Diffraction Theory Diffraction Theory

3. Foundations of Scalar 3. Foundations of Scalar

Diffraction Theory Diffraction Theory

Introduction to Fourier Optics, Chapter 3, J. Goodman

0 0 E

H E H

t H E

t ε

μ

μ ε

∇⋅ =

∇⋅ =

∇× = − ∂

∂

∇× = ∂

∂ G G

G G G G G

G G G

2 2

2

2 2

( , )

( , ) 0,

where,

, 1

o o o

n u P t u P t

c t

n

ε

ε μ ε

∇ − ∂ =

∂

= =

Maxwell’s equations

in the absence of free charge,

Scalar wave equation

in a linear, isotropic, homogeneous, and nondispersive medium,

( ^ln ) ⁰

2

2 2 2

2

=

∂

− ∂

∇

⋅

∇ +

∇ t

E c

n n E

E

G G G G G

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

(See Appendix.)

Helmholtz

Helmholtz Equation Equation

( ) P t A ( ) P [ t ( ) P ]

u , = cos 2 πν + φ

( ) ^{P t} { ^U ( ) ^P ( ^{j t} ) }

u , = Re exp − 2 πν

( ) ^{P A} ( ) ^P [ ^j ( ) ^P ]

U = exp − φ

( ^∇

^{+ k}

) ^{U = 0}

Helmholtz Equation

, 2 2 w h ere

k n

c

ν π

π λ

= =

Complex notation For a monochromatic wave,

The complex function of position (called a phasor)

Helmholtz equation Helmholtz 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

Green

Green ’ ’ s Theorem s 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

( ) ^ds

n G U n

U G d

U G

G U

∫∫

∫∫∫ ^{∇ − ∇ υ = ⎜} _⎝ ^{⎛ ∂} _∂ ^{− ∂} _∂ ^⎟ _⎠ ^⎞

ν

2 2

Where signifies a partial derivative in the outward normal direction at each point on S.

∂n

∂

P V

S

n G

Green

Green ’ ’ s Function s Function

( ) ( ) ^a ( ) ^{x U V} ( ) ^x

dx x dU dx a

U x d

a

+

+

=

( ) ( ) _{( )} ( )

_{( )} (

) (

)

' 2 1

' 2

2

a x G x x x x

dx x x x dG

dx a x x G x d

a − + − = −

− + δ

( ) ^{x G} ( ) ( ) ^{x x}

^{V x}

^dx

U ⁼ ∫ ⁻

“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.

Kirchhoff

Kirchhoff ’ ’ s s Green Green ’ ’ s function s function

( ^{∇ +}

^k

) ^{G = 0}

( ) ( )

01 01 1

exp r P jkr

G =

Kirchhoff’s Green’s function :

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

P

r

V’ : the volume lying between S and S_ε S’ = S + S_ε

Within the volume V’, the disturbance G satisfies

Integral Theorem of

Integral Theorem of Helmholtz Helmholtz and Kirchhoff and Kirchhoff

( ) ( )

s

G U

U G G U d Uk G Gk U d U G ds

n n

2 2 2 2

' ' '

0

ν ν

υ υ ^{⎛ ∂ ∂ ⎞}

∇ − ∇ = − − = = ⎜ ⎝ ∂ − ∂ ⎟ ⎠

∫∫∫ ∫∫∫ ∫∫

( ) ( )

01 01 1

exp r P jkr

G =

Kirchhoff’s Green’s function

'

0

ε

∂ ∂ ∂ ∂ ∂ ∂

⎛ − ⎞ = ⇒ − ⎛ − ⎞ = ⎛ − ⎞

⎜ ∂ ∂ ⎟ ⎜ ∂ ∂ ⎟ ⎜ ∂ ∂ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

∫∫ ∫∫ ∫∫

s S S

G U G U G U

U G d s U G d s U G d s

n n n n n n

V’ : the volume lying between S and S_ε S’ = S + S_ε

P

r

( ) ^ds

n G U n

U G d

U G

G U

s

∫∫

∫∫∫ ^{∇ − ∇ υ = ⎜} _⎝ ^{⎛ ∂} _∂ ^{− ∂} _∂ ^⎟ _⎠ ^⎞

ν

2

Green

Green’ ’s Theorem s Theorem

( ) ( )

01 01 1

exp r P jkr

G =

( ) ( ) ( )

r ds jkr U n

r jkr n

P U U

S

∫∫ ⎭ ⎬ ⎫

⎩ ⎨

⎧ ⎥

⎦

⎢ ⎤

⎣

⎡

∂

− ∂

⎥ ⎦

⎢ ⎤

⎣

⎡

∂

= ∂

01 01 01

01 0

exp exp

4 1

π

P₁

r₀₁

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

(

)

1 01

01 01

( ) 1 exp

cos( ,G G )( ) jkr

G P n r jk

n r r

∂ = −

∂

where represents the cosine of the angle between and .

cos( , G G

)

n r G

n

G r

For a particular case of P₁ on S_ε , and

G G

1 cos( n,r ) = −

( ) ( )

1

exp jk

G P ε

= ε ^{( )}₁ ^exp

( )

( )

G P jk n jk

ε

ε ε

∂ = −

∂ Letting

ε

2 ex p ( ) 1 ( ) ex p ( )

lim lim 4 ( _o) ^o 4 ( _o)

S

G U jk U P jk

U G d s U P jk U P

n n n

ε ε ε

ε ε

πε π

ε ε ε

→ ∞ → ∞

∂

⎡ ⎤

∂ ∂

⎛ − ⎞ = ⎛ − ⎞ − =

⎜ ∂ ∂ ⎟ ⎢ ⎜ ⎟ ∂ ⎥

⎝ ⎠ ⎣ ⎝ ⎠ ⎦

∫∫

Integral theorem of Helmholz and Kirchhoff

Kirchhoff

Kirchhoff ’ ’ s s Formulation of Diffraction Formulation of Diffraction

(A planar screen with a single aperture) (A planar screen with a single aperture)

nˆ

nˆ

R

S P

θ’ θ

Σ

Σ Have infinite screen

with aperture A

Radiation from source, S, arrives at aperture with amplitude

'

'

r E e

E

ikr

=

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

Kirchhoff

Kirchhoff ’ ’ s s Formulation of Diffraction Formulation of Diffraction

( )

1 2

01 0

01

exp( )

1 .

4

jkr

U G

U P G U ds, where G

n n r

π

∂ ∂

⎛ ⎞

= ∫∫ ⎜ ⎝ ∂ − ∂ ⎟ ⎠ =

Kirchhoff boundary conditions

are and

On

- ∑ , U ∂ U/ ∂ n

screen.

no were

there if

as

0.

and 0

, except On

- S

Σ U = ∂ U/ ∂ n =

S

S = S

+ S

From the integral theorem of Helmholz and Kirchhoff

Σ : aperture

Fresnel

Fresnel - - Kirchhoff Kirchhoff ’ ’ s s Diffraction Formula (I) Diffraction Formula (I)

( ) ( ) ( )

01 01 01

01

1

1 exp

,

cos r

jkr jk r

r n n

P

G ⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ −

∂ =

∂ G G

( ) ( )

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ >> → >>

≈

r

k 1

for

λ

01 01 01

, exp

cos r

r r jkr

n

jk G G

( ) ( )

21 21 1

exp r

jkr P A

U =

( ) [ ( ) ] ( ) ( )

r ds n r

n r

r

r r

jk j

P A

U ⎥⎦ ⎤

⎢⎣ ⎡ −

= ∫∫ +

∑

2

, cos ,

cos

exp

01 21

01 21

0

G G G

G λ

For an illumination consisting of a single point source at P

:

Fresnel

Fresnel - - Kirchhoff Kirchhoff ’ ’ s s Diffraction Formula (II) Diffraction Formula (II)

( ) ( ) ( )

r ds P jkr

U P

U

01 01 1

0

∫∫ exp

∑

= ′

( ) ( )

⎥ ⎦

⎢ ⎤

⎣

= ⎡

′

21

21 1

exp 1

r jkr A

P j

U λ

( ) ( )

⎥⎦ ⎤

⎢⎣ ⎡ −

2

, cos ,

cos n G r G

n G r G

restricted to the case ofan 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

Sommerfeld

Sommerfeld radiation condition radiation condition

( )

2

1 0

4 as R

π

∂ ∂

⎛ ⎞

= ∫∫ ⎜ ⎝ ∂ − ∂ ⎟ ⎠ = → ∞

S

U G

U P G U ds

n n

2 2

exp( )

1 exp( )

( )

S

G jkR

R

G jkR

jk jkG

n R R

U U

G jkG U ds G jkU R d

n

n ω

=

∂ ∂ = ⎛ ⎜ ⎝ − ⎞ ⎟ ⎠ ≈

∂ ∂

⎡ − ⎤ = ⎛ ⎜ − ⎞ ⎟

⎢ ∂ ⎥ ∂

⎣ ⎦ ⎝ ⎠

∫∫ ∫

S

lim 0

R

R U jkU

→∞ n

⎛ ∂ ⎞

∴ ⎜ ⎝ ∂ − ⎟ ⎠ =

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 Rayleigh Rayleigh - - Sommerfeld Sommerfeld Solution Solution (When

(When U U at at Σ Σ is known) is known)

( ) ^ds

n U G

n G P U

U

S

∫∫ ^⎜ _⎝ ^{⎛ ∂} _∂ ^{− ∂} _∂ ^⎟ _⎠ ^⎞

= 4

1

0

π

( ) ( ) ( )

01 01 01

01

1

~

exp ~ exp

r r jk r

P jkr

G

= −

( ) ^ds

n U G

P

U

∫∫

∑

−

∂

∂

= − π 4

1

0

( ) ( )

n P G n

P G

∂

= ∂

∂

∂

2

( ) ^ds

n U G

P

U

∫∫

∑

∂

∂

= −

π 2

1

0

Suppose two point sources at P

and P

(mirror image):

Second

Second Rayleigh Rayleigh - - Sommerfeld Sommerfeld Solution Solution (When

(When U U ’ ’ _{n n} at at Σ Σ is known) is known)

( ) ( ) ( )

01 01 01

01

1

~

exp ~ exp

r r jk r

P jkr

G

= +

( ) ^{G ds}

n P U

U

∫∫

^∂ _∂

= 4 π 1

0

G G

= 2

( ) ^Gds

n P U

U

∫∫

∑

∂

= ∂

π 2

1

0

Rayleigh

Rayleigh - - Sommerfeld Sommerfeld Diffraction Formula Diffraction Formula

( ) ( ) ( ) ( ) ^{n r ds}

r P jkr

j U P

U

01 01 1

0

1 exp cos G , G

∫∫

= λ

( ) ( ) ( )

r ds jkr n

P P U

U

01 01 1

0

exp 2

1 ∫∫

∑

∂

= ∂ π

( ) ( )

21 21 1

exp r A jkr P

U =

( ) [ ( ) ] _{( )}

s d r r n

r

r r

jk j

P A

U

∫∫

∑

= +

01 21

01 21

0

exp cos G , G

λ

( ) [ ( ) ] _{( )}

s d r r n

r

r r

jk j

P A

U

∫∫

∑

− +

=

01 21

01 21

0

exp cos G , G

λ

For the case of a spherical wave illumination,

The 1

and 2

R_S solutions are,

Comparison (I) Comparison (I)

( ) ^ds

n U G

n G P U

U ∫∫

∑

⎟ ⎠

⎜ ⎞

⎝

⎛

∂

− ∂

∂

= ∂ π 4

1

0

( ) ∫∫

∑

∂

− ∂

= ds

n U G

P

U

π 2

1

0 1

( ) ^{G ds}

n P U

U

∫∫

∑

∂

= ∂ π 2

1

0

Wow! Surprising!

The Kirchhoff solution is the arithmetic average of

the two Rayleigh-Sommerfeld solutions!

F – K :

1

R-S :

2

R-S :

Comparison (II) Comparison (II)

( ) [ ( ) ]

factor obliquity

, :

exp

01 21

01 21

0

ψ ψ

λ r r ^ds

r r

jk j

P A

U ∫∫

∑

= +

( ) ( )

[ ]

( )

( )

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

−

−

=

21 01

21 01

, cos

, cos

, cos

, 2 cos

1

r n

r n

r n r

n

G G

G G

G G G

G

ψ

Kirchhoff theory

First Rayleigh-Sommerfeld solution Second Rayleigh-Sommerfeld solution

For a normal plane wave illumination, (that means r

Æ infinite)

[ ]

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧ +

= 1

cos

cos 2 1

1 θ

θ ψ

Kirchhoff theory

First Rayleigh-Sommerfeld solution

Second Rayleigh-Sommerfeld solution

For a spherical wave illumination,

Again, remember

Again, remember …… …… After the Huygens After the Huygens- -Fresnel Fresnel 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 theory A very rigorous solution of partial differential wave equation.

The first solution utilizing the electromagnetic theory of light.

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.

Again, let

Again, let ’ ’ s compare the s compare the Kirchhof Kirchhof ’ ’ s s and and Sommerfeld Sommerfeld …… ……

Huygens

Huygens - - Fresnel Fresnel Principle Principle revised by 1

revised by 1 ^{st st} R R – – S solution S solution

( ) ( ) ( )

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

∫∫

,

∑

=

( ) ( ) _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

Generalization to

Generalization to Nonmonochromatic Nonmonochromatic Waves Waves

( ) ^{P t U} ( ^{P ν} ) ( ^{j πν t} ) ^{d ν}

u

, ⁼ ∫

, exp 2

( ) ( )

r ds t

P dt u

d r

r t n

P

u ⎟

⎠

⎜ ⎞

⎝

⎛ −

= ∫∫

∑

πυ

υ

0

,

2

, , cos

G G

( ) P t U ( P ^ν ) ( j ^πν t ) d ^ν u

, ⁼ ∫

, exp 2

At the observation point

At an aperture point

Angular Spectrum Angular Spectrum

( ^{f f} ) ^U ( ^{x y} ) [ ^j ( ^{f x f y} ) ] ^dxdy

A

= ∫ ∫

−

+

∞

−

π 2 0

0 , , exp

; ,

( ) ( ) ^{( )}

) , , k

where

e e

r k j z

y x

P

γ β λ α

π

λ γ β π

λ α π

( ,

exp ,

,

2

2 2

=

=

⋅

=

G G G

( )

( )

1

Y

X

f f f

f β λ γ λ λ

λ

α = = = − −

( x y ) j x y dxdy

U

A ⎥

⎦

⎢ ⎤

⎣

⎡ ⎟

⎠

⎜ ⎞

⎝

⎛ +

−

⎟ =

⎠

⎜ ⎞

⎝

⎛ ∫ ∫

∞

−

λ

β λ

π α λ

β λ

α , ; 0 , , 0 exp 2

Angular spectrum

Propagation of the Angular Spectrum Propagation of the Angular Spectrum

2

= 0 +

∇ U k U

equation, Helmholtz

the satisfy must

U

(

)

U z

A ⎥

⎦

⎢ ⎤

⎣

⎡ ⎟

⎠

⎜ ⎞

⎝

⎛ +

−

⎟ =

⎠

⎜ ⎞

⎝

⎛

∫ ∫

∞

−

λ

β λ

π α λ

β λ

α

( )

λ β λ α λ

β λ

π α λ

β λ

α z j x y d d

A z

y x

U

∫ ∫

∞

− ⎥

⎦

⎢ ⎤

⎣

⎡ ⎟

⎠

⎜ ⎞

⎝

⎛ +

⎟⎠

⎜ ⎞

⎝

= ⎛ , ; exp 2

, ,

⎟ ⎠

⎜ ⎞

⎝

⎛ − −

⎟ ⎠

⎜ ⎞

⎝

= ⎛

⎟ ⎠

⎜ ⎞

⎝

∴ A ⎛ z A j 2 1

z

0 α β

λ π λ

β λ α λ

β λ

α , ; , ; exp

[

]

2 ² _{2 2}

2

2 ⎟ =

⎠

⎜ ⎞

⎝

− ⎛

⎟ −

⎠

⎜ ⎞

⎝ +⎛

⎟⎠

⎜ ⎞

⎝

⎛ z A z

dz A

d , ; , ;

λ β λ β α

λ α π λ

β

λ

α

Propagation of the Angular Spectrum Propagation of the Angular Spectrum

angle different

a at propagates component

wave -

plane each

:

2

2

< 0 + β α

⎟ ⎠

⎜ ⎞

⎝

⎛ − −

⎟ ⎠

⎜ ⎞

⎝

= ⎛

⎟ ⎠

⎜ ⎞

⎝

⎛ z A j z

A 2 1

exp 0

; ,

;

, α β

λ π λ

β λ α λ

β λ α

( ) ⁼ ∫ ∫

^⎜ _⎝ ^{⎛ ⎟} _⎠ ^{⎞ ⎜} _⎝ ^{⎛ − − ⎟} _⎠ ^⎞

∞

−

z j

A z

y x

U 2 1

exp 0

; ,

,

, α β

λ π λ

β λ α

( ^{α β} ) ^{j π α} _λ ^x _λ ^{β y} _⎥ ^{d α} _λ ^d _λ ^β

⎦

⎢ ⎤

⎣

⎡ ⎟

⎠

⎜ ⎞

⎝

⎛ +

+

× circ

exp 2

waves evanescent

:

2

2

> 0

+ β

α

Effect of Aperture Effect of Aperture

( ) ( )

( x y ) ^{: amplitude transmitta nce function of the aperture}

U

y x y U

x t

i t

A

0

0

; ,

; , = ,

⎟ ⎠

⎜ ⎞

⎝

⊗ ⎛

⎟ ⎠

⎜ ⎞

⎝

= ⎛

⎟ ⎠

⎜ ⎞

⎝

⎛

λ β λ α λ

β λ α λ

β λ

α , A , T ,

A

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

⎟ ⎠

⎜ ⎞

⎝

= ⎛

⎟ ⎠

⎜ ⎞

⎝

⎛

λ β λ δ α λ

β λ

α , ,

A

⎟ ⎠

⎜ ⎞

⎝

= ⎛

⎟ ⎠

⎜ ⎞

⎝

⊗ ⎛

⎟ ⎠

⎜ ⎞

⎝

= ⎛

⎟ ⎠

⎜ ⎞

⎝

⎛

λ β λ α λ

β λ α λ

β λ δ α λ

β λ

α , , T , T ,

A

Propagation as a Linear Spatial Filter Propagation as a Linear Spatial Filter

( ) ( ) ( ) ( ) ⎟

⎠ ⎞

⎜ ⎝

⎛ +

= , ; 0 circ

;

,

X

f z A f f f f

f

A λ λ

( ) ( ) _⎥⎦ ^⎤

⎢⎣ ⎡ − −

× exp 2 z 1 f

f

j λ λ

π λ

( ) ^{( ) ( )}

⎪ ⎪

⎩

⎪⎪ ⎨

⎧ ⎥⎦ ⎤

⎢⎣ ⎡ − −

= 0

1 2

exp ,

2 2

Y X

y X

f z f

j f

f H

λ λ λ

π λ

2

1

2

+

〈

X

f

f

otherwise

Transfer function of wave propagation phenomenon in free space

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

( , , ) ] [

)

( z average n x y z n =

⎟⎟ ⎠

⎞

⎜⎜ ⎝

⎛ − −

⎟ ⎠

⎜ ⎞

⎝

= ⎛

⎟ ⎠

⎜ ⎞

⎝

⎛ , ; + , ; exp ( )

( 2 )

( 2 )

λ π β λ

π α λ δ

β λ δ α

λ β λ α

o

h

z z A z j z n z k

A

( ) ⎟

⎠

⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

⎛ +

⎟ ⎠

⎜ ⎞

⎝

⎛ +

= + ∫ ∫

∞

−

λ

β λ

β α λ α

δ π λ

β λ

δ z A α z z j x y d d

z y x

U

2 ( )

exp

; , ,

,

on (x,y) plane

( ^{x y z z} ) ^U ( ^{x y z z} ) [ ^{j n z k z} ]

U , , + δ =

, , + δ exp − δ ( )

δ

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

From Maxiwell’s equations to wave equations

Appendix

Appendix