Reflection spectra and near-field images of a liquid crystalline half-Skyrmion lattice

Jun-ichi Fukuda; Slobodan Žumer

doi:10.1364/OE.26.001174

1. Introduction

Liquid crystals have intrigued physicists of diverse backgrounds because they exhibit various ordered phases and associated topological defects that are accessible and identifiable by optical means [1,2]. Analogies between the structures of liquid crystals and other systems of interest in physics have been particularly invoked, and therefore liquid crystals offer a platform in which to realize important concepts in physics that are hardly accessible in other systems. A notable example of such analogy is a twist-grain-boundary phase of a chiral smectic liquid crystal [3,4] as an analog of a vortex lattice in a type-II superconductor [5]. Another example is a table-top realization of cosmic strings by topological line defects of a nematic liquid crystal [6, 7]; the latter is easily observed by optical microscopy while the former is still elusive.

A more recent example of analogy between liquid crystal and other systems is the formation of a lattice of Skyrmions. Skyrmions are topologically stable swirl-like excitations in a vector field, originally proposed to account for the existence of particle-like entities in a continuous field theory [8]. Now Skyrmions draw considerable attention in the field of condensed matter physics because they have been shown to play an important role in a wide variety of systems including two-dimensional electron gases exhibiting quantum Hall effect [9–11], spinor Bose-Einstein condensates [12–14], superfluid-A phase of ³He [15–17]. In particular, Skyrmions in chiral ferromagnets have been extensively studied since a theoretical proposal [18] and direct experimental observations [19,20] because of their potential for the manipulation of electrons and high-density memory devices [21–25]. Concerning Skyrmions in a liquid crystal, the energetics of an isolated full-Skyrmion (in which the director n rotates by π from the center to the perimeter) was first discussed by Bogdanov et al. [26,27], and several theoretical studies have been devoted to this subject [28,29]. Later Smalyukh and co-workers carried out extensive studies to elucidate the structures of full-Skyrmions and similar solitonic structures in a chiral liquid crystal [30–33]. Chiral liquid crystals accommodate not only full-Skyrmions, and we showed numerically that half-Skyrmions, in which the rotation of n from the center to the perimeter is π/2 instead of π, form a quasi-two-dimensional (2D) hexagonal lattice in a thin cell of a chiral liquid crystal [34]. Note that the structure of this half-Skyrmion lattice closely resembles the in-plane structure of the bulk hexagonal cholesteric blue phases induced by an applied field [35–37], and also that half-Skyrmions are often referred to as merons that were originally discussed in particle physics [38]. The word “meron” is often used also in the study of topological excitations in magnets [39–41]. Recent observations by optical microscope provided direct experimental evidence of the formation of a half-Skyrmion lattice, and identified half-Skyrmions also in an isolated form [42].

In this work, we investigate the optical properties of a hexagonal half-Skyrmion lattice numerically, focusing on its response to normally incident light. One of the great experimental advantages of liquid crystals is that the former allows direct observations of structures by optical measures [42]. Moreover, changing the handedness of the circular polarization of incident light can help investigate the effect of chirality, which has been shown to play an important role in many condensed matter systems exhibiting Skyrmions. Indeed, it is well known that a cholesteric liquid crystal with a given handedness of helical orientational order reflects circularly polarized light of one handedness much more strongly than that of the other handedness [2]. Another feature of interest is that the periodicity of the lattice structure gives rise to non-uniform intensity distribution of evanescent light, which could be detected by near-field optical microscopy (The experiments in [42] relied on conventional (far-field) optical microscopy with unpolarized light and did not take into account evanescent contributions).

This paper is organized as follows: In Section 2, we describe how the profiles of the liquid crystal and their optical properties are calculated. Results and discussions are presented in Section 3. Section 4 concludes the paper.

2. Model

Here we examine the optical properties of the half-Skyrmion lattice shown in Fig. 1(a), and another structure with three-fold rotational symmetry shown in Fig. 1(b). These structures are exhibited by a thin cell of a highly chiral liquid crystal with strong normal liquid crystal alignment (anchoring) at the cell surfaces, and originally obtained in our previous theoretical work [34]. Note that, in contrast to a hexagonal full-Skyrmion lattice in chiral ferromagnets [20], our liquid crystalline half-Skyrmion lattice is embedded in a honeycomb lattice of topological line defects of orientational order. This difference arises from the head-tail symmetry of the director n, i.e., n ≡ −n [34].

Fig. 1 Top (top row) and side (bottom row) view of (a) a hexagonal half-Skyrmion lattice, and (b) a different structure with three-fold rotational symmetry. Red surfaces represent the location of topological defects of orientational order (Topological defects near the confining surfaces are not shown for the clarity of the top views). Short gray rods show the orientational order of the liquid crystal.

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We minimize the Landau-de Gennes free energy in terms of a second-rank symmetric and traceless tensor order parameter Q_αβ, instead of n The advantages of using a tensor order parameter are that the head-tail symmetry of n is naturally incorporated without special treatments, and that topological defects, or singularities in terms of n, need not be considered as singularities in terms of Q_αβ. The total free energy of a thin cell is given by $F = \int d x d y [f_{s} + \int_{0}^{d} d z (f_{local} + f_{el})]$ , where we consider Cartesian coordinates (x, y, z) and the confining surfaces of the cell are located at z = 0 and d. We employ the following expressions for the local free energy density in the bulk f_local, the elastic energy f_el and the surface anchoring energy f_s:

f_{local} {Q_{α β}} = c (T - T^{*}) Q_{α β} Q_{α β} - \sqrt{6} b Q_{α β} Q_{β γ} Q_{γ α} + a {(Q_{α β} Q_{α β})}^{2},

f_{el} {Q_{α β}, \partial_{γ}} = \frac{1}{4} K_{1} {[∊_{α γ δ} \partial_{γ} Q_{δ β} + 2 q_{0} Q_{α β}]}^{2} + \frac{1}{4} K_{0} {[\partial_{β} Q_{α β}]}^{2},

f_{s} {Q_{α β}} = \frac{1}{2} W {(Q - Q_{s})}_{α β}^{2},

where summations over repeated indices are implied and ∊_αγδ is the Levi-Civita symbol. The material parameters chosen are a = 8.0×10⁴ J m⁻³, b = 5.0×10⁴ J m⁻³, c = 3.0×10⁴ J m⁻³ K⁻¹, T − T^* = 0, p ≡ 2π/q₀ = 161 nm (natural pitch of the helical alignment), K₁ = K₀ = 10 pN, W = 1.0 × 10⁻³ J m⁻², and Q_sαβ = Q₀(ν_αν_β − (1/3)δ_αβ), where ν is the surface normal vector along the z-direction, and Q₀ is such that Q_sαβ minimizes f_local. Note that q₀ can be positive and negative, and our present choice of positive q₀ gives rise to local right-handed twist ordering in the terminology of [2] (See Fig. 6.2). The free energy is minimized by the finite-difference method with periodic boundary conditions in the (x, y) plane (in Fig. 1, 2 × 2 unit cells are shown for the clarity of the structures), and solving a simple relaxational equation (∂/∂t)Q_αβ(r) = −δF/δQ_αβ(r) − μ(r)δ_αβ(r) until the left-hand-side becomes small enough for the profile Q_αβ(r) to be regarded as a (meta)stable one. Here μ(r) is a Lagrange multiplier ensuring the the trace of Q_αβ(r) is zero. The profiles shown in Figs. 1(a) and 1(b) are obtained for thicknesses d/p = 0.637 and 0.955, respectively. The pitch p = 161 nm chosen here and in [34] is shorter than that in the experiments in [42] (≃ 360 nm.) However, since the only length scale involved in the Maxwell equations for electromagnetic waves is the wavelength, the actual value of p is not important; what matters is the ratio of the geometrical parameters (say, the cell thickness and the spacing between two neighboring Skyrmions) and the wavelength of the incident light. Therefore we will express all the lengths in units of p. The lattice constant of our Skyrmion lattice, or the distance between the centers of two neighboring Skyrmions excitations, is a = 0.690p.

Our calculation of the optical properties is based on plane-wave expansions, because we can take into account the lateral periodicity of the liquid crystal structure in a natural way. For a non-magnetic medium which we assume the liquid crystal is, the Maxwell equation for the electric field E oscillating with a single frequency ω as ∝ e^−iωt is written as [43]

\nabla \times \nabla \times E - {(ω / c)}^{2} \overset{\leftrightarrow}{∊} E = 0,

with ∊⃡ and c being the dielectric tensor and the speed of light in vacuum, respectively. We assume that the liquid crystal occupies the region 0 ≤ z ≤ d and is sandwiched by glass of dielectric susceptibility ∊_glass = 2.25 (that is, the refractive index is 1.5). We consider the case in which light with frequency ω is incident normally to the liquid crystal from z = −∞. The structure of the liquid crystal is periodic in the (x, y) plane, and therefore, according to the Bloch theorem, E is expanded as

E (r) = {\begin{array}{l} E_{i} exp (i k_{i} \cdot r) + \sum_{m, n} E_{r}^{(m, n)} exp (i k_{r}^{(m, n)} \cdot r) \\ with k_{r}^{(m, n)} \equiv k_{⊥} + G_{⊥}^{(m, n)} + k_{rz}^{(m, n)} \hat{z} (z < 0), \\ \sum_{m, n} E^{(m, n)} (z) exp (i [k_{⊥} + G_{⊥}^{(m, n)}] \cdot r_{⊥}) (0 \leq z \leq d), \\ \sum_{m, n} E_{t}^{(m, n)} exp (i k_{t}^{(m, n)} \cdot r) with k_{t}^{(m, n)} \equiv k_{⊥} + G_{⊥}^{(m, n)} + k_{r t}^{(m, n)} \hat{z} (z > d) . \end{array}

Here r_⊥ = (x, y), k_⊥ (= 0 because of the normal incidence) is the Bloch vector in the (x, y) space, and $G_{⊥}^{(m, n)}$ is a 2D reciprocal vector labeled by integers m and n that is consistent with the periodicity of the liquid crystal structure. The sets (E_i, k_i), ( $E_{r}^{(m, n)}$ , $k_{r}^{(m, n)}$ ) and ( $E_{t}^{(m, n)}$ , $k_{t}^{(m, n)}$ ) characterize, respectively, the incident light, the reflected light and the transmitted light, and satisfy the transverse conditions E_i · k_i = 0, $E_{r}^{(m, n)} \cdot k_{r}^{(m, n)} = 0$ , and $E_{t}^{(m, n)} \cdot k_{t}^{(m, n)} = 0$ for each (m, n). The transverse condition dictates that ${| k_{i} |}^{2} = {| k_{r}^{(m, n)} |}^{2} = {| k_{t}^{(m, n)} |}^{2} = ∊_{glass} {(ω / c)}^{2}$ . We assume a linear relationship between the dielectric tensor of the liquid crystal and the order parameter, that is, ∊_αβ = ∊_iso + ∊_aQ_αβ with ∊_iso = 2.571 and ∊_a = 1.320, so that the average refractive index of the liquid crystal is 1.603. The reflectivity R and the intensity profiles presented in the following are calculated from the z-component of the Poynting vector at z < 0 subtracted by that of the incident wave. Further details of our calculations are described in the Appendix I and also in [42,44,45]. In the following, $λ_{glass} \equiv 2 π c / \sqrt{∊_{glass}} ω$ denotes the wavelength of incident, reflected and transmitted light in the glass, and that in vacuum is $λ \equiv 2 π c / ω = \sqrt{∊_{glass}} λ_{glass}$ .

3. Results and discussions

In Fig. 2, we show the total reflectivity R as a function of λ/p for left/right-circularly polarized incident light (again we follow the terminology of [2] for the handedness of the polarization). The effect of polarization on the reflectivity is almost invisible when λ ≳ λ^* = 0.896p, while light with right-circular polarization is reflected more strongly when λ ≲ λ^*. The latter tendency, although the effect of polarization is much weaker, is consistent with a well known fact [2] that a cholesteric liquid crystal with right-handed helical orientational order reflects right-circularly polarized light much more strongly than left-circularly polarized light. Such dependence of reflectivity on the handedness of circularly polarized incident light has been reported also for chiral sculptured thin films [46].

Fig. 2 Reflectivity of our hexagonal half-Skyrmion lattice [Fig. 1(a)] as a function of λ for different polarizations of the incident light.

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When a 2D lattice in the real space possesses hexagonal symmetry with the lattice spacing being a = 0.690p, the symmetry of its reciprocal lattice is also hexagonal, and its lattice spacing is $b = 4 π / \sqrt{3} a = 10.514 / p = 2 π / λ_{glass}^{*}$ , where $λ_{glass}^{*} = λ^{*} / \sqrt{∊_{glass}} = λ^{*} / 1.5$ , Therefore the threshold $λ_{glass}^{*}$ (or λ^*) is dictated by the geometrical parameter of the hexagonal lattice. Note that, when $λ_{glass} > λ_{glass}^{*}$ , ${| G^{(m, n)} |}^{2} + {(k_{r z}^{(m, n)})}^{2} = {(2 π / λ_{glass})}^{2} < {(2 π / λ_{glass}^{*})}^{2} = b^{2}$ . From the definition of the reciprocal lattice spacing b, |G^(m,n)| > b² unless G^(m,n) = 0, and therefore $k_{r z}^{(m, n)} (= \sqrt{b^{2} - {| G^{(m, n)} |}^{2}})$ is pure imaginary. This means that, when $λ_{glass} > λ_{glass}^{*}$ (or λ > λ^*), the only propagating mode is the normally reflected mode corresponding to G^(m,n) = 0; all the other reflected modes with G^(m,n) ≠ 0 are evanescent.

Because of the nonuniform and anisotropic distribution of the dielectric tensor in the liquid crystal, it is not easy to explain intuitively why the peaks in Fig. 2 are there. However, the leftmost peak at λ/p = 0.896 obviously has relevance to λ^*. The location of the rightmost one at λ/p = 0.953 is close to 0.956 obtained by b = 2π/λ_LC, where λ_LC is the wavelength of light in the liquid crystal under the assumption that the liquid crystal is an isotropic medium with the average refractive index (1.603 as mentioned above, and therefore λ_LC = λ/1.603). The condition b = 2π/λ_LC corresponds to the agreement between the periodicity of the liquid crystal ordering and the wavelength of light propagating in-plane inside the liquid crystal. The middle peak at λ/p = 0.929 can be accounted for by considering the guided-mode resonance of the planar cell [47,48] (see Appendix II).

Figure 3 shows the intensity profiles of reflected light at the planes with different distances from the lower surface of the liquid crystal cell (z = 0) for λ = 0.953p > λ^*, at which the reflectivity exhibits its maximum in Fig. 2. For both polarizations, the intensity profile becomes uniform at a large distance; see Figs. 3(c) and 3(f), and the dependence of the reflectivity R on polarization is small as shown also in Fig 2 (R = 0.1807 for left-circularly polarized light, and R = 0.1906 for right-circularly polarized light). This uniform profile is consistent with the argument above that for λ > λ^* (or $λ_{glass} > λ_{glass}^{*}$ ) the only propagating mode is that reflecting normally. However, the effect of polarization on the intensity profiles for smaller distances is obvious. These results clearly indicate that the evanescent contributions behave differently in response to the polarization of the incident light. Note also that the amplitude of intensity modulation is much larger for right-circularly polarized incident light [49]. From Figs. 3(a) and 3(d), one can see intensity modulations with periodicity smaller than that of the orientational ordering [Fig. 3(g)]. However these evanescent modes with shorter periodicity decay more rapidly because of larger $| k_{r z}^{(m, n)} |$ or shorter decay length (note that $k_{r z}^{(m, n)}$ is pure imaginary), and are invisible for larger distance [Fig. 3(b) and 3(e)].

Fig. 3 (a–f) Intensity profiles of reflected light for incident light with λ = 0.953p and (a–c) left-circular polarization and (d–f) right-circular polarization, for our hexagonal lattice of half-Skyrmions [Fig. 1(a)]. The planes are located at z = (a,d) −0.0796p, (b,e) −0.796p and (c,f) −7.96p. These intensity profiles are viewed from the light source (z = −∞).

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In Fig. 4, we show the intensity profiles for λ = 0.896p at which the reflectivity exhibits its local maximum. In this case, the reflectivity R depends on the polarization of the incident light (0.0325 for left-circular light and 0.0625 for right-circular light), and the polarization affects the intensity profiles as well. Since now λ < λ^* (although the difference is small), in contrast to the previous case of Fig. 3, several modes with nonzero G^(m,n) are of propagating type, yielding non-uniform intensity profiles at far distances (Compare Figs. 3(c) and 4(c) and also Figs. 3(f) and 4(f)). Notice also larger amplitude of intensity modulation for right-circularly incident light as in Fig. 3. As shown in Figs. 3 and 4, and naturally expected, smaller ratio of the wavelength of the incident probe light λ to the characteristic length of the ordering provides more information on the fine structure.

Fig. 4 (a–f) Intensity profiles of reflected light for incident light with λ = 0.896p and (a–c) left-circular polarization and (d–f) right-circular polarization, for our hexagonal lattice of half-Skyrmions [Fig. 1(a)]. The planes are located at (a,d) −0.0796p, (b,e) −0.796p and (c,f) −7.96p.

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Now let us discuss in an intuitive manner why the reflectivity is almost insensitive to the polarization of the incident light for λ > λ^*, while it is not for λ < λ^*. In the case of a cholesteric liquid crystal mentioned above, when the propagating direction of the incident light is parallel to the axis of helical orientational order, the light reacts to the helical order and consequently the reflectivity is highly sensitive to the polarization. However, when λ > λ^*, only the normally reflecting mode contributes to the reflectivity because all the other modes are evanescent. The orientational order is almost uniform along the normal direction except near the confining surfaces and the axes of helical order is parallel to the confining surfaces (Fig. 1(a)); therefore the normally reflecting mode does not “feel” the helical order with definite handedness, which leads to the reflectivity insensitive to the polarization. A tiny dependence of the reflectivity on the polarization can be attributed to the imperfectness in the uniformity along the normal direction. On the other hand, the above argument does not hold for the modes with wavevector $k_{r}^{(m, n)}$ oblique to the normal direction. Therefore the amplitude of such oblique modes is sensitive to the polarization, which results in polarization-dependent intensity profiles from evanescent modes presented in Fig. 3. Moreover, when λ < λ^*, some of the oblique modes can propagate and contribute to the reflectivity that depends on the polarization. We also note that the asymmetry of the reflectivity about λ = λ^* is likely to be attributable to the presence (absence) of the oblique propagating modes at λ < λ^* (λ > λ^*).

Finally, we present the optical properties of a different structure with three-fold symmetry shown in Fig. 1(b). In Figs. 5(a)–5(f), we show the intensity profile for λ = 0.944p, at which the reflectivity exhibits its local maximum (see Fig. 5(g)). 0.944p is larger than but very close to λ^* = 0.943p (derived from the lattice constant a = 0.726p in the real space and b = 9.994/p in the reciprocal space), and therefore the far-field intensity profiles become uniform, although the decay length is large (In Fig. 5, we do not show the uniform profiles). The difference of the intensity profiles arising from the difference of polarization is more evident than the cases in Figs. 3 and 4, a manifestation of a more complex structure of the liquid crystal. Note that in contrast to the previous case of a Skyrmion lattice, the dependence of reflectivity on the polarization is present even when λ > λ^* as can be seen in Fig. 5(g). This is because the structure of the liquid crystal (Fig. 1(b)) is no longer uniform along the normal direction, and therefore the normally reflected mode, the only propagating mode contributing to the reflectivity, can be influenced by the helical ordering with definite handedness. Again there is no intuitive explanation why the peaks are there in Fig. 5(g), but the leftmost peak is, as in the case of Fig. 2, close to λ^* as mentioned above. The location of the rightmost peak, λ/p = 1.003 is close to 1.008 determined by b = 2π/λ_LC, although the small deviation indicates that the liquid crystal can no longer be safely approximated as a homogeneous medium. The location of two peaks expected from guided-mode resonance is λ/p = 0.945 and 0.988 (see Appendix II), agrees well with numerically observed 0.944 and 0.986. Other peaks should be attributed to the complex structure of the liquid crystal and simple explanations are unlikely to exist.

Fig. 5 (a–f) Intensity profiles of reflected light for the structure shown in Fig. 1(b), for incident light with λ = 0.944p and (a–c) left-circular polarization and (d–f) right-circular polarization. The planes are located at z = (a,d) −0.0796p, (b,e) −0.796p and (c,f) −7.96p. (g) Reflectivity as a function of λ for different polarizations of the incident light.

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Although determination of the intensity profiles for evanescent modes is an experimental challenge, the results presented here clearly indicate that the hexagonal half-Skyrmion lattice (Fig. 1(a)) and the more complex structure with three-fold symmetry (Fig. 1(b)) are distinguishable by optical means.

4. Conclusions

To conclude, by numerical calculations based on plane-wave expansions, we investigated how a half-Skyrmion lattice exhibited by a thin cell of a chiral liquid crystal reflect normally incident light. The intensity profile of reflected light, particularly in the near-field regime, depends on the wavelength and the handedness of the circular polarization of the incident light. Surprisingly the reflectivity is almost insensitive to the polarization when the wavelength is larger than a threshold value, although the intensity profiles from the evanescent modes are not. Our results demonstrate that microscopy based on evanescent light can, in cases where the pitch is in the visible range, yield much more details on Skyrmion structures and their lattices than conventional far-field microscopy. Therefore a chiral liquid crystal can be a model system of Skyrmions different from other already known condensed matter systems in that light can be a probe for the investigation of their structures. We hope that our study will promote experimental studies to elucidate the properties of Skyrmions using a liquid crystal as a novel model system.

Appendix I

Here we describe further details of the calculation of eqs. (4) and (5). The unit cell of our system in the calculation is a rectangular parallelepiped with periodic boundary conditions in the (x, y) plane, and the reciprocal lattice vector is simply represented as G^(m,n) = (2mπ/l_x, 2nπ/l_y), where l_x and l_y are the unit cell dimension along the x and y directions, respectively. In our present calculations, we set −16 ≤ m, n ≤ 16.

E^(m,n)(z) inside the liquid crystal is discretized with equal spacings at 0 = z₀ < z₁ < · · · < z_{l_max} = d with l_max = 32. This discretization along the z directions is such that the sum of the total reflectivity R and the total transmittance T equals 1 up to rounding errors [50].

The boundary conditions of the electric field is the continuity of E_x,y and (∇ × E)_x,y at the boundaries between the liquid crystal and the outer space. With the boundary conditions, now we have a set of as many linear equations as unknowns, which is solved with the help of UMFPACK [51] for sparse linear equations.

Appendix II

Here we consider the guided-mode resonance to account for the peaks in the reflection spectra. We assume that the refractive index of the liquid crystal with thickness d to deviate only weakly from its average value n_LC(= 1.603), and let k_p denote the in-plane wavenumber of the guided-mode resonance. We further define $γ = \sqrt{k_{p}^{2} - {| k_{i} |}^{2}}$ and $κ = \sqrt{n_{LC}^{2} {| k_{i} |}^{2} / ∊_{glass} - k_{p}^{2}}$ . Note that |k_i| determines the frequency ω of the light, and the refractive index of the outer space is $\sqrt{∊_{glass}} (= 1.5)$ . Then k_p is given by the solution(s) of [47,48]

\begin{array}{l} tan κ d & = & \frac{2 κ γ}{κ^{2} - γ^{2}} (for the transverse electric (TE) mode) \\ tan κ d & = & \frac{2 n_{LC}^{2} ∊_{glass} κ γ}{∊_{glass}^{2} κ^{2} - n_{LC}^{4} γ^{2}} (for the transverse magnetic (TM) mode) \end{array}

for

1 < k_{p} / | k_{i} | < n_{LC} / \sqrt{∊_{glass}}

.

If k_p is equal to the length of the in-plane reciprocal lattice vector (in the main text, it is denoted by b), resonance inside the liquid crystal occurs and is expected to give rise to strong reflections. In the case of the half-Skyrmion lattice shown in Fig. 1(a) (b = 10.514/p and d = 0.637p), this resonance condition is given by |k_i| = 10.13 (TE mode) and 10.15 (TM mode). This result corresponds to λ = 0.930p (TE mode) and 0.928p (TM mode), close to the location of the middle peak of Fig. 2 at λ = 0.929p, although two peaks are not distinguishable. In the case of the three-fold-symmetric structure shown in Fig. 1(b) (b = 9.994/p and d = 0.955p), the resonance is expected to occur at |k_i| = 9.541 (TE mode) and 9.972 (TM mode), corresponding to λ = 0.988p (TE mode) and 0.945p (TM mode), close to the location of two of the peaks at λ = 0.986p and 0.944p.

Funding

Slovenian Research Agency (ARRS research program P1-0099 and project J1-2335); Center of Excellence NAMASTE; Japan Society for the Promotion of Science (JSPS) KAKENHI (Grant Number JP25400437 and JP17H02947); Cooperative Research Program of “Network Joint Research Center for Materials and Devices” (20171264)

Acknowledgments

The authors thank Dr. Andriy Nych and Professor Igor Muševič for valuable discussions that motivated this work.

References and links

1. P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, 1995). [CrossRef]

2. P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, International Series of Monographs on Physics (Oxford University Press, 1995), 2nd ed.

3. S. R. Renn and T. C. Lubensky, “Abrikosov dislocation lattice in a model of the cholesteric–to–smectic- A transition,” Phys. Rev. A 38, 2132–2147 (1988). [CrossRef]

4. J. W. Goodby, M. Waugh, S. Stein, E. Chin, R. Pindak, and J. Patel, “Characterization of a new helical smectic liquid crystal,” Nature 337, 449–452 (1989). [CrossRef]

5. A. Abrikosov, “Magnetic properties of superconductors of the second group,” Sov. Phys. JETP 5, 1174–1182 (1957).

6. I. Chuang, R. Durrer, N. Turok, and B. Yurke, “Cosmology in the laboratory: defect dynamics in liquid crystals,” Science 251, 1336–1342 (1991). [CrossRef] [PubMed]

7. M. J. Bowick, L. Chandar, E. A. Schiff, and A. M. Srivastava, “The Cosmological Kibble Mechanism in the Laboratory: String Formation in Liquid Crystals,” Science 263, 943–945 (1994). [CrossRef] [PubMed]

8. T. H. R. Skyrme, “A Non-Linear Field Theory,” Proc. Royal Soc. London A 260, 127–138 (1961). [CrossRef]

9. S. L. Sondhi, A. Karlhede, S. A. Kivelson, and E. H. Rezayi, “Skyrmions and the crossover from the integer to fractional quantum Hall effect at small Zeeman energies,” Phys. Rev. B 47, 16419–16426 (1993). [CrossRef]

10. S. E. Barrett, G. Dabbagh, L. N. Pfeiffer, K. W. West, and R. Tycko, “Optically Pumped NMR Evidence for Finite-Size Skyrmions in GaAs Quantum Wells near Landau Level Filling ν = 1,” Phys. Rev. Lett. 74, 5112–5115 (1995). [CrossRef] [PubMed]

11. A. Schmeller, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Evidence for Skyrmions and Single Spin Flips in the Integer Quantized Hall Effect,” Phys. Rev. Lett. 75, 4290–4293 (1995). [CrossRef] [PubMed]

12. U. Al Khawaja and H. Stoof, “Skyrmions in a ferromagnetic bose-einstein condensate,” Nature 411, 918–920 (2001). [CrossRef] [PubMed]

13. A. E. Leanhardt, Y. Shin, D. Kielpinski, D. E. Pritchard, and W. Ketterle, “Coreless vortex formation in a spinor bose-einstein condensate,” Phys. Rev. Lett. 90, 140403 (2003). [CrossRef] [PubMed]

14. L. S. Leslie, A. Hansen, K. C. Wright, B. M. Deutsch, and N. P. Bigelow, “Creation and Detection of Skyrmions in a Bose-Einstein Condensate,” Phys. Rev. Lett. 103, 250401 (2009). [CrossRef]

15. N. D. Mermin and T.-L. Ho, “Circulation and Angular Momentum in the A Phase of Superfluid Helium-3,” Phys. Rev. Lett. 36, 594–597 (1976). [CrossRef]

16. P. W. Anderson and G. Toulouse, “Phase Slippage without Vortex Cores: Vortex Textures in Superfluid ³He,” Phys. Rev. Lett. 38, 508–511 (1977). [CrossRef]

17. V. M. H. Ruutu, J. Kopu, M. Krusius, U. Parts, B. Plaçais, E. V. Thuneberg, and W. Xu, “Critical Velocity of Vortex Nucleation in Rotating Superfluid ³He- A,” Phys. Rev. Lett. 79, 5058–5061 (1997). [CrossRef]

18. U. Rößler, A. Bogdanov, and C. Pfleiderer, “Spontaneous skyrmion ground states in magnetic metals,” Nature 442, 797–801 (2006). [CrossRef]

19. S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. Böni, “Skyrmion Lattice in a Chiral Magnet,” Science 323, 915–919 (2009). [CrossRef] [PubMed]

20. X. Yu, Y. Onose, N. Kanazawa, J. Park, J. Han, Y. Matsui, N. Nagaosa, and Y. Tokura, “Real-space observation of a two-dimensional skyrmion crystal,” Nature 465, 901–904 (2010). [CrossRef] [PubMed]

21. C. Pfleiderer and A. Rosch, “Condensed-matter physics: Single skyrmions spotted,” Nature 465, 880–881 (2010). [CrossRef] [PubMed]

22. A. Fert, V. Cros, and J. Sampaio, “Skyrmions on the track,” Nature Nanotechnology 8, 152–156 (2013). [CrossRef] [PubMed]

23. N. Nagaosa and Y. Tokura, “Topological properties and dynamics of magnetic skyrmions,” Nature Nanotechnology 8, 899–911 (2013). [CrossRef] [PubMed]

24. N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter, K. von Bergmann, A. Kubetzka, and R. Wiesendanger, “Writing and Deleting Single Magnetic Skyrmions,” Science 341, 636–639 (2013). [CrossRef] [PubMed]

25. W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M. B. Jungfleisch, F. Y. Fradin, J. E. Pearson, Y. Tserkovnyak, K. L. Wang, O. Heinonen, S. G. E. te Velthuis, and A. Hoffmann, “Blowing magnetic skyrmion bubbles,” Science 349, 283–286 (2015). [CrossRef] [PubMed]

26. A. Bogdanov and A. Shestakov, “Inhomogeneous two-dimensional structures in liquid crystals,” J. Exp. Theo. Phys. 86, 911–923 (1998). [CrossRef]

27. A. N. Bogdanov, U. K. Rößler, and A. A. Shestakov, “Skyrmions in nematic liquid crystals,” Phys. Rev. E 67, 016602 (2003). [CrossRef]

28. A. O. Leonov, I. E. Dragunov, U. K. Rößler, and A. N. Bogdanov, “Theory of skyrmion states in liquid crystals,” Phys. Rev. E 90, 042502 (2014). [CrossRef]

29. S. Afghah and J. V. Selinger, “Theory of helicoids and skyrmions in confined cholesteric liquid crystals,” Phys. Rev. E 96, 012708 (2017). [CrossRef]

30. I. I. Smalyukh, Y. Lansac, N. A. Clark, and R. P. Trivedi, “Three-dimensional structure and multistable optical switching of triple-twisted particle-like excitations in anisotropic fluids,” Nature Materials 9, 139–145 (2010). [CrossRef]

31. P. J. Ackerman, Z. Qi, and I. I. Smalyukh, “Optical generation of crystalline, quasicrystalline, and arbitrary arrays of torons in confined cholesteric liquid crystals for patterning of optical vortices in laser beams,” Phys. Rev. E 86, 021703 (2012). [CrossRef]

32. P. J. Ackerman, R. P. Trivedi, B. Senyuk, J. van de Lagemaat, and I. I. Smalyukh, “Two-dimensional skyrmions and other solitonic structures in confinement-frustrated chiral nematics,” Phys. Rev. E 90, 012505 (2014). [CrossRef]

33. P. J. Ackerman, J. van de Lagemaat, and I. I. Smalyukh, “Self-assembly and electrostriction of arrays and chains of hopfion particles in chiral liquid crystals,” Nature Communications 6, 6012 (2015). [CrossRef] [PubMed]

34. J. Fukuda and S. Žumer, “Quasi-two-dimensional Skyrmion lattices in a chiral nematic liquid crystal,” Nature Communications 2, 246 (2011). [CrossRef] [PubMed]

35. G. Heppke, B. Jéròme, H.-S. Kitzerow, and P. Pieranski, “Observation of a hexagonal blue phase in systems with negative dielectric anisotropy,” Liq. Cryst. 5, 813–828 (1989). [CrossRef]

36. R. M. Hornreich and S. Shtrikman, “Invited lecture. Hexagonal cholesteric blue phases,” Liq. Cryst. 5, 777–789 (1989). [CrossRef]

37. O. Henrich, D. Marenduzzo, K. Stratford, and M. E. Cates, “Thermodynamics of blue phases in electric fields,” Phys. Rev. E 81, 031706 (2010). [CrossRef]

38. C. G. Callan, R. Dashen, and D. J. Gross, “Toward a theory of the strong interactions,” Phys. Rev. D 17, 2717–2763 (1978). [CrossRef]

39. M. Ezawa, “Compact merons and skyrmions in thin chiral magnetic films,” Phys. Rev. B 83, 100408 (2011). [CrossRef]

40. M. Pereiro, D. Yudin, J. Chico, C. Etz, O. Eriksson, and A. Bergman, “Topological excitations in a kagome magnet,” Nature Communications 5, 4815 (2014). [CrossRef] [PubMed]

41. A. Tan, J. Li, A. Scholl, E. Arenholz, A. T. Young, Q. Li, C. Hwang, and Z. Q. Qiu, “Topology of spin meron pairs in coupled Ni/Fe/Co/Cu(001) disks,” Phys. Rev. B 94, 014433 (2016). [CrossRef]

42. A. Nych, J.-i. Fukuda, U. Ognysta, S. Žumer, and I. Muševič, “Spontaneous formation and dynamics of half-skyrmions in a chiral liquid-crystal film,” Nature Physics 13, 1215–1220 (2017). [CrossRef]

43. L. D. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of continuous media, vol. 8 (elsevier, 1984), 2nd ed.

44. J. Fukuda and S. Žumer, “Exotic Defect Structures and Their Optical Properties in a Strongly Confined Chiral Liquid Crystal,” Mol. Cryst. Liq. Cryst. 594, 70–77 (2014). [CrossRef]

45. J.-i. Fukuda, Y. Okumura, and H. Kikuchi, “Calculation of confocal microscope images of cholesteric blue phases,” Proc. SPIE 9769, 976906 (2016). [CrossRef]

46. V. C. Venugopal, “Three-dimensional periodic chiral sculptured thin films,” J. Nanophotonics 7, 073502 (2013). [CrossRef]

47. D. Marcuse, Theory of Dielectric Optical Waveguides (Second Edition) (Academic Press, 2013).

48. S. S. Wang, R. Magnusson, J. S. Bagby, and M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 7, 1470–1474 (1990). [CrossRef]

49. One might feel it strange that the intensity is negative at some points in the profiles. However, the intensity of the evanescent contributions, when averaged over a two-dimensional plane, is zero, which means that its intensity can be positive and negative unless it is zero throughout the plane. Therefore, local “negative intensity” does not contradict any physics. Note also that the intensity averaged over the plane for given handedness of polarization is independent of the distance, and positive, as it should be.

50. K. B. Dossou and L. C. Botten, “A combined three-dimensional finite element and scattering matrix method for the analysis of plane wave diffraction by bi-periodic, multilayered structures,” J. Comp. Phys. 231, 6969–6989 (2012). [CrossRef]

51. T. A. Davis, “Algorithm 832: Umfpack v4.3—an unsymmetric-pattern multifrontal method,” ACM Trans. Math. Softw. 30, 196–199 (2004). [CrossRef]

Reflection spectra and near-field images of a liquid crystalline half-Skyrmion lattice

Abstract

1. Introduction

2. Model

3. Results and discussions

4. Conclusions

Appendix I

Appendix II

Funding

Acknowledgments

References and links

Cited By

Figures (5)

Equations (6)

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