  We present a theory of the frequency comb generation in the high-Q ring
microresonators with quadratic nonlinearity and normal dispersion and
demonstrate that the naturally large difference of the repetition rates at the
fundamental and 2nd harmonic frequencies supports a family of the bright
soliton frequency combs providing the parametric gain is moderated by tuning
the index-matching parameter to exceed the repetition rate difference by a
significant factor. This factor equals the sideband number associated with the
high-order phase-matched sum-frequency process. The theoretical framework,
i.e., the dressed-resonator method, to study the frequency conversion and comb
generation is formulated by including the sum-frequency nonlinearity into the
definition of the resonator spectrum. The Rabi splitting of the dressed
frequencies leads to the four distinct parametric down-conversion conditions
(signal-idler-pump photon energy conservation laws). The parametric instability
tongues associated with the generation of the sparse, i.e.,
Turing-pattern-like, frequency combs with varying repetition rates are analysed
in details. The sum-frequency matched sideband exhibits the optical Pockels
nonlinearity and strongly modified dispersion, which limit the soliton
bandwidth and also play a distinct role in the Turing comb generation. Our
methodology and data highlight the analogy between the driven multimode
resonators and the photon-atom interaction.
