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Analytical approach and numerical simulation of hydrodynamic forces in immersion optics

Immersion microscopy optics may include liquid droplets (e.g., water) to control the light pathway and the numerical aperture of an optical system. Changing the distances between the optical system and an object slide for image focusing also changes the shape (especially the diameter) of the droplet and the surface energy, thus leading to forces acting on both optics and object slides. We examine these effects analytically and derive a numerical model using numerical integration of a recursive integral to predict the force resulting from a liquid droplet changing its shape in the system. Our solutions show that an alteration of the distance leads to a time-dependency of the droplet surface, which is reflected in the corresponding surface and meniscus energies. With this, we can calculate the hydrostatic force that pulls both optical surfaces closer to each other and simulate the time-dependent equilibration of the system.