Coherent addition of adjacentlasers by forked eigenstate operation

Marc Brunel, Albert Le Floch, Fabien Bretenaker, Jean Marty, and Engin Molva

2402 APPLIED OPTICS

Forked laser eigenstates are shown to provide a powerful tool both to phase lock spatially separated laseroscillators and to add their powers coherently into a TEM00 output beam. Coherent addition of thepowers extracted from two fiber-coupled diode-pumped Nd:YAG channels is theoretically and experi-mentally demonstrated. Pure TEM00 oscillation is obtained with a 20% optical-to-optical conversionefficiency. The coherence of the two-propagation-axis laser is proved, and single-frequency operation isdemonstrated. The scalability of the scheme is discussed. © 1998 Optical Society of America

OCIS codes: 140.0140, 140.3290, 140.3410, 140.3480, 260.5430.

1. Introduction

The development of powerful coherent laser sourcesis mandatory for many applications, such as lasertelemetry, spatial telecommunications, and nonlin-ear optics. The coherent combination of the lightemitted by laser arrays, which has motivated a con-siderable amount of research, would be an ideal so-lution to this objective, avoiding spurious thermaleffects induced by high power densities. With thisaim in view, laser arrays have been shown to oscillatein coupled modes, obtained through partial overlap ofthe adjacent individual emitted fields, as well withsemiconductor laser diodes1–3 and with rare-earth-doped solid-state laser media.4–6 Under specificgeometrical conditions a coherent oscillation can thenbe obtained from the different lasers. However, thespatial diagram of emission is usually multilobe as aresult of the preferential out-of-phase locking be-tween adjacent lasers.4 Other schemes involvingdiffraction mechanisms to couple laser arrays, suchas Talbot cavities7,8 and Fourier-transform resona-tors,9 although they are effective in coherently com-

M. Brunel, A. Le Floch, and F. Bretenaker are with the Labo-ratoire d’Electronique Quantique–Physique des Lasers, UniteMixte de Recherche du Centre National de la Recherche Scienti-fique 6627, Universite de Rennes I, Campus de Beaulieu, F-35042Rennes Cedex, France. J. Marty and E. Molva are with the Labo-ratoire d’Electronique de Technologie et d’Instrumentation, Depar-tement Optronique, Commissariat a l’Energie Atomique, 17 ruedes Martyrs, F-38054 Grenoble Cedex 9, France.

Received 4 August 1997; revised manuscript received 15 Decem-ber 1997.

0003-6935y98y122402-05$15.00y0© 1998 Optical Society of America

y Vol. 37, No. 12 y 20 April 1998

bining the field amplitudes, exhibit multilobe far-field patterns and still require extra external phasefilters to yield an optimal high-brightness beam.Besides, two-dimensional microchip arrays havebeen shown to exhibit a single-lobe far-field patternbut without clear evidence of phase locking.10,11

However, so-called forked eigenstates were recentlyobserved in multiaxis gas lasers.12 These eigen-states consist in intracavity spatially separated par-allel beams that belong to the same laser mode. Onecan hence wonder whether these eigenstates could bea natural solution to the automatic phase-lockingproblem. Consequently our aim in this paper is toinvestigate the possibility for these forked eigen-states to add coherently the powers provided by sev-eral diode-pumped solid-state microlasers.

The two-propagation-axis laser used for the exper-iments is shown in Fig. 1. The active medium is a790-mm-long crystal of 1% at. doped Nd:YAG. Oneface of the crystal is antireflection coated at 1064 nmand transmits 95% of the intensity at 808 nm. Theother face transmits 99% at 1064 nm and reflects 70%at 808 nm. The 125-mm-long resonator is closed bya 200-mm radius-of-curvature concave mirror withtransmission T 5 6% at 1064 nm, which serves as theoutput coupler. A 10-mm-long YVO4 crystal is in-serted inside the cavity. It is cut at 45° of its opticalaxis and is coated for antireflection at 1064 nm onboth faces. Without any other element inside thecavity this birefringent element permits the oscilla-tion of two spatially separated eigenstates, an ordi-nary one with a linear polarization along y and anextraordinary one with a linear polarization along x.They are separated by Dx 5 1 mm in the activemedium and superimposed upon the other side of the

resonator. Owing to the resonator geometry, the or-dinary and the extraordinary beam waists are locatedat the input mirror and have radii w0 5 180 mm.Let us now introduce a quarter-wave plate inside theresonator, between the crystal and the output cou-pler, with its fast axis oriented at an angle C withrespect to the y axis. Depending on the orientationof this plate, two distinct regimes are observed, aspredicted by a self-consistent Jones matrix analysisof the electric field for one round trip inside the cav-ity.12 Let us recall that, in this two-propagation-axis laser, one must use 4 3 4 generalized Jonesmatrices to take into account the transverse shiftbetween the ordinary and the extraordinary beams.To calculate the eigenstates ~eigenpolarizations andeigenfrequencies! one has to solve the followingeigenproblem:

ME 5 lE, (1)

where

M 5

30 0 0 00 2exp~2ife!cos 2C exp i~fe 1 fo!sin 2C 00 exp i~fe 1 fo!sin 2C exp~2ifo!cos 2C 00 0 0 0

4(2)

is the overall Jones matrix for one round trip insidethe cavity starting from the pump input mirror,12 E isthe electromagnetic field four-component eigenvec-tor, and l is the associated eigenvalue. fe and fo arethe phase shifts undergone by the extraordinary andthe ordinary beams, respectively, when they arepropagating through the double-refraction crystal.The threshold condition and the eigenfrequencies can

Fig. 1. Experimental schematic of the two-propagation-axis end-pumped Nd:YAG laser: M, concave mirror; C, double-refractionYVO4 crystal; QWP, quarter-wave plate. C is the angle made bythe fast axis of the quarter-wave plate with respect to the y axis.The laser is pumped by two separate fiber-coupled diode lasers.

be derived from the condition l 5 1. First, when C5 0, one gets the following eigenvectors Ee and Eo:

Ee 5 301004 , (3)

Eo 5 300104 . (4)

The two upper components of these vectors corre-spond to the x and y parts of the electromagnetic fieldpropagating in the extraordinary arm @labeled one inFig. 1#, and the two lower components correspond tothe x and the y parts of the electromagnetic fieldpropagating in the ordinary arm @labeled two in Fig.1#. The results obtained in Eqs. ~3! and ~4! showthat the two eigenstates are spatially separated inthe active medium because each eigenstate has itsnonzero value on either arm of the laser. Hence theeigenstates oscillate simultaneously with indepen-dent phases, like two independent lasers. This isthe same situation as when the quarter-wave plate isremoved from the cavity. Now, on the contrary,when C Þ 0, the two linear eigenpolarizations de-fined by the double-refraction crystal are mixed bythe quarter-wave plate. Indeed, one can see fromEq. ~2! that the Jones matrix for one round trip insidethe cavity becomes nondiagonal, showing that thelaser eigenstates can no longer simply remain anordinary and an extraordinary eigenstate. The res-olution of Eq. ~1! yields the eigenvectors E1 and E2:

E1 51

Î2 301104 , (5)

E2 51

Î2 301

2104 . (6)

Inasmuch as each eigenvector now has one nonzerocomponent in each arm of the laser, the two arms ofthe laser become entangled, with a deterministicphase difference. The laser eigenstates are thencalled forked. In that case the eigenstates have twoarms on the plane mirror side of the cavity, i.e., in theactive medium, which are not only recombined by thedouble-refraction crystal on the other side of the res-onator but are also phase locked. In particular,when C 5 py4, the powers in the two arms of theforked eigenstate are equal. Moreover, there is onlyone beam at the output of the laser, defined by thegeometrical parameters of the resonator. Further-more, it contains the total energy extracted coher-ently from the two gain zones of the active medium.

20 April 1998 y Vol. 37, No. 12 y APPLIED OPTICS 2403

To pump this two-axis laser efficiently we use twofiber-coupled laser diodes, each emitting 420 mW at808 nm. The ends of the fibers, which have 200-mmcore diameters and 0.25 numerical apertures, aresimply placed against the input mirror of the laser.They are separated by 1 mm. As shown in Fig. 2, themeasured threshold pump power is 226 mW and themaximum output power is 160 mW, which corre-sponds to an overall optical-to-optical conversion ef-ficiency of 20%. To quantify the effect of the extralosses introduced by the birefringent elements in-serted inside the cavity we measure the thresholdand maximum output powers of the laser without thedouble-refraction crystal and the quarter-wave plate.As shown in Fig. 2, the optical-to-optical conversionefficiency is 22.5%. It appears hence that the mul-tiaxis laser is an efficient means to extract the energyof both gain zones, with only little cost in efficiency.

Let us now investigate whether this increase inoutput power has left the output TEM00 mode unaf-fected. We check the transverse mode profile at theoutput of the laser. The quarter-wave plate is firstset at C 5 0. In that case the two eigenstates be-have as two independent lasers, an ordinary one andan extraordinary one @see Eqs. ~3! and ~4!#. They areindependently pumped by the two pump lasers. Thetransverse mode profile obtained when only the ordi-nary eigenstate oscillates is reproduced in Fig. 3

Fig. 2. Experimental output power at 1064 nm versus inputpower at 808 nm of the two-propagation axis laser of Fig. 1 ~circles!and of the corresponding single-axis laser when the crystal and thequarter-wave plate have been removed ~squares!.

Fig. 3. Experimental intensity profiles at the output of the laserof Fig. 1 when C 5 0 and only one channel is pumped ~squares! andC 5 py4 and both channels are pumped ~circles!. Solid curves,Gaussian fits.

2404 APPLIED OPTICS y Vol. 37, No. 12 y 20 April 1998

~squares!, together with a Gaussian fit ~solid curve!.Now, with C 5 py4 and with both channels pumpedand phase locked @see Eqs. ~5! and ~6!#, one obtainsthe forked eigenstate profile of Fig. 3 ~circles! to-gether with its Gaussian fit. The radii w of the twomeasured TEM00 profiles are equal to better than 1%and correspond to what is expected from this stablecavity laser, which demonstrates the high quality ofthe spatial mode of the forked eigenstate laser, asexpected. Forked eigenstates are thus emitted in aperfect Gaussian beam with more power than thecorresponding single-axis laser ~without birefringentelements and pumped by a single diode!. Moreover,we insist on the fact that the two eigenstates followthe same Y-shaped path, which we prove by placingan obstacle in the ordinary or the extraordinary path:Indeed, the laser oscillation is then switched off, tothe contrary of what happens in the case of the exis-tence of an ordinary and an extraordinary eigenstatethat do not follow the same path ~when C 5 0!. Wenow focus on the phase behavior of the laser.

The coherence of the laser is evidenced by means ofa two-slit interferometric experiment.5 The x- andthe y-polarized components of the laser output are firstspatially separated by an extra double-refraction crys-tal. They are then directed at a slight difference inangle to a photodiode bar, placed at 5 m, where the twobeams are mixed. The angular separation is per-formed through the combination of a beam splitter anda mirror. Two diaphragms provide the selection ofthe correct polarizations at the output of the beamsplitter. A polarizer is placed in front of the detectorto make the two orthogonal polarizations interfere.This simple interferometric apparatus allows one tomonitor directly the coherence between the two orthog-onally polarized components of the laser. Indeed, thevisibility of the fringe pattern displayed on the photo-diode bar gives a direct quantitative evaluation of thedegree of coherence between the two arms of the laser,as shown in Fig. 4. First, the quarter-wave plate isset at C 5 0. The two components x ~extraordinary!and y ~ordinary! then oscillate with a random phasedifference. As a result, no interference pattern is ob-served @see Fig. 4~a!#. We now set the quarter-waveplate at C 5 py4. The eigenstates of the laser arethen forked: Both x- and y-polarized components arephase locked and belong to the same eigenstate. The100% visibility fringe pattern that results is shown inFig. 4~b! and illustrates the dramatic behavior of theforked eigenstates, which coherently combine the twospatially separated oscillators. Indeed, the result ofFig. 4~a! proves that the 1-mm spatial separation of thetwo eigenstates is large enough for the two lasers toremain unlocked.4,5 The locking obtained in Fig. 4~b!is hence totally different from evanescent laser lockingand is inherent in the nature of forked eigenstates.Note that the interference fringe pattern does not giveany direct information on the number of oscillatingaxial modes of the laser. One can hence wonderwhether this two-axis laser could be really single fre-quency. To check this point we now analyze the lasermode spectrum.

Let us first recall that, when C 5 0, the laseroscillates in two different sets of frequencies, eachassociated with one of the spatially separated eigen-states ~the ordinary one and the extraordinary one!.The competition between the two eigenstates is com-pletely avoided in this case. On the contrary, whenC 5 py4, the eigenstates are forked and superim-posed inside the active medium. Indeed, both E1

and E2 have one nonzero component on each arm@see Eq. ~4!#. Thus the coupling between the twoforked eigenstates must be different from zero.13

We now check these two situations experimentally.When C 5 0, the laser operates with two or threeaxial modes on each arm, depending on the detuningof the cavity. To force single-frequency operation oneach arm we insert an uncoated 2-mm-thick silicaetalon into the one-arm side of the laser. The laserspectrum is monitored by a 7.5-GHz free-spectral-range confocal Fabry–Perot analyzer, as shown inFig. 5~a!. Two eigenstates, an ordinary one and anextraordinary one, oscillate simultaneously, as ex-pected. Their frequency difference depends on thephase shifts introduced by the double-refraction crys-tal. To obtain the forked eigenstate operation wethen rotate the neutral axes of the quarter-waveplate by 45° with respect to the y axis. The corre-sponding laser spectrum is displayed in Fig. 5~b!.Because of the coupling between the two forked ei-genstates, even though it is relatively weak we obtainthe oscillation of only one frequency with approxi-mately the total power contained in the former twoseparated eigenstates @compare Figs. 5~a! and 5~b!#.Besides, because our etalon was not optimized to thatparticular purpose, the laser power drops to 90 mW.Nevertheless, this outcome proves that forked eigen-state operation has two consequences: it phaselocks the two arms of the laser and it adds the powers

Fig. 4. Experimental observation of the interference fringes ob-tained between the two spatially separated orthogonally polarizedx and y components of the laser of Fig. 1. ~a! C 5 0; fringecontrast, 0. ~b! C 5 py4; forked eigenstates; fringe contrast, 1.

initially distributed into several frequencies into asingle TEM00 supermode.

In conclusion, it has been demonstrated that forkedeigenstates provide a unique means to phase locklaser arrays. Emission was experimentally demon-strated in a pure TEM00 mode without additionalphase filters. A two-axis end-pumped Nd:YAG laserwas shown to oscillate with a good efficiency. The100% visibility of the fringes measured on the inter-ference pattern between the two orthogonally polar-ized components was evidence of the phase-lockingprocess. Moreover, single-frequency operation wasdemonstrated in this two-arm laser. The resultingsupermode contains the powers extracted from thetwo arms of the laser, making forked eigenstates anattractive design for high-brightness compact all-solid-state sources. This technique is a priori appli-cable to any laser array geometry and to any kind oflaser medium. Moreover, the scheme presentedhere can be scaled to much higher output powers by,on the one hand, the use of higher power laser diodesand, on the other hand, scaling to a larger number ofpropagation axes by use of extra high-quality double-refraction crystals.12

This research was partially supported by the Di-rection de la Recherche et de la Technologie and bythe Conseil Regional de Bretagne.

References and Notes1. J. E. Ripper and T. L. Paoli, “Optical coupling of adjacent

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Fig. 5. Experimental observation of the optical spectrum of thelaser of Fig. 1 when an etalon is inserted on the one-arm side of thecavity. ~a! C 5 0; one extraordinary eigenstate and one ordi-nary eigenstate oscillate independently. ~b! C 5 py4; only oneforked eigenstate oscillates, combining the overall power of thesystem in a single frequency.

20 April 1998 y Vol. 37, No. 12 y APPLIED OPTICS 2405

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13. The coupling constant between the laser eigenstates has re-cently been directly measured in a Nd:YAG laser: See M.Brunel, M. Vallet, F. Bretenaker, and A. Le Floch, “Differen-tial measurement of the coupling constant between laser ei-genstates,” Appl. Phys. Lett. 70, 2070–2072 ~1997!.