Directional optical switching and transistor functionality using optical parametric oscillation in a spinor polariton fluid
Przemyslaw Lewandowski, Samuel M. H. Luk, Chris K. P. Chan, P. T. Leung, N. H. Kwong, Rolf Binder, Stefan Schumacher
DDirectional optical switching and transistorfunctionality using optical parametric oscillationin a spinor polariton fluid P RZEMYSLAW L EWANDOWSKI , S AMUEL
M. H. L UK , C HRIS
K. P.C
HAN , P. T. L
EUNG , N. H. K
WONG , R OLF B INDER , AND S TEFAN S CHUMACHER Physics Department and Center for Optoelectronics and Photonics Paderborn (CeOPP), UniversitätPaderborn, Warburger Strasse 100, 33098 Paderborn, Germany Department of Physics, University of Arizona, Tucson, AZ 85721, USA Department of Physics, The Chinese University of Hongkong, Hongkong SAR, China College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA * [email protected] Abstract:
Over the past decade, spontaneously emerging patterns in the density of polaritons insemiconductor microcavities were found to be a promising candidate for all-optical switching.But recent approaches were mostly restricted to scalar fields, did not benefit from the polariton’sunique spin-dependent properties, and utilized switching based on hexagon far-field patterns with60 ◦ beam switching (i.e. in the far field the beam propagation direction is switched by 60 ◦ ). Sincehexagon far-field patterns are challenging, we present here an approach for a linearly polarizedspinor field, that allows for a transistor-like (e.g., crucial for cascadability) orthogonal beamswitching, i.e. in the far field the beam is switched by 90 ◦ . We show that switching specificationssuch as amplification and speed can be adjusted using only optical means. © 2018 Optical Society of America OCIS codes: (190.4380) Nonlinear optics, four-wave mixing; (190.4390) Nonlinear optics, integrated optics; (190.4975)Parametric processes; (020.1670) Coherent optical effects.
References and links
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1. Introduction
Exciton-polaritons in semiconductor microcavities are well-known for their unique opticalproperties. Owing to a strong nonlinearity and a long coherence time, they have been investigatedintensely not only in terms of their fundamental principles, but also in regard to possibleoptoelectronic applications [1–10]. A promising system for all-optical switching is that ofstationary optical patterns in the polariton density [1–3, 11–14], which can be viewed assolid-state version of the switching reported for gaseous media in [15, 16]. Above a certaindensity threshold, a spatially homogeneous ensemble of coherently pumped polaritons canbecome optically unstable. Driven by a strong repulsive exciton-exciton interaction, pumppolaritons start to scatter spontaneously into off-axis modes, breaking the system’s translationalsymmetry and giving rise to an optical parametric oscillation (OPO). By their photonic part,these off-axis polaritons partially leak out of the cavity and are then detectable as additionalemitted light beams in the far field. Such patterns and their optical control have been studiedtheoretically [1, 1–3, 15, 16], and a directional switching has been observed experimentally [12],but amplification and transistor-like action has not yet been demonstrated.The motivation for the present work lies in the fact that the previous studies on this topic werebased on directional switching within a hexagon far-field pattern, where the switching results in achange of the beam direction by 60 ◦ . Since hexagonal pattern formation is more challenging thanthat of two-spot patterns, we present in the following a scheme that does not rely on (and indeednot use) any hexagonal pattern. Moreover, the previous switching studies used mainly scalarfields and did not benefit from spin-dependent properties, which lead to (i) the optical spin-Halleffect, where the polariton’s polarization state is affected already in the linear optical regime [17],(ii) a polarization dependent four-wave mixing resonance in the nonlinear regime below theinstability threshold [18], and (iii) - of particular interest here - spin-dependent signatures on theoptical patterns forming above the instability threshold [12, 13]. These phenomena are basedmainly on two underlying mechanisms: (i) Stemming from the photonic part, polaritons haveslightly different dispersions for linear polarization states oriented transverse and longitudinal tothe light’s plane of incidence (TE-TM splitting); (ii) a repulsive (possibly attractive) interactionbetween excitons with parallel (opposite) spins governs a spin-dependent polariton-interaction inthe nonlinear regime. In particular, linearly polarized polaritons get strongly affected by bothmechanisms: TE-TM splitting lifts the azimuthal symmetry, and approaching the instabilitythreshold, pump polaritons scatter preferably into modes oriented parallel or perpendicular tothe pump’s polarization plane [18]. This provides a natural platform for orthogonal-directionig. 1: Sketch of the system and excitation geometry: a quantum well (QW) is embedded betweentwo DBR mirrors, forming a semiconductor microcavity. A linearly polarized pump beam is sentonto the cavity in normal incidence and excites a coherent polariton field. Above the instabilitythreshold, the cavity spontaneously emits two off-axis beams with opposite in-plane momentumparallel to the pump’s polarization plane. This initial far-field pattern can be switched by a weakcontrol beam, polarized orthogonal to the pump polarization, entering the cavity in obliqueincidence in a direction 90 ◦ rotated with respect to the initial emission direction.(i.e., 90 ◦ ) switching, as will be shown below. An important advantage of the 90 ◦ switch overthe previously studied 60 ◦ switch is that the former is not hindered by the TE-TM splitting butactually makes active use of it. In contrast to the 60 ◦ switch, both incoming light fields (pumpand control) are at the same frequency and these are the optimum conditions for the switching tooccur.Figure 1 shows a sketch of the system and the excitation geometry: a linearly polarizedcw-pump in normal incidence excites an ensemble of coherent polaritons. If the polariton densityexceeds the instability threshold, the above-mentioned occupation of off-axis modes appears. Forpump powers only slightly above the threshold, the pattern formation results in two outgoingbeams. Both are cross-linearly polarized to the pump and leave the cavity in a direction parallelto the pump’s linear polarization. Increasing the pump power further, this two-spot pattern couldpossibly transform into a hexagonal pattern as studied in [12, 13], but such hexagon patterns haveturned out to be challenging, especially in experiments. We note that the “parallel” state of theinitial 2-spot pattern, i.e. the emission direction being in a direction parallel to the pump’s linearpolarization, is preferred over the “perpendicular” state, i.e. the emission direction being in adirection perpendicular to the pump’s linear polarization, because of a slightly different densityof states for TM- and TE-modes.A weak, cross-linearly polarized external beam is sent into the cavity with the same angleof incidence as the emission of the outgoing beams, but with a plane of incidence rotated by90 ◦ relative to the plane of the outgoing beams. This “control beam” stimulates a pairwise,phase-matched scattering of pump polaritons into off-axis modes parallel to its incidence planeand “switches on” an optical two-spot pattern in this direction. For sufficiently strong controlbeams, the initial pattern becomes unstable and is switched off. A deeper understanding of thenon-equilibrium phase transitions on the basis of a population-competition model has been givenig. 2: Lower polariton branch in the two-dimensional momentum space and excitation scheme:pump-polaritons with finite momentum are excited 4 meV above the polariton ground-state.Modes on the LPB eligible for a self-sustained OPO are marked red. The spontaneous emergenceof the preferred two-spot pattern oriented along the pump’s tilting axis (x-axis) can be switchedoff triggering an OPO in the y-direction.in [19] for the hexagonal geometry. Switching off the control beam, the OPO in the formerdirection as preferred by the system’s anisotropy emerges again and the initial pattern returns -the switching is reversible. The strength of the anisotropy can easily be enhanced by tilting thepump slightly along the preferred direction: this allows for tailoring of switching attributes likepossible gain or time scale using only optical means.Figure 2 shows the dispersion of the lower polariton branch (LPB) in the two-dimensionalmomentum space ( k x , k y ). For the sake of clarity, the splitting in TE- and TM-branch is notshown here. The x -polarized pump excites the polaritons off-resonantly 4 meV above the polaritonground state and with a finite momentum k pump = ( k pump , ) in x -direction. The polariton densityis adjusted to be slightly above the instability threshold. Using a linear stability analysis [18,20,21],we are able to determine an “effective decay rate” γ eff ( k ) for each mode, considering not only theintrinsic loss rate γ , but also the amplification through stimulated scattering of pump polaritonsinto the given mode. If this amplification exceeds the intrinsic losses, any initial polariton fieldin this mode can start growing exponentially, giving rise to a spontaneous pattern formation asmentioned above. In Fig. 2, the modes eligible for such a self-sustained OPO are marked red:the pump-polaritons can scatter spontaneously either pairwise in x - or in y -direction (parallel orperpendicular to the pump’s polarization plane, respectively). The x-direction is favored through(i) a slightly higher density of state for TM-modes and (ii) a finite tilt of the pump-beam. Tofulfill phase-matching and resonance conditions simultaneously, the polaritons undergo slightfrequency shifts in the course of scattering in this direction. The control beam shown in Fig.1 excites modes eligible for an OPO oriented along the y-axis and, importantly, is at the samefrequency as the pump beam.
2. Results
The pattern formation and all-optical switching discussed above can be studied numerically,computing the polariton’s dynamic in the real-space and time domain. Details of the theoreticalig. 3: The initial far-field pattern, emerging in the cross-linear polarization channel, is shown inthe momentum space (a). Switching on the control beam (indicated by a white square), a twospot-pattern emerges in the y-direction, and the initial pattern is suppressed (b). The intensity ofthe initial state (green) and of the target state (blue) during the switching is shown for a low and ahigh control intensity in c and d, respectively. The switch on and off times of the control beamare marked by vertical lines.approach and the system’s parameters are given below in the methods section. For a pumpintensity slightly above the instability threshold, Fig. 3a shows the density of a stationary far-fieldpattern in the momentum space 2 ns after the onset of pattern formation. The pump-polaritonsare excited 4 meV above the polariton ground-state and with a finite momentum k pump = . µ m − as depicted in Fig. 2. The far-field pattern consisting of two spots is oriented parallel to thepump’s tilt direction and hence to the pump’s polarization plane, and emerges resonantly on theLPB in the cross-linear polarization channel.This initial state can be switched by a control beam sent onto the cavity in oblique incidence,with the same frequency as the pump, but resonant on the LPB. The plane of incidence of thisadditional incoming beam is perpendicular to the orientation of the initial pattern, as shown in Fig.3b. The control beam stimulates a pairwise scattering of pump polaritons. One pump polaritonscatters into the direction of the control beam and amplifies it. To fulfill the phase-matchingcondition, a second polariton scatters in the opposite direction and gives rise to a field withopposite in-plane momentum. After a certain time - the “switching time” - this process resultsin the “target state” of the switching - a two-spot pattern aligned perpendicular to the initialone. With an OPO triggered externally, the OPO in the initial direction is suppressed and theinitial pattern switched off. The applied control power amounts to P control = . µ W and islow compared to the intensity of the initially outgoing beams (each 99.7 µ W). That implies atransistor-like switching with a high gain factor of about 14.ig. 4: (a) Dependence of switching time on the applied control intensity for systems with twodifferent pump momenta. Each minimum control intensity is marked by a vertical line. (b)Dependence of maximum gain and minimum control intensity on pump momentum (tilt awayfrom normal incidence). (c) Back switch time after switching off the control beam (blue line) andspontaneous built-up time of the initial pattern (green line) as a function of pump tilt.To shed light on the switching dynamics, Fig. 3c shows the power (in multiples of P control )of the pattern spots in the initial and target state in the course of the switching process. Theinitial pattern converges after approximately 600 ps after the onset of pattern formation. At t = t = . t = P control = . µ W the gain factor is reduced (down to about 12), but the switching stillremains distinctly transistor-like.As is apparent from Figs. 3c and d, the switching speed depends strongly on the appliedcontrol beam. To discuss this dependency in more detail, we show in Fig. 4a the dependenceof the switching time on the control power P control for the current system (green line). Asvisible there, the switching time exhibits a threshold behavior: below a certain minimum power P control , min = . µ W ), no complete switching is possible, since the applied beam is too lowin power to make the initial pattern unstable and switch it off. Approaching this threshold fromhigher powers, however, the switching time diverges. With larger intensities it decreases asexpected and converges to a minimum at t =
80 ps. The blue graph in Fig. 4a shows a similartime-power dependence for a system driven with a higher pump momentum ( k pump = . µ m − ).With a pump tilted farther away from normal incidence, the system’s anisotropy is increased,and the minimum control power shifts to a higher value ( P control , min = . µ W). The minimumswitching time, however, does not change significantly.The dependence of the minimum control beam power on k pump and hence on the system’sanisotropy, is shown in Fig. 4 b. The power required to switch the system’s initial pattern increasesapproximately quadratically with the pump momentum. The power of each initial pattern, however,is almost constant for each k pump . As a consequence, the gain factor is inverse to P control , min anddecreases dramatically with increasing anisotropy (but is still greater than four in the momentumrange studied here).In contrast to the switching time (the time required to suppress the initial pattern), the backswitch time (i.e., the time it takes for the initial pattern to return after switching off the control) isapproximately independent of the control beam intensity, but strongly dependent on the pumpmomentum, as depicted in Fig. 4c (blue line). For pump momenta below k pump = . µ m − the system remains in the switched state - a back switch into the initial state does not occur. Inthe limit of high pump momenta, the back switch time decreases - the stronger the anisotropy,the faster the system recovers the preferred initial state. In this regime, the back switch time isslightly larger than the time necessary for the spontaneous build-up of the initial pattern (greenline). Approaching k pump = . µ m − from higher momenta, the back switch time divergessince the initial state (two-spot pattern) is not sufficiently stable. The transient dynamics show anontrivial behavior with a small local maximum of the backswitch time at k pump = . µ m − .
3. Conclusions
We have presented the concept of an all-optical switch in which the direction of the far-fieldemission from a semiconductor microcavity is rotated by 90 ◦ . The physics of the switchingmechanism involves optical parametric oscillation and non-equilibrium phase transitions ofpolariton patterns. Furthermore, the 90 ◦ switch utilizes the spin-dependent polariton interactionsand the cavity’s TE-TM splitting. Our theoretical analysis shows that the on and off switchingtimes have different parametric dependencies. The switch-on time can be reduced by increasingthe control beam intensity, and the switch-off time can be reduced by increasing the pump tilt.Nanosecond or sub-ns switching times are possible. The gain, defined as the switched power overthe control power, is found to be between one and two orders of magnitude. The specifications ofthe switching can be adjusted using optical means. Further analysis and optimization, for exampleusing novel cavity design concepts [22], and experimental realization are planned for the future.
4. Appendix – Methods
We compute the coherently pumped polariton field solving the coupled dynamics of the photonicand excitonic constituents. For a single cavity system, our approach is based on a quasi-modeapproximation for the optical field E ± and a microscopic semiconductor theory for the excitonicpolarization field p ± [18], where ± denotes the circular polarization states. The dynamics ofhose fields is given by i (cid:126) ∂∂ t E ± = ( H − i γ c ) E ± + H ± E ∓ − Ω x p ± + E ± pump i (cid:126) ∂∂ t p ± = (cid:0) (cid:15) x − i γ x (cid:1) p ± − Ω x (cid:16) − α PSF | p ± | (cid:17) E ± + T ++ | p ± | p ± + T + − | p ∓ | p ± . Here, H = (cid:15) c − (cid:126) (cid:16) m TM + m TE (cid:17) (cid:16) ∂ ∂ x + ∂ ∂ y (cid:17) denotes the free-particle Hamiltonian, H ± = − (cid:126) (cid:16) m TM − m TE (cid:17) (cid:16) ∂ ∂ x ∓ i ∂∂ y (cid:17) is an off-diagonal element stemming from the TE-TM splitting,and Ω x = . E ± pump is the external pump source.The loss rates of photons and excitons are γ c = . γ x = . m TM = .
215 meV ps µ m − and m TE = . · m TM ,respectively. The excitonic energy dispersion, here assumed as flat dispersion, is given by (cid:15) x . Thephotonic ground state is detuned from the exciton by (cid:15) c = (cid:15) x − T ++ = . · − meV µ m for excitons withparallel spin, and an attractive interaction T + − = − T ++ / α PSF = . · − µ m .After solving the coupled equations in real space and time domain, the results are transformedin the momentum space. The pump intensity outside of the cavity is related to E ± pump by I ± pump = ω pump γ − c | E ± pump | , and the intensity of the outgoing photonic field inside the cavity is I ± = ω pump γ c | E ± | , where (cid:126) ω pump = . P out and I control , respectively), the intensities areintegrated over real space. The gain factor g can be obtained dividing the output pattern power bythe power of the external control beam, g = P out / P control . Funding
The Paderborn group acknowledges financial support from the DFG (SCHU 1980/5 and TRR142) and a grant for computing time at PC2