On the basis of spherical vector wave functions and coordinate rotation theory, the expansion of the fields of an incident Gaussian beam with arbitrary propagation and polarization directions in terms of spherical vector wave functions is investigated, and beam shape coefficients are derived. Using the results of electromagnetic scattering by a uniaxial anisotropic sphere, the analytical expressions of the radiation force and torque exerted on a homogeneous absorbing uniaxial anisotropic sphere by the arbitrary incident Gaussian beam. We numerically analyze and discuss the following: the effects of an anisotropic absorbing dielectric on the axial and transverse radiation forces exerted by an off-axis Gaussian beam on a uniaxial anisotropic sphere; the variations in the axial, transverse, and resultant radiation forces (with incident angle β and polarization angle α) imposed by an obliquely Gaussian beam on a uniaxial anisotropic sphere; and the results on the characteristics of the three components of the radiation forces versus the center-to-center distance between the sphere and beam. Selected numerically results on the radiation torque exerted on a stationary uniaxial anisotropic transparent or absorbing sphere by a linearly polarized Gaussian beam are shown, and the results are compared with those exerted an isotropic sphere. The accuracy of the theory and code is confirmed by comparing the axial radiation forces with the results obtained from references.
© 2012 OSA
Since the pioneering work of Ashkin and associates [1–3], optical tweezers have attracted considerable attention and have been extensively applied in various fields. In principle, small particles (e.g., biological cells) with micrometer and sub-micrometer dimensions can be trapped and non-intrusively manipulated under the radiation force (RF) generated by tightly focused laser beams. Researchers have used various methods to compute the RF exerted on homogeneous isotropic spherical particles. For a particle much smaller than the incident wavelength (), the Rayleigh regime is considered, and RF can be calculated by Rayleigh dipole approximation (RDA) [4, 5]. By expanding the incident field into a plane wave spectrum, Rohrbach et al.  extended the validity range of Rayleigh theory to particle sizes as large ~0.7λ. Conversely, for a particle much larger than the incident wavelength (typically), the geometrical optics regime is considered; in this case, ray optics theory (ROT) is a highly applicable and efficient approach to calculating RF [7–12]. For a particle whose size is of incident wavelength order, both RDA and ROT are inapplicable because diffraction phenomena cannot be neglected in this case. To cover the entire d/λ range, a rigorous electromagnetic theory has been developed for a spherical particle interacting with a laser beam. In this theory, expanding an incident beam in terms of spherical vector wave functions (SVWFs) is necessary. Kim and Lee  used the complex-source-point method to expand the incident field components of Hermite–Gaussian laser beams, and calculated the optical potential well exerted on a homogeneous sphere. Using the surface integral method, Barton expanded the incident field components of a Gaussian beam  and studied the net RF and torque for a spherical particle . Gouesbet et al.  proposed generalized Lorez-Mie theory (GLMT), which employs a set of beam shape coefficients to describe the incident beam . Applying GLMT, Ren et al. [18, 19], Lock et al. , Nahmias et al. , and Martinot-Lagarde et al.  investigated the RF exerted on a spherical particle. Nahmias et al.  and Mao et al.  compared the RFs obtained by ROT and GLMT, and described the limitations of ROT, which confirm that GLMT is a complete beam-scattering theory for particles of all sizes.
Compared with studies on RF calculations for isotropic spherical particles, those devoted to RF prediction for anisotropic particles are considerably rare. Anisotropic medium has drawn increasing attention because of their wide applicability in optical signal processing, and microwave engineering, and so on. In past decades, many scholars have studied the interaction of a plane wave with anisotropic particles [25–29], especially with uniaxial anisotropic spherical particles [30–33]. Researchers from our group recently studied the scattering characteristic of a uniaxial anisotropic sphere illuminated by on-axis, off-axis, and arbitrarily incident Gaussian beams, and discussed the scattered, internal, and incident fields [34, 35]. The RF exerted on an anisotropic sphere may represent varied properties because of the differences in the internal fields of anisotropic and isotropic spheres. Thus, the RF exerted by a focused laser beam on an anisotropic spherical particle is also a critical and interesting study object. The authors have recently investigated the RF exerted by an off-axis Gaussian beam on a uniaxial anisotropic sphere . Since that a uniaxial anisotropic sphere has a primary optical axis, the internal and scattered fields created when the beam’s propagation direction is not parallel to the primary optical axis of the anisotropic sphere will be distinct from the fields created when the beam’s propagation direction is parallel to the primary optical axis. Thus, the RF exhibits different characteristics. Prediction of the RF exerted by an off-axis obliquely incident Gaussian beam on a uniaxial anisotropic sphere is still in its infancy, but is an endeavor of considerable merit. Comparing the RF, the performance the radiation torque causes rotation of a particle is also very useful and interesting phenomenon. Chang and Lee  and Barton  studied the radiation torque exerted on an isotropic sphere by a linearly and circular polarized Gaussian beam. Xu  investigated radiation torque exerted on a spheroid by a Gaussian beam. Because the internal field is made up of two EM waves with different velocities and polarization directions, the radiation torque exerted on a uniaxial anisotropic sphere by an arbitrary direction incident Gaussian beam will be not always zero like that exerted on an isotropic sphere. Thus, it is worthy to research this problem which is still solved.
In this paper, based on the GLMT  and coordinate rotation theory of SVWFs the field components of an incident Gaussian beam with arbitrary propagation and polarization directions are expanded in terms of SVWFs in a fixed rectangular coordinate system. According to the orthogonality of associated Legendre functions and trigonometric functions, we derive the analytical expressions of the transverse and axial RFs and radiation torque exerted on a uniaxial anisotropic sphere in the incident Gaussian beam. Numerical calculations of the radiation force and torque are carried out, and their physical interpretations are presented.
In the subsequent depiction, a time dependence of the form exp(-iωt) is assumed and suppressed, where ω is the circular frequency.
Consider a homogeneous uniaxial anisotropic sphere of radius a centrally located in a spherical coordinate system. The primary optical axis is coincident with the z-axis. As Fig. 1(a) shows, the particle is obliquely illuminated by an x'-polarized Gaussian beam that propagates in the z'-direction in Cartesian coordinate system O'x'y'z', with the beam center located at O'. The coordinates of beam center O' are (x0, y0, z0) in Oxyz. Temporary coordinate system Ox”y”z” is established parallel to O'x'y'z' and is known as the beam system. We denote the propagation direction of the beam as two angles in particle coordinate system Oxyz; that is, incident angle β with reference to the z-axis, and polarization angle α between the x-axis and the propagation direction of the beam on the xoy-plane. Then, Oxyz can be obtained by rotating Ox”y”z” with Euler angles β and α (Fig. 1(b)).
2.1. Description of the incident and scattered fields
In terms of SVWFs, the incident Gaussian beam can be expanded in temporary coordinate system Ox”y”z” as follows :34]. and are the beam shape coefficients obtained by applying the local approximation of GLMT in beam system [40, 41]:
Coefficient slightly differs from that described in [35, 42, 43] because the current study adopts the anti-clockwise rotation of the two coordinate systems, whereas in [35, 42, 43] the clockwise relationship is employed. Moreover, we use only Euler angles α and β to denotes that the incident direction of the Gaussian beam is arbitrary; that is, arbitrarily incident and polarization-oriented β and α, respectively. This depiction not only maintains generality, but also enables the simplification of the problem compared with the depiction in . Substituting Eq. (6) into Eq. (1), we derive the expanded form of the incident field in particle coordinate system Oxyz:Eq. (7).
The scattered fields can be expanded with the SVWFs as follows:
For a uniaxial anisotropic sphere whose permittivity and permeability tensors and are characterized by30, 34, 35]. Utilizing the continuity of the tangential electric and magnetic field components at r = a, the scattering coefficients can be obtained thus [30, 34]:35]. The expressions of coefficients , , , , , and are detailed in .
2.2. Derivation of radiation force
Assuming a steady-state condition, the RF exerted by the beam on the particle is proportional to the net momentum removed from the incident beam. It can be expressed as a close spherical surface integral surrounding particle :
Scattering theory indicates that for spherical particles, the axial components of the incident and scattering fields are very small and can be negligible in the far region. Then, the RF can be written as:
The numerical calculation of the surface integral of Eq. (16) can be avoided by substituting the series expressions for electromagnetic field components and using the approximate expressions of Ricatti-Bessel functions at :Eq. (16) can be directly integrated, and the net force on the particle can be expressed as a series over the expansion coefficients of the incident and scattered fields :
2.3. Derivation of radiation torque
In view of the conservation law of angular momentum, the radiation torque exerted on a uniaxial anisotropic spherical particle equals the average rate at which the angular momentum is transported into the particle. For a steady-state scattering problem, it can be expressed as an integral through the spherical surface centered at the center of the uniaxial anisotropic sphere [15, 37, 38]:
Similarly, substituting the incident and scattered fields into the outward EM fields , and applying the recursion and orthogonality relationships of associated Legendre functions and asymptotic formulae of spherical Bessel functions , the three components of the radiation torque exerted on a uniaxial anisotropic sphere can be derived:
3. Numerical calculations
On the basis of theory developed in the preceding section, we present our numerical results.
3.1. Validation of theory and codes
We perform two comparisons to verify the accuracy of our theory. The scattering coefficients or scattered fields are calculated and compared with the results obtained from  to verify the accuracy of our results. Figure 2 shows the axial RF from our theory, compared with the results in  in which the obliquely incident Gaussian beam is reduced to an on-axis incident Gaussian beam. The accuracy of the results in  is verified by comparison with the experimental results otained form . It can be found they are in excellent agreement, which confirms the validity of the theory and code in the present paper. , and denotes the distance of the sphere center from the beam center. The remaining figures possess similar conditions too. Moreover, the curve denoted by “w0 = 0.4 μm” indicates that the uniaxial anisotropic spherical particle can be captured by a focused Gaussian beam, similar to an isotropic sphere. The negative RF only indicates that the direction of the RF is negative but does not reflect the magnitude of the RF.
3.2. Effects of anisotropic absorbing dielectric
Figures 3(a) and 3(b) shows the variations of the axial RF with exerted on a uniaxial anisotropic sphere with different complex permittivity tensor elements. “A” and “B” denote the imaginary part of permittivity tensor elements and , respectively. It can be found that the influence of the imaginary part of on the axial RF is more sensitive than that of the imaginary part of . The same conclusion can be obtained from wave numbers and in the uniaxial anisotropic sphere . The influence of on the wave vectors in the uniaxial anisotropic sphere is larger than that of . Thus, the influence of on the scattered field and RF may be more sensitive than that of . The axial RF increases with the increase of the imaginary part of when the imaginary part of is invariable. The axial RF also increases with the increase of the imaginary part of when the imaginary part of is invariable. When the imaginary parts of and do not equal zero and increase, a great deal of photons are absorbed, thereby rapidly strengthening the scattering force. Thus, the RF quickly increases, so that the particle which can be originally captured is difficult to be captured.
The effects of the absorption of electric anisotropy on the variations of transverse RF with -x0 and with exerted by an off-axis Gaussian beam on a uniaxial anisotropic sphere are depicted in Fig. 4 . The figure also exhibits the same characteristics illustrated in Fig. 3; that is, the influence of on the scattered field and RF may be more sensitive than that of . When the imaginary part of is large (e.g., A = 0.2 in Figs. 4(a) and 4(b)), the transverse RF only slightly changes provided that the imaginary part of exhibits a little increase (e.g., B from 0.1 to 0.5 in Figs. 4(a) and 4(b)).
It can be seen that the uniaxial anisotropic sphere can be bound in the transverse direction when the anisotropic dielectric sphere is lossless. When the anisotropic dielectric sphere is lossy, the secondary extremum of the transverse RF occurs. Moreover, the value of the secondary extremum increases with the increasing imaginary parts of and , whereas the peak value of the transverse RF decreases with the increasing imaginary parts of and . Thus, when the imaginary parts of and are very large, the uniaxial anisotropic sphere may be not bound in the transverse direction (e.g., A = 0.2 and B = 0.5; A = 0.3 and B = 1.0 in Figs. 4(a) and 4(b)). The variations of the transverse RF with -x0 and with exerted on a uniaxial anisotropic sphere are very similar, since permittivity tensor element and permeability tensor element in the x-direction and y-direction are the same.
3.3. Effects of incident angle and polarization angle
Figure 5(a) shows that the three components of the RF and the resultant RF Fe change with incident angle β exerted on a uniaxial anisotropic sphere under only electric anisotropy. The resultant RF Fe is calculated by . The angles are varied only from 0° to 90° to keep the incident orientation under the xoy-plane. Transverse RF equals zero because polarization α also equals zero. Axial RF exhibits its maximum value at β = 36°, and its minimum value at β = 90°. Note that the minimum is not zero but –0.0742 pN, which results from the effect of . This characteristic is distinct from that of an isotropic sphere. For an isotropic sphere illuminated by a vertical Gaussian beam, axial RF , which is vertical to the propagation direction of the incident wave, is zero. In Fig. 5(a), transverse RF increases with increasing incident angle, and exhibits its maximum value at of β = 90°. The resultant RF has the maximum at β = 50°. Thus, we can conclude that the RF does not remain constant under different incident angles. Figure 5(b) shows the variations of the direction of the resultant RF Fe with the incident angle. and are formed by the direction of the resultant RF with the z-axis and x-axis, respectively. The azimuth angle is zero because transverse RF equals zero. Both transverse RFs and equal zero when β = 0°; in Fig. 5(a), therefore, the values of θ and are not given when β = 0°. is always smaller than the incident angle of the Gaussian beam. When axial RF is negative, is greater than 90°.
Similar to Fig. 5(a), Fig. 6(a) shows the variations of the three components of the RF and the resultant RF Fe with incident angle β exerted on a uniaxial anisotropic sphere with both electric and magnetic anisotropy. The comparison of Figs. 5(a) and 6(a) shows that transverse RF and axial RF are considerably larger when and are not equal to than those when . Moreover, the variations with the incident angle are also more complex. Under many incident angles, transverse RF and axial RF are negative. The variations of the direction of the resultant RF Fe with the incident angle are shown in Fig. 6(b). A comparison of Figs. 6(a) with 6(b) indicates that the resultant RF Fe has the maximum value of 3755.4 pN at β = 90°, and its orientation angle . That is, the maximal resultant RF is very large and occurs almost along the –z-axis. These properties, which are distinct from those of an isotropic sphere, may pertain only to the uniaxial anisotropic sphere and may be applicable to non-intrusive optical detection of optics material through giving the particle approach the surface of optics material a force vertical to the surface, thereby delivering or capturing the particle. Permittivity tensor elements and and permeability tensor elements and impose remarkable influence on the variations of the axial and transverse RFs with the incident angle. Thus, the differences in the permittivity tensor elements and permeability tensor elements may cause distinct variations of the RF with the incident angle.
The three RF components change with polarization angle α for different incident angles β exerted on a uniaxial anisotropic sphere are shown in Fig. 7 . Transverse RFs and equal zero when β = 0°, as expected, and they increase as a whole with increasing incident angle β. However, the shape of the curve exhibits little change. As polarization angle α increases, transverse RF decreases whereas increases. Transverse RFs and yield maximum values at β = 0° and β = 90°, respectively, and exhibit minimum values at β = 90° and β = 0°, respectively. That is, when the incident Gaussian beam is x-axis polarized, has the maximum value but equals zero; when the incident Gaussian beam is y-axis polarized, has the maximum value but equals zero. This phenomenon is highly reasonable. In addition, the variations of the transverse RFs and with the polarization angle are almost oppositely symmetrical. This result may be attributed to the identical permittivity tensor element and permeability tensor element along the x-direction and y-direction. Figure 7(c) shows that polarization imposes a minimal effect on axial RF because the change of the polarization angle causes almost no change in scattering along the propagation direction of the incident Gaussian beam. Note that the value of the axial RF when β = 90° is not zero but –0.10202 pN. This result is similar to that shown in Figs. 7(a) and 7(c).
TiO2, characterized by εt = 5.913ε0 and εz = 7.197ε0, is a typical uniaxial anisotropic medium. The transverse and axial RF exerted on a TiO2 sphere by a Gaussian beam with different incident angles as a function of the coordinates of beam center –x0 and –z0 are given in Figs. 8(a) and 8(b), respectively. Transverse RF equals zero because the polarization equals zero (not plotted here). As incident angle β increases, axial RF Fz decreases, whereas transverse RF generally increases as a whole. When β = 0°, that is, the Gaussian beam propagates along the z-axis, the TiO2 sphere can be captured in the direction of the x-axis (see the curve denoted by “Fx(β = 0°)” in Fig. 8(a)). However, the particle will not be captured in the z-axis direction (refer to the curve denoted by “Fz(β = 0°)” in Fig. 8(b)) because permittivity tensor elements and are very large, while the refractive index of the surrounding medium is small. A small beam waist width and large refractive index of the surrounding medium should be selected to capture a uniaxial anisotropic sphere similar to the TiO2 sphere. The same conclusion is drawn in . When β = 90°, that is, the Gaussian beam vertically illuminates the TiO2 sphere, the uniaxial anisotropic sphere can also be captured in the direction of the z-axis (see the curve denoted by “Fz(β = 90°)” in Fig. 8(b)); it is not captured in the direction of the x-axis (see the curve denoted by “Fx(β = 90°)” in Fig. 8(a)). When the TiO2 sphere is illuminated by an absolutely obliquely incident Gaussian beam (e.g., β = 45°), the TiO2 sphere is very difficult to capture in both the x-axis and z-axis directions.
3.4. Calculation and discussion of radiation torque
In Fig. 9 , x component of the radiation torque Tx is calculated when the beam center moves along y-axis. Here, the y and z components of the radiation torque zero due to the symmetry of the spherical particle, and are not plotted in Fig. 9. When εt = εz, the anisotropic sphere is reduced to an isotropic sphere. For lossless isotropic sphere, the x component of the radiation torque is always equal to zero, as it should be; while for lossless anisotropic sphere the x component of the radiation torque is not always equal to zero. This is because the internal field consists of two EM waves with different velocities and polarization directions, then the orbital angular momentum may be transferred to the anisotropic particle when propagation direction of the beam is not coincident with the primary optical axis of the anisotropic sphere. This radiation torque will cause the anisotropic to rotate with x-axis, thus, the primary optical axis will be changed so that the scattering field and radiation force will be changed. In this case, it may be difficult to capture the uniaxial anisotropic spherical particle in transverse direction. This performance on radiation force and torque is very different from that of an isotropic spherical particle.
For loss isotropic or anisotropic sphere, the orbital angular momentum may be transferred to the particle through absorption, resulting in an orbital torque. Comparing with lossless uniaxial anisotropic sphere, the radiation torques exerted on a positive(εt<εz) or negative(εt>εz) uniaxial anisotropic sphere increase in whole and are larger than those exerted on an lossless anisotropic sphere, and the secondary extremum is also eliminated. As the beam center moves away from the sphere center along y-axis, the lever-arm increases so that the torque increases. When the beam center locates at about the edge of the sphere, the radiation torque procures maxima. The position of the beam center when the maximal radiation torque occurs exerted on an anisotropic sphere is farther from the sphere center than that on an isotropic sphere. Less beam ling flux will incident on the sphere and less angular momentum will be transferred with the increasing of |y0|, thus the radiation torque decreases. Moreover, it can be found that the radiation torques Tx for both isotropic and anisotropic sphere are zero, as expected. .
Figure 10 shows the effect the incident angle β on the radiation torque. Because the spherical particle is illuminated by an on-axis Gaussian beam, the x and z components of the radiation torques are zero and not plotted in Fig. 10. When the anisotropic sphere is reduced to an isotropic sphere, the radiation torque is invariable and equal to zero in despite of the size of the incident angle (see case1). It can be observed that the negative radiation torque will cause the uniaxial anisotropic to rotate anticlockwise when the incident angle 0 <β<90 ; while the positive radiation torque will cause the uniaxial anisotropic to rotate clockwise when the incident angle 90 <β<180 . For the case that the sphere center and the beam center are the same, the distribution radiation torque is symmetrical with β = 90, and it equals zero when the uniaxial anisotropic sphere is vertically illuminated by a Gaussian beam resulting from the uniaxial anisotropic sphere is symmetrical with x-axis. For the case that z0≠0, namely, the sphere center and beam center are not the same, the radiation torque is not equal to zero(see case6) when the incident angle equals 90 resulting from that the symmetrical axis of the uniaxial anisotropic sphere and the propagation direction of the beam are not the same. It is also can be found that the radiation torque decreases in whole for loss both electrical anisotropic or electrical and magnetic anisotropic sphere compared with those lossless anisotropic sphere. It is different from that performance that the radiation torque changes with the location of the beam center along y-axis shown in Fig. 9.
The expansion expressions of an incident Gaussian beam with arbitrary propagation and polarization directions in terms of SVWFs in a fixed coordinate system are derived. The analytical expressions of the Radiation forces and torques exerted on a homogeneous absorbing uniaxial anisotropic sphere are obtained by the incident Gaussian beam. We numerically studied the following: the effects of the anisotropic absorbing dielectric on the variations of axial and transverse components of the RFs exerted on a uniaxial anisotropic sphere, as a function of the center-to-center distance between the beam and sphere; and the variations of the axial, transverse, and resultant RFs with the incident angle and polarization angle exerted by an obliquely Gaussian beam on a uniaxial anisotropic sphere. The numerical results exhibit some interesting phenomena, indicating that the properties of the RF exerted by an obliquely incident Gaussian beam on a uniaxial anisotropic sphere differ from those of the RF exerted on an isotropic sphere. The effects of the location of the beam center and incident angle on the radiation torque exerted on an transparent or absorbing uniaxial anisotropic sphere are numerically analyzed by a linearly polarized Gaussian beam, and the results are compared with those exerted on an isotropic sphere. Due to the length restriction, the more cases such the effects of the polarization model of the beam, size parameter of the particles and so on are not discussed will be studied in our future work. The developed theory and numerical code presented in this paper serve as useful tools for theoretical and experimental studies on trapping or manipulating uniaxial anisotropic spherical particles. RFs exerted by an arbitrary direction incident Gaussian beam on multiple uniaxial anisotropic spheres will be the subject of future research.
The authors gratefully acknowledge support from the National Natural Science Foundation of China under Grant Nos. 61172031 and 60971065 and Fundamental Research Funds for the Central Universities.
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