  Interferometry can completely redirect light, providing the potential for
strong and controllable optical forces. However, small particles do not
naturally act like interferometric beamsplitters, and the optical scattering
from them is not generally thought to allow efficient interference. Instead,
optical trapping is typically achieved via deflection of the incident field.
Here we show that a suitably structured incident field can achieve
beamsplitter-like interactions with scattering particles. The resulting trap
offers order-of-magnitude higher stiffness than the usual Gaussian trap in one
axis, even when constrained to phase-only structuring. We demonstrate trapping
of 3.5 to 10.0~$\mu$m silica spheres, achieving stiffness up to 27.5$\pm$4.1
times higher than is possible using Gaussian traps, and two orders of magnitude
higher measurement signal-to-noise ratio. These results are highly relevant to
many applications, including cellular manipulation, fluid dynamics,
micro-robotics, and tests of fundamental physics.
