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