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Chapter 11: Fraunhofer DiffractionChapter 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
…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 D
diffraction shows how the size/shape of the slits determines the details of the fringe pattern
Josepf von Fraunhofer (1787-1826)
- 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 D
Huygens’ principleevery point on a wavefront may be regarded as a secondary source of wavelets
planar wavefront:
cDt
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
)( tkriLP e
rdsEEd
for each interval ds:
Let r = r0 for wave from center of slit (s=0). Then:
)(
0
0 trkiLP e
rdsEEd D
D
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
)(
0
0sin tkriL
P erbEE
where b is the slit width
and sin21 kb
0E
2
22
0
020
0 sin22
rbEcEcI L
Irradiance:
0I
20 sincII
After integrating:
Recall the sinc function
sinsinc
1 for = 0
zeroes occur when sin = 0
i.e. when mkb sin21
where m = ±1, ±2, ...
Recall the sinc function
sinsinc
0sincossincossin2
dd
maxima/minima when
tan
cossin
Diffraction from a single slit 2
0 sincII
b 2
D
Central maximum:
image of slit
angular width
hence as slit narrows, central maximum spreads
Beam spreadingangular spread of central maximum independent of distance
Aperture dimensions determine pattern
Aperture dimensions determine pattern
220 sincsincII
sin2kb
sin2ka
where
Aperture shape determines pattern
2
10
2
JII
Irradiance for a circular aperture
J1(): 1st order Bessel function
sin21 kDwhere
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 growsD
22.12/1 D
How else can we obstruct a wavefront?
Any obstacle that produces local amplitude/phase variations create patterns in transmitted light
Diffractive optical elements (DOEs)
Diffractive optical elements (DOEs)
Phase plateschange the spatial profile of the light
Demo
ResolutionSharpness of images limited by diffraction
Inevitable blur restricts resolution
Resolutionmeasured from a ground-based telescope, 1978
PlutoCharon
Resolution
http://apod.nasa.gov/apod/ap060624.html
measured from the Hubble Space Telescope, 2005
Rayleigh’s criterionfor just-resolvable images
D
22.1min D where D is the diameter
of the lens
Imaging system (microscope)
DD
ffx 22.1minmin
- where D is the diameter and f is the focal length of the lens
- numerical aperture D/f (typical value 1.2)
minx
Test it yourself!visual acuity
Test it yourself!
Double-slit diffractionconsidering the slit width and separation
Double-slit diffraction
220 cossinc4II
sin21 kb
sin21 ka
single-slitdiffraction
double-slitinterference
Double-slit diffraction
220 cossinc4II
Double-slit diffraction
Multiple-slit diffraction
22
0 )sin(sinsin
NIIP
Double-slit diffraction
2
2
0 cossin4
IIP
single slitdiffraction
multiple beaminterference
single slitdiffraction
two beaminterference
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
ExercisesYou 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