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Beam propagation computations in one and two transverse dimensions
Optics Communications North-Holland
OPTICS COMMUNICATIONS
100 ( 1993) 43-47
Beam propagation computations dimensions
in one and two transverse
Michael Munowitz and David J. Vezzetti Amoco Technology Company, P.O. Box 3011, Naperville, IL 60566, USA Received
25 February
1993
Substitution of slab geometries for two-dimensional index profiles is commonly employed in calculations involving the beam propagation method, and is based on using effective-index theory to replace the full system by a one-dimensional representation with the same propagation constants. Results obtained for sample one-dimensional model systems are compared critically herein with those determined for complete structures described by two transverse coordinates. We show that in most respects the effective-index approach remains qualitatively useful for waveguide design, notwithstanding some clear discrepancies between the one- and two-dimensional results.
1. Introduction The beam propagation method (BPM) has, over the last fifteen years, become one of the most commonly used numerical techniques for modeling electromagnetic fields in dielectric optical waveguides and fibers [ l-61. In this Communication we now address an important technical question pertaining to such studies, namely the validity of substituting slab waveguides for structures where the refractive index actually varies in two transverse dimensions. Practical considerations usually limit calculations to systems with one transverse coordinate, and indeed most of the BPM results reported in the literature make use of the effective-index approximation to reduce the dimensionality [ 7-91. Despite the widespread use of this simplification, however, there seems to have been little systematic attempt to compare the results of beam propagation studies in one and two transverse dimensions. Justification of the effective-index approximation is provided mostly through comparison of propagation constants and static modal fields obtained for both full structures and their corresponding one-dimensional equivalents. By these criteria, the effective-index method is generally found to be very satisfactory for longitudinally invariant systems away from cutoff [ lo] ; what is not quite as clear is how well the mode1 holds 0030-4018/93/$06.00
0 1993 Elsevier
Science Publishers
for bent structures or for complicated radiation patterns. We test some of these questions here by direct comparison of one- and two-dimensional BPM results for certain representative examples. The waveguide parameters are typical for Al,Ga,_,As systems, and we will show that in most respects the effective-index approach remains useful for dealing with such structures.
2. Model systems We take as a point of reference the ridge waveguide illustrated in fig. 1a, specified by the following parameters: Q= 3.385, nc= ns= 3.360, w= 3.0 pm, h,=2.6 pm, h,=0.4 urn, hf=0.3 um,1=0.85 urn. For Al,Ga,_,As at this wavelength, the core and cladding layers would have aluminum fractions of approximately x=0.355 and x=0.4, respectively [ 111. Application of the effective-index method to the ridge waveguide yields an equivalent slab guide with core index equal to 3.3641, cladding index equal to 3.36 12 18, and width equal to w. Both the one- and two-dimensional systems are single-mode for a ridge width of 3.0 urn. For ease of reference we will denote the ridge waveguide by the symbol “R” and its slab equivalent by“‘E1”.
B.V. All rights reserved.
43
Volume
100, number
(b)
OPTICS
COMMUNICATIONS
X
(a)
Substrate
1,2,3,4
nq
1 July 1993
r-
h,
r/
X
I
I31
Fig. 1. Coordinate axes and transverse index profiles for the model systems. (a) Ridge waveguide, nr> n,=n,. (b) Burried-channel waveguide, rectangular in the general case. The notation hr and w is chosen for consistency with the ridge guide, and y is taken as the lateral dimension. nr> n,.
To provide additional comparison we construct two burried-channel waveguides with transverse index profiles as shown in fig. lb, setting the parameters as nf=3.381794, n,=3.361218, hf=0.3 pm for the first (Bl), and +=3.368855, n,=3.361218, hr= 1.O urn for the second (B2 ) . These values have been chosen specifically to yield the same one-dimensional representation under effective-index theory as does the ridge guide described above. Thus when reduced to one transverse dimension, guides R, B 1, and B2 all share a common slab configuration (EI) with core index= 3.3641 and cladding index = 3.36 12 18. Width and wavelength are identical in all structures. Although the fundamental modes of the three twodimensional guides have the same propagation constant, their field profiles (fig. 2) are distinctly different. A single one-dimensional field distribution, though, must suffice to describe all three systems when the one-dimensional version of the beam propagation method is invoked.
44
B2
J Fig. 2. Fundamental mode for each of systems R, Bl, and B2 (as described in the text), shown as equally spaced field contours. Each box is a IO pm x 10 urn square.
3. Computational details The standard Fresnel version of the beam propagation method has been adopted throughout, with Fourier transformation used for the diffraction step. Calculations were performed on the Cray XMP at
Volume 100, number 1,2,3,4
the Minnesota Supercomputer Center, assisted by Cray Scilib subroutines CFFT2 and CFFT2D for oneand two-dimensional fast Fourier transforms. Transverse grids ranging from 128* to 5 12* points were defined over the rectangular domain Lx= 15 umx L,= 50 urn for the two-dimensional computations, while a grid of 4096 points was established over a length Lx= 100 urn for the one-dimensional computations. A longitudinal step length, AZ, of 0.1 A (0.085 urn) was used for both. The background index was taken as the effective index of the local fundamental mode at z=O. Two-dimensional modal fields to be used as initial conditions E(x, y; z=O) were computed using a standard matrix technique based on a double sine expansion [ 12,13 1. Reference fields for local mode analysis (see sect. 5) were obtained in the same way. In all instances, 61 sine waves were sued in the lateral dimension (Lp 50 urn) and 41 sine waves in the vertical (Lx= 15 urn). We now focus the discussion by examining two specific waveguide configurations: the first, a pair of parallel waveguides (as might be used in a directional coupler); the second, a Y junction. These systems are sufficiently complex to provide a critical comparision of propagation under the one- and twodimensional models, and, at the same time, are of some practical interest.
4. Example I - parallel waveguides Consider first the interference between guided and radiation modes in a system of two parallel waveguides with inner separation s. We take as an incident field a normalized, equally weighted combination of the fundamental guided mode of the combined structure (two guides spaced as w--s-w) and an additional, wholly arbitrary, field which will remain unguided and which will beat with the guided component. For the radiative contribution it is convenient to use the fundamental mode of a single guide of width 2w, as illustrated in fig. 3. Figure 4 shows the integrated lateral intensity patterns, defined as
Zb;z)=
s
Z(x,JTz)dx,
1 July 1993
OPTICS COMMUNICATIONS
(b)
(c)
r-l I I
I i
I
L
Fig. 3. Non-modal input for a pair of parallel waveguides. (a) Fundamental mode of the combined system of two guides. (b) Fundamental mode of a guide twice the width, having only partialoverlap with mode (a). (c) Sum of (a) and (b).
that develop in guides R and B 1, and compares these to the intensity Z(x; z) in the one-dimensional guide EI. A grid of 5 12 x 5 12 points has been sued for the two-dimensional computations. Figure 5 provides a closer look at the intensity observed at a single point midway between the parallel guides. The intensity patterns in R and B 1, although arising from waveguides with clearly different geometries and modal fields, are strikingly similar nonetheless. Beat lengths are almost identical, and the contours displayed in fig. 4 have much in common. It is mostly in the liner details of the patterns, such as the height of the local maximum near 400 urn in fig. 5, that differences become apparent. The EI results agree qualitatively with the two-dimensional calculations (perhaps somewhat more so for R than B 1)) but here it is easier to note differences in the contours. The local maxima and minima of the El curve are displaced relative to those of R 45
Volume 100, number 1,2,3,4
OPTICS
I July 1993
COMMUNICATIONS
i
-z
2
0
(b)
IO
200
400
Propagation
61 Fig. 5. Intensity
midway
600 distance,
800
1000
z (pm)
between the parallel guides; from fig. 4.
been used for the one-dimensional computation, and that the two-dimensional results can in no way be regarded as “exact”.
5. Example II - Y junction Here we consider a y branch made from an abrupt angle bend with a half-opening angle .O= 0.5 O. The initial field E(x, y; z=O) is taken as the fundamental mode at z=O, where the combined width of the structure is 2w. The quantity we monitor, Pi (z), is defined at each position along the branch as the fraction of power resident in the fundamental local mode u1 (x, .Y;z):
0
200
400
Propagation
600 distance.
800 2
100
(urn1
Fig. 4. Integrated lateral intensity I(y; z) for parallel guides, following launch of non-modal input into systems (a) R, and (b) BI. (c) Intensity pattern 1(x; z) for equivalent slab, EI. Contours are shown equally spaced.
and Bl, and the beat period is approximately 15% longer than in R. Still one is left with the impression that the effective-index representation does provide a reasonable approximation to the two-dimensional patterns as they evolve over distance. It should be borne in mind also that a liner numerical grid has 46
The local mode is specified for two parallel guides of width w, with inner separation given by s= 2z tan 0. Pi (z) curves for systems R, Bl, B2, and EI are presented in fig. 6a, where, to facilitate comparison, the plots for the two-dimensional systems have been displaced by 30-50 urn so as to bring their local minima into rough coincidence with EI. (Similar to the intensity patterns illustrated in figs. 4 and 5, the first minimum occurs at different values of z for the oneand two-dimensional calculations). Grid dimensions were 512X512. Considering the approximation involved, we would again suggest that the effective-index approach pro-
Volume
100, number
0
OPTICS
1,2,3,4
100
200
300
400
COMMUNICATIONS
500
Branch length, z (pm) (b)
It
1 July 1993
ferences among the solutions, even to a slight extent for the two buried-channel guides with relatively similar modes. These variations must be balanced against the difficulty and cost of obtaining very accurate two-dimensional solutions. Calculations in two transverse dimensions indeed can become so time consuming that it is difficult to establish whether the results are insensitive to further changes in parameters. For the present example, at the very least we need to determine whether the grid spacing is suffkiently fine to yield acceptable values of P,. Power curves computed using grids of 128x128, 256x256, and 512x512 points, shown in fig. 6b, reveal that a minimum of 256 points is needed to assure even a crude convergence for waveguide R. One-dimensional procedures, by contrast, present comparatively modest computational demands, and thus can be pursued at much higher levels of confidence and accuracy. References
100
200
300
Branch length, z (pm) Fig. 6. (a) Point-to-point plots of Pz(z) versus z for systems R, B 1, B2, and EI, following launch of the fundamental mode at z=O. Grid= 5 12 X 5 12. Curves R, B 1, and B2 have been displaced along z to align them with curve EL (b) P,(z) versus z for system R, as determined for grids of different spacing ( 128 x 128,256 x 256, 5 12 x 5 12). Note change in scale.
vides a reasonably satisfactory account of the power disposition in this bent waveguide. Details involving the exact period and depth of the oscillations are not reproduced quantitatively, but it seems that no major design errors will be incurred by using the onedimensional approximation. There are certainly dif-
[ 1] M.D. Feit and J.A. Fleck, Jr., Appl. Optics 17 (1978) 3990. [2] M.D. Feit and J.A. Fleck, Jr., Appl. Optics 18 (1979) 2843. [ 31 J. van Roey, J. van der Donk and P.E. Lagasse, J. Opt. Sot. Am. 71 (1981) 803. [4] L. Thylen, Optical and Quantum Electron. 15 (1983) 433. [ 51 D. Yevick and B. Hermansson, IEEE J. Quantum Electron. QE-25 (1989) 221. [6] D. Yevick and B. Hermansson, Electron. Lett. 25 (1989) 461. [ 71 R.M. Knox and P.P. Toulios, in: Proc. MRI symposium of submillimeter waves, ed. J. Fox (Polytechnic, New York, 1970) p. 497. [ 81 T. Tamir, ed., Guided-wave optoelectronics (Springer, Berlin, 1988). [ 91 G.B. Hacker and WK. Bums, Appl. Optics 16 ( 1977) 113. [ lo] M. Munowitz and D.J. Vezzetti, J. Lightwave Tech. 9 ( 1991) 1068. [ 111 D.W. Jenkins, J. Appl. Phys. 68 ( 1990) 1848. [ 121 C.H. Henry and B.H. Verbeek, J. Lightwave Tech. 7 (1989) 308. [ 13 ] D.J. Vezzetti and M. Munowitz, J. Lightwave Tech. 8 ( 1990) 1228.
47
OPTICS COMMUNICATIONS
100 ( 1993) 43-47
Beam propagation computations dimensions
in one and two transverse
Michael Munowitz and David J. Vezzetti Amoco Technology Company, P.O. Box 3011, Naperville, IL 60566, USA Received
25 February
1993
Substitution of slab geometries for two-dimensional index profiles is commonly employed in calculations involving the beam propagation method, and is based on using effective-index theory to replace the full system by a one-dimensional representation with the same propagation constants. Results obtained for sample one-dimensional model systems are compared critically herein with those determined for complete structures described by two transverse coordinates. We show that in most respects the effective-index approach remains qualitatively useful for waveguide design, notwithstanding some clear discrepancies between the one- and two-dimensional results.
1. Introduction The beam propagation method (BPM) has, over the last fifteen years, become one of the most commonly used numerical techniques for modeling electromagnetic fields in dielectric optical waveguides and fibers [ l-61. In this Communication we now address an important technical question pertaining to such studies, namely the validity of substituting slab waveguides for structures where the refractive index actually varies in two transverse dimensions. Practical considerations usually limit calculations to systems with one transverse coordinate, and indeed most of the BPM results reported in the literature make use of the effective-index approximation to reduce the dimensionality [ 7-91. Despite the widespread use of this simplification, however, there seems to have been little systematic attempt to compare the results of beam propagation studies in one and two transverse dimensions. Justification of the effective-index approximation is provided mostly through comparison of propagation constants and static modal fields obtained for both full structures and their corresponding one-dimensional equivalents. By these criteria, the effective-index method is generally found to be very satisfactory for longitudinally invariant systems away from cutoff [ lo] ; what is not quite as clear is how well the mode1 holds 0030-4018/93/$06.00
0 1993 Elsevier
Science Publishers
for bent structures or for complicated radiation patterns. We test some of these questions here by direct comparison of one- and two-dimensional BPM results for certain representative examples. The waveguide parameters are typical for Al,Ga,_,As systems, and we will show that in most respects the effective-index approach remains useful for dealing with such structures.
2. Model systems We take as a point of reference the ridge waveguide illustrated in fig. 1a, specified by the following parameters: Q= 3.385, nc= ns= 3.360, w= 3.0 pm, h,=2.6 pm, h,=0.4 urn, hf=0.3 um,1=0.85 urn. For Al,Ga,_,As at this wavelength, the core and cladding layers would have aluminum fractions of approximately x=0.355 and x=0.4, respectively [ 111. Application of the effective-index method to the ridge waveguide yields an equivalent slab guide with core index equal to 3.3641, cladding index equal to 3.36 12 18, and width equal to w. Both the one- and two-dimensional systems are single-mode for a ridge width of 3.0 urn. For ease of reference we will denote the ridge waveguide by the symbol “R” and its slab equivalent by“‘E1”.
B.V. All rights reserved.
43
Volume
100, number
(b)
OPTICS
COMMUNICATIONS
X
(a)
Substrate
1,2,3,4
nq
1 July 1993
r-
h,
r/
X
I
I31
Fig. 1. Coordinate axes and transverse index profiles for the model systems. (a) Ridge waveguide, nr> n,=n,. (b) Burried-channel waveguide, rectangular in the general case. The notation hr and w is chosen for consistency with the ridge guide, and y is taken as the lateral dimension. nr> n,.
To provide additional comparison we construct two burried-channel waveguides with transverse index profiles as shown in fig. lb, setting the parameters as nf=3.381794, n,=3.361218, hf=0.3 pm for the first (Bl), and +=3.368855, n,=3.361218, hr= 1.O urn for the second (B2 ) . These values have been chosen specifically to yield the same one-dimensional representation under effective-index theory as does the ridge guide described above. Thus when reduced to one transverse dimension, guides R, B 1, and B2 all share a common slab configuration (EI) with core index= 3.3641 and cladding index = 3.36 12 18. Width and wavelength are identical in all structures. Although the fundamental modes of the three twodimensional guides have the same propagation constant, their field profiles (fig. 2) are distinctly different. A single one-dimensional field distribution, though, must suffice to describe all three systems when the one-dimensional version of the beam propagation method is invoked.
44
B2
J Fig. 2. Fundamental mode for each of systems R, Bl, and B2 (as described in the text), shown as equally spaced field contours. Each box is a IO pm x 10 urn square.
3. Computational details The standard Fresnel version of the beam propagation method has been adopted throughout, with Fourier transformation used for the diffraction step. Calculations were performed on the Cray XMP at
Volume 100, number 1,2,3,4
the Minnesota Supercomputer Center, assisted by Cray Scilib subroutines CFFT2 and CFFT2D for oneand two-dimensional fast Fourier transforms. Transverse grids ranging from 128* to 5 12* points were defined over the rectangular domain Lx= 15 umx L,= 50 urn for the two-dimensional computations, while a grid of 4096 points was established over a length Lx= 100 urn for the one-dimensional computations. A longitudinal step length, AZ, of 0.1 A (0.085 urn) was used for both. The background index was taken as the effective index of the local fundamental mode at z=O. Two-dimensional modal fields to be used as initial conditions E(x, y; z=O) were computed using a standard matrix technique based on a double sine expansion [ 12,13 1. Reference fields for local mode analysis (see sect. 5) were obtained in the same way. In all instances, 61 sine waves were sued in the lateral dimension (Lp 50 urn) and 41 sine waves in the vertical (Lx= 15 urn). We now focus the discussion by examining two specific waveguide configurations: the first, a pair of parallel waveguides (as might be used in a directional coupler); the second, a Y junction. These systems are sufficiently complex to provide a critical comparision of propagation under the one- and twodimensional models, and, at the same time, are of some practical interest.
4. Example I - parallel waveguides Consider first the interference between guided and radiation modes in a system of two parallel waveguides with inner separation s. We take as an incident field a normalized, equally weighted combination of the fundamental guided mode of the combined structure (two guides spaced as w--s-w) and an additional, wholly arbitrary, field which will remain unguided and which will beat with the guided component. For the radiative contribution it is convenient to use the fundamental mode of a single guide of width 2w, as illustrated in fig. 3. Figure 4 shows the integrated lateral intensity patterns, defined as
Zb;z)=
s
Z(x,JTz)dx,
1 July 1993
OPTICS COMMUNICATIONS
(b)
(c)
r-l I I
I i
I
L
Fig. 3. Non-modal input for a pair of parallel waveguides. (a) Fundamental mode of the combined system of two guides. (b) Fundamental mode of a guide twice the width, having only partialoverlap with mode (a). (c) Sum of (a) and (b).
that develop in guides R and B 1, and compares these to the intensity Z(x; z) in the one-dimensional guide EI. A grid of 5 12 x 5 12 points has been sued for the two-dimensional computations. Figure 5 provides a closer look at the intensity observed at a single point midway between the parallel guides. The intensity patterns in R and B 1, although arising from waveguides with clearly different geometries and modal fields, are strikingly similar nonetheless. Beat lengths are almost identical, and the contours displayed in fig. 4 have much in common. It is mostly in the liner details of the patterns, such as the height of the local maximum near 400 urn in fig. 5, that differences become apparent. The EI results agree qualitatively with the two-dimensional calculations (perhaps somewhat more so for R than B 1)) but here it is easier to note differences in the contours. The local maxima and minima of the El curve are displaced relative to those of R 45
Volume 100, number 1,2,3,4
OPTICS
I July 1993
COMMUNICATIONS
i
-z
2
0
(b)
IO
200
400
Propagation
61 Fig. 5. Intensity
midway
600 distance,
800
1000
z (pm)
between the parallel guides; from fig. 4.
been used for the one-dimensional computation, and that the two-dimensional results can in no way be regarded as “exact”.
5. Example II - Y junction Here we consider a y branch made from an abrupt angle bend with a half-opening angle .O= 0.5 O. The initial field E(x, y; z=O) is taken as the fundamental mode at z=O, where the combined width of the structure is 2w. The quantity we monitor, Pi (z), is defined at each position along the branch as the fraction of power resident in the fundamental local mode u1 (x, .Y;z):
0
200
400
Propagation
600 distance.
800 2
100
(urn1
Fig. 4. Integrated lateral intensity I(y; z) for parallel guides, following launch of non-modal input into systems (a) R, and (b) BI. (c) Intensity pattern 1(x; z) for equivalent slab, EI. Contours are shown equally spaced.
and Bl, and the beat period is approximately 15% longer than in R. Still one is left with the impression that the effective-index representation does provide a reasonable approximation to the two-dimensional patterns as they evolve over distance. It should be borne in mind also that a liner numerical grid has 46
The local mode is specified for two parallel guides of width w, with inner separation given by s= 2z tan 0. Pi (z) curves for systems R, Bl, B2, and EI are presented in fig. 6a, where, to facilitate comparison, the plots for the two-dimensional systems have been displaced by 30-50 urn so as to bring their local minima into rough coincidence with EI. (Similar to the intensity patterns illustrated in figs. 4 and 5, the first minimum occurs at different values of z for the oneand two-dimensional calculations). Grid dimensions were 512X512. Considering the approximation involved, we would again suggest that the effective-index approach pro-
Volume
100, number
0
OPTICS
1,2,3,4
100
200
300
400
COMMUNICATIONS
500
Branch length, z (pm) (b)
It
1 July 1993
ferences among the solutions, even to a slight extent for the two buried-channel guides with relatively similar modes. These variations must be balanced against the difficulty and cost of obtaining very accurate two-dimensional solutions. Calculations in two transverse dimensions indeed can become so time consuming that it is difficult to establish whether the results are insensitive to further changes in parameters. For the present example, at the very least we need to determine whether the grid spacing is suffkiently fine to yield acceptable values of P,. Power curves computed using grids of 128x128, 256x256, and 512x512 points, shown in fig. 6b, reveal that a minimum of 256 points is needed to assure even a crude convergence for waveguide R. One-dimensional procedures, by contrast, present comparatively modest computational demands, and thus can be pursued at much higher levels of confidence and accuracy. References
100
200
300
Branch length, z (pm) Fig. 6. (a) Point-to-point plots of Pz(z) versus z for systems R, B 1, B2, and EI, following launch of the fundamental mode at z=O. Grid= 5 12 X 5 12. Curves R, B 1, and B2 have been displaced along z to align them with curve EL (b) P,(z) versus z for system R, as determined for grids of different spacing ( 128 x 128,256 x 256, 5 12 x 5 12). Note change in scale.
vides a reasonably satisfactory account of the power disposition in this bent waveguide. Details involving the exact period and depth of the oscillations are not reproduced quantitatively, but it seems that no major design errors will be incurred by using the onedimensional approximation. There are certainly dif-
[ 1] M.D. Feit and J.A. Fleck, Jr., Appl. Optics 17 (1978) 3990. [2] M.D. Feit and J.A. Fleck, Jr., Appl. Optics 18 (1979) 2843. [ 31 J. van Roey, J. van der Donk and P.E. Lagasse, J. Opt. Sot. Am. 71 (1981) 803. [4] L. Thylen, Optical and Quantum Electron. 15 (1983) 433. [ 51 D. Yevick and B. Hermansson, IEEE J. Quantum Electron. QE-25 (1989) 221. [6] D. Yevick and B. Hermansson, Electron. Lett. 25 (1989) 461. [ 71 R.M. Knox and P.P. Toulios, in: Proc. MRI symposium of submillimeter waves, ed. J. Fox (Polytechnic, New York, 1970) p. 497. [ 81 T. Tamir, ed., Guided-wave optoelectronics (Springer, Berlin, 1988). [ 91 G.B. Hacker and WK. Bums, Appl. Optics 16 ( 1977) 113. [ lo] M. Munowitz and D.J. Vezzetti, J. Lightwave Tech. 9 ( 1991) 1068. [ 111 D.W. Jenkins, J. Appl. Phys. 68 ( 1990) 1848. [ 121 C.H. Henry and B.H. Verbeek, J. Lightwave Tech. 7 (1989) 308. [ 13 ] D.J. Vezzetti and M. Munowitz, J. Lightwave Tech. 8 ( 1990) 1228.
47