W.-Y. Choi

Lect. 14: Diffraction

Optoelectronics and Photonics (18/2)

a

(1) exp( )A jkRR

(2)(1)y

R

( , )totalE R

siny

(2) exp ( sin )A jk R yR

/2( sin )

/2

y ajk R y

y a

A e dyR

/2sin

/2

y ajkR jky

y a

A e e dyR

Since interference is due to phase difference , ignore the constant phase term

/2sin

/2

( , )y a

jkytotal

y a

AE R e dyR

W.-Y. Choi

Lect. 14: Diffraction

Optoelectronics and Photonics (18/2)

/2sin

/2

Evaluate ( , )a

jkytotal

a

AE R e dyR

2 sin( sin )sin 2

A j akR jk

Let ' siny jky

' sin2

'

' sin2

'( , )sin

ay jk

ytotal

ay jk

A dyE R eR jk

sin( sin )2 2sin

akAR k

sin sin2 21

sin

a ajk jkA e eR jk

=> ' sindy jk dy

W.-Y. Choi

Lect. 14: Diffraction

Optoelectronics and Photonics (18/2)

( )(0)

total

total

EE

( ,0)totalE R

( , ) ?( ,0)

total

total

E RE R

sin( sin )2 2( , )sintotal

akAE RR k

0

0

cos( sin ) cos2 2 2

cos

a ak kA

R k

22

A aR

sin( sin )2 2sin

22

akAR k

A aR

sin( sin )

2sin

2

ak

ak

sin( )2

2

y

y

ak

ak sinc function

W.-Y. Choi

Lect. 14: Diffraction

Optoelectronics and Photonics (18/2)

FT relationship

sin( )( ) 2(0)

2

ytotal

totaly

akEaE k

Sourceo

Diffraction

W.-Y. Choi

Lect. 14: Diffraction

Optoelectronics and Photonics (18/2)

1( ) ( )2

j tf t F e d

f( ) ( )t F

( ) ( ) total yE k A y

/2sin

/2

( , )y a

jkytotal

y a

AE R e dyR

From Signals and Systems

-

1( , ) ( ) y

yjk y

total yy

E R k A y e dyR

sinyk k

Diffraction of A(y) is F.T. of A(y)

W.-Y. Choi

Lect. 14: Diffraction

Optoelectronics and Photonics (18/2)

-

1( , ) ( ) y

yjk y

total yy

E R k A y e dyR

Far-field diffraction

Fraunhoffer Diffraction

Jeseph Ritter von Fraunhoffer

(1787-1826)

German physicist

W.-Y. Choi

Lect. 14: Diffraction

Optoelectronics and Photonics (18/2)

a

a

d.o

a

d

o

A(y)

a a

d

sin( )2

2

y

y

ak

akX

2 2y yd djk jk

e e

sin( )( ) 22cos( )(0) 2

2

y

yy

y

k aE k dk k aE

2cos( )2ydk

W.-Y. Choi

Lect. 14: Diffraction

Optoelectronics and Photonics (18/2)

( , )total x yE k k

- -

~ ( , ) yx

y xjk yjk x

y x

A x y e e dxdy

2 2

- -2 2

yx

b ay x

jk yjk x

b ay x

e e dxdy

2 2

-- -2 2

x x

b ay x

jk y jk x

b ay x

e dy e dx

sin( )

2

2

y

y

bk

bk

sin( )2

2

x

x

ak

ak

( , )(0,0)

total x y

total

E k kE

W.-Y. Choi

Lect. 14: Diffraction

Optoelectronics and Photonics (18/2)

Airy pattern

2

1

0

( ) 2 ( )y y

y

I k J k dI k d

d: diameterJ1: Bessel function of first order

For the first dark ring,

3.83yk d sin3.83

2d

1.22d

is often used for imaging resolution

W.-Y. Choi

Lect. 14: Diffraction

Optoelectronics and Photonics (18/2)

Many imaging systems have circular aperture

Larger d Better resolution

sin 1.22d

Smaller d Large

(But smaller depth of field)

W.-Y. Choi

Lect. 14: Diffraction

Optoelectronics and Photonics (18/2)

Augustin-Jean Fresnel(1788~1827)

French Physicist

- It was assumed R>>(far field)

For R~ (near field, or Fresnel diffraction), FT relation cannot be used.

In reality, observation is made at a flat surface. Consequently, there is

additional phase shifts, requiring more complicated analysis

In our analysis,

- Diffraction is observed at locations having same R

W.-Y. Choi

Lect. 14: Diffraction

Optoelectronics and Photonics (18/2)

Homework (Due 11/13)