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Creation of Orbital Angular Momentum States with Chiral Polaritonic Lenses
Robert Dall,1,2 Michael D. Fraser,3 Anton S. Desyatnikov,1 Guangyao Li,1 Sebastian Brodbeck,4 Martin Kamp,4Christian Schneider,4 Sven Höfling,4,5 and Elena A. Ostrovskaya1
1

2

Nonlinear Physics Centre, The Australian National University, Canberra ACT 0200, Australia
AMPL, Research School of Physics and Engineering, The Australian National University, Canberra ACT 0200, Australia
3
Quantum Functional System Research Group, RIKEN Center for Emergent Matter Science,
2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan
4
Technische Physik and Wilhelm-Conrad-Röntgen Research Center for Complex Material Systems,
Universität Würzburg, D-97074 Würzburg, Germany
5
School of Physics and Astronomy, University of St Andrews, St Andrews, Fife, KY16 9SS, United Kingdom(Received 30 June 2014; published 12 November 2014)
Controlled transfer of orbital angular momentum to an exciton-polariton Bose-Einstein condensate
spontaneously created under incoherent, off resonant excitation conditions is a long-standing challenge inthe field of microcavity polaritonics. We demonstrate, experimentally and theoretically, a simple andefficient approach to the generation of nontrivial orbital angular momentum states by using opticallyinduced potentials—chiral polaritonic lenses. These lenses are produced by a structured optical pump witha spatial distribution of intensity that breaks the chiral symmetry of the system.
DOI: 10.1103/PhysRevLett.113.200404

PACS numbers: 03.75.Lm, 67.85.De, 71.36.+cIntroduction.—Recent advances in optical excitation andmanipulation of exciton-polaritons in semiconductor
microcavities lead to the creation and trapping of a polariton Bose-Einstein condensate [1] in optically induced
potentials [2–6]. These potentials are created by incoherent
optical sources of exciton-polaritons due to self-trapping
mechanisms that are inherent in this open-dissipative
system [7–9] and are similar to those at play in opticalsystems with gain and loss [10–12]. The advantage of the“soft”, optically-induced potentials over those “hard wired”
in the microcavity, e.g., by etching process [13], is theability to reconfigure their spatial and energy landscape by
structuring the optical pump.
A long-standing and so far unsolved problem in excitonpolariton physics is the inability to transfer orbital angularmomentum directly from the optical pump to the spontaneously condensed exciton-polaritons. The effective potentials created by an optical pump via an uncondensed
reservoir of high-energy near-excitonic polaritons depend
only on the pump intensity, and all of the phase information
is “scrambled” in the process of energy relaxation. This isin stark contrast to a condensate of ultracold atoms that
admits direct imprinting of quantum states of photons [14]and to coherently driven polaritons in the resonant excitation schemes [15–17]. The solution of this problem holds
the key to controlled creation of quantized orbital angularmomentum states and persistent currents, which could be
employed in the polariton analogue of superconducting
quantum interference device (SQUID) sensors [18–20] andinformation encoding devices [21], as well as in thefundamental studies of vortices [22] and polariton BoseEinstein condensates under rotation. So far, vortices in an
incoherently excited exciton-polariton condensate have
0031-9007=14=113(20)=200404(6)only been generated spontaneously and, in the absenceof total angular momentum in the system, only in the formof vortex-antivortex pairs [23–27] or degenerate spinvortices [28,29].
In this Letter, we show that chiral polaritonic lenses—
potentials with broken chiral symmetries—enable reliable
creation of nontrivial orbital angular momentum statesin spontaneously formed exciton-polariton Bose-Einsteincondensates. Such lenses are formed by an off resonant
optical pump with a chiral distribution of intensity[Fig. 1(a)] structured with a pinhole optical mask.
Chirality of the lens can be accidental due to small beam
shifts relative to the mask [Fig. 1(b)] or engineeredby varying the size or position of pinholes [Figs. 1(d)and 1(e)]. In particular, we demonstrate efficient creationof a single charge vortex by a spiral polaritonic lens in
Fig. 1(e). This method represents a new paradigm in thecontrol of orbital angular momentum states of light andmatter, with the schemes previously demonstrated in optics[30,31], plasmonics [32], ultracold atom physics [14,33],
and polaritonics [16,17] relying on chiral distributions ofphase created by coherent, resonant sources and processes.
Experiment.—In an experiment, we work with an
GaAs=AlGaAs microcavity sandwiched between distributed Bragg reflector mirrors to achieve high confinement ofa photon mode [34]. In the regime of strong coupling
between the photons and excitons confined in 2D quantum
wells imbedded in the microcavity, exciton-polaritons formand can be driven to a Bose-Einstein condensation. The
condensation occurs spontaneously above a threshold
power of the pumping laser.
The cw optical pump, with the excitation energy far
above the exciton or polariton resonance, is a broad

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(a)

(b)

(c)

(d)

(e)

FIG. 1 (color online). Schematics of the optical pump structured via a six-pinhole metal mask. (a) Spatial distribution of theintensity shown together with its mirror image to demonstratechirality; the pump structure cannot be superimposed with its
mirror image by simple rotation. Chirality appears due to an
accidental misalignment as shown in (b); (b) Gaussian pumpbeam shifted relative to the mask center, resulting in the spatialintensity distribution shown in (a); (c)–(e) Engineered intensity
distributions of the (c) nonchiral and (d), (e) chiral polaritonic
lenses tested in experiment. In (a), the intensity of the pump spots
is represented by their size, and the degree of asymmetry in(a), (c)–(e) is exaggerated for clarity.

Gaussian beam spatially modulated in the near field by ametal mask patterned with an arbitrary configuration ofholes, and subsequently, reimaged onto the microcavity
sample surface at normal incidence. It allows us to create
an azimuthal distribution of pump spots responsible fortrapping of the condensate in the center of the effective trap
produced by the polaritonic lens, similarly to the experiments involving spatial light modulators [5,6]. The mask
enables us to create structured potentials of arbitrary shapewith stability limited only by stability of the laser. For thiswork, in particular, we use a six-spot azimuthal intensity
distribution.
As the mask is reimaged on the surface of the sample, thesize of the potential is determined by the depth of focus. In
the current experiment, pump spots of 3 μm at a 7 μm
spacing, with the typical size of the resulting condensate∼5 μm. The imaging system consists of a free-space
microscope with a high numerical aperture objective, whichcollects the photoluminescence from the sample, andallows us to infer both the spatial and momentum distribution of condensed polaritons from emitted photons. The
pump is linearly polarized and the resulting photoluminescence appears to be largely unpolarized. A Michelsoninterferometer is used to analyze the phase structure of thesignal [34].
Control of the polariton flow with a structured opticalpump relies on two experimentally verified processes.
First, several pump spots in close proximity of each othercreate phase-locked condensates [3,4]. Secondly, nonlinear

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interaction between the polaritons and a reservoir creates aneffective trapping potential with the depth proportional to
the strength of the pump [5,6]. The spatial distribution of thecondensate density is affected by the pump intensity. This
could be understood from a simple linear interference
picture. In the far field of each individual pump spot, inthe plane of the quantum well, the polariton matter wave canbe approximated by a wave packet with a radially symmetric phase front and exponentially decaying envelope [8]:
ψ c ∼ expð−ΓrÞ expðikp rÞ. As the intensity of the pumpincreases, so does the chemical potential (energy of thesteady state) of the condensate, μ, [8,35], and both
the wavelength of the polariton matter wave, λp ∼ 1=kp ,and the width of the condensate decrease as fλp ; Γg ∼ μ−1=2[8]. For the perfectly symmetric ringlike configuration of six
identical pump spots, the superposition of long-wavelength
condensate “tails” would tend to localize the density in the
excitation regions for a weaker pump intensity (larger λpand Γ) [Fig. 2(a)]. A higher pump intensity (smaller λp andΓ) would produce a bright spot in the center [Fig. 2(c)]. In
the latter regime, the structured pump works as a lens,focusing the condensate into the center. These spatial
patterns will be amplified due to the stimulated scatteringof polaritons into the regions of high density and therefore,
lead to different spatial structures of the condensate atdifferent pump powers, as seen in the experiment [Figs. 2(b)and 2(d)].
The condensate formed in the center of the polaritonic
lens just above the condensation threshold is almost
perfectly radially symmetric [Fig. 3(a)], and can be thoughtof as a ground state of a radially symmetric 2D potential
well effectively created by the pump through induced
spatial distribution of the reservoir density [5,6]. In thepolar coordinates, the eigenstates of this potential can be
written as ψ n;m ¼ Φn ðrÞ expðimθÞ. The first excited energy
state produced by a stronger pump is a dipole mode
superposition [Fig. 3(b)] of two n ¼ 1 states with thenonzero orbital angular momentum, i.e., quantized vortices
with topological charge m ¼ þ1 and m ¼ −1. The second
excited state is a quadrupole mode superposition (notshown) of n ¼ 2 vortex states with the topological charge
m ¼ þ2 and m ¼ −2 and a state with m ¼ 0 containing aradial node. The total topological charge and orbital
(a)

(b)

(c)

(d)

FIG. 2. Schematic illustration of the linear interference of six
phase coherent polariton condensates (a) at low pump intensity,where the fringes are absent due to the long wavelength; (c) at
high pump intensity (see text). Panels (b), (d) show the respective
experimental real-space images (b) at and (d) above condensationthreshold corresponding to Figs. S3(b) and S3(c).

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(b)

(c)

(d)

(e)

(f)

Intensity (arb. units)

5 μm

Distance (microns)

FIG. 3 (color online). Experimental real space image of cavity
emission intensity showing (a) ground state (jmj ¼ 0), (b) dipole
mode (jmj ¼ 0), and (c), (d) charge one vortices (jmj ¼ 1) createdby the polaritonic lenses in Figs. 1(a) and 1(e). Circles in (a) markpositions of pump spots; (e) Experimental and (inset) theoreticalinterference pattern of retroreflected far-field photoluminescencefrom the charge one vortex in (d). Arrows mark locations of“forks” indicating presence of a single isolated vortex in thecenter of the condensate. (d) Intensity profiles of the vortices in(c), (d) obtained with accidentally (black line) and deliberately
engineered (red line) chiral lens at the pump intensities marked bydashed lines in Figs. 4(a) and 4(b). Images in (a)–(d) are plottedon the same scale, and the interferometric image (e) is magnified
for clarity.

Thus, accidental chirality forces the polariton condensate
into a nonzero orbital angular momentum state.
The mode selection of a vortex state is much more
efficient in polaritonic lenses with an engineered chirality.
In the experiment, we tested both the circular masks withdifferent hole sizes [as shown in Fig. 1(d)] and the spiralmasks with identical holes [Fig. 1(e)]. The efficiency of themode selection is quantified via the energy resolved
measurement of the mode population vs pump power.
As seen in Fig. 4(b), the engineered chiral structurestrongly suppresses formation of the ground state condensate with a zero orbital angular momentum [34]. The
ground state mode clearly visible in Fig. 4(a) at lower
powers, e.g., at ∼10 mW, is very weakly populated inFig. 4(b). The vortex m ¼ 1 state produced by theengineered chiral lens has a stronger admixture of the m ¼
2 state, which results in a much greater contrast, defined
through the minimum and maximum intensity of thephotoluminescence as I c ¼ ðI max − I min Þ=I max . The admixture of the ground state causes the contrast to deviate fromone (for a perfect zero intensity in the vortex core). For thevortex produced with the engineered chiral structure in our
experiments [Fig. 3 (d)], the best contrast is I c ≈ 0.82 [atP ¼ 12.5 mW, dashed line in Fig. 4(b)], whereas the best
contrast achieved for the accidental chirality is I c ≈ 0.76 [at
P ¼ 22.5 mW, dashed line in Fig. 4(a)]. We note that theefficient mode selection by the incoherent polaritonic lensrepresents more than a twofold improvement on thecontrast I c ≈ 0.38 reported in the coherent vortex excitation
experiments via the resonant OPO scheme [16].
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0.8
0.6
0.4
0.2

angular momentum of the superposition states is zero. To
select a mode with nonzero topological charge, one needs
to break the chiral symmetry of the lens.
Even a slight misalignment of the Gaussian pump beamwith the center of the metal mask used to reimage
sophisticated spatial distribution of intensity onto thesurface of the sample can create symmetry breaking in
the polaritonic lens [Figs. 1(c) and 1(d)]. As shown in
Fig. 1, the resulting structures can be nonchiral (c) or chiral(a),(d). At a low pump power, the accidental chirality is
weak, and the resulting higher-order state observed in theexperiment still resembles a dipole mode [Fig. 3(b)].
However, as the pump intensity grows, the mode selection
strongly favors a single charge one vortex, m ¼ 1, clearlyvisible both in real space images [Fig. 3(c)] and interferometric images [Fig. 3(e)] of the photoluminescence signal.

Modal Population

0
Ground

0.8

1st

2nd

3rd

(b)

0.6
0.4
0.2
0

2.5

7.5

12.5

17.5

22.5

27.5

Pump Power (mW)

FIG. 4 (color online). Experimental measurement of relative
populations of the ground (n ¼ 0), first (n ¼ 1), and higher-order(n ¼ 2; 3) energy states of the effective trap induced by thepolaritonic lens with an (a) accidental and an (b) engineeredchirality. The populations are normalized by the total density ofcondensed and uncondensed (“thermal”) polaritons [36]. The
condensation threshold in both cases is Pth ≈ 5 mW. Solid lines
are a guide to the eye. Dashed lines mark vortex states (jmj ¼ 1)
shown in Figs. 3(c) and 3(d).

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

∂Ψ
ℏ2 2
ℏ
iℏ
¼ −
∇ þ Vð~r; tÞ þ i ðRnR − γ c Þ Ψ;
∂t
2
2m ⊥
∂nR
¼ −ðγ R þ RjΨj2 ÞnR ð~r; tÞ þ Pð~rÞ;
∂t

10

8

(c)

(b)

(a)

6

5

4

0
−5

2

|m|=0

−10

|m|=1

|m|=1

0
π

10
5

y (μm)

Modeling.—We consider a spontaneously formed
exciton-polariton condensate under the continuous wave
far off resonant optical excitation injecting free carriers into
the system high above the lower polariton energy. The
model consists of a mean-field equation for the polariton
condensate wave function and a rate equation for the
inhomogeneous density of the reservoir [37]:y (μm)

PRL 113, 200404 (2014)

0

0
−5

ð1Þ

−10
−10

−5

0

x (μm)

where Vð~r; tÞ ¼ gc jΨj2 þ gR nR ð~r; tÞ. Here, Ψ is the condensate wave function, nR is the reservoir density, and Pð~rÞ
is the spatially modulated optical pumping rate. The criticalparameters defining the condensate dynamics are the loss
rates of the polaritons γ c and reservoir polaritons γ R , thestimulated scattering rate R, and the strengths of polaritonpolariton, gc , and polariton-reservoir, gR , interactions. In
what follows, we use the dimensionless form of the modelobtained by using the scaling units of time, energy, andlength: T ¼ 1=γ c , E ¼ ℏγ c , L ¼ ½ℏ=ðmLP γ c Þ1=2 , wheremLP is the lower-polariton effective mass [38].
We model both chiral and nonchiral polaritonic lenses
with very small asymmetries, so that the spot-to-spot
variation of intensity similar to that shown in Fig. 1(a) is
1% ≤ δP ≤ 5%. First, we model the weakly unbalanced
nonchiral lens shown in Fig. 1(c). These are produced, e.g.,
by beam shifts along a symmetry axis of the mask. For thepump intensity at threshold and no symmetry breaking(δP ¼ 0), formation of ground state condensate with peakdensity in the geometrical center was seen in numericalcalculations and in our experiments, and was also observedin recent experiments [5] for similar polaritonic lensesof small radii. In numerical simulations, the thresholdfor condensation is roughly determined by the ratio Pth ¼
γ R γ c =R [37].
Even the weak symmetry breaking results in a drasticallydifferent polariton density distribution. Above the threshold, the condensate favors formation of a steady state withthe dipole-mode structure [Fig. 5(a)]. In simulations, thetwo lobes of the dipole are separated by a phase foldbinding a stable vortex-antivortex pair. Such pairs may
form spontaneously due to nonlinearity-induced instabilities [39] and are destroyed as they move to the periphery of
the condensate, unless a special density profile is engineered to hold them in place [27]. The remarkable feature
here is the survival of a single stable vortex-antivortex pairand the resulting formation of the stationary dipole mode.
We stress that, since the polaritonic lens is nonchiral, thesystem does not distinguish between left- and right-handcirculation of polariton flows. Thus, the symmetry breakingof this kind cannot lead to generation of a single isolated
vortex in the system.

5

10

−10

−5

0

x (μm)

5

10

−10

−5

0

5

10

−π

x (μm)

FIG. 5 (color online). Modeled steady state real-space density(top row) and phase (bottom row) created with (a) the nonchiralhexagonal polaritonic lens shown in Fig. 1(c) for the unbalanced
pump with P=Pth ¼ 1.17; (b) polaritonic lens with accidentalchirality with P=Pth ¼ 1.2; (c) the engineered chiral lens. Insets
show schematic configuration of the polaritonic lenses, and jmj
indicates total topological charge of the resulting structure.

Next, we model the lenses with an accidental [Figs. 1(a)and 1(b)] or an engineered [Figs. 1(d) and 1(e)] chirality.
The drastic consequence of the introduced handedness ofthe system is the formation and stabilization of steady states
containing single vortices [Figs. 5(b) and 5(c)]. We stressthat the vortices appear strictly due to the symmetry
breaking in the chiral polaritonic lens and, in numericalmodeling, no vorticity was “seeded” into the system.
In principle, vortices of various topological charge corresponding to the different degree of asymmetry in thesystem can be generated. Preliminary experimental andtheoretical results on formation of the jmj ¼ 2 state arepresented in the Supplemental Material (section IV) [34].
The detailed study of this process will be reported
elsewhere.
We also note that the model [Eq. (1)] does capture theexperimentally observed resemblance of the collective
polariton modes to eigenstates of a linear potential wellcreated in the middle of a polaritonic lens. Indeed,according to the model, the effective linear potential createdfor polaritons V lin ∼ ðgR γ c =RÞP̄ðrÞ, where P̄ ¼ P=Pth ,takes the form of the strongly repulsive (antitrapping)barrier localized around the periphery of the polaritonic
lens, thus creating a trap in the middle. The selfinduced nonlinear contribution to the potential due topolariton interactions is, to the leading order, V nl ∼gc ½1 − P̄ðrÞðgR γ c Þ=ðgc γ R ÞjΨj2 , and therefore, acts toenhance the trapping potential for the chosen parameters,
which agrees well with previous studies [5,6] and ourexperimental observations.
Conclusions.—We have demonstrated an operation ofchiral polaritonic lenses for the reliable creation andtrapping of incoherently excited polariton condensates with
nonzero orbital angular momentum, containing single

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vortices. The role of the broken symmetry in such lenses is
two fold: first, in the presence of strong polariton interactions, it triggers the development of nonlinear instabilities leading to formation of phase vortices and antivortices,and secondly, the distinct handedness of the system leads toselection of steady states with isolated single vortices andoverall nonzero orbital angular momentum.
The deliberately created isolated vortex states are robust,
reproducible, and persist for hours, i.e., for as long as theexciton-polariton condensate is maintained by the cw
optical pump. Our findings open the way to constructionof all-optical elements for shaping and directing of polariton flows with a well-defined orbital angular momentum,
which could be captured by potentials “hard wired” in a
microcavity, and used for study of vortices and persistentcurrents.
It is tempting to draw parallels between the opticalmanipulation of exciton polaritons in the plane of thequantum well and shaping of radiation by means of surface
plasmon-polariton lenses [32]. However, as stressed in the
introduction, shaping of the optical wave front in thecoherently illuminated nanostructures relies on precise
spiral phase distribution introduced by the asymmetricallyplaced coherent sources. In contrast, formation of vortex
states in an incoherently excited exciton-polariton condensate is a result of the mode selection in an effective
potential induced by a spiral intensity distribution, andtherefore, is much less sensitive to the precise geometry of
the structured pump. Although exciton polaritons allow us
to manipulate light on the microscale rather than a nanoscale, the ultrafast velocities and strong nonlinearities
inherent to exciton polaritons and unavailable in plasmonics could potentially enable novel optoelectronic
devices.
This work was supported by the Australian ResearchCouncil (ARC), the State of Bavaria, and Japan Society for
the Promotion of Science (JSPS) through its “Funding
Program for World-Leading Innovative R&D on Scienceand Technology (FIRST Program). We are indebted to K.
Y. Bliokh for careful reading of the manuscript and in-depth
comments. Discussions with I. V. Shadrivov and Yu. S.
Kivshar are gratefully acknowledged.

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