  We experimentally realize a great precision enhancement in the small tilt
measurement by using a Sagnac interferometer and balanced homodyne detection
(BHD) of high-order optical modes, together with the weak value amplification
(WVA) technique. Smaller minimum measurable tilt (MMT) and higher
signal-to-noise ratio (SNR) can be obtained by using BHD, compared with the
split detection (SD). The precision of 3.8 nrad can be obtained under our
present experimental condition. It is shown that combining WVA technique and
BHD can strengthen each other's advantages and can behave better for some
special application scenarios, such as extremely weak output, wider measurement
bandwidth, etc. Moreover, the precision can be further enhanced by experimental
parameter optimization.
