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International Journal of Advanced Electrical and Electronics Engineering IJAEEE. loss and potentially low cost Moreover the most properties of fiber Bragg grating and its applications in. distinguishing feature of fiber Bragg grating is the optical communication system for high speed modern. flexibility they offer for achieving desired spectral photonic technology The work in this paper. characteristics Numerous physical parameters can be summarizes as follows In section II we describe the. varied including induced index change length physical model of an optical fiber Bragg grating and. apodization period chirp fringe tilt and desired obtained the conditions for dispersion relation This. wavelength section covers the complete analytical model to. characterize the optical wave propagation in fiber Bragg. After the observation of photosensitivity in Ge,grating In Section III a description of photonic. doped silica fiber by Hill et al in 1978 6 and side. bandgap is presented In which most of the incident. exposed holographic technique to make FBG with, frequencies are transmitted and reflected from the. controllable period by Meltz in 1989 7 FBGs has been. grating The filter response of fiber Bragg grating is. used as a critical component for many applications in. investigated in Section IV We also describe the, optical fiber communication and sensor system such as. applications of such spectral response in an optical. optical filters pulse compressor and dispersion, communication system The estimation of bandwidth of. compensator 8 11 etc Recently intensive research and. fiber Bragg grating described in Sections V We provide. development activities are carried out on a variety of. a theoretical expression to calculate the bandwidth of. FBGs which includes uniform chirped tilted apodized. grating by changing the various physical parameter of. FBGs to study their band pass and band rejection,Bragg grating Finally Section VI gives a brief.

responses 12 FBG based optical filters have been, description of the phase response of grating to gives the. widely investigated in photosensitive fibers 13, information about the phase changes at Bragg resonance. semiconductor waveguides 14 and polymer,wavelength in the Bragg grating. waveguides 15 Kashyap et al 16 have developed, ultra steep edge high rejection 74 dB filter using. II PHYSICAL MODEL FOR ANALYSIS OF BRGG, FBG The most significant of such FBG based filter is.

the linewidth of its reflection spectrum which is, relatively narrow As a consequence of this property the. selectivity of FBG is high This feature is an attractive Fiber Bragg grating is defined as a periodic. aspect of FBGs that can be used in an optical fiber as perturbation of the refractive index along the fiber. length This perturbation is formed by exposing the core. notch filters On the other hand the reflection, of the fiber to an intense optical interference pattern. wavelength of FBG is determined by the grating period. Several methods have been adapted to study and analyze. and the effective refractive index Because of this FBG. the reflection and transmission characteristics of FBG. can be used as an inline optical filter to block certain. wavelengths or as a wavelength specific reflector The However in present analytical study we take coupled. modern technological techniques can make the mode theory into consideration According to this. theory it is assumed that at any point along the grating. reflectivity of FBG close to unity over the reflection. within the single mode fiber there is a forward, band 17 21 Moreover the large dispersive behaviors. propagating mode and a backward propagating mode, of the Bragg grating structures make them good devices. Coupled mode theory is described in a number of texts. for linear dispersion comparators optical add drop. multiplexers OADM in wavelength division detailed analysis can be found in 25 29 The notation. multiplexing WDM systems and optical multiplexers in this section follows most closely that of G P Agrawal. 30 Throughout this thesis we assume that the fiber is. demultiplexers with an optical circulator Propagation of. lossless and single mode in the wavelength range of. solitons is another application of FBGs Nonlinear pulse. interest Moreover we assume that the fiber is weakly. propagation and compression have been also reported in. short period FBGs Due to the existence of a stop band guiding i e the difference between the refractive. in the transmission spectrum of FBGs known as a indices in the core and the cladding is very small Then. the electric and magnetic fields are approximately. photonic bandgap PBG the nonlinear pulse, transverse to the fiber axis and we can ignore all.

propagation has many applications such as an optical. polarization effects due to the fiber structure and. switch in them 22 24,consider solely the scalar wave equation. In the present work we have studied analytically, the filter characteristics of fiber Bragg grating using According to the coupled mode theory the total. coupled mode theory We have solved linear coupled field at any value of z can be written as a superposition. of the two interacting modes and the coupling process. mode equations and obtained the expression for, results in a z dependent amplitude of the two coupled. reflectivity of fiber Bragg grating for a CW laser beam. modes It is assumed that any point along the grating. The present paper also covers the fundamental optical. within the single mode fiber has a forward propagating. ISSN Print 2278 8948 Volume 2 Issue 4 2013, International Journal of Advanced Electrical and Electronics Engineering IJAEEE. mode and a backward propagating mode Thus the total and 2 Ab Ab A f 8. field within the core of the fiber is given by z 2. E z F x y Af z exp i B z Ab exp i B z Substituting Ab and A f in Equation 7 8 from. equation 4 and 5 we found the differential equations. Where A f and Ab represents the amplitudes of the in the form. forward and backward propagating modes respectively. B is the Bragg wave number for the first order, grating It is related to the Bragg wavelength through the and.

Bragg condition B 2neff which can be used to z 2, define the Bragg frequency B c neff and Let 2 k 2 q 2. F x y is the transverse modal field distribution The 2 Af. total field given by Equation 1 has to satisfy the wave q 2 Af 0 11. equation given by,and 2 Ab 12,E n 2 z 2 c 2 E 0, In the above formula n z denotes the refractive A general solution of these linear equations takes the. index variation along the FBG and is given by form. 2 A f z A1 exp iqz A2 exp iqz 13,n z neff n2 E n g z 3. And Ab z B1 exp iqz B2 exp iqz 14, Here neff is the average refractive index of the grating. n2 is the Kerr coefficient and These equations show that z dependent parts of the. n g z is the periodic,forward and backward waves in the FBG are.

index variation and E is the electric field propagating exponential with the propagation constant q This. inside the grating Substituting Equation 1 and parameter q representations the linear dispersion. Equation 2 into Equation 3 and considering a slowly relation of fiber Bragg grating and defined as. varying envelope approximation we can obtain the, following coupled mode equations in time 30 q 2 k2 15. A f The constant A1 A2 B1 B2 in Equations 13 14 are. i A f i Ab 4,interdependent and by using Equations 13 14 we. find that these constants satisfy the following four. and Ab i A i A 5 relations,q A1 B1 q B1 A1 16, In the above equations we focus only on the linear. case in which the nonlinear effects are negligible For q B2 A2 q A2 B2 17. such case we neglected the nonlinear parameter in the. One can eliminate A2 and B1 by using Equations 13 to. coupled mode equations In equations 4 and 5 and 14 and write the general solution in terms of an. are detuning parameter and linear coupling coefficient effective reflection coefficient r q as. and nonlinear coefficient respectively and are defined. as Af z A1 exp iqz r q B2 exp iqz 18,1 1 and ng 6 Ab z B2 exp iqz r q A1 exp iqz 19. To solve these equations let us differentiate,Equations 1 16 1 17 with respect to z r q 20.

2 Af A f Ab 7, z 2 z z is the effective reflection coefficient of the fiber Bragg. grating The q dependence of r q and the dispersion. ISSN Print 2278 8948 Volume 2 Issue 4 2013, International Journal of Advanced Electrical and Electronics Engineering IJAEEE. relation 15 indicate that both the magnitude and phase We defined the reflection coefficient of the FBG by. of the backward reflection depend on the frequency the ratio of the amplitude of reflected wave at z 0 to. the amplitude of incident wave at z 0 as,III CONCEPT OF PHOTONIC BANDGAP Ab z 0 B r q A1. A f z 0 A1 r q B2, The dispersion relation of Bragg gratings exhibits. an important property known as the photonic bandgap as. seen clearly in Fig 1 where detuning parameter is, plotted as a function of q for both a uniform medium.

dashed line and a periodic medium solid line, Fig 2 Schematic of a FBG of length L illuminated by. electromagnetic field of amplitude A z,If we use the boundary condition Ab L 0 in Eq 19. B2 r q A1 exp 2iqL 23,Using Equation 20 and Equation 23 in Eq 22 we. obtained the reflection coefficient as,q cos qL i sin qL. The corresponding expression for the reflectivity, Fig 1 Dispersion curves showing variation of with q and.

in the linear regime is found as, the existence of the photonic bandgap for a fiber grating. 2 sin 2 qL, For the uniform medium the slope is constant and thus Rg 25. the dispersion is negligible By introducing a grating q 2 cos 2 qL 2 sin 2 qL. the dispersion relation is modified and if the frequency. At resonance there is no detuning i e 0 hence,detuning of the incident light falls in the range. reflectivity is maximum The expression for the,q becomes purely imaginary Most of the. reflectivity becomes, incident field is reflected in that case since the grating.

does not support a propagating wave The range Rg max rg 0 tanh 2 L. is referred to as the photonic bandgap For this range of. detuning light cannot propagate through the grating and Varing some parameters such as grating length and. undergoes strong reflection It is also called the stop magnitude of induced index change it is possible to. band since light stops transmitting through the grating obtain narrow band transmission as well as high. when its frequency falls within the photonic bandgap reflectivity at the Bragg wavelength Optimization of. these parameters is fundamental when the objective is to. IV FILTER RESPONSE OF BRAGG GRATING use fibre Bragg gratings in band pass filtering. applications such as wavelength, The equation 18 and 19 gives the solution to the multiplexing demultiplexing and add drop optical filter. coupled mode equations in exponential form The Figure 3 shows the spectral response of the Bragg. reflection and transmission coefficient of fiber Bragg grating for the reflected wave condring five different. grating can be calculated by using Eqs 18 and 19 Bragg gratings with increasing lengths 1 L 0 5 mm. applying the appropriate boundary conditions as 2 L 1 0 mm 3 L 2 0 mm 4 L 4 0 mm and 1. Ab z 0 1 and Af z L 0 21 L 8 0 mm In all these cases we assumed effective. index neff 1 451 grating index ng 0 5 10 3 and Bragg. where L is the length of the grating Equation 21 wavelength B 1550 nm From this figure we can. implies that the incident wave has unit amplitude at z identify two different operating regimes Firstly when. 0 and the amplitude of the reflected wave at z L is zero the product L is small compared to 1 the reflectivity. because there is no reflected wave beyond z L The of the Bragg gratings is minimum Since the grating. boundary conditions applying on the FBG structure is begins and ends abruptly and extends for a length L the. shown in Figure 2 spectral response has a characteristic sinc shape. ISSN Print 2278 8948 Volume 2 Issue 4 2013, International Journal of Advanced Electrical and Electronics Engineering IJAEEE. whose bandwidth is inversely proportional to the grating the sidelobes get closer together An undesirable feature. length L We refer to gratings with L 1 as weak seen in Fig 3 is the presence of multiple sidebands. Bragg gratings because in general they only reflect a located on each side of the stop band These sidebands. fraction of the incident light The peak reflectivity in originate from weak reflections occurring at the two. this regime depends upon the value of but the grating ends where the refractive index changes. overall spectral shape and bandwidth is determined only suddenly compared to its value outside the grating. by the grating length A weak Bragg grating does not region Even though the change in refractive index is. make a suitable add drop filter because it only partially typically less than 1 the reflections at the two grating. reflects the input signal However there are cases where ends form a Fabry Perot cavity with its own. the sinc shaped spectral response of a weak grating is wavelength dependent transmission An apodization. desirable In many binary communications systems the technique is commonly used to remove the sidebands. encoded signal has precisely at the same sinc shaped 31. Fig 4 shows the dependence of peak reflectivity R g. Fiber Bragg Grating Filter for Optical the filter characteristics of fiber Bragg grating using coupled mode theory Fiber Bragg grating is defined

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