© The Author(s) 2020

1. Introduction

Electromagnetic (EM) waves carry linear and angular momentum [1]. The linear momentum is associated with the Poynting vector $\vec{S}$, which represents the directional energy flux (the energy transfer per unit area per unit time, measured in W/m²). 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. SAM-state s = 0 corresponds to linear polarization and s = ± 1 to right-handed circular polarization (RHCP) and left-handed 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 so-called topological charge or OAM state number [4]. The OAM per photon is given by l∙ћ, with positive l for right-handed and negative l for left-handed 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 de-multiplexed 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 high-speed, terabit OAM multiplexing wireless communications [5] and in the areas of sensitive optical detection, orientational manipulation (optical tweezers), optical drives of micro-machines, 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 (pitch-fork hologram), birefringent liquid crystals (Q-plate), spiral reflectors, specific array antennas, and anisotropic frequency selective surfaces (meta-surfaces) [8, 9, and references given there]. In the microwave and mm-wave region there are difficulties associated with diffraction losses since the wavelength is much larger compared to those at optical frequencies. Reviews on the state-of-the-art and applications in this frequency region are given in [10, 11]. Record values in high-capacity mm-wave 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 (16-QAM) 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 bit-error rates (BERs) below 3.8x10⁻³.

The present paper shows that gyrotron oscillators and gyro-amplifiers are natural sources of high-power mm-wave beams with very pure higher-order OAM, which can be used for multiplexing in long-range wireless communication. The corresponding mode and helical wavefront sensitive detectors for selective OAM-mode sorting are available and also described in this report. Section 2 gives an overview on gyro-devices and their interaction principles. The OAM and SAM characteristics of rotating gyrotron cavity modes are discussed in Section 3. Section 4 shows examples of high-power, high-order 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. Gyro-devices

Gyro-devices 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

$$\omega - k_{\left|\right|} v_{\left|\right|}=q{\Omega }_c, q=1, 2,\dots ,$$(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 non-relativistic electron cyclotron frequency Ω_co is given by

$${\Omega }_{\mathrm{co}}\left(r\right)=\mathrm{eB}/m_o$$(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

$$\gamma =1+e{V}_b/ m_o{c}^2\approx 1+V_b/511\hspace{0.5em},$$(3)where c is the free-space velocity of light and V_b the acceleration voltage of the electron beam in kV. In the present paper, the field vector $\vec{B}$ 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_|| = 1.2-1.5) [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:

$$\omega \cong q{\Omega }_c, q=1, 2,\dots$$(4)

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]:

$$f_{\mathrm{cut}} \left(\mathrm{GHz}\right)=95.4269 {\chi }_{m,n}/D\left(\mathrm{mm}\right)$$(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 = 2R₀ is the cavity diameter.

Gyrotrons, as fast-wave devices with v_ph > c, are capable of producing very high-power radiation at cm, mm and sub-mm 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, high-power 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 high-order mode (determined by its eigenvalue χ_m,n) with low Ohmic attenuation can be selected just by the cavity radius R₀. 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 $\vec{B}$ 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]

$$R_{\mathrm{b}}={\chi }_{m-\mathrm{1,1}}{R}_0/{\chi }_{m,n}$$(6)the generated TE_m,n mode is co-rotating (right-hand rotating), and for

$$R_{\mathrm{b}}={\chi }_{m+\mathrm{1,1}}{R}_0/{\chi }_{m,n}$$(7)it is counter-rotating (left-hand 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 co-rotating modes can be excited, since always χ_m+1,1 > χ_m,1 (therefore, according to (7): R_b > R₀!).

Bunching of electrons in gyro-devices 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 klystron-type and twystron-type devices). This analogy suggests the correspondence between conventional linear-beam (O-type) devices and various types of gyro-devices. 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/kR₀) relative to the waveguide axis, where k = 2π/λ is the free-space 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 $\vec{S}$ 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

$$R_{\mathrm{c}}=(m/{\chi }_{m,n})R_0$$(8)

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) $\vec{J}$ possessed by such a ray (which is tangential to the caustic of the mode) can be expressed as

$$\vec{J}=J\widehat{z}=\vec{r}\times \vec{p}=R_{\mathrm{c}}\widehat{r}\times p_{\mathrm{\Phi }}\widehat{\Phi }=R_{\mathrm{c}}{p}_{\mathrm{\Phi }}\widehat{z}$$(9)Where $p_{\mathrm{\Phi }}=p\sin {\Theta }_B= ħk\sin {\Theta }_B$, $\sin {\Theta }_B=\frac{k_{\perp }}{k}$ and $k_{\perp }=\frac{{\mathrm{\chi }}_{m,n}}{R_0}$.

Therefore,

$$\vec{J}=R_{\mathrm{c}}{p}_{\mathrm{\Phi }}\vec{z}=\frac{m}{{\mathrm{\chi }}_{m,n}}{R}_0ħk\frac{{\mathrm{\chi }}_{m,n}}{k{R}_0}\widehat{z}=m\mathrm{ħ}\widehat{z}$$(10)

The ratio between the magnitudes of the TAM J_N and the total energy W_N of N photons is given by

$$\frac{J_N}{W_N}=\frac{\mathrm{Nm}ħ}{Nħ\omega }=\frac{m}{\omega }$$(11)where ω is the angular frequency of the operating cavity mode, which is close to the cutoff frequency ω_cut = cχ_m,n/R₀ in the cylindrical gyrotron cavity section. Therefore follows

$$m/\omega =R_c/c$$(12)

This is the result of the photonic (g.o.) approach [16]. Using the time averaged Poynting vector $\vec{S}$ 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 R₀).

• For a given gyrotron cavity radius R₀, the TE_m,n mode with the largest caustic radius R_c has the highest TAM. This is the co-rotating whispering gallery mode (WGM) TE_m,1 with χ_m,1 ≲ kR₀.

Figure 3 shows the analytically calculated amplitude and phase of the time-averaged total electric field of the co-rotating 95 GHz TE_6,2 and TE_10,1 modes, respectively [17]. In Fig. 4, the experimental phase pattern of the co-rotating 140 GHz TE_28,8 mode is plotted (left), generated by a specific low-power 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 Rc [16].

Fig. 3 Theoretical amplitude (a) and phase (b) of the time-averaged total electric field of the co-rotating 95 GHz modes TE6,2 (left) and TE10,1 (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 Low-power (cold) measurement of 140 GHz co-rotating TE28,8 -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 m-1 phase spirals, which means that the OAM mode number of co-rotating TE_m,n-gyrotron modes is l = m-1. 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 counter-rotating 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 counter-rotating 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 co-rotating modes can be excited, since always χ_{m+1,1 >} χ_m,1 (see eq. 7). Co-rotating 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 counter-rotating (LHCP) modes the signs are reversed.

From the above considerations follows:

For possible applications of gyro-devices in long-range 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 counter-rotating with respect to the electron gyration in the cavity magnetic field.

4. Gyrotron orbital angular momentum beam transmitters

Current megawatt-class high-power gyrotrons for electron cyclotron heating, non-inductive current drive, and collective Thomson scattering diagnostics in magnetically confined thermonuclear fusion plasmas are equipped with highly efficient quasi-optical output couplers, which convert the rotating very-high-order 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 higher-order OAM beams in free space results in large diffraction angles, a mode-conserving, non-linear waveguide diameter up-taper 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/Ka-band 10-to-30-kW-class gyrotrons for various technological and industrial applications operate in a low-order 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 high-power OAM techniques. In the early 1990-ties, the Research Center Karlsruhe (FZK, now KIT) and several vacuum electron tube companies performed very successful experiments with 0.5 – 1.0 MW-class 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 long-range wireless communication with OAM diversity. Provided that the tube is equipped with a triode-type 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 counter-rotating 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 long-term 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 non-linear up-taper 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 double-disk FC75 face cooled sapphire window. The measured electronic efficiency of was 31% (without depressed collector) [22]. High-power measurements of the output mode content using a 70 mm diameter wavenumber spectrometer (see Fig. 5) [23] resulted in a purity of the co-rotating TE_10,4 output mode of better than 98% (wrong mode content < -17 dB).

Fig. 5 Measurement of the mode purity of a TE10,4 -mode gyrotron using a wavenumber spectrometer.

Fig. 5 Schematic of the 1.7 MW 165 GHz TE31,17-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 coaxial-cavity gyrotron with axial output

In coaxial gyrotron cavities the existence of the longitudinally-corrugated 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 MW-class cylindrical cavities [24]. Successful activities on the development of 1.5 – 2 MW short-pulse 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 R₀ =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 co-rotating 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 co-rotating 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, non-linear taper to a waveguide diameter of 140 mm was installed after the 100 mm diameter output window.

Fig. 6 Calculated near-field pattern of the non-rotating TE31,17-mode at the up-taper output (140 mm diameter). The caustic radius is Rc = 22.9 mm.

The calculated near-field pattern of the non-rotating TE_31,17-mode at the up-taper 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 near-field 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 high-power far-field pattern of the co-rotating TE_31,17-mode is plotted in Fig. 9. A teflon lens at the up-taper output (140 mm diameter), with a focal length of 165 mm, made the far-field transformation. Fig. 10 shows the measured far-field pattern of the co-rotating TE_31,17-mode [27], where some counter-rotating 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 near-field pattern of the co-rotating TE31,17-mode (l = 30) at the up-taper output (140 mm diameter). The caustic radius is Rc = 22.9 mm. Horizontal polarization (left), vertical polarization (right).

Fig. 8 Measured high-power near-field pattern of the co-rotating 165 GHz TE31,17-mode with OAM state number l = 30 at the up-taper output (140 mm diameter). The caustic radius is Rc = 22.9 mm.

Fig. 9 Measured high-power far-field pattern of the co-rotating 165 GHz TE31,17-mode with OAM l = 30. A teflon lens with a focal length of 165 mm at the up-taper output (140 mm diameter) performed the far-field transformation.

Fig. 10 Measured high-power far-field pattern of the co-rotating 165 GHz TE31,17-mode, where some counter-rotating 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. Gyro-Amplifiers

As shown in Table 1, the family of gyro-devices also consists of different types of amplifiers, the gyro-klystron with a bandwidth < 1%, the gyro-twystron with ≈ 2% bandwidth and the gyro-traveling-wave tube (gyro-TWT) providing ≈ 10% bandwidth [13]. Such amplifiers could be used for long-range wireless communications with OAM diversity. Due to mode competition with absolute instable back-ward waves, mostly very low order modes are employed in the interaction circuits of these amplifiers e.g. TE_1,1 or TE₀₁, which cannot be used for OAM applications (see Table 2). Nevertheless, there are a few investigations on higher-order mode devices, as e.g. the 2^nd harmonic 31.8 GHz inverted gyro-twystron 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.

5. Mode and Helical Wavefront Selective Gyrotron OAM Beam Receivers

Application of rotating high-order OAM gyrotron modes in long-range wireless communication systems requires corresponding mode and helical wavefront sensitive detectors for selective OAM-mode 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 (de-multiplexer). For a number of x OAM modes, 2x receiver channels with 2x-1 beam splitters are required. Those can be realized in oversized circular waveguide or quasi-optical techniques [29–31], which is also the case for the mode and helical wavefront selective detector channels [32]. At low power levels, in long-range applications, spiral phase plates (SPPs), holographic diffraction gratings, birefringent liquid crystals, spiral reflectors, specific array antennas, and anisotropic frequency selective surfaces (meta-surfaces) [8, 9, 10, 11, 12, 33 and references given there] can be used. In the present report, oversized circular waveguide and quasi-optical 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 perturbed-wall 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 m₁ into another mode with azimuthal index m₂ can be achieved by means of a periodic helical structure (∆m ∙ φ − ∆β ∙ z = const.) of the inner waveguide wall:

$$a\left(z,\phi \right)=a_0+a_1\cos \left(∆m\bullet \phi - ∆\beta \bullet z\right)$$(13)under the condition that the geometric period λ_w of the wall perturbations and the unperturbed wavenumbers β₁ and β₂ of the interacting modes have to satisfy the resonance relationship

$$\Delta \beta ={\beta }_1-{\beta }_2=2\pi /{\lambda }_B=2\pi /{\lambda }_w$$(14)And

$$\pm \Delta m=m_1-m_2$$(15)where λ_B = λ₁λ₂ / (λ₂-λ₁) 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 four-step mode conversion series TE_12,2 (right-hand 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 (corkscrew-type) mode converter of the first-step. It consists of 3 periods, where the first and third sections are equipped with linear perturbation amplitude tapers (see Fig.11). The two-step 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 King-type, Southworth-type or Marié-type transducers [20].

Fig. 11 Schematic of the right-hand helical, corkscrew-type TE12,2 -to- TE0,6 mode converter wall contour, including linearly tapered input and output sections [36].

5.2 OAM selective quasi-optical detection systems

Rotation sensitive quasi-optical OAM mode sorting and detection systems are based on quasi-optical mode converters, as they are used in high-power fusion gyrotrons [32], or on mode generators for rotating higher-order 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 quasi-parabolic cylindrical mirror and a gyrotron-type coaxial cavity with perforated, translucent wall for excitation of the desired high-order mode. The correct sense of rotation is achieved by optimized launching of the microwave beam through the cavity-wall 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 highest-order mode, which has been generated using such a system, is the TE_40,23 mode at 204 GHz [18].

Fig. 12 Photo of a quasi-optical mode generator for rotating high-order gyrotron-type modes [43].

6. Conclusions

The orbital angular momentum (OAM) of electromagnetic-wave beams provides further diversity to multiplexing in wireless communication. The present paper shows that higher-order mode gyrotron oscillators and gyro-amplifiers are natural sources of very pure high-power high-order OAM millimeter (mm) wave beams. The well-defined 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 mm-wave beams generated by gyro-devices with axial output in the operating mode of the interaction circuit (no internal mode converter) could be used for long-range wireless communication with OAM diversity. Corresponding mode and helical wavefront sensitive detectors for selective OAM-mode sorting can be realized in oversized circular waveguide or quasi-optical 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 counter-rotating with respect to the electron gyration in the longitudinal magnetic field of the interaction circuit in the gyro-device.

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 two-month stay in 2015 as Visiting Professor at UNIST. In addition, he would like to thank Dietmar Wagner from Max-Planck-Institute 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 quasi-optical 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 high-power near- and far-field 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.

References

All Tables

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 Rc [16].
In the text

	Fig. 3 Theoretical amplitude (a) and phase (b) of the time-averaged total electric field of the co-rotating 95 GHz modes TE6,2 (left) and TE10,1 (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 Low-power (cold) measurement of 140 GHz co-rotating TE28,8 -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 TE10,4 -mode gyrotron using a wavenumber spectrometer.
In the text

	Fig. 5 Schematic of the 1.7 MW 165 GHz TE31,17-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 near-field pattern of the non-rotating TE31,17-mode at the up-taper output (140 mm diameter). The caustic radius is Rc = 22.9 mm.
In the text

	Fig. 7 Calculated near-field pattern of the co-rotating TE31,17-mode (l = 30) at the up-taper output (140 mm diameter). The caustic radius is Rc = 22.9 mm. Horizontal polarization (left), vertical polarization (right).
In the text

	Fig. 8 Measured high-power near-field pattern of the co-rotating 165 GHz TE31,17-mode with OAM state number l = 30 at the up-taper output (140 mm diameter). The caustic radius is Rc = 22.9 mm.
In the text

	Fig. 9 Measured high-power far-field pattern of the co-rotating 165 GHz TE31,17-mode with OAM l = 30. A teflon lens with a focal length of 165 mm at the up-taper output (140 mm diameter) performed the far-field transformation.
In the text

	Fig. 10 Measured high-power far-field pattern of the co-rotating 165 GHz TE31,17-mode, where some counter-rotating 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 right-hand helical, corkscrew-type TE12,2 -to- TE0,6 mode converter wall contour, including linearly tapered input and output sections [36].
In the text

	Fig. 12 Photo of a quasi-optical mode generator for rotating high-order gyrotron-type modes [43].
In the text