Propagating beam theory of optical fiber cross coupling

M. D. Feit; J. A. Fleck

doi:10.1364/JOSA.71.001361

Journal of the Optical Society of America
Vol. 71,
Issue 11,
pp. 1361-1372
(1981)
•https://doi.org/10.1364/JOSA.71.001361

Propagating beam theory of optical fiber cross coupling

M. D. Feit and J. A. Fleck

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M. D. Feit and J. A. Fleck, "Propagating beam theory of optical fiber cross coupling," J. Opt. Soc. Am. 71, 1361-1372 (1981)

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Abstract

Evanescent field coupling between parallel optical waveguides is treated by the propagating beam method. This method utilizes Fourier analysis to generate the modal properties of optical waveguides from numerical solutions to the paraxial-wave equation. Previous applications have been for single waveguides. Detailed results are presented here for a variety of coupled waveguide pairs: identical slab waveguides, identical and nonidentical single-mode optical fibers, and identical few-mode optical fibers. Results include propagation constants and eigenfunctions for the normal modes of the coupled systems. The difference between the propagation constants of the corresponding normal modes determines the coupling length for the mode pair, whereas the eigenfunctions determine the extent of power transfer. The results obtained establish the applicability of the propagating beam method to the study of coupling in a general class of practical waveguides.

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Mode (n)	Δβ^± (cm⁻¹)^a	Δβ^± (cm⁻¹)^b
0	—	—
1	—	—
2	3.27418 × 10⁻¹¹	5.09317 × 10⁻¹¹
3	1.32059 × 10⁻⁹	1.1205 × 10⁻⁹
4	2.1300 × 10⁻⁸	2.14131 × 10⁻⁸
5	2.97900 × 10⁻⁷	2.97943 × 10⁻⁷
6	3.27527 × 10⁻⁶	3.27606 × 10⁻⁶
7	2.93923 × 10⁻⁵	2.93904 × 10⁻⁵
8	2.19158 × 10⁻⁴	2.18996 × 10⁻⁴
9	1.37667 × 10⁻³	1.37675 × 10⁻³
10	7.36793 × 10⁻³	7.36869 × 10⁻³
11	3.3827 × 10⁻²	3.38251 × 10⁻²
12	1.33500 × 10⁻¹	1.33673 × 10⁻¹
13	4.55580 × 10⁻¹	4.55606 × 10⁻¹
14	1.33469 × 10⁰	1.33484 × 10⁰
15	3.31488 × 10⁰	3.31487 × 10⁰
16	6.73585 × 10⁰	6.73585 × 10⁰
17	1.06880 × 10¹	1.06880 × 10¹

Mode (n)	$β_{n}^{' +} ({cm}^{- 1})$ ^a	$β_{n}^{' +} ({cm}^{- 1})$ ^b	Change (cm⁻¹)
0	7.30617 × 10²	7.30617 × 10²	−1.74 × 10⁻⁶
1	6.85889 × 10²	6.85889 × 10²	−2.43 × 10⁻⁶
2	6.42891 × 10²	6.42892 × 10²	−3.06 × 10⁻⁶
3	6.00433 × 10²	6.00433 × 10²	6.02 × 10⁻⁶
4	5.58535 × 10²	5.58535 × 10²	4.12 × 10⁻⁶
5	5.16953 × 10²	5.16952 × 10²	−1.10 × 10⁻⁶
6	4.75700 × 10²	4.75699 × 10²	−1.17 × 10⁻⁵
7	4.34670 × 10²	4.34671 × 10²	3.22 × 10⁻⁶
8	3.93873 × 10²	3.93873 × 10²	3.45 × 10⁻⁵
9	3.53250 × 10²	3.53250 × 10²	4.63 × 10⁻⁶
10	3.12807 × 10²	3.12807 × 10²	2.66 × 10⁻⁵
11	2.72516 × 10²	2.72516 × 10²	6.64 × 10⁻⁶
12	2.32410 × 10²	2.32410 × 10²	3.38 × 10⁻⁴
13	1.92552 × 10²	1.92552 × 10²	3.18 × 10⁻⁵
14	1.53157 × 10²	1.53157 × 10²	1.09 × 10⁻⁴
15	1.14577 × 10¹	1.14578 × 10¹	2.11 × 10⁻⁵
16	7.70496 × 10¹	7.70497 × 10¹	3.80 × 10⁻⁵
17	4.00510 × 10⁰	4.00510 × 10⁰	2.52 × 10⁻⁶

d (μm)	β′⁺ (cm⁻¹)	β′⁻ (cm⁻¹)	Δβ^±(cm⁻¹)
∞	27.50985	27.50985	0
14	29.92722	24.65465	5.27257
10	34.52299	18.96165	15.56134

d (μm)	β′⁺ (cm⁻¹)	β′⁻ (cm⁻¹)	Δβ^±(cm⁻¹)
∞	27.50985	19.09500	8.41485
21	27.53110	18.86732	8.66378
11	30.56741	14.53923	16.02818

Separation d (μm)	Mode		Even Parity $β_{m n}^{' +} ({cm}^{- 1})$	Odd Parity $β_{m n}^{' -} ({cm}^{- 1})$
Separation d (μm)	m	n	Even Parity $β_{m n}^{' +} ({cm}^{- 1})$	Odd Parity $β_{m n}^{' -} ({cm}^{- 1})$
∞	0	0	131.6168	131.6168
	1	0	37.3493	37.3493
	0	1	37.3493	37.3493
20	0	0	131.6725	131.5626
	1	0	38.5693	35.9167
	0	1	37.5600	37.1124
16	0	0	132.0747	131.1756
	1	0	41.4777	32.4936
	0	1	38.2594	36.4229
14	0	0	132.9912	130.3899
	1	0	45.0429	28.2560
	0	1	39.2788	35.4268

Abstract

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Tables (6)

Equations (50)

Journal of the Optical Society of America