1. INTRODUCTION

A. Polarization Control via Homogeneous Uniaxial Retarders

Creating devices that control polarization across wide bandwidths, in either an achromatic or a chromatic manner, has long been a challenge associated with display technologies and remote sensing systems [1,2]. Pancharatnam found that by using stacks of thick birefringent elements, he was able to generate achromatic retardation quarter-wave (and other types of) retarders [3]. Later researchers showed that this approach can also realize chromatic retardation: for example, the well-known Solc and Lyot–Öhman filters [4–6] as well as the lesser-known other polarization interference filters [7–9], which all comprise various combinations of multiple uniaxial retarders and polarizers. While these arrangements have a flexible design space, they are all extremely sensitive to the relative alignment of the multiple plates, sometimes have small clear apertures [4,10], and nearly always result in relatively thick (millimeter to centimeter) systems.

B. Achromatic Multi-Twist Retarders and Polarization Gratings

Another technique for controlling retardation employs a monolithic layer of multiple birefringent layers on a single substrate. Originally demonstrated to achieve an achromatic half-wave retardation for polarization gratings [11] this multi-twist (MT) liquid crystal structure was extended to quarter-wave and half-wave MT retarders (MTRs) [12]. These MT structures comprise two or more chiral nematic doped nematic liquid crystal layers where subsequent layers are aligned spontaneously by the prior layer, without the need for additional alignment processes. The thickness and twist angle of each layer can be chosen independently, for a wide variety of retardation values and bandwidths, without compromising the fabrication complexity. To be clear, until now, MTRs have been developed primarily for achromatic and super-achromatic retardation spectra [11,13,14].

In this work, we examine the ability of MT liquid crystal films to create highly chromatic retardation spectra instead—on both uniform and patterned substrates.

2. CHROMATIC MTR DESIGN

In this section, we adapt the framework developed previously for achromatic MTRs [12] for obtaining the opposite: highly chromatic retardation spectra. We will proceed by summarizing the parameters needed to define an MT liquid crystal film, identifying a target output spectrum for a known input, defining a merit function to evaluate a candidate solution, and using an optimization process to select the closest physically realizable design.

A. MTR Parameter Definition

Here we employ Mueller matrices to describe the optical behavior of MTRs, similar to earlier work on twisted nematic liquid crystal layers [15]. Each MTR layer $m$ is defined by its own individual Mueller matrix ${\mathbf{T}}_m(\lambda )$, listed in Eq. (1) through Eq. (5) in Ref. [12]. This matrix depends on the corresponding layer thickness $d_m$ and twist ${\varphi }_m$. The first layer also has an additional parameter, its starting angle ${\varphi }_0$. Therefore, MTRs with a total of $M$ layers can be completely specified by $2M+1$ parameters. In addition, the material’s birefringence $\mathrm{\Delta }n(\lambda )$ strongly influences the design, which we assume to be $\mathrm{\Delta }n(\lambda )=0.128+8340/{\lambda }^2$ (corresponding to the material used below in experiments).

The Mueller matrix of the whole MTR is therefore ${\mathbf{T}}_{\mathrm{MTR}}={\mathbf{T}}_M\cdots {\mathbf{T}}_2{\mathbf{T}}_1$. The output Stokes vector may be found as ${\mathbf{S}}^o(\lambda )={\mathbf{T}}_{\mathrm{MTR}}(\lambda ){\mathbf{S}}^i(\lambda )$, where ${\mathbf{S}}^i(\lambda )$ is the incident Stokes vector.

B. Target Stokes Vector

In this work, we identify a target Stokes vector ${\mathbf{S}}^t(\lambda )$ that will vary strongly with wavelength. This is the desired output, and the ideal target of the optimization process. Note that this is the primary difference from prior work on achromatic MTRs, where ${\mathbf{S}}^t$ was wavelength-insensitive.

C. Merit Function Definition

Every candidate design is defined by a given set of ($\varphi$, $d$) describing all layers. In order to evaluate any candidate design, a merit function must quantify the difference between its resulting ${\mathbf{S}}^o(\lambda )$ and the target ${\mathbf{S}}^t(\lambda )$. Various merit functions may be employed, depending on the application, but all must result in a scalar value $f$ assessing how close a candidate design is to the target. This may be a direct function of the target and Stokes vectors themselves and may also involve additional Mueller matrices. In the examples below, we use two different merit functions.

D. Optimization Process

We identify and rank the best candidate designs by employing the built-in optimization functions in Matlab (e.g., fmincon and fminsearch), along with multiple randomly generated seeds. This nearly always leads to multiple candidates, which we subsequently review manually to identify those that appear most physically realizable.

3. TWO REPRESENTATIVE EXAMPLES

To investigate the features and limitations of chromatic MT liquid crystal films, we focus on two different examples experimentally: a retarder and a polarization grating. In both, we used the reactive mesogen solution RMS03-001C (Merck Chemicals Ltd.), i.e., a polymerizable liquid crystal mixed with solvent. To achieve twisted layers, we doped this mixture with the chiral molecules CB15 (right-handed, Merck Chemicals Ltd.) and C45S (left-handed, LC Matter Corp.).

A. Example 1: Highly Chromatic Retarder

In some contexts, such as displays and optical remote sensing, it is useful to isolate one wavelength band from others. One way to do this involves an approach known as polarization interference filtering [6–9] or, more simply, as birefringent color filtering. In prior methods, independent films or crystals must be aligned and usually laminated, a process that can be difficult and/or lead to relatively large thicknesses. Here, we imitate these optical principles, but instead achieve the highly chromatic retardation in an MTR as a single self-aligning monolithic film.

As our first representative example, we design a chromatic retarder that aims to pass green (500–570 nm) and block both red (610–660 nm) and blue (430–475 nm) wavelengths. The geometry of this filter, including crossed polarizers, is shown in Fig. 1. In terms of Stokes vectors, we express this as ${\mathbf{S}}^t={[\frac{1}{2},-\frac{1}{2},0,0]}^T$ for the green band and ${\mathbf{S}}^t={[\frac{1}{2},\frac{1}{2}0,0]}^T$ for the red and blue bands, while the input ${\mathbf{S}}^i={[1,0,0,0]}^T$ is unpolarized. The output Stokes vector and transmittance of the whole assembly may be expressed as follows:

 

Fig. 1. Example 1: structure and ideal optical behavior of an MTR with a chromatic retardation that provides half-wave retardation for green (G) light but zero-wave retardation for red (R) and blue (B). This allows G light to pass through crossed polarizers while blocking R and B.

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Applying the procedure outlined in Section 2, we employed a merit function $f=\mathrm{mean}(1-{\mathbf{S}}^o(\lambda )·{\mathbf{S}}^t(\lambda ))/2$, which has a range of [0,1]. This led to an optimal ($\varphi$, $d$) solution for $M=3$ listed in Table 1. This MTR design has physically realizable layer parameters and a low $f$.

Table 1. Green/Magenta Color Filter Retarder (Example 1) Design and Experimental Result

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Fabrication follows the process described in detail in Ref. [12], which we will outline below. First, 1-inch square glass substrates were cleaned. Second, a photo-alignment layer was applied (Rolic ROP-108: spin-coated at 3000 rpm for 1 min, then baked at 180°C for 5 min). Third, the alignment direction was set to $-39.7°$ using a linearly polarized UV exposure ($2.5\mathrm{J}/{\mathrm{cm}}^2$, 365 nm peak). Fourth, the MTR layer $m=1$ was formed using three sublayers, each spin-coated with the mixture (RMS03-001C: CB15: PGMEA at 1.0: 0.00012: 0.0059 by weight) and processing (1030 rpm for 90 s). The MTR layer $m=2$ was created using two sublayers, each spin-coated with the mixture (RMS03-001C: C45S: PGMEA at 1.0: 0.00018: 0.0089 by weight) and processing (1140 rpm for 90 s). The final MTR layer ($m=3$) was created using six sublayers, each spin-coated with the mixture (RMS03-001C: CB15: PGMEA at 1.0: 0.0012: 0.058 by weight) and processing (700 rpm for 90 s). All sublayers were photopolymerized ($1\mathrm{J}/{\mathrm{cm}}^2$, 365 nm peak, in an environment of dry ${\mathrm{N}}_2$) immediately after coating.

The resulting MTR shows clear color filtering when placed between polarizers. As seen in Fig. 2, between crossed polarizers, the color is a fairly saturated green, and between parallel polarizers, the color is a strong magenta.

 

Fig. 2. Fabricated Example 1 MTR between (a) crossed and (b) parallel polarizers.

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The transmittance was measured between linear polarizers in a spectrophotometer. This is shown in Fig. 3, along with the theoretically calculated spectrum. Overall, the experimental retarder spectral response matches the theoretical design, clearly providing half-wave retardation for green with near maximum transmittance while applying very little retardation to blue and red. The experimental contrast ratio between the bands was ${\mathrm{CR}}_{G/R}=10:1$ and ${\mathrm{CR}}_{G/B}=4:1$, which is relatively close to the theoretical prediction of 13:1 and 6:1, respectively. The average green band transmittance was 43% and 5% in the crossed and parallel polarizer measurements, respectively, as compared to 47% and 3% for the ideal design.

 

Fig. 3. Transmittance spectra for Example 1 retarder (green/magenta color filter), between (a) crossed and (b) parallel linear polarizers. The design ${\mathbf{S}}^o(\lambda )$ (black solid lines), measured (circles), and best-fit (dashed lines) spectra are shown, along with the edges of the wavelength bands used as the target.

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In order to estimate MTR parameters actually achieved in the experimental sample, we performed a best fit of the model to the measured data. We employed a merit function minimizing the difference between the data and the Eq. (1) model: $f=\mathrm{rms}(T_{\mathrm{data}}(\lambda )-T_{\mathrm{fit}}(\lambda ))$. The resulting best-fit parameters are listed in Table 1, and the best-fit transmittance is shown in Fig. 3. We note that the data is also shifted slightly toward the blue (by $\sim 13\mathrm{nm}$), which may arise from slightly too-low twist angles ${\varphi }_2$ and ${\varphi }_3$ (see the best-fit parameters in Table 1).

B. Example 2: “Hot” Polarization Grating

In some spectroscopy and astronomy applications, it is useful to have an element engineered to diffract strongly at one wavelength band and simultaneously transmit strongly at others—for example, diffracting infrared (IR) light while transmitting visible (VIS) light. This type of dichroic diffraction grating would be analogous to a “hot” mirror [17]: a specialized thin-film of dielectric layers reflects IR and transmits VIS. While traditional diffraction gratings based on isotropic materials have only limited options to create wide pass and stopbands, the emerging class of geometric-phase holograms [18–20] offers new possibilities. This is because their diffraction efficiency is established almost entirely by the retardation of the birefringent layer forming them [11,12,21]. The simplest hologram of this type is a polarization grating (PG), which creates a linear phase change profile [20,22–24].

As our second representative example, we design a MT liquid crystal film with chromatic retardation that presents half-wave retardation in the infrared band (IR, 1000–2700 nm) and zero-/full-wave retardation for the visible band (VIS, 500–900 nm), all for circularly polarized light. If these targets are achieved, then the zero-order efficiency of the PG will be ${\eta }_0({\lambda }_{IR})=0$ and ${\eta }_0({\lambda }_{\mathrm{VIS}})=1$ for the infrared and visible bands, respectively. Conversely, the first-order efficiencies would be near ${\eta }_{\pm 1}({\lambda }_{IR})=1$ and ${\eta }_{\pm 1}({\lambda }_{\mathrm{VIS}})=0$, where we assume for simplicity that the total first-order efficiency is ${\eta }_{\pm 1}={\eta }_{+1}+{\eta }_{-1}=1-{\eta }_0(\lambda )$. A schematic of this “hot” PG configuration is shown in Fig. 4.

 

Fig. 4. Example 2: structure and ideal optical behavior of a “hot” PG that diffracts IR and passes VIS light. This corresponds to high first-order efficiency ${\eta }_{\pm 1}$ in IR (black arrows) and high zero-order efficiency ${\eta }_0$ for VIS (red and green arrows).

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In PGs, the zero-order efficiency is proportional to the retardation of the film [20,22–24] following ${\eta }_0={\cos }^2(\mathrm{\Gamma }/2)$, where $\mathrm{\Gamma }(\lambda )$ is the film’s retardation. This zero-order efficiency resulting from the retardation seen by circularly polarized light may be thought of as the transmittance of the birefringent film leaking through circular polarizers. If we assume that the input light is unpolarized (${\mathbf{S}}^i={(1,0,0,0)}^T$), i.e., composed of equal parts right- and left-handed circular polarizations, then we can express this leakage as follows:

For optimization, we used the same merit function as in Example 1. This led to an optimal ($\varphi$, $d$) solution for $M=4$ listed in Table 2. This MT design has physically realizable layer parameters and a low $f$. The simulated design spectrum is shown in Fig. 5. Note that this same MT design was previously identified and studied as a homogeneous retarder [21]. In this paper, it is structured into a PG, and its diffractive properties are fully characterized.

Table 2. “Hot” Polarization Grating (Example 2) Design and Experimental Result

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Fig. 5. Diffraction efficiency spectra for Example 2 “hot” PG that diffracts IR light strongly while transmitting nearly all VIS light. The zero-order efficiency ${\eta }_0(\lambda )$ of the design (black solid lines) and the measured (circles) and best-fit (dashed blue line) spectra are shown, as well as the estimated total first-order ${\eta }_{\pm 1}(\lambda )$ efficiency (dashed red line) based on the best fit.

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Fabrication of the PG follows a procedure similar to Example 1, except for the patterning step and the details of the coating conditions. Instead of a uniform exposure of the photo-alignment layer, we exposed an inhomogeneous polarization profile with a direct-write laser scanner [18,25], obtaining an orientation profile $\mathrm{\varphi }(x)=\pi x/\mathrm{\Lambda }$, where $\mathrm{\Lambda }$ is the grating period. Here we set $\mathrm{\Lambda }=28.7\mathrm{\mu m}$, which remains solidly in the Raman–Nath regime (not Bragg) [26,27], with $Q$ values of $\approx 0.09$ and $\approx 0.04$ for 1550 and 633 nm, respectively. The net delivered fluence was $0.2\mathrm{J}/{\mathrm{cm}}^2$ at 355 nm. The MT layer $m=1$ was formed using one sublayer, spin-coated with the mixture (RMS03-001C: CB15: PGMEA at 1.0: 0.006: 0.294 by weight) and processing (1160 rpm for 90 s). The MT layer $m=2$ was created using three sublayers, each spin-coated with the mixture (RMS03-001C: CB15: PGMEA at 1.0: 0.0005: 0.025 by weight) and processing (600 rpm for 90 s). The MT layer ($m=3$) was formed using two sublayers, each spin-coated with the mixture (RMS03-001C: C45S: PGMEA at 1.0: 0.0004: 0.0216 by weight) and processing (700 rpm for 90 s). The final MT layer ($m=4$) was created using one sublayer, spin-coated with the mixture (RMS03-001C: CB15: PGMEA at 1.0: 0.0034: 0.17 by weight) and processing (1000 rpm for 90 s).

The resulting “hot” PG shows a strong difference in the diffraction of IR and VIS light. As seen in Fig. 6, the IR light (1550 nm) diffracts strongly into the first orders with ${\eta }_{\pm 1}=91\%$ efficiency, but the VIS light (measured at 633 nm) passes nearly completely into the zero-order ${\eta }_0=95\%$. The shape of the 635 nm light is elliptical due to the diode laser used. The unaccounted-for signal at 1550 nm is 3.1% which we suspect is haze or imperceptible remaining higher orders.

 

Fig. 6. Fabricated Example 2 “hot” PG. (a) When incident light is within the IR band, the PG diffracts strongly, and (b) when incident light is within the VIS band, the PG directly transmits nearly all the light. The photograph in part (a) is taken by arranging the 1550 nm light output from the PG onto a phosphorescent IR viewing card, which appears red, while the photograph in part (b) is merely the 635 nm light arranged onto a white screen.

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The zero-order efficiency spectra were measured using a spectrophotometer, shown in Fig. 5. The data matches the design fairly closely. Table 2 shows the best fit of the measured data using a minimizing cost function $f=\mathtt{rms}({\eta }_{0,\text{data}}-{\eta }_{0,\text{fit}})$, which takes values [0,100] for the entire range of 500–2700 nm. The spectra of this best fit for the zero and total first orders are also shown in Fig. 5. The best fit (Table 2) suggests that the realized $d_2$ and ${\varphi }_3$ deviate the most from the target design, and its impact appears to be a 5%–10% higher zero-order efficiency in the range of 1000–1500 nm.

This PG has high contrast between the diffraction and transmission bands. The average zero-order efficiency in VIS was ${\eta }_0({\lambda }_{\mathrm{VIS}})=89\%$ and in IR was ${\eta }_0({\lambda }_{\mathrm{IR}})=6.5\%$, from which we estimate the average first-order efficiencies to be ${\eta }_{\pm 1}({\lambda }_{\mathrm{VIS}})\le 10\%$ and ${\eta }_{\pm 1}({\lambda }_{\mathrm{IR}})\ge 90\%$, respectively. The extinction ratio of the first orders ${\eta }_{\pm 1}({\lambda }_{\mathrm{IR}})$ was measured using a 1550 nm linearly polarized laser with a rotated quarter-waveplate to vary between left- and right-handed circular input polarization. It was 864:1 for the +1 order and 2185:1 for the −1 order. The $\pm 1$ order at 1550 nm was highly circular with an ellipticity angle of 44.3° for the −1 order and $-43.8°$for the +1 order. For 633 nm, we measured the zero-order polarization and verified that it was similar to the input. For example, when the input is linear with a degree-of-polarization (DOLP) of 0.995, the zero-order has $\mathrm{DOLP}=0.996$.

4. DISCUSSION

The two examples above clearly demonstrate the ability to create highly chromatic retardation in the monolithic layer with a single alignment step. In Example 1, a passband-type color filter retarder ($M=3$) in VIS was achieved. In Example 2, a “hot” PG ($M=4$) with a very wide IR band was demonstrated with a steep slope between the IR and VIS. Nevertheless, the fabricated samples did deviate from the design spectra. This was because of moderate errors in the ($\varphi$, $d$) parameters of some layers. While this sensitivity presents a challenge to better performance and repeatability, we anticipate that this can be overcome with more sophisticated coating tools and conditions.

We find in general that increasing the number of layers $M$ will enable sharper slopes and smoother bands. As an example of this, we studied three MTRs as color filters (i.e., red/cyan, green/magenta, and blue/yellow), where we varied $M$ from two to four. As a shorthand notation, these correspond to 2TR, 3TR, and 4TR solutions, respectively. The time needed to find the globally optimal solution depends on several factors, including $M$, the merit function, and the quality of initial seeds. On a desktop computer with 12 cores, we can roughly estimate the time needed for 2TR, 3TR, and 4TR optimizations to take less than 10 min, 30 min, and 4 h, respectively. The target spectra and resulting optimal solutions for each condition are shown in Figs. 7(a)–7(c). The corresponding solution parameters are shown in Table 3. Generally, we see that the 4TR solutions match the targets more closely than the lower layer number solutions. Figure 7(d) quantifies the trend in each case showing the rms difference of the optimal solution’s transmission curves as compared to the target spectra sampled from 400 to 700 nm in 2.5 nm divisions. While increasing $M$ will theoretically increase the slope between neighboring wavelength bands, this adds greater complexity to the MT structure that may present practical challenges. Obviously, at some point $M$ becomes so large that the lower layers can no longer precisely align the subsequent layers with low pretilt and uniformity. We are currently investigating the upper limits of this, with MT designs in the range of $M=5$ to 9.

Table 3. Optimal Solutions for the MTR Color Filters of Fig. 7

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Fig. 7. Simulating transmission of three different MT retarders (i.e., color filters) between crossed linear polarizers, (a) red/cyan, (b) green/magenta, and (c) blue/yellow. In (a)–(c), the target spectra (solid colored lines) and optimal solutions for two- (dashed), three- (solid), and four-twist (dot-dashed) cases are shown. Part (d) shows the rms of the difference between the target and corresponding optimal solution.

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5. CONCLUSION

In this paper, MT liquid crystal films are designed and demonstrated to have nontrivial chromatic retardation visible and infrared wavelength ranges. We developed these coatings for two representative examples, a green/magenta color filter retarder and a “hot” polarization grating. Compared to previous methods, MT films allow for very thin (few micrometers) films with only a single alignment step. Of course, all types of color filters are possible, including blue/yellow and red/cyan. Furthermore, all types of geometric-phase holograms, of which the polarization grating is only one type, can be made highly chromatic using the principles here.