L6 Diffraction Theory (Page22 RSvsHK)

8/2/2019 L6 Diffraction Theory (Page22 RSvsHK)

1/40

Diffraction theory

History of diffraction theoryFrom vector to a scalar theorySome mathematical Preliminaries

Hemholtz equationGreens theorem

The Kirchhoff formulation of diffraction by a planar screenThe integral theorem of Helmholtz and Kirchhoff

The Kirchhoff boundary conditionsFresnel-Kirchhoff diffraction formula

The Rayleigh-Sommerfeld (R-S) formulation of diffractionAlternative Greens functionThe R-S diffraction FormulaComparison between Kirchhoff and R-S theories

enera za on o nonmonoc roma c wavesAngular spectrum of a plane wavePhysical interpretation of the angular spectrumPropagation of the angular spectrumEffects of a diffraction aperture on the angular spectrum

1

Propagation phenomenon as a linear spatial filter


2/40

Diffraction

Diffraction is a phenomena in the realm of physical optics.Applicable to all waves such as acoustic and EM waves.

system performance. So we need to understand it to design bettersystems.

Diffraction should not be mistaken with

Refraction: change of direction of propagation of light due to achange in index of refraction of the environment

Penumbra: finite extend of a source causes the light transmittedfrom an aperture to spread away from it. There is no bending oflight involved in Penumbra effect.

Diffraction (Sommerfeld): any deviation of light rays from rectilinear

. Diffraction is caused by confinement of the lateral extend of a wave

(obstruction of the wavefront) and its effects are most pronounced

2

light.


3/40

History of diffraction theory

1665 Grimaldi reported diffraction for the first time. 1678 Huygens attempted to explain the phenomenon Each point on the wavefront of a disturbance is considered to be a new source of a

secondary spherical disturbance. Then the wavefront at later instances can befound by constructing the envelope of the secondary wavelet.

s rogress on wave eory was suppresse y e ac a ew on avore e

corpuscular theory of light (geometrical optics). 1804 Thomas Young introduced the concept of interference to the wave theory of light

(production of darkness from light).

interference theory letting the wavelets interfere mutually to calculate distribution of lightin diffraction patterns with excellent accuracy.

1860 Maxwell identified light as electromagnetic field. .

assumptions about the boundary values of the light incident on surface of an obstaclethat were not absolutely correct but an approximation and constructed a theory thatexhibited excellent agreement with experimental results. He concluded The amplitudes and phases ascribed to the secondary sources of Huygens wavelets

are og ca consequences o e wave na ure o e g . 1892 Poincare; 1894 Sommerfeld proved that the boundary values set by Kirchhoff are

inconsistent with one another. So Kirchhoffs formulation of Huygens-Fresnel principle isregarded as the first approximation although under most conditions it yields excellent

3

. 1896 Sommerfeld modified the Kirchhoffs theory using theory of Greens function. The

result is Rayleigh-Sommerfeld diffraction theory. 1923 Kottler: first satisfactory generalization of the vectorial diffraction theory.


4/40

From vector to a scalar theory I

In all of these theories light is treated as a scalar phenomenon.

At boundaries the various components of the electric and magnetice s are coup e roug axwe equa ons an canno e rea e

independently.

We stay away from those situations when using scalar theory.

Scalar theory yields correct values under two conditions:

The diffracting aperture must be large compared with a

wavelength.

The diffracting fields must not be observed too close to theaperture.

Our treatment is not good for some optical systems such as

diffraction from

high-resolution gratings

4

Read Goodman 3.2 From a Vector to a Scalar Theory


5/40

From vector to a scalar theory II

, , , , ,

or e ec romagne c waves propaga ng n me a w

the following properties, an scalar wave equation

is obeyed by all components of the field vectors .x y z x y zE E E H H H

2 22

2 2

( , )

( , ) 0

( ,

n u P t

u P t c t

u P t

=) , , .is any of the scalar field components at timex y z t

Depth ofpenetration in the

1 2

1 2

( , ) ( , )

( , ) ( , )

linear; if and are solutions to the wave euation, then

is a solution,

u P t u P t

u P t u P t +

wavelengths.Not much effect ontotal wavefront

passing throughthe aperture

,

homogeneous; permitivity is constant throughout the region of propagation,

nondispersive; permitivity is independent of frequency over the region of propagation,

0nonmagne c; magne c permea y s equa oAt the boundaries, the above criteria are not met and coupling betwen electric and

magnetic components of the EM wave happens.

5

We can use the scalar theory if the boundaries are small portion of the

total area through which the wave is passing.


6/40

The Helmholtz equation

2

( )

( , ) ( ) cos[2 ( )] Re{ ( ) }

( ) ( )

where

space dependent part of the field

j t

j P

u P t A P t P U P e

U p A p e

= =

=2 time dependent part of thej te

, ,

field

can be any of the space coordinates or

Substituting the scalar field in scalar wave equation

P x y z

2 2

2

2 2

2 2 2

( , )( , ) 0

j t

n u P t u P t

c t

=

2 2

2 2

2

2

( ) 0

( 2 )( ) ( )

tU P e

c t

n jU P U P e

=

2 2

2

2

(2 ) 20 with wavenumberj t

nk

= = =

c2 2

( ) ( ) 0 .time-independent Helmholtz equation

c

k U P + =

6

or in homogeneous dielectric media has to obey the Helmholtz equation.


7/40

Gausss Theorem

Gauss's theorem:

total outflow of fluxfrom the volume V

=

i inet outflow

totalof flux perunit volume

V S

outflow of fluxfrom the surface

' '

S

7


8/40

Greens Theorem: a mathematical tool

( ) ( )( )

-U G U G U G = +

=

i i i

i i i

2 2( )U G G U U G G U = i

, ,

over the volume and on the surfaceV

2 2

enclosing theS V

U G G U dV U G G U dV = i

2 2

Using the Gauss's theorem convert the volume integral on LHS

V V

U G G U d U G G U dV = si

If we take the gradiant in the outward normal directiS V

||

on the LHS

can be written in a scalar form since at ever oint.

n

n s

8

2 2

( ) ( ) Green's theoremS VG U

U G ds U G G U dV n n

=


9/40

Physical meaning of the Greens function I

Imagine an inhomogeneous linear differential euation2

2 1 02

( ) ( )( ) ( )

( )

is a driving force

d U x dU xa a a U x V x

dx dx

V x

+ + =

( )

( )

is the solution for a known set of boundary conditions (BC)

is a solution to the

U x

G x ( ) ( ')equation with the impulse driving forceV x x x

( ) ( )

( ) ( ') ( - ') '

an e same s.

is an impulse response and we can expand in terms ofG x U x Gs

U x V x G x x dx=

( ) known as the Green's functionG x

( )

of the problem.

may be regarded as an auxiliary function chosen cleverly toG x

9

solve our problem.


10/40

Physical meaning of the Greens function II

2

Imagine an oscillator

2 02( ) ( )

( ) is a driving force

a a U x V x

dxV x

+ =

( )

' - ' '

is impulse responseG x

U x V x G x x dx='

The solutio

x

n is convolution of the driving force with the impulse respons

of the system.

In case of diffraction application of Green's theorem will yeild differentvariations of the diffraction theory based on the choice of Green's function.

10


11/40

Application of Greens Theorem in scalar

Goal: calculation of the complex disterbance at an observation pointU

0

2 2( ) ( )

n space, , us ng reen's eorem.

S V

P

G GU G ds U G G U dV

n n

=

Green's theorem is the prime foundation of the scalar diffraction theory.

To apply it to the diffraction problem we need to have a proper choice of1 an aux ary unct on reen's unct on

2) a close surface

is an arbitrar oint o

G

S

P f observationn

P1V

1

0( )

is an arbitrary point on the surface

We want solution of the wave equation

P S

U PS

P0. ro1

11

0at in terms of the value of the solution

and its derivatives on the surface .

P

S


12/40

The integral theorem of Helmholtz & Kirchhoff' -

01

0 1 1

01

( )

about point (impulse). has to be a solution of the wave equation. At :

2 Treatin the discontinuit at

jkre

P G P G Pr

=

b isolatin it withP S

34' '3

3) New surface & volume: ;

4) Use Green's theorm with Green's function of

S S S V V = + = V

1

r0101

2

1

01

( ) ) 02and Helmholtz equation (

Note both and are the soluti

jkre

G P k U r

G U

= + =

ons of the same

SP

0

.

n

0( )

0

wave euation. After all is the impulse response and

is the disturbance. We want to find or field after the aperture.

At the linit of we get (follow from Goodman page 41)

G

U U P

:

n

( )01 01

0

01 01

1-

4

This result is known as the integral thorem of Helmholtz and Kirchhoff. It has important role in

jkr jkr

S

U e eU P U ds

n r n r =

12

( )0development of the scaler theory of diffraction. ,U P 0the field at point is expressed in terms

of the "boundary values" of the wave on any closed surface surrounding that point.

P


13/40

Fresnel-Kirchhoff diffraction formula I

0

Problem: diffraction of light by an aperture in an infinite opaque screen.

The field at behind the aperture is to be calculated.U P

01.

Assumptions: 01 011/and

Choice of : a plane surface plus a

r k r

S

>> >> S2S1

Waveimpinging

01

1 2spherical cap

Choice of Green's function:

jkr

S S S

eG

= +

=P0.

r01

n

P1

01

1

We apply the Helmholtz-Kirchhoff integral theorem

U G 1 2

0 4 S S n n +

0

Somefeld radiation condition: if the disturbance vanishesU

13

2 0( )

as contribution of to vanishes.R S U P


14/40

Kirchhoffs Boundary conditions

,

The screen is opaque and the aperture is shown by

1) Across the surface the field and its derivatives are exactly

1

the same as they would be in the absence of the screen.

2) Over the portion of S that lies in the geometrical shadow

( )01

4

.

U G

U P G U dsn n

= Condition 1 is not exactly true since close

to the boundaries the field is disturbed.

2

S1

R

Waveimpinging

Condition 2 is not true since theshadow is never perfect and some

P0.r01n

P1

14

.

For this is an OK approximation. >>

1

2

2

:

0

On these portions of S

U UU

n n

= = =


15/40

Fresnel-Kirchhoff diffraction formula IIS2

P.

n

1

R01

011/

1cos ,

Withe above assumptions and we arrive at

jkr

k r

e UU P kU n r ds

>>

=

r01P

1

21

014

If the aperture is illuminated by a single spherical wavejkr

r n

1

21

( ) ,2located at point P aU Pr

=

21

21 1t a distance r from P.

If we can show that (problem 3.3)r >>

( )21 01( )

01 21

0

21 01

cos( , ) cos( , )

2

The Fresnel-Kirchhoff diffraction formula that

jk r rn r n r A e

U P dsj r r

+

=

o s on y or a s ng e po nt source um nat on

0 2

The Fresnel-Kirchhoff diffraction formula

is symmetrical with respect to P and P .

P0. r01n P1P2

r21

15

0A point source at P will produce the same eff .2ect at P

This result is known as: reciprocity theorem of Helmholtz


16/40

Huygens wavelets

-the superposition of secondary waves that produced from a surface

situated between this point and the light source.

If we rewrite

( )21 01( )

01 21

0

cos( , ) cos( , )

jk r rn r n r A e

U P ds+

=

( )01

21 01

0 1

01

'( ) wherejkr

eU P U P ds

r=

21

01 2

1

21

cos( , ) cos( ,1'( )

illuminating wavefrontObservation angle

jkrn r n r Ae

U Pj r

=

1)

2

Illumination angle

0( )

'( ).

Seems like arises from sum of infinite fictitious sources with

amplitudes and phases expressed by

U P

U P

16

We used point source to get this result but it is possible to generalize

this result for any illlumination by using Rayleigh-Sommerfeld theory.


17/40

The Rayleigh-Sommerfeld Formulation of

iffr i nPotential theory: If a two-dimensional potential function and its normal

,

potential function must vanish over the entire plane.This is also true for solution of a three-dimensional wave equation.

1) The Kirchhoff boundary conitions suggests that the diffracted fieldmust e zero everyw ere e n t e aperture. ot true.

2) Also close to aperture the theory fails to produce the observed

.

Inconsistencies of the Kirchhoff theory were removed by Sommerfeld.

He elliminated the need of imposing boundary values on the

17

disturbance and its derivative simultaneously.


18/40

An alternative Greens function I

10

1( )

2

serve e strengt n terms o t e nc ent e an ts norma er vat ves:

(Fresnel-Kirchhoff diffraction formula)S

U GU P G U ds

n n

=

Conditions for validity:

2) Both and are solutions of the homogeneous scalar wave equation

3) The sommerfeld radiation condition holds i.e. if the disturbance

U G

U

2 0( )

as contribution of to vanishes.R S U P

1/ ,

If the Green's function of Kirchhoff theory was modified so that eitheror vanished over entire surface then there is no need to im

G

G n S

/ .

pose

boundar condtions on both andU U n

18


19/40

An alternative Greens function II

0

omer e argue t at one reen's unct on t at meets t ese cr ter a s

composed of two identical point sourcers at two sides of the aperture,

mirror image of each other, oscillating with a 180 phase difference:

01 01

01

1

01

( )

jkr jkr

e eG Pr r

=

01( )

.

Kirchhoff's BC may be applied only on U

U GU P G U ds = 1

1

1 1 2

10we need

n n

S

U GU P G U ds

= + +

= =

01r

P0

.n

P1

.0

P

01r

1 2

2 n n +

1

| 0 0 1 2so if we require only on andSG U

U

= =

2

19

1 20

With the new BC on the U only there is no conflict withthe potential theorem.

n +


20/40

The Rayleigh-Sommerfeld diffraction

F rm l

01 01

With the Green's functionjkr jkr

G

01

1

01

( )

the takes the form

G P

r r

U P

=

( )01

1

0 1 01

01

1( ) cos( , ) or

jkr

I

S

eU P U P n r ds

j r

=

01

/

Assuming

Now applying the Kirchhoff BC only on and not on we g

r

U U n

>>

et

( )01

0 1 01

01

1 ( ) cos( , )I

eU P U P n r dsj r

=

20

.


21/40

Rayleigh-Sommerfeld Diffraction formula'

( )01 01

01

1 0

01

( )

the takes the formjkr jkr

e eG P U P

r r

+

+ = +

( )

01

1

001

( )1

2

Now for the spacial case illumin

r

II

U P eU P ds

n r

=

ation:

01r ..21

2 1

21

( )

/

a diverging spherical wave from point :

We apply the Kirchhoff BC only on and not on and

jkre

P U P Ar

U U n

=

0r01n P1

0

( )21 01( )

0 01

21 01

cos( , )

using we get

jk r r

I

G

A eU P n r ds

j r r

+

=

and and gives

II

G

U

+

( )21 01( )

0 21

21 01

cos( , )

jk r rA e

P n r dsj r r

+

=

21

0

21

21

90 .Where the angle between and is greater than

This is Rayleigh-Sommerfeld Diffraction formula where we assumed

n r

r >>


22/40

Comparison of the Kirchhoff and Rayleigh-

mm rf l R- h r m01 01 01 01 01

01 01

1 1 1

01 01 01

( ) ; ( ) , ( )jkr jkr jkr jkr jkr

K

e e e e eG P G P G P

r r r r r += = + =

reen unc on o eGreen functions of the Sommerfeld formulationKirchhoff formulation

On th 2 2e surface we can show that andK KG

G G Gn

+

= =

( )01

4For the Kirchhoff theory: KK

GUU P G U ds

n n

=

( )

( )

0

0

2For the R-S theory: KI

II

U P U dsn

U P

=

1

KU

G ds

=

0 0( ) ( )

We can see that

I IIU P U P

U P+

=

22

2

Summary: the Kirchhoff solution is the arithmatic average of the two

Rayleigh-Sommerfeld solutions.


23/40

Comparison of the Kirchhoff and R-S theoryKirchhoff theory:

( )21 01 21 01( ) ( )

01 21

0

cos( , ) cos( , )

2

Obliquity factor

jk r r jk r rn r n r A e A e

U P ds dsr r r r

+ +

= =

( )21 01( )

0 01

21 01

cos( , )

Obliquity fac

R-S theory:jk r r

I

A eU P n r j r r

+= 21 01( )

21 01

tor

jk r rA eds dsj r r

+

=

( )21 01 21 01( ) ( )

0 21

21 01 21 01

cos( , )

Obliquity factorjk r r jk r r

II

A e A eU P n r ds ds

j r r j r r

+ +

= =

01r

P0 .r01n P1

. 0PP0. r01n P1

P2r21 .

23


24/40

Comparison of the Kirchhoff and R-S theoryObliquity factor of both Kirchhoff and R-S theory

01 21

01

1[cos( , ) cos( , )]

2

cos( ,

Kirchhof theory

= ) First R-S solution

n r n r

n r

21cos( , ) Second R-S solutionn r

When a point source is at a very far distance

P0

. r01n

P1P2

r21 .

1[1 cos ]

2

cos(

Kirchhof theory

= ) First R-S solution

+

-

In sum

mary: for small angles all three solutions are identical

far away, the angles are small.

R-S solution requires the diffracting screens be planar.

01r P0.r01

n

P1. 0

P

24

.

For most applications both are OK.

We will use the first R-S solution for simplicity.


25/40

Huygens-Fresnel Principle

25


26/40

Generalization to Non-monochromatic waves I

We generalize the R-S's first solution to nonmonochromatic waves (chromatic?)

Monochromatic time function of the disturbance:

1

, e

( , )Time dependent chromatic functions: at the aper

u t e

u P t

=

0

1 0

, ( , )

( , ), ( , ) ' -

ture observation point

in terms of their Fourier transforms. Let's change the variable

u P t

u P t u P t =

2 2 '

1 1 1( , ) ( , ) ( , ') '

j t j tu P t U P e d U P e d

= =

'

0 0( , ) ( , ) (u P t U P e d U P

= =0

, ') 'monochromaticcomplex amplitudeselementaryof the disturbancefunction ofat frequency 'freuency '

e d

1 1 0 0

, , , ,ere an

We see that the cromatic f

u t u t = =

unction is sum

0( , )u P t

1( , )u P t

01r P0

.r01P1. 0P

26

of the monochromatic functions over

different frequencies.

n


27/40

Generalization to Non-monochromatic waves II2 2 '

' 'j t j t

1 1 1

2 2 '

0 0 0

, , ,

( , ) ( , ) ( , ') 'j t j t

u P t U P e d U P e d

= = 0( , )u P t

1( , )u P t

.r01.

e ementarycomplex amplitudefunction ofof the disturbancefreuency '

at frequency '

Now

we introduce the R-S first solution for the diffracted field

01r 0

n0

( )

1

01 0121

0 01 1 01

21 01 01

'

1 1cos( , ) ( ) cos( , )

Obliquity factorjkr jkrjkr

I

Ae e eU P n r ds U P n r ds

j r r j r

= =

( )0 1'

, ' ( , ')

re

U P j U PV

=

01

01

01

cos( , ) / where

This is one frequency component. Summing over all of them we get:

n r ds V c nr

=

( )012 ' /

0 1 01

', ( , ') cos( , )

Complex amplitude at each frequency

j r Ve

u P t j U P n r ds

=

2 ''

Elementaryfunction atthat frequency

j te d

27

01

( )

012 '( )01

0 1

01

cos( , )

, 2 ' ( , ') '2

rj t

Vn r

u P t j U P e d dsVr

=


28/40

Generalization to Non-monochromatic waves III

Next we want to relate the disturbance at the observation oint

( )012 '( )

01

0 1

01

cos( , ), 2 ' ( , ') '

2

r

j tV

n ru P t j U P e d ds

Vr

=

2 '

1 1( , ) ( , ') '

to t e stur ance at t e aperture ocat on

j tu P t U P e d = we use the identity

0( , )u t

1( , )u P t

01r P0

.r01nP1

.0P

2 '

1 1( , ) ( , ') '

j td du P t U P e d

dt dt

=

2 '

1 1( , ) 2 ' ( , ') '

j tdu P t j U P e d

dt

=

01 01cos( , )n r rd

0,

0

The wave disturbanceat P is linearly proportionalto the time derivative of thedisturbance

1

01

,2

1

Incident wave atOver all anglesthe "retarded" timeor the time that theat eah point Pwave was enerated

Vr dt V =

28

In summary the results of the diffration theory for the monochromatic

waves is applicable to the more general case of the chromatic waves.


29/40

The angular spectrum of plane a wave INext we want to formulate the diffraction theory in a framework of

linear, invariant systems.

( , )

incident on a transverse planeAcro

x y0 ( , , 0)ss the planez U U x y= =

( , , )

( , , ) 0

( , , 0)

Across the plane

Objective: to calculate the resulting field down the road

as a function of

z z U U x y z

U U x y z z

U x y

= =

= >

0 ( , ; 0) ( ,FT of the at plane: X YU z A f f U x= =2 ( )

2 ( )

, 0)

, , 0) , ; 0)And

X Y

X Y

j f x f y

j f x f y

y e dxdy

U x A e d d

+

+=

( , ; 0)

What is the physical meaning of these components?

So far we have looked at as the spatial frequency spectrumX YA f f

29

.

What is the direction of propagation of each these components?


30/40

Physical interpretation of angular spectrum

. 2( , , ) ( );

ons er a s mp e p ane wave propagat ng n rect on o :

where andjP x y z e xx yy zz x y z

= = + + = + +k r r k

2 2 2

, . |

1

, and are the direction cosines of Also |

Using , betwe

=

+ + =

k k

en the direction cosines we rewrite

( ) ( )

2 2

2 2

2 2

( ) 2 ( ) 2 1.

; ; 1| | 2 /

X y X Y

x xX Y Z X Y

x

j x y j z j f x f y j f fjk r

f f f f f

P x z e e e e e

+ +

= = = = = = =

= = =

k

x

/ , / 0

( ,

Now with we can writeX Y z

X Y

f f f

A f f

= = =

2 ( ); 0) ( , , 0) asX Yj f x f yU x y e dxdy

+

=

-1

cos-1

2, ; 0 ( , , 0)

( , , 0).is the an ular s ectrum of the disturbance

j x yA U x y e dxdy

U x y

+

=

cos-1

z

30

In summary this results shows that:

each spatial frequency component is propagating at a different angle.

y


31/40

Propagation of the angular spectrum I

2

( , )ons er e angu ar spec rum o e across a p ane para e o

at a distance from it:

j x y

x y

z

+ =, , ,

Our goal is to find the effects of the wave propagati

on on the angular spectrum

2

, ,

( , , ) , ;

We start from

j x y

U x y z A z e d d

+

= must satisfy the Helmholtz equatU 2 2

22

0ion wherever there is no source.

The result is that must satisfy the following differential equation.

U k U

A

+ =

2 2

2 , ; 1 , ; 0

coefficient

The soluti

A z A zdz + =

on can have the form:

31

2 221

, ; , ; 0

j z

A z A e

=


32/40

Propagation of the angular spectrum II2 2

1

, ; , ; 0

For this solution two cases are recognized:

j z

A Z A e

=

2 211) When (true for all direction cosines) the effect of propagation

on the angular spectrum is simply

+

ier transform ofa field distributiion

, ; , ; 0A z A =

2 221

2 22, ; 0 1

A real number

on which BCs of theaperture is imposed

;

j z

ze A e

= = +

Since is a positive real number, thz e wave components are attenuating

as they propagate. They are also called evanescent waves.

32

mm ar to t e case o m crowave wavegu es t ere s a cuto requency.

Below cutoff frequency, these evanescent waves carry no energy away

from the aperture.


33/40

Propagation of the angular spectrum III2 22

1

, ; , ; 0 ( , , )Substituting in the we get:j z

A Z A e U x y z

=

2 221 2

( , , ) , ;x y

j z j x y

U x y z A Z e d d

+

=

, , , ;x y z e e

=

2 2

1 1 x y

+

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L6 Diffraction Theory (Page22 RSvsHK)