Anal. Chem. 1988, 60, 2599-2608
2599
Nonlinear Atomic Spectroscopy of Flames Bruce K. Winker and John C. Wright* Department of Chemistry, University of Wisconsin, Madison, Wisconsin 53706
Two tunable lasers are dlrected Into an atomic absorptlon flame, and nonllnear mlxlng generates an output whose efflclency Is enhanced by resonance with Na or Ca atomlc states. The methods are a potentlal alternatlve to conventlonal atomk spectroscoples. The output frequency Is shlfted from the exciting lasers so there Is no Interference from laser scatter, as there Is In degenerate four-wave mlxlng or phase con)ugatlon methods. Four dlfferent resonance schemes are shown Including resonance wlth the dlfferent dynamlcally Stark split levels of one excited state, resonance wlth two dlfferent electronlc states, a doubly degenerate four-wavemlxlng method lnvolvlng resonance wlth two dlfferent electronlc states, and two-photon-resonance methods lnvolvlng two excited states. The detectlon llmlts are the same as those for fully degenerate four-wave-mlxlng methods and are llmlted by the dynamlc Stark effects from the hlgh peak powers of the pulsed lasers. Theoretlcal models that Include saturation effects and dynamlc Stark effects are obtained and shown to agree excellently wlth the experimental results.
There has been recent interest in exploring new methods for nonlinear laser spectroscopy to determine the potential capabilities for chemical measurement. Most of the interest has been concentrated on applications in molecular spectroscopy (1-3, but there is also interest in expanding the nonlinear methods to atomic spectroscopy (8). Hesselink and Pender introduced the idea when they used phase conjugation to generate a nonlinear signal from Na in a flame (8-11). Two laser beams of the same frequency are directed oppositely through a flame while a third beam of the same frequency is crossed with the others. A new beam is formed that travels oppositely to the third beam and is the conjugate wave. The efficiency of this four-wave mixing is resonantly enhanced if the lasers are all resonant with an atomic state. Each laser is resonant and the overall efficiency of the process is high because the enhancements from each resonance are multiplicative. This approach will be called fuIly degenerate four-wave-mixing spectroscopy. Ramsey and Whitten measured the concentration dependence of the phase conjugation method and found detection limits of 4 ppb for Na (9). Tong et al. presented similar work and obtained detection limits of 5 ppb (10). In both cases, the detection limits were determined by interference from scattering of the exciting lasers. Although the phase conjugate signal is traveling in a different direction from the excitation lasers, it is at the same wavelength, so wavelength discrimination is not possible. The resonances appropriate for fully degenerate four-wave mixing are shown in Figure 1. All the lasers have the same frequency, but two are directed in opposite directions. Phase matching conditions then result in a signal that propagates oppositely to the third laser and can be spatially separated. The signal is enhanced by many orders of magnitude when it becomes resonant with electronic states. Ramsey and Whitten estimated that the three resonance enhancements shown in the figure would produce detection limits in flames of 10 fg/mL if the Rayleigh scatter from flame gases limited the detection (9). Although most nonlinear methods have 0003-2700/88/0360-2599$01.50/0
Table I. Summary of Symbols
-
Ei = energy of state i wij = (Ei - E j ) / f i = angular frequency of i j transition rij = relaxation rate of i j transition WL, w ~ wp , = angular frequency of laser L, S, or P QL, Qs, Qp = Rabi frequency for transition resonant with lasers L, S, or P (e.g. QL = pgIEL/h; QL is absorption, QL* is emission) EL, Es,E p = electric field strength of laser beam at WL, w ~ or, up pij = dipole moment of i j transition p i t = density matrix element that is proportional to induced polarization of i j transition (i # j ) or to the state population (i = j ) , oscillating at frequency A SijA = wij - wA - irij= resonance denominator for i j transition induced by lasers at frequency A
-
-
-
-
detection limits that are determined by nonresonant background, the discrimination between the excitation beams and the signal limits the phase conjugation methods. If one were able to use wavelength discrimination, the laser intensities could be raised to the point where the nonresonant background became the limitation. However, the usable intensity is limited by saturation effects and the dynamic Stark splittings that are induced by the high electric fields of the lasers, and one must properly consider such effects in any study. Fully degenerate four-wave mixing is one case of the more general four-wave-mixing process where the three possible resonances can involve one or more electronic states. In this paper, we explore nonlinear methods that result in signals at different wavelengths from the excitation lasers. Figure 1 shows the resonances that are used for each of the methods. The four diagrams farthest to the left have excited energy levels that represent the Na atom 2P1/zand 2P3/2levels that are involved in the famous D1and Dztransitions, respectively. If the laser of angular frequency WL (labeled L in the figure) is intense, it can induce a dynamic Stark splitting as shown in the second diagram of Figure 1. By use of a laser with a different frequency, ws (labeled S in the figure), the lasers can be made resonant with the dynamically split Stark levels to achieve full resonance. This method will be called dynamic Stark-enhanced nonlinear mixing. A summary of the symbols used in this paper is given in Table I. One can change the frequency of the second laser further to achieve resonance with a different electronic state as shown in the third diagram of Figure 1. This approach will be called nondegenerate nonlinear mixing. The output signal is shifted to a very different frequency from the lasers, but there are only two resonance enhancements for this approach. The signal levels will be lower, but wavelength discrimination will be much better. The fourth diagram in Figure 1shows how the resonances can be changed to once again obtain full resonance enhancement, now involving two different electronic states. This approach is called doubly degenerate nonlinear mixing because two lasers have the same frequency and the signal has the same frequency as the third laser. We show how polarization methods can be combined with spatial discrimination to give larger discriminations against the lasers. Finally, the last diagram in Figure 1 shows how three levels can be used to obtain full resonant enhancement and produce 0 1988 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 60, NO. 23, DECEMBER 1, 1988
-ll-nT '
FULLY DEGENERATE MIXING
DYNAMIC STARK ENHANCED
NONDEGENERAE DOUBLY MIXING DEGENERATE MIXING
TWO PHOTON ENHANCED MIXING
Figure 1. Diagram of the different nonlinear mixing processes discussed in the paper. The letters on the arrows label the frequency of the laser represented by the arrow.
an output frequency that is very different from that of the lasers. The resonances are the same as in two-photon-absorption methods, and this approach will be called two-photon-enhanced nonlinear mixing. All of these methods result in an induced, coherent polarization oscillating at the output frequency, and this polarization launches a new coherent beam that is detected. The intensity of the new beam is a sharp function of whether the driving lasers match resonances in the atomic system. We will explore the theoretical foundations of these methods to understand the factors that control the output signal intensities for all the approaches, including the fully degenerate nonlinear mixing, and we will present an experimental observation of the methods along with a comparison with the theoretical predictions. THEORETICAL SECTION Traditional energy level diagrams such as those shown in Figure 1 are not adequate for communicating the important flows of coherence that control the characteristics of nonlinear methods because they do not include the effects of saturation. That approximation is not appropriate for the experiments reported here because of the dominance of saturation effects. When saturation effects are important, it is not possible to define the number of photons involved in the nonlinear mixing, and terms like four-wave mixing become inappropriate. We will therefore use the term nonlinear mixing. These considerations can be visualized by using a variation of the diagrams introduced by Mukamel for molecular nonlinear spectroscopy (12). The mechanisms for generation of the nonlinear signal are described by density matrix elements, pv = c,cJ* where c, is the amplitude for the ith state. The off-diagonal values p,, are proportional to the dipole moment of the i-j transition while the diagonal matrix elements are proportional to state populations. Interactions with photons oscillating at frequency uAwill induce transitions to different states (state i may evolve to a state 1 or state j may evolve to state k ) and consequent evolution of a plJ to either p l k or plJ (Le. either state can change but not both). All of the work in this paper involves detection of an output (uL- us + up) generated by lasers a t wL, us, and up. The output occurs because each of the lasers induces a polarization of the atomic electron clouds that oscillates at the laser frequency. If the laser electric fields are in the nonlinear regime where the polarization is not proportional to the field, distortions occur in the oscillating polarizations at all of the combinations of the driving frequencies. The size of the output at any given frequency wA is proportional to the induced dipole moment oscillating at that frequency or pvA. We therefore seek diagrams that will show the pathways by which an initial ground-state population, plldC(the superscript dc indicates the oscillation frequency is zero and the population is not oscillating) can evolve by interaction with three lasers to the coherent state producing the output, pvA. The evolution of the output coherence is rigorously described by density matrix theory using the Liouville equation in the manner shown by Bloembergen and Shen (13)and used
22
Figure 2. Density matrices describing fully degenerate nonlinear mixing or phase conjugation are connected by laser interactions labeled by letters indicating the laser frequency responsible for the interaction. The superscripts on the p indicate the induced oscillation frequency while the subscripts indicate the two states.
Figure 3. Density matrix diagram for dynamic Stark-enhanced nonlinear mixing showing relationships induced by two lasers. The L laser is considered strong and the S laser is considered weak.
Figure 4. Density matrix diagram for nondegenerate nonlinear mixing showing relationships between three states and the three lasers that induce interactions between the states. The L and P lasers are considered strong and the S laser is considered weak.
pP-L
=
p;;
- - i" L Lm
7 L
'11
J/":
L
p:1
P2f
Figure 5. Density matrix diagram for doubly degenerate nonlinear mixing showing relationshipsbetween three states and the three lasers that induce interactions between the states. The L and S lasers are considered strong and the P laser is considered weak.
extensively by others (14-20). The diagrams in Figures 2-5 represent the steady-state equations for the first four resonance schemes shown in Figure 1,respectively. Consider the example of fully degenerate nonlinear mixing shown in Figure
ANALYTICAL CHEMISTRY, VOL. 60, NO. 23, DECEMBER 1, 1988
2 (this example corresponds to the diagram at farthest left in Figure 1with q,= ws = wp). One starta from a ground-state population, plldc. Interaction with a field of frequency WL induces a dipole moment proportional to p Z l L corresponding to the transition from state 1to 2. The superscript L indicates that the dipole moment is oscillating at the frequency q,. The strength of the interaction is described by a Rabi frequency for the transition, QL, which is the product of the transition dipole moment and the electric field of the laser, fiZIEL/A.Interaction with a second field (L* in Figure 2) generates either a population plldc that is a correction to the original groundstate population or p22dCthat is an excited-state population. Interaction with a third field g q r e t h e dipole moment proportional to pZlL(a higher order correction to the original pZlL),and it is the oscillating polarization corresponding to this dipole moment that produces the output signal. (Equivalently, we could use the same arguments to produce the complex conjugate of pzlL at p12-L.) When one removes the degeneracy of the fields, the diagram in Figure 2 expands out to make the diagrams in Figures 3-5. The diagrams are defied so that the x , y, and z axes represent interactions with the fields wL, w p , and ws, respectively. Absorptive interactions move one to the right, out of the page, or upward for the three axes while emissive interactions move in the opposite directions. We have made the approximation that two fields are strong and the last field is weak. The field that is weak depends upon which method is used. The diagrams actually extend infinitely far in the two directions of the strong fields, and we are approximating the nonlinear mixing with the contributions shown in the diagrams. An exact solution with the infinite number of equations would require using the supermatrices of the Floquet theory (21,22). These diagrams have collapsed the separate interactions that make pll and p22 (see Figure 2 to see how the different interactions represented by arrows interrelate the p , ) into single arrows since the populations always occur as differences (pll - pz2) in any equation. As an example, we will follow the evolution of coherence from the starting ground-state population plldC to p3lA along the bottom pathway in Figure 4. Interaction with fieid wL causes the initial-state population plldCto evolve to pzlL. At this point, the first interaction has induced an oscillating dipole moment a t frequency W L that is represented by e l w L t . It is possible to return to pl? (or ppdc)by a second interaction with the field wL, and this process causes dynamic Stark splittings. More importantly for this paper, interaction can also occur with the ws field to cause further evolution to pllLs. This interaction represents an emission where the ws field gains energy and the dipole oscillation is represented by ei(wL-ws)t. That difference is indicated by the - si& in the superscript of pLs and by the complex conjugate in the Rabi frequency representing the strength of the interaction, O*s. At this point, the two interactions have induced a ground-state population that is oscillating at the difference frequency of the two lasers, q,- US. Finally, interaction with thet'third laser causes evolution to p3lLs+', which induces the output signal. Note that there are a total of four pathways that can be taken from the initial population pll to the final coherence p3lL-'+' to give the same signal. This diagram omits any contributions from p l j that cannot be resonant, and it neglects any higher order pathways involving interaction with more than three photons. The diagrams are built by determining what resonances are established between different states and using the following rules that define the connections between density matrix elements. Any absorption interaction (like QL) can cause evolution of state i in plJ to a higher state or state j in plJ to a lower state. The most resonant states will be the ones
2601
favored in the evolution. Thus we assume that any one state is not coupled directly to all others, only the ones that are near resonance. Any emissive interaction (like Q*L) can cause evolution of state i in pll to a lower state and or state j to a higher state. The importance of any pathway depends upon the size of each pll along it. The pll is determined by resonances with the lasers producing it. A plJA-B will have a resonance denominator of the form 61)A-B = ( w l J - W A + WB - i r , ) where w,) is the difference in energy between states i and j and rlJis the damping factor that controls the line width of the transition. Clearly each pll will become large when the lasers are set to resonance and the real part of the resonance denominator goes to zero. One of the strengths of the diagrams in Figures 2-5 is that the resonances for each pathway can be determined directly from the diagram since each element in the diagram has a resonance denominator determined completely by its superscript and subscript. The Liouville equations that describe the time evolution of the density matrix elements can be written in the steady state by direct inspection of the diagrams, using the following rules. The left side of an equation is the product of the density matrix of interest and its resonance denominator. The right side of the equation is the s u m of all of the contributions that feed that matrix element in the diagram. Any contribution is the product of half the interaction strength (or Rabi frequency) and the matrix element from which the contribution originates. The' sign of the contribution is positive if state i in pll evolves and negative if state j evolves. As an example, consider the development of p3zps in Figure 4. There are three interactions that produce ~ 3 2 ' ~ :interaction of strength Q*s with ~ 3 1 ' (an emission that causes state 1 to change to state 2 , since ws is resonant between states 1 and 21, interaction of strength Qp with p12-s (an absorption that with causes state 1to change to state 31, and interaction p3lLs+'. By application of the rules just stated, the Liouville equation for p32-'+' would be
If one writes all of the equations indicated in the diagrams and solves them simultaneously, one obtains the solutions shown below. For dynamic Stark enhanced nonlinear mixing
where A212L-S =
.=
and
- 46,,2L-S6
($
+
L-SA
BL-s 11
I 12
L-S QL
11
$i)T lQLI2
2602
ANALYTICAL CHEMISTRY, VOL. 60, NO. 23, DECEMBER 1, 1988
In the limit where wL = us,this equation reduces to the traditional equation for fully degenerate nonlinear mixing or phase conjugation (13-19). For nondegenerate nonlinear mixing
Finally for doubly degenerate nonlinear mixing 02,L-S+P =
where where
A~~L-S+P =
611dC62pdC + 611dcAL PlldC - mdC = alldc6
22
dc
- 6 22dc (AL + AJ
Pll(O'
These equations appear daunting but they can be understood without too much difficulty. Each equation contains the characteristic resonant denominators, Gib,that determine the positions and widths of lines in the spectrum and AijA that determine the saturation effects. The BA factors are simply the ratio of population differences to the lower-state population, both of which are oscillating at frequency A. The first step to gaining insight is to eliminate those terms that have a higher order dependence on the laser intensities and to realize that all the AijA reduce to unity at low intensities. In that approximation, there are different terms that correspond to the different pathways that the coherence evolves from the initial ground-state population, plldc, to the final coherence that produces the output. For example, consider eq 3 and Figure 4, which describe nondegenerate nonlinear mixing.
ANALYTICAL CHEMISTRY, VOL. 60,
NO. 23, DECEMBER 1, 1988 2603
Table 11. Resonances for Each Nonlinear Mixing Method fully degenerate nonlinear mixing dynamic Stark-enhanced nonlinear mixing nondegenerate nonlinear mixing doubly degenerate nonlinear mixing -r MONOCHROMATOR
ENTRANCE SLIT 7.6 cm FL
= US
WL WL
=
US US
r
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I
10 cm FL DIAPHRAGM
I
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WL
,
I
AT OA HEIGHT! ~
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WL
= w21
= wZ1 ws = w21 = 0 ~ 2 1 WQ = ~ 2 % = ~ 2 1 ws = ~ 2
2% - us = w21
WL
WL
I
1 1
wp
=~
- US+ UP = ~
3 1
~p
=
- US
3 1
(e)
22
3 1 U P - WQ = ~ 3 2 WL ~ 3 1 COP- COS = ~ 3 2 WL
A
+ UP = ~
1
x10
JEXCIMERI I 1 308 nrn
1 -
'4
sox
(b)
I
14 rnRod
I\
u3 DL
BEAMSPLITTERJ
~
W1
OPTlCAL AXIS
u4W Figure 6. Experimental diagram of the equipment used for the experiments. The focusing of the three beams for the doubly degenerate nonlinear mixing experiment is shown on the bottom.
- --
- -- -- -- -
There are four pathways to arrive a t ~ 3 1 ~ " :plldc pzlL PllLs P31Ls+p, &ldC PlZ* PllM P31M+p, PlldC -b PIZs P3ZP* P31Ls+p, and p1ldC P3lP P3ZP-' P31LS+P. Correspondingly, there are four terms in the low intensity limit of eq 3 that are the product of the resonance denominators for the three p i p along the pathway. That product determines the importance of each pathway and determines the positions and line widths of the resonances that are seen. One must modify the predictions a t higher intensities by considering the AijAvalues, which change the magnitude of the different resonances and split the resonances by dynamic Stark splittings. For the simple dynamic Stark effect, the splitting will be given by the generalized Rabi frequency, Q' = (a2 A2)l12 where A is the detuning of a laser from resonance (wij - w b r ) . These equations allow one to compare how the different nonlinear methods are related and to optimize their performance. The positions of resonances are summarized in Table II for each of the nonlinear mixing methods. The largest signals will result when three of the resonances that occur in the same term of the equation can all be reached simultaneously. More detailed characteristics of each equation will be examined in the context of the experimental results.
+
EXPERIMENTAL SECTION A block diagram of the instrumentation used for this experiment is shown in Figure 6. It consisted of two dye lasers excited by an excimer laser. The dye lasers had output line widths of 0.006-0.01 nm and 5-ns pulse widths. One beam was split, and the three beams were directed into a slot burner. The beam waists were 1.5 mm. A BOXCARS geometry was used with crossing angles between the beams of 14 mrad as shown in the figure (23). The polarization of the beams was identical with each other for all of the experiments but the doubly degenerate nonlinear mixing. Other polarization combinations could be chosen, and the mixing efficiency would depend on the product of the transition moments for each polarization (the transition moments are part of the Rabi frequencies). A 1-2-mm-diameter aperture was used to isolate the output signal. The output beam was focused onto a 100-pm pinhole to reject any incoherent light and then imaged onto the entrance slit of a 1-m double monochromator.
16985 16990 16995 17000 17005 us (crn-l)
Figure 7. Dynamic Stark-enhanced nonlinear mixing using lasers that are focused into the flame. The scale expansion and the Rabi frequency for each spectra are shown on the right and left, respectively. The vertical arrows indicate the expected position of the resonance peaks, assuming a shift given by the generalized Rabi frequency.
The slot burner used an acetylene-air mixture with fuel flow rates of 0.84-1.3 L/min and oxidant flows of 5.7-8.9 L/min. Atomic densities in the flame were determined by absorption spectroscopy, using the method of equivalent widths (24). The doubly degenerate four-wave-mixing experiments used a similar arrangement except that polarizer5 were placed in the different beams to provide additional discrimination for the output beam against the lasers. A Glan laser polarizer was placed in one beam at ws while the other two beams at WL were polarized perpendicularly to each other by placing a double Fresnel rhomb in one of the two. The geometry of the beams for this case is shown in the figure. RESULTS Dynamic Stark-Enhanced Nonlinear Mixing. Experiments with dynamic Stark-enhanced nonlinear mixing were performed with beams both focused and unfocused into the flame. The focused beams caused resonances in positions that were not expected and line widths that were broader and more asymmetric than those typical of atomic transitions. An example of the spectra is shown in Figure 7. The spectra were obtained for WL pulse energies between 0.08 and 250 pJ, corresponding to power densities of ca.0.1-350 MW/cm2. The Rabi frequency (in reciprocal centimeters) that corresponds is shown to the left of each scan. The pulse energy of us was 0.12 pJ or 160 kW/cm2. The left and right resonances correspond to resonance with the Na Dz and D1 transitions, respectively. Only at the lowest energies (spectra a and b) do the peaks have the expected positions, widths, and symmetry. At the higher energies, the peaks shift as would be expected for higher Rabi, frequencies, but the shifts are not as large as one would expect from the measured intensities of the lasers. The difference is attributed to the spreading in intensities spatially along the focal dimensions and temporally in the laser profile and to the decreased efficiency of nonlinear mixing at high Rabi frequencies. For these reasons, unfocused laser beams were used for the mixing experiments.
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ANALYTICAL CHEMISTRY, VOL. 60, NO. 23, DECEMBER 1, 1988
-.J Ab-
!
I
X5A
:a 1
LII’
I
1
I
I
1
X2.5
I
16955 16960 16965 16970 16975 w (cm-1)
I
fl
~~
- -
-
-_-
Figure g. Diagram of the different combinations of laser frequencies used for nondegenerate nonlinear mixing experiments. The wavy arrow indicates the output frequency and the letters label the laser frequencies.
G=C
I
L~
I
16965 16970 16975
XI
w (crn-1)
Figure 8. Dynamic Stark-enhanced nonlinear mixing using unfocused lasers. The scale expansion is shown on the left of each spectrum. (a) Spectra for constant value of 52, and different detuning, A. The laser intensity was 1.12 MW/cm2,corresponding to a Rabi frequency of OL = 1.46 cm-‘. The arrow indicates the position of wL for each spectrum relative to the D, resonance. (b) Spectra for constant value of A and different values of QL. The detuning was 2.7 1 cm-’ for each spectrum. The 52, values are indicated on the right of each spectrum. The same experiments were performed with unfocused beams with pulse energies in the range 10-500 and 2-3 pJ for the lasers a t WL and us,respectively. Figure 8a shows scans of ws across the D2 resonance for different values of A and constant Rabi frequency QL = 1.46 cm-’ (or intensity of 1.12 MW/cm2) while Figure 8b shows the same scan for different QL values and constant A = 2.71 cm-’. The US laser intensity was 100 kW/cm2 for the experiments. The arrow indicates the value of wL, and the dotted line indicates the position of the D2 resonance. The size of the detuning is shown for each spectrum. In each case, the peak shifts with a dependence that is in agreement with estimates of the generalized Rabi frequency, as will be seen in the discussion section. The large spikes in some of the spectra are caused by particulate scatter in the flame. The detection limit for Na with this method was estimated by measuring the signal-to-noise (S/N) ratio for the nonlinear signal at a known concentration and extrapolating to a S I N = 2. In order to discriminate against laser scatter, the detuning was set to A = 4.67 cm-’ while the laser intensities were fixed a t 1.14 MW/cm2 and 5 kW/cm2 for the wL and ws lasers, respectively. At the acetylene-air flow rates used for this experiment, a M Na solution produces a Na atomic density of 8.3 X lo9 ~ m - ~ . A quadratic signal dependence on the Na concentration was observed from 5 x lo+ to 1 X lo4 M. At the lowest concentration used, the signal consisted of ca. 33 photoelectrons/pulse, which was averaged over 600 pulses. The standard deviation over the 600 pulses was equivalent to h0.5 photoelectrons/pulse (1.5%), about 2 times the shot noise limit. If one extrapolates S I N to a value of 2, assuming a quadratic concentration dependence, the detection limit would be 1.1 X lo-’ M (2.5 ng/mL) or an atomic density of 9 X lo7 ~m-~ The . background fluctuations were mainly caused by weak flame emission, dark counts from the photomultiplier, and a small contribution from laser scatter by flame particulates. Flicker noise was not present. A generalized Rabi frequency of 1.5 cm-l was the minimum required to discriminate effectively against laser scatter. No contribution was ever measured from nonresonant background. Nondegenerate Nonlinear Mixing. Figure 9 shows the four different resonance schemes studied for nondegenerate double electronic four-wave mixing. They differ as to which electronic transitions are resonant with the lasers and the output signal. Some are parametric and some are nonparametric. Figure 10 shows a series of spectra obtained by
m
A -1.01
A= -3.08
p
xA -0.52
-0.26
X12.5
X1.25
16930
16935
16940
16945
Y (cm-9 Figure 10. Nondegenerate nonlinear mixing spectra for a series of detunings, A, of wL from the D, transition and a constant Rabi frequency of 1.27 cm-’. The scale expansions are shown on the left of each spectrum. The x axis gives the frequency of the laser at ws. changing ws to scan across the sodium 2P312 level for a series of detunings A = wZ1 - wL where w21 is the energy of the 2P1!2 level relative to the ground state. When wL - ws + up is resonant with the 2P3/2level, there are two resonances that are nearly symmetrical about the position of the 2P3/2level. The asymmetry that does exist is attributed to small errors and drifts associated with tuning A to exact resonance. As the detuning is changed, the peaks shift and their intensity changes but the position of the most intense peak remains approximately at the position where wL - ws + wp is resonant with the ‘P3/2 level. The symmetrical set of peaks at A = 0 is the result of a dynamic Stark splitting, as can be seen from Figure 11. In this figure, the UL laser position is set for A = 0 while its intensity is changed. The Rabi frequency calculated from the measured intensity of the w~ laser is shown on the right of each spectrum. At low intensities, the peak is not split and is located a t the position of the 2P812resonance. As the intensity is increased, there is a dynamic Stark splitting that results in a 1.34-cm-’ separation a t the highest intensity. The
ANALYTICAL CHEMISTRY, VOL. 60, NO. 23, DECEMBER 1, 1988
2605
1.31
A
x2
x4
1.28
A i\
0*74
X1.25
0.38
-
16935 16940 ws (cm-')
Flgure 11. Nondegenerate nonlinear mixing spectra for a series of different Rabi frequencies, R,, with the wL laser set for resonance with the D, level. The scale expansions are shown on the left.
16972 16974 16976
3P 3P
--
3s Flgure 12. Diagram of the resonances and polarizations of the lasers used for doubly degenerate nonllnear mixing.
splitting expected from the measured laser intensity is 1.28 cm-', only 5% lower than the measured splitting. It is assumed the differences between observed splittings and the splittings predicted from measured laser intensities reflect changes in the spatial and temporal properties of the lasers between experiments. Similar results were obtained when the same experiments were performed with the other wave-mixing schemes shown in Figure 9. The signal levels and the backgrounds observed were similar to each other and to the dynamic Stark-enhanced nonlinear mixing described previously. Detection limits were also very similar and equivalent to the dynamic Stark-enhanced nonlinear mixing experiments. D o u b l y Degenerate D o u b l y Resonant Nonlinear
Mixing. The resonances for doubly degenerate, doubly re-
sonant nonlinear mixing are shown in Figure 12. The bolder arrows indicate the stronger lasers. This approach can be viewed as a variation of the methods sketched in Figure 9 when the output frequency matches one of the laser frequencies. One now has the same difficulty in discriminating against the excitation laser as in the fully degenerate four-wave-mixing methods. The lasers are polarized as shown in the figure in
wp (cm-') Figure 13. Doubly degenerate nonlinear mixing spectra for different Rabi frequencies with oLset for resonance with the D, transition and upscanned across the D, resonance.
order to enhance the discrimination. The orientation of the beams for the three-dimensional phase matching is shown at the bottom of Figure 6. The signal levels for this approach were particularly strong because of the fully resonant nature of the experiment. The example spectra shown in Figure 13 were obtained by monitoring the output signal while the op laser was scanned across the Dztransition at 16 973.4 cm-l. The wL laser was set to be resonant with the D1 transition. The pulse energy of the laser at wp was ca.20 nJ, corresponding to an intensity of 35 W/cm2, while the pulse energy of the other laser varied from 0.075 to 509 pJ, corresponding to intensities of 162 W/cm2 to 1.35 MW/cm2. The scale expansion factors and the Rabi frequencies calculated from these intensities are shown on the left and right of each spectrum, respectively. No splittings are seen from the dynamic Stark effect as was observed in the previous experiments. The maximum signal intensity was obtained for the RL = 0.045 and 0.10 cm-I spectra. If the same experiment is repeated with the value of wp set for the Dz resonance and wL scanned across the D1resonance, the spectra shown in Figure 14 are obtained. Now one sees strong dynamic Stark effects. The point in the scan where wL is resonant becomes a deeper valley as the laser intensity is raised. This behavior is the reason why the maximum signal intensity occurred at wL = 0.045-0.10 cm-' in the previous figure. The asymmetry in the line shape of the top two spectra is the result of contributions to the resonance enhancement from the Dz level. Similar experiments were performed where w~ was scanned acrosa the D, transition for different values of detuning A from the D, transition. The intensity of the laser at W L was maintained at 1.10 MW/cm2 or a Rabi frequency of 1.25 cm-'.
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ANALYTICAL CHEMISTRY, VOL. 60, NO. 23, DECEMBER 1, 1988
J
x1
4
fl=1.24
FWM-0.027 nrn
FWM-0.021
nrn
( FYMM-0.021 * "rn
0
1
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i
(C)
1
1
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585.7
585.8
h= (nm) Flgure 15. Left side of the diagram shows the energy of the lasers used for the two-photon-enhanced nonlinear mixing in Ca. The right-hand spectra show experimental results for scans wp across the 4s6s 'S resonance. The laser intensities and Ca2+ Ca 4s4p 'P and lo-' M concentrations were 3.9 MW/cm2 (w,), 720 W/cm2 (up), for (a); 0.42 MW/cm2 (w,), 720 W/cm2 (w2),and loJ M for (b); and 3.9 MW/cm2 (w,), 720 W/cm2 (up), and M for (c).
-
fi L = O O O 1 A-4
EW-1
/
CM -l
1x144
=To
1, f
A - 2 CM
CM -1
i
= 2 0 ;M
lrZ85
-1
= 4 C CM
1x236)
1x223
1 1
1x223
I , ,
16950
16955
16960
WL(CrIl--') Doubly degenerate nonlinear mixing spectra for different Rabi frequencies with upset for resonance with the D, transition and oLscanned across the D, resonance.
A =O
CM
x434
1
x373
I1
I
,
I
7
-'
x4341
1
x24
1x4941
L
1
15965
Figure 14.
16975
16965
16975
16965
EhERGY (CM
:6975
16965
16975
-:>
Simulatiohs of dynamic Stark-enhanced nonlinear mixing for different Rabi frequencies and different detunings. The scale expansions are given in each spectrum as a logarithm of the scale expansion relative to the maximum signal that could be obtained from a fully degenerate nonlinear mixing experiment. Figure 16.
The resulting spectra did not show any appreciable influence of changing A. The detection limit was determined by measuring the S/N ratio for a known concentration and extrapolating to S I N = and two Ca2+ concentrations. The signal dependence on 2, assuming a quadratic concentration dependence. The laser concentration was not linear at the intensities used for the intensity of the w L laser was 0.92 MW/cm2, giving a Rabi experiment, probably because of thermally induced ionization frequency of 1.14 cm-' and the detuning A = 7.73 cm-'. A 10" from excited Ca states or multiphoton-induced ionization. M solution of Na gave an atomic density of 8.3 X lo9 ~ m - ~ . Detection limits at this point are estimated at 5 X lo8 Ca A measurement of the signal level showed a quadratic deatoms/cm3 in the flame. These experiments demonstrate the pendence on the concentration from l X lo4 to 5 X lo4 M. feasibility of using two-photon-enhanced nonlinear mixing, The 1 X lo4 M solution produced a signal of ca. 140 photobut more work is required to understand the method's electrons/pulse while the fluctuations in the signal and characteristics. background corresponded to ca. 0.8 photoelectrons/pulse near DISCUSSION the S I N = 2 detection limit. Assuming a quadratic dependence of the FWM signal on concentration predicts an estiThe results can be modeled with the equations given premated detection limit of 8 X lo-*M (1.8 ng/mL) or an atomic viously. In order to simulate the spectra, one must have values density of 6 X 10" ~ m - ~The . background signal was due to for the population relaxation rates, rlland FB, and dephasing electronic noise associated with the photomultiplier dark noise rates, rzl,r31, and J?32, of Na in an air-acetylene flame. In and radio frequency pickup from the pulsed laser. The conorder to properly account for the feeding of the ground state tribution from laser scatter, nonresonant background, flame by population decay from the upper states, we set Fll = rZ2. emission, and laser-induced fluorescence was less than 0.2 The population decay rate was measured by Russo and Hieftje photoelectron/pulse. Ultimately, the background level will to have a value of 0.007 37 cm-' (25) while the dephasing rate be determined by the nonresonant background or the exof the Na D1level was estimated to be 0.08 cm-' in an airtinction ratio of the polarizers. acetylene flame (8). We will assume these same values for Two-Photon-Enhanced Nonlinear Mixing. Preliminary all the simulations. experiments were also performed that examined the feasibility For dynamic Stark-enhanced nonlinear mixing, three of the of using two-photon-resonance methods in Ca. The twofour resonances listed in Table I1 will be seen in the experiphoton resonances that were used are sketched in Figure 15. ment where wL = w21 - A and ws is scanned. A t low laser Since the q and ws lasers had the same wavelength, one could intensities, they occur at US = wZ1- 2A, wZ1 - A, and wzl. Figure not achieve full resonance enhancement, but with three lasers, 16 shows a series of simulated spectra for different values of full resonance enhancement would be possible. Figure 15a-c detuning, A, and Rabi frequencies, QL. The scale expansion shows the results of scanning the wp laser over the 4s6p 'P factors indicated in the figure (and subsequent figures) are 4s6s 'S resonance of Ca with the WL laser set to the tworeferenced to the maximum signal level that could theoretically photon 4s2 's 496s 's resonance for two laser intensities be obtained from a fully degenerate nonlinear mixing ex-
-
-
ANALYTICAL CHEMISTRY, VOL. 60, NO. 23, DECEMBER 1, 1988
1, 1 I 1 1 1i 1 1 'yo,1 [ 1 I ,1 -flL=146 C M - l
A
, 4:x
= 7 2 5 CM-l '1310
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I.
nL'149
CM-l
I
274 CM-l
,
x293
1
x216
I
,
8
I
x4,7E14
A =-LO1
CM-1
X6.9E13
057 C M - ~
A =-OS2 CM-l
X1.9E13
I
177 CM-'
x169
16960
x5520
,
16970
A =-0.26 CM-l 034 C M - l
16960 16970
ENERGY (CM
-'
)
Figure 17. Simulations of dynamic Stark-enhanced nonlinear mixing for different Rabi frequencles and different detunings that correspond to the experlmental values in Figure 8. The scale expansions are given in each spectrum relatlve to the maximum signal that could be obtained from a fully degenerate nonlinear mixing experiment.
periment using the same modeling procedure as was used for the methods described. The maximum signal level for fully degenerate nonlinear mixing is limited by saturation at high intensities. In this figure only, the logarithm of the scale expansion factor is given because of the wide dynamic range of signal levels for the figure. In the figure, all three expected resonances are seen, but the outer lines are weak at low laser intensities while the middle line is weak at high laser intensities. The resonances shift as the laser intensity is raised because of dynamic Stark effects. The resonance at ws = wZ1 - 2A is dominant at high intensities and is shifted the largest amount from the laser frequencies. It is the resonance that was observed in Figure 8. The intensity depends upon A and f 2in~ such a way that the optimum A increases for larger QL. The simulations show that the optimum signal level does not increase for larger laser intensities once saturation is reached, but better discrimination against the lasers can be achieved because the optimum occurs at larger values of A. Figure 17 shows a similar set of Simulations where the Rabi frequencies and detunings are chosen to match the experimental results shown in Figure 8. The agreement is excellent. The line shifh predicted by the simulations for different values of A and Q agree excellently with the measurements shown in Figure 8. The relative intensities are also predicted surprisingly well. The relative intensities of the three spectra with QL =: 1.46 cm-' and A = 7.25,4.43, and 2.74 cm-' agree well with experiment. One would not expect quantitative agreement for relative intensities because of the difficulty in experimentally controlling the laser intensities adequately for nonlinear experiments where the signal depends on the third power of the laser intensities. For nondegenerate nonlinear mixing, there will be four resonances seen in scans of ws when q,= wZ1 - A. At low laser intensities, they occur at ws = up1,wzl - A, wzl - wE - A, and wZ1 - a 3 2 - 2A corresponding to the ?il2-', 611L-s,63Zp-', and 631Ls+p resonances, respectively. In the region scanned in Figure 10, only the latter two resonances would be seen. The simulations for this method require an additional parameter, r32. There are no measurements of r32 for Na in an airacetylene flame. The relative intensities of the lines are sensitive to r32, and we found that r 3 2 = 0.46 cm-' produces the best agreement with experiment. This value is almost 6 times larger than the dephasing rates rzland I'31 and suggests that pure dephasing of two closely spaced excited states is substantially larger than that between more distant states. This result agrees with other experiments that measure the different amounts of dephasing between excited states (26-29). Figure 18 shows a series of simulated spectra with WL = wp for different values of detuning, A, and Rabi frequencies, QL.
A = 0.06 CM-l
A = 0,37 CM-l
I
I
x1099
L R 1L X 5 5E 1 4
x1410
I
7.9Ell
XI780
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1
A = 0.55 CM-l
I
A = 1.07 CM-l
A = 2.97 CM-l
I
x639
1
,,
,
110
1
A =-3.08 CM-l
1
4 4 3 CM'l
0.20 CM-'
=,lf
CM-l
A.271
2607
I
x1360
h I
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9.8E13
9.1E14
I
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XlOOO
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x202
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16930 16940
16930 16940
16930 16940
ENERGY ( C M - l ) Figure 18. Simulations of nondegenerate nonlinear mixing for different Rabi frequencles and different detunings. The scale expansions are given in each spectrum relative to the maximum signal that could be obtained from a fully degenerate nonlinear mixing experiment. The middle column of spectra corresponds to the parameter values used for Figure 10.
The 63lLs+' resonance is dominant except for small A where the 63lLs+' and'-'236 resonances become indistinguishable. The optimum signal level occurs at increasing values of A as the Rabi frequency is raised. Saturation again limits the largest value that can be achieved. The overall intensity predicted is an order of magnitude smaller than that for fully degenerate nonlinear mixing. The center column of spectra was simulated with a Rabi frequency equal to that used in the experiment of Figure 10. The agreement is quite good. The line positions agree within experimental error with Figure 10, and the relative intensities of the lines within any one spectrum agree excellently with experiment. The relative intensities of spectra with different values of A agree qualitatively. The weakness of the 63Zp-' resonance in both experiment and theory can be understood by examining the interference that occurs between alternative pathways in Figure 4. In the limit of low laser intensities, the resonances in eq 3 can be written as
The terms in parentheses will change as one approaches the allLs or 63zPs resonances in such a way that the terms in each set of parentheses cancel with each other even as the resonant denominator approaches zero. The net effect is to decrease the 611L-sor 632p-s resonance intensity. If one writes the definition of the G i j A and combines the terms, one obtains
2608
ANALYTICAL CHEMISTRY, VOL. 60, NO. 23, DECEMBER 1 1988
where the are the pure dephasing rates. The combination of r values in the equation above will vanish or become small in the limit of no pure dephasing so the 611L-s and 632p-s resonances are cancelled. The exact cancellation does not occur if the laser intensities cause saturation effects that change the relative sizes of the terms. This cancellation effect is the same behavior that Bloembergen observed in the pressure-induced extra resonances (PIER4) where collisionally induced pure dephasing caused resonances to appear (30). For doubly degenerate nonlinear mixing with wL = ws, one can achieve full resonance enhancement. If wL = wZ1 - A and wp is scanned, Table I1 predicts there are two resonances a t up = w31 and wp = w31 - A. If wp = 031 - A and wL is scanned, one would expect to see two resonances at wL = w21 and wL = q,- A. The resonances that depend on A will have the same cancellation effects observed for the nondegenerate nonlinear mixing. If A = 0, the mixing is triply resonant and the resonances seen in the scan overlap. Figure 19 shows simulations of the spectra where w L is scanned with w p on resonance and spectra where wp is scanned with w L on resonance, respectively. In each case, w L = ws and the spectra are shown for different Rabi frequencies, QL. The intensities are larger than those observed for dynamic Stark-enhanced nonlinear mixing or nondegenerate nonlinear mixing and are similar to those obtained from fully degenerate nonlinear mixing. The up scans are not sensitive to the saturation effects, but the w L scans show very strong dynamic Stark splittings. The optimum detuning of wL again increases with larger Rabi frequencies. The Rabi frequencies used in the simulations match those measured experimentally for Figure 14 if one assumes that the experimental laser intensity is the sum of the wL and os laser intensities used in the simulation. This assumption is justified because the w L and ws photons are supplied by the single laser. The predicted line positions are in agreement with the experimental observations seen in Figure 14, and the relative intensities are in qualitative agreement. The asymmetrical line shape of Figure 14 is believed to result from the additional resonance enhancement of the higher D2 resonance that was not considered in the theoretical model and consequently is not predicted by the simulation. CONCLUSIONS The different nonlinear mixing methods described in this study are excellently described by the theoretical model that was developed for each case. The agreement justifies using the model for optimizing the methods and for comparison with other techniques. The signal levels for the dynamic Starkenhanced and nondegenerate nonlinear mixing methods are perhaps two orders of magnitude lower than that for the fully degenerate nonlinear mixing method, but the output signals fall at a different wavelength than the excitation lasers. The doubly degenerate nonlinear mixing is fully resonant and has the same efficiency as fully degenerate nonlinear mixing, but it has the same output wavelength as one laser. Polarization discrimination provided enough rejection that there was no observable contribution from laser scatter. The reported results were limited by the small absolute number of photons
,036
1 I
x6.1
x11,4 ,0079 2 16970 16978 u p ENERGY (cm-l)
x3.0
I
x11,4
2 16952 16960
u~ ENERGY
(cm-1)
Figure 19. Simulations of doubly degenerate nonlinear mixing for different Rabi frequencies. The scale expansions are glven In each spectrum relative to the maximum signal that could be obtained from a fully degenerate nonlinear mixing experiment. The left-hand spectra correspond to scans of wp while the right-hand spectra correspond to scans of wL.
that were used in the experiments. The detectability was limited by factors that were not dependent on the laser intensity, so one could improve the performance by using lasers with a higher repetition rate and lower pulse energy. The methods might be particularly advantageous for atomic measurements with dense level spacings (e.g. rare earths or transition metals) where one could take advantage of the high selectivity possible with three resonance denominators that can be tuned to different transitions. LITERATURE CITED Lee, S. H.; Steehler, J. K.; Nguyen, D. C.; Wright, J. C. Appl. Spectrosc. 1988, 3 9 , 243. Nguyen, D. C.; Wright, J. C. Appl. Spectrosc. 1985, 3 9 , 230. Nguyen, D. C.; Wright, J. C. Chem. Phys. Len. 1985, 177, 224. Steehler, J. K.; Wright, J. C. J . Chem. Phys. 1985, 83. 3188. Steehler, J. K.; Wrlght, J. C. J . Chem. Phys. 1985, 83, 3200. Riebe, M. T.; Wright, J. C. Chem. Phys. Len. 1987, 738. 565. Riebe, M. T.; Wright, J. C. J . Chem. Phys. 1988, 88, 2981. Pender, J.; Hessellnk, L. Opt. Len. 1985, 70, 264. Ramsey, J. M.; Whitten, W. B. Anal. Chem. 1987, 5 9 , 167. Tong, W. G.; Andrews, J. M.; Wu, 2. Anal. Chem. 1987, 59, 896. Tong, W. G.; Chen, D. A. Appl. Spectrosc. 1987, 4 7 , 586. Mukamel, S.; Loring, R. F. J . Opt. SOC.Am. B : Opt. Phys. 1988, 3 , 595. Bloembergen, N.; Shen, Y. R. Phys. Rev. 1964, 733, A37. Abrams, R. L.; Lind, R. C. Opt. Len. 1978, 2 , 94. Hatter, D. J.; Boyd, R. W. IEEE J . Quantum Electron. 1980, QE-16, 1126. Boyd, R . W.; Raymer, M. G.; Narum, P.; Harter, D. J. Phys. Rev. A 1981, 2 4 , 411. Ducloy, M.; de Oliveira. A. M.; Bloch, D. Phys. Rev. 1985, 3 2 , 1614. Le Boiteux, S . ; Bloch, D.; Ducloy, M. J . Phys. (Les Ulls, Fr.) 1988, 47, 31. Grynberg, G.; Plnard, M.; Verkerk, P. J . Phys. (Les Ulis, Fr.) 1986, 4 7 , 617. Levine. A. M.; Chencinski, N.; Schrelber, W. M.; Weiszmann, A. N.; Prior, Y. Phys. Rev. 1987, 35, 2550. Ho, T. S.;Wang, K.; Chu, S. I . Phys. Rev. 1988, 3 3 , 1798. Wang, K.; Chu, S. I. J . Chem. Phys. 1987, 86, 3225. Eckbreth, A. C. Appl. Phys. Lett. 1978, 3 2 , 421. Corney, A. Atomic and Laser Spectroscopy; Oxford University Press: Oxford, U.K., 1977. Russo, R. E.; Hieftje, G. M. Appl. Spectrosc. 1982. 3 6 , 92. Kibble, B. P.; Copley, G.; Krause, L. Phys. Rev. 1987, 759, 11. Hulpke, E.; Paul, E.; Paul. W. Z . Phys. 1964, 177, 257. Demtrader. W. Z . Phys. 1962, 186, 42. Stopavsky, M.; Krause, L. Can. J . Phys. 1968, 46, 2127. Prior, Y.; Bogdan, A. R.: Dagenais, M.; Bloembergen, N. Phys. Rev. Len. 1981, 43, 111.
RECEIVED for review June 30,1988. Accepted September 1, 1988. This work was supported by the National Science Foundation under Grant CHE8515692.
