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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 23 — Nov. 9, 2009
  • pp: 20806–20815
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An explicit formula for metal wire plasmon of terahertz wave

Jie Yang, Qing Cao, and Changhe Zhou  »View Author Affiliations


Optics Express, Vol. 17, Issue 23, pp. 20806-20815 (2009)
http://dx.doi.org/10.1364/OE.17.020806


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Abstract

An explicit formula for metal wire plasmon of terahertz wave is analytically derived. The derivation is based on the huge relative permittivities of nonmagnetic metals in the spectral region of terahertz wave, some important properties of modified Bessel functions, and a suitable Taylor expansion. The obtained formula is further checked by many numerical tests. We find that, for all 11 tested nonmagnetic metals, for the whole spectral region of terahertz wave, and for the wide radius range from 10 μm to infinity, the relative deviation for the effective index is always smaller than 5%. This good agreement clearly shows that the derived expression can be conveniently used for the analysis and design of metal wire plasmon of terahertz wave.

© 2009 OSA

1. Introduction

Terahertz (THz) wave, locating between the infrared and microwave bands in the electromagnetic spectrum, is one of the hot research topics. It is normally defined as the range from 0.1 to 10 THz (or correspondingly, from 30 μm to 3 mm in wavelength). In recent years, terahertz technology has shown potential applications in many fields, such as in sensing, imaging, and spectroscopy [1

1. B. B. Hu and M. C. Nuss, “Imaging with terahertz waves,” Opt. Lett. 20(16), 1716–1718 ( 1995). [CrossRef] [PubMed]

3

3. M. J. Fitch and R. Osiander, “Terahertz waves for communications and sensing,” Johns Hopkins APL Tech. Dig. 25, 348–355 ( 2004).

]. Among those research works, effective THz waveguides have attracted more and more interests [4

4. K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature 432(7015), 376–379 ( 2004). [CrossRef] [PubMed]

35

35. A. Ishikawa, S. Zhang, D. A. Genov, G. Bartal, and X. Zhang, “Deep subwavelength terahertz waveguides using gap magnetic plasmon,” Phys. Rev. Lett. 102(4), 043904 ( 2009). [CrossRef] [PubMed]

]. In 2004, Wang and Mittleman [4

4. K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature 432(7015), 376–379 ( 2004). [CrossRef] [PubMed]

] reported that a simple metal wire can effectively guide THz wave. Since then, many interesting theoretical and experimental works on metal wire THz waveguide have been carried out [5

5. Q. Cao and J. Jahns, “Azimuthally polarized surface plasmons as effective terahertz waveguides,” Opt. Express 13(2), 511–518 ( 2005). [CrossRef] [PubMed]

21

21. J. A. Deibel, K. Wang, M. Escarra, N. Berndsen, and D. M. Mittleman, “The excitation and emission of terahertz surface plasmon polaritons on metal wire waveguides,” C. R. Phys. 9(2), 215–231 ( 2008). [CrossRef]

].

2. A rough solution to the eigen-equation

For a flat metal-dielectric interface, there exists an electromagnetic bound state which is TM polarization, and this bound state is called surface plasmon (SP) [36

36. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, (Springer, Berlin, 1988).

]. While, in the case of metal wire, SP can also exist at a cylindrical metal-dielectric interface [37

37. M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93(13), 137404 ( 2004). [CrossRef] [PubMed]

,38

38. U. Schröter and A. Dereux, “Surface plasmon polaritons on metal cylinders with dielectric core,” Phys. Rev. B 64(12), 125420 ( 2001). [CrossRef]

]. It has one magnetic field component Hφ, and two electric field components Er and Ez. The only transverse magnetic field component Hφ indicates the TM polarization.

Consider a cylindrical metal wire surrounded by air. We are only interested in axially symmetrical eigenmodes, that is to say, the relations ∂E/∂φ = 0 and ∂H/∂φ = 0 hold in the cylindrical coordinates. For TM polarization of a nonmagnetic metal, by substituting the above-mentioned relations into Maxwell’s equations [39

39. M. Born, and E. Wolf, Principles of Optics, 5th ed. (Pergamon Press, Oxford, 1975).

] and using the continuities of Ez and Hφ at the interface, one can get the following eigen-equation [5

5. Q. Cao and J. Jahns, “Azimuthally polarized surface plasmons as effective terahertz waveguides,” Opt. Express 13(2), 511–518 ( 2005). [CrossRef] [PubMed]

,37

37. M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93(13), 137404 ( 2004). [CrossRef] [PubMed]

,38

38. U. Schröter and A. Dereux, “Surface plasmon polaritons on metal cylinders with dielectric core,” Phys. Rev. B 64(12), 125420 ( 2001). [CrossRef]

]:
εmκmI1(k0κmR)I0(k0κmR)+1κaK1(k0κaR)K0(k0κaR)=0
(1)
where κa = [(neff)2-1]1/2, κm = [(neff)2m]1/2. neff is the effective index of the eigenmode, εm denotes the relative permittivity of the metal. I0(.), K0(.), I1(.) and K1(.) are modified Bessel functions. k0 = 2π/λ0, where λ0 and k0 denotes wavelength and wave number in free space, respectively. R is the radius of the metal wire.

We first consider the first term of Eq. (1). Through a lot of numerical calculations, we find that the effective index neff is always about 1 provided that the radius R is not extremely small. That is to say, neff≈1. On the other hand, the relative permittivity of a metal is huge in the spectral region of THz wave. By taking these two properties into account, one can obtain the following relation.
κm1εm.
(2)
By use of Eq. (2), one can get the following approximation

εmκmI1(k0κmR)I0(k0κmR)εm1εmI1(k0R1εm)I0(k0R1εm).
(3)

From Eq. (3) one can find that the first term of Eq. (1) is approximately a constant for a pre-given R. For convenience, we here define this constant as a:

a=εm1εmI1(k0R1εm)I0(k0R1εm).
(4)

We now further consider the second term of Eq. (1). Unlike κm in the first term, the parameter κa in the second term changes apparently with the change of radius R, so does the ratio K1(k0κaR)/K0(k0κaR). In order to find an approximate expressions for κa, we need first to find a suitable approximate for K1(k0κaR)/K0(k0κaR). For convenience, we define the ratio K1(k0κaR)/K0(k0κaR) as a function f(u):
f(u)=K1(u)K0(u),
(5)
where

u=k0κaR.
(6)

In terms of the parameter a and the function f(u), Eq. (1) can be approximately expressed as

a+f(u)κa=0.
(7)

Equation (7) is the fundament of our further analytical derivation. In other words, our analytical work will be based on Eq. (7).

We denote by fr(u) a rough expression for f(u). We choose fr(u) as the following form
fr(u)=1+αu,
(8)
where α is a constant that needs to be determined. The reasons for such a choice are as follows:

1) By use of the asymptotic properties of modified Bessel functions, one can prove that f (u)~1 + 1/(2u) for very large u. One can further find that both f(u) and fr(u) approach 1 when u approaches infinity.

2) By use of the asymptotic properties of modified Bessel functions, one can prove that K1(u)~1/u, K0(u)~-ln(u/2) for very small u. However, the change of -ln(u/2) is much slower than that of 1/u. As a result, f(u) approaches infinity basically as the function 1/u does when u is very small. One can find that, except for a coefficient α, both f(u) and fr(u) approach infinity with the same functional form when u approaches 0.

The constant α can be optimally chosen. Through many numerical tests, we find that u is basically larger than 0.001 when the radius R is larger than the wavelength in the spectral region of terahertz wave. To optimally control the deviation between fr(u) and f(u) in the wide range from u = 0.001 to u = ∞, we let fr(u) = f(u) at the point u = 0.01. Accordingly, the coefficient α is determined to be 0.2018. In the remainder of this paper, we shall use the relation

α=0.2018.
(9)

Both the functions f(u) and fr(u) in the range of 0.001≤u≤10 are shown in Fig. 1 (a)
Fig. 1 (a) The comparison between f (u) and fr (u) in the range of u from 0.001 to 10. The red curve is f(u), and the black curve is fr(u). (b) The relative deviation between f (u) and fr (u) in the range of u from 0.001 to 10.
. The corresponding relative deviation [f(u)-fr(u)]/f(u) is shown in Fig. 1 (b). One can see that the two functions agree well when u approaches 0.01 or u becomes very large. In the range of 0.01≤u≤10, the maximum relative deviation is about 25%. When u is smaller than 0.01, the deviation becomes apparently. In particular, the deviation becomes about 40% at the point u = 0.001. We consider the deviation of about 40% as an acceptable value because fr(r) is only a rough expression for f(u). Obviously, fr(u) becomes invalid for smaller u, because the relative deviation becomes larger and larger in this case.

We denote by κar a rough expression for κa. Replacing the function f(u) and κa in Eq. (7) by fr(ur) and κar, respectively, one can get
a+fr(ur)κar=0,
(10)
where

ur=k0κarR.
(11)

Substituting Eq. (8) and (9) into Eq. (10), one can obtain the following equation
aκar2+κar+c=0,
(12)
where

c=0.2018k0R.
(13)

Equation (12) has two roots in mathematics. However, only one of them has physical meaning. Because the field distribution of Hφ decays with the increase of radial coordinate r, κar should have a positive real part [5

5. Q. Cao and J. Jahns, “Azimuthally polarized surface plasmons as effective terahertz waveguides,” Opt. Express 13(2), 511–518 ( 2005). [CrossRef] [PubMed]

]. Accordingly, one should choose the following root
κar=114ac2a,
(14)
where a and c are explicitly given by Eq. (4) and Eq. (13), respectively. The corresponding rough solution neffr for neff can be further obtained from the relation

neffr=κar2+1.
(15)

To get an intuitive impression on the rough solution, we calculate the κar values, as a function of R. The metal is chosen to be copper and the frequency is chosen to be 0.5 THz (i.e., λ0 = 0.6 mm). The corresponding εm value is εm = −6.3 × 105 + j2.77 × 106 according to a fitted Drude formula for copper [40]. The scope of metal wire is chosen to be 0.01mm≤R≤104mm. For comparison, the exact values κa are also calculated by numerically solving Eq. (1). The values of κar and κa are shown in Fig. 2 (a)
Fig. 2 (a) The comparison between the rough values κar and the exact values κa, for metal copper and 0.5 THz. The red curves are the exact values, and the black curves are the rough solutions. The dashed curves are Im(κa) and Im(κar), and the solid curves are Re(κa) and Re(κar). (b) The relative deviation of κar. The solid curve represents the relative deviation of Re(κar), and the dashed curve is the relative deviation of Im(κar).
. The relative deviations for the real part and the imaginary part of κar are shown in Fig. 2 (b). The rough values neffr and the exact values neff are shown in Fig. 3 (a)
Fig. 3 (a) The comparison between the rough values neffr and the exact values neff, for metal copper and 0.5 THz. The red curves are the exact values neff, and the black curves are the rough solutions neffr. The dashed curves are Im(neffr) and Im(neff), and the solid curves are Re(neffr)-1 and Re(neff)-1. (b) The relative deviation of neffr. The solid curve represents the relative deviation of Re(neffr)-1, and the dashed curve is the relative deviation of Im(neffr).
. And the relative deviations for the real part and the imaginary part of neffr are shown in Fig. 3 (b).

From the above two figures, we can see that, in the chosen range of 0.01mm≤R≤104mm, the maximum relative deviation of κa is about 20%, and the maximum relative deviation of neffr is about 40%. These deviations are rather large. Therefore, the rough solution needs to be further improved.

3. Transform from the rough solution to an approximate solution

In this Section, we shall use a suitable Taylor expansion and the properties of modified Bessel functions to derive an approximate solution with high accuracy. For convenience, we denote by κaa the approximate expression for κa. Then the corresponding approximate expression ua for u is denoted by

ua=k0κaaR.
(16)

We make the first-order Taylor expansion fa(u) for the function f(u) at the neighborhood of ur, which is given by Eq. (11). Accordingly, the value fa(ua) at the point ua can be written as
fa(ua)=f(ur)+f'(ur)(uaur),
(17)
where the first-order derivative f '(ur) is given by

f'(u)r=K12(ur)K02(ur)K1(ur)K0(ur)ur1.
(18)

Replacing the function f(u) and κa in Eq. (7) by fa(ua) and κaa, respectively, one can get

a+fa(ua)κaa=0.
(19)

Comparing Eq. (19) with Eq. (10), we obtain:

1κaafa(ua)=1κarfr(ur).
(20)

Substituting Eq. (17) into Eq. (20), we finally obtain
κaa=κarf(ur)f'(ur)urfr(ur)f'(ur)ur,
(21)
where f '(ur), ur, and κar are given by Eq. (18), Eq. (11), and Eq. (14), respectively. To this end, we derive an approximately explicit formula for κaa. The corresponding approximate value neffa for the effective index can be further obtained from the relation

neffa=κaa2+1.
(22)

To test the validity of the approximate solution, we compare it with the corresponding exact result, which is obtained by numerically solving Eq. (1). The metal, the frequency, and the range of radius are chosen as those in Section 2. The comparisons between the approximate solution κaa and the exact solution κa are shown in Fig. 4
Fig. 4 (a) The comparison between the accurate values κaa and the exact values κa, for metal copper and 0.5 THz. The dashed curve is Im(κa) and the signs “+” show Im(κaa). The solid curve is Re(κa) and the signs “o” show Re(κaa). (b) The relative deviation of κaa. The solid curve represents the relative deviation of Re(κaa), and the dashed curve is the relative deviation of Im(κaa).
. Similarly, the comparisons between the approximate solution neffa and the exact solution neff are shown in Fig. 5
Fig. 5 (a) The comparison between the approximate values neffa and the exact values neff, for metal copper and 0.5 THz. The dashed curve is Im(neff) and the signs “+” show Im(neffa). The solid curve is Re (neff)-1 and the signs “o” show Re(neffa)-1. (b) The relative deviation of neffa. The solid curve represents the relative deviation of Re(neffa)-1, and the dashed curve is the relative deviation of Im(neffa).
. One can see that, in the chosen range of 0.01mm≤R≤104mm, the maximum relative deviations of κaa and of neffa are only about 2% and 3%, respectively. These deviations are much lower than those in the rough solution.

In addition, to see whether the approximate solution is valid in the whole range of terahertz radiation, we further make more numerical tests on other bands of THz, especially on the two ends. The results of neffa for copper on 0.1THz and 10THz are shown Fig. 6
Fig. 6 The comparison between the approximate values neffa and the exact values neff, for metal copper and 0.1 THz. The dashed curve is Im(neff) and the signs “+” show Im(neffa). The solid curve is Re(neff)-1 and the signs “o” show Re(neffa)-1. (b) The relative deviation of neffa. The solid curve represents the relative deviation of Re(neffa)-1, and the dashed curve is the relative deviation of Im(neffa).
and Fig. 7
Fig. 7 The comparison between the approximate values neffa and the exact values neff, for metal copper and 10 THz. The dashed curve is Im(neff) and the signs “+” show Im(neffa), and the solid curve is Re(neffa)-1 and the signs “o” show Re(neff)-1. (b) The relative deviation of neffa. The solid curve represents the relative deviation of Re(neffa)-1, and the dashed curve is the relative deviation of Im(neffa).
. From them, we can see that the approximate solution performs well in the whole range of terahertz radiation, especially on the higher frequency. The reason is that a higher frequency leads to a larger absolute value of u and better results.

To further test the applicability of our formula on more metals, we make numerical tests on other nonmagnetic metals mentioned in Ref [33

33. M. Nagel, A. Marchewka, and H. Kurz, “Low-index discontinuity terahertz waveguides,” Opt. Express 14(21), 9944–9954 ( 2006). [CrossRef] [PubMed]

]. We find that our formula performs also well for all other nonmagnetic metals of Al, Ag, Au, Mo, W, Pd, Ti, Pb, Pt, V. The maximum relative deviation of the effective index for all 11 tested nonmagnetic metals is smaller than 5% in the whole spectral region of THz wave when the radius of metal wire is in the wide range from 10μm to 104 mm. We do not test the magnetic metals of Co, Fe and Ni because they are beyond the scope of this paper.

It should be pointed out that the approximate solution is actually valid for the wide radius range from 10μm to infinity, though its validity is only directly shown for the radius range of 10μm≤R≤104 mm in Figs. 2-7. In each case, the accuracy of the approximate solution at a radius larger than 104 mm is higher than that at the radius of 104 mm.

4. Conclusions

In conclusion, we have obtained an explicit formula for metal wire plasmon of terahertz wave. This formula is valid for all the tested 11 kinds of nonmagnetic metals, for the whole spectral region of terahertz wave, and for the wide radius range from 10μm to infinity. For all the numerical tests, the relative deviation of the approximate formula for the effective index is smaller than 5%. The obtained formula can be used for the fast analyses and designs of metal wire plasmon of terahertz wave.

References

1.

B. B. Hu and M. C. Nuss, “Imaging with terahertz waves,” Opt. Lett. 20(16), 1716–1718 ( 1995). [CrossRef] [PubMed]

2.

A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69(16), 2321 ( 1996). [CrossRef]

3.

M. J. Fitch and R. Osiander, “Terahertz waves for communications and sensing,” Johns Hopkins APL Tech. Dig. 25, 348–355 ( 2004).

4.

K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature 432(7015), 376–379 ( 2004). [CrossRef] [PubMed]

5.

Q. Cao and J. Jahns, “Azimuthally polarized surface plasmons as effective terahertz waveguides,” Opt. Express 13(2), 511–518 ( 2005). [CrossRef] [PubMed]

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M. Walther, M. R. Freeman, and F. A. Hegmann, “Metal-wire terahertz time-domain spectroscopy,” Appl. Phys. Lett. 87(26), 261107 ( 2005). [CrossRef]

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T.-I. Jeon, J.-Q. Zhang, and D. Grischkowsky, “THz Sommerfeld wave propagation on a single metal wire,” Appl. Phys. Lett. 86(16), 161904 ( 2005). [CrossRef]

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H. Cao and A. Nahata, “Coupling of terahertz pulses onto a single metal wire waveguide using milled grooves,” Opt. Express 13(18), 7028–7034 ( 2005). [CrossRef] [PubMed]

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M. Wächter, M. Nagel, and H. Kurz, “Frequency-dependent characterization of THz Sommerfeld wave propagation on single-wires,” Opt. Express 13(26), 10815–10822 ( 2005). [CrossRef] [PubMed]

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K. Wang and D. M. Mittleman, “Guided propagation of terahertz pulses on metal wires,” J. Opt. Soc. Am. B 22(9), 2001–2008 ( 2005). [CrossRef]

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K. Wang and D. M. Mittleman, “Dispersion of surface plasmon polaritons on metal wires in the terahertz frequency range,” Phys. Rev. Lett. 96(15), 157401 ( 2006). [CrossRef] [PubMed]

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J. A. Deibel, K. Wang, M. D. Escarra, and D. M. Mittleman, “Enhanced coupling of terahertz radiation to cylindrical wire waveguides,” Opt. Express 14(1), 279–290 ( 2006). [CrossRef] [PubMed]

13.

J. A. Deibel, N. Berndsen, K. Wang, D. M. Mittleman, N. C. J. van der Valk, and P. C. M. Planken, “Frequency-dependent radiation patterns emitted by THz plasmons on finite length cylindrical metal wires,” Opt. Express 14(19), 8772–8778 ( 2006). [CrossRef] [PubMed]

14.

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(26), 13021–13029 ( 2006). [CrossRef] [PubMed]

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C. Themistos, B. M. Azizur Rahman, M. Rajarajan, V. Rakocevic, and K. T. V. Grattan, “Finite Element Solutions of Surface-Plasmon Modes in Metal-Clad Dielectric Waveguides at THz Frequency,” J. Lightwave Technol. 24(12), 5111–5118 ( 2006). [CrossRef]

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X. He, J. Cao, and S. Feng, “Simulation of the Propagation Property of Metal Wires Terahertz Waveguides,” Chin. Phys. Lett. 23(8), 2066–2069 ( 2006). [CrossRef]

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J. A. Deibel, M. Escarra, N. Berndsen, K. Wang, and D. M. Mittleman, “Finite-Element Method Simulations of Guided Wave Phenomena at Terahertz Frequencies,” Proc. IEEE 95(8), 1624–1640 ( 2007). [CrossRef]

18.

H. Liang, S. Ruan, and M. Zhang, “Terahertz surface wave propagation and focusing on conical metal wires,” Opt. Express 16(22), 18241–18248 ( 2008). [CrossRef] [PubMed]

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Y. B. Ji, E. S. Lee, J. S. Jang, and T.-I. Jeon, “Enhancement of the detection of THz Sommerfeld wave using a conical wire waveguide,” Opt. Express 16(1), 271–278 ( 2008). [CrossRef] [PubMed]

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P. W. Smorenburg, W. P. E. M. Op ‘t Root, and O. J Luiten, “Direct generation of terahertz surface plasmon polaritons on a wire using electron bunches’,” Phys. Rev. B 78(11), 115415 ( 2008). [CrossRef]

21.

J. A. Deibel, K. Wang, M. Escarra, N. Berndsen, and D. M. Mittleman, “The excitation and emission of terahertz surface plasmon polaritons on metal wire waveguides,” C. R. Phys. 9(2), 215–231 ( 2008). [CrossRef]

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S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fibers,” Appl. Phys. Lett. 76(15), 1987 ( 2000). [CrossRef]

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32.

W. Shi and Y. J. Ding, “Designs of terahertz waveguides for efficient parametric terahertz generation,” Appl. Phys. Lett. 82(25), 4435 ( 2003). [CrossRef]

33.

M. Nagel, A. Marchewka, and H. Kurz, “Low-index discontinuity terahertz waveguides,” Opt. Express 14(21), 9944–9954 ( 2006). [CrossRef] [PubMed]

34.

M. Wächter, M. Nagel, H. Kurz, M. L. Nagel, and H Kurz, “Metallic slit waveguide for dispersion-free low-loss terahertz signal transmission,” Appl. Phys. Lett. 90(6), 061111 ( 2007). [CrossRef]

35.

A. Ishikawa, S. Zhang, D. A. Genov, G. Bartal, and X. Zhang, “Deep subwavelength terahertz waveguides using gap magnetic plasmon,” Phys. Rev. Lett. 102(4), 043904 ( 2009). [CrossRef] [PubMed]

36.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, (Springer, Berlin, 1988).

37.

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93(13), 137404 ( 2004). [CrossRef] [PubMed]

38.

U. Schröter and A. Dereux, “Surface plasmon polaritons on metal cylinders with dielectric core,” Phys. Rev. B 64(12), 125420 ( 2001). [CrossRef]

39.

M. Born, and E. Wolf, Principles of Optics, 5th ed. (Pergamon Press, Oxford, 1975).

40.

M. A. Ordal, R. J. Bell, R. W. Alexander Jr, L. L. Long, and M. R. Querry, “Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W,” Appl. Opt. 24(24), 4493–4499 ( 1985). [CrossRef] [PubMed]

OCIS Codes
(230.7370) Optical devices : Waveguides
(240.6680) Optics at surfaces : Surface plasmons
(260.3090) Physical optics : Infrared, far
(260.3910) Physical optics : Metal optics

ToC Category:
Optics at Surfaces

History
Original Manuscript: September 10, 2009
Revised Manuscript: October 13, 2009
Manuscript Accepted: October 13, 2009
Published: October 29, 2009

Citation
Jie Yang, Qing Cao, and Changhe Zhou, "An explicit formula for metal wire plasmon of terahertz wave," Opt. Express 17, 20806-20815 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-23-20806


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References

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