Now, since r ′ = x ′ x ^ ^ + y ′ y ^ ^ and r ^ ^ = sin ⁡ ⁡ θ θ cos ⁡ ⁡ ϕ ϕ x ^ ^ + sin ⁡ ⁡ θ θ sin ⁡ ⁡ ϕ ϕ y ^ ^ + cos ⁡ ⁡ θ θ z ^ ^ , the expression for the Fraunhofer region field from a planar aperture now becomes Ψ Ψ ( r ) ∝ ∝ e i k r 4 π π r ∬ ∬ a p e r t u r e E i n c ( x ′ , y ′ ) e − − i k sin ⁡ ⁡ θ θ ( cos ⁡ ⁡ ϕ ϕ x ′ + sin ⁡ ⁡ ϕ ϕ y ′ ) d x ′ d y ′ .