Liquid-crystal-based magnetically tunable terahertz achromatic quarter-wave plate

Development of the wideband and tunable quasi-optic terahertz (THz) components is in high demand. In this work, we demonstrate a tunable achromatic quarter-wave plate (AQWP) for the THz frequency range. The phase retardation of this device can be set at 90° ± 9° from 0.20 to 0.50 THz. The operation range from 0.20 to 0.50 THz can be tuned to from 0.30 to 0.70 THz by introducing three nematic liquid crystals phase retarders, of which the birefringence can be magnetically tuned. The frequency-dependent phase retardation is in good agreement with theoretical predictions. © 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement


Introduction
Wave plates are essential for modulating the polarization states of light.The basic wave plate is designed to work at a particular wavelength.Therefore, achromatic wave plate (AWP) with a broad operating wavelength range is required for many applications.Several types of achromatic phase retarders have been demonstrated so far.Mainly, these are for the visible and near-infrared (NIR) wavelength range [1][2][3][4][5].The terahertz (THz) technology and science have been in full swing for over two decades [6].These research and developments urgently need the quasi-optic THz components, such as polarizers [7], phase shifters [8][9][10][11], phase gratings [12][13][14][15], and wave plates [16][17][18][19].Masson et al. reported an achromatic quarterwave plate (AQWP) in the THz frequency range by using six or more pieces of birefringent crystalline quartz plates [17].Owing to the number of quartz plates with precise thickness needed, the fabrication of this kind of AWP is very complicated and bulky.Recently, Zhang et al. also demonstrated an wideband AQWP using silicon grating [16].However, both of these designs cannot adjust the achromatic range.Nematic liquid crystals (LCs) have been used in AWPs because its birefringence is relatively high and can be controlled electrically or magnetically.Previously, our group demonstrated the feasibility of reducing pulse broadening by applying the achromatic half wave plate and AQWP made of two and three LCs cells stacked together [3].In the past years, we have also demonstrated several THz phase shifters based on magnetically or electrically controlled birefringence in nematic LCs [8][9][10][11]20].Most of these devices are capable of being designed for more than 360° of phase shift around 1 THz by increasing the thickness of LCs cell [10,20].Recently, electrically tunable 2π THz phase shifters based on a sandwiched LC cell with indium-tin-oxide nanowhiskers as transparent electrodes have been well demonstrated [10].Further, the magnetically tunable 2π LCs THz phase shifter can be operated over a broad range near room temperature [20].
As described above, by picking up low-loss transparent electrodes which are extremely rare in the THz frequency range [8,10,11,19], the electrically tunable achromatic wave plate can be demonstrated.The advantage of applying the magnetic fields to tune LCs is without the requisite of transparent electrodes.
In this work, we proposed and demonstrated a LCs-based THz AQWP, of which the phase retardation can be tuned by changing the effective refractive index of LCs.The device consists of a standard half-wave plate and two standard third-wave plates.The critical concept is the partial cancellation of the change of retardation with frequency from each wave plate.

Experimental method
To shift THz operation range, we extended the design to a combination of three magnetically controlled LC phase retarders.Nematic LCs E7 are reasonably transparent in the THz range [21], and it also exhibits large birefringence.In THz frequency range, the ordinary and extraordinary refractive indices of E7 are n o = 1.58 and n e = 1.71, respectively, and the corresponding imaginary indices are κ o (0.01) to κ e (0.007) [21].
The LC-based achromatic wave plate has three elements, TR A , TR B and TR C , which are tunable retarders (TRs) and shown in Fig. 1.Each TR consists of a pair of rotatable permanent magnets and a homeotropically aligned LC cell.The LC cell in the TR was constructed with two fused silica substrates and filled with nematic LCs, E7 (Merck).The substrates of LC cells were coated with N, N-dimethyl-n-octadecyl-3aminopropyltrimethoxysilyl chloride (DMOAP) for homeotropic alignment [22].Thicknesses of LC layers in TR A , TR B and TR C were d A = 2.56 mm, d B = 3.86 mm, and d C = 2.56 mm, controlled by Teflon spacers.Besides, the thickness of fused silica is around 1mm.The threshold field required to reorient LC molecules with the magnetic field is less than 0.004 Tesla [23].The maximum magnetic field at cell position in the rotary permanent magnets (sintered Nd-Fe-B) is 0.25 Tesla.Sufficiently large magnetic field is applied for stable homogeneous alignment of LC molecules in the thick LC cell.The rotation axis of rotary magnets is perpendicular to the propagation direction of the THz wave and has an azimuthal angle ρ i with respect to the normal axis of the table (ρ A = 0°, ρ B = 20° and ρ C = 0°).So TR A , TR B , and TR C are used to achieve the desired variable phase retardation, Γ A (θ), Γ B (θ) and Γ C (θ), where θ is defined as the angle between the director of LC molecules and the polarization direction of electromagnetic wave.The TRs were placed between a pair of wiregrid polarizers (Specac, No. GS57204).We set the polarizer at fixed azimuthal angle ρ p = −75°, it made the optic axis of achromatic quarter wave plate has 45° with respect to the polarization direction of THz wave in our photoconductive-antenna (PCA)-based transmission-type THz time-domain spectroscopy (THz-TDS) which has been described well in our previous works for characterization of the devices and materials in the THz frequency range [24].The schematic cartoon of THz emitter and detector is also shown in Fig. 1.The THz-TDS was always purged with dry nitrogen, so the relative humidity was maintained at 5.0 ± 0.5%.In this work, the inclination angles θ = 0° and θ = 30° for all TRs were demonstrated.The corresponding birefringence of LCs is 0.13 and 0.095, respectively.By rotating the magnetic field, the operation frequency range was shifted for that corresponding retardations were equal to quarter wave.
Basically, the purpose to design an THz AQWP can be regarded as finding combinations of retarders causing paths from a linearly polarization state (a fixed point on the equator of the sphere) to a circular polarization state (north of south pole of the sphere) on the Poincaré sphere over a wide THz frequency range.The theoretical model to achieve the goal is based on the analysis of Jones matrices with the phase difference equation, and retrieved total phase retardation from matrices.Each LC wave plate is described by its corresponding Jones matrix J i (i = A, B and C) [23] which the phase retardation Γ i and the orientation angle ψ i with respect to the optical axis are two key parameters in Eq. ( 1).The total Jones matrix of achromatic wave plate is given by [23], and it can be written into where the A, B, -B*, and A* correspond to the matrix components.The total resulting retardation is obtained by

Im Im tan 2 Re Re
For LC wave plates, the inclination angle, θ, is defined as the angle between the director of LC molecules and the polarization direction of electromagnetic wave.The orientation of the LC molecules, which can be described by the angle θ, is proportional to the effective refractive index of LCs [8].The phase retardation, Γ i (θ), due to magnetically controlled birefringence is given by 0 2 ( ) ( , ) , where d is the thickness of LC layer, Δn i is the change of effective birefringence, f is the frequency of the THz waves and c is the speed of light in vacuum.If the magnetic field is strong enough, LC molecules can be reoriented and parallel to the direction of the magnetic field.The phase retardation, Γ i (θ), in Eq. ( 4) can then be re-written as where n o and n e are the ordinary and extra-ordinary refractive indices of the LC.As expected, the tunable operation range of achromatic quarter wave plate is achieved by using three magnetically controlled LC retarders.

Experime
The  In Fig. 3, the transmittance values with different φ (when θ = 0°) are drawn with theoretic prediction at the frequency of 0.44 THz.The main reason we picked up 0.44 THz is the strongest resonant frequency of the THz radiation from PCA-based THz-TDS which is situated there.In other words, the signal-to-noise ratio at 0.44 THz showed the best performance compared with other frequencies.Considering the heavy loss from the LC retarders, we finally decided to show the case of 0.44 THz.The experimental retardations obtained with the combination of the three LC plates at θ = 0° and 30°, depicted in Figs. 4 and 5, respectively.
In Fig. 4, the phase retardation of this device is about 90° from 0.20 to 0.50 THz, when θ = 0°.On the other hand, in Fig. 5, the phase retardation of this device is about 90° from 0.30 to 0.70 THz, when θ = 30°.The operation frequency range can be shifted by rotating the magnet set.The experimental data and the theoretical curves from the calculation of Jones matrices are in very good agreements.The thickness of the LC layer for the THz device should be close to the THz wavelength scale, around hundred μm, which will cause that LC molecules in the middle of cell cannot be aligned well by the magnetic field.The slight deviation between the numerical and the experimental results comes from here.
Fig. 1 retard LC ce respec mm, a The PCAretardation of ellipsometry b retardation of polarized TH electric field E