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Time reversal of optically carried radiofrequency signals
in the microsecond range
Heloïse Linget, Loïc Morvan, Jean-Louis Le Gouët, Anne Louchet-Chauvet
To cite this version:
Heloïse Linget, Loïc Morvan, Jean-Louis Le Gouët, Anne Louchet-Chauvet. Time reversal of optically
carried radiofrequency signals in the microsecond range. Optics Letters, Optical Society of America -
OSA Publishing, 2013. �hal-02106718�
Time-Reversal of Optically-carried
Radiofrequency Signals in the Microsecond Range
H. Linget,
1,2
L. Morvan,
2
J.-L. Le Gou¨et
1
and A. Louchet-Chauvet
1
1
Laboratoire Aim´e Cotton, CNRS-UPR 3321, Bˆatiment 505, 91405 Orsay, France
2
Thales Research and Technology, avenue Fresnel, 91767 Palaiseau, France
Compiled January 22, 2013
The time-reversal protocol we implement in an erbium doped YSO crystal is based on photon-echoes but avoids
the storage of the signal to be processed. Unlike other approaches implying digitizing or highly dispersive
optical fibers, the proposed scheme reaches the µs-range and potentially offers high bandwidth, both required
for RADAR applications. In this letter, we demonstrate faithful reversal of arbitrary pulse sequences with 6 µs
duration and 10 MHz bandwidth. To the best of our knowledge, this is the first demonstration of time reversal
via linear filtering in a programmable material.
c
2013 Optical Society of America
OCIS codes: 000.0000, 999.9999.
When a wave travels through an inhomogeneous
medium, its wavefront is distorted by many phenomena,
such as reflection, diffraction or anisotropy. Specifically
velocity variations inside the medium distort the incident
wavefront, and multireflection paths split it, resulting in
a spatially and temporally poorly focused beam. Time-
reversal (TR) invariance in the wave propagation equa-
tion can be used to counteract these effects. If a wave
with time-varying amplitude s(t) propagates through a
complex medium, the time-reversed waveform s(-t) is so-
lution of the propagation equation too, but converges
with accurate resolution back to the source responsi-
ble for the incident wave. Depending on the wavelength,
different applications arise, from medicine with acoustic
waves, to RAdio Detection And Ranging (RADAR) and
electronic warfare devices with microwaves. Although
digitizing the received signal s(t) is well-fitted to acoustic
waves with limited bandwidth [1], the time-consuming
analog-to-digital conversion excessively limits the band-
width in the microwave domain (e.g. 2 MHz-wide TR
in [2]). For broadband signals, pure analog approach is
possible using optically carried RF signals. In this way,
group delay dispersion (GDD) in optical fibers enables
TR with 18 GHz bandwidth, but is not sufficiently large
to process signals longer than few nanoseconds [3]. While
GDD in an optical fiber is settled by its length, rare earth
ion-doped crystals (REIC) at low temperature offer a
group delay only limited by the homogeneous dephasing
time T
2
of the doping ion, and almost 10
5
better than can
be reached with km-long fibers. With these materials,
we can thus extend the TR-processing to the μs-range
required for RADAR applications, and possibly access
1-100 GHz bandwidth, limited by the inhomogeneous
broadening of the transition. Three pulse photon-echo
(3PE) in REIC has already been considered for TR [4],
but the proposed procedure relied on the encoding of
the RF signal in the active medium. To preserve the en-
graving linearity, one operates with low intensity pulses,
which results in poor-contrast engraving and low pro-
cessing efficiency [5]. In the present work, instead of stor-
ing the RF signal, we program a TR-specific function in
the medium. This allows us to process differently shaped
signals consecutively, after programming the crystal once
and for all. Finally the non storing of the datas relaxes
the low field condition, opening new ways to improve the
programming step.
In a 3PE sequence at frequency ν
1
, if the first two
pulses are separated by t
(1)
12
, the third one is followed
by a t
(1)
12
-delayed atomic emission [Fig.1(a,b)]. Consider-
ing that spectral classes are independently addressable
in a REIC, we can reproduce this 3PE sequence at dif-
ferent frequencies {ν
i
} with different durations {t
(i)
12
}.As
shown in figure 1(b), with a specific choice of parameters
{ν
i
,t
(i)
12
}, the sequence of all the third pulses (time order
: abcd ) is time-reversed in the echo (time order : dcba).
1
2
3
4
5
time
ENGRAVING
a
a
c
c
d
d
b
b
pulse 3
echo pulse
Δν
optic
τ
ν
1
ν
2
ν
3
ν
4
OUTPUT
η * s(-t)
1
2
3
4
5
INPUT
s(t)
time
time
pulse 2pulse 1
(a)
(b)
(c)
LASER
FREQUENCY
LASER
INTENSITY
Laser ON
Laser OFF
Fig. 1. (Color online) 3PE-scheme : (a) laser pulse (solid
line) and echo (dashed line) chirped frequency, (b) 3PEs
at 4 different frequency addresses, (c) TR protocol : con-
tinuous engraving is achieved by pulses 1 and 2. The
waveform - carried by pulse 3 - is time reversed in the
echo.
1
This step-by-step description can be extended to a con-
tinuous one by linearly chirping the laser frequency
over a range Δν
optic
during a time τ. The monochro-
matic pulses of the 3PE are now replaced by three
chirped pulses with respective rates +r, -r and +r where
r=Δν
optic
/τ, resulting in an echo pulse with rate -r
[Fig.1(a)]. Since the photon echo signal intensity varies
linearly with the third pulse intensity, the echo pulse is a
time-reversed image of the input pulse with efficiency η.
Two monochromatic pulses separated by duration t
12
engrave a grating with spectral spacing 1/t
12
in the ab-
sorption profile. In our protocol, the spectral period of
the grating engraved by chirped pulses 1 and 2 [Fig.2(a)]
is frequency-dependent, varying as 1/t
12
(ν). It results in
the encoding of a non-periodic structure over a range
Δν
optic
as sketched in figure 2(b). It is important to no-
tice that the time-reversal function is encoded by this
non-periodic structure : the input pulse containing the
signal will be processed by this function without being
stored in the medium.
0
0.5
0.5 MHz
1
t
12
= 13.5 µs
0.5 MHz
t
12
= 8.9 µs
Δν
optic
time
τ
t
12
t
12
t
12
Pulse 2
Pulse 1
laser frequency
absorption
(i)
(a)
(b)
t
12
= 4.3 µs
0.5 MHz
(ii)
(iii)
0
0.5
1
0
0.5
1
Fig. 2. (Color online) (a) frequency chirps during engrav-
ing, (b) schematic absorption profile, dashed and solid
line: without and with engraving, (i)-(iii) normalized ex-
perimental transmission spectra, black line : sinusoidal
fit. The maximum contrast reaches 15%.
The laser frequency is controlled by an electro-optic
crystal inside the cavity [6]. A mode-hop-free tuning
range Δν
optic
of 1,09 GHz in 6 μs can be reached
(r =1.82 · 10
14
s
−2
). An acousto-optic modulator (AOM)
is used to transpose the RF-signal s(t) on the chirped
optical carrier during the input pulse. The laser beam
propagates parallel to the b axis of a 10mm-long 0.005 %
Er
3+
:YSO crystal cooled at 1.7 K in a liquid helium
cryostat, and is linearly polarized along the extinction
axis D
2
of the crystal (strongest absorption of Er
3+
sub-
stituted in site 1). To reduce spectral diffusion (SD),
a 2-tesla magnetic field B is applied in the plane de-
fined by the extinction axes (D
1
,D
2
) along direction
(B,D
1
) ≈ 135
◦
[7]. The beam waist at the crystal has
been adjusted to 65 μm, representing a trade-off between
high-constrast grating and moderate instantaneous spec-
tral diffusion [8]. To agree with RADAR specifications,
the input signal duration lasts 6 μs, duration for which
instantaneous spectral diffusion has a minor impact con-
sidering our experimental engraving power. The time-
reversed output signal is finally detected by an avalanche
photodiode (APD) placed after an AOM only opened
during the echo pulse.
We have been able to time reverse a 6 μs asymmet-
ric train of Gaussian pulses with a signal-to-noise ra-
tio of 50 in single shot capture, only limited by the
APD dynamic range. The 1.6
0
/
00
efficiency is consistent
with the measured absorption profile modulation con-
trast [Fig.2(i)-(iii)]. The time-reversed waveform over-
all decay [Fig.3(b)] partly reflects the active ion interac-
tion with a fluctuating environment [9]. This also results
from the non-uniformity of the optical depth αLoverthe
scanned range Δν
optic
.
TRANSMITTED INPUT
-8 -6 -4 -2 0 2 4 6 8
-8 -6 -4 -2 0 2 4 6 8
time (μs)
time (μs)
time (μs)
APD intensity
(ua)
(a)
(b)
(c)
τ = 6 μs
OUTPUT
-8 -6 -4 -2 0 2 4 6 8
Input and
perfect TR
resolvable data number : 25
Fig. 3. (Color online) Input and output pulses : (a) in-
put and perfectly time-reversed output, (b) single-shot
experimental transmission, red dashed line : effect of
interaction of Er
3+
with fluctuating environment, red
solid line : effect of optical depth variation over Δν
optic
,
(c) output corrected for the 2 previously cited effects.
At given frequency ν, input and output are separated
by the group delay τ
g
(ν)=τ
g
(ν
0
) − 2(ν − ν
0
)/r.Let
a Fourier transform limited, τ-long, temporal substruc-
ture be injected at frequency ν in the crystal. This pulse
spreads over a spectral interval of order 1/τ.Dueto
group delay dispersion ∂
ν
τ
g
(ν)=−2/r, the pulse under-
goes temporal stretching of order ∂
ν
τ
g
(ν)·1/τ = −2/(rτ)
while travelling through the crystal, and preserves its
initial duration provided τ>> |−2/(rτ)| . Hence,
2/r
represents the duration of the shortest temporal detail
that can propagate throughout the medium without dis-
tortion. In other words, the processor bandwidth is lim-
ited to
r/2. In our experimental conditions, this limits
RF bandwidth to approximately 10 MHz.
Knowing that the spectrum of a sequence of gaussian
pulses of duration t
gauss
is contained in a gaussian en-
velope of spectral width 1/(π · t
gauss
), we can test this
bandwidth by decreasing the parameter t
gauss
of the in-
put pulse, and thus broadening its spectral width Δν
RF
.
On the output pulse shown in figure 4(a), we notice the
progressive appearance in (ii) and (iii) of unwanted os-
cillations due to the distortion of short temporal sub-
2
(b)
1/(π t
gauss
) ∝ Δν
RF
(MHz)
10
0
10
1
-6
-4
-2
0
crosscorrelation (dB)
(ii)
(iii)
-6 dB bandwidth
~ 9.6 MHz
(i)
-8 -6 -4 -2 0 2 4 6 8
time (μs)
(a)
(i)
(ii)
(iii)
APD intensity (ua)
TRANSMITTED INPUT OUTPUT
Fig. 4. (Color online) (a) Normalized input and output pulses for several gaussian pulse durations t
gauss
. Output
corrected for interaction with environment and αL-variation, (i) t
gauss
=95ns, (ii) t
gauss
=46ns, (iii) t
gauss
=39ns,
(b) Bode diagram representing normalized crosscorrelation between input pulse and time-reversed output pulse vs.
input spectral bandwidth Δν
RF
. Input intensity envelope modulations are due to αL-variation over the scanned
range Δν
optic
.
structures, as mentioned above. The faithfulness of the
TR process has been quantified for several values of input
spectral width using crosscorrelation between transmit-
ted input and time-reversed output [Fig.4(b)]. It allows
us to define a -6dB bandwidth of 9.6 MHz for our proto-
col, in agreement with the previously mentioned band-
width. Our engraved TR-function is thus able to faith-
fully process frequencies lower than
r/2, but distortion
occurs for higher spectral components.
In summary we have demonstrated a new time-
reversal protocol dealing with signals up to the μs-range
required for RADAR applications. Since the efficiency
η of the process is proportional to the grating contrast
squared, we can potentially increase our experimental
value 1.6
0
/
00
(to be compared with 0.02
0
/
00
,observed
in [5]) by improving the absorption profile engraving. As
the low-field regime is no longer required , we can imag-
ine techniques other than photon-echo to achieve it, in a
similar approach to the one considered for efficiency opti-
mization of Atomic Frequency Comb engraving [10]. The
limited bandwidth issue can be addressed in the frame-
work of time-space duality. Indeed, true temporal imag-
ing combines a temporal lens with two dispersive lines,
respectively located upstream and downstream from the
lens. These elements are needed to conjugate the tem-
poral object and its time-reversed image [11, 12]. In our
setup, where the chirped carrier and the programmed
crystal respectively play the role of the lens and one
dispersive line, the upstream dispersive element is miss-
ing, resulting in an approximate but simplified temporal
imaging [13]. As a consequence, the s(t) time-reversed
image is blurred, which is reflected in the bandwidth
limitation. Double pass through the same programmed
crystal could provide the two dispersion steps. However
the processing efficiency is presently too weak for double
pass operation. With improved efficiency, true temporal
imaging could be implemented, taking full advantage of
the crystal bandwidth. Finally it can be noticed that our
system is the approximate temporal equivalent of an op-
tical device called camera obscura (literally dark cham-
ber ) which recently gave rise to a heated debate within
the Art history community about its assumed use by
several famous Renaissance painters like Caravaggio or
Vermeer [14].
This research received funding from the People Pro-
gramme (Marie Curie Actions) of the European Union’s
Seventh Framework Programme FP7/2007-2013/ (REA
grant agreement no. 287252).
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4
