3. Foundations of Scalar 3. Foundations of Scalar Diffraction TheoryDiffraction Theory

Introduction to Fourier Optics, Chapter 3, J. Goodman

0

0

E

H

HEt

EHt

ε

μ

μ

ε

∇⋅ =

∇⋅ =

∂∇× = −

∂∂

∇× =∂

2 22

2 2

( , )( , ) 0,

where,

1 ,o o o

n u P tu P tc t

n cεε μ ε

∂∇ − =

∂

= =

Maxwell’s equationsin the absence of free charge,

Scalar wave equationin a linear, isotropic, homogeneous,

and nondispersive medium,

( ) 0ln2 2

2

2

22 =

∂∂

−∇⋅∇+∇tE

cnnEE

If the medium is inhomogeneous with ε(P)that depends on position P,(See Appendix.)

HelmholtzHelmholtz EquationEquation

( ) ( ) ( )[ ]PtPAtPu φπν += 2cos,

( ) ( ) ( ){ }tjPUtPu πν2expRe, −=

( ) ( ) ( )[ ]PjPAPU φ−= exp

( ) 022 =+∇ UkHelmholtz Equation,

2 2

w h ere

k ncν ππ

λ= =

Complex notation

For a monochromatic wave,

The complex function of position (called a phasor)

Helmholtz equationHelmholtz equation

Helmholtz, Hermann von (1821-1894)

Helmholtz sought to synthesize Maxwell's electromagnetic theory of light with the central force theorem. To accomplish this, he formulated an electrodynamic theory of action at a distance in which electric and magnetic forces were propagated instantaneously.

In 3 dimension,

Cartesian

Cylindrical

Spherical

GreenGreen’’s Theorems Theorem

Let U(P) and G(P) be any two complex-valued functions of position, and let S be a closed surface surrounding a volume V.

If U, G, and their first and second partial derivatives are single-valued and continuous within and on S, then we have

( ) dsnUG

nGUdUGGU

s∫∫∫∫∫ ⎟

⎠⎞

⎜⎝⎛

∂∂

−∂∂

=∇−∇ υν

22

Where signifies a partial derivative in the outward normal

direction at each point on S.n∂

∂

P

V

S

n

GreenGreen’’s Functions Function

( ) ( ) ( ) ( )xVUxadxdUxa

dxUdxa =++ 012

2

2

( ) ( ) ( ) ( ) ( ) ( ) ( )''0

'

12

'2

2 xxxxGxadx

xxdGxadx

xxGdxa −=−+−

+− δ

( ) ( ) ( ) ''' dxxVxxGxU ∫ −=

“Green’s function” : impulse response of the system

Consider an inhomogeneous linear differential equation of,

where V(x) is a driving function and U(x) satisfies a known set of boundary conditions.

If G(x) is the solution when V(x) is replaced by the δ(x-x’), such that

Then, the general solution U(x) can be expressed through a convolution integral,

Since the system is linear shift-invariant.

KirchhoffKirchhoff’’ss GreenGreen’’s functions function

( )2 2 0 ∇ + =k G

( ) ( )01

011

exprjkrPG =

Kirchhoff’s Green’s function :A unit amplitude spherical wave expanding about the point Po (called “free-space Green’s function”)

P1

r01

V’ : the volume lying between S and Sε

S’ = S + Sε

Within the volume V’, the disturbance G satisfies

Integral Theorem of Integral Theorem of HelmholtzHelmholtz and Kirchhoffand Kirchhoff

( ) ( )s

G UU G G U d Uk G Gk U d U G dsn n

2 2 2 2

' ' '0

ν νυ υ ∂ ∂⎛ ⎞∇ − ∇ = − − = = −⎜ ⎟∂ ∂⎝ ⎠

∫∫∫ ∫∫∫ ∫∫

( ) ( )01

011

exprjkrPG =

Kirchhoff’s Green’s function

'0

ε

∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞− = ⇒ − − = −⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠∫∫ ∫∫ ∫∫s S S

G U G U G UU G ds U G ds U G dsn n n n n n

V’ : the volume lying between S and Sε

S’ = S + Sε

P1r01

( ) dsnUG

nGUdUGGU

s∫∫∫∫∫ ⎟

⎠⎞

⎜⎝⎛

∂∂

−∂∂

=∇−∇ υν

22GreenGreen’’s Theorems Theorem

( ) ( )01

011

exprjkrPG =

( ) ( ) ( ) dsr

jkrn

Ur

jkrnUPU

S∫∫

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡∂∂

−⎥⎦

⎤⎢⎣

⎡∂∂

=01

01

01

010

expexp41π

P1

r01

Note that for a general point P1 on S’, we have

( )01101

01 01

exp( ) 1cos( , )( )jkrG P n r jk

n r r∂

= −∂

where represents the cosine of the angle between and . 01cos( , )n r n 01rFor a particular case of P1 on Sε , and 01 1cos( n,r ) = −

( ) ( )1

exp jkG P

εε

=( )1 exp( ) 1( )

jkG P jkn

εε ε

∂= −

∂

Letting ε -> 0,

2 ( )exp( ) 1 exp( )lim lim 4 ( ) 4 ( )oo o

S

U PG U jk jkU G ds U P jk U Pn n nε εε

ε επε πε ε ε→ ∞ → ∞

∂⎡ ⎤∂ ∂⎛ ⎞ ⎛ ⎞− = − − =⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦∫∫

Integral theorem of Helmholz and Kirchhoff

KirchhoffKirchhoff’’ss Formulation of DiffractionFormulation of Diffraction(A planar screen with a single aperture)(A planar screen with a single aperture)

n̂

n̂

R

S P

θ’ θ

Σ

Σ

Have infinite screen with aperture A

Radiation from source, S, arrives at aperture with amplitude

'

'

reEE

ikr

o=

Let the hemisphere (radius R) and screen with aperture comprise the surface (Σ)enclosing P.

Since R →∞

E=0 on Σ.

Also, E = 0 on side of screen facing V.

r’r

KirchhoffKirchhoff’’ss Formulation of DiffractionFormulation of Diffraction

( )1 2

010

01

exp( )1 .4 S S

jkrU GU P G U ds, where Gn n rπ +

∂ ∂⎛ ⎞= − =⎜ ⎟∂ ∂⎝ ⎠∫∫

Kirchhoff boundary conditions

are and On - nU/, U ∂∂∑screen. no were thereif as

0. and 0 ,except On - 1 =∂∂=Σ nU/US

S1

S = S1 + S2

From the integral theorem of Helmholz and Kirchhoff

Σ : aperture

FresnelFresnel--KirchhoffKirchhoff’’ss Diffraction Formula (I)Diffraction Formula (I)

( ) ( ) ( )01

01

0101

1 exp1,cosr

jkrr

jkrnnPG

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

∂∂

( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛>>→>>≈

01r1k for λ01

01

0101

exp,cos rr

jkrrnjk

( ) ( )21

211

expr

jkrAPU =

( ) ( )[ ] ( ) ( ) dsrnrnrr

rrjkjAPU ⎥⎦

⎤⎢⎣⎡ −+

= ∫∫∑ 2

,cos,cosexp 2101

0121

01210 λ

For an illumination consisting of a single point source at P2 :

FresnelFresnel--KirchhoffKirchhoff’’ss Diffraction Formula (II)Diffraction Formula (II)

( ) ( ) ( ) dsrjkrPUPU

01

0110

exp∫∫∑

′=

( ) ( )⎥⎦

⎤⎢⎣

⎡=′

21

211

exp1r

jkrAj

PUλ

( ) ( )⎥⎦⎤

⎢⎣⎡ −

2,cos,cos 2101 rnrn

restricted to the case of an aperture illumination consisting of a single expanding spherical wave.

Kirchhoff’s boundary conditions are inconsistent! : Potential theory says that “If 2-D potential function and it normal derivative vanish together along any finite curve segment, then the potential function must vanish over the entire plane”.

Rayleigh-Sommerfeld theory

the scalar theory holds.

Both U and G satisfy the homogeneous scalar wave equation.

The Sommerfeld radiation condition is satisfied.

Huygens-Fresnel

SommerfeldSommerfeld radiation conditionradiation condition

( )02

1 04

as R π

∂ ∂⎛ ⎞= − = → ∞⎜ ⎟∂ ∂⎝ ⎠∫∫S

U GU P G U dsn n

2

2

exp( )

1 exp( )

( )S

jkRGR

G jkRjk jkGn R R

U UG jkG U ds G jkU R dn nΩ ω

=

∂ ⎛ ⎞= − ≈⎜ ⎟∂ ⎝ ⎠∂ ∂⎡ ⎤ ⎛ ⎞− = −⎜ ⎟⎢ ⎥∂ ∂⎣ ⎦ ⎝ ⎠

∫∫ ∫

S1

lim 0R

UR jkUn→∞

∂⎛ ⎞∴ − =⎜ ⎟∂⎝ ⎠

On the surface of S2, for large R,

It is satisfied if U vanishes at least as fast as a diverging spherical wave.

First First RayleighRayleigh--SommerfeldSommerfeld SolutionSolution(When (When UU at at ΣΣ is known)is known)

( ) dsnGUG

nUPU

S∫∫ ⎟

⎠⎞

⎜⎝⎛

∂∂

−∂∂

=1

41

0 π

( ) ( ) ( )01

01

01

011 ~

~expexpr

rjkrjkrPG −=−

( ) dsn

GUPU I ∫∫∑

−

∂∂−

=π41

0( ) ( )

nPG

nPG

∂∂

=∂

∂ − 11 2

( ) dsnGUPU I ∫∫

∑ ∂∂−

=π21

0

Suppose two point sources at P0 and P0 (mirror image):~

Second Second RayleighRayleigh--SommerfeldSommerfeld SolutionSolution(When (When UU’’nn at at ΣΣ is known)is known)

( ) ( ) ( )01

01

01

011 ~

~expexpr

rjkr

jkrPG +=+

( ) dsGnUPU II +

∑∫∫ ∂

∂=

π41

0

GG 2=+

( ) GdsnUPU II ∫∫

∑ ∂∂

=π21

0

RayleighRayleigh--SommerfeldSommerfeld Diffraction FormulaDiffraction Formula

( ) ( ) ( ) ( )dsrnrjkrPU

jPU I 01

01

0110 ,cosexp1

∫∫∑

=λ

( ) ( ) ( ) dsrjkr

nPUPU II

01

0110

exp21

∫∫∑ ∂

∂=

π

( ) ( )21

211

exprjkrAPU =

( ) ( )[ ] ( ) sdrnrr

rrjkjAPU I ∫∫

∑

+= 01

0121

01210 ,cosexp

λ

( ) ( )[ ] ( ) sdrnrr

rrjkjAPU II ∫∫

∑

+−= 21

0121

01210 ,cosexp

λ

For the case of a spherical wave illumination,

The 1st and 2nd R_S solutions are,

Comparison (I)Comparison (I)

( ) dsn

GUGnUPU K

K∫∫∑

⎟⎠⎞

⎜⎝⎛

∂∂

−∂∂

=π41

0

( ) ∫∫∑ ∂

∂−= ds

nGUPU K

π21

01

( ) dsGnUPU KII ∫∫

∑ ∂∂

=π21

0

Wow! Surprising! The Kirchhoff solution is the arithmetic average of

the two Rayleigh-Sommerfeld solutions!

F – K :

1st R-S :

2nd R-S :

Comparison (II)Comparison (II)

( ) ( )[ ]factorobliquity , :exp

0121

01210 ψψ

λds

rrrrjk

jAPU ∫∫

∑

+=

( ) ( )[ ]

( )

( )⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−

−

=

21

01

2101

,cos

,cos

,cos,cos21

rn

rn

rnrn

ψ

Kirchhoff theory

First Rayleigh-Sommerfeld solution

Second Rayleigh-Sommerfeld solution

For a normal plane wave illumination, (that means r21 infinite)[ ]

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧ +

=

1

cos

cos121

θ

θ

ψ

Kirchhoff theory

First Rayleigh-Sommerfeld solution

Second Rayleigh-Sommerfeld solution

For a spherical wave illumination,

Again, remember Again, remember ………… After the After the HuygensHuygens--FresnelFresnel principle 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.

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.

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.

Again, letAgain, let’’s compare the s compare the KirchhofKirchhof’’ss and and SommerfeldSommerfeld …………

HuygensHuygens--FresnelFresnel PrinciplePrinciplerevised by 1revised by 1stst R R –– S solutionS solution

( ) ( ) ( ) 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

Generalization to Generalization to NonmonochromaticNonmonochromatic WavesWaves

( ) ( ) ( ) νπνν dtjPUtPu 200 exp,, ∫∞∞−=

( ) ( ) dsrtPudtd

rrntPu ⎟

⎠⎞

⎜⎝⎛ −= ∫∫

∑ υπυ01

101

010 ,

2,cos,

( ) ( ) ( ) νπνν dtjPUtPu 211 exp,, ∫∞∞−=

At the observation point

At an aperture point

Angular SpectrumAngular Spectrum

( ) ( ) ( )[ ]dxdyyfxfjyxUffA YXYX +−= ∫ ∫∞

∞−π200 exp,,;,

( ) ( ) ( )

) , ,k where

eerkjzyxPzjyxj

γβαλπ

γλπβα

λπ

(,

exp,,2

22

=

=⋅=+

( ) ( )221 YX Y X ff f f λλγλβλα −−===

( ) dxdyyxjyxUA ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +−=⎟

⎠⎞

⎜⎝⎛

∫ ∫∞

∞− λβ

λαπ

λβ

λα 200 exp,,;,

Angular spectrum

Propagation of the Angular SpectrumPropagation of the Angular Spectrum

02 =+∇ UkU equation,Helmholtz the satisfy must U 2

( ) dxdyyxjzyxUzA ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +−=⎟

⎠⎞

⎜⎝⎛ ∫ ∫

∞

∞− λβ

λαπ

λβ

λα 2exp,,;,

( )λβ

λα

λβ

λαπ

λβ

λα ddyxjzAzyxU ∫ ∫

∞

∞−⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +⎟

⎠⎞

⎜⎝⎛= 2exp;,,,

⎟⎠⎞

⎜⎝⎛ −−⎟

⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛∴ zjAzA 22120 βα

λπ

λβ

λα

λβ

λα exp;,;,

[ ] 012 222

2

2

=⎟⎠⎞

⎜⎝⎛−−⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛ zAzA

dzd ;,;,

λβ

λαβα

λπ

λβ

λα

Propagation of the Angular SpectrumPropagation of the Angular Spectrum

angle different a at propagates component wave-plane each : 2 02 <+ βα

⎟⎠⎞

⎜⎝⎛ −−⎟

⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛ zjAzA 2212exp0;,;, βα

λπ

λβ

λα

λβ

λα

( ) ∫ ∫ ⎟⎠⎞

⎜⎝⎛ −−⎟

⎠⎞

⎜⎝⎛=

∞

∞−

zjAzyxU 2212exp0;,,, βαλπ

λβ

λα

( )λβ

λα

λβ

λαπβα ddyxj ⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ++× 2expcirc 22

wavesevanescent : 2 02 >+ βα

Effect of ApertureEffect of Aperture

( ) ( )( ) aperture the of function ncetransmitta amplitude :

yxUyxUyxt

i

tA 0

0;,;,, =

⎟⎠⎞

⎜⎝⎛⊗⎟

⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛

λβ

λα

λβ

λα

λβ

λα ,,, TAA it

• For the case of a unit amplitude plane wave incidence,

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛

λβ

λαδ

λβ

λα ,,iA

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛⊗⎟

⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛

λβ

λα

λβ

λα

λβ

λαδ

λβ

λα ,,,, TTAt

Propagation as a Linear Spatial FilterPropagation as a Linear Spatial Filter

( ) ( ) ( ) ( ) ⎟⎠⎞⎜

⎝⎛ += 22circ 0;,;, YXYXYX ffffAzffA λλ

( ) ( ) ⎥⎦⎤

⎢⎣⎡ −−× 2212exp YX ffzj λλ

λπ

( )( ) ( )

⎪⎪⎩

⎪⎪⎨

⎧⎥⎦⎤

⎢⎣⎡ −−

=

0

12exp,

22YX

yX

ffzjffH

λλλ

πλ122 ⟨+ YX ff

otherwise

Transfer function of wave propagation phenomenon in free space

BPM (Beam Propagation Method)BPM (Beam Propagation Method)

( )],,[)( zyxnaveragezn =

⎟⎟⎠

⎞⎜⎜⎝

⎛−−⎟

⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛ + 2222 )2()2()(exp;,;,

λβπ

λαπδ

λβ

λαδ

λβ

λα

oh kznzjzAzzA

( ) ⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ +⎟

⎠⎞

⎜⎝⎛ +=+ ∫ ∫

∞

∞− λβ

λαβα

λπδ

λβ

λαδ ddyxjzzAzzyxU hh )(2exp;,,,

on (x,y) plane

( ) ( ) [ ]zkznjzzyxUzzyxU oh δδδδ )(exp,,,, −+=+

Invalid for wide angle propagation -> WPM (wave propagation method)

From Maxiwell’s equations to wave equations

AppendixAppendix