Abstract

General theory, which can completely describe the asymmetrical optics in a functional material (FM)-doped 90° twisted nematic liquid crystals (TNLCs), is proposed using Cayley–Hamilton theorem and Jones calculus. The FMs, whose shape and size are similar to those of the adopted NLCs, can be aligned along the long axes of the NLCs. The FMs discussed herein are dichroic dye (DD) and polymer. The experimental results of asymmetrical transmission in DD-doped 90° TNLCs are consistent with the theoretical calculation. Such asymmetrical characterization can be further used in the current applications based on 90° TNLCs in all fields to obtain new potential functions.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The optical behavior of 90° twisted nematic liquid crystals (TNLCs), a versatile optical LC device, has been widely studied in the past decades [1–3]. The polarization state of output light after passing through 90° TNLCs is 90° rotated. The 90° TNLCs satisfying the Gooch–Tarry condition [2-3] have been widely applied in LC display (LCD) industry. The display mode of 90° TNLCs is currently one of the most common display applications in the world. Given their capacity to rotate polarization direction of input linearly polarized (LP) lights for 90°, 90° TNLCs also present potential application in several kinds of photonic devices, such as phase-only spatial light modulator, variable optical attenuator, optical light shutter, 3D device, asymmetrical polarization-dependent light shutter/polarizer, polarization rotator, and optical fiber [1, 3–11]. The considerable advantages of optical devices embedded with 90° TNLCs include simple fabrication, inexpensive cost, fast response, and electrical control [3]. Although 90° TNLC is a versatile optical device, detailed research about its asymmetric optical properties is still limited. In 2017, we reported a complete investigation on polymer network (PN) 90° TNLCs with notable asymmetrical transmission/scattering and reflection [11]. Here, we propose a general theory, which can completely describe the asymmetrical optics in functional material (FM)-doped 90° TNLCs, based on Cayley–Hamilton theorem and Jones calculus. The doped FMs with the shape and size similar to those of the adopted NLCs can be aligned along the directors of NLCs. Such asymmetrical characterization can be further used to the aforementioned applications based on 90° TNLCs to obtain new potential functions [1, 3–10].

Dichroic dye-doped LCs (DDdLCs) can act as guest–host (GH) LCDs, light shutters and optical switches/windows [2-3, 12–17]. The GH effect means that the long axis of the guest (DDs) can be aligned parallel to the director of the host (LCs), thereby indicating that the doped DDs will rotate with the used LCs applied with external fields. When the LC director in a DDdLCs cell is rotated perpendicular to the glass substrates by the application of a suitable field, the absorbance of input lights by DDs is the weakest. Such a state is commonly considered a transparent state. Notably, GH LCDs based on DDd-90° TNLCs have also been investigated [18-19]. However, investigation about the asymmetrical optics of DDd-90° TNLCs is not reported in detail yet. Here, a designed experiment demonstrates that DDd-90° TNLCs can successfully perform asymmetrical transmission. The experimental results will also be adopted to assess our proposed general theory of asymmetrical polarization-dependent optics.

Several optical devices showing asymmetrical optical properties haven been developed [11, 20–24]. Some devices can electrically manipulate light to prevent unwanted LP light from traveling along specifically undesired direction [11, 21, 23, 24]. This ability results in their potential application in laser resonator, optical diode/isolator, and telecommunication [11, 21, 23–26]. FM-doped 90° TNLC has potential to be applied for the aforementioned fields.

2. General theory of FM-doped 90° TNLCs

First, we use DDd-90° TNLCs as example to construct the theory of FM-doped 90° TNLCs. We subsequently extend the results to show that this theory can also be adopted to describe the asymmetrical scattering/reflection in PN-90° TNLCs reported in our previous study [11]. Figure 1 shows the schematic structures of a DDd-90° TNLC cell, where the blue and black rods represent the LC and DD molecules, respectively. The polarization directions of LP light incident from the left (right) side, named as L1in/L2in (L4in/L3in), are perpendicular/parallel to the LC director close to the entrance plane. The rubbing directions of homogeneous planarly aligned films on S1 and S2 sides are perpendicular and parallel to x-axis, respectively. According to the experimental results and theoretical analyses, the light intensity loss of L1in/L4in, which is caused by the DD absorbance, is smaller than that of the L2in/L3in after passing through the whole DDd-90° TNLC cell. Therefore, the transmission of L1out/L4out is higher than that of the L2out/L3out. To analyze the asymmetrical transmission of DDd-90° TNLCs, the following theoretical analysis focuses on the behavior of L1in and L2in by Cayley–Hamilton theorem with Jones calculus.

 figure: Fig. 1

Fig. 1 Schematics of dichroic dye-doped (DDd) 90° twisted nematic liquid crystal (TNLC) structures.

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First, the DDd-90° TNLC plate can be divided into N equally thin plates [2,3] of birefringent material. Each thin plate can be considered as a homogeneous LC layer, of which the slow axis of LC director orients at azimuthal angles 1ρ, 2ρ, 3ρ,…, (N–1)ρ, (ρ = ϕ/N). The thickness of each thin plate is dTN/N. Equation (1) shows the phase retardation matrix (W0) of each thin plate:

W0=[eiΓ2N00eiΓ2N],
where Γ represents the phase retardation provided by the DDd-90° TNLC plate, that is,
Γ=2πΔndTNλ,
where λ, ∆n, and dTN represent the input light wavelength in vacuum, LC birefringence, and thickness of the DDd-90° TNLC plate, respectively. Equation (3) presents the transmission/absorbance parameter matrix (A) of each thin plate:
A=[a00b],
where a=αN and b=βN. The definitions of parameters a and b in Eq. (3) are based on the assumption that the DD absorbance in each layer is identical. The DDs rotate with the LC molecules due to the GH effect. The long axes of the doped DD molecules are parallel to those of LCs [3]. Given that each thin plate can be viewed as a homogenous LC layer (thickness = dTN / N), the transmission parameter [α (β)] is defined as the transmission of the output light after passing through the whole DDd-nematic LCs in a homogeneously aligned LC cell with thickness of dH, when the polarization direction of the input LP light is parallel (perpendicular) to the LC director. The dTN value is the same as that of dH. With regard to positive DDs, α should be smaller than β. When α and β are both zero, the DDs completely absorb the input LP lights. Both of these values are positive and smaller than 1 (≤ 1). The long axes of positive DDs are parallel to those of LCs. Therefore, the overall Jones matrix of DDd-90° TNLC (M) can be written as follows:
M=VNVN1...V3V2V1=m=1NWm=m=1NR(mρ)AW0R(mρ),
where Vm represents the Jones matrix for mth thin plate of birefringent material, and R(mρ) denotes the mth coordinate 2D rotation matrix, which can be written as follows:

R(mρ)=[cos(mρ)sin(mρ)sin(mρ)cos(mρ)].

Under a special case, if the transmission parameters α and β equal 1, no LP lights will be absorbed by the doped DDs. Equation (4) becomes the overall Jones matrix for the N thin plates or the common TNLC Jones matrix [2, 3]. With consideration of a common case, Eq. (4) can be further rewritten as follows:

M=m=1NR(mρ)AW0R(mρ)=R(φ)(cos(φN)aeiΓ2Nsin(φN)aeiΓ2Nsin(φN)beiΓ2Ncos(φN)beiΓ2N)N.

M with the Nth power should be reduced. Nevertheless, the matrix cannot be easily simplified using Chebyshev’s identity because the above Jones matrix is a nonunimodular matrix [2]. Hence, the Cayley–Hamilton theorem is utilized to reduce the Jones matrix [27–29]. We define the matrix H as follows:

H=[DEFG]=[cos(φN)aeiΓ2Nsin(φN)aeiΓ2Nsin(φN)beiΓ2Ncos(φN)beiΓ2N].

Afterward, HN should also be solved. To use the Cayley–Hamilton theorem, we define the characteristic polynomial (p(λ)) as follows:

p(λ)=det(λIH)=det([λ00λ][DEFG])=det[λDEFλG]=λ2(D+G)λ+(DGEF),
where det is the determinant operation, and λ and matrix I are the elements of the characteristic polynomial and the 2×2 unit matrix, respectively. With the use of the Caley–Hamilton theorem, Eq. (8) is transformed as follows:
H2(D+G)H+(DGEF)I=O,
where O is a 2×2 zero matrix. Equation (8) is the characteristic polynomial of matrix H. Accordingly, two eigenvalues, namely, λ1 and λ2, can be obtained from Eq. (8):

λ1=(D+G)+(D+G)24(DGEF)2,
λ2=(D+G)(D+G)24(DGEF)2,

to determine HN via Cayley–Hamilton theorem, Eq. (12) is defined as follows:

HN=KH+J,

with substitution of Eqs. (10) and (11) into Eq. (12), the following equations are obtained:

λ1N=Kλ1+J,
λ2N=Kλ2+J,
where K and J can be expressed by λ1 and λ2 as follows: K=λ1Nλ2Nλ1λ2 and J=λ1λ2λ1N1λ2N1λ1λ2. When K and J are substituted into Eq. (12), HN can be expressed as follows:
(cos(φN)aeiΓ2Nsin(φN)aeiΓ2Nsin(φN)beiΓ2Ncos(φN)beiΓ2N)N=λ1Nλ2Nλ1λ2[cos(φN)aeiΓ2Nsin(φN)aeiΓ2Nsin(φN)beiΓ2Ncos(φN)beiΓ2N]λ1λ2λ1N1λ2N1λ1λ2[1001],
where λ1 and λ2 can be expressed as follows:

λ1=cos(φN)(aeiΓ2N+beiΓ2N)+cos2(φN)(aeiΓ2N+beiΓ2N)24ab2,
λ2=cos(φN)(aeiΓ2N+beiΓ2N)cos2(φN)(aeiΓ2N+beiΓ2N)24ab2,

if the input light is a LP light whose Jones matrix is [10], the light passes through the HN, as shown in Eq. (6), which can be written as the following:

(cos(φN)aeiΓ2Nsin(φN)aeiΓ2Nsin(φN)beiΓ2Ncos(φN)beiΓ2N)N[10]=λ1Nλ2Nλ1λ2[cos(φN)aeiΓ2Nsin(φN)beiΓ2N]λ1λ2λ1N1λ2N1λ1λ2[10]=[(λ1Nλ2N)cos(φN)aeiΓ2Nλ1λ2(λ1N1λ2N1)λ1λ2(λ1Nλ2N)sin(φN)beiΓ2Nλ1λ2]=[Kcos(φN)αNeiΓ2NJKsin(φN)βNeiΓ2N],

the Jones vector of the output light (L1) (Fig. 1) through the DDd-90° TNLC can be expressed as follows:

OL1=R(π2)[Kcos(φN)αNeiΓ2NJKsin(φN)βNeiΓ2N]=[Ksin(φN)βNeiΓ2NKcos(φN)αNeiΓ2NJ],

when the Jones matrix of input LP light is [01], the light passes through the matrix HN in Eq. (6), which can be written as follows:

(cos(φN)aeiΓ2Nsin(φN)aeiΓ2Nsin(φN)beiΓ2Ncos(φN)beiΓ2N)N[01]=λ1Nλ2Nλ1λ2[sin(φN)aeiΓ2Ncos(φN)beiΓ2N]λ1λ2λ1N1λ2N1λ1λ2[01]=[(λ1Nλ2N)sin(φN)aeiΓ2Nλ1λ2(λ1Nλ2N)cos(φN)beiΓ2Nλ1λ2(λ1N1λ2N1)λ1λ2]=[Ksin(φN)αNeiΓ2NKcos(φN)βNeiΓ2NJ],

the Jones matrix of the output light (L2) (Fig. 1) through the DDd-90° TNLC can be expressed as follows:

OL2=R(π2)[Ksin(φN)αNeiΓ2NKcos(φN)βNeiΓ2NJ]=[Kcos(φN)βNeiΓ2N+JKsin(φN)αNeiΓ2N],

accordingly, the general form of the output light L1 transmission (TOL1) can be written as the following:

TOL1=JJ+KKcos2(ϕN)(αN)2αNcos(ϕN)(KJeiΓ2N+JKeiΓ2N)+KKsin2(ϕN)(βN)2,

moreover, the general form of the output light L2 transmission (TOL2) can be written as follows:

TOL2=JJ+KKcos2(ϕN)(βN)2βNcos(ϕN)(KJeiΓ2N+JKeiΓ2N)+KKsin2(ϕN)(αN)2,

the transmission difference (ΔT) between TOL1 [Eq. (22)] and TOL2 [Eq. (23)] is given below:

ΔT=KKcos2(ϕN)((βN)2(αN)2)cos(ϕN)(KJ(βNeiΓ2NαNeiΓ2N)+JK(βNeiΓ2NαNeiΓ2N)),KKsin2(ϕN)((βN)2(αN)2)

3. Experimental preparation

The materials adopted herein were E7 (nematic LCs, Merck) and S428 (DDs, Mitsui). Two homogeneous mixtures, namely, mixtures A and B, were prepared. Mixture A (B) was composed of ~98 (~99.8) wt% E7 and ~2 (0.2) wt% S428. The birefringence of E7 (from Merck) is about 0.211 (for wavelength of 633 nm at 25°C). With regard the empty cells, two indium tin oxide-coated glass substrates coated with homogeneous planarly aligned layers were mechanically rubbed along two orthogonal directions (x- and y-axes, Fig. 1). An empty cell with the cell gap of approximately 18 µm, defined by spacer beads, was fabricated by assembling these two substrates. Finally, the homogeneous mixtures A and B were filled into the empty cells to produce the LC cells with DDd-90° TNLCs. The edges of the LC cells were sealed with epoxy.

4. Results and discussions

4.1. Experimental results on the asymmetric transmission of DDd-90° TNLCs

An experiment was designed to demonstrate the properties of the asymmetrical transmission of the DDd-90° TNLCs using mixture A. Figures 2(a) and 2(b) show the experimental setups for two different observations. The nature light source passed through the polarizer and became a LP light whose polarization direction was set parallel to y-axis [red arrows in Fig. 2(a)/2(b)]. The L1out/L4out, L2out/L3out, and S1/S2 depicted in Fig. 2 are the same as those shown in Fig. 1. Figures 2(c) and 2(d) present the observed images of L1out/L4out and L2out/L3out, which were captured by a digital camera using the experimental setup in Figs. 2(a) and 2(b), respectively. As shown in Fig. 2(a), when the LP light travels to the surface (S2) along + z-axis, the LP backlight light can pass through the DDd-90° TNLCs without evident light absorbance, and the polarization plane will be 90° rotated. If the LP light travels to the surface (S1) along + z-axis [Fig. 2(b)], a large amount of light will be absorbed by the doped DDs. A video (Visualization 1) shows the asymmetrical transmission of the DDd-90° TNLC by rotating the LC cell to 180°.

 figure: Fig. 2

Fig. 2 (a) L1in/out and L4in/out observations (Fig. 1) through DDd-90° TNLCs according to the experimental setup shown in (b). (c) L2in/out and L3in/out observations (Fig. 1) according to the experimental setup shown in (d). Red, black, and blue arrows represent the linear polarization directions of the light source (the transmission axis of the polarizer), rubbing directions of substrate S1, and rubbing directions of substrate S2, respectively (see Visualization 1).

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Figure 3 illustrates the experimental setup for measuring the transmission versus angle of the transmission axis of linear polarizer (T-AoTAoLP) curves of the DDd-90° TNLCs. The unpolarized probed He–Ne laser first passed through a beam expander and subsequently traveled through an iris. After polarization with a linear polarizer, the LP light was split into two beams by a beam splitter (BS). One LP light split from BS was used to monitor the stability of the intensity of the probed He–Ne laser; the other one was used as a probed beam to measure the T-AoTAoLP curves of the LC cell filled with mixtures A and B.

 figure: Fig. 3

Fig. 3 Experimental setup (drawn by Microsoft PowerPoint) for measuring the transmission versus angle of the transmission axis of linear polarizer curves of the DDd-90° TNLCs by rotating the linear polarizer for 360°.

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In Fig. 4, the blue and red curves represent the measured T-AoTAoLP curves of the DDd-90° TNLCs filled with mixtures A and B by rotating the linear polarizer shown in Fig. 3 for 360°, respectively. Both curves show that the maximum/minimum transmission can be obtained as the angle of transmission axis of the linear polarizer is 90° (270°)/0° (180° and 360°) with respect to the rubbing direction of the substrate facing the incident light. With regard to mixture B with low concentration of the doped DDs (0.2 wt%, blue curve), the minimum transmission was measured approximately 0.18. Increasing the concentration of the doped DDs to 2.0 wt% (red curve) can improve the dark state (minimum transmission), but the tradeoff is the decrease of maximum transmission (from ~0.7 to ~0.24). Observations on the maximum and minimum states of the DDd-90° TNLCs with 2.0 wt% DDs are demonstrated in Figs. 2(a) and 2(c), respectively. Figure 2(a) presents the observation on L1in/out and L4in/out cases (Fig. 1) through DDd-90° TNLCs according to the experimental setup shown in Fig. 2(b). Figure 2(c) displays the observation on L2in/out and L3in/out cases (Fig. 1) according to the experimental setup shown in Fig. 2(d). In actual applications, DD concentrations should be further optimized. Theoretical analysis on the asymmetrical transmissions of DDd-90° TNLCs filled with mixtures A and B will be discussed in subsequent section.

 figure: Fig. 4

Fig. 4 Red and blue curves show the plots of the measured transmission of such DDd-90° TNLCs filled with mixtures A and B, respectively, as a function of the degree of polarization direction of the linearly polarized probed light. The corresponding experimental setup is shown in Fig. 3.

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4.2. Theoretical calculation of the asymmetric transmission of DDd-90° TNLCs

4.2.1. Asymmetric transmission of DDd-90° TNLCs (α ≈β)

We first consider the specific dye-doped 90° TNLCs, in which the α and β values of dyes are considerably close. Subsequently, we theoretically analyze the different asymmetrical transmissions between mixtures A and B. Under this condition, the transmission parameter α of the doped DDs is slightly smaller than that of β. Equations (16) and (17) can be rewritten as Eqs. (25) and (26), respectively.

λ1cos(φN)(βNcos(Γ2N))+cos2(φN)(βNcos(Γ2N))2β2N,
λ1cos(φN)(βNcos(Γ2N))cos2(φN)(βNcos(Γ2N))2β2N,
Equation (24) is transformed as follows:
ΔTβNcos(φN)(KJ(2sin(Γ2N))+JK(2sin(Γ2N)))0,
where ΔT is approximately 0. To determine the property of the asymmetrical transmission by DDd-90° TNLCs, the difference between α and β cannot be largely small.

Afterward, the experimental results shown in Fig. 4 are analyzed using the presented theoretical method. Nevertheless, the α and β values of doped DDs are difficult to define because they depend on the types of doped DDs, concentrations and dichroic ratios of doped DDs, absorbance performance, and cell thickness. Notably, the β/α value is not the dichroic ratio of the doped DDs. A set of α and β is first selected to obtain the TOL1 and TOL2 values. Details will be discussed in Sections 4.2.B, 4.2.C, and 4.2.D. Theoretical results were compared with the experimental ones to adjust the values and consequently determine the suitable α and β so that the experimental results of blue and red curves shown in Fig. 4 can be clearly elucidated. All of the following theoretical analyses are based on the following conditions: the split 12 thin plates (N = 12) with the same LC thickness, the 12π of the phase retardation Γ, [Eq. (2)], and the 0.5π of twisted angle (ϕ).

4.2.2. Asymmetric transmission of DDd-90° TNLCs (α = 0.09, β = 0.9)

With consideration that the transmission parameter α is 10 times larger than that of β, that is, β equals 10α, we assume that α and β are 0.09 and 0.9, respectively. With the use of the Cayley–Hamilton theorem, λ1 [Eq. (16)] and λ2 [Eq. (17)] in this case can be calculated to be about 0.99i and −0.819i, respectively. Therefore, K and J are about −0.44i and −0.451, respectively.

Accordingly, the Jones vector [Eq. (18)] of the output light of a LP light, [10] passing through the designed DDd-90° TNLCs can be theoretically obtained as follows:

R(π2)(cos(φN)aeiΓ2Nsin(φN)aeiΓ2Nsin(φN)beiΓ2Ncos(φN)beiΓ2N)12[10]R(π2)(-0.44i)[cos(π24)0.0912ei12π24sin(π24)0.912ei12π24]+R(π2)0.451[10],R(π2){0.44i(0.991)0.818i+0.4510.44i(0.131)0.991i}R(π2)[0.0940.057][0.0570.094]
therefore, the theoretical TOL1 [Eq. (22)] value can be calculated to be about 0.012.

The Jones vector [Eq. (20)] of the output light of a LP light, [01] passing through the designed DDd-90° TNLCs can also be theoretically obtained as follows:

R(π2)(cos(φN)aeiΓ2Nsin(φN)aeiΓ2Nsin(φN)beiΓ2Ncos(φN)beiΓ2N)12[01]R(π2)(-0.44i)[sin(π24)0.0912ei12π24cos(π24)0.912ei12π24]+R(π2)0.451[01],=R(π2){0.44i(0.131)0.818i-0.44i(0.991)0.991i+0.451}R(π2)[-0.0470.883][0.8830.047]
the theoretical TOL2 [Eq. (23)] value can be calculated as about 0.802.

Evidently, the theoretical value of TOL2 (0.802) is much larger than that of TOL1 (0.012). The same procedure can be easily adapted to analyze the TOL3 and TOL4. The reported theoretical analyses can completely support the experimental results of the asymmetrical transmission by DDd-90° TNLCs. Moreover, Eq. (29) shows that the input LP light [01] can be transformed to [0.8830.047]. In addition to the reduced amplitude of the incident light caused by the doped DDs, the linear polarization direction is 90° rotated because of the intrinsic property of the 90° TNLC. In accordance with the red curve shown in Fig. 4, TOL2 and TOL1 represent that the minimum and maximum transmissions, as the angle between the transmission axis of the polarizer and the rubbing direction of the substrate facing the incident light, are 90° (270°) and 0° (180°), respectively. However, experimentally, the maximum transmission of the red curve shown in Fig. 4 is ~0.24, which is inconsistent with the value obtained by theoretical analysis, that is, ~0.802. We infer that the β value should be reduced to increase the absorbance of input L1 in the bulk, which will be discussed in Section 4.2.C. Furthermore, the minimum transmission of the blue curve depicted in Fig. 4 is ~0.18, which is also inconsistent with the theoretical value of ~0.012. We infer that the α value should be enhanced to decrease the absorbance of input L2 in the bulk; details will be discussed in Section 4.2.D.

4.2.3. Asymmetric transmission of DDd-90° TNLCs (α = 0.09, β = 0.55)

To fit the theoretical results with the experimental ones (red curve in Fig. 4), β is increased to 0.55, and α is maintained at 0.09. According to the Cayley–Hamilton theorem, λ1 [Eq. (16)] and λ2 [Eq. (17)] in this case are about 0.951i and −0.819i, respectively. Therefore, K and J are about −0.258i and −0.302, respectively. Therefore, Eq. (28) is transformed into Eq. (30):

R(π2)(cos(φN)aeiΓ2Nsin(φN)aeiΓ2Nsin(φN)beiΓ2Ncos(φN)beiΓ2N)12[10]R(π2)(-0.258i)[cos(π24)0.0912ei12π24sin(π24)0.5512ei12π24]+R(π2)0.302[10],R(π2){0.258i(0.991)0.818i+0.3020.258i(0.131)0.951i}R(π2)[0.093-0.032]=[0.0320.093]

the theoretical TOL1 value [Eq. (22)] can be obtained as about 0.01. Equation (29) is transformed into Eq. (31):

R(π2)(cos(φN)aeiΓ2Nsin(φN)aeiΓ2Nsin(φN)beiΓ2Ncos(φN)beiΓ2N)10[01]R(π2)(-0.258i)[sin(π24)0.0912ei12π24cos(π24)0.5512ei12π24]+R(π2)0.302[01],R(π2){0.258i(0.131)0.818i-0.258i(0.991)0.951i+0.302}R(π2)[-0.0280.545][-0.545-0.028]
the theoretical TOL2 value [Eq. (23)] is approximately 0.298.

The reflections caused by two air-glass boundaries (each reflection is ~4%) and the light loss caused by the nonuniform LC cell (~10%) should be considered. Notably, the light loss herein is about 10% according to an additional experiment (not shown). Accordingly, the transmission values of TOL1 (0.01) and TOL2 (0.298) are reduced to approximately 0.008 and 0.247, respectively. The former (~0.008) and the latter (~0.247) responses to the experimentally obtained minimum transmission (~0.013) and maximum transmission (~0.24) of the red curve are shown in Fig. 4.

4.2.4. Asymmetric transmission of DDd-90° TNLCs (α = 0.5, β = 0.9)

To fit the theoretical results with the experimental ones (blue curve in Fig. 4), β is kept at 0.9, and α is decreased to 0.5. With the use of Cayley–Hamilton theorem, the λ1 [Eq. (16)] and λ2 [Eq. (17)] values in this case are about 0.991i and −0.944i, respectively. Therefore, K and J are about −0.205i and −0.695, respectively. Consequently, Eq. (28) is transformed into Eq. (32):

R(π2)(cos(φN)aeiΓ2Nsin(φN)aeiΓ2Nsin(φN)beiΓ2Ncos(φN)beiΓ2N)12[10]R(π2)(-0.205i)[cos(π24)0.512ei12π24sin(π24)0.912ei12π24]+R(π2)0.695[10],R(π2){0.205i(0.991)0.944i+0.6950.205i(0.131)0.991i}R(π2)[0.503-0.027][0.0270.503]

hence, theoretical TOL1 value [Eq. (22)] is about 0.254. In addition, Eq. (29) is transformed into Eq. (33):

R(π2)(cos(φN)aeiΓ2Nsin(φN)aeiΓ2Nsin(φN)beiΓ2Ncos(φN)beiΓ2N)12[01]R(π2)(-0.205i)[sin(π24)0.512ei12π24cos(π24)0.912ei12π24]+R(π2)0.695[01],=R(π2){0.205i(0.131)0.944i-0.205i(0.991)0.991i+0.695}R(π2)[-0.0250.896][-0.8960.025]
the theoretical TOL2 value [Eq. (23)] becomes about 0.804.

With consideration of the reflections caused by the two air-glass boundaries (each reflection is ~4%) and the estimated light loss caused by the nonuniform cell (~10%), the transmission values of TOL1 (0.254) and TOL2 (0.804) are reduced to about 0.21 and 0.667, respectively. The former (~0.211) and the latter (~0.667) responses to the experimentally obtained minimum transmission (~0.18) and maximum transmission (~0.7) of the blue curve are displayed in Fig. 4.

4.3. General theory for PN-90° TNLCs

According to the presented experimental results and theoretical analyses, a general theory that can completely explain the asymmetric scattering and reflection of PN-90° TNLC is proposed [11]. The matrix in Eq. (3) is replaced with the combined matrix of sR, as described in Eqs. (34) and (35).

s=[s001],
R=[tne00tno],
where s represents the scattering parameter of a thin LC plate when the polarization of the component of a polarized light is parallel to the slow axis of LCs [11]. Therefore, the s denotes the transmission of light passing through a thin LC plate, which scatters parts of the component of a polarized light with its polarization direction parallel to the slow axis of LC. Element s22 (1) in Eq. (34) represents that the scattering only occurs when the component of elliptical polarized lights is parallel to the long axis of LC director. The reason for such condition is discussed in our previous work [11]. tne and tno represent the transmissions after passing through the boundaries between LCs and polymer fibrils when the polarization of the component of a polarized light is parallel to the slow and fast axes of LCs, respectively [11]. Hence, tne and tno denote the transmissions of light passing through a thin LC plate, which reflects parts of the component of a polarized light with its polarization direction parallel to the slow and fast axes of LCs, respectively. When the a and b values in Eq. (3) are replaced with the s and 1 (tneand tno) of Eq. (34) [Eq. (35)], respectively, the theoretical analysis can completely explain the asymmetric scattering/reflectivity of PN-90° TNLCs. tnoand tnecan be calculated by the following equations [2,11,30]:
tno=2nnoLCnnoRM257nnoLC+nnopRM257,
tne=2nneLCnneRM257nneLC+nnepRM257,
where nno-LC, nne-LC, nno-pRM257, and nne-pRM257 are the ordinary refractive index (no) of LC, extraordinary refractive index (ne) of LC, no of pRM257, and ne of pRM257, respectively. pRM257 represents the polymerized RM257 fibril. The scattering/reflection of each thin LC plate is identical because the pRM257 formation through the whole LC bulk is assumed to be uniform [11, 31]. Furthermore, to analyze the transmission when the light passes through the PN-90° TNLCs, Eq. (4) is transformed as follows:
M=m=1NR(mρ)[sR]W0R(mρ),
wheresR means that the reflection and scattering in each thin plate should be considered simultaneously.

4.4. General theory for all FM-doped 90° TNLCs

According to Section 4.3, the general theory for all FM-doped 90° TNLCs is as follows:

M=m=1NR(mρ)[z=1NZz]W0R(mρ),
Z and z depend on the adopted FM; z represents the number of types of asymmetrical properties of such a FM-doped 90° TNLCs, and Z denotes the corresponding matrices. For instance, if the FM is RM257, then the PN-90° TNLCs show asymmetrical scattering and reflection. Accordingly, z is 2, and Z1 and Z2 are s and R, respectively: z=12Zz=Z1Z2=sR. If the FM is DDs, then the PN-90° TNLC shows asymmetrical transmission. Accordingly, z is 1, and Z1 is A, that is,z=11Zz=Z1=A. If the FM (X)-doped TNLCs show asymmetrical transmission, scattering, and reflection, then z = 3, and Z1, Z2, and Z3 are s, R, and A. Additionally,z=13Zz=Z1Z2Z3=sRA.

5. Conclusion

A general theory that can completely describe the asymmetrical optics in all FM-doped 90° TNLCs by using Cayley–Hamilton theorem and Jones calculus is proposed. The asymmetrical transmission in DDd-90° TNLCs is discussed theoretically and experimentally. The experimental results on DDd-90° TNLC support the theoretical calculations. The proposed theory can describe all FM-doped 90° TNLCs. In Eq. (39), z is larger than 3 if FM-doped TNLCs show more than three asymmetrical optics. Such asymmetrical characterization of a FM-doped 90° TNLC can also be added to all the presented applications to gain new potential functions [3–10]. Moreover, the limitation and the detailed effects of the number N, the division of the TNLC plate [Eq. (4)], onto the theoretically analytic results are under investigation. Furthermore, for other cases, if special TNLCs 2 × 2 matrixes with the Nth power, whose four elements in the matrix do not meet the Chebyshev’s identity, are derived, the proposed method based on Cayley–Hamilton theorem and Jones calculus can be a useful reference to simplify the complex matrixes.

Funding

Ministry of Science and Technology of Taiwan (MOST 106-2112-M-008-002-MY3).

Acknowledgments

We sincerely thank the reviewers for their valuable comments and significant suggestions.

References and links

1. C. H. Gooch and H. A. Tarry, “The optical properties of twisted nematic liquid crystal structures with twist angles ≤90 degrees,” J. Phys. D Appl. Phys. 8(13), 1575–1584 (1975). [CrossRef]  

2. P. Yeh and C. Gu, Optics of Liquid Crystals Displays (New York: Wiley, 1999).

3. S. T. Wu and D. K. Yang, Reflective Liquid Crystal Displays (New York: Wiley, 2001).

4. T. H. Barnes, T. Eiju, K. Matusda, and N. Ooyama, “Phase-only modulation using a twisted nematic liquid crystal television,” Appl. Opt. 28(22), 4845–4852 (1989). [CrossRef]   [PubMed]  

5. Y.-Q. Lu, F. Du, Y.-H. Lin, and S.-T. Wu, “Variable optical attenuator based on polymer stabilized twisted nematic liquid crystal,” Opt. Express 12(7), 1221–1227 (2004). [CrossRef]   [PubMed]  

6. A. Serrano-Trujillo, L. E. Palafox, and V. Ruiz-Cortés, “Engineering of cylindrical vector fields with a twisted nematic spatial light modulator,” Appl. Opt. 56(5), 1310–1316 (2017). [CrossRef]  

7. L. Lipton, “Drive method for twisted nematic liquid crystal shutters for stereoscopic and other applications,” US5181133A, Jan. 19, 1993.

8. R. Mazur, W. Piecek, Z. Raszewski, P. Morawiak, K. Garbat, O. Chojnowska, M. Mrukiewicz, M. Olifierczuk, J. Kedzierski, R. Dabrowski, and D. Weglowska, “Nematic liquid crystal mixtures for 3D active glasses application,” Liq. Cryst. 44, 417–426 (2017).

9. X.-T. Wang, H.-H. Hou, P.C. Chang, K.-H. Lin, C.-C. Li, H.-C. Jau, Y.-J. Hung, and T.-H. Lin, “Widely tunable guided-mode resonance filter using 90o twisted liquid crystal cladding,” Opt. Fib. Comm. Conf. Th1G.5 (2017).

10. P. Fiala, C. Dorrer, and K. Marshall, “Twisted-Nematic Liquid Crystal Polarization Rotators for Broadband Laser Applications,” in CLEO: 2015, OSA Technical Digest (online) (Optical Society of America, 2015), paper JW2A.68.

11. C. K. Liu, W. H. Chen, and K. T. Cheng, “Asymmetrical polarization-dependent scattering and reflection in a sole cell of polymer network-90° twisted nematic liquid crystals,” Opt. Express 25(19), 22388–22403 (2017). [CrossRef]   [PubMed]  

12. G. H. Heilmeier, J. A. Castellano, and L. A. Zanoni, “Guest-Host interactions in nematic liquid crystals,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 8, 293–304 (1969).

13. T. Uchida and M. Wada, “Guest-host type liquid crystal displays,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 63(1), 19–43 (1981). [CrossRef]  

14. H. W. Ren and S.-T. Wu, “Anisotropic liquid crystal gels for switchable polarizers and displays,” Appl. Phys. Lett. 81(8), 1432–1434 (2002). [CrossRef]  

15. H.-J. Jin, K.-H. Kim, H. Jin, J. Chang Kim, and T.-H. Yoon, “Dye-doped liquid crystal device switchable between reflective and transmissive modes,” J. Inf. Disp. 12(1), 17–21 (2011). [CrossRef]  

16. C.-T. Wang and T.-H. Lin, “Bistable reflective polarizer-free optical switch based on dye-doped cholesteric liquid crystal,” Opt. Mater. Express 1(8), 1457–1462 (2011). [CrossRef]  

17. S.-W. Oh, J.-M. Baek, and T.-H. Yoon, “Sunlight-switchable light shutter fabricated using liquid crystals doped with push-pull azobenzene,” Opt. Express 24(23), 26575–26582 (2016). [CrossRef]   [PubMed]  

18. T. Uchida, H. Seki, C. Shishid, and M. Wada, “Guest-Host Interactions in Nematic Liquid Crystal Cells with Twisted and Tilted Alignments,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 54(3-4), 161–174 (1979). [CrossRef]  

19. S. T. Wu, “Reflective liquid crystal display and transmissive dye-doped liquid crystal display,” US patent, US 6151094 A, Nov 21 (2000).

20. C. K. Liu, K. T. Cheng, and A. Y. G. Fuh, “Observation of anisotropically reflected colors in chiral monomer-doped cholesteric liquid crystals,” Appl. Phys. Lett. 98(4), 041106 (2011). [CrossRef]  

21. J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4(5), 383–387 (2005). [CrossRef]   [PubMed]  

22. S. Y. Lu and L. C. Chien, “A polymer-stabilized single-layer color cholesteric liquid crystal display with anisotropic reflection,” Appl. Phys. Lett. 91(13), 131119 (2007). [CrossRef]  

23. K. Jamshidi-Ghaleh and F. Moslemi, “Electrically tunable all-optical diode in a one-dimensional photonic crystal structure,” Appl. Opt. 56(14), 4146–4152 (2017). [CrossRef]   [PubMed]  

24. D. Liu, S. Hu, and Y.-H. Gao, “Polarization-independent one-way transmission by silicon annular photonic crystal antireflection structures,” Opt. Commun. 420, 127–132 (2018). [CrossRef]  

25. G. Bendelli and S. Donati, “Optical isolators for telecommunications: Review and current trend,” Europ. Trans. On Telecomm 3(4), 63–69 (1992). [CrossRef]  

26. S. Hua, J. Wen, X. Jiang, Q. Hua, L. Jiang, and M. Xiao, “Demonstration of a chip-based optical isolator with parametric amplification,” Nat. Commun. 7, 13657 (2016). [CrossRef]   [PubMed]  

27. T. Andreescu, Essential Linear Algebra with Applications: A Problem-Solving Approach (New York: Springer, 2014).

28. P. J. Nahin, Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills (Princeton University Press, 2006).

29. G. R. Fowles, Introduction to Modern Optics, (Dover Publications, 2nd edition, 1989).

30. R. Apetz and M. P. B. van Bruggen, “Transparent alumina: a light-scattering model,” J. Am. Ceram. Soc. 86(3), 480–486 (2003). [CrossRef]  

31. A. Y. G. Fuh, W. K. Chen, K. T. Cheng, Y. C. Liu, C. K. Liu, and Y. D. Chen, “Formation of holographic gratings in polymer dispersed liquid crystals using off-resonant light,” Opt. Mater. Express 5(4), 774–780 (2015). [CrossRef]  

References

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  1. C. H. Gooch and H. A. Tarry, “The optical properties of twisted nematic liquid crystal structures with twist angles ≤90 degrees,” J. Phys. D Appl. Phys. 8(13), 1575–1584 (1975).
    [Crossref]
  2. P. Yeh and C. Gu, Optics of Liquid Crystals Displays (New York: Wiley, 1999).
  3. S. T. Wu and D. K. Yang, Reflective Liquid Crystal Displays (New York: Wiley, 2001).
  4. T. H. Barnes, T. Eiju, K. Matusda, and N. Ooyama, “Phase-only modulation using a twisted nematic liquid crystal television,” Appl. Opt. 28(22), 4845–4852 (1989).
    [Crossref] [PubMed]
  5. Y.-Q. Lu, F. Du, Y.-H. Lin, and S.-T. Wu, “Variable optical attenuator based on polymer stabilized twisted nematic liquid crystal,” Opt. Express 12(7), 1221–1227 (2004).
    [Crossref] [PubMed]
  6. A. Serrano-Trujillo, L. E. Palafox, and V. Ruiz-Cortés, “Engineering of cylindrical vector fields with a twisted nematic spatial light modulator,” Appl. Opt. 56(5), 1310–1316 (2017).
    [Crossref]
  7. L. Lipton, “Drive method for twisted nematic liquid crystal shutters for stereoscopic and other applications,” US5181133A, Jan. 19, 1993.
  8. R. Mazur, W. Piecek, Z. Raszewski, P. Morawiak, K. Garbat, O. Chojnowska, M. Mrukiewicz, M. Olifierczuk, J. Kedzierski, R. Dabrowski, and D. Weglowska, “Nematic liquid crystal mixtures for 3D active glasses application,” Liq. Cryst. 44, 417–426 (2017).
  9. X.-T. Wang, H.-H. Hou, P.C. Chang, K.-H. Lin, C.-C. Li, H.-C. Jau, Y.-J. Hung, and T.-H. Lin, “Widely tunable guided-mode resonance filter using 90o twisted liquid crystal cladding,” Opt. Fib. Comm. Conf. Th1G.5 (2017).
  10. P. Fiala, C. Dorrer, and K. Marshall, “Twisted-Nematic Liquid Crystal Polarization Rotators for Broadband Laser Applications,” in CLEO: 2015, OSA Technical Digest (online) (Optical Society of America, 2015), paper JW2A.68.
  11. C. K. Liu, W. H. Chen, and K. T. Cheng, “Asymmetrical polarization-dependent scattering and reflection in a sole cell of polymer network-90° twisted nematic liquid crystals,” Opt. Express 25(19), 22388–22403 (2017).
    [Crossref] [PubMed]
  12. G. H. Heilmeier, J. A. Castellano, and L. A. Zanoni, “Guest-Host interactions in nematic liquid crystals,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 8, 293–304 (1969).
  13. T. Uchida and M. Wada, “Guest-host type liquid crystal displays,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 63(1), 19–43 (1981).
    [Crossref]
  14. H. W. Ren and S.-T. Wu, “Anisotropic liquid crystal gels for switchable polarizers and displays,” Appl. Phys. Lett. 81(8), 1432–1434 (2002).
    [Crossref]
  15. H.-J. Jin, K.-H. Kim, H. Jin, J. Chang Kim, and T.-H. Yoon, “Dye-doped liquid crystal device switchable between reflective and transmissive modes,” J. Inf. Disp. 12(1), 17–21 (2011).
    [Crossref]
  16. C.-T. Wang and T.-H. Lin, “Bistable reflective polarizer-free optical switch based on dye-doped cholesteric liquid crystal,” Opt. Mater. Express 1(8), 1457–1462 (2011).
    [Crossref]
  17. S.-W. Oh, J.-M. Baek, and T.-H. Yoon, “Sunlight-switchable light shutter fabricated using liquid crystals doped with push-pull azobenzene,” Opt. Express 24(23), 26575–26582 (2016).
    [Crossref] [PubMed]
  18. T. Uchida, H. Seki, C. Shishid, and M. Wada, “Guest-Host Interactions in Nematic Liquid Crystal Cells with Twisted and Tilted Alignments,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 54(3-4), 161–174 (1979).
    [Crossref]
  19. S. T. Wu, “Reflective liquid crystal display and transmissive dye-doped liquid crystal display,” US patent, US 6151094 A, Nov 21 (2000).
  20. C. K. Liu, K. T. Cheng, and A. Y. G. Fuh, “Observation of anisotropically reflected colors in chiral monomer-doped cholesteric liquid crystals,” Appl. Phys. Lett. 98(4), 041106 (2011).
    [Crossref]
  21. J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4(5), 383–387 (2005).
    [Crossref] [PubMed]
  22. S. Y. Lu and L. C. Chien, “A polymer-stabilized single-layer color cholesteric liquid crystal display with anisotropic reflection,” Appl. Phys. Lett. 91(13), 131119 (2007).
    [Crossref]
  23. K. Jamshidi-Ghaleh and F. Moslemi, “Electrically tunable all-optical diode in a one-dimensional photonic crystal structure,” Appl. Opt. 56(14), 4146–4152 (2017).
    [Crossref] [PubMed]
  24. D. Liu, S. Hu, and Y.-H. Gao, “Polarization-independent one-way transmission by silicon annular photonic crystal antireflection structures,” Opt. Commun. 420, 127–132 (2018).
    [Crossref]
  25. G. Bendelli and S. Donati, “Optical isolators for telecommunications: Review and current trend,” Europ. Trans. On Telecomm 3(4), 63–69 (1992).
    [Crossref]
  26. S. Hua, J. Wen, X. Jiang, Q. Hua, L. Jiang, and M. Xiao, “Demonstration of a chip-based optical isolator with parametric amplification,” Nat. Commun. 7, 13657 (2016).
    [Crossref] [PubMed]
  27. T. Andreescu, Essential Linear Algebra with Applications: A Problem-Solving Approach (New York: Springer, 2014).
  28. P. J. Nahin, Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills (Princeton University Press, 2006).
  29. G. R. Fowles, Introduction to Modern Optics, (Dover Publications, 2nd edition, 1989).
  30. R. Apetz and M. P. B. van Bruggen, “Transparent alumina: a light-scattering model,” J. Am. Ceram. Soc. 86(3), 480–486 (2003).
    [Crossref]
  31. A. Y. G. Fuh, W. K. Chen, K. T. Cheng, Y. C. Liu, C. K. Liu, and Y. D. Chen, “Formation of holographic gratings in polymer dispersed liquid crystals using off-resonant light,” Opt. Mater. Express 5(4), 774–780 (2015).
    [Crossref]

2018 (1)

D. Liu, S. Hu, and Y.-H. Gao, “Polarization-independent one-way transmission by silicon annular photonic crystal antireflection structures,” Opt. Commun. 420, 127–132 (2018).
[Crossref]

2017 (4)

2016 (2)

S.-W. Oh, J.-M. Baek, and T.-H. Yoon, “Sunlight-switchable light shutter fabricated using liquid crystals doped with push-pull azobenzene,” Opt. Express 24(23), 26575–26582 (2016).
[Crossref] [PubMed]

S. Hua, J. Wen, X. Jiang, Q. Hua, L. Jiang, and M. Xiao, “Demonstration of a chip-based optical isolator with parametric amplification,” Nat. Commun. 7, 13657 (2016).
[Crossref] [PubMed]

2015 (1)

2011 (3)

H.-J. Jin, K.-H. Kim, H. Jin, J. Chang Kim, and T.-H. Yoon, “Dye-doped liquid crystal device switchable between reflective and transmissive modes,” J. Inf. Disp. 12(1), 17–21 (2011).
[Crossref]

C.-T. Wang and T.-H. Lin, “Bistable reflective polarizer-free optical switch based on dye-doped cholesteric liquid crystal,” Opt. Mater. Express 1(8), 1457–1462 (2011).
[Crossref]

C. K. Liu, K. T. Cheng, and A. Y. G. Fuh, “Observation of anisotropically reflected colors in chiral monomer-doped cholesteric liquid crystals,” Appl. Phys. Lett. 98(4), 041106 (2011).
[Crossref]

2007 (1)

S. Y. Lu and L. C. Chien, “A polymer-stabilized single-layer color cholesteric liquid crystal display with anisotropic reflection,” Appl. Phys. Lett. 91(13), 131119 (2007).
[Crossref]

2005 (1)

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4(5), 383–387 (2005).
[Crossref] [PubMed]

2004 (1)

2003 (1)

R. Apetz and M. P. B. van Bruggen, “Transparent alumina: a light-scattering model,” J. Am. Ceram. Soc. 86(3), 480–486 (2003).
[Crossref]

2002 (1)

H. W. Ren and S.-T. Wu, “Anisotropic liquid crystal gels for switchable polarizers and displays,” Appl. Phys. Lett. 81(8), 1432–1434 (2002).
[Crossref]

1992 (1)

G. Bendelli and S. Donati, “Optical isolators for telecommunications: Review and current trend,” Europ. Trans. On Telecomm 3(4), 63–69 (1992).
[Crossref]

1989 (1)

1981 (1)

T. Uchida and M. Wada, “Guest-host type liquid crystal displays,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 63(1), 19–43 (1981).
[Crossref]

1979 (1)

T. Uchida, H. Seki, C. Shishid, and M. Wada, “Guest-Host Interactions in Nematic Liquid Crystal Cells with Twisted and Tilted Alignments,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 54(3-4), 161–174 (1979).
[Crossref]

1975 (1)

C. H. Gooch and H. A. Tarry, “The optical properties of twisted nematic liquid crystal structures with twist angles ≤90 degrees,” J. Phys. D Appl. Phys. 8(13), 1575–1584 (1975).
[Crossref]

1969 (1)

G. H. Heilmeier, J. A. Castellano, and L. A. Zanoni, “Guest-Host interactions in nematic liquid crystals,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 8, 293–304 (1969).

Apetz, R.

R. Apetz and M. P. B. van Bruggen, “Transparent alumina: a light-scattering model,” J. Am. Ceram. Soc. 86(3), 480–486 (2003).
[Crossref]

Baek, J.-M.

Barnes, T. H.

Bendelli, G.

G. Bendelli and S. Donati, “Optical isolators for telecommunications: Review and current trend,” Europ. Trans. On Telecomm 3(4), 63–69 (1992).
[Crossref]

Castellano, J. A.

G. H. Heilmeier, J. A. Castellano, and L. A. Zanoni, “Guest-Host interactions in nematic liquid crystals,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 8, 293–304 (1969).

Chang Kim, J.

H.-J. Jin, K.-H. Kim, H. Jin, J. Chang Kim, and T.-H. Yoon, “Dye-doped liquid crystal device switchable between reflective and transmissive modes,” J. Inf. Disp. 12(1), 17–21 (2011).
[Crossref]

Chen, W. H.

Chen, W. K.

Chen, Y. D.

Cheng, K. T.

Chien, L. C.

S. Y. Lu and L. C. Chien, “A polymer-stabilized single-layer color cholesteric liquid crystal display with anisotropic reflection,” Appl. Phys. Lett. 91(13), 131119 (2007).
[Crossref]

Chojnowska, O.

R. Mazur, W. Piecek, Z. Raszewski, P. Morawiak, K. Garbat, O. Chojnowska, M. Mrukiewicz, M. Olifierczuk, J. Kedzierski, R. Dabrowski, and D. Weglowska, “Nematic liquid crystal mixtures for 3D active glasses application,” Liq. Cryst. 44, 417–426 (2017).

Dabrowski, R.

R. Mazur, W. Piecek, Z. Raszewski, P. Morawiak, K. Garbat, O. Chojnowska, M. Mrukiewicz, M. Olifierczuk, J. Kedzierski, R. Dabrowski, and D. Weglowska, “Nematic liquid crystal mixtures for 3D active glasses application,” Liq. Cryst. 44, 417–426 (2017).

Donati, S.

G. Bendelli and S. Donati, “Optical isolators for telecommunications: Review and current trend,” Europ. Trans. On Telecomm 3(4), 63–69 (1992).
[Crossref]

Du, F.

Eiju, T.

Fuh, A. Y. G.

A. Y. G. Fuh, W. K. Chen, K. T. Cheng, Y. C. Liu, C. K. Liu, and Y. D. Chen, “Formation of holographic gratings in polymer dispersed liquid crystals using off-resonant light,” Opt. Mater. Express 5(4), 774–780 (2015).
[Crossref]

C. K. Liu, K. T. Cheng, and A. Y. G. Fuh, “Observation of anisotropically reflected colors in chiral monomer-doped cholesteric liquid crystals,” Appl. Phys. Lett. 98(4), 041106 (2011).
[Crossref]

Gao, Y.-H.

D. Liu, S. Hu, and Y.-H. Gao, “Polarization-independent one-way transmission by silicon annular photonic crystal antireflection structures,” Opt. Commun. 420, 127–132 (2018).
[Crossref]

Garbat, K.

R. Mazur, W. Piecek, Z. Raszewski, P. Morawiak, K. Garbat, O. Chojnowska, M. Mrukiewicz, M. Olifierczuk, J. Kedzierski, R. Dabrowski, and D. Weglowska, “Nematic liquid crystal mixtures for 3D active glasses application,” Liq. Cryst. 44, 417–426 (2017).

Gooch, C. H.

C. H. Gooch and H. A. Tarry, “The optical properties of twisted nematic liquid crystal structures with twist angles ≤90 degrees,” J. Phys. D Appl. Phys. 8(13), 1575–1584 (1975).
[Crossref]

Heilmeier, G. H.

G. H. Heilmeier, J. A. Castellano, and L. A. Zanoni, “Guest-Host interactions in nematic liquid crystals,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 8, 293–304 (1969).

Hu, S.

D. Liu, S. Hu, and Y.-H. Gao, “Polarization-independent one-way transmission by silicon annular photonic crystal antireflection structures,” Opt. Commun. 420, 127–132 (2018).
[Crossref]

Hua, Q.

S. Hua, J. Wen, X. Jiang, Q. Hua, L. Jiang, and M. Xiao, “Demonstration of a chip-based optical isolator with parametric amplification,” Nat. Commun. 7, 13657 (2016).
[Crossref] [PubMed]

Hua, S.

S. Hua, J. Wen, X. Jiang, Q. Hua, L. Jiang, and M. Xiao, “Demonstration of a chip-based optical isolator with parametric amplification,” Nat. Commun. 7, 13657 (2016).
[Crossref] [PubMed]

Hwang, J.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4(5), 383–387 (2005).
[Crossref] [PubMed]

Ishikawa, K.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4(5), 383–387 (2005).
[Crossref] [PubMed]

Jamshidi-Ghaleh, K.

Jiang, L.

S. Hua, J. Wen, X. Jiang, Q. Hua, L. Jiang, and M. Xiao, “Demonstration of a chip-based optical isolator with parametric amplification,” Nat. Commun. 7, 13657 (2016).
[Crossref] [PubMed]

Jiang, X.

S. Hua, J. Wen, X. Jiang, Q. Hua, L. Jiang, and M. Xiao, “Demonstration of a chip-based optical isolator with parametric amplification,” Nat. Commun. 7, 13657 (2016).
[Crossref] [PubMed]

Jin, H.

H.-J. Jin, K.-H. Kim, H. Jin, J. Chang Kim, and T.-H. Yoon, “Dye-doped liquid crystal device switchable between reflective and transmissive modes,” J. Inf. Disp. 12(1), 17–21 (2011).
[Crossref]

Jin, H.-J.

H.-J. Jin, K.-H. Kim, H. Jin, J. Chang Kim, and T.-H. Yoon, “Dye-doped liquid crystal device switchable between reflective and transmissive modes,” J. Inf. Disp. 12(1), 17–21 (2011).
[Crossref]

Kedzierski, J.

R. Mazur, W. Piecek, Z. Raszewski, P. Morawiak, K. Garbat, O. Chojnowska, M. Mrukiewicz, M. Olifierczuk, J. Kedzierski, R. Dabrowski, and D. Weglowska, “Nematic liquid crystal mixtures for 3D active glasses application,” Liq. Cryst. 44, 417–426 (2017).

Kim, K.-H.

H.-J. Jin, K.-H. Kim, H. Jin, J. Chang Kim, and T.-H. Yoon, “Dye-doped liquid crystal device switchable between reflective and transmissive modes,” J. Inf. Disp. 12(1), 17–21 (2011).
[Crossref]

Lin, T.-H.

Lin, Y.-H.

Liu, C. K.

Liu, D.

D. Liu, S. Hu, and Y.-H. Gao, “Polarization-independent one-way transmission by silicon annular photonic crystal antireflection structures,” Opt. Commun. 420, 127–132 (2018).
[Crossref]

Liu, Y. C.

Lu, S. Y.

S. Y. Lu and L. C. Chien, “A polymer-stabilized single-layer color cholesteric liquid crystal display with anisotropic reflection,” Appl. Phys. Lett. 91(13), 131119 (2007).
[Crossref]

Lu, Y.-Q.

Matusda, K.

Mazur, R.

R. Mazur, W. Piecek, Z. Raszewski, P. Morawiak, K. Garbat, O. Chojnowska, M. Mrukiewicz, M. Olifierczuk, J. Kedzierski, R. Dabrowski, and D. Weglowska, “Nematic liquid crystal mixtures for 3D active glasses application,” Liq. Cryst. 44, 417–426 (2017).

Morawiak, P.

R. Mazur, W. Piecek, Z. Raszewski, P. Morawiak, K. Garbat, O. Chojnowska, M. Mrukiewicz, M. Olifierczuk, J. Kedzierski, R. Dabrowski, and D. Weglowska, “Nematic liquid crystal mixtures for 3D active glasses application,” Liq. Cryst. 44, 417–426 (2017).

Moslemi, F.

Mrukiewicz, M.

R. Mazur, W. Piecek, Z. Raszewski, P. Morawiak, K. Garbat, O. Chojnowska, M. Mrukiewicz, M. Olifierczuk, J. Kedzierski, R. Dabrowski, and D. Weglowska, “Nematic liquid crystal mixtures for 3D active glasses application,” Liq. Cryst. 44, 417–426 (2017).

Nishimura, S.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4(5), 383–387 (2005).
[Crossref] [PubMed]

Oh, S.-W.

Olifierczuk, M.

R. Mazur, W. Piecek, Z. Raszewski, P. Morawiak, K. Garbat, O. Chojnowska, M. Mrukiewicz, M. Olifierczuk, J. Kedzierski, R. Dabrowski, and D. Weglowska, “Nematic liquid crystal mixtures for 3D active glasses application,” Liq. Cryst. 44, 417–426 (2017).

Ooyama, N.

Palafox, L. E.

Park, B.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4(5), 383–387 (2005).
[Crossref] [PubMed]

Piecek, W.

R. Mazur, W. Piecek, Z. Raszewski, P. Morawiak, K. Garbat, O. Chojnowska, M. Mrukiewicz, M. Olifierczuk, J. Kedzierski, R. Dabrowski, and D. Weglowska, “Nematic liquid crystal mixtures for 3D active glasses application,” Liq. Cryst. 44, 417–426 (2017).

Raszewski, Z.

R. Mazur, W. Piecek, Z. Raszewski, P. Morawiak, K. Garbat, O. Chojnowska, M. Mrukiewicz, M. Olifierczuk, J. Kedzierski, R. Dabrowski, and D. Weglowska, “Nematic liquid crystal mixtures for 3D active glasses application,” Liq. Cryst. 44, 417–426 (2017).

Ren, H. W.

H. W. Ren and S.-T. Wu, “Anisotropic liquid crystal gels for switchable polarizers and displays,” Appl. Phys. Lett. 81(8), 1432–1434 (2002).
[Crossref]

Ruiz-Cortés, V.

Seki, H.

T. Uchida, H. Seki, C. Shishid, and M. Wada, “Guest-Host Interactions in Nematic Liquid Crystal Cells with Twisted and Tilted Alignments,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 54(3-4), 161–174 (1979).
[Crossref]

Serrano-Trujillo, A.

Shishid, C.

T. Uchida, H. Seki, C. Shishid, and M. Wada, “Guest-Host Interactions in Nematic Liquid Crystal Cells with Twisted and Tilted Alignments,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 54(3-4), 161–174 (1979).
[Crossref]

Song, M. H.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4(5), 383–387 (2005).
[Crossref] [PubMed]

Takanishi, Y.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4(5), 383–387 (2005).
[Crossref] [PubMed]

Takezoe, H.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4(5), 383–387 (2005).
[Crossref] [PubMed]

Tarry, H. A.

C. H. Gooch and H. A. Tarry, “The optical properties of twisted nematic liquid crystal structures with twist angles ≤90 degrees,” J. Phys. D Appl. Phys. 8(13), 1575–1584 (1975).
[Crossref]

Toyooka, T.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4(5), 383–387 (2005).
[Crossref] [PubMed]

Uchida, T.

T. Uchida and M. Wada, “Guest-host type liquid crystal displays,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 63(1), 19–43 (1981).
[Crossref]

T. Uchida, H. Seki, C. Shishid, and M. Wada, “Guest-Host Interactions in Nematic Liquid Crystal Cells with Twisted and Tilted Alignments,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 54(3-4), 161–174 (1979).
[Crossref]

van Bruggen, M. P. B.

R. Apetz and M. P. B. van Bruggen, “Transparent alumina: a light-scattering model,” J. Am. Ceram. Soc. 86(3), 480–486 (2003).
[Crossref]

Wada, M.

T. Uchida and M. Wada, “Guest-host type liquid crystal displays,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 63(1), 19–43 (1981).
[Crossref]

T. Uchida, H. Seki, C. Shishid, and M. Wada, “Guest-Host Interactions in Nematic Liquid Crystal Cells with Twisted and Tilted Alignments,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 54(3-4), 161–174 (1979).
[Crossref]

Wang, C.-T.

Weglowska, D.

R. Mazur, W. Piecek, Z. Raszewski, P. Morawiak, K. Garbat, O. Chojnowska, M. Mrukiewicz, M. Olifierczuk, J. Kedzierski, R. Dabrowski, and D. Weglowska, “Nematic liquid crystal mixtures for 3D active glasses application,” Liq. Cryst. 44, 417–426 (2017).

Wen, J.

S. Hua, J. Wen, X. Jiang, Q. Hua, L. Jiang, and M. Xiao, “Demonstration of a chip-based optical isolator with parametric amplification,” Nat. Commun. 7, 13657 (2016).
[Crossref] [PubMed]

Wu, J. W.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4(5), 383–387 (2005).
[Crossref] [PubMed]

Wu, S.-T.

Y.-Q. Lu, F. Du, Y.-H. Lin, and S.-T. Wu, “Variable optical attenuator based on polymer stabilized twisted nematic liquid crystal,” Opt. Express 12(7), 1221–1227 (2004).
[Crossref] [PubMed]

H. W. Ren and S.-T. Wu, “Anisotropic liquid crystal gels for switchable polarizers and displays,” Appl. Phys. Lett. 81(8), 1432–1434 (2002).
[Crossref]

Xiao, M.

S. Hua, J. Wen, X. Jiang, Q. Hua, L. Jiang, and M. Xiao, “Demonstration of a chip-based optical isolator with parametric amplification,” Nat. Commun. 7, 13657 (2016).
[Crossref] [PubMed]

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S.-W. Oh, J.-M. Baek, and T.-H. Yoon, “Sunlight-switchable light shutter fabricated using liquid crystals doped with push-pull azobenzene,” Opt. Express 24(23), 26575–26582 (2016).
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H.-J. Jin, K.-H. Kim, H. Jin, J. Chang Kim, and T.-H. Yoon, “Dye-doped liquid crystal device switchable between reflective and transmissive modes,” J. Inf. Disp. 12(1), 17–21 (2011).
[Crossref]

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G. H. Heilmeier, J. A. Castellano, and L. A. Zanoni, “Guest-Host interactions in nematic liquid crystals,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 8, 293–304 (1969).

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[Crossref]

S. Y. Lu and L. C. Chien, “A polymer-stabilized single-layer color cholesteric liquid crystal display with anisotropic reflection,” Appl. Phys. Lett. 91(13), 131119 (2007).
[Crossref]

H. W. Ren and S.-T. Wu, “Anisotropic liquid crystal gels for switchable polarizers and displays,” Appl. Phys. Lett. 81(8), 1432–1434 (2002).
[Crossref]

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[Crossref]

J. Am. Ceram. Soc. (1)

R. Apetz and M. P. B. van Bruggen, “Transparent alumina: a light-scattering model,” J. Am. Ceram. Soc. 86(3), 480–486 (2003).
[Crossref]

J. Inf. Disp. (1)

H.-J. Jin, K.-H. Kim, H. Jin, J. Chang Kim, and T.-H. Yoon, “Dye-doped liquid crystal device switchable between reflective and transmissive modes,” J. Inf. Disp. 12(1), 17–21 (2011).
[Crossref]

J. Phys. D Appl. Phys. (1)

C. H. Gooch and H. A. Tarry, “The optical properties of twisted nematic liquid crystal structures with twist angles ≤90 degrees,” J. Phys. D Appl. Phys. 8(13), 1575–1584 (1975).
[Crossref]

Liq. Cryst. (1)

R. Mazur, W. Piecek, Z. Raszewski, P. Morawiak, K. Garbat, O. Chojnowska, M. Mrukiewicz, M. Olifierczuk, J. Kedzierski, R. Dabrowski, and D. Weglowska, “Nematic liquid crystal mixtures for 3D active glasses application,” Liq. Cryst. 44, 417–426 (2017).

Mol. Cryst. Liq. Cryst. (Phila. Pa.) (3)

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[Crossref]

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[Crossref]

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[Crossref]

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Supplementary Material (1)

NameDescription
» Visualization 1       The visualization directly exhibits the asymmetrical transmission of the dichroic dye-doped 90° twisted nematic liquid crystal by rotating the LC cell to 180°.

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Figures (4)

Fig. 1
Fig. 1 Schematics of dichroic dye-doped (DDd) 90° twisted nematic liquid crystal (TNLC) structures.
Fig. 2
Fig. 2 (a) L1in/out and L4in/out observations (Fig. 1) through DDd-90° TNLCs according to the experimental setup shown in (b). (c) L2in/out and L3in/out observations (Fig. 1) according to the experimental setup shown in (d). Red, black, and blue arrows represent the linear polarization directions of the light source (the transmission axis of the polarizer), rubbing directions of substrate S1, and rubbing directions of substrate S2, respectively (see Visualization 1).
Fig. 3
Fig. 3 Experimental setup (drawn by Microsoft PowerPoint) for measuring the transmission versus angle of the transmission axis of linear polarizer curves of the DDd-90° TNLCs by rotating the linear polarizer for 360°.
Fig. 4
Fig. 4 Red and blue curves show the plots of the measured transmission of such DDd-90° TNLCs filled with mixtures A and B, respectively, as a function of the degree of polarization direction of the linearly polarized probed light. The corresponding experimental setup is shown in Fig. 3.

Equations (39)

Equations on this page are rendered with MathJax. Learn more.

W 0 =[ e i Γ 2N 0 0 e i Γ 2N ],
Γ= 2πΔn d TN λ ,
A=[ a 0 0 b ],
M= V N V N1 ... V 3 V 2 V 1 = m=1 N W m = m=1 N R( mρ ) A W 0 R( mρ ),
R( mρ )=[ cos(mρ) sin(mρ) sin(mρ) cos(mρ) ].
M= m=1 N R( mρ )A W 0 R( mρ )=R( φ ) ( cos( φ N )a e i Γ 2N sin( φ N )a e i Γ 2N sin( φ N )b e i Γ 2N cos( φ N )b e i Γ 2N ) N .
H=[ D E F G ]=[ cos( φ N )a e i Γ 2N sin( φ N )a e i Γ 2N sin( φ N )b e i Γ 2N cos( φ N )b e i Γ 2N ].
p( λ )=det( λIH )=det( [ λ 0 0 λ ][ D E F G ] )=det[ λD E F λG ] = λ 2 ( D+G )λ+( DGEF ),
H 2 ( D+G )H+( DGEF )I=O,
λ 1 = ( D+G )+ ( D+G ) 2 4( DGEF ) 2 ,
λ 2 = ( D+G ) ( D+G ) 2 4( DGEF ) 2 ,
H N =KH+J,
λ 1 N =K λ 1 +J,
λ 2 N =K λ 2 +J,
( cos( φ N )a e i Γ 2N sin( φ N )a e i Γ 2N sin( φ N )b e i Γ 2N cos( φ N )b e i Γ 2N ) N = λ 1 N λ 2 N λ 1 λ 2 [ cos( φ N )a e i Γ 2N sin( φ N )a e i Γ 2N sin( φ N )b e i Γ 2N cos( φ N )b e i Γ 2N ] λ 1 λ 2 λ 1 N1 λ 2 N1 λ 1 λ 2 [ 1 0 0 1 ],
λ 1 = cos( φ N )( a e i Γ 2N +b e i Γ 2N )+ cos 2 ( φ N ) ( a e i Γ 2N +b e i Γ 2N ) 2 4ab 2 ,
λ 2 = cos( φ N )( a e i Γ 2N +b e i Γ 2N ) cos 2 ( φ N ) ( a e i Γ 2N +b e i Γ 2N ) 2 4ab 2 ,
( cos( φ N )a e i Γ 2N sin( φ N )a e i Γ 2N sin( φ N )b e i Γ 2N cos( φ N )b e i Γ 2N ) N [ 1 0 ] = λ 1 N λ 2 N λ 1 λ 2 [ cos( φ N )a e i Γ 2N sin( φ N )b e i Γ 2N ] λ 1 λ 2 λ 1 N1 λ 2 N1 λ 1 λ 2 [ 1 0 ] =[ ( λ 1 N λ 2 N )cos( φ N )a e i Γ 2N λ 1 λ 2 ( λ 1 N1 λ 2 N1 ) λ 1 λ 2 ( λ 1 N λ 2 N )sin( φ N )b e i Γ 2N λ 1 λ 2 ]=[ Kcos( φ N ) α N e i Γ 2N J Ksin( φ N ) β N e i Γ 2N ],
O L1 =R( π 2 )[ Kcos( φ N ) α N e i Γ 2N J Ksin( φ N ) β N e i Γ 2N ]=[ Ksin( φ N ) β N e i Γ 2N Kcos( φ N ) α N e i Γ 2N J ],
( cos( φ N )a e i Γ 2N sin( φ N )a e i Γ 2N sin( φ N )b e i Γ 2N cos( φ N )b e i Γ 2N ) N [ 0 1 ] = λ 1 N λ 2 N λ 1 λ 2 [ sin( φ N )a e i Γ 2N cos( φ N )b e i Γ 2N ] λ 1 λ 2 λ 1 N1 λ 2 N1 λ 1 λ 2 [ 0 1 ] =[ ( λ 1 N λ 2 N )sin( φ N )a e i Γ 2N λ 1 λ 2 ( λ 1 N λ 2 N )cos( φ N )b e i Γ 2N λ 1 λ 2 ( λ 1 N1 λ 2 N1 ) λ 1 λ 2 ]=[ Ksin( φ N ) α N e i Γ 2N Kcos( φ N ) β N e i Γ 2N J ],
O L2 =R( π 2 )[ Ksin( φ N ) α N e i Γ 2N Kcos( φ N ) β N e i Γ 2N J ]=[ Kcos( φ N ) β N e i Γ 2N +J Ksin( φ N ) α N e i Γ 2N ],
T O L1 = J J+ K K cos 2 ( ϕ N ) ( α N ) 2 α N cos( ϕ N )( K J e i Γ 2N + J K e i Γ 2N ) + K K sin 2 ( ϕ N ) ( β N ) 2 ,
T O L2 = J J+ K K cos 2 ( ϕ N ) ( β N ) 2 β N cos( ϕ N )( K J e i Γ 2N + J K e i Γ 2N ) + K K sin 2 ( ϕ N ) ( α N ) 2 ,
ΔT= K K cos 2 ( ϕ N )( ( β N ) 2 ( α N ) 2 ) cos( ϕ N )( K J( β N e i Γ 2N α N e i Γ 2N )+ J K( β N e i Γ 2N α N e i Γ 2N ) ), K K sin 2 ( ϕ N )( ( β N ) 2 ( α N ) 2 )
λ 1 cos( φ N )( β N cos( Γ 2N ) )+ cos 2 ( φ N ) ( β N cos( Γ 2N ) ) 2 β 2 N ,
λ 1 cos( φ N )( β N cos( Γ 2N ) ) cos 2 ( φ N ) ( β N cos( Γ 2N ) ) 2 β 2 N ,
ΔT β N cos( φ N )( K J( 2sin( Γ 2N ) )+ J K( 2sin( Γ 2N ) ) )0,
R( π 2 ) ( cos( φ N )a e i Γ 2N sin( φ N )a e i Γ 2N sin( φ N )b e i Γ 2N cos( φ N )b e i Γ 2N ) 12 [ 1 0 ] R( π 2 )( -0.44i )[ cos( π 24 ) 0.09 12 e i 12π 24 sin( π 24 ) 0.9 12 e i 12π 24 ]+R( π 2 )0.451[ 1 0 ] , R( π 2 ){ 0.44i( 0.991 )0.818i+0.451 0.44i( 0.131 )0.991i }R( π 2 )[ 0.094 0.057 ][ 0.057 0.094 ]
R( π 2 ) ( cos( φ N )a e i Γ 2N sin( φ N )a e i Γ 2N sin( φ N )b e i Γ 2N cos( φ N )b e i Γ 2N ) 12 [ 0 1 ] R( π 2 )( -0.44i )[ sin( π 24 ) 0.09 12 e i 12π 24 cos( π 24 ) 0.9 12 e i 12π 24 ]+R( π 2 )0.451[ 0 1 ] , =R( π 2 ){ 0.44i( 0.131 )0.818i -0.44i( 0.991 )0.991i+0.451 }R( π 2 )[ -0.047 0.883 ][ 0.883 0.047 ]
R( π 2 ) ( cos( φ N )a e i Γ 2N sin( φ N )a e i Γ 2N sin( φ N )b e i Γ 2N cos( φ N )b e i Γ 2N ) 12 [ 1 0 ] R( π 2 )( -0.258i )[ cos( π 24 ) 0.09 12 e i 12π 24 sin( π 24 ) 0.55 12 e i 12π 24 ]+R( π 2 )0.302[ 1 0 ] , R( π 2 ){ 0.258i( 0.991 )0.818i+0.302 0.258i( 0.131 )0.951i }R( π 2 )[ 0.093 -0.032 ]=[ 0.032 0.093 ]
R( π 2 ) ( cos( φ N )a e i Γ 2N sin( φ N )a e i Γ 2N sin( φ N )b e i Γ 2N cos( φ N )b e i Γ 2N ) 10 [ 0 1 ] R( π 2 )( -0.258i )[ sin( π 24 ) 0.09 12 e i 12π 24 cos( π 24 ) 0.55 12 e i 12π 24 ]+R( π 2 )0.302[ 0 1 ] , R( π 2 ){ 0.258i( 0.131 )0.818i -0.258i( 0.991 )0.951i+0.302 }R( π 2 )[ -0.028 0.545 ][ -0.545 -0.028 ]
R( π 2 ) ( cos( φ N )a e i Γ 2N sin( φ N )a e i Γ 2N sin( φ N )b e i Γ 2N cos( φ N )b e i Γ 2N ) 12 [ 1 0 ] R( π 2 )( -0.205i )[ cos( π 24 ) 0.5 12 e i 12π 24 sin( π 24 ) 0.9 12 e i 12π 24 ]+R( π 2 )0.695[ 1 0 ] , R( π 2 ){ 0.205i( 0.991 )0.944i+0.695 0.205i( 0.131 )0.991i }R( π 2 )[ 0.503 -0.027 ][ 0.027 0.503 ]
R( π 2 ) ( cos( φ N )a e i Γ 2N sin( φ N )a e i Γ 2N sin( φ N )b e i Γ 2N cos( φ N )b e i Γ 2N ) 12 [ 0 1 ] R( π 2 )( -0.205i )[ sin( π 24 ) 0.5 12 e i 12π 24 cos( π 24 ) 0.9 12 e i 12π 24 ]+R( π 2 )0.695[ 0 1 ] , =R( π 2 ){ 0.205i( 0.131 )0.944i -0.205i( 0.991 )0.991i+0.695 }R( π 2 )[ -0.025 0.896 ][ -0.896 0.025 ]
s=[ s 0 0 1 ],
R=[ t ne 0 0 t no ],
t no = 2 n noLC n noRM257 n noLC + n nopRM257 ,
t ne = 2 n neLC n neRM257 n neLC + n nepRM257 ,
M= m=1 N R( mρ )[ sR ] W 0 R( mρ ),
M= m=1 N R( mρ )[ z=1 N Z z ] W 0 R( mρ ),

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