OPTICAL BISTABILITY IN FOUR-WAVE MIXING OSCILLATORS

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OPTICAL BISTABILITY IN FOUR-WAVE MIXING OSCILLATORS

G. Grynberg, M. Pinard, D. Grandclément, P. Verkerk

To cite this version:

G. Grynberg, M. Pinard, D. Grandclément, P. Verkerk. OPTICAL BISTABILITY IN FOUR- WAVE MIXING OSCILLATORS. Journal de Physique Colloques, 1988, 49 (C2), pp.C2-471-C2-476.

�10.1051/jphyscol:19882111�. �jpa-00227622�

OPTICAL BISTABILITY IN FOUR-WAVE MIXING OSCILLATORS

G. GRYNBERG, M. PINARD, D. GRANDCL~MENT and P. VERKERK Laboratoire de Spectroscopic Hertzienne de I'ENS, Universit6 Pierre et Marie Curie, 4, Place Jussieu, T-12, E-1,

F-75252 Paris Cedex 05, France

Risume

-

Nous montrons que 12s oscillatsurs a i6lange a 4 ondes sonc intrinsequement bistables. Dss expbriences faites dans le sodiu:?

confirment ces pr~idictions. Toutefois, ies resultats exp6rirnentaux obtenus dans le cas d'une cavit€ en anneau montrsnt qu'ii est encore necessaire d'arniliorer le modsle thborique.

Abstract - We show that four-wave mixing oscillators are intrinsically bistable. Experiments made in sodium vapor cosfirm this prediction.

However, the experimental results obtained in the case of a ring cavity show that it is still necessary to imprsve the theoretical model.

INTRODUCTION

Four-wave mixing oscillators [I] have properties which are very similar to those found in parametric oscillators [2]. In both cases, pump photons are converted into photons emitted by the oscillators (fig. 1). However, there are some differences which are due to the fact that a parametric oscillator is associated to a

x"'

process (one pump photon gives two photons) while the four-wave mixing oscillator is associated to a x t 3 ) process (two pump photons give two photons). In particular, this leads to an intrinsically bistable behaviour for the four-wave mixing oscillator. We report here the results that we have observed in a linear cavity and in a ring cavity.

Fig. 1

-

Gain mechanism for a parametric oscillator (a) and for a four-wave mixing oscillator (b).

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19882111

JOURNAL DE PHYSIQUE

1

-

STATIONNARY REGIME FOR A LINEAR FOUR-WAVE MIXING OSCILLATOR

We consider a nonlinear medium enclosed in a cell of length 1 and interacting with two pump beams of same incident intensity 11, and sans frequency u~ (fig. 2.a). This medium acts as a phase conjugate mirror whose amplitude reflectivity [2j is rc = ikx' 3 ' I~ 1 (we assume

Ire

1

<<

1).

Let us consider a beam E e-" of frequency op at point A and le-, us calculate the value of the reflected beam Er at the same point when a totally reflective mirror is placed behind the nonlinear medium (fig. 2.b.c.d). At zeroth order in r c , we obtain the contribution E;.O1 of the path shown in fig. (2.b) while the first order contributions

EL"

^is

associated with fig. (2.c) and (2.d)

+

is the phase associated with the propagation of light along the path shown in fig. (2.b).

Fig. 2

-

Phase conjugate medium (a). Beam reflected at zeroth order in rc (b) and at first order in rc (c and d).

If we place a't point A a mirror whose reflectivity is r, we must obtain in stationnary regime and at first order in rc

Generally this condition can only be fulfilled when the cavity is almost tuned and we cirll q = Q

-

2nn: the difference between the round trip phase J,

and the c1ose:jt multiple of 2n;. We assume in the following that Iql

<<

1.

Eq. ( 2 ) can then be written

From the imaginary part of ( 3 ) , we deduce

which can also be written

2 - SATURATION DUE TO PUMP BEAM DEPLETION

If the pump beam intensity is sufficient to reach the threshold condition ( 6 ) , the intensity inside the oscillator is determined by saturation mechanisms. We can consider two types of mechanism [3] the atomic saturati.on and the pump beam depletion. We shall focus our attention on this last process in this theoretical analysis.

The intensity of the pump beams after transmission inside the nonlinear medium is Ip - CIe

.

If we can neglect all the losses apart from those due to the four-wave mixing process (fig. l.b), we should have

where I is the intensity of the oscillating beam inside the cavity. If

A Ip

<<

I p , the mean value of the pump beam inside the nonlinear medium is

Ip

-

(A Ip/2) and the phase conjugate reflectivity is equal to

where ro f kx'

"

IIp 1. Using ( 6 ) , ( 7 ) and (8) , we thus deduce the value of the intensity I of the oscillating beam inside the cavity

This resulc which can also be found by more sophisticated argunents [3] is valid when A Ip i< Ip , I. e . wherr I << Ig IT.

3 - OPTICAL BISTABILITY

When the cavity is tuned (71 = 0 ) . the nonlinear phase shift

JOURNAL DE PHYSIQUE

A q N L = kx( ) I 1 is obtained from (9)

We can expect to observe optical bistability if A

CPNL

is of the same order or larger than the width of (9) called q,. We can estimate qo by calculating {:he value for which I = 0 in formula (9)

.

^{We find}

% ^ci J r T / 4 ) and

Apart from a small range above threshold (ro N T/4) , (qo /T) is a quantity which can be as large or larger than T. We can thus estimate that bistability will be easy to achieve in a four-wave mixing oscillator. Such a result is confirmed by a more detailed calculation [3] and is also predicted i n the case of atomic saturation [ 3 ] .

4

-

OPTICAL BISTABILITY IN A LINEAR FOUR-WAVE MIXING OSCILLATOR

The experfimental set-up is shown in fig. (3.a). We use a 5-cm quartz cell with windows at Brewster angles. The temperature of the cell is 150'C.

The light-source is a homemade C.W. dye laser pumped by an ~r' ion laser which delivers about 300 mWatt at 5890 A. Two mirrors Mi and M p are set one on each side of the sodium cell. The transmissions TI and T2 of nirrors M I and M 2 are O.!5 % and 6 %. The maximum output that we have observed is 3 mW C11.

When the pump beam intensity varies, we observe that the intensity of the output beam is different for increasing and decreasing pump beam intensity (fig. 3.b). This is a typical optically bistable behaviour.

Fig. 3 - Experimental set-up (a1 and output intensity versus pump beam intensity (b)

.

When che pump beam intensity is fixed, optical bistability can also be observed by varying the length L of the cavity [4]. We show on fig. (4.a) and (4.b) the variations of I o u t versus L for w p < a, (self-defocusing side

Iout (mw)

t ^{I out} t ^{(kw) I o u t t}

^(mW)

Fig. 4 : Variations of I versus L for w

<

wo (curves (a) and (b)) and for w

out p

>

wo

(curves (c) and ( d ) ) .

5

-

OPTICAL BISTABILITY IN R RING FOUR-WAVE MIXING OSCILLATOR

We describe now some recent results obtained when the linear cavity M I M2 of fig. ( 3 . a ) is replaced by a ring cavity (fig. 5.a). These results have some new characteristics which are not presently fully understood.

We firsc show on fig. (5.b) the variations of the output intensity when the length of the cavity increases. We see that an unusual bistable curve is observed for several emission peaks. Let us first consider the main peak (labelled A on fig. (5.b) ) . Ic presents two sharp variations of intensity.

The upper part of the curve is associated with a degenerate emission, two beams of same frequency w+ = w and w- = o then counterpropagate in che cavity. The lower pare of this emission peak seens to be associated with a nondegenerace oscillation with two beams of frequency w

+

52 and ws

-

12 propagating in opposite direction inside the cavity

( h

being thz free spectral range of the cavity).

Fig. 5 - Scheme of the ring four-wave r!i:<ing oscillator (a). Variation of the output intensity when the lengzh of the cavity increases (b).

JOURNAL DE PHYSIQUE

The curve B also corresponds to a nondegenerate oscillation with two beams of different frequency propagating in opposite direction. We have randomly observed situations where

W+

-

^{W P}

⁺

^{.Q/2 and w- = w - R/2 and}

situations where w+ : w - R / 2 and w- = wD + 11/2. Each of these solutions is very scable and can rexain unaltered for several minutes provided that zhe pump frequency and the length of tha cavizy remain constant. This type of behaviour corresponds to a spontanbous symmetry breaking.

This observation shows that the solution with two travelling waves at frequencies w + R / 2 and wp

-

R / 2 is choosen rather chan che solution with two standing waves cf same frequencies. This suggests that an a t o ~ i c saturacion mec:hanism should be involved in the interpretation of the experimental results.

6 - CONCLUSION

We have shown both theoretically and experimentally that the four-wave mixing oscillators are intrinsically bistable. However it appears that the simple model presented here cannot explain all the features of the experimental observations in izhe case of a ring four-wave mixing oscillator. improvements in the theoretical model, including the possibility to have a multimode operation, should probably be attempted.

REFERENCES

[I] GrandClement D., Grynberg. G and Pinard M., Phys. Rev. Lett. 59, 44 (1987)

.

[2] Shen Y.R., The Principles of Nonlinear Optics (Wiley, New York 1984).

1 3 3 Pinard M., Horowicz R., GrandClkment D. and Grynberg G., IEEE J. of

Quant. El. (to be published in the Special Issues on Nonlinear Phase Conjugation 1988).

[4] Giacobino E., Devaud M., Biraben F. and Grynberg G., Phys. Rev. Lett.

4 5 , 434 (1980).

[5] Gibbs H.M., Optical bistability : controlling light with light (Academic Press, New York 1985).