lecture bundle
TRANSCRIPT
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Lecture "Vibrations and Waves":EM Waves, Polarization
Feridun Ay
Integrated Optical Micro Systems (IOMS)
MESA+ InstituteUniversity of Twente, Enschede, The Netherlands
http://ioms.ewi.utwente.nl
Integrated Optical Micro Systems (IOMS)
slide 2
1600 William Gilbert's De Magnete describe the behavior of magnets.
1729 Stephen Gray discovers electrical conduction.
1784 Pierre Laplace introduces concept of electric potential.
1785 Charles Coulomb announces his law of electrostatics.
1820 Hans Oersted demonstrates electromagnetism.
1821 Michael Faraday demonstrates the principle of the electric motor.
1865 Maxwell's Dynamical Theory of the Electromagnetic Field.
1888 Heinrich Hertz demonstrates the existence of radio waves.
1916 Einstein's general theory of relativity.
1948 John Bardeen, William Brattain and William Shockley producethe transistor.
1948 Feynman introduces his diagrams for quantum electrodynamics.
Introduction (brief history)Feynman 1-28
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(now only dependent on free charges and currents)
Maxwells equations with E, D, B and H
in addition (Lorentz force)
0 B
t
BE
)( BvEF q
f D
tJf
DH
BHED
1
with )()( BHHEDD
in linear casePED 0
(i) (Gausss law)
(ii) (Gausss law for magnetism)
(iii) (Faraday's law)
(iv) (Amperes law with correction of Maxwell)
Feynman 2-18
Integrated Optical Micro Systems (IOMS)
slide 6
8.1.1 The continuity equation:
global and local conservation of charge
J
t
8.1.2. Poyntings theorem (1)
Vem dBEU
2
0
2
0
1
2
1total energy stored in EM field:
Question: How much work, d W, is done by the electromagnetic forcesacting on these charges in the interval dt. Using Lorentz force law, the
work done on a charge q is:
Conservation lawsFeynman II-27, Griffiths 8
Vd
dt
dW
dqmet
dtqdtqddW
)(
;;
)(
JE
Jv
vEvBvElF
EJ: work done per unit time, per unit volume
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in addition (Lorentz force)
0 B
t
BE
(i) (Gausss law)
(ii) (Gausss law for magnetism)
(iii) (Faradays law)
(iv) (Amperes law with correction of Maxwell)
Feynman II,18
(no charges and currents, vacuum)
Maxwells equations in vacuum
0 E
t
EB 00
Integrated Optical Micro Systems (IOMS)
slide 10
Maxwell equations in vacuum
2
2
00
2)()(ttt
EB
BEEE
E satisfies the wave equation2
2
2
2 1
t
EE
00
1
c
because of symmetry of Maxwells equation similar expression can be derived for B
t
BE t
EB 00
EM waves in vacuum: wave equation
0 E 0 B
Feynman II-20;Griffiths 9.2
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EM waves in vacuum: Polarization
EknkrB rk~1~1,
~
0
ceE
ct
ti
nrErk )(
0
~),(
~ tieEt
The polarization vector n defines the plane ofvibration.' Because the waves are transverse n isperpendicular to the direction of propagation:
In terms of the polarization angle,
Thus, the can be considered a superposition oftwo waves-one horizontally polarized, the other
vertically.
Griffiths: 9
Integrated Optical Micro Systems (IOMS)
slide 14
Classification of Polarization
Light in the form of a plane wave inspace is said to be linearly polarized.
If light is composed of two plane wavesof equal amplitude by differing in
phase by 90, then the light is said to
be circularly polarized.
If two plane waves of differing
amplitude are related in phase by 90,
or if the relative phase is other than 90
then the light is said to be elliptically
polarized.http://hyperphysics.phyastr.gsu.edu/hbase/phyopt/
EM waves in vacuum: Polarization
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Elliptical Polarization
Elliptically polarized light consists
of two perpendicular waves of
unequal amplitude which differ in
phase by 90. The illustrationshows right- elliptically polarized
light.
.
http://hyperphysics.phyastr.gsu.edu/hbase/phyopt/
EM waves in vacuum: Polarization
Integrated Optical Micro Systems (IOMS)
slide 18
Polaroid Sunglasses
The polaroid material used in
sunglasses makes use of
dichroism, or selectiveabsorption, to achieve
polarization.
http://hyperphysics.phyastr.gsu.edu/hbase/phyopt/
EM waves in vacuum: Polarization
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Fig. 1-15 from F.T. Ulaby, Applied Electromagnetics, Prentice Hall 1999
The electromagnetic spectrum
Integrated Optical Micro Systems (IOMS)
slide 22
wavelength : 632.8 nm
wavenumber k: K = 2 / ~ 107/m = 10/m
Frequency v: v = c/ ~ 475THz
laser 2 mW:
# fotons/s: N= P/() = 7.5 x 1015/s
I/c; I = 1 mW/mm2 => P = 0.3 x 10-5 N/m2
field: E2 = 2 I/(c 0 )=2 x 103/(3 x 10-3) = 0.6 x 106
=> E ~ 8000 V/m
field in an atom:
E= 1/(4 0) Q/(4 x 10-20)
~ 4 x 1010 V/m
E2-E1=
Helium Neon laser
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Lecture "Vibrations and Waves":Refraction and Dispersion
Markus Pollnau
Integrated Optical Micro Systems (IOMS)
MESA+ Institute
University of Twente, Enschede, The Netherlandshttp://ioms.ewi.utwente.nl
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slide 2
1. Retardation
2. Refractive Index3. Dispersion
4. Absorption
Content
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A field originating in point r with a phase
is retarded in time by
The retardation is a combination of:
1. starting point r2. angular frequency
Retardation
0
12
3
4
5
6
7
-20 -15 -10 -5 0
Distance
Amplitude
crt
crtt
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An electromagnetic wave that travels through a
material forces the atoms in the material to oscillate.The oscillation is an oscillation of the electron cloudsaround their nuclei.
All oscillations occur parallel to the driving force, i.e.
parallel to the electric field Es of the travelling wave.
The oscillating atoms emit an additional wave, i.e.,they create an additional field EA.
This phenomenon can be described macroscopicallyby the refractive index n of the material.
The result is a retardation of the travelling wave.
Refractive Index
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Integrated Optical Micro Systems (IOMS)
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Assume a thin plate of thickness z with refractive
index n slightly larger than 1.
Incoming field from source
located at -z:
Field travels more slowly through the plate:
Retardation time due to refractive index n:
Outgoing field after plate locatedat point P:
Refractive Index
cztiEES /exp0
ES ES+EA
Point P
cznt /1
czntplate /
czncztiEEout /1/exp0
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Integrated Optical Micro Systems (IOMS)
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Retarded field:
Phase shift due to retardation:
Use Taylor expansion for small x:
Rewrite equation for retarded field:
Refractive Index
cznicztiEEout /1exp/exp0
czncztiEEout /1/exp0
czn /1
xx 1exp
cztiEczni
cztiE /exp1
/exp 00
S
EAE
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The additional field due to oscillating atoms
is orthogonal to theincoming field ES(because of factor -i)and leads to a retardation:
Refractive Index
ES
EA
real axis
imaginary
axis
EP=ES+EA
cztiE
c
zniEA /exp
10
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More intuitively,
the incoming energy is absorbed by the atoms.As a result, the atoms start to oscillate.The absorbed energy is re-emitted.
Since the atoms oscillate with the same frequency asthe driving field and emit the energy from the samepoint where it was absorbed, the outgoing field lookslike the incoming field.
The time it takes to absorb and re-emit the energy,leads to a retardation.
Refractive Index
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Assume that the electrons are harmonic oscillators
driven by an external field:
The solution is:
Refractive Index
tiEqxdt
xdm ee exp0
2
02
2
ti
m
Eqx
e
e
exp22
0
0
me = electron massqe = electron charge
= frequency of radiation
0 = resonant frequency ofelectrons in an atom
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Field of a plane of oscillating charges q per unit area:
The velocity of the electrons is
The resulting additional field by the atoms is
Refractive Index
czti
m
Eqi
dt
dxv
e
e
charges /exp220
0
czttvc
qE chargesplane /
2 0
czti
m
Eqi
c
qE
e
eeA /exp
222
0
0
0
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Comparison of
with
results in
Number N of electrons per unit volume is
Refractive index
Refractive Index
czti
m
Eqi
c
qE
e
eeA /exp
222
0
0
0
cztiE
c
zniEA /exp
10
2200
2
21
e
e
m
qzn
zN
2200
2
21
e
e
m
Nqn
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The refractive index varies with the frequency of light:
Dispersion
2200
2
21
e
e
mNqn
-4
-3
-2
-1
0
1
2
3
4
5
6
-25 -20 -15 -10 -5 0 5 10 15 20 25
-
refractiveindex
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Example:
Most gases and other transparent substances (glass):0 is in the ultraviolet region, therefore
0 >> of visible light, and n is nearly constant.
Nevertheless,n increases slowly with the frequency of light.
This phenomenon is called dispersion.
Application:Prism monochromator
Dispersion
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Integrated Optical Micro Systems (IOMS)
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Another example: In the stratosphere, UV light
produces free electrons with density N.
Since free electrons have no restoring force, 0 = 0,and it follows that n < 1.
I.e., the light at a specific frequency travels at speedc/n, which now becomes > c. (A better picture is thatthe electron oscillation is advanced in phasecompared to the driving field.)
However, this does not mean that a signal can betransmitted at a speed >c, because a single sine wavehas no start nor end, it does not transmit information.
Dispersion
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A more complete picture of dispersion:
1. Assume damped oscillation, i.e., the denominatorchanges from to
2. Assume several resonance frequencies (even H
with a single electron has several of them), i.e., theequation for n changes to
Dispersion
22
0 i22
0
kkk
ke
e
i
N
m
qn
22
0
2
21
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At most of the ,the slope is positive (normal dispersion).
Only at a few ,the slope is negative (anomalous dispersion)
Dispersion
kkk
ke
e
i
N
m
qn
22
0
2
21
-5
1
7
0
refractiveindex
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n has now become a complex number:
Rearrange the outgoing field after the plate
such that
Absorption
cznicztiEEout /1exp/exp0
kkk
ke
e
i
N
m
qn
22
0
2
21
imre innn
czniczncztiEE reimout /1exp/exp/exp0
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The additional field due tooscillating atoms is notorthogonal to the
incoming field anymore.
Absorption
cztiEcznicznE reimout /exp/1exp/exp 0
ES
EA
real axis
imaginary
axis
EP=ES+EA
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The first term is new.
It arises due to the fact that we have added the termto the denominator.
This term possesses a real and negative exponent,
i.e., it describes a decrease in amplitude withincreasing length z of the material.
As a result, part of the energy of the wave isabsorbed.
If is close to one of the k, then absorption of lightbecomes the dominant phenomenon in
Absorption
cznim /exp
kk i22
i
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M.P
Integrated
Lecture "Vi
Interferen
Ma
Integrated Op
M
University of Twen
http://
1. Interference of two
2. Young's double-sli
3. Interference of mu
4. Transmission thro5. Diffraction at a sha
Content
Integrated Optical Micro Systems (IOMS)
slide 3
Maxwell's equations without sources (q=0; J=0)
are symmetric in E and B.
A possible solution is an electromagnetic wave, in
which E and B generate each other and are
orthogonal to each other and to the propagation
direction of radiation. Therefore, electromagnetic
waves are transverse waves. Transverse waves are
polarized.
For the E-field, the solution reads:
Repetition
2
2
22
1
t
E
cE w
w
&&
Sinusoidal waves:
Angular frequency:
Period:
Repetition
Z
ZS2T
IZ trkiEntrE&&&&&
exp, 0
IZ tkxEtxE cos, 0
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M.P
Integrated
Huygens principle:
Each element of a wave-fr
secondary disturbance wh
position of the wave-frontwavelets.
Fermats principle:
A beam travels from point
Huygens' and Ferm
Plane wave:
Huygens' Principle
Integrated Optical Micro Systems (IOMS)
slide 7
Phase:
Interference of two waves:
Phase shift at detection point r = 0
(for simplicity, assume same frequency in figure)
is a combination of:
1. starting points ri
2. starting phases i
3. angular frequencies i
Interference of Two Waves
0
1
2
3
4
5
6
7
-20 -15 -10 -5 0
Distance
Amplitude
'I
crt ZDI
Two sinusoidal waves with same amplitude and
frequency but different phase:
Use:
Interference of Two Waves
21 coscos IZIZ tAtAR
CBCBCB11
2
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M.P
Integrated
Angular Frequency:
Amplitude:
Interference term:
Interference of Two
212
1cos2 IIAR
Vector diagram:
(for different amplitud
Interference of Two
212
1cos2 IIAR
Integrated Optical Micro Systems (IOMS)
slide 11
Constructive interference:
= 0
Destructive interference:
=
Interference of Two Waves
0
2
4
6
8
10
12
0 5 10 15 20
Distance
A
mplitude
0
2
4
6
8
10
12
0 5 10 15 20
Distance
Amplitude
Two sinusoidal waves with same frequency but
different amplitude and phase:
Interference of Two Waves
> @ > @2211 expexp IZIZ tiAtiAR
> @ > @ > @tiiAiA ZII expexpexp 2211
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M.P
Integrated
Examples: Two nearb
perpendicular to the p
amplitude but adjusta
Far-field intensities R
d = /2; = 0 d
Interference of Two
21
21cos2 IIAR
d
0
0
44
2
2 2
2
0
2
2
Interference In Thin
Integrated Optical Micro Systems (IOMS)
slide 15
Interference In Thin Films
Using
we obtain
tfi nn TT sinsin0
tt ACAGAE TT sin2/sin
iACAD Tsin
fitt nnADADACAE /sin/sinsin2 0 TTT
FCAEnAEnADn ff 20
EBnBFEBn ff 2'
Interference In Thin Films
Finally, we arrive at
For normal incidence
tftn Tcos2'
0 ti TT tnf2'
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M.P
Integrated
Interference In Thin
Thus we can rewrite the
(no212 ''' tnfnet
Interference In Thin
Example: A soap bubble of 25index of refraction of the soap fil
reflected light? Which colors app
color does the soap film appear
(destructive)
Integrated Optical Micro Systems (IOMS)
slide 19
d
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M.P
Integrated
Interference of Mul
Line source consistin
which emit in phase,
observed in direction
Phase:
Maxima at:
Condition:
m = order of the maxi
Intensity pattern like o
SI sin2 d
SI 2m
Td sin
Interference of a large
diffraction.
Many equally spaced
lines per mm) scatter
considered as emitterShine light through a
Diffraction Gratings
Integrated Optical Micro Systems (IOMS)
slide 23
Diffraction Gratings
grating
incoming beam
with O1, O2
first-order maximum
of outgoing beam
with O1
first-order maximum
of outgoing beam
with O2
Resolving power of a diffraction grating
(Rayleigh's criterion):
Two peaks can be resolved if the minimum of one is
at the maximum of the other.
Solution:
Diffraction Gratings
36
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M.P
Integrated
Diffraction Gratings
OT md sin
TTO
SI ddd cos
2
Condition for constructi
i.e., when phase differen
The closest minimum to
for a phase change of'I
Using the equation for p
OOSS md
n22 '
O
Keep the length of the
more and more emitte
slit (1D) or pinhole
Only one maximum w
many small side maxi
occurs (diffraction)
Transmission Thro
n=50
Integrated Optical Micro Systems (IOMS)
slide 27
Transmission Through a Pinhole
Fresnel diffraction (near-field diffraction):
Far
from
the
slit
zClose
to the
slit
Incident
plane wave
Slit
Transmission Through a Pinhole
slit size >> O
slit size > O
Effect of slit size:
With smaller slit size
diffraction increases
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M.P
Integrated
Edge
Light passing by
an edge
Electrons passing by
an edge
Diffraction at a Sha
Superpose two beams
Oscillation of resultin
Interference of Mul
coscos 21 ZZ AAR
121 2/ ZZZZ |osc
Integrated Optical Micro Systems (IOMS)
slide 31
Beat signal:
Beat Frequency
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4
time
signal
Applications:
Generation of a signal of very low frequency
compared to the two original waves
Measurement of the absolute difference between twovery large frequencies without the need to measure
Beat Frequency
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M.P
Integrated
Principle of interferom
Split a wave into two,
after they have travell
detect their interferen
difference in optical p
waves.
Optical path length:
n = refractive index of
Interferometry
Mach-Zehnder interfe
with interaction sectio
Mach-Zehnder Inte
Pi
Integrated Optical Micro Systems (IOMS)
slide 35
Interferometric sensor:
Mach-Zehnder Interferometer
Sensing section
Measurand
Michelson interferometer:
Michelson Interferometer
Light SourceSample
Mirror
Beam splitter
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M.P
Integrated
Michelson interferom
Example: 11 waveleng
frequency difference
Interference:
Large envelop
signal when all
waves are in phase
The more wavelength
the larger the signal
Resolution ~1/
Michelson Interfero
Michelson fiber interf
Michelson Interfero
Integrated Optical Micro Systems (IOMS)
slide 39
Light source:
Luminescence bandwidth Interferometric resolution
138 nm ~ 2 m
Michelson Interferometer
Optocal Coherence Tomography (OCT):
Michelson Interferometer
skin
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M.P
Integrated
Normal vs. ultrahigh r
(human retina along p
W. Drexler et al., Natu
Michelson Interfero
'O = 30 nm
'O = 250 nm
Short Light Pulses
Superposition of mult
phase relation (you kn
Example: 11 waveleng
frequency difference
Interference:Large envelop
U i i f T Ad d T h l
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Vibrations and Waves
The Fabry-Perot
resonator
University of TwenteAdvanced Technology
Integrated Optical MicroSystems (IOMS) GroupMarkus Pollnau
UT EWI IOMS 2012 Vibrations and Waves Fabry Perot 2
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UT-EWI-IOMS 2012 Vibrations and WavesFabry-Perot 2
The Fabry-Perot Resonator
What is it How does it operate
Important characteristics Transmission & reflection
Spectral shape: Free Spectral Range
Full Width at Half Maximum
Finesse
Q-factor, relaxation time Applications
Wavelength filter
Laser resonator
....
3UT EWI IOMS 2012 Vibrations and Waves Fabry Perot
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3
Fabry-Perot resonatormirror
R2R r
E1tE1
rE1Superposition:
add field amplitudes(accounting for phase)
Ein
ER
EF
EB
ET
r1 r2L
1
LtE e
=a+jb
incidence: t=1r
UT-EWI-IOMS 2012 Vibrations and WavesFabry-Perot
4UT EWI IOMS 2012 Vibrations and Waves Fabry Perot
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F-P transmissionEin
r1 , t1 L r2 , t2
1 2 LinE t t e
first transmission:
2
1 2
Lr r e
each roundtrip:
21 2 1 20
iL LT
in i
Et t e r r e
E
Total transfer:
0
1
1
i
i
xx
j a b where:
1 2 2
1 21
L
L
et t
r r e
UT-EWI-IOMS 2012 Vibrations and WavesFabry-Perot
5UT-EWI-IOMS 2012 Vibrations and Waves Fabry-Perot
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5
F-P as a feedback system
1 2 21 21
LT
Lin
E et t
E r r e
L
a e
1 2Lb r r e
+EinLe
L
e
t2t1
r1 r2
ET
a
b
+x zy
1
z ay az
xy x bz ab
Feedback, general
UT-EWI-IOMS 2012 Vibrations and Waves Fabry-Perot
6UT-EWI-IOMS 2012 Vibrations and Waves Fabry-Perot
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1 2 21 21
LT
Lin
E et t
E r r e
j a b Field response: , with prop. constant:
F-P intensity response
2 41 2 1 2(field) (intensity)
L Lr r e R R ea a
1 2( , , , )A A R R La
1 2( , , , )B B R R La
2
221 sin
2
TTFP
in in
EI AT
I E B
Intensity
transmission:
per roundtrip: 2 2 21 2 1 2
L aL j Lr r e r r e e
b
0
22 2L Ln
b
attenuation:
phase shift:
UT EWI IOMS 2012 Vibrations and Waves Fabry Perot
7
UT-EWI-IOMS 2012 Vibrations and Waves Fabry-Perot
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7
21 2
221 2
(1 )(1 )
1
L
L
e R RA
R R e
a
a
21 2
22
1 2
4
1
L
L
R R eB
R R e
a
a
2 L b
where:
21 sin
2
FP
AT
B
F-P transfer functions
21 sin
2
FP C AAB
1FP FP FPT R A
2
1
1 sin2
FPCR
B
41 2
22
1 2
(1 )(1 )
1
L
L
R R eC
R R e
a
a
m2(m-1)2
Transmission
m2(m-1)2
Reflection
m2(m-1)2
R1=R2= 0.9
aL = 0.01
Absorption
more lossy:R1=R2= 0.7aL = 0.2
UT EWI IOMS 2012 Vibrations and Waves Fabry Perot
8UT-EWI-IOMS 2012 Vibrations and Waves Fabry-Perot
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8
Resonance
2 2
222
(1 )
11 sin2
LFP
L
A e RT A
ReB
a
a
At resonance: = m 2assume:R1=R2=R
no loss: a= 0 1, 0FP FPT A
At resonance:
all forward waves in phase
all backward waves in phase except: direct reflection at first
mirror in antiphase with
transmitted wave fromIB
reflection exactly cancelled
1 0FP FP FP FPT R A R energy conservation
II
L
R
IFI
T
R
IB
RFP= 0 ?
UT EWI IOMS 2012 Vibrations and Waves Fabry Perot
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9
Energy storageAt resonance: = m 2
assume:R1=R2=R
no loss: a= 0
II
L R
IFP IT
R
(1 )1
II T FP FP
II I R I I
R
Large enhancement forR 1
Resonance, IFP>>II
Energy stored inside cavity
TFP = 1,RFP = 0,AFP = 0
UT EWI IOMS 2012 Vibrations and Waves Fabry Perot
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10
Intensity enhancement
Important for lasers, optical amplifiers
other nonlinear optical interactions (nc(3)I)
resonance (e.g. in
Fabry-Perot)
R1=0.99 R2=0.99a=0
F-P100
input
I
I
small waveguide
cross-section
1 mm
1 mm
610
channel
input
I
I
Approaches:
Intensity = Power/cross-section [W/m2]
U a a av a y
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11
Energy leaks from resonator due to transmission through mirrors
absorption in medium
If energy supply is cut (input signal removed):
stored energy ( intensity) decays
Constant factorR1R2e4aL for each roundtrip
Exponential
decay
Relaxation time
FP
t
e
I0I
FP
0I
e
t
1 2
1
12 ln( )
2
FP
nc
R RL
a
y
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12
0
4 42 L n L n L
f
c
b
F-P filterTransfer functions determined by :
Note: n assumed constant
FWHM
2f
cFSR f
n L
Free Spectral Range, FSR,
in terms oforf:
2
2FSR n L
Finesse2
FSRF B
FWHM
FWHM= Full Width @ Half Max.
y
13
UT-EWI-IOMS 2012 Vibrations and WavesFabry-Perot
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13
2F B
21
L
LReF
Re
a
a
Finesse F & Peak width FWHM
FSRFWHMF
2
2FSR
n L
21 2
22
1 2
4
1
L
L
R R eB
R R e
a
a
1 2assume:R R R
R
F a=0
0.01
0.1
1
2 212
L
L
R eFWHM
n L R e
a
a
y
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14
Gain material in Fabry-Perot
2 2
22
(1 )
1
L
FPL
e RT
Re
a
a
in resonance
(=m 2)
R1=R2=R
F-P transmission
2
2(1 )
1
gL
FPgL
e RT
Re
Gain, not loss
our convention:
afield attenuation
gintensity gain
2
0.51
2
gL
gLR eFWHM
n L R e
1 0gLR e If 0FWHM FPT unstable, oscillation
y
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15
2 21 2
L L jG r r e Re e a
1 2( , 2 )r r R L b
Roundtrip gain G
Stability of Fabry-Perot
+EILe
Le
t2t1
r1 r2
ET
j a b
2
1 and 2L
Re ma
System will oscillate at frequencies for which = m 2
(starting from noise)
System becomes
unstable for G = 1:
2 ln( ) / g R La can happen for a< 0,
(withR < 1) if:
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Realisations
Macroscopic
Integrated Optic
Mirrors
(dielectric multilayer
or thin metal layer)
2
2 1
2 1
n nR
n n
InP / GaAs (n 4)
in air R 0.36
Fresnel reflection at facets
R1 R2
Grating (Bragg reflector)L
Wavelength
dependent:0
2 effN
L
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Summary Fabry-Perot
Superposition of multiple reflections multiple interference
Modeling as a feedback system
At resonance, without loss: TFP = 1,RFP = 0 Intensity enhanced inside cavity
Relaxation: FP(L,R,a)
Filter: FSR(L), FWHM(R,
a,
L), F(R,
a,
L) Gain and stability (positive feedback)
Integrated Optic realisations
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Integrated Optical Micro Systems (IOMS)slide 1
Lecture "Vibrations and Waves":Resonators and Scattering
Markus Pollnau
Integrated Optical Micro Systems (IOMS)
MESA+ Institute
University of Twente, Enschede, The Netherlandshttp://ioms.ewi.utwente.nl
Content
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Integrated Optical Micro Systems (IOMS)slide 2
1. Resonators
2. Scattering of light
Content
Energy of an Electromagnetic Wave
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Integrated Optical Micro Systems (IOMS)slide 3
The electric field of a charge with acceleration aat anangle from the axis of the motion at distance ris
The energy of an electromagnetic wave isproportional to the average of its intensity, whichis the square of its electric field.
The energy per unit area per unit timecarried by an electromagnetic wave is
Energy of an Electromagnetic Wave
32
0
2
222
20
16
sin
craqEcS
q
rc
crtqatE
2
04
sin/
q
Power of an Electromagnetic Wave
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Integrated Optical Micro Systems (IOMS)slide 4
Integrated over the whole sphere, this gives aradiated power of
For an acceleration , i.e.,we receive
Power of an Electromagnetic Wave
3
0
22
0
3
3
0
22
6sin
8 c
aqd
c
aqSdAP
qq
ti
exa
02
2
0
42
2
1
xa
3
0
2
0
42
12 c
xqP
Quality Factor of a Resonator
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Integrated Optical Micro Systems (IOMS)slide 5
A charge set to oscillate and then left alone radiatesenergy, i.e., it loses energy
(called energy damping or radiation damping).The slower the oscillator loses energy, the higher is
its quality.
We define the quality factor Qof a resonator as its
total energy Wat any time devided by the energyloss per radian(and using ):
The damping is
Quality Factor of a Resonator
P
W
dtdW
W
ddW
WQ
dtdWdtddtdWddW
QteWWWQPdtdW /0
Quality Factor of an Oscillating Atom
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Integrated Optical Micro Systems (IOMS)slide 6
Total energy of an oscillator:
Eigen frequency (e.g.,yellow line of the sodium atom):
(Feynman lost a factor of in his final equation)
Quality Factor of an Oscillating Atom
7
2
2
0
3
0
2
0
42
2
0
2
105
3
12
2
1
e
cm
cxe
xm
P
WQ
ee
2
0
2
2
1xmW
nmcc 5902
kgme31
101.9 Ase 19106.1
smc /100.3 8 VmAs/106.8 120
04
Lifetime of an Oscillating Atom
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Integrated Optical Micro Systems (IOMS)slide 7
I.e., an atomic oscillator will oscillate for 108 radians
or about 107 oscillations, before its energy falls bya factor 1/e.
Since the oscillation frequency is ,the luminescence lifetime is typically in the rangeof a few ns (10-8 s).
Lifetime of an Oscillating Atom
7
2
2
0
3
0
2
0
42
2
0
2
1053
12
2
1
e
cmcxe
xm
PWQ e
e
11510/
sc
Spectral Linewidth of an Oscillating Atom
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Integrated Optical Micro Systems (IOMS)slide 8
Mechanics and electronics:
, with , the resistance.
With ,
the spectral linewidth of such an atomic oscillation is
This equals
The lineshape is Lorentzian (Fourier transform of anexponential temporal decayinto frequencyspace). For the equation of the lineshape, seeFeynman I-23-2.
Spectral Linewidth of an Oscillating Atom
0
Q
/2 c
mQQccc 140
2
0102.1//2/2/2
MHzHz 10101 7
Scattering of Light
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Integrated Optical Micro Systems (IOMS)slide 9
When the phase difference between two or more lightsources changes rapidly compared to the
detection system, the cosine function of theinterference term averages out, i.e., thephenomenon of interference cannot be observedanymore. In this case, the resulting light intensityis just the sum of the intensities of the
overlapping beams.The same accounts when the light sources are not
perfectly aligned with each other but arerandomly distributed.
Scattering of Light
Scattering of Light
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Integrated Optical Micro Systems (IOMS)slide 10
Since atoms in air excited by an incoming light beamare radiating light as point sources in all
directions and these atoms are randomlydistributed, their light intensities are added up:The light is scattered.
In addition, the atoms move randomly, i.e., even thecosine term of light scattered from a single atom
over time averages out.
As a result of the scattering of sun light, the sun riseand sun set appear red (the higher the airpollution, the more beautiful is the sun set...) andthe sky appears blue.
Scattering of Light
Scattering of Light
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Integrated Optical Micro Systems (IOMS)slide 11
Incoming beam:
Amplitude of oscillating atom:
Total scattered power:
Scattering of Light
tieEE 0
220
0
e
e
m
Eqx
220
2
4
422
0
2
4
2
00
2
3
0
42
163
8
2
1
12
cm
qcEx
c
qP
e
ee
Scattering of Light
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Integrated Optical Micro Systems (IOMS)slide 12
First term: total incident energy
Second term: cross-section for scattering
Classical electron radius:
(Feynman lost )
Scattering of Light
2
00
2
002
1cEEc
22
0
2
4
422
0
2
4
2
00
2
3
0
42
163
8
2
1
12
cm
ecEx
c
qP
e
e
220
2
4
2
022
0
2
4
422
0
2
4
3
8
163
8
r
cm
e
e
s
mcm
er
e
15
2
0
2
01082.2
4
04
Scattering of Light
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In air, the natural frequencies of the oscillators arehigher than the frequencies of visible light, i.e.
and the scattering of light increases with the fourthpower of the frequency of light.
This type of scattering is called Raleigh scattering.
Scatte g o g t
22024
2
0
3
8
r
s
4
0
4
2
0
3
8
rs