  Manipulating underwater pressure waves is crucial for marine exploration, as
electromagnetic signals are strongly absorbed in water. However, the multi-path
phenomenon complicates the accurate capture of acoustic waves by receivers.
Although graded index lenses, based on metamaterials with smoothly varying
properties, successfully focus pressure waves, they tend to have high
directivity, which hinders practical application.
  This work introduces three 2D acoustic lenses made from a metamaterial
composed of solid inclusions in water. We propose an optimization scheme where
the pressure dynamics is governed by Helmholtz's equation, with control
parameters affecting each lens cell's density and bulk modulus. Through an
appropriate cost function, the optimization encourages a broadband,
low-directivity lens. The large-scale optimization is solved using the
Lagrangian approach, which provides an analytical expression for the cost
gradient. This scheme avoids the need for a separate discretization step,
allowing the design to transition directly from the desired smooth refractive
index to a practical lattice structure. As a result, the optimized lens closely
aligns with real-world behavior.
  The homogenized numerical model is validated against finite elements, which
considers acoustic-elastic coupling at the microstructure level. When
homogenization holds, this approach proves to be an effective design tool for
achieving broadband, low-directivity acoustic lenses.
