572 OPTICS LETTERS / Vol. 16, No. 8 / April 15,1991

Dynamics of circularly polarized eigenstates in lasers withnonweak atomic coupling

J. C. Cotteverte, F. Bretenaker, and A. Le Floch

Laboratoire d'Electronique Quantique, Physique des Lasers, U.A. Centre National de la Recherche Scientifique 1202,Universite de Rennes I, Rennes, France

Received October 10, 1990; accepted January 28, 1991

It is shown theoretically and experimentally that the two circularly polarized eigenstates of a laser oscillating on anonweak atomic coupling transition exhibit two types of vectorial bistability. A rotation and an inhibitionmechanism are isolated, depending on the eigenfrequency difference of the two eigenstates. For both mechanisms,the evolution of the hysteresis loop with this frequency difference shows good agreement with a phenomenologicalLandau's potential model.

The study of polarization effects in Zeeman lasers wasan early subject of research in laser physics.1' 2 Re-cently, there has been renewed interest in the dynam-ics of laser eigenstates, especially for the building ofall-optical logic gates with low switching energies3'4and the study of new kinds of instabilities and chaos.5' 6

One must notice that most of these experiments hadbeen performed with J - J + 1 (J > 0) weak atomiccoupling laser transitions, which led to the existence oflinearly polarized eigenstates fixed by the residual an-isotropies of the cavities. The rare early experimentsperformed with strong or neutral atomic coupling linesthat exhibited circularly polarized eigenstates2'7 wereunfortunately made with sealed mirror lasers, pre-venting one from controlling the residual phase andloss anisotropies of the cavity. Recently, a new insta-bility threshold has been predicted in the case of theneutral atomic coupling J = 0 - J = 1 quasi-isotropiclasers.8 The aim of this Letter is to explore the natureand the dynamics of the eigenstates of a quasi-isotropiclaser working on a theoretically neutral (or strong)atomic coupling transition. The direct comparisonwith a usual weak atomic coupling transition will allowus to point out the role of atoms on the possibility ofvectorial chaos in quasi-isotropic lasers.

A laser cavity containing a Faraday rotator thatprovides a nonreciprocal circular-phase anisotropyAki: has a+ and a- eigenstates with eigenfrequenciesgiven by vP = c/2L(k i AO±/ir), where L is the cavitylength, c is the velocity of light, and k is an integer.The evolution of the amplitudes E+ and E_ of the a+and o- eigenstates can be described by self-consistantequations in a form derived from Lamb's theory, 9

= E (aC - +E+2 _ 0,E_2),

= E-(ao - fE _2_0E+2,

(la)

(lb)

where a+ and a- are the net gain coefficients and ,B+,#-, and 0± are the self- and cross-saturation coeffi-cients.9 The coupling constant of the two eigenstatesis defined by C = (0) 2/(f3+fl_) and depends on thefrequencies of the two a eigenstates. When the twoeigenstates have the same frequency and are located at

the center of the gain profile, C reduces to the atomiccoupling Catom that depends only on the total angularmomenta of the levels involved in the laser transition.9

For a J = 1 - J = 1 transition, this atomic coupling istheoretically neutral (Catom = 1). However, because ofcollision effects the atomic coupling is found to beslightly above one,2'9 which leads to possible vectorialbistability between our two circularly polarized eigen-states. The steady-state solutions of Eqs. (1) can beinvestigated in the framework of the associated Lan-dau's potential V,

V(E+,E-) = I a+E+2 -a E 2 + 1 +E+42 2~ 4

+ 1 A E 4 + 1 oE+2E_2, (2)4 2

where E+ and E_ are the two order parameters. Theshape of this potential is represented in Fig. 1. Be-cause C > 1, the two eigenstates exhibit vectorial bi-stability, and the two wells A and B correspond to theoscillation of a single eigenstate. This representationpermits a heuristic approach of the polarization-flip-ping mechanisms.4 On the one hand, when Ak± is low,

V(E+,E_)

E E_:

Fig. 1. Shape of Landau's potential V. The A and B wellscorrespond to the y+ and o- eigenstates, respectively. No-tice the loss barrier along the AMB path.

0146-9592/91/080572-03$5.00/0 © 1991 Optical Society of America

April 15,1991 / Vol. 16, No. 8 / OPTICS LETTERS 573

the barrier between wells A and B (see Fig. 1) becomeslow enough to allow, for example, the a+ eigenstate toslide progressively to the (7 eigenstate without under-going too important losses. In that case, the frequen-cy of the eigenstate slides progressively from v+ to v_while the eigenstate rotates, following the AMB pathand undergoing extra losses. The corresponding sin-gle-frequency Landau's potential is derived in thesame manner as Eq. (2) for a single frequency,

V(8) =-2 a(0)E02 + I fl()Eo4 (3)2 4

where 0 is the angle between the polarization and thepositive axis (see Fig. 1) and where

a(a) = gexp-() - vI 2 C i\ (4I AVD -P LA() (4

A(8) = ,B ep E [v(0) - Vo12

I 1VDJJ

inearly with 0 from I +2 + [v(0) Vo]2} (5)

Here the frequency of the laser v(0) is assumed to varylinearly with 0 from v+ to v-. AVD and y are theinhomogeneous and homogeneous widths of the tran-sition, vo is the central frequency of the transition, pare the average losses per second, Eo = (a/0l)"2 , Daccounts for velocity-changing collisions,'0 and the ex-tra losses in amplitude for one round trip in the cavityat angle 0 are approximated by

AP (0) = 8 AO, 2[02 COS2 0 + (X/2 - 0)2 sin2 0]. (6)

Ir2

The flipping from a+ to a- is obtained when the poten-tial V decreases monotonically when 0 varies from 0 toir/2. This allows us, for a given A0, to find the cavityfrequencies for which the polarization flips. Howev-er, when A10 increases, the extra losses Ap(7r/4) be-come too important for the system to get over thebarrier AMB. The flipping mechanism is then theusual two-frequency inhibition mechanism,9 where,for example, the a+ eigenstate at v+ disappears as thea eigenstate appears at its own eigenfrequency v-.From Eqs. (1), the flipping condition can then be writ-ten as

a- - CYa+ > 0,

perpendicular silica windows. The spherical mirrorhas a radius of curvature R = 1.2 m and transmits 36%of the incident intensity. The plane mirror is totallyreflecting and moved by a piezoelectric transducer.The output power is detected through a quarter-waveplate and a linear polarizer. The axis of the laser iseast-west oriented, in order to minimize the effect ofthe Earth's magnetic field. The He-Ne system at X =3.39 gtm offers a suitable coincidence for our experi-ments. Without any methane in the intracavity cell,the laser oscillates on the usual weak atomic couplingJ = 1 - J = 2 transition at X = 3.3922 Asm that allowsthe two opposite ar polarizations to oscillate simulta-neously. With 10 Torr of methane in the cell, theabsorption extinguishes the X = 3.3922 Am transitionand allows the J = 1 - J = 1 transition at X = 3.3912Am to oscillate." A 1-mm-thick YIG crystal intro-duced into the cavity provides an adjustable circular-phase anisotropy AO± of 0.015'/G. The residual an-isotropies of the cavities are minimized with the usualJ = 1 - J = 2 transition when no methane is intro-duced in the cell, by compensating the residual anisot-ropies with loss and phase plates in order to obtain amaximum rate of rotation of the linear polarization 2

One can then introduce the methane in order to studythe behavior of the J = 1 - J = 1 transition. Withoutany extra magnetic field (isotropic cavity) the oscillat-ing eigenstate is still elliptical with a ratio of the inten-sities along the axes of the ellipse of approximately 1.4.With a small extra magnetic field on the YIG crystal,the circular-phase anisotropy fixes definitely the po-larization of the eigenstates, and one can observe bi-stability between the two a eigenstates. Then, whenone tunes the length of the cavity, the eigenfrequen-cies of the eigenstates are translated through the gainprofile. The experimental output power profiles areshown in Fig. 2 for two different values of Afr+ (0.250and 0.8°). Figures 2(a) and 2(c) display the a+ outputpower, and Figs. 2(b) and 2(d) display the output pow-er recorded when the linear polarizer is aligned alongthe axes of the quarter-wave plate. The peaks in Fig.2(b) show that the polarization rotates continuously

+

(7)

meaning that the gain of the a- eigenstate saturatedby the a+ eigenstate is still positive. The flippingconditions given by relations (3) and (7) lead to impor-tant predictions about the hysteresis loop exhibited bythe system when the cavity length is tuned. In thecase of the rotation (inhibition) mechanism, one pre-dicts that the width of the hysteresis loop increases(decreases) with AO+ and decreases (increases) withthe rate of excitation of the laser.

The verification of these predictions is performedwith a 66-cm-long cavity containing a methane celland a 31-cm-long discharge tube with a 3.5-mm innerdiameter filled with a 7:1 3He:20Ne gas mixture at thetotal pressure of P = 1.1 Torr. It is closed with quasi-

(a) ' N

1450I

(C) V

I I aFr

(b) V (d) V

Fig. 2. The two basic mechanisms: (a) and (c) are the a+output powers versus cavity tuning, and (b) and (d) are theoutput powers versus cavity tuning with the linear polarizeraligned along the axes of the quarter-wave plate. (a) and (b)correspond to the rotation mechanism (A&, - 0.25°), and (c)and (d) correspond to the inhibition mechanism (AO+ -0.8°).

i I I I I I� l � Al t I

i.. ] . .... .... ;� :: �]

I � -T-F-7+ I 1 1 -� + +

-AL ;71--

I

P

FI Or~ I I .

F_

H4-4�:

574 OPTICS LETTERS / Vol. 16, No. 8 / April 15, 1991

0.50

~ 0

-0.-0.5

0 2 4 6 8 10

INCIDENCE (0)

Fig. 3. Relative transmission difference AT between oaand (7 incident waves upon the YIG crystal versus angle ofincidence for AO+ = 0.50.

I I 11 1 I II-A_L" 40h 1(a) -

I LLLI I V I Al !LL

,_4 e (b) -,_ .EES(c) ,

1 1,- J._UAX (d) -

_tl~l ~t ll(e) -_

V

V

V

V

V

Fig. 4. Experimental (left column) and theoretical (rightcolumn) evolutions of the hysteresis loop for five values ofA&, [0.27°, 0.420, 0.65°, 1.02°, and 1.25°, respectively, for(a)-(e)]. (a)-(c) correspond to the rotation mechanism, and(d) and (e) correspond to the inhibition mechanism.

from a+ to (- during the flip for Aq+ = 0.25°, accord-ing to the rotation mechanism. The absence of a peakin Fig. 2(d) shows that for AO+ = 0.80 the polarizationflips according to the inhibition mechanism. We haveconsequently isolated the two basic mechanisms. Letus now explore the evolution of the hysteresis cyclewith Aok in each case. First, Fig. 2 shows that the twoeigenstates do not have the same losses, as the hystere-sis loop is not symmetrical with respect to the center ofthe profiles. This circular dichroism is due to theFabry-Perot effects inside the YIG crystal. Figure 3displays the relative transmission difference AT =

2(T+ - T_)(T+ + T-) versus the angle of incidence ofthe Gaussian beam on the YIG, taking into accountthe Gaussian nature of the beam, the interferencesinside the YIG crystal, and the Faraday effect.'3 Thiseffect is added into Eq. (4) and taken into account inthe theoretical curves of Fig. 4. This figure showsthat, as expected, when Akd. is increased, the hystere-sis cycle increases first [Figs. 4(a)-4(c); the rotationmechanism] up to-a limit width [Fig. 4(c)]. Then,from this point, the width of the hysteresis loop de-creases with Ak± [Figs. 4(d) and 4(c); the inhibitionmechanism]. We have also observed that this width isdiminished in the case of the rotation mechanism andincreased in the case of the inhibition mechanismwhen one increases the excitation, in agreement withthe predictions and with the results already obtainedfor linearly polarized eigenstates.4

In summary, we have shown that unlike what waspredicted by Puccioni et al.

8 for a neutral atomic cou-pling transition, no instability could be observed inour experiments. This research shows that atomsplay a crucial role in laser polarization dynamics.They indeed determine not only the competition be-tween the eigenstates but also the nature of the eigen-states. In particular, no vectorial chaos with a linearpolarization can be obtained from such a laser, unlikein the case of the weak atomic coupling transitions.6

References

1. W. Van Haeringen, Phys. Rev. 158, 256 (1967).2. W. J. Tomlinson and R. L. Fork, Phys. Rev. 164, 466

(1967).3. Y. C. Chen and J. M. Liu, Appl. Phys. Lett. 46,16 (1985);

G. Ropars, A. Le Floch, and R. Le Naour, Europhys.Lett. 3, 695 (1987).

4. A. Le Floch, G. Ropars, J. M. Lenormand, and R. LeNaour, Phys. Rev. Lett. 52,918 (1984).

5. G. P. Puccioni, G. L. Lippi, N. B. Abraham, and R. T.Arrechi, Opt. Commun. 72, 361 (1989).

6. P. Glorieux and A. Le Floch, Opt. Commun. 79, 229(1990).

7. H. De Lang and G. Bouwhuis, Phys. Lett. 20,383 (1966);W. Culshaw and J. Kannelaud, Phys. Rev. 156, 308(1967).

8. G. P. Puccioni, M. V. Tratnik, J. E. Sipe, and G. L. Oppo,Opt. Lett. 12, 242 (1987).

9. M. Sargent III, M. 0. Scully, and W. E. Lamb, Jr., LaserPhysics (Addison-Wesley, Reading, Mass., 1974).

10. P. W. Smith, IEEE J. Quantum Electron. QE-8, 704(1972).

11. C. B. Moore, Appl. Opt. 4, 252 (1965).12. J. C. Cotteverte, F. Bretenaker, and A. Le Floch, Opt.

Commun. 79, 321 (1990).13. J. C. Cotteverte, F. Bretenaker, and A. Le Floch, Appl.

Opt. 30, 305 (1991).

I, I