-dipole approximation , are also used in calculating radiation forces exerted on particles.
So far, most of the studies on radiation force and torque generally consider particles made of isotropic media, which does not always correspond with reality. Effects of different media should be taken into account. We calculated and analyzed the radiation force and torque exerted on a uniaxial anisotropic sphere by a Gaussian beam in [25, 26]. Ambrosio studied the radiation pressure imposed on spherical particles of negative refractive index [27, 28]. The radiation force acting on a chiral sphere in a circularly polarized plane wave was analyzed by Guzatov . This paper discusses the radiation force and torque exerted on a chiral sphere by a Gaussian beam, particularly the effects of the chirality and polarization of the beam. We hope that the study may lend support to research on the optical trapping of biological particles.
The theoretical scheme of this paper is based on GLMT. The problem of plane wave scattering from a chiral sphere was first solved by Bohren , who expanded the internal field of a chiral sphere in terms of spherical vector wave functions (SVWFs). In our previous work, we extended the theory and proposed improved algorithms for beam scattering from chiral spheres . The improved algorithms are applicable to arbitrarily shaped beam scattering if the expansion coefficients of the incident wave are available. The formulations of radiation force and torque are derived from the momentum theorem of electromagnetic field, as is done by Kim  and Barton . Section 2 presents the improved algorithms of beam scattering by a chiral sphere and formulations of radiation force and torque. Section 3 presents numerical results of radiation forces and torque exerted on the chiral sphere, including numerical verification of the theory and the numerical analysis. Section 4 is the conclusion. A time dependence exp(-iωt) is assumed throughout this paper.
2. Theoretical formulations
We summarize the theory of beam scattering by a chiral sphere in Section 2.1. The improved expressions of scattering coefficients, developed in our previous work, are presented. Then, the expansion coefficients of a Gaussian beam are introduced on the basis of literature. In Section 2.2, the expressions of radiation force and torque with respect to the scattering coefficients and expansion coefficients of the incident beam are presented in accordance with the momentum theorem of electromagnetic field.
2.1. Scattering of a Gaussian beam by a chiral sphere
Consider that a chiral sphere of radius is illuminated by a Gaussian beam propagating along the z-axis. The media of the chiral sphere can be described by the following constitutive relations:
On the basis of Bohren’s method  and our previous work , we provide a brief introduction to beam scattering by a chiral sphere. The internal field of a chiral sphere can be expanded in terms of SVWFs  in the following forms:33–35]. The other notations in Eqs. (9)-(12) represent , , , and .
Now consider the expansion coefficients of the incident Gaussian beam. We use a convenient expression presented in  by Doicu and Wriedt for the Davis Gaussian beam . The expression is derived by using a localized approximation for on-axis beams associating with the translational addition theorem for spherical vector wave functions [36, 38]. For an x-polarized Gaussian beam (x-polarized at the waist) propagating in the positive z direction with a beam waist radius and a beam center relative to the sphere center, the expansion coefficients are [36, 39]
If the incident Gaussian beam is on-axis (), Eq. (15) will degenerate into a simple form and will be more convenient in numerical calculations. Note that the above formulations may be inapplicable to a strongly focused beam as a paraxial approximation is used in Davis’s beam theory. If necessary, expansion coefficients of an arbitrary beam, such as a strongly focused beam and a Bessel beam, can be used here. For years many researchers [40–42] have been working on the expansion of an arbitrary beam.
The expansion coefficients of a y-polarized Gaussian beam can be readily obtained from those of an x-polarized Gaussian beam: , . Thus, the coefficients of an RCP Gaussian beam with the same power can be obtained as follows:
2.2. Expressions of radiation force and torque
The radiation force and torque on a sphere illuminated by a beam were studied by many scholars several decades ago [13–20]. On the basis of their methods, we derived the expressions of radiation force and torque with respect to expansion coefficients of incident beam and scattering coefficients [25, 26]. According to electrodynamics, the radiation force exerted on the sphere is equal to the average rate of change in momentum obtained from the beam. This force can be expressed as
Suppose S is a large spherical surface with the center at the origin. Thus, becomes the unit radial vector and the radiation force becomesEqs. (4)-(7) into Eq. (21), as well as using several approximations, yields the radiation force as follows :26]:
3. Numerical results and discussions
According to the expressions above, radiation force and torque are calculated numerically in this section. We focus primarily on the effects of the chirality parameter and beam polarization on the radiation force and torque. Therefore, for simplicity, the permittivity, permeability and chirality parameter are set as real quantities in the succeeding calculations, unless we want to examine the effects of their imaginary parts. In this section, the beam center is regarded as the coordinate origin and the sphere center is located at .
3.1. Verification of radiation force
When the chirality parameter vanishes, the chiral sphere degenerates into an isotropic sphere. In this case, the expressions and codes can be verified by comparing the results with those from literature. Figure 1 shows a comparison of the axial radiation force between our numerical results and those in . The triangles and the line with squares denote the experimental data measured by Bakker Schut  by using static method and those measured by using dynamic method, respectively. The line with stars denotes Schut’s theoretical results, obtained by using the GO model. The red solid line represents results calculated using our method. The parameters used in Fig. 1 are as follows: the refractive index of the sphere is ; the refractive index of the surrounding medium is ; the radius of the sphere is a = 3.75 µm; the power of the x-polarized incident Gaussian beam is P = 0.1 W; the wavelength of the beam in vacuum is λ = 0.488 µm; and the beam waist radius is w0 = 1.8 µm. The experimental results and our numerical results are in good agreement, confirming the validity of the theory and codes in this paper.
3.2. Analysis of radiation force
When the sphere is located on the z-axis, it always experiences a zero transverse radiation force. Figure 2 shows axial radiation force Fz versus sphere location z0 for different chirality parameters, assuming a sphere with radius a = 1.5 μm and refractive index n = 1.59 in a surrounding medium with refractive index n0 = 1.33. The beam center of the x-polarized incident Gaussian beam with wavelength λ = 0.488 μm in vacuum, beam waist radius w0 = 0.5 μm, and power P = 0.1 W is located at the origin. The chirality parameter of the sphere is −0.3, 0.0, 0.3, 0.5, and 0.7, respectively. As shown in Fig. 2, the axial radiation force curve of a chiral sphere is similar to that of an isotropic one. As the sphere moves from the negative z-axis to the positive z-axis, the axial radiation force initially decreases and then increases, during which a stable equilibrium point that can trap the sphere stably appears if negative force occurs. However, all the axial radiation forces in Fig. 2 are positive; thus, the sphere cannot be trapped by this single beam. It seems that a large chirality parameter increases the axial radiation force. Therefore, realizing a negative axial radiation force for a chiral sphere may be more difficult. We find that the curves with chirality parameters −0.3 and 0.3 are coincident. The succeeding numerical results will continue to focus on this and we will try to give some explanations then.
The case of transverse radiation force versus x0 is depicted in Fig. 3 for the same conditions as Fig. 2, except that the sphere now moves along the x-axis. The x-component Fx of transverse radiation force is shown in Fig. 3(a) and the y-component Fy in Fig. 3(b). Figure 3(a) shows that as the chirality parameter increases, Fx varies from positive to negative when the sphere is located on the negative x-axis, and varies from negative to positive when the sphere is on positive x-axis. These findings suggest that transverse radiation force Fx toward the beam axis (z-axis) becomes a repellant force that pushes the sphere away from the z-axis when the chirality parameter is large enough. Similar to the case of axial radiation force in Fig. 2, the curves of transverse radiation force Fx versus x0 with chirality parameters −0.3 and 0.3 are coincident. As we known, the y-component of the radiation force Fy of an isotropic sphere at x-axis is always zero. However, as shown in Fig. 3(b), the absolute value of Fy for a chiral sphere does not vanish but increases as the chirality parameter increases. This result indicates that a chiral sphere at the x-axis experiences a force that pushes it away from the x-axis. Besides, in contrast to the Fx and Fz curves, the Fy curves with chirality parameters −0.3 and 0.3 are not equal but opposite. Thus, the corresponding spheres will be pushed toward different y-directions. We conclude from Fig. 3 that trapping a chiral sphere transversely may be more difficult than trapping an isotropic one.
Radiation forces, as functions of the absolute value of the chirality parameter |κ|, are shown in Figs. 4 and 5 for differently polarized beam incidences. The x-polarized, LCP, and RCP incident Gaussian beams are denoted by “XP,” “LP,” and “RP” in the figures, respectively. Figures 4(a) and 4(b) present axial radiation force Fz versus the chirality parameter at sphere locations of (0 μm, 0 μm, 0 μm) and (0 μm, 0 μm, 10 μm), respectively. Figure 5(a) shows transverse radiation force Fx for a sphere located at (1.0 μm, 0 μm, 0 μm), and Fig. 5(b) shows transverse radiation force Fy for a sphere located at (0.5 μm, 0 μm, 0 μm). Chirality parameter κ varies from −0.9 to 0.9. The other parameters are the same as those above. It can be observed that the axial radiation forces on a chiral sphere are no longer the same for different circularly polarized incident beams. For the chiral sphere located at the beam center, as shown in Fig. 4(a), Fz basically increases with the chirality parameter in all cases. However, Fig. 4(b) shows that for the RCP beam incidence with a negative chirality parameter and the LCP beam incidence with a positive chirality parameter, Fz decreases to its minimum and then increases as chirality |κ| varies from 0.0 to 0.9. For a chiral sphere with an appropriate chirality parameter, therefore, it may be easier to realize a negative axial radiation force by using a circularly polarized beam. Surprisingly, the other Fz curves in Fig. 4(b) show obvious oscillations, which may be due to the special scattering properties of a chiral sphere for circularly polarized incident waves. Similar curves of scattering cross-sections versus the size parameter of a chiral sphere are found in .
From Fig. 5(a) a phenomenon similar to that in Fig. 3(a) can be observed that for the x-polarized beam incidence, RCP beam incidence with negative chirality, and LCP beam incidence with positive chirality, the transverse force Fx on a chiral sphere at x0 = 1.0 μm varies from negative to positive as |κ| increases. In these cases, the chiral sphere with a sufficiently large chirality parameter is pushed away from the x-axis, as the situation shown in Fig. 3(a). However, for the RCP beam incidence with positive chirality, and the LCP beam incidence with negative chirality, Fx is always negative, implying that the force always pulls the chiral sphere toward the beam axis. Thus, to transversely trap a chiral sphere effectively, we should use a beam with appropriate polarization in accordance with the chirality. As shown in Fig. 5(b), a chiral sphere at the x-axis experiences a force in the y-direction for all three types of polarized beam incidences (XP, RP, and LP). As analyzed in Fig. 3(b), this force disrupts the transverse trapping of the sphere, especially when the chirality parameter is large.
We now focus on the radiation forces exerted on chiral spheres with opposite chirality parameters and make an attempt to understand them. Figures 4 and 5 show that, for an RCP beam that illuminates a sphere with chirality parameter κ and a LCP incident beam with -κ, the Fxs and Fzs are equal, respectively, but the Fys are opposite. For an x-polarized beam incidence with opposite chirality parameters, the phenomenon observed is consistent with that depicted in Figs. 2 and 3. It seems that symmetry exists between the radiation forces on chiral spheres with opposite chirality parameters. In fact, as introduced in Section 2, the relationship between the wave numbers of the RCP and LCP waves in chiral media is , which indeed exhibits symmetry with respect to opposite chirality parameters. Thus, an RCP wave scattering from a chiral sphere with chirality parameter κ is symmetric physically to a LCP wave scattering from a chiral sphere with chirality parameter -κ. Therefore, the scattering fields in the two cases are the same in magnitude but contrary in RCP components and LCP components. This attribute may account for the symmetry of radiation forces discussed above.
3.3. Analysis of radiation torque
Figure 6 presents the axial radiation torques exerted on on-axis () chiral and isotropic spheres as functions of axial position z0 for differently polarized Gaussian beam incidences. The parameters are as follows: the refractive index of the sphere is n = 1.59 + 0.0003i; the refractive index of the surrounding medium is n = 1.33; the radius of the sphere is a = 1.5 µm; the power of the incident Gaussian beam is P = 0.1 W; the wavelength of the beam in vacuum is λ = 0.488 µm; and the beam waist radius is w0 = 1.0 μm. It can be observed that in contrast to radiation force, the axial radiation torque increases to its maximum at z0 = 0 and then decreases as the sphere moves from the negative to the positive z-axis. The radiation torques on a chiral sphere with κ = 0.2 and an isotropic one (κ = 0) are almost the same, indicating that the effect of chirality parameter on axial radiation torque is minimal. Instead, polarization of the incident beam has a great influence on the torque. RCP beam incidence and LCP beam incidence rotates the sphere toward different direction, respectively. However, if the chirality parameter is a complex quantity (κ = 0.2 + 0.0002i, for example), as shown in the figure, the curve will be quite different from that for κ = 0.2. Loss of a sphere considerably influences on radiation force and torque. According to the expressions of the wave numbers in chiral media and , the imaginary part of a chirality parameter also affects the absorption characteristics of the media. Thus, the imaginary part of chirality parameter exerts noticeable effects on the radiation torque.
Figure 7 shows three components of radiation torque versus transverse position x0 for differently polarized beam incidences. All the parameters are the same as previously stated, except for the refractive index of the sphere, which is n = 1.59 + 0.0001i. It can be seen that axial torque Nz increases to its maximum at x0 = 0 and then decreases as the sphere moves from the negative to the positive x-axis. However, values of Nz are tiny, compared with transverse torque Nx or Ny. Again it is found that a real-valued chirality parameter minimally influences both axial and transverse radiation torques. For both a general isotropic sphere and a chiral sphere located at the x-axis in an RCP or a LCP beam, the sphere experiences a transverse torque in the x-direction. The sign of torque Nx depends on beam polarization and sphere location. The curves of transverse torque Ny in all cases are close to one another and have large values compared with Nx and Nz as Nys are attributed to the asymmetric intensity of the beam with respect to the sphere.
This paper presents the theory and numerical results for radiation force and torque exerted on a chiral sphere by a Gaussian beam. In some cases, the chirality parameter may enhance the axial radiation force and reduce the transverse force toward the beam axis. Thus, trapping a chiral sphere may be more difficult than trapping a general isotropic sphere. As the scattering patterns of a chiral sphere differ for RCP and LCP beam incidences, polarization of the beam strongly influences radiation force. Thus, an appropriately polarized beam should be considered in trapping a chiral sphere. Symmetry may exist between the radiation forces on chiral spheres with opposite chirality parameters. It is found that variation of a real-valued chirality parameter slightly affects the radiation torque. However, if a complex-valued chirality parameter is considered, its imaginary part noticeably influences the torque.
The authors gratefully acknowledge supports from the National Natural Science Foundation of China under Grant No. 61172031 and the Fundamental Research Funds for the Central Universities.
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