1. Introduction

2. Materials and methods

2.1. Experimental setup

To measure the characteristics of THz surface plasmon polaritons (SPPs), we developed and created a plasmon Michelson interferometer, the schematic diagram of which is shown in [37,38], the results of first testing are presented in [39], detailed description of the SPP interferometer, its technical characteristics and measurement technique are presented in the paper [40]. The source of radiation was a unique installation of Budker Institute of Nuclear Physics SB RAS, the Novosibirsk free electron laser (NovoFEL) [41], which generates a periodic sequence of 100-ps pulses with frequency of 5.6 MHz. The NovoFEL radiation is linearly polarized and fully coherent in the beam cross section; the temporal coherence is 30 ÷100 ps (depending on the operating mode of the laser). The radiation wavelength was λ₀ = 141 µm with line width of less than 1%. The average radiation power at the input to the interferometer was 30 ÷40 W, and the diameter of the Gaussian beam was 12 mm.

The use of the interferometer, similarly to the classical Michelson interferometer, relies on analysis of interferogram formed not by bulk waves, but by collinear SPP beams directed by the surface under study. The real part of the SPP refractive index n_s was determined from comparison of the NovoFEL emission spectrum with the SPP spectrum resulting from the Fourier analysis of interferogram.

In the normal mode of NovoFEL operation, the emission spectrum is not stable enough: the wavelength shift can be of up to 0.2 μm during measurements. For the variations in the laser radiation spectrum to be taken into account, the interferogram formed by the NovoFEL radiation beams was recorded simultaneously with the plasmon interferogram. For this purpose, a Michelson interferometer for bulk waves (BWs) [39] was added to the scheme of the plasmon interferometer (Figure 1).

The p-component corresponding to the SPP polarization was extracted from the NovoFEL radiation beam entering the input of the installation (lithographic polarizer P₁ was used). Polypropylene film beam splitter BS₁ split the linearly polarized radiation into two beams. Cylindrical mirror CL focused the transmitted beam on the edge of the flat substrate of the plasmon interferometer, where, due to diffraction, the radiation was converted into SPPs. Mirror M₅ directed the reflected beam, the intensity of which was controlled by polarizer P₂, to the BW interferometer.

Figure 1. Optical circuit (top view) of THz SPP interferometer.

Figure 1. Optical circuit (top view) of THz SPP interferometer.

In the plasmon interferometer, fixed (M₁) and movable (M₂) mirrors were used (40×20×5 mm³ glass plates with gold coating on the reflecting faces), and 40×25×1 mm³ plane-parallel plate BS₂ of Zeonex polyimide, at an angle of 45° to the incident SPP beam, served as a SPP beam splitter [42]. The lower faces of the mirrors and the beam splitter were optically polished and closely adhered to the substrate, providing optical contact with its surface.

At the exit from the plasmon interferometer, combined collinear SPP beams from both arms passed to the convex surface of the cylindrical out-coupling element adjacent to the end of the substrate for conversion of SPPs into BWs. The element was a 1/8 of a cylinder with curvature radius of 60 mm; its convex polished surface contained a 300 nm layer of gold coated with a ZnS layer 1 μm thick. Having reached the opposite edge of the convex face of the element, the resulting SPP beam diffracted on it and was converted into BWs, registered by the radiation detector. The choice of such specific coupling element is due to the need for spatial separation of the BWs generated by the SPPs at the out-coupling edge of this element from the parasitic BWs arising at 1) conversion of the NovoFEL radiation into SPPs (accompanied by the formation of high-intensity diffracted BWs) [43], 2) the diffraction of the SPPs on the splitter and mirrors of the interferometer [44], and 3) the transition of the SPPs from the substrate to the out-coupling element. Moreover, an additional source of parasitic BWs was the scattering [45] of the SPPs on the roughness and optical inhomogeneities of the substrate surface, leading to the appearance of radiative losses of the SPPs [35]. An additional and effective screen from the parasitic BWs was a foam "gate" (35 mm long and 1 mm high) placed on the substrate at the point of its contact with the conversion out-coupling element.

The detection of the interfering BWs generated by the SPP beams on the free edge of the out-coupling element was done by an optoacoustic detector (Golay cell, TYDEX) of high sensitivity (NEP ≈ 1.4 10^-10 W/Hz^1/2 [46]). The need to use a highly sensitive receiver was due to the low signal intensity, large energy losses of the SPPs during their diffraction on the circuit elements, and SPP attenuation during propagation along the sample surface.

In the BW interferometer, the beams lost energy insignificantly (only in the reflection from splitter BS₃). So, a less sensitive single-pixel pyroelectric receiver MG-33 (Novosibirsk Plant of Semiconductor Devices Vostok) was used to record BW interferograms [47,48]. The radiation directed to its 1×1 mm² sensitive element was collected by a TPX lens with focal length of 50 mm.

Since the receivers applied can register only a time-varying radiation flux, a mechanical obturator was placed at the input of the setup, which modulated the radiation intensity with a frequency of 100 Hz. The signals from each receiver were recorded by two synchronous detectors SR-830 (Stanford Research), from the outputs of which the signals were sent to the two-channel digital oscilloscope (Handyscope 3), which operated in the recorder mode. The measured time dependences were digitized and written to a file.

Movable mirrors M₂ and M₄ were attached to the platforms of motorized stages (Standa), which shifted the mirrors along the x axis during scanning with a step of 2.5 μm. The scanning speed chosen was the highest possible (100 steps per second) at which the detectors had time to correctly register the recorded signals. The recording time for one pair of interferograms corresponding to a displacement of the movable mirrors by 30 mm was about 2 minutes.

2.3. Experimental data processing

Before the terahertz measurements, the optical circuits of both interferometers were adjusted with respect to the collimated beam of a diode laser (λ₀ = 635 nm). The adjustment was done with high accuracy, since the difference in the refractive indices of SPPs on the samples, whose ZnS coatings differed in the thickness by 100 nm, was about 10^-4. In addition, after placement of another sample (with a ZnS layer of a different thickness), it was necessary to re-adjust the SPP interferometer circuit, which took a lot of time because of high accuracy maintained. To reduce the adjustment time and take into account the slight misalignment between the interferometers, we carried out the measurements in two stages. First, two interferograms were measured simultaneously (on the plasmon and BW interferometers) in accordance with the scheme described in Section 2.1, when SPPs propagated along the sample surface. At the second stage of measurements, the NovoFEL radiation in the plasmon interferometer was not converted into SPPs, but was directed straight to beam splitter BS₂ to result in an interferogram formed by the source radiation in the environment (air). Simultaneously with that, an interferogram was recorded on the BW interferometer. Thus, with the two stages of measurements, for each sample (differing in the thickness of the ZnS layer), two pairs of interferograms were obtained; moreover, for collection of statistics, the measurements were repeated four times with each of the samples.

Figure 2a shows an example of SPP and BW interferograms. As the movable mirrors of the interferometers move along the x axis, the amplitudes of the sinusoids first increase, then reach the maximum, and decrease in the final section of the scanning. This is a typical symmetric form of the autocorrelation function for a coherent radiation source [49]. The amplitude envelope has a Gaussian profile, the width of which determines the length (duration) of coherence, which was 27 mm (90 ps) in this experiment. This is close to the maximum duration of the NovoFEL radiation pulse at operation in a stable mode (100 ps).

Figure 2. (a) Example of interferograms obtained with plasmon and bulk wave (BW) interferometers for “Au – ZnS layer 470 nm thick” sample. Left column: interferograms recorded with plasmon interferometer (upper graph, with SPP excitation; lower graph, without SPP excitation); right column: respective patterns obtained with BW interferometer. (b) Fourier spectra of SPPs and NovoFEL radiation generating SPPs; circles: reconstruction from interferograms in Figure 2a; lines: approximation of spectra with Gaussian function.

Figure 2. (a) Example of interferograms obtained with plasmon and bulk wave (BW) interferometers for “Au – ZnS layer 470 nm thick” sample. Left column: interferograms recorded with plasmon interferometer (upper graph, with SPP excitation; lower graph, without SPP excitation); right column: respective patterns obtained with BW interferometer. (b) Fourier spectra of SPPs and NovoFEL radiation generating SPPs; circles: reconstruction from interferograms in Figure 2a; lines: approximation of spectra with Gaussian function.

By means of the Fourier transform, the SPP and NovoFEL radiation spectra were calculated from the interferograms (Figure 2b). The circles indicate the results of calculations of the components of the Fourier spectrum. The number of calculation points was determined by the length of the measured interferograms, which depends on the coherence length of the NovoFEL radiation (l_coh ≈ 27 mm). Next, the spectra were approximated with a Gaussian function of the form $f(x)=A\cdot \exp \left[-\frac{1}{2}{\left(\frac{x-x_{\mathrm{c}}}{w}\right)}^2\right]+f_0$ and normalized to the value (А+f₀). The w value defined the width of the spectrum. The value of the parameter x_c, which describes the position of the central line of the spectrum, corresponded to the desired wavelength. The plots in Figure 3 show the wavelengths (central lines of the spectra): λ_spp1 and λ_0spp2, found from interferograms obtained on the plasmon interferometer with and without SPP excitation, respectively; λ₀₁ and λ₀₂, the corresponding wavelengths of the NovoFEL radiation in air, determined with the BW interferometer.

Since the complex refractive index of the SPPs is defined as the ratio of the SPP wave number k_s to the wave number of the radiation from the source in vacuum k₀ = 2π/λ₀,

$${\tilde{n}}_{\mathrm{s}}=\frac{k_{\mathrm{s}}}{k_0}=n_{\mathrm{s}}+i\cdot {\kappa }_{\mathrm{s}},$$

(1)

the real part n_s was found from the spectra shown in Figure 3 by the following formula:

$$n_{\mathrm{s}}=\left(\frac{{\mathsf{\lambda }}_{\mathrm{spp}1}/{\mathsf{\lambda }}_{0\mathrm{spp}2}}{{\mathsf{\lambda }}_{01}/{\mathsf{\lambda }}_{02}}\right)\cdot \mathrm{Re}\left({\tilde{n}}_{\mathrm{a}}\right),$$

(2)

where the normalization to λ₀₁/λ₀₂ takes into account the shift in the NovoFEL generation spectrum that could occur during the time between recordings of interferograms with and without SPP excitation (see Section 2.1). The factor $\mathrm{Re}\left({\tilde{n}}_{\mathrm{a}}\right)$in expression (2) makes it possible to take into account the fact that the laser radiation propagates not in vacuum, but in air with the complex refractive index ${\tilde{n}}_{\mathrm{a}}=1.0002726+i\cdot 0.0000039$ [50,51]. For each sample, the average value of n_s was found from four sets of interferograms.

Figure 3. SPP intensity attenuation on “Au – ZnS layer 470 nm thick” sample at displacement of mirror М₂ (see Figure 1) in plasmon interferometer, fixed mirror М₁ closed by dump: blue line – experimental data, red line – approximation curve.

Figure 3. SPP intensity attenuation on “Au – ZnS layer 470 nm thick” sample at displacement of mirror М₂ (see Figure 1) in plasmon interferometer, fixed mirror М₁ closed by dump: blue line – experimental data, red line – approximation curve.

To find the κ_s value we measured the attenuation of the SPP intensity at displacement of movable mirror M₂ (see Figure 1) in the plasmon interferometer; in this case, fixed mirror M₁ was covered by an absorbing plate. A typical dependence, measured on a sample with a ZnS layer 470 nm thick is shown in Figure 3. The figure also shows the result of approximation of the measurement results with a function of the form $g(x)=B\cdot \mathrm{e}\mathrm{x}\mathrm{p}\left(-2x/L_{\mathrm{s}}\right)+g_0$, where L_s is the SPP propagation length. The exponent takes into account the 2х increase in the SPP path at displacement of the mirror by х. The noise level g₀ was measured in the experiment and was set as an approximation parameter. For each sample, the average SPP propagation length L_av over four measurements was found and the κ_s value was calculated by the formula [11]

$${\mathsf{\kappa }}_{\mathrm{s}}=\frac{{\mathsf{\lambda }}_0}{4\mathsf{\pi }\cdot L_{\mathrm{av}}},$$

(3)

3. Results

Figure 4. (a) Real part of SPP refractive index n_s (without unity) and (b) imaginary part (SPP absorption index) κ_s vs. thickness d of ZnS layer on the gold surface: experiment (blue circles), approximation curves (red line), numerical solution of equation (5) for ε_m = -7000+i·3000 (green squares), and calculation by Drude model for ε_m = -104700+i·317180) (black line).

Figure 4. (a) Real part of SPP refractive index n_s (without unity) and (b) imaginary part (SPP absorption index) κ_s vs. thickness d of ZnS layer on the gold surface: experiment (blue circles), approximation curves (red line), numerical solution of equation (5) for ε_m = -7000+i·3000 (green squares), and calculation by Drude model for ε_m = -104700+i·317180) (black line).

3.1. Determination of the permittivity of metal surface from the SPP characteristics on a set of samples with a dielectric layer of various thicknesses

In this section, we consider an approximation method for finding the metal permittivity, which uses the ${\tilde{n}}_{\mathrm{s}}$ values found for a set of samples with ZnS coatings of various thicknesses.

To determine the effective permittivity ε_m of a metal surface guiding the SPPs and containing a thin-layer coating of thickness d with permittivity ε_d from the found complex refractive index ${\tilde{n}}_{\mathrm{s}}$ of the SPPs, it is necessary to solve (with respect to ε_m) the SPP dispersion equation for the three-layer structure “metal – dielectric layer – environment” [55]:

$$\tanh \left(k_0\cdot d\cdot \sqrt{{\tilde{n}}_{\mathrm{s}}{}^2-{\epsilon }_{\mathrm{d}}}\right)=-\frac{\sqrt{{\tilde{n}}_{\mathrm{s}}{}^2-{\epsilon }_{\mathrm{d}}}\times \left(\frac{\sqrt{{\tilde{n}}_{\mathrm{s}}{}^2-{\epsilon }_{\mathrm{a}}}}{{\epsilon }_{\mathrm{a}}}+\frac{\sqrt{{\tilde{n}}_{\mathrm{s}}{}^2-{\epsilon }_{\mathrm{m}}}}{{\epsilon }_{\mathrm{m}}}\right)}{{\epsilon }_{\mathrm{d}}\cdot \left(\frac{\sqrt{{\tilde{n}}_{\mathrm{s}}{}^2-{\epsilon }_{\mathrm{a}}}\cdot \sqrt{{\tilde{n}}_{\mathrm{s}}{}^2-{\epsilon }_{\mathrm{m}}}}{{\epsilon }_{\mathrm{a}}\cdot {\epsilon }_{\mathrm{m}}}+\frac{{\tilde{n}}_{\mathrm{s}}{}^2-{\epsilon }_{\mathrm{d}}}{{\epsilon }_{\mathrm{d}}{}^2}\right)},$$

(5)

where ε_a is the permittivity of the environment.

Transcendental equation (5) can be solved numerically, but the solution is very sensitive to small variations in ${\tilde{n}}_{\mathrm{s}}$within the measurement error, which can result in a large error in determination of the metal permittivity ε_m. Therefore, we have developed an ε_m determination method based on the approximation of the (n_s-1)(d) and κ_s(d) dependences measured for a series of samples with different ZnS coating thicknesses.

Since in the THz range ε_m >> ε_d, ε_a, the $k_{0}d<<1/\sqrt{{\epsilon }_{\mathrm{m}}}$ condition is met at d<<λ. Expanding both parts of equation (5) into a Taylor series with respect to a small correction to the refractive index ${\tilde{n}}_{\mathrm{s}}$ because of the dielectric coating on the metal, we obtain an approximate formula for calculating the surface impedance of the three-layer structure (see Appendix A):

$$\xi \approx \frac{i\text{×}\left({\epsilon }_{\mathrm{d}}\cdot \sqrt{-\frac{{\epsilon }_{\mathrm{m}}{}^2}{{\epsilon }_{\mathrm{a}}{+\mathsf{\epsilon }}_{\mathrm{m}}}}\times \left({\epsilon }_{\mathrm{a}}+{\epsilon }_{\mathrm{m}}\right)+k_{0}d\cdot {\epsilon }_{\mathrm{m}}\text{×}\left[{\epsilon }_{\mathrm{a}}\left({\epsilon }_{\mathrm{m}}-{\epsilon }_{\mathrm{d}}\right)-{\epsilon }_{\mathrm{m}}{\epsilon }_{\mathrm{d}}\right]\right)}{{\epsilon }_{\mathrm{d}}\cdot \left({\epsilon }_{\mathrm{a}}+{\epsilon }_{\mathrm{m}}\right)\times \left(k_{0}d\cdot {\epsilon }_{\mathrm{d}}\sqrt{-\frac{{\epsilon }_{\mathrm{m}}{}^2}{{\epsilon }_{\mathrm{a}}+{\epsilon }_{\mathrm{m}}}}+{\epsilon }_{\mathrm{m}}\right)},$$

(6)

where

$${\tilde{n}}_{\mathrm{s}}=\sqrt{{\epsilon }_{\mathrm{a}}-{\left({\epsilon }_{\mathrm{a}}\xi \right)}^2}.$$

(7)

The dependences of n_s -1 and κ_s on the ZnS layer thickness d were found by formulas (6) and (7) for the gold permittivity ε_m calculated according to the Drude model (blue lines in Figure 5). In the figure, the red lines show the results of the numerical solution of Eq. (5). It can be seen that at d ≤ 1500 nm, the approximate and exact solutions practically coincide, and at 1500 nm < d ≤ 3000 nm they differ by no more than 8%.

Figure 5. Real part (without unity) n_s and imaginary part κ_s of SPP refractive index in structure “gold – ZnS layer of thickness d – air” vs. d calculated by approximate formulas (6) and (7) (black lines) and by numerical solution of dispersion equation (5) (blew lines). Calculations used gold permittivity ε_m = -104700+i·317180, calculated by Drude model at λ₀ =141 μm.

Figure 5. Real part (without unity) n_s and imaginary part κ_s of SPP refractive index in structure “gold – ZnS layer of thickness d – air” vs. d calculated by approximate formulas (6) and (7) (black lines) and by numerical solution of dispersion equation (5) (blew lines). Calculations used gold permittivity ε_m = -104700+i·317180, calculated by Drude model at λ₀ =141 μm.

Then the experimental dependences (n_s -1)(d ) and κ_s(d) were approximated by means of a Matlab program, using the least squares method to find the minimum deviations of the experimental values from the corresponding functions $\mathrm{Re}\left({\tilde{n}}_{\mathrm{s}}-1\right)$ and $\mathrm{Im}\left({\tilde{n}}_{\mathrm{s}}\right)$ (${\tilde{n}}_{\mathrm{s}}$was found by formula (7)). The real and imaginary parts of the gold permittivity ε_m were varied as parameters. The search for the solution for both functions was performed in the range of ZnS layer thicknesses 300 nm ≤ d ≤ 1670 nm. The domain of the functions did not include small thicknesses (d < 300 nm), when the SPP decay is governed by the radiative losses, not taken into account by Eq. (5); at d = 3000 nm, the approximate formula gives noticeable deviations from the exact solution.

As a result of the approximation, the permeability of the deposited gold was found: the real part was Re(ε_m) ≈ -7000±3000, and the imaginary was Im(ε_m) ≈ 3000±1300. The approximation dependences (n_s-1)(d) and κ_s(d) are shown by the gray lines in Figure 4. The large errors (of up to 40%) in the determination of ε_m by this method are due to the closeness of the refractive index n_s of the THz SPPs to the refractive index of air, which necessitates a higher accuracy in finding the n_s value.

In Figure 4, the green squares show the results for the numerical solution of dispersion equation (5) by the downhill method [56] at ε_m = -7000+ i·3000. It can be seen that the results of the calculation at d ≤ 1700 nm almost coincide with the curves obtained by approximate formula (7), which indicates the possibility of its effective application for evaluation calculations of the complex refractive index of THz SPPs in a three-layer structure in the presence of a dielectric coating layer of subwavelength thickness on the metal.

It was already noted above that the presence of a dielectric layer on the metal surface leads to increase in the fraction of SPP field energy transferred into the metal. In this case, the depth δ_m of the penetration of the SPP field into the metal is practically independent of the coating thickness, as evidenced by the graphs in Figure 6b, obtained from solution of equation (5) for the "gold - ZnS layer of thickness d - air" structure. It can be seen that with the gold permeability found by the approximation method from the experimental results, the δ_m value is approximately six times greater than that for the permeability calculated by the Drude model. Therefore, for the deposited gold layer to be non-transparent for the THz SPP field (which excludes the effect of the substrate on the SPP characteristics), its thickness must be 150–200 nm at least.

From the plot of κ_s(d) in Figure 6a, it can be seen that at small thicknesses d < 300 nm, the loss of SPPs in the experiment is much greater than the calculations predict. This is explained by the presence of radiative losses of SPPs, which are not taken into account either by the approximate formulas or by the dispersion equation; starting from d ≈ 300 nm (where the minimum of the integral losses of SPPs is reached, and their radiative losses become relatively small compared with the Joule losses), the measured values of the absorption coefficient of SPPs approach the calculated dependence.

Figure 6. (a) Ratio of radiative and Joule losses of SPPs, calculated by formula (8) at λ₀ = 141 μm vs. thickness d of the ZnS layer in “deposited gold – ZnS layer – air” structure; (b) penetration depth δ_m (in terms of intensity) of SPP field into metal for "gold - ZnS layer with thickness d - air" structure at λ₀ = 141 µm and gold permeability ε_m calculated according to Drude model (black line) and ε_m = -7000+i·3000, determined by approximation method from characteristics of SPPs (blew line). Dotted lines: corresponding values of skin layer thickness.

Figure 6. (a) Ratio of radiative and Joule losses of SPPs, calculated by formula (8) at λ₀ = 141 μm vs. thickness d of the ZnS layer in “deposited gold – ZnS layer – air” structure; (b) penetration depth δ_m (in terms of intensity) of SPP field into metal for "gold - ZnS layer with thickness d - air" structure at λ₀ = 141 µm and gold permeability ε_m calculated according to Drude model (black line) and ε_m = -7000+i·3000, determined by approximation method from characteristics of SPPs (blew line). Dotted lines: corresponding values of skin layer thickness.

Based on this interpretation of the existence of minimum of the losses κ_s(d) and the assumption that other mechanisms of losses contribute negligibly to the total SPP decay, the SPP radiative losses can be estimated by the following formula:

$${\kappa }_{\mathrm{rad}}={\kappa }_{\mathrm{s}}-{\kappa }_{\mathrm{Joule}},$$

(8)

where κ_s is the total SPP losses measured in the experiment; κ_Joule is the Joule SPP losses determined from the solution of equation (5).

Figure 6a shows the ratio of radiative losses estimated by formula (8) to Joule losses at different thicknesses d of the ZnS layer, calculated from the solution of equation (5) with substitution of ε_m = -7000+i·3000 (found by the approximation method) into it. In this case, in the absence of a dielectric layer on the metal surface, the radiative losses exceed the Joule losses about four-fold. However, as d grows, this ratio decreases monotonically to zero at d ≥ 960 nm, when the SPP decay is determined mainly by losses in the metal.

3.2. Determination of the effective dielectric constant of metal surface from SPP characteristics on one sample

In Section 3.1, we described an estimation of the dielectric permittivity of metal ε_m via application of the approximation method to the measurements of the characteristics of THz SPPs on a set of samples with different thicknesses of dielectric layers on the metal surface. Despite its rather high reliability, this method is very laborious for a rapid assessment of the dielectric constant of metal (metallized) optical surfaces. Therefore, it is desirable (in view of practical applications) to have a method for determination of ε_m from the complex refractive index ${\tilde{n}}_{\mathrm{s}}=n_{\mathrm{s}}+i{\kappa }_{\mathrm{s}}$ of THz SPPs, found from measurements of its real and imaginary parts on only one sample.

From equation (5), we obtained an approximate formula for calculating ε_m from the measured characteristics of SPPs (see Appendix A):

$${\epsilon }_{\mathrm{m}}\approx -\frac{{\epsilon }_{\mathrm{d}}{}^2}{{\left[k_0\cdot d\cdot \left({\epsilon }_{\mathrm{a}}-{\epsilon }_{\mathrm{d}}\right)+i\cdot {\epsilon }_{\mathrm{d}}\cdot \xi \right]}^2},$$

(9)

where the surface impedance ξ is found from formula (6).

As was established above, radiative losses make a predominant contribution to the SPP decay on a metal without a dielectric coating layer, which leads to large errors in the determination of ε_m. Therefore, it is necessary to find a layer thickness d range in which the error in determination of ε_m is not so large. For this purpose, the permittivity ε_m of deposited gold was calculated by formula (9) with substitution of experimentally determined values of ${\tilde{n}}_{\mathrm{s}}$ for samples with different thicknesses d of the ZnS layer into it.

The found values of the real Re(ε_m) and imaginary Im (ε_m) parts of the gold permittivity are shown by black circles in Figure 7. The errors on the graphs correspond to the maximum permittivity deviations calculated in the ranges of experimental values n_s±Δn_s and κ_s±Δκ_s, where Δn_s and Δκ_s are the corresponding measurement errors. It can be seen that the values of Re(ε_m) and Im(ε_m) at small thicknesses of the ZnS layer ( d < 300 nm) differ greatly from the approximation values, and at d ≥ 300 nm, they approach the approximation values asymptotically (except for Im(ε_m) at d = 3000 nm, where already formula (9) gives insufficiently reliable results). This character of the dependence ε_m(d) can be explained by the fact that formula (9) does not take into account the radiative losses of SPPs; at larger thicknesses of the dielectric layer, when the radiative losses becomes much less than the thermal losses, formula (9) describes the experiment much better.

Figure 7. (a) Real and (b) imaginary parts of gold permittivity ε_m vs. thickness d of ZnS layer (blew circles), calculated by formula (9) with ${\tilde{n}}_{\mathrm{s}}(d)$ values found from results of measurements presented in Figure 4. Red dotted lines: values of both parts of gold permeability, found by approximation method; rectangular areas filled with oblique red lines: corresponding standard deviation ranges.

Figure 7. (a) Real and (b) imaginary parts of gold permittivity ε_m vs. thickness d of ZnS layer (blew circles), calculated by formula (9) with ${\tilde{n}}_{\mathrm{s}}(d)$ values found from results of measurements presented in Figure 4. Red dotted lines: values of both parts of gold permeability, found by approximation method; rectangular areas filled with oblique red lines: corresponding standard deviation ranges.

Note that according to the performed calculations, the real part Re(ε_m) of the permittivity of gold without a ZnS coating layer (d = 0) turned out to be equal to – 800 (see Figure 7). This value of Re(ε_m)₎ is in agreement with the estimates given in earlier publications on THz SPP refractometry of deposited gold without a dielectric coating layer, implemented on a TDS spectrometer [31] and with FEL radiation [24]. However, as follows from the above, this value of Re(ε_m) differs from its real value of -7000 almost 10 times. In our opinion, the erroneous estimation of Re(ε_m) in previous works on THz SPP refractometry was because of the unaccounted significant radiative losses of SPPs on a metal surface free without a coating layer.

It is important to note that the obtained errors for metal effective permittivity (see Figure 7) are very large (reach 50%). It is clear that its more reliable determination by the proposed method of SPP refractometry requires higher accuracy of finding the real part of the SPP refractive index from the recorded interferograms. For the accuracy of determination of the metal permittivity to be 20% (for thicknesses d ≥ 500 nm), it is necessary to determine n_s with an error not worse than 10^-4 (with an actual error of 10% for κ_s, which depends on the signal-to-noise ratio, adjustment, and stability of the intensity of NovoFEL radiation. For such accuracy for n_s, the stability of the radiation source in terms of wavelength during recording of interferogram must be not worse than 0.01 µm (for λ₀ = 141 µm), while the width of the generation line must be an order of magnitude smaller than 1.0 ÷1.5 µm in the normal operation mode. This can be achieved via reduction of the measurement time with faster and more sensitive detectors, as well as sources of coherent THz radiation that are more stable than FELs, such as gas lasers [57], backward wave oscillators [58], gyrotrons [59], and quantum-cascade lasers [60].

Note that it is possible to increase the stability of NovoFEL radiation to temporal variations in the radiation wavelength at operation in the mode of negative detuning of the electron bunch repetition rate from the repetition rate of light pulses inside the FEL optical resonator, accompanied by a decrease in the average radiation power [61]. In operation in this mode, a stable single-mode generation regime was established, the linewidth of which reached the maximum possible narrow value (0.25%) [49], while the wavelength shift in 15 minutes was not more than 0.1 of the width of the generated radiation spectrum (0.04 μm at λ₀ =141 µm), which is close to the desired accuracy. However, to achieve such a generation mode, a long and laborious adjustment of the NovoFEL parameters is necessary.

4. Conclusions

References

1. Ordal, M.A.; Long, L.L.; Bell, R.J.; Bell, S.E.; Bell, R.R.; Alexander, R.W.; Ward, C.A. Optical Properties of the Metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the Infrared and Far Infrared. Appl. Opt. 1983, 22, 1099. [Google Scholar] [CrossRef] [PubMed]
2. Pandey, S.; Gupta, B.; Chanana, A.; Nahata, A. Non-Drude like Behaviour of Metals in the Terahertz Spectral Range. Advances in Physics: X 2016, 1, 176–193. [Google Scholar] [CrossRef]
3. Parshin, V.V.; Myasnikova, S.E. Metals Reflectivity at Frequencies 100-360 GHz. In Proceedings of the 2005 Joint 30th International Conference on Infrared and Millimeter Waves and 13th International Conference on Terahertz Electronics; IEEE: Williamsburg, VA, USA, 2005; Vol. 2, pp. 569–570. [Google Scholar]
4. Naftaly, M.; Dudley, R. Terahertz Reflectivities of Metal-Coated Mirrors. Appl. Opt. 2011, 50, 3201. [Google Scholar] [CrossRef]
5. Johnson, P.B.; Christy, R.W. Optical Constants of the Noble Metals. Phys. Rev. B 1972, 6, 4370–4379. [Google Scholar] [CrossRef]
6. Gatesman, A.J.; Giles, R.H.; Waldman, J. High-Precision Reflectometer for Submillimeter Wavelengths. J. Opt. Soc. Am. B 1995, 12, 212. [Google Scholar] [CrossRef]
7. Handbook of optical constants of solids (V.1). / Ed by E.D. Palik. Academic Press, 2016. 824 P.
8. Reuter, G.E.H.; Sondheimer, E.H. Theory of the Anomalous Skin Effect in Metals. Nature 1948, 161, 394–395. [Google Scholar] [CrossRef]
9. Ordal, M.A.; Bell, R.J.; Alexander, R.W.; Long, L.L.; Querry, M.R. Optical Properties of Au, Ni, and Pb at Submillimeter Wavelengths. Appl. Opt. 1987, 26, 744. [Google Scholar] [CrossRef]
10. Brändli, G.; Sievers, A.J. Absolute Measurement of the Far-Infrared Surface Resistance of Pb. Phys. Rev. B 1972, 5, 3550–3557. [Google Scholar] [CrossRef]
11. Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces; Agranovich, V.M.; Mills, D.L. (Eds.) Modern problems in condensed matter sciences; North-Holland Pub. Co. : Sole distributors for the USA and Canada, Elsevier Science Pub. Co: Amsterdam; New York, 1982; ISBN 978-0-444-86165-8. [Google Scholar]
12. Maier, S.A. Plasmonics: The Promise of Highly Integrated Optical Devices. IEEE J. Select. Topics Quantum Electron. 2006, 12, 1671–1677. [Google Scholar] [CrossRef]
13. Zhu, W.; Agrawal, A.; Nahata, A. Planar Plasmonic Terahertz Guided-Wave Devices. Opt. Express 2008, 16, 6216. [Google Scholar] [CrossRef]
14. Zhang, X.; Xu, Q.; Xia, L.; Li, Y.; Gu, J.; Tian, Z.; Ouyang, C.; Han, J.; Zhang, W. Terahertz Surface Plasmonic Waves: A Review. Adv. Photon. 2020, 2, 1. [Google Scholar] [CrossRef]
15. Pendry, J.B.; Martín-Moreno, L.; Garcia-Vidal, F.J. Mimicking Surface Plasmons with Structured Surfaces. Science 2004, 305, 847–848. [Google Scholar] [CrossRef] [PubMed]
16. Maier, S.A.; Andrews, S.R. Terahertz Pulse Propagation Using Plasmon-Polariton-like Surface Modes on Structured Conductive Surfaces. Appl. Phys. Lett. 2006, 88, 251120. [Google Scholar] [CrossRef]
17. Williams, C.R.; Andrews, S.R.; Maier, S.A.; Fernández-Domínguez, A.I.; Martín-Moreno, L.; García-Vidal, F.J. Highly Confined Guiding of Terahertz Surface Plasmon Polaritons on Structured Metal Surfaces. Nature Photon 2008, 2, 175–179. [Google Scholar] [CrossRef]
18. Kretschmann, E. Die Bestimmung optischer Konstanten von Metallen durch Anregung von Oberflächenplasmaschwingungen. Z. Physik 1971, 241, 313–324. [Google Scholar] [CrossRef]
19. Kitajima, H.; Hieda, K.; Suematsu, Y. Use of a Total Absorption ATR Method to Measure Complex Refractive Indices of Metal-Foils. J. Opt. Soc. Am. 1980, 70, 1507. [Google Scholar] [CrossRef]
20. Regalado, L.E.; Machorro, R.; Siqueiros, J.M. Attenuated-Total-Reflection Technique for the Determination of Optical Constants. Appl. Opt. 1991, 30, 3176. [Google Scholar] [CrossRef]
21. Owner-Petersen, M.; Zhu, B.-S.; Dalsgaard, E. Extreme Attenuation of Total Internal Reflection Used for Determination of Optical Properties of Metals. J. Opt. Soc. Am. A 1987, 4, 1741. [Google Scholar] [CrossRef]
22. Lafait, J.; Abeles, F.; Theye, M.L.; Vuye, G. Determination of the Infrared Optical Constants of Highly Reflecting Materials by Means of Surface Plasmon Excitation-Application to Pd. J. Phys. F: Met. Phys. 1978, 8, 1597–1606. [Google Scholar] [CrossRef]
23. Zhizhin, G.N.; Morozov, N.N.; Moskalova, M.A.; Sigarov, A.A.; Shomina, E.V.; Yakovlev, V.A.; Grigos, V.I. Surface Electromagnetic Wave Absorption on Copper Surfaces with Langmuir Films Using Co2 Laser Excitation. Thin Solid Films 1980, 70, 163–168. [Google Scholar] [CrossRef]
24. Gerasimov, V.V.; Knyazev, B.A.; Nikitin, A.K.; Zhizhin, G.N. A Way to Determine the Permittivity of Metallized Surfaces at Terahertz Frequencies. Appl. Phys. Lett. 2011, 98, 171912. [Google Scholar] [CrossRef]
25. Auston, D.H.; Cheung, K.P. Coherent Time-Domain Far-Infrared Spectroscopy. J. Opt. Soc. Am. B 1985, 2, 606. [Google Scholar] [CrossRef]
26. Grischkowsky, D.; Keiding, S.; Van Exter, M.; Fattinger, Ch. FarInfrared Time-Domain Spectroscopy with Terahertz Beams of Dielectrics and Semiconductors. J. Opt. Soc. Am. B 1990, 7, 2006. [Google Scholar] [CrossRef]
27. Zhou, D.; Parrott, E.P.J.; Paul, D.J.; Zeitler, J.A. Determination of Complex Refractive Index of Thin Metal Films from Terahertz Time-Domain Spectroscopy. Journal of Applied Physics 2008, 104, 053110. [Google Scholar] [CrossRef]
28. Yasuda, H.; Hosako, I. Measurement of Terahertz Refractive Index of Metal with Terahertz Time-Domain Spectroscopy. Jpn. J. Appl. Phys. 2008, 47, 1632–1634. [Google Scholar] [CrossRef]
29. Han, P.Y.; Tani, M.; Usami, M.; Kono, S.; Kersting, R.; Zhang, X.-C. A Direct Comparison between Terahertz Time-Domain Spectroscopy and Far-Infrared Fourier Transform Spectroscopy. Journal of Applied Physics 2001, 89, 2357–2359. [Google Scholar] [CrossRef]
30. Isaac, T.H.; Barnes, W.L.; Hendry, E. Determining the Terahertz Optical Properties of Subwavelength Films Using Semiconductor Surface Plasmons. Appl. Phys. Lett. 2008, 93, 241115. [Google Scholar] [CrossRef]
31. Pandey, S.; Liu, S.; Gupta, B.; Nahata, A. Self-Referenced Measurements of the Dielectric Properties of Metals Using Terahertz Time-Domain Spectroscopy via the Excitation of Surface Plasmon-Polaritons. Photon. Res. 2013, 1, 148. [Google Scholar] [CrossRef]
32. Nazarov, M.M.; Shkurinov, A.P.; Garet, F.; Coutaz, J.-L. Characterization of Highly Doped Si Through the Excitation of THz Surface Plasmons. IEEE Trans. THz Sci. Technol. 2015, 5, 680–686. [Google Scholar] [CrossRef]
33. Kapitulnik, A.; Deutscher, G. Percolation Scale Effects in Metal-Insulator Thin Films. J Stat Phys 1984, 36, 815–826. [Google Scholar] [CrossRef]
34. Tomilina, O.A.; Berzhansky, V.N.; Tomilin, S.V. The Influence of the Percolation Transition on the Electric Conductive and Optical Properties of Ultrathin Metallic Films. Phys. Solid State 2020, 62, 700–707. [Google Scholar] [CrossRef]
35. Gerasimov, V.V.; Knyazev, B.A.; Lemzyakov, A.G.; Nikitin, A.K.; Zhizhin, G.N. Growth of Terahertz Surface Plasmon Propagation Length Due to Thin-Layer Dielectric Coating. J. Opt. Soc. Am. B 2016, 33, 2196. [Google Scholar] [CrossRef]
36. Kumar, M.; Porsezian, K. A Comparative Study of Surface Plasmon Polariton Propagation Characteristics of Various Metals.; Uttar Pradesh, India, 2016; p. 080080.
37. Nikitin, A.K.; Khitrov, O.V.; Gerasimov, V.V.; Khasanov, I.S.; Ryzhova, T.A. In-Plane Interferometry of Terahertz Surface Plasmon Polaritons. J. Phys.: Conf. Ser. 2019, 1421, 012013. [Google Scholar] [CrossRef]
38. Nikitin, A.K.; Khitrov, O.V.; Gerasimov, V.V. Quality Control of Solid Surfaces by the Method of Surface Plasmon Interferometry in the Terahertz Range. J. Phys.: Conf. Ser. 2020, 1636, 012037. [Google Scholar] [CrossRef]
39. Gerasimov, V.V.; Nikitin, A.K.; Khitrov, O.V.; Lemzyakov, A.G. Experimental Demonstration of Surface Plasmon Michelson Interferometer at the Novosibirsk Terahertz Free-Electron Laser. In Proceedings of the 2021 46th International Conference on Infrared, Millimeter and Terahertz Waves (IRMMW-THz); IEEE: Chengdu, China, August 29 2021; pp. 1–2. [Google Scholar] [CrossRef]
40. Gerasimov, V. V.; Nikitin A., K.; Lemzyakov, A. G. Planar Michelson interferometer using terahertz surface plasmons. Instruments and Experimental Techniques 2023, 66, 423–434. [Google Scholar] [CrossRef]
41. Shevchenko, O.A.; Vinokurov, N.A.; Arbuzov, V.S.; Chernov, K.N.; Davidyuk, I.V.; Deichuly, O.I.; Dementyev, E.N.; Dovzhenko, B.A.; Getmanov, Ya.V.; Gorbachev, Ya.I.; et al. The Novosibirsk Free-Electron Laser Facility. Bull. Russ. Acad. Sci. Phys. 2019, 83, 228–231. [Google Scholar] [CrossRef]
42. Islam, M.S.; Cordeiro, C.M.B.; Nine, M.J.; Sultana, J.; Cruz, A.L.S.; Dinovitser, A.; Ng, B.W.-H.; Ebendorff-Heidepriem, H.; Losic, D.; Abbott, D. Experimental Study on Glass and Polymers: Determining the Optimal Material for Potential Use in Terahertz Technology. IEEE Access 2020, 8, 97204–97214. [Google Scholar] [CrossRef]
43. Gerasimov, V.V. , Knyazev, B.A., and Nikitin, A.K. Method for indication of diffraction satellites of surface plasmons in the terahertz range. Pis’ma v ZhTF 2010, 36, no–21. [Google Scholar]
44. Gerasimov, V.V.; Nikitin, A.K.; Lemzyakov, A.G.; Azarov, I.A.; Milekhin, I.A.; Knyazev, B.A.; Bezus, E.A.; Kadomina, E.A.; Doskolovich, L.L. Splitting a Terahertz Surface Plasmon Polariton Beam Using Kapton Film. J. Opt. Soc. Am. B 2020, 37, 1461. [Google Scholar] [CrossRef]
45. Zayats, A.V.; Smolyaninov, I.I.; Maradudin, A.A. Nano-Optics of Surface Plasmon Polaritons. Physics Reports 2005, 408, 131–314. [Google Scholar] [CrossRef]
46. http://www.tydexoptics.com/ru/products/thz_devices/golay_cell/.
47. http://www.nzpp.ru/product/gotovye-izdeli/fotopriemnye-ustroystva/.
48. Paulish, A.G.; Dorozhkin, K.V.; Suslyaev, V.I. Investigation of the spectral characteristics of the sensitivity of a pyroelectric detector based on tetraaminodiphenyl in the terahertz range. In Actual problems of radiophysics: Proceedings of the conference, Tomsk, 2019; pp. 482–485. [Google Scholar]
49. Kubarev, V.V.; Kulipanov, G.N.; Kolobanov, E.I.; Matveenko, A.N.; Medvedev, L.E.; Ovchar, V.K.; Salikova, T.V.; Scheglov, M.A.; Serednyakov, S.S.; Vinokurov, N.A. Modulation Instability, Three Mode Regimes and Harmonic Generation at the Novosibirsk Terahertz Free Electron Laser. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 2009, 603, 25–27. [Google Scholar] [CrossRef]
50. Fizičeskie veličiny: spravočnik; Grigorʹev, I.S. (Ed.) Ėnergoatomizdat: Moskva, 1991; ISBN 978-5-283-04013-4. [Google Scholar]
51. Mathar, R.J. Refractive Index of Humid Air in the Infrared: Model Fits. J. Opt. A: Pure Appl. Opt. 2007, 9, 470–476. [Google Scholar] [CrossRef]
52. E. D. Palik Handbook of Optical Constants of Solids. Academic Press, 1998.
53. Ordal, M.A.; Long, L.L.; Bell, R.J.; Bell, S.E.; Bell, R.R.; Alexander, R.W.; Ward, C.A. Optical Properties of the Metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the Infrared and Far Infrared. Appl. Opt. 1983, 22, 1099. [Google Scholar] [CrossRef] [PubMed]
54. Wang, K.; Mittleman, D.M. Dispersion of Surface Plasmon Polaritons on Metal Wires in the Terahertz Frequency Range. Phys. Rev. Lett. 2006, 96, 157401. [Google Scholar] [CrossRef]
55. Burke, J.J.; Stegeman, G.I.; Tamir, T. Surface-Polariton-like Waves Guided by Thin, Lossy Metal Films. Phys. Rev. B 1986, 33, 5186–5201. [Google Scholar] [CrossRef]
56. Bach, H. On the Downhill Method. Commun. ACM 1969, 12, 675–677. [Google Scholar] [CrossRef]
57. Zhi-Xian, J.; Du-Luo, Z.; Liang, M.; Chun-Chao, Q.; Zu-Hai, C. An Efficient Pulsed CH ₃ OH Terahertz Laser Pumped by a TEA CO ₂ Laser. Chinese Phys. Lett. 2010, 27, 024211. [Google Scholar] [CrossRef]
58. Kozlov, G.; Volkov, A. Coherent Source Submillimeter Wave Spectroscopy. In Millimeter and Submillimeter Wave Spectroscopy of Solids; Grüner, G., Ed.; Topics in Applied Physics; Springer: Berlin Heidelberg, 1998; Vol. 74, pp. 51–109. ISBN 978-3-540-62860-6. [Google Scholar]
59. Idehara, T.; Sabchevski, S.P.; Glyavin, M.; Mitsudo, S. The Gyrotrons as Promising Radiation Sources for THz Sensing and Imaging. Applied Sciences 2020, 10, 980. [Google Scholar] [CrossRef]
60. Wen, B.; Ban, D. High-Temperature Terahertz Quantum Cascade Lasers. Progress in Quantum Electronics 2021, 80, 100363. [Google Scholar] [CrossRef]
61. Kubarev, V.V. Optical systems, diagnostics, and experiments on terahertz and infrared free electron lasers: Dissertation for the degree of doctor of fiz.-mat. sciences, Novosibirsk 2016, 321 pp.
62. Hinderks, L.W.; Maione, A. Copper Conductivity at Millimeter-Wave Frequencies. Bell System Technical Journal 1980, 59, 43–65. [Google Scholar] [CrossRef]
63. Li, Y.; Tantiwanichapan, K.; Swan, A.K.; Paiella, R. Graphene Plasmonic Devices for Terahertz Optoelectronics. Nanophotonics 2020, 9, 1901–1920. [Google Scholar] [CrossRef]
64. Kotelnikov, I.A.; Gerasimov, V.V.; Knyazev, B.A. Diffraction of a Surface Wave on a Conducting Rectangular Wedge. Phys. Rev. A 2013, 87, 023828. [Google Scholar] [CrossRef]