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Chapter 10 DiffractionMarch 9, 11 Fraunhofer diffraction: The single slit
10.1 Preliminary considerationsDiffraction: The deviation of light from propagation in a straight line.There is no essential physical distinction between interference and diffraction.
Huygens-Fresnel Principle: Every unobstructed point of a wave front serves as a source of spherical wavelets. The amplitude of the optical field at any point beyond is the superposition of all these wavelets, taking into account their amplitudes and phases.
Fraunhofer (far field) diffraction: Both the incoming and outgoing waves approach being planar. a2/R<<1 where R is the smaller of the two distances from the source to the aperture and from the aperture to the observation point. a is the size of the aperture. The diffraction pattern does not change when moving the observation plane further away.Fresnel (near field) diffraction: Plane of observation is close to the aperture. General case of diffraction. The diffraction pattern changes when the observation plane moves.
S
P
aR1
R2
2
Coherent line source:
y
x
P (x,y)
dy'
r
-D/2
D/2
' )exp(
),(2/
2/dy
r
ikryxE
D
DLε
L is the source strength per unit length.This equation changes a diffraction problem into an integration (interference) problem.
Mathematical criteria for Fraunhofer diffraction:The phase for the rays meeting at the observation point is a linear function of the aperture variables. S y'
P
y' sinWaves from a point source:Harmonic spherical wave:
),(or ),cos(),( )( tkrier
AtrEtkr
r
AtrE
A is the source strength.
3
10.2 Fraunhofer diffraction10.2.1 The single slit
y
x
P (x,y)
y'r
-D/2
D/2
R
The slit is along the z-axis and has a width of D.
2sin
)0()(
II
sin)2/( ),exp(sin
sin)2/(
]sin)2/sin[()exp(
' )]sin'(exp[
' )exp(
),(
2/
2/
2/
2/
kDikRR
D
kD
kDikR
R
D
dyR
yRik
dyr
ikryxE
L
L
D
DL
D
DL
In the amplitude, r is approximated by R.
In the phase, r is approximated by R-y' sinif D2/R<<1.Fraunhofer diffraction condition
The overall phase is the same as a point source at the center of the slit.
Integrate over z gives the same function.
22
22
'2
cossin'
sin'2')'(
yR
yR
RyyRyr
4
maxima ,...47.3 ,46.2 ,43.1 ,0 ,tan
minima ...3 ,2 , ,0sin
0)sincos(sin2
)0(3
I
d
dI
2sin
)0()(
II sin
2
kD
y
x
P (x,y)
y'r
-D/2
D/2
R
2sin
)0()(
II
I/I(0)=0.047 0.016
(©WIU OptoLab)
DkD
DkD
2/
peaks side theof Widths
2
2/
22
peak central theof Width
7
10.2.2 The double slit z
x
P (x,z)R-a sin
R
ab
)sinexp(1)exp(sin
')]sin'(exp[
')]sin'(exp[),(
2/
2/
2/
2/
ikaikRR
b
dzzRikR
dzzRikR
zxE
L
ba
ba
L
b
b
L
2
2
0 cossin
4)(
II
sin)2/(
sin)2/(
factor.) 1/2 omit the (We slit. oneonly is there whenaxis at theintensity ,Let 2
0
kb
ka
R
bI L
The result is a rapidly varying double-slit interference pattern (cos2) modulated by a slowly varying single-slit diffraction pattern (sin2/2).
March 23 Double slit and many slits
8
(©WIU OptoLab)
sin
cos/sin
)/sin sin()0()( 2
2a
b
bII
Single-slit diffraction Two-slit interference
Envelope Fringes
Question:Which interference maximum coincides with the first diffraction minimum?
b
am
ma
b
sin
sin
“Half-fringe” (split fringe) may occur there. Our author counts a half-fringe as 0.5 fringe.
2
2
0 cossin
4)(
II
half-fringe
9
10.2.3 Diffraction by many slits
z
x
P (x,z)
R-a sin
R
a
b
R-2a sin
)2exp(1
)2exp(1)exp(
sin
])1(2exp[...)4exp()2exp(1)exp(sin
')'(...')'(')'(')'(),(
)]sin'(exp[)'( ,/2/)1(
2/)1(
2/2
2/2
2/
2/
2/
2/
i
NiikRbC
NiiiikRbC
dzzFCdzzFCdzzFCdzzFCzxE
zRikzFRCbaN
baN
ba
ba
ba
ba
b
b
L
22
0 sin
sinsin)(
N
II
sin)2/(
sin)2/(
kb
ka
10
6
4
sin
sinsin22
N
ba
N
22
0 sin
sinsin)(
N
II
Principle maxima:Minima (totally N-1):
Subsidiary maxima (totally N-2):
... ,2 , ,0
)1(
... ,3
,2
, N
N
NNN
2
)32( ..., ,
2
5 ,
2
3
N
N
NN
sin)2/(
sin)2/(
kb
ka
13
10.2.4 The rectangular aperture
Coherent aperture:
Aperture
)exp(dS
r
ikrE A
2
2
2222
222
222
/)(1
/)(21
/)(2/)(1
)()(
RZzYyR
RZzYyR
RZzYyRzyR
zZyYXr
ZYXR
dS=dydz P(Y,Z)rR
x
y
z
Y
Z
Aperture
/)(exp)exp(
),( dSRZzYyikR
ikRZYE A
March 25 Rectangular aperture
Fraunhofer diffraction condition
X
14
).sin2
( 2
'
),sin2
( 2
' ,
'
'sin
'
'sin)exp(
/)(exp)exp(
/)(exp)exp(
),(
2/
2/
2/
2/
Aperture
b
aA
a
a
b
b
A
A
kb
R
Ykb
ka
R
Zka
R
ikRab
dydzRZzYyikR
ikR
dSRZzYyikR
ikRZYE
Rectangular aperture:
22
'
'sin
'
'sin)0(),(
IZYI
dS=dydz P(Y,Z)rR
x
y
z
Y
Za
b
15
22
'
'sin
'
'sin)0(),(
IZYI
kb
Rm YRkbY
2 ... ,2 ,2/' Y minimum:
ka
Rm ZRkaZ
2 ... ,2 ,2/' Z minimum:
.2/'
,2/'
RkbY
RkaZ
17
10.2.5 The circular aperture
Importance in optical instrumentation: The image of a distant point source is not a point, but a diffraction pattern because of the limited size of the lenses.
2
12
222
12
0 0
2
0 0
0
2
0 0
Aperture
/
)/(4
/
)/(2
)exp(
)/(2)exp(
cos/exp)exp(
)cos(/exp)exp(
/)(exp)exp(
),(
)cos(
coscossinsin
Rkaq
RkaqJ
R
AE
Rkaq
RkaqJa
R
ikR
dRkqJR
ikR
ddRqikR
ikR
ddRqikR
ikR
dSRZzYyikR
ikRZYE
q
qqZzYy
A
A
aA
aA
aA
A
2
1)(lim
)()]([
])cos(exp[2
)(
)cosexp(2
1)(
1
0
1
2
0
2
00
u
uJ
uJuuJudu
d
dvvumvii
uJ
dvviuuJ
u
mm
mm
m
m
Bessel functions:.
2
1
2
1
sin
)sin(2)0(
/
)/(2)0()(
ka
kaJI
Rkaq
RkaqJII
March 27 Circular aperture
P(Y,Z)
R
x
y
z
Y
Z
q
a
18
2
1
sin
)sin(2)0()(
ka
kaJII
J0(u)
J1(u)
u
)0(/)( II
sinka
3.83
0.018
q1
lensa for 22.1 ,2
22.11 D
f
a
Rq
Radius of Airy disk:
83.3/ 0)/( 11 RkaqRkaqJ
DP
f
20
March 30 Resolution of imaging systems
10.2.6.0 Equivalence between the far field and the focal plane diffraction pattern
1 1 2
2 1 2
22 22
1
( / ) exp( ) exp[ ( sin )]
( '/ ) exp( ) exp[ ( sin )]
( )exp ( ) constant
( )
(does not depend on )
' ',
E y R C ikR C ik R a
E y f C ikL C ik L a
Eik L R
E
y y y f
R f y R
Two coherent point sources:
P
a sin
Ra
y
• This applies to any number of arbitrarily distributed point sources in space.• Far field and focal plane produce the same diffraction pattern, but with different sizes.• R is replaced by f in the focal plane pattern.A lens pulls a far-field diffraction pattern to its focal plane, reduces the size by f/R.
P'
a sin
a
f
y'L
21
10.2.6 Resolution of imaging systems
Overlap of two incoherent point sources:
Rayleigh’s criterion for bare resolution:The center of one Airy disk falls on the first minimum of the other Airy disk.We can actually do a little better.
Angular limit of resolution:
D
22.1min
Df
qD
fq
22.1
22.1
1
1
D P
f
Image size of a circular aperture:
D
P2S1
S2
P1
far away f
Image size of a far point source:
lensDf
q 22.11 D
P
f
22
Question: Comparing the circular with the square aperture, why does the square aperture produce a smaller diffraction pattern? (/D vs. 1.22 /D)
Wavelength dependence: CD DVD
Angular limit of resolution: D
22.1min
Our eyes: '1mm 2
nm 550min About 1/3000 rad
Spot distance on the retina: 20 mm/3000=6.7mSpace between human photoreceptor cells on the retina: 5-7m.Pixel size of a CCD camera: ~7.5 m.
Pupil diameter Focal length Human cone photoreceptor cells150m
24
Diffraction grating: An optical device with regularly spaced array of diffracting elements.Transmission gratings and reflection gratings.
Grating equation:i
m
a
i
m
a
ma im )sin(sin
Blazed grating: Enhancing the energy of a certain order of diffraction.Blaze angle: Specular reflection: mir 2
April 1, 3 Gratings
m=0
12
-2-1
i r
0
a
specular
0th
25
Grating spectroscopy:
m
i
Na
kaka
N
NII
cos
2
cos
2)sin)(sin2/(
2
sin
sinsin)(
22
0
Angular width of a spectral line due to instrumental broadening. Inversely proportional to Na.
Angular dispersion:
m
mim a
m
d
dma
cos)sin(sin
dm
d
Angular width for a spectral line:N-slit interference
Between two minima, (N-1)/N to (N+1)/N .
26
Limit of resolution:
)sin(sin
2 :for criterion sRayleigh'
coscos dispersion from separationAngular
cos
2)( h wavelengteach of idthAngular w
min
separationwidthmin
separation
width
im
mm
m
m
NamN
a
m
a
m
d
d
Na
min
Resolving power:
Question: Why does the resolving power increase with increasing order number and with increasing number of illuminated slits?
Barely resolved two close wavelengths:
mN min
separation
width
27
Free spectral range:
m
mma fsrim
)1()()sin(sin
sinm
m =1
m =3
m =2
fsr
In higher order diffraction the spectrum is more spread in angle. This results in a higher resolving power but a narrower free spectral range.
mfsr