The previous section introduced the concepts of polarized light and polarizers. This section will show how these ideas are important to liquid crystals.
Liquid crystals are found to be birefringent, due to their anisotropic nature. That is, they demonstrate double refraction (having two indices of refraction). Light polarized parallel to the director has a different index of refraction (that is to say it travels at a different velocity) than light polarized perpendicular to the director. In the following diagram, the blue lines represent the director field and the arrows show the polarization vector.
Thus, when light enters a birefringent material, such as a nematic liquid crystal sample, the process is modeled in terms of the light being broken up into the fast (called the ordinary ray) and slow (called the extraordinary ray) components. Because the two components travel at different velocities, the waves get out of phase. When the rays are recombined as they exit the birefringent material, the polarization state has changed because of this phase difference.
Light traveling through a birefringent medium will
take one of two paths depending on its polarization.
The birefringence of a material is characterized by the difference, Dn, in the indices of refraction for the ordinary and extraordinary rays. To be a little more quantitative, since the index of refraction of a material is defined as the ratio of the speed of light in a vacuum to that in the material, we have for this case, ne = c/V| | and no = c/V^ for the velocities of a wave travelling perpendicular to the director and polarized parallel and perpendicular to the director, so that the maximum value for the birefringence, Dn = ne no. We wont deal here with the general case of a wave travelling in an arbitrary direction relative to the director in a liquid crystal sample, except to note that Dn varies from zero to the maximum value, depending on the direction of travel. The condition ne > no describes a positive uniaxial material, so that nematic liquid crystals are in this category. For typical nematic liquid crystals, no is approximately 1.5 and the maximum difference, Dn, may range between 0.05 and 0.5.
The length of the sample is another important parameter because the phase shift accumulates as long as the light propagates in the birefringent material. Any polarization state can be produced with the right combination of the birefringence and length parameters.
It is convenient here to introduce the concept of optical path in media since for the above two wave components travelling with different speeds in a birefringent material, the difference in optical paths will lead to a change in the polarization state of the wave as it progresses through the medium. We define the optical path for a wave travelling a distance L in a crystal as nL so that the optical path difference for the two wave components mentioned above will be L (ne no) = LDn. The resultant phase difference between the two components (the amount by which the slow, extraordinary component lags behind the fast, ordinary one) is just 2p LDn/lv where lv is the wavelength in vacuum.
The following simulation demonstrates the optical properties of a birefringent material. A linearly polarized light wave enters a crystal whose extraordinary (slow) index of refraction can be controlled by the user. The length of the sample can also be varied, and the outgoing polarization state is shown. The concept of optical path difference and its influence on polarization state can also be explored here. This leads to a discussion of optical retardation plates or phase retarders, in the context of the simulation.
Application to Polarized Light Studies of Liquid Crystals
Consider the case where a liquid crystal sample is placed between crossed polarizers whose transmission axes are aligned at some angle between the fast and slow direction of the material. Because of the birefringent nature of the sample, the incoming linearly polarized light becomes elliptically polarized, as you have already found in the simulation. When this ray reaches the second polarizer, there is now a component that can pass through, and the region appears bright. For monochromatic light (single frequency), the magnitude of the phase difference is determined by the length and the birefringence of the material. If the sample is very thin, the ordinary and extraordinary components do not get very far out of phase. Likewise, if the sample is thick, the phase difference can be large. If the phase difference equals 360 degrees, the wave returns to its original polarization state and is blocked by the second polarizer. The size of the phase shift determines the intensity of the transmitted light.
If the transmission axis of the first polarizer is parallel to either the ordinary or extraordinary directions, the light is not broken up into components, and no change in the polarization state occurs. In this case, there is not a transmitted component and the region appears dark.
In a typical liquid crystal, the birefringence and length are not constant over the entire sample. This means that some areas appear light and others appear dark, as shown in the following microscope picture of a nematic liquid crystal, taken between crossed polarizers. The light and dark areas that denote regions of differing director orientation, birefringence, and length.
Image courtesy of E. Merck Company
The Schlieren texture, as this particular arrangement is known, is characteristic of the nematic phase. The dark regions that represent alignment parallel or perpendicular to the director are called brushes. The next section will describe the textures of liquid crystals in greater detail, but before going there lets see how birefringence can lead to multicolored images in the examination of liquid crystals under polarized white light.
Colors Arising From Polarized Light Studies
Up to this point, we have dealt only with monochromatic light is considering the optical properties of materials. In understanding the origin of the colors which are observed in the studies of liquid crystals placed between crossed linear polarizers, it will be helpful to return to the examples of retarding plates discussed in the Birefringence Simulation. They are designed for a specific wavelength and thus will produce the desired results for a relatively narrow band of wavelengths around that particular value. If, for example, a full-wave plate designed for wavelength is ló is placed between crossed polarizers at some arbitrary orientation and the combination illuminated by white light, the wavelength ló will not be affected by the retarder and so will be extinguished (absorbed) by the analyzer. However, all other wavelengths will experience some retardation and emerge from the full-wave plate in a variety of polarization states. The components of this light passed by the analyzer will then form the complementary color to ló.
Color patterns observed in the polarizing microscope, together with the extinctions already noted in the connection with the Birefringence Simulations are very useful in the study of liquid crystals in many situations, including the identification of textures, of liquid crystal phases and the observations of phase changes.
The following simulation illustrates the role of birefringence in the formation of colored images by a liquid crystal sample located between crossed linear polarizers when observed, for example, in a microscope. The simulation allows you to adjust the birefringence, the length, and the orientation, q, of the liquid crystal sample. Here, q is the angle between the director and the vertical direction (The transmission direction of the polarizer).
Temperature Dependence of Birefringence
Recall that the birefringence of a material results from its anisotropy and the anisotropy of liquid crystals show a strong temperature dependence, vanishing at the nematic to isotropic phase transition. Hence, the birefringence shows a significant temperature dependence. This is discussed and illustrated in the following simulation.