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-1

Fabry-Perot interferometers MIT 2.71/2.710 Optics 10/20/04 wk7-b-2

Relation between r, r’and t, t’ air glass air glass Proof: algebraic from the Fresnel coefficients or using the property of preservation of the field properties upon time reversal Stokes relationships MIT 2.71/2.710 Optics 10/20/04 wk7-b-3

Proof using time reversal air glass air glass MIT 2.71/2.710 Optics 10/20/04 wk7-b-4

Fabry-Perot interferometers reflected transmitted incident 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-5

Calculation of the reflected wave incoming transmitted transmitted reflected reflected transmitted reflected transmitted reflected transmitted reflected air glass air MIT 2.71/2.710 Optics 10/20/04 wk7-b-6

Calculation of the reflected wave Use Stokes relationships MIT 2.71/2.710 Optics 10/20/04 wk7-b-7

Transmission & reflection coefficients MIT 2.71/2.710 Optics 10/20/04 wk7-b-8

Transmission & reflection vs path Path delay Path delay Reflection Transmission Path delay Path delay MIT 2.71/2.710 Optics 10/20/04 wk7-b-9

Fabry-Perot terminology free Spectral range band width Transmission coefficient resonance frequencies Frequency v MIT 2.71/2.710 Optics 10/20/04 wk7-b-10

Fabry-Perot terminology FWHM Bandwidth is inversely proportional to the finesse F (or quality factor) of the cavity Transmission coefficient free spectral range bandwidth finesse MIT 2.71/2.710 Optics 10/20/04 wk7-b-11

Spectroscopy using Fabry-Perot cavity Goal: to measure the specimen’s absorption as function of frequency ω Experimental measurement principle: Scanning stage (controls cavity length L ) Spectrum of light beam is modified by substance Light beam of known spectrum Power meter Transparent windows Container with specimen to be measured Partially-reflecting mirrors (FP cavity) MIT 2.71/2.710 Optics 10/20/04 wk7-b-12

Spectroscopy using Fabry-Perot cavity Goal: to measure the specimen’s absorption as function of frequency ω Experimental measurement principle: Spectrum of light beam is modified by substance Electro-optic (EO) modulator (controls refr. Index n) Light beam of known spectrum Power meter Transparent windows Container with specimen to be measured Partially-reflecting mirrors (FP cavity) MIT 2.71/2.710 Optics 10/20/04 wk7-b-13

Spectroscopy using Fabry-Perot cavity transmissivity Unknown spectrum Sample measured: MIT 2.71/2.710 Optics 10/20/04 wk7-b-14

Spectroscopy using Fabry-Perot cavity aaaFabry–Perot transmissivity Unknown spectrum Sample measured: MIT 2.71/2.710 Optics 10/20/04 wk7-b-15

Spectroscopy using Fabry-Perot cavity aaaFabry–Perot transmissivity Unknown spectrum Sample measured: MIT 2.71/2.710 Optics 10/20/04 wk7-b-16

Spectroscopy using Fabry-Perot cavity unknown spectrum width should not exceed the FSR aaaFabry–Perot transmissivity Unknown spectrum Sample measured: MIT 2.71/2.710 Optics 10/20/04 wk7-b-17

Spectroscopy using Fabry-Perot cavity aaaFabry–Perot transmissivity spectral resolution is determined by the cavity bandwidth Unknown spectrum Sample measured: MIT 2.71/2.710 Optics 10/20/04 wk7-b-18

Lasers MIT 2.71/2.710 Optics 10/20/04 wk7-b-19

Absorption spectra Atmospheric transmission human vision MIT 2.71/2.710 Optics 10/20/04 wk7-b-20

Semi-classical view of atom excitations Energy Atom in ground state Energy Atom in excited state MIT 2.71/2.710 Optics 10/20/04 wk7-b-21

Light generation Energy excited state equilibrium: most atoms in ground state ground state MIT 2.71/2.710 Optics 10/20/04 wk7-b-22

Light generation Energy excited state A pump mechanism (e.g. thermal excitation or gas discharge) ejects some atoms to the excited state ground state MIT 2.71/2.710 Optics 10/20/04 wk7-b-23

Light generation Energy excited state The excited atoms radiatively decay, emitting one photon each ground state MIT 2.71/2.710 Optics 10/20/04 wk7-b-24

Light amplification: 3-level system Energy Super-excited state excited state ground state equilibrium: most atoms in ground state; note the existence of a third, “super-excited” state MIT 2.71/2.710 Optics 10/20/04 wk7-b-25

Light amplification: 3-level system Energy Super-excited state excited state Utilizing the super-excited state as a short-lived “pivot point,” the pump creates a population inversion ground state MIT 2.71/2.710 Optics 10/20/04 wk7-b-26

Light amplification: 3-level system Energy Super-excited state excited state ground state When a photon enters, ... MIT 2.71/2.710 Optics 10/20/04 wk7-b-27

Light amplification: 3-level system Energy Super-excited state excited state When a photon enters, it “knocks” an electron from the inverted population down to the ground state, thus creating a new photon. This amplification process is called stimulated emission ground state MIT 2.71/2.710 Optics 10/20/04 wk7-b-28

w population inversion) Light amplifier Gain medium (e.g. 3-level system w population inversion) MIT 2.71/2.710 Optics 10/20/04 wk7-b-29

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-30

w population inversion) Laser initial photon Gain medium (e.g. 3-level system w population inversion) Partially reflecting mirror Light Amplification through Stimulated Emission of Radiation MIT 2.71/2.710 Optics 10/20/04 wk7-b-31

w population inversion) Laser amplified once initial photon Gain medium (e.g. 3-level system w population inversion) Partially reflecting mirror Light Amplification through Stimulated Emission of Radiation MIT 2.71/2.710 Optics 10/20/04 wk7-b-32

w population inversion) Laser amplified once initial photon Gain medium (e.g. 3-level system w population inversion) reflected Partially reflecting mirror Light Amplification through Stimulated Emission of Radiation MIT 2.71/2.710 Optics 10/20/04 wk7-b-33

w population inversion) Laser amplified once initial photon Gain medium (e.g. 3-level system w population inversion) reflected amplified twice Partially reflecting mirror Light Amplification through Stimulated Emission of Radiation MIT 2.71/2.710 Optics 10/20/04 wk7-b-34

w population inversion) Laser amplified once initial photon Gain medium (e.g. 3-level system w population inversion) reflected output amplified twice reflected Partially reflecting mirror Light Amplification through Stimulated Emission of Radiation MIT 2.71/2.710 Optics 10/20/04 wk7-b-35

w population inversion) Laser amplified once initial photon Gain medium (e.g. 3-level system w population inversion) reflected output amplified twice reflected amplified again etc. Partially reflecting mirror Light Amplification through Stimulated Emission of Radiation MIT 2.71/2.710 Optics 10/20/04 wk7-b-36

Confocal laser cavities diffraction angle waist w0 Beam profile: 2D Gaussian function “TE00mode” MIT 2.71/2.710 Optics 10/20/04 wk7-b-37

Other “transverse modes” (usually undesirable) MIT 2.71/2.710 Optics 10/20/04 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 Optics 10/20/04 wk7-b-39

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-40

Two types of incoherence temporal incoherence spatial incoherence matched paths point source 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-41

Two types of incoherence temporal incoherence spatial incoherence matched paths point source waves with equal paths but from different points on the wave front do not interfere waves from unequal paths do not interfere MIT 2.71/2.710 Optics 10/20/04 wk7-b-42

Coherent vs incoherent beams Mutually coherent: superposition field amplitude is described by sum of complex amplitude Mutually incoherent: superposition field intensity is described by sum of intensities (the phases of the individual beams vary randomly with respect to each other; hence, we would need statistical formulation to describe them properly — statistical optics) MIT 2.71/2.710 Optics 10/20/04 wk7-b-43

Coherence time and coherence length ‧ much shorter than “coherence length” ctc Sharp interference fringes Intensity incoming laser beam ‧ much longer than “coherence length” ctc no interference Michelson interferometer Intensity MIT 2.71/2.710 Optics 10/20/04 wk7-b-44

Coherent vs incoherent beams Coherent: superposition field amplitude is described by sum of complex amplitudes Incoherent: superposition field intensity is described by sum of intensities (the phases of the individual beams vary randomly with respect to each other; hence, we would need statistical formulation to describe them properly — statistical optics) MIT 2.71/2.710 Optics 10/20/04 wk7-b-45

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”): 80-100MHz Typical average power: 1-2W; peak power ~MW-GW MIT 2.71/2.710 Optics 10/20/04 wk7-b-46

Overview of light sources non-Laser Laser Thermal: polychromatic, spatially incoherent (e.g. light bulb) Continuous wave (or cw): strictly monochromatic, spatially coherent (e.g. HeNe, Ar+, laser diodes) Gas discharge: monochromatic, spatially incoherent (e.g. Na lamp) Pulsed: quasi-monochromatic, spatially coherent (e.g. Q-switched, mode-locked) Light emitting diodes (LEDs): monochromatic, spatially incoherent ~nsec ~psec to few fsec pulse duration mono/poly-chromatic = single/multi color MIT 2.71/2.710 Optics 10/20/04 wk7-b-47