CHAPTER 2 BACKGROUND ON DISTRIBUTED AND
3.6 Fibre Bragg Grating Simulation Techniques
3.6.3 FFT Grating Design Algorithm Operation
The FFT model was implemented initially in LabView by Dr Daniel Kitcher, a Research fellow with OTRL at VU. Figure 3.10 shows a simplified version of the Virtual Instrument (vi) block diagram for the FFT grating design model implemented for this project. Shown on the figure are the sub⋅vi’s (“define FBG” and “setup FBG”) called by the FFT.vi for implementing the grating simulation process. The inputs of the sub⋅vi define FBG are the number of grating segments, the number of grating planes per segment, FBG parameters such as ac and dc index modulation, start Bragg wavelength, grating length, chirp rate and exposure function profiles. The output of the sub⋅vi is the
2 builds a waveform 2 points per segment Sine Wave.vi 0.00 0 Array 0.00 0.0 zero-padding Build Array F (0 to 1) Amplitude PhaseJumps R M Phase (radians) Hyperbolic Tangent Amplitude and Phase Spectrum.vi
1.41
grating definition consisting of ac and dc index modulations, grating pitch and phase as a function of position, z, along the grating. The inputs of the sub⋅vi setup FBG are the number of segments and grating definition consisting of a cluster of five elements listed above. The outputs of the sub⋅vi are local Bragg frequencies (FB) and amplitude and
phase of the local coupling coefficient, κ.dz. These are fed into the FFT⋅vi for generating a spectrum of the grating from the known grating parameters.
Figure 3.10 Block diagram for the FFT grating design model in Labview.
The detailed block diagram of the FFT core⋅vi is shown in figure 3.11. The operation of the FFT grating simulation algorithm is described below.
Figure 3.11 Block diagram for the FFT core⋅vi algorithm in Labview.
The amplitudes of the dc refractive index (An) versus position (z) along the grating and
local Bragg periods (ΛB) were used to generate a list of local Bragg frequencies (FB).
The local Bragg frequencies are used to build a set of 2001 segments each containing a number of grating planes. Each segment was defined by the local average refractive
each section is proportional to the local ac coupling coefficient κ.dz. The local ac coupling constant was calculated and integrated over the segment length to generate a set of reflectance contributions at regularly timed intervals. A sine waveform was synthesized at each grating segment based on the amplitude (An =κn.dz) and phase (Φn) of the local average refractive index. The dc refractive index modifies the phase of
the sinusoid in which these samples are generated, resulting in disturbances to the output spectrum representative of the effect of local heating or local strain within the CFBG.
A discrete FFT of the sine waveform was taken to generate the magnitude of the reflective contributions at each frequency. The discrete FFT algorithm is implemented by executing the fast radix-2 FFT routines. Consider the sequence of N complex numbers x0,x1,⋅⋅⋅⋅⋅, xN-1. These are transformed into the sequence of N complex numbers
X0, X1⋅⋅⋅⋅⋅, XN-1 by the radix-2 FFT according to the equation
∑
− = − = 1 0 2 N n nk N i n k x e X π where k =0,⋅⋅⋅⋅⋅, N-1. 3.17The radix-2 FFT algorithm divides the sequence size into two interleaved halves with each recursive stage. It first computes the FFT of the even-indexed numbers
) ,... , ( 0 2 2 2 − = m N m x x x x
e and of the odd-indexed numbers om = x2m+1(x1,x3,...xN−1) and then combines the two results to produce the Fourier transform of the whole sequence. Thus the technique can then be performed recursively to reduce the calculation time. The computing time for the radix-2 FFT is proportional toNlog2(N). The only requirement is that the number of points in the sequence be a power of 2. This is not problematic as the number of sample points can be freely chosen by the user. In this application, the complex sequence consists of the coupling coefficient, κ.dz, defined by the respective amplitude and phase of the local average refractive index of each segment. The output of the FFT is the complex amplitude reflection coefficient,
r(λ). The transformation R(λ)=tanh2r(λ) was applied to the FFT output r(λ) to
Grating Strength (kL) 0 1 0 1 2 3 P ea k R ef lect a n ce R FFT model Grating Strength (kL) 0 1 0 1 2 3 P ea k R ef lect a n ce R FFT model
with coupled mode theory for uniformly exposed gratings. Figure 3.12 shows the peak reflectance of the FFT model output compared with that predicted by the function
) ( tanh2 L
R= κ .
Thus a Fourier transform (FFT) of the contributions to reflectance at each of the N
sections of the grating planes is the core of this grating design model. The FFT method assumes that light propagates the entire length of the grating and is reflected from all grating planes, which is true for weak gratings with less than 50% reflectance investigated in this project. It also has high computation speed as required for grating simulation and on-line processing of distributed measurements. Thus due to its millisecond calculation time and the use of gratings with low reflectance for measurements in this project, the FFT model was implemented for simulating the spectral response of the gratings.
Figure 3.12 Comparison of the peak reflectance of the FFT model output with that predicted by the functionR=tanh2(κL)