#### Transcript Saturation

Saturation Roi Levy Motivation • • To show the deference between linear and non linear spectroscopy To understand how saturation spectroscopy is been applied Motivation Outline • • • • Widths and Profiles of Spectral Lines theory Nonlinear Spectroscopy theory Experimental Schemes Some papers Widths and Profiles of Spectral Lines •Natural Line width •Doppler width •Collisional Broadening •Saturation Broadening Natural Line width • An excited atom can emit its excitation energy as spontaneous radiation • Describing it as dumped harmonic oscillator with dumping constant • As a result the amplitude decreases (in time) and the emitted radiation is no longer monochromatic Natural Line width The square of amplitude A(w) as function of w we can see it has the shape of a Lorentzian. The normalized intensity is 1 L( 0 ) 2 ( 0 ) ( ) 2 2 2 Where the full halfwidth at half-maximum is Natural Line width • Using the uncertainty principle we can relate the mean life time of an exited level i to it’s energy, Ei 1 i i Aik i • Where A is Einstein coefficient for spontaneous emission. ik Doppler Width • One of the major contributions to the spectral line width in gases. • Due to the thermal motion of the absorbing or emitting molecules. • Because Doppler Width is of our main interest we shall present the outline of the derivation. Doppler Width Consider an excited molecule with a velocity v relative to the rest fame of the observer. The central frequency of the emission is w At the rest frame the frequency is 0 e 0 k v The same relation holds for the absorption frequency a 0 k v Doppler Width At thermal equilibrium the molecule of a gas follow a Maxwellian velocity distribution ni (vz )dvz Ni ( e vi 2 ) vp vp n - density of molecules in level E i 2k b T vp m dvz i - the most probable velocity Doppler Width c d and that the Using the relation absorbed radiant power is proportional to the Density we get the intensity of the DopplerBroadened spectral line dv z 0 c( ) 2 0 I ( ) I 0 exp 0 v p Gaussian profile with full halfwidth D 0 8k bT c m ln 2 Doppler Width More detailed consideration will have to include the natural Line width for every molecule. Doppler Width The spectral intensity distribution of the total Absorption or emission is I ( ) I 0 n( ' ) L( ' )d ' This intensity profile is called Voigt profile. Saturation Broadening • At sufficiently large laser intensities the optical pumping rate on an absorbing transition become larger than the relaxation rates. • This saturation causes additional line broadening Saturation of Level Population by Optical pumping For two-level system with population N and N The rate equations are, 1 dN1 dN 2 PN1 R1 N1 PN 2 R2 N 2 dt dt With P B12 () the rate for stimulated emission (absorption), and Ri the relaxation Probability for level i. 2 Saturation of Level Population by Optical pumping Solving for the steady state we get for N1 N1 N P R1 2 P R1 R2 And for the difference between the population of the levels is N 0 N 0 N 1 2P /( R1 R2 ) 1 S Where S the saturation parameter is the ratio of the Pumping rate to the average relaxation rate. Saturation of Level Population by Optical pumping The pump rate due to monochromatic wave with intensity I ( ) is P 12 () I () / so that 2 12 I ( ) S A12 And the saturated absorption coefficient is ( ) 12 N 0 1 S Saturation Broadening of Homogeneous Line Profiles Since the absorption profile of a homogeneously broadened line is Lorentzian, the induced absorption probability is B12 ( ) L( 0 ) And the saturation parameter is B12 ( ) S L( 0 ) S ( 0 ) R 2 2 ( 0 ) 2 2 2 Saturation Broadening of Homogeneous Line Profiles The absorption coefficient will be S ( ) With s 1 S0 0 ( ) 1 S 0 ( 0 ) 2 2 ( 0 ) s 2 2 2 Nonlinear Spectroscopy •Linear and Nonlinear Absorption •Saturation of Inhomogeneous Line Profiles •Saturation Spectroscopy Linear and Nonlinear Absorption Assume that a monochromatic plane lightwave E E0 cos(t kz) With the mean intensity 1 I c 0 E 02 2 passes through a sample of molecules. The Absorption in volume dV is dP I ik NAdz Linear and Nonlinear Absorption In case the incident wave with spectral energy density ( ) I ( ) c And spectral width L Which is large compared to the halfwidth of the Absorption profile L The total intensity becomes I I ( )d I ( 0 ) L Linear and Nonlinear Absorption The absorbed power is then dP N dV I ik ( 0 ) L Remembering that the absorbed power is proportional to the number of absorbed photons n ph We can obtain P Bik ( )NdV h c Bik ik ( )d h Linear and Nonlinear Absorption Let us discuss what happens in open systems Many relaxation channels Also Molecules can diffuse in and out of the excitation volume Linear and Nonlinear Absorption The rate equations dN1 B12 ( N 2 N1 ) R1 N1 C1 dt dN 2 B12 ( N1 N 2 ) R2 N 2 C2 dt Where Ci Rik N k Di k Is the contribution of other levels to the population of level i , Di is the diffusion rate of the molecules in level i into the excitation volume Linear and Nonlinear Absorption Solving these equations under stationary Conditions (dN/dt=0) we get for 0 C 2 R1 C1 R2 N N N R1 R2 0 And for 0 2 0 1 0 N 0 N 0 N 1 B12 ( 1 1 ) 1 S R1 R2 Linear and Nonlinear Absorption The saturation parameter B I S 12* R c R1 R2 R R1 R2 * , The power decrease of the incident light wave from absorption along the length dz N 0 a dP A I 12 dz 1 S L Linear and Nonlinear Absorption In case of incoherent light sources S<<1 a dP P 12N dz L 0 And P is P P0 exp( 12 N z ) P0 e 0 z This is the Lambert-Beer law of linear absorption Linear and Nonlinear Absorption In open systems the saturated population density N Can be very small. 1 I (C1 C 2 ) B12 R2 C1 c N1 I ( R1 R2 ) B12 R2 R1 c C1 C2 N1 ( I ) R1 R2 Where C and C are small compared to R and R 1 2 For close systems 1 N N1 2 2 Saturation of Inhomogeneous broadened Line Profiles Inhomogeneous broadened line profiles such as Doppler-broadening we treat two cases • Hole Burning • Lamb Dip Hole Burning When a monochromatic light wave passes through a gas with Maxwell-Boltzmann velocity distribution the laser frequency in the frame of the molecule is ' k v z With k v z fall within the linewidth ' 0 Hole Burning The absorption cross section for the molecule ( 2) 2 12 ( , vz ) 0 ( 0 kvz ) 2 ( 2) 2 Due to the saturation the population N (v )dv Decreases within the velocity range dvz k and the population N (v )dv increases. Let us right the equations for N and N 1 2 z z 1 2 z z Hole Burning 2 0 S ( 2 ) N 0 0 N1 (, v z ) N1 (v z ) 1 ( 0 kvz ) 2 ( s 2) 2 S0 ( 2) 2 N 0 N 2 (, vz ) N (vz ) 2 ( 0 kvz ) 2 ( s 2) 2 0 2 where 1 2 is the homogeneous width of the transition. And s 1 S 0 Hole Burning Bennet hole (peak) For 1 2 the depth of the hole in N1 is different from the height of the peak in N2. Hole Burning The saturated population difference 2 S ( 2 ) 0 0 N (, v z ) N (v z )1 2 2 ( kv ) ( 2 ) 0 z s The absorption coefficient ( ) N (v z ) 12 ( , v z )dv z Hole Burning Solving the integral we get (0 ) 0 ( ) exp 1 S0 0.6D 0 2 • We can not detect Bennet hole by tuning the laser through the absorption profile. • Something missing ? Hole Burning The Bennet hole can be detected if two lasers are used • The saturating pump laser with the wave vector k1 which is kept at the frequency 1 and which burn the hole • A weak probe laser with the wave vector k2 and frequency tunable across the Voigt profile Hole Burning The absorption coefficient for the probe laser ( ) N (vz ) 12 (2 , vz )dvz N 0 S (1 , 2 ) vp S0 ( 2) 2 ( 2) 2 1 ( 0 k1vz )2 ( s 2) 2 ( 0 k2vz ) ( 2)2 dvZ Hole Burning The absorption coefficient for the probe laser 2 2 S0 0 S (1 , 2 ) ( ) 1 2 1 S0 2 S ( ' ) 2 ' 0 (1 0 ) k2 k1 S s (1 1 S 0 ) ( ' ) 0 ( ' ) S ( ' ) 0 ( ' ) S0 1 S 0 (1 1 S 0 ) Lamb Dip • Pump and probe waves can be generated by a single laser when the incident beam is reflected back into the absorption cell • The absorption profile will be Dopplerbroadened profile with a dip at the center at 0 • This dip is called Lamb dip after W.E.Lamb who first explained it theoretically Lamb Dip The saturated absorption coefficient in case of equal intensities (weak filed S 0 1) S0 0 ( ) ( ) 1 2 s 1 S0 2 2 S 1 2 22 0 S Lamb Dip For strong laser fields S ( ) 0 ( ) 2 2( 0 ) B 1 A B 1 2 2 A ( 0 ) 2 2 2 1 2 B ( 0 ) 2 (1 2S ) 2 2 1 2 Lamb Dip In case the intensity of the reflected wave is very small I 2 I1 S 0 ( ) ( ) 1 0 2 * S 2 2 S 1 2 * 2 2 0 S S 2 Saturation Spectroscopy • Experimental Schemes Saturation Spectroscopy • Experimental Schemes Saturation Spectroscopy • Laser-induced fluorescence Instead of measuring the attenuation of the probe beam the absorption can be monitored by the laser-induced fluorescence. • Advantageous when the density of the absorbing molecule is low. Saturation Spectroscopy Intermodulated fluorescence the pump beam and the probe beam are chopped At different frequencies I1 I10 (1 cos(1t ) I 2 I 20 (1 cos( 2 t ) The fluorescence intensity is I FL CN S ( I1 I 2 ) Saturation Spectroscopy At the center of an absorption line N S N 0 [1 a( I 1 I 2 )] I FL CN 0 [( I1 I 2 ) a( I1 I 2 ) 2 ] Only the term with both intensities contribute to the saturation effect 1 I10 I 20 cos(1t ) cos(2t ) I10 I 20[cos(1 2 )t cos(1 2 )t ] 2 Saturation Spectroscopy Saturation Spectroscopy Saturation Spectroscopy • Cross over signal In case of two center frequencies 1 , 2 which fulfill | 1 2 | D At laser frequency 1 2 the incident 2 Wave saturates the velocity class vZ dvZ 2 1 2k k Saturation Spectroscopy At that frequency we will see an additional saturation signal • Positive for common lower level • Negative for common upper level Beer’s law in presence of saturation effect 0 dI I n * n ( I ) I sat dz 1 I I ess And the optical depth od 0 0 n( x, y, z )dz f ( x, y; *) If f ( x, y; *) * ln Ii Ii I f sat I0 Reference 1. 2. 3. 4. W.Demtroder, Laser Spectroscopy (Springer 1991) T.W.Hansch, et al. P.R.L 27, 707 (1971) G.Reinaudi, et al. OPTICS LETTERS 32,21 (2007) Advance Optics Laboratory Tutorial, University of Colorado at Boulder Deuterium