Evaluation of Eigenmodes of Dielectric Waveguides by a Numerical Method Based on the BPM

Transcription

1 Memirs f the Faculty f EngineeringOkayama UniversityV.29 N.2 pp March 995 Evaluatin f Eigenmdes f Dielectric Waveguides by a Numerical Methd Based n the BPM Anis AHMEDt Shinji TANIGUCHI+ Osami WADAt Ming WANGt and Ryuji KOGAt (Received January ) SYNOPSIS Fr weakly guiding dielectric waveguides the eigenmde field distributins are calculated numerically with a simple algrithm. In this numerical methd the transverse sampling space can be chsen arbitrarily and hence a narrw waveguide can be analyzed. The field satisfying scalar wave equatin is expressed by the discrete Furier transfrm and the mde eigenvalues and eigenfunctins are calculated by slving an eigenvalue equatin numerically. The validity f this methd is checked fr 2-D waveguides having step and parablic r square index distributins. It is fund that fr the well guided TE mdes f the slab waveguide the accuracy f this methd is remarkably gd but sme discrepancies are fund if the mde is near cut ff. In the prblems where the nrmalized guide index b is small cautin shuld be taken in applying the results f this numerical methd.. INTRODUCTION Recently there has been cnsiderable interest in the evaluatin f light prpagatin characteristics f dielectric waveguides used fr the ptical integrated circuits. T get reliable results it is necessary that the field distributin and the prpagatin cnstant shuld be the mde eigenfunctin and eigenvalue f the waveguide respectively. Fr sme knwn refractive index prfiles f a 2-D waveguide [] the reduced wave equatin can be slved analytically. Hwever these index prfiles are nt same as the index prfiles f a practical dielectric waveguide. Fr the realistic index distributin f a guide it is very difficult t get an analytical slutin f the reduced wave equatin. S varius numerical methds have been develped t slve the reduced wave equatin and hence t find the mde characteristics. tdepartment f Electrical and Electrnic Engineering tpresent address: Fujitsu Labratries Ltd. 64 Nishiwaki Ohkub Ch Akashi 674 Japan 49

2 5 Anis AHMED Shinji TANIGUCHI Osami WADA Ming WANG and Ryuji KOGA One f the numerical methds is the beam prpagatin methd (BPM) by which the mde eigenfunctins and eigenvalues are calculated [2]. But this methd needs vast cmputer memry fr the repeated calculatin f the cmplex field up t a reasnable prpagatin distance. Anther imprved numerical methd is develped by expressing the BPM equatin int the cupled mde equatin which is slved numerically as an eigenvalue prblem [3J. In this paper we describe a numerical methd which is simpler than the methd given in Ref. [3]. The present methd is based n the frmulatin f the BPM. All the fundamental apprximatins f this methd are same as that f the BPM but in this methd the transverse sampling space can be chsen arbitrarily. The scalar Helmhltz equatin is first apprximated t get Fresnel equatin [2] whse field is expanded by the discrete Furier transfrm. The scalar Fresnel equatin is then expressed in the frm f an eigenvalue equatin which is slved fr the mde eigenvalues and eigenfunctins. T shw the exactness f the present numerical methd the eigenmdes f a narrw waveguide and als f a wide waveguide with step and parablic r square index distributins are calculated. The btained results are cmpared with that f the exact analytical nes. It is fund that bth results agree well fr the lwer rder eigenmdes but fr higher rder eigenmdes sme discrepancy between the tw results are fund near the waveguide bundaries. Accrding t Ref. [3J these discrepancies are due t the virtual bundaries placed at the waveguide edges fr the cnvenience f the numerical calculatin. In this paper we make it clear that this cnjecture is nt true. The ilbve disagreements are ccurred when the nrmalized guide index (b) is small. Fr simplicity we limit ur discussin fr the TE mdes f a 2-D waveguide. In sectin 2 the derivatin f the eigenvalue equa- tin is given and in sectin 3 we cmpare the numerical results with the analytical nes. 2. DERIVATION OF THE EIGENVALUE EQUATION Let us cnsider that the prpagatin f light thrugh a dielectric slab waveguide is in the z-directin. The field f this waveguide is taken unifrm in the y directin (a/ay = ) ~nd is cnfined in the x-directin nly. The TE mdes f this waveguide cnsist f the field cmpnents E y Hx and Hz. Cnsidering the medium is istrpic surce-free and inhmgeneus the wave equatin fr the TE mdes can be expressed as where 'ljj(x z) = Ey(x z) Cr is the relative dielectric cnstant f the medium and k = w 2 Jlc = 2rr/ A; w is the angular frequency and k A Jl and c are the free space wavenumber wavelength magnetic permeability and dielectric cnstant respectively. In deriving the abve equatin we have cnsidered the time dependence f the field as exp(jwt). Equatin () is als called Helmhltz equatin. The relatins f E y with the ther field cmpnents (Hx and Hz) f the TE mdes are fund frm Maxwell's equatins. Our aim is t slve Eq. () with an apprximated numerical methd which is similar t the BPM. Since we are interested in the calculatin f the eigenmdes f the waveguide we assume that the structure f the waveguide is unifrm in the z-directin. In that case the relative dielectric cnstant C r will be a functin f nly ne crdinate axis x and is written as Cr = n 2 (x). If 8 is the maximum index change f the guide then a general expressin f the index distributin is given by n(x) = n s + 8f(x) (2)

3 Evaluatin f Eigenmdes f Dielectric Waveguides by a Numerical Methd Based n the BPM 5 where f(x) is the index distributin functin. The parameter (j is generally very small in cmparisn with ns. Therefre we can write (3) Let us cnsider that ne type f slutin f Eq. () may be expressed as t/;(xrz) = >(xz)exp{-jknsz} (4) where n s is the refractive index f the substrate. Substituting Eq. (4) int Eq. () and using the apprximatin that the change f the refractive index is very slight ver the distance f ne wavelength we get the parablic r Fresnel wave equatin as [2J 8 >' -j ['t72 k2 {2 2}) -..' ~ = -k-- v t + n - n s If' uz 2 n s (5) where \7~ = 8 2 /8x 2 and >' represents the field distributin f the abve wave equatin which is btained by neglecting the term cntaining the duble derivative f > with respect t z. S hereafter a prime will be used fr the parameters belng t Fresnel equatin. Let the eigenslutins f Eq. () and Eq. (5) are respectively and t/;(x z) = tm(x) exp(-jkneffz) (6) >'(xz) = t:n(x)exp(-jkn~ffz) (7) where Neff and n~ff are the effective indices fund frm Eqs. () and (5) respectively. The effective index Neff fund frm Helmhltz equatin (Eq. ()) is used t define the prpagatin cnstant f3 = kneff The effective index n~ff fund frm Fresnel equatin is related t Neff by the relatin [2J n~ff = (N;ff - n~)/(2ns)' (8) Similarly the field distributins f Fresnel equatin is related t that f Helmhltz equatin by the relatin [2J (9) Thus it is clear that the slutins f Helmhltz and Fresnel equatin will prvide us with the same eigenmde field distributins but their eigenvalues are related by Eq. (8). Substituting Eq. (7) in Eq. (5) we get n~ff >' = -k; [\7~ + k6 {n 2 - nn] >'. () 2 n s The abve equatin can again be mdified t the fllwing frm by substituting Eq. (3) in it n~ff >' = [ ':~ + f( X)] >'. () (j 2k n s {j The apprximated field >' is nw expanded by the discrete Furier transfrm as n (jj'n(z) = ~~ >'(xiz)exp(-jvnxi) (3) wherein = -(N/2-) N/2 Xi = Di/N and the transverse wavenumber Vn = 27rn/Dj N is the ttal sampling pints and D is the length f the calculated regin. In matrix frm Eqs. (2) and (3) are written as </>' F-l~' ~' = F</>' (4) where F = /N. [Fn;] = /N. [exp(-jvnxi)j and F- = [Fi~lJ = [exp(jvnxi)j. Substituting Eq. (4) in Eq. () we get n;t </>' = [qf- K F + R] </>' (5) where matrices K and R are the tw diagnal matrices with elements \7~ = 8 2 / 8x~ = -v; and f(x;) respectively and q is equal t /(2k~ns{j). Let us define the apprximated nrmalized guide index' b' as N 2 2 b' = eff - n s ~ neff (6) n~ - n; {j

4 52 Anis AHMED Shinji TANIGUCHI Osami WADA Ming WANG and Ryuji KOGA where n = n(x = ) is the cre index at x = O. The value f b' may vary frm t. Using the abve equatin we can write Eq. (5) as an eigenvalue equatin i.e. b' 4>' = P 4>' (7) where matrix P = qf- K F + R. Withut calculating the beam prpagatin in the z directin [2] we can calculate the mde eigenfunctins and the eigenvalues at a plane z = by slving Eq. (7). Since we have used the slw index variatin apprximatin the wave equatin ftm mdes can als be made similar t Eq. () fr H y ignring the term cntaining the derivative f c [5]. S the slutins fr the TM mdes will be similar t the TE mdes. 3. NUMERICAL EXAMPLES A. Eigenmdes f Step Index Waveguide The present numerical methd is first applied t a symmetric 2-D waveguide with step index distributin as shwn by the insert f Fig.. The half- and full-width f the waveguide are dented by d and T respectively. Depending n the waveguide width tw types f waveguides are analyzed. Fr a narrw waveguide (where nly fundamental mde exits) Tis taken 2m and fr a wide waveguide (where three mdes exist) T is chsen 8m. The nrmalized frequency V f the narrw waveguide is taken 2 and that f the wide waveguide is chsen 8. The value f f(x) in Eq. (9) is in the cre and in the cladding regin. The ther parameters are n s =. A = m D = 4 pm and N = 256. The maximum index change 8 can then be calculated frm the relatin V = kt(nl- n~)l/2 where nf = n is the cre index fr step index guide. In ur case 8 is fund t be The analytical eigenmde field distributins are fund frm the slutins f the wave equatin as given in Ref. [6]. The eigenmde field distributin f the narrw waveguide btained by this numerical methd is pltted in Fig. l(a) and is shwn by the dtted curve. In the wide waveguide 3 guided mdes are fund and their field distributins are pltted in Fig. (b) and are shwn ~.5.~ <U Z "'C Q).~ <U Ẹ.. Z (a) m=o b' =.4 Transverse directin (xjdj (b) Fig. Mde field distributins btained by the numerical (dtted curves) and analytical methds (slid curves) with step index prfile; (a) narrw waveguide and (b) wide waveguide. 4 m=o b'=.9 m=l b'=.6 m=2 b'=.7 - I--..._I--..._I--..._.L...--'----'

5 Evaluatin f Eigenrrwdes f Dielectric Waveguides by a Numerical Methd Based n the BPM 53 by the dtted curves. The analytical eigenmde field distributins f the narrw as well as the wide waveguide are als pltted in Fig. by the slid curves. It is seen that the dtted curves are cmpletely cincide with the slid curves fr bth types f waveguides. The well agreement f bth slutins cnfirms the validity f Eq. (7). B. Eigenmdes f Truncated Square Index ~aveguide As a secnd example the present numerical methd is applied t a symmetric slab waveguide with truncated square index disti'ibutin (Fig. 2) which is expressed as [IJ (8) / / -d d x Fig. 2 The truncated square index distributin shwn by the slid curve fr which the present calculatin is dne and the typical infinite square index prfile shwn by the dashed curve. The parameter Xn is selected such that the index at a distance Ixl = d will be equal t n s. All the parameters f the square index case except T and V are kept same as thse f the step index case explained in the Example A. Fr a narrw waveguide (single-mde waveguide) T is 4 /lm 'and V is 4 and fr a wide waveguide (multi-mde waveguide) T is chsen 2 /lm and is taken 2. The exact eigenmde field distributins are calculated by the mde matching methd. In the regin Ixl ~ d the exact field is expressed by the parablic cylindrical functin and utside this regin the field will decay expnentially. By matching the tangential field cmpnents f E y and Hz at the bundary Ixl = d the field amplitude and its prpagatin cnstant are fund. Fr a well guided mde the rder f the parablic cylindrical functin is very clse t sme integer value. In that case the parablic functin can be apprximated by Hermite-Gauss functin. Cnsidering the simplicity f the analytical slutin we used Hermite-Gauss functin t express the field in the guide. Thus within the regin Ixl ~ d the exact field is expressed by Hermite-Gauss functin as [IJ where m is the mde number H m ( is the mth rder Hermite functin and w is called the beam mdius given by (2) The parameter w indicates the degree f cnfinement f the fundamental mde. In the regin Ixl ~ d the field is expressed as A exp(-ixl). The amplitude A and the prpagatin cnstant are fund by matching the tangential field cmpnents f E y and Hz at the bundary Ixl = d and they are fund as A = H m (V2d/w) exp(-d 2 /w 2 ) exp(d) (2) = 2d _ 2Vi m Hm _ (Vid/w) (22) w 2 W H m ( Vid/w) In Fig. 3 bth the analytical (slid curves) and the numerically (dtted curves)

6 54 Anis AHMED Shinji TANIGUCHI Osami WADA Ming WANG and Ryuji KOGA btained eigenmde field distributins are pltted. The mde field distributins f the singlemde waveguide and the multi-mde waveguide are shwn in Fig. 3(a) and in Fig. 3(b) respectively. It is seen that the numerical results agree well with the analytical nes fr the lwer rder mdes f a multi-mde waveguide and is cnsistent with Eq. (7). But sme disagreements f these tw results are fund if the mde belng t a square index waveguide. Fr the fundamental mde f a single-mde waveguide as well as fr a higher rder mde f a multi-mde waveguide these discrepancies are due t the beam radius wand the nrmalized guide index b'. Using Eq. (2) the fundamental mde beam radius w f the single-mde and the multi-mde waveguide are fund t be 2. JLm and 3.46 JLm respectively. The ratis f wid fr the single- and multi-mde waveguides are. and.58 respectively. Since w indicates the degree f cnfinement f the fundamental mde smaller value f w in cmparisn t d means better cnfinement f fundamental mde. Thus the cnfinement f the fundamental mde field in the multi-mde waveguide is better than that in the single-mde waveguide. This descriptin is clearly visible in the field distributins frm = shwn in Fig. 3(a) and in Fig. 3(b). Since the guidance f the fundamental mde is weak in the single-mde waveguide sme errrs are fund in the numerical calculatins. With the increase f the mde number the cnfinement f the mdal pwer decreases. S fr higher rder mdes f the multi-mde waveguide the numerical results have als errrs near the bundary f guide. Frm = 2 such field distributin is shwn in Fig. 3(b). Fr the well cnfined mdes b' is large (fr example in ur btained results shwn in Fig. 3(b) b' is.8 when m = ) and fr the weakly cnfined mdes which are near cut ff b' is small (fr example in ur btained results shwn in Fig. 3(b) b' is.6 when m = 2). Thus with the increase f mde number '2 the mde field distributins in the guide spread ut further frm the guide axis and the numerical results are n lnger a gd apprximatin t the exact nes. -s. u a> ii= u a>.~.5 eu EL- z -8-4 (a) m = b'=.5 m=o.5 b'=.8 -s. u a> ii= u a>.~ eue L- z m= b'= Transverse directin (xjd) (b) Fig. 3 Mde field distributins fr the truncated square index prfile btained by the numerical (dtted curves) and analytical calculatins (slid curves); (a) single-mde waveguide and (b) multi~mde waveguide. 8

7 Evaluatin f Eigenmdes f Dielectric Waveguides by a Numerical Methd Based n the BPM 55 Furthermre the gradient f index distributin near the regin Ixl = d is relatively larger than that near the waveguide axis. S the value f matrix P near Ixl = d is nt as accurate as that near the waveguide axis. Thus in calculating the higher rder mde fields at the vicinity f the waveguide edges sme errr may arise in the numerical results. The authrs f Ref. [3J have claimed that the abve disagreements are ccurred due t the virtual bundaries placed at the cre and cladding interfaces f the multi-mde waveguide. But with the present calculatin these discrepancies are present even the virtual bundaries are mved t a distance equal t 3d. Frm the abve discussin it is clear that the btained disagreement between the analytical and numerical slutins are nt due t the virtual bundaries but due t the effects f b' wand "Vn. Since the present numerical methd is based n the BPM it can be inferred that the statinary mde eigenfunctins btained the BPM [2J is nt accurate enugh if the nrmalized guide index b' is small. 4. CONCLUSIONS The eigenmde field distributins f the slab ptical waveguide with step and truncated square index distributins are evaluated numerically with a simple algrithm. Fr the well guided TE mdes f the slab waveguide the accuracy f this methd is remarkably gd but sme discrepancies are fund if the mde is near cut ff. The basic apprximatins f this numerical methd are same as that f the BPM but the transverse sampling space f the present calculatin can be chsen arbitrarily. Using slw index variatin apprximatin the present numerical methd can als be used fr the analysis f the TM mdes [5]. REFERENCES [IJ T. Tamir (ed.) Guided- Wave Optelectrnics Springer-Verlag Berlin Heidelberg (99) Sectin 2.4. [2J M.D. Feit and J.A. Fleck Jr. "Cmputatin f mde eigenfunctins in gradedindex ptical fibers by the prpagating beam methd" Appl. Opt. vl. 9 n. 3 July (98) pp [3] K. On and S. Sawa "Analysis f nrmal mdes f dielectric ptical waveguides based n the prpagating beam methd" Trans. IEIGE vl. J68-C n. 2 Feb. (985) pp (in Japanese). [4] Van Rey J. Van der Dnk J. and Lagasse. P.E. "Beam-prpagatin methd: analysis and assesment" J. Opt. Sc. Am. vl. 7 n. 7 July (98) pp [5J D. Marcuse "The effect f the "Vn 2 term n the mdes f an ptical square-law medium" IEEE J. Quantum Electrn. vl. QE-9 n. 9 Sep. (973) pp [6J D. Marcuse Light Transmissin Optics Van Nstrand Reinhld New Yrk (982) Chap. 8.