### Dimensionality of random light

In the case that only one eigenvalue of the real part *Φ*
^{′}(**r**,*ω*) is nonzero, the electric-field vector fluctuates in just a single direction and the light is considered one dimensional. If, instead, only one eigenvalue of *Φ*
^{′}(**r**,*ω*) is zero, the electric field is restricted to a plane and the light is regarded two dimensional. However, when every eigenvalue is positive, the intensity of each Cartesian field component is nonzero for any orientation of the frame (since *a*
_{3} is the smallest obtainable intensity along a coordinate axis) and the electric-field vector fluctuates in all three dimensions. The physical dimensionality of the light field is thereby determined by the eigenvalues of *Φ*
^{′}(**r**,*ω*) as

$$\begin{array}{*{20}l} \text{1D light:} \quad a_{1}>0, \ a_{2}=0, \ a_{3}=0; \end{array} $$

(4)

$$\begin{array}{*{20}l} \text{2D light:} \quad a_{1}>0, \ a_{2}>0, \ a_{3}=0; \end{array} $$

(5)

$$\begin{array}{*{20}l} \text{3D light:} \quad a_{1}>0, \ a_{2}>0, \ a_{3}>0. \end{array} $$

(6)

We further define isotropic 2D light as one for which *a*
_{1}=*a*
_{2} in Eq. (5) and isotropic 3D light as one that satisfies *a*
_{1}=*a*
_{2}=*a*
_{3} in Eq. (6). In particular, because det*Φ*
^{′}(**r**,*ω*)=*a*
_{1}
*a*
_{2}
*a*
_{3} is invariant under orthogonal transformations, Eqs. (4)–(6) imply that both 1D and 2D light obey det*Φ*
^{′}(**r**,*ω*)=0, while for a genuine 3D light field det*Φ*
^{′}(**r**,*ω*)>0.

We stress that the number of nonnegative eigenvalues of the full complex polarization matrix *Φ*(**r**,*ω*) does not necessarily provide information about the physical dimensionality of light. For instance, the full polarization matrix of a circularly polarized light beam involves just a single nonzero eigenvalue, whereas its real part satisfies *a*
_{1}=*a*
_{2} and *a*
_{3}=0, thereby corresponding to (isotropic) 2D light in view of Eq. (5). Likewise, the complex polarization matrix of an incoherent and orthogonal superposition of a circularly polarized and a linearly polarized beam has two nonnegative eigenvalues, while in this case all three eigenvalues of the real-valued polarization matrix are nonzero. Hence, according to Eq. (6), the superposed field is genuinely 3D in character.

### Polarimetric dimension

Although Eqs. (4)–(6) establish the definitions for the dimensionality of a light field, they do not provide information how 1D-, 2D-, or 3D-like the light in question is. For example, an elliptically polarized beam is formally two dimensional, but from a practical point of view it can be regarded as one dimensional if the polarization ellipse is highly squeezed (cf. linear polarization). Therefore, to characterize the dimensional nature of a light field more quantitatively, we introduce the spectral polarimetric dimension, *D*(**r**,*ω*), via the relation

$$ D(\mathbf{r},\omega)=3-2d(\mathbf{r},\omega), $$

(7)

where *d*(**r**,*ω*) is the distance between the real-valued matrix *Φ*
^{′}(**r**,*ω*) and the identity matrix associated with isotropic 3D light, i.e.,

$$ d(\mathbf{r},\omega)=\sqrt{\frac{3}{2}\left[\frac{\text{tr}\boldsymbol{\Phi}^{\prime2}(\mathbf{r},\omega)} {\text{tr}^{2}\boldsymbol{\Phi}^{\prime}(\mathbf{r},\omega)}-\frac{1}{3}\right]}, $$

(8)

with the scaling chosen so that 0≤*d*(**r**,*ω*)≤1. We remark that an expression formally similar to Eq. (8), with *Φ*(**r**,*ω*) replacing *Φ*
^{′}(**r**,*ω*), has been employed to characterize the degree of polarization of random 3D light fields [20, 21]. The polarimetric dimension is thus a real number that obeys 1≤*D*(**r**,*ω*)≤3. Moreover, it is invariant under orthogonal transformations, but generally not under unitary operations, since the latter may alter the polarization state and, consequently, the dimensionality of the light.

The physical meaning of *D*(**r**,*ω*) becomes more apparent by writing Eq. (7) in terms of the eigenvalues of *Φ*
^{′}(**r**,*ω*), viz.,

$$ {\begin{aligned} D(\mathbf{r},\omega)\,=\,3-\frac{\sqrt{2\left[(a_{1}-a_{2})^{2}+(a_{1}-a_{3})^{2}+(a_{2}-a_{3})^{2}\right]}}{a_{1}+a_{2}+a_{3}}. \end{aligned}} $$

(9)

The above expression indicates that the minimum *D*(**r**,*ω*)=1 is always, and solely, encountered for 1D light (*a*
_{2} = *a*
_{3} = 0), while the maximum *D*(**r**,*ω*) = 3 is reached if, and only if, the field is completely 3D isotropic (*a*
_{1}=*a*
_{2}=*a*
_{3}). For 2D light (*a*
_{3}=0, *a*
_{2}>0), we find that 1<*D*(**r**,*ω*)≤2, with the upper limit taking place when the two principal intensities are equal (*a*
_{1}=*a*
_{2}). Values in the range *D*(**r**,*ω*)>2 are thereby clear signatures of 3D light [note that 3D light may nonetheless assume any value within the interval 1<*D*(**r**,*ω*)≤3].

Since *D*(**r**,*ω*) is generally not an integer, it should not be identified as such with the actual physical dimensionality of the light [specified by Eqs. (4)–(6)], but as an effective dimension characterizing the intensity-distribution spread. Figure 1 provides an interpretative illustration for the polarimetric dimension, in which principal-intensity distributions for three different 3D light fields have been depicted. In the left panel *a*
_{1} is significantly larger than the intensities in the other directions, whereupon the light is effectively one dimensional and thus *D*(**r**,*ω*)≈1. A practical realization of such a field would be a directional surface plasmon polariton beam [22–24]. In the middle panel *a*
_{1}≈*a*
_{2}≫*a*
_{3}, indicating that the light field is virtually 2D isotropic and hence *D*(**r**,*ω*)≈2. An unpolarized or a circularly polarized light beam of high degree of directionality [17, 25] would constitute an example. In the right panel all principal intensities are about equally distributed, which corresponds to isotropic 3D light and thereby yields *D*(**r**,*ω*)≈3, as is the case for instance with blackbody radiation.

### Examples

As concrete examples, we investigate the dimensionality of stationary polarized light and an evanescent wave created in total internal reflection.

#### Polarized random light

Let *E*
_{
α
}(**r**,*ω*) with *α*∈{*x*,*y*,*z*} represent a Cartesian component of the electric-field realization. Furthermore, let

$$ {\begin{aligned} \mu_{\alpha\beta}(\mathbf{r},\omega)&=|\mu_{\alpha\beta}(\mathbf{r},\omega)|e^{i\varphi_{\alpha\beta}(\mathbf{r},\omega)} \\ &=\frac{\left\langle E_{\alpha}^{\ast}(\mathbf{r},\omega)E_{\beta}(\mathbf{r},\omega)\right\rangle}{\sqrt{\left\langle|E_{\alpha}(\mathbf{r},\omega)|^{2}\right\rangle \left\langle|E_{\beta}(\mathbf{r},\omega)|^{2}\right\rangle}}, \quad \alpha,\beta\in\{x,y,z\}, \end{aligned}} $$

(10)

where *φ*
_{
α
β
}(**r**,*ω*) are real-valued phase factors, be the complex correlation coefficient between the *α* and *β* components. Since for polarized light all field components are completely correlated [20], i.e., |*μ*
_{
α
β
}(**r**,*ω*)|=1, we can express the polarization matrix in Eq. (1) of a polarized light field as

$$ \boldsymbol{\Phi}(\mathbf{r},\omega)= \left(\begin{array}{ccc} I_{x} & \sqrt{I_{x}I_{y}}e^{i\varphi_{xy}} & \sqrt{I_{x}I_{z}}e^{i\varphi_{xz}} \\ \sqrt{I_{x}I_{y}}e^{-i\varphi_{xy}} & I_{y} & \sqrt{I_{y}I_{z}}e^{i\varphi_{yz}} \\ \sqrt{I_{x}I_{z}}e^{-i\varphi_{xz}} & \sqrt{I_{y}I_{z}}e^{-i\varphi_{yz}} & I_{z} \\ \end{array} \right)\!\!, $$

(11)

in which the shorthand notations *I*
_{
α
}=〈|*E*
_{
α
}(**r**,*ω*)|^{2}〉 and *φ*
_{
α
β
}=*φ*
_{
α
β
}(**r**,*ω*) have been introduced for convenience, and the phases satisfy [26]

$$ \varphi_{xy}-\varphi_{xz}+\varphi_{yz}=2m\pi, \quad m\in\mathbb{Z}. $$

(12)

By taking the real part of Eq. (11) and utilizing Eq. (12) one then gets that

$$ {\begin{aligned} \text{det}\boldsymbol{\Phi}^{\prime}(\mathbf{r},\omega)=I_{x}I_{y}I_{z}\left[1-\left(c_{xy}^{2}+c_{xz}^{2}+c_{yz}^{2}\right)+2c_{xy}c_{xz}c_{yz}\right]=0, \end{aligned}} $$

(13)

where *c*
_{
α
β
}= cos*φ*
_{
α
β
}, implying that polarized light is necessarily 1D or 2D in nature. In other words, fully polarized 3D light does not exist.

This finding can intuitively be justified as follows. If a random light field with three electric components is spectrally fully polarized, it is at frequency *ω* represented by an ensemble of monochromatic field realizations as

$$\begin{array}{*{20}l} \left\{\mathbf{E}(\mathbf{r},\omega)e^{-i\omega t}\right\}=\{E(\mathbf{r},\omega)\}\mathbf{e}(\mathbf{r},\omega)e^{-i\omega t}, \end{array} $$

(14)

where *E*(**r**,*ω*) is a random scalar variable and **e**(**r**,*ω*) is a deterministic three-component vector. The realizations thus have identical polarization states although their spectral densities may vary, and because the electric-field vector of monochromatic light is necessarily bounded in a plane [27], each realization lies in the same plane. Consequently, the random light that these monochromatic fields represent must fluctuate in this plane as well.

#### Random evanescent wave

As a second example, we consider an optical evanescent wave excited in total internal reflection at a planar dielectric interface (*z*=0) by a stationary beam [28]. Both medium 1 (*z*>0) and medium 2 (*z*<0), having refractive indices *n*
_{1}(*ω*) and *n*
_{2}(*ω*), respectively, are taken lossless, and the *x* axis is chosen to coincide with the surface-propagation direction. Moreover, the incoming beam, generally carrying both an *s*-polarized and a *p*-polarized constituent, hits the boundary at the angle of incidence *θ*(*ω*) that satisfies *θ*
_{c}(*ω*)<*θ*(*ω*)<*π*/2, with \(\theta _{\mathrm {c}}(\omega)=\arcsin {\tilde {n}^{-1}(\omega)}\) being the critical angle and \(\tilde {n}(\omega)=n_{1}(\omega)/n_{2}(\omega)>1\). Under these conditions, the spatial part of the electric-field realization for the evanescent wave takes on in Cartesian coordinates the form [4, 5]

$$ {\begin{aligned} \mathbf{E}(\mathbf{r},\omega)=\frac{1}{\chi(\omega)}\left[ \begin{array}{l} -i\gamma(\omega)t_{p}(\omega)E_{p}(\omega) \\ \chi(\omega)t_{s}(\omega)E_{s}(\omega) \\ \sin{\theta(\omega)}t_{p}(\omega)E_{p}(\omega) \\ \end{array} \right] e^{ik_{1}(\omega)\sin{\theta(\omega)}x}e^{-k_{1}(\omega)\gamma(\omega)z}, \end{aligned}} $$

(15)

where *E*
_{
s
}(*ω*) and *E*
_{
p
}(*ω*) are, respectively, the complex field amplitudes of the *s*- and *p*-polarized components of the incident light. Furthermore,

$$ {\begin{aligned} \chi(\omega)&=\sqrt{\sin^{2}{\theta(\omega)}+\gamma^{2}(\omega)}, \\ \gamma(\omega)&=\tilde{n}^{-1}(\omega)\sqrt{\tilde{n}^{2}(\omega)\sin^{2}{\theta(\omega)}-1}, \end{aligned}} $$

(16)

with *γ*(*ω*) being the decay constant of the evanescent wave, and

$$ \begin{aligned} t_{s}(\omega)&=\frac{2\cos{\theta(\omega)}}{\cos{\theta(\omega)}+i\gamma(\omega)}, \\ t_{p}(\omega)&=\frac{2\tilde{n}^{2}(\omega)\cos{\theta(\omega)}\chi(\omega)}{\cos{\theta(\omega)}+i\tilde{n}^{2}(\omega)\gamma(\omega)} \end{aligned} $$

(17)

are the Fresnel transmission coefficients of the two polarizations, and *k*
_{1}(*ω*) is the wave number in medium 1.

On next calculating the polarization matrix in Eq. (1) for the evanescent wave given by Eq. (15), and then extracting the real part *Φ*
^{′}(**r**,*ω*)=*Φ*
^{′}(*z*,*ω*) from the obtained expression, we find that

$$ {\begin{aligned} \text{det}\boldsymbol{\Phi}^{\prime}(z,\omega)=\frac{\sin^{2}{\theta(\omega)}\gamma^{2}(\omega)}{\chi^{4}(\omega)} w_{s}(z,\omega)w_{p}^{2}(z,\omega)\left[1-|\mu(\omega)|^{2}\right]. \end{aligned}} $$

(18)

Above, \(w_{\nu }(z,\omega)=|t_{\nu }(\omega)|^{2}I_{\nu }(\omega)e^{-2k_{1}(\omega)\gamma (\omega)z}\phantom {\dot {i}\!}\) is proportional to the energy density of the *ν*∈{*s*,*p*} polarized part of the evanescent wave at height *z*, with *I*
_{
ν
}(*ω*)=〈|*E*
_{
ν
}(*ω*)|^{2}〉 being the intensity of the respective component of the incoming beam, and \(\mu (\omega)=\left \langle E_{s}^{\ast }(\omega)E_{p}(\omega)\right \rangle /\sqrt {I_{s}(\omega)I_{p}(\omega)}\) is the correlation coefficient among the *s*- and *p*-polarized constituents of the incident light. Equation (18) especially shows that det*Φ*
^{′}(*z*,*ω*)=0 only when the excitation beam is totally polarized, i.e., *I*
_{
s
}(*ω*)=0, *I*
_{
p
}(*ω*)=0, or |*μ*(*ω*)|=1, and in this case the ensuing evanescent wave is either 1D or 2D in character. Generally, however, when the incident beam is partially polarized [*I*
_{
s
}(*ω*)≠0, *I*
_{
p
}(*ω*)≠0, and |*μ*(*ω*)|≠1], we obtain from Eq. (18) that det*Φ*
^{′}(*z*,*ω*)>0, corresponding to genuine 3D light. This discovery reveals that optical evanescent waves are predominantly 3D light fields, which necessitate a rigorous 3D treatment to fully describe their electromagnetic properties.

Motivated by the above result we further examine how close to isotropic 3D light an evanescent wave can be, and to this end we employ the polarimetric dimension *D*(**r**,*ω*) defined in Eq. (7). Utilizing Eqs. (15)–(17) then yields that the fundamental upper limit that *D*(**r**,*ω*) can attain for such a wave is

$$ D(\mathbf{r},\omega)=3-\frac{2}{\sqrt{1+3\tilde{n}^{4}(\omega)\chi^{4}(\omega)}}, $$

(19)

which is reached when the incident light possesses the properties

$$ {\begin{aligned} |\mu(\omega)|=0, \quad \frac{I_{s}(\omega)}{I_{p}(\omega)}=\left[\frac{\sin^{4}{\theta{(\omega)}}+\gamma^{4}(\omega)}{\chi^{4}(\omega)}\right]\left[\frac{|t_{p}(\omega)|}{|t_{s}(\omega)|}\right]^{2}. \end{aligned}} $$

(20)

For a high refractive-index contrast surface, such as GaP and air with \(\tilde {n}(\omega)\approx 4\) in the optical regime [29], Eq. (19) shows that the polarimetric dimension may be as high as *D*(**r**,*ω*)≈2.96, while for a typical SiO_{2}–air interface the maximum is around *D*(**r**,*ω*)≈2.67.