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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Appl. Opt.

Appl. Phys. Lett.

M.E.J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, �??Optically driven micromachine elements,�?? Appl. Phys. Lett. 78, 547 (2001).
[CrossRef]

Appl. Phys. Letts.

T.A.Wood, H.F. Gleeson, M.R. Dickinson, and A.J. Wright, �??Mechanisms of optical angular momentum transfer to nematic liquid crystalline droplets,�?? Appl. Phys. Letts. 84, 4292 (2004)
[CrossRef]

S. Juodkazis, S. Matsuo, N. Murazawa, I. Hasegawa, and H. Misawa, �??High-efficiency optical transfer of torque to a nematic liquid crystal droplet,�?? Appl. Phys. Letts. 82, 4657 (2003).
[CrossRef]

Electronic J of Differential Equations

J.H. Crichton and P.L. Marston, �??The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,�?? Electronic J of Differential Equations 04, 37 (2000).

J. Appl. Phys.

J.P. Barton, D.R. Alexander and S.A. Schaub, �??Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,�?? J. Appl. Phys. 66, 4594 (1989).
[CrossRef]

J. Mod. Opt.

S. Bayoudh, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, �??Orientation of biological cells using plane-polarized Gaussian beam optical tweezers,�?? J. Mod. Opt. 50, 1581 (2003).

T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, �??Optical measurement of microscopic torques,�?? J. Mod. Opt. 48, 405 (2001)

J. Opt. Soc. Am. B

Nature (London)

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, �??Optical alignment and spinning of laser-trappedmicroscopic particles,�?? Nature (London) 394, 348 (1998); 395, 621(E) (1998).
[CrossRef]

Opt. Express

Phys. Rev. A

A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, �??Optical application and measurement of torque on microparticles of isotropic nonabsorbing material,�?? Phys. Rev. A 68, 033802 (2003).
[CrossRef]

Phys. Rev. E

Z.F. Lin and S.T. Chui, �??Electromagnetic scattering by optically anisotropic magnetic particle,�?? Phys. Rev. E 69, 056614 (2004).
[CrossRef]

E. Higurashi, R. Sawada, and T. Ito, �??Optically induced angular alignment of trapped birefringent micro-objects by linearly polarized light,�?? Phys. Rev. E 59, 3676 (1999).
[CrossRef]

Phys. Rev. Lett

A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, �??Optical microrheology using rotating laser-trapped particles,�?? Phys. Rev. Lett 92, 198104 (2004).
[CrossRef] [PubMed]

Phys. Rev. Lett.

A. La Porta and M. D. Wang, �??Optical torque wrench: angular trapping, rotation, and torque detection of quartz microparticles,�?? Phys. Rev. Lett. 92, 190801 (2004).
[CrossRef] [PubMed]

Z. Cheng, P.M. Chaikin, and T.G. Mason, �??Light streak tracking of optically trapped thin microdisks,�?? Phys. Rev. Lett. 89, 108303 (2002).
[CrossRef] [PubMed]

Physica A

�?. Farsund and B.U. Felderhof, �??Force, torque, and absorbed energy for a body of arbitrary shape and constitution in an electromagnetic radiation field,�?? Physica A 227, 108 (1996).
[CrossRef]

Other

J.D. Jackon, Classical electrodynamics, 3rd edition (Wiley, New York, 1999).

J. Schwinger, L.L. DeRead, K.A. Milton and W-y Tsai, Classical electrodynamics (Perseus Books, Reading, 1998).

J.A. Stratton, Electromagnetic theory (McGraw-Hill, New York, 1941).

C.F. Bohren and D R Huffman, Absorption and scattering of light by small particles (John Wiley & Sons, New York, 1983).

M.J. Weber, Handbook of optical materials (CRC Press, New York, 2002).
[CrossRef]

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Figures (1)

Fig. 1.
Fig. 1.

Geometry of the scattering problem.

Tables (1)

Tables Icon

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

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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