fabry-perot interferometers - massachusetts institute of technology
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MIT 2.71/2.710 Optics10/24/05 wk8-a-1
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 Optics10/24/05 wk8-a-2
Fabry-Perot interferometers
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MIT 2.71/2.710 Optics10/24/05 wk8-a-3
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
MIT 2.71/2.710 Optics10/24/05 wk8-a-4
Proof using time reversal
a
arat
air glassatt’
arat
air glass
atr’
artar2
( )( ) 1
0 22 =′+⇒=′+
′−=⇒=′+
ttrattrarrtrra
⇔
3
MIT 2.71/2.710 Optics10/24/05 wk8-a-5
Fabry-Perot Interferometer
L
incident
reflected
transmitted
Resonance condition: reflected wave = 0⇔ all reflected waves interfere destructively
nmL2λ
=wavelength in free space
refractive index
MIT 2.71/2.710 Optics10/24/05 wk8-a-6
Calculation of the reflected wave
incoming a
reflected ar
transmitted ateiδ
reflected atei2δr’
transmitted ateiδt’
reflected atei3δ(r’)2
transmitted atei2δr’t’
reflected atei4δ(r’)3
transmitted atei3δ(r’t’)2
reflected atei5δ(r’)4
transmitted atei4δ(r’)3t’
air, n=1 glass, n air, n=1 λπδ nL2=
4
MIT 2.71/2.710 Optics10/24/05 wk8-a-7
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−−
=
MIT 2.71/2.710 Optics10/24/05 wk8-a-8
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 ≡
5
MIT 2.71/2.710 Optics10/24/05 wk8-a-9
Transmission & reflection vs pathR=0.95
R=0.5
MIT 2.71/2.710 Optics10/24/05 wk8-a-10
Fabry-Perot terminology
nLmc2
( )nL
cm2
1+ ( )nL
cm2
2+ frequency ν
bandwidth
freespectralrange
resonancefrequencies
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MIT 2.71/2.710 Optics10/24/05 wk8-a-11
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 =
MIT 2.71/2.710 Optics10/24/05 wk8-a-12
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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MIT 2.71/2.710 Optics10/24/05 wk8-a-13
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
electro-optic(EO) modulator
(controls refr.index n)
MIT 2.71/2.710 Optics10/24/05 wk8-a-14
Spectroscopy using Fabry-Perot cavity
ω
I(ω)
unknownspectrum
Fabry–Perottransmissivity
ω1sample measured:
I(ω1)
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MIT 2.71/2.710 Optics10/24/05 wk8-a-15
Spectroscopy using Fabry-Perot cavity
ω
I(ω)
unknownspectrum
Fabry–Perottransmissivity
ω2sample measured:
I(ω2)
MIT 2.71/2.710 Optics10/24/05 wk8-a-16
Spectroscopy using Fabry-Perot cavity
ω
I(ω)
unknownspectrum
Fabry–Perottransmissivity
ω3sample measured:
I(ω3)
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MIT 2.71/2.710 Optics10/24/05 wk8-a-17
Spectroscopy using Fabry-Perot cavity
ω
I(ω)
unknownspectrum
Fabry–Perottransmissivity
ω3sample measured:
I(ω3)
unknown spectrumwidth should not exceed the FSR
MIT 2.71/2.710 Optics10/24/05 wk8-a-18
Spectroscopy using Fabry-Perot cavity
ω
I(ω)
unknownspectrum
Fabry–Perottransmissivity
ω3sample measured:
I(ω3)
spectral resolutionspectral resolutionis determined by thecavity bandwidth
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MIT 2.71/2.710 Optics10/24/05 wk8-a-19
Lasers
MIT 2.71/2.710 Optics10/24/05 wk8-a-20
Absorption spectra
λ (µm)
Atm
osph
eric
tran
smis
sion
human vision
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MIT 2.71/2.710 Optics10/24/05 wk8-a-21
Semi-classical view of atom excitationsEnergy
Energy
Atom in ground stateAtom in ground state
Atom in excited stateAtom in excited state
Ze+e-
MIT 2.71/2.710 Optics10/24/05 wk8-a-22
Light generation
ground state
excited state
equilibrium: most atomsin ground state
Energy
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MIT 2.71/2.710 Optics10/24/05 wk8-a-23
Light generation
ground state
excited state
A pump mechanism (e.g. thermal excitation or gas discharge) ejects some atoms to the excited state
Energy
MIT 2.71/2.710 Optics10/24/05 wk8-a-24
Light generation
ground state
excited state
The excited atoms radiativelydecay, emitting one photon each
Energy
hν
hν
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MIT 2.71/2.710 Optics10/24/05 wk8-a-25
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
MIT 2.71/2.710 Optics10/24/05 wk8-a-26
Light amplification: 3-level system
ground stateUtilizing the super-excited stateas a short-lived “pivot point,” thepump creates a population inversion
Energy super-excited state
excited state
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MIT 2.71/2.710 Optics10/24/05 wk8-a-27
Light amplification: 3-level system
ground stateWhen a photon enters, ...
Energy super-excited state
excited state
hν
MIT 2.71/2.710 Optics10/24/05 wk8-a-28
Light amplification: 3-level system
ground state
Energy super-excited 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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MIT 2.71/2.710 Optics10/24/05 wk8-a-29
Light amplifier
Gain medium(e.g. 3-level system
w population inversion)
Pin
Pout =gPin
MIT 2.71/2.710 Optics10/24/05 wk8-a-30
Light amplifier w positive feedback
Gain medium(e.g. 3-level system
w population inversion)
Pin
Pout =gPin
gΣ+
+
When the gain exceeds the roundtrip losses, the system goes into oscillation
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MIT 2.71/2.710 Optics10/24/05 wk8-a-31
Laser
Gain medium(e.g. 3-level system
w population inversion)
LightAmplification throughStimulatedEmission ofRadiation
initial photon
Partiallyreflecting
mirror
MIT 2.71/2.710 Optics10/24/05 wk8-a-32
Laser
Gain medium(e.g. 3-level system
w population inversion)
LightAmplification throughStimulatedEmission ofRadiation
initial photonamplified once
Partiallyreflecting
mirror
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MIT 2.71/2.710 Optics10/24/05 wk8-a-33
Laser
Gain medium(e.g. 3-level system
w population inversion)
LightAmplification throughStimulatedEmission ofRadiation
initial photonamplified once
reflected
Partiallyreflecting
mirror
MIT 2.71/2.710 Optics10/24/05 wk8-a-34
Laser
Gain medium(e.g. 3-level system
w population inversion)
LightAmplification throughStimulatedEmission ofRadiation
initial photonamplified once
reflected
amplified twice
Partiallyreflecting
mirror
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MIT 2.71/2.710 Optics10/24/05 wk8-a-35
Laser
Gain medium(e.g. 3-level system
w population inversion)
LightAmplification throughStimulatedEmission ofRadiation
initial photonamplified once
reflected
amplified twice
Partiallyreflecting
mirror
reflected
output
MIT 2.71/2.710 Optics10/24/05 wk8-a-36
Laser
Gain medium(e.g. 3-level system
w population inversion)
LightAmplification throughStimulatedEmission ofRadiation
initial photonamplified once
reflected
amplified twice
Partiallyreflecting
mirror
reflected
output
amplified againetc.
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MIT 2.71/2.710 Optics10/24/05 wk8-a-37
Confocal laser cavities
waist w0
diffractionangle
0
tannwπλθ =
Beam profile: 2D Gaussian function
“TE00 mode”
MIT 2.71/2.710 Optics10/24/05 wk8-a-38
Other “transverse modes”
TE10 TE11
(usually undesirable)
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MIT 2.71/2.710 Optics10/24/05 wk8-a-39
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 Optics10/24/05 wk8-a-40
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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MIT 2.71/2.710 Optics10/24/05 wk8-a-41
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)
MIT 2.71/2.710 Optics10/24/05 wk8-a-42
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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MIT 2.71/2.710 Optics10/24/05 wk8-a-43
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)
MIT 2.71/2.710 Optics10/24/05 wk8-a-44
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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MIT 2.71/2.710 Optics10/24/05 wk8-a-45
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)
MIT 2.71/2.710 Optics10/24/05 wk8-a-46
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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MIT 2.71/2.710 Optics10/24/05 wk8-a-47
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
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