Extraction of all propagation constants in a specified region from the transcendental equation of a dispersion relation using the Sakurai–Sugiura projection method
Shingo Sato,1,*
Takao Shimada,2
and Koji Hasegawa1
1Division of Information and Electronic Engineering, Graduate School of Engineering, Muroran Institute of Technology, 27-1 Mizumoto-cho, Muroran 050-8585, Japan
2Department of Electrical and Electronic Engineering, Tsuyama National College of Technology, 624-1 Numa, Tsuyama 708-8509, Japan
Shingo Sato, Takao Shimada, and Koji Hasegawa, "Extraction of all propagation constants in a specified region from the transcendental equation of a dispersion relation using the Sakurai–Sugiura projection method," J. Opt. Soc. Am. A 32, 1216-1221 (2015)
A transcendental equation occurs when we compute the dispersion relations of an electromagnetic waveguide, such as a planar multilayer waveguide. Without an initial guess, the Sakurai–Sugiura projection method (SSM) can obtain solutions to the transcendental equation in a region bounded by a contour integral path in the complex plane. In this paper, a criterion employing the condition number of eigenvalues as a simple index to distinguish physical solutions from spurious ones in the SSM is presented, and a transcendental equation of a multilayer waveguide obtained by the transfer matrix method is solved by the SSM. Numerical results show the usefulness of the index and good agreement with the results of the argument principle method and Newton’s method.
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TM Mode Analysis of a Six-Layer Lossless Dielectric Waveguide (, , , , , and , with ) at : Singular Values of and Solutions for Threshold Values of , , and a
Solution
Singular Value
Mode
1.5947885
1.5549965
1.5549807
1.5549807
1.5018075
Spurious
Spurious
The parameters of SSM are , , , , and .
Table 2.
TM Mode Analysis of the Six-Layer Lossless Dielectric Waveguide (the Same Structure as in Table 1) at : Effective Indices, , Condition Numbers, , and Residuals, , for a
Mode
Inside
2
Spurious
†
Spurious
†
3
Spurious
†
Spurious
†
Spurious
1.4997205
4.2
4
2.8
†
†
1.5549965
†
1.5018075
†
5
1.6200313
†
†
1.5549807
†
†
Spurious
4.4
6
†
1.5947885
†
1.5549807
†
†
Spurious
5.1
Spurious
4.2
7
†
†
†
†
Spurious
†
Spurious
5.1
Spurious
4.2
8
†
†
†
†
Spurious
†
Spurious
Spurious
4.2
Spurious
5.0
The parameters of SSM are , , , and . The symbol “†” in the rightmost column denotes a solution inside .
Table 3.
Effective Indices of TE Guided Modes and Leaky Modes at for the Six-Layer Lossless Dielectric Waveguide (the Same Structure as in Tables 1 and 2)a
The parameters of SSM are , , , , and for the TE guided modes, and and for the TE leaky modes. The symbol “†” in the fifth column denotes a solution inside .
Table 4.
Effective Indices of the TE Modes at for a Three-Layer Metal-Clad Waveguide (, , , and )a
The parameters of SSM are , , , , and . The symbol “†” in the fifth column denotes a solution inside . The effective index in the lowest row is not a number (NaN) due to the fact that in Eq. (14).
Tables (4)
Table 1.
TM Mode Analysis of a Six-Layer Lossless Dielectric Waveguide (, , , , , and , with ) at : Singular Values of and Solutions for Threshold Values of , , and a
Solution
Singular Value
Mode
1.5947885
1.5549965
1.5549807
1.5549807
1.5018075
Spurious
Spurious
The parameters of SSM are , , , , and .
Table 2.
TM Mode Analysis of the Six-Layer Lossless Dielectric Waveguide (the Same Structure as in Table 1) at : Effective Indices, , Condition Numbers, , and Residuals, , for a
Mode
Inside
2
Spurious
†
Spurious
†
3
Spurious
†
Spurious
†
Spurious
1.4997205
4.2
4
2.8
†
†
1.5549965
†
1.5018075
†
5
1.6200313
†
†
1.5549807
†
†
Spurious
4.4
6
†
1.5947885
†
1.5549807
†
†
Spurious
5.1
Spurious
4.2
7
†
†
†
†
Spurious
†
Spurious
5.1
Spurious
4.2
8
†
†
†
†
Spurious
†
Spurious
Spurious
4.2
Spurious
5.0
The parameters of SSM are , , , and . The symbol “†” in the rightmost column denotes a solution inside .
Table 3.
Effective Indices of TE Guided Modes and Leaky Modes at for the Six-Layer Lossless Dielectric Waveguide (the Same Structure as in Tables 1 and 2)a
The parameters of SSM are , , , , and for the TE guided modes, and and for the TE leaky modes. The symbol “†” in the fifth column denotes a solution inside .
Table 4.
Effective Indices of the TE Modes at for a Three-Layer Metal-Clad Waveguide (, , , and )a
The parameters of SSM are , , , , and . The symbol “†” in the fifth column denotes a solution inside . The effective index in the lowest row is not a number (NaN) due to the fact that in Eq. (14).