Synthesis method based on optimization techniques ... - OSA Publishing

Bae et al.

Vol. 23, No. 7 / July 2006 / J. Opt. Soc. Am. B

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Synthesis method based on optimization techniques for designing piecewise-uniform long-period fiber gratings controlled by thermal changes Jinho Bae College of Ocean Science, Cheju National University, 1 Ara 1-dong, Jeju, Jeju-do 690-756, South Korea

Jun Kye Bae, Sang Hyuck Kim, and Sang Bae Lee Photonics Research Center, Korea Institute of Science and Technology, P.O. Box 131 Cheongryang, Seoul 131-650, South Korea

Joohwan Chun Scientific Computing Laboratory, Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology, 373-1 Kusong-dong, Yusong-gu, Taejon 305-701, South Korea

Namkyoo Park Optical Communication Systems Laboratory, School of Electrical Engineering and Computer Science, Seoul National University, Seoul 151-744, South Korea Received April 28, 2005; revised February 22, 2006; accepted February 24, 2006; posted March 8, 2006 (Doc. ID 61807) We propose a new method for synthesizing piecewise-uniform long-period fiber gratings (LPFGs) by using an extended fundamental matrix model with thermal changes. The proposed synthesis method is then applied to the design of the LPFG tuned by the thermal changes for erbium gain equalization by using the simulatedannealing and steepest-descent optimization techniques. We describe how a piecewise-uniform LPFG can be constructed by utilizing the inverted gain spectrum of erbium-doped fiber amplifiers (EDFAs), from the thermal change parameters’ search process. A sensitivity analysis also is done to study the tolerance of our approach against possible error sources, such as the temperature controller, the fabrication of the LPFG, and the EDFA spectrum, by using Monte Carlo simulations. To verify the validity of the proposed synthesis method experimentally, we manufactured the piecewise-uniform LPFG with thermal changes by using a divided coil heater. We observe that the spectrum designed by the proposed synthesis method is close to the corresponding measured spectrum in the wavelength band of interest. We also compare the performance of the proposed method with traditional approaches, such as Newton-like methods. © 2006 Optical Society of America OCIS codes: 060.2340, 050.2770, 060.2430, 230.3120.

1. INTRODUCTION It is a fact that single-mode long-period fiber gratings (LPFGs) have one fundamental core mode and multiple cladding modes that all propagate in the same (forward) direction. Because of their unique features, such as low insertion loss, low backreflection, and excellent polarization insensitivity, LPFGs have attracted great interest for optical telecommunication and sensor applications.1,2 Many researchers have studied various LPFGs, such as the gain equalizers of erbium-doped fiber amplifiers (EDFAs),3–6 wavelength-division-multiplexing isolation fiber filters,7 band rejection filters,1,8 and sensors.9 In particular, some of several scientists have presented temperature sensors9–11 and various tunable filters12–14 by using the temperature sensitivity of LPFGs. In the research on these subjects the physical phenomena of LPFGs with thermal changes10,15 and the manufacture of tunable LPFG filters12,14,16 have been discussed. To accu0740-3224/06/071241-9/$15.00

rately analyze and synthesize the LPFG with thermal changes, we propose the kernel function17 used to calculate the coupling coefficient and the detuning factor changed by temperature. The LPFG can usually be analyzed with the coupledmode equations18–20 and a fundamental matrix model19,21,22 that can be derived from the Maxwell equations. In these models the fundamental matrix model is generally used to analyze and synthesize the nonuniform LPFG.3,19 However, analyzing the LPFG with thermal changes by using these tools is very difficult, so we have proposed an extended fundamental matrix model using the proposed kernel function,17 that can synthesize a piecewise-uniform LPFG with the thermal changes. Synthesis methods for the LPFG have been reported, such as those using a sum of each Gaussian transmission spectrum concatenated by a LPFG,23 a piecewise-uniform LPFG designed by optimization methods,3,24 layer peeling © 2006 Optical Society of America

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methods proposed in the signal-processing fields,25 and Gel’fand–Levitan–Marchenko coupled equations.26 The designing parameters are coupling coefficients, indices, and grating periods for the synthesis of various LPFGs, with the assumption that LPFGs are always maintained at room temperature. Several researchers have manufactured a tunable filter by using the high-temperature sensitivity of a LPFG,12,14 but they have not proposed a synthesis method by using the design parameters for the thermal changes. We have presented a kernel function to translate information about the thermal changes into the coupling coefficients and the detuning factor.17 Thereafter we also proposed an extended fundamental matrix, considering the thermal changes,17 that is accurate and versatile. In this paper we present a way to use the proposed kernel function for the synthesis of a LPFG with thermal changes. The extended fundamental matrix model proposed in Ref. 17 is then used to design the piecewise-uniform LPFG with thermal changes for the inverted spectrum of an EDFA gain profile that uses the optimization method with the simulated-annealing and steepest-descent methods.27–29 The performance of the proposed synthesis method is verified by comparison with Newton-like methods. The sensitivity analysis of the designed structures is also performed by the Monte Carlo simulations.27 To verify the validity of the proposed method, we have manufactured the piecewise-uniform LPFG structure with thermal changes controlled by a divided coil heater for equalization of the nonflat EDFA spectrum, and the experimental results are compared with the theoretical spectrum.

2. EXTENDED FUNDAMENTAL MATRIX MODEL Consider the gratings that are grouped into M sections in such a way that the grating period ⌳i, the refractive index ni, and the absolutely controllable temperature Ti (in units of degrees Celsius) inside each of the M sections are uniform (see Fig. 1), where i = 1 , ¯ , M. The field amplitudes coming into and out of the ith section can be shown17,19,30 to be

冋册 冋 册 Ai

Bi

= Fi

Ai−1

Bi−1

,

共1兲

where Ai and Bi are the complex amplitudes of the fundamental core mode and the pth cladding mode, respectively. The 2 ⫻ 2 complex matrix Fi for the LPFG with codirectional interactions is given by

Fi =

冤

cos共⌼iLi兲 + j

j

␬ Ti ⌼i

␦ Ti ⌼i

sin共⌼iLi兲

sin共⌼iLi兲

␬ Ti ⌼i

sin共⌼iLi兲

cos共⌼iLi兲 − j

␦ Ti ⌼i

sin共⌼iLi兲

冥

,

共2兲

where Li is the section length and def

⌼i = 关共␬Ti兲2 + 共␦Ti兲2兴1/2 is the effective detuning, all for the ith section. The spectrum of an LPFG varies dominantly by the change in the index along the thermal changes.10,15 The coupling coefficient ␬Ti and the detuning factor ␦Ti, which were composed of the index changing with temperature, can be set at approximately17 1 ␬Ti ⬇ 关a共Ti兲X + b共Ti兲兴, ␭

共3兲

␦Ti ⬇ ␦ + c共Ti兲.

共4兲

Here X = ⌬nC is at room temperature (where ⌬n is the induced index change and C is the overall integral factor between the fundamental core mode and the pth cladding co cl − neff,p 兲 − ␲ / ⌳ is the detuning facmode19) and ␦ = ␲ / ␭共neff co is the effective index tor at room temperature (where neff cl of the core, neff,P is the effective index of the cladding for the pth mode, and ⌳ is the grating period). The a共Ti兲, b共Ti兲, and c共Ti兲 in Eqs. (3) and (4) are the functions of temperature and are shown as a共Ti兲
a0Ti

m1

+ a 1T i

m1−1

+ ¯ + am1−1Ti1 + am1 ,

b共Ti兲
b0Ti

m2

+ b 1T i

m2−1

+ ¯ + bm2−1Ti1 + bm2 ,

m3

+ c 1T i

m3−1

+ ¯ + cm3−1Ti1 + cm3 .

c共Ti兲
c0Ti

共5兲

To precisely calculate the coupling coefficients and the detuning factor on the thermal changes, we have designated Eqs. (3) and (4) as the kernel functions.17 A detailed description and derivation of these quantities can be found in Ref. 17. To find a共Ti兲, b共Ti兲, and c共Ti兲 in Eq. (5), we measure the transmission spectra Dq 共q = 1 , 2 , ¯ , 11兲 of the LP05 mode of a single uniform LPFG with a total length of L = 4 cm (grating period, ⌳ = 421.15 ␮m; total number of gratings, 95) along the sampled thermal changes as shown in Fig. 2(a). The spectra are for a single section of F1 only as shown in Fig. 1, where the sampled thermal changes are Ti = T1 = 兵24.9, 35, 45, 55, 65.3, 75, 85, 95.2, 105.3, 115, 125.1其 ° C. The parameters of the fiber used in this paper are as follows: nco = 1.45248, rco = 2.815␮m,

Fig. 1. Block diagram of the extended fundamental matrix model for LPFGs with thermal changes.

j

ncl = 1.44532, rcl = 62.51␮m,

nair = 1, 共6兲

where ncl is the refractive index of the cladding, nair is the refractive index of air, rco is the radius of the core, and rcl is the radius of the cladding. The fiber used to fabricate

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Fig. 2. (Color online) (a) Transmission spectra with thermal changes; (b) X and ␦ 共T1 = 24.9 ° C兲.

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the LPFG is a boron-codoped germanosilicate fiber that has a thermal tuning efficiency that can be significantly enhanced compared with the standard telecommunication fiber in terms of the temperature sensitivity of the LPFG.31 The X in approximation (3) and ␦ in approximation (4) [in the case where T1 = 24.9 ° C, a共T1兲 = 1, b共T1兲 = 0, and c共T1兲 = 0] for D1 as shown in Fig. 2(b) can be computed from ⌳ = 421.15 ␮m, ⌬n = 0.0001652, and the fiber parameters in Eq. (6), where ⌬n has been experimentally determined by fitting the transmission spectrum in Fig. 2(a) by using the simulated-annealing method.28,29 The ␬T1 and ␦T1 for the measured spectra 共D2 , ¯ , D11兲 with the thermal changes in Fig. 2(a) can be calculated by finding the a共T1兲, b共T1兲, and c共T1兲 (rectangles) as shown in Figs. 3(a), 3(c), and 3(e), respectively. The a共T1兲, b共T1兲, and c共T1兲 are determined by fitting the measured transmission spectra Dq 共q = 1 , ¯ , 11兲 as solid and dotted curves in Fig. 2(a) by using the simulated-annealing algorithm. The coefficients am1 共m1 = 0 , 1 , ¯ , 25兲, bm2 共m2 = 0 , 1 , ¯ , 15兲, and cm3 共m3 = 0 , 1 , ¯ , 23兲 in Figs. 3(b), 3(d), and 3(f) for a共T1兲, b共T1兲, and c共T1兲 in Eqs. (5) can be determined from the rectangles in Figs. 3(a), 3(c), and 3(e) by the following steps: (i) To obtain the exact coefficients, we interpolate the dotted curves by using the cubic spline data interpolation27,32 between the rectangles in Figs. 3(a), 3(c), and 3(e). (ii) We find the coefficients as shown in Figs. 3(b), 3(d), and 3(f) from an optimal polynomial curve fitted with dashed curves [see Figs. 3(a), 3(c), and 3(e)] by using a least-squares method.27,32 In this paper we consider the LP05 of the uniform LPFG, but the kernel func-

Fig. 3. (Color online) (a) The a共T1兲 with thermal changes. [For the detailed display, we subtracted 0.9999 from a共T1兲.] (b) Attained coefficients for a共T1兲. (c) The b共T1兲 with thermal changes. (d) Attained coefficients for b共T1兲. (e) The c共T1兲 with thermal changes. (f) Attained coefficients for c共T1兲.

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tion for analyzing the LPFG with the other mode33 has to be calculated from the measured spectra of that mode. The polynomial orders m1, m2, and m3 for a共T1兲, b共T1兲, and c共T1兲, respectively, are optimally determined to match the spectra in Fig. 2(a) exactly. They are used to accurately apply the experimental results of the spectra varying along the thermal changes. The number of the measured spectra is numerically determined along the complexity of the spectra with thermal change. The overall field amplitudes of the entire structure with M uniform sections as shown in Fig. 1 are then

冋册 冋册 AM

A0

=F

BM

B0

F
FMFM−1 ¯ F1


冋

,

Fig. 4.

F11 F12 F21 F22

册

.

共7兲

The overall transmission coefficient for the LPFGs, t
AM / A0 with B0 = 0, is easily seen to be t = F1,1, which is the (1, 1) element of the extended fundamental matrix.

3. SYNTHESIS METHOD USING OPTIMIZATION TECHNIQUES The desired wavelength response of the concatenated LPFGs with thermal changes is the reciprocal of an EDFA gain spectrum over the desired wavelength range. We consider three design parameter vectors ⌰: (i) ⌰1 = 共T1 , T2 , ¯ , TM兲; (ii) ⌰2 = 共⌳1 , ⌳2 , ¯ , ⌳M⌳ , T1 , T2 , ¯ , TM兲, 共M⌳ 艋 M兲; (iii) ⌰3 = 共⌬n1 , ¯ ,⌬nMn , ⌳1 , ¯ , ⌳M⌳ , T1 , ¯ , TM兲, (M⌳ 艋 M and Mn 艋 M). With these vectors ⌬n and ⌳ are used to update X and ␦ in Eqs. (3) and (4), respectively. We select a discrete set of P wavelength points, ␭1 , ¯ , ␭P, where ␭P is given in fractions of the Nyquist rate, and call the magnitude of the desired wavelength response at these wavelength points Ypd. The objective is to obtain ⌰ so that it minimizes the cost function V共⌰兲:

Flow chart for the proposed synthesis method.

using a normally random number in 关−1 , 1兴, converging to the global minimum. The algorithm is an iterative random move along each coordinate direction from an updated point 关⌰共i兲兴. It accepts uphill moves according to a probabilistic criterion (the Metropolis criterion29) to escape the local minimum. After the simulated-annealing algorithm finds a point close enough to the global minimum, we switch to the steepest-descent algorithm to accelerate the iteration process without danger of the algorithm becoming trapped in a local minimum. The steepest-descent algorithm performs a one-dimensional minimization (the line search) along the direction determined by the gradient of the cost function V共⌰兲.27 The required gradient of V共⌰兲 can be computed by3

⳵V共⌰兲 ⳵␪k

P

=2

兺 w 共兩Y共␭ ,⌰兲兩 − Y p

p

d p

兲

⳵兩Y共␭p,⌰兲兩

p=1

⳵␪k

where k = 1 , 2 , ¯ , M + Mn + M⌳, and

⳵兩Y共␭p,⌰兲兩 ⳵␪k

1 =

兩Y共␭p,⌰兲兩

冋

Re Y*共␭p,⌰兲

⳵Y共␭p,⌰兲 ⳵␪k

共9兲

,

册

.

P

V共⌰兲 =

兺 w 共兩Y共␭ ,⌰兲兩 − Y p

p

p

d 2

兲 ,

共8兲

p=1

where wp 共0 ⬍ wp 艋 1兲 is the weight factor for an exact synthesis in local wavelength scopes and Y共␭p , ⌰兲 = F1,1. The sum of squares in Eq. (8) is nonlinear with respect to ⌰; therefore we have to rely on an iterative procedure to determine the optimal ⌰. Our choice for the iterative procedure is the simulated-annealing algorithm followed by the steepest-descent algorithm depicted on the flow chart (see Fig. 4). In this procedure we set N Ⰷ K because the simulated-annealing algorithm is faster than the steepest-descent algorithm and set ⑀1 ⬎ ⑀2 to obtain the exact design parameters. Here N and K are the criterion numbers for terminating the simulated-annealing and the steepest-descent algorithms, respectively, by constraint of iteration, and ⑀1 and ⑀2 are terminating criteria for the cost of stops the simulated-annealing and the steepest-descent algorithms, respectively. The simulatedannealing algorithm can escape from a local minimum in search of the global minimum.27–29 Starting from an initial point ⌰共1兲, the simulated-annealing algorithm iteratively generates a succession of points 关⌰共1兲 , ⌰共2兲 , ¯ 兴 by

A. Comparison of the Proposed and the Newton-like Methods To show the performance of the proposed methods,34 we have compared them with Newtonlike methods as shown in Fig. 5. In Fig. 5 the cost function V共⌰兲 = x2 + 2y2 − 0.3 cos共3␲x兲 − 0.4 cos共3␲x兲 + 0.3+ 0.4, where the local minima and a global minimum is used, where the parameter vector is ⌰ = 共x , y兲 and the global minimum point is (0,0). The performance of the proposed method is shown in Fig. 5(a) for the initial start point (0.8, 0.8); 150 simulated-annealing and 10 steepest-descent iterations were taken. We can see that the proposed method finds the global minimum area exactly through the random moves (circles) of simulated annealing as shown in Fig. 5(a). After the simulated annealing finds the global minimum area, the steepest descent accelerates the iteration process to accurately find an optimal minimum (cross marks) as shown in Fig. 5(a). Figure 5(b) shows the value of V共⌰兲 for the convergence of the proposed algorithm. The performances of Broyden (1970), Fletcher (1970), Goldfard (1970), and Shanno34 (1970) (BFGS) and the steepest-descent algorithms are illustrated in Fig. 5(c) for

Bae et al.

the initial start point (0.8, 0.8). After BFGS and the steepest descent rapidly converged at a local minimum, two Newton-like methods could not find the global minimum during a total of 150 iterations as shown in Fig. 5(c). The costs for BFGS and the steepest descent are shown in Figs. 5(d) and 5(e), respectively.

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B. Theoretical Example 1. EDFA Gain Flattening End Filter We have obtained ⌰ = ⌰2 = 共⌳1 , ¯ , ⌳M⌳ , T1 , ¯ , TM兲 for the concatenated LPFG with 25 sections, where M⌳ = 2, M = 25, each section from Section1 to Section15 contains

Fig. 5. (Color online) (a) Convergence behavior of the proposed optimization method: SA, simulated annealing; SD, steepest decent. (b) The V共⌰兲 value. (c) Convergence behavior of BFGS and SD. (d) Cost of BFGS. (e) Cost of SD.

Fig. 6. (Color online) (a) X at 24.9 ° C for ⌬n = 0.00019; (b) ␦ at 24.9 ° C; (c) attained transmission spectrum curve; (d) corresponding thermal changes (attained periods are ⌳1 = 433.56 ␮m and ⌳2 = 446.9 ␮m).

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Monte Carlo simulations27 of 500 runs are used. Because the grating periods in ⌰ are more exactly controlled, the errors in temperatures in ⌰ are more easily propagated in each section. Now assume that the perturbed parameter ˜ = 共⌳ , ⌳ , T + ⑀ , T + ⑀ , ¯ , T + ⑀ 兲, where the vector ⌰ 1 2 1 1 2 2 25 25 thermal control error ⑀k is generated in bounds, −1.5 ° C ⬍ ⑀k ⬍ 1.5 ° C 共k = 1 , 2 , ¯ , 25兲, when normally distributed random numbers are used. The maximum difference values of the power variations between the designed spectrum for unperturbed thermal changes in Fig. 5(c) and the spectrum for perturbed thermal changes along the wavelength (the maximum power variation, which is important in deciding the performance of gain-flattening filters) calculated by the Monte Carlo simulations are less than 0.8 dB as shown in the histograms in Fig. 7(b).

4. EXPERIMENTAL RESULTS

Fig. 7. (Color online) (a) Transmission spectra calculated by the Monte Carlo simulations; (b) histogram for the maximum power variation.

eight gratings with the attained grating period, ⌳1 = 433.56 ␮m, and each section from Section16 to Section25 contains seven gratings with the attained grating period ⌳2 = 446.9 ␮m. The X at room temperature with ⌬n = 0.00019 is calculated by using the fiber parameters in Eq. (6) as shown in Fig. 6(a). Figure 6(b) shows ␦ for the attained grating periods 433.56 and 446.9 ␮m by using the synthesis method and ␦ for the initial graing periods 436 and 425 ␮m. The coefficients of a共T1兲, b共T1兲, and c共T1兲 are the same as those in Fig. 3. The desired wavelength response curve is sampled at 46 uniformly spaced points over the 1525– 1570 nm range in Fig. 6(c) (circles). Figure 6(d) represents the thermal change profile with the design parameter ⌰. The wavelength response corresponding to the initial vector ⌰共1兲 = 共436⫻ 10−6 , 425⫻ 10−6, 24.9, 24.9, 24.9, 24.9, 24.9, 24.9, 24.9, 24.9, 24.9, 24.9, 24.9, 24.9, 24.9, 24.9, 24.9, 24.9, 24.9, 24.9, 24.9, 24.9, 24.9, 24.9, 24.9, 24.9, 24.9) is depicted by dashed curves in Fig. 6(c). The initial cost is V关⌰共1兲兴 = 1.1882. The optimized minimum cost is V关⌰共⬁兲兴 = 0.0108 (after 1000 simulatedannealing and 25 steepest-descent iterations). The attained wavelength responses are shown in Fig. 6(c) by solid curves. The total length of the entire concatenated LPFGs is 8.331 cm. 2. Sensitivity in Terms of Thermal Changes Monte Carlo methods use stochastic techniques based on random numbers and probability statistics to investigate problems. The thin solid curves in Fig. 7(a) show the transmission spectrum with perturbed parameters when

The experimental methods for the tunable optical fiber gratings with thermal changes that use thin-film heaters35–37 and divided NiCr coil heaters12,14,17 are presented. We have used an experimental setup using the divided NiCr coil heater proposed in Ref. 17, which has approximately a thermal control error of −1.5 ° C ⬍ ⑀k ⬍ 1.5 ° C. We have obtained parameter vectors of ⌰ = ⌰1 = 共T1 , ¯ , TM兲 for the LPFGs with 15 sections 共M = 15兲 equalizing the EDFA gain spectrum. Each section contains six fixed grating periods, ⌳ = 421.15 ␮m. Figure 8(c) shows the X and ␦ in Eqs. (3) and (4). The coefficients of a共T1兲, b共T1兲, and c共T1兲 are shown in Fig. 3, because the parameters of the fiber used are the same. The desired wavelength response curve is sampled at 72 uniformly spaced points over the 1524.1– 1559.6 nm range in Fig. 8(a) (circles). The wavelength responses corresponding to an initial ⌰共1兲 are depicted by a dash–dotted curve in Fig. 8(a) to show each spectrum of a single uniform LPFG at a room temperature of 24.9 ° C. The initial cost is V关⌰共1兲兴 = 9.335. The converged minimum cost is V关⌰共⬁兲兴 = 0.092 (after 1000 simulated-annealing and five steepest-descent iterations). The parameter vector obtained with this procedure is shown in Fig. 8(b). The attained wavelength responses are shown in Fig. 8(a) by solid curves. The total length of the entire concatenated LPFGs is 3.79 cm. To manufacture the gain equalized filter by using the design parameters as shown in Fig. 8(b), we fabricated a single uniform LPFG of 90 gratings with ⌳ = 421.15 ␮m to fit the dotted curve measured in Fig. 8(a). Thereafter we manufactured the gain equalized filters to fit the dashed curve in Fig. 8(a) by tuning the fabricated LPFG by using the designed thermal changes as shown in Fig. 8(b). We can see that the measured and the calculated spectrum curves [see Fig. 8(a)] show a good match in the desired wavelength band, except for a distorted region from a thermal control error in our experimental setup. We show the sensitivity in terms of the fabrication error as shown in Fig. 9, because X and ␦ depend on the fabrication error of a single uniform LPFG used for the initial spectrum shown as squares in Fig. 9(a). The thin solid curves in Fig. 9(a) show transmission spectra with perturbed parameters when the Monte Carlo simulations have 100 runs. We assume that X and ␦ are perturbed as

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˜ = X + ⑀ and ˜␦ = ␦ + ⑀ , respectively, where 10−5 ⬍ ⑀ X X ␦ X ⬍ 10−5 and −5 ⬍ ⑀␦ ⬍ 5 are generated with normally distributed random numbers. The error bounds of ⑀X and ⑀␦ are determined by rule of thumb experimentally. The maximum values of the power variations are less than 25 dB for the initial spectrum [see Fig. 9(b)] and 6 dB for the ideal inverse EDFA gain spectrum [see Fig. 9(c)]. From the results in Fig. 9 we can see that the fabrication error in the initial spectrum is the dominant factor when the filters are manufactured. From this analysis the fabrication error from the manufactured piecewise-uniform LPFG can almost be eliminated, if X and ␦ calculated from the spectrum of the fabricated single uniform LPFG are used in an initial vector ⌰共1兲. Considering the case of parameter vectors ⌰2 and ⌰3, to manufacture the exact filter, the fabricated initial spectrum must be the same as the spectrum calculated by the attained induced index changes and the grating periods in the design parameter vectors.

Fig. 9. (a) Monte Carlo simulations for ⑀X and ⑀␦; (b) histogram of the maximum power variation for the initial spectrum; (c) histogram of the maximum power variation for the fitted spectrum.

Fig. 8. (a) Calculated and attained transmission spectrum curves; (b) corresponding thermal changes with 15 sections; (c) X and ␦ at 24.9 ° C 共⌬n = 0.000197兲.

From this experiment we can see that the manufactured EDFA gain equalized filter shows notable results when compared with those of passive EDFA gain equalized filters5,24 or the acousto-optic filter (AOTF).38 The proposed experimental setup can dynamically adjust the spectrum to an ideal shape through thermal control, whereas a passive EDFA gain equalized filter manufactured by a nonuniform LPFG (Ref. 24) or a phase-shifted LPFG (Ref. 5) can only approach the ideal spectrum. In particular, this experiment shows consistent and accurate results its ability to dynamically control the spectrum, when compared with the case of passive EDFA gain equalized filters using the nonuniform LPFG in Ref. 24. The dynamic gain equalizer of the EDFA using a LiNbO3 integrated AOTF becomes a serious limitation because of the coherent cross-talk problem and excessive insertion loss.38 To overcome such problems, an all-fiber AOTF is demonstrated in Ref. 38. An AOTF is necessary with a

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15.5 cm long interaction length to obtain the band-reject filter of the dip of more than −15 dB.38 The filter size proposed in this paper is shorter than that proposed in Ref. 38, because our band-reject filter requires only 3.79 cm, as seen from the dashed curve in Fig. 8(a).

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10.

11.

5. CONCLUSIONS To calculate a transmission spectrum changed by temperature, the proposed kernel function can be used between 24.9 and 125.1 ° C. We have shown how the proposed analysis method with the kernel function can be applied to synthesizing piecewise-uniform LPFGs with thermal changes. We have determined the design parameters for manufacturing tunable LPFG filters with thermal changes by using the simulated-annealing and steepest-descent optimization techniques with the kernel function. Using the design parameters, obtained we have manufactured the gain equalized filters tuned by thermal changes from a divided NiCr coil heater. We have also shown that the measured spectrum of the manufactured gain flattening filter when the proposed synthesis method is used is experimentally well matched to the desired wavelength response. To exactly manufacture the piecewise-uniform LPFG with thermal changes, we have checked the sensitivity in terms of ⑀X, ⑀␦ by using the Monte Carlo simulations. From this result we have learned that the fabrication error of the initial spectrum is a dominant factor compared with the control error of the thermal changes. E-mail for J. Bae, [email protected].

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