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 1 3 π 2 n 4 λ − 4 Q − 2 L 4 , where n , λ , and Q are its refractive index, eigenmode wavelength, and Q -factor, respectively. For a silica microresonator with Q = 10 8 , this eigenmode has axial speed ∼ 10 − 4 c , where c is the speed of light in vacuum, and its nonuniformity along length L = 100 \unicode{x00B5} m at wavelength λ = 1.5 µ m 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.