Design And Analysis Of All-optical Temporal Integrator Based On Phase-shifted Fiber Bragg Grating

The applications of fiber gratings are more and more popular in optical communication and optical fiber sensors because of its advantages of small cubage, low cost, good compatibility to fiber systems, low insertion loss, simple fabrication and so on. They can be used to make fiber laser, semiconductor laser and ring laser etc with accurate wavelength and power. Fiber gratings can be used in fiber sensor systems as temperature sensors, strain sensors, pressure sensors, refractive index sensors, filtering, devices of dispersion compensation, gain flattening of erbium-doped fiber amplifiers and so on.As the rising volume of global communications and the limited computer data processing capacity, the requirement of AON (All Optical Network) and optical computer for people is very strong. And all-optical temporal integrators are the basic building blocks of optical computer. So the research for the all-optical temporal integrators has become very important.The main work of this thesis is to establish the all-optical temporal integrator model which is based on phase-shifted fiber Bragg grating from theory, using the coupled mode theory and the transfer matrix model was derived transfer function expression, and derived using the results of the matlab term numerical simulation, the final analysis of the results obtained and discussed, draw some important conclusions.We first introduced the design and analysis of all-optical temporal integrator based on the singleπphase-shifted fiber Bragg grating. The singleπphase-shifted fiber Bragg grating structure can be described by the transfer matrix method combined with the couple mode theory. The 2×2 transfer matrix of the singleπphase-shifted fiber Bragg grating is given by (?) (1)Where TF BGand T? are the 2×2 matrices of the fiber Bragg gratings and of the phase shift between the two gratings, respectively. The elements of TF BG are given by (?)Where * denotes the complex conjugation,κis the coupling coefficient,σ= 2πneff (1/λ? 1/λB) is the detuning from the Bragg wavelengthλB,λis the operating wavelength, ne ff is the effective fiber mode index,γ2 =κ2 ?σ2, andτ(τ= ne ffl /c where c is the speed of light in vacuum ) are, respectively, the length and delay of each fiber Bragg grating, andωτ= 2πne ffl/λ. The elements of T? are given by (?)The transfer function of the Tran missive singleπphase-shifted fiber Bragg grating is given by (?)in PSFBG HS(?)Where TP SFBG,22 is an element of TP SFBG, Ei n(ω)is the input electric-field amplitude, and S (ω)is the Tran missive electric-field amplitude. We are interested in the case of ? =πandλ=λBwhere the latter givesσ= 0andγ=κ. Equation (5) can be obtained from Equation (1) when Equations (2)-(4) are substituted into Equation (1). Putting? =π,λ=λB,σ= 0,γ=κand the z-transform parameter, z = exp( jωT) into Equation (5) results in (?)Where T = 2τis the reflection round-trip delay of each fiber Bragg gratings. Note that T = ne ffL /c where L = 2l is the total length of the singleπphase-shifted fiber Bragg grating. Figure (1) shows that transmission (a) and phase (b) responses of the integrator model with (?)Fig.1 Transmission (a) and phase (b) responses of the first-order integrator model with r = 0.999,0.9999,1.0In figure (1), the red curve represents r = 0.999, the blue curve represents r = 0.9999, and the green curve represents r = 1.0. We could find that the spectral peak of the curves becomes sharper with an increase in the r value. And the curves of the phase responses of the integrator almost overlap. Figure (2) shows that the input pulse with a full width at a half maximum (FWHM)=100T into the proposed integrator. The integrated output pulse by the integrator with r = 1.0(the dotted curve) is almost exactly the same as that of the true integral (the solid curve), and in fact these two curves overlap with each other. The integrated pulse with r = 0.9999 still resembles that of the true integral. However, the integrated pulse with r = 0.999 deviates from that of the ideal integrator. (?)Figure.2 Integrated pulses by the integrator with various r values when processing an input pulse with FWHM=100T.For the integrator model of energy efficiency and the integral error rate analysis, we conclude that as the reflectivity increases, the integrator of energy efficiency and the integral error rate also increases, so there is a trade-off between energy efficiency and the integral error rate.We design an N th-order optical temporal integrator model which is based on the first-order optical temporal integrator. An N th-order integrator could be realized by concatenating in series N single first-order integration devices. We can anticipate that an N th-order optical temporal integrator can be implemented using a high-reflectivity uniform fiber Bragg grating incorporating Nπ-phase shifts along the grating profile. We still use the transfer matrix method combined with the couple mode theory to analysis the grating structure. In this paper, we analysis the grating structure when N=2. The multiple-phase-shifted Bragg grating structure can be described by the following matrix product: TΣ= T ( L1 + L2 , L3 )ΦT ( L1 , L2 )Φ(0, L1)··································(7)The transfer function of this device can be easily calculated by the transfer matrix method combined with the couple mode theory just as same as the first-order optical temporal integrator: (?) Figure (3) shows the amplitude of response and the phase of response of the simulated integrator and ideal integrator. (?)Figure.3 The amplitude of response and the phase of response of the simulated integrator and ideal integratorWe can find that there is a very good agreement between the curves which are the amplitude of response and the phase of response of the simulated integrator and ideal integrator. To examine the behavior of this device as a second-order temporal integrator, we have simulated the device's temporal response to an input ideal Gaussian pulse with a full-at-half-maximum (FWHM) duration of 40 ps (FWHM bandwidth≈22-GHz). (?)Figure.3 Time-domain response of the MPS-BG second-order integrator compared with the ideal second time cumulative integrals of the considered input pulse waveforms.The simulated output pulse shape is shown in the same plot with a solid, red curve. As predicted, there is an excellent agreement between the obtained temporal profile at the fiber Bragg grating output and the ideal second-order time cumulative integral of the considered input waveform (Gaussian pulse), which is also shown in the same plot using a dashed, blue curve.From the energy efficiency of the model and the integral error rate on the numerical simulation we have come to raise the integrator model of the maximum reflectivity of each grating will increase its energy efficiency, but also increases the integral error rate. Therefore, there is an important trade-off between energy efficiency and the integral error rate. The two integrator models can perform time integration of arbitrary waveform which time resolution is picoseconds, equivalent to hundreds of GHz processing speed, significantly more than the traditional computer's processing speed.