Fundamental limit of the microresonator field uniformity and slow light enabled angstrom-precise straight-line translation

We determine the fundamental limit of the 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 ${{\frac{1}{3}}}{\pi }^2n^4_r{\lambda }^{-4}Q^{-2}L^4$, where $n_r,\ \lambda $ and $Q$ are its refractive index, the eigenmode wavelength and Q-factor. In the absence of losses ($Q=\infty $), this eigenmode has the amplitude independent of axial coordinate and zero axial speed (i.e., is stopped) within the length $L$. For a silica microresonator with $Q={10}^8$ this eigenmode has the axial speed $\mathrm{\sim}$ 10${}^{-4}$c, where c is the speed of light in vacuum, and its nonuniformity along the length 100 micron at wavelength $\lambda =1.5$ micron is $\mathrm{\sim}$ 10${}^{-7}$. For a realistic fiber with diameter 100 micron and surface roughness 0.2 nm, the smallest eigenmode nonuniformity is $\mathrm{\sim}$ 0.0003. As an application, we consider a bat microresonator evanescently coupled to high Q-factor silica microspheres which serves as a reference supporting the angstrom-precise straight-line translation over the distance $L$ exceeding a hundred microns.