  In this paper, we describe two complementary non-linear spectroscopy methods
which both allow to achieve Doppler-free spectra of atomic gases. First,
saturated absorption spectroscopy is used to investigate the structure of the
$5{\rm S}_{1/2}\to 5{\rm P}_{3/2}$ transition in rubidium. Using a slightly
modified experimental setup, Doppler-free two-photon absorption spectroscopy is
then performed on the $5{\rm S}_{1/2}\to 5{\rm D}_{5/2}$ transition in
rubidium, leading to accurate measurements of the hyperfine structure of the
$5{\rm D}_{5/2}$ energy level. In addition, electric dipole selection rules of
the two-photon transition are investigated, first by modifying the polarization
of the excitation laser, and then by measuring two-photon absorption spectra
when a magnetic field is applied close to the rubidium vapor. All experiments
are performed with the same grating-feedback laser diode, providing an
opportunity to compare different high resolution spectroscopy methods using a
single experimental setup. Such experiments may acquaint students with quantum
mechanics selection rules, atomic spectra and Zeeman effect.
