Fabry-Perot interferometers - Massachusetts Institute of Technology

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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

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Fabry-Perot interferometers

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Relation between r, r’ and t, t’

a

arat

air glass

a’

a’r’

a’t’

12 =′+

−=′

ttrrr Proof: algebraic from the Fresnel coefficients

or using the property of preservation of thepreservation of thefield properties upon time reversalfield properties upon time reversal

glass air

Stokes relationships

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Proof using time reversal

a

arat

air glassatt’

arat

air glass

atr’

artar2

( )( ) 1

0 22 =′+⇒=′+

′−=⇒=′+

ttrattrarrtrra

⇔

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Fabry-Perot Interferometer

L

incident

reflected

transmitted

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

nmL2λ

=wavelength in free space

refractive index

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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’

air, n=1 glass, n air, n=1 λπδ nL2=

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Calculation of the reflected wave

( ){ }

⎭⎬⎫

⎩⎨⎧

′−′′+=

+′+′+′′+=

δδ

δδδ

222

44222reflected

e11e

ee1e

ii

iii

rrttra

rrrttraa L

Use Stokes relationships12 =′+

−=′

ttrrr

( )δ

δ

22

2

reflected e1e1

i

i

rraa−−

=

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Transmission & reflection coefficients

( )δ

δ

22

2

reflected e1 e1

i

i

rraa

−−

= δ22dtransmitte e1 irttaa

−′

=

( )( )

( ) δ

δδ

22

22dtransmitte

22

22reflected

sin411

tcoefficienontransmissi

sin41sin4

tcoefficienreflection

RRR

aa

RRR

aa

+−−

==⎟⎟⎠

⎞⎜⎜⎝

⎛

+−==⎟⎟

⎠

⎞⎜⎜⎝

⎛

2rR ≡

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Transmission & reflection vs pathR=0.95

R=0.5

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Fabry-Perot terminology

nLmc2

( )nL

cm2

1+ ( )nL

cm2

2+ frequency ν

bandwidth

freespectralrange

resonancefrequencies

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Fabry-Perot terminologyFWHM Bandwidth is inversely proportionalto the finessefinesse F (or quality factorquality factor)of the cavity

∆νFWHM

RRF

−≡

1π

∆νFSR

nLFc

2FWHM =∆ν

FFSR

FWHMνν ∆

=∆ ( ) ( )( )finesse

range spectral freebandwidth =

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Spectroscopy using Fabry-Perot cavity

Experimental measurement principle:Experimental measurement principle:

container with specimen to be measured

transparentwindows

light beam ofknown spectrum

spectrum of light beamis modified by substance

partially–reflectingmirrors (FP cavity)

powermeter

Goal: Goal: to measure the specimen’s absorption as function offrequency ω

scanningstage

(controls cavitylength L)

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Experimental measurement principle:Experimental measurement principle:

container with specimen to be measured

transparentwindows

light beam ofknown spectrum

spectrum of light beamis modified by substance

partially–reflectingmirrors (FP cavity)

powermeter

Goal: Goal: to measure the specimen’s absorption as function offrequency ω

electro-optic(EO) modulator

(controls refr.index n)

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ω

I(ω)

unknownspectrum

Fabry–Perottransmissivity

ω1sample measured:

I(ω1)

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ω

I(ω)

unknownspectrum


ω2sample measured:

I(ω2)

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ω

I(ω)

unknownspectrum


ω3sample measured:

I(ω3)

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ω

I(ω)

unknownspectrum


ω3sample measured:

I(ω3)

unknown spectrumwidth should not exceed the FSR

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ω

I(ω)

unknownspectrum


ω3sample measured:

I(ω3)

spectral resolutionspectral resolutionis determined by thecavity bandwidth

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Lasers

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Absorption spectra

λ (µm)

Atm

osph

eric

tran

smis

sion

human vision

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Semi-classical view of atom excitationsEnergy

Energy

Atom in ground stateAtom in ground state

Atom in excited stateAtom in excited state

Ze+e-

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Light generation

ground state

excited state

equilibrium: most atomsin ground state

Energy

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Light generation

ground state

excited state

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

Energy

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Light generation

ground state

excited state

The excited atoms radiativelydecay, emitting one photon each

Energy

hν

hν

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Light amplification: 3-level system

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

Energy super-excited state

excited state

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


excited state

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ground stateWhen a photon enters, ...


excited state

hν

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ground state


excited state

hν

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

hν

hν

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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



Pin

Pout =gPin

gΣ+

+

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

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Laser



LightAmplification throughStimulatedEmission ofRadiation

initial photon

Partiallyreflecting

mirror

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Laser




initial photonamplified once

Partiallyreflecting

mirror

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Laser





reflected

Partiallyreflecting

mirror

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Laser





reflected

amplified twice

Partiallyreflecting

mirror

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Laser





reflected

amplified twice

Partiallyreflecting

mirror

reflected

output

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Laser





reflected

amplified twice

Partiallyreflecting

mirror

reflected

output

amplified againetc.

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Confocal laser cavities

waist w0

diffractionangle

0

tannwπλθ =

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)

1/ν

t

Laser oscillation well approximated by a sinusoid

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)

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Two types of incoherence

d1 d2

Michelson interferometer Young interferometer

1r1r

2r

spatialspatialincoherenceincoherence

temporaltemporalincoherenceincoherence

pointsource

matchedpaths

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

mono-chromatic light(= single color, narrowband)

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Two types of incoherence

d1 d2

1r1r

2r

spatialspatialincoherenceincoherence

temporaltemporalincoherenceincoherence

pointsource

matchedpaths

waves from unequal pathswaves from unequal pathsdo not interferedo not interfere

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

on the wavefronton the wavefrontdo not interferedo not interfere

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Coherent vs incoherent beamsMutually coherent: superposition field amplitudeamplitudeis described by sum of complex amplitudessum of complex amplitudes

1e11φiaa =

2e22φiaa =

221

2

212121 ee

aaaI

aaaaa ii

+==

+=+= φφ

Mutually incoherent: superposition field intensityintensityis described by sum of intensitiessum of intensities1I

21 III +=

2I (the phases of the individual beams vary randomly with respect to each other; hence,we would need statistical formulation todescribe them properly — statistical optics)

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Coherence time and coherence length

incominglaserbeam

l1

l2

Michelson interferometer

Intensity

l1

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

sharp interference fringes

Intensity

l1

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

no interference

0

2I0

I0

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

1e11φiaa =

2e22φiaa =

221

2

212121 ee

aaaI

aaaaa ii

+==

+=+= φφ

Incoherent: superposition field intensityintensityis described by sum of intensitiessum of intensities1I

21 III +=

2I (the phases of the individual beams vary randomly with respect to each other; hence,we would need statistical formulation todescribe them properly — statistical optics)

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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

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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)

mono/poly-chromatic = single/multi color

pulse duration

~nsec ~psec to few fsec

Fabry-Perot interferometers - Massachusetts Institute of Technology

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