Optical parametric oscillation in one-dimensional microcavities

PHYSICAL REVIEW B 87, 155302 (2013)

Optical parametric oscillation in one-dimensional microcavities Timoth´ee Lecomte,1,* Vincenzo Ardizzone,1 Marco Abbarchi,1,2,† Carole Diederichs,1 Audrey Miard,2 Aristide Lemaitre,2 Isabelle Sagnes,2 Pascale Senellart,2 Jacqueline Bloch,2 Claude Delalande,1 Jerome Tignon,1 and Philippe Roussignol1 1

´ Laboratoire Pierre Aigrain, Ecole Normale Sup´erieure, CNRS (UMR 8551), Universit´e Pierre et Marie Curie, Universit´e D. Diderot, F-75231 Paris Cedex 05, France 2 LPN/CNRS, Route de Nozay, F-91460 Marcoussis, France (Received 20 December 2012; published 8 April 2013)

We present a comprehensive investigation of optical parametric oscillation in resonantly excited onedimensional semiconductor microcavities with embedded quantum wells. Such solid-state structures feature a fine control over light-matter coupling and produce a photonic/polaritonic mode fan that is exploited for the efficient emission of parametric beams. We implement an energy-degenerate optical parametric oscillator with balanced signal and idler intensities via a polarization-inverting mechanism. In this paper, we (i) precisely review the multimode photonic/polaritonic structure of individual emitters, (ii) provide a thorough comparison between experiment and theory, focusing on the power and the threshold dependence on the exciton-photon detuning, (iii) discuss the influence of inhomogeneous broadening of the excitonic transition and finite size, and (iv) find that a large exciton-photon detuning is a key parameter to reach a high output power and a high conversion efficiency. Our study highlights the predictive character of the polariton interaction theory and the flexibility of one-dimensional semiconductor microcavities as a platform to study parametric phenomena. DOI: 10.1103/PhysRevB.87.155302

PACS number(s): 68.65.−k, 73.21.−b, 78.67.Pt, 71.36.+c

I. INTRODUCTION

Semiconductor-based microcavities1 (MCs) are of profound interest in several fields of nanophotonics. An emergent subject of investigation is optical parametric phenomena,2,3 such as parametric amplification and oscillation, which are well established in these nanostructures.4–7 More generally, nonclassical states of light in parametric elastic scattering based on semiconductor nanostructures have been investigated both theoretically8,9 and experimentally:10–13 entangled photon pairs,10,11 squeezed light,12 and twin photons13 are important examples demonstrating the possibility to implement several schemes for quantum optics applications. The nonlinear excitonic properties, including its third-order susceptibility χ (3) , enable parametric processes where two incident photons are converted to two correlated photons, which is the paradigm for the realization of optical logic gates in quantum computation and quantum information processing.14 In this framework, the possible exploitation of semiconductor MCs for the realization of compact, efficient, and electrically driven quantum devices makes these systems a very attractive tool for the manipulation of light at the nanoscale. A central issue in this field is the control of the cavity optical-mode properties in order to satisfy the phasematching conditions, the mode parity conservation, and the light coupling with excitonic states. In two-dimensional MCs (2D-MCs) with embedded quantum wells (QWs), different methods have been applied, so far, for the realization of the phase-matching conditions and the efficient production of parametric amplification and optical parametric oscillation (OPO). In the seminal works of Saviddis et al.4 and Baumberg et al.,5 the strong light-matter coupling regime and the consequent creation of polaritonic states (half-light halfmatter quasiparticles) enabled parametric amplification and oscillation. Thanks to the S-shaped lower polariton dispersion band, a triple resonance involving two pump polaritons and 1098-0121/2013/87(15)/155302(11)

two elastically scattered polaritons [signal (S) and idler (C) polaritons] can be obtained in intraband processes, provided that the system is pumped at a nonzero angle from the normal incidence [see, for example, the scheme in Fig. 1(a)]. Due to the strong excitonic nature of the idler polariton, the intensity associated with the twin parametric emission is very unbalanced in intensity, complicating the possibility of generating and exploiting quantum correlated beams. Moreover, the tilted incidence of the excitation beam is a severe drawback for the integration of this system in realistic nanodevices. Finally, the strong-coupling regime in GaAs-based heterostructures is usually achievable at cryogenic temperature only. It is desirable to find a structure where the parametric process would ideally produce intensity-balanced beams for a pump at normal incidence and without being restrained to cryogenic temperatures. Engineering the photonic modes is a promising strategy towards reaching this goal. This can be achieved, for example, by coupling several 2D-MCs.15–17 Lateral etching of the planar structure is another option which has been successfully exploited for the creation of a large typology of nanostructures: passive and active pillar resonators,18,19 quantum dots,20 ring resonators,12 lasers,21 polariton laser,22 and OPO23 are a few examples of the successful application of this top-down approach. Based on that procedure, etching 2D-MCs in only one direction in a wire-shaped structure enables the implementation of 1D-MCs24,25 which are suitable for the generation of OPO.7,26–28 Recently, 1D-MCs have been efficiently used for the generation and manipulation of polariton condensates29 up to room temperature in a MC made of II-VI semiconductors.30 The condensate itself can, in turn, behave as a source for a parametric scattering process.30 Given the versatility of intra- and interbranch elasticscattering channels in both coupled MCs16,17 and 1DMCs,7,24,26,27 a large variety of parametric processes are, in principle, accessible. For example, momentum-degenerate optical parametric oscillation under normal excitation can

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be achieved thanks to the multiband dispersion. The phasematching condition is obtained by tuning the energy of three photonic/polaritonic bands so that they are equidistant,16,17,26 as shown in Fig. 1(b). Relaxing the requirement of the strong-coupling regime, the production of OPO has also been demonstrated in the weak-coupling regime,16,17 allowing for a rise in the operating temperature. However, the beams produced in a momentum-degenerate configuration are still very unbalanced in intensity. Here we will concentrate on the energy-degenerate process obtained in 1D-MCs.27 As described below, signal and idler beams are emitted through interband scattering involving only two bands where pump, signal, and idler share the same exciton-photon detuning, as shown in Fig. 1(c). The twin beams are intrinsically balanced in intensity and can be easily spatially separated in a mirror symmetric emission with respect to the normal-incident pump beam, making the OPO degenerate in energy a good candidate for the observation of twin photon emission and quantum correlations.31,32 We have recently shown that the mechanism is still observed in the weak-coupling regime at 100 K,27 which is an encouraging result towards the realization of III-V–based devices operating at room temperature. In the present paper, we precisely review the multimode photonic/polaritonic structure of individual emitters, focusing on the practical implications for the parametric processes. We explain why the fundamental mode of polaritons in 1D-MCs, with its fine polarization splitting, is appropriate to provide the pump, signal, and idler modes involved in the energy-degenerate parametric process. We derive a model of interacting polaritons for 1D-MCs in order to analyze the power dependence of OPO and determine how it depends on the MC parameters and on the exciton-photon detuning. A thorough comparison between experiment and theory is done, finding a good agreement between theoretical predictions and observations. The effects of the excitonic inhomogeneous linewidth and of the wire lateral size are discussed in regard to the experimental observations. Finally, one main result of the analysis is the discussion of the important parameters that have to be optimized to reach a large output power and a large conversion efficiency. The paper is organized as follows. In Sec. II, we summarize the relevant characteristics of the system under investigation, accounting for the onset of multiple band dispersion, the phase-matching condition, and the consequent production of

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FIG. 2. (Color online) (a) Scheme of a 1D-MC. (b) Scheme of the available wires on the wafer. The wires are oriented perpendicularly  c of the cavity wedge. to the direction ∇L

OPO. In Sec. III, we introduce the general features of OPO generated under resonant excitation. Finally, in Sec. IV, we focus specifically on the power dependence of the OPO output intensity as a function of the MC parameters and the exciton-photon detuning. II. ONE-DIMENSIONAL MICROCAVITIES

The 1D-MCs [see Fig. 2(a)] described in this paper are obtained by etching a 2D-MC27,29,33 grown by molecular-beam epitaxy and structured as a λ/2 cavity made of Ga0.05 Al0.95 As sandwiched between distributed Bragg reflectors (DBRs), made of 26 and 30 pairs of Ga0.05 Al0.95 As / Ga0.80 Al0.20 As layers on top and bottom, respectively. Three stacks of four GaAs 7-nm-thick QWs are grown in the center of the cavity and at the first two antinodes of the electric field. The central cavity is grown with a wedge that is used to adjust the energy detuning between the excitonic transition and the photonic modes. The wire-shaped 1D-MCs [see Fig. 2(b)]  c are etched with their long axis orthogonal to the direction ∇L of the wedge, so that the cavity energy is constant for each emitter. The 1D-MCs length is 1 mm and the available lateral dimensions are 3, 4, 5, 6, and 7 μm. These values represent a good compromise between optical losses (larger in narrower wires) and mode spacing (smaller in wider wires). The energy of the central emission of the excitonic transition is about 1.605 eV at 5 K. A. Photonic modes and coupling with excitons

In 1D-MCs, the wave vector kz in the direction of the growth axis z is quantized:24 kz = (p + 1)π/Lc ; and the lateral discontinuity in the refractive index introduces a supplementary quantization along the in-plane y axis: ky,j = (j + 1)π/Ly , where j is a new quantization number, integer and positive, identifying each sub-branch composing the fan of the photonic modes [see Fig. 3(a) for a sketch of the reference frame of momentum and angles]. The relationship between energy and wave vector in 1D-MCs takes into account these two quantizations and is only continuous along the kx

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direction, h ¯c 2 kx + ky2 + kz2 nc    (j + 1)π 2 1 kx2 = E0 1 + + , 2 2 Ly kz0 kz0

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where E0 = h ¯ ckz0 /nc . For any mode j , the energy is constant along the y direction, while it has a hyperbolic dispersion in the other x direction, as for a planar 2D-MC [see Figs. 3(b) and 3(c)]. Since the stop-band interval has a finite size, the number of bound sub-branch modes confined in the cavity is limited. The electric-field distribution25 in a 1D-MC is given by E(x,y) ∝ exp(ikx x)E(y), where we can approximate the distribution along the y axis with Ej (y) ≈ cos(ky,j y), choosing y in the interval [−Ly /2, + Ly /2]. In Fig. 3(d), the field distributions for the first four confined modes are reproduced, showing lateral lobes as a function of the angle θy . The emission/absorption diagram corresponds to the square modulus of the Fourier transform along y of the field distribution in the real space [see Fig. 3(e)]. Expressing it as a function of ky , it depends on the sum of two sinc functions:      Ly Ly 2 2  + sinc (ky + ky,j ) |Ej (ky )| ∝ sinc (ky − ky,j ) . 2 2 (3) For the lowest mode j = 0, the emission and the absorption are centered on the vertical direction, whereas they are impossible for j = 1. More generally, all of the modes present two main maxima (except for the case of j = 0) and the same parity

as the corresponding index j . This has an important practical consequence that will be discussed at the beginning of Sec. III. In a regime of strong exciton-photon coupling, each photonic mode j is independently coupled to the excitonic mode. The lateral etching has no relevant effect on the exciton energy, with the lateral extension of the 1D-MC (Ly > 1 μm) being much larger than the exciton Bohr radius (aB  11 nm in GaAs). However, the lateral width of the 1D-MCs has a significant effect on the coupling between photonic modes and the excitonic transition. In the planar 2D-MCs, the in-plane translational invariance imposes that the photons having a wave vector kC are only coupled with excitonic states with kX = kC . In the 1D-MCs, the in-plane translational symmetry is kept in the x direction, but is broken in the orthogonal y direction. For this reason, the mode j in a 1D-MC is only coupled with the excitonic modes m having the same lateral symmetry (m = j ).24 The Hamiltonian H describing the system composed by N excitonic states with energies EX (m,KX ) linearly coupled with N photonic modes in the cavity, with energies Ec1D (j,kx ), is block diagonal, composed of 2 × 2 matrices on the main diagonal. It is written as ⎞ ⎛ H0 ⎟ ⎜ .. ⎟ ⎜ . 0 ⎟ ⎜ ⎟ ⎜ ⎟, ⎜ Hn H =⎜ ⎟ ⎟ ⎜ .. ⎟ ⎜ . 0 ⎠ ⎝ HN−1 (4)   h ¯ 1D EX,1s (m = n,kx ) 2 Hn = , h ¯ EC1D (j = n,kx ) 2 where the coupling h ¯ /2 is assumed to be the same for all of the modes j (this is a reasonable approximation for small j ).24 The normal modes, i.e., the so-called polariton states, are obtained by diagonalizing H . B. Nonresonant excitation: Photoluminescence characterization

Before addressing the nonlinear properties and the generation of OPO, we give a detailed description of the photoluminescence properties under nonresonant cw excitation. The experimental setup and the method used to display the relevant spectral features are described in Ref. 27. A cw Ti:sapphire laser is focused at normal incidence through a high numerical aperture (NA) objective lens (NA = 0.4; focal length f = 16 mm) on a single 1D-MC. In this case, the excitation spot is strongly asymmetric: ∼100 μm by ∼4 μm (FWHM) along the x and y directions, respectively. The polarization of the laser is set parallel to the short axis of the cavity (along y) and its energy can be finely controlled. The far-field emission of the single 1D-MCs is investigated by Fourier space imaging: the back focal plane of the main objective lens is focused through a secondary lens (focal length f = 15 cm) on the entrance slit of a 50 cm monochromator and detected by a silicon-based CCD camera enabling a spectral resolution of about 180 μeV (FWHM). The sample is cooled down to 5 K in a cold-finger, liquid-helium cryostat. The sample is pumped at high energy over the stop band of the DBR and we record the emission of the cavity at

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corresponding mode separation is larger, following the relation ky,j = (j + 1)π/Ly . Moreover, the elongated spots of the branches along θ0 are wider and more extended for sharper wires. Indeed, the emission diagram [see Eq. (3)] is a function of (ky ± ky,j )Ly , where Ly plays the role of a scaling factor and displacement for ky . The photonic modes of 1D-MCs present a polarization splitting j previously studied by Dasbach et al. in Ref. 26. We clearly observe this effect, as shown in Fig. 6. We measure a splitting of 0.71 meV between the bottom of the two sub-branches associated with the mode j = 0, 1.12 meV for j = 1 and 1.39 meV for j = 2. We observe an increase of the fine splitting of the modes with increasing mode number j . Moreover, this splitting depends on the wire width; the wider the wire, the smaller j (not shown). We investigated the evolution of the line broadening as a function of the exciton-photonic mode detuning. This behavior will turn out to be important for the resonant processes since it has a strong influence on the oscillation threshold, and will be addressed specifically in Sec. IV B1. Figure 7 reproduces a typical result for a 4-μm-wide 1D-MC: the linewidth of the polaritonic mode j = 0 is obtained in low-excitation power measurements (∼1 mW) as a function of the detuning. In the simple model of coupled oscillators used in our description, this linewidth is γP = X2 γX + C 2 γC , where X and C are the Hopfield coefficients of the polariton state.1 The fit obtained with this model gives γC = 0.1 meV, γX = 2.4 meV, and

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the vicinity of the excitonic transition. A typical emission dispersion obtained on a 5-μm-wide 1D-MC is depicted in Fig 4. In the displayed intensity map, we distinguish six bands associated with the lower polariton transitions and at ∼1.607 eV the much weaker excitonic transition. The angular dependence of each polaritonic mode corresponds to the theoretical dispersion described by Eq. (3) and plotted in Fig. 3(e). Note that the orientation of the long wire axis (x direction) is oriented at about 45◦ with respect to the spectrometer entrance slit. The observation angle selected by the spectrometer entrance slit θ0 is a linear combination of θx and θy , allowing the visualization of modes with different symmetry. The energy dispersion comes from the x component of the emitted field, while the intensity modulation due to the additional quantization comes from the y component. By orienting the wire orthogonally to the spectrometer entrance slit, only the j -even modes would be observed, since the j -odd modes are zero at θy = 0. On the contrary, a parallel orientation would mask the emission of the j -even modes. Starting from the Hamiltonian of Eq. (4), we can fit the measured polariton dispersions extracting the wire width, the Rabi splitting, the energy of the excitonic transition, as well as the position of the lower mode of the cavity (i.e., the excitonphoton detuning). Figure 4 shows a fit by which we obtain the following parameters: E0 = 1.594 eV, EX = 1.606 eV, h ¯  = (10 ± 2) meV, and Ly = 5.6 μm. Figure 5 presents five dispersions obtained for 1D-MCs with different widths. The lateral size Ly influences the far-field emission of the cavity dispersion: the thinner Ly , the larger the mode spacing. For larger confinement, the

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 = 5.2 meV, showing a relevant discrepancy with respect to the correct value of  obtained with the fit of the polariton dispersion:  ∼ 10 meV (see Fig. 4). The coupled oscillator model fails to predict both the linewidths and the energies with the same set of parameters. The presence of an inhomogeneous broadening of the excitonic transition is the source of the failure of the coupled oscillator model with regard to linewidths.34–36 We will come back to this point in Sec. IV B1. III. ENERGY-DEGENERATE PARAMETRIC SCATTERING IN RESONANT EXCITATION: POLARIZATION, PARITY, AND PHASE-MATCHING CONDITIONS

Energy-degenerate processes can take place when the resonant excitation is at normal incidence while achieving the conservation of the wave vector kx . Due to the peculiar field distribution, an efficient excitation at normal incidence is only possible for the modes with a nonzero absorption at ky = 0. In particular, the first mode j = 0 is the only one reaching a maximum of absorption for normal incidence. Thanks to the polarization splitting of this mode j = 0, it is possible to pump on the higher sub-branch while observing the emission on the lower sub-branch. The OPO on the sub-branch j = 0 obtained by exciting the sub-branch j = 0⊥ is formally described by the process (0⊥ ,kx = 0)2 → (0 ,kx = q) × (0 ,kx = −q). Figure 8(a) shows this mechanism on the polariton dispersion. Figure 8(b) reproduces a measured image where the process is observed by resonant excitation of a 3-μm-wide cavity with an incident power of 30 mW. An additional requirement for the generation of OPO is the conservation of the mode parity in the wire.7 The initial state is a product of two pump polaritons, which is always an even state, and the final state is the product of signal and idler states. Thus, the signal and idler must have the same parity (even or odd) in order to obtain an even final state. This selection rule reduces the set of accessible interbranch parametric processes in the momentum-degenerate scattering,26 but it is always satisfied for any energy-degenerate process where we can choose signal and idler belonging to the same mode.27

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FIG. 8. (Color online) (a) Dispersion under nonresonant excitation of a 3 μm cavity detected with polarization parallel to the wire axis at low incident power (4 mW). The full lines represent the fit for the first two polariton modes ; dashed lines are the position of the split modes with opposite orthogonal polarization ⊥. The parametric scattering process (0⊥ ,kx = 0)2 → (0 ,kx = q) × (0 ,kx = −q) is represented by dots and arrows. (b) Emission under resonant excitation at normal incidence of 0⊥ . The laser polarization is set perpendicular to the cavity axis. We observe the emergence of a signal and idler emission on the branch 0 , parallel polarized with respect to the cavity axis. A trace of the reflected laser is still visible. Note that the emission of signal and idler is not strictly degenerate in energy. (c) From the left to the right panel, the images represent the cavity emission corresponding to different incident laser directions at constant energy. Signal and idler (at the left and at the right of the pump, respectively) follow the dispersion of the underlying polaritonic dispersion of the mode 0⊥ . The dashed line represents the normal-incidence direction and the two solid lines represent the two sub-branches 0 and 0⊥ .

To test the parametric nature of the measured mechanism, we study the energy and the wave vector of the signal and idler beams by slightly tilting the laser direction with respect to the normal incidence. Thanks to the finite width of the polariton modes, there is a small interval where the laser is still in resonance with the mode, efficiently pumping the structure when tuning the incidence angle with a constant energy. Figure 8(c) presents some images obtained by tilting the laser incidence. The signal and idler follow the underlying polaritonic dispersion, adjusting their energy to satisfy the selection rules. This is direct evidence of the parametric nature of the emission as opposed to the linear diffusion mechanism (the resonant Rayleigh scattering). It is interesting to visualize the full far-field emission by imaging the Fourier plane of the main objective lens. The signal and idler appear as elongated spots along the ky direction orthogonal to the cavity axis, as shown in Fig. 9(a), while in Fig. 9(b), the corresponding process in the energy-momentum space is shown. The extension in the ky direction corresponds to the width of the central lobe of the emission diagram of the mode j = 0. The width in kx is related to the spectral width of the polariton line. In Fig. 9(a), the structured central spot corresponds to the laser diffraction, meaning that a small part of the excitation pump passes the cross-polarized detection.

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FIG. 9. (Color online) (a) Far-field cavity emission of a 3-μmwide 1D-MC. The angles θX and θY correspond to the horizontal and vertical direction in the laboratory reference frame. The angles θx and θy are the angles in the planes parallel and perpendicular to the wire. The dotted line shows the wire direction in the real space. (b) Scheme of the scattering process in energy-momentum space corresponding to that measured in (a). (c) Far-field emission of a 6-μm-wide 1D-MC. The detuning between the excitation mode and excitonic transition is 5 meV, at 40 mW. The parametric emission is manifold: we identify at least two distinct mechanisms thanks to the number of emission lobes along the ky axis, i.e., the process A corresponding to (0⊥ ,kx = 0)2 → (0 ,kx = q0 ) × (0 ,kx = −q0 ) and the process B corresponding to (0⊥ ,kx = 0)2 → (1 ,kx = q1 ) × (1 ,kx = −q1 ). The three elongated spots at the image center are due to diffraction of the incident laser on the wire border. (d) Scheme of the scattering process in energy-momentum space corresponding to that measured in (c).

(0 ,kx = −q) and (1⊥ ,kx = 0)2 → (0⊥ ,kx = q) × (0⊥ ,kx = −q) were studied. In that case, the negligible small finestructure splitting between the 0⊥ and 0 sub-branches prevented the efficient generation of the process (0⊥ ,kx = 0)2 → (0 ,kx = q) × (0 ,kx = −q), while it enabled the observation of two other scattering channels, with one inverting and one conserving the polarization when pumping on the mode j = 1. The wire lateral width plays a role in the far-field emission of the signal and idler modes: the narrower the cavity, the more extended is its far-field emission in the Fourier plane. Figure 10 shows this effect for two cavities with different sizes. This corresponds to the effect of the size Ly in Eq. (3). To sum up, the lateral size changes the polarization splitting, the mode splitting, and the emission diagram. IV. POWER DEPENDENCE OF OPTICAL PARAMETRIC OSCILLATION

We now focus on the OPO power dependence by comparing the theoretical predictions based on a simple parametric interaction model with the experimental results. We evaluate the effect of the different experimental parameters on the efficiency of the OPO. Figure 11 reproduces signal and idler intensity as a function of the resonant incident laser for a 3-μm-wide 1D-MC when

The long lines parallel to the ky direction composing this spot are attributed to the diffraction of the laser on the cavity edges. Depending on the exciton-photonic mode detuning and the lateral cavity width (i.e., the mode spacing), other energy-degenerate mechanisms are possible in addition to the process (0⊥ ,kx = 0)2 → (0 ,kx = q) × (0 ,kx = −q). In fact, when the splitting within the two polarization components of the mode j = 0 is larger than the energy difference between the two adjacent modes, the process (0⊥ ,kx = 0)2 → (j ,kx = q) × (j ,kx = −q), with j > 0, is allowed. Figure 9(c) reproduces an example of multimode parametric emission from supplementary branches corresponding to the processes schematically described in Fig. 9(d). Although a complete description of these multimode processes is beyond the scope of the present paper, this stresses the richness of the parametric scattering phenomena taking place in 1D-MCs. We also mention that in a previous report,27 the energy-degenerate processes (1⊥ ,kx = 0)2 → (0 ,kx = q) ×

OPO output power (μW)

600 500 400 300 200 100 0 0

50

100 150 200 250 Excitation power (mW)

300

350

FIG. 11. (Color online) Signal and idler output powers as a function of the incident power for a 3-μm-wide wire. The excitation energy is set at −10 meV from the excitonic transition, on 0⊥ . Black circles and red squares are the experimental data relative to signal andidler, respectively. The continuous line is a fit with the formula √ α( Pp − P0 ) [see Eq. (18)], with P0 = 40.0 mW.

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pumping on 0⊥ at −10 meV apart from the excitonic transition. The two measured intensities are obtained by collecting the full far-field emission of each beam on two photodiodes. We note the onset of OPO after a threshold power of about P0 = 40.0 mW. At lower incident power, the parametric gain is too small to compensate for the losses (such as absorption and finite transmission of the DBR). Starting from P0 , the parametric gain compensates the losses and the OPO takes place. We note that the intensities of the signal and idler are very well balanced: the difference is smaller than 10% up to 200 mW. This result corresponds to the particlelike representation where two pump polaritons are scattered in two new polaritons, signal and idler, producing the same number of both types. As the two emissions are at the same energy, their photonic and excitonic components are identical and the observed output intensities are equal. In the case shown in Fig. 11, the output power exceeds 500 μW and, in the best cases, the conversion efficiency reaches ∼1%. A. Polariton interaction model

A simple model of interacting polaritons allows one to describe the OPO power dependence. In Refs. 37 and 38, Ciuti et al. studied the OPO in a planar microcavity in the strongcoupling regime starting from the nonclassical character of the system. Whittaker instead presented a classical treatment of the problem, as described in Ref. 39, showing similar results when considering two coupled oscillators where the exciton is considered as a nonlinear oscillator. Moreover, it has been demonstrated that the results are still valid in a regime of weak coupling.17,27 In the following, we derive a model for the 1D multibranch case, inspired by the work in Refs. 31, 37, and 38. The system is described by a Hamiltonian H = H0 + Hint , where H0 describes the free evolution, while Hint describes the polariton-polariton interaction:  † EP (k)pk pk , (5) H0 = k

Hint

1  † † = Vk,k ,q pk+q pk −q pk pk . 2 k,k ,q

(6)

†

Here, pk (pk ) are the creation (annihilation) operators for the polaritons. k identifies the polariton mode; it includes not only the wave vector but also the index j of the branch and the polarization of the sub-branch, ⊥ or . EP (k) is the mode energy associated with k. Vk,k ,q is the interaction potential composed of two terms, Vk,k ,q = V0 Xk+q Xk −q Xk Xk + 2Vsat (Xk+q Xk −q Ck Xk + Ck+q Xk −q Xk Xk ). (7) Here, V0 and Vsat represent, respectively, the interaction between excitons and the saturation of the photon-exciton coupling. In the present treatment, we take into account three modes taking part in the process under study, identified by kp (pump polaritons), ks (signal polaritons), and kc (idler polaritons). Hint describes all of the possible interactions between these modes. We will only keep the combinations where pkp appears at least twice, which is correct when the

pump power is not too far from the threshold. Moreover, Vk,k ,q depends on the mode position through the Hopfield coefficients, which in turn depends on the detuning between polaritonic and excitonic modes. Since we are interested in energy-degenerate processes where all of the modes have the same energy, we can deal with a single Vk,k ,q = V (E), which will be simply denoted V in the following. Starting from the Hamiltonian H , we can write the Heisenberg equation describing the time evolution of the polariton operators. First, we define the slowly varying operators, ˜

p˜ p,s,c (t) = pp,s,c (t)ei Ep,s,c t/¯h ,

(8)

where E˜ p,s,c are the energies of the involved polaritons (E˜ p is the energy of the pump laser). In the secular approximation, describing the energy conservation during the parametric process (E˜ = E˜ s + E˜ c − 2E˜ p = 0), we obtain the following system of differential equations: i d p˜ s i = − (s + 2V p˜ p† p˜ p )p˜ s − V p˜ c† p˜ p2 − γs p˜ s + Psin , dt h ¯ h ¯ (9a) d p˜ c i i = − (c + 2V p˜ p† p˜ p )p˜ c − V p˜ s† p˜ p2 − γc p˜ c + Pcin , dt h ¯ h ¯ (9b) d p˜ p i i = − (p + 2V p˜ p† p˜ p )p˜ p − 2V p˜ p† p˜ s p˜ c dt h ¯ h ¯ − γp p˜ p + Ppin . (9c) We have introduced here the polariton linewidths γp,s,c (HWHM), the pump terms Pkin , and the detunings p,s,c = EP (kp,s,c ) − E˜ p,s,c corresponding to the energy differences between the naked polariton modes and the blue-shifted polaritonic modes. In the experiment, we always optimize the energy of the laser excitation in order to remain in resonant excitation conditions for every incident power. Thus, p exactly compensates † the blue shift arising from the term 2V p˜ p p˜ p . In addition, the outgoing signal and idler beams will be spontaneously produced in a configuration which minimizes the oscillation threshold. This fixes the values of s and c so that the blue shifts are also compensated in the signal and idler equations. The following model follows these assumptions of perfect resonance. The more general case of a pump laser that can be detuned from the pump polariton mode is presented in Ref. 40 in the case of a 2D-MC. The stationary solutions for the average fields pk can be obtained. Since the system is excited with a single pump, we in in have P s = P c = 0. The system of Eqs. (9) becomes i 0 = − V p∗c p 2p − γs ps , (10a) h ¯ i 0 = − V p∗s p 2p − γc pc , (10b) h ¯ i in 0 = − 2V p∗p ps pc − γp pp + P p . (10c) h ¯ In order to find the solutions, we impose that the determinant of the system is zero, which gives the condition for the pump polariton population |pp |2 at the oscillation

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√ γs γc 2 4 2 . γs γc − (V /¯h) |pp,Thr | = 0 ⇔ |pp,Thr | = (V /¯h)

(11)

At the threshold level, we can neglect the first term in Eq. (10c), which thus gives the relation between the polariton population created by the pump and the pump term,  in 2 P  = γ 2 |pp,Thr |2 . (12) p,Thr p In the approximation of quasimodes,41 the pump term is related to the pump laser field Ain p through the cavity linewidth γC , √ in P p = Cp 2γC Ain (13) p. 2 For the pump laser intensity Ipin = |Ain p | at threshold, we obtain √ γp2 γs γc in . (14) Ip,Thr = 2Cp2 γC (V /¯h)

Equation (14) allows for the discussion of the oscillation threshold dependence of the experimentally accessible parameters. We will see later (see Sec. IV B1) that the oscillation threshold is quite sensitive to the detuning. We note that the oscillation threshold is smaller when the linewidths γp , γs , and γc are small, and when the interaction potential V is large. To obtain a small oscillation threshold, an efficient coupling between the pump and the system is favorable. This corresponds to a large coefficient Cp2 and a large cavity linewidth γC . The population of the signal and idler are given by Eq. (10c),  1/4  in  γc 1 in 2 |ps | = √ (15a) P p − P p,Thr , 2 γs V /¯h γs  1/4  in  γs 1 in 2 |pc | = √ (15b) P p − P p,Thr . 2 γc V /¯h γc Again, in the quasimode approximation,41 it is possible to link these populations to the outgoing electric-field intensities (Ikout = 2γC Ck2 |pk |2 ): √   2γC 3/2 Cs2 Cp γs 1/4  in  in  out Is = Ip − Ip,Thr , (16a) √ γc γs V /¯h √   2γC 3/2 Cc2 Cp γc 1/4  in  in  out Ic = Ip − Ip,Thr . (16b) √ γs γc V /¯h In the special case of the energy-degenerate process that we are interested in, the three polariton populations share the same Hopfield coefficients: Cp = Cs = Cc = C and Xp = Xs = Xc = X. On this basis, we assume that the linewidths are the same for all the spectral components: γp = γs = γc = γC C 2 + γX X2 . Finally, writing the interaction potential [defined in Eq. (7)] as V = V0 X4 + 4Vsat CX3 , we rewrite Eqs. (14) and (16) to evidence the dependence on the exciton-polariton detuning through the Hopfield coefficients: h ¯ (γC C 2 + γX X2 )3 in , (17) Ip,Thr = 2C 2 γC (V0 X4 + 4Vsat CX3 )  √   in 2¯hγC C 3 Ipin − Ip,Thr out out . (18) Is = Ic =  (γC C 2 + γX X2 )(V0 X4 + 4Vsat CX3 )

Equations (17) and (18) will now be confronted with the results of experiments. B. Comparison with experiments

With the model previously developed, we can interpret the results obtained in resonant excitation experiments. We investigate the signal and idler emissions as a function of the incident excitation power and the oscillation threshold dependence on the exciton-photonic mode detuning. Equation (18) predicts that theoutput√powers of signal and idler Ps and Pc behave like Pp − P0 , where Pp is the pump power and P0 is the threshold power. The fit to the experimental data with this functional dependence (see Fig. 11) shows a good agreement with P0 = 40.0 mW. The fit is reasonably good even up to a pump power that is five times larger than the threshold. This is striking, since the model was derived with the assumption that the pump power was not far from that threshold. It turns out that this assumption is valid for a very large range of practical pump powers. Moreover, the signal and idler beams have the same intensity as expected for a process degenerate in energy [Eq. (18)]. This accounts for the reliability of the model developed for the power dependence. 1. Threshold dependence on the detuning

An additional comparison of the model with experiments can be made on the basis of Eq. (17). We studied the threshold dependence on the polariton-exciton detuning. The intensity in Ip,Thr depends on five parameters: the cavity and exciton linewidths γC and γX , the potentials V0 and Vsat of polariton interaction and saturation, and the Rabi splitting  through the Hopfield coefficients C and X. The Rabi splitting acts as a scaling factor on the energy detuning between the polariton and exciton. With the properties of our 1D-MCs, we estimate Vsat /V0 ≈ 0.04.42 V0 is also a scaling factor for the threshold intensity. The theoretical effect of the parameters γC and γX is shown in Fig. 12. We note that the cavity linewidth γC has a negligible impact on the minimal threshold as a function of the detuning. Indeed, the larger γC is, the larger are the losses, but the easier it is to pump the cavity from the outside, so that the effects compensate. Meanwhile, given a smaller γC , we obtain a small threshold on a larger interval of detuning. γX = 0.1 meV γC = 3 meV 1 0.3 0.1 0.03 0.01

103 (a) in Ip,Thr (/V )

threshold:

1

10

10−1

(b)

γC = 0.1 meV γX = 3 meV 1 0.3 0.1

0.03 10−3 -3 -2 -1 0 -3 -2 -1 0 Detuning (Ep − EX )/Ω Detuning (Ep − EX )/Ω

FIG. 12. Threshold intensity from Eq. (17) as a function of the detuning between polariton and exciton, for different values of (a) cavity linewidth γC and (b) exciton linewidth γX .

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102

101

100 -18

-16

-14 -12 -10 -8 -4 -6 Detuning (Ep − EX ) (meV)

-2

0

FIG. 13. (Color online) Oscillation threshold for different wires as a function of the detuning between the polariton and exciton. The points with error bars are the experimental data, while the lines are in . The dashed red line is obtained with plots of Eq. (17) for Ip,Thr  = 5.2 meV, γX = 2.4 meV, and γC = 0.1 meV, which are the parameters obtained from the linewidth study in Sec. II B. The blue dash-dotted line is obtained with  = 10 meV, γX = 2.4 meV, and γC = 0.1 meV. The green dotted line is obtained with  = 5.2 meV, γX = 0.1 meV, and γC = 0.1 meV. Finally, the black solid line is obtained with  = 10 meV, γX = 0.03 meV, and γC = 0.1 meV.

The exciton linewidth γX plays a more crucial role on the threshold power: the smaller threshold is increased by two orders of magnitude when γX is increased by one order of magnitude. In any case, the minimum threshold is found for energies of the order of magnitude of the Rabi splitting or smaller: |Ep − EX |Min.Thr.  . Figure 13 shows the experimental values of the pump power at threshold as a function of the energy detuning between the pump laser energy and the exciton energy, (Ep − EX ). We observe an exponential increase of the threshold power when increasing the detuning. For Ep − EX < −16 meV, the threshold power is larger than that accessible in our experimental setup (∼300 mW). For Ep − EX > −4 meV, the OPO is not detectable, being masked by other processes such as photoluminescence and laser diffusion. The data points for the oscillation threshold presented in Fig. 13 are obtained for wires of different lateral sizes, from 3 to 6 μm. The threshold does not show any evidence of any peculiar tendency on the size. We conclude that the finite lateral size leaves the polariton interaction unchanged, so that it plays no direct role in the power dependence. In Fig. 13, we also represent several evaluations obtained by using Eq. (17). The red dashed line corresponds to the parameter deduced from the linewidth measurements described in Sec. II B, where we obtained  ∼ 5.2 meV, γX ∼ 2.4 meV, and γC ∼ 0.1 meV. This ensemble of parameters shows a poor agreement with the experimental results. In order to improve the fit, it is necessary to give a smaller value to the exciton linewidth. The green dotted line corresponds to γX = 0.1 meV and correctly describes the trend of the experimental data. Alternatively, we can reproduce the data considering a Rabi splitting  = 10 meV, which corresponds to the splitting deduced by the polariton dispersion of Fig. 4, providing an

even smaller exciton broadening: γX = 0.03 meV. The black solid line shows the latter option, which we consider to be the most relevant. We note that the fit of the experimental oscillation thresholds cannot precisely estimate exciton broadenings γX that are very small. Indeed, as can be seen in Fig. 12(b), variations on a small exciton broadening mostly affect the threshold for detunings close to zero, where we have little data because the output intensity is too small. Consequently, we can find a satisfactory fit for γX between 0.01 and 0.03 meV, while keeping  fixed. To conclude, the best agreement is only obtained with an exciton linewidth much smaller than the measured value of 2.4 meV. This discrepancy is explained by the difference between the radiative broadening and its full linewidth, including the inhomogeneous broadening.34–36 We finally deduce that the linewidths measured in Sec. II B display the full exciton linewidth (γX,full ≈ 2.4 meV), whereas nonlinear mechanisms, such as the parametric scattering studied here, are sensible to the homogeneous linewidth only (γX,hom  0.03 meV). In particular, parametric effects are not sensible to the inhomogeneous broadening of the excitonic transition due to small energy differences between the multiple quantum wells in the structure. 2. Output powers dependence on the detuning

From the point of view of applications, it may be desirable to reach a specific amount of output power Isout or a specific efficiency Isout /Ipin . Deciding how far the pump should be detuned from the excitonic transition energy to optimize the output power or the efficiency is a nonintuitive problem. Solving it can have large implications on the choices made when designing the sample. Equation (18) makes this problem tractable. The variation of signal and idler intensities with the pump power and the detuning, as described by Eq. (18), is illustrated in Fig. 14. One could expect that the largest input powers are found for the smallest thresholds and for pump energies relatively close to the excitonic transition (see Fig. 13). In fact, quite the contrary occurs: the beam 105 104

Isout = 100 /V

103 Ipin (/V )

in Pp,Thr (mW)

103

102

10

101

1

100

0.1

−1

0.01

10

0 −2

10

-3.0

-2.5

-1.5 -2.0 -1.0 Detuning (Ep − EX )/Ω

-0.5

0.0

FIG. 14. Signal intensity according to Eq. (18) (abscissa: detuning between pump energy and exciton transition; ordinates: pump intensity). The thick line represents the oscillation threshold; other lines are isolines of the signal intensity. The parameters are γX = 0.1 meV and γC = 0.1 meV. The signal intensity and the pump intensity in the abscissa share the same units of h ¯ /V .

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intensities are progressively larger when pumping further below the excitonic transition. This can be attributed to two effects: First, the further from the excitonic transition the modes are, the larger is their photonic component, making them more visible outside the cavity. This corresponds to the term C 3 in the numerator of Eq. (18). This accounts for the fact that in our experiment, the signal and idler are not visible when Ep − EX > −4 meV: their photonic component is too small. Second, from Eq. (18), Isout also varies with 1/V = 1/(V0 X4 + 4Vsat CX3 ). The excitonic component X tends to zero when the modes are far from the excitonic transition, so that this term diverges. There is no physical inconsistency in that statement since the threshold diverges too, so that the oscillation regime is increasingly harder to reach. We conclude that to reach large output intensities, it is necessary to choose a detuning Ep − EX that is largely negative compared to the Rabi splitting. The only practical limit of that rationale is that the threshold grows too, so large input powers will be necessary. If continuous laser sources are not powerful enough, then using a pulsed laser source to increase the instantaneous pump power can be an effective means to reach the desired regime, as long as the pump power is kept below the damage threshold of the structure. As for the conversion efficiency (Isout + Icout )/Ipin , it is also described by Eq. (18) and illustrated in Fig. 15. The maximum efficiency that we can expect is 1. This corresponds to the conversion of a pair of two pump photons in one signal photon and one idler photon, without any loss. From Fig. 15, we note that this maximum value is achievable when the mode energies are well below the excitonic transition. For example, an efficiency (Isout + Icout )/Ipin of 0.92 can be obtained for Ep − EX < −2.4 . For a fixed detuning, the efficiency has a maximum for a specific pump intensity. This comes from  in  in out the fact that Is varies with Ip − Ip,Thr . Close to the threshold, the output intensity increases superlinearly, then it slows down to a sublinear growth, so that the efficiency 105 104

Efficiency Isout + Icout /Ipin = 0.002

Ipin (/V )

103

0.02

102 101

0.4 0.92

0.2

0.1

0.8

100 10−1 10−2 -3.0

-2.5

-1.5 -2.0 -1.0 Detuning (Ep − EX )/Ω

-0.5

0.0

FIG. 15. Conversion efficiency (Isout + Icout )/Ipin according to Eq. (18) (abscissa: detuning between pump energy and exciton transition; ordinates: pump intensity). The thick line represents the oscillation threshold; other lines are isolines of the efficiency. The parameters are γX = 0.1 meV and γC = 0.1 meV.

ultimately falls down to 0. In our experiments, the maximum efficiency that could be obtained was on the order of 0.012, for Ep − EX ≈ −2  (in Fig. 11, the maximum efficiency is 0.004). The surprisingly large difference between this result and the larger theoretical expectation comes from additional losses: optical losses in the pump and signal beam path, nonoptimal coupling of the input pump beam to the structure, and nonlinear absorption. This suggests that large improvements are possible by focusing on the reduction of these losses. V. CONCLUSION

In conclusion, we have shown a detailed experimental and theoretical characterization of 1D-MCs in the framework of optical parametric oscillation based on polaritonic/excitonic χ (3) nonlinearity. The fan of photonic modes arising from the lateral optical confinement enables the creation of onedimensional polaritonic multimodes where detuning, mode spacing, and polarization splitting can be adjusted. Among the different parametric scattering mechanisms obtainable in this kind of structure, we studied the energy-degenerate process enabled by a triple resonance: (0⊥ ,kx = 0)2 → (0 ,kx = q0 ) × (0 ,kx = −q0 ). Balanced OPO of signal and idler beams are observed under normal-incidence excitation. The general behavior of the average emitted intensities is described according to the exciton interaction model in a strong-coupling regime, on the basis of previous reports on this subject. In this framework, we provided a theoretical description of the parametric interaction leading to the OPO. The threshold intensity shows large variations with the polariton-exciton detuning: it shows a minimum in the vicinity of the Rabi splitting, and exponentially increases when increasing the energy distance from the exciton. This prediction is compared with the experimental data obtained for the oscillation threshold at different detunings. A quantitative agreement with the data is obtained when considering an exciton line broadening that is smaller than that deduced from the photoluminescence linewidth measurements, indicating that the nonlinear parametric processes are sensible to the small homogeneous broadening of the excitonic transition whereas photoluminescence effects involve its inhomogeneous components. Moreover, the model predicts the observed reduction of signal and idler output intensities accounting for the lack of signal at detuning that is less negative than −4 meV. The model indicates that high output powers and large conversion efficiencies are to be obtained for largely negative detunings. There is still significant room for improvement with current samples and setups compared to the theoretically achievable efficiencies. In any case, the stationary characteristics are very well understood and they are forerunners of next steps: room-temperature operation and quantum optic properties of signal and idler correlations. ACKNOWLEDGMENTS

This work was partly supported by the French RENATECH network and by the ANR Contract No. PNANO-07-005 GEMINI. M.A. and V.A. thank the European project EU Network ITN Clermont-4.

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