Abstract

A detailed study of guided modes in a nanoscale metallic rectangular waveguide is presented by using the effective dielectric constant approach. The guided modes, including both traditional waveguide mode and surface plasmon mode, are investigated for the silver rectangular waveguide. The mode evolution in narrow waveguide is also discussed with the emphasis on the dependence of mode dispersion with waveguide height. Finally, the red-shift of the cutoff wavelength of the fundamental mode is observed when the waveguide height decreases, contrary to the behavior of regular metallic waveguide with PEC boundary. The comprehensive analysis can provide some guideline in the design of subwavelength optical devices based on the dispersion characteristics of metallic rectangular bore.

© 2007 Optical Society of America

1. Introduction

Metallic nanostructures have attracted the attention of many scientists and the study of surface plasmon polaritons (SPPs) has become one of the main research branches of nanophotonics[1]. Nanoscale apertures in metallic films show abnormal optical properties and lead to enhanced light transmission and light confinement [2, 3]. According to the waveguide theory in microwave frequencies, the guided modes cannot exist above the cut-off wavelength, which is the same magnitude as the waveguide dimension. However, in optical frequencies, the properties of the metals have to be described by complex permittivity with dispersion and loss. Furthermore, the real parts of the permittivity can be negative below the plasma frequency. This extraordinary mode of light transmission in subwavelength apertures has been investigated for metallic coaxial waveguide [4, 5], circular aperture[6, 7], and rectangular waveguide[8,9]. It was found that such nanoscale apertures can support SPP modes, which can propagate even when the dimension of the aperture is smaller than the guidance wavelength due to fact that such SPP modes have a longer cutoff wavelength.

In analyzing the light transmission through subwavelength hole in metallic films, several methods have been used to model the fields in the apertures. For subwavelength cylindrical and coaxial waveguides, the guided modes can be calculated by conventional waveguide theory. However, for a metallic rectangular waveguide where the metal is modeled as a plasma medium, there is no analytic solution, except when the metal is modeled as PEC. Numerically, the finite-difference method in time domain and frequency domain can be used to analyze the guided modes and transmission characteristics in such metallic rectangular waveguide [10]. Due to the large difference between the permittivities of the metal and the dielectric, the finite-difference method may cause calculation instability when applied to certain nanoscale metallic structures. Recently, the effective dielectric constant (or index) method (EDCM or EIM) [11], which has been successfully used in the analysis of optical waveguide for many years, has been used to analyze the fundamental mode in the metallic rectangular trench and hole [1214].

In this paper, the EDCM is used to analyze both the traditional waveguide (TWG) modes and the SPP modes in the metallic rectangular waveguide where the permittivity of the metal is represented by the Drude model. The mode evolution in narrow waveguide is also discussed with the emphasis on the dependence of mode dispersion with waveguide height. Finally, the red-shift of the cutoff wavelength of the fundamental mode is observed when the waveguide aspect ratio decreases. These results can give some guideline in the design of subwavelength optical devices.

2. Analysis methodology

 

Fig. 1. Cross section of metallic rectangular waveguide (a) and the use of effective dielectric constant method in the x-direction (b) and y-direction (c)

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Figure 1(a) shows a schematic illustration of the metallic rectangular waveguide under consideration. The relative dielectric constant of the rectangular core is assumed to be εr 1, and the relative dielectric constant of the metal at optical frequencies can be approximated by the Drude model

εr2(ω)=εr2(1ωp2ω(ω+iωτ))

where εr 2 is the permittivity at infinite frequency, ωp and ωτ are the plasma and collision frequencies respectively. The collision frequency, typically ωτ<0.1ωp, is related to the metal loss and can be neglected for simplicity.

The concept of effective dielectric constant approach is also shown in Fig.1. The cross section of metallic waveguide can be treated as a combination of two coupled 1D slab waveguides in the x and y directions respectively [Figs. 1(b), 1(c)].

We first consider the Eymn modes (TE-like), in which Ey and Hx are the dominant electromagnetic fields. The wave functions for the x-direction slab can be written as [15]

ϕhx(y)={B1eγy(yb)(yb)A1eikyy+A2eikyy(b<y<b)B2eγy(y+b)(yb)

Applying the boundary conditions at yb: Ez 1=Ez 2 and Hx 1=Hx 2, we get

ikyεr1(A1eikybA2eikyb)=B1γyεr2(ω)
A1eikyb+A2eikyb=B1
ikyεr1(A1eikybA2eikyb)=B2γyεr2(ω)
A1eikyb+A2eikyb=B2

Eliminating the constants in Eq. (3), we can get the guidance condition for the x-direction slab

(1iρy1+iρy)ei2kyb=ei(n1)π(n=1,2,3,)

where ρy=εr1γyεr2(ω)ky and γy2=k02(εr1εr2(ω))ky2.

For real wave number ky, Eq. (4) can be rewritten as

tan(kyb(n1)π2)=ρy(n=1,2,3,)

For imaginary wave number ky, let ky=ik′y, and Eq. (4) can be rewritten as

tanh(kyb)=ρy(Evensolution)
coth(kyb)=ρy(Oddsolution)

The effective dielectric constant for the x-direction slab waveguide can be expressed as εer 1=εr 1-(ky/k 0)2. Thus, the metallic waveguide can be regarded as a slab waveguide [Fig.1 (c)] along y-direction with the effective dielectric constant εer 1 in the core and εr 2(ω) in the cladding and the wave functions can be expressed as

ϕEy(x)={D1eγx(xa)(xa)C1eikxx+C2eikxx(a<x<a)D2eγx(x+b)(xa)

Applying the boundary conditions at xa: Ey 1=Ey 2 and Hz 1=Hz 2, we get

ikx(C1eikxaC2eikxa)=D1γx
C1eikxa+C2eikxa=D1
ikx(C1eikxaC2eikxa)=D2γx
C1eikxa+C2eikxa=D2

Eliminating the constants in Eq. (8), we can get the guidance condition for the y-direction slab:

(1iρx1+iρx)ei2kyb=ei(m1)π(m=1,2,3,)

where ρx=γx/kx and γ 2 x=k 2 0(εer 1-εr 2(ω))-k 2 x.

Since there is no solution when kx is imaginary, Eq. (9) can be rewritten as

tan(kxa(m1)π2)=ρx(m=1,2,3,)

For the Eymn modes, there are two types of guided waves in the metallic rectangular waveguide. The first type is the TWG modes which satisfy the guidance conditions Eq. (5) and Eq. (10), and the second type is the SPP modes which satisfy the guidance conditions Eq. (6) and Eq. (10). The TWG modes are denoted as Eymn, while the SPP modes are denoted as Eyme or Eymo. The subscripts m and n are integers and they correspond to the number of peaks of each field component in the x and y directions respectively, the subscripts e and o refer to the symmetry of the field distribution (even or odd) in x or y directions.

In the similar procedure, we can get the guidance condition for Exmn mode as follows

TWGmodesEmnx:tan(kxa(m1)π2)=εr1εr2(ω)γxkx,tan(kyb(n1)π2)=γyky
SPPmodesEenx:tanh(kxa)=εr1εr2(ω)γxkx,tan(kyb(n1)π2)=γyky
SPPmodesEonx:coth(kxa)=εr1εr2(ω)γxkx,tan(kyb(n1)π2)=γyky

where γ 2 x=k 2 0(εr 1-εr 2(ω))-k 2 x, γ 2 y=k 2 0(εr 1-εr 2(ω))-k 2 x-k 2 y, k x=ik′x, and kx=ik′x(for SPP modes only).

To analyze the dispersion of the guided waves in metallic rectangular waveguide, we solve for kx(or k′x) and ky (or k′y) numerically, and the propagation constant β can be expressed as

β={k02εr1kx2ky2Emnx,Emnyk02εr1+kx2ky2Eenx,Eonxk02εr1kx2+ky2Emey,Emoy

3. Results and discussion

In the calculation, we choose εr 2=3.7 and ωp=7.1085×1015 rad/s to model the permittivity of silver. The parameters were calculated by Sönnichsen [16] from experimental data on the reflection and transmission of 25–50 nm thick silver films for wavelengths from 0.188µm to 1.9µm[17]. Figure 2 shows the dispersion curves for TWG modes and SPP modes in two different silver waveguides. The shaded region beginning from the plasma frequency (fp=1131THz) is the forbidden area where no guided mode exists. This region is delimited with the refractive index of the silver as solid line. It is shown that the SPP modes usually exist below a critical frequency (fc=1003THz) at which the relative permittivity is negative unity, and the lowest order TWG modes (Ex 11, Ey 11) exist above the plasmon frequency, but some higher order TWG modes can exist both below and above the plasmon frequency. All the TWG modes locate in fast wave region (β/k 0<1.0), while the SPP modes can migrate from slow wave region to fast wave region as the operating frequency decreases. Although each mode has its own dispersion characteristics, the dispersion curve of some modes can continue smoothly from one mode to another mode, for example: in both cases, Ey 12 and Ey 1o, Ex 31 and Ex 11 are continuous. Moreover, it is found that some of higher order TWG modes have discontinuous dispersion curves, for example Ex 21 in 400nm×200nm silver waveguide, one part exists below the plasma frequency and connect with the mode Exo 1, and another part exists above the plasmon frequency and connects with Ex 41. Thus, some of TWG modes and SPP modes can coexist at the same frequency with certain waveguide geometries.

 

Fig. 2. Dispersion characteristics of TWG modes and SPPs modes in silver waveguide, the insets show the E field distributions of several modes in the air-core.

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Fig. 3. Dispersion evolution for SPPs mode in silver waveguide with different height

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Figure 3 shows the dispersion curves for the SPP modes in silver waveguides with different waveguide heights. The waveguide width 2a is assumed to be 250nm. For comparison, the dispersion curves for the SPP modes TMe and TMo in the corresponding slab waveguides with the same heights are also shown in Fig. 3. As the waveguide height decreases, the SPP modes Exe 1 and Exo 1 are being suppressed and their cutoff frequencies move up and eventually approach the critical frequency. Also, the slope of curve for the odd SPP mode Ey 1o can change from positive to negative when the waveguide height is very small. This means that there exists backward surface wave in the narrow metal waveguide.

Figure 4 shows the calculated cutoff wavelength dependence of primary SPPs modes on the waveguide aspect ratio. In a rectangular waveguide with the PEC boundary, the waveguide height does not influence the propagation constant for the fundamental TE 10 mode, and the cut-off wavelength of TE 10 is twice the waveguide width. However, the red-shift of cutoff wavelength of fundamental mode Ey 1e appears when waveguide aspect ratio decreases. This allows light with longer wavelength to propagate through the rectangular aperture. Moreover, it is found that when the aspect ratio is less than 0.45, it is more suitable for single mode operation.

 

Fig. 4. Calculated cutoff wavelength for the primary SPPs modes in a silver rectangular waveguide with different aspect ratio

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4. Conclusion

In this paper, the EDCM is used to analyze the modes in the metallic rectangular waveguide. The results show that the metallic waveguide supports not only the TWG modes, but also the SPP modes. The SPP modes usually exist below the critical frequency. As the operation frequency decreases, the SPP modes migrate from slow wave region to fast wave region. When the waveguide height decreases, the SPP modes Exe 1 and Exo 1 are suppressed and the red-shift of cut off wavelength of the fundamental mode is observed. Furthermore, the slope of curve for the odd SPP mode Ey 1 o can change from positive to negative when the waveguide height is very small. This means that there exists backward surface wave in the narrow metal waveguide. Moreover, it is found that when the aspect ratio is less than 0.45, it is more suitable for single mode operation. These results can provide some guideline in the design of nanoscale optical devices based on the dispersion characteristics of metallic rectangular hole.

References and links

1. P. N. Prasad, Nanophotonics (Wiley-Interscience, New Jersey, 2004). [CrossRef]  

2. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003). [CrossRef]   [PubMed]  

3. T. Rindzevicius, Y. Alaverdyan, and B. Sepulveda. “Nanohole plasmons in optically thin gold films,” J. Phys. Chem. C 111, 1207–1212(2007). [CrossRef]  

4. F. I. Baida, A. Belkhir, D. Van Labeke, and O. Lamrous, “Subwavelength metallic coaxial waveguides in the optical range: Role of the plasmonic modes,” Phys. Rev. B 74, 205419 (2006). [CrossRef]  

5. C. K. Chang, D. Z. Lin, and C. S. Yeh, et. al., “Experimental analysis of surface plasmon behavior in metallic circular slits,” Appl. Phys. Lett. 90, 061113 (2007). [CrossRef]  

6. K. Y. Kim, Y. K. Cho, H. S. Tae, and J. H. Lee, “Optical guided dispersions and subwavelength transmissions in dispersive plasmonic circular holes,” Opto-Electron. Rev. 14, 233–241 (2006). [CrossRef]  

7. A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, “Optical transmission properties of a single subwavelength aperture in a real metal,” Opt. Commun. 239, 61–66 (2004). [CrossRef]  

8. R. Gordon, L. K. S. Kumar, and A. G. Brolo, “Resonant light transmission through a nanohole in a metal film,” IEEE Nanotechnology 5, 291–294 (2006). [CrossRef]  

9. F. M. Kong, K. Li, and B. I. Wu, et. al., “Propagation properties of the SPP modes in nanoscale narrow metallic gap, channel, and hole geometries,” Prog. Electromagn. Res. 76, 449–466 (2007) [CrossRef]  

10. E. X. Jin and X. Xu, “Finite-difference time-domain studies on optical transmission through planar nano-apertures in a metal Film,” Jpn. J. Appl. Phys. 43, 407–417(2004). [CrossRef]  

11. K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis (Wiley, Chichester, 2001). [CrossRef]  

12. S. I. Bozhevolnyi, “Effective-index modeling of channel plasmon polaritons,” Opt. Express 14, 9467–9476 (2006). [CrossRef]   [PubMed]  

13. S. Collin, F. Pardo, and J. L. Pelouard, “Waveguiding in nanoscale metallic apertures,” Opt. Express 15, 4310–4320(2007). [CrossRef]   [PubMed]  

14. Y. Satuby and M. Orenstein, “Surface-plasmon-polariton modes in deep metallic trenches-measurement and analysis,” Opt. Express 15, 4247–4252 (2007). [CrossRef]   [PubMed]  

15. B. I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong. “Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability,” J. Appl. Phys. 93, 9386 (2003). [CrossRef]  

16. C. Sönnichsen, “Plasmons in metal nanostructures,” PhD Thesis (Ludwig-Maximilians-Universtät München, München, 2001).

17. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379, (1972). [CrossRef]  

References

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  • |
  • |

  1. P. N. Prasad, Nanophotonics (Wiley-Interscience, New Jersey, 2004).
    [CrossRef]
  2. W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
    [CrossRef] [PubMed]
  3. T. Rindzevicius, Y. Alaverdyan, and B. Sepulveda. "Nanohole plasmons in optically thin gold films," J. Phys. Chem. C 111, 1207-1212 (2007).
    [CrossRef]
  4. F. I. Baida, A. Belkhir, D. Van Labeke, and O. Lamrous, "Subwavelength metallic coaxial waveguides in the optical range: Role of the plasmonic modes," Phys. Rev. B 74, 205419 (2006).
    [CrossRef]
  5. C. K. Chang, D. Z. Lin, C. S. Yeh,  et al., "Experimental analysis of surface plasmon behavior in metallic circular slits," Appl. Phys. Lett. 90, 061113 (2007).
    [CrossRef]
  6. K. Y. Kim, Y. K. Cho, H. S. Tae, and J. H. Lee, "Optical guided dispersions and subwavelength transmissions in dispersive plasmonic circular holes," Opto-Electron.Rev. 14, 233-241 (2006).
    [CrossRef]
  7. A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, "Optical transmission properties of a single subwavelength aperture in a real metal," Opt. Commun. 239, 61-66 (2004).
    [CrossRef]
  8. R. Gordon, L. K. S. Kumar, and A. G. Brolo, "Resonant light transmission through a nanohole in a metal film," IEEE Nanotechnology 5, 291-294 (2006).
    [CrossRef]
  9. F. M. Kong, K. Li, B. I. Wu,  et al., "Propagation properties of the SPP modes in nanoscale narrow metallic gap, channel, and hole geometries," Prog. Electromagn. Res. 76, 449-466 (2007)
    [CrossRef]
  10. E. X. Jin and X. Xu, "Finite-difference time-domain studies on optical transmission through planar nano-apertures in a metal Film," Jpn. J. Appl. Phys. 43, 407-417(2004).
    [CrossRef]
  11. K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis (Wiley, Chichester, 2001).
    [CrossRef]
  12. S. I. Bozhevolnyi, "Effective-index modeling of channel plasmon polaritons," Opt. Express 14, 9467-9476 (2006).
    [CrossRef] [PubMed]
  13. S. Collin, F. Pardo, and J. L. Pelouard, "Waveguiding in nanoscale metallic apertures," Opt. Express 15, 4310-4320 (2007).
    [CrossRef] [PubMed]
  14. Y. Satuby and M. Orenstein, "Surface-plasmon-polariton modes in deep metallic trenches-measurement and analysis," Opt. Express 15, 4247-4252 (2007).
    [CrossRef] [PubMed]
  15. B. I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong. "Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability," J. Appl. Phys. 93, 9386 (2003).
    [CrossRef]
  16. C. Sönnichsen, "Plasmons in metal nanostructures," PhD Thesis (Ludwig-Maximilians-Universtät München, München, 2001).
  17. P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
    [CrossRef]

2007

T. Rindzevicius, Y. Alaverdyan, and B. Sepulveda. "Nanohole plasmons in optically thin gold films," J. Phys. Chem. C 111, 1207-1212 (2007).
[CrossRef]

C. K. Chang, D. Z. Lin, C. S. Yeh,  et al., "Experimental analysis of surface plasmon behavior in metallic circular slits," Appl. Phys. Lett. 90, 061113 (2007).
[CrossRef]

F. M. Kong, K. Li, B. I. Wu,  et al., "Propagation properties of the SPP modes in nanoscale narrow metallic gap, channel, and hole geometries," Prog. Electromagn. Res. 76, 449-466 (2007)
[CrossRef]

S. Collin, F. Pardo, and J. L. Pelouard, "Waveguiding in nanoscale metallic apertures," Opt. Express 15, 4310-4320 (2007).
[CrossRef] [PubMed]

Y. Satuby and M. Orenstein, "Surface-plasmon-polariton modes in deep metallic trenches-measurement and analysis," Opt. Express 15, 4247-4252 (2007).
[CrossRef] [PubMed]

2006

S. I. Bozhevolnyi, "Effective-index modeling of channel plasmon polaritons," Opt. Express 14, 9467-9476 (2006).
[CrossRef] [PubMed]

R. Gordon, L. K. S. Kumar, and A. G. Brolo, "Resonant light transmission through a nanohole in a metal film," IEEE Nanotechnology 5, 291-294 (2006).
[CrossRef]

K. Y. Kim, Y. K. Cho, H. S. Tae, and J. H. Lee, "Optical guided dispersions and subwavelength transmissions in dispersive plasmonic circular holes," Opto-Electron.Rev. 14, 233-241 (2006).
[CrossRef]

F. I. Baida, A. Belkhir, D. Van Labeke, and O. Lamrous, "Subwavelength metallic coaxial waveguides in the optical range: Role of the plasmonic modes," Phys. Rev. B 74, 205419 (2006).
[CrossRef]

2004

A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, "Optical transmission properties of a single subwavelength aperture in a real metal," Opt. Commun. 239, 61-66 (2004).
[CrossRef]

E. X. Jin and X. Xu, "Finite-difference time-domain studies on optical transmission through planar nano-apertures in a metal Film," Jpn. J. Appl. Phys. 43, 407-417(2004).
[CrossRef]

2003

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

B. I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong. "Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability," J. Appl. Phys. 93, 9386 (2003).
[CrossRef]

1972

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Appl. Phys. Lett.

C. K. Chang, D. Z. Lin, C. S. Yeh,  et al., "Experimental analysis of surface plasmon behavior in metallic circular slits," Appl. Phys. Lett. 90, 061113 (2007).
[CrossRef]

IEEE Nanotechnology

R. Gordon, L. K. S. Kumar, and A. G. Brolo, "Resonant light transmission through a nanohole in a metal film," IEEE Nanotechnology 5, 291-294 (2006).
[CrossRef]

J. Appl. Phys.

B. I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong. "Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability," J. Appl. Phys. 93, 9386 (2003).
[CrossRef]

J. Phys. Chem. C

T. Rindzevicius, Y. Alaverdyan, and B. Sepulveda. "Nanohole plasmons in optically thin gold films," J. Phys. Chem. C 111, 1207-1212 (2007).
[CrossRef]

Jpn. J. Appl. Phys.

E. X. Jin and X. Xu, "Finite-difference time-domain studies on optical transmission through planar nano-apertures in a metal Film," Jpn. J. Appl. Phys. 43, 407-417(2004).
[CrossRef]

Nature

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Opt. Commun.

A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, "Optical transmission properties of a single subwavelength aperture in a real metal," Opt. Commun. 239, 61-66 (2004).
[CrossRef]

Opt. Express

Phys. Rev. B

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

F. I. Baida, A. Belkhir, D. Van Labeke, and O. Lamrous, "Subwavelength metallic coaxial waveguides in the optical range: Role of the plasmonic modes," Phys. Rev. B 74, 205419 (2006).
[CrossRef]

Prog. Electromagn. Res.

F. M. Kong, K. Li, B. I. Wu,  et al., "Propagation properties of the SPP modes in nanoscale narrow metallic gap, channel, and hole geometries," Prog. Electromagn. Res. 76, 449-466 (2007)
[CrossRef]

Rev.

K. Y. Kim, Y. K. Cho, H. S. Tae, and J. H. Lee, "Optical guided dispersions and subwavelength transmissions in dispersive plasmonic circular holes," Opto-Electron.Rev. 14, 233-241 (2006).
[CrossRef]

Other

P. N. Prasad, Nanophotonics (Wiley-Interscience, New Jersey, 2004).
[CrossRef]

C. Sönnichsen, "Plasmons in metal nanostructures," PhD Thesis (Ludwig-Maximilians-Universtät München, München, 2001).

K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis (Wiley, Chichester, 2001).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

Cross section of metallic rectangular waveguide (a) and the use of effective dielectric constant method in the x-direction (b) and y-direction (c)

Fig. 2.
Fig. 2.

Dispersion characteristics of TWG modes and SPPs modes in silver waveguide, the insets show the E field distributions of several modes in the air-core.

Fig. 3.
Fig. 3.

Dispersion evolution for SPPs mode in silver waveguide with different height

Fig. 4.
Fig. 4.

Calculated cutoff wavelength for the primary SPPs modes in a silver rectangular waveguide with different aspect ratio

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

ε r 2 ( ω ) = ε r 2 ( 1 ω p 2 ω ( ω + i ω τ ) )
ϕ h x ( y ) = { B 1 e γ y ( y b ) ( y b ) A 1 e i k y y + A 2 e i k y y ( b < y < b ) B 2 e γ y ( y + b ) ( y b )
i k y ε r 1 ( A 1 e i k y b A 2 e i k y b ) = B 1 γ y ε r 2 ( ω )
A 1 e i k y b + A 2 e i k y b = B 1
i k y ε r 1 ( A 1 e i k y b A 2 e i k y b ) = B 2 γ y ε r 2 ( ω )
A 1 e i k y b + A 2 e i k y b = B 2
( 1 i ρ y 1 + i ρ y ) e i 2 k y b = e i ( n 1 ) π ( n = 1 , 2 , 3 , )
tan ( k y b ( n 1 ) π 2 ) = ρ y ( n = 1 , 2 , 3 , )
tanh ( k y b ) = ρ y ( Even solution )
coth ( k y b ) = ρ y ( Odd solution )
ϕ E y ( x ) = { D 1 e γ x ( x a ) ( x a ) C 1 e i k x x + C 2 e i k x x ( a < x < a ) D 2 e γ x ( x + b ) ( x a )
i k x ( C 1 e i k x a C 2 e i k x a ) = D 1 γ x
C 1 e i k x a + C 2 e i k x a = D 1
i k x ( C 1 e i k x a C 2 e i k x a ) = D 2 γ x
C 1 e i k x a + C 2 e i k x a = D 2
( 1 i ρ x 1 + i ρ x ) e i 2 k y b = e i ( m 1 ) π ( m = 1 , 2 , 3 , )
tan ( k x a ( m 1 ) π 2 ) = ρ x ( m = 1 , 2 , 3 , )
TWG modes E mn x : tan ( k x a ( m 1 ) π 2 ) = ε r 1 ε r 2 ( ω ) γ x k x , tan ( k y b ( n 1 ) π 2 ) = γ y k y
SPP modes E en x : tanh ( k x a ) = ε r 1 ε r 2 ( ω ) γ x k x , tan ( k y b ( n 1 ) π 2 ) = γ y k y
SPP modes E on x : coth ( k x a ) = ε r 1 ε r 2 ( ω ) γ x k x , tan ( k y b ( n 1 ) π 2 ) = γ y k y
β = { k 0 2 ε r 1 k x 2 k y 2 E mn x , E mn y k 0 2 ε r 1 + k x 2 k y 2 E en x , E on x k 0 2 ε r 1 k x 2 + k y 2 E me y , E mo y

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