Light propagation, problems

Nonintuitive Light Propagation Effects In Third-Order Experiments. One of the first tasks for a chemist desiring to quantify second- and third-order optical nonlinear polarizability is to gain an appreciation of the quantitative manifestations of macroscopic optical nonlinearity. As will be shown this has been a problem as well for established workers in the field. We will present pictures which hopefully will make these situations more physically obvious. 

Some of the unsolved problems in contemporary electrodynamics draw attention to deeper (more profound) evidence, new ideas and new theories or equations. The aim of this historical introduction is to find the deeper evidence and new basic concepts and connections. The guiding principle is the investigation of light propagation. 

The problem at hand is pictured in Figure 2.10. Here light propagates along the wavevector k, which is taken to be parallel to the unit vector e3. This vector is rotated... 

The preceding analysis has been appropriate to the study of propagation and to the study of continuous-wave optical processes and devices. For comparison, we will review a simpler approach that can be rather securely applied to the interaction of pulsed light fields, but that had better not to be used for study of propagation problems. The modal approach consists in considering annihilation operators Aj(z), j = 1,2, related to the pulses centered at the frequencies coy, j = 1,2. It is assumed that these operators obey the quasiclassical equations... 

We have not discussed here two other problems. One is related to what can be really measured. When we look for a change in the distance, we usually mean that we look for a change in its numerical value in some units. The interpretation would strongly depend on what kind of clocks we use (the measurement of the distance is usually a measurement of light-propagation time) and on our assumptions on what can happen with the value of speed of light c. 

Here, we consider as an example a possible problem with an interpretation of a classical version of the Michelson-Morley experiment. In the experiment some pieces of bulk matter were rotated. It was expected that when rotating their linear scale would not change and comparing the light propagation in different arms of the interferometer we can judge whether the speed of light is the same in different directions. 

The problem of light propagation becomes much more complicated in spatially inhomogeneous liquid crystal layers. There is no general method to solve the Maxwell equations for an arbitrary director distribution or, more generally speaking, for an arbitrary spatial dependence of the dielectric tensor. On the other hand in some important special cases exact solutions were found and useful approximations were worked out for other conditions. In this section we survey these results. 

The mathematical description of light propagation in an arbitrary direction relative to the helix is more laborious. Several authors dealt with this problem. It was found that the central wavelength of the reflection band can be given, in analogy to Bragg s law for X-ray diflFraction, by the relation... 

In order to find such an approximation we return for a moment to the problem of light propagation in a helical structure. In the foregoing discussion the limit of long pitches, A/p C — n, was excluded. As a matter of fact this limit was considered as early as 1911 by Maugin and was treated by de Vries also. They showed that in this regime the normal modes become linearly polarized one is... 

Zheleznyakov et al considered the problem of light propagation in an inhomogeneous cholesteric. They started from the normal modes corresponding to the local pitch of the helix. Spatial variations of the pitch were taken into account as couplings between the normal waves in a similar way as described above. 

A serious problem in the measurement of stresses in glass is the fact that the stress field is rarely uniaxial. One usually needs to consider the effect of a biaxial stress field perpendicular to the direction of light propagation (a stress component along the direction of light propagation has no effect). The stress-optic relation has to be modified to read... 

For arbitrary angle 4) or director axis angular and spatial distributions, and more complicated cell structure, the phase shift, and therefore the transmission of light through the cell and other accompanying polarization selective elements, is not amenable to simple analytical treatment. More sophisticated Jones matrix methods or nmnerical technique such as the finite difference time domain (FDTD) nmnerical methods discussed in the next chapter are needed to solve such a complex propagation problem. 

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