Optical Levitation of Nanodiamonds by Doughnut Beams in Vacuum

Lei-Ming Zhou,1 Ke-Wen Xiao,1 Jun Chen,2 and Nan Zhao1, ∗

1Beijing Computational Science Research Center, Beijing 100193, China2Institute of Theoretical Physics and Collaborative Innovation Center of Extreme Optics, Shanxi University, Shanxi, China

(Dated: August 6, 2018)

Optically levitated nanodiamonds with nitrogen-vacancy centers promise a high-quality hybrid spin-optomechanical system. However, the trapped nanodiamond absorbs energy form laser beams and causesthermal damage in vacuum. It is proposed here to solve the problem by trapping a composite particle (a nanodi-amond core coated with a less absorptive silica shell) at the center of strongly focused doughnut-shaped laserbeams. Systematical study on the trapping stability, heat absorption, and oscillation frequency concludes thatthe azimuthally polarized Gaussian beam and the linearly polarized Laguerre-Gaussian beam LG03 are the op-timal choices. With our proposal, particles with strong absorption coefficients can be trapped without obviousheating and, thus, the spin-optomechanical system based on levitated nanodiamonds are made possible in highvacuum with the present experimental techniques.

Introduction-. By trapping, detecting and manipulatingnano- and micro-particles [1], optical tweezers are widelyused in biophysics [2–4], colloidal sciences [5], chemistry,microfluidic dynamics [6], and fundamental physics [7–15].Because of the wide applicability and high tunablity of theoptically levitated systems, several schemes [16] were pro-posed to realize the ground-state cooling [17], to search fornon-Newtonian gravity [18] and to detect gravitational wave[19]. Particularly, it brings about more interesting phenom-ena and novel applications [20, 21] when the trapped parti-cles have internal degrees of freedom (such as spins or electricdipoles) and enter the quantum regime.

Optically levitated nanodiamonds with nitrogen-vacancy(NV) centers [22–26] are one of the most promising candi-dates for implementing a spin-optomechanical hybrid system.In principle, this system can have both long spin coherencetime and high quality factor of mechanical oscillation in vac-uum. The electron spins of NV centers were shown to havelong spin coherence time (in the order of 102 µs) even innanodiamonds of diameter about 20 nm [27]. When trappedin high-vacuum, the dielectric particles are predicted to haveultra-high quality factor Q larger than 1010 [16, 18, 28]. Re-searchers have trapped diamond particles and observed thesignal from NV centers in liquid [29, 30], in air [31] and veryrecently in vacuum with pressure down to ∼ kPa [24, 25] and∼ 100 Pa [26].

Realizing high quality mechanical oscillation requires trap-ping the particles in high vacuum (e.g, 10−6 Pa) to getQ ∼ 1010. However, the high-vacuum condition usuallycauses the thermal damage problem, and experimentally trap-ping a nanodiamond in high vacuum is still very challeng-ing. Nanodiamonds will absorb energy from the trapping laserbeams due to the intrinsic defects [26] and the inevitable im-perfections or graphitization [32] on diamond surface. Theabsorbed energy can hardly be dissipated in a high-vacuumenvironment, and the nanodiamonds will be quickly heatedup significantly [24–26], which is unfavorable to the defectcenters, or even burns out the diamond particles. Improving

∗ nzhao@csrc.ac.cn

the purity of trapped nanodiamonds is one way to reduce theheat absorption [26]. Here, we study an alternative way by en-gineering the trapping beams, which is in principle applicableto much wider range of particles.

It is proposed here to solve the thermal damage problem bytrapping a silica-coated nanodiamond with doughnut beams[e.g., the Laguerre-Gaussian (LG) beams, see Fig. 1]. Ourproposal is based on the following two observations. Firstly,recent experiments [9] show that micro-sphere made of silicacan be trapped in high-vacuum without strong heat absorp-tion because of the low absorption coefficient. Secondly, it iswell-known that the cross-section intensity distribution of thedoughnut beam has a dark region at the beam center. Nan-odiamonds can be coated with a silica shell [24, 33], form-ing a core-shell structure. When the dark region of the beamcoincides with the diamond core, the heat absorption will besignificantly suppressed. Trapping and manipulating particles

zx

(a)

(b)

xy

(c)

xz 1m

Rr

=1064nm

𝐄 2(𝑉2/𝑚2)

FIG. 1. (a) The schematic illustration of the system. A nanodia-mond coated with a silica shell is levitated in an optical trap formedby two incoherent strongly-focused counter-propagating beams. (b)(c) Front view and side view of the intensity distribution of thetwo focused linearly polarized LG03 incident beams in the focal re-gion. The circles indicate the composite particle with the core radiusr = 100 nm and the shell radius R = 1 µm. Two incident beams arewith total power 100 mW.

arX

iv:1

609.

0867

8v2

[ph

ysic

s.op

tics]

20

Oct

201

6

2

x (7m)-4 -2 0 2 4

U(x

) (

pN 7

m)

-120

-80

-40

0(a)

R=0.507mR=0.707mR=0.787mR=0.907mR=1.107m

R(7m)0.1 0.5 1 1.5 2

. (

pN 7

s/7

m)

0

300

600

900(b) P = 105Pa

P = 103Pa

P = 101Pa

R(7m)0.5 1 1.5 2

. (

pN 7

s/7

m)

0

500

1000

1500

2000 (c) P = 105Pa

P = 103Pa

P = 101Pa

R(7m)0.1 0.5 1 1.5 2

. (

pN 7

s/7

m)

0

500

1000

1500(d) P = 105Pa

P = 103Pa

P = 101Pa

q

.c

(pN 7s/7m) (e)

R(7m)0.5 1 1.5 2

0.7

0.8

0.9

1

10-1

1

101

102

q

.c

(pN 7s/7m) (f)

R(7m)0.1 0.5 1 1.5 2

0.7

0.8

0.9

1

10-1

1

101

102

1

FIG. 2. (a) The transverse optical potential U(x) for a core-shell spherical particle in a single linearly polarized LG03 beam withpower P = 100 mW. (b)-(d) The transverse stability diagram of acore-shell particle trapped in an azimuthally polarized beam, a lin-early polarized LG03 beam and a right circularly polarized LG beam,in turn. The white regions are stable. The Stokes friction coefficientsγ under different pressure P are also plotted with lines for reference.(e) The minimum friction coefficients γc required to stabilize the op-tical trap formed by the dual-beam optical tweezers with the LG03

beams, as functions of the power mismatch ratio q and particle sizeR. The blue ‘×’ denotes a stably trapped sphere for light absorptionsuppression and equilibrium temperature simulation later. (f) Thesame as (e), but for the right circularly polarized LG beams. Otherparameters used here are λ = 1064 nm, NA = 0.95 and each beamwith power P = 100 mW for all figures.

with doughnut beams have been investigated [34–39] in liq-uids. Here, we focus on the problem in vacuum which haslow friction and low dissipation.

The key problem is under what condition that the compositeparticle can be stably trapped in vacuum with the core lyingin the dark region. We numerically solve the electromagneticwave scattering problem and show that the gradient force froma doughnut beam can form a single-well potential, as long asthe particle size exceeds a critical radius Rtrans [see Fig. 2(a)].Furthermore, the doughnut beam usually carries non-zero or-bital angular momentum (OAM) and has different polariza-tions. The OAM and polarization affect the trapping stabilityand heat absorption of the particle, particularly in the caseof the strongly focused beam. We systematically investigatetrapping effect of doughnut beams with various OAM (e.g.,the LG0l beams) and polarization (e.g., linearly or circularlypolarized LG beams and the cylindrical vector beams [40]).By comparing the trapping stability, the heat absorption, and

the oscillation frequency of the composite particle trapped indifferent types of doughnut beams, we conclude that the az-imuthally polarized Gaussian beam and the linearly polarizedLG03 beam are the optimal choices for implementing the hy-brid spin-optomechanical system in vacuum.

Trapping stability-. We consider a dual-beam opticaltweezers system as shown in Fig. 1(a). Two incoherentcounter-propagating laser beams of wavelength λ are focusedby two identical lenses with numerical aperture NA. Whenthe two beams with same parameters except the directions arewell-aligned, the z direction scattering forces from the twobeams cancel each other, and the gradient forces form an op-tical trap near the focal point in three dimensional space.

We start the discussion from considering the strongly fo-cused LG0l beams (l > 0). The focused beams violate theparaxial condition, and we perform numerical calculations ofthe focal field, following the theory developed by Richardsand Wolf [41, 42]. Figures 1(b) and 1(c) show the light in-tensity distribution in the focal region. Similar to the paraxialcase, there is a dark region along the beam propagating axis(the z axis). However, in contrast to the paraxial beams whoseOAM are usually well-defined and separated with the polar-ization degree of freedom, the orbital and polarization degreesof freedom are highly mixed around the focal point. More im-portantly, we will show that the trapping stability and the heatabsorption are sensitive to the choice of the OAM and the po-larization of the focused beams.

The focused beams provide an optical trap to the parti-cle. As an example, Fig. 2(a) shows the trapping potentialU(x) = −

´ x−∞

Fx(x′)dx′ along the x direction for incidentLG03 beams, where Fx is the x component of the optical forceF when the particle is displaced along the x axis [43]. Gener-ally speaking, small particles tend to be trapped at the positionof maximal intensity. For incident LG beams, a small parti-cle (e.g., with radius R λ) will be confined in the regionof the bright ring, corresponding to a double-well potential.With increasing particle size, the trapping potential is gradu-ally changed to a single-well. The transition radius Rtrans fromdouble-well to single-well is comparable to the radius of thebright ring. A composite particle with radius R > Rtrans will betrapped around the equilibrium position with its core locatesin the dark region of the focused LG beams.

The LG beams carry OAM, which accelerates the particlein the azimuthal direction and strongly affects the trapping sta-bility [42]. The trapping stability is determined by the forceconstant matrix K = ∇F of the focused beam at the equilib-rium position (when R > Rtrans). Because of the OAM ofthe LG beams, the y-component of the radiation force Fy isnonzero, when the particle displacement is along the x direc-tion. In this case, the trap stiffness Ki (i.e., the eigen values ofthe force constant matrix K for i = 1, 2 or 3) can be complexnumbers [42]. When Re[Ki] < 0, the trap is single poten-tial well in the focal point. However, if Im[Ki] , 0, the trapis unstable unless the environmental damping is larger than acritical value γc. Figure 2(c) shows a typical stability diagramof the transverse motion of a core-shell particle trapped in asingle LG03 beam. For the applications of optical tweezersin high vacuum, we are interested in the absolutely stable re-

3

r (7m)0.02 0.04 0.06 0.08 0.1

supp

ress

ion 9

10-9

10-6

10-3

100(a) l= 1; 9 9 r0

l= 2; 9 9 r0

l= 3; 9 9 r2

l= 4; 9 9 r4

Azim:; 9 9 r2

NA=0.95NA=0.90NA=0.85NA=0.80

x (7m)-2 -1 0 1 2

|E|2 (

1014

V2 /m

2 )

0

0.5

1(b) l=1

l=2l=3l=4Azim.

FIG. 3. (a) Light absorption suppression ratio ξ for a core-shellsphere at the center of azimuthally and linearly polarized beams(LG0l beams) relative to Gaussian beams, with different core radiusr and numerical aperture NA. Blue solid lines are for our approx-imate analytical results. R = 900 nm, refractive index ndiamond =

2.418 + 0.001i and nsillica = 1.458 (absorption eliminated in thisfigure). (b) Field |E|2 distribution of incident strongly focused az-imuthally and linearly polarized beams (LG0l beams polarized alongx axis). All beams are with power P1 = P2 = 50 mW focused by thelens of NA = 0.95.

gions (ASRs), where the particle can be trapped in the absenceof any damping (γ = 0). In the case of the linearly polarizedLG03 beam, there are several ASRs (e.g., around R = 1.5 µm),

The trapping stability depends on the beam polarization.Similar calculations are made for the azimuthally and circu-larly polarized beams [see Figs. 2(b) and 2(d)]. Being differ-ent from the linearly polarized beams, no ASR appears in thecircularly polarized case. A finite damping rate γ due to theenvironment (e.g., the collisions with the residual molecules)is necessary to maintain a stable trapping. In contrast, theazimuthally polarized beam provides a stable trap as long asthe particle size exceeds the critical radius. This can be un-derstood because the azimuthally polarized beam indeed doesnot carry OAM.

Counter-propagating beams with the same OAM improvethe trapping stability. When the particle is displaced along xdirection, the x components radiation force of the two beamsadd up, while the y components are canceled. Figure 2(e)shows the stability diagram for two counter-propagating LG03beams, with powers P1 and P2 = qP1 (with the power mis-match ratio q). The ASR is enlarged when the ratio q ap-proaches to unity. For example, particles with radius R around0.9 µm and R > 1.3 µm can be stably trapped even whenthe beam intensities are not perfectly matched (e.g., in the re-gion with 0.9 < q ≤ 1). In the circularly polarized case, thecounter-propagating beams reduce the critical damping coeffi-cient. Unfortunately, the ASR appears only when the beam in-tensities are exactly matched (i.e., P1 = P2 or q = 1). Accord-ingly, in terms of trapping stability, we conclude that the az-imuthally polarized Gaussian beam and the linearly polarizedLG beams are appropriate for the dual-beam optical tweezersin high vacuum. The cases for doughnut beams with differentOAM and polarization have been investigated systematicallyand more details can be seen in supplementary information.

Heat absorption-. Having discussed the trapping stability,now we turn to the suppression of the heat absorption. Fig-

P inc (mW)0 200 400 600 800 1000

T (K

)

300

400

500

600(a)

l= 0

P inc (mW)0 200 400 600 800 1000

T (K

)

300

400

500

600(b)

l= 1

P inc (mW)0 200 400 600 800 1000

T (K

)

300

400

500

600(c)

l= 2

P inc (mW)0 200 400 600 800 1000

T (K

)

300

350

400

450(d)

l= 3

Azim:

5=0 5=10-5 5=10-4 5=10-3

FIG. 4. The equilibrium temperature T versus the incident laserpower Pinc for a core-shell sphere trapped in linearly polarized LG0l

beams and azimuthally polarized beam. Cases with (a) l = 0, (b)l = 1, (c) l = 2, (d) l = 3 and (d) azimuthally polarized beam areshown.

ure 3(a) shows the absorption coefficients cabs of the a com-posite particle in azimuthally polarized Gaussian beam andlinearly polarized LG0l beam with different OAM index l. Allthe absorption coefficients are normalized by the absorptioncoefficient cabs,0 of the same particle in a Gaussian beam, andare thus defined as suppression ratio

ξ ≡cabs

cabs,0. (1)

The heat absorption depends on both polarization andOAM. In the azimuthally polarized beam, the heat absorptionof the particle is reduced by a factor of 102 to 103, dependingon the size of the absorptive core. While, for the LG beams,the suppression ratio ξ has qualitatively different behavior fordifferent OAMs. In the cases of l = 1 and l = 2, the suppres-sion effect is relatively weak and the ratio ξ is independent onthe radius of the core. However, for l ≥ 3, the heat absorptionis significantly reduced (ξ < 10−3 for core radius r < 100 nm),and follows a power-law dependence of the core size.

Different behavior of the heat suppression ratio originatesfrom the intensity distribution of the incident strongly focusedbeams. Figure 3(b) shows the intensity distribution along thex axis of the azimuthally polarized beam and LG beams withdifferent OAMs. For the typical core radius r < 100 nm, weexpand the intensity distribution into power series of kx (inthe x direction for example, and kx 1 with the wave num-ber k = 2π/λ) around the equilibrium position (i.e., the beamcenter). We find that, for the azimuthally polarized beam, theintensity exactly vanishes at the beam center (x = 0) and in-creases quadratically as increasing x, i.e., Ia ≈ Aa(kx)2. Forthe strongly focused LG0l modes, the intensity reads

Il(x) ≈ Al(kx)nl , for integer l ≥ 1. (2)

Here, Aa and Al are the expansion coefficients. Being dif-ferent from the paraxial LG beams, the LG01 and the LG02

4

modes have finite intensity at the focal point (i.e., with con-stant leading terms with n1 = n2 = 0). For the LG0l modeswith l ≥ 3, it is absolutely dark at the beam center [Il≥3(0) = 0]and the expansion power index depends on the OAM index las nl = 2(l − 2). This explains the power-law behavior ofthe heat suppression ratio in Fig. 3(a), and suggests that theazimuthally polarized beam and the LG0l modes with l ≥ 3are good candidates for solving the thermal damage problem.Details of the intensity expansion and suppression approxi-mations can be seen in supplementary information. System-atically investigation for right circularly polarized beams hasalso been included.

With the absorption coefficient cabs, we estimate the equi-librium temperature T of the composite particle trapped bydifferent beams. We consider the vacuum environment of tem-perature T0, where radiation is the dominating heat transfermechanism. The absorbed energy from the beams is balancedby the black-body radiation as

cabsPinc + σAT 40 = σAT 4, (3)

where Pinc is the total incident power of the counter-propagating trapping beams, σ is Stefan’s constant and A =

4πR2 is the surface area of the sphere. Here, we have ne-glected the heat dissipation due to the surface conductionj = ksA(T − T0), where ks is the surface conduct coefficientdue to the residual gas. Accordingly, Eq. (3) gives an estima-tion of the upper bound of the equilibrium temperature. Fig-ure 4 shows the equilibrium temperature T of a stably trappedparticle (denoted as a ‘×’ in Fig. 2(e), R = 900nm) as thefunction of incident power Pinc. We assume that the shellof the composite particle is made of silica with absorptioncoefficient 100 dB/km, corresponding to the imaginary partof the refractive index κshell = 2 × 10−9. The diamond corecould be very absorptive due to the intrinsic defects and theimperfect surface. We consider the imaginary part of refrac-tive index of the diamond core ranging from κcore ∼ 10−3 to10−5. For the fundamental Gaussian mode (i.e., l = 0), thetemperature increases dramatically as increasing the incidentpower. However, when trapped by the doughnut beams, par-ticularly the azimuthally polarized beam and the LG03 mode[see Fig. 4(d)], the diamond core has negligible contributionto the temperature increasement. The composite particle canafford much stronger power (up to the order of Watt) of thetrapping beams, without significant heating effect.

In this part, we discuss two figures of merit of the system,namely, the trapping frequency Ω =

√K/M and mechanical

quality factor Q = Ω/Γ of the mechanical oscillation. Here Kis the force constant, M is the mass of oscillator, and Γ = γ/Mis the damping coefficient with Stokes friction coefficient γdue to the residual gas. Figures 5 presents the frequency Ω andthe quality factor Q for the azimuthally polarized beam andthe linearly polarized LG03 beam, respectively. For a givenparticle size, the azimuthally polarized beam provides a trap-ping frequency in the order of 100 kHz, much higher than thatof the LG03 mode. While the later creates an optical trap withnon-degenerate frequencies, in the order 10 kHz, in the x andy directions. The quality factor of the mechanical oscillationQ is inversely proportional to the damping coefficient γ due to

R (7m)0.5 1 1.5 2

+/(

2:)

(Hz)

#104

0

1

2

3

4(a) +

x+

y

R (7m)0 0.5 1 1.5 2

+/(

2:)

(Hz)

#104

0

2.5

5

7.5

10(b) +

R (7m)0.5 1 1.5 2

Q

#1010

0

2

4

6(c) Q

xQ

y

R (7m)0 0.5 1 1.5 2

Q

#1010

0

3

6

9(d) Q

1

FIG. 5. The oscillation frequency Ω/(2π) of the sphere in (a) linearlypolarized LG03 beams and (b) azimuthally polarized beams. Andthe mechanical quality factor Q of the sphere in (c) linearly polar-ized LG03 beams and (d) azimuthally polarized beams under pressure10−6 Pa. All beams are with total power 100 mW and NA = 0.95.

the residual gas. The quality factor reaches Q ∼ 1010, with aresidual pressure P= 10−6 Pa. Reducing the pressure can fur-ther increase the quality factor, to the best figure of merit ofthis opto-mechanical system.

Conclusion-. Optical tweezers achieved great success inthe past years in broad research fields ranging from biology,statistical physics, to microchemistry. However, the heat ab-sorption problem prevents the exciting applications in the low-dissipation vacuum environment. We propose to use dough-nut beams and core-shell particles to implement stable opticaltweezers system in high vacuum. The low-absorptive shell(e.g., the silica shell) plays a role of ‘sample-holder’, whichinteracts with the trapping beam and provides the radiationforce for levitation. While the ‘sample’ could be more generalparticles (not necessarily to be nanodiamonds). Once loadedin the sample-holder, no matter how strong absorptive it is, thecore particle has little chance to see the trapping beam. Witha systematic study of the physical effect of the beam OAMand polarization on the trapping stability and heat absorption,we provide a comprehensive solution to the heat absorptionproblem and make optical tweezers a powerful tool in manydisciplines, especially in optomechanical applications and inthe future quantum technologies.

Appendix A: investigation method: Debye Integral andLorentz-Mie Theory

The investigation method is formulated concisely in thispart. The strongly focused incident laser beam is modeledby the generalized vector Debye integral, and the generalizedLorentz-Mie theory (GLMT) is used to solve the scatteringproblem of light field. Finally, the time-averaged Maxwellstress tensor is used to calculate the force exerted on thesphere particle.

5

The strongly focused beam is affected dramatically by thelens with high numerical aperture (NA), and cannot be de-scribed by the expression of paraxial beams. We model var-ious doughnut beams, including linearly polarized Laguerre-Gaussian (lpLG) beams, right circularly polarized Laguerre-Gaussian (rcLG) beams and azimuthally polarized Gaussian(apG) beam here, by the highly accurate generalized vectorDebye integral theory [41, 44]. The incident field near thefocus is formulated as:

E(ρ, ϕ, z) = −ik f e−ik f

2π

ˆ θmax

0dθsinθ

ˆ 2π

0dφ[E∞(θ, φ)

×eikρsinθ cos(φ−ϕ)+ikzcosθ], (A1)

where E∞(θ, φ) is the electric field vector on the Gaussian ref-erence sphere, and the exponential factor is the phase accumu-lated during the light propagation. This expression has beenused to investigate the properties of strongly focused beamsand shows good accuracy [41, 42].

The scattering of incident beam is calculated by the GLMT[44, 45] and is formulated concisely here. In this theory, theincoming and outgoing fields are expanded in vector sphericalwavefunctions (VSWFs):

Ein =

∞∑n=1

n∑m=−n

[amnMmn(kr) + bmnNmn(kr), (A2)

Eout =

∞∑n=1

n∑m=−n

[pmnMmn(kr) + qmnNmn(kr). (A3)

The coefficients of incoming and outgoing fields are relatedby the T-matrix T of the particle as:(

pq

)= T

(ab

). (A4)

For sphere or multi-layer sphere, T is diagonal and can becalculated directly or by iteration [46, 47].

The force and torque exerted on the sphere particle are cal-culated from time-averaged Maxwell stress tensor:

T =12

Re[εE∗E +B∗Bµ−

12←→1 (εE∗ · E +

B∗ · Bµ

), (A5)

F =z

dS · T , (A6)

Γ =z

dS · (r × T ). (A7)

They are then formulated by the incident and scattered fieldcoefficients, and finally we get, e.g,, the axial force and torque:

Fz =nmP

c2Pc

∞∑n=1

n∑m=−n

m

n(n + 1)Re(a∗mnbmn − p∗mnqmn)

−1

n + 1[n(n + 2)(n − m + 1)(n + m + 1)

(2n + 1)(2n + 3)]1/2

×Im(amna∗mn+1 + bmnb∗mn+1 − pmn p∗mn+1

−qmnq∗mn+1), (A8)

τz =Pkc

2Pc

∞∑n=1

n∑m=−n

[m

×(amna∗mn + bmnb∗mn − pmn p∗mn − qmnq∗mn)]. (A9)

Among the last two equations,

Pc =

∞∑n=1

n∑m=−n

(amna∗mn + bmnb∗mn) (A10)

is proportional to the incident power, nm is the refractive indexof the medium surrounding the particle and P is the power ofthe incident laser beam. Thus from the extinction and scatter-ing coefficients, we get the absorption coefficient of the parti-cle:

cext = −1Pc

∞∑n=1

n∑m=−n

Re(amn p∗mn + bmnq∗mn), (A11)

csca =1Pc

∞∑n=1

n∑m=−n

(pmn p∗mn + qmnq∗mn), (A12)

cabs = cext − csca. (A13)

Appendix B: Phase diagram and Trapping stability in variousdoughnut beams

Systematical investigation of the trapping stability in differ-ent doughnut beams has been made and is shown in this part.For a sphere in an optical trap, the force constant is a tensor Kwith elements [42]

Ki j =∂Fi

∂x j, (B1)

where i, j = x, y, z are the Cartesian coordinates. The eigenvectors of K are the eigen-modes of sphere motion and can beused to analyze its stability. For a sphere in a LG beam, theforce matrix can be written as

K =

a d 0g b 00 0 c

. (B2)

The eigen motion modes along z axis is independent of thoseon the transverse xy plane. Using the theory of the trappingstability by optical vortex beam (i.e., LG beam) [42], variousLG beams with different l and polarizations have been inves-tigated here.

Especially, the case of a single x polarized LG03 beam areshown in Fig. S1(a) for eigen force constant, in Fig. S1(b)for force constant elements and in Fig. 2(c) of the main textfor phase diagram. It is noticed that, different from the phasediagram of LG01 and LG02 beam, there is a region which isalways unstable despite of the environment damping at R ∼1.1µm [see Fig. 2(c)]. It can be seen from Fig. S1(b) that theforce constant b = Kyy is positive at R ∼ 1.1 µm.

The rcLG beams are different from lpLG beam as they haveno stable window when γ = 0, as shown in Fig. S2(b)(e)(h)for l = 1, 2, 3, respectively. For different q, the minimum γcneeded to keep stable is shown in Fig. S2(c)(f)(i).

6

R(7m)0 0.5 1 1.5 2

Ki(p

N/7

m)

-400

-200

0

200

400(a)

Re(Ktran+

)

Re(Ktran-

)

|Im(Ktran

)|

R(7m)0 0.5 1 1.5 2

Kij(p

N/7

m)

-400

-200

0

200

400(b)

adgb

1

FIG. S1. (a) The transverse eigen force constant Ki and (b) force constant element Ki j for a core-shell sphere trapped in a single LG03 beampolarized along x. All beams are with power P = 100 mW, λ = 1064 nm and focused by lens with NA = 0.95. All spheres are with coreradius r = 20 nm.

Radius(7m)0 0.5 1 1.5 2

Ki(p

N/7

m)

-400

-200

0

200

400(a)

Re(Ktran+

)

Re(Ktran-

)

|Im(Ktran

)|

R(7m)0.1 0.5 1 1.5 2

. (

pN 7

s/7

m)

0

500

1000

1500(b) P = 105Pa

P = 103Pa

P = 101Pa

q

.c

(pN 7s/7m) (c)

R(7m)0.1 0.5 1 1.5 2

0.7

0.8

0.9

1

10-1

1

101

102

Radius(7m)0 0.5 1 1.5 2

Ki(p

N/7

m)

-400

-200

0

200

400(d)

Re(Ktran+

)

Re(Ktran-

)

|Im(Ktran

)|

R(7m)0.1 0.5 1 1.5 2

. (

pN 7

s/7

m)

0

500

1000

1500(e) P = 105Pa

P = 103Pa

P = 101Pa

q

.c

(pN 7s/7m) (f)

R(7m)0.1 0.5 1 1.5 2

0.7

0.8

0.9

1

10-1

1

101

102

Radius(7m)0 0.5 1 1.5 2

Ki(p

N/7

m)

-400

-200

0

200

400(g)

Re(Ktran+

)

Re(Ktran-

)

|Im(Ktran

)|

R(7m)0.1 0.5 1 1.5 2

. (

pN 7

s/7

m)

0

500

1000

1500(h) P = 105Pa

P = 103Pa

P = 101Pa

q

.c

(pN 7s/7m) (i)

R(7m)0.1 0.5 1 1.5 2

0.7

0.8

0.9

1

10-1

1

101

102

1FIG. S2. (a) The transverse eigen force constant Ki and (b) the phase diagram of a core-shell sphere in a single right circularly polarizedLG01 beam. (c) Minimum friction γc needed for stable trapping in two counter-propagating rcLG01 beams with different power ratio q. (d-f)The same as the first row, for rcLG02 beams. (g-i) The same as the first row, for rcLG03 beams. All beams are with power P = 100 mW,λ = 1064 nm and focused by lens with NA = 0.95. All spheres are with core radius r = 20 nm.

Appendix C: Approximate expression of the intensity near thefocus

The field of the strongly focused beam near the focus de-termines the absorption coefficients of the levitated core-shellsphere. Although there is no analytic expression of the field ofthe strongly focused beam, an approximate analytical expres-sion near the beam focus will be helpful both for theory andexperimental investigation. Our analytical approximations areshown in this part.

The field distribution of focused LG beam is expressed withvector Debye integral theory with Eq. (A1). When z = 0, and

δ , kρ 1, (C1)

we expand the field near the focus with Taylor series. For anx direction linearly polarized incident LG0l beam, the lightintensity Il = 1

2εmε0vm|El|2 near the focus can be expressed as

Il = Alδnl , nl = 2(l− 2). The absorption coefficients of a small

sphere with radius r can be approximated as

7

cabs =

ˆVηlαlIldV/Pinc, (C2)

where αl is the light attenuation coefficient in sphere, Il is thelight intensity and V is domain of the sphere core with ra-dius r. ηl is a correction factor considering the affection suchas the changing of light field after scattering. The suppres-sion coefficients ratio are defined as ξ = cabs/cabs,0 and followdifferent power-law as ξl = ξl0δ

nl , nl = 2(l − 2). Choosingη0 = 1.0, η1 = 1.3, η2 = 0.9, η3 = 0.45, and η4 = 0.40 whenNA = 1.0, we get ξ10 = 0.21, ξ20 = 0.034, ξ30 = 0.0032, andξ40 = 1.8 × 10−4. The approximate analytical results of ξl areplotted in Fig. 3(a) with blue lines and match the numericalresults well.

Absorption suppression for rcLG beams shows similar be-havior as those in lpLG beams. The absorption scale with rcan also be analyzed as before. The suppression coefficientsfollow different power-law as ξl = ξl0δ

nl , nl = 2l. The approx-imate results are also plotted in Fig. S3(a). Though the sup-pression is stronger in rcLG beams than that in lpLG beams,the minimum γc needed limits quality factor of the mechani-cal oscillation. So, the linear polarized LG beams are betterfor optomechanical application.

The apG beam also have zeros intensity in the beam fo-cus. The phase diagram for a sphere in apG beam is shown

in Fig. 2(b). Above the transition radius Rtran = 0.33 µm, thesphere is always stable. For apG beam, the suppression coef-ficient ratio can be written as ξa = ξa0δ

2. It is also plotted inFig. 3(a).

Appendix D: quality factor Q in various doughnut beams andsome discussions

Doughnut beams have different field distributions ( espe-cially, different radius of their bright rings) and thus havedifferent trapping potential. The mechanical oscillation fre-quency Ω and quality factor Q are investigated systematically.generally, larger radius of bright rings lead to smaller Ω andQ. The case for a spheres in linearly polarized LG0l beamswith different l is shown in Fig. S3.

ACKNOWLEDGMENTS

We are grateful to Prof. Zhifang Lin for sharing the nu-merical code. We thank Prof. Jack Ng for the helpful dis-cussion. This work was supported by the NSFC grants (No.11374032). J.C. is also supported by Shanxi Science andTechnology Department through 2014011008-1, 2014021004and NSFC through 11404201, 11674204.

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8

r (7m)0.02 0.04 0.06 0.08 0.1

supp

ress

ion 9

10-9

10-6

10-3

100

(a)

l= 1

l= 2

l= 3l= 4

Azim:

NA=0.95NA=0.90NA=0.85NA=0.80

R (7m)0.5 1 1.5 2

+/(

2:)

(Hz)

#104

0

3

6

9

(b) l=0l=1l=2l=3

R (7m)0.5 1 1.5 2

Q

#1010

0

3

6

9

(c) l=0l=1l=2l=3

1FIG. S3. (a) Light absorption suppression ratio ξ for a core-shell sphere at the center of azimuthally and right circularly polarized beams(LG0l beams) relative to linearly polarized Gaussian beams, with different core radius r and numerical aperture NA. Red solid lines are forour analytical results. Blue lines are for linearly polarized LG beams. Other parameters are the same as Fig. 3(a). (b) The optomechanicalfrequency Ω/(2π) and (c) mechanical quality factor Q of a sphere in different lpLG beams along x with power 100 mW under pressure 10−6 Pa.

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