Second harmonic light scattering induced by defects in the twist-bend nematic phase of achiral liquid crystal dimers

First we consider the results from polarizing microscopy displayed in Fig. 2; the images shown were obtained after cooling the samples in the applied magnetic field to temperatures below the transition temperature ($T_{TB}$) to the $\mathrm{N_{TB}}$ phase. In the images, the polarizer and analyzer axes are approximately parallel to the field. The effective director $\hat{\mathbf{t}}$ is uniformly oriented along the field, except in the immediate vicinity of parabolic focal conic (PFC) defects. These are delineated in Fig. 2 by dark lines tracing out parabolic trajectories, which are paired together in orthogonal planes such that each parabola in a pair passes through the other’s focus. The paired parabolae resemble the classic PFC texture observed in ordinary smectic LCs under a dilatory strain of the smectic layers Rosenblatt_PFC .

In the images of Fig. 2, typically one parabola lies in the focal plane (parallel to the cell substrates), while the second of the pair occupies a plane normal to the viewing direction. The spacing between the two opposite “arms” within each PFC is $\sim 50\mu$m, while the separation between foci in the core region is about $6~\mu$m. The symmetry axes of the PFCs are well-aligned, and parallel to the $\mathrm{N_{TB}}$ optical axis ($\hat{\mathbf{t}}$) direction. In the thicker (1 mm) cells, the core regions of the PFCs could be resolved in two distinct planes close to the opposing cell boundaries, whereas in the 0.1 mm cells a single focal plane was located at essentially the mid-plane of the cell. Thus, two layers of PFCs form near the opposing substrates in sufficiently thick samples, but are squeezed into a single layer when the cell thickness becomes comparable to the dimensions of the parabolae.

As Fig. 2(c) reveals, the PFCs are absent at temperatures close to (within $\sim 1^{\circ}$ of $T_{TB}$); they appear and their population increases in the range $T_{TB}-T\simeq 1$ to $3^{\circ}$C, and then saturates. By analogy with ordinary smectics, we interpret the PFCs as defects in a “pseudo-layer” structure defined by $\sim 10$ nm thick slabs between planes of constant phase in the heliconical $\mathrm{N_{TB}}$ structure. As the temperature decreases below $T_{TB}$, the “layer” spacing (pitch $t_{0}$) shrinks, resulting in PFC formation at a threshold $t_{0}$ just as a critical dilatory strain does in the case of true smectic layers.

We now turn to our results from SHLS measurements in the $\mathrm{N_{TB}}$ phase. These are summarized in Fig. 3. First consider the case where the incident (fundamental) light is polarized along $\hat{\mathbf{t}}$ and the SH output is polarized perpendicular to $\hat{\mathbf{t}}$ (an $H_{\omega}\rightarrow V_{2\omega}$ process, following the nomenclature introduced above). In both $\mathrm{N_{TB}}$ materials, for $T$ below but within $1^{\circ}$C of the transition ($T_{TB}$) – the range where PFCs are absent – no SH output is detected. Below this range and in the presence of PFCs, a clear SH signal is observed, with a peak in the forward direction. In the KA(0.2) samples, the signal level increases slightly with decreasing $T$ in both thick and thin samples, whereas the signal approximately doubles for DTC5C7. The angular FWHM of the peak (using data from the thin cell where one expects less broadening due to multiple scattering) corresponds to a length scale of $1.22\lambda_{2\omega}/$FWHM $\sim 7~\mu$m, which is comparable to the spacing between foci of the PFCs. The SH signal is clearly associated with the appearance of PFCs in the $\mathrm{N_{TB}}$ phase.

For $V$ polarized input and $V$ output ($V_{\omega}\rightarrow V_{2\omega}$ process), the angular distribution of SH light is again concentrated around the forward direction. However, the peak is split approximately symmetrically about zero scattering angle (compare the thumbnail images in Fig. 3 showing the raw SHLS patterns recorded on the CCD). Also, the temperature dependence is weaker than for the $H_{\omega}\rightarrow V_{2\omega}$ process. On the other hand, for $H$ polarized output ($H_{\omega}\rightarrow H_{2\omega}$ and $V_{\omega}\rightarrow H_{2\omega}$ processes), there is no SH signal. Thus, all detected SH output is polarized perpendicular to the average helical axis of the TB structure or, equivalently, to the axis of the PFC defects.

The peak SH intensity from the 0.1 mm thick samples of KA(0.2) is approximately two times lower than in the 1 mm sample. For incoherent SH generation from the bulk NTB phase, we would normally expect a factor $\sim 10$ decrease. The discrepancy is explained by the SHG originating from the PFC defects: As noted above, there are two “layers” of PFCs, localized near the surfaces, in the thick cell, but only one in the thin cell – hence the factor of $\sim 2$ change observed in SH intensity.

In all cases, the SH signal is quite weak – of order $\sim 10^{-7}$ the level from a 0.5 mm thick Z-cut single crystal quartz reference. Several factors, beyond the possibility of an intrinsically small nonlinear susceptibility, may contribute to this weak level. First, the scattering is incoherent, and there is no phase matching between the fundamental and second harmonic waves. Second, the signal comes only from the PFCs, and the region of strong “pseudo-layer” deformation (defined by the parabolic lines in the images in Fig. 2) occupies only a small fraction of the total volume illuminated by the fundamental laser beam. Third, as we shall describe later in the Discussion section, cancellations in the induced polarization (and effective non-centrosymmetry) occur both at points on the parabolae and between points on opposite sides of them, reducing the SH output.

To confirm that the SH signal arises specifically from deformation of $\mathrm{N_{TB}}$ “pseudo-layers”, we performed similar measurements on the smectic-A phase of an ordinary rod-like LC, octyl cyanobiphenyl (8CB), which also exhibits classical PFCs. As shown in Fig. 4, for both 1 mm and 0.1 mm cells populated with PFC defects of an ordinary smectic layer structure, there is no detectable SH signal under experimental conditions identical to those used for the $\mathrm{N_{TB}}$ samples.

Fig. 5(a) shows a three-dimensional rendering of deformed layers in the core region of a PFC, which is based upon the geometrical construction given in Ref. Rosenblatt_PFC . Although this construction strictly applies in the limit of incompressible layers (i.e., infinite elastic constant for layer compression), it provides a useful starting point.

In a normal smectic-A, curvature of the layers implies a splay of the director field, and thus in principle a flexoelectric polarization normal to the layers and in the direction of $\hat{\mathbf{n}}(\mathbf{\nabla}\cdot\hat{\mathbf{n}})$ Meyer_flexo . However, as already pointed out, no SHG from such a mechanism was observed due to PFCs in the smectic-A phase of the control sample 8CB. For the layer distortion in a PFC defect, the maximum curvature occurs at the conical “cusps”, where the layers “pinch” down to points located along the parabolae (dashed lines in Fig. 5) that define the defect. In order to avoid an infinite splay energy, the molecules must either rotate off the layer normal (relaxing the splay) and adopt a more uniform orientation at the cusp, or the vertex of the cusps must relax from a conical tip (infinite layer curvature) to a bowl shape (finite curvature). In the former case, the splay flexoelectric polarization would be significantly reduced, while in the latter case the curvature of the layer approaching a cusp would have the opposite sign of the curvature at the bottom of the bowl (that replaces the sharp tip of the cusp), again tending to cancel the splay flexoelectricity. These effects can explain the absence of a detectable SH signal in the control sample (ordinary smectic-A).

In the dimers forming the pseudo-layered $\mathrm{N_{TB}}$ phase, the same terminal groups on identical mesogenic units are connected to opposite ends of the flexible spacer; this architecture tends to eliminate an overall longitudinal molecular dipole, and thus negate the conventional splay flexoelectric effect at the molecular level. As will be made clear below, the polarization selection of the SH light observed in our experiment also cannot be explained by conventional splay flexoelectric polarization, which would have its major component along the PFC axis ($\hat{\mathbf{z}}$ in Fig. 5). Moreover, any bend flexoelectricity associated with the equilibrium (heliconical) NTB structure averages out over a length scale much shorter than an optical wavelength.

Thus, we turn to a model that explicitly couples long wavelength distortions of the heliconical director structure to distortions of a helical vector order parameter $\mathbf{P}$ (where $\mathbf{P}$ may originate either from a net transverse dipole moment, or simply from the “shape” polarization, associated with the bent conformation of LC dimers. It is convenient to introduce a dimensionless form for this order parameter, $\mathbf{p}=\mathbf{P}/P_{sat}$, where $P_{sat}$ corresponds to the saturated polarization at low temperature.

The $\mathrm{N_{TB}}$ free energy density may then be expanded in terms of the uniaxial director field $\hat{\mathbf{n}}$ and the polarization field $\mathbf{p}$ as Shamid_PRE

Here, $K_{1}$, $K_{2}$, and $K_{3}$ are the usual Frank elastic constants for splay, twist, and bend distortions of the nematic director $\hat{\mathbf{n}}$. The coefficient $\mu=\mu_{0}(T-T_{0})$ is the temperature-dependent Landau coefficient for the polarization $\mathbf{p}$ ($\mu_{0}$ being a constant), while $\nu>0$ is a higher-order, temperature-independent Landau coefficient. The elastic constant $\kappa$ penalizes spatial distortions in $\mathbf{p}$, and the coefficient $\Lambda$ couples $\mathbf{p}$ with bend distortions. The last term (not included in Ref. Shamid_PRE ), with $\eta>0$, penalizes any component of $\mathbf{p}$ that lies along $\hat{\mathbf{n}}$. Because $\mathbf{p}$ is defined as a dimensionless quantity, the Landau coefficients $\mu$ and $\nu$ carry the same units, and $\kappa$ has the same units as the Frank constants.

In the $\mathrm{N_{TB}}$ phase, the director field has the equilibrium heliconical modulation $\hat{\mathbf{n}}=\hat{\mathbf{z}}\cos\beta+\hat{\mathbf{x}}\sin\beta\cos(q_{0}z)+\hat{\mathbf{y}}\sin\beta\sin(q_{0}z)$, with pitch wavenumber $q_{0}$ and cone angle $\beta$. Likewise, the polarization field has the equilibrium helical modulation $\mathbf{p}=\hat{\mathbf{x}}p_{0}\sin(q_{0}z)-\hat{\mathbf{y}}p_{0}\cos(q_{0}z)$, with magnitude $p_{0}$, perpendicular to $\hat{\mathbf{n}}$ and to the pitch axis $\hat{\mathbf{z}}$, as shown in Fig. 1(b). In the nematic phase, $\beta$ and $p_{0}$ are both zero while $q_{0}$ is undefined; in the $\mathrm{N_{TB}}$ phase, these quantities all become non-zero.

Let us now consider an imposed spatial variation in the phase, $\Phi$, of the helicoidal modulation. This corresponds to a displacement, $u=-\Phi/q_{0}$, of the $\mathrm{N_{TB}}$ “pseudo-layers” along the equilibrium layer normal $\hat{\mathbf{z}}$. Under the imposed distortion, the layer plane varies in orientation, and the directions of $\hat{\mathbf{n}}$ and $\mathbf{p}$ also generally change in order to minimize $F_{NTB}$. To describe such a change in $\hat{\mathbf{n}}$, we introduce the orthogonal triad of unit vectors $\hat{\mathbf{e}}_{1}$, $\hat{\mathbf{e}}_{2}$, $\hat{\mathbf{t}}$, where $\hat{\mathbf{t}}$ is obtained by rotation of $\hat{\mathbf{z}}$ about an arbitrary axis in the $x$-$y$ plane, and $\hat{\mathbf{e}}_{1}$ and $\hat{\mathbf{e}}_{2}$ are the transformed $\hat{\mathbf{x}}$ and $\hat{\mathbf{y}}$ under this rotation. Then the distorted $\hat{\mathbf{n}}$ is

The rotated coordinate unit vectors may be expressed in the $xyz$ system as

where $t_{z}=\sqrt{1-t_{x}^{2}-t_{y}^{2}}$, and $t_{x}$, $t_{y}$ are the $x$, $y$ components of $\hat{\mathbf{t}}$.

Since $\mathbf{p}$ is not a unit vector and need not, in general, remain orthogonal to $\hat{\mathbf{n}}$ under the imposed distortion, we write,

Here $p_{x}$, $p_{y}$, $p_{z}$ are the $xyz$ components of the distortion in $\mathbf{p}$ that does not remain perpendicular to $\hat{\mathbf{n}}$.

The calculation of these components for an imposed “pseudo-layer” displacement $u=u(x)$, which varies along an axis in the equilibrium layer plane (and therefore describes layer bending), and for $u=u(z)$, which varies along the layer normal (describing layer compression), is outlined in the Appendix. For simplicity, the calculation is done only to linear order in $\{p_{x},p_{y},p_{z},t_{x},t_{y}\}$.

The results, assuming large $\eta$ Parsouzi_PRX and small $\beta$ ($\beta\lesssim 10^{\circ}$ close to the $\mathrm{N_{TB}}$-N transition Borshch_Nature ; CMeyer3 , are:

for $u=u(x)$, and

for $u=u(z)$. Here $\mathbf{p}_{\perp}=p_{x}\hat{\mathbf{x}}+p_{y}\hat{\mathbf{y}}$ and similarly for $\mathbf{t}_{\perp}$. The approximate equality signs in Eqs. (5)-(9) reflect the approximation associated with our linear analysis in $\{p_{x},p_{y},p_{z},t_{x},t_{y}\}$.

Now let us apply these results to the cusp region of a PFC observed in the $\mathrm{N_{TB}}$ phase. We employ them with the understanding that they are based on a linear analysis, and thus only represent leading order contributions to $\mathbf{p}$ in the case of “pseudo-layer” distortion associated with a PFC. We also assume that the size of the cusp region is small compared to the size of a homochiral domain (i.e., domain with single handedness of the underlying heliconical director structure) CMeyer2 , so that the polarization $\mathbf{p}$ is not washed out in the cusp region due to mixed left and right-handed helical domains.

As mentioned above and particularly in the case of the $\mathrm{N_{TB}}$ phase, where the “pseudo-layer” compression elastic constant is substantially lower than that for a typical smectic LC Challa ; CMeyer1 , the sharp tips of the cusps should realistically be envisioned as rounded caps. This also means that in addition to “pseudo layer” bending, there will be a compression of the layers in the cusp region. However, as described in the Appendix, for $du/dz\ll 1$, the Euler-Lagrange equation for $\mathbf{t}_{\perp}$ gives $\mathbf{t}_{\perp}=0$ to lowest order, and in that case Eqs. (8) and (9) imply $\mathbf{p}\approx 0$ for “pseudo-layer” compression. Therefore, we will focus on the effect of layer bending.

Consider then a pair of distorted layers equidistant from the center of the PFC, at $+z_{0}$ and $-z_{0}$, along the $z$ axis, as shown in Fig. 5(b), and consider first the behavior in a slice taken through the cusp in the $x-z$ plane for the layer at $-z_{0}$. Along $x$, the displacement $u$ is asymmetric through the cusp region and thus $t_{x}$ will be also; this will lead to a net $p_{y}$ and $p_{z}$, according to Eqs. (6)–(7). On the other hand, for $u=u(x)$, $t_{y}=0$ to lowest order (see Appendix), and thus Eq. (5) implies $p_{x}\approx 0$ in the $-z_{0}$ layer. On opposite sides of the $x$ axis, i.e., $x\rightarrow-x$, the direction of $p_{y}$ is reversed.

Next, for a slice taken in the $y-z$ plane, we may interchange $x$ and $y$ in Eqs. (5)–(7); however, the mirror symmetry across this plane means that only the $p_{z}$ component of the polarization will survive. The net components of $\mathbf{p}$ in the cusp region are shown qualitatively in Fig. 5(b); the relative magnitudes shown for $p_{y}$ and $p_{z}$ will be explained shortly.

For the layer at $+z_{0}$, the results are the same as for $-z_{0}$, except for a $90^{\circ}$ rotation around the $z$ axis, equivalent to an exchange of $p_{x}$ and $p_{y}$ and a reversal in the direction of $p_{z}$ (see Fig. 5(b)).

The relative magnitudes of $p_{y}$ and $p_{z}$ in the $-z_{0}$ layer (or $p_{x}$ and $p_{z}$ in the $+z_{0}$ layer) may be estimated if we consider a harmonic form for $u$ – specifically, a Fourier component whose wavenumber $q_{x}=2\pi/\delta$ corresponds to the length scale of the cusp region of the PFC. That scale can be estimated Rosenblatt_PFC from the width $\delta\simeq 2\mu$m of the birefringent arms of the parabolae (the loci of PFC cusps) in the textures presented in Fig. 2. Then taking $u=u(x)=u_{0}\exp(iq_{x}x)$, the Euler Lagrange equation for $t_{x}$ yields (see Appendix)

Typically, $\beta=0.18$ rad, $q_{0}=2\pi/(0.01\mu\mathrm{m})$, and $q_{x}=2\pi/(2\mu\mathrm{m})$, so $q_{x}^{2}\ll q_{0}^{2}\beta^{2}$. Then the right hand side of the expression for $t_{x}$ may be expanded, yielding

Inserting this result into Eqs. (6) and (7) with $u=u_{0}\exp(iq_{x}x)$, we obtain to lowest order

Then, using the numerical values given above, and taking $K_{1}\simeq 2K_{3}$ (for the bare nematic values) Borshch_Nature , we estimate

Since the SH signal should scale as the square of $\mathbf{p}$ (assuming the components of the second order susceptibility $\mathbf{\chi}^{(2)}$ scale with the magnitude of $\mathbf{p}$), the contribution from $p_{z}$ is negligible. A similar result applies to the layer at $+z_{0}$. In the following subsection, we will therefore concentrate only on the component of $\mathbf{p}$ perpendicular to the axis of the PFC.

Although SHG could possibly arise due to chiral symmetry breaking in the undistorted $\mathrm{N_{TB}}$ state, this structural chirality averages out over a molecular length scale (the heliconical pitch) that is much shorter than the optical wavelength. We detected SH signal from our samples only in the presence of pseudo-layer distortion specifically associated with PFCs. The situation is similar to the chiral smectic-C* phase, where the helical polarization must be partly unwound by an applied field in order to observe a signal FLC_NLO .

Therefore, we propose that the broken centrosymmetry produced by the induced polarization $\mathbf{p}$, over length scales comparable to or longer than the optical scale, is responsible for the observed SHG. As shown in Fig. 5(b), and based on the arguments above, the dominant component of the distortion-induced polarization is normal to the axis of the PFC and to the direction of the applied magnetic field, both of which are along $\hat{\mathbf{z}}$. Eq. (6)and the discussion above reveal that the polarization arises essentially from an electroclinic effect, where the coarse-grain director tilts with respect to the pseudo-layer normal and the polarization develops along the axis perpendicular to the tilt plane. Indeed, for small distortions, the tilt angle is given by the sum $t_{x}+du/dx$ in Eq. (6).

The local symmetry is therefore that of a tilted chiral smectic – namely, $C_{2}$ symmetry, with symmetry axis ($\mathbf{p}$) perpendicular to the axis of the PFC ($\hat{\mathbf{z}}$). As evidenced in Fig. 2, the plane of one parabola of the PFCs is oriented parallel to the substrates. We take this as the $y-z$ plane and the incident (fundamental) light propagation direction as $\hat{\mathbf{k}}_{\omega}=\hat{\mathbf{x}}$. Then the polarizations $V$ or $H$ of the fundamental or second harmonic waves are $V=\hat{\mathbf{y}}$ and $H=\hat{\mathbf{z}}$, and the second harmonic intensities for the various polarization combinations are,

Here $\mathbf{\chi}^{(2)}$ is the second order susceptibility for $C_{2}$ symmetry, with symmetry axis = $\hat{\mathbf{y}}$ ($-z_{0}$ layer in Fig. 5(b)) or $\hat{\mathbf{x}}$ ($+z_{0}$ layer). Assuming Kleinman’s symmetry, the non-vanishing components of $\mathbf{\chi}^{(2)}$ with recurring indices are Boyd_NLO ,

for $-z_{0}$ layers, and

for $+z_{0}$ layers. From these lists and the relations for SH intensity in Eq. (15), we conclude that the intensity for $H$ polarized output vanishes, in agreement with the polarization selection observed in our experiment, and that the SH signal is generated predominantly by the induced polarization in the $-z_{0}$ layers (where $\mathbf{p}\parallel\hat{\mathbf{y}}$) in Fig. 5(b).

Fig. 3 reveals that the SH intensity is systematically lower for a $V_{\omega}\rightarrow V_{2\omega}$ process than for $H_{\omega}\rightarrow V_{2\omega}$, indicating $|\chi_{yzz}^{(2)}|^{2}>|\chi_{yyy}^{(2)}|^{2}$ according to Eq. (15). We can then suggest the following explanation for the splitting of the forward peak in the $V_{\omega}\rightarrow V_{2\omega}$ data: In linear light scattering, we have observed strong depolarized scattering from defect structure at small angles in the $\mathrm{N_{TB}}$ phase. Consider then a process by which small-angle linear scattering converts polarization $V_{\omega}$ to $H_{\omega}$, and then a SH process that converts $H_{\omega}$ to $V_{2\omega}$. This could lead to a slight splitting of the forward peak, as observed in Fig. 3 for the $V_{\omega}\rightarrow V_{2\omega}$ data. On the other hand, the sequence $H_{\omega}\rightarrow V_{\omega}\rightarrow V_{2\omega}$ would be less efficient, producing perhaps a broadening but not a splitting of the peak in the $H_{\omega}\rightarrow V_{2\omega}$ data. Moreover, the peak for $V_{\omega}\rightarrow V_{2\omega}$ splits along the magnetic-field alignment direction – i.e., parallel to the axis of the PFCs. This is because the distortion-induced polarization near the core region will be concentrated more along the PFC axis than normal it.

Finally, we note that the major components of the induced polarization are oppositely directed on opposite sides of the parabola axis (Fig. 5(b)), even if the entire PFC is in a homochiral domain. In this case, there will be no net SH output when the distance across the parabola yields a destructive interference condition, or when the vector sum of polarizations cancels out over a length scale small compared to the coherence length $\lambda_{2\omega}/(n_{2\omega}-n_{\omega})\simeq 5~\mu$m ($n=$ refractive index) Boyd_NLO . As mentioned in the Results section, this effect accounts for the relatively weak signal recorded in the experiment. The phase factor is important, and consequently the signal level is not a direct measure of the amplitude of the helical polarization field.