Expected performances of the phase shifters can be analyzed using the approach that we reported earlier [

3]. In the absence of the external field, the LC molecules were initially in equilibrium and were oriented more or less parallel to the substrates (

Figure 1a). Because of the Fréedericksz transition, when the applied bias was greater than the threshold voltage, the LC molecules will reorient themselves toward the direction of the external field. The threshold voltage for reorienting the LC molecules is given by

V_{th} = π·(

k_{1}/(

ε_{0}·∆

ε))

^{1/2}L/

d [

23], where

ε_{0} = 8.854 × 10

^{−12} F·m

^{−1}, ∆

ε =

ε_{//} −

${\epsilon}_{\perp}$ = 11.2,

k_{1} = 12.6 × 10

^{−12} N, and

L and

d are the free-space permittivity, dielectric anisotropy, splay elastic constants, distance between two electrodes, and thickness of the LC layers, respectively. The maximum tilt angle of the LC director,

θ_{max}, is assumed to be found at the mid-point position of the LC cell, i.e., z =

d/2. Here, the THz wave is assumed to be propagating along the z-direction, normal to the LC cell. The relationship between

θ_{max} and

V can be determined by solving [

23]:

where

θ is the tilt angle of the LC molecule, sin

δ = sin

θ/sin

θ_{max}. We define

η = sin

θ_{max}, then sin

θ (

z) = sin

θ_{max} · sin

δ′(

z),

δ′(

z) being the derivative of the integrating variable. In Equation (1),

ζ = (

k_{3}−

k_{1})/

k_{1} ≈ 0.22,

k_{3} = 15.4 × 10

^{−12} N,

ρ = (

ε_{//} −

${\epsilon}_{\perp}$)/

${\epsilon}_{\perp}$ ≈ 2.55,

${\epsilon}_{\perp}$ = 4.4 for the LC that we used.

V is the applied voltage. Equation (1) can be rewritten and used to find

θ at every

z position in the LC cell, as follows:

where

E is the electric field applied to the device. When considering the effective birefringence of the LC, the total phase shift experienced by the THz wave propagating through the device, ∆

ϕ, is given by

where

f and

c_{0} are the frequency and speed, respectively, of the THz wave in vacuum. The extraordinary and ordinary refractive indices of MDA–00–3461 in a frequency range of 0.3–1.4 THz at 25 °C are essentially frequency-independent:

n_{e} = 1.74 and

n_{o} = 1.54, respectively [

24]. The aforementioned model is suitable for a thin LC cell (thickness of approximately 20 μm or less). However, in this study, the thickness of the LC layer for the 2π THz phase shifter was approximately 1.12 mm, which was approximately 100 times thicker than typical LC devices that were designed in the visible frequency range. In such thick cells, LC molecules more than a few tens of microns away from the cell walls will not be exactly parallel to the rubbed direction in the absence of the applied electric field. Besides, almost all the LC molecules can be easily reoriented in the direction, which is parallel to the external field when the LC cell is biased.

In

Figure 2a, we drew a thick LC cell in the e-ray mode without bias. The LC molecules that are sufficiently away from the substrates, highlighted in yellow, are not oriented toward the rubbed direction. After applying a sufficiently high voltage, e.g.,

V_{max} = 70 V, we can achieve the o-ray mode for the cell shown in

Figure 2b. From the measured phase difference for THz wave propagation through LC cells in these two modes, we find the maximum possible difference in relative phase retardation, ∆

ϕ(

V_{max}) can be significantly different from the total relative phase shift in the ideal case, 2π

f (

n_{e} −

n_{o})

d/

c_{0}. We propose, therefore, to apply a correction factor for the phase shift,

If the LC molecules in the middle of the cell are not perfectly aligned with the rubbing direction because of the weak anchoring ability of the thick LC device, α > 1.

We first measured the effective refractive index,

n_{p}, which is the sum of the refractive indices in the o-ray mode,

n_{o},

_{Exp}, and the birefringence that is caused by the pretilt angle, ∆

n_{E}, in the absence of the external field (see

Figure 2c). The same procedure was repeated with the cell biased at 70 V (rms), as shown in

Figure 2b. The parameters, ∆

ϕ_{E}, and

n_{p} are then related by

where

θ_{0} is the pretilt angle. On the other hand,

n_{p} can also be determined by using the index ellipsoid shown in

Figure 2d,

For the case of β < 1, the LC cell exhibits a pretilt angle in the absence of the bias electric field.