Evanescent-wave and nonlinear transformation analysis of graded-index fibers

Gunnar Jacobsen

doi:10.1364/JOSA.70.001338

Journal of the Optical Society of America
Vol. 70,
Issue 11,
pp. 1338-1344
(1980)
•https://doi.org/10.1364/JOSA.70.001338

Evanescent-wave and nonlinear transformation analysis of graded-index fibers

Gunnar Jacobsen

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Gunnar Jacobsen, "Evanescent-wave and nonlinear transformation analysis of graded-index fibers," J. Opt. Soc. Am. 70, 1338-1344 (1980)

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Abstract

For a general class of graded-index fiber profiles, modal propagation constants and group delays were previously evaluated asymptotically to O(k⁻⁹) by using the systematic calculation scheme of evanescent wave theory. Here, the numerical results obtained directly are improved by using a nonlinear transformation technique developed by Shanks. This technique extracts information from the diverging as well as converging part of an asymptotic series, whereas the direct method utilizes the converging part alone. We obtain accurate results for a profile far from parabolic shape as well as for one of near-parabolic shape. For the profile far from parabolic shape the results quantify the error in group delay for the WKB technique for the lowest-order mode to approximately 30 ps/km. For high-order modes the WKB results agree with the transformed results.

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Corrections

Gunnar Jacobsen, "Evanescent-wave and nonlinear transformation analysis of graded-index fibers: errata," J. Opt. Soc. Am. 71, 788-788 (1981)
https://opg.optica.org/josa/abstract.cfm?uri=josa-71-6-788

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Basis^a	One transform^a	Two transforms^a	Three transforms
2.249 315 633744 705
2.248 940 948007 595*	2.249 109 902 573 268
2.249 248 654 737 377*	2.249 043 262 007 346*	2.249 070 461 044 244
2.248 630 941 370 367	2.249 089 217 609 381*	2.249 061 426 509 419*	2.249 065 241 123 461
2.250 406 458 542 437	2.249 018 907 026 417	2.249 068 028 767 355*	2.249 064 263 054 049
2.244 056 356 399 622	2.249 181 906 884 419	2.249 059 263 803 715
2.270 635 691 704 496	2.248 686 555 023 384
2.144 638 189 025 941

LP₀₀	β₁	β₂	\|Δβ\|/β
Direct	1.499 750	1.499 647	6.87 × 10⁻⁵
Nonlinear transform	1.499 688 381	1.499 688 055	2.17 × 10⁻⁷

Basis^a	One transform^a	Two transforms^a	Three transforms
2.242 471 971 191 753*
2.208 375 569 114 801*	2.237 639 381 313 630*
2.414 846 784 798 286	2.222 819 582 221 667*	2.233 894 242 339 402*
−0.330 102 950 912 366	2.266 642 852 622 586	2.227 801 777 618 433*	2.231 815 231 209 945
47.765 359 851 45 528	1.924 997 694 812 383	2.239 563 053 962 309	2.229 721 674 895 128
−929.890 975 413 054 5	5.893 628 817 836 430	2.179 274 839 473 475
20 919.593 729 071 01	−52.078 159 484 673 05
−501 082.609 749 628 4

LP₅₀	β₁	β₂	\|Δβ\|/β
Direct	1.497	1.486	7.3 × 10⁻³
Nonlinear transform	1.493926	1.493225	4.70 × 10⁻⁴

Basis^a	Second order^a	Third order
2.249 315 633 744 705
2.248 940 948 007 595*
2.249 248 654 737 377*	2.249 074 612 636 790
2.248 630 941 370 367	2.249 059 632 381 893*	2.249 067 265 925 570
2.250 406 458 542 437	2.249 069 831 585 462*	2.249 063 256 038 864
2.244 056 356 399 622	2.249 056 257 286 896
2.270 635 691 704 496
2.144 638 189 025 941

Basis^a	Second order^a	Third order
2.248 300 855 461 353
2.248 261 775 738 927
2.248 263 123 968 625	2.248 2630 224 383 46
2.248 263 010 269 676	2.248 2630 222 249 45	2.248 263 022 240 470
2.248 263 023 998 613	2.248 2630 222 407 49*	2.248 263 022 239 152
2.248 263 021 935 916	2.248 2630 222 390 85*
2.248 263 022 298 611*
2.248 263 022 226 383*

	β₁	β₂	\|Δβ\|/β
LP₀₀
Direct	1.499 420 895 646	1,499 420 895 622	1.60 × 10⁻¹¹
Third-order transform	1.499 420 895 626 160	1.499 420 895 625 856	2.03 × 10⁻¹³
LP₅₀
Direct	1.492 818	1.492 718	6.67 × 10⁻⁵
Third-order transform	1.492 768 576	1.492 768 477	6.60 × 10⁻⁸

μ	ν	β_WKB	β_EWT1	β_EWT2	τ_WKB × 10⁹	τ_EWT1 × 10⁹	τ_EWT2 × 10⁹
0	0	1.499 703 94	1.499 688 38	1.499 688 06	5.003 6814	5.003 713 9	5.003 716 2
0	1	1.499 329 98	1.499 319 64	1.499 317 42	5.004 0255	5.004 039 7	5.004 053 3
0	2	1.498 911 19	1.498 905	1.498 898	5.004 45	5.004 45	5.004 49
0	3	1.498 453	1.498 455	1.498 436	5.004 95	5.004 91	5.005 01
0	4	1.497 957	1.497 973	1.497 934	5.005 52	5.005 43	5.005 60
0	5	1.497 432	1.497 462	1.497 395	5.006 14	5.005 99	5.006 28
0	6	1.496 875	1.496 929	1.496 819	5.006 82	5.006 59	5.007 02
0	7	1.496 290	1.496 374	1.496 208	5.007 55	5.007 20	5.007 84
0	8	1.495 678	1.495 801	1.495 563	5.008 33	5.007 84	5.008 72
0	9	1.495 040	1.495 211	1.494 884	5.009 15	5.008 50	5.009 68
0	10	1.494 378	1.494 606	1.494 171	5.010 02	5.009 18	5.010 70
1	0	1.498 856	1.498 851	1.498 841	5.004 57	5.004 557	5.004 607
1	1	1.498 835	1.498 359	1.4983 34	5.005 16	5.005 102	5.005 219
1	2	1.497 822	1.497 842	1.497 792	5.005 79	5.005 68	5.005 90
1	3	1.497 262	1.497 305	1.497 215	5.006 48	5.006 28	5.006 64
1	4	1.496 676	1.496 747	1.496 605	5.007 21	5.006 91	5.007 46
1	5	1.496 063	1.496 172	1.495 960	5.007 99	5.007 56	5.008 34
1	6	1.495 426	1.495 582	1.495 282	5.008 81	5.008 22	5.009 30
1	7	1.494 764	1.494 978	1.494 570	5.009 68	5.008 90	5.010 32
1	8	1.494 080	1.494 360	1.493 825	5.010 59	5.009 60	5.011 41
1	9	1.493 374	1.493 731	1.493 046	5.011 53	5.010 30	5.012 57
1	10	1.492 646	1.493 092	1.492 235	5.012 51	5.011 03	5.013 79
2	0	1.4977 77	1.497 800	1.497 745	5.005 88	5.005 76	5.005 99
2	1	1.497 178	1.497 226	1.497 126	5.006 64	5.006 42	5.006 82
2	2	1.496 557	1.496 640	1.496 477	5.007 43	5.007 10	5.007 71
2	3	1.495 914	1.496 039	1.495 797	5.008 28	5.007 79	5.008 67
2	4	1.495 248	1.495 427	1.495 084	5.009 15	5.008 49	5.009 69
2	5	1.494 560	1.494 803	1.494 338	5.010 07	5.009 20	5.010 78
2	6	1.493 851	1.494 169	1.493 560	5.011 02	5.009 92	5.011 93
2	7	1.493 121	1.493 525	1.492 749	5.012 01	5.010 65	5.013 15
3	0	1.496 518	1.496 604	1.496 435	5.007 51	5.007 16	5.007 80
3	1	1.495 840	1.495 973	1.495 716	5.008 42	5.007 90	5.008 83
3	2	1.495 142	1.495 335	1.494 966	5.009 35	5.008 65	5.009 93
3	3	1.494 425	1.494 688	1.494 186	5.010 32	5.009 40	5.011 08
3	4	1.493 689	1.494 034	1.493 373	5.011 32	5.010 15	5.012 31
3	5	1.492 934	1.493 372	1.492 529	5.012 36	5.010 91	5.013 60
4	0	1.495107	1.495 304	1.494 927	5.009 42	5.008 70	5.010 00
4	1	1.494 358	1.494 631	1.494 110	5.010 45	5.009 49	5.011 23
4	2	1.493 593	1.493 953	1.493 262	5.011 50	5.010 29	5.012 53
4	3	1.492 809	1.493 270	1.492 384	5.012 59	5.011 07	5.013 88
5	0	1.493 560	1.493 926	1.493 225	5.011 56	5.010 33	5.012 60
5	1	1.492 748	1.493 220	1.492 312	5.012 70	5.011 15	5.014 02

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Equations (18)

Journal of the Optical Society of America