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

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Fabry-Perot interferometers

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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

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Proof using time reversal

air glass air glass

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Fabry-Perot interferometers

reflected

incident

transmitted

Resonance condition: reflected wave = 0

⇔ all reflected waves interfere destructively

wavelength in free space

refractive index

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Calculation of the reflected wave

air glass air

incoming

reflected

transmitted

transmitted

reflected

reflected

reflected

reflected

transmitted transmitted

transmitted

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Calculation of the reflected wave

Use Stokes relationships

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Transmission & reflection coefficients

reflection

coefficient

transmissioncoefficient

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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

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Fabry-Perot terminologyT

rans

mis

sion

coe

ffic

ient

freeSpectral

range

bandwidth

resonancefrequencies

Frequency v

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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

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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)

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Spectroscopy using Fabry-Perot cavity

Fabry–Perottransmissivity

Unknownspectrum

Sample measured:

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Spectroscopy using Fabry-Perot cavity

aaaFabry–Perottransmissivity

Unknownspectrum

Sample measured:

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Spectroscopy using Fabry-Perot cavity

aaaFabry–Perottransmissivity

Unknownspectrum

Sample measured:

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Spectroscopy using Fabry-Perot cavity

aaaFabry–Perottransmissivity

Unknownspectrum

Sample measured:

unknown spectrumwidth should not exceed the FSR

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Spectroscopy using Fabry-Perot cavity

aaaFabry–Perottransmissivity

Unknownspectrum

Sample measured:

spectral resolution is determined by the cavity bandwidth

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Lasers

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Absorption spectra

human vision

Atm

osph

eric

tran

smis

sion

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Semi-classical view of atom excitations

Energy

Energy

Atom in ground state Atom in ground state

Atom in excited state Atom in excited state

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Light generation

Energy excited state

equilibrium: most atomsin ground state

ground state

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Light generation

Energy excited state

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

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Light generation

Energy excited state

ground stateThe excited atoms radiativelydecay, emitting one photon each

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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

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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

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Light amplification: 3-level system

Energy

excited state

Super-excited state

ground stateWhen a photon enters, ...

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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

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Light amplifier

Gain medium(e.g. 3-level system

w population inversion)

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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

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Laser

initial photon

Gain medium(e.g. 3-level system

w population inversion)

Partiallyreflecting

mirror

LightAmplification throughStimulatedEmission ofRadiation

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Laser

initial photon

Gain medium(e.g. 3-level system

w population inversion)

Partiallyreflecting

mirror

LightAmplification throughStimulatedEmission ofRadiation

amplified once

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Laser

initial photon

Gain medium(e.g. 3-level system

w population inversion)

Partiallyreflecting

mirror

LightAmplification throughStimulatedEmission ofRadiation

amplified once

reflected

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Laser

initial photon

Gain medium(e.g. 3-level system

w population inversion)

Partiallyreflecting

mirror

LightAmplification throughStimulatedEmission ofRadiation

amplified once

reflected

amplified twice

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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

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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.

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Confocal laser cavities

diffraction angle

waist w0

Beam profile: 2D Gaussian function

“TE00mode”

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Other “transverse modes”

(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

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 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)

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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

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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)

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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

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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)

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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 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

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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