frustrated total internal reflection of laser eigenstates

Balcou et al. Vol. 13, No. 7 /July 1996/J. Opt. Soc. Am. B 1559

Frustrated total internal reflectionof laser eigenstates

Ph. Balcou, L. Dutriaux*, F. Bretenaker, and A. Le Floch

Laboratoire d’Electronique Quantique–Physique des Lasers, Centre National de la RechercheScientifique, Unite de Recherche Associee 1202, Universite de Rennes I, 35042 Rennes Cedex, France

Received June 2, 1995; revised manuscript received January 16, 1996

The role of frustrated total internal reflection in the dynamics of laser eigenstates is investigated theoreticallyand experimentally. We first derive the Jones matrix of a frustrating element for a realistic Gaussian beamin a single-pass geometry. We point out the existence of three different angular regimes, namely, a purefrustration regime, an intermediate regime, and a quasi-Fabry–Perot regime. We then explore in each casethe nature and the competition between the laser polarization eigenstates. A novel spiraling behavior of thelaser parameters is demonstrated when the frustrating gap is varied, along with puzzling polarization-flippingeffects, that may modify strongly the conditions that optimize the laser-output power. Experimental resultsagree with a theoretical model. 1996 Optical Society of America

1. INTRODUCTIONFrustrated total internal reflection (FTIR) of plane waveshas been extensively studied in single-pass experimentssince the early studies of Quincke1 and Hall2 (for a re-view see Zhu et al.3). It has been used in laser physicsfor intensity control,4 energy extraction,5–7 Q switching,8

and, more recently, optical bistability.9 The cavities usedin these cases usually exhibit strong anisotropies, eitherloss anisotropies or phase anisotropies induced by total in-ternal reflection. To our knowledge, the effects of theseanisotropies on the nature and the dynamics of polariza-tion eigenstates has not been considered so far. Such astudy requires prior determination of the Jones matrix ofthe FTIR element. In the case of tilted plates the Jonesmatrix has been shown to depend on the characteristicsof the incident Gaussian beam.10 The aim of this paperis therefore to determine the actual Jones matrix of afrustrating element for a Gaussian beam and to inves-tigate the related behavior of the polarization eigenstatesof a quasi-isotropic laser containing an element on whichFTIR can occur.

In order to obtain the Jones matrix associated withthe incident Gaussian beam, we perform in Section 2 acomprehensive study of the FTIR of a realistic polarizedGaussian light beam. Usually, in order to ensure an eas-ier control of the frustration spacing, the incident an-gle of the beam is chosen close to the critical angle ofincidence.11 This leads us to expand the beam in plane-wave components, which can be individually either in aFTIR regime or in a Fabry–Perot regime, depending onwhether their incidence angles are larger or smaller thanthe critical angle.12 For the whole Gaussian beam, threedifferent regimes result, whose specific properties are in-vestigated. Then, using the Jones matrix of Section 2,we calculate the eigenstates of a quasi-isotropic laser cav-ity inside which FTIR occurs. We explore the competi-tion between these vectorial eigenstates, which resultsin particular in polarization flipping effects as the dis-

0740-3224/96/071559-10$10.00

tance between the two frustrating prisms is changed. Wediscuss the specific vectorial properties of laser eigen-states in the framework of the energy extraction problem,which is a major issue in the growing field of microlaserdevices.13 We then investigate possible applications forlaser polarization control and writing in a vectorial opticalmemory.

2. JONES MATRIX OF FRUSTRATEDTOTAL INTERNAL REFLECTIONOF GAUSSIAN BEAMSIn order to study the effect of frustration in an activelaser cavity, we need to determine first the Jones matrixof the frustrating element for a Gaussian beam. Thecoefficients of this matrix are calculated theoretically inSubsection 2.A. Subsections 2.B and 2.C then presentand discuss the experimental verifications obtained.

A. Theoretical Study of Frustrated TotalInternal Reflection of a Gaussian BeamWe consider a TE or a TM linearly polarized Gaussianbeam (with a wave vector and a wavelength, in vacuum,of k0 and l0) incident from a medium 1 of index n1 ontoa thin film of index n2, followed by a third medium ofindex n3 (see Fig. 1). The corresponding wave vectorsand wavelengths are denoted by kj and lj in medium j .We consider the case in which n2 , n1, for which total in-ternal reflection occurs between media 1 and 2 for angleslarger than the critical incidence angle ic arcsinsn2yn1d.We also restrict ourselves to the case n3 n1, which isthe most common one in practice. The expression forthe normalized amplitude of the incident Gaussian beampropagating along the z axis in medium 1 is14

Gsrd

ps2k1ypb1d1/2

1 1 2izyb1exp

"2

k1sx2 1 y2db1 1 2iz

#, (1)

1996 Optical Society of America

1560 J. Opt. Soc. Am. B/Vol. 13, No. 7/July 1996 Balcou et al.

where b1 k1w20 is the confocal parameter, w0 is the beam

waist, and r sx, y, zd. We assume a time dependenceof the form exps2ivtd. Gsrd can be expanded as a sumof plane waves by a Fourier transform in the transverse(x, y) plane:

Gsrd 1

2p

Z 1`

2`

Z 1`

2`

Askx, kyd expsik ? rddkxdky , (2)

with

Askx , ky d w0p2p

exp

242w2

0

4

≥k2

x 1 k2y

¥35

w0p2p

exp

0@2u2

x 1 u2y

u20

1A, (3)

where ux kxyk and uy kyyk. u0 is the angularwidth of the plane-wave (Fourier) distribution and cor-responds to the divergence of the beam: u0 lypw0

(half-divergence at 1ye). As shown in Fig. 1, the beamimpinges on the interface in the (x, z) plane with an in-cidence angle i0. The incidence angle of a specific planecomponent (ux, uy) is therefore i0 1 ux at first order. Theamplitude transmission and reflection coefficients for thisplane wave in j TE or TM polarization are given by15

tj si0 1 uxd 1 2 srj d2

1 2 srj d2 expsidd, (4)

rj si0 1 uxd rj f1 2 expsiddg

1 2 srj d2 expsidd. (5)

In these expressions, rj represents the reflection coef-ficient from the first medium on the second one for eitherj TE or TM polarization, as given by the Fresnel for-mulas. d is given either by

d 2k2d

"1 2

n21

n22

sin2si0 1 uxd

# 1/2

(6)

for i0 1 ux , ic (d is then the phase difference betweentwo successively reflected beams in the second medium ofwidth d) or by

d 2ik2d

24n21

n22

sin2si0 1 uxd 2 1

351/2

(7)

for i0 1 ux . ic. The first case corresponds to the par-tial reflection domain, for which the device acts as aFabry–Perot interferometer, with peaks in transmit-ted intensity when the following resonance condition issatisfied:

2d cossird pl2, (8)

where ir is the refraction angle in the air gap, d the spac-ing, and p the resonance order. The second case corre-sponds to the total-internal-reflection domain, for whichthe transmitted intensity decreases almost exponentiallywith the gap.16,17

The transmitted and the reflected amplitudes GjT and

GjR of a realistic beam can now be computed by a Fourier

transformation of the transmitted and the reflected am-plitudes back to the spatial domain:

GjT srd

12p

Z 1`

2`

Z 1`

2`

tj si0 1 kxyk1dAskx, kyd

3 expsik ? rddkxdky , (9)

GjRsrd

12p

Z 1`

2`

Z 1`

2`

rj si0 1 k0x0y k0

10 dAsk0x0 , k0

y 0 d

3 expsik0 ? r0ddk0x0dk0

y 0 , (10)

where we introduce a specularly reflected system ofcoordinates (x0, y 0, z0). From these equations one maypredict the existence of three different regimes: (i) Allthe plane-wave components can have incidence anglesgreater than the critical angle ic (pure frustration regime).(ii) One may have part of the plane waves in a total-internal-reflection region and part in a Fabry–Perot re-gion (intermediate regime). This will happen when thedifference between the mean incidence angle i0 and ic

is less than or of the order of the beam divergence u0.(iii) Finally, all the plane waves can have incidence an-gles smaller than ic (quasi-Fabry–Perot regime). Thisis illustrated in Figs. 2(b), 2(d), 2(d0), and 2(f), whichpresent the computed transmitted intensity, as explainedbefore. Clear differences appear indeed between thethree regimes: One goes from a decaying exponentialfor i . ic [Fig. 2(b)], to an intermediate case that dis-plays a typical revival [Figs. 2(d) and 2(d0)], and finallyto a quasi-Fabry–Perot regime [Fig. 2(f)]. A more thor-ough discussion is presented along with the experimentalresults of Subsection 2.C.

This study is applied to laser resonators containing anelement on which FTIR occurs. It is hence necessary tocalculate the Jones matrix associated with this element inorder to determine the eigenstates of an arbitrary lasercavity. This matrix reads

MFTIR

"rTE 00 rTM

#. (11)

Fig. 1. Experimental setup for Gaussian-beam frustration: P1,deviating prism; P2, frustrating prism; PZT, piezoelectric trans-ducer; BS, beam splitter; L, lens. The light source LS and thedetector D consist either of a red He–Ne laser and a photodiodeor of a white lamp and a spectrometer.


Fig. 2. Comparison between experimental (left column) andtheoretical (right column) transmission rates for a Gaussianbeam with TE (solid curves) or TM (dashed curves) polariza-tion. (a), (b) Incidence angle i0 45.41± (pure frustrationregime). (c), (d), (c0), (d0) i0 45.21± (intermediate regime).(e), (f) i0 44.92± (quasi-Fabry–Perot regime). The confocalparameter is b 147 cm (loose focusing), except for (c0) and (d0),for which b 10 cm (strong focusing).

We need therefore to compute the values of the complexreflection coefficients in amplitude rTE and rTM, in spiteof the fact that the reflected beam is no longer perfectlyGaussian. We define rTE and rTM as the projections ofthe reflected TE and TM beams given by Eqs. (9) and (10)onto the amplitude of the cavity (TEM00) mode:

rTE Z 1`

2`

Z 1`

2`

Gpsx0, y 0dGTER sx0, y 0ddx0dy 0, (12)

rTM Z 1`

2`

Z 1`

2`

Gpsx0, y 0dGTMR sx0, y 0ddx0, dy 0. (13)

It is worthwhile to note that these coefficients do notdepend on the position of the interface with respect to thebeam waist (coordinate z). The coefficients could indeedbe also calculated in the (kx, ky) space, in which neitherthe beam amplitudes [given by Eq. (3)] nor the reflectivi-ties [Eq. (5)] depend on z. rTE and rTM therefore dependonly on the incidence angle i0 and on the Gaussian beamwaist w0.

In spite of the rather complex expressions for rTE andrTM, the resulting Jones matrix can easily be computedas a function of gap spacing, for any incidence angle andGaussian-beam geometry. We performed these calcula-tions to characterize the eigenstates of a laser resonatorin the conditions considered in Section 3. These data arepresented in Section 3 and are compared with experimen-tal results.

B. Single-Pass ExperimentsThe experiment consists of measuring the transmitted in-tensity of a well-controlled Gaussian beam as a functionof the frustration spacing for different values of the in-cidence angle. We focus here on the study of transmis-sion coefficients, in order to check our theoretical analysis.Indeed, the intensity transmitted by frustration drops tozero when the gap increases; thus measurements can beperformed relative to a zero background.

The experimental setup is presented in Fig. 1. Weuse a Gaussian laser beam from a He–Ne laser operat-ing in the near-infrared at l0 3.3922 mm. This wave-length presents the advantage of being roughly five timeslarger than conventional optical wavelengths. Moreover,He–Ne lasers supply almost perfectly Gaussian-beamprofiles when operated on their TEM00 transverse mode.This feature is in contrast with the inherently highly di-vergent profiles typically obtained at microwave frequen-cies, and it allows much more precise comparisons withtheory.

The frustrating device consists of two right-angleprisms P1 and P2, positioned as shown in Fig. 1. Frus-tration is obtained here by approaching two plane inter-faces. The prism material is high-quality fused silica,whose refractive index at 3.39 mm is n1 1.409. Thecritical incidence angle ic is then equal to 45.21±. Auto-collimation of the incident beam on the entrance surfaceof P1 yields the exact position of i0 45±. The coinciden-tal vicinity between this angle and ic thus allows us todetermine the incidence angle in absolute value to within1 arcmin. P2 is maintained on a piezoelectric transducer(PZT), whose maximal nominal extension is 60 mm. Theparallelism between the surfaces of P1 and P2 can be ad-justed by orientation of the axis of PZT, with an accuracybetter than 0.1 mrad. Great care was taken to keep thesurface clean and the setup stable.

The spacing between the prisms is controlled in twosteps. The first one consists of making the prisms par-allel and reducing the spacing to its minimal value. Tothis aim, we directed white light from a lamp (light sourceLS) onto the two parallel interfaces. Retroreflected lightfrom the interfaces is separated by a beam splitter BSand imaged onto a direct vision spectrometer. When thespacing is of the order of a few micrometers or less, veryclear thin-film colors appear, and well-contrasted fringesare observed on the white-light spectrum. This allowsus to ascertain the minimal approach distance, which isequal to 750 nm in all our experiments.

All our experimental data were obtained by recordingof the transmitted intensity when the gap spacing is in-creased from the minimal value to ,50 mm, with simulta-neous determination of the spacing. This was achievedby replacement of the lamp by an He–Ne laser beam


(l 632.8 nm), leading to the appearance of large con-centric fringes monitored by a photodiode. In the exper-iment the transmission signal and the fringe intensityare recorded together by a computer. Numerical treat-ment then allows us to obtain the transmitted intensityas a function of spacing, with an accuracy of the order of50 nm.

A common setup in studies of FTIR in the optical do-main consists of clamping two prisms together with spac-ers and of pushing the second prism P2 until it bendsenough to have its central part nearly collide with P1.18

This allows one indeed to reach spacings of ,0.5 mm, butit presents the drawback of distorting the surfaces and ofcreating stress-induced birefringence in the prisms, whichwould result in spurious effects when we come to intracav-ity studies (see Section 3). In our setup the prisms maycome in contact at the minimal distance, but no stress isapplied.

C. DiscussionExperimental and theoretical results for Gaussian beamsare presented in Fig. 2, on the left and on the right-hand-side columns, respectively. Figures 2(a), 2(c), and 2(e)display the transmitted intensities in TE (solid curve) andTM (dashed curve) polarizations for incidence angles i0

of 45.41±, 45.21±, and 44.92±, respectively, and in a loosefocusing geometry. The confocal parameter b 147 cm(b 2pw2

0yl0, where w0 is the beam waist) is close tothat of the cavity mode of Section 3. Figures 2(c0) and2(d0) show the result of stronger focusing (b 10 cm) onthe intermediate case (i0 ic 45.21±). We now discussthe results for the three angular regimes in turn:

(i) Figures 2(a) and 2(b) present typical data obtainedin the pure frustration regime (i0 45.41±). The theo-retical calculation matches the experimental data. Thecurves display the typical quasi-exponential behavior pre-dicted by the plane-wave formulas for the transmissioncoefficients, which means that the finite character of thebeam profile has little effect on the transmitted intensityin this case.

(ii) The results of the intermediate regime i0 ic areshown in Figs. 2(c) and 2(d). Half of the Fourier com-ponents of the beam are in a frustrated total reflectionregime, and half are in a partial reflection (Fabry–Perot)regime, so that no plane-wave formula exists that could becompared with the data. The first parts of the curves arevery similar to the quasi-exponential decrease obtained inthe pure frustration regime, dropping from one to zero inan interval of roughly two wavelengths. The transmis-sion then remains roughly equal to zero for several wave-lengths and subsequently shows a weak revival that de-cays to zero very slowly in turn. Figures 2(c0) and 2(d0)present results obtained at the same angle but with amuch tighter focusing: b 10 cm (w0 200 mm). Theagreement between the experimental data and the the-oretical predictions is quite good. The revival is due tothe beam Fourier components that are in partial reflectionand may reach a Fabry–Perot resonance when the spac-ing is large enough. In particular, the Gaussian beamstill has Fourier components with appreciable amplitudeat incidence angles of the order of ic 2 u0, where u0 repre-sents the beam divergence. An order of magnitude of the

spacing d necessary to obtain the revival can be given bythe condition

2d cos

8<:arcsin

"n1

n2sinsic 2 u0d

#9=; l2, (14)

which yields

d

√pw0l2

8 cos ic

!1/2

. (15)

The minimal spacing d required therefore decreaseswhen the beam waist decreases, that is, when the di-vergence increases. The waist w0 itself depends on thesquare root of the confocal parameter; thus we actually ob-tain a weak dependence on the focusing conditions. Thisestimate yields d 50 mm and d 28 mm in the con-ditions of Figs. 2(d) and 2(d0), respectively, which are ingood agreement with the maxima of the observed revivals.In contrast, the transmitted intensities at these maximaare roughly equal and seem to be almost independent ofthe beam divergence.

(iii) Figure 2(e) shows the experimental results ob-tained for incidence angles i0 smaller than ic for TE andTM polarizations. Resonance peaks (somehow analogousto Airy peaks) are observed, but their height decreasessteadily when the gap is increased, with simultaneousbroadening of the resonance peaks. This behavior is insharp contrast with the plane-wave case, for which a se-ries of equivalent peaks would be expected, with 100%transmission at resonance. It is, however, well repro-duced by the calculations shown in Fig. 2(f), and can beexplained as follows.

As is usual in the calculation of Fabry–Perot res-onators, one may consider the transmitted beam as acoherent superposition of a directly transmitted beam,plus secondary beams reflected an even number of timeswithin the gap. In the present setup, each round tripinside the gap also causes a lateral shift:

Dx 2d tansird cossid, (16)as compared with the direction of propagation. This re-sults in a decreasing contrast of the interferences betweenthe beams transmitted after internal reflections and thedirectly transmitted one. More precisely, the overlap-ping factor between two successive beams can be shownto be g exps2Dx2y2w2

0d.10 This factor depends onlyon the focusing geometry (beam waist) and not on theposition of the interface relative to the waist.10 In theconditions of Fig. 2(e), one has w0 0.75 mm, whereasDx 0.7 mm for d 50 mm; the overlapping factor gthen drops to 0.65. At resonance, the interference be-tween two successive beams can no longer be perfectlyconstructive (their cross product is decreased by g); thusthe maximum transmitted intensity at resonance Imax de-creases. The minimum intensity Imin at antiresonanceincreases for the same reason. The resulting contrast1 2 IminyImax therefore decreases regularly, as can be seenin Figs. 2(e) and 2(f).

To summarize, the finite width of Gaussian beams hasa major incidence on the Jones matrix of a frustrating el-ement for incidence angles close to or slightly less thanthe critical angle. In all cases our experimental resultsare in good agreement with theory. In Section 3 we in-vestigate how frustration governs the eigenmodes of anactive laser cavity.


Fig. 3. Experimental setup for a laser including frustrated re-flection: M1, M2, cavity end mirrors; active medium, laser gasdischarge; DF , diaphragm; Dp, Df, Brewster plate and stressedplate, inducing controllable loss and phase anisotropies; Po,polarizer; D, photodiode.

3. EIGENSTATE DYNAMICS IN A LASERCONTAINING A FRUSTRATED TOTALINTERNAL REFLECTION ELEMENTWe now turn to the study of active resonators fromwhich light is outcoupled by frustration. We first dis-cuss the nature of the eigenstates; we then describethe competition regime between them and show thatpolarization-flipping effects can be expected. An experi-mental demonstration is presented. We also discuss thebistability properties of the device.

A. Theoretical Discussion of EigenstateCompetition with Frustration

1. Nature of the Polarization EigenstatesWe consider a laser cavity as shown in Fig. 3. The na-ture and the frequencies of the polarization eigenstatesare given by the resonance condition

MJonesE lE, (17)

where MJones is the total Jones matrix for a round tripinside the cavity, including the Jones matrix for FTIR,as determined in Section 2. Assuming that all the Jonesmatrices of intracavity elements are diagonal in the (x, y)basis, one sees from Eq. (11) that the eigenstates corre-spond to TE and TM linearly polarized standing waves.

The dynamics of these eigenstates depend stronglyon the relative values of their loss coefficients pTE andpTM and on phase coefficients FTE and FTM, which havebeen measured in Ref. 7. Figures 4(a)–4(c) thereforepresent the variations of the loss anisotropy pTM 2 pTE

(where pj 1 2 r2j and j TE or TM) versus the phase

anisotropy FTE 2 FTM that is due to frustration, whenthe gap spacing is increased from 0 to 60 mm, for threedifferent angles i0 45.41±, i0 45.21±, and i0 45.14±.The direction of increasing gap spacing is indicated byarrows.

In all cases the curve starts from the origin, with no lossor phase anisotropy when the gap is zero (100% transmis-sion for both polarization). The losses then decrease veryrapidly when the prisms are separated. Transmission byfrustration of the TM-polarized wave is always better, asthe evanescent TM wave extends farther than that of a TEwave. Losses on the TM mode are therefore larger thanthose of a TE mode. The loss anisotropy reaches a max-imum for distances of the order of half a wavelength andsubsequently decreases strongly. The subsequent evolu-tions differ for the three regimes:

(i) In the pure frustration regime [Fig. 4(a)], frustra-tion is no longer effective at large distances; thus theloss anisotropy decays to zero. In the meantime thephase anisotropy reaches a maximum and decays tothe anisotropy predicted by the Fresnel formulas for totalinternal reflection.

(ii), (iii) This limit is no longer obtained in the inter-mediate and the quasi-Fabry–Perot regimes [Figs. 4(b)and 4(c)], for which the curves display instead spirals ofdecreasing extension. For plane waves the reflectivitieswould display a periodic Fabry–Perot behavior for i , ic.The representative point in this diagram should hencedescribe a closed loop, with a period d given by the reso-nance condition (8). We have seen in Section 2 that theFabry–Perot fringes obtained for Gaussian beams have adecreasing contrast; thus the maximal extent of loss andphase variations decrease, leading to this spiraling be-havior. Apart from the origin, the loss anisotropies aremaximal at resonance (point B, which is maximal trans-mission) and minimal at antiresonance (points A and C,which are minimal transmission). For very large spac-ings the separation between successively reflected beams,given by Eq. (16), would eventually exceed the width w0

of the mode; thus only the first (directly reflected) beamwould contribute to the reflectivities as given by Eqs. (12)and (13). The central limit of the spiral therefore corre-

Fig. 4. Reflection loss anisotropy versus phase anisotropy in-duced by frustration for Gaussian beams in a loose focusinggeometry (b 147 cm): (a) i0 45.41±, (b) i0 45.21±, and(c) i0 45.14±. The gap increases from 0mm to 60 mm in thedirection indicated by arrows.


sponds to the anisotropy of single-interface reflectivitiesbetween media 1 and 2.

In the intermediate case [Fig. 4(b)], only the Fouriercomponents slightly less than the critical angle contributeto the spiral; thus the anisotropies only whirl once beforestabilizing to an asymptotic limit. This part of the curvecorresponds to the revival observed of the transmittedintensity data.

The evolutions of these loss and phase anisotropies areshown below to govern that of the gain and saturation pa-rameters of the laser, which we now introduce to describethe competition regime between the eigenstates.

2. Stability of the Polarization EigenstatesIn order to show how FTIR modifies the stability of thelaser eigenstates, we need to recall the equations that de-scribe the competition between polarization modes or lon-gitudinal modes.19 The evolution equations of the elec-tric fields ETE and ETM associated with the TE and theTM polarization eigenstates can be derived from Lamb’stheory to read

ÙETE ETEsaTE 2 bTEE2TE 2 uTE/TME2

TMd, (18)

ÙETM ETMsaTM 2 bTME2TM 2 uTE/TME2

TEd, (19)

where aTE and aTM represent the unsaturated gains of themodes, bTE and bTM are the self-saturation coefficients,and uTE/TM is the cross-saturation coefficient. The unsat-urated gains aj aTE or aTM are given by

aj g0 exp

242

√nj 2 n0

DnD

!235 1

c2L

ln Tcav 1c

2Lln Rj ,

(20)where g0 is the unsaturated gain at line center, nj scy2Ldsn 1 Fjypd is the frequency of mode j (n beingan integer), n0 is the central frequency of the transition,DnD is its Doppler width, L is the cavity length, Tcav isthe intensity transmission factor of light going one waythrough the cavity, which takes into account all lossesexcept outcoupling by the prisms, and Rj is the reflectivityof the FTIR device. The self-saturation and the cross-saturation coefficients are given by

bj b0

h1 1 DLsnj 2 n0d

iexp

242

√nj 2 n0

DnD

!235, (21)

uTE/TM u0

24L

√nTE 2 nTM

2

!1 Lsn0 2 nav d

35exp

242

√n0 2 nav

DnD

!235, (22)

where Lsnd g2ysg2 1 n2d, g is the homogeneous linewidth, nav snTE 1 nTMdy2, D is a Lamb dip-reducingfactor, and j stands again for either TE or TM. Twopolarization-flipping mechanisms have been shown toexist in the bistability regime20: polarization rotationand polarization inhibition. Rotation requires very strin-gent conditions of cavity isotropy to occur, and therefore

is deliberately overlooked in the following. In the in-hibition regime and for a given set of parameters thebehavior of the laser can be deduced from the values ofthe cross-saturated gains Gj , given by19

Gj aj 2 uijai

bi, (23)

with i fi j . Assuming all unsaturated gains to be posi-tive, four situations can occur, depending on the signs ofGTE and GTM (see Fig. 5). First, if GTE . 0 and GTM , 0,only the TE eigenstate is stable and oscillates. Con-versely, if GTE , 0 and GTM . 0, only the TM eigenstate isstable and oscillates. Finally, when GTEGTM . 0, the os-cillation regime is governed by the value of the couplingconstant C u

2TE/TMybTEbTM. If C . 1, the two eigen-

states are in vectorial bistability, i.e., they are both stablebut cannot oscillate simultaneously. On the contrary, ifC , 1, the two eigenstates can oscillate simultaneously.In the case of the laser and the eigenfrequencies we con-sider in the following the coupling constant remains inthe range C . 1; thus the region GTE . 0 and GTM . 0is forbidden. Figure 5 summarizes our discussion of thestability of polarization modes as a function of the cross-saturated gains GTE and GTM. We choose to present herethe axes GTE 0 and GTM 0 rotated because this is themost convenient arrangement for the following theoreti-cal discussion (Fig. 6). The area in which only TE (TM)is stable is represented in horizontal hatches (verticalhatches). The bistability region corresponds to GTE , 0and GTM , 0 (dotted area). The region GTE . 0 andGTM . 0 is forbidden.

The use of this cross-saturated gain diagram now al-lows us to predict the polarization behavior of the laserfor each of the three angular domains, whatever lossand phase anisotropies exist in the cavity. Inspection ofFigs. 4(b) and 4(c) suggests that choosing auxiliary cavityanisotropies opposite the center of the spiral should resultin an especially interesting behavior. We therefore chosesuch conditions to examine the evolution of the laser com-petition regime, both theoretically and experimentally.

Fig. 5. Saturated gain diagram, indicating the polarizationregime of the laser, as a function of the cross-saturated gains ofthe TE and TM modes. Only TE (TM) can oscillate if GTE . 0and GTM , 0 (GTM . 0 and GTE , 0). In the present strongcoupling regime the polarization modes are bistable in the regionin which both gains are negative. The region GTE . 0 andGTM . 0 is forbidden. The axes are rotated to facilitate thecomparison with Fig. 6.


Fig. 6. Theoretical saturated gain diagrams for (a) i0 45.41±,(b) i0 45.21±, and (c) i0 45.14±. We consider thefollowing resonator parameters: (a) pTE 300 3 106 s–1,pTM 296 3 106 s–1, wTE 21 rad, wTM 0 rad, andunsaturated gain at line center g0 334.5 3 106 s–1;(b) pTE 300 3 106 s–1, pTM 318 3 106 s–1, wTE 20.94 rad,wTM 0 rad, and g0 333.6 3 106 s–1; (c) pTE 300 3 106 s–1,pTM 282 3 106 s–1, wTE 0.18 rad, wTM 0.4 rad, andg0 330 3 106 s–1. The value of the coupling constant whenboth eigenstate are at line center is C 1.78.

Figures 6(a), 6(b), and 6(c) present the cross-saturatedgain diagrams computed for the same three angles as inFig. 4 (i0 45.41±, i0 45.21±, and i0 45.14±, respec-tively) and for gap spacings increasing from 0 to 60 mm.The gain (or loss) coefficients are given for one roundtrip and hence are in frequency units (inverse seconds ormegahertz). In particular, the round-trip losses on thereference TE mode are pTE 300 3 106 s–1 in all cases.Open circles are used when the laser is under threshold,and full circles are used when the laser is above thresh-old. We again indicate by arrows the direction of increas-ing spacing. It turns out that the cross-saturated gainsGTE and GTM take nearly opposite values for most of thegap spacings; thus the diagram is given with respect tothe rescaled axes GTE 1 GTM 0 and GTE 2 GTM 0, inagreement with the convention chosen for Fig. 5.

The laser always reaches threshold in the domain inwhich the TE mode dominates because of the larger lossesincurred by the TM mode for short spacings. The valuesof cross-saturated gains are initially very large, whichmeans that the TE mode strongly inhibits the TM mode,and they decrease rapidly until the bistability region isreached. This behavior corresponds to the initial peakin loss anisotropy observed in Figs. 4(a)–4(c), which isfollowed by a return to more isotropic conditions.

(i) Let us analyze first the pure frustration case i0 45.41± [Fig. 6(a)]. The representation point leaves theTE area, crosses the bistability domain, and enters theTM area. No polarization flip happens at the bound-ary between the TE domain and the bistable domain(see Fig. 5), as the TE mode goes on preventing the TMone from oscillating. On the contrary, the polarizationshould flip from TE to TM when it crosses the boundarybetween the bistability and the TM domains, as thelatter becomes dominant at that point, whatever polar-ization state existed before. We therefore predict theexistence of one polarization flip from TE to TM as thespacing is increased. Similar arguments can be used forthe other two angles: In all cases the polarization shouldflip when the system crosses a boundary toward a newdominant mode and never when it crosses a boundary to-ward the bistability region. This rule can also be appliedto analysis of the polarization dynamics in the intermedi-ate and quasi-Fabry–Perot regimes.

(ii) For i0 45.21± [intermediate case Fig. 6(b)], theoscillating point follows a more complex trajectory. Thecross-saturated gains are functions of the loss and thephase coefficients of the modes [Eqs. (20)–(23)]; thus theyfollow qualitatively the variations of the loss and thephase anisotropies presented in Fig. 4, and in particularthey display similar damped spirals. As a result, thesystem first follows the same path as in the previouscase, but subsequently it crosses again the bistabilityregion to reach the TE domain, and eventually it gets backto the bistability region. Two polarization flips shouldtherefore be observed.

(iii) The spiraling behavior is even more obvious fori0 45.14± [quasi-Fabry–Perot regime, Fig. 6(c)], with afinal loop entirely located within the TE domain. Thecavity losses are minimal at antiresonance (points A andC) and maximal at resonance (point B); thus the intra-cavity laser intensity should then be respectively maxi-mal and minimal. The losses are so large at resonancethat the laser actually stops oscillating (open-circle ar-eas). When the prisms are separated, the laser shouldtherefore start oscillating on the TE mode, undergo a po-larization flip to TM, flip again to TE, and stop oscillat-ing. At larger distances, it should start oscillating brieflyagain on the TE mode only.

B. Experimental StudyWe now present experimental verifications of the previousmodel. Figure 3 describes the experimental setup. Weuse a home-made He–Ne laser operating at a wavelengthof 3.3922 mm. This laser is characterized by a stronggain, thus allowing one to study resonators with largeloss coefficients. Our study is therefore mostly relevantto other types of strong gain lasers, such as semiconductorlasers.


The L-shaped resonator consists of a concave mirror M1

that has a 1.2-m radius of curvature, a prism P1, and a flatmirror M2. This setup is similar to that used in Ref. 7.Both mirrors have high reflectivities of 95%. The cavitylength is 1 m and can be finely tuned with a piezoelec-tric transducer. The beam waist of the resulting mode isw0 0.7 mm. The active medium is a 25-cm-long dis-charge tube, filled with a 3He–20Ne 7:1 mixture, at atotal pressure of 1 Torr. A diaphragm is used to selectthe TEM00 transverse mode. The laser operates there-fore on a single longitudinal and transverse mode, whoseprofile was checked to be nearly Gaussian. Light is out-coupled from the cavity by means of FTIR, with the setupfor prisms P1 and P2 being the same as that described inFig. 1. A polarizer Po and a photodiode D allow one tomeasure the output intensity for either TE or TM polar-ization, as a function of the gap spacing d. An additionaltilted window and a stressed plate are inserted inside thecavity to create controllable loss and phase anisotropiesDp and Df.

We studied the power outcoupled from the laser by frus-tration as a function of gap spacing. Figures 7(a)–7(c)present the experimental results in the same conditionsas those considered in our theoretical analysis (Fig. 6).The corresponding theoretical data are presented inFigs. 8(a)–8(c) for comparison. Solid and dashed curvescorrespond to TE and TM polarization, respectively. Wenow discuss the experimental results for the three an-gular regimes.

(i) In the pure frustration regime [i0 45.41±,Figs. 7(a) and 8(a)] the laser reaches threshold on theTE eigenstate for a spacing of ,3 mm. After a rapidincrease in output power, a maximum is reached whenthe frustration losses are roughly equal to the intracavitylosses; for larger spacings the intracavity power remainsalmost constant, and the output decreases proportion-ally to the transmission coefficient of the frustrationsetup, as presented in Fig. 2(a). Up to this point, thelaser behaves as a standard laser with a variable mirrortransmission.14 However, if one further increases thefrustration spacing, the laser flips from the TE to theTM eigenstate (d 6 mm). Surprisingly, a larger max-imum intensity is obtained for the TM than for the TEeigenstate. This occurs when the intracavity losses arelarger for the TE than for the TM eigenstates. Thesefeatures are in agreement with our theoretical model[Fig. 6(a)].

(ii) Two differences arise in the intermediate regime[i0 45.21±, Fig. 7(b)]. The polarization now flips fromTE to TM at a larger distance (d 26 mm) and sub-sequently flips back to TE for d 51 mm. This is ingood agreement with the cross-saturated gain diagramof Fig. 6(b). Moreover, the curve now displays a secondmaximum in TM polarization for spacings of the orderof 35 mm. It is worthwhile to notice that this large fea-ture corresponds to the very faint revival observed in thetransmission coefficients of Fig. 2(c).

(iii) In the quasi Fabry–Perot regime [i0 45.14±,Fig. 7(c)] polarization flips, occurring sooner owing to thespiraling behavior [Fig. 6(a)], are also obtained. Notethat in this case the laser stops oscillating in a large zonearound resonance; oscillation resumes near d 50 mm.

Fig. 7. Laser-output intensity obtained experimentally as afunction of the distance d between the prisms. TE polarization,solid curves; TM polarization, dashed curves. (a) i0 45.41±,(b) i0 45.21±, and (c) i0 45.14 ±. The arrows indicate thatthe curves were recorded when the gap spacing d was increased.The resonator parameters correspond to those given in Fig. 6.

In all cases the agreement between the experimentallymeasured intensities (Fig. 7) and those predicted by the-ory (Fig. 8) is quite satisfactory. The model can there-fore be used to predict the polarization behavior for otherconfigurations with different incidence angles, beam ge-ometry, and cavity anisotropies. Of course, the positionand possibly the number of polarization flips will dependon each specific set of parameters.

C. Bistability PropertiesOur study so far is performed by a monotonic increase inthe gap spacing. One may also study the dynamics of po-larization flips, when the distance between the prisms isvaried periodically. From the cross-saturated gains dia-gram we can predict in particular the occurrence of hys-teresis cycles whenever the system crosses the bistabilityarea of a TE and a TM domain during a round trip. Suchbistability has recently attracted much attention as one ofthe possible means to provide optical memories and log-ical elements.9

We present an experimental demonstration of the po-larization control and bistability in the conditions ofpure FTIR. We choose the experimental parameters of


Figs. 6(a) and 7(a). Figure 9(a) presents the hysteresiscycle on the output intensity, which one obtains by vary-ing periodically the spacing between 4.1 mm and 6 mm,at a 100-Hz frequency. The peak-to-peak amplitude ofthe control voltage applied on the piezo-electric trans-ducer is 1.4 V. In other experiments, Q switching wasobtained by a similar setup with a modulation frequencyof 50 kHz, with a driving voltage of the order of 100 V.5

A very clear hysteresis cycle is obtained, with a con-stant ratio of ,3 between the TM and the TE intensities.When the spacing is increased, the polarization flip oc-curs for the same spacing as in Fig. 7(a) (d 5.7 mm); onthe way back it occurs for d 4.3 mm with a very goodreproducibility. Figure 9(b) shows the time behavior ofthe polarization. The polarization flip time is limitedhere by the cavity decay time.

We obtain polarization control in the present case byvarying the frustration by means of a mechanical device.However, other control methods could also be used to mod-ify the effectiveness of frustration. Instead of varying thedepth of the second medium, one may try to modulate itsrefractive index, for instance, by means of electro-opticmedia. In the case of microchip laser this would allow usto control the polarization with an external signal. An-other possibility would be to induce the polarization flipby injection of an external laser beam, thus realizing anall-optical polarization control.

Fig. 8. Theoretical modeling of the laser-output intensitycurves of Fig. 7: (a) i0 45.41±, (b) i0 45.21±, and(c) i0 45.14±.

Fig. 9. (a) Hysteresis cycle of laser-output intensity as a func-tion of the distance d between the prism. Path (1) is obtainedby an increase in d, path (2) by a decrease in d. The upper statecorresponds to TM polarization, the lower one to TE polarization.(b) Successive switching between TE and TM polarization modes,obtained by modulation of the separation d in time. Experimen-tal conditions are similar to those of Fig. 7(a).

4. CONCLUSIONIn conclusion, we have shown how to determine the Jonesmatrix associated with a FTIR element, taking into ac-count the Gaussian nature of the light beam. This ledus to distinguish between three different regimes depend-ing on the angular domain: (i) a pure frustration regime,(ii) an intermediate regime, and (iii) a quasi-Fabry–Perotregime. In all cases the Jones matrix has been seen todepend only on the beam waist, but not on the position ofthe element along the laser axis.

In an active intracavity configuration the Jones ma-trix has allowed us to derive the loss and the phaseanisotropies induced by FTIR. Novel eigenstate dynam-ics were predicted and experimentally demonstrated ineach of three isolated regimes. This dynamics resultsin peculiar polarization-flipping effects, whose considera-tion may be instrumental to determine the coupling condi-tions that optimize the output intensity. Indeed, the po-larization flips induced by frustration result in dramaticvariations of the output power. Moreover, in the in-termediate and the quasi-Fabry–Perot regimes we haveshown that the laser oscillating point undergoes a spiral-ing evolution in the cross-saturated gain diagram whenthe frustration spacing is varied, which leads to multiplepolarization flips. These features may lead to interest-ing applications for microlaser devices, for which FTIRis one of the few methods available to outcouple light.We have also demonstrated that frustration of total re-flection provides a means to control the polarization ofthe laser, thus realizing an optical bistable device. Weachieved this control by varying the frustration gap; how-


ever, other methods could also be considered, with, forinstance, elecro-optic materials. Alternatively, one mayuse injection of an external light beam by frustration tocommand the flip between the two orthogonal polarizationstates. Moreover, several frustration devices can easilybe implemented on a single resonator. This versatilitymay allow one to reach multiple control on this nonlin-ear system and to obtain more complex optical logical el-ements.

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

*Present address, Lycee Joliot-Curie, 144 Boulevard deVitre, F-35700 Rennes, France.

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frustrated total internal reflection of laser eigenstates

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