  Following the familiar analogy between the optical paraxial wave equation and
the Schr\"odinger equation, we derive the optimal, real-valued wave function
for focusing in one and two space dimensions without the use of any phase
component. We compare and contrast the focusing parameters of the optimal waves
with those of other diffractive focusing approaches, such as Fresnel zones.
Moreover, we experimentally demonstrate these focusing properties on optical
beams using both reflective and transmissive liquid crystal devices. Our
results provide an alternative direction for focusing waves where phase
elements are challenging to implement, such as for X-rays, THz radiation, and
electron beams.
