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Analyzing the multilayer metamaterial waveguide structure with the Kerr-type nonlinear cladding

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Abstract

A general method for analyzing the multilayer metamaterial optical waveguides with the Kerr-type nonlinear cladding was proposed. To prove the accuracy of the proposed general method, a degenerated example was introduced. The analytical and numerical results show excellent agreement. The similar process can also be used to analyze TM waves propagating in the multilayer metamaterial waveguide structure with the Kerr-type nonlinear cladding.

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Authors and Affiliations

  1. Department of Electronic Engineering, National Kaohsiung University of Applied Sciences, Kaohsiung, 807, Taiwan, ROC

    Yaw-Dong Wu & Ming-Hsiung Cheng

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  1. Yaw-Dong Wu
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  2. Ming-Hsiung Cheng
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Correspondence to Yaw-Dong Wu.

Appendix

Appendix

The constants \({\text{A}}_{\text{f}} (n)\), \({\text{A}}_{\text{s}}\), \({\text{A}}_{\text{i}} (n)\), \({\text{X}}_{\text{f}} (n)\), \({\text{X}}_{\text{i}} (n)\) can be expressed as follows:

$${\text{x}}_{\text{f}} ({\text{n}} + 1) = \frac{{ - ({\text{n}} + 1)}}{2}{\text{d}}_{\text{f}} - \frac{\text{n}}{2}{\text{d}}_{\text{i}} + \frac{1}{{{\text{k}}_{ 0} {\text{q}}_{\text{f}} }}{ \tan }^{ - 1} \left( {\frac{{\mu_{f} }}{{\mu_{s} }}\frac{{{\text{q}}_{\text{s}} {\kern 1pt} }}{{{\text{q}}_{\text{f}} }}} \right)$$
(21)
$${\text{x}}_{\text{f}} ({\text{n}} + 1) = \frac{{ - ({\text{n}} + 1)}}{2}{\text{d}}_{\text{f}} - \frac{\text{n}}{2}{\text{d}}_{\text{i}} + \frac{1}{{{\text{k}}_{ 0} {\text{Q}}_{\text{f}} }}{ \tanh }^{ - 1} \left( {\frac{{\mu_{s} }}{{\mu_{f} }}\frac{{{\text{ - Q}}_{\text{f}} {\kern 1pt} }}{{{\text{q}}_{\text{s}} }}} \right)$$
(22)
$${\text{A}}_{\text{f}} {\kern 1pt} (1) = \frac{{{\text{E}}_{\text{c}} }}{{\cos \left[ {{\text{k}}_{ 0} {\text{q}}_{\text{f}} \left( {\frac{{{\text{n}} + 1}}{2}{\text{d}}_{\text{f}} + \frac{\text{n}}{2}{\text{d}}_{\text{i}} - x_{\text{f}} (1)} \right)} \right]}}$$
(23)
$${\text{A}}_{\text{f}} (1) = \frac{{{\text{E}}_{\text{c}} }}{{\sinh \left[ {{\text{k}}_{ 0} {\text{Q}}_{\text{f}} \left( {\frac{{{\text{n}} + 1}}{ 2}{\text{d}}_{\text{f}} + \frac{\text{n}}{2}{\text{d}}_{\text{i}} - x_{\text{f}} ( 1 )} \right)} \right]}}$$
(24)
$${\text{A}}_{\text{f}} ({\text{n}} + 1) = {\text{A}}_{\text{i}} ({\text{n}})\frac{{\cosh \left[ {{\text{k}}_{ 0} {\text{q}}_{\text{i}} \left( {\frac{{{\text{n}} - 1}}{2}{\text{d}}_{\text{f}} + \frac{\text{n}}{2}{\text{d}}_{\text{i}} + {\text{x}}_{i} ({\text{n}})} \right)} \right]}}{{\cos \left[ {{\text{k}}_{ 0} {\text{q}}_{\text{f}} \left( {\frac{{{\text{n}} - 1}}{2}{\text{d}}_{\text{f}} + \frac{\text{n}}{2}{\text{d}}_{\text{i}} + {\text{x}}_{\text{f}} ({\text{n}} + 1)} \right)} \right]}}{\kern 1pt}$$
(25)
$${\text{A}}_{\text{f}} ({\text{n}} + 1) = - {\text{A}}_{\text{i}} ({\text{n}})\frac{{\cosh \left[ {{\text{k}}_{ 0} {\text{q}}_{\text{i}} \left( {\frac{{{\text{n}} - 1}}{2}{\text{d}}_{\text{f}} + \frac{\text{n}}{2}{\text{d}}_{\text{i}} + {\text{x}}_{\text{i}} ({\text{n}})} \right)} \right]}}{{\sinh \left[ {{\text{k}}_{ 0} {\text{Q}}_{\text{f}} \left( {\frac{{{\text{n}} - 1}}{2}{\text{d}}_{\text{f}} + \frac{\text{n}}{2}{\text{d}}_{\text{i}} + {\text{x}}_{\text{f}} ({\text{n}} + 1)} \right)} \right]}}{\kern 1pt}$$
(26)
$${\text{A}}_{\text{i}} (1) = {\text{A}}_{\text{f}} ( 1 )\frac{{{\kern 1pt} \cos \left[ {{\text{k}}_{ 0} {\text{q}}_{\text{f}} \left( {\frac{{{\text{n}} - 1}}{ 2}{\text{d}}_{\text{f}} + \frac{\text{n}}{2}{\text{d}}_{\text{i}} - {\text{x}}_{\text{f}} ( 1 )} \right)} \right]}}{{\cosh \left[ {{\text{k}}_{ 0} {\text{q}}_{\text{i}} \left( {\frac{{{\text{n}} - 1}}{ 2}{\text{d}}_{\text{f}} + \frac{\text{n}}{2}{\text{d}}_{\text{i}} - {\text{x}}_{\text{i}} ( 1 )} \right)} \right]}}$$
(27)
$${\text{A}}_{\text{i}} (1) = {\text{A}}_{\text{f}} ( 1 )\frac{{\sinh \left[ {{\text{k}}_{ 0} {\text{Q}}_{\text{f}} \left( {\frac{{{\text{n}} - 1}}{ 2}{\text{d}}_{\text{f}} + \frac{\text{n}}{2}{\text{d}}_{\text{i}} - {\text{x}}_{\text{f}} ( 1 )} \right)} \right]}}{{\cosh \left[ {{\text{k}}_{ 0} {\text{q}}_{\text{i}} \left( {\frac{{{\text{n}} - 1}}{ 2}{\text{d}}_{\text{f}} + \frac{\text{n}}{2}{\text{d}}_{\text{i}} - {\text{x}}_{\text{i}} ( 1 )} \right)} \right]}}$$
(28)
$${\text{A}}_{\text{s}} = {\text{A}}_{\text{f}} ( {\text{n}} + 1 ) {\text{cos}}\left[ {{\text{k}}_{ 0} {\text{q}}_{\text{f}} \left( {\frac{{{\text{n}} + 1}}{ 2}{\text{d}}_{\text{f}} + \frac{\text{n}}{ 2}{\text{d}}_{\text{i}} + {\text{x}}_{\text{f}} ( {\text{n}} + 1 )} \right)} \right] \cdot { \exp }\left[ {{\text{k}}_{ 0} {\text{q}}_{\text{f}} \left( {\frac{{{\text{n}} + 1}}{ 2}{\text{d}}_{\text{f}} + \frac{\text{n}}{ 2}{\text{d}}_{\text{i}} } \right)} \right]$$
(29)
$${\text{A}}_{\text{s}} = {\text{A}}_{\text{f}} ( {\text{n}} + 1 ) {\text{sinh}}\left[ {{\text{k}}_{ 0} {\text{Q}}_{\text{f}} \left( {\frac{{{\text{n}} + 1}}{ 2}{\text{d}}_{\text{f}} + \frac{\text{n}}{ 2}{\text{d}}_{\text{i}} + {\text{x}}_{\text{f}} ( {\text{n}} + 1 )} \right)} \right] \cdot { \exp }\left[ {{\text{k}}_{ 0} {\text{q}}_{\text{f}} \left( {\frac{{{\text{n}} + 1}}{ 2}{\text{d}}_{\text{f}} + \frac{\text{n}}{ 2}{\text{d}}_{\text{i}} } \right)} \right]$$
(30)

For \({\mathbf{0}} \le {\mathbf{p}} \le {\mathbf{n}} - {\mathbf{1,n}} - {\mathbf{p}} \ne {\mathbf{1}}\)

$${\text{A}}_{\text{f}} ( {\text{n}} - {\text{p)}} = {\text{A}}_{\text{i}} ( {\text{n}} - {\text{p}} - 1 )\frac{{{ \cosh }\left[ {{\text{k}}_{ 0} {\text{q}}_{\text{i}} \left( {\frac{{{\text{n}} - 3}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{{{\text{n}} - 2}}{ 2} - {\text{p)d}}_{\text{i}} + {\text{x}}_{\text{i}} ( {\text{n}} - {\text{p)}} - 1} \right)} \right]}}{{{\text{cos[k}}_{ 0} {\text{q}}_{\text{f}} \left( {\left( {\frac{{{\text{n}} - 3}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{{{\text{n}} - 2}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{i}} + {\text{x}}_{\text{i}} ( {\text{n}} - {\text{p)}}} \right) ]}}$$
(31)
$${\text{A}}_{\text{f}} ( {\text{n}} - {\text{p)}} = - {\text{A}}_{\text{i}} ( {\text{n}} - {\text{p}} - 1 )\frac{{{ \cosh }\left[ {{\text{k}}_{ 0} {\text{q}}_{\text{i}} \left( {\frac{{{\text{n}} - 3}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{{{\text{n}} - 2}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{i}} + {\text{x}}_{\text{i}} ( {\text{n}} - {\text{p}} - 1 ) )} \right]}}{{{ \sinh }\left[ {{\text{k}}_{ 0} {\text{Q}}_{\text{f}} \left( {\left( {\frac{{{\text{n}} - 3}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{{{\text{n}} - 2}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{i}} + {\text{x}}_{\text{i}} ( {\text{n}} - {\text{p)}}} \right)} \right]}}$$
(32)
$${\text{A}}_{\text{i}} ({\text{n}} - {\text{p}}) = {\text{A}}_{\text{f}} ({\text{n}} - {\text{p}})\frac{{\cos \left[ {{\text{k}}_{ 0} {\text{q}}_{\text{f}} \left( {\left( {\frac{{{\text{n}} - 1}}{2} - {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{{{\text{n}} - 2}}{2} - {\text{p}}} \right){\text{d}}_{\text{i}} + {\text{x}}_{\text{f}} ({\text{n}} - {\text{p}})} \right)} \right]}}{{\cosh \left[ {{\text{k}}_{ 0} {\text{q}}_{\text{i}} \left( {\left( {\frac{{{\text{n}} - 1}}{2} - {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{{{\text{n}} - 2}}{2} - {\text{p}}} \right){\text{d}}_{\text{i}} + {\text{x}}_{\text{i}} ({\text{n}} - {\text{p}})} \right)} \right]}}$$
(33)
$${\text{A}}_{\text{i}} ( {\text{n}} - {\text{p)}} = - {\text{A}}_{\text{f}} ( {\text{n}} - {\text{p)}}\frac{{{ \sinh }\left[ {{\text{k}}_{ 0} {\text{q}}_{\text{f}} \left( {\left( {\frac{{{\text{n}} - 1}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{{{\text{n}} - 2}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{i}} + {\text{x}}_{\text{f}} ( {\text{n}} - {\text{p)}}} \right)} \right]}}{{{ \cosh }\left[ {{\text{k}}_{ 0} {\text{q}}_{\text{i}} \left( {\left( {\frac{{{\text{n}} - 1}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{{{\text{n}} - 2}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{i}} + {\text{x}}_{\text{i}} ( {\text{n}} - {\text{p)}}} \right)} \right]}}$$
(34)

Part I:n = odd

$${\mathbf{0}} \le {\mathbf{p}} \le ({\mathbf{n}} - {\mathbf{3)/2 }}$$
$$\begin{aligned} {\text{x}}_{\text{f}} ({\text{n}} - {\text{p}}) &= \left( {\frac{{ - ({\text{n}} - 1)}}{2} + {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{{ - ({\text{n}} - 2)}}{2} + {\text{p}}} \right){\text{d}}_{\text{i}} \\ & + \frac{{\tan^{ - 1} \left\{ {\frac{{\upmu_{f} }}{{\upmu_{i} }}\frac{{{\text{ - q}}_{\text{i}} }}{{{\text{q}}_{\text{f}} }}\tanh \left[ {{\text{k}}_{ 0} {\text{q}}_{\text{i}} \left[ {\left( {\frac{{({\text{n}} - 1)}}{2} - {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{{{\text{n}} - 2}}{2} - {\text{p}}} \right){\text{d}}_{\text{i}} + {\text{x}}_{\text{i}} ({\text{n}} - {\text{p}})} \right]} \right]} \right\}}}{{k_{ 0} {\text{q}}_{\text{f}} }} \\ \end{aligned}$$
(35)
$$\begin{aligned} {\text{x}}_{\text{f}} ( {\text{n}} - {\text{p)}} = \left( {\frac{{ - ( {\text{n}} - 1 )}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{{{\text{ - (n}} - 2 )}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{i}} \\ & \quad+ \frac{{{ \tanh }^{ - 1} \left\{ {\frac{{\upmu_{i} }}{{\upmu_{f} }}\frac{{{\text{Q}}_{\text{f}} }}{{{\text{q}}_{\text{i}} { \tanh }\left[ {{\text{k}}_{ 0} {\text{q}}_{\text{i}} \left[ {\left( {\frac{{ ( {\text{n}} - 1 )}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{{{\text{n}} - 2}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{i}} + {\text{x}}_{\text{i}} ( {\text{n}} - {\text{p)}}} \right]} \right]}}} \right\}}}{{{\text{k}}_{ 0} {\text{Q}}_{\text{f}} }} \\ \end{aligned}$$
(36)
$$\begin{aligned} {\text{x}}_{\text{i}} ( {\text{n}} - {\text{p)}} &= \left( {\frac{{ - ( {\text{n}} - 1 )}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{\text{ - n}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{i}} \\ \quad + \frac{{{ \tanh }^{ - 1} \left\{ {\frac{{\upmu_{\text{i}} }}{{\upmu_{\text{f}} }}\frac{{{\text{ - q}}_{\text{f}} }}{{{\text{q}}_{\text{i}} }}{ \tan }\left[ {{\text{k}}_{ 0} {\text{q}}_{\text{f}} \left[ {\left( {\frac{{ ( {\text{n}} - 1 )}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{\text{n}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{i}} + {\text{x}}_{\text{c}} ( {\text{n}} - {\text{p}} + 1 )} \right]} \right]} \right\}}}{{{\text{k}}_{ 0} {\text{q}}_{\text{i}} }} \\ \end{aligned}$$
(37)
$$\begin{aligned} {\text{x}}_{\text{i}} ( {\text{n}} - {\text{p)}} & = \left( {\frac{{ - ( {\text{n}} - 1 )}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{\text{ - n}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{i}} \\ & \quad + \frac{{{ \tanh }^{ - 1} \left\{ {\frac{{\upmu_{\text{i}} }}{{\upmu_{\text{f}} }}\frac{{{\text{Q}}_{\text{f}} }}{{{\text{q}}_{\text{i}} { \tanh }\left[ {{\text{k}}_{ 0} {\text{Q}}_{\text{f}} \left[ {\left( {\frac{{ ( {\text{n}} - 1 )}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{\text{n}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{i}} + {\text{x}}_{\text{f}} ( {\text{n}} - {\text{p}} + 1 )} \right]} \right]}}} \right\}}}{{{\text{k}}_{ 0} {\text{q}}_{\text{i}} }} \\ \end{aligned}$$
(38)
$${\mathbf{(n}} + {\mathbf{1)/2}} \le {\mathbf{p}} \le ({\mathbf{n}} - {\mathbf{1) }}$$
$$\begin{aligned} {\text{x}}_{\text{f}} ( {\text{n}} - {\text{p)}} & =\left( {\frac{{ - ( {\text{n}} - 1 )}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{{{\text{ - (n}} - 2 )}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{i}} \\ & \quad + \frac{{{ \tan }^{ - 1} \left\{ {\frac{{{\text{ - q}}_{\text{i}} }}{{{\text{q}}_{\text{f}} }}{ \tanh }\left[ {{\text{k}}_{ 0} {\text{q}}_{\text{i}} \left[ {{ - }\left( {\frac{{ ( {\text{n}} - 1 )}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{f}} - \left( {\frac{{{\text{n}} - 2}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{i}} - {\text{x}}_{\text{i}} ( {\text{n}} - {\text{p)}}} \right]} \right]} \right\}}}{{{\text{k}}_{ 0} {\text{q}}_{\text{f}} }} \\ \end{aligned}$$
(39)
$$\begin{aligned} {\text{x}}_{\text{f}} ( {\text{n}} - {\text{p)}} & = \left( {\frac{{ - ( {\text{n}} - 1 )}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{{{\text{ - (n}} - 2 )}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{i}} \\ & \quad + \frac{{{ \tanh }^{ - 1} \left\{ {\frac{{\upmu_{\text{i}} }}{{\upmu_{\text{f}} }}\frac{{{\text{Q}}_{\text{f}} }}{{{\text{q}}_{\text{i}} { \tanh }\left[ {{\text{k}}_{ 0} {\text{q}}_{\text{i}} \left[ {{ - }\left( {\frac{{ ( {\text{n}} - 1 )}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{f}} - \left( {\frac{{{\text{n}} - 2}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{i}} - {\text{x}}_{\text{i}} ( {\text{n}} - {\text{p)}}} \right]} \right]}}} \right\}}}{{{\text{k}}_{ 0} {\text{Q}}_{\text{f}} }} \\ \end{aligned}$$
(40)
$$\begin{aligned} {\text{x}}_{\text{i}} ( {\text{n}} - {\text{p)}} & = \left( {\frac{{ - ( {\text{n}} - 1 )}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{\text{ - n}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{i}} \\ & - \frac{{{ \tanh }^{ - 1} \left\{ {\frac{{\upmu_{\text{i}} }}{{\upmu_{\text{f}} }}\frac{{{\text{ - q}}_{\text{f}} }}{{{\text{q}}_{\text{i}} }}{ \tan }\left[ {{\text{k}}_{ 0} {\text{q}}_{\text{f}} \left[ {{ - }\left( {\frac{{ ( {\text{n}} - 1 )}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{f}} - \left( {\frac{\text{n}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{i}} - {\text{x}}_{\text{f}} ( {\text{n}} - {\text{p}} + 1 )} \right]} \right]} \right\}}}{{{\text{k}}_{ 0} {\text{q}}_{\text{i}} }} \\ \end{aligned}$$
(41)
$$\begin{aligned} {\text{x}}_{\text{i}} ( {\text{n}} - {\text{p)}} & = \left( {\frac{{ - ( {\text{n}} - 1 )}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{\text{ - n}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{i}} \\ & \quad - \frac{{{ \tanh }^{ - 1} \left\{ {\frac{{\upmu_{\text{i}} }}{{\upmu_{\text{f}} }}\frac{{{\text{Q}}_{\text{f}} }}{{{\text{q}}_{\text{i}} { \tanh }\left[ {{\text{k}}_{ 0} {\text{Q}}_{\text{f}} \left[ {{ - }\left( {\frac{{ ( {\text{n}} - 1 )}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{f}} - \left( {\frac{\text{n}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{i}} - {\text{x}}_{\text{f}} ( {\text{n}} - {\text{p}} + 1 )} \right]} \right]}}} \right\}}}{{{\text{k}}_{ 0} {\text{q}}_{\text{i}} }} \\ \end{aligned}$$
(42)

otherwise

$${\text{x}}_{\text{f}} \left( {\frac{{{\text{n}} + 1}}{ 2}} \right) = \frac{ 1}{ 2}{\text{d}}_{\text{i}} - \frac{{\tan^{ - 1} \left\{ {\frac{{\mu_{f} }}{{\mu_{i} }}\frac{{{\text{ - q}}_{\text{i}} }}{{{\text{q}}_{\text{f}} }}\tanh \left[ {{\text{k}}_{ 0} {\text{q}}_{\text{i}} \left( {\frac{ 1}{ 2}{\text{d}}_{\text{i}} - {\text{x}}_{\text{i}} \left( {\frac{{{\text{n}} + 1}}{ 2}} \right)} \right)} \right]} \right\}}}{{{\text{k}}_{ 0} {\text{q}}_{\text{f}} }}$$
(43)
$${\text{x}}_{\text{f}} \left( {\frac{{{\text{n}} + 1}}{ 2}} \right) = \frac{ 1}{ 2}{\text{d}}_{\text{i}} - \frac{{\tanh^{ - 1} \left\{ {\frac{{\mu_{i} }}{{\mu_{f} }}\frac{{{\text{Q}}_{\text{f}} }}{{{\text{q}}_{\text{i}} \tanh \left[ {{\text{k}}_{ 0} {\text{q}}_{\text{i}} \left( {\frac{ 1}{ 2}{\text{d}}_{\text{i}} - {\text{x}}_{\text{i}} \left( {\frac{{{\text{n}} + 1}}{ 2}} \right)} \right)} \right]}}} \right\}}}{{{\text{k}}_{ 0} {\text{Q}}_{\text{f}} }}$$
(44)
$${\text{x}}_{\text{i}} \left( {\frac{{{\text{n}} + 1}}{ 2}} \right) = \frac{ - 1}{ 2}{\text{d}}_{\text{i}} + \frac{{\tanh^{ - 1} \left\{ {\frac{{\mu_{i} }}{{\mu_{f} }}\frac{{{\text{ - q}}_{\text{f}} }}{{{\text{q}}_{\text{i}} }}\tan \left[ {{\text{k}}_{ 0} {\text{q}}_{\text{f}} \left( {\frac{ 1}{ 2}{\text{d}}_{\text{i}} + {\text{x}}_{\text{f}} \left( {\frac{{{\text{n}} + 3}}{ 2}} \right)} \right)} \right]} \right\}}}{{{\text{k}}_{ 0} {\text{q}}_{\text{i}} }}$$
(45)
$${\text{x}}_{\text{i}} \left(\frac{{{\text{n}} + 1}}{ 2}\right) = \frac{ - 1}{ 2}{\text{d}}_{\text{i}} + \frac{{\tanh^{ - 1} \left\{ {\frac{{\mu_{i} }}{{\mu_{f} }}\frac{{{\text{Q}}_{\text{f}} }}{{{\text{q}}_{\text{i}} \tanh [{\text{k}}_{ 0} {\text{Q}}_{\text{f}} (\frac{ 1}{ 2}{\text{d}}_{\text{i}} + {\text{x}}_{\text{f}} (\frac{{{\text{n}} + 3}}{ 2}) )]}}} \right\}}}{{{\text{k}}_{ 0} {\text{q}}_{\text{i}} }}$$
(46)

Part II:n = even

$${\mathbf{0}} \le {\mathbf{p}} \le ({\mathbf{n}} - {\mathbf{2)/2 }}$$
$$\begin{aligned} {\text{x}}_{\text{f}} ( {\text{n}} - {\text{p)}} &= \left( {\frac{{ - ( {\text{n}} - 1 )}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{{{\text{ - (n}} - 2 )}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{i}} \\ & \quad + \frac{{{ \tan }^{ - 1} \left\{ {\frac{{\upmu_{\text{i}} }}{{\upmu_{\text{f}} }}\frac{{{\text{ - q}}_{\text{i}} }}{{{\text{q}}_{\text{f}} }}{ \tanh }\left[ {{\text{k}}_{ 0} {\text{q}}_{\text{i}} \left[ {\left( {\frac{{ ( {\text{n}} - 1 )}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{{{\text{n}} - 2}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{i}} + {\text{x}}_{\text{i}} ( {\text{n}} - {\text{p)}}} \right]} \right]} \right\}}}{{{\text{k}}_{ 0} {\text{q}}_{\text{f}} }} \\ \end{aligned}$$
(47)
$$\begin{aligned} {\text{x}}_{\text{f}} ( {\text{n}} - {\text{p)}} & = \left( {\frac{{ - ( {\text{n}} - 1 )}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{{{\text{ - (n}} - 2 )}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{i}} \\ & \quad + \frac{{{ \tanh }^{ - 1} \left\{ {\frac{{\upmu_{\text{i}} }}{{\upmu_{\text{f}} }}\frac{{{\text{Q}}_{\text{f}} }}{{{\text{q}}_{\text{i}} { \tanh }\left[ {{\text{k}}_{ 0} {\text{q}}_{\text{i}} \left[ {\left( {\frac{{ ( {\text{n}} - 1 )}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{{{\text{n}} - 2}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{i}} + {\text{x}}_{\text{i}} ( {\text{n}} - {\text{p)}}} \right]} \right]}}} \right\}}}{{{\text{k}}_{ 0} {\text{Q}}_{\text{f}} }} \\ \end{aligned}$$
(48)
$$\begin{aligned} {\text{x}}_{\text{i}} ( {\text{n}} - {\text{p)}} & = \left( {\frac{{ - ( {\text{n}} - 1 )}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{\text{ - n}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{i}} \\ & \quad + \frac{{{ \tanh }^{ - 1} \left\{ {\frac{{\upmu_{\text{i}} }}{{\upmu_{\text{f}} }}\frac{{{\text{ - q}}_{\text{f}} }}{{{\text{q}}_{\text{i}} }}{ \tan }\left[ {{\text{k}}_{ 0} {\text{q}}_{\text{f}} \left[ {\left( {\frac{{ ( {\text{n}} - 1 )}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{\text{n}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{i}} + {\text{x}}_{\text{f}} ( {\text{n}} - {\text{p}} + 1 )} \right]} \right]} \right\}}}{{{\text{k}}_{ 0} {\text{q}}_{\text{i}} }} \\ \end{aligned}$$
(49)
$$\begin{aligned} {\text{x}}_{\text{i}} ( {\text{n}} - {\text{p)}}& = \left( {\frac{{ - ( {\text{n}} - 1 )}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{\text{ - n}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{i}} \\ & \quad + \frac{{{ \tanh }^{ - 1} \left\{ {\frac{{\upmu_{\text{i}} }}{{\upmu_{\text{f}} }}\frac{{{\text{Q}}_{\text{f}} }}{{{\text{q}}_{\text{i}} { \tanh }\left[ {{\text{k}}_{ 0} {\text{Q}}_{\text{f}} \left( {\frac{{ ( {\text{n}} - 1 )}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{\text{n}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{i}} + {\text{x}}_{\text{f}} ( {\text{n}} - {\text{p}} + 1 )} \right]}}} \right\}}}{{{\text{k}}_{ 0} {\text{q}}_{\text{i}} }} \\ \end{aligned}$$
(50)
$${\mathbf{n/2}} \le {\mathbf{p}} \le ({\mathbf{n}} - {\mathbf{1) }}$$
$$\begin{aligned} {\text{x}}_{\text{f}} ( {\text{n}} - {\text{p)}} & = \left( {\frac{{ - ( {\text{n}} - 1 )}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{{{\text{ - (n}} - 2 )}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{i}} \\ & \quad - \frac{{{ \tan }^{ - 1} \left\{ {\frac{{\upmu_{\text{i}} }}{{\upmu_{\text{f}} }}\frac{{{\text{ - q}}_{\text{i}} }}{{{\text{q}}_{\text{f}} }}{ \tanh }\left[ {{\text{k}}_{ 0} {\text{q}}_{\text{i}} \left[ {{ - }\left( {\frac{{ ( {\text{n}} - 1 )}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{f}} - \left( {\frac{{{\text{n}} - 2}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{i}} - {\text{x}}_{\text{i}} ( {\text{n}} - {\text{p)}}} \right]} \right]} \right\}}}{{{\text{k}}_{ 0} {\text{q}}_{\text{f}} }} \\ \end{aligned}$$
(51)
$$\begin{aligned} {\text{x}}_{\text{f}} & = \left( {\frac{{ - ( {\text{n}} - 1 )}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{{{\text{ - (n}} - 2 )}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{i}} \hfill \\ & \quad - \frac{{{ \tanh }^{ - 1} \left\{ {\frac{{\upmu_{\text{i}} }}{{\upmu_{\text{f}} }}\frac{{{\text{Q}}_{\text{f}} }}{{{\text{q}}_{\text{i}} { \tanh }\left[ {{\text{k}}_{ 0} {\text{q}}_{\text{i}} \left[ {{ - }\left( {\frac{{ ( {\text{n}} - 1 )}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{f}} - \left( {\frac{{{\text{n}} - 2}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{i}} - {\text{x}}_{\text{i}} ( {\text{n}} - {\text{p)}}} \right]} \right]}}} \right\}}}{{{\text{k}}_{ 0} {\text{Q}}_{\text{f}} }} \hfill \\ \end{aligned}$$
(52)
$$\begin{aligned} {\text{x}}_{\text{i}} ( {\text{n}} - {\text{p)}} & = \left( {\frac{{ - ( {\text{n}} - 1 )}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{\text{ - n}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{i}} \\ & \quad - \frac{{{ \tanh }^{ - 1} \left\{ {\frac{{\upmu_{\text{i}} }}{{\upmu_{\text{f}} }}\frac{{{\text{ - q}}_{\text{f}} }}{{{\text{q}}_{\text{i}} }}{ \tan }\left[ {{\text{k}}_{ 0} {\text{q}}_{\text{f}} \left[ {{ - }\left( {\frac{{ ( {\text{n}} - 1 )}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{f}} - \left( {\frac{\text{n}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{i}} - {\text{x}}_{\text{f}} ( {\text{n}} - {\text{p}} + 1 )} \right]} \right]} \right\}}}{{{\text{k}}_{ 0} {\text{q}}_{\text{i}} }} \\ \end{aligned}$$
(53)
$$\begin{aligned} {\text{x}}_{\text{i}} ( {\text{n}} - {\text{p)}} & = \left( {\frac{{ - ( {\text{n}} - 1 )}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{f}} + \left( {\frac{\text{ - n}}{ 2} + {\text{p}}} \right){\text{d}}_{\text{i}} \\ & \quad - \frac{{{ \tanh }^{ - 1} \left\{ {\frac{{\upmu_{\text{i}} }}{{\upmu_{\text{f}} }}\frac{{{\text{Q}}_{\text{f}} }}{{{\text{q}}_{\text{i}} { \tanh }\left[ {{\text{k}}_{ 0} {\text{Q}}_{\text{f}} \left[ {{ - }\left( {\frac{{ ( {\text{n}} - 1 )}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{f}} - \left( {\frac{\text{n}}{ 2} - {\text{p}}} \right){\text{d}}_{\text{i}} - {\text{x}}_{\text{f}} ( {\text{n}} - {\text{p}} + 1 )} \right]} \right]}}} \right\}}}{{{\text{k}}_{ 0} {\text{q}}_{\text{i}} }} \\ \end{aligned}$$
(54)

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Wu, YD., Cheng, MH. Analyzing the multilayer metamaterial waveguide structure with the Kerr-type nonlinear cladding. Opt Quant Electron 49, 181 (2017). https://doi.org/10.1007/s11082-017-1016-5

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