Polarization fringes in optical systems: a compendium

Abstract

Spectral and spatial fringes in polarized light are produced by the interference of transmitted and reflected waves at the interface between materials with different indexes of refraction. These instrumental artifacts can affect the accuracy of optical designs conceived for high-sensitivity spectroscopy and polarimetry. We review the fundamental mechanisms that are responsible for these artifacts and the possible design pathways that allow us to mitigate them. In order to do so, we also present an approximate treatment of the problem of the transmission and reflection of light through (possibly absorptive) birefringent layers, relying on a few fundamental results that can be found in the already extensive literature on the subject. Unfortunately, many of these results remain the domain of a niche of investigators working in the field of thin films and optical coatings, and are often overlooked even by experienced designers of spectro-polarimetric instrumentation. The treatment presented in this work is limited to isotropic materials and uniaxial crystals, which are the most common types of optics employed in polarimetric instrumentation, and it fundamentally relies on the approximation of small birefringence for its implementation. An extensive set of modeling examples is provided to highlight the salient characteristics of polarization fringes, as well as to assess how approximate treatments such as this compare to exact but more computational expensive formulations of the problem such as Berreman's calculus.

Figures (14)

• Figure 1: The propagation of a beam through a stack of plane-parallel uniaxial crystals, rotated at different angles $\alpha$ with respect to the plane of incidence (PoI) $\langle x,z\rangle$ (in this example, $\alpha_m$ is negative, while $\alpha_{m+1}$ is positive). The $xyz$ axes provide the natural reference for the polarization decomposition of the incoming beam, with $y\parallel\bm{s}$, and $\bm{p}\in\langle x,z\rangle$. The figure shows the case of a stack of uniaxial crystals that have all been cut such that the optic axes ($\bm{e}$) are parallel to the interfaces. This is a common requirement for birefringent optics employed in polarimetric instrumentation, as it maximizes the birefringence experienced by the beam while minimizing polarization aberrations due to the phenomenon of double refraction. For a positive uniaxial crystal, the $\bm{o}$ and $\bm{e}$ axes in the figure correspond, respectively, to the fast and slow axes of the crystal, and they also correspond to orthogonal states of linear polarization for a ray propagating along $z$; for a negative uniaxial crystal, $\bm{o}$ becomes the slow axis and $\bm{e}$ the fast axis. The figure also clarifies the convention followed in this work, where the angle $\alpha$ of the optic is defined by the orientation of the $\bm{o}$ axis. In our treatment, we will always assume the approximation of small birefringence Ye82, so the ordinary (o) and extraordinary (e) rays, into which an incoming ray splits at the interface with a birefringent medium, are assumed to always lie on the PoI, and with polarizations that are orthogonal to each other, regardless of the angle $\phi$ of incidence and the orientation $\alpha$ of the optic.
• Figure 2: Four different examples of birefringent stacks, illustrating their optical and polarization properties, calculated using the formalism presented in this paper. The same models were also considered by Cl04aCl04bCl04c. For each example, we show four panels for the transmittance, reflectance, polarizance, and retardance of the corresponding optical system, as indicated by the labels. Top left: a $\lambda/4$ waveplate at 500 nm, consisting of 13.5 $\mu$m of SiO$_2$. Top right: a $\lambda/2$ waveplate at the same wavelength, consisting of 27 $\mu$m of SiO$_2$. Bottom left: a compound $\lambda/4$ waveplate at the same wavelength, produced by combining the two previous waveplates with crossed principal axes. Bottom right: a compound $\lambda/2$ waveplate consisting of a stack of three $\lambda/2$ waveplates as in the top-right panel, followed by a stack of two identical $\lambda/2$ waveplates at 90$^\circ$ from the first stack. In all cases, we assumed a spectral sampling resolution of 50000 and normal incidence.
• Figure 3: Same plotted quantities and experimental conditions as in Fig. \ref{['fig:examples']}, but for a MgF$_2$ two-plate compound retarder, optimized to approximately behave as a $\lambda/2$ retarder between 130 and 210 nm. The two plates have a bias thickness of 800 $\mu$m.
• Figure 4: Wavelength dependence between 200 and 1000 nm of the (intensity normalized) Mueller matrix of a PCM optimized for full-Stokes polarimetry between 400 and 1000 nm. The polarization fringes are calculated with a spectral resolution of 20000. The PCM design uses three compound retarders in the configuration MgF$_2$-SiO$_2$-MgF$_2$, where the two MgF$_2$ elements are identical. The gray curves plotted over the fringes represent the ideal Mueller matrix from the PCM design.
• Figure 5: Geometric constructs to determine the elemental solid angle (left) and phase delay (right) for a ray incident at an angle $\phi$ on the stack of optics of radius $R$, assuming a spherical wavefront with $\mathrm{f/\#} = 2\tan\phi_R$.