P. Soleillet, “Sur les paramètres caractérisant la polarisation partielle de la lumière dans les phénomènes de fluorescence,” Ph.D. dissertation (École Normale Supérieure, 1929) and published in Annales de Physique 12 (1929), pp. 23–59.

S. Chandrasekhar, “Radiative transfer: a personal account,” in Selected Papers, Vol. 2 of Radiative Transfer and Negative Ion of Hydrogen (University of Chicago, 1989), pp. 511–542.

Interview of S. Chandrasekhar by S. Weart on 18 May 1977, Niels Bohr Library & Archives, American Institute of Physics, College Park, MD, USA, http://www.aip.org/history/ohilist/4551_2.html .

Regrettably, such a confusion affected the V element in three equations in [21]. To comply with the present conventions, these equations should be corrected as follows: 4.100: S3=V=〈SL〉−〈SR〉=+i[〈ExEy*〉−〈Ex*Ey〉], 4.104: S0≔〈〈E|σ0|E〉〉S1≔〈〈E|σ1|E〉〉S2≔〈〈E|σ2|E〉〉S3≔−〈〈E|σ3|E〉〉, 4.116: s⃗3=12(s⃗L−s⃗R)=(0,0,0,1).

Modern treatments of the topic prefer to work with the correlation matrix 〈|Ex|2〉T〈ExEy*〉T〈EyEx*〉T〈|Ex|2〉T, instead of the Stokes parameters. Particularly thorough discussions on polarization statistics are found in [20,36].

M. Born and E. Wolf, “Statistical optics,” in Fundamentals of Photonics, B. E. A. Saleh and M. C. Teich, eds. (Wiley, 1989), Chap. 10, pp. 342–383.

R. C. Jones, “A new calculus for the treatment of optical systems,” series of eight papers published in the J. Opt. Soc. Am. from 1941 to 1956.

We prefer here the “Perrin–Mueller matrix” to the more common “Mueller matrix” not only to give due credit to Francis Perrin, but also because Hans Mueller’s publications on the subject are unfortunately impossible to find (as was confirmed to us by a representative of the libraries of the Massachusetts Institute of Technology, where Mueller worked as a professor). The only written trace of Mueller’s work is a little revealing conference abstract [39]; therefore, we strongly suggest consulting Perrin’s comprehensive 1942 paper [40] for more information on the PM matrix of arbitrary isotropic, symmetric, and asymmetric scatterers. (Credit should be given also to Paul Soleillet [31,41], who got the idea almost two decades before Perrin and Mueller.) Besides, the notation PM recalls the “phase matrix” introduced by Chandrasekhar to treat radiative transfer of polarization.

H. Mueller, “The foundation of optics,” abstract for a contributed paper presented at the 1948 Winter Meeting of the OSA.

R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, E. W. van Stryland, D. R. Williams, and W. L. Wolfe, eds., 2nd ed. (McGraw-Hill, 1995), Vol. 2, Chap. 22, 22.1–22.37.

J. M. Bennett, “Polarizers,” in Handbook of Optics, M. Bass, E. W. van Stryland, D. R. Williams, and W. L. Wolfe, eds., 2nd ed. (McGraw-Hill, 1995), Vol. 2, Chap. 3, 3.1–3.12.

S.-M. F. Nee, “Polarization measurement,” in The Measurement, Instrumentation, and Sensors: Handbook, J. G. Webster, ed. (CRC and IEEE, 1999), Chap. 60, 1655–1679.

It is difficult to ascertain an accurate chronology regarding the emergence of models using the Stokes vector to describe polarized light propagation. Several physicists were particularly prolific to this end, though arguably, long after Sir Stokes’ 1852 paper [30]. Chandrasekhar (apparently unaware of Soleillet’s findings [31]) was the first to reintroduce the then forgotten four real Stokes parameters in the English-speaking community, as part of his pioneering work on radiative transfer in astrophysics [32,33]. A historical overview can also be found in [20].

N. Korolkova, “30.4 Stokes operators questioned: degree of polarization in quantum optics,” in Quantum Information Processing, T. Beth and G. Leuchs, eds., 2nd ed. (Wiley, 2005), pp. 413–416.

More historical and background information on Paul Soleillet and Francis Perrin can be found on Oriol Arteaga’s home page: http://www.mmpolarimetry.com

R. M. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier Science, 1987).

N. G. Parke, “Matrix optics,” Ph.D. dissertation (MIT, 1948).

N. G. Parke, “Matrix algebra of electromagnetic waves,” (MIT Research Laboratory of Electronics, 1948).

N. G. Parke, “Statistical optics: I. Radiation,” (MIT Research Laboratory of Electronics, 1949).

N. G. Parke, “Statistical optics: II. Mueller phenomenological algebra,” (MIT Research Laboratory of Electronics, 1949).

W. A. Shurcliff, “Mueller calculus and Jones calculus,” in Polarized Light (Harvard University, 1962), Chap. 8.

Additional remarks, and references with respect to both matrix methods can be found in the last paragraph of J. M. Bennett, “Polarization,” in Handbook of Optics Volume I, M. Bass, E. W. Van Stryland, D. R. Williams, and W. L. Wolfe, eds., 2nd ed. (McGraw-Hill, 1995), Chap. 5.

H. G. Akarçay, A. Hohmann, A. Kienle, M. Frenz, and J. Rička, “Monte Carlo modeling of polarized light propagation. Part II. Stokes versus Jones,” Appl. Opt.53, 7586–7602 (2014).

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1999).

J. Rička and M. Frenz, “Polarized light: electrodynamic fundamentals” and “From electrodynamics to Monte Carlo simulations,” in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch and M. J. C. van Gemert, eds., 2nd ed. (Springer, 2011), Chap. 4, pp. 65–108 and Chap. 7, pp. 203–266, respectively.

Despite the “quantum” notation, the present treatment remains essentially within classical optics; i.e., we only consider pure and mixed coherent states. For the so-called “nonclassical” polarization states, see, e.g., [23–25].

J. S. Hendricks and T. E. Booth, “MCNP variance reduction overview,” in Monte-Carlo Methods and Applications in Neutronics, Photonics and Statistical Physics (Springer, 1985), pp. 83–92.

This low aperture approximation is also used in the wave-optical treatment in Section 7.5.6 of [21], but the suggested validity range is incorrect. The correct wave-optical approximation, that is valid for moderate apertures up to 0.6, is 〈f|e^D〉=[e^f*·e^D]κ(n^D,k^out)O(n^D,k^out). Wave optical aspects are included in the Fresnel inclination factor κ(n^D,k^out)=(1+k^out·n^D)/2 (as it occurs in Fresnel–Kirchhoff diffraction theory [46,47]) and O(n^D,k^out) is the scalar overlap that describes the effective aperture of the observation. The bracketed quantity [e^f*·e^D] is the same as in geometrical approximation, with e^D from Eq. (18).

M. Born and E. Wolf, “Elements of the theory of diffraction,” in Principles of Optics (Pergamon, 1970), Chap. 8, pp. 370–455.