Show that Fraunhofer diffraction patterns have a center of...

Show that Fraunhofer diffraction patterns have a center of symmetry [i.e., $I(Y, Z)=I(-Y,-Z)]$, regardless of the configuration of the aperture, as long as there are no phase variations in the field over the region of the hole. Begin with Eq. (10.41). We'll see later (Chapter 11) that this restriction is equivalent to saying that the aperture function is real.

Key Concepts

Fraunhofer Diffraction

Fraunhofer diffraction describes the far-field diffraction pattern observed when the light source and observation screen are effectively at infinite distances from the aperture, or when a lens is used to simulate such conditions. This regime allows the use of Fourier analysis to relate the aperture function to the diffraction pattern, simplifying the treatment of diffraction phenomena.

Aperture Function

The aperture function represents how light is transmitted through an opening and typically includes both amplitude and phase information. Its spatial distribution directly influences the resulting diffraction pattern, as the Fourier transform of the aperture function gives the field distribution in the Fraunhofer region.

Real-valued Aperture Function

A real-valued aperture function means that there are no phase variations across the aperture, which implies that the function has no imaginary part. This condition is significant because it leads to specific symmetry properties in the Fourier transform, ensuring that the diffraction pattern exhibits central symmetry.

Fourier Transform and Conjugate Symmetry

The Fourier transform of a real-valued function is conjugate symmetric, meaning that its complex conjugate is mirrored with respect to the origin. As a result, the intensity, which is calculated from the magnitude squared of the Fourier transform, is symmetric about the center. This property directly explains why the Fraunhofer diffraction pattern has a center of symmetry, with I(Y, Z) equaling I(-Y, -Z).

Transcript

Please give Ace some feedback