Generation of tunable three-dimensional polarization in 4Pi focusing system

Wenguo Zhu; Weilong She

doi:10.1364/OE.21.017265

1. Introduction

In recent years, focusing light into a very tight spot is one of the most attractive topics in optics [1–3]. The optimization of the shape and size (intensity and phase distribution) of the focal spot has been frequently discussed [4–6]. The polarization as an important parameter of the focal field, however, does not receive enough attention. Under tightly focusing condition, the longitude component of the focal field plays an equally important role as the transverse one [1]. The field characterized by three-dimensional (3D) polarization is expected to have extensional functionalities and applications [7,8]. For example, by employing 3D polarization, it is possible to directly determine the orientation of anisotropic molecule and crystal [7]. As was demonstrated recently, 3D polarization enables polarization information encryption with ultra-security, because the ‘key’ polarization orientation is unlimited in 3D space [8]. Moreover, this kind of polarization could also find applications in optical manipulation. A light beam with 3D linear polarization can trap small particles with geometrical or optical anisotropy, and make them aligned in the plane of polarization [9]. Different from linear polarization, elliptical polarization including circular one carries spin angular momentum (SAM), which can cause a rotation when acting on a small absorbing or birefringent particle [10–12]. And the rotation is confined in the plane perpendicular to the optical axis for a paraxial trapping beam [10]. In contrast, for a nonparaxial beam with 3D elliptical polarization, the SAM could make the rotation unlimited in 3D space, which significantly enhances the flexibility of optical manipulation [12]. Several methods have been proposed to generate 3D polarization by designing the distribution of input light field [13–15]. Abouraddy and Toussaint use low order azimuthal spatial harmonics of linear polarization as the input field to create 3D polarization in an optical microscopy [13]. However, the 3D polarization will deviate from the desired one as the observation point moves away from the geometrical focus. Besides, the desired polarization does not occur at the peak intensity of the focal field in most case. For producing a diffraction-limited spherical focal spot with arbitrary 3D polarization, Chen and Zhan have combined the electric dipole radiation and vectorial diffraction method to determine the input field at the pupil plane [14,15]. But, their input field is complicated in intensity, phase and polarization distributions.

In this paper, we propose a simple method to create tunable 3D polarization field in a 4Pi focusing system. In this system, a radially polarized beam is focused into longitudinal polarized field in the focal plane [2,4]. When the amplitude of one half of the input beam is tuned from 1 to −1 by an amplitude-phase modulator (APM), which contains an attenuator and a phase shifter, the polarization vector at focus rotates gradually from longitudinal direction to the transverse one that perpendicular to the amplitude jump line due to the vectorial interference effect. Therefore, the polarization vector can take any 3D direction as long as changing the modulator factor and jump line. When one half of the input field is designed to be circularly polarized and another half elliptically polarized, the resultant focal field near focus has 3D elliptical polarization with 3D SAM. The principle for the generation of this 3D elliptical polarization can be easily understood by the superposition of SAM vectors of plane waves [16].

2. The generation of 3D linear polarization

Figure 1 shows the scheme to produce 3D linear polarization, where two counter-propagating radially polarized beams illuminate two opposing high-NA objectives. The APMs before objectives tunes the amplitudes of one half of the input fields, which causes the power conversion from the longitudinal field component to the transverse one. The polarization distributions of an arbitrary 3D polarization state within the central focal spot is shown by the projection on three orthogonal planes. And the resultant polarization vector at the geometrical focus is shown by the light blue arrow in Fig. 1.

Fig. 1 Illustration of the scheme for generating 3D linear polarization. The blue arrows represent the polarization vectors of light field propagating towards the focus, while the light blue arrow represents the resultant polarization vector at geometrical focus. The color rectangle shows the intensity distribution in the xz plane near the focus. And the intensity and polarization distributions within the central focal spot are shown by projections on three orthogonal planes. In calculation, the configure factors (a,ϕ₀) = (0.5,-π/2), NA = 0.95.

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The field after focused by a high-NA objective can be described by Richards-Wolf integral [17]. So, for a radially polarized input field, the three components of the focal field can be written as [1]

[\begin{matrix} E_{x} (r, φ, z) \\ E_{y} (r, φ, z) \\ E_{z} (r, φ, z) \end{matrix}] = \frac{- i C}{π} \int_{0}^{α} \int_{0}^{2 π} X (θ) l_{0} (θ) T (ϕ) \sin θ \exp {i k [z \cos θ + r \sin θ \cos (ϕ - φ)]} [\begin{matrix} \cos θ \cos ϕ \\ \cos θ \sin ϕ \\ \sin θ \end{matrix}] d ϕ d θ,

where C is the normalized factor, α = arcsin(NA/n) with n = 1 in the free space, X(θ) = (cosθ) ^1/2 is pupil apodization function for an aplanatic lens. The function l₀(θ) describes the amplitude distribution of the Bessel-Gaussian beam, which has the form

l_{0} (θ) = \exp [- {(\frac{β \sin θ}{\sin α})}^{2}] J_{1} (\frac{2 β \sin θ}{\sin α}),

with β = 1.5 in our configuration. T(ϕ) is the modulation function, with T(ϕ) = T_L_/_R(ϕ) for the modulators at the left and right sides, respectively. T_R(ϕ) = T_L(ϕ + π), and

T_{L} (ϕ) = {\begin{matrix} 1 ϕ_{0} \leq ϕ < ϕ_{0} + π \\ a ϕ_{0} + π \leq ϕ < ϕ_{0} + 2 π \end{matrix},

where a is the modulator factor ranging from −1 to 1, ϕ₀ is the angle between the amplitude jump line and x-axis,. The electric field near the focus is the superposition of the fields from the left and right objectives. That is [4]

\begin{matrix} E (r, φ, z) = [E_{L}^{x} (r, φ, z) + E_{R}^{x} (r, φ, - z)] e_{x} + [E_{L}^{y} (r, φ, z) + E_{R}^{y} (r, φ, - z)] e_{y} \\ + [E_{L}^{z} (r, φ, z) - E_{R}^{z} (r, φ, - z)] e_{z} . \end{matrix}

It is found that this resultant focal field is linear polarized and can be controlled by the modulation function.

Since two linear polarizations with opposite orientations are actually the same polarization states, We can introduce the polar and azimuthal angles (χ_l,ψ_l) to describe the orientation of 3D linear polarization, where χ_l ranges from 0 to π/2, and ψ_l takes value between 0 and 2π. At the focus, the polarization orientation can be deduced by superposing the fields of plane waves in the angular spectrum from diffractions at two opposing objectives [18]. When a = 1, the APMs will not affect the input radially polarized beams, which result in the focal field being longitudinally polarized [4]. If a decreases from 1 to −1, the polarization vector of the focal field will vary gradually form longitudinal direction to the transverse one perpendicular to the amplitude jump line. That means the polar angle of 3D linear polarization χ_l depends on a. The numerical result is shown in Fig. 2(g), from which one sees that χ_l and a almost have a linear relation. The intensity and polarization distributions in the focal plane are plotted in Figs. 2(a)-2(d) for the case of ϕ₀ = π/2, a = 1, 0.23, −0.3 and −1, which correspond to the polar angles of 0, π/6, π/3, and π/2, respectively. Due to the rotational symmetry of the input radial polarization, rotating APMs in both sides will cause a rotation of the focal field around z axis. After simple analysis, one can easily get that the azimuthal angle of 3D linear polarization ψ_l varies with the rotation angle ϕ₀ according to the law ψ_l = ϕ₀ + π/2. In Figs. 2(d)-2(f), a is fixed to be −1 and ϕ₀ = π/2, π, and 5π/4, which give ψ_l = –π, –π/2, and –π/4. Therefore, by controlling the configure factors a and ϕ₀, arbitrary 3D linear polarization can be created.

Fig. 2 The normalized intensity and polarization distributions in the focal plane (a~f) with polarization orientation (a) (χ_l,ψ_l) = (0,π), (b) (π/6,π), (c) (π/3,π), (d) (π/2,π), (e) (π/2,-π/2), (f) (π/2,-π/4), which correspond to configure factors (a,ϕ₀) = (1,π/2), (0.23,π/2), (−0.3,π/2), (−1,π/2), (−1,π), and (−1,5π/4), respectively. (g) The dependence of χ_l on a. (h) Projection of intensity and polarization distributions on three orthogonal planes within the FWHM region when a = −1, and ϕ₀ = 5π/4. The FWHMs along x, y, and z axes are 0.56λ, 0.82λ, and 0.3λ, respectively. In calculation, NA is chosen to be 0.95.

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The full widths at half-maximum (FWHMs) of the focal spot along x, y, and z axes rely on the modulator factor a, and are equal respectively to 0.56λ and 0.56λ, and 0.3λ at a = 1. When a decreases, The FWHM along x axis will slightly increases, while the FWHMs along y and z axes almost keep the same. It should be noted that the spot size can be significantly reduced by adding an annular aperture [4]. Figure 2(h) shows the projection of intensity and polarization distributions on three orthogonal planes for a = −1 and ϕ₀ = 5π/4, where the polarization vectors are nearly uniformly directed at an angle of −45° to x axis and 90° to z axis. Within the FWHM region, the average intensity ratio between the field component along the dominating polarization direction and the total light field is 95.24%. The ratio is always more than 91% for an arbitrary a. It is also found that the peak intensity of the focal field always occurs at geometrical focus. As demonstrated in Fig. 3, the intensity at geometrical focus only depends on the modulator factor a, and takes its maximal value at a = 1. When a decreases, the intensity decreases gradually; and reaches its minimum at a = −0.07 instead of a = 0 because of the vectorial interference between the modulated and unmodulated parts of the input light field. For the same reason, the intensity at geometrical focus for a positive modulator factor |a| is always larger than that for a negative modulator factor -|a|. This kind of tunable 3D polarization may find important applications in optical imaging [19] and particle manipulation [9].

Fig. 3 The dependence of intensity at geometrical focus on the modulator factor a. The intensity at the case of a = 1 is set to be 1. In calculation, NA is 0.95.

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3. Generation of 3D elliptical polarization

To create 3D elliptical polarization, the polarization state of the input field should be tunable in ellipticity, which can be achieved by letting the input beam successively pass through a polarizer and a liquid crystal variable retarder (LCVR) [20]. The experimental configuration is shown in Fig. 4. The polarizer transmits the electric field along x axis; the LCVR with its fast axis being 45° to x axis leads to a phase retarder δ. The field from the LCVR is cos(δ/2)e_x + isin((δ/2)e_y [21]. When δ = ± π/2, it is right (left) handed circular polarizations e _± = 2^-1/2(e_x ± ie_y). Now, we consider a LCVR, which is divided into two regions (0<ϕ<π and π<ϕ< 2π) having different phase retarders δ₁ and δ₂, where δ₁ = ± π/2 and -π/2≤δ₂≥π/2, respectively. If the polarizer and LCVR have been rotated around z axis by an angle of ϕ₀, the final field becomes

\begin{matrix} E_{L C V R} (ϕ) = A (ϕ) e_{+} + B (ϕ) e_{-} \\ = {\begin{matrix} \frac{e^{- i ϕ_{0}}}{\sqrt{2}} (\cos \frac{δ_{1}}{2} + \sin \frac{δ_{1}}{2}) e_{+} + \frac{e^{i ϕ_{0}}}{\sqrt{2}} (\cos \frac{δ_{1}}{2} - \sin \frac{δ_{1}}{2}) e_{-} ϕ_{0} \leq ϕ < ϕ_{0} + π \\ \frac{e^{- i ϕ_{0}}}{\sqrt{2}} (\cos \frac{δ_{2}}{2} + \sin \frac{δ_{2}}{2}) e_{+} + \frac{e^{i ϕ_{0}}}{\sqrt{2}} (\cos \frac{δ_{2}}{2} - \sin \frac{δ_{2}}{2}) e_{-} ϕ_{0} + π \leq ϕ < ϕ_{0} + 2 π \end{matrix} . \end{matrix}

In the laboratory coordinate system, the input fields at the left and right pupil planes are set to be l₀(θ)[A_L(ϕ)e₊ + B_L₍ϕ)e₊] and l₀(θ)[A_R(ϕ)e₊ + B_R₍ϕ)e₊], respectively, where A_L(ϕ) = A_R(ϕ + π) = A(ϕ), B_L(ϕ) = B_R(ϕ + π) = B(ϕ). Following [22], the three components of the fields after left and right objectives can be respectively expressed as

\begin{matrix} [\begin{matrix} E_{L / R}^{x} (r, φ, z) \\ E_{L / R}^{y} (r, φ, z) \\ E_{L / R}^{z} (r, φ, z) \end{matrix}] = \frac{- i C}{\sqrt{2} π} \int_{0}^{α} \int_{0}^{2 π} X (θ) l_{0} (θ) \sin θ \exp {i k [z \cos θ + r \sin θ \cos (ϕ - φ)] + i ϕ} \\ \times [\begin{matrix} (\cos θ \cos ϕ - i \sin ϕ) A_{L / R} (ϕ) + (\cos θ \cos ϕ + i \sin ϕ) e^{- i 2 ϕ} B_{L / R} (ϕ) \\ (\cos θ \sin ϕ + i \cos ϕ) A_{L / R} (ϕ) + (\cos θ \sin ϕ - i \cos ϕ) e^{- i 2 ϕ} B_{L / R} (ϕ) \\ \sin θ [A_{L / R} (ϕ) + e^{- i 2 ϕ} B_{L / R} (ϕ)] \end{matrix}] d ϕ d θ . \end{matrix}

The resultant electric field near the focus can be obtained by summing these two fields. Since the phases of field components are not identical, the resultant focal field will be 3D elliptically polarized.

Fig. 4 Illustration of the scheme for generating 3D elliptical polarization. The blue arrows represent the SAM vectors of local light fields before and after objectives, while the light blue arrow represents the resultant SAM vector at focus.

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For a paraxial light field with elliptical polarization, the vector normal to the plane of polarization ellipse is limited almost along z-axis. In contrary, the normal vector can take any 3D direction for a nonparaxial field with 3D elliptical polarization. The 3D orientation of elliptical polarization can be characterized by the local SAM vector, which is in the form of

s = ε_{0} / w Im [(E_{y}^{*} E_{z}) e_{x} + (E_{z}^{*} E_{x}) e_{y} + (E_{x}^{*} E_{y}) e_{z}] .

The local SAM vector is normal to the plane of polarization ellipse; and its length, after normalized to the local intensity, is equal to the “third” Stokes parameter, which characterizes the degree of circular polarization [21]. The average spin part of angular momentum per photon is ћσ with

σ = \int_{Ω} s d V / \int_{Ω} E^{*} \cdot E d V,

where the integral region Ω is the whole central spot. Equation (8) is equal to the angular spectrum form given in [16]. |σ| = 1 associates with circular polarization, while |σ| = 0 means linear polarization. As we will show below, the normalized local SAM vector near the geometrical focus and the average SAM vector of the central spot almost have same orientation but different length.

The orientation of local 3D SAM vector of focal field can be analyzed qualitatively by the superposition of SAM vectors of plane waves from objectives, which is similar to that of 3D linear polarization. The resultant SAM vector at focus can take any 3D direction by controlling the configure factors δ₁, δ₂, and ϕ₀. For example, a transverse SAM vector is created at focus with δ₁ = -δ₂ = π/2 and ϕ₀ = -π/2, as shown in Fig. 4. A light field carrying 3D SAM has 3D elliptical polarization. Figures 5(a)-5(c) are the intensity and polarizationdistributions within the central spot, which are projected on three orthogonal planes. When the configure factors (δ₁,δ₂,ϕ₀) are equal to (π/2,π/2,-π/2), (π/2,-π/5,0), and (π/2,-π/2,π/4), the fields at geometrical focus are circularly polarized with normalized local SAM vector (χ_s,ψ_s) = (0,0), elliptically polarized with (χ_s,ψ_s) = (π/4,-π/2), and elliptically polarized with (χ_s,ψ_s) = (π/2,-π/4), respectively. One sees from Figs. 5(a)-5(c) that, the focal fields within the central spot have very similar intensity distributions. The FWHMs of the central spot are 0.64λ and 0.3λ along transverse and longitudinal directions, respectively, although with different configure factors. That means the spot size is always as small as that generated by a circular polarized input beam. In the focal plane, the light field is almost uniformly polarized. However, the polarization will deviate slowly from that at the geometrical focus when the observed point moves away from the focal plane. But the average SAM vectors within the whole central spot almost keep the same direction as that of local one at focus. The difference between polar angle of the average SAM vector in central spot and that of local SAM vector at focus (χ_σ and χ_s) is small, which is evident in Fig. 5(d), where the dependences of χ_σ and χ_s on δ₂ are plotted for δ₁ = ± π/2. The azimuthal angles of the average and local SAM vectors (ψ_σ and ψ_s) are also similar, and $ψ_{s} = ϕ_{0} \mp π / 2$ for δ₁ = ± π/2. The dependences of |s| and |σ| on δ₂ for δ₁ = ± π/2 are shown in Fig. 5(e), from which one sees that, |s(δ₁,δ₂)| = |s(-δ₁,-δ₂)|, |σ(δ₁,δ₂)| = |σ(-δ₁,-δ₂)|, and |σ| is always smaller than |s| because of the polarization deviation. The average SAM |σ| will be much closer to |s| within FWHM region, where the degree of polarization deviation is much smaller. It is well known that, under tightly focusing condition, the longitude component of angular momentum conserves, and SAM will convert to orbital angular momentum [22]. Therefore, the longitude component of average SAM of focal field is smaller than that of input field. But the conservation law of the transverse component of angular momentum does not hold. For example, for the case of δ₁ = -δ₂ = ± π/2, an average zero angular momentum of input field can produce a nonzero transverse SAM in the focal field.

Fig. 5 (a~c) Projection of intensity and polarization distributions on three orthogonal planes within the central spot with configure factors (a) (δ₁,δ₂,ϕ₀) = (π/2,π/2,-π/2), (b) (π/2,-π/5,0), and (c) (π/2,-π/2,π/4), respectively. (d) The dependences of χ_σ (blue) and χ_s (red) on δ₂ for δ₁ = ± π/2. (e) The dependences of |σ| (blue) and |s| (red) on δ₂ for δ₁ = ± π/2. In (d) and (e), the dotted and solid lines correspond to δ₁ = ± π/2, respectively. In calculation, NA is chosen to be 0.95.

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As demonstrated by Allion and associates, a beam with transverse angular momentum will result in a lateral shift of the barycenter of intensity [23]. However, this shift is not observed in focal field of 4Pi focusing system, even though the field carries transverse SAM. This is because the shifts from left and right objectives are canceled with each other. If only one objective is involved, the barycenter shift is obvious, and can be used to study the geometric spin Hall effect of light. Very recently, a 3D rotational motion of dandelion-shaped microparticle is achieved through a combination of holographic ring trap and conventional optical tweezer, where the horizontal rotation is induced by helicity in the ring trap, while the vertical rotation is caused by the radiation pressure of optical tweezer [24]. In a different mechanism, the 3D SAM could also cause a 3D rotational motion of both micro- and nano-particles, which has potential applications in microscopic motor and pump [25].

4. Conclusion

We have presented an approach for 4Pi microscopy to generate 3D polarization by using APMs or the combination of a polarizer and a LCVR. Both APM and LCVR are divided into two regions with different configure factors. When two counter-propagating radially polarized beams are inputted and modulated by APMs, a tunable 3D linear polarization state is generated in the focal field. When the input fields pass through polarizer and LCVR successively and then focused by two opposing high-NA objectives, the focal field has 3D elliptical polarization with 3D SAM. The normalized local 3D SAM vector has been used to characterize the orientation and helicity of local 3D elliptical polarization. The average SAM vector of focal field has been calculated within the whole central spot, and is shown to have the same orientation as that of local SAM vector at the geometrical focus.

Appendix: the local SAM vector

Following [26], the spin part of angular momentum vector has the form

\begin{matrix} s = \frac{ε_{0}}{4 w} r \times [\nabla \times Im (E^{*} \times E)] \\ = \frac{ε_{0}}{2 w} Im [\begin{matrix} 2 E_{y}^{*} E_{z} - \partial_{y} (y E_{y}^{*} E_{z}) - \partial_{z} (z E_{y}^{*} E_{z}) + y \partial_{x} (E_{z}^{*} E_{x}) + z \partial_{x} (E_{x}^{*} E_{y}) \\ 2 E_{z}^{*} E_{x} - \partial_{x} (x E_{z}^{*} E_{z}) - \partial_{z} (z E_{z}^{*} E_{x}) + z \partial_{y} (E_{x}^{*} E_{y}) + x \partial_{y} (E_{y}^{*} E_{z}) \\ 2 E_{x}^{*} E_{y} - \partial_{x} (x E_{x}^{*} E_{y}) - \partial_{y} (y E_{x}^{*} E_{y}) + x \partial_{z} (E_{y}^{*} E_{z}) + y \partial_{z} (E_{z}^{*} E_{x}) \end{matrix}] . \end{matrix}

Obviously, all the derivative terms have no contribution to the average angular momentum. It has been proved in the paraxial region that, the derivative terms do not give rise to any mechanical effect [27]. Therefore, we safely preserve only the first term in each components of SAM vector. As will be shown below, this reduced SAM vector is normal to the plane of polarization ellipse; and its length, after normalized to the local intensity, is equal to the “third” Stokes parameter that characterize the degree of circular polarization.

An arbitrary nonparaxial electric field can be written as

E = E_{x} e_{x} + E_{y} e_{y} + E_{z} e_{z} = a e^{i δ_{x}} e_{x} + b e^{i δ_{y}} e_{y} + c e^{i δ_{z}} e_{z} .

For convenience, we set δ_x = 0, Δδ₁ = δ_z-δ_y, Δδ₂ = δ_z-δ_x, and Δδ₃ = δ_y-δ_x, and rewrite Eq. (10) as

\begin{matrix} E = A [\frac{a}{A} e_{x} + \frac{b \cos Δ δ_{3}}{A} e_{y} + \frac{c \cos Δ δ_{2}}{A} e_{z}] + i B [\frac{b \sin Δ δ_{3}}{B} e_{y} + \frac{c \sin Δ δ_{2}}{B} e_{z}] \\ = A e_{1} + i B e_{2}, \end{matrix}

where

A = \sqrt{a^{2} + b^{2} \cos^{2} Δ δ_{3} + c^{2} \cos^{2} Δ δ_{2}},

B = \sqrt{b^{2} \sin^{2} Δ δ_{3} + c^{2} \sin^{2} Δ δ_{2}},

e₁ and e₂ are two vectors. A new unit vector e₃ orthogonal to e₂ is found to be

e_{3} = \frac{- a B^{2} e_{x} + b c \sin Δ δ_{1} (- c \sin Δ δ_{2} e_{y} + b \sin Δ δ_{3} e_{z})}{B \sqrt{a^{2} B^{2} + b^{2} c^{2} \sin^{2} Δ δ_{1}}},

which locates at the plane determined by e₁ and e₂. Now, the electric field can be written in the basis of e₂ and e₃:

\begin{matrix} E = \frac{1}{2 B} [(b^{2} \sin 2 Δ δ_{3} + c^{2} \sin 2 Δ δ_{2} + i 2 B) e_{2} - 2 \sqrt{a^{2} B^{2} + {(b c \sin Δ δ_{1})}^{2}} e_{3}] \\ = E_{2} e_{2} + E_{3} e_{3} . \end{matrix}

According to [21], the “third” Stokes parameter is

\begin{matrix} s_{3} = \frac{2 Im (E_{2}^{*} E_{3})}{| E_{2} |^{2} + | E_{3} |^{2}} = \frac{2 \sqrt{{(a b \sin Δ δ_{3})}^{2} + {(a c \sin Δ δ_{2})}^{2} + {(b c \sin Δ δ_{1})}^{2}}}{a^{2} + b^{2} + c^{2}} \\ = \frac{2 \sqrt{{[Im (E_{x}^{*} E_{y})]}^{2} + {[Im (E_{y}^{*} E_{z})]}^{2} + {[Im (E_{z}^{*} E_{x})]}^{2}}}{| E_{x} |^{2} + | E_{y} |^{2} + | E_{z} |^{2}} . \end{matrix}

It can be easily verified that

s = (e_{2} \times e_{3}) s_{3} [| E_{x} |^{2} + | E_{y} |^{2} + | E_{z} |^{2}] ε_{0} / 2 w

. Therefore, after normalized to the intensity, |s| is equal to s₃. And s, e₂, and e₃ form a right-handed system.

Acknowledgments

The authors acknowledge the financial support from the National Natural Science Foundation of China (NSFC) (grant 90921009).

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Generation of tunable three-dimensional polarization in 4Pi focusing system

Abstract

1. Introduction

2. The generation of 3D linear polarization

3. Generation of 3D elliptical polarization

4. Conclusion

Appendix: the local SAM vector

Acknowledgments

References and links

Cited By

Figures (5)

Equations (14)

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