Parallel three-wave interaction

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Parallel three-wave interaction

F. Julien, J.-M. Lourtioz, T.A. Detemple

To cite this version:

F. Julien, J.-M. Lourtioz, T.A. Detemple. Parallel three-wave interaction. Journal de Physique, 1986, 47 (5), pp.781-788. �10.1051/jphys:01986004705078100�. �jpa-00210261�

Parallel three-wave interaction (1)

F. Julien

(**),

^{J.-M. Lourtioz}

(*)

^{and T. A.}

DeTemple (**)

(*) ^Institut d’Electronique Fondamentale, ^Bât. 220, Université Paris XI, ⁹¹⁴⁰⁵ Orsay, ^France

(**) Department of Electrical and Computer Engineering, University of Illinois, Urbana, ^IL 61801, ^U.S.A.

(Reçu ^le 3 dicembre 1985, accepti ^{le 13} janvier 1986)

Résumé. ^- Un modèle semi-classique ^{basé sur le} formalisme de la matrice densité est développé pour traiter l’inter- action en proche résonance de trois ondes en parallèle ^{dans un} système atomique ^ou moléculaire à quatre ^niveaux.

Des solutions analytiques ^{sont établies.} Elles contiennent implicitement l’information sur les dédoublements et les déplacements de résonances induits par l’effet Stark dynamique. ^{Le rôle de ces} dédoublements et de ces dépla-

cements est discuté _pour un système où deux ondes sont émises en compétition ^à partir d’un même niveau pompé optiquement; ^{cette situation est} rencontrée dans de nombreux lasers pompés optiquement. L’application ^des

résultats à une compétition Raman-Raman dans l’ammoniac est donnée en illustration.

Abstract. ²⁰¹⁴ A semiclassical density ^{matrix treatment is} developed ^{for the} parallel near-resonant interaction of three waves in a four-level atomic or molecular system. Explicit ^{solutions are} developed ^{and contain} implicit ^AC

Stark-shift and splitting information. The role of splitting and AC Stark-shifts is discussed for a system ^{of two} competing ^{emitted waves} starting ^{from a common} optically pumped level; ^{a situation} encountered _{in many} opti- cally pumped ^lasers. Application of the results to a Raman-Raman competition ^situation in ammonia is illustrated.

Classification Physics ^Abstracts

32.80K - 33.80K

1. Introduction.

The

general

^{treatment of}

multiple

^{coherent waves}

interacting

^{with an atomic or molecular} system ^{is of} considerable interest from the

standpoint

^{of linear}

and non-linear spectroscopy,

optical pumping, optical

ionization

phenomena

^{and laser} induced reactions.

Three-wave interactions in three- or four-level systems have been

intensively

studied in the past, both theore-

tically

^and

experimentally, leading

^{to new}

insights

^for

serial and

parametric

three-waves interactions

[1-3].

This paper is directed ^{at a}

density

^{matrix treatment}

of

parallel

three-wave interactions ^{in a} four-level system. Such interactions exist under various condi- tions and

configurations;

three of which are

graphed

in

figure

^1.

Figure

^{la shows a} very ^common situation in

optically pumped

lasers and illustrates the cases of normal

one-photon pumping

^{with two}

competing

laser- or stimulated Raman-emission lines

starting

from the same upper level of the pump transition.

Figure

^{lb is} analogous ^to

figure

la, ^{but the lower level}

of the pump transition is ^now the common level for the two emission transitions. Such a situation is encountered for instance in

optically pumped NH3.

Figure

^{Ic shows} two-step

pumping

^{which is seen in}

alkali metal _vapours or

optically pumped

^{FIR or MIR}

lasers.

Previous treatments of these situations ^were _gene-

rally

^{based on a}

simple one-photon

^rate

equation analysis. Alternatively,

^a

study

of various effects associated with the simultaneous saturation of two atomic transitions

sharing

^{a common level has been}

presented

in reference

[4] using

the dressed-atom

approach.

However, ^this

approach

^{cannot be}

easily

extended to treat atomic relaxation processes. In contrast, soluble solutions exist for the semi-classical Maxwell-Bloch approach. ^{In section} 2, ^{the basic}

density-matrix

^solutions

appropriate

^{to the}

parallel

three-wave, four-level system ^of

figure

^{1 are outlined.}

Section 3 contains a discussion of the solutions

presented

in section 2 with

emphasis

^{on the AC Stark}

shifts and

splittings

^which may ^occur for strong ^field interactions. Section 4 illustrates one

application

^of

the results to a model of an

experimental parallel

three-wave interaction which occurs in an

optically pumped NH3

^laser.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004705078100

782

Fig. ^{1. - Three} possible configurations ^for parallel ^three-

wave interaction.

Table I. ^-

Sign qf multipliers ai(± 1).

^{See text.}

2.

Density

^{matrix treatment.}

The

density

^matrix p for the

parallel

^three-wave system contains 16 elements, ^four

diagonal

^or popu- lation elements and 12

off-diagonal

elements. The levels are labelled ^as in

figure

^la. The familiar _equa- tions of motion for the

off-diagonal

^{elements are :}

where tij ^{are the}

T2 dephasing

^{times and}

Qij

^{is the}

transition

frequency. Similarly

^the

equations

^{of motion}

for the

population

^{elements are :}

where tii

^{are the}

T1 decay

^{times and}

Prii

^{are the}

equi-

librium

diagonal

^{elements. The} solution of these

coupled equations

^{is made tractable}

assuming

^{a time}

scale

long compared

^{to the}

phenomenological T2 dephasing

^{time, near resonant} interactions and the

implied rotating-wave approximation,

and harmonic

driving

^fields of the form :

with the field envelopes

Ei varying slowly

^{on a}

T2

time scale.

A Floquet

analysis

of the dominant Fourier coef- ficients in equation

(1)

reveals that the

only important

terms for near resonant interactions are of the form :

B

where

Pij

^{are the quasi}

^steady

^state

amplitudes.

Substitutin

these Fourier coefficients into the off

diagonal

equa ions ^and

taking

^{a time} average, ^one obtains a set

coupled algebraic equations,

^twelve

in number, whose

subsequent

solution is tedious but

straightforward.

Because of the symmetry ^{of the}

problem,

^the

resulting

solutions will be

presented

^for

each transition in terms of a

gain

^{which can be written}

in the

following generic

^{form :}

where the Rabi frequencies ^{are defined as}

Ai

⁼

woi ’

* Ei/2 h

^{and where the}

Sik

^terms will be discussed

shortly.

^{If the Rabi}

frequencies,

the transition fre-

quencies, ^{the field}

frequencies,

and the inverse of the relaxation times,

1/tii

^and

1/,rij,

^are

^expressed

^{in units}

of

1/T2,

^the

gain

value deduced from

equation (4)

^is

normalized.

Complex detuning

^{functions are defined as :}

where the

subscripts i and j

^can take the values 1, ²

or 3 and i :0 j. ^The constants ai ⁱⁿ

equations (5)

^take

the value ± ¹

depending

^{on the}

parallel

^three-wave

configuration

which is under consideration. Table I lists the value of these constants for the four confi-

gurations

^{shown in}

figure

^{1. It is to} be noted that the

expressions

^{for the}

gain given

ⁱⁿ

equation (4)

^{for the}

configuration

ⁱⁿ

figure

^{1 a} is also valid for the remain-

ing configurations

^{lb-c within}

slight changes

ⁱⁿ

sign.

With these modifications, ^the

following

^{solutions to}

be discussed below, ^{are valid for all}

configurations

shown in

figure

^1.

In

equation (5),

^the

Loi detunings

^{are related to the}

one-photon

process for wave i, ^{while the}

Rij ^detunings

are related to the

two-photon

process

involving

waves i and j. ^{It is} convenient to set new

detuning

functions :

Using

^these definitions, ^the

general

form of the

Si

coefficients is found to be :

and for i :0 i :

where :

In these, ^the

subscripts i, j, k belong

^{to the}

permutation

group

of (1, 2, 3).

^These solutions, while somewhat non-

transparent, ^are

general

^{in the sense that}

they

^are

equivalent

^to infinite order

perturbation theory

^and

include

strong-field

level shifts and

splittings.

^The

next step of the calculation would entail the solution for the individual

population

^values

by solving

^the

four

equations

of the form of

equation (2)

^{but these}

will not be

presented

^{since most of} the desired spec-

troscopic

information is contained in the

S/

^coef-

ficients.

3. Discussion of solution.

Given the

general

results of

equations (7)

^and

(8),

the location of resonances sensed

by

the various waves

and the field

dependence

^{of the} resonances will now

be considered in selected weakfield limits. Due to the obvious symmetry ^{of the}

parallel

three-wave system, the discussion of the solutions for either the first, the second or the third wave is

equivalent

^{so that the}

following

section will

only

^{be directed at} the influence of an

arbitrary

pump, i.e., ^wave 1, and second wave on a

probe

^{third wave.}

3.1 WEAK FIELD LIMIT. ^- In the limit of very weak

E2

^and

E3

^{fields, the}

parallel

three-wave system ^reduces

to two disconnected three-level systems

involving

waves 2 and 3 and the _pump. As the three-level system has been

extensively

studied in the literature

[5],

^we

only briefly

report here the main results of interest.

The

leading

contribution to the

gain

^{for wave 2 or 3}

may be deduced from

equation (9)

and is found to be :

where i ⁼ 2 or 3. The maxima of the

imaginary

part

of Sll specifies

^{resonance and} may

easily

^{be found}

by seeking

values

of Wi

which minimize the denominator of

equation (10).

^{If the} pump ^wave is on resonance, the

normally degenerate

^{one- and}

two-photon

interactions are

split

^{in a} familiar Autler-Townes doublet

corresponding

^to

the resonance condition in the

sharp

linewidth limit

(T 2 --+ 00) :

If the _pump is off-resonant, there is also a doublet with a component ^near line-centre

given by :

and a component ^{close to the}

two-photon

resonance :

784

The

approximate

^{results on the}

right

hand side remain valid as long ^as the pump Rabi

frequency

^{is weak}

compared

^{to the pump}

detuning

and illustrates the normal AC Stark effect.

As a second

example,

consider the ^case of a strong pump and one additional strong wave, either wave 2

or 3 denoted as the ith wave. If the _pump wave is on resonance, the normal doublet sensed

by

^{a weak ith}

wave contracts to a

singlet

^{at line centre. If the} pump

detuning

^is off-resonant, ^{the resonant} condition for the ith wave

yields

^a doublet. The

frequency

location of the near line centre response is identical to the

previous

^{weak field} value; ^that is, ^{the ith wave} does not shift its own

one-photon

resonance. In contrast, the

two-photon

^{resonance is found to be}

which is valid so

long

^{as the Rabi}

frequencies

^remain

weak

compared

^to the pump

detuning.

^{As seen, the}

pump and the ith wave lead to

opposite

^{Stark shifts}

for the

two-photon

^resonance.

3.2 AC STARK SHIFTS. - In the case of strong pump and emitted field

E2,

^the

degree

^of

shifting

^or

splitting

^{of the resonance sensed}

by

^{third wave can be}

extracted from the

general

form of the

Sij

coefficient.

The desired

spectroscopic

information _may be obtain- ed

by noting

^{that the}

expression

^{for the normalized}

gain

ⁱⁿ

equation (4)

^{is a} linear function of the popu- lation terms.

Applying

^the

superposition principle,

the

expression

^{for the}

gain

^{can be}

simplified by setting

the

population

value of all levels to zero except ^for level 3. Then the

only

coefficient to be considered is

S33

^{which can be}

simplified by keeping

^the

only

^terms

first-order in

A3.

^With these,

S3

^{reduces to :}

and the roots of Re

(No3)

^{= 0}

give

^the resonances.

Calling

^for

simplicity

^{Z, Y and X the}

frequency detuning

^{of wave 1,} 2 and 3 defined as Wi - ai

QOi = Wi - I QOl with i

^{= 1, 2, or 3}

respectively,

^{the root}

equation

^becomes

where the

sharp

line limit has been assumed. Equa-

tion

(16)

is cubic in X and is valid for X :F ^{Z and} Z # Y. The first left-hand side of this

equation clearly specifies

the zero-field one- and

two-photon (1

^and 3,

2 and

3)

^resonance conditions. The other terms must then represent

shifting

^and

splitting.

^The

major implication

^of

equation (16)

^{is that the doublet}

characterizing

^{the resonance} conditions of wave 3 of zero-field is

split

^{in a}

triplet

^{when the second field}

is non-zero. This result is

analogous

^{to that obtained}

for serial three-wave interactions

[1].

^{In the case}

where the second wave

detuning

^is

equal

^to the pump

detuning,

^{Y = Z, the}

previous equation

^{reduces to :}

whose solutions ^{are :}

So for this

particular

^{case, the} resonances

comprise

a doublet. If the _pump field is off-resonant, equa- tion

(18)

^can

easily

^be

recognized

^{as the AC Stark}

shifted

two-photon

^{or Raman} condition while _equa- tion

(19)

^{can be} identified with an AC Stark shifted

one-photon

interaction. If the pump field is ^on resonance, the components ^{of the} doublet

satisfy

the resonance condition :

which is the

two-photon analog

of Autler-Townes

splitting.

3. 3 STRONG-FIELD SOLUTIONS. ^- The shifts and

split- tings

^can be further illustrated for selected cases

by

numerical

evaluating

^{the full set}

of equations, including population

saturation. In the

figures presented

ⁱⁿ

this section, ^all

T2

^{values are set}

equal

and the Rabi

frequencies

^and

detunings

^are

expressed

^{in units of}

1/T2.

^{In this case the}

gain amplitude

ⁱⁿ

equation (4)

is normalized to 1. The

equilibrium populations

^are

assumed to be zero except ^{for level 1. In}

figures

^2-5

are shown the normalized

gain amplitude

^{of the}

third wave as a function of the third wave

detuning,

pump and second ^wave

detunings

and Rabi fre-

quencies. Figures

^{2 and 3}

apply

^{to a resonant}

pumping

case, while

figures

^{4 and 5} correspond ^{to an off}

resonant

pumping

^case.

Figure

² illustrates the

small-signal

^third-wave

tuning

^{curves at} different second-wave

detunings

^for

resonant

pumping.

^{The dashed curves}

give

^the

gain

resonances

predicted by equation (16).

^One may notice the

good

agreement with the full numerical results. Curves

d) and f)

ⁱⁿ

figures

²

emphasize

^a strong ^{reduction of the}

gain amplitude

^and

splitting

when the

detuning

^{of the second wave is close to the}

zero-field wave resonance condition. Curve

d)

^cor-

responds

^{to the}

particular

^case studied in the

previous

section since the second ^wave

detuning

^{is zero and}

equal

^to the pump

detuning.

^The

frequency

^location

of the doublet, ±

11.2/T2,

^{is in}

perfect

agreement with the

predicted

values from

equation (20). Figure

³

is the same as

figure

² except ^{for a} strong ^{Rabi fre-} quency of the third ^wave

illustrating

saturation and power

broadening.

Figure

⁴

corresponds

^{to an} off-resonant _pump

detuning

^of

40/T2

^{with an}

equal

value,

20/T 2,

^{for the}

Rabi

frequencies

^{of the} pump and the second wave

while the Rabi

frequency

of the third wave is zero.

Fig. ^{2. -} Small-signal ^{third wave} tuning ^{curves at different}

second wave detunings ^for on-resonance pumping. ^The

vertical axis is the normalized-to-one gain. ^{The horizontal}

axis gives the third wave detuning in units of 1/T2. ^For

curves a) ^to g), ^{the second wave} detuning ^is varied from

- 14/T2 ^to 14/T2 by step ^of 4.66/T2. ^The conditions for this figure ^are Al ⁼ 10/T2, A2 ⁼ 5/T2 ^and A3 ^{= 0. The}

dashed curves in figure 2 show the evolution of the gain

resonances predicted by equation (16) (see text).

Here also, the evolution of the

gain

^{maxima versus}

the second and third ^wave

detuning

^{is well}

predicted by equation (16) (dashed curves).

^The

asymptotic

resonance conditions

corresponding

^{to zero-field}

second wave should be

X1

^{= - 16.6}

T2

^and

X 2

⁼

48.6/T2.

^{The curves of}

figure

⁴

emphasizes

strong

shifting

^and

splitting especially

^when the second wave

detuning

^{is close to}

X,

^or

X2.

^{The doublet in}

curve fit corresponds

^{to the}

particular

^case studied in the

previous

section since the second ^wave

detuning

^is

equal

^to the pump

detuning.

The location of the doublet is in

perfect

agreement ^{with the}

predicted

values from

equations (18)

^and

(19). Figure

^{5 is the}

same as

figure

⁴ except that the third-wave Rabi

frequency

^is

10/T2.

^This

figure again

illustrates saturation and _power

broadening.

3.4 COMPETITION THRESHOLD. - As it

might

^be

expected, figures

^{2-5 show that} strong

competition

between the second wave and the third wave occurs

when their

detunings

^{are close}

together (X

^{= Y +} s)

where E is a small correction. Consider the third wave

Fig. ^{3. - Same as} figure ² except ^for A3 ⁼ 5/T2.

to be a

probe

^{wave. The}

competition

threshold is said to occur when the second wave is

sufficiently

intense to

split

^the

gain profile

of the third wave into

a resolvable doublet which

requires

^{a doublet} sepa- ration at least

equal

^to the linewidth.

3.4.1 On-resonance pumping. ^- The weak field

resonances for the second and third waves are located at

± A1 (Eq. (11)). Choosing

for instance the

positive

value, ^we may ^{set Y}

= A,

^{and X}

= At

^{+ s. Then, the}

location of the doublet components ^{for the third}

wave can be found from

equation (16), keeping only

the terms lowest order in s. We obtain :

Assuming A2 At,

which is verified a posteriori, the solutions are :

The threshold second wave Rabi

frequency

^for strong

competition

^{is then}

expressed

^{as :}

3.4.2

Off-resonance

pumping

(Z » 1/T 2).

^{- Next}

consider the case of a Raman-Raman

competition.

The

corresponding

weak-field resonance for the second and third waves is located at

1/2(Z

⁺

[Z2

⁺

4Af]1/2) (Eq. (13)).

^In

^analogy

^{with the pre-}

786

Fig. ^{4. -} Small-signal ^{third wave} tuning ^{curves at different} second wave detunings for off-resonance pumping ^case.

For curves a) ^to i), ^{the second wave} detuning is varied from

- 60/T2 ^to 100/T2 by step ^of 20/T2. ^The conditions ^are

40/T2 ^for the pump detuning, At = A2 ⁼ 20/T2 ^and A3 ^{= 0. The dashed curves in} figure 4 show the evolution of the gain resonances predicted by equation (16) (see text).

vious _case, we may ^set Y ⁼

t(Z

⁺

[Z 2

⁺

4 Af]1/2)

and X ⁼ Y + 8. The doublet components ^{for the} third wave are

given by equation (16), again keeping only

^the lower-order, terms in e, ^E satisfied :

Assuming A2 At, the

^{solutions are :}

The threshold second wave Rabi

frequency

^{is deduced}

from the doublet separation :

Fig. ^{5. - Same as} figure ⁴ except ^for A3 ⁼ 10/T2.

which reduces to :

for moderate

pumping (A, Z).

One _may notice that the Raman-Raman _compe- tition ^{is less severe} than the laser-laser one

previously

considered in the case of on-resonance

pumping.

Larger second-wave intensities are

required

^to

perturb

the

gain profile

^{of the}

probed

transition. Equation

(26)

tends to

equation (23) only

^{for very}

high pumping

intensities

(A i » ^Z).

4. Raman-Raman

example.

In this section, ^an

example

^{of the}

application

^{of the}

results to an

experimental parallel

three-wave inter- action, ^{similar to the}

graph

ⁱⁿ

figure

la, ⁱⁿ

NH3 optically pumped by

^{the 9 R16 line} of the TEA

C02

laser is discussed.

Pumping

^{is known to occur}

- 1.3 GHz off-resonance from the aR

(6, 0) absorbing

transition. There are two emission lines

starting

from the same upper level in

the v2

^{state; a FIR 90 gm}

rotational line

corresponding

^{to the}

aR(6, 0)

^transition

in

the v2

^{state and a MIR 12.08} ym

aP(8, 0)

vibrational transition in

the v2

^{band [6, 7,}

^8].

^{The MIR emission}

has been

extensively

studied in resonator

configu-

rations but is undetectable in non-feedback cells.

The FIR line _appears to be _very efficient

[9].

^Both

lines have been found to be

generated by

^{a two-}

photon

process with

parallel polarizations

[6, 9].

This

experimental

^{situation is a}

typical example

^of

a

Raman-Raman

competition.

In order to

investigate

^this situation,

Doppler broadening

^and

m-degeneracy

have been introduced in the

parallel

three-wave model

presented

^above

[10].

The results will be presented ^{in terms of} Beer’s coef- ficient defined as :

where ki

^{is the wave vector, N} the molecular

density, POi(m)

^the

m-dependent dipole

moment, ^v the molecu- lar

velocity, g

^the

polarization

^{unit vector of} wave 4 and a Am ⁼ 0 selection rule is assumed. For numerical

analysis,

^all

T2

^and

T 1

^{values are set}

equal

^{to 1.6 x} 10-8 s.torr. The vibrational

dipole

^{moment is}

0.23

Debye,

^the permanent

dipole

^{moment is 1.23 De-}

bye

^{and the}

equilibrium populations

^{are calculated}

to be 7.3 ^x

10- 5, 0.0113,1.08

^x 10-4and2.63 x 10-3 for levels 0, 1, 2,

3, respectively.

Since the 12.8 _gm emission is

generated by

^{a Raman}

process, the

following

discussion will be restricted to

frequencies

^{close to the}

two-photon

^{resonance condi-}

tion.

Figure

^{6 shows the} 12.8 ym

small-signal

gain ^{as a}

function of

detuning

for different _{90 ym} intensities.

The pump power is 1

MW/cm2

^{and the}

NH3

^pressure

is 5 torr both of which are

typical

conditions. The 90 gm ^wave is assumed to be on

two-photon

^reso-

nance, - 1.64 GHz. Curve

a)

^{is related to a zero-}

field _{90 ym} wave and has a resonance at - 1.64 GHz 12.8 _ym

detuning

^as

expected.

^Curve

b)

is for 100 W/

cm2 90 _tim

intensity,

^{the 12.8} gm

gain

spectra

splits

into a doublet and a

negative gain

^{occurs at the two-}

photon

^{resonance when the} 90 tim

intensity

^reaches

10

kW/CM2 (curve d)

because the 12.8 _gm

two-photon gain

^{is not} sufficient to compensate ^the

wing-absorp-

tion. From

equation (26b) (Sect. 3.4),

^the 90 gm

intensity corresponding

^{to a} situation of strong

competition

^is calculated to be 0.75

kW/cm2.

^This

result is in

perfect

agreement with those obtained from the full numerical simulation.

Since these results are

only

^for the local

gain,

further

insight

^{can be}

gained through

^{the use of a} propagation model. The

simplest propagation

equa- tions are of the form :

where

1 is

^the

intensity

^{of the wave i.}

Figure

^{7 shows}

the results of a numerical solution of the

coupled

^set

of equation

(28)

^{for the}

intensity

^{of the} pump, 90 _gm and 12.8 _ym waves on a

logarithmic

^{scale versus the}

Fig. ^{6. - Raman} tuning ^curves for the 12.8 gm emission line of ammonia optically pumped by ^{the 9 R16} C02 ^line.

The vertical axis is the gain ^{in m-1 while the horizontal}

axis gives ^the 12.8 um detuning ^{in MHz.} The pump detuning

is ^- 1300 MHz while _{the 90 gm} detuning ^{is - 1640 MHz.}

For curve a) ^{no 90} um is present. ^{For curves} b) ^to d), ^the

90 _gm intensity ^is respectively ¹⁰⁰

W/cm2,

¹ kW/CM2 ^and

10 kW/cm2.

Fig. ^{7. -} Propagation ^{curves for} the pump, the 90 um and the 12.8 _um emissions. The vertical axis gives ^{the inten-} sity ⁱⁿ W/cm’ ^{and in} logarithmic scale. The horizontal axis is the propagation length ^{in m. The} injected pump intensity

is assumed to be 1 MW/cm2. ^The frequency detunings ^are

fixed at the entrance of the cell as - 1300 MHz for the pump and ^{as -} 1640 MHz for both 90 um and 12.8 um emission lines.

788

propagation length.

The initial conditions at the entrance of the cell for the MIR and FIR fields are

the

black-body

intensities deduced from the Planck formula and a value of - 1.64 GHz for the MIR and FIR detunings. ^This

figure

emphasizes ^a

rapid

^initial

growth

^{of the FIR wave,} which is consistent with the

large

value of the

small-signal gain.

^{The FIR}

begins

to saturate at intensities greater ^{than 1}

kW/cml,

while the _pump wave shows

only

^{a small}

depletion.

The MIR

intensity

grows slower due to a smaller

gain

and reaches ^a maximum when the FIR

intensity

is around 5

kW/cm2.

^For

longer propagation

distances,

the MIR

intensity

^decreases

slightly

^{due to the nega-}

tive

gain

^{as seen in curve 6-d. It is to} be noted that the MIR

intensity

^{never reaches a} detectable inten-

sity

^{in this} case, in agreement ^with

experimental

observations

[8, 9].

^{The fact that the absence of a}

detectable output ^{at 12.8} ym is not due to pump

depletion

^{is an}

important

^result and demonstrates the

importance

^{of the new} coherent effects contained in the

parallel

three-wave treatement

presented

^here.

5. Conclusion.

A semiclassical treatment of the

parallel

^three-wave

interactions in a four-level system ^{has been} outlined.

The discussion of the solutions focused on a system

composed

^{of two emitted waves}

starting

^{from the}

same

optically pumped

level. When one emitted wave

has a non-zero Rabi

frequency,

^the resonance con-

ditions for the other emitted wave lead to a

triplet.

The

frequency

location of the components ^{of the}

triplet

may be

strongly

^{AC Stark shifted}

according

to the

frequency

of the first emitted _wave, as soon as the Rabi

frequencies

^{of the two emissions are of}

the same order of

magnitude.

This feature could lead to new

triple

^resonance

techniques

for non-linear spectroscopy. ^The

application

of the model to a

competing

situation in ammonia yields ^some

insight

for

explaining

^how one wave can suppress the other

wave. The present ^{results could}

easily

^be

generalized

to other

competiting

situations in ammonia

[11].

In non-selective cavities, the FIR rotational emis- sion suppresses the MIR rovibrational emission.

Acknowledgments.

The authors wish to

acknowledge helpful

discussions with S. J. Petuchowski from Goddard

Space Flight

Center, ^MD.

References

[1] PETUCHOWSKI, ^S. J., OBERSTAR, ^{J. D. and} DETEMPLE,

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[2] WILLENBERG, ^G. D., HEPPNER, ^{J. and} FOOTE, ^F. B.,

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