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We show quantitatively the influence of the dipole density on the resolving capability of a negativerefractive-index slab. A numerical solution for scatter with a model that represents a locally periodic array of electric and magnetic dipoles provides a means for evaluating the influence of periodicity. A Fourier transform of the spatial field solution gives access to the plane wave transfer function in relation to the degree of granularity. Throughout, a quasistatic homogenization procedure is used to ensure that correctly matched effective medium parameters are achieved. The results provide a measure of the fabrication technology necessary to achieve substantially subwavelength resolution.A negative-refractive-index slab provides for evanescent field amplification and subwavelength image resolution ͓1͔. An experiment with a material constructed from an array of electric and magnetic dipoles formed with metallic elements implied a negative refractive index ͓2͔, and there has been enormous subsequent interest. While a perfect lens is impossible with passive material, a lens having substantially improved resolution may be possible ͓3͔, limited by loss ͓4͔ and material mismatch ͓5,6͔. It is clear that any form of material mismatch, including that from the microscopic structure of the synthesized material and its macroscopic geometry, will present some form of limitation on the possible subwavelength resolution. The use of conducting wires and split ring resonators ͓2͔ to couple to the electric and magnetic field, respectively ͑postresonance frequency operation can give rise to a negative index͒, or other such structures defined by lithographic means, place a technology limit on the density of these dipoles with respect to the wavelength. At optical frequencies, both the maximum achievable density with available solid state processing techniques and the losses from use of metals pose practical limits on performance. The properties of such metamaterials have received considerable attention ͑see, for example, ͓7-9͔͒.We demonstrate the influence of the lattice geometry or the degree of granularity on the image resolution of a negative-index lens. This provides a quantitative measure of the necessary dipole density for achieving substantially subwavelength resolution. Our results thus place experimental programs to achieve enhanced resolution at optical frequencies in perspective.A simple measure of the influence of granularity is the finite-difference solution of Maxwell's equations, where a pointwise representation is used. A numerical dispersion relation can be developed for the discrete problem as ͓10͔

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