From (6), the transmitted wave vector k t has magnitude n 2 k 0. Hence, from the geometry, k t = n 2 k 0 ( i sin ⁡ ⁡ θ θ t + j cos ⁡ ⁡ θ θ t ) = k 0 ( i n 1 sin ⁡ ⁡ θ θ i + j n 2 cos ⁡ ⁡ θ θ t ) , where the last step uses Snell's law. Taking the dot product with the position vector, we get k t ⋅ ⋅ r = k 0 ( n 1 x sin ⁡ ⁡ θ θ i + n 2 y cos ⁡ ⁡ θ θ t ) , so that Eq. (7) becomes