Chapter 11: Fraunhofer Diffraction

Diffraction is… Diffraction is… interference on the edge -a consequence of the wave nature of light -an interference effect -any deviation from geometrical optics resulting from obstruction of the wavefront

…on the edge of sea

…on the edge of night

…on the edge of dawn

…in the skies

…in the heavens

…on the edge of the shadows

With and without diffraction

The double-slit experiment interference explains the fringes - narrow slits or tiny holes -separation is the key parameter -calculate optical path difference  diffraction shows how the size/shape of the slits determines the details of the fringe pattern

Josepf von Fraunhofer ( )

-far-field -plane wavefronts at aperture and obserservation -moving the screen changes size but not shape of diffraction pattern Fraunhofer diffraction Next week: Fresnel (near-field) diffraction

Diffraction from a single slit slit  rectangular aperture, length >> width

Diffraction from a single slit  plane waves in - consider superposition of segments of the wavefront arriving at point P - note optical path length differences 

Huygens’ principle every point on a wavefront may be regarded as a secondary source of wavelets planar wavefront: ctct curved wavefront: In geometrical optics, this region should be dark (rectilinear propagation). Ignore the peripheral and back propagating parts! obstructed wavefront: Not any more!!

Diffraction from a single slit for each interval ds: Let r = r 0 for wave from center of slit (s=0). Then: where  is the difference in path length. -negligible in amplitude factor -important in phase factor E L (field strength) constant for each ds Get total electric field at P by integrating over width of the slit

Diffraction from a single slit where b is the slit width and Irradiance: After integrating:

Recall the sinc function 1 for  = 0 zeroes occur when sin  = 0 i.e. when where m = ±1, ±2,...

Recall the sinc function maxima/minima when

Diffraction from a single slit Central maximum: image of slit angular width hence as slit narrows, central maximum spreads

Beam spreading angular spread of central maximum independent of distance

Aperture dimensions determine pattern

where

Aperture shape determines pattern

Irradiance for a circular aperture J 1 (  ) : 1 st order Bessel function where and D is the diameter Friedrich Bessel (1784 – 1846)

Irradiance for a circular aperture Central maximum: Airy disk circle of light; “image” of aperture angular radius hence as aperture closes, disk grows

How else can we obstruct a wavefront? Any obstacle that produces local amplitude/phase variations create patterns in transmitted light

Diffractive optical elements (DOEs)

Phase plates change the spatial profile of the light

Demo

Resolution Sharpness of images limited by diffraction Inevitable blur restricts resolution

Resolution measured from a ground-based telescope, 1978 Pluto Charon

Resolution measured from the Hubble Space Telescope, 2005

Rayleigh’s criterion for just-resolvable images where D is the diameter of the lens

Imaging system (microscope) - where D is the diameter and f is the focal length of the lens - numerical aperture D/f (typical value 1.2)

Test it yourself! visual acuity

Test it yourself!

Double-slit diffraction considering the slit width and separation

Double-slit diffraction single-slit diffraction double-slit interference

Double-slit diffraction

Multiple-slit diffraction Double-slit diffraction single slit diffraction multiple beam interference single slit diffraction two beam interference

If the spatial coherence length is less than the slit separation, then the relative phase of the light transmitted through each slit will vary randomly, washing out the fine- scale fringes, and a one-slit pattern will be observed. Fraunhofer diffraction patterns Good spatial coherence Poor spatial coherence Importance of spatial coherence Max

Imagine using a beam so weak that only one photon passes through the screen at a time. In this case, the photon would seem to pass through only one slit at a time, yielding a one-slit pattern. Which pattern occurs? Possible Fraunhofer diffraction patterns Each photon passes through only one slit Each photon passes through both slits The double slit and quantum mechanics

Each individual photon goes through both slits! Dimming the incident light: The double slit and quantum mechanics

How can a particle go through both slits? “Nobody knows, and it’s best if you try not to think about it.” Richard Feynman

Exercises You are encouraged to solve all problems in the textbook (Pedrotti 3 ). The following may be covered in the werkcollege on 12 October 2011: Chapter 11: 1, 3, 4, 10, 12, 13, 22, 27