Vectorial measurement of the 2nd harmonic response of an FBAR resonator

We have measured the phase and magnitude (“vector”) response of the 2
^nd
harmonic emissions of a thin film bulk acoustic (FBAR) resonator. The phase measurement is the difference in phase between the emitted 2
^nd
harmonic waveform and that of the incident waveform. Although the magnitude has been measured previously by various investigators [1], [2], we are not aware of a phase measurement prior to this work. The phase measurement presents a challenge because it requires an elaborate measurement setup with a sophisticated calibration procedure. Using Agilent's non-linear vector network analyzer (NVNA) we have measured the standard S
₁₁
response of a resonator as well as the vector response of its 2
^nd
harmonic emissions. We have observed that: (1) phase vs. frequency plots of the 2
^nd
harmonic have the same shape as the phase of S
₁₁
and (2) the magnitude vs. frequency plots of the 2
^nd
harmonic are proportional to the group delay of S
₁₁
.