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The spatial frequency domain

MIT 2.71/2.710 Optics

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Recall: plane wave propagation

zz=0

path delay increases

linearly withx

x

x

plane of observation

+

cos2

sin2exp0

z

i

xiE


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Spatial frequency angle of propagation?

zz=0


x

electric field


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zz=0


x


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The cross-section of the optical field with the optical axis

is a sinusoid of the form

+ 00

sin2exp

xiE

i.e. it looks like

( )

sin

where2exp 00 + uxuiE

is called the spatial frequencyspatial frequency


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2D sinusoids

( )[ ]vyuxE +2cos0 ( )[ ]vyuxiE +2exp0corresp. phasor

y

x


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2D sinusoids


x

y

min

max


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2D sinusoids


x

y

1/u1/v

min

max


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2D sinusoids


min

x

y

22

1

tan

vu

vu

+=

=

max


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Periodic Grating /1: vertical


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Space

domain

Frequency(Fourier)

domain

x

y

x

y

u

v

u

v

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Periodic Grating /2: tilted


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Space

domain

Frequency(Fourier)

domain

x

y

x

y

u

v

u

v

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Superposition: multiple gratings

+ +

Space

domain

Frequency(Fourier)

domain

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More superpositions


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Space

domain

Frequency(Fourier)

domain

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Size of object vs frequency content


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Space

domain

Frequency(Fourier)

domain

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Superimposing shifted objects


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Space

domain

Frequency(Fourier)

domain

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Linear shift invariant systems

in the time domain


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F(t)m

k b x(t)

vi(t)

R

C vo(t)

+

( )

( )

( )( )

km

b

m

k

mH

kbimH

tFkxxbxm

2

4

11

1

r

22

r

222

r

22

2

2

==

+=

+=

=++

( )

( )

( )( )

C

H

iH

tvvvRC

=

+=

+=

=+

1

1

1

1

2

2

ioo


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gi(t) go(t)

LSILSI


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gi(t)=(tt0) go(t)=h(tt0)

LSILSI

t

gi(t)

t

go(t)

shift invarianceshift invariance

gi(t)=(t) go(t)=h(t)

LSILSIt

gi(t)

t

go(t)impulse excitation impulse response

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gi(t) go(t)

LSILSI


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gi(t)=(t)+(tt0) go(t)=h(t)+h(tt0)

LSILSI

t

gi(t)

t

go(t)

linearitylinearity

gi(t)=(t) go(t)=h(t)

LSILSIt

gi(t)

t

go(t)impulse excitation impulse response

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gi(t) go(t)

LSILSI

gi(t)=arbitrary go(t)= ??

LSILSIt

gi(t)

t

go(t)

( ) ( ) ( ) 000ii dttttgtg = ( ) ( ) ( ) 000io dttthtgtg =

siftingsifting property of the -function convolution integralconvolution integral


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gi(t) go(t)

LSILSI


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gi(t)=sinusoid go(t)=sinusoid

LSILSIt

gi(t)

t

go(t)

( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( )[ ]

iHH

tHatgtatg

expwhere

coscos 0000o00i

=

==

transfer functiontransfer function

attenuation

phase delay

same frequency

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From time to frequency: Fourier analysis

Can I express an arbitraryg(t)

as a superposition of sinusoids?

t

g(t)


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t

t

t

t

++

++++

t

t

==++++

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The Fourier integral

(aka inverse Fourier transforminverse Fourier transform)

( ) ( ) de 2 tiGtg +=

superposition sinusoids

complex weight,expresses relative amplitude

(magnitude & phase)

of superposed sinusoidsMIT 2.71/2.710 Optics

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The Fourier transform

The complex weight coefficients G(),

aka Fourier transformFourier transform ofg(t)

are calculated from the integral

( ) ( ) ttgGti

de

2

=

t

g(t)

Re[e-i2t]

Re[G()]= dt


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Fourier transform pairspairs

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )[ ]

( ) ( ) ( ) ( ) ( )[ ]

( ) ( ) ( )

( ) ( ) ( )222

2

000

000

00

2exp2

exp

sincrect

2

1

2sin

2

12cos

2exp

1

TTGT

ttg

TTGT

ttg

iGttg

Gttg

Gtitg

Gttg

=

=

=

=

++==

++==

==

==


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2D2D F i t f ii

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2D2D Fourier transform pairspairs

Image removed due to copyright concerns

(from Goodman,

Introduction to

Fourier Optics,

page 14)


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