4-fresnel and fraunhofer diffractionoptics.hanyang.ac.kr/~shsong/4-fresnel and...
TRANSCRIPT
“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
=θ
( ) ( ) ( ) 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.
( ) ( ) ( ) 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
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.
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