# fraunhofer and fresnel diffraction - uw courses incident wave diffracted wave diffraction...

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Diffraction

Fraunhofer and Fresnel Diffraction

Reports due this week

Two light sources have a frequency difference of 4000 GHz. Can

we resolve these two light sources?

A sodium atom has a doublet emission with a frequency difference

of 0.1 GHz. Can we resolve the fine structure?

A Fabry-Perot wave meter. Free spectral range: 3000 GHz.

Finesse 100,000.

Hecht, Pg 312, Fig. 7.34

The square pulse and its transform

Fourier Integrals and Fourier Transforms

0

sin( / 2)( )

/ 2

kLA k E L

kL

The sinc function.

Inverse relationship between widths

FT in single-slit Fraunhofer diffraction

0 0

1( ) ( )cos( ) ( )sin( )f x A k kx dk B k kx dk

( ) ( )cos( )A k f x kx dx

( ) ( )sin( )B k f x kx dx

aperture

incident wave

diffracted wave

Diffraction

Huygens-Fresnel Principle:Every unobstructed point of a wavefront acts

as a source of spherical secondary wavelets.

The field at any point is the superposition of all

of the secondary wavelets.

A

B

P Manifestation of the wave

nature of light light bends

around corners.

When l ~ smallest dim

max AP BP AB

ABl Wave spreads in large angle

ABl Multitude of wavelets

emitted from the aperture

interfere

Analytical Approach

(of aperture)

ikr ikr

o o

S S

e eE E S E dS

r r

Individual Huygens wavelet generated from surface S

Superposition of Huygens wavelets at r.

Time dependence is same for all (same frequency)

Observer

(Detector)

Source

Co-ordinate

System

Eo

Analytical Approach - II

'z z d

The distance between source

plane and detector plane.

Criterion for Fraunhofer

Diffraction

Can ignore quadratic terms if

2 2

12

x yk

d

2wd

l

w is largest dim

of aperture

We assume the incoming wave is homogeneous

Fraunhofer Diffraction: Far field diffraction

1-D Fraunhofer Diffraction Single Slit

Huygens Recipe

Fraunhofer limit

In general, related to

Fourier Transform

of aperture function

SINC sq.

function

Hecht page 452-469

Can determine width w

from minima location

Single Slit Diffraction Pattern

Smaller w, stronger diffraction

Fraunhofer Diffraction Rectangular Aperture

( ' ')ik xx yy

o dE

E e dsd

Hecht page 452-469

Fraunhofer Diffraction Rectangular Aperture

2

1, ( 1/ 2)

(0)

mi

i

In

I

Hecht page 452-469

Fresnel Diffraction

Occurs when w2/dl >> 1

Near Field Effect

Wavefront curvature

is important.

Fresnel Diffraction

rs rd

Fresnel Diffraction

Occurs when w2/dl >> 1

Near Field Effect

Wavefront curvature

is important.

(keep quadratic terms)

rsrd

Fresnel Diffraction

Occurs when w2/dl >> 1

Near Field Effect

Wavefront curvature

is important.

Either source or

Detector in near field

Fresnel Integrals

C

S

I=|P2P1|2

The Cornu Spiral

C

S

I=|P2P1|2

Once width is fixed,

the length of the

curve is fixed

Fresnel Diffraction

All the equations are derived for X = 0.

P

-w/2

w/2

u1

u2P

-w/2+d

w/2+d

u1

u2

P

The Cornu Spiral

Fraunhofer

Diffraction

on Cornu

Spiral

P

-0.5

0.5

u1

u2

u=0

Fraunhofer

Diffraction

on Cornu

Spiral

P

-0.5

0.5

u1 =0

u2 = 1

u=-0.5

Fraunhofer

Diffraction

on Cornu

Spiral

P

-0.5

0.5

u1 =0.5

u2 =1.5

u=-1

Detector is at the u=0.

Start with slit width =1. We continuously increase the slit width.

Detector is at the u=0.

Start with slit width =1. We continuously increase the slit width.

Detector is at the u=0.

Start with normalized slit width =1. We continuously increase the slit

width.

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