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Chapter 11: Fraunhofer Diffraction. Chapter 11: Fraunhofer Diffraction. Diffraction is… . a consequence of the wave nature of light an interference effect any deviation from geometrical optics resulting from obstruction of the wavefront. Diffraction is… interference on the edge.
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Chapter 11: Fraunhofer Diffraction Chapter 11: Fraunhofer Diffraction
Diffraction is… • a consequence of the wave nature of light • an interference effect • any deviation from geometrical optics resulting from obstruction of the wavefront Diffraction is… interference on the edge
The double-slit experiment • interference explains the fringes • -narrow slits or tiny holes • -separation is the key parameter • -calculate optical path difference D • diffraction shows how the size/shape of the slits determines the details of the fringe pattern
Fraunhofer diffraction • far-field • plane wavefronts at aperture and obserservation • moving the screen changes size but not shape of diffraction pattern • 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 D
Huygens’ principle every point on a wavefront may be regarded as a secondary source of wavelets planar wavefront: obstructed wavefront: curved wavefront: cDt In geometrical optics, this region should be dark (rectilinear propagation). Ignore the peripheral and back propagating parts! Not any more!!
Diffraction from a single slit for each interval ds: Let r = r0 for wave from center of slit (s=0). Then: where D is the difference in path length. -negligible in amplitude factor -important in phase factor EL (field strength) constant for each ds Get total electric field at P by integrating over width of the slit
Diffraction from a single slit After integrating: where b is the slit width and Irradiance:
Recall the sinc function 1 for b = 0 zeroes occur when sinb = 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
Irradiance for a circular aperture where and D is the diameter J1(g): 1st order Bessel function 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
Phase plates change the spatial profile of the light
Resolution Sharpness of images limited by diffraction Inevitable blur restricts resolution
Resolution measured from a ground-based telescope, 1978 Charon Pluto
Resolution measured from the Hubble Space Telescope, 2005 http://apod.nasa.gov/apod/ap060624.html
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
Double-slit diffraction considering the slit width and separation
Double-slit diffraction single-slit diffraction double-slitinterference
Double-slit diffraction single slit diffraction two beam interference Multiple-slit diffraction single slit diffraction multiple beam interference
Max Importance of spatial coherence 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
The double slit and quantum mechanics 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 Dimming the incident light: Each individual photon goes through both slits!
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 (Pedrotti3). The following may be covered in the werkcollege on 12 October 2011: Chapter 11: 1, 3, 4, 10, 12, 13, 22, 27
