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.

Hualiang Shi and Ya Yan Lu Opt. Express 23(11) 14618-14629 (2015)

References

You do not have subscription access to this journal. Citation lists with outbound citation links are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

You do not have subscription access to this journal. Article level metrics are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

First Five Zeros ${u}_{n0}^{\text{ref}}$ of the Zeroth-order Bessel Function and Its Numerical Derivative ${u}_{n0}^{\prime \text{ref}}$ Computed from the Eigenvalues $\gamma $ and for $N=30$^{a}

$n$

${u}_{n0}^{\text{ref}}$

${u}_{n0}^{\mathrm{Abra}}$

${u}_{n0}^{\prime \text{ref}}$

${u}_{n0}^{\prime \mathrm{Abra}}$

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 ${u}_{n0}^{\mathrm{Abra}}$ and ${u}_{n0}^{\prime \mathrm{Abra}}$ are comparative values from [16] (pp. 409–411). Values were calculated with ${\rho}_{0}=\lambda $, $\epsilon ={3.5}^{2}$ and ${\partial}_{\theta}=0$.

Table 2.

First Five Zeros ${u}_{n2}^{\text{ref}}$ of the Second-order Bessel Function and Its Numerical Derivative ${u}_{n2}^{\prime \text{ref}}$ Computed from the Eigenvalues $\gamma $ and for $N=30$^{a}

$n$

${u}_{n2}^{\text{ref}}$

${u}_{n2}^{\mathrm{Abra}}$

${u}_{n2}^{\prime \text{ref}}$

${u}_{n2}^{\prime \mathrm{Abra}}$

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 ${u}_{n2}^{\mathrm{Abra}}$ and ${u}_{n2}^{\prime \mathrm{Abra}}$ are comparative values from [16] (pp. 409–411). Values were calculated with ${u}_{1}=\lambda $, $\epsilon ={3.5}^{2}$ and ${\partial}_{\theta}=-2i$.

The effective index of the ${\mathrm{TE}}_{01}$ mode (${\gamma}_{01}^{\text{ref}}/{k}_{0}=1.200502$) of a circular dielectric waveguide with a radius of $\lambda $ is computed with the MMGE with respect to the size of the eigenvalue matrix for three different values of the scaling factor $\chi $. Values were calculated with ${u}_{1}=0.5\lambda $, ${u}_{2}=4\lambda $, ${\epsilon}_{1}=2.5$, and ${\epsilon}_{2}=1$.

Table 4.

Convergence of the Effective Index ${\gamma}_{1}/{k}_{0}=1.14623$ of an Elliptical Dielectric Waveguide (Radii of ${r}_{a}=0.5\lambda $ and ${r}_{b}=0.45\lambda $) with Respect to Both $N$ and $M\mathrm{max}=2M+1$^{a}

$M\mathrm{max}$

$N=11$

$N=13$

$N=15$

$N=17$

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 ${u}_{1}=0.5\lambda $, ${u}_{2}=4{r}_{1}$, ${\epsilon}_{1}=2.5$, and ${\epsilon}_{2}=1$.

Table 5.

Convergence of the Effective Index ${\gamma}_{1}/{k}_{0}=1.24235$ of an Elliptical Dielectric Waveguide (Radii of ${r}_{a}=0.5\lambda $ and ${r}_{b}=0.45\lambda $) with Respect to Both $N$ and $M\mathrm{max}=2M+1$^{a}

$M\mathrm{max}$

$N=11$

$N=13$

$N=15$

$N=17$

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 ${u}_{1}=0.5\lambda $, ${u}_{2}=4{r}_{1}$, ${\epsilon}_{1}=2.5$, and ${\epsilon}_{2}=1$.

Tables (5)

Table 1.

First Five Zeros ${u}_{n0}^{\text{ref}}$ of the Zeroth-order Bessel Function and Its Numerical Derivative ${u}_{n0}^{\prime \text{ref}}$ Computed from the Eigenvalues $\gamma $ and for $N=30$^{a}

$n$

${u}_{n0}^{\text{ref}}$

${u}_{n0}^{\mathrm{Abra}}$

${u}_{n0}^{\prime \text{ref}}$

${u}_{n0}^{\prime \mathrm{Abra}}$

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 ${u}_{n0}^{\mathrm{Abra}}$ and ${u}_{n0}^{\prime \mathrm{Abra}}$ are comparative values from [16] (pp. 409–411). Values were calculated with ${\rho}_{0}=\lambda $, $\epsilon ={3.5}^{2}$ and ${\partial}_{\theta}=0$.

Table 2.

First Five Zeros ${u}_{n2}^{\text{ref}}$ of the Second-order Bessel Function and Its Numerical Derivative ${u}_{n2}^{\prime \text{ref}}$ Computed from the Eigenvalues $\gamma $ and for $N=30$^{a}

$n$

${u}_{n2}^{\text{ref}}$

${u}_{n2}^{\mathrm{Abra}}$

${u}_{n2}^{\prime \text{ref}}$

${u}_{n2}^{\prime \mathrm{Abra}}$

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 ${u}_{n2}^{\mathrm{Abra}}$ and ${u}_{n2}^{\prime \mathrm{Abra}}$ are comparative values from [16] (pp. 409–411). Values were calculated with ${u}_{1}=\lambda $, $\epsilon ={3.5}^{2}$ and ${\partial}_{\theta}=-2i$.

The effective index of the ${\mathrm{TE}}_{01}$ mode (${\gamma}_{01}^{\text{ref}}/{k}_{0}=1.200502$) of a circular dielectric waveguide with a radius of $\lambda $ is computed with the MMGE with respect to the size of the eigenvalue matrix for three different values of the scaling factor $\chi $. Values were calculated with ${u}_{1}=0.5\lambda $, ${u}_{2}=4\lambda $, ${\epsilon}_{1}=2.5$, and ${\epsilon}_{2}=1$.

Table 4.

Convergence of the Effective Index ${\gamma}_{1}/{k}_{0}=1.14623$ of an Elliptical Dielectric Waveguide (Radii of ${r}_{a}=0.5\lambda $ and ${r}_{b}=0.45\lambda $) with Respect to Both $N$ and $M\mathrm{max}=2M+1$^{a}

$M\mathrm{max}$

$N=11$

$N=13$

$N=15$

$N=17$

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 ${u}_{1}=0.5\lambda $, ${u}_{2}=4{r}_{1}$, ${\epsilon}_{1}=2.5$, and ${\epsilon}_{2}=1$.

Table 5.

Convergence of the Effective Index ${\gamma}_{1}/{k}_{0}=1.24235$ of an Elliptical Dielectric Waveguide (Radii of ${r}_{a}=0.5\lambda $ and ${r}_{b}=0.45\lambda $) with Respect to Both $N$ and $M\mathrm{max}=2M+1$^{a}

$M\mathrm{max}$

$N=11$

$N=13$

$N=15$

$N=17$

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 ${u}_{1}=0.5\lambda $, ${u}_{2}=4{r}_{1}$, ${\epsilon}_{1}=2.5$, and ${\epsilon}_{2}=1$.