Lecture 3 Interferometers. Coherence

Today’s summary Different kinds of interferometers Multiple beam interferometers: Fabry-Perot resonators – Stokes relationships – Transmission and reflection coefficients for a dielectric slab – Optical resonance Coherence: spatial / temporal Kinds of lasers

Interferometers

Michelson Interferometer

Mach-Zender Interferometer

Twyman-Green Interferometer

Fabry-Perot interferometers

Relation between r, r’and t, t’ airglassairglass Stokes relationships Proof: algebraic from the Fresnel coefficients preservation of the or using the property of preservation of the field properties upon time reversal

Proof using time reversal airglassairglass

Fabry-Perot interferometers reflected incident transmitted Resonance condition: reflected wave = 0 ⇔ all reflected waves interfere destructively wavelength in free space refractive index

Calculation of the reflected wave airglassair incoming reflected transmitted reflected transmitted

Calculation of the reflected wave

Use Stokes relationships

Transmission & reflection coefficients reflection coefficient transmission coefficient

Transmission & reflection vs path Reflection Transmission Path delay

 Ró ż nica cz ę stotliwo ś ci pomi ę dzy s ą siednimi modami:

 Liczba modów gdzie Δλ jest szeroko ś ci ą po ł ówkow ą linii a λ 0 – długością centralną linii

Fabry-Perot terminology Transmission coefficient free Spectral range band width resonance frequencies Frequency v

Transmission coefficient FWHM Bandwidth is inversely proportional to the finesse F (or quality factor) of the cavity Fabry-Perot terminology

bandwidth free spectral range finesse

Fabry-Perot using options

K K-1 K-2 K+1K-3 K+2 Generation level K+3 Spectrum line Ne Potential mods

Fabry-Perot using options

 Every mod is a superposition of plane waves, which is due to the diffraction losses depend on the x and y coordinates can not give stationary field, and after many reflections the fixed configuration A (x, y) can be achived. The field distribution in the resonator of the two transverse axes of symmetry can be analyzed separately for each axis. Distribution of field for each axis can be described by a function of Hermite- Gaussian  In the above equations show that higher-order transverse modes in addition to the curvature of the wavefront described by kr 2 /2R there are phase jumps for π (change the sign of the amplitude) and then for different modes that occures on different places in the wave front. Number of strokes along the axis of symmetry of the phase corresponds to the values ​​ of the mode index.

 The intensity of higher order modes reach significant values ​​ in a larger area than the primary mode, which means that the laser beam of a higher order takes larger surface on the resonator mirrors, and further has a greater divergence.

Confocal laser cavities diffraction angle waist w 0 Beam profile: 2D Gaussian function “TE 00 mode”

Transverse modes (usually undesirable)

Lasers

Absorption spectra human vision Atmospheric transmission

Absorption spectra

MIT 2.71/2.710 Optics 10/20/04 wk7-b-40 CW (continuous wave lasers) 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” t c Typical coherence times ~20nsec (HeNe), ~10μsec (doubled Nd:YAG)

MIT 2.71/2.710 Optics 10/20/04 wk7-b-41 Two types of incoherence temporalincoherencespatialincoherence point source matched paths Michelson interferometerYoung interferometer poly-chromatic light (=multi-color, broadband) mono-chromatic light (= single color, narrowband)

Coherent vs incoherent beams amplitude Mutually coherent: superposition field amplitude sum of complex amplitude is described by sum of complex amplitude intensity Mutually incoherent: superposition field intensity sum of intensities 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)

Coherence time and coherence length incoming laser beam Michelson interferometer ‧ much shorter than “coherence length” ct c Intensity ‧ much longer than “coherence length” ct c Sharp interference fringes no interference

Coherent vs incoherent beams amplitude Coherent: superposition field amplitude sum of complex amplitudes is described by sum of complex amplitudes intensity Incoherent: superposition field intensity sum of intensities 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)

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”): MHz Typical average power: 1-2W; peak power ~MW-GW

Overview of light sources non-LaserLaser 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

Types of lasers Mode of operation: Continuous wave (cw) Pulsed – Q-switched – mode-locked

Types of lasers Lasing medium: Gas (Ar-ion, HeNe, CO 2 ) Metal-vapour lasers (HeCd, HeHg, HeAg, HeSe …) Solid state (Ruby, Nd:YAG, Ti:Sa) Dye (liquid) Excimer ( 193 nm (ArF), 248 nm (KrF), 308 nm (XeCl), 353 nm (XeF)) Gas dynamic laser FEL Raman laser Semiconductor lasers Diode (semiconductor) Vertical cavity surface-emitting lasers –VCSEL

Types of lasers