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The idea of using a subwavelength metal-coated fibre aperture to investigate a surface with high resolution was proposed initially by Synge4; this has evolved into the technique of near-field optical microscopy5 for two-dimensional (2D) imaging. Here, we present an entirely new imaging approach that involves the use of polarized light, emitted from a tapered optical fibre immersed in an anisotropic medium, in the far field to investigate molecular orientation in three dimensions at nanoscale levels. As there are no significant scattering sources due to dielectric inhomogeneities, the near-field light does not scatter in the customary manner, but instead decays exponentially with distance and is not detected downstream. Instead, the light that reaches the detector from the fibre aperture consists of small-wave-vector Fourier components, and is retarded by a phase δ as it propagates through the continuous birefringent fluid medium. By carrying out in-plane (x y) scans inside the sample at a series of heights zi above the surface, we obtain intensity matrix slices, from which we extract information about the fluid’s local optical properties. The initial image (i=1) is acquired by raster-scanning the fibre at height z1=h1 (Fig. 1) close to the nanopatterned polymer alignment layer. If z1 is sufficiently small, the fluid’s molecular orientation profile will correspond to that at the substrate, and will be nearly spatially uniform through the thickness of the slice. Each subsequent image obtained at larger zi contains information about all slices 1,2,,…, i−1, i. As an example, we examine a nematic liquid crystal, the molecular orientation of which is controlled by a nanoscopic pattern scribed into the underlying polymer-coated substrate. We have selected this system as an ideal test bed for validation of our high-resolution imaging technique because there exist theoretical predictions about liquid crystal’s behaviour, even though extant experimental techniques are unable to examine these small length scales. We empirically find that: (1) the lateral resolution of the fluid’s optical properties is as small as 100 nm and vertical resolution a few tens of nanometres, and (2) the fibre’s perturbation of the fluid structure is small and does not significantly affect the signal. As a consequence, we are able to observe directly the length scale of ∼200 nm over which the molecular orientation relaxes owing to the liquid crystal’s elastic forces, thus demonstrating the utility of polarized optical nanotomography (ONT) for 3D imaging of soft matter.

Figure 1: Schematic diagram of optical nanotomography scheme and scribing pattern on substrate.
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a, Schematic representation of a thin layer of an anisotropic fluid sitting atop a patterned polymer-coated substrate. The aperture of the tapered fibre is shown at three representative heights z=h1, h2 and h3 inside the fluid. The fibre is raster-scanned at each height, illuminating the fluid below with polarized light. The transmitted light is collected and an intensity matrix I versus x,y is obtained at each height. b, Schematic representation of the scribing pattern for two different scribing strengths (2.1 and 5.4 μN of force applied by the stylus) and at two different spacings. Each region of uniform scribing orientation corresponds to a pixel; four pixels are shown, with the pixel boundaries corresponding to the locus of points parallel to the y axis at which an abrupt change in the scribing direction occurs.

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The nematic liquid-crystal phase is characterized by long-range orientational order of the molecules6. The axis of alignment is known as the ‘director,’ and can be manipulated azimuthally (ϕ) by patterning a polymer-coated substrate on length scales as small as 100 nm (refs 1, 2, 7, 8). A critical issue for numerous technologies is the tendency of the liquid-crystal director to follow the imposed pattern, which is known as the ‘easy axis,’ and how spatially rapidly the distortion relaxes. Here, we exploit ONT to visualize in 3D the director profile on the small length scales that are characteristic of orientational relaxation.

The driving force behind this orientational homogenization is the nematic liquid-crystal’s elasticity6, which favours a spatially uniform state. An underlying pattern (Fig. 1b) imposes a spatial variation on the liquid-crystal director, which relaxes azimuthally both in the substrate plane and with increasing height above the substrate. The ‘extrapolation length’ L is the characteristic distance over which the director relaxes to a uniform state from a discontinuity in the boundary condition6,9. Quantitatively, the lowest-order anisotropic part of the interfacial free energy is given by F=1/2W2Δϕ2 (ref. 10), where Δϕ is the angular deviation from the patterned easy axis and W2 is the quadratic ‘anchoring strength coefficient’. The extrapolation length L is given by Ki i/W2, where Ki i corresponds to the appropriate elastic constant K11 for splay, K22 for twist and K33 for bend distortions. For typical values of Ki i∼10−11 N and W2∼10−4–10−5 N m−1 (refs 6, 9), we expect that L∼100–1,000 nm, a distance too small to image by extant methods. Instead, we turn to optical nanotomography, a technique that exploits the propagating far field, for 3D imaging. (In contrast, we note that the very different approach of near-field optical microscopy (ref. 5), which involves scattering of evanescent (near-field) modes as opposed to the propagation and phase retardation associated with the ONT technique, has been used to carry out 2D imaging of liquid crystals11,12,13,14,15.)

To promote an easy-axis pattern, we scribed a polymer-coated substrate with the stylus of an atomic force microscope2 (AFM) in a herringbone configuration (Fig. 1b). Both hard and soft scribing were used and, for each of the two scribing strengths, we used two different line spacings: 100 and 200 nm. Details can be found in Supplementary Information, Methods. Figure 2 shows a sequence of images obtained at several heights above the substrate. Qualitatively, several features are apparent. First, images obtained closer to the substrate are noisier, as would be expected from the weaker signals associated with smaller optical phase retardation δ. A second observation is the appearance of ‘fingers’ spaced at 200 nm within each of the horizontal stripes that was scribed with a spacing of 200 nm; thus, the fingers correspond to individual passes—back and forth—of the AFM stylus. The fingers are due to an inherent asymmetry in the scribing process—the stylus is not conical and the cantilever pushes in one finger and pulls in the adjacent finger, resulting in slightly different groove depths Δz. Observations that: (1) the grooves remain distinct, and (2) the value Δz=(17±3) nm obtained from the ONT intensity contrast due to the difference in δ from one finger to the next is the same as that obtained by an AFM measurement of the topography16 indicate that excellent optical resolution is achievable, even out to z∼500 nm. A further discussion of the optical resolution can be found in Supplementary Information, Discussion.

Figure 2: ONT images at different heights.
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A series of images created from the intensity data matrix collected at heights (i) 115 nm, (ii) 210 nm, (iii) 320 nm and (iv) 480 nm above the polymer-coated substrate, with an uncertainty in height of ±10 nm. The scan dimension is approximately 15×9 μm. The polarizer makes an angle β nominally equal to 7.5∘ and 37.5∘ with respect to the two easy axes.

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We now examine the probe’s influence on the nematic phase. Tadokoro et al. observed that the cladded fibre has negligible influence on the director orientation in the nematic phase11,12, indicating extremely weak anchoring conditions at the fibre. Our own measurements agree with their results. We examined the fibre tip inside two liquid-crystal cells of thickness 300 μm, one treated for planar and one for vertical alignment, finding no observable perturbation of the liquid-crystal director. This is discussed in detail in Supplementary Information, Discussion. These observations suggest that the fibre’s orientational effects are small and will not perturb the director significantly. We also investigated the consequences of the fibre’s motion through the liquid crystal. Images collected from forward and reverse scans are equivalent in detail, thus indicating that the liquid-crystal orientation is largely unaffected by (or very quickly relaxes back to equilibrium after) the fibre’s very slow translational motion of ∼7 μm s−1. See Supplementary Information, Methods.

We will concentrate our data analysis on regions that are scribed with a line spacing of 100 nm, the images of which appear nearly uniform even for the closest approach of the fibre to the substrate; detailed analysis of the images associated with the 200 nm line spacing will be considered elsewhere16. Data were collected at 17 individual heights z. Figure 3 shows intensity profiles for hard- (left) and soft- (right) scribed regions at heights z=115 (the closest approach for which the instrument could be stabilized properly), 210, 320 and 480 nm above the substrate. The profiles represent convolutions of the ‘bare’ intensity profile with the instrument function, which is due mainly to the small spread of light as it passes from the fibre’s aperture to the substrate. The solid lines correspond to calculations for hard and soft scribing, which will be discussed below. We observe that the width w of the region over which the intensity changes from Imin+(1/e)(ΔI/2) to Imin+(ΔI/2) (or equivalently, from Imax−(ΔI/2) to Imax−(1/e)(ΔI/2)), is smaller for the hard-scribed region. Here, Imin and Imax are the minimum and maximum intensities, ΔI=Imax−Imin and e is Euler’s number (2.71828…). As the polarization was chosen so that I is approximately proportional to ϕ close to the substrate, for the data collected at z1=115 nm (Fig. 3a) we can treat w as an upper limit of the extrapolation length L at the substrate, that is, the characteristic distance over which the director approaches its limiting value of ϕ=±15∘ in the interior of the pixel from ϕ=0 at the pixel boundary. That the measured value w is an upper limit for L is because elasticity has already partially relaxed the director profile with increasing height and because the data reflect a convolution of the director profile with the instrument function. Nevertheless, the image in Fig. 2(i) and corresponding data in Fig. 3a represent the first direct visualization of the director relaxation. From the data in Fig. 3a, we find that w≈(220±70) nm for hard scribing and w≈(400±140) nm for soft scribing. Taking K33∼1×10−11 N (ref. 6), we obtain lower limits for the azimuthal anchoring strength (recall that W2∝1/L) W2=(4.5±1.5)×10−5 N m−1 and (2.5±0.8)×10−5 N m−1 for the hard- and soft-scribed regions, respectively. These values of W2 are comparable to those obtained from indirect measurements associated with AFM-scribed polyimides17,18.

Figure 3: Data and theory for optical intensities.
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a–d, Experimental data and theoretical plots at four heights above the substrate: height z is equal to 115 nm (a), 210 nm (b), 320 nm (c) and 480 nm (d). These correspond to images (i), (ii), (iii) and (iv), respectively, in Fig. 2. Results for hard-scribed substrates are plotted on the left, and for soft-scribed substrates on the right. Circles represent experimental data and lines represent calculations for the intensity based on the theoretical calculation for ϕ(x,y,z) in the continuum limit. Note that the intensity scale is in arbitrary units, but is the same in all four panels.

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For images recorded at larger z, we see that the width w increases with z, owing to the twist elasticity- (along z) and bend elasticity- (along y) induced uniformation of the director orientation with increasing z. Well above the substrate, the width w can no longer be associated with the relaxation of the director orientation at that height. Rather, the intensity at each position x,z represents the light propagation through a birefringent medium, the optic axis of which is varying in space.

To better interpret our measurements, we calculated the director orientation in the continuum approximation by minimizing the two-dimensional elastic free energy subject to the boundary condition at the interface in which the elastic torque balances the surface anchoring torque6. Details are presented in Supplementary Information, Methods. Figure 4 shows the orientation ϕ versus x at several heights z for anchoring strength W2=8×10−5 N m−1. Notice that the sharpness of the ϕ curves decreases rapidly with increasing z, that for larger values of z the amplitude of ϕ decreases with increasing z and that by z∼320 nm, ϕ varies nearly sinusoidally with x.

Figure 4: Theoretical azimuthal orientations ϕ.
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Theoretical calculation based on continuum theory for the azimuthal director ϕ(x) at the substrate (z=0) and at five other heights z=115 nm, 210 nm, 320 nm, 480 nm and 600 nm above the substrate. Anchoring strength coefficient Wϕ=8×10−2 dyn cm−1 for the calculation, corresponding to hard scribing.

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To calculate the expected light intensity, we adopt the extremely simple approximation that the collected light has propagated through the birefringent liquid crystal as would a plane wave; this is discussed in Supplementary Information, Discussion. We then carried out a Berreman 4×4 optical calculation19 for light propagation described by our calculations for ϕ(y,z) (see Supplementary Information, Methods.) In this calculation, the liquid crystal is divided into thin layers and both electric and magnetic field boundary conditions are applied at each interface, thus accounting for multiple reflections. The results are shown by the solid lines in Fig. 3. Agreement with the measured intensities at the various heights is superb. It is clear that the intensities measured at a given height z do not wash out as rapidly as the azimuthal orientation ϕ (Fig. 4) at that height, but rather reflect the transmission of light through the thickness between the substrate and the height of the aperture.

The results presented herein clearly demonstrate the excellent spatial resolution associated with optical nanotomography. In particular, we have visualized the extrapolation length for different anchoring conditions and have shown that our optical measurements are consistent with continuum theory. With improved signal-to-noise ratio and a more robust optical model, a fully 3D director profile reconstruction will be possible. Comparing ONT with other imaging techniques such as coherent anti-Stokes Raman scattering20,21 and fluorescence confocal polarizing microscopy22,23, ONT’s strength lies in its ultrahigh resolution within ∼500 nm of the surface, the non-necessity of doping the sample with a dye and the ability to study high-birefringence materials. Of course, ONT has two potential drawbacks: it requires a free upper surface and the fibre can be invasive, depending on the material being studied. Nevertheless, for those applications amenable to ONT, its resolution is unsurpassed, being well below the diffraction limit. In summary, we have established that optical nanotomography is an extremely powerful tool for 3D imaging of liquid crystals, and also may be used for nanoscale reconstruction of a variety of select soft systems such as stacked lipid bilayers, lyotropic liquid crystals composed of macromolecules such as DNA or tobacco mosaic virus and suspensions of carbon nanotubes.