  The unique nonreciprocal responses of superconductors, which stem from the
Cooper pairs' quantum condensation, have been attracting attention. Recently,
theories of the second-order nonlinear response in noncentrosymmetric
superconductors were formulated based on the Bogoliubov-de Gennes theory. In
this paper, we study the mechanism and condition for second-order optical
responses of time-reversal symmetric superconductors. The numerical results
show the characteristic photocurrent and second harmonic generation in the
superconducting state. However, the superconductivity-induced nonlinear optical
responses disappear under some conditions on pair potential. We show that the
coexistence of intraband and interband pairing is necessary for the
second-order superconducting optical responses. In addition, the
superconducting Berry curvature factor, which is related to a component of
Berry curvature in the superconducting state, is essential for the nonlinear
responses. Thus, we derived the microscopic conditions where the
superconducting nonlinear response appears.
