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Abstract
In recent years, optical forces and torques have been investigated in sub-wavelength evanescent fields yielding a rich phenomenology of fundamental and applied interest. Here we demonstrate analytically that guided modes carrying transverse spin density induce optical torques depending on the character, either electric or magnetic, of the dipolar particles. The existence of a nonzero longitudinal extraordinary linear spin momentum suitable to manipulate optical forces and torques modifies optical forces either enhancing or inhibiting radiation pressure. Hybrid modes supported by cylindrical waveguides also exhibit intrinsic helicity that leads to a rich distribution of longitudinal optical torques. Finally, we show that chiral dipolar particles also undergo lateral forces induced by transverse spin density, amenable to chiral particle sorting. These properties are revealed in configurations on achiral and chiral dipolar particles within confined geometries throughout the electromagnetic spectra.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical forces and torques have been a subject of fundamental interest since long ago. These interactions are crucial to achieve trapping of microscopic particles through optical tweezers and nanometric manipulation using different optical configurations [1–10]; optical manipulation in general has fascinating applications not only in optics, but also in atomic and chemical physics, and in biological sciences [3,11–14].
Recall that, for small Rayleigh particles (where higher-order multipolar contributions can be neglected), not only the optical torque but also the radiation pressure forces are connected to the spin angular momentum (SAM) [4,15–19]. Electromagnetic waves carry, apart from the usual linear momentum given by Poynting vector, an extraordinary momentum given by the curl of the spin angular momentum, arising from the existence of a non-linear polarization in a non-homogeneous field [4]. In this context, optical torques and forces explicitly arising from the emergence of transverse SAM in evanescent fields without intrinsic helicity have been outlined in [20–23], and the transversal force induced by these fields in larger particles with simultaneous magneto-electric response has been studied in [24–26]. On the other hand, lateral optical forces based on the helicity properties of the electromagnetic field have been also proposed as means to distinguish chirality of particles or molecules [9,27 –32], those relying on strong lateral forces [27–29] being especially related to the present work.
However, the optical forces and torques induced by guided light inside waveguides have not been studied yet acting on dipolar particles, neither achiral nor chiral. Non-homogeneous electromagnetic fields, with intrinsic helicity, are also found inside waveguides. In fact, planar and cylindrical cavities have been shown to exhibit an interesting phenomenology concerning spin density and angular momenta [33–35], in turn being a suitable configuration to manipulate particles and molecules as microfluidic channels [12]. For instance, extremely interesting optical platforms wherein optical forces and torques in the low frequency domain could be built inside such as water-filled waveguides [36]. Actually, this scenario could also be realized for higher frequencies up to the near-IR regime, which is in turn suitable for optical trapping and manipulation of microscopic particles through optical tweezers [3,5,8]. The lower-refractive index of water (or other liquids) in this electromagnetic regime comes only at the expense of requiring thicker waveguides.
Although transverse optical forces in the surface of waveguides have been studied [33,37–40], the optical interactions inside the waveguide have been not so deeply analyzed, and most of the work focused on the transversal forces in systems with cylindrical symmetry [41]. In this work, we study theoretically the emergence of optical forces and torques on both electric or magnetic dipolar (ED or MD) particles induced by planar and cylindrical guided modes. First, we reveal analytically the peculiar (flipping) optical torque due to transverse spin carried by all guided modes, and the impact of their corresponding extraordinary longitudinal momentum on the enhanced/inhibited radiation pressure; in addition, the intrinsic helicity-induced torque resulting from hybrid modes is also shown. Finally, the effect of transverse spin on chiral particles is explored, inducing lateral forces with implications in chiral sorting. This rich phenomenology is evidenced in planar and cylindrical water-filled waveguides to illustrate optical forces and torques in the mid-IR-to-microwave regime.
2. Optical forces and torques
Let us start with simple planar and cylindrical waveguides (see Fig. 1) consisting of a dielectric layer (core, medium $1$) of thickness $2d$ and a cylinder of radius $R$, with relative dielectric permittivity and magnetic permeability ($\epsilon , \mu$) , surrounded by perfectly conducting walls, so that propagation takes place along the $z$ direction with propagation constant $k_{z}$ and transversal wavevector $k_{t}$. The wavevector $\mathbf {k}$ components in cartesian/cylindrical coordinates of the EM fields inside the waveguide are then given by:
where $\omega$ is the angular frequency.
Fig. 1. (a,b) Dispersion relations for the lowest-order guided modes inside water-filled waveguides with refractive index $n=\sqrt {80}$ and perfectly conducting walls: (a) Transverse electric and magnetic (TE$_i$ and TM$_i$) guided modes (planar); (b) Transverse (TE$_{0l}$, TM$_{0l}$), and hybrid (TE$_{jl}$, TM$_{jl}$) guided modes (cylinder). Circles indicate the specific modes and corresponding wavevectors considered below. Insets: Schematics of the dielectric waveguides considered hereafter: (a) planar, width $2d$, and (b) cylindrical, radius $R$.
Upon imposing boundary conditions, we can obtain the corresponding dispersion relation that determines the wavevectors. For planar waveguides, we find transverse electric (TE) and magnetic (TM) modes that stand for modes with only electric/magnetic field component along the $y$ axis [see Fig. 1(a)]. The dispersion relation in waveguides with perfect conductor walls is given by
for both modes (solutions are degenerate). Therefore, the modes are labeled by the index $i$ (TE$_i$ and TM$_i$) that accounts for the number of nodes that the field shows along the $x$ axis. Bear in mind that, for $i=0$, there is only a TM mode, TM$_0$.In cylindrical waveguides with perfect conductor walls [see Fig. 1(b)], the guided mode dispersion relation reduces to the values of $k_tR$ at which the corresponding Bessel functions vanish [42]; such modes can also be classified as TE or TM, with only magnetic/electric field along the waveguide axis. The modes are labeled by a pair of indices $jl$ where $j = 0, \pm 1, \pm 2, \ldots$ is the azimuthal index and $l = 1, 2, 3, \ldots$ the radial index. Since for $|j|>0$ the modes present intrinsic helicity, we also refer to these modes as hybrid modes, in concordance with the phenomenology showed in waveguides with penetrable walls.
The time-averaged optical force $\mathbf {F}$ and torque $\mathbf {T}$ on a dipole Rayleigh particle with either electric or magnetic polarizability in a non magnetic medium are obtained from the energy density $W$, the linear momentum $\mathbf {P}$ and spin linear momentum $\mathbf {P}^S$ densities, and the spin density $\mathbf {S}$ , as follows [4,43]:
3. Planar waveguide: transverse modes
It has been shown in Ref. [35] that the canonical momentum of transverse modes inside planar waveguides possesses a single component along the propagation direction $\mathbf {P^O}=P^O\mathbf {z}$ that can be simply written as:
where the function $f \equiv f(k_{t}x)=A\cos k_tx$ or $A\sin k_tx$ depending in turn on the electromagnetic mode, TM or TE, $A$ being its amplitude. Nonetheless, we have to consider separately the electric/magnetic contributions to the canonical momentum, which is in turn obtained by subtracting the spin linear momentum, proportional to the curl of the spin angular momentum density on the light field, to the total linear momentum density, proportional to the Poynting vector, which are the ones responsible for inducing radiation pressure to, respectively, electric/magnetic dipolar particles. Since the Poynting vector momentum density $\mathbf {P}$ differs from the canonical momentum only by the spin momentum density, $\mathbf {P}=\mathbf {P^O}+\mathbf {P^S}$, let us first analyze $\mathbf {P^S}$.Actually, the spin densities $\mathbf {S_{e,m}}$ are purely transverse for planar waveguides, and the only nonzero components of the spin densities are $\mathbf {S_{m}} = \mathbf {\hat {y}}S_{m}$ for TE modes and $\mathbf {S_{e}} = \mathbf {\hat {y}}S_{e}$ for TM modes, given by [35]:
where $^{'}$ denotes the derivative with respect to the argument ($\left [|f|^{2}\right ]' \equiv \dfrac {d}{d(k_{t}x)}|f(k_{t}x)|^2$). Therefore, the spin linear momentum density has only magnetic (respectively, electric) nonzero components, $\mathbf {P_e^S}\equiv 0$ (respectively, $\mathbf {P_m^S}\equiv 0$ ), for TE (respectively, TM) modes, indeed oriented along the propagation direction $z$, parallel to $\mathbf {P}$. We can thus write the electric/magnetic contributions to the canonical momentum (removing the vectorial dependence on $\mathbf {\hat {z}}$) that give rise to the radiation pressure to electric/magnetic ($e$/$m$) dipolar particles in Eq. (6), as [35]:Then, in TE/TM modes, the canonical magnetic/electric linear momentum is finally given by
The latter equations for the electric/magnetic canonical momentum densities, Eqs. (10) and (11), and the former spin densities, Eq. (9), along with their impact on scattering forces and torques on achiral dipolar particles, Eqs. (6) and (7), constitute one of the main results of this work. Lateral forces on chiral particles will be addressed later on.
By way of example, we consider a planar metallic waveguide filled with water ($n=\sqrt {80}$ at 1 GHz): particularly, we focus on the TE$_{2}$ and TM$_{2}$ guided modes in planar waveguides with $2d=2$ cm in Fig. 2. We would like to stress the fact that, as pointed out above, qualitatively (and nearly quantitatively) similar results would be obtained for a metallic, water-filled waveguide with widths of the order of a micron operating in the IR ($\lambda \sim 1.55\mu$m). Along with energy and spin densities, radiation pressure forces and torques felt by electric ($p$) and magnetic ($m$) dipole particles (polarizibilities $\alpha _{e,m}$) stemming from, respectively, longitudinal momenta and transverse spin densities, are revealed through arrows and loops.
Fig. 2. (a,d) Energy densities and (b,e) spin densities for TE$_{2}$ (top) and TM$_{2}$ (bottom) modes in a planar waveguide of width $2d$ and $n=\sqrt {80}$ at $\omega d/c=0.42$ (which corresponds to a water-filled layer with thickness $2d=2$ cm at $\nu =\omega /(2\pi )=1$ GHz). Arrows in (a,d) indicate the radiation pressure force felt by dipole particles due to longitudinal momenta, with contributions from the total (Poynting vector) momentum (red) and spin linear momentum (blue). Loops in (b,e) reveal the corresponding torque induced inside the waveguides by transverse spin densities. (c) Normalized radiation force variation, and (f) normalized torque across the waveguide.
First, the rich phenomenology consisting on alternating layers with opposite spins is evident in Figs. 2(b), 2(e), promoting the existence of a modulated torque that changes sign (flips) across the waveguide. Moreover, recall that the spin density may stem from either the electric [$S_e$, Fig. 2(e) or magnetic ($S_m$, Fig. 2(b)] contribution depending on the nature of the transverse mode (TM and TE, respectively). This implies that $p$ and $m$ dipoles will respond differently, exhibiting non-negligible torques [see Fig. 2(f)] only if matching character [5,44,45]. Thus TE (respectively, TM) guided modes exert torque only to magnetic (respectively, electric) dipolar particles, see Fig. 2(b) [respectively, Fig. 2(e)]. On the other hand, both ED/MD particles would experience a transversal conservative force proportional to the gradient of the electromagnetic energy density and a radiation force $F_{scat}$ according to Eq. (6) along the guided mode propagation direction. Nonetheless, this will depart from the expected radiation pressure, proportional to the Poynting vector, wherever the curl of the spin momentum density is non-negligible [cf. Eq. (10)]. In fact, such radiation force $F_{scat}\sim \Im (\alpha _{e,m})P_{e,m}^O$ can be anomalously large (superluminal) or small (subluminal), as observed in Fig. 2(c), where we plot $(P_{e,m}^O-P)/< P > =2k_t^{2}[f^{'2}-f^{2}]/k^{2}$, where $< P >= \int _{-d}^{d}P\textrm {d}x /(2dA^2)$ is the normalized average value of the Poynting vector along the cross section of the waveguide. The maximum value of this magnitude depends on the degree of confinement of the mode, being equal to 2 in the limit of $k_z \rightarrow 0$ and going to zero for $k_z \gg k_t$. Our results are an example of the existence of superluminal forces on dipoles in optical systems different from evanescent waves. Superluminal scattering forces appear when the longitudinal canonical momentum exceeds the momentum of a plane wave with the same local intensity. In evanescent waves this is possible because the local wave vector is larger than $n\omega /c$. However, this is not the case inside waveguides, and the existence of superluminal forces is due to the contribution from the square modulus of the amplitude components of the inhomogeneous electromagnetic field. This extra contribution induces, for some values of $x$, superluminal forces on the dipole.
Dual particles with $\alpha _e=\alpha _m\neq 0$ will experience transverse torques and anomalous radiation pressure for both TM and TE guided modes. Note that, due to the dependence of spin density $S\propto f^{2'}$ and spin momentum density $P_S\propto f^{2''}$, recalling that $W\propto f^{2}+ cte$, the maximum torque across the waveguide is exerted at points of maximum energy density variation [see Fig. 2(f)], whereas the anomalous radiation pressure variation is enforced at local energy density maxima/minima [see Fig. 2(c)]. For the electric/magnetic dipole in a TE/TM mode, the force in the $z$-direction cancels out for the $x$-values where the electric/magnetic energy density is null. However, for TM/TE modes, the scattering force in the $z$-direction never cancels out, because it is impossible to find a value of $x$ at which $f$ and $f^{'}$ are both zero at the same time. It is also interesting to note that, for modes with $k_{z}=k_{t}$, since $f^{2}+f^{'2}=1$, the scattering force is the same all across the waveguide, i.e. it does not depend on $x$.
From the point of view of the gradient forces, in the lateral direction, electric dipoles evolve in a potential well given by the electric energy density, while magnetic dipoles are influenced by the potential well coming from the magnetic energy density. Both potentials have their stable equilibrium points at different locations, and, for that reason, this system is especially suitable for lateral sorting of electric and magnetic dipoles. In the waveguide, for pure electric or magnetic dipolar particles, the gradient forces are related to the scattering forces by the following equation:
4. Cylindrical waveguide: hybrid modes
Now we analyze the role of the optical forces on hybrid modes that carry intrinsic helicity. We focus on the TE$_{12}$ hybrid mode of a cylindrical metallic waveguide, again, filled by water, which exhibits a varied phenomenology. First, radiation forces are shown for both probe particles in Fig. 3(a), with corresponding energy density in Fig. 3(d); as above, recall that radiation forces might be anomalously larger/shorter (super/subluminal) than those expected from the Poynting vector density for longitudinal spin momentum opposite/along the canonical momentum. The transverse spin density is plotted in Fig. 3(e), revealing three rings with flipping torque inside [see Fig. 3(b)] and negligible spin density at the center. Note that the angular torque is proportional to the unit vector in the azimuthal direction, which changes sign on going from $x=r$ to $x=-r$. Nonetheless, despite being hybrid, the transverse spin density contribution of this mode is purely magnetic ($S^{(e)}_{\theta }\equiv 0$), leading to radiation torque acting only on the MD particle $m$. (In this regard, bear in mind that this only occurs for perfectly conducting waveguide walls; otherwise, this mode would carry both electric and magnetic spin densities. Yet its hybrid character is preserved through its intrinsic helicity).
Fig. 3. (a) Normalized radiation force variation, and (b) normalized torque, for a hybrid TE$_{12}$ guided mode across a cylindrical waveguide of radius $R$ and $n=\sqrt {80}$ at $\omega R/c=0.63$ (which corresponds to a water-filled cylinder with thickness $R=3$ cm at $\nu =1$ GHz). (d) Energy, (c) helicity densities, and (e,f) spin densities. Arrows in (d) indicate the radiation pressure force felt by dipole particles due to longitudinal momenta, with contributions from the total momentum (red) and spin linear momentum (blue) ; loops in (e,f) reveal the corresponding torque induced by spin densities.
Next, the helicity density and related longitudinal spin density are shown in Figs. 3(c), 3(f), exhibiting three rings alternating sign. The helicity is defined as $h=\sqrt {\epsilon _0 \mu _0}Im(\mathbf {E}\mathbf {H}^*)/(2\omega )$. It should be mentioned that such helicity-dependent spin density has both electric and magnetic contributions; however, the electric contribution is much larger $S_z^{(e)}\gg S_z^{(m)}$ in this particular case with weak confinement, due to the fact that $S_z^{(e)}/S_z^{(m)} = k^2/k_z^2$. Interestingly, this implies that an electric (respectively, magnetic) dipole particle would undergo in such a waveguide a longitudinal (respectively, transverse) torque. By contrast, if the guided mode were a weakly confined hybrid TM mode, the same would hold upon exchanging electric and magnetic dipole particles. In general, both longitudinal and transverse torques are exerted on either dipolar particle. Indeed, it can be shown to scale as $S_\theta ^{(m)}/S_z^{(m)} = k_t^2 r/(jk_z)$ ($j$ being the mode azimuthal index). Therefore, near the waveguide axis ($r \approx 0$) the longitudinal torque is the dominant contribution; in all other cases, it depends on confinement strength and mode order. Incidentally, we have omitted in Fig. 3 the transverse force induced by the longitudinal spin momentum, which actually exerts no radiation force in the dipole approximation, but does produce a helicity-dependent transverse force in multipolar interactions with larger particles [24].
The actual values of the forces on dipolar particles depend not only on the intensity of the electromagnetic field, but also on the polarizability of the probe. This polarizability depends on the particle’s volume, the dielectric constant of the particle and the dielectric constant of the medium. For a dispersion-less particle in the Rayleigh regime ($ka \rightarrow 0$, $a$ being the particle radius), the change of the force per unit volume with frequency depends on the change of the dielectric medium constant which, for the case of water, can be substantial. As an example, in the case of silica, the scattering force per volume is roughly four times larger in the microwave than in the optical region of the spectrum. The gradient force doubles its value in the microwave region and changes sign due to the inversion of the refractive index contrast. As a final remark on the optical forces, in our work we are not considering forces induced by the scattered field coming from the waveguide surfaces. This contribution is important for dipoles located near the waveguide surface. In Ref. [46] the component of the force on a gold nanoparticle coming from the surface reflections is computed and shown to be negligible for distances from the surface larger than $\lambda /10$. Also, we are not considering interactions between dipoles, which may induce non-trivial dynamics in the system. Our analysis is valid for diluted systems, where interparticle interactions may be neglected [47]. Moreover, the optical forces generated by the guided modes must overcome the Brownian motion generated in the liquid. The Brownian diffusion increases for smaller particles, so these wave guided scattering forces should be more difficult to detect for nanosized probes. We have used Langevin molecular dynamics simulations to analyze the behavior of metal particles with radii about tens of nanometers in the infrared region (far from resonance), and we have confirmed that the reported scattering forces are noticeable for standard laser intensities in optical tweezers.
It should be emphasized that sub-wavelength particles with high-refractive index have been shown to yield strong MD resonances [48], as experimentally demonstrated in the optical and lower-frequency domains [49,50]. These resonances lead to a wealth of phenomenology associated to the high-refractive-index dielectric resonant nanostructures [51,52] and sub-wavelength structures at lower frequency [50], where very large refractive indices are indeed ubiquitous. Recall also that, apart from its impact on induced torques, it has been recently demonstrated that such electric/magnetic spin contributions can be experimentally discerned through high-dielectric-index nanoparticles exhibiting both ED/MD resonances [53]. Therefore, multi-resonant sub-wavelength ED/MD particles would feel electric or magnetic torque depending on the resonant wavelength and guided mode involved (the latter influencing also the spatial dependence of such torque), which overall allows for a rich phenomenology.
5. Optical forces on chiral particles
Finally, we investigate the impact that the rich patterned distribution of transverse SAM discussed above may have on chiral dipolar particles. It is well known that SAM gives rise to strong lateral forces on chiral particles [27–29]. In its simplest form, the response of a dipolar chiral particle can be expressed as [29]:
with $Z_0=(\mu _0/\epsilon _0)^{(1/2)}$, where apart from the electric/magnetic polarizibilities $\alpha _{e,m}$, an off-diagonal term $\chi$ is included connected to the chirality parameter $\kappa$. Chirality introduces new terms depending on the force [29]. The ones depending only on the chirality induced dipoles are zero in this configuration, however the ones resulting from the interference between dipoles are different from zero. In particular, apart from a pure chiral longitudinal force proportional to the Poynting vector, there is a force proportional to the spin angular momentum density given by: for a chiral magnetic dipole in a TE mode, and for a chiral electric dipole in a TM mode. The latter equation reveals that the spin density induces on chiral particles, apart from torque, linear momentum transfer directly proportional to the chiral parameter and pointing to opposite directions for opposite $\chi$ signs. For planar waveguides with transverse spin densities given by Eq. (9): whereas for purely transverse modes in cylindrical waveguides with transverse spin densities given in Ref. [35], the transversal components of the force are given by:Fig. 4. (a) Contour map (in a plane perpendicular to propagation) of the transverse electric spin density and (b) normalized lateral force $F_{int}$ thereby induced on a chiral particle, for an anti-symmetric TM$_{1}$ mode in a planar waveguide of width $2d$ and $\epsilon =80$ for normalized half-width $\omega d/c=0.31$ (which corresponds to a water-filled layer with thickness $2d=1.5$ cm at $\nu =\omega /(2\pi )=1$ GHz).
As an example, we show in Fig. 4 the expected lateral force on a chiral ($\chi \not =0$), electric ($\alpha _m=0$) dipolar particle [Eq. (16b)] in a water-filled planar waveguide supporting the TM$_1$ mode. This mode exhibits a simple antisymmetric spin density profile [see Fig. 4(a)] with only two layers to simplify chiral sorting. Depending on chirality sign ($\pm \chi$), particles will be pushed to opposite directions along the axis ($y$) perpendicular to the $xz$ plane defined by the guided mode propagation direction ($z$) and the confinement direction $x$, allowing for simple means of sorting chiral particles/molecules.
6. Conclusion
To summarize, we have analytically investigated the optical forces and torques induced by guided light in planar and cylindrical waveguides supporting transverse electric/magnetic and hybrid (only in cylinders) modes. We have shown that all (transverse and hybrid) modes exhibit a structured transverse spin density depending basically on guided-mode spatial and polarization symmetry, which gives rise to optical torques for either electric or magnetic dipole rotating in different directions. The direction of rotation flips, alternating in different layers or rings inside the waveguide. Such transverse spin density is shown to carry extraordinary longitudinal spin linear momentum, pointing along or opposite to the Poynting vector inside the waveguide, thus enhancing/inhibiting (super/subluminal) radiation pressure, much in the way of a structured field where the canonical momentum can be larger/shorter than the effective (energy transport related) "luminal" limit. Hybrid modes in cylindrical waveguides exhibit in turn intrinsic helicity which leads to longitudinal optical torque that strongly varies inside the waveguide depending on position and polarization. Finally, the impact of lateral forces acting on chiral particles due to such transverse spin density is explored too, allowing for chirality sorting. This phenomenology could be exploited for optical manipulation acting selectively on electric/magnetic dipolar particles inside e.g. liquid-filled micro- or macro-fluidic cavities [12], bearing in mind that gradient forces will push particles towards higher (electric or magnetic) energy density regions and that other forces (like Casimir forces) may also play a role. Recall also that, apart from its impact on induced torques, it has been recently demonstrated that such electric/magnetic spin contributions can be experimentally discerned through high-dielectric-index nanoparticles exhibiting both ED/MD resonances [53]. Therefore, multi-resonant sub-wavelength ED/MD particles would feel electric or magnetic torque depending on the resonant wavelength and guided mode involved (the latter influencing also the spatial dependence of such torque), which overall allows for an even richer phenomenology.
Funding
Ministerio de Ciencia, Innovación y Universidades (CEX2018-000805-M, FPU15/03566, PGC2018-095777-B-C21, PGC2018-095777-B-C22, UAM-CAM project SI1/PJI/2019-00052).
Disclosures
The authors declare no conflicts of interest.
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