  We develop and demonstrate a spectroscopic method for Rydberg-Rydberg
transitions using a phase-controlled and -modulated, standing-wave laser field
focused on a cloud of cold $^{85}$Rb Rydberg atoms. The method is based on the
ponderomotive (${\bf{A}}^2$) interaction of the Rydberg electron, which has
less-restrictive selection rules than electric-dipole couplings, allowing us to
probe both $nS_{1/2}\rightarrow nP_{1/2}$ and $nS_{1/2}\rightarrow
(n+1)S_{1/2}$ transitions in first-order. Without any need to increase laser
power, third and fourth-order sub-harmonic drives are employed to access
Rydberg transitions in the 40 to 70 GHz frequency range using widely-available
optical phase modulators in the Ku-band (12 to 18 GHz). Measurements agree well
with simulations based on the model we develop. The spectra have prominent
Doppler-free, Fourier-limited components. The method paves the way for optical
Doppler-free high-precision spectroscopy of Rydberg-Rydberg transitions and for
spatially-selective qubit manipulation with $\mu$m-scale resolution in
Rydberg-based simulators and quantum computers.
