Theory of degenerate four-wave mixing in resonant Doppler-broadened systems - I. Angular dependence of intensity and lineshape of phase-conjugate emission

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Theory of degenerate four-wave mixing in resonant Doppler-broadened systems - I. Angular dependence of

intensity and lineshape of phase-conjugate emission

M. Ducloy, D. Bloch

To cite this version:

M. Ducloy, D. Bloch. Theory of degenerate four-wave mixing in resonant Doppler-broadened systems

- I. Angular dependence of intensity and lineshape of phase-conjugate emission. Journal de Physique,

1981, 42 (5), pp.711-721. �10.1051/jphys:01981004205071100�. �jpa-00209057�

Theory ^of degenerate ^four-wave mixing

in resonant Doppler-broadened systems

I. Angular dependence ^of intensity

and lineshape ^of phase-conjugate ^{emission (*)}

M. Ducloy and D. Bloch

Laboratoire de Physique des Lasers (**), Université Paris-Nord, ^F-93430 Villetaneuse, ^France (Rep ^{le 24 novembre} 1980, accepte ^{le 21} janvier 1981)

Résumé.

On présente ^{dans cet article une étude} théorique ^du phénomène d’émission conjuguée par mélange

à quatre ^ondes dégénéré ^{dans un} milieu gazeux résonnant. La polarisation ^non linéaire induite est calculée _par

un développement de la matrice densité atomique ^au troisième ordre en fonction des champs incidents. On trouve que, lorsque ^les fréquences optiques ^sont résonnantes pour ^une transition à un photon, ^{forme de raie et intensité}

de l’émission conjuguée dépendent fortement de l’angle 03B8 ^entre ondes pompe ^et sonde. L’intensité, ^{maximale à}

incidence rasante, ^diminue rapidement ^par plusieurs ^{ordres de} grandeurs quand 03B8 augmente. Simultanément la

largeur ^{de raie} augmente depuis ^la largeur homogène jusqu’à ^la largeur Doppler. ^Des effets semblables sont prédits

dans le cas quasi dégénéré (fréquences pompe ^et sonde différentes). ^En particulier, ^{on montre} que la largeur ^du

filtre optique équivalent ^{subit un} élargissement Doppler proportionnel ^{à sin 03B8. Par contre, aucun} des effets précé-

dents n’est prédit ^{dans le cas} d’une résonance à deux photons pour laquelle l’émission conjuguée ^ne présente

pas de dépendance angulaire.

Abstract.

We present in this paper ^a theoretical study ^{of the} angular dependence ^of phase-conjugate ^emission

via resonant degenerate ^four-wave mixing (FWM) ⁱⁿ Doppler-broadened gas media. The nonlinear optical pola-

rization is calculated through ^an expansion of the atomic density ^matrix up ^to the third order in the incident fields.

One finds that, when the incident frequency ^{is resonant for a} single-photon transition, ^both strength ^and lineshape

of the phase-conjugate ^emission strongly depend ^{on the} angle ^{0 between} standing pump ^wave and probe ^wave.

The emission intensity, ^{maximum at} grazing incidences, ^decreases rapidly by several orders of magnitude ^{when 0}

increases. Simultaneously ^the emission linewidth increases from homogeneous ^to Doppler width. Similar effects

are predicted ^for nearly-degenerate ^FWM (different ^{pump and} probe frequencies). ^In particular, ^{one shows that}

the bandwidth of the equivalent optical reflection filter undergoes ^a Doppler-broadening linearly increasing ^with

sin 03B8. On the opposite, ^none of the above effects is predicted ⁱⁿ two-photon-resonant ^FWM which does not exhibit any angular dependence.

Classification

Physics ^Abstracts

32.80K - 42.50

42.65B

1. Introduction.

Degenerate ^Four-Wave Mixing (DFWM) ^has recently ^received considerable attention

owing ^{to its} ability ^of generating phase-conjugate (time-reversed) optical ^waves [1, 2]. ^{A nonlinear}

medium irradiated by ^a standing ^wave (« pump »)

can generate ^a counter-propagating phase-conjugate replica ^{of a «} probe ^{» beam of same} frequency ^and arbitrary direction. Because phase-matching ^condi-

tions are always fulfilled, ^this process may be used to

replicate wavefronts of arbitrary complexity. ^The phase-conjugate (PC) ^{nature of the reflected wave is} important ⁱⁿ phase distortion corrections [3].

(*) ^Work supported ⁱⁿ part by ^D.R.E.T. (Paris).

Associ6 ^au C.N.R.S., ^LA.282.

DFWM is generally ^{carried out with dense} (solid ^or liquid) ^{nonlinear media,} but it has been shown that DFWM can also be performed ⁱⁿ low-pressure gases, when the nonlinear index is resonantly ^enhanced by tuning the incident frequency within either one-

photon [4-6] ^or two-photon [7] transitions. In these cases, owing ^{to the} standing-wave character of the pump beam, ^the Doppler ^{effect is} partially ^or totally

cancelled. In an experimental study performed ^{on the}

Na resonance line, Liao et al. [5, 6] ^have demonstrated that DFWM with c.w. single-frequency lasers leads to the observation of Doppler-free lineshapes.

As shown in recent experiments [8-10], ^{a sensitive}

way of analysing PC emission consists in using ^a multifrequency ^incident irradiation (« nearly-dege-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01981004205071100

nerate four-wave mixing») ^and detecting ^{the re-’}

emitted electromagnetic (e.m.) ^field by ^its heterodyne beating ^{with one of the} input e.m: fields. Two schemes may be considered :

(i) In the first one (Fig. la) ^both pump and probe

are standing ^{waves, and the} probe frequency, ^{w + 6,}

is detuned from the _pump frequency, ^{co. Thus a PC}

wave I+ re-emitted opposite ^{to either} probe ^{at fre-}

quency ^{OJ -} 6, ^and interferes with the return probe

wave (serving ^{as local} oscillator) ^to yield ^{a beat at} frequency ^{2 6.} Experiments ^{of this} type ^{have been} performed ^{in Ar} discharges [10].

(ii) An alternative scheme consists in overlapping

both running probe (co ⁺ 6) ^and standing pump (OJ)

waves (Fig. lb). ^{Two PC waves are thus} re-emitted at

frequencies ^{o) + 6 and m -} 6, ^{and interfere with the}

return pump ^{wave to} yield ^{a beat} signal ^{at fre-}

quency 6 [8]. Advantages ^{of these} techniques ^{lie in the}

increased sensitivity ^of heterodyne detection and the

possibility ^of reaching ^the signal-shot-noise ^limit by going ^out of the laser noise spectrum ^for large ^b’s [9].,

Fig. ^1.

Schemes for nearly-degenerate ^four-wave mixing. ^In configuration’ (a), ^{four PC} waves are re-emitted [two ^{waves at} frequency ^{w -} 6, opposite ^{to either} (w ⁺ 6) wave, and two waves at frequency ^{w + 2} 6, opposite ^{to either} (co) wave] ^but only ^one

wave is represented ^{in the} figure.

In this series of articles, ^we analyse in detail the

theory of PC emission via resonant DFWM in

Doppler-broadened gas media. The re-emitted e.m.

fields are calculated through ^a third-order pertur- bation expansion in the incident fields. Subsequently

we are not able to take into account optical pumping [6]

or saturation effects [5, 11]. Such saturation pheno.

mena will be considered in later papers.

This first article deals with ^the angular dependence

of the emission strength ^and lineshape ⁱⁿ Doppler-

broadened resonant two-level systems. Similarities and differences with a calculation performed by

Wandzura [12] ^are emphasized. ^{The effect of a} pump

probe frequency detuning ^is studied in conjunction

with proposals ^of using nearly-degenerate ^four-wave mixing (NDFWM) as a means of narrow-band wide-

angle optical filtering [14-16]. Finally two-photon

resonant PC emission is also analysed ^and compared

with single-photon ^resonance.

In a forthcoming paper, ^we shall analyse ^collinear

NDFWM with multifrequency ^incident irradiation and its utilization for Doppler-free heterodyne spec- troscopy ^{in two-} and three-level systems [8-9]. ^The

last article will deal with polarization and collisions effects in degenerate systems (heterodyne ^saturated-

absorption ^and saturated-polarization spectro-

scopy[17]).

2. Wave equation.

^One considers the usual DFWM configuration (Fig. 2) ^{in which a nonlinear}

gas medium is irradiated by ^a standing pump ^wave

(frequency ^o, wavevectors ± ko) ^{and a} probe ^wave

of frequency ^{Q = w + 6} and wavevector, k. 6 is taken to be small enough ^to preserve phase-matching :

Fig. ^2.

Configuration of the incident e.m. fields.

where L is the interaction length. Through ^DFWM

processes, ^a nonlinear macroscopic polarization ^is generated ^at frequency ^{OJr =} co - 6 and wavevector

k,, = - k :

This polarization re-radiates an e.m. field in direction -k

In the slowly-varying field-envelope approximation,

the wave propagation ^is governed by [18-19]

For an optically ^thin sample, ^the input ^{e.m. fields do,}

not vary appreciably along ^{the cell and s can be}

considered as r

independent. Then, ^{in the} steady

state, the re-radiated field amplitude ^{is :}

Obviously, ^this approach ^{cannot account for} strong- field propagation ^effects (saturation ^{of the} amplifi- cation, oscillation, pump depletion). ^To calculate the

macroscopic polarization, ^{we need to} specify ^the

atomic (or molecular) system ^{used as nonlinear} medium. In the next three sections, ^we analyse ^{in detail}

the case of a close resonant two-level system. ^This corresponds ^{to resonance} transitions in which the

global population ^is conserved. It has the interesting peculiarity ^of providing ^{a number of} relatively simple analytical ^results (1). The last section considers two-

photon ^resonant PC emission in three-level systems.

(1) ^{The more general case of an open} two-level system ^with

allowance for collisional relaxation is considered in Appendix.

3. Calculation of the induced polarization ^{in close}

two-level systems (resonance transition).

^We sup- pose that the incident fields are nearly ^{resonant for an}

atomic resonance transition, [ a ) (ground state) -+ [ b ) (resonance state), ^of frequency mo. The equa- tions of motion for the density matrix elements

[p;; _ lp(r,v,t)lj )] ^{can be written as :}

where p ^{is the} hydrodynamic (Boltzmann-type) ^time-

derivative of p

(v is the atomic velocity). ^The conservation of the total number of atoms, N, implies

In (6), gab is the a - b electric dipole ^{matrix element.}

y is the radiative decay ^rate of the level b and _yab is the relaxation rate of the optical dipole, pab. For

a purely ^radiative relaxation,

In presence of phase-changing collisions, _yab > y/2.

Finally, ^{the e.m. field E is the sum of three incident} plane ^waves (2)

The coupling strengths ^{are defined as}

They will be assumed real.

By solving equations (6) ^{with a} perturbation expand sion in the ^e.m. field and making ^{the rotating wave} approximation, ^one gets, for the third-order optical

coherence [18] :

in which

The induced nonlinear polarization ^is given by

in which ) ^means velocity-averaged. Equations (2)-(5) ^and (12)-(14) ^{show that the} frequency ^and

wavevector of the re-emitted field are given by

This e.m. field is radiated in direction - k only

when the following conditions ^are fulfilled

i.e.

The corresponding contribution to the third-order

optical ^{coherence is}

with

Thus the amplitude of the radiated PC field is

with

(We ^{assume a} Maxwell-Boltzmann velocity ^distri-

bution with r.m.s. thermal velocity, u.) ^The velocity- integration ^of 2;ab yields the main features of the (2) ^In the approximation ^{of an} optically ^thin sample (see ^Section 2X

the re-radiated fields (5) ^{are small} compared with the incident fields

(I &’ I , 6v) ^{and they will be} neglected ^{in the} equations ^{of motion}

phase-conjugate ^{wave :} angular dependence ^{of the}

field intensity, ^emission lineshape, optical frequency filtering, ^{etc. For} the sake of simplicity, ^{we consider}

the collisionless case [y ^{= 2 yab, equation} (9)] ⁱⁿ

which Lab has the form :

All the integrations will be carried out in the

fully Doppler-broadened regime :

4. Angular dependence ^of phase-conjugate ^emission

in resonant DFWM. - In this section, ^we analyse

the degenerate ^case (Q

^w

war) ^thus

with

4 .1 PHYSICAL DESCRIPTION OF THE ANGULAR DEPEN- DENCE.

As shown in the analysis ^{of the} previous section, ^the time-ordering ^{of field} interactions (t4 v, À)

needed to generate PC emission is of the form :

± ko, k, ip ko. ^{Such an} example ^of three-interaction processes is schematized in figure ^{3. This} diagram

shows that an important intermediary step ^{in the} nonlinear mixing ^{lies in the} spatial modulation of atomic populations ^induced by the interaction of the absorbing ^{medium with two incident waves} (ko, k), since the intensity of the associated optical

field is spatially ^{modulated at} frequency ko - ^{k :}

Fig. ^3.

Physical process induced in resonant four-wave mixing.

This leads us to interpret ^{DFWM in terms} of popu- lation grating [5, 20] : ^{for a resonant incident} field,

the spatial modulation of population produces ^an absorption modulation (or ^{an index} modulation, off-resonance) ^whose effect is similar to a three- dimensional grating; ^{the return} pump ^wave (- ko)

thus propagates along ^a Bragg diffraction direction and is reflected by ^the grating ⁱⁿ direction - k.

In a gas, the thermal motion tends to mix _up this

grating. This effect depends mainly ^on the relative

importance of the absorber mean free path, ,

compared ^{to the} grating spatial period, ^{d :}

0 is the angle (k, ko).

The grating ^{washes out} for those atoms of such

velocity ^that they ^{run over a} grating period during

their lifetime, ^i.e.

For the atoms travelling along ^the planes ^of equal population (v orthogonal ^{to k -} ko) ^{there is no}

attenuation of the grating ^{contrast. But for the} velocity component perpendicular ^{to these} planes, ^the only important contributions come from such velocity

group that k 2013 ko I v, ;: y. It follows that :

(i) ^{In a collinear} configuration (k II ko, ^{0 -} 0) ^the grating period becomes infinite and all atoms contri- bute.

(ii) ^When probe ^and pump beams are orthogonal (0 ⁼ n/2), ^d is of the order of the wavelength (d

A/,/2-), ^and the re-emission amplitude is decreased

by ^{a factor -} y/ku, because of condition kv_L ;!, ^y.

This effect does not depend ^on the incident frequency,

and then should _appear as a global attenuation

factor, independently of the emission lineshape.

A second factor affecting ^both amplitude ^and lineshape originates ^{in the} velocity ^selection by ^a

monochromatic wave, due to the Doppler effect. The pump beam interacts with velocity groups well defined in directions ± ko, ^{while the} probe selects the

velocity component along ^{k. For 0 -* 0, one tends to £}

« saturated-absorption-type ^» configuration. ^The

interaction is maximum on resonance (cD

wo) ^and

the emission amplitude changes rapidly ^{on a} frequency

scale of the order of _{y. For} 0 ⁼ n/2, ^the velocity

selection is carried out by pump and probe along ^two orthogonal directions : this absence of correlation in the two velocity groups implies ^{that the emission}

linewidth becomes of the order of the Doppler width, ku, ^{and thus the} peak amplitude ^{is further reduced} by

another factor y/ku.

In conclusion, these simple physical arguments predict that, ^{when 0 varies} between 0 and n/2, ^the

linewidth should increase from y ^to ku and the PC

field amplitude ^{on resonance} should decrease by ^a factor - (y/ku)2. ^These predictions ^{are confirmed} by

the complete calculations presented ^below.

4.2 PEAK AMPLITUDE «(0

wo).

^{For v}

0, 2:ib > ^can be written as [Eqs. (2l)-(24)]

where kZ ^{= k cos 0} and kx ^{= k sin 0 are} the compo-

nents of k respectively along ko (Oz axis) and the ortho-

gonal ^direction (Ox axis, ^see Fig. 4). ^{In the} Doppler limit, [Yab ku, Eq. (23)], ^the integration over v2 is easily performed by taking e - v -/u’ -- 1 and using ^the

residue method in the complex plane

Fig. ^4.

Wavevector arrangement [k1,2 = 2 ^{(k ±} ko)].

This integral may be expressed ^{with the} help ^{of the}

Error function [21] ^{and one} finally gets :

where

and [21]

---

An important physical parameter ^{is the} angle

This is the characteristic angle for which the atomic

mean free path [I, Eq. (26)] ^{and the} grating period [d, Eq. (27)] ^{are of the same order of} magnitude. ^{In the} Doppler limit, ^this angle ^{is small} (- ^{1 to 20} mrads.).

(i) ^If 0 >> 0o (large angular separation ^{between k} and ko), ^a

OoA2 tg 0/2) ¹ and Zab > ^{reduces to}

(ii) ^{If 0} 80 (quasi-collinear configuration),

ot Oo/O >> ^{1 and an} expansion off in ^{powers of a-1 I}

yields ^I

The amplitude ^{ratio between 0=0 and} n/2 ^is given by

One gets ^{the result} predicted by ^the previous ^intuitive approach. Figure ^{5 shows the} angular dependence ^of

the relative strength ^{of the PC emission :}

For 00

^10-2 rad, the emission intensity ^varies by

a factor - 10’ ! !

Fig. ^5.

Angular variations of the peak reflectivity (80

^0.01 rad.).

4.3 EMISSION LINESHAPE. - We consider three cases : 0

n/2, 0 Oo, Uo 0 ¹

a) ⁰⁼ n/2

The integration ^of Eab ^is easily performed

because of the separation ^{of the two variables vz and} vx, in (24). ^One easily gets, ^{in the} Doppler ^limit [18]

where

and g is the Dawson’s integral [21] :

Figure ⁶ shows the variations of the conjugate

wave intensity ^{with the} detuning. ^{The emission}

linewidth (full ^{width at half maximum,} FWHM)

is Am = 1.17 ku. As predicted ^{in section 4.1, the} lineshape ^is Doppler-broadened.

Fig. ^6.

^Emission lineshape ^for orthogonal wavevectors.

b) 0 1

For 0 :0 n/2, ^the velocity integration ^{cannot be} performed ^so easily. However, ^{for 0} 1, ^{it is} possible

to find approximate expressions. ^To integrate (24)

over velocity, one uses new reference axes defined by (Fig. 4) :

Then one has With

Then 2:.b ^{can be written as} [cf. Eq. (19)]

Integration ^{over the} velocity distribution is consi- dered in detail in reference [18] ^{for the case} e 1,

i-e-

It is shown that the first term of (45) provides ^the

dominant contribution in the integration ^over vl.

To the lowest order in Yab/k, t4 ^one gets

Let us consider two cases :

This means k2 U 7ab’ An expansion of the inte-

grand ^to the second order in k2 V2/Yab yields ^the following development ⁱⁿ 0/00

For 0 -+ 0, the emission lineshape is lorentzian anq Doppler-free. ^The linewidth is given by

Integral (47) ^{can be} expressed by ^{means of the}

function

which is related to the error function [21]

One can write equation (47) ^{as :}

Condition (51) ^means yab k2 U ^{ku. Then}

’where g ^{is defined} by (41) ^and

Figure ⁷ shows the variations of the conjugate ^wave

intensity ^{with v.} The emission lineshape ^{is now}

Doppler-broadened ^{with a linewidth (FWHM)} given

by ^{Am = 1.19 ku8. I}

Fig. ^7.

^Emission lineshape ^for 00 « 0 1.

Let us summarize the angular dependence ^{of the} linewidth, which is shown in figure ⁸

Fig. ^8.

Angular dependence of the emission linewidth

(00

^0.05 rad.) .

4.4 COMPARISON WITH PREVIOUS WORKS.

Two

important ^{results come out from our} calculations : (i) ^{For 0} >> 0o’ the re-radiated _power varies like

(sin 0)-’. Thus it is better to use grazing incidences in order to get high reflectivity ⁱⁿ phase-conjugation ^with absorbing gas media. Then the sharp 0-dependence sets, forth severe limitations when ^one wants to rectify strongly ^distorted images.

(ii) The emission lineshape ^is Doppler-free ^for 0 gg 00. A Doppler-broadening, approximately pro-

portional ^{to sin 0,} appears for 0 >> 00.

The angular dependence ^{of the} reflectivity ^{has been} previously predicted by ^Wandzura [12]. ^{He also} predicts ^the Doppler-free character of the 0 = 0

lineshape. ^{However, he used a} simple interpolating

formula to analyse ^the angular dependence ^{of the} lineshape. ^{On this} point ^his predictions ^{are in} complete

contradiction with our results [22-24].

Humphrey et ^al. [6] ^have performed ^an experi-

mental study ^{of the} angular dependence ^of phase-

conjugate ^{emission on the} D2 ^line of Na. A compa-.

rison of our predictions with their results is difficult because of the predominant character of the optical pumping ^induced by ^the circularly-polarized light.

In this case, the optical pumping by ^{the ±} ko pump beams strongly alters the velocity distribution, singling

out a particular velocity group [kvz

^± (co - cooi]

in the _mF ⁼ 2 sublevel. This narrow velocity ^distri-

bution leads to Doppler-free contributions, indepen- dently ^{of 0. In a two-level} system, this behaviour can

be predicted only ^{if we take into account saturation}

effects (fifth ^and higher ^order contributions) [25].

5. Nearly-degenerate ^four-wave mixing. ^{- The} application ^of nearly-degenerate ^{FWM to narrow}

bandwidth optical filters has been recently ^discussed.

Pepper and Abrams [14] first showed that the band- width is necessarily ^limited by phase-matching ^condi-

tions. Subsequently Nilsen and Yariv on one hand, ^and

Fu and Sargent ^on the other hand pointed ^{out that in} absorbing ^two-level systems the bandwidth should be limited by ^the dipole relaxation rate [15] ^{or the} population relaxation rate [16] respectively. ^{In this} section, ^{we show} that, ⁱⁿ Doppler-broadened media,

the bandwidth and reflection peak depend ^{on the} incidence angle. ^{First we consider two cases 0 1}

and0 ⁼ n/2, ^for arbitrary values of the _pump detuning,

v ⁼ co - mo. Then we show that, ^{when the} pump field is on resonance, there are some simple expressions ^for

the reflectivity.

5. 1 QUASI-COLLINEAR CONFIGURATION (0 1).

Like in DFWM (section 4.3b). ^the velocity average of Eab is ^more easily performed in the reference ^axes defined by k, ^and k2 [Eq. (42)]. ^{In the same way}

as in 4.3b, ^the dominant contribution comes from the first term of 2:ab ⁱⁿ equation (19) :

where b ⁼ Q - w.

Integration over vi is performed by using ^the relationship

This expression is similar to (47) ^{if we} replace Yab

by ^the complex quantity :

One gets expansion ⁱⁿ 0 similar to equation (49)

For 0

0, ( E., > ^{reduces to}

For v constant (i.e. ^constant pump frequency, m),

this characterizes a reflection coefficient which is

double-peaked ^{at 6}

⁰ (0 ⁼ w) ^and

(i) ^{The 6}

⁰ resonance comes from the modulation of atomic populations ^induced by ^the amplitude-

modulated (at frequency 6) incident field along ko.

In a more general theory (see Appendix), the linewidth 2 yab should be replaced by individual population

relaxation rates, ya ^or yb. ^I

(ii) ^{The 6}

^{2 v} resonance comes from velocity

selection along ko by one-photon ^and three-photon

processes [8] (see Fig. 3).

These two conditions lead to Q

3 W - 2 _Wo.

As shown before [10], ^{NDFWM can be used for an} heterodyne detection of phase conjugation (Fig. 1).

In that case, 6 ^{is constant and} equation (62) ^{leads to a}

single-peaked ^beat signal ^whose intensity ^{is maximum}

for

Heterodyne spectroscopy by ^means of collinear NDFWM will be discussed in more detail in the

following ^articles.

By using ^an expression ^{similar to} (54), ^one gets ^for Lab B [Eq. (60)] :

where W is defined by (52)-(53).

For v constant Eab > ^{still exhibits a double-} peaked reflectivity, but the linewidth of each compo- nent (- kuO) ^{exhibits a} Doppler-broadening linearly increasing with the incidence angle. ^{For w}

wo, the two-components merge into a single ^{one at the line}

centre,

with

5.2 ORTHOGONAL CONFIGURATION.

The velo-

city integration ^of Lab [Eq. (22)] ^is easily performed

because of the separation ^{of the two} variables, v,, and v.,. In the Doppler-limit, ^{one finds}

Note that, ^{for 6}

0, (68) ^is equivalent ^to (39). ^{For a}

constant value of v, Zb characterizes a double-

peaked reflectivity with maxima 6

+ v, ^{and line-}

width - ku. The absence of a preferential axis cancels

the effects of velocity selection, contrary ^{to the case} 0 _1. Thus the two maxima correspond ^{either to a} probe frequency ^{tuned to} the line centre, ^{Q =} (oo;

(6 = - v) ^{or a reflected} frequency ^{tuned to the line}

centre, Wr ⁼ a)o, (b

v). ^The intensity of these maxima decreases with increasing pump detuning [weight

factor, ^W (68) simplifies ^to

5. 3 REFLECTION COEFFICIENT FOR A RESONANT PUMP FIELD (w

(oo).

^{For (9}

cuo, equation (22) gives

The integration ^over Vz ^can be performed ^{like in section}

4.2 [Eqs. (29-30)] ^and yields :

This integral ^is simply ^{related to the error function} [21],

more particularly ^{to W’} [Eq. (67)] :

For 6 ⁼ 0, equation (72) gives ^back equation (31),

because of the relation W’(i0153) ⁼ 2 if(0153)}jn. ^For

0 00 ^{a limited} development ^of W’(z) ⁱⁿ 1/z ^shows

that equation (72) ^is equivalent ^to equation (62) ^with

n ⁼ 0. For 0 >> eo, equation (72) ^becomes

(Eqs. ^{(66)-(69) are two} particular ^cases of the above

expression). ^From (72a), ^one obtains that the band- width (FWHM) ^{of the} equivalent reflection filter is about 1.81 ku sin e.,

In conclusion, ^{we see that} nearly-degenerate ^four-

wave mixing ^is equivalent ^{to an} optical ^{filter of band-}

width limited by the atomic relaxation rates only ^for

0 00. This restricts the field of view quite remarkably.

For 0 » eo, ^the acceptance angle widens, ^{but this}

is done at the expenses of the filter bandwidth which increases like ku sin 0 and of the reflectivity, ^which

decreases like (sin e) - 4.

6. Two-photon phase conjugation.

^The theory

of two-photon phase conjugation ^{has been} previously

considered by several authors [13, 26]. ^{In this section,}

we briefly recall the properties ^of two-photon-resonant

DFWM in order to analyse ^its angular dependence

and to compare it with one-photon-resonant ^DFWM.

We consider a cascade three-level system, ^a-b-c

(Fig. 9), and suppose that the incident laser frequencies

are nearly ^{resonant for the} two-photon ^transition

a-c (2m ^£r (JJac) but far off-resonance from single- photon transition :

Fig. ^9.

Cascade three-level system.

The electric dipole polarization ^{is now} given by

It can be calculated with the equations of motion :

p ^is given by equation (7). yij(yj) ^are the relaxation rates of pij(pjj) ^and nj ^{is the} equilibrium population

of level j in the absence of the incident ^e.m. fields, ^E.

These fields are assumed as above [Eq. (10), Fig. 2].

We suppose that only the lower state is thermally populated (na

N, nb

n,, ⁼ 0).

Because of equation (73), the second-order _popu- lation changes ^are negligible compared ^{with the two-} photon coherence, Par, which is driven at frequencies,

COA ^{+ wv, close to} Wac:

Arguments ^{similar to the ones} developed ^{in section 3}

[Eqs. (12)-(17)] ^{show that} k ⁺ k,

^{0 is a necessary}

condition for generating PC emission. Thus there is ^an

intermediary step corresponding ^to the creation of a

Doppler-free two-photon ^coherence by ^the standing

wave, ± ko [13] :

This _process is schematized in figure ^10.

Fig. ^10.

Physical process induced in two-photon-resonant ^four-

wave mixing.

In contrast with one-photon-resonant DFWM,

in which the intermediate step ^{is the} generation ^{of an} angle-dependent population grating, ^one gets ^{in two-} photon ^{DFWM a} two-photon coherence which is isotropic ^{and in} phase ^over all the irradiated medium.

The irradiation of this coherence by ^the probe ^{wave k}

forces the _gas medium to re-radiate in direction - k.

One easily ^shows [cf. Eqs. (5), (74)] ^{that the} amplitude

of the re-emitted field is

The principal features of two-photon phase conju- gation ^are [27] :

(i) ^The isotropy ^{of the} two-photon ^coherence implies ^a reflectivity independent of ^the angle ^{of inci-} dence, 0. This is important ^for the correction of

strongly ^distorted images. ^But this is achieved at the expenses of the reflection efficiency [28].

(ii) Independently ^of 0, ^{the emission} lineshape ^is Doppler-free, ^{as is the} two-photon ^coherence.

(iii) ^The non-resonant character of the intermediate state eliminates _any resonance condition on the probe

frequency, ^{Q. In that case,} the bandwidth of the equi-

valent optical filter is only ^limited by ^the phase- matching conditions [14].

7. Conclusion.

In this article, ^we have presented

a theory ^of degenerate ^and nearly-degenerate ^four-

wave mixing ^{in resonant} Doppler-broadened gas media. When the incident frequency ^{is resonant for a} single-photon transition, ^{both emission} lineshape ^and strength vary rapidly ^{with the} incidence angle. ^{On the}

other hand, ^{this is not the case for} two-photon ^DFWM.

Nearly-degenerate ^{FWM can be used as an} optical

reflection filter, ^{but an} important ^{result of our} study

is that both filter bandwidth and reflectivity depend

on the incidence angle. ^{In two-level} systems, ^the reflection coefficient is maximum at grazing incidence,

for which the bandwidth is natural-width-limited.

For larger angular separations, ^the reflectivity ^decre-

ases and the bandwidth Doppler-broadens.

Since, ⁱⁿ many instances, ^{there is no} Doppler- broadening ^{in resonant DFWM, it can} be used for

high resolution spectroscopy in gases [5, 6, 13]. ^This presents ^the advantage ^of detecting the emission on zero background, with the related drawback that the

signal ^is generally ^smaller because of the higher ^order

interaction _processes involved. Heterodyne spectro- scopy using ^NDFWM [8-10] ^{allows one to increase}

the signal strength ^and provides ^an alternative tool for sensitive Doppler-free ^{studies. This will be} subject

of the forthcoming ^articles.

Appendix : Effect of collisional relaxation on two- level DFWM. - To account for collisional relaxation

(« open » two-level system), ^{one starts from} equations

similar to equations (75), but reduced to two-level systems [29] :

The _way of solving ^these equations is identical to the one used in section 3. The results are similar if we

replace ^N by na - nb ^{and each} denominator of the type (y ⁺ ix)-1 by 2 ![(Ya ⁺ ix)-1 +(Yb+ ix)-1]. ^For

instance, Lab [Eq. (19)] ^{becomes :}

The integration ^{over the} velocity distribution is discussed in detail in reference [18]. ^The principal

results are the following ^{ones :}

1. 0 >> 00 ^{= 2} Yab/ku. ^{- In that case,} all the results of the simple ^model (Sections 4-5) ^are still valid. Indeed,

since the Doppler-broadening ^{kuO is} larger ^{than the}

atomic relaxation rates (Yab, Ya, Yb), ^one understands

easily ^{that the emission} lineshape ^and intensity ^{do not} depend ^{on the details} of the relaxation model. It

implies that all the quantitative predictions ^concern- ing the behaviour of resonant DFWM (or NDFWM)

for 0 >> 0o still hold independently of the relaxation characteristics.

2. 0 00.

^The velocity integration ^of equation (A. 2) ^is performed like in section 5. 1. An expansion

of Lab in powers of 0 yields [in place ^of Eq. (62)]

with

(i) ^NDFWM (constant pump frequency)

For v constant ’Eal > characterizes a reflectivity double-peaked ^{at 6 = 0 and 6 = 2 v} (see ^section 5 .1. a). ^{But the linewidth} of the 6 = 0 peak ^{is now a}

combination of ya ^and Yb-

(ii) ^DFWM b ⁼ 0)

The behaviour of the emission lineshape is similar to the one predicted in section 4. At 0

0, ^{the line-}

width is _{2 yab} and the emission amplitude ^is propor- tional to [cf. Eq. (36)]

References [1] STEPANOV, ^B. I., IVAKIN, E. V. and RUBANOV, A. S. Dokl.

Akad. Nauk SSSR 196 (1971) ⁵⁶⁷ [Sov. Phys. ^Dokl.

16 (1971) 46] ;

WOERDMAN, ^J. P., Opt. ^{Commun. 2} (1970) ^212.

[2] HELLWARTH, ^R. W., ^J. Opt. Soc. Am. 67 (1977) 1;

YARIV, ^{A. and} PEPPER, D. M., Opt. ^{Lett. 1} (1977) ^16.

[3] See, ^e. g., YARIV, A., IEEE J. Quant. ^Electron. QE ¹⁴ (1978)

650 and references therein.

[4] BLOOM, ^D. M., LIAO, ^{P. F. and} ECONOMOU, ^N. P., Opt. ^Lett.

2 (1978) 58 ;

GRISCHKOWSKY, D., SHIREN, N. S. and BENNET, ^R. J., Appl.

Phys. ^{Lett. 33} (1978) 805 ;

LIND, ^R. C., et al., Appl. Phys. ^{Lett. 34} (1979) ^457.

[5] LIAO, P. F., BLOOM, ^{D.- M. and} ECONOMOU, ^N. P., Appl. Phys.

Lett. 32 (1978) ^813.

[6] HUMPHREY, ^L. M., GORDON, ^{J. P. and} LIAO, ^P. F., Opt. ^Lett. 5

(1980) ^56.

[7] LIAO, ^P. F., ECONOMOU, ^{N. P. and} FREEMAN, ^R. R., Phys. ^Rev.

Lett. 39 (1977) 1473 ;

STEEL, ^{D. G. and} LAM, ^J. F., Phys. ^{Rev. Lett. 43} (1979) ^1588.

[8] RAJ, R. K., BLOCH, D., SNYDER, J. J., CAMY, G. and Du- CLOY, M., Phys. ^{Rev. Lett. 44} (1980) ^1251.

[9] BLOCH, D., RAJ, R. K. and DUCLOY, M., ^« Doppler-free Heterodyne Spectroscopy ^of H03B1. Measurement of the 2 s1/2 collisional quenching ⁱⁿ gas cell », ^{to be} published.

[10] BLOCH, D., RAJ, R. K., SNYDER, ^{J. J. and} DUCLOY, M., ^{« Hete-} rodyne ^{Detection of} Phase-Conjugate Emission in an

Ar Discharge with Low-Power c. w. Laser », J. Physique

Lett. 42 (1981) ^L-31.

[11] WOERDMAN, ^{J. P. and} SCHURMANS, ^M. F., ^J. Opt. ^{Soc. Am.}

70 (1980) ^598.

[12] WANDZURA, S. M., Opt. ^{Lett. 4} (1979) ^208.

[13] MATSUOKA, M., Opt. Commun. 15 (1975) 84; HAUEISEN, D. C., Opt. Commun. 28 (1979) ^183.

[14] PEPPER, D. M. and ABRAMS, R. L., Opt. ^Lett. 3 (1978) ^212.

[15] NILSEN, ^{J. and} YARIV, A., Appl. Opt. ¹⁸ (1979) ^143.

[16] Fu, ^{T.-Y. and} SARGENT, M., Opt. ^{Lett. 4} (1979) ^366.

[17] The calculations presented in these articles are discussed in

more detail in reference [18].

[18] BLOCH, D., Thèse de Troisième cycle, Université de Paris- Nord, March 1980 (unpublished).

[19] LEITE, J. R. R., SHEFFIELD, ^R. L., DUCLOY, M., SHARMA, R. D.

and FELD, ^M. S., Phys. ^{Rev. A 14} (1976) ^1151.

[20] ABRAMS, ^{R. L.} and LIND, R. C., Opt. ^{Lett. 2} (1978) 94; ³ (1978) 205.

[21] ABRAMOWITZ, ^{M. and} STEGUN, ^I. A., ^Handbook of ^Mathema-

tical Functions (Dover, ^New York) 1965, p. 297-329.

[22] ^From equation (23) of reference [12], ^one predicts ^{that the} lineshape Doppler-broadening should be in 03B84 for 03B8 ~ 03B80,

in sin2 03B8 for 03B8 ~ 03B80, and the linewidth at 03B8

03C0/2 ^should be proportional ^to (ku)2/03B3. ^All these results are obviously

incorrect [cf. Eq. (57)].

[23] ^Similar conclusions have been recently ^reached by ^NILSEN

and YARIV (to ^be published) using ^a computer calculation.

[24] ^{ELCI and ROGOVIN} [Opt. ^{Lett. 5 (1980)} 255] ^{have also perform-}

ed a theoretical study ^{of the} Doppler-broadening ^effects.

We did not try ^to compare ^our results with theirs, ^since they ^{did not account} for the vectorial character of the

Doppler-broadening.

[25] ^{It is worth} noticing that, in the absence of the magnetic ^field

necessary ^to keep ^atoms optically pumped ^{in the} mF

⁺ 2 sublevel, ^{HUMPHREY et} al. observed a signal intensity approximately proportional ^to (sin 03B8)-4 (see ^Ref. [6],

note 6), ^{which is} predicted by equation (35). ^{This could}

be connected to an absence of optical pumping (the ^tran-

sitions induced by stray fields tend to cancel velocity-

selective optical pumping effects).

[26] Fu, ^{T.-Y. and} SARGENT, M., Opt. ^{Lett. 5} (1980) ^433.

[27] ^{Let us} remind that our calculations are limited to the third order in the incident fields and do not account for satu- ration effects. In the case of two-photon transitions, ^such saturation phenomena ^as the modification of the linear index induced by ^real two-photon transitions are consi- dered in reference [26]. Related eflects in two-photon dispersion ^are also discussed by GRYNBERG et al., J.

Physique ⁴¹ (1980) ^931.

[28] ^For instance, ^{for 0}

03C0/2, ^the reflectivity ratio between two-

photon ^and one-photon DFWM is of the order of

(ku)6/[4 03C003B32ac (03B403C9)4], which is in general very small, ^because of the large value of the detuning ^03B403C9.

[29] Velocity-changing relaxation effects are not-taken into account.