Multiple Optical Pulse Propagation in Molecular Vapor and Population Transfer

B. K. Clark1, J. M. Standard2, T. S. Boostrom1

 

1 Department of Physics, Illinois State University, Normal, IL 61790-4560

2 Department of Chemistry, Illinois State University, Normal, IL 61790-4160

 

Introduction

 

The topic of pulsed optical pumping of three-level systems has been of interest in chemical physics for the past few decades. Examples of optical pumping are diverse and include but are not limited to optically pumped lasers (OPLs) [1 - 3], electromagnetically induced transparency [4], efficient population transfer [5 - 7], and the generation and propagation of adiabatons [8]. A natural consequence of optically pumped lasers is the movement of population from an initial state to a final state, but the typical work on OPLs does not address the efficiency of the population transfer. Several related works on coherent pulse propagation through vapor does examine the early growth of a weak probe pulse as a strong pump pulse propagates through a vapor, but the effect on final population distributions is not discussed [9].

The efficient transfer of populations of atoms or molecules approximated as three-level systems using stimulated Raman scattering involving adiabatic passage (STIRAP) [5 – 7] is among the many applications of optical pumping. STIRAP has been shown to be more efficient or easier to implement for the purpose of population transfer than the application of pi-pulses, stimulated emission pumping (SEP), or Franck-Condon pumping (FCP) [6]. In its simplest form, STIRAP involves two laser emissions, with a pump laser emission tuned to the |1> to |2> transition and a probe laser emission tuned to the |2> to |3> transition (fig. 1), for example. The laser emissions are introduced into the vapor in the counterintuitive sequence, in which the vapor interacts with the probe laser emission first and the pump laser emission last. This technique has been shown to be very efficient in transferring population between states when the pump and probe laser emission Rabi frequencies are comparable and the emissions are not significantly affected by the population.

 

While not all OPLs are based on stimulated Raman scattering (SRS), the diatomic alkali pulsed OPLs typically are [10]. There are primarily two processes involved in the generation of an OPL in a diatomic alkali system. Employing K2 as an example, the first mechanism is

 

K2X1g+(v", J") + pump K2B1u(v', J'), (1a)

 

K2B1u(v', J') K2X1g+(v'", J'") + ase , (1b)

 

and

 

K2B1u(v', J') + ase K2X1g+(v'", J'") + 2ase , (1c)

 

where pump is the energy of a pump laser photon, and ase is the energy of an emitted photon. Amplified spontaneous emission grows and is observed as a coherent emission with frequency ase. The second mechanism is SRS and represented as

 

K2X1g+(v", J") + pump + ase K2X1g+(v'", J'") + 2ase . (2)

 

 

Even in the absence of the probe laser emission, the forward emission from the vapor is far more intense than the backward emission, which is generally attributed to the domination of the SRS process over the multi-step process of eq. (1).

OPL emissions are generally produced with the pump and probe laser emissions propagating through many Beer’s lengths of vapor, while the STIRAP geometry has been employed for only very thin sections of vapor. One can ask whether a STIRAP type process is involved in OPLs with SRS. Additionally, one can ask what is the population transfer efficiency between |1> and |3> for an OPL that does include SRS. We have become interested in these questions. In this paper we discuss these questions in reference to laser emission interactions with K2 molecular vapor. This system is complicated by the presence of other levels, for example level |4>, that compete with level |3>. The problem is addressed via experimental measurements and computer simulations of the Maxwell-Bloch equations describing the vapor as a four level system.

 

The Experiment

 

Potassium vapor contained in a heat-pipe oven interacts with two pulsed dye lasers. The three-armed oven contains approximately 35 grams of potassium. The longest section of the heat-pipe oven is 81 cm in length and the side arm is 41 cm in length. The vapor is maintained in a region between 6 and 12 cm in length, depending on the amount of heating. For this experiment the oven is operated in the heat-pipe mode at 8 mbar and 684 K. Argon is used as a buffer gas to protect the heat-pipe windows mounted at the Brewster angle for 633 nm.

 

Potassium atoms account for most of the particles in the vapor, and the molecular population is distributed over many ro-vibrational levels. Consequently, while the number density in the vapor is 2 1025 / m3 at 8 mbar in the heat-pipe mode, the number density in the K2X1g+ (v"= 0, J" = 47) ro-vibrational level is N = 3.4 1018 / m3. This level serves as |1> for our work. The number density decreases by 90 % as the vibrational quantum number increases to 12. The fractional population in the levels with which we are concerned is 0.862 in |1>, 0.0 in |2>, 0.075 in |3>, and 0.063 in |4>.

Fig. 2 is a schematic of the experimental apparatus showing the pump and probe dye lasers and the Nd:YAG laser. The pump and probe dye laser emissions are joined by a beam splitter and directed into the heat-pipe oven. Forward emissions from the oven are directed through a 0.3 m Spex monochromator and detected with an S-20 response photomultiplier tube. The entrance to the monochromator is about 0.5 m from the exit window of the heat-pipe oven, and only one 0.05 m focal length imaging lens is in the emission path just at the entrance of the monochromator. Consequently, no fluorescence is detected. The outputs of the photomultiplier tube and hollow cathode lamps are analyzed with gated integrators and recorded with an Intel 80486 based computer. The pump and probe dye lasers were constructed in our laboratory and are pumped with a frequency doubled Spectra Physics Nd:YAG laser. Both dye lasers have grazing-incidence configurations employing 5 cm gratings with 240000 grooves / cm. The oscillator cavities are 11 cm in length so they have free spectral ranges of 1.4 GHz. The total linewidths of the lasers are a result of both the maximum resolution from the grating and the free spectral range. The measured laser linewidths are less than 3 GHz throughout their tuning ranges. Consequently, the laser emissions usually consist of 2 longitudinal modes. The pump laser has a peak emission intensity of 1.6 Mwatt / cm2 using DCM at 633 nm. The probe laser has a peak emission power of 1.7 Kwatt / cm2 using LDS690 at 683 nm. The laser wavelengths are calibrated with hollow cathode lamps. The temporal full width at half maximum (FWHM) pulse length is 3 ns, and each pulse is a train of pulses with a coherence time of approximately 0.4 ns.

 

A coherent emission commonly termed optically pumped laser emission is produced if the pump laser is tuned to induce transitions from |1> to |2>. In K2, coherent emissions usually occur on more than one downward transition from |2>. Two emissions often dominate the observed spectrum. For example, when the pump laser is tuned to the K2B1u X1g+ (v’ = 6, v" = 0) Q(47) transition, emission terminating on the X1g+ (v"= 13 and 14) levels are produced. The addition of the probe laser emission tuned to one of the downward transitions can increase the emission at that frequency and decrease the emission at the competing transition frequency.

 

Computer Model

 

The interaction of the laser emissions and the vapor is based on the Maxwell-Bloch equations, except our equations represent a four level system. We begin with the Heisenberg representation. Further, we make use of the slowly varying envelope approximation and the rotating wave approximation to develop the equations. Our derivation is that of Eberly and colleagues [11, 12] but applied to the four level picture of fig. 1 with zero detunings.

 

The applied electric field is modeled as

 

E(z, t) = Ea(z,t)exp{ia(t-z/c)} + Eb(z,t)exp{ib(t-z/c)} + Ec(z,t)exp{ic(t-z/c)} + c.c. , (3)

 

where Ea, Eb, and Ec are the field amplitudes at the |1> to |2>, |3> to |2>, and |4> to |2> transition frequencies, respectively. Here, a, b, and c are the corresponding resonant transition frequencies. The pump laser emission is resonant with the |1> to |2> transition. The probe laser can be switched between resonance with the |3> to |2> transition and the |4> to |2> transition. For any coherent emission to be generated in the vapor, the associated electric field must begin with some small non-zero value. We have also employed the transformation = t – z/c, so that the coordinate system moves as the speed of light, c. In the K2 system, the transition frequencies are sufficiently different that the laser emissions are coupled to the vapor only through their resonant transitions.

 

We begin with Gaussian shaped pulses for all of the Ei(z,t) (i = a, b, or c) with (FWHM) of 0.4 ns. The peaks of Eb and Ec precede Ea by 0.6 ns. Any coherent emissions will contribute to the electric field at the respective frequencies. The FWHM used in the computer simulation does not agree with the pulse length for the experimental system. The experimental pulses can be viewed as a series of short pulses of varying temporal widths, heights, and phases contained within a 3 ns FWHM envelope. They are best described by an exponentially correlated colored noise model [13, 14]. Such a model complicates the comparison of computer simulation to experiment because it requires far more computer time to complete a single simulation and many computer runs to compare the averaged results of the simulation with the averaged results of the experiment. To simplify the problem we choose simulated pulse lengths that have FWHMs that are equal to the coherence length of the experimental laser beams. The delay between simulated pump and probe laser beams is chosen to yield the most population in |3> when the laser emissions exit the vapor. The actual experimental delay can be any time, t, where 0 < t < 3 ns. This wide range reflects beam jitter and the more complicated pulse structure of the experimental laser emissions.

 

The equations of motion for the density matrix elements for this four level system are given as

 

d11(z,)/d = i/2(a*12 – a21) + f1222 , (4a)

 

d22(z,)/d = - i/2(a*12 –a21 + a*32 – b23 + c*42 – c24) -( + col)22 , (4b)

 

d33(z,)/d = i/2(b*32 – b23) + f3222 , (4c)

 

d44(z,)/d = i/2(c*42 – c24) + f4222 , (4d)

 

d21(z,)/d = -i/2(a*(11 - 22) + b*31 + c*41) - ( + col) 21 , (4e)

 

d23(z,)/d = -i/2(b*(33 - 22) + a*13 + c*43) - ( + col) 23 , (4f)

 

d24(z,)/d = -i/2(b*(44 - 22) + a*14 + c*34) - ( + col) 24 , (4g)

 

d31(z,)/d = -i/2(b21 - c*32) , (4h)

 

d41(z,)/d = -i/2(c21 - b*42) , (4i)

 

and

 

d43(z,)/d = -i/2(c23 - b*42) , (4j)

 

 

where i (i = a, b, or c) = 2diEi/is the Rabi frequency of the electric field. The dipole moment for the respective transitions is di. The radiative transition rate between the B1u and the X1g+ molecular state is = 8 107/s, and the collision rate, col = 3, for all four levels at 8 mbar [15]. The Franck-Condon factor combined with the Honl-London factor for the specific transitions is fi2 (i = 1, 3, or 4) [16]. The initial population distribution matches the experimental conditions.

The input electric fields, Ei, of eq. 3 are modified as they propagate through the vapor according to

 

da/dz = 2ia12 , (5a)

 

db/dz = 2ib32 , (5b)

 

and

 

dc/dz = 2ic42 , (5c)

 

where i (i = a, b, or c) are given by

 

i = N di2i2/ (0c). (6)

 

N = 3.4 1018/ m3 is the number density of molecules in state |1>.

 

The equations are solved on a grid of 10000 time and 10000 space points. The time integration is completed using a fourth order Runge-Kutta algorithm, and the space integration is completed with an Euler algorithm. Increasing or decreasing the number of time and space points does not alter our results.

 

Results and Discussion

The coherent emission intensity as a function of peak probe laser input intensity is shown in fig. 3. For both the computer simulation and the experiment, the pump laser is

tuned to the K2B1u X1g+ (v’ = 6, v" = 0) Q(47) transition (|1> to |2> in fig. 1) and the probe laser is tuned to the K2B1u X1g+ (v’ = 6, v" = 13) Q(47) transition (|2> to |3> in fig. 1). The experimental data, obtained at 8 mbar and 684K, shows an increase in coherent emission by a factor of 12 when the peak input intensity of the probe laser in increased from zero to a maximum of 1700 Watt/cm2 or a Rabi frequency of 4.2 GHz. On the same graph, the computer simulation results show an increase in coherent emission intensity by a factor of 6. The experimental and simulated coherent emission intensities are set equal to each other at 1.7 10-5 Watt / cm2 (10-8 on the graph). The average energy of the pump laser emission was determined with a Scientech calorimeter and power meter. Its peak intensity was in turn determined to be 1.6 Mwatt / cm2 from its temporal length. This is a Rabi frequency of 470 GHz for the |1> to |2> transition. The pump laser emission was then attenuated with neutral density filters until its peak intensity could be compared to the peak intensity of the probe laser emission with the photomultiplier tube and monochromator. The absolute peak input intensity of the probe laser emission measurable to ± 30 %. The peak intensity of the probe emission is 1000 times weaker than the peak intensity of the pump emission, and too weak for our laser power meter to provide a direct measurement.

 

In the absence of the probe laser emission, coherent emission on the K2B1u X1g+ (v’ = 6, v" = 13) Q(47) transition (|2> to |3> in fig. 1) and the K2B1u X1g+ (v’ = 6, v" = 14) Q(47) transition (|2> to |4> in fig. 1) are nearly equal. The coherent emission on the |2> to |4> transition is suppressed as the probe laser emission intensity in increased at the |2> to |3> transition frequency. This is displayed in fig. 4. The

simulation results in this figure were obtained from the same simulations as the results for fig. 3. The experimental results for both figures were recorded under the same conditions. The most significant difference between computer simulation and experiment is that the coherent emission at the |2> to |4> frequency begins to be suppressed at lower probe laser pulse emission intensities. This discrepancy is in part because we are at the limit of our sensitivity when the intensities are reduced to10-1 on fig. 4. We also note that the slope of coherent emission intensity versus probe laser emission intensity is similar over several orders of magnitude.

 

 

Associated with the coherent emission at the |2> to |3> transition frequency, b, is the transfer of population from |2> to |3>. Fig. 5a is a display of the simulated population density in |3> as a function of time at several locations within the vapor. The initial probe emission has a peak intensity of 1700 Watt / cm2 and a Rabi frequency of 4.2 GHz, and the initial pump emission has a peak intensity of 1.6 Mwatt / cm2 and a Rabi frequency of 470 GHz. As the laser and coherent emissions propagate through the vapor they become more efficient at transferring population to |3>. Fig. 5b is a display of the final population density as a function of location within the vapor, for the same input parameters. The simulation shows that nearly 80% of the population ends up in |3>.

 

 

The coherent emission produced at the |2> to |3> transition frequency, b, is distinctively different from the probe laser emission. Fig. 6 is a display of simulated coherent emission pulse shapes (in Rabi frequency) at different locations within the vapor for the same initial conditions of figs. 5a and 5b. Eberly and colleagues [12, 17] have previously shown that in a three level system the coherent emission can grow from both fluorescence and Raman scattering. Their results show that emission growing from Raman scattering can dominate the total emission. In this case the coherent emission moves to earlier times with respect to the input pump laser beam. Our case appears to be a limiting situation where the coherent emission ultimately forms at the leading edge of the input pump laser beam. By 1 cm into the vapor, it is clear where the bulk of the energy will be in the coherent emission.

 

 

Fig. 7 is a display of the Rabi frequencies of the pump laser emission and coherent emission after propagating through 10 cm of vapor for the same initial conditions as figs. 5a and 5b. The amplitude of the coherent emission exceeds that of the pump emission between 1.25 and 1.33 ns. One condition for near unity transfer of population [7] from |1> to |3> is

 

0 1 >> 2 (1 + 10)1/2, (7)

 

where

 

0 = [1/2 (a2 + b2)]1/2. (8)

The value of 1 is the half width of the laser pulses for i (t) = i (0)/e, (i = a or b). The detuning of the lasers from resonance is 0. Our mean experimental detuning is 1.5 GHz, and the simulations are completed for zero detuning. The pulse forms displayed in fig. 7 satisfy the criteria of eq. 7. A further criteria for successfully moving population from |1> to |3> is that the pulse envelope evolves from primarily probe (or coherent) emission tuned to the |3> to |2> transition to the pump emission tuned to the |1> to |2> transition. This can be seen for our simulated pulses in the inset of fig. 7. The effect of meeting these criteria is final population in |3> as shown in figs. 5a and 5b, where the population approaches 80%. The further the emissions propagate through the vapor the more these conditions are met. The conditions can be met sooner with pump and probe emissions that are already of ideal temporal width and relative timing. Alternatively, if the interaction strength between the vapor and the laser emissions is increased by increasing the initial number density in |1> or choosing a molecule with a larger dipole moment, the conditions can be reached earlier in the vapor. In a molecular vapor like K2, the problem with simply increasing the vapor density is that the population of other ro-vibrational levels also increases. Molecules in these levels absorb both pump and probe laser emissions, which results in weaker coherent emission and less population transfer. A further consideration is the re-population of |1> and depopulation of |3> via thermal collisions that other K2 molecules undergo and enter the four level system. The effect of these collisions is not included in the simulation, and should not be significant.

 

Conclusion

We have examined the effect of two laser emissions propagating through a molecular vapor. Computer simulations show that nearly 80% of the population in the system of four levels can be moved to a selected final level via SRS. The further the emissions propagate through the vapor, the more closely the conditions for STIRAP are met. The simulations also show that the coherent emission produced via the interaction between laser emissions and the vapor is narrowed and at the leading edge of the pump laser emission.

 

Acknowledgements

This work was supported in part by research grants from Illinois State University.

 

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