Chapter 5. Diffraction Part 1 - Seoul National...
Transcript of Chapter 5. Diffraction Part 1 - Seoul National...
Changhee Lee, SNU, Korea
Optoelectronics EE 430.423.001
2016. 2nd Semester
1/27
2016. 10. 18.
Changhee Lee School of Electrical and Computer Engineering
Seoul National Univ. [email protected]
Chapter 5. Diffraction Part 1
Changhee Lee, SNU, Korea
Optoelectronics EE 430.423.001
2016. 2nd Semester
2/27
• Diffraction is defined as the bending of light around the corners of an obstacle or aperture into the region of geometrical shadow of the obstacle.
• The essential features of diffraction can be explained qualitatively by Huygens’ principle. The Huygens’ principle states that every point on a wavefront actd as the source of a secondary wave that spreads out in all directions. The envelope of all the secondary waves is the new wave front. Augustin Jean Fresnel (1788-1827) in 1818 explained the diffraction phenomena using the Huygens’ principle and Young’s principle of interference Huygens-Fresnel principle
• We use a more quantitative approach, the Fresnel-Kirchhoff formula to various cases of diffraction of light by obstacles and apertures.
5.1 General description of diffraction
https://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_principle
Diffraction of a plane wave at a slit whose width is several times the wavelength.
Diffraction of a plane wave when the slit width equals the wavelength
Changhee Lee, SNU, Korea
Optoelectronics EE 430.423.001
2016. 2nd Semester
3/27
Green’s theorem
)()()( ,F
,FF theoremDivergence
)()(
2
22
VUVUVUUVVU
dVdA
dVUVUVdAVUUV
n
nn
∇⋅∇+∇=∇⋅∇∇−∇=
⋅∇=∇
∇−∇=∇−∇
∫∫∫∫∫∫∫∫∫∫
5.2 Fundamental theory
2
2
22
2
2
22
1
1
tV
uV
tU
uU
∂∂
=∇
∂∂
=∇
0)( =∇−∇∫∫ dAVUUV nn
If both U and V are wave functions and have a harmonic time dependence of the form eiωt.
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Optoelectronics EE 430.423.001
2016. 2nd Semester
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5.2 Fundamental theory
reVV
tkri )(
0
ω+
=Suppose that we take V to be the wave function
0)()( 2 =Ω∂∂
−∂∂
−∇−∇ ∫∫∫∫ = dr
er
UrU
redA
reUU
re
r
ikrikrikr
nn
ikr
ρρ
Since V becomes infinite at P, we must exclude that point from the integration. Subtract an integral over a small sphere of radius r=ρ centered at P and then let ρ shrink to zero.
PP UdU π4=Ω∫∫
dAUr
er
eUU n
ikrikr
nP ∫∫ ∇−∇−= )(41π
Kirchhoff integral theorem
U = optical disturbance
Changhee Lee, SNU, Korea
Optoelectronics EE 430.423.001
2016. 2nd Semester
5/27
Fresnel-Kirchhoff formula Determine optical disturbance reaching the receiving point P from the source S. V
Two basic simplifying assumptions: (1) The wave function U and its gradient contribute negligible amounts to the integral
except at the aperture opening itself. (2) The values of U and grad U at the aperture are the same as they would be in the
absence of the partition.
'
)'(
0 reUU
tkri ω−
=The wave function U at the aperture
−=
∂∂
=∇
−=
∂∂
=∇
2
''''
2
'')',cos(
'')',cos(
'
),cos(),cos(
re
rikern
re
rrn
re
re
rikern
re
rrn
re
ikrikrikrikr
n
ikrikrikrikr
n
dAr
er
er
er
eeUUikr
n
ikrikr
n
ikrti
P ∫∫ ∇−∇=−
)''
(4
''0
π
ω
Smaller than the 1st term if r, r’>> λ.
Smaller than the 1st term if r, r’>> λ.
Changhee Lee, SNU, Korea
Optoelectronics EE 430.423.001
2016. 2nd Semester
6/27
Fresnel-Kirchhoff formula
[ ]dArnrnrr
eeikUUrrikti
P ∫∫ −−=+−
)',cos(),cos('4
)'(0
π
ω
[ ]
'
,1),cos(4
'0
)(
reUU
dArnr
eUikU
ikr
A
tikriA
P
=
+−= ∫∫− ω
π
Fresnel-Kirchhoff integral formula
Circular aperture, 1)',cos( constant,' −== rnr
=obliquity factor )]',cos(),[cos( rnrn −
Changhee Lee, SNU, Korea
Optoelectronics EE 430.423.001
2016. 2nd Semester
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Complementary apertures. Babinet’s principle If the aperture is divided into two portions A1 and A2 such that A= A1 + A2. The two apertures A1 and A2 are said to be complementary. From the Fresnel-Kirchhoff integral formula, UP= U1P + U2P (Babinet’s principle)
If UP=0, U1P = - U2P The complementary apertures yield identical optical disturbances, except that they differ in phase by 180o. The intensity at P is therefore the same for the two apertures.
Changhee Lee, SNU, Korea
Optoelectronics EE 430.423.001
2016. 2nd Semester
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Babinet’s principle
http://userdisk.webry.biglobe.ne.jp/006/095/15/N000/000/004/136844068624713202721.JPG
Changhee Lee, SNU, Korea
Optoelectronics EE 430.423.001
2016. 2nd Semester
9/27 http://userdisk.webry.biglobe.ne.jp/006/095/15/N000/000/004/136844073251013202889_Corona.JPG
Babinet’s principle
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Optoelectronics EE 430.423.001
2016. 2nd Semester
10/27
5.3 Fraunhofer and Fresnel Diffraction
Fraunhofer diffraction occurs when both the incident and diffracted waves are effectively plane. This will be the case when the distances from the source to the diffracting aperture and from the aperture to the receiving point are both large enough for the curvatures of the incident and diffracted waves to be neglected. If either the source or the receiving point is close enough to the diffracting aperture so that the curvature of the wave front is significant, then one has Fresnel Diffraction.
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Optoelectronics EE 430.423.001
2016. 2nd Semester
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5.3 Fraunhofer and Fresnel Diffraction
...)1'
1(21)
''(
'')()'('
2
22222222
++++=
+−+−+++++=∆
δδ
δδ
dddh
dh
hdhdhdhd
The quadratic term is a measure of the curvature of the wave front. The wave is effectively plane over the aperture if
λδ <<+ 2)1'
1(21
ddCriterion for Fraunhofer diffraction
The variation of the quantity r+r’ from one edge of the aperture to the other is given by
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Optoelectronics EE 430.423.001
2016. 2nd Semester
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5.4 Fraunhofer Diffraction Patterns
[ ]dArnrnrr
eeikUUrrikti
P ∫∫ −−=+−
)',cos(),cos('4
)'(0
π
ω
Simplifying assumptions: (1) The angular spread of the diffracted light is small enough for the obliquity factor not
to vary appreciably over the aperture and to be taken outside the integral. (2) eikr’/r’ is nearly constant and can be taken outside the integral. (3) The variation of eikr/r over the aperture comes principally from the exponential part,
so that the factor 1/r can be replaced by its mean value and taken outside the integral.
dAeCU ikrP ∫∫=
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Optoelectronics EE 430.423.001
2016. 2nd Semester
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5.4 Fraunhofer diffraction patterns for the single slit
F. A. Jenkins and H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, 1957)
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Optoelectronics EE 430.423.001
2016. 2nd Semester
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5.4 Fraunhofer diffraction patterns for the single slit
0for of valuethe
sin
0
0
==
+=
yrr
yrr θFor a single slit of length L and width b, dA=Ldy.
CbLeCkb
C
k
kbLCe
LdyeCeU
ikr
ikr
b
bikyikr
0
0
0
' ,sin21
)sin('
sin
)sin21sin(
2
2
2
sin
==
=
=
= ∫+
−
θβ
ββ
θ
θ
θ
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Optoelectronics EE 430.423.001
2016. 2nd Semester
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5.4 Fraunhofer diffraction patterns for the single slit The irradiance distribution in the focal plane is
The maximum value occurs at θ=0, and minimum
values occur for β=mπ=±π, ±2π, ±3π, …
The 1st minimum, β=π, sinθ=2π/kb=λ/b.
slit theof area 0,for irradiance
)sin(
20
20
2
∝==
==
CbLI
IUI
θ
ββ
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Optoelectronics EE 430.423.001
2016. 2nd Semester
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5.4 Fraunhofer diffraction patterns for the single slit
(Prob. 5.5)
The secondary maxima occur at θ for
which β=tanβ. β=1.43π, 2.46π,
3.47π, ...
F. A. Jenkins and H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, 1957)
Changhee Lee, SNU, Korea
Optoelectronics EE 430.423.001
2016. 2nd Semester
17/27
5.4 Fraunhofer diffraction for the rectangular aperture For a rectangular aperture of width a and height b, dA=dxdy.
220 )sin()sin(
ββ
ααII =
,sin21 ,sin
21 θβφα kbka ==
The minimum values occur for α=±π, ±2π, … and β=±π, ±2π, …
Changhee Lee, SNU, Korea
Optoelectronics EE 430.423.001
2016. 2nd Semester
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5.4 Fraunhofer diffraction for the circular aperture For a circular aperture of radius R, dyyRdA 222 −=
220
21
0 )( where,)(2 RCIJII πρ
ρ=
=
0 as ,2/1/)(kind1st theoffunction Bessel)(
/)(1
sin ,
2
1
1
121
1
22sin0
→→=
=−
==
−=
∫
∫
+
−
+
−
ρρρρ
ρρπ
θρ
ρ
θ
JJ
Jduue
kRRyu
dyyReCeU
ui
R
R
ikyikr
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Optoelectronics EE 430.423.001
2016. 2nd Semester
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5.4 Fraunhofer diffraction for the circular aperture
aperture theofdiameter 2
22.1832.3sin
==
≈==
RD
DkRθλθ
The bright central area is known as the Airy disk.
1st zero of the Bessel function ρ=3.832.
The angular radius of the 1st dark ring is
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Optoelectronics EE 430.423.001
2016. 2nd Semester
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Optical Resolution
The image of a distant point source formed at the focal plane of a camera lens is a Fraunfoffer diffraction pattern for which the aperture is the lens opening D. Thus the image of a composite source is a superposition of many Airy disks. The resolution in the image depends on the size of the individual Airy disks. Rayleigh criterion: minimum angular separation between two equal point sources such that they can be just barely resolved. At this angular separation the central maximum of the image of one source falls on the 1st minimum of the other.
22.1D
λθ ≈
Rayleigh criterion for the resolution
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Optoelectronics EE 430.423.001
2016. 2nd Semester
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5.4 Fraunhofer diffraction for the double slit For a circular aperture of radius R, dyyRdA 222 −=
( )
( )
γβ
β
θγθβ
γβ
β
θ
θ
γβ
θθ
θθθ
θθθ
22
0
sinsin
sinsin)(sin
sin
0
sinsin
cossin
sin21 ,sin
21
cossin2
1sin
1
1sin1
=
==
=
+
−=
−+−=
+=
+
+
∫∫∫
II
khkb
ebe
eik
e
eeeik
dyedyedye
ii
ikhikb
ikbbhikikb
bh
h
ikyb iky
Aperture
iky
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Optoelectronics EE 430.423.001
2016. 2nd Semester
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5.4 Fraunhofer diffraction for the double slit The single-slit factor (sinβ/β)2 appears as the envelope for the interference firnges given by the term cos2γ. Bright fringes occur for γ=0, ±π, ±2π, … The angular separation between fringes is given by ∆γ=π.
2hkhλπθ =≈∆πθθγ =∆=∆ cos
21 kh
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Optoelectronics EE 430.423.001
2016. 2nd Semester
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5.4 Fraunhofer diffraction for the double slit
F. A. Jenkins and H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, 1957)
Changhee Lee, SNU, Korea
Optoelectronics EE 430.423.001
2016. 2nd Semester
24/27
5.4 Fraunhofer diffraction for the double slit
F. A. Jenkins and H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, 1957)
Changhee Lee, SNU, Korea
Optoelectronics EE 430.423.001
2016. 2nd Semester
25/27
Multiple slits, Diffraction gratings
[ ]
θγθβ
γγ
ββ
θ
θ
γβ
θ
θθ
θθθ
θθ
sin21 ,sin
21
sinsinsin
11
sin1
....1sin
1
....
)1(
sin
sinsin
sin)1(sinsin
)1(
)1(
sin2
20
sin
khkb
Nebe
ee
ike
eeik
e
dyedye
Nii
ikh
ikNhikb
hNikikhikb
bhN
hN
ikyhb
h
bh
h
b
Aperture
iky
==
=
−−
⋅−
=
+++−
=
++++=
−
−
+−
−
+
∫∫∫∫∫
22
0 sinsinsin
=
γγ
ββ
NNII
A diffraction pattern of a 633 nm laser through a grid of 150 slits https://en.wikipedia.org/wiki/Diffraction
The factor N has been inserted in order to normalize the expression, so that I=I0 when θ=0.
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2016. 2nd Semester
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Multiple slits, Diffraction gratings
The single-slit factor (sinβ/β)2 appears as the envelope of the diffraction pattern.
Principal maxima occur within the envelope for γ=nπ , n=0, π, 2π, …
θλ sinhn =
Secondary maxima occur for
γ=3π/2Ν, 5π/2Ν, …
Zeros occur for γ=π/Ν, 2π/Ν, …
n=order of diffraction
22
0 sinsinsin
=
γγ
ββ
NNII
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2016. 2nd Semester
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Multiple slits, Diffraction gratings
cos
, cos21
θγλθθθπγ
Nhkh
N=∆∴∆==∆
Resolving power of a grating spectroscope according to the Rayleigh criterion
NnRP =∆
=λ
λ
The angular width of a principal fringe is found by setting the change of Nγ equal to π.
θλ sinhn =
If N is made very large, then ∆θ is very small, and the diffraction pattern consists of a series of sharp fringes corresponding to the different orders n=0, ±π, ±2π, … For a given order the dependence of θ on the wavelength gives by differentiation
θλθ
coshn∆
=∆
For a typical grating with 600 lines/mm ruled over a total width of 10 cm, N=60,000 and the theoretical resolving power can be 60,000n.