Tightly focused optical field with controllable photonic spin orientation

Jian Chen; Chenhao Wan; Ling Jiang Kong; Qiwen Zhan

doi:10.1364/OE.25.019517

1. Introduction

Light possesses intrinsic polarization degrees of freedom, namely the so-called spin angular momentum (SAM), which are associated with the oscillating directions of the electric and magnetic fields. SAM plays important role in spin-orbit interaction (SOI) at subwavelength scales [1–3], leading to promising applications including optical nanoprobing [4], quantum information technology [5], spin controlled directional coupling [6–10], optical trapping [11], microscopy [12], and so on. Hence, there have been significant research efforts devoted to this intriguing phenomenon over the past decade. For example, focused fields with circular polarization at any transverse plane in the focal region of a high numerical aperture (NA) lens was demonstrated by using an interferometric system in combination with spatial light modulators (SLMs) and digital holography [13]. Besides, synthesis of focused beam with arbitrary homogeneous polarization was achieved through employing carefully engineered vectorial optical fields [14]. Both of these two cases address optical focal fields with longitudinal SAM with its spin orientation along the optical axis. On the other hand, optical fields may also rotate around an axis perpendicular to the propagation direction, giving rise to the transverse SAM [15]. Non-diffraction-limited or diffraction-limited focused beam with purely transverse SAM could be realized by focusing polarization tailored vector beams with a high NA lens [16,17]. As for the focal field with hybrid SAM (i.e., simultaneously containing the longitudinal and transverse SAM), optical polarization Möbius strips was generated by tightly focusing the light beam emerging from a liquid crystal q-plate that modifies the polarization of light in a space-variant manner [18]. In general, polarization state of photons has significant influence on the direction of the emissions or scattering from nanostructures. Thus, controlling the photonic spin orientation can be crucial to tailor the radiation patterns arising the nanoscale emitters. For instance, highly asymmetric surface plasmon polariton (SPP) radiation patterns are resulted from an emitter with spin axis tilted with respect to the plane of the interface when optical losses or gains were present in the materials [19]. These effects could be applied to directionally control SPP emission and absorption, as well as to investigate emission and scattering processes close to metal-dielectric interfaces. All-optical control of the emission directivity of a dipole-like nanoparticle was experimentally demonstrated via tuning the dipole moment by the focal field with position dependent local polarization [7]. In a broad sense, SOI in photonic systems showing spin-momentum locking allows mapping the polarization of an optical beam into different amplitudes of guided waves propagating along different optical paths or modes [20], which is very attractive for future optical nanocircuits and on-chip optical links [21]. Hence, the capability of generation of diffraction limited focal spot with arbitrarily controllable and tunable three-dimensional spin orientation may be very useful for a wide range of optical and photonic applications.

In order to fully control the focused optical beam with prescribed spin orientation, the incident field at the pupil plane of a high NA lens needs to be carefully designed. Some schemes have been proposed to develop versatile systems for the creation of vectorial optical fields with desired properties. More specifically, spiral phase plate and azimuth-type polarization analyzer were utilized to generate femtosecond cylindrical vector beams [22,23]. A technique for generating fields with arbitrary polarization and shape distributions was reported through using transmissive SLMs and Mach-Zehnder interferometry setup [24]. A holographic optical technique is employed to modulate the amplitude, phase and states of polarization of a complex field by a single phase-only SLM [25]. Based on two reflective liquid crystal SLMs, a vectorial optical field generator (VOF-Gen) is designed and built to completely manipulate the phase, amplitude, polarization ratio and retardation of an optical field on a pixel-by-pixel basis [26–28]. Additionally, time-reversal method to find pupil fields for focused beams with specified features has been developed based on radiation theory [29,30]. In this work, we develop a method to achieve controllable spin axis orientation within a diffraction limited focused beam produced by a high NA lens. The expressions of the required incident pupil fields are analytically derived through coherently combining the radiation patterns of two electric dipoles placed at the focal point of the high NA lens. The above mentioned VOF-Gen is adopted to completely manipulate the nonuniform amplitude, polarization and phase of the derived pupil fields. And the spin density is calculated to quantitatively analyze the spin axis orientation of the focused beam. Accordingly, this paper is organized as the following: in section 2 the derivation of the pupil field is detailed and the Richard-Wolf vectorial diffraction theory to obtain the electric fields within the focal volume is described. Numerical simulations are given to demonstrate the validity of the presented technique in section 3. The VOF-Gen is reviewed and the experimental results are provided in section 4. Finally, the main conclusions are summarized in section 5.

2. Designing the spin axis orientation

First, two electric dipoles with phase difference of π/2 are placed at the focal point of a high NA aplanatic objective lens to mimic the desired focused beam spinning around an axis oriented at predetermined direction. As shown in Fig. 1, dipole 1 oscillates in the yz plane and has an angle of $θ_{1}$ with respect to the negative direction of the z-axis, while dipole 2 oscillates along the x-axis. Thus, the unit vectors along the oscillation directions of the dipoles 1 and 2 are ${\vec{a}}_{1} = (1, 0, 0)$ and ${\vec{a}}_{2} = (0, s i n θ_{2}, - c o s θ_{2})$ , respectively. Assuming that the direction vector of the spin axis of the focal field is $\vec{b} = (c o s α, c o s β, c o s γ)$ , since $\vec{b} = {\vec{a}}_{1} \times {\vec{a}}_{2}$ , then we can derive the spin axis direction as direction vector of the spin axis of the focal field is $\vec{b} = (c o s α, c o s β, c o s γ)$ , since $\vec{b} = {\vec{a}}_{1} \times {\vec{a}}_{2}$ , then we can derive the spin axis direction as

{\begin{array}{l} α = π / 2 \\ β = θ_{1} \\ γ = π / 2 - θ_{1} \end{array},

where

α

,

β

,

γ

denote the angles formed between the spin axis of the focal field and the positive direction of the x-, y- and z-axis.

Fig. 1 Calculation of the pupil field to obtain focused beam with specific spin axis orientation through coherently superposition of the radiation pattern from two electric dipoles.

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The radiation patterns from the two dipoles are collected and collimated by the lens to form the field distribution in the pupil plane, i.e., the required incident pupil field. According to the dipole antenna theory [31], considering the phase difference π/2, the electric radiation from dipole 1 at point A on the spherical surface Ω could be written as

{\vec{E}}_{1} (θ, φ) = C \cdot e^{j π / 2} \cdot [(\cos θ_{1} \sin θ - \sin θ_{1} \cos θ \sin φ) {\vec{e}}_{θ} - \sin θ_{1} \cos φ {\vec{e}}_{φ}],

where

{\vec{e}}_{θ}

is the unit vector along the direction of the elevation angle

θ

,

{\vec{e}}_{φ}

is the unit vector along the direction of the azimuthal angle

φ

,

C = j ω μ I_{0} l e^{- j k f} / 4 π f

can be seen as a constant,

ω

is the angular frequency,

μ

is the permeability,

I_{0}

is the constant electric current,

l

is the dipole length,

k

is the wave number,

f

is the focal length of the high NA lens.

Similarly, the radiation field from dipole 2 at point A is

{\vec{E}}_{2} (θ, φ) = C \cdot [- \cos θ \cos φ {\vec{e}}_{θ} + \sin φ {\vec{e}}_{φ}] .

Coherent superposition of these two radiation patterns will give the total electric field at point A as

\begin{matrix} {\vec{E}}_{A} (θ, φ) = {\vec{E}}_{1} (θ, φ) + {\vec{E}}_{2} (θ, φ) \\ = C \cdot {[- \cos θ \cos φ + e^{j π / 2} (\cos θ_{1} \sin θ - \sin θ_{1} \cos θ \sin φ)] {\vec{e}}_{θ} \\ + (\sin φ - e^{j π / 2} \sin θ_{1} \cos φ) {\vec{e}}_{φ}} . \end{matrix}

To project the combined radiation pattern in Eq. (4) onto the pupil plane, the bending effect of the high NA lens should be taken into consideration. In this case, a sine condition lens is used, for which the projection function is $P (θ) = \sqrt{\cos θ}$ , thus the required incident pupil field can be found to be

\begin{matrix} {\vec{E}}_{i} (r, φ) = C \cdot {[- \cos θ \cos φ + e^{j π / 2} (\cos θ_{1} \sin θ - \sin θ_{1} \cos θ \sin φ)] {\vec{e}}_{r} \\ + (\sin φ - e^{j π / 2} \sin θ_{1} \cos φ) {\vec{e}}_{φ}} / \sqrt{\cos θ}, \end{matrix}

where

θ = \sin^{- 1} (r / f)

,

{\vec{e}}_{r}

is the radial unit vector. Note that Eq. (5) is expressed in the polar coordinates, the corresponding incident pupil field in the Cartesian coordinates can be shown as

\begin{array}{l} {\vec{E}}_{i} (r, φ) = \frac{C}{\sqrt{\cos θ}} \cdot [A \cdot {\vec{e}}_{x} + B \cdot {\vec{e}}_{y}], \\ A = e^{j π / 2} (\cos θ_{1} \sin θ \cos φ + \sin θ_{1} \cos φ \sin φ - \sin θ_{1} \cos θ \sin φ \cos φ) \\ - \cos θ \cos^{2} φ - \sin^{2} φ, \\ B = e^{j π / 2} (\cos θ_{1} \sin θ \sin φ - \sin θ_{1} \cos^{2} φ - \sin θ_{1} \cos θ \sin^{2} φ) \\ - \cos θ \cos φ \sin φ + \sin φ \cos φ, \end{array}

where

{\vec{e}}_{x}

and

{\vec{e}}_{y}

are the unit vectors along the x- and y-axis, respectively. To verify if the above derived incident pupil field will give rise to the diffraction limited tightly focused spot spinning around a predetermined axis, the Richards-Wolf vectorial diffraction theory is adopted to calculate the electric field distribution of the focused beam [32,33]. After being refracted by the high NA lens, and considering the bending effect of the lens, the electric field in Eq. (6) becomes

\begin{array}{l} {\vec{E}}_{o} (θ, φ) = C \cdot [X \cdot {\vec{e}}_{x} + Y \cdot {\vec{e}}_{y} + Z \cdot {\vec{e}}_{z}], \\ X = e^{j π / 2} (\cos θ_{1} \sin θ \cos θ \cos φ - \sin θ_{1} \cos^{2} θ \sin φ \cos φ \\ + \sin θ_{1} \cos φ \sin φ) - \cos^{2} θ \cos^{2} φ - \sin^{2} φ, \\ Y = e^{j π / 2} (\cos θ_{1} \sin θ \cos θ \sin φ - \sin θ_{1} \cos^{2} θ \sin^{2} φ - \sin θ_{1} \cos^{2} φ) \\ + \sin^{2} θ c o s φ \sin φ, \\ Z = e^{j π / 2} (\cos θ_{1} \sin^{2} θ - \sin θ_{1} \cos θ \sin φ \sin θ) - \cos θ \sin θ \cos φ, \end{array}

where

{\vec{e}}_{z}

is the unit vector along the z-axis. Subsequently, the associated tightly focused optical field is given by

\begin{matrix} \vec{E} (r_{p}, ϕ, z_{p}) = \frac{i}{λ} \int_{0}^{θ_{\max}} \int_{0}^{2 π} {\vec{E}}_{o} (θ, φ) \times e^{j k r_{p} \sin θ \cos (φ - ϕ) + j k z_{p} \cos θ} \sin θ d θ d φ \\ = \frac{i C}{λ} \int_{0}^{θ_{\max}} \int_{0}^{2 π} [X \cdot {\vec{e}}_{x} + Y \cdot {\vec{e}}_{y} + Z \cdot {\vec{e}}_{z}] \times e^{j k r_{p} \sin θ \cos (φ - ϕ) + j k z_{p} \cos θ} \sin θ d θ d φ, \end{matrix}

where

λ

is the optical wavelength,

θ_{\max}

is the maximal angle related with the NA of the lens,

r_{p} = \sqrt{x^{2} + y^{2}}

and

ϕ = \tan^{- 1} (y / x)

are the polar coordinates in the focal volume;

X

,

Y

and

Z

are the same as given in Eq. (7). The polarization distribution can then be further analyzed according to the three components of the focused electric field. Meanwhile, the spin density of the focused beam will be computed to quantitatively evaluate the orientation of its spin axis, and based on the electric field in Eq. (8), the spin density can be expressed explicitly as [18,34]

\vec{S} \propto Im ({\vec{E}}^{*} \times \vec{E}) .

3. Numerical simulations

In this section, we first take $θ_{1} = π / 6$ as an example to illustrate the control of the spin orientation of the focused beam. Then we select several typical values in the range $[0, π / 2]$ to demonstrate the spin axis rotation in the yz plane of the focal volume.

The ideal incident pupil field corresponding to $θ_{1} = π / 6$ is shown in Fig. 2(a) with polarization distribution map, which is denoted by the polarization ellipses. It should be noted that the green ellipses indicate the left-handed states while the blue ellipses represent the right-handed states. Clearly, we can see that the incident pupil field has nonuniform amplitude and polarization distribution, since most of the incident field are in right-handed polarization with different ellipticity and the amplitude is much stronger around the lower edge. Assuming the incident beam is tightly focused by an objective lens with $N A = 0.95$ , using Eq. (8), we can obtain the electric field of the focal spot, as demonstrated in Fig. 2(b) with the intensity and polarization distributions being projected onto three orthogonal planes, which qualitatively reveal the focused beam simultaneously possesses transverse and longitudinal SAM. The line scans of the intensity along the x-, y- and z-axis are depicted in Fig. 2(c), from which the full widths at half-maximum (FWHM) can be found to be 0.606λ, 0.498λ and 1.344λ, respectively. According to the widely used diffraction integral for focused beam of circular lens with plane wave incidence [35], the theoretical FWHMs along the x-, y- and z-axis are 0.54λ, 0.54λ and 1.962λ, which are the so-called diffraction limit along each dimension. It’s obvious that the transverse spot sizes in Fig. 2(b) are very close to their theoretical diffraction limits, and the anisotropy of them is mainly due to the lack of rotational symmetry in the vectorial distribution of the incident pupil field [36]. On the other hand, the axial spot size is much less than the corresponding theoretical diffraction limit, which is mainly attributed to the nonuniform amplitude and polarization distributions of the pupil field. The contributions from the upper and lower parts of the pupil field to the z component of the focused beam are in the opposite directions and consequently compress the FWHM along the z-axis. The focal volume is estimated to be 0.212λ³, which is smaller than the diffraction limited focal volume of 0.3λ³. Therefore, the tightly focused beam in Fig. 2(b) can be seen as diffraction limited in the transverse plane and beyond diffraction limited along the propagation direction. Figure 2(c) also shows that the peak intensity has a tiny shift of 0.024λ along the x-axis, which is brought by the nonzero transverse SAM of the focused beam [37].

Fig. 2 (a) Ideal incident pupil field with polarization map corresponding to $θ_{1} = π / 6$ . (b) Projection of the intensity and state of polarization distributions of the focused beam onto three orthogonal planes in the focal region. (c) Normalized intensity distributions of the focused beam along x-, y- and z-axis.

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To quantitatively analyze the spin axis orientation of the focused beam in Fig. 2(b), without loss of generality, the spin density of the focused beam is calculated with Eq. (9) according to the electric field in the xz plane of the focal volume, as shown in Figs. 3(a)-3(c). The mainlobe in the xz plane, which is defined as the intensity decreases to the $1 / e^{2}$ of the peak value, is marked by the cyan contour to facilitate the analysis of the spin density and orientation. From Figs. 3(a)-3(c), we can find that the spin density is nonuniform through the focal volume and the y- and z-component of the spin density are much stronger than the x-component in the mainlobe. Thus, the spin axis of the focused beam is mainly located in the yz plane. Meanwhile, the spin density distribution also quantitatively reveals that the focused beam simultaneously possesses transverse and longitudinal SAM, especially the SAM along the y- and z-direction. The direction angles of the spin axis can be estimated using the following equation

Fig. 3 Spin density and direction angles of the focused beam in the xz plane of Fig. 2(b). (a)-(c) x-, y- and z-component of the spin density. (d)-(f) The direction angles formed between the spin axis of the focused field and the positive direction of the x-, y- and z-axis.

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{\begin{array}{l} α = \cos^{- 1} (S_{x} / \sqrt{S_{x}^{2} + S_{y}^{2} + S_{z}^{2}}) \\ β = \cos^{- 1} (S_{y} / \sqrt{S_{x}^{2} + S_{y}^{2} + S_{z}^{2}}) \\ γ = \cos^{- 1} (S_{z} / \sqrt{S_{x}^{2} + S_{y}^{2} + S_{z}^{2}}) \end{array} .

The corresponding results are shown in Figs. 3(d)-3(f). It’s clear that the angle $α$ approaches to be a constant close to 90° within the mainlobe, while the angle $β$ and $γ$ manifest some variation, which is mainly caused by the nonuniform distribution of the y- and z-component of the spin density. For the point with maximum y-component of spin density, the spin axis direction is oriented at $(90 °, 36.74 °, 53.26 °)$ , which is very close to the theoretical direction $(90 °, 30 °, 60 °)$ predicted by Eq. (1). The small errors arises from the fact that the NA of the lens is not equal to 1 [29]. If we use a lens with larger NA, the deviation will be reduced. On the other hand, the simulation results of the angle $β$ and $γ$ still satisfy the Eq. (1) despite the small discrepancy. Thus, the proposed method is effective to control the spin axis orientation of the focused beam with a relative high precision.

To demonstrate the spin axis rotation of the focused beam, here we select five typical values of $θ_{1}$ in the range $[0, π / 2]$ , i.e., 0, π/6, π/4, π/3, π/2, to generate the incident pupil fields and the corresponding focused beams (see Visualization 1). Similar to the above method to analyze the spin of the focused beam, we could obtain the direction angles of the spin axis in each case, as shown in Fig. 4(a). Accordingly, Fig. 4(b) visualizes the spin axis rotation in 3D Cartesian coordinates. It’s clear that the spin axes of the focused beams rotate in the yz plane as the $θ_{1}$ varies in the range $[0, π / 2]$ . Specifically, the focused beam is with purely transverse SAM for $θ_{1} = 0$ , and with purely longitudinal SAM for $θ_{1} = π / 2$ . In other cases, the focused beam possesses both longitudinal and transverse SAM as we expect. Meanwhile, both the direction angles $β$ and $γ$ have nonlinear relationships with $θ_{1}$ , which is mainly caused by the NA of the lens less than 1. Nevertheless, this issue could be simply overcome through the precompensation of $θ_{1}$ when generating the pupil field. The spot sizes of the focused beam in each case are given in Fig. 4(c). The FWHM along the z-axis varies around 1.3λ, which is smaller than the theoretical diffraction limit of 1.962λ. And the FWHMs along both the x- and y-axis vary around 0.55λ, which is very close to the theoretical diffraction limit of 0.54λ. Hence, the focused beam maintains as diffraction limited along the transverse directions and beyond the diffraction limit along the z-axis when $θ_{1}$ changes from 0 to π/2.

Fig. 4 (a) Evolution of the direction angles of the spin axis for focused beams corresponding to ideal pupil fields. (b) 3D demonstration of the two dimensional rotation of the spin axis. (c) Evolution of the spot size of the focused beam. (For the incident pupil fields and corresponding focused beams, see Visualization 1).

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4. Experimental results

The derived pupil field for diffraction limited focus spinning around an axis oriented at desired direction could be experimentally generated with a vectorial optical field generator (VOF-Gen). The diagram of the experimental setup is shown in Fig. 5. A collimated horizontally polarized He-Ne laser with wavelength of 632.8 nm is employed as the input. Taking advantage of the HDTV format of the Holoeye HEO 1080P liquid crystal spatial light modulator (SLM), the two SLMs used in the VOF-Gen are divided into four sections for completely controlling all the degree of freedoms to create an arbitrary complex optical field on the pixel level. More specifically, the SLM section 1-4 manipulate the phase, amplitude, polarization ratio, and retardation modulations, respectively. More details about the VOF-Gen can be found in Ref [26]. To reveal the polarization distribution of the generated pupil field, Stokes parameters are measured by inserting a combination of quarter-wave plate and linear analyzer between the lens L6 and the CCD camera.

Fig. 5 Schematic diagram of the experimental setup. HWP, half wave plate; P, polarizer; L, lens; M, mirror; SF, spatial filter; BS, beam splitter.

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Here, we first generate the pupil field for $θ_{1} = π / 6$ with the VOF-Gen through carefully engineering the nonuniform amplitude and polarization distribution. The result is shown in Fig. 6(a), and has a good agreement with the ideal pupil field in Fig. 2(a). If the experimentally generated pupil field is tightly focused by a high NA lens ( $N A = 0.95$ ), using Eq. (8), we can find the electric field distribution in the focal volume, as shown in Fig. 6(b). From the line scans in Fig. 6(c), the FWHMs are calculated to be 0.588λ along the x-axis, 0.495λ along the y-axis, 1.464λ along the z-axis. All these three parameters are very close to the corresponding simulation results, indicating the diffraction limit of the focused beam in Fig. 6(b). Meanwhile, Fig. 6(c) also demonstrates a tiny shift of 0.138λ along the x-axis caused by the nonzero transverse SAM of the focused beam, and the discrepancy compared to the corresponding simulation result mainly arises from the errors between the experimentally generated and ideal pupil field.

Fig. 6 (a) Experimentally generated incident pupil field with polarization map corresponding to $θ_{1} = π / 6$ . (b) Projection of the intensity and state of polarization distributions of the focused beam onto three orthogonal planes in the focal region. (c) Normalized intensity distributions of the focused beam along x-, y- and z-axis.

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The spin density of the focused beam is calculated to quantitatively analyze the spin axis orientation of the focused beam based on the electric field in the xz plane of Fig. 6(b). The results are given in Figs. 7(a)-7(c), and the direction angles of the spin axis are presented in Figs. 7(d)-7(f). Similarly, for the point with the maximum y-component of spin density, its spin axis is oriented at $(92 .47 °, 36.17 °, 53.94 °)$ , which is in a very good agreement with the simulation result in Section 3. The slight discrepancy is mainly due to the errors between the experimentally generated and theoretical pupil field.

Fig. 7 Spin density and direction angles of the focused beam in the xz plane of Fig. 6(b). (a)-(c) x-, y- and z-component of the spin density. (d)-(f) The direction angles formed between the spin axis of the focused field and the positive direction of the x-, y- and z-axis.

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Then, we generate the other four pupil fields in the simulations with the VOF-Gen, and calculate their corresponding focal fields through Eq. (8). All the experimentally generated pupil fields and corresponding foci are animated in Visualization 2. Similar to the above method, we extract the spin axis orientations of the focused beams and plot the direction angles in Fig. 8(a). Combined with the 3D demonstration of spin axis directions, we can find that the spin axes of the focused beams substantially rotate in the yz plane, which are consistent with those results in Section 3. The spot sizes of the focused beams are depicted in Fig. 8(c). The FWHM along the z-axis is close to 1.4λ, which is less than the theoretical axial diffraction limit of 1.962λ. And both the FWHMs along the x- and y-axis approximate 0.53λ, which is very close to the theoretical transverse diffraction limit of 0.54λ. Thus, all the focused beams corresponding to the experimentally generated pupil fields can be regarded as diffraction limited in the transverse plane and beyond the diffraction limit along the propagation direction.

Fig. 8 (a) Evolution of the direction angles of the spin axis for focused beams corresponding to experimentally generated pupil fields. (b) 3D demonstration of the two dimensional rotation of the spin axis. (c) Evolution of the spot size of the focused beam. (For the incident pupil fields and corresponding focused beams, see Visualization 2).

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5. Discussions and conclusions

In this work, we demonstrate an effective method to achieve controllable spin axis orientation within a diffraction limited tightly focused beam. The required pupil fields are analytically derived through reversing the radiation patterns from two electric dipoles located at the focal point of a high NA lens, and are successfully generated with a VOF-Gen. Richard-Wolf vectorial diffraction theory is adopted to calculate the tightly focused electric fields corresponding to both the theoretical and experimentally generated pupil fields. Meanwhile, the spin density of the focused beams are calculated to quantitatively analyze the direction angles of their spin axes. Both the simulation and experimental results show that the spin axis orientations of the focused beams could be adjusted while keeping the focused beam as diffraction limited in the transverse plane and beyond the diffraction limit along the propagation direction. Although the spin orientations are demonstrated in the yz plane, it should be pointed out that by simply rotating the derived pupil field around the z-axis, one can realize arbitrary spin axis orientation of the tightly focused beam in three dimensions. The comprehensive focal field engineering technique presented in this work may find important applications in optical tweezers, microscopy and nano-optics, allowing for tailoring the radiation pattern, robust spin-controlled directional coupling and polarization dependent optical switching and routing in nanophotonic devices.

Funding

National Natural Science Foundation of China (NSFC) (91438108, 61505062), China Scholarship Council.

Acknowledgments

J. Chen and C. Wan are grateful to the Chinese Scholarship Council for supporting their study at the University of Dayton through the Joint Training PhD Program and Visiting Scholar Program.

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Name	Description
Visualization 1	the theoretical incident pupil fields and corresponding focused beams for the spin axis rotation in Fig. 4.
Visualization 2	the theoretical incident pupil fields and corresponding focused beams for the spin axis rotation in Fig. 8.

Tightly focused optical field with controllable photonic spin orientation

Abstract

1. Introduction

2. Designing the spin axis orientation

3. Numerical simulations

4. Experimental results

5. Discussions and conclusions

Funding

Acknowledgments

References and links

Supplementary Material (2)

Cited By

Figures (8)

Equations (10)

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