  This paper presents the experimental realization of an ultrafast electron
microscope operating at a repetition rate of 75 MHz based on a single compact
resonant microwave cavity operating in dual mode. This elliptical cavity
supports two orthogonal TM$_{110}$ modes with different resonance frequencies
that are driven independently. The microwave signals used to drive the two
cavity modes are generated from higher harmonics of the same Ti:Sapphire laser
oscillator. Therefore the modes are accurately phase-locked, resulting in
periodic transverse deflection of electrons described by a Lissajous pattern.
By sending the periodically deflected beam through an aperture, ultrashort
electron pulses are created at a repetition rate of 75 MHz. Electron pulses
with $\tau=(750\pm10)$ fs pulse duration are created with only $(2.4\pm0.1)$ W
of microwave input power; with normalized rms emittances of
$\epsilon_{n,x}=(2.1\pm0.2)$ pm rad and $\epsilon_{n,y}=(1.3\pm0.2)$ pm rad for
a peak current of $I_p=(0.4\pm0.1)$ nA. This corresponds to an rms normalized
peak brightness of $B_{np,\textrm{rms}}=(7\pm1)\times10^6$ A/m$^2$ sr V, equal
to previous measurements for the continuous beam. In addition, the FWHM energy
spread of $\Delta U = (0.90\pm0.05)$ eV is also unaffected by the dual mode
cavity. This allows for ultrafast pump-probe experiments at the same spatial
resolution of the original TEM in which a 75 MHz Ti:Sapphire oscillator can be
used for exciting the sample. Moreover, the dual mode cavity can be used as a
streak camera or time-of-flight EELS detector with a dynamic range $>10^4$.
