# Observation of mutual extinction and transparency in light scattering

Alfredo Rates  Complex Photonic Systems (COPS), MESA+ Institute for
Nanotechnology, University of Twente, P.O. Box 217, 7500 AE Enschede, The
Netherlands  Ad Lagendijk<EMAIL_ADDRESS>Complex Photonic Systems
(COPS), MESA+ Institute for Nanotechnology, University of Twente, P.O. Box
217, 7500 AE Enschede, The Netherlands  Ozan Akdemir  Complex Photonic Systems
(COPS), MESA+ Institute for Nanotechnology, University of Twente, P.O. Box
217, 7500 AE Enschede, The Netherlands  Allard P. Mosk  Debye Institute for
NanoMaterials Science and Center for Extreme Matter and Emergent Phenomena,
Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands  Willem
L. Vos<EMAIL_ADDRESS>Complex Photonic Systems (COPS), MESA+ Institute
for Nanotechnology, University of Twente, P.O. Box 217, 7500 AE Enschede, The
Netherlands

(June 8th, 2021. Intended for PRA.)

###### Abstract

Optical wavefront shaping - where multiple waves with adjustable phases and
amplitudes are sent onto complex media - has revolutionized the study of light
in Nanophotonics. Recently, a new type of wavefront shaping was introduced
where the extinction rather than the scattered intensity is manipulated. The
underlying idea is that upon changing either the phases or the amplitudes of
incident beams, the total extinction will change due to interference described
by the cross terms between the incident beams. Here, we present the first
experimental study of the mutual extinction effect in complex media, in
particular, a human hair and a silicon bar. The mutual extinction is readily
observed with two incident waves at small angles, depending on the relative
phase of the incident beams we observe either nearly zero extinction (mutual
transparency) or almost twice the single-beam extinction (mutual extinction)
in agreement with theory. We use an analytical approximation for the
scattering amplitude starting from a completely opaque object and discuss the
limitations of our approximation. We discuss novel applications of the mutual
extinction and transparency effects in various fields such as non-line-of-
sight communications, microscopy, and biomedical imaging.

##  I Introduction

Random scattering of light inside complex materials such as clouds, paint,
milk, white LEDs, hair, or human tissue is what makes them appear opaque [1,
2, 3, 4, 5]. In these materials, multiple scattering and interference distort
the incident wavefront so strongly that the spatial coherence is immensely
reduced [6]. The invention of optical wavefront shaping (WFS) [7], where
multiple waves are incident with adjustable phases and amplitudes, has
revolutionized the study of scattering of light in Nanophotonics and led to
exciting applications [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25, 26].

In the absence of absorption, the extinguished power is equal to the total
scattered power, a well-known conservation law called the optical theorem
[27]. The standard formulation of the classical optical theorem considers only
a single incident wave [28]. Naively using the single-beam optical theorem for
systems with multiple incident waves, leads to a violation of energy
conservation. Recently we presented a generalized optical theorem that
pertains to the situation of scattering and extinction by multiple incident
waves [29]. This emerged generalized optical theorem has the exciting
consequence that multiple incident waves show mutual extinction and mutual
transparency. These phenomena do not exist in common forward scattering or
self-extinction and are caused by interference between the scattered part of
one beam with the coherent part of the other, which increases or decreases the
total extinction. Furthermore, the mutual extinction and transparency effect
allows us to control the total extinction by manipulating the phase difference
between the two incident beams. Depending on their relative angle, the range
of variation can be as large as 100%.

In the present paper, we present an experimental observation of the mutual
extinction and transparency of two beams crossing in a scattering media that,
to the best of our knowledge, has never been observed before. In our
experiments, we study two different kinds of samples. The first type of sample
is a human hair, which is a biological sample with a naturally cylindrical
cross-section. The second type of sample is a silicon bar with a rectangular
cross-section made from a crystalline-Si wafer as used in CMOS fabrication,
see, e.g., Ref. [30, 31]. We compare our experimental results with the mutual
extinction theory and our analytical approximation, and we discuss when this
approximation fails. Finally, we discuss several applications of the mutual
extinction and transparency effects.

##  II Power flux and mutual extinction

In an experiment, a light detector with area $\cal{A}$ and placed in the
direction of the wavevector $\hat{k}_{{\rm det}}$ detects in far-field the
power flux or Poynting vector [32], which in the scalar-wave approximation is
the current density111For a scalar wave $\psi$, the flux $F$ is equal to
$F\equiv\int_{\cal{A}}{\bm{J}}\rm{d}A\equiv-\int_{\cal{A}}{\rm
Re}(\partial_{t}\psi)^{*}{\bm{\nabla}}\psi\rm{d}A$ ${\bm{J}}$ integrated over
$\cal{A}$ (see Fig. 1). When only beam 1 is incident in the scattering media
with a wave vector $\hat{k}_{{\rm in},1}=\hat{k}_{{\rm det}}$, the power flux
$F$ observed by the detector is equal to [1]

$\displaystyle F_{1,1}=F^{{\rm in}}_{1}-F^{{\rm ext}}_{1}+F^{{\rm scat}}_{1},$
(1)

where $F^{{\rm ext}}_{1}$ is the flux taken out of the incident flux $F^{{\rm
in}}_{1}$ due to interference between the outgoing coherent wave and scattered
waves. Scattered light from beam 1 is present at all angles, where $F^{{\rm
scat}}_{1}$ is the flux of the fraction scattered into area $\cal{A}$ with
wavevector $\hat{k}_{{\rm scat},1}=\hat{k}_{{\rm det}}$.

If a second beam 2 is also present with a wavevector $\hat{k}_{{\rm
scat},2}\neq\hat{k}_{{\rm det}}$, i.e, not in the direction of the detector,
the power flux at the detector becomes

$\displaystyle F_{1,2}=F_{1,1}+F^{{\rm scat}}_{2}+F^{{\rm scat}}_{1,2}+F^{{\rm
cross}}_{1,2}.$ (2)

Similar to $F^{{\rm scat}}_{1}$, $F^{{\rm scat}}_{2}$ is the flux of the
scattered fraction of beam 2 incident into the detector with wavevector
$\hat{k}_{{\rm scat},2}=\hat{k}_{{\rm det}}$. $F^{{\rm scat}}_{1,2}$ is the
cross term describing interference between the scattered waves from both
incident beams, and $F^{{\rm cross}}_{1,2}$ describes interference between the
coherent wave of the incident beam 1 and the scattered wave of the incident
beam 2. This final term of Eq. 2 corresponds to either mutual extinction or
mutual transparency, depending on its sign. In case of destructive
interference, $F^{{\rm cross}}_{1,2}$ is negative and the total extinction is
increased, corresponding to mutual extinction. While in the case of
constructive interference, the total extinction is decreased, corresponding to
mutual transparency.

Figure 1: Diagram of a two-waves scattering system. The two beams are incident
in a scatterer with wavevectors $\hat{k}_{{\rm in},1}$ and $\hat{k}_{{\rm
in},2}$, and the angle between them is $\gamma$. The curved lines represent
the scattered waves, present in any direction, given by the wavevectors
$\hat{k}_{{\rm scat},1}$ and $\hat{k}_{{\rm scat},2}$. The direction of the
scattered waves is represented by the colored arrows. The diagram emphasises
the direction of the scattered waves which lead to the mutual extinction and
transparency. The detector has an area $\cal{A}$. The colors distinguish the
incident and scattered waves and do not represent the wavelength of the light.
Dimensions are not to scale.

We can experimentally obtain $F^{{\rm scat}}_{2}$ if we collect data when only
the incident beam 2 is present, and $F^{{\rm in}}_{1}$ if we collect data
without the scattering medium. Combining the data from the different cases, we
can get the interference term $F^{{\rm cross}}_{1,2}$ from the following
observables

$\displaystyle F^{{\rm cross}}_{1,2}=F_{1,2}-F_{1,1}-F^{{\rm scat}}_{2},$ (3)

where we assume $F^{{\rm scat}}_{1}\ll F^{{\rm in}}_{1}$ (and thus also
$F^{{\rm scat}}_{1},\ F^{{\rm scat}}_{1,2}\approx 0$) which is reasonable
because the measurement direction is equal to the incident direction. Using
this reasonable assumption, we can extract the self-extinction $F^{{\rm
ext}}_{1}$ in a similar way

$\displaystyle F^{{\rm ext}}_{1}=F_{1,1}-F^{{\rm in}}_{1}.$ (4)

We use $F^{{\rm ext}}_{1}$ for normalization, as we want to know how the total
extinction changes due to these interferences with respect to the case when
only self-extinction is considered. Thus, we obtain the normalized total
extinction from

$\displaystyle F^{{\rm TE}}=\frac{F^{{\rm cross}}_{1,2}}{F^{{\rm ext}}_{1}}.$
(5)

Figure 2: Schematic of the experiment. The angle $\gamma$ between the two
beams is controlled by moving and rotating mirror M2 and an LC phase retarder
controls the phase difference ($\Delta\phi$) between the two beams. The inset
shows the position of the sample related to the direction of beam 1. (M:
Mirror; PD: Photodiode; BS: Beamsplitter; LP: Linear polarizer; PR: Phase
retarder.)

##  III Experimental methods

To measure mutual extinction and transparency, we built the experimental setup
shown in Fig. 2. A He-Ne laser ($\lambda=632.8\ \rm{nm}$) is used as a source.
The laser beam is separated into two incoming beams, beam 1 and beam 2, in a
modified Mach-Zehnder configuration with a slight and controllable
misalignment at the outgoing beamsplitter BS2. Before reaching BS2, both beams
pass through linear polarizers, and beam 2 passes through a liquid crystal
phase retarder (LCC1413-B Thorlabs), which we use to control the phase
difference $\Delta\phi$. By carefully moving and rotating mirror M2, we
control the angle $\gamma$ between the two beams. At a fixed location
downstream of BS2, where the sample is located, the two beams cross at an
angle $\gamma$. We use a CCD camera to detect beam 1 by integrating over the
illuminated pixels, and a photodiode at the beginning of the optical circuit
to correct for laser fluctuations. Both the angle variation and the phase
variations are made in beam 2, whereas only beam 1 is detected with the CCD
camera. At every angle $\gamma$, the phase $\Delta\phi$ was varied from $0$ to
$2\pi$ and back. For each phase, we took three consecutive exposures with the
CCD camera to average over environmental noise.

We position the sample in the intersection plane of the two beams, as shown in
the inset of Fig. 2. All samples we study have a bar-like geometry, meaning
that one dimension is much larger than the other two. We define the active
area as the overlapping area of the two beams in the illuminated face of the
sample. Thus, in the x-direction the dimension of the active area is given by
the geometry of the sample, and in the y-direction the dimension is given by
the beam diameter.

Figure 3: Microscope picture of samples used. (a) picture of a human hair,
from which we can extract a diameter $d_{\rm{hair}}=0.062\pm 0.002\ {\rm mm}$.
(b) is separated in a picture from the side (b.1) and from the top (b.2) of
the silicon bar. From such microscopy pictures we extract a thickness
$d_{\rm{Si}}=0.103\pm 0.003\ {\rm mm}$ and a width $w_{\rm{Si}}=0.440\pm
0.002\ {\rm mm}$.

We study two different samples at present: The first sample is a human hair,
which has a natural near-cylindrical shape with a diameter of
$d_{\rm{hair}}=0.062\pm 0.002\ {\rm mm}$ as observed with a microscope (see
Fig. 3(a)). The scattering properties of human hair are of special interest
for the 3D animation industry to obtain a realistic simulation of hair in
animated characters [33]. Furthermore, single human hair fibers are widely
used to teach light diffraction in undergraduate and secondary education,
approximating it to the inverse of a single slit [34]. The second sample is a
thin silicon bar cleaved from a standard Complementary Metal Oxide
Semiconductor (CMOS) wafer, with a thickness $d_{\rm{Si}}=0.103\pm 0.003\ {\rm
mm}$ and a width $w_{\rm{Si}}=0.440\pm 0.002\ {\rm mm}$ (see Fig. 3(b)). The
scattering properties of silicon are highly relevant for the semiconductor
industry since it is the main material used in electronics. To limit the
complexity we use samples that are translational invariant along the y-axis.
To accomplish this, the silicon bar is illuminated from the side and not from
the top.

##  IV Experimental results

Figure 4: Total extinction $F^{{\rm TE}}$ versus phase difference $\Delta\phi$
for an angle $\gamma=0.20^{o}$ between the two beams for the human hair. Dark
red circles are experimental data while the solid blue line is a fit to the
exact mutual extinction model. Dotted line is the effect in case of no mutual
extinction or transparency.

First, let us consider the case where the angle $\gamma$ is fixed. In Fig. 4,
we show the normalized total extinction of a human hair while changing the
phase $\phi$. The angle between incoming beams is $\gamma=0.2^{o}$. We see
that the extinction follows a cosine-like trend, as expected from the
prediction [29]. We also see that near the maximum the extinction nearly
double, while for the minimum the extinction is close to zero so that the
object appears nearly fully transparent. In theory, the total extinction is
minimum for $\Delta\phi=0$ and maximum for $\Delta\phi=\pi$. We see that the
experimental data are shifted in phase, which arises from uncertainty in the
true-zero phase. This effect is due to changes in the optical path of beam 2
when moving and rotating mirror M2 (see Fig. 2). Still, we see that the
periodicity of the observed cosine-like curve agrees well with theory, and
thus this phase offset does not affect the final results.

In Fig. 5, we plot the total extinction of the human hair versus angle
$\gamma$. The corresponds to the maximum and minimum extinctions obtained as a
function of the phase at each angle (see Fig. 4). The observed maximum and
minimum extinctions show an oscillatory behavior versus incident angle
$\gamma$, typical of interference between scattering and coherent beams. Along
with the experimental data, our analytical model is shown to be in good
agreement. This model uses the scattering amplitude of an impenetrable flat
surface (see Appendix A) as an approximation to obtain the variations in the
forward scattering due to the mutual extinction effect. Our model has no
adjustable parameter since the width of the sample $a_{mic}$ is obtained by
microscopy inspection. We see that the data follows a sinc trend similar to
the model, with a slight shift in angle discussed in the next section.

For the silicon bar, the total extinction against angle is shown in Fig. 6.
The data has a similar sinc shape as the human hair, but here the frequency of
the interference fringes is higher. Furthermore, the analytical approximation
agrees even better with the experimental data than for the hair sample.

Figure 5: Total extinction versus angle $\gamma$ between beams for a human
hair. The red circles (green squares) correspond to the maximum (minimum)
extinction obtained from phase variations (see Fig. 4). The dashed lines are
our analytical model with no adjustable parameter (see Appendix A).

##  V Discussion

Figure 6: Total extinction versus angle $\gamma$ between beams for a silicon
bar. The red circles (green squares) correspond to the maximum (minimum)
extinction obtained from phase variations. The dashed lines are our analytical
model with no adjustable parameter.

###  V.1 Observations and model

In both the phase and angle variation experiments, we obtained the trend
predicted by the mutual extinction theory [29], namely a cosine-like trend in
the phase variation experiments and a sinc-like trend in the angle variation
experiments. We see that using our analytical model, we obtain a description
of the mutual extinction, which gives a faithful interpretation of the
observations.

In Fig. 4, we see that the modulation of the total extinction is close to the
full range from 0 to 2. To discern how large this modulation is compared with
the total signal of the beam, we obtained that in the case of the human hair,
the extinguished light amounts to about 15.4% of the incoming light. With the
mutual extinction and transparency effect, the extinguished light varies from
approximately 2.5% to 24%. This is close to the maximum modulation predicted
by theory, which varies from 0% to 30.8%.

In Fig. 5, we observe that the model predicts a slightly lower frequency of
fringes than measured, meaning the nodes of the model are located at larger
angles than the ones from the experimental data. The curve shapes are in
excellent overall agreement. The small deviations emerge from the fact that
the human hair has a cylindrical geometry. Thus, the sample thickness varies
with the lateral position within the incident beams, which is not addressed in
the model, where we assumed a flat impenetrable surface. We can quantify this
deviation using the width of the sample $a$ as a single adjustable parameter
in our model (see Eq. 7), and compare the estimated value $a_{\rm mod}$ with
the width of the sample used originally as input for the model, which we
extracted from optical microscopy inspection $a_{\rm mic}$. In Table 1 we see
from this comparison that the adjusted width does not match with the
independent observation.

Sample |  $a_{\rm mod}$ (mm) |  $a_{\rm mic}$ (mm)
---|---|---
Hair |  $0.072\pm 0.001$ |  $0.062\pm 0.002$
Silicon |  $0.105\pm 0.001$ |  $0.103\pm 0.003$
Table 1: Table of sample dimension extracted from the model ($a_{\rm mod}$),
and extracted from optical microscopy inspection ($a_{\rm mic}$). Error range
of the model is due to the dispersion of the experimental data, while the
error range of the optical microscopy inspection is due to microscope
resolution.

In contrast, for the silicon sample, which has a box-like geometry, $a_{\rm
mod}$ and $a_{\rm mic}$ are equal within the error bars. We see in Fig. 6 that
for small $\gamma$ the analytical model describes the silicon data accurately.
This can lead one to think that this approximation describes the results
completely, but this is not the case. In Fig. 7, we zoom in on the silicon
results to observe in detail the deviation for angles between $\gamma=2$ to
$3^{o}$ degrees. We see that for $\gamma>2.0^{o}$ the trend followed by the
experimental data has some discrepancies with the model: In the beginning, the
nodes start to shift, and finally, the shape of the sinc curve is completely
lost. Importantly, at all these angles the measured phase dependencies keep
following a cosine-like curve as in Fig. 4, and the signals are significantly
larger than the errors. We extended our analytical model by also taking into
account the beam divergence of the laser using a convolution over the Gaussian
beam profile. Nevertheless, we see that the average caused by divergence does
not explain this deviation. The apparent random shape of the total extinction
at larger angles is probably an indication that we are entering the speckle
regime where the variations in the extinction do not depend only on the
dimensions of the sample, but also on the detailed spatial distribution of the
scatterers inside the material or the surface roughness.

Figure 7: Total extinction versus angle $\gamma$ between beams for a silicon
bar. Zoom in for large angles from Fig. 6. The dashed lines are our analytical
model with no adjustable parameter.

When the sample is fully opaque and its size is much larger than the
wavelength, the angular dependence of the scattering amplitudes, and thus of
the mutual extinction and transparency, will follow the conventional
dependence of the Kirchhoff integral [35], which we used to derive our model.
In the situation where the sample is not fully opaque, as in our experiments,
the angular dependence of the scattering amplitudes will become speckle-like,
starting at the large angles $\gamma$. If the sample has a mean free path $l$
much larger than its size ($l\gg L$), the angular speckle will set out in the
immediate vicinity of the first zero-crossing, which itself is determined by
the geometry of the sample. The more transparent the sample is, the more
prominent the speckle will be. But when averaging over many realizations, the
result of the Kirchhoff integral is recovered.

###  V.2 Applications

The mutual extinction and transparency effect can have applications in the
field of ultraviolet communications for non-line-of-sight links [36], which is
based on light scattering in the atmosphere. The scattered light can be
enhanced or reduced if the transmitter uses two beams that cross in the active
area. Furthermore, mutual transparency can be used to reduce losses related to
light attenuation in the atmosphere. Deeper research is needed to translate
the results presented in this paper to a more practical system.

The mutual extinction and transparency effect is a promising tool to use the
observed interference fringes to infer the shape and size of an object,
including free-form samples [37].

Diffusion Wave Spectroscopy (DWS) has become a popular technique for studying
time-dependent optical properties of complex materials and for bio-imaging
[38, 39, 40]. Unfortunately, DWS is not well suited for samples that absorb
rather than scatter. To study the motion of scattering particles, one
currently uses techniques such as Dynamic Light Scattering (DLS) [41], which
is also not suited for samples that absorb rather than scatter. In contrast,
the mutual extinction and transparency effect can be used for samples that
scatter or absorb (or both), as long as there is some detectable intensity
left of the incoming beam. We envisage for those samples that dynamics can be
probed by using a time-dependent mutual extinction technique.

##  VI Conclusions

We measured the total extinction of two beams crossing through a scattering
object, namely a human hair and a silicon bar. Upon varying the relative angle
and phase between the beams, we measured the variations in the total
extinction. When the angle is close to zero, we can manipulate the extinction
in such a way that the scattering object is almost twice as opaque or nearly
fully transparent to the beams. Alternatively, if the angle is larger we
enter, for non-opaque samples, into the speckle regime where fluctuations of
the mutual extinction depend on the precise shape and distribution of the
scatterers distribution in the sample.

Our results are in close agreement with the recently presented mutual
extinction theory [29], turning this experiment into the first demonstration
of this effect. We used an analytical approximation of the scattering
amplitude applicable when the sample is opaque. We have seen that this
approximation is a good model for box-like geometries and small angles, but at
the same time, we see that the mutual extinction effect cannot faithfully be
interpreted with such a simple approximation of the scattering amplitude, for
transparent samples.

##  VII Acknowledgements

We thank Cornelis Harteveld for expert technical support and sample
preparation, Manashee Adhikary and Matthijs Velsink for helpful discussions,
and Catalina García for her contribution. This work was supported by the NWO-
TTW program P15-36 ”Free-form scattering optics” (FFSO) and the MESA+
Institute section Applied Nanophotonics (ANP).

##  Appendix A Theoretical model

In Ref. [29], we have calculated and described the scattering amplitudes of
both a collection of point dipoles and a flat and opaque object. Here, we
compare our experimental results with the latter, where the scattering
amplitude is derived as (see Eq. 8 of Ref. [29])

$\displaystyle
F_{\rm{model}}=\frac{iab}{2\lambda}\rm{sinc}(\alpha)\,\rm{sinc}(\beta)$ (6)

where $\alpha\equiv\frac{a}{2}(\cos\theta_{x,{\rm out}}-\cos\theta_{x,{\rm
in}})$, $\beta\equiv\frac{b}{2}(\cos\theta_{y,{\rm out}}-\cos\theta_{y,{\rm
in}})$, $\theta_{x,{\rm in}}$ and $\theta_{y,{\rm in}}$ are the angles of the
incident waves with respect to the x and y-axes, $\theta_{x,{\rm out}}$ and
$\theta_{y,{\rm out}}$ are the angles of the outgoing waves with respect to
the x and y-axes, and $a$ and $b$ are the dimensions of the sample in the x
and y-directions, respectively.

We can simplify this equation if we consider both incident waves to be only in
the xz-plane. If we define $\gamma$ as the angle formed between the two
incident waves, the scattering amplitude is simplified to

$\displaystyle F_{\rm{model}}\backsimeq
F_{1,\rm{m}}\cdot\textrm{sinc}\left(\frac{2a}{\lambda}\cdot\textrm{sin}\frac{\gamma}{2}\right),$
(7)

where $F_{1,\rm{m}}=\frac{ab}{\lambda}$ is the forward scattering amplitude of
the sample. For the human hair, the dimension $a$ is defined as the diameter
of the hair ($a=d_{\rm{hair}}$), and for the silicon bar it is defined as the
thickness of the bar ($a=d_{\rm{Si}}$).

To take the divergence $\Theta$ of the laser beam into account, we implemented
a convolution between $F_{\rm{model}}$ and the angular momentum profile of a
Gaussian beam $U(\theta)$ given by [42]

$\displaystyle
U(\theta)=\rm{exp}\left(-\rm{tan}^{2}(\theta)/\rm{tan}^{2}(\Theta)\right).$
(8)

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