Interference

Analytical method

Phasor method

Double and single slit

Diffraction at 2-D apertures

Fraunhofer diffraction

Wave propagation (Huygens & Fresnel)

Fresnel-Kirchhoff diffraction integral

Waves and Diffraction

lecture 5

A diffraction pattern for which the phase

of the light at the observation point is a

linear function of the position for all

points in the diffracting aperture is

Fraunhofer diffraction

Fraunhofer Diffraction

eikr −→ eikr0 · ei(!xx+!yy)

Fraunhofer Diffraction

d

δr = |rmax ! d| " a2

8d# λ/8

Fraunhofer Diffraction

!r = !1 + !2 !a2

8

!1ds

+1dp

"" "/8

Fraunhofer Diffraction

! = (k sin ") · y

y

!

aperture

Fraunhofer Diffraction

! = (k sin ") · y

y

!

aperture

f

Fraunhofer Diffraction

! = (k sin ") · y

y

!

aperture

f

illumination

Diffraction in the image plane

A diffraction pattern formed in the

image plane of an optical system is

Fraunhofer diffraction

Fraunhofer Diffraction

what is being imaged?

Fraunhofer

diffraction:

in the image

plane

Fraunhofer Diffraction

Equivalent

lens system:

Fraunhoferdiffractionindependenton aperture position

Fraunhofer Diffraction

Fraunhofer Diffraction & Resolution

! = 1.22 "/D

! ≥ "

peak width !!" f -number

circular

aperture

Rayleigh

criterion

image plane

Fraunhofer Diffraction

• far-field diffraction

• in the image plane

of optical systems

• resolution limit

R! a2/!

!min ! "/a

Bridging the Gap?

• wave propagation

• diffraction in the near field

• Fresnel-Kirchhoff diffraction integral

Fresnel’s Theory of wave propagation

Wave PropagationHuygens’ wavelet ebcid:com.britannica.oec2.identifier.AssemblyIdentifier?assemblyId=...

1 of 1 23.11.2008 19:56 Uhr

Figure 2: Huygens' wavelets. Originating along the fronts of (A) circular waves and (B) plane waves, wavelets recombine to produce the propagating wave front. (C) The diffraction of sound around a corner

arising from Huygens' wavelets.

Huygens’ wavelet

print articles

Huygen‘s wavelets recombine to produce

the propagating wavefront

Huygens secondary sources on wavefront at -z

radiate to point P on new wavefront at z = 0

Fresnel‘s Theory of Wave Propagation

plane-to-plane

up = !

!"(#in, #out)

u0

reikrdS

up!= u0

Fresnel‘s Theory of Wave Propagation

!r =!

q2 + "2 ! q " "2

2q

Phase difference of !

at edge of 1st HPZ

Fresnel‘s Theory of Wave Propagation

λ

2=

ρ2π

2q

!!,n =!

"q · n

Phase difference of !

at edge of 1st HPZ

Fresnel‘s Theory of Wave Propagation

λ

2=

ρ2π

2q

!!,n =!

"q · n1

3

5

7

2

4

6

Elements of

equal area

Fresnel‘s Theory of Wave Propagation

!A = "(#2n+1 ! #2

n)

δr = rn+1 ! rn

" δA

2πq

sub-division of HPZ into annuli

First Half Period Zone

Fresnel‘s Theory of Wave Propagation

R! = 2i!u0"

phasor addition

n!! ⇒ resultant ! " diameter of 1st HPZ

Fresnel‘s Theory of Wave Propagation

R! = i!u0"!= u0

Fresnel‘s Theory of Wave Propagation

Fresnel-Kirchhoff diffraction integral

up = ! i

λ

!η(θin, θout)

u0

reikrdS

η(θin, θout) =12(cos θin + cos θout)

obliquity factor

eikr −→ eikr0 · ei(!xx+!yy)

Fraunhofer diffraction is a special case

Fresnel ! Fraunhofer Diffraction

eikr −→ eikr0 · ei(!xx+!yy)

z < dR = a2/!

near field far field

to 1st

minimum

z > dR

slit shadow

Fraunhofer

diffraction

to 1st

minimum

Fresnel Zone Plate24.11.08 01:40http://upload.wikimedia.org/wikipedia/commons/9/97/Zone_plate.svg

Seite 1 von 1

mask out every

second HPZ

in every

transparent

zone, the phase

is running from

0 to !

Fresnel Zone Plate24.11.08 01:40http://upload.wikimedia.org/wikipedia/commons/9/97/Zone_plate.svg

Seite 1 von 1

mask out every

second HPZ

in every

transparent

zone, the phase

is running from

0 to !

acting as

a focussing

lens

Fresnel Lens

1. HPZ

2. HPZ

3. HPZ

4. HPZ

5. HPZ

sub-division into HPZ

Fresnel Lens

phase jump by !

from HPZ to HPZ

1. HPZ

2. HPZ

3. HPZ

4. HPZ

5. HPZ

sub-division into HPZ

Fresnel Lens

phase jump by !

from HPZ to HPZ

section of a lens

in every HPZ

nearly perfect

focussing lens

1. HPZ

2. HPZ

3. HPZ

4. HPZ

5. HPZ

• Geometrical optics

" No wave effects

Geometrical imaging

• " Scalar diffraction theory:" Analytical methods

" Phasor methods

• Fresnel-Kirchhoff diffraction integral:" Fraunhofer diffraction as a limiting case

Propagation of plane waves ... Fresnel-zone plate

Lectures 1-6: The Story so far

lecture 7