1. INTRODUCTION

Nonconservative fields of optical force on optically trapped particles have long been predicted to occur [1]. These force fields are locally nonpotential in character, so that network is done on a particle even in microscopic closed paths. This differs from, say, electromotive force around a circuit where only the global potential is not definable [2]. In a locally nonconservative force field, external power is continuously coupled to particle motion, leading to dissipation and heating even when the particle is localized about a stable point of the force field.

Radiation pressure on trapped particles can be one source of nonconservative forcing. In recent experiments [3, 4], nonconservative toroidal circulation of optically trapped particles caused by axial radiation pressure was observed and theoretically analyzed. Recent theoretical work [5] has shown how, for nonspherical objects, nonconservative motion arises in coordinates of angle and translation.

Here we show that a simple spatial 3D nonconservative force field, circular rather than toroidal, occurs even in the absence of radiation pressure. As a physical model leading to this force, we describe a sphere trapped by refracting rays. Within any such circular-forcing model, and generalizing to an anisotropic trapping force, we derive the spectrum of motion and the thermal signatures of nonconservative circular forcing.

2. GEOMETRY OF NONCONSERVATIVE FORCES

Before narrowing the discussion to optical trapping, we ask what simple generic statements can be made about nonconservative forces. Powerful classification schemes are available from differential geometry [6, 7, 8] but to use these one must correctly identify forces as fields, not of vectors, but of differential 1-forms. This means that force components such as $f_x$, $f_y$, and $f_z$ properly take their meaning from the work differential, or work 1-form:

In Euclidean space one can reinterpret $f_x$, $f_y$, and $f_z$ as components of a vector. But as a 1-form field, Eq. (1) is constrained by Darboux’s theorem [6, 7], which states that every 1-form field ω can be reduced by some choice of general coordinates ($q_1,q_2,q_3,\dots$) to a shortest canonical form $\omega ={\omega }^{(k)}$ belonging to the sequence

The case $\omega ={\omega }^{(1)}$, i.e., $\omega =d{q}_1$, is exact, i.e., conservative. Not only is $q_1$ a coordinate, but $-q_1$ is the potential function for ω. The case $\omega ={\omega }^{(2)}$, or $\omega =q_{1}d{q}_2$, is reducible to the exact case by an integrating factor $(1/q_1)$, because $(1/q_1)\omega =d{q}_2$. In two dimensions, this exhausts our list, showing that all 1-forms (i.e., differentials) in 2D are either exact or integrable, a well-known and useful fact in thermodynamics.

In 3D, exact (conservative) and integrable force fields of types $\omega ={\omega }^{(1)}$, and $\omega ={\omega }^{(2)}$ can still occur, but the most general possibility is the nonintegrable 1-form ${\omega }^{(3)}$, i.e., $\omega =d{q}_1+q_{2}d{q}_3$. The form of a generic force field is thus more restricted than Eq. (1) would suggest. In the case of optical trapping, the optical force field includes a part derivable from a potential, which we use for our first coordinate: $q_1=-\mathrm{\Phi }$. Following Eq. (4), we write

In the neighborhood of an equilibrium point, where all components of a force field vanish, it is natural to interpret θ in Eq. (5) as an angular variable, motivated by the reduction $-ydx+xdy={\rho }^{2}d\theta$. Nonconservative forces near equilibrium are circulations aligned with some plane that contains the equilibrium point. The nonconservative portion of the force can be taken as strictly circular near the equilibrium point in any coordinates, because a noncircular 2D force pattern in x and y can be written as

Thus, to study a nonconservative 3D force about an equilibrium point, it is sufficient to consider a circular pattern of force in some 2D plane containing the equilibrium point.

From another point of view, this picture is consistent with equilibrium-point analysis of a vector field f, viewed as flow toward a fixed point [9] (Fig. 1). Briefly, with coordinates $q_i$ we have $f_i\approx M_{ij}\delta q_j$ near the fixed point, defining some real matrix M. The eigenvalues of M are roots of a cubic equation with real coefficients, and they generically yield one real and two complex conjugate roots whose real parts, together with their eigendirections, correspond here to a conservative force field. The imaginary part of the complex pair of eigenvalues defines circulation in some plane. The eigendirections of M may be stretched and skewed, corresponding to the nonorthogonal coordinates in Eq. (4).

From yet another point of view, any vector field restricted to the surface of a sphere (about the equilibrium point) will circulate about the sphere in a 2D fashion, according to the familiar “combed hair on a sphere” theorem of mathematics [8] (Fig. 1).

3. NONCONSERVATIVE TRAPPED-SPHERE MODEL

For a “toy model” realization of a circular nonconservative force, we consider an optically trapped sphere much larger than the wavelength of light, so that geometrical optics applies. In our simplified model, we assume no reflections (perhaps due to a graduated-index boundary) and we trap the sphere with only a few rays passing almost centrally through the sphere, so that the refraction and trapping force can be calculated in a paraxial-ray approximation. Our model is certainly contrived, but it is very simple to analyze. In more realistic situations, whenever there is chirality (handedness) of a system of light rays near an equilibrium point, we still expect some degree of circular nonconservative forcing to occur.

We need only consider rays within the plane of Fig. 2a, containing the sphere center. In Fig. 2a, the displacement of the sphere center from the ray is exaggerated. The force of the refracted ray on the sphere due to momentum trans fer is

The system of four rays shown in Fig. 2b results in a purely circular nonconservative force on the sphere. Despite the clockwise circulation of rays, there is no torque on the sphere within this ray-optical model: it is the nonconservative net force on the sphere that we study here. We separate oppositely directed ray pairs by a distance $2b$, exaggerated in the figure, with $b\ll a$ so that our paraxial analysis remains valid. The ray in the $+x$ direction is described by

4. NONCONSERVATIVE MOTION OF A DIFFUSING PARTICLE

To look for spectral features of trapping in a nonconservative field, we consider diffusion through a fluid of a particle with drag coefficient γ. At temperature T, the diffusion coefficient is $\gamma /k_{B}T$, an Einstein relation [11]. Near an equilibrium point we consider 2D motion $\mathbf{r}(t)$ in a plane of circulation where the nonconservative component of force is

We turn to the spectral properties of the diffusing particle, and we first establish formalism [12] with force and motion only in the x direction. For a force $f(t)$ and motion $x(t)$ that extend over all time, the spectral density $(fx{)}_{\omega }$ is defined by [13]

Now include a circular external force. With an isotropic conservative force $-\kappa \mathbf{r}$, the vector equation of motion $\mathbf{N}=\kappa \mathbf{r}-\xi \widehat{\mathbf{z}}\times \mathbf{r}+\gamma \dot{\mathbf{r}}$ gives us

Equation (31) represents the power spectrum of the x motion of a trapped particle with a nonconservative force ($\eta \ne 0$) and 2D anisotropy (${\alpha }_x\ne {\alpha }_y$) both in the $x–y$ plane. Nonconservative forces make this power spectrum non-Lorentzian, even in an isotropic trap. Other interesting effects, such as inertial hydrodynamics and material properties [14, 15], will of course also invalidate a simple Lorentzian spectrum. Carrying out the ω integral in Eq. (24), we obtain

Details of the flow pattern follow from the off-diagonal element of $(\mathsf{r}{\mathsf{r}}^{†}{)}_{\omega }$,

The rate at which work is done on the system is

5. SUMMARY

We have characterized the simplest 3D nonconservative force field, near a stable fixed point, as an anisotropic conservative force plus a nonconservative circular force. We constructed a simple trapping model exhibiting such forcing, and we presented signatures of particle displacement and nonconservative dissipation in the presence of nonconservative forcing. In most optical trapping experiments, simple circular forcing about a stationary point may not be easily observable given the more dominant radiation pressure effects that have been observed by others [3, 4]. Specifically designing chiral trapping geometries for particles and minimizing reflections may allow observation of circular forcing. More generally, however, the geometrical picture we have constructed and the thermal and spectral results we have obtained may be useful in other situations, even outside of optics, where locally nonconservative microscopic forces act.

ACKNOWLEDGMENTS

The authors would like to thank their colleagues in the Department of Physics and Astronomy at Washington State University for support and helpful conversations.

 

Fig. 1 Nonconservative 3D force field near a stable equilibrium point. The circulation of force may be viewed as purely circular in some plane. (a) Vector flow-field point of view. (b) Combed-hair-on-a-sphere point of view.

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Fig. 2 (a) Refraction of a single ray by a sphere. In the paraxial limit, ${\mathbf{r}}_{\perp }$ is much smaller than the radius of the sphere. The force f acts to return the sphere center to the line of $\widehat{\mathbf{k}}$, but it also has a component along $\widehat{\mathbf{k}}$. (b) System of four rays for nonconservative trapping. The limit $b\ll a$ allows a paraxial approximation for each ray.

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Fig. 3 Rms radius $〈r^2{〉}^{1/2}$ (solid curves) of the position distribution $P(x,y)$ of a diffusing particle in a trap with stiffness anisotropy ${\kappa }_y/{\kappa }_x=5$, for various values ξ of counterclockwise circular forcing in the $x–y$ plane. The solid curves are lines of flow, which, for $\xi \ne 0$, do not follow the $\xi =0$ equipotential contour (dotted curves). Circular flow at $\xi =\infty$ (final panel, solid curve) has the rms radius $[2k_{B}T/({\kappa }_x+{\kappa }_y){]}^{1/2}$.

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