Optimal resolution with extreme depth of focus

When the focal depth is required to be much larger than the wavelength, λ, the effective NA of the beam with optimal resolution is much less than unity. An aperture that is much larger than this beam's footprint is then of no consequence. Such beams that maximize the mean encircled energy fraction within a cylindrical focal region are shown to depend on only a single parameter, ω, that is proportional to the ratio of the square of the cylinder's radius to the product of its length and λ. A linear combination of Hermite– or Laguerre–Gaussian modes is used to represent these fields in two and three dimensions, respectively. For small ω, the results are compared both to asymptotic expansions and to optimal Gaussian and Bessel–Gauss beams.