Exact dispersion equation of transverse electric leaky modes for nonhomogeneous optical waveguides

Jianxin Zhu; Jia Zheng

doi:10.1364/JOSAB.32.000092

Journal of the Optical Society of America B
Vol. 32,
Issue 1,
pp. 92-100
(2015)
•https://doi.org/10.1364/JOSAB.32.000092

Exact dispersion equation of transverse electric leaky modes for nonhomogeneous optical waveguides

Jianxin Zhu and Jia Zheng

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Jianxin Zhu and Jia Zheng, "Exact dispersion equation of transverse electric leaky modes for nonhomogeneous optical waveguides," J. Opt. Soc. Am. B 32, 92-100 (2015)

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Table of Contents Category
- Optical Devices
Optics & Photonics Topics
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The topics in this list come from the Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Fiber characterization (060.2270)
- Waveguides, slab (230.7400)

About this Article
History
- Original Manuscript: July 29, 2014
- Revised Manuscript: November 10, 2014
- Manuscript Accepted: November 14, 2014
- Published: December 17, 2014

Abstract

For modal expansion in the nonhomogeneous optical waveguides, it is necessary to develop a high-precision mode solver. In this work, the differential transfer matrix method is applied, and an exact dispersion equation is built generally by the matrix similarity transformation and may be solved iteratively by Newton’s method. In justification of the proposed method, truncation error analysis for the old method is provided. Furthermore, the asymptotic solutions in the form of simpler formulas for the approximated equation are derived. These approximate analytic results are valuable and may well act as initial guesses for numerical methods. More accurate results can be obtained through our method in numerical simulations.

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$m$	$β_{0}$	$\| β_{0} - β_{e} \| / \| β_{e} \|$	$β$	$\| β - β_{e} \| / \| β_{e} \|$	$β_{e}$
1	$13.245744 + 0.301885 i$	$2.61 e - 002$	$13.150245 + 0.184305 i$	$1.49 e - 002$	$13.086932 + 0.000000 i$
2	$12.628709 + 2.169081 i$	$1.41 e - 002$	$12.459165 + 2.117178 i$	$1.22 e - 004$	$12.458250 + 2.115932 i$
3	$10.981862 + 5.208169 i$	$1.65 e - 002$	$10.787234 + 5.241151 i$	$2.16 e - 005$	$10.786977 + 5.241115 i$
4	$9.434126 + 9.559106 i$	$1.36 e - 002$	$9.283916 + 9.661581 i$	$9.66 e - 006$	$9.283798 + 9.661529 i$
5	$8.719804 + 14.394809 i$	$8.68 e - 003$	$8.625922 + 14.506867 i$	$1.62 e - 005$	$8.625761 + 14.507087 i$
6	$8.520377 + 19.100371 i$	$5.65 e - 003$	$8.460290 + 19.203325 i$	$1.05 e - 004$	$8.458786 + 19.201721 i$
7	$8.532366 + 23.620835 i$	$3.86 e - 003$	$8.489366 + 23.707906 i$	$5.48 e - 007$	$8.489363 + 23.707919 i$
8	$8.631097 + 28.003158 i$	$2.94 e - 003$	$8.596620 + 28.082406 i$	$1.60 e - 006$	$8.596667 + 28.082409 i$
9	$8.766241 + 32.288212 i$	$2.26 e - 003$	$8.741486 + 32.359737 i$	$3.65 e - 006$	$8.741607 + 32.359721 i$
10	$8.915734 + 36.503707 i$	$1.74 e - 003$	$8.895643 + 36.566138 i$	$4.82 e - 007$	$8.895631 + 36.566151 i$
11	$9.069212 + 40.668090 i$	$1.45 e - 003$	$9.051350 + 40.725671 i$	$3.52 e - 007$	$9.051340 + 40.725682 i$
12	$9.221606 + 44.793900 i$	$1.21 e - 003$	$9.206962 + 44.847241 i$	$2.23 e - 007$	$9.206953 + 44.847235 i$

$m$	$β_{0}$	$\| β_{0} - β_{e} \| / \| β_{e} \|$	$β$	$\| β - β_{e} \| / \| β_{e} \|$	$β_{e}$
1	$13.138089 + 0.283994 i$	$2.20 e - 002$	$13.034996 + 0.078485 i$	$7.19 e - 003$	$13.086932 + 0.000000 i$
2	$12.269238 + 1.889826 i$	$2.33 e - 002$	$12.650634 + 2.045403 i$	$1.62 e - 002$	$12.458250 + 2.115932 i$
3	$10.346583 + 4.790990 i$	$5.25 e - 002$	$10.970042 + 5.132483 i$	$1.77 e - 002$	$10.786977 + 5.241115 i$
4	$8.556803 + 9.274732 i$	$6.15 e - 002$	$9.408515 + 9.516591 i$	$1.43 e - 002$	$9.283798 + 9.661529 i$
5	$7.821068 + 14.268224 i$	$4.97 e - 002$	$8.697489 + 14.373285 i$	$9.00 e - 003$	$8.625761 + 14.507087 i$
6	$7.653213 + 19.044663 i$	$3.91 e - 002$	$8.503671 + 19.088398 i$	$5.81 e - 003$	$8.458786 + 19.201721 i$
7	$7.692617 + 23.596210 i$	$3.19 e - 002$	$8.519809 + 23.613436 i$	$3.94 e - 003$	$8.489363 + 23.707919 i$
8	$7.810893 + 27.993602 i$	$2.69 e - 002$	$8.621399 + 27.998217 i$	$2.99 e - 003$	$8.596667 + 28.082409 i$
9	$7.959883 + 32.286595 i$	$2.34 e - 002$	$8.758545 + 32.284720 i$	$2.29 e - 003$	$8.741607 + 32.359721 i$
10	$8.119417 + 36.506515 i$	$2.07 e - 002$	$8.909484 + 36.501134 i$	$1.77 e - 003$	$8.895631 + 36.566151 i$
11	$8.280378 + 40.673447 i$	$1.85 e - 002$	$9.064037 + 40.666131 i$	$1.46 e - 003$	$9.051340 + 40.725682 i$
12	$8.438487 + 44.800743 i$	$1.68 e - 002$	$9.217250 + 44.792368 i$	$1.22 e - 003$	$9.206953 + 44.847235 i$

$m$	$β_{0}$	$\| β_{0} - β_{e} \| / \| β_{e} \|$	$β$	$\| β - β_{e} \| / \| β_{e} \|$	$β_{e}$
1	$15.068301 + 0.205683 i$	$2.20 e - 002$	$15.070838 + 0.382987 i$	$3.30 e - 002$	$15.199298 - 0.102563 i$
2	$14.164329 + 1.437404 i$	$5.61 e - 002$	$14.066846 + 1.406596 i$	$4.88 e - 002$	$13.409204 + 1.386958 i$
3	$12.167377 + 3.664102 i$	$8.84 e - 002$	$11.929691 + 3.515885 i$	$7.16 e - 002$	$11.136456 + 3.801401 i$
4	$9.652336 + 7.493346 i$	$9.63 e - 002$	$8.364670 + 8.799125 i$	$5.93 e - 002$	$8.736255 + 8.193539 i$
5	$8.128540 + 12.617504 i$	$5.90 e - 002$	$7.324064 + 13.567342 i$	$2.53 e - 002$	$7.685022 + 13.414923 i$
6	$7.613591 + 17.698697 i$	$3.47 e - 002$	$7.390866 + 18.496712 i$	$7.21 e - 003$	$7.408468 + 18.355140 i$
7	$7.498201 + 22.479868 i$	$2.24 e - 002$	$7.414895 + 23.056290 i$	$2.18 e - 003$	$7.391255 + 23.009323 i$
8	$7.534783 + 27.041442 i$	$1.55 e - 002$	$7.489026 + 27.492572 i$	$7.61 e - 004$	$7.471987 + 27.479187 i$
9	$7.634688 + 31.455824 i$	$1.14 e - 002$	$7.601959 + 31.833066 i$	$2.93 e - 004$	$7.594078 + 31.827600 i$
10	$7.761940 + 35.768926 i$	$8.81 e - 003$	$7.738318 + 36.096069 i$	$1.73 e - 004$	$7.732931 + 36.092662 i$
11	$7.900258 + 40.009679 i$	$6.99 e - 003$	$7.881887 + 40.297209 i$	$8.80 e - 005$	$7.878543 + 40.295836 i$
12	$8.041716 + 44.196966 i$	$5.71 e - 003$	$8.026901 + 44.454810 i$	$5.81 e - 005$	$8.024340 + 44.454245 i$
13	$8.182288 + 48.343592 i$	$4.75 e - 003$	$8.173192 + 48.583269 i$	$1.61 e - 004$	$8.168308 + 48.577026 i$
14	$8.319910 + 52.458544 i$	$3.92 e - 003$	$8.313309 + 52.667457 i$	$1.36 e - 005$	$8.312588 + 52.667400 i$
15	$8.453549 + 56.548319 i$	$3.41 e - 003$	$8.438286 + 56.744977 i$	$3.86 e - 005$	$8.436753 + 56.743383 i$
16	$8.582738 + 60.617733 i$	$3.11 e - 003$	$8.575648 + 60.808384 i$	$2.82 e - 005$	$8.573918 + 60.808295 i$
17	$8.707327 + 64.670436 i$	$2.58 e - 003$	$8.705760 + 64.838373 i$	$1.65 e - 005$	$8.704900 + 64.839029 i$
18	$8.827341 + 68.709247 i$	$2.27 e - 003$	$8.815536 + 68.866067 i$	$8.34 e - 006$	$8.815879 + 68.866533 i$
19	$8.942906 + 72.736378 i$	$2.13 e - 003$	$8.934324 + 72.893332 i$	$1.29 e - 005$	$8.934610 + 72.892429 i$
20	$9.054197 + 76.753592 i$	$1.86 e - 003$	$9.052265 + 76.898880 i$	$2.19 e - 005$	$9.051192 + 76.897562 i$

$m$	$β_{0}$	$\| β_{0} - β_{e} \| / \| β_{e} \|$	$β$	$\| β - β_{e} \| / \| β_{e} \|$	$β_{e}$
1	$15.674293 - 0.109675 i$	$3.13 e - 002$	$13.534376 + 0.005323 i$	$1.10 e - 001$	$15.199298 - 0.102563 i$
2	$14.425679 + 0.554738 i$	$9.75 e - 002$	$13.502559 - 0.035699 i$	$1.06 e - 001$	$13.409204 + 1.386958 i$
3	$11.968575 + 1.998878 i$	$1.69 e - 001$	$13.546133 - 0.161343 i$	$3.94 e - 001$	$11.136456 + 3.801401 i$
4	$8.186195 + 5.322449 i$	$2.44 e - 001$	$13.534376 + 0.005323 i$	$7.92 e - 001$	$8.736255 + 8.193539 i$
5	$5.826948 + 11.297705 i$	$1.82 e - 001$	$12.885342 - 0.018193 i$	$9.32 e - 001$	$7.685022 + 13.414923 i$
6	$5.328745 + 16.912207 i$	$1.28 e - 001$	$9.424429 + 7.062072 i$	$5.80 e - 001$	$7.408468 + 18.355140 i$
7	$5.292687 + 21.931516 i$	$9.76 e - 002$	$7.906905 + 12.425501 i$	$4.38 e - 001$	$7.391255 + 23.009323 i$
8	$5.388623 + 26.620730 i$	$7.91 e - 002$	$7.456618 + 17.605681 i$	$3.47 e - 001$	$7.471987 + 27.479187 i$
9	$5.529878 + 31.113762 i$	$6.68 e - 002$	$7.385978 + 22.426743 i$	$2.87 e - 001$	$7.594078 + 31.827600 i$
10	$5.686433 + 35.480071 i$	$5.79 e - 002$	$7.451040 + 27.007503 i$	$2.46 e - 001$	$7.732931 + 36.092662 i$
11	$5.846119 + 39.759229 i$	$5.12 e - 002$	$7.569801 + 31.432469 i$	$2.16 e - 001$	$7.878543 + 40.295836 i$
12	$6.003592 + 43.975575 i$	$4.60 e - 002$	$7.710135 + 35.751993 i$	$1.93 e - 001$	$8.024340 + 44.454245 i$
13	$6.156461 + 48.144985 i$	$4.18 e - 002$	$7.857900 + 39.996920 i$	$1.74 e - 001$	$8.168308 + 48.577026 i$
14	$6.303722 + 52.278301 i$	$3.84 e - 002$	$8.006408 + 44.187062 i$	$1.59 e - 001$	$8.312588 + 52.667400 i$
15	$6.445053 + 56.383209 i$	$3.53 e - 002$	$8.152388 + 48.335721 i$	$1.47 e - 001$	$8.436753 + 56.743383 i$
16	$6.580478 + 60.465321 i$	$3.29 e - 002$	$8.294251 + 52.452166 i$	$1.36 e - 001$	$8.573918 + 60.808295 i$
17	$6.710189 + 64.528840 i$	$3.09 e - 002$	$8.431280 + 56.543067 i$	$1.27 e - 001$	$8.704900 + 64.839029 i$
18	$6.834461 + 68.576981 i$	$2.88 e - 002$	$- 12.885342 + 0.018193 i$	$1.04 e + 000$	$8.815879 + 68.866533 i$
19	$6.953604 + 72.612247 i$	$2.72 e - 002$	$12.873589 + 0.003209 i$	$9.94 e - 001$	$8.934610 + 72.892429 i$
20	$7.067929 + 76.636622 i$	$2.58 e - 002$	$- 12.885342 + 0.018193 i$	$1.03 e + 000$	$9.051192 + 76.897562 i$

$m$	$β_{0}$	$\| β_{0} - β_{e} \| / \| β_{e} \|$	$β$	$\| β - β_{e} \| / \| β_{e} \|$	$β_{e}$
1	$13.245744 + 0.301885 i$	$2.61 e - 002$	$13.150245 + 0.184305 i$	$1.49 e - 002$	$13.086932 + 0.000000 i$
2	$12.628709 + 2.169081 i$	$1.41 e - 002$	$12.459165 + 2.117178 i$	$1.22 e - 004$	$12.458250 + 2.115932 i$
3	$10.981862 + 5.208169 i$	$1.65 e - 002$	$10.787234 + 5.241151 i$	$2.16 e - 005$	$10.786977 + 5.241115 i$
4	$9.434126 + 9.559106 i$	$1.36 e - 002$	$9.283916 + 9.661581 i$	$9.66 e - 006$	$9.283798 + 9.661529 i$
5	$8.719804 + 14.394809 i$	$8.68 e - 003$	$8.625922 + 14.506867 i$	$1.62 e - 005$	$8.625761 + 14.507087 i$
6	$8.520377 + 19.100371 i$	$5.65 e - 003$	$8.460290 + 19.203325 i$	$1.05 e - 004$	$8.458786 + 19.201721 i$
7	$8.532366 + 23.620835 i$	$3.86 e - 003$	$8.489366 + 23.707906 i$	$5.48 e - 007$	$8.489363 + 23.707919 i$
8	$8.631097 + 28.003158 i$	$2.94 e - 003$	$8.596620 + 28.082406 i$	$1.60 e - 006$	$8.596667 + 28.082409 i$
9	$8.766241 + 32.288212 i$	$2.26 e - 003$	$8.741486 + 32.359737 i$	$3.65 e - 006$	$8.741607 + 32.359721 i$
10	$8.915734 + 36.503707 i$	$1.74 e - 003$	$8.895643 + 36.566138 i$	$4.82 e - 007$	$8.895631 + 36.566151 i$
11	$9.069212 + 40.668090 i$	$1.45 e - 003$	$9.051350 + 40.725671 i$	$3.52 e - 007$	$9.051340 + 40.725682 i$
12	$9.221606 + 44.793900 i$	$1.21 e - 003$	$9.206962 + 44.847241 i$	$2.23 e - 007$	$9.206953 + 44.847235 i$

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Tables (4)

Equations (38)

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Journal of the Optical Society of America B

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