Theory of saturated line shapes in phase-conjugate emission by resonant degenerate four-wave mixing in Doppler-broadened three-level systems

Vol. 73, No. 5/May 1983/J. Opt. Soc. Am. 635

Theory of saturated line shapes in phase-conjugateemission by resonant degenerate four-wave mixing in

Doppler-broadened three-level systems

Daniel Bloch and Martial Ducloy

Laboratoire de Physique des Lasers, Universit6 Paris-Nord, 93430 Villetaneuse, France

Received September 18, 1982

We present a theoretical study of phase-conjugate (PC) emission by backward degenerate four-wave mixing(DFWM) processes in a resonant gas medium. In this study, high-intensity (pump) saturation and Doppler broad-ening are simultaneously taken into account. We develop a general formalism to deal with the saturation effectsin the case of a three-level atom (with degenerate frequencies) and obtain analytical expressions of the PC field am-plitude (after integration over the velocity distribution) under the following conditions: (1) the probe beam andone of the pump beams have intensities weak enough to be treated at the lowest perturbation order, whereas theother pump intensity is arbitrarily high; (2) the probe crosses the standing pump wave at grazing incidence. It isshown that the saturation induced by the forward pump (copropagating with the probe beam) leads to entirely dif-ferent behavior of the nonlinear susceptibility from that induced by the backward pump (counterpropagating withthe probe beam). This saturation anisotropy is specific to the inhomogeneous broadening of the gas medium,which is also responsible for the frequency splitting experimentally observed in the intensity line shape of the PCemission for large pump intensities. At saturation, this splitting is theoretically shown to be proportional to theRabi frequency, with a proportionality coefficient critically dependent on the relaxation processes; in the case ofa single-relaxation model, the splitting is equal to 1.5 times the Rabi frequency. The PC intensity line shape isdemonstrated to be of dispersive origin in the case of backward saturation, whereas for forward saturation, narrowstructures are predicted in both real and imaginary parts of the nonlinear susceptibility, also leading to the intensi-ty line-shape splitting. One also finds that the reflectivity of PC mirrors based on resonant DFWM strongly de-pends on the origin of the saturation. For a saturating forward pump, the PC emission intensity tends toward zerowith increasing pump intensities, and it reaches a finite (nonzero) limit in the case of backward saturation. Mostof these predictions have been observed experimentally [D. Bloch et al., Phys. Rev. Lett. 49, 719 (1982)].

1. INTRODUCTION

The recent development of phase conjugation has opened anew field and stimulated new activities in nonlinear optics(see, e.g., Refs. 1-3). Among the various nonlinear techniques,backward degenerate four-wave mixing (DFWM), proposedand performed independently by several authors, 4 -6 is nowthe technique that is used most to generate a phase-conjugate(PC) wave because it is not limited by phase-matching con-ditions. In this paper, we present a strong-field theoreticalstudy of resonant DFWM in Doppler-broadened gas media.Gas media have the ability to be excited by intense opticalirradiation without irreversible destruction and can serve asnonlinear media offering large PC reflectivities. 7' 8

Let us recall the physical mechanisms involved in low-powerresonant DFWM (Fig. 1). The effect of the probe and onetraveling pump wave is to produce an absorption grating inthe interaction volume. The emission of the PC wave canthus be viewed as the Bragg diffraction of the counterpropa-gating pump beam by this induced grating. In gas media, thevelocity distribution plays a double role: First, the inducedgrating is destroyed by the thermal motion in a time that isof the order of the mean time needed for an atom to travelacross one grating fringe (i.e., a spatial period over the averagethermal velocity). Hence the efficiency of the PC emissionincreases with decreasing angles 0 between pump and probebeam (cf. Fig. 1). In this case, the spatial period of one of thetwo gratings becomes large while the contribution of the other

grating (of spatial period X X/2, with X the optical wave-length) is negligible.

Second, as in saturated absorption, the process responsiblefor the PC emission is Doppler free if there exists a preferen-tial axis along which there is a velocity selection, i.e., for smallcrossing 0. In that case, the PC reflectivity (Rpc, defined asthe PC-to-probe intensity ratio) is a Doppler-free Lorentziancentered at the atomic-resonance frequency, at least for lowincident powers.

All these results can be derived from a third-order calcu-lation9' 10 (lowest-order perturbation theory in which the PCfield is proportional to the product of the three incident fields)and are well known since the first experiments in gasmedia.1 1"12 With increasing pump intensities, saturationeffects produce major changes in the experimental observa-tions. For instance, Humphrey et al. 13 have shown that op-tical-pumping effects seriously alter the angular dependenceof PC reflectivity and line shape, in particular for largecrossing angles 0. For small angular separations, the mostspectacular change affects the line shape, in which a broad-ening and a splitting are observed at large pump intensi-ties-the PC reflectivity tending toward zero at linecenter. 11,14,15

To our knowledge, no rigorous theory has been proposedto interpret the saturated line-shape splitting. It is essentialto point out that these effects essentially originate in the in-homogeneous broadening caused by the velocity distribution.A strong-field calculation performed by Abrams and Lind16

0030-3941/83/050635-12$01.00 ©) 1983 Optical Society of America

D. Bloch and M. Ducloy

636 J. Opt. Soc. Am./Vol. 73, No. 5/May 1983

F Z B

PCFig. 1. Principle of a backward four-wave mixing experiment.

for stationary, resonant two-level absorbers does not predictsuch a double-peak behavior-at least for an optically thinsample in which propagation effects can be neglected.

Two methods have been suggested to interpret these lineshapes. One is based on ac Stark splitting but to our knowl-edge was never supported by any calculations.1 7 Moreover,a careful analysis of the experimental data shows that suchan interpretation cannot hold. For transitions between de-generate levels, one needs to consider the various Rabifrequencies associated with the different transition proba-bilities among sublevels. This would cause substructures inthe PC line shape that have never been observed in all theexperiments performed to date.

A second interpretation, proposed by Woerdman andSchuurmans,14 considers the nonlinear susceptibility XNLgoverning the emission process (in the absence of propagationeffects, the amplitude of the PC field and its intensity areproportional to XNL and I XNLI 2, respectively). This inter-pretation is based on the nonlinear response of stationarytwo-level absorbers to a single plane wave. It is well knownthat, in the saturated regime, the absorption contribution tothe nonlinear response (Im XNL) saturates faster than thedispersion contribution (Re XNL). Indeed, in this case, thenonlinear response is of the form P = xE, with

Y -ibX 2+ Q2 + 62

where y is the natural linewidth, 6 is the frequency detuning,and Q is the Rabi frequency. This simple model predicts thatthe intensity line shape behaves like the square of a dispersioncurve at saturation. However, Woerdman and Schuurmans14ignore the standing-wave character of the pump beam, whichleads to spatial hole burning in XNL (i.e., Q = Q0 cos kx, wherek is the wave vector). When spatial hole burning is taken intoaccount, the dispersive character of the response vanishes inthe spatially averaged value of XNL. This is the result actuallyobtained by Abrams and Lind.16

The purpose of this study is to show the decisive role playedby the atomic-velocity distribution in the interpretation ofthe line shape. Because of the atomic motion, an absorbertravels across several nodes of the standing wave and experi-ences a spatially averaged Rabi frequency (Q). Because ofthis mean value of the Rabi frequency, the dispersive char-acter of the response is not lost. This has been demonstratedin recent experiments1 5 in which, by using heterodyne de-tection of the PC field,' 8 we have obtained direct informationon the complex amplitude of XNL that supports the dispersiveorigin of the line shape. In addition, our experiments haverevealed a remarkable directional anisotropy of the satura-tion.15 When the intensities of the two pumps are markedlydifferent, the saturated line shape depends on which pumpsaturates the medium.38 This anisotropy is a typical effectof the velocity distribution: for stationary absorbers, the twopumps should play an identical role.

In this paper, we investigate the saturation effects affectingthe emission line shape when one of the pump fields is arbi-trarily intense. The atomic system is modeled as a coupled

three-level system with degenerate transition frequencies.'9In Section 2, we describe the general density-matrix formalismfor a three-level system in a semiclassical approach. We givethe general expression of the reemitted field and point out theterms contributing to the PC emission. The theoretical ad-vantages of three-level systems over two-level systems arediscussed, particularly those that concern the saturation ef-fects produced by an intense standing pump wave. Section3 is devoted to the calculation of the contribution of each ve-locity group in the case when only one of the pump fields isintense, the other counterpropagating pump and the probebeing considered as small perturbations and treated at thelowest order. In Section 4, we perform the velocity integrationin the Doppler-limit approximation for grazing incidences(small angular separation between pumps and probe). An-alytical results are obtained for the two cases in which thesaturating pump either copropagates or counterpropagateswith the probe beam. In Section 5, we discuss the main fea-tures of the emission line shape. Physical interpretations areproposed along with simplified analytical formulas using asingle relaxation parameter.

2. GENERAL FORMALISM

We consider a gas medium composed of three-level atoms(ja), Ib), and Ic); see Fig. 2). Their velocity distribution isassumed to be Maxwellian, with mean thermal velocity u:

F(v) = 7r-31 2 u-3 exp(-v 2 /u 2 ). (1)

The li)-Ij) transition frequency is wij, and the relaxationprocesses are described by phenomenological decay rates, 'yjfor the population of level j, and by yij for the i-j atomic co-herence.2 0 The electric-dipole moment of the i-j transitionis I.ij (gab and Acb are taken to be real; jUac = 0 for parity con-siderations).

The incident electromagnetic (e.m.) fields are described asmonochromatic plane waves and are labeled F (forward) andB (backward) for the pump beams and P for the probe beam(see Fig. 1). These three e.m. fields have the same frequencyw, which is nearly resonant for the two atomic transitions a-band c-b. The e.m. field is described as a scalar quantity

E(r, t) = E E, (r, t)Iu=F,B,P

= E 2 {s exp[i(wt - k, . r)] + c.c.), (2)

where kF -kB = k and kp = K. Moreover, we assume thatpump F and probe P interact only with one transition (a-b,for instance), whereas pump B couples b and c only. Such anassumption, although nonphysical in a purely scalar theory,is, however, an adequate description of r+.polarized pump-Fand probe beams and a er-polarized pump B interacting witha J = 1 -J = 0 transition. Straightforward selection rulesimply that the PC emission is a- polarized, i.e., that it caninteract with the Ic) - b ) transition only.

ab 'cb

'a>

IC2Fi g. 2. Schematic of the three-level atom.



The total incident e.m. field interacting with the atomscreates a macroscopic polarization inside the medium, whichis a source of an e.m. field when phase-matching conditionsare fulfilled. To calculate the PC emission, one has to cal-culate the Fourier component of the induced nonlinear com-plex polarization whose space and time dependence is givenby exp i(ct + Kr).2 1 It is shown below that the calculatedfield necessarily has the correct polarization state to interactwith the lc )-l b) transition only. We call Ppc the polarizationterm responsible for the PC emission and define

1Ppc(r, t)= - I7~pC exp[i(wt + Kr)] + c.c.j.

2(3)

For an optically thin sample of length L, and in the slowlyvarying envelope approximation, the incident (monochro-matic) field amplitudes are almost constant along all thesample, so that the reradiated field amplitude is10

&PC = - kLppc.2c0

(4)

Let p(r, t) be the density matrix of the system. If bcb is thecontribution to Peb (Pcb = (c I p(r, t) I b )) that has the adequatespace and time dependence, one has

Ppc(r, t) = Abcpcb(r, t) + c.c.

Pab = (-Yab + j.ab)Pab + O QF exp[i(ot -k r)](Pbb2

-Paa) QB exp[i(Wt + k - r)]Pac2

+- Qp exp[i(cot - K * r)](pbb - Paa),2

Pbc = (-Ycb - iCcb)Pbc + Q2 F exp[-i(cwt - k * r)]pac

+ 2QB exp[-i(wt + k * r)](pcc - Pbb)2

+ 2 Qp exp[-i(cot - K * r)]pac,2

Pac = (-Yac + iCLac)Pac + 2 QF exp[i(wt - k * r)]pbc2

-2QB exp[-i(wt + k * r)]Pab

2

+ - i~p exp[i(wot - K . r)Ipb'c,2

where b is the hydrodynamic (Boltzmann-type) derivative of

PI

(5) p = a +v* V. P

Hence the main problem is the determination of the densitymatrix

p(r, t) = 5d3vF(v)p(r, v, t), (6)

where p(r, v, t) is the density matrix for the velocity group v.p(r, v, t) obeys the following equations of motion:

Paa = 'Ya(na - Paa) QFPab exp[-i(wt - k * r)]2

+ 2 QFPba exp[i(wt - k * r)]2

- QPPab exp[-i(cot -K * r)]2

+ - QPPba exp[i(ct - K r,2

Pbb = Yb(nb - Pbb) + 2QFPab exp[-i(cot -k - r)]2

2 QFPba exp[i(wt - k * r)]2

+ - QBPcb exp[-i(wt + k * r)]2

- QBPbc exp[i(wt + k * r)] + 2QPPab2 2

X exp[-i(cot -K r)] - 2 QPPba exp[i(cot -K r)],2

Pcc = Yc(nc - Pcc) - QBPcb exp[-i(ct + k * r)]2

+ - BPbc exp[i(wot + k -.l2

and where QF = AUab6F/th, Qp = A-ab6p/h, and QB = AcbSB/^h

nj is the equilibrium population of level j in the absence of theincident fields. In Eqs. (7), the usual rotating-wave approx-imation (RWA), ( - Wab - Wcb), has been used to eliminatethe nonresonant terms.

To solve this set of coupled equations, we assume that theprobe intensity is weak enough (Qp << yij, yj) so that we candevelop p in powers of Qp:

p = Z (n~p, (9)

with (n)p . (Qp)n. To get &pc [Eqs. (4) and (5)], one thusneeds to calculate the first-order contribution, (l)p. As aconsequence of the above assumption, our calculation cannotpredict saturation effects induced by an intense probe. In-deed, in most of the experiments, the probe is weak enoughto justify a perturbation expansion.2 2

The RWA implies that the expected values of pjj (as wellas of Pac) are time independent, whereas Pab and Pcb evolveat frequency c. As regards the spatial dependence, p can beFourier developed in eikr and eiKr. For instance, (l)P can beexpressed as

)pjj = y1aj(n)(v)exp[-i(nk + K) * r]

+ Oj(n)(v)exp [i(-nk + K) -r} (odd n),

(')Pac = Zcaa,(n)(v)exp [-i(nk + K) * r]n

+ fa, (n)(v)exp[i(-nk + K) * r]} (odd n),

(Pab = Z-aab(n)(v)exp i[ct - (nk + K) -r]n

+ /3ab(n)(v)exp i[wt -(nk - K) - r]l (even n),

(7)

(8)



(')Pcb = Ejacb (n)(v)exp i[Ot - (nk + K) * r]n

+ Ocbb(n)(v)exp i[wt - (nk - K) * r]1


(even n),(10)

where the a's and O's are the coefficients of the Fourier de-velopment (0j3(n)= aj(-n)*). From Eqs. (3)-(5), it is clear thatthe quantity of interest is the coefficient O

3cb(°)

0 One has

6 PC = - - Abc (I3cb(°)),eO

(11)

where the ( ) indicate the velocity average.At zeroth order (no probe field), one has to find the density

matrix of the atomic system coupled to a standing-wave pumpfield, which is a well-known problem related to gas-lasertheory.2 3 The solution can be also expressed as a Fourierexpansion (j = a, b, c):

(O)pjj = y (0)pj(n)(v)exp(ink - r),(even n)

(°)Pac = 1 (N)pac(n)(v)exp(ink-r),(even n)

(°)Pab = 7 (0 ±b(n)(v)exp[i(wt + nk - r)] (12)(odd n)

and by a similar expansion for (°)Pcb.

The derivation of the recurrence relations between the aand f coefficients is thus straightforward:

[-Ya -i(nk + K) .v]a( )- QFaab(n+1) + 2 QFab(-n+l)* + P NOab(W)]*2 2 2

['Yb -i(nk + K) * v]ab( )2QFab(n+l) QF~ab(-n+l)* +-QB acb(n-1)

2 2 2

2QBI~cb( n P 1-- P()pab~n

[Yc -i(nk + K) . v] ac() b

2 2

[Yab + i(o - Web) - i(nk + K) * aab

= 2 QF[ab(n-) - aa(n-1)]- QB aac(n+1)2 2

+ 2 QP [(°Pb (-n)-(°)Paa (-n)],

['Yab + i(O- Wab) - i(nk - K) * VIab(n)

= p[bb(n+l)* (-n+l)*-(n+ll2 2

[lYcb + i (CO - Wcb) - i(nk + K) - v] axb (n)

2 QF-3ac (-n+,1) + QB [°b (n+l1) - c(n+1)2 2

-- Q p [k°)ac (n]2

['Ycb + i(W - Wcb) - i(nk - K) * Vltcb('n)

= 2 QF2 aac(-n+l) + - QB[tab(-n-1) - a,(-n-1112 2

[Yac - iwac - i(nk + K) . Vica,(n)

= IQFOcb(-n+l) - 2QBa~b (n-1) +-Qp[(O) Ab WI *2 2 2

[Yac - iwac - i(nk - K) - VI#., (n)

( ( n+ 1)* Bb (n-1)2 2

(13)

A considerable simplification of the problem will occur ifEqs. (13) can be reduced to a finite set of linear equations. 2 4

This last point fully justifies the choice of a three-level system,in which pumps F and B do not interact with the same tran-sitions. Indeed, we know that, even if the two pump beamsare intense, the solution of (M)p has a finite number of Fouriercomponents in the case that we analyze because there is nospatial coupling between the saturation effects induced byeach pump beam (no spatial hole burning). (O)Pii (0), (0)&ac (-2),

(°)Pab (1), and (0)Pcb (+1) are the only nonzero Fourier coeffi-cients [Eqs. (12)]. Once (M)p(v) is known, the system of Eqs.(13) becomes a system of nine linear equations coupling nineunknown quantities: aa(a&), ab', a(), aab(°), fOab 2),aeb (/2),

3cb(°), a.C,,M and 13ac(3) (all the other coefficients

cancel). A short demonstration of that major simplificationcan be obtained by considering the limit K = k: The Fourierexpansions of (O)p and (l)p must then have identical form sincepump F and the probe cannot be distinguished from eachother. On the other hand, the calculations for a two-levelsystem are much more complicated if each pump beam hasa saturating intensity. The Fourier expansion of (M)p is infi-nite, and the Fourier coefficients are expressed in terms of acontinued fraction.23 The analogue of Eqs. (13) for (l)p is aninfinite set of linear equations whose solutions are expressedin terms of continued fractions and whose source terms(right-hand-side terms) are continued fractions. The problemis then to integrate over the velocity distribution the coeffi-cient of interest [namely, f

3ab(

0)], which must be calculated by

a kind of double continued fraction.

3. EXPRESSION OF &Pc

Let us come back to our three-level system. (O)p is governedby the system of Eqs. (7) with Qp = 0. Given the general formof (O)p [Eqs. (12)], the six Fourier coefficients of (O)p obey sixlinear coupled equations, the source terms of which are theequilibrium population difference nij = (nj - n1).

Although it is possible in principle to give an analyticalexpression of f 3

cb (0) for arbitrary pump intensities, we restrictthe following analysis to the case of only one saturating pumpwhile we suppose that the intensity of the other counterprop-agating pump wave is kept well below saturation. The maininterest of this restriction is that the velocity integration be-comes analytically tractable. In other words, we consider twocases:

(1) The forward pump has an arbitrarily strong intensity,but the backward intensity is assumed to be so weak that (M)pcan be limited to a first-order contribution in QB.

(2) A symmetrical case with a weak F pump and an in-tense B pump.

Let us note that these two cases are equivalent for the cal-culation of (O)p (the subscripts a and c just need to be inter-changed). In the calculation of (1)p(v), the difference between


cases (1) and (2) lies in the fact that, in case (1), the probe in-teracts with the atomic system by means bf the transition thatexperiences pump saturation, whereas, in case (2), the probeinteracts by means of a nonsaturated transition. Moreover,one should expect a different behavior for cases (1) and (2)after the velocity integration because of nonisotropic veloc-ity-selection effects.

Case (1): Forward Pump of Arbitrary IntensityFrom the solution at zeroth order in Qp,

(°)Pab(-1)(v)

i Q 'Yab-(k v)- 2b~2 fYab 2(1 + SF) +( k6-k*V)2

[(° ) - ( (V)

'Yb2 + (3-k .v)2= Yab2(1 + SF) + (3-k -v)

-i yac + 2ik -v=- B

2[{Ycb + i(6 + k * V)] [Yac + 2ik

J -SF[( 0 )aa - ( 0pbb]tYYab2

X 1lbc - 2'Yb [Yab2 + (6- k- v)2 ]

SF[(O)paa - (°)pbb]'YYab

4(Yac -2ik -V)('Yab + i(b - k * v))J

*v] + 'YYeSF1

'Ycb -i(b + k - v)

{(Tac - 2ik * V)[Ycb - i( + k - v)] + 'YYabSF}I 4X |-ba FQB ['Yab- i(-k * v)]

4 kab2( ( + SF) + (6F + ( V)2

+ QF QBnob4['Ycb - i(0 + k * v)]

8+ QF3QBnbaYab (8'Yb [Ycb -i(b + k -0)][Yab2(j + SF) + (5 k * V)2111

(14)

where y-1 = 1/2(Ya-l + 'Yb-), 5 = W - Wab = W - Wcb (we as-

sume in all the following that wac = 0), and SF is a dimen-sionless saturation parameter:

QIF2SF ='Y'Yab

one finally gets the expression of tcb( 0)(v):

QFQBLac + QFQBAb

4 +1 + 7J ab(6)(a + Ab)]

Ocb(o)(V) = I'Ybc + i(3 + K -v) + 2Lac I11 4I4

--A a.- ( 0N7bb]

2 TYab -i(-K -v) + QF2 (Aa + Ab)

4 [1 + 'F Lab(b)(Aa + Ab)j[41

QFQBLac + QFBAB/4

1 + - -Lab (6) (Aa + Ab)QFOP 4+ (0)(ab(-1 )(Aa + Ab)

I1+ £FLab(5)(a + Ab)I bYab - i( -K * v) + - F 2(Aa + Ab)/4

aba j 1+ K4 Lab(b)(Aa + Ab)J

- QFP La,(°)Pcbb-1) -4

QB0 b(O)pa (-1) '.QB4+ pAb( )Pab +)

4+QF2 1>1ab(3)(Aa A)

(15)

- - h


Ai = LI/i -i(k-K)v]-l (i = a, b). (16)

Note that the system of nine equations with nine unknownquantities that yields the term fcb (°) of (U)p(v) [see Eqs. (13)]is partly decoupled in the approximation of low backward-pump intensity, so that 3cb(°) is formally independent of(°0 )ac (v) and it is required only to solve a linear 6 X 6 systemof equations. 25 More precisely, it should be pointed out that,for consistency of treatment, one must retain in the calculationof the coefficients of (l)p all the terms that are linear in QB:(°)Pbc itself is of first order in QB, but (°)Pab and (°)Pa - (0)Pbb

are zeroth order in QB, so that a term proportional to QB(o)Pab,

or to QB [(°)&aa - (°)Pbb], cannot be neglected in the calculationof 13cb(°). Hence decoupling between f~cb(°) and (O)Pac does notimply that the coherences induced between I a) and I c) do notaffect the emission processes; to do so, they need an additionalinteraction with the probe.

The final expression of &pc can then be obtained by sub-stituting the explicit expression of f3 cb(°) [Eq. (15)] into Eq.(11). The velocity integration is discussed in Section 4.Actually, at this step of the calculation we restrict ourselvesto a configuration of grazing incidence between the pump andthe probe beam, i.e., the angle 0 is small, so that K = k. As iswell known, such an approximation is valid if 0 << yi/k, yij/ku(i, i = a, b, c).10

,26 (It corresponds in a typical experiment to

0 < 10.27) With this approximation, the expressions of thedenominators Lac, £ab(b), and Ai [Eqs. (16)] are notablysimplified (e.g., Ai = 1/,yi). The formal expression of 6 pc isthen reduced to an integral over one velocity component byuse of Eq. (11) and of

(O3cb(°)) i= FQBQP Jdv exp(-v2/u2)

D. Bloch and M. Ducloy640 J. Opt. Soc. Am./Vol. 73, No. 5/May 1983

where

Lac = [Yac + i(k + K)v]-',

-Lab(6) = IYab + i[k - (2k - K)v]h-l,

SB'YYcb('Yc + 2ikv)/4

[Yab - i(b - kv)](y0 c + 2ikv) +SBYYcb4

b X Ya +. ikv'Yb yac + 2ikv

+ QBQF(O)p.b _ 1 + SB~cb12

+ Yb (- bI +1 + + _)-i(b + kv))Yb I (

*iQFQBP[()Pbb- (0)Paa](yac + 2ikv)

8 {[Yab - i(6 - kU)][Yac + 2ikv] + 4Jb

XY1 Y+1i-

|^Yb 7,ac + 2ikuSBYbc

27b [Ybc(1 + 2) -

+ 2ikv) + 'Yab + i(b'Yb-kv)) (Yab2 + (6 - kv)2 - 'YYabSF/4

[Ybc + i(b + kv)](Yac + 2ikv) + YYabSF4

{[Ybc + i(6 + kv)](Yac + 2ikv) + 'YYbSFJ [Y^ab2(1 + SF) + (5 - kv)2 ]2

+ nbc(Yac + 2ikv)|[-Ybc + i (6 + kv)] (7,,c + 2ikv) + 'YabF2

Case (2): Backward Pump of Arbitrary IntensityHere the forward-pump intensity is assumed to be so weakthat &pc has a linear dependence as a function of QF. In suchan approximation, the linear system to be solved for obtaining

the formal dependence of 8cb (0) on (O)p is reduced to a systemof seven equations. Indeed, in the system of Eqs. (13), theequations giving Ilab(2) and Iac(3) are useless. A differencefrom case (1) is that no population terms in (l)p are negligible:

tb(-') 7• 0 and a,(-') $ 0 because of the strong e.m. couplinginduced by the B pump between I b) and Ic ), and aa ('1) X 0because of interactions involving simultaneously the weakfields, F and P. One gets finally, in the grazing-incidenceapproximation (K = k),

Ybc (1+ 2)-'i(b + kv)Ocb(O)(V) = 2

-Yb,2(l + SB) + (5 + kv) 2

X iQB 2 Qp (°)Pac + QB QF( 0)Pcb

4 Ky (1 4(Yac + 2ikv)+

SBYbc27b['Ybc(1 +-) -i(b + kv)]Jj

X nba (Yac

11-i(b + kv)])

(18)

(17)

X _1+


The solutions of (O)7 are analogous to the ones given in case

(1) (Eqs. 14) with respect to the interchange of subscripts aand c and to the sign reversal of k [-y is then defined asy-1 =

(fYca1 + 'Yb)/2]. It should be pointed out that, unlike in case

(1), (N)Pac needs to be calculated according to Eq. (18). The

final expression of epc is then simply deduced by elementary

substitutions [combination of Eqs. (11) and (18)].

4. VELOCITY INTEGRATION IN THEDOPPLER LIMIT

Velocity integration is performed in the limit of large Doppler

broadenings (Doppler-limit approximation), in which theDoppler linewidth ku is assumed to be very large compared

to both the natural linewidth ( yi, yij) and the Rabi frequencyof the saturating pump. The first condition is generally sat-isfied in low-pressure gases, but the second one sets an upper

limit on the intensity of the saturating pump, QF, QB << ku.In the Doppler limit, the Maxwell-Boltzmann factorexp(-v2 /u2 ) can be assumed constant over the range of inte-gration of t3 b(0)(v) if this function decreases at least like v-2for large v's.28,29 The velocity integration of I3cb(0) may thus

be performed by closing the integration contour at v = + - in

the appropriate part of the complex plane (upper-half orlower-half plane) and using the residue theorem. 28 The in-

tegration consists in locating the poles of O3cb(°)

In both case (1) and case (2) the denominators of O3 cb(0) are

almost entirely factorized.

Case (1): Forward SaturationThe denominator of 3Cb(0 )(v) contains the factor

['Ybc + i(6 + kV)](,yac + 2ikv) + SF7ab.

For SF = 0, its two poles are located in the complex upper

half-plane. It can be demonstrated by continuity argumentsthat it remains so whatever the value of the positive saturation

parameter SF is. The remaining denominator is

[Yab2(j + SF) + (6 - kv)2]-2.

The integration is then performed by closing the contour inthe lower half-plane, where only one (double) pole is located

at

-iab- 1+SFk

The integration is thus straightforward:

\/rQF ~B Of(0Cb() ) -= ba

(~~~ku 8 I b( 1 1 1SFX 2- 1 +SF 3

f42 + SFYflab 2 1±l Sp 4(1±S)'4

+YabSF f 2(2 +S) 1 SF

Yb(l + SF) 72b(1 + SF)3/2 (2 + SFYYab)2

tl2+ SF (2l

X fY-Yab( - 1 S) + 'Yab(l - 12f + f2)

(8/f +S 4(1 + S)

+f2 Y + -Yab(2f1 + f2)|1T2_ [ 4S 2 b (1 + SF)1II

(19)

where

Afl = Ycb + y b+S + 2ib,

f2 = 'Yac + 2Yab/SF + 2ib. (20)

Case (2): Backward SaturationThe integration of the terms proportional to (°)Pcb and [(°)Pbb

- (0) Paa] is similar in principle to the integration of case (1).The contour is closed in the lower half-plane, leading to a pole

at-5 iYcb\I 1±SB

V =

k

The denominators of the terms proportional to (°)Pab and (°0 )ac

in Eq. (18) have the following factor:

[Vab + i (6 - kv)](^yac - 2ikv) + -

Here it can be shown that the two poles associated with sucha denominator always lie in the lower half-plane. By inte-

gration in the upper half-plane, the only pole to be consideredis located at

-6 + iYcb +SBk

One may notice that denominators such as

[Ycb(1 + -B i(b + kvj1_

do not contribute any pole because of the presence of similar

factors in the numerators.



The same argument applies to denominators such as (7yc± 2iku)-1 and [Tlab + i(6 - k)]-'.

One finally gets

O} iQFQBQP V f( 3 b b 0 ) = P

8kF *F2* + -SBYYCb+

X [SBTTCbtrb(1 - vTSTB)2 +(l-YSFg]- Yb 2(1-VfTDF 1*X Bbc- -S -~b[ -VI-SW* b ~ B'l' 8Yb2_ycb(1 + SB)312

±y(l1 1- SB) Fl*(4 + SB-4Vi7W+B) + YSB 187b (1 + SB) 8y-b(I + SB) 8zb 2 (1 + SB) [F2 + Yb + 2-Ycb(-1)]1

--S [L 1 (+ 1) + ])+S1 [ TSB [2 -C 2 TcbSB 2(1 - N/m B)FI+ - bc 3 F- - +IFF 2 +SBT7Cb [ cb(T + SB bYb Tyb T

4

+ SB v 2[F2 + 2,ycbN'SB( + 1)] + -+ N-1+ S+ nba SB -( + SBT1 F1T6 2 f 7 TJ 2/ l+Slj

+flb B 2[1 (1+f1 +SB)]

+ nbc SBTTcb (2F1* + F2*) [(J V-1 ypjrg)2Y + (1 Vr- 1 + sBy 2*]~F,*F2 + SB 1 TYcb)2[ 8'Yb2 (1 + SB)

+ SBP (T7cb - 8F1 *2) + nbc32(1 + SB) (YIF2 + SBTTcb)2

< f TSB(2F1 + F2) [27ycbJNi7+F2 2 YcbSB + 2(1 -F1SFl16(1 + SB) I 'Yb2

Tb Y

where the quantities F1 and F2 are the equivalent of fI and f2[Eqs. (20)1:

F, = TYab + lTYbv SB + 2i3,

F2 = TYac + 2,ycbi/fLB + 2ia. (22)

In the above formulas, two limiting cases can be considered:For low intensity, in both configurations (1) and (2), one easilyfinds a Doppler-free Lorentzian line shape, which is predictedby the third-order calculation.' 0 For very high intensities (SF

X or SB - a), a noteworthy effect (which is discussed inmore detail in Section 5) is the striking difference betweenbackward and forward saturation behaviors. With increasingbackward-pump intensities (QF and Qp being kept un-changed), the PC intensity grows toward an upper (nonzero)limit. On the other hand, when the forward-pump intensityalone is increased, the PC intensity, after reaching a maximumvalue, decreases and tends toward zero (the PC field ampli-tude is proportional to SF-1/ 2 at saturation).

At this point, another general feature of these results canbe outlined. Nearly every term of 0,b,(0K)( [Eqs. (17) and(18)] yields a sizable contribution after velocity integration,in opposition to the results of the third-order perturbationcalculation. Let us recall that, at third order, the only effi-cient mechanism has the following ordering: F-pump inter-

action, then probe interaction for creation of a large spacinggrating, and finally interaction with the B pump to read outthe grating. Mathematically, in the nonperturbative formulas[Eqs. (17) and (18)], the denominators to be integrated alwaysown poles on both sides of the real axis, whereas at third orderevery contribution, except one, has all its poles in the samehalf-plane, and they vanish after integration. This shows thatthe physical picture of a unique grating contribution is not welladapted for saturating pumps.3 0 To get an idea of the pos-sible ordering for these additional high-order contributions,it is appropriate to turn to a fifth-order perturbationtheory. 31

5. EMISSION FEATURES IN A SINGLE-RELAXATION-TIME MODEL

We consider here the case when the three-level system isgoverned by a single relaxation rate:

Tij = Tj = Y. (23)

This condition, which might describe a number of cases,such as that of molecular gases when collisional relaxationdominates, yields simplified formulas for the nonlinear sus-ceptibility and the emission line shape.32 In the following,we introduce the saturated nonlinear susceptibility, XNL(%D),

(21)



SB= 0.1

Fig. 3. Theoretical absorption and dispersion line shapes for variousbackward saturation parameters. (Relaxations are described by asingle parameter, y.) The quantities represented on the vertical axis

are Re(V1HXNL) and Im(VHHXNL), which are proportional to thePC e.m. field if the probe and forward pump intensities are keptconstant. The same (arbitrary) units apply to all vertical scales.

SB =100

In the case of backward saturation, condition (23) does notintroduce important simplifications, so we will not write thegeneral expression of XNL, which can be deduced from Eq.(21).

The nonlinear dispersion line shape (Re XNL), the nonlinearabsorption line shape (Im XNL), and the PC reflectivity(SFSB I XNL 12) may easily be computed from the expressionsof XNL- Such calculated line shapes are presented in Figs. 3-6for different values of the saturation parameter (either SF orSB)-

SB=1000

(x 10)

t1o

Fig. 4. Amplitude of the PC field versus frequency detuning for highbackward saturation. (A) and (D) are the absorption component[Im\/ XNL)] and the dispersion component [Re V~HXNL)], re-spectively. Note the sign reversal of the absorption and the changein vertical scale between (A) and (D).

pt = F, B, as a direct generalization for a nonperturbativetreatment of third-order susceptibility X(3)- It must satisfyXNL(&, -1 0) = X(3) and is defined by

An = 4 SEF =20

(x 2) tx 4)

Fig. 5. Absorption and dispersion line shapes in the case of forwardsaturation. (Conditions similar to the ones of Fig. 3.) Note thechange in the vertical scale for SF = 4, 20.

(. 44)

(x 25)

S=20

PPC = - XNL(61)6F6B6P-4

This, in turn, yields for the reradiated field [cf. Eq. (4)]

OPC = -i8 XNL&FOBGP8

and, for the PC reflectivity,

Rpc = |&PC2 = Ih 2 XNL12SFSB.lop tabltbc

(24)

(25)

(26)

Let us recall that S. = (Q,/-y)2 If one assumes that nba = nbc

= N and condition (23), an intense forward pump producesa saturated susceptibility, such as

2(3)(0) 2+ SF -y/ + iL5XNL =YX 2(1 + SF)312 (7Y2 + j5)2 '(27)

where X(3)(0) is the third-order susceptibility on line

center,

B FFig. 6. Emission-intensity line shape (SjXNLI2 versus b) for variousbackward (B) and forward (F) saturation parameters.

X(3>(0) = i Njab j 21 A bc12 I\/X3 ( = Eo'2h3 ku

and yj and Y2 are power-broadened linewidths:

ly 6, l+SF\4Y1 = 21 + 2S

Y2 = 4(1 + 3/ SF).

SBV1

loT

(28)

(29)

(30)

220

S=2

s=1

S= .1


SF = I �

644 J. Opt. Soc. Am./Vol. 73, No. 5/May 1983 D. Bloch and M. Ducloy

Let us first discuss case (2) (backward saturation). Figures3, 4, and 6 show that the origin of the intensity line-shapesplitting lies in the dispersive component of XNL, which be-comes considerably more important than the absorptivecomponent for increasing SB. One shows that, at saturation(SB >> 1), XNL tends toward [cf. Eqs. (11) and (21)-(25)]

XNL 7QB + 8iM 7QB - 8iYX(3)(0) (

3 QB + 4i6)2 (3QB - 4i6)2

9QB2- 24B 6 - 6462(3QB -4i6)4 * (31)

The dispersion linewidth, which yields the frequencysplitting of the two reflectivity peaks, is proportional to thesaturating field amplitude and is equal to 3QB/2 for QB >> -.It is interesting to note that, for a saturation parameter SB =1, the dispersion amplitude is already nearly twice the ab-sorption amplitude and that the central dip on the intensityline shape appears for SB 0.8. With respect to the non-linear absorption, for large values of QB its sign reverses (forQB 9,y, at line center), its amplitude reaches a small as-ymptotic value, and its line shape exhibits a peculiar finestructure (see Fig. 4).33

The major result of this analysis is to demonstrate on a firmtheoretical basis the dispersive character of the intensity lineshape. The existence of an asymptotic value of the PC re-flectivity when only the B-pump intensity is increased (sinceXNL c SB-1/

2 for SB >> 1) can readily be understood. In aDFWM process, the same number of photons is absorbed frompump B and pump F, to be transferred to the probe and con-jugate beams. The limitation in the PC beam intensity is thusdue to the limited number of photons available from pumpF, which is kept at a constant intensity level.

Let us come now to the case of forward saturation. Theconclusions that we derive from Eq. (27), in the case of a singlerelaxation time, can actually be extended to a more generalclass of relaxation processes. Indeed, in the absence of de-phasing collisions [yjj = (,yi + yj)/2], and for ya = Yc, XNLkeeps a form similar to Eq. (27), with yl and 72 defined by(with SF = QF2

/Ya -Yb)

'Ya+y Y+ 2Yb \/TTSF 324 2 2+SF (32)

4 y 4 (3 3 )4 4

Such a model should be valid for a degenerate Zeemantransition governed by a purely radiative relaxation. Equa-tion (27) leads to the computed line shapes of Figs. 5 and 6.

As can be seen directly from Eq. (27), the major effect of theforward saturation is to create substructures in the dispersioncomponent as well as in the absorption component. 34 Thesestructures (which may be understood as coming from high-order couplings between absorption and dispersion) appearfor y2 = 1.5-yl in the absorption (SF = 1.37, i.e., QF = 1.17,yin a single-relaxation model) and for _Y2 = 2,y, in the dispersion(SF = 2.82 or QF = 1.68-y). This leads to a central dip in theintensity line shapes as soon as SF = 1.12 or QF = 1.06y (72> 'Yi \/2). A remarkable feature of forward saturation is thatthis intensity dip is no longer of dispersive origin but is im-posed by the complicated structure of the absorption com-ponent, which exhibits a minimum at line center. The dis-persion-to-absorption amplitude ratio does not exceed 1.1,

contrary to what happens in backward saturation. Anotherconsequence of the appearance of a central dip in the ab-sorption is that, for weak saturations (before the appearanceof the dip, typically SF S 1), the absorption line shape is muchbroader than the one associated with an equivalent backwardsaturation. 35 This feature, which is easily seen on the com-puted curves, has been observed experimentally. 1 5

In the strong-saturation regime (SF >> 1), the nonlinearsusceptibility can be approximated by [see Eqs. (27)-(30)]

XNL 2,y 4i6

-xy3X(3) QF (3QF + 4i6)2

and the PC reflectivity is proportional to

IXNLI2QF 2 4y41X(3 )(0)1 2 + 1662)2

(34)

(35)

Note that the frequency splitting between the two maximaof the saturated reflectivity is equal to 3 QF/2 (the law is similarto the one found for backward saturation).

However, it should be noted that this frequency splittingis characteristic of a single-relaxation-time model. For in-stance, in a purely radiative relaxation model [Eqs. (32) and(33)], the frequency splitting of the two reflectivity peaksbecomes equal to

,y. + 27b - Ya + 2Ybg2 20/ (36)

This feature underlines the limits of a single-relaxationmodel. A second point of considerable importance (alreadymentioned in Section 4) is the behavior of the reflected PCintensity, which decreases like SF-1 , leading therefore to anabsolute maximum of the reflectivity in the case of forwardsaturation (this maximum is reached for SF - 0.4). A com-parison of the peak reflectivity obtained in both forward andbackward saturation configurations is presented in Fig. 7.

A physical interpretation of the striking difference betweenthe two saturation geometries can be found in the character-istic features of saturated absorption in Doppler-broadenedsystems: One can show36 that, when an intense traveling waveand a weak wave counterpropagate inside a resonant-gasmedium, the absorption coefficients of both beams are re-duced by the saturation of the medium while only the re-fractive index of the weak counterpropagating beam experi-ences saturation; the index of the strong beam remains unal-tered. This feature is specifically produced by the atomicmotion. 36 By analogy, the PC field sees its dispersion con-tribution strongly affected by the saturation of the gas me-

rPC REFLECTIVITY

RCm) /R

PCd *auat-

/k.,

II -

10 S

Fig. 7. Peak reflectivity versus saturation parameter: comparisonbetween forward and backward saturation. The peak reflectivityRpc(m) is defined as the maximum value in the curves of Fig. 6. Ro isthe asymptotic value reached by Rpc(m) for infinite backward satu-ration (cf. Ref. 37).

| | -

0.50J

0.2s

l


dium when this is induced by the (counterpropagating) for-ward pump, in contrast with the less important saturation of

the dispersion produced by the (nearly copropagating)backward pump. On the other hand, the absorption contri-butions are strongly reduced by saturation in both cases. This

yields a qualitative interpretation of the large reflectivityobtained for backward saturation and its dispersive origin. It

should be noted that this behavior is not specific to a three-

level system: The dispersive character of the PC intensityline shape for backward saturation can also be demonstrated

for a two-level system, at least in the framework of a fifth-

order perturbation theory.31 Another approach to thesesaturation features is to note that, as the B pump saturatesthe atomic system more and more strongly on line center, it

explores the atomic response further in the line wings, where

the dispersive character is predominant. With forward sat-uration, the efficiency of interaction between the probe and

the gas medium is strongly reduced by the considerable de-crease of the a - b population difference induced by an in-

tense F pump. This saturation effect may be described as a

vanishing contrast of the grating produced by pump F andprobe.

5. CONCLUSION

In this paper, we have theoretically analyzed the origin of some

PC intensity line shapes. All the most interesting featuresthat we pointed out here are typical effects of Doppler inho-

mogeneous broadening, and they demonstrate the lack of

consistency of all the theories of PC emission in a gas medium

that do not deal rigorously with the thermal motion. Among

all the predicted effects, the directional anisotropy of thesaturation is entirely due to the thermal motion and impliessignificant unexpected behavior that we summarize here:

The intensity line shape is of dispersive origin only in the case

of backward saturation. For forward saturation, the PC re-flectivity has a maximum at a rather weak pump intensity and

is much smaller than the reflectivity obtained for backward

saturation. It can then be suggested that the optimizationof the PC reflectivity, which is of considerable importance in

building PC resonators, could be achieved by the choice of an

adequate ratio between pump-F and pump-B intensities.This will require more-complete calculation for confirmation.

We are performing calculations for a three-level system in a

case of two intense pump beams (the main difference between

that study and the present paper is that here the velocity in-tegration has to be performed numerically), and we also intend

to consider the case of a two-level system in which the standing

pump wave is intense, implying creation of spatial holeburning.

This research was supported in part by Direction desRecherches, Etudes et Techniques, Paris. The Laboratoirede Physique des Lasers is associated with the Centre Nationalde la Recherche Scientifique.

REFERENCES

1. Opt. Eng. 21, 155-283 (1982) (special issue on phase-conjuga-tion).

2. M. Ducloy, "Nonlinear optical phase-conjugation," in Fest-

khrperprobleme-Advances in Solid State Physics (Viegweg,Braunschweig, 1982), Vol. XXII, pp. 35-60.

3. Ed. R. A. Fisher, Nonlinear Optical Phase-Conjugation (Aca-demic, New York, 1982).

4. B. I. Stepanov, E. V. Ivakin, and A. S. Rubanov, "Recordingtwo-dimensional and three-dimensional dynamic holograms intransparent substances," Sov. Phys. Dokl. 16, 46-48 (1971).

5. J. P. Woerdman, "Formation of a transient free carrier hologramin Si," Opt. Commun. 2, 212-214 (1970).

6. R. W. Hellwarth, "Generation of time-reversed wavefronts bynonlinear refraction," J. Opt. Soc. Am. 67, 1-3 (1977).

7. D. M. Bloom, P. F. Liao, and N. P. Economou, "Observation ofamplified reflection by degenerate four-wave mixing in atomicsodium vapor," Opt. Lett. 2, 158-160 (1978).

8. D. G. Steel and R. C. Lind, "Multiresonant behavior in nearlydegenerate four-wave mixing: the ac Stark effect," Opt. Lett.6, 587-589 (1981).

9. S. M. Wandzura, "Effects of atomic motion on wave-front con-jugation by resonantly enhanced degenerate four-wave mixing,"Opt. Lett. 4, 208-210 (1979).

10. M. Ducloy and D. Bloch, "Theory of degenerate four-wave mixingin resonant Doppler-broadened systems. I. Angular dependenceof intensity and lineshape of phase-conjugate emission," J. Phys.(Paris) 42, 711-721 (1981).

11. P. F. Liao, D. M. Bloom, and N. P. Economou, "Cw opticalwave-front conjugation by saturated absorption in atomic sodiumvapor," Appl. Phys. Lett. 32, 813-815 (1978).

12. D. G. Steel, R. C. Lind, J. F. Lam, and C. R. Giuliano, "Polariza-tion-rotation and thermal-motion studies via resonant degeneratefour-wave mixing," Appl. Phys. Lett. 35, 376-379 (1979).

13. L. M. Humphrey, J. P. Gordon, and P. F. Liao, "Angular depen-dence of line shape and strength of degenerate four-wave mixingin a Doppler-broadened system with optical pumping," Opt. Lett.5, 56-58 (1980).

14. J. P. Woerdman and M. F. H. Schuurmans, "Effect of saturationon the spectrum of degenerate four-wave mixing in atomic sodiumvapor," Opt. Lett. 6, 239-241 (1981).

15. D. Bloch, R. K. Raj, K. S. Peng, and M. Ducloy, "Dispersivecharacter and directional anisotropy of saturated susceptibilitiesin resonant backward four-wave mixing," Phys. Rev. Lett. 49,719-722 (1982).

16. R. L. Abrams and R. C. Lind, "Degenerate four-wave mixing inabsorbing media," Opt. Lett. 2, 94-96 (1978); errata 3, 205 (1978).In the optically thin sample approximation, the emission intensityline shape is proportional to I K 1 2, in the notation of Abrams andLind.

17. D. J. Harter and R. W. Boyd, in "Nearly degenerate four-wavemixing enhanced by the ac Stark effect," IEEE J. QuantumElectron. QE 16, 1126-1131 (1980)], have actually shown that,in nearly degenerate four-wave mixing, when the pump-probefrequency detuning is varied, the ac Stark splitting is responsiblefor the appearance of structures in the emission line shape. Theirtheory, which does not take into account the atomic motion,cannot apply to DFWM experiments in which the common fre-quency of pump and probe beams is scanned through the atomicresonance (see also Ref. 8).

18. D. Bloch, R. K. Raj, J. J. Snyder, and M. Ducloy, "Heterodynedetection of phase-conjugate emission in an Ar discharge witha low-power c.w. laser," J. Phys. Lett. (Paris) 42, L31-L34(1981).

19. A coupled three-level system is particularly adequate to describea J = 1 J = 0 transition with cross-polarized pumps.' 5

20. In all the calculation, one neglects the cascade effects induced byspontaneous emission among I b), I a), and I c) so that A-type andV-type three-level systems are equivalent.

21. For sake of simplicity, the choice of the space and time origins issuch that the phase term is eliminated in the definition of theincident fields: One can easily verify that the phase of the probefield is sign reversed in the component that we are calculating.

22. In four-wave mixing, the emission of one photon in the PC fieldoccurs simultaneously with the emission of one photon in theprobe field, at the expense of absorption of one photon of eachpump field. A too-high increase in probe intensity does not helpone to increase the reemitted intensity, so the PC reflectivity Rpc(as an efficiency rate) decreases.

23. S. Stenholm and W. E. Lamb, Jr., "Semiclassical theory of a


646 J. Opt. Soc. Am./Vol. 73, No. 5/May 1983 D. Bloch and M. Ducloy

high-intensity laser," Phys. Rev. 181, 618-635 (1969); B. J.Feldman and M. S. Feld, "Theory of a high-intensity gas laser,"Phys. Rev. A 1, 1375-1396 (1970).

24. Actually, the formalism is not specific to the set of polarizationsdiscussed at the beginning of Section 2. It can describe a two-level system if the polarization of EB is identical with the one ofEF and Ep (e.m. field coupling between Ia) and Ib) only; nocoupling between lc) and I b)). The quantity of interest becomes(/

3ab(°)) instead of (6cb(°)), and minor and obvious modifications

must be done to Eqs. (7)-(13).25. The backward pump is too weak to modify the population in level

Ic): acc(-') = 0, and the equationsyieldingacb( 2 ) and aac(3) aredecoupled.

26. D. Bloch and M. Ducloy, "Polarization selection rules and diso-rienting collision effects in resonant degenerate four-wave mix-ing," J. Phys. B 14, L-471-L476 (1981).

27. M. Ducloy, R. K. Raj, and D. Bloch, "Polarization characteristicsof phase-conjugate mirrors obtained by resonant degeneratefour-wave mixing," Opt. Lett. 7, 60-62 (1982).

28. J. R. R. Leite, R. L. Sheffield, M. Ducloy, R. D. Sharma, and M.S. Feld, "Theory of coherent three-level beats," Phys. Rev. A 14,1151-1168 (1976).

29. D. Bloch, "Conjugaison de phase dans les milieux gazeux.Spectroscopie de saturation heterodyne," These de TroisiemeCycle (Universit6 Paris-Nord, Paris, 1980).

30. J. H. Shirley, "Modulation transfer processes in optical hetero-dyne saturation spectroscopy," Opt. Lett. 7, 537-539 (1982).

31. D. Bloch and M. Ducloy, unpublished report (Universit6 Paris-Nord, Paris, 1983).

32. In the case of weak saturation, an approximate justification forthe single-lifetime model can be found if 0 is small but not neg-ligible (O >> y/ku, 0 . dyij/ku). The effective grating lifetime isessentially shortened by the thermal motion and is almost inde-pendent of the relaxation processes (see Ref. 27). For strong

saturation, the situation is complicated by the homogeneousbroadening of the relaxation processes.

33. These structures are predicted only for a high-saturation pa-rameter, so that the conditions for infinite Doppler-width ap-proximation may no longer be fulfilled. This could explain whywe were unable to observe these types of structure experimentally,whereas all the other line-shape features calculated in the presentpaper have been observed experimentally.15

34. These narrow structures have been observed in recent experi-ments in neon (see Ref. 15). It is also probable that the peculiarline shapes observed in the saturated regime of polarizationspectroscopy [H. H. Ritze, V. Stert, and E. Meisel, "High reso-lution polarization spectroscopy in the strong saturation regime,"Opt. Commun. 29, 51-56 (1979)] are somehow related to thestructures observed here.

35. For SF = 1, the width of the absorption line shape is about 2.5times the width at SF = 0 (natural width).

36. B. Couillaud and A. Ducasse, "Les lasers a colorant continus.Leur application a la spectroscopie d'absorption saturee," Thesed'Etat (Universit6 de Bordeaux I, Bordeaux, France, 1978).

37. Let us give a numerical estimation of R 0 , the limit value of thereflectivity for infinite backward saturation. From Eq. (31), onefinds that, at saturation, the maximum value of IXNL/X(3)(0)12is 0.7 SB-1 for 6 = 3QB/4. Let us consider transitions havingoscillator strengths equal to unity and wavelength X = 0.6 um,an atomic density of 10"' atoms/cm 3 (corresponding to p = 3.10-6Torr at 300 K), an interaction length L = 1 cm, and a meanthermal velocity u = 500 m/sec. With these values, one deducefrom Eqs. (26) and (28) that RO/SF = 0.32.

38. A similar saturation anisotropy was recently observed in SF6 [G.P. Agrawal, A. Van Lerberghe, P. Aubourg, and J. L. Boulnois,"Saturation splitting in the spectrum of resonant degeneratefour-wave mixing," Opt. Lett. 7, 540-542 (1982)].

Documents

Theory of saturated line shapes in phase-conjugate emission by resonant degenerate four-wave mixing in Doppler-broadened three-level systems