Photonic band edge and defect modes in 1D cholesteric liquid crystals

Liquid crystals are transparent optically birefringent materials that have the ability to self-assemble into tunable photonic microstructures. They can be modified by adding chiral dopants, by anchoring on confining surfaces, temperature changes, and by external electric or magnetic fields. Cholesteric liquid crystals (CLCs), which have a periodic helical structure, act as photonic crystals and thus partially reflect light with wavelengths comparable to the period of the structure. Possessing these properties, CLCs can be utilized as resonators or even as micro-lasers if doped with organic dye. In this work, we present the findings of a numerical study of light transmission through CLCs with or without isotropic defect layers in different 1D geometries. We also show numerically calculated photonic eigenmodes and their corresponding Q-factors. Overall, this work summarizes the properties of CLC resonators that could be important for the design of liquid crystal micro-lasers and other soft-matter-based photonic devices.


INTRODUCTION
The helical structure of cholesteric liquid crystals (CLCs) 1 leads to a phenomenon known as selective Bragg reflection, where specific wavelengths of circularly polarised light with the same handedness as the structural helix are reflected while others are transmitted.[4][5][6] In the field of photonics, CLCs have been used for reflective displays, diffractive optical elements, 7,8 and tunable filters, [9][10][11] allowing for precise manipulation of transmitted light by adjusting external parameters such as temperature, electric fields, or mechanical stress.0][21] Band edge lasing occurs at frequencies at the outer edge of the photonic band gap, where stimulated emission is enhanced, leading to controlled and tunable laser emission.Defect mode lasing, on the other hand, exploits localized structural defects within the CLC helix, enabling spectrally precise laser emission also at the frequencies inside the photonic band gap.
Propagation of light through different CLC structures has been studied both experimentally and theoretically.Theoretical studies include studies on cholesteric liquid crystal samples with basic helical organisation, [22][23][24] as well as for structures with different defects in the helical structure.Reflectance has been calculated for isotropic 25,26 and anisotropic 27 defect layer between two CLC layers, and also for structures with multiple isotropic layers. 28t has also been shown theoretically that the defect modes emerge also when the helical structure is only locally deformed. 29n this work, light propagation through cholesteric liquid crystals with isotropic defect layers is studied by two methods.The transmission spectra of different geometries are calculated using the finite-difference time-domain (FDTD) method 30 using the Meep software, 31 whereas the eigenmodes with corresponding eigenfrequencies and quality factors are calculated using the finite-difference frequency-domain (FDFD) method. 32Eigenfrequencies and Q-factors of the eigenmodes with frequencies inside and around the photonic band gap are calculated for (i) a structure without a defect layer, (ii) a structure with a single defect layer, (iii) a structure with multiple defect layers, surrounded by cholesteric liquid crystal layers.We present electric field profiles for selected onedimensional eigenmodes across all investigated structures.

Light mode setup
The light is studied in a system for propagation along the helically organised cholesteric liquid crystals (along x axis), with an isotropic optical defect region, as shown in Figure 1.The director field inside a cholesteric liquid crystal layer of thickness D is described as a right-handed helix n where p is the cholesteric pitch and x n is the position of the first boundary of the n-th CLC segment (for example, in Figure 1: In this work, we assume D = k • p, with k being an integer number.The director profile determines the dielectric tensor within the CLC as: where i, j ∈ {x, y, z}, δ ij is the Kronecker delta, and n i (x) are the components of the director n(x).Inside the isotropic defect layer in the centre of the system (index I) and in glass layers (index G), the material is optically isotropic, with dielectric tensor being diagonal and not changing with position:

Methods
We assume that the system extends infinitely along the y and z axes which allows us to perform simulations on a one-dimensional mesh while still accounting for three-dimensional electric and magnetic field vectors, represented as . We assume open boundary conditions at the edges of the system, which is achieved by perfectly matched layers (PMLs). 33, 34

Transmittance spectra
We compute the transmittance spectra of different CLC structures using the finite-difference time-domain (FDTD) method 30 implemented in Meep software. 31The transmittance spectrum T (λ) is calculated as a ratio between the spectrum of a pulse with Gaussian time-dependence transmitted through a CLC sample and a spectrum of an identical pulse transmitted through the isotropic glass.By restricting the duration of the pulse, we ensure that its spectrum encompasses the frequency range of interest.The source, which is placed at x = d G /2 and extends infinitely along y and z, produces plane waves with circular polarisation (left or right) and amplitudes proportional to exp(−iωt − (t − t 0 ) 2 /2w 2 ).The central spectral frequency of the source ω and the spectral width 1/w are set to ω = 1/(pn), where n = (n o + n e )/2 and 1/w = ω so that the spectrum roughly overlaps with the range of frequencies in which reflection of light is observed.The area at which the transmitted flux spectra are measured is positioned at the same distance from the sample as the source but on the other side of the sample, at

Photonic eigenmodes
The electric field in studied structures can be described as a superposition of photonic eigenmodes as E(x, t) = µ Ψ µ (x)e −ikµt , where Ψ µ (x) are the given modes and summation is performed over all modes µ.The modes are determined using the finite-difference frequency-domain method by solving Maxwell's equations as an eigenproblem: 32,35 ∇ where each eigenmode Ψ µ (x) has its corresponding eigenvalue k µ = ω µ /c 0 , sometimes also referred to as an eigenfrequency when c 0 = 1 is assumed.As the dielectric tensor ε(x) is complex-valued with non-zero imaginary components in the PMLs, the electric field profiles Ψ µ (x) and the eigenvalues k µ are complex as well.The absolute value of the imaginary component, |Im(k µ )|, describes the modal loss, and together with the real component |Re(k µ )|, defines the quality factor (Q-factor) of each mode: 36

PHOTONIC BAND EDGE MODES
A particular example of the studied system is a cholesteric liquid crystal without an isotropic defect layer, i.e. when d I = 0.As shown in Figure 2, a cholesteric liquid crystal with a right-handed helix reflects right circularly polarised (RCP) light with wavelengths within the band gap, n o p < λ < n e p, which enters in the direction of the helical axis.On the other hand, the left circularly polarised light is transmitted at all wavelengths.Outside the band gap, oscillations are observed in the transmission spectrum of the RCP light.Their amplitudes decrease with distance from the band gap, but the frequencies (wavelengths) of the maxima in the transmission spectrum, where T (λ) = 1, coincide with the eigenfrequencies of corresponding photonic eigenmodes of the system.
At these frequencies, although all the light effectively passes through the cholesteric liquid crystal (T ≈ 1), the electric and magnetic fields are amplified within the CLC sample due to internal reflections and constructive interference.The cholesteric liquid crystal thus acts as an optical microresonator.The Q-factors of the resonator's eigenmodes increase with the thickness of the sample and also with birefringence ∆n = n e − n o .In thicker resonators, also the number of eigenmodes in a given frequency range is larger, as shown in panels (a-c) of Figure 2. The electric field profiles of some selected band-edge modes from panel (b) of Figure 2  The eigenmodes can be divided into two branches: those on the blue side of the band gap (B1, B2, B3 ... ) and those on the red side of the band gap (R1, R2, R3 ...).B1 and R1, which are closest to the band gap, have the highest Q-factors.It can also be observed that, in general, for the combination of refractive indices n G = 1.5, n o = 1.6 and n e = 1.8, the Q-factors of the modes on the blue side are slightly higher than the Q-factors on the red side of the band gap.Modes B1 and R1 on Figure 2 look very similar, as do B2 and R2, B3 and R3, etc.But a precise analysis reveals that the electric field of mode B1 rotates by 24.5 full angles along the CLC sample with 24 cholesteric pitches, while the electric field of mode R1 rotates by 23.5 full angles.The field rotates by  The E y and E z components of the electric field are shifted against each other by exactly one-quarter of a wavelength along the helical axis, giving the electric field a helical structure.From the profiles of the complex electric field E(x), it is evident that travelling circularly polarised waves propagate out of the CLC in both directions through the glass plates to the boundaries, as the amplitudes of both the imaginary Im(E)(x) and the real part Re(E)(x) of the electric field are equal in these segments.In contrast, inside the resonator, the real components of the electric field of the eigenmodes are almost everywhere significantly larger than the imaginary ones except at the boundaries.This indicates that standing waves prevail within the CLC sample.The ratio between the amplitude of the real and the imaginary part of the field Re(|E(x)|)/Im(|E(x)|) is largest in modes with highest Q-factors (B1 and R1), where standing waves are therefore most dominant.

Single isotropic layer
Inserting a layer of optically isotropic material between two layers of cholesteric liquid crystal (d I > 0) strongly affects the overall optical response of such optical material.Distinctly, for light within the optical band gap (i.e. with wavelengths λ ∈ (n o p, n e p)), there emerge one or more discrete frequencies -the defect modes-within the band gap at which RCP and LCP light are both, at least partly, transmitted.The number of these allowed frequencies increases with d I , as shown in Figure 3.Moreover, it turns out that the transmittance of LCP and RCP is equal at these discrete frequencies (wavelengths), but elsewhere within the interval λ ∈ (n o p, n e p), the transmittance is still zero for RCP and 100% for LCP if the material helix is right-handed.This means that at certain frequencies within the band gap, we can observe maxima in the RCP transmission spectrum and minima in the LCP spectrum, as shown in Figure 3 and in agreement with Yang et al. 25 We also show in Figure 3  that the eigenfrequencies of the eigenmodes coincide with the extrema in the transmission spectra, and that the band-edge eigenfrequencies are also shifted compared to the band-edge eigenfrequencies of the CLC samples without a defect layer.It turns out that the Q-factors of defect modes can be several orders of magnitude larger than the Q-factors of the band edge eigenmodes.Besides that, Q-factors of the defect modes also increase with the thickness of the defect layer d I , as also presented in Figure 3.
In panel (a) of Figure 4, we show the eigenfrequencies and the corresponding Q-factors of the eigenmodes inside and around the photonic band gap for the geometry with thickness D = 12 µm of the right-handed CLC, thickness of the isotropic defect layer d I = 8 µm and pitch length p = 1 µm.The electric fields of some selected modes are shown in panel (b).D1 and D2 are typical examples of defect eigenmodes.The amplitude of the electric field is constant inside the defect isotropic layer.Sinusoidal profiles of E y (x) and E z (x) are shifted by approximately one-quarter of a wavelength, and thus, the electric field profile has a shape of a right-handed helix.However, it is not perfectly circular as amplitudes of components E y and E z are not necessarily equal.The ratio between them can be different in different eigenmodes (compare D1 and D2, for example), but it also depends on the refractive index and thickness of the isotropic layer n I , d I , and also on the thickness of the cholesteric liquid crystal D. Real components of the electric field within the defect layer are significantly larger than imaginary ones, which means that standing waves dominate.The amplitude of the electric field inside the CLC decreases with distance from the boundary of the defect layer towards a constant value, which is the amplitude of the outgoing waves.The outgoing fields are, in general, elliptically polarised, and depend on same parameters as the ratio between E y and E z components inside the defect layer.In glass at the system's boundaries, real and imaginary components of complex fields have equal amplitudes, showing that the travelling waves propagate away from the CLC on both sides of the resonator.
Outside the band gap, there are band-edge modes B1, B2, B3, ... on the blue side and R1, R2, R3, ... on the red side, in which the electric field's amplitude is not highest in the defect layer but rather in CLC layers.The observed electric field profiles are similar to the ones of the usual band-edge modes that appear in cholesteric samples without a defect layer (Figure 2).However, electric fields from the two CLC layers are coupled through the intermediate defect isotropic layer, so the frequencies of the band-edge eigenmodes and their electric field profiles are also altered upon the variation of the defect layer via d I and n I parameters.

Multiple isotropic layers
If two or more, N L ≥ 2, equal isotropic layers (with equal thickness and refractive index) are placed between the CLC layers, defect modes emerge at similar frequencies as in the geometry with a single defect.The difference is that each of these defect modes split into N L defect eigenmodes.These eigenmodes appear at nearly equal frequencies, as shown in panels (a,b,c,d) of Figure 5.The electric field profiles of selected eigenmodes that exist in the geometry with three defect layers of thickness d I = 5 µm between CLC layers of thickness D = 12 µm are shown in panel (e) of Figure 5.The modes with similar frequencies differ significantly in their Q-factors, as well as in their electric field profiles (panel (e) of Figure 5).It is also evident from panels (a-d) of Figure 5 that the Q factors of the defect modes increase with number of isotropic layers N L in the cell.The results from Figure 5 obtained by FDFD method and the mentioned findings are consistent with the transmittance spectra and electric field amplitudes dependence shown in Gao et al. 28 The electric field of the band-edge modes is "trapped" (has maximum amplitudes) not in the defect layers but in the CLC layers.However, the electric field's amplitudes of defect modes are highest in the isotropic defect layers.Moreover, in the modes with the highest Q-factors, the highest amplitude is observed in the innermost isotropic layer.Modes D1, D2 and D4 in panel (e) of Figure 5 are similar but differ in polarisation and frequency.For example, for modes D1 and D4, which differ in frequency, it can be seen from panel (e) of Figure 5 that the electric field inside isotropic defect layers rotates between the boundaries of the defect layer by a little more than 4.5 full angles in mode D1 and by a little less than 4.5 full angles in mode D4.On the other hand, the frequencies of D1 and D2 modes are similar, but they differ in polarisation, as observed from the ratio between the amplitudes of E y and E z components of the electric field.
The spectrum of photonic eigenmodes changes in the same arrangement of CLC layers and isotropic layers if two CLC layers (I and III) are right-handed and the other two (II and IV) are left-handed.Q-factors of the eigenmodes existing in such geometry are shown in panel (a) of Figure 6, whereas some selected corresponding electric field profiles in panel (b).For light with frequencies within the band gap, layers (I) and (III) form a resonator for right-circularly polarised light, whereas layers (II) and (IV) form a resonator for left-circularly polarised light.Effectively, these two resonators overlap in the innermost isotropic layer where standing waves in the form of a right-and a left-handed helix combine into linearly polarised light.Note that the E y (x) and E z (x) components of the electric fields of modes D1, D2, D3, and D4 from panel (b) of Figure 6 are either perfectly in phase (0 • ) or out of phase (180 • ) in this innermost isotropic layer.
One can also observe that the number of eigenfrequencies within the band gap is higher in this case.This can be explained by the fact that in a right-handed resonator, the defect layer effectively extends from the right edge of layer (I) to the left edge of layer (III), even through layer (II) with a left-handed structure that does not reflect RCP light.The same goes for a left-handed resonator between layers (II) and (IV).However, if the defect layers are thicker, a larger number of eigenfrequencies indeed show up inside the band gap.

CONCLUSIONS
In this work, we analyzed the propagation of light through selected effectively one-dimensional structures made of cholesteric liquid crystals and intermediate defect layers of isotropic material.In the absence of an isotropic defect layer, the cholesteric liquid crystal reflects light with wavelengths between n o p and n e p and circular polarisation of the same handedness as the structure, while transmitting the opposite polarisation.If one or more isotropic layers are added between the CLC, photonic modes appear also within the band gap depending on their thickness and refractive index.We present their eigenfrequencies and Q-factors, and for some selected, also the electric field of the corresponding light modes.In general, this work outlines control and possible design of optical properties of photonic microresonators made of cholesteric liquid crystals, which is a contribution to the design of micro-lasers and other soft-matter-based photonic devices.

Figure 1 .
Figure 1.Schematic of the geometry of the studied system.Two layers of the CLC of thickness D with ordinary refractive index no, extraordinary refractive index ne and pitch p are confined between glass plates with refractive index nG at the boundaries and a general isotropic layer of thickness dI with refractive index nI in the centre.Using the PMLs (perfectly matched layers) at the boundaries to provide open boundary conditions, the glass layers perform as effectively infinite on each side.
in CLCs without isotropic defect layers are shown in panel (d) of the same figure.

Figure 2 .
Figure 2. (a,b,c) Q-factors of photonic eigenmodes in a CLC sample without an isotropic defect layer (dI = 0), and the corresponding transmittance spectra for right circularly polarised (RCP) and left circularly polarised (LCP) light.According to the schematic in Figure 1, the thickness of the CLC sample is denoted by 2D which is 14 µm in panel (a) and 24 µm in panel (b) and 50 µm in panel (c).System parameters: p = 1 µm, no = 1.6, ne = 1.8, nG = 1.5.(d) Electric field profiles of selected eigenmodes from panel (b).In accordance with Figure 1, the dark grey background indicates perfectly matched layers (PMLs), the light grey indicates glass, and the blue indicates cholesteric liquid crystal (CLC).

25. 0
full angles in mode B2 and by 23.0 full angles in R2.We can generalise that the electric field of the CLC band-edge modes rotates by an angle 2π(N p + n/2) for modes Bn and by an angle 2π(N p − n/2) for modes Rn, where N p is the number of cholesteric pitches in the sample.

Figure 4 .
Figure 4. (a) Q-factors of resonant photonic eigenmodes with discrete eigenfrequencies in a CLC geometry shown in Figure 1 with isotropic defect layer of thickness dI = 8 µm.The thickness of right-handed CLC layers with pitch p = 1 µm and refractive indices no = 1.6, ne = 1.8 is D = 12 µm.Refractive indices of glass plates and the isotropic defect layer are equal, nG = nI = 1.5.(b) Corresponding electric field profiles of the eigenmodes with highest Q-factors.

Figure 5 .
Figure 5. (a,b,c,d) Q-factors of photonic eigenmodes in a structure with (a) NL = 2, (b) NL = 3, (c) NL = 4, (d) NL = 5 isotropic defect layers of thickness dI = 5 µm surrounded by cholesteric liquid crystal layers of thickness D = 12 µm with pitch p = 1 µm.The refractive indices are no = 1.6, ne = 1.8, nG = nI = 1.5.The handedness of the cholesteric liquid crystal is equal (right-handed) in all CLC layers.(e) Electric field profiles of selected eigenmodes in geometry with three defect layers (NL = 3, panel (b)).The blue background represents the CLC, and the light grey background indicates the glass.

Figure 6 .
Figure 6.(a) Q-factors of photonic eigenmodes in a structure with three isotropic defect layers of thickness dI = 5 µm surrounded by four cholesteric liquid crystal layers of thickness D = 12 µm with pitch p = 1 µm.The refractive indices are no = 1.6, ne = 1.8, nG = nI = 1.5.In this case, the cholesteric liquid crystal is right-handed in layers (I) and (III) and left-handed in layers (II) and (IV) as indicated in panel (b).(b) Electric field profiles of selected eigenmodes.