2.717j/mas.857j optical engineering– transmission and reflection coefficients for a dielectric...

Today’s 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 Optics10/20/04 wk7-b-1

Fabry-Perot interferometers

MIT 2.71/2.710 Optics10/20/04 wk7-b-2

Relation between r, r’ and t, t’

a

arat

air

a’

a’t’

a’r’

glass airglass

Proof: algebraic from the Fresnel coefficientsor using the property of preservation of thepreservation of thefield properties upon time reversal12 =′+

−=′

ttrrr

field properties upon time reversal

Stokes relationships

MIT 2.71/2.710 Optics10/20/04 wk7-b-3

Proof using time reversal

a

arat

air glassatt’

arat

air

MIT 2.71/2.710 Optics10/20/04 wk7-b-4

glass

atr’art

ar2

⇔

( )( ) 1

0 22 =′+⇒=′+

′−=⇒=′+

ttrattrarrtrra

Fabry-Perot Interferometer

incident

reflected

transmitted

L

Resonance condition: reflected wave = 0⇔ all reflected waves interfere destructively

nmL2λ

=wavelength in free space

refractive index

MIT 2.71/2.710 Optics10/20/04 wk7-b-5

Calculation of the reflected wave

incoming a

reflected ar

transmitted ateiδ

reflected atei2δr’transmitted ateiδt’

reflected atei3δ(r’)2

transmitted atei2δr’t’


transmitted atei3δ(r’t’)2


transmitted atei4δ(r’)3t’

λπδ nL2=air, n=1 glass, n air, n=1

MIT 2.71/2.710 Optics10/20/04 wk7-b-6

Calculation of the reflected wave

( ){ }

⎭⎬⎫

⎩⎨⎧

′−′′+=

+′+′+′′+=

δδ

δδδ

222

44222reflected

e11e

ee1e

ii

iii

rrttra

rrrttraa

12 =′+

−=′

ttrrr

Use Stokes relationships

( )δ

δ

22

2

reflected e1e1

i

i

rraa−−

=

MIT 2.71/2.710 Optics10/20/04 wk7-b-7

Transmission & reflection coefficients

( )δ

δ

22

2

reflected e1 e1

i

i

rraa

−−

= δ22dtransmitte e1 irttaa

−′

=

2rR ≡

( )( )

( ) δ

δδ

22

22dtransmitte

22

22reflected

sin411

tcoefficienontransmissi

sin41sin4

tcoefficienreflection

RRR

aa

RRR

aa

+−−

==⎟⎟⎠

⎞⎜⎜⎝

⎛

+−==⎟⎟

⎠

⎞⎜⎜⎝

⎛

MIT 2.71/2.710 Optics10/20/04 wk7-b-8

MIT 2.71/2.710 Optics10/20/04 wk7-b-9

Transmission & reflection vs pathR=0.95

R=0.5

MIT 2.71/2.710 Optics10/20/04 wk7-b-10

Fabry-Perot terminology

nLmc2

( )nL

cm2

1+ ( )nL

cm2

2+ frequency ν

bandwidth

freespectralrange

resonancefrequencies

Fabry-Perot terminology

MIT 2.71/2.710 Optics10/20/04 wk7-b-11

∆νFWHM

FWHM Bandwidth is inversely proportionalto the finessefinesse F (or quality factorquality factor)of the cavity

∆νFSR

RRF

−≡

1π

nLFc

2FWHM =∆ν

FFSR

FWHMνν ∆

=∆ ( ) ( )( )finesse

range spectral freebandwidth =

Spectroscopy using Fabry-Perot cavityGoal: Goal: to measure the specimen’s absorption as function offrequency ω

Experimental measurement principle:Experimental measurement principle:

MIT 2.71/2.710 Optics10/20/04 wk7-b-12

container with specimen to be measured

transparentwindows

light beam ofknown spectrum

spectrum of light beamis modified by substance

scanningstage

(controls cavitylength L)

partially–reflectingmirrors (FP cavity)

powermeter

Spectroscopy using Fabry-Perot cavityGoal: Goal: to measure the specimen’s absorption as function offrequency ω

Experimental measurement principle:Experimental measurement principle:

container with specimen to be measured

transparentwindows

light beam ofknown spectrum

spectrum of light beamis modified by substance

MIT 2.71/2.710 Optics10/20/04 wk7-b-13

partially–reflectingmirrors (FP cavity)

powermeter

electro-optic(EO) modulator

(controls refr.index n)

Spectroscopy using Fabry-Perot cavity

ω

I(ω)

unknownspectrum

Fabry–Perottransmissivity

ω1sample measured:

I(ω1)MIT 2.71/2.710 Optics10/20/04 wk7-b-14


ω

I(ω)

unknownspectrum


ω2sample measured:

I(ω2)MIT 2.71/2.710 Optics10/20/04 wk7-b-15


ω

I(ω)

unknownspectrum


ω3sample measured:

I(ω3)MIT 2.71/2.710 Optics10/20/04 wk7-b-16


ω

I(ω)

unknownspectrum


ω3sample measured:

I(ω3)

unknown spectrumwidth should not exceed the FSR

MIT 2.71/2.710 Optics10/20/04 wk7-b-17


ω

I(ω)

unknownspectrum


ω3sample measured:

I(ω3)

spectral resolutionspectral resolutionis determined by thecavity bandwidth

MIT 2.71/2.710 Optics10/20/04 wk7-b-18

Lasers

MIT 2.71/2.710 Optics10/20/04 wk7-b-19

Absorption spectra

λ (µm)

Atm

osph

eric

tran

smis

sion

human visionMIT 2.71/2.710 Optics10/20/04 wk7-b-20

Semi-classical view of atom excitationsEnergy

MIT 2.71/2.710 Optics10/20/04 wk7-b-21

Ze+e-

Atom in ground stateAtom in ground state

Energy

Atom in excited stateAtom in excited state

Light generation

Energy

MIT 2.71/2.710 Optics10/20/04 wk7-b-22

ground state

excited state

equilibrium: most atomsin ground state

Light generation

MIT 2.71/2.710 Optics10/20/04 wk7-b-23

ground state

Energyexcited state

A pump mechanism (e.g. thermal excitation or gas discharge) ejects some atoms to the excited state

Light generation

MIT 2.71/2.710 Optics10/20/04 wk7-b-24

ground state

Energyexcited state

hν

hν

The excited atoms radiativelydecay, emitting one photon each

Light amplification: 3-level systemEnergy

MIT 2.71/2.710 Optics10/20/04 wk7-b-25

ground state

super-excited state

excited state

equilibrium: most atomsin ground state; note the existenceof a third, “super-excited” state


MIT 2.71/2.710 Optics10/20/04 wk7-b-26

ground state

super-excited state

excited state

Utilizing the super-excited stateas a short-lived “pivot point,” thepump creates a population inversion


MIT 2.71/2.710 Optics10/20/04 wk7-b-27

ground state

super-excited state

excited state

hν

When a photon enters, ...

Light amplification: 3-level system

ground state

Energy super-excited state

excited state

hνhν

hν

MIT 2.71/2.710 Optics10/20/04 wk7-b-28

When a photon enters, it “knocks” an electron from the inverted population down to the ground state, thus creatinga new photon. This amplification process is called stimulated emission

Light amplifier

Gain medium(e.g. 3-level system

w population inversion)

Pin

Pout =gPin

MIT 2.71/2.710 Optics10/20/04 wk7-b-29

Light amplifier w positive feedback



Pin

Pout =gPin

When the gain exceeds the roundtrip losses, the system goes into oscillation

gΣ+

+

MIT 2.71/2.710 Optics10/20/04 wk7-b-30

Laserinitial photon



Partiallyreflecting

mirrorLightAmplification throughStimulatedEmission ofRadiation

MIT 2.71/2.710 Optics10/20/04 wk7-b-31

Laser

MIT 2.71/2.710 Optics10/20/04 wk7-b-32



amplified once initial photon

Partiallyreflecting


Laser

MIT 2.71/2.710 Optics10/20/04 wk7-b-33



amplified once

reflected

initial photon

Partiallyreflecting


Laser



initial photonamplified once

reflected

amplified twice

Partiallyreflecting


MIT 2.71/2.710 Optics10/20/04 wk7-b-34

Laser




reflected

amplified twice

reflected

output

Partiallyreflecting


MIT 2.71/2.710 Optics10/20/04 wk7-b-35

Laser




reflected

amplified twice

Partiallyreflecting

mirror

reflected

output

amplified againetc.

LightAmplification throughStimulatedEmission ofRadiation

MIT 2.71/2.710 Optics10/20/04 wk7-b-36

Confocal laser cavities

MIT 2.71/2.710 Optics10/20/04 wk7-b-37

waist w0

diffractionangle

0

tannwπλθ =

Beam profile: 2D Gaussian function

“TE00 mode”

Other “transverse modes”

TE10 TE11

(usually undesirable)

MIT 2.71/2.710 Optics10/20/04 wk7-b-38

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)

MIT 2.71/2.710 Optics10/20/04 wk7-b-39

CW (continuous wave lasers)

Laser oscillation well approximated by a sinusoid

1/ν

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), ~10µsec (doubled Nd:YAG)MIT 2.71/2.710 Optics10/20/04 wk7-b-40

Two types of incoherencetemporaltemporal

incoherencespatialspatial

incoherenceincoherence incoherence

1r

2r1rmatched

pathspointsource

d2d1

Michelson interferometer Young interferometer

poly-chromatic light(=multi-color, broadband)

mono-chromatic light(= single color, narrowband)

MIT 2.71/2.710 Optics10/20/04 wk7-b-41

Two types of incoherencetemporaltemporal

incoherencespatialspatial

incoherenceincoherence incoherence

1r

2r1rmatched

pathspointsource

d2d1

waves from unequal pathswaves from unequal pathsdo not interfere

waves with equal pathswaves with equal pathsbut from different pointsbut from different points

on the wavefronton the wavefrontdo not interfere

do not interfere

do not interfereMIT 2.71/2.710 Optics10/20/04 wk7-b-42

Coherent vs incoherent beams

MIT 2.71/2.710 Optics10/20/04 wk7-b-43

1e11φiaa =

2e22φiaa =

Mutually coherent: superposition field amplitudeamplitudeis described by sum of complex amplitudessum of complex amplitudes

221

2

212121 ee

aaaI

aaaaa ii

+==

+=+= φφ

Mutually incoherent: superposition field intensityintensityis described by sum 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 todescribe them properly — statistical optics)

2I

Coherence time and coherence length

Intensity

l1

• l1-l2 much shortershorter than “coherence length” ctc

sharp interference fringes

l1

l2

0

2I0

• l1-l2 much longerlonger than “coherence length” ctc

incominglaserbeam no interference

MIT 2.71/2.710 Optics10/20/04 wk7-b-44

Michelson interferometer Intensity

I0

l1

Coherent vs incoherent beams

MIT 2.71/2.710 Optics10/20/04 wk7-b-45

1e11φiaa =

2e22φiaa =

Coherent: superposition field amplitudeamplitudeis described by sum of complex amplitudessum of complex amplitudes

221

2

212121 ee

aaaI

aaaaa ii

+==

+=+= φφ

Incoherent: superposition field intensityintensityis described by sum 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 todescribe them properly — statistical optics)

2I

Mode-locked lasers

Typical sources: Ti:Sa lasers (major vendors: Coherent, Spectra Phys.)Typical mean wavelengths: 700nm – 1.4µm (near IR)

can be doubled to visible wavelengthsor 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”): 80-100MHzTypical average power: 1-2W; peak power ~MW-GW

MIT 2.71/2.710 Optics10/20/04 wk7-b-46

Overview of light sourcesLasernon-Laser

Thermal: 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 Optics10/20/04 wk7-b-47

2.717j/mas.857j optical engineering– transmission and reflection coefficients for a dielectric...

Documents