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Abstract
An electric point dipole is the simplest model for emitters, antennas, and other nanostructures. For a dipole sufficiently small in comparison to the wavelength of the radiated electromagnetic wave, its electric-field intensity and Poynting vector are fully understood. Firstly, both decrease in magnitude with increasing radial distance from the dipole location. Secondly, the Poynting vector increases in magnitude with the sinusoid squared of the polar angle measured from the dipolar axis. We will consider here the degree of circular polarization (CP) of the electromagnetic wave induced by an electric point dipole. Resultantly, we learned that the near-perfect CP states form a pair of hollow-cylindrical surfaces. The radius of those surfaces designating near-perfect CP states is approximately one wavelength detached from the dipolar axis. However, these pictures are appreciably altered for the electromagnetic field induced by an array of electric point linear dipoles.
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1. Introduction
An electric point dipole is the simplest model for optical emitters lying inside and/or near nanostructures [1]. Antennas of varying sizes are also modeled as electric dipoles. When the wavelength of the induced electromagnetic (EM) wave is sufficiently larger than the characteristic feature size of an electric point dipole, the usual formulas for the field variables are valid from nano-photonics through medium-scale antennas to large-scale geophysical situations [2–4].
For generic EM waves, the fundamental parameters include the field variables and the attendant Poynting vector. See the schematic Fig. 1(a), where coordinate systems are introduced. Particularly, the Poynting vector is pointing in the purely radial $r$-direction [1,5]. From a simple physical perspective, a certain EM field can be characterized by its intensity as a quantitative measure and its polarization state as its qualitative measure [6]. Normally, the higher an intensity the more useful an EM field. Meanwhile, the polarization state is linked to direction and/or interferences between field components.
Fig. 1. (a) The coordinate systems centered on an active electric point dipole with a unit vector $\hat{p}$ denoting a linear polarization. (b) A tapered cylinder or a pencil-shaped hull denoting states of near-circular polarization. Comparison of the dipole-induced field to the earth, where the horizontal plane divides the northern hemisphere from the southern hemisphere. (c) The degree of circular polarization (DoCP) on the northern hemisphere along a meridional plane that encompasses five electric linear point dipoles standing in parallel.
There are two issues as regards a state of polarization. Firstly, we need to suppose a certain plane over which two orthogonal polarization components are resolved. Secondly, either linear or circular polarized light is desired depending on an application in mind. For example, one of transverse electric (TE) or transverse magnetic (TM) state is desired. On Fig. 1(a), the pair $\{{{E_z},{E_\rho }} \}$ (axial and cylinder-radial) on a $z\rho$ -plane is more suitable than $\{{{E_r},{E_\theta }} \}$ (sphere-radial and polar) on a $r\theta$-plane depending on emission configurations of interest. For example, the lines of surface gratings determine whether an electric field is TE or TM. Since the EM field induced by a point dipole is essentially spherical, it seems appropriate to take $\{{{E_r},{E_\theta }} \}$ as characterizing the polarization state [1,6].
For an EM field, let us list a set of well-known properties [7,8], that underlies this study. To this goal, let $\{{\vec{E},\vec{H}} \}$ be the electric and magnetic fields propagating through spaces free of electric charges and electric currents. According to the energy conservation law, the energy density of an EM field balances the linear momentum flux, namely, the Poynting vector $\vec{P} \equiv \textrm{Re} ({{{\vec{E}}^\ast } \times \vec{H}} )$. The Poynting vector itself is decomposable into its orbital and spin energy fluxes as $\vec{P} = {\vec{F}^o} + {\vec{F}^s}$. By ${\vec{F}^s} \equiv \nabla \times \left( {{\textstyle{1 \over 2}}\vec{S}} \right)$, the spin energy flux is proportional to the curl of the ‘spin angular momentum density’ $\vec{S}$ (SAM density for short) [8–10]. The polarization state of an EM field is characterized by the SAM density $\vec{S} \equiv {\mathop{\rm Im}\nolimits} ({{{\vec{E}}^\ast } \times \vec{E}} )$ which is in turn serves as the helicity flux according to the helicity conservation law [7]. The orbital angular momentum (OAM) $\vec{r} \times {\vec{F}^o}$ and the spin angular momentum (SAM) $\vec{r} \times {\vec{F}^s}$ are formed by taking cross products respectively of the orbital and spin energy fluxes with the displacement vector $\vec{r}$ from a certain point in space.
Of an OAM and a SAM, an OAM has been known to be a key player in many applications, for instance, as in communication channels. Since we are largely concerned with the polarization states, let us take an example in optical fabrication. The distinction between linearly and circularly polarized EM waves plays a great role in fabricating nanoneedles of helical structures [11]. Of course, the hydrodynamical and thermal deformations of materials into such nanoneedles exert nonnegligible impacts on the final shapes. For instance, polarization-sensitive azo-polymers get moved by the intensity gradients of an EM field along a certain polarization direction [11]. In a certain application, a SAM is effective without the help of an OAM, for instance, in fabricating surface helical relief [11].
In this study, we are concerned particularly with the behaviors of the EM wave induced by an electric point dipole in the near field, which we define to be on the order of only a few wavelengths around the position of a dipole [5]. Not only far-field behaviors but also near-field behaviors of EM waves play crucial roles in wave generation [12], transports of energy and information [13,14], and detection [15–18].
Although one cannot easily recall the spatial distribution of the degree of circular polarization (DoCP) for a dipole-induced EM field, one can look up a desired formula from increasingly prolific yet scattered literatures [5,8,18,19]. However, one would not be sure of the exact locations of the circular polarizations for a dipole-induced EM field. This point will be fully addressed in this study. In this respect, we will present an exact formula for the locations of near-circular polarizations. Resultantly, we found that such locations in space make a couple of surfaces aligned along the dipolar axis as schematically displayed on Fig. 1(b). We have derived that formula for the first time as far as we are aware of.
In addition, we have examined a linear array of five vertical electric dipoles as shown in Fig. 1(c) for its spatial distribution of polarization states. We will compare our results with those of optical needles with respect to field intensities [20,21]. Besides, arrayed Rayleigh scatters or emitters are working in an analogous fashion to give rise to a near-vertical emission upon excitation [9,14]. In this regard, we will make comments on dipoles arranged over a graphene disk [2] or other relevant configurations on metasurfaces [3].
This study is structured in the following way. Section 2 presents well-known fundamental formulas for the EM fields induced by electric point linear dipoles. Section 3 provides a set of essential properties for the degree of circular polarization. Sections 4 handles the exact formula for the near-circular polarizations. Section 5 discussed phase relations among electric fields. Section 6 provides solutions to a simple array of point dipoles. Section 7 offers discussions followed by Section 8 with conclusion.
2. Fundamental analysis
The field point is located at $\vec{r} \equiv r{\hat{e}_r}$ as displayed on Fig. 1(a). It is worth stressing that $\{{\rho ,r} \}$ are respectively the cylinder- and sphere-radial coordinates. Meanwhile, $0 \le \theta \le \pi$, while $0 \le \phi < 2\pi$. Let $\{{t,\omega } \}$ be frequency and time with $\omega > 0$ and $t > 0$ for time-harmonic monochromatic fields. The vacuum wave number is given by $k \equiv {\omega / c}$ with c as the light speed in vacuum. With $\textrm{i} \times \textrm{i} \equiv{-} 1$, $\textrm{exp} ({ - \textrm{i}\omega t} )\to \textrm{exp} ({ - \textrm{i}t} )$ for the temporal factor when making a substitution $\omega \times t \to t$ [13].
To save notations, spatial coordinates are redefined by the substitution $k\vec{r} \to \vec{r}$, according to which $k\nabla \to \nabla$ [18]. Since k is $2\pi$ times the inverse wavelength of an EM field, our near field is henceforth characterized by the dimensionless range on the order of $r = O(1 )$, being $0.3 < r < 3$ for instance. This range is sometimes called an ‘intermediate field’ [5,14,18,19]. In molecule-based photonics [14,18] and nano-photonics [8,9,20], the wavelengths lie in the range of $\mu m = {10^{ - 6}}m$ or less. On an intermediate scale, say, for an EM wave of a frequency $915MHz$ and a wavelength of $33cm$, the near-field range is about $1m$ in size. On a larger scale, the wavelength of radio waves lies in the range of ten meters corresponding to fourteen megahertz.
For the sake of simplicity, we take the medium to be a loss-free dielectric with constant and positive values of $\{{{\varepsilon_0},{\mu_0}} \}$, being the electric permittivity and magnetic permeability. With $\{{\vec{E},\vec{H}} \}$ as the electric and magnetic fields, we make the replacements: the electric displacement $\vec{D} \equiv {\varepsilon _0}\vec{E} \to \vec{E}$ and the magnetic induction $\vec{B} \equiv {\mu _0}\vec{H} \to \vec{H}$. The dimensionless Maxwell equations can thus be formulated into the Ampère–Maxwell law $\nabla \times \vec{H} ={-} \textrm{i}\vec{E}$ and the Faraday–Lenz law $\nabla \times \vec{E} = \textrm{i}\vec{H}$ alongside two Gauss laws $\nabla \cdot \vec{E} = 0$ and $\nabla \cdot \vec{H} = 0$ [19].
For concreteness, we consider a linear dipole $\hat{p} \equiv \alpha {\hat{e}_z}$ aligned along the vertical $z$-axis as shown on Fig. 1(b). With $\textrm{exp} ({ - \textrm{i}t} )$ incorporated, $\hat{p} \equiv \alpha \textrm{exp} ({ - \textrm{i}t} ){\hat{e}_z}$ represents standing waves. Besides, $\alpha ={\pm} 1$ is the spin index, while a unit magnitude $|{\hat{p}} |= 1$ is assumed. Starting with a vector potential $\vec{A} \equiv{-} \textrm{i}{r^{ - 1}}\textrm{exp} ({\textrm{i}r} )\hat{p}$, we obtain $\{{\vec{E},\vec{H}} \}$ by $\vec{H} = \nabla \times \vec{A}$ and $\vec{E} = \textrm{i}\nabla \times \vec{H}$ in sequel as follows [1,4,5,8,13,14,19].
Here, $\{{{{\hat{e}}_x},{{\hat{e}}_y},{{\hat{e}}_z}} \}$, $\{{{{\hat{e}}_r},{{\hat{e}}_\theta },{{\hat{e}}_\phi }} \}$, and $\{{{{\hat{e}}_\rho },{{\hat{e}}_\phi },{{\hat{e}}_z}} \}$ are respective basis-vector triads according to Fig. 1(a). For later uses, we listed here the nonzero field components in two coordinate systems: $\{{{E_r},{E_\theta }} \}$ and $\{{{E_z},{E_\rho }} \}$ both lying on the meridional $r\theta$- or $zx$-plane. Notice from Eq. (1) that the EM field is non-paraxial such that $\vec{E}$ consists of a longitudinal component ${E_r}$ and a transverse component ${E_\theta }$ in view of the sphere-radial propagation represented by $\textrm{exp} ({\textrm{i}r - \textrm{i}t} )$ [4,11,14,19]. Notwithstanding, the transverse component dominates over the longitudinal one in the far field, that is, $|{{{{E_r}} / {{E_\theta }}}} |\to 0$ as $r \to \infty$.
However, ${H_\phi }$ remains the same in both cylindrical and spherical coordinates. Both EM fields lie outside a vanishingly small volume containing a point dipole so that they are free of electric charges and electric currents. One can hence easily verify that $\nabla \cdot \vec{E} = 0$ and $\nabla \cdot \vec{H} = 0$ for $\{{\vec{E},\vec{H}} \}$ presented in Eq. (1).
From Eq. (1), we derive the following energies and the Poynting vector $\vec{P} \equiv \textrm{Re} ({{{\vec{E}}^\ast } \times \vec{H}} )$.
Likewise, we can find ${|{\vec{H}} |^2}$. Throughout this study, we set $\alpha ={+} 1$ except when otherwise stated. Both on the forthcoming Figs. 2, 3, 5 and 6, all plot windows are intentionally confined to $r \equiv \sqrt {{z^2} + {\rho ^2}} \le 10$ on a certain meridional plane as indicated on Fig. 1(a), where abscissa and ordinate refer to $\{{\rho ,z} \}$.
Fig. 2. (a), (b) Streamlines formed by $\textrm{Re} ({\vec{E}} )$ and ${\mathop{\rm Im}\nolimits} ({\vec{E}} )$. (c) A logarithm ${\log _{10}}{|{\vec{E}} |^2}$ of the electric-field intensity. (d) A logarithm ${\log _{10}}{|{\vec{H}} |^2}$ of the magnetic-field intensity. (e) The magnitude $|{\mathrm{\vec{P}}} |$ of the per-mean-intensity Poynting vector. The color bars on the right indicate the respective ranges of the plotted values.
Fig. 3. Two versions of DoCP. (a) $\mathcal{S}^{[z \rho]}$ on the cylindrical basis. (b) Near-circular polarization states with $0.99 \leq\left|\mathcal{S}^{[z \rho]}\right| \leq 1$. (c) A sketch for a pair of pencil-hull-like cylinders with tapered portions in case with $\mathcal{S}^{[z \rho]}$. (d) $\mathcal{S}^{[r \theta]}$ on the spherical basis. The color bar on panel (a) applies both to $\mathcal{S}^{[z \rho]}$ on panels (a) and (b) and to $\mathcal{S}^{[r \theta]}$ on panel (d).
Fig. 4. The normalized determinant $\Delta$ for the perfect circular polarization $\left|\mathcal{S}^{[z \rho]}\right|=1$ with varying r. (b) The projection for the surface of near-circular polarization for $z \ge 0$.
Fig. 5. The distributions of phase differences within the sphere of $r \le 10$ for the electromagnetic field induced by a linear dipole. (a)$\Delta \chi _\theta ^r$ and (b) $\Delta \chi _z^\rho$
Figure 2 displays several properties of the induced EM field. Figure 2(a) presents the real part $\textrm{Re} ({\vec{E}} )$ of the induced field $\vec{E}$, while Fig. 2(b) presents the imaginary part ${\mathop{\rm Im}\nolimits} ({\vec{E}} )$. On both panels vortices and shell- or matryoshka-like structures are visible [2,8,14].
Figure 2(c) shows the electric-field intensity ${\log _{10}}{|{\vec{E}} |^2}$ on logarithmic scale, where radially decreasing levels are found along with a slight polar-angle dependence. Likewise, Figure 2(d) shows the magnetic-field intensity ${\log _{10}}{|{\vec{H}} |^2}$ on logarithmic scale. On both Figs. 2(c) and (d), we have intentionally set the same scale bars to stress that the ${|{\vec{E}} |^2} > {|{\vec{H}} |^2}$ over most of the display windows. Figure 2(e) displays the magnitude $|{\mathrm{\vec{P}}} |$ of the per-mean-field-intensity Poynting vector $\mathrm{\vec{P}} \equiv I_{avg}^{ - 1}\vec{P}$. In all the directions except along the linear-dipolar axis, we find that $|{\mathrm{\vec{P}}} |\to 1$ in the far-field limit as $r \to \infty$, whence subluminal energy velocities prevail over the whole space [8,18].
There is a common feature in drawing constant levels presented in Fig. 1–3 and Figs. 5–7. Here, scientific data are presented only over the zones within the respective semicircles. The purpose is to ensure that all the levels are bounded as specified by the adjoining color bars. Specifically in Figs. 5–7, all levels are bounded from above by a maximum of plus unity and from below by minus unity as indicted by the same color-bar scales.
Fig. 6. The degree of circular polarization (DoCP) along a meridional plane that encompasses an array of five electric linear point dipoles standing in parallel. (a) $\mathcal{S}^{[z \rho]}$ based on $\{{{E_z},{E_\rho }} \}$, and (b) $\mathcal{S}^{[r \theta]}$ based on $\{{{E_r},{E_\theta }} \}$. The pattern in the southern hemisphere is skew-symmetric on (a) and (c), while it is symmetric on (b) and (d). The solid curve on (a) indicated by “from Fig. 4(b)” is the curve of a near-circular polarization for a single dipole.
Fig. 7. The degree of circular polarization (DoCP) on the horizontal plane with $\theta = {\textstyle{1 \over 2}}\pi$ for an array of five electric point linear dipoles pointing in and out of the page (a) $\mathcal{S}^{[z \rho]}$ based on $\{{{E_z},{E_\rho }} \}$, and (b) $\mathcal{S}^{[r \theta]}$ based on $\{{{E_r},{E_\theta }} \}$.
3. Degree of circular polarization (DoCP)
While focusing on the vertical $z$-direction, we form a pair of circular basis vectors $\sqrt 2 {\hat{e}_ \pm } \equiv {\hat{e}_z} \pm \textrm{i}{\hat{e}_\rho }$, for which $\{{{E_ + },{E_ - }} \}$ is accordingly defined by $\sqrt 2 {E_ \pm } \equiv {E_z} \mp \textrm{i}{E_\rho }$ [15,22]. The electric-field intensity ${|{\vec{E}} |^2} \equiv {|{{E_r}} |^2} + {|{{E_\theta }} |^2}$ given in Eq. (2) is then identical to both ${|{{E_z}} |^2} + {|{{E_\rho }} |^2}$ and ${|{{E_ + }} |^2} + {|{{E_ - }} |^2}$. We can define two versions $\left\{\mathcal{S}^{[z \rho]}, \mathcal{S}^{[r \theta]}\right\}$ of the degree of circular polarization (DoCP) [5,8,14,15,18,19].
There are two representative features of the cylindrical DoCP $\mathcal{S}^{[z \rho]}$: (i) the interference between the two electric-field components in $\mathcal{S}^{[z \rho]}$, and (ii) the bounded property $\left|\mathcal{S}^{[z \rho]}\right| \leq 1$ with the help of the Cauchy-Schwarz inequality [18]. $\mathcal{S}^{[z \rho]}$ is nothing but the ratio of the third-to-zeroth Stokes parameters, while $\mathcal{S}^{[z \rho]}$ does not depend on the spin index $\alpha ={\pm} 1$. Discussions on the spherical DoCP $\mathcal{S}^{[r \theta]}$ are soon to follow.
Meanwhile, consider the (electric) spin density $\vec{S} \equiv {\mathop{\rm Im}\nolimits} ({{{\vec{E}}^\ast } \times \vec{E}} )$, which can be easily evaluated from Eq. (1) to be $\vec{S} = {S_\phi }{\hat{e}_\phi }$. Its sole nonzero component ${S_\phi }$ is given as follows respectively in the cylindrical and spherical coordinates [18,19].
It turns out that the numerator $2{\mathop{\rm Im}\nolimits} ({E_z^\ast {E_\rho }} )$ in Eq. (3) is identical to ${S_\phi }$, whence $\mathcal{S}^{[z \rho]}=|\vec{E}|^{-2} S_{\phi}$.
Figure 3 displays the DoCP in two versions. Figure 3(a) displays the cylindrical DoCP $\mathcal{S}^{[z \rho]}$ over a certain meridional plane, thus showing $\mathcal{S}^{[z \rho]}<0$ and $\mathcal{S}^{[z \rho]}>0$ respectively over the northern and southern hemispheres. We thus recovered a half-twist or a four-lobe pattern in the polar direction [18,19,23]. Except right on the dipolar axis, $\mathcal{S}^{[z \rho]}$ exhibits larger values approximately along the linear-dipolar axis.
Based on Figure 3(a), we have drawn Figure 3(b) with a cut-off on $\mathcal{S}^{[z \rho]}$ by enforcing $\mathcal{S}^{[z \rho]}=0$ if $\left|\mathcal{S}^{[z \rho]}\right|<0.99$. Resultantly, the blue curve on the northern hemisphere indicates the approximate locations of $\left|\mathcal{S}^{[z \rho]}\right| \rightarrow 1$, albeit a perfect circular polarization exists nowhere [18]. Because of the rotational symmetry, the pair of curves on Figure 3(b) implies respectively pencil-hull-like tapered cylindrical surfaces as summarily depicted on Fig. 1(b) [1,13]. See Fig. 1(c) of [19], where the zones of large values of $\mathcal{S}^{[z \rho]}$ are off the axis of a linear dipole.
In this respect, Fig. 3(b) shows that $\mathcal{S}^{[z \rho]}=0$ on the horizontal plane at $\theta = {\textstyle{1 \over 2}}\pi$ in confirmation of Eq. (4). That is why the blue and red strips are drawn on Fig. 3(b) such that they run short of neither the origin nor the horizontal plane. These fading portions are called here ‘tapered surfaces’, over which $\mathcal{S}^{[z \rho]} \neq \overrightarrow{0}$ are being slowly established for the EM field. Figure 3(c) illustrates such a pair of curved surfaces obtained on Fig. 3(b).
Figure 3(d) presents the spherical DoCP $\mathcal{S}^{[r \theta]}$ defined in Eq. (3) and formed from Eqs. (1) and (2). This longitudinal-to-transverse electric-field intensity ratio has barely been handled in relevant literature. This ratio is unity in magnitude either along or perpendicular to the dipolar axis. A common feature of $\mathcal{S}^{[r \theta]}$ on Fig. 3(d) and the normalized Poynting vector $\mathrm{\vec{P}}$ on Fig. 2(e) is that both patterns are almost identical except near the origin. However, their far-field limits are hardly definable as $r \to \infty$, since these ratios are highly dependent on the polar angle.
4. Exact formula for near-circular polarizations
The surfaces of near-circular polarizations shown on Fig. 3(b) can be analytically found by setting $\mathcal{S}^{[z \rho]}=1$ in Eq. (3) [18], which is straightforwardly expanded into the following quadratic equation in $\cos ({2\theta } )$.
See Supplement 1 on how to derive Eq. (5).
There is no real-valued solution to $\cos ({2\theta } )$ in the above Eq. (5), because a state of perfect circular polarization is achieved nowhere over the whole space. Notice that $A > 0$ as seen from Eq. (5). Figure 5(a) shows the normalized determinant $\Delta \equiv {A^{ - 2}}({{B^2} - AC} )< 0$ with varying r over the whole $r > 0$. At the minimum location pointed at by the upward dashed line on Fig. 5(a), the relative deviation from $\mathcal{S}^{[z \rho]}=1$ is greatest. However, the error from $|\Delta |= 0$ is still acceptably small. Moreover, the state of circular polarization is approached at the far infinity since $\Delta \to 0$ from below as $r \to \infty$. We took $\cos ({2\theta } )\approx {B / A}$ or $\cos \theta \approx \sqrt {{\textstyle{1 \over 2}}{{({A + B} )} / A}}$ as an approximate solution to the quadratic equation in Eq. (5) by taking $\Delta \to 0$.
Figure 4(b) plots the locus of $\{{\rho ,z} \}\equiv r\{{\sin \theta ,\cos \theta } \}$ on the northern hemisphere with r playing as a parameter. It happens that this surface of near-circular polarizations starts at $\{{\rho ,z} \}= \{{1,0} \}$ and approached $\rho = 2$ as $r \to \infty$. For instance, consider the radial distance $r = 10$ for which $\{{\rho ,z} \}\approx \{{2,9.17} \}$ as marked by the blank black circle on Fig. 5(b). Near the origin, this hollow cylinder becomes approximately a cone [15]. The EM fields are exactly linearly polarized on the three polar angles at $\theta = 0,{\textstyle{1 \over 2}}\pi ,\pi$ as has been indirectly proved by [18]. Indeed, the tapered cylinders as the surfaces of near-circular polarizations lie in between the extremal places reserved for linearly polarized fields.
Concerning the handedness of the cylindrical DoCP , a right-handed near-circular polarization (RCP) prevails along the pencil-hull-like cylinder in the northern hemisphere, whereas a left-handed circular polarization (LCP) prevails along the other cylinder in the southern hemisphere or vice versa [15,18].
Equally interesting is the fact that the spherical DoCP $\mathcal{S}^{[r \theta]}$ displayed on Fig. 3(d) hints at the condition $\mathcal{S}^{[r \theta]}=0$ from Eq. (3) might be almost indistinguishable from the red curve shown on Fig. 3(b) in case with $\mathcal{S}^{[z \rho]}=1$ for the cylindrical DoCP. To see this, we utilize Eq. (5) to express ${\tan ^2}\theta = {({A + B} )^{ - 1}}({A - B} )\equiv {a / b}$ for the afore-mentioned condition for $\Delta = 0$ that approximates $\mathcal{S}^{[z \rho]}=1$. Meanwhile, we express the condition ${|{{E_r}} |^2} = {|{{E_\theta }} |^2}$ for $\mathcal{S}^{[z \rho]}=0$ alternatively by ${\tan ^2}\theta \equiv {c / d}$ from Eq. (1). Resultantly, we reach the following ratio as detailed in Supplement 1 .
Consequently, the numerator and the denominator, differing by the three terms that are higher order in ${r^{ - 2}}$, are convergent at a fast rate over the range $|r |> 1$ to the same value of unity as $r \to \infty$. This is reason why there is an only an unrecognizable discrepancy between the two conditions as shown on Fig. 4(b). In other words, either ${|{{E_ + }} |^2} = 0$ or ${|{{E_ - }} |^2} = 0$ from Eq. (3) is practically identical to ${|{{E_r}} |^2} = {|{{E_\theta }} |^2}$ for $\mathcal{S}^{[r \theta]}=0$.
5. Phase relations for electric fields
Based on Eq. (1), let us express $\{{{E_z},{E_\rho }} \}$ as ${E_z} \equiv |{{E_z}} |\textrm{exp} ({\textrm{i}{\phi_z}} )$ and ${E_\rho } \equiv |{{E_\rho }} |\textrm{exp} ({\textrm{i}{\phi_\rho }} )$ with both $\{{{\phi_z},{\phi_\rho }} \}$ being real. We can reinstate the hitherto assumed temporal factor and the radial dependence to obtain ${E_\rho } \equiv |{{E_\rho }} |\textrm{exp} \{{\textrm{i}[{r + {\phi_\rho }({r,\theta } )- t} ]} \}$ and the like for ${E_z}$ [12,20,21]. The other pair $\{{{E_r},{E_\theta }} \}$ can also be decomposed analogously, for instance, as ${E_r} \equiv |{{E_r}} |\textrm{exp} \{{\textrm{i}[{r + {\phi_r}({r,\theta } )- t} ]} \}$. Therefore, we find that ${\mathop{\rm Im}\nolimits} ({E_z^\ast {E_\rho }} )\propto {\phi _\rho } - {\phi _z}$ for $\mathcal{S}^{[z \rho]}$ defined in Eq. (3). Let us define normalized phase differences as follows.
Figure 5 these phase differences $\{{\Delta \chi_\theta^r,\Delta \chi_z^\rho } \}$, both being dependent on the sphere-radial $r$-coordinate. The phase difference $\Delta \chi _\theta ^r$ on panel Fig. 5(a) is independent of the polar angle due to both rotational and polar symmetries, whereas $\Delta \chi _z^\rho$ on Fig. 5(b) is dependent on the polar angle. The pair of black dashed curves drawn over $\Delta \chi _z^\rho$ on Fig. 5(b) exhibit the characteristic pencil-hull-like shapes as displayed on Fig. 3(b) and 3(c). Other phase angles $\{{{\chi_r},{\chi_\theta },{\chi_\rho },{\chi_z}} \}$ are plotted on Fig. S1 of Supplement 1 .
6. Linear array of five parallel electric point linear dipoles
As a more realistic example, let us consider a one-dimensional array of electric point linear dipoles. For simplicity, we take all the dipoles are parallel and oriented along the vertical $z$-direction. Furthermore, all dipoles are assumed to be of an identical magnitude and of an identical phase. For a set of five electric point linear dipoles that are arranged along the $x$-direction, we are to examine the radiated EM fields. The central dipole is located at $x = 0$, while all five dipoles are separated with an equal inter-dipole distance of two, i.e., $\Delta x = 2$. Therefore, two dipoles are located to the left and two are located to the right of the central dipole. Although this free-standing array appears impractical because of the absence of any supporting substrate, there are some ways to keep micro-scale structures free-standing [24].
Figure 1(c) is reproduced on Fig. 6(a), which shows the cylindrical DoCP $\mathcal{S}^{[z \rho]}$ based on the pair $\{{{E_z},{E_\rho }} \}$ of the electric field. In comparison, Fig. 6(b) shows the spherical DoCP $\mathcal{S}^{[r \theta]}$ based on the pair $\{{{E_r},{E_\theta }} \}$ [22]. See Eq. (1) for both $\{{{E_z},{E_\rho }} \}$ and $\{{{E_r},{E_\theta }} \}$. A necessary algebra for evaluating both $\mathcal{S}^{[z \rho]}$ and $\mathcal{S}^{[r \theta]}$ is presented in Supplement 1 .
Let us make a few observations on Fig. 6. Figure 6(a) shows particularly the two fundamental branches as marked respectively by ‘L (left)’ and ‘R (right)’ right above the display window. These two branches with negative values of $\mathcal{S}^{[z \rho]}$ stem mostly from a single dipole lying at the center. However, the two pairs of off-center dipoles give rise to a pair of slant yellow strips which are in turn flanked by another pair of blue slant strips.
Hence, the cylindrical DoCP $\mathcal{S}^{[z \rho]}$ undergoes alternatingly positive and negative values as the polar $\theta$-angle is increased from the vertical $z$-axis. In addition, we find it interesting that four pairs of blue-red zones are located just above the horizon at $z = 0$. Upon closer look, these pairs are located at the locations of $x ={-} 3, - 1, + 1, + 3$ exactly in between two neighboring dipoles. For instance, we have indicated on Fig. 6(a) the pair at $x ={-} 1$ by a violet slant upward arrow. These inter-dipole activities result from strong interferences between two neighboring dipoles.
Let us turn to Fig. 6(b) showing the spherical DoCP $\mathcal{S}^{[r \theta]}$, which is quite distinct from the cylindrical DoCP $\mathcal{S}^{[z \rho]}$ shown on Fig. 6(a). Notwithstanding, the single branch as marked by ‘C (center)’ resembles that on Fig. 3(d) that has been obtained for a sole dipole. This sole fundamental branch is flanked by a pair of reddish branches indicated respectively by ‘CL (center-left)’ and ‘CR (center-right) in two downward slant arrows in red boxes. As on Fig. 6(a), Fig. 6(b) features two pairs of strong inter-dipole interferences located exactly at $x ={-} 3, - 1, + 1, + 3$, this time being of a three-pronged nature. To get a better feeling about the fundamental and side branches, Fig. 6(c) Fig. 6(d) are prepared as zoomed-out versions respectively of Figs. 6(a) and 6(b) on ten times larger display windows.
Since both DoCPs $\mathcal{S}^{[z \rho]}$ and $\mathcal{S}^{[r \theta]}$ on the preceding Fig. 6 are dependent on the azimuthal $\phi$, it is appropriate to present both DoCPs on the horizontal plane. Figure 7 compares such distributions $\mathcal{S}^{[z \rho]}(\rho, \phi)$ and $\mathcal{S}^{[r \theta]}(\rho, \phi)$ on the display windows perpendicular to the dipolar axes. These patterns are hence complementing the distributions displayed on Fig. 6. Figure 6(a) shows the pair of fundamental branches as marked respectively by ‘L (left)’ and ‘R (right)’, whereas Fig. 6(b) exhibits the central branch ‘C (center)’ along with the two side branches denoted by ‘CL (center-left)’ and ‘CR (center-right). Both patterns are symmetric with respect to the horizontal axes denoting $y = 0$.
The EM fields induced by a linear array of electric point linear dipoles will be more unpredictable in the near field as we make variations in the relative magnitudes and phases among the constituent dipoles. Therefore, the Stokes parameters would become gravely insufficient in properly addressing the state of polarization, partly because the resulting combined spherical waves defy a certain easy-to-define plane [25]. Such three-dimensional (3D) EM fields require, say, the 3D polarization algebra based on the nine Gell-Mann matrices [23,26,27].
7. Discussions
Since $\vec{S} \equiv {\mathop{\rm Im}\nolimits} ({{{\vec{E}}^\ast } \times \vec{E}} )$ can be alternatively cast into $\vec{S} = 2\textrm{Re} ({\vec{E}} )\times {\mathop{\rm Im}\nolimits} ({\vec{E}} )$, of Fig. 2(a) and 2(b) help us to guess at the profiles of $\vec{S}$. Discussions on the so-called internal fluxes $\{{{{\vec{F}}^o},{{\vec{F}}^s}} \}$ of orbital and spin energy fluxes have been made elsewhere for other EM fields [6]. In the future, we hope to make a report on our study on such internal flows for the specific EM field induced by an electric point dipole and its varied arrays.
As an inverse problem, adjusting the DoCPs $\left\{\mathcal{S}^{[z \rho]}, \mathcal{S}^{[r \theta]}\right\}$ in a desired way by controlling the underlying EM fields and the properties of a dipole will be challenging [18]. In addition, measurements of the DoCPs largely involve near-field considerations [3,13,15–17] as realized with near-field polarimetry [18]. For instance, we learned from various numerical experiments that Poynting power flows can be made asymmetric along a one-dimensional array of point dipoles by controlling the phase relationships among various dipoles. In other words, directional power flows can be achieved.
Let us revisit the pencil-hull-like tapered cylindrical surface displayed on Figs. 1(b), 3(b), and 3(c), which we could also dub a ‘hollow needle’ [20]. An optical needle offers a uniform conversion from the orbital energy flux to the spin energy flux from an input position (corresponding to the far field in this study) toward a focal point (corresponding to the dipole position at the origin in this study) [8]. When Eq. (1) is examined along the vertical axis, we have ${E_r} \ne 0$ but ${E_\theta } = 0$ so that a longitudinal electric-field polarization is maintained [21].
For space reasons, we skip other types of arrays of elemental electric point dipoles that have so far been either only envisaged or already realized. Interactions and/or the phase relationships among the multiple emitters will influence the performance as a whole [14]. Of course, dipoles in circular arrays get harder to deal with due to an azimuthal $\phi$-dependence [9,10]. Other array configurations even only with linear dipoles [11,17], are sketched on Fig. S2 in Supplement 1 .
As an immediate extension of this study, one can consider an electric point dipole located from a planar surface with a certain standoff distance. To this goal, one can make use of the formulas for the field solutions provided in Chapter 10 and Appendix C of [1] to evaluate the pertinent DoCP. It will be interesting to find how the pencil-hull-like cylinders are deformed due to the presence of a planar boundary [1,22]. With supporting substrates, considerations of any arrays of point dipoles become often too complicated due to effects of the orientation of a dipole with respect to the planar surface. A similar consideration should be given to a pair of dipoles that make up an electric circular dipole [19].
For a moment, let us turn to an optical needle, where an incoming light is focused onto a certain point along a main propagation axis [20]. These optical needles refer to intensity needles, where a field intensity is focused over an extended length, say sixteen times a participating wavelength [21]. There two modes: (i) a multifocal array consisting of discrete multifocal spots, and (ii) or a continuous line of focal points (as intentional and organized spherical aberrations). If a certain electric-field polarization, say, along that propagation axis is desired in either of the above two configurations, the resulting optical requirements should be more stringent.
Regarding the previous Fig. 3, our task on the measurements winds up in how to spatially resolve the polarization states of the EM field induced by an electric point dipole. In fact, there are several polarimetric techniques, either well-developed or under constant improvements. In the remaining portion of this section, let us examine relevant aspects of differentiating linear from circular polarizations.
Recall Fig. 3(b) concerning the surfaces of the near-circular polarization where circularly polarized light (CPL) dominates over the pencil-hull-like cylindrical surfaces while linearly polarized light (LPL) prevails in the zones within and outside. This distinction has also been confirmed by Fig. 1(c) of [19].
In general, polarization states can be evaluated by measuring electric-field intensities in two orthogonal directions as employed for evaluating the usual Stokes parameters [15–19,28–30]. Furthermore, measurements of polarization states of EM waves essentially involve light-matter interactions, wherein the absorption of the EM energy into matters plays a great role [13,28]. For instance, such an absorption by a nanoparticle is crucial to the measurement of the field’s DoCP as seen from Eq. (5.1) of [30]. In this regard, the afore-mentioned nano-fabrication method in [11] for forming helical structures is strongly dependent not only on the intensity but also on the polarization state of illuminating laser light.
As another example, left- and right-circularly polarized fields could induce distinct electrical currents within the photodetectors if the latter consist of chiral media [16,17,31]. As a proof-of-concept CPL photodetectors, the polarization imaging of [31] shows that their special chiral meta-surfaces respond not to LPL but only to CPL. Such a (disk-shaped) polarization-discriminating meta-surface can be brought onto the location near $z = 10$ on our Fig. 3(b) as marked by a horizontal gray bar. In this manner, we could confirm the circularly polarized states at the cylinder-radial distance $\rho = 2$. A difficulty arises from the scattering of the point-dipolar field by such a disk-shaped sensor as with the near-field polarimetry of [18]. Moreover, those oppositely signed circular polarizations of [31] could be implemented to differentiate between the EM waves in the northern and southern halves on our Fig. 3(b).
Since our findings are valid for generic wavelengths, we suggest employing EM waves of far longer wavelengths, say, millimeter waves for easier measurements. With such a setup, nanoscale measuring devices should not disturb the electric field to any appreciable degrees [19]. Nanoparticles being much smaller than the wavelength of an EM wave would also serve as a good indicator for the polarization state of the embedding EM wave. When a nanoparticle is both electrically and magnetically polarizable, it will be more suitable for measuring the DoCP of an EM wave [29,30]. We could thus map out the pencil-hull-like cylindrical surfaces with larger values of DOCPs.
8. Conclusion
We have examined behaviors of the electromagnetic field induced by an electric point linear dipole by focusing on the polarization properties. To this goal, we have provided an analytic formula for the surfaces of near-circular polarizations. The resulting surfaces consist of a pair of pencil-hull-like hollow cylinders with central transition zones, whereas the common axis of these hollow hulls is aligned with the dipolar axis. Meanwhile, their radius is twice the inverse wave number of the participating electromagnetic wave so that this stand-off distance will play a significant role in the near-field behaviors of antennas and sensing devices. We have also explored basic properties exhibited by the electromagnetic field induced by a one-dimensional array of electric point linear dipoles.
Funding
National Research Foundation of Korea (NRF-2018R1D1A1B07045905).
Acknowledgments
This work was partially supported by the Sun Moon University Research Grant of 2021.
Disclosures
The authors declare no conflicts of interest.
Data availability
There is no underlying data set.
Supplemental document
See Supplement 1 for supporting content.
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