Jones’ calculus

My own introduction to this field came during the course of my graduate studies when my mentor decided that I needed to learn the Jones calculus for treating optical phenomena. The department also possessed a Cary 60 spectrometer system, and in the time between my final oral exam and the beginning of my postdoctoral work I investigated the induction of circular dichroism in several metal complexes of acetylacetone by various chiral agents. Little did I know at the time that this particular work would subconsciously prepare me for the writing of one of the chapters in this book. 

The Jones Calculus is based on the physical fact that any optical element will act upon the electric light vector of the incident wave (Ein) as a linear operator. The operator is expressed in the convenient form of a two-by—two matrix, whose four matrix elements are, in general, complex. 

Liquid crystals are birefringent and they therefore influence the state of polarization of light beams. This interaction is described by the Jones calculus [2], which is briefly outlined below, prior to using it to establish basic results in light modulation by ferroelectric liquid crystals. 

As discussed previously, the electric field vector can assume various polarization states. Jones calculus is a method to treat propagation and evolution of these polarization states in an anisotropic crystal, which will impart various phase shifts to the principal axes components of the electric field. We begin by defining the Jones vector ... 

If A is an integer multiple of tt, then the total wave is linearly polarized in the AxX + Ayf direction. Otherwise, the wave is elliptically polarized with the special case of circular polarization when Ax = Ay. Although Jones calculus cannot represent unpolarized light in a simple way, Jones realized that all of the information in polarized light can be represented by a complex Jones vector, J = or 2 x 1 matrix, where ( ) = ln zl + vf) + Bx + ey)/2. 

Richard Foster Jones, by M. Nicolson.—Essays, by R. F. Jones.—A bibliography of the published writings of Richard Foster Jones.—The invention of the ethical calculus, by L. 

References Ablowitz, M. J., and A. S. Fokas, Complex Variables Introduction and Applications, Cambridge University Press, New York (2003) Asmar, N., and G. C. Jones, Applied Complex Analysis with Partial Differential Equations, Prentice-Hall, Upper Saddle River, N.J. (2002) Brown, J. W., and R. V Churchill, ComplexVariables and Applications, 7th ed., McGraw-Hill, New York (2003) Kaplan, W., Advanced Calculus, 5th ed., Addison-Wesley, Redwood City, Calif. (2003) Kwok, Y. K., Applied Complex Variables for Scientists and Engineers, Cambridge University Press, New York (2002) McGehee, O. C., An Introduction to Complex Analysis, Wiley, New York (2000) Priestley, H. A., Introduction to Complex Analysis, Oxford University Press, New York (2003). 

Transmission by Anisotropic Media The Jones and Mueller Calculus... 

A macroscopic, quantitative way to analyze birefringence is through Jones (33-40) and Mueller (41) calculus and the Poincare sphere (42). Those tools are widely used in the optical industry and references on them abound. Another active field of birefringence applications is the combination of the technique with polymer rheology, called flow-birefringence (43,44). 

Au contraire to the empirical equation of Tait for EOS predictions, theoretical models can be used but generally require an understanding of forces between the molecules. These laws, strictly speaking, need be derived from quantum mechanics. However, Lenard-Jones potential and hard-sphere law can be used. The use of statistical mechanics is an intermediate solution between quantum and continuum mechanics. A canonical partition function can be formulated as a sum of Boltzmann s distribution of energies over all possible states of the system. Necessary assumptions are made during the development of the partition function. The thermodynamic quantities can be obtained by use of differential calculus. For instance, the thermodynamic pressure can be obtained from the partition function Q as follows ... 

R.C. Jones. A new calculus for the treatment of optical systems. Journal of Optical Society America, 31(7) 488-493, July 1941. 

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