Selective Reflection in Cholesteric Liquid Crystals

Cholesteric liquid crystals exhibit a helical structure where the molecular orientation follows a spiral. If the pitch of the helix is denoted as p, the optical periodicity is half of that, p/2. Due to this periodic structure with a p/2 spacing, cholesteric liquid crystals possess a reflection band similar to that of a multi-layered thin film. Let ne be the refractive index along the long axis of the molecules and no that along the short axis. The reflection band for light incident parallel to the helical axis falls within the wavelength range from p* no to p* ne. Since the reflection band corresponds to the p/2 periodicity, it is called as the half-pitch band.

If a reflection band corresponding to the full pitch p (a full-pitch band) were to exist, its wavelength range would be twice that of the half-pitch band. However, in cholesteric liquid crystals, a full-pitch band does not appear. This is because the reflected light from the p/2 and the p periodicity cancel out each other.

If the optical structures at 0° and 180° differ, a full-pitch band can appear. For example, the C-director in SmC liquid crystals has polarity, so there is a distinction between 0° and 180°. However, for light incident perpendicular to the layer, the refractive indices at 0° and 180° are equal, so even in the SmC* phase, only the half-pitch band appears when light is incident from the direction of the layer normal. For light incident at an angle to the layer normal, the refractive indices differ at 0° and 180°, resulting in the appearance of the full-pitch band in addition to the half-pitch band. In antiferroelectric phases such as SmCA* phase, a full-pitch band is not observed, even with oblique incidence. Therefore, the appearance of a full-pitch band with oblique incidence serves as one method to distinguish between SmCA* and SmC* phases.

Cholesteric liquid crystals and SmC* liquid crystals exhibit two key features related to selective reflection. The first is the relationship between the handedness of the liquid crystal’s helix and the reflected circularly polarized light. If the liquid crystal helix is right-handed, it reflects only right circularly polarized light, while left circularly polarized light is transmitted without reflection. Conversely, if the helix is left-handed, it reflects left circularly polarized light and transmits right circularly polarized light. It should be noted that the sign of circular polarization is defined according to the conventions of optical researchers (not the definition by physicist).

The second feature is that the handedness of the reflected circularly polarized light remains the same as that of the incident circularly polarized light. In a typical mirror, right circularly polarized light is reflected as left circularly polarized light and vice versa. However, in cholesteric liquid crystals, a right-handed helix reflects right circularly polarized light as right circularly polarized light, and a left-handed helix reflects left circularly polarized light as left circularly polarized light.

Here, we undertake the bold attempt to explain the selective reflection mechanism of cholesteric liquid crystals without resorting to Maxwell's equations or similar complex mathematical tools.

Reflection by a Rod-like Reflector

Let us consider a rod-shaped reflector. This reflector responds only to the polarization of incident light that is parallel to the long axis of the rod, emitting waves in both forward and backward directions that are in phase with the incident wave.　

In the figure, waves are emitted in both directions with amplitudes nearly equal to that of the incident wave, but this is for clarity. In reality, the reflectivity of a single rod is quite low, and nearly all of the light passes through. However, if we arrange these rods along the direction of light propagation with a spacing of λ/2, interference will cause light of wavelength λ to be reflected. While the rod-like reflector actually emits secondary waves in all directions within the plane perpendicular to the rod, this is ignored here, as it is sufficient for understanding the phase relationship of the reflected light. In the figure above, waves emitted in the same direction as the incident light are also shown, but to simplify the illustration, we will focus only on the backward-reflected light.

Response from Crossed Rods in the Same Plane

Next, let’s consider two rods placed at the origin of the coordinate system, arranged in the x-direction and y-direction, respectively, within the plane perpendicular to the direction of light propagation. Now, linearly polarized light with a 45-degree angle is incident. The phase of the incident light is the same for both x-polarized and y-polarized components, meaning there is no phase difference. Upon reflection, there is no change in the phase difference, so the reflected light remains linearly polarized, with the same orientation as the incident polarization.

Physically, light with the same polarization plane is reflected, but in optical terms, the coordinate system must be maintained as a right-handed system for both incidence and reflection. As a result, one of the coordinate axes is inverted, and the phase of the wave along that axis is also inverted. If the vibration plane of the incident light lies in the first and third quadrants, the vibration plane of the reflected light will lie in the second and fourth quadrants. This functions similarly to a half-wave plate.

Incident Right Circularly Polarized Light

The diagram depicts light propagating towards the xy-plane, with the trajectory of the electric field vector crossing the plane perpendicular to the propagation direction in a counterclockwise manner. When right circularly polarized light is reflected by the rod structure, its phase does not change, but the direction of propagation reverses. As a result, the backward-propagating light maintains the same rotation as the incident light, and the trajectory in the plane parallel to the xy-plane remains counterclockwise. By definition, this circular polarization is now left-handed. The fact that the reflection of linearly polarized light acted like a half-wave plate and that right circularly polarized light was converted into left circularly polarized light both result from the phase inversion along one of the axes, producing a consistent outcome.

Response when One Rod is Shifted by a Quarter Wavelength

Next, we consider the case where the rod in the y-direction is positioned behind the rod in the x-direction by a distance of λ/4. Since the rod in the y-direction is λ/4 behind the x-oriented rod, the returning light from the y-direction is delayed by λ/2 relative to the x-direction.

Let’s analyze the reflection of linearly polarized light with this structure. In the incident light, the x-component and y-component are in phase. In the reflected light, the y-component lags the x-component by 180°. As a result, the vibration plane of the reflected linearly polarized light becomes symmetrically opposite to the incident polarization with respect to the x-axis (or y-axis). Physically, this is equivalent to the action of a half-wave plate. However, due to the sign inversion of one of the coordinate axes, the reflected wave remains in the same tilted plane as the incident wave when considered in the reflected coordinate system. Thus, this structure preserves the polarization state after reflection.

Reflection of Circularly Polarized Light。

Next, we consider the reflection of circularly polarized light. In this case, as with linearly polarized light, the phase of the y-component is delayed by 180°. This delay cancels out the 180° phase shift due to the sign inversion during reflection, so the handedness of the circular polarization remains unchanged. This can explain one of the features of selective reflection in cholesteric liquid crystals. However, in this structure, both right and left circularly polarized light are reflected while preserving their handedness. To explain the other feature of selective reflection, a more complex structural model is needed.

Upon reflection, the structure where the rod in the y-axis is shifted by λ/4 has a mirror plane. To selectively reflect only one type of circular polarization, we need to consider a chiral structure without a mirror plane.

Chiral Structure

Now, let’s consider a structure where a rod is added at a position λ/8 in front of the x-axis rod, at an angle of +135°, and another rod at 3λ/8 at an angle of 45°. We also consider a structure where a rod is added at λ/8 in front of the x-axis rod at an angle of +45°, and another rod at 3λ/8 at an angle of 135°. The first structure corresponds to a right-handed twist, and the second to a left-handed twist. First, let’s examine the relationship between the backward-propagating light from the 0-λ/4 combination and the λ/8-3λ/8 combination for the right-handed twist.

Both return light waves are in phase, indicating that right circularly polarized light incident on this structure is reflected without any issues. Now, let’s consider the left-handed twist.

In this case, while the return light from the λ/8-3λ/8 combination has the same handedness as the return light from the 0-λ/4 combination, their phases differ by 180°, causing them to cancel each other out. This means that the left-handed twist structure does not reflect right circularly polarized light.

Although not shown in the diagram, for left circularly polarized light, the left-handed twist structure returns light in phase, whereas the right-handed twist structure results in phase-inverted reflected light.