Nanoconcentration of Terahertz Radiation in Plasmonic Waveguides

Recent years have seen an explosive research and development of nanoplasmonics in the visible and near-infrared (near-ir) frequency regions. One of the most fundamental effects in nanoplasmonics is nano-concentration of optical energy. Plasmonic nanofocusing has been predicted and experimentally achieved. It will be very beneficial for the fundamental science, engineering, environmental, and defense applications to be able to nano-concentrate terahertz radiation (frequency 1 - 10 THz or vacuum wavelength 300 - 30 microns). This will allow for the nanoscale spatial resolution for THz imaging and introduce the THz spectroscopy on the nanoscale, taking full advantage of the rich THz spectra and submicron to nanoscale structures of many engineering, physical, and biological objects of wide interest: electronic components (integrated circuits, etc.), bacteria, their spores, viruses, macromolecules, carbon clusters and nanotubes, etc. In this Letter we establish the principal limits for the nanoconcentration of the THz radiation in metal/dielectric waveguides and determine their optimum shapes required for this nanoconcentration We predict that the adiabatic compression of THz radiation from the initial spot size of light wavelength to the final size of R = 100 - 250 nm can be achieved with the THz radiation intensity increased by a factor of 10 to 250. This THz energy nanoconcentration will not only improve the spatial resolution and increase the signal/noise ratio for the THz imaging and spectroscopy, but in combination with the recently developed sources of powerful THz pulses will allow the observation of nonlinear THz effects and a carrying out a variety of nonlinear spectroscopies (such as two-dimensional spectroscopy), which are highly informative.


INTRODUCTION
Recent years have seen an explosive research and development of nanoplasmonics in the visible and near-infrared (near-ir) frequency regions. 1 One of the most fundamental effects in nanoplasmonics is nano-concentration of optical energy. Plasmonic nanofocusing has been predicted 2 and experimentally achieved. [3][4][5] Nanoconcentration of optical energy at nanoplasmonic probes made possible optical ultramicroscopy with nanometer-scale resolution [6][7][8] and ultrasensitive Raman spectroscopy. 9 It will be very beneficial for the fundamental science, engineering, environmental, and defense applications to be able to nano-concentrate terahertz radiation (frequency 1 − 10 THz or vacuum wavelength λ 0 = 300 − 30 μm). This will allow for the nanoscale spatial resolution for THz imaging 10 and introduce the THz spectroscopy on the nanoscale, taking full advantage of the rich THz spectra and submicron to nanoscale structures of many engineering, physical, and biological objects of wide interest: electronic components (integrated circuits, etc.), bacteria, their spores, viruses, macromolecules, carbon clusters and nanotubes, etc. There are existing approaches to deep subwavelength THz imaging and probing based on sharp tips irradiated by a THz source, 11 adiabatically-tapered metal-dielectric waveguides 12 similar to optical adiabatic concentrators, 2-5 and nonlinear microscopic THz sources. 13 For the development of the THz nanotechnology, it is extremely important to understand spatial limits to which the THz radiation energy can be concentrated (nanofocused).
A major challenge for the nanoconcentration of the electromagnetic energy in the THz region is the large radiation wavelength in vacuum or conventional dielectrics, λ 0 = 30 μm − 300 μm, where the THz radiation can only be focused to the relatively very large regions of size ∼ λ 0 /2. The developed field of optical energy concentration, which is based on surface plasmon polaritons (SPPs), suggests that one of the ways to solving this problem is to employ the surface electromagnetic waves (SEWs). In the far infrared (ir), the dielectric permittivity of metals has large imaginary part which dominates over its negative real part. 14 This implies that SEWs propagating along a metal-dielectric flat interface in this frequency range, known as Sommerfeld-Zenneck waves, 15,16 are weakly bound to the surface 17 and can hardly be used for the confinement of THz radiation.
It has been suggested that periodically perforating flat surfaces of ideal metals with grooves or holes leads to the appearance of SEWs, which mimic ("spoof") SPPs to be stronger bound to the surfaces 18-20 permitting a better control over the THz fields. It has been predicted that SPPs on an array of parallel grooves cut on the surface of a perfect conductor wire can be localized by adiabatic deepening of the grooves. 21 At a point, where grooves are approximately a quarter of wavelength of light, the highest concentration is achieved to be on the order of tens of micrometers. This method restricts the localization point to a particular frequency, making the concentration very narrow-band. Also, the depth of a groove should be ≈ λ 0 /4, i.e., in the tens to hundred micron range, which precludes completely nanoscale devices.
It is well known from microwave technology that the ideal-metal waveguides with smooth surfaces support TEM waves, where the electric field lines are either infinitely extended or terminate at the metal surfaces normally to them. The latter case requires the waveguide cross-section topology to be more than single-connected; an example may be a coaxial waveguide ("coax"). Such waveguides possess are very wide-band in frequency. The THz waveguides can be adiabatically tapered to concentrate energy. The idea of adiabatic energy concentration comes from ultramicroscopy [22][23][24] and nanoplasmonics, 2 where it has been developed both theoretically and experimentally [3][4][5] and used in ultrasensitive surface enhanced Raman spectroscopy. 9 Employing these ideas of the adiabatic concentration and using a tapered metal-dielectric waveguide, the THz spatial resolution achieved is ∼ 20 μm across the entire THz spectrum. 12 In this Letter, for the first time, we establish the fundamental limits and find the principles of designing the optimum and efficient metal/dielectric waveguides suitable for the THz nanofocusing. The specific examples are for the wide-band concentrators: a plasmonic metal wedge cavity and tapered coax waveguides, which are terminated by funnel-type adiabatic tapers. Such nano-concentrators along with the advent of high-power sources 25,26 and sensitive detectors 27 of THz radiation, will open up an extremely wide range of possible THz applications, in particular, in material diagnostics, probe nanoimaging, biomedical applications etc. -cf. [28][29][30] Note that an alternative approach to the THz energy concentration using doped semiconductor tapers has also been proposed. 31 However, the required heavy doping of the semiconductors may cause fast electron relaxation due to the collisions with the inflicted lattice defects and bring about high losses. Therefore, in this Letter we will pursue the adiabatic nanoconcentration of the THz radiation using metal/dielectric structures.
Conventionally for THz and microwave regions, the metals are considered as ideal which is equivalent to neglecting their skin depth l s = λ 0 /Re √ −ε m , where ε m is the permittivity of the metal (we take into accoint that in the THz region |ε m | 1), and λ 0 = c/ω is the reduced wavelength in vacuum. It is true that in the THz region l s = 30− 60 nm. i.e., l s λ 0 . However, as we show below in this Letter, it is the finite skin depth, though as small as it is, that principally limits the ultimum localization size of the THz fields. For larger waveguides, the THz wave energy is localized mostly in the vacuum (dielectric) and its losses, which occur in the metal's skin layer, are correspondingly small. The effective quality factor (or, figure of merit) of the waveguide, which shows how many periods the wave can propagate without significantly loosing its energy, can be estimated as where a is the characteristic minimum size of the waveguide; this estimate becomes a good approximation for a metal-dielectric-metal planar waveguide [see below Eq. (2)]. When the waveguide size reduces to become on the order of the skin depth, a l s , the THz field is pushed into the metal, and the quality factor reduces to Q 1, which implies strong losses. Qualitatively, this establishes the limit to the nanoconcentration: for upper THz region a l s ≈ 30 nm, while for the 1 THz frequency a l s ≈ 60 nm. These are the practical limits of the THz nanoconcentration for the noble metals (silver, gold, and platinum) and for aluminum.
If one pursues the goal of creating enhanced local fields in a small region, but not necessarily to efficiently transfer the THz energy from the far field to the near field, then the apertureless SNOM approach, where a sharp metal or dielectric tip is irradiated by THz radiation, can, in principle, achieve even higher resolution. 32 However, the efficiency of utilizing the THz energy of the source in this case will be extremely low; the stray, far-field THz energy may create a significant parasitic background.

THEORY OF NANOCONCENTRATION OF TERAHERTZ RADIATION
Here and below in this Letter, we consider examples of the THz adiabatic nanoconcentration quantitatively, where the effect of the specific geometry will become apparent. Consider first a parallel plate waveguide that consists of a dielectric slab of thickness a with dielectric permittivity ε d sandwiched between two thick metal plates (with thickness of at least a few l s , i.e., greater than 200 nm in practical terms) [see Fig.1(a)]. The permittivity of the metal ε m in the THz region has a very large ( 10 6 ) imaginary part that defines the very small skin depth l s 100 nm, which justifies the usual consideration of the metals as perfect conductors. 14 However, as we have already mentioned, for our purposes of the THz nanoconcentration, we need to take into account the field penetration into the metal, which makes the problem plasmonic. In this case, the propagating modes of the system are SPPs, which are TM modes characterized by the symmetry with respect to the reflection in the center plane. We will orient the coordinate system with its z axis normal to the plane and the y axis in the direction of propagation. The symmetric (even) modes have even field components H x and E z and odd E y ; the parity of the antisymmetric (odd) modes is opposite.
From plasmonics it is known that the even modes have a larger fraction of their energy localized in the dielectric and the odd modes in the metal. Therefore, the even modes have much smaller damping and are, therefore, most suitable for the THz energy concentrations. The dispersion relation for the even modes is given in the Appendix as Eq. (5). This equation can be much simplified and solved in a closed analytical form taking into account that we are interested in the subwavelength focusing, i.e., a λ, where λ = λ 0 /ε d is the reduced wavelength in the dielectric; also, in the entire THz region l s λ. This shows that there exists a small parameter in the problem l s a/λ 2 1 [see also Eq. (6)], which allows one to solve analytically the dispersion relation (5) obtaining the modal refraction index n = k/k 0 , where k is the THz wavevector, and k 0 = 1/λ 0 , where the approximate equality is valid for not too tight nanofocusing, i.e., for l s a. From this, we can obtain the quality factor of the waveguide Q = Re n/Im n = 2a/l s , giving a quantitative meaning to the estimate (1).
Plasmonic effects (i.e. those of the finite skin depth) are illustrated in Fig. 1 for silver-vacuum-silver waveguide and frequency of 1 THz. Panels (b) and (c) display the longitudinal electric field E y obtained by the exact solution of the Maxwell equations. Note that this field component is absent for the ideal conductor; here it is relatively small: on the order of 10 −3 of the transverse field. Panel (b) illustrates the case of a relatively wide waveguide (a = 10 μm), where it is evident that the electric field is localized mostly in the dielectric region of the waveguide, and the extinction of the wave is small. In a sharp contrast, for a nanoscopic waveguide (a = 200 nm) in panel (c), the electric field significantly penetrates the metal. In accord with our arguments, there is a very significant extinction of the fields as they propagate; the retardation effects are also evident: the lines of equal amplitude are at an angle relative to the normal (z) direction. The dependence of the modal refraction index on the thickness a of the waveguide obtained from Eq. (2) is plotted in Fig. 1 (d). This index increases as a becomes comparable with the skin depth. While Re n and Im n increase by the same absolute amount, the quality factor Q, obviously, greatly decreases with decrease of a. The mode described above can be used for broadband energy concentration of THz waves.
To introduce the THz nanoconcentration, consider a metal-dielectric-metal waveguide that is slowly (adiabatically) tapered off as a wedge, as illustrated in Fig. 2 (a). Because of the adiabatic change of the parameters, a wave propagating in such a waveguide will adjust to it without reflection or scattering, just as it takes place in nanoplasmonic waveguides. 2 As a result, propagating it will concentrate its energy, conforming to the tapering of the waveguide. The corresponding solution can be obtained from the Maxwell equations using the Wentzel-Kramers-Brillouin (WKB) approximation, similarly to the nanoplasmonic case in the visible, 2 as described in the Appendix. The WKB approximation is applicable under the conditions that where δ is the well-known adiabatic parameter describing how slowly the modal wavelength changes on a distance of its own, and |da/dy| is a parameter describing how adiabatically the transverse size of the confined mode changes along the propagation coordinate.
In the WKB approximation, the behavior of the dominating transverse field component E z as a function of the coordinate y along the propagation direction is shown for the last 6 μm of the propagation toward the edge in Fig. 2 (b). There is a clearly seen spatial concentration of the energy and increase of the field as the wave is guided into the taper. The predicted behavior of the two components of electric field and the magnetic field for the last micron of the propagation is shown in panels (c)-(e). It apparently indicates the adiabatic concentration, without an appreciable loss of the intensity. The THz wave follows the waveguide up to the nanometric size.
The red line in Fig. 2 (f) indicates that the local intensity I as the function of the thickness a of the waveguide for a < 4 μm increases significantly with 1/a, in qualitative accord with the behavior expected for the negligibly low losses. This intensity reaches its maximum for a = 1.6 μm and then starts to decrease as the losses overcome the adiabatic concentration. At smaller thicknesses, a 400 nm, the intensity in Fig. 2 (f) starts to increase again, which is unphysical. The reason is revealed by the behavior of the adiabatic parameter δ shown by the blue line: for a 400 nm, δ becomes relatively large (comparable with 1), i.e., the adiabaticity is violated. This is due to the fact that the fraction of the THz field energy propagating in the metal is dramatically increased for a 400 nm due to the constricted transverse extension of the dielectric in the waveguide. This causes a significant loss per wavelength λ, leading to a rapid change of the wave vector k, breaking down the adiabaticity. This constitutes a fundamental difference from the nanoplasmonic adiabatic concentration in the optical region where the adiabatic parameter is constant, and the adiabaticity holds everywhere including the vicinity of the tip. 2  To provide for the optimum guiding of the THz wave and its concentration on the nanoscale, the terminating (nanoscopic) part of the waveguide should be tapered slower, in a funnel-like manner. That is, one needs to decrease the grading da/dy of the waveguide near the edge in order to keep the adiabaticity parameter δ = d Rek −1 /da × da/dy approximately constant and small enough to prevent the back-reflection. Because for the adiabatic grading (tapering), the derivative d Rek −1 /da does not depend on the grading (it is the same as for the plane waveguide) and is only a function of a, the equation δ = δ(y) is a differential equation for the shape of the waveguide that can be easily integrated. This results in the dependence of the thickness a on the longitudinal coordinate y determined by a simple integral where n(a) is the modal index defined in this case by Eq. (2), and δ(y) is the desired dependence of the adiabatic parameter along the waveguide, which is an arbitrary function of y satisfying the adiabaticity conditions (3).
The geometry of an adiabatically-tapered end of the silver/vacuum waveguide found from Eq. (4) and satisfying Eq. (3) and the corresponding WKB solutions for the 1 THz fields are shown in Fig. 3 (a)-(c). The optimum shape of the waveguide in this case is funnel-like, greatly elongated toward the edge. The nanoconcentration of the field is evident on panels (a)-(c), as well as its penetration into the metal for a 100 nm. As these panels show quantitatively and the red curve on panel (d) qualitatively, the field intensity reaches its maximum at a ≈ 300 nm where it is enhanced with respect to the field at the entrance to the funnel waveguide by a modest factor of 1.2. At the same time, the adiabatic parameter δ decreases toward the tip from 0.07 to 0.05, indicated the applicability of the WKB approximation everywhere. Note that the this funnel-shaped wedge, indeed, continues the linearly-graded wedge waveguide shown in Fig. 2, which yields the enhancement factor of ≈ 8 at a = 2 μm. Sequentially, these two waveguides provide the intensity enhancement by approximately ×10 while compressing the THz wave to the thickness of a = 300 nm and the enhancement by a factor of 3 for a = 100 nm.
Thus, true nanolocalization of THz radiation in one dimension (1d) is possible. The minimum transverse size of this nanolocalization is determined by the skin depth, as we have already discussed qualitatively in the introductory part of this Letter. The obtained 1d beam of the nanoconcentrated THz radiation may be used for different purposes, in particular as a source for the diffraction elements including the nanofocusing zone plates of the type introduced in. 33 The two-dimensional (2d) concentration of the THz radiation can be achieved by using an adiabaticallytapered conical coax waveguide, whose geometry is illustrated in Fig. 4 (a). The central metal wire of radius r is surrounded by a dielectric gap of the radial thickness a, which is enclosed by a thick (∼ 200 nm or thicker) outer metal shell. Both r and a are smooth functions of the longitudinal coordinate y, which describes the tapering of the coax toward the apex (tip) at y = 0. The THz waves propagate from the wide end of the coax toward the apex, adiabatically following the tapering. In the spirit of WKB, for any particular y the wave behavior for the tapered coax is the same as for a cylindrical coax with the values of r and a equal to the local values r(y) and a(y).
The dispersion relation for the coax waveguide that takes into account the plasmonic effects (i.e., the penetration of radiation into the metal and the concurrent losses) is obtained in the Appendix as Eq. (9). Calculated from this expression, the dependence of the modal refractive index n = k(a, r)/k 0 on the dielectric gap a is displayed in Fig. 4 (b) for the frequency of 1 THz, silver as a metal, and vacuum in the dielectric gap. The results are shown for two values of the radius of the central wire: r = 10 μm and r = 60 nm. As one can see, the real part of the modal index practically does not depend on r; it starts growing when a decreases. The imaginary part of the index n increases when the central wire thickness r decreases, but this dependence is very weak. Both Re n and Im n grow dramatically for r l s . This is due to the penetration of the THz field into the metal, i.e., it is a plasmonic effect.
The WKB solution for the radial field E ρ in the cross section of this coax waveguide is shown for the last 3 μm of the propagation toward the tip in Fig. 4 (c). The adiabatic following and energy concentration are evident in this panel. The penetration into the metal of the tangential (to the metal surface) field components E y and H ϕ is noticeable in Figs. 4 (d) and (e). The intensity I of the THz field (relative to the intensity I 0 at the entrance of the waveguide) as a function of the waveguide outer radius R = r + a is shown by the red line in Fig. 4 (f). Dramatically, it shows the adiabatic nanoconcentration and the intensity increase by more than two orders of magnitude for the nanoconcentration from the waveguide radius R = 300 μm, where the THz radiation can be focused, toward R = 300 nm. However, the dramatic increase of the adiabatic parameter δ [plotted by the blue line in Fig. 4 (f)] for R 1 μm shows that these results can only be trusted for R 1 μm.
For the true 2d nanoconcentration of the THz radiation below this micron-scale radius, similar to the 1d case of the wedge, to preserve the adiabaticity, a funnel-like tapering is necessary. Generally, the tapering of the central wire and that of the outer metal shell do not need to be the same. However, we found that better results are obtained when it is the case, i.e., the waveguide is tapered-off self-similarly. In specific calculations, as everywhere in this Letter, we assume that the metal of the waveguide is silver, the dielectric is vacuum, and the frequency is 1 THz. Doing so, we have found the corresponding grading of the waveguide using Eq. (4) and setting δ = 0.05, which is small enough to satisfy the adiabaticity very well. In this case, indeed, we have used the corresponding dispersion relation (9). The obtained shape of the waveguide is a strongly-elongated funnel, as shown in Fig. 5 (a)-(c). These figures display the THz fields that we have calculated in the WKB approximation for this waveguide. As one can see from these figures, within the last half micron of the propagation, the electric and magnetic fields of the THz wave efficiently follow the adiabatically curved waveguide. The penetration into the metal of the tangential (to the metal-dielectric interfaces) field components for y < 400 μm is evident in panels (b) and (c). The longitudinal electric field component E y is significantly localized in the central metal wire [panel(b)], which is a plasmonic effect. The dependence of the THz field intensity in the gap (relative to the intensity I 0 at the entrance of this funnel) on the total radius of the waveguide R is shown in Fig. 5 (d) by the red line. In this case, the adiabatic concentration is very efficient. The intensity of the THz radiation increases by a factor of ×5 when it is compressed from the initial radius of R = 1 μm to the radius R ≈ 250 nm. The penetration of the fields into the metal for smaller values of the radius R (tighter confinement) causes losses that dominate over the effect of the concentration. Again, we remind that this funnel waveguide is a continuation and termination for the straigt cone that yields the field enhancement by ×50 for R = 1 μm [see Fig. 4 (f)]. Consecutively, these two waveguides (the initial cone continued and terminated by the funnel) are very efficient, adiabatically compressing the THz radiation from the initial radius R = 300 μm to the radius R = 250 nm increasing its intensity by a factor ×250. Even for the final radius R = 100 nm, the total THz intensity is increased by a factor of ×10 (which is the products of factors ×50 for the cone part and ×0.2 for the funnel. Thus, the optimally graded plasmonic-metal 2d waveguide is very efficient in the concentration and guidance of the THz fields with the transverse radius of confinement R 100 nm.

CONCLUSION
To discuss the results, we have shown that the THz radiation can be concentrated to the ∼ 100 nm transverse size in adiabatically graded plasmonic (metal/dielectric) waveguides. In the optimum adiabatically-graded, coaxial waveguide, which consists of the initial cone terminated with a funnel, the radiation of a 1 THz frequency whose wavelength is 300 μm, can be compressed to a spot of 250 nm radius, where its intensity increases by a factor of ×250. Even in the case of the extreme compression to a spot of the 100 nm radius, the THz intensity is enhanced by one order of magnitude with respect to the initial intensity of the 300 μm spot at the entrance of the waveguide. The physical process that limits the extent of this spatial concentration is the skin effect, i.e., penetration of the radiation into the metal that causes the losses: the THz field penetrates the depth of l s = 30 − 60 nm of the metal, which by the order of magnitude determines the ultimum localization radius.
The THz nanoconcentration predicted in this Letter for optimally-graded adiabatic plasmonic waveguides provides unique opportunities for THz science and technology, of which we will mention below just a few. The nanoconcentration of the THz radiation will provide the THz ultramicroscopy with a THz source of unprecedented spatial resolution and brightness. The increase of the THz intensity by two orders of magnitude along with the novel high-power THz sources 25 would allow the observation of a wide range of electronic and vibrational nonlinear effects in metal, semiconductors, insulators, and molecules.
These nonlinear THz phenomena can be used to investigate behavior of various materials in ultrastrong fields, for nonlinear spectroscopy (including the multidimensional spectroscopy), and for monitoring and detection of various environmental, biological, and chemical objects and threats such as single bacterial spores and viruses. Such applications will certainly be helped by very large absorption cross sections of various materials in the THz region. A distinct and significant advantage of the adiabatic nanofocusing is that the THz energy is mostly concentrated in the hollow region of the waveguide, whose size can be made comparable with the size of the objects of interest: in the range from 1 micron to 70 nm, which is a typical range for bacteria and their spores, and viruses. This will assure high sensitivity and low background for the objects that are confined inside these waveguides.
Consider as a specific example the spectroscopy or detection of single particles, such as, e.g., anthrax spores, in the air. A sample containing the suspected nanoparticles in a gas, which can be air for the frequencies in the transparency windows, can be pumped through a THz waveguide, and the detection can be made for each particle in the gas separately on the basis of the two-dimensional nonlinear THz spectra that are expected to be highly informative for the detection and elimination of the false-positive alarms. Likewise, many other scientific, technological, environmental, and defense applications may become possible.