Photonic Crystal Molecules: towards an experimentaldemonstration of spontaneous symmetry breaking

S. Haddadi, Juan A. Levenson and Alejandro M. YacomottiLaboratoire de Photonique et de Nanostructures

(CNRS UPR 20), Route de Nozay, 91460 Marcoussis, FranceEmail: alejandro.giacomotti@lpn.cnrs.fr

Abstract—We develop a key element in order to implementPhotonic Crystal Molecules for a demonstration of spontaneoussymmetry breaking: the coupling control. In particular, wenumerically show that the mode splitting in two-evanescentlycoupled Photonic Crystal L3 cavities (three holes missing inthe ΓK direction of the underlying triangular lattice) can becontrolled through barrier engineering. The potential barrier isformed by the air-holes in between the two cavities. By changingthe hole radius of the central row in the barrier up to 30%, thefrequency splitting can be strongly reduced. Moreover, the sign ofthe splitting can be reversed in such a way that the fundamentalmode can be either the symmetric or the anti-symmetric onewithout altering neither the cavity geometry nor the inter-cavitydistance.

I. INTRODUCTION

Spontaneous symmetry breaking (SSB) in optics is arapidly evolving field with applications in Bose-Einstein con-densates, nonlinear optics, polariton physics and so forth [1].SSB phenomena are indeed more general in nature, widespreadfrom particle physics to condensed matter, and refer to a familyof physical mechanisms though which a given symmetricsystem ends up in an asymmetric state. In other words,the equations of motion or the Lagrangian governing thesystem obey certain symmetries, but the emerging (often thelowest energy) states do not. SSB is characterized by a welldefined transition from symmetric to asymmetric states, mostcommonly a Pitchfork bifurcation. In optics, nonlinear opticalcavities are ideal platforms to study these phenomena sincelight matter interaction can be greatly enhanced in high finesseoptical cavities. For instance, SSB has been studied in the pastusing coupled semiconductor laser systems or spatial cavitysolitons [2], [3], [4]. In the context of nanophotonics, althoughfew theoretical studies have addressed SSB in micro and nanodevices such as photonic crystals [5], [6], it has not beendemonstrated in nanophotonic systems so far.

Optical nanocavities in dielectric or semiconductor mate-rials are outstanding devices for light confinement and hencefor enhanced nonlinear phenomena: photons can be efficientlystored provided their quality factor is high enough, and sub-wavelength spatial confinement can be easily achieved in highrefractive index materials. Coupled nanocavities, also calledphotonic molecules [7], are extremely performant devices toachieve laser optimization [8], delay lines [9], dynamic controlof light [10], optical equivalent of EIT [11], as well as atestbed for the exploration of advanced physical (quantum)regimes. Among the diversity of possible geometries, PhotonicCrystal (PhC) nanocavities enable a substantial versatility in

Fig. 1. Schematics of two coupled PhC cavities with modified barrier. Latticeperiod is a = 0.425 nm, hole radius of the PhC triangular lattice is r0 =0.266a (white holes), size of shifted holes is r1 = r0−0.06a (red holes), andradius of modified holes within the barrier (black holes) is r3 = r0(1 + x).

the choice of design parameters. For instance, it is well-knownthat small changes in neighbour holes of a PhC cavity, sucha L3 cavity (three holes missing in the ΓK direction of atriangular lattice), may boost the quality factor by more thatone order of magnitude [12]. In addition, PhC cavities can betailored to improve the beaming properties [13], [14]. In thecontext of coupled cavities, PhCs give a handle to the easyand robust engineering of both the strength and the nature ofthe coupling between the cavities [15]. Note that even periodicarrays of optical defects in PhC lattices (see, e.g., [16]) can beregarded as coupled cavity systems, which somehow extendthe coupled cavity domain to Bloch mode resonators. Themain observable phenomenon associated to the inter-cavityoptical coupling strength is the mode splitting, which has beenlargely studied in a variety of PhC geometries and couplingconfigurations.

Several attempts to control frequency splitting in PhC cou-pled cavities –also called PhC molecules– have been carriedout in the recent years. As a general rule, the distance betweenpoint-defect-cavities, which can be varied in a discrete manner,controls the mode splitting. Therefore, bonding and anti-bonding modes will approach each other for distant cavities,or separate away for close enough defects [17]. In addition,specific coupling configurations may spectrally locate the anti-bonding modes either at the blue or at the red side of thebonding modes. In other words, the coupling sign can bereversed in PhC molecules [18], [19]. The ability of PhCsto give rise to anti-bonding ground states has been recentlydiscussed in [19], in the framework of two-coupled D2 (four

missing holes) PhC cavities.

In this paper we carry out specific barrier engineering oftwo coupled L3 cavities leading to a control of both the cou-pling strength and sign, all this using the same PhC molecule.The fine tuning of the mode splitting together with the controlover its sign provides an additional control parameter usefulfor the investigation of SSB.

II. BARRIER ENGINEERING: COUPLED CAVITY DESIGN

The coupled cavity design implemented here is shown inFig. 1. Two coupled L3 cavities are separated by three rows ofholes. The electromagnetic field is mainly confined into eachdefect, and both cavities mutually interact via their evanescentfields. The lattice period is a = 425 nm, and hole radius of thebackground lattice (white holes on the figure) is r0 = 0.266a.The two end holes of each L3 cavity (red holes in Fig. 1) havebeen separated away by s = 0.16a in order to increase the Q-factor which can be, from 3D Finite-Difference-Time-Domainnumerical simulations (3D-FDTD), as high as Q ∼ 50000. Inaddition, simulations of the coupled system show that the anti-symmetric mode is the ground state, and that the mode splittingis about ∆λ ∼ 4 nm, as it has been previously reported [18].

The so-called ”barrier engineering” is performed as fol-lows. The PhC mirror given by the three rows in betweenthe cavities constitutes the potential barrier. We systematicallymodify the hole radius of the central row (black holes in Fig.1) as r3 = r0(1+x). The parameter x typically varies between-30% and +30%. The modification of optical barrier affects theevanescent fields of the cavities which perturbs the evanescentfield interaction. As a result, a variation in coupling strength,hence in mode splitting, can be expected.

III. NUMERICAL RESULTS

Numerical simulations of the structure sketched in Fig. 1have been carried out by means of the 3D-FDTD method.We have taken into account the thickness of the membrane(e = 265 nm). Radiative losses (leakage) take place out of thePhC plane. The following geometrical parameters have beenused: PhC hole radius r0 = 0.232a , cavity end-hole radiusr1 = r0 − 0.06a , end-hole shift s = 0.16, middle row holesize r3 = r0(1 + x) , and lattice period a = 390 nm. Mirrorsymmetry has been exploited in order to reduce the size of thesimulation volume to 1/8 of the original size. We compute theresonant wavelength of symmetric (λsym) and anti-symmetric(λas) modes. In Fig. 2, the wavelength splitting ∆λ = λas −λsym is plotted as a function of the hole radius of the barriermiddle row [r3 = r0(1 + x)] . Two distinct regions can bedistinguished according to the sign of ∆λ :

• The region x < xc ∼ −0.08 in which ∆λ < 0: thefundamental mode is the symmetric mode.

• The region x > xc where ∆λ > 0: the fundamentalmode is anti-symmetric one.

It can be observed that the mode splitting monotonicallyincreases from ∆λ = −12 nm at x = −0.3, to ∆λ = 9 nm atx = 0.3. Notice that, as a result of the barrier modification, thespectral distance is lower, in modulus, than the non-modifiedcase within the region −0.15 < x < 0, whereas it is larger

Fig. 2. Mode splitting numerically computed as the inter-modal distances∆λ = λas −λsym (in nm) as a function of the modification of the radius inthe middle row of the coupled cavity system (see text). The sign of ∆λindicates that the fundamental mode can be either the antisymmetric one(∆λ > 0), or the symmetric one (∆λ < 0).

elsewhere. This reduced splitting region is interesting for anexperimental demonstration of SSB since bifurcation onsetsare expected to decrease for weak coupling. In addition tothe control of the coupling strength, the inversion of thesymmetry of the fundamental mode is observed for x < xcwith respect to the non-modified case. Finally, interpolatingthe numerical points allows us to infer a mode-crossing pointat x = xc ∼ −0.08. The mode crossing exists in theideal (perfectly symmetrical) case, whereas it it expected tobe blurred out in a realistic (experimental) situation due toimperfect symmetry. From the numerical point of view, weare able to obtain mode splitting as small as (in modulus)∆λ ∼ 10 pm, which is limited by the spectral resolution ofthe algorithm used to compute resonant frequencies.

Several coupled L3 cavities with modified barrier, as theones shown in Fig. 1, have been fabricated in InP-suspendedmembranes containing four InGaAs/InP quantum wells (QWs).Photoluminescence experiments allow to both measure themode splitting and the splitting sign. The latter is carried outusing a Fourier imaging technique after mode filtering [18].Experimental results will be presented elsewhere.

IV. CONCLUSION

We have shown that a barrier engineering technique allowscontrolling the splitting strength and sign in a photonic crystalmolecule. By changing the hole size in the middle row of acoupled cavity system we are able not only to continuouslytune the mode splitting, but also to switch the spectral orderof bonding and ant-bonding modes in the same photonicmolecule. This coupling control is a necessary tool for afuture demonstration of spontaneous symmetry breaking in ananophotonic system.

ACKNOWLEDGMENT

We acknowledge support and funding from CNRS andFrench ANR ”Calin” and ”OptiRoc” Projects.

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