1. INTRODUCTION

Nowadays it is a well-known fact that the special theory of relativity (STR) and one of the basic problems of polarization theory (PT), namely, the interaction of the polarized light with pure (deterministic) orthogonal polarization devices [1–3], have the same mathematical underpinning—the Lorentz transformation. This fact was recognized late in the 1970s by Barakat [4] and Takenaka [5], and until then the two theories developed independently, each of them elaborating its own language and representations.

Due to the wider scientific interest in STR, the algebraic instrument of STR had been more profoundly elaborated, up to the level of group theory, so that after 1970, and even with Takenaka’s papers, the language of group theory applied to the Lorentz transformation in STR was largely imported into PT [6–13].

Nevertheless, there is a mathematical device, a geometrical one, also applicable in both these fields, which was developed in parallel, in the same period, and also very deeply, this time in the PT: the Poincaré sphere. As is well known, in the frame of this geometric representation, a specific method of analysis of the interaction between polarization light and polarization devices, namely, that of the degree of polarization (DoP) surfaces, was elaborated [14–18]. This method can be imported word by word in the STR: a countertransfer, this time, from PT to STR.

The present paper is, first, a contribution to the issue of DoP surfaces, a further step in the analysis of the subtleties of this issue. But I would present it in the light and at the level of this strong kinship between PT and STR. On the other hand, the same approach can be applied in many other branches of optics where the Lorentzian character of the transformation which governs some specific problems has been recognized (e.g., multilayer optics, ray optics, laser cavity physics, quantum optics).

Recently it was established, first for the collinear case [19,20], and then for the general one [21,22], that there is another aspect of this kinship between PT and STR: the Poincaré vector of the light transmitted by an orthogonal (orthogonal eigenvectors) dichroic device can be expressed as a function of the Poincaré vectors of the incoming light and of the device by the same composition law as that of (generally nonconlinear) relativistic allowed velocity, namely [23],

In PT,

Ungar [26] has shown that the relativistic allowed velocities,

Thus the relativistic allowed velocities are the STR gyrovectors, and the polarization Poincaré vectors are the PT gyrovectors. Both are undergone to the constraint [Eq. (5)]; that is, they are confined, or are prisoners, of a unit Poincaré sphere, the Poincaré sphere of the polarization gyrovectors and the Poincaré sphere of the relativistic allowed velocities, respectively. All one can say about one of them can be extended, mutatis mutandis, to the other.

It is straightforward that for $|\mathbf{u}|,|\mathbf{v}|\ll 1$ (classical regime of velocities), as well as for $|{\mathbf{s}}_i|=p_i$, $|{\mathbf{s}}_d|=p_d\ll 1$ ($p_i$: degree of polarization of incident light; $p_d$: degree of dichroism of the device) the composition law, Eqs. (1) and (2), passes in the “classical” composition law:

The gyrovectorial character of the polarization Poincaré vectors—as well as of the relativistic allowed velocities—determines a distortion to their addition (with respect to that of the usual vectors), which is more and more pronounced when their values tend to the upper limit ones. The above aspects refer to the individual “addition” (composition) of two gyrovectors, particularly polarization Poincaré gyrovectors.

This distortion becomes very expressive in the holistic approach (namely, that of the DoP surfaces) to the problem of interaction between polarized light and orthogonal dichroic devices. The characteristics of the corresponding ellipsoids become strongly singular at this limit.

The method of DoP surfaces describes the behavior of a whole sphere of states of optical polarization (SOPs) with the same degree of polarization, a DoP sphere, ${\mathrm{\Sigma }}_2^{p_i}$, under the action of a dichroic device of Poincaré vector ${\mathbf{s}}_d$. This method can be exported in STR too, where it will describe the behavior of a whole sphere of velocities with the same modulus, ${\mathrm{\Sigma }}_2^v$, under the action of a boost $\mathbf{u}$. These spheres will be deformed, and the deformation will be more dramatic as the dichroic device or the boost is much stronger ($p_d$ or $u$ higher). As we shall see in the limit ${|\mathbf{s}}_d|=p_d,|{\mathbf{s}}_i|=p_i\to 1$, or $|\mathbf{v}|,|\mathbf{u}|\to 1$, the characteristics of the ellipsoid become strongly singular.

2. OUTPUT ELLIPSOID CORRESPONDING TO AN INPUT DOP SPHERE

We shall decompose the Poincaré vector of the input SOP, ${\mathbf{s}}_i=p_i{\mathbf{n}}_i$, into two components, parallel and perpendicular to the Poincaré axis of the (orthogonal) partial polarizer, ${\mathbf{n}}_d$ (Fig. 1):

 

Fig. 1. Poincarè sphere notations.

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Their transformation is entirely similar to the relativistic transformation of the components of velocity parallel and perpendicular to the relative velocity of the two IRSs [27,28]:

Let us consider now a whole manifold of input states with the same DoP, $p_i$. In a Poincaré sphere representation they constitute a DoP sphere ${\mathrm{\Sigma }}_2^{p_i}$. The polarization device transforms the polarization states of this sphere in a manifold of states situated, in a Poincaré geometric representation, on a closed surface, the corresponding DoP surface. Each DoP sphere is mapped onto a DoP surface.

It is worth mentioning here that a slightly different language (more precisely denomination) is also used recently in the polarization literature (e.g., [29,30]): $P$-sphere instead of DoP sphere and $P$-surface instead of DoP surface. On one hand, obviously, the states of a DoP (or $P$) sphere all have the same degree of polarization, whereas the states of the corresponding DoP (or $P$) surface in general do not; in this context maybe the latter denomination better avoids a possible confusion. On the other hand, there is another aspect of the action of the polarization devices on the polarized light, referring this time to the gain given by the device, which can be handled also in a geometric holistic approach, in which the term $I$-surface [16], complementary to that of $P$-surface, was introduced. Here is another reason for which this latter language will probably gain ground, all the more so as it was firmly adopted in a very recent prestigious monograph [29].

By eliminating the parameter $\varphi$ between the projections of ${\mathbf{s}}_o$ on the axes $Ox$ and $Oy$ in Eq. (12), and taking advantage of the fact that the whole problem has polar symmetry around the $Ox$ axis, one gets the equation of the ellipsoid corresponding to the action of the partial polarizer $p_d$ on a DoP sphere ${\mathrm{\Sigma }}_2^{p_i}$ [29–31]:

3. BEHAVIOR OF THE SOP ELLIPSOID IN FUNCTION OF PARAMETERS $p_i$ AND $p_d$

The characteristics of the polarization ellipsoid, Eq. (13), namely, its semi-axes, Eqs. (14) and (15), and the displacement of its center with respect to the center of the corresponding DoP sphere, Eq. (16), all depend exclusively on the degree of dichroism of the diattenuator, $p_d$, and on the radius of the DoP sphere, $p_i$. Generally, as we shall see, the behavior of the SOP ellipsoid depends in a complicated and spectacular manner on the competition between $p_d$ and $p_i$. This behavior comes from the strong singularities of $a_x,a_y$, and $\mathrm{\Delta }x$ for $p_i$ and $p_d$ tending together to their maximum value, 1, and it has the ultimate root in the gyrovectorial character of the polarization Poincaré vectors ${\mathbf{s}}_i$, ${\mathbf{s}}_d$, ${\mathbf{s}}_o$, Eqs. (3) and (5).

In this paper, I shall illustrate the behavior of the polarization ellipsoid as a function of the competition between $p_i$ and $p_d$, and will analyze in detail the singularities mentioned above of its characteristics. All this analysis may be transferred into STR, by replacing the polarization Poincaré gyrovectors by the STR specific gyrovectors, namely, the relativistic allowed velocities.

Globally, as the strength of the partial polarizer increases, the ellipsoid becomes smaller and smaller and goes farther and farther toward the wall of the Poincaré sphere (Fig. 2), reaching it for the maximum strength, $p_d=1$, corresponding, evidently, to ideal polarizers. (It is worth pointing out that it is not about a linear polarizer but generally about an elliptical polarizer; the Poincaré axis of the device, ${\mathbf{n}}_d$ in Fig. 1, can point in any direction in the Poincaré sphere). The ellipsoid can never overpass the wall of the Poincaré sphere, because of the physical constraint $p_d\le 1$. In PT this constraint is quite natural.

 

Fig. 2. DoP sphere ${\mathrm{\Sigma }}_2^{p_i}$, $p_i=0.7000$, and the corresponding ellipsoid for (a) $p_d=0.7000$, (b) $p_d=0.9950$.

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An identical behavior has in STR a velocity surface ${\mathrm{\Sigma }}_2^v$ which undergoes a Lorentz boost of rapidity $\beta$, when this rapidity (the strength of the boost) increases to its maximum value, 1. A velocity sphere ${\mathrm{\Sigma }}_2^v$ seen by an observer in his IRS is seen by another observer (moving with respect to the first with rapidity $\beta$) as an ellipsoid (of velocities). As the rapidity of the boost increases, the velocity ellipsoid becomes smaller and smaller and is pushed farther and farther toward the wall of the Poincaré sphere ${\mathrm{\Sigma }}_2^c$ of the relativistic allowed velocities. The ellipsoid cannot protrude through the wall of the Poincaré sphere ${\mathrm{\Sigma }}_2^c$, because of the physical constraint imposed by the second postulate. This STR constraint, unlike the PT one, is counterintuitive.

In the following we shall consider a series of sequences in which $p_d$ and $p_i$ tend, alternatively, to their maximum value, the radius of the Poincaré sphere, 1.

A first sequence is presented in Fig. 3, where the strength of the dichroic device is increased, at a given (moderate) value of $p_i$ ($p_i=0.4$). Figure 3 represents some significant configurations of the corresponding SOP ellipsoid.

 

Fig. 3. Evolution of the ellipsoid at a given value of $p_i=0.40$ for various values of $p_d=0.20$, 0.40, 0.68, 0.80.

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In Fig. 3(b) the rear side of the ellipsoid touches the center of the sphere. In this case $\mathrm{\Delta }x=p_i$, and from Eq. (16) one obtains that $p_i=p_d$. Physically that means the diattenuator is strong enough to annihilate its antagonist SOP of the DoP sphere, namely, ${\mathbf{s}}_i=-{\mathbf{s}}_d$, (i.e., $p_i=p_d$, ${\mathbf{n}}_{\mathbf{i}}=-{\mathbf{n}}_{\mathbf{d}}$). By passing through the device, this state, the most “recalcitrant,” is completely depolarized (it has moved to the center of the Poincaré sphere). All the other polarization states are now oriented on the side of the Poincaré axis of the dichroic device (on the right side of the ${\mathrm{\Sigma }}_2^{p_i}$ sphere).

Here I have to point out that, unlike Fig. 2 which gives a general, oblique perspective on the DoP sphere and the corresponding ellipsoid, in Figs. (3)–(6), I have chosen the $x–y$ view option. The first one is, obviously, more expressive in illustrating the three-dimensionality of the problem, but falsifies or masks the direct visualization of some of its quantitative aspects. As an example, the oblique perspective does not show the tangency of the ellipsoid to the center of the sphere for $p_d=p_i$.

Figure 3(c) corresponds to an increase of the strength, $p_d$, of the dichroic device in such a measure that the SOP ellipsoid becomes tangent exterior to the DoP ${\mathrm{\Sigma }}_2^{p_i}$ sphere. The dichroic device is strong enough to push out from the sphere all the incident SOPs. The DoP of all the incident ${\mathrm{\Sigma }}_2^{p_i}$ SOPs passed by the partial polarizer is now greater than $p_i$. The device succeeds in imposing its polarization structure (${\mathbf{s}}_o={\mathbf{s}}_d$, i.e., $p_o=p_d$, ${\mathbf{n}}_o{=\mathbf{n}}_d$) to the most recalcitrant incident state, its antagonist state.

Figures 3(a) and 3(b) are an expressive illustration of the fact that a partial polarizer can not only increase the degree of polarization of partially polarized light but also decrease it (i.e., having, in this case, a depolarizing effect). All the incident SOPs which correspond to the output SOPs on the rear side of the ellipsoid situated in the ${\mathrm{\Sigma }}_2^{p_i}$ sphere were depolarized by the partial polarizer.

Further increasing the polarization strength of the device, the degree of polarization of output SOPs increases, and these SOPs are gathered together toward the Poincaré axis of the polarizer. The dichroic device pushes all the incident SOPs of the sphere ${\mathrm{\Sigma }}_2^{p_i}$ toward its Poincaré axis, that is, toward its polarization shape (structure, form). The stronger the device (higher $p_d$), the greater its effects on the given Poincaré sphere ${\mathrm{\Sigma }}_2^{p_i}$.

Evidently we can go further in increasing $p_d$ up to its maximum value, 1; the ellipsoid would be pushed further and further up to the walls of the Poincaré sphere, becoming smaller and smaller. But for better illustrating the strange behavior of the ellipsoid when $p_i,p_d\to 1$, it is convenient to stop this sequence at a not-too-high value of $p_d$ [in Fig. 3(d), $p_d=0.8$] and to increase now the value of $p_i$ at this level of $p_d$.

When we keep constant the strength of the dichroic device, at the highest level reached above [$p_d=0.8$, Fig. 3(d)], and gradually increase the radius $p_i$ of the DoP surface, the ellipsoid regresses; it becomes greater and greater and goes back toward the center of the Poincaré sphere (Fig. 4). The higher $p_i$, the feebler the effect of the given ($p_d=0.80$) diattenuator on the corresponding ${\mathrm{\Sigma }}_2^{p_i}$ sphere.

 

Fig. 4. Evolution of the ellipsoid at a given value of $p_d=0.80$ for various values of $p_i=0.45$, 0.50, 0.80, 0.90.

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The tangency of the ellipsoid with the sphere [Fig. 4(b)] occurs now at a higher value of $p_d$, namely, $p_d=0.80$. That means that for converting all the SOPs of the DoP sphere ${\mathrm{\Sigma }}_2^{p_i}$, at this higher level of $p_i$, in polarized states with a degree of polarization greater than $p_i$ and oriented all toward the Poincaré axis of the device, one needs, this time, a stronger polarizer [compare Figs. 4(b) and 3(c)].

Likewise, for the case when the rear side of the ellipsoid touches the center of the sphere, in this new situation one needs a much stronger polarizer $p_d=0.80$ [Fig. 4(c), compared with Fig. 3(b)].

Further increasing $p_i$, the ellipsoid grows further and returns more and more toward the center of the Poincaré sphere. Again, it is convenient to stop the illustration of this process at a not excessively high value of $p_i$ and to restart increasing $p_d$ at this new level of $p_i$ (Fig. 5).

 

Fig. 5. Evolution of the ellipsoid at a higher value of $p_i=0.900$ for various values of $p_d=0.850$, 0.900, 0.994, 0.997.

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Figure 5 is the analog to Fig. 3 at a higher level of $p_i$. Again, the rear side of the ellipsoid touches the center of the corresponding sphere ${\mathrm{\Sigma }}_2^{p_i}$, this time at a higher level of $p_i$ for $p_i=p_d$. The percentage of incident SOPs pushed outward in the sphere is much lower at higher $p_i$ [Fig. 5(b)] than at lower $p_i$ [Fig. 3(b)]. In the cases corresponding to Figs. 5(c) and 3(c), all the incident SOPs are pushed outward in the ${\mathrm{\Sigma }}_2^{p_i}$ sphere toward the Poincaré axis of the diattenuator. The difference, or the ratio, of $p_d$ and $p_i$ for this situation is much reduced at higher levels of $p_i$ and $p_d$ [$p_d-p_i=0.094$ in Fig. 5(c)] than at a lower level of their values [$p_d-p_i=0.28$ in Fig. 3(c)]. The breathing space (the space between the ${\mathrm{\Sigma }}_2^{p_i}$ sphere and the Poincaré sphere) is much reduced at the high level of $p_i$ corresponding to the situation illustrated in Fig. 5 that at the lower level of $p_i$ corresponding to Fig. 3.

Finally let us stop increasing $p_d$ at the (very high this time) value of $p_d=0.997$ and start increasing $p_i$ at this level of $p_d$. Figure 6 is the analog to Fig. 4 at a higher level of $p_d$ (stronger partial polarizer). Increasing $p_i$ and ${\mathrm{\Sigma }}_2^{p_i}$, the corresponding ellipsoid grows back and returns to the center of the Poincaré space. At $p_i=p_d=0.997$, the dichroic device practically has no more “force” in pushing out the SOPs from the sphere ${\mathrm{\Sigma }}_2^{p_i}$ [Fig. 6(b)] as at $p_i=p_d$ [Fig. 4(c)]. Further, for the week predominance of $p_i$ over $p_d$ (of 0.0028), Fig. 6(d), the device has practically no effect on ${\mathrm{\Sigma }}_2^{p_i}$. [Compare Fig. 6(d) with Fig. 4(d).]

 

Fig. 6. Evolution of ellipsoid at a higher value of $p_d=0.997$ for various values of $p_i=0.925$, 0.997, 0.999, 0.9998.

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In Figs. 3 and 4 on the one hand and Figs. 5 and 6 on the other hand, I have illustrated the same interplay (increasing $p_d$ at given $p_i$, followed by a growth of $p_i$ at a given $p_d$), at relatively low, and, respectively, very high levels of the couple ($p_i$, $p_d$). The breathing space is drastically reduced in the second case.

This back-and-forth way can be scaled endlessly at a higher and higher level (of both $p_i$ and $p_d$). Figures 5 and 6 illustrate it in a range of values of $p_i$ and $p_d$ in the neighborhood of their limit value, 1. Comparing Figs. 5 and 6 with Figs. 3 and 4, one can notice that as $p_i$ and the corresponding DoP sphere ${\mathrm{\Sigma }}_2^{p_i}$ increase, the free space of this back-and-forth way (namely, between the sphere ${\mathrm{\Sigma }}_2^{p_i}$ and the Poincaré sphere) is more and more limited. But the process of decreasing of the ellipsoid up to a “point-like” one, when $p_d$ increases at a given $p_i$, and of coming back to a sphere, when $p_i$ increases at a given $p_d$, can be endlessly continued.

In PT this scaled back-and-forth way can be completely and naturally understood in terms of the competition and, in the above illustration, of the interplay between the polarization strength of the dichroic device (its degree of dichroism, $p_d$) and the polarization strength of the incident light (the degree of polarization of the DoP surface). It is also natural that the variation border of this back-and-forth way is more and more limited (between the sphere ${\mathrm{\Sigma }}_2^{p_i}$ and the Poincaré sphere) as $p_i$ increases. The origin or the root of this behavior lies in the fact that all the states of polarization are restricted, or contained in (are “prisoners” of) the Poincaré sphere, which is a quite natural fact (because $p_i,p_d\le 1$).

Exactly the same back-and-forth way may be pursued in STR by replacing the DoP sphere ${\mathrm{\Sigma }}_2^{p_i}$ with a “velocity sphere,” ${\mathrm{\Sigma }}_2^v$, seen by an IRS observer, and $p_d$ with the rapidity $\beta$ of another IRS observer moving with respect to the first one. The latter will see a “velocity ellipsoid” instead of the sphere ${\mathrm{\Sigma }}_2^v$ seen by the first. With this replacement, Figs. 2–6 illustrate the similar behavior in STR. There the roots of this behavior lie in the fact that all the relativistic allowed velocities are confined in (are “prisoners” of) the Poincaré velocity sphere ${\mathrm{\Sigma }}_2^c$, namely, in the second postulate of STR.

The only great difference between these issues, of PT and of STR, is that this confinement is natural in PT and at least counterintuitive in STR. The border of the polarization state vectors manifold is naturally a finite one, the SOP Poincaré sphere, ${\mathrm{\Sigma }}_2^1$, whereas in STR the unlimited space of the velocities manifold was collapsed, counterintuitively, by the second postulate to the Poincaré sphere of relativistic allowed velocities, ${\mathrm{\Sigma }}_2^c$.

 

Fig. 7. $\mathrm{\Delta }x$ as a function of $p_d$, with $p_i$ as parameter: (a) upper line $p_i=0.1$, lower curve $p_i=0.5$; (b) upper curve $p_i=0.95$, lower curve $p_i=0.99$.

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4. STRONG SINGULARITIES OF THE CHARACTERISTICS OF THE SOP ELLIPSOID AT THE MAXIMUM LIMIT VALUE OF ${\mathit{p}}_{\mathit{i}}$ AND ${\mathit{p}}_{\mathit{d}}$

A further insight in this evolution of the SOP ellipsoid when the physical parameters $p_i$ and $p_d$ are varied is obtained by analyzing its characteristics, $a_x,a_y,\mathrm{\Delta }x$, as a function of these parameters, Eqs. (14)–(16). All these functions are strongly singular (they have no limit) when both $p_i$ and $p_d$ tend to their maximum value, unity.

In Fig. 7 $\mathrm{\Delta }x$ is represented as a function of $p_d$, with $p_i$ as a parameter, and in Fig. 8 as a function of $p_i$ with $p_d$ as the parameter. It is remarkable and maybe not insignificant for the mathematics of gyrovectors’ spaces that the expressions of $a_x$ and $\mathrm{\Delta }x$ as functions of $p_i$, $p_d$ can be obtained one from the other by interchanging $p_i$ and $p_d$ [Eqs. (14) and (16)]. That is, the graph of $a_x$ as a function of $p_d$ with $p_i$ as a parameter coincides with the graph of $\mathrm{\Delta }x$ as a function of $p_i$, with $p_d$ as a parameter. Thus the analysis of the behavior of, let us say, $\mathrm{\Delta }x$ and of its singularity for $p_i$, $p_d\to 1$ is completely relevant for all the characteristics of the SOP ellipsoid.

Let us analyze the graph $\mathrm{\Delta }x(p_d)$ with $p_i$ as a parameter (Fig. 7). For low values of the radius $p_i$ of the DoP sphere ${\mathrm{\Sigma }}_2^{p_i}$, $\mathrm{\Delta }x$ grows from 0 to 1 almost linearly with the strength of the dichroic device [Fig. 7(a)]. For a bigger DoP sphere (greater $p_i$), this growth becomes nonlinear: for small values of $p_d$, $\mathrm{\Delta }x$ grows more slowly (the slope of the curve decreases under the unity) and, after some critical value of $p_d$, it increases more rapidly to 1.

 

Fig. 8. $\mathrm{\Delta }x$ as a function of $p_i$, with $p_d$ as parameter: (a) lower curve $p_d=0.1$, upper curve $p_d=0.5$; (b) lower curve $p_d=0.95$, upper curve $p_d=0.99$.

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This behavior becomes more prominent for very large values of $p_i$ (very large DoP sphere, close to the Poincaré wall), let us say $p_i>0.95\text{:}\mathrm{\Delta }x$ increases very slowly with $p_d$ up to the critical value of $p_d$. After this critical value $\mathrm{\Delta }x$ suddenly starts to grow very abruptly with increasing $p_d$ [Fig. 7(b)]. Physically this comes to the fact that at a high value of the radius $p_i$ of the DoP sphere (at high DoP of the incident light) a feeble dichroic device (low value of $p_d$) has a low efficiency in modifying the incident light SOPs, that is, in displacing and deforming the DoP sphere of the incident light. As the strength $p_d$ of the polarizer increases, its action on the DoP sphere is more and more pronounced. The polarizer is now strong enough (and gets stronger and stronger) for giving a substantial growth of $\mathrm{\Delta }x$ and generally a substantial alteration of the DoP sphere into the corresponding ellipsoid.

A similar analysis of the behavior of $\mathrm{\Delta }x$ as a function of $p_i$, with $p_d$ as a parameter, can be pursued on the graphs in Figs. 8(a) and 8(b).

An intriguing aspect of these relationships between the ellipsoid characteristics $a_x$, $a_y$, $\mathrm{\Delta }x$ and the physical parameters $p_i$, $p_d$ is the following: If we judge on the basis of Fig. 7(b), in the limit $p_i$, $p_d\to 1$, we get the limit 1 for $\mathrm{\Delta }x$, whereas if we judge on the basis of Fig. 8(b) for the same extreme case, $p_i$, $p_d\to 1$, one gets for $\mathrm{\Delta }x$ the limit zero.

Likewise, the corresponding graphs for $a_x$ and for $a_y$ lead to the limit 1 and to the limit zero in the same case $p_i$, $p_d\to 1$. In other words, all the functions [Eqs. (14)–(16)] which give the characteristics of the SOP ellipsoid, $a_x$, $a_y$, $\mathrm{\Delta }x$, as functions of the physical parameters of the incident light, $p_i$, and of the dichroic device, $p_d$, are strongly singular functions of these parameters at their limit values ($p_i$, $p_d$ tending both to 1). Figure 9 presents the function $\mathrm{\Delta }x(p_i,p_d)$ in the vicinity of this limit. The singularity appears in all its beauty if we slightly overpass the range of physical acceptability of the parameters $p_i$, $p_d$.

 

Fig. 9. $\mathrm{\Delta }x$ as a function of $p_i$ and $p_d$ in the neighborhood of the limit $p_i\to 1$, $p_d\to 1$.

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All these considerations can be formulated in STR too, for the “velocity ellipsoid,” replacing $p_i$ by $v$ and $p_d$ by $u$.

Generally, all the behavior presented above comes from the gyrovectorial character of the polarization Poincaré vectors in PT and, analogously, of the relativistic allowed velocities in STR. These kinds of 3D vectors, the gyrovectors, are forced to add in such a way [Eqs. (1) and (3)] that their resultant should remain in a spherical volume limited by a Poincaré sphere.

5. CONCLUSIONS

The polarization Poincaré vectors, as well as the relativistic allowed velocities, are both forced, by different physical reasons, to be confined in a closed volume: a Poincaré ball. Therefore their “addition” (composing) differs from that of the ordinary vectors; they are both gyrovectors.

This difference vanishes in the neighborhood of the center of the Poincaré sphere $({\mathbf{s}}_i,{\mathbf{s}}_i\to 0,\mathbf{v},\mathbf{u}\to 0)$ but becomes more and more prominent when the Poincaré (gyro)vectors grow. It becomes dramatic in the limit ${|\mathbf{s}}_i|,|{\mathbf{s}}_i|\to 1,|\mathbf{v}|,|\mathbf{u}|\to 1$, that is, near the walls of the Poincaré sphere.

In PT, unlike STR, we dispose of a global formalism (of the DoP surfaces), which allows an expressive approach to this issue, namely, the analysis of the behavior of the ellipsoids corresponding to various DoP spheres when they approach the walls of the Poincaré ball.

We have seen that the ellipsoid has a somewhat strange and contradictory behavior at the limit $p_i,p_d\to 1$, as $p_i$ or $p_d$ is more advanced in this tendency. The characteristics of the ellipsoid $(a_x,a_y,\mathrm{\Delta }x)$ become strongly singular when $p_i$ and $p_d$ tend together to their maximum limit value, 1.

This behavior illustrates expressively the distortion to which is subjected the world of the Poincaré gyrovectors by the “Procrustes’ bed” constraint of being closed in a limited space—the Poincaré sphere.

What is most important in this analysis is, in my opinion, not so much the specific fact that the polarization Poincaré vectors and the SOP ellipsoids have such a behavior, but rather the following observations.

First, this behavior, which is quite natural in PT, allows us to attenuate the repulsion of our everyday intuition with respect to the absolutely similar, but counterintuitive, behavior of the relativistic allowed velocities.

Second, this behavior is quite natural in many other branches of physics where the Lorentzian character of the transformation has been recognized (multilayer optics [32,33], ray optics [34], laser cavity physics [35], quantum optics [36]). In this respect, I would finish these conclusions coming back to some early assertions of Vigoureux [37], who was probably the first who apprehended the large spectrum of physical problems to which the composition law of what we name today gyrovectors (and by consequence all the results of the present paper) is applicable: “The composition law of velocities, which is usually presented as a specific property of relativity, constitute the expression of a more general and more natural addition law in physics; each time that some physical magnitudes have an upper and a lower bound, the usual addition law should be replaced by an equation of this kind.”