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MIT 2.71/2.710 Optics 10/20/04 wk7-b-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 Optics 10/20/04 wk7-b-2
Fabry-Perot interferometers
MIT 2.71/2.710 Optics 10/20/04 wk7-b-3
Relation between r, r’and t, t’
air glass air glass
Stokes relationships
Proof: algebraic from the Fresnel coefficientsor using the property of preservation of thepreservation of thefield properties upon time reversalfield properties upon time reversal
MIT 2.71/2.710 Optics 10/20/04 wk7-b-4
Proof using time reversal
air glass air glass
MIT 2.71/2.710 Optics 10/20/04 wk7-b-5
Fabry-Perot interferometers
reflected
incident
transmitted
Resonance condition: reflected wave = 0
⇔ all reflected waves interfere destructively
wavelength in free space
refractive index
MIT 2.71/2.710 Optics 10/20/04 wk7-b-6
Calculation of the reflected wave
air glass air
incoming
reflected
transmitted
transmitted
reflected
reflected
reflected
reflected
transmitted transmitted
transmitted
MIT 2.71/2.710 Optics 10/20/04 wk7-b-7
Calculation of the reflected wave
Use Stokes relationships
MIT 2.71/2.710 Optics 10/20/04 wk7-b-8
Transmission & reflection coefficients
reflection
coefficient
transmissioncoefficient
MIT 2.71/2.710 Optics 10/20/04 wk7-b-9
Transmission & reflection vs pathR
efle
ctio
nR
efle
ctio
n
Tra
nsm
issi
onT
rans
mis
sion
Path delay Path delay
Path delay Path delay
MIT 2.71/2.710 Optics 10/20/04 wk7-b-10
Fabry-Perot terminologyT
rans
mis
sion
coe
ffic
ient
freeSpectral
range
bandwidth
resonancefrequencies
Frequency v
MIT 2.71/2.710 Optics 10/20/04 wk7-b-11
Tra
nsm
issi
on c
oeff
icie
nt
FWHM Bandwidth is inversely proportional to the finesse F (or quality factor) of the cavity
Fabry-Perot terminology
bandwidthfree spectral range
finesse
MIT 2.71/2.710 Optics 10/20/04 wk7-b-12
Spectroscopy using Fabry-Perot cavity
GoalGoal: to measure the specimen’s absorption as function offrequency ω
Experimental measurement principle: Experimental measurement principle:
Light beam of known spectrum
Spectrum of light beam is modified by substance
Scanning stage
(controls cavity length L )
Transparent windows Container with
specimen to be measuredPartially-reflecting mirrors (FP cavity)
Power meter
MIT 2.71/2.710 Optics 10/20/04 wk7-b-13
Spectroscopy using Fabry-Perot cavity
GoalGoal: to measure the specimen’s absorption as function offrequency ω
Experimental measurement principle: Experimental measurement principle:
Light beam of known spectrum
Spectrum of light beam is modified by substance
Transparent windows Container with
specimen to be measuredPartially-reflecting mirrors (FP cavity)
Power meter
Electro-optic (EO) modulator
(controls refr. Index n)
MIT 2.71/2.710 Optics 10/20/04 wk7-b-14
Spectroscopy using Fabry-Perot cavity
Fabry–Perottransmissivity
Unknownspectrum
Sample measured:
MIT 2.71/2.710 Optics 10/20/04 wk7-b-15
Spectroscopy using Fabry-Perot cavity
aaaFabry–Perottransmissivity
Unknownspectrum
Sample measured:
MIT 2.71/2.710 Optics 10/20/04 wk7-b-16
Spectroscopy using Fabry-Perot cavity
aaaFabry–Perottransmissivity
Unknownspectrum
Sample measured:
MIT 2.71/2.710 Optics 10/20/04 wk7-b-17
Spectroscopy using Fabry-Perot cavity
aaaFabry–Perottransmissivity
Unknownspectrum
Sample measured:
unknown spectrumwidth should not exceed the FSR
MIT 2.71/2.710 Optics 10/20/04 wk7-b-18
Spectroscopy using Fabry-Perot cavity
aaaFabry–Perottransmissivity
Unknownspectrum
Sample measured:
spectral resolution is determined by the cavity bandwidth
MIT 2.71/2.710 Optics 10/20/04 wk7-b-19
Lasers
MIT 2.71/2.710 Optics 10/20/04 wk7-b-20
Absorption spectra
human vision
Atm
osph
eric
tran
smis
sion
MIT 2.71/2.710 Optics 10/20/04 wk7-b-21
Semi-classical view of atom excitations
Energy
Energy
Atom in ground state Atom in ground state
Atom in excited state Atom in excited state
MIT 2.71/2.710 Optics 10/20/04 wk7-b-22
Light generation
Energy excited state
equilibrium: most atomsin ground state
ground state
MIT 2.71/2.710 Optics 10/20/04 wk7-b-23
Light generation
Energy excited state
ground stateA pump mechanism (e.g. thermal excitation or gas discharge) ejects some atoms to the excited state
MIT 2.71/2.710 Optics 10/20/04 wk7-b-24
Light generation
Energy excited state
ground stateThe excited atoms radiativelydecay, emitting one photon each
MIT 2.71/2.710 Optics 10/20/04 wk7-b-25
Light amplification: 3-level system
Energy
excited state
Super-excited state
ground stateequilibrium: most atomsin ground state; note the existenceof a third, “super-excited” state
MIT 2.71/2.710 Optics 10/20/04 wk7-b-26
Light amplification: 3-level system
Energy
excited state
Super-excited state
ground stateUtilizing the super-excited stateas a short-lived “pivot point,” thepump creates a population inversion
MIT 2.71/2.710 Optics 10/20/04 wk7-b-27
Light amplification: 3-level system
Energy
excited state
Super-excited state
ground stateWhen a photon enters, ...
MIT 2.71/2.710 Optics 10/20/04 wk7-b-28
Light amplification: 3-level system
Energy
excited state
Super-excited state
ground stateWhen 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
MIT 2.71/2.710 Optics 10/20/04 wk7-b-29
Light amplifier
Gain medium(e.g. 3-level system
w population inversion)
MIT 2.71/2.710 Optics 10/20/04 wk7-b-30
Light amplifier w positive feedback
Gain medium(e.g. 3-level system
w population inversion)
When the gain exceeds the roundtrip losses, the system goes into oscillation
MIT 2.71/2.710 Optics 10/20/04 wk7-b-31
Laser
initial photon
Gain medium(e.g. 3-level system
w population inversion)
Partiallyreflecting
mirror
LightAmplification throughStimulatedEmission ofRadiation
MIT 2.71/2.710 Optics 10/20/04 wk7-b-32
Laser
initial photon
Gain medium(e.g. 3-level system
w population inversion)
Partiallyreflecting
mirror
LightAmplification throughStimulatedEmission ofRadiation
amplified once
MIT 2.71/2.710 Optics 10/20/04 wk7-b-33
Laser
initial photon
Gain medium(e.g. 3-level system
w population inversion)
Partiallyreflecting
mirror
LightAmplification throughStimulatedEmission ofRadiation
amplified once
reflected
MIT 2.71/2.710 Optics 10/20/04 wk7-b-34
Laser
initial photon
Gain medium(e.g. 3-level system
w population inversion)
Partiallyreflecting
mirror
LightAmplification throughStimulatedEmission ofRadiation
amplified once
reflected
amplified twice
MIT 2.71/2.710 Optics 10/20/04 wk7-b-35
Laser
initial photon
Gain medium(e.g. 3-level system
w population inversion)
Partiallyreflecting
mirror
LightAmplification throughStimulatedEmission ofRadiation
amplified once
reflected
amplified twice
reflected
output
MIT 2.71/2.710 Optics 10/20/04 wk7-b-36
Laser
initial photon
Gain medium(e.g. 3-level system
w population inversion)
Partiallyreflecting
mirror
LightAmplification throughStimulatedEmission ofRadiation
amplified once
reflected
amplified twice
reflected
output
amplified again etc.
MIT 2.71/2.710 Optics 10/20/04 wk7-b-37
Confocal laser cavities
diffraction angle
waist w0
Beam profile: 2D Gaussian function
“TE00mode”
MIT 2.71/2.710 Optics 10/20/04 wk7-b-38
Other “transverse modes”
(usually undesirable)
MIT 2.71/2.710 Optics 10/20/04 wk7-b-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 Optics 10/20/04 wk7-b-40
CW (continuous wave lasers)
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)
MIT 2.71/2.710 Optics 10/20/04 wk7-b-41
Two types of incoherencetemporal temporal
incoherenceincoherencespatial spatial
incoherenceincoherence
point source
matched paths
Michelson interferometer Young interferometer
poly-chromatic light(=multi-color, broadband)
mono-chromatic light(= single color, narrowband)
MIT 2.71/2.710 Optics 10/20/04 wk7-b-42
Two types of incoherencetemporal temporal
incoherenceincoherencespatial spatial
incoherenceincoherence
point source
matched paths
waves from unequal paths do not interfere
waves with equal paths but from different points
on the wave front do not interfere
MIT 2.71/2.710 Optics 10/20/04 wk7-b-43
Coherent vs incoherent beams
Mutually coherent: superposition field amplitudeamplitude is described by sum of complex amplitudesum of complex amplitude
Mutually incoherent: superposition field intensityintensityis described by sum of intensitiessum of intensities
(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 Optics 10/20/04 wk7-b-44
Coherence time and coherence length
incominglaserbeam
Michelson interferometer
‧ much shorter than “coherence length” ctc
Intensity
Intensity
‧ much longer than “coherence length” ctc
Sharp interference fringes
no interference
MIT 2.71/2.710 Optics 10/20/04 wk7-b-45
Coherent vs incoherent beams
Coherent: superposition field amplitudeamplitudeis described by sum of complex amplitudessum of complex amplitudes
Incoherent: superposition field intensity intensity is described by sum of intensitiessum of intensities
(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 Optics 10/20/04 wk7-b-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 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”): 80-100MHzTypical average power: 1-2W; peak power ~MW-GW
MIT 2.71/2.710 Optics 10/20/04 wk7-b-47
Overview of light sources
non-Laser 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 color
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