Precise frequency measurements of atomic transitions using a Rb-stabilized resonator

We demonstrate a technique for frequency measurements of atomic transitions with a precision of 30 kHz. The frequency is measured using a ring-cavity resonator whose length is calibrated against a reference laser locked to the $D_2$ line of $^{87}$Rb, the frequency of which is known to 10 kHz. We have used this to measure the hyperfine structure in the $5P_{3/2}$ state of $^{85}$Rb. We obtain precise values for the hyperfine constants $A=25.041(6)$ MHz and $B=26.013(25)$ MHz, and a value of 77.992(20) MHz for the isotope shift in the $D_2$ line.