As we have only specified the curl of A, we are free to define the divergence, and choose the following: ∇ ∇ ⋅ ⋅ A = − − j ω ω ε ε Φ Φ which is called the Lorenz gauge condition. The previous expression for A now reduces to ∇ ∇ 2 A + k 2 A = − − J which is the vector Helmholtz equation. The solution of this equation for A is A ( r ) = 1 4 π π ∫ ∫ J ( r ′ ′ ) G ( r , r ′ ′ ) d r ′ ′ where G ( r , r ′ ′ ) is the three-dimensional homogeneous Green's function given by G ( r , r ′ ′ ) = e − − j k | r − − r ′ ′ | | r − − r ′ ′ |