Enhanced optical trapping via structured scattering: Supplementary information

It is well known that the magnitude of optical forces is limited by the momentum flux of the light. This momentum flux is equal to the radiation pressure force on full absorption, and is given by nmP/c, where P is the incident power, nm the medium refractive index, and c the speed of light1. Optical forces are commonly described with the normalized force Q, which is the fraction of the radiation pressure imparted on the particle. Provided the field only interacts once with the particle, this parameter has an upper limit of 2 which is reached for complete reflection of a collimated field. However, this is a propulsion force which always points in the direction of optical propagation. When considering a trapping application, it is most useful to consider the maximum restoring force. The optical force is determined from the difference in the optical momentum before and after the interaction; and since the momentum of the incident light is independent of the particle position, this force can only change with particle position over a maximum range of ∆Q = 2. The most stable optical trap is reached when this range of applicable force is centred about zero, such that the maximum useful trapping force is limited to Q = 1. In addition to this well known limit, there is also a physical limit on the trap stiffness which has not, to our knowledge, previously been discussed. The trap stiffness is defined by the change of the trapping force with the position of the trapped particle, k = − dx . This is constrained because radiation pressure constrains the maximum force F , while the optical wavelength sets the minimum length scale over which a propagating optical trapping fields can significantly change. The main text presents a simple scenario where a beamsplitter is trapped by counter-propagating fields. Displacement of the beamsplitter shifts the relative phases of these fields such that the output intensities are modulated by interference. This can be stably trapped since beamsplitter displacements lead to a relative phase shift on the incident fields. In the limit of normally incident light on a 50/50 beamsplitter, this relative phase shift is given by 4πnmx/λ, with x the displacement around the stable trapping point and λ the vacuum wavelength. The resulting mean power imbalance of the two output fields is then given by 〈∆P 〉 = P sin(4πnmx/λ), (1)