1 electromagnetic waves: interference wednesday october 30, 2002
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Electromagnetic waves: Interference
Wednesday October 30, 2002
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Two-source interference
1r
What is the nature of the superposition of radiation What is the nature of the superposition of radiation from two from two coherentcoherent sources. The classic example of sources. The classic example of this phenomenon is this phenomenon is Young’s Double Slit ExperimentYoung’s Double Slit Experiment
aa
SS11
SS22
xx
LL
Plane wave (Plane wave ())
2r
PP
yy
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Interference terms
12
2
221
*1 ____________________________
rrkioo
P
er
EE
EEP
12
21
221
* ____________________________
rrkioo er
EE
EEPP
12221
**
cos2
_________________2121
r
EE
EEEE
oo
PPPP
1212 rrk
where,where,
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Intensity – Young’s double slit diffraction
122121 cos2 PPPPP IIIII
1212 rrk
PhasePhase difference of beams occurs because of a difference of beams occurs because of a pathpath difference difference!!
12222 cos2
2121
PoPoPoPooPEEEEE
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Young’s Double slit diffraction
I1P = intensity of source 1 (S1) alone
I2P = intensity of source 2 (S2) alone
Thus IP can be greater or less than I1+I2 depending on the values of 2 - 1
In Young’s experiment r1 ~|| r2 ~|| k Hence
Thus r2 – r1 = a sin
122121 cos2 PPPPP IIIII
2112 rrkrkrk
rr22-r-r11
aa
rr11
rr22
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Intensity maxima and minima
m2...4,2,012
2
12...3,12 m
am
2
1sin
Maxima for,Maxima for,
Minima for,Minima for,
...2,1,0sin2sin ma
mormka
If IIf I1P1P=I=I2P2P=I=Ioo
If IIf I1P1P=I=I2P2P=I=Ioo 0cos2 122121 PPPPMIN IIIII
oPPPPMAX IIIIII 4cos2 122121
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Fringe Visibility or Fringe Contrast
MINMAX
MINMAX
II
IIV
To measure the contrast or visibility of these fringes, one To measure the contrast or visibility of these fringes, one may define a useful quantity, the fringe visibility:may define a useful quantity, the fringe visibility:
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Co-ordinates on screen
Use sin ≈ tan = y/L Then
These results are seen in the following Interference pattern
amLy
amLy
2
1min
max
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Phasor Representation of wave addition
Phasor representation of a wave E.g. E = Eosint is represented as a vector of
magnitude Eo, making an angle =t with respect to the y-axis
Projection onto y-axis for sine and x-axis for cosine
Now write,
tEE
as
trkEE
o
o
sin
,
sin
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Phasors
Imagine disturbance given in the form
222
111
sin
sin
tEE
tEE
Po
Po
==φφ22--φφ11
φφ11
φφ22
Carry out addition at t=0Carry out addition at t=0
1212
2121
212
22
12
,
cos2
,
180cos2
rrkwhere
IIIII
or
EEEEE PoPoPoPo
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Other forms of two-source interference
1r
Lloyd’s mirrorLloyd’s mirror
screenscreenSS
S’S’
2r
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Other forms of two source interference
Fresnel BiprismFresnel Biprism
ss22
SS11
SS
dd ss
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Other sources of two source interference
nn
Altering path length for rAltering path length for r22 rr11
rr22
With dielectric – thickness dWith dielectric – thickness d
krkr22 = k = kDDd + kd + koo(r(r22-d)-d)
= nk= nkood+ ko(rd+ ko(r22-d)-d)
= k= koorr22 + k + koo(n-1)d(n-1)d
Thus change in path length = k(n-1)dThus change in path length = k(n-1)d
Equivalent to writing, Equivalent to writing, 22 = = 11 + k + koo(n-1)d(n-1)d
Then Then = kr = kr22 – k – koorr1 1 = k= koo(r(r22-r-r11) + k) + koo(n-1)d(n-1)d
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Incidence at an angle
ii
a sin a sin
a sin a sin ii
1r
2r
Before slitsBefore slitsDifference in path lengthDifference in path length
After slitsAfter slitsDifference in path lengthDifference in path length
= = a sin a sin II in r in r11
= = a sin a sin in r in r22
Now k(rNow k(r22-r-r11) = - k a sin ) = - k a sin + k a sin + k a sin ii
Thus Thus = ka (sin = ka (sin - sin - sinii))
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Reflection from dielectric layer
Assume phase of wave at O (x=0, t=0) is 0
Amplitude reflection co-efficient (n1n2) = 12
(n2 n1) ’=21
Amplitude transmission co-efficient (n1n2) = 12
(n2 n1) ’= 21
Path O to O’ introduces a phase change
nn22nn11 nn11
AA
O’O’
OO
ttx = 0x = 0 x = tx = t
A’A’’’
’’
'cos
2
222
tSk
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Reflection from a dielectric layer
At O: Incident amplitude E = Eoe-iωt
Reflected amplitude ER = Eoe-iωt
At O’: Reflected amplitude Transmitted amplitude
At A: Transmitted amplitude Reflected amplitude
tSkioeE
22' tSki
oeE 22'
tSkioeE
222'' tSki
oeE 222''
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Reflection from a dielectric layer
tSkioA eEE 11''
AA
A’A’
z = 2t tan z = 2t tan ’’
and and ΔΔSS11= z sin = z sin = 2t tan = 2t tan ’ sin ’ sin
•At A’
96.0'2.0'12
12
andnn
nn
Since,Since,
The reflected intensities ~ 0.04IThe reflected intensities ~ 0.04Ioo and both beams (A,A’) will have and both beams (A,A’) will have
almost the same intensity.almost the same intensity.Next beam, however, will have ~ |Next beam, however, will have ~ |||33EEoo which is very small which is very small
Thus assume interference at Thus assume interference at , and need only consider the two , and need only consider the two beam problem.beam problem.
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Transmission through a dielectric layer
At O’: Amplitude ~ ’Eo ~ 0.96 Eo
At O”: Amplitude ~ ’(’)2Eo ~ 0.04 Eo
Thus amplitude at O” is very small
O’O’
O”O”
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Reflection from a dielectric layer
Interference pattern should be observed at infinity
By using a lens the pattern can be formed in the focal plane (for fringes localized at )
Path length from A, A’ to screen is the same for both rays
Thus need to find phase difference between two rays at A, A’.
AA
A’A’
z = 2t tan z = 2t tan ’’
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Reflection from a dielectric surface
tSkioA eEE 11''
AA
A’A’
z = 2t tan z = 2t tan ’’
tSkio
iA eEeE 222'
If we assume If we assume ’ ~ 1’ ~ 1
and since and since ’ = |’ = |||
This is just interference between two sources with equal amplitudes This is just interference between two sources with equal amplitudes
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Reflection from a dielectric surface
cos2 2121 IIIII
'cos2
sin'tan2'cos
2
2
2
12
1122
tkn
tknt
kn
skSk
o
oo
tSkio
iA eEeE 11
' tSki
oA eEE 222'
where,where,
Since Since kk22 = n = n22kkoo kk11=n=n11kkoo
and nand n11sinsin = n = n22sinsin’’ (Snells Law)(Snells Law)
Thus, Thus,
112212 2 skSk
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Reflection from a dielectric surface
Since ISince I11 ~ I ~ I22 ~ I ~ Ioo
Then, I = 2IThen, I = 2Ioo(1+cos(1+cos))
Constructive interferenceConstructive interference
Destructive interferenceDestructive interference
= = 2m 2m = 2ktcos = 2ktcos’ - ’ - (here k=n(here k=n22kkoo))
2ktcos2ktcos’ = ’ = (2m+1)(2m+1)
ktcosktcos’ = ’ = (m+1/2)(m+1/2)
2n2n22coscos’ = ’ = (m+1/2) (m+1/2)oo
2n2n22coscos’ = ’ = m moo
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Haidinger’s Bands: Fringes of equal inclinationdd
nn22
nn11
Beam splitterBeam splitter
ExtendedExtendedsourcesource
PPII PP22
PP
xx
ff
FocalFocalplaneplane
11
11
DielectricDielectricslabslab
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Fizeau Fringes: fringes of equal thickness
Now imagine we arrange to keep cos ’ constant We can do this if we keep ’ small That is, view near normal incidence Focus eye near plane of film Fringes are localized near film since rays diverge
from this region Now this is still two beam interference, but whether
we have a maximum or minimum will depend on the value of t
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Fizeau Fringes: fringes of equal thickness
cos2 2121 IIIII
where, where,
kt
kt
2
'cos2
Then if film varies in thickness we will see fringes as we move our eye.Then if film varies in thickness we will see fringes as we move our eye.
These are termed These are termed Fizeau fringesFizeau fringes..
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Fizeau Fringes
Extended sourceExtended source
Beam splitterBeam splitter
xx nn
nn22
nn
kt
kt
2
'cos2
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Wedge between two plates11 22
glassglassglassglass
airair
DDyy
LL
Path difference = 2yPath difference = 2yPhase difference Phase difference = 2ky - = 2ky - (phase change for 2, but not for 1) (phase change for 2, but not for 1)
Maxima 2y = (m + ½) Maxima 2y = (m + ½) oo/n/n
Minima 2y = mMinima 2y = moo/n/n
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Wedge between two plates
Maxima 2y = (m + ½) Maxima 2y = (m + ½) oo/n/n
Minima 2y = mMinima 2y = moo/n/n
Look at p and p + 1 maximaLook at p and p + 1 maxima
yyp+1p+1 – y – ypp = = oo/2n /2n ΔΔxx
where where ΔΔx = distance between adjacent maximax = distance between adjacent maxima
Now if diameter of object = DNow if diameter of object = D
Then LThen L = D = D
And (D/L) And (D/L) ΔΔx= x= oo/2n or /2n or D = D = ooL/2n L/2n ΔΔxx
airair
DDyy
LL
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Wedge between two platesCan be used to test the quality of surfacesCan be used to test the quality of surfaces
Fringes follow contour of constant yFringes follow contour of constant y
Thus a flat bottom plate will give straight fringes, otherwise Thus a flat bottom plate will give straight fringes, otherwise ripples in the fringes will be seen. ripples in the fringes will be seen.