Optics and Photonics Journal, 2013, 3, 126-130
doi:10.4236/opj.2013.32B031 Published Online June 2013 (http://www.scirp.org/journal/opj)
Multi-TBaud Optical Coding Based on Superluminal
Space-to-Time Mapping in Long Period Gratings
Reza Ashrafi, Ming Li, José Azaña
Institut National de la Recherche Scientifique–Énergie, Matériaux et Télécommunications, Montréal, Québec, Canada
Email: ashrafi@emt.inrs.ca
Received 2013
ABSTRACT
A novel time-domain ultra-fast pulse shaping approach for multi-TBaud serial optical communication signal (e.g.
QPSK and 16-QAM) generation based on the first-order Born approximation in feasible all-fiber long-period gratings is
proposed and numerically demonstrated.
Keywords: Pulse Shaping; Fiber Optics Components; All-optical Devices; Ultra-fast Processing
1. Introduction
Fiber and integrated-waveguide grating structures have
been widely investigated for ultrafast optical pulse shap-
ing and processing applications, including generation and
detection of high-speed complex data streams in tele-
communication systems [1,2]. The advantages of these
solutions are associated with their intrinsic compact, low-
loss all-fiber/waveguide implementations, e.g. in contrast
to widely used programmable linear waveshapers based
on bulk-optics configurations (involving diffraction
gratings and spatial modulators) [3]. In particular, there
has been an important body of work on the use of short-
period (Bragg) fiber/waveguide gratings (BGs) for ultra-
fast optical coding, namely generation of customized
temporal optical data streams under different amplitude
and/or phase coding schemes [1,2]. These solutions are
particularly interesting for applications requiring the
generation of time-limited data streams (composed of a
few consecutive symbols), such as for optical code-divi-
sion multiple access (OCDMA) and optical label-switching
communications [1,2]. Long-period fiber gratings (LPGs)
have recently attracted a great deal of interest for linear
optical pulse shaping and processing applications [4].
However, to date, there are very few published works on
their potential for general optical coding operations;
some interesting LPG designs have been recently re-
ported [5] but they are limited to the synthesis of tempo-
rally symmetric, binary intensity-only (on-off-keying,
OOK) optical codes.
As a general rule, in optical grating-based linear filters,
to achieve a faster temporal signal, a smaller spatial fea-
ture is required in the coupling-coefficient (grating apo-
dization) profile. Previous studies in counter-directional
coupling structures [6,7], e.g. fiber/waveguide BGs, have
revealed that under the first-order Born approximation
(i.e. weak-coupling conditions), the output time-domain
optical field complex envelope variation follows the spa-
tial variation of the complex coupling coefficient. This
phenomenon, referred to as space-to-time mapping, pro-
vides a very straightforward mechanism to synthesize
optical waveforms (e.g. coded communication data streams)
with prescribed complex temporal shapes. However, in
BGs, the ratio (v) between the resolution of the men-
tioned variations in space (Δz) and time (Δt) is necessar-
ily lower than the propagation speed of light in vacuum
(c) [8], i.e. v = Δz/Δt < c, (see the case of BG in Figure 2
and the given numerical example in Table 1). Considering
a typical achievable sub-mm resolution for fiber grating
apodization profiles, fiber BG pulse shapers/ coders are
thus limited to resolutions of at least several picoseconds
[1,2,7].
This work focuses on the use of the first-order Born
approximation in co-directional coupling filters, particu-
larly LPGs. As illustrated in Figure 1(a), similarly to the
case of BGs [6,7], under weak-coupling conditions, the
grating complex (amplitude and phase) apodization pro-
file can be directly mapped into the LPG filter’s temporal
impulse response [8,9]. In contrast to the BG case, the
space-to-time mapping speed (v = Δz/Δt) in LPG filters
can be made much higher than the propagation speed of
light in vacuum. As illustrated in Figure 2, this superlu-
minal space-to-time mapping speed in LPGs enables the
synthesis of waveforms with temporal features several
orders of magnitude faster than those achievable by BGs
*This research was supported in part by the Natural Sciences and En-
gineering Research Council of Canada (NSERC), and le Fonds Qué-
bécois de la Recherche sur la Nature et les Technologies (FQRNT).
Copyright © 2013 SciRes. OPJ
R. ASHRAFI ET AL. 127
(assuming the same spatial resolution in the grating apo-
dization profile). In this work, we numerically demon-
strate the straightforward use of this phenomenon for ultra-
fast optical coding applications, particularly for genera-
tion of customized serial optical communication streams
under any desired complex coding format (e.g. QPSK
and 16-QAM modulation formats in the examples reported
here), well in the TBaud range (femtosecond resolutions)
using readily feasible LPG designs, e.g. with grating
apodization resolutions above the millimeter range.
2. Theory of Superluminal Space-to-Time
Mapping in LPGs
Our theoretical derivations on the superluminal space-to-
time mapping phenomenon in LPGs rely on the standard
coupled-mode equations for the case of co-directional
coupling. The mathematical details of these derivations
will be reported elsewhere [9].
Output
Crosscouplingoperat ionmode
(Apodizationprofi le)
Longperiodgrating
k(z)z
t
(Claddingmode)
Output Output
12
1
Coremodeblocker
2
ShortuniformLP G
(b)
(a)
(Coremode)
Ultrasho r t inputpulse
t
Figure 1. (a) Schematic of the proposed ultra-fast pulse
shaping/coding approach based on superluminal space-to-time
mapping in LPGs; (b) Illustration of a previously demon-
strated fiber-optic approach [4] to transfer the cross-cou-
pling signal from the fiber cladding-mode into the fiber
core-mode by concatenating (1) a core -mode blocker and (2)
a short, strong uniform LPG.
z
Spacetotime
LPG
z
time
input
BG
t
t
eff
v
2
zc
tn
Li
g
htS
p
eed
zc
tN
 Liv
g
ht Speed
out
out
mappi ng
time
CounterdirectionalCoupling
CodirectionalCoupling
(Apodizationprofile)
input
Figure 2. Comparison of the two pulse shaping approaches
based on space-to-time mapping in fiber BGs and LPGs.
The LPG coupling coefficient (apodization) profile, i.e.
k(z) in Figure 1, is a complex function defined as k(z) =
|k(z)|exp[jφ(z)]. The magnitude |k(z)| depends on the am-
plitude of the refractive index modulation along the LPG
length, as illustrated in Figure 3. The grating discrete
phase-shifts and grating period changes along the LPG
length are accounted for in the phase term of the cou-
pling coefficient, i.e. φ(z). Some single phase-shifted
gratings are also illustrated in Figure 3 aimed to induce
the corresponding discrete jumps in the phase of the cou-
pling coefficient profile, i.e. φ(z). Our theoretical studies
[8,9] have shown that under weak-coupling strength con-
ditions (i.e. strictly, cross-coupling power spectral re-
sponse peak < 10%), the complex envelope of the tem-
poral impulse response (let us call it h(t)) of the cross-
coupling transfer function, i.e. core-to-cladding transfer
function in fiber LPGs, is approximately proportional to
the variation of the complex coupling coefficient k(z), as
a function of the grating length z after a suitable space-
to-time scaling [8,9]. In particular, the space-to-time
mapping speed (v), is obtained as v = c/N, where N =
(neff1 - neff2), and neff1 and neff2 are the effective refractive
indices of the two coupled-modes around the wavelength
of interest. Mathematically,
()
()( )/
jz
htk zeztc N
  (1)
Clearly, N can be designed to be much smaller than 1,
and consequently the resulting speed (v) can be made
significantly higher than the speed of light in vacuum.
This superluminal space-to-time mapping speed is also
significantly higher than the corresponding (subluminal)
speed in the case of BG devices, i.e. v = c/(2neff), where
neff is the average effective refractive index of the propa-
gating mode in the grating, see the comparison in Figure
2. This is the key to design optical pulse shapers (e.g.
coders) based on LPGs with impulse responses having
several orders of magnitude faster temporal features than
their counter-directional filter counterparts (BGs).
PhaseShift =2
3
PhaseShift =2
PhaseShift =
Co re-mo de effecti ve refract ive index
z
Uniform
Changing
()kz
Changing
()z
Figure 3. Illustration of variations on the amplitude and
phase of the coupling coefficient profile, i.e. |k(z)| and φ(z)
respectively, along the LPG length. For the phase change
examples, some single phase-shifted gratings to generate the
corresponding discrete jumps in φ(z) are illustrated.
Copyright © 2013 SciRes. OPJ
R. ASHRAFI ET AL.
128
Notice that the LPG’s cross-coupling operation mode
can be practically implemented based on either inte-
grated- waveguide technology (by simply inducing the
coupling between two physically separated waveguides
[10]) or a fiber-optic approach [4]. Figure 1(b) shows a
schematic of a previously demonstrated all-fiber ap-
proach for implementation of the cross-coupling opera-
tion mode in LPGs [4], i.e. to ensure that both the input
and output signals are carried by the fiber core mode. A
core-mode blocker and a short broadband uniform LPG
can be used for undistorted transference of the desired
output signal from the cladding mode into the core mode.
designations.
3. Numerical Comparison between
BG-Based and LPG-Based Pulse Coders
Let us assume a fiber BG working in reflection and a
fiber LPG working in the cross-coupling operation mode,
both made in standard single-mode fiber (Corning SMF28),
see Figure 4. The grating period for the LPG is assumed
to be Λ = 430 μm, which corresponds to coupling of the
fundamental core mode into the LP06 cladding mode at a
central wavelength of 1550 nm. The BG has a period of
528 nm, corresponding to a Bragg wavelength of
1550nm. The average effective refractive index of the
propagating mode in the BG is neff = 1.4684 and for the
LPG: neff1 = 1.4684 and neff2 = 1.4648 [11-13]. Table 1
shows the estimated space-to-time mapping speeds for
these two examples. Let us further assume that the two
considered BG and LPG devices have the same length of
10cm and they are both identically spatially-apodized for
a target optical OOK bit stream pattern generation, as
shown in Figure 4.
()kz
10zcm
1000
11
()ht
100 011
BG
LPG
(ps)t
0.20.4 0.6
Sp eed=5Tbit/s
()ht
100 011
(ps)t
163326 489 652 815 978
Speed=6.1Gbi t / s
0.81.0 1.2
Figure 4. Comparison of the two OOK pulse-coding ap-
proaches based on space-to-time mapping in BGs and LPGs.
Table 1. The estimated space-to-time mapping speed for the
considered BG and LPG made in S MF28 fib er.
Space-to-time mapping speed
BG V
= c / (2 neff) = 1.022 × 108 (m/s)
LPG V
= c / (neff1- neff2) = 833.3 × 108 (m/s)
In both cases, the amount of peak coupling coefficient
is assumed to be low enough to satisfy weak-coupling
conditions. Based on the space-to-time mapping theory,
by launching an ultra-short optical pulse into the consid-
ered optical filters, the target bit stream patterns (i.e. h(t)
in Figure 4) will be generated at the filters’ output port.
As expected from the different space-to-time mapping
speeds, the bit rate of the generated bit stream pattern by
the LPG device should be nearly 1,000 faster than that
generated by the BG filter.
4. Numerical Simulations
Using coupled-mode theory combined with a transfer-
matrix method [13], we have numerically simulated two
different LPG designs for generation of two 8-symbol
optical QPSK and 16-QAM signals, each with a speed of
4TBaud (4TBaud), from an input ultra-short optical
Gaussian pulse with a (full width at 10% of the peak am-
plitude) duration of 100 fs. Figure 5 shows the results of
these numerical simulations. The LPG design parameters
are those defined above and the input optical pulse is
assumed to be centered at the LPG resonance wavelength
of 1550 nm. In the numerical simulations, the following
wavelength dependence has been assumed for the effec-
tive refractive indices of the two interacting (coupled)
modes [12]: neff1(λ) = 1.4884 - 0.031547λ + 0.012023λ2
for the core-mode and neff2(λ) = 1.4806 - 0.025396λ +
0.009802λ2 for the LP06 cladding-mode, where 1.2 < λ <
1.7 is the wavelength variable in μm.
Figsures 5(a) and (b) show the designed amplitude
and phase grating-apodization profiles for the target
QPSK and QAM coding operations, respectively. The
grating designs are relatively straightforward and simple,
just being spatial-domain mapped versions of the respec-
tive targeted complex time-domain optical data streams.
In particular, Figrues 5(g) and (h) show the amplitude
and phase profiles of the time-domain waveforms at the
outputs of the simulated LPG designs, demonstrating
accurate generation of the targeted 4TBaud data streams,
as per the coding formats defined in Figures 5(c) and (d),
respectively, in excellent agreement with the inscribed
grating-apodization profiles.
Notice that considering the superluminal space-to-time
mapping scaling value in the designed LPG (~833.3
108 m/s), each symbol time period of 250 fs corresponds
to a fairly large spatial period of ~2.07 cm. As antici-
pated, time resolutions in the femtosecond regime (e.g.
for the inter-symbol amplitude transitions and discrete
phase jumps) can be achieved based on readily feasible
millimeter grating spatial resolutions. The spectral re-
sponses of the two designed LPG filters are shown in
Figures 5(e) and (f), respectively. It is worth noting the
intrinsic complexity of these responses (also for the
phase, not shown here), which would make it very chal-
Copyright © 2013 SciRes. OPJ
R. ASHRAFI ET AL. 129
lenging for implementation using a frequency-based op-
tical filter design approach, e.g. such as using conven-
tional programmable linear wave-shapers.
-12
-9
-6
-3
0
3
Phase (rad)
-1200 -800-400 0400 800 12000
0.25
0.5
0.75
1
Relative Time
(
fs
)
Am
litude
n.u.
12 1310056 15
/4
-8 -6 -4 -202468
0
6
12
18
24
30
36
Freq. Deviation (THz)
C ross- Cou pling
Transmission Power (% )
0 3 6 912 15 18
0
3
6
9
12
LPG Length (cm)
k (1/m )
Phase shifts
2.07cm
-8
-6
-4
-2
0
2
4
6
8
Phase (rad)
-1200 -800-400 0400800 12000
0.5
1
Relative Time (fs)
mplitude ()An.u.
/2
33
00
22
1
1
-8 -6 -4 -20 2 46 8
0
10
20
30
40
50
60
70
Freq. Devi ati on (TH z)
Cross - Cou plin g
Transmission Power (% )
0369 12 15 18
0
3
6
9
12
LPG Length (cm)
k (1/m)
Phase shifts
2.07cm
3/2
QPSK
/4
3/4
/2
/2
Circular
16QAM
/4
3/4
/2
/2
3/2
Im
Re
2
08
1
3
4
5
67
9
11
12
13
14 15
0
1
Im
Re
01
23
Figure 5. Simulation results of the two designed LPGs (a,b)
to generate 8-symbol optical QPSK (c) and 16-QAM (d)
data stream patterns, i.e. “0”1”3”2”3”0”2”1” and “12”1”3”
10”0”5”6”15” respectively, with a speed of 4TBaud from an
input (full width at 10% of the peak amplitude) 100fs opti-
cal Gaussian pulse. (e,f) The corresponding spectral power
responses of the designed LPGs. (g,h) The corresponding
output temporal amplitude and phase responses.
5. Conclusions
We have proposed and numerically demonstrated a novel
time-domain pulse shaping approach for synthesizing
THz-bandwidth linear optical filters with arbitrary ul-
trafast temporal impulse responses based on the first-
order Born approximation in LPGs. The proposed tech-
nique is particularly useful for generation of multi-TBaud
serial optical communication data streams under complex
(PSK, QAM etc.) coding formats using readily feasible
and simple LPG designs, e.g. with spatial resolutions
above the millimeter range. The corresponding matched-
filtering devices for efficient decoding and detection of
the generated data streams could be also designed and
implemented using this same LPG approach.
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