Post on 21-Mar-2018
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