Synthesis of long-period fiber gratings with a Lagrange multiplier optimization method

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1 Aville online t Optis Communitions 8 (8) Synthesis of long-period fier grtings with Lgrnge multiplier optimiztion method Cheng-Ling Lee, *, Ry-Kung Lee, Yee-Mou Ko Deprtment of Eletro-Optil Engineering, Ntionl United University, Mioli 36, Tiwn, ROC Institute of Photonis Tehnologies, Ntionl Tsing-Hu University, Hhu 3, Tiwn, ROC Deprtment of Physis, Ntionl Chnghu University of Edution, Chnghu 5, Tiwn, ROC Reeived My 7; reeived in revised form August 7; epted 3 August 7 Astrt Bsed on the Lgrnge multiplier optimiztion (LMO) method, new synthesis pproh for designing omplex long-period fier grting (LPG) filters is developed nd demonstrted. The proposed synthesis method is simple nd diret pproh. It ws used to effiiently serh for optiml solutions nd onstrin vrious prmeters of the designed LPG filters ording to prtil requirements. The inverse sttering lgorithm is good tool for designing FBG filters; s well s for designing trnsmission-type fier grting filters like LPGs. Compred to the results of the disrete lyer-peeling (DLP) inverse sttering lgorithm for LPGs, the synthesized LPGs, using the LMO pproh, re more flexile nd workle for the onstrint onditions n e esily set in the user-defined ost funtionl, suh s the limittion on the mximum vlue of the refrtive index modultion nd the prmeters of the initil guess in LMO lgorithm. Liner trnsmission LPG filters nd EDFA gin flttening filters re synthesized nd nlyzed systemtilly y using the proposed method with different prmeters. Moreover, s vrition-sed method, we find tht the onvergene nd the synthesized results of the LMO method re strongly dependent on the initil guess prmeters. Ó 7 Elsevier B.V. All rights reserved. Keywords: Long-period fier grting (LPG); Lgrnge multiplier optimiztion (LMO); Grting synthesis. Introdution * Corresponding uthor. Tel.: ; fx: E-mil ddress: herry@nuu.edu.tw (C.-L. Lee). Long-period fier grtings (LPGs) re trnsmissiontype fier grting devies in whih the guided ore mode is oupled to the forwrd propgting ldding modes through the periodi photo-indued index modultion. LPGs re importnt key omponents in fier ommunition nd fier sensing systems, ting s EDFA gin flttening filters, nd-rejetion filters, mode onverters, nd high sensitivity fier sensors []. Due to lots of pplitions, simple nd effiient pproh to design nd synthesize LPGs with omplex properties is highly demnded. From litertures, the existing synthesis methods for LPGs n e lssified into two tegories: () the inverse sttering method-sed synthesis pprohes [ 8]; nd () the stohsti-sed optimiztion pprohes [9 ]. By using the disrete lyer-peeling (DLP) inverse sttering lgorithm, grting ssisted o-diretionl ouplers (GACCs) nd LPG filters n e synthesized, in whih the oupling oeffiient orresponding to given spetrl trnsmission response is uniquely determined if dditionl ssumptions out the filter properties re used (suh s the under-oupled ssumption) [6]. Even though LPG filters with required spetrl trnsmission shpes n e inversely synthesized with DLP inverse sttering lgorithm, ut there re still numer of disdvntges in using this method for some LPG filter syntheses. These disdvntges inlude: the required grting length is typilly too long to frite, the sptil grting profiles 3-48/$ - see front mtter Ó 7 Elsevier B.V. All rights reserved. doi:.6/j.optom

2 6 C.-L. Lee et l. / Optis Communitions 8 (8) 6 74 (inluding mplitude nd phse) re usully too omplited, nd the phse of the trget trnsmission spetrum of the designed LPG should e undetermined. Due to these disdvntges, the prtil pplition of the LPG devies is severely limited. Therefore, severl LPG synthesized optimiztion methods sed on stohsti methods or Monte Crlo optimiztion pprohes re proposed to diretly synthesize the LPG with some possile onstrint onditions imposed on the grting prmeters [9 ]. These optimiztion methods diretly minimize the differene etween the trgeted nd the synthesized spetr. The minimiztion n e performed y using populr optimiztion lgorithms or in prtiulr the evolutionry lgorithms (EA). Compred to the inverse sttering DLP methods, the optimiztion pprohes do not hve n miguity prolem used y the unknown phse spetrum. Moreover, optimiztion methods hve the potentil pility to otin n index profile tht n e more prtilly implemented y properly imposing dditionl onstrints on the solution to e found. Although the optimiztion method hs the superiorities mentioned ove, however, the lultion time in finding n optimiztion solution is typilly too long nd the designed oupling oeffiient profiles re usully pieewise-defined funtions [9,]. It is not very prtil euse the UV em output from the exposure lser used for fier grting writing is normlly profile with ertin width, whih is not suitle for writing the oupling oeffiients with ly leveled profiles []. Therefore, in this study, new nd simple pproh to the solution of LPG synthesis prolem ws developed. The proposed method is sed on the Lgrnge multiplier optimiztion (LMO) method, whih is vrition-sed pproh nd vrious prmeters of the designed devies n e emedded through user-defined ost funtionl. The LMO method hs een proven to e very useful in designing optil pulse shpes in liner nd non-liner mterils [ 5]. We lso hve demonstrted tht LMO optimiztion lgorithm n e implemented for designing multi-hnnel FBGs (refletion-type) in the DWDM pplitions [6]. An extension of LMO optimiztion method is to synthesize trnsmission-type grtings, in suh wy tht nother rnh of fier grting-sed devies suh s LPGs or diretionl ouplers n lso e synthesized y the proposed simple nd fst method. In generl, ompred to lyer-peering method, our proposed method n e esily emedded with vrious onstrint prmeters. And ompred to Monte Crlo sed pproh, suh s evolutionry lgorithm, our method is diret synthesis method with fst onvergene rte nd without using ny rndom numer genertors [8 ]. For the first time, trnsmission-type LPGs inluding liner LPG nd n EDFA gin flttening LPG filters re suessfully synthesized nd nlyzed y using LMO method. The designed results lso prove tht the presented lgorithm is n effetive method oth for refletion-type nd trnsmission-type fier grting devies.. LMO lgorithm for the synthesis of LPGs For wekly index modultion,the hrteristis of LPGs n e well desried y the oupled mode equtions [9 ]. If only the oupling etween two modes (ore mode nd one ldding mode) in the LPG is onsidered, the oupled mode equtions n e written s da o ðd; zþ ¼ ida o ðd; zþþijðzþa l ðd; zþ; ðþ dz da l ðd; zþ ¼ ida l ðd; zþþij ðzþa o ðd; zþ: ðþ dz Here A o (z) nd A l (z) represent the ore nd the ldding modes respetively, d = (/)[ o l p/k]=pdn eff (/ k /k D ) is the detuning prmeter for LPG, K is the grting period, k D = Dn eff ÆK is the designed wvelength, Dn eff is the differene of the effetive indies for the ore nd ldding modes, o nd l re the propgtion onstnts, nd j(z)=gpdn(z)/k D is the designed oupling oeffiient funtion with Dn(z) eing the envelope funtion of the grting index modultion nd g the overlpping ftor. The min ide of the LMO lgorithm to synthesize LPGs is to find the omplex sptilly oupling oeffiient; j(z), of the grting with tht the orresponding trnsmission spetrum, T o (k), n meet given trget spetrum, T d (k). Therefore, the ojetive funtionl needed to e minimized n e defined y the users, suh s, Z Z L U ¼ ½T o ðkþ T d ðkþš dk þ ½jðzÞŠ dz; ðþ where T o ðkþ ¼jA o ðlþ=a o ðþj is the lulted trnsmission spetrum of output ore mode for LPG, T d (k) is the trget trnsmission spetrum, L is the totl length of the grting, nd is positive numer ting s weighting prmeter for the onstrint ontrol on the sptil oupling oeffiient, j(z). In the defined ost funtionl, Eq. (), the sptilly oupling oeffiient j(z) is used to shpe n output trnsmission spetrum in the ore mode T o (k) for given trnsmission spetrum T d (k), menwhile oth the spetr disrepny nd the norm of the oupling oeffiient profiles re minimized simultneously. In the proposed LMO lgorithm to synthesize LPGs, the Lgrnge multipliers l o nd l l were introdued for the ore nd ldding modes of LPG, respetively. With the ojetive funtionl for LPG inverse design in Eq. (), speifi ost funtionl is expressed s follows: Z L Z J ¼ U þ l o dao R Re dz idao ija l dkdz Z L Z þ l o I Im dao dz idao ija l dkdz Z L Z þ l l dal R Re dz þ idal ij A o dkdz Z L Z þ l l dal I Im dz þ idal ij A o dkdz ð3þ

3 C.-L. Lee et l. / Optis Communitions 8 (8) The ore nd ldding modes, A o (z) nd A l (z), oupling oeffiient, j(z), nd the Lgrnge multipliers, l o nd l l, used in LMO methods re ll omplex numers here. We seprte these omplex vriles into rel nd imginry prts, respetively, i.e. A o ¼ A o R þ iao I, A l ¼ A l R þ ial I, j = j R +ij I, l o ¼ l o R þ ilo I nd l l ¼ l l R þ ill I. To minimize the ost funtionl, J, in Eq. (3) vritionl method ws used with respet to A o R ; Ao I of the ore mode, nd A l R ; Al I of the ldding mode through the Lgrnge multipliers l o (z), l l (z). The resulting equtions of motion for the Lgrnge multipliers n e derived s follows: ol o ðzþ ¼ id l o ðzþþijðzþl l ðzþ; ð4þ oz ol l ðzþ ¼ idl l ðzþþij ðzþl o ðzþ: ð4þ oz The oundry onditions for the oupled equtions of the Lgrnge multipliers, Eq. (4), n e otined y using the vritionl method on the ost funtionl J with respet to A o nd A l t z = L, i.e. oj/oa l (L) =, oj/oa o (L) =. The orresponding oundry onditions for Eq. (4) t z = L re shown s elow: l o ðlþ ¼ A o ðlþd t ; ð5þ x -4 Initil Δn Amplitude of finl Δn.5 x -4 α=5x α=x 3 α=5x 3 α=x To Averge error Trget α=5x.5 α=x 3 α=5x 3 α=x Wvelength (μm) d α=5x α=x 3 α=5x 3 α=x Itertion Fig.. () The initil guess of the podiztion oupling oeffiient for the proposed lgorithm, () finl designed mplitude of the podiztion profiles, () synthesized ore mode trnsmission spetr, nd (d) typil evolution urves of the solute vlue of verge error (verge jd t j) for the designed liner LPGs with different d ho prmeter in the LMO method.

4 64 C.-L. Lee et l. / Optis Communitions 8 (8) 6 74 l l ðlþ ¼; ð5þ where D t = T o (k) T d (k) is the disrepny etween the output nd trget trnsmission spetr. Agin, we use the vrition method on the ost funtionl J with respet to the rel nd imginry prts of the oupling oeffiient funtion j R nd j I, respetively, i.e. dj ¼ j R þ dj R þ Z Z l o R Al I lo I A l R dk l o R Ao I l l I Ao R dk; ð6þ dj ¼ j I þ dj I þ Z Z l o R Al R þ lo I A l I dk l o R Ao R ll I Ao I dk: ð6þ With n initil guess for the oupling oeffiient profile, j(z)=j R (z) +ij I (z), the equtions of motion for the multipliers re solved with the help of Eq. (4) nd the orresponding oundry onditions in Eq. (5). The optimiztion of the LMO method progressed until the onvergene requirement ws stisfied through following itertion proedure: Averge error To Amplitude of finl n Initil n 3 x x -4 3 n =.5x -4 n =x -4 n =.5x -5 n =8x -6 n =.5x -4 n =x -4 n =.5x -5 n =8x Trget n.5 =.5x -4 n =x -4 n =.5x -5 n =8x Wvelength ( m) d n =.5x -4 n =x -4 n =.5x -5 n =8x Itertion Fig.. () The initil guess of the podiztion funtions with different mximum vlues of index modultion Dn, () finl designed mplitude of the podiztion profiles, () synthesized ore mode trnsmission spetr, nd (d) typil evolution urves of the verge error for synthesized liner LPGs.

5 C.-L. Lee et l. / Optis Communitions 8 (8) j R ðzþ ¼j dj RðzÞ R ; dj R ð7þ j I ðzþ ¼j dj IðzÞ I ; dj I ð7þ where R nd I re d ho onstnts for rel nd imginry prts of the oupling oeffiient, respetively. The new oupling oeffiient n e expressed s follows j ðzþ ¼ j R ðzþþij I. The lgorithm we use to implement the LMO methods for designing LPGs n e summrized with following steps, () Guess n initil oupling oeffiient funtion j R (only rel prt is given for simplifying the lgorithm). () Clulte the ore nd ldding modes, A o (z) nd A l (z), with Eq. () from z =toz = L with the given initil onditions, A o () =, A l () = nd otin ll vlues of A o (z), A l (z) from z =toz = L. () If the A o (L), A l (L) re lulted nd known, then the oundry ondition t z = L for the Lgrnge multipliers, l o nd l l, in Eq. (5) re known s well. (d) Compute the propgtion of the Lgrnge-multiplier funtions in Eq. (4) from z = L to z = with the oundry onditions in Eq. (5). (e) Updte the new oupling oeffiient profile with suitle d ho prmeter y using the Eqs. (6) nd (7). Initil Δn -4 x 6 4 Δn=5x -4 Δn=.5x -4 Δn=x -4 Δn=5x -4 Amplitude of finl Δn x 3 Δn=5x -4 Δn=.5x -4 Δn=x -4 Δn=5x To trget Δn=5x -4 Δn=.5x -4 Δn=x -4 Δn=5x Wvelength (μm) Averge error d Δn=5x -4 Δn=.5x -4 Δn=x -4 Δn=5x Itertion Fig. 3. () The initil guess of the podiztion funtions with different mximum vlues of index modultion Dn, () finl designed mplitude of podiztion profiles, () synthesized ore mode trnsmission spetr, nd (d) typil evolution urves of the verge error for synthesized liner LPGs.

6 66 C.-L. Lee et l. / Optis Communitions 8 (8) 6 74 (f) Repet step () to step (e) until dj dj R ¼ nd dj dj I ¼, i.e. the LMO method onverges. 3. Design results nd disussion In this setion, the effetiveness nd fesiility of the proposed synthesis lgorithm for the LPG filters is demonstrted, the designs for oth liner filter nd n EDFA gin flttening filter re given. In the following synthesis demonstrtions, we set the d ho prmeters = R = I, the prmeter to e zero, nd the grting length fixed L = 4 mm for simpliity. The first exmple is LPG filter with liner trnsmission response within ertin rnge of wvelength. Suh liner trnsmission filters hve pplitions in mny fier sensing systems in whih the wvelength modultion of the optil signl is onverted into the intensity modultion. After pssing through the liner LPG, trnsmitted ore mode n e redily deteted y photo-detetor. In this exmple, the mximum trnsmission oeffiient for the ldding mode T mx =.7 t the wvelength k D = mm. The results for liner LPGs re shown systemtilly in Figs. 7 for different d ho prmeter, different initil mximum index modultion, nd different initil podiztion funtion profiles. First of Initil Δn.5 x -4.5 To Amplitude of finl Δn x -4 3 Δn =x -5 Δn =5x -5 Δn =x Trget.5 Δn =x -5 Δn =5x -5 Δn =x Wvelength (μm) Averge error d Δn =x -5 Δn =5x -5 Δn =x Itertion Fig. 4. () The initil guess of the index modultion with different mximum vlue of index modultion Dn, () finl designed mplitude of podiztion profiles, () synthesized ore mode trnsmission spetr, nd (d) typil evolution urves of the verge error for synthesized liner LPGs.

7 C.-L. Lee et l. / Optis Communitions 8 (8) ll, in Fig., n initil guess of podiztion funtion is used, i.e. j(z) =Dn Æ exp{ 4 ln[(z.5l)/ FWHM] }, where Dn is mximum vlue of index modultion in the grting, nd FWHM is the ndwidth of full width t hlf mximum of the funtion (lwys set to e mm). In our lultions, the units of the wvelength k nd the grting length L re in mm. With different d ho vlues, = R = I, the designed mplitude of podiztion profiles of the index modultion nd the synthesized ore mode trnsmission spetr for liner LPGs re shown in Fig. nd, respetively. Fig. d shows the typil evolution urves of the solute vlue of verge error: jd t j with respet to the itertion times, i.e. jd t j is the solute vlue of totl errors divided y the numer of spetrl points. In this liner LPG synthesis se, the est vlue is hosen to e in the rnge For these vlues, ll the simultions quikly onverge only with smll vritions in the finl podiztion profiles, s shown in Fig.. But for lrger vlue of the d ho prmeter, P 5 4, we find tht the lgorithm divergenes for the designed se due to too strong modifition in the vrition method. Bsed on the knowledge of vlue, in Fig., wefix = 4 nd use the sme initil guess profile with the podiztion funtion ut different mximum vlues of the index modultion, Dn, s shown in Fig.. From the resulted podiztion profiles nd the Initil Δn 6 4 x Amplitude of finl Δn 8 x To trget Wvelength (μm) Averge error d Itertion Fig. 5. () Different initil guesses of the podiztion funtion, () finl designed mplitude of podiztion profiles of the index modultion, () synthesized ore mode trnsmission spetr, nd (d) typil evolution urves of the verge error for the designed liner LPGs with fixed mximum vlue of index modultion Dn =.5 4.

8 68 C.-L. Lee et l. / Optis Communitions 8 (8) 6 74 trnsmission spetr in Fig. nd, we n lerly see tht n pproprite mximum vlue of the initil index modultion helps to find etter synthesized result more effiiently. For lrger vlue of the mximum index modultion, the onvergene urve of the verge error delines quikly t the first stte ut osilltes in the long run, s shown in Fig. d. Moreover, due to the intrinsi vrition-sed property of the proposed LMO method, the finl podiztion profile of the index modultion hs very similr shpe with the initil guess one. To prove this point, in the following simultions, we ompre the syntheses of liner LPGs with different mximum vlues of Dn s well s different initil podiztion funtions, in Figs. 6. Exept for the ove mentioned funtion, initil guess index modultions s funtion (SC(z), Fig. 3), funtion (Fig. 4), nd sinusoidl funtion S(z) re lso presented with the expliit formultions of SCðzÞ ¼Dn ½pð z :5L L ÞŠ nd SðzÞ ¼Dn sin½pð z :5L ÞŠ. Due to L the liner trnsmission spetrum of the trget, one n see tht the synthesized LPGs hve etter results when the initil guess funtions re funtions. In this sitution, rod rnge of the initil guess vlue for the mximum vlue of the index modultion Dn, fst nd 4 x x -4.5 To Trget Averge error d Itertion Fig. 6. () Different initil guess of the podiztion funtions, () finl designed mplitude of podiztion profiles of the index modultion, () synthesized ore mode trnsmission spetr, nd (d) typil evolution urves of the verge error for the designed liner LPGs with mximum vlue of index modultion round Dn =.5 5.

9 C.-L. Lee et l. / Optis Communitions 8 (8) monohromti onvergene rte of the verge errors n e otined, s shown in Fig. 3d. Fig. 4 shows the orresponding finl synthesized index modultion profiles, the lulted trnsmission spetr, nd the urve of the verge errors for n initil oupling oeffiient profile. To hieve smooth onvergene rte for the verge error, our lultions lso indite tht the mximum vlue of the initil index modultion Dn should not e too lrger in the LMO lgorithm. In Fig. 5, we ompre the synthesized results with different initil guess of the podiztion funtions (,,, nd sinusoidl) with fixed Dn =.5 4 for liner LPG filters. Agin, the finl podiztion profile of the index modultion hs very similr one with respet to the initil guess. To derese the impt from the initil index modultion profiles, in Fig. 6 we use smller vlue of the mximum modultion index Dn (round 5 ) for different initil podiztion funtions. In this sitution, the impt from the initil guess funtions eomes less dominnt nd lmost the sme results n e otined for suitle d ho prmeter = 4. Bsed on these systemtil lultions of the synthesized liner LPG, we show the typil designed omplex index modultion profiles for liner LPG with totl grting length of 4 m in Fig. 7, inluding the mplitude nd phse of index modultion profiles. It n e lerly seen tht the designed oupling oeffiient profile for liner LPGs sed on the LMO method is smooth one nd not too omplited in their mplitudes s well s phses. In ddition, the totl grting length is not too long to frite y urrent experimentl setups. Another exmple to demonstrte y LMO method is the well known EDFA gin flttening LPG (EDFA-GFLPG). In previous work, vrious design shemes of LPG devies hve een proposed to equlize the gin spetrum of n EDFA [7 ]. The min pprohes re sed on the phse shift LPGs, using multiple LPGs srped together or speilly rrnged LPGs to mth the spetrl shping to fltten EDFA gin spetrum. However, in this work, we diretly synthesize the EDFA-GFLPGs just y using single speifi designed LPG. For the designed EDFA-GFLPGs, the enter wvelength is set t the wvelength k D =.53 3 mm. First of ll, results of the designed EDFA-GFLPGs with different d ho vlues re shown in Fig. 8, where initil guess index modultion profile is used with Dn =.5 5. The omprisons of the finl index modultion profiles, the lulted trnsmission spetr, nd the tendeny of the verge error onvergene re shown in Fig. 8 d. From the lultions, est onvergene of the verge errors ours with the vlue out 5 3. Next we try to use different initil guess of the podiztion funtions to synthesize EDFA-GFLPGs, with fixed Dn =.5 5 nd =5 3. Fig. 9 nd show the initil nd the finl designed profiles of index modultions with,,, nd sinusoidl funtions. In this se, gin the finl synthesized podiztion funtions re shped into similr profiles s the initil guess funtions. For smller vlue of the mximum vlue of Dn =.5 5, the onvergene rtes of the verge errors re ll effiient no mtter wht kind of the initil guess index modultion profile is hosen, s shown in 4 x -5 Finl Δn Amplitude of Δn x -5 Rel prt Imginry prt x Fig. 7. Finl designed omplex index modultion profiles of the synthesizing liner LPG in Fig. with = 4 : () the initil guess index modultion profile, () the rel nd imginry prts, nd () the mplitude nd phse of the finl index modultion profiles.

10 7 C.-L. Lee et l. / Optis Communitions 8 (8) 6 74 Fig. 9d. However, for lrger vlue of Dn, sy out.5 4, the LMO lgorithm eomes divergent for the initil guess profile. In the other hnd, the LMO lgorithm re more stle y using initil guess funtions with nd profiles, s shown in Fig.. From the simultion results for designing EDFA-GFLPGs, we find tht the est results n e otined with the onditions tht the initil guess is funtion, the mximum vlue of the index modultion is smller thn 5 5, nd is out 5 3. For typil result of our synthesizing EDFA-GFLPG filters, its trget nd designed trnsmission spetr, the orresponding flttened gin-profiles of EDFA spetr, nd the ripple devitions in the flttened wvelength region re shown in Fig. in db sles, respetively. It n e seen lerly tht from Fig. the designed refletion spetrum meets well with the trget spetrum with only little devition for rnge of wvelength out 5 nm. And from Fig. the studied spetrum n e flttened to less thn ±.4 db disrepny within the entire C-nd nd the verge ripple devition in the flttened wvelength region is lower thn.5 db. In Fig., the designed Initil Δn 3 x -5 Δn =.5x Amplitude of finl Δn -4 x.5.5 α=x 4 α=5x 3 α=x 3 α=5x To.8.6 Trget.4 α=x 4 α=5x 3. α=x 3 α=5x Wvelength (μm) Averge error d α=x 4 α=5x 3 α=x 3 α=5x Itertion Fig. 8. Designing of 4 m long EDFA-gin flttering LPG with different d ho prmeter in LMO lgorithm; () the initil guess index modultion profile, () finl designed mplitude profiles, () synthesized ore mode trnsmission spetr, nd (d) typil evolution urves of verge error.

11 C.-L. Lee et l. / Optis Communitions 8 (8) omplex index modultion profiles of the EDFA gin flttering LPG synthesized in Fig. re shown with the mplitude nd the phse distriutions of the index modultions. In generl, LPGs re esier to frite thn FBGs due to the periods of LPGs re severl hundreds of lm. One n use the UV-em-snning point y point exposure tehniques [,] to hieve omplex oupling oeffiient j(z) of the grtings. By ontrolling nd djusting grting periods preisely nd ontinuously in every snning step for the ontinuous vrition of the phse Initil Δn 4 x Amplitude of finl Δn.5 x To Trget Wvelength (μm) Averge error d 5 5 Itertion Fig. 9. () Different initil guesses of the podiztion funtion (with fixed Dn =.5 5 nd =5 3 ), () finl designed mplitude of the index modultion, () synthesized ore mode trnsmission spetr, nd (d) typil evolution urves of the verge error for EDFA-GFLPGs synthesis y LMO method.

12 7 C.-L. Lee et l. / Optis Communitions 8 (8) 6 74 of index modultion s required, s well s mplitude of the index modultion, the designed LPGs with omplex index modultion e hieved. Finlly, the lst ut not lest thing, we wnt to emphsize is tht ll of the synthesized LPG filters demonstrted here re lulted with the onstrined prmeter shown in Eq. () set to e zero (no onstrin ondition). The prmeter ts like n lternte nd onstrined prmeter in the ojetive funtion to onstrin the profile of the oupling oeffiient: j(z). It n e set zero for the unonstrined oupling oeffiient of the design ondition nd only the disrepny etween the output nd trget trnsmission spetr of LPGs is onsidered (the first term of Eq. ()), moreover, sed on our previous pulition for refletion-type FBGs, non-zero for further derese the mximum vlue of the index modultion y srifiing the refletivity of the FBGs (i.e. inresing the refletivity error) is the designed se tht the j(z) lwys kept rel numer in FBGs. However, in this present work of optiml LPGs y the LMO method, the phse of the j(z) (omplex) ws introdued to tilor the trget spetrum of LPGs nd from the simultion results, the designed mplitude of 4 - x x -4 To Trget Averge error d Itertion Fig.. () Different initil guesses of the podiztion funtion (with fixed Dn = 4 nd =5 3 ), () finl designed mplitude of index modultion, () synthesized ore mode trnsmission spetr, nd (d) typil evolution urves of the verge error for EDFA-GFLPGs synthesis y LMO method.

13 C.-L. Lee et l. / Optis Communitions 8 (8) To (db) Gin (db) Gin (db) - Trget Designed Unflttened Flttened Wvelength (μm) Fig.. () The omprisons etween the trget nd synthesized ore mode trnsmission spetr (in db sle) of the EDFA-GFLPG filter, () unflttened nd flttened gin-profiles of EDFA gin spetr y the designed gin flttering LPG, nd () ripple devitions in the flttened wvelength region. x-5 Initil Δn Finl Δn Amplitude of Δn x-5 5 Rel prt Imginry prt x Fig.. Finl designed omplex index modultion profiles of the synthesized EDFA gin flttering LPG in Fig. : () the initil guess index modultion profile, () the rel nd imginry prts, nd () the mplitude nd phse of the finl index modultion profiles. Phse of Δn (π) index modultions re under 4, so tht the designed LPGs ould e implemented with the ville ommeril photosensitive fiers. Therefore, in the present pper for the designing optiml LPGs with omplex oupling oeffiient, only unonstrined ondition = is needed to onsider. 4. Conlusion In this pper, for the first time, we hve investigted nd demonstrted simple nd fst LPG synthesis methodsed on the LMO optimiztion lgorithm. Trnsmission-type fier grtings, suh s liner LPG nd EDFA

14 74 C.-L. Lee et l. / Optis Communitions 8 (8) 6 74 gin flttening LPG filters with different prmeters re suessfully synthesized y using the proposed method. Bsed on the simultion results with different d ho prmeter, mximum vlues, nd profiles of initil guess index modultion, the effets of these prmeters in the LMO lgorithm for designing LPGs nd the orresponding onvergene of verge errors re nlyzed systemtilly. It n e seen lerly tht the mximum vlue of initil guess for the index modultion should not e too lrger nd the est vlue of the evolutionry prmeter flls into the rnge of Moreover, the profiles of the finl oupling oeffiients evolve into similr one s the initil guess index modultion funtion. Therefore, n optiml solution might e otined y dding little perturtion on the prtil profile of the index modultion tht will e disussed nd demonstrted in future pper. From this first study, it ppers tht the proposed pproh is n effetive method whih onverges quikly on designing not only refletion-type FBG ut lso trnsmission-type LPG filters. Aknowledgements The uthors thnk Prof. Yinhieh Li for his useful disussions. This reserh is supported y the Ntionl Siene Counil of the Repuli of Chin, NSC 95-- E nd NSC 95--M--6. Referenes [] A.M. Vengsrkr, J.R. Pedrzzni, J.B. Judkins, P.J. Lemire, N.S. Bergno, C.R. Dvidson, Opt. Lett. (996) 336. [] G.H. Song, S.Y. Shin, J. Opt. So. Am. A. (985) 95. [3] E. Perl, J. Cpmny, J. Mrti, IEEE J. Quntum Eletron. 3 (996) 78. [4] L. Wng, T. Erdogn, Eletron. Lett. 37 () 54. [5] G.H. Song, J. Lightwve Teh. 3 (995) 47. [6] R. Feed, M.N. Zervs, J. Opt. So. Am. A 7 () 573. [7] J.K. Brenne, J. Skr, J. Lightwve Teh. (3) 54. [8] G.W. Chern, L.A. Wng, J. Opt. So. Am. A 9 () 77. [9] C.L. Lee, Y. Li, IEEE Photon. Teh. Lett. 4 () 557. [] C.L. Lee, Y. Li, Fier Integrt. Opt. 3 (4) 49. [] C.L. Lee, Opt. Commun. 6 (6) 7. [] N. Wng, H. Ritz, Phys. Rev. A 53 (996) 879. [3] N. Wng, H. Ritz, J. Chem. Phys. 4 (996) 73. [4] R. Buff, Opt. Commun. 53 (998) 4. [5] D.K. Pnt, Ro D. Colson, Mrt I. Hern ndez, Jos e Cmpos- Mrt ınez, J. Lightwve Teh. 6 (998) 9. [6] C.L. Lee, R.K. Lee, Y.M. Ko, Opt. Exp. 4 (6). [7] M. Hrumoto, M. Shigehr, H. Sugnum, J. Lightwve Teh. () 7. [8] J.R. Qin, H.F. Chen, Eletron. Lett. 34 (998) 3. [9] M.G. Xu, R. Msknt, M.M. Ohn, A.T. Alvie, Eletron. Lett. 33 (997) 893. [] A.P. Zhng, X.W. Chen, Z.G. Gun, S.L. He, H.Y. Tm, Y.H. Chung, IEEE Photon. Teh. Lett. 7 (5). [] A.I. Klhev, D.N. Nikogosyn, G. Brmill, J. Lightwve Teh. 3 (5) 568. [] B. Mlo, K.O. Hill, F. Bilodeu, D.C. Johnson, J. Alert, Eletron. Lett. 9 (993) 668.