October 15, 2008 / Vol. 33, No. 20 / OPTICS LETTERS 2371

Electromagnetic torque and force in axiallysymmetric liquid-crystal droplets

István JánossyResearch Institute for Solid State Physics and Optics, PO Box 49, H 1525, Budapest, Hungary (janossy@szfki.hu)

Received May 12, 2008; revised July 21, 2008; accepted August 25, 2008;posted September 16, 2008 (Doc. ID 96004); published October 14, 2008

Circularly polarized light exerts torque on birefringent objects. In the case of axially symmetric particles,however, the moment of radiation force balances the direct optical torque. This explains the observation thatradial liquid-crystal droplets, in contrast to planar droplets, do not spin in circularly polarized light. Theconclusion is in agreement with considerations based on the angular momentum conservation of light [Phys.Rev. Lett. 96, 163905 (2006)]. © 2008 Optical Society of America

OCIS codes: 260.2110, 160.3710.

It is well known that light can carry angular momen-tum [1]. One part of the angular momentum is asso-ciated with the ellipticity of the polarization state ofthe light beam (“spin” angular momentum), while thesecond part is the orbital angular momentum. Circu-larly polarized waves carry a spin angular momen-tum of ±� per photon. This part of the angular mo-mentum was first demonstrated experimentally byBeth [2], who exposed a suspended birefringent ob-ject to polarized light and detected the torque on it. Amore direct evidence of the transfer of angular mo-mentum to matter is found in droplets of anisotropicmaterials, such as liquid crystals, immersed in liq-uids. When such droplets are irradiated with circu-larly polarized light, they are set into continuous ro-tation [3–5]. There are two ways to describe thelatter phenomenon. The first method is to analyzethe optical torque exerted by the electromagneticfield on the material; the second one is to comparethe polarization state of the incoming light beamwith that of the exiting beam.

To illustrate the two methods, we consider a plan-parallel slab of a liquid crystal with thickness d andarea A, illuminated by a homogeneous beam. Foruniaxial media, such as nematic or smectic A phasesthe optical torque density, �, can be written as [6]

� = �P � E� = �0��� − �����n · E��n � E�� �1�

where P is the dielectric polarization, induced in thesubstance by the electric field, E; ��=no

2, ��=ne2 are

the perpendicular and parallel components of the op-tical dielectric tensor; no and ne are the ordinary andextraordinary refractive indices, respectively; n isthe liquid crystal director; and the bracket � � denotestime averaging. In a planar droplet (n homogenousand perpendicular to the light propagation direction,z), the optical torque has only a z component, whichis given as

�z =1

2�0��� − ���E�E� cos ��. �2�

Here, E� and E� are the amplitudes of the electric-field components parallel and perpendicular to the

director, respectively; �� is the phase difference be-

0146-9592/08/202371-3/$15.00 ©

tween them. In the geometrical optics approximation[7] ���z�=k0�ne−no�z+��0, where k0 is the wave vec-tor in vacuum and ��0 is the phase difference be-tween the two polarization components at the en-trance face of the slab �z=0�. The total optical torque,M, acting on the slab has also only a z component:

Mz = A�0

d

�zdz

=A

2�0��� − ���E�E��

0

d

cos�k0�ne − no�z + ��0�dz

= A�0

ne + no

2k0E�E��sin ���d� − sin ��0�. �3�

On the other hand, as shown in [7], in the geometri-cal optics limit inside a birefringent medium, thequantity Lz�z�=�0��ne+no� /2k0�E�E� sin ���z� can beregarded as the spin angular momentum density ofthe beam. Therefore the total torque can be writtenas Mz=A�Lz�d�−Lz�0��; i.e., the torque is equal to thechange of the angular momentum of the light beamacross the sample.

In this Letter we investigate the case of axiallysymmetric droplets. It was pointed out by Marrucciet al. [8] that a slab of nematic liquid crystal orientedradial around the symmetry axis of the slab convertsthe spin angular momentum of light into orbital an-gular momentum. They demonstrated this by consid-ering the example of a half-wave plate. When a (say)left circularly polarized beam passes through theslab, beside becoming right circularly polarized, it be-comes also helical; i.e., the phase of the exiting beamchanges by 2�m going around the optical axis. m isan integer; in the case considered m=2 [8]. Thechange of the spin angular momentum (m�=2� perphoton) is compensated by the change in orbital an-gular momentum, associated with the helicity of thebeam (−2� per photon). The net change of the angu-lar momentum of the light beam is therefore zero.From this fact it follows that the net electromagnetictorque, exerted by the optical field on the sample,should also vanish. However, the direct optical

torque, as determined above for planar cells, does not

2008 Optical Society of America

2372 OPTICS LETTERS / Vol. 33, No. 20 / October 15, 2008

vanish in general for radial droplets. In fact, in thegeometrical optics approximation, the torque is thesame for the radial director configuration as for theuniform one.

In this Letter we prove that the apparent contra-diction between the two approaches of deriving thetotal torque can be resolved by taking into accountthe electromagnetic force acting on the droplet. As itwill be shown, for axially symmetric droplets the zcomponent of the moment of this force balances thedirect optical torque. The total torque is zero henceaxially symmetric droplets are not set into rotationby light beams.

There are different methods to calculate the radia-tion force. We first apply the method developed byGordon [9] for “gaseous” medium, i.e., a mediumwhere the difference between the macroscopic and lo-cal field strengths can be neglected. In this limit, theforce density, f, can be written as

f = f�E� + f�B� with f�E� = �P · ��E, f�B� =P

c� B,

�4�

where B is the magnetic field strength. The first termon the right-hand side of the equation represents theforce acting on dipoles in an electric field gradient,while the second one gives the Lorentz force exertedby the magnetic field on oscillating dipoles. Introduc-ing cylindrical coordinates �, , and z, the electricfield strength can be written as E=E�e�+Ee+Ezez,where e�, e, and ez are unit vectors in the radial,azimuthal, and z directions, respectively. Similar ex-pressions hold for P, f, and �. The azimuthal compo-nent of the force is

f�E� =

1

�PE� +

�E

� + P�

�E

��+ Pz

�E

�z,

f�B� = P� E

�+

�E

��−

1

�

�E�

� + Pz �E

�z−

1

�

�Ez

� .

�5�

In deriving the first relation, the identity �� /��e�

=e was used. For the second relation the Maxwellequation ��E=−�1/c�B was applied; E=�0

t E�t��dt�+C, where C is independent from time (although itdepends on spatial coordinates). For a monochro-matic wave �PiEj�=−�PiEj� and �Pi��Ej /�xk��=−�Pi��Ej /�xk��, where the time averaging is over anumber of full periods. The time average of the azi-muthal force is therefore

�f� =1

��PE� − P�E� +

1

��P�

�E�

�+ P

�E

�+ Pz

�Ez

� .

�6�

Let us consider a circularly polarized beam with anaxially symmetric intensity distribution. Before pen-

etrating into the droplet, the electric-field compo-

nents can be written as Ex=A�� ,z�cos t, Ey=A�� ,z�sin t. Performing a transformation fromrectangular coordinates to cylindrical ones, we obtainE�=A�� ,z�cos�t−�, E=A�� ,z�sin�t−�. Withinthe droplet Ei=Ai cos�t+�i−�, i=� , ,z, where theamplitude Ai and the phase �i depend on the detailsof light propagation. However, as a consequence ofthe axial symmetry of the droplet and the intensitydistribution of the incoming beam, Ai and �i are onlya function of � and z, but not of . Taking into ac-count this fact and using the linear relation betweenP and E, one obtains

�P�

�E�

�+ P

�E

�+ Pz

�Ez

� = 0. �7�

Therefore the z component of the moment of electro-magnetic force density, mz, is

mz = ��f� = �PE� − P�E� = − �z. �8�

As it can be seen, the moment of the force densitycompensates the direct optical torque, even locally.

In the general case the electromagnetic force isusually calculated from the Maxwell stress tensorand the Abraham force [9,10]. In this approach theforce density, fEM, is given as

fEM = DivT +1

c

�

�tE � B, �9�

where the Maxwell stress tensor is, for nonmagneticmaterials, Tij= �Fem+�m�Fem /��m��ij+EiDj+BiBj /�0,with Fem=−�1/2��E ·D+B2 /�0� and �m being the den-sity of the medium. It is shown by Gordon for isotro-pic dielectrics [9] that f and fEM differ by the gradientof a scalar function of the field strengths. To provethat the same holds for anisotropic media, we use theMaxwell equations (assuming no static charges orcurrents): div D=div B=0; ��E=−�1/c�B; ��B= �1/c�D, with D=�0E+P. Taking into account themathematical identities grad�E ·D�= �E ·��D+ �D ·��E+E� ���D�+D� ���E� and 1/2 grad B2

= �B ·��B+B� ���B� one finds from Eqs. (4) and (9)that

fEM − f = grad�m

�Fem

��m−

1

2E · P . �10�

As the difference between the forces can be expressedas a gradient of a scalar function, it follows that�S�fEM,− f�ds=0. The integral is taken along a circlein the �, plane with the center at the z axis. As aconsequence of the axial symmetry, along this path�fEM,− f� must be independent of . For this reasonnot only the integral, but its argument should vanishas well, i.e., �fEM,�= �f�. It is thus shown that Eq. (8)holds in the general case, too.

From the above considerations it follows that incircularly polarized light the total torque acting onradial liquid-crystal droplets vanishes. As a result,

these droplets, in contrast to planar ones, do not spin

October 15, 2008 / Vol. 33, No. 20 / OPTICS LETTERS 2373

in circularly polarized light. This conclusion is inagreement with experimental observations [11,12].

The direct optical torque by itself causes directordeformation, while the electromagnetic force inducesflow. The question that arises is how can they com-pensate each other? To find the answer, we note thatthe initial director configuration is indeed distortedby the optical torque exerted by the light beam. Thisdistortion was directly observed in laser tweezers ex-periments [11,12]. Associated with the director defor-mation, a nonuniform stress field develops in thedroplet [13]. In planar configuration, where the mo-ment of radiation force is zero [5], the force arisingfrom the director stress spins the droplet. In radialdroplets the two forces are equal in magnitude butopposite in direction; therefore the droplet is not ro-tated. This is true regardless of the light-induced de-formation of the director, provided that the directordistribution remains axially symmetric. It should benoted that this conclusion is not based on the geo-metrical optics approximation; it is valid for planewaves as well as for tightly focused beams. It alsoholds for any rheology between the liquid and liquidcrystal. It is possible, nevertheless, that above athreshold intensity symmetry-breaking deformationoccurs in the director configuration of a radial drop-let. Above the threshold the axial symmetry is lost,and therefore the moment of the radiation force doesnot compensate exactly the direct optical torque. Inthis case droplet rotation takes place. Such observa-tions in radial droplets were reported by Murazawaet al. [11,12].

Finally, we discuss two aspects of the above phe-nomenon, both connected with absorption. First, inthe presence of absorption the dielectric polarization,P, is phase shifted with respect to the electric field,E; therefore Eq. (7) is not valid any longer. The exactcompensation between the torques does not hold, sodroplet rotation can occur. The second consequence of

the presence of absorbing dyes is that the optical

torque is greatly enhanced [14,15]. However, asshown in [5], in order to fulfill the law of conservationof angular momentum, the total torque acting on thedroplet, associated with the enhancement, must bezero. Indeed, it was found that dye doping has no in-fluence on the spinning speed of planar droplets[5,16]. In the case of radial droplets, dye-induced en-hancement of the optical torque cannot induce drop-let rotation. On the other hand, it can lower thesymmetry-breaking threshold, just as in the case ofoptical Freedericksz transition in dye-doped homeo-tropic nematic liquid-crystal cells [14].

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