# Ultrafast Optical Signal Processing with Bragg Structures

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

_{g}, the use of slow light with a small v

_{g}implies longer interaction times, and consequently, a more efficient energy conversion [6]. Slow light also offers the possibility to compress optical signals in space, thus reducing the device size [7,8].

## 2. The Bragg-Grating (BG) Structure

^{2}) and long propagation lengths are required to achieve strong nonlinear effects, which may pose serious problems in experiments and applications.

## 3. Optical Signal Processing in Resonantly Absorbing Bragg Reflector (RABR)

#### 3.1. Theoretical Considerations

_{0}is the average refraction index; ω

_{c}is the resonant frequency of the two-level atom; μ is the magnitude of the dipole matrix element ρ is the density of the dopant atoms; $\eta \equiv ({n}_{1}\omega {\tau}_{c})/4$ is the dimensionless coupling constant; P and W are the material polarization and population inversion density, respectively; $\delta \equiv (\omega -{\omega}_{\mathrm{c}})/{\tau}_{\mathrm{c}}$ is the dimensionless detuning; $\tau \equiv t/{\tau}_{c}$ and $\zeta \equiv ({n}_{0}/c{\tau}_{c})z$ are the dimensionless time and spatial coordinates, respectively; and ω is the frequency of the incident light. Here, τ

_{c}is equal to 300 fs [13].

_{R}is the fraction of the nonlinearity arising from molecular vibrations with a typical value of 0.18 [45]; and ${G}_{p,s}\equiv \left({\epsilon}_{0}{c}^{2}{\hslash}^{2}/8{\mu}^{2}{\tau}_{c}\right){g}_{p,s}$ and ${\mathrm{\Gamma}}_{p,s}\equiv {\omega}_{p,s}{n}_{2}{\epsilon}_{0}c{\hslash}^{2}/8{\mu}^{2}{\tau}_{c}$ are the dimensionless nonlinearity and gain coefficients for the pump and Stokes waves, respectively.

_{c}to 5τ

_{c}(τ

_{c}is the cooperative time), into a 2π SIT soliton [49]. Furthermore, input pulses with a multiple-peak shape can be re-shaped to produce a single-peak output. Figure 5 shows the transformation of the three-peak pulse into a single-peak pulse in the RABR structure. In fact, SIT in bulk media can also be applied to compressed optical pulses. However, in that case, the input pulse with single peak is split into multiple ones, rather than morphed into a single-peak SIT soliton, as in RABR. The difference is explained by the fact that reshaping in the RABR originates from multiple reflections in the Bragg structure, while no reflections occur in the the bulk SIT medium. Other interesting phenomena, such as the negatively refracted light in the RABR, have also been discovered [50].

#### 3.2. Experimental Work on RABR

## 4. Optical Signal Processing in Chirped Bragg Structures

_{f}and E

_{b}, the coupled-mode equations are written as [14,56]:

_{0}is the average refractive index with modulation depth Δn, Λ

_{0}is the BG period at the input edge, and C is the chirp. The wavenumber-detuning parameter δ (Figure 6), and the coupling between the forward and backward field, $\kappa =\pi \Delta n/{\Lambda}_{0}$, are functions of the propagation distance.

- (i)
- At low intensities, e.g., 0.65 GW/cm
^{2}, the pulse is almost totally reflected by the Bragg structure due to the presence of the photonic bandgap (Figure 6bd_{1}). Since the pulse’s dispersion is not compensated by the weak nonlinearity, pulse stretching is observed. - (ii)
- At higher intensities, an unstable standing light pulse trapped at the interface is generated. In particular, at ${I}_{P}$ = 2.26 GW/cm
^{2}, the pulse will be reflected after stopping for a short time, as seen in panel (d_{2}) of Figure 6b. Slightly increasing ${I}_{P}$ to 2.29 GW/cm^{2}, in the range of 2.26 to 2.29 GW/cm^{2}, gives rise to a stopping time of ~1.3 ns for the pulse that is eventually reflected, as shown in panel (d_{3}). - (iii)
- A slow moving Bragg soliton can be observed, as shown in panel (d
_{4}), for I_{P}= 2.30 GW/cm^{2}. In this case, the velocity of the pulse is equal to 0.005 c. Further simulations demonstrate that the velocity of the moving soliton increases with a further increase of I_{P}.

_{w}is its width [15].

## 5. Ultrafast Pulse Modulations Based on Cholesteric Liquid Crystal (CLCs) Bragg Gratings

^{−16}cm

^{2}/W; in conjunction with a peak power of kilowatts [26], the resulting nonlinear effective length is on the millimeter scale [60]. In this section, we show that naturally occurring Bragg grating in a highly nonlinear CLCS can enable the same ultrafast (femtoseconds-picoseconds) pulse modulation operations in sub-mm interaction lengths.

_{2}is in the order of 10

^{−14}–10

^{−13}cm

^{2}/W, but owing to the photonic crystal band-edge enhancement, the magnitude of the effective n

_{2}can be as large as 10

^{−12}–10

^{−11}cm

^{2}/W. In comparison to other materials [70] used for ultrafast pulsed laser modulation applications, these n

_{2}values are orders of magnitude larger, and thus, one can envision a tremendous miniaturization possibility.

_{2}can be positive [18] or negative [17]—a common feature found in the electronic nonlinearities of most materials [70]. As shown in Figure 12b for the nematic compound used in [18], the refractive index change induced by the laser causes the bandgap to shift towards the shorter wavelength region (i.e., blue-shit), and consequently gives rise to an increasing intensity dependence since the laser wavelength is located at the long wavelength edge of the CLC (Figure 12b). In Reference [17], the CLC sample with self- defocusing nonlinear properties is used to compress the femtosecond pulse (Figure 13). The nonlinear coefficient of the CLC used in that study was measured to be about −10

^{−11}cm

^{2}/W using a similar intensity dependent transmission measurement. This is four orders of magnitude higher than silica [12]; as a result, the required the thickness of the CLC sample is merely 6 microns in order to compress a 100 fs laser pulse to about 50 fs [17]. If the constituent nematic molecules possess a lower optical nonlinearity, thicker CLC cells of several 100’s microns are required [18], which are nevertheless thin/short compared to other materials used for ultrafast laser pulse modulations.

## 6. Conclusions

**,**including resonantly absorbing Bragg reflectors (RABR), chirped BGs (Bragg gratings), and transparent but highly nonlinear CLCs (cholesteric liquid crystals) that exhibit properties of 1D photonic crystals. The result shows that an ultrafast pulse can be buffered and subsequently released in the RABR Additionally, based on the reduction of the light speed, a nonlinear frequency conversion can be achieved over a short propagation distance. Using a linearly chirped BG concatenated with a uniform one, we have demonstrated the possibility of achieving the efficient slow-down of light with the right mix of forward and backward propagating fields. Such chirped gratings can also function as all-optical photodiodes. Finally, we have discussed a unique class of self-assembled BGs based on CLCs. They feature extraordinarily large and ultrafast-responding optical nonlinearities, which enable direct compressions, as well as the stretching and recompressions of femtoseconds laser pulses over very short (sub-mm) propagation lengths.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**A scheme of RABR with black stripes representing thin layers of two-level atoms. White and gray bands represent a periodically structured nonabsorbing medium (After Reference [11]).

**Figure 2.**(

**a**) The evolution of the population inversion, W, illustrating the creation of an oscillatory standing soliton in the RABR from a sech-shaped input. The pulse width is τ

_{0}= 0.5τ

_{c}, and the injected amplitude is ${\mathrm{\Sigma}}_{0}^{+}=4.3$; (

**b**) The generation of two standing oscillatory solitons, with the pulse widths τ

_{0}= 0.5τ

_{c}and injected amplitude ${\mathrm{\Sigma}}_{0}^{+}=3.6$ [12].

**Figure 3.**A contour plot illustrating the release of a laser pulse stored in the RABR by dint of its collision with an additional pulse injected into the structure [13]. The pulse width is τ

_{0}= 0.5τ

_{c}, and the injected amplitude is ${\mathrm{\Sigma}}_{0}^{+}=3.5$, for the stopped pulse, and ${\mathrm{\Sigma}}_{0}^{+}=3.84$ for the incident one. Both input pulses have a standard sech profile.

**Figure 4.**The evolution of the energy density of (

**a**) pump and (

**b**) stokes pulses in the DRBR under the action of the SRS [7].

**Figure 5.**The input (solid line) and output (dash line) from the RABR, when a three-peak pulse is used as the input [40].

**Figure 6.**(

**a**) The relation between wavenumber detuning and propagation distance in the concatenated BG, which does not include the local defect; (

**b**) The evolution of the pulse with peak intensity equal to (d

_{1}) 0.65 GW/cm

^{2}, (d

_{2}) 2.26 GW/cm

^{2}, (d

_{3}) 2.29 GW/cm

^{2}, and (d

_{4}) 2.30 GW/cm

^{2}[14].

**Figure 7.**Simulations of the pulse propagation in the concatenated BG, with a defect located at the conjunction of the chirped and uniform segments, for different injected peak intensities: (

**a**) 1.94 GW/cm

^{2}; (

**b**) 2.14 GW/cm

^{2}; (

**c**) 2.33 GW/cm

^{2}; and (

**d**) 2.59 GW/cm

^{2}[14].

**Figure 8.**(

**a**) A schematic of the system, built of the linearly chirped BG segment (on the left-hand side) followed by the uniform grating with an inserted periodic array of local defects. The system can be described by parameters (S; d

_{w}; ε), and the definition of S, d

_{w}, and ε are shown in (

**a**); (

**b**) Relations between the trapping position and the initial pulse’s intensity I

_{P}for (S; d

_{w}; ε) = (0.132 cm; 50 μm; 0.04) and (0.132 cm; 50 μm; 0.06) (blue and red curves, respectively) [15].

**Figure 9.**The simulation result of the trapping of three pulses in the BG structure. Here (S; d

_{w}; ε) = (0.132 cm; 50 μm; 0.04). The peak intensities of the three pulses are 3.04, 2.78, and 2.07 GW/cm

^{2}. Such pulses are launched into the gratings at t = 0, t = 3, and t = 6 ns, respectively [15].

**Figure 10.**A typical example of the predicted femtosecond diode effect. The pulses with a width of 100 fs are injected from the left (

**a**) and right (

**b**) hand sides, respectively. The incident peak intensity of the pulse is I = 0.35 GW/cm

^{2}. (

**a**) The femtosecond soliton can propagate from A to B; (

**b**) the soliton bounces back when it is injected from B [16].

**Figure 11.**Observed optical nonlinearity in terms of the nonlinear index coefficients of nematic liquid crystals (including chiral nematics) for several mechanisms and the characteristic relaxation time constants.

**Figure 12.**(

**a**) Observed transmission spectrum of linearly polarized light through the CLC cell used in the experiment; (

**b**) Observed dependence of the transmission of a left-handed circularly polarized laser pulse (λ = 815 nm) on the peak intensity [17].

**Figure 13.**Observed direct pulse compression effect on an initial transform-limited 100 fs pulse (black line) to a 48 fs output pulse (red line); the inset figure corresponds to the simulation results using the measured experimental parameters [17].

**Figure 14.**Observed spectral broadening effect due to the pulse compression for an input pulse peak intensity of 1.04 GW∕cm

^{2}; the inset depict simulation results for the spectrum (solid lines) and spectral phase (dashed lines) [17].

**Figure 15.**(

**a**) Observed direct pulse compression effect on an initial transform-limited 847 fs pulse (open circles) to a 286 fs output pulse (open squares); the inset figure corresponds to the simulation results using the measured experimental parameters; (

**b**) Observed and simulated spectra for the input and output (compressed) pulses [18].

**Figure 16.**Results of pulse modulations by two tandem 550 μm thick CLC cells. Observed pulse shape for the input (upper trace), output after traversing the first cell (middle), and output after traversing the second cell (bottom trace). Initial pulse width: 100 fs laser pulse; wavelength: 780 nm [18].

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**MDPI and ACS Style**

Liu, Y.; Fu, S.; Malomed, B.A.; Khoo, I.C.; Zhou, J. Ultrafast Optical Signal Processing with Bragg Structures. *Appl. Sci.* **2017**, *7*, 556.
https://doi.org/10.3390/app7060556

**AMA Style**

Liu Y, Fu S, Malomed BA, Khoo IC, Zhou J. Ultrafast Optical Signal Processing with Bragg Structures. *Applied Sciences*. 2017; 7(6):556.
https://doi.org/10.3390/app7060556

**Chicago/Turabian Style**

Liu, Yikun, Shenhe Fu, Boris A. Malomed, Iam Choon Khoo, and Jianying Zhou. 2017. "Ultrafast Optical Signal Processing with Bragg Structures" *Applied Sciences* 7, no. 6: 556.
https://doi.org/10.3390/app7060556