R. Clark Jones

A New Calculus for the Treatment of Optical SystemsI. Description and Discussion of the Calculus

The effect of a plate of anisotropic material, such as a crystal, on a collimated beam of polarized light may always be represented mathematically as a linear transformation of the components of the electric vector of the light. The effect of a retardation plate, of an anisotropic absorber (plate of tourmaline; Polaroid sheeting), or of a crystal or solution possessing optical activity, may therefore be represented as a matrix which operates on the electric vector of the incident light. Since a plane wave of light is characterized by the phases and amplitudes of the two transverse components of the electric vector, the matrices involved are two-by-two matrices, with matrix elements which are in general complex. A general theory of optical systems containing plates of the type mentioned is developed from this point of view.

A New Calculus for the Treatment of Optical Systems. IV.

Part IV is divided into two sections. The first is devoted to some additions to the general theory developed in Part I, and the second section to the derivation of the matrices representing two optical elements which were not treated in Parts II and III: (1) plates possessing circular dichroism, and (2) plates cut from crystals of such low symmetry that the principal axes of absorption and refraction are not parallel. In case (2), the discussion is limited to monoclinic and triclinic crystals which do not possess optical activity.

A New Calculus for the Treatment of Optical Systems. VII. Properties of the N-Matrices

The preceding papers of this series have examined the properties of an optical calculus which represented each of the separate elements of an optical system by means of a single matrix M. This paper is concerned with the properties of matrices, denoted by N, which refer not to the complete element, but only to a given infinitesimal path length within the element.If M is the matrix of the optical element up to the point z, where z is measured along the light path, then the N-matrix at the point z is defined by (A)N≡(dM/dz)M-1.Thus one may write symbolically, (B)N=dlogM/dz,and (C)M=M0 exp(∫Ndz).A general introduction is contained in Part I. The definition and general properties of the N-matrices are treated in Part II. Part III contains a detailed discussion of the important special case in which the optical medium is homogeneous, so that N is independent of z; Part III contains in Eq. (3.26) the explicit relation which corresponds to the symbolic relation (C). Part IV describes a systematic method, based on the N-matrices, by which the optical properties of the system at each point may be described uniquely and quantitatively as a combination of a certain amount of linear birefringence, a certain amount of circular dichroism, etc.; the method of resolution is indicated in Table I. Part V treats the properties of the inhomogeneous crystal which is obtained by twisting a homogeneous crystal about an axis parallel to the light path.

A New Calculus for the Treatment of Optical SystemsII. Proof of Three General Equivalence Theorems

The general theory developed in Part I is used to prove three equivalence theorems about optical systems of the type under discussion. We prove that any optical system which contains only retardation plates and rotators is optically equivalent to a system containing only two plates—one a retardation plate, and the other a rotator. We then prove an exactly analogous theorem for systems containing only partial polarizers and rotators. Finally, it is proved that the most general optical system which contains any number of all three types of plates is optically equivalent to a system containing at most four plates—two retardation plates, one partial polarizer, and one rotator.

A New Calculus for the Treatment of Optical SystemsIII. The Sohncke Theory of Optical Activity

Reusch and Sohncke have examined the properties of a system containing a large number n of identical retardation plates, each of which is rotated with respect to the one preceding it through the angle ω. The product of n and ω must be equal to μπ, where μ is an integer. Under certain conditions this system is optically equivalent to a simple rotator. This system, which would be very difficult to examine by ordinary methods, is given a treatment which is completely rigorous and which is more general than any given heretofore.

The General Theory of Bolometer Performance

A general phenomenological theory of the static and dynamic behavior of bolometers is presented. The theory assumes as given the fundamental relations between the temperature and resistance of the bolometer, and the past history of the power dissipated within it. From these basic properties are derived a number of the properties of more immediate interest, such as electrical impedance, responsivity as a function of frequency, and the static load curve.Several equivalent circuits are developed to represent the behavior of the bolometer as a function of frequency at a single operating point. A two-terminal equivalent circuit is described that represents the electrical impedance as a function of frequency. In order to represent the response of the bolometer to incident radiation as a function of frequency, a four-terminal equivalent circuit is described.An electrical bridge is described that permits one to measure by purely electrical means the electrical response that a bolometer would have to radiation of any given time dependence, including radiation that varies sinusoidally. By purely electrical means and without the need of a radiation source (calibrated or otherwise), the bridge provides a precise measurement of the bolometer’s responsivity (output volts per watt of incident radiation) as a function of frequency. An electrical signal S(t) at the input of the bridge produces the same electrical output as would be produced in the normal use of the bolometer by a radiation signal with the same wave form as S(t).The presentation is in three parts: static performance; stability; and dynamic performance.

A New Calculus for the Treatment of Optical SystemsVI. Experimental Determination of the Matrix

A simple, straightforward method is described for determining experimentally the matrix of any, crystalline plate. The method involves three measurements of the state of polarization of light transmitted by the plate, and also a measurement of its transmission factor for natural light. These measurements determine the matrix uniquely, except for a phase factor whose practical significance is small. The conditions are stated for the application of the method to the more general type of optical system described in V. A statement of the content of future papers in this series is included.

New Calculus for the Treatment of Optical Systems. VIII. Electromagnetic Theory

The preceding seven papers of this series present a systematic procedure for computing the effect of an optical system on the state of polarization of the light that passes through it. The M-matrices discussed in the first six papers represent the over-all effect of an optical system; the N-matrices described in Paper VII are essentially path derivatives of the M-matrices and represent the local optical properties at a given point along the light-path.In this paper we suppose that the medium is an anisotropic crystal and note that the description of the local optical properties by the N-matrices must be closely related to the description of the local properties by the dielectric and gyration tensors that are employed in standard crystal optics. We find the exact relation between the N-matrix and the above-mentioned tensors. It is shown how one can compute the dielectric and gyration tensors from a knowledge of the N-matrices for several different directions of the light-path in the crystal. It is also shown how one can compute the N-matrix for any given light direction from a knowledge of the dielectric and gyration tensors; the computation entails finding the square root of a two-by-two complex matrix.Taken together, the eight papers of this series present a compact and systematic procedure for the solution of problems in crystal optics. The N-matrices have the advantage that circular birefringence and circular dichroism are treated in the same framework used for linear birefringence and linear dichroism.

New Method of Describing and Measuring the Granularity of Photographic Materials

The thesis of this paper is that the only fully adequate way to describe the granularity of photographic materials is by means of a film noise spectrum. The film noise spectrum bears the same relation to granularity that the power spectrum (of electrical circuit theory) bears to electrical noise. The film noise spectrum includes all of the information in the previous granularity measures and readily interrelates them. It goes beyond the previous measures in that it leads to the solution of many of the signal-to-noise problems that arise in connection with the detection of target images on photographic materials.The film noise spectrum is defined and discussed in Parts 2 and 3. Its relation to the older methods of description involving granularity coefficients, both ordinary and syzygetic, is the subject of Part 4. The superiority of the new method is explained in detail in Part 5. Part 6 employs the film noise spectrum to derive the relation between ordinary granularity and syzygetic granularity. Part 7 discusses photoelectric scanning of the film and derives general expressions for the signal and for the noise. These expressions are used in Parts 8 and 9 to derive a particularly convenient method for the measurement of the film noise spectrum, and to solve the general problem of maximizing the signal-to-noise ratio in the detection of target images. Part 10 discusses the film noise spectrum of Super-XX film.

The Trapping of Fluorescent Light Produced within Objects of High Geometrical Symmetry

In a fluorescent body (e.g., scintillation counter crystal for detecting nuclear radiation) some of the fluorescent light may be trapped within the body because it is totally internally reflected an indefinitely large number of times. This phenomenon occurs only in bodies whose shapes have high symmetry. Equations are derived for the trapping fractions of rectangular parallelepipeds, plane parallel sheets, and spheres. Illustrative numerical values are given for various values of refractive index. For rectangular parallelepipeds of index 1.225, 1.500, and 2.000, the trapping fractions are 0.0, 0.236, and 0.598 respectively. For a sphere of index 1.50, the trapping fraction is 0.414 or 0.264, depending on whether the index with respect to the exciting radiation is taken as 1.00 or 1.50, these alternatives corresponding essentially to excitation by gamma-radiation or by light, respectively.