Kofi Edee, Mira Abboud, Gérard Granet, Jean Francois Cornet, and Nikolay A. Gippius, "Mode solver based on Gegenbauer polynomial expansion for cylindrical structures with arbitrary cross sections," J. Opt. Soc. Am. A 31, 667-676 (2014)
We present a modal method for the computation of eigenmodes of cylindrical structures with arbitrary cross sections. These modes are found as eigenvectors of a matrix eigenvalue equation that is obtained by introducing a new coordinate system that takes into account the profile of the cross section. We show that the use of Hertz potentials is suitable for the derivation of this eigenvalue equation and that the modal method based on Gegenbauer expansion (MMGE) is an efficient tool for the numerical solution of this equation. Results are successfully compared for both perfectly conducting and dielectric structures. A complex coordinate version of the MMGE is introduced to solve the dielectric case.
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First Five Zeros of the Zeroth-order Bessel Function and Its Numerical Derivative Computed from the Eigenvalues and for a
1
2.4048255577
2.4048255577
3.8317059702
3.8317059702
2
5.5200781103
5.5200781103
7.0155866698
7.0155866698
3
8.6537279129
8.6537279129
10.1734681352
10.1734681351
4
11.7915344390
11.7915344391
13.3236919363
13.3236919363
5
14.9309177085
14.9309177086
16.4706300509
16.4706300509
These values are used as reference values to study the convergence of the numerical results. Note that and are comparative values from [16] (pp. 409–411). Values were calculated with , and .
Table 2.
First Five Zeros of the Second-order Bessel Function and Its Numerical Derivative Computed from the Eigenvalues and for a
1
5.1356223018
5.13562
3.0542369282
3.05424
2
8.4172441404
8.41724
6.7061331942
6.70613
3
11.6198411721
11.61984
9.9694678230
9.96947
4
14.7959517823
14.79595
13.1703708560
13.17037
5
17.9598194950
17.95982
16.3475223183
16.34752
These values are used as reference values to study the convergence of the numerical results. Note that and are comparative values from [16] (pp. 409–411). Values were calculated with , and .
The effective index of the mode () of a circular dielectric waveguide with a radius of is computed with the MMGE with respect to the size of the eigenvalue matrix for three different values of the scaling factor . Values were calculated with , , , and .
Table 4.
Convergence of the Effective Index of an Elliptical Dielectric Waveguide (Radii of and ) with Respect to Both and a
3
1.169860902
1.169590596
1.169551905
1.169546524
5
1.147973728
1.146103877
1.145732438
1.145664811
7
1.147973728
1.146103877
1.145732438
1.145664811
9
1.148409721
1.146645363
1.146295794
1.146232876
11
1.148409721
1.146645363
1.146295794
1.146232877
13
1.148426112
1.146647215
1.146297765
1.146234887
Values were calculated with , , , and .
Table 5.
Convergence of the Effective Index of an Elliptical Dielectric Waveguide (Radii of and ) with Respect to Both and a
3
1.242027987
1.241744579
1.241704827
1.241699366
5
1.242659231
1.242372635
1.242332430
1.242326903
7
1.242659231
1.242372635
1.242332430
1.242326903
9
1.242683061
1.242397679
1.242357753
1.242352297
11
1.242683061
1.242397679
1.242357753
1.242352297
13
1.242683175
1.242397772
1.242357849
1.242352394
Values were calculated with , , , and .
Tables (5)
Table 1.
First Five Zeros of the Zeroth-order Bessel Function and Its Numerical Derivative Computed from the Eigenvalues and for a
1
2.4048255577
2.4048255577
3.8317059702
3.8317059702
2
5.5200781103
5.5200781103
7.0155866698
7.0155866698
3
8.6537279129
8.6537279129
10.1734681352
10.1734681351
4
11.7915344390
11.7915344391
13.3236919363
13.3236919363
5
14.9309177085
14.9309177086
16.4706300509
16.4706300509
These values are used as reference values to study the convergence of the numerical results. Note that and are comparative values from [16] (pp. 409–411). Values were calculated with , and .
Table 2.
First Five Zeros of the Second-order Bessel Function and Its Numerical Derivative Computed from the Eigenvalues and for a
1
5.1356223018
5.13562
3.0542369282
3.05424
2
8.4172441404
8.41724
6.7061331942
6.70613
3
11.6198411721
11.61984
9.9694678230
9.96947
4
14.7959517823
14.79595
13.1703708560
13.17037
5
17.9598194950
17.95982
16.3475223183
16.34752
These values are used as reference values to study the convergence of the numerical results. Note that and are comparative values from [16] (pp. 409–411). Values were calculated with , and .
The effective index of the mode () of a circular dielectric waveguide with a radius of is computed with the MMGE with respect to the size of the eigenvalue matrix for three different values of the scaling factor . Values were calculated with , , , and .
Table 4.
Convergence of the Effective Index of an Elliptical Dielectric Waveguide (Radii of and ) with Respect to Both and a
3
1.169860902
1.169590596
1.169551905
1.169546524
5
1.147973728
1.146103877
1.145732438
1.145664811
7
1.147973728
1.146103877
1.145732438
1.145664811
9
1.148409721
1.146645363
1.146295794
1.146232876
11
1.148409721
1.146645363
1.146295794
1.146232877
13
1.148426112
1.146647215
1.146297765
1.146234887
Values were calculated with , , , and .
Table 5.
Convergence of the Effective Index of an Elliptical Dielectric Waveguide (Radii of and ) with Respect to Both and a