We extend the response theory of optical forces to general electromagnetic systems which can be treated as multi-port systems with multiple mechanical degrees of freedom. We demonstrate a fundamental link between the scattering properties of an optical system to its ability to produce conservative or non-conservative optical forces. Through the exploration of two nontrivial two-port systems, including an analytical Fabry-Perot interferometer and a more complex particle-in-a-waveguide structure, we show perfect agreement between the response theory and numerical first-principle calculations. We show that new insights into the origins of optical forces from the response theory provide clear means of understanding conservative and non-conservative forces in a regime where traditional gradient force picture fails.
© 2011 OSA
Understanding the behavior of optical forces at sub-wavelength scales is crucial in designing optomechanical devices for a broad range of classical and quantum applications [1–5]. Many new applications from sub-nanometer manipulation to ground-state cooling have been recently proposed and experimentally demonstrated [6–20]. However, the understanding of optical forces in externally driven systems has largely been limited to first-principle numerical methods, such as the Maxwell stress tensors [21–23]. Recently a response theory of optical forces (RTOF) was proposed to make rigorous analytical connection between the optical forces and the change in the system response caused by motion of an object . This theory provides an elegant means of examining forces within integrated photonic systems having a handful of input and output ports. Many of these systems can be described by either an analytical response function or a numerically-solved scattering matrix. Simple systems with a single input/output port and a single-degree of freedom have been carefully studied both theoretically and experimentally [8,12,13,24]. However, practical optical systems usually involve more than one output port, due to unavoidable scattering processes. Furthermore, optical actuation is also usually performed over more than one degree of freedom. For example, 3D positioning and rotational control fall within this realm. For these more general systems, there is an urgent need to better understand the relation of the optical forces to the system responses, and to explore new device physics beyond the brute-force calculations using first-principle modeling. One significant challenge in integrated photonic systems is the large index-contrast that renders the conventional gradient force picture inapplicable: the local fields are usually significantly altered by the high-index moving object, such that the incident fields and its gradient becomes a poor approximation of the local fields experienced by the object. Another common structure, involving high-Q resonators, creates even more challenges, because the local fields are significantly enhanced in amplitude when compared to the incident fields. In this paper, we further develop the RTOF theory to understand the behavior of optical forces for multi-port integrated photonic systems, focusing on the design insights as well as efficient modeling and measurement techniques. In addition, we explore the non-conservative nature of the optical force and its origin in these general systems, which have recently been suggested to have significant consequences for many proposed optical trapping schemes [25–27].
This paper is organized as follows: We first review the response theory of optical forces (RTOF) and discuss the distinctly non-conservative nature of optical forces in systems with multiple coherent outputs and multiple degrees of freedom. A Fabry-Perot interferometer, as an analytical example of two-port systems, is studied to illustrate the critical conditions that dictate the conservativeness or non-conservativeness of optical forces. We then focus on a two-dimensional two-port system with two degrees of freedom consisting of a scattering particle moving within a single-mode waveguide. In addition to demonstrating excellent agreement with numerical first-principle calculations, we show that RTOF method provides new understanding of the nonconservative optical forces in a high-index-contrast regime, in which the conventional picture of gradient/scattering forces no longer applies.
2. Response theory of optical forces in single-port and multi-port optical systems
Response Theory of Optical Forces (RTOF), originally proposed in Ref , establishes a new approach of deriving optical forces from the external amplitude and phase responses of an optical system. Specifically, to calculate optical force within any lossless system which satisfies energy and particle conservation, the knowledge of optical responses as a function of mechanical degrees of freedom is sufficient. The mechanical degrees of freedom can include displacement, angle of rotation and other forms. Unlike conventional methods, such as the Maxwell stress tensor [21,22,28] or the Lorentz force density [23,29–31], RTOF method does not rely on the knowledge of the complex electromagnetic fields that are often unique to individual systems. An important implication of the decoupling of the optical forces from the optical field profiles is that it provides a unified understanding of a great many systems. Systems with identical mechanically-variable optical responses produce exactly the same forces, even when they consist of disparate structures and support drastically different internal fields.
To formally describe the RTOF treatment, we use scattering matrix which describes the optical response of a mechanically-variable system as a function of an n-dimensional generalized coordinate q that spans the space of the mechanical degrees of freedom [32,33]. The incident waves are of the same frequency, .The complex amplitude of the output waves at the i-th output port, , is related to that of the incident waves at the j-th input port,, as . The normalized wave amplitudes are defined as and , where and are the input and output power respectively at the j-th port .For single-port systems illustrated in Fig. 1a and 1b, the optical response simplifies to a scalar form, where only the phase response is a function of the mechanical degrees of freedom. For the special case of a single mechanical degree of freedom (Fig. 1a), the generalized coordinate is a scalar q and the resulting optical force is 
For more general systems with n independent degrees of freedom (Fig. 1b), the optical response becomes, where is the k-th component of the generalized coordinate q. The corresponding k-th component of the optical force is
Since F is the gradient of a scalar field, any lossless single-port system produces a conservative force field. This observation has been analytically and numerically verified in Ref . Alternatively, the force field can be summarized by an equivalent optomechanical potential, .
Extending our discussion to systems with multiple input/output ports, the optical force is simply determined by summing over all the output ports. In practice, multi-port systems are far more common than single-port systems, since each scattering pathway essentially constitutes an output port. In an M-port system with a single mechanical degree of freedom (Fig. 1c), the optical force takes the form :
Here, is the phase angle of the wave exiting the j-th output port . Limiting our discussion to systems where q is defined on an open path (or in more precise mathematical terms, q is bounded in a finite range), the force field can again be described by a scalar optical potential .
In a more general case shown in Fig. 1d, systems with multiple degrees of freedom produce a force component of
The existence of both multiple ports and multiple degrees of freedom in the same system has a qualitative impact on the nature of optical forces: they are often non-conservative. In other words, the work done by the optical forces through the motion of the system not only depends on its initial and the final states, but also depends on its path in the coordinate space. Optical forces can do work and transfer energy from the electromagnetic domain to the mechanical domain, even if the trajectory in coordinate-space consists of a closed path beginning and ending at the same point in coordinate-space. A quantitative understanding of such non-conservative force field is therefore important for understanding the stability limit of optical particle trap [2,35,36], or the dynamics of optomechanical systems [8,16], but also for the efficient conversion of energy from light to mechanical motions via nanoscale optical motors [25,26]. Mathematically, a non-conservative force can be identified by evaluating the curl of the force field:
The general results of the analysis presented above are summarized in Table 1 , and apply to systems with any number of input and output ports. Throughout this paper, we will limit our consideration of non-conservative optical forces within more familiar and intuitive two-port systems. We focus on the two-port system since it fully captures the essential properties of the more general class of multi-port systems and provides the simplest starting point for application of multi-port RTOF analysis. The two ports can be broadly categorized as a reflection port (Port 1) and a transmission port (Port 2). To be consistent with many experimental set-ups with single laser source, we make an additional simplification that incident power enters the system only via Port 1: , and . Thus the output waves are determined only by two elements in the scattering matrix:
In what follows, we will examine the optical forces acting on a two-port Fabry-Perot interferometer and a scattering particle inside a single-mode waveguide, which possess one and two mechanical degrees of freedom respectively. Comparison between the RTOF method with first-principle analyses (the Maxwell stress tensor or momentum conservation method) will be made to highlight the additional physical insights and their experimental implication.
3. An analytical example: optical forces in a Fabry-Perot interferometer
In this section, we apply RTOF method to a Fabry-Perot interferometer, using its well-known analytical scattering matrix to calculate the optical forces. We demonstrate the exactness of the RTOF formalism by a comparison to the momentum conservation method. Fabry-Perot interferometers not only serve important roles in cavity optomechanics [3,37], but also prove to be an intuitive starting point to explore two-port systems. The basic structure (Fig. 2 ) of a Fabry-Perot interferometer consists of two parallel mirrors in vacuum separated by a distance l0, corresponding to a round-trip phase shift . For simplicity, the mirrors are taken to be identical and their partial reflectivity is described by scattering matrix
Here the transmissivity t and the reflectivity r are both real numbers . For the scope of this paper, we limit our attention to lossless and reciprocal systems (absent of magneto-optical materials), such that . Note however, the general results of this section are very broadly applicable to a variety of systems if r and t are taken to be frequency dependent, as the partial mirrors considered here can be easily replaced with more complex structures consisting of photonic crystal slabs . It is assumed that an incident wave, of frequency ω, impinges from the left (entering through port-1) setting up an internal optical fields that exert forces on both mirrors.
The internal and external field amplitudes of the Fabry-Perot can be solved after Ref , yielding
Here, a and b represent the amplitudes of the right- and left- going waves respectively between the two mirrors. The optical forces acting on Mirrors 1 and 2, can be found by computing the change in photon momentum induced by each mirror:
Substituting the field amplitudes into these two expressions, the optical forces can be written explicitly as,
Alternatively, one can evaluate the optical forces using RTOF theory. Here we fix the location of the reference planes (), so that the phases of the waves entering or exiting the two-port system are also taken at a fixed location. Solving the scattering matrix at the two reference planes, we have
To calculate the optical force acting on a specific mirror, the location of one mirror is varied while the other mirror is held fixed. For instance, in order to calculate force F1 on Mirror 1, we take l2 to be a constant while l0 is varied, yielding . Substituting this relation into the S-matrix and combining with Eq. (4), one arrives at
Similarly, one can calculate the force on Mirror 2 through RTOF, by keeping l1 as a constant:
Comparison of Eq. (8) with Eqs. (9) and (10) shows exact agreement between RTOF and momentum conservation analysis, demonstrating equivalence between these two treatments in this nontrivial Fabry-Perot system.
Note that the optical force is conservative in the case that only one mirror is allowed to move. This is because the only degree of freedom of the system, l0, is entirely determined by the position of the chosen mirror. The work done by the optical force is path independent, since only one path exists between any two states of the system. In contrast, if both mirrors are allowed to move, two degrees of freedom are present in the system. For simplicity, we take these to be the cavity length, l0, and the position of the aggregate Fabry-Perot system. Since this two-port system is taken to have two degrees of freedom, a seemingly identical Fabry-Perot system produces non-conservative forces.
The ability of optical forces to perform mechanical work on this system can be quite intuitively understood: consider transforming the Fabry-Perot system between highly transmissive and highly reflective states by changing the cavity length l0, while translating the entire system (as a unit) toward and away from the optical source. For example, while moving the cavity towards the source in its transmissive state, no work is done. Conversely, moving the cavity towards the source in its reflective state, work is performed by light. Hence, numerous state-diagrams can be constructed to yield nonzero mechanical work through a closed path in state-space. Hence, this simple system intuitively demonstrates the non-conservative nature of optical forces in optical systems that feature both multiple port and multiple degrees of freedom.
4. A numerical example: a moving cylinder in a single-mode waveguide
Next, we examine non-conservative forces in a more general two-dimensional system, where both the electromagnetic fields and the dielectric functions are non-uniform along the x-y plane. The structure itself is rather simple, consisting of a dielectric particle moving inside a metallic parallel-plate waveguide. However, the field distributions are not analytically solvable, necessitating the use of numerical methods to understand forces. Additionally, despite its resemblance to widely studied optical tweezers [2,39], this structure generates optical forces qualitatively different from the traditional gradient force picture. In this case, we see that the RTOF method is both practically advantageous and conceptually important. From a practical perspective, RTOF is highly efficient because the optical forces acting on the particle can be derived from the phase and amplitude responses, quantities that can be experimentally measured or calculated efficiently using time-domain solvers. Conceptually, RTOF method allows one to understand the stability conditions and predict non-conservative regimes of the forces directly from the external response of the system, which is much more conceptually palatable than field-based analyses using the Maxwell stress tensor.
As depicted in Fig. 3 , we consider a single mode waveguide consisting of an air region between two parallel perfect-electrical-conductor (PEC) plates placed at a distance a. The incident wave is TE-polarized (E-fields along z direction), and the incident frequency is below the cutoff of the second-order spatial mode. A free silicon cylinder () scatters the incident wave and experiences an associated optical force. The entire system is uniform along the z direction. Despite the inaccuracies of the gradient force model, we first examine this system using this model, as it is the paradigm through which a great many non-conservative systems have been examined . Through our examination of this system, we will see that, where the gradient force model fails to predict behaviors, RTOF provides a conceptually simple, exact, and rigorous description of the essential physics. Partitioning optical forces into a gradient force component and a scattering force component is generally a good approximation for low index-contrast systems  and has been considered for certain high index-contrast systems . In this framework, an object experience the gradient force along the lateral direction (y), with a force amplitude of . Here is the polarizability of the object, and I represents the local intensity of the incident beam. Since the fundamental TE mode possesses an intensity profile independent of frequency (Fig. 3e), the gradient force distribution takes on a sinusoidal profile independent of the frequency (Fig. 3d) . At any location, the gradient force points towards the center of the waveguide. On the other hand, the scattering force is generally proportional to the local intensity and takes on the profile shown in Fig. 3e.
This intuitive analysis, however, does not agree with the exact solution (e.g. the Maxwell stress tensor) for the high index-contrast system discussed here. Much of the inaccuracy arises from the dramatic modification of the local electromagnetic fields from the incident fields, even when the cylinder radius is smaller than . This change in the field distributions also varies as a function of the cylinder position. To evaluate the optical forces accurately, one must resort to exact methods such as RTOF or the Maxwell stress tensor. The Maxwell stress tensor is defined locally as .., and a surface integral over a closed surface surrounding the object yields the total optical force :
Unlike the Maxwell stress tensor method, where both local E and B field distributions need to be solved for each position of the moving object, RTOF only requires the knowledge of the power and the phase of the reflected and the transmitted wave. As a consequence of the translational symmetry of this waveguide system, the power and the phase of the transmitted wave, the power of the reflected wave, the axial force and the lateral forces are all invariant with axial (x-) translation of the particle. On the other hand, the phase of the reflected wave , depends explicitly on both x and y. The high degree of symmetry within this system allows us to reduce the total force given by Eq. (6) into its essential components as,Fig. 4a and 4b. It is important to emphasize that, with RTOF method, most time-domain solvers allow the efficient extraction of optical forces over a broad range of frequencies from power and phase response obtained in a single simulation with pulsed excitation . Here, we have used normalized units to allow application of these results to scaled versions of the same system over a wide range of frequencies.
In such waveguide systems, the lateral forces and the axial forces are driven by the optical response from a different set of ports. The axial force, as indicated by Eq. (11), is determined entirely by the reflected power as a function of the cylinder location. This result is in stark contrast with the gradient-force paradigm, which leads one to expect a force which is prescribed by the distribution of the incident power. The axial force is also dependent on the axial k vector of the guided modes, and is consistent with momentum conservation. In other words, a reflected photon acquires a momentum change during scattering, and completely transfers it to the scattering object, regardless of the complexity of the local fields. Consequently, the axial force is bounded, with a maximum value of for a given frequency.
In contrast, Eq. (12) reveals that the lateral forces have nontrivial dependence on phase and amplitude both from the reflected and transmitted waves, as seen from the red and blue curves of Figs. 4d and 4e. Interestingly, no apparent upper bound exists because the lateral phase gradient is not tied to the propagation constant and can be engineered to large values using high-Q resonances. Surprisingly, as seen by Fig. 4e, the lateral force is also seen to repel the cylinder from the center of the waveguide where the incident fields reach intensity maximum. This sign reversal from the gradient force prediction is entirely attributed to the phase gradient term within Eq. (12), since the transmitted power and reflected power are both positive. In generating such repulsive forces, the contribution from the reflected wave is comparable to that of the transmitted wave.
Since the local field modification by the particle depends strongly on the size of the cylinder and the optical frequency, we briefly explore the complex interplay between the contributions from the reflected and the transmitted waves as frequency and particle size vary. In addition to the r = 0.05a geometry explored in Fig. 4, two additional cases with larger cylinder radii: 0.075a and 0.125a are presented in Fig. 5 . The force components are plotted for these three particle radii for frequencies spanning the entire single mode range, from the cut-off frequencies of the 1st-order TE mode to the 2nd-order TE mode. The axial and lateral force components are plotted for positions spanning half of the waveguide, since Faxial is symmetric with respect to the center of the waveguide and Flateral is anti-symmetric. Note, the center of the cylinder cannot reach the edge of the waveguide due to its finite radius. Thus, the positions where the geometry of the system prevents motion are shown as gray shaded regions.
As the particle radius increases, the onset of repellant lateral forces and off-centered maxima of axial forces are seen at lower frequencies. The upper bound to the maximum axial forces, , increases with frequency, and peaks at 1.15 nN/a, independent of the particle size. With increasing frequency, the lateral force, shows a gradual transition from attraction to repulsion with respect to the waveguide center in the case of r = 0.075a. Multiple attraction and repulsion regions can be seen for a larger radius of r = 0.125a. The region where the high-index scatterer reaches stable lateral locations (experiencing zero force in the lateral direction) is shown by the solid white curve in the force plots Fig. 5b, 5f, and 5i. Note, these behaviors contrast starkly with the gradient force model, through which the equilibrium position is predicted to be at the center of the waveguide, as Figs. 5b, 5f, and 5i reveal vanishing lateral forces at locations which are far away from the waveguide center. Interestingly, since the stable particle locations strongly depends on the size and the dielectric constant of the moving object, this phenomenon can be applied to sorting high-index particles in microfluidic waveguides. Further discrepancy from the gradient force model is seen from the fact that, at frequencies where repellent forces exist (blue regions in Fig. 5f and 5i), the center of the waveguide is an unstable equilibrium location. Hence, in the presence of Brownian motion, the particle is unlikely to be found at this position.
As discussed in Section 2, one important property of systems having multiple ports and multiple degrees of freedom is that the optical forces produced by displacement in coordinate are nonconservative. For the waveguide system discussed here, we also consider that at the transmission port, and are parallel in the y direction, because both the power and the phase responses are independent of the axial position, x, of the cylinder, due to the translational symmetry. The curl of the force field is therefore simplified as:
Again we observe the nonconservative force components dictated entirely by the reflected waves. This result is markedly different from the case of low index-contrast trapping in free-space: , where I and are taken from the incident fields. Further simplification can be made by taking into account the relation . Here, the curl is found to be
Hence, the spatial distribution of the nonconservative component of the optical force is entirely determined by the reflected power. Equation (13) exemplifies the capability of RTOF method in rigorously identifying the origin of the non-conservative forces in this nontrivial system, something that cannot be done with the Maxwell stress tensor or the Lorentz force density.
To gain further insights of the nonconservative nature of this system, Fig. 6 shows the curl of the optical force fields numerically calculated for the two cases discussed earlier through examination of Fig. 4. The distributions agree well with Eq. (13). In both cases, zero curl condition is found in the middle of the waveguide. For higher frequencies, two additional zones with vanishing curl exist at the edge of the waveguide. Outside of these regions, a particle undergoing Brownian motion will experience optical force induced heating. In this case, mechanical work is generally performed on the cylinder, for example, along the closed white loop on the left of Fig. 6a. In general, when the lateral equilibrium position (where Flateral = 0) is not at the waveguide center, the curl of the optical forces is also nontrivial and optical heating occurs. In practice, if one seeks to minimize the positional uncertainty, lower frequency is therefore preferred.
The nonconservative nature of optical forces in guided traveling wave systems is more general than the case considered in this paper. As can be seen in Fig. 6c, one can examine the conservativeness of the optical force by considering a rectangular closed path, formed by two sections parallel to the optical axis and two sections normal to the optical axis. The total work done by the optical forces from the reflected photons is generally non-zero, due to the different optical power along the two sections parallel to the optical axis. The only exception is the case where the two sections are symmetrically places on each side of the optical axis. The path-dependent work done by the optical forces therefore indicates nonconservative optical forces for most reflective objects .
In addition to the predictive powers of RTOF, analysis of the experimental observables using RTOF can also provide important insights to experimental measurements of particle trapping systems. For example, Eq. (11) suggests that monitoring the reflection power is equivalent to measuring the axial forces, provided that the constant prefactor is calibrated. In addition, Eq. (13) allows one to identify curl-free regions by monitoring the transmitted power. For particles experiencing Brownian motion with fluctuating y positions, a stable transmitted power indicates a vanishing and therefore a vanishing curl of the force field. Moreover, although our discussion has so far been limited to a two-dimensional system, one can extend this analysis to three-dimensional systems by taking polarization and other spatial modes of the system into account as additional ports. Systems with more degrees of freedom, such as rotation for scattering objects of arbitrary shapes, can also be treated with RTOF analysis. Important application can also be found for structures with known analytical scattering matrices, such as resonant optical systems described by temporal coupled-mode theory. In these cases, it is possible to connect the optical forces directly to the change in the resonant frequency and the quality factor due to the mechanical movement. Our analysis of optical forces on a scattering particle can also be extended to treat multi-mode systems and open systems with radiation loss. In the case of a waveguide supporting N propagating modes, the total port count amounts to 2N, since transmission and reflection mediated through each eigenmode are two independent ports. Especially for those systems with analytical solutions, the M-port analysis presented in Section 2 could yield additional physical intuition. In contrast, for open systems, each radiation mode needs to be tracked as an independent port, for example, through a near-field to far-field transformation . RTOF theory may provide some simplification for special cases with known analytical solutions.
Through this deceptively simple two-dimensional example, we have demonstrated the exactness of RTOF theory, and we have established a fundamental connection between the nature of the optical forces generated within an optical system and its general structure. Applying RTOF theory to multi-port optical systems with multiple mechanical degrees of freedom, we have shown that the optical forces generated within are generally nonconservative. For this reason among others, we have shown that the gradient force picture, as well as the intuition derived from it, break down. Furthermore, in applying RTOF analysis, we have shown that a rigorous and simple understanding of the distinct components of optical force produced by a surprisingly nontrivial structure can be developed from only the phase and amplitudes of the outgoing (scattered) waves. Using this result, optical forces can be very efficiently calculated and we have identified an elegant new way in which optical forces could be directly mapped to experimental observables.
In this paper, we theoretically evaluate radiation pressure in two-port optical systems using the response theory of optical forces (RTOF), where the optical forces are linking to the power and the phase responses of the transmitted and reflected waves. We establish excellent agreement between first-principle calculations and RTOF in analytical modeling of a Fabry-Perot interferometer and numerical modeling of a high-index scatterer in a single-mode waveguide. The RTOF analysis of a two-port system not only properly attributes these behaviors to the experimentally measurable quantities, the phase and the power responses, but also allows one to identify the dominant factors in rendering multi-port optical forces nonconservative. We envision such an analysis will be instructional toward building high-efficiency opto-mechanical rotors that transduce electromagnetic energy into mechanical energy.
Sandia Laboratory is operated by Sandia Co., a Lockheed Martin Company, for the US Department of Energy's NNSA under contract DE-AC04-94AL85000. This work was supported by the DDRE under Air Force contract FA8721-05-C-000, the Meso Dynamic Architectures program at DARPA under the direction of Dr. J.L. Rogers, and Sandia's Laboratory Directed Research and Development program under Dr. W. Hermina. We thank P. Davids for helpful technical discussion.
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