We analyze the electromagnetic spatital distributions and address an important issue of the transmission properties of spherical transverse-electric (TE) and transverse-magnetic (TM) eigenmodes within a tapered hollow metallic waveguide in detail. Explicit analytical expressions for the spatital distributions of electromagnetic field components, attenuation constant, phase constant and wave impedance are derived. Accurate eigenvalues obtained numerically are used to study the dependences of the transmission properties on the taper angle, the mode as well as the length of the waveguide. It is shown that all modes run continuously from a propagating through a transition to an evanescent region and the value of the attenuation increases as the distance from the cone vertex and the cone angle decrease. A strict distinction between pure propagating and pure evanescent modes cannot be achieved. One mode after the other reaches cutoff in the tapered hollow metallic waveguide as the distance from the cone vertex desreases.
©2009 Optical Society of America
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  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 . 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 
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).
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 :
Note that Er,Eθ,Eφ and Hr,Hθ,Hφ 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 Plm(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 Eφ(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
Where the bounary condition n×E = 0 yields Eφ(θ 0) = 0 and Er(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 lmn associated with the TEmn 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 TMmn field mode. The eigenvalues lmn of Eqs. (4) and (7) depend upon the cone half-angle θ 0, such that lmn increases as the magnitude of θ 0 decreases. In the table 1 we give the values of lmn for the first 5 lowest-order modes (TE 11,TM 01,TE 21,TM 11,TE 01) determined by Eqs. (4) and (7).
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 Er,Eθ or Eφ for TE or TM modes, A(θ,φ) is real, and kr 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:
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.
Where H (1) l+½ is the first kind Hankel function of order l + 0.5 with the argument kr . 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 kr , 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 γTEθφ,(r,l), yTMθφ(r,l) and γTMr (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 
where 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 kr with cone half-angle θ 0 as a parameter has been studied and the results are presented in Figs. 2–5.
Figures 2 and 3 show a variation of the attenuation α and phase constants β of Eθ and Eφ 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 Er 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 (Rc). 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 Rc =(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.
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 ∣zTEr∣ and ∣zTMr∣ depend on the cone half-angle θ 0 and kr.
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.
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 .
- We have developed an exact analytical approach for the description of the electromagnetic fields inside a hollow metallic waveguide with a taper. Analytical expressions for the spatital distributions of electromagnetic field components, attenuation constant, phase constant and wave impedance are derived.
- According to our theory the modes configurations inside a tapered hollow metallic waveguide are similar to those in a cylindrical hollow metallic waveguide, but the transmission characteristics and engergy densities distributions along propagating direction have a different behavior. It is shown that all modes run continuously from a propagating through a transition to an evanescent region and the value of the attenuation increases as the distance from the cone vertex and the cone angle desrease. A strict distinction between pure propagating and pure evanescent modes can not be achieved. There is no well-defined cutoff wavelength but rather a cutoff radius. It is interesting to note that the magnitude of the cutoff radius is related to the wavelength and the cone half-angle. The values of attenuation and phase constants for the spherical TE and TM modes inside the tapered hollow metallic waveguide depend on the cone half-angle very seriously. As the cone half-angle decreases, the value of the attenuation increases. The smaller the cone half-angle is, the faster the modes attenuate. This can explain why large taper angle may improve the light throughout in aperture probe which finds an important application in scanning near-field optical microscopy .
- As follows from our calculations, we find that in the propagating region the attentuation of some modes decays faster than those of others, and one mode after the other reaches cutoff in the tapered hollow metallic waveguide as the distance from the cone vertex decreases.
- In the tapered hollow metallic waveguide, light is well confined in the hollow core (air region) because it is reflected back to the core by a metal wall.
This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 10674045, 10576012, 60538010), the National Basic Research Program of China (Grant No. 61359020101-3), and the National High Technology Research and Development Program for Inertial Confinement Fusion of China.
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