Photon delocalization transition in dimensional crossover in layered media.

Abstract

We report a crossover in optical propagation in random layered media from localization towards diffusion as the interaction of the wave with the sample is transformed from one to three dimensional due to nonuniformity in the layer thickness. The crossover occurs at the point that the lateral spread of the wave equals the transverse coherence length in the transmitted speckle pattern.

Full text

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arXiv:0904.1905v1 [cond-mat.mes-hall] 13 Apr 2009
Photon delocalization transition in dimensional crossover in layered media
Sheng Zhang,1Jongchul Park,1Valery Milner,1and Azriel Z. Genack1
1Department of Physics, Queens College, The City University of New York, Flushing, NY 11365, USA
(Dated: April 13, 2009)
We report a crossover in optical propagation in nonuniform random layered media from local-
ization towards diffusion as the interaction of the wave with the sample is transformed from one
to three-dimensional. The crossover occurs at the point that the lateral spread of the wave equals
the transverse coherence length in the transmitted speckle pattern. Delocalization is fostered as the
sample thickness or lateral nonuniformity increases.
PACS numbers: 42.25.Dd, 42.25.Bs, 42.30.Ms
Layered media [1] are ubiquitous in geological, biolog-
ical, electronic, and photonic settings.
transport in these largely one-dimensional structures em-
bedded in three-dimensional space is challenging because
of the critical role of dimensionality in wave propaga-
tion and localization. Both classical and quantum waves
become exponentially peaked or localized [2, 3, 4, 5] in
disordered samples when the number of times a wave
winds its way through typical coherence volumes within
the sample exceeds unity. The return to a point is rein-
forced by the constructive interference of waves following
time reversed paths and is facilitated in low-dimensional
systems which restrict the volume explored by the wave.
As a consequence, localization can always be achieved in
sufficiently large one and two-dimensional samples even
when scattering is weak [3]. In three dimensions, how-
ever, localization can only be realized when scattering
is sufficiently strong that the mean free path, ℓ, is sub-
stantially smaller than the wavelength, which may be ex-
pressed as, kℓ < 1,[6] where k = 2π/λ is the wavevector.
Understanding
Examples of one-dimensional localization abound. Lo-
calization has been observed for acoustic waves along
a wire to which masses are randomly attached [7], mi-
crowave radiation in single-mode metallic waveguides
with random dielectric inserts [8], and infrared radia-
tion in single-mode fibers with random Bragg gratings
[9]. A one-dimensional description [10] is also suitable
in the case of plane wave illumination of an unbounded
medium comprised of parallel layers. It has been used
to describe the localization of electrons in semiconductor
superlattices [11] and photons in parallel dielectric lay-
ers of random thickness [12, 13, 14]. Measurements of
the scaling of average optical transmission, ?T(L)?, for a
normally incident beam in an ensemble of random stacks
of overhead transparencies [13] and glass cover slips [14]
were in accord with 1D simulations. The sample thick-
ness, L, is given in terms of the numbers of glass layers
which alternate with air gaps. Transmission approached
the asymptotic limit, ?T(L)? ∼ exp(−L/2ξ), where ξ is
the calculated average exponential decay length of local-
ized modes within the sample [13, 14]. Transmission in
such samples is mediated by the excitation of states with
single or multiple exponential peaks [5] in the spatial in-
tensity distribution [8, 15]. Localized modes in layered
samples play a particularly important role in amplifying
media since such modes are long-lived by virtue of their
weak coupling to the boundaries. Low-threshold lasing
was demonstrated in a stack of glass slides and dye sheets
when the pump laser and emission spectrum overlapped
localized modes near the center of the sample [14].
Though propagation and lasing in passive and active
layered media has been extensively investigated, the im-
pact of nonuniformity within the layers upon transport
has not been reported. Instead, studies of waves in ran-
dom layered media have focused on their localization per-
pendicular to presumed uniform layers. In this Letter, we
report a crossover from localized towards diffusive propa-
gation with increasing thickness and disorder in random
layered media with nonparallel interfaces. Beyond the
crossover point, transmission departs from 1D simula-
tions and approaches an inverse rather than an exponen-
tial scaling. This reflects a continuous change in dimen-
sionality of wave transport from one to three dimensions
with increasing sample thickness. The crossover occurs
because destructive interference, which results in local-
ization in samples with uniform layers of random thick-
ness, is washed out as wave trajectories spread beyond a
coherence length in the transmitted speckle pattern due
to transverse disorder.
We consider the nonuniformity in thickness within the
layers. We studied transmission of a single frequency
helium-neon laser at 633 nm through stacks of 22-mm2
glass slides with refractive index n = 1.523 and thick-
nesses in the range 125-135µm. Samples are held in place
by two rings of 18-mm inner diameter. Since the thick-
ness of each glass slide and of the air gap between slides
is not uniform, a normally incident beam is scattered
off the normal direction to produce a speckled intensity
pattern at the output. These speckle patterns can be
imaged with a lens upon a CCD camera or scanned with
an optical fiber probe leading to a photodiode detector
(see [16] for details). Examples of measured speckle pat-
terns for samples with 1, 2, 20 and 80 slides are shown
in Fig. 1. The degree of parallelism of the two faces

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FIG. 1: (Color Online) (a) Example of a fringe pattern gen-
erated by a single slide. (b) Typical speckle pattern for two
slides and intervening air gap. (c) and (d) Speckle patterns
generated by samples with 20 and 80 slides, respectively. The
local transmission coefficient indicated by the colorbar may
exceed unity due to the interference of waves with wavevector
components in the layer plane.
of the slide can be ascertained from the fringe patterns
of single slides [see Fig. 1(a)]. The fringes are gener-
ally nearly parallel to the sides of the slide. The fringe
spacing, a, varies from 160 to 6800 µm in a sample of 100
slides, indicating a variation of local wedge angle, θ, from
1.5×105to 2.6×103rad, where 2naθ ≈ λ and n = 1.523
is the refractive index of glass. The air gaps between
slides are nonuniform because of deviations from flatness
of the glass surfaces as well as because of occasional dust
particles. This is reflected from speckle patterns gener-
ated by 2 slides [seen in Fig. 1(b)]. With increasing L,
the speckle patterns at the output are randomized while
the angular distribution of transmitted radiation broad-
ens so that the scale of features in the speckle pattern
shrinks.
The average transmission of a 500-µm-wide collimated
beam directed normal to the glass slides was measured
using an integrating sphere. Transmission was averaged
by translating the sample over a 100-mm2area for 10 dif-
ferent stacks of slides. Measurements are plotted with red
squares in Fig. 2(a) and compared with 1D simulations
for an ensemble of configurations shown as black dots. In
the simulations, transmission and reflection for each layer
is represented by a transfer matrix and transmission for
the entire structure can be obtained from the product of
these matrices. Simulations of ?T(L)? fall exponentially
in the limit of large L. The decay length is predicted to
equal 2ξ, which is equal to the thickness of a stack of 22
slides. ξ is also the exponential decay length ?lnT(L)?
[see inset in Fig. 2(a)]. Measurements of ?T(L)? are in
agreement with simulations up to L = 40 but fall more
FIG. 2: (Color Online) (a) Semi-logarithmic plot of measure-
ments (red squares) and 1D simulations (black dots) of ?T?
versus number of glass slides, L. Simulations of ?lnT? are
shown in the inset. (b) and (c) Schematic of interference be-
tween partial waves following two trajectories, α and β, which
pass through the same layers an equal number of times, in
samples with parallel and nonparallel layers, respectively.
slowly than simulations for larger L.
The departure of measurements of ?T(L)? from 1D
simulations can be understood by comparing the super-
position of corresponding rays in samples with parallel
and nonparallel interfaces shown schematically in Figs.
2(b) and 2(c). The two wave trajectories α and β in the
sample with parallel interfaces shown in Fig. 2(b) pass
through each slide the same number of times and are
therefore of equal length. However, since light reflects
from a higher index medium in one of the two additional
reflections in path β, the partial waves for the two paths
are out of phase by π rad and interfere destructively. As
the number of layers increases, the relative weight of such
pairs of out of phase trajectories increases leading to an
exponential decrease of ?T(L)?. [13]
In a sample in which the interfaces are not parallel,
the trajectories corresponding to those in Fig. 2(b) are
distorted as shown schematically in Fig. 2(c). An ad-
ditional phase difference between the two partial waves
accumulates since trajectories cross the layer at different
points at which the layer thicknesses differ and the an-
gle between trajectories are no longer equal. The phase
difference between such pairs of trajectories is thereby in-
creasingly randomized as the spatial and angular spread
of the beam increases with increasing number of slides
or local wedge angles. The reduced cancellation of trans-
mission of such paired trajectories leads to a slower decay
of ?T(L)?. In the limit in which the correlation between
such pairs of partial waves vanishes, the falloff of ?T(L)?

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FIG. 3: (Color online) (a) Schematic of the average transverse
spread, σ, and the speckle size, d, along x-direction at the
output plane for an incident plane wave. (b) Measurement of
σx and dx versus L. Their crossing at L ≈ 35 is consistent
with the departures of ?T(L)? from results of 1D simulations
beginning at L = 35 [seen in Fig. 2(a)].
becomes diffusive and transmission falls as 1/L [17].
The above considerations make it plain that nonuni-
formity within the layers reduces the impact of localiza-
tion on transmission. The suppression of longitudinal
localization depends upon the relationship between typ-
ical displacements within the plane between trajectories
starting at the same point and the field coherence length,
each of which is influenced by disorder within the layers.
The coherence length is directly exhibited in the inten-
sity speckle pattern and is inversely proportional to the
width of angular spread of the transmitted beam [18].
Wave localization is essentially one dimensional only as
long as the area explored by a wave incident at a point is
smaller than the coherence area of the field. Thus, one-
dimensional localization breaks down once the sample is
thick enough that the characteristic length of the trans-
verse spread of the wave, σ⊥, equals the field correlation
length in the plane, d⊥, σ⊥= d⊥. These lengths along a
single direction are shown schematically in Fig. 3(a).
The field correlation length along the x-direction, dx,
for example, can be determined from the correlation func-
tion of the field on the output surface, Γ(∆x).
in turn is the Fourier transform of the ensemble aver-
aged angular distribution [18] measured in the far-field,
known as the specific intensity, ?I(θx)? [16]. The corre-
lation lengths are taken to be twice the length in which
Re{Γ(∆x)} decays to half its maximum value.
This
The ensemble average of the spread of an incident
beam with intensity profile, Iin(x′,y′), to produce the
intensity distribution at the output surface, ?Iout(x,y)?
may be expressed in terms of the spread function, Pin(x−
x′,y − y′), giving ?Iout(x,y)? =
x′,y − y′)dx′dy′.Pin(x − x′,y − y′) is similar to the
point spread function in three-dimensional diffusive sys-
tems but differs in that it depends upon the wavevec-
tor distribution at the input. Therefore, standard meth-
ods employed to determine the point spread in three-
dimensional random samples such as the direct mea-
surement of the beam profile due to a strongly focused
incident beam [19] or the measurement of the inten-
sity correlation function in the far field as a function
of sample angle [20] cannot be applied in layered sam-
ples. Direct measurement of Iin(x′,y′) and ?Iout(x,y)?
are made by imaging the wave onto a CCD camera (see
[16] for details). Integrating over y, for example, gives,
Iin(x) =?Iin(x,y)dy. Defining σ2
variances based on the functions, Iin(x), ?Iout(x)? and
P(∆x), respectively, and taking the origin as the center
of the incident beam, gives, σ2
σ2
xin. The variance so obtained can be used
to characterize the spread of the wave in the x-direction.
The variations of the widths of the intensity spread
functions and the speckle size along the x-direction as a
function of the number of slides are shown in Fig. 3(b).
Results for σ and d along the y-direction are very close to
those along the x-direction. The measurements of σ and
d allow us to determine the effective number of transverse
modes involved in transmission of the wave over the area
over which the wave spreads, σxσy, N = NxNy, where,
? ?Iin(x′,y′)Pin(x −
xin, σ2
xoutand σ2
xas the
xin=?∞
−∞Iin(x)x2dx, and
x= σ2
xout−σ2
Nx=
?
1,σx/dx< 1;
σx/dx, σx/dx> 1.
and Nyis similarly defined. When N = 1, wave propaga-
tion is essentially one-dimensional. Only a single polar-
ization component of the wave is considered since trans-
mission is highly polarized even in the thickest samples.
The crossing of the curves for σ and d marks a crossover
from one to three-dimensional transport and a transition
from localization to diffusion. Such a crossing occurs at
L ≈ 35 for both the x and y directions [Fig. 3(b)]. Be-
yond this thickness, ?T(L)? departs from 1D simulations
[Fig. 2(a)].
Since transverse disorder leads to both an increased
spread of the wave and to a drop in the coherence length,
we expect that ?T(L)? will depart from 1D simulations
when the degree of nonparallism of the layers increases.
This is confirmed in measurements in samples created by
inserting narrow metal shims at alternating edges of the
glass slides as shown in Fig. 4. The curve in Fig. 4 is
the calculation for photon diffusion utilizing the intensity
reflection coefficient at the air/glass interface, R = [(n −
1)/(n + 1)]2= 0.043 . The falloff of ?T(L)? approaches
the diffusive limit as the wedge angle increases.

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FIG. 4: (Color Online) Comparison of ?T(L)? for samples
with different additional wedge angles. Thin shims with dif-
ferent thicknesses are inserted between layers to introduce av-
erage wedge angles of 0.073◦, 0.218◦and 0.364◦, respectively,
into the air gaps. The side of the slide in which the shims are
placed is alternated so that the average angle of the slides is
not changed. The total transmission could only be measured
up to thicknesses at which the beam spread does not approach
the edges of the slides.
The scaling of transmission in layered samples differs
from scaling observed in samples in which wave propa-
gation is of fixed dimensionality [17, 21, 22]. In such
samples, the wave, once localized, remains localized, and
the scale dependent conductivity or diffusion coefficient
[4, 21, 22, 23] decrease continuously with sample thick-
ness. This is in contrast to wave delocalization observed
here in layered media. Unlike propagation in isotropic
two or three-dimensional random media, for which the
angular distribution of transmission is independent of
thickness for L > ℓ, the angular distribution of the wave
in layered media is highly directional and broadens with
sample thickness and with depth into the sample.
Highly anisotropic angular distributions are also found
in samples in which the index of refraction is uniform in
the longitudinal direction but disordered in the trans-
verse directions [24, 25, 26] The small values of k⊥ in
that case leads to localization in the transverse plane in
relatively short distances even when, kℓ⊥≫ 1, where ℓ⊥
is the transverse mean free path [25]. Such transverse
localization stands in contrast to longitudinal delocaliza-
tion in random layered samples which is exhibited once
the spread of the wave exceeds the transverse coherence
length.
In conclusion, the crossover from localized to diffusive
transport in layered media demonstrates the critical role
of dimensionality in transport in a class of samples which
occurs widely in nature and in photonics and electronic
microstructures.
We thank Victor Kopp for the 1D simulation program,
Samuel Gillman for experimental assistance, and Howard
Rose for the sample holder.
sored by the National Science Foundation under grant
This research was spon-
no. DMR-0538350. V.M. is presently at the Department
of Chemistry and the Laboratory for Advanced Spec-
troscopy and Imaging Research, University of British
Columbia, Vancouver, BC, V6T 1Z1, Canada.
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