We describe a prototype element for use in probing electro-optic retro-reflection in sensor applications, illuminating a planar-aligned nematic liquid crystal electro-optic cell with convergent light having a single, tunable angle of incidence (tunable conical illumination). This illumination is generated using a 100X, high numerical aperture, long working-distance microscope objective under conditions of extreme spherical aberration. The electro-optic effect observed is multiple-beam optical interference between polarized reflections from the two bounding plates of the cell, rendered tunable with voltage-controlled refractive index changes induced by molecular reorientation of the liquid crystal. Characterization of the reflectivity vs. angle of incidence and applied voltage enables identification of conditions of high-contrast, low power, electro-optic reflectivity control applicable to fiber optics.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
A variety of remote sensing applications are enabled by miniature optical sensing devices that generally report spectral changes in absorption or fluorescence and are interrogated at long range by retro-reflected laser light, delivered by fiber optic [1–4] or free space propagation [5–19]. The addition, to these and other remote devices, of a capability to communicate local information by intensity or phase modulation of the retro-reflected light would enable a variety of significant enhancements [20–22]. The implementation of such dynamic retro-reflection in such applications would benefit from high speed, low power, electro-optic methods that can give high contrast modulation of reflected light. Liquid crystal (LC) electro-optic (EO) devices, the basis of the light valves comprising the displays of laptops, pads, and phones, are a promising choice for dynamic retro-reflection applications. Although LC EO devices are of only moderate speed (millisecond to microsecond EO response times), their inherently low power consumption, diverse variety of EO effects, success as a commercial light valve technology, and potential for modulating obliquely incident light, make them attractive for sensing applications [23–26].
Efficient exploration of the EO effects that might be useful in such applications will require techniques whereby the inherently broad range of geometries and conditions available in LC devices, due to their flexibility of alignment, diversity of optical characteristics, and distinctive electro-optic effects, can be effectively probed. Here we present a method in which the reflection characteristics of an LC electro-optic cell, a geometry having a nematic LC layer as a few microns thick, dielectric film in a transparent capacitor, can be conveniently explored as a function of angle of incidence, polarization, applied voltage, and cell geometry. We describe a prototype element for use in probing electro-optic retro-reflection in sensor applications, illuminating a planar-aligned nematic liquid crystal electro-optic cell with convergent light having a single, tunable angle of incidence (tunable conical illumination), generated using a 100X, high numerical aperture, long working-distance microscope objective under conditions of extreme spherical aberration. The electro-optic effect observed is multiple-beam optical interference between polarized reflections from the two bounding plates of the cell, rendered tunable with voltage-controlled refractive index changes induced by molecular reorientation of the liquid crystal.
This geometry would be suitable for information transfer from a sensing device that communicates using light incident from an optical fiber by modulating its reflection back down the fiber. The results show that the electro-optic contrast obtained depends strongly on angle of incidence, cell geometry and thickness, and LC parameters, and enable the design of high-contrast, low power, electro-optic reflectivity control applicable to fiber optics.
2. Experimental geometry
This method is based on a remarkable capability of advanced, infinity-corrected, high performance microscope objectives, afforded to them by the combination of high magnification, long working distance, and large numerical aperture. Specifically, if such a lens is designed to focus incident collimated light to a single point in air but is positioned such that the focal point would be inside a glass plate oriented normal to the objective axis, then a series of on-axis focal points appear, their position varying with the angle of incidence in air, θa, according to the rules of geometrical optics as applied to spherical aberration. If a second surface is introduced at the location of one of these focal points, then we find that the incident light is reflected back through the lens to the field center at infinity in the form of nearly perfect conical reflection, the cone having the angle of incidence corresponding precisely to the calculated spherical aberration at the second surface location. This is a simple geometrical optics result but the high performance of advanced objectives enables it to be achieved with precision and single-degree angular resolution, such that scans of incident angle can be made via z-scans of the objective position, as discussed below.
The basic test setup, shown in Fig. 1, has a collimated 632.8 nm laser beam focused by an infinity-corrected 100X microscope objective (NA = 0.8) onto various optical geometries, with the reflected light collected effectively at infinity by a distant, small area (2 mm2), amplified Si detector, D2. The objective employed (Olympus LMPlanFlI , NA = 0.8) is engineered for imaging at the air/sample interface, producing a diffraction limited focus of the incident beam in air (Fig. 1). Measurement of the intensity at D2 during positional scanning of the objective parallel to the beam (z-scanning, Fig. 2(a)) provides a measurement of the normalized reflected intensity N(z), and an indication of the depth of focus of the diffraction-limited beam, as shown in Fig. 2(b). The focal peak is found to be characterized by a full width at a half height in z of ΔzFWHH = 1.62 µm, which is the diffraction-limited depth of focus for this objective for a plane wave uniformly illuminating its aperture.
3. Conical illumination
Useful LC electro-optic geometries require the liquid crystal material to be between solid plates, enabling alignment of the LC and the application of electric fields, in which case the probe light must pass through some thickness of glass or other transparent optical material. If an objective corrected to focus in air as in Fig. 1 is used, then its behavior when focusing through glass will be affected by aberration, as illustrated in Fig. 2. Aberration produces classic features of a degraded focus, including an asymmetric distribution of on-axis intensity that is much broader (Fig. 2(h)) than that of the aberration-free focus (Fig. 2(b)).
The geometrical optics description of this ascribes the aberration to the nonlinearities inherent in Snell's law, which will cause the rays perfectly focused in air to intersect the system axis in glass at distances increasingly further from the lens as the angle of incidence in air, θa, is increased. If z measures the distance between the first glass interface and the position of the focal spot if the glass were removed, then for an angle of incidence θa(z, tg), the crossing point is a distance t(z) into the glass. If the plate is of finite thickness, so that there is a second reflecting interface a distance tg into the glass, and the crossing point is constrained to be at this interface, (i.e., t(z) = tg, as sketched in Fig. 2(e)), then incident collimated light at the angle of incidence θa(z, tg), will produce reflected collimated light. The relationship between θa and the lens position z is described by Eq. (1):
As z is increased from z = 0, the first rays to be so reflected will be those for the largest θa, θamax, when z = z(θamax), as in Fig. 2(d). As z is increased further, the θa for collimated reflection will decrease until z = z(θa = 0) = tg/ng, where the on-axis rays focus (Fig. 2(f)). Figure 2(g) shows θa(z) vs. z calculated using Eq. (1), for the range of θa corresponding to our objective. Figures 2(h) and 3(a) show Nglass(z), the corresponding collimated reflected intensity from the glass collected at detector D2, normalized such that the front surface reflection has unit peak intensity, i.e., N(z = 0) = 1. The aberrated reflected intensity is largest near θa = 0, where the small-angle rays focus, and becomes smaller with decreasing z until θa ~θamax, beyond which angle the light is cut off by the objective aperture [28–32]. Also shown in Fig. 3(a) are N(z) for the same glass plate with either indium tin oxide (ITO) or ITO coated with polyimide (PI) on the back interface. Since these are of thickness much smaller than the wavelength (tITO = 44 nm and tPI = 17 nm), their reflectivities differ from that of the glass only by a θa-independent factor. Fringes that appear in Nglass(z) for small θa, shown in Figs. 2(h) and 3(a), are a result of the modulation of the field amplitude of the focused light by edge diffraction along the caustic curve evident in the converging light in Fig. 2(i). The intersection of this conical fringe pattern with the lens axis generates the quasi-periodic intensity modulation in N(z) [33–38].
Multiple beam interference as a probe of conical illumination
A diffraction integral calculation of the on-axis intensity I(z) in a geometry of spherical aberration very similar to ours (ng = 1.55, NA = 0.9 [39–41]) is shown in Fig. 3(b). The similarity of the calculated I(z) and Nglass(z) suggests that the measurement of Nglass(z) may be an effective way to probe I(z). In order to test this proposition, and as a prelude to the study of liquid crystal cells, we introduced a third interface to make an air-filled cell of thickness tcell ≈4.5 µm, sketched in Fig. 4. This cell becomes a low finesse, multiple beam interferometer with a path difference for interference that depends on θg = (1/ng) sinθa, the angle of incidence in the glass (Fig. 2(e)). Ncell(z) measured for this air gap cell, plotted along with Nglass(z) in Fig. 3(c), shows distinctive interference minima for z in the range 350 µm < z < 470 µm, corresponding to θa in the range 50° > θa > 0°. The reflected intensity ratio Rcell(z) = Ncell(z)/Nglass(z), plotted in Fig. 3(d), exhibits the following features: (i) constructive interference maxima of R(z) ≈4 for θa < 50°; (ii) destructive interference minima of R(z) ~0.05 for θa < 50°; (iii) reflection without interference from two glass surfaces giving R ~2 for z > 470 µm, where focus is not possible.
For obliquely incident, parallel, monochromatic light, this air gap cell should have a reflected intensity ratio Rcell(θa,na,ng,tcell,δ,z) given by the intensity reflectivity interference function of the cell, r2cell = r2gap, divided by the glass/air Fresnel reflectivity, r2g/a,
The black curve in Fig. 3(d) is a least-squares fit of Eq. (2) to Rcell(θa), showing that this basic interference function depending on a single selected angle of incidence fits the data well in the range of z where there is focus of some selected ray at θa(z). That is, it appears that Δθa(z), the range of angles of incidence selected in the aberrated focus at θa(z) by the detection of collimated reflected far-field light, is much narrower that the θa range corresponding to that of the fringe spacing. The effective range of angles selected δθam in the vicinity of interference minimum m can be probed quantitatively by comparing the Rcell(z) data of Figs. 3(c) and 3(d) in the vicinity of the minimum to a model in which there is a distribution of incident angles about θa(z). For this purpose we performed convolutions of the reflectivity Rcell(θa) calculated using Eq. (2) with the flat-topped distribution: D(θa) = 1/δθa for -δθa/2 < θa < δθa/2, D(θa) = 0 for θa <-δθa/2 and θa > δθa/2. Sample convolutions R(θa) ⊗ D(θa) in the vicinity of the m = 10 minimum are shown in Fig. 3(e) for several δθa values, along with the measured extinction. A least squares fit of the convolution to the m = 10 data (black curve) gives δθa = 0.8°. This value shows that the angular selection of cone angle of illumination in the retro-reflection of aberrated light using the Olympus 100X objective is quite narrow, making the variation of z an effective way of tuning cone angle.
The reflectivity data of Fig. 3(c,d) and the fit Rcell(θa) are shown again in Fig. 5(b), along with Fig. 5(a), a plot of Eq. (4) giving the dependence of the interference order m on z, ncell, and tcell, enabling the determination of m(0) ≡ m(θa = 0). For the air gap cell, tcell ≈4.5 µm and ncell = 1 gives m(0) = 14.6, and minima in θa in the range 50° > θa > 0° for m = 10, 11, 12, 13, and 14, as indicated by the construction of violet lines and violet/white open circles in Figs. 5(a) and 5(b).
4. Liquid crystal electro-optic reflection
The aberrated illumination geometry described in the previous section was applied to the investigation of the retro-reflection characteristics of planar-aligned nematic liquid crystal electro-optic cells, in the geometry shown in Figs. 6(a) and 6(b). The liquid crystal 5CB, a room temperature nematic with uniaxial extraordinary and ordinary refractive indices ne = 1.72 and no = 1.54, was introduced into cells, custom made by Instec, Inc., comprising two glass plates spaced by a tcell ≈4.5 µm gap and coated on the gap faces with ITO electrodes as described above, to make a transparent capacitor with the LC as the dielectric. The electrodes were coated with a rubbed Nissan polyimide layer to generate planar-alignment with the nematic director n(r), the local average mean molecular long axis orientation in the cell, approximately parallel to the plane of the gap surfaces and to the incident optical polarization, as shown in Fig. 6(b). Rubbing of the PI films orients n at the surface along the rubbing direction with a pretilt angle of 5° relative to the surface (Instec specification). Antiparallel rubbing of the two cell plates then gives a uniform director orientation in the cell with ψ(r) ≈5°, where ψ(r) is the angle between n(r) and the x-y plane. Application of an AC voltage (peak-to-peak amplitude, v) across the cell produces a torque on n due to the dielectric anisotropy of the LC, inducing an increase in ψ(r) throughout the cell. Since the cell is homogeneous in the x-y plane, the director orientation with a field applied depends only on s, the distance into the LC from the front cell surface. At high voltage, the orientation of n is nearly along the applied electric field, normal to the cell plates (ψ(s) ≈90° with v = 6V). The orientation in the cell center, ψc = ψ(s = tcell/2), is obtained from Frank elastic energy calculations of the director distortion vs. applied voltage [42,43] using the known parameters for 5CB . The calculations show that the main optical response is in the angular mid-range where ψc is changing rapidly, as expected.
Normalized cell reflectivity z-scans N(z) obtained in the limits of planar orientation (v = 0V) and field-induced homeotropic orientation (v = 6 V), plotted in Fig. 6(b), confirm that N(z) depends strongly on applied voltage. At v = 0 V, the maximum of the reflectivity peak is N(z = 465 µm) ~0.1, which is much smaller than that which would be obtained from an empty (air gap) glass/ITO/PI cell (N(z = 465 µm) ~1) as shown in Fig. 3(c), a result of the reflectivity at the glass/ITO/PI – LC interface being much smaller than that of the glass/ITO/PI – air interface because of index matching. The incident light is polarized along x, parallel to the director orientation n in the absence of field. Applied electric fields rotate the director in the x-z plane, a plane of mirror symmetry of the cell. As a result, the light is not depolarized by the LC birefringence, remaining polarized on average parallel to n, and the electro-optic effect is due principally to variation of the effective refractive index due to the rotation of n and to the resulting changes in the multiple beam interference between reflections at the interior cell surfaces. The field dependence of the refractive index in the cell substantially controls the variation of N(z) with field, and, in particular produces strong variation of N(z) at certain values of z. For example, setting z ≈465 µm, near the reflectivity maximum at θa = 0, yields high contrast variation of N(z) as interference minima in the vicinity are tuned through by varying the applied voltage v, as shown in Fig. 7(a). The field dependence was further explored by tracking the evolution of N(z) vs. applied voltage in the full range of director orientation in the cell, from planar to saturated homeotropic, as shown in Fig. 7. At v = 0 V, N(z) possesses a maximum reflection at position z = 465 μm corresponding to the nematic LC director n essentially parallel to the x-y plane and thus approximately parallel to the optical polarization of the incident conical beam. As v increases, the nematic LC director n reorients toward the field direction and generally away from the optical polarization direction, resulting in a decrease of the effective refractive index of the LC medium. The maximum contrast ratio of N(z) was obtained at the peak position z = 465 μm, where applied voltage can tune the effective LC index to a value that gives extinction at both v = 0.83 V and v = 6.00 V. Interestingly, the high voltage regime, in the range 2.15 V < v < 6.00 V, shows a variation in the interference patterns that is nearly identical to that obtained when the voltage is scanned in the 0 V < v < 0.83 V low voltage regime, as seen by comparing Figs. 7(a) and 7(c).
Voltage dependent reflectivity data R(v,z) are compared to theoretical predictions for a variety of conical scans in Fig. 5. The lower part (Fig. 5(a)) shows plots of θa(z) and m(z) vs. scan position z calculated from Eq. (4) for three cases: the air gap cell (ncell = 1, purple); the planar limit LC cell at v = 0 (ncell = 1.70, black), and the homeotropic limit LC cell at v = 6 V (ncell = 1.54, blue). The corresponding reflectivity curves are shown in Figs. 5(b) and 5(c) with the same color coding, and plotted with the R(v,z) calculated from Eqs. (2) and 4 (black in (b), white in (c)), the features of which are related to the m(z) plots in (a) by the color-coded vertical lines ending in open circles. These serve to illustrate that the minima in R(v,z) calculated for the planar and homeotropic limits correspond to points where the m(z) trajectories cross the lines at which m is integral.
The darker shaded region in between these limits in Fig. 5(a) is the domain of m(v,z) accessed by increasing the voltage applied to the cell, the white arrows indicating the direction of increasing v. Increasing v from 0 to 6 V scans m(v,z) by δm(v,z) ≈-2.5 orders near θa = 0, and by δm(v,z) ≈-3.1 orders near θa = 50°. This near independence of δm(v,z) on z makes R(v,z) a quasi-periodic function of z, with very similar R(z) patterns appearing with voltage differences for which δm = 1 even at intermediate voltages, for example v = 0.83 V, with m(v,z) indicated by the solid blue circles. This is the origin of the similarity between the R(v,z) progressions in the 0 V < v < 0.83 V and the 2.15V < v < 6.00 V ranges as noted above. The reflectivities R(v,z) at the limits of these ranges, shown in Fig. 5(d), are found to be comparable at small θa but to differ substantially at large v and θa. This behavior will be discussed below with the aid of Fig. 9. Figure 5(a) also enables direct assessment of the sensitivity of the interference and electro-optic behavior to changes in refractive index (due to temperature dependence or dispersion) or to changes in gap thickness.
The cell reflectivity R(v,z = 465µm) measured at normal incidence (θa = 0) as a function of voltage, plotted in Fig. 5(e), exhibits interference minima at orders 24, 23, and 22 at voltages v = 0.83 V, 1.20 V, and 6.00 V respectively, at which voltages the measured R(v,z) are very similar, as seen, for example, in Fig. 5(d). Interference maxima are found at v = 0 V and 2.15 V and these are also similar to each other. The 5CB cell thickness was therefore chosen such that there was an on-axis (θa = 0) reflective maximum in the planar alignment case at v = 0 V and an on-axis reflective minimum in the field-induced homeotropic case at v = 6.00 V, the highest voltage employed. The purple curve in Fig. 5(e) shows the field-induced tilt of molecular orientation in the cell center ψ(s = tg/2) associated with the variation of the refractive index that scans m(v,z).
The normalized reflectivity N(v,z) data in Figs. 6 and 7 all show the rapid modulation at small θa due to the fringe pattern in the converging light illustrated in Fig. 2(i), and Figs. 5(b)–5(d) show that experimentally this modulation is not eliminated by normalizing R(v,z) for cells by Rglass(z), i.e., that replacement of the single piece of glass with a sandwich cell fundamentally alters the fringe pattern. Thus Eq. (4) does not describe the short-period fringes at small θa.
R(v,z) scans in the intermediate voltage range 0.83 V < v < 2.15 V, shown in Fig. 8, qualitatively reflect the variations found at high and low voltage but with distinctly shallower intensity minima at extinction. The origin of this difference is summarized in Fig. 9, in which the pattern of illumination obtained with polarized, convergent incident light, and typical sample molecular orientations in the mid-plane of the cell for several applied voltages, are sketched. Along the lens axis, the polarization PTB is vertical (along x) and makes an angle β(s) with the principal molecular orientation a distance s into the cell. At a particular θa, this light experiences a certain integrated phase shift upon passing through the cell depending on path length, β(s), and the birefringence of 5CB. For light entering the lens in the x-y plane, rays entering the lens at the L and R extremities have in general different β(s) values in the cell, βL(s) and βR(s) respectively, as indicated, where βL(s) = βR(s) in the limits of planar (ψ(r) = 0°) or field-induced homeotropic (ψ(r) = 90°) orientation of the LC. For a chosen θa, the ψ(r) = 0° and 90° director orientation conditions therefore minimize the variation in integrated phase shift for different incident azimuthal orientations on the incoming cone of light, so that they have interference minima under the same conditions, leading to high-contrast extinction minima. For the intermediate average molecular orientations appearing at intermediate voltages (Fig. 5(e)), such as those probed in Fig. 8, the L and R optical path lengths in the cell are different from each other and from that of the T and B rays, broadening the interference minima in z.
5. Discussion and conclusion
We have demonstrated an aberrated reflection geometry using a high numerical aperture, infinity-corrected objective that implements conical reflection with high angular resolution and precision very simply, enabling the efficient assessment of the angular dependence of the reflection of liquid crystal cells. Probing the electro-optic reflection of a planar-aligned nematic reveals a complex variation with angle of incidence and applied electric field, the basic features of which can be quantitatively understood, and shows that such a cell can exhibit high contrast electro-optic switching in conical reflection for linearly polarized light at angles of incidence up to θa ~30°. With a restricted azimuthal range of the incident light, useful electro-optic contrast should be obtainable for θa as large as 60°, well into the range required for retro-reflection applications. This research advances our understanding of liquid crystal remote sensing communication, suggesting a lightweight, inexpensive, and easy to employ technology that could become a workhorse in the burgeoning industry of sensors and sensing, as liquid crystals are for portable displays.
A.S.A. set up and carried out the experiment, analyzed the data, created figures, and wrote the manuscript; Z.N. assisted with setting up the experiment; J.E.M. edited the manuscript and contributed to the figures; N.A.C. conceived and designed the experiments, analyzed and modeled the data, and wrote the manuscript.
Soft Materials Research Center under NSF MRSEC Grant No. DMR-1420736 and NSF MRSEC Grant No. DMR-0820579. A.S.A. was supported by Umm Al-Qura University and the Ministry of Education, Saudi Arabia.
We would like to thank Arthur Klittnick and David Engström for their assistance in the early stages of this experiment.
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