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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 5 — Mar. 3, 2008
  • pp: 3326–3333
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Terahertz surface plasmon polaritons on periodically corrugated metal surfaces

Linfang Shen, Xudong Chen, and Tzong-Jer Yang  »View Author Affiliations


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

Surface plasmon polaritons (SPPs) are surface electromagnetic (EM) waves, whose fields can be highly confined to dielectric-metal interfaces [1

1. H. Raether, Surface Plasmons (Springer-Verlag, Berlin, 1988).

]. This confinement leads to a significant enhancement of the EM field at the interface, which is responsible for surface-enhanced optical phenomena such as Raman scattering, second harmonic generation, fluorescence, etc. [2-4

2. H. E. PonathG. I. Stegeman (Eds.), Nonlinear Surface Electromagnetic Phenomena (North-Holland, Amsterdam, 1991).

]. SPPs and localized plasmons at dielectric-metal interfaces open up a previously inaccessible length scale for optical research. The intrinsically two-dimensional nature of SPPs provides great flexibility in engineering photonic circuits with submicron dimensions needed for optical communications and optical computing [5-7

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

]. It would be greatly advantageous to take concepts such as highly localized waveguiding and surface-enhanced nonlinear effects to the terahertz (THz) regime, where a large variety of materials show specific resonances. At THz frequencies, however, metals resemble a perfect electric conductor (PEC), as their plasma frequencies are often in the ultraviolet part of the spectrum, leading to SPPs highly delocalized on metal surfaces [8

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

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]

]. Recently, an idea of engineering surface plasmon at lower frequency was proposed [10

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]

]. That is, by cutting holes or grooves in metal surfaces to increase the penetration of EM fields into the metal, the frequency of existing surface plasmons can be tailored at will [11-13

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]

]. The existence of such geometry-controlled SPPs, named 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]

].

2. Method

HyI(x,z)=nAn(1)eqn(1)zeiβnx,
(1)

where A (1) n are constants, βn=β+2πn/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 qn(1)=(βn)2k02 with k 0 being the wave number in free space. In region III (z<-h), Hy is similarly written as

HyIII(x,z)=nAn(3)eqn(3)(z+h)eiβnx,
(2)

where A (3) n are constants, qn(3)=(βn)2εmk02 , and εm is the relative permittivity of the metal. On the other hand, since the metal thickness between the grooves (da) is much larger than the skin depth for THz frequencies, the fields in the part of the unit cell in region II (-hz≤0) can be treated as a superposition of eigenmodes in an isolated groove waveguide. Thus, Hy in this part of the unit cell is expressed as

HyII(x,z)=m[Am(2)egm(z+h2)+Bm(2)egm(z+h2)]ψm(x),
(3)

1ad2d21ε(x)ψmψndx1a1ε(x)ψmψndx=Nmδmn,
(4)

where ε(x)=1 for |x|≤a/2 and ε(x)=εm elsewhere. Nm are coefficients depending on m.

The nonzero components (Ex and Ez) of the electric field (E) can be obtained straightforwardly from Hy. The parallel components of 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

Am(2)eigmh2Bm(2)eigmh2=ik0q0(1)mUmm[Am(2)eigmh2+Bm(2)eigmh2],
(5)

where U mm are given by Umm=adNmgmk0nq0(1)qn(1)Smn+Smn, with Smn±=1ad2d21ε(x)ψm(x)exp(±iβnx)dx, where the coefficients Nm are defined in Eq. (4). Similarly, matching boundary conditions at the interface z=-h, we obtain

Am(2)eigmh2Bm(2)eigmh2=iεmmVmm[Am(2)eigmh2+Bm(2)eigmh2],
(6)

where V mm are given by Vmm=adNmgmk0ΣnSmn+Smn. Note that qn(3)εmk0 since |εm| is very large for THz frequencies. In the case of our interest, the groove width is much less than the wavelength (λ) of the surface wave, so all higher-order modes (m≥1) in the groove waveguide are evanescent, and their propagation constants gm have a large imaginary part, i.e., Im(gm)≈/a (here Im(·) denotes imaginary part operator), which results in very large values of exp(-igmh/2) in Eqs. (5) and (6) when the grooves are deep. To circumvent this numerical difficulty, we let ζm=(A (2) m-B (2) m)sin(gmh/2) and ηm=(A (2) m+B (2) m)cos(gmh/2), and rewrite Eqs. (5) and (6) as

ζmcot(gmh2)+iηmtan(gmh2)=k0q0(1)mUmm(ζmiηm),
(7)
ζmcot(gmh2)iηmtan(gmh2)=εmmVmm(ζm+iηm),
(8)

respectively, which can be expressed in a matrix form

K1[ζ]+iK[η]=k0q0(1)U([ζ]i[η]),
(9)
K1[ζ]iK[η]=εmV([ζ]+i[η]),
(10)

where the vectors [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 (gmh/2)δ mm, and K -1 is the inverse matrix of K. From Eq. (10), we have approximately

[ζ]+i[η]=1εmV1(K+K1)[ζ].
(11)

Substitution of Eq. (11) into Eq. (9) yields the equation Q[ζ]=(q (1) 0/k 0)[ζ] with Q=Ka1{U+1εm[U+q0(1)k0K]V1Kb}, where Ka=(K -1-K)/2 and Kb=(K -1+K)/2. Evidently, Ka and Kb are a diagonal matrix, and we easily find that {Ka}m,m=cot(gmh) and {Kb}m,m=[1/sin(gmh)]. Finally, we obtain the dispersion relation for spoof SPPs on the corrugated surface in the form

det{Qq0(1)k0I}=0,
(12)

where I is a unit matrix. Note that q0(1)=β2k02 . Equation (12) can be solved numerically via iteration for q (1) 0 or β. If the metal is a PEC, i.e., |εm|→∞, the matrix Q reduces to Q=K -1 aU, and Eq. (12) becomes det{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 λd, 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]

]. The solution of this dispersion equation offers an initial value of iteration for q (1) 0 in solving Eq. (12).

3. Results

Fig. 1. Convergence of the real (a) and imaginary (b) parts of the propagation constant β calculated from Eq. (12) for d=50 µm, h=50 µm, and f=0.8 THz. The corresponding results obtained within the PEC approximation are included as triangles for comparison. The inset shows the geometry of structured surface.

We now analyze the properties of spoof SPPs at THz frequencies on periodically corrugated Al surfaces. Figure 2(a) shows the dispersion curves for spoof SPPs on structured surfaces with d=50 µm. Two groove depths are analyzed: h=d and h=0.5d. 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=0.2d 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

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]

], the corrugated metal surface exhibits relatively high loss for guiding THz wave when the wave field is effectively confined. Evidently, the loss of the spoof SPPs increases when the confinement of the fields of the spoof SPPs is enhanced. The field is more strongly confined to the surface for a larger frequency. This is displayed in Fig. 3, where the distribution of the 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 q0(1)=β2k02 becomes larger, thus increasing the field confinement and the loss of the SPPs.

Fig. 2. (a) Dispersion curves for spoof SPPs. (b) Attenuation coefficients of spoof SPPs. The lattice constant is d=50 µm. The solid and dashed lines correspond to the groove depths h=d and h=0.5d, respectively.

It is interesting to analyze the effect of the lattice constant (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 µm. The groove parameters are a=10 µm and h=50 µm 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.

Fig. 3. Spatial variation of the amplitude of the H field in a unit cell of the surface structure with d=50 µm, a=0.2d, and h=d. For clarity, different length scales are used in the z direction for z≤0 and z>0.
Fig. 4. Dispersion relations (a) and attenuation coefficients (b) of spoof SPPs for different lattice constants d=35, 50, and 75 µm. The parameters of the groove geometry are a=10 µm and h=50 µm. Dotted lines indicate the asymptotic frequencies for three cases.

Fig. 5. (a) Groove depth versus the groove width. (b) Parameter D versus the groove width for f=0.6 THz. Note that for each groove case in (a), the attenuation coefficient of spoof SPPs at f=0.6 THz is always equal to Im(β)=0.045 cm-1. (c)-(d) Spatial variation of the amplitude of the H field at f=0.6 THz for the cases with a=0.2d (h=d) and a=0.6d (h=0.6d), respectively. The lattice constant is d=50 µm.

4. Conclusion

Acknowledgements

This work was supported by the Ministry of Education (Singapore) under Grant No. R263000357112 and R263000357133.

References and links

1.

H. Raether, Surface Plasmons (Springer-Verlag, Berlin, 1988).

2.

H. E. PonathG. I. Stegeman (Eds.), Nonlinear Surface Electromagnetic Phenomena (North-Holland, Amsterdam, 1991).

3.

V. M. Agranovich and D. L. Mills (Eds.), Surface Polaritons (North-Holland, Amsterdam, 1982).

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, S16–S50 (2003). [CrossRef]

5.

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

6.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Physics Reports 408, 131–314 (2005). [CrossRef]

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. 86, 3008–3011 (2001). [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]

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. 97, 176805-1-4 (2006). [CrossRef] [PubMed]

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, 13021–13029 (2006). [CrossRef] [PubMed]

14.

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

15.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

16.

D. R. Lide, CRC Handbook of Chemistry and Physics (CRC Press, Boca Raton, 2004).

17.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976).

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]

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

  1. H. Raether, Surface Plasmons (Springer-Verlag, Berlin, 1988).
  2. H. E. Ponath and G. I. Stegeman eds., Nonlinear Surface Electromagnetic Phenomena (North-Holland, Amsterdam, 1991).
  3. V. M. Agranovich and D. L. Mills eds., Surface Polaritons (North-Holland, Amsterdam, 1982).
  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, S16-S50 (2003). [CrossRef]
  5. W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003). [CrossRef] [PubMed]
  6. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, "Nano-optics of surface plasmon polaritons," Phys. Reports 408, 131-314 (2005). [CrossRef]
  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. 86, 3008-3011 (2001). [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]
  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. 97, 176805-1-4 (2006). [CrossRef] [PubMed]
  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, 13021-13029 (2006). [CrossRef] [PubMed]
  14. A. P. Hibbins, B. R. Evans, and J. R. Sambles, "Experimental verification of designer surface plasmons," Science 308, 670-672 (2005). [CrossRef] [PubMed]
  15. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  16. D. R. Lide, CRC Handbook of Chemistry and Physics (CRC Press, Boca Raton, 2004).
  17. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976).
  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]

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