Here we study birefringent films with highly customizable chromatic retardation spectra, using multi-twist liquid crystal (LC) films. These are made of two or more layers of chiral nematic LC polymer network materials, also known as reactive mesogens, which form a monolithic thin-film wherein the in-plane orientation of subsequent layers is automatically determined by the single alignment layer on the substrate. The multiple layer thicknesses and twists present many degrees of freedom to tailor the retardation. While prior work examined achromatic spectra, here we show how to use Mueller matrix analysis to create highly chromatic spectra. We experimentally demonstrate both a uniformly aligned retarder as a green/magenta color filter and a “hot” polarization grating (PG) that diffracts infrared while passing visible light. The three-twist color filter shows a contrast ratio in transmittance between polarizers as high as 10:1 between the half- and zero-wave retardation bands. The “hot” PG shows an average first-order efficiency of about 90% for 1000–2700 nm and an average zero-order efficiency of about 90% for 500–900 nm. The principles here can be extended to nearly any chromatic retardation spectra, including high/low/bandpass, and to nearly any LC orientation pattern, in general known as geometric-phase holograms.
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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 . 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  this multi-twist (MT) liquid crystal structure was extended to quarter-wave and half-wave MT retarders (MTRs) . 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  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 . Each MTR layer is defined by its own individual Mueller matrix , listed in Eq. (1) through Eq. (5) in Ref. . This matrix depends on the corresponding layer thickness and twist . The first layer also has an additional parameter, its starting angle . Therefore, MTRs with a total of layers can be completely specified by parameters. In addition, the material’s birefringence strongly influences the design, which we assume to be (corresponding to the material used below in experiments).
The Mueller matrix of the whole MTR is therefore . The output Stokes vector may be found as , where is the incident Stokes vector.
B. Target Stokes Vector
In this work, we identify a target Stokes vector 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 was wavelength-insensitive.
C. Merit Function Definition
Every candidate design is defined by a given set of (, ) describing all layers. In order to evaluate any candidate design, a merit function must quantify the difference between its resulting and the target . Various merit functions may be employed, depending on the application, but all must result in a scalar value 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 for the green band and for the red and blue bands, while the input is unpolarized. The output Stokes vector and transmittance of the whole assembly may be expressed as follows:16].
Applying the procedure outlined in Section 2, we employed a merit function , which has a range of [0,1]. This led to an optimal (, ) solution for listed in Table 1. This MTR design has physically realizable layer parameters and a low .
Fabrication follows the process described in detail in Ref. , 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 using a linearly polarized UV exposure (, 365 nm peak). Fourth, the MTR layer 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 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 () 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 (, 365 nm peak, in an environment of dry ) 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.
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 and , 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.
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: . 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 ), which may arise from slightly too-low twist angles and (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 : 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 and for the infrared and visible bands, respectively. Conversely, the first-order efficiencies would be near and , where we assume for simplicity that the total first-order efficiency is . A schematic of this “hot” PG configuration is shown in Fig. 4.
In PGs, the zero-order efficiency is proportional to the retardation of the film [20,22–24] following , where 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 (), i.e., composed of equal parts right- and left-handed circular polarizations, then we can express this leakage as follows:16]. This means we set the target as for the IR band and for the VIS band.
For optimization, we used the same merit function as in Example 1. This led to an optimal (, ) solution for listed in Table 2. This MT design has physically realizable layer parameters and a low . The simulated design spectrum is shown in Fig. 5. Note that this same MT design was previously identified and studied as a homogeneous retarder . In this paper, it is structured into a PG, and its diffractive properties are fully characterized.
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 , where is the grating period. Here we set , which remains solidly in the Raman–Nath regime (not Bragg) [26,27], with values of and for 1550 and 633 nm, respectively. The net delivered fluence was at 355 nm. The MT layer 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 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 () 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 () 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 efficiency, but the VIS light (measured at 633 nm) passes nearly completely into the zero-order . 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.
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 , 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 and 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 and in IR was , from which we estimate the average first-order efficiencies to be and , respectively. The extinction ratio of the first orders 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 order at 1550 nm was highly circular with an ellipticity angle of 44.3° for the −1 order and 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 .
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 () in VIS was achieved. In Example 2, a “hot” PG () 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 (, ) 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 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 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 , 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 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 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 to 9.
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.
National Science Foundation (NSF) (ECCS-0955127).
We thank Eric Stempels and Nikolai Piskunov at Uppsala University for discussions regarding the most relevant requirements for the “hot” PG. The authors declare that there are no conflicts of interest related to this article.
1. A. K. Bhowmik, Z. Li, and P. J. Bos, eds., Mobile Displays: Technology and Applications (Wiley, 2008).
2. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453–5469 (2006). [CrossRef]
3. S. Pancharatnam, “Achromatic combinations of birefringent plates—Part II. An achromatic quarter-wave plate,” Proc. Indian Acad. Sci. A 41, 137–144 (1955). [CrossRef]
4. I. Šolc, “Birefringent chain filters,” J. Opt. Soc. Am. 55, 621–625 (1965). [CrossRef]
5. J. W. Evans, “Solc birefringent filter,” J. Opt. Soc. Am. 48, 142–145 (1958). [CrossRef]
6. A. Gorman, D. W. Fletcher-Holmes, and A. R. Harvey, “Generalization of the Lyot filter and its application to snapshot spectral imaging,” Opt. Express 18, 5602–5608 (2010). [CrossRef]
7. S. E. Harris, E. O. Ammann, and I. C. Chang, “Optical network synthesis using birefringent crystals. I. Synthesis of lossless networks of equal-length crystals,” J. Opt. Soc. Am. 54, 1267–1279 (1964). [CrossRef]
8. G. Sharp, M. Robinson, J. Chen, and J. Birge, “LCoS projection color management using retarder stack technology,” Displays 23, 139–144 (2002). [CrossRef]
9. G. Sharp, “Retarder stacks for polarizing a first color spectrum along a first axis and a second color spectrum along a second axis,” U.S. patent 5,953,083 (Sept. 14, 1999).
10. J. W. Evans, “A birefringent monochromator for isolating high orders in grating spectra,” Appl. Opt. 2, 193–197 (1963). [CrossRef]
11. C. Oh and M. J. Escuti, “Achromatic diffraction from polarization gratings with high efficiency,” Opt. Lett. 33, 2287–2289 (2008). [CrossRef]
12. R. Komanduri, K. Lawler, and M. Escuti, “Multi-twist retarders: broadband retardation control using self-aligning reactive liquid crystal layers,” Opt. Express 21, 404–420 (2013). [CrossRef]
13. N. V. Tabiryan, S. V. Serak, S. R. Nersisyan, D. E. Roberts, B. Y. Zeldovich, D. M. Steeves, and B. R. Kimball, “Broadband waveplate lenses,” Opt. Express 24, 7091–7102 (2016). [CrossRef]
14. G. P. P. L. Otten, F. Snik, M. A. Kenworthy, C. U. Keller, J. R. Males, K. M. Morzinski, L. M. Close, J. L. Codona, P. M. Hinz, K. J. Hornburg, L. L. Brickson, and M. J. Escuti, “On-sky performance analysis of the vector apodizing phase plate coronagraph on MagAO/Clio2,” Astrophys. J. 834, 175 (2017). [CrossRef]
15. S. T. Tang and H. S. Kwok, “Mueller calculus and perfect polarization conversion modes in liquid crystal displays,” J. Appl. Phys. 89, 5288–5294 (2001). [CrossRef]
16. R. A. Chipman, “Ch. 22 polarimetry,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, 1995), Vol. II, pp. 22.1–22.37.
17. A. Thelen, “Design of a hot mirror: contest results,” Appl. Opt. 35, 4966–4977 (1996). [CrossRef]
18. J. Kim, Y. Li, M. N. Miskiewicz, C. Oh, M. W. Kudenov, and M. J. Escuti, “Fabrication of ideal geometric-phase holograms with arbitrary wavefronts,” Optica 2, 958–964 (2015). [CrossRef]
19. C. Yousefzadeh, A. Jamali, C. McGinty, and P. J. Bos, “‘Achromatic limits’ of Pancharatnam phase lenses,” Appl. Opt. 57, 1151–1158 (2018). [CrossRef]
20. S. Nersisyan, N. Tabiryan, D. Steeves, and B. Kimball, “Optical axis gratings in liquid crystals and their use for polarization insensitive optical switching,” J. Nonlinear Opt. Phys. Mater. 18, 1–47 (2009). [CrossRef]
21. K. J. Hornburg, R. K. Komanduri, and M. J. Escuti, “Multiband retardation control using multi-twist retarders,” Proc. SPIE 9099, 90990Z (2014). [CrossRef]
22. L. Nikolova and T. Todorov, “Diffraction efficiency and selectivity of polarization holographic recording,” Opt. Acta 31, 579–588 (1984). [CrossRef]
23. M. J. Escuti, C. Oh, C. Sánchez, C. Bastiaansen, and D. Broer, “Simplified spectropolarimetry using reactive mesogen polarization gratings,” Proc. SPIE 6302, 630207 (2006). [CrossRef]
24. C. Provenzano, P. Pagliusi, and G. Cipparrone, “Highly efficient liquid crystal based diffraction grating induced by polarization holograms at the aligning surfaces,” Appl. Phys. Lett. 89, 121105 (2006). [CrossRef]
25. M. N. Miskiewicz and M. J. Escuti, “Optimization of direct-write polarization gratings,” Opt. Eng. 54, 025101 (2015). [CrossRef]
26. X. Xiang, J. Kim, R. Komanduri, and M. J. Escuti, “Nanoscale liquid crystal polymer Bragg polarization gratings,” Opt. Express 25, 19298–19308 (2017). [CrossRef]
27. T. Gaylord and M. G. Moharam, “Thin and thick gratings: terminology clarification,” Appl. Opt. 20, 3271–3273 (1981). [CrossRef]