Multi-objective optimization of dielectric layer photonic crystal filter

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1 Optic Applict, Vol. XLVII, No. 1, 017 DOI: /o Multi-objective optimiztion of dielectric lyer photonic crystl filter HONGWEI YANG *, CUIYING HUANG, SHANSHAN MENG College of Applied Sciences, Beijing University of Technology, Beijing 10014, Chin * Corresponding uthor: ynghongwei@bjut.edu.cn The weighting fctors method nd the response surfce methodology re used to chieve multi-objective optimiztion of dielectric lyer photonic crystl filter. The size of period nd the trnsmission quntity re considered simultneously nd multi-objective optimiztion model of filter is estblished, which tkes the size of period nd trnsmission quntity to be minimized in stop-bnd s objectives. Globl pproximte expressions of the objective nd the constrint functions re found by response surfce methodology. Then the weighting fctors method is employed to convert the model into qudrtic progrmming model nd the optiml prmeters cn be obtined using sequence qudrtic progrmming. Exmples provide the optimized results in three different weight coefficients. The effect of the weighting fctors on the vlue of the objective function is lso discussed. Results show tht the present method is precise nd efficient for multi-objective optimiztion of dielectric lyer photonic crystl filter. Keywords: filter, photonic crystl, weighting fctors method, response surfce methodology (RSM). 1. Introduction In recent yers, growing demnd cn be observed for dimension nd chrcteristic of filter with the rpid development of microwve techniques. Photonic crystls re periodiclly lyered structures tht re filled with different dielectric mterils nd it is well-known tht they hve specil spectrl structure, the so-clled photonic bnd gp (PBG). This feture cn be employed to design opticl filters. The design of photonic crystl filter hs lredy been undertken by number of reserch works. In [1], the prticle swrm optimiztion method nd the finite-difference time-domin method were used to improve the performnce of two-dimensionl photonic crystl filter. A fbriction process of tunble PBG filter tht cn be tuned in very wide rnge of the centrl pss-bnd wvelength shifting is designed nd simulted in []. In ddition, the optiml design of the dielectric lyer photonic crystl filter using the response surfce methodology is described in [3]. For the works men-

2 30 HONGWEI YANG et l. tioned bove, there re single objective reserches becuse only the property of the filters is considered. However, most rel-world optimiztion problems tht exist in prcticl engineering nd scientific pplictions will be requested to optimize more thn one objective. For the filter, the dimension nd the chrcteristics should be considered to be eqully importnt. In this pper, multi-objective optimiztion model of the photonic crystl filter is proposed, nd the weighting fctors method nd the response surfce methodology (RSM) to solve this model re introduced. In contrst to single objective problem, multi-objective problem is more difficult to solve becuse it hs set of solutions, clled the Preto-optiml set, but there is no limit to n optiml solution. Mny methods for multi-objective optimiztion hve been put forwrd nd hve shown gret progress nd success. The weighting fctors method is the most commonly used technique nd its bsic ide is to trnsform the multi-objective problem into the single objective problem [4]. With the weighting fctors method, we cn issue comprehensive quntittive nlysis of ims nd seek the best vlue to meet the system requirements. RSM stemmed from experimentl design nd ws lter introduced into numericl simultion in relibility ssessment of complex multivrible systems [5, 6]. The bsic ide of RSM is to pproximte the ctul stte function, which my be implicit or very time-consuming to evlute, with the so-clled response surfce function tht is esier to del with complex problems. To construct pproximte model with RSM, no sensitivity nlysis is required, nd thus it is more pplicble to problems with sensitivity difficulty. Besides, response surfce construction involves no informtion inside structurl nlysis procedure. For further reding bout RSM, see [7]. In this pper, we use the weighting fctors method nd the qudrtic RSM to chieve multi-objective optimiztion of the photonic crystl filter. A multi-objective optimiztion model of the filter is estblished first, which tkes the size of period nd trnsmission quntity to be minimized in stop-bnd s objectives. The weighting fctors method is employed to merge two gols into single trget. Then globl pproximte expressions of the objective nd the constrint functions re found by qudrtic RSM. Finlly, the model is converted into qudrtic progrmming model nd the optiml prmeters cn be obtined using sequence qudrtic progrmming. Exmples show its precision nd efficiency.. Bring forwrd the control model Dielectric lyer photonic crystl filter structures in wveguide re shown in Fig. 1. The periodic length, the dielectric thickness d, the reltive permittivity of the dielectric re the three mjor fctors in determining the stop-bnd chrcteristic of the wveguide dielectric lyer photonic crystl structures [8]. As is known to ll, the less the trnsmission quntity in stop-bnd is nd the more trnsmission coefficient beyond stop-bnd is, the better the property of the filter is. When the width of stop-bnd is

3 Multi-objective optimiztion of dielectric lyer d Dielectric Air Fig. 1. Dielectric lyer photonic crystl filter structures in wveguide. fixed, we hope the re surrounded by the trnsmission coefficient curve nd horizontl xis (frequency xis) should be mximum. Here we define (in the stop-bnd) f A sb = ( S 1 )d f (1) s the negtive of trnsmission quntity in the stop-bnd, where S 1 represents the trnsmission coefficient of filter during optimiztion process. Let N sb = A sb, nd thus the mximum vlue problem cn be converted to minimum vlue serching problem. So the less N sb is, the less trnsmission quntity is. In this study, we tke the size of the period nd the trnsmission quntity to be minimized in the stop-bnd s objectives. Estblishing the control model is s follows: Find:, d, Minimize: N sb = A sb = S 1 d f Subject to: d d d d/ k A L A R = = f f 1 f 4 f 3 f 3 f ( S 1 )d f ( S 1 )d f T L T R ()

4 3 HONGWEI YANG et l. where periodic length, dielectric thickness d, reltive permittivity of the dielectric re the design vribles; f nd f 3 re the lower nd upper bounds of the stop-bnd, nd it is obvious tht the bndwidth of the stop-bnd is between f nd f 3 ; f 1 is the lower bound of the concerned bnd on the left of the stop-bnd, f 4 is the upper bound of the concerned bnd on the right of the stop-bnd; A L nd A R re the negtive of trnsmission quntities t corresponding regions; T L nd T R nd re permitted mximum of trnsmission quntity s negtive t corresponding regions.,, d, d,, nd re the lower nd upper bound on the design vribles, d, nd, respectively; k is positive number less 1 becuse d is less thn. Since N sb is negtive, we define B sb = 1/N sb =1/A sb to trnsform the initil problem to the problem for serching positive minimum. Nevertheless, s dimension nd order of mgnitude of the objective functions B sb nd re incomprble to ech other, we normlized them by [B sb ] nd [] which re the estimted verge of B sb nd, respectively. Nmely we tke two dimensionless vlues B sb /[B sb ] nd /[] s objectives simultneously. After merging two gols into single trget by the weighting fctors method, new control model hs been set up s: Find:, d, Minimize: G = α 1 / [ ] + α B sb /[ B sb ] Subject to: d d d d/ k A L A R ( S 1 )d f where α 1 nd α re the weight coefficients of periodic length nd trnsmission quntity in stop-bnd, respectively; α 1 nd α re positive numbers less 1, besides, α 1 + α =1. It is very difficult to deduce n explicit expression of the objective function G with design vrible, d, nd becuse of the strong nonliner chrcteristics of the problem. Fortuntely we cn modify the originl function (3) to n pproximte one nd mke the optimiztion bsed on the pproximte expression. In this study, such pproximtions cn be crried out by RSM. 3. Response surfce methodology = = f f 1 f 4 f 3 ( S 1 )d f T L T R For objective function, the response surfce generlly tkes qudrtic polynomil form. Higher order polynomils generlly re not used for conceptul reson ( com- (3)

5 Multi-objective optimiztion of dielectric lyer puttionl one). In this pper, we use qudrtic form contining the crossing terms. Considering the full qudrtic polynomil form, the response estimted eqution for three designing vribles is given by ỹ = β 0 + β 1 + β + β 3 + β 4 + β 5 + β 6 + β 7 + β 8 + β 9 (4) where β 0, β 1,..., β 9 re 10 coefficients to be determined, nd,, represent, d, nd, respectively. In order to determine ll bets, we should select m (m 10) experimentl points. Putting the coordintes of m experimentl points into Eq. (4), we cn get m estimted response vlues ỹ i = β 0 + β 1 x i1 + β x i + β 3 x i3 + β 4 x i1 + β 5 x i + β 6 x i3 + β 7 x i1 x i + β 8 x i1 x i3 + β 9 x i x i3 (5) where i = 1,..., m, nd x i1, x i, x i3 represent, d, nd of the i-th experimentl point, respectively. In fct, we cn lso get ctul vlues of m experimentl points, represented by y i (i = 1,..., m). Define error ε =(ε 1, ε,..., ε m ) T between the ctul nd the estimted responses, ε i = ỹ i y i, i = 1,..., m (6) Using the lest squre technique, nd minimizing the residul error mesured by the sum of squre devitions between the ctul nd the estimted responses, we hve Let m m S = ε i = 1 i = ( ỹ i y i ) min i = 1 (7) S = 0, β j j = 0,..., 9 (8) Eqution (8) is system of 10 liner equtions with 10 unknowns. Solving Eq. (8), we cn find ll bets nd obtin the qudrtic response function y = β 0 + β 1 + β + β 3 + β 4 + β 5 + β 6 + β 7 + β 8 + β 9 (9) Eqution (9) is the ctul qudrtic response function nd β 0, β 1,..., β 9 re determined. The sequentil qudrtic progrmming is used to obtin the optimum. In the optimiztion process, suppose (l = 1,, 3) is the present designed point of l-th ( ) vrible v

6 34 HONGWEI YANG et l. for l-th vrible. The expres- in v-th itertion, nd specify rtificilly step size sions of move limits re: ( v) ( v) ( v) Δ l ( ) x ( v ) ( v) Δ l l = nd v x = + (l = 1,, 3) (10) l where ( v) x nd ( v) l x represent the lower nd upper bound respectively. The intervl of l l-th designed vrible is ( ), in v-th itertion. v ( v) Furthermore, to improve numericl stbility, it is good prctice to scle ll vribles so tht ech vrible chnges in the rnge [ 1, 1] [9]. Let ζ l (l = 1,, 3), represent the normlized vribles. The trnsformtion formul is s follows [10]: ( v) Δ l ( + x) ζ l = (l = 1,, 3) (11) After the optimiztion, we cn return to initil design vribles nd get their vlue by following trnsformtion: ( )ζ l + = (l = 1,, 3) (1) The choice of the experimentl design cn hve lrge influence on the ccurcy of the pproximtion nd the cost of constructing the response surfce. For qudrtic response models, the centrl composite design (CCD) is n ttrctive lterntive [11]. There re 15 experimentl points in CCD method for three designing vribles, where 8 points re t vertices of qudrilterl, 6 re long the three symmetry xis, nd one is t the center. Figure shows n exmple of CCD for objective response surfce. b Fig.. Design of experiments for objective () nd constrint (b) response surfce.

7 Multi-objective optimiztion of dielectric lyer In the pper, this method is used to choose the experimentl design. This mens tht 15 experiment points (m = 15) re chosen to determine the vlue of bets. After three designed vribles re normlized, in terms of the coordintes the corners of the cube re ( 1, 1, 1), (1, 1, 1), (1, 1, 1), ( 1, 1, 1), ( 1, 1, 1), (1, 1, 1), (1, 1, 1), ( 1, 1, 1); the center point is (0, 0, 0). According to [5], the distnce between xil point nd center point is 1.15, so the xil points re t ( 1.15, 0, 0), (1.15, 0, 0), (0, 1.15, 0), (0, 1.15, 0), (0, 0, 1.15), (0, 0, 1.15). For constrint functions, the response surfces re constructed t the sme vlue of the selected designing prmeters. In this pper, the number of the selection of points for the constrint response is 7 for three vribles. Of which, 6 re symmetricl distribution on the xis nd one is t the center. Figure b shows n exmple of design of experiments for constrint response surfce. 4. The control model used for solving Bsed on the bove discussion, the control model used for solving cn be obtined s follows: Find:, d, Minimize: G = x T Hx/ + f T x Subject to: ( v) v v where x =(, d, ) T, series of coefficient mtrices H, f, A L, B L, A R, B R re obtined by RSM when objective nd constrint functions re pproximtely explicted. This qudrtic progrmming model is solved using qudrtic progrmming nd the optiml prmeters cn be obtined. 5. Numericl results ( ) ( ) d v d/ k v ( ) ( ) v ( ) T A L x + BL T T A R x + BR T L R For dielectric lyer photonic crystl filter structures in wveguide s shown in Fig. 1, the center frequency stop-bnd of this filter is designed t 6 GHz nd the bndwidth (13)

8 36 HONGWEI YANG et l. is GHz. The width nd height of the wveguide re 57 nd 3 mm, respectively. Let the number of the wveguide period be 9 in this pper. Choosing f 1 = 3 GHz, f = 5 GHz, f 3 = 7 GHz, f 4 = 9 GHz, T L = T R =.5, = 0.00 mm, = 100 mm, d = mm, d = 100 mm, = 1.1, = 10, k = 0,9, [B sb ] = 1/40, [] = 0. Three selections of weight coefficients re discussed s follows: α 1 =0 nd α =1; α 1 =0.5 nd α =0.5; α 1 = 0.7 nd α = 0.3. We set up the initil design vribles ccording to the estimted eqution [8] ε e = ε d + 1 r d λ ε e λ g = ( λ/λ c ) = λ g / where ε e is the effective permittivity, λ nd λ g re the wvelength corresponding to the center frequency of the stop-bnd in the vcuum nd wveguide, respectively, λ c is the cutoff wvelength of TE 10 mode in the rectngulr wveguide. Here, ccording to Eq. (14), we choose the initil design vribles s =0mm, d = 15 mm nd = Scenrio 1 In the cse of α 1 = 0 nd α = 1, the multi-objective optimiztion is trnsformed into the single objective optimiztion, which is problem of serching the minimum vlue of the trnsmission quntity in stop-bnd. The optimiztion process is convergent nd stble, which cn be clerly seen in Fig. 3. We see tht the objective function vlue G decreses rpidly t the beginning, fter 7 itertions the vlue strts to converge nd fter 16 itertions the vlue keeps constnt t bout Here we obtin the minimum vlue of the trnsmission quntity, which is bout When the function vlue G converges, the periodic length nd the reltive permittivity of the dielectric hve trend of slow increse nd the dielectric thickness d is still t rte of little decrese, s shown in Fig. 3b. After 16 itertions, we obtin the optimized function vlues, where, d nd re.1 mm, 6.71 mm, nd.80, respectively. Stop-bnd chrcteristics before optimiztion nd fter optimiztion re given in Fig. 3c. It is obvious tht, before optimiztion, the stop-bnd is not deep nd wide, minimum vlue of the trnsmission coefficient nd minimum periodic length re 15 db nd 0 mm, respectively. After optimiztion, the center frequency stop-bnd of this filter is 6 GHz nd the bndwidth is GHz, minimum vlue of the trnsmission coefficient is nerly 39 db nd minimum periodic length is.1 mm. The optiml design is crried out. 5.. Scenrio All the dt re the sme s the scenrio 1 except α 1 =0.5 nd α = 0.5. The optimiztion process is given in Fig. 4 when α 1 = 0.5 nd α = 0.5. Clerly, Figs. 4 nd 4b (14)

9 Multi-objective optimiztion of dielectric lyer Itertion number Object function 0 d Itertion number b After optimiztion Before optimiztion Design vribles [mm] Frequency f [GHz] 1 c Trnsmission coefficient S1 [db] Fig. 3. The optimiztion process nd results for scenrio 1: object function versus itertion numbers when α 1 =0 (), design vribles versus itertion numbers when α 1 =0 (b), nd stop-bnd chrcteristic before nd fter optimiztion when α 1 =0 (c) Object function Itertion number Itertion number d b After optimiztion Before optimiztion Design vribles [mm] Frequency f [GHz] 1 c Trnsmission coefficient S1 [db] Fig. 4. The optimiztion process nd results for scenrio : object function versus itertion numbers when α 1 =0.5 (), design vribles versus itertion numbers when α 1 = 0.5 (b), nd stop-bnd chrcteristic before nd fter optimiztion when α 1 = 0.5 (c).

10 38 HONGWEI YANG et l. show the objective function vlue G is convergent t bout 0.86, nd, d nd re bout 1.15 mm, 8.4 mm, nd.69, respectively, fter 14 itertions. Figure 4c gives the stop-bnd chrcteristic before optimiztion nd fter optimiztion, nd show fter optimiztion tht the stop-bnd of filter is deeper nd wider thn tht before optimiztion, justifying the efficiency of our method. We cn observe more from Fig. 4c tht, fter optimiztion, the minimum vlue of the trnsmission coefficient, which is 38.0 db, is close to the minimum vlue of the trnsmission coefficient when α 1 = 0, which is 39.0 db (see Fig. 3c). Only the trnsmission quntity is tken s the objective when α 1 = 0. This mens tht if we choose the vlues of the weight coefficients α 1 =0.5 nd α = 0.5, the sub-objective, which is the trnsmission quntity, nd the generl objective G cn chieve their optiml vlues simultneously Scenrio 3 Here we let α 1 =0.7 nd α = 0.3. Figures 5 nd 5b give the optimiztion process when α 1 =0.7 nd α = 0.3. After 1 itertions, the objective function vlue G is convergent t bout 0.93, where the optimized function vlue, d nd re bout mm, mm, nd.68, respectively. Figure 5c shows the minimum vlue of the trns- Object function G Design vribles [mm] d b Itertion number Itertion number Trnsmission coefficient S 1 [db] After optimiztion Before optimiztion Frequency f [GHz] 1 c Fig. 5. The optimiztion process nd results for scenrio 3: object function versus itertion numbers when α 1 =0.7 (), design vribles versus itertion numbers when α 1 =0.7 (b), nd stop-bnd chrcteristic before nd fter optimiztion when α 1 = 0.7 (c).

11 Multi-objective optimiztion of dielectric lyer mission coefficient is 30.0 db fter optimiztion. Furthermore, we cn see tht the optimized function vlue is smller thn tht of when α 1 =0, which is.1mm (see Fig. 3b), wheres the minimum vlue of the trnsmission coefficient is lrger thn tht of when α 1 = 0, which is 39.0 db (see Fig. 3c). It is worth noting tht sometimes we merely desire smller size of filter, nd it is not necessry to minimize the trnsmission quntity in stop-bnd. In the cse of this scenrio, choosing α 1 = 0.7 nd α = 0.3, might just fit the bill. 1.6 Object functions /[] G B sb /[B sb ] Weight coefficient α 1 Fig. 6. Object functions versus α 1. The bove discussions imply the process of optimiztion depends strongly on the selection of weight coefficients. Figure 6 shows the curves of the objective functions vrying with the weight coefficient α 1. It is observed tht the vlue of the sub-objective B sb /[B sb ] does not chnge much with smll α 1, while the vlue of the sub-objective /[] decreses grdully. In other words, the vlue of the design vrible keeps chnging slowly when the trnsmission quntity becomes stedy, which is similr to our previous discussion. Wht is more, when the weight coefficient α 1 is smller thn 0.5, the optimized vlue of the sub-objective B sb /[B sb ] is close to 0.65, which is the minimum vlue of the trnsmission quntity in the first scenrio. Thus, it cn be concluded tht within this intervl, the optiml vlue of the sub-objective B sb /[B sb ] is lwys obtined, i.e., the sub-gol B sb /[B sb ] s well s the normlized generl objective G is optimized simultneously. 6. Conclusion Multi-objective optimiztion model of the dielectric lyer photonic crystl filter is proposed, nd the objective functions re the size of period nd the trnsmission quntity in stop-bnd. We use the weighting fctors method in conjunction with the qudrtic RSM to obtin qudrtic progrmming model nd the optiml prmeters cn be obtined using sequence qudrtic progrmming. The optimiztion results demon-

12 40 HONGWEI YANG et l. strte tht the present method is precise nd efficient. According to the discussion on the effect of the weighting fctors on the vlue of objective functions, the conclusions re drwn s follows: 1) When the objective vlue G converges, the periodic length hs trend of slow increse; ) In prctice, we cn choose the corresponding weight coefficients to chieve vrious requirements, including the size of the period nd the trnsmission quntity; 3) When the weight coefficient α 1 is smll, the optimized vlue of the sub-objective B sb /[B sb ] does not chnge drmticlly nd is close to the solution to the model with trnsmission quntity s the single objective. This implies tht, within this intervl, the optiml vlue of the sub-objective B sb /[B sb ] is lwys obtined, i.e., the sub-gol B sb /[B sb ] s well s the normlized generl objective G is optimized simultneously. Acknowledgements This work ws supported by the Ntionl Nturl Science Foundtion of Chin (Grnt Nos , 11700). References [1] BEHNAM SAGHIRZADEH DARKI, NOSRAT GRANPAYEH, Improving the performnce of photonic crystl ring-resontor-bsed chnnel drop filter using prticle swrm optimiztion method, Optics Communictions 83(0), 010, pp [] THUBTHIMTHONG B., CHOLLET F., Design nd simultion of tunble photonic bnd gp filter, Microelectronic Engineering 85(5 6), 008, pp [3] HONGWEI YANG, SHANSHAN MENG, GAIYE WANG, CUIYING HUANG, The optimiztion of the dielectric lyer photonic crystl filter by the qudrtic response surfce methodology, Optic Applict 45(3), 015, pp [4] MIETTINEN K., Nonliner Multiobjective Optimiztion, Kluwer Acdemic Publishers, Boston, [5] ROUX W.J., STANDER N., HAFTKA R.T., Response surfce pproximtions for structurl optimiztion, Interntionl Journl for Numericl Methods in Engineering 4(3), 1998, pp [6] JANSSON T., NILSSON L., REDHE M., Using surrogte models nd response surfces in structurl optimiztion with ppliction to crshworthiness design nd sheet metl forming, Structurl nd Multidisciplinry Optimiztion 5(), 003, pp [7] REN L.Q., Experimentl Optimiztion Technology, Chin Mchine Press, Chin, 1987, pp [8] YAN DUN-BAO, YUAN NAI-CHANG, FU YUN-QI, Reserch on dielectric lyer PBG structures in wveguide bsed on FDTD, Journl of Electronics nd Informtion Technology 6(1), 004, pp [9] HUIPING YU, YUNKAN SUI, JING WANG, FENGYI ZHANG, XIAOLIN DAI, Optiml control of oxygen concentrtion in mgnetic Czochrlski crystl growth by response surfce methodology, Journl of Mterils Science nd Technology (), 006, pp [10] SUI Y., YU H., The Improvement of Response Surfce Method nd the Appliction of Engineering Optimiztion, Science Press, Chin, 010, pp [11] ROBERTO V., Response surfce method for high dimensionl structurl design problems, Ph.D. Disserttion, University of Florid, 000. Received My 19, 016 in revised form July 1, 016

Review of Calculus, cont d

Review of Calculus, cont d Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some