Teorema Fundamental da Trigonometria Demonstração... )θ 1 cos sen 1 0 sen θ cos θ θ ·
Chapter 16. Fraunhofer DiffractionChapter 16. Fraunhofer...
Transcript of Chapter 16. Fraunhofer DiffractionChapter 16. Fraunhofer...
Chapter 16. Fraunhofer DiffractionChapter 16. Fraunhofer Diffraction
Fraunhofer ApproximationFraunhofer Approximation
( ) ( )22201 ηξ −+−+= yxzr
( ) ( ) ( ) ηξηξλ
ddr
jkrUjzyxU exp,, 2
01
01∫∫∑
=
Huygens-Fresnel Principle
Fraunhofer Approximation :
( ) ( ) ( ) ηξηξλπηξ
λddyx
zjU
zjeeyxU
yxz
kjjkz
⎥⎦⎤
⎢⎣⎡ +−= ∫ ∫
∞
∞−
+2exp,,
)(2
22
( )2
max22 ηξ +
⟩⟩kz
)(1)(21
)(21)(1)(
21
21
211
22
2222
22
01
ηξ
ηξηξ
ηξ
yxz
yxz
z
zyx
zyx
zz
zy
zxzr
+−++≈
+++−++=
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −
+⎟⎠⎞
⎜⎝⎛ −
+≈
FT
Fraunhofer DiffractionFraunhofer Diffraction
Fraunhofer diffraction• Specific sort of diffraction
– far-field diffraction– plane wavefront– Simpler maths
(참고) Fresnel Approximation
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −
+⎟⎠⎞
⎜⎝⎛ −
+≈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
( ) ( )22201 ηξ −+−+= yxzr
Fresnel Diffraction• This is most general form of diffraction
– No restrictions on optical layout • near-field diffraction• curved wavefront
– Analysis difficult
Fresnel DiffractionFresnel Diffraction
Screen
Obstruction
16-1. Fraunhofer Diffraction from a Single Slit16-1. Fraunhofer Diffraction from a Single Slit• Consider the geometry shown below. Assume that the slit is very long in
the direction perpendicular to the page so that we can neglect diffraction effects in the perpendicular direction.
Fraunhofer Diffraction from a Single SlitFraunhofer Diffraction from a Single Slit
( )
( )
0
0
0
0
exp
0.
P
P
The contribution to the electric field amplitudeat point P due to the wavelet emanating fromthe element ds in the slit is given by
dEdE i kr tr
Let r r for the source element ds at sThen for any element
dEdEr
ω⎛ ⎞= −⎡ ⎤⎜ ⎟ ⎣ ⎦⎝ ⎠
= =
⎛= ⎜
+ Δ⎝( ){ }0
0
exp
, ., , .
sinL L
i k r t
We can neglect the path difference in the amplitude term but not in the phase termWe let dE E ds where E is the electric field amplitude assumed uniform over the width of the slit
The path difference s
ω
θ
⎞+ Δ −⎡ ⎤⎟ ⎣ ⎦⎜ ⎟
⎠
Δ=
Δ =
( ){ } ( ) ( )
( ) ( )
/ 2
0 0 / 20 0
/ 2
00 / 2
.
exp sin exp exp sin
exp sinexp
sin
bL L
P P b
b
LP
b
Substituting we obtain
E ds EdE i k r s t E i kr t i k s dsr r
i k sEIntegrating we obtain E i kr tr i k
θ ω ω θ
θω
θ
−
−
⎛ ⎞ ⎛ ⎞= + − = −⎡ ⎤ ⎡ ⎤⎜ ⎟ ⎜ ⎟⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎝ ⎠
⎡ ⎤⎛ ⎞= −⎡ ⎤⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠ ⎣ ⎦
∫
Fraunhofer Diffraction from a Single SlitFraunhofer Diffraction from a Single Slit
( ) ( ) ( )
( ) ( ) ( )
( )
00
00
00
exp expexp
sin
1 sin2
exp exp exp2
exp2
LP
LP
L
Evaluating with the integral limits we obtain
i iEE i kr tr i k
where
k b
Rearranging we obtain
E bE i kr t i ir i
E bi kr tr i
β βω
θ
β θ
ω β ββ
ωβ
− −⎡ ⎤⎛ ⎞= −⎡ ⎤⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠ ⎣ ⎦
≡
⎛ ⎞= − − −⎡ ⎤ ⎡ ⎤⎜ ⎟ ⎣ ⎦ ⎣ ⎦⎝ ⎠⎛ ⎞
= −⎡ ⎤⎜ ⎟ ⎣ ⎦⎝ ⎠
( ) ( )00
2 2 2* 2
0 0 0 02 20
sin2 sin exp
1 1 sin sin sinc2 2
L
LP P
E bi i kr tr
The irradiance at point P is given by
E bI = c E E c I Ir
ββ ωβ
β βε ε ββ β
⎛ ⎞= −⎡ ⎤⎜ ⎟ ⎣ ⎦⎝ ⎠
⎛ ⎞= = =⎜ ⎟
⎝ ⎠ ( )θβ sinsinsin 212
02
0 kbcIcII ==
Fraunhofer Diffraction from a Single SlitFraunhofer Diffraction from a Single Slit
2 2 2* 2
0 0 0 02 20
0 0
1 1 sin sin sinc2 2
sinsinc 1 0, lim sinc lim 1
1sin 0, sin 1, 2,2
LP P
The irradiance at point P is given by
E bI = c E E c I Ir
The function is for
The zeroes of irradiance occur when or when k b m m
β β
β βε ε ββ β
ββ ββ
β β θ π
→ →
⎛ ⎞= = =⎜ ⎟
⎝ ⎠
= = =
= = = = ± ± K
( )θβ sinsinsin 212
02
0 kbcIcII ==
Fraunhofer Diffraction from a Single SlitFraunhofer Diffraction from a Single Slit,
sin , 2 / ,
1 22
In terms of the length y on the observation screeny f and in terms of wavelength kwe can write
y b ybf f
Zeroes in the irradiance pattern will occur when
b y m fm yf b
The maximum in the irradiance pattern is at β= 0
θ λ π
π πβλ λ
π λπλ
≅ =
= =
= =
2 2
sin cos sin cos sin 0
sin tancos
.Secondary maxima are found from
dd
β β β β β ββ β β β β
ββ ββ
⎛ ⎞ −= − = =⎜ ⎟
⎝ ⎠
⇒ = =
1.43π
2.46π
3.47π
0
θβ sin21 kb=
β20 sin cII =
Fraunhofer Diffraction from a Single SlitFraunhofer Diffraction from a Single Slit
Note: x- and y-axes switched in book, Figs. 16-5a (here) and Fig. 16-1 do not match.
β20 sin cII =
16-2. Beam Spreading16-2. Beam Spreading
( )
sin
min
1 2
The angular width of the centralmaximum is defined as the angularseparation Δθ between the first minima oneither side of the central maximum,
yf
The first ima in the irradiance patternwill occur when
fm fy Δθb b b
The
θ θ
λλ λ
= ≅
±= = ⇒ =
2
width W of the diffraction pattern thusincreases linearly with distance from the slit,in the regions far from the slit where Fraunhoferdiffraction applies
LW = L Δθbλ
=
L
W
16-3. Rectangular Apertures16-3. Rectangular Apertures
When the length a and width b of therectangular aperture are comparable,a diffraction pattern is observed inboth the x - and y - dimensions, governedin each dimension by the formula wehave already developed. The irradiancepat
( )( )2 20 sinc sinc
1, sin2
tern is
I I
where k a
Zeroes in the irradiance pattern are observed when
m f m fy or xb a
α β
α θ
λ λ
=
=
= =
y
Circular AperturesCircular Apertures
xdsdA =
∫∫=Area
iskAp dAe
rEE θsin
0
222
2⎟⎠⎞
⎜⎝⎛+=
xsR 222 sRx −=
dssRerEE
R
R
iskAp
22sin
0
2−= ∫−
θ
θγ sin ,/ kRRsv ==
{ }⎭⎬⎫
⎩⎨⎧
=−= ∫− γγπγ )(212 1
0
221
10
2 Jr
REdvver
REE AviAp
(the first order Bessel function of the first kind)
Bessel FunctionsBessel Functions
832.3sin21sin === θθγ kDkR (first zero)
θγ sin21 kD=
( ) ( )( )
21
1 212
2 sin( ) 0
sinJ k D
I Ik D
θθ
θ⎡ ⎤
= ⎢ ⎥⎢ ⎥⎣ ⎦
⎭⎬⎫
⎩⎨⎧
=γγπ )(2 1
0
2 Jr
REE Ap
0)at (or, 0 when 21)(1 =→⎭⎬⎫
⎩⎨⎧
→ θγγγJ
1 1min2 2
2sin 3.832
First minimum in the Airy pattern is atDk D k D πθ θ θ
λ⎛ ⎞⎛ ⎞≅ = = ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
Fraunhofer Diffraction from Circular Apertures: The Airy Pattern
Fraunhofer Diffraction from Circular Apertures: The Airy Pattern
)0(/ II
Dλθθ 22.1
21min =Δ=
Airy patternsAiry patterns
Airy Disc
Slit and Circular AperturesSlit and Circular Apertures
Circular aperture
Single slit(sinc ftn)
Sin θ0 λ/D 2λ/D 3λ/D−λ/D−2λ/D−3λ/D
Intensity
16-4. Resolution16-4. Resolution
• Ability to discern fine details of object– Lord Rayleigh in 1896
» resolution is function of the Airy disc.
– Rayleigh: Limit of resolution» Two light sources must be separated by at least the
diameter of first dark band.» Called Rayleigh Criterion
Rayleigh CriterionRayleigh Criterion
Rayleigh Limit
Rayleigh LimitRayleigh Limit
Resolution limit of a lens:
(f = focal length)
NANA
Dfx
λλ
λ
61.02
22.1
22.1min
=⎟⎠⎞
⎜⎝⎛≈
⎟⎠⎞
⎜⎝⎛=
Fraunhofer Diffraction from a Double SlitFraunhofer Diffraction from a Double Slit
Now for the double slit we can imagine that we placean obstruction in the middle of the single slit. Then all that we have to do to calculate the field from the double slit is to change the limits of
( ) ( )( )
( )
( ) ( )( )
( )
( ) ( )( )
( )
/ 2
0 / 20
/ 2
0 / 20
/ 2
00 / 2
exp exp sin
exp exp sin
exp sin expexp
sin
a bL
P a b
a bL
a b
a b
LP
a b
integration.
EE i kr t i k s dsr
E i kr t i k s dsr
Integrating we obtain
i k s iEE i kr tr i k
ω θ
ω θ
θω
θ
+
−
− −
− +
+
−
⎛ ⎞= − +⎡ ⎤⎜ ⎟ ⎣ ⎦⎝ ⎠
⎛ ⎞−⎡ ⎤⎜ ⎟ ⎣ ⎦
⎝ ⎠
⎛ ⎞ ⎡ ⎤= − +⎡ ⎤⎜ ⎟ ⎢ ⎥⎣ ⎦
⎣ ⎦⎝ ⎠
∫
∫
( )( )
( )
( ) ( ) ( )
( ) ( )
( ) ( )
/ 2
/ 2
0
0
0
0
sinsin
exp sin sinexp exp
sin 2 2
sin sinexp exp
2 2
expexp ex
2
a b
a b
L
LP
k si k
i kr t i k a b i k a bEr i k
i k a b i k a b
b i kr tEE ir i
θθ
ω θ θθ
θ θ
ωα
β
− −
− +
⎧ ⎫⎡ ⎤⎪ ⎪⎨ ⎬⎢ ⎥
⎣ ⎦⎪ ⎪⎩ ⎭− ⎧⎡ ⎤ + −⎛ ⎞ ⎡ ⎤ ⎡ ⎤⎪⎣ ⎦= −⎨⎜ ⎟ ⎢ ⎥ ⎢ ⎥
⎪ ⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎩⎫− − − +⎡ ⎤ ⎡ ⎤⎪+ − ⎬⎢ ⎥ ⎢ ⎥⎪⎣ ⎦ ⎣ ⎦⎭
−⎡ ⎤⎛ ⎞ ⎣ ⎦= ⎜ ⎟⎝ ⎠
( ) ( ) ( ) ( ) ( ){ }p exp exp exp exp
sin sin
i i i i i
where k a and k b
β β α β β
α θ β θ
− − + − − −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦
= =
16-5. Fraunhofer Diffraction from a Double Slit16-5. Fraunhofer Diffraction from a Double Slit
( ) ( )( ) ( )
( ) ( )( )
( )
0
0
2 2* 2
0 0 020
exp exp 2 cos
exp exp 2 sin
exp2 cos 2 sin
2
1 1 4sin4cos 4 c2 2 4
LP
LP P
But we know thati i
i i i
Substituting we obtain
b i kr tEE ir i
The irradiance at point P is given by
E bI = c E E c Ir
α α α
β β β
ωα β
β
βε ε αβ
+ − =
− − =
−⎡ ⎤⎛ ⎞ ⎣ ⎦= ⎜ ⎟⎝ ⎠
⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟
⎝ ⎠⎝ ⎠
22
2
2
0 00
sinos
12
LE bwhere I cr
βαβ
ε
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞= ⎜ ⎟
⎝ ⎠
Fraunhofer Diffraction from a Double SlitFraunhofer Diffraction from a Double Slit
The irradiance at point P from a doubleslit is given by the product of thediffraction pattern from single slit and the interference pattern from a double slit.
22
0 2
sin4 cosI I βαβ
⎛ ⎞= ⎜ ⎟
⎝ ⎠
Fraunhofer Diffraction from a Double SlitFraunhofer Diffraction from a Double Slit
Single Slit
Double Slit
16-6. Fraunhofer Diffraction from Many Slits (Grating)
16-6. Fraunhofer Diffraction from Many Slits (Grating)
Now for the multiple slits we just need to againchange the limits of integration. For N even slits with width b evenly spaced a distance a apart, wecan place the origin of the coordinate system at t
( ) ( )( )
( ){/ 2 2 1 / 2
0 2 1 / 210
exp exp sinj N j a b
LP j a b
j
he center obstruction and label the slits with the index j (Note that the diagram does not exactly correspond with this).
EE i kr t i k s dsr
ω θ= − +⎡ ⎤⎣ ⎦
− −⎡ ⎤⎣ ⎦=
⎛ ⎞= −⎡ ⎤⎜ ⎟ ⎣ ⎦⎝ ⎠
∑ ∫
( )( )
( ) }
( ) ( )( )
( )
( )( )
( )
2 1 / 2
2 1 / 2
2 1 / 2/ 2
010 2 1 / 2
2 1 /
2 1 / 2
exp sin
exp sinexp
sin
exp sinsin
j a b
j a b
j a bj NL
Pj j a b
j a b
j a b
i k s ds
Integrating we obtain
i k sEE i kr tr i k
i k si k
θ
θω
θ
θθ
− − +⎡ ⎤⎣ ⎦
− − −⎡ ⎤⎣ ⎦
− +⎡ ⎤⎣ ⎦=
= − −⎡ ⎤⎣ ⎦
− − +⎡ ⎤⎣ ⎦
− − −⎡ ⎤⎣ ⎦
+
⎧⎡ ⎤⎛ ⎞ ⎪= −⎡ ⎤ ⎨⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠ ⎣ ⎦⎪⎩
⎡ ⎤+ ⎢ ⎥⎣ ⎦
∫
∑
( ) ( ) ( )
( ) ( )
2
/ 20
10
exp 2 1 sin 2 1 sinexp exp
sin 2 2
2 1 sin 2 1 sinexp exp
2 2
j NL
j
i kr t i k j a b i k j a bEr i k
i k j a b i k j a b
ω θ θθ
θ θ
=
=
⎫⎪⎬⎪⎭⎧ ⎡ ⎤ ⎡ ⎤− − + − −⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎪⎣ ⎦ ⎣ ⎦ ⎣ ⎦= −⎢ ⎥ ⎢ ⎥⎨⎜ ⎟
⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎪ ⎣ ⎦ ⎣ ⎦⎩⎫⎡ ⎤ ⎡ ⎤− − − − − −⎡ ⎤ ⎡ ⎤ ⎪⎣ ⎦ ⎣ ⎦+ −⎢ ⎥ ⎢ ⎥⎬
⎢ ⎥ ⎢ ⎥⎪⎣ ⎦ ⎣ ⎦⎭
∑
( ) ( ) ( ) ( ) ( )( ) ( ) ( ){ }
( )
/ 20
10
0
0
sin sin
expexp 2 1 exp exp exp 2 1 exp exp
2
exp
j NL
Pj
LP
Substuting using k a and k b and rearranging we obtain
b i kr tEE i j i i i j i ir i
We can rewrite this as
b i kr tEEr
α θ β θ
ωα β β α β β
β
ω
=
=
= =
−⎡ ⎤⎛ ⎞ ⎣ ⎦= − − − + − − − −⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎜ ⎟ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎝ ⎠
−⎡⎛ ⎞ ⎣= ⎜ ⎟⎝ ⎠
∑
( ) ( ) ( ){ }
( ) ( ){ }
( ) ( ) ( ) ( ) ( ){ }
/ 2
1
/ 2
010
/ 2
010
2 sin exp 2 1 exp 2 12
sinexp Re exp 2 1
sinexp Re exp exp 3 exp 5 exp 1
j N
j
j NL
j
j NL
j
i i j i ji
E b i kr t i jr
E b i kr t i i i i Nr
The last te
β α αβ
βω αβ
βω α α α αβ
=
=
=
=
=
=
⎤⎦ − + − −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦
⎛ ⎞ ⎛ ⎞= − −⎡ ⎤ ⎡ ⎤⎜ ⎟ ⎜ ⎟⎣ ⎦ ⎣ ⎦
⎝ ⎠⎝ ⎠⎛ ⎞ ⎛ ⎞
= − + + + + −⎡ ⎤ ⎡ ⎤⎜ ⎟ ⎜ ⎟⎣ ⎦ ⎣ ⎦⎝ ⎠⎝ ⎠
∑
∑
∑ L
( ) ( ) ( ) ( ){ }/ 2
1
2*
0 00
sinRe exp exp 3 exp 5 exp 1sin
. int
1 1 sin2 2
j N
j
LP P P
rm is a geometric series that converges to
Ni i i i N
The details of the last step are outlined in the book The irradiance at po P is given by
E bI c E E cr
αα α α αα
βε εβ
=
=
+ + + + − =⎡ ⎤⎣ ⎦
⎛ ⎞= = ⎜ ⎟
⎝ ⎠
∑ L
2 22 2
0sin sin sin
sinN NIα β αα β α
⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠
Fraunhofer Diffraction from Multiple SlitsFraunhofer Diffraction from Multiple Slits
2 2 22 2*
0 0 00
1 1 sin sin sin sin2 2 sin sin
sin, . , ' 'sin
lim
LP P P
m
The irradiance at point P is given by
E b N NI c E E c Ir
NWhen m the term is a maximum For this condition from L Hospital s rule
α
β α β αε εβ α β α
αα πα
→
⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠
=
( )
( )
sinsin coslim lim
sinsin cos
1 sin sin 0, 1, 2,2
, .
m m
d NN N Nd N
dd
The principal maxima the irradiance pattern occur forp pka a m mN N
For large N the principal maxima are bright and well separated This ana
π α π α π
αα αα
αα αα
π πα θ θ πλ
→ →= = = ±
= = = = = = ± ± K
,
sin
lysis givesus the grating equation
a mθ λ=
2sin
⎟⎟⎠
⎞⎜⎜⎝
⎛ββ
2
sinsin
⎟⎠⎞
⎜⎝⎛
ααN
N2
λθ ma =sin
m=1
m=2
m=0
2 2
0sin sin
sinNI I β α
β α⎛ ⎞ ⎛ ⎞= ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠
Diffraction grating equationDiffraction grating equation
λθ ma =sin m=1
m=2
m=0
m=1
m=0
Fraunhofer Diffraction from Multiple SlitsFraunhofer Diffraction from Multiple Slits
N = 2
N = 3
N = 4
N = 5