Yu et al. Vol. 16, No. 8 /August 1999 /J. Opt. Soc. Am. B 1261

Theory of forward degenerate four-wave mixingin two-level saturable absorbers

Dai-Hyuk Yu, Jai-Hyung Lee, and Joon-Sung Chang

Department of Physics, Seoul National University, 151-742, Korea

Jae Won Hahn

Optical High Temperature Measurement Group, Division of Quantum Metrology,Korea Research Institute of Standards and Science, Taejon 305-606, Korea

Received May 5, 1998; revised manuscript received September 8, 1998

A theoretical description for a forward degenerate four-wave mixing process in homogeneously broadened two-level absorbers with arbitrary intensities of all four beams is presented. Assuming no pump absorption anddepletion, we obtain an analytical solution of forward degenerate four-wave mixing signal intensity in the limitof the strong pumps and weak probe and signal. In the case of low absorption or large detuning, this solutionis shown to become equivalent to that of the phase-conjugate degenerate four-wave mixing derived by Abramsand Lind [Opt. Lett. 2, 94 (1978)]. For arbitrary beam intensities and absorption parameters, the signal in-tensity is numerically calculated by solution of the coupled equations of complex wave amplitudes. Compar-ing the results of the numerical calculation with those of the analytical solution, we discuss the validity ofusing the analytical solution in practical experiments and show the saturation behavior that is due to thestrong probe beam. The line shapes under the various ratios among the input beams and absorption param-eters are also obtained and discussed. © 1999 Optical Society of America [S0740-3224(99)01601-X]

OCIS codes: 190.0190, 190.4380, 300.2570, 120.1740, 280.2470.

1. INTRODUCTIONIn recent years, degenerate four-wave mixing (DFWM)has emerged as an important technique in nonlinear op-tics and has found many applications in various fieldssuch as atomic, molecular, and solid-state spectroscopy,1

generation of phase-conjugate waves,2 optical imageprocessing,3 and combustion diagnostics.4,5 For DFWMexperiments, two kinds of phase-matching geometries—phase-conjugate geometry and forward geometry—arepopular, and these are shown in Fig. 1. In the phase-conjugate geometry the pump beams are counterpropa-gating. The signal beam is generated at the same wave-length as those of the input beams, and it propagatesbackward with a conjugate phase along the same path ofthe probe beam. In the forward geometry all the laserbeams are incident from the same side and are arrangedin a three-dimensional box-type geometry. The gener-ated signal beam propagates in the direction that fulfillsthe phase-matching condition. DFWM with phase-conjugate geometry (PC-DFWM) has been used in variousfields, as noted above. Also, DFWM with forward geom-etry (FDFWM) has found increasing usage, for example,in the gas phase detection.6–12

In the theoretical developments, most of the researchhas been concentrated on understanding PC-DFWM. Ananalytical model of PC-DFWM in stationary two-levelsaturable absorbers was presented by Abrams and Lind.1

In their model they assumed that the probe and signalbeams were much weaker than the pump beams, and theabsorption and depletion of the pump beams were ne-glected. Extension of the Abrams–Lind model was made

0740-3224/99/081261-08$15.00 ©

to handle arbitrary-intensity input beams for the case inwhich the polarizations of the pump beams wereorthogonal.13 Most recently, a numerical calculation forPC-DFWM with arbitrary-intensity input beams thathave the same polarization was reported by Ai andKnize.14 These studies on PC-DFWM focused on the ho-mogeneously broadened absorbers that include energytransfer among the beams while the beams are propagat-ing through the absorbers. Efforts to include inhomoge-neous broadening, especially Doppler broadening in thegas phase, have also been made in the limit of weakbeams by perturbation expansion.1 The PC-DFWMtheory was extended to include saturation effects that aredue to one strong pump beam through the assumptionthat the other pump beam and probe beam were weakenough to be treated with perturbation expansion.15 Theauthors solved density-matrix equations for a specific ve-locity group of absorbers in the steady state. The DFWMsignal field amplitude, including the Doppler broadeningeffect, was obtained after integration of all velocitygroups, which has a Maxwell–Boltzmann velocity distri-bution. Recently, Lucht et al.16 numerically calculatedthe signal intensities and line shapes by solving the time-dependent density-matrix equations that interacted withthree arbitrary intensity laser fields. The electric-fieldamplitude of the signal was determined by numericalFourier decomposition. In this way they calculated thenon-steady-state signal when the input beams had Gauss-ian pulse shapes.

For the FDFWM, however, few theories have beendeveloped.6,8,11 Klein et al. reported a theory that in-

1999 Optical Society of America

1262 J. Opt. Soc. Am. B/Vol. 16, No. 8 /August 1999 Yu et al.

cluded saturation effects with Doppler broadening, as-suming that only one of the pump beams had arbitrary in-tensity, but they mainly investigated the nearly DFWMcase.6 Attal-Tretout et al. dealt with a L-type three-levelmodel with orthogonal polarization of pump beams.11

Two pump beams were assumed to interact with two dif-ferent sublevels and one common upper level, owing totheir orthogonal polarizations. A density-matrix equa-tion was solved, and Doppler broadening was includedunder the condition that two pump beams had arbitraryintensities and the probe beam was nonsaturating. How-ever, all the FDFWM models mentioned above calculatedthe signal directly from the off-diagonal element of thedensity matrix by assuming an optically thin medium.

In this paper we obtain a theoretical model of FDFWMin homogeneously broadened two-level absorbers forarbitrary-intensity beams that have parallel polariza-tions. We calculate the FDFWM signal intensity fromthe full Maxwell equation by using the nonlinear polar-ization obtained from the density-matrix equations.Thus we can deal with arbitrary absorption parameterswithout the theoretical assumption of an optically thinmedium. For a simple description of the FDFWM pro-cess and practical usage, we derive the analytical solutionof a FDFWM signal, assuming that the pump beams havearbitrary and constant intensities and that the probe andsignal beams are much weaker than the pump beams.The validity of using the analytical solution in practicalexperimental situations is discussed. For various inputbeam intensities, absorption parameters, and detunings,we also perform the numerical calculations to investigatethe FDFWM signal behavior. In Section 2 the generalformalism of the theoretical description of FDFWM signalis presented. In Section 3 the analytical form of the FD-

Fig. 1. Phase-matching configurations for DFWM: (a) DFWMwith phase-conjugate geometry, (b) DFWM with forward geom-etry.

FWM signal intensity is obtained under limited condi-tions. In Section 4 the procedure for numerical calcula-tion without the conditions assumed in Section 3 ispresented. Then results of signal behavior under variousparameters are presented and discussed in Section 5.Concluding remarks constitute Section 6.

2. DERIVATION OF COUPLED-WAVEEQUATIONS: GENERAL FORMALISM FORBOTH PC-DFWM AND FDFWMThe Cartesian coordinates used in these calculations areshown in Fig. 1. The two pump beams (beams 1 and 2)and one probe beam (beam 3) are introduced into the en-semble of many identical homogeneously broadened two-level absorbers. The signal beam (beam 4) is generatedalong the fourth direction. Since all four waves are as-sumed to have the same polarization and the angles be-tween the wave vectors and the z axes are small enough,we use only amplitudes of electric-field vectors for sim-plicity:

E~r, t ! 5 E~r!exp~2ivt ! 5 (j51

4

Ej~r!exp~ikj • r 2 ivt !.

(1)

Using the slowly varying envelope approximation andthe paraxial approximation and neglecting the portion ofthe Laplacian operator that involves derivatives in direc-tions transverse to the wave vectors, we can obtain thefollowing equation from the Maxwell equation:

(j51

4

i2kj • ¹Ej~r!exp~ikj • r! 5 24pk2P~r!, (2)

where Ej(r) and kj are the complex amplitude and thewave vector for the jth beam, respectively. P(r) is thecomplex amplitude of the polarization field and is givenby

P~r! 52e0a0

k

i~1 2 id !

1 1 d 2

E~r!

1 1 uE~r!u2/ES2~d !

, (3)

where e0 is the permittivity of vacuum, k is the wavenumber, and a0 is the linear field absorption coefficient atline center. d is the normalized detuning parameter de-fined as (v0 2 v)T2 , where v0 is the resonance fre-quency and T2 is the transverse relaxation time. ES(d )is the saturation electric field defined by

ES2~d ! 5

h2~1 1 d 2!

T1T2m2 , (4)

where T1 is the longitudinal relaxation time and m is thedipole moment. If Eq. (2) is multiplied by the conjugateof each of the exponential factors, exp(2ikj • r), and is in-tegrated over one period of the gratings produced by theinterference of the probe and pump beams, Eq. (2) is con-verted to the following equation:

i2k • ¹Ej~r! 5 P̂j~r! ~ j 5 1, 2, 3, 4 !. (5)

The quantity P̂j(r) is the Fourier components of the po-larization P(r) and is given by

Yu et al. Vol. 16, No. 8 /August 1999 /J. Opt. Soc. Am. B 1263

P̂j~r! 5 24pk2

l1l2E

2l1/2

l1/2

dx8E2l2/2

l2/2

dy8exp~2ikj • r8!

3 P~r 1 r8! ~ j 5 1, 2, 3, 4 !. (6)

In Eq. (6), l1 and l2 are grating periods along the direc-tions of the gratings (x axes and y axes in Fig. 1):

l1 5l

2 sinu1

2

, l2 5l

2 sinu2

2

, (7)

where u1 (or u2) is the angle between k1(r) [or k2(r)] andk3(r). It can easily be shown that P̂j(r) has the sameform for both FDFWM and PC-DFWM. Therefore wenote that the two-point boundary-value problem found inthe PC-DFWM calculation has now changed into the ini-tial value problem found in the FDFWM calculation, ow-ing to the different directions of the beam propagationvectors.

3. ANALYTICAL CALCULATION OF THEFDFWM SIGNAL INTENSITY UNDERLIMITED CONDITIONSIn this section we assume that the probe and signal fieldamplitudes are much smaller than those of the pumpfields. Neglecting the depletion of the pump fields, weobtain simplified coupled equations of probe and signalfield amplitudes. From these equations we ultimatelyderive the analytical expression of the FDFWM signal in-tensity. First, we describe the polarization given in Eq.(3) with a first-order perturbation of the probe and signalfield amplitudes. Next, we use the expanded polariza-tion given in Eq. (6) to find out the Fourier components.Finally, from Eq. (5), the following coupled equations forprobe and signal field amplitudes are obtained:

ddz

E3~z ! 5 2aE3~z ! 1 k* E4* ~z !, (8a)

ddz

E4* ~z ! 5 2a* E4* ~z ! 1 kE3~z !. (8b)

In these equations a accounts for the saturated absorp-tion and dispersion of the weak fields (in the presence ofstrong fields) and is given by

a 5 a0

1 2 id

1 1 d 2

1 1 ~I1 1 I2!/IS

$@1 1 ~I1 1 I2!/IS#2 2 4I1I2 /IS2%3/2 ,

(9)

where Ij is the intensity of the jth beam. The couplingcoefficient k is given by

k 5 a0

1 2 id

1 1 d 2

2~I1I2!1/2/IS

$@1 1 ~I1 1 I2!/IS#2 2 4I1I2 /IS2%3/2 ,

(10)

where IS 5 IS0(1 1 d 2) and IS0 is the saturation inten-sity at the line center.

For the FDFWM, the complex amplitudes of E3(0) andE4(0) are specified. In this case the solution of Eqs. (8) is

E4* ~z ! 5 H Fcosh~gz ! 1 iaIm

gsinh~gz !GE4* ~0 !

1k

gsinh~gz !E3~0 !J exp~2aRez !, (11)

where aRe and aIm are the real and the imaginary parts ofa, respectively, and g 5 (uku2 2 aIm

2)1/2. Changing thesubscript 4 to 3 in Eq. (11) yields E3(z). Usually there isno signal input, i.e., E4(0) 5 0, and then the intensity ofthe FDFWM signal can be expressed as follows:

E4* ~L ! 5 exp~2aRe L !Fk

gsinh~gL !GE3~0 !, (12)

where L is the interaction length along the propagatingdirection (z axes in Fig. 1).

The FDFWM signal in Eq. (12) consists of two parts.The second part represents the energy transfer from theother beams, and the exponential function describes thesaturated absorption (a0 . 0) or gain (a0 , 0). Wenote that, if uku2 . aIm

2, g is a positive real number, withthe signal intensity showing hyperbolic behavior, andthat, if uku2 , aIm

2, the signal intensity is sinusoidal, withan exponential decrease (a0 . 0) or increase (a0 , 0).When uku2 . aIm

2, the signal intensity becomes a simpleexponential function. In this paper, we deal with the ab-sorption case (a0 . 0). On resonance (d 5 0), aIm be-comes zero, so the signal intensity behaves hyperbolically.In this case it can be shown from Eq. (12) that the signalintensity reaches its maximum at some z position andthen decreases. The maximum signal-intensity positionmeets the following equation:

tanh~gz ! 5 g/aRe . (13)

This means that on resonance we cannot obtain the maxi-mum signal intensity if tanh(gL) . g/aRe . With detun-ing, the signal intensity can be sinusoidal. If two pumpbeams have equal intensities, the signal intensity is sinu-soidal when the normalized pump intensity is below agiven intensity, as shown below:

0 , I1 /IS0 , d ~1 1 d 2!~d 1 A1 1 d 2!/2. (14)

In this case two kinds of behaviors are possible. If gL. p/2, the signal intensity oscillates with the amplitude,decaying exponentially. If gL < p/2, the signal intensityshows a monotonic increase. Therefore, in this case, wecannot obtain the maximum signal intensity if gL. p/2. With intensities outside the range shown in re-lation (14), the signal intensity shows hyperbolic behavioras presented above.

We also note that, with low absorption, very strongpump intensity, or large detuning, the FDFWM signal inEq. (12) becomes equivalent to the PC-DFWM signal ofthe Abrams–Lind model. This means that, if absorptionis practically negligible, the directions in which the beamsare propagating are not important. This equivalence hasan important meaning. The PC-DFWM theory of theAbrams–Lind model has often been adopted to explainthe results of FDFWM experiments performed in gaseousmedia, without rigorous verification.10,12 From the re-sults of this section, however, such usage is valid only un-der limited conditions, as explained above.

1264 J. Opt. Soc. Am. B/Vol. 16, No. 8 /August 1999 Yu et al.

4. NUMERICAL CALCULATION OF THEFDFWM SIGNAL INTENSITY IN THEGENERAL CASEFor calculation without any assumptions, we first calcu-late the Fourier components of the polarization given inEq. (6) without making any assumptions, such as that ofweakness of the probe and signal beams and that of noabsorption and depletion of the pump beams. After this,we numerically solve the four coupled equations of thecomplex amplitudes given in Eq. (5). Obtaining the ana-lytical expression of each of the Fourier components iscrucial for fast numerical computation. Since the Fou-rier components of the polarization relevant for this prob-lem was already obtained for the PC-DFWM signal [Eqs.(41), (48), and (53) in Ref. 14], a detailed derivation is notgiven in this paper.

We use a Richardson extrapolation and the Bulirsch–Store method to solve the differential equationsnumerically.17 If we know the initial values of the de-pendent variables (complex amplitudes of fields) at somestarting point and wish to obtain the values at a targetpoint, the applied numerical method consists of dividingthe step with increasing numbers of substeps to find outthese values through extrapolations of the values of infi-nite substeps. With step size control, we get the finalvalues at the last point of the assumed interaction length.The boundary conditions for the coupled equations at z5 0 are given by the three input beams and one signalbeam E4(0) 5 0.

5. RESULTS OF NUMERICALCALCULATIONA. FDFWM Signal IntensitiesFor convenience and simplicity, all the intensities are ex-pressed in units of the saturation intensity at the linecenter, IS0 . We use the linear intensity absorption coef-ficient aI(aI 5 2a0) instead of the linear field absorptioncoefficient a0 , since in typical experimental situations theintensity absorption rather than the field absorption canbe measured directly.

Figure 2 shows signal versus total input intensity forthe case in which the linear intensity absorption coeffi-cient satisfies aIL 5 1.0 and the laser is tuned to the linecenter; d 5 0. In practice, since we usually use threebeams divided from the same laser source for a DFWMexperiment, we vary the ratios among the input beam in-tensities with a given total input intensity. The firstthree cases [Figs. 2(a), 2(b), and 2(c)] are such that theprobe beam has the same intensity as at least one of theinput beams and the ratio of the two pump beams is var-ied. In the next three cases [Figs. 2(d), 2(e), and 2(f )] theprobe beam intensity is smaller than the two pump beamsand the ratio of the two pump beams is varied. In theregime in which the total input intensity is smaller thanthe saturation intensity, the signal intensity becomes themaximum value for the given total intensity when threeinput beams have equal intensities. This result agreeswell with a simple grating description. According to thegrating description, the FDFWM signal is generatedthrough the scattering of one of the pump beams off the

grating made by the probe and the other pump beam.The diffraction efficiency h is correlated with the modula-tion depth of the interference fringe pattern written bythe two contributing beams, and it can be written as

h ij } u~Ei 1 Ej!2 2 ~Ei 2 Ej!

2u2 } IiIj ,

~i 5 1, j 5 3 or i 5 2, j 5 3 !, (15)

where Ii and Ij are the intensities of the beams that makethe grating. Therefore, the intensity of the generatedsignal is

I4 } h13 I2 1 h23 I1 } I1I2I3 . (16)

The overall multiplication of three variables of which thesum is fixed has the maximum value when all three vari-ables have equal values. Furthermore, using relation(16), we can approximately calculate the relative signalintensities among various kinds of intensity distribution.However, in Fig. 2 we can observe the change in relativesignal intensity at a large input intensity, which is com-pletely different from the simple calculation with relation(16). This means that the simple grating description (ne-glecting the saturation effect) becomes invalid as thesaturation strongly affects the signal generation. Thesignal intensity in the saturation regime shows two dif-ferent tendencies. When the intensity of one of the inputbeams becomes much bigger than those of the other twobeams, the signal intensity shows strong saturation,which causes the signal intensity to drop sharply as thetotal input intensity increases [Figs. 2(b), 2(e), and 2(f )].However, if the two input beams have the same intensityand the intensity of the other beam is the same as orsmaller than that, the saturation effect is relatively not so

Fig. 2. FDFWM signal intensity versus total input intensity forvarious intensity distributions. aIL 5 1.0, and d 5 0.

Yu et al. Vol. 16, No. 8 /August 1999 /J. Opt. Soc. Am. B 1265

strong; thus the signal intensity shows a slight decreaseas the total input intensity increases [Figs. 2(a), 2(c), and2(d)]. One can qualitatively understand this differencein overall trends to see the saturation of the absorbers inthe medium. If three beams are mixed in the mediumand one of them is too strong, the intensity even at the dipof a three-beam interference pattern is strong enough tosaturate the two-level absorbers, since in this case the dipcannot go to zero value. This results in saturation of theabsorbers all through the medium, and there is actuallyno absorber that can generate signal. When the inputbeams have mostly uneven intensities as in Fig. 2(e), thisabsorber saturation is the strongest, owing to the shallowmodulation depth and high intensity even at the dip ofthe interference pattern, and thus the smallest signal isgenerated. However, in the other three cases mentionedabove, it can be found that there are dips of the interfer-ence pattern at which the intensity approaches zero.Thus the absorbers at this point cannot be saturated, andsignal can still be generated effectively. Therefore it isbetter to use at least a set of two equal input beam inten-sities rather than unequal intensities of all three inputbeams. From now on in this paper, two pump beam in-tensities are set equal unless otherwise noted.

In Fig. 3 we compare the results of the analytical solu-tion obtained in Section 3 with those of the numerical cal-culation. Figure 3(a) shows efficiency versus pump beamintensity for various probe-to-pump-beam intensity ra-tios. ‘‘Efficiency’’ as used in this figure is defined by theintensity ratio of the signal beam to the probe beam, andit is analogous to the reflectivity of the PC-DFWM in the

Fig. 3. (a) FDFWM efficiency versus pump beam intensity forvarious probe-to-pump-beam intensity ratios in logarithmicscale. The efficiency is defined to be the signal-to-probe-beamintensity ratio. aIL 5 1.0, d 5 0, and the two input pumpbeams are set to have the same intensity. (b) Weak beam inten-sity regime of (a) in normal scale.

Abrams–Lind model.1 Figure 3(a) shows the role of theprobe beam in saturation as the probe-to-pump-beam in-tensity ratio is increased. In the nonsaturating regime,where the pump beam is weak, the numerically calcu-lated efficiency is lower than that obtained from the ana-lytically solved solution. This is due to pump beam ab-sorption, which is neglected in the analytical solution. Inthe saturating regime the numerically calculated effi-ciency is also smaller than that obtained from the analyti-cal solution, and the difference grows as the probe-to-pump-beam ratio increases. To estimate the pump beamabsorption effect in this saturating regime, we numeri-cally calculated the efficiency without pump beam deple-tion. It was found that, in the case of I3 5 0.001I1 , therewas no difference in the efficiency in the strong intensityregime, regardless of whether the pump beam depletionwas neglected in the numerical calculation. This meansthat, in the saturating regime, strong beams bleach theabsorbers, and the absorption can be neglected. There-fore the difference is not caused by the pump beam ab-sorption. By contrast, when we calculated the efficiencyfor the case in which the input probe beam intensity is al-ways smaller than the saturation intensity (I35 0.000001IS0), the results showed perfect agreementwith those of the analytical solution in this saturating re-gime. From these results we can infer that the differencebetween the numerical calculation and the analytical so-lution is due mainly to the probe beam saturation. Evenwhen the probe beam intensity is 1% of the pump beamintensity, the probe beam intensity can exceed the satu-ration intensity in the strongly saturating regime and cancontribute to the absorber saturation. Therefore, weshould be careful in using the analytical solution in thesaturating regime, since the analytical solution cannot re-veal the saturation effect caused by the probe beam.

In Fig. 3(b) we show in detail, with normal scale, thesignal behavior in the weak-field regime of Fig. 3(a). Allthe signals are normalized to its maximum value. Thiskind of calculation is frequently used in the typicalDFWM experiment, for example, to find out the satura-tion intensity of the medium. One can obtain the satu-ration intensity by fitting the analytically obtained signalintensity [Eq. (12) for FDFWM] to the experimentally ob-tained signal intensity with the fitting parameter of thesaturation intensity. As shown in Fig. 3(b), however,there is a large difference between the analytical solutionand the numerical calculation. Even in case of the small-est probe beam intensity considered here, which issmaller than the pump beam intensity by an order of 2,there is still a notable difference in magnitude, as in thestrong beam intensity regime. Several curves in Fig. 3(b)show maximum values at some pump beam intensitiesand a monotonic decrease after the maximum, which isnot predicted by the analytical solution.

In Fig. 4 we present the numerically calculated signalintensities at resonance in terms of total input intensityfor various absorption parameters. All three inputbeams are set to have the same intensity. The maximumsignal intensity increases according to the absorption pa-rameter. In the regime in which the total input intensityis weak, the signal obtained with the absorption param-eter aIL 5 1.0 is stronger than that obtained with the ab-

1266 J. Opt. Soc. Am. B/Vol. 16, No. 8 /August 1999 Yu et al.

sorption parameter aIL 5 0.1, but the signal is extremelyreduced, by an order of 3 or more, with the absorption pa-rameter aIL 5 10.0. This is due to the competition oftwo different mechanisms resulting from the absorption.The increase of the absorber number density, which givesa larger absorption parameter, can generate a strongersignal, but this signal should experience stronger absorp-tion. In the strong-field regime, however, since strongfields bleach the absorbers so that absorption becomesnegligible, the signal always becomes stronger with theincrease in the absorption parameter.

In Fig. 5 we show signal intensity versus total input in-tensity for various detunings. The intensity absorptionparameters used in the calculations are 0.1 and 10.0 forFigs. 5(a) and 5(b), respectively. All the input beams areset to have equal intensities. Since the saturation inten-sity becomes larger when detuning increases, a larger in-tensity is needed to obtain the maximum signal, andhence the total intensity yielding the maximum signal be-comes larger as the detuning increases. In Fig. 5(a), inthe case of aIL 5 0.1, there occurs a change in the orderof the signal intensity in the three cases as the total inputbeam intensity increases. However, in Fig. 5(b), a differ-ent behavior can be seen when the absorption becomesstrong. In the large-intensity regime the signal-intensitybehavior is similar to that of the low absorption caseshown in Fig. 5(a), but in the small-intensity regime thesignal intensity with no detuning is the smallest of all.This result is also caused by the competition between thetwo different mechanisms. With the small detuning, alarge signal can be generated because of the strong inter-action between the beam and the absorber, but all thebeams experience strong absorption. However, withstrong intensity, absorption is negligible because ofbleaching, and a behavior similar to that of the cases inwhich the absorption parameter is small is restored.Therefore this leads to the optimum total intensity for ob-taining the maximum signal at a given total input beamintensity.

Fig. 4. FDFWM signal intensity versus total input intensity forvarious absorption parameters aIL. d 5 0, and I1 5 I2 5 I3 .

B. FDFWM Signal Line ShapesIn Fig. 6 we plot the line shapes for various pump beamintensities. The probe beam intensity is set to 0.01IS0 ,and the absorption parameters are given as 0.1 and 10.0for Figs. 6(a) and 6(b), respectively. All the signal inten-sities are normalized to each of their maximum values.

In Fig. 6(a), a dip begins to appear at resonance whenthe pump beam intensity is larger than 10IS0 . We havealready discussed the detuning effect on the signal gen-eration when the absorption parameter is small [Fig.5(a)]. With low pump beam intensity, the maximum sig-nal was obtained on resonance. However, when the inci-dent beam intensity is much larger than the saturationintensity at line center, the strong beam saturates the ab-sorbers and causes the decrease in the signal intensity onresonance. Also, if we detune the incident beam, we canavoid the saturation of absorbers that produces the largesignal, and consequently a dip appears on the top of thespectral line. The dips in Fig. 6(b) are more complex inshape. In the case of strong absorption, the absorption ofthe signal passing through the medium is introduced inaddition to the saturation. When the incident beam in-tensity is relatively small [I3 5 0.1IS0 or I3 5 IS0 in Fig.6(b)], the dip is due mainly to the strong absorption, andthe dip becomes shallow as the pump intensity increases.However, with a relatively strong pump beam [I35 10IS0 or I3 5 100IS0 in Fig. 6(b)], the dip again be-comes deeper, and this dip is due mainly to saturation asexplained above, since in this case absorption becomesnegligible because of the bleaching of absorbers.

Fig. 5. FDFWM signal intensity versus total input beam inten-sity for various detunings. I1 5 I2 5 I3 .

Yu et al. Vol. 16, No. 8 /August 1999 /J. Opt. Soc. Am. B 1267

In the weak absorption case, we can observe the powerbroadening of the linewidth as the pump beam intensityincreases. In the strong absorption case, however, thedip results mainly from the absorption, for I3 5 0.1IS0and I3 5 IS0 , and the increased intensity is needed toovercome the strong absorption and generate a larger sig-

Fig. 6. FDFWM line shapes for various pump beam intensities.I3 5 0.01IS0 .

Fig. 7. FDFWM line shapes for various ratios between the twopump beam intensities. All the intensities are normalized tothe maximum value of the four cases considered. The additionof two input pump beam intensities is fixed to 2IS0 , I35 0.01IS0 , and aIL 5 0.1.

nal than before. For this reason, the linewidth becomessmaller (not larger as in power broadening) as the pumpbeam intensity increases. For I3 5 10IS0 and I35 100IS0 , the line shape is affected mainly by the satu-ration as noted above, the linewidth shows broadening asthe pump beam intensity increases, and the dip at theresonance again becomes deeper.

In all the results given thus far, two pump beams areset to have equal intensities. Now we calculate the FD-FWM signal line shapes for various ratios between thetwo pump beams at low absorption (aIL 5 0.1), and theresults are shown in Fig. 7. The addition of two pumpbeam intensities equals two times the saturation inten-sity at the line center, and the probe beam intensity is setto 0.01IS0 . All the line shapes in the figure are normal-ized to the maximum value of the signal obtained withequal pump beam intensities. A dip at resonance beginsto appear when the ratio between the two pump beam in-tensities is approximately 5. This dip can be explainedwith the results shown in Fig. 2. In Fig. 2 we can seethat, if the total intensity of the three input beams is fixedbut the intensity of one input beam is much stronger thanthe others, the saturation effect that appears is strong,because the saturation occurs all through the medium.Therefore, as the ratio between the two pump beam inten-sities increases and hence one of the pump beams growslarger, the signal intensity at the line center becomessmaller, and the dip begins to appear.

6. CONCLUSIONSWe have presented a theory of FDFWM in homogeneouslybroadened two-level absorbers for arbitrary-intensity in-put beams. An analytical solution of FDFWM is derivedby assuming no pump beam depletion, strong pumps, anda weak probe. The shape of the analytical solution of FD-FWM is different from that of PC-DFWM as derived byAbrams and Lind, but it becomes equivalent to that ofPC-DFWM in the case of low absorption or large detun-ing. Also, the validity of applying the PC-DFWM theoryto FDFWM experiments is discussed.

For arbitrary beam intensities and absorption param-eters, one obtains the signal intensity by numericallysolving the coupled equations of complex amplitudes.The results of the numerical calculation and the analyti-cal solution of FDFWM have been compared for variousprobe-to-pump-beam intensity ratios. Even when theprobe beam is smaller than the pump beam by an order of2, there are notable differences between the numericalcalculation and the analytical solution, owing to the probebeam saturation effects on the signal generation. Thevalidity of using the analytical solution of FDFWM underpractical experimental conditions is also discussed. Inthe strong absorption case, the numerically calculatedspectral line shape is shown to be rather complex. Inthis case a dip appears on the top of the line shape, owingto two different mechanisms: absorption and saturation.Furthermore, the linewidth is also found to show twokinds of behavior. When the absorption is dominant, thelinewidth grows smaller with an increase in the input in-tensity, but when the saturation is dominant typicalpower broadening is restored.

1268 J. Opt. Soc. Am. B/Vol. 16, No. 8 /August 1999 Yu et al.

ACKNOWLEDGMENTSThis research was supported by the Korea Ministry of Sci-ence and Technology. D.-H. Yu, J.-H. Lee, and J.-S.Chang also acknowledge the support of a grant from theBasic Science Research Institute Program, Ministry ofEducation [BSRI-97-2421 (1997)].

REFERENCES1. See, for example, R. A. Fisher, ed., Optical Phase Conjuga-

tion (Academic, New York, 1983) and H. J. Eichler, P.Gunter, and D. W. Pohl, Laser-Induced Dynamic Gratings(Academic, New York, 1984).

2. P. Yeh, ‘‘Scalar phase conjugator for polarization correc-tion,’’ Opt. Commun. 51, 195–197 (1984).

3. I. Biaggio, J. P. Partanen, B. Ai, R. J. Knize, and R. W. Hell-warth, ‘‘Optical image processing by an atomic vapor,’’ Na-ture (London) 371, 318–320 (1994).

4. R. L. Farrow and D. J. Rakestraw, ‘‘Detection of trace mo-lecular species using degenerate four-wave mixing,’’ Sci-ence 257, 1894–1900 (1992).

5. K. Kohse-Hinghaus, ‘‘Laser techniques for the quantitativedetection of reactive intermediates in combustion systems,’’Prog. Energy Combust. Sci. 20, 203–279 (1994).

6. A. Klein, M. Oria, D. Bloch, and M. Ducloy, ‘‘Saturation be-havior and dynamic stark splitting of nearly degeneratefour-wave and multiwave mixing in a forward boxcar con-figuration,’’ Opt. Commun. 73, 111–116 (1989).

7. G. Meijer and D. W. Chandler, ‘‘Degenerate four-wave mix-ing on weak transitions in the gas phase using a tunableeximer laser,’’ Chem. Phys. Lett. 192, 1–4 (1992).

8. H. Bervas, S. Le Boiteux, L. Labrunie, and B. Attal-Tretout,

‘‘Rotational line strengths in degenerate four wave mixing,’’Mol. Phys. 79, 911–941 (1993).

9. K. Nyholm, M. Kaivola, and C. G. Aminoff, ‘‘Detection of C2and temperature measurement in a flame by using degen-erate four-wave mixing in a forward geometry,’’ Opt. Com-mun. 107, 406–410 (1994).

10. A. P. Smith, G. Hall, B. J. Whitaker, A. G. Astill, D. W.Neyer, and P. A. Delve, ‘‘Effects of inert gases on the degen-erate four-wave-mixing spectrum of NO2,’’ Appl. Phys. B:Photophys. Laser Chem. 60, 11–18 (1995).

11. B. Attal-Tretout, H. Bervas, J. P. Taran, S. LeBoiteux, P.Kelley, and T. K. Gustafson, ‘‘Saturated FDFWM line-shapes and intensities: theory and application to quanti-tative measurements in flames,’’ J. Phys. B 30, 497–522(1997).

12. K. Nyholm, ‘‘Two-dimensional imaging of OH in a flame byusing degenerate four-wave mixing in a forward geometry,’’Appl. Phys. B 64, 707–712 (1997).

13. W. P. Brown, ‘‘Absorption and depletion effects on degener-ate four-wave mixing in homogeneously broadened absorb-ers,’’ J. Opt. Soc. Am. 73, 629–634 (1983).

14. B. Ai and R. J. Knize, ‘‘Degenerate four-wave mixing intwo-level saturable absorbers,’’ J. Opt. Soc. Am. B 13,2408–2419 (1996).

15. G. Grynberg, M. Pinard, and P. Verkerk, ‘‘Saturation in de-generate four-wave mixing: theory for a two-level atom,’’J. Physique 47, 617–630 (1986).

16. R. P. Lucht, R. L. Farrow, and D. J. Rakestraw, ‘‘Saturationeffects in gas-phase degenerate four-wave mixing spectros-copy: nonperturbative calculations,’’ J. Opt. Soc. Am. B 19,1508–1520 (1993).

17. See, for example, W. H. Press, S. A. Teukolsky, W. T. Vet-terling, and B. P. Flannery, Numerical Recipes in C, 2nd ed.(Cambridge U. Press, New York, 1992).