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We investigate the propagation properties of terahertz plasmon of a metallic nanowire with sub-skin-depth diameter. By taking the small radius and the huge relative permittivity into account, we establish an approximate analytical description for this kind of surface plasmon. It is shown that the main propagation properties are closely related to the product of the radius of the metallic nanowire and the complex wave number of the metal. In addition, when the radius of the metal wire is smaller than the skin-depth, the size of the modal field is simply proportional to the radius of the metal wire. We also carefully verify these analytical predictions with rigorous numerical simulations.
©2010 Optical Society of America
1. Introduction
Terahertz (THz) wave, locating between the infrared and microwave bands of 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 [1–3]. Among those research works, THz waveguides have attracted more and more interests [4–43]. In 2004, Wang and Mittleman [4] reported that a simple metal wire can effectively guide THz wave. It was quickly shown that the THz waveguide effect of metal wire comes from the azimuthally polarized surface plasmon [5]. Up to now, many interesting theoretical and experimental works on metal wire THz waveguide have been carried out [5–27].
A very important size for a plasmon in the spectral region of terahertz wave is the skin-depth. Recently, some research works on metallic nano-slit with sub-skin-depth width have been carried out [42,43]. It was reported that terahertz field can be enhanced by a metallic nano-slit. However, until now, only few papers have touched the topics of terahertz plasmon of a metal nanowire with a sub-skin-depth diameter [24]. The investigation on this kind of plasmon is just at the very beginning. Therefore, a lot of research works are needed to be done in this interesting area.
In this paper, we shall develop an elegant analytical description to reveal the physical mechanism that governs the propagation properties of the terahertz plasmon of a metallic nanowire with a sub-skin-depth diameter. The paper is organized as follows. In Section 2, we shall derive an approximate analytical description for the effective index and show that the main propagation properties are closely related to the product of the radius of the metallic nanowire and the complex wave number of the metal. In addition, we shall also reveal that the size of the modal field is simply proportional to the radius of the metal wire when the radius of metal wire is smaller than the skin-depth. In Section 3, we shall carefully test the analytical predictions with rigorous numerical simulations. And in Section 4, we shall conclude this paper and discuss some related problems. For simplicity, we shall only consider the case of nonmagnetic metals whose relative magnetic permeabilities are always 1.
2. Approximate analytical description for THz plasmon of a metallic nanowire with a sub-skin-depth diameter
Consider a cylindrical metal wire surrounded by air. We are interested in the axially symmetric eigenmode. Namely, the relations ∂E/∂φ = 0 and ∂H/∂φ = 0 always hold. For TM polarization of a nonmagnetic metal, by substituting the above-mentioned relations into Maxwell’s equations [44] and using the continuities of Ez and Hφ at the interface, one can get the following eigen-equation [5,23,25,45,46]:
where κa = [(neff)2-1]1/2, κm = [(neff)2-εm]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 denote wavelength and wave number in free space, respectively, and R is the radius of the metal wire.The effective index neff is an important parameter for the plasmon. It is hidden in Eq. (1). We now derive an approximate expression for the effective index neff. By use of the property |neff 2|<<|εm|, one can get κm≈(-εm)1/2. In addition, the relation K1(k0κaR)≈(k0κaR)−1 holds for small R. Substituting these two approximations into Eq. (1), one can further get
whereThe left-hand side of Eq. (2) has two factors, one is κa 2, the other is K0(k0κaR). The former factor κa 2 changes fast and the latter factor K0(k0κaR) changes very slowly. Basically, the factor K0(k0κaR) changes as slowly as a logarithmic function. By use of these properties, we develop the following recurrence formula
where n = 1,2,3…, κa,n is the corresponding approximate value for κa after n times recurrences. The recurrence solution κa,n of Eq. (4) is simply given byTo implement the recurrence formula of Eq. (5), one needs an initial input κa,0. Simply letting the factor K0(k0κaR) of Eq. (2) be 1, one can get the initial rough value κa,0, which is given by
For many important analyses, one does not need a highly accurate κa,n value with a large n. As we shall show below, the approximate value κa,2 is actually good enough for this kind of analysis. After two times uses of Eq. (5), one can get
where K0(.) is a modified Bessel function. The corresponding approximate value neff,2 can be further obtained from the relationFrom Eq. (7) one can see that κa,2 has two factors. The leading factor a1/2 changes much faster than the other. The leading factor a1/2 is uniquely dependent on the product of k0R(-εm)1/2 = jkR, where k = εm 1/2k0, is the complex wave number. The other factor of κa,2 changes very slowly with R. And even this slowly varying factor is also dependent on the parameter a, which is a single variable function of jkR. It is well known that the absorption and the dispersion of the THz plasmon of a metallic nanowire are directly related to the imaginary part and the real part of the effective index, respectively. And the effective index is related to the parameter κa. As we shall show below, the value of κa can be well approximated by κa,2, which is closely related to the product jkR. Therefore, the main propagation properties of a THz metal wire plasmon working in the sub-skin-depth region are closely related to the product of the radius R of the metallic nanowire and the complex wave number k of the metal.
We now discuss the size of the modal field. For simplicity, we discuss the unique magnetic field component Hφ(r). As we know, the Hφ field is proportional to K1(k0κar) in the air, and proportional to I1(k0κmr) in the metal [5]. The maximum absolute value of Hφ appears at the surface of metal wire. Outside the metal wire, the absolute value of Hφ can be normalized as |Hφ(r)|/ |Hφ(R)| = |K1(k0κar)|/|K1(k0κaR)|. As we know, the relation K1(k0κaR)≈(k0κaR)−1 holds for small R. Accordingly, |Hφ(r)|/ |Hφ(R)| = R/r outside the metal. If we use the full width at half maximum (FWHM) to describe the size of the modal field, then the corresponding position is r = 2R outside the metal. Inside the metal wire, the absolute value of Hφ field can be normalized as |Hφ(r)|/|Hφ(R)| = |I1(k0κmr)|/|I1(k0κmR)|. As we know, the relation I1(k0κmR)≈(k0κmR/2) holds for small R. Accordingly, |Hφ(r)|/|Hφ(R) = r/R inside the metal. According to the definition of FWHM, the corresponding position is r = 0.5R inside the metal. As a whole, the size of the modal field is simply given by 2R-0.5R = 1.5R. Accordingly, we find that the size of the modal field is simply proportional to the radius R of the metal wire.
3. Numerical tests
To get an intuitive impression on the capability of our analytical expression, we calculate the approximate values neff,2 and the exact values neff in the radius range from 5 nm to 500 nm. 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 [47]. The approximate neff,2 values and the exact neff values are shown in Fig. 1(a) . And the relative deviations for the real part and the imaginary part of neff,2 are shown in Fig. 1(b). One can see that the maximum relative deviation of neff,2 is less than 1% in the considered radius range. It should be pointed out that the radius R = 500 nm is already far larger than the skin depth δ, which is 72.5 nm according to the definition of δ = λ0/[2πIm(εm 1/2)].
Fig. 1 (a) Comparison between the approximate values neff,2 and the exact values neff, for metal copper and 0.5 THz. The dashed curve is Im(neff) and the signs “+” show Im(neff,2). The solid curve is Re(neff) and the signs “o” show Re(neff,2). (b) The relative deviation of neff,2. The solid curve represents the relative deviations of Re(neff,2), and the dashed curve is the relative deviations of Im(neff,2).
We also test the applicability of the approximate formula for neff,2 for other nonmagnetic metals mentioned in Ref [47]. We find that the approximate formula for neff,2 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 formula neff,2 is always less than 3% for all 11 tested nonmagnetic metals in the whole spectral region of THz wave when the radius of metal wire is in the wide range from 5 nm to 500 nm.
The attenuation coefficient α is related to the imaginary part of the effective index and is given by α = k0Im(neff). To see the attenuation of the propagating mode at few nanometer diameter wires, we calculate the exact values and the approximate values of the attenuation coefficients in the radius range from 1 nm to 10 nm. The metal is chosen to be copper, and the frequency is chosen to be 0.5 THz (i.e., λ0 = 0.6 mm). The results are shown in Fig. 2 . One can see that the approximate values agree well with the exact values. It is also shown that the attenuation coefficient increases with the decrease of the radius of the metallic wire.
Fig. 2 Comparison between the approximate values and the exact values of the attenuation coefficient, for metal copper and 0.5 THz. The red curve denotes the exact values and the black curve denotes the approximate values. Note that the red curve and the black curve cannot be distinguished.
To get an intuitive impression on the property that the size of the modal field is simply proportional to the radius of the metal wire, we calculate the normalized Hφ fields for three different values of R. The metal is still chosen to be copper and the frequency is still chosen to be 0.5 THz (i.e., λ0 = 0.6 mm). The corresponding εm value is still εm = −6.3 × 105 + j2.77 × 106 according to a fitted Drude formula for copper. And the skin depth is still 72.5 nm. For comparison, we choose R = 5 nm, 50 nm and 500 nm as three different metal wire radii. The range of radial coordinate r is chosen to be 0≤r≤4R. The corresponding results for R = 5 nm, R = 50 nm, and R = 500 nm are shown in Fig. 3(a) , 3(b), and 3(c), respectively. From Fig. 3(a) and 3(b) one can see that the exact positions (dashed lines) of half maximum agree well with the analytical predictions of r = 2R and of r = 0.5R (solid lines). These results confirm the validity of the analytical prediction for metal nanowires with sub-skin-depth diameters. We note that the predicted position of r = 0.5R for half maximum is different from the actual position of half maximum for the case of R = 500 nm. This result is expected because the radius of 500 nm is much larger than the skin depth of 72.5 nm. We also note that the predicted position of r = 2R for half maximum is still accurate for the case of R = 500 nm. The reason is that the absolute value of κa is much smaller than that of κm. As a consequence, the absolute value of k0κaR is still far smaller than 1 in the case of R = 500 nm, even though the absolute value of k0κmR is not far smaller than 1 any longer in this case. In fact, the predicted position of r = 2R for the half maximum in the modal field outside the metal has much larger validity range. The reason is that the absolute value of κa decreases with the increase of R. As a consequence, the absolute value of k0κaR continuously keeps a very small value for a very broad radius range of R. However, we don’t further discuss this issue here, because it is beyond the scope of this paper.
Fig. 3 Comparison between the approximate sizes of the modal fields and the exact sizes of the modal fields, for metal copper and 0.5 THz. The red curves are the exact values of Hφ field. The black solid lines denote the approximate sizes of the modal fields and black dashed lines show the exact sizes of the modal fields. (a) The case for a copper wire with a radius of 5 nm. (b) The case for a copper wire with a radius of 50 nm. (c) The case for a copper wire with a radius of 500 nm. Note that the dashed lines and the solid lines cannot be distinguished for the positions of r = 2R in (a), (b), and (c); and for the position of r = 0.5R in (a).
4. Conclusions and discussions
In conclusion, we have established an approximate analytical description for the terahertz plasmon of a metallic nanowire with a sub-skin-depth diameter. It is shown that the main propagation properties are closely related to the product of the radius of the metallic nanowire and the complex wave number of the metal. In addition, when the radius of the metal wire is smaller than the skin-depth, the size of the modal field is simply proportional to the radius of the metal wire. The obtained results can be used for the fast analyses and design of the terahertz plasmon of a metallic nanowire with a sub-skin-depth diameter.
It is worth mentioning that Ref [24]. has touched the topics of terahertz plasmon of a metal nanowire with a sub-skin-depth diameter before our current work. However, the main contents of our current work are quite different from those of Ref [24]. The new results of our current paper are mainly as follows.
- 2) We show that the main propagation properties of a sub-skin-depth plasmon of a terahertz wave are closely related to the product of the radius of the metallic nanowire and the complex wave number of the metal. This important result is very helpful for understanding the physical mechanism of a sub-skin-depth plasmon.
- 3) We show that the FWHM of the model field is simply equal to 1.5R. This result presents an intuitive and quantitative picture for the width of the modal field.
For potential practical applications, the excitation, coupling and detection of sub-skin-depth plasmon are also important. Recently, Refs [26,27]. experimentally investigated the excitation, coupling and detection of subwavelength plasmon. We guess that the methods developed in Refs [26,27]. might be also valid for the excitation, coupling and detection of sub-skin-depth plasmon. In particular, we strongly suggest the use of radially polarized beam as the input beam to increase the coupling efficiency because this kind of beam matches well the polarization of the terahertz plasmon of a metallic nanowire [5].
Acknowledgment
The authors are indebted to the reviewer for the comments and suggestions for improving the paper, in particular for the suggestion of differentiating our current work from that of Ref. [24].
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