1. Introduction

Wave plates are one of the most fundamental passive optics in current advanced photonics. They are made of quartz and MgF₂ crystals for the near-infrared to the ultraviolet frequency region, and of transparent birefringent semiconductors for the mid-infrared frequency region [1]. In the far-infrared frequency region, a multi-order wave plate based on a single quartz plate has been used in conventional frequency domain spectroscopy, which is available at discrete frequencies. However, the ultrashort optical pulse technique allows THz time-domain spectroscopy to be performed [2], which requires a new method for polarization control. Because few-cycle THz pulses are coherently detected, polarization optics effective over a wide frequency region are required. There are many potential applications of polarization control and analysis. There are several types of achromatic wave plates, including wave plates based on Fresnel reflectors [3,4], form birefringence gratings [5–8], achromatic wave plates based on a stack of several quartz plates [9], and metamaterials [10,11].

Recently, we proposed an achromatic wave plate in the THz frequency region based on parallel metal plate waveguides (PPWGs) with a through-hole array [12]. When the electromagnetic wave is incident with a polarization parallel to the metal plates (θ = 0°), the light propagates in a transverse electric (TE) waveguide mode in the PPWGs. Above the cutoff frequency of ν_c = c/2g, the wave propagates with the phase velocity v = c/$\sqrt{1-{\left({\nu }_c/\nu \right)}^2}$ faster than the light velocity, c, introducing a negative phase shift [13,14]. When the electromagnetic wave is incident with a polarization perpendicular to the stacked metal plates (θ = 90°), light propagates in a transverse magnetic (TM) waveguide mode with a slower phase velocity below the resonance of the through-hole array, ν₀. These propagation properties can be controlled by changing the gap distance, g, of the PPWGs and the size of the through-hole array. There is a minimum phase difference between the frequencies of ν_c and ν₀, and an approximately constant phase shift can be obtained in this frequency region. We formed steel plates with a through-hole array by chemical etching and experimentally demonstrated that the optimized artificial medium functions as an achromatic quarter-wave plate in the frequency region from 0.67 to 1.37 THz.

This artificial medium exhibits controllable dispersion, and has the advantages of cost, size, versatility, and available bandwidth. The available frequency of this achromatic wave plate can be designed along the scaling direction. For tuning the wave plate to a higher frequency, the size and pitch of the through-holes array must be reduced. Consequently, thinner metal sheets must be used for maintaining the aspect ratio of the through-hole formed by chemical etching, which causes problematic bending of the metal sheets. Therefore, in this paper instead of through-hole arrays, we focus on periodic rough structures on metal sheets, including corrugations, periodic diaphragms, grooves, and arrays of pillars [13,15–19]. These cause phase retardation of the TM waveguide mode below the resonance frequency, and can be fabricated easily, even on thick metal sheets.

In this paper, we performed a finite-difference time-domain (FDTD) simulation to optimize the birefringence of PPWGs with the simplest periodic rough structures for an achromatic wave plate. Using the optimal wave plate design, we fabricated a low-cost achromatic wave plate using a stack of parallel metal plates with periodic rough structures. We showed that the PPWGs with a pillar array function as an achromatic wave plate in the frequency region from 2.0 to 3.1 THz.

2. Simulations

The PPWGs in our FDTD simulations are illustrated in Fig. 1. We assumed that the metal sheets were 50 μm thick on both periodic boundaries with the conductivity dispersion of gold. A gap distance of g = 900 μm, corresponding to a cutoff frequency of the lowest TE mode of 0.17 THz. For a simple simulation, we evaluated a THz pulse traveling in a waveguide with simple rectangular grooves. We also assumed the top width of a, and the fixed period of the grooves of b = 43 μm, which corresponded to the resonant frequency ν₀ = c/2b = 3.4 THz. We chose a groove depth, d, that was much shorter than the wavelength. In the numerical simulation of the THz wave propagation, we used commercial FDTD software (RSOFT, FULLWAVE). A 0.5-ps-width Gaussian pulse with a center frequency of 2 THz was used as the excitation source. The polarization of the THz electric field was parallel (θ = 0°) and perpendicular (θ = 90°) to the metal plate, corresponding to the TE and TM waveguide modes, respectively. The mesh sizes near and far from the metal surface in the FDTD simulations were 0.5 and 2 μm, respectively. Because we reported the 10-mm-long PPWGs with a through-hole array for an achromatic wave plate around 1 THz [12], we initially set the waveguide length to 5 mm for a wave plate operating around 2 THz along the scaling.

 

Fig. 1 Schematic of the PPWGs used in our FDTD simulations. The wave source is denoted by the blue line, and the 20 observable points are denoted by the green circles.

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Figure 2 shows the field images of the traveling TE and TM pulses at a given time. Although the field lies at the center of the gap in TE waveguide mode, it is near the metal plates in TM waveguide modes. This implies that the periodic structures on the metal plates mainly affect the propagation properties of the TM mode. Therefore, we discuss the dependence of the TM waveguide mode on the shape of the structures.

 

Fig. 2 Spatial distribution of the field for (a) TE and (b) TM pulses traveling in the + z direction. The parameters of the periodic grooves are a = 20 μm, b = 43 μm, and d = 4 μm. In this plot, the line source is at z = 0. (c) Time profiles of the TE and TM waves evaluated at the position x = 450 (red curves) and 950 μm (blue curves). Gray curves show the time profile of the TEM wave propagating in the free space.

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Figure 2(b) shows that the field lies near the metal sheets with the retarded phase for a TM mode. This implies that the phase of the TM wave is retarded near the periodic grooves. In our analysis, we evaluated the temporal profiles of the plane wave components at a finite-number of observable points by using the FDTD method. As shown in Fig. 1, 20 laterally arranged observable points were placed 0.5 mm from the terminal of the 5-mm-long PPWGs. Because 0.5 mm is longer than the center wavelength of the THz pulse, the propagating wave components could be extracted rather than the strongly localized components.

Figure 2(c) shows the time profiles of the evaluated TE and TM waves. These are monitored at the positions z = 5500 μm, x = 450 μm (red curves), and x = 950 μm (blue curves). We add the profile of the TEM wave propagating in the free space as the gray curves. Obviously, the profile of the TE wave at the position x = 450 μm shows the negative phase shift while that of the TM wave at the position x = 950 μm shows the positive phase shift. They are the expected phase behavior. However, the phase of both TE and TM waves depends on the lateral position. It produces a convex wave front as shown in Figs. 2(a) and (b), resulting in the enhancement of a diffraction outside of the PPWGs. To extract the zero-order diffracted component in our analysis, we obtained the averaged time profiles of the THz pulses at the 20 observable points with (E_s) and without (E_r) passing through the PPWGs before evaluating the Fourier transformed transmittances, E_s/E_r = T e^iϕ.

Figure 3(a) shows the phase shift, ϕ, of the TE (dashed curves) and TM (solid curves) waveguide mode with a fixed top width of a = 20 μm and groove depths of d = 0, 2, 4, 6, and 8 μm. The phase shift of the TE waveguide mode diverges to –∞ with the frequency approaching the cutoff frequency of the lowest TE mode. It is not affected by changing the value of d because there is little field distribution near the metal plates. For the TM waveguide mode, the phase increases with frequency, which is dramatically enhanced with the value of d. The inset shows ϕ for the TM waveguide mode at 2.0 THz as a function of d. A similar groove depth dependence of the phase retardation has been reported in [15]. A large-d PPWG functions as a multi-order wave plate, and the large deviation in the phase shift for the TM waveguide modes may enhance the non-zero-order diffraction outside the PPWGs. It results in the undesirable reduction of the transmissivity for the optics. Therefore, we conclude that the optimal grooves are several micrometers deep.

 

Fig. 3 (a) Phase shift, ϕ, of the TE (dashed curves) and TM (solid curves) waveguide modes with a fixed top width of a = 20 μm and a groove depth of d = 0, 2, 4, 6, and 8 μm. The inset shows ϕ for the TM waveguide mode at 2.0 THz as a function of d. (b) Phase shift of the TE (dashed curves) and TM (solid curves) waveguide mode at a fixed groove depth of d = 4 μm and top widths of a = 20, 10, and 8 μm.

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The wave propagating in a waveguide is usually calculated by solving the approximate boundary condition [20], where the structures can be treated as the equivalent thin dielectric layer. However, the mixing of the vertically localized standing wave modes at the frequency of c/4d [19] strongly modulates the dispersion of the TM waveguide mode in the case d > b, which drastically behaves as the spoof surface plasmon polariton [17]. We investigated the top width a dependence of the phase shift, because the phase velocity of the spoof surface plasmon polariton below the frequency of c/4d decreases with the duty ratio (b-a)/b of the grooves [21]. Figure 3(b) shows the phase spectra of the TE and TM waves with a fixed groove depth of d = 4 μm and top widths of a = 8, 12, and 20 μm. The phase of the TM waveguide mode is slightly smaller for a = 8 μm, but the phase shift for the TE and TM waveguide modes is relatively insensitive to a. Therefore, we conclude that the phase retardation of the TM wave is mainly due to the resonance of ν₀ = c/2b.

Although the corresponding surface impedance shows d dependence of the phase retardation, it simultaneously causes the loss and scattering near the resonance of ν₀. The reduction of the transmittance near the resonance of ν₀ was observed for PPWGs containing a through-hole array [12]. Therefore, d must be small for a small phase shift of the TM wave combined with a longer waveguide length for a quarter-wave plate.

3. Experiments

We fabricated artificial media based on the PPWGs with periodic rough structures by using the optimized design for the quarter-wave plate. Initially, we formed PPWGs with simple periodic grooves by chemical etching. However, the groove width on the metal plates fluctuated by 10 μm, which restricted the size of the structures. The narrow bottom of the grooves may have caused inhomogeneity in the solution concentration during the chemical etching. To avoid the solution stagnating, we fabricated an array of pillars on the metal sheets, which have a wider bottom area. A large pillar array is generally cumbersome from the viewpoint of electromagnetic design. The waves in this waveguide should be treated as three-dimensional structures in the numerical simulation, which requires a long calculation time and too much memory [22]. However, the structure with the same effective surface impedance value should result in similar phase retardation for TM waveguide modes.

Figure 4(a) shows a photograph of the metal plate with pillars in an artificial medium assembly. We fabricated a periodic array of chemically etched pillars on both sides of a 0.1-mm-thick steel plate. The pillar diameter and pitch were a = 20 μm and p = 50 μm, respectively. In a triangular lattice, the effective period is b = $\sqrt{3}$p/2 = 43 μm. The position of the pillars was the same on both sides of the metal sheet because the patterns were written on two superposed plastic masks at the same time and the metal sheet was inserted between the plastic masks during the photoresist process. Figure 4(b) shows the profile of the pillars on one metal plate, which was taken with a confocal microscope (Lasertec, OPTELICS H1200). The height of the pillars was measured at different positions as h = 3.7 ± 0.5 μm. The sheets were shaped with exact dimensions of 50 × 10 mm. Figure 4(c) shows that the sheets have two holes with a diameter of 3.1 mm to allow exact arrangement, and they were stacked with the same gap distance of g = 0.9 mm using metal washers. Figure 4(d) shows the assembly of the PPWGs. The cutoff and resonant frequencies were ν_c = c/2g = 0.17 THz and ν₀ = c/$\sqrt{3}$p = 3.4 THz, respectively. Because the phase retardation of the TM waveguide mode for the pillar array was smaller than that for the thorough-hole array, we set the thickness of the artificial medium of 10 mm, which was twice the length of the PPWGs in the FDTD simulation.

 

Fig. 4 (a) Photograph of the pillars on one metal plate in one artificial medium assembly. (b) Profile of the pillars on one metal plate. (c) Photograph of one metal plate. (d) Assembly of the wave plate based on the PPWGs.

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To evaluate the phase retardation, we performed conventional THz time-domain spectroscopy. The details of the experimental setup are available in our previous paper [12]. The original beam from an Yb:doped fiber laser (IMRA America Inc., FCPA μJewel D-1000) with a wavelength of 1.04 μm, a pulse duration of 0.48 ps, a repetition of 100 kHz, and a pulse energy of 10 μJ was split into two beams, one for THz pulse generation and the other for detection. The excitation beam with a tilted pulse front, controlled by using a grating and a lens pair, was incident on a Mg:LiNbO₃ prism and was used as an emitter. The generated linearly polarized THz pulses with a broadened bandwidth were collimated and focused on a 0.4-mm-thick (110)-oriented GaP crystal as a detector, instead of CdTe, which was used in our previous report [12]. The amplitude of the electric field was measured by electro-optic sampling. For the detection of broadband THz pulses, the pulse width of the sampling pulse was compressed to 0.08 ps by using a high-index optical fiber and a negatively dispersive optical fiber [23]. Because higher-order nonlinear processes broaden the spectral bandwidth of the generated THz pulses, the bandwidth of the generated THz pulse was expanded and the available frequency for spectroscopy was 0.5–3.5 THz. In the transmission measurement, the collimated THz pulse was incident into the artificial medium with a polarization of θ. We measured the time profiles of the THz pulse without (E_r) and with (E_s) the artificial medium.

Figure 5(a) shows the complex transmittance, E_s/E_r = T e^iϕ, for the artificial medium with a pillar array with a height of h = 3.7 μm. The phase of the transmittance, ϕ, for θ = 0° (TE waveguide mode) is negative. For θ = 90° (TM waveguide mode), the transmittance, T, decreases at the resonance frequency of 3.4 THz. This also results in the phase increasing with frequency below 3.4 THz. Although the phase retardation for the TM waveguide mode is smaller than that in the PPWGs with a through-hole array [12], it was large enough to produce the minimum phase difference, Δϕ, between the phase for θ = 0° and 90°. We added Δϕ to the lower panel of Fig. 5(a). For the artificial medium with a pillar height of 3.7 μm, the minimum for Δϕ was approximately π/2 at 2.5 THz. The green dashed lines denote a 7% deviation from π/2. The frequency region where the phase shows a deviation of 7% from the ideal π/2 retardation ranges from 2.0 to 3.1 THz. The ratio of the transmittances for θ = 0° and 90° is T₉₀/T₀ > 0.85 in this frequency region, which gives the ellipticity of the circularly polarized THz wave converted from the linearly polarized THz wave using this π/2 phase shifter. Therefore, this artificial medium based on the stacked parallel metal plates with a pillar array functions as an achromatic quarter-wave plate over a wide frequency range.

 

Fig. 5 Complex transmittances, Te^iϕ, of the artificial medium for the angle between the metal plates and the polarization of the incident THz pulse of θ = 0° (dashed curves) and 90° (solid curves). The phase difference, Δϕ, is shown in the lower panel. The height of the pillars is h = 3.7 μm, and the gap distances are g = 0.9 mm. (b,c) Phase differences, Δϕ, of artificial media with different gap distances, g. The heights of the pillar are (b) h = 3.7 μm and (c) h = 8.5 μm. In these graphs, we plot the phase difference at the frequency where the ratio of the transmittances is T₉₀/T₀ > 0.85.

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Recently, we briefly reported similar PPWGs, where an array of the pillars with the same p and a lies on one side of a metal sheet [24]. The phase spectra for PPWGs with 8-μm-high pillars on one side of the metal sheets are almost same as for PPWGs with 3.7-μm-high pillars on the both sides of the metal sheets.

The conduction loss on the metal sheet may reduce the transmittance even for the short waveguide because it is critical for the high-frequency devices. We coated both sides of the PPWG steel plates with gold to improve their surface conductivity, which was the same as that in the numerical simulation. However, the measured transmittance was not improved. The transmittance for our artificial media was mainly determined by the reflection and scattering losses and diffraction after the wave passed through the PPWGs, rather than the conduction loss of the metal sheets.

Next, we compare the phase difference between the numerical simulations and experimental results. The simulated phase dispersions of the 10-mm-long PPWGs with the grooves are shown as the thin curves in Fig. 5(a). The depth and the top width of the groove were 4 and 8 μm, respectively. Although the effective surface impedance on the metal depended on the structures of the PPWGs, the phase spectra imply similar effective surface impedances.

The minimum Δϕ for g = 0.9 mm was 7% lower than π/2 at 2.5 THz, beside which there were the frequencies for Δϕ = π/2. By making the spacers thinner, so that g = 0.8 mm, the minimum Δϕ along the phase dispersion changed for the TE waveguide mode and the minimum Δϕ approached π/2, as shown in Fig. 5(b). We also fabricated the PPWGs with a pillar height of h = 8.5 μm. While the results of the FDTD simulation with g = 0.9 mm indicate that the large phase dispersion of TM waveguide mode for large-d PPWGs does not allow it to function as a quarter-wave plate, it operated as a quarter-wave plate when larger spacers were used (g = 1.1 mm), as shown in Fig. 5(c). However, large d reduces the transmittance for the TM waveguide mode above 2.1 THz, reducing the available bandwidth.

This achromatic wave plate allows easy conversion of few-cycle linearly polarized THz pulses with adequate bandwidth. However, we used few-cycle THz pulses with frequencies from 0.5 to 3.5 THz in our experiment, which is partly beyond the available frequency of our achromatic wave plate. To obtain few-cycle circularly polarized THz pulses, we restricted the bandwidth of incident THz pulses using a wire-grid polarizer (Murata Manufacturing Co., Ltd., MWG40-IV) [23] formed by chemical etching. Because it had a gap of 50 μm between the metal wires, it worked as a polarizer below a cutoff frequency of 3.0 THz, where its transmissivity was low for the polarization parallel to the direction of the wire. Above 3.0 THz, it worked as a grating, and the transmissivity corresponding to zero-order diffraction efficiency is about 50% [25]. Therefore, this wire-grid polarizer functioned as a high-pass filter. Figure 6(a) shows the time profile of the THz pulse after passing through the wire-grid polarizer, and the corresponding power spectrum is shown in Fig. 6(b). While the spectral component of the original THz pulses below 2.5 THz was much more intense than that above 3.0 THz, we obtained the narrowband THz pulses with the center frequency of 2.5THz. Its bandwidth is within the available frequency region for our achromatic wave plate with g = 0.9 mm and h = 3.7 μm. In this experimental demonstration, we inserted two wire-grid polarizers to extract the different polarization components. Details are described in [12].

 

Fig. 6 (a) Time profile of the THz pulse after passing through the wire-grid polarizer, (b) corresponding power spectrum, (c,d) temporal profiles of the THz pulses after passing through the achromatic wave plate (g = 0.9 mm and h = 3.7 μm) with a configuration of (c) θ = −45° and (d) θ = 45°.

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Figures 6(c) and (d) show the temporal profiles of the THz pulses after passing through the achromatic wave plate with θ = −45° and 45°. In these plots, we show the two polarization field profiles of E_x and E_y, and the trajectories of the electric field for the incidence of the few-cycle THz pulses. We set the directions of E_x and E_y as the polarization direction of the incident THz pulse as (E_x, E_y) = (1, 1), denoted by the green arrows. The field profiles of E_x for θ = −45° and E_y for θ = 45° are similar to that of the incident THz pulse, which are characteristic for the TM waveguide mode. On the other hand, the phase of E_y for θ = −45° and E_x for θ = 45° is different by -π/2 from that of the other polarization components, which are characteristic for the TE waveguide modes. The trajectories of the electric field are characteristic for the rightward (clockwise) circularly polarized pulse for θ = −45° and the leftward (counterclockwise) circularly polarized pulse for θ = 45°, as defined from the point of view of the source. Therefore, we succeeded in converting high-frequency few-cycle THz pulses into circularly polarized pulses.

4. Summary

We demonstrated an achromatic wave plate based on PPWGs over a wide region of high THz frequencies. The results of the FDTD simulation indicate that periodic rough structures several micrometers in height are required for the phase retardation of the wave plates. We fabricated PPWGs with a pillar array and experimentally demonstrated that they operate as an achromatic quarter-wave plate from 2.0 to 3.1 THz. The operating frequency of this achromatic wave plate can be expanded in the mid-infrared region by scaling the periodic rough structures and the gap between the PPWGs.