Diffraction and the Wave Theory of Light — Question 7

Question 7

A lens cancels the quadratic phase at its focal plane

A thin lens multiplies a field by $\exp[-iku^2/(2f)]$. Propagation a distance L contributes the Fresnel factor $\exp[ik(x-u)^2/(2L)]$. Assume one transverse coordinate, paraxial propagation, and a lens large enough to avoid clipping.

1. Expand the combined exponent and determine the distance at which its quadratic dependence on u vanishes.

2. At that distance, identify the spatial frequency sampled by detector coordinate x and evaluate it for $f=0.25\,\mathrm m$, $x=1.0\,\mathrm{mm}$, $\lambda=500\,\mathrm{nm}$.

Question 7 - Solution

A lens cancels the quadratic phase at its focal plane

(a) Method and calculation

$$\Phi=\frac{kx^2}{2L}-\frac{kxu}{L}+\frac{ku^2}{2}\left(\frac1L-\frac1f\right).$$$$\boxed{L=f}\Longrightarrow U(x,f)\propto e^{ikx^2/(2f)}\int A(u)e^{-ikxu/f}du.$$

(b) Interpretation and further calculation

$$\boxed{q=kx/f=2\pi x/(\lambda f)}.$$$$\boxed{q=16000\pi\,\mathrm{m^{-1}}=5.02655\times10^4\,\mathrm{m^{-1}}}.$$

Physical check

The remaining quadratic factor depends only on detector position, so it changes phase but not the focal-plane intensity.