## 3 Fibre Bragg Grating  
Fiber Bragg gratings are optical interference filters which are inserted in a glass carrier. They are composed of dielectric layers with alternating high and low refractive index. The thickness of the layers is selected so that partially reflected light waves constructively interfere at the interfaces of the layers. The refractive index provides a sinusoidal relationship of  
\[n(z)=n_{0}+\Delta n\sin\left(\frac{2\pi}{\Lambda}z\right), \tag{1}\]  
which can be periodically varied. \(\Lambda\) is the period length, \(\Delta n\) is the modulation amplitude and \(n_{0}\) is the average refractive index in the modulated range. Figure 3 shows the schematic structure together with the description of the path difference in an FBG. When a light wave of higher refractive index enters, it is partially reflected, and this reflection becomes maximum when the phase difference between the waves is a multiple of the wavelength. In the case of an FBG, the path difference becomes  
\[\Delta=2\Lambda\cos\theta. \tag{2}\]  
This leads to the Bragg condition,  
\[2n_{0}\Lambda\cos\theta=m\lambda_{m},m=1,2,3,..., \tag{3}\]  
where \(\Lambda\) is a given lattice constant, \(\theta\) is the angle of incidence, and \(\lambda_{m}\) is the wavelength at which the reflection is maximum. The sinusoidal refractive index profile can be viewed for the first order \(m=1\), from where the Bragg wavelength condition can be calculated as  
\[\lambda_{B}=2n_{0}\Lambda\cos\theta. \tag{4}\]  
The Bragg wavelength \(\lambda_{B}\) is highly dependent on external influences. Mechanical elongation and temperature change \(\Delta T\) particularly affect the Bragg wavelength. As a simple approximation, this dependence can be stated as  
\[\frac{\Delta\lambda_{B}}{\lambda_{B}}=C_{\mathbb{S}}\varepsilon_{\mathbb{S}}+ C_{\mathbb{T}}\Delta T, \tag{5}\]  
where \(C_{\mathbb{S}}\) is the strain sensitivity, \(C_{\mathbb{T}}\) is the temperature coefficient, and \(\varepsilon_{\mathbb{S}}=\frac{\Delta L}{L}\) is the relative elongation of the fibre [46]. The strain may be used as an adjustable parameter for the fibre Bragg grating. This dependence of the Bragg wavelength is used to realize the concept as a tunable optical filter [40; 41; 47]. In the course of the work, however, the temperature effects are not negligible, and represent a particular challenge in terms of control. The FBGs have an important function in the system design, as they help to form narrowband wavelength selective mirrors in the system. These are needed to generate the two laser modes (wavelengths) with the semiconductor optical amplifier. Therefore, the special structure of the FBGs used is presented here, and afterwards, how they are controlled in the overall system. The internal structure of the tunable FBG from Advanced Optics Solutions [48] is schematically shown in Figure 4. The FBG is embedded into a holding device whose length can be controlled (stretched or elongated) by a stepper motor. As shown in Equation (5), the Bragg wavelength is dependent on the elongation of the fibre. From the rotation of the stepper motor, this results in a direct change of the Bragg wavelength. An integrated controller realizes the communication between the stepper motor (through the stepper motor's driver) and a computer. This system is used as an interface to the Universal Serial Bus (USB). Due to the high temperature sensitivity of FBGs (again in accordance with Equation (5)), there is additionally a temperature sensor in the housing of the adjustable FBG (not shown in Figure 4), in order to allow temperature compensation. The parameters of the temperature compensation can be determined by the vendor during its manufacturing phase, and stored in a look-up table [48; 49; 50]. The temperature compensation of the FBG used is therefore performed by calling a function and using the the look-up table that is provided by the vendor. Figure 4: Schematic representation of the internal structure of the tunable fibre Bragg grating.