A plane wave propagating in the +x-direction with electric field E₀ strikes a thin metal screen at x = 0 and diffracts from a long and narrow horizontal slit (width a) cut out of the screen. The scattering vector k lies in the x-y plane (perpendicular to the long direction of the slit) at an angle φ from the forward direction. Evaluate Smythe’s formula (21.90) using Kirchoff’s approximation and the far-field limit of the free-space Green function in two dimensions, G⁽²⁾₀ (ρ, ρ') = (i/4)H₀⁽¹⁾ (k|ρ − ρ'|), to show that the diffracted electric field is

a (, ) = = -2kx -k x (x x Eo), 2 kp sin(ka sin ) ka sin exp[i (kp/4)]-