Exact solutions for nondiffracting beams . I . The scalar theory

We present exact, nonsingular solutions of the scalar-wave equation for beams that are nondiffracting. This means that the intensity pattern in a transverse plane is unaltered by propagating in free space. These beams can have extremely narrow intensity profiles with effective widths as small as several wavelengths and yet possess an infinite depth of field. We further show (by using numerical simulations based on scalar diffraction theory) that physically realizable finite-aperture approximations to the exact solutions can also possess an extremely large depth of field. Any field of wavelength X initially confined to a finite area of radius r in a transverse plane will be subject to diffractive spreading as it propagates outward from that plane in free space. The characteristic distance beyond which diffractive spreading becomes increasingly noticeable is r 2 /X, the Ray-leigh range. For this reason it is commonly thought that any beamlike field (i.e., one whose intensity is maximal along the axis of propagation and that tends to zero with an increasing transverse coordinate) must eventually undergo diffractive spreading as it propagates. This is certainly true, for example , of Gaussian beams: a Gaussian beam having a spot size r diverges at an angle proportional to X/r at distances z >> r 2 / X from the beam waist. 1 We present here free-space, beamlike, exact solutions of the wave equation that are not subject to transverse spreading (diffraction) after the plane where the beam is formed. These solutions are nonsingular and, like plane waves, have finite energy density rather than finite energy. Most importantly , they can have sharply defined intensity distributions as small as several wavelengths in every transverse plane, independent of propagation distance. We begin with the wave equation for free space: (V2-12A-2) E(r,t) 0. (1) One can easily verify that an exact solution of Eq. (1) for scalar fields propagating into the source-free region z 2 0 is E(x, y, z 2 0, t) r21 = exp[i(3z-_ t)] J A(MO) exp[ia(x cos 0 + y sin o)]do, (2) where i 2 + a 2 = (W/C) 2 and A(0) is an arbitrary complex function of 4. When f3 is real, Eq. (2) represents a class of fields that are nondiffracting in the sense that the time-averaged intensity profile at z = 0, I(x, y, z 2 0) =1/2 OEr, t) 12 =I(x, y, z = 0), (3) is exactly reproduced for all …