1. Introduction

Radiation pressure has been thoroughly studied for optical trapping and manipulation of microscopic objects [1, 2, 3] and parametric processes in interferometers [4, 5, 6, 7]. With recent advances in nanophotonics, the mass and the dimensions of optical devices have been miniaturized to the degree that device tuning through optical actuation is possible at micro- to milli-watt power levels [8, 9, 10, 11, 12]. In many of these cases, optically induced forces can scale to large values when optical fields are enhanced through high-Q resonances [13, 14, 15]. Such observations have sparked significant scientific interest in light-driven mechanically variable systems that can perform trapping [13, 16], actuation [10, 12, 14, 15, 17], transduction [8, 9, 11, 17, 18] and manipulation [13, 19] of nanoscale objects. Since the mechanical state of such systems is intimately linked to its optical state, these mechanical functions can lead to variable directional couplers [10, 16], parametric optical processes [8, 9, 12, 14, 15] in cavities, ultra-widely tunable microcavities [13], and microcavity athermalization through self-adaptive optomechanical behaviors [13, 20]. A general analytical formalism capable of handling such complex optical systems is therefore essential in tailoring optical forces at nanoscales.

It is typically believed that the knowledge of the full electromagnetic field distribution in such mechanically variable optical systems is a prerequisite for the computation of optical forces. In these electromagnetic field-based calculations, the Maxwell stress-tensor (MST) is numerically integrated over a closed surface surrounding the movable components in the system to compute the optical forces acting on them [21]. While the MST method is reliable, it offers little intuition for the design of a system with a desired optical force profile as system is tuned through different optomechanical states. In addition, no simplification can be made through MST methods to unify systems with similar optical response but different field distributions. With the trend of increasing complexity in nanooptical systems, a full numerical approach is computationally intensive, and therefore limits the scale of the system that can be studied (since the electromagnetic field distributions at every mechanical configuration must be solved to evaluate the force).

In this paper, we develop a simple analytical approach to calculating optically induced forces and potentials in open, lossless, mechanically variable optical systems (consisting of only linear media) which possess a single mechanical degree of freedom. By “open” systems, we refer to systems with optical input and output ports, which allow electromagnetic energy to enter and exit the system. In such systems, we show that the optical force acting on the mechanical degree of freedom is determined by the optical response of the system, and is completely independent of structural implementation. In contrast to the computation of complex electromagnetic field distributions, it is generally straightforward to derive an analytical expression for the optical response of nontrivial optomechanically variable systems (involving ring resonators, photonic crystals, waveguides, etc.) using temporal coupled mode theory (CMT) and/or scattering matrix (S-matrix) methods [22, 23, 24, 25, 26]. Given the relative ease with which the response of optical systems can be treated by these and other methods, our formalism, which we term the Response Theory of Optical Forces (or RTOF), provides a very unique and powerful analytical alternative to MST methods. Furthermore, the RTOF method can leverage analytical theories through which optical system response can be synthesized (independent of the underlying electromagnetic complexity) [27], yielding a means of engineering optical forces profiles as well. Moreover, since the optical response captures the salient physics of optomechanical systems, universal designs which are independent of structural implementation are possible using the RTOF method.

In what follows, we first discuss the definition of “open” mechanically variable optical systems and describe a formalism for energy and photon-number conservation in the context of such systems. The optical forces are first derived for single-input single-output systems under monochromatic excitation and are reduced to a simple analytical expression of the optical phase and amplitude response of the system. As an extension of this result, a scalar optical potential can be defined from the force profile, allowing formulation of a generalized optical trapping potential in these systems. We then expand the theory to include open systems under polychromatic excitation and open systems with multiple inputs and multiple outputs. Examples involving waveguides (possessing a continuum of modes) and resonant systems (supporting a discrete set of modes) demonstrate a perfect agreement between our analytical theory and MST methods.

 

Fig. 1. (a) Schematic showing a generic open photonic system that can be mechanically varied through displacement of q. Optical power flows into and out of system. The response of the device, S̃(ω,q), is also a function of frequency ω. (b) An example of a mechanically-variable open system in the form of an ideal lossless Gires-Tournois interferometer with a fixed partial mirror M1 and a movable mirror M2. Here, q is taken to be the separation of mirror M2 from M1.

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2. Definition of an open mechanically variable optical system

We define an open photonic (or electromagnetic) system as one which exchanges electromagnetic energy with the environment through input and output ports. The power in-flux is determined by optical sources, typically lasers, at a fixed frequency and power level. At steady state, the output power can be related to the input power by the optical response of the system, for example, as expressed by a scattering matrix [22]. We focus on systems with few input and output ports in the forms of single-mode waveguides or collimated Gaussian beams in free space, as in typical experimental settings.

The optical response of these systems can be varied by mechanical movement of a sub-component. Here we consider simple cases where the movement can be characterized by a change in the scalar coordinate, q. A one-port example is illustrated in Fig. 1(a), in which the mechanical degree of freedom q, represented iconically as a “knob”, affects the optical response S̃(ω,q). Here, the optical response, S̃(ω,q), is a 1×1 scattering-matrix [22], which relates the complex-valued amplitudes of the transmitted wave (b) to that of the incident waves (a) at a steady state: b̃=S̃(ω,q)ã. (Note, while the specific examples used in this paper exhibit linear displacements through motion of q, we may consider q to be a generalized coordinate [28]. Within a Lagrangian dynamics framework, the generalized coordinate might represent a rotation, translation, or motion along an arbitrary path along which the degree of freedom is constrained to move [28, 29].) In a dynamic regime (i.e., where q is time-varying) the electro-magnetic energy in the system can be converted into/from mechanical energy within the system. In general, such a conversion is accompanied by a change in the frequency of transiently stored photons, a change in the frequency of the photons exiting the system through its output ports, and a change in the output power level. We show that the energy conversion, and the related optical forces, can be analytically calculated from the changes in the optical response of the system alone, provided that the changes occur adiabatically. (For definition of an adiabatic process, see [30].)

While closed system analyses of optical forces are proven and widely applied, a formalism for the treatment of open systems has many clear advantages when analyzing experimentally relevant physical systems. For more on closed systems, see appendices A and B. The limitations of closed system analyses stem from their primary assumptions, which are: (1) no electromagentic energy is exchanged with the outside world, (2) the number of photons in the system is a constant, (3) photons reside only in energy eigenstates (or resonances) of the system, and (4) the system is lossless. As a consequence, one must often apply complicated constraints to closed systems in order to approximate coupling to the outside world (if even possible).

An open system analysis eliminates these difficulties, as it explicitly treats exchange of energy and photons with the outside world. The open system analysis developed here will reveal that the optically generated forces within an optomechanically variable system are related to the coordinate-dependent optical response of the system, S̃(ω,q). The ability to properly treat external coupling is useful since, in reality, optomechanically variable systems, such as tunable optical cavities, exchange both photons and electromagnetic energy with the outside world when tuned through resonance with an incident laser signal of constant power. More generally, the group delay of an open system may be modified while tuning the mechanical state of the system, thereby changing the stored energy and number of photons. We show that these complex behaviors of the open system can be treated analytically, provided that S̃(ω,q) is known.

Furthermore, the optical forces generated within real-world systems are highly dependent on the conditions by which they are coupled to the outside world. For example, an open system can be driven by external source(s) with, not only a discrete set, but also a continuum of frequencies. In contrast to closed systems, open systems can also operate at a frequency far detuned from its natural eigenfrequencies (or resonances). Additional degrees of freedom inherent in open systems (e.g., multiple input and output ports driven by different external sources), require an analytical theory capable of taking these variables directly into account.

It is worth emphasizing that the analytical treatment that we will develop here (or RTOF method) extends the existing analytical models, such as scattering matrix and coupled mode theory to yield analytical solutions of optical forces. The optical response, S̃(ω,q), of open systems can be treated very precisely and completely through use of S-matrix and/or temporal coupled mode theory (CMT) methods in common nano-photonic systems, such as ring resonators, photonic crystals and coupled waveguides [22, 23, 26, 24, 25]. Simple analytical models of these nontrivial open systems can be constructed in terms of these formalisms, capturing all aspects of energy storage in that system (which can be related to the group delay of an optical system – a quantity which is naturally computed through these methods). Furthermore, both S-matrix and CMT methods are capable of treating the external degrees of freedom associated with coupling to the outside world.

As a concrete example of an open mechanically variable optical system, we consider a lossless Gires-Tournois interferometer (seen in Fig. 1(b)) excited by a monochromatic source. When the movable mirror (M2) of the Gires-Tournois interferometer is displaced, a cavity resonance, of frequency ω_o(q), can be tuned through the source frequency, allowing the cavity to store a large number of photons at a range of positions coinciding with the resonant excitation of the cavity. Through this process, photons and energy are exchanged with the outside world. Resonant systems of this type, possessing high Q (quality factor), are also interesting for their ability to generate large optical force. In this system, the optical forces could perform mechanical work in this system (i.e., convert energy for electromagnetic to mechanical domains), by displacing the movable mirror, M2.

Through coupling to the outside world, the optical nature of the forces allows one to drive the system in a number of unique ways. For instance one could excite the system with a laser that is off resonance with the cavity, with several lasers at various frequencies and various power levels, or with a source consisting of a continuum of frequencies (such as a white light source). These inherent flexibilities, as we will see later, are compatible with the assumptions in our open-system analysis, and do not require any approximation or equivalent systems necessary in a closed system analysis.

 

Fig. 2. Schematic showing generic optomechanically-variable open-system within a closed surface forming the boundary to the volume, V. This system can be seen as a reflectionless one-port system. Optical power flows into and out of system (no power is reflected). Here q represents the mechanical degree of freedom of the system, which impacts the optical response of the system in some manner. Through the motion of q against internally generated optically-induced forces, work can be done on the electromagnetic field.

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3. Energy conservation and photon-number conservation

The conservation of energy and photon number are critical in establishing the time-dependent form of the energy conversion (between electromagnetic and mechanical domains) and energy exchange (i.e., electromagnetic energy exchange with the outside world) derived in this paper. We will show that, photon and energy conservation dictate that the optical forces in a system are uniquely determined by the optical response of the system. We first consider the conservation of energy, known as Poynting’s theorem in classical electromagnetics [21]. For linear systems containing movable components [31], Poynting’s theorem can be written as

Here S⃗ is the Poynting vector, ∂V is the bounding surface to the volume (V) under consideration, U_in is defined as the total electromagnetic energy contained within the volume, V, and J⃗·E⃗ represents the mechanical work done per unit time per unit volume by the field. Here J⃗ represents the polarization currents generated by dielectric moving parts, and the surface charge currents generated by moving parts consisting of perfect electric conductors [31]. We note that, although there are several different formulations of the Maxwell’s equations and constitutive relations for moving media [31], one can show that Eq. 1 is applicable to any lossless system absent of magnetic materials.

First, we consider a reflectionless one-input, one-output system (Fig. 2) enclosed within a volume V, outside of which the electromagnetic fields associated with the system are negligible. A net power of P_i (P_o) enters (exits) the system through its input (output) port. The energy converted from the electromagnetic to mechanical domains (corresponding to J⃗·E⃗) must be equivalent to the work done through displacement of, q, by the optically induced force (F_opt). Thus, the volume integral over J⃗·E⃗ can be equated to F_opt·q̇, allowing us to express Poynting’s theorem as

Defining W as the work done, through displacement of q, on the electromagnetic field, we have ∂W/∂t=-F_opt·_q̇. Here, F_opt is taken to be the instantaneous optical-force component in the direction of displacement of the generalized coordinate (subject to the constraints on the generalized coordinate). Thus, Eq. 2 can be expressed as

Above, we have replaced the surface integral over S⃗ with (P_o(t)-P_i), where P_i and P_o(t) are defined as the electromagnetic powers entering and exiting the system respectively. Note, P_i is assumed to be a constant (even as the mechanical state of the system changes). In contrast, P_o can be time dependent, because the motion of q changes the frequency of the photons transiently stored in the system and the energy storage capacity (or group delay) of the system.

It is important to note that, in general, the instantaneous optical force, F_opt, expressed on the coordinate, q, will be velocity dependent (i.e., F_opt=F_opt (q,q̇)) when q evolves rapidly. This can be seen from the fact that the optically induced force generated by an optomechanically-variable system depends on the instantaneous electromagnetic-field distribution produced within the system (e.g., F_opt acting on the movable component can be related to the fields through MST methods). Thus, if q evolves on a time-scale that is rapid with respect to the photon lifetime of the system, the internal field distribution may significantly differ from the steady-state field distribution, implying that F_opt is velocity dependent. However, in the gradual (or adiabatic) limit of motion, the instantaneous field distribution approaches the steady-state field distribution, meaning that the optically induced forces become state-dependent (or strictly a function of q), approaching those produced within the static system. In this paper, we seek only to develop an analytical method of computing the forces in the static-limit. Thus, while the velocity dependence of the forces should be noted, treatment of them beyond the scope of this paper. As we will show later, in the adiabatic limit of motion, the optical forces (F_opt) are conservative, and can be expressed in terms of an effective (opto-)mechanical potential (U_opt (q)) as F_opt=-∂U_opt/∂_q.

In section 4 we will derive an explicit relationship for the optical forces (F_opt) in the steady state using Eq. 3. Key to these derivations are the further simplifications that can be made to Eq. 3 by expressing energy and power in terms of the photon construct. In the classical limit (i.e., in the limit of large photon flux), one can generally express the powers entering and exiting the device as P_i=Φ_i·ħω and P_o=Φ_o(t)·ħω′(t) respectively, through the photon-picture. Here ħ is the Planck constant, while, ω (ω′) and Φ_i (Φ_o) are the mean frequency and the flux of incident (transmitted) photons respectively. The electromagnetic energy transiently stored in volume V can be expressed as U_in=Nħω_in, where N is the number of photons transiently stored at a mean frequency ω_in. Note that ω_in and ω′ must be interpreted as a mean photon frequency, since energy conservation is considered for the entire system (which we treat as an ensemble). This subtlety is important because dynamic variation in q can shift the frequencies of the transiently stored and outflowing photons (For further discussion, see Section 4).

Through semiclassical treatment of this time-varying system (i.e., as q evolves), we assume that photons experience no inelastic scattering [32, 33, 34], meaning that photon number is conserved. (Note, photon conservation is known to be a valid assumption in many time-varying optomechanical systems provided that the motion of the mechanical degree of freedom, q, is sufficiently gradual [32, 33, 34].) Thus, photon conservation requires that

In other words, the difference between the incident and transmitted photon fluxes must be accounted for by the rate of change of transiently stored photons (dN/dt), if photon number is conserved. For simplicity, we focus on the case when the system is driven by a monochromatic source of constant frequency and power. Mathematically, this corresponds to a constant flux of photons (Φ_i) of fixed frequency (ω) entering the system.

4. A lossless one-port system driven at a single frequency

Having established the mathematical forms of energy conservation and photon number conservation in a mechanically tunable optical system, we next analyze the conversion of energy (between electromagnetic and mechanical domains) as q varies. By relating the energy and the power flow to the optical response of the system, an analytical form will be derived, allowing us to express optically induced forces and potentials as a function of the optical response and the mechanical coordinate, q.

For the most general form of a lossless, reflectionless system with one input and one output (seen in Fig. 3), at steady-state (i.e., assuming q is constant and U_in is constant), the incident wave simply experiences a coordinate-dependent phase-shift, ϕ(q,ω), in traversing the system:

Since power and photon number are conserved at steady state in this lossless system, the amplitude of the transmitted wave and the transmitted photon flux are constant when q is static. We assume that the incident photon flux (Φ_i) is a constant given by Φ. Thus, ω=ω′ and Φ_i=Φ_o≡Φ in the static case. (In other words, the steady-state response can be expressed as S̃(ω,q)=exp[iϕ(ω,q)].)

In the case when q is time varying, however, the behavior of the system is somewhat different. In general, both the amplitude and phase of the wave exiting the system will take on a nontrivial time dependence. We define the phase imparted by the time-varying system as

Generally speaking, determination of the functional forms of the transmitted wave amplitude and phase (ψ(t,ω)) requires solution of complex nonlinear differential equations, which is beyond the scope of this paper. Nevertheless, in an adiabatic condition of motion, we will show that the time-varying phase (ψ(t,ω)) differs from the time dependent version of the steady-state response, ϕ(q(t),ω), only by a small correction (which will be discussed in more detail). As a general consequence of the nontrivial time-varying phase, ψ(t,ω), light exiting the system is no longer monochromatic. To take this into account through our photon model, we simply interpret ω′ as the mean photon frequency exiting the system. While photon number must be conserved in the framework of this semi-classical model (for gradual evolution of q), photon flux is not necessarily conserved. This is because the stored energy (or number of stored photons) may vary with time. As a consequence, the transmitted photon flux (Φ_o(t)), and the number of transiently stored photons (N) are generally time dependent.

 

Fig. 3. Schematic showing lossless optomechanically variable open system consisting of linear media. Optical power flows into and out of system. Here q represents a generalized coordinate which changes the response of the device, S̃(ω,q). Here it is assumed that this is a reflectionless system.

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To examine the energetics of this system, we assume that q takes on explicit time dependence, and seek a general relation for the work done on the electromagnetic field dW in unit time dt. Energy conservation (Eq. 3) allows us to express the time-varying electromagnetic energy as

Above, we have recast Eq. 3 in terms of N,Φ_i, Φ_o, ω, ω_in and ω′ through the semiclassical model developed here. It is critical to note that, to arrive at the above expression, we have used the relation Φ_o(t)=Φ_i-dN/dt, which results from photon conservation in the time varying-case.

Through integration of Eq. 7, we now seek an expression for the static optomechanical potential as the energy of the system changes through gradual (or adiabatic) motion of the spatial coordinate. In a manner similar to that employed to derive the quantum mechanical adiabaticity theorem [30], we consider the work done on the electromagnetic field (ΔW) by a small change of our spatial coordinate, Δq, over a time, Δt, where q(t) is defined as q(t)=q_i+f (t)·Δq. Here, f (t) is continuous function defining the rate of coordinate change along the interval [0,Δt], with values f (t)=0 for t : (-∞,0], f (t)=[0,1] for t : [0,Δt] and f (t)=1 for t : [Δt,∞). Integrating Eq. 7 over a time interval t : [0,Δt+T], one finds

Here, δω(t)≡ω′(t)-ω≡-ψ̇, where -ψ̇ is the instantaneous frequency shift produced by the time-varying phase, allowing us to express Eq. 9 as

In general, one does not know the time-varying phase of the transmitted wave, ψ(t), which is necessary to evaluate the above integral. However, if we allow the system to relax following the interval of motion (i.e., in the limit as T≫τ_p, where τ_p is the photon-lifetime of the system), it must return to steady state, meaning that, at the end-points of integration, ψ(t,ω) can be replaced with ϕ(q(t),ω), the steady-state phase response. Steady-state behavior also implies that the frequency of the photons transiently stored in the system (ω_in) must be equal in frequency to the incident photon frequency (ω), requiring the last term of Eq. 10 to vanish, yielding

Furthermore, if we require that |d f/dt|≪1/τ_p, and |d ² f/dt ²|≪(1/τ_p)|d f/dt| for all times of motion, ψ(t) can, in general, be expressed as ψ(t)=ϕ(q(t))+Δψ(t), where Δψ(t) is a small correction to the phase response (of order Δq in smallness). Subject to these requirements, the number of transiently stored photons, N(t), can also be expressed as N(t)=N_q(q(t))+ΔN(t), where N_q(q) is the number of photons stored by the system at steady-state, and ΔN(t) is a small correction (of order Δq in smallness). Note, N_q(q) is related to the optical response as N_q(q)=Φ_i·τ_g(q), where τ_g(q) is the coordinate-dependent group-delay of the system. Finally, we assume that |(∂ϕ/∂q)·Δq|≪1 and |(∂N_q/∂q)·Δq|≪N_q(q). In other words, the system response is not drastically effected by the small change of coordinate, Δq. Given these constraints on f (t) and Δq, one can express Eq. 11 as

Here, the higher order terms (H.O.T.) involve Δψ(t) and ΔN(t), and are of order (Δq)² in smallness. Dividing the above relation by Δq, and taking the limit as Δq→0, all terms except the first vanish, yielding the following exact relationship for the optically induced force

Here, F_q should be interpreted as the time-averaged optical force acting on q (i.e., averaged over an optical cycle). Note, since this derivation can be performed for any initial value, q_i, the optically induced force can be expressed as

at a fixed ω, for any value of q. Integration of Eq. 14 reveals that the change in electromagnetic energy induced through motion of the coordinate, q, can be expressed as

Here, q_o is an arbitrary point of origin. Since the change in energy of the system corresponds to mechanical work performed through motion along the generalized coordinate, q, ΔW(q) can be interpreted as the (opto-)mechanical potential U_opt (q) of the system. Dropping the superfluous constant term, Φ·ħϕ(q_o), the time-averaged static potential is

for any fixed frequency of excitation, ω. Thus, in a lossless system consisting of linear media, we see that the exact optomechanical potential is given by the phase change imparted on the transmitted wave as the generalized coordinate varies. It follows from the above that the optically induced forces are conservative, and can always be expressed in terms of an effectiveopto-mechanical potential as F_q=-∂U_opt/∂q, provided that the corresponding trajectory of the generalized coordinate, q, is single valued in a multidimensional space. (Otherwise, q could map to multiply-defined locations in state-space, making valid formulation of the potential difficult.) Note also, these expressions for optical force and potential (or Eqs. 14 and 16) can be generalized for the treatment of systems with an arbitrary number of mechanical degrees of freedom. In the following sections, we will show that the forces computed through this formalism (which we term the Response Theory of Optical Forces, or the RTOF method) are found to be identical to those computed through exact closed-system and Maxwell stress tensor analyses of the equivalent physical systems, indicating that this formulation is consistent with all of the essential physics necessary to describe static forces and potentials.

Finally, although no assumption is made about the phase response of the lossless linear system examined here, a sharp resonance can greatly enhance the forces (both attractive and repulsive) at various positions in space, enabling the creation of nontrivial, and tailorable potential wells [13]. Such potentials can be adiabatically transformed by varying the conditions of excitation, enabling ultra-precise manipulation of nanomechanical systems [13]. While Maxwell stress tensor methods could be employed to numerically examine nontrivial behaviors of this form, analysis of them becomes exceedingly complicated to examine and highly computationally intensive. In contrast, the formalism derived here provides tremendous simplicity and insight as it offers an analytical means of describing the optically induced force and potential in terms the optical response of the system.

5. Tailoring potentials via polychromatic excitation and use of optical resonances

Through use of the RTOF method (and the relations derived above), it becomes apparent that an open system (of the form we have analyzed here) offers numerous unique degrees of freedom with which one can tailor force and potential profiles. For instance, through degrees of freedom offered by external coupling, one can think of simultaneously exciting the system with a superposition of wavelengths, or varying the wavelength of excitation of a monochromatic source to produce adiabatic transformation of the potential [13]. Since the forces and potentials are additive for distinct wavelengths, multiple excitation frequencies yield a time-averaged (in this case, averaging over the period of ploychromatic oscillation) potential of the form

Above, we have assumed that N distinct sources of frequency ω_l, and flux Φ_l simultaneously drive the system. The above sum can be made into a continuous integral, enabling the treatment of broadband sources as well.

6. Generalization to lossless multi-port systems

Next, we generalize the RTOF method from the case of single-input single-output (or reflectionless one-port) systems to the more general case of multi-input- multi-output-port optomechanically variable systems. The multi-port system under consideration is schematically illustrated in Fig. 4(a), showing N-independent input and output ports in the device. (However, the analysis presented here would be identical for N bi-directional ports). Similar to the one-port system examined in Section 5, we assume that the steady-state response of the system is variable through motion of the generalized coordinate, q, shown as a rotational degree of freedom in Fig. 4(a). Figure 4(b) shows a simple example of an optomechanically variable Fabry-Perot cavity, where the generalized coordinate, q, is taken to be the mirror separation. In contrast to Fig. 4(a) this system has two bi-directional optical ports. However, since the RTOF method is based on power conservation, the orientation of the incoming and outgoing waves is unimportant in the multi-port treatment shown here.

 

Fig. 4. (a) Sketch of a generic lossless and linear optomechanically-variable open-system with N inputs and N outputs. Light of constant intensity, frequency and phase flows into system. Here q represents a generalized coordinate which changes the response of the device, which can be expressed in terms of a scattering matrix of the form S̃_l,m(ω,q). (b) A lossless Fabry-Perot interferometer, which serves as a specific example of a multi-port system of this form.

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Through this analysis of force and potential, we assume that N input signals, of fixed frequency and amplitude, enter the multi-port system from the left with powers specified by P _i,k=Φ_i,k·ħω. This implies that the incident photon fluxes, Φ_i,k, must be fixed; however, a change in q will, in general, effect the output photon fluxes, Φ_o,k(q), as power can be redistributed among the output ports at steady-state in a multi port system. Thus, the steady-state power exiting the k^th output port can be expressed as P _o,k(q)=Φ_o,k(q)·ħω. We assume that the system is lossless; thus, at steady state, photon flux is conserved, requiring that ∑_kΦ_i,k=∑_kΦ_t,k(q)=Φ_tot is satisfied for all values of q.

To examine F_q(q) and U_opt (q), we again assume that q is time dependent, and integrate ∂W/∂t=(P_o(t)-P_i)+∂U_in/∂t. However, in the multi-port case, the incident power (P_i) is given by P_i=ħ·∑_kΦ_i,k·ω=Φ_tot ·ħω and transmitted (P_o) powers is P_o(t)=ħ·∑_kΦ_o,k(t)·ω′_k (t). Note that the instantaneous frequency (ω′_k(t)) of the photons exiting the k^th output port will, in general, be distinct in a multi-port system since the time-dependent phase from each port can be different. Through a similar derivation to that used to analyze the single port system, the optically induced force is seen to be

Integration of Eq. 18, yields the multi-port potential

Here, ϕ _o,k(q) is defined as the phase response of the k^th output port. The above is a general form of the optomechanical potential for a lossless optomechanically variable system with N inputs and N outputs, and having an arbitrary optical excitation of the inputs – all at a single fixed frequency, ω. Apparently, no explicit knowledge of the field distribution generated within is necessary to compute the force and potential created by light. For fixed-input conditions, one need only know the steady-state amplitude and phase response of the system as the generalized optomechanical coordinate, q, is varied.

Since the optical response of multi-port systems are most often expressed in terms of scattering matrices [22, 26], we illustrate how the optical forces and potentials (calculated using the RTOF method) can be also expressed in terms of scattering-matrix elements. Through the scattering-matrix formalism, the steady-state response of the system can be expressed as

where ã_k (b̃_k) is the complex wave amplitude entering (exiting) the k^th input port seen on the left (right) of the Fig. 4(a). The scattering amplitudes can be related to the photon fluxes entering the k^th input-port as Φ_i,k=|ã_k|²/ħω and exiting the k^th output-port as Φ_o,k(q)=|b̃_k|²/ħω. Furthermore, the steady-state phase of the exiting wave is given by

Through use of these simple relations, we see that, provided the scattering matrix (S̃_l,m(ω,q)) of the lossless system of interest is known, the optically induced force and potential can be computed in a straightforward manner.

7. Demonstration of equivalence with maxwell stress tensor methods

Next, we apply the RTOF-method to calculate optical forces in several systems with exact analytical solutions. Through these examples, we are able to demonstrate exact equivalence between the forces computed through Maxwell stress tensor analysis and the RTOF formalism derived here. We begin by examining the optical forces (or gradient forces) produced in mechanically variable waveguide systems whose guided modes are modified by a geometric change. We also explore, resonantly enhanced forces generated within an optical cavity through analysis of a lossless Gires-Tournois interferometer.

7.1. Optical forces produced in optomechanically variable waveguide geometries

An attractive or repulsive optical force can be generated by the light guided in two parallel evanescently coupled waveguides, depending on the symmetry of the compound mode which is excited (see Figs. 5(b)–(d)). The attractive and repulsive forces in such waveguide systems were first studied by Povinelli et. al (for more details on such systems, see ref [10]). Examples of two different optomechanically variable waveguide geometries of Fig. 5. Figures 5(b)–(d) show schematics of the waveguide cross-section and mode-profiles for the coupled dual-waveguide system examined here. These waveguide systems can be treated as reflectionless and lossless optomechanically variable one-port devices for treatment with the RTOF method, provided that light is coupled into a single waveguide eigenmode (coupling into more that one mode would require a multi-port analysis). In both cases, the generalized coordinate (q), which effects the response of the system, is taken to be the separation between the two bodies. As a first application of our open-system formulation of force and potential, we show that perfect agreement is found between MST methods and the analytical RTOF method derived here.

Open system treatment of optical forces from guided modes

In applying the RTOF method to optomechanically variable waveguide modes, we consider the forces generated between the waveguide segments (of length L) enclosed in the surface seen in Fig. 5. To evaluate the optically induced force through Eq. 14, we must determine the photon flux, Φ, entering (and exiting) the closed surface through the waveguide mode, and the coordinate-dependent phase-response, ϕ(ω,q), that our optomechanically variable waveguide segment (within the surface) imparts on the transmitted wave. If we define P as the power entering our waveguide, the photon flux is Φ=P/(ħω). Thus, the phase response of the system is simply dictated by the optical path length of the system:

 

Fig. 5. (a) Diagram of an optomechanically variable dual-waveguide system within a closed surface of length L. (b–d) and (e–f) show two different optomechanically variable waveguide geometries which might be examined in a nearly identical manner. (b–d) are schematics of the waveguide cross-section and mode-profiles for the coupled dual-waveguide system treated in ref [10], and (d–e) is a schematic representation of a waveguide mode whose effective index is modified through evanescent perturbation by a uniform dielectric body (treated in ref [18]). In the latter system it is assumed that n ₁>n ₂.

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Here, n_eff(ω,q) is defined as the effective index of the waveguide mode of interest, L is the waveguide length, and c represents the speed of light in vacuum. Thus, applying Eq. 14 yields a relationship for the optically induced force of the form

Above, F^o_q (ω,q) represents the optically induced force generated between the two bodies seen in either Fig. 5(b) or Fig. 5(e). Note that if the symmetric mode of Fig. 5(c) or the perturbed mode in Fig. 5(f) is excited for, ∂n/∂q<0, then the force tends to be attractive. Interestingly, as illustrated by Povinelli et. al, the anti-symmetric mode can yield ∂n/∂q>0 over some range of motion, generating repulsive forces between the waveguides.

Comparison of RTOF and Maxwell stress-tensor methods

In the previous section, we derived a simple and general analytical relation (Eq. 23) for computing the optical forces produced within optomechanically variable waveguide systems using the RTOF method. Next, we compute the optical forces generated in a specific optomechanically variable waveguide system using Maxwell stress tensor method, and compare with those found using the Eq. 23.

 

Fig. 6. Shows the computed forces (pN/µm/mW) generated by a symmetric waveguide mode on the dual waveguide system. Forces computed by RTOF method (dashed line) and Maxwell stress-tensor method (circles) are over-layed, revealing perfect agreement. Inset shows and intensity-map of the E_x electric-field component computed with a full-vectorial mode-solver.

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Through this example, we examine the forces generated through excitation of the symmetric waveguide mode seen in the dual waveguide system shown in Fig. 5. The waveguide system considered here assumes optically coupled waveguides of width, w=450nm, height h=200nm, core (cladding) refractive index of 3.5 (1.0), and a free-space optical wavelength of λ=1.55µm. As before, q is defined to be the waveguide separation. A computed mode profile (corresponding to the E_x component of symmetric mode) generated by this structure is shown in the inset of Fig. 6. The optically induced forces between the waveguides (shown as circles in Fig. 6) were found by integrating the Maxwell stress tensor over a closed surface containing one of the waveguides at a number of waveguide separations. For comparison, the forces were also computed using Eq. 23 (corresponding to the dashed line). Upon examination of Fig. 6, perfect agreement is seen between the two methods. Thus we see that the RTOF method provides a valid and greatly simplified means of computing optically induced forces in this dual-waveguide system, and numerous other complex waveguide topologies. In addition, we note that an analytical expression which is exactly equivalent to Eq. 23 can be derived through closed-system energetics, confirming the validity of this expression in an independent manner. (For further details, see Appendices A and B.)

7.2. Optical forces in optomechanically variable interferometers

Next we demonstrate the utility of the RTOF method by computing the optically induced forces generated on a mirror within a lossless Gires-Tournois interferometer (GTI). The GTI under consideration is shown in Fig. 7(a), and consists of partially reflecting mirror (M ₁) and a perfectly reflecting mirror (M ₂) separated by a distance q. We assume that a plane wave with complex amplitude E_i·exp[i(k_z-ωt)] is incident from the left. Since the system is lossless, the reflected wave can be written as E_o·exp[i(-k_z-ωt+ϕ(ω,q))], where E_i=E_o=E. We assume that M ₁ has an amplitude reflectivity, r, and consider the optically induced forces acting on M ₂. Note that the electromagnetic energy stored in the GTI, which is proportional to the number of photons transiently stored in the interferometer, may vary by orders of magnitude as the mirror separation tunes the cavity through resonance with the incident plane wave.

 

Fig. 7. (a) and (b) are schematics of the same lossless Gires-Tournois interferometer. In both diagrams, the generalized coordinate q is taken to be the separation between mirrors M ₁ and M ₂. (a) Shows the incident (E_i) and exiting (E_o) field amplitudes, while (b) shows the internal fields impinging on (E ⁺ _in) and receeding from (E ^- _in) mirror M ₂.

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Optical forces in a Gires-Tournois interferometer via the RTOF method

Since the GTI is a lossless system with a single input and output, we can simply compute the phase response of the system, ϕ(ω,q), and evaluate the optically induced force on M ₂ using Eq. 14. Numerous methods can be employed to compute the phase response of a GTI interferometer [35, 22]. For instance, the general scattering matrix for a two-mirror interferometer is derived in Ref. [22]. From this result, one can show that the phase response of our GTI is given by

Here δ≡2q(ω/c). To evaluate the force using Eq. 14, we must also compute the incident photon flux. From the time averaged Poynting vector, one can show that the incident power per unit area is given by P_i/A=|E_i|²/(2µ_oc). Here, µ_o is the magnetic permeability of vacuum, and A is the area under consideration. Thus, the incident photon flux is

Substituting Eq. 25 and Eq. 24 into Eq. 14, we find that the force per unit area acting on M ₂ is given by

Notice that, as one might expect, on resonance (i.e. when δ=2π·m), the forces on M ₂ reach a maximum value.

Forces in a Gires-Tournois interferometer via Maxwell stress tensor methods

Next, we compute the forces on M2 via Maxwell stress tensor (MST) methods. The Maxwell stress tensor (in vacuum) is defined as T _i,j=ε_o[E_i E_j-(1/2)δ _i,j|E|²]+[B_iB_j-(1/2)δ _i,j|B|²]=µ_o [21], and can be related to the force on M ₂ through the surface integral

Here, S represents the closed surface, seen in Fig. 7(b), which consists of parts S ₁–S ₄. If we take M ₂ to be a perfect electric conductor of infinite extent, the only nonvanishing contribution to the integral will come from S ₁. Within the interferometer, we represent the light impinging on M ₂ as a plane wave of the form E _in e ^ikz. Therefore, taking the front face of M ₂ to be at position z=0, we can express the electric and magnetic fields near M ₂ as

Using the above field amplitudes, and evaluating the Maxwell stress tensor over S ₁, we see that the time-averaged force per unit area acting on M ₂ is given by

Here, P_in is defined as the net power impinging on M ₂ from within the GTI. It is convenient to express the force in terms of the incident optical power since we can use the interferometer scattering matrix analysis presented in [22] to compute P_in. Using the results of Ref. [22] we find that

Using Eq. 30 to evaluate the time-averaged force per unit area on M ₂, we have

In comparing Eq. 32 and Eq. 26, we see that identical relations are found, illustrating exact equivalence between the RTOF method (derived here) and the Maxwell stress tensor analysis, through examination of this nontrivial resonant system. Furthermore, one can validate the multi-port formalism derived in Section 6 by assuming that M ₂ is partially reflecting, and evaluating the force using the multi-port relation (Eq. 18) derived in the previous section. Through this exercise, equivalence is again found between the two formalisms.

8. Summary and conclusions

In this paper, we have derived a general formalism for the treatment of optically induced forces and potentials (within a Lagrangian mechanics framework) in the context of lossless optomechancially variable open systems with a single mechanical degree of freedom (consisting of linear media). The proposed and derived RTOF method allows the optically induced forces to be calculated solely from the optical response of the system, thereby offering tremendous simplicity and insight when compared to computationally intensive Maxwell stress tensor methods, which require explicit computation of complex internal field distributions. Through application of the RTOF method to examples having exact analytical solutions, we have shown that, although simple, this method yields exact correspondence with conventional Maxwell stress tensors methods.

 

Fig. 8. Examples of single- and multi-port systems which are analytically treatable with RTOF method. (a) and (b) show an all-pass ring resonator which is optomechanically tunable via dielectric perturbation, after Ref. [36]. (c) and (d) show the analagous optomechanically variable geometry for a photonic cyrstal defect cavity. (e) shows a tunable resonant structure utilizing photonic crystal guided-slab resonances (see Ref. [25].

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A key insight of the RTOF method is that, provided the scattering matrix response of a lossless optomechanically variable system (i.e., an S-matrix of the form S _i,j(ω,q)) is known, the force and potential energy expressed on the generalized coordinate can be computed analytically. An important corollary of the RTOF method is that, provided any two systems have an identical response (or s-matrix S _i,j(ω,q)), identical forces are generated within. Thus, equivalent systems can be created in the context of photonic crystals, microrings and free-space optics, revealing that the general property which determines the optical force is the optical response, not the complex field distributions (an insight that MST methods leave obscured).

Finally, while the examples explored through use of the RTOF method were chosen for their simplicity, this method is applicable to any lossless open-system with a single mechanical degree of freedom which consists of linear media and conserves photon number. For example, Fig. 8(a)–(c) shows optomechanically variable ring resonator and photonic crystal structures that could be treated using the RTOF method. Provided these topologies (which are more complex that a Gires-Tournois) abide by the basic assumptions of the RTOF method, and response of these systems are known, one can compute the resulting force and potential profiles exactly. Lastly, because this formalism establishes a direct correspondence between the optical response of a system and the forces and potentials produced by light, well-established theories of optical filter systems [27] could therefore be extended to provide tools for synthesizing complex optical force and potential profiles.

9. Appendix A: Closed-system energetics

It has been demonstrated that a simple closed-system analysis of energy can be adapted to analytically compute optically induced forces in a number of physical systems, yielding exact equivalence with Maxwell stress tensor methods [14, 10, 18]. To be precise, we define an optical closed system as one which is lossless and does not exchange electromagnetic energy with the outside world. A generic closed system is illustrated in Fig. 9(a) whose optical properties are assumed to be variable through motion of the spatial coordinate, q, shown as a rotational degree of freedom (illustrated iconically as a “knob”). A specific example of a closed system is shown

in Fig. 9(b) as an idealized lossless Fabry-Perot cavity where the generalized coordinate, q, is taken to be the mirror separation.

 

Fig. 9. (a) Schematic showing generic optomechanically-variable closed system. Here q represents a generalized coordinate through which work can be done on the system, changing the electromagnetic energy within. (b) A specific example of a closed system in the form of an idealized and lossless electromagnetically-closed cavity. Here, the generalized coordinate q is taken to be the mirror separation.

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Since a closed system is lossless and doesn’t exchange electromagnetic energy with the outside world, the primary simplifying assumption one makes when treating a closed system is that the number of photons within remains fixed for all of time. Of course, the photons must reside in energy eigenstates of the system. Therefore, if we assume that N photons populate an optical eigenfrequency ω_o, the energy of the closed system can be expressed as U_c^EM=Nħω_o. If the eigenfrequencies of the system depend on the generalized coordinate, q, (as is the case in optomechanically variable systems) the energy of the system becomes U^c_EM(q)=Nħω_o(q). Note, through this expression for the energy, it is assumed that the energy states of the system evolve adiabatically through motion of the coordinate q. Thus, using energy conservation, one can show that the force expressed on the generalized coordinate is

(33)$$F^c\left(q\right){\mid }_N=-N\hslash \frac{d{\omega }_o\left(q\right)}{dq}.$$

Through use of this simple expression, perfect agreement can be demonstrated with Maxwell stress tensor methods [14, 10, 18].

10. Appendix B: Equivalence between open- and closed-system analyses

Through closed-system energetics, we apply Eq. 33 to compute the optically induced forces. Initially, it is not immediately obvious that Eq. 33 will lead to an equivalent expression to Eq. 23. However, through a few simple manipulations, we show that the two results are equivalent. Through closed system analysis of an open system, one seeks to construct a closed system which is analogous to the open system of interest. In the case of this waveguide system, we seek to mimic the electromagnetic energy density with in our open-system waveguide through proper choice of the number of photons in our closed system (i.e., to ensure the electromagnetic field distributions within both systems is identical). Based on the incident photon flux, the mean number of photons, 〈N〉, in our open system would simply be 〈N〉=Φ·τ_g. Here τ_g represents the group delay associated with a waveguide of length L. Thus, we let N=〈N〉 through our formulation of an equivalent closed system. In this case, N can be written explicitly as

(34)$$N=\Phi ·{\tau }_g=\frac{P}{\hslash \omega }\frac{L}{{\nu }_g}=\frac{P}{\hslash \omega }\frac{PL{n}_g}{c}.$$

Above, n_g represents the group index of the guided mode of interest, and c is the speed of light in vacuum.

Next, we must relate ∂ω/∂q to the waveguide effective index. For the purposes of our closed-system analysis, we can define the energy eigenstates of our system by applying periodic boundary conditions to our waveguide segment. In this case, ϕ(ω,q), represents the round-trip phase of the system, yielding the resonance condition ϕ(ω,q)=2π·m for the m^th eigenfrequency. Differential expansion of the resonance condition reveals that dϕ(ω,q)=(∂ϕ/∂ω)·dω+(∂ϕ/∂q)·dq=0. Using this relation, to solve for ∂ω/∂q [36, 24], we have

(35)$$\frac{\partial \omega }{\partial q}=-\frac{\omega }{n_g}\frac{\partial n_{eff}}{\partial q}.$$

Substituting Eqs. 34 and 35 into Eq. 33, we find the optically induced force to be

(36)$$F_q^c(\omega ,q)=L·\frac{P}{c}\frac{\partial n_{eff}}{\partial q}.$$

Since, Eq. 36 is of the exact same analytical form as Eq. 23, exact equivalence is demonstrated between the RTOF method derived here and closed system energetics. Furthermore, perfect agreement is found with Maxwell stress tensor methods through numerous published examples [14, 10, 17] using this expression.

Acknowledgments

We acknowledge the generous support and encouragement of M. Soljačić, E. P. Ippen, Y. Fink and J. D. Joannopoulos. We thank P. Davids for helpful technical discussions regarding Poynting’s theorem and feedback through the writing of this manuscript. In addition, we thank C. E. Rakich for help in preparing this manuscript. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. This work was supported in part by the office of the Director of Defense Research and Engineering under Air Force contract FA8721-05-C-0002.

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