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Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 2 — Jan. 24, 2005
  • pp: 511–518
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Azimuthally polarized surface plasmons as effective terahertz waveguides

Qing Cao and Jürgen Jahns  »View Author Affiliations


Optics Express, Vol. 13, Issue 2, pp. 511-518 (2005)
http://dx.doi.org/10.1364/OPEX.13.000511


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Abstract

Quite recently, it was found that metal wires can effectively guide terahertz radiation. Based on the fact that the absolute values of the relative permittivities of metals in the spectral region of terahertz radiation are huge, we here analyse the properties of this kind of waveguide and explain the related experimental results. In particular, we show that the observed waveguiding is due to the propagation of an azimuthally polarized surface plasmon along the wire. Some related aspects, such as the choice of metal and the slowly decaying modal field, are also discussed. In particular, we show that, if a copper wire with a radius of 0.45 mm is used, the attenuation coefficient is smaller than 2×10-3 cm-1 in the whole range of 0.1~1 THz.

© 2005 Optical Society of America

1. Introduction

The terahertz (THz) region of the electromagnetic spectrum is located between microwave and optical frequencies, normally defined as the range from 0.1 to 10 THz (or correspondingly, from 30 µm to 3 mm in wavelength). In recent years, THz radiation has attracted a lot of interests [1

1. D. M. Mittleman, ed. Sensing with Terahertz Radiation (Springer, Heidelberg, 2002).

9

9. S. Wang and X. -C. Zhang, “Pulsed terahertz tomography,” J. Phys. D 37, R1–R36 (2004). [CrossRef]

], because it offers significant scientific and technological potential in many fields, such as in sensing, in imaging, and in spectroscopy. However, waveguiding in this intermediate spectral region is still a challenge [10

10. R. W. McGowan, G. Gallot, and D. Grischkowsky, “Propagation of ultrawideband short pulses of THz radiation through submillimeter-diameter circular waveguides,” Opt. Lett. 24, 1431–1433 (1999). [CrossRef]

16

16. S. Coleman and D. Grischkowsky, “A THz transverse electromagnetic mode two-dimensional interconnect layer incorporating quasi-optics,” Appl. Phys. Lett. 83, 3656–3658 (2003). [CrossRef]

]. Both of the conventional metal waveguides for microwave radiation and the dielectric fibers for visible radiation cannot be used to effectively guide THz radiation. The obstacles come from the high loss from the finite conductivity of metals or the high absorption coefficient of dielectric materials in this spectral range. Quite recently, Wang and Mittleman [17

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

] found that a simple metal wire can be used as effective THz waveguide. This finding paves the way for a wide range of new applications for THz sensing and imaging. In Ref. [17

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

], the authors focused on the report of experimental observations, however, without a deep theoretical explanation.

2. Explanations for THz metal wire waveguides

It is well known that there exists an electromagnetic bound state at a flat metal-dielectric interface. This bound state can only exist for the TM polarization and is called surface plasmon (SP) [18

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

, 19]. SPs can be excited by periodic structures like metal gratings [20

20. Q. Cao and Ph. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88, 057403 (2002). [CrossRef] [PubMed]

], and can propagate along flat metal-dielectric interfaces [21

21. S. I. Bozhevolnyi, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86, 3008–3011 (2001). [CrossRef] [PubMed]

]. Similar to THz radiation, SPs have also attracted much attention in recent years [19].

It is relatively less well known that SP can also exist at a cylindrical metal-dielectric interface [22

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

24

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

]. For an SP at a flat metal-dielectric interface, there are one magnetic field component Hy, and two electrical field components Ex and Ez. The only transverse magnetic field component Hy indicates the TM polarization. For an SP at a cylindrical metal-dielectric interface, however, there are one magnetic field component Hϕ and two electric field components Er and Ez. The only transverse magnetic field component Hϕ indicates the TM polarization. The latter kind of SP can be reasonably called APSP, because the only magnetic field component Hϕ is angular.

Consider the eigenproblem of a cylindrical metal wire surrounded by air. We denote by ε and µ the relative permittivity and the relative magnetic permeability, respectively. For the metal, we use εm and µm. And for air, ε=µ=1. We denote by λ0, c, and R the wavelength in free space, the light speed in free space, and the radius of the metal wire, respectively. We assume a temporal factor exp(-jωt), where ω=2πc/λ0, is the angular frequency.

We are interested in axially symmetrical eigenmodes. For them, the relations ∂E/∂ϕ=0 and ∂H/∂ϕ=0 hold in the cylindrical coordinates. By use of these relations, one can separate the electromagnetic field into two families. One is the TE polarization, and the other is the TM polarization. For the former, there are three components Eϕ, Hr and Hz. For the latter, there are another three components Hϕ, Er and Ez. The two families of TE and TM polarizations are decoupling in this case. We now focus on the TM polarization. By substituting the relations ∂E/∂ϕ=0 and ∂H/∂ϕ=0 into Maxwell’s equations [25

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

], one obtains

Hϕ=jμk0(ErzEzr),
(1)
Ez=jεk01rr(rHϕ),
(2)
Er=jεk0Hϕz.
(3)

All of Hϕ, Er and Ez are functions of the two variables r and z. For the formulation of the eigenproblem, they can be written as Hϕ(r,z)=Hϕ(r)exp(jk0neffz), Er(r,z)=Er(r)exp(jk0neffz), and Ez(r,z)=Ez(r)exp(jk0neffz), respectively, where neff is the effective index of the eigenmode. Then the operator ∂/∂z can be replaced by jk0neff. In the remainder of this paper, we refer to Hϕ, Er and Ez specifically as Hϕ(r), Er(r) and Ez(r), respectively. By using the relation ∂/∂z=jk0neff in Eq. (3), one gets

Er=ε1neffHϕ.
(4)

Substituting Eq. (4) into Eq. (1), and using the relation ∂/∂z=jk0neff again, one can obtain

Hϕ=jεk0(μεneff2)dEzdr,
(5)

Further substituting Eq. (5) into Eq. (2), one can obtain

ρ2d2Ezdρ2+ρdEzdρρ2Ez=0,
(6)

where ρ=k0[(neff)2-µε]1/2r. The suitable solution for Ez has the form I0(k0κmr) in the metal and K0(k0κar) in air, because this solution approximately decays exponentially from the interface, where I0(.) and K0(.) are the generalized Bessel functions [26

26. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, UK1966).

], κm=[(neff)2mεm]1/2 and κa=[(neff)2-1]1/2. Substituting this kind of solution into Eq. (5), and using the relations [26

26. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, UK1966).

] dI0(k0κmr)/dr=k0κmI1(k0κmr) and dK0(k0κar)/dr=-k0κaK1(k0κar), one can determine the solution form of Hϕ. By using the continuities of Ez and Hϕ at the interface r=R, one can obtain the following eigenvalue equation

εmκmI1(k0κmR)I0(k0κmR)+1κaK1(k0κaR)K0(k0κaR)=0.
(7)

The existence of the solution of Eq. (7) results from the negative real part of εm. This negative value is due to the electron plasma contribution when the frequency is lower than the plasma frequency. Accordingly, this eigenmode is traditionally called SP. In particular, when R→∞, the relations I1(k0κmR)/I0(k0κmR)=1 and K1(k0κaR)/K0(k0κaR)=1 hold. By use of these properties, one can obtain, for very large R,

neff=εm(εmμm)(1+εm)(εm1).
(8)

Eq. (8) is exactly valid for an SP at a flat metal-air interface. It can be expected that Eq. (8) is also approximately valid for an APSP at the interface of a very thick metal wire. In particular, for nonmagnetic metals with µm=1, Eq. (8) reduces to neff=[εm/(1+εm)]1/2. Eq. (8) is already enough for some qualitative analyses, though the exact neff value of the APSP should be obtained by numerically solving Eq. (7).

Fig. 1. Normalized modal field of an APSP of a copper wire. The red curve is the modal field outside the metal, and the blue curves are the modal field in the metal. (a) The total profile. (b) The detailed distribution of the very small penetration of the modal field in the metal.

1). It is well known that the absolute values of εm of metals are huge in the spectral region of THz radiation. This property can be deduced from the Drude model [25

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

, 27

27. M. A. Ordal, R. J. Bell, R. W. Alexander, 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, 4493–4499 (1985). [CrossRef] [PubMed]

]. Typically [27

27. M. A. Ordal, R. J. Bell, R. W. Alexander, 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, 4493–4499 (1985). [CrossRef] [PubMed]

], -Re(εm) is larger than 104 and Im(εm) is larger than 105, where Re(.) and Im(.) indicate the real and the imaginary parts, respectively. By using the properties |εm|≫1 and |εm|≫µm in Eq. (8), one can find that neff≈1. This result explains the low loss of metal wire waveguides because Im(neff) is very small.

2). The relation neff≈1 holds for a very wide bandwidth, because |εm| is huge in the whole range of THz radiation. This property explains the very low dispersion of metal wire waveguides.

4). Because the longitudinal field Ez is negligible, the remaining only electric field component in air is Er. Then it looks like that the electric field is radially polarized.

5). The electrical field of the light source is linearly polarized, but the electric field of the APSP is approximately radially polarized. The polarization mismatch between them leads to a very low coupling efficiency, though some additional setup was used to help the coupling.

8). The metal wire is axially symmetric. For axially symmetric electromagnetic fields, TM and TE polarizations are decoupling. Thus, the radial polarization of the electric field can be maintained during propagation, even after propagating off the end of the metal wire.

Clearly, these excellent explanations confirm that the waveguide effect of a metal wire is indeed due to the propagation of an APSP. It is worth mentioning that, besides the zero-order SP (i.e., APSP) mentioned above, there also exist higher-order SPs [23

23. C. A. Pfeiffer, E. N. Economou, and K. L. Ngai, “Surface polaritons in a circularly cylindrical interface: surface plasmons,” Phys. Rev. B 10, 3038–3051 (1974). [CrossRef]

]. However, they are not really guided modes. In fact, those higher-order modes were called virtual radiative SPs in Ref. [23

23. C. A. Pfeiffer, E. N. Economou, and K. L. Ngai, “Surface polaritons in a circularly cylindrical interface: surface plasmons,” Phys. Rev. B 10, 3038–3051 (1974). [CrossRef]

], in contrast to APSP, which is called real nonradiative SP in Ref. [23

23. C. A. Pfeiffer, E. N. Economou, and K. L. Ngai, “Surface polaritons in a circularly cylindrical interface: surface plasmons,” Phys. Rev. B 10, 3038–3051 (1974). [CrossRef]

]. Therefore, the existence of those virtual radiative SPs does not affect our analyses.

3. Further discussions

In Section 2 we have explained the experimental results of Ref. [17

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

]. In this Section we discuss some further problems.

As we state above, the low loss and the very low dispersion of an APSP come from the huge absolute values |εm|. In the spectral region of THz radiation, different metals still have different |εm| [27

27. M. A. Ordal, R. J. Bell, R. W. Alexander, 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, 4493–4499 (1985). [CrossRef] [PubMed]

] though those values are all huge. For example, the value |εm| of copper is more than 10 times higher than that of iron [27

27. M. A. Ordal, R. J. Bell, R. W. Alexander, 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, 4493–4499 (1985). [CrossRef] [PubMed]

]. Generally speaking, the larger the value |εm|, the better the waveguide. In addition, the relative magnetic permeability µm also plays a role.

By use of a Taylor expansion of Eq. (8) for huge εm, one can prove that neff≈1-µm/(2εm). From this expression, one can deduce that, for the same εm, a magnetic metal will increase the loss by µm times. Therefore, the use of ferromagnetic metal [25

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

, 28], such as iron and nickel, should be avoided. We suggest the use of nonmagnetic metals with very large |εm|. Copper, silver and gold are good candidates. Copper, of course, would be preferrable because it is much cheaper than silver and gold. We notice that stainless steel wires are used in the experiments of Ref. [17

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

]. Stainless steels can be magnetic or “nonmagnetic” [29, 30

30. C. Weber and J. Fajans, “Saturation in “nonmagnetic” stainless steel,” Rev. Scientific Instruments 69, 3695–3696 (1998). [CrossRef]

], depending on the kinds. In Ref. [17

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

], there is no report about the µm value of the used stainless steel. Therefore, it is difficult to concretely calculate the waveguide parameters of the stainless steel wires used there [17

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

]. However, the µm value should be not smaller than 1 [30

30. C. Weber and J. Fajans, “Saturation in “nonmagnetic” stainless steel,” Rev. Scientific Instruments 69, 3695–3696 (1998). [CrossRef]

]. Also, the |εm| value of a stainless steel should be smaller than that of copper, because the |εm| value of iron, which is the main chemical composition of stainless steel, is at least 10 times smaller than that of copper [27

27. M. A. Ordal, R. J. Bell, R. W. Alexander, 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, 4493–4499 (1985). [CrossRef] [PubMed]

]. Therefore, a copper wire waveguide is superior to a stainless steel wire waveguide.

As we state above, the exact neff value can only be obtained by numerically solving the eigenvalue equation (7), though Eq. (8) is very useful for qualititative analyses. It is known that both the functions I1(.) and I0(.) approximately increase exponentially with the increase of the real part of the argument. Unfortunately, the real part of the argument k0κmR is quite large in the investigated case. For example, in the example of Fig. 1, the argument k0κmR is as large as 6.21×103-j4.95×103. Because the argument k0κmR is quite large, numerical overflow will happen if one directly calculates the functions I1(k0κmR) and I0(k0κmR) that appear in Eq. (7). To solve this problem, we use the asymptotic expression [26

26. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, UK1966).

]

Iν(u)exp(u)(2πu)12[14ν218u+(4ν21)(4ν29)2(8u)2]

Fig. 2. Change of effective index with the radius R of metal wire. The red curves are the calculated results, and the black curves are the values given by Eq. (8). The dashed curves are Im(neff), and the solid curves are Re(neff)-1.

To see the influence of the finite radius of the metal wire, we calculate the effective indices neff of the APSPs at the interfaces of copper wires with different R. The frequency is chosen to be 0.5 THz (i.e., λ0=0.6 mm). The εm value is determined to be εm=-6.3×105+j2.77×106 according to a fitted Drude formula for copper [27

27. M. A. Ordal, R. J. Bell, R. W. Alexander, 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, 4493–4499 (1985). [CrossRef] [PubMed]

]. The relation µm=1 is used. The calculated neff values, as a function of R, are shown in Fig. 2. One can see that, just as expected, neff approach to the value determined by Eq. (8) for very large R. One can also see that, both the values of Im(neff) and Re(neff)-1 increase with the decrease of the radius R. Thick metal wires are sometimes inconvenient for certain applications, but with lower attenuation. Therefore, one should balance between the size and the attenuation when designing a concrete metal wire waveguide.

To see the dependency on the frequency, we calculate the neff values, as a function of frequency. The metal is chosen to be copper, and the radius R is chosen to be 0.45 mm. The frequency-dependent εm of copper is obtained from the corresponding Drude formula of Ref. [27

27. M. A. Ordal, R. J. Bell, R. W. Alexander, 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, 4493–4499 (1985). [CrossRef] [PubMed]

]. The calculated neff, as a function of frequency is shown in Fig. 3(a). By use of the relation α=k0Im(neff), one can further obtain the attenuate coefficient α. The calculated α value is shown in Fig. 3(b). One can see that the attenuation is smaller than 2×10-3 cm-1 in the range of 0.1~1THz. This result explicitly shows the superiority of a copper wire waveguide.

Fig. 3. (a) Change of effective index with the frequency. The red curve is Im(neff), and the black curve is Re(neff)-1. (b) Change of attenuation with the frequency.

4. Conclusions

References and links

1.

D. M. Mittleman, ed. Sensing with Terahertz Radiation (Springer, Heidelberg, 2002).

2.

P. R. Smith, D. H. Auston, and M. C. Nuss, “Subpicosecond photoconducting dipole antennas,” IEEE J. Quant. Electron. 24, 255–260 (1998). [CrossRef]

3.

M. Exter and D. Grischkowsky, “Characterization of an optoelectronic terahertz beam system,” IEEE Trans. Microwave Theory Tech. 38, 1684–1691 (1990). [CrossRef]

4.

P. U. Jepsen, R. H. Jacobsen, and S. R. Keiding, “Generation and detection of terahertz pulses from biased semiconductor antennas,” J. Opt. Soc. Am. B 13, 2424–2436 (1996). [CrossRef]

5.

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, “T-ray imaging,” IEEE J. Select.Top. Quant. Electron. 2, 679–692 (1996). [CrossRef]

6.

R. H. Jacobsen, D. M. Mittleman, and M. C. Nuss, “Chemical recognition of gases and gas mixtures with terahertz waves,” Opt. Lett. 21, 2011–2013 (1996). [CrossRef] [PubMed]

7.

R. M. Woodward, V. P. Wallace, D. D. Arnone, E. H. Linfield, and M. Pepper, “Terahertz pulsed imaging of skin cancer in the time and frequency domain,” J. Biol. Phys. 29, 257–261 (2003). [CrossRef]

8.

K. Kawase, Y. Ogawa, and Y. Watanabe, “Non-destructive terahertz imaging of illicit drugs using spectral fingerprints,” Opt. Express 11, 2549–2554 (2003). [CrossRef] [PubMed]

9.

S. Wang and X. -C. Zhang, “Pulsed terahertz tomography,” J. Phys. D 37, R1–R36 (2004). [CrossRef]

10.

R. W. McGowan, G. Gallot, and D. Grischkowsky, “Propagation of ultrawideband short pulses of THz radiation through submillimeter-diameter circular waveguides,” Opt. Lett. 24, 1431–1433 (1999). [CrossRef]

11.

G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Terahertz waveguides,” J. Opt. Soc. Am. B 17, 851–863 (2000). [CrossRef]

12.

R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguides,” J. Appl. Phys. 88, 4449–4451 (2000). [CrossRef]

13.

S. P. Jamison, R. W. McGown, and D. Grischkowsky, “Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fiber,” Appl. Phys. Lett. 76, 1987–1989 (2000). [CrossRef]

14.

R. Mendis and D. Grischkowsky, “Undistorted guided-wave propagation of subpicosecond terahertz pulses,” Opt. Lett. 26, 846–848 (2001). [CrossRef]

15.

R. Mendis and D. Grischkowsky, “THz interconnect with low loss and low group velocity dispersion,” IEEE Microwave Wireless Comp. Lett. 11, 444–446 (2001). [CrossRef]

16.

S. Coleman and D. Grischkowsky, “A THz transverse electromagnetic mode two-dimensional interconnect layer incorporating quasi-optics,” Appl. Phys. Lett. 83, 3656–3658 (2003). [CrossRef]

17.

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

18.

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

19.

http://www.surfaceplasmonoptics.org

20.

Q. Cao and Ph. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88, 057403 (2002). [CrossRef] [PubMed]

21.

S. I. Bozhevolnyi, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86, 3008–3011 (2001). [CrossRef] [PubMed]

22.

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

23.

C. A. Pfeiffer, E. N. Economou, and K. L. Ngai, “Surface polaritons in a circularly cylindrical interface: surface plasmons,” Phys. Rev. B 10, 3038–3051 (1974). [CrossRef]

24.

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

25.

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

26.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, UK1966).

27.

M. A. Ordal, R. J. Bell, R. W. Alexander, 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, 4493–4499 (1985). [CrossRef] [PubMed]

28.

http://hyperphysics.phy-astr.gsu.edu/hbase/tables/magprop.html#c1

29.

http://www.stainless-rebar.org/grades.htm

30.

C. Weber and J. Fajans, “Saturation in “nonmagnetic” stainless steel,” Rev. Scientific Instruments 69, 3695–3696 (1998). [CrossRef]

31.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]

32.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003). [CrossRef] [PubMed]

33.

Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10, 324–331 (2002). [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:
Research Papers

History
Original Manuscript: December 15, 2004
Revised Manuscript: January 12, 2005
Published: January 24, 2005

Citation
Qing Cao and Jürgen Jahns, "Azimuthally polarized surface plasmons as effective terahertz waveguides," Opt. Express 13, 511-518 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-2-511


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References

  1. D. M. Mittleman, ed. Sensing with Terahertz Radiation (Springer, Heidelberg, 2002).
  2. P. R. Smith, D. H. Auston, and M. C. Nuss, �??Subpicosecond photoconducting dipole antennas,�?? IEEE J. Quant. Electron. 24, 255-260 (1998). [CrossRef]
  3. M. Exter and D. Grischkowsky, �??Characterization of an optoelectronic terahertz beam system,�?? IEEE Trans. Microwave Theory Tech. 38, 1684-1691 (1990). [CrossRef]
  4. P. U. Jepsen, R. H. Jacobsen, and S. R. Keiding, �??Generation and detection of terahertz pulses from biased semiconductor antennas,�?? J. Opt. Soc. Am. B 13, 2424-2436 (1996). [CrossRef]
  5. D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, �??T-ray imaging,�?? IEEE J. Select.Top. Quant. Electron. 2, 679-692 (1996). [CrossRef]
  6. R. H. Jacobsen, D. M. Mittleman, and M. C. Nuss, �??Chemical recognition of gases and gas mixtures with terahertz waves,�?? Opt. Lett. 21, 2011-2013 (1996). [CrossRef] [PubMed]
  7. R. M. Woodward, V. P. Wallace, D. D. Arnone, E. H. Linfield, and M. Pepper, �??Terahertz pulsed imaging of skin cancer in the time and frequency domain,�?? J. Biol. Phys. 29, 257-261 (2003). [CrossRef]
  8. K. Kawase, Y. Ogawa, and Y. Watanabe, �??Non-destructive terahertz imaging of illicit drugs using spectral fingerprints,�?? Opt. Express 11, 2549-2554 (2003). [CrossRef] [PubMed]
  9. S. Wang and X. �??C. Zhang, �??Pulsed terahertz tomography,�?? J. Phys. D 37, R1-R36 (2004). [CrossRef]
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