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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-

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