Laser Physics 5 Inhomogeneous broadening

Transcription

1 Laser Physics 5 Inhomogeneous broadening Fabien Bretenaker Fabien.Bretenaker@u-psud.fr Laboratoire Aimé Cotton Orsay - France

2 Homogeneous vs Inhomogeneous Homogeneous broadening: all the atoms have the same behavior, are described by the same cross sec8on σ(ν) α( ν ) = σ( ν )Δn Broadening mechanism: short coherence life8me (dephasing collisions, phonons, magne8c fields, electric fields, ) Inhomogeneous broadening: the atoms behave slightly differently, resul8ng in slightly shibed resonances This dispersion in resonance leads to an overall broadening of the transi8on If this dispersion is much broader than the homogeneous profile, the width is referred to as purely inhomogeneous. 2

3 Example: Gaussian inhomogeneous profile Sta8s8cal nature of the physical origin oben (but not always!) leads to a Gaussian profile: Example: Doppler broadening. Frequency shib due to atom velocity: Probability distribu8on for the velocity component along the z axis is Gaussian at thermal equilibrium 3

4 Example: Gaussian inhomogeneous profile The probablity for the resonance to lie in the interval [ν, ν + ν] seen along the z axis is propor8onal to this velocity distribu8on This leads to the following normalized profile and Doppler linewidth (M is molar mass, R is the ideal gas constant) Other examples of inhomogeneous broadening: ions in a crystalline host, isotopic mixture, etc 4

5 Unsaturated amplifica;on coefficient In the general case, distribu8on of resonance frequencies ν i denoted as P(ν i - ν 0 ), centered in ν 0 Assume pumping is the same for all atom class ( n 0 does not depend on ν i ) Popula8on inversion for atoms whose resonance frequencies lie in the range [ν i, ν i + dν i ] : 5

6 Unsaturated amplifica;on coefficient For the atom with resonance frequency ν i, the cross-sec8on σ i (ν) is given by the normalized homogeneous profile g(ν - ν i ) Overall unsaturated gain obtained by summing over all classes of atoms 6

7 Unsaturated amplifica;on coefficient: limi;ng cases In the limi8ng cases: purely homogeneous or purely inhomogeneous, becomes 7

8 Spectral hole burning Now let us consider satura8on of popula8on inversion by a wave at frequency ν. The satura8on intensity at frequency ν for atoms in the class i is I sat0 = hν/2*σ 0 τ is the satura8on intensity at resonance. Popula8on inversion for this class of atoms is therefore: 8

9 Spectral hole burning Unsaturated spectral population inversion Saturation term Satura8on term is nonzero only close to resonance Burn a hole with rela8ve depth given by 9

10 Spectral hole burning for Lorentzian homogeneous profile For a Lorentzian homogeneous profile The satura8on term reads In the purely inhomogenous approxima8on, this leads to a hole of width Saturation broadening Only the atoms close to resonance par8cipate, the others are passive spectators 10

11 Saturated amplifica;on coefficient Gain is obtained by summing over all classes of atoms Leading to = + σi( )dδni α( ν ) ν In the purely inhomogeneous case with P = G and g = L, one can assume P(ν i - ν 0 ) P(ν - ν 0 ), and take it out of the integral Satura8on reproduces the inhomogeneous profile, but reduces the gain 11

12 Laser opera;on in purely inhomogenous case In a mostly inhomogeneously broadened gain medium, several longitudinal modes may oscillate simultaneously Indeed each mode oscilla8ng at ν q correponds to a class of atoms that interact only with this mode if the homogeneous width is much smaller than the FSR of the laser Mul8ple spectral holes are burnt in the popula8on inversion 12

13 Laser opera;on in purely inhomogenous case If I q is the intensity of mode q, we write the equality of gain and losses for each mode Leading to the following expression for I q Intensity of the mode varies like the square of the inhomogeneous profile. Output power is 13

14 Mode compe;;on CW opera8on of a mul8mode laser leads to several new physical problems Mode compe88on: do the mode oscillate simultaneously, or does one of them oscillate alone? Mode intensi8es: what happens to the previous result when the broadening is not purely inhomogeneous Mode phase: what is the rela8ve phase of simultaneously oscilla8ng modes, and what does it imply? 14

15 Spectral hole burning in a linear cavity For a single-mode gas laser in a linear cavity, the light interacts with two velocity classes d N/dv z d N/dv z v z Δv c z = Δ ν ν h v z v - v z v + v z 15

16 Spectral hole burning in a linear cavity The laser extracts energy from two velocity classes If ν = ν 0, the two holes merge, and atoms with zero longitudianl velocity interact with both counterpropaga8ng waves Neglec8ng spa8al hole burning, the gain is As a consequence, when ν = ν 0, the total power decreases by a factor of 2, over a width equal to the homogeneous linewidth of the gain medium This is known as the Lamb dip 16

17 Reverse Lamb dip The same phenomenon exists in absorp8on In a laser including a saturable absorber, an increase in intensity is observed when the laser frequency coincides with the resonance of the absorbing cell. Due to bleaching of the zero velocity class Used to stabilize lasers in frequency 17

18 Mode selec;on Mode selec8on in a laser can be performed by inser8ng a filter in the cavity (introducing frequency-dependent losses): Birefringent filter (Lyot filter) Prism or gra8ng Fabry Perot Etalon Subcavity ac8ng as filters Once single mode laser opera8on is reached, stabiliza8on is required for various applica8ons using an extenral reference (stabilized cavity, atomic resonance) and an ac8ve feedback loop on the laser cavity 18