After decades of development, laser technology has spread widely into a variety of applications like atmospheric detection, biotechnology, chemical technology, laser optical tweezers and laser radar detection. In order to precisely describe and manipulate lasers, the demand for an in depth understanding of the interaction between laser beam and material has become increasingly urgent. The chiral particle, especially the chiral sphere, has become one of the widely used theoretical models due to its ability to accurately describe light-particle interactions. Many macromolecules and biological structures in nature, including haze and aerosol, are, in general, composed of chiral media, which leads to multiple intriguing effects when interacting with light. Light scattering of symmetric chiral sphere under different forms of light beams is important for laser radar transmission mode selection, target recognition, microwave technology, atmospheric science, laser tweezers and laser field acquisition efficiency research. Using vector wave functions, Bohren analyzed the scattering of plane waves by spheres with inherent optical activity.^[1] Since then, structured laser beams have been utilized to investigate various fields with practical significance, like optical capture and manipulation,^[2] atomic optics,^[3] and imaging.^[4] The scattering of Hermite–Gaussian beams from the chiral sphere has been studied by calculating the multipole field under the incidence of conventional Hermite-Gaussian beams.^[5] The analytical solutions for the scattering of chiral spheres irradiated by Gaussian and Laguerre–Gaussian beams have been documented.^[6–8] In all these papers, the incident electromagnetic (EM) fields were expanded into an infinite series of partial or spherical vector wave functions (SVWFs), which are, however, usually difficult to obtain, especially for a large number of structured laser beams. In this paper, a general expression of EM field scattered by a chiral sphere being irradiated by an arbitrary monochromatic laser beam is given, for which the incident EM field only needs to be explicitly described.

The rest of this paper is organized as follows. In Section 2, a theoretical calculation method for the scattered field of chiral spheres irradiated by monochromatic laser is proposed. In Section 3, numerical results of the normalized internal and near-surface field intensity distribution (FID) are given for Gaussian, Hermite-Gaussian, doughnut mode and zero-order Bessel beams (ZOBB). The effects of different light fields on propagation are analyzed. Finally, we draw some conclusions in Section 4.

As a schematic diagram, Figure 1 shows a chiral sphere irradiated by a monochromatic laser beam, which propagates in free space and along the positive axis in the Cartesian coordinate system (laser beam coordinate system). A chiral sphere is attached to the system Oxyz (parallel to ), with origin O located at ( ) in . In this paper, the time-dependent part of the EM field is assumed to be .

Fig. 1.

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	Fig. 1. Chiral sphere illuminated by monochromatic laser beam.

The scattered fields by the chiral sphere can be expanded by using the SVWFs of the third kind with respect to the system Oxyz in the following form: 1 2 where and are the free space wave number and characteristic impedance, respectively, and a_mn and b_mn are the expansion coefficients.

The constitutive relations of a chiral medium can be described by^[6,7] 3 4 where κ, , and are the chirality parameter, relative permeability, and permittivity of chiral medium, respectively.

As discussed in Refs. [6–8], the SVWFs of the first kind can be used to expand the fields within the chiral sphere (internal fields) as follows: 5 6 where and .

The SVWFs of the first or third kind in Eqs. (1), (2), and Eqs. (5) and (6) indicate that the radial dependence functions in them are respectively the spherical Bessel or Hankel functions of the first kind.^[9]

The boundary conditions over the sphere surface can be represented by 7 where and are the incident electric and magnetic field, and r₀ is the sphere radius.

Substituting Eqs. (1), (2), (5), (6) into Eq. (7), we have 8 9

By substituting spherical harmonic functions into the vector spherical harmonics (VSHs) denoted as and , the latter can be written as 10 11

If equations (8) and (9) are multiplied (dot product) by the VSHs and , and then are integrated over the sphere surface (the projection method), we can obtain 12 13 14 15 where , and 16 17 18 19 20 21 In deriving Eqs. (12)–(15), the following formula are used: 22 23

The validity of theoretical procedures above can be easily proved. The incident EM fields are expanded, as usual, in terms of the SVWFs of the first kind as follows: 24 25

By substituting Eqs. (24) and (25) into Eqs. (12)–(15) and then considering Eqs. (22) and (23), the relationship can be readily obtained among the expansion coefficients of the incident laser beam, internal and scattered fields, which is equivalent to the exact analytical solution following the generalized Lorenz–Mie theory (GLMT),^[6,10,11], i.e., Eqs. (20)–(23) in Ref. [6].

The integrations over the sphere surface in Eqs. (18)–(21) are numerically evaluated, e.g., by Simpson’s 13 rule in our programming with MATLAB, for which the integration expressions are provided for two examples as shown below. 26 27 where and are respectively the θ and ϕ component of .

From Eqs. (12)–(15), the expansion coefficients can be determined as follows: 28 where 29 30 31 32 33 34 35

By substituting a_mn and b_mn into Eqs. (1) and (2), and c_mn and d_mn into Eqs. (5) and (6), the scattered field and the internal fields can be obtained.

In this paper, the normalized internal and near-surface FID are calculated, which are defined by 36 and 37 where the subscripts r, θ, and ϕ denote the r, θ, and ϕ components of the electric fields, respectively.

When studying the scattering of laser beams by spherical objects,^[6–8] the usual analytical method needs to expand the incident beams with SVWFs. As common laser beams, such as Gaussian beam, Hermite–Gaussian beam, Laguerre–Gaussian beam or Bessel beam, can be only described approximately to a higher degree to satisfy Maxwell equations, which makes them very difficult expanded with SVWFs. According to previous researches, in this paper we present the projection method, i.e., multiplying both sides of the EM boundary conditions by VSHs and integrating them on the spherical surface. It is only necessary to know the expression of the beam description, rather than to obtain the SVWF expansion of the beam. The results obtained by the projection method are in agreement with those obtained by the analytical method, which have been verified by the theoretical and numerical results of Gaussian beams with known SVWF expansions. The scattering of common beams (Gaussian beam, Hermite–Gaussian beam, Laguerre–Gaussian beam, and Bessel beam) by chiral spherical objects is calculated. Unlike the scattered far-field reported in most of published papers, the near-field and internal-field characteristics of the chiral sphere irradiated by EM beam are studied in detail in this paper. These properties are very important for studying the properties of chiral media, and also for manipulating the optical tweezers and imaging the particles with super-resolution.

For the incidence of a fundamental Gaussian beam (TEM mode focused laser beam) described by the Davis–Barton third-order corrected expressions (see Appendix A),^[12–14] the normalized internal and near-surface FIDs for a chiral sphere are shown in Fig. 2. Numerical results show that the incident Gaussian beam is focused within the chiral sphere when propagating. Comparing with Fig. 2(b), the chiral sphere in Fig. 2(c) deviates from the Gaussian beam center. Although the focal position is unchanged within the chiral sphere, the normalized FIDs become asymmetrical and the maximum intensity is reduced by about 70%, which is suitable for the stable capture of particles in the optical tweezer system.

Fig. 2.

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	Fig. 2. (a) Normalized FID for TEM mode focused laser beam ( ) in plane. Normalized FIDs in plane for chiral sphere ( , , , illuminated by the above focused laser beam respectively with (b) ( ) and (c) ( , ).

As discussed in Ref. [14], expressions of EM components for a higher-order Hermite–Gaussian beam can be derived by taking transverse derivatives of TEM as

Figure 3 shows the normalized internal and near-surface FID for a chiral sphere under the illumination of a TEM mode Hermite–Gaussian beam. It is demonstrated in Fig. 3(c) that the FID profile is symmetrical about origin , while for the case of a dielectric sphere the FID profile is symmetrical about the axis .^[13]

Fig. 3.

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	Fig. 3. (a) Normalized FID for TEM mode Hermite–Gaussian beam ( ) in the plane. Normalized FIDs (b) in the plane and (c) in the plane for the same model as in Fig. 2 ( , , , ), illuminated by the above Hermite–Gaussian beam ( ).

Any Laguerre–Gaussian beam can be expressed in a sum of Hermite–Gaussian beams,^[14] in which various doughnut mode beams are of particular interest. Figure 4 shows the normalized internal and near-surface FID for a chiral sphere illuminated by a radial doughnut mode beam created by TEM . Comparing with the normalized FID in Fig. 4(a) for the doughnut mode beam, in Fig. 4(c) periodical max and min field distributions along the radius direction arise in the FID.

Fig. 4.

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	Fig. 4. (a) Normalized FID for TEM doughnut mode beam ( ) in plane. Normalized FID (b) in the plane and (c) in the plane for the same model as in Fig. 2 ( , , , ), illuminated by the above doughnut mode beam ( ).

Figures 2(a)–4(a) display the intensity distributions for different types of beams. For each case, the beams described by Figs. 2(a)–4(a) have the same total power, and the beam waist is assumed to be . Figure 2(a) shows that the maximum intensity occurs at the point ( , and Fig. 3(a) at the point ( , ) and Fig. 4(a) at a radius of from origin .

The EM field components of a ZOBB propagating along the positive axis in have been given in ^[15] and ^[16]. Figure 5 shows the normalized internal and near-surface FID in the plane for a chiral sphere illuminated by a ZOBB with a half-cone angle ψ. From Fig. 5 we can see that the focal region outside the chiral sphere is far from the chiral sphere, and more focal regions appear as well. Thus, the gradient distribution of field intensity is more evident, which is beneficial to cancelling the scattering force and realizing particle capture.

Fig. 5.

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	Fig. 5. (a) Normalized FID in plane for ZOBB. (b) Normalized FID in plane for the same model as in Fig. 22 ( , , , ), for chiral sphere illuminated by ZOBB ( , ).

In Figs. 3–5 the chiral sphere is located in the middle of incident EM field. As a result, a symmetrical distribution of the field intensity on both sides of the axis is observed. The numerical results based on Fig. 3 show that the focused intensity of Hermite-Gaussian beam decreases and its intensity gradient increases obviously when compared with the scenarios of Gaussian beam. The symmetrical distribution of the light field on both sides of the central axis is beneficial to the formation of a balanced light field. Numerical results based on Fig. 4 show that the intensity of focused beam is between the intensity of Hermite–Gaussian beam and the intensity of Gaussian beam, and the gradient distribution of intensity is stronger than that of Gaussian beam and weaker than that of Hermite–Gaussian beam. Numerical results based on Fig. 5 show that compared with the fundamental Gaussian laser beam, Hermite–Gaussian beam and doughnut mode beam, the ZOBB beam has a focal region that is located in the center of chiral sphere, and the intensity of the light field decreases. For a ZOBB, the gradient distribution of the intensity is more obvious, and the resultant gradient force potential trap enables us to capture the particle conveniently.

According to the field expansions in terms of the SVWFs, EM boundary conditions and projection method, a general approach to the accurate calculation of scattering from a chiral sphere is presented and verified. The normalized FIDs of internal and near-surface field are calculated for a variety of structured beams, e.g., the fundamental Gaussian beam, Hermite–Gaussian beam, doughnut mode beam and ZOBB. The focusing position and intensity distribution due to the illumination of different light field are analyzed. This work provides a general solution to arbitrarily structured laser beam scattering from a chiral sphere, and is of significance in relevant fields of atmospheric detection, biotechnology, chemical technology, laser optical tweezers, laser radar detection, etc.

Third-order corrected expressions for the EM field components of a fundamental Gaussian laser beam (TEM mode focused beam) in , developed by Barton and Alexander,^[12–14] are written as A1 A2 A3 A4 A5 A6 where , , , , , , and are the non-dimensional parameters, and w₀ is the beam waist radius.

By considering the duality principle, i.e., replacing in Eqs. (A1)–(A6) by , by , by , and by , third-order corrected expressions for a TEM mode focused Gaussian beam can be conveniently obtained.