Optical forces through the effective refractive index

Abstract

The energy-based methods as the Dispersion Relation (DR) and Response Theory of Optical Forces (RTOF) have been largely applied to obtain the optical forces in the nano-optomechanical devices, in contrast to the Maxwell Stress Tensor (MST). In this work, we apply first principles to show explicitly why these methods must agree with the MST formalism in linear lossless systems. We apply the RTOF multi-port, to show that the optical force expression on these devices can be extended to analyze multiple light sources, broadband sources, and multimode devices, with multiple degrees of freedom. We also show that the DR method, when expressed as a function of the derivative of the effective index performed at a fixed wave vector, may be misinterpreted and lead to overestimated results. Optical (transverse gradient) forces between two adjacent dielectric structures, due to the overlap of the evanescent field of the guided modes, were proposed by Povinelli et al. [1]. The optical forces can be rigorously calculated using the Maxwell Stress Tensor (MST) formalism [2] or, alternatively, in linear lossless dielectric materials, it may be obtained directly from the device's dispersion relation as a function of one of the structure's degrees of freedom (gap) [1,3,4]. Due to its computational simplicity and physical insights, the latter method has been more used than the former [3,4]. Furthermore, the Dispersion Relation (DR) method can be further simplified to express the optical force as a function of the mode effective (refractive) index derivative, with respect to the gap, leading to two expressions: one where the derivative must be performed at a fixed wave vector [5], and another where it must be performed at a fixed angular frequency [6][7][8]. The latter expression version can also be obtained in an alternative formal manner, by using the recently developed Response Theory of Optical Force (RTOF) method, proposed by Rakich et al. [9,10]. Besides that, all these expressions must agree with the MST formalism [1,9].In this letter, we apply first principles to explicitly show why these energy-based methods must agree with the MST formalism. We analyze a typical nano-optomechanical device to show that the DR method, expressed in terms of the effective index with derivative performed at a fixed wave vector, may overestimate the optical force if the correct transformations are not used, thus disagreeing with the MST. We also use the RTOF theory to extend the correct expression to more general cases.For a nano-optomechanical system composed of pure dielectric materials, the conservation of linear momentum states that [2,11]:where EM is the electromagnetic momentum density, ME is the mechanical momentum (or the force) density, and ⃡ is the Maxwell Stress Tensor (MST), which represents the momentum flux, given by [2,11]: where ( , ) and ( , ) are the electric and magnetic field, respectively, 0 and 0 are the vacuum electric permittivity and magnetic permeability, respectively, and the symbol * denotes their complex conjugate, an...

Optical force between two coupled identical parallel optical nanofibers

“…Indeed, the forces of the even modes are negative (see the upper parts of the figures), while the forces of the odd modes are positive (see the lower parts of the figures). Thus, the optical forces between the nanofibers are attractive for the fields in the symmetric modes and repulsive for the fields in the antisymmetric modes, in agreement with the results for other systems of coupled waveguides and cavities [4][5][6][7][8][9][10][11][12][13]. These features are the consequences of the fact that, for increasing fiber separation distance d, the propagation constant β and, hence, the effective refractive index n eff = β/k decrease in the case of the even array modes and increase in the case of the odd array modes [34].…”

“…2, where the nanofibers are expected to be accelerated towards the regions with a stronger field. Note that the typical order of magnitude of the power-normalized force per unit length F/P for coupled nanofibers is 1 pN µm −1 mW −1 , similar to that for coupled silicon strip waveguides [4], a silicon waveguide suspended over a silica substrate [8], and coupled silicon slab waveguides [9]. According to Ref.…”

“…( 2) are identical to that of the calculations from Eq. ( 5) [4,8,9]. The refractive index of the fibers is n1 = n2 = 1.4533 and that of the surrounding medium is n0 = 1.…”

“…Two coupled waveguides are essential components of several optical devices such as optical directional couplers, multicore fibers, polarization splitters, interferometers, and ring resonators [1][2][3]. It is known that the overlap of the modes of evanescently coupled waveguides or cavities results in an optical gradient force [4][5][6][7][8][9], and examples of devices where this is exploited include coupled strip waveguides [4], a waveguide suspended over a silica substrate [8], coupled slab waveguides [9], coupled whispering-gallery-mode microspheres [10,11], coupled guiding mirrors [12], and coupled microring resonators [13]. It has already been shown that the optical gradient force between two coupled dielectric structures is attractive or repulsive depending on whether a symmetric or antisymmetric mode is excited [4][5][6][7][8][9][10][11][12][13].…”

“…It is known that the overlap of the modes of evanescently coupled waveguides or cavities results in an optical gradient force [4][5][6][7][8][9], and examples of devices where this is exploited include coupled strip waveguides [4], a waveguide suspended over a silica substrate [8], coupled slab waveguides [9], coupled whispering-gallery-mode microspheres [10,11], coupled guiding mirrors [12], and coupled microring resonators [13]. It has already been shown that the optical gradient force between two coupled dielectric structures is attractive or repulsive depending on whether a symmetric or antisymmetric mode is excited [4][5][6][7][8][9][10][11][12][13]. The optical gradient forces between coupled macroscopic structures have been experimentally observed for a waveguide coupled to a high-Q microresonator [14] or to a dielectric substrate [15], coupled nanophotonic waveguides [16,17], and coupled ring resonators [18].…”