aa r X i v : . [ phy s i c s . op ti c s ] A p r Dissipation Effect on Optical Force and Torque nearInterfaces
Daigo Oue ‡ Division of Frontier Materials Science, Osaka University, 1-3 Machikaneyama,Toyonaka, Osaka, Japan 560-8531E-mail: [email protected]
March 2019
Abstract.
The Fresnel-Snell law, which is one of the fundamental laws in opticsand gives insights on the behaviour of light at interfaces, is violated if there existsdissipation in the transmitting media. In order to overcome this problem, we extendthe angle of refraction from a real number to a complex number. We use this complex-angle approach to analyse the behaviour of light at interfaces between lossy mediaand lossless media. We reveal that dissipation makes the wavenumber of the lightexceed the maximum allowed at lossless interfaces. This is surprising because, ingeneral, dielectric loss only change the intensity profiles of the light, so this excesswavenumber cannot be produced in the bulk even if there exists dielectric loss.Additionally, anomalous circular polarisation emerges with dissipation. The directionof the anomalous circular polarisation is transverse, whereas without dissipation thedirection of circular polarisation has to be longitudinal. We also discuss how the excesswavenumber can increase optical force and how the anomalous circular polarisation cangenerate optical transverse torque. This novel state of light produced by dissipationwill pave the way for a new generation of optical trapping and manipulation.
1. Introduction
Optical force and optical linear momentum were theoretically predicted by Maxwell [1]and observed by Nichols and Hull [2, 3]. The existence of optical torque and opticalangular momentum was theoretically proposed by Poynting [4], and experimentallyconfirmed by Beth [5]. Although the interesting fact that light can push and rotateobjects attracted many scientists, not much progress was made in the study of opticalforce and torque until the availability of a high-intensity light source for generating largeforces or torques. Around 1950, the laser was invented in Bell laboratories. Ashkin,who also belonged to Bell laboratories at that time, demonstrated manipulation ofmicroparticles with laser beam [6, 7]. After Ashkin’s experiments, many interestingsetups for optical manipulation have been proposed. Some of them function in ‡ Present address: Department of Physics, Imperial College London Prince Consort Road, Kensington,London SW7 2AZ, UK. issipation Effect on Optical Force and Torque near Interfaces
2. Electromagnetic fields near interfaces with dissipation
In order to calculate the electromagnetic field near dissipative interfaces, we furtherdevelop a complex-angle approach, which is originally proposed by Bekshaev et. al. todeal with total internal reflection [21]. FIG. 1 shows the situation we consider in thispaper. We consider interfaces between dissipative media and non-dissipative media.
Figure 1.
Refraction of light at interface. We consider light incidence on interfacebetween non-dissipative media ( n ) and dissipative media (˜ n = n + iκ ). Let theangle of incidence θ and the angle of refraction ˜ θ . The field is incident on the interface from the lower side, and we take θ for the angleof incidence. In dissipative media, the refractive index must be a complex value withnon-zero imaginary part (˜ n ∈ C ).˜ n = n + iκ . (1)Considering Snell’s law, n sin θ = ˜ n sin ˜ θ . (2)In this equation, the left hand side is real ( n , sin θ ∈ R ), while ˜ n has imaginary part.Thus, sin ˜ θ is not a real number any longer, but it is a complex number with non-zeroimaginary part (cid:16) sin ˜ θ ∈ C (cid:17) , and the angle of refraction must be a complex numberwith non-zero imaginary part,˜ θ = θ + iψ ( ψ < , (3) issipation Effect on Optical Force and Torque near Interfaces ( Re(˜ n sin ˜ θ ) = n sin θ , (4a)Im(˜ n sin ˜ θ ) = 0 . (4b)Once we obtain the angle of refraction, we can calculate the explicit expressions ofthe transmitted field by rotating + z -propagating plane wave E = E exp( i k · r ) towardsthe directions of refraction. Here, E = E p E s , k = n k . (5) k is the wavenumber in vacuum. For the rotation, we use R ( θ ) = cos θ θ − sin θ θ . (6)By substituting the complex-angle of refraction ˜ θ into the rotation matrix, we geta complex-angle rotation matrix which gives the wavevector of electromagnetic field indissipative media: k → ˜ k = R (˜ θ ) k = k + i η , (7)where k = k n sin θ cosh ψ − κ cos θ sinh ψ n cos θ cosh ψ + κ sin θ sinh ψ , (8) η = k − n sin θ sinh ψ + κ cos θ cosh ψ . (9)The imaginary part η represents the decay of the field, and the real part k representsthe propagation of the field. Note that, from the second equation of the modified Snell’slaw, η x ∝ n cos θ sinh ψ + κ sin θ cosh ψ = 0 . (10)FIG. 2 shows the iso-frequency curves followed by the real part of the wavevectorfor fixed k near the interface between a dissipative medium and various lossless media.The solid black curve is the curve with the presence of dissipation (˜ n = 1 . . i ), andthe grey curve is without dissipation (˜ n = 1 . i ). It can be said that the curve withdissipation is greater than the non-dissipative curve, which means that higher (spatial)frequency field is produced by dissipation and that this could enlarge the optical force onparticle in the field. More excess wavenumber is produced for larger angle of incidence.Physically, this excess wavenumber in the x -direction is produced at the expense of thewavenumber in the z -direction. issipation Effect on Optical Force and Torque near Interfaces Figure 2.
Excess wavevector is produced by dissipation. The grey curve is theiso-frequency curve without the presence of dissipation (˜ n = 1 . i ), while the blacksolid curve is the iso-frequency with the presence of dissipation (˜ n = 1 . . i ). By the complex-angle rotation, we can also calculate the electric field vector, E → ˜ E = R (˜ θ ) T p E p T s E s (11)= T p E p (cos θ cosh ψ − i sin θ sinh ψ ) T s E s −T p E p (sin θ cosh ψ + i cos θ sinh ψ ) . (12)Here, T p and T s are Fresnel coefficients. Finally, we get the explicit representation ofthe refracted field near an interface with dissipation:˜ E = ˜ E exp (cid:16) i ˜ k · r (cid:17) (13)= ˜ E exp ( i k · r ) · exp ( − η · r ) . (14)In this paper, we are interested in p-polarisation since it has non-trivial longitudinalfield vector, so from here we set E s = 0 and for simplicity set E p = 1 and n = 1,but it is straightforward to discuss the general case ( E s , E p = 0, and n ≥
1) using ourapproach. Note that (8) to (14) can also be derived directly from Maxwell’s equations.We can see that there is phase difference between the transverse z component of thefield vector and the longitudinal x component. That is because the continuity conditionsfor the tangential component of field and for the normal component are different, andthis induces phase difference between the two components, and causes rotation of thefield. To describe the degree of circular polarisation and direction of the field rotation, issipation Effect on Optical Force and Torque near Interfaces Figure 3.
Anomalous circular polarisation (transverse y component of s ) emergeswith dissipation ( ˜ n = 1 . . i ). we can use a psuedovector s = g ω Im ( E ∗ × E ) , (15)where g = 1 / π is a Gaussian-unit factor. For our field, we have s = g ω sinh ψ cosh ψ exp ( − η · r ) e y . (16)Here, e y is the unit vector in the direction of y . We can observe if we flip the sign of θ ,then, from (4b), the sign of ψ is also flipped, and thus the direction of the psuedovector s is flipped. This is one kind of spin-momentum locking [22], in which the direction ofcircular polarisation and that of propagation are tied to each other. In FIG. 3, we plotthe transverse y component of the psuedovector. This implies that the field is rotatingin the transverse direction and that we can exert transverse optical torque on a particlein the field.
3. Optical force and torque near dissipative interfaces
In this section, we discuss the effect of dissipation on optical forces and torques. Weconsider the situation shown in FIG. 4. There is a small probe particle on the interfacebetween dissipative media and lossless transparent media. For the calculation of theoptical force and torque, we assume that the particle is small enough compared to thewavelength of the field so that it can be regarded as a point dipole. The time averagedoptical force and torque exerted on a point dipole is given by [23, 24] F = (cid:28) ( d e · ∇ ) E + ( d m · ∇ ) H + 1 c ˙ d e × B − c ˙ d m × D (cid:29) ∝ Im( α e ) k , (17) T = h d e × E + d m × Hi ∝ Im( α e ) s . (18) issipation Effect on Optical Force and Torque near Interfaces Figure 4.
Schematic image of the setup for calculating optical force and torque. Wehave a small probe particle on the interface between dissipative media and transparentmedia. The diameter of the particle is 100 nm.
Here, E and H are real fields associated with complex fields, E = Re ( E e − iωt ) and H = Re ( H e − iωt ). We do not consider higher order contributions, which could causeelectric-magnetic dipolar interaction force [23] and torque [25, 26], but focus on thefundamental order by the dipole approximation. These complex fields satisfy themonochromatic Maxwell’s equations: ∇ · E = ∇ · H = 0 , (19) ∇ × E = i ωc µ H , (20) ∇ × H = − i ωc ε E . (21)Complex flux densities, D and B , are characterized by permittivity ε and permeability µ : D = ε E , B = µ H , and these give the real fields: D = Re ( D e − iωt ), B =Re ( B e − iωt ). We also use complex dipole moments d e , d m to give real dipole moments: d e ( r , t ) = Re ( d e ( r ) e − iωt ), d m ( r , t ) = Re ( d m ( r ) e − iωt ). Electric and magnetic dipoles arecharacterized by electric polarisability α e and by magnetic polarisability α m : d e = α e E , d m = α m H .In FIG. 5, we compare the scattering radiation force on a 100 nm gold nanosphere by650 nm excitations with and without dissipation. The left figure is the case without anydissipation (˜ n = 1 . i ), and the right top and bottom figures are with dissipation(˜ n = 1 . . i and 1 . . i , respectively). It can be said that dissipation assistsoptical force in both x and z direction. At large θ , the radiation force reaches 0. Thisis because the larger angle of refraction is, the smaller Fresnel coefficient is. When theangle of incidence is π/
2, the transmission coefficient is zero.To clarify the excess wavenumber effect, we plot optical force on particle per unitpower in FIG. 6 and compare the dissipative cases and the non-dissipation case. We canconfirm in both x and z directions optical force per intensity is assisted by dissipation.The enhancement of the force which we can see at θ = 0 is a contribution from thepolarisability of the probe particle. The radiation force is not only proportional to thewavevector of the field but also to the polarisability of the probe particle, which increaseswith the imaginary part of the refractive index of surrounding media. This is anotherfactor of the the enhancement of the radiation force. issipation Effect on Optical Force and Torque near Interfaces Figure 5.
Dissipation-assisted optical force. The left figure is the plot of scatteringradiation forces exerted on the gold nanosphere in the case without dissipation(˜ n = 1 . i ). The right top and bottom figures are the case with dissipation(˜ n = 1 . . i and 1 . . i , respectively). In these graphs, the purple lines indicatethe radiation force in the x -direction F x , and the green lines the radiation force in the z -direction F z . The excitation wavelength is 650 nm in all plots. FIG. 7 shows transverse optical torques on the gold nanosphere induced byanomalous circular polarisation, which cannot be generated without dissipation. Asthe anomalous circular polarisation vanishes at θ →
0, the transverse optical torquevanishes. This confirms that the anomalous polarisation causes the transverse torque.The optical force and torque are proportional to the real part of the wavevectorand the spin vector, respectively. Both of these vectors become larger as the dissipationparameter increases. Since there also exists Fresnel coefficient contribution, it cannotsimply be said that the force and the torque increase with the dissipation parameter.However, optical force and torque per unit intensity do monotonically increase with thedissipation. The physical meaning of this statement is that large dissipation causesstrong compression of light near the interface and results in enhancements of themomentum and the spin of photons. issipation Effect on Optical Force and Torque near Interfaces Figure 6.
Optical forces per unit intensity for various dissipation parameters(˜ n = 1 . i, . . i, . . i ). The left figure shows the radiation force inthe x -direction, and the right figure is the plot for the radiation force per in the z -direction. It is clear that in both x and z directions, the optical radiation force isassisted. Figure 7.
Transverse torques exerted by anomalous circular polarisation on 100 nmgold nanosphere. The excitation wavelength is 650 nm. The purple curve and the greencurve are the y -components of the optical torques T y with dissipation (˜ n = 1 . . i and 1 . . i , respectively), while the blue curve is the torque without any dissipation(˜ n = 1 . i ). It is obvious that there is non-zero transverse torque with dissipation,whereas there is no torque without dissipation since the field does not rotate in thetransverse direction without dissipation. issipation Effect on Optical Force and Torque near Interfaces
4. Conclusion
To sum up, we utilised a complex-angle approach for calculating electromagneticfield near interfaces with dissipation, and revealed dissipation-induced extraordinarybehaviours of the field: production of excess wavevector and generation of anomaloustransverse circular polarisation. We also studied what kind of effects these behaviourscause on optical force and torque. Excess wavevector assisted optical force, andanomalous circular polarisation generated transverse optical torque. Since the effectsdiscussed in this paper can be produced simply by adding dissipation, they are easyto explore in experiments. These effects add additional degrees of freedom for opticaltrapping and manipulation.
Acknowledgments
I thank Professor Hajime Ishihara and Professor Tomohiro Yokoyama for fruitfuldiscussions on optical force and torque. I thank Thomas Hodson, Samuel Palmer, andYuki Kondo for proofreading and helping me to improve this manuscript.
Appendix: Polarisability of a small particle
According to literatures [19, 23, 24, 27], we can use the formula below to calculate thepolarisability of a subwavelength spherial particle. α e ≃ ε ( ε p − ε ) ε p + 2 ε a . Here, ε p and ε are the permittivity of the particle and that of the surrounding media,respectively. a is the radius of the particle. References [1] Maxwell J C 1865
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