Theoretical generation of arbitrarily homogeneously 3D spin-orientated optical needles and chains

Li Hang; Peifeng Chen; Ying Wang

doi:10.1364/OE.27.006047

1. Introduction

Over the past few years, super-resolved optical fields have always been attracting intensive research interest for their novel properties and potential applications in many fields including particle acceleration [1,2], optical data storage [3,4], optical coherence tomography [5,6], fluorescent imaging [7,8] and so on. Optical needles, also called non-diffraction beams, and optical chains are typically two-dimensional (2D) and three-dimensional (3D) super-resolved optical fields, respectively. These special optical fields are achieved and optimized by focusing modulated incident beams with different focusing systems by various researchers [9–21]. To extend the applications of the super-resolved optical fields, some research is focusing on controlling their spin angular momentum (SAM) [22,23]. The gradient force derived from intensity distribution can only trap particles [24] but the SAM can make the particles rotated. A circularly polarized beam only has longitudinal SAM while transverse SAM can be generated under certain conditions, including plasmonic, evanescent wave and tightly focused beams [22, 25–27]. Among the aforementioned several methods, the simplest way to obtain super-resolved optical fields with controllable SAM is to focus modulated incident beams with a high numerical aperture (NA) focusing system.

In these regards, there are still some obstacles remaining in the tight focusing realm. First, it is hard to realize homogeneous spin-orientation distribution in a 3D region. In the literature [23], only in a plane (such as xz plane) can the spin-orientation distribution be homogeneous. Second, there is no standard to evaluate spin-orientation homogeneity quantitatively. It is important to develop a standard because the spin-orientation cannot be totally homogeneous for an optical field except a few special occasions such as a circularly polarized plane wave. Third, the obtained 3D super-resolved optical field is usually a single focal spot because it is difficult to make multiple spots in different positions have the same spin-orientation. To solve these problems, we choose a thin annular beam as the incident beam to obtain an optical needle (non-diffraction beam) so that the longitudinal distribution and the transverse distribution of the focal field can be separated. In this way, the spin-orientation will be homogeneous in a 3D region as long as it is homogeneous in a plane which is perpendicular to the propagating direction. With the help of a 4π focusing system, a needle can be tailored to a chain [28]. Besides, we propose a conception called “spin-orientated homogeneity purity” (SOHP) to evaluate spin-orientation homogeneity quantitatively. SOHP of the obtained super-resolved optical fields in this paper are all beyond 0.996.

2. Theory and configuration

Figure 1 shows schematic diagrams of tight focusing systems. Figure 1(a) is a parabolic mirror, which can be utilized to generate optical needles [15]. Figure 1(b) is a 4 π system composed of two confocal aplanatic lenses, which can be used to generate optical chains [28]. A parabolic mirror’s NA can reach 1 (i.e. θ₀ = 90°), which means it can generate a smaller focal spot than an aplanatic lens [15].

Fig. 1 Schematic diagrams of tight focusing systems. (a) a parabolic mirror; (b) a 4π system. The incident beams are collimated modulated thin annular beams. These systems are azimuthally symmetric.

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In general, a collimated incident beam can be express as

{\vec{E}}_{i} (θ, φ) = \sum_{m = - \infty}^{+ \infty} f_{m} (θ) exp (i m φ) {\vec{e}}_{ρ} + \sum_{m = - \infty}^{+ \infty} g_{m} (θ) exp (i m φ) {\vec{e}}_{φ},

where f_m, g_m are complex amplitude factors of radially and azimuthally polarized components when the topological charge is m, respectively. When the incident beam is thin, f_m, g_m can be regarded as constants. Such annular light can be generated by an axicon and a lens [15]. Considering a single aplanatic focusing system such as Fig. 1(a), according to the Richards-Wolf vector diffraction theory [29], the electric field near the focus can be expressed as

\begin{array}{l} \vec{E} (r, φ, z) = A \int_{θ_{0} - \frac{Δ θ}{2}}^{θ_{0} + \frac{Δ θ}{2}} P (θ) sin θ e^{i k z cos θ} \sum_{m = - \infty}^{\infty} i^{m} e^{i m ϕ} \times \\ [\begin{matrix} - i cos θ f_{m} (- e^{- i ϕ} J_{m - 1} + e^{i ϕ} J_{m + 1}) {\vec{e}}_{x} + g_{m} (e^{- i ϕ} J_{m - 1} + e^{i ϕ} J_{m + 1}) {\vec{e}}_{x} \\ - cos θ f_{m} (- e^{- i ϕ} J_{m - 1} + e^{i ϕ} J_{m + 1}) {\vec{e}}_{y} + i g_{m} (e^{- i ϕ} J_{m - 1} - e^{i ϕ} J_{m + 1}) {\vec{e}}_{y} \\ 2 sin θ f_{m} J_{m} {\vec{e}}_{z} \end{matrix}] d θ, \end{array}

where A is a constant related to wavelength λ, k = 2π/λ is wave number, θ₀ represents the angular position of the central ray of the incident beam and Δθ is the angular thickness of the incident beam as shown in Fig. 1(a). P(θ) is apodization factor, obtained from energy conservation. For a parabolic mirror, P(θ) = 2/(1 − cos θ); for an aplanatic lens,

P (θ) = \sqrt{cos (θ)}

. J_n = J_n(kr sin θ), where J_n(·) denotes the nth-order Bessel function of the first kind. e⃗_x, e⃗_y and e⃗_z are Cartesian unit vectors. It is easy to verify that Eq. (2) fulfills the equations ∇⃗ · E⃗ = 0 and (∇⃗² + k²)E⃗ = 0.

With some approximations when Δθ << θ₀ [28], Eq. (2) can be expressed as

\begin{array}{l} \vec{E} (r, φ, z) = AP (θ_{0}) \frac{2 sin (k z sin θ_{0} sin \frac{Δ θ}{2}) exp (i k z cos θ_{0} cos \frac{Δ θ}{2})}{k z} \sum_{m = - \infty}^{\infty} i^{m} e^{i m ϕ} \times \\ [\begin{matrix} - i cos θ_{0} f_{m} (- e^{- i ϕ} J_{m - 1} + e^{i ϕ} J_{m + 1}) {\vec{e}}_{x} + g_{m} (e^{- i ϕ} J_{m - 1} + e^{i ϕ} J_{m + 1}) {\vec{e}}_{x} \\ - cos θ_{0} f_{m} (- e^{- i ϕ} J_{m - 1} + e^{i ϕ} J_{m + 1}) {\vec{e}}_{y} + i g_{m} (e^{- i ϕ} J_{m - 1} - e^{i ϕ} J_{m + 1}) {\vec{e}}_{y} \\ 2 sin θ_{0} f_{m} J_{m} {\vec{e}}_{z} \end{matrix}], \end{array}

where J_n = J_n(kr sin θ₀). Equation (3) shows that the longitudinal distribution and the transverse distribution of the focused field are separated and a non-diffraction beam is obtained. To solve the equation

{| \vec{E} (r, ϕ, z) |}^{2} = \frac{1}{2} {| \vec{E} (r, ϕ, 0) |}^{2}

, it is easy to gain that the longitudinal full width at half maximum (LFWHM) or depth of focus is about

\frac{0.8743 λ}{sin θ_{0} Δ θ}

[28].

When r = 0, only J₀ ≠ 0. J_n of non-zero orders represent hollow fields and will make resolution worse. To achieve a super-resolved optical field, only J₀ is desired. Hence, m should only be 0 or ±1. So only f₋₁, f₀, f₁, g₋₁ and g₁ can be retained. A larger sin θ₀ (θ₀ is near 90°) will cause a smaller focal spot [15, 18, 28], so a large sin θ₀ will be adopted in this paper. In the extreme case θ₀ = 90°, f₁ and f₋₁ will not generate J₀ component in Eq. (3), so we choose f₁, f₋₁ = 0. Now there are still six degrees of freedom (because f₀, g₋₁ and g₁ are complex numbers), but the direction of SAM only has two degrees of freedom. We fix the phase and add an amplitude restriction and then f₀, g₋₁ and g₁ can be expressed as

\begin{array}{l} f_{0} = - cos α / (2 sin θ_{0}) \\ g_{- 1} = - i sin α cos β \\ g_{1} = sin α sin β, \end{array}

where α ∈ [0, π], β ∈ [0, 2π]. The amplitude weighting factors of three components of the incident beam can be controlled by α and β. So f₀, g₋₁ and g₁ only have two degrees of freedom, which can be used to control spin-orientation. The factor (2 sin θ₀)⁻¹ is for simplifying the calculation. Such a beam can be realized by a vectorial optical field generator, and α and β can be adjusted by loading the proper phase patterns onto spatial light modulators [30]. Equation (4) represents the input beams which can generate optical needles. Substituting Eq. (4) into Eq. (3), the electric field can be expressed as

\begin{array}{l} \vec{E} (r, φ, z) = AP (θ_{0}) \frac{2 sin (k z sin θ_{0} sin \frac{Δ θ}{2}) exp (i k z cos θ_{0} cos \frac{Δ θ}{2})}{k z} \times \\ [\begin{matrix} [- J_{0} sin α e^{- i β} + i J_{1} cos α cos ϕ cot θ_{0} + J_{2} sin α (i sin β e^{i 2 ϕ} - cos β e^{- i 2 ϕ})] {\vec{e}}_{x} \\ [i J_{0} sin α e^{i β} + i J_{1} cos α sin ϕ cot θ_{0} + J_{2} sin α (sin β e^{i 2 ϕ} - i cos β e^{- i 2 ϕ})] {\vec{e}}_{y} \\ [- J_{0} cos α] {\vec{e}}_{z} \end{matrix}] . \end{array}

According to the reference [31], the SAM density can be given explicitly as

\vec{s} \propto Im ({\vec{E}}^{*} \times \vec{E}),

where E⃗^* is the complex conjugation of E⃗, and Im(·) denotes the imaginary part. To describe the direction of the SAM density s⃗, we define a unit vector γ⃗, which is expressed as

\vec{γ} = \frac{\vec{s}}{\sqrt{\vec{s} \cdot \vec{s}}} (\vec{s} \neq \vec{0}) .

It is not easy to calculate an analytic result of γ at an arbitrary point. And the result is too complex to give a distinct physical picture. But it is easy to calculate γ⃗ at the origin r = 0, z = 0. According to Eqs. (5)–(7), γ⃗(0, 0, 0) is

\vec{γ} (0, 0, 0) = [\begin{matrix} \frac{cos α cos β}{\sqrt{{cos}^{2} α + {sin}^{2} α {cos}^{2} 2 β}} {\vec{e}}_{x} \\ \frac{- cos α sin β}{\sqrt{{cos}^{2} α + {sin}^{2} α {cos}^{2} 2 β}} {\vec{e}}_{y} \\ \frac{- sin α cos 2 β}{\sqrt{{cos}^{2} α + {sin}^{2} α {cos}^{2} 2 β}} {\vec{e}}_{z} \end{matrix}] (sin α \neq 0) .

When sin α = 0, s⃗ is 0⃗, which will not be discussed in this paper.

γ⃗(0, 0, 0) can be regarded as a “theory value” or “target value”. It is controllable by adjusting α and β. To evaluate the spin-orientation homogeneity quantitatively, we define SOHP as

\begin{array}{l} η_{S} (\vec{a}) = \frac{\int_{S} \vec{s} \cdot \vec{a} d S}{\int_{S} | \vec{s} | | \vec{a} | d S} (| \vec{a} | \neq 0) \\ η_{V} (\vec{a}) = \frac{\int_{V} \vec{s} \cdot \vec{a} d V}{\int_{V} | \vec{s} | | \vec{a} | d V} (| \vec{a} | \neq 0), \end{array}

where a⃗ is a constant vector, η_S and η_V denote SOHP with respect to a⃗ in a 2D region and a 3D region, respectively. When s⃗ is always parallel to a⃗ in the integral domain, η = 1. When s⃗ is always inversely parallel to a⃗, η = −1. When s⃗ is always perpendicular to a⃗, η = 0. When s⃗ is parallel to a⃗ in half space of the integral region and antiparallel to a⃗ in the rest half space and |s⃗| is constant, η = 0. Only when η is close to 1, the homogeneity of spin-orientation is satisfactory. These basic properties indicate that this definition is logical. In the following, a⃗ = γ⃗(0, 0, 0).

3. Numerical simulation and discussion

3.1. Optical needle

The angular position θ₀ and the angular width Δθ of the incident beam can influence LFWHM and the transverse full width at half maximum (TFWHM) [28]. Equation (3) shows that Δθ has little influence on electric field in a transverse profile. So it will be hardly discussed in the following. In this subsection, θ₀ = 90° and Δθ = 0.01.

Figure 2 shows an axial profile at r = 0 and three lateral profiles at z = 0 of an optical needle by two methods with $α = \frac{π}{4}$ and $β = \frac{π}{4}$ . The solid blue lines are generated from Eqs. (2) and (4) by numerical integration; the dashed red lines are produced by the approximation Eq. (5). These two methods lead to consistent results, which implies that the approximation is accurate. Figures 3(a)–(c) are coherent with Figs. 2(b)–(d), respectively. They display that the longitudinal distributions are similar while the transverse distributions are different when φ is different. The former is from the diffraction of the thin annular incident beam, which is very similar to Fraunhofer diffraction at the slit [32]. The latter is due to the interference of two azimuthally polarized components with different topological charges, which will lead to TFWHM as a function of φ. Equation (5) can be re-expressed as E⃗(r, φ, z) ≈ F(z) G⃗(r, φ). So |F(z)|² and |G⃗(r, φ)|² represent longitudinal and lateral distribution of electric energy density, respectively. With the major premise θ₀ = 90°, |G⃗(r, φ)|² will be independent of φ only when sin² α sin 2β = 0 (i.e. g₁g₋₁ = 0).

Fig. 2 Distribution of the normalized electric energy density along z-axis and r-axis with $θ_{0} = 90 ° = \frac{π}{2}$ , Δθ = 0.01, $α = \frac{π}{4}$ and $β = \frac{π}{4}$ . Figure 2(a) is the longitudinal profile and Figs. 2(b)–(d) are the transverse profiles with φ = 0, $\frac{π}{2}$ , $\frac{π}{4}$ , respectively. The solid blue lines are generated from Eqs. (2) and (4); the dashed red lines are produced by Eq. (5). This figure shows that numerical integration and the approximation by Eq. (5) lead to consistent results for optical needles.

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Fig. 3 Distribution of the normalized electric energy density in r − z plane with $α = \frac{π}{4}$ and $β = \frac{π}{4}$ . Figures 3(a)–(c) are the profiles with φ = 0, $\frac{π}{2}$ , $\frac{π}{4}$ , respectively. These images correspond to Eq. (5).

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Solving the equation ${| \vec{E} (r, ϕ, z) |}^{2} = \frac{1}{2} {| \vec{E} (0, ϕ, z) |}^{2}$ , TFWHMs as a function of φ are shown in Figs. 4(a1) and (a2) with $β = \frac{π}{4}$ , 0, , respectively. Figures 4(b1) and (b2) are relevant transverse profiles to Figs. 4(a1) and (a2). Figures 4(a2) and (b2) show TFWHM=0.366λ is a constant, owing to g₁ = 0. On the contrary, when $β = \frac{π}{4}$ or $β = \frac{3}{4} π$ (i.e.|g₁|/|g₋₁| = 1), the interference between two incident azimuthally polarized components reaches the maximum, which results in the maximal deviation of TFWHM when φ varies. In spite of the deviation, TFWHM at φ = 0 or $\frac{π}{2}$ is a constant when β changes. Besides, Fig. 4(a1) indicates that TFWHM at φ = 0 or $\frac{π}{2}$ is a mean value. Therefore, it is suitable to use TFWHM at φ = 0 or $\frac{π}{2}$ as a mean TFWHM.

Fig. 4 (a1) and (a2) display TFWHM as a function of φ with $β = \frac{π}{4}$ , 0, respectively. (b1) and (b2) are transverse profiles of the normalized electric energy at z = 0 with $β = \frac{π}{4}$ and 0, respectively. These images are generated from Eq. (5) with $α = \frac{π}{4}$ .

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The parameters α and β are used to control the spin-orientation. But before the spin-orientation is discussed, the size of the focal spot should be given. Figure 5 shows the mean TFWHM as a function of α and β. It implies that the mean TFWHM is independent of β and will increase when sin α increases. This is because the radially polarized component of the incident beam only causes J₀ while the azimuthally polarized components cause J₀ and J₂. Hence, the mean TFWHM reaches the maximum when the radially polarized component vanishes (i.e. $α = \frac{π}{2}$ ), and drops to the minimum when the azimuthally polarized components vanish (i.e. α = 0, π). As shown in Fig. 5, the mean TFWHM varies from 0.36λ to 0.37λ when α and β vary. To make the calculation simpler, we will choose 0.366λ as the diameter of the focal spot in the following in this subsection.

Fig. 5 Mean TFWHM as a function of α and β. This figure is generated from Eq. (5).

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Figure 6 shows distribution of SAM density in two extreme cases. For Figs. 6(a1) and (b1), the parameters $α = \frac{π}{4}$ , $β = \frac{π}{4}$ cause $\vec{γ} (0, 0, 0) = \frac{\sqrt{2}}{2} {\vec{e}}_{x} - \frac{\sqrt{2}}{2} {\vec{e}}_{y}$ , which means the spin-orientation at the origin is purely transverse. On the contrary, the parameters $α = \frac{π}{2}$ , β = 0 cause γ⃗(0, 0, 0) = e⃗_z in Figs. 6(a2) and (b2), which leads to purely longitudinal spin-orientation at the origin. Figures 6(a1) and (a2) both display almost perfect spin-orientated homogeneity in the focal region. Though the 3D pictures are visual, the results are rough and usually depend on the angle of the observation. Their 2D projections Figs. 6(b1) and (b2) can help to imagine their 3D spin-orientation.

Fig. 6 Distribution of SAM density and related 2D projections. For Figs. 6(a1) and (b1), $α = β = \frac{π}{4}$ . For Figs. 6(a2) and (b2), $α = \frac{π}{2}$ , β = 0. These images are generated from Eqs. (5) and (6).

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To evaluate spin-orientated homogeneity quantitatively, Fig. 7 shows that SOHP is always beyond 0.996 when α and β vary. The integral domain is r ≤ 0.183λ. It should be noted that in Fig. 7 α ∈ [0.01, π − 0.01] instead of α ∈ [0, π]. This is because the target value will be meaningless when sin α = 0. When $α = \frac{π}{2}$ , the radially polarized component will vanish, leading to the focal field with zero longitudinal component. Hence, purely longitudinal spin-orientation will be obtained in the focal region and SOHP will equal to 1 as shown in Fig. 7.

Fig. 7 SOHP as a function of α and β. This figure is produced by Eqs. (5)–(9). The integral domain is r ≤ 0.183λ.

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3.2. Optical chain

When a 4π confocal system replaces a single focusing system, an optical chain will appear. The angular position θ₀ can influence the aspect ratio of every spot of the chain [28]. In order to obtain a chain with spherical spot, the aspect ratio should equal 1. According to the mathematical method in [28], θ₀ = 55.66° = 0.9715 is used in this section. Besides, Δθ = 0.01 is still used.

The field near the confocal spot can be expressed as

{\vec{E}}_{total} (r, ϕ, z) = {\vec{E}}_{L} (r, ϕ, z) + {\vec{E}}_{R} (r, ϕ, z),

where E⃗_L and E⃗_R denote the beam from the left and the right, respectively. The left input beam is the same as the input beam to generate optical needles. As for the right one, its azimuthally components are the same as the left while its radially polarized component should have an opposite polarization. Then E⃗_R can be expressed as

\begin{array}{l} {\vec{E}}_{R} (r, ϕ, z) = A \int_{θ_{0} - \frac{Δ θ}{2}}^{θ_{0} + \frac{Δ θ}{2}} d θ P (θ) sin θ e^{- i k z cos θ} \times \\ [\begin{matrix} [- J_{0} sin α e^{- i β} - i J_{1} cos α cos ϕ cos θ / sin θ_{0} + J_{2} sin α (i sin β e^{i 2 ϕ} - cos β e^{- i 2 ϕ})] {\vec{e}}_{x} \\ [i J_{0} sin α e^{i β} - i J_{1} cos α sin ϕ cos θ / sin θ_{0} + J_{2} sin α (sin β e^{i 2 ϕ} - i cos β e^{- i 2 ϕ})] {\vec{e}}_{y} \\ [- J_{0} cos α sin θ / sin θ_{0}] {\vec{e}}_{z} \end{matrix}] \\ \approx AP (θ_{0}) \frac{2 sin (k z sin θ_{0} sin \frac{Δ θ}{2}) exp (- i k z cos θ_{0} cos \frac{Δ θ}{2})}{k z} \\ [\begin{matrix} [- J_{0} sin α e^{- i β} - i J_{1} cos α cos ϕ cot θ_{0} + J_{2} sin α (i sin β e^{i 2 ϕ} - cos β e^{- i 2 ϕ})] {\vec{e}}_{x} \\ [i J_{0} sin α e^{i β} - i J_{1} cos α sin ϕ cot θ_{0} + J_{2} sin α (sin β e^{i 2 ϕ} - i cos β e^{- i 2 ϕ})] {\vec{e}}_{y} \\ [- J_{0} cos α] {\vec{e}}_{z} \end{matrix}] . \end{array}

It is easy to verify that Eq. (11) also respects the equations ∇⃗ · E⃗_R = 0 and (∇⃗² + k²)E⃗_R = 0 before the approximation.

Figure 8 shows the profiles of the chain in x − z plane (a) and y − z plane (b). The figure displays 17 spherical spots, in which the maximum value of the rightmost/leftmost spot is 0.9886. The number of bright spots is inversely proportionate to Δθ [28]. When Δθ = 0.1, there are only 3∼5 bright spots. Mean TFWHM and LFWHM of the chain is about $\frac{λ}{4 cos θ_{0}} = 0.043 λ$ . The focal volume is estimated to be 0.046λ³, which is far samller than the result 0.222λ³ in [22]. To make the calculation simpler, we will choose 0.444λ as the diameter of the focal spot in this section.

Fig. 8 Distribution of the normalized electric energy density in x − z plane and y − z plane with $α = \frac{π}{4}$ and $β = \frac{π}{4}$ . These images are generated from Eqs. (5), (10) and (11).

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The existence of J₁ leads to the fact that the longitudinal distribution and the transverse distribution of the confocal electric field cannot be separated completely. Therefore, SOHP in a 3D region should be considered.

Figure 9 shows SOHPs for the spots of the chain. The integral domains of Figs. 9(a)–(c) are r² + z² ≤ 0.222²λ², r² + (z − z₀)² ≤ 0.222²λ² and r² + (z − 8z₀)² ≤ 0.222²λ², where $z_{0} = \frac{λ}{2 cos θ_{0}}$ . They represent the first, the second and the ninth spot with respect to the origin. Like the occasion of the needles, α varies from 0.01 to π-0.01 and SOHPs are always beyond 0.996.

Fig. 9 SOHP as a function of α and β. This figure is produced by Eqs. (5)–(11). The integral domains of (a)–(c) are the first, the second and the ninth spot with respect to the origin in the chain. The related mathematical expressions are r² + z² ≤ 0.222²λ², r² + (z − z₀)² ≤ 0.222²λ² and r² + (z − 8z₀)² ≤ 0.222²λ², respectively.

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4. Conclusion

In conclusion, we theoretically investigate the generation of homogeneously 3D spin-orientated optical needles and optical chains. The spin-orientation are continuously tunable by adjusting two parameters of the incident beam. Meanwhile, SOHP can keep beyond 0.996. This research can be also used to generate magnetization needles and magnetization chains with controllable orientation, because the magnetization field induced by the inverse Faraday effect is proportional to SAM density of the electric field [28].

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Theoretical generation of arbitrarily homogeneously 3D spin-orientated optical needles and chains

Abstract

1. Introduction

2. Theory and configuration

3. Numerical simulation and discussion

3.1. Optical needle

3.2. Optical chain

4. Conclusion

References

Cited By

Figures (9)

Equations (11)

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