What Does a Partially Polarized Wave Look Like?
Or why an ellipse is no longer enough to tell the whole story.
There is something wonderfully simple about representing a polarized electromagnetic wave. Draw two perpendicular axes, let the electric field oscillate, and the tip of its vector traces an ellipse. That ellipse may flatten into a straight line or, conversely, become perfectly circular. Between these two extremes lies an entire world of elliptical polarization states, each characterized by an orientation, an ellipticity, and a sense of rotation.
But here is a question that optics textbooks tend to address more readily with equations than with drawings: what does a partially polarized wave actually look like?
This question came back to me recently while reading a doctoral thesis. The electric field was mathematically expressed as the superposition of two contributions, one fully polarized and the other partially polarized.
At first sight, I found this rather puzzling. After all, when the two transverse components of an electric field maintain a fixed amplitude ratio and phase difference, the field describes a well-defined polarization state. Partial polarization only emerges when these relationships cease to remain fixed over the observation ensemble. Whether these variations are random or arise from deterministic temporal evolution, their effect must be described statistically.
Simply adding two deterministic electric fields does not, by itself, explain partial polarization.
And yet, there was something appealing about the original idea. If a wave can be only partially polarized, why not attribute a fraction of its energy to a perfectly organized component and leave the remainder to fluctuations?
In fact, such a decomposition makes perfect sense at the level of intensity, provided that we distinguish a fully polarized contribution from a completely unpolarized one. The intensity is a quadratic quantity, obtained from the statistical average of the squared field amplitude. This is precisely what makes it possible to express the total intensity as the sum of two statistically uncorrelated contributions.
The difficulty appears when we attempt to translate this decomposition of intensity back into a decomposition of the electric field itself. If the polarized contribution carries a fraction P of the total intensity, its field amplitude must be multiplied by √P, not by P. Similarly, the unpolarized contribution must be weighted by √(1-P), assuming that both fields are properly normalized and statistically uncorrelated.
This seemingly minor algebraic detail points to a much deeper issue: we are trying to represent, through the addition of electric fields, a property that fundamentally belongs to their statistics.
When the Ellipse Becomes a Statistical Property
Consider the two complex transverse components of an optical field, Aₓ and Aᵧ. If their amplitude ratio and phase difference remain constant, the polarization state is perfectly defined. A Jones vector e=(Aₓ,Aᵧ)ᵗ is sufficient to describe it, and the corresponding ellipse retains an unchanging shape, regardless of the overall optical phase.
But now suppose that these amplitudes and phases fluctuate. Over a short interval, the electric field may trace one ellipse. A little later, another. And if we observe the field for sufficiently long, no single ellipse will be able to characterize the entire ensemble.
This is where a particularly elegant mathematical object comes into play: the coherence matrix, sometimes called the polarization matrix.
This matrix brings together the four correlations between the field components. It no longer describes a particular realization, but rather the electromagnetic information that survives statistical averaging.
Its trace gives the total intensity. Its determinant, together with its trace, allows us to calculate the degree of polarization:
When P=1, the light is fully polarized. When P=0, it is completely unpolarized, in the sense of this second-order statistical description. Between these two extremes, the light is partially polarized.
But can we give this matrix a geometric representation?
Two Ellipses Hidden in a Matrix
The answer lies in an operation familiar to physicists: diagonalization. Since the coherence matrix Γ is Hermitian and positive semidefinite, it possesses two real, nonnegative eigenvalues, λ₁ and λ₂, associated with two orthogonal eigenvectors. In other words, it can always be decomposed into two contributions corresponding to fully polarized states:
Each of these eigenvectors corresponds to a polarization ellipse. The two states are orthogonal in the sense of the complex Jones inner product. For noncircular elliptical states, their major axes are perpendicular, and their senses of rotation are opposite.
The degree of polarization can then be deduced directly from the contrast between the eigenvalues:
And there we have our first picture: a partially polarized wave can be represented statistically by two orthogonal polarization ellipses whose intensities add together.
When the two eigenvalues become equal, the degree of polarization vanishes. Neither eigenstate is then privileged, and the eigenvectors themselves cease to be uniquely defined.
Conversely, when one of the eigenvalues vanishes, only one ellipse remains: that of a fully polarized state.
However, these two ellipses should not be interpreted as two deterministic oscillations that can simply be added point by point. It is their contributions to the coherence matrix that add together. This distinction is essential.
What If We Superimposed All the Ellipses?
A second approach is to abandon the search for a single ellipse and instead represent a multitude of possible realizations of the electric field.
Imagine simulating a hundred, a thousand, or even ten thousand random Jones vectors, all drawn from a statistical ensemble characterized by the same coherence matrix. For each realization, we draw the corresponding polarization ellipse, making it almost transparent.
Gradually, the curves overlap. Some regions become more densely populated than others. A pattern begins to emerge, suggesting a statistical envelope around a preferred direction.
Bringing Time Continuity into the Picture
Rather than considering isolated, discrete snapshots of the electric field, we may wish to represent it as a continuous process evolving over time.
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We can then trace the trajectory of the tip of the electric field vector over a given time interval, either in two dimensions, by projecting it onto the plane transverse to the direction of propagation, or in three dimensions.
An important distinction remains, however: the degree of polarization tells us how the field behaves statistically, but not how rapidly its polarization state fluctuates. Two waves may have exactly the same coherence matrix while exhibiting very different temporal dynamics. To generate a continuous trajectory, we must therefore introduce an additional assumption about the temporal correlations of the field.
Poincaré Had Another Idea
Henri Poincaré proposed a representation whose elegance lies precisely in its ability to bring together every possible state of polarization within a single geometric object.
On the sphere that now bears his name, each fully polarized state corresponds to a point on its surface. Linear polarizations lie along the equator, circular polarizations occupy the poles, and elliptical polarizations fill the regions in between.
A partially polarized wave, on the other hand, is represented by a normalized Stokes vector whose endpoint lies inside the sphere. The distance of this point from the center is precisely the degree of polarization, P.
But rather than drawing only this mean point, why not represent all the instantaneous polarization states as a cloud of points on the surface of the sphere?
Each realization of the electric field has its own polarization state, and therefore its own point on the sphere. The average of these points, weighted by their respective intensities, gives us back the Stokes vector of the partially polarized wave.
Yet a subtlety emerges. The same coherence matrix can correspond to very different point clouds.
This may seem surprising. After all, diagonalizing the coherence matrix provides us with two eigenvalues and two orthogonal eigenstates, which are uniquely determined. Why should these not be sufficient to describe the distribution?
Because these two eigenstates constitute one particular decomposition of the matrix, rather than necessarily being the only two states actually encountered by the field.
Consider completely unpolarized light. We can produce it by alternating between two orthogonal polarization states with equal probability. On the Poincaré sphere, the realizations then concentrate at two antipodal points.
But we could just as well imagine polarization states distributed uniformly over the entire sphere. Their average is also zero, and the resulting coherence matrix is exactly the same.
And this is not restricted to completely unpolarized light. Consider, for instance, a degree of polarization of P=0.5. The same mean Stokes vector can be obtained either by combining two orthogonal states with probabilities of 75% and 25%, or by distributing pure states uniformly along a circle of constant latitude on the Poincaré sphere.
Different distributions of fully polarized states can possess the same mean Stokes vector and, consequently, the same coherence matrix.
We have reached a fundamental limitation of our representation. The coherence matrix contains all the second-order polarimetric information, but it does not, by itself, determine the complete statistical distribution of the field.
The ambiguity we have just encountered is not specific to the Poincaré sphere. In fact, it applies to every representation we have explored whenever we attempt to reconstruct individual realizations of the electric field from its coherence matrix.
The two eigenstates, their respective intensities, and the mean Stokes vector are uniquely determined by the coherence matrix, apart from possible degeneracies. But the distribution of individual Jones vectors is not. Whether we draw a cloud of points on the Poincaré sphere, a superposition of polarization ellipses, or a continuously fluctuating electromagnetic wave, we must introduce additional assumptions about the statistics of the field.
This is precisely what we have done in our simulations by choosing complex Gaussian statistics. Another statistical model, compatible with the same coherence matrix, could produce very different pictures.
The coherence matrix tells us what remains statistically invariant across realizations, but not which realizations actually occurred.
Could Color Help Us Visualize Partial Polarization?
Perhaps we could go one step further and let color carry some of the information that an ellipse alone cannot convey.
The idea is to borrow the familiar HSV color representation, but give each of its three components a physical meaning. Hue (H) encodes the orientation of the polarization ellipse, saturation (S) measures its degree of linearity, and value (V) represents the intensity-weighted density of field-tip positions. A linearly polarized state is therefore represented by a vivid color, while a circularly polarized state becomes achromatic, since its ellipse has no preferred orientation.
We can now return to our statistical ensemble of Jones vectors. Rather than drawing all the corresponding ellipses in the same color, we assign each realization its own hue and saturation, according to its orientation and ellipticity. By superimposing their contributions, we obtain a colored statistical portrait of the electric field.
The results are rather revealing. When the wave is fully polarized (P=1), all realizations share the same polarization ellipse, even though their amplitudes may fluctuate. The resulting image therefore exhibits a single characteristic color. For a circularly polarized wave, this color disappears altogether: the distribution becomes gray, reflecting the absence of any preferred orientation.
As the degree of polarization decreases, the picture gradually transforms. Different elliptical states contribute their own colors, which overlap and mix. What was initially a single, well-defined polarization state becomes a rich distribution of orientations and ellipticities. The change is particularly striking when starting from circular polarization: an initially achromatic distribution progressively develops a whole palette of colors.
Of course, this representation is not unique. The resulting color distribution depends on the statistical model chosen to generate the Jones realizations. But it offers an appealing way of bringing together two complementary aspects of partial polarization: the geometry of individual polarization states and the statistical distribution of the electric field.
When Polarization Meets Speckle
Let us finally consider what happens when coherent light illuminates a rough surface. The interference of the waves scattered by its microscopic irregularities produces the familiar granular intensity pattern known as speckle.
Under fully developed speckle conditions, each independent complex Gaussian polarization mode produces exponentially distributed intensity fluctuations. When two orthogonal polarization modes contribute to the detected field, however, their intensities add together, and the resulting intensity distribution depends on their relative statistical weights.
If the scattered light is fully polarized, the intensity contrast, defined as the ratio of its standard deviation to its mean, is equal to one. But when the scattered field becomes partially polarized, two statistically uncorrelated, orthogonally polarized contributions may coexist, each producing its own speckle pattern. Since a polarization-insensitive detector records the sum of their intensities, the fluctuations are partially averaged out. The resulting contrast decreases according to C=√{(1+P²)/2}, reaching 1/√2 ≈ 0.707 for completely unpolarized light.
Surprisingly, this reduction is not particularly striking to the naked eye: even unpolarized light produces a strongly contrasted speckle pattern! Nor do the grains become larger or optically blurred, since their characteristic dimensions remain governed by the aperture and the spatial coherence properties of the field.
In this spatially resolved representation, we retain the same definitions for hue and saturation, but assign brightness to the local total intensity rather than to the statistical density of field-tip positions.
Yet our HSV representation reveals something that ordinary intensity images largely conceal. By assigning hue to the orientation of each local polarization ellipse, saturation to its degree of linearity, and brightness to the total intensity, we uncover the diversity of polarization states hidden within the speckle pattern. A fully polarized field produces grains sharing the same polarization color, although their intensities fluctuate considerably. As the degree of polarization decreases, different grains exhibit increasingly varied colors, revealing the changing polarization structure of the scattered field. What appears in intensity as a relatively modest reduction in contrast becomes, in color, a remarkable transformation of the underlying electromagnetic field.
So, What Does a Partially Polarized Wave Really Look Like?
An ellipse describes an individual polarization state; two orthogonal ellipses reveal the eigenstructure of its coherence matrix; a cloud of realizations makes its statistical nature visible; and a continuous trajectory reminds us that the electric field never stops evolving. The Poincaré sphere captures what these realizations share, while color allows us to explore the diversity they conceal. Each representation reveals something different, and each leaves something out.
Further reading:
Julien Fade, PhD thesis (2009);
Franco Gori, Matrix treatment for partially polarized, partially coherent beams, 1998;
Edward Collett and Beth Schaefer, Visualization and calculation of polarized light, 2008;
Luo et al., Singular-value decomposition and electromagnetic coherence of optical beams, 2022.
José J. Gil (2014) — Interpretation of the coherency matrix for three-dimensional polarization states. Physical Review A, 90, 043858.
Joseph W. Goodman (1976) — Some fundamental properties of speckle. Journal of the Optical Society of America, 66, 1145–1150.
