We study spontaneous emission from a rubidium atom into the fundamental and higher-order modes of a vacuum-clad ultrathin optical fiber. We show that the spontaneous emission rate depends on the magnetic sublevel, the type of modes, the orientation of the quantization axis, and the fiber radius. We find that the rate of spontaneous emission into the TE modes is always symmetric with respect to the propagation directions. Directional asymmetry of spontaneous emission into other modes may appear when the quantization axis does not lie in the meridional plane containing the position of the atom. When the fiber radius is in the range from 330 nm to 450 nm, the spontaneous emission into the HE$_{21}$ modes is stronger than into the HE$_{11}$, TE$_{01}$, and TM$_{01}$ modes. At the cutoff for higher-order modes, the rates of spontaneous emission into guided and radiation modes undergo steep variations, which are caused by the changes in the mode structure. We show that the spontaneous emission from the upper level of the cyclic transition into the TM modes is unidirectional when the quantization axis lies at an appropriate azimuthal angle in the fiber transverse plane.
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