Continuous wave terahertz common path phase shifting interference using a liquid crystal phase shifter

Figure 1 shows the configuration of the common path PSI system using an LC phase shifter. In the case of crossed Nicols, the transmitted intensity of the THz wave I is given as follows;

$$\begin{eqnarray}&&I={I}^{{\prime} }+I^{\prime\prime} \left\{1-\,\cos \left({\phi }_{{\rm{s}}}+{\phi }_{{\rm{LC}}}\right)\right\}.\end{eqnarray} \tag{ 1 }$$

Here, I' and I'' are the light intensity factors, which include noise signals, ϕ_s is the phase difference distribution between the extraordinary and ordinary rays of the test sample, and ϕ_LC is the phase shift introduced by the LC phase shifter. In this study, we adopt a four-step algorithm for data analysis. ³⁰⁾ The four output light intensity distribution data (i.e. I₁ to I₄) obtained corresponding to the phase shift values of ϕ_LC = π/2, π, 3π/2, and 2π, respectively, are given as follows;

$$\begin{eqnarray}&&{I}_{1}={I}^{{\prime} }+I^{\prime\prime} \left\{1-\,\cos \left({\phi }_{{\rm{s}}}+\displaystyle \frac{\pi }{2}\right)\right\},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{I}_{2}={I}^{{\prime} }+I^{\prime\prime} \left\{1-\,\cos \left({\phi }_{{\rm{s}}}+\pi \right)\right\},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{I}_{3}={I}^{{\prime} }+I^{\prime\prime} \left\{1-\,\cos \left({\phi }_{{\rm{s}}}+\displaystyle \frac{3\pi }{2}\right)\right\},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{I}_{4}=I^{\prime} +I^{\prime\prime} \left\{1-\,\cos \left({\phi }_{{\rm{s}}}+2\pi \right)\right\}.\end{eqnarray} \tag{ 5 }$$

Through the simple calculation using Eqs. (2)–(5), the light intensity factors I' and I'', which include noise signals, are successfully eliminated, leaving behind only the phase data, which can be extracted as follows;

$$\begin{eqnarray}&&{\phi }_{{\rm{s}}}={\tan }^{-1}\left(\displaystyle \frac{{I}_{1}-{I}_{3}}{{I}_{2}-{I}_{4}}\right).\end{eqnarray} \tag{ 6 }$$

If we know the sample thickness d_s, the birefringence of sample Δn_s is given as follows:

$$\begin{eqnarray}&&{\rm{\Delta }}{n}_{s}=\displaystyle \frac{{\phi }_{{\rm{s}}}\lambda }{2\pi {d}_{{\rm{s}}}}\end{eqnarray} \tag{ 7 }$$

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where λ is the wavelength of the incident THz wave. Note that the LC phase shifter should control the phase without changing the intensity of the incident wave, i.e. an LC material exhibiting no dichroism is essential for this method.

Figure 2(a) shows the structure of the electrically tunable LC phase shifter. An alignment of the molecules parallel to the substrate can be achieved by performing a suitable surface treatment or the coating of substrates. In this study, hydrogen-bonded LC material (PC29, LCC Inc., Japan) is injected into a sandwich cell where both inner surfaces are treated via parallel rubbing. This rubbing is conducted after coating of the planar alignment polyimide (SE2170: Nissan Chemical Industries, Japan) with the aim of obtaining a homogeneous alignment. The hydrogen-bonded LC shows a thermotropic LC phase due to the intermolecular hydrogen bonding between identical molecules [see Fig. 2(b)]. The THz properties of PC29 were reported in our previous work. ²⁹⁾ The cell thickness is determined by sheet spacers. Moreover, the direction of LC molecules can be controlled by an external electrical field. To maintain the high transmittance of the THz wave, we use z-cut quartz substrates and poly (3,4-ethylenedioxythiophene) polystyrene sulfonate (PEDOT/PSS) electrodes. PEDOT/PSS exhibits the high transmittance of the THz wave and acts as a good transparent electrode. ^31–33)

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Figure 3 shows the optical configuration of our PSI process realized by using an LC phase shifter. An optically pumped gas laser was used as the CW-THz source. This laser can generate a CW-THz wave with sufficient power for precise measurements. A CO₂ laser was used for pumping of the CH₂F₂ gas and a wavelength of 117.7 μm (2.5 THz) was used for our measurements. Here, the propagation direction of the THz wave is in the z direction as shown in Fig. 3 and the intensity of the transmitted THz wave was detected by the pyroelectric detector. The CW-THz wave of the optically pumped gas laser is Gaussian beam and the output beam diameter is about 15 mm. This beam diameter is small enough to investigate the common path PSI. Two wire grid polarizers were used for transmission measurements and the angle of the polarizer or optical axis is defined as the inset of Fig. 3. The LC device and sample were placed between polarizers of crossed Nicols (Ψ_P = 0° and Ψ_A = 90°). The angle between the polarizing direction of the THz wave and the optic axis of LC phase shifter was 45° (Ψ_LC = 45°). Moreover, the optic axis of a sample is perpendicular to the optic axis of the phase shifter (Ψ_s = −45°).

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We calculate the transmittance of THz wave in the common path PSI by using the Jones matrix method. ³⁴⁾ The electric fields of the transmitted THz waves through the LC device and sample (E_xout and E_yout, see Fig. 3) are described as follows:

$$\begin{eqnarray}&&\left[\begin{array}{c}{E}_{{\rm{xout}}}\\ {E}_{{\rm{yout}}}\end{array}\right]={{\boldsymbol{P}}}_{{\bf{A}}}{{\boldsymbol{W}}}_{{\bf{s}}}{{\boldsymbol{W}}}_{{\bf{L}}{\bf{C}}}\left[\begin{array}{c}\cos \,{{\rm{\Psi }}}_{{\rm{P}}}\\ \sin \,{{\rm{\Psi }}}_{{\rm{P}}}\end{array}\right],\end{eqnarray} \tag{ 8 }$$

where, P _A , W _s , and W _LC represent the Jones matrix of the analyzer, sample, and LC phase shifter, respectively. Here P _A is calculated from:

$$\begin{eqnarray}{{\boldsymbol{P}}}_{{\bf{A}}}={\boldsymbol{R}}\left({{\rm{\Psi }}}_{{\rm{A}}}\right)\left[\begin{array}{cc}1 & 0\\ 0 & 0\end{array}\right]{\boldsymbol{R}}\left(-{{\rm{\Psi }}}_{{\rm{A}}}\right),\end{eqnarray} \tag{ 9 }$$

where Ψ_P and Ψ_A are the angle of the polarizers as defined in Fig. 3 and R (Ψ) is the rotation matrix,

$$\begin{eqnarray}{\boldsymbol{R}}\left({\rm{\Psi }}\right)=\left[\begin{array}{cc}\cos \,{\rm{\Psi }} & -\,\sin \,{\rm{\Psi }}\\ \sin \,{\rm{\Psi }} & \cos \,{\rm{\Psi }}\end{array}\right].\end{eqnarray} \tag{ 10 }$$

The Jones matrix of the sample ( W _s ) is expressed as follows:

$$\begin{eqnarray}{{\boldsymbol{W}}}_{{\bf{s}}}={\boldsymbol{R}}\left({{\rm{\Psi }}}_{{\rm{s}}}\right)\left[\begin{array}{cc}A & 0\\ 0 & B\end{array}\right]{\boldsymbol{R}}\left(-{{\rm{\Psi }}}_{{\rm{s}}}\right),\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&A=\exp \left(-i\displaystyle \frac{\pi {\rm{\Delta }}{n}_{{\rm{s}}}{d}_{{\rm{s}}}}{\lambda }\right)\exp \left(-\displaystyle \frac{2\pi {n}_{{\rm{es}}}^{{\prime\prime} }{d}_{{\rm{s}}}}{\lambda }\right),\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&B=\exp \left(i\displaystyle \frac{\pi {\rm{\Delta }}{n}_{{\rm{s}}}{d}_{{\rm{s}}}}{\lambda }\right)\exp \left(-\displaystyle \frac{2\pi {n}_{os\,}^{{\prime\prime} }{d}_{s}}{\lambda }\right),\end{eqnarray} \tag{ 13 }$$

where Ψ_s is the angle of the optic axis of the sample, Δn_s is the birefringence and d_s is the thickness. Moreover, n''_es and n''_os are the imaginary parts of the complex refractive indices associated with the extraordinary direction and the ordinary direction of the sample, respectively.

Furthermore, the Jones matrix of the LC phase shifter ( W _LC ) is given as

$$\begin{eqnarray}{{\boldsymbol{W}}}_{{\bf{L}}{\bf{C}}}={\boldsymbol{R}}\left({{\rm{\Psi }}}_{{\rm{LC}}}\right){\boldsymbol{Q}}\left[\begin{array}{cc}C & 0\\ 0 & D\end{array}\right]{\boldsymbol{QR}}\left(-{{\rm{\Psi }}}_{{\rm{LC}}}\right),\end{eqnarray} \tag{ 14 }$$

where Ψ_LC is the angle of the optic axis of the LC phase shifter. The Jones matrix of the z-cut quartz substrate in the LC phase shifter Q may be expressed as follows:

$$\begin{eqnarray}{\boldsymbol{Q}}=\exp \left(-\displaystyle \frac{2\pi {n}_{{\rm{q}}}^{{\prime\prime} }{d}_{{\rm{q}}}}{\lambda }\right)\left[\begin{array}{cc}1 & 0\\ 0 & 1\end{array}\right],\end{eqnarray} \tag{ 15 }$$

where d_q and n''_q are the thickness and imaginary part of the refractive index corresponding to the z-cut quartz substrate, respectively, and λ is the wavelength of the THz wave. Here, C and D are given as

$$\begin{eqnarray}&&C=\exp \left(-i\displaystyle \frac{\pi {\rm{\Delta }}{n}_{{\rm{LC}}}{d}_{{\rm{LC}}}}{\lambda }\right)\exp \left(-\displaystyle \frac{2\pi {n}_{{\rm{eLC}}}^{{\prime\prime} }{d}_{{\rm{LC}}}}{\lambda }\right),\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&D=\exp \left(i\displaystyle \frac{\pi {\rm{\Delta }}{n}_{{\rm{LC}}}{d}_{{\rm{LC}}}}{\lambda }\right)\exp \left(-\displaystyle \frac{2\pi {n}_{\mathrm{oLC}\,}^{{\prime\prime} }{d}_{{\rm{LC}}}}{\lambda }\right),\end{eqnarray} \tag{ 17 }$$

where Δn_LC is the birefringence and d_LC is the thickness. In addition, n''_eLC and n''_oLC are the imaginary parts of the complex refractive indices associated with the extraordinary direction and the ordinary direction of the LC layer, respectively. The transmitted intensity of the THz wave I is given as follows:

$$\begin{eqnarray}&&I={\left|{E}_{{\rm{xout}}}\right|}^{2}+{\left|{E}_{{\rm{yout}}}\right|}^{2}.\end{eqnarray} \tag{ 18 }$$

Here, we assume that the initial orientation condition and reorientation of the LC molecule by an external electric field in the LC phase shifter are as shown in Fig. 4. In this model, the directors of the LC molecules are uniformly distributed with an average angle of θ, which increases with increasing external electric field. The LC layer is very thick in the THz phase shifter, and hence this simple model can explain the THz phase shift properties of the LC phase shifter. In this case, the birefringence of the LC phase shifter Δn_LC is given as follows:

$$\begin{eqnarray}&&{\rm{\Delta }}{n}_{{\rm{LC}}}={n}_{{\rm{eLC}}}^{{\prime} }-\displaystyle \frac{{n}_{{\rm{oLC}}}^{{\prime} }{n}_{{\rm{eLC}}}^{{\prime} }}{\sqrt{{n}_{{\rm{oLC}}}^{{\prime} 2}{\sin }^{2}\theta +{n}_{{\rm{eLC}}}^{{\prime} 2}{\cos }^{2}\theta }},\end{eqnarray} \tag{ 19 }$$

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where n'_oLC and n'_eLC are the real parts of the complex refractive indices associated with the extraordinary direction and the ordinary direction of the LC, respectively.

To investigate the phase property of the LC phase shifter, the transmittance is measured without the sample. Figure 5(a) shows the experimental transmittance of the LC phase shifter using PC29 as a function of the applied voltage at 2.5 THz. Here, the phase shifter is placed between the polarizers of crossed Nicols (Ψ_P = 0° and Ψ_A = 90°) as shown in Fig. 3. The angle between the polarizing direction of the THz wave and the optic axis of the phase shifter Ψ_LC is 45°. The vertical axes of Fig. 5(a) is normalized by data of THz intensity without sample and LC phase shifter.

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On the other hand, the THz wave transmittance calculated using the Jones matrix method ³⁴⁾ is plotted as a function of the average LC molecular angle θ (see Fig. 4) as shown in Fig. 5(b). The transmittance is calculated from Eq. (18) and the transmitted THz intensity is 1 in the case without sample and LC phase shifter in the light path. Here, we set the angle of the polarizers and the optic axis of the LC phase shifter to Ψ_P = 0°, Ψ_A = 90°, and Ψ_LC = 45°. We consider the result without the sample, and hence we set Δn_s = 0 in the calculation. Furthermore, we set the thickness (d_q) and imaginary part of the refractive index associated with the z-cut quartz substrate in the LC phase shifter (n''_q) to 500 μm and 0.00047, respectively. ²⁹⁾ The relationship between the average LC molecular angle θ and the birefringence of the LC phase shifter Δn_LC is determined from Eq. (19). From our previous work, ²⁹⁾ the birefringence of PC29 is 0.17 at 2.5 THz. The absolute value of refractive indices associated with PC29 at 2.5 THz is currently being investigated and will be reported in the future. Therefore, in this study, we use the value of the ordinary refractive index in the visible region (n'_oLC = 1.47 at 589 nm) as the ordinary refractive index of PC29 at 2.5 THz. The corresponding birefringence is 0.17, and hence the extraordinary refractive index of PC29 is set to n'_eLC = 1.64 and the birefringence of the LC phase shifter Δn_LC is determined from Eq. (19). Here, when the ordinary refractive index n'_oLC changes by 0.2, the birefringence of LC phase shifter Δn_LC changes by <2%. The value of Δn_LC changes only slightly with the value of n'_oLC. Therefore, we believe that the calculation results shown in Fig. 5(b) are sufficiently valid for discussing the experimental result.

The variation in the transmittance determined via experiments [see Fig. 5(a)] is similar to that determined via calculations [see Fig. 5(b)]. Our experimental and calculation results indicate that applied voltage levels of 50–150 V correspond to director angles of 0°–70°. Furthermore, a non-zero transmittance at 150 V [see Fig. 5(a)] suggests that the LC molecules are only partly parallel to the applied electric field. This result indicates that our circulation method, which uses a simple LC orientation model (as shown in Fig. 4), is suitable for phase calculations of the THz LC phase shifter.

For four-step PSI, the phase shift values ϕ_LC = π/2, π, 3π/2, and 2π are required for determining the phase value of the sample [dashed lines in Fig. 5(b) denote the calculated results corresponding to the aforementioned shift values]. The ϕ_LC = π and 2π phase shift values correspond to the maximum transmittance and the minimum transmittance, respectively. On the other hand, the transmittance of half maxima corresponds to shift values of ϕ_LC = π/2 and 3π/2 as shown in Fig. 5(b). Using these calculation results, we determine the applied voltage required for four-step PSI [see the experimental results shown in Fig. 5(a)]. The lines in Fig. 5(a) indicate the points associated with phase shift values of ϕ_LC = π/2, π, 3π/2, and 2π, which correspond to applied voltages of V₁ = 112 V, V₂ = 99 V, V₃ = 90 V, and V₄ = 78 V, respectively. The operation speed of the LC phase shifter is important because the measurement time of the PSI depends on the operation speed of the LC phase shifter. In this study, the thickness of the LC layer in our LC phase shifter is 800 μm and we need few minutes to obtain the data of Fig. 5(a). This measurement time is not equal to the measurement time of the PSI because only four data are needed to obtain the phase of sample in the four-step PSI. However, the LC phase shifter is not fast in this stage. We think that the introduction of the electrospun nanofiber [35] is a good approach to improve the operation speed of the LC phase shifter.

Figure 6 shows the experimental results of the common path PSI using hydrogen-bonded LC (PC29) phase shifter. In Fig. 6, the transmittance of a THz wave through the LC phase shifter and x-cut crystalline quartz is plotted as a function of the voltage applied to the LC phase shifter. In this study, two types of x-cut quartz substrates are measured. Here, we place the sample at the position shown in Fig. 3, and we set Ψ_P = 0°, Ψ_A = 90°, Ψ_LC = 45°, and Ψ_s = −45°. Figure 6(a) shows the result of sample 1, which is an x-cut quartz substrate made for a λ/4 wave plate at λ = 214.6 μm. The thickness (d_s) of this sample 1 is 1141 μm in the catalog, and the corresponding birefringence is 0.047. Figure 6(b) shows the result of sample 2, which is made for a transparent quartz window. The thickness of sample 2 is 500 μm, but the birefringence is unknown. As shown in Figs. 6(a) and 6(b), the V–I curves with the sample data differ from those without the data [see Fig. 5(a)]. That is, the phase of the transmitted THz wave is affected by the birefringence of the samples.

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Table I shows the results of the common path PSI performed on sample 1 and 2. Here, we adopt the four-step algorithm for data analysis (see Sect. 2.2) and I₁ to I₄ correspond to the phase shift values for ϕ_LC = π/2, π, 3π/2, and 2π, respectively. These shift values are obtained by applying V₁ = 112 V, V₂ = 99 V, V₃ = 90 V, and V₄ = 78 V to the LC phase shifter (see Fig. 6). As shown in Table I, the phase difference distribution between extraordinary and ordinary rays of samples 1 and 2, ϕ_s1 = 0.25 and ϕ_s2 = −1.26 radian, is calculated from Eq. (6). The birefringence of the samples estimated from Eq. (7) is also shown in Table I. In the case of sample 1, phase unwrapping is needed to estimate the accurate value of the birefringence because of the large optical length of sample 1. Considering the experimental configuration (especially the optic axis of the LC phase shifter and sample), subtracting nπ (where n is an integer) from ϕ_s1 is reasonable for phase unwrapping. Here, the unwrapped phase of ϕ'_s1 = ϕ_s1 − π is used to estimate the value of the birefringence (see Table I). The estimated birefringence of sample 1 and 2 are −0.05 and the negative values are consistent with our experimental configuration. Furthermore, the estimated birefringence of sample 1 agrees with the catalog data of 0.047 and the estimated birefringence of sample 2 is consistent with the result of sample 1 and the reported value. ^35,36)

Then, we calculate the transmittance of THz wave in the common path PSI by using the Jones matrix method. ³⁴⁾ Here, the transmittance of the THz wave intensity through the LC phase shifter and the sample is calculated. Figure 7 shows the calculated THz intensity as a function of the average LC molecular angle θ. The intensity is calculated from Eq. (18) and in the calculation, we set Ψ_P = 0°, Ψ_A = 90°, Ψ_LC = 45°, and Ψ_s = −45°. In addition, we use the Δn_s and d_s of sample 1 and sample 2 as shown in Table I. Here, we assume the imaginary part of reflective indices of samples n''_os = n''_es = 0.00047, which is the same value as that of the z-cut quartz substrates used in the LC phase shifter. Figures 7(a) and 7(b) show the calculation results of sample 1 and sample 2, respectively. The experimental THz intensity in Figs. 6(a) and 6(b) is, nevertheless, smaller than the calculated transmittance in Figs. 7(a) and 7(b). These results indicate that the bias noise was included in our experimental data. We think that this bias noise is most likely from reflection at the surface of the sample. However, this kind of attenuation is included in the signal of I'' [see Eq. (1)], which is eliminated during the calculation process as shown in Sect. 2.1. Therefore, the phase values of samples obtained from our experimental results are unaffected by reflection at the sample surface. These results indicate that the phase of samples is successfully obtained and using the hydrogen-bonded LC is effective in common path PSI associated with the THz region. For future applications such as detecting the unique chemical or biological agent, it is important to extend the measurement frequency of THz common path PSI. Since the LC materials with no dichroism are essential in the THz common path PSI using LC phase shifter, the broadband THz properties of hydrogen-bonded LC will be reported in our next paper.

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