Measuring the spatial extent of individual localized photonic states

Abstract

We measure the spatial extent of individual localized photonic states in a
slow-light photonic crystal waveguide. The size of the states is measured by
perturbing each state individually through a local electromagnetic interaction
with a near-field probe. We find localized states which are not observed in
transmission and show that these states are shorter than the waveguide. We also
directly obtain near-field measurements of the participation ratio, from which
the size of the states can be derived, in quantitative agreement with the size
measured with the perturbation method.

Full text

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Measuring the spatial extent of individual localized photonic states
Marko Spasenovi´ c,1Daryl M. Beggs,1Philippe Lalanne,2Thomas F. Krauss,3and L. Kuipers1
1Center for Nanophotonics, FOM Institute for Atomic and Molecular Physics (AMOLF),
Science Park 104, 1098 XG Amsterdam, The Netherlands
2Laboratoire Charles Fabry de l’Institut d’Optique, CNRS, Univ Paris-Sud,
Campus Polytechnique, RD128, 91127 Palaiseau cedex, France.
3School of Physics and Astronomy, University of St Andrews, North Haugh, Fife, KY16 9SS, UK.
We measure the spatial extent of individual localized photonic states in a slow-light photonic
crystal waveguide. The size of the states is measured by perturbing each state individually through
a local electromagnetic interaction with a near-field probe. We find localized states which are not
observed in transmission and show that these states are shorter than the waveguide. We also directly
obtain near-field measurements of the participation ratio, from which the size of the states can be
derived, in quantitative agreement with the size measured with the perturbation method.
Waves in disordered media can undergo multiple scat-
tering, resulting in the formation of Anderson-localized
states with an associated impeded wave transport. An-
derson localization is a universal wave phenomenon, with
manifestations in electron transport [1], sound [2], matter
waves [3, 4] and light [5]. The spatial extent of localized
states, or localization length, is of primary importance.
For example, in 1D systems, when this length is larger
than the length of the sample, disorder has little effect
on wave propagation [6]. Conversely, when this length is
smaller than the sample length, strongly confined states
occur and wave transport is severely disrupted. The lo-
calization length is an ensemble averaged quantity, usu-
ally measured for a series of localization instances, by
averaging over frequency or many realizations of disor-
der. To date, Anderson localization in optical systems
has only been observed in transport [5] as narrow reso-
nances in transmission spectra [7–10], or using the mod-
ified spontaneous emission rates of embedded emitters
[11], or an increase in out-of-plane scattering [7, 12, 13].
These methods all share the feature that they measure
the ensemble average of an observable. From this av-
eraged quantity, one determines the most likely value
for some transport property, in the type of system being
studied. However, since disorder is stochastic, one does
not know how light in a specific structure at a specific
frequency will behave. In a sense, the problem is com-
parable to molecular spectroscopy prior to the success of
single-molecule detection, which enabled the study of in-
dividual molecules, rather than ensemble averages only
[15].
Here we measure the spatial extent of individual local-
ized photonic states for a single realization of disorder in
a photonic crystal waveguide. To emphasize the differ-
ence between the ensemble averaged localization length
and the measured length of an individual localized state
for a single optical frequency for a single realization of
disorder, we will use the symbol Lindfor the value that
we measure. We measure Lindby perturbing the states
locally through the light-matter interaction with a near-
field probe. We also determine Lind by measuring the
individual inverse participation ratio, obtaining values
for Lind that agree with those obtained with the local
perturbation method.
The idea that disorder in photonic crystals can be used
as a model system for the study of localization is as old
as the research field of photonic crystals itself [20]. Pho-
tonic crystal waveguides have been shown to be a good
example of true disordered systems in which Anderson
localization occurs [7, 11]. For 1D periodic systems, lo-
calization always occurs for any non-zero disorder, a fact
well established for electrons in 1D potentials [1, 21]. As
a model system, we investigate for a range of frequen-
cies (wavelengths) near the band edge, a 1-dimensional
photonic crystal waveguide (PhCW), formed by a sin-
gle row of missing holes from a hexagonal lattice of air
holes perforating a thin dielectric membrane, as depicted
in fig. 1(a). We introduce a near-field probe into the
evanescent field above the waveguide and observe an in-
teraction between the probe and the electromagnetic field
which changes the spectrum of the probed field. From
the change in the spectrum, we measure Lindof individ-
ual photonic states. We also measure Lindwith a method
based on the inverse participation ratio (IPR). The IPR is
a quantity related to Lind, often used as a measure of dis-
order. For example, the ensemble-averaged IPR has been
used to measure quantum eigenfunctions in ensembles of
metallic grains [14], disorder in 2-dimensional photonic
lattices [17], and to diagnose malignancy in biological
cells [18, 19]. The Lind measured with our probe-field
interaction method agrees with that measured with the
IPR method. We identify states for which Lindis smaller
than the length of the waveguide. We show that such
states do not contribute to transmission significantly.
Photonic crystal waveguides are a versatile tool for
controlling the propagation of light.
rameters (such as the size of the holes or thickness of
the membrane) control the dispersion of the waveguide
modes, which describes the relation between frequency
and wavevector of light in the waveguide. The calculated
dispersion of our PhCW is shown in fig. 1(b). Impor-
tant related quantities are the group velocity of light,
Geometric pa-
arXiv:1111.5942v1 [physics.optics] 25 Nov 2011

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FIG. 1: Schematic representation of the sample
and near-field probe. a, An aluminum-coated
near-field probe (inset) is scanned over the surface and
collects a fraction of the light. The magnetic field of
light interacts with the aluminum coating to shift the
resonance of localized states. b, Calculated dispersion
curve for the nominal design.
given by the slope of the dispersion (vg ≡ dω/dk), and
the density of photonic states. At wavelengths close to
1567.5nm, the slope of the dispersion becomes shallow,
indicating a decreased group velocity. Conversely, the
density of states (DOS) increases until the band-edge at
ka/2π = 0.5 is reached and (in theory) it diverges to in-
finity. In practice, however, disorder puts a lower limit
on the group velocity (and an upper limit on the DOS)
achievable in real systems [6, 7, 12, 22].
Although our PhCW is fabricated with state-of-the-
art techniques to nominally ideal designs, some residual
disorder always remains, resulting in photon scattering.
The amount of backscattering into the waveguide scales
with the group index squared [23], increasing near the
band edge. Since the localization length decreases with
increasing backscattering [6], near the band edge local-
ization length decreases. When the localization length
becomes shorter than the waveguide length, Lwav, pho-
ton transport in the waveguide is severely affected. For
device lengths typically used in practice (≈ 100µm), this
transition occurs, depending on the fabrication quality,
at group indices ng≈ 65[7].
The resonance behaviour of localized states of light
closely resembles that of photonic microcavities [24]. In
photonic crystals, such microcavities are deliberately en-
gineered defects which confine light to small volumes,
with high quality factors and well defined resonance fre-
quencies [25]. Anderson localization can be viewed as a
formation of such “cavities” through the wave interfer-
ence resulting from multiple scattering by random con-
figurations of disorder in the waveguide. As such, the
“cavities” occur at unpredictable positions, with unpre-
dictable quality factors and resonance frequencies. We
probe the near field of these Anderson localized states
using the electrically and magnetically mediated light-
matter interactions with a metallodielectric probe.
Our near-field probe is a tapered glass fibre of diameter
∼ 200nm, coated with aluminum of thickness ∼ 150nm
(Fig. 1a, inset). Usually, in near-field microscopy, such a
probe is introduced into the evanescent field of an opti-
cal mode and the electric field of light in photonic eigen-
states is detected with high resolution without signifi-
cantly perturbing the states [26]. Above a PhC cavity,
however, the probe may perturb the states due to the
light-matter interaction between it and the light in the
cavity. The electric part of the interaction typically leads
to an increase in the effective volume of the cavity, re-
sulting in a red-shift of the resonance, the magnitude
of which is inversely proportional to the volume of the
photonic state [27]. In the magnetic part of the interac-
tion, the magnetic field of light drives a current through
the metal coating, and, in a nanoscopic manifestation
of Lenz’s law, the magnetic field caused by the current
opposes that of the cavity [28, 29]. The result is that
the magnetic interaction shifts the resonance frequency
of the cavity to the blue, again with a magnitude in-
versely proportional to the volume of the state. In our
measurements of the PhCW, the magnetic interaction is
dominant, like for earlier measurements on a cavity [28],
although also the electric interaction plays a role. We
insert the probe in the near-field of the localized states
and determine the state volume and length from the re-
sultant blue-shift. Hence we measure Lind directly on
each instance of localization.
Firstly, we measure a transmission spectrum of the un-
perturbed waveguide, with the probe far from the sample
(Fig. 2a). Three spectral regions can be discerned. For
a free space wavelength λ∼
is high, as light is transported through the modes of the
waveguide. For 1561.6nm∼
sion drops, but many large and wide peaks are apparent.
Finally for λ∼
with a few narrow and sparse peaks appearing, a sign of
Anderson localization [8, 30].
Next, we bring the near-field probe close to the sam-
ple and measure the electric field above the waveguide.
The amplitude of the electric field, as a function of laser
wavelength and probe position, is depicted in Fig. 2b.
Measuring the electric field at the exit of the PhCW (at
the position x = 82 µm), one notes that the magnitude of
the measured electric field correlates with the measured
transmission for all wavelengths. For λ∼
of the image), we operate far above the band edge, and
light propagates in the Bloch modes of the periodic struc-
ture. The electric field amplitude at the exit is similar
to that at the entrance. For 1561.6nm∼
the magnitude of the measured electric field at the exit
decreases on average. However, we also occasionally ob-
serve sharp increases in the amplitude in the waveguide,
corresponding to the first localization peaks; for example
close to λ ≈ 1562.0nm. Another example of an extended
<1561.6nm, the transmission
<λ∼
<1563.2nm the transmis-
> 1563.2nm, the transmission is very low,
<1561.6nm (top
<λ∼
<1563.2nm,

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localization instance is seen at λ = 1563.35nm. Such
features seem to be spatially extended over the length of
the waveguide. At longer wavelengths (λ∼
populated states become sparse, but we observe light in-
side the waveguide at wavelengths at which transmission
is negligible. At these wavelengths, the light seems to
have more intensity close to the entrance of the waveg-
uide than at the exit, indicating states which are shorter
than the waveguide and which are populated by the light
incident from the input waveguide on the left. It is note-
worthy that the periodic amplitude modulation owing
to beating between forward- and backward-propagating
Bloch modes persists throughout the localization regime,
an indication that at all frequencies the light populates
modes of the nominal, perfect structure.
> 1563.2nm),
FIG. 2: Waveguide transmission, near-field ampli-
tude, and dispersion. a, Transmission spectrum, T,
of the PhCW. b, Near-field amplitude collected through
the probe, as a function of wavelength and probe position
along the waveguide. Light enters the PhCW from the
left, at position x = 0. c, Band structure of the waveg-
uide. Waveguide modes appear as lines in the spectrum.
A parabolic fit is depicted by the dashed yellow line. The
group index, ng, is obtained by taking the derivative of
this line. Localized states appear as horizontal lines in
the Fourier spectrum aligning with peaks in transmission.
The almost vertical line around k = 0.33 is the light line
in the silicon slab. d, Closeup of the near-field region
indicated with a dotted white box in part b).
Figure 2(c) depicts the dispersion of the waveguide
mode, obtained via a fast Fourier transform of the real-
space data to reciprocal space [31]. The dispersion curve
of the PhC waveguide follows the behavior expected from
calculation (Fig. 1b). It is noteworthy that the measured
bandstructure follows that of the nominal, perfect struc-
ture throughout the localization regime.
structure, localized states appear as horizontal lines span-
ning reciprocal space; the broad distribution of wavevec-
tors results from the high spatial confinement.
from the band edge (top of the image), the dispersion
curve can be approximated by a parabola. A parabolic
fit to this part of the curve results in the dashed yellow
line. The group index, shown on the right-hand axis,
is obtained by taking the first-order derivative of the fit.
Theory and measurements involving statistical averaging
[7] have shown that the average length of localization in-
stances becomes of the order of the waveguide length at
ng ∼ 65. This group index corresponds very well with
our first indication of localization, at λ = 1562nm.
The near-field probe collects the electric field with high
resolution, but at the same time we observe its interac-
tion with the electric and magnetic fields as shifts in the
spectral position of the localized resonances. This shift-
ing is most evident for long wavelengths (λ > 1563.5nm),
at the bottom of Fig. 2b. We observe that for certain spa-
tial positions of the probe, the maximum of the electric
field is shifted from its original spectral position towards
shorter wavelengths. One such position is indicated by
a white dot in the image. A closeup of the region with
the strongest shifts, indicated with a dotted white box in
Fig. 2b, is given in Fig. 2d. The shift occurs periodically
with spatial position, as a result of interference between
forward- and backward-propagating Bloch modes.
In engineered PhC microcavities, the spectral shift was
shown to be given by [27, 28]:
In the band
Away
∆λ(rpr)
λ
≈ αe
eff
αe
|E0|2
U0
+ αm
eff
|B0|2
U0
=
eff
2Vcav
αe
eff
2Vcav
|E(rpr)|2
max[?(rpr)|E|2]+
αm
eff
2VcavβH,
αm
eff
2Vcav
|Hz(rpr)|2
max[µ0|Hz|2]
=βE+ (1)
where U0is the energy of the unperturbed cavity field; E0
and B0are, respectively, the total electric and magnetic
fields of the unperturbed cavity; αe
larisability of the probe; αm
effis the magnetic polarisabil-
ity of the probe; Vcavis the cavity volume (area*length);
?(rpr) is the electric permittivity at the position of the
probe; µ0is the magnetic constant; max|E| is the max-
imum magnitude of the electric field in the sample;
|E(rpr)| is the magnitude of the electric field at the po-
sition of the probe; |Hz(rpr)| is the magnitude of the
component of the magnetic field which points into the
effis the effective po-

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probe at the position of the probe, and max|Hz| is the
maximum magnitude of Hzin the sample.
The electric and magnetic polarisabilities of our probes
were calculated by Burresi et al. [28] to have values of
αe
eff= 3 x 10−21m3?0and αm
spectively. To estimate the other parameters in equation
(1), we calculated the eigenmodes of the unperturbed
photonic crystal waveguide by using the MIT Photonic
Bands Package [32]. The eigenmodes at the wavelengths
of our study have an effective cross sectional area A of
1.12a2, with a the period of the PhC lattice. The ratios
substituted by variables βEand βHat the height of the
probe are both equal to 0.22. We are now in a position
to estimate Lindas
eff= −12 x 10−21m3/µ0, re-
Lind= 0.22λ
∆λ
αe
eff+ αm
2A
eff
, (2)
where we have used Vcav = LindA and take A to be
the same as that of the waveguide mode. Applying eq.
2 to clearly identifiable localized states from fig. 2 re-
sults in a determination of their localization length. We
depict these measured localization lengths as a function
of wavelength and group index by the blue points in
fig. 3. To the best of our knowledge, this is the first
time localized state length has been measured directly
on individual localized states in a photonic system. The
localization length generally becomes smaller at higher
ng’s, as the backscattering of waves increases [23] and
using our method we can precisely measure that for this
particular waveguide Anderson localization sets in, when
Lind< Lwav, at ng> 80, at which point the waveguide is
no longer useful for reliable optical transport. At higher
values of ng, we observe states in the near field of the
waveguide, but they are shorter than the waveguide and
do not contribute significantly to transmission.
Using eq. 2 we can determine Lindby measuring the
perturbation of localized states via their spectral shift.
We can also determine the degree of localization by mea-
suring IPR [33], also in the near field. The IPR is defined
as [33, 34]
IPR =
?
|E(r)|4dr, (3)
and is proportional to localization length [34, 35]. In 1D,
and normalised to the length of the waveguide, the IPR
takes the form
IPR =
Lloc
Lwav
=
?|E(r)|4dr
(?|E(r)|2dr)2, (4)
where Llocis the ensemble averaged localization length.
As we have measured the electric field as a function of
position and we know Lwav, we determine Lindusing eq.
FIG. 3: Dispersion of the localization length. The
blue dots are points measured with the near-field
interaction method. The green data points are
measured with the IPR method. The horizontal dashed
line indicates Lwav, the length of the waveguide.
4, assuming that the equation holds for measurements
of individual states as well as ensemble averages. We
plot the Lind obtained with this IPR method as green
points in Fig. 3, together with Lindobtained by measur-
ing the spectral shift of localized states. The agreement
between the two methods is quantitative. On the right
side of the plot, where Lindbecomes shorter than Lwav,
the two methods give the same results. The localiza-
tion length decreases towards longer wavelengths (higher
group indices), dropping to ≈ Lwav/3 at the lowest en-
ergy localized state of this study.
is decreased away from the band edge, the localization
length rises, and becomes on the order of the waveguide
length at λ ≈ 1563.5nm (ng ≈ 80). At shorter wave-
lengths, Lind ≈ Lwav, with sharp dips at the positions
of localization instances. At λ < 1561nm (ng < 55),
Lind> Lwav, indicating extended states.
Taking figures 2 and 3 as a whole, we now conclude
that three transport regimes can be identified.
λ∼
the waveguide is hardly affected by disorder. For higher
wavelength the localization length decreases, until for
1561.6nm∼
Lind≈ Lwav. In this regime photon transport becomes
unpredictable, with almost equal probability of high or
low transmission at any given wavelength. Finally, for
the longest wavelengths (λ∼
length has decreased well below the waveguide length,
and photons can only be transmitted in sporadic cases.
Populated localized states do exist in this regime, how-
ever, but their length is shorter than the length of the
waveguide. It is exactly these states which are useful
As the wavelength
For
< 1561.6nm, Lind > Lwav and photon transport in
< λ∼
< 1563.2nm, we have a situation where
>1563.2nm), the localization

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for applications requiring strongly localized fields, such
as quantum optics [11]. In all regimes of transport, our
method of measuring the Lindfrom the probe-field inter-
action gives the same result as the IPR method, which
is commonly used to measure the statistically averaged
degree of disorder in various systems [17–19].
Our work shows that a key parameter in localization
studies can be measured directly for a single realisation
of disorder. The Lind of a single instance cannot be
measured by observing transmission, and states for
which Lind
< Lwav are often not even observable
in transmission.Our measurement approach can be
extended to investigate and understand the behavior
of any disordered system where electromagnetic waves
are present at the surface.
be extended to finite size 2D systems, which are far
from trivial, and in which the interplay between local-
ization and out-of-plane scattering still has to be fully
understood. Notably, we do not need to perform the
measurement over the entire length or volume of the
state in order to determine Lind. Our method could thus
be used to determine the volume of localized states just
below the surface of a 3D disordered (photonic crystal)
structure. From the application point of view, a recent
study shows that localization in photonic crystal waveg-
uides can initiate lasing from quantum dots [16]. The
gain of random lasers has been theoretically predicted
to depend exponentially on localization length [40], but
shown numerically to have a power-law dependence on
localization length [41]. Our technique provides an ideal
method to resolve such dual predictions, by allowing
studies of gain and state length for each individual
localized state. We further expect our local perturbation
method to also work for other wave phenomena, such
as acoustics. The fact that we exploit the magnetically
induced light-matter interaction also reveals the fasci-
nating opportunity to study disorder in systems with
a strong magnetic field at optical frequencies, such as
metamaterials [38, 39].
The technique can easily
We thank Ad Lagendijk, Femius Koenderink, and
Sanli Faez for useful discussions. This work is part of the
research program of the Stichting voor Fundamenteel
Onderzoek der Materie (FOM), which is financially
supported by the Nederlandse Organisatie voor Weten-
schappelijk Onderzoek (NWO). We thank the EC Marie
Curie Scheme (contract MEST-CT-2005-021000).
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