Remind! Huygens-Fresnel principle

(1)

4. Fresnel and Fraunhofer diffraction

(2)

Remind! Diffraction under paraxial approx.

(3)

“Every unobstructed point of a wavefront, at a given instant in time, serves as a source of secondary wavelets (with the same frequency as that of the primary wave).

The amplitude of the optical field at any point beyond is the superposition of all these wavelets (considering their amplitude and relative phase).”

Huygens’s principle:

By itself, it is unable to account for the details of the diffraction process.

It is indeed independent of any wavelength consideration.

Fresnel’s addition of the concept of interference

Remind! Huygens-Fresnel principle

(4)

Fresnel’s shortcomings :

He did not mention the existence of backward secondary wavelets,

however, there also would be a reverse wave traveling back toward the source.

He introduce a quantity of the obliquity factor, but he did little more than conjecture about this kind.

Gustav Kirchhoff : Fresnel-Kirhhoff diffraction theory

A more rigorous theory based directly on the solution of the differential wave equation.

He, although a contemporary of Maxwell, employed the older elastic-solid theory of light.

He found K(χ) = (1 + cosθ )/2. K(0) = 1 in the forward direction, K(π) = 0 with the back wave.

Arnold Johannes Wilhelm Sommerfeld : Rayleigh-Sommerfeld diffraction theory A very rigorous solution of partial differential wave equation.

The first solution utilizing the electromagnetic theory of light.

Remind! After the Huygens-Fresnel principle

(5)

( ) ( ) ( )

r ds P jkr

j U P

U θ

λ ^cos

exp 1

01 01 1

0

∫∫

∑

=

( ) P h ( P P ) ( ) U P ds U

₀

∫∫

₀

,

₁ ₁

∑

=

( ) ( ) _θ

λ ^cos

exp , 1

01 01 1

0

r

jkr P j

P

h =

) 90 by phase incident

the (lead 1

:

1

o

phase j

amplitude ν

λ ^∝

∝

Each secondary wavelet has a directivity pattern cosθ

Remind! Huygens-Fresnel Principle revised by 1 ^st R – S solution

Impulse response function H-F principle

(6)

( ) ( ) ( )

r ds P jkr

j U P

U 1 exp cos

01 01 1

0

θ

λ ∫∫

∑

=

01

cos r

= z θ

( ) ( _ξ _η ) ( ) _ξ _η

λ

r ^d ^d

U jkr j

y z x

U exp

,

, ₂

01

∫∫

01

∑

=

( ) (

²

)

²

2

01

= z + x − ξ + y − η

r

Note! So far we have used only two assumptions (R-S solution) : Scalar wave equation + ( r

₀₁

>> λ )

Huygens-Fresnel Principle Huygens-Fresnel Principle

Screen (x,y) Aperture (ξ,η)

r₀₁

(7)

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ ⎟

⎠

⎜ ⎞

⎝ + ⎛ −

⎟ ⎠

⎜ ⎞

⎝ + ⎛ −

≈

2 2

01

2 1 2

1 1

z y z

z x

r ξ η

( ) _λ ( ) ^ξ ^η [ ( ^x ^ξ ) ( ^y ^η ) ] ^d ^ξ ^d ^η

z j k z U

j y e

x U

jkz

exp 2 ,

,

² ²

⎭ ⎬

⎫

⎩ ⎨

⎧ − + −

= ∫ ∫

^∞

∞

−

( ) ⁽ ⁾ ( ) ^ξ ^η ⁽ ⁾

⁽ ⁾

^ξ ^η

λ

η λ ξ

η π

ξ

e d d

e U

z e j y e

x

U

^z ^j ^z ^x ^y

j k y

z x j k

jkz

∫ ∫

^∞

∞

−

+

− +

+

⎭ ⎬

⎫

⎩ ⎨

=

² ² ²

⎧ ,

² ² ² ²

,

( )

( ) ⁽ ⁾

z y f z x f z

j k y

z x j k jkz

Y X

e U

z e j y e

x U

λ λ

η

ξ

λ ξ

/ ,

/ 2

2

2 2 2

2

, )

, (

=

= +

+

⎭ ⎬

⎫

⎩ ⎨

= F ⎧

Since k is 10

Since k is 10⁷⁷ order, the quadratic terms in rorder, the quadratic terms in r₀₁₀₁ must be considered in the exponent.must be considered in the exponent.

Fresnel diffraction integral

Fresnel (paraxial) Approximation

(8)

This is most general form of diffraction – No restrictions on optical layout

• near-field diffraction

• curved wavefront

– Analysis somewhat difficult

Fresnel Diffraction Fresnel Diffraction

Screen

z z

Curved wavefront (diverging wavelets)

Fresnel (near-field) diffraction Fresnel (near-field) diffraction

( )

²

⁽

² ²

⁾

( , ) ,

j k

U x y _{≈ ⎨} ^⎧ U ξ η e

z ^{ξ η}⁺

^⎫ _⎬

⎩ ⎭

F

(9)

y

Wavefront emitted

earlier

z Wavefront emitted

later

θ

z y

k G

later

earlier

(

₀₁

)

exp jkr

( )

_⎥⎦^⎤

⎢⎣⎡ ² + ²

exp 2 x y

z j k

(

₀₁

)

exp − jkr

( )

_⎥⎦^⎤

⎢⎣⎡− ² + ²

exp 2 x y

z j k

(

j2

πα

y

)

exp

(

^j² ^y

)

exp −

πα

z=0

z=0 diverging

converging

Positive phase and Negative phase

(10)

λ

16 D

2

z ≥

( ) ( )

[ ]

²max

2 3 2

4 ξ η

λ

π ₋ ₊ ₋

〉〉 x y

z

• Accuracy can be expected for much shorter distances

( , ) 2 4

for U ξ η smooth & slow varing function; x − = ≤ ξ D λ z

Fresnel approximation

Accuracy of the Fresnel Approximation

(11)

( ^x ^y ) ^U ( ^ξ ^η ) ( ^h ^x ^ξ ^y ^η ) ^d ^ξ ^d ^η

U , = ∫ ∫

^∞

, − , −

∞

−

( ) ⁼ _⎢⎣ ^⎡ (

²

⁺

²

) _⎥⎦ ^⎤

exp 2

, x y

z jk z

j y e

x h

jkz

λ

This convolution relation means that Fresnel diffraction is linear shift-invariant.

( ) _λ ( ) ^ξ ^η [ ( ^x ^ξ ) ( ^y ^η ) ] ^d ^ξ ^d ^η

z j k z U

j y e

x U

jkz

exp 2 ,

,

² ²

⎭ ⎬

⎫

⎩ ⎨

⎧ − + −

= ∫ ∫

^∞

∞

−

Impulse Response and Transfer Functions of Fresnel diffraction

Impulse response function of Fresnel diffraction

(12)

(

X^, Y

)

^exp ² ¹

(

X

) (

² Y

)

²

H f f j π z λ f λ f

λ

⎡ ⎤

= ⎢⎣ − − ⎥⎦

( ) ^{ ^}

( ) ( )

,

2 2 2 2

( , )

exp exp

X Y

jkz

X Y

H f f h x y

e j x y e j z f f

j z z

π πλ

λ λ

≡

⎧ ⎡ ⎤ ⎫ ⎡ ⎤

= ⎨ ⎩ ⎢ ⎣ + ⎥ ⎦ ⎬ ⎭ = ⎣ − + ⎦

F F

( ) (

²

)

²

( ) (

²

)

²

1 1

2 2

X Y

f f

f f λ λ

λ λ

− − ≈ − −

Under the paraxial approximation, the transfer function in free space becomes

Impulse Response and Transfer Functions of Fresnel diffraction

(

X^, Y^;

) (

X^, Y^{;0 exp}

)

² ¹

(

X

) ( )

² Y ²

f f z f f j z f f

ε ε π λ λ

λ

⎡ ⎤

= ⎢⎣ − − ⎥⎦

Remind! Transfer function of wave propagation phenomenon in free space

(

^X^, ^Y

)

^j² ^z

^exp (

^X² ^Y²

)

H f f e j z f f

π

λ

^⎡ πλ ^⎤

= ⎣ − + ⎦

We confirm again that the Fresnel diffraction is paraxial approximation

(13)

z y z

z x

r ξ η

−

01 ≈

( ) ( ) ^ξ ^η ⁽ ⁾ ^ξ ^η

λ

η λ ξ

π

d d e

z U j

y e x

U

^j ^z ^x ^y

jkz ∞ − +

∞

∫ ∫

−

= ,

²

,

( )

{ }

_f _x _z _f _y _z

jkz

Y X

z U j

e

λ

η

λ

λ ξ ^,

⁼ ^/ ^, ⁼ ^/

= F

( ) (

²

)

²

2

01

= z + x − ξ + y − η r

Fresnel Diffraction between Confocal Spherical Surfaces Fresnel Diffraction between Confocal Spherical Surfaces

The Fresnel diffraction makes the exact Fourier transforms between the two spherical caps.

(14)

In summary, Fresnel diffraction is …

(15)

( ) ( ) ( ^ξ ^η ) ^ξ ^η

λ η π

λ U ξ j z ^x ^y ^d ^d

z j e y e

x U

y z x

j k jkz

⎥⎦ ⎤

⎢⎣ ⎡ − +

= ∫ ∫

^∞

∞

−

+

2 exp ,

,

) 2 (

2 2

( )

2

max 2

2

η

ξ +

〉〉 k z

( )

{ }

_f _x _z _f _y _z

y z x

j k jkz

Y X

z U j

e e

λ

η

λ

λ ξ

^/ ^, ^/

) 2 (

,

2 2

=

= +

= F

formular s

designer' anttena

D :

z λ

2

>

Fraunhofer Approximation Fraunhofer Approximation

01

1 x y

r z

z z

ξ η

⎡ ⎤

≈ ⎢ ⎣ − − ⎥ ⎦

(16)

• Specific sort of diffraction – far-field diffraction

– plane wavefront – Simpler maths

Fraunhofer (far-field) diffraction Fraunhofer (far-field) diffraction

( )

^,

(

^,

)

^exp ²

( )

U x y U j x y d d

z

ξ η π ξ η ξ η

λ

∞

−∞

⎡ ⎤

≈

∫ ∫

⎢⎣− + ⎥⎦

Plane waves

(17)

(18)

Circular aperture ¹

1 r a circ r

r a a

⎧ ≤

⎛ ⎞ = ⎨

⎜ ⎟ >

⎝ ⎠ ⎩

(19)

Circular Aperture

Circular aperture

Single slit (sinc ftn)

Sin θ

0 λ/D 2λ/D 3λ/D

−λ/D

−2λ/D

−3λ/D

Intensity

(20)

(21)

(22)

(23)

(24)

Airy Disc

• Similar pattern to single slit – circularly symmetrical

• First zero point when ½kD sinθ = 3.83 – sinθ = 1.22

^λ

/

_D

– Central spot called “Airy Disc”

• Every optical instrument

• microscope, telescope, etc.

– Each point on object

• Produces Airy disc in image.

(25)

(26)

(27)

(Appendix : Step and repeat function)

(28)

Two-dimensional array of rectangular apertures

(29)

Two-dimensional array with irregular spacing

randomized

(30)

Diffraction from an array of finite size

(31)

Diffraction from an array of finite size (2) : for large C

unit cell (a x l)

c

(32)

Double-slit diffraction

( ) ( ) ( ) ( )

[ ]

β α α β

β ε β

ε

β α β β

θ α

θ β

θ

β β

α β

β α

θ θ

2 2

2 2 2

2

2 1 2

1

2 / sin ) ( 2

/ sin ) ( 2

/ sin ) (

2 / sin ) (

2 ) (

2 ) ( 2 sin

) (

2 ) (

sin

sin cos 4

sin cos 2

2 2

sin cos ) 2

( )

2 (

sin ,

sin sin

1

⎟⎟⎠

⎜⎜ ⎞

⎝

= ⎛

⎟⎟⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

⎟⎛

⎠

⎜ ⎞

⎝

= ⎛

⎟⎠

⎜ ⎞

⎝

= ⎛

=

− +

−

=

≡

− +

−

=

⎟⎟⎠

⎜⎜ ⎞

⎝ +⎛

⎟⎟⎠

⎜⎜ ⎞

⎝

= ⎛

−

− +

−

− +

−

+

−

− +

−

∫

o o

L o

o R

o i L

i i i

i i o

R L

b a ik b

a ik b

a ik o

L

b a

iks o

b L a

b a

iks o

R L

r I b E c

irradiance

; c E

I

r b e E

e e

i e b r

E E

ka

kb

e e

e ik e

r E

ds r e

ds E r e

E E

a b

b (a-b)/2 (a+b)/2

(33)

Double-slit diffraction

m b a p

: (2)

&

(1) satisfying or

orders, missing

for Condition

2, 1, 0, m , θ b

m : minima n

diffractio

(2) - - - - - - - - - b

b

: term envelope n

diffractio The

2, 1, 0, p , θ a

p : maxima ce

interferen

(1) - - - - - - - - - a

: term ce interferen The

r b E I c

I

o L o o

o

⎟⎠

⎜ ⎞

⎝

= ⎛

±

=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

±

=

⎥⎦⎤

⎢⎣⎡

=

⎟⎟⎠

⎜⎜ ⎞

⎝

⎟⎛

⎠

⎜ ⎞

⎝

= ⎛

⎟⎟⎠

⎜⎜ ⎞

⎝

= ⎛

"

sin sin sin sin

sin cos sin

cos , 2 sin cos

4

2 2

2

2 2

2

λ λ

θ π λ

θ π

λ

λ θ α π

α ε β

β

(34)

Multiple-slit diffraction

[[ ]]

{ }

2 2 2 /

1

2 ) ) 1 2 (

sin sin

sin ⎟

⎠

⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

= ⎛

+

=

∑ ∫ ∫

=

+

−

+

−

α α β

β

θ θ

I N I

ds e

ds r e

E E

o

N

j

b a j

iks iks

o R L

integers other

all p

for occur minima

2N, N,

0, p for occur maxima

principal

2, 1, 0, p , θ N a

p or N , : p

minima

N N N

: N magnitude

2, 1, 0, m , θ a

m or , m :

maxima principal

(1) - - - - - - - - - N

: term ce interferen The

lim

m m

2

=

±

=

⇒

±

=

±

=

±

=

→

⎟⎠

⎜ ⎞

⎝

⎛

→

"

sin

cos cos sin

sin

sin sin

sin

λ α π

α α α

α

λ π

α α

α

π α π

α

(35)

Examples of Fraunhofer Diffraction

⎟ ⎠

⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

⎥⎦ ⎛

⎢⎣ ⎤

⎡ +

= ω

η ω

ξ ξ

π 2 2

2 2 2

1 m f rect rect

t

_A

cos(

_o

)

Thin sinusoidal amplitude grating

(36)

25 .

0

= 0 η

% .

/ 16 6 25

2

1

= m ≤

η

+

% .

/ 16 6 25

2

1

= m ≤ η

−

Thin sinusoidal amplitude grating

⎭⎬

⎫

⎩⎨

⎧ ⎟

⎠

⎜ ⎞

⎝

⎛ −

⎟+

⎠

⎜ ⎞

⎝

⎛ +

⎟+

⎠

⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

⎥⎦ ⎛

⎢⎣ ⎤

= ⎡ 2 ( )

4 sin )

2 ( 4 sin

sin 2 sin 2

) 2 ,

( ²

2 2

z f z x

m c z

f z x

m c z

c x z

c y z

y A x

I _o _oλ

λ λ ω

λ ω λ

ω λ

) , (

) , ( )

cos(

_o _X _Y _X _o _Y

m f

_X

f

_o

f

_Y

f f m f

FT ⎥⎦ ⎤ = + + + −

⎢⎣ ⎡ + π ξ δ δ δ

4 4

2 2 1

2 2

1 m=1

(37)

( ) ( ) ⎟

⎠

⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

⎥⎦ ⎛

⎢⎣ ⎤

= ⎡

rect w rect w

m f j t

_A

2 2 2

2

⁰

η ξ ξ

π η

ξ , exp sin

( ) ( ) ⎟

⎠

⎜ ⎞

⎝

⎥⎦ ⎛

⎢⎣ ⎤

⎡ −

⎟ ⎠

⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

≈ ⎛ ∑

^∞

−∞

=

z

z wy qf

z x w J m

z y A

x I

q

λ λ

sinc 2 sinc 2

,

²

2

² ₀ ²

2

Thin sinusoidal phase grating

m/2=4

) , (

) sin(

exp

_X _o _Y

q

o

m f qf f

J m f

j

FT ⎟ −

⎠

⎜ ⎞

⎝

= ⎛

⎭ ⎬

⎫

⎩ ⎨

⎧ ⎥⎦ ⎤

⎢⎣ ⎡

∑

^∞

−∞

=

δ

ξ

π 2

2 2

⎟ ⎠

⎜ ⎞

⎝

= ⎛

2

m

J

_q

η

q

% .

.

% . .

min ,

max ,

27 4

2 2 4

2 2

8 33 8

2 1

2 1 2

0 0

2 1 1

⎟ =

⎠

⎜ ⎞

⎝

⎛ =

⎟ =

⎠

⎜ ⎞

⎝

⎛ =

=

⎟ =

⎠

⎜ ⎞

⎝

⎛ =

=

±

J m where 0,

m J

m

J η

η

(38)

Fresnel Diffraction by Square Aperture

(b) Diffraction pattern at four axial positions marked by the arrows in (a) and

corresponding to the Fresnel numbers N_F=10, 1, 0.5, and 0.1. The shaded area represents the geometrical shadow of the slit. The dashed lines at represent the width of the Fraunhofer pattern in the far field. Where the dashed lines coincide with the edges of the geometrical shadow, the Fresnel number N_F=0.5.

( ^D)^d

x = λ/

Fresnel Diffraction from a slit of width D = 2a. (a) Shaded area is the geometrical shadow of the aperture. The dashed line is the width of the Fraunhofer diffracted beam.

Remind! Huygens-Fresnel principle

4. Fresnel and Fraunhofer diffraction

4. Fresnel and Fraunhofer diffraction

Remind! Diffraction under paraxial approx.

Remind! Diffraction under paraxial approx.