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Optical pulling force on a uniaxial anisotropic sphere by a high-order Bessel (vortex) beam

Zheng Jun Li

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Abstract

Based on the generalized Lorenz-Mie theory (GLMT) and the scattering theory of uniaxial spheres, a theoretical approach is introduced to study the axial radiation force (AOF) exerted on a uniaxial anisotropic sphere illuminated by an on-axis high-order Bessel (vortex) beams (HOBVBs). Applying Maxwell’s stress tensor, an analytical expression of the AOF on a uniaxial anisotropic sphere by the on-axis HOBVB is derived. The correctness of the theoretical and numerical results is verified by comparing the AOF on an isotropic sphere by a zero-order Bessel beam (ZOBB) with those results by a plane wave, Gaussian beam, and ZOBB. The focus of this study is to determine some conditions of the tractor beam, so as to realize the inverse motion of an anisotropic sphere through a Bessel beam. The range of optical pulling force (OPF) that can pull particles in reverse motion generated by zero-order and first-order Bessel beams is extended from isotropic spherical particles to anisotropic spherical particles. The effects of the sphere radius, conical angle, and especially electromagnetic anisotropy parameters on the OPF in water or a vacuum environment are discussed in detail. Moreover, the OPF exerted on the uniaxial anisotropic sphere illuminated by a HOBVB with $l = {2}$, 3, and 4 is also exhibited. It indicates that the HOBVB with $l = {2}$, 3 is also a good tractor beam for the uniaxial anisotropic sphere. The OPF generated by Bessel beams on uniaxial anisotropic spherical particles is not only affected by the conical angle and radius but is also significantly influenced by anisotropic parameters and topological charges. These properties of the OPF are different from those on an isotropic sphere. The theory and results are hopeful to provide an effective theoretical basis for the study of optical micromanipulation of biological and anisotropic complex particles by optical tractor (vortex) beams.

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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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Figures (12)
Fig. 1.
Fig. 1. The schematic diagram of a uniform uniaxial anisotropic sphere is illuminated by an on-axis HOBVB.

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Fig. 2.
Fig. 2. Variation of AOF ${F_z}$ on an isotropic sphere (${\varepsilon _t} = {\varepsilon _z} = 1.21{\varepsilon _0}$) illuminated by a ZOBB and Gaussian beam.

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Fig. 3.
Fig. 3. Axial optical force ${F_z}$ on a uniaxial anisotropic sphere illuminated by a ZOBB with $\alpha$ (${\varepsilon _t} = 4$, ${\mu _t} = {\mu _z} = 1$, $a = 0.5\,{\unicode{x00B5}{\rm m}}$, ${n_0} = 1.33$).

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Fig. 4.
Fig. 4. Variation of the AOF ${F_z}$ on a uniaxial anisotropic sphere illuminated by a HOBVB with $\alpha$: (a) $l = 1$; (b) $l = 2$; (c) $l = 3$; (d) $l = 4$ (${\varepsilon _t} = 4$, ${\mu _t} = {\mu _z} = 1$, $a = 0.5\,{\unicode{x00B5}{\rm m}}$, ${n_0} = 1.33$).

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Fig. 5.
Fig. 5. Variation of the AOF ${F_z}$ on a uniaxial anisotropic sphere with sphere radius illuminated by a ZOBB: (a) ${n_0} = 1.33$; (b) ${n_0} = 1$ (${\varepsilon _t} = 2.0,{\varepsilon _z} = 2.4$, ${\mu _t} = 2.0,{\mu _z} = 2.8$).

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Fig. 6.
Fig. 6. Variation of the OPF on a uniaxial anisotropic sphere with sphere radius $a$ and conical angle $\alpha$ illuminated by a HOBVB: (a) $l = 0$; (b) $l = 1$ (${\varepsilon _t} = 5.913,{\varepsilon _z} = 7.197$, ${\mu _t} = {\mu _z} = 1.0$, ${n_0} = 1.33$).

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Fig. 7.
Fig. 7. Variation of the OPF on a uniaxial anisotropic sphere with $a$ and $\alpha$ illuminated by a HOBVB: (a) $l = 2$; (b) $l = 3$ (${\varepsilon _t} = 5.913,{\varepsilon _z} = 7.197$, ${\mu _t} = {\mu _z} = 1.0$, ${n_0} = 1.33$).

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Fig. 8.
Fig. 8. Variation of the OPF on a uniaxial anisotropic sphere with ${\varepsilon _t}$ and ${\varepsilon _z}$ illuminated by a ZOBB: (a) ${n_0} = 1.33$; (b) ${n_0} = 1.0$ (${\mu _t} = {\mu _z} = 1.0$, $\lambda = 1.064\,{\unicode{x00B5}{\rm m}}$, $a = 0.2\,{\unicode{x00B5}{\rm m}}$, $\alpha = {70^ \circ}$).

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Fig. 9.
Fig. 9. Variation of the OPF ${F_z}$ on a uniaxial anisotropic sphere with ${\varepsilon _t}$ and ${\varepsilon _z}$ illuminated by a HOBVB: (a) $l = 1$; (b) $l = 2$; (c) $l = 3$ (${\mu _t} = {\mu _z} = 1.0$, $\alpha = 0.3\,\,\unicode{x00B5}\rm m$, $\alpha = {80^ \circ}$, ${n_0} = 1.33$).

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Fig. 10.
Fig. 10. Variation of the OPF ${F_z}$ on a uniaxial anisotropic sphere with ${\varepsilon _t}$ and ${\varepsilon _z}$ illuminated by a HOBVB: (a) $l = 1$; (b) $l = 2$; (c) $l = 3$ (${\mu _t} = {\mu _z} = 1.0$, $a = 0.3\,{\unicode{x00B5}{\rm m}}$, $\alpha = {80^ \circ}$, ${n_0} = 1.0$).

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Fig. 11.
Fig. 11. Variation of the OPF ${F_z}$ on a uniaxial anisotropic sphere with ${\varepsilon _t}$ and ${\varepsilon _z}$ illuminated by a ZOBB: (a) ${\mu _t} = {\mu _z} = 2.0$; (b) ${\mu _t} = {\mu _z} = 4.0$; (c) ${\mu _t} = {\mu _z} = 6.0$ ($a = 0.2\,{\unicode{x00B5}{\rm m}}$, $\alpha = {70^ \circ}$, ${n_0} = 1.33$).

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Fig. 12.
Fig. 12. Variation of the OPF ${F_z}$ on a uniaxial anisotropic sphere with ${\mu _t}$ and ${\mu _z}$ illuminated by a HOBVB: (a) $l = 0$; (b) $l = 1$; (c) $l = 2$; (d) $l = 3$ (${\varepsilon _t} = 8.0,{\varepsilon _z} = 5.0$, $a = 0.2\,{\unicode{x00B5}{\rm m}}$, $\alpha = {80^ \circ}$, ${n_0} = 1.33$).

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Equations (14)

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( E x ′ E y ′ E z ′ ) = 1 2 E 0 exp [ i ( k z ′ z ′ + l ϕ ′ ) ] { ( 1 + k z ′ k 0 − k R 2 x ′ 2 k 0 2 R ′ 2 + l ( l − 1 ) ( x ′ − i y ′ ) 2 k 0 2 R ′ 4 ) J l ( k R ′ R ′ ) − k R ( y ′ 2 − x ′ 2 − 2 i l x ′ y ′ ) k 0 2 R ′ 3 J l + 1 ( k R ′ R ′ ) ( l ( l − 1 ) ( 2 x ′ y ′ + i ( x ′ 2 − y ′ 2 ) ) − x ′ y ′ k R 2 R ′ 2 k 0 2 R ′ 4 ) J l ( k R ′ R ′ ) + k R ′ [ 2 x ′ y ′ + i l ( y ′ 2 − x ′ 2 ) ] k 0 2 R ′ 3 J l + 1 ( k R ′ R ′ ) i k 0 R ′ ( 1 + k z ′ k 0 ) [ l ( x ′ − i y ′ ) R ′ J l ( k R ′ R ′ ) − x ′ k R J l + 1 ( k R ′ R ′ ) ] } ,
( H x ′ H y ′ H z ′ ) = 1 2 ε 0 μ 0 E 0 exp [ i ( k z ′ z ′ + l ϕ ′ ) ] { ( l ( l − 1 ) ( 2 x ′ y ′ + i ( x ′ 2 − y ′ 2 ) ) − x ′ y ′ k R 2 R ′ 2 k 0 2 R ′ 4 ) J l ( k R ′ R ′ ) + k R ′ [ 2 x ′ y ′ + i l ( y ′ 2 − x ′ 2 ) ] k 0 2 R ′ 3 J l + 1 ( k R ′ R ′ ) ( 1 + k z ′ k 0 − k R 2 y ′ 2 k 0 2 R ′ 2 + l ( l − 1 ) ( y ′ + i x ′ ) 2 k 0 2 R ′ 4 ) J l ( k R ′ R ′ ) − k R ′ ( x 2 − y 2 + 2 i l x y ) k 0 2 R ′ 3 J l + 1 ( k R ′ R ′ ) i k 0 R ′ ( 1 + k z ′ k 0 ) [ ( l ( y ′ + i x ′ ) R ′ ) J l ( k R ′ R ′ ) − y ′ k R J l + 1 ( k R ′ R ′ ) ] } ,
E i n c = E 0 ∑ n = 1 ∞ ∑ m = − n n [ a mn i n c M mn ( 1 ) ( r , k ) + b mn i n c N mn ( 1 ) ( r , k ) ] , H i n c = E 0 k i ω μ 0 ∑ n = 1 ∞ ∑ m = − n n [ a mn i n c N mn ( 1 ) ( r , k ) + b mn i n c M mn ( 1 ) ( r , k ) ] ,
( a mn i n c b mn i n c ) = ( μ 0 / ε 0 1 ) 2 n + 1 ( k 0 a ) 2 E 0 4 π n ( n + 1 ) ψ n ( k 0 a ) ( n − m ) ! ( n + m ) ! × ∫ 0 2 π ∫ 0 π ( i H r = a i n c E r = a i n c ) P n m ( cos ⁡ θ ) e − i m ϕ sin ⁡ θ d θ d ϕ ,
a mn i n c = i n − m 2 n + 1 n ( n + 1 ) ( n − m ) ! ( n + m ) ! 1 + cos ⁡ α 4 e − i k z z 0 × { [ m π n m ( cos ⁡ α ) + τ n m ( cos ⁡ α ) ] δ m , ( l + 1 ) − [ m π n m ( cos ⁡ α ) − τ n m ( cos ⁡ α ) ] ] δ m , ( l − 1 ) } , b mn i n c = i n − m 2 n + 1 n ( n + 1 ) ( n − m ) ! ( n + m ) ! 1 + cos ⁡ α 4 e − i k z z 0 × { [ m π n m ( cos ⁡ α ) + τ n m ( cos ⁡ α ) ] δ m , ( l + 1 ) + [ m π n m ( cos ⁡ α ) − τ n m ( cos ⁡ α ) ] δ m , ( l − 1 ) } ,
π n m ( cos ⁡ α ) = P n m ( cos ⁡ α ) sin ⁡ α , τ n m ( cos ⁡ α ) = d P n m ( cos ⁡ α ) d α .
E s = E 0 ∑ n = 1 ∞ ∑ m = − n n [ a mn s M mn ( 3 ) ( r , k 0 ) + b mn s N mn ( 3 ) ( r , k 0 ) ] , H s = E 0 k 0 i ω μ 0 ∑ n = 1 ∞ ∑ m = − n n [ a mn s N mn ( 3 ) ( r , k 0 ) + b mn s M mn ( 3 ) ( r , k 0 ) ] ,
∇ × ( μ ¯ ¯ − 1 ⋅ ∇ × E ) − ω 2 ε ¯ ¯ ⋅ E = 0 ,
ε ¯ ¯ = ε 0 [ ε t 0 0 0 ε t 0 0 0 ε z ] , μ ¯ ¯ = μ 0 [ μ t 0 0 0 μ t 0 0 0 μ z ] ,
E I = ∑ q = 1 2 ∑ n = 1 ∞ ∑ m = − n n ∑ n ′ = 1 ∞ 2 π G m n ′ q ∫ 0 π [ A mnq e M mn ( 1 ) ( r , k q ) + B mnq e N mn ( 1 ) ( r , k q ) + C mnq e L mn ( 1 ) ( r , k q ) ] P n ′ m ( cos ⁡ θ k ) k q 2 sin ⁡ θ k d θ k , H I = ∑ q = 1 2 ∑ n = 1 ∞ ∑ m = − n n ∑ n ′ = 1 ∞ 2 π G m n ′ q ∫ 0 π [ A mnq h M mn ( 1 ) ( r , k q ) + B mnq h N mn ( 1 ) ( r , k q ) + C mnq h L mn ( 1 ) ( r , k q ) ] P n ′ m ( cos ⁡ θ k ) k q 2 sin ⁡ θ k d θ k ,
a mn s = 1 h n ( 1 ) ( k 0 a ) [ ∑ q = 1 2 ∑ n ′ = 1 ∞ 2 π G m n ′ q ∫ 0 π A mnq e j n ( k q r ) × P n ′ m ( cos ⁡ θ k ) k q 2 sin ⁡ θ k d θ k − a mn i n c j n ( k 0 a ) ∑ n ′ = 1 ∞ ] , b mn s = 1 h n ( 1 ) ( k 0 a ) i ω μ 0 k 0 [ ∑ q = 1 2 ∑ n ′ = 1 ∞ 2 π G m n ′ q ∫ 0 π A mnq h j n ( k q r ) × P n ′ m ( cos ⁡ θ k ) k q 2 sin ⁡ θ k d θ k − b mn i n c j n ( k 0 a ) ∑ n ′ = 1 ∞ ] ,
F = 1 2 Re ∫ 0 2 π ∫ 0 π [ 1 2 ε E r E + μ H r H − 1 2 ( ε 0 E 2 + μ 0 H 2 ) e ^ r r 2 sin ⁡ θ d θ d ϕ | r > a ] ,
F z = n 0 I 0 c k 0 2 Re ∑ n = 1 ∞ ∑ m = − n n [ i n ( n + 2 ) ( n − m + 1 ) ( n + m + 1 ) ( 2 n + 1 ) ( 2 n + 3 ) N mn − 1 N m n + 1 − 1 ( a m n + 1 i n c a mn s ∗ + a m n + 1 s a mn i n c ∗ + b m n + 1 i n c b mn s ∗ + b m n + 1 s b mn i n c ∗ + 2 a m n + 1 s a mn s ∗ + 2 b m n + 1 s b mn s ∗ ) − m N mn − 2 ( a mn i n c b mn s ∗ + b mn i n c a mn s ∗ + 2 a mn s b mn s ∗ ) ] ,
N mn = ( 2 n + 1 ) ( n − m ) ! 4 π ( n + m ) ! ( m = 0 , ± 1 , ⋯ , ± n ) .
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