Let u(x, y, z, t ) be a monochromatic, Gaussian beam solution of the scalar wave equation

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Let u(x, y, z, t ) be a monochromatic, Gaussian beam solution of the scalar wave equation in the paraxial approximation. The beam propagates in the z-direction. Use a suitably chosen Lorenz gauge vector potential to confirm the statement made in the text that the electric and magnetic fields of this beam in the same approximation can be written 

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Confirm also that ∇ · E = 0 and ∇ · B = 0 within the paraxial approximation.

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