32nd URSI GASS, Montreal, 19-26 August 2017

Force Tracing and its Application in Optical Manipulation

Alireza Akbarzadeh(1)* and Christophe Caloz(2)

(1) Foundation for Research and Technology-Hellas (FORTH), Heraklion, Crete, Greece 71110 (2) Poly-Grames Reseach Center, Polytechnique Montréal, Montreal, Quebec, Canada H3T 1J4

Abstract The recently developed method of force tracing is reviewed and it is described, through several examples and time-flying animations, how the Lorentz force fields can be traced along the ray trajectories in a medium. Furthermore, to show the applicability of force tracing in practice, this technique is employed to design various graded-index systems for optical manipulation. 1. Introduction It is known from the theory of electrodynamics that energy and momentum are elements of the universal stress-energy tensor and related to each other through the conservation equations [1]. In the realm of geometrical optics, with the use of Lagrangian calculus, the energy transport is described by the language of geometry [2]. Similarly, it is interpreted that, given the relation between energy and momentum, one should also be able to describe the momentum transport or momentum variation (optical force) through the geometrical approach. This aspect of geometrical optics, which has not been much considered in the literature, led to the recently developed method of force tracing [3]. In principle, force tracing is a technique to trace the optical force arrows along the light ray trajectories in a medium. It is worth mentioning that the main motivation of developing this method was to ease the complexities related to the optical force full-wave computation, and also manipulate optical force according to one’s interests in a reverse engineering style. Owing to the simplicity and versatility it brings to the computations and design procedures, force tracing can be invoked in many different applications. This paper reviews the fundamentals of force tracing, and demonstrates, through various examples and time-flying animations, how force tracing can be applied in isotropic and anisotropic media. Furthermore, several interesting designs for optical manipulation assisted by this method will be reviewed. 2. Formulation of Force Tracing The time-averaged bulk Lorentz force density applied to bound current and charges by electromagnetic fields in a source-free region is [4-7]

( ) ( ){

}0

0 0 0

1Re

2f P E M H

i P H i M E

μ

ω μ ωμ ε

∗ ∗

∗ ∗

= −∇ ⋅ + − ∇ ⋅

− × + ×

(1)

where, P

and M

are polarization and magnetization vectors. Considering quasi-plane wave expressions with rapidly varying phases and slowly varying amplitudes

( ) ( ){ } { } ( )0 0 0, , , , expE r t H r t E H ik k r i tω= ⋅ −

for the

electric and magnetic fields and using Hamilton’s ray equations, equation (1) for isotropic media with refractive index n simplifies considerably as,

4

1knormalized

f k Ln

= ×

(2)

where ( ) ( )ˆk z x y y xL e k dk d k dk dτ τ = −

, k

is the

wave vector, ˆze is the unit vector coplanar with 0E

and

0H

, τ is the ray tracing parameter, and normalization has

been carried out with respect to 2

0 0 2Eε

. Some

algebraic manipulations show that equation (2) is directly proportional to the curvature of ray trajectories, which is an important geometrical aspect of the optical force [3]. A similar geometrical-optics-based approach can be used to simplify the bulk Lorentz force density for each polarization of the electromagnetic waves in anisotropic media, although the obtained expressions are much more complicated than that of the isotropic case. Force tracing formulation in anisotropic media with diagonal constitutive tensors can be found in [3]. As an example of force tracing in isotropic media, consider a Luneburg lens with radially symmetric graded

index ( ) 22n r r= − and outer radius 0 1r = . It should be

noted that all the radii and distances in this paper are in arbitrary units of length (a.u.l), which must satisfy the conditions required by the geometrical optics at the wavelength of operation. Shown in Figure 1(a) is the traced optical force field along the corresponding ray trajectories, which has been obtained with equation (2). As seen in Figure 1(a), for rays near the boundary of a Luneburg lens the optical force is stronger (thicker arrows), due to the fact that the rays with larger impact parameters experience larger curvatures. For the same reason, the ray crossing the center of the lens does not feel any perturbation, i.e. zero curvature, and hence imparts no force onto the lens. In order to show the validity of force tracing analysis, the full-wave simulation results for the optical force in the Luneburg lens are depicted in Figure 1(b), where a good agreement between the full-wave and force tracing analysis is seen. As another example, the result of force tracing performance for the invisible cloak

[7] is illustrated in Figure 1(c,d), where an acceptable agreement between the force tracing results and the analytical results given in [7] is seen. As highlighted by the green arrows in Figure 1(c,d), at the boundary of the cloak, the incident rays undergo refraction, and, due to the changing momentum at the boundary, surface forces appear [3].

(a)

(b)

(c)

(d)

Figure 1. Force density arrows along the rays in the Luneburg lens computed by (a) force tracing, and (b) full-wave analysis. Bulk (black) and surface (green) force density arrows in a cloak of inner radius of 0.25 (a.u.l.) and outer radius of 1 (a.u.l.) computed by (c) force tracing, and (d) analytical calculations.

3. Force Tracing for Optical Manipulation As seen in Figure 1(a), the collimated rays apply a positive force (i.e. pushing force) onto the Luneburg lens. Alternatively, if the rays enter the lens from a single point, they impart a negative force (i.e. pulling force) onto the lens [8]. As a consequence, under the illumination of a collimated light beam, two sufficiently close Luneburg lenses attract one another. The profiles of optical forces on the two lenses in the vicinity of each other can be calculated through the use of force tracing. Then, with the assumption of elastic collision between the two lenses and the presence of damping, the equations of motion for each of the lenses is obtained [see Figure 2(a)]. As seen in Figure 2(a), the lenses undergo a sequence of damped translational and oscillatory motions. The magnitudes of the two motions keep decreasing after each collision till the ultimate point, where the lenses stop moving and remain in contact. The corresponding equations of motion for the case of fully inelastic collision between lenses can be obtained in a similar way [8]. The inelastic collision makes it possible to arrange the Luneburg lenses in interesting configurations, among which two are shown in Figure 3. Figure 3(a) compares two Luneburg lenses that are complementary to each other, so that full transparency

is achieved by cascading two pairs of Luneburg lenses. Another interesting configuration is depicted in Figure 3(b) with vertical and horizontal illumination of collimated beams on four Luneburg lenses. This specific inelastic parallel arrangement of Luneburg lenses leads to the construction of an isolated space, which is out of the reach of rays and circumscribed by the lenses. Both the elastic and inelastic collision mechanisms and the described systems are offering versatile opportunities for various types of optical manipulation such as particle imaging and trapping, particle transport, bio-sensing, accurate measurements, beam guiding, and space isolation.

Figure 2. (a) Distance profile between two Luneburg lenses elastically colliding under the illumination of a collimated light beam. (b) Optical force fields (black arrows) within two neighboring Luneburg lenses. The collimated beam is shining from the left side.

Figure 3. (a) Full transparency, and (b) space isolation obtained through fully inelastic collision between four Luneburg lenses.

Another interesting design is the optical dimer formed by two graded-index lenses with identical masses

1 2m m m= = and radii 1 2 2R R R= = . In this design, the

optical force is used to cancel out the gravity and form a potential well by the lower lens [see Figure 4(a)] to trap the higher lens. The profile indices of lenses may be designed such that the lenses would float freely in space without touching each other upon illumination from the bottom. According to the described design objectives and requirements, a reverse engineering approach combined by the tomographic technique in geometrical optics [9] could be followed to compute the refractive index profiles. The details of the reverse engineering approach will be explained during the presentation and can be found in [10].

Figure 4. (a) Three dimensional schematic of the proposed dimer. (b) The paths of the lenses and the corresponding center of mass ( )cx t versus time.

After determining the properties of lenses, the optical force distribution, and hence the resulting equations of motion would be obtained with the use of force tracing technique. As shown in Figure 4(b), after a transient phase of damped collision-less oscillatory and translational motions, the system converges to its equilibrium state, in which the lenses stand floating in space at a distance from each other, forming a stationary dimer. The other important feature to study is the stability of dimer and its sensitivity to the lateral misalignment between the lenses, which is likely to appear in practice. The force tracing analysis is performed to inspect how a lateral distance like

yΔ between the lenses may affect the overall

performance of the designed dimer. As seen in Figure 5(a,b), if the lateral distance does not exceed a critical value like cyΔ , i.e. cy yΔ < Δ , the lateral restoring force

yF aligns the lenses and leads the systems to come back

to its equilibrium state. The proposed dimer is a potential platform for several applications such as gravitational sensing, bio-sensing, and accurate distance measurement. Furthermore, the designed dimer in addition to the other devices reviewed in this summary are initial steps towards the more sophisticated optically self-aligning systems, in which a set of randomly arranged lenses are programmed to align themselves along an arbitrary grid of interest. Attaining such intelligent self-aligning systems would open avenues for various interesting applications and offer new possibilities for optical manipulation.

Figure 5. (a) Force tracing of the dimer with the lateral misalignment. (b) The restoring force versus the lateral distance ( 9.8g = is the acceleration gravity).

4. References 1. J. D. Jackson, Classical Electrodynamics. New York, NY, USA: John Wiley, 1998. 2. M. Born and E. Wolf, Principles of Optics. Cambridge, UK: Cambridge University Press, 1999. 3. A. Akbarzadeh, M. Danesh, C. W. Qiu, and A. J. Danner, “Tracing Optical Force Fields within Graded-index Media,” New. J. Phys. 16, 2014, p. 053035. 4. R. M. Fano, L. J. Chu, and R. B. Adler, Electromagnetic Fields, Energy and Forces. Cambridge, MA, USA: MIT Press, 1968. 5. B. A. Kemp, “Macroscopic Theory of Optical Momentum,” Prog. Opt. 60, 2015, pp. 437-488. 6. B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Ab Initio Study of the Radiation Pressure on Dielectric and Magnetic Media,” Opt. Express, 13, 2005, pp. 9280-9291. 7. H. Chen, B. Zhang, Y. Luo, B. A. Kemp, J. Zhang, L. Ran, and I. B. Wu, “Lorentz Force and Radiation Pressure on a Spherical Cloak,” Phys. Rev. A, 80, 2009, p. 011808(R).

8. A. Akbarzadeh, J. A. Crosse, M. Danesh, C. W. Qiu, A. J. Danner, and C. M. Soukoulis, “Interplay of Optical Force and Ray-optic Behavior between Luneburg Lenses,” ACS Photon. 2, 2015, pp. 1384-1390. 9. U. Leonhardt and T. G. Philbin, Geomtry and Light: The Science of Invisibility. New York, NY, USA: Dover Publication, 2010. 10. A. Akbarzadeh, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Graded-index Optical Dimer Formed by Optical Force,” Opt. Express, 24, 2016, pp. 11376-11386.