## Terahertz surface plasmon polaritons on periodically corrugated metal surfaces

Optics Express, Vol. 16, Issue 5, pp. 3326-3333 (2008)

http://dx.doi.org/10.1364/OE.16.003326

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### Abstract

Based on a modal expansion of electromagnetic fields, a rigorous method for analyzing surface plasmon polaritons (SPPs) on a periodically corrugated metal surface has been formulated in this paper. This method takes into account the finite conductivity of the metal as well as higher-order modes within the grooves of the surface structure, thus is able to accurately calculate the loss of these spoof SPPs propagating along the structured surface. In the terahertz (THz) frequency range, the properties of the dispersion and loss of spoof SPPs on corrugated Al surfaces are analyzed. For spoof SPPs at THz frequencies, the strong confinement of the fields is often accompanied with considerable absorption loss, but the performance of both low-loss propagation and subwavelength field confinement for spoof SPPs can be achieved by the optimum design of surface structure.

© 2008 Optical Society of America

## 1. Introduction

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

8. J. F. OHara, R. D. Averitt, and A. J. Taylor, “Terahertz surface plasmon polariton coupling on metallic gratings,” Opt. Express **12**, 6397–6402 (2004). [CrossRef]

9. J. Saxler, J. Gomez Rivas, C. Janke, H. P. M. Pellemans, P. Haring Bolivar, and H. Kurz, “Time-domain measurements of surface plasmon polaritons in the terahertz frequency range,” Phys. Rev. B **69**, 155427-1-4 (2004). [CrossRef]

10. J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science **305**, 847–848 (2004). [CrossRef] [PubMed]

11. F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A: Pure Appl. Opt. **7**, S97–S101 (2005). [CrossRef]

*spoof*SPPs, has recently been verified experimentally in the microwave regime [14

14. A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science **308**, 670–672 (2005). [CrossRef] [PubMed]

10. J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science **305**, 847–848 (2004). [CrossRef] [PubMed]

11. F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A: Pure Appl. Opt. **7**, S97–S101 (2005). [CrossRef]

## 2. Method

*a*, depth

*h*, and lattice constant

*d*is drilled, as illustrated in the inset of Fig. 1(b). We are interested in H-polarized surface waves propagating in the x direction along the surface. The fields of these waves have the form of

**H**=

*ŷH*and

_{y}**E**=

*x̂E*+

_{x}*ẑE*. Based on a modal expansion of the fields,

_{z}*H*in region I (

_{y}*z*>0) can be written as

*A*

^{(1)}

_{n}are constants,

*β*=

_{n}*β*+2

*πn*/

*d*(here the propagation constant of the surface wave

*β*lies in the first Brillouin zone, i.e., |Re(

*β*)|≤

*π*/

*d*, where Re(·) denotes real part operator), and

*k*

_{0}being the wave number in free space. In region III (

*z*<-

*h*),

*H*is similarly written as

_{y}*A*

^{(3)}

_{n}are constants,

*ε*is the relative permittivity of the metal. On the other hand, since the metal thickness between the grooves (

_{m}*d*–

*a*) is much larger than the skin depth for THz frequencies, the fields in the part of the unit cell in region II (-

*h*≤

*z*≤0) can be treated as a superposition of eigenmodes in an isolated groove waveguide. Thus,

*H*in this part of the unit cell is expressed as

_{y}*A*

^{(2)}

_{m}and

*B*

^{(2)}

_{m}are constants, and the modal field profiles

*ψ*are given by [15]

_{m}*f*(

*ξ*)=cos(

*ξ*) for symmetric modes with even

*m*(

*m*=0, 2, ….), or

*f*(

*ξ*)=sin(

*ξ*) for asymmetric modes with odd

*m*(

*m*=1, 3, …). Here,

*g*and

_{m}*p*are governed by the dispersion relation [15]: tan (

_{m}*p*/2)=

_{m}a*γ*/(

_{m}*ε*) for even

_{m}p_{m}*m*, or tan(

*p*/2-

_{m}a*π*/2)=

*γ*/(

_{m}*ε*) for odd

_{m}p_{m}*m*. Note that

*ψ*have the property of orthogonality [15]

_{m}*ε*(

*x*)=1 for |

*x*|≤

*a*/2 and

*ε*(

*x*)=

*ε*elsewhere.

_{m}*N*are coefficients depending on

_{m}*m*.

*E*and

_{x}*E*) of the electric field (

_{z}**E**) can be obtained straightforwardly from

*H*. The parallel components of

_{y}**E**and

**H**must be continuous at the interface between regions I and II and the one between regions II and III. Matching boundary conditions at the interface

*z*=0, we obtain

*U*

_{mm′}are given by

*N*are defined in Eq. (4). Similarly, matching boundary conditions at the interface

_{m}*z*=-

*h*, we obtain

*V*

_{mm′}are given by

*ε*| is very large for THz frequencies. In the case of our interest, the groove width is much less than the wavelength (

_{m}*λ*) of the surface wave, so all higher-order modes (

*m*≥1) in the groove waveguide are evanescent, and their propagation constants

*g*have a large imaginary part, i.e., Im(

_{m}*g*)≈

_{m}*mπ*/

*a*(here Im(·) denotes imaginary part operator), which results in very large values of exp(-

*ig*/2) in Eqs. (5) and (6) when the grooves are deep. To circumvent this numerical difficulty, we let

_{m}h*ζ*=(

_{m}*A*

^{(2)}

_{m}-

*B*

^{(2)}

_{m})sin(

*g*/2) and

_{m}h*ηm*=(

*A*

^{(2)}

_{m}+

*B*

^{(2)}

_{m})cos(

*g*/2), and rewrite Eqs. (5) and (6) as

_{m}h*ζ*] and [

*η*] have a length equal to the number of modes included in Eq. (3). The diagonal matrix

*K*is defined as

*K*

_{mm′}=tan (

*g*/2)

_{m}h*δ*

_{mm′}, and

*K*

^{-1}is the inverse matrix of

*K*. From Eq. (10), we have approximately

*ζ*]=(

*q*

^{(1)}

_{0}/

*k*

_{0})[

*ζ*] with

*K*=(

_{a}*K*

^{-1}-

*K*)/2 and

*K*=(

_{b}*K*

^{-1}+

*K*)/2. Evidently,

*K*and

_{a}*K*are a diagonal matrix, and we easily find that {

_{b}*K*}

_{a}_{m,m}=cot(

*g*) and {

_{m}h*K*}

_{b}_{m,m}=[1/sin(

*g*)]. Finally, we obtain the dispersion relation for spoof SPPs on the corrugated surface in the form

_{m}h*I*is a unit matrix. Note that

*q*

^{(1)}

_{0}or

*β*. If the metal is a PEC, i.e., |

*ε*|→∞, the matrix

_{m}*Q*reduces to

*Q*=

*K*

^{-1}

*, and Eq. (12) becomes det{*

_{a}U*K*

^{-1}

_{a}

*U*-(

*q*

^{(1)}

_{0}/

*k*

_{0})

*I*}=0. If we further neglect the effect of higher-order modes within grooves, i.e., only the lowest-order (

*m*=0) mode is included in the expansion, the dispersion equation is simplified to

*q*

^{(1)}

_{0}=(

*a/d*)

*k*

_{0}tan(

*k*

_{0}

*h*) in the limit

*λ*≫

*d*, which is identical to Eq. (9) of Ref. [11

11. F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A: Pure Appl. Opt. **7**, S97–S101 (2005). [CrossRef]

*q*

^{(1)}

_{0}in solving Eq. (12).

## 3. Results

*d*=50

*µ*m, and the groove depth is

*h*=

*d*. Two groove widths are considered: (1)

*a*=0.2

*d*, (2)

*a*=0.6

*d*. The asymptotic frequencies for two cases, which are evaluated at Re(

*β*)=

*π*/

*d*(i.e., the border of the first Brillouin zone), are

*f*=1.275 and 1.165 THz, respectively. The frequency of the surface wave is set to

_{s}*f*=0.8 THz. The calculated results are plotted in Fig. 1, where the corresponding results obtained within the PEC approximation are included as triangles for comparison. As seen from Fig. 1, both Re(

*β*) and Im(

*β*) converge rapidly. The results of Re(

*β*) in the singlemode (

*m*= 0) approximation are accurate within 3% for both cases. The difference between the values of Re(

*β*) for Al and those for PEC are almost negligible, meaning that the model with the PEC approximation is valid for analyzing the dispersion of spoof SPPs at THz frequencies. The result of Im(

*β*) obtained within the single-mode approximation is always larger than the accurate value (see Fig. 1(b)), and its relative error is 2.4% for the case with

*a*=0.2

*d*and nearly 10% for the case with

*a*=0.6

*d*. Our analysis indicates that in the latter case, the relative error becomes 88.7% when

*f*=1.15 THz. Evidently, a few modes with higher orders become important when the groove width is large, especially at frequencies close to the asymptotic frequency of the surface structure.

*d*=50

*µ*m. Two groove depths are analyzed:

*h*=

*d*and

*h*=0.5

*d*. For each groove depth, several values of the groove width are shown. As seen from Fig. 2(a), the larger the groove depth, the lower the asymptotic frequency. This is also the situation for the groove width. But compared with the groove depth, the groove width weakly affects the asymptotic frequency. Figure 2(b) shows the losses of these spoof SPPs as a function of frequency. The loss of spoof SPPs grows significantly with increasing frequency. For example, in the case of

*a*=0.2

*d*and

*h*=

*d*, the attenuation coefficient of the spoof SPPs is 0.045 cm

^{-1}for

*f*=0.6 THz, and it becomes 0.16 cm

*-1*for

*f*=0.8 THz and 0.7 cm

^{-1}for

*f*=1 THz. Compared to purely dielectric THz waveguides reported in Refs. [18

18. M. Nagel, A. Marchewka, and H. Kurz, “Low-index discontinuity terahertz waveguides,” Opt. Express **14**, 9944–9954 (2006). [CrossRef] [PubMed]

19. S. Q. Lou, T. Y. Guo, H. Fang, H. L. Li, and S. S. Jian, “A new type of terahertz waveguides,” Chin. Phys. Lett. **23**, 235–238 (2006). [CrossRef]

**H**field in a unit cell of the surface structure is plotted for various frequencies. As shown in Fig. 3, the field of the surface wave is almost concentrated in the structured surface when

*f*=1 and 1.2 THz. Figure 2(b) clearly shows that for a given frequency, a larger groove width or depth correspond to a larger loss of spoof SPPs. As a larger groove width or depth correspond to a lower asymptotic frequency, leading to a larger departure of the propagation constant for a given frequency from the light line (see Fig. 2(a)). Correspondingly, the z component of the wave vector

*d*) on the dispersion and loss of spoof SPPs. Figure 4(a) shows the dispersion curves for spoof SPPs on corrugated surfaces with different lattice constants

*d*=35, 50, and 75

*µ*m. The groove parameters are

*a*=10

*µ*m and

*h*=50

*µ*m for all cases. The losses of spoof SPPs for three cases are plotted in Fig. 4(b). As seen from Figs. 4(a) and 4(b), a smaller lattice constant corresponds to a larger asymptotic frequency (indicated by dotted line in the Fig. 4(a)), but it corresponds to a larger loss of spoof SPPs for a given frequency. Evidently, a structured surface with a smaller lattice constant possesses a higher asymptotic frequency only because it allow spoof SPPs to have a larger range of propagation constant. But the asymptotic frequency doesn’t seem to be sensitive to the lattice constant. However, as shown in Fig. 4(b), the loss of the spoof SPPs is quite sensitive to the lattice constant. For a given frequency, an increase of the lattice constant may lead to a considerable reduction in the loss of spoof SPPs.

## 4. Conclusion

## Acknowledgements

## References and links

1. | H. Raether, |

2. | H. E. PonathG. I. Stegeman
(Eds.), |

3. | V. M. Agranovich and D. L. Mills (Eds.), |

4. | A. V. Zayats and I. I. Smolyaninov, “Near-field photonics: surface plasmon polaritons and localized surface plasmons,” J. Opt. A: Pure Appl. Opt. |

5. | W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature |

6. | A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Physics Reports |

7. | S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M.W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. |

8. | J. F. OHara, R. D. Averitt, and A. J. Taylor, “Terahertz surface plasmon polariton coupling on metallic gratings,” Opt. Express |

9. | J. Saxler, J. Gomez Rivas, C. Janke, H. P. M. Pellemans, P. Haring Bolivar, and H. Kurz, “Time-domain measurements of surface plasmon polaritons in the terahertz frequency range,” Phys. Rev. B |

10. | J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science |

11. | F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A: Pure Appl. Opt. |

12. | S. A. Maier, S. R. Andrews, L. Martin-Moreno, and F. J. Garcia-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett. |

13. | Y. Chen, Z. Song, Y. Li, M. Hu, Q. Xing, Z. Zhang, L. Chai, and C. Y. Wang, “Effective surface plasmon polaritons on the metal wire with arrays of subwavelength grooves,” Opt. Express |

14. | A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science |

15. | A. W. Snyder and J. D. Love, |

16. | D. R. Lide, |

17. | N. W. Ashcroft and N. D. Mermin, |

18. | M. Nagel, A. Marchewka, and H. Kurz, “Low-index discontinuity terahertz waveguides,” Opt. Express |

19. | S. Q. Lou, T. Y. Guo, H. Fang, H. L. Li, and S. S. Jian, “A new type of terahertz waveguides,” Chin. Phys. Lett. |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(240.6690) Optics at surfaces : Surface waves

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: August 6, 2007

Revised Manuscript: September 24, 2007

Manuscript Accepted: September 24, 2007

Published: February 27, 2008

**Citation**

Linfang Shen, Xudong Chen, and Tzong-Jer Yang, "Terahertz surface plasmon polaritons on periodically corrugated metal surfaces," Opt. Express **16**, 3326-3333 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-5-3326

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### References

- H. Raether, Surface Plasmons (Springer-Verlag, Berlin, 1988).
- H. E. Ponath and G. I. Stegeman eds., Nonlinear Surface Electromagnetic Phenomena (North-Holland, Amsterdam, 1991).
- V. M. Agranovich and D. L. Mills eds., Surface Polaritons (North-Holland, Amsterdam, 1982).
- A. V. Zayats and I. I. Smolyaninov, "Near-field photonics: surface plasmon polaritons and localized surface plasmons," J. Opt. A: Pure Appl. Opt. 5, S16-S50 (2003). [CrossRef]
- W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003). [CrossRef] [PubMed]
- A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, "Nano-optics of surface plasmon polaritons," Phys. Reports 408, 131-314 (2005). [CrossRef]
- S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M.W. Skovgaard, and J. M. Hvam, "Waveguiding in surface plasmon polariton band gap structures," Phys. Rev. Lett. 86, 3008-3011 (2001). [CrossRef] [PubMed]
- J. F. OHara, R. D. Averitt, and A. J. Taylor, "Terahertz surface plasmon polariton coupling on metallic gratings," Opt. Express 12, 6397-6402 (2004). [CrossRef]
- J. Saxler, J. Gomez Rivas, C. Janke, H. P. M. Pellemans, P. Haring Bolivar, and H. Kurz, "Time-domain measurements of surface plasmon polaritons in the terahertz frequency range," Phys. Rev. B 69, 155427-1-4 (2004). [CrossRef]
- J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004). [CrossRef] [PubMed]
- F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, "Surfaces with holes in them: new plasmonic metamaterials," J. Opt. A: Pure Appl. Opt. 7, S97-S101 (2005). [CrossRef]
- S. A. Maier, S. R. Andrews, L. Martin-Moreno, and F. J. Garcia-Vidal, "Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires," Phys. Rev. Lett. 97, 176805-1-4 (2006). [CrossRef] [PubMed]
- Y. Chen, Z. Song, Y. Li, M. Hu, Q. Xing, Z. Zhang, L. Chai, and C. Y. Wang, "Effective surface plasmon polaritons on the metal wire with arrays of subwavelength grooves," Opt. Express 14, 13021-13029 (2006). [CrossRef] [PubMed]
- A. P. Hibbins, B. R. Evans, and J. R. Sambles, "Experimental verification of designer surface plasmons," Science 308, 670-672 (2005). [CrossRef] [PubMed]
- A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
- D. R. Lide, CRC Handbook of Chemistry and Physics (CRC Press, Boca Raton, 2004).
- N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976).
- M. Nagel, A. Marchewka, and H. Kurz, "Low-index discontinuity terahertz waveguides," Opt. Express 14, 9944-9954 (2006). [CrossRef] [PubMed]
- S. Q. Lou, T. Y. Guo, H. Fang, H. L. Li, and S. S. Jian, "A new type of terahertz waveguides," Chin. Phys. Lett. 23, 235-238 (2006). [CrossRef]

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