Multi-TBaud Optical Coding Based on Superluminal Space-to-Time Mapping in Long Period Gratings

doi:10.4236/opj.2013.32B031

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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: [email protected]

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

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).

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

Cross‐couplingoperat ionmode

(Apodizationprofi le)

Long‐periodgrating

k(z)z

(Cladding‐mode)

Output Output

‐Coremodeblocker

‐ShortuniformLP G

(b)

(a)

(Core‐mode)

Ultra‐sho r t inputpulse

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

Space‐to‐time

LPG

time

input

t

eff

v



Li

htS

eed



 Liv

ht Speed

out

mappi ng

time

Counter‐directionalCoupling

Co‐directionalCoupling

(Apodizationprofile)

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,





()

()( )/

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



PhaseShift =2



PhaseShift =



Co re-mo de effecti ve refract ive index

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.

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

()ht

100 011

LPG

(ps)t

0.20.4 0.6

Sp eed=5Tbit/s

()ht

100 011

(ps)t

163326 489 652 815 978

Speed=6.1Gbi 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-

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

Phase (rad)

-1200 -800-400 0400 800 12000

0.25

0.5

0.75

Relative Time

(

)

litude

(

n.u.

)

12 1310056 15



-8 -6 -4 -202468

Freq. Deviation (THz)

C ross- Cou pling

Transmission Power (% )

0 3 6 912 15 18

LPG Length (cm)

k (1/m )

Phase shifts

2.07cm

-8

-6

-4

-2

Phase (rad)

-1200 -800-400 0400800 12000

0.5

Relative Time (fs)

mplitude ()An.u.



-8 -6 -4 -20 2 46 8

Freq. Devi ati on (TH z)

Cross - Cou plin g

Transmission Power (% )

0369 12 15 18

LPG Length (cm)

k (1/m)

Phase shifts

2.07cm

3/2



QPSK



3/4



Circular

16‐QAM



3/4



3/2



14 15

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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