Experimental and Theoretical Study of Transient Human Thermal Comfort Response in Convective and...

Optics Communications 230 (2004) 337–345

www.elsevier.com/locate/optcom

Experimental and theoretical study of the transient rotationof isotropic transparent microparticles in astigmatic

optical tweezers

Antonio Ambrosio, Bruno Piccirillo, Antonio Sasso, Enrico Santamato *

INFM, Dipartimento di Scienze Fisiche dell’Universit�aa di Napoli ‘‘Federico II’’, via Cintia 80126 Napoli, Italy

Received 28 April 2003; received in revised form 20 November 2003; accepted 21 November 2003

Abstract

Micrometer-sized rod-like non-birefringent transparent particles trapped by an astigmatic laser beam having el-

liptical intensity profile spontaneously rotate until they become aligned with the major axis of the beam cross-section.

The observed rotation confirms a simple picture based on orbital angular momentum transfer from the light beam to

the trapped particle. A model which regards the particle as a thin cylindrical lens with effective focal lengths f1 and f2has allowed to calculate the applied optical torque and, hence, the time law governing the particle alignment. The time

evolution of the particle alignment is studied for different rod-like particles by analysing the sequence of frames re-

corded by a CCD camera. The time evolution of the angle a formed between the main axes of the body and the laser

shape results in good agreement with the theory for bodies fully immersed in the spot size of the laser. Discrepancies

observed for rod-like particles longer than the minor elliptical axis of the beam cross-section, were removed by de-

veloping a more general model taking into account the partial overlapping between the body and the laser profile.

� 2003 Elsevier B.V. All rights reserved.

PACS: 42.50.Vk

Keywords: Optical tweezers; Orbital angular momentum of light

1. Introduction

Manipulation of microscopic objects without

mechanical contact with accuracy below the mi-

* Corresponding author. Tel.: +39-081-676359; fax: +39-081-

676346.

E-mail address: [email protected] (E. Santam-

ato).

0030-4018/$ - see front matter � 2003 Elsevier B.V. All rights reserv

doi:10.1016/j.optcom.2003.11.057

crometric range has been achieved by optical

tweezers since years [1]. Optical tweezers exploit

optical gradient forces from few tens of fN up to

hundreds of pN to produce potential wells whereparticles are trapped. Optical tweezers have found

many applications for the manipulation of bio-

logical materials [2] and for the investigation of

weak forces on microscopic scale [3–5].

There are many experimental situations, how-

ever, where individual biological particles have to

ed.

mail to: [email protected]

338 A. Ambrosio et al. / Optics Communications 230 (2004) 337–345

be oriented or where measuring or inducing me-

chanical torques it is important. Recently, Sacconi

et al. [6] have combined optical tweezers and a

magnetic trap to rotate super-paramagnetic beads.

However, when non-magnetic biological objects

are envisaged, they have to be attached to themagnetic beads, which is a difficult task. It seems

therefore interesting developing all-optical systems

able to simultaneously trap and rotate the particles

under investigation. In this case optical tweezers

are used as a tool to transfer angular momentum

from light to the trapped particles. As it is well

known the angular momentum carried by light can

be distinguished by the spin angular momentumassociated to the circular polarization and the or-

bital angular momentum associated with the spa-

tial distribution of the wavefront [7]. For circularly

polarized light the photon spin is well known to be

��h for r�, respectively. It has been demonstrated

[8] that helical wave fronts with azimuthal phase

term expðil/Þ give rise to an orbital angular mo-

mentum for linearly polarized light of l�h perphoton. As it is well known the rotation of mac-

roscopic bodies by angular momentum of light

was demonstrated for the first time on 1936 by

Beth [9] in an elegant and famous experiment.

Direct angular momentum transfer can be

achieved in two ways: (a) by photon absorption

and (b) by photon scattering. In the former case,

photons are absorbed along with their own angu-lar momentum. In the latter case, to the body is

transferred only the difference in the angular mo-

mentum between the incident and scattered pho-

tons. This last process was first observed in liquid

crystals by Santamato et al. [10] which named such

a phenomena as self-induced stimulated light

scattering (SISLS). The SISLS was later used to

realize liquid-crystal-based molecular motors [11].In recent times, spin angular momentum was

transferred to rotate birefringent [12,13] particles

trapped by optical tweezers. In another experiment

Friese et al. [14] have demonstrated that absorbing

particles trapped in a Gaussian beam were set into

rotation by elliptically polarized light and rotate in

a direction that depends on the handledness of the

ellipticity. Very recently, the optical rotation ofelongated rods made of transparent isotropic ma-

terial was achieved using the light polarization,

because of the so-called ‘‘form birefringence’’ of

the trapped particles [15]. In these works involving

optical tweezers only the spin of photons was ex-

ploited. New class of experiments where the orbital

angular momentum is transferred was first realized

by Sato et al. [16] which rotated red blood cellsdragging them with a laser beam prepared in a

higher-order Hermite–Gauss mode. In that ex-

periment a rectangular aperture was placed inside

the laser cavity to force the laser to oscillate in

TEM�01. Blood cells were aligned along the major

axis of the pattern mode and were oriented in the

transverse plane by rotating the rectangular aper-

ture. In a more recent paper O�Neil and Padgett[17] revisited Sato�s idea with a simplified method

by putting a rectangular aperture simply along the

path of a Gaussian laser beam. They showed that

an asymmetric object trapped in the beam waist

was aligned along the aperture axis. One alterna-

tive approach to achieve rotation was very recently

performed by our group [18] by means of an

astigmatic laser beam created by a cylindricallens.

Direct observation of transfer of angular mo-

mentum to absorptive particles from a laser beam

with a phase singularity (LG03 doughnut mode

obtained by computer generated hologram) was

demonstrated by Friese and coworkers [19,20].

Simultaneous transfer of spin and angular mo-

mentum has also been studied by Simpson et al.[21] in absorbing particles.

Several schemes have been recently proposed

where objects were rotated in the plane perpen-

dicular to the direction of the trapping beam,

thus adding a new useful degree of control.

Some of them are based on the peculiar shape of

either the trapped objects [22,23] or the trapping

optical field pattern. For example, Paterson et al.[24,25] drove the trapped particles around a

circle by rotating the intensity pattern created by

interference of a laser beam prepared in a La-

guerre–Gauss mode with a second Gaussian laser

beam. Optical torque created by induced polar-

ization in dielectric material by linearly polarized

laser beam has been demonstrated to be able to

manipulate and control the rotation of micro-objects [26,27]. In a very recent paper, Bingelyte

et al. [28] demonstrated that microscopic objects

A. Ambrosio et al. / Optics Communications 230 (2004) 337–345 339

held simultaneously in a double-optical tweezers

could be set into controlled rotation about any

axis of choice.

All the experiments where light angular mo-

mentum is transferred to trapped particles belong

to two main fields of applications: either to inducepermanent rotation like a micro-motor [29], or just

for alignment. Of course, especially for biological

applications which need to rotate particles or to

exert on macromolecules torques with a high de-

gree of control the latter case is much more useful.

While quantitative measurements of rotation fre-

quency of particles in an optical spanner have been

reported in several papers [13,20,21], according toour knowledge, no studies have been performed

regarding the time law of microparticles motion

during their alignment along the optical trap.

In the present work, we studied in details the

angular time law obeyed by the motion of elon-

gated trapped particles under the action of an as-

tigmatic laser trap. In particular, we focused on

the effects due to the shape of the particle and toits overlapping with the incident light intensity

profile.

2. Theory

The calculation of the orbital angular momen-

tum transferred from an elliptical laser beam whenpassing through a cylindrical lens has been already

discussed with great details in [30,31]. In the present

work the optical torque transferred to the lens was

calculated with a simplified theoretical description

with respect to that reported in [18]. When passing

through a transparent body each photon is scattered

from a state with orbital angular momentum �hr� k

to a state with orbital angular momentum �hr� k0,where k and k0 are the incident and scattered photon

wavevectors, respectively. The body receives an

amount l ¼ �hr� ðk0 � kÞof angularmomentumper

photon. The overall torqueM acting on the body is

given by M ¼Rl/ðx; yÞdxdy, where /ðx; yÞ is the

photon flux in the xy-plane transverse to the beam.

The light intensity Iðx; yÞ is related to the photon

flux by Iðx; yÞ ¼ hm/ðx; yÞ, m being the optical fre-quency. In the paraxial approximation, the optical

wavevector is related to the optical phase w by

k ¼ rw. We may therefore write down M as

M ¼ P2pm

hr�rðw0 � wÞi; ð1Þ

where P is the incident optical power and h�imeans

the intensity-weighted average over the spatial

beam profile. The difference w0 � w is the phasechange that the optical wave suffers in traversing

the body, that we suppose perfectly transparent.

For a body of refractive index n immersed in a

fluid of refractive index n0, we have

w0 � w ¼ ð2p=kÞðn� n0Þdðx; yÞ; ð2Þ

where dðx; yÞ is the local thickness of the body and

k ¼ c=m is the optical wavelength in vacuum. For

the sake of simplicity, we assume the intensity

profile of the laser beam in the trapping region

Gaussian with different radii wx and wy (1=e2 in-tensity) along the x and y directions, respectively.

The body is regarded as a thin cylindrical lens with

effective focal lengths f1 and f2 and oriented at

angle a with respect to the x-axis (see Fig. 1).

Without loss of generality, we assume the major

axis of the intensity profile directed along x (i.e.,

wx > wy). Under these assumptions, the phase

change in Eq. (2) becomes

w0 � w ¼ � pkf1

ðx cos aþ y sin aÞ2

� pkf2

ðx sin a� y cos aÞ2: ð3Þ

Taking the z-component of Eq. (1), we obtain

Mz ¼ �P ðf1 � f2Þðw2

x � w2yÞ

4cf1f2sin 2a

¼ �A sin 2a: ð4Þ

Here Mz is the z-component of the orbital an-

gular momentum that the trapped body extracts

from the light beam per unit time and, hence, itprovides the mechanical torque acting on the

body. Eq. (4) shows that in the case of spherical

particles (f1 ¼ f2) or circularly-shaped beams

(wx ¼ wy) the torque Mz is zero. Moreover, Mz

vanishes when a ¼ 0 or a ¼ p=2. Under the torque

Mz the body starts rotating around the z-axis untilit reaches its stable equilibrium position a ¼ 0

Fig. 1. The trapped particle can be considered as a cylindrical lens acting on the beam. Beam refraction results in a couple of forces on

the lens.


when A > 0, or a ¼ p=2 when A < 0. In the case of

overdamped rotator, the equation of motion for

the body is

�c _aa ¼ A sin 2a; ð5Þwhere c is the viscosity coefficient involved in the

rotation. For a cylinder of length L and diameter

d, c is given by [32]

c ¼ pgL3

3½ln p � dðpÞ� ; ð6Þ

where g is the viscosity of the surrounding fluid,

p ¼ L=d and dðpÞ is a tabulated function ac-

counting for end-effects [32]. Eq. (5) can be solved

yielding, for A > 0,

tan aðtÞ ¼ e�t=s tan að0Þ; ð7Þwith time constant s ¼ c=2A.

3. The experimental apparatus

We realized optical tweezers to trap and orient

micro-particles immersed in distilled water. A

block scheme of our apparatus is shown in Fig. 2.

In our previous paper [18] we used a TEM00

doubled-Nd:YVO4 laser (k ¼ 532 nm, impinging

power 100–300 mW) whose intensity profile was

made elliptical by using a cylindrical lens. In the

present work instead as laser source we used a

semiconductor diode laser emitting in the nearinfrared (k ¼ 785 nm, impinging power ranging

from 10 to 30 mW). The beam profile of the beam

emerging from the diode laser exhibited an ellip-

tical shape with radii Wx ¼ 3 mm and Wy ¼ 1 mm

at 1=e2 intensity. The beam was collimated

through a laser diode objective with focal length

8.5 mm. The intensity profile of the trapping beam

in the focal region of the microscope was mademore elliptical by using the cylindrical lens L1, with

focal length fy ¼ 80 mm and cylindrical axis along

the x direction. The focal plane of the lens L1 was

set at the standard distance of 160 mm from the

objective plane of the microscope. A polarizing

cube beam-splitter was inserted between the eye-

piece and the microscope objective so to observe

the particles through the same optical port whichthe laser beam was injected into. The focusing lens

L1 assured proper filling of the back pupil of the

objective to achieve high trapping efficiency. The

trap used a 40� (NA¼ 0.65 in water) and a 100�(NA¼ 1.25 in water) microscope objectives. The

beam radii (1=e2 intensity) at the trapping position

were wx ¼ 44� 2 lm, and wy ¼ 2:8� 0:1 lm with

Fig. 2. Astigmatic optical tweezers setup: a cylindrical lens in-

jects a TEM00 laser mode into the microscope objective.


the 40� objective and wx ¼ 17� 1 lm, and

wy ¼ 1:1� 0:1 lm with the 100� objective. The

cylindrical lens could be rotated around the beam

axis so to change the orientation of the intensity

profile. The image of the sample was projected

through the same focusing objective at the en-

trance of an eye-piece (10�). The magnified image

was observed with a CCD video camera on amonitor and recorded with a personal computer

for further analysis. An appropriate sharp cut-off

filter was used to block the laser beam before im-

aging. A hot-mirror was inserted between the il-

luminating lamp and the sample in order to reduce

heating and undesired convective motions inside

the water solution. With this device we trapped

small glass fragments obtained by crushing a smallpiece of optical fiber deprived of its cover (the

fragments were 10–100 lm across).

4. Results and discussion

In our previous paper concerning the subject of

the present work [18] we manipulated cylindrical

objects by an astigmatic trap. In the present ex-periment we handled more irregular particles

having approximately rod-like cylindrical shape.

Efficient trapping was found close to the bottom

of the sample chamber where spherical aberra-

tions are reduced. However, since our objects

were quite heavy, trapping was possible only very

close to the glass surface. Therefore, we can ef-

fectively consider that our glass fragments wereconfined only laterally and they lied on the mi-

croscope slide (2D trapping). That, apart from

possible friction due to the surface contact, does

not affect our investigation since our interest was

to study the transient orientation of the fragments

in the transverse plane. Fig. 3 shows typical se-

quence frames concerning the orientation of two

such particles trapped with the 40� objective andwith the 100� objective. The measured power

impinging into the trapping cell was 30 and 12

mW in the two cases, respectively. Once the par-

ticles had been stopped they could be oriented at

will by slowly rotating the cylindrical lens L1.

Although this dragging procedure may be very

useful to achieve accurate angular control of the

particle orientation, in this work we were mainlyinterested in the analysis of the transient motion

of the particles.

To have a quantitative proof that the orbital

angular momentum of light was really the re-

sponsible for rotation, we reconstructed the mo-

tion of our glass fragments from the frames

recorded by the CCD camera. For each recorded

frame, we marked the particle edges with twocoloured points. The slope b of the line connecting

the two markers was determined by suitable

computer software. The angles b and a are related

by

aðtÞ ¼ bðtÞ � b1; ð8Þ

where b1 is the slope angle which the body as-ymptotically reaches when it becomes aligned with

the laser profile. We fitted the data with the time

law (see Eq. (7))

Fig. 3. (a) Rotational motion of a rod-like particle trapped by the 40� objective. The particle dimensions were L ¼ 37 lm and d ¼ 6

lm. The power impinging onto the particle was 30 mW and the laser beam waists were wx ¼ 44 lm and wy ¼ 2:8 lm. (b) Rotational

motion of a rod-like particle trapped by the 100� objective. The particle dimensions were L ¼ 39 lm and d ¼ 15:6 lm. The power

impinging onto the particle was 12 mW and the laser beam waists were wx ¼ 17 lm and wy ¼ 1:1 lm. In both sequences (a) and (b) the

major axis of the laser elliptical spot is horizontal (not visible in the figure).


logðtanðbðtÞ � b1ÞÞ � logðtanðb0 � b1ÞÞ ¼ � ts;

ð9Þ

where b0 was the starting angular position of the

particle with respect to the x-axis. We used b0 and

s as fitting parameters. We analysed first the mo-

tion of the particle shown in Fig. 3(a). The result

of best fit is shown in Fig. 4(a) where the dots

represent the experimental data and the solid line

is the fitting curve obtained from Eq. (9). Simi-larly, Fig. 4(b) shows the fit between Eq. (9) and

Fig. 4. (a) Fit of Eq. (7) to the motion data of the particle shown in

f ðaðtÞÞ given by the term on the left of Eq. (7). (b) The same for the

the motion of the particle in Fig. 3(b). As it can be

seen, both fits are very poor. We ascribed this

discrepancy to the fact that for elongated rod-like

particles the fraction of the power intercepted by

the rod changes during its rotation. In particular,when the trapped rod is perpendicular to the el-

liptical laser cross-section the overlapping is min-

imum while, when it is parallel, the overlapping is

maximum. A simple way to account for this geo-

metric effect is depicted in Fig. 5. Let us assume an

elliptical laser cross-section with radius rðaÞ givenby

Fig. 3(a). On the ordinate axis we plotted f ðaðtÞÞ=f ðað0ÞÞ, withparticle in Fig. 3(b).

Fig. 5. This figure shows how the intersection area between the

particle and the laser beam was calculated. The ellipses is the

laser beam profile and the rectangle is the trapped rod-like

particle.


rðaÞ ¼ ab

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 sin2 aþ b2 cos2 a

p ; ð10Þ

where a � 2wx and b � 2wy are the major and

minor axes of the ellipse. As shown in Fig. 5, when

the cylinder is at angle a, it intersects the beam

cross-section over an area given approximately by2drðaÞ (gray area in Fig. 5). This simple approxi-

mation holds true when the dimensions L and d of

the rod obey LP a and d 6 b. Therefore, the

amount of the total laser power P intercepted by

the rod is

P ðaÞ ¼ 8drðaÞpab

� �P : ð11Þ

Replacing P with P ðaÞ in Eq. (4) yields the motion

law

Fig. 6. (a) Fit of Eq. (13) to the motion data of the particle shown in

gðaðtÞÞ given by the term on the left of Eq. (13). (b) The same for th

c _aa ¼ � 4ffiffiffi2

pdA

psin 2affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2 þ b2 � ða2 � b2Þ cos 2ap : ð12Þ

Eq. (12) is solved to

1þ DðaÞ1� DðaÞ

� �DðaÞ � mDðaÞ þ m

� �m

¼ 1þ Dða0Þ1� Dða0Þ

� �Dða0Þ � mDða0Þ þ m

� �m

e�t=s0 ; ð13Þ

where m ¼ b=a,

s0 ¼ pac16dA

¼ pa8d

� �s ð14Þ

and

DðaÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ m2 � ð1� m2Þ cos 2a

2

r: ð15Þ

Eq. (13) requires an estimation of m that in our

case is m ¼ 0:065 for both the 40� and the 100�objectives. As shown in Fig. 6, the agreement be-

tween the experimental data and the time law in

Eq. (13) is now very good over the full time range

for both particles. From the fit to the data we

calculated s0 ¼ 1:9� 0:01 s for the particle inFig. 3(a) and s0 ¼ 0:87� 0:01 s for the particle in

Fig. 3(b). The constant A, defined in Eq. (4), can be

estimated from Eq. (13) calculating c from Eq. (6)

and taking g ¼ 8:9� 10�3 N s/m2 at T ¼ 25 �C for

water. For the particle in Fig. 3(a) we estimated

p ¼ L=d ¼ 6:16, dðpÞ ¼ 0:51, c ¼ 3:6� 10�16 Nm s,

Fig. 3(a). On the ordinate axis we plotted gðaðtÞÞ=gðað0ÞÞ, withe particle in Fig. 3(b).


A ¼ 5:5� 10�16 Nm.For the particle in Fig. 3(b) we

estimated p ¼ L=d ¼ 2:5, dðpÞ ¼ 0:30, c ¼ 9:0�10�16 Nm s,A ¼ 4:6� 10�16 Nm.From the value of

A we also estimated the order of magnitude of the

torque involved during the rotation as Mz � 4ffiffiffi2

pdA=ðp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2

pÞ � 6:7� 10�17 N m, corres-

ponding to an optical force acting on the rotating

rod F � Mz=L � 1:8 pN for the particle in Fig. 3(a).

Similarly, for the particle in Fig. 3(b), we found

Mz � 3:7� 10�17 Nmand F � 9:3 pN. These forces

are of the right order of magnitude of the forces

usually involved in optical tweezers.

5. Conclusions

We presented a study on the transfer of the

orbital angular momentum of light to transparent

isotropic bodies trapped in optical tweezers with

laser beam having an elliptical profile. We studied,

in the transient regime, the laser-induced orienta-

tion process of particles having elongated rod-likeshapes and different sizes. A simple time law based

on angular momentum conservation was found in

poor agreement with the observed motion. We

ascribed this discrepancy to the fact that a time

varying fraction of the laser power is intercepted

by the particle during its rotation. A corrected

motion law accounting for this geometric effect

was proposed and found in excellent agreementwith our experimental findings. Having a proper

time law for particle rotation may be very useful

when dynamic rather than static torque measure-

ments are envisaged. Dynamic torque measure-

ments, in fact, do not require anchoring the

particle to a fixed body (e.g., the sample walls or

glass beads [6,33]). Fixing the particle, in fact, may

be a difficult task, especially in biological appli-cations. The dynamic technique presented here,

moreover, does not require the exotic laser sources

commonly used to transfer the orbital angular

momentum of light as, for example, Laguerre–

Gauss beams.

Acknowledgements

We thank INFM (Istituto Nazionale per la Fi-

sica della Materia) and MIUR (Ministero

dell�Istruzione, Universit�aa e Ricerca) for financial

support.

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