Random light fields -- commonly known as speckles -- demonstrate Rayleigh intensity statistics and only possess local correlations: which occur within the individual speckle grains. In this work, we develop an experimental method for customizing the intensity probability density function (PDF) of speckle patterns while simultaneously introducing non-local spatial correlations among the speckle grains. The various families of tailored speckle patterns -- created by our method -- can exhibit radically different topologies, statistics, and variable degrees of spatial order. Irrespective of their distinct statistical properties, however, all of these speckles are created by appropriately encoding high-order correlations into the phase front of a monochromatic laser beam with a spatial light modulator. In addition to our experimental demonstration, we explore both the theoretical and practical limitations on the extent to which the intensity PDF and the spatial intensity correlations can be manipulated concurrently in a speckle pattern. This work provides a versatile methodology for creating complex light fields and controlling their statistical properties with varied applications in microscopy, imaging, and optical manipulation.
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