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2
cos16.0 20
II p
o
rpp II
2
0 cos16.0
o
p
otnII
22
0
2cos16.0n t
2
2cos16.0 2
0
o
p
tnII
Wavelength (nm)
R
t = 400 nm, n = 1.515, = 0
0
0.05
0.1
0.15
0.2
400 450 500 550 600 650 700
Eo
Reflection coefficient:
Transmission coefficient:
o
r
EE
r
o
t
E
Et
for internal:o
r
EE
r '
rEo
tEo tr’E
o
tt’r’Eo tt’r’3Eo tt’r’5Eott’r’7Eo
for internal:o
t
E
Et '
nd
Multiple Beam Interference
Amplitude of each reflection:
t
Modify amplitude and phase for each reflected ray:
tioerEE 1
tioeErttE ''2
233 '' ti
oeErttE 35
4 '' tioeErttE
132'' Ntio
NN eErttE
For N >= 2
Phase lag between reflections:
0cos222 t
orp
o
nd
(difference between the first and the rest will be covered by reflection amplitudes r and r’)
2
132''N
Ntjo
NtjoR eErtterEE
Total reflected field:
factor
2
242'''N
NjNjtjoR ererttreEE
2
22'''N
NjjtjoR ererttreEE
combine
...1 2
2
2
xxxN
N
Which converges to:x
11
for |x|<1
This makes a geometric series:
j
jtjoR er
erttreEE 2'11
''
By the way:
21' rtt
'rr Stoke’s Relations
j
jtj
oR erer
eEE 211
…but when using complex notation for fields:
221
RoR cEI
2
21
RoR EcI …which means…*
21
RRoR EcEI
…where * means the complex conjugate ( j -> -j )
j
jtj
j
jtj
ooR erer
eerer
ecEI 22
221
11
11
422
22
21
111rerer
eercEI jj
jj
ooR
1cos2
cos2224
22
21
rrr
cEI ooR
ii ee cos2Use:
221
ooo cEI oR IIR
1cos2cos22
24
2
rrr
R
Double Beam
Multiple Beam
0
0.05
0.1
0.15
0.2
400 450 500 550 600 650 700
R
t = 500 nm, n = 1.515, = 0
Wavelength (nm)
t = 500 nm, n = 1.515, = 0
R
0
0.05
0.1
0.15
0.2
400 450 500 550 600 650 700
Double Beam
Multiple Beam
Add 50% reflective surfaces!
Wavelength (nm)
t = 500 nm, n = 1.515, = 0
00.20.40.60.81
1.2
400 450 500 550 600 650 700
R
00.20.40.60.81
1.2
400 450 500 550 600 650 700
t = 500 nm, n = 1.515, = 0
R
Phasors: Describe amplitude and phase with vector length and direction.
A
B
A+B
constructive destructive
A
B
A+B
A
B
A+B
Phasors: Describe amplitude and phase with vector length and direction.
in between
00.20.40.60.81
1.2
400 450 500 550 600 650 700
destructive:
constructive:
/20 off constructive:
2
2cos16.0 2
o
tnR
/20 off destructive
destructive
00.20.40.60.81
1.2
400 450 500 550 600 650 700
R
tioerEE 1
tioeErttE ''2
233 '' ti
oeErttE 35
4 '' tioeErttE
Double Beam
Multiple Beam
Changing phase, not time
00.20.40.60.81
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
R
t = 500 nm, n = 1.515, = 600 nm
0 45 90
i
Multiple beam interference also causes sharp peaks with angle of incidence:
11cos2
cos2224
2
T
rrr
1TR
1cos2cos22
1cos21cos2
24
2
24
24
rrr
rrrr
T
1cos212
24
24
rr
rrT
1cos2124
22
rrr
T
-0.20
0.20.40.60.81
400 450 500 550 600 650 700
t = 500 nm, n = 1.515, = 0
T
Multiple beams interfere constructively and destructively with a much sharper phase dependence than double beam interference since the multiple interfering components dephase at different rates.
