  Experiments involving optical traps often require careful control of the ac
Stark shifts induced by strong confining light fields. By carefully balancing
light shifts between two atomic states of interest, optical traps at the magic
wavelength have been especially effective at suppressing deleterious effects
stemming from such shifts. Highlighting the power of this technique, optical
clocks today exploit Lamb-Dicke confinement in magic-wavelength optical traps,
in some cases realizing shift cancellation at the ten parts per billion level.
Theory and empirical measurements can be used at varying levels of precision to
determine the magic wavelength where shift cancellation occurs. However, lasers
exhibit background spectra from amplified spontaneous emission or other lasing
modes which can easily contaminate measurement of the magic wavelength and its
reproducibility in other experiments or conditions. Indeed, residual light
shifts from laser background have plagued optical lattice clock measurements
for years. In this work, we develop a simple theoretical model allowing
prediction of light shifts from measured background spectra. We demonstrate
good agreement between this model and measurements of the background light
shift from an amplified diode laser in an Yb optical lattice clock.
Additionally, we model and experimentally characterize the filtering effect of
a volume Bragg grating bandpass filter, demonstrating that application of the
filter can reduce background light shifts from amplified spontaneous emission
well below the $10^{-18}$ fractional clock frequency level. This demonstration
is corroborated by direct clock comparisons between a filtered amplified diode
laser and a filtered titanium:sapphire laser.
