A crossed waveguide grating is presented that can extract the total guided-mode power emitted by a pointsource dipole located in the structure. Results obtained with rigorous numerical simulations are compared with a simple graphic analysis to facilitate an understanding of the far-f ield radiation pattern of such a luminescent device

A Sentenac

F Lemarchand

H Giovannini

H Rigneault

CiteSeerX

148 OPTICS LETTERS / Vol. 24, No. 3 / February 1, 1999Extraction of light from sources located inside
waveguide grating structures
H. Rigneault, F. Lemarchand, A. Sentenac, and H. Giovannini
Laboratoire d’Optique des Surfaces et des Couches Minces, Unité de Recherche Associée au Centre National de la Recherche
Scientifique, Unité Propre de Recherche de l’Enseignement Supérieur A 6080, Microcavities and Light Scattering Groups,
Ecole Nationale Supérieure de Physique de Marseille, Domaine Universitaire de St Jérôme, 13397 Marseille Cedex 20, FranceReceived October 6, 1998
A crossed waveguide grating is presented that can extract the total guided-mode power emitted by a point-
source dipole located in the structure. Results obtained with rigorous numerical simulations are compared
with a simple graphic analysis to facilitate an understanding of the far-f ield radiation pattern of such a
luminescent device.  1999 Optical Society of America
OCIS codes: 050.2770, 300.2140, 310.2790, 350.3950, 260.3800, 260.2110.The problem of extracting the total electromagnetic
power provided by a localized source located inside a
solid-state microcavity structure has been addressed
by many authors since Purcell1 discussed the pos-
sibility of modifying the spontaneous-emission rate
by tailoring the electromagnetic environment into
which the source can radiate. Microcavities with
different geometries have been considered, starting
from the simple planar Fabry–Perot stack2 to some
more-advanced three-dimensional structures such as
microspheres,3 microdisks,4 micropillars,5 and pho-
tonic wires.6 In the case of sources located inside
planar microcavities it is now well established that
although the extracted light can be directive the
extraction coefficient is less than a few tens of percent
(usually ,30%), with the remaining fraction of the
total power trapped in the substrate or in guided
modes.7 The problem is somehow more complex for
three-dimensional microcavities, which can have low
extraction coefficients (usually associated with high-Q
rotating modes) or broad-angle emission (owing to
their small output surfaces). A source located inside
a corrugated structure is a suitable alternative for ex-
tracting the electromagnetic power. Kitson et al. have
shown that it is possible to scatter light from non-
radiative modes by exciting dye molecules on a metallic
grating.8 These molecules can relax by generating
surface plasmon polaritons. It is even possible to pro-
hibit particular modes, causing a full photonic bandgap
effect.9,10
In this Letter we present a solid-state device that
can extract more than 80% of the total emitted
light in specific directions. The structure [Fig. 1(b)]
is a waveguide grating structure (WGS) with pe-
riodicity along two orthogonal directions (crossed
grating).
Before reporting rigorous numerical results we de-
scribe the physical idea that led us to use grating
structures to achieve strong extraction coefficients.
Consider first a simple dielectric slab waveguide whose
refractive index ng is higher than the indices of the
surrounding air na and the substrate ns [Fig. 1(a)].0146-9592/99/030148-03$15.00/0For simplicity we assume that this waveguide sup-
ports a single guided mode. It was shown11 that when
a luminescent source is introduced inside this slab
most of the provided electromagnetic power is car-
ried by the guided mode. Furthermore, this light
is trapped for every polar f direction in the film
plane [Fig. 1(a)]. It is then natural to address the
following question: Is it possible to extract 100% of
the trapped light by use of a bidirectional corrugated
surface?
The use of a grating to couple out guided light has
been extensively treated,12 including sources directly
located under corrugated surfaces.13 Nevertheless,
only gratings with periodicity along one direction,
which in fact are unable to extract the guided power
for each f direction, have been considered.8 This re-
sult is quite easy to see if we consider plane vec-
tor kp, where kp ­ kxx 1 kyy is the orthogonal
projection of k in the sOxyd plane. Figure 2(a) shows
this plane for a slab waveguide supporting a single
guided mode whose wave-vector modulus is kg. For
waves with wave-vector modulus k , k0 (free-space
range), where k0 ­ 2pyl0, the light can emerge out-
side the structure [shaded area in Fig. 2(a)], whereas
the guided mode solid circle with radius kg . k0 re-
mains trapped in the slab. To couple the guided
mode to the propagating waves running outside the
guide it is interesting to use a shallow corrugated
surface with periodicity dx along the x axis. In this
case a specific running wave of planar wave vector
kp0 ­ kx0x 1 ky 0y is coupled through the grating
Fig. 1. (a) Planar slab and polar angle f and (b) biperiodic
WGS considered in the computation and source location. 1999 Optical Society of America
ebruary 1, 1999 / Vol. 24, No. 3 / OPTICS LETTERS 149F
Fig. 2. Planar kp space for (a) a slab and (b) a corrugated
surface along the x axis (darker circles) and along the x and
y axes (darker and lighter circles). The solid circle denotes
the guided mode.
to all the waves of planar wave vectors kpm with
kpm ­ kp0 1 mKxx, where Kx ­ 2pydx. An alter-
native way to describe this phenomenon is as the pe-
riodic duplication of the free-space disk along the kx
axis. This situation is depicted in Fig. 2(b), where
only the three darker circles along the kx axis must
be considered. In this case the guided-mode solid
circle sections that overlap the darker circles can be
extracted outside the slab because of the m ­ 11 and
m ­ 21 f irst grating orders. On the other hand, the
remaining guided-mode directions, which do not over-
lap the darker free-space circles, are still trapped in
this Ox corrugated surface. To couple out the total
guided mode outside the structure it is necessary to
use a periodic surface along both the x and the y
directions with periods dx and dy. In this case, as
is schematically depicted in Fig. 2(b), where all the
lighter and darker circles must now be considered, the
guided-mode solid circle always intercepts the first-
order free space circles. Furthermore, for this fig-
ure we have chosen Kx ­ Ky ­ kg. Note that this
condition ensures that dx and dy are smaller than l
and that the crossed WGS has a subwavelength scale.
For the particular points Gi si ­ 1, 2, 3, 4d, wave-
guide vector kg is coupled with the normally incident
wave vector skp0 ­ 0d. More generally, any planar
wave vector kp0 can be coupled through the grating
to a kpm, n wave if kpm, n ­ kp0 1 mKxx 1 nKyy .
We now consider a point-source dipole located at
r0 ­ x0x 1 y0y 1 z0z inside the WGS. This source
emits free-space and trapped electromagnetic power
for both polarization states. We know already from
the simple previous analysis that the guided mode
is totally extracted from the structure because the
guided-mode solid circle is entirely intercepted by the
free-space circles. Furthermore, we expect the guided
light to escape from the structure following these in-
tersections, giving rise to specific arc-shaped direc-
tions that concentrate the electromagnetic extracted
power.
Aside from these qualitative considerations and to
get some insight into this emission extraction process,
it is interesting to model rigorously the emission from
a dipole located in a WGS. In straightforward terms,
our method is based on a Fourier description of thesource, the structure, and the fields. More precisely,
our point source is expressed in the Fourier plane as
P sx, y, zd ­ P0dsx 2 x0dds y 2 y0ddsz 2 z0d
­
Z 1`
2`
dkx
Z 1`
2`
dky
P0dsz 2 z0d
s2pd2
3 exp ifkxsx 2 x0d 1 kys y 2 y0dg
­
Z 2p/dx
0
dkx
Z 2p/dy
0
dky
P0dsz 2 z0d
s2pd2
3
1X̀
n­2`
1X̀
m­2`
exp ifskx 1 mKxd sx 2 x0d
1 sky 1 nKy d s y 2 y0dg . (1)
The last term in Eq. (1) is a coherent superposition
of periodic sources that have different phases but the
same periodicity as the grating structure along both
x and y axes. Because of the superposition princi-
ple in linear electromagnetic theory, the overall prob-
lem of a point source radiating inside a WGS reduces
therefore to an infinite number of problems for pe-
riodic sources radiating inside the WGS. This prob-
lem is simpler because emitters, structure, and fields
have the same periodicity, which allows us to solve
the emission problem by the differential method14
in which the source is treated as a field disconti-
nuity in the numerical integration.15 The outgoing
wave’s boundary conditions enable us to compute the
total power radiated by the source at infinity by
means of Poynting f lux calculations through an Oxy
parallel plane.
To confirm the simple physical picture given in
Fig. 2 we now present the result of computations
concerning a realistic structure in which a biperiodic
grating is etched onto a homogeneous guiding layer
[Fig. 1(b)]. We assume a deltalike emission at l ­
650 nm for a point-source dipole oriented along the x
axis and located in the middle of the guiding layer.
This location ensures good positioning of the source
with regard to the fundamental guided modes of this
WGS. More precisely, this structure supports two
guided modes (one for each polarization state). The
periods are chosen such that the TE guided mode
satisfies the condition kgTE ­ Kx ­ Ky [configuration
of Fig. 2(b)], whereas the TM mode with kgTM , kgTE
intercepts the free-space disks for every f polar angle
as well. This situation ensures the total extraction of
the guided-mode’s power for both polarization states.
Figure 3 shows in planar kp space the result of com-
putation for the Poynting f lux power density per unit
solid angle dPydV in air [Fig. 3(a)] and in the substrate
[Fig. 3(c)]. The darker curves are those of maxi-
mum power. As expected, light escapes from the
structure in arc-shaped directions. These patterns
150 OPTICS LETTERS / Vol. 24, No. 3 / February 1, 1999Fig. 3. Energy-density patterns from a biperiodic struc-
ture holding a point source. Computations are made for
(a) air and (c) the substrate. The shapes of the arcs
can be deduced from the guided modes TE0 (broad solid
curves) and TM0 (thin solid curves) of the homogenous
structure that intercept the free-space disks [(b) air and
(d) substrate].
can be easily understood if we plot the intersections be-
tween the guided-mode solid circles and the free-space
disks [Figs. 3(b) and 3(d)]. These intersections corre-
spond to the guided-mode resonance directions of the
structure without any source. It is interesting to men-
tion that this simple analysis holds although the grat-
ing is not shallow (the groove depth is equal to half of
the guiding layer thickness). Indeed, because of the
strong refractive index of the homogeneous layer and
the small effective index of the corrugated layer, the
guide is weakly perturbed by the grating. Neverthe-
less, for a stronger corrugation of the guide, the guided
mode kg may depend on f and the guided closed line
may no longer be a circle. In the latter case, only
rigorous computation can give the accurate radiation
pattern at infinity.
Not given by simple graphic considerations are the
intensities and widths of the various arcs. Numerical
integration in the planar k space of dPydV shows that
only the TE mode carries signif icant power, not the
TM mode (thinner arcs in Figs. 3(b)–3(d)]. From the
calculation we find that 65% of the total emitted light
is contained in the arcs, whereas the remaining 35%
is emitted in all directions and appears as a quasi-
isotropic background. All in all, the light emission is
divided into one third for air and two thirds for the
substrate. From Fig. 3(c) it is possible to see the total
ref lection circle of unit diameter (dashed curve). For
k greater than this circle’s radius the light is trapped inthe substrate. Although it is coupled out by the WGS,
this trapped light is 30% of the total power emitted into
the substrate.
In conclusion, we have presented a crossed WGS
that is able to extract the total guided-mode power,
whatever the polar direction f is. We have presented
a simple graphic analysis of the arc-shaped radiation
pattern at infinity and compared it with a rigorous
electromagnetic calculation of a point-source dipole
emitting into the structure. Although 80% of the light
escapes from the WGS through running waves through
the two sides of the structure, some unwanted light
still remains at a large angle and may be difficult
to collect. Studies of cancellation of these arc-shaped
light concentrations at large angles by use of strong
corrugated gratings are in progress. This remaining
problem a side, this type of structure is of potential
interest for use in building high-extraction-coefficient
luminescent devices.
H. Rigneault’s e-mail address is herve.rigneault@
enspm.u-3.mrs.fr.
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Extraction of Light from Sources Located inside Waveguide Grating Structures.

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Extraction of Light from Sources Located inside Waveguide Grating Structures.

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