Issue 
TST
Volume 13, Number 1, March 2020



Page(s)  1  21  
DOI  https://doi.org/10.1051/tst/2020131001  
Published online  31 March 2020 
Invited Paper
Gyrodevices – natural sources of highpower highorder angular momentum millimeterwave beams
^{1}
Institute for Pulsed Power and Microwave Technology (IHM), Karlsruhe Institute of Technology (KIT), Kaiserstr. 12, 76131 Karlsruhe, Germany
^{2}
Institute of Radiofrequency Engineering and Electronics (IHE), Karlsruhe Institute of Technology (KIT)Kaiserstr. 12, 76131 Karlsruhe, Germany
^{*} Email: manfred.thumm@kit.edu
Received:
23
January
2020
The Orbital Angular Momentum (OAM) carried by light beams with helical phasefront (vortex beams) has been widely employed in many applications such as optical tweezers, optical drives of micromachines, atom trapping, and optical communication. OAM provides an additional dimension (diversity) to multiplexing techniques, which can be utilized in addition to conventional multiplexing methods to achieve higher data rates in wireless communication. OAM beams have been thoroughly studied and used in the optical regime but in the mmwave and THzwave region, they are still under investigation. In these frequency bands, there are difficulties associated with beamsplitting and beamcombining processes as well as with the use of spiral phase plates and other methods for OAM generation, since the wavelength is much larger compared to those at optical frequencies, leading to higher diffraction losses.
The present paper describes the natural generation of highpower OAM modes by gyrotype vacuum electron devices with cylindrical interaction circuit and axial output of the generated rotating higherorder transverse electric mode TE_{m,n}, where m > 1 and n are the azimuthal and radial mode index, respectively. The ratio between the total angular momentum (TAM) J_{N} and total energy W_{N} of N photons is given by m/ω, where ω is the angular frequency of the operating mode, which in a gyrotron oscillator is close to the TE_{m,n}mode cutoff frequency in the cavity. Therefore, m/ω = R_{c}/c, where R_{c} is the caustic radius and c the velocity of light in vacuum. This means that the OAM is proportional to the caustic radius and at a given frequency the same for all modes with the same azimuthal index m. Righthand rotation (corotation with the electrons) corresponds to a positive value of m and lefthand rotation to negative m. The corresponding OAM mode number (topological charge) is l = m – 1. Circularly polarized TE_{1n} modes only possess a Spin Angular Momentum (SAM: s = ±1). TE_{0n} modes have neither SAM nor OAM.
This is the result of the photonic (quasioptical) approach to derive the TAM of modes generated in gyrotrons. The same result follows from the electromagnetic (EM) wave approach for the TAM within a given waveguide volume per total energy of the EM wave in the same volume.
Such highpower output beams with very pure higherorder OAM, generated by gyrotron oscillators or amplifiers (broadband) could be used for multiplexing in longrange wireless communications. The corresponding mode and helical wavefront sensitive detectors for selective OAMmode sorting are available and described in the present paper.
Key words: Orbital angular momentum (OAM) / Spin angular momentum (SAM) / Millimeterwave and THzwave vortex beams / Gyrotron / Gyroamplifiers / Longrange wireless communication / Multiplexing / Diversity
© The Author(s) 2020
This is an Open Access article distributed under the terms of the Creative Commons Attribution License CCBY (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, except for commercial purposes, provided the original work is properly cited.
1. Introduction
Electromagnetic (EM) waves carry linear and angular momentum [1]. The linear momentum is associated with the Poynting vector , which represents the directional energy flux (the energy transfer per unit area per unit time, measured in W/m^{2}). The total angular momentum (TAM) of paraxial EM wave beams can be described as the sum of spin angular momentum (SAM) and orbital angular momentum (OAM), where the SAM is associated with the polarization state [2, 3]. The SAM per photon is given by s ∙ћ, where ћ is the Planck constant divided by 2π and 1 ≤ s ≤ 1. SAMstate s = 0 corresponds to linear polarization and s = ± 1 to righthanded circular polarization (RHCP) and lefthanded circular polarization (LHCP), respectively. In between, the wave beam is elliptically polarized. The OAM is related to the spatial wave front with the azimuthal phase distribution exp(jlφ), where φ is the azimuthal angle and l is any integer, the socalled topological charge or OAM state number [4]. The OAM per photon is given by l∙ћ, with positive l for righthanded and negative l for lefthanded phase rotation. EM wave beams carrying OAM (often called vortex beams) exhibit unique features like azimuthal phase changes in a plane perpendicular to the beam axis (helical phase fronts) and an amplitude singularity (annular doughnut shape) in the propagation direction. The amount of phase front twisting is determined by the OAM state number l (l phase spirals). EM wave beams with different OAM are mutually orthogonal, allowing them to be multiplexed together at the same carrier frequency along the same propagation axis and demultiplexed with low crosstalk. Therefore, in addition to conventional multiplexing methods to achieve higher data rates, OAM can be an effective means to increase the channel capacity and spectral efficiency in wireless communication technology by a factor equal to the number of available OAM states (OAM diversity) [5].
OAM beams have been thoroughly studied and used in the optical regime for highspeed, terabit OAM multiplexing wireless communications [5] and in the areas of sensitive optical detection, orientational manipulation (optical tweezers), optical drives of micromachines, atom trapping, quantum information processing and biology microscopy [6, 7].
Despite the various applications of OAM, it is still a challenging task to generate OAM beams with pure and controllable topological charge state l. Many methods have been developed to realize such OAM modes, including spiral phase plates (SPPs), holographic diffraction gratings (pitchfork hologram), birefringent liquid crystals (Qplate), spiral reflectors, specific array antennas, and anisotropic frequency selective surfaces (metasurfaces) [8, 9, and references given there]. In the microwave and mmwave region there are difficulties associated with diffraction losses since the wavelength is much larger compared to those at optical frequencies. Reviews on the stateoftheart and applications in this frequency region are given in [10, 11]. Record values in highcapacity mmwave communications with OAM multiplexing have been reported in [12]. In this article eight multiplexed 28 GHz OAM beams (l = ± 1 and ± 3 on each of two orthogonal polarizations), generated by SPPs, each carrying a 4×1 Gbit/s (4 bits per symbol) quadrature amplitude modulation (16QAM) signal, thereby achieving a capacity of 32 Gbit/s, were transmitted over a distance of 2.5 m with a spectral efficiency of ≈ 16 bit/(s∙Hz). All the eight OAM channels were recovered with biterror rates (BERs) below 3.8x10^{−3}.
The present paper shows that gyrotron oscillators and gyroamplifiers are natural sources of highpower mmwave beams with very pure higherorder OAM, which can be used for multiplexing in longrange wireless communication. The corresponding mode and helical wavefront sensitive detectors for selective OAMmode sorting are available and also described in this report. Section 2 gives an overview on gyrodevices and their interaction principles. The OAM and SAM characteristics of rotating gyrotron cavity modes are discussed in Section 3. Section 4 shows examples of highpower, highorder gyrotron OAM beam transmitters and Section 5 introduces to corresponding mode and helical wavefront selective receivers for OAM mode sorting. Finally, Section 6 gives the conclusions and summarizes the present results.
2. Gyrodevices
Gyrodevices are electron cyclotron masers in which electromagnetic (EM) waves with the angular frequency ω are generated by relativistic electrons gyrating in an external longitudinal magnetic field with a perpendicular velocity v_{⊥} [13]. The resonance condition between the periodic rotation of the electrons and the EM wave (TE mode) in the interaction circuit is
(1)where k_{} is the characteristic axial wavenumber of the EM wave in the interaction structure, v_{} is the longitudinal electron drift velocity and Ω_{c} = Ω_{co}/γ is the relativistic electron cyclotron frequency.
The nonrelativistic electron cyclotron frequency Ω_{co} is given by
(2)where e and m_{o} are the electrical charge and rest mass of an electron and B is the magnitude of the cavity magnetic field. The relativistic Lorentz factor γ can be simply calculated from
(3)where c is the freespace velocity of light and V_{b} the acceleration voltage of the electron beam in kV. In the present paper, the field vector is pointing to the output of the interaction circuit.
Gyrotrons are oscillators, which mainly employ weakly relativistic electron beams (V_{b} ≤ 80 kV, γ ≤ 1.16) with high transverse momentum (velocity ratio α = v_{⊥}/v_{} = 1.21.5) [13, 14]. The wave vector of the EM wave in the cavity is almost transverse to the direction of the longitudinal magnetic field (k_{⊥} >> k_{} and the Doppler shift is small), resulting, according to eq. (1), in wave generation near the electron cyclotron frequency or one of its harmonics:
In gyrotrons with cylindrical waveguide cavity, the operating TE_{m,n} mode is close to cutoff, which means v_{ph} = ω/k_{z} >> c and the frequency mismatch ω  qΩ_{c} is small but positive in order to achieve correct phasing, i.e. keeping the azimuthal electron bunches in the retarding phase to transfer energy to the TE wave. The Doppler term k_{}v_{} is of the order of the gain width and is small compared to the radiation frequency.
The frequency of a gyrotron is approximately given by the cutoff frequency f_{cut} of the cavity mode [15]:
(5)where χ_{m,n} is the n^{th} root of the derivative of the corresponding Bessel function J’_{m} (TE_{mn} mode) and D = 2R_{0} is the cavity diameter.
Gyrotrons, as fastwave devices with v_{ph} > c, are capable of producing very highpower radiation at cm, mm and submm wavelengths [13, 14], since the use of large cavity and output waveguide cross sections reduces wall losses as well as the danger of breakdown and permits the passage of large, highpower electron beams. In contrast to klystrons, the resonance frequency of a gyrotron cavity is not determined by the characteristic cavity size, but by the strength of the magnetic field (see eqs. 1 and 3). Then, according to eq. (5) for a given frequency, operation in a rotating highorder mode (determined by its eigenvalue χ_{m,n}) with low Ohmic attenuation can be selected just by the cavity radius R_{0}. Operation at the q^{th} cyclotron harmonic frequency reduces the required magnetic field for a given frequency by the factor q.
Whether the direction of cavity mode rotation is co or counter with respect to the electron gyration in the cavity magnetic field depends on the positioning of the annular electron beam. If the electron beam radius R_{b} for fundamental frequency excitation (q = 1) is chosen as [15]
(6)the generated TE_{m,n} mode is corotating (righthand rotating), and for
(7)it is counterrotating (lefthand rotation). The corresponding beam radii are a little bit smaller and a little bit larger than the radius of the intensity maximum of the cavity TE_{m,n} mode. In the case of the circular symmetric TE_{0,n} modes both radii are the same and identical with the radius of the maximum of the electric field. There is no mode rotation. In the case of TE_{m,1} modes, only the corotating modes can be excited, since always χ_{m+1,1} > χ_{m,1} (therefore, according to (7): R_{b} > R_{0}!).
Bunching of electrons in gyrodevices has much in common with that in conventional linear electron beam devices, namely, monotron, klystron, TWT, twystron and BWO [13]. In both cases the primary energy modulation of electrons gives rise to bunching (azimuthal or longitudinal), which is inertial. The bunching continues even after the primary modulation field is switched off (in the drift sections of klystrontype and twystrontype devices). This analogy suggests the correspondence between conventional linearbeam (Otype) devices and various types of gyrodevices. Table 1 presents the schematic drawings of devices of both classes [14].
3. OAM and SAM of rotating gyrotron modes
Helically propagating rotating TE_{m,n} modes in a circular waveguide can be decomposed into a series of cylindrical plane waves, each propagating at the Brillouin angle θ_{B} = arcsin(χ_{m,n}/kR_{0}) relative to the waveguide axis, where k = 2π/λ is the freespace wavenumber. The requirement of a zero azimuthal electric field at the waveguide wall defines their relative phases. In the geometric optical (g.o.) limit a plane wavefront is represented by one ray. Its transverse location is defined by the requirement that at a particular point of interest the ray direction must coincide with the direction of the Poynting vector of the original TE_{m,n} mode field distribution. If the point of interest is located at the waveguide wall the ray has the distance
from the waveguide center. Hence if all plane waves are presented by g.o. rays, they form a caustic at the radius R_{c}. In an unperturbed circular symmetric waveguide the density of the rays along the caustic is uniform [16].
The total angular momentum (TAM) possessed by such a ray (which is tangential to the caustic of the mode) can be expressed as
Therefore,
The ratio between the magnitudes of the TAM J_{N} and the total energy W_{N} of N photons is given by
(11)where ω is the angular frequency of the operating cavity mode, which is close to the cutoff frequency ω_{cut} = cχ_{m,n}/R_{0} in the cylindrical gyrotron cavity section. Therefore follows
This is the result of the photonic (g.o.) approach [16]. Using the time averaged Poynting vector and the expressions of the electric and magnetic fields of rotating TE_{m,n} modes, equation (11) also follows from the EM wave approach for the TAM of a gyrotron cavity mode within a given waveguide volume per time averaged total energy of the EM wave in the same volume [16].
From relation (12), the following features of rotating gyrotron TE_{m,n} modes can be found:
At a given frequency f = ω2π, the TAM increases with increasing azimuthal mode number m, whereas for fixed m it decreases with increasing frequency.
At a given frequency all gyrotron modes with the same azimuthal mode number m have the same TAM (same caustic radius R_{c}, but different cavity radius R_{0}).
For a given gyrotron cavity radius R_{0}, the TE_{m,n} mode with the largest caustic radius R_{c} has the highest TAM. This is the corotating whispering gallery mode (WGM) TE_{m,1} with χ_{m,1} ≲ kR_{0}.
Figure 3 shows the analytically calculated amplitude and phase of the timeaveraged total electric field of the corotating 95 GHz TE_{6,2} and TE_{10,1} modes, respectively [17]. In Fig. 4, the experimental phase pattern of the corotating 140 GHz TE_{28,8} mode is plotted (left), generated by a specific lowpower mode generator (see Section 5) and measured using a standard rectangular waveguide probe antenna [18]. Along an azimuthal circle, 27 phase spirals can be counted (right). In the center of the pattern the phase is random because the very small E_{r} and E_{φ} fields.
Fig. 1 (a) Ray propagation for a rotating wave in a cylindrical gyrotron cavity with consecutive reflections. (b) Set of rays forming a caustic with radius R_{c} [16]. 
Fig. 3 Theoretical amplitude (a) and phase (b) of the timeaveraged total electric field of the corotating 95 GHz modes TE_{6,2} (left) and TE_{10,1} (right) (in the Fresnel zone at 10 mm distance from a 20 mm diameter aperture) [17]. The propagation direction is perpendicular to the measured pattern, coming out of the plane. 
Fig. 4 Lowpower (cold) measurement of 140 GHz corotating TE_{28,8} mode phase pattern (left), taken with a fundamental rectangular waveguide probe for vertical polarization. Along an azimuthal circle, 27 phase spirals can be counted (right) [18]. 
All the phase patterns presented in Figs. 3 and 4 show m1 phase spirals, which means that the OAM mode number of corotating TE_{m,n}gyrotron modes is l = m1. The mode number of the TAM is m = l + 1. The SAM with s = 1 has the same sense of rotation as the OAM. For counterrotating modes the signs are reversed.
Table 1 summarizes the SAM and OAM features of different types of gyrotron TE modes. A special case are the circular symmetric modes (m = 0), where the optimum electron beam radii R_{b} for excitation of “co and counterrotating modes” are identical (radius of electric field maximum, see eqs. 6 an 7). Here m = l + s = 0, with no rotation (l = 0, no OAM), and since the electric field is at each transverse coordinate linearly polarized: s = 0 (no SAM). The circularly polarized TE_{1,1} mode has s = 1 (RHCP) and l = 0. Circularly polarized (rotating) TE_{1,n} modes (n >1) show a SAM (s = ±1) but have no OAM (l = 0). In the case of whispering gallery modes TE_{m,1} modes (WGMs) with m > 1, only the corotating modes can be excited, since always χ_{m+1,1 >} χ_{m,1} (see eq. 7). Corotating TE_{m,n} gyrotron modes with m,n >1 have the OAM mode number l = m  1 (number of phase spirals) and SAM (s = 1), since their electric fields are at each transverse coordinate circularly polarized with the same sense of rotation. For counterrotating (LHCP) modes the signs are reversed.
From the above considerations follows:
For possible applications of gyrodevices in longrange wireless communication with OAM diversity, the TE_{2,2} mode is the lowest order OAM mode, carrying the topological charges l = ± 1, depending whether it is co or counterrotating with respect to the electron gyration in the cavity magnetic field.
4. Gyrotron orbital angular momentum beam transmitters
Current megawattclass highpower gyrotrons for electron cyclotron heating, noninductive current drive, and collective Thomson scattering diagnostics in magnetically confined thermonuclear fusion plasmas are equipped with highly efficient quasioptical output couplers, which convert the rotating veryhighorder TE_{m,n}cavity mode to a fundamental, linearly polarized Gaussian output beam with transversal tube output window [13–15, 19]. Of course, the output of future gyrotron OAM beam transmitters must not use such output mode converters. In this case, directly the rotating interaction circuit mode should be used as tube output through an axially installed dielectric window [20].
Since propagation of higherorder OAM beams in free space results in large diffraction angles, a modeconserving, nonlinear waveguide diameter uptaper section should be added after the interaction circuit to reduce the Brillouin angle θ_{B} of the output wave. In addition, focusing lenses should be installed to reduce beam diffraction.
Several todays Ku/Kaband 10to30kWclass gyrotrons for various technological and industrial applications operate in a loworder rotating cavity mode, e.g. TE_{2,2}, TE_{3,2}, or TE_{2,3}, which also serves as axial (longitudinal) output mode [13]. Such tubes could be used for highpower OAM techniques. In the early 1990ties, the Research Center Karlsruhe (FZK, now KIT) and several vacuum electron tube companies performed very successful experiments with 0.5 – 1.0 MWclass axial output gyrotrons. Those operated in the following OAM modes [13]:
84 GHz, 110 GHz and 140 GHz, TE_{15,2} (VARIAN, now CPI),
110 GHz, TE_{12,2} and TE_{22,2} (VARIAN, now CPI),
110 GHz, TE_{6,4} (THOMSON, now THALES),
120 GHz, TE_{12,2} and TE_{22,12} (JAERI, TOSHIBA, now QST, CANON)
140 GHz, TE_{10,4} and TE_{28,16} (FZK, now KIT),
165 GHz, TE_{31,17} (FZK, now KIT).
The output of such gyrotrons could be used for longrange wireless communication with OAM diversity. Provided that the tube is equipped with a triodetype magnetron injection electron gun (MIG) with modulation anode and that it operates with suitable electron beam radius R_{b}, it may be electronically switched between the co and counterrotating cavity modes (right and left handed OAM) at the same operating frequency. To achieve high frequency stability and a narrow line width, a phase locking system can be utilized to control the electron beam energy. The longterm stability can be guaranteed by a reference clock. The relative width of the frequency spectrum and the frequency stability obtained of a 263 GHz, 100 W gyrotron, operating in the TE_{5,3} cavity mode are 4 ∙ 10^{−12} (approximately 1 Hz line width!) and 10^{−10}, respectively [21].
4.1. KIT 500 kW, 140 GHz TE_{10,4} mode gyrotron with axial output
This tube was equipped with a nonlinear uptaper from the cavity (16.22 mm diameter) to the 70 mm diameter output waveguide which also served as collector. The tube output was through a frequency tunable doubledisk FC75 face cooled sapphire window. The measured electronic efficiency of was 31% (without depressed collector) [22]. Highpower measurements of the output mode content using a 70 mm diameter wavenumber spectrometer (see Fig. 5) [23] resulted in a purity of the corotating TE_{10,4} output mode of better than 98% (wrong mode content < 17 dB).
Fig. 5 Measurement of the mode purity of a TE_{10,4} mode gyrotron using a wavenumber spectrometer. 
Fig. 5 Schematic of the 1.7 MW 165 GHz TE_{31,17}mode gyrotron with coaxial cavity (left), inverse magnetron injection electron gun (right) and axial output through a 100 mm diameter collector waveguide and window [27]. 
4.2. KIT 1.2 MW, 165 GHz TE_{31,17} mode coaxialcavity gyrotron with axial output
In coaxial gyrotron cavities the existence of the longitudinallycorrugated inner rod reduces the problems of mode competition and limiting current, thus allowing one to use even higher order modes at 2 MW power level with the same Ohmic attenuation as in 1 MWclass cylindrical cavities [24]. Successful activities on the development of 1.5 – 2 MW shortpulse gyrotrons with coaxial cavities and axial output window were conducted at IAP Nizhny Novgorod (TE_{28,16}  mode at 140 GHz [25]) and KIT (FZK) Karlsruhe (modes TE_{28,16} at 140 GHz and TE_{31,17} at 165 GHz) [24, 26]. In preliminary 1 ms short pulse operation of the 165 GHz tube with R_{0} =27.38 mm cavity radius and R_{b} = 9.57 mm electron beam radius, at B = 6.61 T, V_{b} = 84.8 kV and I_{b} =52 A, a maximum RF output power in the corotating TE_{31,17}mode (OAM state l = 30) as high as 1.2 MW was achieved with an efficiency of 27 % [27]. In this gyrotron cavity the corotating TE_{90,1} mode at 163.4 GHz would have the highest OAM state number (l = 89). Fig. 6 shows a schematic of this gyrotron in its superconcucting magnet and a photo of the utilized inverse magnetron injection gun (IMIG). For mode purity measurements, an optimized, nonlinear taper to a waveguide diameter of 140 mm was installed after the 100 mm diameter output window.
Fig. 6 Calculated nearfield pattern of the nonrotating TE_{31,17}mode at the uptaper output (140 mm diameter). The caustic radius is R_{c} = 22.9 mm. 
The calculated nearfield pattern of the nonrotating TE_{31,17}mode at the uptaper output is shown in Fig. 6. In Fig. 7, the corresponding pattern of the calculated rotating mode is plotted, the horizontal polarization (left) and the vertical polarization (right). The complete measured nearfield pattern of the rotating mode, recorded at high power using a PVC target plate and an infrared camera, is shown in Fig. 8. The measured highpower farfield pattern of the corotating TE_{31,17}mode is plotted in Fig. 9. A teflon lens at the uptaper output (140 mm diameter), with a focal length of 165 mm, made the farfield transformation. Fig. 10 shows the measured farfield pattern of the corotating TE_{31,17}mode [27], where some counterrotating mode was generated intentionally by means of reflection at a longitudinal quartz plate on the axis of propagation. In azimuthal direction, 62 intensity maxima and minima can be counted (see also Fig. 6).
Fig. 7 Calculated nearfield pattern of the corotating TE_{31,17}mode (l = 30) at the uptaper output (140 mm diameter). The caustic radius is R_{c} = 22.9 mm. Horizontal polarization (left), vertical polarization (right). 
Fig. 8 Measured highpower nearfield pattern of the corotating 165 GHz TE_{31,17}mode with OAM state number l = 30 at the uptaper output (140 mm diameter). The caustic radius is R_{c} = 22.9 mm. 
Fig. 9 Measured highpower farfield pattern of the corotating 165 GHz TE_{31,17}mode with OAM l = 30. A teflon lens with a focal length of 165 mm at the uptaper output (140 mm diameter) performed the farfield transformation. 
Fig. 10 Measured highpower farfield pattern of the corotating 165 GHz TE_{31,17}mode, where some counterrotating mode was intentionally produced by means of reflection at a quartz plate, positioned longitudinally on the axis of propagation [27]. In azimuthal direction, 62 intensity maxima and minima can be counted (see Fig. 6). 
4.3. GyroAmplifiers
As shown in Table 1, the family of gyrodevices also consists of different types of amplifiers, the gyroklystron with a bandwidth < 1%, the gyrotwystron with ≈ 2% bandwidth and the gyrotravelingwave tube (gyroTWT) providing ≈ 10% bandwidth [13]. Such amplifiers could be used for longrange wireless communications with OAM diversity. Due to mode competition with absolute instable backward waves, mostly very low order modes are employed in the interaction circuits of these amplifiers e.g. TE_{1,1} or TE_{01}, which cannot be used for OAM applications (see Table 2). Nevertheless, there are a few investigations on higherorder mode devices, as e.g. the 2^{nd} harmonic 31.8 GHz inverted gyrotwystron of NRL Washington D.C. operating in the TE_{4,2} mode (l = ± 3), with 160 kW output power at 25% efficiency, 33 dB gain, and 1.3% bandwidth [28]. Further investigation on such devices is needed in order to develop broadband amplifiers for OAM communications.
Intensity on axis I_{0}, spin angular momentum (SAM) and orbital angular momentum (OAM) of rotating gyrotron TE modes in circular waveguides.
5. Mode and Helical Wavefront Selective Gyrotron OAM Beam Receivers
Application of rotating highorder OAM gyrotron modes in longrange wireless communication systems requires corresponding mode and helical wavefront sensitive detectors for selective OAMmode sorting, which are described in the present section. The first needed components are beam splitters with low crosstalk to provide the necessary receiver channels, which are matched to right and left  hand rotation of the utilized OAM modes (demultiplexer). For a number of x OAM modes, 2x receiver channels with 2x1 beam splitters are required. Those can be realized in oversized circular waveguide or quasioptical techniques [29–31], which is also the case for the mode and helical wavefront selective detector channels [32]. At low power levels, in longrange applications, spiral phase plates (SPPs), holographic diffraction gratings, birefringent liquid crystals, spiral reflectors, specific array antennas, and anisotropic frequency selective surfaces (metasurfaces) [8, 9, 10, 11, 12, 33 and references given there] can be used. In the present report, oversized circular waveguide and quasioptical techniques are discussed, which can be used at higher power levels.
5.1. OAM selective oversized waveguide detection systems
Oversized waveguide OAM mode sorting and detection systems are based on sequences of periodic perturbedwall mode converters for rotating modes [34–38]. In overmoded circular waveguides with average radius a_{o} a selective transformation of one specific mode with azimuthal mode index m_{1} into another mode with azimuthal index m_{2} can be achieved by means of a periodic helical structure (∆m ∙ φ − ∆β ∙ z = const.) of the inner waveguide wall:
(13)under the condition that the geometric period λ_{w} of the wall perturbations and the unperturbed wavenumbers β_{1} and β_{2} of the interacting modes have to satisfy the resonance relationship
(15)where λ_{B} = λ_{1}λ_{2} / (λ_{2}λ_{1}) is the beat wavelength of the two modes. The radial wavenumber of the mode is changed by scattering the incoming wave at the periodic structure of the waveguide wall which acts as a diffraction grating. Fourier integral transformation theory yields the general conclusion, that the length of the mode converter must be at least of the order of λ_{B} of the two considered modes. The coherence condition (14) guarantees that the conversion to other unwanted modes, which are also coupled by the waveguide perturbations (same Δm), suffers destructive interference. The requirements for low parasitic mode amplitudes are ±Δβ ≠ β_{1,2} – β_{p} or ±Δm ≠ m_{1,2} – m_{p} where β_{p} is the wavenumber and m_{p} the azimuthal mode index of the parasitic mode. Mode transformers consisting of a large number N of geometrical periods create low levels of unwanted parasitic modes but inherently exhibit narrow bandwidth [39].
Here, as an example, the fourstep mode conversion series TE_{12,2} (righthand rotating) – TE_{0,6} – TE_{0,4} – TE_{0,1} – TE_{1,0} (standard rectangular w.g.) at 120 GHz and average waveguide radius a_{o} = 8.74 mm (waveguide type designation C120) will be described [36, 38]. Table 3 summarized the geometrical parameters of the helical (corkscrewtype) mode converter of the firststep. It consists of 3 periods, where the first and third sections are equipped with linear perturbation amplitude tapers (see Fig.11). The twostep conversion TE_{0,6} – TE_{0,4} – TE_{0,1} can be done employing highly efficient (98%) circular symmetric, periodically rippled wall mode transducers [40]. Finally, the TE_{0,1} – TE_{1,0} (standard rectangular w.g.) conversion can be performed by using adiabatic Kingtype, Southworthtype or Mariétype transducers [20].
Fig. 11 Schematic of the righthand helical, corkscrewtype TE_{12,2} to TE_{0,6} mode converter wall contour, including linearly tapered input and output sections [36]. 
5.2 OAM selective quasioptical detection systems
Rotation sensitive quasioptical OAM mode sorting and detection systems are based on quasioptical mode converters, as they are used in highpower fusion gyrotrons [32], or on mode generators for rotating higherorder modes [41, 42]. Figure 12 shows the photo of such a mode generator [43]. It consists of a Gaussian mode launching horn antenna, fed by a standard rectangular waveguide, a system of cylindrical lenses for proper beam shaping, a quasiparabolic cylindrical mirror and a gyrotrontype coaxial cavity with perforated, translucent wall for excitation of the desired highorder mode. The correct sense of rotation is achieved by optimized launching of the microwave beam through the cavitywall perforation to the caustic radius of the desired rotating mode [18, 44]. Of course, for OAM mode detection, this arrangement would be used the other way round. Careful alignment of such a mode generator system leads to suppression of mode power with the wrong sense of rotation by 23 dB. The highestorder mode, which has been generated using such a system, is the TE_{40,23} mode at 204 GHz [18].
6. Conclusions
The orbital angular momentum (OAM) of electromagneticwave beams provides further diversity to multiplexing in wireless communication. The present paper shows that higherorder mode gyrotron oscillators and gyroamplifiers are natural sources of very pure highpower highorder OAM millimeter (mm) wave beams. The welldefined total angular momentum (TAM) of rotating gyrotron modes operating close to the cutoff frequency of the cylindrical interaction circuit can be derived by photonic and electromagnetic (EM) wave approaches. Rotating (circularly polarized) TE_{m,p} modes exhibit the TAM mode number m where m = l + 1. Their OAM with mode number (topological charge) l = m – 1 (l spirals of the helical phase pattern), and spin angular momentum (SAM) with s = 1 have the same sense of rotation. Such very pure OAM mmwave beams generated by gyrodevices with axial output in the operating mode of the interaction circuit (no internal mode converter) could be used for longrange wireless communication with OAM diversity. Corresponding mode and helical wavefront sensitive detectors for selective OAMmode sorting can be realized in oversized circular waveguide or quasioptical beam waveguide technologies. The rotating TE_{2,2} mode is the lowest order OAM mode, carrying the topological charges l = ± 1, depending whether it is co or counterrotating with respect to the electron gyration in the longitudinal magnetic field of the interaction circuit in the gyrodevice.
Acknowledgments
The author wishes to express his deep gratitude to EunMi Choi and Ashwini Sawant from Ulsan National Institute of Science and Technology (UNIST), Ulsan, South Korea, for the fruitful and excellent collaboration in this field of gyrotron OAM modes, starting from his twomonth stay in 2015 as Visiting Professor at UNIST. In addition, he would like to thank Dietmar Wagner from MaxPlanckInstitute for Plasma Physics, Garching, Germany, for calculating the mode patterns shown in Figures 6 and 7 as well as Stefan Illy and Tobias Ruess, KIT, for very fruitful discussions on rotating gyrotron modes and quasioptical OAM mode detectors. Finally, the author is very glad to acknowledge the help of Oliver Braz, EnNet GmbH, Wemding, Germany, and Vladimir Malygin, Institute of Applied Physics (IAP) of the Russian Academy of Sciences (RAS), Nizhny Novgorod, Russia, who performed in 1998 the highpower near and farfield measurements of the rotating gyrotron TE_{31,17}mode plotted in Figures 8 and 9, working as PhD student, respectively visiting scientist in the Karlsruhe Gyrotron Team. Regrettably, V. Malygin passed away on April 12, 2019.
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All Tables
Intensity on axis I_{0}, spin angular momentum (SAM) and orbital angular momentum (OAM) of rotating gyrotron TE modes in circular waveguides.
All Figures
Fig. 1 (a) Ray propagation for a rotating wave in a cylindrical gyrotron cavity with consecutive reflections. (b) Set of rays forming a caustic with radius R_{c} [16]. 

In the text 
Fig. 3 Theoretical amplitude (a) and phase (b) of the timeaveraged total electric field of the corotating 95 GHz modes TE_{6,2} (left) and TE_{10,1} (right) (in the Fresnel zone at 10 mm distance from a 20 mm diameter aperture) [17]. The propagation direction is perpendicular to the measured pattern, coming out of the plane. 

In the text 
Fig. 4 Lowpower (cold) measurement of 140 GHz corotating TE_{28,8} mode phase pattern (left), taken with a fundamental rectangular waveguide probe for vertical polarization. Along an azimuthal circle, 27 phase spirals can be counted (right) [18]. 

In the text 
Fig. 5 Measurement of the mode purity of a TE_{10,4} mode gyrotron using a wavenumber spectrometer. 

In the text 
Fig. 5 Schematic of the 1.7 MW 165 GHz TE_{31,17}mode gyrotron with coaxial cavity (left), inverse magnetron injection electron gun (right) and axial output through a 100 mm diameter collector waveguide and window [27]. 

In the text 
Fig. 6 Calculated nearfield pattern of the nonrotating TE_{31,17}mode at the uptaper output (140 mm diameter). The caustic radius is R_{c} = 22.9 mm. 

In the text 
Fig. 7 Calculated nearfield pattern of the corotating TE_{31,17}mode (l = 30) at the uptaper output (140 mm diameter). The caustic radius is R_{c} = 22.9 mm. Horizontal polarization (left), vertical polarization (right). 

In the text 
Fig. 8 Measured highpower nearfield pattern of the corotating 165 GHz TE_{31,17}mode with OAM state number l = 30 at the uptaper output (140 mm diameter). The caustic radius is R_{c} = 22.9 mm. 

In the text 
Fig. 9 Measured highpower farfield pattern of the corotating 165 GHz TE_{31,17}mode with OAM l = 30. A teflon lens with a focal length of 165 mm at the uptaper output (140 mm diameter) performed the farfield transformation. 

In the text 
Fig. 10 Measured highpower farfield pattern of the corotating 165 GHz TE_{31,17}mode, where some counterrotating mode was intentionally produced by means of reflection at a quartz plate, positioned longitudinally on the axis of propagation [27]. In azimuthal direction, 62 intensity maxima and minima can be counted (see Fig. 6). 

In the text 
Fig. 11 Schematic of the righthand helical, corkscrewtype TE_{12,2} to TE_{0,6} mode converter wall contour, including linearly tapered input and output sections [36]. 

In the text 
Fig. 12 Photo of a quasioptical mode generator for rotating highorder gyrotrontype modes [43]. 

In the text 
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