Fourier optics

Remarks

Combining these rules in the paraxial approximation gives the Fourier transforming property of a lens. Let $u_0$ be the field in the front focal plane, and let the lens aperture be large. Then the field in the back focal plane is $$u_f(x, y) = \frac{1}{i \lambda f} \, \widehat{u_0}\left( \frac{x}{\lambda f}, \frac{y}{\lambda f} \right),$$ up to a constant phase factor. Example: a 4f system, two lenses of focal length $f$ separated by $2f$, maps $u_0(x, y)$ to $u_0(-x, -y)$. A mask $M$ in the common focal plane turns it into a filter that multiplies the spectrum by $M(\lambda f f_x, \lambda f f_y)$. In coherent imaging, the image is the convolution of the ideal image with the amplitude point spread function, whose Fourier transform is a scaled copy of the pupil function. The model ignores polarisation and treats elements as infinitely thin; thick or strongly scattering elements fall outside it.