D ESIGN CON SID ERATION S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Bll/ding a Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Beam Steering 254 Trapping Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 T RAPPIN G TH EORY . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . ... . . . . 260 Ray-Optics Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 Electromagnetic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 F ORCE M EASUREM EN T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Measurement of Trap Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 Physics of Trup Stiffness Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 B�ownia,! Motion During Force Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 272 PlcotenslOmeters 273 Other Applications of Picotensiometry .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Determinants of Trapping Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

Biological applications of optical forces

Radiation forces on a dielectric sphere in the Rayleigh scattering regime

Metallic objects reflect light and have generally been considered poor candidates for optical traps, particularly with optical tweezers, which rely on a gradient force to provide trapping. We demonstrate that stable trapping can occur with optical tweezers when they are used with small metallic Rayleigh particles. In this size regime, the scattering pictures for metals and dielectrics are similar, and the larger polarizability of metals implies that trapping forces are greater. The latter fact makes the use of metal particles attractive for certain biological applications. Comparison of trapping forces for latex and gold spheres demonstrates that the gradient force is the major determinant of trapping strength and that competing effects, such as scattering or radiometric forces, are relatively minor.

Optical trapping of metallic Rayleigh particles.

We briefly review basic properties of dielectric whispering gallery mode resonators that are important for applications of the resonators in optics and photonics.

Optical resonators with whispering-gallery modes-part I: basics

We present a comprehensive overview of sensor technology exploiting optical whispering gallery mode (WGM) resonances. After a short introduction we begin by detailing the fundamental principles and theory of WGMs in optical microcavities and the transduction mechanisms frequently employed for sensing purposes. Key recent theoretical contributions to the modeling and analysis of WGM systems are highlighted. Subsequently we review the state of the art of WGM sensors by outlining efforts made to date to improve current detection limits. Proposals in this vein are numerous and range, for example, from plasmonic enhancements and active cavities to hybrid optomechanical sensors, which are already working in the shot noise limited regime. In parallel to furthering WGM sensitivity, efforts to improve the time resolution are beginning to emerge. We therefore summarize the techniques being pursued in this vein. Ultimately WGM sensors aim for real-world applications, such as measurements of force and temperature, or alternatively gas and biosensing. Each such application is thus reviewed in turn, and important achievements are discussed. Finally, we adopt a more forward-looking perspective and discuss the outlook of WGM sensors within both a physical and biological context and consider how they may yet push the detection envelope further.

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Whispering gallery mode sensors

Series expressions for the net radiation force and torque for a spherical particle illuminated by an arbitrarily defined monochromatic beam are derived utilizing the spherical‐particle/arbitrary‐beam interaction theory developed in an earlier paper. Calculations of net force and torque are presented for a 5‐μm‐diam water droplet in air optically levitated by a tightly focused (2 μm beam waist diameter) TEM00‐mode argon‐ion (λ=0.5145 μm) laser beam for on and off propagation axis, and on and off structural resonance conditions. Several features of these theoretical results are related to corresponding experimental observations.

Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam

Fifth‐order corrected expressions for the electromagnetic field components of a monochromatic fundamental Gaussian beam (i.e., a focused TEM00 mode laser beam) propagating within a homogeneous dielectric media are derived and presented. Calculations of relative error indicate that the fifth‐order Gaussian beam description provides a significantly improved solution to Maxwell’s equations in comparison with commonly used paraxial (zeroth‐order) and first‐order Gaussian beam descriptions.

Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam

Theoretical expressions for the internal and external electromagnetic fields for an arbitrary electromagnetic beam incident upon a homogeneous spherical particle are derived, and numerical calculations based upon this theoretical development are presented. In particular, spatial distributions of the internal and near‐surface electric field magnitude (source function) for a focused fundamental (TEM00 mode) Gaussian beam of 1.06 μm wavelength and 4 μm beam waist diameter incident upon a 5‐μm‐diam water droplet in air are presented as a function of the location of the beam focal point relative to the sphere center. The calculations indicate that the internal and near‐surface electric field magnitude distribution can be strongly dependent upon relative focal point positioning and may differ significantly from the corresponding electric field magnitude distribution expected from plane‐wave irradiation.

Internal and near‐surface electromagnetic fields for a spherical particle irradiated by a focused laser beam

The spherical particle/arbitrary beam interaction theory developed in an earlier paper is used to investigate the dependence of structural resonance behavior on focal point positioning for a spherical particle illuminated by a tightly focused (beam diameter less than sphere diameter), linearly polarized, Gaussian‐profiled laser beam. Calculations of absorption efficiency and distributions of normalized source function (electric field magnitude) are presented as a function of focal point positioning for a particle with a complex relative index of refraction of n=1.33+5.0×10−6i and a size parameter of α≊29.5 at both nonresonance and resonance conditions. The results of the calculations indicate that structural resonances are not excited during the on‐center focal point positioning of such a tightly focused beam but structural resonances can be excited by proper on‐edge focal point positioning. Electric wave resonances were found to be excited by moving the focal point from on‐center towards the edge of the sphere parallel to the direction of the incident beam electric field polarization. Magnetic wave resonances were found to be excited by moving the focal point from on‐center towards the edge of the sphere perpendicular to the direction of the incident beam electric field polarization.

Internal fields of a spherical particle illuminated by a tightly focused laser beam: Focal point positioning effects at resonance

A theoretical procedure in which a spheroidal coordinate separation-of-variables solution is used is developed for the determination of the internal and the near-surface electromagnetic fields for an arbitrary monochromatic field that is incident upon a homogeneous spheroidal particle. Calculations are presented for both the prolate and the oblate geometries, demonstrating the effects of particle size, particle axis ratio, and the orientation and character (plane-wave and focused Gaussian beam) of the incident field on the resultant internal and near-surface electromagnetic-field distributions.

John P. Barton

Papers

Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam

Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam

Internal and near‐surface electromagnetic fields for a spherical particle irradiated by a focused laser beam

Internal fields of a spherical particle illuminated by a tightly focused laser beam: Focal point positioning effects at resonance

Internal and near-surface electromagnetic fields for an absorbing spheroidal particle with arbitrary illumination