  Muon pairs can be produced in the annihilation of ultra-high energy (UHE, $E
\gtrsim 10^{18} \,\mathrm{eV}$) photons with low energy cosmic background
radiation in the intergalactic space, giving birth to neutrinos. Although the
branching ratio of muon pair production is low, products of other channels,
which are mainly electron/positron pairs, will probably transfer most of their
energies into the new generated UHE photon in the subsequent interaction with
the cosmic background radiation via Compton scattering in deep Klein-Nishina
regime. The regeneration of these new UHE photons then provides a second chance
to produce the muon pairs, enhancing the neutrino flux. We investigate the
neutrino production in the propagation of UHE photons in the intergalactic
space at different redshifts, considering various competing processes such as
pair production, double pair production for UHE photons, and triplet production
and synchrotron radiation for UHE electrons. Regarding the least energetic
outgoing particles as energy loss, we obtain the effective penetration length
of the leading particle, as well as energy loss rate including the neutrino
emission rate in the cascade process. Finally, we find that an extra component
of UHE neutrinos will arise from the propagation of UHE cosmic rays due to the
generated UHE photons and electron/positrons. However, the flux of this
component is quite small, with a flux of at most $10\%$ of that of the
conventional cosmogenic neutrino at a few EeV, in the absence of a strong
intergalactic magnetic field and a strong cosmic radio background. The precise
contribution of extra component depends on several factors, e.g., cosmic radio
background, intergalactic magnetic field, the spectrum of proton, which will be
discussed in this work.
