The interaction of arbitrary three-dimensional light beams with optical elements is described by the generalized Jones calculus, which has been formally proposed recently [Azzam, J. Opt. Soc. Am. A 28, 2279 (2011)]. In this work we obtain the parametric expression of the differential generalized Jones matrix (dGJM) for arbitrary optical media assuming transverse light waves. The dGJM is intimately connected to the Gell-Mann matrices, and we show that it provides a versatile method for obtaining the macroscopic GJM of media with either sequential or simultaneous anisotropic effects. Explicit parametric expressions of the GJM for some relevant optical elements are provided.
©2013 Optical Society of America
In the widely-used Jones calculus, completely polarized light beams are characterized by the Jones vector. Such description assumes transverse plane waves traveling along a fixed propagation direction parallel to the axis, which is a valid approach in a broad range of physical situations. However, during the last years there is a growing interest in applications that entail propagation direction changes and/or non-transverse electric fields. Some of these applications are light propagation in turbid media, high numerical aperture focusing, near-field optics and light interaction with nanoparticles [1–3]. Completely polarized light beams in such situations can be characterized by the generalized Jones vector (GJV), also called 3D Jones vector [4,5]. A generalized Jones calculus has been recently proposed for describing the interaction of GJVs with anisotropic optical media and devices . However, no generalized Jones matrices for optical devices or media have been given so far.
In this work we present a method for obtaining the generalized Jones matrices (GJMs) of arbitrary optical media and devices. Our development is based on the differential formulation of the generalized Jones calculus, and assumes transverse light waves. We present the explicit form of the differential generalized Jones matrix (dGJM), which extends the well-known differential Jones matrix to the three-dimensional framework. The GJM of samples with either sequential or simultaneous optical effects can be readily obtained from the dGJM. In order to illustrate the applicability of the proposed method, we provide explicit expressions of the GJM for the major types of optical devices, namely linear retarders and dichroic absorbers, and optically-active rotators.
2. Generalized Jones calculus and the differential generalized Jones matrix
According to the generalized Jones calculus , the equation that describes the elastic interaction of an input generalized Jones vector with a sample is:
The GJM characterizes an optical element as a whole, basically modeling light-sample interactions as an input-output mechanism. The differential formulation enables to further describe the continuous evolution of polarized light propagation through anisotropic media. In the conventional Jones calculus, the differential Jones matrix completely characterizes the anisotropic properties of an infinitesimal slab of the medium . The differential Jones matrix is intimately connected with the Pauli matrices [7,8], the generators of the group SU(2), due to the fact that the Jones matrix constitutes a representation of the Lorentz group . Specifically, the differential Jones matrix can be expressed as:7,8]. The well-known general expression for the differential Jones matrix is thus:6], which we denote here as . In particular, and are the isotropic retardation and absorption, while parameters and account for birefringence and dichroism effects. Subscripts indicate the specific type of anisotropy, where the convention has been adopted.
From this theoretical approach, the differential calculus can be extended to the three-dimensional formulation. In order to do that, it is necessary to replace the Pauli matrices by the Gell-Mann matrices [4,8], which form the complete set of infinitesimal generators of the Lie group SU(3):Eq. (7). The rest of parameters complement the three-dimensional anisotropy characterization: and are the linear and circular birefringence and dichroism in the plane, while and are analogously defined for the plane. Finally, and quantify the difference in linear retardance and absorption between the direction and the plane.
The dGJM presents some remarkable characteristics. Firstly, comparing Eq. (7) and Eq. (11), it can be observed that the upper-left block coincides with , apart from parameters and , which contribute to the expression in the three-dimensional framework. Additionally, the symmetries of the dGJM are similar to those shown by the differential Jones matrix. In the extended matrix , the positions occupied by the differential parameters for anisotropic effects in the plane are restricted to elements and , while those for the plane are contained in and .
The equation describing the GJV evolution of a transversally polarized light wave along the propagation direction is:8,10,11], and enables to obtain the GJM of media showing multiple simultaneous optical effects. In the case of sequential optical elements, each of them characterized by , the total GJM is given by:
3. Generalized Jones matrices of basic polarization devices
The presented approach is now applied to obtain explicit expressions of the GJMs for the most relevant classes of optical elements. A right-handed laboratory Cartesian coordinate system is assumed (Fig. 1 ). According to the convention used in  and adopted in this work, any other local coordinate system is univocally defined by (i.e. by the polar and azimuthal angles of the axis), being and . The effect of the coordinate system transformation on the GJV is:12]. However, a vast number of optical devices and samples are made of uniaxial materials. Uniaxial media show two equal refractive indices (the ordinary indices) different to the third one (called the extraordinary index). In such situations, the index ellipsoid becomes an ellipsoid of revolution, whose orientation can be completely defined by the optic axis (which is aligned with the extraordinary principal axis) as shown in Fig. 1.
Considering a linear birefringent medium in which the optic axis is parallel to the axis (Fig. 1, left), the dGJM in the laboratory system is the following diagonal matrix:Eq. (17) into Eq. (13), the GJM of this device is readily obtained:Fig. 1, right). Obviously, the GJM in the local coordinate system verifies . As a consequence, the simplest way to find the GJM of an arbitrarily oriented LR is to perform a coordinate system change, multiply by , and finally apply the inverse coordinate change:Eq. (19) parallels the approach used in Jones calculus . As a particular example, the GJM of such device for is:Eq. (22) into Eq. (19). The GJM of this optical element for is hereby provided for illustrative purposes:
In this work, a method for obtaining the GJM of non-depolarizing anisotropic samples has been presented. Our approach is based on the dGJM, which is obtained by the extension of the differential Jones matrix to the three-dimensional framework. The dGJM provides a versatile method for obtaining the GJM of arbitrary anisotropic optical elements in a simple and elegant way. Explicit parametric expressions of the GJM for some relevant devices have been given. Transverse totally polarized light waves have been assumed. Such approach is useful for applications that entail propagation direction changes (e.g. scattering in the far field), and constitutes a necessary first step toward the development of the generalized Jones calculus, which presents a foreseeable potential for a wide range of applications.
References and links
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