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Radiation torque on a spherical birefringent particle in the long wave length limit: analytical calculation

Nian Ji, Mengkun Liu, Jihao Zhou, Zhifang Lin, and S. T. Chui

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Abstract

We present an analytical calculation of the radiation torque on a spherical birefringent particle illuminated by plane electromagnetic wave of arbitrary polarization mode and direction of propagation in the small particle limit. The calculation is based on the extended Mie theory and the Maxwell stress tensor formalism. It is found that, even in the small particle limit, the torque is not always normal to the external electric field for the linearly polarized light. For different incident directions and polarization modes of the incident light, the radiation torque τ may exhibit different types of power law dependence on the particle radius a, τ~a γ , with the exponent γ=3, 5, and 6. In the presence of viscous drag, the extraordinary axis of the illuminated particle may be aligned by the optical torque with the incident electric field, the incident magnetic field, or, the incident wave vector, depending on the incident polarization mode and material birefringence of the particle.

©2005 Optical Society of America

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Figures (1)
Fig. 1.
Fig. 1. Geometry of the scattering problem.

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Tables (1)

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Table 1. Final orientation of the extraordinary axis by the radiation torque

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Equations (70)

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τ = q ̂ ( χ o − χ e ) τ 0 sin 2 ϕ e
ε r ε 0 ε ↔ = ε r ε 0 ( 1 0 0 0 1 0 0 0 1 + u a ) ,
∇ × ∇ × ( ε ↔ − 1 · D I ) − k s 2 D I = 0 ,
ε ↔ − 1 = ( 1 0 0 0 1 0 0 0 1 1 + u a ) .
D I = ∑ n , m E mn [ c mn M mn ( 1 ) ( k , r ) + d mn N mn ( 1 ) ( k , r ) ] ,
C mn = [ 2 n + 1 n ( n + 1 ) ( n − m ) ! ( n + m ) ! ] 1 ⁄ 2 .
ε ↔ − 1 · M mn ( 1 ) = ∑ v = 0 2 ∑ u = − v + v [ g ˜ uv mn M uv ( 1 ) + e ˜ uv mn N uv ( 1 ) + f ˜ uv mn L uv ( 1 ) ] ,
ε ↔ − 1 · N mn ( 1 ) = ∑ v = 0 2 ∑ u = − v + v [ g ¯ uv mn M uv ( 1 ) + e ¯ uv mn N uv ( 1 ) + f ¯ uv mn L uv ( 1 ) ] ,
g ˜ uv mn = δ nv δ mu + ( n 2 + n − m 2 ) u a δ nv δ mu n ( n + 1 )
e ˜ uv mn = i ( n + m ) m u a δ n − 1 , v δ mu n ( 2 n + 1 ) + i ( n − m + 1 ) m u a δ n + 1 , v δ mu ( n + 1 ) ( 2 n + 1 )
f ˜ uv mn = − i ( n + m ) m u a δ n − 1 , v δ mu ( 2 n + 1 ) + i ( n − m + 1 ) m u a δ n + 1 , v δ mu ( 2 n + 1 )
g ¯ uv mn = − i ( n + m ) ( n + 1 ) m u a δ n − 1 , v δ mu n ( n − 1 ) ( 2 n + 1 ) − i ( n − m + 1 ) n m u a δ n + 1 , v δ mu ( n + 1 ) ( n + 2 ) ( 2 n + 1 )
e ¯ uv mn = δ nv δ mu + [ ( 2 n 2 + 2 n + 3 ) m 2 + ( 2 n 2 + 2 n − 3 ) n ( n + 1 ) ] u a δ nv δ mu n ( n + 1 ) ( 2 n − 1 ) ( 2 n + 3 )
f ¯ uv mn = − ( n 2 + n − 3 m 2 ) u a δ n v δ mu ( 2 n − 1 ) ( 2 n + 3 ) + ( n + 1 ) ( n + m − 1 ) ( n + m ) u a δ n − 2 , v δ mu ( 2 n − 1 ) ( 2 n + 1 )
∑ v = 1 2 ∑ u = − v + v E uv E mn [ g ˜ mn uv c uv + g ¯ mn uv d uv ] = λ c mn ,
∑ v = 1 2 ∑ u = − v + v E uv E mn [ e ˜ mn uv c uv + e ¯ mn uv d uv ] = λ d mn .
V l = − i ε 0 ε r λ l ∑ n = 1 2 ∑ m = − n n E mn [ c mn , l M mn ( 1 ) ( k l , r ) + d mn , l N mn ( 1 ) ( k l , r ) ]
∇ · V l = 0 ,
∇ × ∇ × ( ε ↔ − 1 · V l ) − k s 2 V l = 0 .
D I = ∑ l = 1 2 n d α l V l ,
E I = 1 ε 0 ε r ε ↔ − 1 · D I = − ∑ n = 1 2 ∑ m = − n n i E mn ∑ l = 1 2 n d α l [ c mn , l M mn ( 1 ) ( k l , r ) + d mn , l N mn ( 1 ) ( k l , r ) + w mn , l λ l L mn ( 1 ) ( k l , r ) ]
+ ∑ l = 1 2 n d i α l [ w 00 , l λ l L 00 ( 1 ) ( k l , r ) ]
H I = − i ω μ 0 ∇ × E I = − 1 ω μ 0 ∑ n = 1 2 ∑ m = − n n E mn ∑ l = 1 2 n d k l α l [ d mn , l M mn ( 1 ) ( k l , r ) + c mn , l N mn ( 1 ) ( k l , r ) ]
E s = ∑ n = 1 2 ∑ m = − n n i E mn [ a mn N mn ( 3 ) ( k 0 , r ) + b mn M mn ( 3 ) ( k 0 , r ) ]
H s = k 0 ω μ 0 ∑ n = 1 2 ∑ m = − n n E mn [ b mn N mn ( 3 ) ( k 0 , r ) + a mn M mn ( 3 ) ( k 0 , r ) ]
k 0 = k 0 ( sin θ k cos φ k e x + sin θ k sin φ k e y + cos θ k e z ) ,
E inc = E 0 p ̂ e i k 0 · r = E 0 ( p θ θ ̂ k + p φ φ ̂ k ) e i k 0 · r ,
H inc = E 0 k 0 × p ̂ ω μ 0 e i k 0 · r = k 0 ω μ 0 E 0 ( p θ φ ̂ k − p φ θ ̂ k ) e i k 0 · r ,
k ̂ 0 × θ ̂ k = φ ̂ k , θ ̂ k × φ ̂ k = k ̂ 0 , φ ̂ k × k ̂ 0 = θ ̂ k .
E inc = − ∑ n = 1 2 ∑ m = − n n i E mn [ p mn N mn ( 1 ) ( k 0 , r ) + q mn M mn ( 1 ) ( k 0 , r ) ]
H inc = − k 0 ω μ 0 ∑ n = 1 2 ∑ m = − n n E mn [ q mn N mn ( 1 ) ( k 0 , r ) + p mn M mn ( 1 ) ( k 0 , r ) ] ,
p mn = [ p θ τ mn ( cos θ k ) − i p φ π mn ( cos θ k ) ] e − i m φ k ,
q mn = [ p θ π mn ( cos θ k ) − i p φ τ mn ( cos θ k ) ] e − i m φ k ,
π mn ( cos θ ) = C mn m sin θ P n m ( cos θ ) ,
τ mn ( cos θ ) = C mn d d θ P n m ( cos θ ) ,
1 m s ∑ l = 1 2 n d 1 k ¯ l λ l j n ( k ¯ l m s η ) w mn , l α l + ξ n ′ ( η ) a mn + 1 m s ∑ l = 1 2 n d 1 k ¯ l ψ n ′ ( k ¯ l m s η ) d mn , l α l = ψ n ′ ( η ) p mn
ξ n ( η ) b mn + 1 m s ∑ l = 1 2 n d 1 k ¯ l ψ n ( k ¯ l m s η ) c mn , l α l = ψ n ( η ) q mn
ξ n ( η ) a mn + μ 0 μ s ∑ l = 1 2 n d ψ n ( k ¯ l m s η ) d mn , l α l = ψ n ( η ) p mn
ξ n ′ ( η ) b mn + μ 0 μ s ∑ l = 1 2 n d ψ n ′ ( k ¯ l m s η ) c mn , l α l = ψ n ′ ( η ) q mn
η = k 0 a , m s = 1 + u a k s k 0 , k ¯ l = k l k s , λ l = k s 2 k l 2 = 1 k ¯ l 2 ,
ψ n ( z ) = z j n ( z ) , ξ n ( z ) = z h n ( 1 ) ( z ) .
E e = ∑ n = 1 2 ∑ m = − n n i E mn [ a mn N mn ( 3 ) ( k 0 , r ) + b mn M mn ( 3 ) ( k 0 , r ) − p mn N mn ( 1 ) ( k 0 , r ) − q mn M mn ( 1 ) ( k 0 , r ) ]
H e = k 0 ω μ 0 ∑ n = 1 2 ∑ m = − n n E mn [ b mn N mn ( 3 ) ( k 0 , r ) + a mn M mn ( 3 ) ( k 0 , r ) − q mn N mn ( 1 ) ( k 0 , r ) − p mn M mn ( 1 ) ( k 0 , r ) ]
T ̂ = 1 2 Re [ E e D e * + H e B e * − 1 2 ( E e · D e * + H e · B e * ) I ̂ ]
τ = − ∫ d S · K ̂ = − ∫ [ e r · K ̂ ] d S
K ̂ = T ̂ · [ r × I ̂ ] = T ̂ × r .
τ = − ∫ [ e r · ( T ̂ × r ) ] dS = ∫ [ r × ( T ̂ · e r ) ] dS = r 3 ∫ e r × [ T ̂ · e r ] d Ω ,
ξ n ( ρ ) ∼ ( − i ) n + 1 exp ( i ρ ) , ζ n ( ρ ) ∼ i n + 1 exp ( − i ρ ) , ψ n ( ρ ) ∼ [ ξ n ( ρ ) + ζ n ( ρ ) ] ⁄ 2 .
τ x = Re [ 𝒩 1 ] , τ y = Im [ 𝒩 1 ] , τ z = − Re [ 𝒩 2 ] ,
𝒩 1 = 2 π ε 0 ∣ E 0 ∣ 2 k 0 3 ∑ n = 1 2 ∑ m = − n n − 1 ρ mn [ a mn a m 1 n * + b mn b m 1 n *
− 1 2 ( a mn p m 1 n * + p mn a m 1 n * + b mn q m 1 n * + q mn b m 1 n * ) ]
𝒩 2 = 2 π ε 0 ∣ E 0 ∣ 2 k 0 3 ∑ n = 1 2 ∑ m = − n n m [ a mn a m n * + b mn b m n *
− 1 2 ( a mn p m n * + p mn a m n * + b mn q m n * + q mn b m n * ) ]
𝒩 1 = c 3 η 3 + c 5 η 5 + c 6 η 6 + o ( η 7 ) ,
c 3 = 3 p θ u a ε r [ Re p φ − i p θ cos θ k ] sin θ k 4 π 2 ( 2 + ε r ) ( u a ε r + ε r + 2 ) ε 0 λ 0 3 ∣ E 0 ∣ 2
c 5 = [ i ∣ p φ ∣ 2 u a ε r ( ε r − 1 ) 2 f 0 sin 2 θ k + i p θ 2 u a ε r ( f 1 + f 2 cos 2 θ k ) sin 2 θ k
+ u a ε r p θ [ Re p φ ] ( f 3 − f 2 cos 2 θ k ) sin θ k ] ε 0 λ 0 3 ∣ E 0 ∣ 2
c 6 = 3 p θ u a 2 ε r 2 [ Im p φ ] sin θ k 2 π 2 ( 2 + ε r ) 2 ( u a ε r + ε r + 2 ) 2 ε 0 λ 0 3 ∣ E 0 ∣ 2
τ x = 6 π p θ [ Re p φ ] u a ε r sin θ k ( 2 + ε r ) ( u a ε r + ε r + 2 ) ε 0 a 3 ∣ E 0 ∣ 2 + 96 π 4 p θ [ Im p φ ] u a 2 ε r 2 sin θ k ( 2 + ε r ) 2 ( u a ε r + ε r + 2 ) 2 λ 0 3 ε 0 a 6 ∣ E 0 ∣ 2 ,
τ y = − 3 π p θ 2 u a ε r sin 2 θ k ( 2 + ε r ) ( u a ε r + ε r + 2 ) ε 0 a 3 ∣ E 0 ∣ 2 + 4 π 3 ∣ p φ ∣ 2 u a ε r ( ε r − 1 ) 2 sin 2 θ k 15 ( 2 ε r + 3 ) ( 2 ε r + 3 + u a ε r ) λ 0 3 ε 0 a 5 ∣ E 0 ∣ 2 ,
τ x = 6 π p θ [ Re p φ ] u a ε r sin θ k ( 2 + ε r ) ( u a ε r + ε r + 2 ) ε 0 a 3 ∣ E 0 ∣ 2 ,
τ y = − 3 π p θ 2 u a ε r sin 2 θ k ( 2 + ε r ) ( u a ε r + ε r + 2 ) ε 0 a 3 ∣ E 0 ∣ 2 ,
cos θ τ = − cos θ k Q ( p θ 1 − q − ∣ Re p φ ∣ 1 + q )
q = p θ 2 − ∣ p φ ∣ 2 { ( p θ 2 − ∣ p φ ∣ 2 ) 2 + 4 p θ 2 [ Re p φ ] 2 } 1 ⁄ 2 and Q = { 2 p θ 2 cos 2 θ k + 2 [ Re p φ ] 2 } 1 ⁄ 2 .
τ = − 6 π u a ε r p θ sin θ k ( 2 + ε r ) ( 2 + ε r + u a ε r ) ε 0 a 3 ∣ E 0 ∣ 2 e z × ( p θ θ ̂ k + p φ φ ̂ k )
= 3 π ( χ e − χ o ) sin 2 ϕ e ( 2 + ε r ) ( 2 + ε r + u a ε r ) ε 0 a 3 ∣ E 0 ∣ 2 q ̂
τ 0 = 3 π ( 2 + ε r ) ( 2 + ε r + u a ε r ) ε 0 a 3 ∣ E 0 ∣ 2 .
τ = τ y e y = 4 π 3 ( ε r − 1 ) 2 ε r u a sin 2 θ k 15 ( 2 ε r + 3 ) ( 3 + 2 ε r + u a ε r ) λ 0 2 ε 0 a 5 ∣ E 0 ∣ 2 e y ,
τ = τ x e x = 96 π 4 p θ Im ( p φ ) u a 2 ε r 2 sin θ k ( 2 + ε r ) 2 ( u a ε r + ε r + 2 ) 2 λ 0 3 ε 0 a 6 ∣ E 0 ∣ 2 e x .
t E ≡ ( e z × E inc ) · τ e z · E inc = 6 π ε r u a ( p φ 2 + p θ 2 cos 2 θ k ) ( 2 + ε r ) ( 2 + ε r + u a ε r ) ε 0 a 3 ∣ E 0 ∣ 2 .
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