  There has been substantial interest in miniaturizing optical systems by flat
optics. However, one essential optical component, free space, fundamentally
cannot be substituted with conventional local flat optics with space-dependent
transfer functions, since the transfer function of free space is
momentum-dependent instead. Overcoming this difficulty is important to achieve
utmost miniaturization of optical systems. In this work, we show that free
space can be substituted with nonlocal flat optics operating directly in the
momentum domain. We derive the general criteria for an optical device to
replace free space, and provide a concrete design of a photonic crystal slab
device. Such a device can substitute much thicker free space with a compression
ratio as high as 144. Our work paves the way for utmost miniaturization of
optical systems using combination of local and nonlocal flat optics.
