24Liquid Crystal Cells

24.1Introduction

Liquid crystal cells are optical elements that manipulate polarized light by rotating liquid crystal molecules. Liquid crystal cells are used as electrically controllable retarders, polarization controllers, and spatial light modulators. Liquid crystal displays (LCDs) incorporate liquid crystal cells with illumination and electronics for displaying information. LCDs have traditionally been utilized in flat screen televisions, mobile phone screens, and computer screens.

LCDs comprise a large fraction of the polarization economy. LCDs were introduced in the 1970s and have dominated the display business since the late 1980s for many reasons.

• LCDs operate at low voltages, typically less than 5 V.

• LCDs have large etendue, a combination of large area and large numerical aperture; thus, LCDs can utilize a large fraction of the light from an incandescent, fluorescent, or LED illumination system.

• LCDs can be fabricated in arrays with large numbers of small pixels; pixel sizes below 2 μm are common.

• LCDs can be produced at low cost through the use of complex foundries.

The image quality of early LCDs was poor. To attain their position of dominance, LCDs had to overcome many obstacles, including absorption, scattering, low contrast, muddy colors, long switching times, uniformity for larger display, limited viewing angle, disclinations within pixels, and polarization aberration. LCDs had to compete with the installed base of cathode ray tube televisions and displays. Other new technologies also threatened the early LCD industry. Over time, billions of dollars of investment in production technologies has addressed these issues and LCDs became the dominant display technology. This chapter discusses the construction, operation, and polarization aberrations of common liquid crystal devices.

Several common types of liquid crystal cells are listed and described in Table 24.1. The construction of these liquid crystal cells is explained and fabrication issues are discussed, along with more detailed discussions of the types of liquid crystal cells and their evolution. The testing of liquid crystals is described using imaging polarimetry. Finally, the modeling of cells’ polarization aberrations and their compensation with multilayer biaxial films are shown.

Table 24.1Common Types of Liquid Crystal Cells

1. Fréedericksz, untwisted nematic cell

   • Retardance and phase modulators, polarimetry, signal processing

2. Twisted nematic (TN)

   • Most common display technology

3. Super twisted nematic (STN)

   • Displays

4. Vertically aligned mode (VAN mode)

   • Displays

5. Hybrid aligned

   • Displays

6. In-plane switching, fast field switching (FFS), and fringe-field-switching (FFS)

   • iPhone/iPad/high-end monitors/televisions

7. Multi-domain vertically aligned (MVA), PSVA, UV2A, etc.

   • Televisions

24.2Liquid Crystals

Liquid crystals (LCs) are a state of matter with properties between solid and liquid. When typical substances are heated and melt, they transform directly from a highly ordered crystalline solid structure (Figure 24.1, left) into a disordered isotropic liquid form (Figure 24.1, right). In contrast, when a solid LC material melts, it transitions through a partially ordered liquid state (Figure 24.1, middle) before becoming an isotropic liquid. Thus, a material in a liquid crystal state is an ordered fluid that exhibits less order than found in solids, but more order than found in liquids. LC molecules are not ordered by position, but the orientation of the molecule is correlated to the orientation of neighboring molecules. The shape of the LC molecule is highly anisotropic, which affects the preferred orientation of its nearby molecules. Thus, it has physical properties of a liquid but with a short range of crystal-like order.

Figure 24.1State of matter between solid and liquid.

A botanist in Austria, Friedrich Reinitzer, first described liquid crystals in 1888. He encountered a material, cholesteryl benzoate, that exhibited a mesophase between the solid and liquid states. When heating through 145°C, it melted into a viscous white fluid. At 179°C, it transitioned into a clear isotropic liquid. He shared his discovery with a physics professor at the Technical University Karlsruhe, Otto Lehmann, describing the two melting points. Lehmann observed that in the mesophase, the liquid demonstrated the double refraction effect, a characteristic of a crystal. Since cholesteryl benzoate showed both liquid and crystal characteristics, he named it a “fliessende crystal.” This translated into the English name “liquid crystal.” Hence, the molecules themselves are not the liquid crystals. Liquid crystal refers to the phase, the state between solid and liquid.

The most common molecules used in LC cells are the nematic liquid crystals, which are rod-like positive uniaxial molecules. The orientation of the LC molecules is specified by the director, which is the direction of the extraordinary axis, also called the optic axis. An example of a typical molecule used in LC cells is shown in Figure 24.2. This molecule is elongated or rod-like, and has a large permanent dipole moment. Note the three fluorine atoms attached to a benzene ring on one end. These three highly electronegative fluorine atoms draw electrons from the benzene ring and beyond toward their end of the molecule, leaving the other end positively charged, setting up a significant dipole moment.

Figure 24.2Chemical structure of 4′-Propyl-bicyclohexyl-4-carboxylic acid 3,4,5-trifluoro-phenyl ester (C22H29F3O2), one of the common liquid crystal molecules used in displays. The director of this molecule, indicating the axis along which the dipole is oriented, is horizontally oriented.

When LC cells are drawn, lines or ellipsoids are drawn to show the variation of the director throughout the cell, as seen in Figure 24.3. The sign of the director is not used; the director has unit length, since its magnitude is not important. Because the material is liquid, the director can vary throughout the cell. The molecules locally strive to remain aligned with each other to reduce their energy, but mechanical and electrical forces cause a change in the director orientation. Rotating the LC molecules provides a tunable birefringence. For nematic LCs, the ordinary refractive index is usually about nO = 1.5 and the birefringence Δn varies from 0.05 to 0.5.

Figure 24.3Temperature influences the order in nematic LC molecules. At high temperature, (right) the LCs are in an isotropic phase where no order is present. (Middle) Lowering the temperature leads to orientation order in the nematic phase, and when cooled to crystallization, (left) the molecules assume long-range positional and orientation order.

Molecules used in LC devices are fabricated in a great variety of shapes. For LC displays, two shape families dominate; the nematic LC, with a rod-shaped molecule as shown in Figure 24.4, and the discotic LC, with a disk-like or pancake-like molecule.

Figure 24.4Schematics for nematic molecules (left) and discotic molecules (right).

24.2.1Dielectric Anisotropy

LC molecules have an intrinsic dipole moment; the center of a molecule’s positive charge is displaced from the center of the negative charge. One end of the molecule has an excess of electrons. When an electric field is applied, a torque is induced, which rotates the molecules. For molecules with positive dielectric anisotropy, the directors rotate toward aligning parallel with the electric field, as shown in Figure 24.5. Rod-like molecules with positive dielectric anisotropy have an excess of charge at one end of the rod. Such molecules have a larger retardance without a field and near-zero retardance when large voltage is applied. For negative dielectric anisotropy, the torque tends to orient the directors perpendicular to the field. These molecules have an excess of electrons near the middle of the rod on one side.

Figure 24.5Molecules with (left) positive dielectric anisotropy and (right) negative dielectric anisotropy are shown. The LCs are at their rest state at 0 V at the top row. The electric field lines are shown in blue, with number of lines indicating field strength.

24.3Liquid Crystal Cells

A liquid crystal cell is a device that packages a layer of liquid crystal with electrodes in order to modulate the distribution of directors and thus modulate the phase and polarization of the transmitted light. Liquid crystal cells allow the electrical control of light at low voltages, over large areas, and with large numbers of pixels. The modulation schemes are divided into three classes:

1. Intensity modulators, commonly used for the display of information

2. Polarization modulators, for polarization manipulation and control

3. Phase modulators, for wavefront control, interferometry, diffractive optical elements, and so on

Consider the earliest type of liquid crystal cell, the Fréedericksz cell shown in Figure 24.6. The space between two glass plates is filled with a thin layer of nematic liquid crystal with positive anisotropic anisotropy, usually between 0.5 and 7 μm thick depending on the application. When no voltage is applied (Figure 24.6, left), the directors are all oriented parallel to the glass plates (x-axis in figure). In this orientation, the LC cell acts like an A-plate, a conventional waveplate, and each layer contributes the same retardance to the cells overall retardance. The electric field across the cell is adjusted by charging electrodes outside the glass, and because of their positive dielectric anisotropy, the rod-like LC molecules rotate toward the z-axis, the normal to the glass and the nominal direction of light propagation. As the LC molecules rotate, the birefringence along the light propagation direction is reduced, and the overall retardance (along the optical axis) is reduced. As the voltage increases (toward the right side of Figure 24.6), the cell’s retardance is minimized. At higher voltages, most of the molecules are aligned along the z-axis, except for the thin layers along the glass that cannot rotate as these are attached or stuck to the cell faces. The distributions of directors will be described by a twist angle ϕ and a tilt angle θ as shown in Figure 24.7.

Figure 24.6Variation of director orientation (purple rods) in a Fréedericksz cell with light propagating from top, through a transparent electrode and glass plate (orange sheet), through the liquid crystal layer, and out the bottom glass and electrode (orange sheet). The applied voltage increases from 0 V on the left to the maximum voltage on the right, and the nematic liquid crystals with positive dielectric anisotropy rotate closer to the field as the electric field increases. The LC liquid crystal cell modulates from an A-plate retarder (like a waveplate) at low voltages to almost a C-plate (optic axis along the direction of light propagation) as the voltage increases. Thus, this LC cell is a retardance modulator.

Figure 24.7Tilt θ is the angle measured from the z-axis to the vector. Twist ϕ is the angle measured from the x-axis to the projected vector on the xy plane.

Figure 24.8 shows a typical retardance versus voltage curve for the Fréedericksz cell. Thus, this cell can operate as a linear retardance modulator with axes along x and y. As the cell modulates, the refractive index for the x-component of the light changes; the molecules are rotating in the x–z plane, but the y-refractive index remains nearly constant. If a polarizer is placed above the cell with the transmission axis along x, the light that is transmitted through the cell is in an eigenpolarization at every layer of the LC and thus does not change polarization. The refractive index is however changing, so the phase will vary with voltage. Thus, this liquid crystal cell can be used as a phase modulator. Finally, an intensity modulator can be constructed by placing a polarizer at 45° before the cell and a polarizer at 135° after the cell. The cell thickness and liquid crystal birefringence are selected; thus, the cell’s retardance modulates from about ½ wave of retardance to close to 0 retardance. At high voltages, the LC retardance is small, the polarization is not changed much since the cell is acting like a C-plate, and the transmission through the cell and crossed polarizers is nearly dark. At low voltages, the LC retardance is about 180° and the 45° light transmitted through the first polarizer is rotated to 135°, aligned with the second polarizer creating a bright state. Intermediate voltages provide light levels from light to dark.

Figure 24.8A typical retardance versus voltage curve for a Fréedericksz cell. Below a threshold voltage of around 2 V, the LC molecules in this cell barely rotate. Further increasing the voltage reduces the retardance, which approaches but does not reach zero at higher voltages.

This concept of varying the output polarization state from being aligned with the output polarizer to being orthogonal to the output polarizer as a function of voltage is the basis of all polarization-based LCDs, from calculators, to watches, ATMs, computer displays, projectors, and televisions. Almost all liquid crystal cells are built for intensity modulation since the human eye is sensitive to changes in intensity and blind to changes in light polarization and phase. LC cells for polarization modulation and phase modulation, while important as optical components, comprise much less than 1% of the LC cell market, which is dominated by displays.

24.3.1Construction of Liquid Crystal Cells

Figure 24.9 shows a typical LC cell configuration. The LC cell provides a means for transmitting light through a thin layer of LC and controlling the orientation of the layers with an applied electric field. The LC layer is 1 to 7 μm thick and contained between two parallel thin glass plates. These glass plates, which mechanically hold the LC cell, must have very low birefringence so they do not change the polarization modulation due to the LC layer. The inside layer of each glass plate, the surface against the LC, has a thin layer of transparent conductor such as indium tin oxide (ITO) about 100 nm thick. Over the electrode is a thin alignment layer, a soft plastic layer in which grooves can easily be formed, typically made of polyimide, which controls the orientation of the LC layers adjacent to the glass substrate. For retardance modulators (polarization controller) and phase modulators, such as Figure 24.9a, this is all that is required; no polarizers are included. A liquid crystal display (LCD) includes several additional components to perform intensity modulation and color modulation. Outside the glass substrates, additional retardation films, often called biaxial multilayer films, are usually applied to optimize the color of the display and minimize the variation of color with angle. Over these films, film polarizers are glued. The film polarizers are needed to change the cell from a polarization modulator into an intensity modulator. Electrodes are connected to the transparent electrodes to apply and hold the voltages. One or both electrodes will be pixelated with associated transistors and capacitors to set the voltage per pixel each using an individual patch of ITO film. For color displays, an array of color filters, red, green and blue, will be placed outside the glass plates.

Figure 24.9(a) A typical LC cell configuration and an LCD configuration are shown. The LC cells are illuminated from the top. The modulated light exits at the bottom of the cell. (b) For a color display, adjacent pixels incorporate red, green, and blue color filters. Intensity modulation is obtained by the placement of polarizers on top and bottom.

24.3.2Restoring Forces

The intermolecular forces between nearby LC molecules can be visualized as tiny springs. The elastic constant k of the springs defines how hard the LC molecules “push” against each other. There are three types of k values: the “splay” direction push, the” twist” direction push, and the “bend” direction push as shown in Figure 24.10.

Figure 24.10(Top) Director distributions with splay, (middle) bend, (bottom) and twist.

LC cells are voltage controlled retardance modulators that change their retardance by rotating anisotropic LC molecules in an electric field. The LC cell is analyzed and understood as a spatially varying anisotropic material with spatially varying retardance. The spatial variation of the director is easily perturbed by electric fields and magnetic fields and by the shape of the bounding surfaces. In the absence of voltage, the liquid crystal seeks its lowest free energy configuration; it mechanically relaxes into the lowest energy state, the ground state. Then, applied fields cause the molecules to rotate, increasing the volume somewhat and increasing the spring-like energy between the molecules. When the field is removed, the molecules push each other back into the ground state. The type of cell is typically named after the configuration of directors in this ground state.

Figure 24.11 shows a vertically aligned nematic (VAN) LC cell with negative dielectric anisotropy sandwiched between two transparent electrodes. The LC molecules against the outer surfaces are attached perpendicular to the glass plates, which set the director distribution throughout the cell parallel to normal incident light at zero voltage. Thus, the directors line up with the electric field. As the voltage increases, the negative dielectric anisotropy LCs rotate perpendicular to the applied electric field as a function of the applied voltage. At the highest applied voltage shown, the directors at the center of the cell are twisted 90°. The top and bottom layers of the LCs do not rotate, since the ends of the molecules have been anchored to the surfaces of the cell. Figure 24.11 (Bottom) The typical transmittance for a VAN mode cell between crossed polarizers.

Figure 24.11A VAN mode LC cell with negative dielectric anisotropy is shown. (Top left) All LC’s layers are aligned at zero voltage. As the voltage increases, (toward the right) the LC’s molecules rotation increases. (Top right) At the highest voltage, most of the directors are nearly perpendicular to the field. (Bottom) Typical transmittance vs. voltage.

Dozens of different “modes” of LC cells have been invented for different purposes such as the following:

• Wider viewing angle

• Lower electrical power

• Higher contrast

• Brighter

• Blacker “blacks”

• Better color

• Lower manufacturing cost

• Unaffected by squeezing for the purpose of touch screen capability

24.3.3Liquid Crystal Display: High Contrast Ratio Intensity Modulation

Most LC devices are operated as intensity modulators, not as phase or polarization modulators. The liquid crystal cell is placed between two polarizers, usually linear polarizers. When the voltage applied to the cell changes, the cell’s retardance and exiting polarization state changes, modulating the intensity. Retarders are often used in conjunction with the linear polarizers if they improve performance. However, cost is a driving factor for high volume production, and simplicity is highly valued.

One metric for evaluating display performance is the contrast ratio C, defined as the maximum flux Imax divided by the minimum flux Imin,

$$C=\frac{I_{\text{max}}}{I_{\text{min}}}.$$24.1

24.1

A high Imax is desirable for a bright display, but usually a small Imin and a good black is far more important and more difficult to achieve. Thus, achieving a very low Imin often is a driving factor in the display design. Some of the reasons why a cell might have a poor Imin are the following:

• Misalignment between analyzer and exiting polarization state

• Spectral variation of exiting polarization state

• Variation of polarization state with angle through the LC cell

• Scattering and depolarization

• Polarizer leakage, poor-quality polarizers

Typical contrast from an LC projector is about 100 and an average LC computer screen has a contrast ratio of about 1000. The contrast of most LC-based TV sets is much greater than 1000, and some models can reach a contrast ratio of 10,000 to 30,000. In general, the more expensive the TV, the better the black, and the higher the contrast.

One ideal configuration consists of a liquid crystal cell as a variable linear retarder oriented at 45° to the first polarizer, which can modulate over a range of at least a half of a wavelength. This variable retarder brings 0° incident light through a series of elliptical polarizations, through circularly polarized light when δ = π/2 into vertically linearly polarized light. This polarization evolution is well described by a spectral trajectory on the Poincarè sphere along a circle of longitude about the variable retarders’ fast and slow axes. The analyzer can be oriented at either 0° for a dark state at low retardance or 90° for a dark state at high retardance. The choice is usually driven by which configuration produces the better dark state. Retardance modulation of greater than half a wavelength generally does not benefit the performance of an intensity modulator, since the device has already spanned its entire intensity range.

24.4Configurations of Liquid Crystal Cells

Most transmissive LC displays share a basic configuration: a pair of polarizers with an adjustable LC retarder in between them. The LC retarder varies between on and off states. At the bright state, the LC retarder transforms light exiting the incident polarizer so it aligns with the exiting polarizer, thus allowing maximum light to pass. At the dark state, the light is orthogonal to the exiting polarizer. Most commercial LCD designs are variants of this approach. Throughout the evolution of LC cells, complexity has continually increased.1

Many different LC configurations are manufactured. Configurations differ in the director distribution functions, electrode locations, and subpixel structures. One of the oldest LC cell designs is the untwisted Fréedericksz cell, which is preferred for polarization control and acts well as an adjustable retarder. The twisted nematic cells are very popular in the LCD industry; the three major families of commercial LC cell technologies used are the twisted nematic, the in-plane switching, and the vertical-aligned nematic. Many more types of LC cells have been developed. The following are three of the most important families for displays:

• Twisted nematic cell

  • Inexpensive

  • Most common

  • Significant polarization and color variation with angle

• Vertically aligned [nematic] (VA, VAN, MVA, PVA, S-PVA, etc.)

  • Good switching speed, off-axis viewing (wide viewing angle), intrinsic color gamut (range), relative to TN; black level overall

  • Intrinsic image quality not as good as IPS, but many incremental improvements have been introduced

  • Popular for TVs

• In-plane switching (IPS, S-IPS, AS-IPS, H-IPS, A-TW-IPS, etc.)

  • High intrinsic image quality; reference standard for color output, and off-axis performance (wide viewing angle)

  • Initial obscuration and switching speed handicap, largely overcome by design and process evolution

  • Costly

24.4.1The Fréedericksz Cell

The first LC cell developed was the Fréedericksz cell, also known as the untwisted nematic cell, or planar aligned nematic cell, as shown in Figure 24.12. The directors align and rotate in a single plane; thus, the label untwisted. At 0 V in the off state, all directors are parallel and aligned to the electrodes, like an A-plate retarder, with a small pretilt. As the voltage increases, the LCs rotate in the x–z plane toward the direction of light propagation, thus reducing the retardance. At high voltages, the directors rotate to almost vertical at the middle layer of the cell, while the molecules at either end are anchored at the glass plates. Figure 24.13 shows a small single-pixel Fréedericksz cell retardance modulator with a pair of wires for the driving voltage.

Figure 24.12Director orientations in a Fréedericksz cell. The applied voltage increases from (left) 0 to 5 V (right). The LCs modulate from an A-plate-like retarder configuration to almost a C-plate.

Figure 24.13A small Fréedericksz cell retardance modulator for adjustable linear retardance. The LC is sealed between glass plates.

Since the directors are untwisted and only rotate in one plane, the Fréedericksz cell is always a linear retarder with constant orientation; the retardance axis usually oriented at 45° to the edge of the cell. The cell is often driven by 500–10,000 Hz square wave at low voltage (1–13 V). It is a desirable 180° phase modulator. The Fréedericksz cells are small and inexpensive compared to other retardance modulators, such as electro-optical modulators, photo elastic modulators, magneto-optical modulators, or retarders mounted on rotating motors. However, the parallel aligned LC cells have many issues; they are often too slow and have large variation of color and retardance with field of view.

24.4.290° Twisted Nematic Cell

The twisted nematic cell (TN cell) was the first LC technology in production for LC displays, introduced by Schadt and Helfrich and Fergason in 1971. The TN technology still has a large market share. It is often found in notebook PC screens, desktop LCD monitors, and some low-end cell phones. Because of decades of experience of manufacturing TN cells, the production cost is relatively low. The TN cell operates at a lower voltage and switches considerably faster than the Fréedericksz cell.

The TN directors twist steadily from window to window at zero voltage, while remaining parallel to the glass plates (no tilt). In the off state, the directors are twisted by 90° around the direction of light propagation, as shown in Figure 24.14, and the pixel has an approximately 180° of retardance. That pixel becomes bright when placed between crossed polarizers. When a TN pixel goes dead, the transmission gets stuck on and appears white. As shown in Figure 24.15, when voltage is applied, the LC molecules rotate toward the z-axis, becoming normal to the cell, and the retardance reduces, since birefringence goes to zero for propagation along the director. At the high range of applied voltage, the retardance becomes almost zero, as shown in Figures 24.15 (right) and 24.16. Then, the pixel is dark when placed between crossed polarizers. At the dark state, the retardance at the top and the bottom of the cell cancels each other, thus providing a larger FOV.

Figure 24.14The helix distribution of the liquid crystal directors in a 90° TN-LC cell with a left-handed twist is indicated by the distribution of the rods. The arrows on the ends indicate the rubbing directions in the alignment layer.

Figure 24.15Director distributions for a 90° twisted nematic LC cell and the variation of the directors with voltage. The entering and exiting windows are indicated in orange. The leftmost image is at 0 V, where all molecules are parallel to the windows but twist through 90°. The voltage increases from left to right in arbitrary increments. At high voltages, above 5 V, the molecules have mostly rotated to the vertical direction except for the thin layers near the top and bottom alignment layers.

Figure 24.16Resultant retardance as a function of voltage across the 90° TN cell.

For a TN cell, as the voltage begins increasing from 0 V, very little LC molecular motion occurs, until suddenly at around 1 V, enough torque is applied for the molecules to overcome their intermolecular attraction, and they begin to rotate. As the voltage increases, the central layer of the molecules rotates to vertical and then more layers become vertical; most of the cell is approaching a C-plate, until finally most of the cell volume has a vertical orientation except for the layers at the very top and bottom, which are not free to rotate. The elliptical eigenpolarizations of the LC cell are depicted in Figure 24.17. The eigenpolarizations are the states that exit the cell in the incident polarization state. The phase delay between the two orthogonal eigenpolarizations is the retardance. The structure of the LC cell is like slices of linear retarders rotating through 90°. Since misaligned linear retarders give circular retardance, the eigenpolarizations are elliptically polarized. As the wavelength and the incident angle vary, these eigenpolarizations and the retardance also change in magnitude and orientation. The minimum retardance is asymptotically approached at high voltage state, above 5 V as shown in Figure 24.16, since most of the directors are vertical. They are like the C-plates with no retardance. Since the retardance does not decrease all the way down to zero due to small residual retardance from layers against the cell windows, sometimes a weak retarder is added in series, a trim retarder perhaps with about 5° of retardance, to bring the retardance down to zero. Because of retardance dispersion, a shorter wavelength produces a larger retardance modulation while a longer wavelength gives a smaller retardance modulation.

Figure 24.17The polarization ellipses of the normal modes in a TN-LC.

The shortcomings of the TN cell motivated the development of other cell designs to provide higher contrast and better display. A common defect of TN cells is crossed-polarizer leakage due to linear state obliquity. Another issue is that for the dark state, the TN cell converts the input linear state to a slightly elliptical state that the output polarizer cannot completely block.

The retardance of TN liquid crystal cells has a large variation with angle of incidence, which is simulated in Section 24.5.4. Figure 24.35 maps the retardance versus angle of incidence for a typical TN cell with increasing voltage.

24.4.3Super Twisted Nematic Cell

The super twisted nematic cell (STN) is a variation of the TN cell, which was commonly found in displays in the 1990s and 2000s.2 The term super refers to a twist angle larger than the 90° twist typical of TN cells. This lets the STN cell switch faster at lower voltages. The angle of incidence characteristics are better but the cells are more expensive to fabricate. The schematics of the directors for the STN cell with 180° and 270° twist at 0 V are shown in Figure 24.18.

Figure 24.18Director orientations through the cell for STN cells with (left) 180° and (right) 270° rotations.

In Figure 24.19 (left), with no applied voltage, the directors twist by 180°. At maximum voltage (right), the director orientation approaches a C-plate configuration, with the directors substantially aligned along the direction of light propagation.

Figure 24.19The 180° supertwisted nematic liquid crystal cell’s directors at several voltages. At low voltage (left), the director twists in the plane of the cell, in this example by 180°. At high voltages (right), the directors tilt by almost 90° and are aligned along the direction of light propagation except for thin layers against the alignment layers (orange planes).

24.4.4Vertically Aligned Nematic Cell

The vertically aligned nematic (VAN) LC was a common configuration for TVs in the early 2000s, often replacing the TN cell in this application. The VAN LC cell was preferred for its high contrast in the off state and good viewing angle performance.3 The VAN mode cell uses a negative dielectric anisotropy molecule. In the off state, the directors are tilted vertically, called homeotropic alignment, perpendicular to the electrodes, like a C-plate with small pretilt. As shown in Figure 24.20, in the ground state with 0 V, the directors are near vertical. The applied voltage then rotates the directors in the x–z plane.

Figure 24.20The VAN cell’s directors at several voltages. It gives a dark state at 0 V shown on the left, and a bright state at high voltage shown on the right.

At 0 V, the VAN cell is basically a C-plate providing an extremely good dark state between crossed polarizers with a wide field of view. As a function of angle of incidence, the ordinary and extraordinary modes are shown in Figure 24.21. At normal incidence, the two modes are degenerate, and the LC cell has zero retardance.

Figure 24.21VAN-mode eigenpolarizations in a cell without pretilt, (left) ordinary mode is tangentially polarized, and (right) extraordinary mode is radially polarized. At normal incidence, the modes are degenerate; the ordinary and extraordinary modes have the same refractive index.

The VAN mode cell functions well as a phase modulator. The directors of the VAN cell remain in one plane as voltage is applied as is seen in Figure 24.20. If the cell is illuminated with light in this plane of polarization, the light remains in an eigenpolarization as it propagates through the cell; hence, the polarization state emerges unchanged. But the refractive index changes from nO toward nE as the molecules rotate, thus changing the phase without changing the polarization.

The dark state of the VAN cell is an improvement from the TN and STN cells. The wide-angle contrast is still limiting by the leakage in the dark state. But this can be well compensated by additional retarders before and after the LC cell. The nematic LC molecules are normal to the glass interfaces, which can produce stability issues—moving picture image sticking (MPIS). In some severe cases, the image sticking remains for a short time after the image has changed, as shown in Figure 24.22. Because of disclination, the LC director orientations are undefined and can easily flip the wrong way. These lead to transmission variability and slower response time.

Figure 24.22Example of motion picture image sticking that appears in VA panels and does not appear in S-IPS mode panels.

The patterned vertical-aligned cell (PVA) was developed by Samsung in 1996. The PVA mode tackles disclination by having a non-normal field applied by patterning the electrodes on planar layers bounding the LC so they are offset from each other, as shown in Figure 24.23. This configuration produces multiple domains, which can address the asymmetric behavior of the overall display when arranged correctly and give excellent viewing angle. In PVA, the electrodes are arranged as zigzag chevron geometry (Figure 24.23, right).

Figure 24.23(Left) The director distribution in a PVA cell in its off state resembles a VAN mode cell. (Middle) The director distribution in the on state is perpendicular to the electric field lines (red). (Right) Each surface’s electrode distribution within each pixel has two regions (indicated by the blue line) with electrodes patterned in a different direction.

Another method to treat disclinations in VAN mode cells is the multi-domain vertically aligned (MVA) cell. In the MVA mode, structures with protrusions are formed in the layers bounding the LC, which makes the director non-normal to the display axis in the off state, as shown in Figure 24.24 with pyramids on two sides or in Figure 24.25. Multiple domains are produced in a symmetric pattern, which overcomes any asymmetry in transmission characteristics. Another variation of the MVA configuration replaces the protrusions on one substrate with patterned ITO slits. This reduces the number of manufacturing steps and increases the contrast ratio because the residual birefringence around the protrusions on one substrate is removed.

Figure 24.24The configuration for MVA panels uses arrays of pyramids or ridges (left) for alignment and anchoring. (Middle) Directors in the off state. (Right) Directors in the on state.

Figure 24.25A variation of the MVA concept with pyramids on only one surface in (left) off and (right) on state.

24.4.5In-Plane Switching Cell

The in-plane switching (IPS) LC cell was developed by Hitachi in 1996 to increase the viewing angle, improve the color reproduction, and provide stable image quality.4,5 The IPS mode is most commonly found in high-end TVs and cell phone displays, including the Apple Cinema and Thunderbolt Display.

The IPS cell configuration has both electrodes on the same substrate, as shown in Figure 24.26. Therefore, the electric field is predominantly parallel to the glass plates. The molecules are not anchored at the boundaries. The applied voltage rotates the directors along the applied field direction, which is in the plane of the glass plates. The directors always remain perpendicular to the display normal, so the LC molecules change very little with switching.6 The IPS cell acts like a rotating half-waveplate. At low voltage, the directors are aligned with one of the polarizers to give a dark state. The applied voltage twists the directors about the z-axes. At the highest voltage shown, the director is 45° from the polarizers, providing a bright white state. Unlike the TN cell, when a pixel fails, IPS gives a dark dead pixel. The dark and white states could be reversed by orienting the polarizers parallel to the directors in the off state.

Figure 24.26(Left) The in-plane switching (IPS) LC cell has two electrodes, shown in yellow, both on one of the glass plates. The applied voltage increases from the left figure to the right. The off state of the IPS cell is shown on the left, where, with crossed polarizers, indicated by gray lines, no light transmits through the cell. As the applied voltage increases, the directors rotate in the x–y plane. As the directors twist, the brightness increases. (Right) At the highest voltage, the directors are rotated 45° to the polarizers, the cell thickness chosen for a half wave of retardance, and the highest brightness is observed.

Pressure applied through touching the screen causes little change to the transmission. This provides an easy way to identify an IPS display, shown in Figure 24.27, and has contributed to IPS becoming the preferred technology for touch screens.

Figure 24.27(Left) The transmittance through an IPS cell is not affected by applying pressure to an IPS, (right) whereas with a VAN cell, pressure is easily visible.

The IPA technology rapidly became a mainstream technology. Because of the director arrangement, the IPA mode is inherently robust at large viewing angle. The initial production of Super-IPS (S-IPS) mode in 1998 provided a minimal color shift and has a truly wide viewing angle 178° in all directions. The IPS color reproduction and accuracy have always surpassed other LC modes. The S-IPS shows no image sticking even when touching a moving image.

One of the major drawbacks in the original IPS was slow response times. It was originally so slow, at around 60 ms, that it was unsuitable for viewing motion pictures. In 2005, LG. Phillips (now LG. Display) adapted the overdrive circuitry technology and produced the enhanced-IPS (E-IPS), often called the advanced S-IPS (AS-IPS), which improved response times to 5 ms.

The original low contrast ratio of IPA has also improved significantly in the modern IPS generation by using innovative electrode geometries, such as herringbone patterned electrodes and novel pixel structures. The obscuration from the co-planar electrodes initially limited the effective transmittance and has been reduced through development. Compared to the non-planar structures of MVA, the IPS structure is relatively simple, but the IPS electrode is more complex.

The high color saturation, sufficient black levels, touch insensitivity, and the large viewing angle allowed IPS to initially penetrate the high-end markets, such as medical imaging. IPS then rapidly expanded market share in consumer applications, such as tablets, smartphones, and Apple’s Retina displays. IPS provides good color performance at high pixel densities and works well for touch screens. Compared to TN displays, IPS requires more power and is more expensive to manufacture, so TN has maintained some market share.

Other common defects of the IPS display include persistence image (a faint remnant of the old image stays after a new image replaces it) after the display has been left on for a long period of time. The IPS monitor also shows slight color and brightness shifts from edge to edge even with the high pixel resolution.

24.4.6Liquid Crystal on Silicon Cells

Liquid crystal on silicon (LCoS) cells are pixelated liquid crystal cells fabricated over a mirrored surface on a silicon integrated circuit. LCoS are reflective displays; all the other LC cell configurations described here are transmissive. As shown in Figure 24.28, the chip provides a compact electronic means of addressing the LCoS panel, delivering the pixilated electric field to the liquid crystal. The optical components are a transparent conductive electrode, such as ITO, a thin layer of liquid crystal about 5 μm thick, and a reflective surface over the silicon structure.

Figure 24.28Schematic of an LCoS cell. Light enters from the top, transmitting through the LC layer, reflecting from a mirror coated directly on a circuit addressing the electrodes. The light double passes the LC layer and exits out the top.

Figure 24.28 shows an LCoS panel schematic with a cover glass with an ITO layer, a liquid crystal layer, and a mirror over Si integrated circuit. LCoS are used for many compact systems, such as head up displays where weight and volume are at a premium.

Figure 24.29 shows the configuration for a typical projector. Light from the lamp is shaped by illumination optics and the color is modulated by a spinning color wheel. Since the LCoS is a reflective device, it is generally illuminated on-axis using a polarizing beam splitter (PBS). The illuminating light passes through a pre-polarizer and is reflected from the PBS. The LCoS is illuminated with linear polarized light, which transmits through the ITO and liquid crystal layer, reflects, and then makes a second pass through the liquid crystal and ITO before exiting the device. The change in polarization state introduced by the LCoS panel is transformed into intensity variations by an analyzer following the LCoS. In the dark state, the light reflects without polarization change and is reflected back into the illumination system by the PBS. In the on state, the LCoS rotates the polarization 90°, and this reflected light transmits through the PBS and onto the viewing screen. The PBS is typically a MacNeille-type cube beam splitter or a wire grid beam polarizing splitter. Other projector configurations may utilize one, two, or three panels for different wavelengths in a variety of configurations.

Figure 24.29For a typical LCoS projector, polarized light reflecting from a PBS is incident on the LCoS panel. As the retardance of the pixels is modulated, the polarization state from each pixel changes. The light coupled into the orthogonal polarization transmits through the PBS and is projected onto a screen. Color images can be generated by rotating a color wheel with three color filters in the incident beam, and creating red, green, and blue frames at three times the overall frame rate.

24.4.7Blue Phase LC Cells

Another configuration for LCD uses the blue phase (BP) mode, a highly twisted cholesteric LC configuration with a regular cubic structure. The BP can switch in approximately 0.1 ms. BP has been developed for 3D TV, which, due to the need to project alternating images to the left and right eyes, must project twice the number of images and operate at twice the speed. The blue phase is an example of the very complex director configurations that can now be fabricated.

In BP, the directors of a cholesteric LC are arranged in a double twisted structure, the double twist cylinder, where the molecules twist in two dimensions simultaneously. A view of the orientation distribution of the directors is shown in Figure 24.30a.7 At the rim of a slice of the cylinder, all directors lie tangent to the cylinder and 45° from the circumference, as shown in Figure 24.30b. Along each radial axis on that slice, the directors have a 90° twist, as shown in Figure 24.30c. The directors rotate from one end of the slice, at 45° relative to the circumference, to the center axis of the cylinder, where all directors are aligned vertically, and then they continue to rotate in the same direction and reach the other side of the circumference with a total rotation of 90°. The structure of the slice continues down the cylinder as shown in Figure 24.30d. The cylinder is stable up till the 45° twist, which has a distance in the order of 100 nm from the central axis.

Figure 24.30(a) Top view of a blue phase mode double twist cylinder showing the orientation variation of the directors on one of the layers. (b) 3D view of the director orientation around the circumference. (c) The orientations of the directors are shown in two orthogonal axes on a slice of the cylinder. (d) All directors are rotating about the central axis of the double twist cylinder.

Different configurations of double twist cylinders are constructed by stacking the cylinders orthogonally with interlaced disclinations. These defects appear at regular distances throughout the 3D structure. Two types of the BP are shown in Figure 24.31. The periodic cubic structure induces Bragg reflection since the BP structure contains periodic disclination defects on the order of the visible wavelength. Light of a particular color diffracts with the color controlled by an applied electric field. The applied voltage induces birefringence in the liquid crystal through the Kerr effect.

Figure 24.31(Left) Basic double twist cylinders stacking on each other, where disclination occurs at where they are in contact. (Middle) BP II has a simple cubic structure. (Right) BP I has a face-centered cubic structure.

The BP mode has been used in IPS structures, which is modeled with the Kerr effect in a macroscopic scale. Research has focused on lowering the driving voltage of BP LCs, by using materials with high Kerr constants or with optimized structure design.

24.5Polarization Models

The design and analysis of liquid crystal cells require the calculation of color and polarization. Liquid crystal cells are spatially varying anisotropic materials. Simulation of the LC cells is done to design and tolerance the cells and understand their optical properties. Computer simulation of LC cells involves two stages:

1. Determining the distribution of LC directors given the boundary conditions and electric and magnetic fields

2. Given the director distribution, to determine the optical properties for light transmitted through the cell, such as the Jones matrix as a function of angle of incidence and wavelength

For many of the most common LC cells, such as the twisted nematic and VAN mode cells, the cell has a planar structure and can be divided into layers parallel to the faces with a constant director in each layer, similar to a multi layer thin film. The optical properties of such cells can be analyzed by the generalization of multi layer films to anisotropic layers. The IPS cell is not a plane parallel structure and cannot be analyzed so simply as a multilayer film.

24.5.1Extended Jones Matrix Model

The extended Jones matrix method of cell simulation divides the LC into many thin retarder layers with varying optic axis direction (Figure 24.32) and calculates a Jones matrix for each layer.8,9 Consider an arbitrary ray propagating straight through a cell, with an x and y defined in its transverse plane for Jones matrix definition. When the director for a layer is perpendicular to the ray, the birefringence is maximum, nE − nO. For a thickness Δt and director twisted by ϕ from the x-axis, the layer’s Jones matrix is

Figure 24.32(Left) A light ray passes through an LC cell that is modeled as a series of retarders. (Right) Model of an LC cell as a stack of thin birefringent plates with twisting and tilting directors, that is, as a series of discrete layers of retarders.

$$\mathbf{\text{J}}(\varphi )=\mathbf{\text{R}}(\varphi )\cdot \mathbf{\text{LR}}\cdot \mathbf{\text{R}}(-\varphi )=\left(\begin{array}{ll}\text{cos}\varphi & -\text{sin}\varphi \\ \text{sin}\varphi & \text{cos}\varphi \end{array}\right)\cdot \left(\begin{array}{ll}{e}^{-i{\Theta }_O}& 0\\ 0& e^{-i{\Theta }_E}\end{array}\right)\cdot \left(\begin{array}{ll}\text{cos}\varphi & \text{sin}\varphi \\ -\text{sin}\varphi & \text{cos}\varphi \end{array}\right).$$24.2

24.2

LR is a linear retarder Jones matrix where the ordinary component’s phase changes by ΘO = 2π nO Δt/λ across the layer, and the extraordinary phase changes by ΘE = 2π nE Δt/λ. If the director is tilted by θ from the ray (π/2 − θ from the transverse plane), the effective refractive index of the extraordinary mode, ne(θ), polarized in the plane of the director, is reduced from its maximum value nE to ne,

$$\frac{1}{n_e{(\theta )}^2}=\frac{{\text{cos}}^2\theta }{n_O^2}+\frac{{\text{sin}}^2\theta }{n_E^2}.$$24.3

24.3

The birefringence, nO − ne(θ), encountered by the ray is projected onto the transverse plane of the ray and an equivalent retarder Jones matrix is calculated using the thickness for each layer. Light propagating along a director, θ = 0, experiences no birefringence and no retardance; light propagating perpendicular to a director, θ = π/2, experiences the full birefringence, nO − nE. The overall cell Jones matrix is calculated by multiplying the Jones matrices for the individual layers.

The extended Jones matrix methods can use Jones matrices since the light is propagating in nearly a straight line through the cell.

24.5.2Single Pass with Polarization Ray Tracing Matrices

The single pass calculation can also be performed in three dimensions using the P polarization ray tracing matrices two ways: (1) propagating a single ray through the cell, or (2) propagating two rays, ordinary and extraordinary. Method (1) works best for cell designs with significant twist, rotation about the z-axis, and parallels the extended Jones matrix method, producing an equivalent result. Method (2) is more accurate for cells without twist, such as Fréedericksz cells and VAN-mode cells, where most of the light remains in either the E-mode throughout the cell or the O-mode.

Consider Method (1). As light enters the cell, the P for refraction into the glass cell windows is calculated based on the dielectric tensor ε1 for the first layer. Next, the light refracts into the first LC layer, q = 1, where the high and low index modes have slightly different propagation directions, k1,H and k1,L, and refractive indices, n1,H and n1,L. An average k is calculated weighted by the fraction of light in each mode. The birefringence in the transverse plane Δn1 is calculated and a corresponding pure retarder polarization ray tracing matrix P1 is constructed as well as an average optical path length, OPL1. The ray continues on a straight path through the LC cell and is divided into Q segments. The dielectric tensors are calculated at the centers of each ray segment εq from the director distribution. Then, a pure retarder P matrix, Pq, is written for each short ray segment. The P’s are multiplied together to describe the polarization change through the cell as a sequence of retarders, yielding a net PTotal for propagation through the LC cell,

$${\mathbf{\text{P}}}_{\mathrm{Total}}={\mathbf{\text{P}}}_Q{\mathbf{\text{P}}}_{Q-1}\cdots {\mathbf{\text{P}}}_q\cdots {\mathbf{\text{P}}}_2{\mathbf{\text{P}}}_1={\displaystyle \prod _{q=1}^Q{\mathbf{\text{P}}}_{Q-q+1}},$$24.4

24.4

from which the retardance is calculated. The average optical path length is found from the sum

$$OP{L}_{\mathrm{Total}}={\displaystyle \prod _{q=1}^{Q}OP{L}_q}.$$24.5

24.5

Sets of OPLTotal for ray grids are used to calculate wavefront aberration functions for focusing through cells.

Method (2). As light refracts into the first LC layer, q = 1, the refractive indices, n1,H and n1,L, and two slightly different propagation directions, k1,H and k1,L, are used to generate two rays, one for each of the high and low index modes. Two polarization ray tracing matrices are constructed for the first modeled layer, P1,H and P1,L. These two rays are continued through the cell and two polarization ray tracing matrices are constructed for each additional layer Pq,H and Pq,L Finally, the layer matrices are multiplied together to yield two end-to-end matrices, one for the low index mode PL,Total and one for the high index mode PH,Total,

$${\mathbf{\text{P}}}_{L,\mathrm{Total}}={\mathbf{\text{P}}}_{\text{L},Q}{\mathbf{\text{P}}}_{\text{L},Q-1}\cdots {\mathbf{\text{P}}}_{\text{L},q}\cdots {\mathbf{\text{P}}}_{\text{L},2}{\mathbf{\text{P}}}_{\text{L},1}={\displaystyle \prod _{q=1}^Q{\mathbf{\text{P}}}_{\text{L},Q-q+1}}$$24.6

24.6

and

$${\mathbf{\text{P}}}_{\text{H},\mathrm{Total}}={\mathbf{\text{P}}}_{\text{H},Q}{\mathbf{\text{P}}}_{\text{H},Q-1}\cdots {\mathbf{\text{P}}}_{\text{H},q}\cdots {\mathbf{\text{P}}}_{\text{H},2}{\mathbf{\text{P}}}_{\text{H},1}={\displaystyle \prod _{q=1}^Q{\mathbf{\text{P}}}_{\text{H},Q-q+1}}.$$24.7

24.7

Because these are single mode P, each has the properties of polarizers. The differences in phase yield the retardance and the differences in transmission yield the diattenuation for the ray path. This two-mode method is most applicable to untwisted cells, such as the Fréedericksz cell and IPS cell, where, once started, little coupling occurs between the low and high modes.

The extended Jones matrix method ignored the difference between k1,H and k1,L and used the average propagation direction for the ray direction within the LC.

24.5.3Multilayer Interference Models

The previous two methods analyze the LC cell as a series of retarders in a single a pass. Because of the refractive index variations along the path, some light is reflected into the reverse direction continuously along the path. The cell can be considered as a multilayer structure similar to a thin film with interference occurring between different layers, layers that are anisotropic. The first full multilayer interference model was attributed to Berreman in 1972 based on a four-element vector (Ex, Ey, Bx, By) containing the electric and magnetic fields.10,11 A 4 × 4 matrix is developed for each layer, and matrix multiplication yields the relationship between the amplitude, phase, and polarizations of the incident and exiting plane waves. A different multilayer interference model was developed by Mansuripur.12 For most cells, the single pass methods and multilayer interference methods produce almost the same values, and since LC cells are relatively thin, the differential refractive index is not changing very rapidly, and only multiple low energy reflected light is present.

These models use practical simplifications of the simulation, (1) dividing the continuously varying dielectric tensor into a set of separate layers, and (2) assuming the modes propagate along the same ray path from the LC entrance to exit. When light refracts from glass into the LC, the light refracts at different angles into the high and low index modes. The next layer has a slightly rotated dielectric tensor, so most of the high mode refracts into the high mode, but a tiny bit couples into the low mode. Similarly, a tiny fraction of the low mode refracts into the high mode; these tiny fractions have a third and fourth propagation direction. This continues at each of the layers generating new small beams at additional angles. LC display cells are thin. The result is a large set of rays that exit from almost the same ray intercept with almost the same propagation direction, but a small amount of spreading in location and angle occurs. This spreading is not accounted for in the models presented here.

When setting up an LC model, it is not initially clear how many layers the model should use to divide the cell. As the number of layers is increased, the LC’s polarization matrix changes until it asymptotically approaches its final value. One common technique is to start with a small number of layers and keep doubling the number of layers until the calculation converges.

24.5.4Calculation for Liquid Crystal Cell ZLI-1646

An example of a twisted nematic liquid crystal cellsimulated with a multilayer interference model is performed using Polaris-M (Airy Optics Inc.) by modeling the cell as a stack of interfering biaxial thin films. The cell is formed from the liquid crystal mixture ZLI-1646 at a wavelength at 589 nm. This replicates the configuration from the reference Optics of Liquid Crystal Displays, 2nd edition by Pochi Yeh and Claire Gu. The directors are anchored to the top and bottom surfaces of the cell. The orientation of the directors through the LC cell is defined by the twist angle ϕ, rotation in the plane parallel to the glass plates, and the tilt angle θ (rotation on the plane orthogonal to the glass plates). The total twist angle is defined by the angles of the grooves on the two alignment layers, Φ = 90°. The maximum tilt angle θmax is a function of the applied voltage and occurs halfway through the cell. The cell gap has a thickness of d, and the position in the cell is z/d. The cell is assumed isotropic in x and y; it is a planar structure. Figure 24.33 shows the director orientation for three applied voltages.

Figure 24.33(Left) The tilt angle and (middle) the twist angle of a 90° TN cell for three different applied voltages are shown for 5.03 V (turquoise), 2.2 V (pink), and 1.3 V (purple). (Right) The corresponding director orientations through the LC cell.

By increasing the applied voltage from 0 V (right), the LC cell switches from behaving as a stack of thin A-plates with a 90° twist to a stack of C-plates (left). The polarization properties of this LC cell are calculated over an angle of incidence range of 0.08 NA to study the LC performance variation with incident angle. Figure 24.34 maps the magnitude and orientation of the angle of incidence across the beam. Figure 24.35 maps the variation of the retardance versus angle of incidence with increasing voltage. Ellipses indicate the extraordinary eigenpolarization (slow state) and the size of the ellipse indicates the magnitude of the retardance. Ideally, in each figure, all the ellipses would be identical, so the behavior of the cell would be independent of angle. As the voltage increases from 0 V, the on-axis retardance changes from about one-half wave (3.14 radians) to almost zero, while the ellipticity of the fast state increases toward circular polarization. At 1.3 V, the fast axis rotates 180° around normal incidence with some ellipticity present.

Figure 24.34AOI map for 0.08 NA with a key for the scale at the lower right.

Figure 24.35Retardance maps versus field of view for twisted nematic LC for 5.03 V, 2.2 V, and 1.3 V. The label at the bottom right of each map shows the maximum retardance in radians over the field.

A small amount of diattenuation is also observed in the LC cell model in Figure 24.36. One source of diattenuation occurs refracting into and out of the glass windows due to the Fresnel equations. Since the polarization state evolves from the front, through the cell, to the back window, these two outer surface diattenuation contributions are no longer aligned and may substantially add or cancel depending on the LC setting. Interference between LC layers also generates additional diattenuation. As is seen in Figure 24.36, LC cell diattenuation is generally minor, less than 0.02 in this case.

Figure 24.36Diattenuation map for 5.03 V, 2.2 V, and 1.3 V. The label at the bottom right of each map shows the maximum diattenuation over the field.

24.6Issues in the Construction of LC Cells

This section examines the construction of LC cells. Following their invention, LC displays took a long time to come to market due to their complexity and difficulty of manufacture. Among the earliest commercial LC applications were digital watches and calculator displays, applications where speed and accurate color were not requirements. As the technology steadily solved the many technical challenges, LCs moved into more and more demanding applications. Laptops and conference room projectors, for example, needed more pixels than calculators, but these applications still did not require great speed or fine color control. By the early 2000s, LC displays were finally competitive as high-end TVs.

24.6.1Spacers

Fabricating and maintaining a constant thickness of the thin liquid crystal layer is challenging. To maintain proper distance between the glass plates, spacers, such as precision glass beads or plastic spheres, are placed between the glass plates to fix the separation, as seen in Figure 24.37. When a voltage is applied across the LC cell, the two electrodes and plates are electrostatically attracted toward each other, squeezing the cell, attempting to reduce its volume. The spacers hold the plates apart, but between the spacers, the glass plates are pulled toward each other and the cell thickness varies. This variation of cell thickness can be seen in the retardance map of the untwisted nematic cell in Figure 24.38. More recent LC cell designs use pillars or pyramids micro-lithographically fabricated onto one of the glass faces as spacers.

Figure 24.37Typically, the glass substrate is 0.7–1.1 mm thick, the alignment layers are 10–20 nm thick, the ITO electrodes are about 100 nm thick, and the spacer is about 1–7 μm.

Figure 24.38(Left) The retardance of a single-pixel polarization controller, a Fréedericksz cell, shows non-uniformities in horizontal stripes due to cell thickness variations. These periodic variations of about one degree of retardance result from the periodic spacing between rows of spacers. In between rows of spacers, the glass plate bows due to the attractive force between the electrodes. The resulting thickness variations cause these retardance variations.

24.6.2Disclinations

Disclinations are locations in the LC where the director abruptly changes its orientation; they are defects in the positional order of the directors. Consider the schematic in Figure 24.39 (right), which shows a disclination in the Fréedericksz cell. In Figure 24.12 (left) at 0 V, the directors lie parallel to the glass surfaces. When a voltage is applied, the molecules have an equal probability of rotating clockwise or counterclockwise. Ideally, the whole pixel’s directors should rotate together either clockwise or counterclockwise. If one region of a pixel rotates clockwise, and another rotates counterclockwise as seen in Figure 24.39 (left), a disclination will form at the boundary, and the pixel will appear non-uniform from different angles. Figure 24.39 (right) shows a disclination in a VAN mode cell. In nematic LCs, disclinations usually appear as a line defect, but a variety of topological forms can occur. Disclinations are undesirable, causing visible defects and reducing specifications. Chiral dopants, which are helical molecules, can be added to ensure a uniform rotation direction, either clockwise or counterclockwise.

Figure 24.39The disclination between two volumes of (left) a parallel aligned cell and (right) a VAN LC cell occurs where the director abruptly changes direction. In both figures, the left four columns of directors have rotated clockwise and the right five columns have rotated counterclockwise. Thus, the disclinations occur between columns four and five. Such discontinuous changes of director cause visible lines within the corresponding pixels when used with pairs of polarizers.

24.6.3Pretilt

To avoid disclinations, most LC cells introduce a pretilt in the alignment surfaces as shown in Figure 24.40. Small alignment grooves are placed in a thin layer of soft material, typically polyimide (PI layer).13 Polyimide can be buffed unidirectionally with velvet-like cloth to create small alignment grooves. The LC molecules at the boundary layer fit into the grooves. When the surface layer of molecules has a small pretilt of 2°–4°, the LC molecules have already started their rotation at 0 V; this breaks the balance between rotating clockwise and counterclockwise when a voltage is applied. Now, when a small field is applied, the directors continue to rotate in the direction of the pretilt, avoiding the disclination, and keeping the pixel from breaking up into different domains. Buffed polyimide has been steadily replaced by UV photo aligned layers with grooves created interferometrically.14

Figure 24.40Pretilt examples set the tilt of the boundary layer of the LC axes. The pretilt is created in the alignment layers by rubbing. The rubbing directions for different cell designs are indicated by the arrows. (Left) Parallel alignment, (middle) splay cell, and (right) bend cell.

24.6.4Oscillating Square Wave Voltage

The LC material filling a cell always contains some free ions, such as sodium and calcium ions. When a DC voltage is applied across an LC cell for a long time, these ions drift through the LC and accumulate near one of the electrodes. Groups of positive ions form at one side of the cell while the negative ions collect on the other side of the cell. Their concentration buildup degrades the liquid crystal performance and frequently leads to cell failure.

To avoid this ion buildup problem, LC cells are driven with square wave voltages, as shown in Figure 24.41. The exact frequency is not of importance since the LC works for either DC or square wave voltage; the oscillation is what is necessary to prevent ion buildup. Typical square wave frequencies lie between 500 and 5000 Hz. The LC directors cannot move far in the brief time between positive and negative voltages, so the directors remain nearly fixed.

Figure 24.41The driving voltages to a liquid crystal cell should be square waves centered on 0 V, not DC voltages.

24.7Limitations on LC Cell Performance

The ideal LCD has many target specifications. It should be very bright for high settings and have a very dark state for low settings. When all the pixels are set to the same level, the color and brightness should be constant across the display. The color should also be constant over viewing angle. The colors should be saturated. The display should switch fast, and modulation at 120 Hz would be ideal. Performance should be consistent over a large range of temperatures. For smartphones, the display should be insensitive to touch and to forces on the front surface. For a mass market, the cell should be inexpensive to manufacture in high volume with excellent yield.

Real liquid crystal cells used in displays must find a balance between these many often conflicting objectives. For LC televisions, two of the driving specifications are the quality of the dark state, and the contrast ratio, the ratio between the bright and dark state levels. Some of the limitations on the dark state and the contrast ratio arise from the following:

• Dispersion of retardance and spectral bandwidth

• Variation of retardance with angle and angular bandwidth

• Scattering and depolarization

• Misaligned polarizers

• Variation of the retardance axis

The retardance magnitude uniformity is limited by the spatial uniformity of the applied electric field and temperature. The retardance orientation uniformity is affected by the alignment layer. Many LC retarders are slightly elliptical retarders, weak diattenuators, and weak depolarizers. Figure 24.42 shows the polarization properties of a single-pixel Fréedericksz cell, an electrically adjustable LC retarder, where less uniformity is observed at low voltage.

Figure 24.42Retardance magnitude, retarder ellipticity, depolarization index, and retardance orientation of an LC retarder with 0 V (left) and 2.5 V (right) applied voltage.

The issues and problems of this small LC component include the following: a slow response time of around 50–80 ms, a fairly high temperature variation about 0.5% per °C, and depolarization of around 1%–10%, which produces scattering. It is very challenging to maintain a uniform thickness and temperature of the LC cell. The non-uniformity shown in Figure 24.13 has a magnitude retardance variation of 14°.

24.7.1LC Cell Speed

The LC cell speed is characterized by the response time measured in milliseconds that a pixel takes to transform from one value to another (typically from 10% to 90% brightness) and then back to the original value, as shown in Figure 24.43. It includes the rise and fall times as the pixel changes from state to state. Since liquid crystal cells modulate retardance by rotating molecules, this is an intrinsically slow modulation mechanism, particularly compared to electro-optical modulators, such as LiNbO3 modulators used in fiber optics, which only need to move electrons within molecules.

Figure 24.43The response time of a liquid crystal can be defined as the time the cell takes to change from 10% to 90% brightness. To increase the response time, the voltage can be increased briefly above the target voltage to accelerate the change. This may result in an overshoot in the target retardance.

When an object is moving across the screen, if the trailing pixels behind the object have a slow response time and do not shift their color quickly, a shadow-trail artifact is observed following the moving object, as shown in Figure 24.44. A slow response time is not acceptable for motion pictures, sports, and video games. The typical response times for current LCD are 8–16 ms for a transition from black-to-white-to-black, and 2–6 ms for a transition from gray-to-gray. Human eyes can perceive differences in response times greater than 5 ms. Because of limitation of the eye (for a 60-Hz frame rate), response times below 10 ms are hard to perceive.

Figure 24.44Example of motion blur due to slow response time.

Several technologies can reduce the response time, motion blurring, and trailing. In the response time compensation (RTC) and the over driving circuit (ODC) technologies, an over-voltage is applied to force the LC molecules to rotate into position faster as shown in Figure 24.45, producing artifacts like Figure 24.46. Overdrive significantly reduces pixel transition time. Double overdrive technology improves the transition times by applying overdrive to both the rise time and the fall time. However, if overdrive is applied aggressively or is controlled poorly, an overdrive color trailing (a pale or dark halo) appears behind the moving object due to the intervening state (Figure 24.47). Without overdrive, typical black-white-black response times are 5 ms for the TN mode, 12 ms for the VA mode, and 16 ms for the IPS mode, where the gray-to-gray transition times are higher. With overdrive, the typical times are 2 ms for the TN mode, 6 ms for the VA, and 5 ms for the IPS.

Figure 24.45(Left) Drive voltage and response times without overdrive and (right) with overdrive.

Figure 24.46(top) A frame from a video of a moving car, and (bottom) the corresponding view from a display with a poor response time blurs the moving objects.

Figure 24.47An example of the artifacts due to a large amount of RTC overshoot. In this example, the RTC overshoot causes a pale halo behind the red moving object, and a dark halo behind the white and yellow moving objects.

24.7.2Spectral Variation of Exiting Polarization State

The refractive index of a liquid crystal is a function of wavelength and polarization state. Figure 24.48 shows the dispersion of a common liquid crystal cell. Each material has a unique dispersion characteristic. Consider an example modulator constructed from a variable linear retarder oriented at 45° to the polarization state generator. On the Poincaré sphere, the polarization state is stretched out along the trajectory and so a polarizer cannot extinguish the whole spectrum at once.

Figure 24.48(Left) Dispersion measurement of an example Fréedericksz cell. (Right) The output polarization state expressed as a Stokes spectrum is shown as a function of wavelengths on a Poincaré sphere.

For maximum polarization state variation, the incident polarization state is 45° from the generator. In this situation, each degree of retardance variation moves the polarization state 1° around a great circle about the retardance axis at 45°. The polarization of the exiting wavelengths will be spread out along the great circle. Thus, each wavelength has a different extinction, the net extinction being an integral over the weighted spectrum. Therefore, for contrast ratio of 100, the dispersion of the retardance cannot be much greater than about twice 5°. Similarly, for a contrast ratio of 10,000, the retardance dispersion cannot be much larger than about 1°, a very tough specification.

For an LC cell, the dispersion is likely to be less at the end of the operating range with the smaller retardance. For a Fréedericksz cell, the dispersion is greater at the low voltage end of its operating range. In contrast, a VAN mode cell is intrinsically near zero retardance at 0 V, so the low voltage setting will be preferable for lowest retardance dispersion.

24.7.3Variation of Retardance with Angle of Incidence

Most liquid crystal devices have substantial angular polarization aberrations. This angular response can be measured by focusing light through the LC cell in an imaging Mueller matrix polarimeter. Consider a monochromatic spherical wave illuminating an LC cell. The LC layer in an LC display is illuminated through a polarizer. If the polarizer’s absorption axis is along y, then the x- and z-components are transmitted, so the incident light has a linear polarization state like latitude vectors in a region about the equator, since latitude vectors have no component along the y-axis. The LC layer behaves as a series of thin birefringent layers, and each angle sees different projections of the director. The retardance varies as a function of angle of incidence, so the exiting polarization state will also show variation with angle. These LC retardance variations can be as large as several degrees of retardance per degree of angle of incidence. This resulting spatially varying polarization state is incident on the analyzing polarizer leading to transmission variations with angle. Because of the LC’s variation of retardance with wavelength, the transmission of colors also varies with wavelength. The underlying variation of retardance with angle and wavelength becomes visible as a variation of color with angle, which is very noticeable to the eye.

The variation of retardance with angle is also observed as a variation of contrast with angle. Hence, contrast is commonly measured as a function of angle of incidence after integrating over red, green, and blue spectral bands. Also, if the transmission axes of the front and back polarizers are not parallel (they are most commonly crossed), the relative angle of the polarizers will vary as the incident beams move diagonally in angle with respect to the polarizers, as shown in Figure 1.16.

For example, the retardance variation measured for a Fréedericksz LC shown in Figure 24.49 is rather large, a 60° linear retardance across the 30° field of view. Each of the three retardance parameters shown has a linear variation with angle. The linear retardance magnitude varies most rapidly in the 45° plane and the retardance orientation rotates most rapidly in the 135° plane. At normal incidence and in the 45° plane, the retardance is linear (the circular retardance is near zero) and the orientation varies in two different directions.

Figure 24.49Variation of retardance of a Fréedericksz cell as a function of angle of incidence over a 30° cone, (a) linear retardance magnitude, (b) linear retardance orientation, and (c) circular retardance magnitude. The cell is illuminated with an NA = 0.55 numerical aperture microscope objective and the transmitted light is collected with another objective. The whole data set is measured simultaneously as a Mueller matrix image.

Figure 24.50 shows the linear retardance magnitude measured for two LC single-pixel untwisted cells at three different applied voltages. The mean values and the peak-to-peak linear retardance of the measurement are shown in Table 24.2, where the first LC is a little less uniform than the second one.

Figure 24.50Linear retardance magnitude plot of two LCs.

Table 24.2The Mean Linear Retardance Magnitude and Peak-to-Peak Values of the Measurement Shown in Figure 24.50

	1900 mV	2700 mV	5500 mV
LC1	Mean value = 193.6° 22.0° peak-to-peak	Mean value = 116.4° 17.2° peak-to-peak	Mean value = 21.7° 9.7° peak-to-peak
LC2	Mean value = 180.8° 7.8° peak-to-peak	Mean value = 79.4° 9.1° peak-to-peak	Mean value = −4.7° 5.0° peak-to-peak

24.7.4Compensating LC Cells’ Polarization Aberrations with Biaxial Films

To address the variation of retardance with angle, the technology of discotic field-widening films was developed. Discotic LC molecules are disk-like or pancake-like negative uniaxial molecules. They are manufactured into multilayer films and used as compensating films to compensate for angle of incidence and wavelength variations of LC cells. The structure of typical discotic molecules is shown in Figure 24.51. In multilayer films, they can be engineered to twist and tilt from layer to layer, as in the example of Figure 24.52, representing a twisted discotic film.

Figure 24.51Typical discotic molecules as a derivative of a hexabenzocoronene and 2,3,6,7,10,11-hexakishexyloxytriphenylene.

Figure 24.52(Left) Schematic of a uniform film of discotic molecules, represented as cylinders. (Right) Schematic of multilayer wide-field film with discotic molecules rotating through 90° from the top to the bottom of the film.

When used as a compensating film, the discotic layers are oriented to cancel the retardance aberration of the LCs. A discotic LC layer paired with a nematic LC layer with opposite uniaxial properties can cancel their polarization, yielding a polarization matrix which is an identity matrix at all angles. Typically, the eigenpolarization of the discotic compensating film has oriented orthogonal to the eigenpolarization of the corresponding LC layer. Figure 24.53 (left) shows a layer of the compensating discotic film (above blue plane) canceling the retardance aberration for a layer of the nematic LC. By pairing the layers as shown by the red arrows (Figure 24.53, middle), near zero retardance is obtained for each pair of layers, yielding near zero retardance for the compensating discotic film plus LC cell pair. Thus, the polarization aberrations can be compensated throughout the entire thickness of an LC cell.

Figure 24.53(Left) Compensating film layer of discotic molecules is shown compensating the adjacent nematic LC layer. (Middle) Each layer of the compensating film compensates on the layer of the LC, reducing the combination to an identity matrix. (Right) Compensating films can be applied to both sides of the LC cell, with each film correcting the aberration of half the cell.

24.7.5Polarizer Leakage

The contrast ratio is also limited by the non-ideal contrast of polarizers. An LC device using polarizers with a contrast ratio of 100:1 cannot exceed a contrast ratio of 100:1. Dichroic polarizers are absorptive, so the extinction depends on the thickness. Polarizing films constructed have several relevant defects. The polarizer material may have wrinkles and pinholes that leak unpolarized light. The polarizer may have areas of thinner dichroic material and reduced contrast ratio. The orientation of the front and rear polarizing films is never parallel. The alignment between the generator and analyzer polarizers will have small spatial variations contributing to a loss of extinction ratio. Figure 24.54 shows the Mueller matrix image of a low-quality polarizing film. The red and blue streaks in the M03, M13, M30, and M31 images indicate variation of the transmission axis.

Figure 24.54Mueller matrix image of poor-quality polarizing film, showing spatial variations of the polarizer’s orientation in vertical streaks.

24.7.6Depolarization

Depolarization is the reduction of the degree of polarization when polarized light interacts with a sample. Depolarization is particularly critical to minimize in LC cells because half of any depolarized light will leak through the analyzer and thus has a particular impact on the dark state and the contrast ratio specification.

Depolarization occurs in all materials, including liquid crystals, due to scattering. Liquid crystals are moderate sized molecules that naturally have some scattering. This scattering is minimized by optimizing the mixtures of LCs and solvents used. Liquid crystals have an additional rather unique contribution to depolarization. Since cells are driven with square wave voltages (Section 24.6.4), time-varying fluctuations in the electric field’s magnitude allow the molecules to vibrate. If the directors vibrate, the retardance vibrates, allowing a time-dependent retardance. A time-averaged retardance and a time-averaged Mueller matrixmanifest themselves as depolarization. Figure 24.55 shows measured depolarization data as a degree of polarization map for a Fréedericksz cell. Such levels of depolarization in the 1% to 3% range are unacceptable in cell phone and LC television displays. For more on depolarization, see Chapter 6.

Figure 24.55A degree of polarization map for a Fréedericksz cell on a flattened Poincaré sphere shows the exiting degree of polarization varies from 0.97 (3% depolarization) for 45°, right, 135°, and left circularly polarized light to 0.99 (1% depolarization) for 0° and 90° light.

Depolarization adversely affects LC display performance differently from incorrect retardance or retardance non-uniformity. When some depolarization is present, a fraction of the exiting light, the depolarized component, can be treated as unpolarized light. Fifty percent of the depolarized light will pass through the analyzer and 50% will be blocked. In the dark state, the leaked depolarized light increases the dark state intensity and, if significant, has a severe effect on the contrast ratio. Thus, for high contrast to be achieved, the LC displays must have very low levels of depolarization. In the white state, half of the depolarized light is blocked by the analyzer decreasing the white state brightness, a less critical problem than dark state leakage. Scattering is a common cause of depolarization in liquid crystals. Liquid crystal depolarization also arises from spatial averaging; micron-scale retardance variations cause adjacent parts of the beam to emerge with different polarization states that average at the polarimeter, resulting in a depolarized component in the measurement. An imaging polarimeter measures the average retardance within each of the polarimeters’ pixels and any subpixel retardance variations are measured as depolarization. Temperature variations, electric field variations, edge effects in pixels, and disclinations in the LC all contribute to depolarization.

24.8Testing Liquid Crystal Cells

Mueller matrix polarimeter testing has become widespread in the LC industry. Polarimeters are used to test the glass for LC cells for the presence of any small birefringence; small birefringences can significantly alter the cell performance, appearing as defects in color and uniformity. Polarimeters and ellipsometers verify that the LC cell’s retardance and orientation are within specifications, so that when the polarizers are attached, the intensity and color modulate properly.15 Mueller matrix polarimeters are used when setting up production, to verify that the production processes stay within specification and to assist in failure analysis when problems do occur.

The cell gap, twist angle, pretilt, and rubbing direction for liquid crystal cells can be measured by measuring retardance as a function of angle of incidence and then fitting these parameters using optimization algorithms. Mueller matrix polarimeters are used to measure the retardance magnitude and retardance eigenstates. The diattenuation and depolarization are also determined and can be helpful in understanding cell performance.

Figure 24.56 shows commercial Mueller matrix polarimeters used for liquid crystal cell testing (left, center) of angle of incidence variations and (right) of spatial uniformity. On the production lines for LC displays and LC TVs, entire LC TV panels, large pieces of films, and large glass plates must be tested. Figure 24.57 shows a Mueller matrix polarimeter for integration into a production line with a capability of scanning over 1.5 m while simultaneously measuring Mueller matrices at three angles of incidence.

Figure 24.56(Left) A Mueller matrix spectropolarimeter for testing LC cells requires a capability to adjust angle of incidence. The top blue component is the polarization state generator. A fiber enters the top, bringing light from a monochromator. The bottom blue component is the polarization state analyzer. The sample is placed on the black tray, which tilts to vary angle of incidence. (Center) A closeup of the pair of rotation stages for angle of incidence testing. The silver rotary stage on the right adjusts the angle of incidence. The black round rotary stage on the left side varies the azimuth of the angle of incidence, allowing tests to be performed in different planes. (Right) A Mueller matrix spectropolarimeter configured for spatial uniformity mapping has xy- translation stages (silver) to move the sample holder, the black rectangle). (Courtesy of Axometrics Inc., Huntsville, Alabama.)

Figure 24.57(Left) A Mueller matrix polarimeter for production testing of large-screen LCD TVs and for the stress birefringence testing of the associated glass. The head can scan over 1.5 m while the part under test is carried through the fixture from front to back. (Right) Closeup of a three-head polarimeter, for high-speed testing of large-screen LCD TVs. Testing is necessary on- and off-axis, so using three heads is far faster than scanning a single head in angle of incidence. Rotary stages above and below the polarimeter heads allow the fixtures to rotate about the vertical axis to vary the azimuth. (Courtesy of Axometrics Inc., Huntsville, Alabama.)

24.8.1Twisted Nematic Cell Example

An example of LC cell testing is shown for “cell A” in Figure 24.58. The Mueller matrix spectrum of the cell at normal incidence has been measured (not shown) and the retardance properties have been calculated. The retardance eigenstates are plotted on the Poincarè sphere; for best cell performance, ideally the eigenstates would all be located at left circularly polarized light, but cells typically have some spectral variation. To measure the cell gap and director orientations through the cell, the Mueller matrix is measured as the cell is tilted through normal incidence in multiple planes. The retardance eigenstates of “cell B” shown in Figure 24.59 have a quadratic variation with superior spectral performance. These functions can be fitted by optimizing the cell gap and twist angles in an optimization program to measure the cell’s gap and twist angle.

Figure 24.58Retardance eigenstates of cell A, a twisted nematic (TN) cell as a function of wavelength, are designed to pass through left circularly polarized light unchanged. Green circularly polarized light is unchanged, but other wavelengths vary linearly with wavelength. (Courtesy of Axometrics Inc., Huntsville, Alabama.)

Figure 24.59Tilt test where color indicates the variation of angle of incidence as the cell rotates in a fixture in the polarimeter (see Figure 24.56 left and center). The retardance eigenstates for a TN cell (left) at 500 nm and (right) at 550 nm shows quadratic and very small variation of their retardance eigenstates. (Courtesy of Axometrics Inc., Huntsville, Alabama.)

24.8.2IPS Tests

An example of a Mueller matrix spectropolarimetric test of an IPS cell at normal incidence is shown in Figure 24.60. The configuration of the cell is shown in Figure 24.26 (left). All the LC directors are parallel, so the retardance spectrum as a function of wavelength is constant at low voltage. Figure 24.61 shows tilt scans of the retardance for a cell without pretilt, compared to Figure 24.62 for a cell with pretilt, indicating how such tests are used to set up and maintain production machinery.

Figure 24.60(Left) Retardance eigenstates of a well-aligned IPS cell at normal incidence are constant with wavelength. (Right) Introducing a small tilt off normal incidence introduces a slight ellipticity at all wavelengths. (Courtesy of Axometrics Inc., Huntsville, Alabama.)

Figure 24.61Retardance measurements of tilt scans of an IPS cell with photo aligned alignment layers (left) along the alignment direction and (right) perpendicular to the rubbing direction are centered at 0°; this symmetry indicates no pretilt in the polyimide alignment layer. (Courtesy of Axometrics Inc., Huntsville, Alabama.)

Figure 24.62Retardance measured from tilt scans of an IPS cell with rubbed alignment layers (left) along the rubbing direction and (right) perpendicular to the rubbing direction is not centered at 0°; the asymmetry indicates a pretilt in the polyimide alignment layer in this case of about 2°. (Courtesy of Axometrics Inc., Huntsville, Alabama.)

24.8.3VAN Cell

Another example of pretilt determination using polarimetry is shown in Figure 24.63 for a vertically aligned cell, similar to Figure 24.20 (left). The cell’s retardance is measured as a function of angle of incidence. The cell of Figure 24.61 has a pretilt of 90°; thus, the retardance is symmetric. The cell of Figure 24.62 has a pretilt of 89° and an asymmetry in the retardance about the origin is observed.

Figure 24.63Retardance versus tilt angle for a VAN cell (left) with a pretilt of 90° is symmetric, while (right) a pretilt of 89° yields an asymmetric tilt scan. (Courtesy of Axometrics Inc., Huntsville, Alabama.)

24.8.4MVA Cell Test

By simulating the measurement using methods shown in Section 24.5 and converting the result to Mueller matrices, the measured data can be fitted to the simulation result. The best-fit model parameters (cell gap, twist angle, etc.) are calculated iteratively for the smallest RMS difference between the simulated and the measured Mueller matrices.

The multi-angle Mueller matrix imaging technique can determine the 3D structure of the LC directors’ arrangement in the LC cell. There is fundamentally no limit to the cell gap and pretilt ranges that can be measured. The cell can be positioned in any orientation. By scanning the LC cell with a high-resolution imaging polarimeter, the polarization behavior within an LCD pixel can be studied. It is possible to design a multi-domain pixel, investigate the LC behavior near patterned electrodes, and analyze bad and damaged pixels. Figure 24.64 shows the parameter map of an MVA LCD pixel.

Figure 24.64The measured parameters of an MVA LCD pixel. (Left) The intensity image with four subpixel regions indicated. (Center) An LC layer thickness map calculated from multi-angle Mueller matrix images. (Right) The orientation map for the alignment layers. (Courtesy of Axometrics Inc., Huntsville, Alabama.)

24.8.5Sheet Retarder Defect

The retarders incorporated into LC displays are usually produced at high speed in wide sheets by stretching plastic or by spraying self-aligning molecules onto a moving substrate. For quality control, retardance must be monitored for defects. Figure 24.65 shows an example of how a microscopic defect on a compensating film can appear in a false color retardance image due to non-uniform deposition of the retarding layer.

Figure 24.65Appearance of a defect in an LC compensating film produced on a continuous web process shown in false color retardance. (Courtesy of Axometrics Inc., Huntsville, Alabama.)

24.8.6Misalignment between Analyzer and Exiting Polarization State

Extinction of a polarized state is described by a generalization of Malus’ law. On the Poincaré sphere, the transmission of a state through an ideal polarizer is given by sin2ψ, where ψ is the angle between the polarization state and the analyzer’s extinction axis. As the output state moves past the extinction axis, it is unlikely to pass exactly through the extinction axis but is likely to have some minimum ψ. Thus, getting a good dark state and consistently high contrast ratio involves keeping the polarization trajectory close to the extinction axis. For a contrast ratio of 100, ψ must be less than 0.1 radians or 5.7°, which is not difficult to achieve. A contrast ratio of 10−4 requires a ψ less than 0.01 radians.

24.9Problem Sets

1. Address the following performance factors of LC cells.

   1. What is the principal factor that limits the speed of liquid crystal cells?

   2. Why do liquid crystal cells have some depolarization?

   3. Why are liquid crystal cells cheaper than other retardance modulators?

   4. How can liquid crystal cells be configured as phase-only modulators?

   5. List five disadvantages of liquid crystal cells.

   6. List five advantages of liquid crystal cells.

2. A certain twisted nematic liquid crystal cell can be modeled in its off state by 91 parallel layers, each with a retardance of 0.060463 radians, with the fast axis of the first layer at 0°, the second layer at 1°, the third layer at 2°, and so on, until the 91st layer is oriented at 90°.

   1. Calculate the Jones matrix J for transmission through the cell.

   2. What are the eigenpolarizations of the cell?

   3. How can this LC cell and two linear polarizers be oriented to produce a dark state?

   4. How can this LC cell and two linear polarizers be oriented to produce a bright, fully transmitting state?

   5. Find the Jones matrix J1, eigenpolarizations, and retardance for the first half of the cell, layers 1 to 45. Draw the ellipse for at least one of the eigenpolarizations.

   6. Find the Jones matrix J2, eigenpolarizations, and retardance for the second half of the cell, layers 46 to 91. Draw the ellipse for at least one of the eigenpolarizations.

   7. Compare the polarization of the first and second half of the cell. Explain with Pauli matrices how the Jones matrices J1 and J2 combine to form J.

3. Compare the expression for a series of weak polarization elements with the exact equation for the following liquid crystal structure. Light is propagating in the +z-direction. A twisted nematic cell is modeled as 60 layers of linear retarders. Each layer is a linear retarder with retardance π/300. The retarder axes (the directors) rotate in the x–y plane through 360° such that the axis is along the line (cos θq, sin θq, 0), θq = q × 6° for q = 1 to 60.

   1. List the Jones matrices in the order they will be multiplied. Then, multiply the linear retarders in the correct order and calculate the Jones matrix JLC for the entire sequence.

   2. Express JLC as in normalized Pauli matrix form.

   3. Express the first two retarders J1 and J2 in weak polarization element form. These have 6° and 12° directors.

   4. Make a table of the σ1 components for each of the 60 retarders. Sum the components.

   5. Make a table of the σ2 components for each of the 60 retarders. Sum the components.

   6. Obtain a weak polarization element expression for the sequence of weak Jones matrices. Compute the polarization properties of the sequence.

Acknowledgment

The authors wish to thank Jon Herlocker, David Serrano, and Matt Smith, who provided materials and assistance in the preparation of this chapter.

References

1J. A. Castellano, Liquid Gold: The Story of Liquid Crystal Displays and the Creation of an Industry, World Scientific (2005).

2C. H. Gooch and H. A. Tarry, The optical properties of twisted nematic liquid crystal structures with twist angles ≤ 90 degrees, J. Phys. D 8.13 (1975): 1575.

3L. Vicari, (ed.), Optical Applications of Liquid Crystals, CRC Press (2016).

4N. Konishi, K. Kondo, and H. Mano, 34-cm Super TFT-LCD with wide viewing angle, Hitachi Rev. 45.4 (1996): 165–172.

5S. Aratani et al., Complete suppression of color shift in in-plane switching mode liquid crystal displays with a multidomain structure obtained by unidirectional rubbing. Jpn. J. Appl. Phys. 36.1A (1997): L27.

6Y. Momoi, O. Sato, T. Koda, A. Nishioka, O. Haba, and K. Yonetake, Surface rheology of rubbed polyimide film in liquid crystal display, Opt. Mater. Express 4 (2014): 1057–1066.

7S. He, J.-H. Lee, H.-C. Cheng, J. Yan, and S.-T. Wu, Fast-response blue-phase liquid crystal for color-sequential projection displays, J. Disp. Technol. 8 (2012): 352–356, 10.1109/JDT.2012.2189434.

8P. Yeh, and C. Gu, Optics of Liquid Crystal Displays, Chapter 8, 2nd edition, John Wiley & Sons (2010).

9D.-K. Yang and S.-T. Wu, Fundamentals of Liquid Crystal Devices, 2nd Edition, John Wiley & Sons (2015).

10D. W. Berreman, Optics in stratified and anisotropic media: 4 × 4-matrix formulation, J. Opt. Soc. Am. 62(4), 502–510 (1972).

11I. J. Hodgkinson and Q. H. Wu. Birefringent Thin Films and Polarizing Elements, World Scientific, Singapore (1997).

12M. Mansuripur, Effects of high-numerical-aperture focusing on the state of polarization in optical and magneto-optic data storage systems, Appl. Opt. 30.22 (1991): 3154–3162.

13V. G. Chigrinov, V. M. Kozenkov, and H.-S. Kwok, Photoalignment of Liquid Crystalline Materials: Physics and Applications, Vol. 17, John Wiley & Sons (2008).

14F. S. Yeung, J. Y. Ho, Y. W. Li, F. C. Xie, O. K. Tsui, P. Sheng, and H. S. Kwok, Variable liquid crystal pretilt angles by nanostructured surfaces, Appl. Phys. Lett. 88(5) (2006): 051910.

15S. T. Tang and H. S. Kwok, Transmissive liquid crystal cell parameters measurement by spectroscopic ellipsometry, J. Appl. Phys. 89.1 (2001): 80–85.