Degenerate four-wave mixing line shapes in sodium vaporunder pulsed excitation
Prem Kumar
Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Received July 30, 1984; accepted November 15,1984
Degenerate four-wave mixing (DFWM) line shapes are investigated in sodium vapor near the D 2 resonance lineusing nearly Fourier-transform-limited pulses. At low pump intensities sub-Doppler resolution is obtained.When the Rabi frequency associated with the pump intensity becomes comparable with the ground-state hyperfinefrequency separation of sodium, each component of the double-peaked DFWM spectrum splits further into twocomponents. An adiabatic-following model explains the near-resonant intensity dependence of the DFWM signal.A qualitative explanation of the observed splitting is also given.
Resonant degenerate four-wave mixing (DFWM) hasbeen shown to be an efficient technique for the gener-ation of phase-conjugate wave fronts.' In backwardconfigurations, in which the two pump beams as well asthe two signal (probe and conjugate) beams counter-propagate to each other, the cw DFWM spectrum isDoppler free and has been used for high-resolutionspectroscopy.2 However, true spectral information isobtained only when the pump-beam intensities aresmall compared with the saturation intensity of theresonant medium. At higher pump intensities, com-plicated line shapes are observed; in particular, the linebroadens and splits into two components. Severalstudies of this behavior have appeared in recent litera-ture. 3
The early experiments on resonant DFWM employedpulsed dye lasers.4 7 The DFWM spectrum in this caseis also complicated and shows a dip to zero at the linecenter. Bloom et al.4 were the first to observe such aline shape. In the pulsed case, fluctuations (pulse-to-pulse-energy, pulse-shape, and center-frequency)and other nonlinear effects such as self-focusing com-plicate the line shape even further. Recently we mea-sured the DFWM line shapes with a pulsed dye-lasersystem that was stabilized in center frequency andbandwidth. Our control of the fluctuations in theDFWM signal was essential for the measurement of thephoton-counting statistics of light generated byDFWM.8 In this Letter we describe the results of astudy of DFWM line shapes with such a source.
The experimental setup is shown in Fig. 1. A stabi-lized cw ring dye laser (submegahertz linewidth) isamplified through a chain of pulsed dye-laser amplifierspumped by the smoothed output of a Nd:YAG laser.9
The output pulses of 4-nsec duration with typically10-20% energy fluctuations and a total linewidth of 200MHz, which is approximately twice the Fourier-trans-form-limited linewidth, are used to perform DFWM.Backward DFWM geometry is employed with orthog-onally polarized pump beams.6 A phase-conjugate(PC) signal whose polarization is orthogonal to theprobe beam (PB) is separated uising a polarization beamsplitter (PBS) and directed onto a photomultiplier(PMT) whose output is sent to a boxcar integrator.The output of the boxcar, which is proportional to the
PC pulse energy, goes to the y axis of a chart recorderwhose x axis is swept with the dye-laser frequency.
DFWM is performed in sodium vapor generated ina heat-pipe oven at 3100C, implying a sodium-vapordensity of 4 X 1014 atoms/cm 3 for the measurementsreported in this Letter. 2.2 Torr of helium is used as thebuffer gas. All the beams arrive in time coincidence atthe sodium cell. The dye-laser frequency is scannedover 10 GHz across the sodium D 2 (2S1/2-2P3/ 2 ) line.The spectra thus obtained are shown in Fig. 2 as afunction of the pump-beam intensities, which are keptnearly the same (to within 10%) for both the pumps.Appropriate neutral-density filters are introduced inthe PB path to avoid saturation of the PMT. Theprobe-pulse energy is kept less than a few percent of thepump-pulse energy in all cases to avoid pump deple-tion.
At the lowest pump intensities used [0.3 kW/cm 2 (allrelative intensity measurements are ±10%) (Ref. 10)],the line shape consists of two peaks, which are 4.5 GHzapart as shown in Fig. 2(a). Also shown is the fluores-cence observed in a direction making a small angle(- 1°) with the PC beam. The widths (FWHM) are
0.74 and 0.83 GHz for the lower- and the higher-fre-(a) 0.3 quency peaks, respectively. Therefore sub-Doppler
kW/cm2 resolution is obtained at these pump intensities evenx 5 though the peaks occur away from the center of the
fI I >*>; <^2 Doppler-broadened line of width 1.8 GHz.As the pump intensities are increased to 0.6 kW/cm 2 ,
A .6 the higher-frequency peak splits into two components( b ) / \ 0.6 separated by 1.3 GHz, as shown in Fig. 2(b). A further
increase of the pump intensities to 1.9 kW/cm2 leads tox2 a splitting in the lower-frequency peaks as well, as
l_, I \ Is | j GHZ I +_1 t shown in Figs. 2(c) and 2(d). In Fig. 2(d) the highest
reflectivity is 0.7%. The reason for such a low reflec-(C ) / \ 0.9 tivity is that a relatively large angle was chosen between
x I the pump and the probe beams. 2 This choice was dic-tated by the low-background requirement in our
I I ~e1 ,\ IJ 1 ; A, I DFWM quantum-noise measurements reported ear-lier.8 With smaller angle and higher pump intensities
1.9 /' 7 t(at least an order of magnitude higher than those re-(d) \ ;X~t I \ ported herein), we have observed reflectivities as large
as 700. Under these conditions the conjugate-pulseduration is significantly shorter than the probe-pulseduration. 11
At still higher pump intensities of 3-10 kW/cm2 , onlytwo peaks remain, as shown in Figs. 2(e) and 2(f). A
Xl further increase of the pump intensities leads to abroadening and weakening of the lower-frequency peak,
k- y 1.2 -A \ J t 9which is consistent with the observations of earlier;sv _, g ,GHZ g W t l L ! workers. A slight broadening and weakening is already
observable in Fig. 2(g). Jabr et al.7 did not observe thelower-frequency peak when they made the same choice
(e) 3.0 of pump- and probe-beam polarizations that we employ.Their pump intensities were at least an order of mag-
X I x t l \ nitude larger than ours. At high pump intensities,self-defocusing of the pump beams on the lower-fre-quency side of the resonance causes a reduction of theactual pump intensities in the mixing medium, whichresults in a lowering of the PC signal.
The splitting of the DFWM line shape [Figs. 2(b)-2(d)] was not observed in earlier experiments. 4 -7 This,we believe, is because we have used stable-center-fre-quency nearly Fourier-transform-limited pulses for the
(f) 5 9 above measurements.9 Furthermore, at low intensities2.5 \/ \sub-Doppler lines whose widths are limited by power/ \ . 2.5 / \ broadening are observed.
The nonlinearity responsible for DFWM in sodiumvapor is due to the resonantly enhanced electronic Kerreffect. Grischkowky et al. 5 used the adiabatic-following(AF) model for a two-level atom under pulsed excitationto derive an expression for the DFWM reflectivity as afunction of the pump intensities and detuning. When
(g) 9.4 the AF conditions are satisfied,12 the third-order non-/,, .5 At \ ~~~~~linear susceptibility is given by
X(3) Np4A 2
E X Is 1 1; |PI X h3(v-vo)3(1 +A2/A,2)3/20 5 10 where A is the electric-dipole moment of the two-level
DYE LASER FREQUENCY (GHz) atom, A's are the pump-field amplitudes (assumed
Fig. 2. DFWM line shapes at various pump intensities. equal for both the pumps), v - vo is the detuning of theVertical scale is arbitrary and is linearly proportional to the DFWM frequency from the atomic line-center fre-DFWM signal; plotter scale factors are labeled. The fluo- quency, N is the total number of atoms per unit volume,rescence spectra are superimposed in (a), (b), and (g). Pump and A,2 is the normalized saturation intensity given byintensities are labeled in kilowatts per square centimeter in A, hl (v - Po)l /jg. The DFWM reflectivity is given byeach plot. Start frequency in (f) and (g) is slightly shifted. R = tan 2 KL K2L2 , where K = 2rvX(3 )/2cn, n is the
Fig. 3. Dependence of the DFWM signal on the pump in-tensities.
linear refractive index of the medium, L is the interac-tion length, and the approximation is valid under con-ditions of weak reflectivity.
As the pump intensities are increased, the DFWMsignal saturates because of the saturation of X(3)- Thisis verified for a detuning of 2.3 GHz for which the AFconditions are satisfied, as shown in Fig. 3. DFWMreflectivities were measured from Fig. 2 at a detuningmarked by the arrows. Dots are the experimental data,and the solid line is a fit to Eq. (1) for IS = 7.8 kW/cm 2 ,which agrees well with I, = coc[h(v - Vo)/gt]2 /2 = 2.1kW/cm2. Thus a good agreement is obtained with theAF model in its region of validity. The detailed lineshapes cannot be predicted using the AF model, whichis not valid sufficiently close to resonance. A qualita-tive explanation can, however, be given.13
The splitting of the DFWM line shape as shown inFigs. 2(b)-2(d) occurs when the Rabi frequency corre-sponding to the pump intensities is close to theground-state hyperfine splitting of the sodium atom,which is 1.77 GHz. Moreover, the splitting of thelower-frequency peak occurs at approximately twice thepump intensities than that of the higher-frequencypeak. This suggests that the multiple-level nature ofthe sodium atom is playing a role. This is not surprisingbecause the spectrum of the pulses used in this experi-ment is much narrower than the ground-state hyperfinesplitting of sodium.
For our 200-MHz, 4-nsec pulses, sodium can be wellmodeled as a three-level atom in the vicinity of the D2line. The ground state consists of two hyperfine splitsublevels F = 1 and F = 2. Both the lower-frequencyF = 2 to {F'I and the higher-frequency F = 1 to JFltransitions enhance the nonlinearity responsible forDFWM. At low pump intensities [Fig. 2(a)], the twopeaks would then be the lower-frequency peak of thefirst transition and the higher-frequency peak of thesecond. This assumes that the nonlinearity is of adispersive nature and that, at frequencies between thetwo transitions, the dispersion from one cancels that ofthe other because of its opposite sign. At higher pumpintensities [Figs. 2(b)-2(d)], the dispersions are powerbroadened, with the lower-frequency broadening 5/3times that of the higher frequency. When this broad-ening becomes comparable with the hyperfine fre-
quency separation, the cancellation is incomplete, andthe two peaks that are due to each line appear. Thisalso qualitatively explains why the splitting of thehigher-frequency peak occurs at roughly half of thepump intensities of the lower-frequency peak. At stillhigher pump intensities [Figs. 2(e)-2(g)], the hyperfinesplitting is inconsequential, and a two-peak spectrumis obtained as expected for a two-level system.
The above explanation can be considered only qual-itative. Woerdman and Schuurmans3 used similarreasoning to explain the ow DFWM spectrum in theirexperiments at much lower pump intensities (50 W/cm2), but they ignored the effect of spatial averaging.Spatial averaging plays a consequential role in deter-mining DFWM line shapes, as shown by Agarwal etal.3
In conclusion, we have observed a novel structure inpulsed DFWM line shapes and have given a qualitativeexplanation for it on the basis of dispersive adiabaticnonlinearity.
The author wishes to acknowledge discussions withR. S. Bondurant. This research was supported in partby the U.S. Office of Naval Research.
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