Terahertz field enhancement to the MV/cm regime in a tapered parallel plate waveguide.

We investigate field enhancement properties of a tapered parallel plate waveguide for ultrashort terahertz (THz) pulses. We use two independent methods, air biased coherent detection inside the waveguide and free-space electro-optic sampling, respectively, which enables a calibrated, quantitative measurement of the field strength at the output of the waveguide. Field enhancement factors greater than 20 are demonstrated and record-high field strengths of > 1.4 MV/cm are reached. We find an excellent agreement between the two independent methods of field measurement and a numerical 3D full-vectorial time-domain simulations.


Introduction
Terahertz (THz) radiation allows for analysis of various properties of materials, including for example electronic states in semiconductors, dielectric properties of glasses and plastics, and vibrational modes of crystals [1,2]. In some cases, sensing applications require detection of small amounts of substances, and thus a tight confinement of the radiation to the volume of the sample would be advantageous. In classical free-space wave optics the confinement of the electromagnetic wave is lower-limited by diffraction effects to approximately λ /2. It has been recently shown that this limitation can be overcome by use of plasmonic structures and metamaterials [3][4][5][6][7][8]. A significant effort in achieving deep subwavelength confinement has been focused on the use of tapered waveguides [9][10][11][12][13][14][15]. Zhan et al. [13] demonstrated that a tapered parallel plate waveguide (TPPWG) supports superfocusing of THz waves below λ /250. In this scheme the wave confinement is limited by the skin depth of radiation in the metal [15] and beam spots as small as 60 nm were demonstrated for tapered transmission lines illuminated with 10 μm radiation [16]. Tapering leads to compression of the electromagnetic energy carried by the electromagnetic wave, and in this way opens doors for significant field enhancement. In this paper we investigate the field enhancement properties of a TPPWG. The paper is organized in the following way. In section 2 we describe the experimental setup and we give detailed information about the waveguide. In section 3 we develop a simple analytical approach to the estimation of the THz electric field enhancement inside a TPPWG. A full-wave numerical time-domain simulation of THz wave propagation is compared with the analytical estimation. Next two independent experimental methods, free-space electro-optic sampling (FSEOS) and air biased coherent detection (ABCD), are then used to measure the THz electric field at the output of the waveguide. The two experimental methods are compared with each other and with the numerical simulations.  Figure 1 shows the schematics of the experimental setup. The TPPWG consists of two electrically isolated, fine polished aluminum plates of varying width (input 3 mm, output 49 μm) and adjustable plate separation. The plate separation B in at the input facet of the waveguide is set to 1 mm, while the output plate separation B out can be varied between 0 and 500 μm by a micrometric stage. Aluminum was chosen as material for the waveguide plates because of its combination of high electrical conductivity and mechanical hardness, needed for low propagation loss and for the manufacturing of narrow and well-defined tips. The total length of the TPPWG is 25.4 mm. Strong broadband THz pulses are generated by tilted pulsefront optical rectification in LiNbO 3 [18] using near-infrared (NIR) 0.6 mJ pump pulse from a 1 kHz, 90 fs, 800 nm regenerative Ti:sapphire femtosecond laser amplifier (Spectra-Physics Hurricane). The intensity front of fs pulses is tilted by a 2000 line/mm holographic grating and imaged by a 63 mm-focal length lens with demagnification factor of 2 onto the front surface of a stoichiometric LiNbO 3 crystal at room temperature. THz pulse energies up to 0.5 μJ were measured directly after the LiNbO 3 with a calibrated pyroelectric detector. A pair of 50.8 mm-diameter, 101.6 mm and 76.2 mm focal length off-axis parabolic mirrors is used to collimate and focus the THz radiation at the input of the TPPWG. A pyroelectric detector with a 0.25 mm-diameter aperture mounted on an x-y motorized translation stage was used to image the THz field intensity distribution at the input plane of the waveguide. The THz beam has an elliptical shape, as shown in Fig. 1(c), with intensity FWHM of 1.6 mm along the x-direction and 0.7 mm along the y-direction. The THz wave couples into the TPPWG, propagates between the aluminium plates in the fundamental TEM mode and then couples out into free space at the narrow exit facet of the waveguide. Attenuated NIR probe pulses from the same laser amplifier are focused using a 50 mm-focal length lens onto a 300 μm-thick [110] gallium phosphide (GaP) crystal. The NIR probe enters the GaP crystal and is reflected from the back surface of the electro-optic crystal and then copropagates with the THz pulse. The beam waist at the focus point has been estimated to be approximately 20 μm. The lens is mounted on a micrometric x-y-z translation stage for accurate alignment of the focus spot on the GaP crystal. Due to geometrical restrictions the distance between GaP crystal and waveguide output cannot be shorter than 300 μm. The THz electric field induces a phase retardation between polarization components of the NIR probe. This phase retardation is measured by the balanced detection. The same setup was used to detect THz field at the input plane of the waveguide. Figure 2 shows the output intensity distributions of the THz electric field measured using a pyroelectric detector placed 0.8 mm away from the waveguide tip for output gaps of 0 μm, 20 μm and 200 μm with the input gap B in = 1 mm. For the closed waveguide no THz field is transmitted directly through the output. The blurred halo observed around the center of the scan is THz waves that have leaked around the waveguide tip. The vertical lines visible in the picture are THz waves that propagate on the outside vertical walls of the waveguide. For the open gaps the major part of the THz wave propagates through the gap and dominates the pictures. These strong bursts of THz radiation, even for output gaps that are more than one order of magnitude smaller than the THz wavelength, indicate that high THz intensities are focused well below the diffraction limit.

Analytical and numerical field estimation
First we present a simple analytical estimation of the field enhancement at the output of the TP-PWG. In the case of illumination of the waveguide with a Gaussian elliptical beam (as depicted in Fig. 1(c)), the THz electric field E (z) along the waveguide can be expressed by where E 0 is the peak electric field of the Gaussian input mode at the entrance of the waveguide, FE (z) is the field enhancement, and C is the coupling constant between the Gaussian input excitation beam E G (x, y) and the uniform field distribution E U (x, y) of the TEM eigenmode of the TPPWG at the input of the waveguide. The coupling constant C can be calculated using the mode overlap integral [19]: For the elliptical Gaussian beam with intensity FWHM of 1.6 mm and 0.7 mm and a uniform TEM mode, the coupling constant C has been calculated to be 0.877.
The field enhancement FE (z) along the waveguide is determined by two main factors: energy squeezing due to geometrical wave confinement and propagation losses due to finite conductivity of the metal. Assuming, for simplicity, the absence of radiation losses and that the uniform TEM mode profile is sustained during wave propagation, and that the whole electric field is confined within the volume between waveguide plates, we can write that and B (z) are the electric field, cross-sectional area of the waveguide, impedance of the waveguide, its width and plate separation, respectively. By neglecting changes of impedance along the waveguide (which is reasonable since Z in ≈ 126 Ω and Z out ≈ 153 Ω for B out = 20 μm, calculated using Z W G = 120πb/w [20]) and by taking the logarithm of the last equation, we get Differentiating both sides we derive a differential equation for the THz electric field squeezing: In the case of our tapered parallel plate waveguide the width W and plate separation B are linear functions of position along the waveguide where W in , B in , W out , B out are waveguide width and plate separation at the input and the output of the waveguide. L WG is the total length of the waveguide. Also the decrease of the THz field due to Ohmic losses can be described by a differential equation: where α is the attenuation constant for the TEM mode and can be expressed by [21] R s = π f μ/σ is the waveguide surface resistivity, n is the refractive index of the material filling the waveguide, Z 0 is the free space impedance and σ is the electrical conductivity of the metal. The final equation describing the field evolution in this simple model is a sum of Eq. (4) and Eq. (7).
Integrating this equation with the width and plate separation coordinate dependence given by Eq. (5) and Eq. (6), and inserting the solution into Eq. (1), we get an expression for the electric field along the waveguide, The term FS = W in B in W (z)B(z) describes the field squeezing due to geometrical restrictions, while the second term, which we will call field reduction FR, is attributed to the conduction losses inside the metal.  Figure 3(a) shows the calculated field reduction FR at the output of a TPPWG as a function of the output gap B out for the field frequencies of 0.5, 1.0 and 2.0 THz. It can be seen in the figure that the field reduction depends strongly on the output plate separation B out , in a way that the larger the plate separation the smaller the losses. Hence the attenuation coefficient α increases with the frequency of the propagating THz field as a √ f , therefore the field reduction FR is also frequency dependent. Figure 3(b) shows the field enhancement FE = E (L) /E 0 at the output of the TPPWG, calculated using Eq. (10). The field enhancement increases with decreasing output plate separation. For B out smaller than 20 μm values of FE above 45 are achieved. The presented calculations account neither for diffraction losses nor reflection from the non-perfect impedance matching, and therefore the actual values of FE will be lower. The derivation of Eq. (10) also does not take into account that the mode profile of the guided wave may depend on plate width and separation.
The time-domain numerical simulations are performed with CST Microwave Studio for a full size TPPWG made of aluminium treated as a lossy metal. The TPPWG is meshed with 40 · 10 6 mesh cells. Perfectly matched layer (PML) boundary conditions are used. Simulations are performed on a 12-core processor workstation with 48 GB of memory and each requires ∼ 4.5 hours of computing time. The THz beam at the input to the waveguide has elliptical Gaussian shape, with intensity FWHM of 1.6 mm along x-direction and 0.7 mm along y-direction, as determined in our experiments. The THz time waveform measured at the input faced of the waveguide was used for simulation. Figure 4 shows the peak electric field at the central point between the metal plates along the TPPWG for output gaps B out of 20, 40, 60, 100, 140 and 200 μm. The electric field is normalized to the maximum of the Gaussian field at the input to the waveguide. The input of the waveguide is at z = −25.4 mm and the output is at z = 0.0 mm. During the first 7 mm of the waveguide the THz peak electric field oscillates. Those changes are likely caused by the mode conversion between the free-space Gaussian mode and the TEM mode of finite-width waveguide. In the last 18 mm of the TPPWG we observe a gradual increase of the peak THz electric field. A field enhancement higher than a factor of 20 is achieved for the output gap of 20 μm. As the THz wave arrives at the output facet of the TPPWG part of it is reflected due to the impedance mismatch between the waveguide and free space. The interference between the incident and reflected wave causes oscillations of the peak THz electric field observed in vicinity of the waveguide output [22]. At z = 0 mm THz field couples out of the TPPWG, and for z > 0 mm the peak electric field decreases due to strong diffraction. The smaller the output gap the bigger the diffraction angle in the vertical direction and therefore the faster the decrease of the peak electric field. Also the reflection coefficient is larger for smaller gaps, so less energy couples out when B out is narrow. Figure 5 shows the calculated cross-sectional distributions of the y-component of the peak THz electric field along the TPPWG for few chosen positions z. The field at the input (z = −25.1 mm) of the waveguide has an elliptical gaussian distribution. The general trend visible from the presented field distributions is that the major part of the energy of the THz field is confined within the area between metal plates. The confinement is tighter for smaller plate separations. At the position of z = −9.1 mm the field has a distribution more resembling the uniform field distribution of the TEM mode, except in the proximity of the edges of the waveguide, where it is enhanced. This effect has previously been reported by Zhan et al. [13], and can be explained by the presence of propagating edge plasmons. Once the THz transient couples out of the TPPWG the THz beam rapidly diffracts. In Fig. 5(j) we see a blurred halo around the center of the image and the vertical lines that originate from the THz waves that propagate on the outside vertical walls of the waveguide, in close agreement with the measurements shown in Figs. 2(a) and 2(b).  Figure 6 shows the ratio between the THz field enhancement simulated using CST Microwave Studio (FE sim ) and calculated analytically (FE an ) using Eq. (10), at the output of the TPPWG for different sizes of the output gap B out . The main difference between FE sim and FE an is that the analytical quantity does not account for radiation losses, so FE sim FE an is a good representation of radiation losses during propagation along the TPPWG. It is clear from the Fig. 5 that diffraction losses increase with increasing output gap separation. This effect can be explained using the contrast between the effective waveguide impedance Z W G and the free space impedance Z 0 . Narrowing the plate separation increases the effective waveguide impedance Z W G . Higher contrast between Z W G and Z 0 leads to higher reflection coefficient in the x-direction, which means that the THz wave is tighter confined within the waveguide. This effect is analogous to solid core optical fibers, where high contrast in refractive index gives tight confinement of the optical wave. An exponential function fits the data points in Fig. 6 accurately, with parameters shown in the figure. This observation is in agreement with the work of Zhan et al. [13], who have shown experimentally that the field confinement in the area between the plates of a PPWG decreases exponentially with increasing plate separation.

THz electric field calibration
To experimentally quantify the predicted THz field enhancement in the TPPWG, two independent methods are used. The first method is to detect THz radiation using free space electro-optic (EO) sampling [23-25] outside the TPPWG, as shown in Fig. 1(b), and then to use outcoupling coefficients, delivered by numerical simulation, to compute the field strengths at the output of the waveguide. In the EO detection the THz electric field was calculated using the equation E T Hz = ΔI   presented in the inset to Fig. 7(a). Outcoupling coefficients are calculated by considering the average of the peak THz electric field over the length of the GaP crystal divided by the peak value of the field at the tip of the waveguide. Figure 7(b) shows the amplitude spectra of the detected THz radiation obtained from a Fourier transform. The amplitude spectrum extends from 0.1 THz up to 2.5 THz. Dips in the spectra are due to absorption of THz radiation by water vapor in the air.  The second method for calibrating the electric field strength of the THz wave at the output of the TPPWG is based on the air-photonic measurement the THz induced second harmonic (SH) intensity [22]. Figure 8(a) shows the simplified schematics of the air photonic setup for THz field measurement inside a tapered parallel plate waveguide. A high voltage source is connected to the waveguide plates and can deliver an electric bias field between the metal plates. A NIR beam from the same laser amplifier as used for THz generation is used for probing the THz field. The probe beam is focused in the center of the volume between the plates with a 19 mm-focal length lens and recollimated after the waveguide. The beam is sent through 400 nm bandpass filters and then to a photomultiplier tube (PMT) which detects the 400-nm light generated in the nonlinear process. The signal from the PMT is measured by a lock-in amplifier referenced to an optical chopper modulating the THz beam. The polarizations of the four interacting electric fields (terahertz, fundamental, second harmonic and bias) are perpendicular to the waveguide plates. Further experimental details concerning setup are presented elsewhere [22]. According to Karpowicz et al. [27] the interaction between the laser NIR pulse (intensity I ω ) and the THz electric field E T Hz in the presence of the external electric field E bias leads to generation of a second harmonic beam with intensity I 2ω of where χ xxxx is the third-order nonlinear susceptibility of the air. If an optical chopper is modulating the THz beam, a lock-in amplifier referenced to the chopper frequency detects following SH intensity If two measurements of SH intensity (I 0 2ω without and I bias 2ω with known external DC bias) are performed, then the absolute amplitude and sign of the THz electric field can be determined: where I mod is the relative modulation depth of the generated second harmonic. Figure 8(b) shows measured intensity of the time-dependent THz induced SH without and with external 10 kV/cm bias at the output of TPPWG with an output gap of 100 μm. Because of the bipolar shape of the single-cycle THz transient [ Fig. 7(a)] of the THz transient one peak of the induced SH decreases in the presence of the external bias field while the other one increases. Using the method outlined above, the peak of the THz electric field has been measured to be 295 kV/cm. Figure 9 presents a summary of our measurements of the peak THz field at the output of the TPPWG for different output gaps B out . Black squares are the results of the direct measurement using THz-induced second harmonic generation. Unfortunately, due to the high intensity of the probe beam (though still lower than the plasma ionization threshold), measurements of the THz field in this way for B out smaller than 60 μm resulted in a high chance of damaging the waveguide, so values are presented only for B out ≥ 60μm. The red circles represent values of the peak THz electric field measured using electro-optic detection in GaP crystal corrected by outcoupling coefficients. The blue dashed line shows the peak values of the THz field from CST Microwave Studio simulations, using the previously described elliptical beam with the experimentally determined value of the peak THz field of 68 kV/cm as input. All the three values of the peak electric field at the tip of the TPPWG agree well with each other over a wide range of B out . The small differences can have various origins. For the THz field obtained by air-photonic method they can be caused by non-uniform distribution of the bias field within the THz beam. Deviations can also come from the fact that local field enhancement at the edges of the waveguide were probed. The uncertainty of the electro-optic values are increasing with decreasing the output gap size, since the outcoupling coefficients are smaller. Error bars in the Fig. 9 are dominated by the uncertainty of the distance between the waveguide tip and GaP crystal. In any case the good agreement between different measurements indicate that very high values of the THz field at the tip of the waveguide were achieved. For the output gap of 20 μm we estimated a value of over 1.4 MV/cm. Even higher values are expected for smaller output gaps.

Conclusions
We have investigated field enhancement properties of the TPPWG and we have showed that a field enhancement of over 20 is possible, resulting in the THz peak electric field exceeding 1.4 MV/cm for output gaps of 20 μm. The field values have been measured using two methods: free-space electro-optic sampling and THz enhanced second harmonic generation. The obtained values agree with each other and also with predictions from numerical simulations. Even higher values of the peak THz field are expected for smaller output gaps. The tight concentration of high intensity THz radiation can be potentially applied in exploration of nonlinear THz phenomena, deep-subwavelength imaging, and allow for development of compact optoelectronic devices.