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Todays summary

Multiple beam interferometers: Fabry-Perot resonators

Stokes relationships

Transmission and reflection coefficients for a dielectric slab Optical resonance

Principles of lasers

Coherence: spatial / temporal

MIT 2.71/2.710 Optics

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Fabry-Perot interferometers


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Relation between r, r and t, t

a

ar

at

air

a

at

a

r

glass airglass

Proof: algebraic from the Fresnel coefficients

or using the property ofpreservation of thepreservation of the

field properties upon time reversal12 =+

=

ttrrr

field properties upon time reversal

Stokes relationships


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Proof using time reversal

a

ar

at

air glass

att

ar

at

air


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glass

atr

art

ar2

( )

( ) 10

22 =+=+

==+

ttrattra

rrtrra

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Fabry-Perot Interferometer

incident

reflected

transmitted

L

Resonance condition: reflected wave = 0

all reflected waves interfere destructively

n

mL

2

=

wavelength in free space

refractive index


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Calculation of the reflected wave

incoming a

reflected ar

transmitted atei

reflected atei2

r

transmitted ateit

reflected atei3(r)2

transmitted atei2rt

reflected atei4(r)3

transmitted atei3(rt)2

reflected atei5(r)4

transmitted atei4(r)3t

nL

2=

air, n=1 glass, n air, n=1


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Calculation of the reflected wave

( ){ }

+=

++++=

22

2

44222

reflected

e11e

ee1e

i

i

iii

rrttra

rrrttraa "

12 =+=ttr

rrUse Stokes relationships

( )

22

2

reflectede1e1

i

i

rraa=


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Transmission & reflection coefficients

( )

22

2

reflected

e1

e1i

i

r

raa

= 22dtransmitte e1 ir

ttaa

=

2rR

( )

( )( )

22

22

dtransmitte

22

22

reflected

sin41

1

tcoefficien

ontransmissi

sin41

sin4

tcoefficien

reflection

RR

R

a

a

RR

R

a

a

+

==

+==


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Transmission & reflection vspath

R=0.95

R=0.5

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Fabry-Perot terminology

nLmc2

( )nL

cm2

1+ ( )nL

cm2

2+ frequency

band

width

free

spectral

range

resonance

frequencies

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Fabry-Perot terminology


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FWHM

FWHM Bandwidth is

inversely proportional

to thefinessefinesse F(orquality factorquality factor)

of the cavity

FSR

R

R

F

1

nLF

c

2FWHM =

F

FSRFWHM

= ( )

( )( )finesse

rangespectralfreebandwidth =

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Spectroscopy using Fabry-Perot cavity

Goal:Goal: to measure the specimens absorption as function of

frequency

Experimental measurement principle:Experimental measurement principle:


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container with

specimen to be measured

transparent

windows

light beam of

known spectrum

spectrum of light beam

is modified by substance

scanning

stage

(controls cavity

lengthL)

partiallyreflecting

mirrors (FP cavity)

power

meter

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Goal:Goal: to measure the specimens absorption as function of

frequency

Experimental measurement principle:Experimental measurement principle:

container with

specimen to be measured

transparent

windows

light beam of

known spectrum

spectrum of light beam

is modified by substance


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partiallyreflecting

mirrors (FP cavity)

power

meter

electro-optic

(EO) modulator

(controls refr.

index n)

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I()

unknownspectrum

FabryPerot

transmissivity

1

sample measured:

I(1)MIT 2.71/2.710 Optics

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I()

unknownspectrum

FabryPerot

transmissivity

2

sample measured:


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I()

unknownspectrum

FabryPerot

transmissivity

3

sample measured:


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I()

unknownspectrum

FabryPerot

transmissivity

3

sample measured:

I(3)

unknown spectrum

width should not

exceed the FSR


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I()

unknownspectrum

FabryPerot

transmissivity

3

sample measured:

I(3)

spectral resolutionspectral resolutionis determined by the

cavity bandwidth


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Lasers


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Absorption spectra

(m)

Atmospherictransm

ission

human visionMIT 2.71/2.710 Optics

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Semi-classical view of atom excitations

Energy


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Ze+e-

Atom in ground stateAtom in ground state

Energy

Atom in excited stateAtom in excited state

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Light generation

Energy


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ground state

excited state

equilibrium: most atoms

in ground state

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Light generation


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ground state

Energyexcited state

A pump mechanism (e.g. thermal

excitation or gas discharge) ejects

some atoms to the excited state

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Light generation


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ground state

Energyexcited state

h

h

The excited atoms radiatively

decay, emitting one photon each

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Light amplification: 3-level system

Energy


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ground state

super-excited state

excited state

equilibrium: most atoms

in ground state; note the existence

of a third, super-excited state

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Energy


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ground state

super-excited state

excited state

Utilizing the super-excited state

as a short-lived pivot point, the

pump creates a population inversion

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Energy


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ground state

super-excited state

excited state

h

When a photon enters, ...

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ground state

Energysuper-excited state

excited state

hh

h


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When a photon enters, it knocks

an electron from the inverted population

down to the ground state, thus creating

a new photon. This amplification processis called stimulated emission

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Light amplifier

Gain medium

(e.g. 3-level system

w population inversion)

Pin

Pout =gPin


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Light amplifier w positive feedback

Gain medium



Pin

Pout =gPin

When the gain exceeds the roundtrip losses, the system goes

into oscillation

g++


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Laser

initial photon

Gain medium



Partiallyreflecting

mirrorLight

Amplification through

StimulatedEmission of

Radiation


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Laser


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Gain medium



amplified once initial photon

Partiallyreflecting

mirrorLight



Radiation

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Laser


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Gain medium



amplified once

reflected

initial photon

Partiallyreflecting

mirrorLight



Radiation

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Laser

Gain medium



initial photonamplified once

reflected

amplified twice

Partiallyreflecting

mirrorLight



Radiation


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Laser

Gain medium




reflected

amplified twice

reflected

output

Partiallyreflecting

mirrorLight



Radiation


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Laser

Gain medium




reflected

amplified twice

Partiallyreflecting

mirror

reflected

output

amplified again

etc.

Light



Radiation


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Confocal laser cavities


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waist w0

diffraction

angle0tan

nw

=

Beam profile:

2D Gaussian function

TE00 mode

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Other transverse modes

TE10 TE11

(usually undesirable)


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Types of lasers

Continuous wave (cw)

Pulsed

Q-switched

mode-locked

Gas (Ar-ion, HeNe, CO2)

Solid state (Ruby, Nd:YAG, Ti:Sa)

Diode (semiconductor)

Vertical cavity surface-emitting lasers VCSEL (also sc)

Excimer (usually ultra-violet)


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CW (continuous wave lasers)

Laser oscillation well approximated by a sinusoid

1/

t

Typical sources:

Argon-ion: 488nm (blue) or 514nm (green); power ~1-20W

Helium-Neon (HeNe): 633nm (red), also in green and yellow; ~1-100mW

doubled Nd:YaG: 532nm (green); ~1-10W

Quality of sinusoid maintained over a time duration known as

coherence time tc

Typical coherence times ~20nsec (HeNe), ~10sec (doubled Nd:YAG)MIT 2.71/2.710 Optics

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Two types of incoherence

temporaltemporal

incoherence

spatialspatial

incoherenceincoherence incoherence

1r

2r1r

matched

pathspoint

sourced2d1

Michelson interferometer Young interferometer

poly-chromatic light

(=multi-color, broadband)

mono-chromatic light

(= single color, narrowband)


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T f i h

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Two types of incoherence

temporaltemporal

incoherence

spatialspatial

incoherenceincoherence incoherence

1r

2r1r

matched

pathspoint

sourced2d1

waves from unequal pathswaves from unequal paths

do not interfere

waves with equal pathswaves with equal paths

but from different pointsbut from different points

on the wavefronton the wavefrontdo not interfere

do not interfere

do not interfereMIT 2.71/2.710 Optics

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C h t i h t b

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Coherent vs incoherent beams


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1e11iaa =

2e22iaa =

Mutually coherent: superposition field amplitudeamplitude

is described bysum of complex amplitudessum of complex amplitudes

2

21

2

2121

21

eeaaaI

aaaaa

ii

+==

+=+=

Mutually incoherent: superposition field intensityintensity

is described bysum of intensities1I

sum of intensities

21 III+=

(the phases of the individual beams vary

randomly with respect to each other; hence,

we would need statistical formulation to

describe them properly statistical optics)

2I

C h ti d h l th

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Coherence time and coherence length

Intensity

l1

l1-l2 much shortershorter than

coherence length ctcsharp interference fringes

l1

l2

0

2I0

l1-l2 much longerlonger than

coherence length ctc

incoming

laser

beamno interference


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Michelson interferometer Intensity

I0

l1

C h t i h t b

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Coherent vs incoherent beams


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1e11iaa =

2e22iaa =

Coherent: superposition field amplitudeamplitude

is described bysum of complex amplitudessum of complex amplitudes

2

21

2

2121

21

eeaaaI

aaaaa

ii

+==

+=+=

Incoherent: superposition field intensityintensity

is described bysum of intensities1I

sum of intensities

21 III+=

(the phases of the individual beams vary

randomly with respect to each other; hence,

we would need statistical formulation to

describe them properly statistical optics)

2I

M d l k d l

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Mode-locked lasers

Typical sources: Ti:Sa lasers (major vendors: Coherent, Spectra Phys.)

Typical mean wavelengths: 700nm 1.4m (near IR)can be doubled to visible wavelengths

or split to visible + mid IR wavelengths using OPOs or OPAs

(OPO=optical parametric oscillator;

OPA=optical parametric amplifier)Typical pulse durations: ~psec to few fsec

(just a few optical cycles)

Typical pulse repetition rates (rep rates): 80-100MHz

Typical average power: 1-2W; peak power ~MW-GWMIT 2.71/2.710 Optics

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O i f li ht

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Overview of light sources

Lasernon-LaserThermal:polychromatic,

spatially incoherent

(e.g. light bulb)

Gas discharge: monochromatic,

spatially incoherent(e.g. Na lamp)

Light emitting diodes (LEDs):

monochromatic, spatially

incoherent

Continuous wave (or cw):

strictly monochromatic,

spatially coherent(e.g. HeNe, Ar+, laser diodes)

Pulsed: quasi-monochromatic,spatially coherent

(e.g. Q-switched, mode-locked)

~nsec ~psec to few fsec

pulse duration

mono/poly-chromatic = single/multi colorMIT 2.71/2.710 Optics

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Documents

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