PHYSICS OF THE SODIUM ATOM

The sodium atom has complex interactions with the radiation field and its environment. These interactions can change the photon return/atom by over a factor ten, depending on the precise spectral and temporal format of the laser. In this page we discuss a simplified model of the atom that captures the essential physics of these interactions, enables us to predict the photon returns and suggests ways to maximize the return for a given laser technology.
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Sodium Cell
TABLE OF CONTENTS

Optical Pumping
Interaction of the Sodium Atom with its Environment
       Doppler and Natural Line Widths
        Collision Lifetime
        Radiation Pressure
        Effect of the Earth's Magnetic Field
       
Depopulation of the Upper Ground State of the Sodium Atom
Spin Exchange  
      Effect of Spin Exchange on Photon Return/Watt
Monte Carlo Simulations
Saturation
Single Frequency CW Laser Photon Returns
Increasing the Photon Return/Watt
        Chirping
        Backpumping

Predicted Chirping and Backpumping Enhancements
                                                                   


                   

Optical PumpingEnergy levels of sodium atom

The sodium atom has two ground levels, an upper F=2 state and a lower F = 1 state. Under conditions of thermal equilibrium, 5/8 of the atoms are in the F=2 state. For the D2 transition, the upper F levels is split into 4 hyperfine sub levels each separated by some tens of MHz. All levels are additionally divided into different M levels that have the same energy, but different cross-sections, to the radiation field. A simplified energy diagram of the atom is shown in figure 1. Initially, the sodium atoms will be in thermal equilibrium with an equal probability of being in any given M level. If we have a single frequency laser tuned exactly to the transition between the F =2 ground state and F = 3 upper state the average cross-section of the atoms in thermal equilibrium for the F=2 ground state will be 7.717 x 10^-14 m^2. When an atom in a given M level is excited, the M level in the upper state is determined by the polarization of the laser light. Right handed circularly polarized photon of the appropriate frequency can only excite a sodium atom in the  (2,M) ground state to the upper (3,M+1) level,  but it can then decay to the (2,M), (2,M+1) or (2,M+2) ground level. If this is repeated, and in the absence of other effects, the atom will then be optical pumped into a stable transition between the (2,2) and (3,3) state after a small number of cycles.

This transition has the highest cross-section to the radiation field (1.6537 x 10^-13 m^2) and highest backscattering efficiency to ground, so that this optical pumping mechanism significantly increases the photon return/sodium atom. We should note that the higher photon return is not only because of the enhanced backscatter return for this transition (a factor of 1.5 compared to isotropic scattering) but also because the cross-section is a factor of about 2 higher than the cross-section averaged over all M levels. We expect an increase in photon return of a factor of about 3 and this has been observed at MIT/Lincoln Labs with a suitably tailored laser spectral format and a well resolved spot.

     However, we can also see from figure 1 that the photon return can be significantly reduced by depopulation of the F=2 ground state by another optical pumping process. If the sodium atom and laser frequency are slightly detuned, the energy diagram may favor transitions from the F =2 ground state to either the F=1 or F=2 upper state. For these excited atoms there is now a choice of ground state; an atom starting in the same ground (2,0) level may get excited to the upper (2,1) or even the (1,1) level and on decay may go either to the original F=2 ground state or the F=1 ground state. Because the energy separation between the two ground states (1.77 GHz) is substantially larger than the Doppler linewidth (500MHz HWHM), an atom  once pumped to the F=1 state will have minimal photon interaction with a single frequency laser tuned to pump atoms in the F=2 ground level and the photon return will be correspondingly reduced. There are thus two competing optical pumping processes at work, one pumping the atom to a favorable and stable (2,2) to (3,3) transition (GOOD) and one pumping atoms depopulating the upper ground state so that they no longer interact with a single frequency radiation field (BAD).
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Interaction of the Sodium Atom with its Environment

Doppler and Natural Linewidths

 The natural linewidth of the D2 line, set by the lifetime in the upper state, is 10 MHz FWHM. This linewidth is much narrower than the Doppler width of 1 GHz for atoms in the mesosphere . This is shown to scale in figure 2. Because the natural line-width is so narrow, only a few per cent of the atoms have a strong interaction with the radiation field. An atom whose (2,2) to (3,3) transition is exactly tuned to the laser frequency will absorb and spontaneously re emit one photon every 170 nsec for a laser beam intensity of 10 watts/m^2 and will continue to re emit photons until it collides with an air molecule. It will then usually move in different direction and only minimally interact with the radiation field, until it finds itself again with another near-zero line of sight velocity after yet another collision. When excited by a single frequency laser, a given Sodium atom will therefore emit photons in bursts that occur when the atom is moving in the right direction.

                       Doppler and natural linewidths
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Collision Lifetime

The conventional model of a gas is one in which hard spheres ( aka "billiard balls") bounce off each other. The time between collisions is set determined by the size of the particles, their speeds and their density. Most calculations of the photon return from the mesosphere assume that collisions occur between hard spheres with radii derived from viscosity measurements. For this class of particles, the scattering is isotropic and independent of the collision energy between particles. For momentum-changing collisions, which are important for viscosity, the hard sphere and more advanced calculations using, for instance, the Lennard-Jones model, produce similar results. Lennard-Jones Scattering between sodium and nitrogenHowever, for our case, a near collision in which the velocity of the sodium atom is changed by only a few percent in the orthogonal direction of its motion can totally change the interaction cross-section between the sodium atom and the light. Atmospheric molecules are neither hard nor spherical and long range interactions between the sodium atom and other molecules have a significant effect, especially because the intermolecular potential energy between nitrogen and sodium is substantially higher than the temperature of molecules in the mesosphere. 


The scattering angle as a function of reduced impact parameters is shown in figure 3 using a Lennard Jones model. These calculations only approximate the full quantum mechanical calculation, which should certainly be used for low scattering angles, but suggest that the near-miss collision cross-section, defined as introducing a 5 MHz velocity change in the line of sight Doppler velocity, is over three times that given by the hard sphere model. We should note here that the more correct model is one in which the sodium atom makes small random walks with near misses and small velocity shifts in positive and negative directions until it experiences a significant direction changing collision. The effective collision time for this process appears to be about half that of the hard sphere scattering model. The collision lifetime is thus another difficult parameter to estimate and may require empirical determination from the data.
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Radiation Pressure


 Every time a sodium atom absorbs a photon there is momentum transfer in the direction of the laser beam. This shift amounts, on average, to a red shift of 50 kHz/photon absorption.  If the sodium atom is initially moving in a direction giving the maximum return, the rate of photon production is therefore reduced by a factor of 2 in 100 cycles. For single frequency line of intensity of 20 watt/m^2, this occurs in about 100 microseconds, so that radiation pressure significantly reduces the photon return. This effect becomes even more significant for the long collision times in the upper parts of the mesosphere. Because the mean collision time changes by a factor of over ten across the mesosphere it is not in general possible to use an mean collision time for interactions, rather we must calculate the photon return as a function of height and sodium density and integrate the total photon return through the mesosphere. Radiation pressure can be used to increase the photon return by chirping the laser.
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Effect of the Earth's Magnetic Field

    Another effect is the Larmor precession of the atom in the Earth's magnetic field. In the absence of a radiation field, Larmor precession shifts the M level state of the atom in a periodic manner so that an atom initially in the preferred (2,2) ground state will get shifted to a neighboring M level in a time scale of about 1 microsec, returning to its original level in about 6 microsec for zenith pointing in a typical northern hemisphere site. If the atom is cycling in the radiation field at much higher rates than this, the atom will tend to remain in the (2,2) state and the photon return/watt will be high. If, however, the cycle time is much lower than 1 microsecond, the M levels will tend to be scrambled and the photon return reduced because both now both the mean cross-section and backscattering efficiency are reduced. This is one reason for the low return of the Gemini laser - a multi-line CW laser spreads its power between the lines so that while more sodium atoms are excited at any one time, (because there are more lines available for interaction), the cycle time of atoms is increased and can no longer maintain atoms in the (2,2) ground state.
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Depopulation of the Upper Ground State of the Sodium Atom

    A typical collision time is between 10 and 100 microseconds, depending on the position of the atom in the mesosphere. During this time, some sodium atoms will get excited into the upper F=1 or F=2 states and then, instead of returning automatically to the upper ground state, has about an even chance of ending up in the lower F=1 ground level.  Transitions to these upper F=1 and F=2 levels enable Depopulation os sodium atomssubstantial loss of atoms in the F=2 ground state to occur. It is important to realize that these transitions effect the total population of sodium atoms, because even atoms moving in a direction hundreds of MHz from the laser line still have a small chance of interaction with the radiation field. Over a collision lifetime (100 microsecond), can get pumped to the lower level. A cycle time of 50 microsecond is all that may be needed for a given velocity class to be depopulated. This is shown in figure 4, which presents the number of atoms in the upper level as a function of Doppler velocity after a collision lifetime of 100 microsecond. In this simulation, the laser has an intensity a 200 w/m^2, which should be typical for next generation Laser Guide Star AO facilities. If there were no depopulation pumping, the curve would follow the conventional Doppler Gaussian curve, with a FWHM of about 1 GHz. In reality, a single frequency laser depopulates about 40% of the atoms in the upper ground state in 100 microseconds (figure 4a). The situation is much worse for multi-line lasers, such as are used at the Gemini Observatory, essentially all the atoms being pumped to the lower level after 100 microseconds (figure 4b).
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Spin Exchange

Collisions of sodium atoms with nitrogen molecules cannot bring the ground states back into thermal equilibrium. If this was the only mechanism for rethermalization we would be dependent on the flow of new atoms into the laser beam either by diffusion or mesospheric wind, both of which have typical time scales of a few milliseconds. This time scale should be compared to depopulation lifetime, which is similar to the collision lifetime (100 microseconds). Under these conditions the photon return would be quickly reduced to a fraction of its initial value.

Fortunately there is another mechanism for repopulating the upper ground level available. The energy separation between the two ground states occurs because there can be two spin alignments between the nucleus and electron - in the F=2 ground state the spins are parallel, in the F=1 state antiparallel. Oxygen atoms or molecules have a similar energy separations. If the sodium atom has a close encounter with an oxygen atom or molecule, the two particles can exchange electron spins and, after a small number of interactions, oxygen can pump the sodium ground back into thermal equilibrium, whatever the initial population of the two ground states. The spin exchange cross-section is of the same order as the momentum transfer collision cross-section, so that we would expect spin exchange to occur about 1 collision with a air molecule in ten (20% of the molecules are oxygen and half of these have spins opposite to that of the sodium atom, a condition needed to change spin). There are now two time scales at work, one defining the time between strong interactions of the sodium atom with the radiation field and the other being the time for the ground states to return to thermal equilibrium by interaction with oxygen.
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Effect of Spin Exchange on Photon Return/Watt

The relative values of the two time constants determine the mean population in the upper ground level. For a CW laser, the first time scale is of order the (Doppler width/Saturated linewidth), which is about 50 collisions, whereas the second timescale is fixed, and is of order 10 to 30 collisions. The sodium atom is therefore able to experience a number of spin exchange interactions before it interacts significantly with the radiation field again, so that the sodium atom population will be approximately in thermal equilibrium. However, a multiline CW laser, which consists of a number of single frequencies spaced about 100 MHz apart will have more frequent, if less intense, interactions with the radiation field and therefore less time to rethermalize between interactions.  For this laser, the appropriate time between interactions will now be of order the (Mode Spacing/Linewidth), or less than 10 hard sphere collisions, while the rethermalization time will still be of order 30 collision lifetimes.  There will therefore be significant depopulation of the upper ground level.

One reason for the significantly lower photon return of the CW mode locked laser compared to the SOR laser is due to this effect, there are just less sodium atoms available in the upper ground state to interact with the laser because of wide-scale depopulation pumping in all velocity groups. This conclusion is not valid for a long pulse laser, such as the one in operation at Palomar. Although this laser has a similar spectral format, the time between pulses (a few millisecond) is more than sufficient to allow essentially complete rethermalization before the next pulse. For this type of pulsed laser we expect a similar photon return/watt to that of the single frequency CW laser and that both will have a significantly higher return than CW multiline lasers. This appears to be born out experimentally.
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Monte Carlo Simulations

Optical Bloch Equations (OBE) are usually used to  calculate photon returns for the sodium atom, suitable modified to take account of magnetic fields, radiation pressure and collisions [ e.g.Milonni and Thode, Appl.Opt 31 785 (1992); L.C. Bradley, JOSAB,9  1931 (1992); Temkin, JOSAB 10, 830 (1993); Gavrielides and Peterson, Opt. Communications 104, 46 (1993); Kruger Z.Phys D31 13 (1994); Holzlohner et al (2009)]. OBEs solve the time averaged photon return over the ensemble of sodium atoms rather than tracking the interactions of individual atoms and have some problems capturing the full physics of the interaction of the sodium atom with its environment, in particular the interaction between radiation pressure and collision lifetime. We have used instead a more intuitive, although more computationally intensive, Monte Carlo approach based on the use of rate equations. These rate equations give similar results to OBEs when the pulse length is long compared to the natural lifetime (16 nsec) or, for short pulses, when the atom is not highly saturated. Rate equations also do not model intralevel coherences within a level, but again these are small for conditions of interest ( see Morris [JOSAA 11, 832 (1994)]). They capture the other physical processes directly and completely. Because the collision lifetime changes by more than a factor of ten across the mesosphere we have to calculate the return/atom as a function of height and integrate this data, suitably weighted with the density of sodium atoms, across the mesosphere.
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Saturation

One of the first results of the simulations was the effect of saturation on the photon return due to the finite lifetime of the sodium atom in the upper level. It is conventional to assume that the photon return/watt is reduced by a factor of two at a photon intensity level of 62.4 watts/m^2. Effect of saturationHowever, this saturation intensity is only valid for the peak return (line center) of the (2,2) to (3,3) transition. At higher intensities linewidth increases by an amount equal to saturation formula   so that the total fraction of atoms in the Doppler velocity population which interact with the radiation field is also increased. This reduces the effective saturation of the photon return/watt when integrated over the total population of sodium atoms. Photon return saturation effects are changed by radiation pressure and magnetic fields. Figure 5 shows that saturation of the line center and that of the photon return. The first has a saturation intensity of 62.4 watts/m^2 as expected, the second, the more important saturation parameter, is approximately 250 watts/m^2 for conditions at the Zenith at the SOR facility.
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Single Frequency CW Laser Photon Returns 
           
Calculated photon returns v SOR data
In figure 6 we show the predicts return for the SOR data in circular and linear polarized light using an earlier code which assumes a single collision lifetime chosen to fit the data. We can use the differences in return between circular and linear polarized light to estimate a collision lifetime. The curvature of the lines depends on the spin exchange time.The best estimate of the collision time was 40 microseconds, with a spin relaxation time of 120 microseconds.
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Increasing the Photon Return/Watt


Chirping
Radiation pressure and chirping
Tom Jeys and co-workers at MIT/Lincoln suggested that radiation pressure could be used to increase the photon return from the mesosphere. How this might happen is illustrated in the panel on the right. The Upper left figure shows the photon return as a function of time for a sodium atom initially tuned to the laser frequency ( Dn= 0 MHz). The photon production rate at first increases, due to optical pumping of the sodium population to the (2,2) to (3,3) state and then decreases as radiation pressure pushes the atom away from the laser frequency.
If the atom is traveling in a slightly different direction, radiation pressure can initially push it towards the laser frequency, but then radiation pressure pushes the atom away. This is shown in the upper right for Dn= 10 MHz.

If the atom is traveling so that there is a -30 MHz frequency difference, the photon production rate is much reduced. There is an increase in photon production rate with time but due to optical pumping and the effect of radiation pressure is less important than the low cycle time of the atom. However if we change the frequency of the laser at the right rate, we find that radiation pressure starts to move the atom so as to reduce the frequency difference. The atom speeds up until it is moving exactly at the velocity needed to match the chirp rate of the laser and the atom produces a strong return until it is collisionally de-excited.

Chirping uses radiation pressure to corral a velocity range of atoms that would not normally interact with the radiation field to be swept up into a red shifted velocity set by the chirp frequency. It does not just reduce the effect of radiation pressure but enables a much greater fraction of the total population of sodium atoms to interact with the laser beam.  Theory predicts that chirping only works successfully if the laser intensity is sufficiently high that the chirp rate is of > 0.5 MHz/microsecond, implying a cycle time of ≤ 100 nsec. Chirping can increases in the photon return/sodium atom in the mesosphere by a factor of 3 larger for suitable laser spectral and pulse formats. The results of chirping experiments with the Chicago sum frequency pulsed laser are shown on the laser guide star page.
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Backpumping

Backingpumping energy levelsThe second technique proposed by MIT/LL was backpumping. In this technique, a second frequency, blue shifted 1.71 GHz from the primary frequency, is used to optically pump atoms that had been depopulated back from the lower ground level to the upper level. Steady state calculations were also carried out in Chicago in the early 1990’s suggesting that, provided the laser intensity was sufficiently high, only a few per cent of the total intensity would be needed to repump the sodium atoms so that most atoms are in the upper F=2 ground state.  Once an atom has been pumped from the F=1 to F=2 ground state by the secondary frequency it has a good chance of being pumped by the primary frequency into the (2,2) to (3,3) where it will remain. Because this process also requires high laser intensities/natural line-width it can only be effectively used by CW narrow line or multi-line pulsed lasers.
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Predicted Chirping and Backpumping Enhancements


Predicted enhancements due to chirping and backpumpingThe Monte Carlo code was used to make predictions on the enhancement in photon return due to the effects of chirping and backpumping for the Sum-Frequency Laser in operation at Palomar. Also shown are experimental returns. The Chirping experiment was done under good seeing conditions and gave an enhancement of 1.8 at an average laser intensity of 40 Watts/m^2. Seeing was much worse for the backpumping experiment. Also note that the scales has slightly shifted in this figure. The blue and green lines actually go to an enhancement factor of 1.0. This figure shows that very significant enhancements, of order a factor 3+, can be obtained by a combination of backpumping and chirping but that high laser intensities are required. The bandwidth of the laser was 1 GHz with a duty cycle of 6%. An average power of 100 w/m^2 is equivalent to a light intensity of 170 watts/m^2/natural linewidth during the laser pulse.
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7:10 pm Feb 23 2010 Edward Kibblewhite