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Abstract
In this paper, we propose a novel (to our knowledge) vector beam by combining the radially polarized beams with the different polarization orders, which is called the grafted polarization vector beam (GPVB). Compared with the tight focusing of traditional cylindrical vector beams, GPVB can present more flexible focal field patterns by adjusting the polarization order of two (or more) grafted parts. Moreover, because the GPVB possesses the non-axisymmetrical polarization state distribution, which will lead to the spin-orbit coupling in its tight focusing, it can obtain the spatial separation of spin angular momentum (SAM) and orbital angular momentum (OAM) in the focal plane. The SAM and the OAM are well modulated by adjusting the polarization order of two (or more) grafted parts. Furthermore, we also find the on-axis energy flow in the tight focusing of the GPVB can be changed from positive to negative by adjusting its polarization order. Our results provide more modulation freedom and potential applications in optical tweezers and particles trapping.
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1. Introduction
The polarization plays an important role in optical field modulation and the light-matter interaction. In recent years, the polarization modulation has attracted much attentions owe to its extensive applications in optical information [1,2], near (or far) field optics [3,4], nonlinear optics [5,6], optical trapping and manipulation of particles [7–10].
The vector beams possess the spatial variant polarization state in its phase front [11–15]. The cylindrical vector beams, such as the radially (or azimuthally) polarized beam, is a kind of vector beam with the axisymmetric polarization state distribution. Because of the spin-orbit coupling in the tight focusing of the radially polarized beam, some interesting phenomena appear in the focal region, such as the SAM to OAM conversion [16–19], the on-axis negative energy flow [20–22], the spin-orbit Hall effect [23] etc., which have led to extensive applications, such as optical tweezers [24–26], optical chain [27], optical micro-fabrication [28], optical cage and needle [29,30], and optical micro-manipulation [31–33]. Besides the cylindrical vector beams, some non-axisymmetrical vector beams also become a hot research topic in recent years, such as the power-exponent azimuthal-variant vector beams [34,35], the fractional order radially polarized beam [36,37], and both of these beams have the non-axisymmetrical polarization state distribution. Because of these beams’ broken symmetry of polarization state distribution, the spatial separation of SAM occurs in the tight focusing of the non-axisymmetrical vector beams, it can be considered as a manifestation of the optical spin Hall effect.
In this paper, we propose a novel kind of vector beam by combining the grafted radially polarized beams with the different polarization orders, which is called the grafted polarization vector beam (GPVB). Because its polarization order is different azimuthally, the GPVB possess the non-axisymmetrical polarization state distribution. We theoretically study the tight focusing of the GPVB, and find that the focal field intensity pattern can be well modulated by adjusting the polarization order of two (or more) grafted parts. It shows the spatial separation of SAM and OAM occurs in its tight focusing, and the patterns of SAM and OAM also can be well modulated by adjusting the polarization order of the grafted parts. Furthermore, we find the on-axis energy flow in the focal region can be changed from positive to negative by adjusting its polarization order. Our results provide more modulation freedoms and potential applications in optical tweezers and particles trapping.
2. Theoretical model
The GPVB can be constructed by combining two (or more) grafted radially polarized beams with different polarization order, just as shown in Fig. 1. In Fig. 1, the GPVB is composed of two radially polarized parts, one half with the polarization order ${m_1}$, the other half with the polarization order ${m_2}$, and the polarization state maintains the continuity in the combination position. Then, the polarization state of the radially polarized beam can be expressed as
Fig. 1. The schematic diagram of construction process of the GPVB. (a) the combination of two half parts with different polarization order ${m_1}$ and ${m_2}$, (b) the polarization state distribution of GPVB.
Based on the Richards-Wolf diffraction integral formula [38], the normalized intensity patterns of tightly focused GPVB is shown in Fig. 2. It has known that, the strong longitudinal field component can be obtained in the tight focusing of the traditional radially polarized beams [39]. From Fig. 2, we see that the relative intensity of the transverse and longitudinal field components can be changed by adjusting the polarization order of two grafted parts, both the magnitude and the pattern of the longitudinal and the transverse field are modulated synchronously. Comparing with the focal field of the traditional radially polarized beam, the focal spot pattern of the GPVB can be flexibly controlled by adjusting the polarization order of the GPVB, which is shown in Fig. 3. Moreover, we find the focal intensity distribution displays the asymmetric property or is split into two spots, this is closely related to the asymmetric polarization state distribution of the GPVB. We have known that, if the vector beam possesses the broken axial symmetrical polarization state distribution, such as the power-exponent azimuthal-variant vector beam [34,35] and the fractional order radially polarized beam [36,37], the asymmetric spin-dependent splitting occurs during its propagation or tight focusing, it can be considered as a manifestation of the optical spin Hall effect induced by the spin-orbit coupling. In next, we will investigate the optical angular momentum property in the tight focusing of GPVB.
Fig. 2. Normalized intensity patterns of GPVB in the focal plane, the first column is the total intensity, second column is the intensity of the transverse field component, and third column is the intensity of the longitudinal field component.
Fig. 3. Normalized total intensity patterns in the focal plane of tightly focused GPVB with different polarization order ${m_1}$ and ${m_2}$.
3. Optical angular momentum separations in the focal plane
According to the definition of the SAM and OAM density which can be expressed as follows [40]:
Fig. 4. Normalized longitudinal SAM (first row) and OAM (second row) densitys in the focal plane of tight focusing of GPVB with different polarization order ${m_1}$ and ${m_2}$.
Fig. 5. Normalized transverse SAM (first row) and OAM (second row) structures in the focal plane of tight focusing of GPVB with different polarization order ${m_1}$ and ${m_2}$.
The on-axis energy flow in the focal plane also can be controlled by the grafted polarization state. According to the expression of the longitudinal energy flow ${S_z} = \textrm{Re} (E_x^ \ast {H_y} - E_y^ \ast {H_x})$, Fig. 6 shows the longitudinal energy flow density in the focal plane. In previous works [18,19], it knows that there is the on-axis negative energy flow in the tight focusing of the radially polarized beam when the polarization order equal to 2. We find that, by adjusting the polarization order of two grafted parts, the longitudinal energy flow distribution doesn’t maintain the cylindrical symmetry pattern in the focal plane, and the on-axis energy flow in the focal plane can change from the positive to the negative with the change of the polarization order. From Fig. 7, we see that the on-axis negative energy flow can be obtained when the polarization order ${m_1}$ and ${m_2}$ take the different values, and the magnitude of the on-axis negative energy flow can be modulated by the changing the polarization order ${m_1}$ and ${m_2}$. The maximum on-axis negative energy flow is obtain when the polarization order ${m_1} = {m_2} = 2$. In fact, the GPVB returns to the traditional second-order radially polarized beam when ${m_1} = {m_2} = 2$. It means the occurrence of the on-axis negative energy flow is closely related to the second-order radial polarization state. This provides a more flexible modulation method of the on-axis energy flow in the tight focusing of vector beams.
Fig. 6. The longitudinal energy flow density in the focal plane of tight focusing GPVB with different polarization order ${m_1}$ and ${m_2}$.
Fig. 7. Evolution of longitudinal energy flow density in the focal plane of tight focusing GPVB with different polarization order ${m_1}$ and ${m_2}$.
4. Conclusions
In this paper, we construct a novel vector beam by combining the radially polarized beams with the different polarization orders, which is called the GPVB. In some previous works [41–44], researchers present the generation method of vector beams based on the sectored optical elements. The GPVB is composed of two (or more) sectors with the different polarization order, then it may be generated based on the sectored optical elements experimentally. Because the polarization order of the GPVB is different azimuthally, namely, the GPVB also can be considered as a kind of hybrid vector beams. In previous works, it has known the broken symmetry of polarization state also can lead to the spin-orbit coupling in some optical systems, and induce the optical spin Hall effect. We investigate the tight focusing of the GPVB and show some interesting phenomena induced by the spin-orbit coupling, such as the flexible and adjustable focal field pattern, the spatial separation of the SAM and the OAM, the controllable on-axis negative energy flow. By adjusting the polarization order of two (or more) grafted parts, the focal field properties can be well modulated. We provide a more flexible modulation technique of the tight focusing of vector beam and the potential applications in optical modulation and particles trapping.
Funding
National Natural Science Foundation of China (11974101, 11974102).
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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