1. Introduction

The advent of high–purity ultra-low-loss silica fiber as a transmission medium in the late 1970s provided a basis for the modern optical communication infrastructure. Although highly successful, silica waveguides have fundamental limitations in the their attenuation and nonlinearities that result from the interaction of light with a dense, material-filled core. A different approach to waveguiding circumvents these problems by confining light in a hollow core using highly reflective walls. This approach is exemplified by hollow waveguides with perfectly conducting metallic wall [1] and omnidirectional reflector claddings [2, 3]. Prior to the emergence of sillica fiber, the hollow conical waveguides (horns) have got successful applications in many fields such as millimeter-wave communication, microwave radio-relay links, and satellite communication [4, 5]. They have been mainly used as a simple and compact antenna structure. In recent years, hollow metallic waveguides with a conical taper have been investigated in view of their applicability as couplers. These waveguides find some important applications in optical communication, scanning near-field optical microscopy, and laser-wakefield accelerators [6–9]. A theroretical treatment of a tapered optical waveguide presents formidable mathematical difficulties. One can not but , therefore, have recourse to approximate methods. A typical approach is to assume the tapered hollow metallic waveguides as a stack of a large number of optical waveguides of increasing cross-sectional size arranged end to end [10]. However, the actual electromagnetic fields configurations, their transmission characteristics, as welll as impedance and fields intensities distributions have not been thoroughly investigated.

In the present paper, we report the results of a theoretical investigation of a tapered hollow metallic waveguide from a exact analytical approach. Some numerical solutions based on a discretization are used to overcome the difficulties involed in finding the accurate eigenvalues of this boundary value problem and the evaluation of the associated eigenfunctions. A detailed study of the electromagnetic fields configurations, their transmission characteristics, as well as impedance and fields intensities distributions inside a conical hollow metallic waveguides is facilitated by accurate computation of eigenvalues. Explicit expressions for the field components, attenuation constant, phase constant, wave impedance and field intensity are obtained for the spherical TE and TM modes in a perfectly conducting conical hollow waveguide.

2. Spatital distributions of electromagnetic fields for the spherical TE and TM modes

Here we discuss the behaviour of electromagnetic fields inside a truncated cone of circular cross-sectional shape with perfectly conducting metallic walls and with a loss-free air core. A meridional and a transverse section of such a waveguide can be seen in Figs. 1(a) and 1(b). Throughout this paper we consider time-harmonic dependence of the fields. Further, we omit the factor exp(-iωt) from all expressions. In spherical coordinates (r,θ,φ) the basic equation for the Hertz function U(r, θ, φ) of electromagnetic fields in conical waveguide can be written as [11]

Here r is the distance from the cone vertex, θ and φ are polar and azimuthal angles, respectively. k = 2π/λ = ω/v is wave number (ω and v are the frequency and the velocity of light respectively, and λ. denotes its wavelength in air).

 

Fig. 1. Schematic illustrating a conical hollow metallic waveguide. (a) Meridional section of hollow metallic waveguide with a taper. The apex of the cone coincides with the origin of the spherical coordinate system. 2θ ₀ is the cone angle, z ₀ is the longitudinal coordinate at the waveguide exit, 2R ₀ is the aperture diameter in the exit plane, r_in and z_in = r_in cos θ ₀ are the radial and longitudinal coordinates at the waveguide entrance, respectively. (b) Transverse section of hollow metallic waveguide with a taper. R presents transverse section radius and R = r sin θ ₀.

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If the apex of the cone coincides with the origin of the spherical coordinate system and polar axis is the longitudial axis of symmetry of the cone, then conical surface will coincide with surface of constant colatitude as shown in Fig. 1(a). The field components of the permissible TE modes to r can be expressed in terms of the Hertz function U by means of the following relations [11]:

Note that E_r,E_θ,E_φ and H_r,H_θ,H_φ denote the components of the electric E and magnetic H fields associated with the corresponding spherical coordinates r,θ,φ . Where R(r) = kr · h ⁽¹⁾ _l (kr) , h ⁽¹⁾ _l(kr) is the spherical Bessel function of the third kind with a noninteger index l , corresponding to the inward-traveling wave, and P_l^m(cosθ) is the associated Legendre function of order l and degree m. The azimuthal symmetry sets m as an integer. l is determined by the bounary condition n×E = 0 which yields E_φ(θ ₀) = 0 for the tangential component of the electric field E on the boundary between a hollow core and a perfectly conducting metallic surface of a cone. With the help of Eq. (2), the boundary condition for TE waves can be rewritten in terms of the associated Legendre function

For TM modes to r, one can use the above notation for the Hertz function U , where U obeys the Eq. (1). In this case the field components inside conical waveguides can be written as [11]

Where the bounary condition n×E = 0 yields E_φ(θ ₀) = 0 and E_r(θ ₀) = 0 for the tangential component of the electric field E at an interface. With the help of Eq. (5), the boundary condition for TM waves can be rewritten in terms of the associated Legendre function

The Eq. (4) yields a set of eigenvalues l_mn associated with the TE_mn field modes. For a given azimuthal number m (m = 0,1,2,…), the index n (n = 1,2,3,…) denotes the corresponding root of Eq. (4). Similarly, each choice of a pair (m, n) in Eq. (7) determined a possible TM_mn field mode. The eigenvalues l_mn of Eqs. (4) and (7) depend upon the cone half-angle θ ₀, such that l_mn increases as the magnitude of θ ₀ decreases. In the table 1 we give the values of l_mn for the first 5 lowest-order modes (TE ₁₁,TM ₀₁,TE ₂₁,TM ₁₁,TE ₀₁) determined by Eqs. (4) and (7).

Table 1. The eigenvalues lmn for the TE11,TM01,TE21,TM11,TE01 modes as functions of the cone half-angle 90.

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3. Transmission characteristics of the spherical TE and TM modes inside conical hollow metallic waveguide

The transmission properties of the guide modes are governed by a number of physical quantities. The important parameters are the attenuation and phase constants which are defined as the logarithmic rate of decrease of amplitude and phase, respectively, of a field component in the direction of propagation. More interesting, however, is the electric field component. For all electric field components of the spherical TE and TM modes mentioned previously, one can expressed them as

where E represents E_r,E_θ or E_φ for TE or TM modes, A(θ,φ) is real, and k_r is the propagation constant in the direction r and complex. With the help of Eq. (8), we can obtain

Further, we introduce the following quantity:

Substituting this into Eqs. (8) and (9), we can obtain

where α is defined as the attenuation and β as the phase constant and α, β are real. If β < 0, we have inward-traveling waves which are attentuated or augmented according as α is positive or negative.

For TE or TM modes inside the conical hollow metallic waveguide, and with the help of Eqs. (2), (5) and (10), one obtains

and

Where H ⁽¹⁾ _l+½ is the first kind Hankel function of order l + 0.5 with the argument k_r . H′⁽¹⁾ _l+½,H″⁽¹⁾ _l+½ and H′′′⁽¹⁾ _l+½ are the first, the second and the third derivatives of the Hankel function with the argument k_r , respectively. The Hankel function function H ⁽¹⁾ _l+½ (kr) and its derivative can be expressed as complex quantities and the real and imaginary parts of γ^TE_θφ,(r,l), y^TM_θφ(r,l) and γ^TM_r (r,l) give the attenuation and phase constants, respectively, for the TE to r and TM to r modes.

Another parameter of particular interest associated with the mode transmission is the wave impedance, defined by [11]

With the help of Eqs. (2), (5), (15) and (16), one can obtain

where $\eta =\sqrt{\genfrac{}{}{0.1ex}{}{\mu }{\epsilon }}$ is called the intrinstic impedance of the medium.

4. The space variation of the attenuation and phase constants of the spherical TE and TM modes

Based on the Eqs. (12), (13) and (14), a variation of the attenuation α and phase constants β for the spherical TE and TM modes as a function of k_r with cone half-angle θ ₀ as a parameter has been studied and the results are presented in Figs. 2–5.

 

Fig. 2. The attenuation constant (α) of E_θ and E_φ for the TE and TM modes as a function of kr.

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Fig. 3. The phase constant (β) of E_θ and E_φ for the TE and TM modes as a function of kr.

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Fig. 4. The attenuation constant (α) of E_r for TM modes as a function of kr .

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Fig. 5. The phase constant (β) of E_r for the TM modes as a function of kr .

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Figures 2 and 3 show a variation of the attenuation α and phase constants β of E_θ and E_φ for the first 5 lowest-order modes as a function of kr with cone half-angle θ ₀ as a parameter in the propagation direction, respectively. Figures 4 and 5 present α and β of E_r for the TM ₀₁ and TM ₁₁ modes as a function of kr with cone half-angle θ ₀ as a parameter, respectively. As kr desreases all modes run continuously from a propagating through a transition to an evanescent region. We note that a strict distinction between pure propagating and pure evanescent modes can not be achieved. The value of the attenuation α increases and the phase constants β changes from ≈ -k to 0 as kr desreases. β < 0 represents inward-traveling wave and β > 0 represents outward-traveling wave which describes the reflection effect inside the conical hollow metallic waveguide. There is no well-defined cutoff wavelength but rather a cutoff radius (R_c). All modes have a cutoff when β = 0. However, it is interesting to note that the magnitude of the cutoff radius is related to the wavelength and the cone half-angle. At about kr = 10, for example, the TE ₁₁ reaches cutoff inside the tapered hollow waveguide with cone half-angle θ ₀ = π/24. By computation, we can obtain the cutoff radius R_c =(5λ/π)· sin π/24 ≈ 0.2077λ. That is, the mode reaches cutoff at subwavelength-sized aperture. In the Figs 2–5, we observe the values of attenuation α and phase constants β depend on the cone half-angle θ ₀ very seriously. As θ ₀ decreases, the value of the attenuation increases. The smaller the cone half-angle is, the faster the modes attenuate. It is necessary to stress that our analysis is restrict to perfect metallic conductor only. While this approximation of real metallic walls with perfect electric conductors is applicable only for near infrared and lower frequencies, because metallic wavuguides become lossy at high frequencies due to the finite condutivity of metals.

Figure 6 shows α and β for the first 5 lowest-order modes as a function of kr with fixed cone half-angle θ ₀ = π/6 in the propagation direction. Note that a logarithmic scale is used in Fig. 6(a). In the propagating region the attentuation of some modes decays faster than those of others. At kr = 10, for example, the TM ₁₁ mode shows the lowest attenuation. The sequence of modes of the cylindrical hollow metal waveguide starts with TE ₁₁,TM ₀₁,TE ₂₁,TM ₁₁,TE ₀₁, and the latter two modes TM ₁₁ and TE ₀₁ are degenerate. Whereas for the conical hollow metallic waveguide we can obtain sequence TM ₁₁,TM ₀₁,TE ₁₁,TE ₂₁,TE ₀₁ beyond kr ≈ 7.5. The TM ₁₁ and TE ₀₁ modes are not degenerate. From the Fig. 6(b), we observe that one mode after the other reaches cutoff in the tapered hollow metallic waveguide as kr decreases.

 

Fig. 6. Variation of the attenuation α and phase constants β for the first 5 lowest-order modes as a function of kr with cone half-angle θ ₀ = π/6 as a parameter.

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5. Radial dependences of the wave impedance for the spherical TE and TM modes

We shall consider below the main features in the behaviour of the wave impedance as a function of kr for the spherical TE and TM modes inside the conical hollow metallic waveguide. Based on the Eqs. (17) and (18), a variation of the the wave impedance for TE ₁₁ and TM ₀₁ modes as a function of kr with cone half-angle θ ₀ as a parameter has been studied and the results are presented in Fig. 7. By numerical simulations we find the magnitude of the absolute values for the wave impedances ∣z^TE_r∣ and ∣z^TM_r∣ depend on the cone half-angle θ ₀ and kr.

 

Fig. 7. The absolute value of the wave impedance ∣Z_r^TE∣ and ∣Z_r^TM∣ as a function of kr for the spherical TE and TM modes inside the conical hollow metallic waveguide.

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6. Electromagnetic fields configurations and fields intensities distributions within tapered hollow metal waveguide

In this section we consider the electromagnetic fields configurations and fields intensities distributions inside tapered hollow metal waveguide. To provide a better understanding of the fields configurations for the spherical TE and TM modes, in Fig. 8 the electric-field time-average engergy density ½ε∣E∣² is plotted for the TE ₁₁,TM ₀₁,TE ₂₁,TM ₁₁,TE ₀₁ modes at a wavelength λ = 1.06μm , the distance from the cone vertex r = 20mm and a cone half-angle θ ₀ = π/6. We find that the modes configurations are similar to those of the cylindrical hollow metal waveguide for which completely transverse modes are obtained.

 

Fig. 8. The electric-field time-average engergy density for the first 5 modes is shown in the spherical cross section with the radius r = 20mm. The color scheme is such that the engergy density goes from minimum (green) to maximum (yellow).

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Fig. 9. The fields intensities distributions within tapered hollow metal waveguide. The color scheme is such that the engergy density goes from minimum (green) to maximum (yellow). (a) Intensity distributions of TE ₁₁ mode propagating from r = 20mm to r = 15mm inside tapered hollow metal waveguide with θ ₀ = π/6 . (b) Intensity distributions of TM ₀₁ mode propagating from r = 20mm to r = 10mm inside tapered hollow metal waveguide with θ ₀ = π/6.

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Figures 9(a) and 9(b) show distinct intensity distributions for TE ₁₁ and TM ₀₁ modes inside tapered hollow metal waveguide with θ ₀ = π/6 , respectively. Figure 9(a) reprents the intensity distributions of TE ₁₁ mode propagating from r = 20mm to r =15mm. Figure 9(b) reprents the intensity distributions of TM ₀₁ mode propagating from r = 20mm to r = 10mm. In order to capture the fields intensities distributions in the hollow core along the propagating direction, we only plot the intensities distributions in the azimuthal angles range from 0 to 3 π/2. In the Fig. 9 we observe that all modes appear to be very well confined to the hollow core, i.e., the tapered hollow metallic waveguide can focus a laser beams into a small beam spot. Utilizing the characteristic of the tapered hollow metallic waveguide, some researchs find the waveguide is useful for medial and dental applications [12].

7. Conclusions