Fraunhofer Diffraction

Fraunhofer Diffraction • Last Lecture • Dichroic Materials • Polarization by Scattering • Polarization by Reflection from Dielectric Surfaces • Birefringent Materials • Double Refraction • The Pockel’s Cell • This Lecture • Fraunhofer versus Fresnel Diffraction • Diffraction from a Single Slit • Beam Spreading • Rectangular and Circular Apertures

Optical Diffraction •Diffraction is any deviation from geometric optics that results from the obstruction of a light wave, such as sending a laser beam through an aperture to reduce the beam size. Diffraction results from the interaction of light waves with the edges of objects. • The edges of optical images are blurred by diffraction, and this represents a fundamental limitation on the resolution of an optical imaging system. • There is no physical difference between the phenomena of interference and diffraction, both result from the superposition of light waves. Diffraction results from the superposition of many light waves, interference results from the interference of a few light waves.

Optical Diffraction Hecht, Optics, Chapter 10

Fraunhofer versus Fresnel Diffraction •The passage of light through an aperture or slit and the resulting diffraction patterns can be analyzed using either Fraunhofer or Fresnel diffraction theory. In Fraunhofer (far-field) diffraction theory the source is far enough from the aperture that the wavefronts are planar at the aperture, and the image plane is far enough from the aperture that the wavefronts are planar at the image plane. • If the curvature of the optical waves must be taken into account at the aperture or image plane, then we must use Fresnel (near-field) diffraction theory. • The Huygens-Fresnel principle is used in diffraction theory, in that every point of a given wavefront of light can be considered as a source of secondary wavelets. To analyze two-slit interference, we assumed that the slits were point sources. To analyze diffraction, we need to consider the generation of wavelets at different spatial positions within the slit.

Fraunhofer versus Fresnel Diffraction •We can move to the Fraunhofer regime by placing lenses on each side of the aperture. Lens 1 is place a focal length away from the point source so that the wavefronts are planar at the aperture. The observation screen is in the focal plane of lens 2 so that the diffraction pattern is imaged at infinity.

Fraunhofer Diffraction from a Single Slit •Consider the geometry shown below. Assume that the slit is very long in the direction perpendicular to the page so that we can neglect diffraction effects in the perpendicular direction.

Fraunhofer vs. Fresnel diffraction In Fraunhofer diffraction, both incident and diffracted waves may be considered to be plane (i.e. both S and P are a large distance away) If either S or P are close enough that wavefront curvature is not negligible, then we have Fresnel diffraction S P Hecht 10.2 Hecht 10.3

Fraunhofer Vs. Fresnel Diffraction Now calculate variation in (r+r’) in going from one side of aperture to the other. Call it 

Fraunhofer diffraction limit • If aperture is a square -  X  • The same relation holds in azimuthal plane and 2 ~ measure of the area of the aperture • Then we have the Fraunhofer diffraction if, Fraunhofer or far field limit

Fraunhofer, Fresnel limits • The near field, or Fresnel, limit is

Fraunhofer Diffraction from a Single Slit

Fraunhofer Diffraction from a Single Slit Note: x- and y-axes switched in book, Figs. 16-5a (here) and Fig. 16-1 do not match.

Beam Spreading

Rectangular Apertures x y

Square Apertures

Fraunhofer Diffraction from General Apertures

Fraunhofer Diffraction from Circular Apertures

Fraunhofer Diffraction from Circular Apertures: Bessel Functions

Fraunhofer Diffraction from Circular Apertures: The Airy Pattern

Circular Apertures

Fraunhofer Diffraction