Design optimization of two concatenated long period waveguide grating devices for an application specific target spectrum Girish Semwal1,* and Vipul Rastogi2 1
Instrument Research and Development Establishment, Dehradun 248008, India
2
Department of Physics, Indian Institute of Technology Roorkee, Roorkee 247667, India *Corresponding author:
[email protected] Received 23 December 2014; revised 2 March 2015; accepted 9 March 2015; posted 9 March 2015 (Doc. ID 231323); published 3 April 2015
We propose a global optimization method to optimize the parameters of two concatenated long period waveguide gratings (LPWGs) for generating a desired target spectrum. The design consists of two concatenated LPWGs with different grating periods inscribed in the guiding films of a four-layer planar waveguide with finite over cladding. We have used the transfer matrix method to compute the modes of the structure and the coupled mode theory to compute the spectrum of the device. The adaptive particle swarm optimization method has been used to optimize the parameters of LPWGs to generate symmetric as well as asymmetric target spectra. Two concatenated gratings of different lengths and periods have been used to generate the target spectra. To demonstrate the method of optimization we have designed a variety of wavelength filters including a rectangular shape rejection band filter, asymmetric band rejection filters, band rejection filters for flattening the amplified spontaneous emission (ASE) spectrum of an erbium doped fiber amplifier (EDFA), and a gain equalization filter for an erbium doped waveguide amplifier (EDWA) in the C-band. Seven parameters of the proposed LPWG structure have been optimized to achieve the desired spectra. We have obtained an ASE flattening with 0.8 dB peakto-peak ripple in case of the EDFA and gain flattening with 0.4 dB peak-to-peak ripple in case of an EDWA. The study would be useful in the design of wavelength filters for specific applications. © 2015 Optical Society of America OCIS codes: (130.0130) Integrated optics; (130.7408) Wavelength filtering devices; (350.2770) Gratings. http://dx.doi.org/10.1364/AO.54.003141
1. Introduction
Long period fiber gratings (LPFG), being the transmission gratings, have been preferred in the field of optical communication and sensor technologies due to their ease of fabrication as compared to fiber Bragg gratings [1]. Optical communication devices based on the LPFG have been designed and developed for band rejection filters, dispersion compensation, wavelength division multiplexing, and gain equalization of erbium doped fiber amplifiers (EDFAs) [2–5]. 1559-128X/15/113141-09$15.00/0 © 2015 Optical Society of America
Similarly, the sensing devices using LPFG have been developed for single and multi-sensing applications. The LPFG sensors for temperature, strain as well as chemical sensing, have been developed and demonstrated successfully [6–10]. The LPFG couples the core mode with the co-propagated cladding mode in the presence of grating and the characteristics of the rejection band depend on the grating parameters (index modulation, period, length, and apodization of grating). The rejection band generated by LPFG is, in general, symmetric around the resonance wavelength. However, there are certain applications that require asymmetric wavelength rejection band spectrum. Attempts have been made to 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS
3141
tailor the asymmetric rejection band by using phase shifted fiber grating, apodized phase shifted fiber gratings, and step changed LPFG [11–13]. LPFG has limitations due to material and geometrical constraints. These constraints restrict the degree of freedom in the design process of LPFG devices. To overcome these limitations the concept of long period waveguide grating (LPWG) has been theoretically proposed and experimentally demonstrated [14–16]. Fabrication techniques and applications of LPWG in the field of optical communication and sensing technologies have been reported in literature [17–20]. Although LPWG has the advantage of the degree of freedom with respect to the material and geometry, still the generation of asymmetric and application specific spectra is a challenging task. A method has been proposed to introduce asymmetry in the rejection band spectrum by controlling the cladding layer thickness in the longitudinal direction [21,22]. Recently, we reported the optimization of cladding layer parameters and grating parameters using adaptive particle swarm optimization (APSO) for obtaining desired wavelength rejection bands [23]. Experimental control of the cladding layer thickness in the longitudinal direction is a difficult task however. Another approach for generating an asymmetric rejection band can be the cascading two long period gratings (LPGs) in a uniform cladding waveguide. However, cascading of gratings has two drawbacks: (1) practical realization of the structure with a desired phase shift at the junction of two gratings can be difficult; (2) optimization of the structure becomes more cumbersome and time consuming due to phase shift at the junction. In the present study we have carried out the design optimization of two concatenated LPWGs on the single substrate to achieve the asymmetric rejection band spectra. The APSO method has been used to optimize the design parameters of two concatenated gratings. The particle swarm optimization (PSO) has also been implemented in the field of optical communication to optimize the design parameters [24,25]. The present design approach of concatenated grating is free from the phase mismatch issue at the junction of two gratings. The method optimizes the required design parameters of two concatenated waveguide gratings simultaneously. As specific examples we have demonstrated optimization of the parameters of two concatenated LPWGs based amplified spontaneous emission (ASE) flattening filters of EDFA and gain flattening filters for erbium doped waveguide amplifier (EDWA) using this method. 2. LPWG Structure and Analysis
A four-layer waveguide structure has been used for the present study and analysis. The four-layer waveguide structure is a guiding structure that consists of a thick layer of substrate of refractive index ns , the guiding film of thickness df , refractive index nf , cladding layer of thickness dcl , a refractive index ncl , and an over-clad layer of refractive nex as shown in 3142
APPLIED OPTICS / Vol. 54, No. 11 / 10 April 2015
Fig. 1. Four-layer waveguide structure. (a) Index profile and (b) typical corrugated grating structure.
Fig. 1(a). The thickness of substrate and over-clad are extended infinitely. The guiding film thickness df is selected to support only the fundamental mode and the higher-order modes propagate in the cladding layer of waveguide. Two concatenated LPGs with grating periods Λ1 and Λ2 , lengths L1 and L2 , and corrugation heights h1 and h2 have been embedded in the guiding layer of the waveguide as shown in Fig. 1(b). The LPGs act as perturbing agents and lead to coupling between a cladding mode and the fundamental mode. A combination of two different LPGs results in an asymmetric rejection band whose shape can be controlled by the waveguide and grating parameters. In this study we have considered seven parameters namely Λ1, Λ2 , L1 , L2 , h1 , h2 , and dcl to control the shape of the rejection band for desired application. The modes of the waveguide have been obtained by using the same approach as in [14]. The transmission spectrum of the structure has been computed by using the transfer matrix method (TMM) to solve coupled mode equations [26]. TMM allows considering the effect of one grating at a time. The coupled mode equations for core mode and co-propagating cladding mode in the presence of an LPG are given as follows [27]: dA κBeiΓz ; dz
(1)
dB −κAe−Γz ; dz
(2)
where Az and Bz are the z-dependent amplitude coefficients, Γ is phase mismatch, and κ is the coupling coefficient between the fundamental mode and the co-propagating cladding mode in the presence of grating. In case of corrugated grating having corrugation height h, the phase mismatch and the coupling coefficients are given as Γ β0 − βm −
κ
k0 Δn2 2πcμ0
Z
2π ; Λ
df
df −h
E0y Emy dx:
(3)
(4)
E0y x and Emy x are the power normalized transverse electric fields for guided and cladding TE modes, β0 and βm are corresponding propagation constants, c is velocity of light, k0 2π∕λ0 is the freespace wave number, λ0 is the free-space wavelength, μ0 is the free-space permeability, and Δn2 n2f − n2cl is the index modulation in the corrugated region. In this paper we have considered coupling between the TE0 and TE1 mode to obtain the desired spectrum. 3. APSO
PSO has been proposed by Kennedy and Eberhart to understand the socio-cognitive behavior of metaphor [28,29]. It is a population based heuristic system that governs its dynamics by self-learning process. PSO has a tendency to acquire its optimum position in the domain of interest. The domain is defined by the minimum and maximum values of the variables. The best value of the variable is searched within the domain. PSO is simpler to implement and only a few governing equations are iterated with suitable socio-cognitive parameters as compared to the other global optimizations like genetic algorithm, antcolony optimization, or simulated annealing process. The governing equations for PSO are given as [30,31] xi0 xmin randxmax − xmin ;
(5)
xmin randxmax − xmin ; Δt
(6)
vi0
g plk − xik pk − xik c2 rand ; c1 rand Δt Δt (7)
vik1
wvik
xik1 xik vik Δt;
(8)
where xik ; i 1; 2…N, k 1; 2…:M, represent the variables to be optimized, and k is the time step (iteration number) for forward motion of particles. xmin and xmax are minimum and maximum values of xi, respectively. Equations (5) and (6) have been used to generate the initial random population of swarm
and their initial random velocities within the predefined boundaries of variables. plk and pgk are the local and global minimum of xi parameter in kth iteration. Local minimum is the minimum of a variable in total population for current iteration while global minimum is the minimum of the variable in all previous iterations including the current iteration. Δt is the time interval and has been taken as 1 for the computation. Equations (7) and (8) govern the dynamics of optimizer during the optimization process. The dynamics of the optimizer toward its optimum position is dependent on the three socio-cognitive parameters w, c1 , and c2 . The PSO optimizer is a self-learning system and the optimizer becomes intelligent through learning process from its society with the progress of time. Hence these governing parameters are dependent on the social behavior of swarm. w is known as the inertia parameter and has a tendency to retain the previous memory of dynamics. If this parameter is very large, it generates large inertia in the dynamics of the swarm to change the state of the swarm. Similarly, in case of a very small value of w, swarms lose their previous memory and move randomly. c1 and c2 are called the acceleration parameters. c1 has a tendency to drag the swarm toward its local minimum and c2 has tendency to drag the swarm toward its global minimum. Suitable value of these socio-cognitive parameters is required to achieve the convergence of the optimization process. The parameter selection and convergence analysis of PSO has been analyzed in detail [32]. The objective of all research related to PSO is to evolve the techniques that accelerate the convergence of optimization process. To achieve the goal, different variants of PSO have originated. The research related to the different variants of PSO algorithms and their applications are described in literature [33]. All variants of PSO are developed for fast convergence and prevent the particles from being trapped in local minima and converge at global minima. Variants are broadly categorized into two classes: (1) hybrid PSO, which consists of the PSO in conjunction with other evolutionary algorithms such as genetic algorithm, evolutionary programming, and other heuristic optimization process; and (2) adaptive PSO in which the parameters of PSO change adaptively for fast convergence toward its global minima. The generation based adaptation in inertia weight, k-mean clustering based adaptation, and fuzzy logic adaptation in both inertia weight and acceleration coefficients are a few popular adaptation methods developed. Performance has been tested [34,35]. In the present study we have followed the fuzzy logic-based adaptation procedure in PSO. The details of implementation of the adaptation procedure and convergence analysis for different benchmark functions are given in [35]. The PSO is a population based method of optimization. Hence, the initial population for all the parameters is generated with the help of a random generator. The boundary of parameters is defined 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS
3143
by assigning their minimum and maximum values. The space between minimum and maximum value of parameter is called its domain. Initially the parameters are randomly scattered in the entire domain. The total number of parameters to be optimized is called the dimension of PSO. The scattering of particles in the domain is described by the mean distances of particles. The mean distance of the ith particle, with respect to all other particles is given as v N1 u N uX X 1 t di xl − xlj 2 ; N 1 − 1 j1;j≠i l1 i
(9)
where di is the mean distance of ith particle. N 1 is the total population size, and N is the dimension of PSO. dmin and dmax are the minimum and maximum mean distances of particles in each iteration and dg is the mean distance of global best particle with respect to all other particles. The normalized value of dg is defined as the evolutionary factor as follows: f
dg − dmin : dmax − dmin
(10)
Initially, the swarms are scattered in the complete domain; hence, dg has maximum deviation from dmax to dmin . Evolutionary factor f has maximum value initially and then starts decreasing with the number of iterations. Once the function is optimized, the value of the evolutionary factor vanishes. The complete dynamics of f are divided into four groups. These groups are named as exploration (S1 ), exploitation (S2 ), convergence (S3 ), and jumping out (S4 ). The classification has been carried out on the basis of careful observation of dynamics of swarms with different benchmark functions. The membership functions developed for four groups are known as fuzzy classifiers. These classifiers are the function of f . Here we have used the membership functions as described in [35]. The adaptation in inertia weight factor is given as w
1 : 1 1.5e−2.6f
(11)
The criteria for the adaptation in c1 and c2 have been classified on the basis of these four fuzzy groups S1 ; S2 ; S3; S4 . c1 and c2 are initialized and a small amount of decrement or increment in c1 and c2 during adaptation depends on the group in which it falls. It is decided by the factor f as detailed below. (1) S1 is the exploration state. In the exploration state, randomly generated swarms are scattered in complete domain and have a tendency to acquire the local minima. c1 has a tendency to drag the swarms in the local minima. Hence, if swarms are in the state S1 , increment in c1 and decrement in c2 is introduced. 3144
APPLIED OPTICS / Vol. 54, No. 11 / 10 April 2015
(2) S2 is the exploitation state. Using the local information, the swarms group into the local minima. Small increment in c1 and small decrement in c2 is introduced in the state S2 . (3) S3 is the convergence state. In this state swarms have a tendency to acquire global minima. The influence of c2 can be increased by slightly increasing c1 and c2 . This helps to drag the swarms in the global minima. (4) S4 is the jumping out state. In this state swarms jump out from local minima and move towards a better optimum position. Hence the influence of c1 is reduced and c2 is increased by decreasing the value of c1 and increasing the value of c2. The amount of increment and decrement is obtained by generating the random number in the interval [0.05, 0.1]. The small increment or small decrement is obtained by generating a random number between the interval [0.025, 0.05]. When the sum of these two acceleration coefficients exceeds 4 then c1 and c2 are normalized by following the constraint relation as ci
ci 4.0 c1 c2
i 1; 2:
(12)
The value of parameters may drift outside of the defined boundary of variables. In such cases boundary conditions have been imposed on the particles to drag them within the predefined boundaries of variables. Different types of boundary conditions have been proposed and implemented in the APSO. Here a damped boundary condition has been used to retain the swarm within the computational domain [36]. In a damped boundary condition, particle positions are computed by Eq. (8), and if they fall outside the predefined limits of the variable they are replaced by the randomly generated value of the variable within two extreme limits of variables. This boundary condition does not affect the convergence of APSO and only prevents the swarm from escaping from the domain of interest. 4. Design Optimization of LPWG by APSO
LPWG has been proposed to increase the degree of freedom in the design process. LPWG has four degrees of freedom. Apart from choice of material, variable parameters of grating are length (L), period (Λ), corrugation height (h), and cladding thickness (dcl ). These parameters govern the nature of the resonance rejection band [37]. Suitable combinations of these parameters generate the desired resonance rejection band. The present LPWG structure has two concatenated gratings. The structure has seven design parameters, namely grating lengths L1; L2 , periods Λ1 ; Λ2 , corrugation heights h1 ; h2 , and cladding thickness (dcl ). Application specific target spectrum is generated by suitable combinations of the above mentioned grating parameters. The suitable combination of grating parameters is obtained by applying the APSO in
coupled mode equations. The brief steps of procedures are given as follows. (1) Define the upper and lower bound of variables corresponding to the grating parameters in the APSO. These bounds are decided on the basis of the physical nature and fabrication limits of LPWG. (2) Randomly generate the population (variable values) between predefined boundary as per step (1) using Eqs. (5) and (6). Population size is defined as per problem under consideration. Large population size increases the computational time and small population size decreases the convergence rate. The optimum population size should be taken for optimization process. The APSO is initialized with randomly generated population. The population forms the sets of grating parameters. (3) Solve the coupled mode equations, Eqs. (1) and (2), for each set of parameters and compute the transmission spectrum. (4) Compare the computed spectrum (Sc ) with target spectrum (St ) and compute the fitness as follows: Fitness
q St − Sc 2 :
(13)
(5) Find the set of parameters corresponding to the minimum value of the fitness function. The minimum value in the first iteration is stored as local and global minima. (6) Population of parameters for subsequent iterations is generated using Eqs. (7)–(12). The maximum number of iterations depends on the complexity of the problem under consideration. Hence, for any problem, it could be fixed with initial trial runs and convergence analysis of the problem. (7) Compute local minimum in each iteration. If local minimum is smaller than the global minimum in any particular iteration then the global minimum is replaced by the lowest value of local minimum. (8) Follow steps (2)–(7) until the maximum number of iteration is achieved. The set of parameters corresponding to the global minimum are the optimum value of grating parameters to generate the required target spectra.
have been fixed to 300 and population size has been taken as 20. The population size and number of iterations have been decided on the basis of initial trial run and careful examination of the rate of convergence toward the optimum values of parameters. Once the parameters of APSO have been fixed for the optimization of LPWG device design, APSO algorithms have been implemented to design LPWG for given target spectra. We have demonstrated optimization of LPWG parameters for a variety of target spectra including a rectangular wavelength rejection filter, an asymmetric wavelength rejection filter, a filter to flatten ASE spectrum of an EDFA, and a gain flattening filter for EDWA. The following waveguide parameters have been used for design of LPWG: ns 1.444;
nf 1.53;
ncl 1.50;
nex 1;
df 2 μm:
(14)
These parameters are typical of polymer waveguide on glass substrate [22]. Figure 2 shows a rectangular wavelength rejection band used as the target spectrum. Such a rejection band spectrum can be generated by the merging of two resonance rejection bands of two concatenated gratings. Parameters of concatenated gratings have been optimized by using APSO in order to achieve the target spectrum. The target spectrum as shown in Fig. 2 by a solid line has been stored in St for the fitness computation in the optimization process. The optimizer searches the suitable combination of parameters within the pre-defined search domain of parameters. Search domain is defined by lower and upper bounds of the parameter’s values. In the present simulation, lower and upper boundaries of the search domain for grating length (L) have been taken as 2 and 20 mm, respectively. The typical value of grating period (Λ) for the LPG ranges from 50 to 800 μm. Upper and lower bounds of the grating period should be taken between these two extreme limits. The corrugation height (h) has been taken
5. Numerical Simulation
Numerical simulations for the design of LPWG for the application-specific target spectra have been carried out using APSO. Parameters (w, c1 , c2 , total iterations, population size) of APSO have been fixed before its implementation in the LPWG parameters optimization. Weight factor w and acceleration coefficients c1 and c2 are the dynamics governing parameters of APSO, which control the rate of convergence in the optimization process. These parameters change adaptively during the optimization. Hence, only the bounds of the parameters and initial values have been assigned in the optimization process. w varies according to Eq. (11) between 0.9 and 0.4, respectively. Initial values of c1 and c2 have been taken as 1.6 and 2.4, respectively. The maximum iterations
Fig. 2. Ideal rectangular spectrum. Target spectrum St and computed spectrum Sc . 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS
3145
between 10 and 120 nm and the range for cladding thickness (dcl ) has been defined as 3–15 μm. APSO randomly generates 20 sets of values of various parameters in the above defined respective ranges. The coupled-mode equations have been solved for each set of parameter values and the computed spectrum has been generated. The computed spectrum thus obtained has been designated by Sc. During the optimization process, the values of sociocognitive parameters varied adaptively. The variations of w, f , and c1 , c2 are shown in Figs. 3(a), 3(b), and 4(a), respectively. The variation in c1 and c2 is rapid initially and represents the complex dynamics. Figure 3 shows that w and f initially had a tendency to produce the large inertia but acquired the lowest value for optimum adaptation. The total number of optimization parameters are seven in the present case, which makes the optimizer take a longer time to converge into global minimum. The convergence curves of c1 and c2 shows smoother convergence toward the value 2 after 150 generation as can be seen in Fig. 4(a). The fitness function has been evaluated for computed spectra corresponding to each set of parameters. The set of parameters corresponding to minimum value of cost
Fig. 4. Acceleration coefficients and fitness. (a) Variation of c1 and c2 with number of iterations. (b) Fitness functions change with number of iterations.
Fig. 3. Adaptation of APSO parameters. (a) Variation of inertia factor (w) with number of iterations. (b) Variation of evolutionary factor (f ) with number of iterations. 3146
APPLIED OPTICS / Vol. 54, No. 11 / 10 April 2015
function in the first iteration has been stored as local and global minima. The minimum of the fitness function for subsequent iterations has been computed until the maximum predefined iterations were achieved. The variation of fitness functions for local and global minima with respect to the number of iterations have been shown in Fig. 4(b). The plots show that there is variation in local minimum but the global minimum also has a decreasing trend. The mean value of the fitness function is also plotted in the Fig. 4(b). The mean fitness function is the average of the fitness function for all populations in an individual iteration. The decreasing trend of mean value of fitness indicates that all particles are clustering around the global minimum. Fitness curves show that the global and local fitness curves merge after 150 iterations but the mean fitness has decreasing tendency. The decreasing tendency of mean fitness indicates that all sets of parameters are converging to the global minimum. It signifies that the optimization process is converging properly toward its optimum value (global minimum). The set of parameters corresponding to the lowest value of cost function in all iterations is the optimum set of parameters of the grating device. The optimum values of parameters for rectangular rejection filter
Table 1.
Optimized Gating Parameters of Two Concatenated Gratings for Different Targetsa
Targets
L1 (mm)
Λ1 (μm)
h1 (nm)
L2 (mm)
Λ2 (μm)
h2 (nm)
dcl (μm)
Optimization Time (h)
Fig. 2 Fig. 5(a) Fig. 5(b) ASE EDWA
4.9 4.8 3.2 4.3 10.1
71.91 72.19 67.05 74.24 75.55
107.0 70.0 60.9 97.2 76.6
4.7 2.3 2.3 4.3 6.5
71.18 74.42 64.13 74.35 111.89
105.0 58.0 74.5 46.6 125.0
4.41 4.43 3.63 4.73 4.91
7.0 7.7 7.7 4.2 3.0
a
Figure 2 corresponds to the rectangular target. Figures 5(a) and 5(b) correspond to two asymmetric targets, ASE and EDWA for targets of ASE of EDFA flattening and EDWA gain flattening. The optimization time corresponds to the Matlab code executed on PC (Core i7).
are shown in Table 1 and tagged as Fig. 2 in the target column. Values of parameters of two concatenated gratings are very close to each other. The rejection band generated by these optimized parameters is shown in Fig. 2 by dotted lines. The asymmetric spectra, as shown by solid lines in Fig. 5, have been considered for further analysis. A pair of concatenated gratings with suitable grating parameters has been utilized to generate the asymmetry in the transmission spectra. The procedure described for the rectangular rejection band has been followed to optimize the LPWG parameters for generating the spectra shown in Fig. 5. The optimized design parameters as obtained by using the APSO are shown in Table 1 and are indicated by Figs. 5(a) and 5(b) in the target column. The computed spectra corresponding to the optimized grating parameters have been shown by dotted lines in Fig. 5. We see a good agreement between the target and computed spectra. Finally, we demonstrate the optimization of LPWG parameters for flattening the ASE spectrum of EDFA
shown by the solid line in Fig. 6(b) and gain spectrum of EDWA shown by the solid line in Fig. 7(b). The target spectrum to flatten the ASE of EDFA is shown by the solid line in Fig 6(a). The optimized LPWG parameters to generate this target spectrum are shown in Table 1 and the computed spectrum of the optimized LPWG is shown by the dotted line in Fig. 6(a). Corresponding flattened ASE spectrum is shown by the dotted line in Fig. 6(b). The simulation results show that the ASE is flattened up to 0.8 dB peak-to-peak ripples from 1530 to 1560 nm spectrum band. This is slightly better than the 1 dB ripple obtained by using the segmented cladding approach [23]. In the case of EDWA, gain
Fig. 5. (a),(b) asymmetric target spectra (St ) and corresponding computed spectra (Sc ).
Fig. 6. (a) Target spectra (St ) and computed spectra (Sc ) for ASE flattening of EDFA. (b) ASE curve of EDFA and flattened curve. 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS
3147
from 7.7 to 3 h. In the present work we have proposed the capability of APSO in integrated optics for the design optimization. However, the computation time can be brought down if the code is adapted for a high-end machine with parallel processing. The optimization efficiency and accuracy are interlinked in the present problem. The computational efficiency depends on the number of parameters to be optimized and the selected range of the search domain xmax ; xmin . Using two concatenated gratings in the present design increases the accuracy at the cost of decreased efficiency due to the increased number of optimization parameters. 6. Conclusion
Fig. 7. (a) Target spectra (St ) and computed spectra (Sc ) for EDWA flattening. (b) Gain curve of EDWA and flattened curve.
flattening the target and computed spectra are shown in Fig. 7 and the optimized LPWG parameters are shown in Table 1. The flattening in this case could be obtained with a 0.4 dB ripple as compared to a 0.2 dB ripple by using the segmented cladding approach [23]. Although the flattening in the case of EWDA could not be improved, the concatenated grating approach is practically more feasible as compared to the segmented cladding approach from the point of view of fabrication. The optimization time corresponding to each design is shown in the last column of the Table 1. Optimization time depends on the amount of computation involved in each iteration. In the present optimization problem we are optimizing a total of seven numbers of parameters including the cladding thickness. There are 20 sets of these parameters (population size) in each iteration. The mode solver has to numerically compute the effective indices and modal field profile of the modes of the waveguide at all wavelengths. There are 166 wavelengths in case of asymmetric spectra and 50 wavelengths in case of EDWA. The computation of the coupling coefficient and solution of the coupled mode equation also has to be carried out for all sets of parameters involving all the wavelengths. This requires a huge amount of computation. Hence the optimization time is varied 3148
APPLIED OPTICS / Vol. 54, No. 11 / 10 April 2015
The APSO has been implemented to design the LPWG with a pair of concatenated gratings to generate a variety of spectra. The results show that it is an efficient method to optimize the design parameters to achieve various target spectra including rectangular and asymmetric wavelength rejection spectra. The application of this method has been demonstrated by designing a pair of concatenated gratings in a four-layer planar waveguide geometry for the flattening of the ASE curve of the EDFA and the gain curve of the EDWA. The predefined target spectra such as the rectangular rejection-band spectrum, ASE flattening of the EDFA, and the EDWA gain flattening has been obtained by segmented cladding grating in our previous work [22]. In the present design we have optimized seven parameters; however, in the segmented cladding grating only six design parameters have been optimized. The design and fabrication complexity increases by increasing the number of design parameters. Although the number of controlling parameters increase in the present design, the fabrication of concatenated grating is easier than the segmented grating. Hence the proposed design is better due to ease of fabrication. The results obtained using the present design are better than the segmented cladding grating in case of rectangular filter and the ASE flattening of the EDFA and comparable in case of gain flattening of the EDWA. References 1. S. A. Vasiliev and O. I. Medvedkov, “Long-period refractive index fiber gratings: properties, applications and fabrication techniques,” Proc. SPIE 4083, 212–223 (2000). 2. A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. Sipe, “Long period fiber grating as band rejection filter,” J. Lightwave Technol. 14, 58–65 (1996). 3. M. Das and K. Thyagarajan, “Dispersion compensation in transmission using uniform long period fiber grating,” Opt. Commun. 190, 159–163 (2001). 4. M. Das and K. Thyagarajan, “Wavelength division multiplexing isolation filter using concatenated chirped long period gratings,” Opt. Commun. 197, 67–71 (2001). 5. A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, and C. R. Devidson, “Long period fiber grating based gain equalizers,” Opt. Lett. 21, 336–338 (1996). 6. A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, P. K. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
7. V. Bhatia, “Application of long period gratings to single and multi-parameters sensing,” Opt. Express 4, 457–466 (1999). 8. B. Lee, “Review of present status of optical fiber sensors,” Opt. Fiber Technol. 9, 57–79 (2003). 9. H. Hochreiner, M. Cada, and P. D. Wentzell, “Modeling the responses of a long-period fiber grating to ambient refractive index change in chemical sensing application,” J. Lightwave Technol. 26, 1986–1992 (2008). 10. G. Yin, S. Lou, Q. Li, and H. Zou, “Theory analysis of mode coupling in tilted long period fiber grating based on the full vector complex coupled mode theory,” Opt. Laser Technol. 48, 60–66 (2013). 11. K. S. Chiang and J. H. Peng, “Analysis of phase shifted long period fiber gratings,” IEEE Photon. Technol. Lett. 10, 1596–1598 (1998). 12. F. Y. M. Chan and K. S. Chiang, “Analysis of apodized phased shifted long period fiber gratings,” Opt. Commun. 244, 233–243 (2005). 13. B. Guan, A. Zhang, H. Tam, L. W. Chan, C. L. Choy, X. M. Tao, and M. S. Demokan, “Step-changed long period fiber gratings,” IEEE Photon. Technol. Lett. 14, 657–659 (2002). 14. V. Rastogi and K. S. Chiang, “Long period grating in planar optical waveguide,” Appl. Opt. 41, 6351–6355 (2002). 15. Q. Liu, K. S. Chiang, and V. Rastogi, “Analysis of corrugated long period gratings in the slab waveguide and their polarization dependence,” J. Lightwave Technol. 21, 3399–3405 (2003). 16. K. S. Chiang, C. K. Chow, Q. Liu, H. P. Chan, and K. P. Lor, “Band-rejection filter with widely tunable centre wavelength and contrast using metal long period grating on polymer waveguide,” IEEE Photon. Technol. Lett. 18, 1109–1111 (2006). 17. Y. M. Chu, K. S. Chiang, and Q. Liu, “Widely tunable optical band pass filter by use of polymer long-period waveguide gratings,” Appl. Opt. 45, 2755–2760 (2006). 18. K. S. Chiang, K. P. Lor, C. K. Chow, H. P. Chan, V. Rastogi, and Y. M. Chu, “Widely tunable long-period grating fabricated in polymer-clad ion-exchanged glass waveguide,” IEEE Photon. Technol. Lett. 15, 1094–1096 (2003). 19. Q. L. Kin, K. S. Chiang, and K. P. Lor, “Tailoring the transmission characteristics of polymer long-period waveguide gratings by UV irradiation,” IEEE Photon. Technol. Lett. 17, 2340–2342 (2005). 20. C. J. Brooks, G. L. Vossler, and K. A. Winick, “Integrated optics dispersion compensator that uses chirped gratings,” Opt. Lett. 20, 368–370 (1995). 21. M. S. Kwon and S. Y. Shin, “Spectral tailoring of cladding thickness control,” Opt. Commun. 250, 41–47 (2005). 22. Q. Liu and K. S. Chiang, “Design of long-period waveguide grating filter by control of waveguide cladding profile,” J. Lightwave Technol. 24, 3540–3546 (2006).
23. G. Semwal and V. Rastogi, “Design of application specific long period waveguide grating filters using adaptive particle swarm optimization algorithms,” J. Opt. 16, 015504 (2014). 24. H. Jiang, K. Xie, and Y. Wang, “Shooting algorithm and particle swarm optimization based Raman fiber amplifiers gain spectra design,” Opt. Commun. 283, 3348–3352 (2010). 25. H. Jiagn, K. Xie, and Y. Wang, “Pump scheme for gainflattened Raman fiber amplifiers using improved particle swarm optimization and modified shooting algorithm,” Opt. Express 18, 11033–11045 (2010). 26. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997). 27. A. Yariv, “Coupled mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973). 28. J. Kennedy and R. Eberhart, “Particle swarm optimization,” in Proceedings of IEEE International Conference on Neural Networks 4, 1995, pp. 1942–1948. 29. R. Eberhart and J. Kennedy, “A new optimizer using particle swarm theory,” in Proceedings of the 6th Symposium on Micro Machine Human Science (1995), pp. 39–43. 30. R. Hassan, B. Cohanim, and O. Weck, “A comparison of particle swarm optimization and the genetic algorithm,” in 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference 2005, pp. 1–13. 31. S. Das, A. Abraham, and A. Konar, “Particle swarm optimization and differential evolution algorithms: technical analysis, applications and hybridization perspectives,” in Advances of Computational Intelligence, (Springer, 2008) Vol. 38, pp. 1–38. 32. I. C. Trelea, “Particle swarm optimization algorithm: convergence analysis and parameters selection,” Info. Process. Lett. 85, 317–325 (2003). 33. Y. D. Valle, G. K. Venayagamoorthy, S. Mohgheghi, J. C. Hernandez, and R. G. Harley, “Particle swarm optimization: Basic concepts, variant and application in power systems,” IEEE Trans. Evol. Comput. 12, 171–194 (2008). 34. Z. H. Zhan, J. Xiao, J. Zhang, and W. N. Chen, “Adaptive control of acceleration coefficients for particle swarm optimization based on clustering analysis,” in IEEE Congress on Evolutionary Computation (IEEE, 2007), pp. 3276–3282. 35. Z. H. Zhan, J. Zhang, Y. Li, and H. S. Chung, “Adaptive particle swarm optimization,” IEEE Trans. Syst., Man, Cybern. B 39, 1362–1382 (2009). 36. T. Huang and A. S. Mohan, “A hybrid boundary condition for robust particle swarm optimization,” IEEE Antennas Wireless Propag. Lett. 4, 112–117 (2005). 37. S. Pal and B. R. Singh, “Analysis and Design of Corrugated long period gratings in silica-on-silicon planar waveguides,” J. Lightwave Technol. 25, 2260–2267 (2007).
10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS
3149
