Full-vectorial analysis of high-index-contrast coupled channel waveguides

Full-vectorial analysis of high-index-contrastcoupled channel waveguides

Md. Zahed M. Khan* and Husain Ali JamidKing Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

*Corresponding author: [email protected]

Received 9 March 2009; revised 15 June 2009; accepted 28 June 2009;posted 14 July 2009 (Doc. ID 108421); published 27 July 2009

Coupling of two identical channel waveguides separated by an air gap is analyzed. The coupled structureexhibits a strong refractive index contrast in both the transverse and longitudinal dimensions, whichnecessitates the use of a full-vectorial model. The 3D full-vectorial bidirectional method-of-lines beampropagation is utilized for this purpose. The effect of the transverse and longitudinal displacements onthe modal reflectivity and modal transmissivity of the fundamental TE-like and TM-like modes is re-ported. Numerical results are presented for both the full-vectorial model and the approximate semivec-torial model. A significant difference between the predictions of these two models is seen. © 2009Optical Society of America

OCIS codes: 130.2790, 230.7380, 130.0130.

1. Introduction

Guided wave structures are essential elements in op-tical communications for the purpose of filtering, cou-pling, switching, modulating, power splitting, etc.They have been investigated widely in 2D work-spaces utilizing various numerical techniques and inboth low-high-index-contrast material systems [1–7].However, it is well known that high-index-contraststructures cannot be well approximated by using 2Dmodels and thus require a rigorous 3D workspace foranalysis. The advent of fast processors and largememory storage capabilities has initiated the devel-opment and improvement of various numerical tech-niques to model different waveguide problems in a3D workspace rather than the 2D environment. Thisincludes the waveguide facets and step discontinu-ities utilizing the finite difference time domain meth-od [8], radiation modes method [9], transfer matrixmethod [10], a field-based method [11], finite ele-ment method [12] and bidirectional method-of-linesbeam propagation method (MoL-BPM) [13]. More-over, multidiscontinuity problems, for instance,coupled waveguides, tapered waveguides, and polar-

ization rotators, have been addressed by employingthe free space radiation mode method [14], modematching method, [15] and method of lines [16].However, these reported waveguide problems arebased in general on low-index-contrast material sys-tem under the 3D semivectorial approximation. Im-proved 3D numerical techniques that are based onthe transverse magnetic formulation and the full-vectorial nature of the field rather than semivectorialapproximation have recently been reported [17,18].These MoL-BPM techniques account rigorously forthe longitudinal boundary condition and are, in gen-eral, numerically efficient. With these promising fea-tures, they can be applied to both low-and high-index-contrast waveguide problems. They have beenapplied to high-index-contrast waveguide facets [17],coupled waveguides, and polarization rotators [18] toshow the effectiveness of the techniques rather thaninvestigating them, in general.

In this work, we propose to analyze the high-index-contrast coupled channel waveguides utilized in [18].The arrangement consists of two aligned raisedchannel waveguides separated by an air gap formingtwo strong longitudinal discontinuities. The struc-ture may represent the practical problem of deter-mining the transmission from a laser, across an air

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4468 APPLIED OPTICS / Vol. 48, No. 22 / 1 August 2009

gap, and into a waveguide. The influence of the trans-verse separation (vertical and lateral) on the modalreflectivity and modal transmissivity of the coupledwaveguides will be presented, which closely simu-lates a misaligned laser with a waveguide. To thebest of our knowledge, such an investigation has notbeen reported in the literature. Numerical results ofthe fundamental TE-like and TM-like modes utiliz-ing the 3D full-vectorial MoL-BPM numerical tech-nique [18] will be reported. Moreover, the resultsobtained from both the full-vectorial and the semi-vectorial formulations will be shown for comparisonand to highlight the difference between the predic-tions of both formulations.

2. Background

A brief outline of the numerical approach that isbased on the transverse magnetic field formulationis presented. The reader is referred to [17,18] for de-tails. Consider the 3D wave equation that upon dis-cretization in the transverse dimensions x and y,using p and q sample points, respectively, yields thefollowing equation [17]:

d2

dz2�HþC2 �H ¼ �0: ð1Þ

The general solution of the above equation is �H ¼ej C z �Aþ e−jCz�B. �H ¼ ð�Hx

�HyÞT , where the column vec-tors �Hx and �Hy contain the magnetic field componentHx and Hy in discrete form, respectively. The squarematrix C2 is the characteristic matrix that accountsfor the spatial derivatives in the x–y plane as well asthe coupling between the transverse magnetic fieldcomponents Hx and Hy [17]. The square matricese�jCz account for the forward and the backward fields,respectively. The transverse electric field compo-nents, in discrete form, can be obtained from thetransverse magnetic field components by using thefollowing equation [18]:

�E ¼ S�H; ð2Þwhere �E ¼ ð�Ey − �ExÞT and where �Ex and �Ey containthe discretized values of the transverse electric fieldcomponents Ex and Ey, respectively. The square ma-

trix S ¼�jk−10

ffiffiffiffiffiffiffiffiffiffiffiffiμ0ε�1

0

q �N−2ðO − C2ÞðjCÞ−1, where N2

contains the square of the refractive index distribu-tion and O accounts for the transverse differentialoperators in discretized form. The square matrixðjCÞ−1 corresponds to the integration operator thatappears in the transverse electric field relation. Itis noteworthy that the size of the square matrix Swillbe pq × pq for the semivectorial formulation and2pq × 2pq for the full-vectorial model.Now, consider two abrupt longitudinal discontinu-

ities placed at z ¼ 0 and z ¼ d, shown in Fig. 1, withthe input transverse magnetic field assumed to be in-cident from the left at z ¼ 0. The transmitted and thereflected magnetic fields of the structure shown in

Fig. 1 can be obtained by applying the continuityof the transverse magnetic and electric fields at theselongitudinal boundaries (z ¼ 0 and z ¼ d). This re-sults in a set of four equations that can be cast intoa simple linear matrix relation as shown [18]:

Γ �X ¼ �Λ; ð3Þwhere

Γ ¼

2666664ðS1 þ S2Þ −2S2P2 0 0

0 ðS2 þ S3Þ ðS3 − S2ÞP2 0b0 ðS1 − S2ÞP2 ðS1 þ S2Þ 0b0 0 −2S2P2 ðS2 þ S3Þ

3777775

�Λ ¼

266664ðS1 − S2Þ�A1

�0

2S1 �A1

�0

377775:

The column vector �X ¼ ½�B1�B2�A2�A3�T contains therequired reflected (�B1) and transmitted (�A3) mag-netic fields in addition to the fields within the discon-tinuous structure. The term �A1 appearing in theknown column vector �Λ represents the incident mag-netic field, while the square matrix P2 ¼ ejC2d is thepropagation matrix. The square matrix C, which isrequired for obtaining the matrix S, has been ob-tained by using Padé approximants of the squareroot operator, and the matrix P has been obtained byusing Padé approximants of the exponential operator[7,18]. Equation (3) is a simple matrix relation thatcan be solved iteratively utilizing the biconjugatestabilized algorithm (BI-CGSTAB)[19]. However, sol-ving Eq. (3) directly degrades the performance of BI-CGSTAB, leading to an unacceptably large numberof iterations or even failure to achieve convergence.To overcome this difficulty, Eq. (3) is preconditionedin amanner similar to the one reported in [18], beforeapplication of BI-CGSTAB.

3. Numerical Results

In this section, the numerical approach is first de-monstrated for accuracy and later applied to analyzethe coupled channel waveguides utilized in [18]. Theresults of both the semivectorial and full-vectorial

Fig. 1. Two abrupt longitudinal discontinuities placed at z ¼ 0and z ¼ d.

1 August 2009 / Vol. 48, No. 22 / APPLIED OPTICS 4469

models are shown for comparison, assuming the fun-damental TE-like (TE00) and TM-like (TM00) modesto be incident on the first longitudinal discontinuityplaced at z ¼ 0. The Padé orders associated with thesquare root and the propagation operators are fixedat 4 and 8, respectively [18]. In addition, the wave-guides are surrounded by a perfectly matched ab-sorbing layer from all sides to absorb the radiativefield [20].Since the analysis requires calculating the funda-

mental TE-like and TM-like mode transmissivity(the ratio of the transmitted mode power to the inci-dent mode power) and mode reflectivity (the ratio ofthe reflected mode power to the incident modepower), the procedure used for calculating these va-lues will be briefly explained. The modal transmis-sivity Tm0m, associated with the transmitted modem0 and the incident mode m, is given by

Tm0m ¼ Ptm0=Pi

m; ð4Þ

where the incident mode power Pim and the trans-

mitted mode power Ptm0, respectively, are given by

Pim ¼ 0:5

ZZReðEi

x;mHi�y;m − Ei

y;mHi�x;mÞdxdy; ð5Þ

Ptm0 ¼ 0:5

ZZReðEt

x;m0Ht�y;m0 − Et

y;m0Ht�x;m0 Þdxdy: ð6Þ

The superscripts i and t refer to the incident andtransmitted fields, respectively, and the asterisk in-dicates the complex conjugate. The transverse elec-tric and magnetic field components appearing inEqs. (5) and (6) are modal field components, as indi-cated by the subscripts m and m0. The modal fieldassociated with the incident mode m is readily avail-able, because the incident field is purely modal. How-ever, the transmitted modal field needs to beextracted from the transmitted field before applica-tion of Eq. (6). This can be done by first finding theassociated transmitted mode amplitude αtm0 and thenmultiplying it by a suitably normalized version of thetransmitted modal field. A simple numerical proce-dure that can be used to calculate the mode ampli-tudes has been reported in [16]. This method mayalso be used to calculate the modal reflectivity Rm0massociated with the reflected mode m0 and the inci-dent mode m.Now, consider the coupled buried waveguide struc-

ture shown in Fig. 2(a). It consists of two identicalburied low-index-contrast waveguides separated byan air gap of width L. All the relevant waveguideparameters are shown in the figure. The arrange-ment has been previously reported under the 3Dsemivectorial approximation [14,15]. Figure 3 showsthe variation of the modal transmissivity of the TE-like and TM-like modes of the coupled waveguides asa function of the air gap width L, based on the semi-vectorial model. The results reported in [14] and our

results are observed to be virtually indistinguish-able. This establishes the accuracy of the presenttechnique. However, it is observed that the low-in-dex-contrast material system of the selected struc-ture does not show any difference between thesemivectorial and the full-vectorial representationsof the field.

To further ascertain the accuracy of the numericalapproach under the high-index-contrast environ-ment, another coupled channel waveguide shown inFig. 2(b) is considered [18]. The structure consists oftwo identical channel waveguides separated by anair gap of width L. The arrangement exhibits strongdiscontinuities in both the longitudinal and thetransverse dimensions. All the relevant waveguideparameters are indicated in the same figure, andthe air gap width is fixed at L ¼ 0:1 μm. In this case,the waveguide parameters are unaltered except forthe waveguide core width wx, which is varied from1.0 to 5:0 μm. The calculated result of the fundamen-tal TE-like and TM-like mode reflectivity and trans-missivity as a function of the core width wx is shownin Fig. 4. The results of both the semivectorial andthe full-vectorial models are shown in the figure. No-tice that when the waveguide core width wx in-creases, the modal transmissivity and reflectivityvalues obtained by using the semivectorial andfull-vectorial models approach each other. They sub-sequently approach the calculated results of the cor-responding 2D model, as indicated by the arrowsshown in the figure. The results shown in Fig. 4further ascertain the accuracy of the technique. Itis noteworthy to mention that when the waveguidewidth wx becomes small, the results obtained fromboth the semivectorial and full-vectorial modelsshow a substantial difference in the modal transmis-sivity values [see Fig. 4(b)]. The deviation in the re-sults of the semivectorial and full-vectorial models isclearly due to the full-vectorial model’s accountingfor the minor field as well. However, the modal reflec-tivity values as predicted by bothmodels remain gen-erally in good agreement for the entire range of thewaveguide core width wx [see Fig. 4(a)].

The numerical approach demonstrated above willnow be utilized to analyze the coupled high-index-contrast channel waveguides shown in Fig. 2(b).We will first consider the effect of the air gap widthL on the fundamental TE-like and TM-like mode re-sponses. As mentioned earlier, this arrangementmay simulate power coupling from a laser, throughair, and into the optical waveguide. The waveguidecore width is fixed at wx ¼ 1:0 μm, keeping all the re-maining parameters of the structure unaltered. Thefull-vectorial transverse magnetic field patterns ofthe fundamental TE-like and TM-like modes of theinput waveguide are shown in Fig. 5. The corre-sponding calculated mode effective indices are3.45647 and 3.46508, respectively. Figures 6 and 7,respectively, show the calculated fundamental TE-like and TM-like mode responses as a function ofthe air gap width L. These two figures show the


calculated mode reflectivity, transmissivity, and frac-tion of radiated power (FPR) under the semivectorialand full-vectorial assumptions. Since the channelwaveguides depicted in Fig. 2(b) are single mode,the FPR may be calculated by using the relationshipFPR ¼ 1 −modal reflectivity −modal transmissivity,by assuming negligible polarization cross coupling.Again, a substantial difference in the calculated re-sults obtained from the semivectorial and the full-vectorial models are noticeable for the entire rangeof the air gap width under consideration. As shownin Figs. 6 and 7, the modal reflectivities associatedwith the TE-like and TM-like fundamental modes,respectively, attain their maximum values for anair gap width L ≈ 0:5 μm, which is due to constructiveinterference (Fabry–Perot cavity effect). Throughoutthis work the maximum longitudinal step size is setto Δz ¼ 0:1 μm. In addition, convergent results typi-cally require a transverse mesh density of 20 meshpoints per micrometer. The BI-CGSTAB routine re-quires about 15 to 20 iterations to converge.

Fig. 2. (a) Two identical coupled buried waveguides and (b) two identical coupled raised channel waveguides.

Fig. 3. Calculated semivectorial results of the fundamental TE-like and TM-like mode transmissivity corresponding to thecoupled waveguides shown in Fig. 2(a) at the operating wave-length λ ¼ 0:86 μm.


Further to the investigation of the coupled channelwaveguides, if the right channel waveguide (outputwaveguide) of Fig. 2(b) is displaced from its originalposition in the transverse direction (horizontal orvertical), the power coupled to the output waveguideis expected to decrease with an increase in the modalreflectivity and FPR. This may again emulates thepractical problem of a misaligned laser with an opti-cal waveguide. In order to examine this assumption,the vertical axis (y axis) of the output waveguide ismoved in the horizontal direction (along the x axis)with respect to the vertical axis of the input wave-guide. This displacement is termed the horizontaldisplacementΔx throughout this paper. With this ar-rangement, the mode reflectivity and transmissivityof the coupled channel waveguides [see Fig. 2(b)] iscalculated as a function of the horizontal displace-

mentΔx. The analysis is carried out under two situa-tions. The first situation considered is when theinput and the output waveguides have no longitudi-nal separation (i.e., L ¼ 0 μm). In the second situa-tion, the longitudinal separation between the twochannel waveguides is fixed at L ¼ 0:1 μm, resultingin a small air gap. The remaining parameters of thewaveguides are kept unchanged. The calculated re-sults are shown in Figs. 8 and 9 which correspondto the fundamental TE-like and the TM-like modes,respectively. As seen in Fig. 8, a disagreement in themodal reflectivity values of the fundamental TE-likemode is observed in general at relatively large valuesofΔx, as predicted by both the semivectorial and full-vectorial models for both situations. This discre-pancy can be easily noticed whenΔx > 0:3 μm for thefirst situation (L ¼ 0 μm) and Δx > 0:8 μm for thesecond situation (L ¼ 0:1 μm). On the other hand,the modal transmissivities as predicted by boththe semivectorial and full-vectorial models are

Fig. 4. Fundamental TE-like and TM-like (a) mode reflectivityand (b) mode transmissivity, corresponding to the coupled channelwaveguides shown in Fig. 2(b) at the operating wavelength λ ¼1:55 μm and a fixed air gap width L ¼ 0:1 μm.

Fig. 5. Magnitude of the incident fundamental TE-like and TM-like modes corresponding to the coupled channel waveguidesshown in Fig. 2(b) at the operating wavelength λ ¼ 1:55 μm andcore width wx ¼ 1 μm.

Fig. 6. Fundamental TE-like mode response corresponding to thecoupled channel waveguides shown in Fig. 2(b).

Fig. 7. Fundamental TM-like mode response corresponding tothe coupled channel waveguides shown in Fig. 2(b).


generally in good agreement throughout the range ofthe horizontal displacement Δx for the case of di-rectly coupled waveguides (L ¼ 0 μm). However, inthe case of waveguide coupling through an air gap(L ¼ 0:1 μm), the modal transmissivity values calcu-lated by the semivectorial and the full-vectorial mod-el show some discrepancy when Δx < 0:4 μm. Thefundamental TM-like mode response of the structure(Fig. 9) again depicts similar behavior of the modaltransmissivity and reflectivity curves as predictedby the semivectorial and full-vectorial models forboth situations. In Figs. 8 and 9, the modal reflectiv-ity tends to stabilize at a nonzero value when Δx ex-ceeds a value of roughly 0:8 μm, while the modaltransmissivity tends to zero in this same range.The reason for this behavior is obvious, since beyondthis relatively large horizontal displacement, theoutput waveguide is effectively replaced by air, caus-

ing the stabilized nonzero modal reflectivity. The out-put waveguide also becomes sufficiently displacedaway from the input modal field, resulting in negli-gible power coupling. Because the waveguide struc-ture is symmetrical with respect to the vertical axis;the results associated with negative values of thehorizontal displacement Δx are expected to be iden-tical to these obtained using positive values of Δx.

Another possibility for transverse separation iswhen the horizontal axis of the output waveguideshown in Fig. 2(b) is displaced vertically with respectto the horizontal axis (x axis) of the input waveguide.This displacement is termed the vertical displace-ment Δy throughout this paper. The modal reflectiv-ity and transmissivity variation with Δy are shownin Figs. 10 and 11, corresponding to the TE-likeand the TM-like modes, respectively. Again, twosituations (L ¼ 0 μm and L ¼ 0:1 μm) are consideredhere. In this case, the arrangement is asymmetric

Fig. 8. Fundamental TE-like mode response corresponding to thecoupled channel waveguides shown in Fig. 2(b) when the output(right) waveguide is displaced in the horizontal direction.

Fig. 9. Fundamental TM-like mode response corresponding tothe coupled channel waveguides shown in Fig. 2(b) when the out-put (right) waveguide is displaced in the horizontal direction.

Fig. 10. Fundamental TE-like mode response corresponding tothe coupled channel waveguides shown in Fig. 2(b) when the out-put (right) waveguide is displaced in the vertical direction.

Fig. 11. Fundamental TM-like mode response corresponding tothe coupled channel waveguides shown in Fig. 2(b) when the out-put (right) waveguide is displaced in the vertical direction.


with respect to horizontal axis; the results of both thenegative and the positive vertical displacements areshown in Figs. 10 and 11. Both of these figures showa significant variation in the modal reflectivity andtransmissivity values corresponding to L ¼ 0 μmand L ¼ 0:1 μm, as predicted by both the semivector-ial and full-vectorial models. In general, the modalreflectivity value is noticeably large when the verti-cal displacement Δy is negative (i.e., downwards dis-placement of the output waveguide). This is due tothe input fundamental mode seeing air as the secondregion whenΔy becomes more negative, whereas, forpositive values of Δy, the modal reflectivity valuereaches a steady-state value. For the situation wherethe waveguides are coupled by an air gap, the steady-state values of the fundamental TM-like and the TE-likemode reflectivities are, respectively, 0.15 and 0.1.The steady-state modal reflectivity value corre-sponding to the direct coupling of the two wave-guides, for both the TE-like mode and the TM-likemode, is observed to be very low (≈10−4). This is cer-tainly due to the input mode’s seeing the substrate ofthe output waveguide as the second region, whichcorresponds to a low-index-contrast longitudinal in-terface. In general, the modal transmissivity valuecorresponding to L ¼ 0 μm is identical, as predictedby both the semivectorial and full-vectorial models,from Figs. 10 and 11. However, the modal trans-missivity corresponding to the coupling of twowaveguides with an air gap (L ¼ 0:1 μm) shows sig-nificant difference in the range −0:2 μm ≤ Δy ≤

0:2 μm. The modal transmissivity values predictedby the full-vectorial model are observed to be some-what lower than the prediction of the semivector-ial model.

4. Conclusion

In this paper, two coupled raised channel waveguidesthat exhibit a high index contrast in both the long-itudinal and transverse dimensions has been ana-lyzed. The numerical results of both longitudinal andtransverse misalignments showed a substantial dif-ference in the power coupled to the output waveguideand the power reflected back into the input wave-guide, as predicted by both the semivectorial andthe full-vectorial models. The difference in the calcu-lated results obtained by both models is observed tobe somewhat larger for the TE-like case when com-pared to the TM-like case.

The authors gratefully acknowledge King FahdUniversity of Petroleum and Minerals, Dhahran,Saudi Arabia, for supporting this research workthrough research grant SB 070017.

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Full-vectorial analysis of high-index-contrast coupled channel waveguides

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