# Intense Usage In Wavelength Division Multiplexed Systems Biology Essay

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FBGs attracts great interest because of its intense usage in wavelength division multiplexed systems. a single sampled FBG works greatly in the filtering operation or dispersion compensation in the multiple channels .the sampled grating was initially proposed for the semiconductors lasers [31].there were many methods that were used to design a sampled FBG but every method had some issue. Some of the methods and there issues were:

The most simple sampling function is a periodic sequence of rectangular functions. In spectral domain, the rectangular sampling function corresponds to a sinc envelope, which modulates the amplitudes of the multiple channels, those results in a high channel nonuniformity. The sinc sampling can overcome the no non uniformity problem [32]. But the problem was that sinc-sampled FBG fabrication requires a precise control of both amplitude and phase in the grating, which was very difficult in the phase mask side-writing approach. The inherent problem of both rect and sinc amplitude sampling is that the regions between two consecutive samples in the fiber are not written with gratings and because of this they have no contribution to the grating reflection. This leads to a requirement for an high index modulation in the fiber that realistically doesn't exist.

In 2003 hungpu li (shizouko university) proposed binary and multilevel phase only sampling function [33] for the sampled FBGs. It required less refractive index modulation as compared to amplitude (rect and sinc methods) sampling methods. There design uses the simulated quenching optimization with the temperature rescaling. It results in the high channel uniformity and minimum out of band energy. They demonstrated five channel nonlinearly chirped multilevel phase only sample FBG for the chromatic dispersion compensation.

In 2006 the further improvement was done when continuous phase only sampling function was proposed that was capable of producing 81 channel FBG [34] with fantastic uniformity and high inband energy efficiency. They used simulated annealing algorithm and phase only linearly chirped FBGs for dispersion compensation was used.

Again in 2007 the further advance step was when hungpu li enable to design a multichannel FBG where the spectral response of each channel was identical or non identical[35]. They demonstrated the 9 channel non linearly chirped FBG, which is used as a simultaneous dispersion and slope dispersion compensation.

## Nine channel binary phase-sampled FBG

## Amplitude of binary phased-only sampling function for Nine channels

## Phase of binary phased-only sampling function for eight channels

Random -sample

## binary phased-only sampling function

## Fourier transform of binary phased-only sampling function

The Damman sampling function introduces phase segments in each sampling period of the grating. The phase values added are either 0 or pi. The segment widths are initially selected at random. However, when the algorithm is applied on the cost function, it returns a set of phase transition points. If the phase shifts are introduced at these transition points, the cost function will be minimum. Thus, minimizing our cost function is equivalent of finding optimum phase transition points. If we introduce phase shifts at these points the difference between the Fourier coefficients for the desired number of channels will be minimum. In figure below for instance, initially the phase is ï°.. These phase segments vary in lengths. The designed grating has eight Fourier coefficient of equal amplitude. That implies that grating has a constant response for nine-channels. The out-of-band channels amplitude is relatively smaller and the response is non-constant.

## Reflection Spectrum of Nine uniform channels using transfer matrix analysis

## Reflection Spectrum in dB scale of Nine uniform channels using transfer matrix analysis

## Phase of binary phased-only sampling function for 16 channels.

## Amplitude of binary phased-only sampling function for 16 channels

## Fourier transform of binary phased-only sampling function.

## Reflection Spectrum of 16 uniform channels using transfer matrix analysis

## Reflection Spectrum in dB scale of 16 uniform channels using transfer matrix analysis

## (a) Amplitude of binary phased-only sampling function for 39 channels

## (b): Phase of binary phased-only sampling function for 39 channnels.

## (c): Fourier transform of binary phased-only sampling function.

## (d): Reflection Spectrum of 39 uniform channels using transfer matrix analysis

## (e): Reflection Spectrum in dB scale of 39 uniform channels using transfer matrix analysis.

The optimised transition points for 9, 16, 39 channels are as follows

Transition points

9 channels

16 channels

39 channels

Z0

0

0

0

Z1

0.1779

0.0576

0.0320

Z2

0.3446

0.1845

0.0640

Z3

0.4093

0.2113

0.1085

Z4

0.4770

0.2621

0.1449

Z5

0.8759

0.2788

0.1892

Z6

1

0.3251

0.2738

Z7

0.4185

0.2824

Z8

0.4568

0.3117

Z9

0.5

0.4635

Z10

0.5576

0.4929

Z11

0.6845

0.5833

Z12

0.7113

0.6176

Z13

0.7621

0.6592

Z14

0.7788

0.6783

Z15

0.8251

0.6995

Z16

0.7621

0.7321

Z17

0.7788

0.7548

Z18

0.8251

0.7879

Z19

0.91185

0.8621

Z20

0.9568

0.9307

Z21

1

0.9711

Z22

1

As we have said that the acceptance probability decreases as the temperature is lowered. Below we show, it actually does. The plot is taken for a 24-channel grating.

## Change in the acceptance probability during quenching iterations

The following plot shows the diffraction efficiency achieved. We try to minimise our cost function whilst maintaining a high diffraction efficiency and our success is depicted in the following figure. The plot is taken for a 24-channel grating.

## Diffraction efficiency of the system as it undergoes optimization

The pattern of temperature variation and the rescaling process is depicted in the following

## Temperature variation in the quenching process, Jumps reflect the rescaling process.

To let the reader appreciate the success of our results, below, we show the results achieved by us and by Hongpu Li et al. in [10].

## Fourier Transform of the 8-channel sampling function designed by us

## Fourier Transform of the 8-channel sampling function as in [5]

## Figure 5-13 (a): Fourier Transform of the 16-channel sampling function designed by us.

## Fourier Transform of the 16-channel sampling function as in [5]

Chirp parameter

We used linear chirp in the grating period. For this, we substituted:

where we set = 1 for 1 nm/cm chirp in the grating period. and were substituted in to simulate chirped FBG.

Apodization

Equation was used in the transfer matrix equation to simulate the apodization profile with g(z) being the raised-cosine apodization profile:

## 9-channel Binary-level chirp

## 16-Channel Binary Level Chirp

## 39-Channel binary level Chirp

## Chirped Fiber Bragg Gratings

THE SIMULATION RESULTS OF THE SPECTRAL RESPONSE

Linear chirped gratings with different chirp variables

Linear chirped gratings with different lengths

The above figure shows the reflection spectrum with the value of the chirp variable -1(nm / cm)

The above figure shows the reflection spectrum with the value of the chirp variable 1(nm / cm)

The above figure show the reflectance spectrum of two chirped gratings with an equal chirp of opposite signs. The values of the chirp variables are -1(nm / cm) (Blue) and = 1(nm/cm) (Red), with the following parameters: L =3000(m) , =1.447, n = 0.0004 , 1.550( m) .

If is positive, the period of the linear chirp grating increases along the propagation direction. Where as if is negative, the period of the linear chirp grating reduces along the propagation direction. If is negative, the center wavelength of the grating moves to the left hand side (shorter ). If is positive, the center wavelength of the grating moves to the right hand side (longer

Chirp reflection spectrum for L=3mm

Chirp reflection spectrum for L=3mm

Chirp delay spectrum for L=3mm

Chirp dispersion spectrum for L=3mm

Chirp reflection spectrum for L=4mm

Chirp reflection spectrum for L=4mm

Chirp Delay spectrum for L=4mm

Chirp Dispersion spectrum for L=4mm

Chirp reflection spectrum for L=5mm

Chirp reflection spectrum for L=5mm

Chirp delay spectrum for L=5mm

Chirp dispersion spectrum for L=5mm

The reflectance spectrum of three linear chirped gratings with different lengths: L= 3000(um) (Blue), L=4000(um) (Red) and L= 5000(um) (Green) and the following parameters: =1.447, n = 0.0004, 1.550( m) .

The above figure shows the reflectance spectrum of linear chirped gratings with different lengths, and with the same "chirp parameter". The reflectance is increased when the length of the grating is increased. At the same time, the bandwidth of the spectrum is reduced.

The above graph shows the steepness of gradient for the a particular channel with length variable L=3mm

The dispersion for the variable length chirp with Length=3mm is shown above

The above graph shows the reduced steepness in the gradient of Delay graph of chirp with the variable length L=4mm

The above graph shows further zoomed delay due chirp with Length=4mm

The above graph shows the dispersion of chirped grating with Length=4mm

The above graph shows the reduced steepness in the gradient of the chirped grating delay graph with Length=5mm

The above graph shows the dispersion for the Length=5mm

From the above graphs, we can deduce that with increasing the length of the grating reflection is increased and the gradient of the delay graph becomes less steep and reduces which leads to the conclusion that as the length of the grating increases so does the dispersion reduces.