Lecture Bundle

7/31/2019 Lecture Bundle

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5/25/2012

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Integrated Optical Micro Systems (IOMS)

slide 1

Lecture "Vibrations and Waves":EM Waves, Polarization

Feridun Ay


MESA+ InstituteUniversity of Twente, Enschede, The Netherlands

http://ioms.ewi.utwente.nl

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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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slide 5

(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

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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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slide 9

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

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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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slide 13

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

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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/



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slide 17

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/


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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/



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slide 21

Fig. 1-15 from F.T. Ulaby, Applied Electromagnetics, Prentice Hall 1999

The electromagnetic spectrum

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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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slide 1

Lecture "Vibrations and Waves":Refraction and Dispersion

Markus Pollnau


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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slide 3

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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slide 4

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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slide 5

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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slide 6

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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slide 7

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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slide 8

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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slide 9

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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slide 10

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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slide 11

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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slide 12

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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slide 13

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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slide 14

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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slide 15

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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slide 16

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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slide 17

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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slide 18

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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slide 19

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

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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

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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:


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

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slide 11

Constructive interference:

= 0

Destructive interference:

=


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:


> @ > @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

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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


Thus we can rewrite the

(no212 ''' tnfnet


Example: A soap bubble of 25index of refraction of the soap fil

reflected light? Which colors app

color does the soap film appear

(destructive)

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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

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slide 23


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:


36


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M.P

Integrated


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

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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

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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

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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

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slide 39

Light source:

Luminescence bandwidth Interferometric resolution

138 nm ~ 2 m


Optocal Coherence Tomography (OCT):


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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4

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


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



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6

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




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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 ?




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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


10UT-EWI-IOMS 2012 Vibrations and WavesFabry-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



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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


MESA+ Institute

University of Twente, Enschede, The Netherlandshttp://ioms.ewi.utwente.nl

Content


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1. Resonators

2. Scattering of light

Content

Energy of an Electromagnetic Wave


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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 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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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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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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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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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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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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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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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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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

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