  We investigate experiments that are sensitive to the scalar and parity-odd
coefficients for Lorentz violation in the photon sector of the Standard Model
Extension (SME). We show that of the classic tests of special relativity,
Ives-Stilwell (IS) experiments are sensitive to the scalar coefficient, but at
only parts in 10^5 for the state-of-the-art experiment. We then propose
asymmetric Mach-Zehnder interferometers with different electromagnetic
properties in the two arms, including recycling techniques based on travelling
wave resonators to improve the sensitivity. With present technology we estimate
that the scalar and parity odd coefficients may be measured at sensitivity
better than parts in 10^11 and 10^15 respectively.
