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The electromagnetic coupling and mechanical interaction between a plane wave and dispersive gain chiral structures are investigated using the Auxiliary Differential Equation Finite Difference Time Domain (ADE-FDTD) method. Utilizing the constitutive relations containing frequency-dependent Lorentzian models and a Condon model, the wave equations and time-averaged Lorentz force density for the magneto-electric coupling chiral media are presented. Numerical results show that the cross-polarized transmission coefficient is larger than the co-polarized transmission coefficient for a gain chiral slab with certain thickness. The gradient force engendered by bound currents of the cross-polarized waves in chiral media is larger than the scattering force to pull the slab towards the incident source. The complicated optical pulling or pushing force density among slabs, which is illuminated by a normally incident plane wave, containing chiral materials with different medium parameters is achieved.
© 2016 Optical Society of America
1. Introduction
Generally, the optical force resulting from the radiation pressure, exerted by the electromagnetic radiation, is usually described in terms of scattering, gradient, and binding force components. The scattering force tends to push and gradient force try to trap a particle [1–11]. Since the first experimental observation of Ashkin [1], the optical manipulation and trapping of particles have been a topic of great interest for four decades. Research on this subject [2,3] has included structured beams, cavity optomechanics, special materials, and geometries, etc. The optical trapping of microparticles can be realized by means of the broken mirror symmetry (namely, chirality) [4–11]. Moocarme et al. demonstrate the plasmon-induced Lorentz forces of nanowire chiral hybrid modes [5]. Ding et al. [8] realize the optical pulling force via the coupling of the linear momentum of a chiral particle with the spin angular momentum of light and Tkachenko et al. [10] discuss the mechanical separation of chiral dipoles by chiral light.
We consider various contributions to the electromagnetic force from the magnetoelectric coupling constitutive relations, dispersive properties. However, we find the radiation pressure is positive for wave propagation in homogeneous passive chiral media [12], even when both the real parts of the relative permittivity and permeability are negative. The numerical experiment results in this paper show that the radiation pressure can be negative as a gain chiral medium illuminated by a plane wave. Several studies have been focused on gain media with a positive imaginary part of a complex permittivity [12,13], whereas the wave propagation in and electromagnetic forces on gain chiral media with Im(ε)Im(µ)/(ε0µ0)<Im2(κ) have received little attention.
The Finite-Difference Time-Domain (FDTD) method [14–19] can be used to simulate wave propagation in natural [14], man-made, and effective [15] chiral media, compared with other analytical and numerical approaches, such as the Mie series and Method of Moment. Though the chiral media can be simulated by the Bi-Isotropic FDTD method [18], which obeys the conventional FDTD analysis techniques, the set of decomposed wavefields in dispersive chiral media computed with BI-FDTD method remain uncoupled. The fields and currents equations containing dispersive and magneto-electric coupling constitute relations can be directly obtained and discretized using the Auxiliary Differential Equation (ADE) FDTD method. The Maxwell stress tensor and Lorentz force are generally applied to calculate the optical forces based on the electromagnetic field distributions of particles. The radiation pressure on dielectric or magnetic dispersive achiral media [20,21] can be computed with the electromagnetic fields and induced electric current (or magnetic current). However, both induced and coupled electromagnetic fields and currents have to be implemented into calculating the Lorentz force on dispersive and magneto-electric coupling chiral media.
In this paper, a plane wave propagation and the Lorentz force density in gain chiral structures are researched with the ADE-FDTD method. Based on the magnetoelectric coupling constitutive relation containing dispersive Lorentzian and Condon models, wave equations and the Lorentz force density for electric and magnetic dispersive chiral media are given. After verifying the accuracy of the methods, the positive or negative radiation pressure on slabs containing chiral materials contributed by co- and cross-polarized waves is simulated.
2. Theory for chiral media
2.1 Constitutive relations
In this paper, we consider chiral media that are described by the following constitutive relations [22]:
in which D and B represent the electric and magnetic flux density vectors, respectively. E and H represent the electric and magnetic fields. The convention for the time harmonics, ejωt, is used. ε(ω), μ(ω), and κ(ω) are frequency-dispersive permittivity, permeability, and chirality parameter. The permittivity and permeability are characterized by the Lorentzian models, and the chirality parameter is characterized by a Condon model, that is In these equations, ε∞, μ∞, εs, and μs are the permittivity and permeability at infinite and zero frequencies, respectively. ωe, ωh, ωκ, ξe, ξh, and ξκ represent resonance angular frequencies and damping factors, respectively. τκ is a characteristic time constant measuring the magnitude of the chirality parameter.For a passive chiral medium, the imaginary parts of the permittivity, permeability, and chirality parameter satisfy the following conditions [22],
If the medium parameters violate any of the conditions in Eqs. (6), the chiral medium becomes a gain medium [12,13]. Natural large chiral multifunctionalized molecules [23], artificial chiral boron nitride nanotubes [24], and chiral metamaterials [25] with giant optical activity may be potential gain chiral material candidates.
2.2 1D ADE-FDTD formulation
By introducing the induced and coupled electric and magnetic currents, the electromagnetic field and current equations used to simulate the interaction between the electromagnetic wave and chiral media in the time domain are given as [15]
where J and K are induced electric and magnetic currents, Jc and Kc are coupled electric and magnetic currents. The field and current equations in Eq. (7) can be numerically solved via the ADE-FDTD method [7,15]. For the sake of simplicity, only a one dimensional plane wave propagating in chiral media along the z direction is considered in this paper. The fields Ex and Hy, as well as currents Ky, Jcy, Jx, and Kcx are the components of the incident co-polarized light, whereas Ey, Hx, Kx, Jcx, Jy, and Kcy are the components of the induced cross-polarized light. The Mur first order absorbing boundary condition is implemented in this paper. Both Ex and Ey are absorbed at the boundaries.2.3 Lorentz force density
Because the electromagnetic fields and currents are the functions of the time and space, the force density distribution in chiral media can be calculated by time-averaging the Lorentz force density in one period of the stable time harmonic electromagnetic fields as [20,21]
where the bound electric charge density is ρe_bound and magnetic charge density is ρm_bound. The bound electric current Je_bound and magnetic current Jm_bound areThe bound electric and magnetic currents for chiral media are affected by both the induced and coupled electric and magnetic currents, compared with normal electric or magnetic dispersive medium without coupled currents [21].
3. 1D FDTD simulation of the chiral structures
3.1 Verification of the ADE-FDTD method
In order to validate the accuracy of the ADE-FDTD method used to simulate the dispersive chiral medium, Fig. 1 shows the co-polarized and cross-polarized propagation properties of a chiral slab. The medium parameters for the 15mm-thickness slab are εs = 4.4, ε∞ = 3.5, μs = 1, μ∞ = 1, ωe = ωh = ωk = 2π × 8 GHz, ξe = ξh = 0.07ωe, ξk = 0.09, τk = 10−12 s. Very good agreement between the numerical results computed with the ADE-FDTD method in this paper and the exact results in [16] is obtained.
Fig. 1 Co- polarized reflection, co- and cross-polarized transmission coefficients of a chiral slab.
3.2 Electromagnetic wave and force densities in a gain chiral slab
In section 3.2 and 3.3, the center frequency of interest is chosen to be f0 = 468.75THz, corresponding to a free-space wavelength λ0 = 640 nm. The FDTD cell size is δ = 5 nm, time step Δt = δ/2c, and the electric field Ein = sin(2πf0t) V/m at normal incidence. The amplitude of the incident magnetic field is H0 = 2.6544 × 10−3A/m. The initial Poynting vector in the z direction is Sz = 0.5E0H0 = 1.3mW/m2.
Figure 2 illustrates the interaction between a plane wave and a 110nm-thickness chiral slab simulated with the ADE-FDTD method. The medium parameters for the slab in Eqs. (3)-(5) are chosen to be εs = 1.2, ε∞ = 1, μs = 1.2, μ∞ = 1, ωe = ωh = 2π × 250 THz, ωk = 2π × 320 THz, ξe = 0.07ωe, ξh = 0.07ωh, ξk = 0.1, τk = 4 × 10−16 s. The values of the relative permittivity εr, permeability μr, and chirality parameter κr at f0 = 468.75THz are approximately 0.92–j0.008, 0.92–j0.008, and –0.97–j0.25. The solid and dash lines in Fig. 2(a) plot the time histories of co- and cross-polarized electric fields in the middle of the chiral slab. The electric fields exhibit steady time-harmonic oscillations after several periods. The magnitude of the coupled electric field Ey is much smaller than that of the incident electric field Ex at z = 0.
Fig. 2 FDTD computed co-polarized and cross-polarized electric fields, scattering coefficients and Lorentz force densities Fz (per unit cross-sectional area) in a chiral slab. (a) versus timestep, in the middle of the chiral slab, (b) versus z, at time t = 18000Δt, (c) reflection and transmission coefficients, (d) force densities.
Figure 2(b) depicts the co- and cross-polarized electric fields propagating through the chiral slab suspended in the free space. The fields are measured at time t = 18000Δt. The purple dash lines marked in Fig. 2(b) specify the two faces of the slab and the green dash line denotes the connective boundary where the incident wave is generated. The amplitude of the co-polarized electric field Ex, which is smaller than that of the cross-polarized electric filed Ey after penetrating the chiral slab, obviously decreases versus the z axis. It is because the chiral slab in Fig. 2(b) is a gain medium with ε0μ0Im2(κ)>Im(μ)Im(ε), which differs from that with negative Im(μ) or Im(ε) [12,13].
The co- and cross-polarized reflection and transmission coefficients of the chiral slab are given in Fig. 2 (c). If the chirality parameter is set to be zero, the chiral slab computed in Fig. 2 will become an impedance matched medium to the free space. Because chiral medium is reciprocal, thus the polarization state of the reflected wave is nearly changed by the chiral slab [22]. The cross-polarized reflection coefficient is so small that it can be negligible. Because of ε0μ0Im2(κ)>Im(μ)Im(ε) [22], the cross-polarized transmission coefficient is larger than the co-polarized transmission coefficient. For one-dimensional electromagnetic problem, no induced bound electric and magnetic charges exist at the interfaces between two adjacent media as the chiral slab is illuminated by the normally incident plane wave [20]. Thus, the propagation characteristics of the electromagnetic wave in the chiral slab are simply influenced by the bound electric and magnetic currents. The mean values of the medium parameters, rather than those of the electromagnetic fields and currents, for two adjacent media are applied across the interfaces between the chiral medium and free space.
Figure 2 (d) shows that the time averaged co-polarized Lorentz force density is positive and the cross-polarized force density is negative in the chiral slab. Whereas both the absolute values of the co-polarized and cross-polarized spatial force densities first increase and then decrease in Fig. 2 (d), the negative net (co-polarized plus cross-polarized) force density keeps decreasing in the chiral slab along the z axis. The integrated net, co- and cross-polarized radiation pressure can be obtained by integrating three kinds of the local densities on the chiral slab and are ∫Fet(z)dz = –0.35 pN/(mW/cm2), ∫Fco(z)dz = 2.35 pN/(mW/cm2), and ∫Fcr(z)dz = –2.7 pN/(mW/cm2), respectively. As the electromagnetic wave propagate through the gain medium slab in Fig. 2 (b), cross-polarized waves are continuously coupled from the incident co-polarized plane wave because of the large optical activity of the chiral medium. Therefore, the gradient force engendered by the continuous increase of the cross-polarized waves can exceed the scattering force to attract the single gain slab towards the wave source in Fig. 2 (d).
3.3 Positive or negative radiation pressure on slabs containing chiral medium
The ADE-FDTD method simulated the distribution of net, co-polarized, and cross-polarized force densities in two 20nm-apart and 110nm-thickness dispersive chiral medium and normal dielectric medium with εr = 4 slabs are shown in Fig. 3(a). The medium parameters for the chiral slab in Fig. 3(a) are same to those in Fig. 2. The integrated net, co-polarized, and cross-polarized Lorentz force densities along the z axis for the first chiral slab are –0.49 pN/(mW/cm2), 2.97 pN/(mW/cm2), –3.46 pN/(mW/cm2) and the second dielectric slab are 2.2 pN/(mW/cm2), 0.6 pN/(mW/cm2), 1.6 pN/(mW/cm2). The negative radiation pressure is found to impose on the chiral slab, whereas the positive radiation pressure is exerted on the dielectric slab. The mutual electromagnetic force between the gain chiral slab and the dielectric slab is a pushing force. Even if the thickness of the second slab are very small (not shown in this paper), the radiation pressure on the second normal dielectric slab is still positive under the similar condition. That is, the cross-polarized wave generated by the chiral medium cannot attract a normal dielectric object.
Fig. 3 FDTD predicted co-polarized, cross-polarized and net force densities inside two slabs containing a chiral medium. (a) repulsive force densities, (b) attractive force densities.
The co- and cross-polarized transmission waves of single chiral slab are directly absorbed by the absorbing boundary of the FDTD method in Fig. 2. However, the co- and cross-polarized transmission waves of the first chiral slab are reflected by the second dielectric slab and reenter the first chiral slab in Fig. 3(a). Then the co- and cross-polarized incident wave and reflected wave mutually couple and superpose in the first chiral slab. One can find that the existence of the second dielectric slab in Fig. 3(a) makes the absolute values of the integrated net, co- and cross-polarized force densities of a single chiral slab in Fig. 2(d) increase. The radiation pressure of a single 110nm-thickness dielectric slab with εr = 4 at f0 = 468.75 THz is 2.5 pN/m2 in this paper and in [20], which validates the correctness of the Lorentz force density with the ADE-FDTD method. However, the radiation pressure acting on the same dielectric slab in Fig. 3(a) is 2.9 pN/m2, the increase is produced by the multiple reflections between the chiral and dielectric slabs.
Figure 3(b) presents the computed net, co- and cross-polarized force densities inside two 55nm-thickness and 25nm-apart chiral slabs suspended in the free space. Most of the medium parameters for the two slabs are same to those in Fig. 2 except the much smaller characteristic time constant τk = 1.8 × 10−16 s and relative chirality parameter κr = –0.43–j0.11 at f0 = 468.75 THz. The absolute values for the three kinds of force densities increase versus z axis as the plane wave propagates through the chiral slabs. The radiation pressure in the free space is zero. The positive integrated co-polarized force density is larger than the absolute value of the negative integrated cross-polarized force density in the first chiral slab; however, the former is smaller than the latter in the second chiral slab. As can be seen in Fig. 3(b), the force densities at the front face of the second chiral slab and the back face of the first chiral slab are continuous to a certain degree. The integrated net, co- and cross-polarized radiation pressure for the first chiral slab are 0.008 pN/(mW/cm2), 0.2 pN/(mW/cm2), –0.192 pN/(mW/cm2) and for the second chiral slab are –0.03 pN/(mW/cm2), 0.51 pN/(mW/cm2), –0.54 pN/(mW/cm2). As shown in Fig. 3(b), the electromagnetic interaction force between the chiral slabs is a pulling force. By appropriately selecting the thicknesses and medium parameters, a zero, repulsive, or attractive force between two chiral slabs can be got.
4. Concluding remarks
We present the interaction mechanism between a plane wave and dispersive gain chiral structures in this paper. By introducing the frequency-dependent electric permittivity, magnetic permeability, and chirality parameter, the time-averaged Lorentz force density exerted on magneto-electric coupling chiral media contributed by the bound electric and magnetic currents is presented and incorporated by the ADE-FDTD method. It should be noted that the bound current density gives rise to the induced electric current and coupled magnetic current. Simultaneously, the bound magnetic current arises from the induced magnetic current and coupled electric current in chiral media. Our research reveals that the negative force density keeps increasing as the cross-polarized waves are continuously coupled out by the chiral medium. The sign of the radiation pressure on a chiral slab is influenced by the medium parameters of the chiral and surrounding media. We believe that this work provides guidelines for manipulating gain chiral particles and delivers a contribution towards a comprehensive understanding of trapping forces produced by gain chiral media.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (41104097, 41304119, 61201007, and 11504252), the Specialized Research Fund for the Doctoral Program of Higher Education (20120185120012).
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