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Abstract
Manipulation of light energy flow within the tight focus not only is important to the fundamental study of light-matter interactions but also underpins significant practical applications. However, the coupling between the electric and the magnetic fields of a focused light beam sets a fundamental barrier for independent control of these field components, restricting the focal energy flow primarily in the axial direction. In this paper, a 4π microscopic configuration is theoretically proposed to untangle the tight relation between the electric field and the magnetic field in a subwavelength-scale focal voxel. By independently altering the amplitudes of different field components in the focal region, energy flow with three-dimensionally unlimited orientation and ultra-high orientation purity (more than 90%) can be generated. This result expands the flexibility of energy flow manipulations and holds great potential in nanophotonics such as light scattering and optical force at subwavelength dimensions.
© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Being a characteristic attribute of light, energy flow, as the name implies, reveals the propagation of the light energy. It can be denoted by the Poynting vector S which is normal to the electric field E and the magnetic field H in the corresponding space. Except for its fundamental physical meaning, energy flow also plays a pivotal role in the applications related to light scattering and optical force [1–5]. For example, energy flow can be applied to manipulate and transport absorbing particles [6,7]. In an optical tweezer system, the motion trajectories of the absorbing particles coincide with the orientations of the energy flow, and the velocities of the particle motion are proportional to the corresponding flow modulus [8–10]. Therefore, manipulation of light energy flow is not only fundamentally interesting, but has significant practical application value.
For paraxial light beams, the orientation of the energy flow is mainly consistent with the propagating direction of the beams, which is known as the longitudinal energy flow, due to the nearly negligible longitudinal field components of the paraxial electromagnetic field. With the rapid development of cylindrical vector beams and tight focusing manners, advantageous depolarization effect [11] becomes pronounced in the focal region of focal lenses with high numerical apertures (NAs). In this scenario, strong longitudinal field components can be generated and the corresponding field energy can be constrained in a volume with a smaller lateral size [12–16]. With these merits, a number of more complicated behaviors of energy flow at the subwavelength scale have been unveiled, such as transverse toroidal energy flow formed by strongly focused vortex beams, Poincaré beams, light beams with hybrid polarizations, etc. [8,9,17–22], longitudinal energy backflow [14,23–27] circulating around the infinitesimal singularities, and curved trajectories of both transverse and longitudinal energy flows in nano-interferometric field structures [28].
In order to further control the orientation of energy flow at will, more flexible design of the electromagnetic field is necessary. However, the tailoring of the electric field is always accompanied by some undesired changes of the magnetic field and vice versa, which indicates that independent adjustments of the electric field and the magnetic field are unprocurable under the conventional focal condition. Therefore, arbitrary orientation manipulation of energy flow still remains challenging.
4π microscopic configuration which has been applied to generate various functional light field structures, such as steerable photonic spin structures in three dimensions [29], provides a potential way to control electric field and magnetic field separately in the focal region. With this configuration, independent manipulations of orthogonally polarized electric-field components have been demonstrated by designing the constructive or destructive interferences of different field components [30]. In this paper, two counter-propagating vector beams consisting of coherently configured linear and radial components are tightly focused under the 4π focal condition. Through controlling the constructive or destructive interferences of the electric-field and the magnetic-field components independently, the orientation of the energy flow can be manipulated arbitrarily in three dimensions at the subwavelength scale. Furthermore, the flow orientation purity is higher than 90% within the focal volume defined by the full widths at half maximum (FWHMs).
2. Theoretical simulation
As shown in Figs. 1(a)–1(c), the magnetic field (Hx) and the electric field (Ey and Ez) in the center of the focal region can be modulated separately by configuring the linear and radial components of the counter-propagating incident beams under the 4π focal condition. The incident electric field (Ein), magnetic field (Hin), and the propagating direction of each incident beam satisfy the right-handed spiral relationship. High-purity Hx and Ey can be obtained by focusing the linear components of the counter-propagating incident beams with an electric-field phase difference of π and 0, respectively. Ez can be similarly obtained by focusing the radially polarized electric-field components with a phase difference of π. Figures 1(d)–1(f) indicate the relationship between the orientation of the energy flow, denoted by the Poynting vector S, and the corresponding configured electromagnetic field. β is intersection angle between S and the z direction. Eyz is the total electric field in the center of the focal region which can be obtained by the vector composition of Ey and Ez and its orientation can be tuned arbitrarily in the y-z plane by altering the amplitude ratio between Ey and Ez. As the orientation of the magnetic field (Hx) is fixed in the x direction, the orientation of time-averaged Poynting vector < S > in the center of the focal region can be tuned to any direction in the y-z plane correspondingly according to the relationship [31]
Fig. 1. Schematic illustration of the manipulation of energy flow with arbitrary three-dimensional orientations by controlling the electric-field and magnetic-field components independently under the 4π microscopic configuration. (a)-(c) The generation of high-purity Hx, Ey, and Ez components in the focal region by tailoring the linear and radial components of the counter-propagating incident beams. (d)-(f) The relationship between the orientation of the Poynting vector S and the configured electromagnetic field.
Based on the Debye diffraction theory [11,32–35], the focal electric fields and magnetic fields of the forward-propagating incident beams (incident from the left-hand side) as shown in Figs. 1(a)–1(c) can be expressed as
The polarization properties of the incident beams from the left-hand side in Figs. 1(a) and 1(b) are the same, and the corresponding focal electric field and magnetic field can be expressed by E1f and H1f, respectively. E2f and H2f indicate the focal electric field and magnetic field of the forward-propagating radially polarized incident beam as shown in Fig. 1(c), respectively. The focal lenses are supposed to be illuminated with plane waves and hence A is a complex constant. μ0 is the permeability of vacuum and c is the speed of light in vacuum. r, φ, and z are the cylindrical coordinates in the focal region. θ is the converging angle, which varies from 0 to α. α is the maximum converging angle of the focal lenses which is equal to arcsin(NA). k is the wave vector of the beams. J0, J1, and J2 denote the Bessel functions of the first kind. Equations (2) and (3) are represented by the unit base vectors ex, ey, and ez, while Eqs. (4) and (5) are represented by er, eφ, and ez.
According to the interferential characteristic of the light field in the focal region of a 4π microscopic configuration [30,36,37], the total electric field and magnetic field can be expressed as
When C2 = 0 and C1 = C3, which indicate Ey = 0 and Eyz = Ez, high-purity uniformly oriented transverse energy flow is obtained as shown in Fig. 2. Figures 2(a) and 2(b) show the normalized distributions of the energy density of the electromagnetic field and the corresponding time-averaged Poynting vector in the y-z plane which passes through the center of the focal region, respectively, when NA = 0.95. The energy density W = 1/2(ε0|Ef|2 + μ0|Hf|2) [38], where ε0 is the permittivity of vacuum. As shown in Fig. 2(b), the projections of the orientations of < S > on the y-z plane shown as the white streamlines are along the y direction in the central region, namely β = 90°. Cross sections of the normalized |<S>| along the x, y, and z axes are illustrated in Fig. 2(c), which are denoted by the black, red, and blue curves, respectively. |<S>| is the modulus of < S > and |<S>|max is the maximum of |<S>|. It can be seen that the FWHMs of |<S>| along these three axes are 0.45λ, 0.84λ, and 0.3λ, respectively, which are below the wavelength of the incident electromagnetic waves. Especially, the FWHM along the z axis is far beyond the diffraction limit. Due to the depolarization effect, the orientation of < S > beyond the center of the focal region may not be along the y direction strictly. In this regard, the orientation purity of < S > which can be expressed as |< S >β|2/|<S>|2 is calculated. < S >β = <S > 90 or < S>y in this scenario. Figure 2(d) shows the dependence of the orientation purity of < S > on the values of the contour surface of |<S>|. It can be seen that the orientation purity is higher than 90% within a volume where |<S>|/|<S>|max ≥ 0.1. The orientation purity is still higher than 80% even at places where |<S>|/|<S>|max < 0.1. This indicates that high-purity uniformly orientated transverse energy flow can be obtained at the subwavelength scale in whole directions through the proposed 4π microscopic configuration.
Fig. 2. Generation of high-purity uniformly oriented transverse energy flow when NA = 0.95. (a) The normalized distribution of the energy density of the electromagnetic field in the y-z plane which passes through the center of the focal region when C2 = 0 and C1 = C3. The FWHMs of the electromagnetic field distribution along the x, y, and z axes are 0.76λ, 0.68λ, and 0.76λ, respectively. (b) The normalized distribution of the corresponding time-averaged Poynting vector < S > in the y-z plane which passes through the center of the focal region. (c) Cross sections of the normalized |<S>| along the x, y, and z axes. (d) Dependence of the orientation purity of < S > on the values of the contour surface of |<S>|.
In order to demonstrate the validity of generating arbitrarily oriented energy flow by this 4π method, normalized distributions of < S > with other orientations are calculated in detail. The orientation angle β can be tuned continuously from 0° to 90° by altering the amplitude ratio between Ez and Ey from 0 to +∞, as tanβ = |Ez|/|Ey|. Figures 3(a)–3(f) show the normalized distributions of < S > with β = 0°, 15°, 30°, 45°, 60°, and 75°, respectively, when NA = 0.95. When Ez = 0 and Eyz = Ey, namely C3 = 0 and C1 = C2, the 4π configuration degrades into a single y-polarized incident beam as illustrated in Figs. 1(a) and 1(b) and the energy flow orients consistently with the propagating direction of the incident beam as shown in Fig. 3(a). By increasing the proportion of Ez, the in-plane component of < S > increases accordingly and the orientation of < S > can be altered gradually as shown in Figs. 3(b)–3(f). C1 = C3 in this changing process, except for the state as shown in Fig. 3(a), to ensure the constructed energy flow with the expected orientation is dominantly distributed in the focal region.
Fig. 3. The normalized distributions of < S > in the y-z plane which passes through the center of the focal region for (a) β = 0°, (b) β = 15°, (c) β = 30°, (d) β = 45°, (e) β = 60°, and (f) β = 75° by altering the amplitude ratio between Ey and Ez, when NA = 0.95.
Figure 4(a) shows the dependences of the FWHMs of < S > along the x, y, and z axes on the orientation angle β, which are denoted by the black, red, and blue curves, respectively, when NA = 0.95. It can be seen that the FWHMs of < S > along the x and z directions decrease with the increment of β. Especially, they decrease below the diffraction limit (< 0.5λ) when β > 30° and β > 65°, respectively. Although the FWHM along the y direction is diffraction-limited and increases with the increment of β, it keeps at the subwavelength scale for any β. Especially, when β = 90°, at which angle the FWHM along the y direction reaches its maximum, the focal volume defined by the FWHMs of < S > is still smaller than that of the conventional diffraction-limited focal light field which can be estimated by (4π/3)(0.5λ)3. Figure 4(b) shows the dependence of the orientation purity of < S > on the orientation angle and the values of the contour surface of |<S>| when NA = 0.95. It can be seen that the orientation purity is higher than 90% within a volume where |<S>|/|<S>|max ≥ 0.125 for any β. It is still higher than 80% approximately even at places where |<S>|/|<S>|max < 0.125. Especially, the orientation purity is higher than 90% within the whole distribution of < S > when β ≤ 50° approximately. Figure 4(c) shows the dependence of the orientation purity of < S > on the orientation angle and NA when |<S>|/|<S>|max = 0.5. It can be seen that the orientation purity is higher than 90% when NA ≥ 0.75 for any β or under any NA condition when β ≤ 30° approximately.
Fig. 4. (a) Dependences of the FWHMs of < S > along the x, y, and z axes on the orientation angle (β) of < S > when NA = 0.95. (b) Dependence of the orientation purity of < S > on the orientation angle and the values of the contour surface of |<S>| when NA = 0.95. (c) Dependence of the orientation purity of < S > on the orientation angle and NA when |<S>|/|<S>|max = 0.5.
3. Conclusion
In conclusion, three-dimensional orientation manipulation of light energy flow at the subwavelength has been demonstrated through tightly focusing two counter-propagating vector beams under the 4π microscopic configuration. By controlling the electric field and the magnetic field in the focal region independently, arbitrarily oriented energy flow with high orientation purity (more than 90% within the volume defined by the FWHMs of |<S>|) can be constructed. Especially, the FWHM of transversely oriented < S > along the longitudinal direction is far beyond the diffraction limit. These results are important in manipulations of uniformly oriented energy flow at small dimensions. As energy flow may be associated with mechanical properties in the light-mater interactions, manipulation of energy flow is hence intriguing in the field of optical force and may provide a potential way to enrich the movement behaviors of micro-/nano-sized particles controlled by tightly focused light fields in optical tweezer systems. Apart from conventional far-field optical tweezers [39], near-field plasmonic tweezers are also powerful tools for manipulating nanostructures and materials with higher precision [35,40,41]. In this case, the corresponding energy flow distribution may also contribute to the total trapping manipulations. Therefore, energy flow manipulations of plasmonic waves would be another interesting topic in the field of optical tweezers, which is worthy of study in the future.
Funding
Ministry of Science and Technology of the People's Republic of China (MOST) (2016YFA0300802, 2018YFE0109200); National Natural Science Foundation of China (NSFC) (61975066, 62075085, 62005097); Guangdong Basic and Applied Basic Research Foundation (2019A1515010864, 2021A1515011586, 2020B1515020058, 2020A1515011529); Guangzhou Science and Technology Program (202002030258); Fundamental Research Funds for the Central Universities (21620413, 21620446); Guangdong Provincial Innovation and Entrepreneurship Project (2016ZT06D081); Science and Technology Planning Project of Guangzhou (202007010002); Guangzhou Basic and Applied Basic Research Project (Ph.D. Young Scientists) (202102020999).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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