(Solved): 4. [4 marks] Given the Gaussian solution to the paraxial wave equation is (in complex form): WO 1 ...

4. [4 marks] Given the Gaussian solution to the paraxial wave equation is (in complex form): WO 1 U (x, y, z) = U? — -exp-(x² + y²). 2iz 2iz Wo + + k k By considering exp[] term in this expression, show that defining the Rayleigh range as zo = kwo²/2 leads to a Gaussian shaped beam of width w given by *-*+-6] w² = w 1+