Light as a Wave — Question 3

Question 3

A phase ramp across an aperture steers the emerging wavefront

A thin transparent plate in air has thickness $t(y)=t_0+ay$, index $n=1.50$, and slope $a=0.020$. A normally incident plane wave acquires transmission phase $\Phi(y)=k_0(n-1)t(y)$. The outgoing field has phase $k_xx+k_yy-\omega t$. Neglect reflection and use the local phase-screen model.

1. Determine the outgoing transverse wave-vector component and angle from the positive x axis.

2. Find the transverse distance over which the plate contributes one full phase cycle for $\lambda_0=500\,\mathrm{nm}$.

Question 3 - Solution

A phase ramp across an aperture steers the emerging wavefront

(a) Method and calculation

$$k_y=\frac{d\Phi}{dy}=k_0(n-1)a,\qquad k_y=k_0\sin\theta.$$$$\boxed{\sin\theta=0.010,\quad\theta=0.5730^\circ}.$$

(b) Interpretation and further calculation

$$\Phi(y+\Lambda_y)-\Phi(y)=k_0(n-1)a\Lambda_y=2\pi.$$$$\boxed{\Lambda_y=\frac{\lambda_0}{(n-1)a}=50.0\,\mu\mathrm m}.$$

Physical check

The positive thickness gradient produces positive transverse phase gradient under the stated phase convention.