A simple three dimensional wide-angle beam propagation method

Transcription

1 A sipe three diensiona wide-ange ea propagation ethod Changao Ma and Edward Van Keuren Departent of hsics Georgetown Universit Washington DC 57 Astract: The deveopent of three diensiona 3-D waveguide structures for chip scae panar ightwave circuits LCs is hapered the ack of effective 3-D wide-ange WA ea propagation ethods BMs. We present a sipe 3-D wide-ange ea propagation ethod WA-BM using Hoekstra s schee aong with a new 3-D wave equation spitting ethod. The appicaiit accurac and effectiveness of our ethod are deonstrated apping it to siuations of wide-ange ea propagation and coparing the with anatica soutions. 6 Optica Societ of Aerica OCIS codes:.3 Nuerica approtion and anasis; 3.79 Integrated optics: Guided waves; ropagation. References and inks. M. D. Feit and J. A. Feck Jr. Light propagation in graded-inde optica fiers App. Opt D. Yevick A guide to eectric fied propagation techniques for guided-wave optics Opt. uantu Eectron. 6 S85-S R. Scaroino A. Gopinath R. rega and S. Hefert Nuerica techniques for odeing guided-wave photonic devices IEEE J. Se. Top. uantu Eectron J. Van Roe J. van der Donk and. E. Lagasse Bea-propagation ethod: anasis and assessent J. Opt. Soc. A W. H. ress B.. Fanner S. A. Teukosk and W. T. Vettering Nuerica recipes: The art of scientific coputing Caridge Universit ress New York G. R. Hade Transparent oundar condition for ea propagation Opt. Lett Y. Chung and N. Dagi Assessent of finite difference ea propagation Opt. Lett M. D. Feit and J. A. Feck Jr. Anasis of ri waveguides and coupers the propagation ethod J. Opt. Soc. A. A C. Vassao Reforuation for the ea-propagation ethod J. Opt. Soc. A. A J. Gerdes and R. rega Bea-propagation agorith ased on the ethod of ines J. Opt. Soc. Aer. A R.. Ratowsk and J. A. Feck Jr. "Accurate nuerica soution of the Hehot equation iterative Lancos reduction" Opt. Lett C. Lee D. Schu and E. Voges Three-diensiona finite difference ea propagation agoriths for photonic devices J. Lightwave Techno C. Lee and E. Voges Three-diensiona sei-vector wave-ange ea propagation ethod J. Lightwave Techno A. Shara and A. Agrawa New ethod for nonpara ea propagation J. Opt. Soc. A. A S. L. Chui and Y. Y. Lu "Wide-ange fu-vector ea propagation ethod ased on an aternating direction ipicit preconditioner" J. Opt. Soc. A. A G. R. Hade Wide-ange ea propagation using adé appront operators Opt. Lett G. R. Hade Mutistep ethod for wide-ange ea propagation Opt. Lett H. J. W. M. Hoekstra G. J. M. Krinen and. V. Laeck On the accurac of the finite difference ethod for appications in ea propagating techniques Opt. Coun Z. Ju J. Fu and E. Feng A sipe wide-ange ea-propagation ethod for integrated optics Microwave Opt. Techno. Lett #695 - $5. USD Received 3 March 6; revised Ma 6; accepted 5 Ma 6 C 6 OSA 9 Ma 6 / Vo. No. / OTICS EXRESS 668

2 . J. Yaauchi J. Shaa O. Saito O. Ucha and H. Nakano Iproved finite-difference eapropagation ethod ased on the generaied Dougas schee and its appication to seivector anasis J. Lightwave Techno S. Suecki. Sewe T. M. Benson and. C. Kenda Nove ea propagation agoriths for tapered optica structures J. Lightwave Techno Y.. Chiou and H. C. Chang Bea-propagation ethod for anasis of -dependent structures that uses a oca oique coordinate sste Opt. Lett D. Z. Durdevic T. M. Benson et. a Fast and accurate anasis of 3-D curved optica waveguide coupers J. Lightwave Techno Introduction The finite-difference ea propagation ethod FD-BM is an effective and wide used technique for the stud of optica waveguide devices [-]. It is ased on a nuerica soution of the scaar Hehot equation using the Crank-Nichoson schee [5] which is unconditiona stae. In addition the FD-BM akes use of the high efficient Thoas agorith [5] the transparent oundar condition TBC [6] which efficient suppresses the refections at the edges of the finite coputationa window and can e easi ipeented the aternating-direction ipicit finite difference ethod ADIFDM [5] which can great sipif the siuations for 3-D structures spitting 3-D equation into two steps of -D siuation and the sow varing enveope approtion SVEA. The SVEA sustant sipifies the agorith negecting the second-order derivative with respect to in the wave equation. However it iits the siuations to para eas aong the ais and ow refractive inde contrast ratio etween the core and cadding of the waveguide [3 7-9]. Fortunate severa schees have een proposed to rea these iitations for wide-ange WA siuations such as the ethod of ines BM [] the Lancos reduction ethod [] the Eigenode epansion BM [] and the atri epansion BM [] the power series epansion ethod [3] the coocation ethod [] the BM ased on aternating direction ipicit preconditioner [5] the adé appront schee and the utistep ethod [6 7] and the BM using Hoekstra s schee [8]. Each of the has soe advantages over the others ut aso has drawacks. For eape the Lancos ethod has convergence proes the ethod of ines requires that Eigenfunction epansions e cacuated for each different cross section structure the adé appront schee increases the atri andwidth in higher-order adé operators etc. A these puished ethods can e appied to -D wideange siuations ut on a few have een proposed epicit for 3-D structures. Even in coercied software packages such as Bearop TM 3-D wide-ange cacuation can on e perfored using up to adé appront and for ore accurate cacuation on saer step sies is suggested. Because of the utiit of 3-D waveguide structures espec in the deveopent of chip scae panar ightwave circuits effective 3-D agoriths need to e deveoped. In this artice we propose a sipe and practica 3-D BM using Hoekstra s schee aong with a new 3-D wave equation spitting ethod.. Foruation The 3-D scaar Hehot equation is given E E E k n E where k denotes the free-space wavenuer E ep iω t is the eectric fied coponent with anguar frequencω and n is the refractive inde profie. If we assue the sow varing enveope approtion SEVA then E can e separated into two parts: the cope fied apitude the a sow varing enveope #695 - $5. USD Received 3 March 6; revised Ma 6; accepted 5 Ma 6 C 6 OSA 9 Ma 6 / Vo. No. / OTICS EXRESS 669

3 ter and a propagation factor ep-ik n the rapid varing phase ter. The fied is then epressed as ep n ik E where n is the refractive inde in the cadding. Sustituting Eq. into Eq. ieds 3 With n k a and n n k. In the para approtion one assues / / >> and so negects the second order derivative / in Eq. 3 ieding The FD-BM cacuates how an input optica fied distriution propagates through a waveguide first discretiing the fied into a grid in the pane perpendicuar to the propagation direction. The fied is then nuerica propagated aong the -direction to the net section. This enaes the wave equation to e soved nuerica for waveguide structures which cannot e soved anatica. The eectric fied at the grid point of and is epressed. First we discretie Eq. 3 in the direction as 5 In order to spit the equation to the stage where the tridgona atri equations a e fored we app the approtion / where / is the interedte fied etween the steps of and in the propagation direction. Eq. 5 can then e rewritten as / / / / / / 6 Using the aternating-direction ipicit finite difference ethod ADIFDM [5] this can e further spit into two equations: / / / / 7a / / / / 7 In the -D agorith [8 9] / can e repaced the epression otained fro the para wave equation. However in the 3-D agorith if / in Eqs. 7a and 7 is direct repaced the para wave Eq. oth and wi e present in a steps at direction which resuts in the equations ecoing intractae with the efficient Thoas ethod. Therefore the para equation needs to e separated. Equation. can e rewritten as #695 - $5. USD Received 3 March 6; revised Ma 6; accepted 5 Ma 6 C 6 OSA 9 Ma 6 / Vo. No. / OTICS EXRESS 67

4 8 Using the sae approtion for the haf-step fied / Eq. 8 ecoes / / / 9 Again using the ADIFDM Eq. 9 can e spit into two equations / / / a / / / Fina Eq. is separated into the foowing three equations a / / / c Eqs. were aso derived J. Shaa in a different wa and successfu appied to iprove the accurac of BM []. The phsica significance is that the effects fro the second order derivatives in and are separated into two distinct haf-step propagations. Repacing Eq. 7 Eq. we get the foowing two wide-ange equations / / / a / / / where a i ± and h a i a i 8 ± ± ± with or and h for / for and for respective. In this agorith the second-order derivative with respect to is shown as the difference etween two first-order derivatives with respect to which are fina repaced the second-order derivatives in transverse directions. As a resut the truncation error of this agorith is aintained at O O in transverse directions This however is not ensured at the optica interfaces as is in the cassica BM. The eectric fied at the interedte step / is first cacuated using Eq. a and then the fied at can e otained using the resuts of Eq. a and Eq.. Since Eqs. a and are tridgona the can e soved efficient using the Thoas ethod coined with Transparent Boundar Conditions TBC. Higher accurac is epected ecause the second-order derivative with respect to is incuded in the new schee. 3. Siuations and discussions In order to show the appicaiit accurac and effectiveness of our ethod we first copare the siuations of a 3-D Gaussn ea propagating in free space unit refractive inde #695 - $5. USD Received 3 March 6; revised Ma 6; accepted 5 Ma 6 C 6 OSA 9 Ma 6 / Vo. No. / OTICS EXRESS 67

5 with a 3-degree tit with respect to ais and aong a direction id wa etween the and aes. Both our technique and the cassica para BM are copared with the anatica resuts. The free space waveength for this siuation is.85 µ and the input Gaussn ea profie has a waist radius w 3. µ at. The siuations are perfored in a window of [- µ 55 µ] and [- µ 55 µ] with the step sies of.5 µ and. µ. Figure shows the input at and the outputs at 6 µ in the sae siuation window in order to faciitate the coparison of their reative positions and odui. As is shown our resut in green has the coser oduus to the eact soution in ue with a reative error in L nor of.% ut with a sight shift of 3.66% in position whie the para BM cacuation in red propagates at the wrong ange and incorrect preserves the setric Gaussn shape rather than the spread out asetric profie. In our ethod the para approtion is sti ipicit at each step of the siuation even though the second-order derivative with respect to is incuded in the derivation which causes the sight isatch etween our resut and the eact soution. It woud Fig.. Moduus profies of the input and outputs resuting fro the propagation of a 3-D Gaussn ea with a 3-degree phase tit a distance of 6 µ through free space. The ea is init centered at as shown in eow with pink eshes. The para output is in red the wide-ange output is in green and the eact anatica output is in ue. We aso anae the propagation in a singe-ode channe waveguide again tited at 3- degrees with respect to ais and 5 degrees with respect to the and aes. The diensions of the channe are µ µ and the refractive inde of the core and the cadding is.55 and.56 at the waveength of.85 µ respective. The siuations are perfored using the sae step sies and the sae siuation window as in the case of the Gaussn ea propagation aove. Figure shows the input at and the outputs at 6 µ in the sae siuation window. Our resut in green has a sir profie to the anatica cacuation in ue with a reative error in L nor of 5.% with again a sight shift of.9% in position. The red profie is the resut of the para BM. Sir to the case with the free space propagation it is not propagating at the right direction and its profie is spreading out ore quick even inside a waveguide channe. #695 - $5. USD Received 3 March 6; revised Ma 6; accepted 5 Ma 6 C 6 OSA 9 Ma 6 / Vo. No. / OTICS EXRESS 67

6 Fig.. Moduus profies of the input and outputs resuting fro the propagation of the ode in a singe-ode waveguide channe with a 3-degree tit with respect to ais a distance of 6 µ. The ea is init centered at as shown in eow with pink eshes. The para output is in red the wide-ange output is in green and the eact anatica output is in ue. Siuations were perfored using different is set to equa and. For a fied in the eapes aove changes in in the range fro.5 µ to.5µ on resut in sight or even indistinguishae differences in output. Figure 3 shows the dependences of the reative L nor errors on the grid spacing of the output fied odui and the reative position shift of the output fied profie with respect to the eact soutions at 6 µ of the Gaussn ea propagation and ode propagation in the waveguide. The siuation cacuated using our wide-ange BM has uch etter reative L nor errors and reative position shifts than the cassica BM for the Gaussn ea propagation in unifor ed. It Reative Error % icrons a osition Shift % icrons Fig. 3. a The reative L nor error; the reative position shifts. A are cacuated with respect to the eact soutions of the fied oduus at 6 µ using different is set to equa. The cacuations using our wide-ange BM are in green and cacuations using the cassica BM are in red. The pots with trnges are for the Gaussn ea propagation and those with squares are for the ea propagation in waveguide. #695 - $5. USD Received 3 March 6; revised Ma 6; accepted 5 Ma 6 C 6 OSA 9 Ma 6 / Vo. No. / OTICS EXRESS 673

7 aso has etter resuts for the waveguide ea propagation. The errors cacuated for the waveguide ea propagation using the cassica BM change irreguar ecause the cassica BM does not proper account for the wide-ange ter the second-order derivative with respect to which causes an asetrica output rather than the setric anatica profie. The truncation error in this case is overwheed and the L nor error no onger varies in a predictae fashion. The pots aso suggest sa transverse step sies are preferred. Both the errors and position shifts otained using our wide-ange BM for the ea propagation in the waveguide are higher than for the Gaussn propagation in unifor ed and ecoe worse for higher refractive inde contrasts etween the core and cadding. This is ecause the truncation error O O of the agorith is not ensured at the transverse discontinuities. Our agorith can e direct appied to a sei-vector schee to overcoe this proe starting with the sei-vector Fresne wave equation foowing a sir derivation as aove. The etension to a fu-vector treatent is not triv and is current under investigation. A coparison with other vartions of the BM such as the eapes siuated in Ref. [5] and references therein is difficut ecause these eapes have -varnt structures which cannot e accurate cacuated on using wide-ange BMs incuding our ethod. It is coon known that cassica BMs and wide-ange BMs cannot suppress the nuerica refections and the staircasing approtion for -varnt structures such as tapers and Y-ranches which require spec BM agoriths [-3]. These agoriths tpica invove nonorthogona coordinate sstes. It is ike that a coined use of these structure reated agoriths and wide-ange agoriths woud iprove the accurac even for wideange -varnt structures. The wide-ange BMs ased on adé appront operators and the utistep ethod deveoped G. R. Hade [6 7] are the ost coon epoed techniques to iprove the nuerica accurac for wide-ange siuations of -D structures. However when the are appied to 3-D structures arge sparse atrices wi e invoved. The atri ageraic equation is no onger tridgona and needs to e soved using iterative ethods such as the i-conugate gradients ethod [5]. Iterative ethods have convergence proes for iconditioned sparse atrices and cannot e appied to ver wide-ange structures which usua resut in i-conditioned sparse atrices. In suar we derived a sipe and practica 3-D wide-ange BM and deonstrated its utiit for two cases: the wide-ange propagation of a Gaussn ea with a 3 degree tit in free space and the wide-ange propagation in a singe-ode waveguide channe 3 degree tited with respect to ais. Our resuts have uch etter atches in oth position and oduus with the eact soutions than the para cacuation. It can e easi ipeented ecause its tridgona for is sae as the cassica 3-D para BM ecept for the coefficients. This ater is ased upon work supported the Nationa Science Foundation under Grant No The authors aso epress gratitude to rofessor A Liu in the Departent of hsics at Georgetown Universit for her hepfu discussion. #695 - $5. USD Received 3 March 6; revised Ma 6; accepted 5 Ma 6 C 6 OSA 9 Ma 6 / Vo. No. / OTICS EXRESS 67