1. Introduction

Circular polarizers are critical optical elements that are used in a broad range of fields such as microscopy [1], control of chiral objects [2], quantum optics [3], and astronomy [4]. The action of such polarizers on a fully coherent electromagnetic beam is well understood [5,6]. However, in many cases of practical interest the assumption of full coherence is not justified. One example of this are beams generated by multi-mode lasers. Also, in remote sensing applications, a beam may start out as fully coherent and fully polarized but, due to atmospheric turbulence, these properties are lost on propagation. To the best of our knowledge, circular polarizers have not been analyzed for such more general, random beams. The traditional matrix methods of polarization optics [7,8], Jones calculus and Mueller calculus, deal with one-point quantities, namely Jones vectors and Stokes vectors. They cannot be used to describe how coherence properties, which are described by two-point correlation functions, are affected by passage through a linear optical element. In other words, these methods cannot be used for partially coherent beams. A recently developed generalized matrix formalism [9–11] can handle beams that are partially coherent and partially polarized. Here, we employ it to study the transmission efficiency of circular polarizers, as well as the changes in coherence they produce in a random electromagnetic beam. Because the state of coherence of a field determines its interference capabilities and how it interacts with a material object [12–14], it is both of foundational and practical importance to understand how spatial coherence is changed by passage through a circular polarizer.

2. Generalized transfer matrices

A random electromagnetic beam, propagating along the $z$-axis, is characterized, in a cross-sectional plane, by a cross-spectral density (CSD) matrix of the form [13]

The spectral degree of polarization (DOP) is given by [13]

3. Gaussian Schell-model beams

From now on we take the incident field to be of the Gaussian Schell-model type, with an intensity profile that is uniform across the polarizer. In that case the CSD matrix elements are [13]

We next turn our attention to the spectral degree of coherence, which is defined as [13]

This quantity is a measure of the fringe visibility that is seen in Young’s interference experiment with pinholes placed at $\boldsymbol {\rho }_1$ and $\boldsymbol {\rho }_2$. Alternative measures of coherence have been proposed in [16]. In the following we choose the first reference point to be on the beam axis, i.e., $\boldsymbol {\rho }_1 =(0,0)$. The spectral degree of coherence is then symmetric about the $z$-axis and, for the input beam described by Eqs. (9) we find that

4. Numerical examples

We illustrate our findings with two examples. We choose $B_{xy}= \pm {\rm i} 0.7$, and set the coherence radii as $\delta _{xx}=1\,{\rm cm}$, $\delta _{yy}=2\,{\rm cm}$, and $\delta _{xy}=1.6\,{\rm cm}$. It is readily verified that these parameter values satisfy the constraints (10) and (11). In Fig. 1 the spectral degree of coherence $\eta (\rho _2)$ is shown both for the incident beam and the transmitted beam. We see a marked difference between the three curves. To quantify this, let us define an axial coherence area $\cal A$ as the disk on which the field at every point is significantly correlated with the field on the beam axis:

 

Fig. 1. The spectral degree of coherence $\eta ^{\rm (i)}$ of the incident beam (black, dashed), and that of the transmitted field, $\eta ^{\rm (o)}$, for $B_{xy}= {\rm i} 0.7$ (blue) and $B_{xy}=- {\rm i} 0.7$ (red).

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We next examine the influence of the radius $\delta _{xy}$ on the state of coherence. The point of observation ${\boldsymbol \rho }_2$ is fixed at the radius of the coherence area of the input beam, i.e., $\eta ^{\rm (i)} (\rho _2) = 0.5$. We now vary ${\rm Im}\,B_{xy}$, together with $\delta _{xy}$, over a selected interval for which they both satisfy the inequalities (10) and (11). The resulting output $\eta ^{\rm (o)} (\rho _2)$ is shown in Fig. 2. As a visual aid, the reference level $\eta ^{\rm (o)} =0.5$ is indicated by the gray surface. It is seen that if ${\rm Im}\,B_{xy}>0$ then $\eta ^{\rm (o)} (\rho _2) > 0.5$, implying that the coherence area after passage through the polarizer has increased. The coherence area decreases if ${\rm Im}\,B_{xy}<0$. This effect is especially pronounced for higher values of $\delta _{xy}$. This means that the coherence area increases after passage through the left-circular polarizer when the incident beam has left-handed polarization. The opposite effect occurs when the beam has right-handed polarization.

 

Fig. 2. The spectral degree of coherence, $\eta (\rho _2)$, at a fixed value of $\rho _2 = 1.28\,{\rm cm}$, of the transmitted beam, as a function of $\delta _{xy}$ and ${\rm Im} \{ B_{xy} \}$. The value of $\rho _2$ is chosen such that $\eta ^{\rm (i)} (\rho _2)= 0.5$. The reference level 0.5 is indicated by the gray surface. In this example $\delta _{xx} = 1.0\, {\rm cm}$ and $\delta _{yy} = 1.2\, {\rm cm}$.

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5. Conclusions

In this study a generalized matrix method, previously applied to linear polarizers [11], was used to study how a partially coherent, partially polarized beam is affected by passage through a circular polarizer. The use of a generalized matrix formalism is necessary, because the traditional Jones and Mueller methods cannot handle two-point correlation functions. In other words, they cannot be used for partially coherent beams.

The incident field was assumed to be of the broad class of Gaussian Schell-model beams. The transmission efficiency was shown to depend only on the state of polarization. In contrast, the state of coherence behind the polarizer depends on polarization as well as coherence. The coherence area, a disk on which the field at each point is significantly correlated with the field on the beams axis, can undergo a significant change. It can either be increased or decreased by the polarizer. Since the state of coherence of a field determines its propagation and interference, we expect that our findings may be of relevance for the many applications in which circular polarizers are used.