1. Introduction

Uniaxial materials are anomalous anisotropic materials possessing a distinctive symmetry axis [1–3]. Electromagnetic waves can propagate through these materials with two mutually orthogonal states of polarization and with different phase velocities [4]. Uniaxial materials naturally exist in the world, e.g., calcite, zircon and rutile [5]. Uniaxial materials can also be assembled artificially, e.g., thin electrical sheets of various isotropic materials arranged as a layered stack [6,7]. Uniaxial materials have attracted peculiar attention due to their application in several aberrant phenomenon, e.g., negative-refraction and sub-wavelength imaging [8]. Although uniaxial material are prevalently employed in fabrication of optical devices, they are also utilized in nano imaging [9].

In past various authors have been studied radiation emitted from a point magnetic-dipole laying in uniaxial materials [4]. Salem and Caloz have analyzed the electromagnetic radiations generated from an arbitrary current carrying loop antenna embedded in the isotropic material [10]. Pozar has studied the radiation and scattering of electromagnetic waves exuding from a microstrip antenna laying in uniaxial-dielectric substrate [11]. Li et al., has computed the electromagnetic fields radiated by a thin circular loop-antenna in both near and far zones by adopting the dyadic Greens functions technique [12]. Ghaffar et al., has studied the focusing of electromagnetic waves into a uniaxial crystal such that the waves are refracted by an inhomogeneous slab [13]. Moreover, Hayat and Faryad have studied the radiation emitted from a Hertzian dipole having finite length placed in a uniaxial dielectric material. They have adopted the dyadic Green function approach in frequency domain to evaluate radiation fields generated by a current loop in near and far fields [14]. Recently, closed-form expressions for electromagnetic waves generated by a current loop in a uniaxial dielectric medium in the far zone were studied by us [15].

To effectively model real radiation sources in uniaxial materials such as quantum dots, wires, and sheets, it is necessary to consider magnetic dipoles. Consequently, we aim to establish the radiation characteristics of an electric current loop, which serves as an effective representation of a magnetic dipole. This paper assumes an arbitrary radius for the loop, making the problem suitable for simulating real extended radiation sources. Additionally, employing a current loop to model the magnetic dipole not only facilitates its direct realization but also extends the field generalization from an isotropic medium to the simplest form of anisotropic medium, namely, a uniaxial dielectric-magnetic medium. In the current study, we establish closed-form expressions for electromagnetic fields emitted from a limited sized loop, where the axis of symmetry of loop is either aligned with or orthogonal to the optic axis of the material.

The article’s overview is as follows. The far-zone approximated dyadic Green functions for the uniaxial dielectric-magnetic material are simulated in Section 2. The analytical results for the fields generated by the current loop when its axis of symmetry is aligned with the optic axis of the uniaxial dielectric-magnetic material, are evaluated in Section 3. The same analytical findings for the instance when loop’s axis (it stands for the axis normal to the disk enclosed by the loop, which is the $z$ axis) of symmetry is orthogonal to the optic axis are presented in Section 4. In Section 5 a comparison for the radiation pattern of a current loop against that of point magnetic dipole has been presented. The final outcomes and conclusion are presented in Sections 6. and 7 respectively.

The time dependence $\exp (i\omega t)$ is implied in this article, where $i=\sqrt {-1}$ and $\omega$ represents angular frequency. Furthermore, $\mu _o$, $\epsilon _o$ and $k_o=\omega \sqrt {\epsilon _o \mu _o}$ symbolize the vacuum permeability, vacuum permittivity and wave number respectively. Vectors are represented by letters with arrow heads and doubly underlined symbols represent the dyadics. The identity dyadic is denoted by $\underline {\underline {I}}$.

2. Dyadic Green functions

The relevance between a vector source and corresponding vector field is explicated in terms of dyadic Green function and can be represented as [4]

Here volume covered by current density $\vec {J}_e{(\vec {r}')}$ is denoted by $V'$ with

The materials whose permittivity and permeability dyadics have a common uniaxiality axis are known as uniaxial dielectric-magnetic materials [4]. The permittivity and permeability dyadics of the uniaxial dielectric-magnetic materials are given as [16]

Since we are concerned with the fields that a current loop exudes in the far-zone, so here the far-zone approximation of the dyadic Green functions are relevant only, and their exact expressions are given in Refs. [4,17]. The approximated dyadic Green functions in the far-zone are given as [4,17]

3. Optic axis and loop axis are parallel

Let us suppose a current loop having radius $a$ carrying constant current $I_o$ laying in xy plane as illustrated in Fig. 1. In that case the loop axis and the optic axis are directed parallel to the $z$-axis.

 

Fig. 1. Illustration of a current loop having radius $a$, with a constant current flowing through it and laying in a uniaxial dielectric-magnetic material, such that loop axis and the optic axis are aligned in the same direction.

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In spherical coordinates electric current density phasor can be expressed as

The electric field radiated from a current loop in the far-zone can be evaluated using Eqs. (6) and (13) in Eq. (1), so that we get

Since spherical coordinates are related with the Cartesian coordinates as follows

As the net electric field and magnetic field are given by:

Here we are not providing the detailed calculation because these expressions can be obtained by the same method that we already provided in Ref. [15]. Following the same procedure given in Ref. [15] the final expression for electrically extraordinary wave that can be extracted from Eq. (14) is given by

It shows that the extraordinary electric field emitted by the current loop has a smaller amplitude, approximated to zero, if the loop axis is aligned with the optic axis of the material.

Similarly the final expression for magnetic part of electric field that can be separated from Eq. (14) is given as

Repeating the same procedure as explained earlier for electric field and also given in Ref. [15] the expression for $\vec {H}_{e}$ is given by

Since the final extraordinary magnetic field contains term proportional to $1/r^2$ only, so we can write

Hence, Eqs. (21) and (24) reveal that only magnetic part of the extraordinary waves is exuded in the far-zone when the loop axis and the optic axis are aligned in the same direction, i.e., parallel to $z$-axis. Moreover, the emanated fields satisfy the orthogonality condition, similar to the behaviour of the ordinary electromagnetic radiations.

The time averaged-power, emitted from the current loop in the far-zone is given as [18]

Using Eqs. (21) and (24) in Eq. (24), and simplifying the expression we can write

The final result of magnetic power emanated from a current loop in far-zone do not include $\epsilon ^{\delta }$, i.e., magnetic waves are polarized in a direction orthogonal to the optic axis.

4. Optic axis and the loop axis are mutually perpendicular

Now, let us assume that the optic axis of the material is turned along $x$-axis while axis of the current loop is still in $z$-direction, i.e., both are perpendicular to each other, the schematic diagram of this configuration is depicted in Fig. (2). Here we are going to use the same electric and magnetic current densities described in the Eq. (13), while we choose ${\bf {\hat {c}}}={\bf {\hat {x}}}$ for a uniaxial dielectric-magnetic material.

 

Fig. 2. Illustration of a current loop having radius $a$, with a constant current flowing through it and laying in a uniaxial dielectric-magnetic material, such that loop axis and the optic axis are mutually perpendicular.

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Our main goal is to compute the waves emitted from that current loop at a very far away from point P. Since the procedure for computing the expressions for waves and time-averaged was already discussed in Ref. [15] so we will not provide the detailed calculations. We will only present the final result and skip the details. Following the same procedure as followed earlier for Sec. 3 the expressions for $\vec {E}_{e}$, $\vec {E}_{m}$, $\vec {H}_{e}$, and $\vec {H}_{m}$ are given as follows:

Furthermore, it is observed that in the far-zone, the radiated extraordinary wave consists of both electrically extraordinary wave and magnetically extraordinary wave parts for the instance when axis of the loop is orthogonal to the optic axis of the material. We can add the powers of both parts of the extraordinary wave to get the total power radiated per unit time per unit solid angle, given as

In order to calculate time-averaged power of the extraordinary waves emitted by the current loop in the far-zone using Eqs. (26) and (27) in Eq. (32) and switching the variables into spherical coordinates we can write

The time-averaged power of the magnetically extraordinary wave emitted by the current loop in the far-zone can be computed by using Eqs. (27) and (28) within Eq. (33) we can obtain

Now, the total power emitted by the current loop per unit time per unit solid angle can be computed by using Eqs. (34) and (35) in Eq. (31) we obtain

On substituting $\epsilon ^{\delta }=1$ (which gives $\Phi _e=1$ and $\rho _d=1$) and $\mu ^{\delta }=1$ (which means that $\Phi _m=1$ and $\sigma _d=1$ ) in the Eq. (36) and by substituting $\Theta _m =1$ and $\mu ^{\delta }=1$ into Eq. (25) we observe that both equations are equivalent, displaying same result as for the power radiated by the current loop in an isotropic material [10].

5. Validation of work

A point magnetic dipole can be effectively emulated by a small electric current loop when the loop’s radius $a$ is very short indicating that $|k_o n_o a|<<1$. Under this approximation, it becomes possible to replicate the outcomes described in Secs. 3 and 4 for a point magnetic dipole. These achieved results can then be compared to those presented in Chapter 3, Section 3.5.3 of Ref. [4]. Additionally, by selecting values such as $\epsilon ^{\delta }=1$, $\mu ^{\delta }=1$, $\Theta _m=1$, $\Phi _e=1$ and $\Phi _m=1$, a point magnetic dipole placed within an isotropic material can be obtained. Notably, these findings align with those detailed in Section 2.1.4 of Chapter 2 in Ref. [4].

6. Numerical results and discussion

The main outcomes of this article were the analytical expressions of the electromagnetic waves emitted by a current loop in the far-zone, represented in Eqs. (25) and (34)–(36). The far-zone magnetically extraordinary wave emanated by a current loop when the loop axis is aligned in the direction of the optic axis of the material is illustrated in Fig. (3) for $a=0.1\lambda _o$, and for $a=0.3\lambda _o$. To demonstrate their applications, we give illustrative numerical results for radiations emitted by different current loops having various radii, while they are placed in a uniaxial dielectric-magnetic material having parameters as follows $\mu ^{\parallel }=1.5$, $\mu ^{\perp }=1.2$, $\epsilon ^{\parallel }=1.8$, $\epsilon ^{\perp }=2.5$, $I_o=0.1 A$ and $\lambda _o=0.584$ $\mu m$ [19] for the current loops with radii $a=0.1\lambda _o$ and $a=0.3\lambda _o$.

 

Fig. 3. Far-zone radiation patterns for the time averaged power of the magnetically extraordinary wave emitted by a current loop that is computed in Eq. (25), when the loop axis is aligned with optic axis of the material. The closed-patterns are sketched for parameters $\epsilon ^{\parallel }=1.8$, $\epsilon ^{\perp }=2.5$, $\mu ^{\perp }=1.2$, $\mu ^{\parallel }=1.5$ and $I_o=0.1 A$ with wavelength of $\lambda _o=0.584$ $\mu m$ in angle ranges $\theta \in [0,\pi ]$ and $\phi \in [\pi /2 , 3\pi /2]$ for current loop having radius $a=0.1\lambda _o$ (left) and $a=0.3 \lambda _o$ (right).

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For more prominent visualization we sketched Eq. (25) for ranges $\theta \in [0,\pi ]$ and $\phi \in [\pi /2 , 3\pi /2]$. The plots display that wave patterns are analogous with the plot for radiations emitted by a point magnetic-dipole, when laying in an isotropic material.

Moreover, with the increase in the loop size, i.e., $a=0.3 \lambda _o$ emitted radiations are suppressed along the loop and spread along axis of the current loop. Also no radiations are emitted along the optic axis.

When the optic axis of the material is at right angles to the axis of the current loop, the extraordinary electric part emission pattern are illustrated in Fig. 4. Since the pattern is symmetric about $yz$-plan so we are sketching Eq. (34) for ranges $\theta \in [0,\pi ]$ and $\phi \in [\pi /2 , 3\pi /2]$ for two different radii of the current loop, i.e., for $a=0.1\lambda _o$ and $a=0.3 \lambda _o$.

 

Fig. 4. Far-zone radiation patterns for the time averaged power of the electrically extraordinary wave emitted by a current loop that is computed in Eq. (34), when the loop axis is orthogonal to the optic axis of the material. The closed-patterns are sketched for parameters $\epsilon ^{\parallel }=1.8$, $\epsilon ^{\perp }=2.5$, $\mu ^{\perp }=1.2$ and $I_o=0.1 A$ with wavelength of $\lambda _o=0.584$ $\mu m$ in angle ranges $\theta \in [0,\pi ]$ and $\phi \in [\pi /2 , 3\pi /2]$ for current loop having radius $a=0.1\lambda _o$ (left) and $a=0.3 \lambda _o$ (right).

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These emission patterns reveal that for small loop size $a=0.1 \lambda _o$ radiations are emitted in a direction that is perpendicular to both loop axis and optic axis as plotted in Fig. 4(a). Whereas Fig. 4(b) depicts emission pattern of a large sized loop $a=0.3 \lambda _o$, illustrating that emitted radiations are entirely suppressed along loop axis and the plane perpendicular to it, with the maximum suppression in a direction normal to both the loop axis and the optic axis.

The emission pattern of magnetically extraordinary wave, evaluated in Eq. (35), exuded by the current loop when the loop axis and optic axis of the material are orthogonal to each other are illustrated in Fig. 5 for $a=0.1\lambda _o$ and $a=0.3 \lambda _o$.

 

Fig. 5. Identical to Fig. 4, but the pattern is for the extraordinary wave’s magnetic part as evaluated in Eq. (35).

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The radiations are pointed perpendicular to the loop’s axis as well as the optic axis of the material, when the loop size is minimal. However, when the loop’s size increases, the radiations are wholly suppressed along the loop’s axis and the plane which is orthogonal to it, with the sharpest suppression being in a direction that is orthogonal to both of optic axis and the loop’s axis.

7. Conclusions

Analytical results were computed to comprehend the patterns of radiations exuded in the far-zone from a uniform current loop laying in a uniaxial dielectric-magnetic material. The corresponding numerical findings were reported for a selected uniaxial dielectric-magnetic material. Similar far-field results of emitted fields and power were evaluated for a point magnetic-dipole. The loop axis was assumed to be identical or orthogonal to optic axis of the material. When loop axis and optic axis were aligned in same direction only magnetically extraordinary wave were emanated in far-zone. Both electrically extraordinary and magnetically extraordinary waves were exuded from the current loop in far-zone when its axis is laying normal to the direction of optic axis. Although both electrically extraordinary wave and magnetically extraordinary waves were squashed in the direction of optic axis of material. In all of these instances it is obvious that the enlargement of loop size appreciably increased the directionality of emitted radiations patterns. For limiting case, i.e., for loop of very small radius exuded radiation patterns were similar with that of a point magnetic-dipole laying in same uniaxial dielectric-magnetic material. It is evidently noticed that radiation patterns of a current loop have an accurate resemblance with that of a point magnetic-dipole, solely if the loop size is quite tiny.