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J. Opt. Soc. Am. A / Vol. 32, No. 1 / January 2015
H. Yu and W. She
Rotation dynamics of particles trapped in a rotating beam Huachao Yu and Weilong She* State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-Sen University, Guangzhou 510275, China *Corresponding author:
[email protected] Received September 8, 2014; revised October 28, 2014; accepted October 28, 2014; posted November 4, 2014 (Doc. ID 222624); published December 11, 2014 The rotation dynamics of particles trapped in a rotating beam is theoretically investigated. We find that there is a critical angular speed for the rotating beam. If the angular speed of the rotating beam is smaller than the critical value, the angular velocity of the trapped particle is nearly the same as that of the rotating beam, which is in accord with existing experimental observation. Otherwise, the angular velocity of the trapped particles will become periodic or quasi-periodic with time, depending on the beam polarization, which, to the best of our knowledge, has not been previously reported. Moreover, we also propose some methods to determine the ratio between the beam power and the maximal angular speed of the trapped particle, which can be used to estimate the minimum power required to rotate the particle at a given angular speed. © 2014 Optical Society of America OCIS codes: (290.4020) Mie theory; (350.4855) Optical tweezers or optical manipulation. http://dx.doi.org/10.1364/JOSAA.32.000090
1. INTRODUCTION Since the invention of optical tweezers [1], many methods for rotating particles with light have been proposed [2–16], thus offering potential applications in optically driven machines [17], cell manipulation [18], and microfluidics [19]. Among these methods, the one using a rotating beam to set particles into rotation has drawn particular attention from researchers [5–11,20], since it does not rely on intrinsic properties of the particles and can be applied to any particle that can be optically trapped [5]. Thus, as a result of this method, many improved ways for generating a rotating beam have been proposed [21–24]. In the theoretical aspect, some works concerning the angular momentum of the rotating beam have been developed [25–29], but research on the mechanical effect of the rotating beam on the microtrapped particle, to the best of our knowledge, has never before been reported. To gain a better understanding of such effect, in the present paper, we will make a theoretical investigation on the rotation dynamics of the particle trapped in a rotating beam. It has been experimentally reported that the angular velocity of the trapped particle is the same as that of the rotating beam [5,7,9,20]. However, our theory shows there is a critical angular speed for the rotating beam. When the angular speed of the rotating beam is smaller than this critical angular speed, it becomes possible to rotate the particle synchronously with the rotating beam. Otherwise, the angular velocity of the particle will become periodic or quasi-periodic with time, depending on the beam polarization. The polarization dependence of the particle’s angular velocity will gradually become significant as the angular speed of the rotating beam increases. In addition, our theory also indicates that the ratio between the beam power and the particle’s maximal angular speed can be used for predicting the minimum power required to drive the particle at a given angular speed. Further, some methods for determining this ratio have been proposed. It has been found 1084-7529/15/010090-11$15.00/0
that the theoretically predicted minimum power is close to the experimental measurement.
2. RADIATION FORCE ON PARTICLE TRAPPED IN ROTATING BEAM As the azimuthal radiation force drives the rotation of the particle trapped in a rotating beam, we need to know the radiation force first in order to study the rotation dynamics of the particle. In this section, we will present the calculation of the radiation force. Generally, a rotating beam can be an arbitrary shape. But for simplicity, here we only consider the one generated by introducing an angular frequency shift into two copropagating vortex beams with opposite azimuthal index [21,22]. If the two vortex beams are Laguerre–Gaussian beams with azimuthal index m0 , then the rotating beam pattern looks like those shown in Fig. 1. In order to hold particles in a rotating beam, the beam must be focused to form a trap. Here, we assume the beam is propagating along the z axis and is focused by an aplanatic lens obeying the sine condition. If the frequency shift between the two supposition vortex beams is 2ω0 , then the electric field of the focused rotating beam has the form −iωω0 t E0 xe−iω−ω0 t ; E0 x; t 2 Re E0 − xe
(1)
where E0 x are the electric phasor fields of the focused vortex beams with frequency ω ω ω0 . If the transverse mode functions of the two unfocused vortex beams with opposite azimuthal index m0 , in the cylindrical coordinates of the spatial variable x, are written as Ψm0 ρ; ϕ; z; x0 , where x0 is the position of the origin with respect to the focal point of the focused beam, then, according to the vector diffraction theory [30,31], E0 x are expressed as © 2015 Optical Society of America
H. Yu and W. She
Vol. 32, No. 1 / January 2015 / J. Opt. Soc. Am. A
C m0 ≈
p !jm0 j1 v u 2π 2f u ; t 2 w0 γ jm j 1; 2f sin2 ϑ 0
w20
91
(6)
0
where m0 ≠ 0, and γm; z is the incomplete Gamma function [32]. If we introduce Fig. 1. Patterns of rotating even Laguerre–Gaussian modes with azimuthal index (a) m0 1, (b) m0 2, and (c) m0 3. The angular velocity of the rotating beam is Ω ω0 ∕m0 , where 2ω0 is the angular frequency shift between the two Laguerre–Gaussian modes with opposite azimuthal index.
E0 x
Z
π 2
0
Z dϑ
2π
0
dφ
2 k ¯ ik ·x ; cos ϑ sin ϑE ϑ; φ · Ie 2π (2)
where k are the wave vectors of the fields with frequency ω , k ; ϑ; φ are the spherical coordinates of k in the reciprocal space, I¯ is the unit tensor, and E ϑ; φ are the angular spectrums of the forms p iω cos ϑ E ϑ; φ A eiφ e A− e−iφ e− eik ·x0 k cos ϑ × Ψm0 f sin ϑ; φ π; zen ; 0Θϑ0 − ϑ;
(3)
p where cos ϑ, f , and 2ϑ0 are the apodization factor, focal length, and angular aperture of the lens, respectively; zen is the axial position of the entrance of the lens, Θz is the Heaviside step function, A are the normalized amplitudes of the p two circularly polarized components, and e eϑ ieφ ∕ 2 are the circular polarization bases in the reciprocal spherical coordinates. If the two unfocused vortex beams are Laguerre– Gaussian beams with focal planes at the entrance of the lens, then the transverse mode function takes the form Ψm0 ρ; ϕ; zen ; 0 C m0
jm j 2 ρ 0 ρ exp − 2 eim0 ϕ−π ; f w0
(4)
where w0 is the Gaussian waist size, and C m0 is the normalization constant. To form a rotating beam, the electric fields of the two unfocused vortex beams should have the same magnitudes, namely, jω Ψm0 j jω− Ψ−m0 j. If choosing a proper zero point of time, we have ω C m0 ω− C −m0 , where C m0 are positive. Usually, the introduced angular frequency shift is far smaller than the central frequency of the beam, j2ω0 j ≪ ω. Under this condition, the normalization constants C m0 are chosen so that the time average power of the incident beam and the normalized amplitudes A satisfy the relation P−
ω 2π
Z
2π ω
0
dt0
I ∂V 0
d2 Σ · S0 x; t t0
σ
jA j2 − jA− j2 jA j2 jA− j2
7
to describe the degree of polarization of the unfocused beam, then the normalized amplitude of each circularly polarized component can be expressed as
A e
iδ∓χ
r 1 σμ0 μP ; 4ωk
(8)
where δ is an arbitrary global phase, and 2χ is the phase difference between the normalized amplitudes of the circularly polarized components. Note that the major axis of the polarization ellipse of the unfocused beam is oriented at angle χ with respect to the x axis. The radiation force on the particle trapped in a rotating beam is now calculated with the generalized Lorenz–Mie theory [33,34], which is rigorous and valid for particles with arbitrary size. Usually there are multiple particles manipulated in a rotating beam, but, for simplicity, in our theoretical approach, the radiation force is calculated with a singleparticle model, where the particle is treated as a sphere. The particle and the ambient fluid are assumed to be dispersionless at the vicinity of the central frequency of the beam. In addition, the origin of the coordinate system is chosen to be the center of the particle, as shown in Fig. 2. The radius and ¯ respectively. The refractive index of the particle are R and n, refractive index of the fluid is n. The electric field E outside the spherical particle is the supposition of the incident field E0 and the scattered field E1 . Note that the expression of the scattered field is similar to Eq. (1), except for the superscript. In the case of absence of free charge, the phasors of the incident and scattered electric fields are solenoid. They can be expanded in the solenoid vector spherical wave functions as [34]
2ωk jA j2 jA− j2 ; μ0 μ (5)
where S0 x; t E0 x; t × H 0 x; t is the Poynting vector of the incident field, and V 0 is the semi-infinite region z ≥ −R. It can be deduced from Eq. (5) that the normalization constants are
Fig. 2. Spherical particle with radius R and refractive index n¯ is immersed in fluid with refractive index n and illuminated by a focused beam rotating axially with angular velocity Ω. The origin of coordinates is at the center of the particle. The focal point of the beam is located at −x0 .
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J. Opt. Soc. Am. A / Vol. 32, No. 1 / January 2015
Ej x
H. Yu and W. She
8 ∞ X l < X l1
s μ0 μk P h j a;lm M j il1 ω C m0 lm k ; x : 2ll 1ω m−l
j bj ;lm N lm k ; x
9 i= ;
;
(9)
where j 0 or 1, μ0 is the vacuum permeability, μ is the rel ative permeability of the fluid, a lm and blm are, respectively, the multipole coefficients of the TE and TM waves, and j M j lm k; x and N lm k; x are the solenoid vector spherical wave functions [35]: i j M j lm k; x − p x × ∇hl krY lm θ; ϕ; ll 1 1 j Nj lm k; x ∇ × M lm k; x; k
(10)
where r; θ; ϕ are the spherical coordinates of x, h0 l kr is the spherical Bessel function j l kr, and Y lm θ; ϕ is the normalized spherical harmonic. According to Faraday’s law of induction, the phasors of the incident and scattered magnetic fields can be represented as ∇ × Ej x iω s ( ∞ X l X μ0 μk P j il k C m0 bj ;lm M lm k ; x 2ll 1ω l1 m−l )
Bj x
j aj ;lm N lm k ; x :
(11)
By applying the boundary conditions of the electromagnetic fields at the surface of the spherical particle, it can be shown that the multipole coefficients of the incident and scattered fields satisfy the relations a1 ;lm −
μj l k Rk¯ Rj l k¯ R0 − μj ¯ l k¯ Rk Rj l k R0 a0 ;lm 1 0 μh k Rk¯ Rj l k¯ R − μj ¯ l k¯ Rk Rh1 k R0 l
0
2π ω
dt0
I ∂V
nP ¯ f; d2 Σ · Θx; t t0 ≡ c
(14)
where V is a concentric spherical volume containing the particle, as shown in Fig. 2, d2 Σ is the area element directed along the outward normal to the surface of V , f is the dimensionless ¯ is the Maxwell stress tensor: radiation force, and Θ 1 ¯ Θx; t Dx; t · Ex; t Bx; t · Hx; tI¯ 2 − Dx; tEx; t − Bx; tHx; t:
(15)
Then, by substituting Eqs. (9) and (11) into Eq. (14), one can obtain the expression of the dimensionless transverse radiation force, which is a bit lengthy and given in Eq. (A1). ¯ 0 Rj ≪ c, we can make the folHowever, if jnω0 Rj ≪ c and jnω lowing approximation for the scattering coefficients: TE ¯ ¯ S TE l k R; k R ≈ S l kR; kR;
(16)
TM ¯ ¯ S TM l k R; k R ≈ S l kR; kR;
(17)
¯ where k nω∕c and k¯ nω∕c. Then Eq. (A1) can be simplified to 0 0 0 f x if y ≈ Φa0 lm ; alm ; blm ; blm ;
(18)
0 where a0 lm and blm are, respectively, given in Eqs. (A2) and (A3), and the function Φ is given in Eq. (A4). The radial and azimuthal components of the dimensionless radiation force with respect to the beam axis can be obtained from the formula
f ρ if ϕ e−iϕ0 f x if y ;
(19)
where ϕ0 is the azimuthal angle of x0 . Equation (A1) [or (18)] shows that the radiation force can be determined if the multipole coefficients of the incident beam are known. These coefficients can be calculated with the method presented in Appendix B.
(12)
3. ROTATION DYNAMICS OF PARTICLE TRAPPED IN ROTATING BEAM
εj l k Rk¯ Rj l k¯ R0 − ε¯ j l k¯ Rk Rj l k R0 0 b1 b;lm ;lm − 1 εh k Rk¯ Rj l k¯ R0 − ε¯ j l k¯ Rk Rh1 k R0 l
Z
l
0 ¯ ≡S TE l k R; k Ra;lm ;
0 ¯ ≡S TM l k R; k Rb;lm ;
ω F− 2π
l
(13)
¯ ∕c, ε is the relative permittivity of the fluid, ε¯ where k¯ nω and μ¯ are, respectively, the relative permittivity and per0 meability of the trapped particle, ξhj l ξ denotes the derivaj ¯ tive of ξhl ξ with respect to ξ, and S TE l k R; k R and TM ¯ S l k R; k R are, respectively, the scattering coefficients of the TE and TM waves. The time average of the radiation force on the particle over an optical period can be calculated as [31]
If the radiation force on the particle trapped in a rotating beam is known, the rotation dynamics of the particle can be studied with its equation of motion in the azimuthal direction: d2 ϕ dρ dϕ dϕ nP M ρ0 20 2 0 0 γρ0 0 f x ; t; c ϕ 0 dt dt dt dt
(20)
where M is the mass of the particle, and γ is the drag force coefficient. Now we will determine the form of the dimensionless azimuthal force. As the trapped particle considered here is in micrometers, and ω0 ∕π < 1 kHz, the dimensionless force can be evaluated with Eq. (18). It can be shown from Eqs. (A2), 0 (A3), (B8), (B9), (B13), and (B14) that a0 lm and blm take the forms
H. Yu and W. She
Vol. 32, No. 1 / January 2015 / J. Opt. Soc. Am. A
iδ −imϕ0 im0 ϕ0 −ω0 t −im0 ϕ0 −ω0 t a0 C m0 A0 C −m0 A0 ; lm −e e ;lm e −;lm e
(21) 0 iδ −imϕ0 C B0 eim0 ϕ0 −ω0 t C −im0 ϕ0 −ω0 t ; b0 m0 ;lm −m0 B−;lm e lm −e e
(22)
where the Stokes’ law γ 6πηR [36] has been used in the last step, and η is the viscosity of the fluid. The radial position of the trapped particle usually remains unchanged during the particle’s rotation, so ρ0 can be regarded as a timeindependent quantity. Then by properly choosing the initial value of ϕ0 , we obtain an approximate solution for ϕ0 , which is 1 tan m0 ϕ0 − ω0 t argV − V − 2 p p Δ tan ΔΩ0 t − jV − V − j ; ImV 0 − ωΩ00
where p p 0 iϕ0 −χ 1 σ − K 0 e−iϕ0 −χ 1 − σ ; A0 ;lm G;lm e ;lm
(23)
p p 0 iϕ0 −χ 1 σ K 0 e−iϕ0 −χ 1 − σ : B0 ;lm G;lm e ;lm
(24)
Accordingly, the dimensionless transverse force has the form f ρ if ϕ V 0 V ei2m0 ϕ0 −ω0 t V − e−i2m0 ϕ0 −ω0 t ;
(26)
0 0 0 jC 2m0 jΦA0 −;lm ; A;lm ; B−;lm ; B;lm ;
(27)
0 0 0 V − ≈ jC 2m0 jΦA0 ;lm ; A−;lm ; B;lm ; B−;lm :
(28)
V ≈
(33)
Ω0 t
dϕ0 t ω0 dt m0
p
Ω0 ImV 0 − ω0 Δ sec2 ΔΩ0 t 1 p p
: m0 ImV 0 − ωΩ00 2 Δ tan ΔΩ0 t − jV − V − j 2 (34)
In deriving the above formulas, we have used the fact that C m0 ≈ C −m0 . It can be shown from Eqs. (26), (27), and (28) that V 0 and V are generally dependent on ϕ0 , except in the case of σ 1, where the unfocused beam is circularly polarized. Next, we will consider the rotation of the trapped particle according to the value of σ. A. Rotation of Particle in Rotating Beam with jσj 1 First, let us investigate the rotation of the trapped particle in the case of jσj 1. The dimensionless azimuthal force, namely, the imaginary part of Eq. (25), is f ϕ ImV 0 jV − V − j sin2m0 ϕ0 − 2ω0 t argV − V − : (29) Obviously, f ϕ is a periodic function of ϕ0 . The period is π∕jm0 j. However, even in this case, the equation of motion in Eq. (20) still cannot be solved analytically. Nevertheless, in experiments, the trapped particle is usually immersed in a low-Reynolds-number fluid, where inertia force is far smaller than the drag force [36], so the first term in the left-hand side of Eq. (20) can be ignored. The equation of motion in Eq. (20) becomes dϕ0 Ω0 f ; dt m0 ϕ
(30)
m0 nP m0 nP ; γρ0 c 6πηRρ0 c
(31)
where Ω0
ω 2 Δ ImV 0 − 00 − jV − V − j2 : Ω
Taking the time derivative of Eq. (32), we obtain the angular velocity of the trapped particle:
0 0 0 V 0 ≈ jC 2m0 jΦA0 ;lm ; A;lm ; B;lm ; B;lm 0 0 0 jC 2m0 jΦA0 −;lm ; A−;lm ; B−;lm ; B−;lm ;
(32)
where
(25)
where
93
Note that the time evolution of the angular velocity depends on the sign of Δ. If Δ < 0, or equivalently, ImV 0 − jV − V − jΩ0 < ω0 < ImV 0 jV − V − jΩ0 ; (35) the angular velocity, as time increases, will tend to a constant: Ω0 ∞
ω0 Ω: m0
(36)
Thus, in this case the angular velocity of the trapped particle is the same as the rotating beam. In addition, Eq. (36) also shows that, for a given angular frequency shift, the larger jm0 j is, the slower the particle rotates. From Eqs. (34) and (35), it can be shown that the angular velocity has a maximal value of nP 6πηRρ0 c
(37)
nP : 6πηRρ0 c
(38)
Ω0;max ImV 0 jV − V − j and a minimal value of Ω0;min ImV 0 − jV − V − j
These extrema values seem to be independent of m0 . But, actually, it can be seen from Fig. 3(a) that ImV 0 ≈ 0 and the maximum of jV − V − j decreases with m0 . Thus, the maximal and minimal values of the angular velocity are implicitly dependent on m0 . In Fig. 3(b), the variation of the maximal angular velocity of a silica sphere with its radial position is shown. One can see that when kρ0 < 6, the maximal angular velocity decreases with m0 . But when kρ0 > 9, the maximal angular velocity increases with m0 . The maximum point of
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H. Yu and W. She
The average value of the angular velocity is 1 Ω0 ImV 0 − ω0 p 0 ¯ ω Ω0 ΔΩ : m0 0 jΩ0 ImV 0 − ω0 j
Fig. 3. (a) ImV 0 and jV − V − j as functions of kρ0 . (b) Maximal angular velocity of the trapped particle as a function of kρ0 . In the two figures, the power, wavelength, and degree of polarization of the beams are 1 mW, 1064 nm, and 1, respectively. The trapped particle is a silica sphere with n¯ 1.450 [37] and R 500 nm. The fluid surrounding the particle is water with n 1.325 [37] and η 0.8937 mPa · s [38]. The angular aperture and focal length of the lens are ϑ0 1.125 and f 3w0 , respectively. As a particle is usually trapped near the focal plane, we have set kz0 0.
(44)
Note that when Δ < 0, Eqs. (43) and (44) become meaningless. So we need another way to define the average angular velocity. According to Eqs. (34) and (36), the angular velocity will rapidly tend to ω0 ∕m0 as time increases from zero; thus, ¯ 0 as ω0 ∕m0 for Δ < 0. The above results, we can define Ω Eqs. (43) and (44), together with Eq. (33), show that when Δ > 0, as the magnitude of the angular frequency shift or the angular speed of the rotating beam increases, the period ¯ 0 j will decrease. This is T and the average angular speed jΩ because, when the beam is rotating very fast, the azimuthal force will be not large enough to make the trapped particle rotate synchronously with the beam. In Figs. 4(a) and 4(b), the variation of the angular velocity with time and the variation of the average angular velocity with the angular frequency shift are shown, respectively. One can see that they are really in accord with the above discussion. Moreover, it can be inferred from Fig. 4 that, when Δ < 0, the angular velocity of the trapped particle can be linearly controlled with ω0 . It also can be inferred from Eq. (31) and Fig. 4(b) that, as the beam power increases, the linear control range also increases.
Ω0;max kρ0 also increases with m0 . Also, it can be verified that ImV 0 and jV − V − j have the properties ImV 0 jσ−1 −ImV 0 jσ1 ;
(39)
jV − V − jσ−1 jV − V − jσ1 :
(40)
So, rigorously speaking, the rotation of the particle in the case of σ 1 is not the same as that in the case of σ −1. However, since usually ImV 0 ≈ 0, the difference between these two cases is small. If Δ > 0, namely, ω0 < ImV 0 − jV − V − jΩ0
(41)
ω0 > ImV 0 jV − V − jΩ0 ;
(42)
or
the angular velocity will become a periodic function with period π T p 0 : ΔΩ
(43)
Fig. 4. (a) Time variation of the angular velocity of the trapped particle. (b) Average angular velocity of the trapped particle as a function of the angular frequency shift. The azimuthal index and power of the beam are m0 1 and P 1 mW, respectively. The parameters of the particle and the fluid are the same as those used in Fig. 3. The radial position of the particle is ρ0 500 nm.
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B. Rotation of Particle in Rotating Beam with jσj ≠ 1 Second, let us consider the rotation of the trapped particle for the case of jσj ≠ 1. Obviously, V 0 and V are dependent on ϕ0 in this case. They are of the forms V i V i0 V i ei2ϕ0 −χ V i− e−i2ϕ0 −χ ;
(45)
where V ij are given in Appendix C. Accordingly, the dimensionless transverse force has the form f ρ if ϕ V 00 V 0 ei2m0 ϕ0 −ω0 t V −0 e−i2m0 ϕ0 −ω0 t V ei2m0 2ϕ0 −ω0 t−χ V −− e−i2m0 2ϕ0 −ω0 t−χ V − ei2m0 −2ϕ0 −ω0 tχ V − e−i2m0 −2ϕ0 −ω0 tχ V 0 ei2ϕ0 −χ V 0− e−i2ϕ0 −χ :
(46)
By taking the imaginary part of the above formula, we obtain the dimensionless azimuthal force: f ϕ ImV 00 jV 0 − V −0 j sin2m0 ϕ0 − 2ω0 t argV 0 − V −0 jV 0 − V 0− j sin2ϕ0 − 2χ argV 0 − V 0− jV − V −− j × sin2m0 2ϕ0 − ω0 t − χ argV − V −− jV − − V − j × sin2m0 − 2ϕ0 − ω0 t χ argV − − V − :
(47)
From the expressions p of V ij [Eqs. (C2) to (C5)], it can be seen that V i ∝ 1 − σ 2 . Thus, when σ 1, in the right-hand side of Eq. (47), the last three terms vanish, and the first two terms reduce to the right-hand side of Eq. (29). For σ 0, the variations of ImV 00 and the coefficients of the sine functions in Eq. (47) with kρ0 are plotted in Fig. 5, from which one can infer that ImV 00 0, and the term proportional to jV 0 − V −0 j in Eq. (47) is the dominant contribution to f ϕ . Also, it can be shown from Eqs. (40), (45), and (C4) that jV 0 − V −0 j are independent of σ. So we have jV 0 − V −0 j jV − V − jσ1 :
(48)
Therefore, we can expect that the average motion of the particle in the case of jσj ≠ 1 should be nearly the same as the one described by Eq. (34). Actually, if ignoring the subordinate terms, namely, the last three terms in the right-hand side of Eq. (47), we obtain an approximate analytical solution of Ω0 jσ0 , which is similar to Ω0 jσ1 given in Eq. (34) but with V i replaced by V i0 and Δ replaced by ω 2 Δ0 ImV 00 − 00 − jV 0 − V −0 j2 : Ω
(49)
Note that Δ Δ0 jσ1 . According to Eq. (48) and the fact that ImV 00 0 ≈ ImV 0 jσ1 , we have Δ0 ≈ Δ. Thus, the approximate analytical solution of Ω0 jσ0 is nearly the same as Ω0 jσ1 . If the last three terms in the right-hand side of Eq. (47) are retained, the equation of motion in Eq. (30) does not have analytical solutions and must be solved numerically. A comparison of the approximate analytical solution and the numerical solution of Ω0 jσ0 is made in Fig. 5. It can be seen from Fig. 6 that the parameter Δ0 defined in
Fig. 5. ImV 00 and jV ij − V kl j as functions of kρ0 . The light wavelength, as well as the parameters of the particle and the fluid, is the same as those used in Fig. 3. The degree of polarization of the beams is 0. Note that the legend in (a) also applies to the other figures.
the approximate analytical solution still can be used to distinguish different rotational states of the particle for the case of σ 0. When Δ0 < 0, as shown in Fig. 6(a), the approximate analytical solution can basically reflect the average motion of the trapped particle. The effect of the last three terms in the right-hand side of Eq. (47) is that they just introduce some periodic fluctuations to the angular velocity of the trapped particle. As the angular speed of the rotating beam increases, the frequencies and magnitudes of these periodic fluctuations increase. However, when Δ0 > 0, as shown in Fig. 6(b), the approximate analytical solution becomes unable to reflect the average motion of the particle. The numerical solution shows that Ω0 jσ0 is a quasi-periodic function, whose quasiperiod is smaller than the period predicted by the approximate analytical solution. Since the approximate analytical
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Fig. 6. Time variation of the angular velocity of the trapped particle in a rotating beam with m0 1 for (a) Δ0 < 0 and (b) Δ0 > 0, where the solid line labels the numerical solution of Ω0 jσ0 , and the dashed line labels the approximate analytical solution of Ω0 jσ0 . The wavelength and power of the beam, as well as the parameters of the particle and the fluid, are the same as those used in Fig. 3. The radial position of the particle is ρ0 500 nm, and we have set χ 0.
solution of Ω0 jσ0 is nearly the same as Ω0 jσ1 , the noticeable difference between the approximate analytical solution and the numerical solution of Ω0 jσ0 for Δ0 > 0 indicates that the polarization effect on the rotation of the trapped particle becomes significant when Δ0 > 0. Interestingly, we find that if the beam polarization also is rotating, there will be another polarization effect on the rotation of the trapped particle. Note that the rotating polarization can be generated by introducing a frequency shift 2ω00 between the two circularly polarized components of the beam with a device such as an electro-optic modulator [39] or a pair of acousto-optic modulators [40]. Properties of such a rotating beam can be found in [41]. The angular velocity of the rotating polarization, which is equal to ω00 , is determined by the time derivative of the polarization orientation angle, namely, ω00 dχ∕dt. If ω00 is constant, then χt χ0 ω00 t, and the time evolution of Ω0 jσ0 for different values of ω00 is shown in Fig. 7, which is obtained by solving Eq. (47) numerically. Figure 7(a) shows that, for Δ0 < 0, if ω00 ω0 , the periodic fluctuation of the angular velocity will be weaker than that in the case of ω00 0. But if ω00 differs from ω0 by a large amount, the periodic fluctuation will become stronger than that in the case of ω00 0, as indicated by the curve labeled ω0 −10 rad∕s. This is because the last two terms in the righthand side of Eq. (47), which account for the fluctuation of the angular velocity, will have higher and higher oscillatory frequency as jω00 − ω0 j increases. Moreover, Fig. 7(b) shows that
Fig. 7. Time variation of the angular velocity of the particle trapped in a beam with a rotating pattern and rotating polarization for (a) Δ0 < 0, (b) and (c) Δ0 > 0. The azimuthal index of the beam is m0 1. The solid lines in (a) are the same as those plotted in Fig. 6(a). The dashed line in (c) is the same as the one plotted in Fig. 6(b). The wavelength and power of the beam, as well as the parameters of the particle and the fluid, are the same as those used in Fig. 6.
for Δ0 > 0, if ω00 ω0 33.4 rad∕s, the quasi-periodic angular velocity shown in Fig. 6(b) will become a periodic one with reduced fluctuation. But this conclusion is only valid for jω0 j, which is slightly larger than the critical angular speed. If jω0 j is large enough, the angular velocity of the particle will remain quasi-periodic, as indicated by the curve labeled ω00 ω0 37 rad∕s. In Fig. 7(c), the time evolution of Ω0 jσ0 for large ω00 is shown. One can see that it is similar to the approximate analytical solution of Ω0 jσ0 for ω00 0 except for the highly oscillatory fluctuation. Actually, this is not coincidence. Notice that, when jω00 j is large compared with jω0 j, the last three terms in the right-hand side of Eq. (47) become highly oscillatory terms. These terms will fluctuate out
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in a short period 2π∕jω00 j due to their nearly zero average values. Accordingly, they will play a negligible role in the average motion of the particle. Recall that the approximate analytical solution is just obtained by neglecting the last three terms in the right-hand side of Eq. (47). So the average motion of the particle can be approximately described by the approximate analytical solution when jω0 j ≪ jω00 j.
4. DISCUSSION In the previous section, we developed a theory to analyze the rotation dynamics of the particle trapped in a rotating beam. Next, we will discuss the application of the theory. As implied by Eqs. (31) and (35) to (38), for jσj 1, the ratio between the beam power P and the maximal angular speed of the trapped particle, jΩ0 jmax , can be used for predicting the minimum power P min required to rotate the trapped particle at a given angular speed Ωg according to P min
Ωg P : jΩ0 jmax
(50)
Now we propose a method to determine the ratio P∕jΩ0 jmax . Since ImV 0 ≈ 0, one can infer from Eqs. (37) and (38) that the maximal angular speed of the particle is about jΩ0 jmax jV − V − j
nP 1 jV − V − jΩ0 : 6πηRρ0 c m0
(51)
By setting jω0 j large enough so that the angular velocity becomes time periodic, it can be deduced from Eqs. (33) and (44) that jΩ0 jmax
s ¯ 0 2ω0 − Ω ¯0 : Ω m0
(52)
Thus, the jΩ0 jmax can be obtained by measuring the average angular velocity of the trapped particle. Then it is not difficult to obtain the parameter P∕jΩ0 jmax . When Δ < 0, the approximate analytical solution of Ω0 is nearly the same as jΩ0 jσ1 , so the parameter P∕jΩ0 jmax obtained with the above method also can be used to evaluate P min for the case of jσj ≠ 1. Apart from the above method, P min also can be estimated with the present calculation. For example, if there are two silica spheres with R 500 nm trapped at ρ0 500 nm in each spot of rotating even Laguerre–Gaussian modes with m0 1, it can be inferred from Fig. 3(b) that, when the beam power is 1 mW, the maximal angular speeds of the spheres at ρ0 500 nm or kρ0 3.9 are about 32.9 rad∕s. So, according to Eq. (50), the minimum beam power required to rotate the spheres at a given angular speed, say 7 Hz or 14π rad∕s, is 14π × 1∕32.9 ≈ 1.3 mW. This theoretical result is needed to be verified by further experiment. While in existing experiments it has been reported that in a rotating beam similar to the above one, but formed by interfering a plane wave and a Laguerre–Gaussian beam with m0 2, the minimum beam power required to rotate two silica spheres with R 500 nm around the beam axis at 7 Hz is 1 mW [5], which is close to the above theoretical result. As the two aforementioned rotating beams have the same number of spots and the same angular velocity for a given frequency shift [42],
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the above comparison could basically reflect the accuracy of our theory.
5. SUMMARY In this paper, we have made a theoretical investigation on the rotation dynamics of particles trapped in a rotating beam. Our theory indicates that there is a critical angular speed for the rotating beam. If the angular speed of the rotating beam is smaller than the critical one, the trapped particle can be rotated synchronously with the beam, which suggests an efficient way to control the particle’s rotation with the angular frequency shift of the beam. Otherwise, the angular velocity of the trapped particle becomes periodic or quasi-periodic, depending on the beam polarization, and, as the angular speed of the rotating beam increases, the average angular speed of the trapped particle decreases. This implies that the angular speed of the particle does not always increase with that of the rotating beam. When the beam is rotating fast enough, it is found that the maximal angular speed of the particle can be determined from the frequency shift of the beam and the average angular velocity of the particle. Further, it is shown that the ratio between the beam power and the particle’s maximal angular speed can be used to estimate the minimum beam power required to rotate the particle at a given angular speed. The present work may provide useful insight into the rotational manipulation with rotating beams and the orbital angular momentum transfer from rotating beams to matter.
APPENDIX A: EXPRESSION OF DIMENSIONLESS TRANSVERSE RADIATION FORCE With the use of the fact that jω0 j ≪ ω, it can be shown from Eq. (14) that the dimensionless transverse radiation force has the form (p ∞ X l l − ml m 1 1X f x if y − 2 l1 m−l l2 l 12 h 0 0 1 1 1 1 0 × a1 l;m1 blm al;m1 blm 2al;m1 blm bl;m1 alm s i l m 1l m 2 0 1 1 1 bl;m1 alm 2bl;m1 alm l 14 2l 12l 3 0 0 1 1 1 × a1 l1;−m−1 al;−m al1;−m−1 al;−m 2al1;−m−1 al;−m 0 0 1 1 1 − a1 l1;m1 alm − al1;m1 alm − 2al1;m1 alm 0 0 1 1 1 − b1 l1;m1 blm − bl1;m1 blm − 2bl1;m1 blm 0 b1 l1;−m−1 bl;−m
1 b0 l1;−m−1 bl;−m
1 2b1 l1;−m−1 bl;−m
) ; (A1)
where j −iω0 t iω0 t C −m0 aj ; aj lm C m0 a;lm e −;lm e
(A2)
j j −iω0 t C iω0 t : bj −m0 b−;lm e lm C m0 b;lm e
(A3)
Under the approximation in Eqs. (16) and (17), Eq. (A1) can be simplified to Eq. (18). The function Φ in Eq. (18) is defined as
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H. Yu and W. She
0 ~ 0 0 Φa~ 0 lm ;alm ; blm ;blm p ∞ X l l−mlm1 EM 0 0 1X 0 Rl a~ l;m1 blm REM b~ 0 l l;m1 alm 2 l1 m−l l2 l12 s lm1lm2 TE 0 0 TE ~ 0 R al1;−m−1 a~ 0 l;−m −Rl a l1;m1 alm l14 2l12l3 l 0 TM ~ 0 ~ 0 RTM b0 ; (A4) l l1;−m−1 bl;−m −Rl bl1;m1 blm
where TE TE ¯ ¯ RTE l −S l1 kR; kR − S l kR; kR TE ¯ ¯ − 2S TE l1 kR; kRS l kR; kR;
RTM l
¯ −S TM l1 kR; kR −
−
−
(B6)
b0 ;lm
s Z Z 2π π ik2 2 2 dϑ dφ πC m0 μ0 μω k P 0 0
∂Y ϑ; φ eϑ − imY lm ϑ; φeφ · Eϑ; φ : × cos ϑ sin ϑ lm ∂ϑ (B7)
By substituting Eq. (3) into Eqs. (B6) and (B7), we obtain the multipole coefficients of the incident beam: (A6)
¯ − S TM kR; kR ¯ −S TE kR; kR REM l l l TM ¯ ¯ kR; kRS 2S TE l l kR; kR:
s Z Z 2π π k2 2 2 dϑ dφ πC m0 μ0 μω k P 0 0
∂Y ϑ; φ × cos ϑ sin ϑ lm eφ imY lm ϑ; φeϑ · Eϑ; φ ; ∂ϑ
(A5)
¯ S TM l kR; kR
TM ¯ ¯ 2S TM l1 kR; kRS l kR; kR;
a0 ;lm
(A7)
p p iδ G −iχ 1 σ − K iχ 1 − σ ; a0 ;lm e ;lm e ;lm −e
(B8)
p p iδ b0 G;lm e−iχ 1 σ K ;lm eiχ 1 − σ ; ;lm −e
(B9)
where we have used the fact that jω0 j ≪ ω and
APPENDIX B: DERIVATION OF MULTIPOLE COEFFICIENTS OF INCIDENT BEAM To obtain the multipole coefficients of the incident beam, one can substitute in Eq. (2) the expansion of the dyadic plane wave [43]: ¯ ik·x 4π Ie
∞ X l X l0 m−l
4π
∞ X l X l1 m−l
ϑ0
Z
dϑ
2π
dφ ∂Y ϑ; φ sin ϑ lm − mY lm ϑ; φ ∂ϑ 0 0 2π p 2 2 × e−iφ cos ϑsin ϑjm0 j e−α sin ϑ eim0 φ eik ·x0 ; (B11) Z
K ;lm
il−1 P lm ϑ; φL0 lm k; x
dφ ∂Y ϑ; φ sin ϑ lm mY lm ϑ; φ ∂ϑ 0 0 2π p 2 2 (B10) × eiφ cos ϑsin ϑjm0 j e−α sin ϑ eim0 φ eik ·x0 ; Z
G;lm
ϑ0
Z
dϑ
2π
il C lm ϑ; φM 0 lm k; x
il−1 B lm ϑ; φN 0 lm k; x;
(B1)
where α f ∕w0 . With the help of the recurrence formulas of the spherical harmonic:
where
sin ϑ Plm ϑ; φ Y lm ϑ; φek ;
1 L0 lm k; x ∇j l krY lm ϑ; φ; k
(B2)
∂Y lm ϑ; φ lΛl1;m Y l1;m ϑ; φ ∂ϑ − l 1Λlm Y l−1;m ϑ; φ;
where Λlm and (B9) to
p l2 − m2 ∕4l2 − 1, one can simplify Eqs. (B8)
(B3) G;lm lΛ l1;m Q;l1;m − l 1Λ lm Q;l−1;m mQ;lm 0 ≡ G;lm eim0 −m1ϕ0 ;
ik ∂Y ϑ; φ Blm ϑ; φ p lm ; ∂k ll 1
(B4)
im0 −m−1ϕ0 ; ≡ K 0 ;lm e
(B5)
Then, after balancing the coefficients of the vector spherical wave functions in Eqs. (2) and (9), one can obtain:
(B13)
K ;lm lΛ l1;m Z ;l1;m − l 1Λ lm Z ;l−1;m − mZ ;lm
where i ∂ Clm ϑ; φ p × kY lm ϑ; φ: ll 1 ∂k
(B12)
Q;lm
Z
(B14)
h p 2 2 dϑ Y lm ϑ; 0 cos ϑsin ϑjm0 j e−α sin ϑ ijm0 −m1j 0 i × eim0 −m1ϕ0 eik z0 cos ϑ J jm0 −m1j k ρ0 sin ϑ ; ϑ0
(B15)
H. Yu and W. She
Z
Z ;lm
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h p 2 2 dϑ Y lm ϑ; 0 cos ϑsin ϑjm0 j e−α sin ϑ ijm0 −m−1j 0 i × eim0 −m−1ϕ0 eik z0 cos ϑ J jm0 −m−1j k ρ0 sin ϑ ; ϑ0
(B16) 0 G;lm
and where J n z is the Bessel function. Note that the 0 K ;lm defined in the right-hand side of Eqs. (B11) and (B12) are independent of ϕ0 . In deriving Eqs. (B13) and (B14), we have used the Jacobi–Anger expansion of a plane wave [32]: eik·x0 eikz0 cos ϑ
∞ X n−∞
eikz0 cos ϑ
∞ X n−∞
in J n kρ0 sin ϑeinϕ0 −φ ijnj J jnj kρ0 sin ϑeinϕ0 −φ ;
(B17)
where ρ0 ; ϕ0 ; z0 are the cylindrical coordinates of x0 , and J −n z −1n J n z has been used in the last equality.
APPENDIX C: EXPRESSIONS OF V ij In Eq. (45), V ij are quantities that can be determined from Eqs. (21), (22), and (26)–(28). They have the following expressions: 0 0 0 V 00 1 σjC 2m0 jΦG0 ;lm ; G;lm ; G;lm ; G;lm 0 0 0 ΦG0 −;lm ; G−;lm ; G−;lm ; G−;lm 0 0 0 1 − σjC 2m0 jΦ−K 0 ;lm ; −K ;lm ; K ;lm ; K ;lm 0 0 0 Φ−K 0 −;lm ; −K −;lm ; K −;lm ; K −;lm ;
V 0
(C1)
p 0 0 0 1 − σ 2 jC 2m0 jΦ−K 0 ;lm ; G;lm ; K ;lm ; G;lm 0 0 0 Φ−K 0 −;lm ; G−;lm ; K −;lm ; G−;lm ;
(C2)
p 0 0 0 1 − σ 2 jC 2m0 jΦG0 ;lm ; −K ;lm ; G;lm ; K ;lm
V 0−
0 0 0 ΦG0 −;lm ; −K −;lm ; G−;lm ; K −;lm ;
(C3)
0 0 0 V 0 jC 2m0 j1 σΦG0 ∓;lm ; G;lm ; G∓;lm ; G;lm 0 0 0 1 − σΦ−K 0 ∓;lm ; −K ;lm ; K ∓;lm ; K ;lm ;
(C4)
p 0 0 0 1 − σ 2 jC 2m0 jΦ−K 0 ∓;lm ; G;lm ; K ∓;lm ; G;lm ;
(C5)
p 0 0 0 1 − σ 2 jC 2m0 jΦ−G0 ∓;lm ; −K ;lm ; G∓;lm ; K ;lm ;
(C6)
V
V −
ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China (grants nos. 11274401 and U0934002),
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the National Basic Research Program of China (grant no. 2010CB923200), and the Ministry of Education of China (grant no. V200801).
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