Fundamental limit of microresonator field uniformity and slow light enabled ultraprecise displacement metrology

We determine the fundamental limit of microresonator field uniformity. It can be achieved in a specially designed microresonator, called a bat microresonator, fabricated at the optical fiber surface. We show that the relative nonuniformity of an eigenmode amplitude along the axial length L of an ideal bat microresonator cannot be smaller than 13π²n⁴λ^-4Q^-2L⁴, where n,λ, and Q are its refractive index, eigenmode wavelength, and Q-factor, respectively. For a silica microresonator with Q=10⁸, this eigenmode has axial speed ∼10^-4c, where c is the speed of light in vacuum, and its nonuniformity along length L=100\unicode{x00B5}m at wavelength λ=1.5µm is ∼10^-7. For a realistic fiber with diameter 100 µm and surface roughness 0.2 nm, the smallest eigenmode nonuniformity is ∼0.0003. As an application, we consider a bat microresonator evanescently coupled to high Q-factor silica microspheres, which serves as a reference supporting ultraprecise straight-line translation.