The complex Jacobi iterative method for threedimensional wide-angle beam propagation

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1 The complex Jacob teratve method for threedmesoal wde-agle beam propagato Kha Q. Le R. Godoy-Rubo eter Bestma ad G. Roald Hadley 3 Departmet of Iformato Techology Ghet Uversty-IMEC St- eterseuwstraat 4 B-9 Ghet Belgum Departameto de Igeería de Comucacoes Uversty of Malaga 97 Spa 3 Sada Natoal Laboratores Albuquerque NM 8785 USA Correspodg author: ha.le@tec.uget.be Abstract: A ew complex Jacob teratve techque adapted for the soluto of three-dmesoal (3D) wde-agle (WA) beam propagato s preseted. The beam propagato equato for aalyss of optcal propagato wavegude structures s based o a ovel modfed adé() approxmat operator whch gves evaescet waves the desred dampg. The resultg approach allows more accurate approxmatos to the true Helmholtz equato tha the stadard adé approxmat operators. Furthermore a performace comparso of the tradtoal drect matrx verso ad ths ew teratve techque for WA-beam propagato method s reported. It s show that complex Jacob terato s faster ad better-suted for large problems or structures tha drect matrx verso. 8 Optcal Socety of Amerca OCIS codes: (.443) Geeral: Numercal approxmato ad aalyss; (.56) Optcal desg ad fabrcato: ropagatg methods; (35.55) Other areas of optcs: ropagato. Refereces ad Ls. R. Scarmozzo A. Gopath R. regla ad S. Helfert Numercal techques for modelg guded-wave photoc devces IEEE J. Sel. Top. Quatum Electro ().. C. Ma ad E. V. Keure A smple three dmesoal wde-agle beam propagato method Opt. Express (6). 3. W. H. ress B.. Flaery S. A. Teuolsy ad W. T. Vetterg Numercal recpes: The art of scetfc computg (Cambrdge Uversty ress New Yor 986). 4. M. D. Fet ad J. A. Flec Jr. Aalyss of rb wavegudes ad couplers by the propagato method J. Opt. Soc. Am. A (99). 5. F. A. Mlazzo C. A. Zala ad G. H. Brooe Ratoal square root approxmatos for parabolc equato algorthms J. Acoust. Soc. Am (997). 6. Y. Y. Lu ad. L. Ho Beam propagato method usg a [(p-)/p] adé approxmat of the propagator Opt. Lett (). 7. T. Aada T. Hoazoo T. Hraoa J.. Hsu T. M. Beso ad. Sewell Very-wde-agle beam propagato methods for tegrated optcal crcuts IEICE Tras. Electro. E8-C (999). 8. G. R. Hadley Wde-agle beam propagato usg adé approxmat operators Opt. Lett (99). 9. S. Suec Wde-agle fte-dfferece beam propagato oblque coordate system J. Opt. Soc. Am (7).. C. Ma ad E. V. Keure A three-dmesoal wde-agle beam propagato method for optcal wavegude structures Opt. Express (7).. G. R. Hadley A complex Jacob teratve method for the defte Helmholtz equato J. Comp. hys (5).. Y. Y. Lu A complex coeffcet ratoal approxmato of x Appl. Numer. Math (998). 3. G. R. Hadley Low-trucato-error fte dfferece equatos for photocs smulato I: beam propagato J. Lghtwave Techol (998). 4. T.A. Davs Drect Methods for Sparse Lear Systems (SIAM 6). 5. J.. Béreger A perfectly matched layer for the absorpto of electromagetc waves J. Comput. hys (994). # $5. USD Receved Aug 8; revsed 7 Sep 8; accepted Oct 8; publshed Oct 8 (C) 8 OSA 3 October 8 / Vol. 6 No. / OTICS EXRESS 7

2 6.. Vadersteege B. Maes. Bestma ad R. Baets Usg the complex Jacob method to smulate Kerr o-lear photoc compoets Opt. Quatum Electro (6). 7. Z. Ju J. Fu ad E. Feg A smple wde-agle beam-propagato method for tegrated optcs Mcrowave Opt. Techol. Lett (997). 8.. C. Lee ad E. Voges Three-dmesoal sem-vectoral wde-agle beam propagato method J. Lghtwave Techol. 5 5 (994) 9. Y. Tsu M. Koshba ad T. Shrash Fte elemet beam propagato method for three-dmesoal optcal wavegude structures J. Lghtwave Techol (997).. G. R. Hadley Multstep method for wde-agle beam propagato Opt. Lett (99).. Itroducto The beam propagato method (BM) has become oe of the most wdely used techques for the study of optcal wavegude devces []. For the paraxal approxmato or Fresel equato used BM ths s usually accomplshed by splt-step methods cludg the alteratg-drecto mplct (ADI) methods that are both fast ad easy to mplemet [3]. However the splt-step schemes are oly frst-order-accurate the step sze ad a drect matrx verso s requred for acceptable accuracy f WA propagato s eeded. Besdes the lmtato to paraxal beams these methods also restrct the smulatos to a low refractve dex cotrast rato betwee the core ad claddg of the wavegude [4]. Efforts have bee tae to relax the lmtatos for WA smulatos. Dfferet treatmets of WA-BM based o the slowly varyg evelope approxmato have bee developed cludg the ratoal approxmats of the square root operator [5] the oe-way propagator [6] the expoetal of the square root operator [7] ad the adé approxmat operator [8] for rectagular coordates as well as a oblque coordate system [9]. The adé-approxmat-based WA- BM s oe of the most commoly used techques for modelg optcal wavegude structures. However the method was orgally lmted to D structures due to the lac of effcet solvers. A 3D WA-BM based o Hoestra s scheme was recetly developed usg the effcet Thomas algorthm ad the splttg of the 3D Fresel wave equato to three D wave equatos []. However ths may cause splttg errors. Recetly C. Ma et al. [] preseted a ew 3D WA-BM also based o Hoestra s scheme that does ot requre the splttg of the Fresel wave equato or the use of the ADI method. By usg a techque for shftg the smulato wdow to reduce the dmeso of the umercal equato ad a threshold techque to further esure ts covergece ths approach shows accuracy ad effectveess. However the resultat propagato scheme ca be very slow f ether the problem sze s large or the structure or the boudary codtos are chagg as the propagato proceeds requrg frequet reversos of the propagato matrx. Thus t s mperatve to fd more effcet soluto methods for 3D WA-BM. Recetly the complex Jacob teratve method a ew teratve techque for soluto of the defte Helmholtz equato was troduced []. It s based o a complex geeralzato of the pot relaxato techque proposed by Jacob 845 ad has bee show to coverge at a rate depedet oly upo the grd sze ad effectve absorpto coeffcet. For beam propagato of wave profles wth a D cross secto the wde agle beam propagato equato ca be recast terms of a Helmholtz equato wth source term ad the effectve absorpto coeffcet appearg ths equato s hgh leadg to rapd covergece of the method. I ths wor we apply the complex Jacob teratve (CJI) method for beam propagato ad show t to be hghly effcet for the soluto of large problems compared wth exstg techques. The beam propagato equato for aalyss of optcal propagato wavegude structures s based o a modfed adé() approxmat operator we propose here whch gves evaescet waves the desred dampg. Our modfed adé approxmat propagato operators allow more accurate approxmato to the true Helmholtz equato ad faster covergece whe the CJI method s used for the modfed adé approxmat-based WA- # $5. USD Receved Aug 8; revsed 7 Sep 8; accepted Oct 8; publshed Oct 8 (C) 8 OSA 3 October 8 / Vol. 6 No. / OTICS EXRESS 7

3 BM tha those of the stadard adé approxmat approach [8]. Furthermore sce the utlty of the CJI techque depeds mostly upo ts executo speed comparso wth the drect matrx verso (DMI) method we also preset several speed comparsos. Numercal mplemetatos are carred out for both D ad 3D optcal wavegude structures. The rest of ths paper s structured as follows: I Secto we wll descrbe the ovel modfed adé approxmat operators. Next WA-BM formulatos usg the CJI ad DMI methods are preseted Secto 3. I Secto 4 the vestgato of covergece rate of CJI s dscussed. I Secto 5 the bechmar results performed by WA-BM usg CJI ad DMI are preseted followed by coclusos Secto 6.. Modfed adé approxmat operators The scalar Helmholtz equato obtaed by usg the slowly varyg evelope approxmato s gve by [8] H H H () where ( ref) ( ) ref wth ref the refractve dex x y profle ref the referece refractve dex the vacuum wavevector. We may formally rewrte Eq. () the form Equato () suggests the recurrece relato H H. () Hadley [8] proposed the ratoal approxmato of WA beam propagato usg adé approxmat operators wth tal value of. For ths gves us the well-ow adé () approxmat-based WA beam propagato formula as follows: where. (3) H H. (4) 4 If Eq. (4) s compared wth a formal soluto of Eq. () wrtte the well-ow form. X H ( ) H ( we obta the approxmato formula X ) H (5) # $5. USD Receved Aug 8; revsed 7 Sep 8; accepted Oct 8; publshed Oct 8 (C) 8 OSA 3 October 8 / Vol. 6 No. / OTICS EXRESS 73

4 X 4 X. X 4 Sce the operator X has a real spectrum t s useful to cosder the approxmato of X by the adé approxmat propagato operator. Fgure shows the absolute value of X ad ts frst-order adé approxmat called Hadley () as a fucto of X. However as the deomator of Hadley () gradually approaches zero ts absolute value approaches as ca clearly be see Fg.. hyscally ths meas that the stadard adé approxmat correctly propagates the evaescet modes. To crcumvet ths problem we propose a modfed adé approxmat operator. Frst of all followg [] by multplyg both sdes of Eq. () wth we obta We may rewrte Eq. (7) as follows X f ( X ) (8) f ( X ). (6) (7) where f ( X ). Eq. (8) suggests the recurrece relato f ( X ) X f ( X ) for... (9) Y. Y. Lu [] has proved that Eq. (9) ca provde a good approxmato to X wth the tal value of () f ( X ) β where β. > Subsequetly we use ths fact to go bac to the orgal recurrece relato (3) ad costruct modfed adé approxmat operators by usg a dfferet tal value of β. For the frst-order modfed adé() approxmat operator s gve as follows: H () H β 4 ( ) # $5. USD Receved Aug 8; revsed 7 Sep 8; accepted Oct 8; publshed Oct 8 (C) 8 OSA 3 October 8 / Vol. 6 No. / OTICS EXRESS 74

5 The absolute value of the modfed adé() approxmat of X s also depcted Fg.. It s see that our modfed adé approxmat operator (wth β) allows more accurate approxmatos to the true Helmholtz equato tha the stadard adé approxmat operator. Furthermore the stadard ratoal adé approxmat correctly propagates the evaescet modes as ther deomator gradually approaches zero whle the modfed adé approxmat gves the waves propagatg the evaescet rego the desred dampg as clearly see Fg.. Fg.. The absolute values of (X) / - (sold le) ts frst-order stadard adé approxmat (X/)/(X/4) (sold le wth crcles) ad modfed adé approxmat (X/)/ (X/{4(beta/)}) (dotted le). # $5. USD Receved Aug 8; revsed 7 Sep 8; accepted Oct 8; publshed Oct 8 (C) 8 OSA 3 October 8 / Vol. 6 No. / OTICS EXRESS 75

6 Fg.. The absolute value of (X) / - (sold le) the frst-order stadard (X/)/(X/4) (sold le wth crcles) ad modfed (X/)/(X/{4(beta/)}) (dotted le) adé approxmat of (X) / - wth respect to X. 3. WA-beam propagato formulato 3. Basc equato By usg the modfed adé() approxmat the 3D semvectoral WA beam propagato equato ca be wrtte as follows [3]: ( ) ( ) () where z the complex cougate of ad z the propagato step. 4 ( β / ) WA-BM usg CJI By dvdg both sdes of Eq. () by t may be wrtte as a homogeeous Helmholtz equato or ( ( ref ) ) ( ) (3) ( β z / ) β / z ( ( ref ) 4 o ref ) source term. (4) z ( βz / ) # $5. USD Receved Aug 8; revsed 7 Sep 8; accepted Oct 8; publshed Oct 8 (C) 8 OSA 3 October 8 / Vol. 6 No. / OTICS EXRESS 76

7 Thus the beam propagato ca be cast as a D Helmholtz equato wth source term a effectve medum wth loss of ( ) / 4 z z z ref ref β. Ths loss s hgh for a typcal choce of z. Ths s a codto that favors rapd covergece for the CJI method. 3.3 WA-BM usg DMI By dscretzg Eq. (3) we fd that. E D C B A E D C B A (5) where. ( ) ( ) ( ) ( y x C y E D x B A y x C y E D x B A ref ref o (6) Actually the coeffcets Eq. (6) are the same as the result whe applyg Hoestra s scheme to the 3D wave equato []. Eq. (5) s a M by M matrx equato for a M by M mesh grd. However each row of the coeffcet matrx has o more tha fve o-zero values. As a result ths sparse matrx equato ca be effcetly solved usg varous methods [3]. I our calculatos the sparse matrx solver-umfack pacage has bee used [4]. 4. Covergece studes of CJI I ths secto we vestgate the covergece rate of the CJI method for 3D WA-BM based o the modfed adé() approxmat ad Hadley(). We smulate the propagato of a 3D Gaussa beam through a symmetrc Y-brach wavegude. The structure parameters of the Y- brach wavegude are the same as [7] wth m d μ ad m d μ. I Fg. 3 we show that the CJI method coverges faster wth WA-BM based o the modfed adé() approxmat tha those of Hadley(). However t suffers from the fact that the terato cout betwee two successve D cross sectos creases throughout the propagato drecto. To overcome ths problem we propose the use of a perfectly matched layer (ML) whch ca absorb cdet radato wthout ay addtoal parastc reflectos regardless of wavelegth cdet agle or polarzato as boudary codtos [5-6]. It s see that the CJI techque becomes more stable wth the use of a ML as show Fg. 3. # $5. USD Receved Aug 8; revsed 7 Sep 8; accepted Oct 8; publshed Oct 8 (C) 8 OSA 3 October 8 / Vol. 6 No. / OTICS EXRESS 77

8 Fg. 3. The terato cout per propagato step for propagato through a symmetrc Y-brach wavegude obtaed usg the CJI method for stadard adé approxmat-based WA BM wthout () ad wth () ML ad the CJI method for the modfed adé wthout (3) ad wth (4) ML. (um s defed as μm) 5. Bechmar results We ow employ the WA-BM usg the ew CJI ad the tradtoal DMI methods to perform bechmar tests o both D ad 3D optcal wavegude structures. The smulatos were all ru o a oteboo C usg Matlab. For the D case we cosder a -degree tlted wavegude [7] ad Y-brach wavegude [7]. I the tlted wavegude the fudametal mode for the slab of wdth w.µm s propagated through 3µm at wavelegth λ.55µm a medum of refractve dex 3.4 wth the propagato step sze of z.µm. Wth a very strct propagato error tolerace of -9 the CJI method oly too 84.8 secods whereas the DMI method too 96.5 secods. I the Y-brach wavegude the parameters eeded for calculato are the same as [7]. Wth a small propagato step sze z.µm (requrg frequet matrx verso) the DMI method performed the propagato 44.8 secods whle the CJI method too oly 584. secods. It s obvous that for these D wavegude structures the CJI method s faster tha DMI. For the 3D case we cosder Gaussa beam propagato a straght rb wavegude [8-9] ad guded-mode propagato a Y-brach rb wavegude. The wdth ad heght of the straght rb wavegude are wµm ad h.µm as see Fg. 4 of [9]. The gudg core has a dex f 3.44 ad a thcess t.µm whle the refractve dex of substrate ad cover s s 3.4 ad c respectvely. The Gaussa beam wth a wast radus w.3µm has bee ected to the rb wavegude at wavelegth λ.55µm. Due to the large memory requred for DMI the small computatoal wdow of xµm s dscretzed wth a grd sze of x y.µm ad the short path legth of µm s dscretzed wth a propagato step sze z.µm. The resultg rutme of DMI s 77.9 secods whle rutme for CJI s oly 4.7 secods. # $5. USD Receved Aug 8; revsed 7 Sep 8; accepted Oct 8; publshed Oct 8 (C) 8 OSA 3 October 8 / Vol. 6 No. / OTICS EXRESS 78

9 For a Y-brach the tal rb wavegude s splt to two 5-degree tlted wavegudes as show Fg. 4. The logtudal dmeso s h µm. The other structure parameters are the same as the above straght rb wavegude. The fudametal TE mode of the rdge wavegude of wdth wµm at.55µm wavelegth s used as the excted feld at z. The propagato step sze s z.µm. The feld patter at z 3µm calculated by DMI ad CJI are depcted Fg. 5(a) ad 5(b) respectvely. Due to the hgh effectve loss the propagato medum the complex Jacob method performed the propagato oly 5.9 secods whle DMI requred 68.9 secods. Table. Quattatve comparso of rutmes of the drect matrx verso ad the complex Jacob terato for WA beam propagato wavegude (WG) structures Structure D 3D Tlted WG Y-ucto Straght rb WG Y-brach rb Method WG WG DMI 96.5 s 44.8 s 77.9 s 68.9 s CJI 84.8 s 584. s 4.7 s 5.9 s Table shows the performace of the two methods for the optcal wavegude structures chose here. It s clearly see that the rutme of the teratve method s substatally lower tha that of the DMI method. For large problems requrg very large storage space ad also for structures wth a log path legth wth small propagato step sze that requre frequet matrx versos the DMI techque s umercally very tesve. I cotrast for typcal choces of z the CJI techque offers rapd covergece ad shorter rutmes. Fg. 4. Y-brach optcal rb wavegude # $5. USD Receved Aug 8; revsed 7 Sep 8; accepted Oct 8; publshed Oct 8 (C) 8 OSA 3 October 8 / Vol. 6 No. / OTICS EXRESS 79

10 (a) (b) Fg. 5. Itesty cotours for TE mode propagatg a 3D Y-brach rb wavegude at z 3µm calculated by (a) DMI ad (b) the CJI method. 4. Coclusos A ew complex Jacob teratve method adapted for the soluto of 3D WA beam propagato has bee preseted. We proposed the modfed adé approxmat of the WA beam propagato operator that gves the waves propagatg the evaescet rego the desred dampg. The resultg approach allows accurate approxmatos to the true Helmholtz equato. Furthermore a quattatve comparso of rutmes betwee the tradtoal drect matrx verso ad the ew complex Jacob teratve method for both D ad 3D WA beam propagato demostrates covcgly that the complex Jacob teratve method s very compettve for demadg problems. Ths soluto techque wll also eable a developmet of hgher order 3D adé approxmat-based WA-beam propagato algorthms usg the multstep method [] whch wll be preseted a future publcato. Acowledgmets arts of ths wor were performed wth the cotext of the Belga IA proect hotocs@be. R. Godoy-Rubo would le to tha the Spash CICYT roect TEC6-868 the Adalusa CICYE roect TIC-946 ad the Jua de la Cerva" Natoal Fellowshp program. # $5. USD Receved Aug 8; revsed 7 Sep 8; accepted Oct 8; publshed Oct 8 (C) 8 OSA 3 October 8 / Vol. 6 No. / OTICS EXRESS 73