Inferring periodic orbits from spectra of simple shaped micro-lasers
Inferring periodic orbits from spectra of simple shaped micro-lasers
M. Lebental1,2, N. Djellali1, C. Arnaud1, J.-S. Lauret1, J. Zyss1
R. Dubertrand2, C. Schmit2 , and E. Bogomolny2∗
CNRS, Ecole Normale Supérieure de Cachan, UMR 8537,
Laboratoire de Photonique Quantique et Moléculaire,
94235 Cachan, France
CNRS, Université Paris Sud, UMR 8626,
Laboratoire de Physique Théorique et Modèles Statistiques,
91405 Orsay, France
(Dated: October 26, 2018)
Dielectric micro-cavities are widely used as laser resonators and characterizations of their spectra
are of interest for various applications. We experimentally investigate micro-lasers of simple shapes
(Fabry-Perot, square, pentagon, and disk). Their lasing spectra consist mainly of almost equidistant
peaks and the distance between peaks reveals the length of a quantized periodic orbit. To measure
this length with a good precision, it is necessary to take into account different sources of refractive
index dispersion. Our experimental and numerical results agree with the superscar model describing
the formation of long-lived states in polygonal cavities. The limitations of the two-dimensional
approximation are briefly discussed in connection with micro-disks.
PACS numbers: 42.55.Sa, 05.45.Mt, 03.65.Sq
I. INTRODUCTION
Two-dimensional micro-resonators and micro-lasers
are being developed as building blocks for optical
telecommunications [1, 2]. Furthermore they are of in-
terest as sensors for chemical or biological applications
[2, 3, 4] as well as billiard toy models for quantum chaos
[5, 6]. Towards fundamental and applied considerations,
their spectrum is one of the main features. It was used,
for instance, to experimentally recover some information
about the refractive index [7] or geometrical parameters
In this paper we focus on cavities much larger than
the wavelength and propose to account for spectra in
terms of periodic orbit families. Cavities of the simplest
and most currently used shapes were investigated: the
Fabry-Perot resonator, polygonal cavities such as square
and pentagon, and circular cavities.
Our experiments are based on quasi two-dimensional
organic micro-lasers [9]. The relatively straightfor-
ward fabrication process ensures good quality and repro-
ducibility as well as versatility in shapes and sizes (see
Fig. 1). The experimental and theoretical approaches
developed in this paper can be easily extended to more
complicated boundary shapes. Moreover this method is
useful towards other kinds of micro-resonators, as it de-
pends only on cavity shape and refractive index.
The paper is organized as follows. In Section II a de-
scription of the two-dimensional model is provided to-
gether with its advantages and limitations. In Section III
micro-lasers in the form of a long stripe are investigated
as Fabry-Perot resonators to test the method and eval-
∗Electronic address: lebental@lpqm.ens-cachan.fr
FIG. 1: Optical microscope photographs of some organic
micro-lasers: stripe (partial view, used as Fabry-Perot res-
onator), square, pentagon, disk, quasi-stadium, and cardioid.
Typical dimension: 100 µm.
uate its experimental precision. This protocol is then
further applied to polygonal cavities. In Section IV the
case of square cavities is discussed whereas in Section V
dielectric pentagonal cavities are investigated. The the-
oretical predictions based on a superscar model are com-
pared to experiments as well as numerical simulations
and a good agreement is found. Finally, in Section VI
the case of several coexisting orbits is briefly dealt with
on the example of circular cavities.
II. PRELIMINARIES
Dielectric micro-cavities are quasi two-dimensional ob-
jects whose thickness is of the order of the wavelength
but with much bigger plane dimensions (see Fig. 1). Al-
http://arxiv.org/abs/0704.1950v2
mailto:lebental@lpqm.ens-cachan.fr
though such cavities have been investigated for a long
time both with and without lasing, their theoretical de-
scription is not quite satisfactory. In particular, the au-
thors are not aware of true three-dimensional studies of
high-excited electromagnetic fields even for passive cavi-
ties. Usually one uses a two-dimensional approximation
but its validity is not under control.
Within such approximation fields inside the cavity and
close to its two-dimensional boundary are treated differ-
ently. In the bulk, one considers electromagnetic fields as
propagating inside an infinite dielectric slab (gain layer)
with refractive index ngl surrounded by medias with re-
fractive indices n1 and n2 smaller than ngl. In our ex-
periments, the gain layer is made of a polymer (PMMA)
doped with a laser dye (DCM) and sandwiched between
the air and a polymer (SOG) layer (see Fig. 2 (a) and [9]).
It is well known (see e.g. [10] or [11]) that in such ge-
ometry there exist a finite number of propagating modes
confined inside the slab by total internal reflection. The
allowed values of transverse momentum inside the slab,
q, are determined from the standard relation
e2ihqr1r2 = 1 (1)
where h is the slab thickness and r1,2 are the Fresnel re-
flection coefficients on the two horizontal interfaces. For
total internal reflection
ri = exp(−2iδi) (2)
where
δi = arctan
sin2 θ − n2i
ngl cos θ
 (3)
Here θ is the angle between the direction of wave prop-
agation inside the slab and the normal to the interface.
The νi parameter is 1 (resp. (ngl/ni)
2) when the mag-
netic field (resp. the electric field) is perpendicular to
the slab plane. The first and second cases correspond
respectively to TE and TM polarizations.
Denoting the longitudinal momentum, p = nglk sin θ,
as p = neffk, the effective refractive index, neff , is de-
termined from the following dispersion relation
= arctan
− n21
+ arctan
− n22
+ lπ , l ∈ N . (4)
This equation has only a finite number of propagating
solutions which can easily be obtained numerically. Fig. 3
presents possible propagating modes for our experimental
setting n1 = 1 (air), n2 = 1.42 (SOG) [28] and ngl = 1.54
deduced from ellipsometric measurements (see Fig. 2 (b))
in the observation range.
0.800.700.600.50
Wavelength (µm)
FIG. 2: (a) Notations for refractive indexes and propagation
wavenumbers. From top to bottom, the layers of our samples
[9] are the air (n1 = 1), a polymer (PMMA) doped with a
laser dye (DCM) (ngl = 1.54), and another polymer (SOG)
(n2 = 1.42) or silica (n2 = 1.45). (b) Refractive index of the
gain layer versus the wavelength inferred from ellipsometric
measurements.
FIG. 3: Effective refractive index versus the thickness over
wavelength variable calculated from Eq. (4). The refractive
indices are assumed to be constant: 1 for air, 1.42 for SOG,
and 1.54 for the gain layer (horizontal black lines). The TE
polarization is plotted with solid blue lines and TM polariza-
tion with dotted red lines. Integer l (see (4)) increases from
left to right starting from l = 0.
The Maxwell equations for waves propagating inside
the slab are thus reduced to the two-dimensional scalar
Helmholtz equation:
∆+ n2effk
Ψin(x, y) = 0 . (5)
Ψ represents the field perpendicular to the slab, i.e. the
electric field for TM and the magnetic field for TE po-
larization [29].
This equation adequately describes the wave propa-
gation inside the cavity. But when one of these propa-
gating modes hits the cavity boundary, it can partially
escape from the cavity and partially be reflected inside
it. To describe correctly different components of electro-
magnetic fields near the boundary, the full solution of
the three dimensional vectorial Maxwell equations is re-
quired, which to the authors knowledge has not yet been
adressed in this context. Even the much simpler case of
scalar scattering by a half-plane plate with a small but
finite thickness is reduced only to numerical solution of
the Wiener-Hopf type equation [12].
To avoid these complications, one usually considers
that the fields can be separated into TE and TM po-
larization and obey the scalar Helmholtz equations (5)
∆+ n2in,outk
Ψin,out(x, y) = 0 . (6)
with nin is the neff effective index inferred from Eq. (4)
and nout the refractive index of the surrounding media,
usually air so nout = 1. This system of two-dimensional
equations is closed by imposing the following boundary
conditions
Ψin|B = Ψout|B , νin
|B = νout
∂Ψout
|B . (7)
Here ~τ indicates the direction normal to the boundary
and ν depends on the polarization. When the electric
(resp. magnetic) field is perpendicular to the cavity
plane, called TM polarization (resp. TE polarization),
νin,out = 1 (resp. νin,out = 1/n
in,out). Notice that these
definitions of ν are not the same for horizontal and ver-
tical interfaces.
We consider this standard two-dimensional approach
keeping in mind that waves propagating close to the
boundary (whispering gallery modes) may deviate
significantly from two-dimensional predictions. In
particular leakage through the third dimension could
modify the life-time estimation of quasi-stationary states.
Our polymer cavities are doped with a laser dye and
uniformly pumped one by one from above [9], so that
the pumping process induces no mode selection. The
complete description of such lasing cavities requires the
solution of the non-linear Maxwell-Bloch equations (see
e.g. [13, 14, 15] and references therein). For clarity,
we accept here a simplified point of view (see e.g. [1]
Sect. 24) according to which true lasing modes can be
represented as a linear combination of the passive modes
which may lase (i.e. for which gain exceeds losses)
Ψlasing =
CmΨm . (8)
From physical considerations, it is natural to consider the
Ψm modes as the quasi-stationary states of the passive
cavity. Though this choice leads to well known difficulties
(see e.g. [1]) it is widely noticed and accepted at least
for modes with small losses (cf. [14, 15, 16]).
For each individual lasing mode, the Cm coefficients
could be determined only after the solution of the full
Maxwell-Bloch equations. But due to the statistical na-
ture of fluorescence the lasing effect starts randomly and
independently during each pump pulse. So it is quite nat-
ural to average over many pump pulses. Then the mean
spectrum exhibits peaks at frequencies of all possible las-
ing modes. The experimental data studied in this paper
are recorded after integration over 30 pump pulses and
agree with this simple statistical model. More refined
verifications are in progress.
III. FABRY-PEROT RESONATOR
The Fabry-Perot configuration is useful for the calibra-
tion control of further spectral experiments due to the
non ambiguous single periodic orbit family which sus-
tains the laser effect.
A long stripe can be considered to a good approxima-
tion as a Fabry-Perot resonator. In fact the pumping
area is very small compared to the length (see Fig. 4
(a)) and the material is slightly absorbing, so that reflec-
tions at far extremities can be neglected. Moreover the
pumping area is larger than the width of the stripe, thus
the gain is uniformly distributed over the section. For a
Fabry-Perot cavity, the emission is expected along both
θ = 0 and θ = π directions (see Fig. 4 (a) for notations).
Fig. 4 (b) shows that this directional emission is observed
experimentally which confirms the validity of our set-up.
Long stripe
Detector
-10 -5 0 5 10
θ (degrees)
FIG. 4: (a) Diagram summarizing the main features of the
Fabry-Perot experiment. (b) Detected intensity versus θ angle
for a Fabry-Perot experiment.
The experimental spectrum averaged over 30 pump
pulses is made up of almost regularly spaced peaks (see
Fig. 5 (a)) which is typically expected for a Fabry-
Perot resonator. In fact, due to coherent effects, the k
wavenumbers of quasi-bound states of a passive Fabry-
Perot cavity are determined from the quantization con-
dition along the only periodic orbit of L = 2W length as
for (1):
r2ei L k neff (k) = 1 (9)
where r is the Fresnel reflection coefficient and neff is
the effective refractive index (4). The solutions of this
equation are complex numbers: the imaginary part cor-
responds to the width of the resonance and the real part
(called km afterwards) gives the position of a peak in the
spectrum and verify
L km neff (km) = 2π m , m ∈ N . (10)
With δkm = km+1−km assumed to be small, the distance
between adjacent peaks is constrained by
δkm[neff (km) + km
∂neff
(km)] L = 2π . (11)
We call
nfull = neff (km) + km
∂neff
(km) (12)
the full effective refractive index. It is a sum over
two terms: one corresponding to the phase velocity,
neff (km), and the other one to the group velocity,
∂neff
(km). If nfull is considered as a constant over
the observation range, which is true with a good accu-
racy, δk can be retrieved from the experimental spec-
trum. For instance, the Fourier transform of the spec-
trum (intensity versus k) is made up of regularly spaced
peaks (Fig. 5 (b) inset), with the first one (indicated with
an arrow) centered at the optical length (L nfull) and the
others at its harmonics. So the geometrical length of the
periodic orbit can be experimentally inferred from the
knowledge of nfull which is independently determined as
described below. For the Fabry-Perot resonator, the ge-
ometrical length is known to be 2W , thus allowing to
check the experimental precision. The relative statistical
errors on the W width is estimated to be less than 3 %.
The error bars in Fig. 5 (b) are related to the first peak
width of the Fourier transform and are less than 5 % of
the optical length.
The full effective refractive index, nfull, is indepen-
dently inferred from ellipsometric measurement (Fig. 2
(b)) and standard effective index derivation described in
the previous Section. Depending on the parameter h/λ
(thickness over wavelength), one or several modes are al-
lowed to propagate. Our samples are designed such as
only one TE and TM modes exist with neff effective
refractive index according to Eq. (4).
In Fig. 3 the refractive index of the gain layer, ngl,
is assumed to be constant: ngl = 1.54 in the middle of
the experimental window, λ varying from 0.58 to 0.65
µm. From Eq. (4) a neff = 1.50 is obtained in the ob-
servation range with a h = 0.6 µm thickness, and corre-
sponds to the phase velocity term. The group velocity
term km
∂neff
(km) is made up of two dispersion contri-
butions: one from the effective index (about 4 %) and
the other from the gain medium (about 7 %). The de-
pendance of ngl with the wavelength is determined with
the GES 5 SOPRA ellipsometer from a regression with
the Winelli II software (correlation coefficient: 0.9988)
and plotted on Fig. 2 (b). Taking into account all contri-
butions (that means calculating the effective refractive
index with a dispersed ngl), the nfull full effective re-
fractive index is evaluated to be 1.645 ± 0.008 in the
observation range. So the group velocity term made up
625620615610605
Wavelength (nm)
22020018016014012010080
Width (µm)
2.0x10
1.00.0
Optical length (nm)
FIG. 5: (a) Experimental spectrum of a Fabry-Perot res-
onator with W = 150 µm. (b) Optical length versus Fabry-
Perot width W . The experiments (red points) are linearly
fitted by the solid red line. The dashed blue line corresponds
to the theoretical prediction without any adjusted parameter.
Inset: Normalized Fourier transform of the spectrum in (a)
expressed as intensity versus wavenumber.
of the two types of refractive index dispersion contribute
for 10 % to the full effective index, which is significant
compared to our experimental precision. The nfull in-
dex depends only smoothly on polarization (TE or TM),
and on the h thickness, which is measured with a surface
profilometer Veeco (Dektak3ST). Thus, the samples are
designed with thickness 0.6 µm and the precision is re-
ported on the full effective index which is assumed to be
1.64 with a relative precision of about 1 % throughout
this work.
Considering all of these parameters, we obtain a satis-
factory agreement between measured and calculated op-
tical lengths, which further improves when taking into ac-
count several Fabry-Perot cavities with different widths
as shown on Fig. 5 (b). The excellent reproducibil-
ity (time to time and sample to sample) is an addi-
tional confirmation of accuracy and validity. With these
Fabry-Perot resonators, we have demonstrated a spectral
method to recover the geometrical length of a periodic
615610605600595
Wavelength (nm)
1401201008060
Width (µm)
2.0x10
1.00.0
Optical length (nm)
FIG. 6: (a) Experimental spectrum of a square-shaped micro-
laser of 135 µm side width. (b) Optical length versus a square
side width. The experiments (red points) are linearly fitted
by the solid red line. The dashed blue line corresponds to the
theoretical prediction (diamond periodic orbit) without any
adjusted parameter. Top inset: Two representations of the
diamond periodic orbit. Bottom inset: Normalized Fourier
transform of the spectrum in (a) expressed as intensity versus
wavenumber.
orbit which can now be confidently applied to different
shapes of micro-cavities.
IV. SQUARE MICRO-CAVITY
In the context of this paper square-shaped micro-
cavities present a double advantage. Firstly, they are
increasingly used in optical telecommunications [2, 17].
Secondly, the precision and validity of the parameters
used above can be tested independently since there is
only one totally confined periodic orbit family. In fact
the refractive index is quite low (about 1.5), so the di-
amond (see Fig. 6 (b), top inset) is the only short-
period orbit without refraction loss (i.e. all reflection
angles at the boundary are larger than the critical angle
χc = arcsin(1/n) ≈ 42◦.)
In a square-shaped cavity light escapes mainly at the
corners due to diffraction. Thus the quality design of cor-
ners is critical for the directionality of emission but not
for the spectrum. Indeed for reasonably well designed
squared micro-cavity (see Fig. 1), no displacement of the
spectrum peaks is detectable by changing the θ obser-
vation angle. The spectra used in this paper are thus
recorded in the direction of maximal intensity.
Fig. 6 (a) presents a typical spectrum of a square-
shaped micro-cavity. The peaks are narrower than in
the Fabry-Perot resonator spectrum, indicating a better
confinement, as well as regularly spaced, revealing a sin-
gle periodic orbit. Data processing is performed exactly
as presented in the previous Section: for each cavity the
Fourier transform of the spectrum is calculated (Fig. 6
(b), bottom inset) and the position of its first peak is
located at the optical length. Fig. 6 (b) summarizes the
results for about twenty different micro-squares, namely:
the optical length inferred from the Fourier transform
versus the a square side width. These experimental
results are fitted by the solid red line. The dotted blue
line corresponds to an a priori slope given by nfull
(1.64) times the geometrical length of the diamond
periodic orbit (L = 2
2a). The excellent agreement
confirms that the diamond periodic orbit family provides
a dominant contribution to the quantization of dielectric
square resonator.
This result is far from obvious as square dielectric cav-
ities are not integrable. At first glance the observed dom-
inance of one short-period orbit can be understood from
general considerations based on trace formulae which are
a standard tool in semiclassical quantization of closed
multi-dimensional systems (see e.g. [18, 19] and refer-
ences therein). In general trace formulae express the
density of states (and other quantities as well) as a sum
over classical periodic orbits. For two-dimensional closed
cavities
d(k) ≡
δ(k − kn) ≈
ikLp−iµp + c.c. (13)
where k is the wavenumber and kn are the eigenvalues
of a closed cavity. The summation on the right part is
performed over all periodic orbits labeled by p. Lp is the
length of the p periodic orbit, µp is a certain phase accu-
mulated from reflection on boundaries and caustics, and
amplitude cp can be computed from classical mechanics.
In general for integrable and pseudo-integrable systems
(e.g. polygonal billiards) classical periodic orbits form
continuous periodic orbit families and in two dimensions
where Ap is the geometrical area covered by a periodic or-
bit family (see the example of circular cavities in Section
Non-classical contributions from diffractive orbits and
different types of creeping waves (in particular, lateral
waves [19]) are individually smaller by a certain power
of 1/k and are negligible in semiclassical limit k → ∞
compared to periodic orbits.
There exist no true bound states for open systems.
One can only compute the spectrum of complex eigen-
frequencies of quasi-stationary states. The real parts
of such eigenvalues give the positions of resonances and
their imaginary part measure the losses due to the leak-
age from the cavity.
For such systems it is quite natural to assume that the
density of quasi-stationary states
d(k) ≡ 1
Im(kn)
(k − Re(kn))2 + Im(kn)2
can be written in a form similar to (13) but the contri-
bution of each periodic orbit has to be multiplied by the
product of all reflection coefficients along this orbit (as it
was done in a slightly different problem in [19])
d(k) ≈
r(j)p
 eikLp−iµp + c.c. . (16)
Here Np is the number of reflections at the boundary and
p is the value of reflection coefficient corresponding to
the jth reflection for the p periodic orbit.
When the incident angle is larger than the critical
angle the modulus of the reflection coefficient equals 1
(see Eq. (2)), but if a periodic orbit hits a piece of
boundary with angle smaller than the critical angle, then
|rp| < 1 thus reducing the contribution of this orbit.
Therefore, the dominant contribution to the trace for-
mula for open dielectric cavities is given by short-period
orbits (cp ∝ 1/
Lp) which are confined by total internal
reflection. For a square cavity with n = 1.5 the diamond
orbit is the only confined short-period orbit which ex-
plains our experimentally observation of its dominance.
Nevertheless, this reasoning is incomplete because the
summation of contributions of one periodic orbit and its
repetitions in polygonal cavities does not produce a com-
plex pole which is the characteristics of quasi-stationary
states.
In order to better understand the situation, we have
performed numerical simulations for passive square cav-
ities in a two-dimensional approximation with TM po-
larization (see Section II and [20]). Due to symmetries,
the quasi-stationary eigenstates can be classified accord-
ing to different parities with respect to the square diago-
nals. In Fig. 7 (a), the imaginary parts of wavenumbers
are plotted versus their real part for states antisymmet-
ric according to the diagonals (that means obeying the
Dirichlet boundary conditions along the diagonals) and
called here (− −) states.
These quasi-stationary states are clearly organized in
families. This effect is more pronounced when wave func-
tions corresponding to each family are calculated. For in-
stance, wave functions for the three lowest families with
30 50 70 90
 Re(a k)
−0.35
30 80 130 180 230
Re (a k)
−0.14
−0.07
10 30 50 70
FIG. 7: (a) Imaginary parts versus real parts of the wavenum-
bers of quasi-stationary states with (− −) symmetry for a
dielectric square resonator with neff = 1.5 surrounded by air
with n = 1. (b) The same as in (a) but for the states with the
smallest modulus of the imaginary part (the most confined
states). Inset. Empty triangles: the difference (20) between
the real part of these wavenumbers and the asymptotic ex-
pression. Filled circles: the same but when the correction
term (21) is taken into account.
(− −) symmetry are presented in Fig. 8. The other mem-
bers of these families have similar patterns. The existence
of such families was firstly noted in [21] for hexagonal di-
electric cavities, then further detailed in [20].
One can argue that the origin of such families is anal-
ogous to the formation of superscar states in pseudo-
integrable billiards discussed in [22] and observed ex-
perimentally in microwave experiments in [23]. In gen-
eral, periodic orbits of polygonal cavities form continuous
families which can be considered as propagating inside
(a) (b) (c)
FIG. 8: Squared modulus of wave functions with − − symmetry calculated with numerical simulations. (a) ak = 68.74− .026 i,
(b) ak = 68.84 − .16 i, (c) ak = 69.18 − .33 i.
(a) (b) (c)
FIG. 9: Squared modulus of wave functions calculated within the superscar model (17) and corresponding to the parameters
of Fig. 8.
FIG. 10: Unfolding of the diamond periodic orbit. Thick lines
indicate the initial triangle.
straight channels obtained by unfolding classical motion
(see Fig. 10). These channels (hatched area Fig. 10) are
restricted by straight lines passing through cavity cor-
ners. In [22] it was demonstrated that strong quantum
mechanical diffraction on these singular corners forces
wave functions in the semiclassical limit to obey simple
boundary conditions on these (fictitious) channel bound-
aries. More precisely it was shown that for billiard prob-
lems Ψ on these boundaries take values of the order of
k) → 0 when k → ∞. This result was obtained
by using the exact solution for the scattering on peri-
odic array of half-planes. No such results are known for
dielectric problems. Nevertheless, it seems natural from
semiclassical considerations that a similar phenomenon
should appear for dielectric polygonal cavities as well.
Within such framework, a superscar state can be con-
structed explicitly as follows. After unfolding (see Fig.
10), a periodic orbit channel has the form of a rectan-
gle. Its length equals the periodic orbit length and its
width is determined by the positions of the closest sin-
gular corners. The unfolded superscar state corresponds
to a simple plane wave propagating inside the rectangle
taking into account all phase changes. It cancels at the
fictitious boundaries parallel to the x direction and is
periodic along this direction with a periodicity imposed
by the chosen symmetry class. This procedure sets the
wavenumber of the state and the true wavefunction is
obtained by folding back this superscar state.
Superscar wave functions with (− −) symmetry asso-
ciated with the diamond orbit (see Fig. 10) are expressed
as follows:
Ψ(− −)m,p (x, y) = sin
κ(−)m x
+ sin
κ(−)m x
′ − 2δ
where x′ and y′ are coordinates symmetric with respect
to square side. In coordinates as in Fig. 10
x′ = y , y′ = x
In (17) m and p are integers with p = 1, 2, . . . , and
m ≫ 1. l =
2a is the half of the diamond periodic
orbit length [30], δ is the phase of the reflection coeffi-
cient defined by r = exp(−2iδ). For simplicity, we ig-
nore slight changes of the reflection coefficient for differ-
ent plane waves in the functions above. So δ is given by
(3) with ν = 1 for TM polarization and θ = π/4. And
m is the momentum defined by
κ(−)m l− 4δ = 2πm . (18)
This construction conducts to the following expression
for the real part of the wavenumbers [31]
neff lRe(km,p) = 2π
δ)2 + p2
= 2π(m+
δ) +O( 1
) . (19)
To check the accuracy of the above formulae we plot in
Fig. 9 scar wave functions (17) with the same parameters
as those in Fig. 8. The latter were computed numerically
by direct solving the Helmholtz equations (6) but the
former looks very similar which supports the validity of
the superscar model.
The real part of the wavenumbers is tested too. In
Fig. 7 (b) the lowest loss states (with the smallest mod-
ulus of the imaginary part) with (− −) symmetry are
presented over a larger interval than in Fig. 7 (a). The
real parts of these states are compared to superscar pre-
dictions (19) with p = 1, leading to a good agreement.
To detect small deviations from the theoretical formula,
we plot in the inset of Fig. 7 (b) the difference between
a quantity inferred from numerical simulations and its
superscar prediction from (19).
δ)2 + p2
. (20)
From this curve it follows that this difference tends to
zero with m increasing, thus confirming the existence of
the term proportional to p2. By fitting this difference
with the simplest expression
(a) (b)
FIG. 11: Simplest whispering gallery periodic orbit family
for a pentagonal cavity. (a) Solid line indicates the inscribed
pentagon which is an isolated periodic orbit. A periodic orbit
in its vicinity is plotted with dashed line. It belongs to the
family of the five-pointed star periodic orbit. The fundamen-
tal domain is indicated in grey. (b) Boundary of the family
of the five-pointed star periodic orbit.
we find that c ≃ −6.9. By subtracting this correction
term from the difference (20), one gets the curve indi-
cated with filled circles in inset of Fig. 7 (b). The result
is one order of magnitude smaller than the difference it-
self.
All these calculations confirm that the real parts of
resonance wavenumbers for square dielectric cavities are
well reproduced in the semiclassical limit by the above
superscar formula (19) and our experimental results can
be considered as an implicit experimental confirmation
of this statement.
V. PENTAGONAL MICRO-CAVITY
The trace formula and superscar model arguments can
be generalized to all polygonal cavities. The pentagonal
resonator provides a new interesting test. In fact, due to
the odd number of sides, the inscribed pentagonal orbit
(indicated by solid line in Fig. 11 (a)) is isolated. The
shortest confined periodic orbit family is twice longer.
It is represented with a dashed line in Fig. 11 (a) and
can be mapped onto the five-pointed star orbit drawn in
Fig. 11 (b) by continuous deformation. In this Section
we compare the predictions of the superscar model for
this periodic orbit family with numerical simulations and
experiments.
Due to the C5v symmetry, pentagonal cavities sustain
10 symmetry classes corresponding to the rotations by
2π/5 and the inversion with respect to one of the sym-
metry axis. We have studied numerically one symmetry
class in which wave functions obey the Dirichlet bound-
ary conditions along two sides of a right triangle with
angle π/5 (see Fig. 11 (a) in grey). The results of these
computations are presented in Fig. 12.
As for the square cavity, lowest loss states are orga-
nized in families. The wave functions of the three lowest
loss families are plotted in Fig. 13 and their superscar
structure is obvious.
The computation of pure superscar states can be per-
40 60 80 100
 Re (a k)
8 12 16 20
−0.25
FIG. 12: (a) Wave numbers for a pentagonal cavity. a is the
side length of the cavity. The three most confined families
are indicated by solid, dashed and long-dashed lines. (b) The
difference (28) between the real part of quasi-energies and
superscar expression (25) for the three indicated families in
formed as in the previous Section. The five-pointed star
periodic orbit channel is shown in Fig. 15. In this case
boundary conditions along horizontal boundaries of pe-
riodic orbit channel are not known. By analogy with su-
perscar formation in polygonal billiards [22], we impose
that wave functions tend to zero along these boundaries
when k → ∞.
Therefore, a superscar wave function propagating in-
side this channel takes the form
Ψscar(x, y) = exp(iκx) sin(
py)Θ(y)Θ(w − y) , (22)
where w is the width of the channel (for the five-pointed
star orbit w = a sin(π/5) where a is the length of the pen-
tagon side). Θ(x) is the Heavyside function introduced
here to stress that superscar functions are zero (or small)
outside the periodic orbit channel.
The quantized values of the longitudinal momentum,
κ, are obtained by imposing that the function (22) is
periodic along the channel when all phases due to the
reflection with the cavity boundaries are taken into ac-
count
κL = 2π
. (23)
Here M is an integer and L is the total periodic orbit
length. For the five-pointed star orbit (see Fig. 11)
L = 10a cos(
) , (24)
and δ is the phase of the reflection coefficient given by
(3) with ν = 1 (for TM polarization) and θ = 3π/10.
For these states the real part of the wavenumber is the
following
nLRek = 2π
+O( 1
) . (25)
Wave function inside the cavity are obtained by folding
back the scar function (22) and choosing the correct rep-
resentative of the chosen symmetry class. When Dirich-
let boundary conditions are imposed along two sides of a
right triangle passing through the center of the pentagon
(see Fig. 15), M must be written as M = 5(2m) if p is
odd and M = 5(2m − 1) if p is even. Then the wave
function inside the triangle is the sum of two terms
Ψm,p(x, y) = sin(κmx) sin(
py)Θ(y)Θ(w − y) +
+ sin(κmx
′ − 2δ) sin( π
py′)Θ(y′)Θ(w − y′) (26)
where the longitudinal momentum is
= 2π(m+
δ − ξ) (27)
with ξ = 0 for odd p and ξ = 1/2 for even p. x′ and y′
in (26) are coordinates of the point symmetric of (x, y)
with respect to the inversion on the edge of the pentagon.
In the coordinate system when the pentagon edge passes
through the origin (as in Fig. 15)
x′ = x cos 2φ+ y sin 2φ , y′ = x sin 2φ− y cos 2φ
and φ = π/5 is the inclination angle of the pentagon
side with respect to the abscissa axis. Wavefunctions ob-
tained with this construction are presented in Fig. 14.
They correspond to the first, second, and third perpen-
dicular excitations of the five-star periodic orbit family
(p = 1, 2, and 3).
To check the agreement between numerically computed
real parts of the wavenumbers and the superscar predic-
tion (25) and (27), we plot in Fig. 12 (b) the following
(a) (b) (c)
FIG. 13: Squared modulus of wave functions for pentagonal cavity with (− −) symmetry calculated with numerical simulations.
(a) ak = 104.7 − 0.017 i, (b) ak = 102.2 − 0.05 i, (c) ak = 105.0 − 0.12 i.
(a) (b) (c)
FIG. 14: Squared modulus of wave functions calculated within the superscar model and corresponding to the parameters of
Fig. 13.
FIG. 15: Unfolding of the five-star periodic orbit for a pen-
tagonal cavity. Thick lines indicate the initial triangle.
difference
n cosφ
Rekanum
δ − ξ
2 tanφ
. (28)
For pure scar states ζ = p2. As our numerical sim-
ulations have not reached the semiclassical limit (see
scales in Figs. 7 and 12), we found it convenient to fit
numerically the ζ constant. The best fit gives ζ ≈ 0.44,
2.33, and 5.51 for the three most confined families (for
pure scar functions this constant is 1, 4, 9 respectively).
The agreement is quite good with a relative accuracy
of the order of 10−4 (see Fig. 12 (b)). Irrespective of
precise value of ζ the total optical length, nL, is given
by (25) and leads to an experimental prediction twice
longer than the optical length of the inscribed pentagon,
which is an isolated periodic orbit and thus can not base
superscar wavefunctions.
Comparison with experiments confirms the superscar
nature of the most confined states for pentagonal res-
onators. In fact, the spectrum and its Fourier transform
in Fig. 16 correspond to a pentagonal micro-laser with
side a = 80 µm, and show a periodic orbit with optical
length 1040 ± 30 µm to be compared with the five-star
optical length nfull10a cos(π/5) = 1061 µm. The agree-
ment is better than 2%.
This result is reproducible for cavities with the same size.
Other sizes have been tested as well. For smaller cavities,
the five-pointed star orbit is not identifiable due to lack
of gain, whereas for bigger ones it is visible but mixed
with non confined periodic orbits. This effect, not spe-
cific to pentagons, can be assigned to the contribution of
different periodic orbit families which become important
when the lasing gain exceeds the refractive losses. We will
describe this phenomenon in a future publication [24].
The good agreement of numerical simulations and ex-
periments with superscar predictions gives an additional
credit to the validity of this approach even for non-trivial
configurations.
620615610605600
Wavelength (nm)
3.0x10
2.01.00.0
Optical length (nm)
FIG. 16: Experimental spectrum of a pentagonal micro-laser
of 80 µm side length. Inset: Normalized Fourier transform of
the spectrum expressed as intensity versus wavenumber.
VI. MICRO-DISKS
Micro-disk cavities are the simplest and most widely
used micro-resonators. In the context of this work, they
are of interest because of the coexistence of several pe-
riodic orbit families with close lengths. For low index
cavities (n ∼ 1.5) each regular polygon trajectory with
more than four sides is confined by total internal reflec-
tion.
In the two-dimensional approximation passive circu-
lar cavities are integrable and the spectrum of quasi-
stationary states can be computed from an explicit quan-
tization condition
J ′m(nkR)
Jm(nkR)
m (kR)
m (kR)
. (29)
Here R is the radius of the disk, n the refractive index of
the cavity, and ν = 1 (resp. ν = n2) for the TM (resp.
TE) polarization. For each angular quantum number m,
an infinite sequence of solutions, km,q, is deduced from
(29). They are labeled by the q radial quantum number.
(a) (b)
FIG. 17: (a) Two examples of periodic orbits: the square and
the pentagon. (b) Two representations of the square periodic
orbit and the caustic of this family in red.
For large |k| the km,l wavenumbers are obtained from a
semiclassical expression (see e.g. [25]) and the density of
quasi-stationary states (15) can be proved to be rewrit-
ten as a sum over periodic orbit families. The derivation
of this trace formula assumes only the semi-classical ap-
proximation (|k|R ≫ 1) and can be done in a way similar
to that of the billiard case (see e.g. [27]), leading to an
expression closed to (16)
d(k) ∝
|rp|Np cos(nLpk−Np (2δp +
Here the p index specifies a periodic orbit family.
This formula depends on periodic orbit parame-
ters: the number of bounces on the boundary, Np,
the incident angle on the boundary, χp, the length,
Lp = 2NpR cos(χp), and the area covered by periodic
orbit family, Ap = πR
2 cos2(χp), which is the area
included between the caustic and the boundary (see Fig.
17 (b)). 2δp is the phase of the reflection coefficient at
each bounce on the boundary (see Eq. (3)) and |rp| is
its modulus.
For orbits confined by total internal reflection δp does
not depend on kR in the semi-classical limit, and rp is
exponentially close to 1 [25, 26]. From (30) it follows
that each periodic orbit is singled out by a weighing
coefficient cp =
|rp|Np . Considering the experi-
mental values |k|R ∼ 1000, |rp| can be approximated
to unity with a good accuracy for confined periodic
orbits, and thus cp =
depends only on geometrical
quantities. Fig. 18 shows the evolution of cp for polygons
when the number of sides is increasing. As the critical
angle is close to 45◦, the diameter and triangle periodic
orbits are not confined and the dominating contribution
comes from the square periodic orbit. So we can
reasonably conclude that the spectrum (15) of a passive
two-dimensional micro-disk is dominated by the square
periodic orbit.
The experimental method described in the previous
Sections has been applied to disk-shaped micro-cavities.
A typical experimental spectrum is shown on Fig. 19 (a).
The first peak of its Fourier transform (see Fig. 19 (b)
5.0 5.5 6.0 6.5
Geometrical length
FIG. 18: Vertical red sticks: cp coefficient for polygons con-
fined by total internal reflection (square, pentagon, hexagon,
etc...). The dotted blue line indicates the position of the
perimeter.
inset) has a finite width coming from the experimental
conditions (discretization, noise, etc...) and the contribu-
tions of several periodic orbits. This width is represented
as error bars on graph 19 (b). The continuous red line
fitting the experimental data is surrounded by the dashed
green line and the dotted blue line corresponding to the
optical length of the square and hexagon respectively,
calculated with nfull = 1.64 as in the previous Sections.
The perimeter (continuous black line) overlaps with a
large part of the error bars which evidences its contribu-
tion to the spectrum, but it is not close to experimental
data.
These experimental results seem in good agreement
with the above theoretical predictions. But actu-
ally these resonances, usually called whispering gallery
modes, are living close to the boundary. Thus both
roughness and three-dimensional effects must be taken
into account. At this stage it is difficult to evaluate and
to measure correctly such contributions for each periodic
orbit. For micro-disks with a small thickness (about 0.4
µm) and designed with lower roughness, the results are
more or less similar to those presented in Fig. 19 (b).
VII. CONCLUSION
We demonstrate experimentally that the length of the
dominant periodic orbit can be recovered from the spec-
tra of micro-lasers with simple shapes. Taking into ac-
count different dispersion corrections to the effective re-
fractive index, a good agreement with theoretical predic-
tions has been evidenced first for the Fabry-Perot res-
onator. Then we have tested polygonal cavities both
with experiments and numerical simulations, and a good
agreement for the real parts of wavenumbers has been
obtained even for the non trivial configuration of the pen-
tagonal cavity.
The observed dominance of confined short-period orbits
is, in general, a consequence of the trace formula and the
2000I
630620610600
Wavelength (nm)
7060504030
Radius (µm)
2.0x10
1.00.0
Optical length (nm)
FIG. 19: (a) Experimental spectrum of a micro-disk of 30 µm
radius. (b) Optical length versus radius. The experiments
(red points) are linearly fitted by the solid red line. The
other lines correspond to theoretical predictions without any
adjusted parameters: the dashed green line to the square, the
dotted blue line to the hexagon, and the solid black line to
the perimeter. Inset: Normalized Fourier transform of the
spectrum in (a) expressed as intensity versus wavenumber.
formation of long-lived states in polygonal cavities is re-
lated to strong diffraction on cavity corners.
Finally, the study of micro-disks highlights the case of
several orbits and the influence of roughness and three-
dimensional effect.
Our study opens the way to a systematic exploration of
spectral properties by varying the shape of the boundary.
In increasing the experimental precision even tiny details
of trace formulae will be accessible. The improvement of
the etching quality will suppress the leakage due to sur-
face roughness and lead to a measure of the diffractive
mode losses which should depend on symmetry classes.
From the point of view of technology, it will allow a better
prediction of the resonator design depending on the ap-
plications. From a more fundamental physics viewpoint,
it may contribute to a better understanding of open di-
electric billiards.
VIII. ACKNOWLEDGMENTS
The authors are grateful to S. Brasselet, R. Hierle, J.
Lautru, C. T. Nguyen, and J.-J. Vachon for experimental
and technological support and to C.-M. Kim, O. Bohi-
gas, N. Sandeau, J. Szeftel, and E. Richalot for fruitful
discussions.
[1] A.E. Siegman, Lasers, (University Science Books, Mill
Valley, California, 1986).
[2] K. Vahala (ed.), Optical microcavities (World Scientific
Publishing Company 2004).
[3] E. Krioukov, D.J. W. Klunder, A. Driessen, J. Greve,
and C. Otto, Opt. Lett. 27, 512 (2002).
[4] A. M. Armani and K. J. Vahala, Opt. Lett. 31, 1896
(2006).
[5] C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel,
A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science
280, 1556 (1998).
[6] M. Lebental, J.-S. Lauret, J. Zyss, C. Schmit, and E.
Bogomolny, Phys. Rev. A 75, 033806 (2007).
[7] R. C. Polson, G. Levina, and Z. V. Vardeny, Appl. Phys.
Lett. 76, 3858 (2000).
[8] D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K.
J. Vahala, Nature 421, 925 (2003).
[9] M. Lebental, J.-S. Lauret, R. Hierle, and J. Zyss, Appl.
Phys. Lett. 88, 031108 (2006).
[10] C. Vassallo, “Optical waveguide concepts”, (Eslevier,
Amsterdam-Oxford-New York-Tokyo, 1991).
[11] P. K. Tien, Appl. Opt. 10, 2395 (1971).
[12] D.S. Jones, Proceedings of the Royal Society of London.
Series A, Mathematical and Physical Sciences, 217, 153
(1953).
[13] T. Harayama, P. Davis, and K.S. Ikeda, Phys. Rev. Lett.,
90, 063901 (2003).
[14] H.E. Türeci, A.D. Stone, and B. Collier, Phys. Rev. A
74, 043822 (2006).
[15] H.E. Türeci, A.D. Stone, and Li Ge, arXiv:
cond-mat/0610229 (2006).
[16] S. Shinohara, T. Harayama, H.E. Türeci, and A.D. Stone,
Phys. Rev. A 74, 033820 (2006).
[17] C. Y. Fong and A. Poon, “Planar corner-cut square mi-
crocavities: ray optics and FDTD analysis,” Optics Ex-
press 12, 4864 (2004).
[18] M.J. Giannoni, A. Voros, and J. Zinn-Justin (ed.), Chaos
and Quantum physics, les Houches Summer School Lec-
tures LII (North-Holland, Amsterdam, 1991).
[19] R. Blümel, T.M. Antonsen Jr., B. Georgeot, E. Ott, and
R. E. Prange, Phys. Rev. E, 53, 3284 (1996).
[20] J. Wiersig, Phys. Rev. A 67, 023807 (2003).
[21] I. Braun, G. Ihlein, F. Laeri, J. U. Nöckel, G. Schulz-
Ekloff, F. Schüth, U. Vietze, O. Weiß, D. Wöhrle, Appl.
Phys. B 70, 335 (2000).
[22] E. Bogomolny and C. Schmit, Phys. Rev. Lett. 92,
244102 (2004).
[23] E. Bogomolny, B. Dietz, T. Friedrich, M. Miski-Oglu, A.
Richter, F. Schäfer, and C. Schmit, Phys. Rev. Lett. 97,
254102 (2006).
[24] M. Lebental, N. Djellali, J.-S. Lauret, J. Zyss, R. Duber-
trand, C. Schmit, and E. Bogomolny, in preparation.
[25] J. Nöckel, PhD dissertation (1997).
http://www.eng.yale.edu/stonegroup/publications.html
[26] H. Schomerus and M. Hentschel, Phys. Rev. Lett. 96,
243903 (2006).
[27] M. Brack, R.K. Bhaduri, Semiclassical physics,
(Addison-Wesley Publishing Company, 1997).
[28] For some samples, the underlying layer is silica with re-
fractive index n2 = 1.45, so neff is slightly different.
[29] This definition is consistent all over the paper. In the
literature, these names are sometimes permutated.
[30] For a given symmetry class, the length entering the quan-
tization condition may be a part of the total periodic
orbit length.
[31] The estimation of the imaginary parts of these states as
well as the field distribution outside the cavity is beyond
the scope of this paper and will be discussed elsewhere.
http://arxiv.org/abs/cond-mat/0610229
http://www.eng.yale.edu/stonegroup/publications.html
