Focusing coherent light through opaque strongly
I. M. Vellekoop* and A. P. Mosk
Complex Photonic Systems, Faculty of Science and Technology and MESA? Research Institute,
University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
*Corresponding author: email@example.com
Received March 6, 2007; revised June 14, 2007; accepted June 21, 2007;
posted June 28, 2007 (Doc. ID 80762); published August 2, 2007
We report focusing of coherent light through opaque scattering materials by control of the incident wave-
front. The multiply scattered light forms a focus with a brightness that is up to a factor of 1000 higher than
the brightness of the normal diffuse transmission. © 2007 Optical Society of America
OCIS codes: 290.1990, 290.4210, 030.6600.
Random scattering of light is what makes materials
such as white paint, milk, or human tissue opaque.
In these materials, repeated scattering and interfer-
ence distort the incident wavefront so strongly that
all spatial coherence is lost . Incident coherent
light diffuses through the medium and forms a vol-
ume speckle field that has no correlations on a dis-
tance larger than the wavelength of light. The com-
plete scrambling of the field makes it impossible to
control light propagation using the well-established
wavefront correction methods of adaptive optics (see
We demonstrate focusing of coherent light through
disordered scattering media by the construction of
wavefronts that invert diffusion of light. Our method
relies on interference and is universally applicable to
scattering objects regardless of their constitution and
scattering strength. We envision that, with such ac-
tive control, random scattering will become benefi-
cial, rather than detrimental, to imaging  and com-
Figure 1 shows the principle of the experiment.
Normally, incident light is scattered by the sample
and forms a random speckle pattern [Fig. 1(a)]. The
goal is to match the incident wavefront to the sample
so that the scattered light is focused in a specified
target area [Fig. 1(b)]. The experimental setup for
constructing such wavefronts is shown in Fig. 2.
Light from a 632.8 nm HeNe laser is spatially modu-
lated by a liquid-crystal phase modulator and focused
on an opaque, strongly scattering sample. The num-
ber of degrees of freedom of the modulator is reduced
by grouping pixels into a variable number ?N? of
square segments. A CCD camera monitors the inten-
sity in the target focus and provides feedback for an
algorithm that programs the phase modulator.
We performed first tests of inverse wave diffusion
using rutile ?TiO2? pigment, which is one of the most
strongly scattering materials known. The sample
consists of an opaque, 10.1-?m-thick layer of rutile
 with a transport mean free path of 0.55±0.10 ?m
measured at ?=632.8 nm. Since in this sample the
transmitted light is scattered hundreds of times,
there is no direct relation between the incident wave-
front and the transmitted image [7,8].
In Fig. 3 we show the intensity pattern of the
transmitted light. In Fig. 3(a) we see the pattern that
was transmitted when a plane wave was focused onto
the sample. The light formed a typical random
speckle pattern with a low intensity. We then opti-
mized the wavefront so that the transmitted light fo-
cused to a target area with the size of a single
speckle. The result for a wavefront composed of 3228
individually controlled segments is seen in Fig. 3(b),
where a single bright spot stands out clearly against
the diffuse background. The focus was over a factor of
1000 more intense than the nonoptimized speckle
pattern. By adjusting the target function used as
feedback it is also possible to optimize multiple foci
simultaneously, as is shown in Fig. 3(c) where a pat-
tern of five spots was optimized. Each of the spots
has an intensity of approximately 200 times the
original diffuse intensity. In Fig. 3(d) we show the
phase of the incident wavefront corresponding to Fig.
3(c). Neighboring segments are uncorrelated, which
indicates that the sample fully scrambles the inci-
The algorithm that constructs the inverse diffusion
wavefront uses the linearity of the scattering process.
The transmitted field in the target, Em, is a linear
combination of the fields coming from the N different
segments of the modulator:
where Anand ?nare, respectively, the amplitude and
phase of the light reflected from segment n. Scatter-
Fig. 1. (Color online) Design of the experiment. (a) A plane
wave is focused on a disordered medium, and a speckle pat-
tern is transmitted. (b) The wavefront of the incident light
is shaped so that scattering makes the light focus at a pre-
August 15, 2007 / Vol. 32, No. 16 / OPTICS LETTERS
0146-9592/07/162309-3/$15.00 © 2007 Optical Society of America
ing in the sample and propagation through the opti-
cal system is described by the elements tmnof the un-
known transmission matrix. Clearly, the magnitude
of Emwill be the highest when all terms in Eq. (1) are
in phase. We determine the optimal phase for a
single segment at a time by cycling its phase from 0
to 2?. For each segment we store the phase at which
the target intensity is the highest. At that point the
contribution of the segment is in phase with the al-
ready present diffuse background. After the measure-
ments have been performed for all segments, the
phase of the segments is set to their stored values.
Now the contributions from all segments interfere
constructively and the target intensity is at the glo-
bal maximum. A preoptimization with a small num-
ber of segments significantly improves the signal-to-
noise ratio. This method is generally applicable to
linear systems and does not rely on time reversal
symmetry or absence of absorption. Although math-
ematically this algorithm is the most efficient, in
noisy experimental conditions adaptive learning al-
gorithms  might be more effective, and an investi-
gation of such algorithms is on its way.
The maximum intensity enhancement that can be
reached is related to the number of segments that are
used to describe the incident wavefront. For a disor-
dered medium the constants tmnare statistically in-
dependent and obey a circular Gaussian distribution
[8,10–12], and the expected enhancement ?, defined
as the ratio between the optimized intensity and the
average intensity before optimization, can be calcu-
4?N − 1? + 1.
It was assumed that all segments of the phase modu-
lator contribute equally to the total incident inten-
sity. We expect the linear scaling behavior to be uni-
versal as Eq. (2) contains no parameters. Also, since
we are free to choose the basis for Eq. (1), we expect
to find the same enhancement regardless of whether
the target is a focus or a far-field beam and regard-
less of how the shaped wavefront is projected onto
the sample. Interesting correlations between the
transmission matrix elements, which will cause cor-
rections on Eq. (2), are predicted when N approaches
the total number of mesoscopic channels [11,12].
With our current apparatus we are far from this re-
gime and no deviation from Eq. (2) is expected.
We tested the universal scaling behavior implied
by Eq. (2) by changing N. Using the same TiO2
sample as before, the algorithm was targeted to con-
struct a collimated beam. In Fig. 4 the enhancement
is plotted as a function of the number of segments for
different focusing conditions. The linear relation be-
tween the enhancement and the number of segments
is evident until the enhancement saturates at ?
=1000. All measured enhancements were slightly be-
low the theoretical maximum. This is understand-
able since all perturbations move the system away
from the global maximum. The main reason for de-
viations from the optimal wavefront is residual am-
plitude modulation in the phase modulator, which in-
troduced an uncontrolled bias in the field amounting
to 14% of the total intensity.
The saturation of the enhancement is the result of
slow changes in the speckle pattern. This instability
effectively limited the number of segments for which
the optimal phase could be measured. We estimate
that the effective enhancement decreases to ?eff
=?/?1+NT/Tp?, where T=1.2 s is the time needed for
one measurement and the persistence time Tp
=5400 s is the time scale at which the speckle pat-
tern of the TiO2sample remains stable. Depending
on the environmental conditions, Tpcan be consider-
632.8 nm HeNe laser beam is expanded and reflected off a
Holoeye LR-2500 liquid crystal spatial light modulator
(SLM). Polarization optics select a phase mostly modula-
tion mode. The SLM is imaged onto the entrance pupil of
the objective with a 1:3 demagnifying lens system (not
shown). The objective is overfilled; we use only segments
that fall inside the pupil. The shaped wavefront is focused
on the strongly scattering sample (S), and a CCD camera
images the transmitted intensity pattern. ?/4, quarter-
wave plate; ?/2, half-wave plate; M, mirror; BS, 50% non-
polarizing beam splitter; P, polarizer.
(Color online) Schematic of the apparatus. A
Fig. 3. Transmission through a strongly scattering sample
consisting of TiO2pigment. (a) Transmission micrograph
with an unshaped incident beam. (b) Transmission after
optimization for focusing at a single target. The scattered
light is focused to a spot that is 1000 times brighter than
the original speckle pattern. (c) Multibeam optimization.
The disordered medium generates five sharp foci at the de-
fined positions. (a)–(c) are presented on the same logarith-
mic color scale that is normalized to the average transmis-
wavefront used to form (c).
OPTICS LETTERS / Vol. 32, No. 16 / August 15, 2007
ably higher and enhancements of over 2000 have
been measured overnight.
To verify the universal applicability of inversion of
wave diffusion, we used a variety of materials of
natural origin. Table 1 lists the intensity enhance-
ment for different materials we used. Although the
samples vary in thickness, composition, and scatter-
ing strength, they were all able to focus a properly
prepared wavefront to a sharp spot. The intensity en-
hancement varies between 60 and 1000. The main
reason for this variation is that the persistence time
is not the same for all materials.
In summary, our results show that precise control
of diffuse light is possible using an optimal, nonitera-
tive algorithm; light can be directed through opaque
objects to form one or multiple foci. The brightness of
the focal spot is explained by a model based on sta-
tistical optics. We expect inverse wave diffusion to
have applications in imaging and light delivery in
scattering media, possibly including metal nano-
structures . Dynamic measurements in biological
tissue are possible when the time required for achiev-
ing a focus can be reduced to below 1 ms per segment
[14,15]; we estimate that this time scale is techno-
logically possible with the use of fast phase modula-
tors . Furthermore, the high degree of control
over the scattered light should permit experimental
verification of random matrix theories for the trans-
port of light [11,12].
We thank Ad Lagendijk for valuable discussions,
Willem Vos and Vinod Subramaniam for a critical
reading of the manuscript, and the Photon Scattering
group of the Institute for Atomic and Molecular Phys-
ics (AMOLF) for providing samples. This work is part
of the research program of the Stichting voor Funda-
menteel Onderzoek der Materie (FOM), which is fi-
nancially supported by the Nederlandse Organisatie
voor Wetenschappelijk Onderzoek (NWO).
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Table 1. Measured Intensity Enhancement for Dif-
Daisy petal, fresh
Daisy petal, dried
aL, sample thickness ??surface roughness?; ?, maximum en-
hancement reached; N, number of segments used by the algorithm
to describe the wavefront.
a function of the number of segments. Squares, sample in
focus; triangles, sample 100 ?m behind focus; solid curve,
ideal enhancement [Eq. (2)], dotted curve, corrected for re-
sidual amplitude modulation and finite persistence time of
Tp=5400 s. The experimental uncertainty is of the order of
the symbol size.
(Color online) Measured intensity enhancement as
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