Simple THz Phase Retarder Based on Mach–Zehnder Interferometer for Polarization Control

Coherent excitation of collective quasiparticles using light is an intriguing direct approach for investigating condensed matter physics and tailoring emergent phenomena at ultrafast timescales. Abundant types of collective excitations in matters lie in the terahertz (THz) regime, e.g., molecular vibrations, phonons, magnons, and plasmons, which motivated recent technological development in laser science. Intense terahertz pulses allow us to resonantly create a large amplitude modulation in quasiparticle dynamics, which may result in the induction of a thermally inaccessible metastable state [1‐3], stabilizing an almost energetically degenerate excited state [4‐6], or nonlinear coupling with another quasiparticle excitation, e.g., exemplified by magnon-phonon coupling [7, 8] or nonlinear phononics [9, 10].

Very recently, a strong demand for intense circularly polarized THz pulses has appeared due to the rapidly growing field of chiral phononics [11, 12]. Resonantly driving degenerate phonon modes with a circularly polarized THz radiation create atomic rotations and an associated effective magnetic field via the phonon inverse Faraday effect [13‐15], which may require coupling with an electronic subsystem or quantum effects for enhancement in the induced magnetic field strength [16‐21]. The most convenient way to achieve the phase retardation of λ/4 that converts a linear polarization from a THz laser source, e.g., a free-electron laser [22], a laser-driven optical rectification [23], and difference frequency generation [24], into circular polarization may be the use of a birefringent crystal, which is typically x- or y-cut quartz. However, it does not convert a broadband THz pulse into circular polarization, as it achieves a perfect circular polarization only at a single wavelength in the terahertz range for a given thickness [14]. Besides, the working frequency range of a quartz waveplate is only below ~ 5 THz. Several mechanisms of achromatic THz waveplates have been suggested, e.g., silicon gratings [25], stacked parallel metal plates [26], dielectric artificial birefringence grating filled with polymer dispersed liquid crystal [27], magnetic tuning of molecular orientation in liquid crystals [28], metamaterial [29], total internal reflections in a Fresnel’s rhomb [30], and stacked waveplates with appropriate thicknesses and orientations [31], though they are typically limited to a narrow frequency range or with low transmission, as found in Table 1.

Here, we demonstrate a simple wavelength-variable phase retarder based on the Mach–Zehnder interferometer, i.e., using delay lines for two split identical beams. This setup allows us to convert the linear polarization of a THz pulse to circular polarization with wavelength tunability, high transmission (~ 76%), and wide operation wavelength range. Compared to the method using two THz pulses with orthogonal linear polarizations generated by optical rectification of signal and idler beams from optical parametric amplification to combine with an appropriate phase shift [32], a significant advantage is the simplicity and convenience. The performance of the setup is manifested by the achieved large circular polarization degree (> 99.9%).

We built up the phase retarder and tested it for two different wavelengths of narrowband THz pulses from the free-electron laser FELBE (free-electron lasers at the ELBE (electron linear accelerator with high brilliance and low emittance)) in Helmholtz-Zentrum Dresden-Rossendorf, Germany. Figure 1a shows a schematic picture of the phase retarder. A p-polarized FEL beam is split into two beams with orthogonal ± 45° polarizations with equal intensities for transmission and reflection by a free-standing wire grid polarizer tilted by 45° from the normal incidence, acting as a polarizing beam splitter. The polarizer consists of tungsten wires with 25 µm thickness and 10 µm spacing (GS57205, Specac Ltd.), and the beams travel along the different interferometer arms. The difference in the arm lengths (d₁ and d₂) results in a relative phase shift acquired before recombining the beams at another free-standing wire grid polarizer with the same specifications as the first one. When the relative phase shift is π/2 or − π/2, the obtained polarization is circular, i.e., C + or C −. The interferometer arms must be almost perfectly balanced, e.g., d₁ ≈ d₂, for ultrashort laser pulses in contrast to a monochromatic continuous THz source. Otherwise, pulses from the interferometer arms would not interfere. One of the gold mirrors is mounted on a translation stage with a micrometer to optimize the phase shift. A perfect polarization state is achieved only at a single wavelength but practically remains clean with a wavelength-dependent circular polarization degree |S₃| [C + (S₃ = + 1) or C- (S₃ = − 1)] of > 99.5% for the narrowband FELBE beam (~ 2%) within ± 3σ from the optimized central frequency, as shown in Fig. 1b. Here, σ stands for the standard deviation of a Gaussian distribution.

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Using the misalignment angles of the two wire grid polarizers from the nominal value ± 45° as Δθ₁ for the first one and Δθ₂ for the second one, their Jones matrices are expressed asfor transmission and reflection, respectively, of the first wire grid polarizer with the + 45° grid direction with respect to the horizontal plane, andfor reflection and transmission, respectively, of the second wire grid polarizer with the − 45° grid direction with respect to the horizontal plane. Here, T and R refer to the intrinsic transmittance and reflectance of the wire grid polarizers at the incidence angle of 45° (ideally 50% each). Creating a phase shift of δ between the two optical paths (d₁ and d₂) is represented by a Jones matrix

The Jones vectors along the first and second optical paths (d₁ and d₂) after the second wire grid polarizer are obtained as

In the ideal case, i.e., \(\Delta {\theta }_{1}=\Delta {\theta }_{2}=0 \text{and }\delta =\pi /2\). This leads to a perfect circular polarization state \(\mathbf{J}=TR\left(\begin{array}{c}1+i\\ 1-i\end{array}\right)\). Misalignment in the wire grid polarizer angles and the phase shift both result in elliptical polarization but along different axes.

We characterized the setup by measuring the transmission intensity as a function of the polarization analyzer angle (\(\varphi\)) placed right after the setup and optimized the polarization degree by adjusting the interferometer arm length difference after balancing the beam intensities along the two beam paths by adjusting the wire grid polarizer angles (Δθ₁ and Δθ₂). Figure 2a and b show polar plots of the measured intensity after the analyzer for two wavelengths, 118.2 µm (2.54 THz) and 201 µm (1.49 THz), respectively. The experimental data \(I\left(\varphi \right)\) fits well withwhere \(\varphi\) represents the analyzer rotation angle, \({\varphi }_{0}\) is an offset in the origin of \(\varphi\) due to experimental errors, and \({A}_{\text{p}\left(\text{s}\right)}\) denotes the THz transmission power amplitude along the horizontal (vertical) direction. The fitted parameters allow us to obtain the linear polarization degree (\(\sqrt{{{S}_{1}}^{2}+{{S}_{2}}^{2}}\)) and circular polarization degree (\(\left|{S}_{3}\right|\)) using the so-called Stokes parameters S = (S₁, S₂, S₃) based on the following equations:

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Here, the respective Stokes parameters represent the degree of vertical or horizontal linear polarization [s (S₁ = + 1) or p (S₁ = –1)], ± 45° linear polarization [+ 45° (S₂ = + 1) or –45° (S₂ = –1)], and circular polarization, while keeping \(\left|\mathbf{S}\right|=1\). Table 2 tabulates the linear and circular polarization degrees for respective nominal polarization states at the two wavelengths. For figures of merit, the achieved circular polarization degrees are larger than 99.9% for all the settings we tested, which demonstrates the high efficiency of our simple setup based on the Mach-Zender interferometer as a phase retarder. Note that the typically obtained linear polarization degree of nominal linear polarization is lower than the circular polarization degree of nominal circular polarization since the phase shift was not as thoroughly optimized, but it is still more than 90%. Since our experimental geometry allows us to tweak the interferometer arm length difference in the order of micrometer, one should be able to achieve a similar performance up to 60 THz, which is already deep in the range of commercially available waveplates. Thus, the setup overcomes the issue of a limited frequency range of operation (see Table 1) and well covers the gap frequency range. The transmission of the setup is more than ~ 76%. The possible sources of transmission loss are intrinsic transmittance and reflectance of the wire grid polarizers (T and R), reflectance of the gold mirrors, and imperfections in their orientation or alignment, resulting in a small portion of the beam leakage in the other direction of the interferometer arms. Since the misalignment in the wire grid polarizer angles and the interferometer arm length difference result in elliptical polarization with different elliptical axes, which cannot compensate for each other, the high circular polarization degree suggests that this is not likely the main source of transmission loss. Therefore, we believe the dominant origin of the transmission loss is in the intrinsic transmittance and reflectance of the wire grid polarizers.

A previously reported setup for a THz phase retarder is based on the Martin-Puplett interferometer and consists of a wire grid polarizer for beam splitting of orthogonal linear polarizations, two roof prism mirrors for 180° back reflections, and the same wire grid polarizer for recombination. It achieves similar performance in terms of the circular polarization degree at a particular frequency [33]. The back reflections from the roof prism mirrors hit a different spot on the wire grid polarizer. Since the grazing incidence angle (45°) significantly enlarges the THz beam footprint on the polarizer, our setup using two independent wire grid polarizers easily handles a larger THz beam with a longer wavelength compared to the Martin-Puplett interferometer. Intensity balancing between the two interferometer arms is crucial to obtain circular polarization. The alignment may be easier in our setup because our setup has independent splitting and recombining sections.

We have demonstrated the high efficiency of our phase retarder based on the Mach–Zehnder interferometer to convert the polarization of a THz pulse from linear to circular. Even though perfect polarization is achieved only at a single wavelength, the wavelength-dependent polarization degree variation is not relevant for the narrowband THz pulse like the one at the FELBE. The advantages of our setup inherent to the Mach–Zehnder design are (1) variable wavelength, (2) wider operation wavelength range, both shorter or longer sides, (3) small losses (~ 24%), (4) simplicity, and (5) high polarization degree. A quartz waveplate suffers significant absorption above ~ 5 THz due to optical phonons. However, our setup can operate at any frequency unless the precision of the adjustment in the split delay lines limits. Our simple phase retarder allows us to explore material dynamics triggered or monitored by polarization-controllable THz pulses.