–diffraction€¦ · experimental physics 3 -diffraction 3 fresnels formulation of the huygens...
TRANSCRIPT
Experimental Physics 3 - Diffraction 1
Experimental Physics EP3 Optics
– Diffraction –
https://bloch.physgeo.uni-leipzig.de/amr/
Experimental Physics 3 - Diffraction 2
From geometric optics to wave optics
Christiaan Huygens
( )0
001 krtier
E -= w
Experimental Physics 3 - Diffraction 3
Fresnels formulation of the Huygens principle
F
P
ra
Ø F is an arbitrary surface enclosing light sources.Ø Each point on F is a source of the secondary waves propagating in all directions.
Ø All secondary waves are coherent.
Ø Light field due to interference of the secondary waves in the space out of F does coincide with that of due to the real light sources.
( ){ }ò -=F
dFktiaE rwrexp
O
F
D
0rj
Pr
ra ( ) 0EKa a=
( ){ }ò --=F
dFkkrtirKE rwra
00
exp)(
( )ap
a cos14
)( +=ikKKirchhoff:
dF
𝒂𝟎
Experimental Physics 3 - Diffraction 4
Spherical waves in free space
O
F
D0r
jPr
ra
dFjjp drdF sin2 2
0=jd
( )( )2022 arrb -++=r
jj cossin 00 rarb ==
( )constrrrrr
dd=
+=
0,00
sin rrjj( ){ }ò --=F
dFkkrtirKE rwra
00
exp)(
( ) rrp rw deKerr
E ikrr
r
krti -+
- ò+=
0
0
2
0
)(2 Further derivations require some assumptions about K(r)
( )( )2022
02 cos1sin jjr -++= rrr
ba
Experimental Physics 3 - Diffraction 5
2/l+r
Fresnel zones
O
F
DPr
)2/(2 l+r
( ) rrp rw deKerr
E ikrr
r
krti -+
- ò+=
0
0
2
0
)(2
( ) ikrnnnr
nr
ikn e
ikKdeK -+
+
-+
- -=ò21 1
2/
2/)1(
rl
l
r
Because l is small, one may reasonably assume that K(r) is
constant within a given Fresnel zone.
( ) ( )( )å å= =
+-+
+-==
N
n
N
n
rrktinnn e
rrikKEE
1 1 0
1 0
)(41 wp
21 NEEE +
=
211 ffE +=
322 ffE +=
1++= NNN ffE
( ) 11
1 1 ++-+= N
N ffE
11 2 fE »
( ) 1112 ++-» N
NN fE ( )
( )( )0
0
11 22
rrktierrikKEE +-
+== wp
( )02
cos14 rrNikE
+=+=
ra
p0=
Experimental Physics 3 - Diffraction 6
Diffraction in a hole
A
B
E Fo
S P
( ) aOFD 22/ 2 ´»
a b( ) bOED 22/ 2 ´»
÷øö
çèæ +=
baDEF 118
2
÷øö
çèæ +=
baDm 114
2
l
lmbaabDm +
=4
a=b=1 m; l=600 nm
D1»1.1 mm; D2»1.44 mm; D3»1.9 mm;
Experimental Physics 3 - Diffraction 7
Arago-Poisson spot
O
F
DP
D = 2 mm D = 1 mm
( )( )( )0
0
11 22
rrktierrikKEE +-
+== wp
a=b=1 m; l=633 nm
D = 4 mm
16 m
m
Experimental Physics IIa - Diffraction 8
Ø The Huygens principle says that light field due to interference of
the secondary waves coincides with that due to the real light sources.
Ø The Fresnel zones are constructed in such a way that the distance
between two adjacent secondary wave zones differs by the half-
wavelength measured from the point of consideration.
Ø This approach is not strict, but helps to solve
diffraction-related problems in many cases.
Ø The light field due to the first Fresnel zone
is twice of that due to the real source.
Ø Diffraction is responsible for the formation of
the Arago-Fresnel spot.
To remember!
Experimental Physics 3 - Diffraction 9
Classification of diffraction phenomena
Diffraction in parallel rays
Fraunhofer diffraction
Far field
Diffraction in non-parallel rays
Fresnel diffraction
Near field
2/l+r
O
F
DPr
lbaabD+
=4
1
dFresnel number
lbdF2
=
1³F1<<F
21
2
~DdF
aperture diameter
diameter of the 1st Fresnel zone
Experimental Physics 3 - Diffraction 10
Babinet’s principle
dF
0rr
'0r'r
R
( )òÎ
--=hole
',11 '
)(dFe
rrKE
E krkrtiout wa ( )òÏ
--=hole
',22 '
)(dFe
rrKE
E krkrtiout wa
dF
0rr
'0r'r
R
inout EE 1a= inout EE 2a=
121 =+aacomplementary screens
EEE =+ 21 - without screen
outE
inE
( ){ }ò --=F
dFkkrtirKE rwra
00
exp)(
Experimental Physics 3 - Diffraction 11
Fraunhofer diffraction on a slit
qdxO
dxeaEb
b
ikxò-
=2/
2/
sinq
aasinbE =
qa sin21 kbº
a20sincII =
pa m=lq mb =sin
-3p -2p -1p 0p 1p 2p 3p
-0.2
0.0
0.2
0.4
0.6
0.8
1.0 sinc(a) sinc2(a)
Experimental Physics 3 - Diffraction 12
An arbitrarily shaped hole
O dF ),( yxr!
s!
( )dFeE srikò ×=!!
s! - unit vector
Rectangular hole (a, b)
( ){ }dxdyysxsikEa
a
b
byxò ò
- -
+=2/
2/
2/
2/
exp
bb
aa sinsinabE =
ba 220 sincsincII = l
pb
lpa
y
x
bs
as
=
=
Experimental Physics 3 - Diffraction 13
Circular aperture diffraction
aa )(12 JaE =
( )dFeE srikò ×=!!
qa ka=
am
mlq úûù
êëé -
+»2161.0min,
Minimum Maximum Imax
q1 = 0.61 q1 = 0 1q2 = 1.12 q2 = 0.81 0.0175q3 = 1.62 q3 = 1.33 0.0042q4 = 2.12 q4 = 1.851 0.0016
Experimental Physics 3 - Diffraction 14
Diffraction grating
q
a
b
dba =+
l
qsindl = qj sinkdkl ==
asinc1 bE = jieEE 12 =j2
12ieEE =
( )å-
=
=1
01
N
k
kieEE jj
j
i
iN
eeE--
=11
1
( )( )
( ) 2/11 2/sin
2/sin j
jj -= NieNEE
÷øö
çèæ
÷øö
çèæ
=
2sin
2sin
1 j
jNAA
2
1
2sin
2sin
÷÷÷÷
ø
ö
çççç
è
æ
÷øö
çèæ
÷øö
çèæ
=j
jNII
Experimental Physics 3 - Diffraction 15
Diffraction grating
Experimental Physics 3 - Diffraction 16
( )( )
2
1 2/sin2/sin÷÷ø
öççè
æ=
jjNII
q
a
b
dba =+
l
Diffraction grating
qj sinkd=
0=q 1NAA= 12INI =
pj m=2/ lq md =sin ( ),...2,1,0 ±±=m
Condition for main maxima Order
( ) 02/;2/ ¹+= jpj pNmN
( )lq Npmd /sin += ( )1,...2,1 -= Np
Condition for minima
Between this minima the secondary maxima are formed.
Experimental Physics IIa - Diffraction 17
Ø If only a part of the first Fresnel zone contributes to diffraction,
then it is called Fraunhofer diffraction. Otherwise it is referred to as
Fresnel diffraction.
Ø The Babinet’s principles says that light intensities scattered due to
diffraction by two complementary screens do coincide for directions
different from that of the incident light beam.
Ø The light intensity for Fraunhofer diffraction
due to a slit is given by square of sinc function.
Ø Diffraction grating, i.e. a periodic combination
of many slits, does provide much sharper spectra.
Ø In the latter case there are main and secondary
maxima in the light intensity.
To remember!