Ultra-Wide-Angle Beam Propagation Method Based on High-Order Finite-Difference

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1 INTENATIONA JOUNA OF MICOWAVE AND OPTICA TECHNOOGY 9 VO., NO., JUNE Ultra-Wde-Angle Beam Propagaton Method Based on Hgh-Order Fnte-Derence Hua Zhang *, We-Png Huang, and Nng-Nng Feng Department o Electrcal and Computer Engneerng, McMaster Unversty, Hamlton, ON, 8S 4K, Canada Tel: ext.78; Fax: ; Emal: zhangh3@mcmaster.ca Materals Processng Center, Massachusetts Insttute o Technology, Cambrdge, MA 39 USA Abstract- The true ourth-order nte-derence scheme s ncorporated nto an ultra wde-angle beam propagaton method (BPM) based on Padé seres expanson. A comprehensve study or the accuracy o varous nte-derence (FD) ormulas n lterature or the wde-angle beam propagaton method s presented. It s demonstrated that the new hgh-order FD wde-angle BPM s hghly accurate or ecent smulaton o wde-angle eld propagaton and s superor to the other FD schemes n the case o hgh ndex contrast and/or ultra wde-angle propagaton. Index Terms- Beam propagaton method (BPM), dscontnutes, nte derence methods, stepndex optcal wavegudes. I. INTODUCTION Beam propagaton method (BPM) based on nte-derence (FD) schemes s one o the most popular numercal technques or smulaton o electromagnetc eld propagaton n optcal wavegudes and photonc ntegrated crcuts. The wde-spread acceptance and applcaton o the FD-based BPM are prmarly due to ts relatve ease o mesh mplementaton n comparson wth other methods such as the nte element methods (FEM) []. It s well-known that one o the man lmtng actors or the accuracy o the FD-BPM arses rom the nte derence schemes or the second-order dervatves n the transverse drecton. The conventonal central derencng [] s o second-order n the transverse mesh dscretzaton (.e., Δx and Δ y ). Several mproved ormulatons wth lower truncaton errors have been derved or the optcal wavegude mode solvers [3]-[]. Parallel development has also occurred to the beam propagaton methods, where most o the applcatons o hgh-order FD ormulas have been restrcted to the paraxal approxmatons [7]-[9]. Yamauch et al. rst ntroduced the Douglas scheme to a wde-angle FD-BPM [], [] n whch the truncaton error s reduced to the ourth order or the case o graded-ndex wavegudes. The dscusson s lmted to the TE polarzaton only. Vassallo ormulated a wde-angle algorthm based on hgh-order ormulas by usng Taylor seres expansons to approxmate the exponental uncton o the square root operator [], [3]. The treatment o dscontnutes n the reractve ndex results n the second-order accurate FD ormula even at the dscontnutes as long as the ndex nteraces le mdway between the adjacent grd ponts. Hadley proposed a quas-ourth-order scheme [4] or step-ndex wavegudes n whch the grd ponts concde wth the delectrc boundary, and appled ths scheme or the wdeangle beam propagaton usng the Padé (,) approxmaton [5]. The hgher-order eld dervatves are evaluated through the Helmholtz propagaton equaton and complex averagng technques. So ar, no attempt has been made to generalze the wde-angle algorthm to the ultra wde- angle propagaton schemes nvolvng hgher-order Padé approxmaton. Chou et al. derved a ourth-order accuracy FD ormula by the Taylor seres expanson and matchng the nterace condtons or a step-ndex prole regardless o the exstence o the nterace []. Derent rom the prevous hgh-order nte derencng schemes, the ormulas developed n [] mantans the ourth-order IJMOT---79 ISAMT

2 INTENATIONA JOUNA OF MICOWAVE AND OPTICA TECHNOOGY 93 VO., NO., JUNE truncaton error even n the presence o ndex dscontnutes. Because o ts general and smple algebrac orm, Chou s truly ourth-order scheme can be appled to both mode solvers and propagaton analyses. In ths work, by takng advantage o the Padé approxmaton to the square root operator [7], we have successully ncorporated the truly ourth-order ormula nto an ultra-wde-angle scheme based on Padé seres expanson. In the case o D wavegude structures, the hgh-order FD dscretzaton and the mplementaton o the PM numercal boundary condtons lead to a trdagonal matrx and can be readly solved. Both TE and TM polarzatons are consdered or wavegude structures wth hgh ndex contrast and/or ultrawde angle propagaton. For the sake o comparson, the conventonal central-derencng scheme and several other hgh-order ntederence ormulas are also ncorporated nto the BPM and ther scope o valdty and degree o accuracy are accessed n a systematc ashon. Secton II gves the procedure or the combnaton o hgh-order ormulas wth the ultra-wde-angle algorthm. Varous ormulas are assessed through numercal results n secton III. Secton IV summarzes the man conclusons. A. Governng Equaton II. FOMUATIONS For the sake o smplcty, we conne our dscussons to two-dmensonal structures throughout ths paper. I the reractve ndex vares slowly along z (the propagaton drecton), the governng Helmholtz equaton n terms o the transverse electrc elds s gven by [8] z + P E = () where E represents E y or TE polarzaton (y polarzed transverse E eld) or E x or TM polarzaton (x polarzed transverse E eld), and the operator P s dened as + kn x P = ( n ) + kn x n x TE polarzaton TM polarzaton () where k s the vacuum wave vector and n = n( x, s the reractve ndex o the medum. The tme dependence o the eld s assumed to j t be e ω. By assumng the wave propagates along + z drecton, the eld E ( x, can be expressed as jknz E( x, ( x, e = (3) where ( x, represents the slow varyng eld and n s the reerence reractve ndex. Substtutng (3) nto (), we obtan the Helmholtz equaton wrtten n terms o the slow varyng eld jk n + knp = z z where the operator P takes the orm o + k ( n n) kn x TE P = ( n ) k + ( n n) TM kn x n x (4). (5) Ignorng the backward eld yelds the ollowng one-way Helmholtz equaton [5] = jk z ( + ). () n P B. Ultra- Wde-Angle Scheme Based on Padé Seres Approxmatons Note that () contans the square root operator whch s not amendable to drect numercal soluton wthout ratonalzaton. In ths respect, IJMOT---79 ISAMT

3 INTENATIONA JOUNA OF MICOWAVE AND OPTICA TECHNOOGY 94 VO., NO., JUNE several approaches have been proposed to resolve ths problem n both underwater acoustcs [7], [9] and ntegrated optcs [5], [], []. Snce the emphass o ths paper s the applcaton o the hgh-order FD ormulas to the ultra wdeangle scheme, we employ the ecent mult-step method based on the Padé seres approxmaton []. Gven the eld at the propagaton dstance z, the analytcal soluton o () at z + Δz s ( z+δ = exp jknδ z( + P ) (.(7) The square root operator on the rght sde o (7) s approxmated by [] where a Χ + Χ + (8) a k, m = sn m + m k, m k= + bk, mχ kπ m + (9) ollowng approxmate equaton n whch only the transverse derental operator s retaned + c P + Δ = + Δ + c P where km, [ z k z/ m] [ z ( k ) z/ m] * km, c k, m bk, m ak, m (3) jk n Δz =. (4) C. Hgh-Order FD Formulas To obtan the numercal soluton o (3), FD ormulas are requred to replace derental operators. Followng the procedure proposed by Chou et al. [], we derve the E-eld ormulaton as ollows. Consder the three consecutve ponts shown n Fg., where reractve ndex dscontnutes exst between samplng ponts. b k, m kπ = cos. () m + The mult-step scheme s acheved by rewrtng (7) as m akm, P ( z+δ = exp jknδz ( k bkm, P = +, () m akm, P = exp( jknδz ) ( + b P k = km, rom whch the kth step takes the orm [ z+ kδz/ m]. () [ ] a P = jk n Δz z + k Δz m km, exp( ) ( ) / + bkm, P Wth the Crank-Ncholson scheme [3] or the exponental uncton n (), we obtan the Fg.. Schematcs o grd ponts wth dscontnutes. Usng the Taylor seres expanson wthn a unorm medum, s expanded n terms o as 3 3 p p p = ! x! x 3! x. (5) p p Oh 4 5 4! x 5! x Smlarly, + s expressed n terms o as IJMOT---79 ISAMT

4 INTENATIONA JOUNA OF MICOWAVE AND OPTICA TECHNOOGY 95 VO., NO., JUNE q q q = + + +! x! x 3! x q q Oh 4 5 4! x 5! x () To guarantee the ourth-order accuracy, we let the reractve ndex dscontnuty le mdway between samplng ponts,.e., p = q = h /. The boundary condtons requre that = θ (7) where k ( n n ) η = +. By substtutng (7), (8), and ()-(5) nto () and approprately derentatng (5), + can be expressed n terms o as = ' '' (3) + 3 (4) (5) O( h ). () The ollowng expresson can be smlarly obtaned: where = (8) ' ' = e + e + e + e ' '' (3) 3 + e + e + O h (4) (5) 4 5 (7) TE polarzaton θ =. (9) n / n+ TM polarzaton The hgher-order dervatves o and are connected by the one-dmensonal Helmholtz equaton β = + k n. () x From (7) and (), we have + kn + = + kn θ () x x or '' '' = θ + η. () Smlarly, the hgher-order dervatves o are connected as (3) (3) ' and = + η (3) (4) (5) (4) '' ( + η + η ) = θ (4) = (5) + (3) ' + η η (5) where the coecents are gven n Appendx. The ' '' derence orms o and are derved as ollows by gnorng the hgher-order terms contanng,, and n () and (7) (3) (4) (5) + e e e = + Oh ' + e e D x + e e e = + Oh '' + e e Dx ' Elmnatng and and (7), we obtan (8). (9) (5) smultaneously n () + e e e e e = '' (3) (4) 4 g g O( h ) '' 4 g g O h = x x (3) where g and g are gven n Appendx. eplacng the rst and second derental operators n the brackets o (3) wth (8) and (9), respectvely, we nally obtan the ourth- IJMOT---79 ISAMT

5 INTENATIONA JOUNA OF MICOWAVE AND OPTICA TECHNOOGY 9 VO., NO., JUNE order FD ormula: '' Dx 4 = + O( h ) (3) + g D + g D x x whch s reerred to as the true FD4 scheme n ths paper. I we gnore the reractve ndex dscontnutes, (3) s reduced to '' δ (3) Δx ( + δ /) where δ = + +. Equaton (3) s o the 4 th -order accuracy only or transverse wavegude structures wth graded ndex prole and/or weak ndex derence and thereore reerred to as the quas-fd4. Ths FD ormulas are employed n [], [], and [3]. The second-order FD ormula (reerred to as the FD) can be obtaned when the hgh-order terms up to thrd dervatve are retaned n the dervaton. Dong so leads to the ollowng expresson: b + b a a b a = + Oh '' + ab ba. (33) The coecents n (33) are summarzed n Appendx. By neglectng the reractve ndex dscontnuty n the coecents o (33), the conventonal central derence (CD) schemes [4] are obtaned as: + Δx '' + or TE polarzaton and a b Δx + c '' + or TM polarzaton, where (34) (35) n a = n + n (3) b = + n (37) n + n n + n+ n c = n n. (38) All these FD ormulas,.e., the true FD4, the quas-fd4, the FD, as well as the conventonal CD schemes, wll be appled to the wde-angle beam propagaton method and compared n terms o accuracy under derent stuatons. III. NUMEICA ESUTS AND COMPAISONS As the rst example or comparson, we smulated the radaton rom a lne source n ree space, whch propagates n all drectons evenly. As such, ths example provdes us wth an excellent opportunty to check the accuracy o the derent numercal solutons ntutvely and precsely. The exctaton s generated by the Hankel uncton [5]. Smulatons are carred out based on the central derencng (CD) and the ourth-order derencng (FD4) by usng the wde-angle BPMs wth derent Padé orders. Snce the reractve ndex s unorm, there s no derence between the true and quas FD4 and between the CD and the FD schemes, respectvely. As the ocus o ths study s on perormance o the FD-based wde-angle schemes, both coarse mesh and ne mesh are used or the sake o comparson. The computaton wndow s chosen as µm, n whch a PM [] o 5 µm s placed adjacent to the edge o the wndow. At the edge o the wndow, the conventonal transparent boundary condton s employed. In the smulaton, we propagate the eld orgnated rom the lne source along +z drecton or total o µm wth a longtudnal step sze equal to Δ z =.5 µm. The wavelength s λ =.55 µm. Fgures -(d) show the eld patterns smulated by the paraxal scheme n whch the CD and the IJMOT---79 ISAMT

6 INTENATIONA JOUNA OF MICOWAVE AND OPTICA TECHNOOGY 97 VO., NO., JUNE FD4 are used wth coarse ( Δ x =.4 µm) and ne ( Δ x =.5 µm) meshes, respectvely. It s noted that the paraxal schemes do not reproduce the correct radaton patterns as expected. Further, we see that, whle the ne mesh smulatons o both FD schemes yeld the same results, consderable derence s noted or the coarse mesh calculatons. The errors nherent n the paraxal approxmaton are reduced successvely by ncreasng the Padé order n the wde-angle scheme and the smulaton results o Padé (,) or the CD and FD4 ormulas are llustrated n Fg. 3-(d) or the coarse and ne meshes, respectvely. The wde-angle BPMs can produce the expected radaton patterns or the ne mesh case, whereas the FD4 s seen to be more accurate than the CD or the coarse mesh. In ths sense, the hgh-order FD scheme does produce more accurate results than the conventonal CD scheme or the ultra-wde angle BPMs. (c) (d) Fg.. Feld dstrbutons o the radaton rom a lne source obtaned by the paraxal BPM. CD n coarse mesh; FD4 n coarse mesh; (c) CD n ne mesh; (d) FD4 n ne mesh. Coarse mesh Δx=.4 µm, ne mesh: Δx=.5 µm, and longtudnal step-sze: Δz=.5 µm. IJMOT---79 ISAMT

7 INTENATIONA JOUNA OF MICOWAVE AND OPTICA TECHNOOGY 98 VO., NO., JUNE To gan some quanttatve results or the comparson, we calculate the relatve errors or the ntenstes and the phases o the radaton eld as unctons o Padé orders. The results obtaned by the two derencng schemes are compared n Fg. 4. It s observed that the relatve errors decrease rapdly as the Padé order ncreases and become stablzed at Padé (3,3). The conventonal central derencng (CD) and the ourth-order derencng (FD4) produce smlar results n the case o ne mesh ( Δ x =.5 µm). In contrast, the ourth-order derence scheme yelds more accurate results than the central derence scheme when the relatvely coarse mesh ( Δ x =.4 µm) s employed. Ths concluson s consstent wth the observaton made or the eld patterns n Fg. and 3. (c) (d) Fg. 3. Feld dstrbutons o lne source obtaned by Padé (,) wde-angle BPM. CD n coarse mesh; FD4 n coarse mesh; (c) CD n ne mesh; (d) FD4 n ne mesh. The other parameters are the same as n Fg.. Fg. 4. The comparson o CD and FD4 or coarse and ne meshes, respectvely. The relatve error or the ntensty; The relatve error or the phase. IJMOT---79 ISAMT

8 INTENATIONA JOUNA OF MICOWAVE AND OPTICA TECHNOOGY 99 VO., NO., JUNE As the second example or the comparson, we consder a step-ndex slab wavegude. It s well known that the mproper choce o reerence reractve ndex causes severe errors n the paraxal beam propagaton analyss, due to the ast phase varatons n the complex eld ( x,. A wde-angle scheme based on Padé approxmatons can allevate ths problem to some degree snce the second dervatve o ( x, s ncluded n the governng equaton. To assess varous FD ormulas, we calculate the relatve errors n the propagaton constants due to the ast oscllaton n ( x, along z or both TE and TM polarzatons by choosng n to be derent rom the mode eectve ndex. The relatve error n the propagaton constants s dened as βcalculated βexact ε = (39) β exact PM boundary condton [] s employed at the edges o the computaton wndow. The electrc elds o the undamental mode are launched as the ncdent elds. Fg. 5 shows the eects o the varaton n the reerence reractve ndex on the relatve errors o the propagaton constants or the paraxal and wde-angle schemes. The results obtaned usng the CD scheme, FD, and FD4 are presented n the same gure or comparson. Fg. 5 and are or the cases o TE and TM polarzatons, respectvely. For each polarzaton, t can be seen that the best results are obtaned wth the ourth-order FD ormula or both paraxal and wde-angle schemes. As expected, the mult-step wde-angle scheme yelds accurate results over a wder range o reerence ndex values, compared to paraxal approxmatons. For each curve the best results are acheved when n s chosen to be n e. where β exact and βcalculated are the exact and the numercally calculated propagaton constants, respectvely. β can be ound as ollows: calculated βcalculated Δφ = k n + (4) Δ where Δ φ s the phase sht and Δ s the propagaton dstance. The phase sht s extracted rom the overlap ntegral between the nput and the output elds expressed as dx output nput. (4) The reractve ndces o the core and the claddng are n =. 5 and n =., respectvely. The wavelength s λ =. µm. The core wdth s chosen to be D =. 47 µm so as to orm a sngle-mode wavegude. The eectve ndces or TE and TM polarzatons o ths wavegude are n e.33 and n e. 495, respectvely. The step szes are chosen to be Δx =. 75 µm and Δz =. µm. The wdth o the computaton wndow s W = µm. The IJMOT---79 ISAMT

9 INTENATIONA JOUNA OF MICOWAVE AND OPTICA TECHNOOGY VO., NO., JUNE Fg. 5. Eect o the varaton n the reerence ndex on the relatve error o the propagaton constant. TE mode; TM mode. We now proceed to study the convergence o varous FD ormulas as a uncton o transverse step sze Δ x. The reerence ndex s xed at n =.8 and other parameters are the same as n Fg. 5. The relatve errors o the propagaton constant or TE and TM modes versus Δx are shown n Fg. and. In general, the dscretzaton errors ncreases wth the ncrease o Δ x as llustrated n the curves correspondng to the wde-angle scheme based on CD and FD. It should be noted that the wde-angle scheme based on FD4 mantans hgh accuracy even wth very coarse grds. In paraxal cases, t s observed that the results obtaned wth varous FD ormulas are nearly supermposed. Ths ndcates that the man source o error arses rom the paraxal approxmaton. Fg.. elatve error n the propagaton constant as a uncton o the transverse step sze TE mode; TM mode. To emphasze the eectveness o treatment o nterace condtons, we compare the quas-fd4 and the true FD4 schemes based on Padé (3,3) approxmaton as shown n Fg. 7 n a hgher ndex contrast wavegude wth a derent core ndex n =.. The claddng ndex n s xed to be.. We take λ =. µm and D =. 75 µm so that only the undamental mode propagates. The longtudnal step sze s Δz =. µm. The reerence ndces or TE and TM polarzatons are chosen to be.95 and., respectvely, whch are ar derent rom ther respectve eectve ndces. For both TE and TM polarzatons, the relatve error o the propagaton constants observed or FD4 s much smaller than that or quas-fd4 when the coarse grds are employed. Snce these two ormulas are obtaned rom the same order Taylor seres expansons, the mprovement o FD4 n accuracy s contrbuted to the proper treatment o nterace condtons. Fg 7. elatve error n the propagaton constant as a uncton o the transverse step sze or quas-fd4 and true FD4. TE mode; TM mode. IJMOT---79 ISAMT

10 INTENATIONA JOUNA OF MICOWAVE AND OPTICA TECHNOOGY VO., NO., JUNE Fnally, the comparson o quas-fd4 and true FD4 scheme s conducted by calculatng the mode-msmatch loss [8] as a uncton o the relatve reractve ndex derence ( Δ n/ n ) between the core and claddng regons. The wavelength s λ =. µm, the claddng ndex n s., and the relatve reractve ndex derence s set successvely at Δ n/ n =.,.,., and.. The sngle-mode condton s guaranteed by choosng the wavegude wdth to x the normalzed requency around.5. The modemsmatch loss obtaned wth paraxal BPM s plotted versus Δ n/ n n Fg. 8 at the propagaton dstance o µm wth Δ z =. µm or both coarse mesh ( Δ x =.895 µm) and ne mesh ( Δ x =.895 µm). It s conrmed that the strongly gudng structures suer larger modemsmatch loss. In the case o ne mesh, the two FD schemes appear to converge as the gudance becomes weak. In the case o coarse mesh, bg derence between the results obtaned by the two schemes s observed rom weak gudance to strong gudance, whch ndcates that the FD4 scheme s able to provde accurate results wth hgh ecency. Fg. 8. The mode-msmatch loss as a uncton o relatve reractve ndex derence or quas-fd4 and true FD4. TE mode; TM mode. IV. CONCUSION We have successully appled the ourth-order nte-derence ormula to an ecent ultrawde-angle scheme based on Padé approxmatons, n whch the Padé approxmatons can go to any hgher order. For the D cases, the resultng FD equatons are trdagonal n the orm o matrx and solvable by the standard solver such as Thomas algorthm. We have compared the accuracy o varous FD ormulas by smulatng the propagaton o a cylndrcal wave n ree space and a TE/TM mode n a step-ndex slab wavegude. It s demonstrated that the ourth-order ormulaton takes nto account nterace condtons and thereore oers hghly accurate results, especally when smulatng wavegude structures o hgh ndex contrast wth relatvely coarse grds. APPENDIX The coecents used n () are as ollows: 4 q η q η = θ O( h ) 4 (A) IJMOT---79 ISAMT

11 INTENATIONA JOUNA OF MICOWAVE AND OPTICA TECHNOOGY VO., NO., JUNE 4 pq η pq η = θ p+ + + q qη qη Oh 4 p q p q η q η = θ pq η + pq + + O h η η 3 = θ p pq p q pq p q q p qη qη Oh p pq q = θ pq pq Oh (A) (A3) (A4) (A5) q η 4 b = θ + + O( h ) (A9) 3 pq η q η 4 b = θ p + + q + + O( h ) (A) p q 4 b = θ + + pq + O( h ) (A) 3 3 p pq p q q 4 b 3 = θ O( h ) (A) The coecents a j s used n (33) can be obtaned by replacng p, q and n + wth c, d and n, respectvely. EFEENCES 5 = θ p pq pq pq 3 5 pq q Oh (A) The coecents used n (7) can be obtaned by replacng p, q and n + wth c, d and n, respectvely. The coecents used n (3) are expressed as g g e e 3 3 = (A7) e e e e 4 4 = (A8) e e The coecentsb s used n (33) are gven by j [] M. Koshba, Y. Tsuj, A wde-angle nteelement beam propagaton method, IEEE Photon. Technol. ett. vol. 8, pp. 8-, 99. []. D. chtmeyer and K. W. Morton, Derence Methods or Intal Value Problems, nd ed. New York: Interscence-Wley. [3] C. Vassallo, Improvement o nte derence methods or step-ndex optcal wavegudes, Inst. Elect. Eng. Proc.-J., vol.39, pp. 37-4, 99. [4] Y. C. Chang, Y. P. Chou, H. C. Chang, Improved ull-vectoral nte-derence mode slover or optcal wavegudes wth step-ndex proles, J. ghtwave Technol., vol., pp. 9-8, Aug.. [5] G.. Hadley, Hgh-accuracy nte-derence equatons or delectrc wavegude analyss I: Unorm regons and delectrc nteraces, J. ghtwave Technol., vol., pp. -8, Jul.. [] G.. Hadley, Hgh-accuracy nte-derence equatons or delectrc wavegude analyss II: Delectrc corners, J. ghtwave Technol., vol., pp. 9-3, Jul.. [7]. Sun and G.. Yp, Moded nte-derence beam-propagaton method based on Douglas scheme, Opt. ett., vol. 8, no. 5, pp. 9-3, 993. IJMOT---79 ISAMT

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