correlation between polarization and intensity in argon-ion lasers

2276 J. Opt. Soc. Am. B/Vol. 11, No. 11/November 1994

Correlation between polarizationand intensity in argon-ion lasers

Pierre Frere, Andre Daud6, and Albert Le Floch

Laboratoire d'Electronique Quantique-Physiques des Lasers, Unite Associ~e Centre Nationalde la Recherche Scientifique 1202, Universit6 de Rennes I, 35042 Rennes Cedex, France

Fabien Bretenaker

Laboratoire d'Electronique Quantique-Physique des Lasers, Unit6 Associee Centre National de la RechercheScientifique 1202, Universit6 de Rennes I, 35042 Rennes Cedex, France, and Soci6t6 d'Applications

G6ndrales d'Electricit6 et de Mecanique, 72-74, rue de la Tour Billy, B.P. 72, 95101 Argenteuil Cedex, France

Received February 8, 1994; revised manuscript received June 7, 1994

The nature and behavior of the eigenstates of a multimode argon-ion laser containing a variable linear lossanisotropy and submitted to a longitudinal magnetic field are theoretically and experimentally investigated.In particular it is shown that for a given value of the loss anisotropy a transition occurs for a given value of thelongitudinal magnetic field. This defines the concept of a critical magnetic field for multimode lasers. Forthis value of the magnetic field the eigenstates of the multimode laser pass from linearly polarized eigenstatesto circularly polarized eigenstates. The shapes of the output spectra for the different behaviors are discussedand observed, and the correlation between the nature of these eigenstates and the output power of the laseris investigated. The puzzling output power enhancement caused by the insertion of a Brewster window isinterpreted.

1. INTRODUCTION

In typical laser models the laser intensity can be calcu-lated for a given polarization eigenstate of the laser.For example, Lamb's theory1 applies particularly wellto the case of a laser that is linearly polarized. Be-cause the light-matter interaction depends on itspolarization,2 '3 the laser intensity generally dependson the polarization of the laser light.4 Consequently,when one intends to build a laser, the following ques-tions arise: which polarization must be chosen to op-timize laser operation? What happens to the laser ifan external parameter changes the polarization eigen-state? In particular, does this change induce a modifi-cation of the laser output power? In many cases, suchas in the frequency tuning of lasers5 7 or in improv-ing the efficiency of the pumping electric discharge,8-1 1

transverse or longitudinal magnetic fields have beenused to improve the performance of lasers. In par-ticular, since the early developments of multimodeargon-ion lasers, a strong longitudinal magnetic fieldhas been used to confine the electrons at the centerof the discharge tube. 8 '9 This longitudinal magneticfield gives rise to strong Zeeman and Faraday effects.Depending on the presence of a Brewster plate insidethe laser cavity, the output of the laser is composed ofeither linearly or circularly polarized longitudinal modes.Moreover, an old puzzling effect remains, to the best ofour knowledge, unexplained: the output power of thelaser increases in the presence of the Brewster plate.'' 12

This power enhancement is paradoxical because the in-sertion of an element inside the cavity usually leads toan increase of the losses and a reduction of the output

power. The order of magnitude of the relative output-power augmentation that is due to the introduction of theBrewster window is -10%. Such a correlation betweenintensity and polarization seems to have many pointsin common with the bifurcations observed in monomodequasi-isotropic lasers. 3 Our aim in this paper is conse-quently to investigate the bifurcation from linearly po-larized modes to circularly polarized modes in argon-ionlasers relative to a variable longitudinal magnetic fieldand, in contrast to previous experiments,8 9 a variableintracavity loss anisotropy. In Section 2 no intracavityloss anisotropy is introduced so that we may investigatethe nature of the modes of the isotropic laser submit-ted to the longitudinal magnetic field. In Section 3 weintroduce a variable loss anisotropy inside the cavity toinvestigate the transition between linearly polarized andcircularly polarized modes in such a multimode laserwhen the magnetic field is changed. This allows usto investigate the evolution of the laser output powerrelative to the nature of the polarization. Section 4 il-lustrates what happens when the loss anisotropy is toosignificant for the transition to occur, as in the case of acavity containing a Brewster plate.

2. EIGENSTATES OF THEISOTROPIC CAVITY

The experimental arrangement is schematized in Fig. 1.The argon-ion laser oscillates on the 4p2 D;12 - 4s2D3,2transition at A = 488 nm and consists of a 94-cm-longcavity closed by spherical mirror M1 (radius of curvatureR = 3 m) and plane mirror M2. A tilted silica windowcan be introduced inside the cavity with a variable angle

0740-3224/94/112276-06$06.00 ©1994 Optical Society of America

Frere et al.

Vol. 11, No. 11/November 1994/J. Opt. Soc. Am. B 2277

lV11 V M 2 P

ACTIVE MEDIUM 2 J4

OUTPUT X k

POWER MODE SPECTRUMt z ANALYSIS

Fig. 1. Experimental setup. The laser tube contains a re-movable tilted plate whose angle i can be changed to tunethe x-y loss anisotropy. A longitudinal magnetic field B isapplied on the discharge. The beam at the output of mirrorM2 is analyzed through a circular polarizer (quarter-wave plateand linear polarizer P) and confocal Fabry-Perot cavity. D's,detectors.

of incidence i. A longitudinal magnetic field B can be ap-plied to the discharge tube (bore diameter, 3 mm) thanksto a solenoid. The laser power is measured at the outputfrom mirror Ml, and the laser spectrum is observed witha confocal Fabry-Perot scanning interferometer. In ad-dition, to avoid saturation of the detector and spuriousbackreflections from the interferometer, we use an at-tenuator, and the axis of the Fabry-Perot is slightlytilted. In this section we remove the intracavity tiltedplate from the beam path. When no magnetic field is ap-plied, the eigenstates of the laser are mutually perpendic-ular, linearly polarized eigenstates fixed by the residualanisotropies of the cavity. We then choose the dischargecurrent (I = 6 A) to be just above threshold, so the laseroscillates on only two successive longitudinal modes, ascan be seen from the spectrum of the laser [see the toptrace of Fig. 2(a)]. As is the case of typical gas lasersoscillating on a J - J ± 1 transition' 4 (weak atomiccoupling 5 ), these two longitudinal modes oscillate on thetwo perpendicular polarizations corresponding to the twolinearly polarized eigenstates of the laser to reduce thecoupling between the two modes. This can be seen inFig. 2(a), for which a linear polarizer is placed in front ofthe Fabry-Perot interferometer.

When a longitudinal magnetic field is applied to the ac-tive medium, the Zeeman sublevels of the upper and thelower levels of the laser transition are split by the Zeemaneffect. In the case of the 4p2D;/2 - 4s2D3 /2 transitionat A = 488 nm, although the Land6 factors of the upper(g = 1.241) and the lower (g = 1.334) levels of the transi-tion are slightly different,8 we can approximate the result-ing gain curve as two frequency-split gain curves for thear+ and o- transitions. The frequency splitting betweenthese two gain curves is then equal to 3.5 MHz/G. For agiven frequency the active medium of the laser is conse-quently equivalent to a circular dichroism associated withthe nonreciprocal Faraday effect caused by the splitting ofthe a-+ and o- dispersion curves associated with the gaincurves. For a given longitudinal mode the Jones matrixfor one round trip in the cavity is consequently given by

M=GFR, (1)

where G and F are the gain and the Faraday-effect ma-trices, respectively, and

R =[r 0]

represents the isotropic losses of the cavity (mirror losses,diffraction losses, and so on). The dichroic gain matrixG of the active medium for one round trip is given by:

1 [ (g+ + g-)2 i(g+- g-)

-i(g+ - g-) 1(g+ + g-) (2)

The a+ and o- gain coefficients g+ and g- at angularfrequency are given by

g±(w) = exp 2goL exp[ ( ) ]j (3)

where go is the unsaturated gain at line center, o+ ando_ are the center angular frequencies of the or- and o-

gain curves, respectively, ku is the Doppler width of thetransition, and L is the cavity length. The Faraday-effectmatrix for one round trip is simply a rotation matrix:

F=[ cosk -sin 1sin 0 cos j (4)

where, for a given frequency, the Faraday rotation angle Ocan be considered roughly proportional to the longitudinalmagnetic field B (Ref. 13) and is given by

0 = k(n -n+)L,

I t(a) I t

(b) l

(c) I tIt

(5)

Cf B =0O GI I A

~EItVo

- -- -B=700G

V

1 1 ~B = 500 GVI

Fig. 2. Analysis of the laser output through the confocalFabry-Perot cavity. No intracavity tilted plate is introducedinside the cavity, and the laser is maintained just abovethreshold to sustain the oscillation of only two longitudinalmodes. (a) B = 0 G. The laser eigenstates are two linearlyx- and y-polarized eigenstates fixed by the residual anisotropiesof the cavity. One of the two longitudinal modes is along thex eigenstate, and the other is along the y eigenstate. Thequarter-wave plate is removed. The linear polarizer is removedin the first spectrum and reinserted to select the x and the ypolarizations in the second and third spectra, respectively (v axis,190 MHz per division). (b) B = 700 G. The laser eigenstatesare o-' and o- circularly polarized eigenstates. One of the twolongitudinal modes is along the a+- eigenstate, and the other isalong the o- eigenstate. The circular analyzer is successivelyremoved and aligned along the a+- polarization and along theo-- polarization (v axis, 730 MHz per division). (c) Observationof the frequencies of the two circularly polarized modes for twovalues of the longitudinal magnetic field. The circular analyzerhas been removed (v axis, 730 MHz per division).

Frere et al.


where k is the wave number and n and n are therefractive indices for the o-+ and the a- polarizations,respectively. The values of n can be obtained from alinear approximation:

go(w - o) (6)

The multiplication in Eq. (1) leads to

M =r r g+ exp(-ib) + g- exp(iq5)2 i[g+ exp(-iok) - g. exp(io)]

fact that the two gain curves are split by the magneticfield can be seen in Fig. 2(c) for two values of the magneticfield. In this figure the longitudinal mode that oscillateson the a+- eigenstate follows the a-+ gain curve, and themode that oscillates on the - eigenstate follows the -gain curve.

When the laser is strongly multimode, which is usu-ally the case in argon lasers, one obtains the experimen-tal spectra of Fig. 3 (discharge current I = 10 A). From

-i[g+ exp(-io) - g- exp(iqs)]1 (7)g+ exp(-io) + g- exp(ik) I

The eigenvectors and eigenvalues of this matrix lead toa -+ eigenstate with gain g+ and a a- eigenstate withgain g-. The frequencies of these eigenstates for the pthlongitudinal mode are given by

c (q0=L P - (8)

where c is the velocity of light and L the cavity length.Now that the eigenstates of each longitudinal modeare known, we must consider the stability of theseeigenstates. We first make the approximation that thecoupling between the successive longitudinal modes issufficiently weak to allow us to consider that the twoeigenstates of a given longitudinal mode are independentof the other modes. This assumption is based on the factthat the laser transition is inhomogeneously broadenedso that each longitudinal mode interacts essentially withits own atomic velocity class. The problem of the sta-bility of the two eigenstates of one longitudinal mode isthen similar to the problem of the stability of the two lin-early polarized eigenstates of a monomode laser.'7 Fora given cavity length and a given longitudinal mode, thetwo circularly polarized eigenstates are submitted to again anisotropy (equivalent to the linear loss anisotropyfor the two linearly polarized eigenstates of Ref. 17) thatis due to the Zeeman effect and to a phase anisotropythat is due to the Faraday effect (equivalent to the linearphase anisotropy of Ref. 17). One then knows that thetwo eigenstates can be simultaneously stable if the effectof the phase anisotropy overcomes the effect of the dichro-ism. In the case of argon lasers, the maximum value ofthe Faraday effect 0 is only a few degrees.9 Further-more, because the magnetic fields considered are strong(as high as 1000 G), the important splitting of the o-+and the a- gain curves leads to a strong gain anisotropy,except near the middle of the two gain curves. Conse-quently, except in the vicinity of the center frequency ofthe initial gain curve (without the magnetic field), one canexpect the effect of the gain anisotropy to overcome thedecoupling induced by the frequency difference betweenthe two eigenstates. Hence only the circularly polarizedeigenstate with the stronger gain will be stable.

This prediction can be tested experimentally thanksto our two-longitudinal-mode laser. When a longitudinalmagnetic field of 700 G is applied, the spectra of Fig. 2(b)are obtained, showing that the laser output consists oftwo longitudinal modes, each one oscillating on one of thecircularly polarized eigenstates and centered on the gaincurve corresponding to the polarization considered. The

Fig. 3(a) one can confirm that, as expected, the longitu-dinal modes are distributed in two packets of oppositecircularly polarized modes. As expected, the modes forwhich the a+ gain is more important that the a- gain os-cillate exclusively on the -+ eigenstates, and vice versa.Moreover, Fig. 3(b) illustrates the splitting of the twoMaxwellian gain curves of the active medium that is dueto the longitudinal magnetic field. These results with astrongly multimode laser show that our assumption thatthe behaviors of the successive longitudinal modes areroughly independent is valid.

3. TRANSITION BETWEEN LINEARLYAND CIRCULARLY POLARIZED MODES

Let us now introduce inside the cavity a tilted silica platewith an angle of incidence i, with the x axis being in theplane of incidence (see Fig. 1). The Jones matrix M forone round trip inside the cavity then becomes

M = TGFTR, (9)

where G and F are given in Eqs. (2) and (4). T is theJones matrix for one pass through the tilted plate and isgiven by

T =t 2]

I t(a) t

Lt

(b)I

(10)

HB = 800 GI I I I I I 1.1 I I

I I I lV'

I 1 .1 I I i I I IT 1 1121.L I I B=800G

I 1_ I IB=50OGk

Fig. 3. Analysis of the laser output through the confocalFabry-Perot cavity. No intracavity tilted plate is introducedinside the cavity, and the laser is now strongly multimode.(a) B = 800 G. The laser eigenstates are or+ and o- circularlypolarized eigenstates. Some of the longitudinal modes are alongthe o-+ eigenstate, and the others are along the o- eigenstates.The circular analyzer is successively removed and aligned alongthe oA polarization and along the o- polarization (v axis,960 MHz per division). (b) Observation of the frequencies ofthe two packets of circularly polarized modes for two values ofthe longitudinal magnetic field. The circular polarizer has beenremoved (v axis, 960 MHz per division).

I " A.1 j_.jWU I II I I '�. I I I I I

I 01PIPLI II I I . - I . I . I

i I I I 1 11111111 1J1 1. 1 1 1

I _. l - I .

Frere et al.

-ji

I


where the electric-field transmission coefficients t and tytake the two air-silica interfaces of the tilted plate intoaccount and are given by Fresnel's laws.'8 The calcula-tion of M leads to

[g+ exp(-io) + g- exp(i 0)]t. 2

i[g+ exp(-io) - g- exp(io)]t~ty

The two eigenstates of this matrix are two ellipticallypolarized eigenstates with the same ellipticity. Let ebe the eccentricity of the eigenstate electric field ellipse.Then Figs. 4(a) and 4(b) show the evolution of e ver-sus B for three different values of the detuning and forAt/t = (to - t)/t, equal to 0.014 and 0.023, respectively.The values of the other parameters are ku/27r = 2.1 GHzand go = 0.037 m-1. These figures clearly exhibit thetransition, for a critical value of the Faraday effect, froma quasi-linearly polarized eigenstate (e - 0) to a quasi-circularly polarized eigenstate (e = 1), as previously ob-served in quasi-isotropic lasers.13 These figures showthat this transition appears for larger values of 'k as At/tis increased, as expected. Owing to the presence of acircular dichroism, the polarization of the eigenstate isalways elliptic (e 0), contrary to that for the quasi-isotropic monomode laser.13 However, one can see inFig. 4 that there always exists a transition from a quasi-linearly polarized eigenstate (E 0) to a quasi-circularlypolarized eigenstate (e 1) for the same critical valueof the magnetic field for all values of wt. Here we can-not really call this soft transition a bifurcation anymore.However, the behavior of the eigenstate polarization re-mains qualitatively the same as in the absence of circulardichroism. Consequently, for a given angle of the intra-cavity window, i.e., for a given value of At/t, we expectour laser to exhibit two different behaviors, dependingon the value of the applied magnetic field. On the onehand, for a value of the magnetic field below the criticalmagnetic field B, corresponding to the critical Faraday ef-fect, we expect our laser to have quasi-linearly polarizedeigenstates. Then, because of the difference between thelosses of these two eigenstates, we expect only one linearpolarization to be stable for every longitudinal mode. Onthe other hand, for a value of the magnetic field abovethe critical magnetic field Bc, we expect the laser to ex-hibit quasi-circularly polarized eigenstates. Then, for agiven longitudinal mode, only one of these two eigenstateswill remain stable, depending on the sign of g+ - g forthat particular longitudinal mode. The spectrum of thelaser will consequently be split in one packet of quasi-a-+ eigenstates and one packet of quasi-ar- eigenstates,as in the case of the laser with no Brewster window(Section 2). As for the output power, if our assumptionthat the dynamics of the successive longitudinal modesare independent is valid, we expect the active medium tobe more strongly saturated in the case of circularly po-larized eigenstates than in the case of linearly polarizedeigenstates, as was shown in Ref. 4. We consequentlyexpect the output power to diminish at the transitionfrom the quasi-linearly polarized eigenstates to the quasi-circularly polarized eigenstates.

These predictions are tested experimentally for anangle of incidence of the tilted window of i = 200, corre-

sponding to At/t = 0.014. Figure 5(a) shows the evolu-tion of the output power versus the longitudinal magneticfield. As expected, a kink appears in this curve forB= = 450 G. The observed reduction of the output power

-i[g+ exp(-i) - g- exp(ik)]t tY . 11)[g+ exp(-i) + g- exp(i)]ty2 j

can be estimated to be 20% around B - 600 G rela-tive to an extrapolation of the curve without the kink[dashed curve of Fig. 5(a)]. This reduction is comparablewith those reported in Refs. 8 and 9. To confirm that thiskink corresponds to the critical magnetic field, i.e., to thetransition from linearly polarized eigenstates to circularlypolarized eigenstates, we analyze the output spectrum ofthe laser for B < B [Fig. 5(b)] and for B > B, [Fig. 5(c)].As expected, for B < B [Fig. 5(b)] the laser modes are lin-early polarized and for B > B, [Fig. 5(c)] the laser spec-trum consists of two packets of opposite circularly polar-ized modes. The width of the transition between the tworegimes in Fig. 5(a) is due to (i) the width of the transi-tion for each longitudinal mode (see Fig. 4) and (ii) thedifference between the values of B, for the successive lon-gitudinal modes that do not experience exactly the sameFaraday effect. These results show that our assumptionthat the successive longitudinal modes are roughly inde-pendent permits us to explain the dynamics of the wholespectrum and the fact that the intensity difference be-tween linearly and circularly polarized eigenstates is dueto the difference between the saturation behavior of thetransition in both cases, just as in Ref. 4. We now con-firm that the critical magnetic field B, increases with thevalue of the loss anisotropy At/t. As in Fig. 5(a), Fig. 6shows the evolution of the output power as a function ofthe longitudinal magnetic field for two values of the tiltangle of the intracavity window: i = 200 (At/t = 0.014)and i = 250 (At/t = 0.023). As expected, the value of

0.8.

0.6.

0.4-

0.2-rs

0

01

0,

01

(a).

-0 1) .Att=O.0 14.0 0 80 .20 1. 0 . .

0 400 800 1200 1600 20C

B (G)

1 . . . . . . . . . .

6 * 400 800 1200 1600 2600

B (G)Fig. 4. Theoretical evolution of the eccentricity e of the po-larization ellipse versus longitudinal magnetic field B for (a)At/t = 0.014 and (b) At/t = 0.023. In both cases the threecurves denoted (1), (2), and (3) correspond to three values of thecavity detuning, namely, 10, 50, and 100 MHz, respectively.

.8 (b)A.

4]. (3) () 1)

2±n. - A =O0.023,

Frere et al.

00


the critical field B, is larger for i = 250 than for i = 200.Moreover, we notice that the ratio of the two successivevalues of B (1.5) is approximately the same as the ra-tio of the successive values of At/t (1.6), showing thatthe critical magnetic field indeed evolves linearly with theloss anisotropy, 3 a fact also consistent with Fig. 4. Forboth values of i one can see the kink corresponding to thetransition between linearly and circularly polarized eigen-states. Notice that the intensity reduction correspondingto the transition is relatively important (-20%).

4. CAVITY CONTAINING ABREWSTER PLATE

Users of argon-ion lasers usually prefer a purely linearlypolarized output beam. This is why one or two Brewsterwindows are often introduced inside the cavity. When

(a)

(b)

I I I I I I I IX I

I 111 I I I AI

0 B

1 - ] 1 I I' l -

V

I I L&I hj~ I (c) I,4 _

I l1 I i

B = 415 G

B = 830 G

V

Fig. 5. The intracavity tilted plate is introduced inside thecavity with an angle of incidence i = 200, and the laser isstrongly multimode. (a) Solid curve, total output power versuslongitudinal magnetic field (B axis, 200 G per division). Noticethe kink for the critical magnetic field B, - 450 G. Dashedcurve, extrapolation of the curve if there were no kink. (b), (c)Analysis of the laser output through the confocal Fabry-Perotcavity; (b) B = 415 G: before the transition. The laser eigen-states are x-polarized eigenstates. The quarter-wave plate isremoved, and the linear polarizer is successively aligned alongthe x and the y polarizations (v axis, 960 MHz per division).(c) B = 830 G: after the transition. The laser eigenstates area-+ and o-- circularly polarized eigenstates. Some of the lon-gitudinal modes are along the -+ eigenstate and the others arealong the a- eigenstates. The circular analyzer is successivelyremoved and aligned along the a-+ polarization and along the -polarization (v axis, 960 MHz per division).

B,(201); 4-c(25')

I I 11

0 BFig. 6. Total output power versus longitudinal magnetic field(B axis, 200 G per division) for two values of the tilt angle of theintracavity window (i = 200 and i = 250).

I(a)

(b Ixt

(b)

_ I IM ZiL- _B = 500 G

I I I I I I I I B = 800 G

I I ., I I I I IV_

Fig. 7. The intracavity tilted plate is introduced inside thecavity at the Brewster angle, and the laser is strongly multimode.Analysis of the laser output through the confocal Fabry-Perotcavity for two values of the longitudinal magnetic field: (a)B = 500 G and (b) B = 800 G. The laser eigenstates are alwaysx-polarized eigenstates. The quarter-wave plate is removed,and the linear polarizer is successively aligned along the x and ypolarizations (v axis, 960 MHz per division).

only one such intracavity window is tilted at the Brew-ster angle, the loss anisotropy already becomes impor-tant. In our experiment the Brewster window leads toAt/t = 0.148. This means that the value of the corre-sponding critical magnetic field also becomes important.For example, here, if the Faraday effect evolves linearlywith B, it should be B, = 4700 G. Anyway, such val-ues of the longitudinal magnetic field are never used, be-cause they lead to a decrease of the gain of visible ar-gon lasers.8 9 This fortunately prevents the appearanceof circularly polarized modes and the associated outputpower decrease in the presence of an intracavity Brew-ster window. We can confirm this experimentally forour laser. Typical output spectra corresponding to thisusual case of a Brewster window are shown in Fig. 7 forB = 500 G and B = 800 G, showing that the laser outputis always linearly polarized. The transition still existsin principle, but one cannot create a Faraday effect suf-ficient to reach it.

5. CONCLUSION

To summarize, the assumption that the successive longi-tudinal modes of a multimode argon-ion laser are roughlyindependent permits us to understand the transition be-tween linearly and circularly polarized eigenstates in alaser containing a loss anisotropy and submitted to astrong longitudinal magnetic field. This understandingpermits the prediction of a critical magnetic field for mul-timode lasers corresponding to this transition, whose ex-perimental value has been shown to increase linearly withthe loss anisotropy. Moreover, this approach explainsthe puzzling difference between the intensities of the laserwith and without an intracavity Brewster window as adependence of the nonlinear saturation of the laser gainon the polarization of the oscillating eigenstate. Finally,this correlation between polarization and intensity andthe fact that the critical magnetic field increases with thelinear loss anisotropy lead to a combination of two ad-vantages for the cavity containing a Brewster window: alinearly polarized output beam and a larger output powerfor usual values of the magnetic field.

U

i i i I I I Ii i i i i 4- IV_

I I 1 6 [A I I I II I I " I I I I �� I I

i i i i i ! i i i

Ii I I I. I I I

Frere et al.

I

II


ACKNOWLEDGMENTS

This research was partially supported by the Directiondes Recherches, Etudes et Techniques and the ConseilR6gional de Bretagne.

REFERENCES

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

2. A. Dienes, "On the physical meaning of the 'two non-degenerate levels' atomic model in nonlinear calcula-tions," IEEE J. Quantum Electron. QE-4, 260-263 (1968);"Polarization-dependent gain saturation and nonlinearity-induced anisotropy in the 3.39-/pk He-Ne laser amplifier,"IEEE J. Quantum Electron. QE-5, 162-166 (1969).

3. M. Sargent III, W. E. Lamb, Jr., and R. L. Fork, "Theory ofa Zeeman laser," Phys. Rev. 164, 436-465 (1967).

4. F. Bretenaker and A. Le Floch, "Nonlinear intensity effectsin a laser generating the three main standing waves," Phys.Rev. A 43, 3704-3709 (1991).

5. M. A. Kelly, J. K. McIver, R. F. Shea, and G. D. Hager, "Fre-quency tuning of a cw atomic iodine laser via the Zeemaneffect," IEEE J. Quantum Electron. 27, 263-273 (1991).

6. G. D. Hager, D. Kopf, B. S. Hunt, B. Anderson, C. Woolhiser,and P. Crowell, "The chemical oxygen iodine laser in thepresence of a magnetic field I: gain measurements and po-larization effects," IEEE J. Quantum Electron. 29, 933-943(1993).

7. G. D. Hager, D. Kopf, B. S. Hunt, B. Anderson, and P.Crowell, "The chemical oxygen iodine laser in the presenceof a magnetic field II: frequency tuning behavior," IEEEJ. Quantum Electron. 29, 944-953 (1993).

8. E. F. Labuda, E. I. Gordon, and R. C. Miller, Continuous-duty argon ion lasers," IEEE J. Quantum Electron. QE-1,273-279 (1965).

9. D. C. Sinclair, "Polarization characteristics of an ionized-gaslaser in a magnetic field," J. Opt. Soc. Am. 56, 1727-1731(1966).

10. S. Kobayashi, T. Kamiya, T. Hayashi, and T. Goto, "Powerenhancement for argon ii narrow tube lasers by a transversemagnetic field," Appl. Phys. Lett. 46, 925-927 (1985).

11. S. Kobayashi, T. Kamiya, and K. Ujihara, "Power enhance-ment for argon II narrow tube lasers due to Zeeman effectby a transverse magnetic field," IEEE J. Quantum Electron.QE-23, 633-640 (1987).

12. A. L. Bloom, Gas Lasers (Wiley, New York, 1968).13. J. C. Cotteverte, F. Bretenaker, and A. Le Floch, "Study

of the dynamical behaviour of the polarization of a quasi-isotropic laser in the Earth's magnetic field," Opt. Commun.79, 321-327 (1990), and references therein.

14. R. Balhorn, H. Kunzmann, and F. Lebowsky, "Frequencystabilization of internal-mirror helium-neon lasers," Appl.Opt. 11, 742-744 (1972); T. M. Niebauer, J. E. Faller,H. M. Godwin, J. L. Hall, and R. L. Barger, "Frequencystability measurements on polarization-stabilized He-Nelasers," Appl. Opt. 27, 1285-1289 (1988).

15. J. C. Cotteverte, F. Bretenaker, and A. Le Floch, "The dy-namics of circularly polarized eigenstates in lasers with non-weak atomic coupling," Opt. Lett. 16, 572-574 (1991), andreferences therein.

16. V. S. Letokhov and V. P. Chebotayev, Nonlinear Laser Spec-troscopy (Springer-Verlag, Berlin, 1977).

17. F. Bretenaker, A. Le Floch, J. Davit, and J. M. Chiquier,"One- and two-eigenstate stability domains in laser sys-tems," IEEE J. Quantum Electron. 28, 348-354 (1992).

18. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon,Oxford, 1993).

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correlation between polarization and intensity in argon-ion lasers

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