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Analytic solutions of the normal modes and light transmission of a cholesteric liquid crystal cell

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Abstract

Cholesteric liquid crystals, consisting of chiral molecules, form self-assembled periodic structures exhibiting a photonic bandgap. Their selective reflectivity makes them well suited for a variety of applications; their optical response is therefore of considerable interest. The reflectance and transmittance of finite cholesteric cells is usually calculated numerically. Evanescent modes in the bandgap make the calculations challenging; existing matrix propagation methods cannot describe the reflection and transmission coefficients of thick cholesteric cells accurately. Here we present analytic solutions for the electromagnetic fields in cholesteric cells of finite thickness, and use them to calculate the transmission and reflection spectra. The use of analytic solutions allows for the accurate description of arbitrarily thick cholesteric cells, which would not be possible with only direct numerical methods.

© 2013 Chinese Laser Press

1. INTRODUCTION

Cholesteric liquid crystals (CLCs) are one-dimensional, self-assembled photonic band-gap materials. Their helical structure, easily deformable by external fields, produces a modulation of the refractive indices along the helical axis. This induces Bragg reflection in a range of wavelengths close to the helical pitch, for circularly polarized light of the same handedness as the CLC structure [1]. In addition to its unique optical properties, CLC is one of the two photonic band gap structures for which exact analytic solutions of the Maxwell’s equations exist for light propagating along the helical axis [2]. The analytic solution for CLCs was first derived by Mauguin in 1911 [3], for the case when the pitch is much larger than the incident light wavelength, and was later formulated more generally by Oseen in 1933 [4] and de Vries in 1951 [5]. Several authors have since proposed analytic solutions for infinite or semi-finite bulk CLCs [6,7], solving Maxwell’s equations in more general anisotropic stratified systems, such as smetic liquid crystals [8], or for oblique incidence [9].

Selective reflection and its easily deformable structure makes CLC slabs suitable for a variety of applications, such as tunable filters, mirrors, polarizers, smart windows [10], etc. Depending on the application, the thickness of a CLC slab would typically vary from a few to hundreds of micrometers. Thicker slabs are required for devices operating at longer wavelengths, such as infrared reflectors. The reflection and transmission spectra of a finite CLC slab have not been derived analytically. Typically, numerical methods, such as the Berreman 4×4 method [11,12] or the Jones matrix method [13], are used for such calculations. However, these propagation matrix methods are not suitable for thick samples since they tend to become unstable [13,14]. Numerical instabilities appear in the reflection band due to terms increasing exponentially with the cell thickness. For instance, double-precision computation results in diverging errors inside the band-gap for samples thicker than 100 pitches of the cholesteric [13].

Following preliminary work [15], we present here analytic expressions for the radiation fields of the transmitted and reflected light for normal incidence on a CLC slab of finite thickness. These allow us to overcome the problems of numerical errors. Indeed, the reflected and transmitted waves can be calculated for a sample of arbitrary thickness, enabling accuracy far beyond the limitation of numerical propagation matrix methods.

2. METHODOLOGY

The wave equation describing the electric field inside a CLC is obtained from Maxwell’s equations. It is solved in a rotating frame that follows the CLC’s helical structure. The solution is then transformed into the lab coordinate system.

For the sake of completeness we derive the solution from the first principles rather than citing existing results.

Typically, CLCs consist of molecules whose average orientation depends on position. The molecules are assumed to be effectively cylindrically symmetric; the orientation of a molecule is taken to be the orientation of the axis of symmetry. The average direction of the symmetry axes of the molecules, the nematic director, is described by a unit vector n^, which is perpendicular to the helical axis of the CLC. The helical structure of the CLC is shown in Fig. 1.

 figure: Fig. 1.

Fig. 1. Schematic representation of the helical structure of the CLC. The orientation of the ellipsoids represents the average orientation of the molecules. The coordinate axes of the lab frame and the rotating frame are shown.

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The lab coordinate system (x^,y^,z^) is chosen so that the helical axis points along the z^ direction. In this frame the director’s coordinates are (cos(qz),sin(qz),0), where q is the helical wave number, related to the helical pitch p by q=2π/p.

In this work we assume that the director traces out a right-handed helix. Similar procedure can be used for a left-handed CLC. The rotating coordinate system (η^,ξ^,z^) follows the rotation of the director n^. The rotating frame is related to the lab frame via the transformation

η^=cos(qz)x^+sin(qz)y^ξ^=sin(qz)x^+cos(qz)y^.

In the rotating frame the director is constant, pointing along η^=(1,0,0). In this frame, the dielectric tensor is diagonal, given by ε¯¯r=εI¯¯+(εε)η^η^, where ε and ε are the principal values of the dielectric constant parallel and perpendicular, respectively, to the director.

We assume a plane wave solution with the electric field in the x^y^ (ξ^η^) plane: E=E(z)ei(kzωt). The wave equation is then reduced to

E(z)+2ikEk2E(z)=ω2c2ε¯¯rE(z),
where the prime indicates derivative with respect to the z-coordinate.

Assuming that the light in the rotating frame is elliptically polarized with constant amplitude, i.e., E(z)=Eη^+Eξ^, and noting that η^=qξ^ and ξ^=qη^, one obtains

(α2ε+n2)E=2inαE(α2ε+n2)E=2inαE,
where α=λ0/p is the dimensionless wavelength and n is the refractive index. Solving for the eigenmodes gives
n2=ε+ε2+α2±[(εε)24+2(ε+ε)α2]12.

This indicates that there exist two propagating normal modes, with two waves propagating in the positive and two in the negative z-direction:

n1±=±[ε¯+α2+(δ2+4ε¯α2)12]12,
n2±=±[ε¯+α2(δ2+4ε¯α2)12]12,
where ε¯=(1/2)(ε+ε) and δ=(1/2)(εε).

Positive (negative) roots of expressions (5) and (6) correspond to propagation in the positive (negative) direction along the z^ axis. Having determined the refractive indices, one can write the solution for the electric fields inside the CLC. In the rotating frame we denote η^=[10], ξ^=[01], then

E1±=E±[1±iA]ei(±k1zωt),
E2±=E±[iB1]ei(±k2zωt).
The “+” and “” superscripts indicate propagation in the z^ and z^ directions, respectively. A and B are directly related to the ellipticities of the normal modes and are given by
A=2α2εε2+[(εε)24+2(ε+ε)α2]122α{ε+ε2+α2+[(εε)24+2(ε+ε)α2]12}12,
B=2α2+εε2[(εε)24+2(ε+ε)α2]122α{ε+ε2+α2[(εε)24+2(ε+ε)α2]12}12,
or, which is the same, as
A=n12+α2ε2n1α,B=n22+α2ε2n2α.

The corresponding wave vectors are given by k1,2=2πn1,2/λ0. Here n1=n1+ and n2=n2+.

Equation (6) shows that n22 is negative when pε<λ0<pε, which implies that n2 is imaginary and E2 is a nonpropagating mode. This wavelength range corresponds to the bandgap, the selective reflection band of the CLC, for circularly polarized light of the same handedness as the CLC structure.

In the lab frame, we denote x^=[10], y^=[01], and the normal modes are as follows:

E1+=E1+eiωt((1+A)[1i]ei(k1z+qz)+(1A)[1i]ei(k1zqz)),
E1=E1eiωt((1+A)[1i]ei(k1z+qz)+(1A)[1i]ei(k1zqz)),
E2+=E2+eiωt((1+B)[i1]ei(k2z+qz)+(1B)[i1]ei(k2zqz)),
E2=E2eiωt((1+B)[i1]ei(k2z+qz)+(1B)[i1]ei(k2zqz)),
where E1+, E1, E2+, and E2 are constants. In the lab frame, each normal mode is thus a linear combination of left and right circularly polarized light, with different amplitudes and wavenumbers.

Having identified the normal modes, we next solve the boundary problem for a CLC cell of finite thickness d. The CLC is confined between parallel plane substrates with refractive indices nsi on the side of the incident light and nst on the other. We assume that our substrates are semi-infinite; this is to avoid additional Fresnel reflections from air-substrate interfaces. Fresnel reflections are discussed in [16]. The helix axis is perpendicular to the substrate surfaces. The lab coordinate is chosen so that the director is along the x^ direction, and z=0 at the CLC surface where light is incident (see Fig. 2).

 figure: Fig. 2.

Fig. 2. Schematic representation of a CLC film sandwiched between two substrates, the helical configuration in the lab coordinates systems and the electric fields inside and outside the CLC.

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The complex electric fields in the lab frame, corresponding to the incident Ei, reflected Er, and transmitted Et light waves, can be written as follows:

Ei=[EixEiy]ei(kszωt)Er=[ErxEry]ei(kszωt)Et=[EtxEty]ei(ks(zdp)ωt).
The magnetic fields can be expressed in terms of the electric fields as H=i×E/(ωμ0).

The boundary conditions are the continuity of the tangential components of the electric and magnetic fields. The first set of equations relating Ei, Er and the fields inside the CLC is obtained from satisfying the boundary conditions at the interface z=0. Similarly, the second set of equations relating Et and the fields inside the CLC is obtained at the interface z=d. Eliminating the fields inside the slab gives the relations between the components of the incident, reflected, and transmitted waves:

[EixEiyErxEry]=[P¯¯0¯¯Q¯¯0¯¯][EtxEty00].
The transfer matrices P¯¯ and Q¯¯ are 2×2 matrices. Their elements are given in the Appendix A. 0¯¯ denotes a zero 2×2 matrix.

The transmitted and reflected fields can be obtained explicitly from the incident field to give

[EtxEty]=P¯¯1[EixEiy],
[ErxEry]=Q¯¯P¯¯1[EixEiy].
Analytic expressions for the elements of P¯¯1 and Q¯¯P¯¯1 matrices together with the detP¯¯, needed to invert P¯¯, are also given in the Appendix A. These analytic expressions relating the incident, reflected and transmitted fields are one of our key results.

3. RESULTS AND DISCUSSIONS

Figure 3 shows the transmitted and reflected intensities from light incident on a CLC slab of thickness d=100p. If the intensities of reflected and transmitted light are calculated numerically, say by evaluating P¯¯1 and Q¯¯P¯¯1, numerical errors arise, which make the solution in the bandgap unreliable, as indicated in Fig. 3(a). We discuss the origin of these errors below. If we use our analytic results for P¯¯1 and Q¯¯P¯¯1, no such errors arise, enabling the accurate determination of the response of even very thick cells accurately, as shown in Figs. 3(b) and 4.

 figure: Fig. 3.

Fig. 3. Reflection and transmission spectra of a CLC slab: (a) numerical inversion of matrix P¯¯ used and (b) P¯¯1 and Q¯¯P¯¯1 evaluated from analytic expressions.

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 figure: Fig. 4.

Fig. 4. Reflection and transmission spectra of the CLC slab. Cell thickness to the helical pitch ratio is 104.

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Figure 4 shows the spectra of a CLC cell with d=10,000p, and in Figs. 5 and 6, the reflection spectrum of the same cell is shown in details at the center of the reflection band and near its edge. The parameters used are nsi=nst=ns=1.6, ε=2.25, ε=2.89, p=380nm. We have chosen the principal values of the dielectric tensor so that they would correspond to typical refractive index values in nematics. In our paper, the refractive indices corresponding to the dielectric constant values 1.5 and 1.7 (in 5CB, the values are approximately 1.53 and 1.72 at room temperature). We have chosen the substrate index to be the average of these, 1.6, in order to minimize cavity effects. The incident electric field was chosen to be linearly polarized in the x^ direction for these calculations. Equivalent calculations can be carried out for other polarizations.

 figure: Fig. 5.

Fig. 5. Reflection spectrum of the CLC slab near the edge of the reflection band. Cell thickness to the helical pitch ratio is 104.

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 figure: Fig. 6.

Fig. 6. Reflection spectrum of the CLC slab at the center of the reflection band. Cell thickness to the helical pitch ratio is 104.

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We now turn to the origin of numerical errors and how they are avoided by use of our analytic results. Each element of the transfer matrices P¯¯ and Q¯¯ (see Appendix A) contains terms of the form eik2d that diverge when n2 is imaginary, ik2d is positive, and d is sufficiently large. These become the dominant terms and determine the order of magnitude of the elements of the two propagation matrices. Since every element contains eik2d, the detP¯¯ might be expected to contain ei2k2d as a dominant term. Yet analytic calculations show that terms ei2k2d in detP¯¯ always cancel out, making eik2d the dominant term. This is a consequence of the specific structure of the propagation matrices. One can thus see that P¯¯ and detP¯¯ are of the same order of magnitude and P¯¯1 is typically of the order of unity. Errors result from the direct numerical calculation of detP¯¯, where products are formed of ei2k2d and a number that is not exactly zero, but is within machine precision of zero. For a thick cell, ei2k2d can be orders of magnitude larger than the machine epsilon, adding significant errors to the final value of the determinant. Similar behavior occurs during the calculation of Q¯¯P¯¯1: the dominant term eik2d is canceled out if the product is taken analytically, but does not cancel exactly numerically. For nonabsorbing CLCs, the reflected light intensity is simply the difference of the incident and transmitted intensities; thus calculating only P¯¯1 is sufficient. To calculate reflection from an absorbing cell, one necessarily needs to use both P¯¯1 and Q¯¯P¯¯1.

Numerical propagation methods, such as the classical Berreman 4×4 method [11], show errors of a similar origin, making numerical modeling of thick CLC samples problematic [17]. We note that for thinner cells, such as d=25p, the Berreman 4×4 method and analytic solution give an identical spectrum.

4. CONCLUSION

Analytic solutions of the wave equation in the bulk, together with boundary conditions, allow analytic expressions for the transmitted and reflected fields for CLC cells in terms of the incident field. Although equations are long and cumbersome, our approach allows the calculation of the reflection and transmission spectra for arbitrarily thick CLC cells in the case of normally incident light. This scheme overcomes the limitations of existing numerical propagation methods.

APPENDIX A: ANALYTIC EXPRESSIONS FOR PROPAGATION MATRICES

[ExiEyiExrEyr]=[P¯¯0¯¯Q¯¯0¯¯][ExtEyt00],
P¯¯=[eiδ1p11(1)+eiδ1p11(2)+eiδ2p11(3)+eiδ2p11(4)i(eiδ1p12(1)+eiδ1p12(2)+eiδ2p12(3)+eiδ2p12(4))i(eiδ1p21(1)+eiδ1p21(2)+eiδ2p21(3)+eiδ2p21(4))eiδ1p22(1)+eiδ1p22(2)+eiδ2p22(3)+eiδ2p22(4)],
Q¯¯=[eiδ1q11(1)+eiδ1q11(2)+eiδ2q11(3)+eiδ2q11(4)i(eiδ1q12(1)+eiδ1q12(2)+eiδ2q12(3)+eiδ2q12(4))i(eiδ1q21(1)+eiδ1q21(2)+eiδ2q21(3)+eiδ2q21(4))eiδ1q22(1)+eiδ1q22(2)+eiδ2q22(3)+eiδ2q22(4)],
where δ1=k1d, δ2=k2d, 0¯¯=[0000], and
p11(1)=18(1γiA+N1i)[(eiqdeiqd)(BN2tXγtX+BY)+(eiqd+eiqd)(XγtBY+N2tY)],
p11(2)=18(1+γiAN1i)[(eiqdeiqd)(BN2tX+γtX+BY)+(eiqd+eiqd)(XγtBY+N2tY)],
p11(3)=18(Bγi+BN2i)[(eiqdeiqd)(N1tX+γtAXY)+(eiqd+eiqd)(AXAN1tY+γtY)],
p11(4)=18(B+γiBN2i)[(eiqdeiqd)(N1tXγtAXY)+(eiqd+eiqd)(AXAN1tY+γtY)],
p12(1)=18(1γiA+N1i)[(eiqdeiqd)(X+γtBYN2tY)+(eiqd+eiqd)(BN2tX+γtXBY)],
p12(2)=18(1+γiAN1i)[(eiqdeiqd)(X+γtBYN2tY)+(eiqd+eiqd)(BN2tXγtXBY)],
p12(3)=18(Bγi+BN2i)[(eiqdeiqd)(AX+AN1tYγtY)+(eiqd+eiqd)(N1tXγtAX+Y)],
p12(4)=18(B+γiBN2i)[(eiqdeiqd)(AX+AN1tYγtY)+(eiqd+eiqd)(N1tX+γtAX+Y)],
p21(1)=18(Aγi+AN1i)[(eiqdeiqd)(BN2tX+γtXBY)+(eiqd+eiqd)(X+γtBYN2tY)],
p21(2)=18(A+γiAN1i)[(eiqdeiqd)(BN2tX+γtX+BY)+(eiqd+eiqd)(XγtBY+N2tY)],
p21(3)=18(1γiB+N2i)[(eiqdeiqd)(N1tXγtAX+Y)+(eiqd+eiqd)(AX+AN1tYγtY)],
p21(4)=18(1+γiBN2i)[(eiqdeiqd)(N1tXγtAXY)+(eiqd+eiqd)(AXAN1tY+γtY)],
p22(1)=18(Aγi+AN1i)[(eiqdeiqd)(X+γtBYN2tY)+(eiqd+eiqd)(BN2tX+γtXBY)],
p22(2)=18(A+γiAN1i)[(eiqdeiqd)(XγtBY+N2tY)+(eiqd+eiqd)(BN2tX+γtX+BY)],
p22(3)=18(1γiB+N2i)[(eiqdeiqd)(AX+AN1tYγtY)+(eiqd+eiqd)(N1tXγtAX+Y)],
p22(4)=18(1+γiBN2i)[(eiqdeiqd)(AXAN1tY+γtY)+(eiqd+eiqd)(N1tXγtAXY)],
q11(1)=18(1+γiAN1i)[(eiqdeiqd)(BN2tXγtX+BY)+(eiqd+eiqd)(XγtBY+N2tY)],
q11(2)=18(1γiA+N1i)[(eiqdeiqd)(BN2tX+γtX+BY)+(eiqd+eiqd)(XγtBY+N2tY)],
q11(3)=18(B+γiBN2i)[(eiqdeiqd)(N1tX+γtAXY)+(eiqd+eiqd)(AXAN1tY+γtY)],
q11(4)=18(Bγi+BN2i)[(eiqdeiqd)(N1tXγtAXY)+(eiqd+eiqd)(AXAN1tY+γtY)],
q12(1)=i8(1+γiAN1i)[(eiqdeiqd)(X+γtBYN2tY)+(eiqd+eiqd)(BN2tX+γtXBY)],
q12(2)=i8(1γiA+N1i)[(eiqdeiqd)(X+γtBYN2tY)+(eiqd+eiqd)(BN2tXγtXBY)],
q12(3)=i8(B+γiBN2i)[(eiqdeiqd)(AX+AN1tYγtY)+(eiqd+eiqd)(N1tXγtAX+Y)],
q12(4)=i8(Bγi+BN2i)[(eiqdeiqd)(AX+AN1tYγtY)+(eiqd+eiqd)(N1tX+γtAX+Y)],
q21(1)=i8(A+γiAN1i)[(eiqdeiqd)(BN2tX+γtXBY)+(eiqd+eiqd)(X+γtBYN2tY)],
q21(2)=i8(Aγi+AN1i)[(eiqdeiqd)(BN2tX+γtX+BY)+(eiqd+eiqd)(XγtBY+N2tY)],
q21(3)=i8(1+γiBN2i)[(eiqdeiqd)(N1tXγtAX+Y)+(eiqd+eiqd)(AX+AN1tYγtY)],
q21(4)=i8(1γiB+N2i)[(eiqdeiqd)(N1tXγtAXY)+(eiqd+eiqd)(AXAN1tY+γtY)],
q22(1)=18(A+γiAN1i)[(eiqdeiqd)(X+γtBYN2tY)+(eiqd+eiqd)(BN2tX+γtXBY)],
q22(2)=18(Aγi+AN1i)[(eiqdeiqd)(XγtBY+N2tY)+(eiqd+eiqd)(BN2tX+γtX+BY)],
q22(3)=18(1+γiBN2i)[(eiqdeiqd)(AX+AN1tYγtY)+(eiqd+eiqd)(N1tXγtAX+Y)],
q22(4)=18(1γiB+N2i)[(eiqdeiqd)(AXAN1tY+γtY)+(eiqd+eiqd)(N1tXγtAXY)].
Here γi=α/nsi, γt=α/nst, N1i=n1/nsi, N1t=n1/nst, N2i=n2/nsi, N2t=n2/nst, A=(n12+α2ε)/2n1α, B=(n22+α2ε)/2n2α, X=1/(ABN2tN1t) and Y=1/(ABN1tN2t), α=λ0/p.
detP¯¯=P11P22P12P21=(p11(1)p22(2)+p11(2)p22(1)+p11(3)p22(4)+p11(4)p22(3)+p12(1)p21(2)+p12(2)p21(1)+p12(3)p21(4)+p12(4)p21(3))+ei(δ1+δ2)(p11(1)p22(3)+p11(3)p22(1)+p12(1)p21(3)+p12(3)p21(1))+ei(δ1δ2)(p11(1)p22(4)+p11(4)p22(1)+p12(1)p21(4)+p12(4)p21(1))+ei(δ1δ2)(p11(2)p22(3)+p11(3)p22(2)+p12(2)p21(3)+p12(3)p21(2))+ei(δ1+δ2)(p11(2)p22(4)+p11(4)p22(2)+p12(2)p21(4)+p12(4)p21(2))+ei2δ1(p11(1)p22(1)+p12(1)p21(1))+ei2δ1(p11(2)p22(2)+p12(2)p21(2))+ei2δ2(p11(3)p22(3)+p12(3)p21(3))+ei2δ2(p11(4)p22(4)+p12(4)p21(4)).
The following equalities are an important result of this work:
p11(1)p22(1)+p12(1)p21(1)=0,
p11(2)p22(2)+p12(2)p21(2)=0,
p11(3)p22(3)+p12(3)p21(3)=0,
p11(4)p22(4)+p12(4)p21(4)=0.
These show that the leading terms in the determinant cancel out:
detP¯¯=P11P22P12P21=(p11(1)p22(2)+p11(2)p22(1)+p11(3)p22(4)+p11(4)p22(3)+p12(1)p21(2)+p12(2)p21(1)+p12(3)p21(4)+p12(4)p21(3))+ei(δ1+δ2)(p11(1)p22(3)+p11(3)p22(1)+p12(1)p21(3)+p12(3)p21(1))+ei(δ1δ2)(p11(1)p22(4)+p11(4)p22(1)+p12(1)p21(4)+p12(4)p21(1))+ei(δ1δ2)d(p11(2)p22(3)+p11(3)p22(2)+p12(2)p21(3)+p12(3)p21(2))+ei(δ1+δ2)(p11(2)p22(4)+p11(4)p22(2)+p12(2)p21(4)+p12(4)p21(2)).
For imaginary k2 detP¯¯ scales as eik2d, not as ei2k2d, i.e., P¯¯ and detP¯¯ are of the same order of magnitude and P¯¯1 is typically of the order of unity.
P¯¯1=P˜detP¯¯,
P˜=[P22P12P21P11].
For nsi=nst=ns detP¯¯ is simplified to
detP¯¯=116(ABN1N2)(ABN2N1)(8[A2γN1A(γ2+N121)+γN1][B2γN2B(γ2+N221)+γN2]+ei(δ1+δ2){(AB1)[γ2(N1+1)(N2+1)]+γ(AB)(N1N2)}2ei(δ1δ2){(AB+1)[γ2+(N1+1)(N21)]γ(A+B)(N1+N2)}2ei(δ1δ2){(AB+1)[γ2+(N11)(N2+1)]γ(A+B)(N1+N2)}2+ei(δ1+δ2){(AB1)[γ2+(N11)(N21)]γ(AB)(N1N2)}2),
Q¯¯P¯¯1=Q¯¯P˜detP¯¯.
For imaginary k2 Q¯¯P˜ scales as eik2d, just like detP¯¯, thus Q¯¯P¯¯1 is typically of the order of unity.
[Q¯¯P˜]11=(q11(1)p22(2)+q11(2)p22(1)+q11(3)p22(4)+q11(4)p22(3)+q12(1)p21(2)+q12(2)p21(1)+q12(3)p21(4)+q12(4)p21(3))+ei(δ1+δ2)(q11(2)p22(4)+q11(4)p22(2)+q12(2)p21(4)+q12(4)p21(2))+ei(δ1+δ2)(q11(1)p22(3)+q11(3)p22(1)+q12(1)p21(3)+q12(3)p21(1))+ei(δ1δ2)(q11(2)p22(3)+q11(3)p22(2)+q12(2)p21(3)+q12(3)p21(2))+ei(δ1δ2)(q11(1)p22(4)+q11(4)p22(1)+q12(1)p21(4)+q12(4)p21(1)),
[Q¯¯P˜]12=i[(q11(1)p12(2)q11(2)p12(1)q11(3)p12(4)q11(4)p12(3)+q12(1)p11(2)+q12(2)p11(1)+q12(3)p11(4)+q12(4)p11(3))+ei(δ1+δ2)(q11(2)p12(4)q11(4)p12(2)+q12(2)p11(4)+q12(4)p11(2))+ei(δ1+δ2)(q11(1)p12(3)q11(3)p12(1)+q12(1)p11(3)+q12(3)p11(1))+ei(δ1δ2)(q11(2)p12(3)q11(3)p12(2)+q12(2)p11(3)+q12(3)p11(2))+ei(δ1δ2)(q11(1)p12(4)q11(4)p12(1)+q12(1)p11(4)+q12(4)p11(1))],
[Q¯¯P˜]21=i[(q21(1)p22(2)+q21(2)p22(1)+q21(3)p22(4)+q21(4)p22(3)q22(1)p21(2)q22(2)p21(1)q22(3)p21(4)q22(4)p21(3))+ei(δ1+δ2)(q21(2)p22(4)+q21(4)p22(2)q22(2)p21(4)q22(4)p21(2))+ei(δ1+δ2)(q21(1)p22(3)+q21(3)p22(1)q22(1)p21(3)q22(3)p21(1))+ei(δ1δ2)(q21(2)p22(3)+q21(3)p22(2)q22(2)p21(3)q22(3)p21(2))+ei(δ1δ2)(q21(1)p22(4)+q21(4)p22(1)q22(1)p21(4)q22(4)p21(1))],
[Q¯¯P˜]22=(q21(1)p12(2)+q21(2)p12(1)+q21(3)p12(4)+q21(4)p12(3)+q22(1)p11(2)+q22(2)p11(1)+q22(3)p11(4)+q22(4)p11(3))+ei(δ1+δ2)(q21(2)p12(4)+q21(4)p12(2)+q22(2)p11(4)+q22(4)p11(2))+ei(δ1+δ2)(q21(1)p12(3)+q21(3)p12(1)+q22(1)p11(3)+q22(3)p11(1))+ei(δ1δ2)(q21(2)p12(3)+q21(3)p12(2)+q22(2)p11(3)+q22(3)p11(2))+ei(δ1δ2)(q21(1)p12(4)+q21(4)p12(1)+q22(1)p11(4)+q22(4)p11(1)).

ACKNOWLEDGMENTS

We acknowledge support from the NSF under grant IIP 1114332.

REFERENCES

1. P. Collings and M. Hird, Introduction to Liquid Crystals: Chemistry and Physics, the Liquid Crystals Book Series (Taylor & Francis, 1997).

2. V. Beliakov, Diffraction Optics of Complex-Structured Periodic Media, Partially Ordered Systems (Springer-Verlag, 1992).

3. C. Mauguin, “Sur la représentation géométrique de Poincaré relative aux propriétés optiques des piles de lames,” Bull. Soc. Fr. Mineral. Cristallogr. 34, 6–15 (1911).

4. C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–899 (1933). [CrossRef]  

5. H. de Vries, “Rotatory power and other optical properties of certain liquid crystals,” Acta Crystallogr. 4, 219–226 (1951). [CrossRef]  

6. R. Dreher and G. Meier, “Optical properties of cholesteric liquid crystals,” Phys. Rev. A 8, 1616–1623 (1973). [CrossRef]  

7. H. L. Ong, “Wave propagation in cholesteric and chiral smectic-C liquid crystals: exact and generalized geometrical-optics-approximation solutions,” Phys. Rev. A 37, 3520–3529 (1988). [CrossRef]  

8. C. Oldano, “Electromagnetic-wave propagation in anisotropic stratified media,” Phys. Rev. A 40, 6014–6020 (1989). [CrossRef]  

9. C. Oldano, E. Miraldi, and P. T. Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A 27, 3291–3299 (1983). [CrossRef]  

10. M. Mitov and N. Dessaud, “Going beyond the reflectance limit of cholesteric liquid crystals,” Nat. Mater. 5, 361–364 (2006). [CrossRef]  

11. D. W. Berreman, “Optics in stratified and anisotropic media: 4x4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972). [CrossRef]  

12. M. Xu, F. Xu, and D.-K. Yang, “Effects of cell structure on the reflection of cholesteric liquid crystal displays,” J. Appl. Phys. 83, 1938–1944 (1998). [CrossRef]  

13. D.-K. Yang and X.-D. Mi, “Modelling of the reflection of cholesteric liquid crystals using the Jones matrix,” J. Phys. D 33, 672–676 (2000). [CrossRef]  

14. Z. Lu, “Mathematical methods in physics; numerical approximation and analysis; anisotropic optical materials,” Opt. Lett. 33, 1948–1950 (2008). [CrossRef]  

15. W. Cao, “Fluorescence and lasing in liquid crystalline photonic bandgap materials,” Ph.D. thesis (Kent State University, 2005).

16. L. B. Glebov, J. Lumeau, S. Mokhov, V. Smirnov, and B. Y. Zeldovich, “Reflection of light by composite volume holograms: Fresnel corrections and Fabry–Perot spectral filtering,” J. Opt. Soc. Am. A 25, 751–764 (2008). [CrossRef]  

17. H. Wöhler, G. Haas, M. Fritsch, and D. A. Mlynski, “Faster 4x4 matrix method for uniaxial inhomogeneous media,” J. Opt. Soc. Am. A 5, 1554–1557 (1988). [CrossRef]  

References

  • View by:

  1. P. Collings and M. Hird, Introduction to Liquid Crystals: Chemistry and Physics, the Liquid Crystals Book Series (Taylor & Francis, 1997).
  2. V. Beliakov, Diffraction Optics of Complex-Structured Periodic Media, Partially Ordered Systems (Springer-Verlag, 1992).
  3. C. Mauguin, “Sur la représentation géométrique de Poincaré relative aux propriétés optiques des piles de lames,” Bull. Soc. Fr. Mineral. Cristallogr. 34, 6–15 (1911).
  4. C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–899 (1933).
    [Crossref]
  5. H. de Vries, “Rotatory power and other optical properties of certain liquid crystals,” Acta Crystallogr. 4, 219–226 (1951).
    [Crossref]
  6. R. Dreher and G. Meier, “Optical properties of cholesteric liquid crystals,” Phys. Rev. A 8, 1616–1623 (1973).
    [Crossref]
  7. H. L. Ong, “Wave propagation in cholesteric and chiral smectic-C liquid crystals: exact and generalized geometrical-optics-approximation solutions,” Phys. Rev. A 37, 3520–3529 (1988).
    [Crossref]
  8. C. Oldano, “Electromagnetic-wave propagation in anisotropic stratified media,” Phys. Rev. A 40, 6014–6020 (1989).
    [Crossref]
  9. C. Oldano, E. Miraldi, and P. T. Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A 27, 3291–3299 (1983).
    [Crossref]
  10. M. Mitov and N. Dessaud, “Going beyond the reflectance limit of cholesteric liquid crystals,” Nat. Mater. 5, 361–364 (2006).
    [Crossref]
  11. D. W. Berreman, “Optics in stratified and anisotropic media: 4x4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [Crossref]
  12. M. Xu, F. Xu, and D.-K. Yang, “Effects of cell structure on the reflection of cholesteric liquid crystal displays,” J. Appl. Phys. 83, 1938–1944 (1998).
    [Crossref]
  13. D.-K. Yang and X.-D. Mi, “Modelling of the reflection of cholesteric liquid crystals using the Jones matrix,” J. Phys. D 33, 672–676 (2000).
    [Crossref]
  14. Z. Lu, “Mathematical methods in physics; numerical approximation and analysis; anisotropic optical materials,” Opt. Lett. 33, 1948–1950 (2008).
    [Crossref]
  15. W. Cao, “Fluorescence and lasing in liquid crystalline photonic bandgap materials,” Ph.D. thesis (Kent State University, 2005).
  16. L. B. Glebov, J. Lumeau, S. Mokhov, V. Smirnov, and B. Y. Zeldovich, “Reflection of light by composite volume holograms: Fresnel corrections and Fabry–Perot spectral filtering,” J. Opt. Soc. Am. A 25, 751–764 (2008).
    [Crossref]
  17. H. Wöhler, G. Haas, M. Fritsch, and D. A. Mlynski, “Faster 4x4 matrix method for uniaxial inhomogeneous media,” J. Opt. Soc. Am. A 5, 1554–1557 (1988).
    [Crossref]

2008 (2)

2006 (1)

M. Mitov and N. Dessaud, “Going beyond the reflectance limit of cholesteric liquid crystals,” Nat. Mater. 5, 361–364 (2006).
[Crossref]

2000 (1)

D.-K. Yang and X.-D. Mi, “Modelling of the reflection of cholesteric liquid crystals using the Jones matrix,” J. Phys. D 33, 672–676 (2000).
[Crossref]

1998 (1)

M. Xu, F. Xu, and D.-K. Yang, “Effects of cell structure on the reflection of cholesteric liquid crystal displays,” J. Appl. Phys. 83, 1938–1944 (1998).
[Crossref]

1989 (1)

C. Oldano, “Electromagnetic-wave propagation in anisotropic stratified media,” Phys. Rev. A 40, 6014–6020 (1989).
[Crossref]

1988 (2)

H. L. Ong, “Wave propagation in cholesteric and chiral smectic-C liquid crystals: exact and generalized geometrical-optics-approximation solutions,” Phys. Rev. A 37, 3520–3529 (1988).
[Crossref]

H. Wöhler, G. Haas, M. Fritsch, and D. A. Mlynski, “Faster 4x4 matrix method for uniaxial inhomogeneous media,” J. Opt. Soc. Am. A 5, 1554–1557 (1988).
[Crossref]

1983 (1)

C. Oldano, E. Miraldi, and P. T. Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A 27, 3291–3299 (1983).
[Crossref]

1973 (1)

R. Dreher and G. Meier, “Optical properties of cholesteric liquid crystals,” Phys. Rev. A 8, 1616–1623 (1973).
[Crossref]

1972 (1)

1951 (1)

H. de Vries, “Rotatory power and other optical properties of certain liquid crystals,” Acta Crystallogr. 4, 219–226 (1951).
[Crossref]

1933 (1)

C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–899 (1933).
[Crossref]

1911 (1)

C. Mauguin, “Sur la représentation géométrique de Poincaré relative aux propriétés optiques des piles de lames,” Bull. Soc. Fr. Mineral. Cristallogr. 34, 6–15 (1911).

Beliakov, V.

V. Beliakov, Diffraction Optics of Complex-Structured Periodic Media, Partially Ordered Systems (Springer-Verlag, 1992).

Berreman, D. W.

Cao, W.

W. Cao, “Fluorescence and lasing in liquid crystalline photonic bandgap materials,” Ph.D. thesis (Kent State University, 2005).

Collings, P.

P. Collings and M. Hird, Introduction to Liquid Crystals: Chemistry and Physics, the Liquid Crystals Book Series (Taylor & Francis, 1997).

de Vries, H.

H. de Vries, “Rotatory power and other optical properties of certain liquid crystals,” Acta Crystallogr. 4, 219–226 (1951).
[Crossref]

Dessaud, N.

M. Mitov and N. Dessaud, “Going beyond the reflectance limit of cholesteric liquid crystals,” Nat. Mater. 5, 361–364 (2006).
[Crossref]

Dreher, R.

R. Dreher and G. Meier, “Optical properties of cholesteric liquid crystals,” Phys. Rev. A 8, 1616–1623 (1973).
[Crossref]

Fritsch, M.

Glebov, L. B.

Haas, G.

Hird, M.

P. Collings and M. Hird, Introduction to Liquid Crystals: Chemistry and Physics, the Liquid Crystals Book Series (Taylor & Francis, 1997).

Lu, Z.

Lumeau, J.

Mauguin, C.

C. Mauguin, “Sur la représentation géométrique de Poincaré relative aux propriétés optiques des piles de lames,” Bull. Soc. Fr. Mineral. Cristallogr. 34, 6–15 (1911).

Meier, G.

R. Dreher and G. Meier, “Optical properties of cholesteric liquid crystals,” Phys. Rev. A 8, 1616–1623 (1973).
[Crossref]

Mi, X.-D.

D.-K. Yang and X.-D. Mi, “Modelling of the reflection of cholesteric liquid crystals using the Jones matrix,” J. Phys. D 33, 672–676 (2000).
[Crossref]

Miraldi, E.

C. Oldano, E. Miraldi, and P. T. Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A 27, 3291–3299 (1983).
[Crossref]

Mitov, M.

M. Mitov and N. Dessaud, “Going beyond the reflectance limit of cholesteric liquid crystals,” Nat. Mater. 5, 361–364 (2006).
[Crossref]

Mlynski, D. A.

Mokhov, S.

Oldano, C.

C. Oldano, “Electromagnetic-wave propagation in anisotropic stratified media,” Phys. Rev. A 40, 6014–6020 (1989).
[Crossref]

C. Oldano, E. Miraldi, and P. T. Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A 27, 3291–3299 (1983).
[Crossref]

Ong, H. L.

H. L. Ong, “Wave propagation in cholesteric and chiral smectic-C liquid crystals: exact and generalized geometrical-optics-approximation solutions,” Phys. Rev. A 37, 3520–3529 (1988).
[Crossref]

Oseen, C. W.

C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–899 (1933).
[Crossref]

Smirnov, V.

Valabrega, P. T.

C. Oldano, E. Miraldi, and P. T. Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A 27, 3291–3299 (1983).
[Crossref]

Wöhler, H.

Xu, F.

M. Xu, F. Xu, and D.-K. Yang, “Effects of cell structure on the reflection of cholesteric liquid crystal displays,” J. Appl. Phys. 83, 1938–1944 (1998).
[Crossref]

Xu, M.

M. Xu, F. Xu, and D.-K. Yang, “Effects of cell structure on the reflection of cholesteric liquid crystal displays,” J. Appl. Phys. 83, 1938–1944 (1998).
[Crossref]

Yang, D.-K.

D.-K. Yang and X.-D. Mi, “Modelling of the reflection of cholesteric liquid crystals using the Jones matrix,” J. Phys. D 33, 672–676 (2000).
[Crossref]

M. Xu, F. Xu, and D.-K. Yang, “Effects of cell structure on the reflection of cholesteric liquid crystal displays,” J. Appl. Phys. 83, 1938–1944 (1998).
[Crossref]

Zeldovich, B. Y.

Acta Crystallogr. (1)

H. de Vries, “Rotatory power and other optical properties of certain liquid crystals,” Acta Crystallogr. 4, 219–226 (1951).
[Crossref]

Bull. Soc. Fr. Mineral. Cristallogr. (1)

C. Mauguin, “Sur la représentation géométrique de Poincaré relative aux propriétés optiques des piles de lames,” Bull. Soc. Fr. Mineral. Cristallogr. 34, 6–15 (1911).

J. Appl. Phys. (1)

M. Xu, F. Xu, and D.-K. Yang, “Effects of cell structure on the reflection of cholesteric liquid crystal displays,” J. Appl. Phys. 83, 1938–1944 (1998).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Phys. D (1)

D.-K. Yang and X.-D. Mi, “Modelling of the reflection of cholesteric liquid crystals using the Jones matrix,” J. Phys. D 33, 672–676 (2000).
[Crossref]

Nat. Mater. (1)

M. Mitov and N. Dessaud, “Going beyond the reflectance limit of cholesteric liquid crystals,” Nat. Mater. 5, 361–364 (2006).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (4)

R. Dreher and G. Meier, “Optical properties of cholesteric liquid crystals,” Phys. Rev. A 8, 1616–1623 (1973).
[Crossref]

H. L. Ong, “Wave propagation in cholesteric and chiral smectic-C liquid crystals: exact and generalized geometrical-optics-approximation solutions,” Phys. Rev. A 37, 3520–3529 (1988).
[Crossref]

C. Oldano, “Electromagnetic-wave propagation in anisotropic stratified media,” Phys. Rev. A 40, 6014–6020 (1989).
[Crossref]

C. Oldano, E. Miraldi, and P. T. Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A 27, 3291–3299 (1983).
[Crossref]

Trans. Faraday Soc. (1)

C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–899 (1933).
[Crossref]

Other (3)

P. Collings and M. Hird, Introduction to Liquid Crystals: Chemistry and Physics, the Liquid Crystals Book Series (Taylor & Francis, 1997).

V. Beliakov, Diffraction Optics of Complex-Structured Periodic Media, Partially Ordered Systems (Springer-Verlag, 1992).

W. Cao, “Fluorescence and lasing in liquid crystalline photonic bandgap materials,” Ph.D. thesis (Kent State University, 2005).

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Figures (6)

Fig. 1.
Fig. 1. Schematic representation of the helical structure of the CLC. The orientation of the ellipsoids represents the average orientation of the molecules. The coordinate axes of the lab frame and the rotating frame are shown.
Fig. 2.
Fig. 2. Schematic representation of a CLC film sandwiched between two substrates, the helical configuration in the lab coordinates systems and the electric fields inside and outside the CLC.
Fig. 3.
Fig. 3. Reflection and transmission spectra of a CLC slab: (a) numerical inversion of matrix P¯¯ used and (b) P¯¯1 and Q¯¯P¯¯1 evaluated from analytic expressions.
Fig. 4.
Fig. 4. Reflection and transmission spectra of the CLC slab. Cell thickness to the helical pitch ratio is 104.
Fig. 5.
Fig. 5. Reflection spectrum of the CLC slab near the edge of the reflection band. Cell thickness to the helical pitch ratio is 104.
Fig. 6.
Fig. 6. Reflection spectrum of the CLC slab at the center of the reflection band. Cell thickness to the helical pitch ratio is 104.

Equations (68)

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η^=cos(qz)x^+sin(qz)y^ξ^=sin(qz)x^+cos(qz)y^.
E(z)+2ikEk2E(z)=ω2c2ε¯¯rE(z),
(α2ε+n2)E=2inαE(α2ε+n2)E=2inαE,
n2=ε+ε2+α2±[(εε)24+2(ε+ε)α2]12.
n1±=±[ε¯+α2+(δ2+4ε¯α2)12]12,
n2±=±[ε¯+α2(δ2+4ε¯α2)12]12,
E1±=E±[1±iA]ei(±k1zωt),
E2±=E±[iB1]ei(±k2zωt).
A=2α2εε2+[(εε)24+2(ε+ε)α2]122α{ε+ε2+α2+[(εε)24+2(ε+ε)α2]12}12,
B=2α2+εε2[(εε)24+2(ε+ε)α2]122α{ε+ε2+α2[(εε)24+2(ε+ε)α2]12}12,
A=n12+α2ε2n1α,B=n22+α2ε2n2α.
E1+=E1+eiωt((1+A)[1i]ei(k1z+qz)+(1A)[1i]ei(k1zqz)),
E1=E1eiωt((1+A)[1i]ei(k1z+qz)+(1A)[1i]ei(k1zqz)),
E2+=E2+eiωt((1+B)[i1]ei(k2z+qz)+(1B)[i1]ei(k2zqz)),
E2=E2eiωt((1+B)[i1]ei(k2z+qz)+(1B)[i1]ei(k2zqz)),
Ei=[EixEiy]ei(kszωt)Er=[ErxEry]ei(kszωt)Et=[EtxEty]ei(ks(zdp)ωt).
[EixEiyErxEry]=[P¯¯0¯¯Q¯¯0¯¯][EtxEty00].
[EtxEty]=P¯¯1[EixEiy],
[ErxEry]=Q¯¯P¯¯1[EixEiy].
[ExiEyiExrEyr]=[P¯¯0¯¯Q¯¯0¯¯][ExtEyt00],
P¯¯=[eiδ1p11(1)+eiδ1p11(2)+eiδ2p11(3)+eiδ2p11(4)i(eiδ1p12(1)+eiδ1p12(2)+eiδ2p12(3)+eiδ2p12(4))i(eiδ1p21(1)+eiδ1p21(2)+eiδ2p21(3)+eiδ2p21(4))eiδ1p22(1)+eiδ1p22(2)+eiδ2p22(3)+eiδ2p22(4)],
Q¯¯=[eiδ1q11(1)+eiδ1q11(2)+eiδ2q11(3)+eiδ2q11(4)i(eiδ1q12(1)+eiδ1q12(2)+eiδ2q12(3)+eiδ2q12(4))i(eiδ1q21(1)+eiδ1q21(2)+eiδ2q21(3)+eiδ2q21(4))eiδ1q22(1)+eiδ1q22(2)+eiδ2q22(3)+eiδ2q22(4)],
p11(1)=18(1γiA+N1i)[(eiqdeiqd)(BN2tXγtX+BY)+(eiqd+eiqd)(XγtBY+N2tY)],
p11(2)=18(1+γiAN1i)[(eiqdeiqd)(BN2tX+γtX+BY)+(eiqd+eiqd)(XγtBY+N2tY)],
p11(3)=18(Bγi+BN2i)[(eiqdeiqd)(N1tX+γtAXY)+(eiqd+eiqd)(AXAN1tY+γtY)],
p11(4)=18(B+γiBN2i)[(eiqdeiqd)(N1tXγtAXY)+(eiqd+eiqd)(AXAN1tY+γtY)],
p12(1)=18(1γiA+N1i)[(eiqdeiqd)(X+γtBYN2tY)+(eiqd+eiqd)(BN2tX+γtXBY)],
p12(2)=18(1+γiAN1i)[(eiqdeiqd)(X+γtBYN2tY)+(eiqd+eiqd)(BN2tXγtXBY)],
p12(3)=18(Bγi+BN2i)[(eiqdeiqd)(AX+AN1tYγtY)+(eiqd+eiqd)(N1tXγtAX+Y)],
p12(4)=18(B+γiBN2i)[(eiqdeiqd)(AX+AN1tYγtY)+(eiqd+eiqd)(N1tX+γtAX+Y)],
p21(1)=18(Aγi+AN1i)[(eiqdeiqd)(BN2tX+γtXBY)+(eiqd+eiqd)(X+γtBYN2tY)],
p21(2)=18(A+γiAN1i)[(eiqdeiqd)(BN2tX+γtX+BY)+(eiqd+eiqd)(XγtBY+N2tY)],
p21(3)=18(1γiB+N2i)[(eiqdeiqd)(N1tXγtAX+Y)+(eiqd+eiqd)(AX+AN1tYγtY)],
p21(4)=18(1+γiBN2i)[(eiqdeiqd)(N1tXγtAXY)+(eiqd+eiqd)(AXAN1tY+γtY)],
p22(1)=18(Aγi+AN1i)[(eiqdeiqd)(X+γtBYN2tY)+(eiqd+eiqd)(BN2tX+γtXBY)],
p22(2)=18(A+γiAN1i)[(eiqdeiqd)(XγtBY+N2tY)+(eiqd+eiqd)(BN2tX+γtX+BY)],
p22(3)=18(1γiB+N2i)[(eiqdeiqd)(AX+AN1tYγtY)+(eiqd+eiqd)(N1tXγtAX+Y)],
p22(4)=18(1+γiBN2i)[(eiqdeiqd)(AXAN1tY+γtY)+(eiqd+eiqd)(N1tXγtAXY)],
q11(1)=18(1+γiAN1i)[(eiqdeiqd)(BN2tXγtX+BY)+(eiqd+eiqd)(XγtBY+N2tY)],
q11(2)=18(1γiA+N1i)[(eiqdeiqd)(BN2tX+γtX+BY)+(eiqd+eiqd)(XγtBY+N2tY)],
q11(3)=18(B+γiBN2i)[(eiqdeiqd)(N1tX+γtAXY)+(eiqd+eiqd)(AXAN1tY+γtY)],
q11(4)=18(Bγi+BN2i)[(eiqdeiqd)(N1tXγtAXY)+(eiqd+eiqd)(AXAN1tY+γtY)],
q12(1)=i8(1+γiAN1i)[(eiqdeiqd)(X+γtBYN2tY)+(eiqd+eiqd)(BN2tX+γtXBY)],
q12(2)=i8(1γiA+N1i)[(eiqdeiqd)(X+γtBYN2tY)+(eiqd+eiqd)(BN2tXγtXBY)],
q12(3)=i8(B+γiBN2i)[(eiqdeiqd)(AX+AN1tYγtY)+(eiqd+eiqd)(N1tXγtAX+Y)],
q12(4)=i8(Bγi+BN2i)[(eiqdeiqd)(AX+AN1tYγtY)+(eiqd+eiqd)(N1tX+γtAX+Y)],
q21(1)=i8(A+γiAN1i)[(eiqdeiqd)(BN2tX+γtXBY)+(eiqd+eiqd)(X+γtBYN2tY)],
q21(2)=i8(Aγi+AN1i)[(eiqdeiqd)(BN2tX+γtX+BY)+(eiqd+eiqd)(XγtBY+N2tY)],
q21(3)=i8(1+γiBN2i)[(eiqdeiqd)(N1tXγtAX+Y)+(eiqd+eiqd)(AX+AN1tYγtY)],
q21(4)=i8(1γiB+N2i)[(eiqdeiqd)(N1tXγtAXY)+(eiqd+eiqd)(AXAN1tY+γtY)],
q22(1)=18(A+γiAN1i)[(eiqdeiqd)(X+γtBYN2tY)+(eiqd+eiqd)(BN2tX+γtXBY)],
q22(2)=18(Aγi+AN1i)[(eiqdeiqd)(XγtBY+N2tY)+(eiqd+eiqd)(BN2tX+γtX+BY)],
q22(3)=18(1+γiBN2i)[(eiqdeiqd)(AX+AN1tYγtY)+(eiqd+eiqd)(N1tXγtAX+Y)],
q22(4)=18(1γiB+N2i)[(eiqdeiqd)(AXAN1tY+γtY)+(eiqd+eiqd)(N1tXγtAXY)].
detP¯¯=P11P22P12P21=(p11(1)p22(2)+p11(2)p22(1)+p11(3)p22(4)+p11(4)p22(3)+p12(1)p21(2)+p12(2)p21(1)+p12(3)p21(4)+p12(4)p21(3))+ei(δ1+δ2)(p11(1)p22(3)+p11(3)p22(1)+p12(1)p21(3)+p12(3)p21(1))+ei(δ1δ2)(p11(1)p22(4)+p11(4)p22(1)+p12(1)p21(4)+p12(4)p21(1))+ei(δ1δ2)(p11(2)p22(3)+p11(3)p22(2)+p12(2)p21(3)+p12(3)p21(2))+ei(δ1+δ2)(p11(2)p22(4)+p11(4)p22(2)+p12(2)p21(4)+p12(4)p21(2))+ei2δ1(p11(1)p22(1)+p12(1)p21(1))+ei2δ1(p11(2)p22(2)+p12(2)p21(2))+ei2δ2(p11(3)p22(3)+p12(3)p21(3))+ei2δ2(p11(4)p22(4)+p12(4)p21(4)).
p11(1)p22(1)+p12(1)p21(1)=0,
p11(2)p22(2)+p12(2)p21(2)=0,
p11(3)p22(3)+p12(3)p21(3)=0,
p11(4)p22(4)+p12(4)p21(4)=0.
detP¯¯=P11P22P12P21=(p11(1)p22(2)+p11(2)p22(1)+p11(3)p22(4)+p11(4)p22(3)+p12(1)p21(2)+p12(2)p21(1)+p12(3)p21(4)+p12(4)p21(3))+ei(δ1+δ2)(p11(1)p22(3)+p11(3)p22(1)+p12(1)p21(3)+p12(3)p21(1))+ei(δ1δ2)(p11(1)p22(4)+p11(4)p22(1)+p12(1)p21(4)+p12(4)p21(1))+ei(δ1δ2)d(p11(2)p22(3)+p11(3)p22(2)+p12(2)p21(3)+p12(3)p21(2))+ei(δ1+δ2)(p11(2)p22(4)+p11(4)p22(2)+p12(2)p21(4)+p12(4)p21(2)).
P¯¯1=P˜detP¯¯,
P˜=[P22P12P21P11].
detP¯¯=116(ABN1N2)(ABN2N1)(8[A2γN1A(γ2+N121)+γN1][B2γN2B(γ2+N221)+γN2]+ei(δ1+δ2){(AB1)[γ2(N1+1)(N2+1)]+γ(AB)(N1N2)}2ei(δ1δ2){(AB+1)[γ2+(N1+1)(N21)]γ(A+B)(N1+N2)}2ei(δ1δ2){(AB+1)[γ2+(N11)(N2+1)]γ(A+B)(N1+N2)}2+ei(δ1+δ2){(AB1)[γ2+(N11)(N21)]γ(AB)(N1N2)}2),
Q¯¯P¯¯1=Q¯¯P˜detP¯¯.
[Q¯¯P˜]11=(q11(1)p22(2)+q11(2)p22(1)+q11(3)p22(4)+q11(4)p22(3)+q12(1)p21(2)+q12(2)p21(1)+q12(3)p21(4)+q12(4)p21(3))+ei(δ1+δ2)(q11(2)p22(4)+q11(4)p22(2)+q12(2)p21(4)+q12(4)p21(2))+ei(δ1+δ2)(q11(1)p22(3)+q11(3)p22(1)+q12(1)p21(3)+q12(3)p21(1))+ei(δ1δ2)(q11(2)p22(3)+q11(3)p22(2)+q12(2)p21(3)+q12(3)p21(2))+ei(δ1δ2)(q11(1)p22(4)+q11(4)p22(1)+q12(1)p21(4)+q12(4)p21(1)),
[Q¯¯P˜]12=i[(q11(1)p12(2)q11(2)p12(1)q11(3)p12(4)q11(4)p12(3)+q12(1)p11(2)+q12(2)p11(1)+q12(3)p11(4)+q12(4)p11(3))+ei(δ1+δ2)(q11(2)p12(4)q11(4)p12(2)+q12(2)p11(4)+q12(4)p11(2))+ei(δ1+δ2)(q11(1)p12(3)q11(3)p12(1)+q12(1)p11(3)+q12(3)p11(1))+ei(δ1δ2)(q11(2)p12(3)q11(3)p12(2)+q12(2)p11(3)+q12(3)p11(2))+ei(δ1δ2)(q11(1)p12(4)q11(4)p12(1)+q12(1)p11(4)+q12(4)p11(1))],
[Q¯¯P˜]21=i[(q21(1)p22(2)+q21(2)p22(1)+q21(3)p22(4)+q21(4)p22(3)q22(1)p21(2)q22(2)p21(1)q22(3)p21(4)q22(4)p21(3))+ei(δ1+δ2)(q21(2)p22(4)+q21(4)p22(2)q22(2)p21(4)q22(4)p21(2))+ei(δ1+δ2)(q21(1)p22(3)+q21(3)p22(1)q22(1)p21(3)q22(3)p21(1))+ei(δ1δ2)(q21(2)p22(3)+q21(3)p22(2)q22(2)p21(3)q22(3)p21(2))+ei(δ1δ2)(q21(1)p22(4)+q21(4)p22(1)q22(1)p21(4)q22(4)p21(1))],
[Q¯¯P˜]22=(q21(1)p12(2)+q21(2)p12(1)+q21(3)p12(4)+q21(4)p12(3)+q22(1)p11(2)+q22(2)p11(1)+q22(3)p11(4)+q22(4)p11(3))+ei(δ1+δ2)(q21(2)p12(4)+q21(4)p12(2)+q22(2)p11(4)+q22(4)p11(2))+ei(δ1+δ2)(q21(1)p12(3)+q21(3)p12(1)+q22(1)p11(3)+q22(3)p11(1))+ei(δ1δ2)(q21(2)p12(3)+q21(3)p12(2)+q22(2)p11(3)+q22(3)p11(2))+ei(δ1δ2)(q21(1)p12(4)+q21(4)p12(1)+q22(1)p11(4)+q22(4)p11(1)).

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