Fraunhofer vs Fresnel approximation - NIU

Fraunhofer vs Fres

• Consider an incoming wave f(x,y)

• Now assume the function is confin
• We also consider d to be large en

small
• Then

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snel approximation

) that propagates in free space (d)

ned to is
nough so that

YS 630 – Fall 2008

Diffra

• Consider an incoming wave f(x,y)
• The wave is intercepted by an ape

also called “pupil function”
• Then propagates a distance d
• The final complex amplitude of the

p(x,y)
f(x,y)

d

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action

)
erture with transmission p(x,y)
e wave is g(x,y)

g(x,y)

d

YS 630 – Fall 2008

Fraunhofer

• In Fraunhofer diffraction, the comp
the aperture is computed using th

• This is valid if the Fresnel number
• Consider an incoming plane wave
• Downstream of the aperture with t

• And after a drift of length d

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r diffraction

plex wave amplitude downstream of
he Fraunhofer approximation

r is <<1
e
transmission p(x,y) we have

where

YS 630 – Fall 2008

Fraunhofer diffraction:

• Consider an incoming wave interc
size Dx and Dy. What is the intens

• We have
• Using the previous slides

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rectangular aperture I

cepted by a rectangular aperture of
sity of the diffraction pattern?

YS 630 – Fall 2008

Fraunhofer diffraction:

• We finally obtain

y
Dx=Dy
x

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rectangular aperture II

y
Dx=2Dy
x

YS 630 – Fall 2008

Fraunhofer diffractio

• Now we take a circular aperture o

• The Fourier transform of the trans

• Here we change to cylindrical coo
symmetry. Introducing

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on: circular aperture I

of radius a
smission function is

ordinates because of the cylindrical

YS 630 – Fall 2008

Fraunhofer diffraction

• We can write

• So finally

And the diffraction pattern intensity is

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n: circular aperture II

YS 630 – Fall 2008

Fraunhofer diffraction

• Diffraction pattern intensity
y
x

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n: circular aperture III

J1(r)/r

r

[J1(r)/r]2

r

YS 630 – Fall 2008

Fresnel di

• In Fresnel diffraction, the complex
the aperture is computed using th

• The intensity is given by

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iffraction I

x wave amplitude downstream of
he Fresnel approximation

YS 630 – Fall 2008

Fresnel di

• Written in a normalized coordinate

• This is the convolution of the trans
considered aperture with the func

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iffraction II

e system

size ~ a/[λd]1/2

smission function p(X,Y) of the
ction

cos(πx2)

sin(πx2)

x

YS 630 – Fall 2008

Fresnel dif

• In the equation

The result of this convolution is go
NF=a2/(λd)
• If NF is large the convolution is go
p(X,Y).
• In the limit NF→∞, ray optics is ap
the shadow of the aperture
• In the opposite limit Fresnel diffrac
Fraunhofer pattern.

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ffraction III

size ~ a/[λd]1/2

overned by the Fresnel number
oing to yield a function similar to
pplicable (λ→0) and the pattern is
ction pattern converge to the

YS 630 – Fall 2008

Fresnel diffractio

• Consider a slit infinitely long in the

• From we n

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on: slit aperture

e y-direction then
need to compute g(X)

YS 630 – Fall 2008

Fresnel diffractio

• Fresnel patterns for different Fres

Nf=10

Nf=1

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on: slit aperture II

snel number

Nf=0.5

Nf=0.1

YS 630 – Fall 2008

Summ

• In the order of increasing distance
pattern is

• A shadow of the aperture.

• A Fresnel diffraction patter
“normalized” aperture func

• A Fraunhofer diffraction p
absolute value of the Fouri
function. The far field has a
to λ/D where D is the diam

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mary

e from the aperture, diffraction

.
rn, which is a convolution ot the
ction with exp[-iπ(X2+Y2)].
pattern, which is the squared-
ier transform of the aperture
an angular divergence proportional
meter of the aperture.

YS 630 – Fall 2008