P. Piot, PHYS 630 – Fall 2008• Resonator• Interaction of Atoms and photons• Amplification• Generation of laser lightLasers (5-6 lessons)P. Piot, PHYS 630 – Fall 2008Resonator: introduction• Resonator are the main ingredientof lasers.• Used to increase the optical powerassociated to a mode.• Boundary conditions implies theexistence of “eigenmodes” andeigenfrequencies.• Simplest model (to understand thephysics and the mathematical des-cription) is one-dimensional.P. Piot, PHYS 630 – Fall 2008Resonator: standing wave approach• Take the case of a plane-parallel resonator configuration• Consider an optical wave with complex amplitude• The boundaries conditions imposed by the planar mirror givesrise to a “quantification” of the wave vectorwith associated mode having a complex amplitudes• The mode frequency separation isdP. Piot, PHYS 630 – Fall 2008Resonator: traveling wave approach• Modes can be determine by following a travelingwave as it travels back and forth between two mirrors• The phase shift imparted by a round trip isP. Piot, PHYS 630 – Fall 2008Density of Modes• The number of modes per unit frequency is the inverse of thefrequency spacing between the mode.• The density of mode is the number of mode per unit of frequencyand unit of resonator length.• The number of mode per unit of frequency is• For one-dimensional resonator this isFactor 2 to account for 2 orthogonalpolarizationP. Piot, PHYS 630 – Fall 2008Losses & resonance spectral width• In a realistic resonator losses are present (mirror refection non unity,or due to the medium composing the resonator)• Consider the loss per round trip to be• Then the complex amplitude summed over an infinite number ofpasses is• For one-dimensional resonator this isP. Piot, PHYS 630 – Fall 2008Finesse I• The latter expression can be written• Where• Expliciting φ givesP. Piot, PHYS 630 – Fall 2008Finesse IIν/νfI/Imax550500P. Piot, PHYS 630 – Fall 2008Losses I• Consider a resonator with losses• And rewrite as a distributed loss• Identify to getwhereMirror 1Mirror 2Distributed lossesIgnore diffractionlossesP. Piot, PHYS 630 – Fall 2008Losses II• Now specialize to identical mirrors• The finesse is thenso finesse inversely proporti-onal to losses• “Photon lifetime”αrdfinesseP. Piot, PHYS 630 – Fall 2008Q-factor• Figure-of-merit for microwave, optical and electronic oscillatorsystems• Defined as• soFinesse is proportional to QConsidered frequencyP. Piot, PHYS 630 – Fall 2008Spherical-mirror resonator I• Difference with planar-mirror resonator is transverse focusing effects.• Naively would think that beam can be over-focussed and eventuallydiverges• The pass n is related to pass n-1 via the matrix equationwhereP. Piot, PHYS 630 – Fall 2008Spherical-mirror resonator II• The condition for stability is Tr(M)<2 which givesLine correspondingto symmetric resonatorg1g2P. Piot, PHYS 630 – Fall 2008Gaussian Modes• Gaussian beams are the modes associated to spherical mirrorresonator• Given the phase• The on-axis phase at the two mirror locations areso that the total round-trip phase shift is• From which the resonator frequencies are obtainedP. Piot, PHYS 630 – Fall 2008• Boundary conditions are imposedalong all directions• Resonator frequencieswith• Mode density3d rectangular resonatorsP. Piot, PHYS 630 – Fall 2008Whispering gallery• Example of a 2-Dmicro-resonatorP. Piot, PHYS 630 – Fall 2008Whispering mode gallery• Dynamical resonator using a dropletof oil.Optics Letters, Sept. 1, 2007, pp.