Table Of ContentSpatio-temporal Control of Light Transmission
through a Multimode Fiber with Strong Mode Coupling
Wen Xiong1, Philipp Ambichl2, Yaron Bromberg1, Brandon Redding1, Stefan Rotter2, Hui Cao1∗
1Applied Physics Department, Yale University, New Haven CT 06520, USA
2Institute for Theoretical Physics, Vienna University of Technology, A-1040 Vienna, Austria
We experimentally generate and characterize the eigenstates of the Wigner-Smith time-delay
matrix,calledprincipalmodes,inamultimodefiberwithstrongmodecoupling. Theuniquespectral
and temporal properties of principal modes enable a global control of the temporal dynamics of
optical pulses transmitted through the fiber, despite random mode mixing. Our analysis reveals
thatthewell-defineddelaytimeoftheeigenstatesareformedbymulti-pathinterference,whichcan
be effectively manipulated by the spatial degrees of freedom of the input wavefront. This study is
6 essentialtocontrollingthedynamicsofwavescattering,pavingthewayforcoherentcontrolofpulse
1 propagation through complex media.
0
2 PACSnumbers: 42.65.Sf,42.25.-p,42.81.Cn
n
a
J The temporal dynamics of wave scattering in complex ItremainsobscurehowPMsareformedwithwell-defined
8 systems has been widely studied in quantum mechanics, delaytimesandwhatpropertiestheypossessinthepres-
1 nuclearphysics,acousticsandoptics. Mostofthesestud- ence of non-perturbative mode mixing. Here we report
ies, e.g., electromagnetic or ultrasonic wave propagation on a demonstration of PMs in a MMF with strong mode
]
s in billiards [1–4], electron transport through quantum coupling. Our analysis uncovers that the well-defined
c dots [5, 6], and light scattering in random media [7–10] delay time of a PM can be explained by multi-path in-
i
t focused on the statistics of delay times, i.e., eigenvalues terference that is tailored by spatial degrees of freedom
p
of the Wigner-Smith time-delay matrix [11–13]. Despite of the input wavefront. This multi-path interference also
o
. innumerable trajectories the wave could take through an determines the spectral bandwidth of PMs, which limits
s
opencomplexsystem,aneigenstateoftheWigner-Smith the temporal width of optical pulses that can be trans-
c
i matrix remarkably has a well-defined delay time. Some mitted through the fiber without distortion.
s
y of the eigenstates are particlelike with their wavefunc- The Wigner-Smith time-delay matrix is defined as
h tions concentrating on a single trajectory [4], and hence Q≡−iS−1dS/dω, where S is the scattering matrix of a
p a definite transit time is expected. Most of the states, system [12, 13]. In the absence of backscattering in the
[
however, consist of enormous trajectories with various fiber, it can be expressed as Q ≡ −iT−1dT/dω [16, 18],
1 lengths such that it is concealed how well-defined delay inwhichS isreplacedbythetransmissionmatrixT. We
v times can be attributed to these states. experimentallymeasurethefieldtransmissionmatrixofa
6
Largelyinparallel,theWigner-Smitheigenstateswere MMFasafunctionoffrequencyinanoff-axisholographic
4
introduced for multimode optical fibers (MMFs), which setup shown schematically in Fig. 1(a). To introduce
6
4 attractedmuchattentioninrecentyearsduetotherapid strongmodecouplinginaone-meter-longfiber,weapply
0 development of space-division multiplexing for telecom- stress to the fiber using clamps. The field transmission
1. munications [14]. Inherent imperfections and external matrix is measured in momentum space and converted
0 perturbations introduce random coupling of the guided to the mode basis. Figure 1(b) shows the amplitudes of
6 modes in the fiber and cause temporal broadening and a measured transmission matrix. Whichever mode the
1
distortion of transmitted pulses. As a generalization of inputlightislaunchedinto,theoutputfieldspreadsover
:
v principal states of polarization in a single-mode fiber all modes, although higher order modes have lower am-
i [15], the Wigner-Smith eigenstates, also called principal plitude due to stronger loss. The transmission matrix
X
modes(PMs)ofMMFs,wereproposedtosuppressmodal is very different from that in the weak coupling regime,
r
a dispersion [16]. which has larger elements closer to the matrix diagonal,
Advances in wavefront shaping techniques now make confirmingthemodesarestronglycoupledinthecurrent
it possible to probe a single Wigner-Smith eigenstate in fiber.
optics. Recently PMs were observed experimentally in TheeigenstatesofQ, whichisconstructedfromtrans-
a few-mode fiber with weak mode coupling [17]. In this mission matrices measured at different frequencies, give
regime, mode coupling in the fiber is only perturbative the input fields of PMs. They are unique input states
and hence the PMs are similar to the eigenmodes of a that produce frequency-independent output field pat-
perfect fiber. In the strong mode coupling regime, how- terns to the first order of frequency variation [16]. In
ever,allmodesarestronglymixedandthemultiplescat- anidealMMF,thePMsaresimplythelinearlypolarized
tering of light between different guided modes generates modes,whicharetheeigenmodesofthefiberintheweak
numerous paths for light to propagate through the fiber. guidingapproximation. Intheweakmodecouplinglimit,
2
(a)
(a) M2
SLM Laser
C1
M1
BS1
L
O
MMF
CCD BS2 C2
(b) (c) (d)
0.3
20
e 0.25
mod 40 de 0.2 (b)
Output 8600 Amplitu0.01.51 B) 0
110200 20 4In0pu6t 0m8o0de100120 0.005 20 40 60 80100120 ns(d-2 Prminocdipeal
Mode number o
0.05 0.2 0.35 ti
a-4
Figure 1. (color online) (a) Schematic of the simplified ex- el
r
perimentalsetupformeasurementofthetransmissionmatrix or-6 Random
ofaMMF.Thecontinuous-waveoutputfromatunablelaser C input
source is collimated by a collimator (C1), and split into two
-314 -157 0 157 314
armsbyabeamsplitter(BS1). Thelightinonearmismod-
ω-ω =Δω(GHz)
ulated by the SLM and imaged to the fiber facet by a lens 0
(L) and an objective (O). The output field from the fiber is
collimated(C2)andcombinedwiththelightintheotherarm Figure 2. (color online) (a) Output field amplitude for the
at a second beam-splitter (BS2). By offsetting the BS2 to input wavefront of a PM at ω0 = 1219 THz (top row), or
introduce a phase tilt between the two wavefronts, interfer- a random superposition of linearly polarized modes (bottom
ence fringes are formed. From the interferogram recorded by row). The input frequency is ω−ω0 = −157 GHz (left col-
thecamera(CCD),theoutputfieldisextracted. Themirrors umn), 0 (middle column), and 157 GHz (right column). The
(M1,M2)areusedtomatchthepath-lengthofthetwoarms outputfieldpatternsforthePMinputaresimilarwhilethose
of the interferometer. The MMF is one meter long with 50 for random input are totally different. (b) Spectral correla-
µm core diameter and 0.22 numerical aperture. (b) Ampli- tion function C(∆ω) of the output field pattern, measured
tude of the measured transmission matrix at λ = 1550 nm experimentallyforaPM(redsolidcurve),orcalculatedfrom
(ω=1219 THz). (c) Amplitude profile of the output field of themeasuredtransmissionmatrixandinputspatialprofileof
a PM. (d) Decomposition of the PM in (c) by the linearly the same PM (green dotted curve). For comparison, C(∆ω)
polarized modes, revealing that it consists of many modes. forarandominputisalsoshown(bluedashedcurve). C(∆ω)
is normalized to one at ∆ω=0. The agreement between the
red and green curves illustrates the accuracy of the measure-
ment. TheoutputfieldpatternforthePMdecorrelatesmuch
the fiber length is less than the correlation length (the
slower with frequency than the random input, and C(∆ω)
distance beyond which the spatial field profile becomes
displays a plateau at ∆ω=0.
uncorrelated [19]), and each PM consists of a few modes
withsimilarpropagationconstants. However,ifthefiber
length well exceeds the correlation length, all modes are ure 2(a) shows the far-field patterns at three frequencies
thoroughly mixed and the PMs are expected to be dis- (toprow),andtheyarenearlyidentical. Forcomparison,
tinct from those in the weak coupling regime. We use a a random superposition of modes at the input results in
spatial light modulator (SLM) to generate input wave- different output profiles at these three frequencies [bot-
fronts of individual PMs of this MMF with strong mode tom row of Fig. 2(a)]. This striking difference illustrates
coupling. To modulate both amplitude and phase of the thattheoutputfieldpatternofthePMdecorrelatesmuch
inputfieldwithaphase-onlySLM,acomputer-generated slower with frequency.
phase hologram is employed [20]. Figure 1(c) shows the Tobemorequantitative,wecalculatethespectralcor-
output pattern of a PM, which is speckled and does not relation function C(∆ω ≡ ω −ω ) ≡ |Ψ(ω )∗ ·Ψ(ω)|,
0 0
resembleanylinearlypolarizedmodeofthefiber. Modal where Ψ(ω) is a vector representing the output fields
decomposition of the pattern reveals that the PM is a in all spatial channels with its magnitude normalized to
mixture of many modes [Fig. 1(d)], in contrast to the unity. AsshowninFig.2(b),C(∆ω)forthePMissignif-
PM in the weak mode coupling regime. icantlylargerthanthatfortherandominput. Itdisplays
To investigate the spectral property of PMs, we scan a broad plateau at ∆ω =0. To understand the shape of
the frequency ω while keeping the input field pattern to the correlation curve, we denote C(∆ω) = cos[θ(∆ω)],
that of a PM at ω . The output field pattern is mea- whereθ istheanglebetweenthetwooutputfieldvectors
0
suredateachfrequencyandcomparedtothatatω . Fig- at ω and ω . Since θ(0) = 0, the first-order derivative
0 0
3
(a) (b) (c) of the invariance of output field pattern with frequency.
0.5 Input pulse Namely, the output field vector at frequency ω can be
mplitude ensity0.3 RPaMndom input wtorritΦtencoarsreΨsp(oωn)d=stφo(ωa)PTM(ωa)tΦω,0w.hIefrtehtehienpinuptubtafinedldwivdetch-
A nt is less than the spectral correlation width of the PM,
I
T(ω)Φ≈α(ω)Ψˆ ,whereΨˆ isaunitvectorrepresenting
0.1 0 0
thenormalizedoutputfieldprofileforthePMatω ,and
-0.2 -0.1 0 0.1 0.2 -0.2 -0.1 0 0.1 0.2 -0.1 0 0.1 0.2 0
Relative delay time (ps) Relative delay time (ps) Relative delay time (ps) α(ω)isacomplexnumberthatmayvarywithfrequency.
TheFouriertransformofΨ(ω)givestheoutputfieldvec-
Figure 3. (color online) (a,b) Temporal variation of the out- torΨ(t)=φ˜(t)Ψˆ ,whereφ˜(t)=(cid:82) φ(ω)α(ω)e−iωtdtrep-
0
put field amplitude in three spatial channels (three speckles
resents the output pulse shape. Hence, the spatial and
grains)whenanopticalpulseislaunchedintoarandomsuper-
temporalvariationsoftheoutputfieldbecomedecoupled
positionoffibermodes(a)oraPMatω =194THz(b). The
0
spatial profile of the output field is recorded in a frequency for the PM. The temporal traces in all output channels
range of 400 GHz with a step size of 2.5 GHz. The Fourier areidenticaluptoaconstantfactorgivenbytheelements
transform is then performed to obtain the field evolution in ofΨˆ . Thespatialprofileoftheoutputfieldremainscon-
0
time. The temporal traces of individual spatial channels are stant in time, allowing the spatial and temporal distor-
totallydifferentfortherandominput,butnearlyidenticalfor
tionstobecorrectedseparately. Forexample,theoutput
the PM input. (c) Spatially integrated intensity of the input
pulse shape can be tailored by modulating the spectral
(black dotted curve) and the output pulses when a Gaussian
phaseofinputspectrumφ(ω). Sincetheoutputfieldsare
pulseisinjectedtotheMMFwithrandomspatialprofile(blue
dashed curve) or with the profile of a PM (red solid curve). spatiallycoherent,aspatialmaskcanconverttheoutput
to any desired pattern or focus to a diffraction-limited
spot.
of C with respect to ∆ω vanishes at ∆ω =0 for any in- Let us consider a simple case, α(ω) (cid:39) α eiη(ω), where
0
put wavefront. The second-order derivative at ∆ω = 0 α isaconstantamplitudeandthephaseη(ω)(cid:39)η(ω )+
0 0
is proportional to [θ(cid:48)(0)]2, where θ(cid:48) ≡ dθ/d∆ω. For the η(cid:48) (ω−ω ), where η(cid:48) is the value of dη/dω at ω . Then
0 0 0 0
PM,θ(cid:48)(0)=0,becausetheoutputspatialprofileremains theoutputpulse,φ˜(t)∝φ(t−η(cid:48)),hasthesametemporal
0
unchangedtothefirstorderoffrequencyvariation. Thus shape as the input one. This is confirmed by synthesiz-
the second-order derivative vanishes for the PM, leading ing a pulse with Gaussian spectrum and flat phase at
to a plateau of the correlation curve, that is absent for the input. The output intensity, summed over all spatial
the random input. channels, is plotted in Fig. 3(c) together with the input
In a next step we probe the temporal dynamics of a pulseintensity. Theoutputpulsehasnegligiblebroaden-
singlePM.Likeotheropenchaoticsystems,thetransmis- ingandshapedistortion,despitestrongmodecouplingin
sion of a pulse through a MMF with strong mode cou- thefiber. Incontrast,thesamepulse,butwitharandom
pling involves spatial and temporal distortions. Strong input pattern, suffers from severe broadening as seen in
modemixingresultsinhoppingofthelightamongmodes Fig. 3(c). PMs can thus compensate for the temporal
with different propagation constants. Thus the light can distortion induced by modal dispersion in an MMF.
take many paths of varying lengths through the fiber. The unique spectral and temporal properties of PMs
Theoutputineachspatialchannel(e.g. specklegrain)is hold only within a finite frequency range. It is hence im-
asumofwaveswithdifferentpaths,eachassociatedwith portant to determine the bandwidth of PMs. Since the
a respective time delay, leading to temporal broadening spectral decorrelation of the output pattern for any in-
anddistortionoftheinputpulse. Typically,thetemporal put wavefront depends on the fiber properties, such as
trace varies from one channel to another, since the com- the fiber length and numerical aperture, we consider be-
binationofpathsdiffers. Thisisconfirmedbysimulating low the ratio of the bandwidth of the PM to the average
the propagation of a pulse, φ(t) = (cid:82) φ(ω)e−iωtdω, with bandwidth of random inputs. Figure 4(a) plots the ex-
a field spectrum φ(ω). The pulse is launched into a ran- perimentallymeasuredbandwidthofallPMsversustheir
domsuperpositionofmodesattheinputofthefiber,and delay time. The shorter the delay, the larger the band-
the output field patterns are recorded experimentally as width.
a function of frequency. We perform the Fourier trans- To obtain a physical understanding of PMs and their
formtoobtainthetemporalevolutionoftheoutputfield bandwidth, we resort to the intuitive picture of optical
in each spatial channel. Figure 3(a) shows the temporal paths in the fiber. The output field pattern is a re-
traces of field magnitude in three spatial channels. They sult of interference of waves following innumerable pos-
are very different from each other, due to strong mode sible trajectories in the MMF created by strong mode
scrambling in the fiber. coupling. As the input frequency changes, the relative
However, if the input light is coupled to a PM, the phase accumulated along trajectories of different length
output fields in different spatial channels are synchro- varies, modifying the output field pattern. More specif-
nized,asshowninFig.3(b). Thisisadirectconsequence ically, the output field in the m-th spatial channel can
4
2.5 (a) (b) paths, the destructive interference of different trajecto-
h
widt 0.12 rieswiththesamelengthmakesU(l)vanishforthelonger
nd 2 path-length. The opposite happens to the slow PM. The
a
malized b1.5 ()Ul00..0048 rdtieemdteiesrtmirniibnaueMtdioMbnyFotfwhieinttihnenpstsurittoynwgaamvmeoofnrdogendtc.ioffuTeprhleeinnrtegfpoisaretdh,e-ttlheerenmgdtienhlaeidys
Nor by the multi-path interference effect, which can be ef-
1 0
0 0.04 0.08 0.12 -0.024 0 0.024 0.048 fectively controlled by spatial degrees of freedom of the
Delay time (ns) Relative path-length (m)
input wavefront.
2.5 (c) (d) The final question we address here is why the fast PM
h 0.08
dt has a narrower path-length distribution. To answer this
dwi 2 question, we perform numerical simulations using the
n
ed ba ()Ul0.04 ccoonncsiadteernaatpeldanwaarvweagvueidgueidmeowdietlh[a213]0.0µFmorcsoirmepalnicditay,0.w22e
z1.5
ali Without Loss numerical aperture, supporting 86 guided modes. The
m With Loss
or one-meter-long waveguide is composed of 20 segments,
N
1 0
0 0.04 0.08 0.12 -0.048-0.024 0 0.0240.048 ineachofwhichlightpropagateswithoutmodecoupling.
Delay time (ns) Relative path-length (m) Between adjacent segments, the guided modes are ran-
domlycoupled,assimulatedbyaunitaryrandommatrix.
Figure 4. (color online) (a) Measured spectral correlation
To include mode-dependent loss, we introduce a uniform
width ∆ω of PMs with different delay times. ∆ω , given
c c absorption coefficient to each waveguide segment. The
by |C(∆ω )|=0.9|C(0)|, is normalized by the average band-
c
higher-order modes, with smaller propagation constants,
width of random inputs. The shortest delay time is set to
0. (b) Intensity distribution over the path-length U(l) for have longer transit time, thus experiencing more atten-
three measured PMs with the delay time = 0, 0.06, 0.12 uation. In terms of optical path, the longer paths have
ns. The relative path-length l is obtained by subtracting the more loss than the shorter ones.
average path-length of random inputs. U(l) is normalized:
Figure 4(c) plots the bandwidth of all PMs in the
(cid:82)
U(l)dl = 1. (c) Calculated spectral correlation width of
absence of mode-dependent loss (black crosses). The
thePMswith(redcircles)andwithout(blackcrosses)mode-
fast and slow PMs have almost identical bandwidth, as
dependent loss. (d) Intensity distribution over the path-
length for two PMs with the delay time = 0 ns (blue line), thecorrespondingintensitydistributionamongthepath-
0.12 ns (black line), in the presence (dashed line) or absence length exhibits similar spread at different mean values
(solidline)ofmode-dependentloss. Themode-dependentloss [Fig. 4(d), solid curves]. With the introduction of mode-
narrows (broadens) the path-length distribution for the fast dependent loss, the bandwidth of fast PMs increases,
(slow)principlemode,therebyincreasing(reducing)thespec-
while the bandwidth of slow ones decreases, leading to
tral correlation width.
agreementwiththeexperimentaldata[comparereddots
in Fig. 4(a) with those in Fig. 4(c)]. This rearrangement
can be explained by the change in intensity distribution
(cid:82)
be written as Ψm = um(l)dl, where um(l) is a sum of over the path-length, U(l), that is plotted in Fig. 4(d).
fields taking all possible paths with the same length l. The distribution for a fast PM, which concentrates on
With a small frequency detuning ∆ω, the output field shortpaths, becomesnarrower, becausethelongerpaths
becomes Ψm(∆ω)=(cid:82) um(l)ei∆ωl/cdl, in the weak guid- are further suppressed by the loss. For the slow PM, the
ing approximation. Thus um(l) can be obtained exper- stronger attenuation of longer paths not only shifts the
imentally from the Fourier transform of Ψm(∆ω). Ac- peak of U(l) to smaller l, but also broadens the distribu-
countingforallspatialchannels,U(l)=(cid:80)|um(l)|2 gives tion. Qualitatively,thechangeofPMbandwidthinduced
the intensity distribution over the path-length, which is bythemode-dependentlossisnotsensitivetothekindof
determined by the input wavefront. How fast the out- loss the fiber experiences, as long as higher-order modes
put field decorrelates is determined by how broad the have more loss, as expected for the MMF.
intensity is distributed over the path-length spectrum.
Insummary,weexperimentallyprobeindividualeigen-
The narrower the distribution, the weaker the dephasing
states of the Wigner-Smith time-delay matrix of a mul-
amongdifferentpath-lengthsbyfrequencydetuning,and
timode fiber with strong mode coupling. We find that
the smaller the change in the interference pattern at the
thewell-defineddelaytimesoftheeigenstatesareformed
output. Hence, the PMs with shorter delay times have
by multi-path interference, which can be manipulated
larger spectral correlation widths due to the narrower
by the spatial degrees of freedom of the input wave-
path-length distribution.
front. The multi-path interference also determines the
Figure4(b)comparesU(l)forthreePMswithdifferent frequencyrangeoverwhichtheuniquespectralandtem-
delay times. The fast PM has intensity concentrated on poral properties of the Wigner-Smith eigenstates pre-
shorterpaths. Althoughthewavescantakemanylonger serve. Within the bandwidth, the spatial and temporal
5
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