We analyze radiation produced by an ultrarelativistic charge as it exits the open end of a cylindrical waveguide with a dielectric lining. The end of the waveguide can be either orthogonal to the structure axis or skewed. To obtain terahertz radiation from waveguides with centimeter or millimeter radii, we consider high order TM0m modes driven by the beam. We obtain an integral representation which describes the radiation produced by a single waveguide mode in the Fraunhofer zone. We perform a series of numerical calculations for structures which look promising for generation of THz radiation. It is shown that for a mode with large mode number, the aperture of the vacuum channel gives the main contribution to the field if the skew angle of the waveguide aperture is not too small. Simple expressions for the angle of the main pattern lobe maximum are obtained.
© 2014 Optical Society of America
Filling the Terahertz gap with powerful and efficient sources of radiation is a challenging problem of contemporary physics. A number of promising applications from physics, chemistry, and biology are waiting for a convenient tool for their investigations, with THz radiation being the most suitable candidate . Moreover, different practical applications in environment monitoring, medical diagnostics and others require sources of THz radiation .
Different techniques have been developed for generating THz radiation: laser driven THz emitters, solid state oscillators (high frequency diodes), quantum cascade laser-based and electron beam driven devices (see  and references therein). Recently reported free electron THz schemes use certain classical radiation processes such as synchrotron radiation (for accelerator facilities with circular rings) or transition radiation, diffraction radiation and Smith-Purcell radiation (for linear accelerators) [1, 4, 5].
In this paper, we consider the generation of THz waves in a form of the wakefield left behind by a charged particle beam (Cherenkov radiation). This mechanism allows generation of a high peak power, narrow bandwidth and high energy THz pulse. Several experiments have demonstrated generation of THz radiation in dielectric loaded structures [6, 7, 8] and corrugated waveguides . Most notably, the E201 collaboration at the Facility for Advanced Accelerator Tests (FACET) at SLAC National Accelerator Laboratory generated a 40 mJ signal with 0.3% bandwidth at 1.2 THz by passing an electron beam through a 10 cm long silica capillary. It should be underlined that dielectric loaded cylindrical waveguides for THz generation are significantly easier to manufacture than corrugated metal structures. For this reason, in this paper we focus on dielectric loaded waveguides only.
Recently, a few techniques allowing production of sub-picosecond electron bunch trains have been developed and demonstrated (see [10, 11] and references therein). Such a bunch train can selectively drive a mode in the wakefield structure with the frequency corresponding to the periodicity of bunches. This fact allows efficient generation of THz radiation via high order TM0m – like modes in a large (∼1 mm) dielectric loaded structure. The larger aperture allows easier beam transmission and yields narrow bandwidth radiation as high order modes tend to have a low (a few % of the speed of light) group velocity.
In this paper we consider the problem of extraction of high order modes (TM0m– like) from a dielectric loaded waveguide. We suppose that the waveguide end is made at arbitrary angle with respect to the waveguide axis. This allows clarifying preferences and shortcomings of using the skewed end. Moreover, it should be taken into account that accurate orthogonal end is difficult for manufacturing in the case of thin dielectric structure. Note that vacuum waveguides with slant end (so called Vlasov antennas) are used for transformation of waveguide modes into high power microwave radiation .
2. Problem statement, general results and approximations
We consider a cylindrical waveguide of radius a with an axisymmetric vacuum channel of radius b and a cylindrical layer of nondispersive dielectric (described by permittivities ε and μ) of thickness d = a − b. One end of the waveguide is open and skewed at an angle α (see Fig. 1(a)). It is convenient to introduce the Cartesian coordinate frame x, y, z with the z axis directed along the waveguide axis, the corresponding cylindrical frame , φ, z, the Cartesian frame x′, y′, z′ with the x′ − y′ plane coinciding with the plane of the wedge and the corresponding spherical frame R′, θ′, ϕ′. We suppose that a single waveguide mode propagates in the waveguide. For example, this mode can be generated by a point charge q moving along the z axis on the trajectory x = y = 0, z = υt, where υ = βc and c is the light speed in vacuum. In the case of an infinite waveguide (without an open end), the rigorous expressions for the field components are known . One can show that for motion of an ultra-relativistic charge, where γ = (1 − β2)−1/2 ≫ 1, and under the additional conditions, ωmbc−1 ≫ 1, ωmdc−1 ≫ 1, smb ≫ 1, the Cherenkov radiation field of a waveguide mode with number m takes the form
Note that one of the approximate methods (based on the laws of ray optics) for open structures was developed in the paper . However this approach is not suited for the problem under consideration because the field frequently oscillates on the dielectric surface of the aperture but has other behaviors on the vacuum surface. Therefore we will use another approach known in antenna theory , that can be applied independently to the field behavior on the aperture. Moreover, this method allows calculating the field at any distance from the aperture (including the Fraunhofer area where ray optics fail). According to this method, the radiation field in the Fraunhofer zone is a spherical wave, with its electrical vector being described by the following integral over the surface of the waveguide aperture Sa = Sav + Sad (see Fig. 1(b)):(2) is valid in the Fraunhofer zone where: Eq. (3) for m = 10 (λ10 ≈ 0.6mm, ν10 = c/λ10 ≈ 500GHz) R′ ≫ 15mm and for m = 20 (λ20 ≈ 0.3mm, ν20 ≈ 1THz) R′ ≫ 30mm.
To use formula (2), we should know the Ea, Ha fields on the outer (with respect to the waveguide) surface Sa. Despite the rich literature on open-ended waveguides (see for example ), the authors could not find any detailed treatment of the field in such a complicated structure (circular geometry coupled with layering and obliquity). Therefore, to find Ea and Ha we will utilize the following simplest approximation. The field on the vacuum channel aperture Sav is supposed to be given by Eq. (1) for r < b. After a series of calculations, we get for θ′- components of the field E produced by the aperture Sav (subscripts “e” and “m” correspond to the first and second summands of Eq. (2)):
The field on the dielectric aperture Sad is calculated by decomposing the functions sin and cos in Eq. (1) for b < r < a into a sum of two exponents. Two obtained waves refract through the aperture surface and form the field on the aperture Sad. In the issue we get:Eq. (6) from 17]. Here we consider a high order mode that should result in a better accuracy of the calculation.
In the case of an orthogonal waveguide flange (α = π/2), only θ- component of electric field is non vanishing, and we obtain:
3. Numerical results and discussions
To present the results, we will use the angle θ1 instead of usual θ (this is made traditionally in antenna theory), where for given polar angle ϕ0 ∈ [0, π], θ1 = θ for ϕ = ϕ0 and θ1 = −θ for ϕ = ϕ0 + π. Figure 2(a) shows the dependence of the electric field Eθ on θ1 in the case of an orthogonal waveguide end (α = π/2). Note that the results of the direct calculation of aperture integrals Eqs. (4) and (8) were found to be in full agreement with the analytical results given by Eqs. (11) and (12). As one can see, the vacuum channel aperture gives the main contribution to the field while radiation from dielectric is negligible (around three orders of magnitude smaller). As simple calculations show, this fact is connected with the total internal reflection of waves at the interface of the dielectric layer aperture. In total, radiation for θ = 0 is absent, the radiation pattern is symmetrical with respect to the z -axis and exhibits maxima for θ1 ≈ ±15° in agreement with Eq. (11).
For the case of non-orthogonal waveguide end (α = 60°, see Fig. 2(b)), the radiation patterns are generally similar to those for the case of the orthogonal waveguide end. The main contribution to the field comes from the vacuum channel aperture. However, the two lobes of the radiation pattern slightly differ in magnitude, a number of lateral lobes appear and a small field (around 2% compared with the maximum field) exists for θ = 0. As our calculations show, the essential change arises for α = 30° and lower, where radiation from dielectric aperture become comparable with that from the vacuum channel aperture. This means that the waves propagating in a dielectric layer leave the layer without total internal reflection at the aperture surface. The main lobe situated in the region θ1 > 0 become essentially larger (around three times) compared with the lobe located in the region θ1 < 0. This difference in the field magnitude corresponds to around one order of magnitude difference in the radiated energy.
We have developed the theory of the electromagnetic field radiated by a relativistic beam exiting the open end of a dielectric lined cylindrical waveguide and generating a high order radial mode. We have shown that this scheme is a good candidate technology for the development of intense accelerator based THz sources.
According to a traditional antenna theory approach, the field in the Fraunhofer zone (at distances exceeding several centimeters for the waveguide sizes considered here) has been obtained in terms of integrals over the aperture of the waveguide exit. These integrals can be easily calculated numerically for arbitrary structure parameters. Moreover, in the case of an orthogonal end, we have obtained simple analytical formulas for the field patterns and angles of the main radiation lobes. In particular, the main portion of the radiation comes from the vacuum channel aperture and radiation is absent in the direction along the waveguide axis.
We have also investigated numerically how the radiation pattern changes with an increase in deviation from the orthogonal end of the waveguide. Particularly, the radiation pattern becomes asymmetrical, and the main maximum increases with a decrease in the angle of the waveguide end. However, in this case the essential radiation comes from the dielectric part of the aperture, therefore the radiation pattern exhibits many narrow lobes.
Very acute waveguide obliquity is marginally suitable for THz generation because the radiated energy is distributed over a wide range of angles. Rather, even in the case of the orthogonal end, the main lobe of the radiation pattern (which possesses cylindrical symmetry) is directed at a sufficiently large angle with respect to the bunch trajectory. This allows convenient energy capture without a need to eliminate the parasitic bunch influence. Nevertheless, some deviation from orthogonal obliquity results in the loss of pattern symmetry and a consequently stronger concentration of radiated energy in certain directions.
The authors would like to thank A. Kanareykin and P. Schoessow for fruitful discussions. This work was supported by the Grant of the President of Russian Federation (No. 273.2013.2).
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