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4. Fresnel and Fraunhofer diffraction

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4. Fresnel and Fraunhofer diffraction4. Fresnel and Fraunhofer diffraction

Remind! Diffraction under paraxial approx.Remind! Diffraction under paraxial approx.

“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 superpositionof 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 principleRemind! Huygens-Fresnel principle

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 theoryA 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 theoryA very rigorous solution of partial differential wave equation.The first solution utilizing the electromagnetic theory of light.

Remind! After the Huygens-Fresnel principleRemind! After the Huygens-Fresnel principle

( ) ( ) ( ) dsrjkrPU

jPU θ

λcosexp1

01

0110 ∫∫

=

( ) ( ) ( )dsPUPPhPU 1100 ,∫∫∑

=

( ) ( ) θλ

cosexp1,01

0110 r

jkrj

PPh =

)90by phaseincident the(lead 1 :

1

o

jphase

amplitude νλ∝∝

Each secondary wavelet has a directivity pattern cosθ

Remind! Huygens-Fresnel Principlerevised by 1st R – S solution

Remind! Huygens-Fresnel Principlerevised by 1st R – S solution

Impulse response function

H-F principle

( ) ( ) ( ) dsr

jkrPUj

PU cosexp1

01

0110 θ

λ ∫∫∑=

01

cosrz

( ) ( ) ( ) ηξηξλ

ddr

jkrUjzyxU exp,, 2

01

01∫∫∑

=

( ) ( )22201 ηξ −+−+= yxzr

Note! So far we have used only two assumptions (R-S solution) :

Scalar wave equation + ( r01 >> λ )

Huygens-Fresnel PrincipleHuygens-Fresnel Principle

Screen (x,y)Aperture (ξ,η)

r01

⎥⎥⎦

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

+⎟⎠⎞

⎜⎝⎛ −

+≈22

01 21

211

zy

zxzr ηξ

( ) ( ) ( ) ( )[ ] ηξηξηξλ

ddyxz

kjUzj

eyxUjkz

2

exp,, 22

⎭⎬⎫

⎩⎨⎧ −+−= ∫ ∫

∞−

( ) ( ) ( ) ( ) ( )ηξηξ

ληξ

λπηξ

ddeeUezj

eyxUyx

zj

zkjyx

zkjjkz

∫ ∫∞

∞−

+−++

⎭⎬⎫

⎩⎨⎧

=2

222222

,,

( ) ( ) ( )

zyfzxf

zkjyx

zkjjkz

YX

eUezj

eyxUλλ

ηξηξ

λ/,/

222222

,),(==

++

⎭⎬⎫

⎩⎨⎧

= F

Since k is 10Since 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) ApproximationFresnel (paraxial) Approximation

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

• near-field diffraction• curved wavefront

– Analysis somewhat difficult

Fresnel DiffractionFresnel Diffraction

Screen

zz

Curved wavefront(diverging wavelets)

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

( ) ( )2 2

2( , ) ,kjzU x y U eξ η

ξ η+⎧ ⎫

≈ ⎨ ⎬⎩ ⎭

F

y

Wavefront emitted earlier

z Wavefront emitted

later

θ

z

y k

Wavefront emitted

later

Wavefront emitted earlier

( )01exp jkr

( )⎥⎦⎤

⎢⎣⎡ + 22

2exp yx

zkj

( )01exp jkr−

( )⎥⎦⎤

⎢⎣⎡ +− 22

2exp yx

zkj

( )yj πα2exp

( )yj 2exp πα−

z=0

z=0

diverging

converging

Positive phase and Negative phasePositive phase and Negative phase

λ16

2Dz ≥

( ) ( )[ ]2max223

4ηξ

λπ

−+−⟩⟩ yxz

• Accuracy can be expected for much shorter distances

( , ) 2 4for U smooth & slow varing function; x D zξ η ξ λ− = ≤

Fresnel approximation

Accuracy of the Fresnel ApproximationAccuracy of the Fresnel Approximation

( ) ( ) ( ) ηξηξηξ ddyxhUyxU , ,, −−= ∫ ∫∞

∞−

( ) ( )⎥⎦⎤

⎢⎣⎡ += 22

2exp, yx

zjk

zjeyxh

jkz

λ

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

( ) ( ) ( ) ( )[ ] ηξηξηξλ

ddyxz

kjUzj

eyxUjkz

2

exp,, 22

⎭⎬⎫

⎩⎨⎧ −+−= ∫ ∫

∞−

Impulse Response and Transfer Functions of Fresnel diffraction

Impulse Response and Transfer Functions of Fresnel diffraction

Impulse response function of Fresnel diffraction

( ) ( ) ( )2 2, exp 2 1X Y X YzH f f j f fπ λ λλ

⎡ ⎤= − −⎢ ⎥⎣ ⎦

( ) { }

( ) ( ),

2 2 2 2

( , )

exp exp

X Y

jkzjkz

X Y

H f f h x y

e j x y e j z f fj z z

π πλλ λ

⎧ ⎫⎡ ⎤ ⎡ ⎤= + = − +⎨ ⎬⎢ ⎥ ⎣ ⎦⎣ ⎦⎩ ⎭

F

F

( ) ( ) ( ) ( )2 22 21 1

2 2X Y

X Y

f ff f

λ λλ λ− − ≈ − −

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

Impulse Response and Transfer Functions of Fresnel diffraction

Impulse Response and Transfer Functions of Fresnel diffraction

( ) ( ) ( ) ( )2 2, ; , ;0 exp 2 1X Y X Y X Yzf f z f f j f fε ε π λ λλ

⎡ ⎤= − −⎢ ⎥⎣ ⎦

Remind! Transfer function of wave propagation phenomenon in free space

( ) ( )2

2 2, exp

j z

X Y X YH f f e j z f fπλ πλ⎡ ⎤= − +⎣ ⎦

We confirm again that the Fresnel diffraction is paraxial approximation

zy

zxzr ηξ

−−≈01

( ) ( ) ( )ηξηξ

ληξ

λπ

ddeUzj

eyxUyx

zjjkz +−

∞−∫ ∫=

2

,,

( ){ }zyfzxf

jkz

YXU

zje

λληξ

λ /,/,

=== F

( ) ( )22201 ηξ −+−+= yxzr

Fresnel Diffraction between Confocal Spherical SurfacesFresnel Diffraction between Confocal Spherical Surfaces

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

In summary, Fresnel diffraction is …

( ) ( ) ( ) ηξηξλπηξ

λddyx

zjU

zjeeyxU

yxz

kjjkz

⎥⎦⎤

⎢⎣⎡ +−= ∫ ∫

∞−

+2exp,,

)(2

22

( )2

max22 ηξ +

⟩⟩kz

( ){ }zyfzxf

yxz

kjjkz

YXU

zjee

λληξ

λ /,/

)(2

,

22

==

+

= F

formular sdesigner' anttena : Dzλ

22>

Fraunhofer ApproximationFraunhofer Approximation

01 1 x yr zz zξ η⎡ ⎤≈ − −⎢ ⎥⎣ ⎦

• Specific sort of diffraction– far-field diffraction– plane wavefront– Simpler maths

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

( ) ( ) ( )2, , expU x y U j x y d dzπξ η ξ η ξ ηλ

−∞

⎡ ⎤≈ − +⎢ ⎥⎣ ⎦∫ ∫

Plane waves

Circular aperture 1 1

r arcircr aa≤⎧⎛ ⎞ = ⎨⎜ ⎟ >⎝ ⎠ ⎩

Circular Aperture

Circular aperture

Single slit(sinc ftn)

Sin θ0 λ/D 2λ/D 3λ/D−λ/D−2λ/D−3λ/D

Intensity

Airy Disc• Similar pattern to single slit

– circularly symmetrical• First zero point when ½kD sinθ = 3.83

– sinθ = 1.22λ/D– Central spot called “Airy Disc”

• Every optical instrument• microscope, telescope, etc.

– Each point on object• Produces Airy disc in image.

(Appendix : Step and repeat function)

Two-dimensional array of rectangular apertures

Two-dimensional array with irregular spacing

randomized

Diffraction from an array of finite size

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

unit cell (a x l)

c

Double-slit diffraction

( ) ( ) ( ) ( )[ ]

[ ]

αββ

αββε

ε

αββ

β

θαθβ

θ

ββαββα

θθθθ

θθ

22

222

2

21

21

2/sin)(2/sin)(2/sin)(2/sin)(

2)(

2)(sin2)(

2)(sin

cossin4cossin22

2

cossin2)()(

2

sin,sin

sin1

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛=

⎟⎠

⎞⎜⎝

⎛=

=−+−=

≡≡

−+−=

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎞⎜⎜⎝

⎛=

−−−

−+−−+−

+

−−

+− ∫∫

oo

Lo

Ro

o

Liiiiii

o

LR

baikbaikbaikbaik

o

L

ba

baiks

o

Lba

baiks

o

LR

Ir

bEc

irradiance ; Ec

I

rbE

eeeeeeib

rE

E

ka kb

eeeeikr

E

dserE

dserE

E

a

b

b

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

Double-slit diffraction

bmpa :(2) & (1) satisfying or orders,missing for Condition

2, 1, 0,m ,θbm :minima ndiffractio

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

b

: termenvelope ndiffractio The

2, 1, 0,p ,θap :maxima ce interferen

(1) --------- a : termce interferen The

rbEc

I IIo

Looo

⎟⎠

⎞⎜⎝

⎛=

±±==

⎟⎟⎟⎟

⎜⎜⎜⎜

±±==

⎥⎦

⎤⎢⎣

⎡=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛=⎟⎟

⎞⎜⎜⎝

⎛=

sin

sin

sinsin

sin

sincoscos

2,cossin4

2

22

22

2

λλ

θπλ

θπ

λλ

θπα

εα

ββ

Multiple-slit diffraction

[ ][ ]

[ ][ ]{ }

22

2/

1

2))12(

2))12(

2))12(

2))12(sinsin

sinsinsin

⎟⎠

⎞⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

+= ∑ ∫ ∫=

+−−

−−−

+−

−−

αα

ββ

θθ

NII

dsedserE

E

o

N

j

baj

baj

baj

bajiksiks

o

LR

integers other all p for occurminima 2N, N, 0,p for occurmaxima principal

2, 1, 0,p ,θaNp or ,

Np :minima

NNNN :magnitude

2, 1, 0,m ,θam or ,m :maxima principal

(1) --------- N : termce interferen The

limlimmm

2

=±±=⇒

±±===

±==

±±==→

⎟⎠

⎞⎜⎝

→→

sin

coscos

sinsin

sinsin

sin

λπα

αα

αα

λπααα

παπα

Examples of Fraunhofer Diffraction

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎥⎦⎤

⎢⎣⎡ +=

ωη

ωξξπ

222

221 rectrectfmt oA )cos(

Thin sinusoidal amplitude grating

25.00 =η

%./ 2561621 m ≤=+η

%./ 2561621 m ≤=−η

Thin sinusoidal amplitude grating

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ ++⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎥⎦⎤

⎢⎣⎡= )(2sin

4)(2sin

42sin2sin

2),( 2

22

222

2

zfxz

cmzfxz

cmzxc

zyc

zAyxI oo λ

λωλ

λω

λω

λω

λ

),(),(),()cos( YoXYoXYXo fffmfffmfffmFT −+++=⎥⎦⎤

⎢⎣⎡ + δδδξπ

44212

221

m=1

( ) ( ) ⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎥⎦⎤

⎢⎣⎡=

wrect

wrectfmjt A 22

22 0

ηξξπηξ sinexp,

( ) ( ) ⎟⎠⎞

⎜⎝⎛

⎥⎦⎤

⎢⎣⎡ −⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛≈ ∑

−∞= zwyzqfx

zwmJ

zAyxI

qq λ

λλλ

2sinc2sinc2

, 20

222

Thin sinusoidal phase grating

m/2=4

),()sin(exp YoXq

qo fqffmJfmjFT −⎟⎠⎞

⎜⎝⎛=

⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ ∑

−∞=δξπ

22

2

⎟⎠⎞

⎜⎝⎛=

22 mJqqη

%..

%..

min,

max,

27422

422

833812

21

200

211

=⎟⎠⎞

⎜⎝⎛ ==⎟

⎠⎞

⎜⎝⎛ ==

=⎟⎠⎞

⎜⎝⎛ ==

±

±±

mJ where0, mJ

mJ

η

η

Fresnel Diffraction by Square Aperture

(b) Diffraction pattern at four axial positions marked by the arrows in (a) and corresponding to the Fresnel numbers NF=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 NF=0.5.

( )dDx /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.

number Fresnel : zwNF λ/=

2w

Talbot Images( ) ( )[ ]LmtA /2cos1

21, πξηξ +=

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛+=

Lxm

Lx

LzmyxI πππλ 2cos2coscos21

41, 22

2

Appendix : Shah function (Impulse train function)

Appendix : Sampling, Step and Repeat Function