998 OPTICS LETTERS / Vol. 23, No. 13 / July 1, 1998
Beam-propagation method for analysis of z-dependent
structures that uses a local oblique coordinate system
Yih-Peng Chiou and Hung-chun Chang
College of Electrical Engineering, National Taiwan University, Taipei 106-17, Taiwan Received March 27, 1998
In the most current beam-propagation method (BPM), the coupling between the transverse and the longitudinal fields in z-dependent structures is ignored under the staircase approximation, which results in violation of power conservation. We propose a novel BPM that is derived in a local oblique coordinate system to analyze z-dependent structures accurately and efficiently without taking the staircase approximation. The coupling between the transverse and the longitudinal fields is automatically included in the local oblique coordinate system, and power is thus conserved if the ref lection is neglected. 1998 Optical Society of America
OCIS codes: 260.2110, 350.5500, 230.7370, 350.7420.
The beam-propagation method (BPM) has been widely used in the investigation of optoelectronic devices for years and has been improved by various modif ications. A comprehensive review can be found in Ref. 1. In the conventionally used BPM’s, wave propagation along or near the z axis is assumed, which does not cor-respond to the physical propagation direction of the field in the tilted-waveguide case. Wide-angle BPM’s were proposed to extend the application to off-axis wave propagation.2,3
Nonetheless, to apply the cur-rent BPM’s, which were developed in the rectangular coordinate system (RCS), to z-dependent structures, we need to make the staircase approximation. Conse-quently, small meshes must be employed to minimize the discretization error, which requires large memory and much computation time. Recently, BPM’s based on the oblique coordinate system (OCS BPM’s) were proposed4,5
to solve tilted-waveguide problems eff i-ciently. Since the staircase approximation is avoided and rapid f ield variation in the transverse directions is removed by the slowly varying approximation, the ac-curacy and eff iciency of the OCS BPM’s are much bet-ter than those of methods using rectangular coordinate systems (RCS BPM’s).4,5
However, the interfaces be-tween the waveguide core and cladding must be paral-lel to the f ixed OCS in the OCS BPM’s, which restricts their applications, and consequently they cannot be ap-plied to such frequently used structures as tapered or curved waveguides.
Besides, in the current RCS BPM’s, derived with only the transverse f ields taken into account, the derivative of the refractive index along the z axis, or equivalently the coupling between the transverse and the longitudinal fields, is ignored for z-varying structures under the staircase approximation.6
Such coupling is not signif icant in most calculations, and therefore consideration of it is not vital. However, in some cases the coupling can accumulate con-siderably during propagation, for instance, in a whispering-gallery mode in a curved waveguide with a suff iciently small curvature for true adiabatic propa-gation without radiation leakage. Ignoring such coupling would result in the breakdown of power
con-servation. This problem can be solved by use of the conformal transformation from the curved waveguide to a straight one or by rotation of the coordinate system such that the propagation direction always coincides with the tangent to the waveguide axis at each propa-gation step.7
However, the interfaces of the wave-guide are required to be parallel to the new propagation direction. If the curved waveguide were tapered, the coupling would still arise because of nonparallel inter-faces and could not be treated by simple rotation of the coordinate system.
To alleviate such a problem, we propose a novel BPM derived in the local oblique coordinate system
(LOC BPM). Instead of using a global OCS, we
def ine a local OCS for every sampled point, and there-after all local OCS’s are combined into a global ensem-ble. This proposed method not only saves numerical effort as previous OCS BPM’s do but also is not re-stricted to interface-parallel waveguides. In particu-lar, the OCS BPM’s can be considered special cases of the LOC BPM in which the tilted angles are the same, and the RCS BPM’s are those with no tilted angles. Furthermore, since the coupling between the transverse and the longitudinal fields is included automatically through the coordinate transformation, the power-conservation problem in the current BPM’s is avoided.
Considering an interface with tilted angle u from the z axis, as shown in Fig. 1, we make a coordinate transformation def ined by
j苷 x 2 z tan u, z苷 z , (1)
such that locally the refractive-index distribution
n共x, z兲 苷 n共j兲 and ≠n共j兲兾≠z 苷 0, and hence the
staircase approximation can be avoided in the dis-cretization. We assume that the backward field can be neglected. Instead of assuming z propagation by E共x, z兲 苷 C共x, z兲exp共2jk0¯nz兲, we assume oblique
propagation along the interface by
E共x, z兲 苷 F共x, z兲exp关2jk0n¯共z cos u 1 x sin u兲兴 (2) 0146-9592/98/130998-03$15.00/0 1998 Optical Society of America
July 1, 1998 / Vol. 23, No. 13 / OPTICS LETTERS 999
Fig. 1. Sampled points near the interface T .
under the slowly varying envelope approximation, and the Helmholtz equation then becomes
Ω ≠2 ≠z2 1 ≠2 ≠x2 2 2jk0n¯ µ cos u ≠ ≠z1 sin u ≠ ≠x ∂ 1 k02关n2共x, z兲 2 ¯n2兴 æ F共x, z兲 苷 0 , (3) where k0is the wave number in free space and ¯nis the
reference refractive index. Note that with the slowly varying envelope F, not only the rapid longitudinal but also the rapid transverse variations are removed, which accounts for the high accuracy and efficiency of the OCS BPM in the simulation of tilted waveguides. Making use of the coordinate transformation, we find that Eq. (3) in the local OCS is
Ω ≠2 ≠z2 1 sec 2u ≠2 ≠j2 2 2jk0¯n cos u ≠ ≠z 2 2 tan u ≠2 ≠j≠z 1 k02关n2共j兲 2 ¯n2兴 æ F共j, z兲 苷 0 . (4) Neglecting ≠2兾≠z2 and taking the central difference
scheme as in the common RCS BPM’s, we f ind that the f ield at the sth longitudinal step and the mth transverse point, Fms, is expressed in terms of that in
the previous step as 共Ms
mFms 苷 Mms21Fs21m 兲L, (5)
where L denotes local. Since F are functions of local variables in their corresponding OCS’s, they are com-bined through Eq. (2) into the global f ield E:
共Ms mH s mE s m苷 M s21 m H s21 m E s21 m 兲L, (6)
where the phase factor Hm 苷 exp关jk0¯n共zmcos um 1
xmsin um兲兴 is restored. Combining all the sampled
points, we obtain an ensemble form 共Ps
Es苷 Ps21Es21兲G (7)
similar to that in the RCS BPM’s, where P 苷
共M0H0M1H1· · · MNHN兲 and G denotes global.
Equa-tion (7) can be solved eff iciently by such techniques as the Thomas algorithm. As to the truncated boundary conditions, efficient algorithms8
used in the RCS BPM can be applied.
The BPM’s using the global OCS were success-fully applied to solve tilted waveguides with parallel
interfaces in Refs. 4 and 5. The undesirable mode-mismatch loss when the OCS BPM’s are used can even be several orders smaller than that experienced when the RCS BPM’s are used. Besides, this spurious loss when one is using the RCS BPM’s accumulates to a sig-nif icant amount, whereas the loss when one is using the OCS BPM’s remains almost constant.
The widely used RCS BPM’s suffer from an L2
para-dox6
that is due to the staircase approximation, so power conservation is usually taken as P1苷
R
jEj2dx. P1can be used in z-invariant structures, but we need a more rigorous def inition of power, P0苷 Re
R
EⴱHdx, in z-dependent structures. To demonstrate the appli-cability of the LOC BPM, we consider a whispering-gallery mode with negligible radiation loss in a curved slab waveguide, shown as an inset in Fig. 2. The waveguide radius R is taken to be共Ri1 Ro兲兾2, where
Ri and Ro are the inner and outer radii at the
in-terfaces between the core and the claddings, respec-tively. Note that the two interfaces are not parallel to each other for z . 0, especially for small radii, mak-ing the equivalent waveguide width D0increase along the z direction and D0苷 共R
022 z2兲1/22共Ri22 z2兲1/2苷
共R02 2 R2sin2 u兲1/2 2 共Ri22 R2sin2u兲1/2 艐 D sec u.
Since the interfaces are not parallel, BPM’s with global OCS’s are not applicable, and here we resort to the lo-cal OCS.
First, we consider a waveguide with such small cur-vature that the f ield distribution is almost the same as the normal mode of a straight waveguide. The power
P0共u兲 at any u should always be equal to P0共z 苷 0兲,
which would lead to P1共u兲 艐 sec uP1共z 苷 0兲 after the
field propagates z苷 R sin u.6
Figure 2 shows the cal-culated normalized power P0共z兲兾P0共z 苷 0兲 when the
RCS BPM and the LOC BPM are used for a curved waveguide with R苷 4000 mm, wavelength l 苷 1.0 mm, waveguide width D 苷 2 mm, cladding and core refrac-tive indices nclad苷 1.00 and ncore苷 1.03, respectively,
and under the fundamental mode incidence. Although the RCS BPM satisf ies the nonphysical L2
conserva-tion, it gives physical power that decreases with cos u as the f ield propagates. The calculated power with the LOC BPM remains almost constant, and the small deviation is ,1024 even after the f ield propagates z苷 R cos 60±.
Fig. 2. Normalized power variation as the guided mode propagates along the curved waveguide.
1000 OPTICS LETTERS / Vol. 23, No. 13 / July 1, 1998
Fig. 3. Field distributions of the guided mode af-ter it propagates (a) z 苷 4000 sin 60±
mm and ( b) z 苷 25 sin 60±
mm. Dashed curves, theoretical results; solid curves, results calculated with the LOC BPM. The thick and the thin curves are the f ield-intensity amplitude and its corresponding real part, respectively. The oscillat-ing behavior of the real parts accounts for the transverse phase difference.
Table 1. Field-Distribution Error
ncore R 1.0 2 CR共u 苷 30±兲 1.0 2 CR共u 苷 60±兲 1.03 4000l 9.24 3 1027 6.94 3 1026 1.03 2000l 3.97 3 1026 3.12 3 1025 1.03 1000l 1.24 3 1025 1.21 3 1024 1.20 100l 2.32 3 1024 1.49 3 1023 1.50 50l 8.37 3 1024 5.85 3 1023 2.00 25l 2.73 3 1023 2.53 3 1022
Figure 3(a) shows the calculated and theoretical nor-malized f ield distributions as functions of the trans-verse distance when the LOC BPM is used after the field propagates z苷 R sin 60±. The f ield-distribution error ef 苷 1.0 2 CR is merely 6.94 3 1026. CR is
the cross correlation between the calculated f ield and the theoretical f ield, ˜F共x兲 苷 F 共x cos u兲exp共2jb sin ux兲,
where F共x兲 is the field distribution of the incident whispering-gallery mode and b is the propagation con-stant. It can be seen that not only the f ield intensities but also the phases are almost indistinguishable. On the other hand, the calculated CR when the RCS BPM is used is as small as 7.43 3 1027. Physically, the f ield
distribution is under a phase variation f 苷 b sin ux along the transverse direction. However, the
calcu-lated f ield distribution when the RCS BPM is used is almost of equal phase, which accounts for the consider-ably low CR when the RCS BPM is used.
Our calculations for similar structures with differ-ent parameters also yield excelldiffer-ent results. Several results are given in Table 1 for different values of ncore
and R; the other parameters are the same as in the case of Fig. 3. To prevent radiation losses, we take
ncoreto be large enough for smaller radii. For strongly curved waveguides共R , 100 mm兲, the peak of field for the whispering-gallery mode will be noticeably shifted to the outer interface, and thus the f ield distribution is asymmetric. Figure 3( b) shows the calculated and the theoretical field distributions for R 苷 25 mm and u 苷 60±. Theoretically, the propagation constant b varies with 1兾r, where r is the radius shown in Fig. 2, because ≠兾≠u 苷 2jrb is constant for the whispering-gallery mode with negligible loss. If R is small, the propagation constants at the inner and the outer inter-faces will be noticeably different, which accounts for the larger effor smaller radii and the larger field deviation
farther away from the waveguide center. The LOC BPM can also be applied to other z-dependent struc-tures, such as tapered waveguides.
In summary, we have developed a novel BPM de-rived in the local oblique coordinate system to treat
z-dependent structures. The staircase approximation is avoided in this method, so more-accurate results than those of other methods are obtained. The power-conservation problem suffered in the conventionally used BPM is alleviated. This proposed method was numerically validated by examination of the field dis-tribution and the power conservation for whispering-gallery modes in curved waveguides.
This study was supported by the National Science Council of the Republic of China under grant NSC87-2215-E-002-008.
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