Vol 450 | 22 November 2007 | doi:10.1038/nature06334

LETTERS
Coherent zero-state and p-state in an
exciton–polariton condensate array
C. W. Lai1,2,3, N. Y. Kim1,2, S. Utsunomiya3,4, G. Roumpos1, H. Deng1, M. D. Fraser1, T. Byrnes2,3, P. Recher1,2,
N. Kumada4, T. Fujisawa4 & Y. Yamamoto1,3

shown in Fig. 1a (see also Methods). The GaAs-based microcavity
structure contains 12 quantum wells, exhibiting a vacuum Rabi splitting of ,14 meV. For the given microcavity structure under the
metallic layer, the cavity resonance energy EC (for kjj 5 0) is expected
to increase by ,400 meV according to the transfer matrix method27aθ

Au/Ti strips

k

kz

a

k||
DBR
λ/2 AlAs cavity

Three stacks of four
GaAs quantum wells

GaAIAsAIAs

DBR
Substrate

b

c

Au/Ti

E
Cavity + Au

CavityphotonCavity only
Quantum well
exciton~2U0100
δE0 (µeV)The effect of quantum statistics in quantum gases and liquids
results in observable collective properties among many-particle
systems. One prime example is Bose–Einstein condensation,
whose onset in a quantum liquid leads to phenomena such as
superfluidity and superconductivity. A Bose–Einstein condensate
is generally defined as a macroscopic occupation of a singleparticle quantum state, a phenomenon technically referred to as
off-diagonal long-range order due to non-vanishing off-diagonalcomponents of the single-particle density matrix1–3. The wavefunction of the condensate is an order parameter whose phaseis essential in characterizing the coherence and superfluid
phenomena4–11. The long-range spatial coherence leads to the existence of phase-locked multiple condensates in an array of superfluid helium12, superconducting Josephson junctions13–15 or atomic
Bose–Einstein condensates15–18. Under certain circumstances, aquantum phase difference of p is predicted to develop among
weakly coupled Josephson junctions19. Such a meta-stable p-state
was discovered in a weak link of superfluid 3He, which is characterized by a ‘p-wave’ order parameter20. The possible existence of sucha p-state in weakly coupled atomic Bose–Einstein condensates has
also been proposed21, but remains undiscovered. Here we report theobservation of spontaneous build-up of in-phase (‘zero-state’) andantiphase (‘p-state’) ‘superfluid’ states in a solid-state system; anarray of exciton–polariton condensates connected by weak periodic
potential barriers within a semiconductor microcavity. These inphase and antiphase states reflect the band structure of the onedimensional polariton array and the dynamic characteristics ofmetastable exciton–polariton condensates.
The strong coupling between quantum well excitons and microcavity photons allows studies of cavity quantum electrodynamic
effects in a solid-state system. The reversible energy exchange
between the microcavity photons and quantum well excitons leads
to the anti-crossing of eigenstates, known as the upper and lower
exciton–polaritons (UPs and LPs)22. Unlike exciton–polaritons inbulk semiconductors, cavity exciton–polaritons involve no transport
and are fully characterized by the in-plane wavenumber kjj, enabling a
direct measurement of the exciton–polariton dispersion relation andmomentum distribution of particles through angularly resolved
spectroscopy23. The exciton–polaritons have been shown to condense
in momentum space and exhibit spontaneous build-up of macroscopic spatial and temporal coherence24–26. Using Young’s double-slit interference experiment, we measured a spatial coherence
length of the condensate up to ,20 mm, limited by the pumping spot
size (see Supplementary Information).
In our experiments, an array of one-dimensional cigar-shaped
exciton–polaritons is created by the deposition of periodic strips ofa metallic thin film on the top surface of the microcavity structure, as

50

LP
U0
0

k||

0
0

1

2

3

X/a

Figure 1 | Formation of an exciton–polariton array. a, A schematic of acavity polariton array formed by depositing periodic thin metallic strips (Au/Ti) on top of a microcavity structure. The 1.4-mm-wide strips are equally
spaced with a pitch distance a 5 2.8 mm. The microcavity structure consistsof a l/2 AlAs cavity (indicated by red lines) sandwiched by two distributed
Bragg reflectors with alternating GaAlAs/AlAs l/4 layers, where l is thecavity resonance wavelength (varying around the quantum well exciton
resonance, ,776 nm with tapering). Three stacks of four GaAs quantum
wells are positioned at the central three antinodes of the microcavity photon
field (the schematic black oscillatory curve). The distributed Bragg reflectors(DBR) consist of repeated GaAIAs/AIAs layers and are symbolized by thetwo short dotted vertical lines. b, Dispersion curves of the cavity photon
mode (dashed lines) and LP (solid lines). The black dashed lines represent
the uncoupled cavity photon mode and quantum-well heavy-hole exciton
state, which are in resonance (EC 5 EX) at k | | 5 0 in this case. The addition of
the metallic layer shifts the LP energy (red solid line) by U0 < 200 meV.
c, Spatial LP energy modulation dE0 measured by scanning a pinhole acrossthe lattice at the first conjugate real-image plane (see Fig. 2 and Methods).

1
E. L. Ginzton Laboratory, Stanford University, Stanford, California 94305, USA. 2Institute of Industrial Science, University of Tokyo, Meguro-ku, Tokyo 153-8505, Japan. 3National
Institute of Informatics, Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan. 4NTT basic research laboratories, NTT Corporation, Atugi, Kanagawa 243-0198, Japan.

529©2007 Nature Publishing GroupLETTERS

NATURE | Vol 450 | 22 November 2007

Considering a periodic and coherent array of condensates aligned
along the x (where x is X or Y, see below) axis, we can approximate theorder parameter in momentum space as28:
Xyðkx Þ~y0 ðkx Þe i n ð kx azwÞ(see Methods). When the cavity photon (EC) is near resonancewith the quantum well exciton (EX) at kjj 5 0, the shift of thelower-polariton energy induced by the metallic layer is ,200 meV,
approximately half of the cavity photon resonance shift (Fig. 1b).
Therefore, the LPs are expected to be trapped in the gap region where
the LP energy is lower. The measured spatial modulation of LP energy
is ,100 meV, which is less than the theoretical prediction owing to thediffraction-limited spatial resolution (,2 mm) of our optical detection system (Fig. 1c). We confirm by an independent measurementthat the LP energy in a bare microcavity is ,200 meV lower than the
LP energy under a uniform metallic layer of similar thickness for
EC < EX.
The microcavity is excited by a mode-locked Ti:Sapphire laser witha ,2.5 ps pulse near the quantum well exciton resonance at an incident angle of 60u, corresponding to an in-plane wavenumber
kjj < 7 3 104 cm21 in air. The large kjj of the pump ensures that thecoherence of the pump laser is lost by multiple phonon emissions
before the polaritons scatter into kjj < 0 states. The LP distributions
in coordinate space (near-field) and momentum space (far-field) aremeasured by micro-photoluminescence spectroscopy and imaging,using the set-up shown in Fig. 2a (see Methods). The near-field images
illustrate the spatial LP distribution, while the far-field images reveal
the in-plane momentum LP distribution. The spectroscopy inmomentum space (that is, angularly resolved spectroscopy) provides
a direct measurement of the LP dispersion relation and effective mass.
For a system of two condensates, the interference fringes can beobserved even in the absence of a locked relative phase in a single shotmeasurement7. In the case of an array of condensates, however, interference fringes appear only if the relative phases between consecutive
condensates are locked28. The high degree of interference fringe
visibility in momentum space can thus be used as an indication ofa long-range spatial coherence across the whole array of multiple
condensates. Such an interference measurement is commonly performed in atomic Bose–Einstein condensates by releasing condensates in an optical lattice.
aYn~0,+1:::+nMsinðN kx a=2Þ~y0 ðkx Þsinðkx a=2Þ w~0

where n labels the condensate array element, N 5 2nM 1 1 is the total
number of periodic array elements, a is the pitch distance between
two neighbouring elements, and y0 ðkx Þ is the momentum-space
wavefunction of an individual element. The overall momentum distribution, rðkx Þ~jyðkx Þj2 , reflects the coherence properties andnature of the condensate arrays through distinctive interference
patterns. In the presence of the lattice, the momentum distribution
displays narrow peaks at kx 5 m 3 2p/a (where m is an integer) withan envelope function given by r0 ðkx Þ~jy0 ðkx Þj2 .
If all the elements are locked in-phase (w 5 0), at a LP wavelengthl 5 780 nm and a grating pitch distance a 5 2.8 mm, a Fraunhofer
diffraction pattern has a central peak at h 5 0u and two side lobes at
h 5 sin21(l/a) < 616u (kjj 5 62p/a; m 5 61 first-order diffraction). The LP emissions from an exciton–polariton condensate arrayare expected to display such an interference pattern in momentum
space, just like an atomic Bose–Einstein condensate experiment15–18.
The observed LP distribution in both coordinate and momentum
space of the condensate array are shown in Fig. 2b and c. Below thecondensation threshold (pump power P 5 20 mW), the near-field
image reveals the cigar-shaped LP emissions through the 1.4-mmwide gaps between the metallic strips. The corresponding LP
momentum distribution is broad (DhFWHM < 36u) and isotropic,
independent of the periodic intensity modulation and the ellipticallyshaped pumping spot. The result indicates that there is negligible
phase coherence among different cigar-shaped LPs. With an increasing pumping rate, two strong and two weak side lobes emerge at

Double slits/pinhole

CCD/
spectrometer

X
L1

ObjectL2

qP
k-spacebr-spacek-space

70 mW

200 mW∆Y = 50 µm

20 mW∆X = 60 µm

10–2

10–1

k|| = K sin q

c
q

8º

0º

16º

0º

∆qY = 60º

24º

ð1Þ

∆qX = 60º, ∆kX = 2π/l = 8.1 µm–1

530
©2007 Nature Publishing Group1

Figure 2 | Imaging and spectroscopy of
exciton–polariton distribution in coordinate andmomentum space. a, Schematic of the microphotoluminescence set-up for imaging andspectroscopy in coordinate and momentum
space (see Methods). hP < 60u is the incident
angle of the pump beam. b, Near-field images
showing the LP distribution across a polariton
array in coordinate space under pumping powersof 20, 70 and 200 mW (left to right) at T 5 20 K.
Here the condensation threshold pumping power
is ,45 mW (intensity ,1,600 W cm22). The
white dashed lines indicate selected locations of
the 1.4-mm-wide metallic strips. The colour scaleis the normalized LP emission intensity.
c, Corresponding far-field images showing the LP
distribution in momentum space. When passing
through the threshold, lobes at 68u and 624u
emerge out of the isotropic background of thethermal polariton gas, which is indicative of the
p-state. The strong central lobe at 0u and two
weak lobes at 616u appear at higher pumping
rates, indicative of the zero-state.

LETTERS

NATURE | Vol 450 | 22 November 2007

‘trap’ of polaritons in momentum space. When the spatial potential
modulation is introduced, a meta-stable dynamic condensate canoccur near the bottom of the second band (point A in Fig. 3b). At
these points, polaritons experience a relaxation bottleneck, resulting
in metastable condensates. Eventually, the LP system will relax to thelowest-energy in-phase state (point C in Fig. 3b) as the pumping rateis increased.
To understand the spatial distributions of LPs in the array (nearfield images shown in Fig. 2b), we consider the Bloch wavefunctions.
The Bloch wavefunctions for these eigenstates exhibit not only
opposite relative phases between adjacent elements in the array but
also different characteristics. Schematic Bloch wavefunctions forselected eigenstates labelled as A, B and C in Fig. 3b are shown in
Fig. 3d. The zero-state at point C corresponds to the ‘s-like’ state withmaximal amplitude in potential wells and identical phase between
adjacent elements across the array. The wavefunctions of A and B at
kjj 5 G0/2 both exhibit a relative p-phase difference between adjacentelements. The lower-energy state at point B corresponds to antibonding of ‘s-states’, while the higher-energy state at point A
corresponds to bonding of ‘p-states’. Like the atomic p-state, theLP density vanishes at the centre of the potential well for this p-state.
This is fully confirmed by the near-field imaging shown in Fig. 2b. We
note that under the weak potential modulation, the Bloch wavefunctions are superpositions of forward-and-backward travelling planewaves at the zone boundary and have a relatively broad distribution
in space compared to the standard tight-binding model of a normal
condensed-matter system.
The metastability of the p-state is a unique property for dynamic
polariton condensates. Under strong pumping rates, the zero-state
dominates eventually, owing to enhanced cooling by stimulated
polariton–polariton scattering at high densities. A typical evolution
a

10–3

10–2

10–1

b

1

4

3

2

2

1st BZ

3

4

1.5955 eV

8 meV

n=3

A

n=2
n=1
–3/2

–16º

–16º

∆k|| = 3.5 × (2π/a) ≈ 7.85 µm–1

Probability amplitudedk||/G01.5955 eVAC

B–3/2 –1 –1/2 0 1/2 1 3/2

c

6 meV

h < 68u and 624u, which correspond to kjj 5 6p/a andkjj 5 63p/2a, respectively. The diffraction pattern suggests that thephase difference of adjacent condensates in the array is locked exactly
to p (see w 5 p in equation (1)). Moreover, the corresponding nearfield image (in Fig. 2b; 70 mW) reveals that strong LP emissions are
generated from under the metallic strips rather than the gaps. The
contrast is better observed near the boundary of the central condensate and the outside thermal LP emissions, which come through thegaps. The dominant LP emission through the metallic layers despite
the lower transmission indicates that LP condensates are strongly
localized under the metallic strips. With a further increasing pumping rate, the intensity of the central peak near h 5 0u with weak side
lobes at h < 616u (Fig. 2c) gradually surpasses the peaks at h < 68uand 624u as shown in Fig. 2c. The corresponding near-field image ofthe condensate also recovers a standard spatial modulation (near thecentral area indicated by the white arrow in Fig. 2b).
These two distinct interference patterns suggest the transition
from an antiphase state to an in-phase state. We refer to the in-phasestate as the ‘zero-state’ and the antiphase state as the ‘p-state’ on thebasis of their relative phase differences between adjacent elements.
Both the zero-state and p-state can be observed for an exciton–polariton condensate array consisting of more than 30 strips ofcondensates (total array dimension ,100 mm). These collective states
are manifestations of the long-range spatial coherence as well as ofphase-locking across the array. Next, we examine the phase-locking
mechanism for the zero-state and p-state through the LP dispersion
relation.
In Fig. 3a, we show the energy versus in-plane momentum
spectra for an LP array near resonance EC < EX and below threshold.
From the dispersion curve, we deduce the LP kinetic energy at
kjj 5 6G0/2 5 p/a (half of the primitive reciprocal lattice vector),
Ek ~B2 ðG0 =2Þ2 =2m <500 meV, where the LP effective mass
m* 5 9 3 1025me (where me is the free electron mass). The kinetic
energy Ek is larger than the spatial LP energy modulation,
U0 < 200 meV. Therefore, we expect the exciton–polariton condensate to be only weakly perturbed by the periodic potential and thecoherence is maintained across the array. In such a large tunnelling
limit, we neglect the polariton–polariton interaction and deduce theband structure of the polariton array by assuming a ‘nearly freepolariton’ in the presence of the periodic square-well potential.
Given a one-dimensional periodic potential U0(X) with a lattice
constant a, the band structure can be obtained using the standard
Bloch wavefunction formalism. Only those polaritons with in-plane
momentum close to the Bragg condition (kjj 5 m G0/2) are subject tobackscattering, owing to the periodic potential, and gaps appear at
the Bragg planes where standing waves are formed. Similar to thestandard extended-zone scheme29, the band structure can be constructed by starting with an original parabola and displaced parabolas
ELP(kjj 6 mG0), as shown in Fig. 3b. Here the size of the circles represents the expected relative LP emission intensity from each band
(Methods). We observe this in the LP energy versus in-plane
momentum with a below-threshold pumping rate (Fig. 3a). The first
bandgap, between the first and second bands near the zone boundaries (Bragg planes), is of the order of the barrier potential, jU0j. This
small gap is masked in the measured spectra by the large inhomogeneous broadening of the LP emission lines (linewidth ,500 meV).
In Fig. 3c, we show the energy versus in-plane momentum of apolariton condensate array above threshold and EC 2 EX <16 meV.
Above threshold, LP emissions occur in two states with an energy
difference of about 1 meV. These two states with emission peaks at
kjj 5 0, 6G0 and at kjj 5 6G0/2, 63G0/2 correspond to the zero-stateand p-state, respectively.
On referring to the multi-valley band structure (Fig. 3b), thedynamic condensation process of the exciton–polariton array
becomes more transparent. Without the spatial modulation, thequasi-stationary state of the LP condensate is the lowest-energy state
at kjj 5 0. Here the bottom of the LP dispersion parabola serves as a

–1/2

0

~|U0|
B
1/2

3/2 (G0)

a

C
B

A

Figure 3 | Band structure and ‘superfluid’ states in an exciton–polariton
condensate array. a, Time-integrated energy versus in-plane momentum (Eversus k | | ) near resonance EC < EX below threshold (P 5 10 mW) at T < 7 K.
The central bright parabola corresponds to the dispersion curve for thepolaritons in the absence of a periodic potential. Two additional parabolas
displaced by 6G0 5 62p/a cross the central dispersion curve at k | | 5 6G0/
2. b, Extended-zone scheme of the band structure. Anti-crossing occurs at
the boundary between the first and second Brillouin zones (BZ). The size ofthe open circles represents the expected relative emission intensity (logscale) of each energy band. Condensation occurs at the valleys labelled bysolid red (point A) and blue (point C) circles. c, E versus k | | at blue detuning
EC 2 EX < 6 meV above threshold (P 5 40 mW). The lobes at k | | 5 0, 6G0
correspond to the in-phase zero-state, whereas the lobes at k | | 5 6G0/2,
63G0/2 correspond to the antiphase p-state. d, Schematic Bloch-wave
functions for states labelled as A, B or C in b.
531©2007 Nature Publishing GroupLETTERS

NATURE | Vol 450 | 22 November 2007

b
20 mW
25 mW
30 mW
40 mW
100 mW

Intensity (arbitrary units)

103
102
101
100
10–1
–1.5 –1.0 –0.5

0
0.5
k||/G0

1.0

1.5

30 mW
35 mW
40 mW
60 mW
100 mW
200 mW

103
Intensity (arbitrary units)

a

102
101
100
10–1
–1.5

–1.0

–0.5

0
k||/G0from the metastable p-state to the stationary zero-state is illustrated
by the LP momentum distribution profiles for various pumping rates
in Fig. 4. At a high pumping rate, the LP emission from the zero-state
surpasses that from the p-state by more than an order of magnitude.
The transition between these dynamic condensates depends on the
coupling strength between adjacent elements in the array (seeSupplementary Information).
Thus we observed a metastable condensate in a p-state (antibonding of p-states) as well as a stable condensate in a zero-state(bonding of s-states) in an exciton–polariton array. Such a coherent
linear superposition state formed over many condensates in periodical sites is referred to as a ‘superfluid’ state. However, some common superfluidity properties such as the frictionless flow over a long
distance beyond the pumping region are not necessarily meant. Our
experiments constitute a first step towards the experimental studyof a dynamical Bose–Hubbard model in a solid-state system. The
polariton system allows the direct observation of the spatial andmomentum distributions as well as the dispersion characteristics ofthe condensates. For instance, full control of the on-site interaction,
barrier potential and occupation number may allow us to study the
dynamical quantum phase transition properties of a Mott insulator.

6.

METHODS SUMMARY
The periodic spatial modulation of LP energy ,200 meV is induced by theperiodic thin metallic strips (a ,20-nm-thick Au /3-nm-thick Ti layer) with apitch distance a 5 2.8 mm deposited on top of the microcavity. The relativetransmission through the top distributed Bragg reflectors (DBR) with and without the metallic layer is ,40–50%.
The imaging and spectroscopy of LPs in the near-field (coordinate space) andfar-field (momentum space) are performed by positioning a charged-coupleddevice (CCD) camera or an imaging spectrometer in a plane conjugate to thereal-image or Fourier (back focal) plane of the objective. The collection efficiency
is approximately uniform to a full angle of ,60u.
More details about the preparation of the metallic grating layer and its effects
on the far-field interference pattern, the experimental settings, including thedetermination of the in-plane wavevector in momentum space, and the expected
LP diffraction patterns and intensities of the zero-state and p-state are given in
the Methods section and Supplementary Information.
Full Methods and any associated references are available in the online version of
the paper at www.nature.com/nature.
Received 23 May; accepted 20 September 2007.
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1.5

Figure 4 | Evolution of the ‘zero-state’ and ‘pstate’. a, Profiles of the p-state as a function of inplane momentum taken at a cross-section in thedispersion curve (exemplified by the white dottedlines in the inset) for an exciton–polariton
condensate array for various pumping rates. The
inset shows the energy versus in-plane momentum
spectra (axes and colour scale same as Fig. 3c, but
only the top 2 meV is shown in the inset) forP 5 100 mW, where relaxation from the p-state tozero-state is visible. The blue dashed line displays
the uniform collection efficiency of the opticalsystem up to | k | | | 5 3G0/2. b, Profiles of the zerostate. The intensity of the zero-state surpasses thatof the p-state at around P 5 45 mW.

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Supplementary Information is linked to the online version of the paper at
www.nature.com/nature.
Acknowledgements This work was supported by the JST/SORST programme andby Special Coordination Funds for Promoting Science and Technology in Japan. The
high-quality microcavity cavity sample is courtesy of G. S. Solomon, R. Hey, K. Ploog
and A. Forchel. We thank T. Maruyama for support and S. Sasaki for the device
fabrication. C.W.L. thanks Y. R. Shen for comments and discussions.
Author Information Reprints and permissions information is available atwww.nature.com/reprints. Correspondence and requests for materials should be
addressed to C.W.L. (cwlai@stanford.edu) or Y.Y. (yyamamoto@stanford.edu).

532©2007 Nature Publishing Groupdoi:10.1038/nature06334

METHODS
Exciton–polariton array. The thin metallic film deposited on top of the microcavity modifies the boundary condition of the top DBR structure. For a baremicrocavity structure, the resonant photon mode exhibits an anti-node at thesemiconductor/free-space interface. In the presence of a metallic layer, the antinode shifts towards the centre of the cavity, owing to suppression of the electromagnetic field inside the metallic layer. As a result, the energy of the cavity
photon resonance increases (blue shift). For a 20-nm-thick Au/3-nm-thick Ti
layer on top of the microcavity structure, the cavity resonance energy increases
by ,400 meV according to the calculation by the transfer matrix method27. The
associated LP energy shift is of the order of ,200 meV near-resonance EC < EX,approximately half of the cavity resonance shift. This is consistent with theexperimentally measured energy shift under a uniform metallic layer of similar
thickness.
The relative transmittance through the top DBR with and without the metallic
layer is ,40–50%, so the optical pumping and LP emission through the periodic
metallic strips are spatially modulated. The initial spatial modulation of LP
generation by the metallic strips is mostly washed out by internal diffusion ofpolaritons through the cooling process. However, the detected LP emission is
subject to intensity modulation by the periodic metallic strips. This effect must
be separately considered for understanding the far-field interference patternfrom the spontaneously formed ‘zero-state’ and ‘p-state’ under the built-in
periodic potential.
Experimental techniques. The exciton–polaritons are excited by a p-polarized
Ti:sapphire laser of 2.5 ps duration (0.5 meV spectral width) at a 76 MHz repetition rate. The pulses are focused to an elliptical spot (,60 mm 3 30 mm or,100 mm 3 50 mm) onto the sample at an oblique incidence angle of ,60u
(kjj < 7 3 104 cm21 in air). The elliptical pumping spot covers approximately
20–30 periodic elements.
LPs are trapped in the cavity mostly until they relax to states of lower energy
and in-plane momentum. The emitted luminescence is collected by a microphotoluminescence set-up composed of a 503 objective lens (effective focal
length, fobj 5 4 mm and a numerical aperture, NA 5 0.55, corresponding to afull collection angle of ,67u in air) and a pair of relay lenses of focal lengthfL 5 10 cm (L1, which is removable, and L2) as shown in Fig. 2. The collection
efficiency is approximately uniform up to a full angle Dh < 60u (Dkjj < 2p/l).
The imaging and spectroscopy of LPs in the near-field (coordinate space)and far-field (momentum space) are performed by positioning a chargedcoupled-device (CCD) camera or an imaging spectrometer in a plane
conjugate to the real-image or Fourier (back focal) plane of the objective. The
optical axis of the collection optical system is perpendicular to the surface ofthe sample. For imaging and spectroscopy in coordinate space (near-field),the repositionable lens L1 is removed. The Fourier (back focal) plane of thesample located in the object plane, with coordinates X and Y, is thus imaged
onto to a CCD camera directly or through an imaging spectrometer. This
arrangement allows determination of the LP distribution in momentum space(kX, kY) (far-field imaging) as well as the LP dispersion relation ELP(kjj).
Coordinate space is correlated to momentum
space bya Fourier transform:ðX, Y Þ~ðu=M, v=M Þ<ðkX , kY Þ~ k|u=fobj , k|v=fobj , where M is the magnification (M 5 fL/fobj < 25), (u, v) are coordinates of the near-field or far-field
images, and kjj ~kX ,Y ~k sin h~ð2p=lÞ sin h in air. Near-field and far-field
images are measured with a CCD with a pixel size of ,8 3 8 mm2, giving adiffraction-limited spatial resolution of ,2 mm and a pixel-size-limited angularresolution of dh 5 0.11u. The energy versus in-plane momentum relation of LPs
and LP arrays are determined by spectroscopy in momentum space (angleresolved spectroscopy). The spectroscopic system is a 75-cm spectrometerequipped with an 1,800 g mm21 grating and a liquid-nitrogen-cooled CCD witha pixel size of 26 3 26 mm2. The system has an angular resolution of dh < 0.4u anda spectral resolution of ,20 meV, limited by the pixel size.
LP emission intensity from the array. The probability amplitude of LPs in thepresence of a periodic potential is represented by the Bloch wave function yn,kjj ,where kjj is the in-plane lattice momentum, and n the index of the band. The
expected leakage LP emission
intensity
is proportional to the overlapping
D
E2




probability density r~ wkjj jyn,kjj  , where wkjj ~ exp i kjj :r is the plane wave
corresponding to the free-moving polaritons (that is, photons leaking out fromthe cavity) with in-plane momentum kjj. The calculated relative r of each energy
band is represented by the size of the open circles (log scale) in Fig. 3b.

©2007 Nature Publishing Group