The authors of [1] report on a “polarization coherence theorem.” Although their result is technically sound, we dispute its status as a new theorem.

It is well known that unequal field amplitudes reduce the achievable visibility in the interferogram produced by their superposition. The visibility is restored to its maximal value by equalizing the field amplitudes. This fact was recognized early on by Zernike for Young’s double-slit interference [2]: “By definition the ‘degree of coherence’ of two light-vibrations shall be equal to the visibility of the interference fringes that may be obtained from them under the best circumstances, that is, when both intensities are made equal…” (italics in the original). The same fact was noted by Wolf in the context of polarization [3]: “Thus there exists a pair of directions for which the two intensities are equal. For this pair of directions the degree of coherence $|{\mu }_{xy}|$ of the electric vibrations has its maximum value and this value is equal to the degree of polarization of the wave.” The same conclusion was later reached by Mandel [4] through a different line of thought.

These well-known findings can be summarized quantitatively in a simple manner. The coherence properties of any two-dimensional degree of freedom of an optical field, whether polarization [3], scalar field amplitudes at two points [2,5], or two spatial modes [6], are captured by a $2\times 2$ Hermitian semi-positive coherence matrix

Any such matrix $\mathbf{G}$ can be written as

The normalized intensity imbalance between the two field components (or distinguishability) is $D=\frac{{\lambda }_1-{\lambda }_2}{{\lambda }_1+{\lambda }_2}|\cos \varphi |$, and is in turn not a unitary invariant. When $E_1$ and $E_2$ are the field amplitudes at two points, then $D$ is the normalized difference in their intensity. When $E_1$ and $E_2$ are two polarization components, $D=|S_1|/S_0$ is the normalized difference between the statistical averages of their squared magnitudes in the horizontal/vertical basis.

From these two definitions we have $V^2+D^2={(\frac{{\lambda }_1-{\lambda }_2}{{\lambda }_1+{\lambda }_2})}^2$. Because the eigenvalues are unitary invariants, this result applies to any field described by a $2\times 2$ coherence matrix. Historically, the quantity $P=\frac{{\lambda }_1-{\lambda }_2}{{\lambda }_1+{\lambda }_2}=\sqrt{1-4\frac{\det \mathbf{G}}{{(\mathrm{Tr}\mathbf{G})}^2}}$ has been used as a definition for the degree of polarization and is given by $P=\sqrt{S_1^2+S_2^2+S_3^3}/S_0$ in terms of the Stokes parameters, but it can equally quantify coherence for any other degree of freedom described by the model in Eqs. (1) and (2). The relation $V^2+D^2=P^2$, which was presented in [1] as a new theorem, readily follows.

Trivially, the maximum visibility $V_{\max }=\frac{{\lambda }_1-{\lambda }_2}{{\lambda }_1+{\lambda }_2}$ for the field at two points corresponds to $\varphi =\frac{\pi }{2}$ in Eq. (2), whereupon $D=0$. That is, the unitary transformation that equalizes the field magnitudes (“the best circumstances” referred to in [2]) necessarily produces the maximum visibility. Similarly, the maximum distinguishability $D_{\max }=\frac{{\lambda }_1-{\lambda }_2}{{\lambda }_1+{\lambda }_2}$ corresponds to $\varphi =0$ in Eq. (2), whereupon $\mathbf{G}$ is diagonalized and $V=0$. That is, the unitary transformation that maximizes the difference between the field squared magnitudes also eliminates the interference visibility. These basic results quantify the common notions underpinning the above-cited quotes by Zernike and Wolf. Finally, we note that similar formulae to those presented in [1] were previously reached by Luis [8] (Eq. 11).

We commend the authors of [1] on re-introducing this fact, but emphasize that presenting it as a new polarization-specific relation may be confusing.