Introduction
In next-generation wireless networks, emerging applications such as immersive extended reality and the metaverse impose stringent requirements on reliability, latency, and data rate, while simultaneously demanding that the wireless infrastructure adapts to diverse user locations and propagation conditions [1 ] . To meet these requirements, flexible-antenna architectures have been introduced as a means of dynamically modifying the antenna configuration and, consequently, the wireless channel [2 ] . In more detail, these architectures exploit spatially reconfigurable radiating elements whose locations can be adapted according to the communication objectives. However, realizing large-scale spatial reconfigurability through conventional antenna architectures introduces critical challenges in terms of feeding complexity, deployment cost, and propagation losses, particularly at high frequencies [2 ] . In such settings, transporting the signal through low-loss guiding structures before radiating it becomes particularly attractive, leading to the need for antenna architectures that can jointly exploit guided-wave propagation and dynamically configurable radiation points [3 ] . Therefore, enabling flexible radiation locations while preserving an efficient signal distribution mechanism becomes important for realizing spatially adaptive wireless networks.
Among the technologies envisioned to support such flexible antenna deployments, pinching-antenna systems (PAS) have emerged as a promising approach for providing reconfigurable wireless connectivity [3 , 4 , 5 ] . Specifically, in PAS, the radio-frequency signal propagates along a dielectric waveguide and is radiated into free space through PAs that can be activated at different positions along it, thereby enabling the radiation points to be established close to the served users and facilitating favorable line-of-sight links [4 ] , [5 ] . At the same time, the selection of the PA locations directly affects both the free-space propagation distances and the phase shifts of the corresponding communication links, rendering the achievable performance inherently dependent on the spatial configuration of the activated PAs [6 ] , [7 ] . However, the signal radiated by each PA is ultimately generated through its physical interaction with the guided electromagnetic field, which determines how the incident guided power is redistributed between the signal remaining in the waveguide and the field radiated into the surrounding environment [8 ] , [9 ] . As a result, the communication performance of PAS does not only depend on where the PAs are activated, but also on how their physical response modifies the guided and radiated signals. Consequently, understanding the extent to which the physical PA response affects the achievable communication performance becomes critical, motivating the need to systematically characterize PAS under physically consistent models.
State-of-the-Art
An increasing amount of literature has focused on the communication-level performance and design of PAS. In this direction, [10 ] investigated PA-position and resource-allocation optimization, while [11 ] characterized the frequency-selective behavior of multi-PA transmission and introduced an OFDMA-based framework. Furthermore, [12 ] analyzed the achievable array gain and the impact of the number and spacing of the activated PAs, whereas [7 ] characterized the outage probability, average rate, and optimal PA location. More recently, PAS deployments have been extended beyond conventional horizontal straight waveguides, with [13 ] considering vertical PAS for UAV communications and [14 ] investigating curved waveguide deployments. Therefore, these studies show that the achievable PAS performance is strongly affected by the PA configuration and waveguide deployment, motivating an accurate description of the physical mechanisms affecting PAS performance.
The design and performance implications of PAS have also been examined from a hardware and power-coupling perspective, providing further insight into the physical interaction between the guiding structure and the activated PAs. In more detail, [8 ] developed a physics-based hardware model in which the PA is represented as an open-ended directional coupler and coupled-mode theory is employed to characterize the corresponding power transfer, while [15 ] proposed an adjustable power-radiation model in which the radiated power is controlled through the coupling spacing between the PA and the waveguide. More recently, [16 ] established a physically consistent multiport-network representation for reconfigurable PAs, providing a network-level description of their guided and radiated signal interactions. In parallel, the radiation characteristics of PAs have been investigated through electromagnetic analysis, full-wave simulations, measurements, and measured-pattern-based system models. Specifically, [9 ] demonstrated that the PA geometry substantially affects the radiation, [17 ] developed a radiation-pattern model based on a two-dimensional dielectric slab, coupled-mode theory, and a two-dimensional radiation integral, and [18 ] introduced a polarization-aware full-wave electromagnetic model for multi-mode PAS. Finally, [19 ] incorporated an externally obtained measured radiation pattern together with coupling-dependent power extraction for PA placement and multi-PA design. Taken together, these studies have progressively established the importance of jointly accounting for PA power coupling, network response, and directional radiation when relating the physical PA behavior to communication-level performance.
Motivation & Contribution
The practical realization of a PA requires a local perturbation of the guiding structure to couple part of the guided electromagnetic energy into free space [8 , 15 ] . In particular, depending on the applied perturbation, it can modify the local geometric characteristics of the waveguide and, consequently, affect both the guided-wave propagation and the corresponding radiation response [9 ] . As a result, the PA geometry can jointly determine the radiated power, the amplitude and phase of the signal remaining in the waveguide, and the spatial radiation characteristics, thereby directly affecting the complex signal transmitted along the downstream waveguide and the resulting communication performance. However, existing studies have mainly characterized these effects through power-coupling or radiation-oriented formulations [8 , 9 , 15 , 17 , 18 ] . Therefore, to the best of the authors’ knowledge, no existing work has jointly related the same finite physical PA perturbation to its complete complex guided-wave scattering response and corresponding three-dimensional radiation characteristics within a unified electromagnetic framework.
In this paper, we develop an electromagnetic model of a finite deformation-based PA that directly relates the applied waveguide perturbation to its guided-wave and radiation responses. Specifically, starting from the same finite PA geometry, the proposed model jointly derives the complex reflection and transmission coefficients and the corresponding three-dimensional radiation characteristics, thereby determining both the complex signal remaining in the waveguide and the field radiated into free space. Furthermore, validation against full-wave electromagnetic simulations performed using MEEP shows that the pinching ratio and pinched-section length jointly affect the transmitted power and phase, radiation power-fraction, and radiation pattern, thus establishing these geometric parameters as design degrees of freedom for shaping the PA radiation response. Finally, the proposed model is evaluated in terms of the received SNR, showing that minimizing the PA–user distance or maximizing the total radiation power-fraction does not necessarily yield the highest received SNR, since the applied perturbation also determines the realized gain and, consequently, the effectiveness of a given PA position. Therefore, the proposed electromagnetic model directly connects the finite deformation-based PA geometry to its complex downstream guided-wave response and three-dimensional radiation characteristics, providing the physical quantities required for geometry-aware PAS design.
The remainder of the paper is organized as follows. Section II develops the guided-wave propagation and PA scattering model, while Section III derives the corresponding far-field radiation characteristics and realized gain. Section IV presents the numerical results, full-wave validation, and received-SNR evaluation, whereas Section V concludes the paper.
Waveguide Propagation and Pinching-Antenna Response
In this section, the propagation of the guided signal along the dielectric waveguide and its modification by an activated PA are characterized. The fundamental H E 11 \mathrm{HE}_{11} HE 11
Propagation in a Cylindrical Dielectric Waveguide with Pinching Antennas
In a PAS, the guided signal propagates along a dielectric waveguide and is locally modified by the activated PAs. To isolate and quantify the electromagnetic effect of the mechanical perturbation introduced by pinching, an axisymmetric reduced-radius representation is adopted, in which the k k k a a a b k = κ k a b_k = \kappa_k a b k = κ k a L P , k L_{P,k} L P , k κ k \kappa_k κ k L P , k L_{P,k} L P , k [20 ] , [21 ] . The resulting structure therefore consists of a transition from a a a b k b_k b k b k b_k b k a a a
Under the considered single-mode regime, both the unperturbed and pinched sections support the fundamental hybrid H E 11 \mathrm{HE}_{11} HE 11 [22 ] . Accordingly, for a cylindrical dielectric waveguide section of radius r ∈ { a , b k } r \in\{a,b_k\} r ∈ { a , b k }
V ( r ) = k 0 r n 1 2 − n 2 2 = k 0 r ε r − 1 , (1) V(r)=k_0r\sqrt{n_1^2-n_2^2}=k_0r\sqrt{\varepsilon_r-1}, \tag*{(1)} V ( r ) = k 0 r n 1 2 − n 2 2 = k 0 r ε r − 1 , (1) where k 0 = 2 π / λ 0 k_0=2\pi/\lambda_0 k 0 = 2 π / λ 0 n 1 = ε r n_1=\sqrt{\varepsilon_r} n 1 = ε r n 2 = 1 n_2=1 n 2 = 1 V ( r ) V(r) V ( r ) H E 11 \mathrm{HE}_{11} HE 11 u ( r ) = r k 0 2 ε r − β 2 ( r ) u(r)=r\sqrt{k_0^2\varepsilon_r-\beta^2(r)} u ( r ) = r k 0 2 ε r − β 2 ( r ) w ( r ) = r β 2 ( r ) − k 0 2 w(r)=r\sqrt{\beta^2(r)-k_0^2} w ( r ) = r β 2 ( r ) − k 0 2 u 2 ( r ) + w 2 ( r ) = V 2 ( r ) u^2(r)+w^2(r)=V^2(r) u 2 ( r ) + w 2 ( r ) = V 2 ( r ) w ( r ) = V 2 ( r ) − u 2 ( r ) w(r)=\sqrt{V^2(r)-u^2(r)} w ( r ) = V 2 ( r ) − u 2 ( r )
β ( r ) = k 0 2 + w 2 ( r ) r 2 . (2) \beta(r)=\sqrt{k_0^2+\frac{w^2(r)}{r^2}}. \tag*{(2)} β ( r ) = k 0 2 + r 2 w 2 ( r ) . (2) Thus, for a specified radius r r r w ( r ) w(r) w ( r ) β ( r ) \beta(r) β ( r ) u ( r ) u(r) u ( r ) ρ = r \rho=r ρ = r H E 11 \mathrm{HE}_{11} HE 11 [22 ]
[ J 1 ′ ( u ( r ) ) u ( r ) J 1 ( u ( r ) ) + K 1 ′ ( w ( r ) ) w ( r ) K 1 ( w ( r ) ) ] [ n 1 2 J 1 ′ ( u ( r ) ) u ( r ) J 1 ( u ( r ) ) + n 2 2 K 1 ′ ( w ( r ) ) w ( r ) K 1 ( w ( r ) ) ] = ( β ( r ) k 0 ) 2 ( 1 u 2 ( r ) + 1 w 2 ( r ) ) 2 , (3) \left[\frac{J_1'(u(r))}{u(r)J_1(u(r))}+\frac{K_1'(w(r))}{w(r)K_1(w(r))}\right]
\left[n_1^2\frac{J_1'(u(r))}{u(r)J_1(u(r))}+n_2^2\frac{K_1'(w(r))}{w(r)K_1(w(r))}\right]
=\left(\frac{\beta(r)}{k_0}\right)^2\left(\frac{1}{u^2(r)}+\frac{1}{w^2(r)}\right)^2, \tag*{(3)} [ u ( r ) J 1 ( u ( r )) J 1 ′ ( u ( r )) + w ( r ) K 1 ( w ( r )) K 1 ′ ( w ( r )) ] [ n 1 2 u ( r ) J 1 ( u ( r )) J 1 ′ ( u ( r )) + n 2 2 w ( r ) K 1 ( w ( r )) K 1 ′ ( w ( r )) ] = ( k 0 β ( r ) ) 2 ( u 2 ( r ) 1 + w 2 ( r ) 1 ) 2 , (3) where J 1 ( ⋅ ) J_1(\cdot) J 1 ( ⋅ ) K 1 ( ⋅ ) K_1(\cdot) K 1 ( ⋅ ) ( ⋅ ) ′ (\cdot)' ( ⋅ ) ′ w ( r ) = V 2 ( r ) − u 2 ( r ) w(r)=\sqrt{V^2(r)-u^2(r)} w ( r ) = V 2 ( r ) − u 2 ( r ) u ( r ) ∈ ( 0 , V ( r ) ) u(r)\in(0,V(r)) u ( r ) ∈ ( 0 , V ( r )) H E 11 \mathrm{HE}_{11} HE 11 w ( r ) w(r) w ( r ) β ( r ) \beta(r) β ( r ) r = a r=a r = a r = b k r=b_k r = b k β ( a ) \beta(a) β ( a ) β ( b k ) \beta(b_k) β ( b k )
At each radius transition, the local guided response is determined by the H E 11 \mathrm{HE}_{11} HE 11 a a a b k b_k b k ( ρ , ϕ , z ) (\rho,\phi,z) ( ρ , ϕ , z ) z z z ρ \rho ρ ϕ \phi ϕ r ∈ { a , b k } r\in\{a,b_k\} r ∈ { a , b k } H E 11 \mathrm{HE}_{11} HE 11 [22 ]
E z ( ρ , ϕ , z ; r ) = C ( r ) Ψ ( ρ ; r ) cos ϕ e − j β ( r ) z , (4) E_z(\rho,\phi,z;r)=C(r)\Psi(\rho;r)\cos\phi\,e^{-j\beta(r)z}, \tag*{(4)} E z ( ρ , ϕ , z ; r ) = C ( r ) Ψ ( ρ ; r ) cos ϕ e − j β ( r ) z , (4) H z ( ρ , ϕ , z ; r ) = D ( r ) Ψ ( ρ ; r ) sin ϕ e − j β ( r ) z , (5) H_z(\rho,\phi,z;r)=D(r)\Psi(\rho;r)\sin\phi\,e^{-j\beta(r)z}, \tag*{(5)} H z ( ρ , ϕ , z ; r ) = D ( r ) Ψ ( ρ ; r ) sin ϕ e − j β ( r ) z , (5) where the corresponding radial field dependence is
Ψ ( ρ ; r ) = { J 1 ( u ( r ) ρ / r ) J 1 ( u ( r ) ) , 0 ≤ ρ < r , J 1 ( u ( r ) ) K 1 ( w ( r ) ) K 1 ( w ( r ) ρ / r ) , ρ ≥ r . (6) \Psi(\rho;r)=
\begin{cases}
\dfrac{J_1(u(r)\rho/r)}{J_1(u(r))}, & 0\leq\rho<r,\\[6pt]
\dfrac{J_1(u(r))}{K_1(w(r))}K_1(w(r)\rho/r), & \rho\geq r.
\end{cases} \tag*{(6)} Ψ ( ρ ; r ) = ⎩ ⎨ ⎧ J 1 ( u ( r )) J 1 ( u ( r ) ρ / r ) , K 1 ( w ( r )) J 1 ( u ( r )) K 1 ( w ( r ) ρ / r ) , 0 ≤ ρ < r , ρ ≥ r . (6) Since the H E 11 \mathrm{HE}_{11} HE 11 [22 ]
D ( r ) C ( r ) = − β ( r ) ω μ 0 1 u 2 ( r ) + 1 w 2 ( r ) J 1 ′ ( u ( r ) ) u ( r ) J 1 ( u ( r ) ) + K 1 ′ ( w ( r ) ) w ( r ) K 1 ( w ( r ) ) , (7) \frac{D(r)}{C(r)}=-\frac{\beta(r)}{\omega\mu_0}
\frac{\dfrac{1}{u^2(r)}+\dfrac{1}{w^2(r)}}
{\dfrac{J_1'(u(r))}{u(r)J_1(u(r))}+\dfrac{K_1'(w(r))}{w(r)K_1(w(r))}}, \tag*{(7)} C ( r ) D ( r ) = − ω μ 0 β ( r ) u ( r ) J 1 ( u ( r )) J 1 ′ ( u ( r )) + w ( r ) K 1 ( w ( r )) K 1 ′ ( w ( r )) u 2 ( r ) 1 + w 2 ( r ) 1 , (7) where ω = 2 π f 0 \omega=2\pi f_0 ω = 2 π f 0 μ 0 \mu_0 μ 0 [22 ]
E t ( ρ , ϕ , z ; r ) = − j e − j β ( r ) z k t 2 ( ρ ; r ) { ρ ^ cos ϕ [ β ( r ) C ( r ) d Ψ ( ρ ; r ) d ρ + ω μ 0 D ( r ) Ψ ( ρ ; r ) ρ ] − ϕ ^ sin ϕ [ β ( r ) C ( r ) Ψ ( ρ ; r ) ρ + ω μ 0 D ( r ) d Ψ ( ρ ; r ) d ρ ] } , (8) \mathbf{E}_t(\rho,\phi,z;r)=-\frac{je^{-j\beta(r)z}}{k_t^2(\rho;r)}
\left\{
\hat{\boldsymbol{\rho}}\cos\phi
\left[\beta(r)C(r)\frac{\mathrm{d}\Psi(\rho;r)}{\mathrm{d}\rho}
+\omega\mu_0D(r)\frac{\Psi(\rho;r)}{\rho}\right]
-\hat{\boldsymbol{\phi}}\sin\phi
\left[\beta(r)C(r)\frac{\Psi(\rho;r)}{\rho}
+\omega\mu_0D(r)\frac{\mathrm{d}\Psi(\rho;r)}{\mathrm{d}\rho}\right]
\right\}, \tag*{(8)} E t ( ρ , ϕ , z ; r ) = − k t 2 ( ρ ; r ) j e − j β ( r ) z { ρ ^ cos ϕ [ β ( r ) C ( r ) d ρ d Ψ ( ρ ; r ) + ω μ 0 D ( r ) ρ Ψ ( ρ ; r ) ] − ϕ ^ sin ϕ [ β ( r ) C ( r ) ρ Ψ ( ρ ; r ) + ω μ 0 D ( r ) d ρ d Ψ ( ρ ; r ) ] } , (8) while the transverse magnetic field follows as
H t ( ρ , ϕ , z ; r ) = − j e − j β ( r ) z k t 2 ( ρ ; r ) { ρ ^ sin ϕ [ β ( r ) D ( r ) d Ψ ( ρ ; r ) d ρ + ω ε ( ρ ; r ) C ( r ) Ψ ( ρ ; r ) ρ ] + ϕ ^ cos ϕ [ β ( r ) D ( r ) Ψ ( ρ ; r ) ρ + ω ε ( ρ ; r ) C ( r ) d Ψ ( ρ ; r ) d ρ ] } , (9) \mathbf{H}_t(\rho,\phi,z;r)=-\frac{je^{-j\beta(r)z}}{k_t^2(\rho;r)}
\left\{
\hat{\boldsymbol{\rho}}\sin\phi
\left[\beta(r)D(r)\frac{\mathrm{d}\Psi(\rho;r)}{\mathrm{d}\rho}
+\omega\varepsilon(\rho;r)C(r)\frac{\Psi(\rho;r)}{\rho}\right]
+\hat{\boldsymbol{\phi}}\cos\phi
\left[\beta(r)D(r)\frac{\Psi(\rho;r)}{\rho}
+\omega\varepsilon(\rho;r)C(r)\frac{\mathrm{d}\Psi(\rho;r)}{\mathrm{d}\rho}\right]
\right\}, \tag*{(9)} H t ( ρ , ϕ , z ; r ) = − k t 2 ( ρ ; r ) j e − j β ( r ) z { ρ ^ sin ϕ [ β ( r ) D ( r ) d ρ d Ψ ( ρ ; r ) + ω ε ( ρ ; r ) C ( r ) ρ Ψ ( ρ ; r ) ] + ϕ ^ cos ϕ [ β ( r ) D ( r ) ρ Ψ ( ρ ; r ) + ω ε ( ρ ; r ) C ( r ) d ρ d Ψ ( ρ ; r ) ] } , (9) Moreover, the radial derivative required in (8) and (9) follows directly from (6) as
d Ψ ( ρ ; r ) d ρ = { u ( r ) r J 0 ( u ( r ) r ρ ) − 1 ρ J 1 ( u ( r ) r ρ ) , 0 ≤ ρ < r , − J 1 ( u ( r ) ) K 1 ( w ( r ) ) [ w ( r ) r K 0 ( w ( r ) r ρ ) + 1 ρ K 1 ( w ( r ) r ρ ) ] , ρ ≥ r . (10) \frac{\mathrm{d}\Psi(\rho;r)}{\mathrm{d}\rho}=
\begin{cases}
\dfrac{u(r)}{r}J_0\left(\dfrac{u(r)}{r}\rho\right)-\dfrac{1}{\rho}J_1\left(\dfrac{u(r)}{r}\rho\right), & 0\leq\rho<r,\\[8pt]
-\dfrac{J_1(u(r))}{K_1(w(r))}
\left[\dfrac{w(r)}{r}K_0\left(\dfrac{w(r)}{r}\rho\right)+\dfrac{1}{\rho}K_1\left(\dfrac{w(r)}{r}\rho\right)\right], & \rho\geq r.
\end{cases} \tag*{(10)} d ρ d Ψ ( ρ ; r ) = ⎩ ⎨ ⎧ r u ( r ) J 0 ( r u ( r ) ρ ) − ρ 1 J 1 ( r u ( r ) ρ ) , − K 1 ( w ( r )) J 1 ( u ( r )) [ r w ( r ) K 0 ( r w ( r ) ρ ) + ρ 1 K 1 ( r w ( r ) ρ ) ] , 0 ≤ ρ < r , ρ ≥ r . (10) while ε ( ρ ; r ) \varepsilon(\rho;r) ε ( ρ ; r )
ε ( ρ ; r ) = { ε 0 ε r , 0 ≤ ρ < r , ε 0 , ρ ≥ r , (11) \varepsilon(\rho;r)=
\begin{cases}
\varepsilon_0\varepsilon_r, & 0\leq\rho<r,\\
\varepsilon_0, & \rho\geq r,
\end{cases} \tag*{(11)} ε ( ρ ; r ) = { ε 0 ε r , ε 0 , 0 ≤ ρ < r , ρ ≥ r , (11) which determines the corresponding transverse wavenumber as
k t 2 ( ρ ; r ) = ω 2 μ 0 ε ( ρ ; r ) − β 2 ( r ) = { u 2 ( r ) / r 2 , 0 ≤ ρ < r , − w 2 ( r ) / r 2 , ρ ≥ r . (12) k_t^2(\rho;r)=\omega^2\mu_0\varepsilon(\rho;r)-\beta^2(r)=
\begin{cases}
u^2(r)/r^2, & 0\leq\rho<r,\\
-w^2(r)/r^2, & \rho\geq r.
\end{cases} \tag*{(12)} k t 2 ( ρ ; r ) = ω 2 μ 0 ε ( ρ ; r ) − β 2 ( r ) = { u 2 ( r ) / r 2 , − w 2 ( r ) / r 2 , 0 ≤ ρ < r , ρ ≥ r . (12) Hence, evaluating (4) –(12) for r = a r=a r = a r = b k r=b_k r = b k 11 _{11} 11 β − ( r ) = − β + ( r ) \beta^-(r)=-\beta^+(r) β − ( r ) = − β + ( r ) C − ( r ) = − C + ( r ) C^-(r)=-C^+(r) C − ( r ) = − C + ( r ) D − ( r ) = D + ( r ) D^-(r)=D^+(r) D − ( r ) = D + ( r ) [21 ] . Hence, at the reference plane, the transverse fields satisfy E ‾ t − = E ‾ t + \overline{\mathbf{E}}_t^-=\overline{\mathbf{E}}_t^+ E t − = E t + H ‾ t − = − H ‾ t + \overline{\mathbf{H}}_t^-=-\overline{\mathbf{H}}_t^+ H t − = − H t + u ( r ) u(r) u ( r ) w ( r ) w(r) w ( r ) Ψ ( ρ ; r ) \Psi(\rho;r) Ψ ( ρ ; r ) β 2 ( r ) \beta^2(r) β 2 ( r )
With the radius-dependent HE11 _{11} 11 C ( r ) C(r) C ( r ) D ( r ) D(r) D ( r ) r = a r=a r = a r = b k r=b_k r = b k 11 _{11} 11 r r r [22 ] , [23 ]
P g ( r ) = 1 2 ∣ ℜ { ∫ 0 2 π ∫ 0 ∞ [ E t ( ρ , ϕ , z ; r ) × H t ∗ ( ρ , ϕ , z ; r ) ] ⋅ z ^ ρ d ρ d ϕ } ∣ . (13) P_g(r)=\frac{1}{2}\left|\Re\left\{\int_0^{2\pi}\int_0^\infty[\mathbf{E}_t(\rho,\phi,z;r)\times\mathbf{H}_t^*(\rho,\phi,z;r)]\cdot\hat{\mathbf{z}}\,\rho\,d\rho\,d\phi\right\}\right|. \tag*{(13)} P g ( r ) = 2 1 ℜ { ∫ 0 2 π ∫ 0 ∞ [ E t ( ρ , ϕ , z ; r ) × H t ∗ ( ρ , ϕ , z ; r )] ⋅ z ^ ρ d ρ d ϕ } . (13) Consequently, the corresponding unit-power modal fields are
E ‾ ( ρ , ϕ , z ; r ) = E ( ρ , ϕ , z ; r ) P g ( r ) , (14) \overline{\mathbf{E}}(\rho,\phi,z;r)=\frac{\mathbf{E}(\rho,\phi,z;r)}{\sqrt{P_g(r)}}, \tag*{(14)} E ( ρ , ϕ , z ; r ) = P g ( r ) E ( ρ , ϕ , z ; r ) , (14) H ‾ ( ρ , ϕ , z ; r ) = H ( ρ , ϕ , z ; r ) P g ( r ) , (15) \overline{\mathbf{H}}(\rho,\phi,z;r)=\frac{\mathbf{H}(\rho,\phi,z;r)}{\sqrt{P_g(r)}}, \tag*{(15)} H ( ρ , ϕ , z ; r ) = P g ( r ) H ( ρ , ϕ , z ; r ) , (15) where E = z ^ E z + E t \mathbf{E}=\hat{\mathbf{z}}E_z+\mathbf{E}_t E = z ^ E z + E t H = z ^ H z + H t \mathbf{H}=\hat{\mathbf{z}}H_z+\mathbf{H}_t H = z ^ H z + H t E ‾ t \overline{\mathbf{E}}_t E t H ‾ t \overline{\mathbf{H}}_t H t 11 _{11} 11 E ‾ t − = E ‾ t + \overline{\mathbf{E}}_t^-=\overline{\mathbf{E}}_t^+ E t − = E t + H ‾ t − = − H ‾ t + \overline{\mathbf{H}}_t^-=-\overline{\mathbf{H}}_t^+ H t − = − H t + 11 _{11} 11
Pinching Antenna Response
In a PAS, activating a PA locally modifies the waveguide geometry and redistributes the incident guided power among the signal transmitted to the downstream waveguide section, the signal reflected toward the source, and radiation. Hence, the PA cannot be characterized through an independently assigned radiation coefficient, but through the scattering response induced by its finite geometry. In particular, the k k k a a a b k b_k b k b k b_k b k L P , k L_{\mathrm{P},k} L P , k b k b_k b k a a a
Within the considered single-mode operating regime, the pinched section supports the fundamental HE11 _{11} 11 β ( b k ) \beta(b_k) β ( b k )
At each radius transition, the change in waveguide radius modifies the transverse field distribution of the supported HE11 _{11} 11 [20 ] , and retaining the HE11 _{11} 11
W k = 1 2 ∫ 0 2 π ∫ 0 ∞ [ E ‾ t ( ρ , ϕ , 0 ; a ) × H ‾ t ∗ ( ρ , ϕ , 0 ; b k ) ] ⋅ z ^ ρ d ρ d ϕ , (16) W_k=\frac{1}{2}\int_0^{2\pi}\int_0^\infty\left[\overline{\mathbf{E}}_t(\rho,\phi,0;a)\times\overline{\mathbf{H}}_t^*(\rho,\phi,0;b_k)\right]\cdot\hat{\mathbf{z}}\,\rho\,d\rho\,d\phi, \tag*{(16)} W k = 2 1 ∫ 0 2 π ∫ 0 ∞ [ E t ( ρ , ϕ , 0 ; a ) × H t ∗ ( ρ , ϕ , 0 ; b k ) ] ⋅ z ^ ρ d ρ d ϕ , (16) Q k = 1 2 ∫ 0 2 π ∫ 0 ∞ [ E ‾ t ∗ ( ρ , ϕ , 0 ; b k ) × H ‾ t ( ρ , ϕ , 0 ; a ) ] ⋅ z ^ ρ d ρ d ϕ , (17) Q_k=\frac{1}{2}\int_0^{2\pi}\int_0^\infty\left[\overline{\mathbf{E}}_t^*(\rho,\phi,0;b_k)\times\overline{\mathbf{H}}_t(\rho,\phi,0;a)\right]\cdot\hat{\mathbf{z}}\,\rho\,d\rho\,d\phi, \tag*{(17)} Q k = 2 1 ∫ 0 2 π ∫ 0 ∞ [ E t ∗ ( ρ , ϕ , 0 ; b k ) × H t ( ρ , ϕ , 0 ; a ) ] ⋅ z ^ ρ d ρ d ϕ , (17) where W k W_k W k Q k Q_k Q k 11 _{11} 11 11 _{11} 11 11 _{11} 11 a r , k a_{r,k} a r , k a t , k a_{t,k} a t , k 11 _{11} 11 11 _{11} 11 [21 ] , and using the forward–backward field convention, gives ( 1 + a r , k ) W k = a t , k (1+a_{r,k})W_k=a_{t,k} ( 1 + a r , k ) W k = a t , k ( 1 − a r , k ) Q k = a t , k (1-a_{r,k})Q_k=a_{t,k} ( 1 − a r , k ) Q k = a t , k a r , k a_{r,k} a r , k a t , k a_{t,k} a t , k
a r , k = Q k − W k Q k + W k , (18) a_{r,k}=\frac{Q_k-W_k}{Q_k+W_k}, \tag*{(18)} a r , k = Q k + W k Q k − W k , (18) a t , k = 2 W k Q k Q k + W k . (19) a_{t,k}=\frac{2W_kQ_k}{Q_k+W_k}. \tag*{(19)} a t , k = Q k + W k 2 W k Q k . (19) Therefore, a t , k a_{t,k} a t , k a r , k a_{r,k} a r , k 11 _{11} 11 11 _{11} 11 a a a b k b_k b k b k b_k b k a a a a t , k a_{t,k} a t , k a r , k a_{r,k} a r , k − a r , k -a_{r,k} − a r , k [21 ] .
The local step response obtained above characterizes the guided HE11 _{11} 11 [24 ] , [25 ] . Since the two radius transitions are separated by the finite distance L P , k L_{\mathrm{P},k} L P , k 11 _{11} 11 [26 ] . To characterize this interaction, the additional fields in the uniform pinched section of radius b k b_k b k ϕ \phi ϕ m = 1 m=1 m = 1 11 _{11} 11 m = 1 m=1 m = 1 q ≥ 0 q \ge0 q ≥ 0 q 2 + β q 2 = k 0 2 q^2+\beta_q^2=k_0^2 q 2 + β q 2 = k 0 2
β q = { k 0 2 − q 2 , 0 ≤ q < k 0 , − j q 2 − k 0 2 , q > k 0 , (20) \beta_q =
\begin{cases}
\sqrt{k_0^2-q^2}, & 0 \le q < k_0,\\
-j\sqrt{q^2-k_0^2}, & q > k_0,
\end{cases} \tag*{(20)} β q = { k 0 2 − q 2 , − j q 2 − k 0 2 , 0 ≤ q < k 0 , q > k 0 , (20) where q < k 0 q<k_0 q < k 0 q > k 0 q>k_0 q > k 0 u q = b k k 0 2 ϵ r − β q 2 u_q=b_k\sqrt{k_0^2\epsilon_r-\beta_q^2} u q = b k k 0 2 ϵ r − β q 2 [27 ] , for each q q q m = 1 m=1 m = 1 s ∈ { 1 , 2 } s\in\{1,2\} s ∈ { 1 , 2 }
E z , q , s ( ρ , ϕ , z ; b k ) = Ψ E , q , s ( ρ ; b k ) cos ϕ e − j β q z , (21) E_{z,q,s}(\rho,\phi,z;b_k)=\Psi_{E,q,s}(\rho;b_k)\cos\phi\,e^{-j\beta_q z}, \tag*{(21)} E z , q , s ( ρ , ϕ , z ; b k ) = Ψ E , q , s ( ρ ; b k ) cos ϕ e − j β q z , (21) H z , q , s ( ρ , ϕ , z ; b k ) = Ψ H , q , s ( ρ ; b k ) sin ϕ e − j β q z , (22) H_{z,q,s}(\rho,\phi,z;b_k)=\Psi_{H,q,s}(\rho;b_k)\sin\phi\,e^{-j\beta_q z}, \tag*{(22)} H z , q , s ( ρ , ϕ , z ; b k ) = Ψ H , q , s ( ρ ; b k ) sin ϕ e − j β q z , (22) where
Ψ E , q , s ( ρ ; b k ) = { C q , s J 1 ( u q ρ / b k ) , 0 ≤ ρ < b k , A q , s J 1 ( q ρ ) + B q , s Y 1 ( q ρ ) , ρ ≥ b k , (23) \Psi_{E,q,s}(\rho;b_k)=
\begin{cases}
C_{q,s}J_1(u_q\rho/b_k), & 0\le\rho<b_k,\\
A_{q,s}J_1(q\rho)+B_{q,s}Y_1(q\rho), & \rho\ge b_k,
\end{cases} \tag*{(23)} Ψ E , q , s ( ρ ; b k ) = { C q , s J 1 ( u q ρ / b k ) , A q , s J 1 ( q ρ ) + B q , s Y 1 ( q ρ ) , 0 ≤ ρ < b k , ρ ≥ b k , (23) and
Ψ H , q , s ( ρ ; b k ) = { D q , s J 1 ( u q ρ / b k ) , 0 ≤ ρ < b k , F q , s J 1 ( q ρ ) + G q , s Y 1 ( q ρ ) , ρ ≥ b k . (24) \Psi_{H,q,s}(\rho;b_k)=
\begin{cases}
D_{q,s}J_1(u_q\rho/b_k), & 0\le\rho<b_k,\\
F_{q,s}J_1(q\rho)+G_{q,s}Y_1(q\rho), & \rho\ge b_k.
\end{cases} \tag*{(24)} Ψ H , q , s ( ρ ; b k ) = { D q , s J 1 ( u q ρ / b k ) , F q , s J 1 ( q ρ ) + G q , s Y 1 ( q ρ ) , 0 ≤ ρ < b k , ρ ≥ b k . (24) where Y 1 ( ⋅ ) Y_1(\cdot) Y 1 ( ⋅ ) C q , s C_{q,s} C q , s D q , s D_{q,s} D q , s A q , s A_{q,s} A q , s B q , s B_{q,s} B q , s F q , s F_{q,s} F q , s G q , s G_{q,s} G q , s E z E_z E z H z H_z H z E ϕ E_\phi E ϕ H ϕ H_\phi H ϕ ρ = b k \rho=b_k ρ = b k [27 ] . These longitudinal fields then determine the transverse fields directly from Maxwell’s equations, consistently with (8) and (9) , as
E t , q , s ( ρ , ϕ , z ; b k ) = − j [ β q ∇ t E z , q , s − ω μ 0 z ^ × ∇ t H z , q , s ] k t , q 2 ( ρ ) , (25) \mathbf{E}_{t,q,s}(\rho,\phi,z;b_k)=-\frac{j\left[\beta_q\nabla_t E_{z,q,s}-\omega\mu_0\hat{\mathbf{z}}\times\nabla_t H_{z,q,s}\right]}{k_{t,q}^2(\rho)}, \tag*{(25)} E t , q , s ( ρ , ϕ , z ; b k ) = − k t , q 2 ( ρ ) j [ β q ∇ t E z , q , s − ω μ 0 z ^ × ∇ t H z , q , s ] , (25) H t , q , s ( ρ , ϕ , z ; b k ) = − j [ β q ∇ t H z , q , s + ω ϵ ( ρ ; b k ) z ^ × ∇ t E z , q , s ] k t , q 2 ( ρ ) , (26) \mathbf{H}_{t,q,s}(\rho,\phi,z;b_k)=-\frac{j\left[\beta_q\nabla_t H_{z,q,s}+\omega\epsilon(\rho;b_k)\hat{\mathbf{z}}\times\nabla_t E_{z,q,s}\right]}{k_{t,q}^2(\rho)}, \tag*{(26)} H t , q , s ( ρ , ϕ , z ; b k ) = − k t , q 2 ( ρ ) j [ β q ∇ t H z , q , s + ω ϵ ( ρ ; b k ) z ^ × ∇ t E z , q , s ] , (26) where ∇ t = ρ ^ ∂ ∂ ρ + ϕ ^ 1 ρ ∂ ∂ ϕ \nabla_t=\hat{\boldsymbol{\rho}}\frac{\partial}{\partial\rho}+\hat{\boldsymbol{\phi}}\frac{1}{\rho}\frac{\partial}{\partial\phi} ∇ t = ρ ^ ∂ ρ ∂ + ϕ ^ ρ 1 ∂ ϕ ∂
k t , q 2 ( ρ ) = { u q 2 / b k 2 , 0 ≤ ρ < b k , q 2 , ρ ≥ b k . (27) k_{t,q}^2(\rho)=
\begin{cases}
u_q^2/b_k^2, & 0\le\rho<b_k,\\
q^2, & \rho\ge b_k.
\end{cases} \tag*{(27)} k t , q 2 ( ρ ) = { u q 2 / b k 2 , q 2 , 0 ≤ ρ < b k , ρ ≥ b k . (27) To use these fields to determine how strongly each cylindrical component is excited at a radius transition, their arbitrary modal scaling must first be fixed. Accordingly, the cylindrical solutions are scaled using the radiation-mode normalization in [27 ] , while both the propagating and decaying components of the continuous spectrum are retained consistently with the radiation-spectrum treatment in [25 ] , and the resulting transverse fields are denoted by E t , q , s \mathbf{E}_{t,q,s} E t , q , s H t , q , s \mathbf{H}_{t,q,s} H t , q , s [21 ] and the radiation-spectrum treatment of dielectric-waveguide discontinuities in [25 ] , their coupling to the guided HE11 _{11} 11
W k , s ( q ) = 1 2 ∫ 0 2 π ∫ 0 ∞ [ E t + ( ρ , ϕ , 0 ; a ) × H t , q , s ( ρ , ϕ , 0 ; b k ) ] ⋅ z ^ ρ d ρ d ϕ , (28) W_{k,s}(q)=\frac{1}{2}\int_0^{2\pi}\int_0^\infty\left[\mathbf{E}_t^+(\rho,\phi,0;a)\times\mathbf{H}_{t,q,s}(\rho,\phi,0;b_k)\right]\cdot\hat{\mathbf{z}}\rho\,d\rho\,d\phi, \tag*{(28)} W k , s ( q ) = 2 1 ∫ 0 2 π ∫ 0 ∞ [ E t + ( ρ , ϕ , 0 ; a ) × H t , q , s ( ρ , ϕ , 0 ; b k ) ] ⋅ z ^ ρ d ρ d ϕ , (28) Q k , s ( q ) = 1 2 ∫ 0 2 π ∫ 0 ∞ [ E t , q , s ( ρ , ϕ , 0 ; b k ) × H t + ( ρ , ϕ , 0 ; a ) ] ⋅ z ^ ρ d ρ d ϕ . (29) Q_{k,s}(q)=\frac{1}{2}\int_0^{2\pi}\int_0^\infty\left[\mathbf{E}_{t,q,s}(\rho,\phi,0;b_k)\times\mathbf{H}_t^+(\rho,\phi,0;a)\right]\cdot\hat{\mathbf{z}}\rho\,d\rho\,d\phi. \tag*{(29)} Q k , s ( q ) = 2 1 ∫ 0 2 π ∫ 0 ∞ [ E t , q , s ( ρ , ϕ , 0 ; b k ) × H t + ( ρ , ϕ , 0 ; a ) ] ⋅ z ^ ρ d ρ d ϕ . (29) The common factor − j -j − j [27 ] , making the transverse modal profiles real at z = 0 z=0 z = 0 11 _{11} 11 11 _{11} 11 b k b_k b k [21 ] , their corresponding cross-projections vanish. Consequently, projecting the electric- and magnetic-field continuity relations onto the cylindrical modes under the same forward–backward convention yields the corresponding sum and difference of spectral amplitudes [21 , 25 ] . Hence, for unit HE11 _{11} 11
c k , s ( 1 ) ( q ) = 1 2 [ ( 1 + a r , k ) W k , s ( q ) + ( 1 − a r , k ) Q k , s ( q ) ] . (30) c_{k,s}^{(1)}(q)=\frac{1}{2}\left[(1+a_{r,k})W_{k,s}(q)+(1-a_{r,k})Q_{k,s}(q)\right]. \tag*{(30)} c k , s ( 1 ) ( q ) = 2 1 [ ( 1 + a r , k ) W k , s ( q ) + ( 1 − a r , k ) Q k , s ( q ) ] . (30) Similarly, for unit HE11 _{11} 11 11 _{11} 11 a t , k a_{t,k} a t , k 11 _{11} 11 − a r , k -a_{r,k} − a r , k 11 _{11} 11 b k b_k b k
c k , s ( 2 ) ( q ) = a t , k 2 [ W k , s ( q ) − Q k , s ( q ) ] . (31) c_{k,s}^{(2)}(q)=\frac{a_{t,k}}{2}\left[W_{k,s}(q)-Q_{k,s}(q)\right]. \tag*{(31)} c k , s ( 2 ) ( q ) = 2 a t , k [ W k , s ( q ) − Q k , s ( q ) ] . (31) Thus, c k , s ( 1 ) ( q ) c_{k,s}^{(1)}(q) c k , s ( 1 ) ( q ) c k , s ( 2 ) ( q ) c_{k,s}^{(2)}(q) c k , s ( 2 ) ( q ) ( q , s ) (q,s) ( q , s ) 11 _{11} 11 [21 ] . Once excited, each cylindrical component propagates or decays between the two radius transitions according to e − j β q L p , k e^{-j\beta_qL_{p,k}} e − j β q L p , k 11 _{11} 11 [26 ] . Consequently, combining the coupling at one transition, propagation through the finite pinched section, and coupling at the other transition over the cylindrical spectrum gives
μ k = ∑ s = 1 2 ∫ 0 ∞ [ c k , s ( 1 ) ( q ) ] 2 e − j β q L p , k d q , (32) \mu_k=\sum_{s=1}^{2}\int_0^\infty\left[c_{k,s}^{(1)}(q)\right]^2e^{-j\beta_qL_{p,k}}\,dq, \tag*{(32)} μ k = s = 1 ∑ 2 ∫ 0 ∞ [ c k , s ( 1 ) ( q ) ] 2 e − j β q L p , k d q , (32) ν k = ∑ s = 1 2 ∫ 0 ∞ c k , s ( 1 ) ( q ) c k , s ( 2 ) ( q ) e − j β q L p , k d q , (33) \nu_k=\sum_{s=1}^{2}\int_0^\infty c_{k,s}^{(1)}(q)c_{k,s}^{(2)}(q)e^{-j\beta_qL_{p,k}}\,dq, \tag*{(33)} ν k = s = 1 ∑ 2 ∫ 0 ∞ c k , s ( 1 ) ( q ) c k , s ( 2 ) ( q ) e − j β q L p , k d q , (33) γ k = ∑ s = 1 2 ∫ 0 ∞ [ c k , s ( 2 ) ( q ) ] 2 e − j β q L p , k d q , (34) \gamma_k=\sum_{s=1}^{2}\int_0^\infty\left[c_{k,s}^{(2)}(q)\right]^2e^{-j\beta_qL_{p,k}}\,dq, \tag*{(34)} γ k = s = 1 ∑ 2 ∫ 0 ∞ [ c k , s ( 2 ) ( q ) ] 2 e − j β q L p , k d q , (34) where μ k \mu_k μ k 11 _{11} 11 a a a ν k \nu_k ν k 11 _{11} 11 a a a 11 _{11} 11 b k b_k b k γ k \gamma_k γ k H E 11 \mathrm{HE}_{11} HE 11 b k b_k b k a t , k a_{t,k} a t , k a r , k a_{r,k} a r , k μ k \mu_k μ k ν k \nu_k ν k γ k \gamma_k γ k H E 11 \mathrm{HE}_{11} HE 11 L P , k L_{\mathrm{P},k} L P , k H E 11 \mathrm{HE}_{11} HE 11
Proposition 1. For the k k k κ k = b k / a \kappa_k=b_k/a κ k = b k / a L P , k L_{\mathrm{P},k} L P , k
S 11 , k P A = a r , k + p k [ − a r , k p k ( a t , k 2 + ν k 2 ) + 2 ( 1 − p k γ k ) a t , k ν k ] Δ k , (35) S_{11,k}^{\mathrm{PA}}=a_{r,k}+\frac{p_k\left[-a_{r,k}p_k\left(a_{t,k}^{2}+\nu_k^{2}\right)+2\left(1-p_k\gamma_k\right)a_{t,k}\nu_k\right]}{\Delta_k}, \tag*{(35)} S 11 , k PA = a r , k + Δ k p k [ − a r , k p k ( a t , k 2 + ν k 2 ) + 2 ( 1 − p k γ k ) a t , k ν k ] , (35) S 21 , k P A = μ k + p k [ ( 1 − p k γ k ) ( a t , k 2 + ν k 2 ) − 2 a r , k p k a t , k ν k ] Δ k , (36) S_{21,k}^{\mathrm{PA}}=\mu_k+\frac{p_k\left[\left(1-p_k\gamma_k\right)\left(a_{t,k}^{2}+\nu_k^{2}\right)-2a_{r,k}p_ka_{t,k}\nu_k\right]}{\Delta_k}, \tag*{(36)} S 21 , k PA = μ k + Δ k p k [ ( 1 − p k γ k ) ( a t , k 2 + ν k 2 ) − 2 a r , k p k a t , k ν k ] , (36) respectively, where Δ k = ( 1 − p k γ k ) 2 − a r , k 2 p k 2 \Delta_k=\left(1-p_k\gamma_k\right)^2-a_{r,k}^{2}p_k^{2} Δ k = ( 1 − p k γ k ) 2 − a r , k 2 p k 2 p k = e − j β ( b k ) L P , k p_k=e^{-j\beta(b_k)L_{\mathrm{P},k}} p k = e − j β ( b k ) L P , k H E 11 \mathrm{HE}_{11} HE 11 ∣ S 11 , k P A ∣ 2 \left|S_{11,k}^{\mathrm{PA}}\right|^2 S 11 , k PA 2 ∣ S 21 , k P A ∣ 2 \left|S_{21,k}^{\mathrm{PA}}\right|^2 S 21 , k PA 2
Proof. Consider a unit-amplitude H E 11 \mathrm{HE}_{11} HE 11 x 1 , k x_{1,k} x 1 , k x 2 , k x_{2,k} x 2 , k p k x 1 , k p_kx_{1,k} p k x 1 , k p k x 2 , k p_kx_{2,k} p k x 2 , k H E 11 \mathrm{HE}_{11} HE 11 H E 11 \mathrm{HE}_{11} HE 11 ν k \nu_k ν k H E 11 \mathrm{HE}_{11} HE 11 γ k \gamma_k γ k x 1 , k x_{1,k} x 1 , k x 2 , k x_{2,k} x 2 , k a t , k a_{t,k} a t , k − a r , k -a_{r,k} − a r , k ( 1 − p k γ k ) x 1 , k + a r , k p k x 2 , k = a t , k \left(1-p_k\gamma_k\right)x_{1,k}+a_{r,k}p_kx_{2,k}=a_{t,k} ( 1 − p k γ k ) x 1 , k + a r , k p k x 2 , k = a t , k a r , k p k x 1 , k + ( 1 − p k γ k ) x 2 , k = ν k a_{r,k}p_kx_{1,k}+\left(1-p_k\gamma_k\right)x_{2,k}=\nu_k a r , k p k x 1 , k + ( 1 − p k γ k ) x 2 , k = ν k
x 1 , k = ( 1 − p k γ k ) a t , k − a r , k p k ν k Δ k , (37) x_{1,k}=\frac{\left(1-p_k\gamma_k\right)a_{t,k}-a_{r,k}p_k\nu_k}{\Delta_k}, \tag*{(37)} x 1 , k = Δ k ( 1 − p k γ k ) a t , k − a r , k p k ν k , (37) x 2 , k = ( 1 − p k γ k ) ν k − a r , k p k a t , k Δ k . (38) x_{2,k}=\frac{\left(1-p_k\gamma_k\right)\nu_k-a_{r,k}p_ka_{t,k}}{\Delta_k}. \tag*{(38)} x 2 , k = Δ k ( 1 − p k γ k ) ν k − a r , k p k a t , k . (38) In addition, the guided field transmitted through the second transition consists of the coupling contribution μ k \mu_k μ k x 1 , k x_{1,k} x 1 , k x 2 , k x_{2,k} x 2 , k
S 21 , k P A = μ k + p k [ a t , k x 1 , k + ν k x 2 , k ] . (39) S_{21,k}^{\mathrm{PA}}=\mu_k+p_k\left[a_{t,k}x_{1,k}+\nu_kx_{2,k}\right]. \tag*{(39)} S 21 , k PA = μ k + p k [ a t , k x 1 , k + ν k x 2 , k ] . (39) Therefore, substituting (37) and (38) into (39) gives (36). Similarly, the reflected guided field consists of the direct reflection a r , k a_{r,k} a r , k x 1 , k x_{1,k} x 1 , k x 2 , k x_{2,k} x 2 , k
S 11 , k P A = a r , k + p k [ a t , k x 2 , k + ν k x 1 , k ] . (40) S_{11,k}^{\mathrm{PA}}=a_{r,k}+p_k\left[a_{t,k}x_{2,k}+\nu_kx_{1,k}\right]. \tag*{(40)} S 11 , k PA = a r , k + p k [ a t , k x 2 , k + ν k x 1 , k ] . (40) Thus, substituting (37) and (38) into (40) gives (35), which concludes the proof. ■
Remark 1: The pinched-section length L P , k L_{\mathrm{P},k} L P , k p k p_k p k μ k \mu_k μ k ν k \nu_k ν k γ k \gamma_k γ k L P , k L_{\mathrm{P},k} L P , k L P , k → 0 L_{\mathrm{P},k}\to0 L P , k → 0
Remark 2: The adoption of the fundamental H E 11 \mathrm{HE}_{11} HE 11 H E 11 \mathrm{HE}_{11} HE 11 μ k \mu_k μ k ν k \nu_k ν k γ k \gamma_k γ k
Far-Field Radiation and Pinching Antenna Gain
As the guided signal reaches the k k k H E 11 \mathrm{HE}_{11} HE 11 κ k \kappa_k κ k L P , k L_{\mathrm{P},k} L P , k H E 11 \mathrm{HE}_{11} HE 11
Relative to the unperturbed radius-a a a b k ≤ ρ ≤ a b_k\leq\rho\leq a b k ≤ ρ ≤ a L P , k L_{\mathrm{P},k} L P , k [28 ] . Accordingly, a local coordinate system is centered at the k k k z z z r = R u ^ ( θ , ϕ ) \mathbf{r}=R\hat{\mathbf{u}}(\theta,\phi) r = R u ^ ( θ , ϕ ) R R R u ^ ( θ , ϕ ) \hat{\mathbf{u}}(\theta,\phi) u ^ ( θ , ϕ ) k k k
Proposition 2: For the k k k κ k = b k / a \kappa_k=b_k/a κ k = b k / a L P , k L_{\mathrm{P},k} L P , k
E P A , k f f ( R , θ , ϕ ) = e − j k 0 R R [ F θ , k ( θ , ϕ ) θ ^ + F ϕ , k ( θ , ϕ ) ϕ ^ ] (41) \mathbf{E}^{\mathrm{ff}}_{\mathrm{PA},k}(R,\theta,\phi)=\frac{e^{-jk_0R}}{R}\left[F_{\theta,k}(\theta,\phi)\hat{\boldsymbol{\theta}}+F_{\phi,k}(\theta,\phi)\hat{\boldsymbol{\phi}}\right] \tag*{(41)} E PA , k ff ( R , θ , ϕ ) = R e − j k 0 R [ F θ , k ( θ , ϕ ) θ ^ + F ϕ , k ( θ , ϕ ) ϕ ^ ] (41) where
F θ , k ( θ , ϕ ) = ω 2 μ 0 ϵ 0 ( 1 − ϵ r ) 4 cos ϕ [ A k + Z k + ( θ ) R θ , k + ( θ ) − A k − Z k − ( θ ) R θ , k − ( θ ) ] (42) F_{\theta,k}(\theta,\phi)=\frac{\omega^2\mu_0\epsilon_0(1-\epsilon_r)}{4}\cos\phi\left[A_k^+Z_k^+(\theta)R_{\theta,k}^+(\theta)-A_k^-Z_k^-(\theta)R_{\theta,k}^-(\theta)\right] \tag*{(42)} F θ , k ( θ , ϕ ) = 4 ω 2 μ 0 ϵ 0 ( 1 − ϵ r ) cos ϕ [ A k + Z k + ( θ ) R θ , k + ( θ ) − A k − Z k − ( θ ) R θ , k − ( θ ) ] (42) F ϕ , k ( θ , ϕ ) = ω 2 μ 0 ϵ 0 ( 1 − ϵ r ) 4 sin ϕ [ A k + Z k + ( θ ) R ϕ , k + ( θ ) − A k − Z k − ( θ ) R ϕ , k − ( θ ) ] . (43) F_{\phi,k}(\theta,\phi)=\frac{\omega^2\mu_0\epsilon_0(1-\epsilon_r)}{4}\sin\phi\left[A_k^+Z_k^+(\theta)R_{\phi,k}^+(\theta)-A_k^-Z_k^-(\theta)R_{\phi,k}^-(\theta)\right]. \tag*{(43)} F ϕ , k ( θ , ϕ ) = 4 ω 2 μ 0 ϵ 0 ( 1 − ϵ r ) sin ϕ [ A k + Z k + ( θ ) R ϕ , k + ( θ ) − A k − Z k − ( θ ) R ϕ , k − ( θ ) ] . (43) with A k + = x 1 , k A_k^+=x_{1,k} A k + = x 1 , k A k − = p k x 2 , k A_k^-=p_kx_{2,k} A k − = p k x 2 , k Z k ± ( θ ) = ( e j [ k 0 cos θ ∓ β ( b k ) ] L P , k − 1 ) / ( j [ k 0 cos θ ∓ β ( b k ) ] ) Z_k^\pm(\theta)=\left(e^{j[k_0\cos\theta\mp\beta(b_k)]L_{\mathrm{P},k}}-1\right)/\left(j[k_0\cos\theta\mp\beta(b_k)]\right) Z k ± ( θ ) = ( e j [ k 0 c o s θ ∓ β ( b k )] L P , k − 1 ) / ( j [ k 0 cos θ ∓ β ( b k )] ) R θ , k ± ( θ ) R_{\theta,k}^\pm(\theta) R θ , k ± ( θ ) R ϕ , k ± ( θ ) R_{\phi,k}^\pm(\theta) R ϕ , k ± ( θ )
R θ , k ± ( θ ) = J 1 ( u ( b k ) ) P g ( b k ) K 1 ( w ( b k ) ) { ± b k 2 k 0 w 2 ( b k ) ( ϵ r − cos 2 θ ) [ ( B θ ( θ ) ω μ 0 D ( b k ) − A θ ( θ ) β ( b k ) C ( b k ) cos θ ) I 1 , k + ( B θ ( θ ) β ( b k ) C ( b k ) − A θ ( θ ) ω μ 0 D ( b k ) cos θ ) I 2 , k ] + C ( b k ) A θ ( θ ) I 0 , k } (44) R_{\theta,k}^\pm(\theta)=\frac{J_1(u(b_k))}{\sqrt{P_g(b_k)}K_1(w(b_k))}\left\{\frac{\pm b_k^2}{k_0w^2(b_k)(\epsilon_r-\cos^2\theta)}\left[\left(B_\theta(\theta)\omega\mu_0D(b_k)-A_\theta(\theta)\beta(b_k)C(b_k)\cos\theta\right)I_{1,k}+\left(B_\theta(\theta)\beta(b_k)C(b_k)-A_\theta(\theta)\omega\mu_0D(b_k)\cos\theta\right)I_{2,k}\right]+C(b_k)A_\theta(\theta)I_{0,k}\right\} \tag*{(44)} R θ , k ± ( θ ) = P g ( b k ) K 1 ( w ( b k )) J 1 ( u ( b k )) { k 0 w 2 ( b k ) ( ϵ r − cos 2 θ ) ± b k 2 [ ( B θ ( θ ) ω μ 0 D ( b k ) − A θ ( θ ) β ( b k ) C ( b k ) cos θ ) I 1 , k + ( B θ ( θ ) β ( b k ) C ( b k ) − A θ ( θ ) ω μ 0 D ( b k ) cos θ ) I 2 , k ] + C ( b k ) A θ ( θ ) I 0 , k } (44) R ϕ , k ± ( θ ) = J 1 ( u ( b k ) ) P g ( b k ) K 1 ( w ( b k ) ) { ∓ b k 2 k 0 w 2 ( b k ) ( ϵ r − cos 2 θ ) [ ( B ϕ ( θ ) ω μ 0 D ( b k ) + A ϕ ( θ ) β ( b k ) C ( b k ) cos θ ) I 1 , k + ( B ϕ ( θ ) β ( b k ) C ( b k ) + A ϕ ( θ ) ω μ 0 D ( b k ) cos θ ) I 2 , k ] + C ( b k ) A ϕ ( θ ) I 0 , k } . (45) R_{\phi,k}^\pm(\theta)=\frac{J_1(u(b_k))}{\sqrt{P_g(b_k)}K_1(w(b_k))}\left\{\frac{\mp b_k^2}{k_0w^2(b_k)(\epsilon_r-\cos^2\theta)}\left[\left(B_\phi(\theta)\omega\mu_0D(b_k)+A_\phi(\theta)\beta(b_k)C(b_k)\cos\theta\right)I_{1,k}+\left(B_\phi(\theta)\beta(b_k)C(b_k)+A_\phi(\theta)\omega\mu_0D(b_k)\cos\theta\right)I_{2,k}\right]+C(b_k)A_\phi(\theta)I_{0,k}\right\}. \tag*{(45)} R ϕ , k ± ( θ ) = P g ( b k ) K 1 ( w ( b k )) J 1 ( u ( b k )) { k 0 w 2 ( b k ) ( ϵ r − cos 2 θ ) ∓ b k 2 [ ( B ϕ ( θ ) ω μ 0 D ( b k ) + A ϕ ( θ ) β ( b k ) C ( b k ) cos θ ) I 1 , k + ( B ϕ ( θ ) β ( b k ) C ( b k ) + A ϕ ( θ ) ω μ 0 D ( b k ) cos θ ) I 2 , k ] + C ( b k ) A ϕ ( θ ) I 0 , k } . (45) Moreover, I 0 , k I_{0,k} I 0 , k I 1 , k I_{1,k} I 1 , k I 2 , k I_{2,k} I 2 , k [ f ( ρ ) ] b k a = f ( a ) − f ( b k ) [f(\rho)]_{b_k}^{a}=f(a)-f(b_k) [ f ( ρ ) ] b k a = f ( a ) − f ( b k ) A ν ( θ ) A_\nu(\theta) A ν ( θ ) B ν ( θ ) B_\nu(\theta) B ν ( θ ) ν ∈ { θ , ϕ } \nu\in\{\theta,\phi\} ν ∈ { θ , ϕ } H 1 ( 2 ) ( ⋅ ) H_1^{(2)}(\cdot) H 1 ( 2 ) ( ⋅ )
Proof: The proof can be found in Appendix A.
Having characterized the complex far-field response of the k k k [23 ] , the radiation intensity of the k k k ( θ , ϕ ) (\theta,\phi) ( θ , ϕ )
U k ( θ , ϕ ) = 1 2 η 0 [ ∣ F θ , k ( θ , ϕ ) ∣ 2 + ∣ F ϕ , k ( θ , ϕ ) ∣ 2 ] . U_k(\theta,\phi)=\frac{1}{2\eta_0}\left[\left|F_{\theta,k}(\theta,\phi)\right|^2+\left|F_{\phi,k}(\theta,\phi)\right|^2\right]. U k ( θ , ϕ ) = 2 η 0 1 [ ∣ F θ , k ( θ , ϕ ) ∣ 2 + ∣ F ϕ , k ( θ , ϕ ) ∣ 2 ] . Since the incident H E 11 \mathrm{HE}_{11} HE 11 k k k
η r a d , k = ∫ 0 2 π ∫ 0 π U k ( θ , ϕ ) sin θ d θ d ϕ . (46) \eta_{\mathrm{rad},k}=\int_0^{2\pi}\int_0^\pi U_k(\theta,\phi)\sin\theta\,d\theta\,d\phi. \tag*{(46)} η rad , k = ∫ 0 2 π ∫ 0 π U k ( θ , ϕ ) sin θ d θ d ϕ . (46) Accordingly, the directivity of the k k k D k ( θ , ϕ ) = 4 π U k ( θ , ϕ ) η r a d , k D_k(\theta,\phi)=\frac{4\pi U_k(\theta,\phi)}{\eta_{\mathrm{rad},k}} D k ( θ , ϕ ) = η rad , k 4 π U k ( θ , ϕ ) G k ( θ , ϕ ) = η r a d , k D k ( θ , ϕ ) G_k(\theta,\phi)=\eta_{\mathrm{rad},k}D_k(\theta,\phi) G k ( θ , ϕ ) = η rad , k D k ( θ , ϕ )
G k ( θ , ϕ ) = 2 π η 0 [ ∣ F θ , k ( θ , ϕ ) ∣ 2 + ∣ F ϕ , k ( θ , ϕ ) ∣ 2 ] . (47) G_k(\theta,\phi)=\frac{2\pi}{\eta_0}\left[\left|F_{\theta,k}(\theta,\phi)\right|^2+\left|F_{\phi,k}(\theta,\phi)\right|^2\right]. \tag*{(47)} G k ( θ , ϕ ) = η 0 2 π [ ∣ F θ , k ( θ , ϕ ) ∣ 2 + ∣ F ϕ , k ( θ , ϕ ) ∣ 2 ] . (47) Hence, the pinching ratio κ k \kappa_k κ k L P , k L_{\mathrm{P},k} L P , k η r a d , k \eta_{\mathrm{rad},k} η rad , k D k ( θ , ϕ ) D_k(\theta,\phi) D k ( θ , ϕ ) G k ( θ , ϕ ) G_k(\theta,\phi) G k ( θ , ϕ )
Simulation Results
In this section, the accuracy and validity of the proposed electromagnetic model are evaluated through full-wave electromagnetic simulations performed using MEEP [29 ] . Specifically, the considered lossless cylindrical waveguide is operated at f 0 = 28 f_0=28 f 0 = 28 ϵ r = 2.56 \epsilon_r=2.56 ϵ r = 2.56 [30 ] . Moreover, an unperturbed waveguide radius a = 2.75 a=2.75 a = 2.75 0.75 ≤ κ k ≤ 1 0.75\leq\kappa_k\leq1 0.75 ≤ κ k ≤ 1 L P , k / λ 0 ∈ { 2 , 4 , 10 } L_{\mathrm{P},k}/\lambda_0\in\{2,4,10\} L P , k / λ 0 ∈ { 2 , 4 , 10 } P t = 1 P_t=1 P t = 1
Figure 1 illustrates the reflected and transmitted power fractions as a function of the pinching ratio κ \kappa κ κ \kappa κ ∣ S 21 ∣ 2 \lvert S_{21}\rvert^2 ∣ S 21 ∣ 2 H E 11 \mathrm{HE}_{11} HE 11 ∣ S 11 ∣ 2 \lvert S_{11}\rvert^2 ∣ S 11 ∣ 2 − 25 -25 − 25 L P , k L_{\mathrm{P},k} L P , k κ = 0.75 \kappa=0.75 κ = 0.75 L P , k L_{\mathrm{P},k} L P , k 2 λ 0 2\lambda_0 2 λ 0 4 λ 0 4\lambda_0 4 λ 0 ∣ S 21 ∣ 2 \lvert S_{21}\rvert^2 ∣ S 21 ∣ 2 0.842 0.842 0.842 0.822 0.822 0.822 10 λ 0 10\lambda_0 10 λ 0 0.870 0.870 0.870
Figure 1. Scattering responses versus κ \kappa κ L P , k L_{P,k} L P , k ∣ S 21 ∣ 2 |S_{21}|^2 ∣ S 21 ∣ 2 ∣ S 11 ∣ 2 |S_{11}|^2 ∣ S 11 ∣ 2
Figure 2 illustrates the phase of S 21 S_{21} S 21 κ \kappa κ κ \kappa κ
Figure 2. Arg { S 21 } \operatorname{Arg}\{S_{21}\} Arg { S 21 } κ \kappa κ L P , k L_{\mathrm{P},k} L P , k L P , k = 2 λ 0 L_{\mathrm{P},k}=2\lambda_0 L P , k = 2 λ 0 L P , k = 4 λ 0 L_{\mathrm{P},k}=4\lambda_0 L P , k = 4 λ 0 L P , k = 10 λ 0 L_{\mathrm{P},k}=10\lambda_0 L P , k = 10 λ 0
I 0 , k = [ ρ k 0 2 ( ϵ r − cos 2 θ ) + w 2 ( b k ) / b k 2 { k 0 ϵ r − cos 2 θ J 2 ( k 0 ρ ϵ r − cos 2 θ ) K 1 ( w ( b k ) b k ρ ) − w ( b k ) J 1 ( k 0 ρ ϵ r − cos 2 θ ) K 2 ( w ( b k ) b k ρ ) b k } ] b k a , (48) I_{0,k}=\left[\frac{\rho}{k_0^2(\epsilon_r-\cos^2\theta)+w^2(b_k)/b_k^2}\left\{k_0\sqrt{\epsilon_r-\cos^2\theta}J_2\left(k_0\rho\sqrt{\epsilon_r-\cos^2\theta}\right)K_1\left(\frac{w(b_k)}{b_k}\rho\right)-\frac{w(b_k)J_1\left(k_0\rho\sqrt{\epsilon_r-\cos^2\theta}\right)K_2\left(\frac{w(b_k)}{b_k}\rho\right)}{b_k}\right\}\right]_{b_k}^{a}, \tag*{(48)} I 0 , k = k 0 2 ( ϵ r − cos 2 θ ) + w 2 ( b k ) / b k 2 ρ ⎩ ⎨ ⎧ k 0 ϵ r − cos 2 θ J 2 ( k 0 ρ ϵ r − cos 2 θ ) K 1 ( b k w ( b k ) ρ ) − b k w ( b k ) J 1 ( k 0 ρ ϵ r − cos 2 θ ) K 2 ( b k w ( b k ) ρ ) ⎭ ⎬ ⎫ b k a , (48) I 1 , k = [ ρ K 1 ( w ( b k ) b k ρ ) ∂ ∂ ρ J 1 ( k 0 ρ ϵ r − cos 2 θ ) ] b k a + k 0 2 ( ϵ r − cos 2 θ ) I 0 , k , (49) I_{1,k}=\left[\rho K_1\left(\frac{w(b_k)}{b_k}\rho\right)\frac{\partial}{\partial\rho}J_1\left(k_0\rho\sqrt{\epsilon_r-\cos^2\theta}\right)\right]_{b_k}^{a}+k_0^2(\epsilon_r-\cos^2\theta)I_{0,k}, \tag*{(49)} I 1 , k = [ ρ K 1 ( b k w ( b k ) ρ ) ∂ ρ ∂ J 1 ( k 0 ρ ϵ r − cos 2 θ ) ] b k a + k 0 2 ( ϵ r − cos 2 θ ) I 0 , k , (49) I 2 , k = [ K 1 ( w ( b k ) b k ρ ) J 1 ( k 0 ρ ϵ r − cos 2 θ ) ] b k a . (50) I_{2,k}=\left[K_1\left(\frac{w(b_k)}{b_k}\rho\right)J_1\left(k_0\rho\sqrt{\epsilon_r-\cos^2\theta}\right)\right]_{b_k}^{a}. \tag*{(50)} I 2 , k = [ K 1 ( b k w ( b k ) ρ ) J 1 ( k 0 ρ ϵ r − cos 2 θ ) ] b k a . (50) A ν ( θ ) = − 2 [ J 1 ′ ( k 0 a sin θ ) − J 1 ( k 0 a sin θ ) H 1 ( 2 ) ′ ( k 0 a sin θ ) H 1 ( 2 ) ( k 0 a sin θ ) ] j Δ ( θ ) × { J 1 ( k 0 a ϵ r − cos 2 θ ) ϵ r − cos 2 θ − J 1 ( k 0 a ϵ r − cos 2 θ ) H 1 ( 2 ) ′ ( k 0 a sin θ ) sin θ H 1 ( 2 ) ( k 0 a sin θ ) , ν = θ , cos θ J 1 ( k 0 a ϵ r − cos 2 θ ) a k 0 ( 1 ϵ r − cos 2 θ − 1 sin 2 θ ) , ν = ϕ . (51) A_\nu(\theta)=-\frac{2\left[J_1'(k_0a\sin\theta)-\frac{J_1(k_0a\sin\theta)H_1^{(2)\prime}(k_0a\sin\theta)}{H_1^{(2)}(k_0a\sin\theta)}\right]}{j\Delta(\theta)}\times\begin{cases}
\frac{J_1(k_0a\sqrt{\epsilon_r-\cos^2\theta})}{\sqrt{\epsilon_r-\cos^2\theta}}-\frac{J_1(k_0a\sqrt{\epsilon_r-\cos^2\theta})H_1^{(2)\prime}(k_0a\sin\theta)}{\sin\theta H_1^{(2)}(k_0a\sin\theta)}, & \nu=\theta,\\
\frac{\cos\theta J_1(k_0a\sqrt{\epsilon_r-\cos^2\theta})}{ak_0}\left(\frac{1}{\epsilon_r-\cos^2\theta}-\frac{1}{\sin^2\theta}\right), & \nu=\phi.
\end{cases} \tag*{(51)} A ν ( θ ) = − j Δ ( θ ) 2 [ J 1 ′ ( k 0 a sin θ ) − H 1 ( 2 ) ( k 0 a s i n θ ) J 1 ( k 0 a s i n θ ) H 1 ( 2 ) ′ ( k 0 a s i n θ ) ] × ⎩ ⎨ ⎧ ϵ r − c o s 2 θ J 1 ( k 0 a ϵ r − c o s 2 θ ) − s i n θ H 1 ( 2 ) ( k 0 a s i n θ ) J 1 ( k 0 a ϵ r − c o s 2 θ ) H 1 ( 2 ) ′ ( k 0 a s i n θ ) , a k 0 c o s θ J 1 ( k 0 a ϵ r − c o s 2 θ ) ( ϵ r − c o s 2 θ 1 − s i n 2 θ 1 ) , ν = θ , ν = ϕ . (51) B ν ( θ ) = − 2 [ J 1 ′ ( k 0 a sin θ ) − J 1 ( k 0 a sin θ ) H 1 ( 2 ) ′ ( k 0 a sin θ ) H 1 ( 2 ) ( k 0 a sin θ ) ] j Δ ( θ ) × { cos θ J 1 ( k 0 a ϵ r − cos 2 θ ) a k 0 ( 1 ϵ r − cos 2 θ − 1 sin 2 θ ) , ν = θ , − [ ϵ r J 1 ′ ( k 0 a ϵ r − cos 2 θ ) ϵ r − cos 2 θ − J 1 ( k 0 a ϵ r − cos 2 θ ) H 1 ( 2 ) ′ ( k 0 a sin θ ) sin θ H 1 ( 2 ) ( k 0 a sin θ ) ] , ν = ϕ . (52) B_\nu(\theta)=-\frac{2\left[J_1'(k_0a\sin\theta)-\frac{J_1(k_0a\sin\theta)H_1^{(2)\prime}(k_0a\sin\theta)}{H_1^{(2)}(k_0a\sin\theta)}\right]}{j\Delta(\theta)}\times\begin{cases}
\frac{\cos\theta J_1(k_0a\sqrt{\epsilon_r-\cos^2\theta})}{ak_0}\left(\frac{1}{\epsilon_r-\cos^2\theta}-\frac{1}{\sin^2\theta}\right), & \nu=\theta,\\
-\left[\frac{\epsilon_r J_1'(k_0a\sqrt{\epsilon_r-\cos^2\theta})}{\sqrt{\epsilon_r-\cos^2\theta}}-\frac{J_1(k_0a\sqrt{\epsilon_r-\cos^2\theta})H_1^{(2)\prime}(k_0a\sin\theta)}{\sin\theta H_1^{(2)}(k_0a\sin\theta)}\right], & \nu=\phi.
\end{cases} \tag*{(52)} B ν ( θ ) = − j Δ ( θ ) 2 [ J 1 ′ ( k 0 a sin θ ) − H 1 ( 2 ) ( k 0 a s i n θ ) J 1 ( k 0 a s i n θ ) H 1 ( 2 ) ′ ( k 0 a s i n θ ) ] × ⎩ ⎨ ⎧ a k 0 c o s θ J 1 ( k 0 a ϵ r − c o s 2 θ ) ( ϵ r − c o s 2 θ 1 − s i n 2 θ 1 ) , − [ ϵ r − c o s 2 θ ϵ r J 1 ′ ( k 0 a ϵ r − c o s 2 θ ) − s i n θ H 1 ( 2 ) ( k 0 a s i n θ ) J 1 ( k 0 a ϵ r − c o s 2 θ ) H 1 ( 2 ) ′ ( k 0 a s i n θ ) ] , ν = θ , ν = ϕ . (52) Δ ( θ ) = { cos θ J 1 ( k 0 a ϵ r − cos 2 θ ) a k 0 ( 1 ϵ r − cos 2 θ − 1 sin 2 θ ) } 2 − { J 1 ′ ( k 0 a ϵ r − cos 2 θ ) ϵ r − cos 2 θ − J 1 ( k 0 a ϵ r − cos 2 θ ) H 1 ( 2 ) ′ ( k 0 a sin θ ) sin θ H 1 ( 2 ) ( k 0 a sin θ ) } × { ϵ r J 1 ′ ( k 0 a ϵ r − cos 2 θ ) ϵ r − cos 2 θ − J 1 ( k 0 a ϵ r − cos 2 θ ) H 1 ( 2 ) ′ ( k 0 a sin θ ) sin θ H 1 ( 2 ) ( k 0 a sin θ ) } . (53) \Delta(\theta)=\left\{\frac{\cos\theta J_1(k_0a\sqrt{\epsilon_r-\cos^2\theta})}{ak_0}\left(\frac{1}{\epsilon_r-\cos^2\theta}-\frac{1}{\sin^2\theta}\right)\right\}^2-\left\{\frac{J_1'(k_0a\sqrt{\epsilon_r-\cos^2\theta})}{\sqrt{\epsilon_r-\cos^2\theta}}-\frac{J_1(k_0a\sqrt{\epsilon_r-\cos^2\theta})H_1^{(2)\prime}(k_0a\sin\theta)}{\sin\theta H_1^{(2)}(k_0a\sin\theta)}\right\}\times\left\{\frac{\epsilon_rJ_1'(k_0a\sqrt{\epsilon_r-\cos^2\theta})}{\sqrt{\epsilon_r-\cos^2\theta}}-\frac{J_1(k_0a\sqrt{\epsilon_r-\cos^2\theta})H_1^{(2)\prime}(k_0a\sin\theta)}{\sin\theta H_1^{(2)}(k_0a\sin\theta)}\right\}. \tag*{(53)} Δ ( θ ) = { a k 0 cos θ J 1 ( k 0 a ϵ r − cos 2 θ ) ( ϵ r − cos 2 θ 1 − sin 2 θ 1 ) } 2 − { ϵ r − cos 2 θ J 1 ′ ( k 0 a ϵ r − cos 2 θ ) − sin θ H 1 ( 2 ) ( k 0 a sin θ ) J 1 ( k 0 a ϵ r − cos 2 θ ) H 1 ( 2 ) ′ ( k 0 a sin θ ) } × { ϵ r − cos 2 θ ϵ r J 1 ′ ( k 0 a ϵ r − cos 2 θ ) − sin θ H 1 ( 2 ) ( k 0 a sin θ ) J 1 ( k 0 a ϵ r − cos 2 θ ) H 1 ( 2 ) ′ ( k 0 a sin θ ) } . (53) signal accumulates a geometry-dependent phase while propagating through the PA, which is subsequently carried by the transmitted HE11 _{11} 11 L P , k L_{P,k} L P , k L P , k = 10 λ 0 L_{P,k}=10\lambda_0 L P , k = 10 λ 0 [ 0 , 2 π ) [0,2\pi) [ 0 , 2 π ) κ \kappa κ
In Figure 3 , the normalized far-field radiation patterns are illustrated for representative PA geometries, where the close agreement between the analytical and MEEP results validates the ability of the proposed model to capture the angular redistribution of the radiated power. In more detail, for the same pinched-section length, varying κ \kappa κ L P , k L_{P,k} L P , k L P , k = 10 λ 0 L_{P,k}=10\lambda_0 L P , k = 10 λ 0
Figure 3. Normalized far-field radiation patterns in the ϕ = 0 ∘ \phi= 0^\circ ϕ = 0 ∘ κ = 0.75 \kappa= 0.75 κ = 0.75 L P , k = 2 λ 0 L_{P,k} = 2\lambda_0 L P , k = 2 λ 0 κ = 0.95 \kappa= 0.95 κ = 0.95 L P , k = 2 λ 0 L_{P,k} = 2\lambda_0 L P , k = 2 λ 0 κ = 0.75 \kappa= 0.75 κ = 0.75 L P , k = 10 λ 0 L_{P,k} = 10\lambda_0 L P , k = 10 λ 0
In Figure 4 , the radiation power-fraction η r a d , k \eta_{\mathrm{rad},k} η rad , k κ \kappa κ κ \kappa κ H E 11 \mathrm{HE}_{11} HE 11 η r a d , k \eta_{\mathrm{rad},k} η rad , k κ → 1 \kappa\to1 κ → 1 L P , k L_{\mathrm{P},k} L P , k
Figure 4. Radiation power-fraction versus κ \kappa κ L P , k L_{P,k} L P , k L P , k = 2 λ 0 L_{P,k} = 2\lambda_0 L P , k = 2 λ 0 L P , k = 4 λ 0 L_{P,k} = 4\lambda_0 L P , k = 4 λ 0 L P , k = 10 λ 0 L_{P,k} = 10\lambda_0 L P , k = 10 λ 0
Finally, Figure 5 shows the impact of the PA radiation characteristics on the received SNR, where, following the deployment considered in [7 ] , a 30 × 10 m 2 30 \times10\ \mathrm{m}^{2} 30 × 10 m 2 ( 0 , 5 , 3 ) m (0,5,3)\ \mathrm{m} ( 0 , 5 , 3 ) m ( 30 , 5 , 3 ) m (30,5,3)\ \mathrm{m} ( 30 , 5 , 3 ) m 10 4 10^{4} 1 0 4 z z z x x x x x x H E 11 \mathrm{HE}_{11} HE 11 z z z γ r = γ t G ( θ , ϕ ) ( λ 0 4 π R ) 2 \gamma_{\mathrm{r}}=\gamma_{\mathrm{t}}G(\theta,\phi)\left(\frac{\lambda_0}{4\pi R}\right)^2 γ r = γ t G ( θ , ϕ ) ( 4 π R λ 0 ) 2 γ t = 10 9 \gamma_{\mathrm{t}}=10^{9} γ t = 1 0 9 R R R G ( θ , ϕ ) G(\theta,\phi) G ( θ , ϕ ) Figure 5 correspond to the received SNR averaged over the 10 4 10^{4} 1 0 4 Case 1 considers an isotropic antenna located at the access point ( 0 , 5 , 3 ) m (0,5,3)\ \mathrm{m} ( 0 , 5 , 3 ) m Case 2 follows the distance-minimizing PA placement considered in [7 ] , such that the PA is positioned at the waveguide point closest to the user and κ \kappa κ L P , k L_{\mathrm{P},k} L P , k η r a d , k \eta_{\mathrm{rad},k} η rad , k Case 3 jointly
Figure 5. Received SNR for the considered transmission cases.
optimizes the PA position, κ \kappa κ L P , k L_{\mathrm{P},k} L P , k κ ∈ [ 0.75 , 0.995 ] \kappa\in[0.75,0.995] κ ∈ [ 0.75 , 0.995 ] L P , k / λ 0 ∈ [ 2 , 10 ] L_{\mathrm{P},k}/\lambda_0 \in[2,10] L P , k / λ 0 ∈ [ 2 , 10 ] 0.005 0.005 0.005 0.05 0.05 0.05 Case 2 minimizes the PA–user distance and maximizes the total radiated-power fraction, it remains 8.94 8.94 8.94 Case 1 , since the resulting PA radiates only a fraction of the incident guided power and neither criterion accounts for how this power is angularly distributed toward the intended user. However, by jointly adapting the PA position and geometry, Case 3 additionally accounts for the corresponding directional gain, yielding improvements of 2.38 2.38 2.38 11.32 11.32 11.32 Cases 1 and 2 2 2
Conclusion
In this paper, we developed an electromagnetic model that directly relates the finite perturbation applied to a dielectric waveguide to the guided-wave and radiation responses of the resulting PA. The proposed formulation jointly characterizes the complex reflection and transmission coefficients and the three-dimensional radiation characteristics, and its accuracy was validated through full-wave MEEP simulations. The results showed that the pinching ratio and pinched-section length jointly control the transmitted power and phase, radiation power-fraction, and radiation pattern, with their interaction producing non-monotonic responses and establishing the PA geometry as a physical design degree of freedom for shaping the radiation characteristics. Moreover, the considered single-PA received-SNR evaluation demonstrated that minimizing the PA–user distance or maximizing the total radiation power-fraction does not necessarily provide the highest received SNR, since the applied perturbation also determines the directional gain and, consequently, the effectiveness of a given PA position. Therefore, the proposed framework establishes a direct connection between the finite deformation-based PA geometry and PAS design, and can provide a basis for extending physics-compliant PAS modeling to multi-PA deployments, multi-mode operation, and other physically realizable waveguide perturbations.
Proof of Proposition 2
The unperturbed radius-a a a H E 11 \mathrm{HE}_{11} HE 11 k k k J e q , k \mathbf{J}_{\mathrm{eq},k} J eq , k [28 ]
E P A , k ( r ) = − j ω μ 0 ∭ V P , k G ‾ a ( r , r ′ ) ⋅ J e q , k ( r ′ ) d V ′ , r ∉ V P , k , (54) \mathbf{E}_{\mathrm{PA},k}(\mathbf{r})=-j\omega\mu_0\iiint_{V_{P,k}}\overline{\mathbf{G}}_a(\mathbf{r},\mathbf{r}')\cdot\mathbf{J}_{\mathrm{eq},k}(\mathbf{r}')\,\mathrm{d}V',\quad\mathbf{r}\notin V_{P,k}, \tag*{(54)} E PA , k ( r ) = − j ω μ 0 ∭ V P , k G a ( r , r ′ ) ⋅ J eq , k ( r ′ ) d V ′ , r ∈ / V P , k , (54) where r ′ \mathbf{r}' r ′ G ‾ a ( r , r ′ ) \overline{\mathbf{G}}_a(\mathbf{r},\mathbf{r}') G a ( r , r ′ ) V P , k V_{P,k} V P , k b k ≤ ρ ≤ a b_k\leq\rho\leq a b k ≤ ρ ≤ a 0 ≤ ϕ ′ < 2 π 0\leq\phi'<2\pi 0 ≤ ϕ ′ < 2 π 0 ≤ z ≤ L P , k 0\leq z\leq L_{P,k} 0 ≤ z ≤ L P , k r = R u ^ ( θ , ϕ ) \mathbf{r}=R\hat{\mathbf{u}}(\theta,\phi) r = R u ^ ( θ , ϕ ) e − j k 0 R / R e^{-jk_0R}/R e − j k 0 R / R ν ^ ∈ { θ ^ , ϕ ^ } \hat{\nu}\in\{\hat{\theta},\hat{\phi}\} ν ^ ∈ { θ ^ , ϕ ^ }
F ν , k ( θ , ϕ ) = lim R → ∞ [ R e j k 0 R ν ^ ⋅ E P A , k ( R u ^ ( θ , ϕ ) ) ] , ν ∈ { θ , ϕ } . (55) F_{\nu,k}(\theta,\phi)=\lim_{R\to\infty}\left[Re^{jk_0R}\hat{\nu}\cdot\mathbf{E}_{\mathrm{PA},k}\bigl(R\hat{\mathbf{u}}(\theta,\phi)\bigr)\right],\quad\nu\in\{\theta,\phi\}. \tag*{(55)} F ν , k ( θ , ϕ ) = R → ∞ lim [ R e j k 0 R ν ^ ⋅ E PA , k ( R u ^ ( θ , ϕ ) ) ] , ν ∈ { θ , ϕ } . (55) Defining the far-field Green tensor of the reference waveguide as G ‾ a ∞ ( θ , ϕ ; r ′ ) = lim R → ∞ R e j k 0 R G ‾ a ( R u ^ ( θ , ϕ ) , r ′ ) \overline{\mathbf{G}}_a^\infty(\theta,\phi;\mathbf{r}')=\lim_{R\to\infty}Re^{jk_0R}\overline{\mathbf{G}}_a\bigl(R\hat{\mathbf{u}}(\theta,\phi),\mathbf{r}'\bigr) G a ∞ ( θ , ϕ ; r ′ ) = lim R → ∞ R e j k 0 R G a ( R u ^ ( θ , ϕ ) , r ′ )
F ν , k ( θ , ϕ ) = − j ω μ 0 ∭ V P , k ν ^ ⋅ G ‾ a ∞ ( θ , ϕ ; r ′ ) ⋅ J e q , k ( r ′ ) d V ′ . (56) F_{\nu,k}(\theta,\phi)=-j\omega\mu_0\iiint_{V_{P,k}}\hat{\nu}\cdot\overline{\mathbf{G}}_a^\infty(\theta,\phi;\mathbf{r}')\cdot\mathbf{J}_{\mathrm{eq},k}(\mathbf{r}')\,\mathrm{d}V'. \tag*{(56)} F ν , k ( θ , ϕ ) = − j ω μ 0 ∭ V P , k ν ^ ⋅ G a ∞ ( θ , ϕ ; r ′ ) ⋅ J eq , k ( r ′ ) d V ′ . (56) Since reciprocity holds for the reference waveguide, the source and observation roles in (52) can be interchanged. Hence, according to the mixed-reciprocity relation, the far-field Green tensor can be mapped to the local field established within the reference waveguide under a reciprocal plane-wave illumination from the observation direction as [31 ]
4 π ν ^ ⋅ G ‾ a ∞ ( θ , ϕ ; r ′ ) ⋅ J e q , k ( r ′ ) = J e q , k ( r ′ ) ⋅ E a , ν ( r ′ ) , (57) 4\pi\hat{\nu}\cdot\overline{\mathbf{G}}_a^\infty(\theta,\phi;\mathbf{r}')\cdot\mathbf{J}_{\mathrm{eq},k}(\mathbf{r}')=\mathbf{J}_{\mathrm{eq},k}(\mathbf{r}')\cdot\mathbf{E}_{a,\nu}(\mathbf{r}'), \tag*{(57)} 4 π ν ^ ⋅ G a ∞ ( θ , ϕ ; r ′ ) ⋅ J eq , k ( r ′ ) = J eq , k ( r ′ ) ⋅ E a , ν ( r ′ ) , (57) E P , k ( ρ , ϕ ′ , z ) = J 1 ( u ( b k ) ) P g ( b k ) K 1 ( w ( b k ) ) { j b k 2 w 2 ( b k ) [ A k + e − j β ( b k ) z + A k − e j β ( b k ) z ] [ ρ ^ cos ϕ ′ ( − β ( b k ) C ( b k ) w ( b k ) b k K 0 ( w ( b k ) ρ b k ) + ω μ 0 D ( b k ) − β ( b k ) C ( b k ) ρ K 1 ( w ( b k ) ρ b k ) ) + ϕ ^ sin ϕ ′ ( ω μ 0 D ( b k ) w ( b k ) b k K 0 ( w ( b k ) ρ b k ) + ω μ 0 D ( b k ) − β ( b k ) C ( b k ) ρ K 1 ( w ( b k ) ρ b k ) ) ] + z ^ C ( b k ) K 1 ( w ( b k ) ρ b k ) cos ϕ ′ [ A k + e − j β ( b k ) z − A k − e j β ( b k ) z ] } . (58) \mathbf{E}_{P,k}(\rho,\phi',z)=\frac{J_1(u(b_k))}{\sqrt{P_g(b_k)}K_1(w(b_k))}\left\{\frac{j b_k^2}{w^2(b_k)}\left[A_k^+e^{-j\beta(b_k)z}+A_k^-e^{j\beta(b_k)z}\right]\left[\hat{\boldsymbol{\rho}}\cos\phi'\left(-\frac{\beta(b_k)C(b_k)w(b_k)}{b_k}K_0\left(\frac{w(b_k)\rho}{b_k}\right)+\frac{\omega\mu_0D(b_k)-\beta(b_k)C(b_k)}{\rho}K_1\left(\frac{w(b_k)\rho}{b_k}\right)\right)+\hat{\boldsymbol{\phi}}\sin\phi'\left(\frac{\omega\mu_0D(b_k)w(b_k)}{b_k}K_0\left(\frac{w(b_k)\rho}{b_k}\right)+\frac{\omega\mu_0D(b_k)-\beta(b_k)C(b_k)}{\rho}K_1\left(\frac{w(b_k)\rho}{b_k}\right)\right)\right]+\hat{\mathbf{z}}C(b_k)K_1\left(\frac{w(b_k)\rho}{b_k}\right)\cos\phi'\left[A_k^+e^{-j\beta(b_k)z}-A_k^-e^{j\beta(b_k)z}\right]\right\}. \tag*{(58)} E P , k ( ρ , ϕ ′ , z ) = P g ( b k ) K 1 ( w ( b k )) J 1 ( u ( b k )) { w 2 ( b k ) j b k 2 [ A k + e − j β ( b k ) z + A k − e j β ( b k ) z ] [ ρ ^ cos ϕ ′ ( − b k β ( b k ) C ( b k ) w ( b k ) K 0 ( b k w ( b k ) ρ ) + ρ ω μ 0 D ( b k ) − β ( b k ) C ( b k ) K 1 ( b k w ( b k ) ρ ) ) + ϕ ^ sin ϕ ′ ( b k ω μ 0 D ( b k ) w ( b k ) K 0 ( b k w ( b k ) ρ ) + ρ ω μ 0 D ( b k ) − β ( b k ) C ( b k ) K 1 ( b k w ( b k ) ρ ) ) ] + z ^ C ( b k ) K 1 ( b k w ( b k ) ρ ) cos ϕ ′ [ A k + e − j β ( b k ) z − A k − e j β ( b k ) z ] } . (58) where E a , ν r ( r ′ ) \mathbf{E}^{r}_{a,\nu}(r') E a , ν r ( r ′ ) a a a ν \nu ν V P , k V_{P,k} V P , k ϵ r ϵ 0 \epsilon_r\epsilon_0 ϵ r ϵ 0 ϵ 0 \epsilon_0 ϵ 0 [28 ]
J e q , k ( r ′ ) = j ω ϵ 0 ( 1 − ϵ r ) E P , k t o t ( r ′ ) , r ′ ∈ V P , k , (59) \mathbf{J}_{\mathrm{eq},k}(r')=j\omega\epsilon_0(1-\epsilon_r)\mathbf{E}^{\mathrm{tot}}_{P,k}(r'),\qquad r'\in V_{P,k}, \tag*{(59)} J eq , k ( r ′ ) = j ω ϵ 0 ( 1 − ϵ r ) E P , k tot ( r ′ ) , r ′ ∈ V P , k , (59) where E P , k t o t \mathbf{E}^{\mathrm{tot}}_{P,k} E P , k tot
F ν , k ( θ , ϕ ) = ω 2 μ 0 ϵ 0 ( 1 − ϵ r ) 4 π ∭ V P , k E P , k t o t ( r ′ ) ⋅ E a , ν r ( r ′ ) d V ′ . (60) F_{\nu,k}(\theta,\phi)=\frac{\omega^2\mu_0\epsilon_0(1-\epsilon_r)}{4\pi}\iiint_{V_{P,k}}\mathbf{E}^{\mathrm{tot}}_{P,k}(r')\cdot\mathbf{E}^{r}_{a,\nu}(r')\,dV'. \tag*{(60)} F ν , k ( θ , ϕ ) = 4 π ω 2 μ 0 ϵ 0 ( 1 − ϵ r ) ∭ V P , k E P , k tot ( r ′ ) ⋅ E a , ν r ( r ′ ) d V ′ . (60) Thus, the far-field radiation pattern is reduced to the overlap between the electric field established over the pinched section and the reciprocal field established over the same volume by an excitation incident from the observation direction.
Within the adopted single-guided-state representation, the electric field over the finite radius-b k b_k b k H E 11 \mathrm{HE}_{11} HE 11 H E 11 \mathrm{HE}_{11} HE 11 [25 , 26 ] . Accordingly, this interaction is incorporated into the internal guided amplitudes x 1 , k x_{1,k} x 1 , k x 2 , k x_{2,k} x 2 , k ν k \nu_k ν k γ k \gamma_k γ k H E 11 \mathrm{HE}_{11} HE 11 H E 11 \mathrm{HE}_{11} HE 11 b k b_k b k
E P , k ( ρ , ϕ ′ , z ) ≃ A k + E b k + ( ρ , ϕ ′ , z ) + A k − E b k − ( ρ , ϕ ′ , z ) , (61) \mathbf{E}_{P,k}(\rho,\phi',z)\simeq A_k^+\mathbf{E}_{b_k}^+(\rho,\phi',z)+A_k^-\mathbf{E}_{b_k}^-(\rho,\phi',z), \tag*{(61)} E P , k ( ρ , ϕ ′ , z ) ≃ A k + E b k + ( ρ , ϕ ′ , z ) + A k − E b k − ( ρ , ϕ ′ , z ) , (61) where A k + = x 1 , k A_k^+=x_{1,k} A k + = x 1 , k A k − = p k x 2 , k A_k^-=p_kx_{2,k} A k − = p k x 2 , k [21 ] , for which β − ( b k ) = − β + ( b k ) \beta^-(b_k)=-\beta^+(b_k) β − ( b k ) = − β + ( b k ) C − ( b k ) = − C + ( b k ) C^-(b_k)=-C^+(b_k) C − ( b k ) = − C + ( b k ) D − ( b k ) = D + ( b k ) D^-(b_k)=D^+(b_k) D − ( b k ) = D + ( b k ) b k ≤ ρ ≤ a b_k\leq\rho\leq a b k ≤ ρ ≤ a
The reciprocal field E a , ν r \mathbf{E}^{r}_{a,\nu} E a , ν r ν \nu ν a a a [32 ] , this field can be expanded in cylindrical harmonics, while the first-order azimuthal dependence of E P , k \mathbf{E}_{P,k} E P , k m = ± 1 m=\pm1 m = ± 1 ρ < a \rho<a ρ < a q i ( θ ) = k 0 ϵ r − cos 2 θ q_i(\theta)=k_0\sqrt{\epsilon_r-\cos^2\theta} q i ( θ ) = k 0 ϵ r − cos 2 θ q 0 ( θ ) = k 0 sin θ q_0(\theta)=k_0\sin\theta q 0 ( θ ) = k 0 sin θ
E a , z , ν r = e j k 0 z cos θ { A θ ( θ ) J 1 ( q i ( θ ) ρ ) cos ( ϕ ′ − ϕ ) , ν = θ , − A ϕ ( θ ) J 1 ( q i ( θ ) ρ ) sin ( ϕ ′ − ϕ ) , ν = ϕ , (62) E^{r}_{a,z,\nu}=\frac{e^{jk_0z\cos\theta}}{ }\begin{cases} A_\theta(\theta)J_1(q_i(\theta)\rho)\cos(\phi'-\phi), & \nu=\theta,\\ -A_\phi(\theta)J_1(q_i(\theta)\rho)\sin(\phi'-\phi), & \nu=\phi,\end{cases} \tag*{(62)} E a , z , ν r = e j k 0 z c o s θ { A θ ( θ ) J 1 ( q i ( θ ) ρ ) cos ( ϕ ′ − ϕ ) , − A ϕ ( θ ) J 1 ( q i ( θ ) ρ ) sin ( ϕ ′ − ϕ ) , ν = θ , ν = ϕ , (62) and
H a , z , ν r = e j k 0 z cos θ η 0 { B θ ( θ ) J 1 ( q i ( θ ) ρ ) sin ( ϕ ′ − ϕ ) , ν = θ , − B ϕ ( θ ) J 1 ( q i ( θ ) ρ ) cos ( ϕ ′ − ϕ ) , ν = ϕ . (63) H^{r}_{a,z,\nu}=\frac{e^{jk_0z\cos\theta}}{\eta_0}\begin{cases} B_\theta(\theta)J_1(q_i(\theta)\rho)\sin(\phi'-\phi), & \nu=\theta,\\ -B_\phi(\theta)J_1(q_i(\theta)\rho)\cos(\phi'-\phi), & \nu=\phi.\end{cases} \tag*{(63)} H a , z , ν r = η 0 e j k 0 z c o s θ { B θ ( θ ) J 1 ( q i ( θ ) ρ ) sin ( ϕ ′ − ϕ ) , − B ϕ ( θ ) J 1 ( q i ( θ ) ρ ) cos ( ϕ ′ − ϕ ) , ν = θ , ν = ϕ . (63) where η 0 \eta_0 η 0
E a , t , ν r = j q i 2 ( θ ) [ k 0 cos θ ∇ t E a , z , ν r + ω μ 0 z ^ × ∇ t H a , z , ν r ] , (64) \mathbf{E}^{r}_{a,t,\nu}=\frac{j}{q_i^2(\theta)}\left[k_0\cos\theta\nabla_tE^{r}_{a,z,\nu}+\omega\mu_0\hat{\mathbf{z}}\times\nabla_tH^{r}_{a,z,\nu}\right], \tag*{(64)} E a , t , ν r = q i 2 ( θ ) j [ k 0 cos θ ∇ t E a , z , ν r + ω μ 0 z ^ × ∇ t H a , z , ν r ] , (64) H a , t , ν r = j q i 2 ( θ ) [ k 0 cos θ ∇ t H a , z , ν r − ω ϵ r ϵ 0 z ^ × ∇ t E a , z , ν r ] . (65) \mathbf{H}^{r}_{a,t,\nu}=\frac{j}{q_i^2(\theta)}\left[k_0\cos\theta\nabla_tH^{r}_{a,z,\nu}-\omega\epsilon_r\epsilon_0\hat{\mathbf{z}}\times\nabla_tE^{r}_{a,z,\nu}\right]. \tag*{(65)} H a , t , ν r = q i 2 ( θ ) j [ k 0 cos θ ∇ t H a , z , ν r − ω ϵ r ϵ 0 z ^ × ∇ t E a , z , ν r ] . (65) Using z ^ × ρ ^ = ϕ ^ \hat{\mathbf{z}}\times\hat{\boldsymbol{\rho}}=\hat{\boldsymbol{\phi}} z ^ × ρ ^ = ϕ ^ z ^ × ϕ ^ = − ρ ^ \hat{\mathbf{z}}\times\hat{\boldsymbol{\phi}}=-\hat{\boldsymbol{\rho}} z ^ × ϕ ^ = − ρ ^
E a , ρ , ν r = j q i 2 ( θ ) [ k 0 cos θ ∂ E a , z , ν r ∂ ρ − ω μ 0 ρ ∂ H a , z , ν r ∂ ϕ ′ ] , E^{r}_{a,\rho,\nu}=\frac{j}{q_i^2(\theta)}\left[k_0\cos\theta\frac{\partial E^{r}_{a,z,\nu}}{\partial\rho}-\frac{\omega\mu_0}{\rho}\frac{\partial H^{r}_{a,z,\nu}}{\partial\phi'}\right], E a , ρ , ν r = q i 2 ( θ ) j [ k 0 cos θ ∂ ρ ∂ E a , z , ν r − ρ ω μ 0 ∂ ϕ ′ ∂ H a , z , ν r ] , E a , ϕ , ν r = j q i 2 ( θ ) [ k 0 cos θ ρ ∂ E a , z , ν r ∂ ϕ ′ + ω μ 0 ∂ H a , z , ν r ∂ ρ ] , E^{r}_{a,\phi,\nu}=\frac{j}{q_i^2(\theta)}\left[\frac{k_0\cos\theta}{\rho}\frac{\partial E^{r}_{a,z,\nu}}{\partial\phi'}+\omega\mu_0\frac{\partial H^{r}_{a,z,\nu}}{\partial\rho}\right], E a , ϕ , ν r = q i 2 ( θ ) j [ ρ k 0 cos θ ∂ ϕ ′ ∂ E a , z , ν r + ω μ 0 ∂ ρ ∂ H a , z , ν r ] , H a , ϕ , ν r = j q i 2 ( θ ) [ k 0 cos θ ρ ∂ H a , z , ν r ∂ ϕ ′ − ω ϵ r ϵ 0 ∂ E a , z , ν r ∂ ρ ] . (66) H^{r}_{a,\phi,\nu}=\frac{j}{q_i^2(\theta)}\left[\frac{k_0\cos\theta}{\rho}\frac{\partial H^{r}_{a,z,\nu}}{\partial\phi'}-\omega\epsilon_r\epsilon_0\frac{\partial E^{r}_{a,z,\nu}}{\partial\rho}\right]. \tag*{(66)} H a , ϕ , ν r = q i 2 ( θ ) j [ ρ k 0 cos θ ∂ ϕ ′ ∂ H a , z , ν r − ω ϵ r ϵ 0 ∂ ρ ∂ E a , z , ν r ] . (66) The amplitudes A ν ( θ ) A_\nu(\theta) A ν ( θ ) B ν ( θ ) B_\nu(\theta) B ν ( θ ) ρ = a \rho=a ρ = a E z E_z E z H z H_z H z E ϕ E_\phi E ϕ H ϕ H_\phi H ϕ [32 ] , which gives (45) and (46). Consequently, substituting (58) and (59) into (62), while using the identity ∂ ∂ ρ J 1 ( q i ( θ ) ρ ) = q i ( θ ) J 1 ′ ( q i ( θ ) ρ ) \frac{\partial}{\partial\rho}J_1(q_i(\theta)\rho)=q_i(\theta)J_1'(q_i(\theta)\rho) ∂ ρ ∂ J 1 ( q i ( θ ) ρ ) = q i ( θ ) J 1 ′ ( q i ( θ ) ρ ) ω μ 0 / η 0 = k 0 \omega\mu_0/\eta_0=k_0 ω μ 0 / η 0 = k 0
E a , ν r = e j k 0 z cos θ { z ^ A θ ( θ ) J 1 ( q i ( θ ) ρ ) cos ( ϕ ′ − ϕ ) + j ρ ^ sin ( ϕ ′ − ϕ ) q i 2 ( θ ) [ k 0 cos θ A θ ( θ ) q i ( θ ) J 1 ′ ( q i ( θ ) ρ ) − k 0 B θ ( θ ) ρ J 1 ( q i ( θ ) ρ ) ] + j ϕ ^ sin ( ϕ ′ − ϕ ) q i 2 ( θ ) [ − k 0 cos θ A θ ( θ ) ρ J 1 ( q i ( θ ) ρ ) + k 0 B θ ( θ ) q i ( θ ) J 1 ′ ( q i ( θ ) ρ ) ] , ν = θ , − z ^ A ϕ ( θ ) J 1 ( q i ( θ ) ρ ) sin ( ϕ ′ − ϕ ) − j ρ ^ sin ( ϕ ′ − ϕ ) q i 2 ( θ ) [ k 0 cos θ A ϕ ( θ ) q i ( θ ) J 1 ′ ( q i ( θ ) ρ ) + k 0 B ϕ ( θ ) ρ J 1 ( q i ( θ ) ρ ) ] − j ϕ ^ cos ( ϕ ′ − ϕ ) q i 2 ( θ ) [ k 0 cos θ A ϕ ( θ ) ρ J 1 ( q i ( θ ) ρ ) + k 0 B ϕ ( θ ) q i ( θ ) J 1 ′ ( q i ( θ ) ρ ) ] , ν = ϕ . (67) \mathbf{E}^{\mathrm{r}}_{a,\nu}=e^{jk_0z\cos\theta}
\left\{
\begin{aligned}
&\hat{\mathbf{z}}A_\theta(\theta)J_1(q_i(\theta)\rho)\cos(\phi'-\phi)\\
&+\frac{j\hat{\boldsymbol{\rho}}\sin(\phi'-\phi)}{q_i^2(\theta)}
\left[k_0\cos\theta A_\theta(\theta)q_i(\theta)J_1'(q_i(\theta)\rho)-\frac{k_0B_\theta(\theta)}{\rho}J_1(q_i(\theta)\rho)\right]\\
&+\frac{j\hat{\boldsymbol{\phi}}\sin(\phi'-\phi)}{q_i^2(\theta)}
\left[-\frac{k_0\cos\theta A_\theta(\theta)}{\rho}J_1(q_i(\theta)\rho)+k_0B_\theta(\theta)q_i(\theta)J_1'(q_i(\theta)\rho)\right],
\qquad\nu=\theta,\\
&-\hat{\mathbf{z}}A_\phi(\theta)J_1(q_i(\theta)\rho)\sin(\phi'-\phi)\\
&-\frac{j\hat{\boldsymbol{\rho}}\sin(\phi'-\phi)}{q_i^2(\theta)}
\left[k_0\cos\theta A_\phi(\theta)q_i(\theta)J_1'(q_i(\theta)\rho)+\frac{k_0B_\phi(\theta)}{\rho}J_1(q_i(\theta)\rho)\right]\\
&-\frac{j\hat{\boldsymbol{\phi}}\cos(\phi'-\phi)}{q_i^2(\theta)}
\left[\frac{k_0\cos\theta A_\phi(\theta)}{\rho}J_1(q_i(\theta)\rho)+k_0B_\phi(\theta)q_i(\theta)J_1'(q_i(\theta)\rho)\right],
\qquad\nu=\phi.
\end{aligned}
\right. \tag*{(67)} E a , ν r = e j k 0 z c o s θ ⎩ ⎨ ⎧ z ^ A θ ( θ ) J 1 ( q i ( θ ) ρ ) cos ( ϕ ′ − ϕ ) + q i 2 ( θ ) j ρ ^ sin ( ϕ ′ − ϕ ) [ k 0 cos θ A θ ( θ ) q i ( θ ) J 1 ′ ( q i ( θ ) ρ ) − ρ k 0 B θ ( θ ) J 1 ( q i ( θ ) ρ ) ] + q i 2 ( θ ) j ϕ ^ sin ( ϕ ′ − ϕ ) [ − ρ k 0 cos θ A θ ( θ ) J 1 ( q i ( θ ) ρ ) + k 0 B θ ( θ ) q i ( θ ) J 1 ′ ( q i ( θ ) ρ ) ] , ν = θ , − z ^ A ϕ ( θ ) J 1 ( q i ( θ ) ρ ) sin ( ϕ ′ − ϕ ) − q i 2 ( θ ) j ρ ^ sin ( ϕ ′ − ϕ ) [ k 0 cos θ A ϕ ( θ ) q i ( θ ) J 1 ′ ( q i ( θ ) ρ ) + ρ k 0 B ϕ ( θ ) J 1 ( q i ( θ ) ρ ) ] − q i 2 ( θ ) j ϕ ^ cos ( ϕ ′ − ϕ ) [ ρ k 0 cos θ A ϕ ( θ ) J 1 ( q i ( θ ) ρ ) + k 0 B ϕ ( θ ) q i ( θ ) J 1 ′ ( q i ( θ ) ρ ) ] , ν = ϕ . (67) longitudinal component according to
E a , ν r = ρ ^ E a , ρ , ν r + ϕ ^ E a , ϕ , ν r + z ^ E a , z , ν r , ν ∈ { θ , ϕ } , (68) \mathbf{E}^{\mathrm{r}}_{a,\nu}=\hat{\boldsymbol{\rho}}E^{\mathrm{r}}_{a,\rho,\nu}+\hat{\boldsymbol{\phi}}E^{\mathrm{r}}_{a,\phi,\nu}+\hat{\mathbf{z}}E^{\mathrm{r}}_{a,z,\nu},\qquad\nu\in\{\theta,\phi\}, \tag*{(68)} E a , ν r = ρ ^ E a , ρ , ν r + ϕ ^ E a , ϕ , ν r + z ^ E a , z , ν r , ν ∈ { θ , ϕ } , (68) yields the explicit reciprocal electric field in (67) .
With E P , k \mathbf{E}_{P,k} E P , k E a , ν r \mathbf{E}^{\mathrm{r}}_{a,\nu} E a , ν r (60) can be evaluated directly. Expressing the PA volume in cylindrical coordinates gives
F ν , k ( θ , ϕ ) = ω 2 μ 0 ϵ 0 ( 1 − ϵ r ) 4 π ∫ 0 L P , k { ∫ b k a ρ [ ∫ 0 2 π E P , k ( ρ , ϕ ′ , z ) ⋅ E a , ν r ( ρ , ϕ ′ , z ) d ϕ ′ ] d ρ } d z . (69) F_{\nu,k}(\theta,\phi)=\frac{\omega^2\mu_0\epsilon_0(1-\epsilon_r)}{4\pi}\int_0^{L_{P,k}}\left\{\int_{b_k}^{a}\rho\left[\int_0^{2\pi}\mathbf{E}_{P,k}(\rho,\phi',z)\cdot\mathbf{E}^{\mathrm{r}}_{a,\nu}(\rho,\phi',z)\,d\phi'\right]d\rho\right\}dz. \tag*{(69)} F ν , k ( θ , ϕ ) = 4 π ω 2 μ 0 ϵ 0 ( 1 − ϵ r ) ∫ 0 L P , k { ∫ b k a ρ [ ∫ 0 2 π E P , k ( ρ , ϕ ′ , z ) ⋅ E a , ν r ( ρ , ϕ ′ , z ) d ϕ ′ ] d ρ } d z . (69) The integral in (69) is evaluated separately for the two reciprocal polarizations ν ∈ { θ , ϕ } \nu\in\{\theta,\phi\} ν ∈ { θ , ϕ }
ν = θ \nu=\theta ν = θ θ \theta θ (58) and the ν = θ \nu=\theta ν = θ (67) into (69) yields
F θ , k ( θ , ϕ ) = ω 2 μ 0 ϵ 0 ( 1 − ϵ r ) 4 π J 1 ( u ( b k ) ) P g ( b k ) K 1 ( w ( b k ) ) ∫ 0 L P , k e j k 0 z cos θ { − b k 2 k 0 w 2 ( b k ) q i 2 ( θ ) [ A k + e − j β ( b k ) z + A k − e j β ( b k ) z ] × [ − β ( b k ) C ( b k ) w ( b k ) b k K 0 ( w ( b k ) b k ρ ) + ω μ 0 D ( b k ) − β ( b k ) C ( b k ) ρ K 1 ( w ( b k ) b k ρ ) ] × ( k 0 cos θ A θ ( θ ) q i ( θ ) J 1 ′ ( q i ( θ ) ρ ) − k 0 B θ ( θ ) ρ J 1 ( q i ( θ ) ρ ) ) + sin ϕ ′ sin ( ϕ ′ − ϕ ) ( ω μ 0 D ( b k ) w ( b k ) b k K 0 ( w ( b k ) b k ρ ) + ω μ 0 D ( b k ) − β ( b k ) C ( b k ) ρ K 1 ( w ( b k ) b k ρ ) ) × ( − k 0 cos θ A θ ( θ ) ρ J 1 ( q i ( θ ) ρ ) + k 0 B θ ( θ ) q i ( θ ) J 1 ′ ( q i ( θ ) ρ ) ) + C ( b k ) A θ ( θ ) K 1 ( w ( b k ) b k ρ ) J 1 ( q i ( θ ) ρ ) cos ϕ ′ cos ( ϕ ′ − ϕ ) × [ A k + e − j β ( b k ) z − A k − e j β ( b k ) z ] } d ϕ ′ d ρ d z . (70) \begin{aligned}
F_{\theta,k}(\theta,\phi)
={}&\frac{\omega^2\mu_0\epsilon_0(1-\epsilon_r)}{4\pi}
\frac{J_1(u(b_k))}{\sqrt{P_g(b_k)}K_1(w(b_k))}
\int_0^{L_{P,k}}e^{jk_0z\cos\theta}\Bigg\{
-\frac{b_k^2k_0}{w^2(b_k)q_i^2(\theta)}
\left[A_k^+e^{-j\beta(b_k)z}+A_k^-e^{j\beta(b_k)z}\right]\\
&\times\left[
-\frac{\beta(b_k)C(b_k)w(b_k)}{b_k}K_0\left(\frac{w(b_k)}{b_k}\rho\right)
+\frac{\omega\mu_0D(b_k)-\beta(b_k)C(b_k)}{\rho}K_1\left(\frac{w(b_k)}{b_k}\rho\right)
\right]\\
&\times\left(k_0\cos\theta A_\theta(\theta)q_i(\theta)J_1'(q_i(\theta)\rho)-\frac{k_0B_\theta(\theta)}{\rho}J_1(q_i(\theta)\rho)\right)\\
&+\sin\phi'\sin(\phi'-\phi)
\left(\frac{\omega\mu_0D(b_k)w(b_k)}{b_k}K_0\left(\frac{w(b_k)}{b_k}\rho\right)
+\frac{\omega\mu_0D(b_k)-\beta(b_k)C(b_k)}{\rho}K_1\left(\frac{w(b_k)}{b_k}\rho\right)\right)\\
&\times\left(-\frac{k_0\cos\theta A_\theta(\theta)}{\rho}J_1(q_i(\theta)\rho)+k_0B_\theta(\theta)q_i(\theta)J_1'(q_i(\theta)\rho)\right)\\
&+C(b_k)A_\theta(\theta)K_1\left(\frac{w(b_k)}{b_k}\rho\right)J_1(q_i(\theta)\rho)\cos\phi'\cos(\phi'-\phi)\\
&\times\left[A_k^+e^{-j\beta(b_k)z}-A_k^-e^{j\beta(b_k)z}\right]
\Bigg\}\,d\phi'\,d\rho\,dz. \tag*{(70)}
\end{aligned} F θ , k ( θ , ϕ ) = 4 π ω 2 μ 0 ϵ 0 ( 1 − ϵ r ) P g ( b k ) K 1 ( w ( b k )) J 1 ( u ( b k )) ∫ 0 L P , k e j k 0 z c o s θ { − w 2 ( b k ) q i 2 ( θ ) b k 2 k 0 [ A k + e − j β ( b k ) z + A k − e j β ( b k ) z ] × [ − b k β ( b k ) C ( b k ) w ( b k ) K 0 ( b k w ( b k ) ρ ) + ρ ω μ 0 D ( b k ) − β ( b k ) C ( b k ) K 1 ( b k w ( b k ) ρ ) ] × ( k 0 cos θ A θ ( θ ) q i ( θ ) J 1 ′ ( q i ( θ ) ρ ) − ρ k 0 B θ ( θ ) J 1 ( q i ( θ ) ρ ) ) + sin ϕ ′ sin ( ϕ ′ − ϕ ) ( b k ω μ 0 D ( b k ) w ( b k ) K 0 ( b k w ( b k ) ρ ) + ρ ω μ 0 D ( b k ) − β ( b k ) C ( b k ) K 1 ( b k w ( b k ) ρ ) ) × ( − ρ k 0 cos θ A θ ( θ ) J 1 ( q i ( θ ) ρ ) + k 0 B θ ( θ ) q i ( θ ) J 1 ′ ( q i ( θ ) ρ ) ) + C ( b k ) A θ ( θ ) K 1 ( b k w ( b k ) ρ ) J 1 ( q i ( θ ) ρ ) cos ϕ ′ cos ( ϕ ′ − ϕ ) × [ A k + e − j β ( b k ) z − A k − e j β ( b k ) z ] } d ϕ ′ d ρ d z . (70) Since the dependence on ϕ ′ \phi' ϕ ′ (70) is confined to cos ϕ ′ cos ( ϕ ′ − ϕ ) \cos\phi'\cos(\phi'-\phi) cos ϕ ′ cos ( ϕ ′ − ϕ ) sin ϕ ′ sin ( ϕ ′ − ϕ ) \sin\phi'\sin(\phi'-\phi) sin ϕ ′ sin ( ϕ ′ − ϕ ) [33 2.532.6] and [33 2.532.4] , (70) can be rewritten as
F θ , k ( θ , ϕ ) = ω 2 μ 0 ϵ 0 ( 1 − ϵ r ) 4 π J 1 ( u ( b k ) ) cos ϕ P g ( b k ) K 1 ( w ( b k ) ) ∫ 0 L P , k e j k 0 z cos θ { − b k 2 k 0 w 2 ( b k ) q i 2 ( θ ) [ A k + e − j β ( b k ) z + A k − e j β ( b k ) z ] × ∫ b k a ρ K 0 ( w ( b k ) b k ρ ) [ B θ ( θ ) ω μ 0 D ( b k ) − A θ ( θ ) β ( b k ) C ( b k ) cos θ ] J 1 ′ ( q i ( θ ) ρ ) d ρ + w ( b k ) b k ∫ b k a K 0 ( w ( b k ) b k ρ ) J 1 ( q i ( θ ) ρ ) d ρ + q i ( θ ) [ ω μ 0 D ( b k ) − β ( b k ) C ( b k ) ] [ A θ ( θ ) cos θ + B θ ( θ ) ] ∫ b k a K 1 ( w ( b k ) b k ρ ) J 1 ′ ( q i ( θ ) ρ ) d ρ − [ ω μ 0 D ( b k ) − β ( b k ) C ( b k ) ] [ A θ ( θ ) cos θ + B θ ( θ ) ] ∫ b k a w ( b k ) b k K 1 ( w ( b k ) b k ρ ) J 1 ( q i ( θ ) ρ ) d ρ + C ( b k ) A θ ( θ ) [ A k + e − j β ( b k ) z − A k − e j β ( b k ) z ] ∫ b k a ρ K 1 ( w ( b k ) b k ρ ) J 1 ( q i ( θ ) ρ ) d ρ } d z . (71) \begin{aligned}
F_{\theta,k}(\theta,\phi)
={}&\frac{\omega^2\mu_0\epsilon_0(1-\epsilon_r)}{4\pi}
\frac{J_1(u(b_k))\cos\phi}{\sqrt{P_g(b_k)}K_1(w(b_k))}
\int_0^{L_{P,k}}e^{jk_0z\cos\theta}\Bigg\{
-\frac{b_k^2k_0}{w^2(b_k)q_i^2(\theta)}
\left[A_k^+e^{-j\beta(b_k)z}+A_k^-e^{j\beta(b_k)z}\right]\\
&\times\int_{b_k}^{a}\rho K_0\left(\frac{w(b_k)}{b_k}\rho\right)
\left[B_\theta(\theta)\omega\mu_0D(b_k)-A_\theta(\theta)\beta(b_k)C(b_k)\cos\theta\right]
J_1'(q_i(\theta)\rho)\,d\rho\\
&+\frac{w(b_k)}{b_k}\int_{b_k}^{a}K_0\left(\frac{w(b_k)}{b_k}\rho\right)
J_1(q_i(\theta)\rho)\,d\rho\\
&+q_i(\theta)\left[\omega\mu_0D(b_k)-\beta(b_k)C(b_k)\right]
\left[A_\theta(\theta)\cos\theta+B_\theta(\theta)\right]
\int_{b_k}^{a}K_1\left(\frac{w(b_k)}{b_k}\rho\right)J_1'(q_i(\theta)\rho)\,d\rho\\
&-\left[\omega\mu_0D(b_k)-\beta(b_k)C(b_k)\right]
\left[A_\theta(\theta)\cos\theta+B_\theta(\theta)\right]
\int_{b_k}^{a}\frac{w(b_k)}{b_k}K_1\left(\frac{w(b_k)}{b_k}\rho\right)J_1(q_i(\theta)\rho)\,d\rho\\
&+C(b_k)A_\theta(\theta)\left[A_k^+e^{-j\beta(b_k)z}-A_k^-e^{j\beta(b_k)z}\right]
\int_{b_k}^{a}\rho K_1\left(\frac{w(b_k)}{b_k}\rho\right)J_1(q_i(\theta)\rho)\,d\rho
\Bigg\}\,dz. \tag*{(71)}
\end{aligned} F θ , k ( θ , ϕ ) = 4 π ω 2 μ 0 ϵ 0 ( 1 − ϵ r ) P g ( b k ) K 1 ( w ( b k )) J 1 ( u ( b k )) cos ϕ ∫ 0 L P , k e j k 0 z c o s θ { − w 2 ( b k ) q i 2 ( θ ) b k 2 k 0 [ A k + e − j β ( b k ) z + A k − e j β ( b k ) z ] × ∫ b k a ρ K 0 ( b k w ( b k ) ρ ) [ B θ ( θ ) ω μ 0 D ( b k ) − A θ ( θ ) β ( b k ) C ( b k ) cos θ ] J 1 ′ ( q i ( θ ) ρ ) d ρ + b k w ( b k ) ∫ b k a K 0 ( b k w ( b k ) ρ ) J 1 ( q i ( θ ) ρ ) d ρ + q i ( θ ) [ ω μ 0 D ( b k ) − β ( b k ) C ( b k ) ] [ A θ ( θ ) cos θ + B θ ( θ ) ] ∫ b k a K 1 ( b k w ( b k ) ρ ) J 1 ′ ( q i ( θ ) ρ ) d ρ − [ ω μ 0 D ( b k ) − β ( b k ) C ( b k ) ] [ A θ ( θ ) cos θ + B θ ( θ ) ] ∫ b k a b k w ( b k ) K 1 ( b k w ( b k ) ρ ) J 1 ( q i ( θ ) ρ ) d ρ + C ( b k ) A θ ( θ ) [ A k + e − j β ( b k ) z − A k − e j β ( b k ) z ] ∫ b k a ρ K 1 ( b k w ( b k ) ρ ) J 1 ( q i ( θ ) ρ ) d ρ } d z . (71) Moreover, by using the relation K ν ′ ( x ) = − K ν − 1 ( x ) − ν K ν ( x ) / x K_\nu'(x)=-K_{\nu-1}(x)-\nu K_\nu(x)/x K ν ′ ( x ) = − K ν − 1 ( x ) − ν K ν ( x ) / x [33 8.486.12] , with ν = 1 \nu=1 ν = 1 (71) can be rewritten as
F θ , k ( θ , ϕ ) = ω 2 μ 0 ϵ 0 ( 1 − ϵ r ) 4 J 1 ( u ( b k ) ) cos ϕ P g ( b k ) K 1 ( w ( b k ) ) ∫ 0 L P , k e j k 0 z cos θ { b k 2 k 0 w 2 ( b k ) q i 2 ( θ ) [ A k + e − j β ( b k ) z + A k − e j β ( b k ) z ] × [ B θ ( θ ) ω μ 0 D ( b k ) − A θ ( θ ) β ( b k ) C ( b k ) cos θ ] ∫ b k a [ ∂ ∂ ρ K 1 ( w ( b k ) b k ρ ) ∂ ∂ ρ J 1 ( q i ( θ ) ρ ) + 1 ρ 2 K 1 ( w ( b k ) b k ρ ) J 1 ( q i ( θ ) ρ ) ] d ρ + [ B θ ( θ ) β ( b k ) C ( b k ) − A θ ( θ ) ω μ 0 D ( b k ) cos θ ] ∫ b k a [ ∂ ∂ ρ K 1 ( w ( b k ) b k ρ ) J 1 ( q i ( θ ) ρ ) + K 1 ( w ( b k ) b k ρ ) ∂ ∂ ρ J 1 ( q i ( θ ) ρ ) ] d ρ + C ( b k ) A θ ( θ ) [ A k + e − j β ( b k ) z − A k − e j β ( b k ) z ] ∫ b k a ρ K 1 ( w ( b k ) b k ρ ) J 1 ( q i ( θ ) ρ ) d ρ } d z . (72) \begin{aligned}
F_{\theta,k}(\theta,\phi)
={}&\frac{\omega^2\mu_0\epsilon_0(1-\epsilon_r)}{4}
\frac{J_1(u(b_k))\cos\phi}{\sqrt{P_g(b_k)}K_1(w(b_k))}
\int_0^{L_{P,k}}e^{jk_0z\cos\theta}
\Bigg\{
\frac{b_k^2k_0}{w^2(b_k)q_i^2(\theta)}
\left[A_k^+e^{-j\beta(b_k)z}+A_k^-e^{j\beta(b_k)z}\right]\\
&\times\left[B_\theta(\theta)\omega\mu_0D(b_k)-A_\theta(\theta)\beta(b_k)C(b_k)\cos\theta\right]
\int_{b_k}^{a}\left[\frac{\partial}{\partial\rho}K_1\left(\frac{w(b_k)}{b_k}\rho\right)\frac{\partial}{\partial\rho}J_1(q_i(\theta)\rho)+\frac{1}{\rho^2}K_1\left(\frac{w(b_k)}{b_k}\rho\right)J_1(q_i(\theta)\rho)\right]d\rho\\
&+\left[B_\theta(\theta)\beta(b_k)C(b_k)-A_\theta(\theta)\omega\mu_0D(b_k)\cos\theta\right]
\int_{b_k}^{a}\left[\frac{\partial}{\partial\rho}K_1\left(\frac{w(b_k)}{b_k}\rho\right)J_1(q_i(\theta)\rho)
+K_1\left(\frac{w(b_k)}{b_k}\rho\right)\frac{\partial}{\partial\rho}J_1(q_i(\theta)\rho)\right]d\rho\\
&+C(b_k)A_\theta(\theta)\left[A_k^+e^{-j\beta(b_k)z}-A_k^-e^{j\beta(b_k)z}\right]
\int_{b_k}^{a}\rho K_1\left(\frac{w(b_k)}{b_k}\rho\right)J_1(q_i(\theta)\rho)\,d\rho
\Bigg\}\,dz. \tag*{(72)}
\end{aligned} F θ , k ( θ , ϕ ) = 4 ω 2 μ 0 ϵ 0 ( 1 − ϵ r ) P g ( b k ) K 1 ( w ( b k )) J 1 ( u ( b k )) cos ϕ ∫ 0 L P , k e j k 0 z c o s θ { w 2 ( b k ) q i 2 ( θ ) b k 2 k 0 [ A k + e − j β ( b k ) z + A k − e j β ( b k ) z ] × [ B θ ( θ ) ω μ 0 D ( b k ) − A θ ( θ ) β ( b k ) C ( b k ) cos θ ] ∫ b k a [ ∂ ρ ∂ K 1 ( b k w ( b k ) ρ ) ∂ ρ ∂ J 1 ( q i ( θ ) ρ ) + ρ 2 1 K 1 ( b k w ( b k ) ρ ) J 1 ( q i ( θ ) ρ ) ] d ρ + [ B θ ( θ ) β ( b k ) C ( b k ) − A θ ( θ ) ω μ 0 D ( b k ) cos θ ] ∫ b k a [ ∂ ρ ∂ K 1 ( b k w ( b k ) ρ ) J 1 ( q i ( θ ) ρ ) + K 1 ( b k w ( b k ) ρ ) ∂ ρ ∂ J 1 ( q i ( θ ) ρ ) ] d ρ + C ( b k ) A θ ( θ ) [ A k + e − j β ( b k ) z − A k − e j β ( b k ) z ] ∫ b k a ρ K 1 ( b k w ( b k ) ρ ) J 1 ( q i ( θ ) ρ ) d ρ } d z . (72) Moreover, applying integration by parts to the first radial integral and the product rule to the second, while utilizing [33 5.54.1,8.407.1] for the remaining Bessel-product integral together with the order-one Bessel differential identity ∂ ∂ ρ [ ρ ∂ ∂ ρ J 1 ( q i ( θ ) ρ ) ] = [ 1 ρ − q i 2 ( θ ) ρ ] J 1 ( q i ( θ ) ρ ) \frac{\partial}{\partial\rho}\left[\rho\frac{\partial}{\partial\rho}J_1(q_i(\theta)\rho)\right]=\left[\frac{1}{\rho}-q_i^2(\theta)\rho\right]J_1(q_i(\theta)\rho) ∂ ρ ∂ [ ρ ∂ ρ ∂ J 1 ( q i ( θ ) ρ ) ] = [ ρ 1 − q i 2 ( θ ) ρ ] J 1 ( q i ( θ ) ρ ) (72) reduces to
F θ , k ( θ , ϕ ) = ω 2 μ 0 ϵ 0 ( 1 − ϵ r ) cos ϕ 4 [ A k + R k + ( θ ) ∫ 0 L P , k e j [ k 0 cos θ − β ( b k ) ] z d z − A k − R k − ( θ ) ∫ 0 L P , k e j [ k 0 cos θ + β ( b k ) ] z d z ] . (73) \begin{aligned}
F_{\theta,k}(\theta,\phi)
={}&\frac{\omega^2\mu_0\epsilon_0(1-\epsilon_r)\cos\phi}{4}
\left[
A_k^+R_k^+(\theta)
\int_0^{L_{P,k}}e^{j[k_0\cos\theta-\beta(b_k)]z}\,dz
-A_k^-R_k^-(\theta)
\int_0^{L_{P,k}}e^{j[k_0\cos\theta+\beta(b_k)]z}\,dz
\right]. \tag*{(73)}
\end{aligned} F θ , k ( θ , ϕ ) = 4 ω 2 μ 0 ϵ 0 ( 1 − ϵ r ) cos ϕ [ A k + R k + ( θ ) ∫ 0 L P , k e j [ k 0 c o s θ − β ( b k )] z d z − A k − R k − ( θ ) ∫ 0 L P , k e j [ k 0 c o s θ + β ( b k )] z d z ] . (73) where I 0 , k I_{0,k} I 0 , k I 1 , k I_{1,k} I 1 , k I 2 , k I_{2,k} I 2 , k (48) , (49) , and (50) , respectively, while R θ , k + ( θ ) R_{\theta,k}^{+}(\theta) R θ , k + ( θ ) R θ , k − ( θ ) R_{\theta,k}^{-}(\theta) R θ , k − ( θ ) (44) . The minus sign associated with the backward contribution follows from the adopted forward–backward HE11 _{11} 11 R θ , k − ( θ ) R_{\theta,k}^{-}(\theta) R θ , k − ( θ )
∫ 0 L P , k e j [ k 0 cos θ + β ( b k ) ] z d z = Z k + ( θ ) , (74) \int_{0}^{L_{\mathrm{P},k}} e^{j[k_{0}\cos\theta+\beta(b_k)]z}\,\mathrm{d}z=Z_k^{+}(\theta), \tag*{(74)} ∫ 0 L P , k e j [ k 0 c o s θ + β ( b k )] z d z = Z k + ( θ ) , (74) whose substitution into (73) yields (42) .
ν = ϕ \nu=\phi ν = ϕ ϕ \phi ϕ (58) and the ν = ϕ \nu=\phi ν = ϕ (67) into (69) gives the azimuthal terms cos ϕ sin ( ϕ ′ − ϕ ) \cos\phi\sin(\phi'-\phi) cos ϕ sin ( ϕ ′ − ϕ ) sin ϕ cos ( ϕ ′ − ϕ ) \sin\phi\cos(\phi'-\phi) sin ϕ cos ( ϕ ′ − ϕ ) [33 2.532.5] , their integration over 0 ≤ ϕ ′ < 2 π 0\leq\phi'<2\pi 0 ≤ ϕ ′ < 2 π − π sin ϕ -\pi\sin\phi − π sin ϕ π sin ϕ \pi\sin\phi π sin ϕ sin ϕ \sin\phi sin ϕ K ν ′ ( x ) = − K ν − 1 ( x ) − ν K ν ( x ) / x K_{\nu}'(x)=-K_{\nu-1}(x)-\nu K_{\nu}(x)/x K ν ′ ( x ) = − K ν − 1 ( x ) − ν K ν ( x ) / x ν = 1 \nu=1 ν = 1 ν = θ \nu=\theta ν = θ I 0 , k I_{0,k} I 0 , k I 1 , k I_{1,k} I 1 , k I 2 , k I_{2,k} I 2 , k (48) –(50) . Collecting the corresponding terms through R ϕ , k + ( θ ) R_{\phi,k}^{+}(\theta) R ϕ , k + ( θ ) R ϕ , k − ( θ ) R_{\phi,k}^{-}(\theta) R ϕ , k − ( θ ) (45) therefore gives
F ϕ , k ( θ , ϕ ) = ω 2 μ 0 ϵ 0 ( 1 − ϵ r ) sin ϕ 4 [ A k + R ϕ , k + ( θ ) ∫ 0 L P , k e j [ k 0 cos θ − β ( b k ) ] z d z − A k − R ϕ , k − ( θ ) ∫ 0 L P , k e j [ k 0 cos θ + β ( b k ) ] z d z ] . (75) \begin{aligned}
F_{\phi,k}(\theta,\phi)
={}&\frac{\omega^{2}\mu_{0}\epsilon_{0}(1-\epsilon_{r})\sin\phi}{4}
\left[
A_{k}^{+}R_{\phi,k}^{+}(\theta)
\int_{0}^{L_{\mathrm{P},k}}e^{j[k_{0}\cos\theta-\beta(b_k)]z}\,\mathrm{d}z \right.\\
&\left.{}-A_{k}^{-}R_{\phi,k}^{-}(\theta)
\int_{0}^{L_{\mathrm{P},k}}e^{j[k_{0}\cos\theta+\beta(b_k)]z}\,\mathrm{d}z
\right]. \tag*{(75)}
\end{aligned} F ϕ , k ( θ , ϕ ) = 4 ω 2 μ 0 ϵ 0 ( 1 − ϵ r ) sin ϕ [ A k + R ϕ , k + ( θ ) ∫ 0 L P , k e j [ k 0 c o s θ − β ( b k )] z d z − A k − R ϕ , k − ( θ ) ∫ 0 L P , k e j [ k 0 c o s θ + β ( b k )] z d z ] . (75) The remaining longitudinal integrations are the same Z k + ( θ ) Z_k^{+}(\theta) Z k + ( θ ) Z k − ( θ ) Z_k^{-}(\theta) Z k − ( θ ) (74) , whose substitution into (75) yields (43) .
After deriving the two polarization components, the far-field radiation pattern is obtained by combining the θ \theta θ ϕ \phi ϕ
F k ( θ , ϕ ) = F θ , k ( θ , ϕ ) θ ^ + F ϕ , k ( θ , ϕ ) ϕ ^ . (76) \mathbf{F}_k(\theta,\phi)=F_{\theta,k}(\theta,\phi)\hat{\boldsymbol{\theta}}+F_{\phi,k}(\theta,\phi)\hat{\boldsymbol{\phi}}. \tag*{(76)} F k ( θ , ϕ ) = F θ , k ( θ , ϕ ) θ ^ + F ϕ , k ( θ , ϕ ) ϕ ^ . (76) Substituting (42) and (43) into this expression and restoring the common far-field dependence e − j k 0 R / R e^{-jk_{0}R}/R e − j k 0 R / R (41) , which concludes the proof.
References 
