  We demonstrate simultaneous control of both the phase and amplitude of light
using a conjugate gradient minimisation-based hologram calculation technique
and a single phase-only spatial light modulator (SLM). A cost function which
incorporates the inner product of the light field with a chosen target field
within a defined measure region is efficiently minimised to create high
fidelity patterns in the Fourier plane of the SLM. A fidelity of $F=0.999997$
is achieved for a pattern resembling an $LG^{0}_{1}$ mode with a calculated
light-usage efficiency of $41.5\%$. Possible applications of our method in
optical trapping and ultracold atoms are presented and we show uncorrected
experimental realisation of our patterns with $F = 0.97$ and $7.8\%$ light
efficiency.
